# Tutorial 9 – Plane Turbulent Channel Flow

## Plane Turbulent Channel Flow

This tutorial describes how to set up and run the Plane-Turbulent-Channel-Flow test case. We will learn how to use the split form DG method to guarantee non-linear stability of the turbulent channel flow. In a second step, we add the sub grid scale model of Smagorinsky combined with Van Driest type damping to run stable wall-bounded turbulent flows with explicit small scale dissipation. The tutorial assumes that you are familiar with the general FLEXI and HOPR work flow (please finish the previous tutorials first if this sounds strange to you).

### Flow description

The flow is calculated in a plane channel with half-height $\delta=1$, streamwise (x coordinate) length $2\pi$ and span (z coordinate) width $\pi$ with periodic boundaries in the x- and z-directions as well as no-slip walls at the top and the bottom of the domain. As initial conditions an analytical mean turbulent velocity profile is used superimposed with sinus perturbations in the u, v and w velocity components and a constant density of $\rho=1$. The superimposed perturbations lead to rapid production of turbulent flow structures. Since the wall friction would slow down the flow over time, a constant pressure source term imposing a pressure gradient $\frac{dp}{dx}=-1$ is added as a volume source. While the test case is incompressible in principle, we solve it here in a compressible setting. The chosen Mach number with respect to the bulk velocity in the field is $Ma=0.1$ according to the Moser channel test case. In this setting, the wall friction velocity $\tau$ will always be equal to $1$. We can the define a Reynolds number based on the channel half-height and the wall friction velocity as $Re_{\tau}=1/\nu$.

### Compiler options

Make sure that FLEXI is compiled with the CMake options listed in the following table.

Option Value Comment
CMAKE_BUILD_TYPE Release
FLEXI_EQYNSYSNAME navierstokes
FLEXI_PARABOLIC ON
FLEXI_MPI ON optional
FLEXI_EDDYVISCOSITY ON optional
FLEXI_NODETYPE GAUSS-LOBATTO
FLEXI_SPLIT_DG ON
FLEXI_TESTCASE channel
FLEXI_BUILDPOSTI ON
POSTI_BUILD_CHANNEL_FFT ON

Table: CMake options for the plane turbulent channel flow test case simulation.

For all other CMake options you may keep the default values. Compile the code.

### Mesh Generation with HOPR

We use a Cartesian mesh with 4 cells per direction for the tutorial. The mesh is stretched in the wall-normal direction to accommodate for the straining of the vortexes close to the wall. In case you want to generate other meshes the parameter file for HOPR is included in the tutorial directory (parameter_hopr.ini),
the default mesh is included. Using 4 cells with a polynomial degree of $N=5$, means we use a large eddy simulation setup of $24$ DOFs per direction.

### Tutorial – Flow at $Re_{\tau}=180$

Copy the plane_turbulent_channel_flow tutorial folder to your working directory.

cp -r $FLEXI_TUTORIALS/plane_turbulent_channel_flow .  Step into the folder. In case you do not want to generate the mesh files yourself, a default mesh has already been provided. #### Preparing the Flow Simulation with FLEXI The simulation setup is already defined in parameter_flexi.ini. ##### Output In this tutorial we don’t look at the flow visualization of the instantaneous state files. Here, we will rather post process consecutive, instantaneous state files with the posti_channel_fft tool. As an output, we receive mean velocity and Reynolds stress profiles as well as turbulent energy spectra at different locations normal to the channel wall. ##### Interpolation / Discretization parameters In this tutorial we use the split form DG method to guarantee non-linear stability of the turbulent channel flow simulation. As already specified in the CMake options table above, the FLEXI_SPLIT_DG option has to be switched ON in combination with the FLEXI_NODETYPE GAUSS-LOBATTO. FLEXI has several split flux formulations implemented. Therefore, a specific split flux formulation has to be set in the parameter_flexi.ini file. In this tutorial the pre-defined split flux formulation by Pirozzoli is used, which results in a kinetic energy preserving DG scheme. ! ================================================ ! ! SplitDG ! ================================================ ! SplitDG = PI ! SplitDG formulation to be used: SD, MO, DU, KG, PI  To switch on Smagorinsky’s model set the eddyViscType to$1$in the paramerter_flexi.ini file. In addition, the following parameters have to be set. CS is the Smagorinsky constant usually chosen around$0.11$for wall bounded turbulent flows and the turbulent Prandtl number is commonly set to$0.6$. To ensure the correct behavior of the eddy viscosity provided by Smagorinsky’s model when approaching a wall, Van Driest type damping has to be switched on. ! ================================================ ! ! LES MODEL ! ================================================ ! eddyViscType = 0 ! Choose LES model, 1:Smagorinsky VanDriest = T ! Van Driest damping for LES viscosity (channel flow only) CS = 0.11 ! Smagorinsky constant PrSGS = 0.6 ! turbulent Prandtl number  #### Running the Simulation and Results Now run the simulation, either using flexi parameter_flexi.ini  or mpirun -np XX flexi parameter_flexi.ini  when you want to use more than one processor. Once the simulation finished state files can be post processed by the posti_channel_fft tool which was build by the POSTI_BUILD_CHANNEL_FFT CMake option. To run the postprocessing, the standard command is posti_channel_fft parameter_channel_fft.ini [State1 State2 ...]  where the parameter_channel_fft.ini file is given in the tutorial folder and the amount of statefiles is specified by the user. In this tutorial we use all state files with a timestamp between$t=10.0$and$t=15.0$. As an output you receive three different files. One containing the mean velocity profiles as well as the Reynolds stress profiles and the other two files contain turbulent energy spectra. To visualize those files you can run the python script plotChannelFFT.py in the tools/testcases folder with the following command in your simulation directory python$FLEXIROOT/tools/testcases/plotChannelFFT.py -p $PROJECTNAME -t$POSTITIME


where $PROJECTNAME specifies the project name specified in the parameter_flexi.ini file and $POSTITIME the timestamp of your output files from the posti_channel_fft tool.

#### Part I: SplitDG iLES

First, we run FLEXI without Smagorinsky’s model which we call an implicit LES (iLES), as no explicit sub-grid scale dissipation model is added. The resulting mean velocity and Reynolds stress profiles as well as turbulent energy spectra close to the center of the channel are given in the following figure. Mean velocity and Reynolds stress profiles (left) as well as turbulent energy spectra close to the center of the channel (right} of an implicit LES at $Re_{\tau}=180$.

#### Part II: SplitDG with explicit LES model

In a second step, we run FLEXI with Smagorinsky’s model and Van Driest damping which needs to be switched on in the parameter file as described above. The resulting mean velocity and Reynolds stress profiles as well as turbulent energy spectra close to the centre of the channel are given in the figure below. In comparison to the previous simulation you might recognize the effect of the explicit damping on the Reynolds stress profile $\overline{u’u’}$ close to the maximum, most. To further study the influence of Smagorinsky’s model play around with the spatial resolution both in terms of grid resolution as well as the polynomial degree N. You can also increase the Reynolds number to $Re_{\tau}=395$ or $Re_{\tau}=590$ and compare the results to DNS results from Moser et al. (Moser, Kim and Mansour 1999). Mean velocity and Reynolds stress profiles (left) as well as turbulent energy spectra close to the centre of the channel (right} of a LES with Smagorinsky’s model and vanDriest damping at $Re_{\tau}=180$.