Proximal Bodies in Hypersonic Flow

Problem Description

In this problem, we seek to determine the forces experienced by a secondary body in the shocked region created by a primary body travelling at hypersonic speeds. The problem is considered in two and three dimensions - in the first case bodies are circular cylinders; in the second, spheres. The flow is created by a ramping up of flow speed at the left boundary to a freestream Mach number of 10. The downstream displacement of the secondary body is fixed (at 4 and 3 primary body radii, center-to-center, for the cylindrical and spherical cases respectively), while the axial displacement is varied. These computations are to be compared to a series of experiments in the T5 Hypervelocity Shock Facility. [source codes]

• Configuration for the spherical case:
  

Numerical Simulation

• Euler equations in two- and three-dimensions, using an ideal gas (air with %$\gamma=1.4$%)
• Van-Leer flux vector splitting with Godunov dimensional splitting

3-D

• AMR-computation with coarse grid of 40x40x32; up to four additional refinement levels, each with refinement factor 2
• approximately 1400 timesteps with CFL-No. 0.8 to t=3 (approx. 1200 timesteps to t=2.5 for computation with 4 additional refinement levels)
• computational domain 1.25x1.25x1; sphere radii 0.16 and 0.08 respectively
• Timing information ...

Results

• Force coefficients on second body as a function of axial displacement for simulations with 3 additional refinement levels:
  

• Drag coefficient on first sphere: %$\displaysize C_D = \frac{F_D}{\frac{1}{2} \rho u^2 \pi r^2}}=0.8785 $%

• Color plot of pressure distribution in 3D:
  

Refinement study

A refinement study has been carried out, in which the axial displacement of the secondary body is 2.5 primary body radii. Values for the drag and lift coefficients of the second body for different levels of refinement are given in the accompanying table. The %$\Delta$% value is the difference between the current level value and that from the previous level. The attached figure shows the time histories of the drag and lift profiles. The dashed vertical line indicates the point after which the time averages are taken.

Additional levels	%$C_D$%	%$\Delta C_D$%	%$C_L$%	%$\Delta C_L$%
1	1.264		-0.176
2	1.442	0.178	-0.019	0.157
3	1.423	-0.019	0.052	0.071
4	1.408	-0.015	0.087	0.035

As may be seen, in this configuration the force coefficients (lift expecially) are quite sensitive to changes in the refinement level. This is due to the improved shock resolution for higher refinement levels, as may be seen in the following schlieren image, which shows the position of the shock for different additional refinement levels (2=blue, 3=red, 4=black). This improved resolution leads to a noticeable change in the impingement point of the shock on the secondary body.

This configuration thus represents a 'worst case' scenario - convergence is significantly improved if the secondary body is entirely inside the primary shock. For the computation with 3 additional refinement levels we estimate the error in the drag and lift coefficients to be 2% and 5% respectively of the drag coefficient value. The error in the experimental measurements is expected to be of the order of 5%.

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-- StuartLaurence - 23 Jan 2005