Second‐moment closures and length scales for weakly stratified turbulent shear flows

Homogeneous sheared stratified turbulence was simulated using a DNS code. The initial turbulent Reynolds numbers (Re) were 22, 44, and 89, and the initial dimensionless shear rate (S*) varied from 2 to 16. We found (similarly to Rogers (1986) for unstratified flows) the final value of S* at high Re to be ∼ 11, independent of initial S*. The final S* varies at low Re, in agreement with Jacobitz et al. (1997). At low Re, the stationary Richardson number (Ris) depends on both Re and S*, but at higher Re, it varies only with Re. A scaling based on the turbulent kinetic energy equation which suggests this result employs instantaneous rather than initial values of flow parameters.At high Re the dissipation increases with applied shear, allowing a constant final S*. The increased dissipation occurs primarily at high wavenumbers due to the stretching of eddies by stronger shear. For the high-Re stationary flows, the turbulent Froude number (Frt) is a constant independent of S*. An Frt-based scaling predicts the final value of S* well over a range of Re. Therefore Frt is a more appropriate parameter for describing the state of developed stratified turbulence than the gradient Richardson number.

Transient development of perturbations in inviscid stratified shear flow is investigated. Use is made of closed form analytic solutions that allow concise identification of optimally growing plane-wave solutions for the case of an unbounded flow with constant shear and stratification. For the case of channel flow, variational techniques are employed to determine the optimally growing disturbances. The maximum energy growth attained over a specific time interval decreases continuously with increasing stratification, and no special significance attaches to Ri = 0.25. Indeed, transient growth can be substantial even for Ri = O(1). A general lower bound on the energy growth attained by an optimal perturbation in a stratified flow over a given time interval is the square root of the growth attained by the corresponding perturbation in unstratified flow. Enhanced perturbation persistence is found for mean-flow stratification lying in the range Ri between 0.1 and 0.3. Small but finite perturbations in mean flow with Ri less than 0.4 produce regions with locally negative total density gradient, which are expected to overturn. Although the perturbations are of wave form, buoyancy fluxes mediate transfer between perturbation kinetic and potential energy during transient development, thus implying that buoyancy flux is not a determinative diagnostic for distinguishing between waves and turbulence in stratified flows.

Abstract

A generalized Osborn–Cox model for estimating fluxes in nonequilibrium stably stratified turbulent shear flows

We present some results and comments on DNS of freely decaying horizontal parallel flows when initially affected by a turbulent velocity field in a strongly stratified fluid. Turbulent shear flows in stratified media have already been simulated by Jacobitz et al. [2] for instance. Experiments have been carried out in a thermally stratified wind tunnel by Piccirillo et al. [3]. However, these numerical and experimental studies generally focus on the influence of a horizontal mean flow on the stratified turbulence. In the present paper, we focus on the accelerating effect of stratified turbulence onto the horizontal mean flow. We solve the 3D Navier-Stokes equations under the Boussinesq approximation with a pseudo-spectral direct simulation code (O. Thual [4]). The flow evolves in a periodic cubic box with no external forcing, the initial condition being the sum of a horizontal parallel flow \( \vec U \)(z, 0) in the x-direction and a homogeneous, isotropic turbulent velocity field \( \vec{u}' \) (x, y, z, 0). The Reynolds number is 67 initially (based on the rms value u 0 of u’, the integral lengthscale l and the kinematic viscosity). We choose \( \vec{U} \) (z,0) as a cosine with amplitude U 0 and a wavelength equal to the size of the box L b . The value of U 0 is chosen such that the mean flow energy is initially one third of the total energy. In the stratified case, the Prandtl number is taken equal to unity and there is no potential energy initially. The initial Froude number Fr = u 0 /Nl (where N is the Brunt-Väisälä pulsation) is 0.13. Such a low value is typical of strongly stratified turbulence.

