Turbulent physics: Zero Equation Turbulence Model

Sym v1.2.0

This tutorial walks you through the process of adding a algebraic (zero equation) turbulence model to the Modulus Sym simulations. In this tutorial you will learn the following:

  1. How to use the Zero equation turbulence model in Modulus Sym.

  2. How to create nodes in the graph for arbitrary variables.


This tutorial assumes that you have completed the Introductory Example tutorial on Lid Driven Cavity Flow and have familiarized yourself with the basics of the Modulus Sym APIs.

In this tutorial you will add the zero equation turbulence for a Lid Driven Cavity flow. The problem setup is very similar to the one found in the Introductory Example. The Reynolds number is increased to 1000. The velocity profile is kept the same as before. To increase the Reynolds Number, the viscosity is reduced to 1 × 10−4 \(m^2/s\).

The case set up in this tutorial is very similar to the example in Introductory Example. It only describes the additions that are made to the previous code.


The python script for this problem can be found at examples/ldc/ldc_2d_zeroEq.py

Importing the required packages

Import Modulus Sym’ ZeroEquation to help setup the problem. Other import are very similar to previous LDC.


import os import warnings from sympy import Symbol, Eq, Abs import torch import modulus.sym from modulus.sym.hydra import to_absolute_path, instantiate_arch, ModulusConfig from modulus.sym.utils.io import csv_to_dict from modulus.sym.solver import Solver from modulus.sym.domain import Domain from modulus.sym.geometry.primitives_2d import Rectangle from modulus.sym.domain.constraint import ( PointwiseBoundaryConstraint, PointwiseInteriorConstraint, ) from modulus.sym.domain.monitor import PointwiseMonitor from modulus.sym.domain.validator import PointwiseValidator from modulus.sym.domain.inferencer import PointwiseInferencer from modulus.sym.eq.pdes.navier_stokes import NavierStokes from modulus.sym.eq.pdes.turbulence_zero_eq import ZeroEquation from modulus.sym.utils.io.plotter import ValidatorPlotter, InferencerPlotter from modulus.sym.key import Key

Defining the Equations, Networks and Nodes

In addition to the Navier-Stokes equation, the Zero Equation turbulence model is included by instantiating the ZeroEquation equation class. The kinematic viscosity \(\nu\) in the Navier-Stokes equation is a now a sympy expression given by the ZeroEquation. The ZeroEquation turbulence model provides the effective viscosity \((\nu+\nu_t)\) to the Navier-Stokes equations. The kinematic viscosity of the fluid calculated based on the Reynolds number is given as an input to the ZeroEquation class.

The Zero Equation turbulence model is defined in the equations below. Note, \(\mu_t = \rho \nu_t\).

(144)\[\mu_t=\rho l_m^2 \sqrt{G}\]
(145)\[G=2(u_x)^2 + 2(v_y)^2 + 2(w_z)^2 + (u_y + v_x)^2 + (u_z + w_x)^2 + (v_z + w_y)^2\]
(146)\[l_m=\min (0.419d, 0.09d_{max})\]

Where, \(l_m\) is the mixing length, \(d\) is the normal distance from wall, \(d_{max}\) is maximum normal distance and \(\sqrt{G}\) is the modulus of mean rate of strain tensor.

The zero equation turbulence model requires normal distance from no slip walls to compute the turbulent viscosity. For most examples, signed distance field (SDF) can act as a normal distance. When the geometry is generated using either the Modulus Sym’ geometry module/tesselation module you have access to the sdf variable similar to the other coordinate variables when used in interior sampling. Since zero equation also computes the derivatives of the viscosity, when using the PointwiseInteriorConstraint, you can pass an argument that says compute_sdf_derivatives=True. This will compute the required derivatives of the SDF like sdf__x, sdf__y, etc.


