Engineering calculators
Quick calculators for engineers, grouped by topic. Results are initial estimates; check them against the model or measured data.
Mesh, boundary-condition and time-step parameters for a CFD model, in set-up order.
Reynolds number and hydraulic diameter
Flow regime, friction factor and pressure drop in pipes and ducts.
What is it?
The Reynolds number compares the inertia of the flow with the viscosity of the fluid. Low Re: smooth flow with layers sliding past each other (laminar). High Re: strongly mixing flow (turbulent).
Why it matters
Re tells you whether a turbulence model is needed. It is also the input to most other estimates: friction factor, y+ and inlet turbulence.
How to use it
Choose the cross-section and enter the bulk velocity and dimensions. Non-circular sections are converted to a hydraulic diameter Dh. Drag the velocity slider to see the flow change from laminar to turbulent.
- Hydraulic diameter Dh
- Darcy friction factor f
- Pressure drop Δp/L
- Entrance length Le
Formulas and assumptions
- Dh = 4A/P: round pipe D; rectangle 2ab/(a + b); annulus Do − Di. Re = ρUDh/μ.
- Pipe flow regime: laminar for Re < 2 300, transitional 2 300–4 000, turbulent for Re > 4 000.
- Laminar: f = 64/Re. Turbulent: 1/√f = −1.8log10[(ε/Dh/3.7)1.11 + 6.9/Re] (Haaland). Linear interpolation in the transitional range.
- Δp/L = (f/Dh)·½ρU². Entrance length: laminar 0.05·Re·Dh; turbulent 4.4·Re1/6·Dh.
- Typical roughness: commercial steel 0.045 mm, galvanised steel 0.15 mm, plastic 0.0015 mm.
Mach number and compressibility
Check the incompressible assumption and choose a suitable density model.
What is it?
The Mach number is the ratio of the flow velocity to the speed of sound in the fluid. Ma = 1 means the flow moves as fast as sound.
Why it matters
At low Ma the gas density hardly changes and can be treated as constant, which keeps the model simple and fast. At high Ma the gas is noticeably compressed; ignoring this gives wrong pressure, temperature and flow rate.
How to use it
Enter the largest velocity in the domain, usually at a nozzle, a narrow gap or a blade tip. Below 0.3: constant density; from 0.3 upwards: the ideal-gas model.
- Speed of sound a
- Stagnation density change Δρ/ρ
- Dynamic pressure q = ½ρU²
Formulas and assumptions
- Ideal gas: a = √(γRT), γ = 1.4, R = 287 J/(kg·K). Water: speed of sound from Marczak.
- Ma = U/a; gas: ρ0/ρ = [1 + (γ − 1)Ma²/2]1/(γ−1); liquid: Δρ/ρ ≈ Ma²/2.
- Ma < 0.3: density change below 5 %, constant density is acceptable. 0.3–0.8: ideal gas. 0.8–1.2: transonic. Above 1.2: supersonic.
y+ and first cell height
Near-wall cell height for a target y+, estimated from a skin-friction correlation.
What is it?
Next to a solid surface (a pipe wall, a fan blade, an equipment casing) the flow slows to zero within a very thin layer. y+ measures the distance from the wall to the centre of the first mesh cell on the scale of that layer: the smaller y+, the closer the cell is to the wall.
Why it matters
Turbulence models treat the near-wall region in one of two ways: resolve it down to the wall (y+ ≈ 1) or estimate it with wall functions (y+ from 30 to 300). If the first cell falls in the wrong range, friction, heat transfer and separation are all predicted wrongly.
How to use it
Choose y+ ≈ 1 when heat transfer or separation must be accurate; choose 30–100 when pressure drop and flow rate are the main results. Enter Δy₁ as the first layer height in your software.
- Cell-centre distance yP
- Reynolds number ReL
- Skin friction coefficient Cf
- Wall shear stress τw
- Friction velocity uτ
- Boundary layer thickness δ
Formulas and assumptions
- ReL = ρUL/μ; τw = ½CfρU²; uτ = √(τw/ρ); yP = y⁺μ/(ρuτ); Δy1 = 2yP.
