This tutorial can be run as a Jupyter notebook. Try it on our cloud resources!
Compressible Flow Simulation
Learning objectives
Welcome to the tutorial on the Compressible Flow simulation in Nektar++ framework. This tutorial aims to show how to setup and run a simulation for inviscid compressible flows (governed by Euler equations) using a well known geometry of NACA0012 airfoil.
After completetion of this tutorial you will be familiar with:
- Configuring a session file suitable for compressible Euler equations in Nektar++
- Solving the compressible flows explicitly and implicitly
**Note:** It is suggested to download the cases to your local machine or cluster to complete the tutorial.
Introduction
In this tutorial we walk through the user how to configure a session file for solving the compressible Euler equations using a simple flow example. We consider the 2D flow over the NACA0012 airfoil.
Governing equation
The compressible solver implemented in Nektar++ uses the conservative form of the governing equations.
For Navier-Stokes equations: \begin{equation} \label{eq::NS} \frac{\partial \mathbf{Q}}{\partial t} + \frac{\partial \mathbf{F}^c_1}{\partial x_1} + \frac{\partial \mathbf{F}^c_2}{\partial x_2} + \frac{\partial \mathbf{F}^c_3}{\partial x_3} = \frac{\partial \mathbf{F}^v_1}{\partial x_1} + \frac{\partial \mathbf{F}^v_2}{\partial x_2} + \frac{\partial \mathbf{F}^v_3}{\partial x_3} \end{equation}
For Euler equations: \begin{equation} \label{eq::Euler} \frac{\partial \mathbf{Q}}{\partial t} + \frac{\partial \mathbf{F}^c_1}{\partial x_1} + \frac{\partial \mathbf{F}^c_2}{\partial x_2} + \frac{\partial \mathbf{F}^c_3}{\partial x_3} = 0 \end{equation}
In the above $x_i$ denotes the $x$, $y$, and $z$-directions for $i=1,2,3$, respectively, $\mathbf{Q}$ is the vector of conservative variables, $\mathbf{F}^c_i = \mathbf{F}^c_i(\mathbf{Q})$ and $\mathbf{F}^v_i = \mathbf{F}^v_i(\mathbf{Q},\nabla \mathbf{Q})$ are vectors for convective flux and viscous flux, respectively \begin{equation} \label{eq::NS_vectors} \mathbf{Q} =\left( \begin{array}{c} \rho \ \rho u \ \rho v \ \rho w \ E \end{array} \right), \hspace{0.5 cm} \mathbf{F}^c_i =\left( \begin{array}{c} \rho u_i \ \rho u_1 u_i + p \delta_{1i} \ \rho u_2 u_i + p \delta_{2i} \ \rho u_3 u_i + p \delta_{3i} \ u_i (E + p) \end{array} \right), \hspace{0.5 cm} \mathbf{F}^v_i =\left( \begin{array}{c} 0 \ \tau_{1i} \ \tau_{2i} \ \tau_{3i} \ u_j\tau_{ji} + \kappa \partial T / \partial x_i \end{array} \right) \end{equation} where $\rho$ is the density, $u_i$ is the velocity component in the $x_i$-direction, $p$ is the pressure, $\delta_{ij}$ is the Kronecker delta function, and $E$ is the total energy per unit volume. As for the viscous flux vectors, $T$ is the temperature, $\kappa$ is the thermal conductivity, $\tau_{ij}$ is stress tensor component.
Problem description
In this tutorial we simulate a 2D compressible flow over NACA0012 airfoil. The simulation is non-dimesionalized so that the chord length is scaled to a unit, $L=1.0$, and the following freestream parameters are used:
| \(\gamma\) | \(Ma\) | \(R\) | \(T_{\infty}\) | \(\rho_{\infty}\) | \(U_{\infty}\) | \(\alpha\) |
|---|---|---|---|---|---|---|
| \(1.4\) | \(0.4\) | \(\frac{1}{\gamma Ma^2}\) | \(1.0\) | \(1.0\) | \(1.0\) | \(0.0\) |
The flow domain is a rectangle of sizes $[-5,10] \times [-5,5]$ as shown in the figure below.
