Two-layer Shear Flow Research Articles

Singular effect of interfacial slip for an otherwise stable two-layer shear flow: analysis and computations

We consider instability of the flat interface in a two-layer Couette flow model developed earlier (Kalogirou & Papageorgiou, 2016, J. Fluid Mech. 802 , 5–36; Katsiavria & Papageorgiou, 2022, Wave Motion 114 , 103018. ( doi:10.1016/j.wavemoti.2022.103018 )) for a thin layer near one of the walls. For the case when the less viscous fluid resides next to the moving wall, we find that even a small slip effect at the interface can destabilize an otherwise highly stable flow to the Turing-type instability. The singular effect of small slip in an otherwise very stable configuration may have important ramifications in physical and technological applications. The neutral points of the dispersion relation give rise to travelling wave solutions that are continued to finite amplitude numerically and their linear stability properties identified for a set of parameter values for disturbances that include subharmonic modes with twice the wavelength of the nonlinear travelling wave. We determined Hopf and regular bifurcation points of travelling waves and rigorously justified their existence for some set of parameter values. Weakly nonlinear analysis close to bifurcation from a flat state is also presented for small amplitude waves in general. We also present global existence and regularity results for periodic initial conditions without any restriction on parameters.

Nonlinear dynamics of unstably stratified two-layer shear flow in a horizontal channel

The Rayleigh–Taylor instability at the interface of two sheared fluid layers in a horizontal channel is investigated in the absence of inertia. The dynamics of the flow is described by a nonlinear lubrication equation which is solved numerically for adverse density stratifications. The early-time dynamics features a number of finger-like protrusions of different heights at the interface. The fingers travel at different speeds leading to a sequence of merging events after which the interface eventually settles to a near-saturated state, comprising only one finger that includes most of the lower fluid. For sufficiently large density stratifications, the final state spans the height of the channel and includes two thin fluid films at each wall, both of which undergo chaotic dynamics, but finite-time touch-down/touch-up is shown to be precluded by the shear flow. An asymptotic analysis in the large-Bond-number limit (intense density stratification) reveals the finer structure of the final state including Landau–Levich-type connection regions. The asymptotic solutions are compared with numerical results of the lubrication model as well as direct numerical simulations, and excellent agreement is observed between the three in terms of interfacial structure, wave speed and film thicknesses.

Обоснование математической модели многослойного сдвигового течения бетонной смеси по наклонной поверхности откоса канала

In assessing the hydrodynamic parameters of concrete mixtures placed in monolithic channel linings, the most important point is the development of mathematical models that describe the processes of concrete mixture slippage, taking into account the dynamics of speed reduction and increase in the viscosity of the concrete mixture over time. The main regularities of the hydrodynamic multilayer shear flow of a concrete mixture on the surface of a flat channel slope, where a multilayer plane-parallel viscous flow moves under the action of gravity, are investigated. The boundary value problem of a two-layer flow of a concrete mixture is solved, a system of two layers of viscous incompressible liquids is considered with the properties of each, respectively, the thickness of the bottom and upper layers, taking into account the sliding velocity of particles of a viscous liquid located one above the other. The obtained analytical solution of the problem in the stationary regime of a two-layer shear flow of two different immiscible viscous fluids with different properties makes it possible to formulate the main laws of such flows and describe the processes of sliding of a multilayer concrete mixture, taking into account the dynamics of speed reduction and increase in the viscosity of the concrete mixture over time. A mathematical model has been developed where the dynamic and kinematic structures of a multilayer shear flow in both viscous layers are presented separately.

