Integrodifferential Hyperbolic Equations and its Application for 2-D - PowerPoint PPT Presentation

Integrodifferential Hyperbolic Equations and its Application for 2-D Rotational Fluid Flows Integrodifferential Hyperbolic Equations and its Application for 2-D Rotational Fluid Flows Alexander Chesnokov Lavrentyev Institute of Hydrodynamics Novosibirsk, Russia chesnokov@hydro.nsc.ru July 14, 2006

Integrodifferential Hyperbolic Equations and its Application for 2-D Rotational Fluid Flows Outline 1. Integrodifferential hyperbolic models. 2. Unsteady interaction of uniformly shear flows in a narrow channel. 2.1. Mathematical model. 2.2. Hyperbolicity conditions. 2.3. Riemann problem. 2.4. The equation of the two-layer flow with constant vorticity. 2.5. Wave of flows interaction. Particles trajectories. 2.6. Discontinuous solutions. 2.7. Conservation laws. General case. 3. Exact solutions for the shear shallow water equations. 3.1. Shallow water equations for shear flows. 3.2. Traveling waves. 3.3. The sequences of the smooth solutions converging to the discontinuous. 3.4. Self-similar solutions. 4. Conclusions.

Integrodifferential Hyperbolic Equations and its Application for 2-D Rotational Fluid Flows 1. Integrodifferential hyperbolic models. 1. Integrodifferential hyperbolic models. A number of problems of theoretical hydrodynamics can be deduced to the integrodifferential equations which can be written as U t + A ( U ) � U x � = 0 (1) Here U ( t, x, λ ) - desired vector and A ( U ) is a nonlocal matrix operator acting over the variable λ . For example, 1) long-wave models for 2-D rotational (shear) flows of an ideal homogeneous and barotropic fluid with free surface and in channels; 2) Vlasov type 1-D kinetic equations, in particular bubbly flow kinetic models (written in Euler-Lagrangian coordinates) are belong to this class. Theoretical analysis of such systems is based on a generalization of the concept of hyperbolicity and characteristics for equations with operator coefficients suggested by V.M. Teshukov . It was established that integrodifferential models contain both discrete and continuous spectra of the characteristics velocities. Hiperbolicity conditions for this type equations were formulated. The theory of simple waves was developed.

Integrodifferential Hyperbolic Equations and its Application for 2-D Rotational Fluid Flows 1. Integrodifferential hyperbolic models. However, general theory of integrodifferential equations is under development now, and great number of the problems is not solved including development of numerical schemes, theory of discontinuous solutions, etc. This paper deals with models of type (1) describing in long-wave approximation plane-parallel shear flows of a perfect fluid. There are two main goals: 1. To study characteristic properties and to solve the Riemann problem for the equations describing shear plane-parallel flows of an ideal incompressible liquid in a narrow channel; 2. To construct the sequences of the exact smooth solutions converging to discontinues for the shear shallow water equations.

Integrodifferential Hyperbolic Equations and its Application for 2-D Rotational Fluid Flows 2. Unsteady interaction of uniformly shear flows in a narrow channel. 2. Unsteady interaction of uniformly shear flows in a narrow channel. 2.1. Mathematical model. The Euler equations and boundary conditions in dimensionless form (an ideal incompressible fluid in a narrow straight channel): u t + uu x + vu y + ρ − 1 p x = 0 , ε 2 ( v t + uv x + vv y ) + ρ − 1 p y = − g, u x + v y = 0 , v ( t, x, 0) = 0 , v ( t, x, h 0 ) = 0 . Long-wave approximation ε = 0 : � y � h 0 u t + uu x + vu y + p ∗ u x dy ′ , x = 0 , v = − u x dy = 0 . (2) 0 0

Integrodifferential Hyperbolic Equations and its Application for 2-D Rotational Fluid Flows 2. Unsteady interaction of uniformly shear flows in a narrow channel. Note that flow rate Q in the channel does not depend on the cross section � h 0 Q ( t ) = u ( t, x, y ) dy. 0 Using equations for velocities at the walls u it + u i u ix + p ∗ x = 0 , ( u 0 = u ( t, x, 0) , u 1 = u ( t, x, 1)) we can eliminate p ∗ ( t, x ) (pressure at the upper boundary of the channel). As result we have evolutionary equation for relative velocity w = u − u 1 : w t + ( w 2 / 2 + wu 1 ) x + vw y = 0 , � y � h 0 � u 1 = 1 � w x ( t, x, y ′ ) dy ′ , v = − yu 1 x − Q ( t ) − w ( t, x, y ) dy . h 0 0 0 We pass to the Euler-Lagrangian coordinates ( x, λ ) by the substitution y = Φ( t, x, λ ) : Φ t + u ( t, x, Φ)Φ x = v ( t, x, Φ) , Φ(0 , x, λ ) = Φ 0 ( x, λ ) .