 |
 |
| Full domain and mesh | Near wall mesh |
We apply the following boundary conditions (BCs):
- No-penetration condition on the airfoil surface;
- Riemann boundary condition at the boundary of the other four boundaries (inflow, outflow, top, and bottom)
**Note:** For the top and bottom boundaries, a periodic boundary condition in the $y$-direction can be used instead.
Session file setup: explicit Euler
In this section we go through the settings in the session file (.xml) for explicit simulations. Please open the session file session_naca_ex.xml. First, let us complete the <EXPANSIONS> tags.
Expansion bases
Task 1: Please write the missing parts in <EXPANSIONS> tag. A completed <EXPANSIONS> tag has the following standard form:
<EXPANSIONS>
<E COMPOSITE="C[1]" NUMMODES="2" FIELDS="rho,rhou,rhov,E" TYPE="MODIFIED" /> <!--quad elements-->
<E COMPOSITE="C[2]" NUMMODES="2" FIELDS="rho,rhou,rhov,E" TYPE="MODIFIED" /> <!--tri elements-->
</EXPANSIONS>`
Here,
COMPOSITErefers to the ID assigned to the geometric domain (or sub-domain) defined in the geometry xml file.NUMMODESis used to indentify the polynomial order of the expansion basis functions as P =NUMMODES-1.TYPEdenotes the basis functions. Possible choices areMODIFIED,GLL_LAGRANGE,GLL_LAGRANGE_SEM. One may note that the first basis type uses modal expansion whereas the remaining two consider nodal expansion.
Conditions setup
Now, we will complete the <CONDITIONS> tag. There are five sections in this tag:
- SolverInfo
- Parameters
- Variables
- Boundary Conditions
- Initial Conditions
Solver info
Task 2: Please complete the properties EQTYPE, TimeIntegrationMethod, and UpwindType as shown below,
<I PROPERTY="EQType" VALUE="EulerCFE" />
<I PROPERTY="TimeIntegrationMethod" VALUE="RungeKutta2_SSP" />
<I PROPERTY="UpwindType" VALUE="Roe" />
where
EQTYPEsets the kind of equations we want to solve. Here we use the Euler equations;TimeIntegrationMethodsets the integration method for time advancing. Here we use the 2nd order Runge-Kutta;UpwindTypesets the Riemann solver numerical inviscid flux computation. Here we use the approximate Riemann solver of Roe.
Parameters
In <PARAMETERS> tag, we can define the parameters present in the flow problem as well as the parameters required by various solvers of Nektar++. Hence a parameter can either be defined in the context of the problem we are solving (dummay variable), or be predefined within Nektar++ (solver parameter). One should note that the names of the solver parameters can’t be altered.
Task 3: Assign values to the following parameters as
<P> Ma = 0.4 </P>
<P> GasConstant = 1.0/(Gamma*Ma^2) </P>
<P> AoA = 0.0/180.0*PI </P>
<P> Timestep = 5.0e-5 </P>
<P> NumSteps = 40000 </P>
Here,
Mais the Mach number. It is not necessary but set to compute non-dimensionalizedGasConstantconveniently;GasConstantis the gas constant predefined in Nektar++.TimeStepsets the time-step size for the time integration. One should select magnitude ofTimeStepstrategically to avoid the situation of CFL (Courant-Friedrichs-Lewy) number to be too large. However, in the following implicit session file, this value can be much larger!NumStepssets the number of time-steps required for the simulation.
Variables
In <VARIABLES> tag, we define the number (and name) of solution variables. We prescribed a unique ID to each variable.
Task 4: Provide a ID for the conservative variable $\rho v$ as
<V ID="2"> rhov </V>
Boundary conditions
Boundary conditions are defined by two tags. First we define the boundary regions in the <BOUNDARYREGIONS> tag in terms of composite C[ ] entities from the <GEOMETRY> tag section of the mesh xml file. Each boundary region has a unique ID. For the current computational domain there are seven composites (see towards the bottom section of naca.xml file) marked as C[1] to C[7].
