Fluids Of Different Densities Research Articles

Theoretical study on finite-thickness effect on harmonics in Richtmyer-Meshkov instability for arbitrary Atwood numbers

The finite-thickness effect of two superimposed fluids on harmonics in the Richtmyer-Meshkov instability (RMI) for arbitrary Atwood numbers is investigated by using weakly nonlinear analysis up to the third order. When the thickness of the two fluids tends to be infinity, our results can reproduce the classical results where RMI happens at the interface separating two semi-infinity-thickness fluids of different densities. It is found that the thickness has a large influence on the amplitudes of the first three harmonics compared with those in classical RMI. On the one hand, the thickness effect encourages or reduces the amplitudes of the first three harmonics, and on the other hand, it changes the phases of the second and the third harmonics.

Elementary symmetrization of inviscid two-fluid flow equations giving a number of instant results

We consider two models of a compressible inviscid isentropic two-fluid flow. The first one describes the liquid–gas two-phase flow. The second one can describe the mixture of two fluids of different densities or the mixture of fluid and particles. Introducing an entropy-like function, we reduce the equations of both models to a symmetric form which looks like the compressible Euler equations written in the nonconservative form in terms of the pressure, the velocity and the entropy. Basing on existing results for the Euler equations, this gives a number of instant results for both models. In particular, we conclude that all compressive shock waves in these models exist locally in time. For the 2D case, we make the conclusion about the local-in-time existence of vortex sheets under a “supersonic” stability condition. In the sense of a much lower regularity requirement for the initial data, our result for 2D vortex sheets essentially improves a recent result for vortex sheets in the liquid–gas two-phase flow.

Conditions for the development of Rayleigh-Taylor instability on the spherical flame front

The instability of the flame front is one of the actual problems of modern fluid and gas mechanics. Depending on the processes determining the growth of inhomogeneities, the hydrodynamic (Darrieus-Landau), diffusive-thermal, and Rayleigh-Taylor instabilities are distinguished for a free spherical flame. The present work is devoted to the conditions for the development of the latter. Rayleigh-Taylor instability arises at the interface of two fluids of different density in the presence of acceleration directed from the lighter to the heavier one. When the flame propagates, combustion products with less density move with acceleration in a denser gas. When the acceleration vector and the density gradient at the flame front are co-directed, instability develops. To investigate the development of instabilities, a series of experiments was carried out. Transparent latex shells were filled with a pre-prepared hydrogen-air mixture. In different series of experiments, the percentage of hydrogen changed. The flame was ignited by a spark discharge with energy of 1 mJ, a spark gap located in the center of the shell. Registration of the flame front propagation was performed using the schlieren method implemented in the shadow device IAB-451 and the high-speed VideoSprint camera. The video was shot at a frame rate of 500 to 1000 fps with an exposure of 500 μs. To automate the processing of images in the Matlab environment, a program is written that converts a set of images of the expanding flame front into a dependence of the mean radius on time. It is found that at the initial stage of propagation there are both acceleration and deceleration of the flame front. Experimentally obtained propagation parameters of the flame front are supplemented by calculations carried out on an analytical model from the literature.

Analysis of dynamics, stability, and flow fields' structure of an accelerated hydrodynamic discontinuity with interfacial mass flux by a general matrix method

We develop a general matrix method to analyze from a far field the dynamics of an accelerated interface between incompressible ideal fluids of different densities with interfacial mass flux and with negligible density variations and stratification. We rigorously solve the linearized boundary value problem for the dynamics conserving mass, momentum, and energy in the bulk and at the interface. We find a new hydrodynamic instability that develops only when the acceleration magnitude exceeds a threshold. This critical threshold value depends on the magnitudes of the steady velocities of the fluids, the ratio of their densities, and the wavelength of the initial perturbation. The flow has potential velocity fields in the fluid bulk and is shear-free at the interface. The interface stability is set by the interplay of inertia and gravity. For weak acceleration, inertial effects dominate, and the flow fields experience stable oscillations. For strong acceleration, gravity effects dominate, and the dynamics is unstable. For strong accelerations, this new hydrodynamic instability grows faster than accelerated Landau-Darrieus and Rayleigh-Taylor instabilities. For given values of the fluids' densities and their steady bulk velocities, and for a given magnitude of acceleration, we find the critical and maximum values of the initial perturbation wavelength at which this new instability can be stabilized and at which its growth is the fastest. The quantitative, qualitative, and formal properties of the accelerated conservative dynamics depart from those of accelerated Landau-Darrieus and Rayleigh-Taylor dynamics. New diagnostic benchmarks are identified for experiments and simulations of unstable interfaces.

