The continuity equation (∂ρ/∂t + ∇·(ρu) = 0) expresses conservation of mass. It states that the rate of change of mass in a control volume equals the net mass flux through its boundaries.

The FVM integrates governing equations over each cell. Because the flux computed at a shared face is identical for both cells, what leaves one cell must enter the neighbor. This guarantees discrete conservation.

Upwind schemes use information from the upstream direction, mimicking the physics of convection. This prevents spurious oscillations that central differencing can produce at high Peclet numbers, ensuring bounded solutions.

The Peclet number compares convective transport (ρu) to diffusive transport (Γ/Δx). When Pe >> 1, convection dominates and upwinding is necessary. When Pe << 1, central differencing is adequate.

Reynolds averaging the Navier-Stokes equations produces the Reynolds stress tensor (-ρu'ᵢu'ⱼ), adding 6 new unknowns but no new equations. Turbulence models like k-ε provide closure by relating these stresses to mean flow quantities.

SST (Shear Stress Transport) blends k-omega near walls (better for adverse pressure gradients and separation) with transformed k-epsilon in the freestream (avoids sensitivity to inlet ω). This makes it robust for external flows with separation.

Low-Re models resolve the viscous sublayer directly, requiring the first cell centroid to be within y+ < 1. This contrasts with wall functions that bridge the near-wall region and require y+ = 30-300.

SIMPLE solves momentum with a guessed pressure (p*), which generally violates continuity. The pressure correction equation is derived by substituting velocity corrections into continuity, ensuring the final velocity field is divergence-free.

On collocated grids, the pressure gradient (pE - pW)/(2Δx) skips the cell-center value. A checkerboard pattern (alternating high/low) has zero gradient, yet is non-physical. Rhie-Chow interpolation fixes this.

Basic iterative methods quickly remove high-frequency errors but struggle with smooth (low-frequency) errors. Multigrid restricts residuals to coarser grids where smooth errors appear high-frequency and are eliminated efficiently, then prolongates corrections back.

GCI provides a 95% confidence band for discretization error. Using results from at least 3 grids with different refinement levels, GCI estimates how close the fine-grid solution is to the grid-independent (exact) solution.

Verification asks 'Are we solving the equations correctly?' (code correctness, grid convergence). Validation asks 'Are we solving the right equations?' (comparison with experiments, physical accuracy).

Below Mach 0.3, density variations are typically less than 5%, and incompressible flow assumptions are valid. Above Ma 0.3, compressibility effects become significant and density-based or pressure-based compressible solvers are needed.

Periodic residual oscillations often indicate the flow has inherent unsteadiness (like vortex shedding behind a bluff body). Forcing steady-state convergence is inappropriate; switch to a transient simulation.

Meshing typically consumes about 30% of CFD project effort. Creating a quality mesh that captures relevant physics, satisfies y+ requirements, and enables grid convergence is one of the most time-consuming tasks in practical CFD.