diff --git a/doc/source/examples/incompressible-flow/2d-backward-facing-step/2d-backward-facing-step.rst b/doc/source/examples/incompressible-flow/2d-backward-facing-step/2d-backward-facing-step.rst index f4f335638e..6bf5828d91 100644 --- a/doc/source/examples/incompressible-flow/2d-backward-facing-step/2d-backward-facing-step.rst +++ b/doc/source/examples/incompressible-flow/2d-backward-facing-step/2d-backward-facing-step.rst @@ -300,7 +300,7 @@ where the final value of :math:`x_r` is :math:`2.893`. We notice from the graph .. image:: image/Reynolds100-error-analysis.png -The reference value used in the error analysis is taken from Erturk (2008) `[1] `_. +The reference value used in the error analysis is taken from Erturk (2008) [#erturk2008]_. Higher Reynolds Number (:math:`Re=1000`) @@ -324,11 +324,11 @@ Validation and Comparison Reattachment Length ~~~~~~~~~~~~~~~~~~~ -In this section, the solutions obtained with Lethe are compared with data that can be found in the scientific literature (Erturk (2008) `[1] `_, Armaly and al. (1983) `[2] `_ and Velivelli and Bryden (2015) `[3] `_). Several studies include datasets of :math:`x_r/h = f(Re)` (reattachment length) either experimentally or numerically. The next figure illustrates some of them in comparison with *Lethe*. +In this section, the solutions obtained with Lethe are compared with data that can be found in the scientific literature (Erturk (2008) [#erturk2008]_, Armaly and al. (1983) [#armaly1983]_ and Velivelli and Bryden (2015) [#velivelli2015]_). Several studies include datasets of :math:`x_r/h = f(Re)` (reattachment length) either experimentally or numerically. The next figure illustrates some of them in comparison with *Lethe*. .. image:: image/xr-comparison.png -First, the results provided by Lethe are identical or so to all of the three selected studies for low Reynolds numbers (:math:`Re \leq 400`). After that point, both results form *Lethe* and from Erturk (2008) `[1] `_ diverge from the experimental data of Armaly and al. (1983) `[2] `_. According to `[1] `_, this error is due to 3D effects that are more potent as the flow becomes more and more turbulent. Furthermore, there is also a less significant but clearly noticeable error between *Lethe* and Erturk (2008) `[1] `_: the fact that certain tolerances have been set higher for higher Reynolds number cases might have underestimated the reattachment length. Also, first order elements have been used throughout the whole simulation process. Using second order elements for velocity, for instance, could yield better results for higher Reynolds numbers, however, at a higher computational cost. The following table illustrates the error at :math:`Re = 600` for first and second order velocity elements. +First, the results provided by Lethe are identical or so to all of the three selected studies for low Reynolds numbers (:math:`Re \leq 400`). After that point, both results form *Lethe* and from Erturk (2008) [#erturk2008]_ diverge from the experimental data of Armaly and al. (1983) [#armaly1983]_. According to [#erturk2008]_, this error is due to 3D effects that are more potent as the flow becomes more and more turbulent. Furthermore, there is also a less significant but clearly noticeable error between *Lethe* and Erturk (2008) [#erturk2008]_: the fact that certain tolerances have been set higher for higher Reynolds number cases might have underestimated the reattachment length. Also, first order elements have been used throughout the whole simulation process. Using second order elements for velocity, for instance, could yield better results for higher Reynolds numbers, however, at a higher computational cost. The following table illustrates the error at :math:`Re = 600` for first and second order velocity elements. +---------------+----------------+----------------+ | Order | :math:`x_r/h` | Error | @@ -369,11 +369,11 @@ Possibilities for Extension References ---------- -`[1] `_ E. Erturk, “Numerical solutions of 2-D steady incompressible flow over a backward-facing step, Part I: High Reynolds number solutions,” *Comput. Fluids*, vol. 37, no. 6, pp. 633–655, Jul. 2008, doi: 10.1016/j.compfluid.2007.09.003. +.. [#erturk2008] \E. Erturk, “Numerical solutions of 2-D steady incompressible flow over a backward-facing step, Part I: High Reynolds number solutions,” *Comput. Fluids*, vol. 37, no. 6, pp. 633–655, Jul. 2008, doi: `10.1016/j.compfluid.2007.09.003 `_\. -`[2] `_ B. F. Armaly, F. Durst, J. C. F. Pereira, and B. Schönung, “Experimental and theoretical investigation of backward-facing step flow,” *J. Fluid Mech.*, vol. 127, pp. 473–496, Feb. 1983, doi: 10.1017/S0022112083002839. +.. [#armaly1983] \B. F. Armaly, F. Durst, J. C. F. Pereira, and B. Schönung, “Experimental and theoretical investigation of backward-facing step flow,” *J. Fluid Mech.*, vol. 127, pp. 473–496, Feb. 1983, doi: `10.1017/S0022112083002839 `_\. -`[3] `_ A. C. Velivelli and K. M. Bryden, “Domain decomposition based coupling between the lattice Boltzmann method and traditional CFD methods – Part II: Numerical solution to the backward facing step flow,” *Adv. Eng. Softw.*, vol. 82, pp. 65–74, Apr. 2015, doi: 10.1016/j.advengsoft.2014.11.006. +.. [#velivelli2015] \A. C. Velivelli and K. M. Bryden, “Domain decomposition based coupling between the lattice Boltzmann method and traditional CFD methods – Part II: Numerical solution to the backward facing step flow,” *Adv. Eng. Softw.*, vol. 82, pp. 65–74, Apr. 2015, doi: `10.1016/j.advengsoft.2014.11.006 `_\. diff --git a/doc/source/examples/incompressible-flow/2d-flow-around-cylinder/2d-flow-around-cylinder.rst b/doc/source/examples/incompressible-flow/2d-flow-around-cylinder/2d-flow-around-cylinder.rst index 0501bb5074..38d8abb21a 100644 --- a/doc/source/examples/incompressible-flow/2d-flow-around-cylinder/2d-flow-around-cylinder.rst +++ b/doc/source/examples/incompressible-flow/2d-flow-around-cylinder/2d-flow-around-cylinder.rst @@ -31,7 +31,7 @@ All files mentioned below are located in the example's folder (``examples/incomp ----------------------- Description of the Case ----------------------- -We simulate the flow around a fixed cylinder with a constant upstream fluid velocity. The following schematic describes the geometry with its relevant quantities (taken from the article by Blais *et al.* `[1] `_): +We simulate the flow around a fixed cylinder with a constant upstream fluid velocity. The following schematic describes the geometry with its relevant quantities (taken from the article by Blais *et al.* [#blais2016]_): .. image:: images/geometry-description.png :alt: The geometry @@ -283,4 +283,4 @@ Possibilities for Extension Reference ---------- -`[1] `_ B. Blais, M. Lassaigne, C. Goniva, L. Fradette, and F. Bertrand, “A semi-implicit immersed boundary method and its application to viscous mixing,” *Comput. Chem. Eng.*, vol. 85, pp. 136–146, Feb. 2016, doi: 10.1016/j.compchemeng.2015.10.019. \ No newline at end of file +.. [#blais2016] \B. Blais, M. Lassaigne, C. Goniva, L. Fradette, and F. Bertrand, “A semi-implicit immersed boundary method and its application to viscous mixing,” *Comput. Chem. Eng.*, vol. 85, pp. 136–146, Feb. 2016, doi: `10.1016/j.compchemeng.2015.10.019 `_\. diff --git a/doc/source/examples/incompressible-flow/2d-lid-driven-cavity-flow/lid-driven-cavity-flow.rst b/doc/source/examples/incompressible-flow/2d-lid-driven-cavity-flow/lid-driven-cavity-flow.rst index 9356d37c31..1aeac9f254 100644 --- a/doc/source/examples/incompressible-flow/2d-lid-driven-cavity-flow/lid-driven-cavity-flow.rst +++ b/doc/source/examples/incompressible-flow/2d-lid-driven-cavity-flow/lid-driven-cavity-flow.rst @@ -21,8 +21,8 @@ Files Used in This Example All files mentioned below are located in the example's folder (``examples/incompressible-flow/2d-lid-driven-cavity``). - Base case parameter file (:math:`Re=400`): ``cavity.prm`` -- Experimental data file from Ghia `et al.` (1982) `[1] `_: ``ref-2d-ghia-u.txt`` -- Experimental data file from Erturk `et al.` (2005) `[2] `_: ``ref-2d-erturk-u.txt`` +- Experimental data file from Ghia `et al.` (1982) [#ghia1982]_: ``ref-2d-ghia-u.txt`` +- Experimental data file from Erturk `et al.` (2005) [#erturk2005]_: ``ref-2d-erturk-u.txt`` - Higher-Reynolds case parameter file (:math:`Re=7500`): ``Reynolds_7500/cavity.prm`` - Postprocessing Python script for the :math:`Re=400` case: ``post_process_Reynolds_400.py`` - Postprocessing Python script for the :math:`Re=7500` case: ``Reynolds_7500/post_process_Reynolds_7500.py`` @@ -299,7 +299,7 @@ Increasing the number of cells by a factor 4 (to :math:`\approx` 65k cells) allo Possibilities for Extension ---------------------------- -- **Validate at even higher Reynolds numbers:** The Erturk `[2] `_ data within the example investigates this case up to a Reynolds number of 20000. It is an interesting exercise to simulate these more complex cases using the adjoint time-stepping ``steady_bdf`` scheme. +- **Validate at even higher Reynolds numbers:** The Erturk [#erturk2005]_ data within the example investigates this case up to a Reynolds number of 20000. It is an interesting exercise to simulate these more complex cases using the adjoint time-stepping ``steady_bdf`` scheme. - **High-order methods:** Lethe supports higher order interpolation. This can yield much better results with an equal number of degrees of freedom than traditional second-order (Q1-Q1) methods, especially at higher Reynolds numbers. @@ -310,6 +310,6 @@ Possibilities for Extension References ----------- -`[1] `_ U. Ghia, K. N. Ghia, and C. T. Shin, “High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method,” *J. Comput. Phys.*, vol. 48, no. 3, pp. 387–411, Dec. 1982, doi: 10.1016/0021-9991(82)90058-4. +.. [#ghia1982] \U. Ghia, K. N. Ghia, and C. T. Shin, “High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method,” *J. Comput. Phys.*, vol. 48, no. 3, pp. 387–411, Dec. 1982, doi: `10.1016/0021-9991(82)90058-4 `_\. -`[2] `_ E. Erturk, T. C. Corke, and C. Gökçöl, “Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers,” *Int. J. Numer. Methods Fluids*, vol. 48, no. 7, pp. 747–774, 2005, doi: 10.1002/fld.953. +.. [#erturk2005] \E. Erturk, T. C. Corke, and C. Gökçöl, “Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers,” *Int. J. Numer. Methods Fluids*, vol. 48, no. 7, pp. 747–774, 2005, doi: `10.1002/fld.953 `_\. diff --git a/doc/source/examples/incompressible-flow/2d-naca0012-low-reynolds/2d-naca0012-low-reynolds.rst b/doc/source/examples/incompressible-flow/2d-naca0012-low-reynolds/2d-naca0012-low-reynolds.rst index 8644d347fa..d50d053a17 100644 --- a/doc/source/examples/incompressible-flow/2d-naca0012-low-reynolds/2d-naca0012-low-reynolds.rst +++ b/doc/source/examples/incompressible-flow/2d-naca0012-low-reynolds/2d-naca0012-low-reynolds.rst @@ -249,7 +249,7 @@ The following average pressure and velocity fields are obtained for an angle of .. image:: image/average_velocity.png -It is already noticeable that the higher the angle of attack, the greater the pressure gradient. Following this observation, the lift coefficient :math:`C_L` is expected to increase with the angle of attack until stall condition is reached. The variation of the lift and drag coefficients are given below with a comparison to the work of Kouser *et al.* `[1] `_. Both coefficients are computed using the following formula: +It is already noticeable that the higher the angle of attack, the greater the pressure gradient. Following this observation, the lift coefficient :math:`C_L` is expected to increase with the angle of attack until stall condition is reached. The variation of the lift and drag coefficients are given below with a comparison to the work of Kouser *et al.* [#kouser2021]_. Both coefficients are computed using the following formula: .. math:: C_L = \frac{F_L}{0.5\rho_{\infty} u_{\infty}^2 S} \; \; \; \; \; C_D = \frac{F_D}{0.5\rho_{\infty} u_{\infty}^2 S} @@ -258,7 +258,7 @@ with :math:`F_L` and :math:`F_D`, respectively, the lift and drag forces. Those .. image:: image/cl_cd_results_plot.png -The results obtained fit the drag and lift coefficients found by Kouser *et al.* `[1] `_. Note that the value given for the :math:`C_D` and :math:`C_L` coefficients are Root Mean Squared (RMS) values. The time span considered is 25s long (between 15 :math:`\text{s}` and 40 :math:`\text{s}`). The first 15 seconds were not considered to let the system reach a pseudo-steady state. +The results obtained fit the drag and lift coefficients found by Kouser *et al.* [#kouser2021]_. Note that the value given for the :math:`C_D` and :math:`C_L` coefficients are Root Mean Squared (RMS) values. The time span considered is 25s long (between 15 :math:`\text{s}` and 40 :math:`\text{s}`). The first 15 seconds were not considered to let the system reach a pseudo-steady state. One can also see the low-velocity zones on the upper part of the airfoil, which corresponds to the recirculating zone: the ``noslip`` condition on the NACA imposes a zero velocity condition on the fluid. The following streamline representation helps to see the movements of the fluid inside the recirculating zone: @@ -285,7 +285,7 @@ In order to retrieve the frequency of the vortex shedding, one can look at the f .. image:: image/plot_cl_time.png -The best mathematical tool available to make a spectral analysis is a Fourier transform, which is performed below, with literature results (Kouser *et al.* (2021) `[1] `_) for comparison: +The best mathematical tool available to make a spectral analysis is a Fourier transform, which is performed below, with literature results (Kouser *et al.* (2021) [#kouser2021]_) for comparison: .. image:: image/fft_cl_comparison.png @@ -300,17 +300,17 @@ Possibilities for Extension - **Going 3D**: the mesh can be extruded into the third dimension. Some modifications will be required in the boundary conditions, and getting the correct boundaries id is not trivial. However, with periodic boundary conditions set on the sides of the box, spanwise effects can be taken into account, which should yield much better results. -- **Validate for higher Reynolds numbers**: Literature is available for comparison at :math:`Re=10000` at Yamaguchi *et al.* (2013) `[2] `_ and :math:`Re=23000` at Kojima *et al.* (2013) `[3] `_. +- **Validate for higher Reynolds numbers**: Literature is available for comparison at :math:`Re=10000` at Yamaguchi *et al.* (2013) [#yuta2013]_ and :math:`Re=23000` at Kojima *et al.* (2013) [#kojima2013]_. ---------- References ---------- -`[1] `_ T. Kouser, Y. Xiong, D. Yang, and S. Peng, “Direct Numerical Simulations on the three-dimensional wake transition of flows over NACA0012 airfoil at Re=1000,” *Int. J. Micro Air Veh.*, vol. 13, p. 17568293211055656, Jan. 2021, doi: 10.1177/17568293211055656. +.. [#kouser2021] \T. Kouser, Y. Xiong, D. Yang, and S. Peng, “Direct Numerical Simulations on the three-dimensional wake transition of flows over NACA0012 airfoil at Re=1000,” *Int. J. Micro Air Veh.*, vol. 13, p. 17568293211055656, Jan. 2021, doi: `10.1177/17568293211055656 `_\. -`[2] `_ Y. Yuta, O. Tomohisa, and M. Akinori, “1201 Pressure Distribution on a Naca0012 Airfoil at Low Reynolds Numbers,” *Proc. Int. Conf. Jets Wakes Separated Flows ICJWSF*, vol. 2013.4, p. 1201-1 - 1201-5 , 2013, doi: 10.1299/jsmeicjwsf.2013.4._1201-\1_. +.. [#yuta2013] \Y. Yuta, O. Tomohisa, and M. Akinori, “1201 Pressure Distribution on a Naca0012 Airfoil at Low Reynolds Numbers,” *Proc. Int. Conf. Jets Wakes Separated Flows ICJWSF*, vol. 2013.4, p. 1201-1 - 1201-5 , 2013, doi: `10.1299/jsmeicjwsf.2013.4._1201-\1_ `_\. -`[3] `_ R. Kojima, T. Nonomura, A. Oyama, and K. Fujii, “Large-Eddy Simulation of Low-Reynolds-Number Flow Over Thick and Thin NACA Airfoils,” *J. Aircr.*, vol. 50, no. 1, pp. 187–196, Jan. 2013, doi: 10.2514/1.C031849. +.. [#kojima2013] \R. Kojima, T. Nonomura, A. Oyama, and K. Fujii, “Large-Eddy Simulation of Low-Reynolds-Number Flow Over Thick and Thin NACA Airfoils,” *J. Aircr.