Almost limiting configurations of steady interfacial overhanging gravity waves

We study progressive gravity waves at the interface between two unbounded fluids of different densities. The main concern is to find almost limiting configurations for the so-called overhanging waves. The latter were first computed by Meiron & Saffman (J. Fluid Mech., vol. 129, 1983, pp. 213–218). By means of the Hopf lemma, we rigorously prove that, if $\unicode[STIX]{x1D703}$ is the angle between the tangent line to the interfacial curve and the horizontal direction, then $-\unicode[STIX]{x03C0}<\unicode[STIX]{x1D703}<\unicode[STIX]{x03C0}$. This inequality allows us to put forward a criterion of proximity of the interface to the limiting configuration, namely, the angle $|\unicode[STIX]{x1D703}|_{max}$ must be close to $\unicode[STIX]{x03C0}$ but may not exceed $\unicode[STIX]{x03C0}$. We develop a new numerical method of computing interfacial waves based on the representation of a piecewise-analytic function to be found in such a manner that only the shape of the interface is unknown. All other hydrodynamic quantities can be expressed analytically in terms of functions describing this shape. Using this method, we compute almost limiting configurations of interfacial waves with $|\unicode[STIX]{x1D703}|_{max}>179.98^{\circ }$. Analysing the results of computations, we introduce two new concepts: an inner crest, and an inner solution near the inner crest. These concepts allow us to make a well-grounded prediction for the shapes of limiting interfacial configurations and confirm Saffman & Yuen’s (J. Fluid Mech., vol. 123, 1982, pp. 459–476) conjecture that the waves are geometrically limited.

A conformal-mapping model for intrusion

The conformal-mapping approach is used to formulate a minimal exactly solvable model for studying the lock-exchange gravity currents in an infinite horizontal channel. The distinctive feature of such flows is a pressure jump which arises at the interface between two incompressible fluids of different densities at once after removing a lock gate. Because of this, the process of intrusion does not fit into Benjamin’s scenario but rather is like propagation of two shock waves. Being identical in shape and speed, they move in opposite directions along the upper and lower channel boundaries, reaching a steady state as t → ∞.

Simulation of fluid dynamics through complicated networks of channels with cellular automata

To simulate the behavior of a liquid flowing through complicated regions of channels and porous media, we propose to simplify the problem using cellular automata (CA) simulation, in order to study qualitatively the behavior of the fluid dynamics. CA allows us to work with complicated border conditions by simply considering, or not, a particular cell to be part of the fluid flow. The fluid flow in the CA is achieved by setting the following rules of behavior: it is considered that the movement of fluid particles can only be due to gravity (vertically) and/or horizontally (for space availability) using probability values; upward movements can only be due to the balance of hydrostatic pressure outside the framework of CA; also, the particles will not be able to move to spaces already occupied by the fluid or spaces that form the walls of the structure. It was further possible to simulate the simultaneous behavior of two fluids of different density that cannot be mixed due to the programming simplicity of CA. Several case studies are discussed.

Oscillatory instability in a reaction front separating fluids of different densities.

Reaction fronts described by the Kuramoto-Sivashinsky (KS) equation can exhibit complex behavior as they separate reacted from unreacted fluids. If the fluid of higher density is above a fluid of lower density, then the Rayleigh-Taylor instability can lead to fluid motion. In the reverse situation, where the lighter fluid is on top, gravitationally driven forces can stabilize a convectionless flat front inhibiting the complex front propagation described by the KS equation. In these cases, a critical density difference is required to provide stability to the flat front. A linear stability analysis shows that the transition from stable to unstable flat fronts can be oscillatory for viscous fluid motion. Once the transition takes place, the fronts exhibit oscillatory convection resulting in oscillations of the shape and speed of the front.

