.gitattributes

-lbmpy/_version.py export-subst
+src/lbmpy/_version.py export-subst

.gitlab-ci.yml

 stages:
   - pretest
   - test
+  - nightly
+  - docs
   - deploy
 
+# --------------------------  Templates ------------------------------------------------------------------------------------
+
+# Base configuration for jobs meant to run at every commit
+.every-commit:
+  rules:
+    - if: $CI_PIPELINE_SOURCE != "schedule"
+
+# Configuration for jobs meant to run on each commit to pycodegen/pystencils/master
+.every-commit-master:
+  rules:
+    - if: '$CI_PIPELINE_SOURCE != "schedule" && $CI_PROJECT_PATH == "pycodegen/lbmpy" && $CI_COMMIT_BRANCH == "master"'
+
+
+# Base configuration for jobs meant to run at a schedule
+.scheduled:
+  rules:
+    - if: $CI_PIPELINE_SOURCE == "schedule"
 
 # --------------------------  Pre Tests --------------------------------------------------------------------------------
 
 # Normal test - runs on every commit all but "long run" tests
 tests-and-coverage:
   stage: pretest
-  except:
-    variables:
-      - $ENABLE_NIGHTLY_BUILDS
-  image: i10git.cs.fau.de:5005/pycodegen/pycodegen/full
+  extends: .every-commit
+  image: i10git.cs.fau.de:5005/pycodegen/pycodegen/full:cupy12.3
   script:
     # - pip install sympy --upgrade
     - export NUM_CORES=$(nproc --all)
@@ -62,9 +79,7 @@ tests-and-coverage-with-longrun:
 
 minimal-conda:
   stage: pretest
-  except:
-    variables:
-      - $ENABLE_NIGHTLY_BUILDS
+  extends: .every-commit
   image: i10git.cs.fau.de:5005/pycodegen/pycodegen/minimal_conda
   script:
     - pip install git+https://gitlab-ci-token:${CI_JOB_TOKEN}@i10git.cs.fau.de/pycodegen/pystencils.git@master#egg=pystencils
@@ -76,9 +91,7 @@ minimal-conda:
 # Linter for code formatting
 flake8-lint:
   stage: pretest
-  except:
-    variables:
-      - $ENABLE_NIGHTLY_BUILDS
+  extends: .every-commit
   image: i10git.cs.fau.de:5005/pycodegen/pycodegen/full
   script:
     - flake8 src/lbmpy
@@ -91,9 +104,7 @@ flake8-lint:
 # pipeline with latest python version
 latest-python:
   stage: test
-  except:
-    variables:
-      - $ENABLE_NIGHTLY_BUILDS
+  extends: .every-commit
   image: i10git.cs.fau.de:5005/pycodegen/pycodegen/latest_python
   before_script:
     - pip install git+https://gitlab-ci-token:${CI_JOB_TOKEN}@i10git.cs.fau.de/pycodegen/pystencils.git@master#egg=pystencils
@@ -133,9 +144,7 @@ latest-python:
 
 minimal-sympy-master:
   stage: test
-  except:
-    variables:
-      - $ENABLE_NIGHTLY_BUILDS
+  extends: .every-commit
   image: i10git.cs.fau.de:5005/pycodegen/pycodegen/minimal_conda
   before_script:
     - pip install -e .
@@ -151,9 +160,7 @@ minimal-sympy-master:
 
 ubuntu:
   stage: test
-  except:
-    variables:
-      - $ENABLE_NIGHTLY_BUILDS
+  extends: .every-commit
   image: i10git.cs.fau.de:5005/pycodegen/pycodegen/ubuntu
   before_script:
     # - apt-get -y remove python3-sympy
@@ -210,10 +217,41 @@ pycodegen-integration:
     - cuda11
     - AVX
 
+# -------------------- Scheduled Tasks --------------------------------------------------------------------------
+
+
+nightly-sympy:
+  stage: nightly
+  extends: .scheduled
+  image: i10git.cs.fau.de:5005/pycodegen/pycodegen/latest_python
+  before_script:
+    - pip install -e .
+    - pip install git+https://gitlab-ci-token:${CI_JOB_TOKEN}@i10git.cs.fau.de/pycodegen/pystencils.git@master#egg=pystencils
+    - pip install --upgrade --pre sympy
+  script:
+    - env
+    - pip list
+    - export NUM_CORES=$(nproc --all)
+    - mkdir -p ~/.config/matplotlib
+    - echo "backend:template" > ~/.config/matplotlib/matplotlibrc
+    - mkdir public
+    - pytest -v -n $NUM_CORES -m "not longrun" --junitxml=report.xml
+  tags:
+    - docker
+    - AVX
+    - cuda
+  artifacts:
+    when: always
+    reports:
+      junit: report.xml
+
+
 # -------------------- Documentation and deploy ------------------------------------------------------------------------
 
 build-documentation:
-  stage: test
+  stage: docs
+  needs: []
+  extends: .every-commit
   image: i10git.cs.fau.de:5005/pycodegen/pycodegen/documentation
   before_script:
     - pip install -e .
@@ -232,7 +270,9 @@ build-documentation:
 
 pages:
   image: i10git.cs.fau.de:5005/pycodegen/pycodegen/full
+  extends: .every-commit-master
   stage: deploy
+  needs: ["tests-and-coverage", "build-documentation"]
   script:
     - ls -l
     - mv coverage_report html_doc
@@ -242,5 +282,3 @@ pages:
       - public
   tags:
     - docker
-  only:
-    - master@pycodegen/lbmpy

COPYING.txt

-------------------------   Important ---------------------------------
-
-lbmpy is under the following GNU AGPLv3 license. 
-This license holds for the sources of lbmpy itself as well 
-as for all kernels generated with lbmpy i.e. 
-the output of lbmpy is also protected by the GNU AGPLv3 license. 
-
-----------------------------------------------------------------------
-
-
-
-                    
                     GNU AFFERO GENERAL PUBLIC LICENSE
                        Version 3, 19 November 2007
 

conftest.py

@@ -19,9 +19,10 @@ try:
     import pyximport
 
     pyximport.install(language_level=3)
+
+    from lbmpy.phasefield.simplex_projection import simplex_projection_2d  # NOQA
 except ImportError:
     pass
-from lbmpy.phasefield.simplex_projection import simplex_projection_2d  # NOQA
 
 SCRIPT_FOLDER = os.path.dirname(os.path.realpath(__file__))
 sys.path.insert(0, os.path.abspath('lbmpy'))
@@ -106,18 +107,25 @@ class IPyNbTest(pytest.Item):
 
         # disable matplotlib output
         exec("import matplotlib.pyplot as p; "
+             "p.close('all'); "
              "p.switch_backend('Template')", global_dict)
 
         # in notebooks there is an implicit plt.show() - if this is not called a warning is shown when the next
         # plot is created. This warning is suppressed here
+        # Also animations cannot be shown, which also leads to a warning.
         exec("import warnings;"
-             "warnings.filterwarnings('ignore', 'Adding an axes using the same arguments as a previous.*');",
+             "warnings.filterwarnings('ignore', 'Adding an axes using the same arguments as a previous.*');"
+             "warnings.filterwarnings('ignore', 'Animation was deleted without rendering anything.*');",
              global_dict)
         with tempfile.NamedTemporaryFile() as f:
             f.write(self.code.encode())
             f.flush()
             runpy.run_path(f.name, init_globals=global_dict, run_name=self.name)
 
+        #   Close any open figures
+        exec("import matplotlib.pyplot as p; "
+             "p.close('all')", global_dict)
+
 
 class IPyNbFile(pytest.File):
     def collect(self):
@@ -140,10 +148,19 @@ class IPyNbFile(pytest.File):
         pass
 
 
-def pytest_collect_file(path, parent):
-    glob_exprs = ["*demo*.ipynb", "*tutorial*.ipynb", "test_*.ipynb"]
-    if any(path.fnmatch(g) for g in glob_exprs):
-        if pytest_version >= 50403:
-            return IPyNbFile.from_parent(fspath=path, parent=parent)
-        else:
-            return IPyNbFile(path, parent)
+if pytest_version >= 70000:
+    #   Since pytest 7.0, usage of `py.path.local` is deprecated and `pathlib.Path` should be used instead
+    import pathlib
+
+    def pytest_collect_file(file_path: pathlib.Path, parent):
+        glob_exprs = ["*demo*.ipynb", "*tutorial*.ipynb", "test_*.ipynb"]
+        if any(file_path.match(g) for g in glob_exprs):
+            return IPyNbFile.from_parent(path=file_path, parent=parent)
+else:
+    def pytest_collect_file(path, parent):
+        glob_exprs = ["*demo*.ipynb", "*tutorial*.ipynb", "test_*.ipynb"]
+        if any(path.fnmatch(g) for g in glob_exprs):
+            if pytest_version >= 50403:
+                return IPyNbFile.from_parent(fspath=path, parent=parent)
+            else:
+                return IPyNbFile(path, parent)

