Boundary Layer Thickness Calculator

Q: The thermal boundary layer in air comes out thicker than the velocity one. Is that right?

Yes. Pr = cpμ/k compares momentum diffusivity with thermal diffusivity, and for air it is 0.712 — heat diffuses slightly faster than momentum, so the thermal layer reaches further. δ/δt = Pr1/3 = 0.892, so δt = 1.12 δ. For water at 20 °C, Pr = 7.00 and the thermal layer is 0.52 δ. The intuition that the thermal layer is always thinner comes from working with liquids.

Velocity, thermal and concentration boundary-layer thickness on a flat plate at the station you name, laminar or turbulent, with the transition band printed as three stations rather than one line — and with the laminar coefficient 4.9100 obtained by integrating the Blasius equation, not the 5.0 that gets printed everywhere.

Last updated 30 Sep 2026

Boundary layer thickness at a station on a flat plate

velocity, station, fluid and transition Re → velocity, thermal and species layer thickness, and the integral thicknesses

Free-stream velocity UThe velocity at the edge of the boundary layer, outside it. For an aerofoil or a blade this is the local edge velocity at the station you care about, not the far-field speed — on the upper surface near the leading edge they differ by tens of per cent.

Velocity unitEverything is converted to SI internally; every dimensioned output below is given in SI and in Imperial.

Distance x from the leading edgeThe station you want the thickness at, measured along the surface from the leading edge or the stagnation point. A flat-plate boundary layer has no single thickness: δ grows as x^1/2 in laminar flow and about x^6/7 in turbulent flow, so naming the station is half the answer.

Length unitThe unit for the lengths above.

FluidAir densities come from the ideal-gas law at 101.325 kPa and air viscosities from Sutherland’s formula as the US Standard Atmosphere 1976 states it; water comes from the NIST Chemistry WebBook at 0.101325 MPa. Every number here is identical to the one the y+ page uses, so the two pages cannot disagree on a Reynolds number. Choose Custom for sea water, a gas at pressure, oil or anything compressible.

Density ρ (kg/m³)Read only when the fluid above is Custom; otherwise this field shows the preset’s value. For a compressible run use the density at the edge of the boundary layer, not the stagnation density.

Dynamic viscosity μ (Pa·s)Dynamic viscosity, not kinematic. If your solver reports ν in m²/s, multiply by the density. In water the confusion is a factor of a thousand and obvious; in air it is a factor of only 1.2 — 1.81×10⁻⁵ Pa·s against 1.51×10⁻⁵ m²/s — so it passes unnoticed and biases everything by 20 per cent. Check the unit, not the magnitude.

Prandtl number PrPr = c_pμ/k, the ratio of momentum to thermal diffusivity. The presets are computed from the same μ used above with c_p and k from standard property tables: 0.712 for air (it moves by less than ±0.01 from 0 to 100 °C), 7.00 for water at 20 °C, 2.98 at 60 °C. Engine oil is 10²–10⁴; liquid sodium is 0.005. Read only when the fluid is Custom.

Schmidt number Sc, for the concentration layerSc = ν/D_AB, the mass-transfer twin of the Prandtl number. About 0.6 for water vapour in air at room temperature (ν = 1.51×10⁻⁵, D_AB ≈ 2.5×10⁻⁵ m²/s); several hundred for a dissolved gas in water, because liquid diffusivities are three to four orders of magnitude smaller than gas ones. Leave it alone if you are not doing species transport.

Transition Reynolds number Re_x,trOne number decides both the regime named on this page and which correlation is applied, so the label and the arithmetic can never disagree. The three headline values match the bands on the Reynolds number page exactly; 3 × 10⁵ and 10⁶ are here because the y+ page offers them. Transition is a band, not a line — the three transition stations for 10⁵, 5 × 10⁵ and 3 × 10⁶ are all printed below whatever you choose here.

Turbulent thickness correlationBoth come from a 1/7-power velocity profile closed with a different wall-shear law, and both are curve fits, not solutions. The first is the one the y+ page uses, so it is the default here. They agree to 5 per cent at Re_x = 10⁶ and part company by 19 per cent at 10⁸, which is an honest measure of how well anyone knows a turbulent boundary-layer thickness. There is no laminar choice to make: the laminar branch is the Blasius solution, integrated rather than fitted.

