Dimensionless numbers
Calculate 27 dimensionless numbers and see what each value means for your CFD set-up.
Reynolds number
Reynolds number Re
How to use the dimensionless numbers calculator
1. Pick a number. The numbers are grouped by what they describe: flow regime, forces and losses, heat transfer, multiphase flow and particles, mass transfer, and waves and combustion.
2. Enter your values. Each field shows its unit; use SI units throughout. The defaults are a worked example, for instance water flowing at 1 m/s through a pipe of 1 m diameter for the Reynolds number.
3. Read the result and what it means. The note under the value gives the regime, for example laminar or turbulent pipe flow, or whether a lumped thermal model is valid. Copy the value with four significant figures.
Tip: the names in the definitions table below open each number directly in the calculator.
Why dimensionless numbers matter
A dimensionless number is a ratio of two effects, such as inertia and viscosity, in which all units cancel. Two flows with the same dimensionless numbers behave the same way whatever their size. This is dynamic similarity, and it is why a scale model in a wind tunnel can predict the flow around a full-size car.
The Buckingham π theorem tells you how many independent groups a problem has: the number of variables minus the number of base units. Pipe flow described by ρ, U, D and μ has four variables and three units (mass, length, time), so it has one group: the Reynolds number.
In CFD the same numbers decide which physics you switch on and which mesh and time step you need. The next section lists the checks to make before you set up a case.
Which numbers matter in a CFD set-up
Check these numbers before you choose the models in Simcenter STAR-CCM+ or any other solver:
| Number | Check against | What it changes in the set-up |
|---|---|---|
| Reynolds Re | About 2300 in a pipe, 5 × 105 on a flat plate | Laminar or turbulent model; with a turbulence model, the near-wall mesh and its y+ |
| Mach Ma | About 0.3 | Ideal-gas density instead of constant density; above about 0.8, a mesh that resolves shocks |
| Knudsen Kn | About 0.01 | Slip walls or a particle method instead of no-slip Navier–Stokes |
| Courant C | About 1 | The time step of an unsteady run; explicit schemes need C ≤ 1 |
| Prandtl Pr | Far from 1 | How thin the thermal boundary layer is compared with the velocity boundary layer, and so the near-wall mesh |
| Richardson Ri | Above about 0.1 (0.1–10 mixed, above 10 natural convection) | Switch on gravity and a buoyancy-driven density model |
| We, Ca, Eo | Order 1 or below | Whether surface tension must be modelled in a VOF or other multiphase set-up |
| Stokes Stk | About 0.1–1 | Whether particles act as passive tracers or need their own momentum (Lagrangian tracking) |
Once the regime is clear, the turbulence model selector, the y+ calculator and the turbulence boundary conditions calculator take it from there.
Definitions of all 27 numbers
Each number with its definition and the ratio it expresses. Select a name to open it in the calculator.
| Number | Definition | Ratio of |
|---|---|---|
| Flow regime | ||
| Reynolds number | Re = ρUL/μ | Inertial to viscous forces |
| Mach number | Ma = U/a | Flow speed to the speed of sound |
| Froude number | Fr = U/√(gL) | Inertia to gravity |
| Knudsen number | Kn = λ/L | Molecular mean free path to the length scale |
| Strouhal number | St = fL/U | Flow time L/U to the oscillation period |
| Courant number | C = UΔt/Δx | Distance travelled in one time step to the cell size |
| Forces and losses | ||
| Drag coefficient | CD = 2FD/(ρU2A) | Drag force to dynamic pressure times area |
| Lift coefficient | CL = 2FL/(ρU2S) | Lift force to dynamic pressure times area |
| Pressure coefficient | Cp = (p − p∞)/(ρ∞U∞2/2) | Pressure difference to dynamic pressure |
| Euler number | Eu = Δp/(ρU2) | Pressure drop to the inertial pressure ρU2 |
| Fanning friction factor | f = τw/(ρU2/2) | Wall shear stress to dynamic pressure |
| Heat transfer | ||
| Nusselt number | Nu = hL/k | Convection to conduction in the fluid |
| Prandtl number | Pr = cpμ/k = ν/α | Momentum diffusivity to thermal diffusivity |
| Biot number | Bi = hLc/ks | Conduction resistance in the solid to convection resistance at its surface |
| Péclet number | Pe = LUρcp/k = Re Pr | Heat carried by the flow to heat conducted |
| Eckert number | Ec = U2/(cpΔT) | Kinetic energy to enthalpy difference |
| Grashof number | Gr = gβΔTL3/ν2 | Buoyancy to viscous forces |
| Rayleigh number | Ra = gβΔTL3/(να) = Gr Pr | Buoyancy to viscous and thermal diffusion |
| Richardson number | Ri = gβΔTL/U2 = Gr/Re2 | Buoyancy to inertia |
| Multiphase and particles | ||
| Weber number | We = ρU2L/σ | Inertia to surface tension |
| Capillary number | Ca = μU/σ | Viscous forces to surface tension |
| Eötvös (Bond) number | Eo = ΔρgL2/σ | Gravity to surface tension |
| Stokes number | Stk = t0U0/L0 | Particle response time to the flow time scale |
| Mass transfer | ||
| Schmidt number | Sc = ν/D | Momentum diffusivity to mass diffusivity |
| Sherwood number | Sh = hmL/D | Convective to diffusive mass transfer |
| Waves and combustion | ||
| Ursell number | Ur = Hλ2/h3 | Wave steepness to the cube of the relative depth (nonlinearity of long waves) |
| Weaver flame speed number | Wea = 100 w/wH | Laminar flame speed of a gas to that of hydrogen |
Conventions vary between textbooks. Some define the Euler number with ρU2/2, which makes it the pressure loss coefficient K, and the Darcy friction factor of the Moody chart is four times the Fanning factor in the table. The calculator uses the forms in the table.
