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VIIIth International Symposium on Stratified
Flows, San Diego 2016
UC San Diego
Title:
Acceleration-driven variable- density turbulent flow
Journal Issue:
International Symposium on Stratified Flows, 1(1)
Author:
Gat, Ilana
, Caltech
Matheou, Georgios
, JPL, Caltech
Chung, Daniel
, U. Melbourne;
Dimotakis, Paul
, Applied Physics, Caltech
Publication Date:
2016
Permalink:
http://escholarship.org/uc/item/61d722q6
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Acceleration-driven variable-density turbulent flow
Ilana Gat
,
1
Georgios Matheou,
2
Daniel Chung,
3
and Paul E. Dimotakis
4
1
Graduate Aerospace Laboratories,
2
Jet Propulsion Laboratory,
California Institute of Technology
California Institute of Technology
igat@caltech.edu
matheou@caltech.edu
3
Mechanical Engineering,
4
Graduate Aerospace Laboratories,
University of Melbourne
California Institute of Technology
daniel.chung@unimelb.edu.au
dimotakis@caltech.edu
Abstract
We discuss turbulent dynamics and mixing of a variable-density flow subject to a uniform-
acceleration field. The flow resulting from initial misalignments of pressure and density
gradients is investigated for small to large density ratios, with evidence that the small-
density ratio flow is described by the Boussinesq approximation. A new shear-layer growth
rate is reported. Spectra collapse when properly scaled for variable density.
1 Introduction
Turbulence and mixing between fluids of different densities responding to an externally
imposed acceleration field, such as gravity, occur in many applications ranging from geo-
physics to astrophysics. The present study focuses on flow dynamics resulting from a body
force,
ρ
g
, with
ρ
the local fluid density and
g
=
−
̂
z
g
the imposed uniform-acceleration
field, in the zero Mach number limit.
Many flows can be treated as incompressible with small density variations, ∆
ρ/ρ
1, and
the Boussinesq approximation can often adequately describe the flow physics (e.g., Peltier
and Caulfield, 2003, and references therein). Even though the Boussinesq linearization
only accounts for the body force in the momentum equation (Batchelor et al., 1992) for
small density variations, the Boussinesq linearization may capture the dynamics of mis-
aligned hydrostatic pressure and density gradients in flows with small density variations.
For the large density ratios studied, however, the Boussinesq linearization approximation
cannot be used.
To study the effects of baroclinic torques, a flow configuration is considered in which two
different gas-phase fluids are initialized with their density gradient perpendicular to the
uniform-acceleration field. Density ratios in the range of 1
.
005
≤
R
≡
ρ
1
/ρ
2
≤
10 are
considered. In particular, it is presently found that even if this flow is initialized with
near-unity density ratios, i.e.,
R
= 1+
, which one would expect to tend to the Boussinesq
approximation for small
, its flow dynamics are statistically similar to large-density-ratio
cases when properly scaled.
Generically, such flows are encountered in Rayleigh-Taylor instability, inertial-confinement
fusion, as well as astrophysical and geophysical environments, such as in katabatic winds
and Antarctic bottom-water (AABW) formation.
VIII
th
Int. Symp. on Stratified Flows, San Diego, USA, Aug. 29 - Sept. 1, 2016
1
2 Problem formulation
The conservation of mass, momentum, and species-transport equations, absent species
sources and sinks, are solved in the presence of the externally imposed uniform-acceleration
field.
∂ρ
∂t
+
∇·
(
ρ
u
) = 0
(1a)
∂ρ
u
∂t
+
∇·
(
ρ
uu
) =
−
(
Γ
+
∇
p
)
−
ρg
̂
z
+
∇·
τ
(1b)
∂ρY
α
∂t
+
∇·
[
ρY
α
(
u
+
v
α
)] = 0
(1c)
where
ρ
=
ρ
(
x
,t
) is the local binary-mixture density,
u
(
x
,t
) is the velocity vector,
p
(
x
,t
)
is the pressure,
Γ
is the uniform component of the pressure gradient,
Y
α
(
x
,t
) is the mass
fraction of the
α
-species, and
v
α
(
x
,t
) is the
α
-species diffusion velocity (e.g., Dimotakis,
2005). A Newtonian viscous stress tensor,
τ
(
x
,t
), is assumed for monatomic gases (zero
bulk viscosity).
The flow evolves with initial pure-fluid densities
ρ
1
,
ρ
2
, and
ρ
1
> ρ
2
. In the limit of zero
Mach number, the only part in the species-diffusion velocity is that for Fickian transport,
i.e.,
ρY
α
v
α
=
−
ρ
D∇
Y
α
, which with equations 1a and 1c yields the density-evolution
equation,
∂ρ
∂t
+
u
·∇
ρ
=
−
ρ
∇·
u
=
ρ
∇·
(
D
ρ
∇
ρ
)
.
(1d)
Gas-phase molecular diffusion (
Sc
=
ν/
D ≈
1) is assumed, with a uniform dynamic vis-
cosity,
μ
, leading to a variable diffusion coefficient,
D
(
x
,t
) =
μ/ρ
(
x
,t
). Flow simulations
set
Γ
=
−
ρ
0
g
̂
z
, with
ρ
0
=
βρ
1
+ (1
−
β
)
ρ
2
, where
β
is the volume fraction of high-density
fluid in the domain. This choice ensures constant volume-averaged momentum.
This flow is studied by direct numerical simulation (DNS) in a triply-periodic cubic do-
main, initialized with high-density fluid between regions of low-density fluid, subject to an
imposed uniform vertical acceleration field (Figure 1). A Fourier pseudo-spectral spatial-
discretization method is used with a Helmholtz-Hodge decomposition of the pressure
(Chung and Pullin, 2010) and a semi-implicit Runge-Kutta method (Spalart et al., 1991)
for time integration.
3 Flow parameters
The flow domain is a triply periodic cube with a (dimensional) spatial extent scaled to
L
= 4
π
. The characteristic time,
τ
, and other parameters are scaled as,
τ
= 4
π
√
`
A
g
(2)
`
=
L
2
(3)
A
=
R
−
1
R
+ 1
(4)
R
=
ρ
1
ρ
2
(5)
VIII
th
Int. Symp. on Stratified Flows, San Diego, USA, Aug. 29 - Sept. 1, 2016
2