doc/notebooks/06_tutorial_modifying_method_smagorinsky.ipynb

 %% Cell type:code id: tags:
 ``` python
 from lbmpy.session import *
 from lbmpy.relaxationrates import *
 ```
 %% Cell type:markdown id: tags:
 # Tutorial 06: Modifying a LBM method: Smagorinsky model
 In this demo, we show how to modify a lattice Boltzmann method. As example we are going to add a simple turbulence model, by introducing a rule that locally computes the relaxation parameter dependent on the local strain rate tensor. The Smagorinsky model is implemented directly in *lbmpy* as well, however here we take the manual approach to demonstrate how a LB method can be changed in *lbmpy*.
 ## 1) Theoretical background
 Since we have *sympy* available, we want to start out with the basic model equations and derive the concrete equations ourselves. This approach is less error prone, since the calculations are done by the computer algebra system, and oftentimes this approach is also more general and easier to understand.
 ### a) Smagorinsky model
 The basic idea of the Smagorinsky turbulence model is to safe compute time, by not resolving the smallest eddies of the flow on the grid, but model them by an artifical dissipation term.
 The energy dissipation of small scale vortices is taken into account by introducing a "turbulent viscosity". This additional viscosity depends on local flow properties, namely the local shear rates. The larger the local shear rates the higher the turbulent viscosity and the more artifical dissipation is added.
 The total viscosity is
 $$\nu_{total} = \nu_0 + \underbrace{(C_S \Delta)^2 |S|}_{\nu_{t}}$$
 where $\nu_0$ is the normal viscosity, $C_S$ is the Smagorinsky constant, not to be confused with the speed of sound! Typical values of the Smagorinsky constant are between 0.1 - 0.2. The filter length $\Delta$ is chosen as 1 in lattice coordinates.
 The quantity $|S|$ is computed from the local strain rate tensor $S$ that is given by
 $$S_{ij} = \frac{1}{2} \left( \partial_i u_j + \partial_j u_i \right)$$
 and
 $$|S| = \sqrt{2 S_{ij} S_{ij}}$$
 ### b) LBM implementation of Smagorinsky model
 To add the Smagorinsky model to a LB scheme one has to first compute the strain rate tensor $S_{ij}$ in each cell, and compute the turbulent viscosity $\nu_t$ from it. Then the local relaxation rate has to be adapted to match the total viscosity $\nu_{total}$ instead of the standard viscosity $\nu_0$.
 A fortunate property of LB methods is, that the strain rate tensor can be computed locally from the non-equilibrium part of the distribution function. This is somewhat surprising, since the strain rate tensor contains first order derivatives. The strain rate tensor can be obtained by
 $$S_{ij} = - \frac{3 \omega_s}{2 \rho_{(0)}} \Pi_{ij}^{(neq)}$$
 where $\omega_s$ is the relaxation rate that determines the viscosity, $\rho_{(0)}$ is $\rho$ in compressible models and $1$ for incompressible schemes.
 $\Pi_{ij}^{(neq)}$ is the second order moment tensor of the non-equilibrium part of the distribution functions $f^{(neq)} = f - f^{(eq)}$ and can be computed as
 $$\Pi_{ij}^{(neq)} = \sum_q c_{qi} c_{qj} \; f_q^{(neq)}$$
 %% Cell type:markdown id: tags:
 We first have to find a closed form for $S_{ij}$ since in the formula above, it depends on $\omega$, which should be adapated according to $S_{ij}$.
 So we compute $\omega$ and insert it into the formula for $S$:
 %% Cell type:code id: tags:
 ``` python
 τ_0, ρ, ω, ω_total, ω_0 = sp.symbols("tau_0 rho omega omega_total omega_0", positive=True, real=True)
 ν_0, C_S, S, Π = sp.symbols("nu_0, C_S, |S|, Pi", positive=True, real=True)
 Seq = sp.Eq(S, 3 * ω / 2 * Π)
 Seq
 ```
 %% Output
     $\displaystyle |S| = \frac{3 \Pi \omega}{2}$
           3⋅Π⋅ω
     |S| = ─────
             2
 %% Cell type:markdown id: tags:
 Note that we left of the minus, since we took the absolute value of both tensor. The absolute value is defined as above, with the factor of two inside the square root. The $\rho_{(0)}$ has been left out, remembering that $\Pi^{(neq)}$ has to be divided by $\rho$ in case of compressible models|.
 Next, we compute $\omega$ from the total viscosity as given by the Smagorinsky equation:
 %% Cell type:code id: tags:
 ``` python
 relaxation_rate_from_lattice_viscosity(ν_0 + C_S ** 2 * S)
 ```
 %% Output
     $\displaystyle \frac{2}{6 C_{S}^{2} |S| + 6 \nu_{0} + 1}$
               2
     ─────────────────────
          2
     6⋅C_S ⋅|S| + 6⋅ν₀ + 1
 %% Cell type:markdown id: tags:
 and insert it into the equation for $|S|$
 %% Cell type:code id: tags:
 ``` python
 Seq2 = Seq.subs(ω, relaxation_rate_from_lattice_viscosity(ν_0 + C_S **2 * S ))
 Seq2
 ```
 %% Output
     $\displaystyle |S| = \frac{3 \Pi}{6 C_{S}^{2} |S| + 6 \nu_{0} + 1}$
                    3⋅Π
     |S| = ─────────────────────
                2
           6⋅C_S ⋅|S| + 6⋅ν₀ + 1
 %% Cell type:markdown id: tags:
 This equation contains only known quantities, such that we can solve it for $|S|$.
 Additionally we substitute the lattice viscosity $\nu_0$ by the original relaxation time $\tau_0$. The resulting equations get simpler using relaxation times instead of rates.
 %% Cell type:code id: tags:
 ``` python
 solveRes = sp.solve(Seq2, S)
 assert len(solveRes) == 1
 SVal = solveRes[0]
 SVal = SVal.subs(ν_0, lattice_viscosity_from_relaxation_rate(1 / τ_0)).expand()
 SVal
 ```
 %% Output
     $\displaystyle - \frac{\tau_{0}}{6 C_{S}^{2}} + \frac{\sqrt{72 C_{S}^{2} \Pi + 4 \tau_{0}^{2}}}{12 C_{S}^{2}}$
                   ___________________
                  ╱       2         2
         τ₀     ╲╱  72⋅C_S ⋅Π + 4⋅τ₀
     - ────── + ──────────────────────
            2                2
       6⋅C_S           12⋅C_S
 %% Cell type:markdown id: tags:
 Knowning $|S|$ we can compute the total relaxation time using
 $$\nu_{total} = \nu_0 +C_S^2 |S|$$
 %% Cell type:code id: tags:
 ``` python
 τ_val = 1 / (relaxation_rate_from_lattice_viscosity(lattice_viscosity_from_relaxation_rate(1/τ_0) + C_S**2 * SVal)).cancel()
 τ_val
 ```
 %% Output
     $\displaystyle \frac{\tau_{0}}{2} + \frac{\sqrt{18 C_{S}^{2} \Pi + \tau_{0}^{2}}}{2}$
             _________________
            ╱       2       2
     τ₀   ╲╱  18⋅C_S ⋅Π + τ₀
     ── + ────────────────────
     2             2
 %% Cell type:markdown id: tags:
 To compute $\Pi^{(neq)}$ we use the following functions:
 %% Cell type:code id: tags:
 ``` python
 def second_order_moment_tensor(function_values, stencil):
     assert len(function_values) == len(stencil)
     dim = len(stencil[0])
     return sp.Matrix(dim, dim, lambda i, j: sum(c[i] * c[j] * f for f, c in zip(function_values, stencil)))
 def frobenius_norm(matrix, factor=1):
     return sp.sqrt(sum(i*i for i in matrix) * factor)
 ```
 %% Cell type:markdown id: tags:
 In the next cell we construct equations that take an standard relaxation rate $\omega_0$ and compute a new relaxation rate $\omega_{total}$ according to the Smagorinksy model, using `τ_val` computed above
 %% Cell type:code id: tags:
 ``` python
 def smagorinsky_equations(ω_0, ω_total, method):
     f_neq = sp.Matrix(method.pre_collision_pdf_symbols) - method.get_equilibrium_terms()
     return [sp.Eq(τ_0, 1 / ω_0),
             sp.Eq(Π, frobenius_norm(second_order_moment_tensor(f_neq, method.stencil), factor=2)),
             sp.Eq(ω_total, 1 / τ_val)]
 smagorinsky_equations(ω_0, ω_total, create_lb_method())
 ```
 %% Output
     $\displaystyle \left[ \tau_{0} = \frac{1}{\omega_{0}}, \  \Pi = \sqrt{4 \left(- f_{5} + f_{6} + f_{7} - f_{8} - u_{0} u_{1}\right)^{2} + 2 \left(- \frac{\delta_{\rho}}{3} + f_{1} + f_{2} + f_{5} + f_{6} + f_{7} + f_{8} - u_{1}^{2}\right)^{2} + 2 \left(- \frac{\delta_{\rho}}{3} + f_{3} + f_{4} + f_{5} + f_{6} + f_{7} + f_{8} - u_{0}^{2}\right)^{2}}, \  \omega_{total} = \frac{1}{\frac{\tau_{0}}{2} + \frac{\sqrt{18 C_{S}^{2} \Pi + \tau_{0}^{2}}}{2}}\right]$
     ⎡                  ___________________________________________________________
     ⎢                 ╱
     ⎢     1          ╱                                2     ⎛  δᵨ
     ⎢τ₀ = ──, Π =   ╱   4⋅(-f₅ + f₆ + f₇ - f₈ - u₀⋅u₁)  + 2⋅⎜- ── + f₁ + f₂ + f₅ +
     ⎢     ω₀      ╲╱                                        ⎝  3
     ⎢
     ⎢
     ⎢
     ⎣
     
