NPARC Alliance Validation Archive
Validation Home   >   Archive   >   Prandtl-Meyer 15 Degree Corner Expansion at Mach 2.5

Prandtl-Meyer 15 Degree Corner Expansion at Mach 2.5

Figure 1 is described in the surrounding text
Figure 1. Mach number contours for inviscid, Prandtl-Meyer Mach 2.5 flow over a 15 degree expansion.

Introduction

This verification case involves the steady, inviscid, adiabatic Mach 2.5 flow over a 15 degree expansion corner. A centered expansion wave sets up at the corner. This flow is a classic, Prandtl-Meyer supersonic flow whose analytic solution is exact and can be found in any compressible flow textbook, such as the book by Anderson.

Supersonic Conditions

The supersonic conditions are presented in Table 1 and represent the reference conditions. The choice of pressure and temperature are arbitrary and do not change the basic character of the flow for perfect gases.

Table 1. Supersonic conditions.
Mach Pressure (psia) Temperature (R) Angle-of-Attack (deg) Angle-of-Sideslip (deg)
2.5 12.0 550.0 0.0 0.0

Geometry

The geometry is a straight plate with a 15 degree expansion corner located starting at x = 1.0 ft extending to a distance of x = 2.0 ft. The farfield boundary is located at y = 1.0 ft.

Comparison Data

An analytic solution for the inviscid, supersonic, steady, adiabatic flow over a 15 degree expansion is well known as a Prandtl-Meyer flow and presented in any compressible flow text book. The centered expansion wave is focused at the corner. For an inflow of Mach 2.5, the Mach wave at the head has an angle of 23.5782 degrees. The Mach wave at the tail has an angle of 17.9955 degrees. Table 3 presents the conditions across the wave. The subscript 1 refers to the conditions prior to the expansion, while the subscript 2 refers to the conditions on the wall past the expansion.

Computational Domain, Grids, and Boundary Conditions

Since the flow is supersonic, the inflow boundary I1 of the domain is specified as FROZEN. The outflow boundary IMAX is specified as an OUTFLOW boundary and flow is also expected to be supersonic. Thus, an extrapolation will be specified. The J1 boundary is a slip wall and is specified as an INVISCID WALL. The JMAX boundary is the farfield supersonic flow boundary and is specified as FROZEN.

The two-dimensional grid was generated using the ICEM CFD geometry modeling and grid generation package. The grid size is (71x61). The grid is clustered near the surface of the plate and corner. The grid file is pm15.x and is in Plot3d format (unformatted, multi-zone, 3D, whole, single-precision). This is then converted to the common grid file format as pm15.cgd.

The boundary conditions are implemented through GMAN as

gman < gman.com

Initial Conditions

The initial flow conditions are the freestream conditions as specified in Table 1.

Computation Strategy

The computation is performed using the time-marching capabilities of WIND to march to a steady-state (time asymptotic) solution. Local time stepping is used at each iteration. The time-marching is performed until convergence criteria is achieved.

Input Parameters and Files

The input data file for the WIND computation is pm15.dat. Much of the default settings are used. The CFL number is 0.5 and the computation is run for 2000 iterations.

Computation

The output files for the WIND and NPARC computations are listed in Table 2.

Table 2. Output files.
File WIND
Solution pm15.cfl
List pm15.lis

Convergence

The residual information was read from the WIND list files using the RESPLT utility and plotted using CFPOST,

resplt < resplt.nsl2.com
cfpost < cfpost.nsl2.com

The plot of the solution residual history with respect to cycles is presented in Fig. 2.

Figure 2 is described in the surrounding text
Figure 2. Solution residual history with cycles.

Post-Processing

The CFPOST utility was used to obtain information from the solution.

Flowfield Mach Number Contours. The Mach number contours for the flow field can be generated by

cfpost < cfpost.mach.com

Properties at J1. The properties along J1, which is the axial direction, are output to the GENPLOT file J1.gen by

cfpost < cfpost.J1.com

The average properties aft of the corner on the surface are computed using the Fortran program avg.f. It simply sums up the values after x of 1.2 and averages them. These values are expected be constant.

avg < J1.gen

Comparisons of the Results

A comparison of the properties after the compression fan are provided in Table 3 below. As can be seen the WIND computations compare fairly well with the exact solution for pressure and total temperature, but suffers for other quantities, especially total pressure.

Table 3. Analytic solution and CFD results.
Computation Exact WIND Error%
M2 3.2368 3.04866 -5.81
p2 / p1 0.3274 0.32727 0.04
pt2 / pt1 1.0 0.75679 -24.32
rho2 / rho1 0.4505 0.42127 -6.49
T2 / T1 0.7269 0.77687 6.87
Tt2 / Tt1 1.0 0.98709 -1.29

WIND Check Procedures

Modification of the WIND code requires the developer to check that no harm has been done to the code. This verification case can be run as one check. A typical procedure is listed below:

wind -runinplace -dat pm15
cfpost < cfpost.J1.com
avg < J1.gen

The average values computed by avg can be compared to the exact solution.

Download tar File

All the files of this case are available in the compressed tar file pm15.tar.Z. The files can then be extracted by the unix command

uncompress -c pm15.tar.Z | tar xvof -

References

Anderson, J.D., Modern Compressible Flow , McGraw Hill Inc., New York, 1982.

Contact Information

Questions or comments about this case can be sent be emailed to John W. Slater,

NASA John H. Glenn Research Center, MS 86-7
21000 Brookpark Road
Cleveland, Ohio 44135
Phone: (216) 433-8513
e-mail: John.W.Slater@grc.nasa.gov
Last Updated: Tuesday, 08-Jul-2008 07:34:39 EDT