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The UTSD shock reflection problem Glancing weak Mach reflection

Glancing weak Mach reflection

Allen M. Tesdall John K. Hunter HYP2012, Padova, June 2012

Allen M. Tesdall, John K. Hunter HYP2012

The UTSD shock reflection problem Glancing weak Mach reflection

Outline

1

The UTSD shock reflection problem

2

Glancing weak Mach reflection

Allen M. Tesdall, John K. Hunter HYP2012

The UTSD shock reflection problem Glancing weak Mach reflection

Shock reflection

Basic reflection patterns: regular reflection and Mach reflection.

M

triple point

Regular reflection Mach reflection

M

Physically, the reason for the reflection is that the flow changes direction across the shock, and must return to being tangential to the solid boundary.

Allen M. Tesdall, John K. Hunter HYP2012

The UTSD shock reflection problem Glancing weak Mach reflection

Experimental shock reflection

Regular reflection Mach reflection

From M. VAN DYKE, "An Album of Fluid Motion", 1982.

Allen M. Tesdall, John K. Hunter HYP2012

The UTSD shock reflection problem Glancing weak Mach reflection

Weak shocks

Von Neumann triple point paradox: for sufficiently weak shocks, Mach reflection is impossible, but experiments show a pattern that appears to be Mach reflection. 1

(?)

2

Allen M. Tesdall, John K. Hunter HYP2012

The UTSD shock reflection problem Glancing weak Mach reflection

The UTSD shock reflection problem

Weak shock Mach reflection described asymptotically by problem for the unsteady transonic small disturbance equation (UTSDE) in y > 0: ut + 1 2u2

• x

+ vy = 0, uy − vx = 0. u(x, y, 0) = if x > ay, 1 if x < ay, v(x, y, t) = 0 if x > s(y, t), v(x, 0, t) = 0. Here, x = s(y, t) is the location of the incident and Mach shocks.

Allen M. Tesdall, John K. Hunter HYP2012

The UTSD shock reflection problem Glancing weak Mach reflection

The UTSD shock reflection problem

The problem is self-similar. It depends on a single parameter a, given by a = θw

• 2(M2 − 1)

where M is the Mach number of the incident shock and θw is the wedge angle.

Allen M. Tesdall, John K. Hunter HYP2012

The UTSD shock reflection problem Glancing weak Mach reflection

Numerical solution of UTSD shock reflection problem

x/t y/t

1 1.005 1.01 0.505 0.51 0.515 0.52

Figure: u-contours, showing a sequence of triple points, expansion waves and reflected shocks in a tiny region. a = 0.5.

Allen M. Tesdall, John K. Hunter HYP2012

The UTSD shock reflection problem Glancing weak Mach reflection

Weak shock reflection for the full Euler equations

x/t y/t

• 2
• 1

1 0.5 1 1.5 2

Pressure, p

Figure: M=1.075, wedge angle θ = 15 degrees.

Allen M. Tesdall, John K. Hunter HYP2012

The UTSD shock reflection problem Glancing weak Mach reflection

Weak shock reflection for the full Euler equations

x/t y/t

• 2
• 1

1 0.5 1 1.5 2

Mach number, M

Figure: M=1.075, wedge angle θ = 15 degrees.

Allen M. Tesdall, John K. Hunter HYP2012

The UTSD shock reflection problem Glancing weak Mach reflection

Solution near reflection point

x/t y/t

1.0748 1.075 1.0752 1.0754 0.41 0.4102 0.4104 0.4106

Figure: Mach number contours, M, showing a sequence of triple points, expansion waves and reflected shocks in a tiny region. M=1.075, wedge angle θ = 15 degrees.

Allen M. Tesdall, John K. Hunter HYP2012

The UTSD shock reflection problem Glancing weak Mach reflection

Solution near reflection point

x/t y/t

1.0748 1.075 1.0752 1.0754 0.41 0.4102 0.4104 0.4106

Figure: Contour plot (left) and surface plot (right) near reflection point. Call structure “Guderley Mach reflection” (GMR).

Allen M. Tesdall, John K. Hunter HYP2012

The UTSD shock reflection problem Glancing weak Mach reflection

Pattern occurs in tiny region

x/t y/t

• 2
• 1

1 0.5 1 1.5 2

x/t y/t 1.06 1.08 1.1 0.4 0.42 0.44 x/t y/t 1.074 1.076 1.078 0.406 0.408 0.41 0.412 0.414 Size of previous fiigure

Allen M. Tesdall, John K. Hunter HYP2012

The UTSD shock reflection problem Glancing weak Mach reflection

The glancing limit: a ≪ 1.

(?)

