Fluids Of Different Densities Research Articles

Plasma Instability Modeling With Mixed Wave/Diffusion Field Behavior

Recent efforts have extended fluid dynamic modeling to include low-electrical-conductivity (wavelike) electromagnetic field effects in plasmas. Here, we present results of a new study to explore and illustrate the capability of resolving such effects in the development of plasma instabilities. Recent work[1, 2] has explored the development of an efficient numerical procedure for simulating the coupled fluid dynamic and full electromagnetic (Maxwell) equations. This system of equations is more general than the resistive magnetohydrodynamic (MHD) model. The physical advantage to this system is that the inclusion of the displacement current and charge separation effects extends fluid dynamic modeling of plasmas to include regions of extremely low (or zero) electrical conductivity, where the electromagnetic fields possess a wave nature. Capturing the correct physical behavior of plasma instabilities in the limit of extremely low electrical conductivity is of significance for modern aerospace applications, including energy bypass arrangements for scramjet propulsion using magnetohydrodynamic augmentation, due to the extremely low electrical conductivity of the aerospace environment. The results depicted in this paper illustrate the potential for the numerical method realized in Ref. 1 to effectively resolve this behavior in cases where the MHD approximation may not be valid due to an exceedingly low electrical conductivity. A new study was initiated to explore more fully the capabilities to resolve regions of both low and high electrical conductivity using this numerical scheme. The problem selected for this study was the classic MHD KelvinHelmholtz instability. The initial and boundary conditions are provided in Ref. 2. A periodic box is initialized with two fluids of different density moving in opposite directions. An initial small perturbation is applied to the interfaces between the fluids, seeding an instability. A 256x256 grid size using a Roe solver for the coupled system of Navier-Stokes and full Maxwell equations was used, as detailed in [1, 2]. Three cases were explored: Case A was assigned a zero electrical conductivity, σ = 0 (purely hydrodynamic), Case B was given a small electrical conductivity of σ = 1, and Case C was given a very high conductivity of σ = 10 6 . Case A provides a hydrodynamic benchmark. Case B permits an investigation of a simulation that includes the effects of electromagnetic wave propagation in the plasma evolution. Case C effectively realizes the magnetohydrodynamic approximation. Results for the simulations are presented in a unique fashion in Fig. 1. For each subfigure, three spacetime isosurfaces of the density (blue, green and orange corresponding to the color range) have been depicted. Isosurfaces are the same for all three presented subfigures. Time flows from right (initial conditions) to left (final solution). For Case A, the fluid achieves a highly chaotic solution. The propagation of the initial disturbance at the interfaces between the two fluids is clear, as well as their development into a highly turbulent flow. Case B possesses an extremely low electrical conductivity, and therefore is not subject to the magnetic diffusion approximation of MHD. In fact, electromagnetic wave propagation was visible in this solution. The presence of a finite electrical conductivity provides some diffusion, but it does not match very well to the MHD prediction. However, the conductivity is large enough to abate the low-wavelength instabilities somewhat, and provide stability to the flow. Case C realizes the magnetohydrodynamic limit, where the magnetic field tends to stabilize the flow and strongly suppresses low-wavelength instabilities. A much more organized series of flow structures evolve through time to the final state. The magnetic field is effectively “frozen” to the fluid with such a large electrical conductivity, and the field strongly diffuses with little wave propagation.

On sharp interface limits for diffuse interface models for two-phase flows

We discuss the sharp interface limit of a diffuse interface model for a two-phase flow of two partly miscible viscous Newtonian fluids of different densities, when a certain parameter \epsilon > 0 related to the interface thickness tends to zero. In the case that the mobility stays positive or tends to zero slower than linearly in \epsilon we will prove that weak solutions tend to varifold solutions of a corresponding sharp interface model. But, if the mobility tends to zero faster than \epsilon^3 we will show that certain radially symmetric solutions tend to functions, which will not satisfy the Young-Laplace law at the interface in the limit.

Dynamics of nonlinear three-dimensionalwaves on the interface between two fluids in a channel with low-sloping bottom and top

A combined approach is proposed to describe the transformation of three-dimensional disturbances of the interface between two incompressible immiscible fluids of different densities contained in a channel with fixed rigid top and bottom. It is assumed that the wavelengths are moderately large, the amplitudes are small but finite, the top and the bottom can be gently sloping, and capillary effects are small. The system of equations derived is applicable for modeling disturbances simultaneously scattering in arbitrary horizontal directions. Some typical wave problems are numerically solved and the effect of governing parameters is shown.

