Page 211 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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CONVECTION HEAT TRANSFER
where ˜u 1 and ˜u 2 are the intermediate momentum variables. It is obvious that the CG scheme
can now be applied, as the above equations are very similar to the convection–diffusion
equations of the previous section. If the CG procedure is applied to the above equations, a
semi-discrete from of the equations is obtained, namely,
intermediate x 1 momentum equation
! n
n n n 2 2
˜ u 1 − u 1 ∂u 1 ∂u 1 ∂ u 1 ∂ u 1
=−u 1 − u 2 + ν +
t ∂x 1 ∂x 2 ∂x 2 ∂x 2
1 2
t ∂ ∂u 1 n ∂u 1 n
+ u 1 u 1 + u 2
2 ∂x 1 ∂x 1 ∂x 2
t ∂ ∂u 1 n ∂u 1 n
+ u 2 u 1 + u 2 (7.129)
2 ∂x 2 ∂x 1 ∂x 2

intermediate x 2 momentum equation

! n
n n n 2 2
˜ u 2 − u 2 ∂u 2 ∂u 2 ∂ u 2 ∂ u 2
=−u 1 − u 2 + ν +
t ∂x 1 ∂x 2 ∂x 2 ∂x 2
1 2
t ∂ ∂u 2 n ∂u 2 n
+ u 1 u 1 + u 2
2 ∂x 1 ∂x 1 ∂x 2
t ∂ ∂u 2 n ∂u 2 n
+ u 2 u 1 + u 2 (7.130)
2 ∂x 2 ∂x 1 ∂x 2

Step 2 Pressure calculation: The pressure field is calculated from a pressure equation of
the Poisson type. The pressure equation is derived from the fact that the intermediate
velocities at the first step need to be corrected. If the pressure terms are not removed from
the momentum equations, then the correct velocities are obtained, but with the loss of
some advantages. If the semi-discrete form of the momentum equations are written without
removing the pressure terms, then

semi-discrete x 1 momentum equation
n
!
n+1 n n n 2 2 n
u 1 − u 1 ∂u 1 ∂u 1 ∂ u 1 ∂ u 1 1 ∂p
=−u 1 − u 2 + ν + −
t ∂x 1 ∂x 2 ∂x 2 ∂x 2 ρ ∂x 1
1 2
t ∂ ∂u 1 n ∂u 1 n 1 ∂p n
+ u 1 u 1 + u 2 +
2 ∂x 1 ∂x 1 ∂x 2 ρ ∂x 1
t ∂ ∂u 1 n ∂u 1 n 1 ∂p n
+ u 2 u 1 + u 2 + (7.131)
2 ∂x 2 ∂x 1 ∂x 2 ρ ∂x 1