Page 196 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
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where φ is a scalar variable, k is a diffusion coefficient (thermal conductivity if φ = T ),
u i are the convection velocity components and Q is a source term. In the above equation,
the first term is a transient term, the second and third terms are convection terms and the
fourth term is the diffusion term. For a one-dimensional problem, the above equation is
reduced to
∂φ ∂φ ∂u 1 ∂ ∂φ
+ u 1 + φ − k + Q = 0 (7.72)
∂t ∂x 1 ∂x 1 ∂x 1 ∂x 1
If the convection velocity u 1 is assumed to be constant, we can rewrite Equation 7.72
as follows:
∂φ ∂φ ∂ ∂φ
+ u 1 − k + Q = 0 (7.73)
∂t ∂x 1 ∂x 1 ∂x 1
A one-dimensional convection equation without a source term is obtained by neglecting
the diffusion and source terms as follows:
∂φ ∂φ
+ u 1 = 0 (7.74)
∂t ∂x 1
Note that an appropriate solution for the above equation is valid for any similar equations
such as the energy equation.

7.4.1 Finite element solution to convection–diffusion equation

Unlike the conduction equation, a numerical solution for the convection equation has to
deal with the convection part of the governing equation in addition to diffusion. For most
conduction equations, the finite element solution is straightforward, as discussed in the
previous chapters. However, if a similar Galerkin type approximation was used in the
solution of convection equations, the results will be marked with spurious oscillations in
space (see the example discussed later in this section) if certain parameters exceed a critical
value (element Peclet number). This problem is not unique to finite elements as all other
spatial discretization techniques have the same difficulties. In a finite difference formulation,
the spatial oscillations are reduced, or suppressed, by a family of discretization methods
called upwinding schemes (Fletcher 1988; Spalding 1972). In the finite element method,
procedures such as Petrov–Galerkin (Zienkiewicz and Taylor 2000) and Streamline Upwind
Petrov Galerkin (SUPG) (Brooks and Hughes 1982) are equivalent upwinding schemes with
the specific purpose of eliminating spatial oscillations. In these methods, the basic shape
function is modified to obtain the upwinding effect.
For time-dependent equations, however, a different kind of approach is followed. The
finite difference Lax–Wendroff (Hirsch 1989) scheme has an equivalent in the finite element
method, which is referred to as the Taylor–Galerkin (TG) scheme (Donea 1984). Another
similar method, which is widely used, is known as the Characteristic Galerkin (CG) scheme
(Zienkiewicz and Taylor 2000). For scalar variables, the CG and TG methods are identical
(L¨ ohner et al. 1984). In this book, we follow the Characteristic Galerkin (CG) approach
to deal with spatial oscillations due to the discretization of the convection transport terms.