CHAPTER 5. RUDIMENTS OF HYDRODYNAMIC INSTABILITY

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1 1 Lecture Notes on Fluid Dynamics (1.63J/.1J) by Chiang C. Mei, 00 CHAPTER 5. RUDIMENTS OF HYDRODYNAMIC INSTABILITY References: Drazin: Introduction to Hydrodynamic Stability Chandrasekar: Hydrodynamic and Hydromagnatic Instability Stuart: Hydrodynamic Stability, in Rosenhead (ed) : Laminar boundary layers Lin : Theory of Hydrodynamic Stability Drazin and Reid : Hydrodynamic Stability Schlichting: Boundary Layer Theory C.S. Yih, 1965, Dynamics of Inhomogeneous Fluids, MacMillan. Kundu : Fluiid Mechanics. Various factors can be crucial to hydrodynamic instability.: shear, gravity, surface tension, heat, centrifugal force, etc. Some instabilities lead to a different flow; some to turbulence. We only discuss the linearized analysis of instability of a few sample problems. More examples will be discussed in later chapters. Let us take the parallel flow as the example and outline the standard procedure of linearized analysis as follows: 1. Assume the flow domain to be of infinite extent in the x direction. Let the basic flow have velocity U(z) in the x direction.. Derive the linearized equation for infinitestimal perturbations and homogeneous boundary conditions. 3. Consider sinusoidal wave- like disturbances so that all unknowns are of the form f i = f i (y, z) e i(kx ωt) 4. Deduce the partial differential equation(s) in the transverse plane (y, z) and obtain an eignevalue problem. 5. Solve for the eigenvalue ω for given real k. 6. If the eigenvalue is complex, ω = ω r +iω i, find the condition under which ω i > 0. Since e iωt = e i(ωr+iω i)t = e iωrt e ω it, the flow is unstable ω i > 0 as time grows, stable if ω i < 0, and neutrally stable if ω i = 0.

2 5.1 Kelvin-Helmholtz instability of flow with discontinuous shear and stratification Nonlinear formulation Consider two immiscible fluids. The upper (z > 0) and lower (z < 0) fluids have different densities ρ 1 and ρ. Before any perturbations appear the lower fluid is stationary, while the upper fluid moves at the steady and uniform velocity q o = U i. With gravity, this is called the Kelvin-Helmholtz instability problem. For the time being, let us assume that there is no gravity. Now let there be a disturbance on the interface whose vertical displacement is z = ζ(x, t). The corresponding disturbances of velocity and pressure are q 1 and p 1 in the upper fluid and q and p in the lower fluid. Let us first formulate the nonlinear problem. In the upper fluid, we have and the Bernoulli equation, q 1 = Φ 1, and Φ 1 = 0, z > 0 (5.1.1) In the lower fluid, we have : Φ 1 + ( Φ 1 ) = p 1 ρ 1 + C 1 (5.1.) with the Bernoulli equation, and q = Φ, and Φ = 0 z < 0 (5.1.3) Φ + ( Φ ) On the interface F = z ζ(x, t) = 0 the kinematic conditions are or, more explicitly, The dynamic condition is = p ρ + C (5.1.4) F + Φ 1 F = 0 (5.1.5) F + Φ F = 0 (5.1.6) + Φ 1 x x = Φ 1, z = ζ, (5.1.7) + Φ x x = Φ, z = ζ, (5.1.8) p 1 = p, or z = ζ. (5.1.9)

3 ρ 1 ( Φ1 + ( Φ 1 ) Far away from the interface, C 1 ) = ρ ( Φ + ( Φ ) 3 ) C, or z = ζ. (5.1.10) Φ 1 U 1 i, z, (5.1.11) Φ U i, z. (5.1.1) Note that in the absence of any transient disturbances, Φ 1 = U 1 i, Φ = U i, so that 5.1. Linearization ( ρ 1 C 1 U 1 ) ( = ρ C U ) (5.1.13) We now limit ourselves to infinitesimal disturbances. Specificlly, the interface displacement is infinitesimal when compared to other length scales. Φ 1 = U 1 x + φ 1, and Φ = U x + φ, (5.1.14) with φ n Φ n, n = 1, and linearize all the conditions. We then get Φ 1 = 0, z > 0, (5.1.15) nad Φ = 0, z < 0, (5.1.16) Note that On the inteface we have ther kinematic conditions and the dynamic condition f(x, ζ, t) = f(x, 0, t) + ζ f z=0 +, (5.1.17) ρ 1 ( φ1 + U 1 where (5.1.13) has been used. + U 1 x = φ 1, z = 0, (5.1.18) + U x = φ, z = 0, (5.1.19) ) ( φ 1 φ = ρ x + U ) φ, z = 0. (5.1.0) x

