Kelvin-Helmholtz Instability and Roll-up
- Chihiro Matsuoka, Department of Physics, Graduate School of Science and Technology, Ehime University
Kelvin-Helmholtz Instability (KHI) (Helmoltz 1868, Kelvin 1871) is a hydrodynamic instability in which immiscible, incompressible, and inviscid fluids are in relative and irrotational motion. In KHI, the velocity and density profiles are uniform in each fluid layer, but they are discontinuous at the (plane) interface between the two fluids. This discontinuity in the (tangential) velocity, i.e., the shear flow, induces vorticity at the interface; as a result, the interface becomes an unstable vortex sheet that rolls up into a spiral. In order to simplify the theoretical and numerical calculations, the effects of gravity, surface tension, and the density difference between the two fluids are not considered in most cases; however, KHI can occur as long as a uniform velocity shear exists when a lighter fluid is superposed on a heavier fluid (Chandrasekhar 1981, Drazin 1970, Nayfeh and Saric 1972, Weissman 1979). When a heavier fluid is superposed on a lighter fluid, the instability is distinguished as Rayleigh-Taylor instability (RTI) (Chandrasekhar 1981, Andrew and David 2009). RTI is an instability caused by gravity, in which the uniform shear flow is not considered.
Linear stability theory
The physical mechanism underlying KHI is shown in Figure 1. The interface is regarded as a vortex sheet in the two-dimensional plane \((x,y)\ ,\) at which a horizontal velocity discontinuity exists. In Figure 1, the physical quantities for \(y < 0\) and \(y > 0\) are represented by the subscripts 1 and 2, respectively. The uniform velocity distribution in each fluid is known as basic flow, which is given by \({\mathbf{ U}_1} = (U_1,0)\) for \(y < 0\) and \({\mathbf{ U}_2} = (U_2,0)\) for \(y > 0\) in the figure. The system is governed by the normal velocity continuity condition (the kinematic boundary condition) and pressure continuous condition at the interface. The former condition implies that the normal component of the interfacial velocity \({\mathbf{u}}_{int}\)is equivalent to the normal velocity of each fluid; i.e., \[\tag{1} {\mathbf{n}} \cdot {\mathbf{u}}_{int} = {\mathbf{n}}\cdot {\mathbf{u}}_1 ={\mathbf{n}} \cdot {\mathbf{u}}_2, \]
where \[ {\mathbf{n}} = \frac{1}{\sqrt{1+\left(\frac{\partial \eta}{\partial x}\right)^2}}\left(-\frac{\partial \eta}{\partial x},1 \right) \] is the unit normal to the interface and the fluid velocity \({\mathbf{u}}_i\) in each fluid \(i\) \((i=1,2)\) is related to the velocity potential \(\phi_i\) as \( {\mathbf{u}}_i = {\mathbf{U}}_i+\nabla \phi_i\ .\) The deviation of the interface \(y = \eta(x,t)\ ,\) where \(t\) denotes time, is measured from \(y=0\ .\) Then, (1) can be rewritten as \[ \frac{\partial \phi_1}{\partial y} =\frac{\partial \eta}{\partial t} + \frac{\partial \phi_1}{\partial x}\frac{\partial \eta}{\partial x}, \] \[\tag{2} \frac{\partial \phi_2}{\partial y} = \frac{\partial \eta}{\partial t} + \frac{\partial \phi_2}{\partial x}\frac{\partial \eta}{\partial x}. \]
The condition that the pressure \(p_i\) in fluid \(i\) \((i=1,2)\) is continuous at the interface, i.e.; \(p_1 = p_2\) at \(y=\eta(x,t)\ ,\) gives \[\tag{3} \rho_1\left\{\frac{\partial \phi_1}{\partial t} + \frac{1}{2}\left[\left(\frac{\partial \phi_1}{\partial x}\right)^2 + \left(\frac{\partial \phi_1}{\partial y}\right)^2 \right]\right\} + (\rho_1 - \rho_2)g\eta =\left\{\rho_2\frac{\partial \phi_2}{\partial t} + \frac{1}{2}\left[\left(\frac{\partial \phi_2}{\partial x}\right)^2 + \left(\frac{\partial \phi_2}{\partial y}\right)^2 \right]\right\} + \frac{\sigma\left(\frac{\partial^2 \eta}{\partial x^2}\right)}{\left[1+\left(\frac{\partial \eta}{\partial x}\right)^2\right]^{3/2}} , \]
where \(\rho_i\) is the density in fluid \(i\) \((i=1,2)\ ,\) \(g\) is gravity, and \(\sigma\) is surface tension coefficient at the interface. In the case of KHI, \(\rho_1 \geq \rho_2\) is assumed. Equation (3) is known as the Bernoulli equation. Equations (2) and (3) are the governing equations of KHI.