*, vol. 50, no. 1, pp. 187–196, Jan. 2013, doi: `10.2514/1.C031849 `_\. diff --git a/doc/source/examples/incompressible-flow/2d-taylor-couette-flow-nitsche/2d-taylor-couette-flow-nitsche.rst b/doc/source/examples/incompressible-flow/2d-taylor-couette-flow-nitsche/2d-taylor-couette-flow-nitsche.rst index 162c2d18e7..c9c53211f7 100644 --- a/doc/source/examples/incompressible-flow/2d-taylor-couette-flow-nitsche/2d-taylor-couette-flow-nitsche.rst +++ b/doc/source/examples/incompressible-flow/2d-taylor-couette-flow-nitsche/2d-taylor-couette-flow-nitsche.rst @@ -2,7 +2,7 @@ Taylor-Couette Flow Using Nitsche Immersed Boundary ======================================================== -This example revisits the same taylor-couette flow problem in :doc:`../2d-taylor-couette-flow/2d-taylor-couette-flow`, +This example revisits the same taylor-couette flow [#bird2006]_ problem in :doc:`../2d-taylor-couette-flow/2d-taylor-couette-flow`, now using immersed boundaries to represent the inner cylinder. This example demonstrates some of the capabilities of Lethe to simulate the flow around complex geometries without meshing them explicitly with a conformal mesh, but instead by using the Nitsche immersed boundary method available within Lethe. @@ -384,4 +384,4 @@ Possibilities for Extension Reference ------------ -[1] R. B. Bird, W. E. Stewart, and E. N. Lightfoot, *Transport Phenomena*, vol. 1. John Wiley & Sons, 2006. +.. [#bird2006] \R. B. Bird, W. E. Stewart, and E. N. Lightfoot, *Transport Phenomena*, vol. 1. John Wiley & Sons, 2006\. diff --git a/doc/source/examples/incompressible-flow/2d-taylor-couette-flow/2d-taylor-couette-flow.rst b/doc/source/examples/incompressible-flow/2d-taylor-couette-flow/2d-taylor-couette-flow.rst index bb5adc2a6c..f17ac4c064 100644 --- a/doc/source/examples/incompressible-flow/2d-taylor-couette-flow/2d-taylor-couette-flow.rst +++ b/doc/source/examples/incompressible-flow/2d-taylor-couette-flow/2d-taylor-couette-flow.rst @@ -2,7 +2,7 @@ Taylor-Couette Flow ================================== -This example showcases another classical fluid mechanics problem, the Taylor-Couette flow. This example introduces the usage of analytical solution and monitors the convergence of the CFD solver by using progressively refined meshes. +This example showcases another classical fluid mechanics problem, the Taylor-Couette flow [#bird2006]_. This example introduces the usage of analytical solution and monitors the convergence of the CFD solver by using progressively refined meshes. --------- @@ -293,4 +293,4 @@ Possibilities for Extension References ------------ -[1] R. B. Bird, W. E. Stewart, and E. N. Lightfoot, *Transport Phenomena*, vol. 1. John Wiley & Sons, 2006. \ No newline at end of file +.. [#bird2006] \R. B. Bird, W. E. Stewart, and E. N. Lightfoot, *Transport Phenomena*, vol. 1. John Wiley & Sons, 2006\. diff --git a/doc/source/examples/incompressible-flow/2d-transient-flow-around-ahmed-body/2d-transient-flow-around-ahmed-body.rst b/doc/source/examples/incompressible-flow/2d-transient-flow-around-ahmed-body/2d-transient-flow-around-ahmed-body.rst index b33b1c22c5..34a133d3d3 100644 --- a/doc/source/examples/incompressible-flow/2d-transient-flow-around-ahmed-body/2d-transient-flow-around-ahmed-body.rst +++ b/doc/source/examples/incompressible-flow/2d-transient-flow-around-ahmed-body/2d-transient-flow-around-ahmed-body.rst @@ -42,7 +42,7 @@ The following schematic describes the simulation. * bc = 1 (u = 1; flow in the x-direction) * bc = 2 (Slip boundary condition) -The basic geometry for the Ahmed body is given below, as defined in Ahmed et al. `[1] `_, with all measures in mm. +The basic geometry for the Ahmed body is given below, as defined in Ahmed et al. [#ahmed1984]_, with all measures in mm. .. image:: images/ahmed-geometry.png :alt: Geometry detailed description @@ -236,4 +236,4 @@ Possibilities for Extension ---------- Reference ---------- -`[1] `_ Ahmed, S. R., et al. “Some Salient Features of the Time -Averaged Ground Vehicle Wake.” *SAE Transactions*, vol. 93, 1984, pp. 473–503. http://www.jstor.org/stable/44434262. +.. [#ahmed1984] \Ahmed, S. R., et al. “Some Salient Features of the Time -Averaged Ground Vehicle Wake.” *SAE Transactions*, vol. 93, pp. 473–503, 1984. http://www.jstor.org/stable/44434262\. diff --git a/doc/source/examples/incompressible-flow/2d-transient-flow-around-cylinder/2d-transient-flow-around-cylinder.rst b/doc/source/examples/incompressible-flow/2d-transient-flow-around-cylinder/2d-transient-flow-around-cylinder.rst index fa08c87f89..ba5d8e01fa 100644 --- a/doc/source/examples/incompressible-flow/2d-transient-flow-around-cylinder/2d-transient-flow-around-cylinder.rst +++ b/doc/source/examples/incompressible-flow/2d-transient-flow-around-cylinder/2d-transient-flow-around-cylinder.rst @@ -28,7 +28,7 @@ All files mentioned below are located in the example's folder (``examples/incomp Description of the Case ----------------------- -We simulate the flow around a fixed cylinder with a constant upstream fluid velocity. We re-use the geometry and mesh presented in `2D Flow around a cylinder `_, which were taken from Blais *et al.* `[1] `_: +We simulate the flow around a fixed cylinder with a constant upstream fluid velocity. We re-use the geometry and mesh presented in `2D Flow around a cylinder `_, which were taken from Blais *et al.* [#blais2016]_: .. image:: images/geometry-description.png :alt: The geometry @@ -280,11 +280,11 @@ The obtained values of the drag and lift coefficients as well as the Strouhal nu - 1.396 :math:`\pm` 0.048 - -0.003 :math:`\pm` 0.072 - 0.2 - * - Lethe Sharp `[2] `_ + * - Lethe Sharp [#barbeau2022]_ - 1.395 :math:`\pm` 0.047 - :math:`\pm` 0.071 - 0.2 - * - Braza et al. `[3] `_ + * - Braza et al. [#braza1986]_ - 1.400 :math:`\pm` 0.050 - :math:`\pm` 0.075 - 0.2 @@ -316,8 +316,8 @@ Possibilities for Extension References ---------- -`[1] `_ B. Blais, M. Lassaigne, C. Goniva, L. Fradette, and F. Bertrand, “A semi-implicit immersed boundary method and its application to viscous mixing,” *Comput. Chem. Eng.*, vol. 85, pp. 136–146, Feb. 2016, doi: 10.1016/j.compchemeng.2015.10.019. +.. [#blais2016] \B. Blais, M. Lassaigne, C. Goniva, L. Fradette, and F. Bertrand, “A semi-implicit immersed boundary method and its application to viscous mixing,” *Comput. Chem. Eng.*, vol. 85, pp. 136–146, Feb. 2016, doi: `10.1016/j.compchemeng.2015.10.019 `_\. -`[2] `_ L. Barbeau, S. Étienne, C. Béguin, and B. Blais, “Development of a high-order continuous Galerkin sharp-interface immersed boundary method and its application to incompressible flow problems,” *Comput. Fluids*, vol. 239, p. 105415, May 2022, doi: 10.1016/j.compfluid.2022.105415. +.. [#barbeau2022] \L. Barbeau, S. Étienne, C. Béguin, and B. Blais, “Development of a high-order continuous Galerkin sharp-interface immersed boundary method and its application to incompressible flow problems,” *Comput. Fluids*, vol. 239, p. 105415, May 2022, doi: `10.1016/j.compfluid.2022.105415 `_\. -`[3] `_ M. Braza, P. Chassaing, and H. H. Minh, “Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder,” *J. Fluid Mech.*, vol. 165, pp. 79–130, Apr. 1986, doi: 10.1017/S0022112086003014. +.. [#braza1986] \M. Braza, P. Chassaing, and H. H. Minh, “Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder,” *J. Fluid Mech.*, vol. 165, pp. 79–130, Apr. 1986, doi: `10.1017/S0022112086003014 `_\. diff --git a/doc/source/examples/incompressible-flow/3d-flow-over-periodic-hills/3d-flow-over-periodic-hills.rst b/doc/source/examples/incompressible-flow/3d-flow-over-periodic-hills/3d-flow-over-periodic-hills.rst index cd04a3ae75..95a41aa4bf 100644 --- a/doc/source/examples/incompressible-flow/3d-flow-over-periodic-hills/3d-flow-over-periodic-hills.rst +++ b/doc/source/examples/incompressible-flow/3d-flow-over-periodic-hills/3d-flow-over-periodic-hills.rst @@ -24,7 +24,7 @@ Files Used in This Example Description of the Case ----------------------- -In this case a well-defined flow passes over a series of hills which repeat along a channel in a periodic fashion as it can be seen in the following figure (taken from ERCOFTAC `[1] `_): +In this case a well-defined flow passes over a series of hills which repeat along a channel in a periodic fashion as it can be seen in the following figure (taken from ERCOFTAC [#ercoftac2010]_): .. image:: images/geometry-description.jpg :alt: The geometry @@ -232,7 +232,7 @@ Due to the complexity of this example we recommend that you run this example usi ---------------------- Results and Discussion ---------------------- -To summarize, a coarse mesh of 250K cells was simulated, using a time step of 0.1 and taking average quantities between 207s and 1000s. The results are compared against established test data from both experiments and another CFD simulation. The experimental data corresponds to the data obtained from Rapp `[2] `_ and the computational data is extracted from the results of the LESOCC CFD code by Breuer et al. `[3] `_. +To summarize, a coarse mesh of 250K cells was simulated, using a time step of 0.1 and taking average quantities between 207s and 1000s. The results are compared against established test data from both experiments and another CFD simulation. The experimental data corresponds to the data obtained from Rapp [#rapp2009]_ and the computational data is extracted from the results of the LESOCC CFD code by Breuer et al. [#breuer2009]_. The following image shows the average velocity profiles in the x-direction: @@ -266,15 +266,15 @@ Possibilities for Extension - **High-order elements**: It would be interesting to observe the effect of high-order elements in the simulation of the periodic hills flow. For example, Q2-Q2 elements. The only part of the parameter file that would need to change would be the ``FEM`` section. -- **High Reynolds numbers**: The example could be run at higher Reynolds numbers. In fact, one can find experimental and numerical results in the literature for Reynolds numbers equal to 10600 or 37000. This comes of course with a higher computational effort. +- **High Reynolds numbers**: The example could be run at higher Reynolds numbers. In fact, one can find experimental and numerical results in the literature for Reynolds numbers equal to 10600 [#breuer2009]_ or 37000. This comes of course with a higher computational effort. ---------- References ---------- -`[1] `_ ERCOFTAC. "File: hill3d.jpg". 2010. https://kbwiki.ercoftac.org/w/index.php/File:Hill3d.jpg. +.. [#ercoftac2010] \ERCOFTAC. "File: hill3d.jpg". 2010. https://kbwiki.ercoftac.org/w/index.php/File:Hill3d.jpg\. -`[2] `_ C. R. Rapp, "Experimentelle studie der turbulenten strömung über periodische hügel", PhD dissertation, Technische Universität München, Munich, Germany, 2009. [Online] Available: https://mediatum.ub.tum.de/doc/677970/677970.pdf +.. [#rapp2009] \C. R. Rapp, "Experimentelle studie der turbulenten strömung über periodische hügel", PhD dissertation, Technische Universität München, Munich, Germany, 2009. [Online] Available: https://mediatum.ub.tum.de/doc/677970/677970.pdf\. -`[3] `_ M. Breuer, N. Peller, Ch. Rapp, and M. Manhart, “Flow over periodic hills – Numerical and experimental study in a wide range of Reynolds numbers,” *Comput. Fluids*, vol. 38, no. 2, pp. 433–457, Feb. 2009, doi: 10.1016/j.compfluid.2008.05.002. +.. [#breuer2009] \M. Breuer, N. Peller, Ch. Rapp, and M. Manhart, “Flow over periodic hills – Numerical and experimental study in a wide range of Reynolds numbers,” *Comput. Fluids*, vol. 38, no. 2, pp. 433–457, Feb. 2009, doi: `10.1016/j.compfluid.2008.05.002 `_\. diff --git a/doc/source/examples/incompressible-flow/3d-mixer-using-single-rotating-frame/3d-mixer-using-single-rotating-frame.rst b/doc/source/examples/incompressible-flow/3d-mixer-using-single-rotating-frame/3d-mixer-using-single-rotating-frame.rst index f6d01e2d71..8c44a014db 100644 --- a/doc/source/examples/incompressible-flow/3d-mixer-using-single-rotating-frame/3d-mixer-using-single-rotating-frame.rst +++ b/doc/source/examples/incompressible-flow/3d-mixer-using-single-rotating-frame/3d-mixer-using-single-rotating-frame.rst @@ -74,7 +74,7 @@ The dimensions of the system are listed in the following table: | :math:`D` | Impeller diameter | :math:`27` cm | +-------------------------+----------------------------------+-------------------------+ -To simulate the flow in such complex geometry, we take advantage of the system's symmetry and opt for a Lagrangian reference frame. Instead of observing the velocity profile from an Eulerian reference frame (or "lab reference frame"), we place ourselves on the impeller's reference making it static and inducing a no-slip boundary condition. This way, the cylindrical tank will be the only moving geometry in our system. The figure below illustrates the difference between the Eulerian and Lagrangian reference frames. +To simulate the flow in such complex geometry, we take advantage of the system's symmetry and opt for a Lagrangian reference frame. Instead of observing the velocity profile from an Eulerian reference frame (or "lab reference frame"), we place ourselves on the impeller's reference making it static and inducing a no-slip boundary condition. This way, the cylindrical tank will be the only moving geometry in our system. The figure below illustrates the difference between the Eulerian and Lagrangian reference frames [#delacroix2020]_. .. image:: images/eulerian-vs-langrangian-reference.jpg :alt: Eulerian and Lagrangian reference frames @@ -503,4 +503,4 @@ It could be interesting to simulate at higher Reynolds numbers in a transient si Reference ----------- -`[1] `_ B. Delacroix, A. Bouarab, L. Fradette, F. Bertrand, and B. Blais, “Simulation of granular flow in a rotating frame of reference using the discrete element method,” *Powder Technol.*, vol. 369, pp. 146–161, Jun. 2020, doi: 10.1016/j.powtec.2020.05.006. +.. [#delacroix2020] \B. Delacroix, A. Bouarab, L. Fradette, F. Bertrand, and B. Blais, “Simulation of granular flow in a rotating frame of reference using the discrete element method,” *Powder Technol.*, vol. 369, pp. 146–161, Jun. 2020, doi: `10.1016/j.powtec.2020.05.006 `_\. diff --git a/doc/source/examples/incompressible-flow/3d-nitsche-mixer-with-pbt-impeller/nitsche-mixer-with-pbt-impeller.rst b/doc/source/examples/incompressible-flow/3d-nitsche-mixer-with-pbt-impeller/nitsche-mixer-with-pbt-impeller.rst index ced1d6dd31..6f0ac34826 100644 --- a/doc/source/examples/incompressible-flow/3d-nitsche-mixer-with-pbt-impeller/nitsche-mixer-with-pbt-impeller.rst +++ b/doc/source/examples/incompressible-flow/3d-nitsche-mixer-with-pbt-impeller/nitsche-mixer-with-pbt-impeller.rst @@ -7,7 +7,7 @@ Simulation of mixing in stirred-tanks is one important industrial application wh This example illustrates how the transient flow in a stirred-tank can be simulated by Lethe using the Nitsche Immersed Boundary (NIB) formulation. .. seealso:: - This example is related to the article *A parallel and adaptative Nitsche immersed boundary method to simulate viscous mixing* by Joachim *et al.* `[1] `_ + This example is related to the article *A parallel and adaptative Nitsche immersed boundary method to simulate viscous mixing* by Joachim *et al.* [#joachim2023]_ -------- @@ -35,7 +35,7 @@ All files mentioned below are located in the example's folder (``examples/incomp Description of the Case ----------------------- -We simulate the flow generated by a pitched blade turbine (PBT) in a stirred tank. The PBT is an axial impeller, which generates a flow pattern that amplifies axial circulation within the vessel (in opposition to radial impellers, e.g. Rushton turbines) `[2] `_. +We simulate the flow generated by a pitched blade turbine (PBT) in a stirred tank. The PBT is an axial impeller, which generates a flow pattern that amplifies axial circulation within the vessel (in opposition to radial impellers, e.g. Rushton turbines) [#paul2003]_. The setup that we wish to simulate is schematized in the following figure: @@ -391,7 +391,6 @@ Possibilities for Extension Reference --------- -`[1] `_ J. Joachim, C.-A. Daunais, V. Bibeau, L. Heltai, and B. Blais, “A parallel and adaptative Nitsche immersed boundary method to simulate viscous mixing,” *J. Comput. Phys.*, vol. 488, p. 112189, Sep. 2023, doi: 10.1016/j.jcp.2023.112189. +.. [#joachim2023] \J. Joachim, C.-A. Daunais, V. Bibeau, L. Heltai, and B. Blais, “A parallel and adaptative Nitsche immersed boundary method to simulate viscous mixing,” *J. Comput. Phys.*, vol. 488, p. 112189, Sep. 2023, doi: `10.1016/j.jcp.2023.112189 `_\. -`[2] `_ E. L. Paul, V. A. Atiemo-Obeng, and S. M. Kresta, -*Handbook of Industrial Mixing*, John Wiley & Sons, Ltd, 2003. doi: 10.1002/0471451452. +.. [#paul2003] \ E. L. Paul, V. A. Atiemo-Obeng, and S. M. Kresta, *Handbook of Industrial Mixing*, John Wiley & Sons, Ltd, 2003, doi: `10.1002/0471451452 `_\. diff --git a/doc/source/examples/incompressible-flow/3d-taylor-green-vortex/3d-taylor-green-vortex.rst b/doc/source/examples/incompressible-flow/3d-taylor-green-vortex/3d-taylor-green-vortex.rst index 4dbc0f962c..0cdb3dd942 100644 --- a/doc/source/examples/incompressible-flow/3d-taylor-green-vortex/3d-taylor-green-vortex.rst +++ b/doc/source/examples/incompressible-flow/3d-taylor-green-vortex/3d-taylor-green-vortex.rst @@ -28,7 +28,7 @@ All files mentioned below are located in the example's folder (``examples/incomp Description of the Case ----------------------- -The Taylor–Green vortex is an unsteady flow of a decaying vortex, which has an exact closed form solution of the incompressible Navier–Stokes equations in Cartesian coordinates. It is named after the British physicist and mathematician Geoffrey Ingram Taylor and his collaborator A. E. Green `[1] `_. In the present case, we simulate one Taylor-Green vortex at a Reynolds number of 1600 in a domain :math:`\Omega = [-\pi,\pi]\times[-\pi,\pi]\times[-\pi,\pi]` using periodic boundary conditions. +The Taylor–Green vortex is an unsteady flow of a decaying vortex, which has an exact closed form solution of the incompressible Navier–Stokes equations in Cartesian coordinates. It is named after the British physicist and mathematician Geoffrey Ingram Taylor and his collaborator A. E. Green [#wikipedia2023]_. In the present case, we simulate one Taylor-Green vortex at a Reynolds number of 1600 in a domain :math:`\Omega = [-\pi,\pi]\times[-\pi,\pi]\times[-\pi,\pi]` using periodic boundary conditions. The three velocity components :math:`[u_x,u_y,u_z]^T` and the pressure :math:`p` are specified at time :math:`t=0` by: @@ -49,7 +49,7 @@ In this case, the vortex, which is initially 2D, will decay by generating smalle E_k &= \frac{1}{\Omega} \int_{\Omega} \frac{\mathbf{u}\cdot \mathbf{u}}{2} \mathrm{d}\Omega \\ \mathcal{E} &= \frac{1}{\Omega} \int_{\Omega} \frac{\mathbf{\omega}\cdot \mathbf{\omega}}{2} \mathrm{d}\Omega -where :math:`\mathbf{\omega}=\nabla \times \mathbf{u}` is the vorticity. The results we obtain are compared to reference spectral results extracted from Wang *et al.