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) At≪O(1) or (ii) Frc2≪O(1), where At is the ratio of density variation to the mean density and Frc is the ratio of the phase speed to the long wave speed. The second clause implies that, in contrast to the commonly accepted notion, a flow with large density variations can also be Boussinesq. Indeed, we show that deep water surface gravity waves are Boussinesq, while shallow water surface gravity waves are not. However, in the presence of moderate/strong shear, the Boussinesq approximation implies the conventionally accepted At≪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 include 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 interaction perspective.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) At≪O(1) or (ii) Frc2≪O(1), where At is the ratio of density variation to the mean density and Frc is the ratio of the phase speed to the long wave speed. The second clause implies that, in contrast to the commonly accepted notion, a flow with large density variations can also be Boussinesq. Indeed, we show that deep water surface gravity waves are Boussinesq, while shallow water surface gravity waves are not. However, in the presence of moderate/strong shear, the Boussinesq approximation implies the conventionally accepted At≪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 ...

It is commonly assumed that heat flux and temperature diffusivity coefficients obtained in steady-state measurements can be used in the derivation of the heat conduction equation for fluid flows. Meanwhile it is also known that the steady-state heat flux as a function of temperature gradient in stably stratified turbulent shear flow is not monotone: at small values of temperature gradient the flux is increasing, whereas it is decreasing after a certain critical value of the temperature gradient. Therefore the problem of heat conduction for large values of temperature gradient becomes mathematically ill-posed, so that its solution (if it exists) is unstable. In the present paper it is shown that a well-posed mathematical model is obtained if the finiteness of the adjustment time of the turbulence field to the variations of temperature gradient is taken into account. An evolution-type equation is obtained for the temperature distribution (a similar equation can be derived for the concentration if the stratification is due to salinity or suspended particles). The characteristic property which is obtained from a rigorous mathematical investigation is the formation of stepwise distributions of temperature and/or concentration from continuous initial distributions.

Direct and large eddy simulations of stratified homogeneous shear flows

The relative strength of the stabilizing effect of buoyancy and the destabilizing effect of velocity shear in a stratified shear flow, such as a stable atmospheric boundary layer, is measured by the gradient Richardson number, Rig. The boundary layer static stability, as described by the buoyancy frequency, N, can be calculated from the virtual potential temperature gradient derived from RASS temperature profiles. The mean wind profiles from a sodar can be used to calculate the mean vertical velocity shear. In combination these profilers are potentially a powerful tool for the remotely sensing the dynamic stability of the boundary layer. However, experience shows that the combinations of two experimentally derived quantities, like N and shear, may give highly variable results. On the other hand, a simple sensitivity analysis shows that reasonable estimates of Rig are achievable over a range of conditions in the stable nocturnal boundary layer. To test this conclusion, high spatial and temporal resolution temperature and velocity soundings were obtained above 50 m in the stable nocturnal boundary layer using a 920 MHz continuous wave Radio Acoustic Sounding System (RASS) and 1.875 kHz and 5.00 kHz Doppler sodars. Examples of the evolution of Rig are presented from 24 hours of observations of the boundary layer in Canberra, on the tablelands in south- eastern Australia. Most of the boundary layer had Rig between 0.1 and 1. Thus, it was marginally dynamically stable, even with the gradient Richardson number calculated from finite differences over a vertical interval of 68 m. A comparison of the results from the two sodars showed that the velocity shear increased significantly when the vertical differencing interval was decreased from 68 m to 20 m.

A simple subgrid turbulent diffusion model based on an analogy to the von Neumann–Richtmyer artificial viscosity is explored for use in modelling mixing in turbulent stratified shear flow. The model may be more generally applicable to multicomponent turbulent hydrodynamics and to subgrid turbulent transport of momentum, composition and energy. As in the case of the von Neumann artificial viscosity and many subgrid-scale models for large-eddy simulation, the turbulent diffusivity explicitly depends on the grid size and is not based on a quantitative model of the unresolved turbulence. In order to address the issue that it is often not known a priori when and where a flow will become turbulent, the turbulent diffusivity is set to zero when the flow is expected to be stable on the basis of a Richardson/Rayleigh–Taylor stability criterion, in analogy to setting the von Neumann artificial viscosity to zero in expanding flows. One-dimensional predictions of this model applied to a simple shear flow configuration are compared to those obtained using a K–ε model. The density and velocity profiles predicted by both models are shown to be very similar.

