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
+Ω
2
z
−
Ω
3
y
∂u
∂x
=
∂
∂x
U
2
+Ω
3
x
−
Ω
1
z
U
3
+Ω
1
y
−
Ω
2
x
⎛
⎞
0
−
Ω
3
Ω
2
⎝
⎠
.
=
Ω
3
0
−
Ω
1
−
Ω
2
Ω
1
0
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