Glancing weak Mach reflection Allen M. Tesdall John K. Hunter - PowerPoint PPT Presentation

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 The UTSD shock reflection problem 1 Glancing weak Mach reflection 2 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 M triple point Regular reflection Mach reflection 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. V AN D YKE , "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. 2 1 (?) 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: � 1 � 2 u 2 u t + + v y = 0 , x u y − v x = 0 . � 0 if x > ay , u ( x , y , 0 ) = 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 θ w a = � 2 ( M 2 − 1 ) where M is the Mach number of the incident shock and θ w 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 0.52 0.515 y/t 0.51 0.505 1 1.005 1.01 x/t 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 2 1.5 y/t 1 0.5 0 -2 -1 0 1 x/t 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 2 1.5 y/t 1 0.5 0 -2 -1 0 1 x/t 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 0.4106 0.4104 y/t 0.4102 0.41 1.0748 1.075 1.0752 1.0754 x/t 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 0.4106 0.4104 y/t 0.4102 0.41 1.0748 1.075 1.0752 1.0754 x/t 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 0.414 0.44 Size of previous fiigure 0.412 0.42 y/t 0.41 y/t 0.408 0.4 0.406 1.074 1.076 1.078 1.06 1.08 1.1 x/t x/t 2 1.5 y/t 1 0.5 0 -2 -1 0 1 x/t 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 θ w a = � 2 ( M 2 − 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) � 1 � 2 u 2 u t + + v y = 0 , u y − v x = 0 x 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 U t + V y = 0 , U y − V x = 0 � θ √ ρ 3 ρ 3 / 2 + 1 2 θ 2 √ ρ − 2 � if ρ > 0 , if ρ > 0 , U = V = 0 if ρ < 0 , 0 if ρ < 0 , where ( ρ, θ ) are self-similar variables t + y 2 � x � θ = y ρ = − , t . 4 t 2 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 � 1 � 2 u 2 u t + + v y = 0 , u y − v x = 0 , x with initial/far-field conditions as t → 0 + u ∼ α U , v ∼ α V where ( U , V ) is linearized UTSDE solution and α is a small (non-removable) parameter with √ α = 4 2 a 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: � 1 � 2 u 2 − ξ u ξ − η u η + + v η = 0 , ξ u η − v ξ = 0 . Sonic line: ξ + η 2 4 = u ( ξ, η ) . u < ξ + η 2 / 4 hyperbolic (supersonic flow) → u > ξ + η 2 / 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 4 η 2 , θ = η, τ = log t , v = v − 1 ¯ ¯ u = u − r , 2 θ u , we get � 1 � v θ + 3 u + 1 u 2 ¯ 2 ¯ + ¯ 2 ¯ u τ + 2 r = 0 , r u θ − ¯ ¯ v r = 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 4 η 2 , θ = η, τ = log t , v = v − 1 ¯ ¯ u = u − r , 2 θ u , we get � 1 � v θ + 3 u + 1 u 2 ¯ 2 ¯ + ¯ 2 ¯ u τ + 2 r = 0 , r u θ − ¯ ¯ v r = 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 Inner y/t region x/t u = 0 Outer linearized solution Lower boundary y/t=0 corresponds to incident/Mach shock. Allen M. Tesdall, John K. Hunter HYP2012

The UTSD shock reflection problem Glancing weak Mach reflection Numerical solutions 4 3 y/t 2 1 0 -10 -8 -6 -4 -2 0 x/t Figure: Contours of u -velocity, α = 0 . 3. Solved on numerical domain with parabolic boundaries. The grid contains 5 . 6 × 10 6 grid points. Allen M. Tesdall, John K. Hunter HYP2012

The UTSD shock reflection problem Glancing weak Mach reflection Closeup near the reflection point 0.25 0.2 y/t 0.15 0.1 0.05 0 -0.1 -0.05 0 0.05 0.1 0.15 0.2 x/t 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