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Long waves on 3D shear flows: hyperbolicity and discontinuous - - PowerPoint PPT Presentation

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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

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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

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Introduction

Shallow water equations ut + uux + vuy + ghx = 0, vt + uvx + vvy + ghy = 0, ht + (hu)x + (hv)y = 0. Long Wave Approximation Vertical shear Simplification Mathematics

Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 2 / 26

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Euler Equations

Euler equations: ut + (u · ∇)u + ρ−1∇p = g, div u = 0, (1) 0 z h(t, x, y). Boundary conditions: z = 0 : w = 0. z = h : w = ht + uhx + vhy, p = p0. Initial data: u|t=0 = u0, h|t=0 = h0.

Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 3 / 26

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Long Wave Approximation

ε = H0/L0 → 0 ut + uux + vuy + wuz + px/ρ = 0, vt + uvx + vvy + wvz + py/ρ = 0, ux + vy + wz = 0. (2) Hydrostatic pressure: pz = −ρg.

Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 4 / 26

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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: Φt + u(t, x, y, Φ) Φx + v(t, x, y, Φ) Φy = w(t, x, y, Φ), (4) Φ|t=0 = Φ0(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

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Equations of Shear Flows

The governing equations: ut + uux + vuy + ghx = 0, vt + uvx + vvy + ghy = 0, (6) Ht + (uH)x + (vH)y = 0 for unknown functions u(t, x, y, λ), v(t, x, y, λ), H(t, x, y, λ). Here H(t, x, y, λ) = Φλ(t, x, y, λ), h = 1 H dλ.

Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 6 / 26

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Integrodifferential Form

Ht + (uH)x + (vH)y = 0, ut + uux + vuy + g 1 Hx dλ = 0, vt + uvx + vvy + g 1 Hy dλ = 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

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Outline

1

Hyperbolic Properties Generalized Characteristics Hyperbolicity Conditions

2

Rankine–Hugoniot Relations Jump Conditions Two Flows Interaction

Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 8 / 26

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Hyperbolic Properties

1

Hyperbolic Properties Generalized Characteristics Hyperbolicity Conditions

2

Rankine–Hugoniot Relations

Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 9 / 26

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Hyperbolic Properties Generalized Characteristics

Stationary 3D flows

(uH)x + (vH)y = 0, uux + vuy + g 1

0 Hx dλ = 0,

uvx + vvy + g 1

0 Hy dλ = 0.

System with operator coefficients AUx + BUy = 0, (7) where U(x, y, λ) = (H, u, v)T, and A =   u H g 1

0 . . . dλ

u u   , B =   v H v g 1

0 . . . dλ

v   .

Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 10 / 26

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Hyperbolic Properties Generalized Characteristics

Generalized Characteristics

AUx + BUy = 0 (7) [V. M. Teshukov, 1985] Eigen value problem F, (ξA + ηB)ϕ = 0. (8) Characteristic normal: (ξ, η) Eigen functional: F Test function: ϕ(λ) Characteristic relations F, AUx + BUy = 0 (9)

Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 11 / 26

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Hyperbolic Properties Hyperbolicity Conditions

Characteristic Equation

χ(γ) ≡ 1 − g 1 H dλ q2 sin2(ϑ − γ) = 0 (10) where (ξ, η) = (cos γ, sin γ), (u, v) = q(cos ϑ, sin ϑ). Characteristic curves Discrete spectrum: γ1, γ2 Continuous spectrum: γλ = ϑ(λ)

Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 12 / 26

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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

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Rankine–Hugoniot Relations

1

Hyperbolic Properties

2

Rankine–Hugoniot Relations Jump Conditions Two Flows Interaction

Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 14 / 26

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Rankine–Hugoniot Relations Jump Conditions

Equivalent System

(uH)x + (vH)y = 0, uux + vuy + ghx = 0, uvx + vvy + ghy = 0 is equivalent to (uH)x + (vH)y = 0, u2 + v2 2

  • λx

− (v(vx − uy))λ = 0, u2 + v2 2

  • λy

+ (u(vx − uy))λ = 0, 1 Hu2 dλ + gh2 2

  • x

+ 1 Huv dλ

  • y

= 0, 1 Huv dλ

  • x

+ 1 Hv2 dλ + gh2 2

  • y

= 0. (11)

Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 15 / 26

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Rankine–Hugoniot Relations Jump Conditions

Assumptions

Assume: u, v, H, uλ, vλ, Hλ and vx − uy, (vx − uy)λ are bounded and discontinuous across Γ : S(x, y) = 0. Then [uτ] = 0. (12) where uτ = u · τ, τ is tangent vector to Γ.

Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 16 / 26

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Rankine–Hugoniot Relations Jump Conditions

Rankine–Hugoniot Conditions

  • Hun
  • = 0,

(13)

  • uτ
  • = 0,

(14)

  • (u2

n) λ

  • = 0,

(15) 1 Hu2

n dλ + gh2

2

  • = 0,

(16) u2

n

2 + gh

  • 0.

(17) (un = u · n, n — normal vector to Γ)

Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 17 / 26

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Rankine–Hugoniot Relations Jump Conditions

Flow behind the jump

If parameters of the flow uτ1, un1, H1,

  • n one side of the jump and a position of the jump are known,

then the flow parameters on the other side of the jump uτ2, un2, H2 can be uniquely determined.

Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 18 / 26

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Rankine–Hugoniot Relations Jump Conditions

Proof

From the Rankine–Hugoniot relations we obtain F(K) = F(0), where F(K) = 1 H1un1

  • u2

n1 − K dλ + g

2 1 H1un1

  • u2

n1 − K

dλ 2 , 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

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Rankine–Hugoniot Relations Jump Conditions

Particular Solutions

qλ = 0, ϑλ/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) = ϑ0 + Az.

Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 20 / 26

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Rankine–Hugoniot Relations Jump Conditions

Weak Solutions

Relations qλ = 0, ϑλ/H = A. (18) are preserved across the jump.

Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 21 / 26

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Rankine–Hugoniot Relations Two Flows Interaction

ϑ–h diagrams

Relations between ϑ0

2 and h2:

(q2

1h1+gh2 1)(1+tan2 α)+ q2 1

2A

  • 2 tan α(cos 2Ah1−1)+(tan2 α−1) sin 2Ah1
  • =

= (q2

2h2 + gh2 2)(1 + tan2 α) + q2 2

2A

  • 2 tan α
  • cos 2(ϑ0

2 + Ah2) − cos 2ϑ0 2

  • +

+ (tan2 α − 1)

  • sin 2(ϑ0

2 + Ah2) − sin 2ϑ0 2

  • ,

where tan α = − cos Ah1 tan ϑ0

2 − sin Ah1 + cos Ah2 tan ϑ0 2 + sin Ah2

2 sin Ah1 tan ϑ0

2

±

  • (cos Ah1 tan ϑ0

2 + sin Ah1 − cos Ah2 tan ϑ0 2 − sin Ah2)2

−4 sin Ah1 tan ϑ0

2(cos Ah1−cos Ah2+sin Ah2 tan ϑ0 2)

1/2 (2 sin Ah1 tan ϑ0

2).

Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 22 / 26

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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

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Rankine–Hugoniot Relations Two Flows Interaction

Two Shear Flows Interaction

“5” is a developable surface: ϑ = ϑ2′(λ) = ϑ3′(λ) = ϑ0

2′ + Az

Alexander Khe (Novosibirsk, Russia) Long waves on 3D shear flows HYP-2012 (Padua, Italy) 24 / 26

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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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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) 26 / 26