Role of non-zero bulk viscosity in three-dimensional Rayleigh-Taylor instability: Beyond Stokes’ hypothesis

Highlights

• The present work questions the validity of the constitutive relation, i.e. the Stokes’ hypothesis for compressible, three-dimensional (3D) Rayleigh-Taylor instability (RTI).

• A model is used to approximate the bulk viscosity of air, previously reported for 2D RTI in Roles of bulk viscosity on Rayleigh-Taylor instability: Non-equilibrium thermodynamics due to spatio-temporal pressure fronts. Phys. Fluids, 29, 019101 (2016).

• It is shown that including bulk viscosity into the formulation is essential to capture all pertinent flow physics for 3D RTI, particularly in the onset stage.

• The mixing layer height and growth rates are compared against existing experimental estimates, showing a very good match in cases where non-zero bulk viscosity has been used.

Abstract

Three-dimensional direct numerical simulations (DNS) of Rayleigh-Taylor instability (RTI) at the interface of two masses of air with a sharp temperature gradient of 149 K are performed by solving the compressible Navier-Stokes equation (NSE). The flow is studied in an isolated box with non-periodic walls along the three directions. A non-conducting interface separating the two air masses is impulsively removed at the onset of the instability. No external perturbation has been used at the interface to instigate the instability at the onset, corresponding to practical scenarios in experiments. Computations have been carried out for the two configurations reported by Read (Experimental investigation of turbulent mixing by Rayleigh-Taylor instability, Physica D. 12, 45–58 (1984)). The compressible formulation is free from the Boussinesq approximation commonly used for solving the incompressible NSE. The role of non-zero bulk viscosity is quantified by using a model from acoustic attenuation measurements for the bulk viscosity of air. Effects of Stokes’ hypothesis on the onset of RTI and the growth of mixing layer are reported. The incipient stage is shown to have a strong dependence on the constitutive relation used. The small-scale billowing motion is only observed for non-zero bulk viscosities. Following this stage, the growth rates for bubbles and spikes in the mixing layer are found to be underpredicted by 12% with the use of Stokes’ hypothesis. The results imply that the evolution of RTI from the onset to the fully turbulent regime is best captured by using non-zero values of the bulk viscosity.

Introduction

The Rayleigh-Taylor instability (RTI) is a buoyancy-driven event that occurs when the equilibrium condition of a heavier (colder) fluid resting atop a lighter (hotter) one is studied in the presence of downward-acting gravitational force. The unstably stratified system has a large initial temperature difference, thus one cannot use the Boussinesq approximation, which is commonly adopted for problems of mixed convection. The fluid system starts from a static condition, but convection sets in due to creation of vortices by the baroclinic source term [1], $-\frac{1}{{\rho }^2}(\nabla p\times \nabla \rho )$. The source term is triggered by the misalignment in the pressure and density gradients due to background disturbances in the flow. This demonstrates that the equilibrium arrangement is not stable and the instability is triggered by unsettling moment on any control volume at the interface caused by buoyancy, once the insulated partition separating the fluid masses is removed. The vorticity generated by the instability is initially located near the interface between the two fluids, but for miscible fluids the region over which there are large density and temperature gradients increases with time, and so does the region generating vorticity.

An important application of RTI is in the design of capsules for inertial confinement fusion [2]. In this process, a capsule containing the fuel mixture is bombarded with energy by lasers to initiate the fusion reaction. RTI occurs at two instances in this operation: (i) during the initial implosion of the target and (ii) deceleration between the high temperature gas mixture and outer, colder layer of ice. Eventually, turbulent mixing caused by RTI brings cold fuel from an outer layer to the hot centre, a process which may suppress ignition altogether [3]. Thus, an understanding of the underlying RTI is important to design more efficient capsules. Other examples of RTI include the mushroom-shaped clouds formed in volcanic eruptions, nuclear explosions and in oceanography.

Rayleigh [4] was the first to study RTI between fluids with different densities in his famous experimental work. Theoretical studies by Taylor [5] were performed almost 70 years later for two inviscid incompressible fluids separated by a horizontal interface, with small sinusoidal perturbations. This was later expanded to include effects of viscosity [6]. These focused on RTI from a single-mode perturbation analysis point-of-view.

Interest in RTI was revitalized with experiments [7], [8] investigating the evolution of RTI due to small, random perturbations which is more relevant for practical applications. One such experiment by Read [7] visualized the mixing process in RTI in two three-dimensional (3D) configurations. This experiment confirmed that the instability follows quadratic growth in time, which was later shown numerically by Youngs [9]. Gravitationally accelerated experiments [10], [11], [12], [13] were performed, where a tank with an initially heavy over light fluid configuration is separated by a horizontal barrier that is removed, thus creating RTI. Unlike other experiments which employed high acceleration to trigger RTI, these experiments allowed the analysis of the interior of the mixing layer. In the present compressible flow simulations, we use the same configuration, i.e. of a heavy air mass resting on top a lighter air mass, initially separated by an insulated barrier without any external artificial perturbation to trigger RTI. The difference in density is brought about by imposing a large temperature difference. A recent experimental study [14] determines the effects of forced experiments vis-a-vis unforced ones on the development of RTI. Comprehensive reviews of the basic properties of the flow, turbulence and mixing induced by RTI are provided by Zhou [15], [16].