* `[2] `_ +where :math:`\mathbf{\omega}=\nabla \times \mathbf{u}` is the vorticity. The results we obtain are compared to reference spectral results extracted from Wang *et al.* [#wang2013]_. -------------- @@ -361,6 +361,6 @@ Possibilities for Extension References ------------ -`[1] `_ “Taylor–Green vortex,” *Wikipedia*. Dec. 01, 2023. Available: https://en.wikipedia.org/wiki/Taylor%E2%80%93Green_vortex +.. [#wikipedia2023] \“Taylor–Green vortex,” *Wikipedia*. Dec. 01, 2023. Available: https://en.wikipedia.org/wiki/Taylor%E2%80%93Green_vortex\. -`[2] `_ Z. J. Wang *et al.*, “High-order CFD methods: current status and perspective,” *Int. J. Numer. Meth. Fluids*, vol. 72, no. 8, pp. 811–845, 2013, doi: 10.1002/fld.3767. \ No newline at end of file +.. [#wang2013] \Z. J. Wang *et al.*, “High-order CFD methods: current status and perspective,” *Int. J. Numer. Meth. Fluids*, vol. 72, no. 8, pp. 811–845, 2013, doi: `10.1002/fld.3767 `_\. diff --git a/doc/source/examples/incompressible-flow/3d-turbulent-taylor-couette/3d-turbulent-taylor-couette.rst b/doc/source/examples/incompressible-flow/3d-turbulent-taylor-couette/3d-turbulent-taylor-couette.rst index 9eea9b0518..fb815dbcb7 100644 --- a/doc/source/examples/incompressible-flow/3d-turbulent-taylor-couette/3d-turbulent-taylor-couette.rst +++ b/doc/source/examples/incompressible-flow/3d-turbulent-taylor-couette/3d-turbulent-taylor-couette.rst @@ -20,16 +20,16 @@ Files Used in This Example All files mentioned below are located in the example's folder (``examples/incompressible-flow/3d-turbulent-taylor-couette``). - Parameter file: ``tc-matrix-free.prm`` -- Reference data files from Wang and Jourdan (2021): `[1] `_ (``enstrophy_wang_p%.dat``) +- Reference data files from Wang and Jourdan (2021): [#wang2021]_ (``enstrophy_wang_p%.dat``) - Postprocessing Python scripts: ``tc-postprocessing.py`` and ``tc-functions.py`` ------------------------ Description of the Case ------------------------ -The Taylor-Couette flow occurs in the annular space between two coaxial cylinders with different angular velocities. For a laminar flow, an analytical solution exists (see `Taylor-Couette Flow `_). As the Reynolds number increases, the flow undergoes a transition where Taylor vortices emerge (symmetrical vortices in the radial-vertical plane). Eventually, as the flow becomes fully turbulent, a chaotic vortex structure appears with intense fluid agitation `[2] `_ . +The Taylor-Couette flow occurs in the annular space between two coaxial cylinders with different angular velocities. For a laminar flow, an analytical solution exists (see `Taylor-Couette Flow `_). As the Reynolds number increases, the flow undergoes a transition where Taylor vortices emerge (symmetrical vortices in the radial-vertical plane). Eventually, as the flow becomes fully turbulent, a chaotic vortex structure appears with intense fluid agitation [#wikipedia2024]_ . -This example is drawn from a case study by Wang and Jourdan `[1] `_. It simulates a turbulent Taylor-Couette flow with a Reynolds number of 4000. It incorporates initial conditions based on a modified version of the laminar solution to generate specific vortical structures, inspired by the Taylor-Green vortex. +This example is drawn from a case study by Wang and Jourdan [#wang2021]_. It simulates a turbulent Taylor-Couette flow with a Reynolds number of 4000. It incorporates initial conditions based on a modified version of the laminar solution to generate specific vortical structures, inspired by the Taylor-Green vortex. The inner cylinder rotates counterclockwise at a constant angular velocity :math:`\omega`, while the outer cylinder remains fixed. Periodic boundary conditions are applied to the upper and lower openings of the annular section. The following figure illustrates the geometry of this case: @@ -334,6 +334,6 @@ Possibilities for Extension References ------------ -`[1] `_ Z. J. Wang and E. Jourdan, “Benchmark for scale-resolving simulation with curved walls: the Taylor Couette flow,” Advances in Aerodynamics, vol. 3, no. 1, Jun. 2021, doi: 10.1186/s42774-021-00071-0. +.. [#wang2021] \Z. J. Wang and E. Jourdan, “Benchmark for scale-resolving simulation with curved walls: the Taylor Couette flow,” Advances in Aerodynamics, vol. 3, no. 1, Jun. 2021, doi: `10.1186/s42774-021-00071-0 `_\. -`[2] `_ “Taylor–Couette flow,” *Wikipedia*. Feb. 15, 2024. Available: https://en.wikipedia.org/wiki/Taylor%E2%80%93Couette_flow \ No newline at end of file +.. [#wikipedia2024] \“Taylor–Couette flow,” *Wikipedia*. Feb. 15, 2024. Available: https://en.wikipedia.org/wiki/Taylor%E2%80%93Couette_flow\. diff --git a/doc/source/examples/multiphysics/3d-dam-break/3d-dam-break.rst b/doc/source/examples/multiphysics/3d-dam-break/3d-dam-break.rst index dbb151bb34..86f4bd8215 100644 --- a/doc/source/examples/multiphysics/3d-dam-break/3d-dam-break.rst +++ b/doc/source/examples/multiphysics/3d-dam-break/3d-dam-break.rst @@ -2,7 +2,7 @@ 3D Dam-Break with an Obstacle =============================== -This example simulates a dam-break experiment from the Maritime Research Institute Netherlands (MARIN) `[1] `_. +This example simulates a dam-break experiment from the Maritime Research Institute Netherlands (MARIN) [#issa2002]_. .. warning:: This example displays the need for improvement of low-viscosity fluid flow simulation of the current numeric model. Further work will be done to improve this aspect of the model. @@ -278,4 +278,4 @@ References ----------- -`[1] `_ R. Issa and D. Violeau, “Test-case 2, 3D dambreaking, Release 1.1,” *ERCOFTAC SPH Eur. Res. Interest Community SIG Électricité Fr. Lab. Natl. Hydraul. Environ.*, 2006, Accessed: Dec. 07, 2022. [Online]. Available: https://www.spheric-sph.org/tests/test-02 \ No newline at end of file +.. [#issa2002] \R. Issa and D. Violeau, “Test-case 2, 3D dambreaking, Release 1.1,” *ERCOFTAC SPH Eur. Res. Interest Community SIG Électricité Fr. Lab. Natl. Hydraul. Environ.*, 2006, Accessed: Dec. 07, 2022. [Online]. Available: https://www.spheric-sph.org/tests/test-02\. diff --git a/doc/source/examples/multiphysics/air-bubble-compression/air-bubble-compression.rst b/doc/source/examples/multiphysics/air-bubble-compression/air-bubble-compression.rst index af4a8466b9..f0fdeb5b9c 100644 --- a/doc/source/examples/multiphysics/air-bubble-compression/air-bubble-compression.rst +++ b/doc/source/examples/multiphysics/air-bubble-compression/air-bubble-compression.rst @@ -3,7 +3,7 @@ Air Bubble Compression ================================ This example simulates the compression of an air bubble by surrounding liquid. -The problem is inspired by the test case of Caltagirone *et al.* `[1] `_ +The problem is inspired by the test case of Caltagirone *et al.* [#caltagirone2011]_ -------- @@ -337,4 +337,4 @@ The following figures present the comparison between the analytical results and References ---------- -`[1] `_ J.-P. Caltagirone, S. Vincent, and C. Caruyer, “A multiphase compressible model for the simulation of multiphase flows,” *Comput. Fluids*, vol. 50, no. 1, pp. 24–34, Nov. 2011, doi: 10.1016/j.compfluid.2011.06.011. \ No newline at end of file +.. [#caltagirone2011] \J.-P. Caltagirone, S. Vincent, and C. Caruyer, “A multiphase compressible model for the simulation of multiphase flows,” *Comput. Fluids*, vol. 50, no. 1, pp. 24–34, Nov. 2011, doi: `10.1016/j.compfluid.2011.06.011 `_\. diff --git a/doc/source/examples/multiphysics/capillary-wave/capillary-wave.rst b/doc/source/examples/multiphysics/capillary-wave/capillary-wave.rst index e77d718d5b..3a6a5f61f3 100644 --- a/doc/source/examples/multiphysics/capillary-wave/capillary-wave.rst +++ b/doc/source/examples/multiphysics/capillary-wave/capillary-wave.rst @@ -2,7 +2,7 @@ Capillary Wave ================================ -This example simulates the damping of a small amplitude capillary wave for different time-steps allowing us to study the capillary time-step constraint. The problem is inspired by the test case of Denner *et al.* `[1] `_ +This example simulates the damping of a small amplitude capillary wave for different time-steps allowing us to study the capillary time-step constraint. The problem is inspired by the test case of Denner *et al.* [#denner2022]_ -------- @@ -84,7 +84,7 @@ Since, the phase fraction (:math:`\phi`) is treated explicitly, the temporal res .. math:: \Delta t_\sigma = \frac{\Delta x}{\sqrt{2} c_\sigma} = \sqrt{\frac{\hat{\rho}}{2\pi\sigma}{{\Delta x}^3}} -with the shortest unambiguously resolved capillary wave having a wavelength of :math:`\lambda_\sigma = 2 \Delta x` `[2] `_. +with the shortest unambiguously resolved capillary wave having a wavelength of :math:`\lambda_\sigma = 2 \Delta x` [#denner2015]_. Therefore, in order to get stable simulation results, :math:`\Delta t < \Delta t_\sigma` should be respected. In this example, different time-steps will be used to explore the stability limit of Lethe's current implementation. @@ -258,7 +258,7 @@ to run the simulation using four CPU cores. Feel free to use more CPU cores. Results ------- -We compare the relative amplitude :math:`\left(\frac{a}{a_0} \right)` of the wave at :math:`x=0` with the analytical solution (equation 22) proposed by Prosperetti `[3] `_. +We compare the relative amplitude :math:`\left(\frac{a}{a_0} \right)` of the wave at :math:`x=0` with the analytical solution (equation 22) proposed by Prosperetti [#prosperetti1981]_. The analytical solution csv file can be generated using: @@ -373,8 +373,8 @@ We would like to thank Prof. Fabian Denner for sharing his time and knowledge th References ---------- -`[1] `_ F. Denner, F. Evrard, and B. van Wachem, “Breaching the capillary time-step constraint using a coupled VOF method with implicit surface tension,” *J. Comput. Phys.*, vol. 459, p. 111128, Jun. 2022, doi: 10.1016/j.jcp.2022.111128. +.. [#denner2022] \F. Denner, F. Evrard, and B. van Wachem, “Breaching the capillary time-step constraint using a coupled VOF method with implicit surface tension,” *J. Comput. Phys.*, vol. 459, p. 111128, Jun. 2022, doi: `10.1016/j.jcp.2022.111128 `_\. -`[2] `_ F. Denner and B. G. M. van Wachem, “Numerical time-step restrictions as a result of capillary waves,” *J. Comput. Phys.*, vol. 285, pp. 24–40, Mar. 2015, doi: 10.1016/j.jcp.2015.01.021. +.. [#denner2015] \F. Denner and B. G. M. van Wachem, “Numerical time-step restrictions as a result of capillary waves,” *J. Comput. Phys.*, vol. 285, pp. 24–40, Mar. 2015, doi: `10.1016/j.jcp.2015.01.021 `_\. -`[3] `_ A. Prosperetti, “Motion of two superposed viscous fluids,” *Phys. Fluids*, vol. 24, no. 7, pp. 1217–1223, Jul. 1981, doi: 10.1063/1.863522. \ No newline at end of file +.. [#prosperetti1981] \A. Prosperetti, “Motion of two superposed viscous fluids,” *Phys. Fluids*, vol. 24, no. 7, pp. 1217–1223, Jul. 1981, doi: `10.1063/1.863522 `_\. diff --git a/doc/source/examples/multiphysics/concentric-heat-exchanger/concentric-heat-exchanger.rst b/doc/source/examples/multiphysics/concentric-heat-exchanger/concentric-heat-exchanger.rst index 7de54d0fcc..14b3809faf 100644 --- a/doc/source/examples/multiphysics/concentric-heat-exchanger/concentric-heat-exchanger.rst +++ b/doc/source/examples/multiphysics/concentric-heat-exchanger/concentric-heat-exchanger.rst @@ -40,7 +40,7 @@ Heat exchangers are common unit operations used in many types of industries to t We consider copper concentric tubes with radii of :math:`R_0=1\text{mm} ,R_1=2\text{mm},R_2=3\text{mm}` in which water circulates. We consider a counter-current flow with an inner tube velocity of :math:`u_i=10\text{mm/s}` and an outer tube velocity of :math:`u_o=-4\text{mm/s}`. The inlet temperature within the inner tube is :math:`100^\circ C` and it is :math:`0^\circ C` in the outer tube. We do not formulate the problem in SI units, but instead we express the fundamental length in mm. This ensures that most variables of interest are close to unit value and this leads to a system matrix with an improved condition number. -We will compare the results we obtain with the CFD simulations with results obtained using the Number of Transfer Unit (NTU) approach (see `[1]`_). Since the flow within both pipes is not developed, the Nusselt number in the inner pipe can be estimated as: +We will compare the results we obtain with the CFD simulations with results obtained using the Number of Transfer Unit (NTU) approach [#incropera2006]_. Since the flow within both pipes is not developed, the Nusselt number in the inner pipe can be estimated as: .. math:: @@ -262,6 +262,4 @@ Possibilities for Extension References ---------------------------- -.. _[1]: - -[1] F. P. Incropera, D. P. DeWitt, T. L. Bergman, and A. S. Lavine, *Fundamentals of heat and mass transfer*, 6th ed. John Wiley & Sons, 2006. +.. [#incropera2006] \F. P. Incropera, D. P. DeWitt, T. L. Bergman, and A. S. Lavine, *Fundamentals of heat and mass transfer*, 6th ed. John Wiley & Sons, 2006\. diff --git a/doc/source/examples/multiphysics/dam-break/dam-break.rst b/doc/source/examples/multiphysics/dam-break/dam-break.rst index 540fbdeb1f..634eb2df67 100644 --- a/doc/source/examples/multiphysics/dam-break/dam-break.rst +++ b/doc/source/examples/multiphysics/dam-break/dam-break.rst @@ -2,7 +2,7 @@ Dam-Break ========================== -This example simulates the dam break experiments of Martin and Moyce `[1] `_. +This example simulates the dam break experiments of Martin and Moyce [#martin1952]_. ---------------------------------- @@ -258,7 +258,7 @@ Run to execute this post-processing code, where ``./output`` is the directory that contains the simulation results. In post-processing, the maximum dimensionless lateral position of the liquid phase is tracked -through time and compared with the experiments of Martin and Moyce (1952) `[1] `_. +through time and compared with the experiments of Martin and Moyce (1952) [#martin1952]_. The following figure shows the result of the post-processing, with a good agreement between the simulation and the experiment: .. image:: images/xmax-t.png @@ -280,4 +280,4 @@ and refines the meshes on the interface. References ---------------------------- -`[1] `_ J. C. Martin *et al.*, “Part IV. An experimental study of the collapse of liquid columns on a rigid horizontal plane,” *Philos. Trans. R. Soc. Lond. Ser. Math. Phys. Sci.*, vol. 244, no. 882, pp. 312–324, Mar. 1952, doi: 10.1098/rsta.1952.0006. \ No newline at end of file +.. [#martin1952] \J. C. Martin *et al.*, “Part IV. An experimental study of the collapse of liquid columns on a rigid horizontal plane,” *Philos. Trans. R. Soc. Lond. Ser. Math. Phys. Sci.*, vol. 244, no. 882, pp. 312–324, Mar. 1952, doi: `10.1098/rsta.1952.0006 `_\. diff --git a/doc/source/examples/multiphysics/laser-melt-pool/laser-melt-pool.rst b/doc/source/examples/multiphysics/laser-melt-pool/laser-melt-pool.rst index 7e1fe59d0f..1e2acc5f9c 100644 --- a/doc/source/examples/multiphysics/laser-melt-pool/laser-melt-pool.rst +++ b/doc/source/examples/multiphysics/laser-melt-pool/laser-melt-pool.rst @@ -2,7 +2,7 @@ Laser Melt Pool ========================== -This example simulates a two-dimensional melt pool with a laser `[1] `_. +This example simulates a two-dimensional melt pool with a laser [#li2022]_. ---------------------------------- @@ -165,7 +165,7 @@ The ``multiphysics`` subsection enables to turn on (``true``) and off (``false`` Laser Parameters ~~~~~~~~~~~~~~~~ -In the ``laser parameters`` section, the parameters of the laser model are defined. The exponential decaying model `[2] `_ is used to simulate the laser heat source. In the exponential decaying model, the laser heat flux is calculated using the following equation: +In the ``laser parameters`` section, the parameters of the laser model are defined. The exponential decaying model [#liu2018]_ is used to simulate the laser heat source. In the exponential decaying model, the laser heat flux is calculated using the following equation: .. math:: q(x,y,z) = \frac{\eta \alpha P}{\pi r^2 \mu} \exp{\left(-\eta \frac{r^2}{R^2}\right)} \exp{\left(- \frac{|z|}{\mu}\right)} @@ -306,6 +306,7 @@ The following animation shows the temperature distribution in the simulations do References ----------- -`[1] `_ E. Li, Z. Zhou, L. Wang, Q. Zheng, R. Zou, and A. Yu, “Melt pool dynamics and pores formation in multi-track studies in laser powder bed fusion process,” *Powder Technol.*, vol. 405, p. 117533, Jun. 2022, doi: 10.1016/j.powtec.2022.117533. +.. [#li2022] \E. Li, Z. Zhou, L. Wang, Q. Zheng, R. Zou, and A. Yu, “Melt pool dynamics and pores formation in multi-track studies in laser powder bed fusion process,” *Powder Technol.*, vol. 405, p. 117533, Jun. 2022, doi: `10.1016/j.powtec.2022.117533 `_\. -`[2] `_ S. Liu, H. Zhu, G. Peng, J. Yin, and X. Zeng, “Microstructure prediction of selective laser melting AlSi10Mg using finite element analysis,” *Mater. Des.*, vol. 142, pp. 319–328, Mar. 2018, doi: 10.1016/j.matdes.2018.01.022 \ No newline at end of file +.. [#liu2018] \S. Liu, H. Zhu, G. Peng, J. Yin, and X. Zeng, “Microstructure prediction of selective laser melting AlSi10Mg using finite element analysis,” *Mater. Des.