Rowan-Gollan-PhD-Thesis - Mechanical Engineering - University of ...

30 Flow Solver Chapter 3
For low-order reconstruction, the adjacent cell centre properties are taken as the interface val-
ues. Based on the flow properties at cell interfaces, the inviscid flux is calculated using a flux
calculator. Inmbcns there are five flux calculator options: 1) an approximate Riemann problem
solver developed by Jacobs [94], 2) the advection upstream splitting method (AUSM) by Liou
and Steffen Jr. [95], 3) the AUSMDV calculator by Wada and Liou [96], 4) the equilibrium flux
method (EFM) by Macrossan [59], and 5) an adaptive calculator which uses EFM near shocks
and AUSMDV elsewhere.
The viscous transport update is based on the calculation of viscous derivatives at the cell
centres. The cell-centred derivatives are calculated as an average of the derivatives evaluated
at the cell vertices. Those cell-vertex derivatives are calculated by application of Gauss’ diver-
gence theorem to convert a surface integral around a secondary volume to the derivative value
within that volume, that is, at the vertex.
The remaining pieces of physics in the timestep-splitting algorithm are the subject of this
thesis: the change of state due to chemical reactions is presented in Chapter 4; and the change
of state due to thermal relaxation is the subject of Chapter 5.
Prior to the development work in this thesis, mbcns had gas models for a variety of gases
of interest in hypersonic flows, such as, perfect gases, equilibrium gases (via a look-up table),
nitrogen in vibrational equilibrium and some exotic gases for comparison to rarefied gas dy-
namics calculations. The work in this thesis has contributed the following gas models to the
above list: a mixture of ideal gases, a mixture of thermally perfect gases and a mixture of gases
with nonuniform internal energy excitement (a multi-temperature gas mixture).
3.2 Verification of the flow solver
As the flow solver, mbcns, is the foundation for the high-temperature modelling presented in
this thesis, it was deemed a worthy exercise to verify and validate the code. The terms “ver-
ify” and “validate” take on the technical meanings used by Roache [97]. Roache cites Boehm
and Blottner when he defines verification as “solving the equations right” and validation as
“solving the right equations.” In the context of this work, the code solves the set of partial
differential equations given in Section 2.1. Verification is the act of testing that this claim is
true in a rigorous manner: a check of the solution methodology by assessing the consistency of
the order of accuracy as mesh discretisation tends to zero (this can include a temporal mesh).
Validation, on the other hand, is the act of assessing if indeed the governing equations (partial
differential equations) are appropriate for the problem of interest and how well those equations
model the problem. Also in this section, the verification that is referred to is verification of the
code. This is distinct from verification of a calculation as pointed out by Roache.
Two verification cases are presented. The first case uses the Method of Manufactured Solu-
tions to generate an analytical solution for a purely supersonic flow field based on “manufac-
tured” source terms. This case is used to demonstrate the formal order of spatial accuracy of the
code without the complications of embedded shocks. The Method of Manufactured Solutions
is not easily applied to test flows with embedded shocks, that is, the shock-capturing aspect of