Search

The paper discusses the formation and characteristics of stratified air–water downflow in the mesodomain. The stratified pattern, observed at low phase velocities, is characterized by air and water flowing side by side and both wetting the conduit wall, similar to stratified distributions in horizontal conduits. Such flow segregation without the influence of gravity is counterintuitive and no detailed investigation on the formation and flow physics of this distribution is reported till date. We have performed extensive experiments in glass conduits of 2.5–12.5 mm diameter where the two phases are introduced through T and Y junctions with different included angles between the entry arms. Our experiments reveal that stratified flows are formed up to a critical angle ϕc subtended by the edges of the circumferential liquid film and beyond the critical wetting angle, the flow pattern is annular with liquid completely wetting the conduit wall. We further note ϕc to depend on liquid properties, included angle of Y-entry, conduit diameter, and phase flow rates. Based on experimental observations, we propose a simplistic analysis to relate the liquid properties, conduit dimension, and Y-entry included angle and the liquid and gas flow rates to describe the formation of stratified and annular flow at the junction. The analysis also estimates the in situ void fraction during stratified gas–liquid downflow.

Taylor's approach on the dispersion phenomenon is generalized for solute transport in a two-phase laminar flow of immiscible fluids in a slit. The reduced-order models for solute transport are derived using Reynolds decomposition and averaging techniques from which the exact analytical expressions for all elements of the dispersion tensor and the matrix of coefficients of the advection term are derived. It is shown that the dispersion tensor is generally not symmetric, and the asymmetry originates from the presence of an interface between the two fluids. We also discussed conditions at which the solute transport in a two-phase laminar flow in a slit lead to dispersion barrier, osmotic dispersion, and reverse dispersion. The results provide a thorough insight into modeling solute transport across an interface/film in two-phase stratified flows and find applications in the design and optimization of microfluidic devices where two fluids flow in laminar contact.

Particles in microfluidic channels experience two dominant lift forces in the direction transverse to the flow—the shear gradient lift force and the wall lift force. These forces contribute to the lift experienced by the particle and cause their cross stream migration until they attain an equilibrium position where the net lift force in the transverse direction is zero. Stratified coflow of two liquids with different viscosities is a stable flow-regime observed under some operating conditions. The presence of the second fluid alters the shear gradient induced lift force and the wall force acting on the particle at each point, changing the final equilibrium position. These positions can be tuned and controlled by altering the viscosity or the flow rates of the two fluids so that the particles focus in one fluid. A numerical method based on the combined Immersed Boundary-Lattice Boltzmann Method is used to study inertial focusing of neutrally buoyant particles in stratified Couette flows and pressure driven flows. We analyze how different factors such as the Reynolds number, flow rate ratio, viscosity ratio of the fluids, and particle size affect the particle migration in two-dimensional (2D) and three-dimensional (3D) geometries. Our study shows that in Couette flows, the particle focuses in the low viscosity fluid when the interface is at the center. We also found that a critical viscosity ratio exists beyond which particle focusing in low viscous fluid is guaranteed, for a given flow rate ratio in pressure driven flows.

Analyzing a large database of high-resolution three-dimensional direct numerical simulations of decaying rotating stratified flows, we show that anomalous mixing and dissipation, marked anisotropy, and strong intermittency are all observed simultaneously in an intermediate regime of parameters in which both waves and eddies interact nonlinearly. A critical behavior governed by the stratification occurs at Richardson numbers of order unity and with the flow close to being in a state of instability. This confirms the central dynamical role, in rotating stratified turbulence, of large-scale intermittency, which occurs in the vertical velocity and temperature fluctuations, as an adjustment mechanism of the energy transfer in the presence of strong waves.Analyzing a large database of high-resolution three-dimensional direct numerical simulations of decaying rotating stratified flows, we show that anomalous mixing and dissipation, marked anisotropy, and strong intermittency are all observed simultaneously in an intermediate regime of parameters in which both waves and eddies interact nonlinearly. A critical behavior governed by the stratification occurs at Richardson numbers of order unity and with the flow close to being in a state of instability. This confirms the central dynamical role, in rotating stratified turbulence, of large-scale intermittency, which occurs in the vertical velocity and temperature fluctuations, as an adjustment mechanism of the energy transfer in the presence of strong waves.

In this paper, we first revisit the celebrated Boussinesq approximation in stratified flows. Using scaling arguments we show that when the background shear is weak, the Boussinesq approximation yields either (i) $A_t\ll \mathcal{O}(1)$ or (ii) $Fr_c^2 \ll \mathcal{O}(1)$, where $A_t$ is the ratio of density variation to the mean density and $Fr_c$ is the ratio of the phase speed to the long wave speed. The second clause implies, contrary to the commonly accepted notion, that a flow with large density variations can also be Boussinesq. Indeed, we show that deepwater surface gravity waves are Boussinesq while shallow water surface gravity waves are not. However, in the presence of moderate/strong shear, Boussinesq approximation implies the conventionally accepted $A_t\ll \mathcal{O}(1)$. To understand the inertial effects of density variation, our second objective is to explore various non-Boussinesq shear flows and study different kinds of stably propagating waves that can be present at an interface between two fluids of different background densities and vorticities. Furthermore, three kinds of density interfaces - neutral, stable and unstable - embedded in a background shear layer, are investigated. Instabilities ensuing from these configurations, which includes Kelvin-Helmholtz, Holmboe, Rayleigh-Taylor and triangular-jet, are studied in terms of resonant wave interactions. The effects of density stratification and the shear on the stability of each of these flow configurations are explored. Some of the results, e.g. the destabilizing role of density stratification, stabilizing role of shear, etc. are apparently counter-intuitive, but physical explanations are possible if the instabilities are interpreted from wave interactions perspective., Physics of Fluids