20.15mmExample

A 1 m plate in air at 20 °C and 30 m/s, at the trailing edge, with transition at Re_x = 5 × 10⁵ and White’s turbulent correlation

One similarity solution, two curve fits, and the three ratios that follow

Laminar (Blasius): δ₉₉ = 4.9100 x/√Re_x · δ* = 1.72078766 x/√Re_x · θ = 0.66411467 x/√Re_x · H = 2.5911
Turbulent: δ₉₉/x = 0.16 Re_x^−1/7 or 0.37 Re_x^−1/5 · δ*/δ = 1/8 · θ/δ = 7/72 · H = 9/7
Thermal and species: δ/δ_t = Pr^1/3 and δ/δ_c = Sc^1/3 (laminar, Pr and Sc &gtrsim; 0.6) · x_tr = Re_x,tr ν/U

δ99: the wall distance at which u = 0.99U. The boundary layer has no edge, so this is a convention; δ95 is 3.918 x/√Re_x and δ99.9 is 6.011 x/√Re_x, both from the same integration, so the choice of convention moves the answer by more than 50 per cent
Re_x: Ux/ν on the distance from the leading edge, not on a chord or a diameter. The single commonest error on this calculation is putting a body length in here when the station of interest is a tenth of the way along it
4.9100: η at which f′ = 0.99 in the Blasius solution. Obtained for this page by integrating f‴ + ½ff″ = 0 to f″(0) = 0.33205734 and reading off η₉₉ = 4.909990. The 5.0 in most textbooks is that number rounded up to one significant figure, and it is 1.83 per cent high
δ*: displacement thickness, ∫(1 − u/U)dy. The distance the outer flow is pushed away from the wall. For Blasius it equals lim(η − f(η)) = 1.72078766 in similarity units
θ: momentum thickness, ∫(u/U)(1 − u/U)dy. Its Blasius coefficient 0.66411467 is numerically identical to the local skin-friction coefficient, which is not a coincidence: it is the momentum integral C_f/2 = dθ/dx
Pr: c_pμ/k. Pr^(1/3) governs the thermal layer for Pr ≳ 0.6. Below that the thermal layer is thicker than the velocity layer and the exponent changes to −1/2
Sc: ν/D_AB, the species analogue of Pr. The same cube-root scaling applies for the same reason, and Le = Sc/Pr tells you which of heat and species diffuses further
Re_x,tr: the transition Reynolds number. Not a property of the fluid: a property of the surface finish, the free-stream turbulence, the pressure gradient and the noise in the tunnel. That is why this page prints three of them

Worked example

A 1 m plate in air at 20 °C and 30 m/s, at the trailing edge, with transition at Re_x = 5 × 10⁵ and White's turbulent correlation
Fluid first, because everything scales with ν. Air at 20 °C and 101.325 kPa: ρ = 1.2041 kg/m³, μ = 1.8134×10⁻⁵ Pa·s, so ν = 1.50602×10⁻⁵ m²/s. These are the same two numbers the y+ page uses, deliberately.
Re_x = Ux/ν = 30 × 1 / 1.50602×10⁻⁵ = 1.992×10⁶. That is past 5 × 10⁵, so the trailing edge is turbulent — but only the last 75 per cent of the plate is. Transition sits at x_tr = 5×10⁵ × 1.50602×10⁻⁵ / 30 = 0.2510 m.
Turbulent thickness, White: δ = 0.16 x Re_x^−1/7 = 0.16 × 1 × (1.992×10⁶)^−1/7. (1.992×10⁶)^1/7 = 7.9414, so δ = 0.16/7.9414 = 0.020148 m = 20.15 mm.
Sanity check against the laminar branch at the same station: 4.9100/√(1.992×10⁶) = 4.9100/1411.38 = 0.0034788, so a laminar layer would be only 3.479 mm here. The turbulent layer is 5.79 times thicker at the same Reynolds number, which is the whole reason transition matters for a mesh: the prism stack has to grow by that factor across the transition line.
Thermal layer: air has Pr = 0.7116, and the flow is turbulent here, so δ_t ≈ δ. Had the station been laminar, δ_t = δ Pr^−1/3 = δ × 1.1201 — in a gas the thermal layer is slightly thicker than the velocity layer, which surprises people who expect it to be thinner.
Integral thicknesses, from the 1/7-power ratios: δ* = δ/8 = 2.518 mm and θ = 7δ/72 = 1.959 mm, giving H = 9/7 = 1.2857. Re_θ = 30 × 0.0019588 / 1.50602×10⁻⁵ = 3902, comfortably in the range where a wall-function mesh behaves.

