A FREE BOUNDARY PROBLEM FOR TWO-DIMENSIONAL GAS DYNAMICS EQUATIONS Katarina Jegdić Department of Computer and Mathematical Sciences University of Houston Downtown Sunčica Čanić, University of Houston Barbara Lee Keyfitz, Ohio State University Texas PDE Seminar University of Texas Pan American March 26-27, 2011
Reflection of a shock by a wedge INTRODUCTION two-dimensional Riemann problems for systems of conservation laws steady / unsteady transonic small disturbance equations nonlinear wave system gas dynamics equations (isentropic, adiabatic) shock reflection regular reflection (strong, weak) Mach reflection Guderley reflection self-similar coordinates 8 < : mixed type (hyperbolic, elliptic) system free boundary value problem nonlinear problem theory of second order elliptic equations with mixed boundary conditions (Dirichlet and oblique) and fixed point theory Result Riemann problem for the isentropic gas dynamics equations strong regular reflection existence of a local solution in weighted Holder spaces A free boundary problem for two-dimensional gas dynamics equations 1
SHOCK REFLECTION PSfrag replacements flow reflected shock incident shock t < 0 t = 0 t > 0 ments motivating problem: airflow past object types of reflection strong regular reflection weak regular reflection Mach reflection inc. shock refl. shock refl. shock inc. shock refl. shock subsonic inc. shock subsonic subsonic region region region Mach stem supersonic region A free boundary problem for two-dimensional gas dynamics equations 2
ements TWO-DIMENSIONAL RIEMANN PROBLEMS U t + F (U) x + G(U) y = 0, U : R 2 [0, ) R m y similarity variables ξ = x/t and η = y/t U 1 reduced system: U 0 U1 y = kx ξ (F ξu) + η (G ηu) = 2U = kx initial data becomes boundary data at infinity x the reduced system changes type: hyperbolic in the far field, mixed type near the origin resolve one-dimensional discontinuities in the far field formulate a free boundary problem for the reflected shock and the subsonic state theory of second order elliptic equations with mixed boundary conditions (Gilbarg, Trudinger, Lieberman) fixed point theorems A free boundary problem for two-dimensional gas dynamics equations 3
adiabatic gas dynamics equations E = 1 γ 1 p + u2 +v 2 ρ 2 ρ t + (ρu) x + (ρv) y = 0 (ρu) t + (ρu 2 + p) x + (ρuv) y = 0 (ρv) t + (ρuv) x + (ρv 2 + p) y = 0 (ρe) t + (u(ρe + p)) x + (v(ρe + p)) y = 0 isentropic gas dynamics equations nonlinear wave system p = p(ρ) ρ t + (ρu) x + (ρv) y = 0 (ρu) t + (ρu 2 + p) x + (ρuv) y = 0 (ρv) t + (ρuv) x + (ρv 2 + p) y = 0 p = p(ρ) ρ t + (ρu) x + (ρv) y = 0 (ρu) t + p x = 0 (ρv) t + p y = 0 unsteady transonic small disturbance equation u t + uu x + v y = 0, v x + u y = 0 Euler equations for potential flow, pressure gradient system A free boundary problem for two-dimensional gas dynamics equations 4
ISENTROPIC GAS DYNAMICS EQUATIONS lacements ρ t + (ρu) x + (ρv) y = 0 y (ρu) t + (ρu 2 + p) x + (ρuv) y = 0 x = ky (ρv) t + (ρuv) x + (ρv 2 + p) y = 0 U 0 U 1 x p = p(ρ), c 2 (ρ) := p (ρ) > 0 and increasing x = ky U = (ρ, u, v) Initial data: U 0 = (ρ 0, u 0, 0), U 1 = (ρ 1, 0, 0) - fix ρ 0 > ρ 1 > 0 - fix k (k C (ρ 0, ρ 1 ), ) - set u 0 = 1 + k 2 q (ρ 0 ρ 1 )(p(ρ 0 ) p(ρ 1 )) ρ 0 ρ 1 A free boundary problem for two-dimensional gas dynamics equations 5
