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7.2.3 Outline of the Integral Approximate Method
For a hydrodynamic boundary layer:

Step 1: assume the velocity profile, such as

u = a + by
u = a + by + cy
2
3
u = a + by + cy
2
+ dy
4
u = a + by + cy
2
+ dy
3
+ ey
Step 2: from the velocity BCs to determine the coefficients a, b, c, d, and e
Step 3: put the velocity profile into momentum integral to solve δ(x)
Step 4: put δ(x) back to the velocity profile
Step 5: the friction factor
τ w
μ (∂u/∂y)
C fx =
=
0
can be determined
(1/2)ρU 2
(1/2)ρU 2
∞
∞
Similarly, for a thermal boundary layer:

Step 1: assume a temperature profile in one of the following forms:

T = a + by
T = a + by + cy
2
3
T = a + by + cy
2
+ dy
4
T = a + by + cy
2
+ dy
3
+ ey
Step 2: from the temperature BCs to determine the coefficients a, b, c, d, and e
Step 3: put the temperature profile into energy integral to solve δ T (x)
Step 4: put δ T (x) back to the temperature profile and
""
q
−k(∂T/∂y) 0
h =
w
=
can be determined.
T w − T ∞
T w − T ∞
153
External Forced Convection
Examples
7.1 Assume third-order velocity and temperature profiles for the boundary layer
flow to satisfy the BCs: