Arbitrary Density Ratio Research Articles

General vorticity‐streamfunction formulation for incompressible binary flow with arbitrary density ratio

AbstractThe classical vorticity‐streamfunction formulation (VSF) can avoid the difficulty in the calculation of pressure gradient term of the Navier Stokes equation via eliminating pressure gradient term from the theoretical basis. Within this context we propose a general VSF, together with redefined vorticity and streamfunction, so as to realize numerically stable and reliable simulations of binary fluids with an arbitrary density contrast. By incorporating the interface‐tracking phase‐field model based on the conservative Allen‐Cahn equation [Phys. Rev. E 94, 023311 (2016)], the binary flow simulation framework is established. Numerical tests are conducted using the Lattice Boltzmann method (LBM), which is usually regarded as an easy‐to‐use tool for solving the Navier–Stokes equation but generally suffers from the drawback of not being capable of enforcing incompressibility. The LBM herein functions as a numerical tool for solving the vorticity transport equation, the streamfunction equation, and the conservative Allen‐Cahn equation. Three two‐dimensional benchmark cases, i.e., the Capillary wave, the Rayleigh–Taylor instability, and the droplet splashing on a thin liquid film, are discussed in detail to verify the present methodology. Results show good agreements with both analytical predictions and literature data, as well as good numerical stability in terms of high density ratio and high Reynolds number. Overall, the general VSF inherits the intrinsic superiority of the classical VSF in enforcing incompressibility, and offers a useful and reliable alternative for binary flow modeling.

Quantitative theory for spikes and bubbles in the Richtmyer-Meshkov instability at arbitrary density ratios

It is well known that Goncharov's theory based on Layzer's approach for the Richtmyer-Meshkov instability provides good predictions for bubbles, but could give qualitatively incorrect predictions for spikes. This leads to the belief that Layzer's approach is not applicable to spikes. We show that, by incorporating the distinctive behaviors of spikes and bubbles properly, Layzer's approach is actually applicable to both spikes and bubbles, and can provide accurate predictions for both spikes and bubbles, for systems with arbitrary density ratios, and for the entire evolution process of unstable fingers.

Experimental study of bubble competition and spike competition in Richtmyer–Meshkov flows

Shock-tube experiments on various two-bubble and two-spike interfaces are performed to examine the dependence of bubble competition and spike competition on the initial spectra and density ratio of the interface. The differences in the influences of bubble competition and spike competition on the Richtmyer–Meshkov instability are highlighted for the first time. The bubble-competition effect is mainly dependent on the initial spectra of the two-bubble configuration. In contrast, the spike-competition effect is determined by both the initial spectra and density ratio. The extended buoyancy–drag model is adopted to explain the variation of the drag force imposed on the long-wavelength and short-wavelength structures as the initial conditions change. Based on the spectrum analysis, it is found that the constituent modes of two-bubble and two-spike interfaces have different contributions to the long-wavelength and short-wavelength perturbation growths. A generalised, nonlinear, analytical model is then established to quantify the bubble-competition effect and spike-competition effect considering arbitrary initial spectra and density ratio. The bubble-competition effect is believed to be stronger than the spike-competition effect at a high density ratio because it suppresses the high-frequency perturbation growth more evidently.

Quantitative theory for spikes and bubbles in the Richtmyer−Meshkov instability at arbitrary density ratios in three dimensions

It is known that conducting numerical simulations and experiments for the shock-induced Richtmyer–Meshkov instability in three dimensions is much more difficult and time-consuming than that in two dimensions. Therefore, theories can play a more important role in the study of three-dimensional Richtmyer–Meshkov instability. We present analytical formulas for predicting the behavior of growth rate and amplitude of fingers at the unstable Richtmyer–Meshkov interface. Our theory is for both spikes and bubbles, for the arbitrary density ratio between the two fluids, and for the entire development process from early to late times.