*, vol. 142, pp. 319–328, Mar. 2018, doi: `10.1016/j.matdes.2018.01.022 + `_\. diff --git a/doc/source/examples/multiphysics/melting-cavity/melting-cavity.rst b/doc/source/examples/multiphysics/melting-cavity/melting-cavity.rst index a877c8cbf9..ce30134206 100644 --- a/doc/source/examples/multiphysics/melting-cavity/melting-cavity.rst +++ b/doc/source/examples/multiphysics/melting-cavity/melting-cavity.rst @@ -2,7 +2,7 @@ Melting Cavity ========================== -This example simulates a two-dimensional gallium melting cavity and is inspired by the work of Blais and Ilinca `[1] `_. +This example simulates a two-dimensional gallium melting cavity and is inspired by the work of Blais and Ilinca [#blais2018]_. .. _two-dimensional gallium melting cavity: https://www.sciencedirect.com/science/article/pii/S0045793018301415 @@ -53,7 +53,7 @@ where :math:`\beta` and :math:`T_{ref}` denote thermal expansion coefficient and A two-dimensional block of gallium (initially in solid phase) is heated from its left wall at :math:`t = 0` s. Its initial temperature is close to (but slightly smaller than) the melting point and the temperature of the left wall is higher than the melting point. Hence, the block starts melting from the left wall. In the melted zone, close to the left wall, the buoyant force (natural convection) creates vortices inside the liquid. -The simulation parameters are selected according to the references `[1] `_, `[2] `_ to satisfy the desired values for the dimensionless numbers +The simulation parameters are selected according to the references [#blais2018]_ [#gau1986]_ to satisfy the desired values for the dimensionless numbers .. math:: \text{Ra} = \frac{\rho^2 \beta g (T_w - T_m) L^3 c_p}{k \mu} = 10^5 @@ -238,7 +238,7 @@ A python post-processing code `(melting-cavity.py)` is added to the example folder to post-process the results. Run ``python3 ./melting-cavity.py ./output`` to execute this post-processing code, where ``./output`` is the directory that -contains the simulation results. In post-processing, the position of the solid-liquid interface at the top, center and bottom of the cavity, as well as the melted volume fraction are plotted and compared with experiments of Gau and Viskanta `[2] `_. Note that the discrepancies in the interfaces are attributed to the two-dimensional simulations and they were also observed and reported by Blais and Ilinca `[1] `_. +contains the simulation results. In post-processing, the position of the solid-liquid interface at the top, center and bottom of the cavity, as well as the melted volume fraction are plotted and compared with experiments of Gau and Viskanta [#gau1986]_. Note that the discrepancies in the interfaces are attributed to the two-dimensional simulations and they were also observed and reported by Blais and Ilinca [#blais2018]_. .. image:: images/xmax-t.png @@ -359,6 +359,6 @@ The following graph shows the evolution of the liquid fraction as a function of References ----------- -`[1] `_ B. Blais and F. Ilinca, “Development and validation of a stabilized immersed boundary CFD model for freezing and melting with natural convection,” *Comput. Fluids*, vol. 172, pp. 564–581, Aug. 2018, doi: 10.1016/j.compfluid.2018.03.037. +.. [#blais2018] \B. Blais and F. Ilinca, “Development and validation of a stabilized immersed boundary CFD model for freezing and melting with natural convection,” *Comput. Fluids*, vol. 172, pp. 564–581, Aug. 2018, doi: `10.1016/j.compfluid.2018.03.037 `_\. -`[2] `_ C. Gau and R. Viskanta, “Melting and Solidification of a Pure Metal on a Vertical Wall,” *J. Heat Transf.*, vol. 108, no. 1, pp. 174–181, Feb. 1986, doi: 10.1115/1.3246884. +.. [#gau1986] \C. Gau and R. Viskanta, “Melting and Solidification of a Pure Metal on a Vertical Wall,” *J. Heat Transf.*, vol. 108, no. 1, pp. 174–181, Feb. 1986, doi: `10.1115/1.3246884 `_\. diff --git a/doc/source/examples/multiphysics/rayleigh-benard-convection/rayleigh-benard-convection.rst b/doc/source/examples/multiphysics/rayleigh-benard-convection/rayleigh-benard-convection.rst index ab747d2729..f34528cea0 100644 --- a/doc/source/examples/multiphysics/rayleigh-benard-convection/rayleigh-benard-convection.rst +++ b/doc/source/examples/multiphysics/rayleigh-benard-convection/rayleigh-benard-convection.rst @@ -2,7 +2,7 @@ Rayleigh-Bénard Convection ========================== -This example simulates two-dimensional Rayleigh–Benard convection `[1] `_ at Rayleigh numbers of :math:`10^4` and :math:`2.5 \times 10^4` . +This example simulates two-dimensional Rayleigh–Benard convection [#venturi2010]_ [#mpi2022]_ at Rayleigh numbers of :math:`10^4` and :math:`2.5 \times 10^4` . ---------------------------------- @@ -187,6 +187,6 @@ Note that at Ra=10000, two vortices exist in the fluid, while an extra (relative References ----------- -`[1] `_ D. Venturi, X. Wan, and G. E. Karniadakis, “Stochastic bifurcation analysis of Rayleigh–Bénard convection,” *J. Fluid Mech.*, vol. 650, pp. 391–413, May 2010, doi: 10.1017/S0022112009993685. +.. [#venturi2010] \D. Venturi, X. Wan, and G. E. Karniadakis, “Stochastic bifurcation analysis of Rayleigh–Bénard convection,” *J. Fluid Mech.*, vol. 650, pp. 391–413, May 2010, doi: `10.1017/S0022112009993685 `_\. -`[2] `_ “Optimal transport.” https://www.mis.mpg.de/applan/research/optimal-transport.html (accessed Jul. 06, 2022). \ No newline at end of file +.. [#mpi2022] \“Rayleigh-Bénard Convection” *Max Planck Institute*, Accessed: 17 Jul. 2024, Available: https://archive.ph/XrJXx\. diff --git a/doc/source/examples/multiphysics/rayleigh-plateau-instability/rayleigh-plateau-instability.rst b/doc/source/examples/multiphysics/rayleigh-plateau-instability/rayleigh-plateau-instability.rst index bcaf326024..89ae716b36 100644 --- a/doc/source/examples/multiphysics/rayleigh-plateau-instability/rayleigh-plateau-instability.rst +++ b/doc/source/examples/multiphysics/rayleigh-plateau-instability/rayleigh-plateau-instability.rst @@ -2,7 +2,7 @@ Rayleigh-Plateau Instability ================================ -This example simulates the transition of a continuous jet to a droplet regime under the influence of a perturbation. The case simulated in this example corresponds to the case J1 in absence of gravity from the work of Denner *et al.* `[1] `_ with the Weber number :math:`We = 20` and the Ohnesorge number :math:`Oh = 0.1`. +This example simulates the transition of a continuous jet to a droplet regime under the influence of a perturbation. The case simulated in this example corresponds to the case J1 in absence of gravity from the work of Denner *et al.* [#denner2022]_ with the Weber number :math:`We = 20` and the Ohnesorge number :math:`Oh = 0.1`. **** @@ -104,7 +104,7 @@ The ``multiphysics`` subsection is used to enable the VOF solver. Physical Properties ~~~~~~~~~~~~~~~~~~~~ -In the ``physical properties`` subsection, we define the jet fluid (``fluid 1``) as presented for case J1 in Denner *et al.* `[1] `_ The viscosity is deduced from the imposed Ohnesorge number :math:`\left(Oh=\frac{\mu_1}{\sigma\rho_1 R_\mathrm{inlet}} \right)` value of :math:`0.1`. The ambient fluid (``fluid 0``) is defined such that the density :math:`\left(\frac{\rho_1}{\rho_0} = 10^3 \right)` and dynamic viscosity :math:`\left(\frac{\mu_1}{\mu_0} = 10^2\right)` ratios are respected. A ``fluid-fluid`` type of material interaction is also defined to specify the ``surface tension model``. In this case, it is set to ``constant`` (default value) with the ``surface tension coefficient`` (:math:`\sigma`) set to :math:`0.0674 \; \mathrm{N \, m^{-1}}`. +In the ``physical properties`` subsection, we define the jet fluid (``fluid 1``) as presented for case J1 in Denner *et al.* [#denner2022]_ The viscosity is deduced from the imposed Ohnesorge number :math:`\left(Oh=\frac{\mu_1}{\sigma\rho_1 R_\mathrm{inlet}} \right)` value of :math:`0.1`. The ambient fluid (``fluid 0``) is defined such that the density :math:`\left(\frac{\rho_1}{\rho_0} = 10^3 \right)` and dynamic viscosity :math:`\left(\frac{\mu_1}{\mu_0} = 10^2\right)` ratios are respected. A ``fluid-fluid`` type of material interaction is also defined to specify the ``surface tension model``. In this case, it is set to ``constant`` (default value) with the ``surface tension coefficient`` (:math:`\sigma`) set to :math:`0.0674 \; \mathrm{N \, m^{-1}}`. .. code-block:: text @@ -288,7 +288,7 @@ This 3D simulation was simulated using the ``3D-delta0_30/rayleigh-plateau-J1-3D Code to Code Comparison ~~~~~~~~~~~~~~~~~~~~~~~ -We compare the dimensionless breakup length :math:`\left(\frac{L_\mathrm{b}}{R_\mathrm{jet}}\right)` with the simulation results from Denner *et al.* `[1] `_ :math:`L_\mathrm{b}` is the breakup length defined as **the shortest distance from the nozzle (inlet) to the tip of the continuous jet**. +We compare the dimensionless breakup length :math:`\left(\frac{L_\mathrm{b}}{R_\mathrm{jet}}\right)` with the simulation results from Denner *et al.* [#denner2022]_ :math:`L_\mathrm{b}` is the breakup length defined as **the shortest distance from the nozzle (inlet) to the tip of the continuous jet**. The results can be postprocessed using the provided Bash script (``rayleigh-plateau-postprocess.sh``). Make sure that the file has executable permissions before calling it with: @@ -317,7 +317,7 @@ The script then calculates an average :math:`L_\mathrm{b}` which is used to eval +-------------------------------------------------------------------------------------------------------------------+ As it can be seen above, for :math:`\delta_0 \leq 0.1`, we observe no breakup. The jet stabilizes despite the perturbation. An additional case was studied at :math:`\delta_0 = 0.12` to check the increasing stabilizing tendency of the jet for lower excitation amplitude values. -We also observe that none of the other evaluation points match with the work of Denner *et al.* `[1] `_ However, a similar trend in values is observed for :math:`\delta_0 \in [0.2,0.5]`. At :math:`\delta_0 = 0.6`, a huge difference is observed. This is due to the way the satellite droplets are formed. As opposed to previous simulations, the satellite droplets are formed from the broken-off part of the jet, decreasing significantly :math:`L_\mathrm{b}` as displayed in the video below. This might have not been the case in the work of Denner *et al.* `[1] `_ +We also observe that none of the other evaluation points match with the work of Denner *et al.* [#denner2022]_ However, a similar trend in values is observed for :math:`\delta_0 \in [0.2,0.5]`. At :math:`\delta_0 = 0.6`, a huge difference is observed. This is due to the way the satellite droplets are formed. As opposed to previous simulations, the satellite droplets are formed from the broken-off part of the jet, decreasing significantly :math:`L_\mathrm{b}` as displayed in the video below. This might have not been the case in the work of Denner *et al.* [#denner2022]_ .. raw:: html @@ -329,4 +329,4 @@ We also observe that none of the other evaluation points match with the work of References ---------- -`[1] `_ F. Denner, F. Evrard, A. A. Castrejón-Pita, J. R. Castrejón-Pita, and B. van Wachem, “Reversal and Inversion of Capillary Jet Breakup at Large Excitation Amplitudes,” *Flow Turbul. Combust.*, vol. 108, no. 3, pp. 843–863, Mar. 2022, doi: 10.1007/s10494-021-00291-w. +.. [#denner2022] \F. Denner, F. Evrard, A. A. Castrejón-Pita, J. R. Castrejón-Pita, and B. van Wachem, “Reversal and Inversion of Capillary Jet Breakup at Large Excitation Amplitudes,” *Flow Turbul. Combust.*, vol. 108, no. 3, pp. 843–863, Mar. 2022, doi: `10.1007/s10494-021-00291-w `_\. diff --git a/doc/source/examples/multiphysics/rayleigh-taylor-instability/rayleigh-taylor-instability.rst b/doc/source/examples/multiphysics/rayleigh-taylor-instability/rayleigh-taylor-instability.rst index b60edf4f03..2f2d65ab6b 100644 --- a/doc/source/examples/multiphysics/rayleigh-taylor-instability/rayleigh-taylor-instability.rst +++ b/doc/source/examples/multiphysics/rayleigh-taylor-instability/rayleigh-taylor-instability.rst @@ -2,7 +2,7 @@ Rayleigh-Taylor Instability ============================ -This example simulates the dynamic evolution of the single-mode Rayleigh-Taylor instability `[1] `_ by density contrast. +This example simulates the dynamic evolution of the single-mode Rayleigh-Taylor instability [#he1999]_ by density contrast. -------- @@ -56,7 +56,7 @@ which result in Reynolds and Atwood numbers equal to At = \frac{\rho_r - 1}{\rho_r + 1} = 0.5 -A perturbed interface defined as :math:`y = 2H + 0.1 H \cos{(2 \pi x / H)}` separates the fluids. At the top and bottom boundaries, a ``no-slip`` boundary condition is applied, while on the left and right walls, a ``periodic`` boundary condition is used. The temporal evolution of the interface is visually compared with the simulations of Garoosi and Hooman `[2] `_ at dimensionless times (:math:`t^* = t \sqrt{\mathbf{g} / H}`) of :math:`1.5`, :math:`2.5`, :math:`3.5`, :math:`4.0` and :math:`4.5`. The temporal evolution of the spike and the bubble positions are then compared to the results of He *et al.* `[1] `_ The term "spike" refers to the lowest point of ``fluid 1`` and the term "bubble" refers to the highest point of ``fluid 0``. +A perturbed interface defined as :math:`y = 2H + 0.1 H \cos{(2 \pi x / H)}` separates the fluids. At the top and bottom boundaries, a ``no-slip`` boundary condition is applied, while on the left and right walls, a ``periodic`` boundary condition is used. The temporal evolution of the interface is visually compared with the simulations of Garoosi and Hooman [#garoosi2022]_ at dimensionless times (:math:`t^* = t \sqrt{\mathbf{g} / H}`) of :math:`1.5`, :math:`2.5`, :math:`3.5`, :math:`4.0` and :math:`4.5`. The temporal evolution of the spike and the bubble positions are then compared to the results of He *et al.* [#he1999]_ The term "spike" refers to the lowest point of ``fluid 1`` and the term "bubble" refers to the highest point of ``fluid 0``. -------------- @@ -315,7 +315,7 @@ The following animation shows the results of this simulation: -In the following figure, we compare the simulation results with that of Garoosi and Hooman (2022) `<[2]_>`_. +In the following figure, we compare the simulation results with that of Garoosi and Hooman (2022) [#garoosi2022]_. .. image:: images/comparison.png @@ -331,9 +331,9 @@ By invoking the ``rayleigh-taylor_postprocess.py`` postprocessing script found w python3 rayleigh-taylor_postprocess.py ./output/adaptive/ -we compare the position of the spike and the bubble with the results of He *et al.* `[1] `_ +we compare the position of the spike and the bubble with the results of He *et al.* [#he1999]_ -In the figure below, it can be seen that as :math:`t^*` increases, there is a growing difference between the spike position of the current simulation and that of He *et al.* `[1] `_ Nevertheless, the bubble position follows the same evolution as the reference. +In the figure below, it can be seen that as :math:`t^*` increases, there is a growing difference between the spike position of the current simulation and that of He *et al.* [#he1999]_ Nevertheless, the bubble position follows the same evolution as the reference. +---------------------------------------------------------------------------------------+ | .. image:: images/spike_and_bubble_evolution_He_et_al_comparison.png | @@ -369,6 +369,6 @@ The following figures show the mass of ``fluid 1`` throughout the simulation wit References ----------- -`[1] `_ X. He, S. Chen, and R. Zhang, “A Lattice Boltzmann Scheme for Incompressible Multiphase Flow and Its Application in Simulation of Rayleigh–Taylor Instability,” *J. Comput. Phys.*, vol. 152, no. 2, pp. 642–663, Jul. 1999, doi: 10.1006/jcph.1999.6257. +.. [#he1999] \X. He, S. Chen, and R. Zhang, “A Lattice Boltzmann Scheme for Incompressible Multiphase Flow and Its Application in Simulation of Rayleigh–Taylor Instability,” *J. Comput. Phys.*, vol. 152, no. 2, pp. 642–663, Jul. 1999, doi: `10.1006/jcph.1999.6257 `_\. -`[2] `_ F. Garoosi and K. Hooman, “Numerical simulation of multiphase flows using an enhanced Volume-of-Fluid (VOF) method,” *Int. J. Mech. Sci.*, vol. 215, p. 106956, Feb. 2022, doi: 10.1016/j.ijmecsci.2021.106956. +.. [#garoosi2022] \F. Garoosi and K. Hooman, “Numerical simulation of multiphase flows using an enhanced Volume-of-Fluid (VOF) method,” *Int. J. Mech. Sci.*, vol. 215, p. 106956, Feb. 2022, doi: `10.1016/j.ijmecsci.2021.106956 `_\. diff --git a/doc/source/examples/multiphysics/rising-bubble/rising-bubble.rst b/doc/source/examples/multiphysics/rising-bubble/rising-bubble.rst index 3d266f0c5f..5dc19c6fd9 100644 --- a/doc/source/examples/multiphysics/rising-bubble/rising-bubble.rst +++ b/doc/source/examples/multiphysics/rising-bubble/rising-bubble.rst @@ -2,7 +2,7 @@ Rising Bubble ========================== -This example simulates a two-dimensional rising bubble `[1] `_. +This example simulates a two-dimensional rising bubble [#zahedi2012]_. -------- @@ -261,7 +261,7 @@ to run the simulation using eight CPU cores. Feel free to use more. Results and Discussion ----------------------- -The following image shows the shape and dimensions of the bubble after :math:`3` seconds of simulation, and compares it with results of `[1] `_. +The following image shows the shape and dimensions of the bubble after :math:`3` seconds of simulation, and compares it with results of [#zahedi2012]_. .. image:: images/bubble.png :alt: bubble @@ -277,7 +277,7 @@ Run python3 ./rising-bubble.py output to execute this post-processing code, where ``output`` is the directory that -contains the simulation results. The results for the barycenter position and velocity of the bubble are compared with the simulations of Zahedi *et al.* `[1] `_ and Hysing *et al.