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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:
    Computational schlieren image of two spheres in a Mach 10 freestream

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:
    Force coefficients on second body as a function of axial displacement

  • 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:
    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


Time histories for different refinement levels

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.

Levels

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%.


-- StuartLaurence - 23 Jan 2005


I Attachment sort Size Date Who Comment
spheres_drag.mpg 1021.5 K 02 Feb 2005 - 06:02 StuartLaurence Movie of flow startup with force histories on the two spheres
stable_schlieren.mpg 4937.7 K 04 Mar 2005 - 17:29 StuartLaurence Schlieren movie of stable flow over two cylinders
schlieren_3d.mpg 808.4 K 04 Mar 2005 - 21:51 StuartLaurence Computational schlieren movie of startup process over two spheres
unstable_schlieren.mpg 8743.3 K 10 Mar 2005 - 00:29 StuartLaurence Schlieren movie of unstable flow over two cylinders
stability_comparison.mpg 16295.4 K 10 Mar 2005 - 01:25 StuartLaurence Movie of schlieren superimposed on refinement level sets showing stable and unstable flow development

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