Incompressible 3-D Laminar Flow Boundary Layer Equations
Assumptions/Approximations
• x and z coordinates lie along the body surface; y coordinate is normal to the body surface.
• The fluid is either an incompressible liquid or a nearly incompressible ideal gas at very low Mach number.
• Gravity is neglected.
Scale Factors or Stretching Factors
Define x
dR
hdx
≡
G
, y
dR
hdy
≡
G
, and z
dR
hdz
≡
G
, where (,) (,)
R
rxz ynxz≡+
G
G
G is the distance from the fixed origin to a
point inside the boundary layer, is the distance from the fixed origin to the body surface, and
r
Gn
G
is the unit
outward normal (in the y-direction away from the body surface).
Note: hy is always unity when y is normal to the body surface, as assumed above.
General 3-D Boundary Layer Equations
Continuity:
() ()
10
zx
xz
v
hu hw
hh x z y
∂∂ ∂
⎡⎤
++
⎢⎥
∂∂ ∂
⎣⎦
=
,
x-momentum: 22
xz 2
1
x z xz xz x
hh
uu wu u uw w p u
v
hxhz yhh z hh x hx y
ν
ρ
∂∂
∂∂∂ ∂∂
+++ − =− +
∂∂∂ ∂ ∂ ∂∂
,
y-momentum: 0
p
y
∂=
∂ (same as 2-D B. L., as long as
δ
<< any local radius of curvature of the body),
z-momentum: 22
xz 2
1
x z xz xz z
hh
uw ww w u uw p w
v
hxhz yhh z hh x hz y
ν
ρ
∂∂
∂∂∂ ∂∂
++− + =− +
∂∂∂ ∂ ∂ ∂∂
.
Euler Equations for the Irrotational Outer Flow (U and W along the body surface)
Note: These expressions can replace the pressure gradient terms in the x- and z-momentum equations above, so
that the boundary layer equations can be written in terms of the known outer flow velocity field at the wall, U(x,z)
and W(x,z).
x-momentum: 2
xz
1
x z xz xz x
hh
UU WU UW W p
hxhzhhzhhx hx
ρ
∂∂
∂∂
++ − =−
∂∂ ∂ ∂
∂
∂
,
z-momentum: 2xz
1
x z xz xz z
hh
UW WW U UW p
hxhzhhzhhx hz
ρ
∂∂
∂∂
+− + =−
∂∂ ∂ ∂
∂
∂
.
Boundary Conditions
• No slip conditions: u = v = w = 0 at y = 0 (at the surface of the body) for all x and z.
• Edge conditions: u → U and w → W as y → ∞ (outside the BL) for all x and z.
• Starting conditions: Must specify u and w profiles at some x and z locations (starting profiles) to begin the
calculations.
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