* `[2] `_. The following images show the results of these comparisons. The agreement between the two simulations is remarkable considering the coarse mesh used within this example. +contains the simulation results. The results for the barycenter position and velocity of the bubble are compared with the simulations of Zahedi *et al.* [#zahedi2012]_ and Hysing *et al.* [#hysing2009]_. The following images show the results of these comparisons. The agreement between the two simulations is remarkable considering the coarse mesh used within this example. .. image:: images/ymean-t.png :alt: ymean_t @@ -299,8 +299,8 @@ Animation of the rising bubble example: References ----------- -`[1] `_ S. Zahedi, M. Kronbichler, and G. Kreiss, “Spurious currents in finite element based level set methods for two-phase flow,” *Int. J. Numer. Methods Fluids*, vol. 69, no. 9, pp. 1433–1456, 2012, doi: 10.1002/fld.2643. +.. [#zahedi2012] \S. Zahedi, M. Kronbichler, and G. Kreiss, “Spurious currents in finite element based level set methods for two-phase flow,” *Int. J. Numer. Methods Fluids*, vol. 69, no. 9, pp. 1433–1456, 2012, doi: `10.1002/fld.2643 `_\. -`[2] `_ S. Hysing *et al.*, “Quantitative benchmark computations of two-dimensional bubble dynamics,” *Int. J. Numer. Methods Fluids*, vol. 60, no. 11, pp. 1259–1288, 2009, doi: 10.1002/fld.1934. +.. [#hysing2009] \S. Hysing *et al.*, “Quantitative benchmark computations of two-dimensional bubble dynamics,” *Int. J. Numer. Methods Fluids*, vol. 60, no. 11, pp. 1259–1288, 2009, doi: `10.1002/fld.1934 `_\. -`[3] `_ J. U. Brackbill, D. B. Kothe, and C. Zemach, “A continuum method for modeling surface tension,” *J. Comput. Phys.*, vol. 100, no. 2, pp. 335–354, Jun. 1992, doi: 10.1016/0021-9991(92)90240-Y. +.. [#brackbill1992] \J. U. Brackbill, D. B. Kothe, and C. Zemach, “A continuum method for modeling surface tension,” *J. Comput. Phys.*, vol. 100, no. 2, pp. 335–354, Jun. 1992, doi: `10.1016/0021-9991(92)90240-Y `_\. diff --git a/doc/source/examples/multiphysics/sloshing-in-rectangular-tank/sloshing-in-rectangular-tank.rst b/doc/source/examples/multiphysics/sloshing-in-rectangular-tank/sloshing-in-rectangular-tank.rst index fd1fd839e9..f5fadfe040 100644 --- a/doc/source/examples/multiphysics/sloshing-in-rectangular-tank/sloshing-in-rectangular-tank.rst +++ b/doc/source/examples/multiphysics/sloshing-in-rectangular-tank/sloshing-in-rectangular-tank.rst @@ -2,7 +2,7 @@ Sloshing in a Rectangular Tank ================================ -This example simulates the damping of a small amplitude wave for Reynolds number of (:math:`2`, :math:`20`, :math:`200` and :math:`2000`). The problem is inspired by the test case of Carrica *et al.* `[1] `_ +This example simulates the damping of a small amplitude wave for Reynolds number of (:math:`2`, :math:`20`, :math:`200` and :math:`2000`). The problem is inspired by the test case of Carrica *et al.* [#carrica2007]_ -------- @@ -179,7 +179,7 @@ to run the simulation using eight CPU cores. Feel free to use more. Results ------- -We compare the relative height of the free surface at :math:`x=0` with an analytical solution proposed by Wu *et al.* `[2] `_ For the Reynolds number of :math:`2`, :math:`20` and :math:`200`, data were directly extracted from Carrica *et al.* `[1] `_, whereas for the Reynolds of :math:`2000`, the simplified analytical expression of Wu *et al.* `[2] `_ is used. The results for Reynolds number of :math:`2`, :math:`20`, :math:`200` and :math:`2000` can be post-processed by invoking the following command from the folder of the Reynolds number of interest (:math:`Re=20` in the example below): +We compare the relative height of the free surface at :math:`x=0` with an analytical solution proposed by Wu *et al.* [#wu2001]_ For the Reynolds number of :math:`2`, :math:`20` and :math:`200`, data were directly extracted from Carrica *et al.* [#carrica2007]_, whereas for the Reynolds of :math:`2000`, the simplified analytical expression of Wu *et al.* [#wu2001]_ is used. The results for Reynolds number of :math:`2`, :math:`20`, :math:`200` and :math:`2000` can be post-processed by invoking the following command from the folder of the Reynolds number of interest (:math:`Re=20` in the example below): .. code-block:: text :class: copy-button @@ -217,7 +217,7 @@ The following table presents a comparison between the analytical results and the References ---------- -`[1] `_ P. M. Carrica, R. V. Wilson, and F. Stern, “An unsteady single-phase level set method for viscous free surface flows,” *Int. J. Numer. Methods Fluids*, vol. 53, no. 2, pp. 229–256, 2007, doi: 10.1002/fld.1279. +.. [#carrica2007] \P. M. Carrica, R. V. Wilson, and F. Stern, “An unsteady single-phase level set method for viscous free surface flows,” *Int. J. Numer. Methods Fluids*, vol. 53, no. 2, pp. 229–256, 2007, doi: `10.1002/fld.1279 `_\. -`[2] `_ G. X. Wu, R. Eatock Taylor, and D. M. Greaves, “The effect of viscosity on the transient free-surface waves in a two-dimensional tank,” *J. Eng. Math.*, vol. 40, no. 1, pp. 77–90, May 2001, doi: 10.1023/A:1017558826258. \ No newline at end of file +.. [#wu2001] \G. X. Wu, R. Eatock Taylor, and D. M. Greaves, “The effect of viscosity on the transient free-surface waves in a two-dimensional tank,” *J. Eng. Math.*, vol. 40, no. 1, pp. 77–90, May 2001, doi: `10.1023/A:1017558826258 `_\. diff --git a/doc/source/examples/multiphysics/static-bubble/static-bubble.rst b/doc/source/examples/multiphysics/static-bubble/static-bubble.rst index bb41609962..24e3d7f7c6 100644 --- a/doc/source/examples/multiphysics/static-bubble/static-bubble.rst +++ b/doc/source/examples/multiphysics/static-bubble/static-bubble.rst @@ -2,7 +2,7 @@ Static Bubble ========================== -This example simulates a two-dimensional static bubble `[1] `_. +This example simulates a two-dimensional static bubble [#zahedi2012]_. ---------------------------------- @@ -43,13 +43,13 @@ Surface Tension Force When including the surface tension force in the resolution of the Navier-Stokes equations, the numerical computation of the curvature can give rise to parasitic flows near the interface between the two fluids, as presented in :doc:`../../../theory/multiphase/cfd/vof` theory guide. -The static bubble case is a relevant case to study the parasitic currents, since the analytical solution is zero for the velocity. Therefore, non-zero velocities in the computed velocity field are considered parasitic currents `[1] `_. The analytical pressure drop between the interior (:math:`p_{int}`) and exterior (:math:`p_{ext}`) of the bubble is given by the Young-Laplace relation: +The static bubble case is a relevant case to study the parasitic currents, since the analytical solution is zero for the velocity. Therefore, non-zero velocities in the computed velocity field are considered parasitic currents [#zahedi2012]_. The analytical pressure drop between the interior (:math:`p_{int}`) and exterior (:math:`p_{ext}`) of the bubble is given by the Young-Laplace relation: .. math:: \Delta p = p_{int} - p_{ext} = \sigma \kappa -with the analytical curvature of the 2D bubble : :math:`\kappa = 1/R`. This example is based on the static droplet case reported in `[1] `_, where :math:`\sigma = 1.0`, :math:`R = 0.5` and :math:`\kappa = 2.0`. +with the analytical curvature of the 2D bubble : :math:`\kappa = 1/R`. This example is based on the static droplet case reported in [#zahedi2012]_, where :math:`\sigma = 1.0`, :math:`R = 0.5` and :math:`\kappa = 2.0`. -------------- Parameter File @@ -257,6 +257,6 @@ Finally, the time evolution of the :math:`\mathcal{L}^2` norm of the error on th References ----------- -`[1] `_ S. Zahedi, M. Kronbichler, and G. Kreiss, “Spurious currents in finite element based level set methods for two-phase flow,” *Int. J. Numer. Methods Fluids*, vol. 69, no. 9, pp. 1433–1456, 2012, doi: 10.1002/fld.2643. +.. [#zahedi2012] \S. Zahedi, M. Kronbichler, and G. Kreiss, “Spurious currents in finite element based level set methods for two-phase flow,” *Int. J. Numer. Methods Fluids*, vol. 69, no. 9, pp. 1433–1456, 2012, doi: `10.1002/fld.2643 `_\. -`[2] `_ J. U. Brackbill, D. B. Kothe, and C. Zemach, “A continuum method for modeling surface tension,” *J. Comput. Phys.*, vol. 100, no. 2, pp. 335–354, Jun. 1992, doi: 10.1016/0021-9991(92)90240-Y. +.. [#brackbill1992] \J. U. Brackbill, D. B. Kothe, and C. Zemach, “A continuum method for modeling surface tension,” *J. Comput. Phys.*, vol. 100, no. 2, pp. 335–354, Jun. 1992, doi: `10.1016/0021-9991(92)90240-Y `_\. diff --git a/doc/source/examples/multiphysics/stefan-problem/stefan-problem.rst b/doc/source/examples/multiphysics/stefan-problem/stefan-problem.rst index 6cf68479f9..98bce1a574 100644 --- a/doc/source/examples/multiphysics/stefan-problem/stefan-problem.rst +++ b/doc/source/examples/multiphysics/stefan-problem/stefan-problem.rst @@ -2,7 +2,7 @@ Stefan Problem: Melting of a Solid ==================================== -This example simulates the Stefan `[1] `_ problem following the approach taken by Blais and Ilinca (2018) `[2] `_. +This example simulates the Stefan [#wikipedia2023]_ problem following the approach taken by Blais and Ilinca (2018) [#blais2018]_. ---------------------------------- @@ -29,7 +29,7 @@ Both files mentioned below are located in the example's folder (``examples/multi Description of the Case ------------------------- -The Stefan problem describes the melting or the solidification of a pure substance by conduction in a 1D semi-infinite domain. This classical problem, extensively described in the literature `[3] `_, is often used as the core test case to establish the accuracy and the robustness of numerical models for phase change. Although it is established for a semi-infinite domain, it can be solved on a finite domain provided that it is sufficiently long. This problem is illustrated in the following figure: +The Stefan problem describes the melting or the solidification of a pure substance by conduction in a 1D semi-infinite domain. This classical problem, extensively described in the literature [#wiesche2007]_, is often used as the core test case to establish the accuracy and the robustness of numerical models for phase change. Although it is established for a semi-infinite domain, it can be solved on a finite domain provided that it is sufficiently long. This problem is illustrated in the following figure: .. image:: images/stefan-problem-illustration.png :alt: problem_illustration @@ -67,7 +67,7 @@ the diffusivity coefficient in the liquid phase and :math:`\delta (t)` the melti Although simple, this problem can be challenging to solve numerically because of the sharp impact of the phase change on the specific heat within the solidification interval. Even if this problem is inherently a 1D problem, we analyse it in 2D by generating structured quadrilateral meshes on a [0, 0]X[1, 0.1] domain. The number of nodes in the y direction is kept at 2 (one cell), but it is adjusted to 101 in the x direction which is the direction in which the heat transfer occurs and in which the interface displaces. -Lethe uses a specific heat phase change model to solve this type of problem. This model is quasi-identical to the one described by Blais and Ilinca (2018) `[2] `_. It is also described in the :doc:`../../../parameters/cfd/physical_properties` section of the documentation. +Lethe uses a specific heat phase change model to solve this type of problem. This model is quasi-identical to the one described by Blais and Ilinca (2018) [#blais2018]_. It is also described in the :doc:`../../../parameters/cfd/physical_properties` section of the documentation. -------------- @@ -148,7 +148,7 @@ Next, we define the physical properties: end end -This subsection defines the various parameters of the specific heat model for phase change. Key parameters to note are the solidus and liquidus temperatures. These parameters define the phase change interval, that is the temperature interval over which the phase change occurs. For pure substance, this interval should, in theory, be infinitely small. However, this leads to a numerically unstable solution. Consequently, we set a finite value which should be relatively small, but not too small as to lead to numerical instabilities. In the present case, we set this interval to 0.02C, which is sufficient to guarantee a high degree of accuracy while maintaining numerical stability. The impact of this parameter on the stability and the accuracy of the model has been studied in depth by Blais and Ilinca (2018) `[2] `_. +This subsection defines the various parameters of the specific heat model for phase change. Key parameters to note are the solidus and liquidus temperatures. These parameters define the phase change interval, that is the temperature interval over which the phase change occurs. For pure substance, this interval should, in theory, be infinitely small. However, this leads to a numerically unstable solution. Consequently, we set a finite value which should be relatively small, but not too small as to lead to numerical instabilities. In the present case, we set this interval to 0.02C, which is sufficient to guarantee a high degree of accuracy while maintaining numerical stability. The impact of this parameter on the stability and the accuracy of the model has been studied in depth by Blais and Ilinca (2018) [#blais2018]_. Simulation Control ~~~~~~~~~~~~~~~~~~ @@ -212,8 +212,8 @@ Possibilities for Extension References ---------------------------- -`[1] `_ “Stefan problem,” Wikipedia. Jul. 29, 2023. Accessed: Feb. 19, 2022. [Online]. Available: https://en.wikipedia.org/wiki/Stefan_problem +.. [#wikipedia2023] \“Stefan problem,” Wikipedia. Jul. 29, 2023. Accessed: Feb. 19, 2022. [Online]. Available: https://en.wikipedia.org/wiki/Stefan_problem\. -`[2] `_ B. Blais and F. Ilinca, “Development and validation of a stabilized immersed boundary CFD model for freezing and melting with natural convection,” *Comput. Fluids*, vol. 172, pp. 564–581, Aug. 2018, doi: 10.1016/j.compfluid.2018.03.037. +.. [#blais2018] \B. Blais and F. Ilinca, “Development and validation of a stabilized immersed boundary CFD model for freezing and melting with natural convection,” *Comput. Fluids*, vol. 172, pp. 564–581, Aug. 2018, doi: `10.1016/j.compfluid.2018.03.037 `_\. -`[3] `_ S. aus der Wiesche, “Numerical heat transfer and thermal engineering of AdBlue (SCR) tanks for combustion engine emission reduction,” *Appl. Therm. Eng.*, vol. 27, no. 11, pp. 1790–1798, Aug. 2007, doi: 10.1016/j.applthermaleng.2007.01.008. \ No newline at end of file +.. [#wiesche2007] \S. aus der Wiesche, “Numerical heat transfer and thermal engineering of AdBlue (SCR) tanks for combustion engine emission reduction,” *Appl. Therm. Eng.*, vol. 27, no. 11, pp. 1790–1798, Aug. 2007, doi: `10.1016/j.applthermaleng.2007.01.008 `_\. diff --git a/doc/source/examples/multiphysics/water-injection-in-a-closed-cell/water-injection-in-a-closed-cell.rst b/doc/source/examples/multiphysics/water-injection-in-a-closed-cell/water-injection-in-a-closed-cell.rst index fb0826fc63..a588b50207 100644 --- a/doc/source/examples/multiphysics/water-injection-in-a-closed-cell/water-injection-in-a-closed-cell.rst +++ b/doc/source/examples/multiphysics/water-injection-in-a-closed-cell/water-injection-in-a-closed-cell.rst @@ -3,7 +3,7 @@ Water Injection in a Closed Cell ================================ This example simulates the compression of air in a closed cell by injection of water. -The problem is inspired by the test case of Caltagirone *et al.* `[1] `_ +The problem is inspired by the test case of Caltagirone *et al.* [#caltagirone2011]_ -------- @@ -303,4 +303,4 @@ The following figures present the comparison between the analytical results and References ---------- -`[1] `_ J.-P. Caltagirone, S. Vincent, and C. Caruyer, “A multiphase compressible model for the simulation of multiphase flows,” *Comput. Fluids*, vol. 50, no. 1, pp. 24–34, Nov. 2011, doi: 10.1016/j.compfluid.2011.06.011. \ No newline at end of file +.. [#caltagirone2011] \J.-P. Caltagirone, S. Vincent, and C. Caruyer, “A multiphase compressible model for the simulation of multiphase flows,” *Comput. Fluids*, vol. 50, no. 1, pp. 24–34, Nov. 2011, doi: `10.1016/j.compfluid.2011.06.011 `_\. diff --git a/doc/source/examples/sharp-immersed-boundary/3d-rbf-static-mixer/3d-rbf-static-mixer.rst b/doc/source/examples/sharp-immersed-boundary/3d-rbf-static-mixer/3d-rbf-static-mixer.rst index c3b093b7c3..b0544e2190 100644 --- a/doc/source/examples/sharp-immersed-boundary/3d-rbf-static-mixer/3d-rbf-static-mixer.rst +++ b/doc/source/examples/sharp-immersed-boundary/3d-rbf-static-mixer/3d-rbf-static-mixer.rst @@ -31,7 +31,7 @@ All files mentioned below are located in the example's folder (``/examples/sharp **RBF preparation files:** * Parameter file: ``rbf_generation/RBF.param``; -* Surface grid file: ``rbf_generation/helix.stl``. This surface grid was taken from `[1] `_ under CC BY 4.0. +* Surface grid file: ``rbf_generation/helix.stl``. This surface grid was taken from Thingiverse [#thingiverse]_ under CC BY 4.0. **Lethe's fluid simulation files:** @@ -369,4 +369,4 @@ As the plot shows, the mass conservation is constant after only a few time steps Reference --------- -`[1] `_ Group 9., «Helix Static Mixer» on Thingiverse. +.. [#thingiverse] "Group 9., Helix Static Mixer," *Thingiverse* Available: https://www.thingiverse.com/thing:3915237 diff --git a/doc/source/examples/sharp-immersed-boundary/sedimentation-1-particle/sedimentation-1-particle.rst b/doc/source/examples/sharp-immersed-boundary/sedimentation-1-particle/sedimentation-1-particle.rst index d4ff283c17..b9a3fbf356 100644 --- a/doc/source/examples/sharp-immersed-boundary/sedimentation-1-particle/sedimentation-1-particle.rst +++ b/doc/source/examples/sharp-immersed-boundary/sedimentation-1-particle/sedimentation-1-particle.rst @@ -2,7 +2,7 @@ Sedimentation of One Particle ============================================================================== -This example aims to numerically reproduce the results obtained by Ten Cate `et al.