Page 192 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
184
Energy equation
2
2
∂T
+ u 1 ∂T + u 2 ∂T = α ∂ T + ∂ T ! (7.53)
∂t ∂x 1 ∂x 2 ∂x 2 ∂x 2
1 2
where ν = µ/ρ is the kinematic viscosity. To obtain a set of non-dimensional equations,
let us consider three different cases of convective heat transfer. We start with the forced
convection problem followed by the ‘natural’ and ‘mixed’ convection problems. For each
case, we discuss one set of non-dimensional scales. There are several other ways of scaling
the equations. Some of these are discussed in the latter part of the chapter and others can
be found in various other publications listed at the end of this chapter.
7.3.1 Forced convection
In forced convection problems, the following non-dimensional scales are normally employed:

x 1 x 2 tu a
∗ ∗ ∗
x = ; x = ; t = ;
1
2
L L L
u 1 u 2 p
∗ ∗ ∗
u = ; u = ; p = 2 ;
1
2
u a u a ρu a
T − T a
∗
T = (7.54)
T w − T a
Where ∗ indicates a non-dimensional quantity, L is a characteristic dimension, the
subscript a indicates a constant reference value and T w is a constant reference temperature,
for example, wall temperature. The density ρ and viscosity µ of the fluid are assumed to
be constant everywhere and equal to the inlet value.
Substitution of the above scales into the dimensional Equations 7.50 to 7.53 leads to
the following non-dimensional form of the equations:
Continuity equation
∂u ∗ ∂u ∗
1 2
+ = 0 (7.55)
∂x ∗ ∂x ∗
1 2
x 1 momentum equation
!
2 ∗
2 ∗
∂u ∗ ∂u ∗ ∂u ∗ ∂p ∗ 1 ∂ u ∂ u
1 + u ∗ 1 + u ∗ 1 =− + 1 + 1 (7.56)
∂t ∗ 1 ∂x ∗ 2 ∂x ∗ ∂x ∗ Re ∂x ∗2 ∂x ∗2
1 2 1 1 2
x 2 momentum equation
!
2 ∗
∂u ∗ ∂u ∗ ∂u ∗ ∂p ∗ 1 ∂ u ∂ u
2 ∗
2 + u ∗ 2 + u ∗ 2 =− + 2 + 2 (7.57)
∂t ∗ 1 ∂x ∗ 2 ∂x ∗ ∂x ∗ Re ∂x ∗2 ∂x ∗2
1 2 2 1 2
Energy equation
!
2
2
∂T ∗ ∂T ∗ ∂T ∗ 1 ∂ T ∗ ∂ T ∗
+ u ∗ 1 + u ∗ 2 = + (7.58)
∂t ∗ ∂x ∗ ∂x ∗ ReP r ∂x ∗2 ∂x ∗2
1 2 1 2