Page 208 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
200

N i
t
n
+ u 2


2 u 1 N j  ∂N i ∂N j ∂N k {φ} d n 2
∂x 1
∂x 1
∂x 1
0
 
t N i ∂N i ∂N j ∂N k n
+ u 2 u 2 N j  {φ} d n 2

2 0 ∂x 2 ∂x 2 ∂x 2
 n

u 1 (b i φ i + b j φ j + b k φ k ) + u 2 (c i φ i + c j φ j + c k φ k )
u 1 t
=  u 1 (b i φ i + b j φ j + b k φ k ) + u 2 (c i φ + c j φ j + c k φ k )  n 1
2A 2 2
0
 n

u 1 (b i φ i + b j φ j + b k φ k ) + u 2 (c i φ i + c j φ j + c k φ k )
u 2 t
+  u 1 (b i φ i + b j φ j + b k φ k ) + u 2 (c i φ + c j φ j + c k φ k )  n 2
2A 2 2
0
(7.118)
The assembled equation for a two-dimensional analysis takes a form similar to the one-
dimensional Equation 7.102. Once again, the boundary terms from Equation 7.118 may be
neglected in the calculations.
7.5 Stability Conditions
The stability conditions for a given time discretization may be derived using a Von Neu-
mann or Fourier analysis for either the convection- or the convection–diffusion equations.
However, for more complicated equations such as the Navier–Stokes equations, the deriva-
tion of the stability limit is not straightforward. A detailed discussion on stability criteria is
not within the scope of this book and readers are asked to refer to the relevant text books
and papers for details (Hirsch 1989; Zienkiewicz and Codina 1995). A stability analysis
will give some idea about the time-step restrictions of any numerical scheme.
In general, for fluid dynamics problems, the time-step magnitude is controlled by two
wave speeds. The first one is due to the convection velocity and the second to the real
diffusion introduced by the equations. In the case of a convection–diffusion equation, the

√
2
2
convection velocity is u i u i ,which is u + u =|u|. The diffusion velocity is 2k/h
1 2
where h is the local element size. The time-step restrictions are calculated as the ratio of
the local element size and the local wave speed. It is therefore correct to write that the
time step is calculated as
t = min( t c , t d ) (7.119)
where t c and t d are the convection and diffusion time-step limits respectively, which
are
h
t c =
|u|
h 2
t d = (7.120)
2k