Page 221 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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CONVECTION HEAT TRANSFER
7.7 Artificial Compressibility Scheme
As mentioned before, convection heat transfer calculations can be carried out using a fully
explicit Artificial Compressibility (AC) scheme. In AC schemes, an artificial compressibility
is introduced at Step 2 of the CBS scheme, that is,
!
2
2
1 ∂p ∂ p ∂ p ∂ ˜u 1 ∂ ˜u 1
− t + + + = 0 (7.170)
β ∂t ∂x 2 ∂x 2 ∂x 1 ∂x 2
2
1 2
where β is an artificial compressibility parameter. The above equation can be derived by
assuming a density variation in the continuity equation by substituting
∂ρ 1 ∂p
≈ (7.171)
2
∂t c ∂t
where c is the speed of sound, which, for incompressible flows, approaches infinity. How-
ever, c can be replaced by an artificial compressibility parameter β, as given in Equation
7.170, for the purpose of introducing an explicit scheme. In the artificial-compressibility-
based CBS scheme, Step 2 will be replaced with
1 { p} n 1
[M] + [K]{p} =− [G 1 ]{˜ u 1 }+ [G 2 ]{˜ u 2 } +{f 3 } (7.172)
β 2 t t
n
where {p} = {p n+1 − p }. The artificial compressibility parameter can be chosen as
β = max(c o ,u conv ,u diff ,u therm ) (7.173)
where c o is a small constant (between 0.1 to 0.5) and u conv , u diff and u therm are respectively
the convection, diffusion and thermal velocities, which may be defined as

2
u conv = u + u 2
1 2
2ν
u diff =
h
2α
u therm = (7.174)
h
All other steps of the CBS scheme remain the same. However, for the solution of
transient problems, a dual time-stepping procedure has to be introduced. In this dual time-
stepping procedure, a transient problem is split into several instantaneous steady states and
integrated via a real global time-step. Further details on the dual time-stepping procedure
can be found in references (Malan et al. 2002; Nithiarasu 2003).

7.8 Nusselt Number, Drag and Stream Function

The two important quantities of interest in many heat transfer applications are the rate
of heat transfer (Nusselt number) and the flow resistance offered by a surface (drag). A
stream function is often used to draw streamlines in order to better understand the flow
pattern around a body. In this section, a brief summary is given on how to calculate these
quantities.