     ______________________________________________________________________
                         2                                               2
                       2⎞      ⎛  δᵨ                                   2⎞
      f₆ + f₇ + f₈ - u₁ ⎟  + 2⋅⎜- ── + f₃ + f₄ + f₅ + f₆ + f₇ + f₈ - u₀ ⎟  , ωₜₒₜₐₗ
                        ⎠      ⎝  3                                     ⎠
     
     
     
     
     
                                 ⎤
                                 ⎥
                    1            ⎥
      = ─────────────────────────⎥
                _________________⎥
               ╱       2       2 ⎥
        τ₀   ╲╱  18⋅C_S ⋅Π + τ₀  ⎥
        ── + ────────────────────⎥
        2             2          ⎦
 %% Cell type:markdown id: tags:
 ## 2) Application: Channel flow
 Next we modify a *lbmpy* scenario to use the Smagorinsky model.
 We create a MRT method, where we fix all relaxation rates except the relaxation rate that controls the viscosity.
 %% Cell type:code id: tags:
 ``` python
-lbm_conifg = LBMConfig(stencil=Stencil.D2Q9, method=Method.MRT, force=(1e-6, 0),
+lbm_config = LBMConfig(stencil=Stencil.D2Q9, method=Method.MRT, force=(1e-6, 0),
                        force_model=ForceModel.LUO, relaxation_rates=[ω, 1.9, 1.9, 1.9])
-method = create_lb_method(lbm_config=lbm_conifg)
+method = create_lb_method(lbm_config=lbm_config)
 method
 ```
 %% Output
     <lbmpy.methods.momentbased.momentbasedmethod.MomentBasedLbMethod at 0x7f745d5c3b20>
 %% Cell type:markdown id: tags:
 Only the collision rule has to be changed. Thus we first construct the collision rule, add the Smagorinsky equations and create a normal scenario from the modified collision rule. To avoid that the macroscopic quantity symbols in the Smagorinsky equations fall prey to optimization, we must disable simplification:
 %% Cell type:code id: tags:
 ``` python
 optimization = {'simplification' : False}
 collision_rule = create_lb_collision_rule(lb_method=method, optimization=optimization)
 collision_rule = collision_rule.new_with_substitutions({ω: ω_total})
 collision_rule.subexpressions += smagorinsky_equations(ω, ω_total, method)
 collision_rule.topological_sort(sort_subexpressions=True, sort_main_assignments=False)
 collision_rule
 ```
 %% Output
     AssignmentCollection: d_6, d_5, d_2, d_3, d_0, d_1, d_7, d_4, d_8 <- f(f_2, f_4, f_7, f_5, omega_total, f_6, f_3, f_1, f_0, f_8)
 %% Cell type:markdown id: tags:
 In the next cell the collision rule is simplified by extracting common subexpressions
 %% Cell type:code id: tags:
 ``` python
 from pystencils.simp import sympy_cse
 #collision_rule = sympy_cse(collision_rule)
 ```
 %% Cell type:markdown id: tags:
 A channel scenario can be created from a modified collision rule:
 %% Cell type:code id: tags:
 ``` python
 ch = create_channel((300, 100), force=1e-6, collision_rule=collision_rule,
                     kernel_params={"C_S": 0.12, "omega": 1.999})
 ```
 %% Cell type:code id: tags:
 ``` python
 #show_code(ch.ast)
 ```
 %% Cell type:code id: tags:
 ``` python
 ch.run(5000)
 ```
 %% Cell type:code id: tags:
 ``` python
 plt.figure(dpi=200)
 plt.vector_field(ch.velocity[:, :])
 np.max(ch.velocity[:, :])
 ```
 %% Output
     $\displaystyle 0.00504266401703371$
     0.00504266401703371
 %% Cell type:markdown id: tags:
 ## Appendix: Strain rate tensor formula from Chapman Enskog
 The connection between $S_{ij}$ and $\Pi_{ij}^{(neq)}$ can be seen using a Chapman Enskog expansion. Since *lbmpy* has a module that automatically does this expansions we can have a look at it:
 %% Cell type:code id: tags:
 ``` python
 from lbmpy.chapman_enskog import ChapmanEnskogAnalysis, CeMoment
 from lbmpy.chapman_enskog.chapman_enskog import remove_higher_order_u
 compressible_model = create_lb_method(stencil=Stencil.D2Q9, compressible=True, zero_centered=False)
 incompressible_model = create_lb_method(stencil=Stencil.D2Q9, compressible=False, zero_centered=False)
 ce_compressible = ChapmanEnskogAnalysis(compressible_model)
 ce_incompressible = ChapmanEnskogAnalysis(incompressible_model)
 ```
 %% Cell type:markdown id: tags:
 The Chapman Enskog analysis yields expresssions for the moment
 $\Pi = \Pi^{(eq)} + \epsilon \Pi^{(1)} +  \epsilon^2 \Pi^{(2)} \cdots$
 and the strain rate tensor is related to $\Pi^{(1)}$. However the best approximation we have for $\Pi^{(1)}$ is
 $\Pi^{(neq)}$. For details, see the paper "Shear stress in lattice Boltzmann simulations" by Krüger, Varnik and Raabe from 2009.
 Lets look at the values of $\Pi^{(1)}$ obtained from the Chapman enskog expansion:
 %% Cell type:code id: tags:
 ``` python
 Π_1_xy = CeMoment("\\Pi", moment_tuple=(1,1), superscript=1)
 Π_1_xx = CeMoment("\\Pi", moment_tuple=(2,0), superscript=1)
 Π_1_yy = CeMoment("\\Pi", moment_tuple=(0,2), superscript=1)
 components = (Π_1_xx, Π_1_yy, Π_1_xy)
 Π_1_xy_val = ce_compressible.higher_order_moments[Π_1_xy]
 Π_1_xy_val
 ```
 %% Output
     $\displaystyle \frac{\rho u_{0}^{2} {\partial^{(1)}_{0} u_{1}} + 2 \rho u_{0} u_{1} {\partial^{(1)}_{0} u_{0}} + 2 \rho u_{0} u_{1} {\partial^{(1)}_{1} u_{1}} + \rho u_{1}^{2} {\partial^{(1)}_{1} u_{0}} - \frac{\rho {\partial^{(1)}_{1} u_{0}}}{3} - \frac{\rho {\partial^{(1)}_{0} u_{1}}}{3} + u_{0}^{2} u_{1} {\partial^{(1)}_{0} \rho} + u_{0} u_{1}^{2} {\partial^{(1)}_{1} \rho}}{\omega}$
         2                                                    2          ρ⋅D(u_0)
     ρ⋅u₀ ⋅D(u_1) + 2⋅ρ⋅u₀⋅u₁⋅D(u_0) + 2⋅ρ⋅u₀⋅u₁⋅D(u_1) + ρ⋅u₁ ⋅D(u_0) - ──────── -
                                                                            3
     ──────────────────────────────────────────────────────────────────────────────
                                                                ω
     
      ρ⋅D(u_1)     2                  2
      ──────── + u₀ ⋅u₁⋅D(rho) + u₀⋅u₁ ⋅D(rho)
         3
     ─────────────────────────────────────────
     
