Investigation of Stable Stratifications

A stratified flow is a phenomenon that depends on density differences and buoyancy effects.

Two fluids with different densities, like cold and hot water or air and helium, can be stratified.

If the lighter fluid (ρ1) is on top of the heavier fluid (ρ2) they form a stable stratification (Fig.

9). It is stable, because naturally the lighter fluid tends to be on top of the heavier fluid due to buoyancy.

Fig. 9: Scheme of a stable stratification g

ρ₂ ρ₁

An important characteristic such a stratification is its stability. It takes some effort to disturb or destroy it. If a disturbance occurs normal to the stratification, for example an impinging jet (Fig. 9), the stratification tries to restore its original shape. Another relevant characteristic is the non-isotropy associated with the stratification. The density gradient in the direction of gravity is very steep. At the same time there is almost no density gradient normal to the direction of gravity.

Density stratifications occur on very different occasions, e.g. large-scale geophysical phenomena, temperature stratifications in lakes or the forming of a light-gas cloud in the course of a loss-of-coolant accident.

Ivey et al. [57] performed laboratory-scale experiments as well as numerical calculations to investigate the turbulence mixing of density stratifications in oceans. They identified mixing due to turbulent patches that grow and decay over time. Those patches are the result of energy transport by internal gravity waves.

The interaction of a plume of warm air with a stable temperature stratification in the atmosphere above a city was investigated by Noto [58]. In laboratory-scale experiments, different flow pattern have been found. All observed flows have in common, that the stable stratification is suppressing the height of the plume and that a vortex pair is produced near the top of the plume. The flow pattern depend on the stability of the stratification and the heat rate of the plume.

Noto et al. [59] also performed a direct numerical simulation of a hot plume originating from a heated plate in a stable stratification. The quality of the DNS has been demonstrated with several energy spectra. It turned out that turbulence is suppressed at any degree of stable stratification. In a weaker stratification, turbulence is generated near the centre of the plate.

But due to the suppressive effect of the stable stratification, the flow becomes laminar again.

In a strong stratification the suppression of turbulence is so dominating that the flow stays laminar. Noto concluded, that the plume behaviour can be controlled with the degree of the stratification.

A stably stratified free surface open channel flow was investigated by Taylor et al. [60] with LES. The stable stratification is a result of a constant heat flux at the free surface. It was found, that a sufficient increase of the friction Richardson number, the density layer or pycnocline can change from turbulent to laminar. Another effect of the increase of the friction Richardson number is the increase of the bulk Reynolds number as well as a strengthening of the pycnocline.

The effect of the gradient Richardson number on a stable stratification as a measure of flow laminarisation was investigated by Galperin et al. [61]. It turned out, that a single critical Richardson number smaller than one at which turbulence is totally suppressed and laminarised, does not exist. Therefore, the critical Richardson number should be avoided as a criterion of turbulence extinction.

Stretch et al. [62] investigated the mixing efficiency in stratified flows with direct numerical simulations and rapid distortion theory calculations. The aim of this research was the determination of the mixing efficiency of decaying, homogeneous, stably-stratified turbulence as a function of the initial turbulence Richardson number. The investigated stratification is

caused by different salt concentrations in water. It was found, that for small Richardson numbers, the mixing efficiency can be increased by increasing the Richardson number. For larger Richardson numbers, the mixing efficiency becomes constant. In the experiments, this means Ri > 1. Further investigation on the numerical part is suggested due to quantitative deviations of both, the direct numerical simulation and the rapid distortion theory calculations compared to the experimental data.

The effect of stable stratification on turbulence anisotropy was investigated by Sarkar [63]

with direct numerical simulations. Two flows have been investigated, a flow with horizontal mean shear as well as one flow with vertical mean shear. The results show that the horizontal and vertical velocity fluctuations remain coupled for either flow. One consequence of this coupling is, that vertical mixing is induced by horizontal mean shear. Another consequence is, that the vertical mixing is larger when the mean shear is horizontal, because the gravity has no damping effect on the turbulence production.

Lin et al. [64] carried out experiments to investigate the entrainment due to a turbulent fountain at a density interface. Fig. 10 shows the set-up of the experiment. Qp is the volume flux of the plume, Qf is the volume flux of the fountain, QE is the volume flux of the entrainment, QEU is the volume flux of the entrainment and fluid from the upper layer, and Qout

is the volume flux leaving the tank. Mf is the momentum flux of the fountain. B is the buoyancy flux of the plume source. g₁^' Is the reduced gravity of the upper layer and g₂^' is the reduced gravity of the lower layer. H is the height of the tank and h is the depth of the interface.

The tank is filled with water. A downward plume on the left hand side of the tank is filling the lower vessel with salt water. Because of the greater density of salt water a stable stratification is the result. A downward water fountain on the right hand side of the tank is eroding the density layer. The permanent supply with salt water and the outlet in the middle of the ceiling

Fig. 10: Steady state mixing of a density stratification [64]

enable a steady state situation.

The interaction between stratification and fountain in the experiments of Lin is comparable to the TH20 flow (Fig. 4). The major difference is that the mixing is induced by a fountain of the less dense fluid in the direction of gravity whereas in the TH20 case a jet of the denser fluid against the direction of gravity is responsible for the mixing.