An arbitrary Lagrangian-Eulerian method for simulating interfacial dynamics between a hydrogel and a fluid

Hydrogels are crosslinked polymer networks swollen with an aqueous solvent, and play central roles in biomicrofluidic devices. In such applications, the gel is often in contact with a flowing fluid, thus setting up a fluid-hydrogel two-phase system. Using a recently proposed model (Young et al. [41] 2019), we treat the hydrogel as a poroelastic material consisting of a Saint Venant-Kirchhoff polymer network and a Newtonian viscous solvent, and develop a finite-element method for computing flows involving a fluid-hydrogel interface. The interface is tracked by using a fixed-mesh arbitrary Lagrangian-Eulerian method that maps the interface to a reference configuration. The interfacial deformation is coupled with the fluid and solid governing equations into a monolithic algorithm using the finite-element library deal.II. The code is validated against available analytical solutions in several non-trivial flow problems: one-dimensional compression of a gel layer by a uniform flow, two-layer shear flow, and the deformation of a Darcy gel particle in a planar extensional flow. In all cases, the numerical solutions are in excellent agreement with the analytical solutions. Numerical tests show second-order convergence with respect to mesh refinement, and first-order convergence with respect to time-step refinement.

Finite volume Ghost Fluid Method implementation of interfacial forces in PISO loop

In this work, we implement interfacial forces (surface tension, hydrostatic and viscous forces) by enlisting the finite volume discretization of GFM (Ghost Fluid Method) using A-CLSVOF (Algebraic Coupled Level-Set/Volume-of-Fluid) method for the mass conservative, and smooth interface description. The widely used PISO momentum solution to resolve the pressure–velocity coupling is presented along with the present GFM discretization and its placement within the PISO loop. The pressure jump at the interface due to the interfacial forces is made sharp via direct calculation of the modified pressure matrix coefficients corresponding to targeted interfacial cells, and as a source term for the jump value itself. The Level-Set field is enlisted for curvature computation in A-CLSVOF and for the interpolation and weighting of the relative contribution of the capillary force in adjacent for the matrix coefficients in the FV framework. To assess the A-CLSVOF/GFM performance, four canonical cases were studied. In the case of a static droplet in suspension, A-CLSVOF/GFM produces a sharp and accurate pressure jump compared to the traditional CSF implementation of A-CLSVOF. The interaction of viscous and capillary forces is proven to be accurate and consistent with theoretical results for the classical capillary wave. For the linear two-layer shear flow, GFM sharp treatment of the viscosity captured the velocity gradient across the interface and removed the diffusion of the viscous stresses caused by the discontinuous material properties. Finally, the combination of all GFM improvements proposed in this study are compared to experimental findings of terminal velocity for a gaseous bubble rising in a viscous fluid. GFM outperforms CSF with errors of 4.6% and 14.0% respectively.

Waves in two-layer shear flow for viscous and inviscid fluids

A two-layer shear flow is studied for inviscid and viscous fluids. Here, the layers flow between two horizontal walls and are buoyantly stable. Each layer contains a finite amount of shear and the horizontal velocity is specified such that it is continuous when unperturbed. The interface between the two layers is given a small sinusoidal perturbation and the subsequent response of the system is studied. Different solution techniques are employed for the inviscid and viscous flows. These both rely on linearizing the governing equations for each of these flows. In particular, the viscous flow is constrained to remain within a small perturbation of the unperturbed flow as it evolves. This assumption is justified since standing wave behaviour is expected in the inviscid case. Solutions are presented for a variety of different values of the shear parameters and the way these parameter choices affect the interaction between vorticity and density in the viscous case is investigated in detail. These linearized solutions are confirmed by comparison with fully non-linear results obtained numerically.

