Simulation of Kelvin Helmholtz Instability with Flux-Corrected Transport Method

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1 Commun. Theor. Phys. (Beijing, China) 51 (2009) pp c Chinese Physical Society and IOP Publishing Ltd Vol. 51, No. 5, May 15, 2009 Simulation of Kelvin Helmholtz Instability with Flux-Corrected Transport Method WANG Li-Feng, 1,2 YE Wen-Hua, 2,3,4, FAN Zheng-Feng, 2 and LI Ying-Jun 1 1 China University of Mining and Technology (Beijing), Beijing , China 2 Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, P.O. Box , Beijing , China 3 Department of Physics, Zhejiang University, Hangzhou , China 4 Center for Applied Physics and Technology, Peking University, Beijing , China (Received July 25, 2008) Abstract The sixth-order accurate phase error flux-corrected transport numerical algorithm is introduced, and used to simulate Kelvin Helmholtz instability. Linear growth rates of the simulation agree with the linear theories of Kelvin Helmholtz instability. It indicates the validity and accuracy of this simulation method. The method also has good capturing ability of the instability interface deformation. PACS numbers: Ft, Py, Lf Key words: Kelvin Helmholtz instability, flux-corrected transport algorithm, numerical simulation 1 Introduction The Kelvin-Helmholtz (KH) instability arises from a horizontally stratified heterogeneous fluid when the different layers are in relative motion. [1] The KH instability is of great interest in many astrophysical and geophysical situations, ranging from the interaction of the solar wind with the earth s magnetosphere [2,3] and cometary tails [4] to the jets in nuclei extragalactic radio sources [5,6] and young stellar objects. [7] The KH instability is also of great importance in fields such as turbulence study, [8] small-scale mixing in Rayleigh Taylor (RT), and Richtmyer Meshkov (RM) instabilities, [9,10] etc. In the final stage of RT and RM instabilities, KH instability is initiated and results in mixing of fluid on small-scale, because the speed difference of lighter and heavier fluids along the sides of the spike is enlarged. [11] The appearance of KH instability aggravates the development of final nonlinearity of RT instability or RM instability, and quickens the process of fluid flock mixing round the interface. Therefore, analysis on the developmental rule of this instability is of significance in hydrodynamics and practical application. The direct numerical simulation (DNS) is the main method in analysis hydrodynamic instability. When the hydrodynamic instability enters the nonlinear stage, the fluid interface will twist severely. Lagrange method has the mesh intersecting difficulty. Nowadays, Euler method is extensively used in studying hydrodynamic instability. Hydrodynamic instability often occurs at the region where the density gradient is large, so high order Euler method should be used in simulations. The numerical method is often used in hydrodynamic instability including the fluxcorrected transport (FCT), [12,13] total variation diminishing (TVD) [14] and Godunov. [15,16] The TVD and Godunov methods are mainly used in solving conservative equations set and has difficult in solving nonconservative equations set. The FCT algorithm is the first specifically monotone, positivity-preserving technique developed at Naval Research Laboratory (NRL). The goal of FCT is to maximally utilize the flux computed from high-order scheme and without creating overshot or underdamping phenomena. Its fundamental idea is that we first defined fluxes difference of the high- and low-order scheme as antidiffusive fluxes, then use the solution deriving from low- order scheme to determine the correction to the antidiffusion coefficients, and further limit and amend the antidiffusive fluxes to make it satisfy local monotone. The FCT algorithm has high resolution for steep gradients. It is important in reactive flows, where the gradients at the interface in multiphase must be accurately represented. The FCT method computes generalized continuity equations singly, which has the characters of relative high accuracy, flexibility and utility, etc. The FCT method has been successfully used in simulation ablative RT instability and RM instability. [17,18] We select the FCT method to simulate KH instability. 