Lock-exchange flows in inclined pipes: the relevance of the Prandtl mixing length model

We examine the applicability of the Prandtl mixing length model to transverse momentum and mass flux in strongly confined, stably stratified turbulent shear flows. These fluxes were measured in the vertical diametral plane of lock-exchange flows in an inclined pipe by the simultaneous use of planar laser-induced fluorescence and particle image velocimetry at local Reynolds numbers ranging from Re = 580 to 1770 and Richardson numbers ranging from Ri = 0.26 and 1.6. Measurements indicate that the eddy diffusivities of mass and momentum are symmetric about the pipe axis, with their maximum at the axis. The corresponding Prandtl mixing lengths decrease with increasing distance from the pipe axis within the central 60% of the pipe cross-section. Within the range of experimental conditions, the mixing lengths at the axis increase linearly with Ri so that the corresponding turbulent Prandtl number Pr_t decreases with Ri. In contrast, Pr_t and the mixing lengths do not display a systematic dependence on Re. Comparison with unbounded and semi-bound shear flows suggests that the strong confinement imposed by the pipe wall may be constraining the integral length scale and Prandtl mixing lengths.     

Unstable Surface Waves in Running Water

We consider the stability of periodic gravity free-surface water waves traveling downstream at a constant speed over a shear flow of finite depth. In case the free surface is flat, a sharp criterion of linear instability is established for a general class of shear flows with inflection points and the maximal unstable wave number is found. Comparison to the rigid-wall setting testifies that the free surface has a destabilizing effect. For a class of unstable shear flows, the bifurcation of nontrivial periodic traveling waves is demonstrated at all wave numbers. We show the linear instability of small nontrivial waves that appear after bifurcation at an unstable wave number of the background shear flow. The proof uses a new formulation of the linearized water-wave problem and a perturbation argument. An example of the background shear flow of unstable small-amplitude periodic traveling waves is constructed for an arbitrary vorticity strength and for an arbitrary depth, illustrating that vorticity has a subtle influence on the stability of free-surface water waves.     

A nonlinear PSE method for two-fluid shear flows with complex interfacial topology

A nonlinear stability method is developed for laminar two-fluid shear flows which undergo changes in the interface topology. The method is based on the nonlinear parabolized stability equations (PSE) and incorporates a scalar-based interface capturing (IC) scheme in order to track complex deformations of the fluid interface. In doing so, the formulation retains the flexibility and physical insight of instability-wave based methods, while providing hydrodynamic modeling capabilities similar to direct numerical calculations: the new formulation, referred to as the IC-PSE, can capture the nonlinear physical mechanisms responsible for generating large-scale, two-fluid structures, without incurring heavy computational costs. This approach is valid for spatially developing, laminar two-fluid shear flows which are convectively unstable, and can naturally account for the growth of finite amplitude interfacial waves, along with changes to the interfacial topology. We demonstrate the accuracy of the IC-PSE against direct Navier–Stokes calculations for two-fluid mixing layers with density and viscosity stratification. The comparisons show that the IC-PSE can predict the dynamics of the instability waves and capture the formation of Kelvin–Helmholtz vortex rolls and large scale liquid structures, at an order of magnitude less computational cost than direct calculations. The role of surface tension in the IC-PSE formulation is shown to be valid for flows in which Re/We ? 1, and the method accurately predicts the formation and non-linear evolution of flow structures in this limit. This is demonstrated for spatially developing mixing layers which lead to vortex roll-up and ligaments, prior to droplet formation. The pinch-off process itself is a high surface tension phenomenon and in not considered herein. The method also accurately captures the effect of interfacial waves on the mean flow, and the topology changes during the non-linear evolution of the two-fluid structures.     

Control of gradient-driven instabilities using shear Alfvén beat waves

A new technique for manipulation and control of gradient-driven instabilities through nonlinear interaction with Alfvén waves in a laboratory plasma is presented. A narrow, field-aligned density depletion is created in the Large Plasma Device, resulting in coherent, unstable fluctuations on the periphery of the depletion. Two independent shear Alfvén waves are launched along the depletion at separate frequencies, creating a nonlinear beat-wave response at or near the frequency of the original instability. When the beat wave has sufficient amplitude, the original unstable mode is suppressed, leaving only the beat-wave response, generally at lower amplitude.     

