Fluids Of Different Densities Research Articles

SUPERSONIC SHEAR INSTABILITIES IN ASTROPHYSICAL BOUNDARY LAYERS

Disk accretion onto weakly magnetized astrophysical objects often proceeds via a boundary layer that forms near the object's surface, in which the rotation speed of the accreted gas changes rapidly. Here we study the initial stages of formation for such a boundary layer around a white dwarf or a young star by examining the hydrodynamical shear instabilities that may initiate mixing and momentum transport between the two fluids of different densities moving supersonically with respect to each other. We find that an initially laminar boundary layer is unstable to two different kinds of instabilities. One is an instability of a supersonic vortex sheet (implying a discontinuous initial profile of the angular speed of the gas) in the presence of gravity, which we find to have a growth rate of order (but less than) the orbital frequency. The other is a sonic instability of a finite width, supersonic shear layer, which is similar to the Papaloizou-Pringle instability. It has a growth rate proportional to the shear inside the transition layer, which is of order the orbital frequency times the ratio of stellar radius to the boundary layer thickness. For a boundary layer that is thin compared to the radius of the star, the shear rate is much larger than the orbital frequency. Thus, we conclude that sonic instabilities play a dominant role in the initial stages of nonmagnetic boundary layer formation and give rise to very fast mixing between disk gas and stellar fluid in the supersonic regime.

FDS 기법과 HCIB법을 이용한 3차원 내면파 수치 모사

A code developed using the flux-difference splitting scheme on the hybrid Cartesian/immersed boundary method is applied to simulate three-dimensional internal waves. The material interface is regarded as a moving contact discontinuity and is captured on the basis of mass conservation without any additional treatment across the interface. Inviscid fluxes are estimated using the flux-difference splitting scheme for incompressible fluids of different density. The hybrid Cartesian/immersed boundary method is used to enforce the boundary condition for a moving three-dimensional body. Immersed boundary nodes are identified within an instantaneous fluid domain on the basis of edges crossing a boundary. The dependent variables are reconstructed at the immersed boundary nodes along local normal lines to provide the boundary condition for a discretized flow problem. The internal waves are simulated, which are generated by an pitching ellipsoid near an material interface. The effects of density ratio and location of the ellipsoid on internal waves are compared.

Kelvin-Helmholtz Instability in a Fluid Layer Bounded Above by a Porous Layer and Below by a Rigid Surface in Presence of Magnetic Field

We study the stability of an interface between two fluids of different densities flowing parallel to each other in the presence of a transverse magnetic field. A simple theory based on fully developed flow approximations is used to de-rive the dispersion relation for the growth rate of KHI. We replace the effect of boundary layer with Beavers and Joseph slip condition. The dispersion relation is derived using suitable boundary and surface conditions and results are discussed graphically. The magnetic field is found to be stabilizing and the influence of the various parameters of the problem on the interface stability is thoroughly analyzed. These are favorable to control the surface instabilities in many practical applications discussed in this paper.

A numerical study of the triggering mechanism of a lock-release density current

A numerical study on the effects induced by the impulsive vertical removal of a lock-gate at the interface between two fluids of different densities is presented. This configuration represents the typical setup of those experiments commonly employed for investigating density currents in the laboratory. Experimentally induced effects resulting from opening the lock-gate are expected to occur, but the evaluation of these dynamics and their impact on the evolution of the laboratory density current produced in such a manner are not easy to estimate. Despite the fact that numerical studies are often concerned with lock-release density currents, the triggering mechanism which occurs in the early stages of the evolution of the fluid flow is commonly neglected. Here a comparison is established between the case when the triggering mechanism is completely neglected and a series of cases where, in contrast, this effect is taken into account. The withdrawal of the lock-gate is modeled either by employing a zero-thickness lock-gate or by accounting for the volumetric nature of the lock-gate. Subsequently the influence of speed on the withdrawal of the lock-gate is assessed. The numerical results suggest that the density current is mainly affected by the constraining effect of the lock-gate on the flow and by the responses of the submerged fluid and the free surface to the displacement of the lock-gate. These differences lead to improved physical modeling and numerical simulation validation in the case where the physics of the lock-gate is accounted for. Such differences can be very important particularly in particulate-laden flows, where small changes in initial conditions may lead to longer-term divergence as a result of positive feedback effects. The work has significant implications for physical modeling of density currents and a series of recommendations are made for the standardization of experimental protocols. Finally, the approach adopted here for the moving gate is applicable to civil and environmental engineering problems including dam-break flows and sluice gate modeling.

