Compressible Rayleigh Research Articles

A lattice Boltzmann model for thermal compressible flows at low Mach numbers beyond the Boussinesq approximation

A modified weakly compressible lattice Boltzmann model is developed for simulating low Mach number thermal compressible flows. Using the low Mach number approximation (LMNA) and splitting the pressure into two parts, namely thermodynamic and hydrodynamic pressure, the Navier–Stokes equations at low Mach number limits with variable density are recovered from the kinetic model. The fluid density is governed by the chosen equation of state. In order to satisfy the continuity equation at large temperature variations, which leads to large density differences, a suitable compensation for the finite divergence of the velocity field is introduced via a forcing term. This relationship correlates the total derivatives of the temperature and thermodynamic pressure to the velocity field divergence. The present model is successfully validated against well-known benchmarks. Numerical experiments are conducted to predict the critical Rayleigh number in a compressible Rayleigh–Benard convection test case, where the Boussinesq approximation is no longer valid due to the large temperature differences across the domain. The results are in good agreement with theory and previously published works. Moreover, two-dimensional natural convection heat transfer in a square cavity with large temperature differences is simulated and the results agree with available literature data. One interesting aspect of the presented algorithm is its capability to be extended to two-phase liquid-gas flows to model the gas phase as thermally compressible for low Mach numbers flows.

Compressible Rayleigh–Taylor turbulent mixing layer between Newtonian miscible fluids

Rayleigh–Taylor instability induced turbulence between two compressible miscible Newtonian fluids is studied in a strongly stratified configuration at a moderate Atwood number. A direct numerical simulation has been carried out with an auto-adaptive multidomain Chebyshev–Fourier–Fourier numerical method. The spatial resolution is increased up to $(9\times 100)\times 1000^{2}=900M$ collocation points. These numerical data are compared with those obtained from a simulation carried out at a lower Reynolds number and at the same Atwood number, and those obtained from a simulation carried out within the Boussinesq approximation at the same Reynolds number. A comprehensive data analysis is reported. Physical-variable mean profiles – density, concentration, temperature, entropy, velocity, vorticity, helicity and palinstrophy – are provided. Anisotropy is studied in the spectral space. The intermediate-scale isotropy and the small-scale anisotropy are exhibited for the scalars, i.e. concentration and temperature. Velocity is anisotropic at all scales but this anisotropy is more marked at small scales. The data are also analysed with the Favre-averaged equations. Sources of the turbulent kinetic energy, mass flux, root-mean-square density and energy equations are analysed. Compressibility effects are discussed in particular with the Kovàsznay-mode decomposition. A statistical study is reported where skewnesses, flatnesses and probability density functions (PDFs) are displayed and commented. A flow visualization is also given. Finally, the temperature field appears to be the slave of the mixing. This conclusion is drawn from the comparison of power spectra, anisotropy spectra, skewnesses, flatnesses, PDFs and correlation coefficients. There is however a significant time lag between the density and temperature evolution.

A Validation Study of the Compressible Rayleigh–Taylor Instability Comparing the Ares and Miranda Codes

The compressible Rayleigh–Taylor (RT) instability is studied by performing a suite of large eddy simulations (LES) using the Miranda and Ares codes. A grid convergence study is carried out for each of these computational methods, and the convergence properties of integral mixing diagnostics and late-time spectra are established. A comparison between the methods is made using the data from the highest resolution simulations in order to validate the Ares hydro scheme. We find that the integral mixing measures, which capture the global properties of the RT instability, show good agreement between the two codes at this resolution. The late-time turbulent kinetic energy and mass fraction spectra roughly follow a Kolmogorov spectrum, and drop off as k approaches the Nyquist wave number of each simulation. The spectra from the highest resolution Miranda simulation follow a Kolmogorov spectrum for longer than the corresponding spectra from the Ares simulation, and have a more abrupt drop off at high wave numbers. The growth rate is determined to be between around 0.03 and 0.05 at late times; however, it has not fully converged by the end of the simulation. Finally, we study the transition from direct numerical simulation (DNS) to LES. The highest resolution simulations become LES at around t/τ ≃ 1.5. To have a fully resolved DNS through the end of our simulations, the grid spacing must be 3.6 (3.1) times finer than our highest resolution mesh when using Miranda (Ares).