The linear stability of two-phase stratified flows through soft-gel-coated walls is studied in this work. A consistent nonlinear neo-Hookean model describes the soft-gel. The base state is characterised by a first normal stress difference in this model. This results in a significantly different effect on the stability of two-phase flows as compared to when a linear viscoelastic model is used for the soft-gel. Both fluids are assumed Newtonian and incompressible. We analyze the system in the absence and presence of a soluble surfactant. A linear stability analysis is carried out to identify different instability modes in the system. The linearised equations result in a generalised eigenvalue problem which is numerically solved employing a Chebyshev collocation technique. Three distinct instabilities are identified in the absence of a soluble surfactant. A long wave interfacial instability, a Tollmien-Schlichting wave type instability, and a gel-liquid instability arise in the system. Two distinct additional instabilities are identified in the presence of soluble surfactants in the system, a Marangoni driven long wave instability and a liquid-liquid short wave mode. It is shown that all instabilities except the Tollmien-Schlichting shear instability can be stabilised by the soft-gel layers for a suitable choice of parameters. Insights into the physical mechanism driving the different instabilities are discussed.The linear stability of two-phase stratified flows through soft-gel-coated walls is studied in this work. A consistent nonlinear neo-Hookean model describes the soft-gel. The base state is characterised by a first normal stress difference in this model. This results in a significantly different effect on the stability of two-phase flows as compared to when a linear viscoelastic model is used for the soft-gel. Both fluids are assumed Newtonian and incompressible. We analyze the system in the absence and presence of a soluble surfactant. A linear stability analysis is carried out to identify different instability modes in the system. The linearised equations result in a generalised eigenvalue problem which is numerically solved employing a Chebyshev collocation technique. Three distinct instabilities are identified in the absence of a soluble surfactant. A long wave interfacial instability, a Tollmien-Schlichting wave type instability, and a gel-liquid instability arise in the system. Two distinct additiona...

The large-scale structures in the ocean and the atmosphere are in geostrophic balance, and a conduit must be found to channel the energy to the small scales where it can be dissipated. In turbulence, this takes the form of an energy cascade, whereas a possible mechanism in a balanced flow is through the formation of fronts, a common occurrence in geophysics. We show that an iconic configuration in laboratory and numerical experiments for the study of turbulence, the so-called Taylor-Green or von Karman swirling flow, can be suitably adapted to domains with large aspect ratios, leading to the creation of an imposed large-scale vertical shear. To this effect, we use direct numerical simulations of the Boussinesq equations without net rotation and with no small-scale modeling. Various grid spacings are used, up to 20482 × 256 spatial points. The grids are always isotropic, with box aspect ratios of either 1:4 or 1:8. We find that when shear and stratification are comparable, the imposed shear layer resulting from the forcing leads to the formation of fronts and filaments which destabilize and evolve into a turbulent flow in the bulk, with a sizable amount of dissipation and mixing, following a cycle of front creation, instability, and development of turbulence. The results depend on the vertical length scales of shear and stratification.

The linear stability of two-layer plane Couette flow of Newtonian fluids (designated by labels A and B) of thicknesses (1−β)R and βR, and viscosities μa and μb past a soft, deformable linear viscoelastic solid of thickness HR, shear modulus G, and viscosity ηw is determined using a combination of low wavenumber asymptotic analysis and a numerical method. There are two qualitatively different interfacial modes in this system, viz., the two-fluid interfacial mode due to viscosity stratification [“mode 1;” C. S. Yih “Instability due to viscosity stratification,” J. Fluid Mech. 27, 337 (1967)], and the fluid–solid interfacial mode [“mode 2;” Kumaran, Fredrickson, and Pincus, “Flow induced instability of the interface between a fluid and a gel at low Reynolds number,” J. Phys II 4, 893 (1994)]. The respective effects of solid layer deformability and fluid viscosity stratification on mode 1 and mode 2 are analyzed in detail using both asymptotic and numerical methods. Results of our low wavenumber asymptotic an...

Partial modeling of the hydrodynamic evolution of the supernovae is one of the prominent applications of laboratory astrophysics. In particular the role of Rayleigh–Taylor instability (RTI) in supernova evolution needs to be explained. In this paper we analyze the compressibility effects on the RTI in the linear regime. We compare the compressible isothermal and stratified incompressible RTI growth rates and analyze the vorticity generation at the interface. We show that for several configurations the effect of compressibility can be significant in supernovae remnants.

We study the effect of buoyancy on shear flow stability. The vertical body force is induced by a vertical, constant, and positive thermal gradient. A linear stability analysis is carried out, in the spatial framework, focusing on both exponential and transient growth. In both cases, positive thermal stratification is found to stabilize the disturbances.

A method is developed to describe the linear internal wavefield generated by an oscillating source in horizontally moving, depth-dependent background. Ray theory is used to approximate the vertical eigenfunctions. A spatial solution is then obtained by inverse Fourier transform. This is a practical approach with a more convenient range of validity than the method of spatial ray tracing. The solutions for a stationary source and for a vertically oscillating source in a vertically sheared background current are given for illustration.

After having previously demonstrated the occurrence of ageostrophic, anticyclonic instability (AAI) for several other steady flows, in this paper we further test the hypothesis that rotating, stably stratified, shear flows are generally subject to AAI at finite Rossby (Ro) and Froude (Fr) numbers by calculating the inviscid, incompressible normal modes of a geostrophic, barotropic boundary current that vanishes toward the interior of the domain. We focus on a background current profile with no inflection point and nonvanishing absolute vorticity to exclude other known instability types. The hypothesis is again confirmed. The instability occurs for finite Ro only in an anticyclonic background flow. Its modal growth rate steeply decreases (approximately inverse exponentially, ∼e−c/Ro) as Ro→0. The downstream phase velocity of the unstable mode lies within the speed range for the background current, so the eigenmode has a near-critical layer within the zone of strong background shear. Its unstable eigenmode has an aspect ratio of downstream and vertical wave numbers, l/m, on the order of the ratio of the Coriolis and Brunt-Vaisalla frequencies (as is typical for rotating, stratified flows). The maximum growth rate is independent of Fr for small Fr (the usual geophysical regime), and it decreases with large Fr, disappearing when there is no stratification. The eigenmode has a weakly decaying, oscillatory, interior far-field structure similar to an inertia-gravity wave. This supports the interpretation that the modal instability arises through the coalescence (i.e., resonance) of two stable normal-mode branches, one associated with the background shear flow and the other an inertia-gravity or Kelvin wave mode.