Rayleigh-Taylor instability of steady fronts described by the Kuramoto-Sivashinsky equation.

We study steady thin reaction fronts described by the Kuramoto-Sivashinsky equation that separates fluids of different densities. This system may lead to hydrodynamic instabilities as buoyancy forces interact with the propagating fronts in a two-dimensional slab. We use Darcy's law to describe the fluid motion in this geometry. Steady front profiles can be flat, axisymmetric, or nonaxisymmetric, depending on the slab width, the density gradient, and fluid viscosity. Unstable flat fronts can be stabilized having a density gradient with the less dense fluid on top of a denser fluid. We find the steady front solutions from the nonlinear equations executing a linear stability analysis to determine their stability. We show regions of bistability where stable nonaxisymmetric and axisymmetric fronts can coexist. We also consider the stability of steady solutions in large domains, which can be constructed by dividing the domain into smaller parts or cells.

Low Mach number fluctuating hydrodynamics of diffusively mixing fluids

We formulate low Mach number fluctuating hydrodynamic equations appropriate for modeling diffusive mixing in isothermal mixtures of fluids with different density and transport coefficients. These equations eliminate the fluctuations in pressure associated with the propagation of sound waves by replacing the equation of state with a local thermodynamic constraint. We demonstrate that the low Mach number model preserves the spatio-temporal spectrum of the slower diffusive fluctuations. We develop a strictly conservative finite-volume spatial discretization of the low Mach number fluctuating equations in both two and three dimensions and construct several explicit Runge-Kutta temporal integrators that strictly maintain the equation of state constraint. The resulting spatio-temporal discretization is second-order accurate deterministically and maintains fluctuation-dissipation balance in the linearized stochastic equations. We apply our algorithms to model the development of giant concentration fluctuations in the presence of concentration gradients, and investigate the validity of common simplifications such as neglecting the spatial non-homogeneity of density and transport properties. We perform simulations of diffusive mixing of two fluids of different densities in two dimensions and compare the results of low Mach number continuum simulations to hard-disk molecular dynamics simulations. Excellent agreement is observed between the particle and continuum simulations of giant fluctuations during time-dependent diffusive mixing.

Nonlinear Rayleigh–Taylor Instability of a Cylindrical Interface in Explosion Flows

The nonlinear growth of instabilities of an outward propagating, but decelerating, cylindrical interface separated by fluids of different densities is investigated. Single mode perturbations are introduced around the contact-surface, and their evolution is studied by conducting inviscid 2D and 3D numerical simulations. In the past, a significant amount of work has been carried out to model the development of the perturbations in a planar context where the contact surface is stationary or in a spherical context where a point-source blast wave is initiated at the origin. However, for the finite-source cylindrical blast-wave problem under consideration, there is a need for a framework which includes additional complexities such as compressibility, transition from linear to nonlinear stages of instability, finite thickness of the contact interface (CI), and time-dependent deceleration of the contact surface. Several theoretical potential flow models are presented. The model which is able to capture the above mentioned effects (causing deviation from the classical Rayleigh–Taylor Instability (RTI)) is identified as it compares reasonably well with the DNS results. Only for higher wavenumbers, the early development of secondary instabilities (Kelvin–Helmholtz) complicates the model prediction, especially in the estimation of the high-density fluid moving into low-density ambient.

Fluid injection into a confined porous layer

AbstractWe present a theoretical and experimental study of viscous flows injected into a porous medium that is confined vertically by horizontal impermeable boundaries and filled with an ambient fluid of different density and viscosity. General three-dimensional equations describing such flows are developed, showing that the dynamics can be affected by two separate contributions: spreading due to gradients in hydrostatic pressure, and that due to the pressure drop introduced by the injection. In the illustrative case of a two-dimensional injection of fluid at a constant volumetric rate, the injected fluid initially forms a viscous gravity current insensitive both to the depth of the medium and to the viscosity of the ambient fluid. Beyond a characteristic time scale, the dynamics transition to being dominated by the injection pressure, and the injected fluid eventually intersects the second boundary to form a second moving contact line. Three different late-time asymptotic regimes can emerge, depending on whether the viscosity of the injected fluid is less than, equal to or greater than that of the ambient fluid. With a less viscous injection, the flow undergoes a slow decay towards a similarity solution in which the two contact lines extend linearly in time with differing prefactors. Perturbations from this long-term state are shown to decay algebraically with time. Equal viscosities result in both contact lines approaching the same leading-order asymptotic position but with a first-order correction to the distance between them that expands as $t^{1/2}$ due to gravitational spreading. For a more viscous injection, the distance between the contact lines approaches a constant value, with perturbations decaying exponentially. Data from a new series of laboratory experiments confirm these theoretical predictions.

