Page 220 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
212
Hot fluid Porous material
Heat convection
h c

T a
Air flow

Figure 7.14 Example of convection boundary condition

The initial conditions, which describe the initial state of the fluid (temperature, pressure,
velocity and properties), are employed at the onset of the heat convection calculations. These
conditions are problem-dependent and are discussed for various applications in the latter
sections of this chapter.

7.6.4 Steady and transient solution methods

A steady state solution for a problem can be obtained, using the CBS scheme, by time-
stepping to achieve a steady state. This can be done by fixing a tolerance criterion as
follows:
nnodes n+1 n
φ − φ
i i ≤ (7.169)
t
i=1
where φ i is any heat convection variable at a node, n nodes is the total number of nodes and
is a prescribed tolerance, which will tend to zero as the solution approaches steady state.
A transient solution can be of two types. The first type is the ‘real’ time variation of
the solution for problems in which a steady state solution exists. The second category is
one that has no real steady state, for instance, vortex shedding behind a cylinder or Bernard
convection. In the first type, the calculations commence with prescribed initial conditions
and progress with a suitable time-stepping algorithm until a steady state is reached. The
time history of the variables need to be stored and monitored as the transient solution
progresses in order to study the behaviour of the solution. In the second type of problems,
that is, Bernard convection and vortex shedding, the steady state tolerance of Equation
7.169 is not applicable and steady state is never reached. The time history of these types
of problems needs to be followed as long as the user is interested in the solution.