The structure and origin of confined Holmboe waves

Finite-amplitude manifestations of stratified shear flow instabilities and their spatio-temporal coherent structures are believed to play an important role in turbulent geophysical flows. Such shear flows commonly have layers separated by sharp density interfaces, and are therefore susceptible to the so-called Holmboe instability, and its finite-amplitude manifestation, the Holmboe wave. In this paper, we describe and elucidate the origin of an apparently previously unreported long-lived coherent structure in a sustained stratified shear flow generated in the laboratory by exchange flow through an inclined square duct connecting two reservoirs filled with fluids of different densities. Using a novel measurement technique allowing for time-resolved, near-instantaneous measurements of the three-component velocity and density fields simultaneously over a three-dimensional volume, we describe the three-dimensional geometry and spatio-temporal dynamics of this structure. We identify it as a finite-amplitude, nonlinear, asymmetric confined Holmboe wave (CHW), and highlight the importance of its spanwise (lateral) confinement by the duct boundaries. We pay particular attention to the spanwise vorticity, which exhibits a travelling, near-periodic structure of sheared, distorted, prolate spheroids with a wide ‘body’ and a narrower ‘head’. Using temporal linear stability analysis on the two-dimensional streamwise-averaged experimental flow, we solve for three-dimensional perturbations having two-dimensional, cross-sectionally confined eigenfunctions and a streamwise normal mode. We show that the dispersion relation and the three-dimensional spatial structure of the fastest-growing confined Holmboe instability are in good agreement with those of the observed confined Holmboe wave. We also compare those results with a classical linear analysis of two-dimensional perturbations (i.e. with no spanwise dependence) on a one-dimensional base flow. We conclude that the lateral confinement is an important ingredient of the confined Holmboe instability, which gives rise to the CHW, with implications for many inherently confined geophysical flows such as in valleys, estuaries, straits or deep ocean trenches. Our results suggest that the CHW is an example of an experimentally observed, inherently nonlinear, robust, long-lived coherent structure which has developed from a linear instability. We conjecture that the CHW is a promising candidate for a class of exact coherent states underpinning the dynamics of more disordered, yet continually forced stratified shear flows.

A turbulent diffusion model of Rayleigh-Taylor mixing

A simple turbulent diffusion model is described for predicting late term mixing due to Rayleigh-Taylor instability at an interface that initially separates two semi-infinite regions of fluid of different densities. The one-dimensional, transient species conservation equation is combined with an available vertical dispersion equation which relates the turbulent diffusivity to the local density gradient and a characteristic mixing length. The similarity solution of the species conservation equation yields the well-known long time result for the vertical extent of the mixing region and shows that the dimensionless growth constant that appears in the mixing thickness equation is simply a measure of the size of the characteristic mixing length. A simple polynomial relation is derived for the heavy fluid volume fraction profile within the mixing region which is in close agreement with the profile obtained from a previously reported numerical simulation of Rayleigh-Taylor mixing. It is demonstrated how the model can be readily applied to predict Rayleigh-Taylor mixing in a finite fluid region between top and bottom boundaries.

Oscillations of oblate drop between heterogeneous plates under uniform electric field

The forced oscillations of the incompressible fluid drop under the action of the uniform electric field are considered. In equilibrium, the drop has the form of a circular cylinder bounded axially by the parallel solid planes; the contact angle is right. An incompressible fluid of different density surrounds the drop. The external electric field acts as an external force that causes motion of the contact line. In order to describe this contact line motion, the modified Hocking boundary condition is applied: the velocity of the contact line is proportional to the deviation of the contact angle and the speed of the fast relaxation processes, whose frequency is proportional to twice the frequency of the electric field. The case of heterogeneous plates is investigated. We assume that the Hocking parameter depends on the polar angle in this case. The function describing the change in the coefficient of the interaction between the plate and the fluid (the contact line) is expanded in a series of the Laplace operator eigenfunctions.