The influence of electric fields and surface tension on Kelvin–Helmholtz instability in two-dimensional jets

We consider nonlinear aspects of the flow of an inviscid two-dimensional jet into a second immiscible fluid of different density and unbounded extent. Velocity jumps are supported at the interface, and the flow is susceptible to the Kelvin–Helmholtz instability. We investigate theoretically the effects of horizontal electric fields and surface tension on the nonlinear evolution of the jet. This is accomplished by developing a long-wave matched asymptotic analysis that incorporates the influence of the outer regions on the dynamics of the jet. The result is a coupled system of long-wave nonlinear, nonlocal evolution equations governing the interfacial amplitude and corresponding horizontal velocity, for symmetric interfacial deformations. The theory allows for amplitudes that scale with the undisturbed jet thickness and is therefore capable of predicting singular events such as jet pinching. In the absence of surface tension, a sufficiently strong electric field completely stabilizes (linearly) the Kelvin–Helmholtz instability at all wavelengths by the introduction of a dispersive regularization of a nonlocal origin. The dispersion relation has the same functional form as the destabilizing Kelvin–Helmholtz terms, but is of a different sign. If the electric field is weak or absent, then surface tension is included to regularize Kelvin–Helmholtz instability and to provide a well-posed nonlinear problem. We address the nonlinear problems numerically using spectral methods and establish two distinct dynamical behaviors. In cases where the linear theory predicts dispersive regularization (whether surface tension is present or not), then relatively large initial conditions induce a nonlinear flow that is oscillatory in time (in fact quasi-periodic) with a basic oscillation predicted well by linear theory and a second nonlinearly induced lower frequency that is responsible for quasi-periodic modulations of the spatio-temporal dynamics. If the parameters are chosen so that the linear theory predicts a band of long unstable waves (surface tension now ensures that short waves are dispersively regularized), then the flow generically evolves to a finite-time rupture singularity. This has been established numerically for rather general initial conditions.

Diffuse-interface approach to rotating Hele-Shaw flows

When two fluids of different densities move in a rotating Hele-Shaw cell, the interface between them becomes centrifugally unstable and deforms. Depending on the viscosity contrast of the system, distinct types of complex patterns arise at the fluid-fluid boundary. Deformations can also induce the emergence of interfacial singularities and topological changes such as droplet pinch-off and self-intersection. We present numerical simulations based on a diffuse-interface model for this particular two-phase displacement that capture a variety of pattern-forming behaviors. This is implemented by employing a Boussinesq Hele-Shaw-Cahn-Hilliard approach, considering the whole range of possible values for the viscosity contrast, and by including inertial effects due to the Coriolis force. The role played by these two physical contributions on the development of interface singularities is illustrated and discussed.

Axisymmetric boundary integral formulation for a two‐fluid system

SUMMARYA 3D axisymmetric Galerkin boundary integral formulation for potential flow is employed to model two fluids of different densities, one fluid enclosed inside the other. The interface variables are the velocity potential and the normal velocity, and they can be solved for separately, the second linear system being symmetric. The algorithm is validated by comparing with the analytic solutions for a static interior spherical drop over a range of values for the relative densities of exterior and interior fluids and various boundary conditions. For time‐dependent simulations utilizing a level set method for the interface tracking, the accuracy has been checked by comparing against the known oscillation frequency of the sphere. Pinch‐off profiles corresponding to an initial two‐lobe geometry drop and D = 6 are also presented. Published in 2011 by John Wiley & Sons, Ltd.

Asymptotic models for the generation of internal waves by a moving ship, and the dead-water phenomenon

This paper deals with the dead-water phenomenon, which occurs when a ship sails in a stratified fluid, and experiences an important drag due to waves below the surface. More generally, we study the generation of internal waves by a disturbance moving at constant speed on top of two layers of fluids of different densities. Starting from the full Euler equations, we present several nonlinear asymptotic models, in the long wave regime. These models are rigorously justified by consistency or convergence results. A careful theoretical and numerical analysis is then provided, in order to predict the behaviour of the flow and in which situations the dead-water effect appears.