A multiscale methodology for small-scale bubble dynamics in turbulence

We formulate in this paper a multiscale numerical framework that handles small-scale bubble dynamics in turbulence. Our framework involves bubbles with arbitrary density ratios with the carrier phase. We use a Moving Reference Frame method that follows a bubble to deal with a fast rising of bubbles present at high density ratios between the phases. We use a Proportional Integral Derivative controller to handle an additional acceleration term in the governing equations that stems from the change of a coordinate system from a fixed to a non-inertial one. Our framework accounts for the fact that the dynamics of bubbles are significantly influenced by the unsteadiness of the small-scale turbulent liquid fluctuations that modify the bubble shapes and alter their motion. In addition, we improve and speed up, with at least two orders of magnitude in computational time, the numerical framework recently proposed by Milan et al. (2020). The developed numerical framework can capture processes occurring at time scales even smaller than the Kolmogorov times. It can be applied to droplets, bubbles or particle systems in both laminar and turbulent flows, using any general DNS technique that handles two-phase flows.

Mixed mass of classical Rayleigh-Taylor mixing at arbitrary density ratios

The accurate prediction of mixed mass induced by turbulent Rayleigh-Taylor mixing is of fundamental importance for many natural phenomena and engineering applications; however, no quantitative theory based on it has been established yet. In this study, we establish a quantitative theory to predict its evolution at arbitrary density ratios by combining the theory of density-ratio-invariant mean species profiles, which was recently developed by us, and the closure model for the turbulent fluctuations presented in this letter. The transformation formula between mixed mass and mixing width is obtained; in addition, the quantitative variation of normalized mixed mass with respect to the density ratio is derived. The theoretical results agreed very well with the direct numerical simulations at varied density ratios. The study sheds light on the quantitative prediction of mixed mass in practical engineering applications such as inertial confinement fusion.

A fictitious domain method for particulate flows of arbitrary density ratio

In this paper, our previous direct-forcing fictitious domain method is modified for particle-laden flows of arbitrary density ratio. The instability of original scheme for low density ratio is circumvented by a simple modification in the particle motion equations, with the key idea of increasing the coefficient of particle inertial term. Three schemes are proposed, which are shown to be equally accurate. The modified DF/FD method is extensively validated with various tests, and it is shown that the accuracy of our method is comparable to previous methods. In addition, our method is applied to the direct numerical simulations of the motion of finite-size particles (or bubbles) in a vertical turbulent channel flow of the density ratio 0.001. Our results indicate that the effects of particle-fluid density ratio on turbulent channel flow are negligibly small when the density ratio is smaller than unity and the gravity effect is not considered. The fluid Reynolds shear stress and the wall friction are enhanced in upflow where the particles are aggregated near the wall, whereas they are reduced in downflow where the particles move towards the channel center.

Quantitative Theory for the Growth Rate and Amplitude of the Compressible Richtmyer-Meshkov Instability at all Density Ratios.

Theoretical treatment of the Richtmyer-Meshkov instability in compressible fluids is a challenging task due to the presence of compressibility and nonlinearity. In this Letter, we present a quantitative theory for the growth rate and the amplitude of fingers in Richtmyer-Meshkov instability for compressible fluids based on the methods of the two-point Padé approximation and asymptotic matching. Our theory covers the entire time domain from early to late times and is applicable to systems with arbitrary fluid density ratios. The theoretical predictions are in good agreement with data from several independent numerical simulation methods and experiments.

Gravity currents produced by lock-release: Theory and experiments concerning the effect of a free top in non-Boussinesq systems

We present an experimental and theoretical analysis of non-Boussinesq inertial (large Reynolds number flow) gravity currents (GCs) flowing in rectangular cross-sections. Attention is focused on the effects of the open upper boundary, which become pronounced when the density contrast between the current and the ambient is significant (non-Boussinesq case). The study is conducted deriving first a two-layer shallow-water (SW) model for the release of a constant fluid volume into an ambient of given height with a free-surface boundary condition, with an arbitrary density ratio r between the ambient and the intruding fluid. The jump conditions at the front are provided by the novel extension of Benjamin (1968) analysis , subject to energy-dissipation and sub-critical speed requests, as detailed in Ungarish (2017). The resulting SW lock-release problem is solved via a finite difference approach. Lock-release experiments are then conducted in this configuration with r=0.837−0.950, full-depth and part-depth locks, and two different lock lengths. Experimental results obtained for the front position and speed of the intruding current, its thickness, and the free-surface depression of the ambient fluid (a signature of these experiments marking the position of the intruding front) agree, to various degrees, with their theoretical counterparts, with a better agreement for the front position and depression than for the other quantities, but demonstrating in all cases the consistency of the trend. Some of the experiments were repeated with a top-lid boundary condition, to discern its effect on the current propagation. These effects turn out to be relatively minor except for the obvious absence of interface-height depression. The range of density ratio r examined is further extended by conducting a numerical simulations with r=0.7 using a numerical commercial Computational Fluid Dynamics code (CFD) based on Reynolds Averaged Navier-Stokes equations (RANS). A comparison among SW theory, experiments, and the numerical simulations is conducted in terms of the trough behind the current nose, leading to an overestimation of theoretical results with respect to experiments. The length of the through adequately scales with the lock-length.

