Page 240 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

232
Conservation of mass
∂u 1
(7.197)
+ ∂u 2 = 0 CONVECTION HEAT TRANSFER
∂x 1 ∂x 2
Conservation of momentum, x 1 component
1 ∂p ∂ ∂
∂u 1 ∂u 1 ∂u 1 1 ∂τ 11 1 ∂τ 12
+ u 1 + u 2 =− + + − (u u ) − (u u )
1 2
1 1
∂t ∂x 1 ∂x 2 ρ ∂x 1 ρ ∂x 1 ρ ∂x 2 ∂x 1 ∂x 2
(7.198)
Conservation of momentum, x 2 component
∂u 2 ∂u 2 ∂u 2 1 ∂p 1 ∂τ 21 1 ∂τ 22 ∂ ∂
+ u 1 + u 2 =− + + − (u u ) − (u u )
2 1
2 2
∂t ∂x 1 ∂x 2 ρ ∂x 2 ρ ∂x 1 ρ ∂x 2 ∂x 1 ∂x 2
(7.199)
All the terms of the above equations are very similar to the ones derived in the begin-
ning of this chapter, with averaged quantities appearing as the main variable. The major
difference, however, is due to the extra terms appearing in the equations, which are con-
cerned with the turbulent eddy process. These extra terms are normally modelled using
turbulence modelling techniques, in order to obtain time-averaged quantities. Therefore, to
model the turbulence, it is necessary to consider the widely used Boussinesq hypothesis,
namely,
∂u i ∂u j 2 ∂u k 2
u u = τ R = ν t + − δ ij − κδ ij (7.200)

i j ij
∂x j ∂x i 3 ∂x k 3
where τ R is the so-called Reynolds stress, ν t is the turbulent eddy viscosity and κ is the
ij
turbulent kinetic energy.
On substituting Equation 7.200 into the time-averaged momentum Equations 7.198 and
7.199, we obtain the final form of the averaged momentum equations as
Conservation of momentum, x 1 component
∂u 1 ∂u 1 ∂u 1 1 ∂p 1 ∂τ 11 1 ∂τ 12 ∂τ 11 R ∂τ 12 R
+ u 1 + u 2 =− + + + + (7.201)
∂t ∂x 1 ∂x 2 ρ ∂x 1 ρ ∂x 1 ρ ∂x 2 ∂x 1 ∂x 2
Conservation of momentum, x 2 component
1 ∂p R R
∂u 2 ∂u 2 ∂u 2 1 ∂τ 21 1 ∂τ 22 ∂τ 21 ∂τ 22
+ u 1 + u 2 =− + + + + (7.202)
∂t ∂x 1 ∂x 2 ρ ∂x 2 ρ ∂x 1 ρ ∂x 2 ∂x 1 ∂x 2
A closer examination of the time-averaged continuity Equation 7.197 and the momen-
tum Equations 7.201 and 7.202, shows that the extra parameters which remain and require
determination, are the turbulent eddy viscosity ν t and the turbulent kinetic energy κ.
The turbulent eddy viscosity may be calculated from several turbulence models. The
accuracy of such turbulence models can vary, but in this case a one-equation turbulence
model will be considered, which employs one transport equation in the calculation of the
turbulent eddy viscosity. The turbulent eddy viscosity relation is given as

ν t = C µ 1/4 1/2 l m (7.203)
κ