Page 104 - Introduction to Computational Fluid Dynamics

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4.6 BOUNDARY CONDITIONS
with the notion that the variation in in the transverse direction asymptotically 11:7 83
approaches a value ∞ (say) there. Thus, the fictitious notion of a boundary layer
thickness is associated with
− I
= A, (4.62)
∞ − I
where suffix I refers to the inner boundary (wall or symmetry) and A is typically
taken to be 0.99 by convention. Note, however, that this boundary layer thickness
willbedifferentfordifferentmeaningsof andthemagnitudeofthicknesstypically
2
depends on the Prandtl number Pr defined as
ν
Pr ≡ . (4.63)

The Prandtl number is a property of the fluid. In fact, in Table 4.1, we may replace
.
k/C p by µ/Pr T and ρ D k by µ/Pr ω k
There is one further notion associated with the free stream. If we assume the
E boundary to be the free boundary (see Figure 4.1), the flow region above the
boundary can be taken to be a region in which there is no transverse convection or
diffusion and
b = ∞ (x), (4.64)

where ∞ (x) is specified. However, the physical location where this boundary
condition is to be applied is not a priori known because of the asymptotic nature
of variation of in the vicinity of this boundary. To circumvent this problem,
Patankar and Spalding [50] relied on estimating the entrainment rate (− ˙ m E ) into
the boundary layer that occurs from the fluid above the E-boundary.
Thus, as previously mentioned, since there is no net flux of in the transverse
direction, from Equation 4.17, it follows that
∂ ∂

∂
(a + b ω) E = c . (4.65)
∂ω ∂ω ∂ω
E E
However, at the E boundary, ω = 1. Therefore,
−1
(a + b ω) E = a + b =−ψ EI ∂ψ E . (4.66)
∂x
Thus, Equation 4.65 can be written as
−1
∂ψ E ∂ ∂ ∂ ∂ ∂
=−ψ EI c =−ψ EI c
∂x ∂ω ∂ω ∂ω ∂ ∂ω
∂ ∂
=− r
=−r E ˙ m E . (4.67)
∂ ∂y
2 The term Prandtl number applies to variables T and h. When = ω k , the appropriate dimensionless
number is called the Schmidt number (Sc). For velocity variables, of course, Pr = 1. We thus use
Pr generically to cover all s.