03ER25575, and the National Science Foundation, grant a Research supported by the Department of Energy, grant DE-FG02-

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1 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 1 What Studying Quasi-Steady Problems Can Tell Us About Steady Transonic Flow November, 3 Barbara Lee Keyfitz a Department of Mathematics University of Houston Houston, Texas joint work with Sunčica Čanić and Eun Heui Kim computations: Alexander Kurganov, Maria Lukáčova, Richard Sanders, Manuel Torrilhon students: Tim Morant, Namrata Shirude a Research supported by the Department of Energy, grant DE-FG- 3ER5575, and the National Science Foundation, grant

2 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 Dedication Sans doute vous serez célèbre Par les grands calculs de l algèbre Où votre esprit est absorbé J oserais m y livrer moi même: Mais Hélas! A + D B N est pas = à je vous aime. Jean François Marie AROUET = Voltaire ( ) poem written about Émilie du Châtelet, Thanks to Luc Tartar

3 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 3 Connections: Steady TS Flow and Multidimensional Conservation Laws Motivating problem: Steady airflow past object Variables: Density ρ, Velocity (u, v) Pressure p = p(ρ) = ργ γ Sonic Line State variable (ρ, u, v) = U(x, y, t) Wing Profile Unsteady equations (- or 3-D CL): ρ t + (ρu) x + (ρv) y = (ρu) t + (ρu + p(ρ)) x + (ρuv) y = (ρv) t + (ρuv) x + (ρv + p(ρ)) y = ρ (u, v) Assumptions: ideal gas; no viscosity, no energy transfer BC: no slip (u, v) n = Issues: Does a steady flow exist? Is it smooth? Existence for unsteady problem? Flow regimes: sub/supersonic & transonic 8M Stagnation Point Supersonic Region Shock Slip Line

4 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 4 The Unsteady Problem: Isentropic Compressible Euler Equations Quasilinear form linearize: t ρ u ρ u + c /ρ u x v u c (ρ) = dp dρ > ρ v ρ ρ u + v y u = v c /ρ v v Euclidean/Galilean symmetry: O() Normals Acoustic c (q)>q 3 Acoustic 1 Surfaces u u u, v v v x x u t, y y v t Characteristics: det τi + µa + νb = Strict hyperbolicity: τ = µu νv, τ = µu νv ± c(ρ) µ + ν τ ν Shear µ t y Shear x 4

5 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 5 Potential Flow Equations; Steady Potential Flow Velocity q = (u, v) Vorticity ω = curl q = u y v x Proposition: If ω(x, y, ) = and if U is smooth, then ω for all t. Introduce velocity potential φ in momentum equations, φ = q: u t + uu x + p x /ρ + vu y = v t + uv x + vv y + p y /ρ = ( φ t + 1 φ + c ) (ρ) ρ = i(ρ) = c /ρ = p (ρ)/ρ, i > : φ t + 1 φ + i(ρ) = f(t) = C u ρ v 1_ p = g h Steady flow: i(ρ) = 1 (ˆq q ); ρ = ρ(q) Bernoulli s law: still waters run deep Replaces momentum equations x Shallow Water Equations ρ y

6 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 6 Structure of the Steady Transonic Equations (ρ( φ ) φ ) = Alternatively (ρu) x + (ρv) y =, u y v x = with i(ρ) = 1 (ˆq q ) (c φ x)ρ xx φ x φ y φ xy + (c φ y)φ yy = ρ stag c(q) ρ(q) q Sing. at stag and cav; ch. type at q : c(q ) = q qρ(q) q * q cav q Compare with time-dependent problem (no change of type) Normals Surfaces Normals Surfaces Acoustic c (q)>q 3 Acoustic Acoustic c (q)<q 3 Acoustic 1 1 τ 4 6 Shear t 1 vs. τ 4 6 Shear t 1 8 ν µ 3 4 y Shear x 4 8 ν µ 3 4 y Shear x 4

7 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 7 Two Themes in Cathleen Morawetz s Early Work 1. Well-posed problems for change of type (or mixed ) equations. Existence/non-existence of classical/weak solutions Hodograph map (linearizes equation): A x u +B y u = A(u, v), B(u, v) det J = v v u x v x u y v y à x u + B v x = Ã(u, v), B(u, v); K(v)xuu +x vv = y y Elliptic D + D l D r u Hyperbolic v - Tricomi: K continuous, vk(v) C - Data on C only - Uniqueness nonexistence - Existence of weak solutions (airfoil data)?