The structural equilibrium behavior of the general linear second-moment closure model in a stably stratified, spanwise rotating homogeneous shear flow is considered with the aid of bifurcation analysis. A closed form equilibrium solution for the anisotropy tensor aij, dispersion tensor Kij, dimensionless scalar variance θ2/k(S/Sθ)2, and the ratio of mean to turbulent time scale e/Sk is found. The variable of particular interest to bifurcation analysis, e/Sk is shown as a function of the parameters characterizing the body forces: Ω/S (the ratio of the rotation rate to the mean shear rate) for rotation and Rig (the gradient Richardson number) for buoyancy; it determines the bifurcation surface in the e/Sk−Ω/S−Rig space. It is shown, with the use of the closed form solution, that the Isotropization of Production model does not have a real and stable equilibrium solution when rotational and buoyant forces of certain magnitudes are simultaneously imposed on the flow. When this occurs, time integration of the ...

A theoretical analysis for the onset of gas pull-through (entrainment) during discharge from a stratified two-phase region through two vertically aligned side branches has been developed in this paper. Initially, a simplified point-sink model was developed, which was then followed by the acquisition of a more accurate finite-branch model. The prediction of the critical height at the onset of gas entrainment was found to be a function of the corresponding Froude number of each branch (Fr1 and Fr2), as well as the vertical distance between the centerlines of the two branches (L/d). The predicted values of the critical height were found to be consistent with the corresponding experimental data for different values of Fr1, Fr2 and L/d. From the basis of the present models, it was found that by increasing the flow through the lower branch, the critical height increases for all values of Fr1 and L/d. Furthermore, by increasing the vertical distance between the two branches, the effect of the lower branch on the determination of the critical height was decreased.

Similarity solutions for laminar two-fluid jets and wakes are derived in the boundary-layer approximation. Planar and axisymmetric fan jets as well as classical and momentumless planar wakes are considered. The interface between the immiscible fluids is stabilized by the action of gravity, with the heavier fluid, taken to be a liquid, placed beneath the lighter fluid. Velocity profiles for the jets and the classical wake depend intimately, but differently, on the parameter χ=ρ1μ1/ρ2μ2, where ρi and μi are, respectively, the density and absolute viscosity of the fluid in the upper (i=1) and lower (i=2) fluid domains, while the momentumless wake profile depends on the parameter Ω=ρ1μ23/ρ2μ13. Generally, all interfaces deflect from horizontal except the fan jet. However, while the interface for the classical planar two-fluid wake is never flat, the interfaces for the planar jet and the momentumless wake become flat in the particular case μ1=μ2. Velocity profiles illustrating the strongly asymmetrical jet and...

Effects of stratification on the three dimensional instability of elliptic flow having a horizontal axis are investigated. It is revealed that the stratification works fundamentally to suppress the instability. However, its responses to increasing stratification are different depending on situations of the elliptic flow in a vertical plane. Generally, the instability tends to be dumped further when the longer axis of the elliptic flow is oriented vertical rather than horizontal.

We analyze the three-dimensional stability properties of monochromatic, large-amplitude, nonhydrostatic inertio-gravity waves propagating in a rotating stratified fluid with no mean shear. The ratio of Coriolis parameter f to the Brunt-Vaisala frequency N is fixed at a value f/N=0.01, chosen to be representative of the Earth’s atmosphere and upper oceans in extra-tropical latitudes. By using Floquet theory combined with suitable coordinate transformations, our analysis provides an exact representation of the spatial and temporal periodicity of the basic wave state. Growth rates are computed for several discrete phase-elevation angles 80°⩽θ⩽89.7°, spanning the frequency range in which rotational effects become important, and the basic wave state changes from fully overturned to suboverturned. For each basic wave, we identify the growth rate peaks in disturbance wavenumber space—these are the principal modes. At the lowest phase-elevation angle considered, θ=80°, the dominant principal mode is similar to it...

The quasi-cylindrical approximation is used to describe numerically the structure of a submerged swirling jet for subcritical values of the swirl ratio S⩽Sc. The emerging flow structure is affected by the swirling motion, which enhances the entrainment rate of the jet and induces an adverse pressure gradient that reduces its momentum flux. The effect is more pronounced as the swirl ratio S is increased, yielding for sufficiently large values of S a jet with an annular structure. The integration describes the smooth transition towards the far-field self-similar solution for all values of S smaller than a critical value S=Sc, at which the numerical integration fails to converge at a given downstream location. The comparisons with previous experimental results confirm the correspondence between the onset of vortex breakdown and the failure of the quasi-cylindrical approximation.

The physical mechanism of instability in a superposed two-layer liquid film flow down an incline plane is analyzed. If the layer adjacent to the wall is sufficiently thin and less viscous in certain two-layer parallel Newtonian liquid flows of the same density with an interface but without a free surface, the flows are stable with respect to long waves. This is the so-called “thin lubrication layer effect.” However, when a free surface exists in the two-layered flow, the flow becomes unstable even when the Reynolds number approaches zero. Thus the thin-layer lubrication effect is lost due to the presence of the free surface, and inertialess instability occurs. The reason for the loss of the lubrication effect and the mechanism of inertialess instability are explained by use of the energy budget in the mechanical energy equation. Contrary to the case of two-layer flows without a free surface, the flow with a free surface is stable if the layer adjacent to the solid wall is more viscous. The stabilization i...

In this paper we compare the wakes of various bluff bodies in a stratified fluid at moderately high Froude numbers (F≡2UB/ND>8) and Reynolds numbers (Re≈5000). The size and amplitude of the long-lasting wakes clearly depend on the shape of the bluff body, the wake width being small for a streamlined object and large for an object with sharp edges. However, the wake width can be collapsed when it is normalized by an effective diameter based on the drag force, often called the momentum thickness. General laws for the wake width, the velocity defect, and the Strouhal number are thus deduced and fit the data well. Finally, the cross-fluctuations of the velocity and the turbulent kinetic energy are analyzed. Their amplitudes and widths are proportional to those of the mean profile. Thus, the wake remembers only the momentum flux given by the bluff body to the fluid and not any other aspects of its geometry.

