Abstracts - KTH Mechanics

Stratified turbulence
G. Brethouwer ∗ , P. Billant † and E. Lindborg ∗
Stratified fluids have been investigated in many theoretical, experimental and
numerical studies owing to its importance for understanding geophysical flows. It
remains a topic of dispute if the dynamics of stratified turbulence at low Froude
number Fr (strong stratification) is essential two-dimensional with a inverse cascade
of energy or three-dimensional with a forward cascade. The problem we are facing
in laboratory and numerical experiments is that it is impossible to reach the same
combination of Re (Re is the Reynolds number) and Fr as in geophysical flows.
Stratified flows studied by laboratory and numerical experiments are perhaps not
fully representful for the flows found outdoors.
When Fr ≪ 1andReF r 2 ≫ 1, analysis of the governing equations suggests the
scaling lv ∼ U/N for the vertical length scale lv (U is a horizontal velocity scale and N
is the Brunt Vaisala frequency) if the dynamics of the flow imposes the vertical length
scale and not any other externally imposed condition 1 . Because of the sharp vertical
gradients in stratified flows implied by this scaling, the advection terms involving the
vertical velocity are of the same order as the other terms, although the vertical velocity
is much smaller than the horizontal velocity, implying three-dimensional dynamics in
strongly stratified fluids. Lindborg 2 adopted the scaling lv ∼ U/N in his theoretical
work which led him to the hypothesis of three-dimensional turbulence in strongly
stratified fluids, but with a highly anisotropic forward cascade of energy. The scaling
lv ∼ U/N was confirmed by numerical simulations of strongly stratified fluids in
very shallow boxes and computed transfer functions of spectral energy convincingly
showed a forward cascade. The scaling lv ∼ U/N exist only if ReF r 2 ≫ 1. A different
scaling exist if ReF r 2 ≪ 1, a case which is often considered in laboratory experiments.
Further analysis suggests that the dynamics is two-dimensional and strongly affected
by viscosity if ReF r 2 ≪ 1.
The aim of our study is to investigate systematically the influence of Re and
Fh on the dynamics of strongly stratified fluids, in particular on turbulence, length
scales and instabilities using and direct numerical simulations (DNS) and validate
our scaling analysis with the DNS. In this way, we want to contribute to a better
understanding and a clearer interpretation of the dynamics of strongly stratified fluids.
DNS of homogeneous turbulence with a linear stratification are carried out. Energy
is injected at the large scales by means of a forcing technique. Consequently, the flow
attains a statistical stationary state. A series of DNS is performed at a fixed Re and
with varying Fr at the moment with resolutions up to 1024 × 1024 × 192 grid points.
The parameters will be chosen so that both regimes, ReF r 2 > 1 but and ReF r 2 < 1,
are covered by the DNS. Preliminary results reveal the existence of anisotropic threedimensional
turbulence with a forward cascade of energy if ReF r 2 > 1 and viscously
dominated, predominantly horizontal turbulence with large pancake-like structures
and with almost no overturning of the density field if ReF r 2 < 1.
∗ KTH Mechanics OB 18, SE-100 44 Stockholm, Sweden.
† LadHyX, Ecole Polytechnique, F-91128 Palaiseau Cedex, France.
1 Billant & Chomaz Phys. Fluids 13, 1645 (2001).
2 Lindborg, J. Fluid Mech., Inpress
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