Dimensionless Numbers Reference & Calculator — ThermoFluidCalc
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🔢 Engineering Dimensionless Parameters

Dimensionless Numbers Matrix & Solvers

Physical significance, governing ratios, critical thresholds, and instant interactive calculators for thermal fluid engineering.

Re Reynolds Number
Fluid Dynamics
Physical Ratio: $\frac{\text{Inertial Forces}}{\text{Viscous Forces}} = \frac{\rho u L}{\mu}$
$$Re = \frac{\rho u D_h}{\mu} = \frac{u D_h}{\nu}$$

Predicts flow regime transitions. Pipe flow: Laminar for $Re < 2300$, Turbulent for $Re > 4000$. Flat plate boundary layer: Critical transition at $Re_x \approx 5 \times 10^5$.

Calculated Reynolds Number: 49,479
Turbulent Flow
Pr Prandtl Number
Heat Transfer
Physical Ratio: $\frac{\text{Momentum Diffusivity}}{\text{Thermal Diffusivity}} = \frac{\nu}{\alpha}$
$$Pr = \frac{c_p \mu}{k} = \frac{\nu}{\alpha}$$

Compares velocity boundary layer thickness to thermal boundary layer thickness: $\delta / \delta_t \approx Pr^{1/3}$. Air: $Pr \approx 0.71$, Water @ 20°C: $Pr \approx 7.0$.

Calculated Prandtl Number: 0.714
Gas Regime ($\delta \approx \delta_t$)
Nu Nusselt Number
Convective Heat Transfer
Physical Ratio: $\frac{\text{Convective Heat Transfer}}{\text{Pure Conductive Heat Transfer}} = \frac{h L}{k_{fluid}}$
$$Nu_L = \frac{h L}{k_{fluid}}$$

Dimensionless temperature gradient at the wall surface. For pure conduction without fluid motion, $Nu = 1$. Dittus-Boelter correlation for turbulent pipe flow: $Nu_D = 0.023 Re_D^{0.8} Pr^n$.

Calculated Nusselt Number: 230.77
Strong Convection ($230\times$ conduction)
CFL Courant-Friedrichs-Lewy Condition
CFD Numerical Stability
Physical Ratio: $\frac{\text{Fluid Distance in Step }\Delta t}{\text{Mesh Cell Size }\Delta x} = \frac{u \Delta t}{\Delta x}$
$$CFL = \frac{u \Delta t}{\Delta x} \le CFL_{max}$$

Fundamental stability criterion in explicit time-marching CFD and wave propagation. Explicit schemes require $CFL \le 1.0$ to prevent numerical divergence.

Calculated Courant Number: 0.500
Stable (CFL ≤ 1)
Ma Mach Number
Compressible Flow
Physical Ratio: $\frac{\text{Flow Velocity}}{\text{Speed of Sound}} = \frac{u}{a} = \frac{u}{\sqrt{\gamma R T}}$
$$Ma = \frac{u}{a} = \frac{u}{\sqrt{\gamma R T}}$$

Compressibility criterion. $Ma < 0.3$: Incompressible (constant density assumption valid). $0.8 < Ma < 1.2$: Transonic. $Ma > 1.0$: Supersonic with shock waves.

Calculated Mach Number: 2.00
Supersonic (Shock Waves Present)
Bi Biot Number
Transient Conduction
Physical Ratio: $\frac{\text{Internal Conductive Resistance}}{\text{External Convective Resistance}} = \frac{h L_c}{k_{solid}}$
$$Bi = \frac{h L_c}{k_{solid}} = \frac{h (V/A)}{k_{solid}}$$

Lumped Capacitance Method criterion. If $Bi < 0.1$, the solid has nearly uniform internal temperature, simplifying transient analysis to a 0D ordinary differential equation.

Calculated Biot Number: 0.0025
Lumped Capacitance Valid (Bi < 0.1)
Ra Rayleigh Number
Natural Convection
Physical Ratio: $\frac{\text{Buoyancy Forces}}{\text{Viscous Thermal Damping}} = Gr \times Pr$
$$Ra_L = Gr_L \cdot Pr = \frac{g \beta (T_s - T_\infty) L^3}{\nu \alpha}$$

Governs natural (buoyancy-driven) free convection regime on vertical and horizontal plates. Laminar for $Ra < 10^9$, Turbulent buoyant plumes for $Ra > 10^9$.