Task 5: Define the inflow region using C[4] as shown below
<B ID="3"> C[4] </B> <!-- inflow -->
The next tag we define is <BOUNDARYCONDITIONS> tag by which we actually specify the boundary conditions for each boundary ID specified in the above<BOUNDARYREGIONS> tag.
Task 6: Define the the conservative variables at inflow as the farfield quantities as
<REGION REF="3">
<D VAR="rho" VALUE="rhoInf" />
<D VAR="rhou" VALUE="rhoInf * uInf" />
<D VAR="rhov" VALUE="rhoInf * vInf" />
<D VAR="E" VALUE="pInf / (Gamma - 1) + 0.5 * rhoInf * (uInf * uInf + vInf * vInf)"/>
</REGION>
Initial conditions
Task 7: Provide the initial conditions as the freestream conditions in the “InitialConditions” FUNCTION as
<E VAR="rho" VALUE="rhoInf" />
<E VAR="rhou" VALUE="rhoInf * uInf" />
<E VAR="rhov" VALUE="rhoInf * vInf" />
<E VAR="E" VALUE="pInf / (Gamma - 1) + 0.5 * rhoInf * (uInf * uInf + vInf * vInf)"/>
Running the solver
The two xml files provided in this tutorial are:
- Geometry file: naca.xml
- Condition file: session_naca.xml
The flow solvers in Nektar++ can be run either in serial or parallel. However, for this case it is suggessted to run in parallel on your local machine. Here we use ! (the jupyter notebook syntax for bash command) to run the solver in this online interactive tutorial which used Docker imange and Binder to run Nektar++ in the background.
Task 8: Go to the next cell and click the Run icon to run the solver. In this example we use 4 physical cores to solve the problem.
**Note:** Use of high number of processors might not work in the current online tutorial. Also, parallel run might be slower than the serial run due to resource constrain.
**Note:** To run the solver in serial, remove the `mpirun -np N` part.
!mpirun -np 4 CompressibleFlowSolver naca.xml session_naca_ex-Complete.xml
Command output
gs_setup: 290 unique labels shared
pairwise times (avg, min, max): 1.24138e-06 1.2205e-06 1.2558e-06
crystal router : 2.245e-06 2.2254e-06 2.27103e-06
all reduce : 2.77909e-06 2.72729e-06 2.8288e-06
used all_to_all method: pairwise
handle bytes (avg, min, max): 3756 3092 4516
buffer bytes (avg, min, max): 2320 1888 2816
gs_setup: 290 unique labels shared
pairwise times (avg, min, max): 1.15705e-06 1.14245e-06 1.19517e-06
crystal router : 2.25196e-06 2.23555e-06 2.25799e-06
all reduce : 2.64696e-06 2.60947e-06 2.66479e-06
used all_to_all method: pairwise
handle bytes (avg, min, max): 3756 3092 4516
buffer bytes (avg, min, max): 2320 1888 2816
gs_setup: 290 unique labels shared
pairwise times (avg, min, max): 1.26855e-06 1.23549e-06 1.29472e-06
crystal router : 2.22756e-06 2.19028e-06 2.25045e-06
all reduce : 2.62342e-06 2.61217e-06 2.6295e-06
used all_to_all method: pairwise
handle bytes (avg, min, max): 3756 3092 4516
buffer bytes (avg, min, max): 2320 1888 2816
gs_setup: 290 unique labels shared
pairwise times (avg, min, max): 1.189e-06 1.1649e-06 1.21007e-06
crystal router : 2.22535e-06 2.20472e-06 2.24514e-06
all reduce : 2.6423e-06 2.62968e-06 2.66181e-06
used all_to_all method: pairwise
handle bytes (avg, min, max): 3756 3092 4516
buffer bytes (avg, min, max): 2320 1888 2816
=======================================================================
EquationType: EulerCFE
Session Name: naca
Spatial Dim.: 2
Max SEM Exp. Order: 2
Num. Processes: 4
Expansion Dim.: 2
Riemann Solver: Roe
Advection Type:
Projection Type: Discontinuous Galerkin
Advection: explicit
AdvectionType: WeakDG
Diffusion: explicit
Time Step: 5e-05
No. of Steps: 40000
Checkpoints (steps): 5000
