## Goals

Upon completing this tutorial, the user will be familiar with performing a simulation of external, laminar flow over a flat plate. The solution will provide a laminar boundary layer on the surface, which can be compared to the Blasius solution as a validation case for SU^{2}. Consequently, the following capabilities of SU^{2} will be showcased in this tutorial:

- Steady, 2D, Laminar Navier-Stokes equations
- Multigrid
- Roe (Second-Order) numerical scheme in space
- Euler implicit time integration
- Inlet, Outlet, Symmetry, and Navier-Stokes Wall boundary conditions

The intent of this tutorial is to introduce a common viscous test case which is used to explain how different equation sets can be chosen in SU^{2}. We also introduce some details on the numerics and a new type of convergence criteria which monitors the change of a specific objective, such as lift or drag, in order to assess convergence.

## Resources

The resources for this tutorial can be found in the *TestCases/navierstokes/flatplate* directory. You will need the configuration file (lam_flatplate.cfg) and the mesh file (mesh_flatplate_65x65.su2).

## Tutorial

The following tutorial will walk you through the steps required when solving for the flow over a flat plate using SU^{2}. It is assumed you have already obtained and compiled the SU2_CFD code for a serial computation. If you have yet to complete these requirements, please see the Download and Installation pages.

__Background__

In his Ph.D dissertation in 1908, H. Blasius obtained what is now referred to as the Blasius equation for incompressible, laminar flow over a flat plate:

The third-order, ordinary differential equation can be solved numerically using a shooting method resulting in the well-known laminar boundary layer profile. Using the numerical solution, an expression for the skin friction coefficient along the flat plate can also be derived:

where Re_{x} is the Reynolds number along the plate. In this tutorial, we will perform a solution of *nearly* incompressible (low Mach number) laminar flow over a flat plate and compare our results against the analytical Blasius solutions for the profile shape and skin friction coefficient along the plate. This problem has become a classic validation case for viscous flow solvers. More detail on the Blasius solution and the similarity variables can be found in Chapter 18 of *Fundamentals of Aerodynamics* (Fourth Edition) by John D. Anderson, Jr. and most other texts on aerodynamics.

__Problem Setup__

This problem will solve the for the flow over the flat plate with these conditions:

- Inlet Stagnation Temperature = 300.0 K
- Inlet Stagnation Pressure = 100000.0 N/m
^{2} - Inlet Flow Direction, unit vector (x,y,z) = (1.0, 0.0, 0.0)
- Outlet Static Pressure = 97250.0 N/m
^{2} - Resulting free-stream Mach number = 0.2
- Reynolds number = 1301233.166 for a plate length of 0.3048 m (1 ft)

#### Mesh Description

The computational mesh for the flat plate is structured (quadrilaterals) with 65 nodes in both the *x*- and *y*-directions. The flat plate is along the lower boundary of the domain (*y* = 0) starting at *x* = 0 m and is of length 0.3048 m (1 ft). In the figure of the mesh, this corresponds to the Navier-Stokes (no-slip) boundary condition highlighted in green. The domain extends a distance upstream of the flat plate, and a symmetry boundary condition is used to simulate a free-stream approaching the plate in this region (highlighted in purple). Axial stretching of the mesh is used to aid in resolving the region near the start of the plate where the no-slip boundary condition begins at *x* = 0 m, as shown in Figure (1).

Figure (1): Figure of the computational mesh with boundary conditions.

Because the flow is subsonic and disturbances caused by the presence of the plate can propagate both upstream and downstream, the characteristic-based, subsonic inlet and outlet boundary conditions are used for the flow entrance plane (red) and the outflow regions along the upper region of the domain and the exit plane at *x* = 0.3048 m (blue). In any simulation of viscous flow, it is important to capture the behavior of the boundary layer. Doing so requires an appropriate level of grid refinement near the wall. In this mesh, the vertical spacing is such that approximately 30 nodes lie within the boundary layer, which is typical for laminar flows of this nature.

#### Configuration File Options

Several of the key configuration file options for this simulation are highlighted here. Here, we set the problem definition for a viscous flow:

To compute viscous flows, the Navier-Stokes governing equations are selected. In conjunction with selecting Navier-Stokes as the problem type, the type of turbulence model must also be specified. Laminar flows can be computed by entering "NONE" as the option for the type of turbulence model. For turbulent flows, SU^{2} also has the Spalart-Allmaras model (SA) and the Shear Stress Transport (SST) model of Menter available for use. If this were an inviscid flow problem, the user would enter "EULER" for the problem type. SU^{2} supports other governing equations, as well, and the user is invited to review the configuration page for a description of the possible options.

Defining a no-slip boundary condition can be accomplished in one of two ways:

An adiabatic, no-slip boundary condition can be selected by using the MARKER_HEATFLUX option with the value of the heat flux set to 0.0. An isothermal wall condition is also available with a similar format.

Here, we define some of the typical numerical methods chosen for calculating viscous flows with SU2:

For this problem, we are choosing a typical set of numerical methods. However, it is advised that users should experiment with various numerical methods for their own problems. The gradients are calculated via weighted least squares. The 2nd-order Roe upwind method with the Venkatakrishnan limiter is used for computing convective fluxes, and the viscous terms are computed with the corrected average of gradients method (by default). A limiter coefficient of 0.3 is typical, but the user is also advised to experiment with the level of limiting for their particular application (note that a *smaller* value of the coefficient implies *larger* limiting of the gradients).

SU2 features multiple ways to assess convergence:

Rather than achieving a certain order of magnitude in the density residual to judge convergence, what we call the Cauchy convergence criteria is chosen for this problem. This type of criteria measures the change in a specific quantity of interest over a specified number of previous iterations. With the options selected above, the flat plate solution will terminate when the change in the drag coefficient (CAUCHY_FUNC_FLOW) for the plate over the previous 100 iterations (CAUCHY_ELEMS) becomes less than 1E-6 (CAUCHY_EPS). A convergence criteria of this nature can be very useful for design problems where the solver is embedded in a larger design loop and reliable convergence behavior is essential.

__Running SU__^{2}

^{2}

The flat plate simulation for the 65x65 node mesh is small and will execute relatively quickly on a single workstation or laptop in serial. To run this test case, follow these steps at a terminal command line:

- Move to the directory containing the config file (lam_flatplate.cfg) and the mesh file (mesh_flatplate_65x65.su2). Make sure that the SU
^{2}tools were compiled, installed, and that their install location was added to your path. - Run the executable by entering "SU2_CFD lam_flatplate.cfg" at the command line.
- SU
^{2}will print residual updates with each iteration of the flow solver, and the simulation will terminate after reaching the specified convergence criteria. - Files containing the results will be written upon exiting SU
^{2}. The flow solution can be visualized in ParaView (.vtk) or Tecplot (.dat for ASCII).

__Results__

Results are here given for the SU^{2} solution of laminar flow over the flat plate. The results show excellent agreement with the analytic Blasius solution.

Figure (2): Mach contours for the laminar flat plate.

Figure (3): Velocity data was extracted from the exit plane of the mesh (x = 0.3048 m) near the wall, and the boundary layer velocity profile was plotted compared to and using the similarity variables from the Blasius solution.

Figure (4): A plot of the skin friction coefficient along the plate created using the values written in the surface_flow.csv file and compared to Blasius.