- Flat plate: Cf = [2log10Re − 0.65]−2.3 (Schlichting, Re < 10⁹); δ ≈ 0.37 L Re−0.2.
- Pipe, duct: Cf = f/4 with f from Haaland for a smooth pipe; δ is taken as Dh/2.
- STAR-CCM+ and Fluent evaluate y+ at the centre of the wall cell, so the cell height is twice yP.
- The correlations assume developed turbulent flow over a smooth wall. At stagnation points, in separated and accelerating regions the actual y+ differs; check the y+ distribution after the first run.
- Air: ideal gas at 101 325 Pa, Sutherland viscosity. Water 0–100 °C: Kell density, Vogel viscosity.
Prism layer mesh
Total prism layer thickness, outermost layer and transition to the core mesh.
What is it?
Prism layers are thin cells stacked parallel to the surface, growing thicker away from the wall. They capture the steep velocity change inside the boundary layer, where an ordinary mesh is too coarse.
Why it matters
Layers that are too thin do not cover the boundary layer. Layers that grow too fast, or a last layer very different from the core cells, increase errors and make the solution harder to converge.
How to use it
Take Δy₁ and δ from the y+ calculator, then adjust the growth ratio r and the number of layers N until the total thickness covers the boundary layer and the last layer is close to the core cell size. The remarks below show whether the set-up is acceptable.
- Last layer thickness hN
- Ratio hN / core cell
- Coverage T / δ
Formulas and assumptions
- T = Δy1(rᴺ − 1)/(r − 1); hN = Δy1rN−1.
- Typical growth ratios are 1.1–1.3. The prism layer should cover the boundary layer (T ≥ δ).
- An hN/core ratio of about 0.3–1, i.e. core cells no larger than about three times the last layer, gives a smooth size transition.
Turbulence boundary conditions
Inlet k, ε and ω from the turbulence intensity and length scale.
What is it?
In turbulent flow the velocity always fluctuates around its mean. The k-ε and k-ω models describe these fluctuations with k (their energy) and ε or ω (the rate at which that energy dissipates).
Why it matters
At an inlet the software needs to know how turbulent the incoming flow already is. Values that are too high or too low distort the mixing, heat transfer and losses downstream of the inlet.
How to use it
Estimate the turbulence intensity I, the size of the fluctuations relative to the mean velocity, from the real conditions; for pipe flow, press "Use" for a suggested value. Enter k, ε and ω as the inlet boundary condition.
- Dissipation rate ε
- Specific dissipation rate ω
- Length scale ℓ
- Velocity fluctuation u′
- Turbulent viscosity ratio μt/μ
- Reynolds number based on Dh
Formulas and assumptions
- u′ = I·U; k = 1.5(I·U)².
- ε = Cμ3/4k3/2/ℓ; ω = k1/2/(Cμ1/4ℓ); μt/μ = ρCμk²/(με); Cμ = 0.09.
- ℓ = 0.07·Dh for fully developed pipe flow; downstream of a screen or perforated plate, take ℓ from the hole size.
- Typical intensity: below 1 % (wind tunnel, quiescent air), 1–5 % (pipes, ducts), 5–20 % (downstream of fans, valves, mixers).
- STAR-CCM+ takes intensity and turbulent viscosity ratio directly; Fluent takes intensity with a length scale or Dh.
Time step from the Courant number
Time step for transient simulation and the number of steps per flow-through of the domain.
What is it?
The Courant number (CFL) tells you how many mesh cells the flow crosses in one time step. CFL = 1 means exactly one cell per step.
Why it matters
With too large a time step the flow skips several cells per step: the solution loses detail or diverges. With too small a step the run takes longer without a better result.
How to use it
Enter the largest velocity and the smallest cell in the region of interest, and choose a CFL suited to the solution method. The number of steps per flow-through helps estimate the total run time.