 %% Cell type:markdown id: tags:
 This term has lots of higher order error terms in it. We assume that $u$ is small in lattice coordinates, so if we neglect all terms in $u$ that are quadratic or higher we get:
 %% Cell type:code id: tags:
 ``` python
 remove_higher_order_u(Π_1_xy_val.expand())
 ```
 %% Output
     $\displaystyle - \frac{\rho {\partial^{(1)}_{1} u_{0}}}{3 \omega} - \frac{\rho {\partial^{(1)}_{0} u_{1}}}{3 \omega}$
       ρ⋅D(u_0)   ρ⋅D(u_1)
     - ──────── - ────────
         3⋅ω        3⋅ω
 %% Cell type:markdown id: tags:
 Putting these steps together into a function, we can display them for the different cases quickly:
 %% Cell type:code id: tags:
 ``` python
 def get_Π_1(ce_analysis, component):
     val = ce_analysis.higher_order_moments[component]
     return remove_higher_order_u(val.expand())
 ```
 %% Cell type:markdown id: tags:
 Compressible case:
 %% Cell type:code id: tags:
 ``` python
 tuple(get_Π_1(ce_compressible, Pi) for Pi in components)
 ```
 %% Output
     $\displaystyle \left( - \frac{2 \rho {\partial^{(1)}_{0} u_{0}}}{3 \omega}, \  - \frac{2 \rho {\partial^{(1)}_{1} u_{1}}}{3 \omega}, \  - \frac{\rho {\partial^{(1)}_{1} u_{0}}}{3 \omega} - \frac{\rho {\partial^{(1)}_{0} u_{1}}}{3 \omega}\right)$
     ⎛-2⋅ρ⋅D(u_0)   -2⋅ρ⋅D(u_1)     ρ⋅D(u_0)   ρ⋅D(u_1)⎞
     ⎜────────────, ────────────, - ──────── - ────────⎟
     ⎝    3⋅ω           3⋅ω           3⋅ω        3⋅ω   ⎠
 %% Cell type:markdown id: tags:
 Incompressible case:
 %% Cell type:code id: tags:
 ``` python
 tuple(get_Π_1(ce_incompressible, Pi) for Pi in components)
 ```
 %% Output
     $\displaystyle \left( \frac{2 u_{0} {\partial^{(1)}_{0} \rho}}{3 \omega} - \frac{2 {\partial^{(1)}_{0} u_{0}}}{3 \omega}, \  \frac{2 u_{1} {\partial^{(1)}_{1} \rho}}{3 \omega} - \frac{2 {\partial^{(1)}_{1} u_{1}}}{3 \omega}, \  \frac{u_{0} {\partial^{(1)}_{1} \rho}}{3 \omega} + \frac{u_{1} {\partial^{(1)}_{0} \rho}}{3 \omega} - \frac{{\partial^{(1)}_{1} u_{0}}}{3 \omega} - \frac{{\partial^{(1)}_{0} u_{1}}}{3 \omega}\right)$
     ⎛2⋅u₀⋅D(rho)   2⋅D(u_0)  2⋅u₁⋅D(rho)   2⋅D(u_1)  u₀⋅D(rho)   u₁⋅D(rho)   D(u_0
     ⎜─────────── - ────────, ─────────── - ────────, ───────── + ───────── - ─────
     ⎝    3⋅ω         3⋅ω         3⋅ω         3⋅ω        3⋅ω         3⋅ω       3⋅ω
     
     )   D(u_1)⎞
     ─ - ──────⎟
          3⋅ω  ⎠
 %% Cell type:markdown id: tags:
 In the incompressible case has some terms $\partial \rho$ which are zero, since $\rho$ is assumed constant.
 Leaving out the error terms we finally obtain:
 $$\Pi_{ij}^{(neq)} \approx \Pi_{ij}^{(1)} = -\frac{2 \rho_{(0)}}{3 \omega_s} \left( \partial_i u_j + \partial_j u_i \right)$$

doc/sphinx/lbmpy.bib

@@ -276,4 +276,42 @@ journal = {Communications in Computational Physics}
     url = {https://doi.org/10.1063/1.1399290},
 }
 
+@Article{rozema15,
+  doi       = {10.1063/1.4928700},
+  year      = {2015},
+  month     = {aug},
+  publisher = {AIP Publishing},
+  volume    = {27},
+  number    = {8},
+  author    = {Wybe Rozema and Hyun J. Bae and Parviz Moin and Roel Verstappen},
+  title     = {Minimum-dissipation models for large-eddy simulation},
+  journal   = {Physics of Fluids}
+}
+
+@article{Han2021,
+   doi = {10.1088/1873-7005/ac1782},
+   url = {https://dx.doi.org/10.1088/1873-7005/ac1782}                    ,
+   year = {2021},
+   month = {aug},
+   publisher = {IOP Publishing},
+   volume = {53},
+   number = {4},
+   pages = {045506},
+   author = {Mengtao Han and Ryozo Ooka and Hideki Kikumoto},
+   title = {A wall function approach in lattice Boltzmann method: algorithm and validation using turbulent channel flow},
+   journal = {Fluid Dynamics Research}
+}
+
+@article{Maronga2020,
+ author = {Maronga, Bj{\"o}rn and Knigge, Christoph and Raasch, Siegfried},
+ year = {2020},
+ title = {{An Improved Surface Boundary Condition for Large-Eddy Simulations Based on Monin--Obukhov Similarity Theory: Evaluation and Consequences for Grid Convergence in Neutral and Stable Conditions}},
+ pages = {297--325},
+ volume = {174},
+ number = {2},
+ issn = {0006-8314},
+ journal = {{Boundary-layer meteorology}},
+ doi = {10.1007/s10546-019-00485-w}
+}
+
 @Comment{jabref-meta: databaseType:bibtex;}

pyproject.toml

@@ -11,7 +11,7 @@ authors = [
 ]
 license = { file = "COPYING.txt" }
 requires-python = ">=3.10"
-dependencies = ["pystencils>=1.3", "sympy>=1.6,<=1.11.1", "numpy>=1.8.0", "appdirs", "joblib"]
+dependencies = ["pystencils>=1.3", "sympy>=1.9,<=1.12.1", "numpy>=1.8.0", "appdirs", "joblib"]
 classifiers = [
     "Development Status :: 4 - Beta",
     "Framework :: Jupyter",
@@ -69,8 +69,7 @@ tests = [
 [build-system]
 requires = [
     "setuptools>=69",
-    "versioneer>=0.29",
-    "tomli; python_version < '3.11'",
+    "versioneer[toml]>=0.29",
 ]
 build-backend = "setuptools.build_meta"
 
@@ -87,7 +86,7 @@ namespaces = false
 # resulting files.
 VCS = "git"
 style = "pep440"
-versionfile_source = "lbmpy/_version.py"
+versionfile_source = "src/lbmpy/_version.py"
 versionfile_build = "lbmpy/_version.py"
 tag_prefix = "release/"
 parentdir_prefix = "lbmpy-"

src/lbmpy/__init__.py

@@ -7,11 +7,12 @@ from .creationfunctions import (
     LBMConfig,
     LBMOptimisation,
 )
-from .enums import Stencil, Method, ForceModel, CollisionSpace
+from .enums import Stencil, Method, ForceModel, CollisionSpace, SubgridScaleModel
 from .lbstep import LatticeBoltzmannStep
 from .macroscopic_value_kernels import (
     pdf_initialization_assignments,
     macroscopic_values_getter,
+    strain_rate_tensor_getter,
     compile_macroscopic_values_getter,
     compile_macroscopic_values_setter,
     create_advanced_velocity_setter_collision_rule,
@@ -38,9 +39,11 @@ __all__ = [
     "Method",
     "ForceModel",
     "CollisionSpace",
+    "SubgridScaleModel",
     "LatticeBoltzmannStep",
     "pdf_initialization_assignments",
     "macroscopic_values_getter",
+    "strain_rate_tensor_getter",
     "compile_macroscopic_values_getter",
     "compile_macroscopic_values_setter",
     "create_advanced_velocity_setter_collision_rule",
@@ -54,7 +57,5 @@ __all__ = [
 ]
 
 
-from ._version import get_versions
-
-__version__ = get_versions()["version"]
-del get_versions
+from . import _version
+__version__ = _version.get_versions()['version']

src/lbmpy/advanced_streaming/communication.py

 import itertools
 from pystencils import CreateKernelConfig, Field, Assignment, AssignmentCollection
-from pystencils.slicing import shift_slice, get_slice_before_ghost_layer, normalize_slice
-from lbmpy.advanced_streaming.utility import is_inplace, get_accessor, numeric_index, \
-    Timestep, get_timesteps, numeric_offsets
+from pystencils.slicing import (
+    shift_slice,
+    get_slice_before_ghost_layer,
+    normalize_slice,
+)
+from lbmpy.advanced_streaming.utility import (
+    is_inplace,
+    get_accessor,
+    numeric_index,
+    Timestep,
+    get_timesteps,
+    numeric_offsets,
+)
 from pystencils.datahandling import SerialDataHandling
 from pystencils.enums import Target
 from itertools import chain
@@ -10,22 +20,28 @@ from itertools import chain
 
 class LBMPeriodicityHandling:
 
-    def __init__(self, stencil, data_handling, pdf_field_name,
-                 streaming_pattern='pull', ghost_layers=1,
-                 cupy_direct_copy=True):
+    def __init__(
+        self,
+        stencil,
+        data_handling,
+        pdf_field_name,
+        streaming_pattern="pull",
+        ghost_layers=1,
+        cupy_direct_copy=True,
+    ):
         """
-            Periodicity Handling for Lattice Boltzmann Streaming.
-
-            **On the usage with cuda:**
-            - cupy allows the copying of sliced arrays within device memory using the numpy syntax,
-            e.g. `dst[:,0] = src[:,-1]`. In this implementation, this is the default for periodicity
-            handling. Alternatively, if you set `cupy_direct_copy=False`, GPU kernels are generated and
-            compiled. The compiled kernels are almost twice as fast in execution as cupy array copying,
-            but especially for large stencils like D3Q27, their compilation can take up to 20 seconds.
-            Choose your weapon depending on your use case.
+        Periodicity Handling for Lattice Boltzmann Streaming.
+
+        **On the usage with cuda:**
+        - cupy allows the copying of sliced arrays within device memory using the numpy syntax,
+        e.g. `dst[:,0] = src[:,-1]`. In this implementation, this is the default for periodicity
+        handling. Alternatively, if you set `cupy_direct_copy=False`, GPU kernels are generated and
+        compiled. The compiled kernels are almost twice as fast in execution as cupy array copying,
+        but especially for large stencils like D3Q27, their compilation can take up to 20 seconds.
+        Choose your weapon depending on your use case.
         """
         if not isinstance(data_handling, SerialDataHandling):
-            raise ValueError('Only serial data handling is supported!')
+            raise ValueError("Only serial data handling is supported!")
 