Integration Type: RungeKutta
=======================================================================
Initial Conditions:
- Field rho: rhoInf
- Field rhou: rhoInf * uInf
- Field rhov: rhoInf * vInf
- Field E: pInf / (Gamma - 1) + 0.5 * rhoInf * (uInf * uInf + vInf * vInf)
Writing: "naca_0.chk" (0.00510452s, XML)
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renaming "naca_1.chk" -> "naca_1.bak2.chk"
Writing: "naca_1.chk" (0.0152179s, XML)
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renaming "naca_2.chk" -> "naca_2.bak2.chk"
Writing: "naca_2.chk" (0.0151992s, XML)
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renaming "naca_3.chk" -> "naca_3.bak2.chk"
Writing: "naca_3.chk" (0.0161381s, XML)
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renaming "naca_4.chk" -> "naca_4.bak2.chk"
Writing: "naca_4.chk" (0.016031s, XML)
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renaming "naca_5.chk" -> "naca_5.bak2.chk"
Writing: "naca_5.chk" (0.016035s, XML)
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renaming "naca_6.chk" -> "naca_6.bak2.chk"
Writing: "naca_6.chk" (0.0157765s, XML)
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renaming "naca_7.chk" -> "naca_7.bak2.chk"
Writing: "naca_7.chk" (0.0162489s, XML)
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renaming "naca_8.chk" -> "naca_8.bak1.chk"
Writing: "naca_8.chk" (0.0158526s, XML)
Time-integration : 142.1s
renaming "naca.fld" -> "naca.bak4.fld"
Writing: "naca.fld" (0.0158572s, XML)
-------------------------------------------
Total Computation Time = 158s
-------------------------------------------
L 2 error (variable rho) : 12.2441
L inf error (variable rho) : 1.08473
L 2 error (variable rhou) : 12.2431
L inf error (variable rhou) : 1.40827
L 2 error (variable rhov) : 0.0787936
L inf error (variable rhov) : 0.617216
L 2 error (variable E) : 142.775
L inf error (variable E) : 12.6828
Post-processing
To visualise the flowfield first we convert the .fld file (generated by the solver) to a .vtu format file by using the FieldConvert ulitity available in the Nektar++.
Task 9: Run the following code cell to convet the .fld into a .vtu file.
!FieldConvert -f naca.xml session_naca_ex-Complete.xml naca.fld flowfield_ex.vtu
Command output
Writing: "flowfield.vtu"
Written file: flowfield.vtu
**Note:** `-f` is supplied to force output, otherwise the user would be prompted whether to overwrite an existing file.
**Note:** It is strongly recommended to use `-f` option when one is running the commands in Jupyter notebook cells.
Task 10: Run the following python script which uses pyvista combined with itkwidgets to visualise the flowfield in-browser.
import pyvista as pv
# First read the VTK file
mesh = pv.read('flowfield_ex.vtu')
# Now create an itkwidget to visualise this in-browser.
pl = pv.PlotterITK()
pl.add_mesh(mesh, scalars=mesh['u'], smooth_shading=True)
pl.show(True)
Viewer(geometries=[{'vtkClass': 'vtkPolyData', 'points': {'vtkClass': 'vtkPoints', 'name': '_points', 'numberO…
Viewer(geometries=[{'vtkClass': 'vtkPolyData', 'points': {'vtkClass': 'vtkPoints', 'name': '_points', 'numberO…
 |
| The velocity field solving by set `NUMMODES=4` |
Session file setup: implicit Euler
In Nektar++ the compressible flows can also be solved implicitly. The majority of the session file is the same as the one for explicit solving while the differences only sit in the SOLVERINFO and PARAMETERS tag. Please open the session file session_naca_im.xml and complete the following steps.