Substituting the basic flow \({\mathbf{U}}_i\) \((i=1,2)\) into (2) and (3), and taking up to the first order of the small disturbance, one obtains \[ \frac{\partial \eta}{\partial t} - \frac{\partial \phi_1}{\partial y} = -U_1 \frac{\partial \eta}{\partial x}, \] \[\tag{4} \frac{\partial \eta}{\partial t} - \frac{\partial \phi_2}{\partial y} = -U_2 \frac{\partial \eta}{\partial x}, \]
and \[\tag{5} \rho_1\left( \frac{\partial \phi_1}{\partial t} + U_1 \frac{\partial \phi_1}{\partial x}\right) + (\rho_1 - \rho_2)g\eta = \rho_2\left(\frac{\partial \phi_2}{\partial t} + U_2\frac{\partial \phi_2}{\partial x}\right) + \sigma\frac{\partial^2 \eta}{\partial x^2}, \]
where all quantities with respect to \(\phi_i\) are taken at \(y=0\ .\) From the incompressibility condition, Laplace equation, i.e., \(\triangle \phi_i = \) holds in each fluid \(i\ .\) Thus, one gets solutions of (4) and (5) as \[ \eta(x,t) = {\rm Re}\left[A_0{\rm e}^{i (k x - \omega t)}\right], \] \[ \phi_1(x,t) = {\rm Re}\left[iA_0\left(U_1 - \frac{\omega}{k}\right){\rm e}^{i(kx-\omega t)}\right]{\rm e}^{ky} \quad (y < 0), \] \[\tag{6} \phi_2(x,t) ={\rm Re}\left[-iA_0\left(U_2 - \frac{\omega}{k}\right){\rm e}^{i(kx-\omega t)}\right]{\rm e}^{-ky} \quad (y > 0), \]
where \(k\) is the wave number, \({\rm Re}\) denotes the real part, \(A_0\) is a constant that corresponds to the initial amplitude of the interface, and \(\tag{17} \omega = \frac{\rho_1 U_1 + \rho_2 U_2}{\rho_1 + \rho_2}k \pm \sqrt{ \frac{-\rho_1\rho_2(U_1 - U_2)^2}{(\rho_1 + \rho_2)^2}k^2 + \frac{\rho_1 - \rho_2}{\rho_1 + \rho_2}gk + \frac{\sigma}{\rho_1 + \rho_2}k^3}. \)
Here, the condition that the disturbance vanishes at \(y=\pm \infty\) is imposed. The system grows exponentially under linear approximation when \[ \frac{\rho_1\rho_2(U_1 - U_2)^2}{(\rho_1 + \rho_2)^2}k^2 - \frac{\rho_1 - \rho_2}{\rho_1 + \rho_2}gk - \frac{\sigma}{\rho_1 + \rho_2}k^3 > 0. \] The left-hand side of this inequality denotes the (linear) growth rate of KHI. When \(\rho_2 > \rho_1\) and \(U_1 = U_2 =0\ ,\) RTI occurs. The detailed calculations of the linear stability theory for KHI have been provided by Batchelor (Batchelor 2000) (for the case that \(\rho_1 = \rho_2\) and \(\sigma=0\)) and Chandrasekhar (Chandrasekhar 1981).