2 FCT Algorithm The continuity equation can be written as the explicit three-point approximation, i = a i ρ n i+1 + b i ρ n i + c i ρ n i 1. (1) Conservation of ρ in Eq. (1) constrains the coefficients a i, b i, and c i by the condition a i + b i + c i = 1. (2) The project supported by the National Basic Research Program (973 Program) under Grant No. 2007CB815100, and the Research Fund for the Doctoral Program of Higher Education of China under Grant No Corresponding author, ye wenhua@iapcm.ac.cn

2 910 WANG Li-Feng, YE Wen-Hua, FAN Zheng-Feng, and LI Ying-Jun Vol. 51 Positivity of for { i } all possible positive profiles {ρ n i } requires that {a i }, {b i }, and {c i } be positive for all i. Equation (1), in conservative form, again becomes i = ρ n i 1 2 [ε i+1/2(ρ n i+1 + ρ n i ) ε i+1/2 (ρ n i+1 + ρ n i )] + [ν i+1/2 (ρ n i+1 ρ n i ) ν i+1/2 (ρ n i+1 ρ n i )], (3) where t ε i+1/2 υ i+1/2 x. (4) The {ν i+1/2 } are nondimensional numerical diffusion coefficients and ρ represents density, momentum or energy. The requirements of positivity and accuracy seem to be mutually exclusive, a positivity numerical algorithm must involve a certain amount of numerical diffusion to assure positivity and stability at each timestep. Therefore, in order to increase accuracy, antidiffusion stage is indispensable. Simple antidiffusion can generate ripple or new maxima and minima, so the antidiffusion fluxes should be corrected. The FCT algorithm employs the nonlinear flux-correction formula. These methods use the stabilizing ν = 1/2ε 2 diffusion where monotonicity is not threatened, and increase ν to values approaching ν = 1/2 ε when required to assure that the solution remains monotone. Different criteria are imposed in the same timestep at different locations on the computational grid according to the local profile of the physical solution. The dependence of the local smoothing coefficients ν on the solution profile makes the overall algorithm nonlinear. The FCT algorithm has four essential steps. They are as follows. (i) Applying finite volume method to one-dimensional continuity equation = 1 r α 1 r (rα 1 ρυ) 1 r α 1 r (rα 1 D 1 ) 1 D 3 + C 2 r α 1 r (rα 1 D 2 ) + C 3 r + D 4, (5) where D 1, C 2, D 2, C 3, D 3, and D 4 are various form source terms and α = 1, 2, 3 represent Cartesian or planar geometry, cylindrical geometry and spherical geometry, respectively. It can be written as discrete scheme, Λ n i ρ T i = Λ n i ρ n i t i+1/2 A i+1/2 υ i+1/2 + t i 1/2 A i+1/2 υ i+1/2 + γ i+1/2 Λ i+1/2 (ρ n i+1 ρ n i ) γ i 1/2 Λ i 1/2 (ρ n i ρn i 1 ) 1 2 ta i+1/2(d 1,i+1 + D 1,i ) ta i+1/2(d 1,i1 + D 1,i 1 ) tc 2,i[A i+1/2 (D 1,i+1 + D 1,i ) A i 1/2 (D 2,i + D 1,i 1 )] tc 3,i(A i+1/2 + A i 1/2 ) (D 3,i+1 D 3,i 1 ) + tλ n i D 4, (6) where {Λ i }, {A i+1/2 }, and {υ i+1/2 } are the basic cell volumes, the interface areas and velocity difference of fluid and cell, respectively. This step has the effect of stability and its result is used to compute uncorrected antidiffusive fluxes. (ii) The diffusion stage of this FCT algorithm also includes the cell volume change when the grid is moving, Λ n+1 i ρ i = Λ n i ρ T i + ν i+1/2 Λ i+1/2 (ρ n i+1 ρ n i ) ν i 1/2 Λ i 1/2 (ρ n i ρ n i 1). (7) The diffusion coefficients can be chosen to reduce phase errors. This step can offset some scheme viscosity. (iii) The diffusion has decreased the accuracy, and the antidiffusion skill should be taken. The transported but not diffused values, {ρ T i } are used to calculate the raw, uncorrected antidiffusive fluxes, f ad i+1/2 = µ i+1/2λ i+1/2 (ρ T i+1 ρ T i ). (8) (iv) The antidiffusion stage should not introduce oscillation or nonphysical solutions. Fluxes corrected is indispensable. The corrected antidiffusion