Effects of Heterogeneity, Connectivity, and Density Variations on Mixing and Chemical Reactions Under Temporally Fluctuating Flow Conditions and the Formation of Reaction Patterns

AbstractSolute mixing, spreading, and fast chemical reactions in aquifers are strongly influenced by spatial variability of the hydraulic properties, temporal flow fluctuations, and fluid density differences. We study the coupling of heterogeneity, transient forcing, and density‐driven flow on mixing and chemical reactions between two fluids of different density under a stable stratification. We consider the reaction of the fast dissolution of calcite. We find that temporal fluctuations and heterogeneity cause strong local enhancement of the mixing and reaction rates and this impact increases with the degree of connectivity of hydraulic conductivity. The global mixing and reactivity, however, are on the order of or smaller than their homogeneous counterparts due to heterogeneity‐induced fluid segregation. The local maxima of the mixing and reaction rates are found to be located around strongly stretched regions corresponding to high velocity zones where dispersive mass transfer mechanisms are increased by dispersion. We also find that density variations compress the interface, which in turn emphasizes local maxima in mixing and reaction rates. Numerical results provide evidence that the stretching of the interface induced by spatial heterogeneity and transient effects coupled with density variations lead to the formation of complex patterns of reactive hotspots, zones of enhanced reaction efficiency, and that its distribution is directly linked to the deformation properties and topology of the flow field. These results provide new insights into the role of spatial and temporal variability on the mixing and reaction efficiency as well as the formation of reactive geochemical patterns in actual environmental systems.

What is the final size of turbulent mixing zones driven by the Faraday instability?

Miscible fluids of different densities subjected to strong time-periodic accelerations normal to their interface can mix due to Faraday instability effects. Turbulent fluctuations generated by this mechanism lead to the emergence and the growth of a mixing layer. Its enlargement is gradually slowed down as the resonance conditions driving the instability cease to be fulfilled. The final state corresponds to a saturated mixing zone in which the turbulence intensity progressively decays. A new formalism based on second-order correlation spectra for the turbulent quantities is introduced for this problem. This method allows for the prediction of the final mixing zone size and extends results from classical stability analysis limited to weakly nonlinear regimes. We perform at various forcing frequencies and amplitudes a large set of homogeneous and inhomogeneous numerical simulations, extensively exploring the influence of initial conditions. The mixing zone widths, measured at the end of the simulations, are satisfactorily compared to the predictions, and bring a strong support to the proposed theory. The flow dynamics is also studied and reveals the presence of sub-harmonic as well as harmonic modes depending on the initial parameters in the Mathieu phase diagram. Important changes in the flow anisotropy, corresponding to the large scale structures of turbulence, occur. This phenomenon appears directly related to the orientation of the most amplified gravity waves excited in the system, evolving due to the enlargement of the mixing zone.

Resonant interfacial capillary–gravity waves in the presence of damping effects

An investigation is made into the waves of small and moderate amplitude which may occur at the interface of two inviscid fluids of different densities. The external forces are those of gravity and surface tension and the waves are due to the resonant interaction between the M th and N th harmonics of the fundamental mode. In contrast to previous studies, damping effects are taken into account. Important parameters in the problem are the velocity and density ratios. A pair of coupled nonlinear Schrodinger-type partial differential equations for the wave amplitudes is derived which model the evolution of the waves, correct up to third order. A wide variety of sinusoidal solutions to the equations is shown to exist, irrespective of the values assigned to the parameters. The stability of these solutions to small modulational perturbations is considered. It is found that when the damping is due to dissipation then the waves are stabilised.

Two-phase gravity currents resulting from the release of a fixed volume of fluid in a porous medium

We consider the instantaneous release of a finite volume of fluid in a porous medium saturated with a second, immiscible fluid of different density. The resulting two-phase gravity current exhibits a rich array of behaviours due to both the residual trapping of fluid as the current recedes and the differing effects of surface tension between advancing and receding regions of the current. We develop a framework for the evolution of such a current with particular focus on the large-scale implications of the form of the constitutive relation between residual trapping and initial saturation. Pore-scale hysteresis within the current is represented by families of scanning curves relating capillary pressure and relative permeability to saturation. In the resulting vertically integrated model, all capillary effects are incorporated within specially defined saturation and flux functions specific to each region. In the long-time limit, when the height of the current and the saturations within it are low, the saturation and flux functions can be approximated by mathematically convenient power laws. If the trapping model is approximately linear at low saturations, the equations admit a similarity solution for the propagation rate and height profile of the late-time gravity current. We also solve the governing partial differential equation numerically for the nonlinear Land’s trapping model, which is commonly used in studies of two-phase flows. Our investigation suggests that for trapping relations for which the proportion of trapped to initial fluid saturation increases and tends to unity as the initial saturation decreases, both of which are properties of Land’s model, a gravity current slows and eventually stops. This trapping behaviour has important applications, for example to the ultimate distance contaminants or stored carbon dioxide may travel in the subsurface.

