Baolin Tian

Evolution of mixing width induced by general Rayleigh-Taylor instability

A theory determining the evolution of general Rayleigh-Taylor mixing fronts is established to reproduce firstly all of the documented experiments conducted for diverse acceleration histories and all density ratios. The theory is established in terms of the fundamental conservation and symmetry principles, with special consideration given to the symmetry breaking of the density fields occurring in actual flows. The results reveal the sensitivity/insensitivity of the evolution of a mixing front neighbouring light/heavy fluid to the degree of symmetry breaking, and also explain the distinct evolutions in two experiments with the same configurations.Turbulent mixing induced by Rayleigh-Taylor (RT) instability occurs ubiquitously in many natural phenomena and engineering applications. As the simplest and primary descriptor of the mixing process, the evolution of mixing width of the mixing zone plays a notable role in the flows. The flows generally involve complex varying acceleration histories and widely varying density ratios, two dominant factors affecting the evolution of mixing width. However, no satisfactory theory for predicting the evolution has yet been established. Here a theory determining the evolution of mixing width in general RT flows is established to reproduce, first, all of the documented experiments conducted for diverse (i.e., constant, impulsive, oscillating, decreasing, increasing, and complex) acceleration histories and all density ratios. The theory is established in terms of the conservation principle, with special consideration given to the asymmetry of the volume-averaged density fields occurring in actual flows. The results reveal the sensitivity or insensitivity of the evolution of a mixing front of a neighboring light or heavy fluid to the degree of asymmetry and thus explain the distinct evolutions in two experiments with the same configurations.

Preventing numerical oscillations in the flux-split based finite difference method for compressible flows with discontinuities

In simulating compressible flows with contact discontinuities or material interfaces, numerical pressure and velocity oscillations can be induced by point-wise flux vector splitting (FVS) or component-wise nonlinear difference discretization of convection terms. The current analysis showed that the oscillations are due to the incompatibility of the point-wise splitting of eigenvalues in FVS and the inconsistency of component-wise nonlinear difference discretization among equations of mass, momentum, energy, and even fluid composition for multi-material flows. Two practical principles are proposed to prevent these oscillations: (i) convective fluxes must be split by a global FVS, such as the global Lax-Friedrichs FVS, and (ii) consistent discretization between different equations must be guaranteed. The latter, however, is not compatible with component-wise nonlinear difference discretization. Therefore, a consistent discretization method that uses only one set of common weights is proposed for nonlinear weighted essentially non-oscillatory (WENO) schemes. One possible procedure to determine the common weights is presented that provided good results. The analysis and methods stated above are appropriate for both single- (e.g., contact discontinuity) and multi-material (e.g., material interface) discontinuities. For the latter, however, the additional fluid composition equation should be split and discretized consistently for compatibility with the other equations. Numerical tests including several contact discontinuities and multi-material flows confirmed the effectiveness, robustness, and low computation cost of the proposed method.

Formula for growth rate of mixing width applied to Richtmyer-Meshkov instability

The mixing zone width and its growth rate are of great significance in the study of the Richtmyer-Meshkov instability (RMI). In this paper, a formula for the growth rate of the mixing width is proposed for analysis of the RMI-induced mixing process. A new definition of the mixing width h , based on the mass fraction ϕ, is used to derive the formula of the growth rate of the mixing width, h . In the derivation, the velocity field and the diffusion term are concisely introduced into the formula by using the mass equation and mass fraction equation. This formula is used together with two-dimensional (2D) and three-dimensional (3D) numerical data to quantitatively study the effects of compressibility and the diffusion process on the development of the RMI. The results based on our simulations show the following. After a shock, the magnitudes of the contributions of compressibility and diffusion to h increase initially, and in the middle stage of the RMI, they appear to attain a maximum value, around 10%...

Characteristic-based and interface-sharpening algorithm for high-order simulations of immiscible compressible multi-material flows

The present work focuses on the simulation of immiscible compressible multi-material flows with the MieGrneisen-type equation of state governed by the non-conservative five-equation model [1]. Although low-order single fluid schemes have already been adopted to provide some feasible results, the application of high-order schemes (introducing relatively small numerical dissipation) to these flows may lead to results with severe numerical oscillations. Consequently, attempts to apply any interface-sharpening techniques to stop the progressively more severe smearing interfaces for a longer simulation time may result in an overshoot increase and in some cases convergence to a non-physical solution occurs. This study proposes a characteristic-based interface-sharpening algorithm for performing high-order simulations of such flows by deriving a pressure-equilibrium-consistent intermediate state (augmented with approximations of pressure derivatives) for local characteristic variable reconstruction and constructing a general framework for interface sharpening. First, by imposing a weak form of the jump condition for the non-conservative five-equation model, we analytically derive an intermediate state with pressure derivatives treated as additional parameters of the linearization procedure. Based on this intermediate state, any well-established high-order reconstruction technique can be employed to provide the state at each cell edge. Second, by designing another state with only different reconstructed values of the interface function at each cell edge, the advection term in the equation of the interface function is discretized twice using any common algorithm. The difference between the two discretizations is employed consistently for interface compression, yielding a general framework for interface sharpening. Coupled with the fifth-order improved accurate monotonicity-preserving scheme [2] for local characteristic variable reconstruction and the tangent of hyperbola for the interface capturing scheme [3] for designing other reconstructed values of the interface function, the present algorithm is examined using some typical tests, with the MieGrneisen-type equation of state used for characterizing the materials of interest in both one- and two-dimensional spaces. The results of these tests verify the effectiveness of the present algorithm: essentially non-oscillatory and interface-sharpened results are obtained.

Consistent implementation of characteristic flux-split based finite difference method for compressible multi-material gas flows

In order to prevent velocity, pressure, and temperature spikes at material discontinuities occurring when the interface-capturing schemes inconsistently simulate compressible multi-material flows(when the specific heats ratio is variable),various non-conservative or quasi-conservative numerical models have been proposed. However, designing a consistent numerical algorithm, especially using the high-order characteristic flux-split based finite-difference method (CFS-FDM) is still an open question. In this study, a systematical analysis of previous algorithms of the consistent implementing the high-order CFS-FDM for such flows is performed, and the reasons of special treatments in these algorithms are revealed. Based on this analysis, a new general numerical methodology that successfully avoids any special treatments as those required in previously reported algorithms, is derived. In this new algorithm, we rewrite the non-conservative term as a conservative term with a source term containing velocity divergence. By consistently treating the advection velocity in the conservative term and velocity divergence in the source term by imposing a new additional criterion, specifically, that a multi-fluid algorithm should have the ability of maintaining a pure single-fluid, we finally derive a new general algorithm that does not need any special treatment, and is very convenient to implement. The results of some benchmark tests show that the final algorithm not only maintains the velocity, pressure, and temperature equlibria, but is also suitable for problems regarding the interaction of interfaces and strong shock and rarefaction waves.

Characteristics of turbulent mixing at late stage of the Richtmyer-Meshkov instability

We present a model for the mixing width, h, at self-similar stage of the Richtmyer-Meshkov instability (RMI). The derivation of the model is based on the formula of the growth rate of mixing width [Gao et al., Phys. Fluids 28, 114101, (2016)], which is essentially equals to the first principle (Navior-Stokes equation). The model predicts that, in the self-similar mixing stage of the RMI h2 is linearly proportional to time. This is substantially different from the classical h∼τ𝜃 description. The linearity is validated by various experimental and numerical data. The exponent, 𝜃, in the classical relation is also discussed according to the present model.