Page 212 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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semi-discrete x 2 momentum equation
n
!
n+1 n n n CONVECTION HEAT TRANSFER
n
2
2
u 2 − u 2 ∂u 2 ∂u 2 ∂ u 2 ∂ u 2 1 ∂p
=−u 1 − u 2 + ν + −
t ∂x 1 ∂x 2 ∂x 2 ∂x 2 ρ ∂x 2
1 2
t ∂ ∂u 2 n ∂u 2 n 1 ∂p n
+ u 1 u 1 + u 2 +
2 ∂x 1 ∂x 1 ∂x 2 ρ ∂x 2
t ∂ ∂u 2 n ∂u 2 n 1 ∂p n
+ u 2 u 1 + u 2 + (7.132)
2 ∂x 2 ∂x 1 ∂x 2 ρ ∂x 2
The real velocity field may be directly obtained if the above equations are utilized.
Subtracting Equation 7.129 from 7.131 and 7.130 from 7.132 results in the following two
equations:
u n+1 −˜u 1 1 ∂p n t ∂ 1 ∂p n t ∂ 1 ∂p n
1
=− + u 1 + u 2
t ρ ∂x 1 2 ∂x 1 ρ ∂x 1 2 ∂x 2 ρ ∂x 1
u n+1 −˜u 2 1 ∂p n t ∂ 1 ∂p n t ∂ 1 ∂p n
2
=− + u 1 + u 2 (7.133)
t ρ ∂x 2 2 ∂x 1 ρ ∂x 1 2 ∂x 2 ρ ∂x 2
It is obvious that if the pressure terms can be calculated from another source, the
intermediate velocities of Step 1 can be corrected using Equation 7.133. However, an
independent pressure equation is required in order to substitute the pressure values into
the above equation. In order to do this, u n+1 terms have to be eliminated from the above
i
equation. This can be done via the continuity equation if we differentiate the first equation
with respect to x 1 and the second equation with respect to x 2 and adding these together,
that is, (neglecting third-order terms)
n+1 n+1 2 2 ! n
∂u ∂u ∂ ˜u 1 ∂ ˜u 2 t ∂ p ∂ p
1 2
+ − − =− 2 + 2 (7.134)
∂x 1 ∂x 2 ∂x 1 ∂x 2 ρ ∂x ∂x
1 2
Note that from the continuity equation
∂u n+1 ∂u n+1
1 2
+ = 0 (7.135)
∂x 1 ∂x 2
On substituting the above equation into Equation 7.134, we obtain the pressure equation
as follows:
n
2 2 !
1 ∂ p ∂ p 1 ∂ ˜u 1 ∂ ˜u 2
+ = + (7.136)
ρ ∂x 2 ∂x 2 t ∂x 1 ∂x 2
1 2
It should be noted that there are no transient or convection terms present in the above
equation. Although this equation does not require any special treatment in order to stabilize
the oscillations, the absence of a transient term leads to a compulsory implicit treatment
solution procedure. In other words, a matrix solution method is necessary in order to obtain