The lift on a small sphere in a linear shear flow near the interface of two immiscible fluids

AbstractAnalytical expressions have been derived which predict, to lowest order, the inertial lift and the lateral migration velocity of a rigid sphere translating and rotating in a linear shear flow field near the flat interface of two immiscible fluids. This asymptotic analysis is primarily based on the assumption that the two Reynolds numbers defined by the gap width between the interface and the sphere, the shear rate and the translational slip velocity with which the spherical particle moves parallel to the interface are small. Furthermore, the radius of the sphere is assumed to be small compared to the gap width. To leading order in this creeping flow regime, the linear Stokes equations are obtained and a symmetry argument can be used to show that the Stokes solution does not predict any lift force. The transverse force experienced by the sphere and its migration velocity are due to the small but finite inertial terms in the Navier‐Stokes equations, which can be studied by perturbation techniques. By applying a Green's function approach and matched asymptotic methods, which also incorporate the effects of the outer Oseen‐like flow regime, the three components comprising the lift velocity have been calculated in closed form: the one induced by the shear rate only, the purely slip induced one and the one due to the interaction of the slip velocity with the shear flow field. The thus obtained expressions for the case of two immiscible fluids with arbitrary density and viscosity ratios extend the results that already exist in the literature for other flow configurations, such as an unbounded shear flow field [1] or a wall‐bounded one, where the wall lies either within the leading order Stokes region [2] or in the outer Oseen region [3]. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Radial fingering under arbitrary viscosity and density ratios

The role of inertia in interfacial pattern formation in radial viscous fingering is studied for arbitrary viscosity and density ratios. A surprising effect of inertia is found: At finite Reynolds numbers the tip-splitting regime is replaced by a ``dendriticlike'' regime with side-branching structures.

A non-iterative immersed boundary method for spherical particles of arbitrary density ratio

In this paper an immersed boundary method with semi-implicit fluid–solid coupling for mobile particles of arbitrary density ratio is developed. The new scheme does not require any iterations to balance fluid forces and particle forces at the interface. A new formulation of the particle equations of motion is proposed which not only accounts for the particle itself but also for a Lagrangian layer surrounding the particle surface. Furthermore, it is shown by analytical considerations that the six equations for the linear and angular velocity of the spherical particle decouple which allows their sequential solution. On this basis a new time integration scheme is obtained which is unconditionally stable for all fluid–solid density ratios and enables large time steps, with Courant numbers around unity. The new scheme is extensively validated for various test cases and its convergence is assessed. An appealing issue is that compared to existing immersed boundary methods the new scheme only alters coefficients in the particle equations and the order of the steps, making it easy to implement in present codes with explicit coupling. This substantially extends the field of application of such methods.

A relativistic PIC model of nonlinear laser absorption in a finite-size plasma with arbitrary mass and density ratios

AbstractIn this paper, we have studied the nonlinear laser absorption in a finite-size plasma, using a fully kinetic relativistic 1½-dimensional particle-in-cell simulation code, and shown that the rate in which laser absorption increases depends on the mass ratio (δ = m+/m−) and the equilibrium density ratio (α = n0+/n0− = z−/z+), respectively. This can be attributed to the essential effects of these parameters on the nonlinear phenomena related to the laser absorption such as phase-mixing and laser-scattering process. We have indicated that the reduction of the defined mass and density ratios increases the absorption rate in the finite-size plasma as long as the laser scattering from the plasma is not considerable. Moreover, the study of kinetic phase-space distributions indicates in a specific range of mass and density ratios a large amount of laser energy leads to arise a population of charged particles having directed kinetic energy along the laser direction. These simulation results are expected to be relevant to the experiments on the intense laser–plasma interactions with applications to the particle acceleration and fast ignition concept as well as to the astrophysical environments.