` `[1] `_ for the E4 experience. This experience measures the velocity of the sedimentation of a 1.5 cm particle in a container filled with a viscous fluid. The container is sufficiently small to impact the particle sedimentation. +This example aims to numerically reproduce the results obtained by Ten Cate `et al.` [#tencate2002]_ for the E4 experience. This experience measures the velocity of the sedimentation of a 1.5 cm particle in a container filled with a viscous fluid. The container is sufficiently small to impact the particle sedimentation. .. warning:: @@ -64,7 +64,7 @@ Simulation Control * The ``time step`` is set to 0.0025. This ensures a low error due to the time discretization for this case. -* The ``time end`` is set to 1.3. This is slightly longer than the experimental results of Ten Cate `et al.` `[1] `_. This ensures that the entire trajectory of the particle has been simulated. +* The ``time end`` is set to 1.3. This is slightly longer than the experimental results of Ten Cate `et al.` [#tencate2002]_. This ensures that the entire trajectory of the particle has been simulated. Physical Properties ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ @@ -348,7 +348,7 @@ First, we look at a slice of the velocity profile during the acceleration phase. :alt: flow_field_acceleration :align: center -We can also compare the results obtained for the velocity in time with the results proposed by the article of Ten Cate `et al.` `[1] `_ +We can also compare the results obtained for the velocity in time with the results proposed by the article of Ten Cate `et al.` [#tencate2002]_ .. image:: images/velocity-comparison.png :alt: flow_field_acceleration @@ -359,5 +359,5 @@ We can also compare the results obtained for the velocity in time with the resul Reference --------------- -`[1] `_ A. ten Cate, C. H. Nieuwstad, J. J. Derksen, and H. E. A. Van den Akker, “Particle imaging velocimetry experiments and lattice-Boltzmann simulations on a single sphere settling under gravity,” *Phys. Fluids*, vol. 14, no. 11, pp. 4012–4025, Oct. 2002, doi: 10.1063/1.1512918. +.. [#tencate2002] \A. ten Cate, C. H. Nieuwstad, J. J. Derksen, and H. E. A. Van den Akker, “Particle imaging velocimetry experiments and lattice-Boltzmann simulations on a single sphere settling under gravity,” *Phys. Fluids*, vol. 14, no. 11, pp. 4012–4025, Oct. 2002, doi: `10.1063/1.1512918 `_\. diff --git a/doc/source/examples/sharp-immersed-boundary/sedimentation-64-particles/sedimentation-64-particles.rst b/doc/source/examples/sharp-immersed-boundary/sedimentation-64-particles/sedimentation-64-particles.rst index 5c32fb796b..a6821d5865 100644 --- a/doc/source/examples/sharp-immersed-boundary/sedimentation-64-particles/sedimentation-64-particles.rst +++ b/doc/source/examples/sharp-immersed-boundary/sedimentation-64-particles/sedimentation-64-particles.rst @@ -354,9 +354,4 @@ The results are shown in the animation below. We can see the complex motion of t ---------------- -Reference ---------------- - -`[1] `_ A. ten Cate, C. H. Nieuwstad, J. J. Derksen, and H. E. A. Van den Akker, “Particle imaging velocimetry experiments and lattice-Boltzmann simulations on a single sphere settling under gravity,” *Phys. Fluids*, vol. 14, no. 11, pp. 4012–4025, Oct. 2002, doi: 10.1063/1.1512918. diff --git a/doc/source/parameters/cfd/dynamic_flow_control.rst b/doc/source/parameters/cfd/dynamic_flow_control.rst index 68753a9009..fa3ec290f8 100644 --- a/doc/source/parameters/cfd/dynamic_flow_control.rst +++ b/doc/source/parameters/cfd/dynamic_flow_control.rst @@ -6,7 +6,7 @@ The purpose of this subsection is to enable dynamic flow control. It is importan velocity on a specific boundary (CFD) or the whole domain (for CFD-DEM). To control the average velocity of the flow, the code calculates a :math:`\beta` coefficient at each time step that is used as a source term in the momentum equation to keep the average velocity at a targeted value. -The main controller of the average velocity is the following equation and is based on approach of Wang `[1] `_: +The main controller of the average velocity is the following equation and is based on approach of Wang [#wang2023]_: .. math:: \beta^{n+1} = \beta^n + \frac{\alpha}{\Delta t} \left[ \bar{U}^{0} - 2\bar{U}^{n} + \bar{U}^{n-1} \right] @@ -56,4 +56,4 @@ The default parameters are: Reference --------- -`[1] `_ W. Wang, “A non-body conformal grid method for simulations of laminar and turbulent flows with a compressible large eddy simulation solver,” Ph.D., Ann Arbor, United States, 2009. Accessed: May 4, 2023. [Online]. Available: https://www.proquest.com/docview/304905306 +.. [#wang2023] \W. Wang, “A non-body conformal grid method for simulations of laminar and turbulent flows with a compressible large eddy simulation solver,” Ph.D., Ann Arbor, United States, 2009. Accessed: May 4, 2023. [Online]. Available: https://www.proquest.com/docview/304905306\. diff --git a/doc/source/parameters/cfd/laser_heat_source.rst b/doc/source/parameters/cfd/laser_heat_source.rst index b151d4e95d..46cb36442c 100644 --- a/doc/source/parameters/cfd/laser_heat_source.rst +++ b/doc/source/parameters/cfd/laser_heat_source.rst @@ -39,7 +39,7 @@ If a laser heat source is present in a simulation, it can be added in this secti * The ``power`` parameter sets the power of the laser :math:`[ML^2T^{-3}]`. -* The ``absorptivity`` parameter is defined as the fraction of incident radiation that is absorbed by the surface, and it is measured using diffuse reflectance spectroscopy (DRS). Generally, a constant value in the range of :math:`0.3`-:math:`0.8` (for welding processes with titanium) is used in the literature. However, recent studies show that it varies with powder particle size distribution and the angle of incidence that changes due to the dynamic melt pool surface `[1] `_. +* The ``absorptivity`` parameter is defined as the fraction of incident radiation that is absorbed by the surface, and it is measured using diffuse reflectance spectroscopy (DRS). Generally, a constant value in the range of :math:`0.3`-:math:`0.8` (for welding processes with titanium) is used in the literature. However, recent studies show that it varies with powder particle size distribution and the angle of incidence that changes due to the dynamic melt pool surface [#zhang2019]_. * The ``penetration depth`` parameter determines the penetration depth of the laser in the simulation domain in the direction of emission. @@ -97,7 +97,7 @@ Laser types q(x,y,z) = \frac{|\nabla \psi| \alpha P}{\pi R^2} -* When the ``type`` parameter is set to ``exponential_decay``, the exponential model from Zhang *et al.* `[2] `_ is used to simulate the laser heat source: +* When the ``type`` parameter is set to ``exponential_decay``, the exponential model from Liu *et al.* [#liu2018]_ is used to simulate the laser heat source: .. math:: q(x,y,z) = \frac{\eta \alpha P}{\pi R^2 \mu} \exp{\left(-\eta \frac{r^2}{R^2}\right)} \exp{\left(- \frac{|z|}{\mu}\right)} @@ -117,6 +117,6 @@ Laser types ----------- References ----------- -`[1] `_ Z. Zhang *et al.*, “3-Dimensional heat transfer modeling for laser powder-bed fusion additive manufacturing with volumetric heat sources based on varied thermal conductivity and absorptivity,” *Opt. Laser Technol.*, vol. 109, pp. 297–312, Jan. 2019, doi: 10.1016/j.optlastec.2018.08.012. +.. [#zhang2019] \Z. Zhang *et al.*, “3-Dimensional heat transfer modeling for laser powder-bed fusion additive manufacturing with volumetric heat sources based on varied thermal conductivity and absorptivity,” *Opt. Laser Technol.*, vol. 109, pp. 297–312, Jan. 2019, doi: `10.1016/j.optlastec.2018.08.012 `_\. -`[2] `_ S. Liu, H. Zhu, G. Peng, J. Yin, and X. Zeng, “Microstructure prediction of selective laser melting AlSi10Mg using finite element analysis,” *Mater. Des.*, vol. 142, pp. 319–328, Mar. 2018, doi: 10.1016/j.matdes.2018.01.022. +.. [#liu2018] \S. Liu, H. Zhu, G. Peng, J. Yin, and X. Zeng, “Microstructure prediction of selective laser melting AlSi10Mg using finite element analysis,” *Mater. Des.*, vol. 142, pp. 319–328, Mar. 2018, doi: `10.1016/j.matdes.2018.01.022 `_\. diff --git a/doc/source/parameters/rpt/detector_parameters.rst b/doc/source/parameters/rpt/detector_parameters.rst index d9162f9b22..eb3645e545 100644 --- a/doc/source/parameters/rpt/detector_parameters.rst +++ b/doc/source/parameters/rpt/detector_parameters.rst @@ -25,7 +25,7 @@ This subsection contains the specific information of the detector. ``Detector po Options: Any positive float *(default value: 1)* -The following parameters are variables in the gamma-ray Monte-Carlo model from Beam *et al.* (1978) `[1] `_: +The following parameters are variables in the gamma-ray Monte-Carlo model from Beam *et al.* (1978) [#beam1978]_: - ``activity``: Radioactive source activity of the tracer [Beq] Options: Any positive float *(default value: 1)* @@ -37,5 +37,5 @@ The following parameters are variables in the gamma-ray Monte-Carlo model from B References ~~~~~~~~~~~ -`[1] `_ G. B. Beam, L. Wielopolski, R. P. Gardner, and K. Verghese, “Monte Carlo calculation of efficiencies of right-circular cylindrical NaI detectors for arbitrarily located point sources,” *Nucl. Instrum. Methods*, vol. 154, no. 3, pp. 501–508, Sep. 1978, doi: 10.1016/0029-554X(78)90081-2. +.. [#beam1978] \G. B. Beam, L. Wielopolski, R. P. Gardner, and K. Verghese, “Monte Carlo calculation of efficiencies of right-circular cylindrical NaI detectors for arbitrarily located point sources,” *Nucl. Instrum. Methods*, vol. 154, no. 3, pp. 501–508, Sep. 1978, doi: `10.1016/0029-554X(78)90081-2 `_\. diff --git a/doc/source/parameters/rpt/parameter_tuning.rst b/doc/source/parameters/rpt/parameter_tuning.rst index c4d35baea1..434b5753a7 100644 --- a/doc/source/parameters/rpt/parameter_tuning.rst +++ b/doc/source/parameters/rpt/parameter_tuning.rst @@ -2,7 +2,7 @@ Parameter Tuning ================ -This subsection contains information regarding the tuning parameters with NOMAD. Enable tuning parameters requires the ``verbosity parameter`` in the subsection ``rpt parameters`` to be disabled by setting it as ``quiet`` otherwise it will interact with `NOMAD `_ since it needs the cost function value only. So far there are 3 types of cost functions implemented, one from Larachi *et al.* (1994) `[1] `_, the L1 function, and the L2 function. To tune parameters, the cost function compares the calculated counts with the Monte Carlo technique and the measured counts that are provided in the ``.experimental`` file. The three parameters ``dead time``, ``activity`` and ``attenuation coefficient reactor`` seen in the `detector parameter subsection <./detector_parameters.html>`_ of the ``.prm`` file are obtained using NOMAD. The second example `Tuning Parameters with NOMAD <../../examples/rpt/tuning-parameters-with-nomad/tuning-parameters-with-nomad.html>`_ explains how we can obtain the values of these parameters using NOMAD. +This subsection contains information regarding the tuning parameters with NOMAD. Enable tuning parameters requires the ``verbosity parameter`` in the subsection ``rpt parameters`` to be disabled by setting it as ``quiet`` otherwise it will interact with `NOMAD `_ since it needs the cost function value only. So far there are 3 types of cost functions implemented, one from Larachi *et al.* (1994) [#larachi1994]_, the L1 function, and the L2 function. To tune parameters, the cost function compares the calculated counts with the Monte Carlo technique and the measured counts that are provided in the ``.experimental`` file. The three parameters ``dead time``, ``activity`` and ``attenuation coefficient reactor`` seen in the `detector parameter subsection <./detector_parameters.html>`_ of the ``.prm`` file are obtained using NOMAD. The second example `Tuning Parameters with NOMAD <../../examples/rpt/tuning-parameters-with-nomad/tuning-parameters-with-nomad.html>`_ explains how we can obtain the values of these parameters using NOMAD. .. code-block:: text @@ -43,5 +43,5 @@ This subsection contains information regarding the tuning parameters with NOMAD. References ~~~~~~~~~~~ -`[1] `_ F. Larachi, G. Kennedy, and J. Chaouki, “A γ-ray detection system for 3-D particle tracking in multiphase reactors,” *Nucl. Instrum. Methods Phys. Res. Sect. Accel. Spectrometers Detect. Assoc. Equip.*, vol. 338, no. 2, pp. 568–576, Jan. 1994, doi: 10.1016/0168-9002(94)91343-9. +.. [#larachi1994] \F. Larachi, G. Kennedy, and J. Chaouki, “A γ-ray detection system for 3-D particle tracking in multiphase reactors,” *Nucl. Instrum. Methods Phys. Res. Sect. Accel. Spectrometers Detect. Assoc. Equip.*, vol. 338, no. 2, pp. 568–576, Jan. 1994, doi: `10.1016/0168-9002(94)91343-9 `_\. diff --git a/doc/source/parameters/rpt/rpt_parameters.rst b/doc/source/parameters/rpt/rpt_parameters.rst index accf67d89e..beeefde928 100644 --- a/doc/source/parameters/rpt/rpt_parameters.rst +++ b/doc/source/parameters/rpt/rpt_parameters.rst @@ -39,7 +39,7 @@ This subsection contains the general information required for the photon count c Options: Any positive float *(default value: 0.1)* -The following parameters are variables in the gamma-ray Monte-Carlo model from Beam *et al.* (1978) `[1] `_: +The following parameters are variables in the gamma-ray Monte-Carlo model from Beam *et al.* (1978) [#beam1978]_: - ``peak-to-total ratio``: The proportion of the events appearing in the full energy peak to the total number of events [] Options: Any positive float *(default value: 1)* @@ -54,5 +54,5 @@ The following parameters are variables in the gamma-ray Monte-Carlo model from B References ~~~~~~~~~~~ -`[1] `_ G. B. Beam, L. Wielopolski, R. P. Gardner, and K. Verghese, “Monte Carlo calculation of efficiencies of right-circular cylindrical NaI detectors for arbitrarily located point sources,” *Nucl. Instrum. Methods*, vol. 154, no. 3, pp. 501–508, Sep. 1978, doi: 10.1016/0029-554X(78)90081-2. +.. [#beam1978] \G. B. Beam, L. Wielopolski, R. P. Gardner, and K. Verghese, “Monte Carlo calculation of efficiencies of right-circular cylindrical NaI detectors for arbitrarily located point sources,” *Nucl. Instrum. Methods*, vol. 154, no. 3, pp. 501–508, Sep. 1978, doi: `10.1016/0029-554X(78)90081-2 `_\. diff --git a/doc/source/parameters/sharp-immersed-boundary/sharp-immersed-boundary.rst b/doc/source/parameters/sharp-immersed-boundary/sharp-immersed-boundary.rst index 2c453fdc99..67bea4e7dd 100644 --- a/doc/source/parameters/sharp-immersed-boundary/sharp-immersed-boundary.rst +++ b/doc/source/parameters/sharp-immersed-boundary/sharp-immersed-boundary.rst @@ -185,7 +185,7 @@ This subsection contains the parameters related to the sharp immersed boundary s * The ``lubrication range min`` parameter defines the minimal distance used in the lubrication force calculation. The range is defined as a multiple of the smallest cell. This limits the force that can be applied on a particle since the lubrification force has a singularity when the distance between 2 particles is 0. We use this parameter to define a lower bound on the distance between 2 particles for the force calculation to avoid this singularity. Physically, this distance can be interpreted as the surface roughness of the particles. .. note:: - The lubrication force between two particles is expressed by the equation :math:`\mathbf{F_{lub_{ij}}} = \frac{3}{2} \pi \mu_f \left(\frac{d_{p_i} d_{p_j}}{d_{p_i}+d_{p_j}}\right)^2 \frac{1}{y}(\mathbf{v_{ij}}\cdot \mathbf{e_{ij}})\mathbf{e_{ij}}`. Where :math:`\mu_f` is the fluid viscosity, :math:`d_{p_i}` the diameter of the first particle, :math:`d_{p_j}` the diameter of the second particle, :math:`y` the gap between the two particles, :math:`\mathbf{v_{ij}}` the relative velocity of the two particles, :math:`\mathbf{e_{ij}}` the unit vector along the line that joint the centroide of the two particles. In the case of particle wall lubrication force we take the diameter of the second particle to be infinity `[1] `_. + The lubrication force between two particles is expressed by the equation :math:`\mathbf{F_{lub_{ij}}} = \frac{3}{2} \pi \mu_f \left(\frac{d_{p_i} d_{p_j}}{d_{p_i}+d_{p_j}}\right)^2 \frac{1}{y}(\mathbf{v_{ij}}\cdot \mathbf{e_{ij}})\mathbf{e_{ij}}`. Where :math:`\mu_f` is the fluid viscosity, :math:`d_{p_i}` the diameter of the first particle, :math:`d_{p_j}` the diameter of the second particle, :math:`y` the gap between the two particles, :math:`\mathbf{v_{ij}}` the relative velocity of the two particles, :math:`\mathbf{e_{ij}}` the unit vector along the line that joint the centroide of the two particles. In the case of particle wall lubrication force we take the diameter of the second particle to be infinity [#kim2005]_. This model requires a constant viscosity and density of the fluid. * The ``particle nonlinear tolerance`` parameter controls particle dynamics' nonlinear tolerance. The nonlinear solver