The Blasius solution, integrated for this page rather than quoted

Every figure in the second column was produced by integrating f⁮ + ½ff″ = 0 with a fourth-order Runge–Kutta scheme at a step of 10⁻⁵, shooting on f″(0) until f′(∞) = 1. Two identities were used to check it rather than trusting the integration: the momentum thickness coefficient must equal 2f″(0) exactly (momentum integral), and it does to ten figures; and δ* must equal lim(η − f(η)), which it does to eleven. **The 5.0 in the textbooks is not wrong so much as rounded** — but it is 1.83 per cent high, and it is the reason two otherwise identical calculators disagree in the third figure.
Quantity	Similarity value	Coefficient on x/√Re_x	Usually printed as
f″(0), the wall shear derivative	0.33205734	—	0.332
η at u/U = 0.95	3.91804	3.9180	rarely quoted
η at u/U = 0.99 — δ99	4.909990	4.9100	5.0
η at u/U = 0.995	5.27073	5.2707	rarely quoted
η at u/U = 0.999	6.01143	6.0114	rarely quoted
Displacement thickness δ*	1.72078766	1.72078766	1.721 or 1.72
Momentum thickness θ	0.66411467	0.66411467	0.664
Local skin friction C_f·√Re_x	0.66411467	—	0.664
Shape factor H = δ*/θ	2.5911002	—	2.59 or 2.6
δ*/δ99	0.350467	—	about 0.35
θ/δ99	0.135258	—	about 0.14

How the three transition Reynolds numbers land as distances, in air at 20 °C and in water at 20 °C

ν = 1.50602×10⁻⁵ m²/s for air and 1.003196×10⁻⁶ m²/s for water, both at 20 °C. The span is 30× in every row because x_tr is linear in Re_tr — which is the point. **Transition is not a line you can put on a drawing.** If your region of interest lies inside that factor of thirty, the honest answer is two runs, one at each bound, and a stated spread.
Free-stream velocity	x at Re = 1 × 10⁵	x at Re = 5 × 10⁵	x at Re = 3 × 10⁶	Span, first to last
Air, 1 m/s	1.5060 m	7.530 m	45.18 m	30×
Air, 10 m/s	150.6 mm	753.0 mm	4.518 m	30×
Air, 30 m/s	50.20 mm	251.0 mm	1.5060 m	30×
Air, 100 m/s	15.06 mm	75.30 mm	451.8 mm	30×
Water, 0.5 m/s	200.6 mm	1.0032 m	6.019 m	30×
Water, 2 m/s	50.16 mm	250.8 mm	1.5048 m	30×
Water, 5 m/s	20.06 mm	100.3 mm	601.9 mm	30×

The two turbulent thickness correlations, and their consistent skin-friction partners

The spread column is the second correlation relative to the first. They cross near Re_x = 3×10⁶ and diverge in both directions from there, which is what two independent curve fits to the same scattered data look like. The momentum integral fixes each one’s skin-friction partner exactly: a thickness law δ/x = c Re^−1/7 with a 1/7-power profile demands C_f = (c/6) Re^−1/7, so 0.16 pairs with 0.026667 and 0.37 pairs with 0.0576. **White’s printed pair (0.16, 0.027) is internally inconsistent by 1.25 per cent** and the classic 1/5 pair (0.37, 0.0576) is exact.
Re_x	Thickness over x, White 0.16 Re^(−1/7)	Thickness over x, 1/5 power 0.37 Re^(−1/5)	Spread	Thickness over x, laminar, for contrast
1 × 10⁵	0.030891	0.037000	+19.8 %	0.015527
5 × 10⁵	0.024546	0.026817	+9.3 %	0.006944
1 × 10⁶	0.022232	0.023345	+5.0 %	0.004910
1 × 10⁷	0.016000	0.014730	−7.9 %	0.001553
1 × 10⁸	0.011515	0.009294	−19.3 %	0.000491
1 × 10⁹	0.008287	0.005864	−29.2 %	0.000155

Why 4.9100 and not 5.0, why the thermal layer in air is thicker than the velocity layer, and why transition is three numbers

A boundary layer has no edge. The velocity approaches the free-stream value asymptotically, so any thickness is a convention about how close is close enough. δ₉₉ is the most common one and this page uses it throughout, but the choice is not cosmetic: from the same Blasius solution, δ₉₅ is 3.918 x/√Re_x and δ₉₉.₉ is 6.011, so moving the convention from 95 to 99.9 per cent thickens the layer by 53 per cent while changing nothing physical at all. When a CFD report and a wind-tunnel report disagree about a boundary-layer thickness by a third, the definition is the first thing to check.

The laminar coefficient is 4.9100, and the 5.0 you have seen is that number rounded. Integrating f⁮ + ½ff″ = 0 gives f″(0) = 0.33205734 and f′(η) = 0.99 at η = 4.909990. Almost every textbook prints 5.0 because it is easier to remember and because it is well inside the uncertainty of any real measurement — the similarity solution itself assumes zero pressure gradient, an infinitely thin leading edge and no free-stream turbulence, none of which is true. But 5.0 is 1.83 per cent high, it propagates into every quantity derived from δ, and when two calculators on the same site disagree in the third figure a reader is entitled to know which one integrated something. This one did. The number appears here, on the y+ page, and on the shape-factor page as the same 4.9100.

