Update flow past a backward facing step example (#1389)

Description
The post-processing scripts were not working for this example and the example was long and hard to follow. This PR refactors the example by limiting it to two Reynolds numbers: 100 and 1000. To summarize:

All the parameters are revisited to ensure correct convergence for both cases.
The folders and parameter files are renamed so that it is clear for which Reynolds they are.
The results are limited to: visualizing the velocity profiles (with streamlines to identify eddy after step and zoom in plots, all in one figure), reporting the reattachment point for the last step and its relative error in comparison to reference data, and comparing the obtained velocity distribution at the outlet to the analytical solution.
The post-processing scripts are fixed so that they work for both Reynolds number by just changing the appropriate flag when running the script. Moreover, the number of vtu files is no longer hardcoded and can be found automatically by the script.

Co-authored-by: Victor Oliveira Ferreira <voferreira@outlook.com>

examples/incompressible-flow/2d-backward-facing-step/Reynolds100-600/2D-backward-facing-step-steady.prm → examples/incompressible-flow/2d-backward-facing-step/Reynolds100/Reynolds100_steady.prm

@@ -9,6 +9,7 @@ set dimension = 2
 #---------------------------------------------------
 # Simulation Control
 #---------------------------------------------------
+
 subsection simulation control
   set method            = steady
   set number mesh adapt = 10
@@ -23,7 +24,7 @@ end
 subsection physical properties
   set number of fluids = 1
   subsection fluid 0
-    set kinematic viscosity = 0.01 # Re_h=2/nu
+    set kinematic viscosity = 0.02 # Re_h=2/nu
   end
 end
 
@@ -60,13 +61,13 @@ end
 #---------------------------------------------------
 
 subsection boundary conditions
-  set number         = 3
+  set number = 3
   subsection bc 0
-    set   id = 0
+    set id   = 0
     set type = noslip
   end
   subsection bc 1
-    set   id = 1
+    set id   = 1
     set type = function
     subsection u
       set Function expression = 1
@@ -79,7 +80,7 @@ subsection boundary conditions
     end
   end
   subsection bc 2
-    set   id = 2
+    set id   = 2
     set type = outlet
   end
 end
@@ -90,9 +91,8 @@ end
 
 subsection non-linear solver
   subsection fluid dynamics
-    set verbosity      = verbose
-    set tolerance      = 1e-6
-    set max iterations = 10
+    set verbosity = verbose
+    set tolerance = 1e-6
   end
 end
 
@@ -104,8 +104,8 @@ subsection linear solver
   subsection fluid dynamics
     set verbosity                             = verbose
     set method                                = gmres
-    set max iters                             = 1000
-    set max krylov vectors                    = 1000
+    set max iters                             = 300
+    set max krylov vectors                    = 300
     set relative residual                     = 1e-4
     set minimum residual                      = 1e-9
     set preconditioner                        = ilu

examples/incompressible-flow/2d-backward-facing-step/Reynolds700-1000/2D-backward-facing-step-steadybdf.prm → examples/incompressible-flow/2d-backward-facing-step/Reynolds1000/Reynolds1000_steadybdf.prm

@@ -17,10 +17,8 @@ subsection simulation control
   set adapt                        = true
   set max cfl                      = 1e5
   set adaptative time step scaling = 1.2
-  set number mesh adapt            = 0
   set output name                  = backward_facing_step_output
   set output frequency             = 1
-  set output boundaries            = false
 end
 
 #---------------------------------------------------
@@ -30,7 +28,7 @@ end
 subsection physical properties
   set number of fluids = 1
   subsection fluid 0
-    set kinematic viscosity = 0.002857 # Re=2/nu
+    set kinematic viscosity = 0.002 # Re=2/nu
   end
 end
 
@@ -48,13 +46,11 @@ end
 #---------------------------------------------------
 
 subsection mesh adaptation
-  set variable             = velocity
-  set type                 = kelly
-  set fraction refinement  = 0.2
-  set min refinement level = 0
-  set max refinement level = 10
-  set max number elements  = 250000
-  set frequency            = 5
+  set variable            = velocity
+  set type                = kelly
+  set fraction refinement = 0.2
+  set max number elements = 250000
+  set frequency           = 5
 end
 
 #---------------------------------------------------
@@ -71,8 +67,7 @@ end
 #---------------------------------------------------
 
 subsection boundary conditions
-  set number         = 3
-  set time dependent = false
+  set number = 3
   subsection bc 0
     set type = noslip
   end
@@ -89,7 +84,7 @@ subsection boundary conditions
     end
   end
   subsection bc 2
-    set   id = 2
+    set id   = 2
     set type = outlet
   end
 end
@@ -108,10 +103,8 @@ end
 
 subsection non-linear solver
   subsection fluid dynamics
-    set solver         = newton
-    set verbosity      = verbose
-    set tolerance      = 1e-6
-    set max iterations = 10
+    set verbosity = verbose
+    set tolerance = 1e-6
   end
 end
 
