Page 54 - Introduction to Computational Fluid Dynamics

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2.7 DEALING WITH NONLINEARITIES
In the following, we discuss methods for dealing with nonlinearities through 10:49 33
modification of Su i and Sp i .
2.7.1 Nonlinear Sources
Consider a pin fin losing heat to its surroundings under steady state by convection

with heat transfer coefficient h. Then, q will be given by
h i P i x i (T i − T ∞ )

q =− , (2.48)
i
A i x i
where P i is the local fin perimeter. Therefore,

q V i =−h i P i x i (T i − T ∞ ). (2.49)
i
When this equation is included in Equation 2.43, it is obvious that T i will now
appear on both sides of the equation. One can therefore write the total source term
as

Source term = Su i + h i P i x i (T ∞ − T i ). (2.50)

This prescription can be accommodated by updating Su i and Sp i as

Su i = Su i + h i P i x i T ∞ ,

Sp i = Sp i + h i P i x i , (2.51)

where Su i and Sp i on the RHSs are the original quantities given in Equation 2.47.
Note that, in this case, the updated Sp i is positive and, therefore, there is no
danger of rendering AP i + Sp i negative. Thus, Scarborough’s criterion cannot be
violated. However, if we considered dissipation of heat due to an electric current or
chemical reaction (as in setting of cement) then, because heat is generated within
m
the medium, q = a + bT , where b is positive. In this case, Su i = Su i + a V i
i i
and Sp i = Sp i − bT m−1 V i . But now, there is a danger of violating Scarbor-
i

ough’s criterion and, therefore, one simply sets Su i = Su i + q V i and Sp i is not
i
updated.
Accounting for the source term in the manner of Equation 2.51 is called source
term linearization [49]. We shall discover further advantages of this form when
dealing with the application of boundary conditions.

2.7.2 Nonlinear Coefficients

Coefficients AE i and AW i can become functions of temperature owing to thermal
2
conductivity as in k = a + bT + cT . Thus, k i+1/2 in AE i (see Equation 2.45),