         self.stencil = stencil
         self.dim = stencil.D
@@ -56,12 +72,16 @@ class LBMPeriodicityHandling:
         self.comm_slices = []
         timesteps = get_timesteps(streaming_pattern)
         for timestep in timesteps:
-            slices_per_comm_dir = get_communication_slices(stencil=stencil,
-                                                           comm_stencil=copy_directions,
-                                                           streaming_pattern=streaming_pattern,
-                                                           prev_timestep=timestep,
-                                                           ghost_layers=ghost_layers)
-            self.comm_slices.append(list(chain.from_iterable(v for k, v in slices_per_comm_dir.items())))
+            slices_per_comm_dir = get_communication_slices(
+                stencil=stencil,
+                comm_stencil=copy_directions,
+                streaming_pattern=streaming_pattern,
+                prev_timestep=timestep,
+                ghost_layers=ghost_layers,
+            )
+            self.comm_slices.append(
+                list(chain.from_iterable(v for k, v in slices_per_comm_dir.items()))
+            )
 
         if self.target == Target.GPU and not cupy_direct_copy:
             self.device_copy_kernels = list()
@@ -81,11 +101,11 @@ class LBMPeriodicityHandling:
             arr[dst] = arr[src]
 
     def _compile_copy_kernels(self, timestep):
+        assert self.target == Target.GPU
         pdf_field = self.dh.fields[self.pdf_field_name]
         kernels = []
         for src, dst in self.comm_slices[timestep.idx]:
-            kernels.append(
-                periodic_pdf_copy_kernel(pdf_field, src, dst, target=self.target))
+            kernels.append(periodic_pdf_gpu_copy_kernel(pdf_field, src, dst))
         return kernels
 
     def _periodicity_handling_gpu(self, prev_timestep):
@@ -100,7 +120,12 @@ class LBMPeriodicityHandling:
 
 
 def get_communication_slices(
-        stencil, comm_stencil=None, streaming_pattern='pull', prev_timestep=Timestep.BOTH, ghost_layers=1):
+    stencil,
+    comm_stencil=None,
+    streaming_pattern="pull",
+    prev_timestep=Timestep.BOTH,
+    ghost_layers=1,
+):
     """
     Return the source and destination slices for periodicity handling or communication between blocks.
 
@@ -116,7 +141,9 @@ def get_communication_slices(
     if comm_stencil is None:
         comm_stencil = itertools.product(*([-1, 0, 1] for _ in range(stencil.D)))
 
-    pdfs = Field.create_generic('pdfs', spatial_dimensions=len(stencil[0]), index_shape=(stencil.Q,))
+    pdfs = Field.create_generic(
+        "pdfs", spatial_dimensions=len(stencil[0]), index_shape=(stencil.Q,)
+    )
     write_accesses = get_accessor(streaming_pattern, prev_timestep).write(pdfs, stencil)
     slices_per_comm_direction = dict()
 
@@ -130,7 +157,9 @@ def get_communication_slices(
             d = stencil.index(streaming_dir)
             write_index = numeric_index(write_accesses[d])[0]
 
-            origin_slice = get_slice_before_ghost_layer(comm_dir, ghost_layers=ghost_layers, thickness=1)
+            origin_slice = get_slice_before_ghost_layer(
+                comm_dir, ghost_layers=ghost_layers, thickness=1
+            )
             src_slice = _fix_length_one_slices(origin_slice)
 
             write_offsets = numeric_offsets(write_accesses[d])
@@ -138,13 +167,15 @@ def get_communication_slices(
 
             # TODO: this is just a hotfix. _trim_slice_in_direction breaks FreeSlip BC with adjacent periodic side
             if streaming_pattern != "pull":
-                src_slice = shift_slice(_trim_slice_in_direction(src_slice, tangential_dir), write_offsets)
+                src_slice = shift_slice(
+                    _trim_slice_in_direction(src_slice, tangential_dir), write_offsets
+                )
 
             neighbour_transform = _get_neighbour_transform(comm_dir, ghost_layers)
             dst_slice = shift_slice(src_slice, neighbour_transform)
 
-            src_slice = src_slice + (write_index, )
-            dst_slice = dst_slice + (write_index, )
+            src_slice = src_slice + (write_index,)
+            dst_slice = dst_slice + (write_index,)
 
             slices_for_dir.append((src_slice, dst_slice))
 
@@ -152,10 +183,10 @@ def get_communication_slices(
     return slices_per_comm_direction
 
 
-def periodic_pdf_copy_kernel(pdf_field, src_slice, dst_slice,
-                             domain_size=None, target=Target.GPU):
-    """Copies a rectangular array slice onto another non-overlapping array slice"""
-    from pystencils.gpucuda.kernelcreation import create_cuda_kernel
+def periodic_pdf_gpu_copy_kernel(pdf_field, src_slice, dst_slice, domain_size=None):
+    """Generate a GPU kernel which copies all values from one slice of a field
+    to another non-overlapping slice."""
+    from pystencils import create_kernel
 
     pdf_idx = src_slice[-1]
     assert isinstance(pdf_idx, int), "PDF index needs to be an integer constant"
@@ -176,18 +207,28 @@ def periodic_pdf_copy_kernel(pdf_field, src_slice, dst_slice,
     def _stop(s):
         return s.stop if isinstance(s, slice) else s
 
-    offset = [_start(s1) - _start(s2) for s1, s2 in zip(normalized_from_slice, normalized_to_slice)]
-    assert offset == [_stop(s1) - _stop(s2) for s1, s2 in zip(normalized_from_slice, normalized_to_slice)], \
-        "Slices have to have same size"
-
-    copy_eq = AssignmentCollection(main_assignments=[Assignment(pdf_field(pdf_idx), pdf_field[tuple(offset)](pdf_idx))])
-    config = CreateKernelConfig(iteration_slice=dst_slice, skip_independence_check=True)
-    ast = create_cuda_kernel(copy_eq, config=config)
-    if target == Target.GPU:
-        from pystencils.gpucuda import make_python_function
-        return make_python_function(ast)
-    else:
-        raise ValueError('Invalid target:', target)
+    offset = [
+        _start(s1) - _start(s2)
+        for s1, s2 in zip(normalized_from_slice, normalized_to_slice)
+    ]
+    assert offset == [
+        _stop(s1) - _stop(s2)
+        for s1, s2 in zip(normalized_from_slice, normalized_to_slice)
+    ], "Slices have to have same size"
+
+    copy_eq = AssignmentCollection(
+        main_assignments=[
+            Assignment(pdf_field(pdf_idx), pdf_field[tuple(offset)](pdf_idx))
+        ]
+    )
+    config = CreateKernelConfig(
+        iteration_slice=dst_slice,
+        skip_independence_check=True,
+        target=Target.GPU,
+    )
+
+    ast = create_kernel(copy_eq, config=config)
+    return ast.compile()
 
 
 def _extend_dir(direction):
@@ -196,10 +237,10 @@ def _extend_dir(direction):
     elif direction[0] == 0:
         for d in [-1, 0, 1]:
             for rest in _extend_dir(direction[1:]):
-                yield (d, ) + rest
+                yield (d,) + rest
     else:
         for rest in _extend_dir(direction[1:]):
-            yield (direction[0], ) + rest
+            yield (direction[0],) + rest
 
 
 def _get_neighbour_transform(direction, ghost_layers):

src/lbmpy/boundaries/__init__.py

 from lbmpy.boundaries.boundaryconditions import (
-    UBB, FixedDensity, DiffusionDirichlet, SimpleExtrapolationOutflow,
+    UBB, FixedDensity, DiffusionDirichlet, SimpleExtrapolationOutflow, WallFunctionBounce,
     ExtrapolationOutflow, NeumannByCopy, NoSlip, NoSlipLinearBouzidi, QuadraticBounceBack, StreamInConstant, FreeSlip)
 from lbmpy.boundaries.boundaryhandling import LatticeBoltzmannBoundaryHandling
+from lbmpy.boundaries.wall_function_models import MoninObukhovSimilarityTheory, LogLaw, MuskerLaw, SpaldingsLaw
 