Subgrid-scale dynamics and model test in a turbulent stratified jet with coexistence of stable and unstable stratification

The subgrid-scale dynamics of stratified flows is studied in a horizontally introduced turbulent jet with coexistence of stable and unstable stratification of a low Richardson number case and a high Richardson number case. The positive production of subgrid-scale kinetic energy and the production of scalar variance suggest the forward energy cascade. The subgrid-scale buoyant destruction plays a role as a sink of subgrid-scale kinetic energy in the stable stratification while holds a role of turbulent generation in the unstable stratification. The role-switch of buoyant destruction in the stable stratification of high-Ri case implies the occurrence of a destabilising process triggered by the coupled instability mechanisms. The energy balance assumption related to the production of and the dissipation of subgrid-scale kinetic energy as well as the subgrid-scale buoyant destruction may fail. The a-priori test suggests the scale-invariant dynamic and standard Smagorinsky models not to work properly here, while the scale-dependent dynamic model gives a decent performance but with restrictions of the ratio between two testing filter scales.     

Mixing-induced global modes in open active flow

We describe how local mixing transforms a convectively unstable active field in an open flow into absolutely unstable. Presenting the mixing region as one with a locally enhanced effective diffusion allows us to find the linear transition point to an unstable global mode analytically. We derive the critical exponent that characterizes weakly nonlinear regimes beyond the instability threshold and compare it with numerical simulations of a full two-dimensional flow problem.     

Asymptotic Models of Diffusion Effects due to Nonlinear Resonant Interaction of Waves with Flows

We develop an asymptotic theory describing nonlocal effects caused by weak-diffusion processes in the case of resonant interaction of quasi-harmonic waves of small but finite amplitudes with flows of various physical nature in the case of an arbitrary relation between the nonlinearity and diffusion.We analyze the interaction of internal gravity waves with plane-parallel stratified shear flows in the nonlinearly-dissipative critical layer (CL) formed in the vicinity of the resonance level where the flow velocity is equal to the phase velocity of the wave. It is shown that the combined effect of the radiation force in the inner region of the CL and vorticity diffusion to the outer region results in the formation of a flow in which the asymptotic values of average vorticity at different sides of the CL are constant but different. If the criterion of the linear dynamic stability is satisfied (the Richardson number Ri>1/4), the resulting vorticity steps are comparable to the unperturbed vorticity. As a result, a wave reflected from the vorticity inhomogeneity in the CL is formed. As the amplitude of the incident wave increases, the average vorticity at the incidence side approaches the linear-stability threshold (Richardson number Ri > 1/4), and the reflection coefficient tends to -1.In the regime of nonlinear dissipative CL, we study the quasi-stationary asymptotic behavior of the flow formed by an internal gravity wave incident on a dynamically stable flow with velocity and density stratification, whose velocity at some level is equal to the phase velocity of the wave. It is shown that the vorticity diffusion results in the formation of a nonlocal transition region between the CL and the unperturbed flow, which we call the diffusive boundary layer (DBL). In this case, the CL is shifted toward the incident wave. We obtain a self-similar solution for the average fields, which is valid in the case of a constant vorticity step in the CL, and determine its parameters depending on the inner Reynolds number in the CL which describes the relation between the nonlinear and diffusive effects for the wave field in the resonance region. We determine the structure and temporal dynamics of the DBL formed by a rough surface streamlined by a stratified fluid whose velocity changes direction at some level.It is shown that in the case of the nonlinear resonance interaction of plasma electrons with a Langmuir wave, the electron diffusion in the velocity space leads to a significant nonlocal distortion of the electron distribution function outside the trapping region. We determine the distorted distribution function and calculate the rate of the nonlinear Landau damping of a finite-amplitude wave for an arbitrary ratio of the electron collision rate and the oscillation period of trapped electrons.     

内孤立波沿缓坡地形传播特性的实验研究

以南中国海东北部海域底部缓坡地形为背景, 在大型重力式分层流水槽中模拟了下凹型内孤立波沿缓坡地形传播过程中的浅化、破碎、分裂等现象, 利用分层染色标识方法和多点组合探头阵列技术对内孤立波沿缓坡地形演化特征进行了定性分析和定量测量. 实验表明: 浅化效应使内孤立波传播速度减小, 对大振幅内孤立波具有抑制作用, 对小振幅波具有放大效应; 浅化效应可导致内孤立波的剪切失稳及破碎, 还可导致大振幅内孤立波的分裂. 利用Miles稳定性理论可定性描述内孤立波沿缓坡地形传播时发生不稳定状态的位置, 实验结果与理论分析相符合. 关键词： 分层流 缓坡地形 内孤立波 不稳定性     

Unstable fingering patterns of Hele-Shaw flows as a dispersionless limit of the Kortweg-de vries hierarchy

We show that unstable fingering patterns of two-dimensional flows of viscous fluids with open boundary are described by a dispersionless limit of the Korteweg-de Vries hierarchy. In this framework, the fingering instability is linked to a known instability leading to regularized shock solutions for nonlinear waves, in dispersive media. The integrable structure of the flow suggests a dispersive regularization of the finite-time singularities.     

