1. Notation Conventions#

This guide uses a single notation convention throughout all chapters:

  1. Scalars use italic symbols, e.g. \(p\), \(\rho\), \(T\), \(Y_k\).

  2. Vectors use arrow notation, e.g. \(\vec{u}\), \(\vec{A}\), \(\vec{q}\).

  3. Second-order tensors use double-underline notation, e.g. \(\Matrix{\tau}\), \(\Matrix{S}\).

  4. Unit vectors use hats, e.g. \(\hat{\vec{n}}\).

  5. Dyadic products use \(\otimes\), e.g. \(\vec{u}\otimes\vec{u}\).

Face fluxes and area vectors use the stored cl-to-cr orientation. In a balance for cell \(P\), multiply each stored face contribution by its incidence sign \(\sigma_{Pf}\). A face-only subscript denotes the stored orientation; a cell-face subscript, where used, denotes the quantity after this sign conversion. The complete convention is defined in Loci Face Orientation and Cell Incidence.

For a transported algebraic row, \({}^{(\phi)}\!a_P\) is the diagonal in cell \(P\) and \({}^{(\phi)}\!a_{P,N}\) is its coupling to neighbor \(N\). The diagonal of the neighboring cell is \({}^{(\phi)}\!a_N\), a different quantity. The complete known RHS is \({}^{(\phi)}\!b_P\); any pressure, history, or source term removed from that RHS is identified explicitly. For momentum, a vector RHS collects the three component RHS values.

Physical time levels use \(n\) and \(n+1\), while \(k\) labels an outer nonlinear iteration within a time step. A star identifies the provisional momentum stage and a prime identifies a pressure-induced correction. A superscript \(\mathrm{pred}\) on a coefficient identifies the diagonal or RHS used by the relaxed predictor; it is not another physical time level.

1.1. Derivation Style Used In This Guide#

For each major equation family (transport equations, pressure-correction equation, and time-discretization updates), we follow the same sequence:

  1. Start from the continuous equation.

  2. Integrate over the control volume (and time level, where needed).

  3. State the approximation used (upwinding, linearization, neglected higher-order terms, etc.).

  4. Show the algebraic form used in implementation.

When a derivation uses a closure or approximation, that assumption is stated explicitly next to the corresponding equation.