This Brief Communication investigates laminar Craya–Curtet flows, formed when a jet with moderately large Reynolds number discharges into a coaxial ducted flow of much larger radius. It is seen that the Craya–Curtet number, C=(Jc/Jj)1/2, defined as the square root of the ratio of the momentum flux of the coflowing stream to that of the central jet, arises as the single governing parameter when the boundary-layer approximation is used to describe the resulting steady slender jet. The numerical integrations show that for C above a critical value Cc the resulting streamlines remain aligned with the axis, while for C

Many viscous flows are mechanically incompressible, yet thermally expand and shrink. Approximations of the compressible Navier–Stokes equations are routinely utilized to model diverse phenomena that share these properties, with a primary goal to remove rapid timescales associated with sound waves. Most models are derived from thermodynamic assumptions coupled with application-specific, scale separation assumptions in time and space. The Boussinesq model for laboratory-scale, buoyancy-driven thermal convection patterns [Spiegel and Veronis, Astrophys. J. 131, 442 (1960)] and the anelastic model for atmospheric-scale, density-stratification phenomena [Emanuel, Atmospheric Convection (Oxford University Press, New York, 1994)] are two important examples. Some engineering models of thermal expansion are fluid specific, e.g., for molten glasses and polymers, and rest upon thermodynamic assumptions alone. These models postpone specification of flow conditions, since applications range from slow confined flows fo...

An exhaustive identification is made of the steady parallel base states which might occur in a thermogravitational column, with a constant horizontal thermal gradient, and a Soret coefficient depending both on temperature and composition, ST(T,C). It is shown that the horizontal dynamics is mainly governed by the Soret thermal sensitivity, and the vertical separation by its compositional one. Published ST(T,C) empirical laws are used to define a generic model which allows one to carry out a “normal mode” analysis of these flows. To account for the possible complete species vertical separation, the Boussinesq approximations are updated. To the leading order in the Soret coefficient amplitude, the horizontal dynamics is seen to be controlled by the solutal Peclet number, Pe, and a vertical stratification parameter, both drifting very slowly along the vertical. Stabilizing and potentially unstable vertical stratifications of the species are predicted, and, for each, the steady parallel flows are described for every thermal sensitivity of the Soret coefficient. Then an accurate determination of the vertical stratification profile is given for a constant and linear C-dependence of ST. The determinant peculiarity of the C-dependence is shown to be the number of zeroes that ST possesses in C∈]0,1[. Some suggestions are made regarding the way the Soret coefficient measurement could benefit from this first analysis. It is shown, in particular, that the vertical separation slope may be misleading in real molecular mixtures, such as the 0.5 N NaCl solution about T=12.09 °C, where the Soret term is mainly linear with temperature.

In this paper we investigate the linear instability of two superposed inviscid fluids between two rotating concentric cylinders. In the hydrostatic state the two fluids rotate as a rigid body with the same angular velocity as the annulus. Two problems are investigated. First, calculations are made of the instability of the liquid layer coating the outside of a cylinder under rotation. Second, results are obtained of the instability of two stratified fluids between two concentric cylinders. The first case is the most unstable condition of the second case when the outer fluid is absent. Therefore, in the second case only the instability of a system with a heavier fluid located outside the interface was investigated. It was found that the introduction of the inner cylinder produces new and interesting results about the axial and azimuthal modes not previously published in the literature. In particular, when a liquid layer coats the outside of a cylinder it was found that the azimuthal mode, n=1, is the most important in a range of the nondimensional surface tension where the purely axial mode dominated in the absence of the inner cylinder. Additionally, it was found that some azimuthal spiral modes with a finite wavenumber appear as the most unstable in the presence of the inner cylinder. It was shown that the larger the centrifugal force is, the larger the nondimensional radius of the inner cylinder should be to make the instability more sensitive to the presence of the inner cylinder. An equation for the growth rate was obtained in the limit of very thin liquid layers and it was found that the maxima of growth rate agree very well with the experimental results of a fluid layer coating the outside of a rotating cylinder.

It is shown that vortices rising through a high-Reynolds number boundary layer can interact with the mean shear so as to yield a localized structure similar to the bulges seen in turbulent boundary layers. These bulges evolve so as to entrain irrotational flow into the boundary layer. It is also shown that vortices can initiate the evolution of detrainment, in which rotational fluid is ejected into the outer flow. The results of this work support the idea that horseshoe vortices can lead to the evolution of large-scale structure in the boundary layer.

The dynamics of quasi-horizontal motions in a stably stratified fluid have been simulated for Froude numbers of order 1, so that the flows are strongly affected by the stable density stratification, and for a range of Reynolds numbers. It is found that the horizontal scales of the motion grow continuously in time. The vertical scales decrease and the vertical shearing increases with time, maintaining the Richardson number of order 1, as suggested by Lilly [J. Atmos. Sci. 40, 749 (1983)] and Babin et al. [Theor. Comput. Fluid Dyn. 9, 223 (1997)]. Small-scale instabilities and turbulent-like motions are observed to occur in the high shearing regions, while the larger-scale motions appear to evolve somewhat independently of the Reynolds number. The results suggest that the larger-scale, quasi-horizontal motions would be a continuous source of smaller-scale turbulence until the local Reynolds number drops below a critical value, which is estimated. Finally, a Froude number based upon a vertical differential scale and used in previous scaling arguments and theories is estimated in terms of other parameters.