A Model for Strongly Nonlinear Long Interfacial Waves with Background Shear

The Miyata–Choi–Camassa (MCC) system of equations describing long internal nonhydrostatic and nonlinear waves at the interface between two layers of inviscid fluids of different densities bounded by top and bottom walls is mathematically ill‐posed despite the fact that physically stable internal waves are observed matching closely those of MCC. A regularization to the MCC equations that yields a computationally simple well‐posed system for time‐dependent evolution is proposed here. The regularization is accomplished by keeping the full hyperbolic part of MCC and exchanging spatial and temporal derivatives in one of the linearized dispersive terms. Solitary waves of MCC over a wide range of parameters are used as a benchmark to check the accuracy of the model. Our model includes the possibility of a background shear, and we show that, contrary to the no shear case, solitary waves can cross the midlevel between the top and the bottom walls and may have different polarity from the case with no background shear. Time‐dependent solutions of the regularization stable model are presented, including interactions of its solitary waves, and classical and modified Korteweg‐de Vries equations for small amplitude waves with the inclusion of background shear are derived. Throughout the paper, the Boussinesq approximation is taken, although the results can be extended to the non‐Boussinesq case.

Simulación directa de turbulencia en corrientes de gravedad con efecto Coriolis

Gravity currents are flows generated by horizontal pressure gradients resulting from the effect of gravity on fluids of different density. When they occur in nature, gravity currents have a strong nonlinear behavior and have a wide range of temporal and spatial scales. In addition, the rotation of the earth raises the level of complexity of gravity currents due to the effect of the Coriolis force. This work addresses rotating gravity currents in planar geometry by direct numerical simulation (DNS). The simulations allow for a detailed analysis of the flow development, macroscopic parameters of the flow and turbulence structure. Simulations were performed using a pseudospectral code that uses Fourier expansions in the two horizontal directions and Chebyshev expansions in the vertical direction. This work documents in detail the code developed and the validation performed by comparing a simulation with experimental observations and theoretical predictions. This work also reports on two simulations with different rotation speeds. It was found that the flow rotation restricts the development of the current in the propagating direction, and induces oscillations in the front position. The frequency of these oscillations varies linearly with the rotation speed. Finally, this work also reports on Kelvin-Helmholtz-like turbulent structures at the front of the current produced by the rotation of the system.

Interfacial instability in a time-periodic rotating Hele-Shaw Cell

The effect of time-periodic angular velocity on the interfacial instability of two immiscible, viscous fluids of different densities and confined in an annular Hele-Shaw cell is investigated. An inviscid linear stability analysis of the viscous and time dependent basic flow leads to a periodic Mathieu oscillator describing the evolution of the interfacial amplitude. We show that the relevant parameters that control the interface are the Bond number, viscosity ratio, Atwood number and the frequency number.

Numerical and Experimental Analysis of the Growth of Gravitational Interfacial Instability Generated by Two Viscous Fluids of Different Densities

In the geophysical context, there are a wide variety of mechanisms which may lead to the formation of unstable density stratification, leading in turn to the development of the Rayleigh-Taylor instability and, more generally, interfacial gravity-driven instabilities, which involves moving boundaries and interfaces. The purpose of this work is to study the level set method and to apply the process to study the Rayleigh-Taylor instability experimentally and numerically. With the help of a simple, inexpensive experimental arrangement, the R-T instability has been visualized with moderate accuracy for real fluids. The same physical phenomenon has been investigated numerically to track the interface of two fluids of different densities to observe the gravitational instability with the application of level set method coupled with volume of fraction replacing the Heaviside function. Good agreement between theory and experimental results was found and growth of instability for both of the methods has been plotted.

Numerical simulation of the three-dimensional Rayleigh–Taylor instability

The Rayleigh–Taylor instability is a fundamental instability of an interface between two fluids of different densities, which occurs when the light fluid is pushing the heavy fluid. During the nonlinear stages, the growth of the Rayleigh–Taylor instability is greatly affected by three-dimensional effects. To investigate three-dimensional effects on the Rayleigh–Taylor instability, we introduce a new method of computation of the flow of two incompressible and immiscible fluids and implement a time-dependent pressure boundary condition that relates to a time-dependent density field at the domain boundary. Through numerical examples, we observe the two-layer roll-up phenomenon of the heavy fluid, which does not occur in the two-dimensional case. And by studying the positions of the bubble front, spike tip, and saddle point, we show that the three-dimensional Rayleigh–Taylor instability exhibits a stronger dependence on the density ratio than on the Reynolds number. Finally, we perform a long time three-dimensional simulation resulting in an equilibrium state.