8 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 8 v Elliptic D + C D l D r u Nonexistence: Tools Hyperbolic Maximum principle Method of Characteristics K. O. Friedrichs, Symmetric Positive Linear Operators N. Popivanov, degenerate equations D. Lupo K. Payne, properties independent of type R. DiPerna, L. Tartar, D. Serre, compensated compactness

9 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 9 Our program Self-similar solutions: a tool for multi-dimensional CL. Points in this talk: 1. Self-similar problems resemble steady problems. NLWS as simplified model for gas dynamics 3. Need for/usefulness of elliptic estimates 4. Bifurcation problems/von Neumann paradox/supersonic patch 5. Application of Morawetz s nonexistence result? 6. Lessons for steady flow

10 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 1 Similarity Analysis of Two-Dimensional Systems U t +F (U) x +G(U) y =, ( ) Data: U(x, y, ) = f Examples: -D Riemann problems Shock reflection by a wedge Flow x y Incident Shock S= Σ t Reflected Shock t< t= t> X= Ξ t U R n Wedge y Sectorially Constant Data Reduce + 1-D problem in space-time to -D quasi-steady problem Method: resolve 1-D far-field discontinuities; solve as IV/BVP in -D Type Changes: hyperbolic in far field; subsonic region near origin x

11 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 11 Related work -Majda et al on multi-d shock stability -SX Chen et al on steady conical shocks -Brio-Hunter, Tabak-Rosales, Tesdall-Hunter on UTSDE -GQ Chen & Feldman on steady shock stability in pot eqn -CK & Lieberman on TSDE -Zhang & Zheng on self-sim wave interaction structures in GD -Zheng et al on self-sim pressure-gradient system -GQ Chen et al on divergence-measure fields

12 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 1 Acoustic-type Structure: Degenerate/Non-degenerate Characteristics U t + AU x + BU y = ; det Iτ + Aµ + Bν = (n l i σ ) σ T Q N σ i=1 6 4 Characteristic Normals σ=(µ,ν,τ) Acoustic Envelopes in Physical Space 3 Shear Acoustic σ = (µ, ν, τ) ξ = x/t, η = y/t ( (A ξi) ξ + (B ηi) η ) U = τ 4 6 ν µ Shear t y 4 x n i=1 CHANGE OF TYPE THEOREM Reduced equation hyperbolic iff x = (1, ξ, η) outside acoustic wave cone C W = { x T Q 1 N x = }. RP in + 1 dim CP w. data at Ξ = (ξ, η) dual vector α = (α, β) l i ( α Ξ, α, β) p(σ( α, Ξ), U) }{{} p( α,ξ,u) η ξ SUBSONIC REGION: ELLIPTIC OR MIXED DEGENERATE CHARACTERISTICS τ τ + NONDEGENERATE SPACELIKE CURVE Ξ CHARACTERISTICS SUPERSONIC REGION: HYPERBOLIC L

13 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 13 The Search for Prototype Systems: NLWS Isentropic Gas Dynamics: p = ρ γ /γ ρ t + (ρu) x + (ρv) y = (ρu) t + (ρu + p) x + (ρuv) y = (ρv) t + (ρuv) x + (ρv + p) y = Nonlinear Wave System: ρ t + m x + n y = m t + p x = m = ρu n t + p y = n = ρv Self-sim nd-order equation for nonlinear characteristic variable (ρ): ( ) ( ) (c (ρ) U )ρ ξ UV ρ η ( ) ξ+ (c (ρ) ξ )ρ ξ ξηρ η ξ (c (ρ) V )ρ η UV ρ ξ +... = + ( (c (ρ) η )ρ η η ξηρ ξ )η U = u ξ, V = v η ( pseudo-vel. ) +ξρ ξ + ηρ η = Transport equation for linear characteristic variable: W = V ξ U η = v ξ u η = vorticity UW ξ + V W η + (U ξ + V η + 1)W = w = n ξ m η = vorticity (ξ, η) w + w = ; w t = w = n x m y = 1 t (n ξ m η )

14 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 14 Comparison of Self-Similar Isentropic Gas Dynamics & NLWS Similarities and differences Transonic equation for ρ: GD: coupled with U = ξ u, V = η v ( (c (ρ) U )ρ ξ UV ρ η )ξ + ( (c (ρ) V )ρ η UV ρ ξ ) NLWS: uncoupled ( (c (ρ) ξ )ρ ξ ξηρ η )ξ + ( (c (ρ) η )ρ η ξηρ ξ ) Evolution of vorticity: GD: UW ξ + V W η + (U ξ + V η + 1)W = NLWS: w t = η +... = η + ξρ ξ + ηρ η =

15 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 15 Interacting Shock Data A Bifurcation Problem for NLWS y x= κ a y U 1 =(ρ 1,m 1,n 1 ) U =(ρ,m,n ) x=κ a y -state data: U, U 1 Data give shocks Far field soln: 4 waves η Shock U 1b Linear Wave U 1 U Shock U 1a Linear Wave x Symmetric prototype for converging sector boundaries Weak shock reflection (no slip line), von Neumann paradox Features 1. parameters: ρ /ρ 1 > 1 and κ a (Mach # and cot(θ w )). Incident shocks: ξ = κ a η χ, ξ = κ a η + χ 3. Small κ: two local solutions weak and strong regular reflection 4. Large κ: curved shock, weak reflected wave 5. Intermediate values of κ: no sol n from shock polars (Q1D RP) ξ

16 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 16 Interacting Shock Problem for the Nonlinear Wave System S b l b : ξ= κ a η + : ξ= κa η χ a U 1b U 1 η C 1 U C U 1a S a : ξ=κa η+χ a l a : ξ=κ a η Q 1 D RP C: Q-1-D RP solvable S b + : ξ= κa η χ a U 1b l b : ξ= κ a η U 1 η C 1 U C U 1a ξ S a : ξ=κa η+χ a l a : ξ=κ a η A+A*: Shock meets C ξ θ w =cot 1 (κ) Region C Region B κ * ρ /ρ 1 Region A* Region A Three regions: A+A* Weak MR possible C RR possible B neither possible

17 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 17 Theory and Numerical Simulations of the Solution Region A (Weak Mach Reflection): Large κ, Small θ w Contour Plot of Density ρ. Data U = (64,, ); U 1 = (1,,); κ a = 8; κ b = η axis ξ axis Sonic circle C = {ξ + η = c (ρ )} Supersonic soln known U continuous at C U/ r singular Simulation by Alex Kurganov c (ρ) = ρ γ 1

18 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 18 Same Case: κ a = 8, ρ = 64, Momentum Component n Simulation of full field and close-up near (, ) Logarithmic singularity in n at (, )

19 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 19 Analysis of Weak Mach Stem Problem (Large κ) Degenerate Elliptic Free Boundary Problem Existence theorem for global problem for NLWS Q ( ) (c (ρ) ξ )ρ ξ ξηρ η + ( (c (ρ) η )ρ ξ η ξηρ ξ )η + ξρ ξ + ηρ η Q(ρ) = (degenerate elliptic) in Ω Σ c (ρ) = ξ + η = c (ρ ) on σ Free (degenerate boundary, continuous solution) ρ ξ = (symmetry) on Σ Free boundary from RH equations: Σ Ω σ N(ρ) β ρ = (oblique deriv) on Σ Neumann Dirichlet dη dξ = η s ξη + s (ξ + η s ) ρ = ρ max at Σ Σ (part of D. bdry) s = [p] [ρ] Approach: Fixed Point Theorem (CK & Lieberman, CKK) New difficulties: N not uniformly oblique; est. at degenerate corner

