Long waves on 3D shear flows: hyperbolicity and discontinuous solutions Alexander Khe Lavrentyev Institute of Hydrodynamics Novosibirsk, Russia 14th International Conference on Hyperbolic Problems: Theory, Numerics, Applications Padua, June 25–29, 2012 Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 1 / 26
Introduction Shallow water equations Long Wave Approximation Vertical shear u t + uu x + vu y + gh x = 0 , Simplification v t + uv x + vv y + gh y = 0 , Mathematics h t + ( hu ) x + ( hv ) y = 0 . Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 2 / 26
Euler Equations Euler equations: u t + ( u · ∇ ) u + ρ − 1 ∇ p = g , (1) div u = 0 , 0 � z � h ( t , x , y ) . Boundary conditions: z = 0 : w = 0 . z = h : w = h t + uh x + vh y , p = p 0 . Initial data: u | t =0 = u 0 , h | t =0 = h 0 . Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 3 / 26
Long Wave Approximation ε = H 0 / L 0 → 0 u t + uu x + vu y + wu z + p x /ρ = 0 , v t + uv x + vv y + wv z + p y /ρ = 0 , (2) u x + v y + w z = 0 . Hydrostatic pressure: p z = − ρ g . Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 4 / 26
Eulerian–Lagrangian Coordinates Change of variables: x = x ′ , y = y ′ , z = Φ( t , x ′ , y ′ , λ ) , λ ∈ [0 , 1] . (3) Function Φ( t , x , y , λ ) is a solution to the Cauchy problem: Φ t + u ( t , x , y , Φ) Φ x + v ( t , x , y , Φ) Φ y = w ( t , x , y , Φ) , (4) Φ | t =0 = Φ 0 ( x , y , λ ) , (5) and Φ | λ =0 = 0 , Φ | λ =1 = h ( t , x , y ) . Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 5 / 26
Equations of Shear Flows The governing equations: u t + uu x + vu y + gh x = 0 , v t + uv x + vv y + gh y = 0 , (6) H t + ( uH ) x + ( vH ) y = 0 for unknown functions u ( t , x , y , λ ) , v ( t , x , y , λ ) , H ( t , x , y , λ ) . Here � 1 H ( t , x , y , λ ) = Φ λ ( t , x , y , λ ) , h = H d λ. 0 Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 6 / 26
Integrodifferential Form H t + ( uH ) x + ( vH ) y = 0 , � 1 u t + uu x + vu y + g H x d λ = 0 , 0 � 1 v t + uv x + vv y + g H y d λ = 0 0 Models with similar form of the system: plane-parallel flows, gas dynamics, kinetic models, flows in elastic tubes, horizontally sheared flows. (V. M. Teshukov, B. N. Elemesova, A. A. Chesnokov, V. Yu. Liapidevskii, et al.) Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 7 / 26
Outline Hyperbolic Properties 1 Generalized Characteristics Hyperbolicity Conditions Rankine–Hugoniot Relations 2 Jump Conditions Two Flows Interaction Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 8 / 26
Hyperbolic Properties Hyperbolic Properties 1 Generalized Characteristics Hyperbolicity Conditions Rankine–Hugoniot Relations 2 Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 9 / 26
Hyperbolic Properties Generalized Characteristics Stationary 3D flows ( uH ) x + ( vH ) y = 0 , � 1 uu x + vu y + g 0 H x d λ = 0 , � 1 uv x + vv y + g 0 H y d λ = 0 . System with operator coefficients A U x + B U y = 0 , (7) where U ( x , y , λ ) = ( H , u , v ) T , and u H 0 v 0 H � 1 , . 0 0 A = B = v g 0 . . . d λ u 0 � 1 0 0 0 . . . d λ 0 u g v Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 10 / 26
Hyperbolic Properties Generalized Characteristics Generalized Characteristics A U x + B U y = 0 (7) [V. M. Teshukov, 1985] Eigen value problem � F , ( ξ A + η B ) ϕ � = 0 . (8) Characteristic normal: ( ξ, η ) Eigen functional: F Test function: ϕ ( λ ) Characteristic relations � F , A U x + B U y � = 0 (9) Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 11 / 26
Hyperbolic Properties Hyperbolicity Conditions Characteristic Equation � 1 H d λ χ ( γ ) ≡ 1 − g q 2 sin 2 ( ϑ − γ ) = 0 (10) 0 where ( ξ, η ) = (cos γ, sin γ ), ( u , v ) = q (cos ϑ, sin ϑ ). Characteristic curves Discrete spectrum: γ 1 , γ 2 Continuous spectrum: γ λ = ϑ ( λ ) Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 12 / 26