Nonlinear theory
Birkhoff-Rott equation
The nonlinear dynamics of the interface is described in terms of the dynamics of a vortex sheet. The temporal evolution of a sheet is expressed by an integro-differential equation known as the Birkhoff-Rott equation, which is derived from the Biot-Savart law (Birkhoff 1962, Rott 1956). For simplicity, we set \(\rho_1 = \rho_2\ ,\) \(g=0\ ,\) and \(\sigma=0\ .\) By denoting the interface as the complex form: \(z(\theta,t) = x(\theta,t) + iy(\theta,t)\ ,\) the Birkhoff-Rott equation is expressed as \[\tag{7} \frac{\partial z^*(\theta, t)}{\partial t} = \frac{1}{2\pi i}{\rm {P.V.}}\int_{-\infty}^\infty \frac{\gamma (\theta')}{z(\theta, t)-z(\theta', t)}d\theta' , \]
where \(\theta\) is a Lagrangian parameter that parameterizes the interface, \(z^*\) is the complex conjugate of \(z\ ,\) \({\rm {P.V.}}\) denotes the principal value integral, and \( \gamma\) is the vortex sheet strength defined as \( \gamma = d\Gamma/d\theta\) through the circulation \( \Gamma\ .\) Equation (7) gives the velocity of the sheet induced by the vorticity at the interface, and the configuration of the interface is obtained from the temporal integration of (7). When density stratification does not exist, i.e., when the fluid density in the system is uniform, the circulation \(\Gamma\) is conserved along Lagrangian particle paths (Kelvin's circulation theorem; see Batchelor 2000, Kelvin 1869) and \( \gamma\) does not depend on time. The Birkhoff-Rott equation is equivalent to the 2D Euler system expressed by (2) and (3) (in a sense of weak solutions) when the initial interface is sufficiently smooth (Lopes Filho et al. 2007).
Moore's curvature singularity
Taking the Lagrangian parameter \(\theta\) as the circulation \(\Gamma\ ,\) i.e., \(\theta = \Gamma\) \((-\infty < \Gamma < \infty)\ ,\) Moore investigated the analyticity of the Birkhoff-Rott equation (7) (Moore 1979). Setting the initial condition as \[ z(\Gamma,0) = \Gamma + i\epsilon \sin \Gamma \quad (\epsilon \ll 1) \] and assuming the form \[\tag{8} z(\Gamma,t) = \Gamma + 2i\sum_{n=1}^\infty A_n(t)\sin n\Gamma, \]
he obtained the asymptotic expansion for \(n \gg 1\) and \(t \gg 1\) as \[\tag{9} \epsilon^n A_{n0} \sim t^{-1}(2\pi)^{-1/2}(1+i)n^{-5/2}\exp\left[n\left(1 + \frac{1}{2} + \ln \frac{1}{4}\epsilon t \right) \right], \]
where \(A_n\) is expanded as \(A_n = \epsilon^n A_{n0} + \epsilon^{n+2} A_{n2} + \epsilon^{n+4} A_{n4} + \cdots\ .\) The asymptotic solution to the Birkhoff-Rott equation loses its analyticity at a finite time \(t=t_c\) such that the condition \(1 + t_c/2 + \ln t_c = \ln (4/\epsilon)\) is satisfied. Then, the shape of the sheet is given by \[\tag{10} z(\Gamma,t_c) \sim \Gamma + \sum_{n=- \infty}^\infty C n^{-5/2} {\rm e}^{in\Gamma} \]
and \[ z(\Gamma,t) = \Gamma + \frac{2\sqrt{3}}{3t}(1+i)\left[(1-{\rm e}^{i\Gamma}\epsilon\Theta)^{3/2} - (1-{\rm e}^{-i\Gamma}\epsilon\Theta)^{3/2}\right] \] at \(\Gamma = \Gamma_c =2n\pi\) \((n=0, \pm 1, \pm2, \cdots)\ ,\) where \(C\) is a constant and \(\Theta = t/4 \exp\left(t/2+1 \right)\ .\) Since \(\epsilon\Theta \to 1\) as \(t \to t_c\ ,\) the curvature of the sheet diverges as \(|\Gamma - \Gamma_c|^{-1/2}\) in the neighborhood of the singular point \(\Gamma = \Gamma_c\ .\) The true sheet strength \(\kappa\ ,\) defined as \(\kappa = {d \Gamma}/{d s} = \gamma/s_\theta\ ,\) has the cuspidal form \(\kappa(\Gamma,t) \propto |\Gamma - \Gamma_c|^{1/2}\ ,\) where \(s_\theta = \sqrt{x_\theta^2 + y_\theta^2}\ ,\) the subscript \(\theta\) denotes differentiation with respect to that variable, and \(d\) denotes differentiation along the fluid particles.