Continuum models for particles sedimenting in a fluid often assume that the diffusivity is a local function of the particulate volume fraction. Since the hydrodynamically induced diffusivity is a result of the velocity fluctuations of particles, the recent identification [e.g., Tee et al., Phys. Rev. Lett. 89, 054501 (2002)] of particle density stratification as a controlling parameter for the velocity fluctuations also extends to the diffusivities. In particular, the stratification control strongly affects the diffusivity in the vicinity of the falling sediment front between particle-laden fluid below and clarified fluid above. The resulting scaling for stratification-controlled diffusivities in creeping flow sedimentation is presented and compares favorably with measurements from dilute-limit particle simulations. Steadily falling concentration profiles for dilute sedimentation with these diffusivities are then presented, and an extension of the model to higher volume fractions is discussed.

An asymptotic analysis of the equations for quasi-two-dimensional flow in stratified fluids is conducted, leading to a model for the diffusion of pancake-like vortices in cyclostrophic balance. This analysis permits one to derive formally the model for the diffusion of an axisymmetric monopole proposed by Beckers et al. [J. Fluid Mech. 433, 1 (2001)], and to extend their results. The appropriate parameter for the perturbation analysis is identified as the square of the vertical Froude number Fv=U/LvN, where U is the horizontal velocity scale, N is the Brunt–Vaisala frequency, and Lv the vertical length scale. The physical mechanisms involved in the vortex decay are examined under the light of the asymptotic analysis results. In particular we discuss the effects of the Schmidt number, Sc, which measures the balance between the diffusion of momentum and the diffusion of the stratifying agent. Remarkably, the vertical transport due to the slow cyclostrophic adjustment is shown to slowdown the velocity decay when Sc is larger than unity whereas it accelerates it when Sc is smaller than unity.

Passive and active scalar fluxes in unsteady strongly stratified turbulence are analyzed using the rapid distortion theory (RDT). Analytical solutions of the RDT equations show that the initial conditions, i.e., the initial cross correlation between passive scalar and active scalar (density) and the initial potential energy, make a difference between the passive and active scalar flux, giving the difference between the turbulent diffusion coefficients for passive and active scalars. In other words even the initial zero correlation gives the complete correlation in the long time development, in contrast to the previous discussions in the literature based on the equations for stationary turbulence. The difference appears in the components slowly oscillating at half frequency N (N: Brunt–Vaisala frequency) in the passive scalar flux, while the active scalar flux has only the rapidly oscillating components which oscillate at frequency 2N. It is also found that the correlation coefficient between passive and a...

We present linear stability results based on the three-dimensional Stokes equations for chemically reacting, propagating fronts giving rise to an unstable density stratification in a Hele-Shaw cell. The results are compared with the experiments in M. Bockmann and S. C. Muller [Phys. Rev. Lett. 85, 2506 (2000)], as well as with a corresponding linear stability analysis based on the Darcy equations that was performed in A. De Wit [Phys. Rev. Lett. 87, 054502 (2001)]. The reason for the good agreement between these earlier Darcy data and the experimentally observed growth rates is found in the relatively low experimental value of the Rayleigh number, Ra=79, for which the flow is approximately of Poiseuille type. Already for Ra values as low as 300, we observe a discrepancy between the stability results based on the Darcy and Stokes equations, respectively, with the Darcy results overpredicting both the most amplified wavenumber, as well as the corresponding growth rate, by about a factor of two. This indicates that three-dimensional effects quickly gain importance as Ra increases, so that the stability analysis needs to be based on the full, three-dimensional Stokes equations. The stability results based on the Stokes equations furthermore demonstrate the stabilizing influences of both an increasing interfacial thickness, as well as increasing frontal propagation velocities, confirming the earlier Darcy-based findings by De Wit. An argument in terms of vorticity is forwarded to explain the latter effect. A more rapidly advancing front deposits vorticity over a wider layer of fluid particles, so that the concentrated regions of vorticity needed for rapid instability growth cannot form. Somewhat surprisingly, however, slowly propagating fronts are seen to be more unstable than nonreacting fronts of equivalent thickness, as the chemical reaction leads to the formation of more compact perturbations in the interfacial region.

The aim of this paper is to present the results of experiments conducted in a stratified fluid with a momentum source modeled by a moving jet in the asymptotic case when the size of the source is negligibly small but the resulting force (momentum flux), which acts on the fluid, remains finite. In this case the forcing may be considered as a “point” force moving in the fluid and the problem has no obvious characteristic length scale, compared to the case of a bluff body. The latter, towed horizontally in the stratified fluid, frequently leaves behind a highly organized vortex street, the characteristics of which strongly depend on the body diameter. It is shown that a similar type of the vortex streets may be generated by a “point” momentum source and conditions when this occurs are found. Experimental data on the characteristics of the vortex streets are presented and compared with the towed body case.

Two-fluid natural-convection flow in the horizontal cylindrical annulus and its effect on mass transfer through the liquid-liquid interface of two immiscible fluids are studied numerically. The liquids are stratified by gravity, with the denser one occupying the lower part of the annulus. The convective motion is driven by heating of the inner or outer cylindrical boundary. It is shown that the mass transfer of a passive scalar (say, a protein) through the interface can be significantly enhanced by the convective flow. Varying the radii ratio from 0.1 to 0.5, it is found that the mass transfer is more intensive in annuli with smaller radii ratio. No significant difference in the mass transfer rates was found between the heating of either inner or outer cylinder. A possibility of further mass transfer enhancement using more complicated temperature distribution on the boundaries is demonstrated. The problem is related to the search for novel bioseparator devices.

The evolution of initially axisymmetric shielded pancake-like vortices in a nonrotating linearly stratified fluid has been investigated experimentally and numerically. The evolution process and the shape of tripoles in laboratory experiments depend on the experimental parameter values. In order to investigate this phenomenon we have considered the influence of the Reynolds (Re) and Froude (F) numbers on the tripole formation process. Also, the role of the (absolute) ratio [gamma] between the vorticity values of the satellite vortices and the core vortex on the tripole evolution has been investigated. Additionally, a set of numerical simulations has been performed to enable an examination of the role of Reynolds numbers (up to Re = 10 000) and Froude numbers (F = 0.1, 0.2, 0.4, and 0.8) outside the experimentally accessible range. The steepness parameter [alpha] was varied between 2 and 8 in order to estimate the relative importance of the different modes constituting the perturbation. From this study we conclude that tripole formation and dipole splitting in a linearly stratified fluid can be well described in terms of the parameter set (Re,F, [alpha] ).