Two-equation and multi-fluid turbulence models for Rayleigh–Taylor mixing

This paper presents a new, improved version of the K–L model, as well as a detailed investigation of K–L and multi-fluid models with reference to high-resolution implicit large eddy simulations of compressible Rayleigh–Taylor mixing. The accuracy of the models is examined for different interface pressures and specific heat ratios for Rayleigh–Taylor flows at initial density ratios 3:1 and 20:1. It is shown that the original version of the K–L model requires modifications in order to provide comparable results to the multi-fluid model. The modifications concern the addition of an enthalpy diffusion term to the energy equation; the formulation of the turbulent kinetic energy (source) term in the K equation; and the calculation of the local Atwood number. The proposed modifications significantly improve the results of the K–L model, which are found in good agreement with the multi-fluid model and implicit large eddy simulations with respect to the self-similar mixing width; peak turbulent kinetic energy growth rate, as well as volume fraction and turbulent kinetic energy profiles. However, a key advantage of the two-fluid model is that it can represent the degree of molecular mixing in a direct way, by transferring mass between the two phases. The limitations of the single-fluid K–L model as well as the merits of more advanced Reynolds-averaged Navier–Stokes models are also discussed throughout the paper.

Turbulent convective flows in the solar photospheric plasma

The origin of the 22-year solar magnetic cycle lies below the photosphere where multiscale plasma motions, due to turbulent convection, produce magnetic fields. The most powerful intensity and velocity signals are associated with convection cells, called granules, with a scale of typically 1 Mm and a lifetime of a few minutes. Small-scale magnetic elements (SMEs), ubiquitous on the solar photosphere, are passively transported by associated plasma flows. This advection makes their traces very suitable for defining the convective regime of the photosphere. Therefore the solar photosphere offers an exceptional opportunity to investigate convective motions, associated with compressible, stratified, magnetic, rotating and large Rayleigh number stellar plasmas. The magnetograms used here come from a Hinode/SOT uninterrupted 25-hour sequence of spectropolarimetric images. The mean-square displacement of SMEs has been modelled with a power law with spectral index ${\it\gamma}$. We found ${\it\gamma}=1.34\pm 0.02$ for times up to ${\sim}2000~\text{s}$ and ${\it\gamma}=1.20\pm 0.05$ for times up to ${\sim}10\,000~\text{s}$. An alternative way to investigate the advective–diffusive motion of SMEs is to look at the evolution of the two-dimensional probability distribution function (PDF) for the displacements. Although at very short time scales the PDFs are affected by pixel resolution, for times shorter than ${\sim}2000~\text{s}$ the PDFs seem to broaden symmetrically with time. In contrast, at longer times a multi-peaked feature of the PDFs emerges, which suggests the non-trivial nature of the diffusion–advection process of magnetic elements. A Voronoi distribution analysis shows that the observed small-scale distribution of SMEs involves the complex details of highly nonlinear small-scale interactions of turbulent convective flows detected in solar photospheric plasma.

The effects of plasma diffusion and viscosity on turbulent instability growth

We perform two-dimensional simulations of strongly–driven compressible Rayleigh–Taylor and Kelvin–Helmholtz instabilities with and without plasma transport phenomena, modeling plasma species diffusion, and plasma viscosity in order to determine their effects on the growth of the hydrodynamic instabilities. Simulations are performed in hydrodynamically similar boxes of varying sizes, ranging from 1 μm to 1 cm in order to determine the scale at which plasma effects become important. Our results suggest that these plasma effects become noticeable when the box size is approximately 100 μm, they become significant in the 10 μm box, and dominate when the box size is 1 μm. Results suggest that plasma transport may be important at scales and conditions relevant to inertial confinement fusion, and that a plasma fluid model is capable of representing some of the kinetic transport effects.

The effect of high viscosity on compressible and incompressible Rayleigh–Plesset-type bubble models

Free oscillations of a single spherical gas bubble in glycerol have been examined numerically and experimentally at different ambient temperatures and pressures. The bubble was generated using a Q-switched Nd:YAG laser and the unsteady radius measurement was based on a shadowing technique of a He–Ne laser beam. The measurements were compared to computations obtained from two models, first taking into consideration the liquid compressibility and then assuming an incompressible liquid domain, respectively. In both cases the temperature fields inside and outside the bubble were computed by solving the energy equation in both phases as the thermodynamic processes have great importance to the bubble behavior. For high amplitude oscillations the incompressible model provides poor agreement with the measurements and the modeling of the liquid compressibility becomes necessary. In contrast to the standard method, a practical region of applicability for the incompressible approach was determined as a function of the instantaneous Mach and Reynolds numbers, rather than specifying a simple threshold Mach number.

