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
Outline 1. Integrodifferential hyperbolic models. 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.
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.
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.
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, Long-wave approximation ε = 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. u t + uu x + vu y + p x = 0, y v = u x dy, 0 h0 0 u x dy = 0. (2)
Note that flow rate Q in the channel does not depend on the cross section Q(t) = h0 Using equations for velocities at the walls 0 u(t, x, y) dy. 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 v = yu 1x w x (t, x, y ) dy, u 1 = 1 ) h0 (Q(t) w(t, x, y) dy. 0 h 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).
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) ) u 1 = 1 h 0 (Q(t) 1 0 wh dλ 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)
Equation defining characteristic velocity takes form: 1 χ(k) = ω 1 (u 1 k) + 1 ω 0 (u 0 k) + 1 0 ( ) 1 Integrodifferential model (3) has continuous characteristic spectrum k λ (t, x) = u(t, x, λ). ω λ dλ u k = 0. 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 ).
2.3. Riemann problem. As was shown above, the solution of the equation v = yu 1x w t + (w 2 /2 + wu 1 ) x + vw y = 0, (4) y 0 w x (t, x, y ) dy, u 1 = 1 h0 w(t, x, y) dy. h 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 w t=0 = r (y) u r (h 0 ), x > 0 u l (y) u l (h 0 ), x < 0 0
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 0, ul (y) = ω 2 y + u l 0, which correspond to the interaction of flows with constant vorticities. In this case the solution of the Eqs. (2) takes form { Ω1 y + u u(t, x, y) = 0 (t, x), 0 y h(t, x) Ω 2 (y h 0 ) + u 1 (t, x), h(t, x) y h 0 { yu0x, 0 y h(t, x) v(t, x, y) = (h y)u 1x hu 0x, 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
2.4. The equation of the two-layer flow with constant vorticity. scalar conservation law h t + ϕ x = 0, ϕ(h) = Ω 1 Ω 2 h 3 + 2h 0 with initial data h(0, x) = ( Ω 2 Ω 1 2 ) { 0 or h0, x < 0 h 0 or 0, x > 0 h 2 Ω 2h 0 h. (5) 2 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 h(k) = (2ω 2 ω 1 )h 0 3(ω 1 ω 2 ) + (2ω 2 ω 1 ) 2 h 2 0 + h 0(2k + ω 2 h 0 ) 9(ω 1 ω 2 ) 2 3(ω 1 ω 2 )
2.5. Wave of flows interaction. Particles trajectories. 2.5. Wave of flows interaction. Particles trajectories. Liquid particles in the flow move along the trajectories x = x(t) and y = y(t): ẋ = u(t, x, y), ẏ = v(t, x, y). It is convenient to use variables (k, y). Then we have equations ( dk dy = u(k, y) k, = V (k, y) s = ln t, V = tv, k = x ) ds ds t 0 t The singular points (where u(k, y) k = 0, V (k, y) = 0) of the system are the points A and C. Let us separate out the linear part of the system in a neighborhood of the point A (C) and calculate the eigenvalues of the coefficient matrix. This analysis shows that A is a saddle and C is a node (figure a ).
2.5. Wave of flows interaction. Particles trajectories. It is evident that the flow in the region of interaction of the flows has a substantially 2-D nature. Along the line of contact of the two rotating flows, jet flows is formed that is directed to the upper wall or the bottom, depending on the ratio of the vorticities. Figure: Relative velocity field (u k, V ). Figure: Velocity field (u, V ).
2.6. Discontinuous solutions. 2.6. Discontinuous solutions. If condition 2 1 < α 0 < 2, (α 0 = Ω 1 /Ω 2 ) is fell then smooth solution of the equation (5) does not exists. To construct solution of the equation h t + ϕ(h) x = 0 we should combine simple wave and shock.
2.6. Discontinuous solutions. Oleinik conditions ϕ(h) ϕ(h ) h h D (h < h < h + ) ϕ(h) ϕ(h + ) h h + D (h + < h < h ) Possible configuration of the simple wave and shock are shown. Let 0 < ω 2 < ω 1, ω 1 /ω 2 > 2. Discontinuous solution 0, x < Dt h(t, x) = h(k), Dt x ϕ (h 0 )t h 0, x > ϕ (h 0 )t