Solver info
Task 11: Please complete the properties EQTYPE, AdvectionAdvancement, and TimeIntegrationMethod as shown below,
<I PROPERTY="EQType" VALUE="EulerImplicitCFE"/>
<I PROPERTY="AdvectionAdvancement" VALUE="Implicit" />
<I PROPERTY="TimeIntegrationMethod" VALUE="DIRKOrder2" />
where EQTYPE is set to be the implicit type of Euler equations, AdvectionAdvancement and TimeIntegrationMethod of the implicit versions are used accordingly.
Parameters
Since we are now solving the Euler equations implicitly, the restrictions on time step is very much relexed. However, to start from the uniform freestream condition, we still use a smaller time step to keep convergence.
Task 12: Assign values to the following parameters as
<P> Timestep = 1.0e-4 </P>
<P> NumSteps = 100 </P>
Running the solver: starting
Task 13: Go to the next cell and click the Run icon to run the impicit solver.
!mpirun -np 4 CompressibleFlowSolver naca.xml session_naca_im.xml
Command output
gs_setup: 290 unique labels shared
pairwise times (avg, min, max): 1.24138e-06 1.2205e-06 1.2558e-06
crystal router : 2.245e-06 2.2254e-06 2.27103e-06
all reduce : 2.77909e-06 2.72729e-06 2.8288e-06
used all_to_all method: pairwise
handle bytes (avg, min, max): 3756 3092 4516
buffer bytes (avg, min, max): 2320 1888 2816
gs_setup: 290 unique labels shared
pairwise times (avg, min, max): 1.15705e-06 1.14245e-06 1.19517e-06
crystal router : 2.25196e-06 2.23555e-06 2.25799e-06
all reduce : 2.64696e-06 2.60947e-06 2.66479e-06
used all_to_all method: pairwise
handle bytes (avg, min, max): 3756 3092 4516
buffer bytes (avg, min, max): 2320 1888 2816
gs_setup: 290 unique labels shared
pairwise times (avg, min, max): 1.26855e-06 1.23549e-06 1.29472e-06
crystal router : 2.22756e-06 2.19028e-06 2.25045e-06
all reduce : 2.62342e-06 2.61217e-06 2.6295e-06
used all_to_all method: pairwise
handle bytes (avg, min, max): 3756 3092 4516
buffer bytes (avg, min, max): 2320 1888 2816
gs_setup: 290 unique labels shared
pairwise times (avg, min, max): 1.189e-06 1.1649e-06 1.21007e-06
crystal router : 2.22535e-06 2.20472e-06 2.24514e-06
all reduce : 2.6423e-06 2.62968e-06 2.66181e-06
used all_to_all method: pairwise
handle bytes (avg, min, max): 3756 3092 4516
buffer bytes (avg, min, max): 2320 1888 2816
=======================================================================
EquationType: EulerCFE
Session Name: naca
Spatial Dim.: 2
Max SEM Exp. Order: 2
Num. Processes: 4
Expansion Dim.: 2
Riemann Solver: Roe
Advection Type:
Projection Type: Discontinuous Galerkin
Advection: explicit
AdvectionType: WeakDG
Diffusion: explicit
Time Step: 5e-05
No. of Steps: 40000
Checkpoints (steps): 5000
Integration Type: RungeKutta
=======================================================================
Initial Conditions:
- Field rho: rhoInf
- Field rhou: rhoInf * uInf
- Field rhov: rhoInf * vInf
- Field E: pInf / (Gamma - 1) + 0.5 * rhoInf * (uInf * uInf + vInf * vInf)
Writing: "naca_0.chk" (0.00510452s, XML)
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renaming "naca_1.chk" -> "naca_1.bak2.chk"
Writing: "naca_1.chk" (0.0152179s, XML)
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renaming "naca_2.chk" -> "naca_2.bak2.chk"
Writing: "naca_2.chk" (0.0151992s, XML)
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renaming "naca_3.chk" -> "naca_3.bak2.chk"
Writing: "naca_3.chk" (0.0161381s, XML)
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renaming "naca_4.chk" -> "naca_4.bak2.chk"
Writing: "naca_4.chk" (0.016031s, XML)
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renaming "naca_5.chk" -> "naca_5.bak2.chk"
