For a typical case, x o = 0, δ T = 0,
δ T = 0.975Pr −1/3 � Pr −1/3
δ
For the nonheating leading length problem, x o > 0,
hx
0.323 Re
1/2 Pr 1/3
x
Nu x =
= )
(7.39)
k
3 1 − (x 0 /x) 3/4
If x 0 = 0, go back to the typical case.
157
External Forced Convection
Remarks
For the integral method, students are expected to know how to sketch and
derive momentum and energy integral equations from mass, force, and heat
balance across the boundary layer. Students are also expected to know how
to solve velocity boundary-layer thickness and the friction factor from the
derived momentum integral equation by assuming any velocity profile to
satisfy velocity BCs across the boundary layer; as well as how to solve thermal boundary thickness and the heat transfer coefficient from the derived
energy integral equation by assuming any temperature profile to satisfy thermal BCs (given surface temperature or surface heat flux) across the thermal
boundary layer. Note that one will get a slightly different velocity boundarylayer thickness (and friction factor) by using different velocity profiles across
the boundary layer, and a slightly different thermal boundary-layer thickness (and a heat transfer coefficient) by using different temperature profiles
across the thermal boundary layer. This is the nature of the integral method.
Another note is that velocity and thermal boundary-layer thickness is the
same if Prandtl number unity is assumed.
PROBLEMS
7.1. Consider a steady, incompressible, low-speed 2-D laminar
boundary-layer flow (at U ∞ , T ∞ ) over a flat plate at a uniform wall temperature T W . Assume that there exist no body
force and constant thermal and fluid properties. The similarity
momentum and energy equations are listed here for reference:
f """ + (1/2)ff "" = 0 and θ "" + (1/2)P r f θ " = 0.
a. From the Blasius solution of the above similarity equations,
sketch the relations between the similarity functions (f " , θ) and
the similarity variable (η) for both water and air (i.e., sketch f "
versus η for both water and air on the same plot; and θ versus
η for both water and air on the same plot). Explain why they
δ T = 0.975Pr −1/3 � Pr −1/3
δ
For the nonheating leading length problem, x o > 0,
hx
0.323 Re
1/2 Pr 1/3
x
Nu x =
= )
(7.39)
k
3 1 − (x 0 /x) 3/4
If x 0 = 0, go back to the typical case.
157
External Forced Convection
Remarks
For the integral method, students are expected to know how to sketch and
derive momentum and energy integral equations from mass, force, and heat
balance across the boundary layer. Students are also expected to know how
to solve velocity boundary-layer thickness and the friction factor from the
derived momentum integral equation by assuming any velocity profile to
satisfy velocity BCs across the boundary layer; as well as how to solve thermal boundary thickness and the heat transfer coefficient from the derived
energy integral equation by assuming any temperature profile to satisfy thermal BCs (given surface temperature or surface heat flux) across the thermal
boundary layer. Note that one will get a slightly different velocity boundarylayer thickness (and friction factor) by using different velocity profiles across
the boundary layer, and a slightly different thermal boundary-layer thickness (and a heat transfer coefficient) by using different temperature profiles
across the thermal boundary layer. This is the nature of the integral method.
Another note is that velocity and thermal boundary-layer thickness is the
same if Prandtl number unity is assumed.
PROBLEMS
7.1. Consider a steady, incompressible, low-speed 2-D laminar
boundary-layer flow (at U ∞ , T ∞ ) over a flat plate at a uniform wall temperature T W . Assume that there exist no body
force and constant thermal and fluid properties. The similarity
momentum and energy equations are listed here for reference:
f """ + (1/2)ff "" = 0 and θ "" + (1/2)P r f θ " = 0.
a. From the Blasius solution of the above similarity equations,
sketch the relations between the similarity functions (f " , θ) and
the similarity variable (η) for both water and air (i.e., sketch f "
versus η for both water and air on the same plot; and θ versus
η for both water and air on the same plot). Explain why they