Inverse scattering for stratified elastic media

The trace method is applied to recover the Lamé parameters λ and μ and the mass density ϱ of a stratified elastic medium.     

Kinematic simulation for stably stratified and rotating turbulence

The properties of one-particle and particle-pair diffusion in rotating and stratified turbulence are studied by applying the rapid distortion theory (RDT) to a kinematic simulation (KS) of the Boussinesq equation with a Coriolis term.Scalings for one- and two-particle horizontal and vertical diffusions in purely rotating turbulence are proposed for small Rossby numbers.Particular attention is given to the locality-in-scale hypothesis for two-particle diffusion in purely rotating turbulence both in the horizontal and the vertical directions. It is observed that both rotation and stratification decrease the pair diffusivity and improve the validity of the locality-in-scale hypothesis. In the case of stratification the range of scales over which the locality-in-scale hypothesis is observed is increased.It is found that rotation decreases the diffusion in the horizontal direction as well as, though to a much lesser extent, in the vertical direction.     

One-dimensional two-fluid equations for horizontal stratified two-phase flow

Advanced computer codes for water reactor loss-of-coolant analysis are based on the use of the two-fluid model of two-phase flow, in which conservation equations are solved for the gas and liquid phases separately. The standard two-fluid equations, however, sometimes predict the growth of instabilities in the flow, and occasionally become improperly posed. These difficulties have in the past led to the proposal of several different forms for the conservations equations.To help resolve these uncertainties a widely accepted form of the one-dimensional two-fluid equations is used to calculate wave propagation speeds, and stability limits, for the illustrative case of a frictionless horizontal stratified gas-liquid flow. Calculated propagation velocities are shown to agree with the appropriate limit of an exact solution, and the predicted stability limits are found consistent with available observations on the stability of the stratified flow regime.These comparisons help improve confidence in the ability of the two-fluid equations to analyse more complex problems in transient two-phase flow.     

Subgrid length-scales for large-eddy simulation of stratified turbulence

The influence of buoyancy on the length-scales for the dissipation rate of kinetic energy, and for momentum, heat, and other scalar transport has to be known for subgrid-scale (SGS) models in a large-eddy simulation (LES). For the inertial subrange, Lilly (1967) has shown that grid spacing is the relevant length-scale for SGS effects. Deardorff (1980) proposed to reduce all the length-scales for stable stratification. Numerical and experimental data show, however, that the dissipation length-scale may strongly increase in stable layers with little shear. Lumley's (1964) theory for the energy spectrum in a stratified fluid also suggests such an increase. In this paper we apply the analysis of previous algebraic second-order closure SGS models, parameter studies with different length-scale models in LES, and the analysis of direct simulations of sheared and unsheared stably stratified homogeneous turbulence. These analyses show advantages of first-order closures for LES and suggest that the limiting effect of stratification should only be applied to the length-scales of vertical eddy-diffusivities of heat and scalars but not to those of momentum and dissipation.Dedicated to Professor J.L. Lumley on the occasion of his 60th birthday.This work was supported by the Deutsche Forschungsgemeinschaft.     

Moving grid method for numerical simulation of stratified flows

We develop a second‐order accurate Navier–Stokes solver based on r‐adaptivity of the underlying numerical discretization. The motion of the mesh is based on the fluid velocity field; however, certain adjustments to the Lagrangian velocities are introduced to maintain quality of the mesh. The adjustments are based on the variational approach of energy minimization to redistribute grid points closer to the areas of rapid solution variation. To quantify the numerical diffusion inherent to each method, we monitor changes in the background potential energy, computation of which is based on the density field. We demonstrate on a standing interfacial gravity wave simulation how using our method of grid evolution decreases the rate of increase of the background potential energy compared with using the same advection scheme on the stationary grid. To further highlight the benefit of the proposed moving grid method, we apply it to the nonhydrostatic lock‐exchange flow where the evolution of the interface is more complex than in the standing wave test case. Naive grid evolution based on the fluid velocities in the lock‐exchange flow leads to grid tangling as Kelvin–Helmholtz billows develop at the interface. This is remedied by grid refinement using the variational approach. Copyright © 2012 John Wiley & Sons, Ltd.     

Acoustic speeds for stratified two-phase fluids in rectangular ducts

A two-dimensional, time harmonic, free-mode analysis has been applied to the stratified regime, assuming a planar interface at constant height separating two fluids. The resulting dispersion relation has been solved asymptotically in the common case of one fluid being much lighter than the other, and a set of acoustic modes, apparently unnoticed previously, has been highlighted.     