LARGE AMPLITUDE THEORY OF A SHOCK-ACCELERATED INSTABILITY IN COMPRESSIBLE FLUIDS

The interface between fluids of different densities is unstable under acceleration by a shock wave. A previous small amplitude linear theory for the compressible Euler equation failed to provide a quantitatively correct prediction for the growth rate of the unstable interface. In this paper, to include dominant nonlinear effects in a large amplitude regime, we present high-order perturbation equations of the Euler equation, and boundary conditions for the contact interface and shock waves.

Measuring single-cell density

We have used a microfluidic mass sensor to measure the density of single living cells. By weighing each cell in two fluids of different densities, our technique measures the single-cell mass, volume, and density of approximately 500 cells per hour with a density precision of 0.001 g mL(-1). We observe that the intrinsic cell-to-cell variation in density is nearly 100-fold smaller than the mass or volume variation. As a result, we can measure changes in cell density indicative of cellular processes that would be otherwise undetectable by mass or volume measurements. Here, we demonstrate this with four examples: identifying Plasmodium falciparum malaria-infected erythrocytes in a culture, distinguishing transfused blood cells from a patient's own blood, identifying irreversibly sickled cells in a sickle cell patient, and identifying leukemia cells in the early stages of responding to a drug treatment. These demonstrations suggest that the ability to measure single-cell density will provide valuable insights into cell state for a wide range of biological processes.

The Stability of Large‐Amplitude Shallow Interfacial Non‐Boussinesq Flows

The system of equations describing the shallow‐water limit dynamics of the interface between two layers of immiscible fluids of different densities is formulated. The flow is bounded by horizontal top and bottom walls. The resulting equations are of mixed type: hyperbolic when the shear is weak and the behavior of the system is internal‐wave like, and elliptic for strong shear. This ellipticity, or ill‐posedness is shown to be a manifestation of large‐scale shear instability. This paper gives sharp nonlinear stability conditions for this nonlinear system of equations. For initial data that are initially hyperbolic, two different types of evolution may occur: the system may remain hyperbolic up to internal wave breaking, or it may become elliptic prior to wave breaking. Using simple waves that give a priori bounds on the solutions, we are able to characterize the condition preventing the second behavior, thus providing a long‐time well‐posedness, or nonlinear stability result. Our formulation also provides a systematic way to pass to the Boussinesq limit, whereby the density differences affect buoyancy but not momentum, and to recover the result that shear instability cannot occur from hyperbolic initial data in that case.

Two-phase gravity currents in porous media

We develop a model describing the buoyancy-driven propagation of two-phase gravity currents, motivated by problems in groundwater hydrology and geological storage of carbon dioxide (CO2). In these settings, fluid invades a porous medium saturated with an immiscible second fluid of different density and viscosity. The action of capillary forces in the porous medium results in spatial variations of the saturation of the two fluids. Here, we consider the propagation of fluid in a semi-infinite porous medium across a horizontal, impermeable boundary. In such systems, once the aspect ratio is large, fluid flow is mainly horizontal and the local saturation is determined by the vertical balance between capillary and gravitational forces. Gradients in the hydrostatic pressure along the current drive fluid flow in proportion to the saturation-dependent relative permeabilities, thus determining the shape and dynamics of two-phase currents. The resulting two-phase gravity current model is attractive because the formalism captures the essential macroscopic physics of multiphase flow in porous media. Residual trapping of CO2 by capillary forces is one of the key mechanisms that can permanently immobilize CO2 in the societally important example of geological CO2 sequestration. The magnitude of residual trapping is set by the areal extent and saturation distribution within the current, both of which are predicted by the two-phase gravity current model. Hence the magnitude of residual trapping during the post-injection buoyant rise of CO2 can be estimated quantitatively. We show that residual trapping increases in the presence of a capillary fringe, despite the decrease in average saturation.

