Shock Reflection-Diffraction and Multidimensional Conservation Laws Gui-Qiang Chen Department of Mathematics, Northwestern University Email: gqchen@math.northwestern.edu Website: http://www.math.northwestern.edu/˜gqchen/preprints Mikhail Feldman University of Wisconsin-Madison NSF-FRG 2003-07: S. Canic, C. M. Dafermos, J. Hunter, T.-P. Liu C.-W. Shu, M. Slemrod, D. Wang, Y. Zheng Website: http://www.math.pitt.edu/˜dwang/FRG.html 12th International Conference on Hyperbolic Problems University of Maryland at College Park, June 9–13, 2008 Gui-Qiang Chen (Northwestern) Shock Reflection-Diffraction Hyp08, June 12, 2008 1 / 45
Bow Shock in Space generated by a Solar Explosion Gui-Qiang Chen (Northwestern) Shock Reflection-Diffraction Hyp08, June 12, 2008 2 / 45
Shock Waves generated by Blunt-Nosed and Shape-Nosed Supersonic Aircrafts Gui-Qiang Chen (Northwestern) Shock Reflection-Diffraction Hyp08, June 12, 2008 3 / 45
Blast Wave from a TNT Surface Explosion Gui-Qiang Chen (Northwestern) Shock Reflection-Diffraction Hyp08, June 12, 2008 4 / 45
Shock Wave from an Underwater Nuclear Explosion Gui-Qiang Chen (Northwestern) Shock Reflection-Diffraction Hyp08, June 12, 2008 5 / 45
����� ���� ��� ��� ��� ? Shock Wave Patterns Around a Wedge (airfoils, inclined ramps, · · · ) Complexity of Reflection-Diffraction Configurations Was First Identified and Reported by Ernst Mach 1879 Experimental Analysis: 1940s= ⇒ : von Neumann, Bleakney, Bazhenova Glass, Takyama, Henderson, · · · Gui-Qiang Chen (Northwestern) Shock Reflection-Diffraction Hyp08, June 12, 2008 6 / 45
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0.4108 0.4106 Subsonic 0.4104 y/t Supersonic 0.4102 0.41 0.4098 1.0746 1.0748 1.075 1.0752 1.0754 1.0756 x/t Guderley Mach Reflection : A. M. Tesdall and J. K. Hunter: TSD, 2002 A. M. Tesdall, R. Sanders, and B. L. Keyfitz: NWE, 2006; Full Euler, 2008 B. Skews and J. Ashworth: J. Fluid Mech. 542 (2005), 105-114 Gui-Qiang Chen (Northwestern) Shock Reflection-Diffraction Hyp08, June 12, 2008 11 / 45
Shock Reflection-Diffraction Patterns Gabi Ben-Dor Shock Wave Reflection Phenomena Springer-Verlag: New York, 307 pages, 1992. Experimental results before 1991 Various proposals for transition criteria Milton Van Dyke An Album of Fluid Motion The parabolic Press: Stanford, 176 pages, 1982. Various photographs of shock wave reflection phenomena Richard Courant & Kurt Otto Friedrichs Supersonic Flow and Shock Waves Springer-Verlag: New York, 1948. Gui-Qiang Chen (Northwestern) Shock Reflection-Diffraction Hyp08, June 12, 2008 12 / 45
Scientific Issues Structure of the Shock Reflection-Diffraction Patterns Transition Criteria among the Patterns Dependence of the Patterns on the Parameters wedge angle θ w , adiabatic exponent γ ≥ 1 incident-shock-wave Mach number M s · · · · · · Interdisciplinary Approaches: Experimental Data and Photographs Large or Small Scale Computing Colella, Berger, Deschambault, Glass, Glaz, .... Anderson, Hindman, Kutler, Schneyer, Shankar, ... Yu. Dem’yanov, Panasenko, .... Asymptotic Analysis : Keller, Lighthill, Hunter, Majda, Rosales, Tabak, Gamba, Harabetian, Morawetz.... Rigorous Mathematical Analysis (Global Analysis?) Existence, Stability, Regularity, Bifurcation, · · · · · · Gui-Qiang Chen (Northwestern) Shock Reflection-Diffraction Hyp08, June 12, 2008 13 / 45
2-D Riemann Problem for Hyperbolic Conservation Laws x = ( x 1 , x 2 ) ∈ R 2 ∂ t U + ∇ · F ( U ) = 0 , U 3 U 2 t=0 U 1 0 U N U N-1 Books and Survey Articles Glimm-Majda 1991, Chang-Hsiao 1989, Li-Zhang-Yang 1998 Zheng 2001, Chen-Wang 2002, Serre 2005, Chen 2005, · · · Numerical Simulations Glimm-Klingenberg-McBryan-Plohr-Sharp-Yaniv 1985 Schulz-Rinne-Collins-Glaz 1993, Chang-Chen-Yang 1995, 2000 Lax-Liu 1998, Kurganov-Tadmor 2002, · · · Gui-Qiang Chen (Northwestern) Shock Reflection-Diffraction Hyp08, June 12, 2008 14 / 45
Riemann Solutions I Gui-Qiang Chen (Northwestern) Shock Reflection-Diffraction Hyp08, June 12, 2008 15 / 45
Riemann Solutions II Gui-Qiang Chen (Northwestern) Shock Reflection-Diffraction Hyp08, June 12, 2008 16 / 45