On the stability of capillary waves on the surface of a space-charged dielectric liquid jet moving in a material medium

We have derived and analyzed the dispersion equation for capillary waves with an arbitrary symmetry (with arbitrary azimuthal numbers) on the surface of a space-charged cylindrical jet of an ideal incompressible dielectric liquid moving relative to an ideal incompressible dielectric medium. It has been proved that the existence of a tangential jump of the velocity field on the jet surface leads to a periodic Kelvin–Helmholtz- type instability at the interface between the media and plays a destabilizing role. The wavenumber ranges of unstable waves and the instability increments depend on the squared velocity of the relative motion and increase with the velocity. With increasing volume charge density, the critical value of the velocity for the emergence of instability decreases. The reduction of the permittivity of the liquid in the jet or an increase in the permittivity of the medium narrows the regions of instability and leads to an increase in the increments. The wavenumber of the most unstable wave increases in accordance with a power law upon an increase in the volume charge density and velocity of the jet. The variations in the permittivities of the jet and the medium produce opposite effects on the wavenumber of the most unstable wave.     

Hall Currents and the Rayleigh-Taylor Instability of a Finitely Conducting Plasma in the Presence of Coriolis Forces

We have studied the effect of rotation on the development of Rayleigh-Taylor instability of an incompressible, viscous, Hall, finitely conducting plasma of variable density. The solution is developed, through variational methods, for a semi-infinite plasma in which the density varies exponentially along the vertical. It is found that the system is unstable for all wave numbers when the effects of magnetic resistivity are included. The effects of coriolis forces and viscosity on the growth rate of the unstable system are found to be stabilizing while that of Hall currents is destabilizing. Finite conductivity affects the growth rate of the unstable mode differently for the smaller and larger values of the wave numbers, destabilizing for the waves of large wave length and stabilizing for waves of small wave length.     

Hydrodynamic singularities and clustering in a freely cooling inelastic gas

We employ hydrodynamic equations to follow the clustering instability of a freely cooling dilute gas of inelastically colliding spheres into a well-developed nonlinear regime. We simplify the problem by dealing with a one-dimensional coarse-grained flow. We observe that at a late stage of the instability the shear stress becomes negligibly small, and the gas flows solely by inertia. As a result the flow formally develops a finite-time singularity, as the velocity gradient and the gas density diverge at some location. We argue that flow by inertia represents a generic intermediate asymptotic of unstable free cooling of dilute inelastic gases.     

Understanding the sub-critical transition to turbulence in wall flows

In contrast with free shear flows presenting velocity profiles with inflection points which cascade to turbulence in a relatively mild way, wall bounded flows are deprived of (inertial) instability modes at low Reynolds numbers and become turbulent in a much wilder way, most often marked by the coexistence of laminar and turbulent domains at intermediate Reynolds numbers, well below the range where (viscous) instabilities can show up. There can even be no unstable mode at all, as for plane Couette flow (pCf) or for Poiseuille pipe flow (Ppf) that are currently the subject of intense research. Though the mechanisms involved in the transition to turbulence in wall flows are now better understood, statistical properties of the transition itself are yet unsatisfactorily assessed. A widely accepted interpretation rests on non-trivial solutions of the Navier-Stokes equations in the form of unstable travelling waves and on transient chaotic states associated to chaotic repellors. Whether these concepts typical of the theory of temporal chaos are really appropriate is yet unclear owing to the fact that, strictly speaking, they apply when confinement in physical space is effective while the physical systems considered are rather extended in at least one space direction, so that spatiotemporal behaviour cannot be ruled out in the transitional regime. The case of pCf will be examined in this perspective through numerical simulations of a model with reduced cross-stream (y) dependence, focusing on the in-plane (x, z) space dependence of a few velocity amplitudes. In the large aspect-ratio limit, the transition to turbulence takes place via spatiotemporal intermittency and we shall attempt to make a connection with the theory of first-order (thermodynamic) phase transitions, as suggested long ago by Pomeau.      

Shock-wave mach-reflection slip-stream instability: a secondary small-scale turbulent mixing phenomenon

Theoretical and experimental research, on the previously unresolved instability occurring along the slip stream of a shock-wave Mach reflection, is presented. Growth rates of the large-scale Kelvin-Helmholtz shear flow instability are used to model the evolution of the slip-stream instability in ideal gas, thus indicating secondary small-scale growth of the Kelvin-Helmholtz instability as the cause for the slip-stream thickening. The model is validated through experiments measuring the instability growth rates for a range of Mach numbers and reflection wedge angles. Good agreement is found for Reynolds numbers of Re 2 x 10(4). This work demonstrates, for the first time, the use of large-scale models of the Kelvin-Helmholtz instability in modeling secondary turbulent mixing in hydrodynamic flows, a methodology which could be further implemented in many important secondary mixing processes.     

