Graphics Reference
In-Depth Information
-9-
Tu r b u l e n c e
This chapter takes a look at methods aimed to capture more of the fine-
scale swirly motion characteristic of turbulence. This is far from a scientific
examination of turbulence, and in fact scientific work on the subject tends
to concentrate on averaging or smoothing over the details of turbulent ve-
locity fields—whereas we want to get to those details as cheaply as possible,
even if they fall short of true accuracy.
9.1 Vorticity
Our first stop is getting at a precise measurement of the “swirliness” char-
acteristic of turbulent flow. That is, at any point in space, we would like
to measure how the fluid is rotating. In Chapter 8 on viscosity we saw how
the gradient of the velocity field gives a matrix whose symmetric part mea-
sures deformation—independent of rigid body motions. It's not surprising
then that what's left over, the skew-symmetric part, gives us information
about rotation. (And of course, u itself without any derivatives tells us the
translational component of the motion.)
Let's take a look at a generic rigid motion velocity field in 3D:
u ( x )= U + Ω
×
x.
Here U is the translation, and Ω is the angular velocity measured around
the origin. Let's work out the gradient of this velocity field in three dimen-
sions to see how we can extract the angular velocity:
U 1 2 z
Ω 3 y
∂u
∂x =
∂x
U 2 3 x
Ω 1 z
U 3 1 y
Ω 2 x
0
Ω 3
Ω 2
.
=
Ω 3
0
Ω 1
Ω 2
Ω 1
0
127
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