2.6. Discontinuous solutions. Particles trajectories in the discontinuous solution (0 < ω 2 < ω 1, ω 1 /ω 2 > 2).
2.7. Conservation laws. General case. 2.7. Conservation laws. General case. The Riemann problem for arbitrary monotonic velocity profiles u l (y) and u r (y) can be solve numerically. We propose to use the conservation laws (y = Φ(t, x, λ), H = Φ λ ) w t + (w 2 /2 + wu 1 ) x = 0, H t + (uh) x = 0, ( ) 1 1 1 u 1 = H dλ wh dλ, u = w + u 1 0 Then we divide interval [0, 1] into N intervals 0 = λ 0 < λ 1 <... < λ N = 1 and introduce the designations 0 y i = Φ(t, x, λ i ), η i = y i y i 1, u i = u(t, x, λ i ) ω i = (u i u i 1 )/η i, u ci = (u i + u i 1 )/2. (such approach was used by Teshukov, Russo, Chesnokov (2004) for shear shallow water equation). Taking into account the equality H dλ and using the piecewise linear approximation for horizontal velocity u = ω i (y y i 1 ) + u i 1, y [y i 1, y i ]
2.7. Conservation laws. General case. we arrive to the differential conservation laws. In this case unknown vector satisfies the following 2 2N system η i t + x (u ciη i ) = 0, (η 1,..., η N, ω 1 η 1,..., ω N η N ) ( N ) 1 N u 0 = η i i=1 t (ω iη i ) + x (u ciω i η i ) = 0, u ci = u 0 + ω iη i 2 + i=1 i 1 j=1 ( i 1 ωi η i η i 2 + j=1 ω j η j, ω j η j ). Equation (η 1 +...η N ) t = 0 (which means that depth of the channel is equal to h 0 = const) is the sequence of the system of equations.
3. Exact solutions for the shear shallow water equations. 3. Exact solutions for the shear shallow water equations. 3.1. Shallow water equations for shear flows. The solution of the problem u t + uu x + vu y + p x = 0, ε 2 (v t + uv x + vv y ) + p y = g, u x + v y = 0; h t + u(t, x, h)h x = v(t, x, h), v(t, x, 0) = 0, p(t, x, h) = p 0 ; (ε = H 0 /L 0 0). describes plane-parallel free-boundary flows of an ideal incompressible fluid in a gravity field (dimensionless variables).
3. Exact solutions for the shear shallow water equations. Ignoring terms of order ε 2, we can find pressure p = g(h y) + p 0. After transformations, one obtains the equations for unknown u(t, x, y) and h(t, x) describing shear shallow water flows: ( ) h u t +uu x +vu y +gh x = 0, h t + u dy 0 We formulate the equations using Euler-Lagrangian coordinates: y = Φ(t, x, λ), (0 λ 1) x y = 0, v = u x dy. 0 Φ t + u(t, x, Φ)Φ x = v(t, x, Φ), Φ(0, x, λ) = λh(0, x). Substitution is reversible if Φ λ 0. The functions u(t, x, λ) and H(t, x, λ) = Φ λ satisfy the system u t + uu x + g 1 Eqs. (6) belong to the class (1). 0 H x dλ = 0, H t + (uh) x = 0. (6)
3. Exact solutions for the shear shallow water equations. The dimensionless vorticity of the flow is equal to ω = u y + ε 2 v x satisfies the equation ω t + uω x + vω y = 0. In the long-wave theory ω = u y and also conserve along the trajectories. Vlasov-like formulation can be obtain for flows with ω 0: W t + uw x gh x W u = 0, h = u1 u 0 W du, u 1t + u 1 u 1x + gh x = 0, u 0t + u 0 u 0x + gh x = 0. (7) Here W = ω 1 (analog of the distribution function) is considered as an unknown function depending on t, x, u; functions u 0 (t, x) and u 1 (t, x) are horizontal velocities at the bottom and at free surface, respectively. It follows from equation (7) that the quantity W (t, x, u) is conserved along the trajectories dx dt = u, du dt = gh x.