Two-phase gravity currents in porous media

We develop a model describing the buoyancy-driven propagation of two-phase gravity currents, motivated by problems in groundwater hydrology and geological storage of carbon dioxide (CO2). In these settings, fluid invades a porous medium saturated with an immiscible second fluid of different density and viscosity. The action of capillary forces in the porous medium results in spatial variations of the saturation of the two fluids. Here, we consider the propagation of fluid in a semi-infinite porous medium across a horizontal, impermeable boundary. In such systems, once the aspect ratio is large, fluid flow is mainly horizontal and the local saturation is determined by the vertical balance between capillary and gravitational forces. Gradients in the hydrostatic pressure along the current drive fluid flow in proportion to the saturation-dependent relative permeabilities, thus determining the shape and dynamics of two-phase currents. The resulting two-phase gravity current model is attractive because the formalism captures the essential macroscopic physics of multiphase flow in porous media. Residual trapping of CO2 by capillary forces is one of the key mechanisms that can permanently immobilize CO2 in the societally important example of geological CO2 sequestration. The magnitude of residual trapping is set by the areal extent and saturation distribution within the current, both of which are predicted by the two-phase gravity current model. Hence the magnitude of residual trapping during the post-injection buoyant rise of CO2 can be estimated quantitatively. We show that residual trapping increases in the presence of a capillary fringe, despite the decrease in average saturation.

The bifurcation of interfacial capillary-gravity waves under O(2) symmetry

The evolution of an interface between two fluids of different densities is considered. The particular case under examination is when the motion is due to an interaction between the Mth and Nth harmonics of the fundamental mode. By means of a hodograph transformation the problem is cast as an operator equation between two suitable function spaces. Classical techniques are used to reduce the problem to a finite system of algebraic equations. Solutions are found which exhibit a rich variety of behaviour including primary, secondary and multiple bifurcation.

Droplet and bubble pinch-off computations using level sets

A two fluid potential flow model is employed to analyze the pinching characteristics of an inviscid fluid immersed in a second inviscid fluid of different density. The system behavior is controlled by the relative density of the two fluids, D=ρE/ρI, where D=0 corresponds to droplets in air, and D=100 to bubbles in water. The numerical method employed combines the level set method for advancing the free surface position and boundary condition, together with a 3D axisymmetric boundary integral formulation to obtain fluid velocities. This approach provides a numerical methodology to analyze the pinch-off behavior up to and beyond the initial break-up of the inner fluid. The combined algorithm is validated using the analytical solution for an oscillating sphere. A series of numerical experiments, up to and beyond the initial break-up of the inner fluid, have been carried out for the two extremes, D=0 and D=100. The calculated scaling exponents match the theoretical values, and the computed front profiles are in good agreement with recent experimental findings.

Flow structure and momentum transport for buoyancy driven mixing flows in long tubes at different tilt angles

Buoyancy driven mixing of fluids of different densities (ρ1 and ρ2) in a long circular tube is studied experimentally at the local scale as a function of the tilt angle from vertical (15°≤θ≤60°) and of the Atwood number [10−3≤At=(ρ2−ρ1)/(ρ2+ρ1)≤10−2]. Particle Image Velocimetry (PIV) and Laser Induced Fluorescence (LIF) measurements in a vertical diametral plane provide the velocity and the relative concentration (and, hence, density) fields. A map of the different flow regimes observed as a function of At and θ has been determined: as At increases and θ is reduced, the regime varies from laminar to intermittent destabilizations and, finally, to developed turbulence. In the laminar regime, three parallel stable layers of different densities are observed; the velocity profile is linear and well predicted from the density profile. The thickness of the intermediate layer can be estimated from the values of At and θ. In the turbulent regime, the density varies slowly with z in the core of the flow: there, transverse turbulent momentum transfer is dominant. As At decreases and θ increases, the density gradient β in the core (and, hence, the buoyancy forces) becomes larger, resulting in higher extremal velocities and indicating a less efficient mixing. While the mean concentration varies with time in the turbulent regime, the mean velocity remains constant. In the strong turbulent regime (highest At and lowest θ values), the transverse gradient of the mean concentration and the fluctuations of concentration and velocity remain stationary, whereas they gradually decay with time when turbulence is weaker.