flux is f c i+1/2 = S i+1/2 max{0, min[s i+1/2 Λ n+1 i+1 ( ρ i+1 ρ i ), f ad i+1/2, S i+1/2λ n+1 i ( ρ i+1 ρ i )]}, (9) where S i+1/2 = sign ( ρ i+1 ρ i ). The final density at the new time is i = ρ i 1 (fi+1/2 c fc i 1/2 ). (10) Λ n+1 i The antidiffusion stage should not generate new maxima or minima in the solution, nor accentuate already existing extrema. The accuracy of the phase error impact the distortion of fluid wave profile. In simulation, we find that the fourth-order phase error has more distortion in computing the instability interface. Here, we select the vales of µ, ν, and γ that have the sixth-order accurate phase error and find that the resolution of the instability interface has been greatly improved, ν i+1/2 = ε2 i+1/2, γ i+1/2 = ε2 i+1/2, µ i+1/2 = ε2 i+1/2. (11) The FCT algorithm is a high-order scheme in solving generalized continuity equations. The important properties of FCT are that it is a high-order, monotone, conservative, positivity preserving algorithm. This means that the algorithm is accurate and resolves steep gradients, allowing grid scale numerical resolution. 3 Simulation Method In Sec. 2, we described the monotone FCT algorithm for integrating a single continuity equation. We now extend the approach to solving coupled continuity equations.

3 No. 5 Simulation of Kelvin Helmholtz Instability with Flux-Corrected Transport Method 911 For the two-dimensional hydrodynamic equation sets, we apply timestep splitting scheme to create a multidimensional monotone calculation. The ideal two-dimensional Euler equation sets in Cartesian (x-y) geometry are: = u x υ u υ = uu x uυ x, = uυ x υυ + P)u x The pressure and energy are related by (E + P)υ y. (12) E = ε ρ(u2 + υ 2 ), (13) where ε P/(γ 1). The right sides of Eqs. (12) are separated into two parts, the x-direction terms and the y-direction terms. This arrangement in each of the four equations separates the x-derivatives and the y-derivatives in the divergence and gradient terms into parts that can be treated sequentially by a general one-dimensional continuity equation solver. Each x-direction column in the grid is integrated using the one-dimensional FCT algorithm to solve the four coupled continuity equations (12) from time t + t to. The x-direction split-step equations to be solved are = u x, υ = uυ x, u = uu x P x, + P)u x. (14) Equations (14) are in the form of the general continuity equation (5) with for α = 1 planar geometry. Because the x gradients and fluxes are being treated together, the one dimensional integration connects those cells which are influencing each other through the x-component of convection. The changes due to the derivatives in the y-direction must now be included. This is done in a second split step of one-dimensional integrations along each x-column, = υ υ = υυ u = uυ + P)υ y, (15) where α = 1 in Eq. (5) for planar geometry. The x and y integrations are alternated, each pair of sequential integrations constituting a full convection timestep. Thus a single optimized algorithm for a reasonably general continuity equation can be used to build up multidimensional fluid dynamics models. This approach is second-order accurate. In order to get the second-order accuracy in time, we employ the second-order Runge Kutta method. Solving Eqs. (14) or (15) is best done by determining the timestep, then integrating from the old time t o forward a half timestep to t + t/2, and then integrating from t o to the full timestep t + t. The results of the halfstep integration are used to evaluate time-centered spatial derivatives and fluxes. In our simulation, the fine uniform grids cluster near the vicinity of the interface and the width of grids on both sides of fine uniform grids is gradually enlarged. Typical initial perturbations are a superposition of modes. We initiate each mode at time t = 0 by taking a interface perturbation in the form of η(x, t = 0) = η 0 cos kx, (16) and apply periodic boundary conditions at x-direction, where k is the perturbation wave