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4.5 DETERMINATION OF ω, y, AND r
for j = 2, 3,..., JN − 1. Note that superscript d is now dropped for 11:7 79
convenience.
4.5 Determination of ω, y, and r
Equation 4.43 represents a set of algebraic equations at a streamwise location x d .
u
These equations can be solved by TDMA when values of at x u are known along
j
with the two boundary conditions at x d (i.e., at j = 1 and j = JN ). Thus, starting
with x = x 0 (say), one can execute a marching procedure taking step x. This
situation is very much like the unsteady conduction problem in which the marching
procedure is executed with time step t.
Thus, at x = x 0 , the u j ∼ y j relationship is assumed to have been prescribed
either from experimental data or from an analytical solution. One can use this
prescription to set ω j once and for all. Let

ω j = ω P , ω c, j = ω s , ψ j = ψ P , ψ c, j = ψ s ,
y j = y P , y c, j = y s , r j = r P , r c, j = r s , (4.44)

where, at x = x 0 , y j ( j = 1, 2,..., JN ) are known. Thus, one can set y c,1 = y c,2 =
y 1 where y 1 refers to the I boundary and y JN to the E boundary. Now, from the
geometry of Figure 4.1, it follows that r j and r c, j can be evaluated from the formula

r = r I + y cos (α), (4.45)

where α is function of x. This completes the grid specification at x = x 0 .
For evaluation of ω j , we first calulate ψ j . Thus, setting ψ 1 = ψ c,1 = ψ I (say),
where ψ I is arbitrarily chosen, one can use Equation 4.5 to set all other ψ j . The
relevant discretised equations are

ψ c, j = ψ c, j−1 + (ρ ru) j−1 (y c, j − y c, j−1 ), j = 2, 3,..., JN, (4.46)

ψ j = ψ j−1 + 0.5 (ρ ru) j + (ρ ru) j−1 (y j − y j−1 ), j = 2, 3,..., JN.
(4.47)

It is now a simple matter to evaluate ω j and ω c, j using definition (4.2). Thus,
ω j at y j represents the ratio of streamwise mass flow rate from y 1 = y I to y j to the
total mass flow rate from y I to y E at x = x 0 . It is now assumed that this ratio remains
intact at all values of x and thus the ω j distribution does not change throughout the
domain in the x direction.
Note, however, that the physical distance y (and therefore r) must go on changing
at different values of x as the boundary layer grows or shrinks. We thus seek the
y j ∼ ω j relationship applicable to every x.