Writing: "naca_5.chk" (0.016035s, XML)
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renaming "naca_6.chk" -> "naca_6.bak2.chk"
Writing: "naca_6.chk" (0.0157765s, XML)
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renaming "naca_7.chk" -> "naca_7.bak2.chk"
Writing: "naca_7.chk" (0.0162489s, XML)
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renaming "naca_8.chk" -> "naca_8.bak1.chk"
Writing: "naca_8.chk" (0.0158526s, XML)
Time-integration : 142.1s
renaming "naca.fld" -> "naca.bak4.fld"
Writing: "naca.fld" (0.0158572s, XML)
-------------------------------------------
Total Computation Time = 158s
-------------------------------------------
L 2 error (variable rho) : 12.2441
L inf error (variable rho) : 1.08473
L 2 error (variable rhou) : 12.2431
L inf error (variable rhou) : 1.40827
L 2 error (variable rhov) : 0.0787936
L inf error (variable rhov) : 0.617216
L 2 error (variable E) : 142.775
L inf error (variable E) : 12.6828
Task 14: Use the command in the next block to rename the output naca.fld file as naca_im_start.fld
!mv naca.fld/ naca_im_start.fld/
Now we can use a much larger time step restarting from the previous result. Please follow these step to re-run the solver.
Task 15: Assign values to the following parameters as
<P> Timestep = 2.0e-3 </P>
<P> NumSteps = 1000 </P>
Task 16: Set the “initialConditions” FUNCTION as the output of the starting
<FUNCTION NAME="InitialConditions">
<F VAR="rho,rhou,rhov,E" FILE="naca_im_start.fld"/>
</FUNCTION>
Task 17: Go to the next cell and click the Run icon to restart the impicit solver.
!mpirun -np 4 CompressibleFlowSolver naca.xml session_naca_im.xml
Command output
gs_setup: 290 unique labels shared
pairwise times (avg, min, max): 1.24732e-06 1.215e-06 1.27451e-06
crystal router : 2.30079e-06 2.26684e-06 2.32877e-06
all reduce : 2.83667e-06 2.79583e-06 2.86503e-06
used all_to_all method: pairwise
handle bytes (avg, min, max): 3756 3092 4516
buffer bytes (avg, min, max): 2320 1888 2816
gs_setup: 290 unique labels shared
pairwise times (avg, min, max): 1.30166e-06 1.27563e-06 1.32388e-06
crystal router : 2.29748e-06 2.26544e-06 2.33203e-06
all reduce : 2.64156e-06 2.61944e-06 2.65921e-06
used all_to_all method: pairwise
handle bytes (avg, min, max): 3756 3092 4516
buffer bytes (avg, min, max): 2320 1888 2816
gs_setup: 290 unique labels shared
pairwise times (avg, min, max): 1.20776e-06 1.17887e-06 1.22907e-06
crystal router : 2.32121e-06 2.30866e-06 2.33734e-06
all reduce : 2.6498e-06 2.63294e-06 2.6634e-06
used all_to_all method: pairwise
handle bytes (avg, min, max): 3756 3092 4516
buffer bytes (avg, min, max): 2320 1888 2816
gs_setup: 290 unique labels shared
pairwise times (avg, min, max): 1.20704e-06 1.18352e-06 1.23838e-06
crystal router : 2.2938e-06 2.26917e-06 2.31992e-06
all reduce : 2.65795e-06 2.64859e-06 2.66964e-06
used all_to_all method: pairwise
handle bytes (avg, min, max): 3756 3092 4516
buffer bytes (avg, min, max): 2320 1888 2816
=======================================================================
EquationType: EulerImplicitCFE
Session Name: naca
Spatial Dim.: 2
Max SEM Exp. Order: 2
Num. Processes: 4
Expansion Dim.: 2
Riemann Solver: Roe
Advection Type:
Projection Type: Discontinuous Galerkin
Advection: implicit
AdvectionType: WeakDG
Diffusion: explicit
Time Step: 0.002
No. of Steps: 1000
Checkpoints (steps): 500
Integration Type: DIRK
=======================================================================
Initial Conditions:
- Field rho: from file naca_im_start.fld
- Field rhou: from file naca_im_start.fld
