Example of Scale Factors for 3-D Boundary Layer Equations – Flow over a Cone
Geometry
Y
r y
α
x
α
α
R
G
r0
P
Z
z or
θ
n
G
X X-Z plane
z or
θ
r0
θ
y
P
Y
X
r
X
Y
3-D view, in x-y plane (X-Z plane is tilted back as shown) View from the rear (X-Y plane)
Definition of scale factors or stretching factors
Recall, x
dR
hdx
≡
G
, y
dR
hdy
≡
G
, and z
dR
hdz
≡
G
, where
(
)
,,
R
XYZ≡
G
is the distance from a fixed origin to a point P
inside the boundary layer. Consider a simple cone of half-angle
α
as an example (axisymmetric about the Z-axis).
Let the fixed origin be the apex (tip) of the cone. Here, let r0 be the perpendicular distance from the Z-axis to the
body surface. Let r be the perpendicular distance from the Z-axis to point P. (Note that this is a different r than what
we defined previously for a general 3-D coordinate system. It is the same r, however, that we defined previously
when we discussed axisymmetric boundary layers.)
The boundary layer coordinates are (x,y,z) where x is a straight line along the body surface from the origin (a ray), y
is normal to the body surface, and z is the angle along the body in the
θ
-direction, measured from the X-axis. (In fact,
we can let z =
θ
.) From trig, we see that X = r⋅cos(
θ
), Y = r⋅sin(
θ
), and Z = x⋅cos(
α
) - y⋅sin(
α
). Also, we see that r =
x⋅sin(
α
) + y⋅cos(
α
).
Now, by definition,
22
x
dR X Y Z
hdx x x x
∂∂∂
⎛⎞⎛⎞⎛⎞
≡= + +
⎜⎟⎜⎟⎜⎟
∂∂∂
⎝⎠⎝⎠⎝⎠
G2
, and similarly for the other scale factors, i.e.,
22
y
dR X Y Z
hdy y y y
⎛⎞⎛⎞⎛⎞
∂∂∂
≡= + +
⎜⎟⎜⎟⎜⎟
∂∂∂
⎝⎠⎝⎠⎝⎠
G2
and
22
z
dR X Y Z
hdz z z z
∂∂∂
⎛⎞⎛⎞⎛⎞
≡= + +
⎜⎟⎜⎟⎜⎟
∂∂∂
⎝⎠⎝⎠⎝⎠
2
G
.
In class we will solve for these scale factors for this example. These can then be plugged into the 3-D boundary layer
equations.
Note: If we assume that the boundary layer is thin with respect to r0, and that there is no swirl, these equations should
reduce to the Mangler equations for axisymmetric flow!
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