Inviscid linear stability analysis of two vertical columns of different densities in a gravitational acceleration field

We study the inviscid linear stability of a vertical interface separating two fluids of different densities and subject to a gravitational acceleration field parallel to the interface. In this arrangement, the two free streams are constantly accelerated, which means that the linear stability analysis is not amenable to Fourier or Laplace solution in time. Instead, we derive the equations analytically by the initial-value problem method and express the solution in terms of the well-known parabolic cylinder function. The results, which can be classified as an accelerating Kelvin–Helmholtz configuration, show that even in the presence of surface tension, the interface is unconditionally unstable at all wavemodes. This is a consequence of the ever increasing momentum of the free streams, as gravity accelerates them indefinitely. The instability can be shown to grow as the exponential of a quadratic function of time.

Compressibility effects on outflows in a two-fluid system. 2. Point source in spherical geometry

Fluid outflow from a point source into a surrounding fluid of different density is considered. A sharp interface separates the two fluids, and its mean radius increases with time due to the mass produced by the source. The inner fluid is incompressible and inviscid, but the outer fluid is weakly compressible and is modelled using a Boussinesq approximation. A linearized theory is presented, and it assumes that disturbances to the overall outflow remain small in amplitude. A spectral scheme for solving the non-linear problem is discussed. The results demonstrate that compressibility acts to suppress the Rayleigh–Taylor type instability of the interface, which would occur if both fluids were incompressible. In addition, the compressibility of the outer fluid forces the source within the inner incompressible fluid to behave in a more complicated manner while still preserving the overall ejected mass flux. This is confirmed in both the linearized and non-linear solutions.

The interaction of thermocapillary and vibrational instabilities in a two-layer system subjected to the tangential vibrations

The paper deals with the investigation of interaction of thermocapillary, parametric and frozen wave instability mechanisms in a two-layer system of immiscible incompressible viscous fluids of different densities (more dense fluid is located below). External boundaries are rigid. They are maintained at constant different temperatures. The system is subjected to the gravity force and horizontal vibrations of finite frequency and amplitude. The study is performed taking into account the interface deformations. The thermal expansion of fluids is neglected. It is found that vibrations make strong destabilizing effect on the longwave instability mode related to the interface deformations and stabilizing effect on Pearson instability mode.

The effects of fluid motion on oscillatory and chaotic fronts

We study oscillatory and chaotic reaction fronts described by the Kuramoto-Sivashinsky equation coupled to different types of fluid motion. We first apply a Poiseuille flow on the fronts inside a two-dimensional slab. We show regions of period doubling transition to chaos for different values of the average speed of Poiseuille flow. We also analyze the effects of a convective flow due to a Rayleigh-Taylor instability. Here the front is a thin interface separating two fluids of different densities inside a two-dimensional vertical slab. Convection is caused by buoyancy forces across the front as the lighter fluid is under a heavier fluid. We first obtain oscillatory and chaotic solutions arising from instabilities intrinsic to the front. Then, we determine the changes on the solutions due to fluid motion.

Optimal control of time discrete two‐phase flow governed by a diffuse interface model

AbstractWe present results on optimal control of two‐phase flows. The fluid is modeled by a thermodynamically consistent diffuse interface model and allows to treat fluids of different densities and viscosities. In earlier work we proposed an energy stable time discretization for this model that we now employ to derive existence of optimal controls for a time discrete optimal control problem. The control aim is to obtain a desired distribution of the two phases in the system. For this we investigate three control actions. We use tangential Dirichlet boundary control and distributed control. We further consider the inverse problem of finding an initial distribution such that the evolution over a given time horizon starting from this value is close to a desired distribution. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)