A study of nonlinear waves and resonance in intrusion flows

A stratified intrusion flow is considered in which there are three moving (horizontal) fluid layers and two interfaces. The top and bottom layers move with different speeds and may even move in opposite directions, producing an exchange flow. The middle layer is in motion relative to the outer two, and possesses shear so that the speed in the three-fluid system is continuous when the interfaces are both unperturbed. The flow configuration supports the propagation of periodic waves. A linearized analysis for small wave amplitudes is presented. This is compared to some nonlinear periodic solutions found numerically using a Fourier technique. Such solutions permit nonlinear resonances between the various solution modes and these have been computed extensively. References P. O. Rusas and J. Grue. Solitary waves and conjugate flows in a three-layer fluid. European Journal of Mechanics, B/Fluids, 21:185--206, 2002. doi:10.1016/S0997-7546(01)01163-3. L. K. Forbes, G. C. Hocking, and D. E. Farrow. An intrusion layer in stationary incompressible fluids: Part 1: Periodic waves. Euro. Jnl of Applied Mathematics, 17:557--575, 2006. doi:10.1017/S0956792506006711. G. C. Hocking and L. K. Forbes. A note on the flow of a homogeneous intrusion into a two-layer fluid. Euro. Jnl of Applied Mathematics, 18:181--193, 2007. doi:10.1017/S0956792507006924. H. Michallet and F. Dias. Non-linear resonance between short and long waves. Proc. of the 9th International Offshore and Polar Engineering Conference, pages 193--198, 1999. http://cat.inist.fr/?aModele=afficheN&cpsidt=1174804. D. I. Pullin and R. H. J. Grimshaw. Interfacial progressive gravity waves in a two-layer shear flow. Phys. Fluids, 26:1731--1739, 1983. doi:10.1063/1.864372.

Kelvin–Helmholtz Instability as a Boundary-Value Problem

The Kelvin–Helmholtz (KH) instability is traditionally viewed as an initial-value problem, wherein wave perturbations of a two-layer shear flow grow over time into billows and eventually generate vertical mixing. Yet, the instability can also be viewed as a boundary-value problem. In such a framework, there exists an upstream condition where a lighter fluid flows over a denser fluid, wave perturbations grow downstream to eventually overturn some distance away from the point of origin. As the reverse of the traditional problem, this flow is periodic in time and exhibits instability in space. A natural application is the mixing of a warmer river emptying into a colder lake or reservoir, or the salt-wedge estuary. This study of the KH instability from the perspective of a boundary-value problem is divided into two parts. Firstly, the instability theory is conducted with a real frequency and complex horizontal wavenumber, and the main result is that the critical wavelength at the instability threshold is longer in the boundary-value than in the initial-value situation. Secondly, mass, momentum and energy budgets are performed between the upstream, unmixed state on one side, and the downstream, mixed state on the other, to determine under which condition mixing is energetically possible. Cases with a rigid lid and free surface are treated separately. And, although the algebra is somewhat complicated, both end results are identical to the criterion for complete mixing in the initial-value problem.

Spontaneous generation and impact of inertia‐gravity waves in a stratified, two‐layer shear flow

Inertia‐gravity waves exist ubiquitously throughout the stratified parts of the atmosphere and ocean. They are generated by local velocity shears, interactions with topography, and as geostrophic (or spontaneous) adjustment radiation. Relatively little is known about the details of their interaction with the large‐scale flow, however. We report on a joint model/laboratory study of a flow in which inertia‐gravity waves are generated as spontaneous adjustment radiation by an evolving large‐scale mode. We show that their subsequent impact upon the large‐scale dynamics is generally small. However, near a potential transition from one large‐scale mode to another, in a flow which is simultaneously baroclinically‐unstable to more than one mode, the inertia‐gravity waves may strongly influence the selection of the mode which actually occurs.

Higher-order Korteweg-de Vries models for internal solitary waves in a stratified shear flow with a free surface

Abstract. A higher-order extension of the familiar Korteweg-de Vries equation is derived for internal solitary waves in a density- and current-stratified shear flow with a free surface. All coefficients of this extended Korteweg-de Vries equation are expressed in terms of integrals of the modal function for the linear long-wave theory. An illustrative example of a two-layer shear flow is considered, for which we discuss the parameter dependence of the coefficients in the extended Korteweg-de Vries equation.