The longevity and surprising stability of late wakes emerging from an initially-turbulent state in a stably stratified ambient has been noted frequently in the literature. Here, the streamwise spacing is proposed and detailed as a diagnostic for the structure generated by a towed sphere. The wake Strouhal number and its evolution in time is reported over 2 orders of magnitude range in internal Froude number F∈[2,200]. Over the entire range of F, the decay rates adhere closely to a constant St∼Nt−1/3 power law that can be argued from simple geometric grounds. There is no tendency for values of St to aggregate at, or close to a constant value, such as St=0.175, as would be expected for a near-wake spiral shedding mode. The collective Nt−1/3 evolution is a statistical product of individual, nearest-neighbor interactions. Characteristic departures from the general rule in the lifetime of specific like-signed vortex pairs show pairing (or tripling) interactions with an asymmetry in the spacing distribution evo...

Direct numerical simulations are performed of spatial, three-dimensional, laminar jets of different inlet geometric configurations for the purpose of quantifying the characteristics of the flows; both single-phase (SP) and two-phase (TP) free jets are considered. The TP jets consist of gas laden with liquid drops randomly injected at the inlet. Drop evaporation ensues both due to the gaseous flow being initially unvitiated by the vapor species corresponding to the liquid drops, and to drop heating as the initial drop temperature is lower than that of the carrier gas. The conservation equations for the TP flow include complete couplings of mass, momentum, and energy based on thermodynamically self-consistent specification of the vapor enthalpy, internal energy, and latent heat of vaporization. Inlet geometries investigated are circular, elliptic, rectangular, square, and triangular. The results focus both on the different spreading achieved according to the inlet geometry, as well as on the considerable change in the flow field due to the presence of the drops. The most important consequence of the drop interaction with the flow is the production of streamwise vorticity that alters entrainment and species mixing according to the inlet geometry. Similar to their SP equivalent, TP jets are shown to reach steady-state entrainment; examination of the flows at this time station shows that the potential cores of TP jets are shorter by an order of magnitude than their SP counterpart. Moreover, whereas the TP circular jet exhibits a symmetric entrainment pattern well past the streamwise location of the potential core, noncircular jets display at the same location strong departures from symmetry. Furthermore, the SP-jet phenomenon of axis switching is no longer present in TP jets. The distributions of drop-number density, liquid mass, and evaporated species are compared for different inlet cross sections and recommendations are made regarding the optimal choice for different applications.

International audience; In mixing-layers between two parallel streams of different densities, shear and gravity effects interplay; buoyancy acts as a restoring force and the Kelvin-Helmholtz mode is known to be stabilized by the stratification. If the density interface is sharp enough, two new instability modes, known as Holmboe modes, appear, propagating in opposite directions. This mechanism has been studied in the temporal instability framework. The present paper analyzes the associated spatial instability problem. It considers, in the Boussinesq approximation, two immiscible inviscid fluids with a piecewise linear broken-line velocity profile. We show how the classical scenario for transition between absolute and convective instability should be modified due to the presence of propagating waves. In the convective region, the spatial theory is relevant and the slowest propagating wave is shown to be the most spatially amplified, as suggested by intuition. Predictions of spatial linear theory are compared with mixing-layer [C.G. Koop and F.K. Browand, J. Fluid Mech. 93, 135 (1979)] and exchange flow [G. Pawlak and L. Armi, J. Fluid Mech. 376, 1 (1999)] experiments. The physical mechanism for Holmboe mode destabilization is analyzed via an asymptotic expansion that predicts the absolute instability domain at large Richardson number. © 2002 American Institute of Physics.

This study aims to develop an asymptotic theory for the slow spreading of a thin layer of viscous immiscible dense liquid on the bottom of a waterway under the combined effects of surface waves and density current. By virtue of the sharply different length and time scales (wave periodic excitation being effective at fast scales, while gravity and streaming currents at slow scales), a multiple-scale perturbation analysis is conducted. Evolution equations are deduced for the local and global profile distributions of the dense liquid layer as functions of the slow-time variables. When reflected waves are present, the balance between gravity and streaming will result, on a time scale one order of magnitude longer than the wave period, in an undulating water/liquid interface whose displacement amplitude is much smaller than the thickness of the dense liquid layer. On the global scale, the streaming current can predominate and drive the dense liquid to propagate with a distinct pattern in the direction of the surface waves.

Direct numerical simulation is used to investigate stratified turbulent flow over a series of prescribed moving water waves at a bulk Reynolds number Re=8000 and waveslope ak=0.1. Unstable, neutral, and stable stratifications are considered for a range of wave phase speeds c. Stratification is shown to significantly alter the mean vertical profiles of velocity and temperature, turbulence variances, wave-induced flow fields, and surface form stress. For the range of conditions considered, the surface form stress (drag) and flow patterns (critical-layer height and streamlines) are well correlated with the friction velocity u*, which therefore contains the essential information about stratification influences. Nonseparated sheltering [Belcher and Hunt, Annu. Rev. Fluid. Mech. 30, 507 (1998)], which determines the drag in neutral flow over stationary topography, is modified by stratification and the movement of the underlying waves. The variation of the form stress with phase speed is correlated with the movement of the critical layer above the surface. Compared to neutral flow at a given phase speed, the flow patterns with unstable stratification are similar to the flow patterns over slower moving waves while stable stratification results in flow patterns typical of faster moving waves. This behavior is qualitatively captured by the wave age parameter c/u*. The wave-induced temperature field responds to the wave-induced velocity fields by forming positive and negative patches over the wave crests and troughs, respectively, with the resulting wave-induced heat flux as much as 15% of the total surface heat flux. Estimates of wave growth from the DNS are in reasonable agreement with field observations and laboratory experiments, and they are larger than predictions from high Reynolds-number, second-order closure models for c/u* 10, the present calculations predict less negative form stress (or less damping) of the waves compared to second-order closure models.