Adaptive wavelet collocation method simulations of Rayleigh–Taylor instability

Numerical simulations of single-mode, compressible Rayleigh–Taylor instability are performed using the adaptive wavelet collocation method (AWCM), which utilizes wavelets for dynamic grid adaptation. Due to the physics-based adaptivity and direct error control of the method, AWCM is ideal for resolving the wide range of scales present in the development of the instability. The problem is initialized consistent with the solutions from linear stability theory. Non-reflecting boundary conditions are applied to prevent the contamination of the instability growth by pressure waves created at the interface. AWCM is used to perform direct numerical simulations that match the early-time linear growth, the terminal bubble velocity and a reacceleration region.

Numerical simulations of compressible Rayleigh–Taylor turbulence in stratified fluids

We present the results of our numerical simulations of the Rayleigh–Taylor turbulence, performed using a recently proposed (Sbragaglia et al 2009 J. Fluid Mech. 628 299, Scagliarini et al 2010 Phys. Fluids 22 055101) lattice Boltzmann method that can describe consistently a thermal compressible flow subjected to an external forcing. The method allowed us to study the system in both the nearly Boussinesq regime and the strongly compressible regime. Moreover, we show that when the stratification is important, the presence of the adiabatic gradient causes the arrest of the mixing process.

Lattice Boltzmann methods for thermal flows: Continuum limit and applications to compressible Rayleigh–Taylor systems

We compute the continuum thermohydrodynamical limit of a new formulation of lattice kinetic equations for thermal compressible flows, recently proposed by Sbragaglia et al. [J. Fluid Mech. 628, 299 (2009)]. We show that the hydrodynamical manifold is given by the correct compressible Fourier–Navier–Stokes equations for a perfect fluid. We validate the numerical algorithm by means of exact results for transition to convection in Rayleigh–Bénard compressible systems and against direct comparison with finite-difference schemes. The method is stable and reliable up to temperature jumps between top and bottom walls of the order of 50% the averaged bulk temperature. We use this method to study Rayleigh–Taylor instability for compressible stratified flows and we determine the growth of the mixing layer at changing Atwood numbers up to At∼0.4. We highlight the role played by the adiabatic gradient in stopping the mixing layer growth in the presence of high stratification and we quantify the asymmetric growth rate for spikes and bubbles for two dimensional Rayleigh–Taylor systems with resolution up to Lx×Lz=1664×4400 and with Rayleigh numbers up to Ra∼2×1010.

Compressible magnetic Rayleigh–Taylor instability in stratified plasmas: Comparison of analytical and numerical results in the linear regime

Magnetic Rayleigh–Taylor instability is addressed in compressible hydrostatic media. A full model is presented and compared to numerical results from a linear perturbation code. A perfect agreement between both approaches is obtained in a wide range of parameters. Compressibility effects are examined and substantial deviations from classical Chandrasekhar growth rates are obtained and confirmed by the model and the numerical calculations.

Using Linear Stability Analysis as a Tool to Evaluate Jet and Cavity Flow Control Situations

Flow control has the potential to improve the efficiency of aerospace systems. The design of a flow control system requires knowledge of flow phenomena including information on the evolution of naturally occurring disturbances in the flow. Natural flow disturbances can evolve at various frequencies and growth rates and can assume various mode shapes. Knowledge of the behavior of natural flow instabilities could aid actuator selection and strategy for flow control. In this paper we use linear stability analysis as a tool to analyze jet and cavity flow control situations. We present results from a linear stability study of the resonant acoustic mechanisms in subsonic impinging jets and subsonic cavity flows using the compressible Rayleigh equation. In an impinging jet, the subsonic instabilities are known to be responsible for the acoustic feedback, which is an essential link in setting up the resonances associated with this flow. The instabilities of the helical type are also relevant from the standpoint o...

Compressible Rayleigh–Taylor Instability with Preheat inInertial Confinement Fusion

The compressible Rayleigh–Taylor instability of accelerated ablationfront is analysed in consideration of the preheat effects, and thecorresponding eigen-problem is solved numerically using thefourth-order accurate two-point compact difference scheme. Both thegrowth rate and perturbation profiles are obtained, and the obtainedgrowth rate is close to the results of direct numerical simulation. Ourresults show that the growth rate is more reduced and the cutoffwave length becomes longer as preheat increases.

Asymptotic calculation of inviscidly absolutely unstable modes of the compressible boundary layer on a rotating disk

In this work a long-wavelength asymptotic approach is used to analyze the region of absolute instability in the compressible rotating disk boundary layer flow. Theoretically determined values of branch points for the occurrence of absolute instability in the compressible flow are shown to match onto the ones which are obtained via a numerical solution of the linear inviscid compressible Rayleigh equations.