- Field rhov: from file naca_im_start.fld
- Field E: from file naca_im_start.fld
renaming "naca_3.chk" -> "naca_3.bak0.chk"
Writing: "naca_3.chk" (0.0277014s, XML)
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renaming "naca_4.chk" -> "naca_4.bak0.chk"
Writing: "naca_4.chk" (0.0154979s, XML)
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renaming "naca_5.chk" -> "naca_5.bak0.chk"
Writing: "naca_5.chk" (0.0159675s, XML)
Time-integration : 82.5745s
renaming "naca.fld" -> "naca.bak6.fld"
Writing: "naca.fld" (0.0158489s, XML)
-------------------------------------------
Total Computation Time = 83s
-------------------------------------------
L 2 error (variable rho) : 12.2441
L inf error (variable rho) : 1.08474
L 2 error (variable rhou) : 12.2431
L inf error (variable rhou) : 1.40795
L 2 error (variable rhov) : 0.0787586
L inf error (variable rhov) : 0.617043
L 2 error (variable E) : 142.775
L inf error (variable E) : 12.6824
Post-processing
To visualise the flowfield first we convert the .fld file (generated by the solver) to a .vtu format file by using the FieldConvert ulitity available in the Nektar++.
Task 18: Run the following code cell to convet the .fld into a .vtu file.
!FieldConvert -f naca.xml session_naca_im.xml naca.fld flowfield_im.vtu
Command output
Writing: "flowfield_im.vtu"
Written file: flowfield_im.vtu
import pyvista as pv
# First read the VTK file
mesh = pv.read('flowfield_im.vtu')
# Now create an itkwidget to visualise this in-browser.
pl = pv.PlotterITK()
pl.add_mesh(mesh, scalars=mesh['u'], smooth_shading=True)
pl.show(True)
Viewer(geometries=[{'vtkClass': 'vtkPolyData', 'points': {'vtkClass': 'vtkPoints', 'name': '_points', 'numberO…
Viewer(geometries=[{'vtkClass': 'vtkPolyData', 'points': {'vtkClass': 'vtkPoints', 'name': '_points', 'numberO…
Extra tasks
Task 19: A better resolution can be obtained by increasing the expansion order under the <EXPANSIONS> tag. We can set the NUMMODES=5 and check the result. Please noted that a higher order expansion may require the time step to be smaller to avoid divergence.
**Note:** To run the Navier-Stokes simulation, under the `` tag change `EulerCFE` or `EulerImplicitCFE` to be `NavierStokesCFE` or `NavierStokesImplicitCFE` for an explicit or implicit simulation, respectively . `DiffusionType`, `DiffusionAdvancement`, and `ViscosityType` need to be specified as well. Other differences include the parameters related to viscosity and heat transfer under `` tag and the type of wall boundary condition under `` tag. An example for implicitly solving Navier-Stokes equations is provided in [**session_naca_NS-Complete.xml**](session_naca_NS-Complete.xml). More information can also be found in the [tutorial](https://doc.nektar.info/tutorials/latest/cfs/CylinderSubsonic_NS/cfs-CylinderSubsonic_NS.pdf)
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# Take-a-ways
It is worthy to keep in mind the following points:
- The comprssible flows can be solved either explicitly or implicitly;
- The explicit solving requires a smaller time step to satisfy the CFL condition while the restriction is very much relexed for the implicit solving;
# References
1. Nektar++: Spectral/hp Element Framework, Version 5.0.1, [User guide](http://www.nektar.info/downloads), January, 2021.
2. Kirby R, Sherwin S. Stabilisation of spectral/hp methods through spectral vanishing viscosity: Application to fluid mechanics modelling. Comput. Methods Appl. Mech. Engrg. 2006; 195: 3128–3144
## End of the tutorial.
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