Interfacial instability of two superimposed immiscible viscous fluids in a vertical Hele-Shaw cell under horizontal periodic oscillations

The effect of horizontal periodic oscillations on the interfacial instability of two immiscible, viscous fluids of different densities, confined in a vertical Hele-Shaw cell, is investigated. An inviscid linear stability analysis of the viscous basic flow leads to the periodic Mathieu oscillator describing the evolution of interfacial amplitude. We examine mainly the effect of the periodic oscillations and the influence of the viscosity on the stability of the interface. The results show that a decrease in the viscosity contrast has a stabilizing effect on the Kelvin-Helmholtz instability, which is displaced towards the long-wave region. The effects of other parameters such as the frequency number and the Weber number are also examined.

Laminar flow of two miscible fluids in a simple network

When a fluid comprised of multiple phases or constituents flows through a network, nonlinear phenomena such as multiple stable equilibrium states and spontaneous oscillations can occur. Such behavior has been observed or predicted in a number of networks including the flow of blood through the microcirculation, the flow of picoliter droplets through microfluidic devices, the flow of magma through lava tubes, and two-phase flow in refrigeration systems. While the existence of nonlinear phenomena in a network with many inter-connections containing fluids with complex rheology may seem unsurprising, this paper demonstrates that even simple networks containing Newtonian fluids in laminar flow can demonstrate multiple equilibria. The paper describes a theoretical and experimental investigation of the laminar flow of two miscible Newtonian fluids of different density and viscosity through a simple network. The fluids stratify due to gravity and remain as nearly distinct phases with some mixing occurring only by diffusion. This fluid system has the advantage that it is easily controlled and modeled, yet contains the key ingredients for network nonlinearities. Experiments and 3D simulations are first used to explore how phases distribute at a single T-junction. Once the phase separation at a single junction is known, a network model is developed which predicts multiple equilibria in the simplest of networks. The existence of multiple stable equilibria is confirmed experimentally and a criterion for existence is developed. The network results are generic and could be applied to or found in different physical systems.

A line vortex in a two-fluid system

This paper considers the classical problem of a line vortex in planar flow of a fluid. However, an interface is present at some finite radius from the line vortex, and beyond that is a second fluid of different density. The interface is therefore subject to shearing-type instabilities and may overturn as time progresses. A linearized inviscid theory is developed and reveals unstable behaviours, dependent on the parameters in the system. The non-linear inviscid problem is solved by a spectral method, and high-frequency modes are regularized by a type of filtering. In addition, a Boussinesq viscous model is presented and allows the overturning interface to fold. Results are discussed and compared with the predictions of the inviscid theory.

Viscous Fingering through Two Dimensional Porous Layer in Microgravity Environment

Slow fluid-fluid displacement in a porous medium under the influence of real microgravity field is going to reveal important fluid physics associated with hydrology, chemical engineering and the physics of disordered media. Most systems of practical importance include fluids of different densities. Therefore it is important to study the effect of gravity on the front/displacement-structure (fingering: due to the nonlinear interactions among viscous, capillary and gravitational forces). CFD(Computational Fluid Dynamics) analysis glycerin/water mixture through two dimensional single layer aniso-tropic artificial porous layer have been performed in ground level condition as well as microgravity modifying the body force source term in the momentum equation through UDF(user defined functions) written in C. Ground level experiment to capture fingering has also been performed to validate the CFD results. Fingering structures in the microgravity and g-jitter conditions have been predicted using the validated CFD model.

Simultaneous Absolute Determination of Particle Size and Effective Density of Submicron Colloids by Disc Centrifuge Photosedimentometry

Disc centrifuge photosedimentometry (DCP) with fluids of different densities is used to simultaneously determine the particle size and effective density of spherical silica particles. Incorporation of a calibrated infrared pyrometer into a DCP instrument is shown to enhance the measurement capability of the DCP technique by correcting for the temperature dependence of the spin fluid's density and viscosity. Advantages of absolute DCP determinations for size and density analysis relative to standardized DCP measurements include the elimination of instrument standardization with a particle of known density and measurements or estimation of the effective particle density. The reliability of diameter determinations provided by absolute DCP was confirmed using silica particles with nominal diameters ranging from 250 to 700 nm by comparison of these analyses with a diameter determination by transmission electron microscopy for silica particle size standards. Effective densities determined by absolute DCP for the silica particles ranged from 2.02 to 2.34 g/cm(3). These findings indicate that the silica particles have little or no porosity. The reported characterization of colloidal silica using absolute DCP suggests applicability of the technique to a variety of particle types including colloidal materials other than silica, core-shell particles, compositionally heterogeneous mixtures of nanoparticles, and irregularly shaped, structured colloids.