20 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 Procedure: Find the Free Boundary as a Fixed Point Formulate as nd order PDE for density, ρ (not potential); Rewrite RH conditions as (1) evolution eqn for shock and () ODBC for ρ Problem is quasilinear, degenerate elliptic PDE, mixed BC Regularize PDE (parameter ε) Step 1 Fix approx. η = η(ξ), defines Σ K ε H 1+α1 (Hölder) Step Solve (fixed) mixed BVP for ρ Lieberman s Mixed BVP theory + linearization + modifications for loss of obliqueness Step 3 Map η η = Jρ by other RH condition (shock evolution) Schauder F. P. Thm: Compactness fixed pt for J J : K H 1+α1 K H 1+α, α > α 1 Step 4 Send ε, show limits η and ρ exist and solve the problem.

21 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 1 A Priori Estimates (following Lieberman, Zheng, Canic&Kim, CKL) Regularize: Q ε ρ = Qρ + ε ρ. Obliqueness fails at Ξ s. Step 1: for FBP for Q ε ; a priori bds on ρ ε, η ε (ξ) (FB Σ) unif. in ε. Step : Local lower barrier for ρ ε independent of ε, unif. local ellipticity local compactness. Step 3: Convergent subsequence; limit solves the problem in Ω \ {Ξ }: applying regularity, compactness and a diagonalization. Step 4: Convergence at Ξ requires more. Ξ Ξ s c Σ η Σ c Ξ Ω σ ξ

22 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 Details of Step 1: Holder and Schauder bounds for ρ L ε ρ = D i (a ij (Ξ, w)d j ρ) + ε ρ + b i (Ξ)D i ρ = in Ω Nρ = β i D i ρ = β i (Ξ, w)d i ρ = on Σ = {η = η(ξ)} H (b) a { ρ (b) a sup δ> δ a+b ρ a,ω\{σ(δ) ΩV (δ)} < } V= Ξ ρ = ρ on σ, ρ ξ = on Σ ρ(ξ s ) = ρ s Σ Free Ω Ξ V (δ) W H ( γ 1) : w ( γ 1) K; w α,ω K loc s Σ(δ) K H 1+α1 ([, ξ ]): η(ξ ) = η, etc Basic estimates: Σ Ω σ ρ γ,ωv (d ) C 1 ρ, γ γ V and Neumann ρ +α,ω loc C ρ, α α Ω Without obliqueness at Ξ s : Dirichlet ρ 1+p,Σ(d) C(ε, α 1, γ 1, K, d ) u, d d combining these estimates gives T (w) = ρ: T (W) W

23 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 3 Example Σ When Obliqueness Fails BC is u x = f Let φ = u x Ω Then φ = in Ω φ = on Σ u = in Ω β u = f on Σ, β = (1, ) Ordinary Dirichlet condition Good ests. on φ & on u Nonlinear version

24 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 4 The Upper Barrier at Ξ Barrier construction (supersolution): Qψ in Ω(a, h), Nψ, ψ ρ on Ω(a, h) Upper barrier: b (, 1) ψ(r, θ) = ρ + A(c r) b + B(θ 1 θ) Singular barriers: Choi-McKenna &Canic-Kim. Conjecture: sing. at σ; b = 1/ optimal Σ θ r Ξ σ Ω (a,h) On Σ, ODBC: if N 1 (ρ ε )ψ β(ρ ε ) ψ, then ψ ρ ε >. Now ψ = (ψ r, ψ θ ) as r c : ψ n AND ψ t : usual barrier construction fails. But N 1 ψ = β r ψ r + β θ ψ θ, Now β 1 ξ + β η (η η ξ)(ξ + η η) }{{} > by geometry β r = 1 r (β 1ξ + β η), ( r (c (ρ) + 3s (ρ, ρ 1 )) 4c s ). }{{} > at r=c,ρ=ρ A priori bound: need β 1 ξ + β η > for all ρ (ρ, ρ M ). But ρ M = ρ + O(1/κ a ). If ρ >> ρ 1, then κ a 8 suffices.