Response to “Comment on ‘Compressible Rayleigh–Taylor instabilities in supernova remnants’” [Phys. Fluids 17, 069101 (2005)

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Comment on “Compressible Rayleigh–Taylor instabilities in supernova remnants” [Phys. Fluids 16, 4661 (2004)

Comment on “Compressible Rayleigh–Taylor instabilities in supernova remnants” [Phys. Fluids 16, 4661 (2004)

The Stability of a Compressible Rayleigh Layer

The aim of this work is to determine the linear stability of a compressible Rayleigh layer and to ascertain what role unsteady effects play. A Rayleigh layer is formed when an infinite flat plate is impulsively set in motion in its own plane with constant velocity beneath an initially quiescent fluid. When the fluid is compressible there is a motion both parallel and normal to the plate. The classical boundary-layer scaling is employed to determine solutions which are expressed in terms of a similarity variable and are valid for a large range of Mach, Prandtl and Reynolds numbers. Solutions are presented for both an adiabatic and iso-thermal temperature boundary condition at the plate. The temporal stability of the flow is considered by solving an Orr–Sommerfeld system: here the underlying flow is calculated at a certain time and the instantaneous stability to viscous travelling waves is determined. The stability is seen to be altered by changing the Mach number (an increase of which decreases the stability of the flow), and also by cooling and heating the wall. These results are limited by the fact that the growth of the layer in time is not taken into account. To include this we consider the large Reynolds number limit and use a triple-deck structure to determine the modes’ characteristics. The triple-deck approach is used to determine an asymptote to the lower branch of the neutral curve and unsteady effects can be included in a consistent manner. For the upper branch, however, a five-deck structure is required due to the fact that the critical layer is now distinct from the viscous sublayer. The upper-branch stability is only calculated to the first order which is sufficient to give an insight into the stability characteristics.

An Auto-Adaptive Multidomain Spectral Technique for Linear Stability Analysis: Application to Viscous Compressible Flows

An auto-adaptive multidomain pseudo-spectral technique is considered in order to solve the linear stability problem of viscous compressible flows. Both the locations of the interfaces and the parameters of the mappings in each subdomain are adapted by minimizing the H2ω-norm of the calculated solution. Such method provides automatically—this is the key point—the best polynomial interpolation of the basic state the stability of which is studied. It turns out that the whole procedure is needed to obtain reliable results. The method is first validated against results available in the literature (both viscous incompressible and inviscid compressible Rayleigh–Taylor configurations). The efficiency of the numerical method is illustrated with results on the linear stability of the compressible viscous diffusive Rayleigh–Taylor flow where no analytical or numerical results are available. New results showing the influence of stratification, viscosity, diffusity between species and thermal diffusivity are presented.

2D pseudo-spectral parallel Navier–Stokes simulations of compressible Rayleigh–Taylor instability

A 2D Fourier–Chebyshev pseudo-spectral numerical method for the full Navier–Stokes equations with a dynamical domain decomposition technique has been parallelized on a SPMD machine, a CRAY-T3E. The parallelism is based on the distribution of data among processors and on simultaneous computations on each subdomain. The SHMEM paradigm is used for communication between processors. Comparisons between the vectorial and the parallel versions of spectral derivatives with Fourier and Chebyshev expansions are presented. Performances versus the number of processors and collocation points are also studied. The numerical code is then used to simulate single and multimode Rayleigh–Taylor flows. The influence of the physical diffusion between species is emphasized on a single mode configuration. A simulation with 22 initial perturbation modes is also carried out and analyzed.

Absolute and Convective Instabilities in the Compressible Boundary Layer on a Rotating Disk

A numerical study has been undertaken to investigate the nature of inviscid instability of the three-dimensional compressible boundary layer flow due to a rotating disk. The compressible Rayleigh equation is integrated using a spectral Chebyshev-collocation method together with a fourth-order Runge–Kutta integrator. In the context of spatio-temporal stability analysis, the singularities of the resulting dispersion relation are determined and the ones that satisfy the Briggs–Bers pinching criterion have been selected. In certain finite parameter regions of eigenvalues (wave numbers and wave angles, for instance) it is found that by varying the Mach number, absolute instability occurs in the compressible boundary layer on a rotating disk. The range corresponding to the incompressible flow case given in Lingwood (1995) (e between 14.615° and 38.114°) is verified. The results of Cole (1995) are also verified. The overall effect of compressibility is to reduce the extent of absolute instability at higher Mach numbers. The effect of heating the wall is to enhance the absolute instability properties, however, cooling the wall is found to decrease greatly the region of absolute instability regime for the range of Mach numbers studied. It is also shown in this study that for non-insulated walls a direct spatial resonance of the eigenmodes is possible and this raises the possibility of large local algebraic growth of perturbations being important in some instances.