161
External Forced Convection
energy equations are listed here for reference: f """ + (1/2)ff "" = 0
and θ "" + (1/2) P r f θ " = 0.
a. Define the similarity variable (η), the similarity function for
temperature (θ), and the derivative of the similarity function for velocity (f " )? Write the BCs that can be used for this
problem?
b. From the Blasius solution of the above similarity equations,
sketch the relations between the similarity functions (f " , θ) and
the similarity variable (η) for water, air, and liquid metal (i.e.,
sketch f " versus η for water, air, and liquid metal on the same
plot; and θ versus η for water, air, and liquid metal on the
same plot). Explain why they have differences, if any. Explain
briefly how to determine the local velocity (u) and temperature
(T) from the sketches? You do not need to solve the above
equations.
c. Explain briefly how to determine the local heat transfer coefficient from the sketches in (b)? Answer whether water or liquid
metal will provide a higher convective heat transfer coefficient
from the surface, if both are at the same free-stream velocity
(U ∞ )? Explain why?
7.8. 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.
a. Based on the integral method, sketch and write down the
momentum balance as well as the energy balance across the
boundary layers? (If you cannot remember, derive it.)
b. Assuming a uniform velocity profile inside the boundary layer
that is, u = U ∞ , and assuming a linear temperature profile
inside the thermal boundary layer as T = a + b × y, determine
the local thermal boundary-layer growth and the local Nusselt
number distribution along the flat plate (i.e., δ t versus X and
Nu versus X).
c. Consider a uniform blowing through the wall (i.e., v = v 0 ), and
comment on whether the local Nusselt number distribution
along the flat plate will be higher, the same, or lower than that
without boundary-layer blowing? Explain why.
7.9. 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.
a. Based on the integral method, write down the final form of
momentum integral equation. (If you cannot remember, please
derive it.)
b. Based on the integral method, can you remember to write
down the final form of the energy integral equation? (If you
cannot remember, please derive it.)
c. Assuming a uniform velocity profile inside the boundary layer
that is, u = U ∞ , and assuming a linear temperature profile
External Forced Convection
energy equations are listed here for reference: f """ + (1/2)ff "" = 0
and θ "" + (1/2) P r f θ " = 0.
a. Define the similarity variable (η), the similarity function for
temperature (θ), and the derivative of the similarity function for velocity (f " )? Write the BCs that can be used for this
problem?
b. From the Blasius solution of the above similarity equations,
sketch the relations between the similarity functions (f " , θ) and
the similarity variable (η) for water, air, and liquid metal (i.e.,
sketch f " versus η for water, air, and liquid metal on the same
plot; and θ versus η for water, air, and liquid metal on the
same plot). Explain why they have differences, if any. Explain
briefly how to determine the local velocity (u) and temperature
(T) from the sketches? You do not need to solve the above
equations.
c. Explain briefly how to determine the local heat transfer coefficient from the sketches in (b)? Answer whether water or liquid
metal will provide a higher convective heat transfer coefficient
from the surface, if both are at the same free-stream velocity
(U ∞ )? Explain why?
7.8. 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.
a. Based on the integral method, sketch and write down the
momentum balance as well as the energy balance across the
boundary layers? (If you cannot remember, derive it.)
b. Assuming a uniform velocity profile inside the boundary layer
that is, u = U ∞ , and assuming a linear temperature profile
inside the thermal boundary layer as T = a + b × y, determine
the local thermal boundary-layer growth and the local Nusselt
number distribution along the flat plate (i.e., δ t versus X and
Nu versus X).
c. Consider a uniform blowing through the wall (i.e., v = v 0 ), and
comment on whether the local Nusselt number distribution
along the flat plate will be higher, the same, or lower than that
without boundary-layer blowing? Explain why.
7.9. 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.
a. Based on the integral method, write down the final form of
momentum integral equation. (If you cannot remember, please
derive it.)
b. Based on the integral method, can you remember to write
down the final form of the energy integral equation? (If you
cannot remember, please derive it.)
c. Assuming a uniform velocity profile inside the boundary layer
that is, u = U ∞ , and assuming a linear temperature profile