Page 186 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
178
∂[(ru )u ]
2
1
+ ....
(ru )u +∆x
2
2
1
∂x
2
∂[(ru )u ]
(ru )u 1 ∆x 2 (ru )u +∆x 1 1 1 + ....
1
1
1
∂x 1
∆x 1
(ru )u 1
2
Figure 7.4 Infinitesimal control volume in a flow field. Derivation of conservation of
momentum in x 1 direction. Rate of change of momentum
Since the momentum equation is a vector equation, the momentum in the x 1 direction
will also have a contribution in the x 2 direction. The momentum entering the bottom face
in the x 1 direction is
ρu 2 x 2 u 1 = ρu 1 u 2 x 1 (7.15)
A Taylor expansion is employed to work out the x 1 momentum, leaving the control
volume. In the x 1 direction, we have
2
∂(ρu )
2 1
ρu x 2 + x 2 x 1 (7.16)
1
∂x 1
Similarly, the x 1 momentum leaving the x 2 direction (top surface) is
∂(ρu 1 u 2 )
ρu 1 u 2 x 1 + x 1 x 2 (7.17)
∂x 2
Note that the second- and higher-order terms in the previous Taylor expansion are
neglected. The rate of change of momentum within the control volume due to the x 1
component is written as
∂(ρu 1 )
x 1 x 2 (7.18)
∂t
The net momentum of the control volume is calculated as the ‘momentum exiting the
control volume − momentum entering the control volume + rate of change of the momen-
tum, which is
2
∂(ρu ) ∂(ρu 1 u 2 ) ∂(ρu 1 )
1
x 1 x 2 + + (7.19)
∂x 1 ∂x 2 ∂t
For equilibrium, the above net momentum should be balanced by the net force acting
on the control volume. In order to derive the net force acting on the control volume, refer
to Figure 7.5. From the figure, the total pressure force acting on the control volume in the
x 1 direction is written as (positive in the positive x 1 direction and negative in the negative
x 1 direction)
∂p ∂p

p x 2 − p + x 1 x 2 =− x 1 x 2 (7.20)
∂x 1 ∂x 1