10 Degree Cone at Mach 2.35

Figure 1. Mach number contours for inviscid Mach 2.35 flow past a 10 degree cone.

Introduction

This verification case involves the steady, inviscid, adiabatic Mach 2.35 flow over a cone with a semi-vertex angle of 10 degrees. The flow is a classic conical flowfield with an attached shock at the apex of the cone with conical rays of constant properties eminating from the apex. This case contains results from the WIND and NPARC CFD codes.

Freestream Conditions

The freestream conditions are presented in Table 1. These conditions correspond to a Reynolds number of 2.5 million / ft, which is a typical operating point of the NASA Glenn 10x10 wind tunnel.

Geometry

The geometry is a cone with a semi-vertex angle of 10 degrees and a length of approximately 1.0 ft.

Comparison Data

An analytic solution for the inviscid, supersonic, steady, adiabatic flow over a cone is available through the Taylor-Maccoll differential equation as described in most compressible flow textbooks. The program HAP provided as part of the text book "Hypersonic Airbreathing Propulsion" by W.H. Heiser and D.T. Pratt solves this equation for the conditions just after the conical shock and on the cone surface. The subscript 1 refers to the freestream conditions. The subscript 2 refers to the conditions behind the shock. The subscript 3 refers to the conditions one the surface of the cone. The angle of the shock is 27.1843 degrees.

Computational Domain, Grids, and Boundary Conditions

Since the flow is supersonic, the inflow boundary I1 of the domain starts - 0.2 ft ahead of the apex of the cone and flow conditions are FROZEN. The outflow boundary IMAX is at approximately 1.0 ft with supersonic outflow, and so, is specified as an OUTFLOW boundary with extrapolation. The J1 boundary contains the axis-of-symmetry and the cone surface and is specified as an INVISCID WALL. The JMAX boundary is the supersonic freestream, and so, is specified as FROZEN. For three-dimensional computations, the K1 and KMAX boundaries are specified as REFLECTION planes.

The grids for the axisymmetric and three-dimensional domains are cone10.axi.x and cone10.3D.x, respectively, and are in Plot3d format (unformatted, multi-zone, 3D, whole, single-precision). These are converted to the common grid file format as cone10.axi.cgd and cone10.3D.cgd, respectively. The grid size of the grids is (121,81,5) with only one K plane used for the planar axisymmetric computations.

The boundary conditions are implemented through GMAN as

gman < gman.axi.com
gman < gman.3D.com

The restart file (fort.2) for NPARC is created through the Fortran program create2.f.

Initial Conditions

The initial flow conditions are the freestream conditions. WIND by default creates this. The Fortran program create2.f is used to create the initial restart file for NPARC.

Computation Strategy

The computation is performed using the time-marching capabilities of WIND and NPARC 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. WIND is computed using both planar axisymmetric and three-dimensional domains. NPARC is computed using only planar axisymmetric domains.

Input Parameters and Files

The input data file for the planar axisymmetric WIND computation is cone10.waxi.dat. A 5 degree section is used. The CFL number is 0.5. Grid sequencing is used to converge through 3 grid sizes each twice as fine as the previous. Each level is brought to convergence as part of a grid convergence study. Other settings use the default.

The input data file for the three-dimensional WIND computation is cone10.w3D.dat. A CFL number of 1.5 is used.

The input data file for the planar axisymmetric NPARC computation is cone10.nparc.in. A DTCAP of 0.5 is used.

Computation

The planar axisymmetric WIND computation was run with 3 levels of grid sequencing and converged solutions were obtained at each level as part of the grid convergence study. Each level required 600 cycles (3000 iterations) to converge. The final fine grid files are listed in Table 3.

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 convergence was checked by acknowledging that the L2 norm of the residuals had leveled off and then by the constancy of the average Mach number on the cone surface.

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 IMAX. The properties at IMAX are output to the GENPLOT file IMAX.gen by

cfpost < cfpost.IMAX.com

Properties at J1. The properties along J1, which includes the cone surface, are output to the GENPLOT file J1.gen by

cfpost < cfpost.J1.com

The Mach number on the cone surface can be output to the GENPLOT file machcone.gen by

cfpost < cfpost.machcone.com

The average Mach number (or other average property) can be computed using the Fortran program mavg.f. It simply sums up the values after x of 0.2 and averages them. The values should be fairly constant.

Comparisons of the Results

A comparison of the properties on the cone surface are provided below. As can be seen the WIND computations and the NPARC computations compare fairly well with each other and with the exact solution.

Grid Convergence Study

A grid convergence study was performed using the 3 levels of grids of the WIND axisymmetric computations. The Fortran program verify is used to perform the computations of the grid convergence study. The output is printed below. The results indicted that the solutions are well within the asymptotic range.

 --- VERIFY: Performs verification calculations ---
  
 Number of data sets read =  3
  
      Grid Size     Quantity
  
      1.000000      2.146783
      2.000000      2.146728
      4.000000      2.146115
  
 Order of convergence using first three finest grid 
 and assuming constant grid refinement (Eqn. 5.10.6.1) 
 Order of Convergence, p =  3.47636485
  
 Richardson Extrapolation: Use above order of convergence
 and first and second finest grids (Eqn. 5.4.1) 
 Estimate to zero grid value, f_exact =  2.1467886
  
 Grid Convergence Index on fine grids. Uses p from above.
 Factor of Safety =  1.25
  
   Grid     Refinement            
   Step      Ratio, r       GCI(%)
   1  2      2.000000      0.000680
   2  3      2.000000      0.007564
  
 Checking for asymptotic range using Eqn. 5.10.5.2.
 A ratio of 1.0 indicates asymptotic range.
  
  Grid Range    Ratio
  12 23      1.000000
  
 --- End of VERIFY ---

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. Some of the code features this case can check is 1) shock resolution, 2) axisymmetric / 3D consistency. A typical procedure is listed below:

cp cone10.3D.cgd run.cgd
cp cone10.w3D.dat run.dat (modify to run another 100 cycles)
cp cone10.w3D.cfl run.cfl
cp cone10.w3D.lis run.lis
wind -runinplace -dat run
cfpost < cfpost.machcone.com
mavg < machcone.gen
resplt < resplt.nsl2.com
cfpost < cfpost.nsl2.com

The value of the average Mach number on the cone surface can be compared to the value listed above to see if things changed signficantly. It should only change to the fifth decimal place. The L2 norm of the residual can also be checked to see that a large change is not present.

References

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

Heiser, W.H. and D.T. Pratt, Hypersonic Airbreathing Propulsion, AIAA Education Series, Washington, D.C., 1994.

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