Formulate the problem in self-similar coordinates: ξ = x/t, η = y/t (ρu) ξ + (ρv ) η + 2ρ = 0 (U, V ) U + U + p ξ /ρ = 0 (U, V ) V + V + p η /ρ = 0 - here, PSfragU replacements := u ξ and V := v η - when linearized about a constant state U = (ρ, u, v ), the system is hyperbolic outside the circle C U : (u ξ) 2 + (v η) 2 = c 2 (ρ ) η U 0 l 1 C Ub U a S 1 U 1 ξ U 0 = (ρ 0, 0, n 0 ), U 1 = (ρ 1, 0, 0) S a : ξ = kη + χ a, l a : ξ = kη S b : ξ = kη + χ a, l b : ξ = kη ρ 0, m a, n a ), U b = (ρ 0, m a, n a ) m a, n a, χ a - functions of ρ 0, ρ 1, k C U 1 C U a U b l 2 S 2 A free boundary problem for two-dimensional gas dynamics equations 6
Sfrag replacements Solution in the hyperbolic part of the domain η U 0 l 1 PSfrag replacements 1.4 k C Ub U a U 1 1.2, U 1 = (ρ 1, 0, 0) χ a, l a : ξ = kη a, l b : ξ = kη = (ρ 0, m a, n a ) nctions of ρ 0, ρ 1, k C U1 C Ua S 1 U 1 U a 0.6 Ξ s U b C U1 C 0.4 S U a 2 C U Ub b U 0 = (ρ 0, 0, n 0), U 1 = (ρ 1, 0, 0) 0.2 S a : ξ = kη + χ a, l a : ξ = kη S l b : ξ = kη + χ a, l b : ξ = kη 2 U a = (ρ 0, m a, n a), U b = (ρ 0, m a, n a) m a, n a, χ a - functions of ρ 0, ρ 1, k ξ U 0ξ η S 1 1 S 2 l1 0.8 l 2 Ξ s 2 4 6 8 Ρ 0 Ρ 1 If k (k C (ρ 0, ρ 1 ), ), then: 1) Ξ s is outside the sonic circles C U 1, C U a, C U b 2) quasi-1-d Riemann problem at Ξ s has a solution (shock shock) - two solutions: U F = (ρ F, u F, 0) and U R = (ρ R, u R, 0) Ξ s is subsonic w.r.t. U F when k (k C, ) Ξ s is subsonic w.r.t. U R when k (k C, k ) Ξ s is supersonic w.r.t. U R when k (k, ) A free boundary problem for two-dimensional gas dynamics equations 7
Two types of regular reflection η ρ 0 η ρ 0 ements ρ ρ 1 ρ ρ 1 Ξ s ξ Ξ s ξ C U F C U R Case 1: Ξ s is subsonic w.r.t. the solution at Ξ s strong regular reflection Case 2: Ξ s is supersonic w.r.t. the solution at Ξ s weak regular reflection A free boundary problem for two-dimensional gas dynamics equations 8
Numerical simulations for the adiabatic gas dynamics equations using CLAWPACK software transonic (strong) and supersonic (weak) regular reflection 1 q(1) at time 0.2000 1 q(1) at time 0.2000 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 y 0.5 y 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x A free boundary problem for two-dimensional gas dynamics equations 9
S b l a Free boundary lb problem describing transonic regular reflection η reduced system in Ω Ω σ Ω Σ 0 Σ U a Ξ s U 1 ξ jump conditions along Σ symmetry conditions along Σ 0 Dirichlet condition for ρ on σ Dirichlet conditions at Ξ s Theorem: Main idea. There exists a local solution ρ H ( γ) 1+α, U, V H( γ) 1+ɛ and Σ H 1+α of the above free boundary problem. (0 < ɛ < α < γ < 1) j 2nd order equation for ρ in Ω reduced system in Ω two 1st order transport equations for U and V in Ω 8 < oblique derivative boundary condition for ρ along Σ jump conditions on Σ Dirichlet conditions for U and V along Σ : shock evolution equation dη/dξ = Ψ(ρ, ξ, η) A free boundary problem for two-dimensional gas dynamics equations 10