Air entrainment due to a translating axisymmetric laminar water jet plunging into an otherwise quiescent pool of water was studied experimentally. The jet impact diameter, Dj, and velocity, Vj, ranged from 0.33 to 1.21 cm and 104 to 365 cm/s, respectively. For all of these jet flow conditions, the water surface around the jet impact site of the stationary jet is smooth and no air is entrained. When the jet is moving horizontally with velocity Vt, a depression of the free surface forms directly downstream of the jet. In any set of experiments with constant Vj and Dj, as Vt is increased, the depth and streamwise length of this depression increases and a cusp forms at the bottom of its upstream edge. Air entrainment first occurs in the form of discrete small bubbles with diameters of about 0.05 cm that are injected from the cusp when the Froude number (Fr=Vt/gDj, where g is the acceleration of gravity) exceeds a critical value of about 1.4 for Vj/Vt>5. At higher values of Vt, a separate mode of entrainment occurs where large pockets of air are injected from the cusp into the flow in the form of intermittent bursts. At still higher values of Vt, the high-entrainment-rate condition occurs continuously. High-speed flow visualization movies of the entrainment process show that in all entrainment regimes, the bubbles enter the water when vortices from the jet shear layer pass over the leading-edge cusp of the depression and that these bubbles are initially trapped within the vortices. The boundaries between the three entrainment regimes are determined and presented on a plot of Froude number versus velocity ratio (Vj/Vt).

Internal wave effects generated by a solid body moving uniformly at constant depth through a density-stratified fluid of uniform depth are investigated. In particular an averaged potential energy Γ per unit track length is defined and analyzed under the assumptions of linear theory, uniform stratification, and rigid lid boundary conditions. A general expression is found for Γ in terms of the intrinsic causal structure of the fluid system and the geometric properties of the generating object. It is found that, as the speed of the body decreases to zero, Γ passes through an infinite sequence of infinite-amplitude resonances. A source function approximation is used to model the shape of the generating object and numerical evaluations are made for the case of a sphere. A consideration of the infinite-depth limit allows one to see Γ in terms of the resonance-inducing effects of the rigid lid boundary conditions in conjunction with the relatively smooth and finite “background resonances” associated with pure buoyancy effects.

Transcritical, or resonant, flow of a stratified fluid over an obstacle is studied using a forced extended Korteweg–de Vries model. This model is particularly relevant for a two-layer fluid when the layer depths are near critical, but can also be useful in other similar circumstances. Both quadratic and cubic nonlinearities are present and they are balanced by third-order dispersion. We consider both possible signs for the cubic nonlinear term but emphasize the less-studied case when the cubic nonlinear term and the dispersion term have the same-signed coefficients. In this case, our numerical computations show that two kinds of solitary waves are found in certain parameter regimes. One kind is similar to those of the well-known forced Korteweg–de Vries model and occurs when the cubic nonlinear term is rather small, while the other kind is irregularly generated waves of variable amplitude, which may continually interact. To explain this phenomenon, we develop a hydraulic theory in which the dispersion ter...

This is the second paper in a two part series in which we analyze two diffuse interface models to study pinchoff and reconnection in binary fluid flow in a Hele-Shaw cell with arbitrary density and viscosity contrast between the components. Diffusion between the components is limited if the components are macroscopically immiscible. In one of the systems (HSCH), the binary fluid may be compressible due to diffusion. In the other system (BHSCH), a Boussinesq approximation is used and the fluid is incompressible. In this paper, we focus on buoyancy driven flow and the Rayleigh–Taylor instability. In the fully nonlinear regime before pinchoff, results from the HSCH and BHSCH models are compared to highly accurate boundary-integral simulations of the classical sharp interface system. In this case, we find that the diffuse interface models yield nearly identical results and we demonstrate convergence to the boundary-integral solutions as the interface thickness vanishes. We find that the break-up of an unstabl...

This paper is devoted to an experimental characterization of the collapse under buoyancy forces of the turbulent motions in the wake of a sphere in a linearly stable saline stratification. This phenomenon is part of the transition between two commonly recognized regions of the wake, the near wake and the far wake. It is shown here that the wake evolves in four distinct stages: the three-dimensional near wake, the collapse, a transition region, and finally the far wake. Hot-film measurements in the near wake and the region beyond show that initially the mean defect velocity decreases as in a nonstratified fluid, but then increases briefly before continuing to decrease again. The short period of mean defect-velocity increase or mean-flow acceleration characterizes the collapse and results in the higher levels of kinetic energy found in stratified wakes as compared to nonstratified wakes. The far-wake region of the flow was studied by a digital particle image velocimetry technique yielding decay laws for the defect velocity as well as the vertical component of vorticity for both laminar and turbulent wake regimes. The decay of these far-wake quantities was found not to depend on the regime considered, implying that the decay rates are independent of the collapse. However, it was found that the characteristic longitudinal scales do depend on the Froude number, implying a selective memory of the initial conditions in spite of collapse.

We revisit the Rayleigh–Taylor instability when the two fluids are miscible and in the geometry of a Hele-Shaw cell. We provide analytical dispersion relations for the particular cases of either a sharp front between the two fluids or of a uniform density gradient stratification and for various fluid flow models, including an unbounded geometry, a two-dimensional gap-averaged Navier–Stokes–Darcy equation, and an effective porous medium. The results are compared to three-dimensional lattice BGK simulations, based on which the relevance of the various models in different wavelength regimes is discussed.

Criticality plays a central role in the study of reductions and stability of hydrodynamical systems. At critical points, it is often the case that nonlinear reductions with dispersion arise to govern solution behavior. By considering when such models become bidirectional and lose their initial dispersive properties, it will be shown that higher order dispersive models may be supported in hydrodynamical systems. Precisely, this equation is a two-way Boussinesq equation with sixth order dispersion. The case of two layered shallow water is considered to illustrate this, and it is reasoned why such an environment is natural for such a system to emerge. Further, it is demonstrated that the regions in the parameter space for nontrivial flow, which admit this reduction, are vast and in fact form a continuum. The reduced model is then numerically simulated to illustrate how the two-way and higher dispersive properties suggest more exotic families of solitary wave solutions can emerge in stratified flows.

