5.2. Freestream
Unlike the previous tutorial, which dealt with simpler equations, we will now consider the behavior of a compressible fluid flow described by the Navier-Stokes equations. The simplest valid flow solution imaginable is a freestream scenario under conditions of pressure, density, and velocity. For the current setup, we specify a freestream scenario with constant pressure \(p=101325.0\ \mathrm{Pa}\), density \(\rho=1.225\ \mathrm{kg/m^3}\) and velocity vector \(\boldsymbol{U}=(1,1,1)^T\ \mathrm{m/s}\). This configuration provides a baseline for analyzing how the solver handles simple flow conditions and sets the stage for more challenging simulations.
5.2.1. Mesh Generation
In the tutorial directory, we provide the necessary mesh file, cartbox_mesh.h5, along with a parameter file for HOPR to generate this mesh. You can recreate the mesh by running the following command.
hopr parameter_hopr.ini
5.2.2. Build Configuration
FLEXI should be compiled with the freestream preset using the following commands.
cmake -B build --preset freestream
cmake --build build
5.2.3. Simulation Parameters
The parameter file to run the simulation is supplied as parameter_flexi.ini. The parameters specific to the Navier-Stokes equation system can be found in the EQUATION section of the file.
! ============================================================ !
! EQUATION
! ============================================================ !
IniExactFunc = 1
IniRefState = 1
RefState = (/1.225,1.0,1.,1.,101325./)
The initial condition is set via the variable vector RefState which represents the solution vector \((\rho, u, v, w, p)^T\). FLEXI permits for multiple RefState vectors, allowing each to be referenced by its corresponding cardinal number for its order number within the .ini file. In this example, only a single RefState vector is defined, referenced as \(1\). Thus, the selection process for the solution vector looks like the following.
IniRefState = 1: the initial condition usesRefState 1for the initial flow field solution.IniExactFunc = 1: the employed exact function routine usesRefState 1, such as for calculating the \(L_2\) error norm.
The material properties of the fluid medium, such as the ideal gas constant, are given in Table 5.2 and define the gas behavior in combination with the ideal gas law \(p=\rho R T\).
Variable |
Value |
Description |
|---|---|---|
mu0 |
1.8547e-5 |
Dynamic viscosity \(\mu\) |
R |
276 |
Ideal gas constant \(R\) |
kappa |
1.4 |
Isentropic coefficient \(\kappa\) |
The Discontinuous Galerkin (DG) solution is represented by piecewise polynomials on the computational mesh. In this tutorial, the polynomial degree \(N\) is chosen as \(N=3\). The remaining numerical parameters are outlined in Table 5.3.
Variable |
Value |
Description |
|---|---|---|
N |
3 |
Polynomial degree |
MeshFile |
cartbox_mesh.h5 |
Mesh file to be used |
tend |
1e-6 |
End time of the simulation |
Analyze_dt |
1e-6 |
Time interval for analysis |
CFLscale |
0.99 |
Scaling for the theoretical CFL number |
DFLscale |
0.4 |
Scaling for the theoretical DFL number |
5.2.4. Simulation and Results
We proceed by running the code with the following command.
flexi parameter_flexi.ini
Running the code prints all output to STDOUT. If the run completes successfully, the last lines should appear similar to the following (condensed) output.
==============================================================
INITIALIZATION DONE! [ 0.01 sec ]
==============================================================
--------------------------------------------------------------
Sys date : 06.11.2024 14:08:54
#GridCells : 8.0000000E+00
#DOFs : 5.1200000E+02
#Procs : 1.0000000E+00
#DOFs/Proc : 5.1200000E+02
WRITING INITIAL SOLUTION:
--------------------------------------------------------------
Initial Timestep : 1.0000000E-06
--------------------------------------------------------------
Errors of initial solution:
Sim time : 0.000E+00
L_2 : 2.902E-16 2.902E-16 2.902E-16 2.902E-16 3.815E-11
L_inf : 6.661E-16 6.661E-16 6.661E-16 6.661E-16 1.164E-10
--------------------------------------------------------------
Time = 0.0E+00 dt = 0.1E-05 ETA [d:h:m] 0:00:00:00 |> | [ 0.00%]
==============================================================
FLEXI RUNNING cartbox... [ 0.01 sec ] [ 0:00:00:00 ]
==============================================================
CALCULATION RUNNING...
--------------------------------------------------------------
Sys date : 06.11.2024 14:08:54
CALCULATION TIME PER STAGE/DOF: [ 7.05792E-07 sec ]
EFFICIENCY: CALCULATION TIME [s]/[Core-h]: [ 5.90480E-01 sec/h ]
Timestep : 1.063E-04
#Timesteps : 1.000E+00
Sim time : 1.000E-06
L_2 : 2.902E-16 2.521E-15 2.477E-15 2.513E-15 3.815E-11
L_inf : 6.661E-16 1.643E-14 1.421E-14 1.443E-14 1.164E-10
--------------------------------------------------------------
Time = 0.1000E-05 dt = 0.1000E-05 ETA [d:h:m]:<1 min |==>| [100.00%]
==============================================================
FLEXI RUNNING cartbox... [ 0.02 sec ] [ 0:00:00:00 ]
==============================================================
==============================================================
FLEXI FINISHED! [ 0.02 sec ] [ 0:00:00:00 ]
==============================================================
Error
If the output does not look like the one above, check for any error messages to diagnose the issue.
After completing the simulation, examine the contents of the working directory. For a successful run, the directory should contain additional generated files named <PROJECTNAME>_State_<TIMESTAMP>.h5. Each file stores the solution vector of the conserved variables at each interpolation node at a specific time, corresponding to multiples of Analyze_dt.
freestream
├── cartbox_mesh.h5
├── cartbox_State_0000000.000000000.h5
├── cartbox_State_0000000.000001000.h5
├── parameter_convert.ini
├── parameter_flexi.ini
├── parameter_hopr.ini
├── parameter_postiVisu.ini
5.2.5. Visualization
FLEXI relies on ParaView for visualization. In order to visualize the FLEXI solution, its format has to be converted from the HDF5 format into another format suitable for Paraview. FLEXI provides a post-processing tool posti_visu which generates files in VTK format with the following command.
posti_visu parameter_postiVisu.ini parameter_flexi.ini cartbox_State_0*