number, and η 0 is the initial perturbation amplitude. We analyze the spectrum by first calculating the area density integral ρ(x, y)dy, and then take the discrete Fourier transformation. There are two reasons for this: First of all, the direct measurement of ICF laser light on the background of positive photographic is the density perturbation amplitude. [19] In order to compare with the experimental results, the density perturbation amplitude must also be given in numerical simulation. Moreover, at the linear stage and the weakly nonlinear stage, the interface instability amplitude is small, and sometimes it is difficult to observe directly, but the density perturbations amplitude can give quantitative results. 4 Simulation Results and Analysis We program a two-dimensional hydrodynamic code applying the sixth-order accurate phase error FCT algorithm. In linear stage, there is analytic solution to compare. In nonlinear stage, we show the physical interface. 4.1 Linear Growth Rate of Single Mode KH Instability At linear stage, if we give a perturbation at the initial, the perturbation amplitude will satisfy this formula η = 1 2 η 0[exp(γt) + exp( γt)], γ = k (0) u A u (0) ρa ρ B B, ρ A + ρ B where η 0, γ, ρ i, and u (0) i are the initial perturbation amplitude, linear growth rate, density and velocity of the two layers, respectively. The term of exp( γt) will attenuate as the time increase. At relative late time in linear stage, the perturbation amplitude will cater the formula of η = 1/2η 0 exp(γt). In order to verify our code quantitatively, we introduce a velocity potential perturbation to depress the development of the terms of exp( γt). In fact, in our DNS, the terms of exp( γt) don not grow at all. In the DNS, the

4 912 WANG Li-Feng, YE Wen-Hua, FAN Zheng-Feng, and LI Ying-Jun Vol. 51 upper fluid has a velocity of 2 cm µs 1 along x-axis and a density of 0.1 g cm 3. The lower fluid has a velocity of 2 cm µs 1 along the negative x-axis and a density of 1 g cm 3. The initial perturbation amplitudes are within 0.1 µm 0.2 µm. We show the comparison results in Fig. 1. heavier fluids is lost. At last, the perturbation enters the strong nonlinear stage. From the Figures, we can see that our code has good ability in capturing instability interface. Fig. 3 The physical interface at µs. Fig. 1 The linear growth rate. Our DNS gives almost the exact results and the relative error are within 2%, which means that our code is reliable in simulating KH instability in incomprehensible fluid. 4.2 The Onlinear Growth of Single Mode KH Instability When the perturbation amplitude grows to a certain amount, the perturbation enters into nonlinear phase.in Fig. 2, we show the amplitude of fundamental mode at different time. Form Fig. 2, we can see that the perturbation enters the nonlinear phase at about µs. Fig. 4 The physical interface at µs. Fig. 5 The physical interface at µs. Fig. 2 The amplitude of fundamental mode vs. time. We give the physical interface at the time of µs, µs, and µs in Figs. 3 5, respectively. At early stage, the initial sinusoidal perturbation of the interface between the fluids grows exponentially in time. As the amplitude of the instability becomes larger, nonlinear effects become important and a great deal of higher harmonics is initiated and the symmetry between lighter and 5 Conclusions There exits the large interface deformation difficulty in KH instability. In this paper, we introduce the FCT algorithm and program a two-dimensional hydrodynamic code to simulate the KH instability. The DNS results of linear growth rates agree with the linear theories of KH instability. At the nonlinear phase, the physical interface is given, which indicate that our code can well capture the instability interface. Our DNS indicate that the FCT algorithm is suitable for computing the KH instability in incomprehensible fluids and our two-dimensional hydrodynamic code is reliable in the simulation of KH instability.

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