Page 66 - Introduction to Computational Fluid Dynamics

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2.9 PROBLEMS FROM RELATED FIELDS
slope of the temperature profile at the point of discontinuity (x = 0.0125 m) in 10:49 45
conductivity. Finally, by assigning different values to k 2 , k 3 , r 2 , r 3 , and t, it would
be possible to carry out a parametric study to aid optimisation of fin volume and
economic cost in a separate design study.
2.9 Problems from Related Fields
Quite a few problems from the fields of fluid mechanics, convective heat transfer,
and diffusion mass transfer are governed by equations that bear similarity with
Equation 2.5. Only the dependent variable, the coefficients, and the source term
need to be interpreted appropriately. We discuss such problems next.

Fully Developed Laminar Flow
Steady, fully developed laminar flow in a tube is governed by
∂ ∂u dp
µ2π r − 2π r = 0, (2.77)
∂r ∂r dz
where u is velocity parallel to the tube axis and the pressure gradient is a negative
constant. Since velocity u is directed in the z direction, it can be treated as a scalar
with respect to the r direction. Comparison with Equation 2.5 shows that T ≡ u,

∂x ≡ ∂r, A ≡ 2πr, k ≡ µ, andq ≡−dp/dz. For a circular tube, u = 0atr = R
(tube radius) and ∂u/∂r = 0 at the tube axis r = 0. Equation 2.77 is also applicable
to an annulus with boundary conditions u = 0at r = R i and r = R o . Similarly, the
equation is applicable to flow between parallel plates if we set A = 2π r = 1 and
∂x ≡ ∂r ≡ ∂y, where y is measured from the symmetry axis.

Fully Developed Turbulent Flow
In this case, if Boussinesq approximation is considered valid then the axial velocity
is governed by
∂ ∂u
dp
(µ + µ t )2π r − 2π r = 0, (2.78)
∂r ∂r dz

where the turbulent viscosity µ t = ρ l 2 ∂u with
m ∂r
y y

⎧ +
⎪ κ y 1 − exp − for < y l ,
⎪
26 R
⎨
l m = (2.79)
y
⎪
⎪
⎩ 0.085 R for > y l ,
R
√
where κ = 0.41, y = R − r, y l
0.2, and y = y τ w /ρ /ν with τ w the shear
+
stress at the wall (i.e., τ w = µ∂u/∂y | y=0 ). Clearly, Equation 2.78 can be solved
iteratively by estimating the turbulent viscosity distribution from the velocity
gradient.