-__all__ = ['NoSlip', 'NoSlipLinearBouzidi', 'QuadraticBounceBack', 'FreeSlip',
+__all__ = ['NoSlip', 'NoSlipLinearBouzidi', 'QuadraticBounceBack', 'FreeSlip', 'WallFunctionBounce',
            'UBB', 'FixedDensity',
            'SimpleExtrapolationOutflow', 'ExtrapolationOutflow',
            'DiffusionDirichlet', 'NeumannByCopy', 'StreamInConstant',
-           'LatticeBoltzmannBoundaryHandling']
+           'LatticeBoltzmannBoundaryHandling',
+           'MoninObukhovSimilarityTheory', 'LogLaw', 'MuskerLaw', 'SpaldingsLaw']

src/lbmpy/boundaries/boundaries_in_kernel.py

@@ -67,7 +67,7 @@ def boundary_conditional(boundary, direction, streaming_pattern, prev_timestep,
 
     assignments = []
     for direction_idx in dir_indices:
-        rule = boundary(f_out, f_in, direction_idx, inv_dir, lb_method, index_field=None)
+        rule = boundary(f_out, f_in, direction_idx, inv_dir, lb_method, index_field=None, force_vector=None)
 
         #   rhs: replace f_out by post collision symbols.
         rhs_substitutions = {f_out(i): sym for i, sym in enumerate(lb_method.post_collision_pdf_symbols)}

src/lbmpy/boundaries/boundaryhandling.py

+from dataclasses import replace
 import numpy as np
 
 from pystencils import Assignment, CreateKernelConfig, create_kernel, Field, Target
 from pystencils.boundaries import BoundaryHandling
 from pystencils.boundaries.createindexlist import numpy_data_type_for_boundary_object
+from pystencils.field import FieldType
 from pystencils.simp import add_subexpressions_for_field_reads
 from pystencils.stencil import inverse_direction
 
@@ -156,22 +158,31 @@ class LatticeBoltzmannBoundaryHandling(BoundaryHandling):
 
 def create_lattice_boltzmann_boundary_kernel(pdf_field, index_field, lb_method, boundary_functor,
                                              prev_timestep=Timestep.BOTH, streaming_pattern='pull',
-                                             target=Target.CPU, **kernel_creation_args):
+                                             target=Target.CPU, force_vector=None, **kernel_creation_args):
 
     indexing = BetweenTimestepsIndexing(
         pdf_field, lb_method.stencil, prev_timestep, streaming_pattern, np.int32, np.int32)
 
+    dim = lb_method.stencil.D
     f_out, f_in = indexing.proxy_fields
     dir_symbol = indexing.dir_symbol
     inv_dir = indexing.inverse_dir_symbol
 
-    boundary_assignments = boundary_functor(f_out, f_in, dir_symbol, inv_dir, lb_method, index_field)
-    boundary_assignments = indexing.substitute_proxies(boundary_assignments)
-
-    config = CreateKernelConfig(index_fields=[index_field], target=target, default_number_int="int32",
+    config = CreateKernelConfig(target=target, default_number_int="int32",
                                 skip_independence_check=True, **kernel_creation_args)
 
     default_data_type = config.data_type.default_factory()
+
+    if force_vector is None:
+        force_vector_type = np.dtype([(f"F_{i}", default_data_type.c_name) for i in range(dim)], align=True)
+        force_vector = Field.create_generic('force_vector', spatial_dimensions=1,
+                                            dtype=force_vector_type, field_type=FieldType.INDEXED)
+
+    config = replace(config, index_fields=[index_field, force_vector])
+
+    boundary_assignments = boundary_functor(f_out, f_in, dir_symbol, inv_dir, lb_method, index_field, force_vector)
+    boundary_assignments = indexing.substitute_proxies(boundary_assignments)
+
     if pdf_field.dtype != default_data_type:
         boundary_assignments = add_subexpressions_for_field_reads(boundary_assignments, data_type=default_data_type)
 

src/lbmpy/boundaries/wall_function_models.py

+import sympy as sp
+from abc import ABC, abstractmethod
+
+from pystencils import Assignment
+
+
+class WallFunctionModel(ABC):
+    def __init__(self, name):
+        self._name = name
+
+    @abstractmethod
+    def shear_stress_assignments(self, density_symbol: sp.Symbol, shear_stress_symbol: sp.Symbol,
+                                 velocity_symbol: sp.Symbol, wall_distance, u_tau_target):
+        """
+        Computes a symbolic representation for the log law.
+
+        Args:
+            density_symbol: symbol density, should be provided by the LB method's conserved quantity computation
+            shear_stress_symbol: symbolic wall shear stress to which the calculated shear stress will be assigned
+            velocity_symbol: symbolic velocity that is taken as a reference in the wall functions
+            wall_distance: distance to the wall, equals to 0.5 in standard cell-centered LBM
+            u_tau_target: in implicit wall functions, a target friction velocity can be provided which will be used as
+                          initial guess in the Newton iteration. This target friction velocity can be obtained, e.g.,
+                          from the target friction Reynolds number
+        """
+        pass
+
+
+# end class WallFunctionModel
+
+
+class ExplicitWallFunctionModel(WallFunctionModel, ABC):
+    """
+    Abstract base class for explicit wall functions that can be solved directly for the wall shear stress.
+    """
+
+    def __init__(self, name):
+        super(ExplicitWallFunctionModel, self).__init__(name=name)
+
+
+class MoninObukhovSimilarityTheory(ExplicitWallFunctionModel):
+    def __init__(self, z0, kappa=0.41, phi=0, name="MOST"):
+        self.z0 = z0
+        self.kappa = kappa
+        self.phi = phi
+
+        super(MoninObukhovSimilarityTheory, self).__init__(name=name)
+
+    def shear_stress_assignments(self, density_symbol: sp.Symbol, shear_stress_symbol: sp.Symbol,
+                                 velocity_symbol: sp.Symbol, wall_distance, u_tau_target=None):
+        u_tau = velocity_symbol * self.kappa / sp.ln(wall_distance / self.z0 + self.phi)
+        return [Assignment(shear_stress_symbol, u_tau ** 2 * density_symbol)]
+
+
+class ImplicitWallFunctionModel(WallFunctionModel, ABC):
+    """
+    Abstract base class for implicit wall functions that require a Newton procedure to solve for the wall shear stress.
+    """
+
+    def __init__(self, name, newton_steps, viscosity):
+        self.newton_steps = newton_steps
+        self.u_tau = sp.symbols(f"wall_function_u_tau_:{self.newton_steps + 1}")
+        self.delta = sp.symbols(f"wall_function_delta_:{self.newton_steps}")
+
+        self.viscosity = viscosity
+
+        super(ImplicitWallFunctionModel, self).__init__(name=name)
+
+    def newton_iteration(self, wall_law):
+        m = -wall_law / wall_law.diff(self.u_tau[0])
+
+        assignments = []
+        for i in range(self.newton_steps):
+            assignments.append(Assignment(self.delta[i], m.subs({self.u_tau[0]: self.u_tau[i]})))
+            assignments.append(Assignment(self.u_tau[i + 1], self.u_tau[i] + self.delta[i]))
+
+        return assignments
+
+
+class LogLaw(ImplicitWallFunctionModel):
+    """
+    Analytical model for the velocity profile inside the boundary layer, obtained from the mean velocity gradient.
+    Only valid in the log-law region.
+    """
+
+    def __init__(self, viscosity, newton_steps=5, kappa=0.41, b=5.2, name="LogLaw"):
+        self.kappa = kappa
+        self.b = b
+
+        super(LogLaw, self).__init__(name=name, newton_steps=newton_steps, viscosity=viscosity)
+
+    def shear_stress_assignments(self, density_symbol: sp.Symbol, shear_stress_symbol: sp.Symbol,
+                                 velocity_symbol: sp.Symbol, wall_distance, u_tau_target=None):
+        def law(u_p, y_p):
+            return 1 / self.kappa * sp.ln(y_p) + self.b - u_p
+
+        u_plus = velocity_symbol / self.u_tau[0]
+        y_plus = (wall_distance * self.u_tau[0]) / self.viscosity
+
+        u_tau_init = u_tau_target if u_tau_target else velocity_symbol / sp.Float(100)
+
+        wall_law = law(u_plus, y_plus)
+        assignments = [Assignment(self.u_tau[0], u_tau_init),  # initial guess
+                       *self.newton_iteration(wall_law),  # newton iterations
+                       Assignment(shear_stress_symbol, self.u_tau[-1] ** 2 * density_symbol)]  # final result
+
+        return assignments
+
+
+class SpaldingsLaw(ImplicitWallFunctionModel):
+    """
+    Single formula for the velocity profile inside the boundary layer, proposed by Spalding :cite:`spalding1961`.
+    Valid in the inner and the outer layer.
+    """
+
+    def __init__(self, viscosity, newton_steps=5, kappa=0.41, b=5.5, name="Spalding"):
+        self.kappa = kappa
+        self.b = b
+
+        super(SpaldingsLaw, self).__init__(name=name, newton_steps=newton_steps, viscosity=viscosity)
+
+    def shear_stress_assignments(self, density_symbol: sp.Symbol, shear_stress_symbol: sp.Symbol,
+                                 velocity_symbol: sp.Symbol, wall_distance, u_tau_target=None):
+        def law(u_p, y_p):
+            k_times_u = self.kappa * u_p
+            frac_1 = (k_times_u ** 2) / sp.Float(2)
+            frac_2 = (k_times_u ** 3) / sp.Float(6)
+            return (u_p + sp.exp(-self.kappa * self.b) * (sp.exp(k_times_u) - sp.Float(1) - k_times_u - frac_1 - frac_2)
+                    - y_p)
+
+        u_plus = velocity_symbol / self.u_tau[0]
+        y_plus = (wall_distance * self.u_tau[0]) / self.viscosity
+
+        u_tau_init = u_tau_target if u_tau_target else velocity_symbol / sp.Float(100)
+
+        wall_law = law(u_plus, y_plus)
+        assignments = [Assignment(self.u_tau[0], u_tau_init),  # initial guess
+                       *self.newton_iteration(wall_law),  # newton iterations
+                       Assignment(shear_stress_symbol, self.u_tau[-1] ** 2 * density_symbol)]  # final result
+
+        return assignments
+
+
+class MuskerLaw(ImplicitWallFunctionModel):
+    """
+    Quasi-analytical model for the velocity profile inside the boundary layer, proposed by Musker. Valid in the inner
+    and the outer layer.
+    Formulation taken from :cite:`malaspinas2015`, Equation (59).
+    """
+
+    def __init__(self, viscosity, newton_steps=5, name="Musker"):
+
+        super(MuskerLaw, self).__init__(name=name, newton_steps=newton_steps, viscosity=viscosity)
+
+    def shear_stress_assignments(self, density_symbol: sp.Symbol, shear_stress_symbol: sp.Symbol,
+                                 velocity_symbol: sp.Symbol, wall_distance, u_tau_target=None):
+        def law(u_p, y_p):
+            arctan = sp.Float(5.424) * sp.atan(sp.Float(0.119760479041916168) * y_p - sp.Float(0.488023952095808383))
+            logarithm = (sp.Float(0.434) * sp.log((y_p + sp.Float(10.6)) ** sp.Float(9.6)
+                                                  / (y_p ** 2 - sp.Float(8.15) * y_p + sp.Float(86)) ** 2))
+            return (arctan + logarithm - sp.Float(3.50727901936264842)) - u_p
+
+        u_plus = velocity_symbol / self.u_tau[0]
+        y_plus = (wall_distance * self.u_tau[0]) / self.viscosity
+
+        u_tau_init = u_tau_target if u_tau_target else velocity_symbol / sp.Float(100)
+
+        wall_law = law(u_plus, y_plus)
+        assignments = [Assignment(self.u_tau[0], u_tau_init),  # initial guess
+                       *self.newton_iteration(wall_law),  # newton iterations
+                       Assignment(shear_stress_symbol, self.u_tau[-1] ** 2 * density_symbol)]  # final result
+
+        return assignments