Solution to Rayleigh-Taylor instabilities: Bubbles, spikes, and their scalings.

When a fluid pushes on and accelerates a heavier fluid, small perturbations at their interface grow with time and lead to turbulent mixing. The same instability, known as the Rayleigh-Taylor instability, operates when a heavy fluid is supported by a lighter fluid in a gravitational field. It has a particularly deleterious effect on inertial-confinement-fusion implosions and is known to operate over 18 orders of magnitude in dimension. We propose analytic expressions for the bubble and spike amplitudes and mixing widths in the linear, nonlinear, and turbulent regimes. They cover arbitrary density ratios and accelerations that are constant or changing relatively slowly with time. We discuss their scalings and compare them with simulations and experiments.

A monotonicity preserving conservative sharp interface flow solver for high density ratio two-phase flows

This paper presents a novel approach for solving the conservative form of the incompressible two-phase Navier–Stokes equations. In order to overcome the numerical instability induced by the potentially large density ratio encountered across the interface, the proposed method includes a Volume-of-Fluid type integration of the convective momentum transport, a monotonicity preserving momentum rescaling, and a consistent and conservative Ghost Fluid projection that includes surface tension effects. The numerical dissipation inherent in the Volume-of-Fluid treatment of the convective transport is localized in the interface vicinity, enabling the use of a kinetic energy conserving discretization away from the singularity. Two- and three-dimensional tests are presented, and the solutions shown to remain accurate at arbitrary density ratios. The proposed method is then successfully used to perform the detailed simulation of a round water jet emerging in quiescent air, therefore suggesting the applicability of the proposed algorithm to the computation of realistic turbulent atomization.

Reduced equations of motion of the interface of dielectric liquids in vertical electric and gravitational fields

The dynamics of the interface between two dielectric fluids in the presence of vertical electric and gravitational fields is studied theoretically. It is shown that, in the particular case where the rate of change of the electric field is proportional to the effective gravitational acceleration, a special flow regime can be realized for which the velocity and electric potentials are linearly dependent functions. This means that there exists a frame of reference in which liquids move along the electric field lines. We derive and analyze the corresponding reduced equations of motion of a liquid-liquid interface. For small density ratio, they turn into the equations describing the Laplacian growth. In the case of two spatial dimensions, we show that these equations determine the asymptotic behavior of the system. For arbitrary density ratios, the Laplacian growth equations adequately describe the initial (weakly nonlinear) stage of the interface instability development. The integrability of these equations makes it possible to investigate the evolution of nonlinear waves at the boundary and, in particular, to demonstrate the tendency to the formation of singularities (cusps).

Mathematical Modeling of Powder‐Snow Avalanche Flows

Powder‐snow avalanches are violent natural disasters which represent a major risk for infrastructures and populations in mountain regions. In this study we present a novel model for the simulation of avalanches in the aerosol regime. The second scope of this study is to get more insight into the interaction process between an avalanche and a rigid obstacle. An incompressible model of two miscible fluids can be successfully employed in this type of problems. We allow for mass diffusion between two phases according to the Fick's law. The governing equations are discretized with a contemporary fully implicit finite volume scheme. The solver is able to deal with arbitrary density ratios. Several numerical results are presented. Volume fraction, velocity, and pressure fields are presented and discussed. Finally, we point out how this methodology can be used for practical problems.

