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## Normal Shock at Mach 1.3

Figure 1. Mach number contours for inviscid Mach 1.3 flow through a normal shock.

### Introduction

This verification case involves the steady, inviscid, adiabatic Mach 1.3 flow with a normal shock. Supersonic flow enters the normal shock and subsonic flow exits the shock. This flow is a classic, fundamental 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)
1.3 10.0 520.0 0.0 0.0

### Geometry

The geometry is a straight duct of length 2.0 ft with a diameter of 1.0 ft.

### Comparison Data

An analytic solution for the inviscid, supersonic, steady, adiabatic flow through a normal shock is well known and presented in any compressible flow text book. Table 3 presents the solution for a Mach 1.3 supersonic flow. The subscript 1 refers to the supersonic side of the shock, while the subscript 2 refers to the subsonic side of the shock.

### 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 where the Mach number is specified. The J1 boundary is a singular axis and is specified as an INVISCID WALL. The JMAX boundary is the casing of the duct and is specified as an INVISCID WALL. The K1 and KMAX boundaries are specified as REFLECTION planes.

The grid is generated using the Fortran 90 program normic.f90 with a grid size of (202x11x5). The grid is uniformly spaced in all directions. The grid file is norm.x and is in Plot3d format (unformatted, multi-zone, 3D, whole, single-precision). A formatted Plot3d file (formatted, multi-zone, 3D, whole, double-precision) is available as norm.x.fmt. This is converted to the common grid file format as norm.cgd.

The boundary conditions are implemented through GMAN as

gman < gman.com

### Initial Conditions

The initial flow conditions are created through the Fortran program normic.f90 to correspond to the analytic solution for the normal shock. The shock is placed at x = 1.0 ft and the even number of grid points assures that the shock is initially placed between two grid points.

### Computation Strategy

The computation was performed using the time-marching capabilities of Wind-US 3.0 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 norm.dat. Much of the default settings are used. The CFL number was 1.5 and the computation was run for 1000 iterations.

### Computation

The input and output files for the Wind-US computation are listed in Table 2.

Table 2. Input and Output files.
File Wind-US 3.139
Grid norm.cgd
Input Data File norm.dat
Solution norm.cfl
List / Residual norm.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

### 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 shock 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.wind.gen
avg < J1.wind.gen
avg < J1.nparc.gen

### Comparisons of the Results

A comparison with theoretical values of the properties behind the shock are provided below. Results from Wind-US are listed along with results from an earlier version of WIND and NPARC. All computations provide excellent agreement with the theoretical values.

Table 3. Analytic solution and CFD results.
Computation Exact Wind-US 3.139 WIND 2.172 NPARC 3.0
M2 0.7860 0.7860 0.7860 0.7860
p2 / p1 1.8050 1.8050 1.8049 1.8049
pt2 / pt1 0.9794 0.9794 0.9794 0.9794
rho2 / rho1 1.5157 1.5159 1.5156 1.5156
T2 / T1 1.1909 1.1907 1.1909 1.1908
Tt2 / Tt1 1.0000 1.0000 1.0000 1.0000

### References

Anderson, J.D., Modern Compressible Flow , McGraw Hill Inc., New York, 1984.