src/lbmpy/creationfunctions.py

@@ -20,21 +20,21 @@ and belongs to pystencils, not lbmpy. This can be found in the pystencils module
 of the generated code is specified.
 
 1. *Method*:
-         the method defines the collision process. Currently there are two big categories:
+         the method defines the collision process. Currently, there are two big categories:
          moment and cumulant based methods. A method defines how each moment or cumulant is relaxed by
          storing the equilibrium value and the relaxation rate for each moment/cumulant.
 2. *Collision/Update Rule*:
          Methods can generate a "collision rule" which is an equation collection that define the
          post collision values as a function of the pre-collision values. On these equation collection
          simplifications are applied to reduce the number of floating point operations.
-         At this stage an entropic optimization step can also be added to determine one relaxation rate by an
+         At this stage an entropic optimisation step can also be added to determine one relaxation rate by an
          entropy condition.
          Then a streaming rule is added which transforms the collision rule into an update rule.
          The streaming step depends on the pdf storage (source/destination, AABB pattern, EsoTwist).
          Currently only the simple source/destination  pattern is supported.
 3. *AST*:
         The abstract syntax tree describes the structure of the kernel, including loops and conditionals.
-        The ast can be modified e.g. to add OpenMP pragmas, reorder loops or apply other optimizations.
+        The ast can be modified, e.g., to add OpenMP pragmas, reorder loops or apply other optimisations.
 4. *Function*:
         This step compiles the AST into an executable function, either for CPU or GPUs. This function
         behaves like a normal Python function and runs one LBM time step.
@@ -63,7 +63,7 @@ import pystencils.astnodes
 import sympy as sp
 import sympy.core.numbers
 
-from lbmpy.enums import Stencil, Method, ForceModel, CollisionSpace
+from lbmpy.enums import Stencil, Method, ForceModel, CollisionSpace, SubgridScaleModel
 import lbmpy.forcemodels as forcemodels
 from lbmpy.fieldaccess import CollideOnlyInplaceAccessor, PdfFieldAccessor, PeriodicTwoFieldsAccessor
 from lbmpy.fluctuatinglb import add_fluctuations_to_collision_rule
@@ -78,7 +78,7 @@ from lbmpy.methods.momentbased.entropic import add_entropy_condition, add_iterat
 from lbmpy.relaxationrates import relaxation_rate_from_magic_number
 from lbmpy.simplificationfactory import create_simplification_strategy
 from lbmpy.stencils import LBStencil
-from lbmpy.turbulence_models import add_smagorinsky_model
+from lbmpy.turbulence_models import add_sgs_model
 from lbmpy.updatekernels import create_lbm_kernel, create_stream_pull_with_output_kernel
 from lbmpy.advanced_streaming.utility import Timestep, get_accessor
 from pystencils import CreateKernelConfig, create_kernel
@@ -145,7 +145,7 @@ class LBMConfig:
     """
     delta_equilibrium: bool = None
     """
-    Determines whether or not the (continuous or discrete, see `continuous_equilibrium`) maxwellian equilibrium is
+    Determines whether or not the (continuous or discrete, see `continuous_equilibrium`) Maxwellian equilibrium is
     expressed in its absolute form, or only by its deviation from the rest state (typically given by the reference
     density and zero velocity). This parameter is only effective if `zero_centered` is set to `True`. Then, if
     `delta_equilibrium` is `False`, the rest state must be reintroduced to the populations during collision. Otherwise,
@@ -175,7 +175,7 @@ class LBMConfig:
     """
     A list of lists of modes, grouped by common relaxation times. This is usually used in
     conjunction with `lbmpy.methods.default_moment_sets.mrt_orthogonal_modes_literature`.
-    If this argument is not provided, Gram-Schmidt orthogonalization of the default modes is performed.
+    If this argument is not provided, Gram-Schmidt orthogonalisation of the default modes is performed.
     """
 
     force_model: Union[lbmpy.forcemodels.AbstractForceModel, ForceModel] = None
@@ -239,12 +239,17 @@ class LBMConfig:
     omega_output_field: Field = None
     """
     A pystencils Field can be passed here, where the calculated free relaxation rate of
-    an entropic or Smagorinsky method is written to
+    an entropic or subgrid-scale method is written to
     """
-    smagorinsky: Union[float, bool] = False
+    eddy_viscosity_field: Field = None
     """
-    set to Smagorinsky constant to activate turbulence model, ``omega_output_field`` can be set to
-    write out adapted relaxation rates. If set to `True`, 0.12 is used as default smagorinsky constant.
+    A pystencils Field can be passed here, where the eddy-viscosity of a subgrid-scale model is written.
+    """
+    subgrid_scale_model: Union[SubgridScaleModel, tuple[SubgridScaleModel, float], tuple[SubgridScaleModel, int]] = None
+    """
+    Choose a subgrid-scale model (SGS) for large-eddy simulations. ``omega_output_field`` can be set to
+    write out adapted relaxation rates. Either provide just the SGS and use the default model constants or provide a
+    tuple of the SGS and its corresponding model constant.
     """
     cassons: CassonsParameters = False
     """
@@ -371,8 +376,8 @@ class LBMConfig:
         if not self.compressible and self.method in (Method.MONOMIAL_CUMULANT, Method.CUMULANT):
             raise ValueError("Incompressible cumulant-based methods are not supported (yet).")
 
-        if self.zero_centered and (self.entropic or self.fluctuating):
-            raise ValueError("Entropic and fluctuating methods can only be created with `zero_centered=False`.")
+        if self.zero_centered and self.entropic:
+            raise ValueError("Entropic methods can only be created with `zero_centered=False`.")
 