Instability due to second normal stress jump in two-layer shear flow of the Giesekus fluid

The two-layer Couette flow of superposed Giesekus liquids is examined. In order to emphasize the effect of a jump in the second normal stress difference, the analysis is focused on flows where the shear rate and first normal stress difference are continuous across the interface. In this case, the flow is neutrally stable to streamwise disturbances, but can be unstable for spanwise disturbances driven by a jump in the second normal stress difference. Whether the long and order one waves are stable or not depends on the sign of this difference. Short waves are unstable. In the case of order one wave instability, the mode of maximum growth rate gives rise to stationary ripples perpendicular to the flow. The eigenvalue problem for purely spanwise wave vectors can in principle be solved analytically, although, in general, the analytical solution is too complicated to obtain. In most cases, however, a simplifying assumption can be made which makes analytical solutions feasible. We present such solutions and compare them with purely numerical solutions.

Stability of a potential vorticity front: from quasi-geostrophy to shallow water

The linear stability of a simple two-layer shear flow with an upper-layer potential vorticity front overlying a quiescent lower layer is investigated as a function of Rossby number and layer depths. This flow configuration is a generalization of previously studied flows whose results we reinterpret by considering the possible resonant interaction between waves. We find that instabilities previously referred to as ‘ageostrophic’ are a direct extension of quasi-geostrophic instabilities.Two types of instability are discussed: the classic long-wave quasi-geostrophic baroclinic instability arising from an interaction of two vortical waves, and an ageostrophic short-wave baroclinic instability arising from the interaction of a gravity wave and a vortical wave (vortical waves are defined as those that exist due to the presence of a gradient in potential vorticity, e.g. Rossby waves). Both instabilities are observed in oceanic fronts. The long-wave instability has length scale and growth rate similar to those found in the quasi-geostrophic limit, even when the Rossby number of the flow is O(1).We also demonstrate that in layered shallow-water models, as in continuously stratified quasi-geostrophic models, when a layer intersects the top or bottom boundaries, that layer can sustain vortical waves even though there is no apparent potential vorticity gradient. The potential vorticity gradient needed is provided at the top (or bottom) intersection point, which we interpret as a point that connects a finite layer with a layer of infinitesimal thickness, analogous to a temperature gradient on the boundary in a continuously stratified quasi-geostrophic model.

Experimental study of interfacial long waves in a two-layer shear flow

Interfacial stability of two-layer Couette flow was investigated experimentally in a channel bent into an annular ring. This paper is focused on the supercritical long-wave instability which arises for a broad range of flow parameters. Above the critical upper plate velocity, a slowly growing long wave appears with wavelength equal to the perimeter of the channel. Transients of this wave were studied within the theoretical frame of amplitude equations obtained from the long-wave interface equation. Near the onset of instability, the unstable fundamental harmonic is described by the Landau–Stuart equation, and the nonlinear dynamics of the harmonics closely follows the central and slaved modes analysis. For the higher upper plate velocity, harmonics gain some autonomy but they eventually are enslaved by the fundamental, through remarkable collapses of amplitudes and phase jumps leading to wave velocity and frequency locking. Dispersive effects play a crucial role in the nonlinear dynamics. Far from the threshold, the second harmonic becomes unstable and bistability appears: the saturated wave is dominated either by the fundamental harmonic, or by the even harmonics, after periodic energy exchange.

Experimental investigation of turbulent entrainment in an annulus with moving sidewalls

The establishment of a turbulent mixed layer in a two-layer stratified shear flow, and the rate of entrainment into that layer were studied experimentally in a modified annulus. The modification of the conventional annulus was made by replacing the upper rotating screen with inner rotating sidewalls, extending over the upper half of the channel, so that the flow in the upper layer was nearly uniform and almost laminar, while the bottom layer was quiescent. Vertical density profile measurements were conducted using single electrode conductivity probes. The flow was visualized during the various stages of the experiment using the hydrogen bubble technique.