Characteristic scales are a choice. State which length and speed you use (diameter or radius, chord or span, bulk or centreline velocity), because the value of a number and its critical thresholds depend on it.
References and changelog
References
F. M. White, Fluid Mechanics, 8th ed., McGraw-Hill, 2016: chapter 5, dimensional analysis and similarity.
F. P. Incropera, D. P. DeWitt, T. L. Bergman and A. S. Lavine, Fundamentals of Heat and Mass Transfer, 6th ed., Wiley, 2007: the dimensionless groups of heat and mass transfer.
R. Courant, K. Friedrichs and H. Lewy, “Über die partiellen Differenzengleichungen der mathematischen Physik”, Mathematische Annalen 100, 32–74, 1928: the Courant condition.
G. Karniadakis, A. Beskok and N. Aluru, Microflows and Nanoflows, Springer, 2005: Knudsen number flow regimes.
H. Schlichting and K. Gersten, Boundary-Layer Theory, 9th ed., Springer, 2017: transition Reynolds numbers for pipes and flat plates.
Review
Reviewed by Christoffer Johansson, M.Sc., Senior Application Specialist at Volupe. Version 2.0.
Changelog
October 2026: rebuilt as a live calculator. Automated checks confirm the same results as the previous version, except for these fixes:
– The lift coefficient divides by ρU²S; the old formula left out one factor of the speed, so it was only right at 1 m/s.
– The Biot inputs were relabelled; the heat transfer coefficient and the length had swapped labels.
– The Prandtl number describes k as the thermal conductivity, and its example uses air (k = 0.026 W/(m·K)) instead of 19 W/(m·K).
– The pressure coefficient describes p∞ as the free-stream static pressure, with 101 325 Pa as the example.
– Schmidt units read m²/s, the Knudsen symbol is Kn, the Euler and Eckert descriptions were corrected, and results are rounded to three significant figures.
New: the Courant, Grashof, Rayleigh and Richardson numbers, what each value means, Simcenter STAR-CCM+ hints and realistic examples. The first version of this page was based on Wikipedia’s list of dimensionless numbers in fluid mechanics.
Related tools
All CFD Handbook tools →k–ω SST · k–ε · S–A
Interactive guide
Turbulence model selector
Pick a turbulence model and wall treatment for your case, with a y+ target and prism layer count.
Open the selector →
y+ = uτy / ν
Calculator
y+ calculator
Once you know the Reynolds number, size the first cell for a target y+.
Open calculator →
h = Nu k / L
Calculator
Nusselt number calculator
Turn Re, Ra and Pr into a heat transfer coefficient for plates, cylinders and pipes.
Open calculator →
Questions about these numbers
Calculator reviewed by
Christoffer Johansson
Senior Application Specialist, M.Sc.
Not sure which regime your case is in? We reply within one working day.
Ask a technical question →
What is a dimensionless number?
A ratio of two quantities with the same units, so the units cancel. In fluid mechanics most are ratios of forces or of transport rates: the Reynolds number compares inertia with viscosity, the Prandtl number momentum diffusion with heat diffusion. Flows with equal dimensionless numbers behave alike at any scale.
Which Reynolds number means turbulent flow?
It depends on the geometry and on the length you use. Pipe flow is laminar below about 2300 and usually turbulent above about 4000. On a flat plate the boundary layer starts to turn turbulent near Rex ≈ 5 × 105. Disturbances, roughness and inlet turbulence can move these values.
What Courant number should I aim for?
Explicit time stepping is only stable for C ≤ 1. Implicit solvers, such as implicit unsteady in Simcenter STAR-CCM+, stay stable at higher values, but scale-resolving simulations such as LES and DES usually aim for C ≤ 1 where the flow detail matters. Whatever you choose, check that the result does not change when you halve the time step.
When does compressibility matter?
When the Mach number exceeds about 0.3, density changes exceed about 5 % and you should use an ideal-gas density model. Below that, constant density is usually accurate, unless large temperature differences change the density; then check the Richardson number.
What is the difference between the Fanning and Darcy friction factors?
The Darcy friction factor, used in the Moody chart and in Δp = fD(L/D)½ρU², is four times the Fanning friction factor f = τw/(½ρU²). Check which one a correlation or a solver report uses before you compare values.
Do I have to use SI units?
The numbers themselves have no units, so any consistent unit system gives the same value. The calculator expects SI units in every field; mixing, for example, millimetres with metres gives a wrong result.
Can Volupe help me set up my simulation in Simcenter STAR-CCM+?
Yes. Our application engineers help you choose models, meshes and time steps in Simcenter STAR-CCM+ for your flow regime. Ask a technical question and we reply within one working day.
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