Hyperbolic Properties Hyperbolicity Conditions Characteristic Function The system in question is hyperbolic, if χ ( γ ∗ ) > 0 . Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 13 / 26
Rankine–Hugoniot Relations Hyperbolic Properties 1 Rankine–Hugoniot Relations 2 Jump Conditions Two Flows Interaction Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 14 / 26
Rankine–Hugoniot Relations Jump Conditions Equivalent System ( uH ) x + ( vH ) y = 0 , uu x + vu y + gh x = 0 , uv x + vv y + gh y = 0 is equivalent to ( uH ) x + ( vH ) y = 0 , � u 2 + v 2 � − ( v ( v x − u y )) λ = 0 , 2 λ x � u 2 + v 2 � + ( u ( v x − u y )) λ = 0 , 2 (11) λ y �� 1 �� 1 Hu 2 d λ + gh 2 � � + Huv d λ = 0 , 2 0 0 x y �� 1 �� 1 Hv 2 d λ + gh 2 � � Huv d λ + = 0 . 2 0 0 x y Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 15 / 26
Rankine–Hugoniot Relations Jump Conditions Assumptions Assume: u , v , H , u λ , v λ , H λ and v x − u y , ( v x − u y ) λ are bounded and discontinuous across Γ : S ( x , y ) = 0. Then [ u τ ] = 0 . (12) where u τ = u · τ , τ is tangent vector to Γ. Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 16 / 26
Rankine–Hugoniot Relations Jump Conditions Rankine–Hugoniot Conditions � � = 0 , (13) Hu n � � u τ = 0 , (14) � ( u 2 � n ) λ = 0 , (15) �� 1 n d λ + gh 2 � Hu 2 = 0 , (16) 2 0 � u 2 � n 2 + gh � 0 . (17) ( u n = u · n , n — normal vector to Γ) Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 17 / 26
Rankine–Hugoniot Relations Jump Conditions Flow behind the jump If parameters of the flow u τ 1 , u n 1 , H 1 , on one side of the jump and a position of the jump are known, then the flow parameters on the other side of the jump u τ 2 , u n 2 , H 2 can be uniquely determined. Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 18 / 26
Rankine–Hugoniot Relations Jump Conditions Proof From the Rankine–Hugoniot relations we obtain F ( K ) = F (0) , where � 1 �� 1 � 2 n 1 − K d λ + g H 1 u n 1 � u 2 F ( K ) = H 1 u n 1 d λ , 2 � u 2 0 0 n 1 − K which can be solve provided that the flow ahead of the jump is supercritical (i.e. the equations are hyperbolic). Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 19 / 26
Rankine–Hugoniot Relations Jump Conditions Particular Solutions q λ = 0 , ϑ λ / H = A ( A = const) . (18) q ( x , y , λ ), ϑ ( x , y , λ ) — magnitude and angle of u = ( u , v ). In Eulerian coordinates: ϑ ( z ) = ϑ 0 + Az . Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 20 / 26
Rankine–Hugoniot Relations Jump Conditions Weak Solutions Relations q λ = 0 , ϑ λ / H = A . (18) are preserved across the jump. Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 21 / 26
Rankine–Hugoniot Relations Two Flows Interaction ϑ –h diagrams Relations between ϑ 0 2 and h 2 : 1 )(1+tan 2 α )+ q 2 � 2 tan α (cos 2 Ah 1 − 1)+(tan 2 α − 1) sin 2 Ah 1 � ( q 2 1 h 1 + gh 2 1 = 2 A 2 )(1 + tan 2 α ) + q 2 � = ( q 2 2 h 2 + gh 2 2 cos 2( ϑ 0 2 + Ah 2 ) − cos 2 ϑ 0 � � 2 tan α + 2 2 A + (tan 2 α − 1) �� sin 2( ϑ 0 2 + Ah 2 ) − sin 2 ϑ 0 � , 2 where tan α = − cos Ah 1 tan ϑ 0 2 − sin Ah 1 + cos Ah 2 tan ϑ 0 2 + sin Ah 2 2 sin Ah 1 tan ϑ 0 2 � (cos Ah 1 tan ϑ 0 2 + sin Ah 1 − cos Ah 2 tan ϑ 0 2 − sin Ah 2 ) 2 ± � 1 / 2 � − 4 sin Ah 1 tan ϑ 0 2 (cos Ah 1 − cos Ah 2 +sin Ah 2 tan ϑ 0 (2 sin Ah 1 tan ϑ 0 2 ) 2 ) . Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 22 / 26
Rankine–Hugoniot Relations Two Flows Interaction ϑ –h diagrams ϑ – h diagrams are analogous to ϑ – p polars in 2D gas dynamics. Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 23 / 26
Rankine–Hugoniot Relations Two Flows Interaction Two Shear Flows Interaction “5” is a developable surface: ϑ = ϑ 2 ′ ( λ ) = ϑ 3 ′ ( λ ) = ϑ 0 2 ′ + Az Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 24 / 26
Rankine–Hugoniot Relations Two Flows Interaction Two Shear Flows Interaction Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 25 / 26
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