Integrodifferential Hyperbolic Equations and its Application for 2-D Rotational Fluid Flows 2. Unsteady interaction of uniformly shear flows in a narrow channel. As result, we obtain following system which belong to the class (1): w t + ( w 2 / 2 + wu 1 ) x = 0 , H t + (( w + u 1 ) H ) x = 0 , (3) � 1 � � u 1 = 1 Q ( t ) − wH dλ h 0 0 Here w = u − u 1 , H = Φ λ , λ ∈ [0 , 1] . The equation h t = 0 , which expresses the fact that the upper boundary is fixed, is a consequence of the system. 2.2. Hyperbolicity conditions. The characteristics of the system (1) is determined by the equation x ′ ( t ) = k ( t, x ) , where k is eigenvalue of the operator A ∗ . System (1) is hyperbolic if all the eigenvalues k are real and set of equations for the characteristics ( F, U t + AU x ) = 0 ( F is eigenfunctional acting on functions of the variable λ ) is equivalent to equations (1)

Integrodifferential Hyperbolic Equations and its Application for 2-D Rotational Fluid Flows 2. Unsteady interaction of uniformly shear flows in a narrow channel. Equation defining characteristic velocity takes form: � 1 � � 1 1 1 dλ χ ( k ) = − ω 1 ( u 1 − k ) + ω 0 ( u 0 − k ) + u − k = 0 . ω 0 λ Integrodifferential model (3) has continuous characteristic spectrum k λ ( t, x ) = u ( t, x, λ ) . Hiperbolicity conditions are formulated in terms of the limiting values χ ± of the complex function χ ( z ) from upper and lower half-planes on the real axis. For flows with monotonic velocity depth profile u λ � = 0 , conditions △ arg( χ + ( u ) /χ − ( u )) = 0 , χ ± ( u ) � = 0 are necessary and sufficient for Eq. (3) to be hyperbolic (functions u and H are sufficiently smooth, ω = u λ /H , and the argument increment is calculated for u varied from u 0 to u 1 ).

Integrodifferential Hyperbolic Equations and its Application for 2-D Rotational Fluid Flows 2. Unsteady interaction of uniformly shear flows in a narrow channel. 2.3. Riemann problem. As was shown above, the solution of the equation w t + ( w 2 / 2 + wu 1 ) x + vw y = 0 , (4) � y � h 0 u 1 = − 1 w x ( t, x, y ′ ) dy ′ , v = − yu 1 x − w ( t, x, y ) dy. h 0 0 0 describes plane-parallel shear motions of an ideal fluid in a narrow channel ( Q = 0 ). Equation (4) admits particular solutions of the form w = w ( y ) , that correspond to steady shear flows. A natural generalization of the formulation of the Riemann problem in the case of Eq. (4) is the Cauchy problem � u r ( y ) − u r ( h 0 ) , x > 0 w | t =0 = u l ( y ) − u l ( h 0 ) , x < 0

Integrodifferential Hyperbolic Equations and its Application for 2-D Rotational Fluid Flows 2. Unsteady interaction of uniformly shear flows in a narrow channel. 2.4. The equation of the two-layer flow with constant vorticity. 2.4. The equation of the two-layer flow with constant vorticity. First, we consider special initial data u r ( y ) = ω 1 y + u r u l ( y ) = ω 2 y + u l 0 , 0 , which correspond to the interaction of flows with constant vorticities. In this case the solution of the Eqs. (2) takes form � Ω 1 y + u 0 ( t, x ) , 0 ≤ y ≤ h ( t, x ) u ( t, x, y ) = Ω 2 ( y − h 0 ) + u 1 ( t, x ) , h ( t, x ) ≤ y ≤ h 0 � − yu 0 x , 0 ≤ y ≤ h ( t, x ) v ( t, x, y ) = ( h − y ) u 1 x − hu 0 x , h ( t, x ) ≤ y ≤ h 0 Here Ω 1 = ω 1 , Ω 2 = ω 2 if ω i > 0 (figure ” a ”) and Ω 1 = ω 2 , Ω 2 = ω 1 if ω i < 0 (figure ” b ”). The form of the solution and condition that the velocity vector is continuous on the boundary of the layers y = h ( t, x ) allow us to obtain following

Integrodifferential Hyperbolic Equations and its Application for 2-D Rotational Fluid Flows 2. Unsteady interaction of uniformly shear flows in a narrow channel. 2.4. The equation of the two-layer flow with constant vorticity. scalar conservation law � � ∂h ∂t + ∂ϕ ϕ ( h ) = Ω 1 − Ω 2 Ω 2 − Ω 1 h 2 − Ω 2 h 0 h 3 + ∂x = 0 , h. (5) 2 h 0 2 2 with initial data � 0 or h 0 , x < 0 h (0 , x ) = h 0 or 0 , x > 0 The function ϕ ( h ) is convex if 2 − 1 < α 0 < 2 , ( α 0 = Ω 1 / Ω 2 ) . In this case continuous solution (simple wave of flows interaction) exists. Since the equations and boundary conditions are invariant with respect to uniform stretching of the variables t and x , we seek self-similar solution h = h ( k ) , k = x/t . Let 0 < ω 2 < ω 1 , 1 < ω 1 /ω 2 < 2 (we should take Ω 1 = ω 1 , Ω 2 = ω 2 ). Integrating Eq. (5) we obtain � (2 ω 2 − ω 1 ) 2 h 2 h ( k ) = − (2 ω 2 − ω 1 ) h 0 + h 0 (2 k + ω 2 h 0 ) 0 + 9( ω 1 − ω 2 ) 2 3( ω 1 − ω 2 ) 3( ω 1 − ω 2 )