Page 179 - Introduction to Computational Fluid Dynamics

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S L 2D CONVECTION – CARTESIAN GRIDS
S
T
2B

L

X
2
X
1
Figure 5.29. Flow in a channel containing rods.

21. Consider fully developed turbulent flow in a pipe of radius R. Assuming that
+
the inner layer extends up to y = 100 from the wall, estimate the inner layer
thickness as a fraction of R for Re = 5,000, 25,000, 75,000, and 100,000.
22. Air at 30 C enters a tube (diameter D = 5.0 cm) of a solar air-heater with a
◦
uniform velocity of 10 m/s. The tube is 2.1 m long. The tube wall tempera-
ture is 90 C. Determine the exit bulk temperature and the pressure drop. Also
◦
determine the length-averaged Nusselt number. Use the HRE model.
23. Repeat Exercise 22 assuming that the tube is rough with roughness height
y r /D = 0.01. Use the HRE model. For a rough surface, the velocity profile
near a wall is given by [65]
1 y
+
u = ln + 8.48.
κ y r
This equation can be cast in the form of Equation 5.86 so that
1 exp(8.48κ)
+ +
u = ln E r y , E r = .
κ y +
r
Thus, the wall-function treatment remains valid with E replaced by E r . Simi-
larly, PF (Equation 5.88) must be replaced by PF r = 5.19 Pr 0.44 y + 0.2 − 8.48
r
with Pr t = 1 [22]. (Hint: You will need to modify the BOUND subroutine and
STAN function in the Library file in Appendix C to account for y r .)
24. Consider steady turbulent flow in a two-dimensional plane channel (see Fig-
ure 5.29) containing an array of rods (of diameter D). Flow enters at x 1 = 0
with uniform velocity u 1,in . It is of interest to determine the pressure drop
over length L. To reduce the computational effort in this densely filled flow
situation, model the flow as a porous-body flow in which it is assumed that the