3. Exact solutions for the shear shallow water equations. 3.2. Traveling waves. 3.2. Traveling waves. A solution of the form W = W (ζ, u), u i = u i (ζ), where ζ = x Dt, describes a travelling wave propagating at constant velocity D. For this class of solutions, Eq. (7) takes the form (u D)W ζ gh (ζ)w u = 0, h = u1 u 0 W du, (u i D)u i (ζ) + gh (ζ) = 0, (i = 0, 1). Previous equations are integrated: W = Φ(η), η = u 2 2Du + 2gh, u 0 u u 1, u 2 i 2Du i + 2gh = η i. Let us consider the Cauchy problem ζ = ζ 0 : W = W 0 (u), u 0 = u 00, u 1 = u 10
3. Exact solutions for the shear shallow water equations. 3.2. Traveling waves. The solution takes constant values on the characteristics η = (u D) 2 + 2gh D 2 = const Designations: η 0 = 2gh 0 D 2, η 00 = (u 00 D) 2 2gh 0 D 2, η 10 = (u 10 D) 2 2gh 0 D 2 ; (u 00 < D < u 10 ). The solution of the Cauchy problem is uniquely determined from initial data in the domains Ω 1 and Ω 2. Ω 1 = {(u, h) 0 < h < h 0, u 0 (h) < u < u (h)} Ω 2 = {(u, h) 0 < h < h 0, u (h) < u < u 1 (h)} Ω 3 = {(u, h) 0 < h < h 0, u (h) < u < u (h)}
3. Exact solutions for the shear shallow water equations. 3.2. Traveling waves. Here u 0 (h) = D 2g(h 0 h) + (u 00 D) 2 u 1 (h) = D + 2g(h 0 h) + (u 10 D) 2 u (h) = D 2g(h 0 h), We determine the functions u (h) = D + 2g(h 0 h). Φ (η) = W 0 (D η 2gh 0 + D 2 ), Using functions Φ ±, we obtain the following solution in the domains Ω 1 and Ω 2 W = Φ (u 2 2Du + 2gh), W = Φ + (u 2 2Du + 2gh). To construct the solution in the domain Ω 3, we transform the relation h = u1 u 0 W du, W = Φ(η); s = 2gh D 2 to an integral equation for the function Φ(η):
3. Exact solutions for the shear shallow water equations. 3.2. Traveling waves. η 0 s F (s) = s + D2 2g Φ(η) dη η s = F (s), (s = 2gh D 2 ) 1 2 η 00 η 0 Φ 0 (η) dη 1 η s 2 η 10 The solution of the Abel equation yields Φ(η) = 1 ( η0 ) F (η0 ) π η0 η F (s) ds s η η η 0 Φ + 0 (η) dη. η s 3.3. The sequences of the smooth solutions converging to the discontinuous. Traveling wave solution involve arbitrary function h = h(x Dt). Form the sequence smooth functions h n converging to the discontinuous solution. Let we have following initial data W (ζ 0, u) = W 0 (u) = ω 1 =const, u 0 (ζ 0 ) = u 00, u 1 (ζ 0 ) = u 10 (u 00 < D < u 10 ), corresponding to flow u = ωy + u 00, v = 0, h = (u 10 u 00 )/ω.
3. Exact solutions for the shear shallow water equations. 3.2. Traveling waves. Then Φ (η) = Φ + (η) = ω 1, Φ(η) = 1 ω 1 ( ) η 0 η arcsin πω η 00 η + arcsin η 0 η 1 η0 η. η 10 η πg The solution ω 1, W (h, u) = Φ(u 2 2Du + 2gh), ω 1, u 0 (h) u u (h) u (h) < u < u (h) u (h) u u 1 (h) Here u 0,1 (h) = D 2g(h 0 h) + (u 00 D) 2 u (h) = D 2g(h 0 h), u (h) = D + 2g(h 0 h). Let us take D = 0, h 0 = 0, h min = 0.55, h n (x) = 2 1 (h 0 h min )(2π 1 arctg(nx) + 1) + h min.
Integrodifferential Hyperbolic Equations and its Application for 2-D Rotational Fluid Flows 3. Exact solutions for the shear shallow water equations. 3.2. Traveling waves.
3. Exact solutions for the shear shallow water equations. 3.2. Traveling waves. Streamlines and velocity field Figure: Velocity field (u, v) in the traveling wave. Figure: Streamlines of the flow. Flow with critical layer is considered (there is line, where fluid velocity coincides with traveling wave velocity).
3. Exact solutions for the shear shallow water equations. 3.2. Traveling waves. Flows without critical layer (D < u 00 < u 10 or u 00 < u 10 < D): Streamlines and velocity field Figure: Velocity field (u, v). Figure: Streamlines in the traveling wave.
3. Exact solutions for the shear shallow water equations. 3.4. Self-similar solutions. 3.4. Self-similar solutions. This algorithm also works for construction self-similar solutions (at least in the case, when critical layer is absent) W = W (ξ, u), u 0 = u 0 (ξ), u 1 = u 1 (ξ), ξ = x/t : (u ξ)w ξ gh W u = 0, (u i ξ)u i + gh = 0, h = u1 u 0 W du The solution takes constant values along the integral curves of the equation du/dξ = gh (ξ)/(u ξ)
4. Conclusions. 4. Conclusions. The Riemann problem (the problem of decay of an arbitrary discontinuity) for the system of equationd describing shear plane-parallel flows of an ideal incompressible liquid in a channel is solved in a long-wave approximation. Exact solutions of the shallow water equations for shear flows for the class of traveling waves and simple waves are derived. The sequence of smooth solutions converging to discontinuous solution is constructed.