Most numerical simulations for RTI have been performed with weak stratification parameters [17] by solving either the Euler [9], [18], [19] or the NSE [17], [20], [21], [22]. Youngs [9] solved two-dimensional (2D) Euler equations to investigate the turbulent mixing due to RTI and found the mixing layer to be insensitive to the initial condition, which is not the case in practical flow scenarios. Tryggvason [18] employed two Lagrangian-Eulerian vortex methods to simulate RTI for inviscid fluids of different densities in a rectangular box with periodic boundary conditions. The onset of RTI is at the juncture of the interface with the side-walls of the box, as this is the location where the baroclinic source attains its maximum value [1], [7], [20], [21] and thus, use of periodic boundary conditions [19], [23] disrupts the onset of RTI. Reckinger et al. [17] simulated an infinite domain RTI such that the pressure waves approaching the boundaries did not reflect into the domain and interact with and disturb the growth of the instability. Furthermore, the authors added a buffer layer to damp pressure waves in the vertical direction. Here, we will impose non-periodic, no-slip boundary conditions on the walls to ensure that we are studying an isolated system which does not artificially attenuate disturbances, as has been previously reported for studying the non-equilibrium thermodynamics of RTI [20], [21], [24].

Large eddy simulations are carried out by Cook et al. [22] to solve the 3D incompressible equations for RTI and a relationship for the growth rate of the mixing layer was proposed. Monotone implicit large eddy simulations using a finite-volume technique was implemented for the study of the influence of initial perturbation on turbulent 3D single-mode [19] and multi-mode [23] RTI. Olson and Cook [25] in their compressible RTI simulations report strong compressibility effects via the formation of shock waves in the upper heavy fluid. Total entropy of RTI in an isolated system has been studied in a non-equilibrium framework [20], [24], [26], [27] using high-resolution dispersion relation preserving schemes [28] with compressible flow formulation. Sengupta et al. [29] investigated effects of error metrics specifically identified for simulating RTI. The authors reported a difference in the onset of RTI, with computations for two different CFL numbers, and traced the difference to a very seemingly insignificant difference in the value of numerical amplification factor for these two CFL numbers. Evolution of RTI is very sensitive to the imposed density stratification and initial conditions as well as to the definitions of the mixing layer widths [30], [31]. Liang et al. [32], [33] conducted 3D study of RTI with low Rayleigh-Taylor instability (RTI) in a long square duct using a multiple-relaxation-time lattice Boltzmann models to investigate Reynolds number effects on the interfacial dynamics.

The present study deals with the evolution of RTI in a confined box with non-periodic boundary conditions. The inhomogeneous fluid system consists of two air masses with a temperature difference of 149 K, separated by a non-conducting interface, placed at the centre of the box at the onset of the numerical experiment. The large temperature difference does not allow the use of Bousinessq approximation as it would induce large error. Mikaelian [34] studied the Bousinessq approximation for RTI and found that it underpredicts bubbles (fluid structures of light fluid growing into heavier fluid) by about $14\%$. Here, we will not make use of the Boussinesq approximation. As we compute RTI for single phase flow of air, the problem of incorporating species transport-diffusion equation is also circumvented. Although, here we have considered a zero-thickness interface, there have been studies showing the dependence of RTI evolution on thickness of the interface [35], [36], [37]. In experimental investigation of RTI, Read [7] studied the evolution of instability without any interface perturbations, which is relevant to practical problems. Here, we attempt to validate the present computations against these experimental results and hence we also do not introduce any external perturbations in the computation and use no-slip boundary conditions for all the walls of the domain.

The utility of the Stokes’ hypothesis [38] has often been questioned in fluid dynamic and thermodynamic communities [39], [40], [41], [42], [43], [44], [45]. Here, we will perform computations without this hypothesis, incorporating the non-zero values of bulk viscosity (${\mu }_b$) obtained by regression analysis of the sound dispersion absorption data and acoustic attenuation measurements of Ash et al. [46] by solving the 3D compressible NSE. The role of bulk viscosity has been explored for 2D compressible flow simulations of RTI [21] and for transonic flow over airfoils [47], showing the unambiguous need for operating without the Stokes’ hypothesis. Here, we will explore non-zero bulk viscosity effects for the 3D RTI problem for the first time and determine whether Stokes’ hypothesis is a valid assumption for buoyancy driven flows, such as in RTI.

The paper is formatted in the following manner. In the next section, the formulation of the RTI problem is described with the introduction of governing equations, auxiliary conditions and numerical methodology. In Section III, the computed solution with Stokes’ hypothesis is compared against the experimental results of Read [7]. The method of incorporating bulk viscosity into the governing NSE by using results of Ash et al. [46] is described in Section IV. The computed solution is validated with the experimental results [7] by using this method. In Section V, the RTI problem in a 2D arrangement of the experiment [7] is computed without Stokes’ hypothesis and numerical validation is performed. The significant physical events during the evolution of RTI are explored in Section VI. In Section VII, the characteristics of the turbulent mixing layer are provided and comparisons with existing experiments are made. The paper closes with a summary and conclusions section.