4 Normal mode analysis Assume a sinusoidal disturbance ζ = Ae i(kx ωt) (5.1.1) We follow the usual convention that the real part of the final solution is to be taken to represent physical quaitities. To satisfy Laplace equation we take φ 1 = φ 1 e kz e i(kx ωt) (5.1.) φ = φ e kz e i(kx ωt) (5.1.3) Here A, φ 1, φ are unknown constants. Substituting into boundary conditions on the interface eqs ( ) we get three homogeneous alegebric equations for A, φ 1, φ, ( iω + iku 1 )A = k φ 1 (5.1.4) ( iω + iku )A = k φ (5.1.5) which can be written in matrix form ρ 1 ( iω + iku 1 )φ 1 = ρ ( iω + iku )φ iω + iku 1 k 0 iω + iku 0 k 0 ρ 1 ( iω + iku 1 ) ρ ( iω + iku ) Ā φ 1 φ = 0 In order to get non trivial solutions the coefficient determinant of the algebraic equations must vanish, leading to the eigenvalue condition ρ 1 ( ω + ku 1 ) + ρ ( ω + ku ) = 0 or (ρ 1 + ρ )ω k(ρ 1 U 1 + ρ U )ω + k (ρ 1 U 1 + Uρ U ) = 0 The solution is ω = k ρ 1U 1 + ρ U ρ 1 + ρ ρ1 ρ U 1 U ± ik (5.1.6) ρ 1 + ρ The lower sign corresponds to instability. Thus a velocity discontinuity is always unstable. This is the limiting case of the continuous velocity profile with a point of inflection.

5 5.1.4 Physical interpretation Let us use the mathematical results to see the physics ( Batchelor, p 516ff). For simlicity we choose ρ 1 = ρ = ρ and U 1 = U = U > 0 so that (5.1.6) reduces to ω = ±iku (5.1.7) and is purely imaginary. The positive root (upper sign) signifies instability. The basic flow has a vortex sheet at z = 0. Due to the disturbance, the vorticity distribution is changed. Across the interface the total vorticity is approximately 0+ 0 u dz = u 1 (z = 0 +) u (z = 0 ) = φ 1 x 0 φ x 0 Note that a positive vorticity vector points into the paper (as does the y axis); the circulation is clockwise in the x z plane. From the solution (5.1.4) and (5.1.5) we get φ 1 x 0 = ik φ 1 e ikx iωt = i( iω + iku 1 )Ae ikx iωt = i( iω + iku 1 )ζ and φ x 0 = ik φ e ikx iωt = i( iω + iku )Ae ikx iωt = i( iω + iku )ζ Therefore the vorticity along the interface is [ ω + k(u 1 + U )]ζ = ikuζ = kua exp(ikx iπ/) ± kut) Consider only the upper signs for the unstable solution, [ ( ikuζ = kua exp ik x π ) ] [ ( + kut = kua exp ik x λ ) ] + kut k 4 5 since k = π/λ where λ is th wave length. In real form the vorticity disturbance varies as whereas the interface varies as kua cos [ k ( x λ 4 A cos kx e kut The vorticity disturbance varies along the interface and leads the interface displacement ζ in phase by π/ or one quarter of a wavelength. As sketched in the upper part of Figure 5.1.1, the vorticity disturbance is at the positive (clockwise) maximum at points like A where ζ = 0 and ζ x < 0 and at the negative (counterclockwise) maximum at points like B where ζ = 0 and ζ x > 0, being zero at the crests and troughs of the interface. All these vortices tend to lift the interface crests and supress the troughs. )] e kut

6 6 Figure 5.1.1: Disturbance vorticity along a velocity dicontinuity. U 1 = U > 0, U = U < 0. Moreover the disturbances are convected by the average velocity φ 1 x 0 + φ x 0 = k(u 1 U )ζ = Uζ which is positive near the crests and negative near the troughs. Therefore clockwise vorticity disturbances accumulate around A, while the counterclockwise disturbances are swept away from and thinned out around B. Because of this assymetry the wavy interface not only tend to amplify but to curl up in the forward direction as sketched in the lower part of Figure After some time the disturbance grows so large that nonlinearity must be accounted for. Numerical nonlinear analysis as well as laboratory experiments show that the interface curls up.

7 Homework K-H instability: Show that when gravity and surface tension are both present, the eigenvalue condition is ω k = ρ [ 1U 1 + ρ U ± ρ 1ρ (U 1 U ) + g ρ ρ 1 + ρ 1 + ρ (ρ 1 + ρ ) k ρ 1 + ρ T k ρ 1 + ρ 7 ] 1/ (5.1.8) Discuss the condition for instability. Remark: The special case of U 1 = U = 0 but g 0, i.e., the instability of normally accelerated density discontinuity, is called the Rayleigh-Taylor instability. It is of interest when an interface is acelerated in the normal direction, and has a wide range of industrial and astrophysical applcations. Remark: The K-H instabilty problem can be generalized in a number of ways, by including viscosity and continuous density and/or velocity profiles, see Chandrasekhar.

Kelvin Helmholtz Instability

Kelvin Helmholtz Instability A. Salih Department of Aerospace Engineering Indian Institute of Space Science and Technology, Thiruvananthapuram November 00 One of the most well known instabilities in fluid