         #   Check or infer delta-equilibrium
         if self.delta_equilibrium is not None:
@@ -702,8 +707,8 @@ def create_lb_collision_rule(lb_method=None, lbm_config=None, lbm_optimisation=N
                                                      limiter=cumulant_limiter)
 
     if lbm_config.entropic:
-        if lbm_config.smagorinsky or lbm_config.cassons:
-            raise ValueError("Choose either entropic, smagorinsky or cassons")
+        if lbm_config.subgrid_scale_model or lbm_config.cassons:
+            raise ValueError("Choose either entropic, subgrid-scale or cassons")
         if lbm_config.entropic_newton_iterations:
             if isinstance(lbm_config.entropic_newton_iterations, bool):
                 iterations = 3
@@ -714,14 +719,23 @@ def create_lb_collision_rule(lb_method=None, lbm_config=None, lbm_optimisation=N
         else:
             collision_rule = add_entropy_condition(collision_rule, omega_output_field=lbm_config.omega_output_field)
 
-    elif lbm_config.smagorinsky:
+    elif lbm_config.subgrid_scale_model:
         if lbm_config.cassons:
-            raise ValueError("Cassons model can not be combined with Smagorinsky model")
-        smagorinsky_constant = 0.12 if lbm_config.smagorinsky is True else lbm_config.smagorinsky
-        collision_rule = add_smagorinsky_model(collision_rule, smagorinsky_constant,
-                                               omega_output_field=lbm_config.omega_output_field)
-        if 'split_groups' in collision_rule.simplification_hints:
-            collision_rule.simplification_hints['split_groups'][0].append(sp.Symbol("smagorinsky_omega"))
+            raise ValueError("Cassons model can not be combined with a subgrid-scale model")
+
+        model_constant = None
+        sgs_model = lbm_config.subgrid_scale_model
+
+        if isinstance(lbm_config.subgrid_scale_model, tuple):
+            sgs_model = lbm_config.subgrid_scale_model[0]
+            model_constant = lbm_config.subgrid_scale_model[1]
+
+        collision_rule = add_sgs_model(collision_rule=collision_rule, subgrid_scale_model=sgs_model,
+                                       model_constant=model_constant, omega_output_field=lbm_config.omega_output_field,
+                                       eddy_viscosity_field=lbm_config.eddy_viscosity_field)
+
+    if 'split_groups' in collision_rule.simplification_hints:
+        collision_rule.simplification_hints['split_groups'][0].append(sp.Symbol("sgs_omega"))
 
     elif lbm_config.cassons:
         collision_rule = add_cassons_model(collision_rule, parameter=lbm_config.cassons,

src/lbmpy/db.py

@@ -4,7 +4,7 @@ import inspect
 
 from pystencils.runhelper.db import PystencilsJsonEncoder
 from pystencils.simp import SimplificationStrategy
-from lbmpy import LBStencil, Method, CollisionSpace
+from lbmpy import LBStencil, Method, CollisionSpace, SubgridScaleModel
 from lbmpy.creationfunctions import LBMConfig, LBMOptimisation
 from lbmpy.methods import CollisionSpaceInfo
 from lbmpy.forcemodels import AbstractForceModel
@@ -16,7 +16,7 @@ class LbmpyJsonEncoder(PystencilsJsonEncoder):
     def default(self, obj):
         if isinstance(obj, (LBMConfig, LBMOptimisation, CollisionSpaceInfo, CassonsParameters)):
             return obj.__dict__
-        if isinstance(obj, (LBStencil, Method, CollisionSpace)):
+        if isinstance(obj, (LBStencil, Method, CollisionSpace, SubgridScaleModel)):
             return obj.name
         if isinstance(obj, AbstractForceModel):
             return obj.__class__.__name__

src/lbmpy/enums.py

@@ -219,5 +219,20 @@ class ForceModel(Enum):
     """
     CENTRALMOMENT = auto()
     """
-    See :class:`lbmpy.forcemodels`
+    See :class:`lbmpy.forcemodels.CentralMoment`
+    """
+
+
+class SubgridScaleModel(Enum):
+    """
+    The SubgridScaleModel enumeration defines which subgrid-scale model (SGS) is used to perform
+    Large-Eddy-Simulations (LES).
+    """
+    SMAGORINSKY = auto()
+    """
+    See :func:`lbmpy.turbulence_models.add_smagorinsky_model`
+    """
+    QR = auto()
+    """
+    See :func:`lbmpy.turbulence_models.add_qr_model`
     """

src/lbmpy/flow_statistics.py

+import sympy as sp
+import pystencils as ps
+
+from pystencils.field import Field
+
+
+def welford_assignments(field, mean_field, sum_of_squares_field=None, sum_of_cubes_field=None):
+    r"""
+    Creates the assignments for the kernel creation of Welford's algorithm
+    (https://en.wikipedia.org/wiki/Algorithms_for_calculating_variance#Welford's_online_algorithm).
+    This algorithm is an online algorithm for the mean and variance calculation of sample data, here implemented for
+    the execution on scalar or vector fields, e.g., velocity.
+    The calculation of the variance / the third-order central moments is optional and only performed if a field for
+    the sum of squares / sum of cubes is given.
+
+    The mean value is directly updated in the mean vector field.
+    The variance / covariance must be retrieved in a post-processing step. Let :math `M_{2,n}` denote the value of the
+    sum of squares after the first :math `n` samples. According to Welford the biased sample variance
+    :math `\sigma_n^2` and the unbiased sample variance :math `s_n^2` are given by
+
+    .. math ::
+        \sigma_n^2 = \frac{M_{2,n}}{n}
+        s_n^2 = \frac{M_{2,n}}{n-1},
+
+    respectively.
+
+    Liekwise, to the 3rd-order central moment(s) of the (vector) field, the sum of cubes field must be divided
+    by :math `n` in a post-processing step.
+    """
+
+    if isinstance(field, Field):
+        dim = field.values_per_cell()
+        welford_field = field.center
+    elif isinstance(field, Field.Access):
+        dim = field.field.values_per_cell()
+        welford_field = field
+    else:
+        raise ValueError("Vector field has to be a pystencils Field or Field.Access")
+
+    if isinstance(mean_field, Field):
+        welford_mean_field = mean_field.center
+    elif isinstance(mean_field, Field.Access):
+        welford_mean_field = mean_field
+    else:
+        raise ValueError("Mean vector field has to be a pystencils Field or Field.Access")
+
+    if sum_of_squares_field is None:
+        # sum of products will not be calculated, i.e., the covariance matrix is not retrievable
+        welford_sum_of_squares_field = None
+    else:
+        if isinstance(sum_of_squares_field, Field):
+            welford_sum_of_squares_field = sum_of_squares_field.center
+        elif isinstance(sum_of_squares_field, Field.Access):
+            welford_sum_of_squares_field = sum_of_squares_field
+        else:
+            raise ValueError("Sum of squares field has to be a pystencils Field or Field.Access")
+
+        assert welford_sum_of_squares_field.field.values_per_cell() == dim ** 2, \
+            (f"Sum of squares field does not have the right layout. "
+             f"Index dimension: {welford_sum_of_squares_field.field.values_per_cell()}, expected: {dim ** 2}")
+
+    if sum_of_cubes_field is None:
+        # sum of cubes will not be calculated, i.e., the 3rd-order central moments are not retrievable
+        welford_sum_of_cubes_field = None
+    else:
+        if isinstance(sum_of_cubes_field, Field):
+            welford_sum_of_cubes_field = sum_of_cubes_field.center
+        elif isinstance(sum_of_cubes_field, Field.Access):
+            welford_sum_of_cubes_field = sum_of_cubes_field
+        else:
+            raise ValueError("Sum of cubes field has to be a pystencils Field or Field.Access")
+
+        assert welford_sum_of_cubes_field.field.values_per_cell() == dim ** 3, \
+            (f"Sum of cubes field does not have the right layout. "
+             f"Index dimension: {welford_sum_of_cubes_field.field.values_per_cell()}, expected: {dim ** 3}")
+
+    # for the calculation of the thrid-order moments, the variance must also be calculated
+    if welford_sum_of_cubes_field is not None:
+        assert welford_sum_of_squares_field is not None
+
+    # actual assignments
+
+    counter = sp.Symbol('counter')
+    delta = sp.symbols(f"delta_:{dim}")
+
+    main_assignments = list()
+
+    for i in range(dim):
+        main_assignments.append(ps.Assignment(delta[i], welford_field.at_index(i) - welford_mean_field.at_index(i)))
+        main_assignments.append(ps.Assignment(welford_mean_field.at_index(i),
+                                              welford_mean_field.at_index(i) + delta[i] / counter))
+
+    if sum_of_squares_field is not None:
+        delta2 = sp.symbols(f"delta2_:{dim}")
+        for i in range(dim):
+            main_assignments.append(
+                ps.Assignment(delta2[i], welford_field.at_index(i) - welford_mean_field.at_index(i)))
+        for i in range(dim):
+            for j in range(dim):
+                idx = i * dim + j
+                main_assignments.append(ps.Assignment(
+                    welford_sum_of_squares_field.at_index(idx),
+                    welford_sum_of_squares_field.at_index(idx) + delta[i] * delta2[j]))
+
+        if sum_of_cubes_field is not None:
+            for i in range(dim):
+                for j in range(dim):
+                    for k in range(dim):
+                        idx = (i * dim + j) * dim + k
+                        main_assignments.append(ps.Assignment(
+                            welford_sum_of_cubes_field.at_index(idx),
+                            welford_sum_of_cubes_field.at_index(idx)
+                            - delta[k] / counter * welford_sum_of_squares_field(i * dim + j)
+                            - delta[j] / counter * welford_sum_of_squares_field(k * dim + i)
+                            - delta[i] / counter * welford_sum_of_squares_field(j * dim + k)
+                            + delta2[i] * (2 * delta[j] - delta2[j]) * delta[k]
+                        ))
+
+    return main_assignments