Some catastrophe properties of two-layer shear flow

In this paper a simple current system which consists of two stratified incompressible layers is examined. For the basic equations of the motion of fluid a lower order spectrum model is established by means of Galerkin method. Adopting the difference of wind velocity between the upper and lower layers,\(\Delta \bar u = \frac{{\rho _1 \bar u_1 - \rho _2 \bar u_2 }}{{\rho _2 }}\) as a control parameter, the bifurcation and stability of the solution of the dynamical system are discussed. It is found that the flow states in the lower layer will have a catastrophe, when\(\left| {\Delta \bar u} \right| > 2\frac{{\delta ^2 }}{{f^2 }}C_g^3 \) where Cg is the phase velocity of the internal inertio-gravitational wave in a geostrophic current. These results may give a reasonable explanation for the mechanism of the catastrophe phenomena, including the “pressure-jump” in the atmosphere.

Experimental Study on the Stability of a Two-layer Shear Flow

Experimental Study on the Stability of a Two-layer Shear Flow

Interfacial progressive gravity waves in a two-layer shear flow

Nonlinear interfacial gravity waves in a two-layer Boussinesq fluid are studied. In a previous paper, nonlinear waves on a vortex sheet separating two layers, each of constant density and velocity were considered. In the present paper the basic flow model consists of a constant vorticity upper layer bounded by a rigid surface and an irrotational lower layer of infinite depth with continuity of the unperturbed velocity at the density interface. Numerical solutions obtained from an exact formulation in terms of a complex-valued integral equation for the shape and local vortex-sheet strength of the wave profile are compared with results from a second-order Stokes expansion. It is found that the wave of maximum amplitude displays different geometrical features depending on the unperturbed flow parameters. These include waves containing an S-shaped section, waves with cusped crests and surface-constrained waves with long flat crests. Wave integral properties calculated, including the flux of momentum and energy in the wave propagation direction, showed monotonic variation with increasing wave amplitude.

Stability Criterion for Two-Layer Shear Flow

Stability Criterion for Two-Layer Shear Flow

A shear instability induced by a composite internal wave

Experiments have been performed in a thermally stratified two-layer shear flow with a nonzero mean velocity to observe the effects of artifically introduced low frequency internal waves on higher frequency, Kelvin–Helmholtz waves growing spatially on the density interface. Phillips’ model for wave-induced shear across an interface is extended to the two first-mode wave trains which can propagate in such a flow. It is shown that the Richardson number becomes a function of the phase of the internal composite waves and that its minimum value is either at the front or at the back face of the waveform depending on the sign of the background vorticity. Measurements are presented of the phase-averaged temperature and velocity fields of generated internal waves and show the waves breaking down on the face where the Richardson number was found to be at a minimum. The breakdown is characterized by the rapid growth of high frequency waves which appear to be phase locked on the parent internal gravity wave. Possible instances where this type of breakdown could occur in nature are discussed.

Development of radiation fields and baroclinic eddies in a β-plane

The evolution of an eddy in a two-layer shear flow and the radiation field generated by the eddy are investigated by solving the initial value problem. The solution indicates that two wave regimes with different characteristics exist in the x / t plane, where t is the time and x is the distance from the initial disturbance. The near-field (small x / t ) behaviour depends critically on the stability of the shear. In a baroclinically stable shear flow, the radiation field of the baroclinic waves is confined to a region the boundary of which expands linearly with time. If the shear flow is baroclinically unstable, the eddy gradually evolves from an isotropic structure into a wave packet that grows exponentially and expands as the square root of time. The far field (large x / t ) consists mainly of long barotropic Rossby waves. As x / t increases, the effect of the mean currents becomes weaker and the motion becomes increasingly barotropic in character. If we consider the strong boundary current as a source of baroclinic energy in the ocean, the far-field results show that the mid-ocean eddies can reach large amplitudes independent of the mean currents. The transient motion (small x and t ) of the eddy is examined by numerically computing the stream function from the solution. The results show that the combined actions of baroclinic instability and wave dispersion cause an initially isotropic eddy to remain roughly isotropic for a relatively long period of time. This suggests that the initial development of unstable eddies cannot be neglected in studying the meridional scale of the eddy field in the atmosphere. Application of the model to study the dispersion of wind-generated Rossby waves is discussed.