CFD Glossary — Dimensionless Numbers, Flow States & Boundary Conditions

Quick-reference guide for computational fluid dynamics. Covers the six most important dimensionless numbers with formulas and physical meaning, laminar/turbulent flow regimes, the three families of turbulence simulation (DNS, LES, RANS), and the standard boundary condition types used in CFD solvers.

Dimensionless Numbers

Dimensionless numbers are ratios of competing physical forces. Two flows with identical dimensionless numbers behave identically regardless of scale — the principle behind wind tunnel testing.

Re — Reynolds Number

Re = (ρ \cdot U \cdot L) / μ = (inertial forces) / (viscous forces) ρ = fluid density (kg/m³), U = velocity (m/s) L = characteristic length (m), μ = dynamic viscosity (Pa\cdots)

Physical meaning: How turbulent is the flow? Low Re (≪ 2300 in a pipe) → laminar viscous flow. High Re (≫ 4000) → turbulent. The transition region is unpredictable and sensitive to perturbations. A bacterium swims at Re ≈ 0.01 (viscosity dominates). A cargo ship runs at Re ≈ 10⁹ (inertia dominates).

Pr — Prandtl Number

Pr = (μ \cdot c_p) / k = (momentum diffusivity) / (thermal diffusivity) c_p = specific heat (J/kg\cdotK), k = thermal conductivity (W/m\cdotK)

Physical meaning: Relative thickness of velocity boundary layer vs temperature boundary layer. Pr < 1 (liquid metals, Pr ≈ 0.01): heat diffuses much faster than momentum. Pr ≈ 1 (air: Pr ≈ 0.71). Pr ≫ 1 (oil: Pr ≈ 1000): very thick thermal boundary layer.

Pe — Péclet Number

Pe = Re \cdot Pr = (U \cdot L) / α α = thermal diffusivity = k / (ρ \cdot c_p) (m²/s)

Physical meaning: Advective vs diffusive heat transport. High Pe → heat is carried by flow (advection-dominated), numerically stiff in CFD. Low Pe → conduction is dominant and spatially smooth.

Ma — Mach Number

Ma = U / c c = speed of sound in fluid (m/s) = \sqrt(γ \cdot R \cdot T) for ideal gas

Physical meaning: Compressibility. Ma < 0.3 → incompressible approximation valid (density changes < 5%). Ma > 1 → supersonic; shockwaves appear. Ma > 5 → hypersonic (entry vehicles).

Fr — Froude Number

Fr = U / \sqrt(g \cdot L) g = gravitational acceleration (m/s²)

Physical meaning: Inertial vs gravitational forces. Used in open-channel flows and ship hydrodynamics. Fr < 1 (subcritical): gravity waves can propagate upstream. Fr > 1 (supercritical): flow is faster than surface waves; hydraulic jumps form.

We — Weber Number

We = (ρ \cdot U² \cdot L) / σ σ = surface tension coefficient (N/m)

Physical meaning: Inertia vs surface tension. We ≪ 1 → droplets stay spherical. We ≫ 1 → droplets break up (atomization, spray). Critical in inkjet printing (We ≈ 1–100), fuel injection, and ocean spray.

Flow Regimes

Regime	Re Range (pipe)	Characteristics	Examples
Laminar	Re < 2300	Smooth parallel streamlines, parabolic velocity profile, viscous dominated	Blood in capillaries, oil in microchannels
Transitional	2300–4000	Intermittent bursts of turbulence, highly sensitive to disturbances	HVAC ducts at low speed
Turbulent	Re > 4000	Chaotic, 3D eddies at many scales, enhanced mixing, flat velocity profile	Most engineering flows, rivers, wind

Note on Re thresholds: The 2300/4000 values apply specifically to smooth circular pipe flow. For other geometries: flow over a flat plate goes turbulent at Re_x ≈ 5×10⁵; a sphere drags less above Re ≈ 5×10⁵ (drag crisis). Always use the geometry-specific critical Re.

Simulation Methods: DNS / LES / RANS

Method	What is Resolved	Cost	Best For
DNS Direct Numerical Simulation	All scales of turbulence down to Kolmogorov scale η	O(Re³) — extremely expensive	Fundamental research at Re < 10,000; validation datasets
LES Large Eddy Simulation	Large eddies (energy-carrying); small eddies modelled by SGS model	O(Re) – O(Re²) — expensive but tractable	Complex geometry, unsteady flows, aeroacoustics
RANS Reynolds-Averaged Navier-Stokes	Time-averaged flow only; all turbulence modelled	O(1) relative to DNS — fast, industry standard	Steady-state industrial CFD, automotive, HVAC

Hybrid: DDES/SAS-SST — uses RANS near walls (where LES is too expensive) and LES in the free stream. Common in high-Re automotive and aerospace simulations.

Common RANS turbulence models: k-ε (robust, good wake predictions), k-ω SST (better near walls and adverse pressure gradients), Spalart-Allmaras (one equation, fast, popular in aerospace).

Boundary Conditions

BC Type	Applied Where	What It Specifies	Notes
No-slip wall	Solid surfaces	u = v = w = 0 (fluid velocity = wall velocity)	Physical for viscous flow; requires fine mesh near wall for RANS
Slip wall	Frictionless surfaces, LBM outer BC	Normal velocity = 0, tangential = free	Used in inviscid approximations, microflow (Kn > 0.01)
Inlet (Dirichlet)	Flow entry face	Velocity profile prescribed (uniform, parabolic, mapped)	Also set turbulence intensity (TI) and length scale for RANS
Outlet (Neumann)	Flow exit face	Zero normal gradient: ∂u/∂n = 0, pressure = P_ref	Pressure-outlet in Fluent; fails if backflow is strong
Periodic	Pairs of faces	Inlet state = outlet state of periodic pair	Used for channel flow, turbine blade passages; halves domain size
Symmetry	Plane of symmetry	Normal velocity = 0; gradients of other vars = 0 across plane	Valid only if flow truly symmetric; saves 50% domain
Moving wall	Rotating machinery	u = ω × r (wall velocity)	Applied in MRF (Moving Reference Frame) or sliding mesh

Quick Reference — All Dimensionless Numbers

Symbol	Name	Formula	Key Ratio	Critical Values
Re	Reynolds	ρUL/μ	Inertia / Viscosity	Pipe: 2300 (lam), 4000 (turb)
Ma	Mach	U/c	Flow / Sound speed	0.3 (compressible), 1.0 (sonic)
Fr	Froude	U/√(gL)	Inertia / Gravity	1.0 (critical open-channel)
We	Weber	ρU²L/σ	Inertia / Surface tension	~1 (droplet deformation onset)
Pr	Prandtl	μc_p/k	Momentum / Thermal diff.	Air: 0.71, Water: 7, Oil: 1000
Pe	Péclet	Re·Pr = UL/α	Advection / Diffusion	>1 advection-dominated
Nu	Nusselt	hL/k	Convective / Conductive HT	1 = pure conduction
St	Strouhal	fL/U	Oscillation / Advection	Vortex shedding ≈ 0.2
Ca	Capillary	μU/σ	Viscosity / Surface tension	Microfluidics: <1
Kn	Knudsen	λ/L	Mean free path / Scale	>0.01 continuum breaks down