Shear instability in fluids with a density-dependent viscosity

We present a shear instability, which can be triggered in compressible fluids with density-dependent viscosity at shear rates above critical. The instability mechanism is generic: It is based on density-dependent viscosity, compressibility, as well as flow two-(three-)dimensionality that provides coupling between streamwise and transversal velocity components and density variations. The only factor stabilizing the instability is fluid elasticity. The corresponding eigenvalue problem for a plane Couette flow is solved analytically in the limiting cases of large and small wave numbers.     

On stochastic stabilization of the Kelvin-Helmholtz instability by three-wave resonant interaction

An analytical investigation of the effect of three-wave resonant interactions with the linearly unstable wave is proposed. We consider the waves in the Kelvin-Helmholtz model, consisting of two fluid layers with different densities and velocities. We suppose that the velocity shear is weakly supercritical, the instability is of the algebraic type, i.e., the amplitude of the unstable wave grows linearly, and the instability occurs within the framework of a single mode. The amplitudes of two other waves taking part in the nonlinear interaction are assumed to be stable. The initial amplitudes of these waves are supposed to be small in comparison with the initial amplitude of the unstable wave. We present an analysis of the system of amplitude equations derived for this case using JWKB-method. As a result, we obtain equations that couple solutions pre- and post-passing the singular point, i.e., the point where the amplitude of the unstable wave has a local minimum. These equations give us the transformation rule of a parameter that characterizes the phase shift between fast and slow waves and defines the behavior of the system. This parameter is constant between two singular points and varies by chance at a singular point. As long as it stays positive, the amplitude of the wave remains limited and performs stochastic oscillations. If this parameter passes over zero, then we leave the region of stabilization and turn out in the region, where the amplitude grows infinitely. Accordingly, the transition to the region of instability happens stochastically. However, if the time interval, when the amplitude remains bounded, is large enough, the proposed scenario can be treated as a partial stabilization of instability.     

Dynamics of local density perturbation in stably stratified fluid: Results of numerical experiments

Linear and nonlinear numerical models of dynamics of local density perturbation in a stably stratified medium are constructed. The influence of viscosity on the process of generation and propagation of internal waves generated by the local density perturbation in a pycnocline is evaluated. The problem on the dynamics of local density perturbation in the presence of wave background is considered.     

Stability of periodic progressive gravity wave solutions of the Whitham equation in the presence of vorticity

The modulational instability of two-dimensional nonlinear traveling-wave solutions of the Whitham equation in the presence of constant vorticity is considered. It is shown that vorticity has a significant effect on the growth rate of the perturbations and on the range of unstable wavenumbers. Waves with kh greater than a critical value, where k is the wavenumber of the solution and h is the fluid depth, are modulationally unstable. This critical value decreases as the vorticity increases. Additionally, it is found that waves with large enough amplitude are always unstable, regardless of wavelength, fluid depth, and strength of vorticity. Furthermore, these new results are in qualitative agreement with those obtained by considering fully nonlinear solutions of the water-wave equations.     

Modulation instability of long internal waves with moderate amplitudes in a stratified horizontally inhomogeneous ocean

It has been shown that the known effect of the modulation instability of wavepackets can occur for long internal waves with a moderate amplitude in a stratified horizontally inhomogeneous ocean under certain conditions on the vertical structure of the density field and flows. The numerical calculations that have been performed for the transformation of wavepackets in some regions of the World Ocean indicate the possibility of the appearance of rogue waves in the bulk of the Ocean.     

Weakly nonlinear non-Boussinesq internal gravity wavepackets

Internal gravity wavepackets induce a horizontal mean flow that interacts nonlinearly with the waves if they are of moderately large amplitude. In this work, a new theoretical derivation for the wave-induced mean flow of internal gravity waves is presented. Using this we examine the weakly nonlinear evolution of internal wavepackets in two dimensions. By restricting the two-dimensional waves to be horizontally periodic and vertically localized, we derive the nonlinear Schrödinger equation describing the vertical and temporal evolution of the amplitude envelope of non-Boussinesq waves. The results are compared with fully nonlinear numerical simulations restricted to two dimensions. The initially small-amplitude wavepacket grows to become weakly nonlinear as it propagates upward due to non-Boussinesq effects. In comparison with the results of fully nonlinear numerical simulations, the nonlinear Schrödinger equation is found to capture the dominant initial behaviour of the waves, indicating that the interaction of the waves with the induced horizontal mean flow is the dominant mechanism for weakly nonlinear evolution. In particular, due to modulational stability, hydrostatic waves propagate well above the level at which linear theory predicts they should overturn, whereas strongly non-hydrostatic waves, which are modulationally unstable, break below the overturning level predicted by linear theory.