The power law \(n^{-5/2}\) for the Fourier coefficient in (10) has been verified by several researchers via numerical (Krasny 1986, Shelley 1992) and analytical (Caflisch and Semmes 1990, Cowley et al. 1999, Meiron et al. 1982) calculations. Baker et al. (Baker et al. 1993) extended Moore's asymptotic theory for curvature singularity to the case of Atwood number \(A \ne 0\) and found that a family of exact solutions for which singularities develop on the fluid interface, where \(A\) is defined by the fluid densities \(\rho_1\) and \(\rho_2\) as \(A = (\rho_2 - \rho_1)/(\rho_1 + \rho_2)\ .\) They showed theoretically and numerically that these singularities reach the real axis in the complex plane in finite time for \(A \ne 1\ ,\) and then, Moore's curvature singularity appears in the physical plane. They also predicted that these singularities never reach the real axis in finite time when \(A=1\ ,\) which suggests that the curvature singularity does not appear for \(A = 1\ .\) Although the rigorous mathematical proof does not exist, their prediction for finite Atwood numbers including \(A=1\) is supported theoretically (Tanveer 1993) and numerically (Matsuoka and Nishihara 2006).
Numerical method
Zero surface tension case
Suppose that the initial disturbance is \(2\pi\)-periodic in the \(x\)-direction; \(z(\theta + 2\pi, t) = 2\pi + z(\theta,t)\ ,\) \(\gamma(\theta+2\pi) = \gamma(\theta)\ .\) Then, the Birkhoff-Rott equation (7) is rewritten as
\[\tag{11}
\frac{\partial z^*(\theta, t)}{\partial t} = \frac{1}{4\pi i}{\rm {P.V.}}\int_0^{2\pi}\gamma(\theta')\cot\frac{1}{2}\left[ z(\theta, t)-z(\theta', t) \right]d\theta' .
\]
Highly accurate calculations are required to detect singularity formation. Shelley (Shelley 1992) carried out the spatial integration expressed by (11) with spectral accuracy by using the alternate point quadrature method (Sidi and Israeli 1988), and numerically verified the \(-5/2\)-type power law and curvature singularity predicted by Moore. This method does not include the error by the spatial integration, i.e., it is exact within the machine accuracy; however, the accuracy is typically lost at the critical time. Although it is first order accurate, the point vortex method presented by Krasny (Krasny 1986) remains consistent at the critical time at which the Moore's curvature singularity appears. For the mathematical discussion on the relation between the point vortex approximation and the critical time, refer to the reference (Caflisch et al. 1999).
Curvature singularity is an unphysical phenomenon. In real systems, the interface is a layer with a finite thickness; in addition, viscosity or surface tension affects the interface. Owing to these effects, singularity formation is prevented in reality; instead, the roll-up of a sheet is observed. In order to perform long-time calculations beyond curvature singularity, Krasny introduced a small parameter \(\delta\) and regularized the Cauchy integral in the Birkhoff-Rott equation (11) (Krasny 1987). For the periodic case, this regularization yields \[\frac{\partial x}{\partial t}(\theta, t) = -\frac{1}{4 \pi}\int_{0}^{2\pi}\frac{\sinh\left(y(\theta, t) - y(\theta', t)\right)\gamma(\theta', t)}{\cosh\left(y(\theta, t) - y(\theta', t)\right)-\cos\left(x(\theta, t)-x(\theta', t)\right) + \delta^2}d\theta', \] \[\tag{12} \frac{\partial y}{\partial t}(\theta, t) = \frac{1}{4 \pi}\int_{0}^{2\pi}\frac{\sin\left(y(\theta, t) - y(\theta', t)\right)\gamma(\theta', t)}{\cosh\left(y(\theta, t) - y(\theta', t)\right)-\cos\left(x(\theta, t)-x(\theta', t)\right) + \delta^2}d\theta'. \]
This approach is known as the vortex blob method. The parameter \(\delta\) plays an important role in cutting higher-order Fourier modes. When \(\delta \ne 0\ ,\) (12) does not coincide with the 2D Euler system (Holm et al. 2006); however, this equation effectively describes real interfacial motion such as roll-up. The spatial integration in (12) can be carried out using the conventional trapezoidal rule.
