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TECHNICAL PAPERS

Transpiration Induced Shock Boundary-Layer Interactions

[+] Author and Article Information
B. Wasistho

Center for Simulation of Advanced Rockets, University of Illinois at Urbana-Champaign, Urbana, IL 61801

J. Fluids Eng 128(5), 976-986 (Jan 28, 2006) (11 pages) doi:10.1115/1.2236127 History: Received March 22, 2005; Revised January 28, 2006

Steady and unsteady shock boundary-layer interactions are studied numerically by solving the two-dimensional time-dependent Navier-Stokes equations. To validate the numerical method, the steady interaction is compared with measurements and other numerical results reported in the literature. The numerical study of the steady interaction leads to a suitable method for transpiration boundary conditions. The method applies to unsteady flows as well. Using the validated numerical method, we show that an unsteady shock boundary-layer interaction can occur in a supersonic flow over a flat plate subjected to suction and blowing from the opposite side of the plate, even though the imposed transpiration is steady. Depending on the Mach number, the Reynolds number, the distance of the transpiration boundary to the lower wall, and the transpiration profile, the unsteadiness can be inviscid or viscous dominated. The viscous effect is characterized by the occurrence of self-excited vortex shedding. A criterion for the onset of vortex shedding for internal compressible flows is also proposed.

Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 14

Skin friction coefficient of time-averaged flow resulting from the resolution 115×32(dashed),191×64 (solid), and 247×96 (dashed dotted)

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Figure 15

Mach-field of the flow corresponding to domain height 120, 90, 60, and 40, from above to below. Dashed lines denote subsonic regions.

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Figure 19

Separation bubble height as function of domain height, Mach, Re, and suction and blowing amplitude

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Figure 22

Vortex shedding parameter defined by Eq. 23 as function of Re (fixed h), h (fixed Re), and Rh for M=1.3

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Figure 23

Vortex shedding parameter defined by Eq. 23 (left figure) and Eq. 24 (right figure) including data at different Mach numbers

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Figure 1

Three different upper boundary conditions under study: B1: characteristic method, B2: underprescription, B3: prescription of analytical solution. Bold arrows denote prescribed variables while thin arrows denote extrapolations from the interior domain.

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Figure 2

Impinging steady shock generated by a wedge. The computational domain is bounded by the dashed line (a) and enlarged on (b). σ is the shock angle and 1, 2, and 3 denote the inviscid area upstream of the impinging shock, the area between the impinging shock and the reflected shock, and the area downstream of the reflected shock, respectively.

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Figure 3

Mach-field of steady SBLI using upwind scheme in both directions. The shock is induced by wedge/corner.

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Figure 4

Displacement thickness and separation bubble of time-averaged flow: reference domain (*), shifted outflow (dashed), and shifted inflow (solid)

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Figure 5

Displacement thickness and separation bubble of time-averaged flow: reference domain (*), lower (dashed) and higher upper boundary (solid)

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Figure 6

Wall pressure (above) and skin friction coefficient (below) for p3∕p1=1.4: B3 upwind (solid), B1 upwind (dashed dotted), B1 mixed (dashed), Katzer’s numerical result (*), and Hakkinen ’s experiment (o)

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Figure 7

Mach-field of steady SBLI using central discretization in the x2 direction and upwind in the x1 direction

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Figure 8

Temperature along the upper boundary from boundary condition B2 (dashed dotted) and B1 (dashed) compared to theory (solid)

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Figure 9

Wall pressure (above) and skin friction coefficient (below) for p3∕p1=1.25: B3 upwind (solid) and Hakkinen’s experiment (o). Dashed line corresponds to zero pressure gradient solution.

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Figure 10

u1 profiles for p3∕p1=1.40: B3 upwind (solid) and Hakkinen’s experiment (o)

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Figure 11

u1 profiles for p3∕p1=1.25: B3 upwind (solid) and Hakkinen’s experiment (o)

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Figure 12

Characteristics of prescribed u2 along the upper boundary

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Figure 13

Shock sensor of reference case with threshold value 0.2 and statistically stationary state beginning at t≈5000

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Figure 16

Vorticity contours corresponding to the domain heights in Fig. 1. Solid line is negative vorticity, dashed line positive.

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Figure 17

Average shock strength dependence on domain height, Mach, Re, and suction and blowing amplitude

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Figure 18

Shock unsteadiness dependence on domain height, Mach, Re, and suction and blowing amplitude

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Figure 20

Minimum cf as function of domain height, Mach, Re, and suction and blowing amplitude

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Figure 21

Vortex shedding parameter as function of Rh, where ∘ denotes separation with vortex shedding, × without vortex shedding, and ⊗ a transition between the two, as defined by Eq. 20 (left figure) and Eq. 21 (right figure) for M=1.3. Area between dash lines denotes overlapping regime.

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