Differential equation for a stratified fluid and its particular solutions

The plane steady motion of a stratified ideal incompressible fluid in a gravity field is examined. Considering that the parameter characterizing the fluid particles — their density ₀ — is constant along the streamline, it is convenient to take the stream function as one of the independent variables and, in view of the presence of the gravity force, the Cartesian coordinate as the other. In this study, on the basis of Lavrent'eva's equation [1, 2, 3], the differential equation is derived for a single unknown function — the vertical displacement of the streamline y(y₀, x), where y₀ is its initial position and x is the horizontal coordinate. The particular solutions corresponding to a wave guide, cnoidal and solitary waves and, moreover, waves of the type corresponding to a smooth ascent to a new level are presented. A similar coordinate system was used in [4], but there the problem was reduced to a system of partial differential equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 83–87, September–October, 1986.The authors are grateful to A. A. Barmin for discussing their results.     

Structural and interfacial stability of multiple solutions for stratified flow

Prediction of the liquid level in stratified two-phase upwards flow shows that one may have multiple solutions. In this case it is necessary to determine which solutions will actually occur and whether hysteresis is possible, namely whether it is possible to have two or more solutions for the same operating conditions. In this work the stability of the solutions for stratified flow is considered using two types of stability analyses: (1) structural stability analysis; and (2) interfacial stability analysis (Kelvin—Helmholtz, K—H). For the K—H stability analysis we used two methods: an approximate simplified method suggested by Taitel & Dukler; and a more rigorous method suggested by Barnea, which is based on a combination of the viscous K—H and inviscid K—H analyses. The results show that whenever three solutions exist only the first, i.e. the solution with the thinnest liquid level, is stable. The middle solution is always structurally unstable (linearly), whereas the third solution is structurally unstable to large disturbances (non-linear stability). The third solution is usually also unstable to the K—H type of instability. As a result it is concluded that hysteresis is not possible and that only the thinnest solution will be observed practically.     

On the Squire's transformation for stratified two-phase flows in inclined channels

The applicability of the Squire's transformation for stability analysis of stratified two-phase flow in horizontal and inclined channels is examined. It is shown that for the considered flow such a transformation requires some additional constraints on the change of the inclination angle and flow rates of each of the phases. While the Squire's theorem (on the two-dimensionality of the critical disturbances) rigorously holds for the horizontal two-phase flow, for the inclined flow an exact mathematical theorem cannot be formulated. Nevertheless, it has been proven that 2D perturbations are the critical ones also for the case of inclined channel, since the transformation of a 3D stability problem to its 2D analog is associated with a stabilizing effect of reducing the system inclination, in addition to the reduction of the phases flow rates as in the case of horizontal flows.     

A structure-based model for transport in stably stratified homogeneous turbulent flows

We present an extension that allows a recently proposed structure-based model for turbulent scalar transport to account for buoyancy effects. The proposed model is based on a generalization of the Interactive Particle Representation Model (IPRM) and is accompanied by a four-equation transport model that provides the turbulence scales needed for the closure of the complete structure-based model (SBM). The structure tensors and their invariants are used to model the additional buoyancy terms that emerge in the four-equation transport equations. Model parameters are set by matching the asymptotic decay exponents in decaying turbulence. The validity of the model is considered for a large number of different types of stably stratified flows at different Richardson numbers (Ri), showing encouraging results. The complete structure-based model achieves fair agreement with LES and DNS predictions for vertical shear in the presence of vertical mean stratification, while the structure tensors are shown to be suitable for use as diagnostic tools for the morphology of highly anisotropic turbulent structures. Additionally, the proposed model is shown to be sensitive to the variation of the inclination angle θ between the direction of the mean velocity gradient and the orientation of the mean scalar gradient. Furthermore, the model correctly predicts that the evolution of the inverse shear parameter is insensitive to the choice of inclination angle, yielding a turbulent Prandtl number close to unity, in accordance with DNS results.     

Method of narrow strips for nonlinear problems of a stratified fluid

Plane steady flow is considered for an ideal incompressible stratified fluid in a gravitational field of force. It is a characteristic feature of these flows that the density is constant and Bernoulli's constant remains the same along a streamline. Internal waves arise because of ponderability in the stratified fluid; they are not due to the presence of a free surface. These wave motions are studied in detail in the linear formulation, but flows of the solitary wave type can be described only by nonlinear equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 174–178, March–April, 1986.     

The analytical solution for Helmholtz boundary problem in non-horizontally stratified domains

There are N domains D_j(j=0,1,...,N-1) of different physical parameters in the whole space and their interfaces S_j,i+1 are non-horizontally smooth curved surfaces. The following boundary problem is called Hclinholiz boundary problem:V²H⁽¹⁾+K_jH⁽¹⁾=0 (j=0,1,…,N-1)(H⁽⁰⁾-H⁽¹⁾)=δ(S) (δ(S):generalized function)(H¹-Hⁱ⁺¹)=0 (j=0,1,…,N-2)The analytical solution of the above problem is given in this paper.