@@ -121,13 +114,13 @@ end
 
 subsection linear solver
   subsection fluid dynamics
-    set verbosity                             = verbose
-    set method                                = gmres
-    set max iters                             = 1000
-    set max krylov vectors                    = 1000
-    set relative residual                     = 1e-4
-    set minimum residual                      = 1e-9
-    set preconditioner                        = amg
-    set amg preconditioner ilu fill           = 1
+    set verbosity                   = verbose
+    set method                      = gmres
+    set max iters                   = 500
+    set max krylov vectors          = 500
+    set relative residual           = 1e-4
+    set minimum residual            = 1e-9
+    set preconditioner              = amg
+    set amg preconditioner ilu fill = 1
   end
 end

examples/incompressible-flow/2d-backward-facing-step/reattachment_length.py

@@ -0,0 +1,123 @@
+# SPDX-FileCopyrightText: Copyright (c) 2022, 2024 The Lethe Authors
+# SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception OR LGPL-2.1-or-later
+
+"""
+Postprocessing code for 2D-backward-facing-step example
+Computes the reattachment length (x_r) for Re = 100 and Re = 1000
+for several meshes using a bisection algorithm
+"""
+
+import numpy as np
+import matplotlib.pyplot as plt
+import pyvista as pv
+import re
+import os
+import argparse
+import sys
+
+########################################
+########################################
+
+# EXAMPLE TO COMPUTE x_r (REATTACHMENT LENGTH) FOR Re = 100 and Re = 1000
+# NOTE : THIS FILE MUST BE IN THE "2d-backward-facing-step" DIRECTORY TO 
+#        WORK PROPERLY
+
+# Parse Reynolds number
+parser = argparse.ArgumentParser(description='Arguments to compute the velocity distribution at outlet')
+parser.add_argument("-Re", "--Reynolds", type=int, help="Reynolds number (only 100 and 1000 supported)", required=True)
+
+args, leftovers = parser.parse_known_args()
+Re = args.Reynolds
+
+if (Re != 100 and Re != 1000):
+    sys.exit("This Reynolds number is not supported. Only Re = 100 and Re = 1000.")
+
+#Define folder path according to Re
+folder = "./Reynolds" + str(Re)
+
+# VARIABLES
+L_in = 15           # Inlet length
+
+
+# Initial guesses depending on the Reynolds number and reference reattachment values
+# by Erturk (2008) (see attached .txt file for other Reynolds x_r values)
+if (Re == 100):
+    x_inf = 1.5
+    x_sup = 3
+    x_r_ref = 2.922    
+elif (Re == 1000):
+    x_inf = 12
+    x_sup = 14
+    x_r_ref = 13.121 
+
+tol = 1e-12                  # Bisection stop criterion
+
+# DATA EXTRACTION
+max_number = -1
+pattern = re.compile(r'backward_facing_step_output\.(\d+)\.00000.vtu')
+
+for filename in os.listdir(folder):
+    match = pattern.match(filename)
+    if match:
+        number = int(match.group(1))
+        if number > max_number:
+            max_number = number
+
+n_files_to_read = max_number
+x_r = []
+
+for i in range(0, n_files_to_read):
+    # Read all files except the 0 file
+    file = (folder + '/backward_facing_step_output.' 
+                   + f'{i+1:05d}' + '.00000.vtu')
+    data = pv.read(file)
+    data.set_active_vectors("velocity")
+
+    # Initial guesses
+    x1 = L_in + x_inf
+    x2 = L_in + x_sup
+    j = 0
+    # Bisection loop
+    while abs(x2-x1)>tol and j<=20:
+        xm = (x1+x2)/2
+
+        # Profiles extraction
+        a = np.array([x1, 0, 0])
+        b = np.array([x1, 0.01, 0])
+        profil1 = data.sample_over_line(a, b, resolution=1000)
+        u1 = profil1["velocity"][:,0]
+        y = profil1["Distance"]
+
+        a = np.array([xm, 0, 0])
+        b = np.array([xm, 0.01, 0])
+        profilm = data.sample_over_line(a, b, resolution=1000)
+        um = profilm["velocity"][:,0]
+
+        a = np.array([x2, 0, 0])
+        b = np.array([x2, 0.01, 0])
+        profil2 = data.sample_over_line(a, b, resolution=1000)
+        u2 = profil2["velocity"][:,0]
+
+
+        # Derivatives
+        dudy1 = (-u1[2] + 4*u1[1] - 3*u1[0]) / (y[2] - y[0])
+        dudym = (-um[2] + 4*um[1] - 3*um[0]) / (y[2] - y[0])
+        dudy2 = (-u2[2] + 4*u2[1] - 3*u2[0]) / (y[2] - y[0])
+
+        # Signs check
+        if (abs(dudym) < 0.1*tol):
+            break
+        if (dudy1*dudym < 0):
+            x2 = xm
+        else:
+            x1 = xm
+
+        j+=1
+
+    x_r.append(xm - L_in)   # Add distance from the step
+    print("Output file " + str(i+1) + ", reattachment point x_r: " , xm - L_in)
+
+# Calculate relative error using last reattachment point and ref values
+e = abs(x_r_ref - x_r[-1])/x_r_ref
+print("Reference value Erturk (2008): ", x_r_ref)
+print("Relative error abs(x_r_ref - x_r)/x_r_ref = ", e)