The evolution of a localized nonlinear wave of the Kelvin–Helmholtz instability with gravity

At the interface between two fluids of different density and in the presence of gravity, there are well known periodic surface waves which can propagate for long distances with little attenuation, as it is for instance the case at the surface of the sea. If wind is present, these waves progressively accumulate energy as they propagate and grow to large sizes—this is the Kelvin–Helmholtz instability. On the other hand, we show in this paper that for a given wind strength, there is potential for the growth of a localized nonlinear wave. This wave can reach a size such that the hydrostatic pressure drop from top to bottom equals the stagnation pressure of the wind. This process for the disruption of the flat interface is localized and nonlinear. We study the properties of this wave using numerical simulations of the Navier–Stokes equations.

Numerical simulation of particle-driven gravity currents

Particle-driven gravity currents frequently occur in nature, for instance as turbidity currents in reservoirs. They are produced by the buoyant forces between fluids of different density and can introduce sediments and pollutants into water bodies. In this study, the propagation dynamics of gravity currents is investigated using the FLOW-3D computational fluid dynamics code. The performance of the numerical model using two different turbulence closure schemes namely the renormalization group (RNG) \({k-\epsilon}\) scheme in a Reynold-averaged Navier-Stokes framework (RANS) and the large-eddy simulation (LES) technique using the Smagorinsky scheme, were compared with laboratory experiments. The numerical simulations focus on two different types of density flows from laboratory experiments namely: Intrusive Gravity Currents (IGC) and Particle-Driven Gravity Currents (PDGC). The simulated evolution profiles and propagation speeds are compared with laboratory experiments and analytical solutions. The numerical model shows good quantitative agreement for predicting the temporal and spatial evolution of intrusive gravity currents. In particular, the simulated propagation speeds are in excellent agreement with experimental results. The simulation results do not show any considerable discrepancies between RNG \({k-\epsilon}\) and LES closure schemes. The FLOW-3D model coupled with a particle dynamics algorithm successfully captured the decreasing propagation speeds of PDGC due to settling of sediment particles. The simulation results show that the ratio of transported to initial concentration Co/Ci by the gravity current varies as a function of the particle diameter ds. We classify the transport pattern by PDGC into three regimes: (1) a suspended regime (ds is less than about 16 μm) where the effect of particle deposition rate on the propagation dynamics of gravity currents is negligible i.e. such flows behave like homogeneous fluids (IGC); (2) a mixed regime (16 μm 40 μm) where the PDGC rapidly loses its forward momentum due to fast deposition. The present work highlights the potential of the RANS simulation technique using the RNG \({k-\epsilon}\) turbulence closure scheme for field scale investigation of particle-driven gravity currents.

Cylindrical effects on Richtmyer-Meshkov instability for arbitrary Atwood numbers in weakly nonlinear regime

When an incident shock collides with a corrugated interface separating two fluids of different densities, the interface is prone to Richtmyer-Meshkov instability (RMI). Based on the formal perturbation expansion method as well as the potential flow theory, we present a simple method to investigate the cylindrical effects in weakly nonlinear RMI with the transmitted and reflected cylindrical shocks by considering the nonlinear corrections up to fourth order. The cylindrical results associated with the material interface show that the interface expression consists of two parts: the result in the planar system and that from the cylindrical effects. In the limit of the cylindrical radius tending to infinity, the cylindrical results can be reduced to those in the planar system. Our explicit results show that the cylindrical effects exert an inward velocity on the whole perturbed interface, regardless of bubbles or spikes of the interface. On the one hand, outgoing bubbles are constrained and ingoing spikes are accelerated for different Atwood numbers (A) and mode numbers k'. On the other hand, for ingoing bubbles, when |A|k'3/2≲1, bubbles are considerably accelerated especially at the small |A| and k'; otherwise, bubbles are decelerated. For outgoing spikes, when |A|k'≳1, spikes are dramatically accelerated especially at large |A| and k'; otherwise, spikes are decelerated. Furthermore, the cylindrical effects have a significant influence on the amplitudes of the ingoing spike and bubble for large k'. Thus, it should be included in applications where the cylindrical effects play a role, such as inertial confinement fusion ignition target design.