Riemann Solutions vs General Entropy Solutions Asymptotic States and Attractors Observation ( C–Frid 1998) : Let R ( x t ) be the unique piecewise Lipschitz continuous Riemann solution with Riemann data: R | t =0 = R 0 ( x | x | ) Let U ( t , x ) ∈ L ∞ be an entropy solution with initial data: U | t =0 = R 0 ( x R 0 ∈ L ∞ ( S d − 1 ) , P 0 ∈ L 1 ∩ L ∞ ( R d ) | x | )+ P 0 ( x ) , The corresponding self-similar sequence U T ( t , x ) := U ( Tt , Tx ) is loc ( R d +1 compact in L 1 ) + � for any Ω ⊂ R d = ⇒ ess lim | U ( t , t ξ ) − R ( ξ ) | d ξ = 0 t →∞ Ω Building Blocks and Local Structure Local structure of entropy solutions Building blocks for numerical methods Gui-Qiang Chen (Northwestern) Shock Reflection-Diffraction Hyp08, June 12, 2008 17 / 45
Full Euler Equations (E-1): ( t , x ) ∈ R 3 + := (0 , ∞ ) × R 2 ∂ t ρ + ∇ · ( ρ v ) = 0 ∂ t ( ρ v ) + ∇ · ( ρ v ⊗ v ) + ∇ p = 0 ∂ t (1 (1 2 ρ | v | 2 + ρ e ) + ∇ · 2 ρ | v | 2 + ρ e + p ) v � � = 0 Constitutive Relations : p = p ( ρ, e ) ρ –density, v = ( v 1 , v 2 ) ⊤ –fluid velocity, p –pressure e –internal energy, θ –temperature, S –entropy γ = 1 + R For a polytropic gas: p = ( γ − 1) ρ e , e = c v θ, c v κ p = p ( ρ, S ) = κρ γ e S / c v , γ − 1 ρ γ − 1 e S / c v , e = e ( ρ, S ) R > 0 may be taken to be the universal gas constant divided by the effective molecular weight of the particular gas c v > 0 is the specific heat at constant volume γ > 1 is the adiabatic exponent, κ > 0 is any constant under scaling Gui-Qiang Chen (Northwestern) Shock Reflection-Diffraction Hyp08, June 12, 2008 18 / 45
Euler Equations: Isentropic or Isothermal (E-2) � ∂ t ρ + ∇ · ( ρ v ) = 0 ∂ t ( ρ v ) + ∇ · ( ρ v ⊗ v ) + ∇ p = 0 where the pressure is regarded as a function of density with constant S 0 : p = p ( ρ, S 0 ) . For a polytropic gas, p ( ρ ) = κ 0 ρ γ , γ > 1 ( γ = 2 also for the shallow water equations) For an isothermal gas, p ( ρ ) = κ 0 ρ ( i . e . γ = 1) where κ 0 > 0 is any constant under scaling Gui-Qiang Chen (Northwestern) Shock Reflection-Diffraction Hyp08, June 12, 2008 19 / 45
Euler Equations for Potential Flow (E-3): v = ∇ Φ � ∂ t ρ + ∇ · ( ρ ∇ Φ) = 0 , γ − 1 = ρ γ − 1 2 |∇ Φ | 2 + ρ γ − 1 ∂ t Φ + 1 γ − 1 ; 0 or, equivalently, � � ∂ t ρ ( ∇ Φ , ∂ t Φ , ρ 0 ) + ∇ · ρ ( ∇ Φ , ∂ t Φ , ρ 0 ) ∇ Φ = 0 , with − ( γ − 1)( ∂ t Φ + 1 1 ρ γ − 1 γ − 1 . 2 |∇ Φ | 2 ) � � ρ ( ∇ Φ , ∂ t Φ , ρ 0 ) = 0 Celebrated steady potential flow equation of aerodynamics : ∇ · ( ρ ( ∇ Φ , ρ 0 ) ∇ Φ) = 0 . This approximation is well-known in transonic aerodynamics. We will see the Euler equations for potential flow is actually EXACT in an important region of the solution to the shock reflection problem. Gui-Qiang Chen (Northwestern) Shock Reflection-Diffraction Hyp08, June 12, 2008 20 / 45
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Initial-Boundary Value Problem: 0 < ρ 0 < ρ 1 , v 1 > 0 Initial condition at t = 0: � (0 , 0 , p 0 , ρ 0 ) , | x 2 | > x 1 tan θ w , x 1 > 0 , ( v , p , ρ ) = ( v 1 , 0 , p 1 , ρ 1 ) , x 1 < 0; Slip boundary condition on the wedge bdry: v · ν = 0 . X 2 (0) (1) Shock X 1 Invariant under the Self-Similar Scaling: ( t , x ) − → ( α t , α x ) , α � = 0 Gui-Qiang Chen (Northwestern) Shock Reflection-Diffraction Hyp08, June 12, 2008 22 / 45
Seek Self-Similar Solutions ( ξ, η ) = ( x 1 t , x 2 ( v , p , ρ )( t , x ) = ( v , p , ρ )( ξ, η ) , t ) ( ρ U ) ξ + ( ρ V ) η + 2 ρ = 0 , ( ρ U 2 + p ) ξ + ( ρ UV ) η + 3 ρ U = 0 , ( ρ UV ) ξ + ( ρ V 2 + p ) η + 3 ρ V = 0 , ( U (1 γ − 1)) ξ + ( V (1 γ p γ − 1)) η + 2(1 γ p γ p 2 ρ q 2 + 2 ρ q 2 + 2 ρ q 2 + γ − 1) = 0 , √ U 2 + V 2 and ( U , V ) = ( v 1 − ξ, v 2 − η ) is the pseudo-velocity. where q = λ ± = UV ± c √ q 2 − c 2 λ 0 = V Eigenvalues : U (repeated) , , U 2 − c 2 � where c = γ p /ρ is the sonic speed When the flow is pseudo-subsonic: q < c , the system is hyperbolic-elliptic composite-mixed Gui-Qiang Chen (Northwestern) Shock Reflection-Diffraction Hyp08, June 12, 2008 23 / 45
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