- Flow-through time L/U
- Steps per flow-through
Formulas and assumptions
- CFL = UΔt/Δx, so Δt = CFL·Δx/U. Use the largest velocity and the smallest cell in the region of interest.
- Flow-through time t = L/U; steps = t/Δt. Transient runs usually cover several flow-through times to wash out the initial condition.
- Explicit schemes need CFL ≤ 1. Implicit schemes remain stable at higher CFL; LES and DES need CFL ≈ 1 in the region of interest, URANS usually tolerates a CFL of a few units.
Physical properties of air, water and gas mixtures against temperature, as input data for simulation and design calculations.
Properties of air
Density, viscosity, specific heat and thermal conductivity of dry air from −50 to 1 500 °C.
- Dynamic viscosity μ
- Kinematic viscosity ν
- Specific heat cp
- Thermal conductivity k
- Prandtl number Pr
- Thermal diffusivity α
- Heat capacity ratio γ
- Speed of sound a
- Thermal expansion coefficient β
Formulas and assumptions
- Ideal gas: ρ = p/(RT), R = 287.05 J/(kg·K); β = 1/T.
- μ and k from Sutherland: μ0 = 1.716×10⁻⁵ Pa·s, S = 110.4 K; k0 = 0.0241 W/(m·K), S = 194 K (T0 = 273.15 K).
- cp: fourth-order polynomial in T, within about 1 % of air property tables over 250–2 000 K.
- Pr = μcp/k; α = k/(ρcp); γ = cp/(cp − R); a = √(γRT).
- Dry air. At ordinary pressures, pressure affects only ρ, ν and α.
Properties of gas mixtures
Properties of flue gas, exhaust gas and fuel gas from their N₂, O₂, CO₂, H₂O, Ar, CH₄, CO and H₂ content, with the dew point.
- Molar mass M
- Dynamic viscosity μ
- Kinematic viscosity ν
- Specific heat cp
- Thermal conductivity k
- Prandtl number Pr
- Thermal diffusivity α
- Dew point
Formulas and assumptions
- Ideal gas: ρ = pM/(RuT); composition in volume % (mole fraction), normalised to 100 %.
- cp of each species from NASA 7-coefficient polynomials (GRI-Mech 3.0); mixture cp = Σxicp,i/M.
- μ and k of each species: Sutherland for N₂, O₂, CO₂, Ar, CH₄, CO and H₂ (k of CO₂, CH₄ and H₂ refitted to tables); the IAPWS dilute-gas correlations for water vapour. Mixing by Wilke (μ) and Mason–Saxena (k).
- Dew point: saturation temperature at the water-vapour partial pressure (Antoine, 1–100 °C). Flue gas containing SO₃ reaches its acid dew point at a higher temperature.
- Compositions marked "typical" are literature values and vary widely with fuel and operation; enter measured data when available. Species not in the list (heavier hydrocarbons, H₂S, NH₃, tar) are ignored and the rest is normalised to 100 %. Errors are typically a few per cent, up to about 10 % for k at high temperature.
Properties of water
Properties of liquid water from 0 to 100 °C, with saturation pressure and surface tension.
- Dynamic viscosity μ
- Kinematic viscosity ν
- Specific heat cp
- Thermal conductivity k
- Prandtl number Pr
- Thermal diffusivity α
- Thermal expansion coefficient β
- Saturation pressure psat
- Surface tension σ
- Speed of sound a
Formulas and assumptions
- ρ from Kell (1975); μ = 2.414×10⁻⁵·10247.8/(T − 140) (Vogel); β = −(1/ρ)dρ/dT.
- cp and k interpolated from NIST tables at 101.325 kPa in 10 °C steps.
- psat from Antoine; σ from IAPWS; speed of sound from Marczak.
- Liquid water at atmospheric pressure; pressure has a negligible effect on these properties at ordinary pressures.
Technical documents — a guide to applying CFD in HVAC design and a simulation problem checklist, sent by email on request.
See documentsReview your model set-up
Send a description of the problem and the intended parameters; CFDWAYS engineers will comment on the mesh, boundary conditions and physics models.