cfd-fundamentals

name	cfd-fundamentals
description	CFD fundamentals — Navier-Stokes equations, discretization (FVM/FDM/FEM), pressure-velocity coupling (SIMPLE), solver settings, boundary conditions, post-processing, y+ wall treatment.
metadata	{"priority":8,"promptSignals":{"phrases":["CFD","computational fluid dynamics","Navier-Stokes","SIMPLE","Fluent","OpenFOAM","pressure velocity coupling","boundary condition CFD","flow simulation"],"minScore":3}}

CFD Fundamentals — Complete Skill

Governing Equations (Navier-Stokes, Incompressible)

Continuity (Mass Conservation)

∇·u = 0 [incompressible: ρ = constant] ∂ρ/∂t + ∇·(ρu) = 0 [compressible]

Momentum (Newton's 2nd Law)

ρ(∂u/∂t + u·∇u) = -∇p + μ∇²u + ρg

Each direction: x: ρ(∂u/∂t + u∂u/∂x + v∂u/∂y + w∂u/∂z) = -∂p/∂x + μ(∂²u/∂x² + ∂²u/∂y² + ∂²u/∂z²)

Energy (with Heat Transfer)

ρc_p(∂T/∂t + u·∇T) = ∇·(k∇T) + Q_source + viscous dissipation

Discretization Methods

Finite Volume Method (FVM) — Industry Standard (Fluent, OpenFOAM)

Integrate governing equations over each control volume (cell) Conservative by construction: flux through face = same for adjacent cells Fluxes at cell faces: linear interpolation or upwind schemes ∫_V ∂ρu/∂t dV + ∮_S ρu(u·n)dA = ∮_S (τ·n)dA - ∮_S pn dA

Finite Difference Method (FDM) — simple, structured grids

Replace derivatives with Taylor series approximations ∂u/∂x ≈ (u_{i+1} - u_{i-1})/(2Δx) [2nd order central difference] ∂u/∂x ≈ (u_{i+1} - u_i)/Δx [1st order upwind, diffusive but stable]

Finite Element Method (FEM) — COMSOL, structural CFD

Weak (Galerkin) form; complex geometry; less common in pure CFD

Pressure-Velocity Coupling

SIMPLE (Semi-Implicit Method for Pressure-Linked Equations)

1. Guess pressure field p*
2. Solve momentum equations → u* (using p*)
3. Solve pressure correction equation → p'
4. Update: p = p* + αp × p', u = u* + u' (from p')
5. Iterate until convergence

Relaxation factors: α_p = 0.3 (pressure), α_u = 0.7 (velocity) — standard Smaller factors → more stable, slower convergence At convergence: continuity and momentum residuals < 10⁻³ to 10⁻⁶

SIMPLEC: better pressure correction (α_p up to 1.0 possible) PISO: better for transient (non-iterative per time step) Coupled solver (Fluent): solves p-u simultaneously → fewer iterations, more memory

Boundary Conditions

Inlet

Velocity inlet: specify U, turbulence intensity (I = u'/U, typ. 1-5%), length scale L_t Pressure inlet: specify total pressure p₀ = p + ½ρU² (good for internal flows with known p₀) Mass flow inlet: specify ṁ [kg/s] (useful for compressors, fans)

Turbulence at inlet: k = 1.5(U×I)², ε = C_μ^(3/4) k^(3/2) / L_t (L_t ≈ 0.07 × D_hydraulic)

Outlet

Pressure outlet: specify static gauge pressure p = 0 (common for external flows) Outflow: parabolic/fully-developed (don't use if backflow expected) Pressure far-field: compressible, specifies M, T, p at free stream

Walls

No-slip: u = 0 (stationary wall) or u = u_wall (moving wall, rotating) Thermal: specified T, specified q̇ [W/m²], or convection h,T_inf Symmetry: ∂u_n/∂n = 0, u_n = 0 (mirror BC, reduces domain)

Periodic

Translational: inlet = outlet shifted (duct flow without entrance effects) Rotational: fan/pump sector models

Wall Treatment and y⁺

y⁺ (Dimensionless Wall Distance)

y⁺ = y × u_τ / ν u_τ = √(τ_w/ρ) [friction velocity] τ_w = C_f × ½ρU² [wall shear stress, C_f from flat plate correlation]

Near-Wall Regions

y⁺ < 5: viscous sublayer (linear: u⁺ = y⁺) 5 < y⁺ < 30: buffer layer (transition) y⁺ > 30: log-law region: u⁺ = (1/κ)ln(y⁺) + B, κ=0.41, B=5.2

Wall Treatment Choice

Low-Reynolds (resolve sublayer): y⁺ < 1, full boundary layer resolution

• Required for: k-ω SST, Spalart-Allmaras
• Best accuracy for separation, heat transfer, adverse pressure gradient

Wall functions (log-law): y⁺ = 30-300, boundary layer NOT resolved

• Works with: k-ε realizable, k-ε standard
• Faster (fewer cells), adequate for attached flows

First cell height (for y⁺ = 1): y₁ = L × Re_L^(-7/8) × ... exact from flat plate friction: y₁ ≈ L × y⁺ × √(0.037/Re_L^0.2) / U

Convergence Criteria

Residuals: scaled residual < 10⁻³ (adequate), 10⁻⁴ (good), 10⁻⁶ (excellent)

• Always monitor: continuity, x-velocity, y-velocity, z-velocity, k, ε (or ω)
• Energy residual: 10⁻⁷ (heat transfer simulations)

Monitor additional quantities: drag coefficient C_d, lift C_l, outlet temperature, pressure drop Convergence: residuals flat AND monitored quantities flat (last 100 iterations)

Stalling residuals: stuck at 10⁻³ → reduce relaxation, check mesh quality, check BCs

Discretization Schemes

First-order upwind: stable but diffusive (adds numerical viscosity), use for initial run Second-order upwind: more accurate, needed for final results (default for accurate production runs) QUICK (3rd order): good for structured hex meshes Bounded central differencing: LES and scale-resolving

Pressure interpolation: Standard (p-u decoupled), PRESTO! (curved domains, rotating), Body-Force Weighted (gravity)

CFD Software Guide

Software	Type	Strength
ANSYS Fluent	Commercial	Industry standard, all physics, GUI + scripting
ANSYS CFX	Commercial	Coupled solver, rotating machinery, multiphase
OpenFOAM	Open source	Full control, HPC, scripting
STAR-CCM+ (Siemens)	Commercial	Best CAD integration, overset mesh
COMSOL	Commercial	Multiphysics, FEM-based
Simscale	Cloud	Web-based, OpenFOAM backend

Post-Processing

Streamlines/pathlines: flow visualization, recirculation zones Velocity vectors/contours: flow field understanding Wall y⁺ plot: verify y⁺ is in target range everywhere Pressure coefficient: Cp = (p - p∞)/(½ρU∞²) Drag/lift: integrate pressure + wall shear over surfaces Heat flux: q" = -k(∂T/∂y)|_wall Q-criterion: vortex identification (Q > 0 where rotation > strain)

Output

Provide: governing equations for flow regime (laminar/turbulent/compressible), BC setup, turbulence model recommendation, y⁺ target, solver settings, convergence criteria, key post-processing quantities.