won't have converged until the residual on the dynamics equations of all the particles is smaller than this threshold. @@ -342,4 +342,4 @@ Therefore, the near-particle zone around each particle is refined ``mesh``:``ini Reference --------------- -`[1] `_ S. Kim and S. J. Karrila, *Microhydrodynamics: Principles and Selected Applications*. Courier Corporation, 2005. +.. [#kim2005] \S. Kim and S. J. Karrila, *Microhydrodynamics: Principles and Selected Applications*. Courier Corporation, 2005. Available: https://books.google.ca/books?id=_8llnUUGo0wC&lpg=PP1&hl=pt-BR&pg=PP1#v=onepage&q&f=false\. diff --git a/doc/source/theory/multiphase/cfd/cahn-hilliard.rst b/doc/source/theory/multiphase/cfd/cahn-hilliard.rst index 270c8febdc..916ca2c914 100644 --- a/doc/source/theory/multiphase/cfd/cahn-hilliard.rst +++ b/doc/source/theory/multiphase/cfd/cahn-hilliard.rst @@ -2,7 +2,7 @@ Cahn-Hilliard Method ================================ -The Cahn-Hilliard system of equations `[1] `_ is a model used to describe the process of phase separation based on the principle of free energy minimization. The key idea at the heart of the Cahn-Hilliard equation is that the system try to achieve a state where the free energy is minimized while competing with the energy cost associated with creating new interfaces between phases. Let us introduce those concepts formally. +The Cahn-Hilliard system of equations [#cahn1958]_ is a model used to describe the process of phase separation based on the principle of free energy minimization. The key idea at the heart of the Cahn-Hilliard equation is that the system try to achieve a state where the free energy is minimized while competing with the energy cost associated with creating new interfaces between phases. Let us introduce those concepts formally. Let :math:`\Omega = \Omega_0 \cup \Omega_1` be the domain formed by two fluids, namely fluid :math:`0` and :math:`1`, with :math:`\Gamma` the boundaries of the system. Like in :doc:`vof`, we define a scalar function :math:`\phi` as a phase indicator such that: @@ -142,9 +142,8 @@ This tensor is added to the usual viscous stress tensor to take into account the \mathbf{f_\sigma} & = \nabla \cdot (\lambda(\nabla \phi \otimes \nabla \phi))\\ & = \eta\nabla\phi + \nabla\Psi \end{align} - -We then define a modified pressure :math:`\hat{p}`, which corresponds to the usual pressure with the additional :math:`\Psi` term. This new pressure is the same in the bulk phases and varies more smoothly in the interface `[2] `_. -Then, to take into account the change of momentum of the system due to the diffusive flux of species, we add the following term into the :doc:`usual momentum equation<../../multiphysics/fluid_dynamics/navier-stokes>`: + +We then define a modified pressure :math:`\hat{p}`, which corresponds to the usual pressure with the additional :math:`\Psi` term. This new pressure is the same in the bulk phases and varies more smoothly in the interface [#lovric2019]_. Then, to take into account the change of momentum of the system due to the diffusive flux of species, we add the following term into the :doc:`usual momentum equation<../../multiphysics/fluid_dynamics/navier-stokes>`: .. math:: (\mathbf{\tilde{J}}\cdot \nabla)\mathbf{v} = (\frac{\rho_0-\rho_1}{2}\mathbf{J}\cdot \nabla)\mathbf{v} @@ -165,7 +164,7 @@ The Cahn-Hilliard-Navier-Stokes momentum equation solved in Lethe is: & - \nabla \cdot \left(\mu(\phi)(\nabla\mathbf{u} + \nabla\mathbf{u}^\mathbf{T})\right) + \nabla \hat{p} - \eta\nabla\phi = 0 \\ \end{align} -With an adequate choice of definition of velocity (see `[3] `_), the velocity field remains divergence-free: +With an adequate choice of definition of velocity [#abels2011]_, the velocity field remains divergence-free: .. math:: \nabla \cdot \mathbf{u} = 0 @@ -191,6 +190,9 @@ References +.. [#cahn1958] \J. W. Cahn and J. E. Hilliard, ‘Free Energy of a Nonuniform System. I. Interfacial Free Energy’, The Journal of Chemical Physics, vol. 28, no. 2, pp. 258–267, Feb. 1958, doi: `10.1063/1.1744102 `_\. - +.. [#lovric2019] \A. Lovrić, W. G. Dettmer, and D. Perić, ‘Low Order Finite Element Methods for the Navier-Stokes-Cahn-Hilliard Equations’. arXiv, Nov. 15, 2019. doi: `10.48550/arXiv.1911.06718 `_\. + +.. [#abels2011] \H. Abels, H. Garcke, and G. Grün, ‘Thermodynamically Consistent, Frame Indifferent Diffuse Interface Models for Incompressible Two-Phase Flows with Different Densities’. arXiv, Apr. 07, 2011. doi: `10.48550/arXiv.1104.1336 `_\. diff --git a/doc/source/theory/multiphase/cfd/vof.rst b/doc/source/theory/multiphase/cfd/vof.rst index 08946d2e9d..57cfc8f713 100644 --- a/doc/source/theory/multiphase/cfd/vof.rst +++ b/doc/source/theory/multiphase/cfd/vof.rst @@ -4,7 +4,7 @@ The Volume of Fluid (VOF) Method Numerous examples of flow encountered in engineering involve multiple fluids: sloshing of fuel in aircraft tanks, mixing of bread dough, and motion of droplets and bubbles to name a few. In these cases, the involved fluids can be immiscible, and we are interested in the evolution of the interfaces between those fluids. -Let :math:`\Omega = \Omega_0 \cup \Omega_1` be the domain formed by two fluids, namely fluid :math:`0` and :math:`1`, with :math:`\Gamma` denoting their interface and :math:`\partial \Omega`, the remaining boundaries, as illustrated in the figure below. In the VOF method `[1] `_, we define the scalar function :math:`\phi` as a phase indicator such that: +Let :math:`\Omega = \Omega_0 \cup \Omega_1` be the domain formed by two fluids, namely fluid :math:`0` and :math:`1`, with :math:`\Gamma` denoting their interface and :math:`\partial \Omega`, the remaining boundaries, as illustrated in the figure below. In the VOF method [#hirt1981]_, we define the scalar function :math:`\phi` as a phase indicator such that: .. math:: \phi = @@ -84,7 +84,7 @@ Find :math:`\phi^h \in \Phi^h \times [0,T]` such that Stabilization -------------- -The numerical resolution of the advection equation requires stabilization because of its purely advective character, which makes the equation hyperbolic. Furthermore, a second stabilization term is added to improve the capturing of the interface due to sharp gradient across :math:`\Gamma`. Since SUPG only adds diffusion along the streamlines, crosswind oscillations may occur if no appropriate shock capturing scheme is used. To that end, a Discontinuity-Capturing Directional Dissipation (DCDD) shock capturing scheme is used `[2] `_: +The numerical resolution of the advection equation requires stabilization because of its purely advective character, which makes the equation hyperbolic. Furthermore, a second stabilization term is added to improve the capturing of the interface due to sharp gradient across :math:`\Gamma`. Since SUPG only adds diffusion along the streamlines, crosswind oscillations may occur if no appropriate shock capturing scheme is used. To that end, a Discontinuity-Capturing Directional Dissipation (DCDD) shock capturing scheme is used [#tezduyar2003]_: .. math:: @@ -140,7 +140,7 @@ The phase fraction limiter above will update the phase fraction if it failed to \end{cases} where :math:`c` denotes the sharpening threshold, which defines -a phase fraction threshold (generally :math:`0.5`), and :math:`\alpha` corresponds to the interface sharpness, which is a model parameter generally in the range of :math:`(1,2]`. This interface sharpening method was proposed by Aliabadi and Tezduyar (2000) `[3] `_. +a phase fraction threshold (generally :math:`0.5`), and :math:`\alpha` corresponds to the interface sharpness, which is a model parameter generally in the range of :math:`(1,2]`. This interface sharpening method was proposed by Aliabadi and Tezduyar (2000) [#aliabadi2000]_. """""""""""""""""""""""""""""""" @@ -157,7 +157,7 @@ where :math:`\phi'` is the filtered phase fraction value, and :math:`\beta` is a Surface Tension --------------- -When two immiscible fluids are in contact, surface tension tends to deform their interface (also called the free surface) into a shape that ensures a minimal energy state. An example would be the force that drives a droplet into its spherical shape `[4] `_. +When two immiscible fluids are in contact, surface tension tends to deform their interface (also called the free surface) into a shape that ensures a minimal energy state. An example would be the force that drives a droplet into its spherical shape [#brackbill1992]_. Resolution of the interface motion via the advection equation allows to compute the surface tension term and add its effect to the Navier-Stokes momentum equations. @@ -167,7 +167,7 @@ As its name suggests, the surface tension :math:`\bf{f_{\sigma}}` is a surface f {\bf{f_{\sigma}}} = \sigma \kappa {\bf{n}} -where :math:`\sigma` is the surface tension coefficient, :math:`\kappa` is the curvature and :math:`\bf{n}` is the unit normal vector of the free surface. Here, :math:`{\bf{f_{\sigma}}}` is a force per unit of area. To account for its effect in the Navier-Stokes equations, the surface force is transformed in a volumetric surface force :math:`\bf{F_{\sigma}}` using the continuous surface force (CSF) model `[4] `_, that is: +where :math:`\sigma` is the surface tension coefficient, :math:`\kappa` is the curvature and :math:`\bf{n}` is the unit normal vector of the free surface. Here, :math:`{\bf{f_{\sigma}}}` is a force per unit of area. To account for its effect in the Navier-Stokes equations, the surface force is transformed in a volumetric surface force :math:`\bf{F_{\sigma}}` using the continuous surface force (CSF) model [#brackbill1992]_, that is: .. math:: @@ -191,7 +191,7 @@ and the unit normal vector of the free surface, pointing from fluid 0 to 1, is o \bf{n} = \frac{\nabla \phi}{|\nabla \phi|} -When including the surface tension force in the resolution of the Navier-Stokes equations, the numerical computation of the curvature can give rise to parasitic flows near the interface between the two fluids. To avoid such spurious currents, the phase fraction gradient and curvature are filtered using projection steps `[5] `_, as presented in section :ref:`Normal and curvature computations`. +When including the surface tension force in the resolution of the Navier-Stokes equations, the numerical computation of the curvature can give rise to parasitic flows near the interface between the two fluids. To avoid such spurious currents, the phase fraction gradient and curvature are filtered using projection steps [#zahedi2012]_, as presented in section :ref:`Normal and curvature computations`. .. _Normal and curvature computations: @@ -227,12 +227,12 @@ where :math:`\alpha` and :math:`\beta` are user-defined factors, and :math:`h` i References ----------- -`[1] `_ C. W. Hirt and B. D. Nichols, “Volume of fluid (VOF) method for the dynamics of free boundaries,” *J. Comput. Phys.*, vol. 39, no. 1, pp. 201–225, Jan. 1981, doi: 10.1016/0021-9991(81)90145-5. +.. [#hirt1981] \C. W. Hirt and B. D. Nichols, “Volume of fluid (VOF) method for the dynamics of free boundaries,” *J. Comput. Phys.*, vol. 39, no. 1, pp. 201–225, Jan. 1981, doi: `10.1016/0021-9991(81)90145-5 `_\. -`[2] `_ T. E. Tezduyar, “Computation of moving boundaries and interfaces and stabilization parameters,” *Int. J. Numer. Methods Fluids*, vol. 43, no. 5, pp. 555–575, 2003, doi: 10.1002/fld.505. +.. [#tezduyar2003] \T. E. Tezduyar, “Computation of moving boundaries and interfaces and stabilization parameters,” *Int. J. Numer. Methods Fluids*, vol. 43, no. 5, pp. 555–575, 2003, doi: `10.1002/fld.505 `_\. -`[3] `_ S. Aliabadi and T. E. Tezduyar, “Stabilized-finite-element/interface-capturing technique for parallel computation of unsteady flows with interfaces,” *Comput. Methods Appl. Mech. Eng.*, vol. 190, no. 3, pp. 243–261, Oct. 2000, doi: 10.1016/S0045-7825(00)00200-0. +.. [#aliabadi2000] \S. Aliabadi and T. E. Tezduyar, “Stabilized-finite-element/interface-capturing technique for parallel computation of unsteady flows with interfaces,” *Comput. Methods Appl. Mech. Eng.*, vol. 190, no. 3, pp. 243–261, Oct. 2000, doi: `10.1016/S0045-7825(00)00200-0 `_\. -`[4] `_ J. U. Brackbill, D. B. Kothe, and C. Zemach, “A continuum method for modeling surface tension,” *J. Comput. Phys.*, vol. 100, no. 2, pp. 335–354, Jun. 1992, doi: 10.1016/0021-9991(92)90240-Y. +.. [#brackbill1992] \J. U. Brackbill, D. B. Kothe, and C. Zemach, “A continuum method for modeling surface tension,” *J. Comput. Phys.*, vol. 100, no. 2, pp. 335–354, Jun. 1992, doi: `10.1016/0021-9991(92)90240-Y `_\. -`[5] `_ S. Zahedi, M. Kronbichler, and G. Kreiss, “Spurious currents in finite element based level set methods for two-phase flow,” *Int. J. Numer. Methods Fluids*, vol. 69, no. 9, pp. 1433–1456, 2012, doi: 10.1002/fld.2643. \ No newline at end of file +.. [#zahedi2012] \S. Zahedi, M. Kronbichler, and G. Kreiss, “Spurious currents in finite element based level set methods for two-phase flow,” *Int. J. Numer. Methods Fluids*, vol. 69, no. 9, pp. 1433–1456, 2012, doi: `10.1002/fld.2643 `_\. diff --git a/doc/source/theory/multiphase/cfd_dem/dem.rst b/doc/source/theory/multiphase/cfd_dem/dem.rst index cb75484422..45d9d581e4 100644 --- a/doc/source/theory/multiphase/cfd_dem/dem.rst +++ b/doc/source/theory/multiphase/cfd_dem/dem.rst @@ -2,7 +2,7 @@ Discrete Element Method (DEM) ==================================== -In this guide, we summarize the theory behind DEM. For further details, we refer the reader to the article by Blais *et al.* `[1] `_ and the one by Golshan *et al.* `[2] `_ +In this guide, we summarize the theory behind DEM. For further details, we refer the reader to the article by Blais *et al.* [#blais2019]_ and the one by Golshan *et al.* [#golshan2023]_ .. math:: @@ -51,7 +51,7 @@ Where: :align: center :alt: particle-particle_collision - Representation of a typical particle-particle contact. [2] + Representation of a typical particle-particle contact. [#golshan2023]_ The contact normal vector :math:`\mathbf{n}_{ij}` is computed as: @@ -87,7 +87,7 @@ The spring and damping constants for the linear and nonlinear viscoelastic model * - Parameters - Linear model definitions - - Nonlinear viscoelastic model definitions + - Nonlinear viscoelastic model definitions [#garg2012]_ * - Normal spring constant - :math:`k_n = \frac{16}{15}\sqrt{R_{e}}Y_{e}\left(\frac{15m_{e}V^2}{16\sqrt{R_{e}}Y_{e}}\right)^{0.2}` - :math:`k_n = \frac{4}{3}Y_{e}\sqrt{R_{e}\delta_n}` @@ -198,7 +198,7 @@ The parameters are computed as follows: Cohesive force models ----------------------- -Lethe supports two cohesive force models: the Johnson-Kendall-Roberts (JKR) and the Derjaguin-Muller-Toporov (DMT). Both models describe attractive forces due to van der Waals effects. Choosing the right model can be based on the Tabor parameter :math:`\mathbf{\tau}` which represents the ratio between the normal elastic deformation caused by adhesion and the distance at which adhesion forces occur. `[4] `_ +Lethe supports two cohesive force models: the Johnson-Kendall-Roberts (JKR) and the Derjaguin-Muller-Toporov (DMT). Both models describe attractive forces due to van der Waals effects. Choosing the right model can be based on the Tabor parameter :math:`\mathbf{\tau}` which represents the ratio between the normal elastic deformation caused by adhesion and the distance at which adhesion forces occur. [#grierson2005]_ This parameter can be described as: @@ -212,7 +212,7 @@ Where :math:`\mathbf{z_{o}}` is the equilibrium separation of the surfaces and : Johnson-Kendall-Roberts force model ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -The Johnson-Kendall-Roberts (JKR) model describes attractive forces due to van der Waals effects. `[5] `_ +The Johnson-Kendall-Roberts (JKR) model describes attractive forces due to van der Waals effects. [#coetzee2023]_ This model modifies the Hertz formulation by defining a larger contact path radius (:math:`\mathbf{a}`) and by taking into account the effective surface energy (:math:`\mathbf{\gamma}_{e}`). The model is defined by: @@ -228,7 +228,7 @@ The effective surface energy can be computed as: \gamma_{e} = \gamma_{i} + \gamma_{j} - 2\gamma_{i,j} Where :math:`\gamma_{i}` and :math:`\gamma_{j}` are the surface energy of each material (particle or wall) and where :math:`\gamma_{i,j}` is the interface energy which is equal to zero when both surfaces are the same material. -The interface energy term is approximated using `[6] `_: +The interface energy term is approximated using [#israelachvili–289]_: .. math:: \gamma_{i,j} \approx \left( \sqrt{\gamma_{i}} - \sqrt{\gamma_{j}} \right)^{2} @@ -243,7 +243,7 @@ This equation can be rewritten as a fourth-order polynomial function with two co .. math:: 0 = a^{4} - 2R_{e}\delta_{n}a^{2} - 2\pi\gamma_{e}R_{e}^{2}a + R_{e}^{2}\delta_{n}^{2} -Since we are always solving for the same real root, a straightforward procedure, described by Parteli et al. can be used `[7] `_: +Since we are always solving for the same real root, a straightforward procedure, described by Parteli et al. can be used [#parteli2014]_: .. math:: c_{0} &= R_{e}^{2}\delta_{n}^{2} \\ @@ -268,9 +268,9 @@ Finally, the :math:`\mathbf{F_{n}^{JKR}}` can be computed as follows: The normal damping, tangential damping and tangential spring constants need to be computed using the same procedure as the nonlinear model. -A simplified version of the JKR model (SJKR-A) is implemented in Lethe. Please refer to C. J. Coetzee and O. C. Scheffler for more information on the different versions of the JKR model and their specific features `[5] `_. +A simplified version of the JKR model (SJKR-A) is implemented in Lethe. Please refer to C. J. Coetzee and O. C. Scheffler for more information on the different versions of the JKR model and their specific features [#coetzee2023]_. -A modified Coulomb's limit, based on the work of C. Thornton `[9] `_, is used for the JKR model. Using the usual limit can result in permanent slip since the total normal force can be equal to zero even when there is a substantial overlap between particles. +A modified Coulomb's limit, based on the work of C. Thornton [#thornton1991]_, is used for the JKR model. Using the usual limit can result in permanent slip since the total normal force can be equal to zero even when there is a substantial overlap between particles. The modified Coulomb's criterion is breached when the following condition is broken during a collision: @@ -287,7 +287,7 @@ Where :math:`\mathbf{F_{po}}` is the pull-off force, which can be computed as fo Derjaguin-Muller-Toporov force model ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -The Derjaguin-Muller-Toporov (DMT) model describes attractive forces due to van der Waals effects. This model is more suitable for particles with smaller diameter, lower surface energy and higher Young's modulus. In Lethe, the DMT model is implemented using the Maugis approximation which simply adds an adhesion term :math:`\mathbf{F_{ad}^{DMT}}` to the normal force calculation `[8] `_. +The Derjaguin-Muller-Toporov (DMT) model describes attractive forces due to van der Waals effects. This model is more suitable for particles with smaller diameter, lower surface energy and higher Young's modulus. In Lethe, the DMT model is implemented using the Maugis approximation which simply adds an adhesion term :math:`\mathbf{F_{ad}^{DMT}}` to the normal force calculation [#violano2018]_. .. math:: \mathbf{F_{ad}^{DMT}} = -2\pi\gamma_{e}R_{e} \mathbf{n}_{ij} @@ -321,20 +321,20 @@ And velocity Verlet method is calculated with half-step velocity as: References ------------- -`[1] `_ Bruno Blais, David Vidal, Francois Bertrand, Gregory S. Patience and Jamal Chaouki, “Experimental Methods in Chemical Engineering: Discrete Element Method—DEM,” *Can. J. Chem. Eng.