Competitions between Rayleigh–Taylor instability and Kelvin–Helmholtz instability with continuous density and velocity profiles

In this research, competitions between Rayleigh–Taylor instability (RTI) and Kelvin–Helmholtz instability (KHI) in two-dimensional incompressible fluids within a linear growth regime are investigated analytically. Normalized linear growth rate formulas for both the RTI, suitable for arbitrary density ratio with continuous density profile, and the KHI, suitable for arbitrary density ratio with continuous density and velocity profiles, are obtained. The linear growth rates of pure RTI (γRT), pure KHI (γKH), and combined RTI and KHI (γtotal) are investigated, respectively. In the pure RTI, it is found that the effect of the finite thickness of the density transition layer (Lρ) reduces the linear growth of the RTI (stabilizes the RTI). In the pure KHI, it is found that conversely, the effect of the finite thickness of the density transition layer increases the linear growth of the KHI (destabilizes the KHI). It is found that the effect of the finite thickness of the density transition layer decreases the “effective” or “local” Atwood number (A) for both the RTI and the KHI. However, based on the properties of γRT∝A and γKH∝1−A2, the effect of the finite thickness of the density transition layer therefore has a completely opposite role on the RTI and the KHI noted above. In addition, it is found that the effect of the finite thickness of the velocity shear layer (Lu) stabilizes the KHI, and for the most cases, the combined effects of the finite thickness of the density transition layer and the velocity shear layer (Lρ=Lu) also stabilize the KHI. Regarding the combined RTI and KHI, it is found that there is a competition between the RTI and the KHI because of the completely opposite effect of the finite thickness of the density transition layer on these two kinds of instability. It is found that the competitions between the RTI and the KHI depend, respectively, on the Froude number, the density ratio of the light fluid to the heavy one, and the finite thicknesses of the density transition layer and the velocity shear layer. Furthermore, for the fixed Froude number, the linear growth rate ratio of the RTI to the KHI decreases with both the density ratio and the finite thickness of the density transition layer, but increases with the finite thickness of the velocity shear layer and the combined finite thicknesses of the density transition layer and the velocity shear layer (Lρ=Lu). In summary, our analytical results show that the effect of the finite thickness of the density transition layer stabilizes the RTI and the overall combined effects of the finite thickness of the density transition layer and the velocity shear layer (Lρ=Lu) also stabilize the KHI. Thus, it should be included in applications where the transition layer effect plays an important role, such as the formation of large-scale structures (jets) in high energy density physics and astrophysics and turbulent mixing.

Embedding sharp interfaces within the lattice Boltzmann method for fluids with arbitrary density ratios

Current lattice Boltzmann methods for simulating two fluids create a diffuse interface between the fluids. In this work, we develop a novel technique for embedding sharp interfaces between fluids with unbounded density ratios for the LB method. Distribution functions streamed across an interface are transformed so that the receiving node is passed information corresponding to its fluid phase. Two different methods are employed to determine the transformation. The first uses analytical distribution functions from steady Poiseuille flow to determine the jump in moments of the distribution functions across the interface. The second uses approximate expansions of distribution functions to determine jumps in distribution functions. The accuracy and stability of the methods are examined in simulations of Poiseuille-Couette flows with an interface parallel to the walls. Both methods show linear convergence to the analytical solution.

Effect of Schmidt number on the structure and propagation of density currents

The results of a numerical study of two- and three-dimensional Boussinesq density currents are described. They are aimed at exploring the role of the Schmidt number on the structure and dynamics of density driven currents. Two complementary approaches are used, namely a spectral method and a finite-volume interface capturing method. They allow for the first time to describe density currents in the whole range of Schmidt number 1 ≤ Sc ≤ ∞ and Reynolds number 102 ≤ Re ≤ 104. The present results confirm that the Schmidt number only weakly influences the structure and dynamics of density currents provided the Reynolds number of the flow is large, say of O(104) or more. On the contrary low- to moderate-Re density currents are dependant on Sc as the structure of the mixing region and the front velocities are modified by diffusion effects. The scaling of the characteristic density thickness of the interface has been confirmed to behave as (ScRe)−1/2. Three-dimensional simulations suggest that the patterns of lobes and clefts are independent of Sc. In contrast the Schmidt number is found to affect dramatically (1) the shape of the current head as a depression is observed at high-Sc, (2) the formation of vortex structures generated by Kelvin–Helmholtz instabilities. A criterion is proposed for the stability of the interface along the body of the current based on the estimate of a bulk Richardson number. This criterion, derived for currents of arbitrary density ratio, is in agreement with present computed results as well as available experimental and numerical data.