Vorticity and Enstrophy Changes
due to Stretching/Compression
in a Turbulent Mixing Layer
Patrick Rabenold
ENME 656 Midterm Project
April 14, 2005
Abstract
A joint probability density function (JPDF) and covariance integrand
analysis is performed on experimental mixing layer data in order to better
understand the term in the transport equations for instantaneous vortic
ity and enstrophy that represents stretching and compression of vortic
ity filaments due to reorientation.
The data is found to contain several
anomalies which must be corrected, at least approximately.
Using this
data, strong correlations are found between Ω
j
and
∂U
i
/∂x
j
, which help
in understanding correlations between Ω
i
and Ω
j
∂U
i
/∂x
j
, which result in
a net increase in the rate of change of enstrophy due to vorticity filament
stretching or compression.
It is found that vorticity filament stretching
dominates compression.
1
Transport Equations
The vorticity field,
Ω
≡ ∇×
u
, can be very useful in understanding many aspects
of turbulent flows. Therefore, it is helpful to derive its own transport equation
by taking the curl of the incompressible NavierStokes equation, which results
in
∂
Ω
i
∂t
+
U
j
∂
Ω
i
∂x
j
= Ω
j
∂U
i
∂x
j
+
ν
∂
2
Ω
i
∂x
j
∂x
j
.
(1)
The enstrophy,
1
2
Ω
i
Ω
i
, is a scalar measure of the magnitude of vorticity, similar
to kinetic energy for velocity. The enstrophy transport equation is derived from
equation (1) as
∂
(
1
2
Ω
i
Ω
i
)
∂t
+
U
j
∂
(
1
2
Ω
i
Ω
i
)
∂x
j
= Ω
i
Ω
j
∂U
i
∂x
j
+
ν
∂
2
(
1
2
Ω
i
Ω
i
)
∂x
j
∂x
j
−
ν
∂
Ω
i
∂x
j
∂
Ω
i
∂x
j
.
(2)
The terms in both of these equations represent, from left to right, the Eulerian
time rate of change of vorticity or enstrophy, advection, stretching and com
pression due to reorientation, viscous diffusion, and viscous dissipation for the
enstrophy.
1
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A physical interpretation of the stretching/compression term can be under
stood as follows.
If a vortex filament is aligned in the
x
direction, then the
term Ω
x
∂U/∂x
represents a gain or loss of Ω
x
due to pure stretching or com
pression, depending on the signs of Ω
x
and
∂U/∂x
. However, the Ω
y
equation
contains the term Ω
x
∂V/∂x
, which represents pure rotation of the filament
into the
y
direction.
An arbitrarily oriented vortex filament can be thought
of as undergoing simultaneous stretching and compression due to reorientation
by the velocity gradients.
This physical description can be extended to the
stretching/compression term in the enstrophy equation, by considering a vortex
filament projected onto a coordinate plane. All combinations of the signs of Ω
i
,
Ω
j
, and
∂U
i
/∂x
j
determine whether there is an increase due to stretching or a
decrease due to compression in the contribution to the change in enstrophy.
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 Spring '11
 chandra
 Fluid Dynamics, Derivative, probability density function, vorticity, Ωz, vorticity equation

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