Holder spaces Let S R 2, u : S R. Define u 0;S := sup u(x) [u] α;s := sup x y u(x) u(y) x y α u α;s := u 0;S + [u] α;s supermum norm α-holder seminorm α-holder norm u k+α;s := P k j=0 Dj u 0;S + [D k u] α;s (k + α)-holder norm Let T S and δ > 0. Let S δ;t := {x S : dist(x, T ) > δ}, and for a > 0 and b such that a b 0, define u ( b) a;s\t := sup δ a b u a;sδ;t δ>0 weighted interior Holder norm A free boundary problem for two-dimensional gas dynamics equations 11
Theorem: There exists a local solution ρ H ( γ) of the free boundary problem 1+α, U, V H( γ) 1+ɛ and Σ H 1+α a ij (ρ, U, V )D ij ρ + b i (ρ, U, V )D i ρ + c ij (ρ, U, V )D i ρd j ρ = 0 (U, V ) U + U + p ξ /ρ = 0 (U, V ) V + V + p η /ρ = 0 β(ρ, U, V ) ρ = F (ρ, U, V ) U = G(ρ, ξ, η) V = H(ρ, ξ, η) dη dξ = Ψ(ρ, ξ, η) 9 >= >; ρ η = U η = V = 0 on Σ 0 ρ = f on σ U(Ξ s ) = U F, η(ξ s ) = 0 Remark. We introduce cut-off functions to ensure that: on Σ : η = η(ξ) the 2nd order equation for ρ is uniformly elliptic in Ω the condition for ρ on Σ is uniformly oblique the shock evolution equation is well-defined 9 = ; in Ω A free boundary problem for two-dimensional gas dynamics equations 12
Step 1. given Σ m K H 1+α, find ρ m, U m, V m solving the fixed BP in Ω m ff a ij (ρ, U, V )D ij ρ + b i (ρ, U, V )D i ρ + c ij (ρ, U, V )D i ρd j ρ = 0 (U, V ) U + U + p ξ /ρ = 0, (U, V ) V + V + p η /ρ = 0 in Ω β(ρ, U, V ) ρ = F (ρ, U, V ) on Σ, ρ η = 0 on Σ 0, ρ σ = f, ρ Ξs = ρ s U Σ = G(ρ), U Ξs = u F ξ s, U η = 0 on Σ 0 V Σ = H(ρ), V Ξs = 0, V = 0 on Σ 0 fix ω, W, Z H ( γ) 1+ɛ linearize the second order problem for density using ω, W, Z in the coefficients show that there exists a solution ρ H ( γ) 1+α show that the map ω ρ has a fixed point ρ[w, Z] of this linear problem linearize the first order problem for pseudo-velocities using W, Z, ρ[w, Z] show that there exists a solution U, V H ( γ) 1+ɛ of this linear problem show that the map (W, Z) (U, V ) is a contraction Step 2. find Σ m+1 using the shock evolution equation dη/dξ = Ψ(ρ m, ξ, η) show that the map Σ m Σ m+1 has a fixed point Σ A free boundary problem for two-dimensional gas dynamics equations 13
RELATED WORK steady transonic small disturbance equation shock perturbation: Čanić-Keyfitz-Lieberman unsteady transonic small disturbance equation strong regular reflection: Čanić-Keyfitz-Kim weak regular reflection: Čanić-Keyfitz-Kim Guderley reflection: Tesdall-Hunter nonlinear wave system Mach reflection: Čanić-Keyfitz-Kim, Sever strong regular reflection: Jegdić-Keyfitz-Čanić weak regular reflection: Jegdić Guderley reflection: Tesdall-Sanders-Keyfitz pressure-gradient system weak regular reflection: Y.Zheng-D.Wang Euler equations for potential flow weak regular reflection: G.Q.Chen-M.Feldman adiabatic/isentropic gas dynamics equations T.Chang-G.Q.Chen, S.X.Chen et al., T.Zhang-Y.Zheng, T.P.Liu-V.Elling, A.Tesdall-R.Sanders A free boundary problem for two-dimensional gas dynamics equations 14
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