Parameterization of turbulent fluxes and scales using homogeneous sheared stably stratified turbulence simulations

Abstract

Laboratory experiments on stably stratified grid turbulence have suggested that turbulent diffusivity $\kappa_\rho$ can be expressed in terms of a turbulence activity parameter $\epsilon/\nu N^2$, with different power-law relations appropriate for different levels of $\epsilon/\nu N^2$. To further examine the applicability of these findings to both a wider range of the turbulence intensity parameter $\epsilon/\nu N^2$ and different forcing mechanisms, DNS data of homogeneous sheared stratified turbulence generated by Shih et al. (2000) and Venayagamoorthy et al. (2003) are analysed in this study. Both scalar eddy diffusivity $\kappa_\rho$ and eddy viscosity $\kappa_\nu$ are found to be well-correlated with $\epsilon/\nu N^2$, and three distinct regimes of behaviour depending on the value of $\epsilon/\nu N^2$ are apparent. In the diffusive regime $D$, corresponding to low values of $\epsilon/\nu N^2$ and decaying turbulence, the total diffusivity reverts to the molecular value; in the intermediate regime $I$, corresponding to $7 \,{<} \epsilon/\nu N^2 \,{<}\, 100$ and stationary turbulence, diffusivity exhibits a linear relationship with $\epsilon/\nu N^2$, as predicted by Osborn (1980); finally, in the energetic regime $E$, corresponding to higher values of $\epsilon/\nu N^2$ and growing turbulence, the diffusivity scales with $(\epsilon/\nu N^2)^{1/2}$. The dependence of the flux Richardson number $R_f$ on $\thing$ explains the shift in power law between regimes $I$ and $E$. Estimates for the overturning length scale and velocity scales are found for the various $\epsilon/\nu N^2$ regimes. It is noted that $\epsilon/\nu N^2 \,{\sim}\, \hbox{\it Re}/\hbox{\it Ri}\,{\sim}\,\hbox{\it ReFr}^2$, suggesting that such Reynolds–Richardson number or Reynolds–Froude number aggregates are more descriptive of stratified turbulent flow conditions than the conventional reliance on Richardson number alone.

Robust identification of dynamically distinct regions in stratified turbulence

“…Such sensitivity is also observed in the results reported in both Hebert & de Bruyn Kops (2006a) and Hebert & de Bruyn Kops (2006b). More recently, de Bruyn Kops (2015) found that the dynamics of stratified flows are different at Gn = 48 and Gn = 220, which suggests another threshold at Gn ∼ O(100), consistent with the findings of Shih et al (2005) and Salehipour & Peltier (2015).…”

“…This intermediate range is also of particular interest as different dynamical regimes have been identified with different ranges of Gn. Specifically, turbulence is expected to be suppressed leading to largely quiescent (though not exactly laminar) flow when Gn ∼ O(1) (Shih et al 2005). In the flows considered by Bartello & Tobias (2013), they found that the turbulence dynamics is very sensitive to Gn unless Gn is larger than order O(10), which is consistent with some of the earliest scaling analyses on the subject (Gargett et al 1984;Gibson 1986;Rohr et al 1988;Itsweire et al 1993).…”

A statistical mechanics approach to mixing in stratified fluids

“…Even if it is natural to expect that the dissipation rate of buoyancy and kinetic energy become independent from the value of molecular viscosity and diffusivity when they are sufficiently weak, numerical and laboratory experiments are often performed in intermediate regimes for which those parameters may influence the mixing efficiency, see e.g. Shih et al (2005); Lozovatsky & Fernando (2013) ;Bouffard & Boegman (2013); Salehipour & Peltier (2015).…”

Parameterizing individual effects of shear and stratification on mixing in stably stratified shear flows

“…[3] The effects of stratification on turbulence are often analyzed using the turbulent Froude number Fr or the gradient Richardson number Ri g in laboratory studies [e.g., Rohr et al, 1984Rohr et al, , 1988bYoon and Warhaft, 1990], Direct Numerical Simulation (DNS) or Large Eddy Simulation (LES) studies [e.g., Holt et al, 1992;Kaltenbach et al, 1994;Shih et al, 2000Shih et al, , 2005, and field studies [e.g., Peters et al, 1995;Stacey et al, 1999;Yeates, 2008]. These nondimensional numbers are defined as…”

“…(Note that the terminal value tends to be ≈9 in unstratified shear flows [Rohr et al, 1988a;Tavoularis and Karnik, 1989], but ≈11 in stratified shear flows [Rohr et al, 1988b;Shih et al, 2000Shih et al, , 2005.) However, Sh is included in the analysis of stratified shear flows relatively recently [Jacobitz et al, 1997;Piccirillo and van Atta, 1997;Shih et al, 2000Shih et al, , 2005, and as far as I am aware, there is no mixing parameterization in terms of both Sh and Fr (or Ri g ) based on experimental results. Stability functions for twoequation closure schemes [e.g., Kantha and Clayson, 1994;Canuto et al, 2001] are examples of such parameterizations, but they are inconsistent with experimental data under strong stratification, as shown later in this study.…”