Experimental validation of non-invasive and fluid density independent methods for the determination of local wave speed and arrival time of reflected wave

The relationship between the vessel diameter ( D) and fluid velocity ( U) in arteries and flexible tubes has been recently characterized as linear in the absence of wave reflections. This relationship allowed for determining local wave speed ( C DU ) using the ln DU-loop method. Using C DU , it was possible to separate U and D waveforms into their forward and backward components. It was also possible to calculate wave intensity ( dI DU ), using D and U, from which the arrival time of reflected wave (Trw DU ) could be determined. These techniques are fluid density independent and require only non-invasive measurements of D and U. In this work we experimentally validate the relative accuracy of these new techniques in vitro, by comparing their results of C DU and Trw DU to those determined by the established techniques, PU-loop and wave intensity analysis, C and Trw, respectively. We generated a single semi-sinusoidal wave in long flexible tubes, and simultaneously measured pressure ( P), D, and U at the same site. Sequentially in time, we repeated this experiment at three sites along each of the flexible tubes, which were made of different materials and sizes, and three fluids of different densities. C DU compared well with that C and likewise Trw DU was very similar to Trw. Varying fluid density did not appreciably change the difference between the results of the two techniques. We conclude that the new techniques for determining C DU and Trw DU , although independent of density, provide relatively accurate estimates of wave speed and arrival times of reflected waves in vitro. The new techniques require only non-invasive measurements of D and U, and further in vivo validation is required to establish its advantage in the clinical setting.

A note on Taylor’s stability rule

Analyzing the stability of the interface between two fluids of different densities which is accelerating normal to its surface, Taylor concluded a rule that “this surface is stable or unstable according to whether the acceleration is directed from the heavier to the lighter fluid or vice versa.” We show by following Taylor’s analysis that in the range where the acceleration of the interface is in the direction of the gravitational acceleration and its magnitude is smaller, this rule reverses direction. The interface is stable or unstable according to whether the acceleration is from the lighter to the heavier fluid or vice versa.

Viscous lock-exchange in rectangular channels

In a viscous lock-exchange gravity current, which describes the reciprocal exchange of two fluids of different densities in a horizontal channel, the front between two Newtonian fluids spreads as the square root of time. The resulting diffusion coefficient reflects the competition between the buoyancy-driving effect and the viscous damping, and depends on the geometry of the channel. This lock-exchange diffusion coefficient has already been computed for a porous medium, a two-dimensional (2D) Stokes flow between two parallel horizontal boundaries separated by a vertical height H and, recently, for a cylindrical tube. In the present paper, we calculate it, analytically, for a rectangular channel (horizontal thickness b and vertical height H) of any aspect ratio (H/b) and compare our results with experiments in horizontal rectangular channels for a wide range of aspect ratios (1/10 to 10). We also discuss the 2D Stokes–Darcy model for flows in Hele-Shaw cells and show that it leads to a rather good approximation, when an appropriate Brinkman correction is used.

Transient buoyancy-driven ventilation: Part 1. Modelling advection

The unsteady development of the vertical temperature profile in a ventilated space containing a heat source is modelled. The buoyant fluid released from the heat source is modelled as a turbulent buoyant plume, using a standard integral plume model with a fixed entrainment coefficient. Two types of natural ventilation flow are considered, with the flow driven entirely by the density contrast between the fluid inside and outside the space (stack effect). The ventilation types are (a) classic displacement ventilation, with outflow of warm air through upper openings and inflow of cool air through lower openings; and (b) doorway ventilation, with an exchange flow through the doorway. An improved version of the doorway exchange flow model is given as compared to previous studies. The boundaries of the space are considered to be perfectly insulating, so that heat is transported entirely by the fluid motion. The temporal stratification that develops within the space (outside the plume) is calculated using a modified filling-box model, with successive layers added to the top of the space over time. Laboratory experiments giving reduced-scale simulations of the flows were also conducted, where saline solution and fresh water are used to model fluid of different density. The developing density profiles in the laboratory experiments compare very well with the model predictions. The use of this type of model, capturing the main physical flow features, allows rapid and accurate calculations of transient stratification in ventilated spaces.