# make geometry height = 0.1 width = 0.1 x, y = Symbol("x"), Symbol("y") rec = Rectangle((-width / 2, -height / 2), (width / 2, height / 2)) # make list of nodes to unroll graph on ze = ZeroEquation(nu=1e-4, dim=2, time=False, max_distance=height / 2) ns = NavierStokes(nu=ze.equations["nu"], rho=1.0, dim=2, time=False) flow_net = instantiate_arch( input_keys=[Key("x"), Key("y")], output_keys=[Key("u"), Key("v"), Key("p")], cfg=cfg.arch.fully_connected, ) nodes = ( ns.make_nodes() + ze.make_nodes() + [flow_net.make_node(name="flow_network")] )

Setting up domain, adding constraints and running the solver

This section of the code is very similar to LDC tutorial, so the code and final results is presented here.


# make ldc domain ldc_domain = Domain() # top wall top_wall = PointwiseBoundaryConstraint( nodes=nodes, geometry=rec, outvar={"u": 1.5, "v": 0}, batch_size=cfg.batch_size.TopWall, lambda_weighting={"u": 1.0 - 20 * Abs(x), "v": 1.0}, # weight edges to be zero criteria=Eq(y, height / 2), ) ldc_domain.add_constraint(top_wall, "top_wall") # no slip no_slip = PointwiseBoundaryConstraint( nodes=nodes, geometry=rec, outvar={"u": 0, "v": 0}, batch_size=cfg.batch_size.NoSlip, criteria=y < height / 2, ) ldc_domain.add_constraint(no_slip, "no_slip") # interior interior = PointwiseInteriorConstraint( nodes=nodes, geometry=rec, outvar={"continuity": 0, "momentum_x": 0, "momentum_y": 0}, batch_size=cfg.batch_size.Interior, compute_sdf_derivatives=True, lambda_weighting={ "continuity": Symbol("sdf"), "momentum_x": Symbol("sdf"), "momentum_y": Symbol("sdf"), }, ) ldc_domain.add_constraint(interior, "interior") # add validator file_path = "openfoam/cavity_uniformVel_zeroEqn_refined.csv" if os.path.exists(to_absolute_path(file_path)): mapping = { "Points:0": "x", "Points:1": "y", "U:0": "u", "U:1": "v", "p": "p", "d": "sdf", "nuT": "nu", } openfoam_var = csv_to_dict(to_absolute_path(file_path), mapping) openfoam_var["x"] += -width / 2 # center OpenFoam data openfoam_var["y"] += -height / 2 # center OpenFoam data openfoam_var["nu"] += 1e-4 # effective viscosity openfoam_invar_numpy = { key: value for key, value in openfoam_var.items() if key in ["x", "y", "sdf"] } openfoam_outvar_numpy = { key: value for key, value in openfoam_var.items() if key in ["u", "v", "nu"] } openfoam_validator = PointwiseValidator( nodes=nodes, invar=openfoam_invar_numpy, true_outvar=openfoam_outvar_numpy, batch_size=1024, plotter=ValidatorPlotter(), requires_grad=True, ) ldc_domain.add_validator(openfoam_validator) # add inferencer data grid_inference = PointwiseInferencer( nodes=nodes, invar=openfoam_invar_numpy, output_names=["u", "v", "p", "nu"], batch_size=1024, plotter=InferencerPlotter(), requires_grad=True, ) ldc_domain.add_inferencer(grid_inference, "inf_data") else: warnings.warn( f"Directory{file_path}does not exist. Will skip adding validators. Please download the additional files from NGC https://catalog.ngc.nvidia.com/orgs/nvidia/teams/modulus/resources/modulus_sym_examples_supplemental_materials" ) # add monitors global_monitor = PointwiseMonitor( rec.sample_interior(4000), output_names=["continuity", "momentum_x", "momentum_y"], metrics={ "mass_imbalance": lambda var: torch.sum( var["area"] * torch.abs(var["continuity"]) ), "momentum_imbalance": lambda var: torch.sum( var["area"] * (torch.abs(var["momentum_x"]) + torch.abs(var["momentum_y"])) ), }, nodes=nodes, requires_grad=True, ) ldc_domain.add_monitor(global_monitor) # make solver slv = Solver(cfg, ldc_domain) # start solver slv.solve() if __name__ == "__main__": run()

The results of the turbulent lid driven cavity flow are shown below.


Fig. 65 Visualizing variables from Inference domain


Fig. 66 Comparison with OpenFOAM data. Left: Modulus Sym Prediction. Center: OpenFOAM, Right: Difference

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