The curvature singularity in KHI and the roll-up of a sheet are shown in Figure 2 and Figure 3, respectively. Here, the initial condition is taken as \(x(\theta,0) =\theta + 0.05\sin\theta\) and \(y(\theta,0) = -0.05\sin\theta\) for both cases. For the regularized parameter \(\delta = 0\ ,\) roll-up does not occur. The point \(\theta = \theta_c\) (\(\Gamma_c\) in Moore's analysis) at which the curvature diverges is known as the vortex core. The \(N\)-th derivative (\(N \geq 2\)) with respect to \(\theta\) diverges at the vortex core when \(\delta=0\ .\) When \(\delta \ne 0\ ,\) the roll-up occurs and the vortex core becomes the center of a spiral. The speed of the roll-up slows down for larger values of \(\delta\ .\) In addition to the vortex blob method, there are some alternative regularization approaches for describing the roll-up of a vortex sheet (Baker and Pham 2006, Holm et al. 2006).
Singularity formation is also detected in 3D vortex motion (Hou et al. 2003, Ishihara and Kaneda 1995, Ishihara and Kaneda 1996, Nitsche 2001, Sakajo 2002). In this case, the vortex-induced velocity corresponding to the Birkhoff-Rott equation (11) is expressed by the Biot-Savart law.
Finite surface tension case
When the surface tension is taken into account, the roll-up can occur even though the regularized parameter \(\delta = 0\ .\) Hou et al. numerically verified this fact for the case of the Atwood number \(A=0\ ,\) i.e., \(\rho_1 = \rho_2\) (Hou et al. 1994, Hou et al. 1997). Their method is generalized to the case of \(A \ne 0\) (Ceniceros and Hou 1998, Matsuoka 2009). The roll-up for \(\sigma \ne 0\) and \(A \ne 0\) is shown in Figure 4, where the initial condition is selected as \(x(\theta,0)= \theta\) and \(y(\theta,0) = 0.1\sin\theta\ .\) For the roll-up with finite surface tension, a phenomenon called "pinching singularity" is observed instead of Moore's curvature singularity (Hou et al. 1994). This pinching phenomenon disappears as the surface tension coefficient increases (Matsuoka 2009). When \(\rho_1 > \rho_2\) (\(A < 0\ ;\) the lighter fluid is superposed on the heavier fluid) and the relative shear velocity \(|U_1 - U_2| = 0\ ,\) the system is linearly stable and the roll-up does not occur.
Effects of viscosity
The vortex sheet model is not a realistic representation of shear layers in nature. In reality, a boundary layer between which the velocity shear exists has a finite thickness and the basic flow \(U\) is a function with respect to \(y\ ;\) \(U=U(y)\ .\) KHI is a terminology that describes inviscid fluids, however, it is frequently used even in viscous fluids when the roll-up is found.
Orr-Sommerfeld equation
When viscosity is included in the system, the two-dimensional Navier-Stokes equations in vorticity form are given by \[\tag{13} \left[ \frac{\partial}{\partial t} + \frac{\partial \Psi}{\partial y}\frac{\partial}{\partial x} - \frac{\partial \Psi}{\partial x}\frac{\partial}{\partial y}\right]\nabla^2\Psi = -\frac{1}{R}\nabla^4 \Psi, \quad {\mathbf{ \Omega} }= -\nabla^2 \Psi {\mathbf{ e}_z}, \]
where \({\mathbf{ \Omega}}\) is vorticity field and the stream function \(\Psi(x,y,t)\) is related to the fluid velocity \({\mathbf{ u}}\) as \({\mathbf{ u}} = u{\mathbf{ e}_x} + v{\mathbf{ e}_y} = {\partial \Psi}/{\partial y} {\mathbf{ e}_x} - {\partial \Psi}/{\partial x} {\mathbf{ e}_y}\ ,\) in which \({\mathbf{ e}_x}\ ,\) \({\mathbf{ e}_y}\ ,\) and \({\mathbf{ e}_z}\) are unit vectors in the \(x\ ,\) \(y\ ,\) and \(z\) directions, respectively. Here, the Reynolds number \(R\) is defined by a length \(L\) and velocity scale \(V\) typical of the basic flow as \(R=LV/\nu\ ,\) where \(\nu\) is viscosity.
