5.22 Solving for energy

Examples with energy conservation in this chapter assume eqn is constant to produce the transport equation for eqn in Eq. (2.65 ). When eqn cannot be assumed to be constant, it is common to solve an equation for internal energy eqn instead, e.g. Eq. (2.60 ) with the material derivative replaced using Eq. (2.14 )

@--e-+ r ( ue) r rT = 0: @t \relax \special {t4ht=
This equation can be discretised to form a matrix equation eqn, from which eqn can be solved. The challenge is that the diffusion term eqn is not expressed in terms of eqn, so is discretised explicitly, which adversely affects convergence.

To improve convergence, an implicit term is introduced which is similar in form and scale to the problem term. For energy Eq. (5.39 ), we use eqn , where the diffusivity eqn is calculated from the eqn function of eqn.

The extra term is both “added and subtracted” in implicit and explicit form. This has the effect of increasing the contributions to matrix coefficients, while cancelling out a large part of the explicit contribution from eqn, as illustrated below

@---e @t + r ( ue) r ere + r ere r rT = 0: |--------i---m-{pzli----ci---t-----} |----------------------e---x{plz-ic----it--------------------} \relax \special {t4ht=
The overall solution procedure involves first updating eqn from eqn from thermodynamic relationships, e.g. Eq. (2.62 ). The equation for eqn is then solved, and the subsequent solution for eqn is converted back to eqn.

If eqn is expressed as a polynomial function of eqn, Eq. (2.64 ), then eqn is converted to eqn using the analytical integral of Eq. (2.62 )


From temperature to energy

The conversion from eqn to eqn is more complex because eqn cannot be made the subject of Eq. (5.40). Instead, it can be “inverted” using an iterative scheme such as the Newton-Raphson method,8 which for this problem is:

T ℬ T e(T-)---e = T e(T-)---e: @e=@T cV(T ) \relax \special {t4ht=
eqn is updated from the current eqn using the evaluated eqn and eqn polynomials, Eq. (5.40 ) and Eq. (2.64 ), respectively. One iteration is often sufficient for the error eqn to fall within a prescribed tolerance, but further iterations of Eq. (5.41) can be applied if necessary.

Boundary conditions

The boundary conditions for the energy equation are generally specified in terms of eqn. But since they must be applied to the variable being solved, they must be reposed in terms of eqn. A fixed value condition eqn is converted to an equivalent condition eqn, e.g. by Eq. (5.40 ). A fixed gradient condition eqn for eqn is converted to a fixed gradient eqn for eqn by

rneb cVrnTb : \relax \special {t4ht=

8Isaac Newton,De analysi per aequationes numero terminorum infinitas, 1669 and Joseph Raphson, Analysis aequationum universalis, 1690.

Notes on CFD: General Principles - 5.22 Solving for energy