Scalar transport and mixing in turbulent stratified shear flow

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 θ2̄/k(S/Sθ)2, and the ratio of mean to turbulent time scale ε/Sk is found. The variable of particular interest to bifurcation analysis, ε/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 ε/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 turbulence model results in a diverging solution. A new set of scalar model coefficients that is consistent with experimental data, predicts turbulence decay past the critical gradient Richardson number Rigcr=0.25, and ensures the existence of stable, real solutions for all combinations of rotation and buoyancy is proposed.

We present a deep probabilistic convolutional neural network (PCNN) model for predicting local values of small-scale mixing properties in stratified turbulent flows, namely the dissipation rates of turbulent kinetic energy and density variance, $\varepsilon$ and $\chi$ . Inputs to the PCNN are vertical columns of velocity and density gradients, motivated by data typically available from microstructure profilers in the ocean. The architecture is designed to enable the model to capture several characteristic features of stratified turbulence, in particular the dependence of small-scale isotropy on the buoyancy Reynolds number $Re_b:=\varepsilon /(\nu N^2)$ , where $\nu$ is the kinematic viscosity and $N$ is the background buoyancy frequency, the correlation between suitably locally averaged density gradients and turbulence intensity and the importance of capturing the tails of the probability distribution functions of values of dissipation. Empirically modified versions of commonly used isotropic models for $\varepsilon$ and $\chi$ that depend only on vertical derivatives of density and velocity are proposed based on the asymptotic regimes $Re_b\ll 1$ and $Re_b\gg 1$ , and serve as an instructive benchmark for comparison with the data-driven approach. When trained and tested on a simulation of stratified decaying turbulence which accesses a range of turbulent regimes (associated with differing values of $Re_b$ ), the PCNN outperforms assumptions of isotropy significantly as $Re_b$ decreases, and additionally demonstrates improvements over the fitted empirical models. A differential sensitivity analysis of the PCNN facilitates a comparison with the theoretical models and provides a physical interpretation of the features enabling it to make improved predictions.

The theme of this chapter will be a more detailed discussion of various kinds of turbulent shear flows, which are (at least at the beginning of the period of interest) well past the state of marginal stability. Some knowledge of the properties of a turbulent shear flow in a homogeneous fluid must be assumed (see, for example, Townsend 1956), and we consider here the additional effects introduced by the presence of density gradients. Turbulent flows in which gravity plays an essential role in driving the mean motion (e.g. turbulent gravity currents) will be treated separately in chapter 6. These flows will be discussed against the background of the classification introduced in §4.3.1, emphasizing the turbulent features, but also referring to the wave aspects when necessary. We consider first a shear flow near a horizontal boundary, and the effect of a vertical density gradient on the velocity and density profiles, and on the rates of transport. Next we discuss the few theoretical results which are available to describe ‘boundary’ and ‘interior’ turbulence in stratified shear flows. Finally, we present and discuss laboratory and larger scale observations which can be used to test these ideas. Velocity and density profiles near a horizontal boundary The most important example of a shear flow near a boundary is the wind near the ground, in a shallow enough layer for the effects of the earth's rotation to be ignored.

Differential diffusion of a passive scalar and an active scalar/density in stratified shear flow is considered when both the density and the passive scalar have a mean vertical gradient. The vertical diffusion coefficient of the passive scalar {ie397-01} (c: passive scalar fluctuation, \-c(x3): mean passive scalar distribution) has been obtained by the rapid distortion theory (RDT), and the comparison with the density diffusion coefficient {ie397-02} (ρ: density fluctuation, {ie397-03} mean density distribution) shows that the difference between Kc and Kρ comes from the initial density fluctuations. Namely, passive scalar flux has a ‘slow mode’ which oscillates at half frequency of the density flux, whose magnitude is proportional to the initial potential energy due to density fluctuations.