Decomposing the stream function \(\Psi\) into basic and perturbation terms
\[\tag{14}
\Psi(x,y,t) = \int U(y) dy + \psi(x,y,t),
\]
and substituting (14) into Eq. (13), one obtains the Orr-Sommerfeld equation \[\tag{15} \left[U-c \right]\left[ \frac{d^2 {\tilde \psi}}{d y^2} - k^2{\tilde \psi}\right] - \frac{d^2 U}{d y^2}{\tilde \psi} = \frac{1}{ikR}\left(\frac{d^2}{d y^2} - k^2 \right)^2{\tilde \psi}, \]
where \(c \equiv \omega/k = c_r + i c_i\) is the complex phase velocity associated with the complex frequency \(\omega\) \((\in \mathbb C)\ ,\) in which the real part \(c_r\) determines the wave speed and the imaginary part \(c_i\) determines the damping or growth rate of the disturbance. The function \({\tilde \psi}={\tilde \psi}(y)\) is related to the velocity components \(u\) and \(v\) through the perturbation stream function \(\psi(x,y,t) = {\rm Re}\left[{\tilde \psi}(y) {\rm e}^{i(kx - \omega t)} \right]\) as \[ u(x,y,t) = {\rm Re}\left[ \frac{d {\tilde \psi}}{dy}(y) {\rm e}^{i(kx - \omega t)}\right], \] \[\tag{16} v(x,y,t) = -{\rm Re}\left[ik{\tilde \psi}(y) {\rm e}^{i(kx - \omega t)}\right]. \]
When four boundary conditions are imposed, the Orr-Sommerfeld equation (15) forms an eigenvalue problem for the eigenfunction \({\tilde \psi}(y)\) with eigenvalues \(c_r\) and \(c_i\ .\) The system is unstable if \(c_i > 0\ .\) The solution of the Orr-Sommerfeld equation describes the stability problem in viscous flows such as flows of the boundary layer-type. The analytical solution of Eq. (15) is obtained by the WKBJ approximations (Nayfeh 1973), in which the solutions of the inner and outer viscous layers are connected using the "matched asymptotic expansions" (Drazin and Reid 2004). The solution (6) corresponds to the outer solution in the inviscid limit of \(R \to \infty\ .\) The detailed derivation of the Orr-Sommerfeld equation and its stability analysis are presented in the references (Drazin and Reid 2004, Huerre and Rossi 1998).
Roll-up in viscous flow
As described above, the vortex sheet is a model of infinitesimally thin layer. In realistic systems in which viscosity is included, a vortex sheet becomes a finite thickness vortex layer, and that enables us to regularize the problem past the curvature singularity. Tryggvason et al. (Tryggvason et al. 1991) numerically solved full viscous Navier-Stokes equations (13) and concluded that the viscous solution of a thin vorticity layer with high Reynolds number almost becomes the same one obtained by a vortex sheet model with relatively small regularization parameter \(\delta\ .\) Forbes et al. presented accurate numerical scheme for solving viscous flow without using the vortex sheet model (Chen and Forbes 2011) and found the roll-up of a thin viscous layer in a weakly compressible system (Forbes 2011). These results suggest that viscosity acts as a regularization effect in the roll-up.
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- M. A. Weissman Nonlinear wave packets in the Kelvin-Helmholtz instability Philos. Trans. Roy. Soc. A 290(1979), 639-685. (online)
Further reading
- A. J. Majda and A. L. Bertozzi "Vorticity and incompressible flow" Cambridge Universiy Press, 2002. ISBN 0-521-63057-6.
- P. G. Saffman, "Vortex Dynamics" Cambridge Universiy Press, 1992. ISBN 0-521-47739-5.
Internal references
- W. Cook Andrew and Youngs David (2009) Rayleigh-Taylor instability and mixing Scholarpedia, 4(2):6092. (online)
- Oleg Schilling and Jeffrey W. Jacobs (2008) Richtmyer-Meshkov instability and re-accelerated inhomogeneous flows Scholarpedia, 3(7):6090. (online)