25 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 5 Completing the Solution: m and n Ξ Ξ s c Σ η Σ c Ξ Ω Transport σ ξ m r = 1 r c (ρ)ρ ξ n r = 1 r c (ρ)ρ η Integrate from Ω. Ω = {r = r (θ)} σ: ρ C(Ω) m, n in D R ξρ ξ ηρ η + m ξ + n η Q(ρ) = ξ,η R + R = BC at R. Singular behavior at

26 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 6 Bifurcation Diagram Region A*: (conjecture) Weak reflected shock or reflected shock with zero strength at triple point (conjecture) θ w =cot 1 (κ) Region C Region B Region A*. κ * ρ /ρ 1 Region A Contour Plot of Density ρ. Data U = (64,, ); U 1 = (1,,); κ a = ; κ b = η axis ξ axis

27 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 7 η Possible Behavior at Triple Point, Region B u S b S c u 1 =(ρ 1,,) S a u =(ρ 1,,n ) ξ Proposition: NO nontrivial sol ns to R-H eq ns for constant states {u, u 1, u } separated by shock lines S a, S b, S c. η S b u 3 u S c u 1 =(ρ 1,,) S a u =(ρ 1,,n ) ξ Proposition: nontrivial sol ns to R-H eq ns for states {u, u 1, u, u 3 } separated by shock lines S a, S b, S c + linear wave. - States u and u 3 must be subsonic (causality) - Only discont. supp. in sub. reg. is lin. wave - Only lin. waves are those in data

28 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 8 Supersonic Patch Numerical results of Tesdall and Hunter on UTSD eqn SIAP, 3 Quasi-steady simulation Cascade of embedded supersonic regions ALLEN M. TESDALL AND JOHN K. HUNTER y/t y/t x/t x/t

29 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 9 Scenario for a Triple Point in NLWS: Embedded Supersonic Region 3 Physical Space with Supersonic Region 1 Mach Stem Formation: Downstream Loci ρ.8 9 η axis c M c m Mach Stem Ξ M U m c S a + U 1 U Refl. Shock U M Ξ m ξ axis; Ξ M = (.33541,.696); Ξ c = (,.361) Sonic Line for Ξ c S (U 1 ) U M U m S(U 1a ) U =U 1a Sonic Line U 1 m ρ =4; ρ =1,κ=1; Ξ = (.33541,.696); Ξ = (,.361) 1 M c Construction of states U M (sonic), U m (supersonic) One parameter family, param. by ξ M (det. by far field) Supersonic bubble not a domain of determinacy (analysis needed) Analysis of cascade? Numerical evidence lacking for NLWS

30 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 3 Diverging Rarefactions: Sonic Free Boundary Problem Data: U, U 1 : far field rarefaction cont. soln ρ to Q(ρ) = : U 1 U 1 =(ρ 1,m 1,n 1 ) Σ T y x= κ a y U =(ρ,m,n ) x=κ a y rarefaction Σ L Ω Σ R Γ rarefaction x Σ B U - candidate for supersonic region not a domain of determinacy - Γ outlines smooth dom. of det. for far field - solution ρ in Ω does not extend to (m, n) - weak solution : possible nonexistence of cont s solution

31 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 31 η axis Contour Plot of Density ρ. Data U = (16, 4, 4); U 1 = (1,,); κ a = Inf; κ b = Simulations κ = ±1: shock κ >> 1: inconclusive Existence of shock-free solution: nontrivial sol n of degen. Goursat prob. singular shock-free ρ but... C ξ axis Σ T D Γ o B u= u= D D 1 V u= Σ R A

32 The XXIst Annual Courant Lectures, Courant Institute, NYU, November, 3 3 Concluding Slide Steady and Quasisteady Flow are Similar: Quasisteady equations change type like steady TS equations Shock is free boundary in both cases Quasisteady Equations are More Difficult: Potential formulation does not generally apply Hodograph transform does not linearize equation Quasisteady Equations are Easier: Change of type is natural Cavitation and stagnation are not issues Expect well-posedness Techniques from elliptic theory have been applied