Question: still get GMR?

Interested in the limit as a → 0+, where a = θw

• 2(M2 − 1)

Allen M. Tesdall, John K. Hunter HYP2012

The UTSD shock reflection problem Glancing weak Mach reflection

UTSDE

Inner problem near reflection point consists of unsteady transonic small disturbance equation (UTSDE) (asymptotic reduction of compressible Euler equations) ut + 1 2u2

• x

+ vy = 0, uy − vx = 0 where (u, v) correspond to (x, y) velocity perturbations. Pressure perturbations proportional to u. Supplemented with far-field/initial conditions obtained by matching with outer linearized solution.

Allen M. Tesdall, John K. Hunter HYP2012

The UTSD shock reflection problem Glancing weak Mach reflection

Linearized UTSDE solution

Matching data near reflection point is solution of linearized UTSDE equation Ut + Vy = 0, Uy − Vx = 0 U = θ√ρ if ρ > 0, if ρ < 0, V =

• − 2

3ρ3/2 + 1 2θ2√ρ

if ρ > 0, if ρ < 0, where (ρ, θ) are self-similar variables ρ = − x t + y2 4t2

• ,

θ = y t .

Allen M. Tesdall, John K. Hunter HYP2012

The UTSD shock reflection problem Glancing weak Mach reflection

UTSDE problem

Inner problem near reflection point consists of UTSDE ut + 1 2u2

• x

+ vy = 0, uy − vx = 0, with initial/far-field conditions u ∼ αU, v ∼ αV as t → 0+ where (U, V) is linearized UTSDE solution and α is a small (non-removable) parameter with α = 4 √ 2a π

Allen M. Tesdall, John K. Hunter HYP2012

The UTSD shock reflection problem Glancing weak Mach reflection

The Self-Similar UTSDE

This problem is self-similar, so its solution depends only on ξ = x/t and η = y/t. We write: −ξuξ − ηuη + 1 2u2

• ξ

+ vη = 0, uη − vξ = 0. Sonic line: ξ + η2 4 = u(ξ, η). u < ξ + η2/4 → hyperbolic (supersonic flow) u > ξ + η2/4 → elliptic (subsonic flow)

Allen M. Tesdall, John K. Hunter HYP2012

The UTSD shock reflection problem Glancing weak Mach reflection

Solving the Self-Similar Equations

Making the change of variables r = ξ + 1 4η2, θ = η, τ = log t, ¯ u = u − r, ¯ v = v − 1 2θu, we get ¯ uτ + 1 2 ¯ u2

• r

+ ¯ vθ + 3 2 ¯ u + 1 2r = 0, ¯ uθ − ¯ vr = 0.

Allen M. Tesdall, John K. Hunter HYP2012

The UTSD shock reflection problem Glancing weak Mach reflection

Solving the Self-Similar Equations

Making the change of variables r = ξ + 1 4η2, θ = η, τ = log t, ¯ u = u − r, ¯ v = v − 1 2θu, we get ¯ uτ + 1 2 ¯ u2

• r

+ ¯ vθ + 3 2 ¯ u + 1 2r = 0, ¯ uθ − ¯ vr = 0. Can solve using standard transonic finite difference techniques.

Allen M. Tesdall, John K. Hunter HYP2012

The UTSD shock reflection problem Glancing weak Mach reflection

UTSDE glancing reflection

Outer linearized

u = 0

y/t x/t Inner region

to incident/Mach shock. Lower boundary y/t=0 corresponds

solution Allen M. Tesdall, John K. Hunter HYP2012

The UTSD shock reflection problem Glancing weak Mach reflection

Numerical solutions

x/t y/t

• 10
• 8
• 6
• 4
• 2

1 2 3 4

Figure: Contours of u-velocity, α = 0.3. Solved on numerical domain with parabolic boundaries. The grid contains 5.6 × 106 grid points.

Allen M. Tesdall, John K. Hunter HYP2012

The UTSD shock reflection problem Glancing weak Mach reflection

Closeup near the reflection point

x/t y/t

• 0.1
• 0.05

0.05 0.1 0.15 0.2 0.05 0.1 0.15 0.2 0.25

Figure: Contours of u-velocity, α = 0.3, showing GMR.

Allen M. Tesdall, John K. Hunter HYP2012

The UTSD shock reflection problem Glancing weak Mach reflection

Conclusions

We have obtained a preliminary solution which suggests GMR occurs in the glancing limit. Question: What happens as α → 0? Next step: repeat the derivation of an inner problem for glancing Mach reflection, beginning with the Euler equations instead of with the UTSDE.

Allen M. Tesdall, John K. Hunter HYP2012