Page 123 - Introduction to Computational Fluid Dynamics

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U o 0 521 85326 5 2D BOUNDARY LAYERS
Φ

Figure 4.11. Flow over a spinning cone.

6. It is desired to calculate turbulent boundary layer development so that the initial

velocity profile may be given by Equation 4.116. Choose a distribution of y j
(0 < y <δ) such that (ω j+1 − ω j )/(ω j − ω j−1 ) = 1.2 for all j.
7. Consider flow across a long horizontal cylinder of radius R. It is desired to cal-
culate boundary layer development near the forward stagnation point. Specify
variation of α and r I with x. Also specify the starting velocity profile.
8. In Exercise 7, it is of interest to calculate the mass transfer of an inert substance
in the forward stagnation region. Specify the starting mass fraction profile and
select the appropriate boundary conditions for the mass-fraction variable ω and
u. (Hint: Use the integral method to specify the ω profile.)
9. It is desired to calculate boundary layer development over a cone spinning with
angular velocity (see Figure 4.11). Write the governing equations and the
boundary conditions at the I and E boundaries for this problem. Also provide
initial conditions. (Hint: Assume that the spinning rate is high so that centrifugal
and Coriolis forces must be considered. Also, ∂p/∂r is not negligible. Hence,
dp/dx will vary with y.)
10. Consider an adiabatic wall 2 m high, as shown in Figure 4.12. The bottom 1 m
is covered with a thick layer of highly volatile solid material having latent heat
λ fu . The fuel burns in stagnant dry air under natural convection conditions.
Assume SCR (4.86) with reaction rate given by (4.87).

(a) Write all relevant equations governing the phenomenon of burning along
with their source terms. (Hint: Use the Boussinesq approximation for the
buoyancy term.)

(b) Write boundary conditions at the I boundary to determine the burning rate.
AlsowriteconditionsattheEboundary.[Hint:Inthisproblem,theadiabatic
condition implies that T b = T T . Further, the burning surface temperature
will equal the evaporation (or boiling) point temperature T bp and is a known
property. Further, the SCR assumption implies that ω fu = ω ox = 0 at the
burning surface.]