*, 97: 1964-1973. https://doi.org/10.1002/cjce.23501. +.. [#blais2019] \B. Blais, D. Vidal, F. Bertrand, G. S. Patience and J. Chaouki, “Experimental Methods in Chemical Engineering: Discrete Element Method—DEM,” *Can. J. Chem. Eng.*, vol. 97, pp. 1964-1973, 2019, doi: `10.1002/cjce.23501 `_\. -`[2] `_ S. Golshan, P. Munch, R. Gassmöller, M. Kronbichler, and B. Blais, “Lethe-DEM: an open-source parallel discrete element solver with load balancing,” *Comput. Part. Mech.*, vol. 10, no. 1, pp. 77–96, Feb. 2023, doi: 10.1007/s40571-022-00478-6. +.. [#golshan2023] \S. Golshan, P. Munch, R. Gassmöller, M. Kronbichler, and B. Blais, “Lethe-DEM: an open-source parallel discrete element solver with load balancing,” *Comput. Part. Mech.*, vol. 10, no. 1, pp. 77–96, Feb. 2023, doi: `10.1007/s40571-022-00478-6 `_\. -`[3] `_ R. Garg, J. Galvin-Carney, T. Li, and S. Pannala, “Documentation of open-source MFIX–DEM software for gas-solids flows,” Tingwen Li Dr., p. 10, Sep. 2012. +.. [#garg2012] \R. Garg, J. Galvin-Carney, T. Li, and S. Pannala, “Documentation of open-source MFIX–DEM software for gas-solids flows,” Tingwen Li Dr., p. 10, Accessed: Sep. 2012, Available: https://mfix.netl.doe.gov/doc/mfix-archive/mfix_current_documentation/dem_doc_2012-1.pdf\. -`[4] `_ D. S. Grierson, E. E. Flater, and R. W. Carpick, “Accounting for the JKR–DMT transition in adhesion and friction measurements with atomic force microscopy,” Journal of Adhesion Science and Technology, vol. 19, no. 3–5, pp. 291–311, Jan. 2005, doi: 10.1163/1568561054352685. +.. [#grierson2005] \D. S. Grierson, E. E. Flater, and R. W. Carpick, “Accounting for the JKR–DMT transition in adhesion and friction measurements with atomic force microscopy,” *Journal of Adhesion Science and Technology*, vol. 19, no. 3–5, pp. 291–311, Jan. 2005, doi: `10.1163/1568561054352685 `_\. -`[5] `_ C. J. Coetzee and O. C. Scheffler, “Review: The Calibration of DEM Parameters for the Bulk Modelling of Cohesive Materials,” Processes, vol. 11, no. 1, Art. no. 1, Jan. 2023, doi: 10.3390/pr11010005. +.. [#coetzee2023] \C. J. Coetzee and O. C. Scheffler, “Review: The Calibration of DEM Parameters for the Bulk Modelling of Cohesive Materials,” *Processes*, vol. 11, no. 1, Art. no. 1, Jan. 2023, doi: `10.3390/pr11010005 `_\. -`[6] `_ J. N. Israelachvili, “Chapter 13 - Van der Waals Forces between Particles and Surfaces,” in Intermolecular and Surface Forces (Third Edition), Third Edition., J. N. Israelachvili, Ed., Boston: Academic Press, 2011, pp. 253–289. doi: https://doi.org/10.1016/B978-0-12-391927-4.10013-1. +.. [#israelachvili2011] \J. N. Israelachvili, “Chapter 13 - Van der Waals Forces between Particles and Surfaces,” in *Intermolecular and Surface Forces*, 3rd ed., J. N. Israelachvili, Ed., Boston: Academic Press, 2011, pp. 253–289, doi: `10.1016/B978-0-12-391927-4.10013-1 `_\. -`[7] `_ E. J. R. Parteli, J. Schmidt, C. Blümel, K.-E. Wirth, W. Peukert, and T. Pöschel, “Attractive particle interaction forces and packing density of fine glass powders,” Sci Rep, vol. 4, no. 1, Art. no. 1, Sep. 2014, doi: 10.1038/srep06227. +.. [#parteli2014] \E. J. R. Parteli, J. Schmidt, C. Blümel, K.-E. Wirth, W. Peukert, and T. Pöschel, “Attractive particle interaction forces and packing density of fine glass powders,” *Sci Rep*, vol. 4, no. 1, Art. no. 1, Sep. 2014, doi: `10.1038/srep06227 `_\. -`[8] `_ Violano, Guido, Giuseppe Pompeo Demelio, and Luciano Afferrante. “On the DMT Adhesion Theory: From the First Studies to the Modern Applications in Rough Contacts.” Procedia Structural Integrity, AIAS 2018 international conference on stress analysis, 12 (January 1, 2018): 58–70. https://doi.org/10.1016/j.prostr.2018.11.106. +.. [#violano2018] \G. Violano, G. P. Demelio, and L. Afferrante, “On the DMT Adhesion Theory: From the First Studies to the Modern Applications in Rough Contacts.” *Procedia Structural Integrity*, vol. 12, pp. 58–70, Jan. 2018, doi: `0.1016/j.prostr.2018.11.106 `_\. -`[9] `_ C. Thornton, “ Interparticle sliding in the presence of adhesion,” Journal of Physics D: Applied Physics, vol. 24, no. 11, pp. 1942–1946, 1991, https://doi.org/10.1088/0022-3727/24/11/007. \ No newline at end of file +.. [#thornton1991] \C. Thornton, “ Interparticle sliding in the presence of adhesion,” *Journal of Physics D: Applied Physics*, vol. 24, no. 11, pp. 1942–1946, 1991, doi: `10.1088/0022-3727/24/11/007 `_\. diff --git a/doc/source/theory/multiphase/cfd_dem/unresolved_cfd-dem.rst b/doc/source/theory/multiphase/cfd_dem/unresolved_cfd-dem.rst index d45264dbde..4a92fcd5ef 100644 --- a/doc/source/theory/multiphase/cfd_dem/unresolved_cfd-dem.rst +++ b/doc/source/theory/multiphase/cfd_dem/unresolved_cfd-dem.rst @@ -12,7 +12,7 @@ The micro-meso scale approach allows for particle-fluid simulations involving la :name: geometry :scale: 40 -In this guide, we summarize the theory behind Unresolved CFD-DEM. For further details, we refer the reader to the articles by Bérard *et al.* `[1] `_, and Zhou *et al.* `[2] `_. +In this guide, we summarize the theory behind Unresolved CFD-DEM. For further details, we refer the reader to the articles by Bérard *et al.* [#berard2020]_, and Zhou *et al.* [#zhou2010]_. Particles ---------- @@ -28,7 +28,7 @@ where: * :math:`m_i` is the mass of the particle; * :math:`\mathbf{v}_i` is the velocity vector; * :math:`\mathbf{f}_{c,ij}` are the contact forces between particles :math:`i` and :math:`j` (detailed in the DEM section of this guide); -* :math:`\mathbf{f}_{nc,ij}` are the non-contact forces between particles :math:`i` and :math:`j`, such as lubrication forces `[3] `_ (**not yet available on Lethe**); +* :math:`\mathbf{f}_{nc,ij}` are the non-contact forces between particles :math:`i` and :math:`j`, such as lubrication forces [#nitsche1994]_ (**not yet available on Lethe**); * :math:`\mathbf{f}_{pf,i}` is the force exerted by the surrounding fluid over particle :math:`i`; * :math:`\mathbf{f}_{g,i}` is the gravitational force; * :math:`I_i` is the moment of inertia; @@ -72,7 +72,7 @@ where: * :math:`\mathbf{u}` is the the fluid velocity vector; * :math:`\varepsilon_f` is the void fraction. -Models A and B differ from each other in the way the momentum equation is calculated. In Model A, we consider that the pressure and the viscous shear stress are in both phases, while for Model B both are only in the fluid `[2] `_: +Models A and B differ from each other in the way the momentum equation is calculated. In Model A, we consider that the pressure and the viscous shear stress are in both phases, while for Model B both are only in the fluid [#zhou2010]_: Model A: @@ -107,7 +107,7 @@ Lethe is capable of simulating unresolved CFD-DEM cases with both Models A and B Void Fraction -------------- -Determining the void fraction is an important step in unresolved CFD-DEM, as can be noted by the VANS equations and the drag models `[4] `_. There exist several methods for the calculation of the void fraction in a CFD-DEM simulation. Some are approximations while others are analytical approaches. In the finite element method, the void fraction is initially calculated inside a cell but must then be projected to the mesh nodes so that one can assemble the system of equations. This is done by :math:`\mathcal{L}^2` projection `[6] `_: +Determining the void fraction is an important step in unresolved CFD-DEM, as can be noted by the VANS equations and the drag models [#rong2013]_. There exist several methods for the calculation of the void fraction in a CFD-DEM simulation. Some are approximations while others are analytical approaches. In the finite element method, the void fraction is initially calculated inside a cell but must then be projected to the mesh nodes so that one can assemble the system of equations. This is done by :math:`\mathcal{L}^2` projection [#larson2013]_: .. math:: \min_{\varepsilon_f \in \mathbb{R}} \frac{1}{2} \sum_i \left (\sum_j \varepsilon_{f,j} \varphi_j - \varepsilon_{f,i} \right ) @@ -130,7 +130,7 @@ where :math:`L` is the smoothing length, used as parameter in Lethe unresolved C The Particle Centroid Method ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -The Particle Centroid Method (PCM) `[5] `_ is simple and the most popular method. It consists of tracking the position of the centroid of each particle and applying the total volume of the particle to the calculation of the void fraction of the cell. This means that in either of the following situations the void fraction of the colored cell is the same: +The Particle Centroid Method (PCM) [#peng2014]_ is simple and the most popular method. It consists of tracking the position of the centroid of each particle and applying the total volume of the particle to the calculation of the void fraction of the cell. This means that in either of the following situations the void fraction of the colored cell is the same: .. image:: images/void_frac1.png :width: 49% @@ -165,7 +165,7 @@ where :math:`n_{sp}` is the number of pseudo-particles j belonging to particle i The Quadrature Centered Method ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -The Quadrature Centered Method (QCM) `[7] `_ is an analytical method that decouples the averaging volume from the mesh cells. It constructs an averaging sphere centered at each quadrature point in a given cell, and it calculates the void fraction directly in the averaging volume at the quadrature point. Since the sphere-sphere (particle-averaging sphere) intersection is analytically easier to calculate than sphere-polyhedron (particle-mesh cell), this method is less expensive than other analytical methods as the intersection does not involve the calculation of trigonometric functions at each CFD time step. The advantage of this method is that the void fraction varies within a cell. Additionally, particles in neighboring cells can affect the void fraction of the current cell. This allows the method to be continuous in both space and time. This is advantageous, especially in solid-liquid systems where the term :math:`\rho_f \frac{\partial \epsilon_f}{\partial t}` of the continuity equation is very stiff and unstable, when there exist even small discontinuities in the void fraction, and where it explodes when :math:`\Delta t_{CFD} \to 0`. +The Quadrature Centered Method (QCM) [#geitani2023]_ is an analytical method that decouples the averaging volume from the mesh cells. It constructs an averaging sphere centered at each quadrature point in a given cell, and it calculates the void fraction directly in the averaging volume at the quadrature point. Since the sphere-sphere (particle-averaging sphere) intersection is analytically easier to calculate than sphere-polyhedron (particle-mesh cell), this method is less expensive than other analytical methods as the intersection does not involve the calculation of trigonometric functions at each CFD time step. The advantage of this method is that the void fraction varies within a cell. Additionally, particles in neighboring cells can affect the void fraction of the current cell. This allows the method to be continuous in both space and time. This is advantageous, especially in solid-liquid systems where the term :math:`\rho_f \frac{\partial \epsilon_f}{\partial t}` of the continuity equation is very stiff and unstable, when there exist even small discontinuities in the void fraction, and where it explodes when :math:`\Delta t_{CFD} \to 0`. An averaging volume sphere is constructed around each quadrature point. All particles lying in the sphere will contribute to the void fraction value of this quadrature point. Therefore, a cell will be affected by the particles lying in it and in its neighboring cells. @@ -183,19 +183,19 @@ where :math:`V^N_{sphere}` is the normalized volume of the volume averaging sphe .. math:: \frac{h_{\Omega}}{2} \leq R_s \leq h_{\Omega} - -Reference + +References ----------- -`[1] `_ A. Bérard, G. S. Patience, and B. Blais, “Experimental methods in chemical engineering: Unresolved CFD-DEM,” *Can. J. Chem. Eng.*, vol. 98, no. 2, pp. 424–440, 2020, doi: 10.1002/cjce.23686. +.. [#berard2020] \A. Bérard, G. S. Patience, and B. Blais, “Experimental methods in chemical engineering: Unresolved CFD-DEM,” *Can. J. Chem. Eng.*, vol. 98, no. 2, pp. 424–440, 2020, doi: `10.1002/cjce.23686 `_\. -`[2] `_ Z. Y. Zhou, S. B. Kuang, K. W. Chu, and A. B. Yu, “Discrete particle simulation of particle–fluid flow: model formulations and their applicability,” *J. Fluid Mech.*, vol. 661, pp. 482–510, Oct. 2010, doi: 10.1017/S002211201000306X. +.. [#zhou2010] \Z. Y. Zhou, S. B. Kuang, K. W. Chu, and A. B. Yu, “Discrete particle simulation of particle–fluid flow: model formulations and their applicability,” *J. Fluid Mech.*, vol. 661, pp. 482–510, Oct. 2010, doi: `10.1017/S002211201000306X `_\. -`[3] `_ L. C. Nitsche, “Microhydrodynamics: Principles and selected applications. By Sangtae Kim and Seppo J. Karrila, Butterworth-Heinemann, Boston, 1991” *AIChE J.*, vol. 40, no. 4, pp. 739–743, 1994, doi: 10.1002/aic.690400418. +.. [#nitsche1994] \L. C. Nitsche, “Microhydrodynamics: Principles and selected applications. By Sangtae Kim and Seppo J. Karrila, Butterworth-Heinemann, Boston, 1991” *AIChE J.*, vol. 40, no. 4, pp. 739–743, 1994, doi: `10.1002/aic.690400418 `_\. -`[4] `_ L. W. Rong, K. J. Dong, and A. B. Yu, “Lattice-Boltzmann simulation of fluid flow through packed beds of uniform spheres: Effect of porosity,” *Chem. Eng. Sci.*, vol. 99, pp. 44–58, Aug. 2013, doi: 10.1016/j.ces.2013.05.036. +.. [#rong2013] \L. W. Rong, K. J. Dong, and A. B. Yu, “Lattice-Boltzmann simulation of fluid flow through packed beds of uniform spheres: Effect of porosity,” *Chem. Eng. Sci.*, vol. 99, pp. 44–58, Aug. 2013, doi: `10.1016/j.ces.2013.05.036 `_\. -`[5] `_ Z. Peng, E. Doroodchi, C. Luo, and B. Moghtaderi, “Influence of void fraction calculation on fidelity of CFD-DEM simulation of gas-solid bubbling fluidized beds,” *AIChE J.*, vol. 60, no. 6, pp. 2000–2018, 2014, doi: 10.1002/aic.14421. +.. [#peng2014] \Z. Peng, E. Doroodchi, C. Luo, and B. Moghtaderi, “Influence of void fraction calculation on fidelity of CFD-DEM simulation of gas-solid bubbling fluidized beds,” *AIChE J.*, vol. 60, no. 6, pp. 2000–2018, 2014, doi: `10.1002/aic.14421 `_\. -`[6] `_ M. G. Larson and F. Bengzon, *The Finite Element Method: Theory, Implementation, and Applications*. Springer Science & Business Media, 2013. +.. [#larson2013] \M. G. Larson and F. Bengzon, *The Finite Element Method: Theory, Implementation, and Applications*. Springer Science & Business Media, 2013. https://link.springer.com/book/10.1007/978-3-642-33287-6\. -`[7] `_ T. El Geitani and B. Blais, “Quadrature-Centered Averaging Scheme for Accurate and Continuous Void Fraction Calculation in Computational Fluid Dynamics–Discrete Element Method Simulations,” *Ind. Eng. Chem. Res.*, vol. 62, no. 12, pp. 5394–5407, Mar. 2023, doi: 10.1021/acs.iecr.3c00172. \ No newline at end of file +.. [#geitani2023] \T. El Geitani and B. Blais, “Quadrature-Centered Averaging Scheme for Accurate and Continuous Void Fraction Calculation in Computational Fluid Dynamics–Discrete Element Method Simulations,” *Ind. Eng. Chem. Res.*, vol. 62, no. 12, pp. 5394–5407, Mar. 2023, doi: `10.1021/acs.iecr.3c00172 `_\.