The long-time evolution of the grid-generated turbulence in a stably stratified air flow is investigated using a duct that has both a vertical contraction and a lateral expansion. Through this duct a Brunt–Vaisala frequency N increases, while a constant mean velocity U is maintained, so that a nondimensional buoyancy time (Nt/2π≈1.3) longer than any other previous wind tunnel experiments is obtained. Two complete cycles of the normalized vertical buoyancy flux −wθ¯/w′θ′ oscillation have been observed for the first time experimentally. The present experiments show that the period of the vertical buoyancy flux oscillation is independent of the initial conditions, Prandtl number, Froude number, and Reynolds number, showing that it is determined by the linear processes as described by rapid distortion theory. Initial anisotropy produced by the distorting duct also does not change this period. At a stratification with N⩽2.5 s−1, we found that the oscillation of the vertical buoyancy flux with the higher Reynolds number and the Froude number decays faster, which may be explained by the nonlinear effect. Combination of the strongest stratification (N=3.0 s−1) with the smallest mesh size (M=2.54 cm) gave an internal wave behavior which could be identified by the phase spectra of Phwθ≈90° and by the vertical buoyancy flux of −wθ¯≈0. The ratio of potential energy to vertical kinetic energy becomes about 0.8 at the most downstream measuring point. The effect of mean strain on the stratified turbulence is investigated by comparing the present experiment for laterally distorted flow with the previous experiments for the nondistorted or longitudinally distorted flow. Strong countergradient flux is observed in both the laterally and longitudinally distorted flows. We found that the maximum value of the ratio of potential energy to vertical kinetic energy of the distorted flow becomes three times as large as that of the nondistorted flow. The spectral analysis shows that this large ratio has a significant effect on the strong countergradient flux in the distorted flow.The long-time evolution of the grid-generated turbulence in a stably stratified air flow is investigated using a duct that has both a vertical contraction and a lateral expansion. Through this duct a Brunt–Vaisala frequency N increases, while a constant mean velocity U is maintained, so that a nondimensional buoyancy time (Nt/2π≈1.3) longer than any other previous wind tunnel experiments is obtained. Two complete cycles of the normalized vertical buoyancy flux −wθ¯/w′θ′ oscillation have been observed for the first time experimentally. The present experiments show that the period of the vertical buoyancy flux oscillation is independent of the initial conditions, Prandtl number, Froude number, and Reynolds number, showing that it is determined by the linear processes as described by rapid distortion theory. Initial anisotropy produced by the distorting duct also does not change this period. At a stratification with N⩽2.5 s−1, we found that the oscillation of the vertical buoyancy flux with the higher Reynol...

Vortex double layers and vortex projectiles (VPs) are the essential coherent structures which emerge in the shock excited planar-parallel “curtain” (s/f/s) simulations of a two-dimensional shock tube with the piecewise parabolic method. These opposite signed layers, formed by shock induced baroclinic deposition of vorticity, “bind” and are strongly affected by secondary reflected shocks and vortex interactions. This versatile configuration, easily obtained in the laboratory, allows us to study the formation, evolution, and reacceleration of VPs and stratified turbulence and mixing.

It is shown theoretically that an electric field can be used to control and suppress the classical Rayleigh-Taylor instability found in stratified flows when a heavy fluid lies above lighter fluid. Dielectric fluids of arbitrary viscosities and densities are considered and a theory is presented to show that a horizontal electric field (acting in the plane of the undisturbed liquid-liquid surface), causes growth rates and critical stability wavenumbers to be reduced thus shifting the instability to longer wavelengths. This facilitates complete stabilization in a given finite domain above a critical value of the electric field strength. Direct numerical simulations based on the Navier-Stokes equations coupled to the electrostatic fields are carried out and the linear theory is used to critically evaluate the codes before computing into the fully nonlinear stage. Excellent agreement is found between theory and simulations, both in unstable cases that compare growth rates and in stable cases that compare frequencies of oscillation and damping rates. Computations in the fully nonlinear regime supporting finger formation and roll-up show that a weak electric field slows down finger growth and that there exists a critical value of the field strength, for a given system, above which complete stabilization can take place. The effectiveness of the stabilization is lost if the initial amplitude is large enough or if the field is switched on too late. We also present a numerical experiment that utilizes a simple on-off protocol for the electric field to produce sustained time periodic interfacial oscillations. It is suggested that such phenomena can be useful in inducing mixing. A physical centimeter-sized model consisting of stratified water and olive oil layers is shown to be within the realm of the stabilization mechanism for field strengths that are approximately 2 × 104 V/m.

The aim of this contribution is to present the results of experiments conducted to elucidate the conditions of formation and evolution of planar dipolar eddies in a stratified fluid in the presence of a vertical background shear. The flow regime diagram is derived and conditions under which the formation of dipolar eddies is possible are found and explained. The estimates for the dipole lifetime are given. Possible applications include oceanic mushroom-like currents, late stratified wakes and other cases when dipolar or quasi-dipolar eddies are observed in a stratified shear flow.

At sufficiently high values of the Reynolds number (Re⩾4.5×103) and internal Froude number (F⩾4), initially turbulent bluff body wakes evolve in the presence of a stable background density gradient with wake-averaged mean and turbulence length and velocity scales that are independent of Re and F for at least two orders of magnitude extension in both parameters. The way in which the initially three-dimensional motions transition to the characteristic (and Re- and F-independent) late wakes (where vertical velocities, w≪u,v) is both of great practical interest, and complex, hence somewhat unclear. Here, digital particle imaging velocimetry type measurements on towed-sphere wakes are described, so that the development of anisotropy can be measured by the time development of turbulence profiles in horizontal and vertical centerplanes. The observed anisotropies can be associated with energy transfer to internal wave modes, and suppression of other vertical displacements, that contrasts with sphere wakes at simi...