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a. Draw the velocity distribution from Martinelli Universal
Velocity Profile (i.e., law of the wall) for air flow at 1 atm, 25 ◦ C,
and Re D = 30, 000.
b. Calculate the laminar sublayer thickness.
10.7. Consider the turbulent boundary-layer flow heat transfer: Air at
300 K, 1 atm, flows at 12 m/s along a flat plate maintained at 600 K.
Plot the temperature profile T(y) across the boundary layer for the
following two cases:
a. At a location x = 0.1 m for a laminar boundary layer.
b. At a location x = 1.0 m if the transition Reynolds number is
10 5 .
Plot both profiles on the same graph to show significant
differences.
∗
""
c. Determine C fx , τ s , u , St x , Nu x , h x , and q for case (b).
s
10.8. Consider a 2-D incompressible turbulent flow in a pipe.
a. Specialize (simplify) the given continuity and Navier–Stokes
equations for a fully developed turbulent flow in a pipe.
Write appropriate BCs to solve the flow equations for a fully
developed turbulent flow. Do not attempt to solve the problem.
b. The velocity profile (u(r)) for a fully developed turbulent flow
in a pipe is given by
� 1/7
u
R − r
=
u max
R
where R is the pipe radius, r is the radial distance measured
from the pipe axis, and U max is the maximum velocity. Calculate mean or the bulk velocity (U m ) for a fully developed flow
in terms of U max .
c. Obtain an expression for the skin friction coefficient (Fanning
friction factor) for a fully developed turbulent flow in terms
of Re D , where Re D is the Reynolds number based on the pipe
hydraulic diameter.
10.9. Consider a steady low-speed, constant-property, fully turbulent boundary-layer flow over a flat surface at constant wall
temperature.
a. Based on the Reynolds time-averaged concept, derive the following momentum and energy equations (make necessary
assumptions):
∂u
∂u
∂
∂u
u
+ v
=
(ν + ε M )
∂x
∂y
∂y
∂y
∂T
∂T
∂
ν
ε M ∂T
u
+ v
=
+
∂x
∂y
∂y
Pr
Pr t ∂y
+
+
b. Define dimensionless parameters, u , T + , and y , respectively, for universal velocity and temperature profiles for turbulent flow and heat transfer problems. Based on the Prandtl’s
218
Analytical Heat Transfer
�
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a. Draw the velocity distribution from Martinelli Universal
Velocity Profile (i.e., law of the wall) for air flow at 1 atm, 25 ◦ C,
and Re D = 30, 000.
b. Calculate the laminar sublayer thickness.
10.7. Consider the turbulent boundary-layer flow heat transfer: Air at
300 K, 1 atm, flows at 12 m/s along a flat plate maintained at 600 K.
Plot the temperature profile T(y) across the boundary layer for the
following two cases:
a. At a location x = 0.1 m for a laminar boundary layer.
b. At a location x = 1.0 m if the transition Reynolds number is
10 5 .
Plot both profiles on the same graph to show significant
differences.
∗
""
c. Determine C fx , τ s , u , St x , Nu x , h x , and q for case (b).
s
10.8. Consider a 2-D incompressible turbulent flow in a pipe.
a. Specialize (simplify) the given continuity and Navier–Stokes
equations for a fully developed turbulent flow in a pipe.
Write appropriate BCs to solve the flow equations for a fully
developed turbulent flow. Do not attempt to solve the problem.
b. The velocity profile (u(r)) for a fully developed turbulent flow
in a pipe is given by
� 1/7
u
R − r
=
u max
R
where R is the pipe radius, r is the radial distance measured
from the pipe axis, and U max is the maximum velocity. Calculate mean or the bulk velocity (U m ) for a fully developed flow
in terms of U max .
c. Obtain an expression for the skin friction coefficient (Fanning
friction factor) for a fully developed turbulent flow in terms
of Re D , where Re D is the Reynolds number based on the pipe
hydraulic diameter.
10.9. Consider a steady low-speed, constant-property, fully turbulent boundary-layer flow over a flat surface at constant wall
temperature.
a. Based on the Reynolds time-averaged concept, derive the following momentum and energy equations (make necessary
assumptions):
∂u
∂u
∂
∂u
u
+ v
=
(ν + ε M )
∂x
∂y
∂y
∂y
∂T
∂T
∂
ν
ε M ∂T
u
+ v
=
+
∂x
∂y
∂y
Pr
Pr t ∂y
+
+
b. Define dimensionless parameters, u , T + , and y , respectively, for universal velocity and temperature profiles for turbulent flow and heat transfer problems. Based on the Prandtl’s
218
Analytical Heat Transfer