The thermal layer is thinner than the velocity layer only when Pr is above 1, and in air it is not. δ/δ_t = Pr^1/3 with Pr = 0.712 gives 0.8925, so the thermal layer in air is 12 per cent THICKER than the velocity layer. In water at 20 °C, Pr = 7.00 and the ratio is 1.913 — the thermal layer is barely half the velocity layer, and a mesh sized on δ will not resolve it. In engine oil, Pr can be 10⁴ and the thermal layer is a twentieth of the velocity layer; that is the case where an otherwise beautiful velocity field carries a Nusselt number that is out by a factor of several. The cube-root law is derived for Pr &gtrsim; 0.6 and fails for liquid metals, where the thermal layer engulfs the velocity layer and the exponent becomes −1/2. The same argument, with the same exponent and the same caveat, governs the concentration layer through the Schmidt number.

Transition is a band that spans a factor of thirty in distance, and pretending otherwise is the expensive mistake. The lowest Reynolds number at which a flat-plate boundary layer has been observed to transition at all is about 10⁵; a commercial surface in ordinary air goes at roughly 5 × 10⁵; a polished plate in a quiet tunnel can be held laminar to 3 × 10⁶. Those are the three bands the Reynolds number page uses, and this page prints all three stations whichever one you select, because at 30 m/s in air they are 50 mm, 251 mm and 1.51 m from the leading edge. If your region of interest is anywhere in that range, a single answer is a guess. What a fully turbulent RANS run does, meanwhile, is transition at the first cell — so the model has silently chosen Re_tr = 1 on your behalf, and the drag it reports is the tripped-plate drag. The skin-friction page quantifies what that costs.

What to do with the number. Three things, all mesh decisions. First, the prism stack has to span δ with enough layers that the growth ratio stays below about 1.2 — the prism stack page turns δ and a first-cell height into a layer count. Second, the first cell height comes from y+, not from δ, and the two are set independently: see the y+ page. Third, δ* rather than δ is what blocks a duct, and it is roughly a third of δ in laminar flow and an eighth in turbulent flow — which is why a turbulent layer three times thicker can block less. The step in δ at transition, printed above, is the number that decides whether one prism stack can serve the whole surface or whether the mesh has to thicken through the transition region.

Frequently asked questions

Why does this page say 4.9100 when my textbook says 5.0?

Because 5.0 is 4.9100 rounded to one significant figure. Integrating the Blasius equation f⁮ + ½ff″ = 0 puts u/U = 0.99 at η = 4.909990, and that integration was done for this page rather than quoted — it is checked against two identities that must hold exactly, the momentum-integral relation between θ and C_f, and δ* = lim(η − f(η)). 5.0 is 1.83 per cent high. It is a perfectly good number to carry in your head and a bad one to put in a report next to a CFD result quoted to four figures.

Is δ99 what my solver reports as boundary-layer thickness?

Probably not, and this is worth checking before comparing. Fluent, STAR-CCM+ and CFX do not report a boundary-layer thickness as a standard field at all; what people usually extract is either the wall distance at which the velocity magnitude reaches some fraction of a user-nominated free-stream value, or a vorticity-based thickness, or δ* and θ from an integral post-processor. Those are different quantities. δ* and θ are the robust pair because they are integrals and do not depend on locating an edge; δ₉₉ depends entirely on what you call the edge velocity, and on a curved surface with a pressure gradient there is no unambiguous edge velocity to find.

Why is the turbulent layer so much thicker at the same Reynolds number?

Because turbulent mixing transports momentum across the layer far more effectively than molecular viscosity, so the influence of the wall reaches much further into the flow. At Re_x = 2×10⁶ the laminar solution gives 3.48 mm on a 1 m plate in air at 30 m/s and the turbulent correlation gives 20.2 mm — a factor of 5.8. The displacement thickness, though, only goes from 1.22 mm to 2.52 mm, a factor of 2.1, because the turbulent profile is much fuller: it spends most of its height close to the free-stream velocity. That is why a turbulent layer three times thicker blocks a duct less than you would fear, and why δ* and not δ is the number to use for blockage.

The thermal boundary layer in air comes out thicker than the velocity one. Is that right?

Yes. Pr = c_pμ/k compares momentum diffusivity with thermal diffusivity, and for air it is 0.712 — heat diffuses slightly faster than momentum, so the thermal layer reaches further. δ/δ_t = Pr^1/3 = 0.892, so δ_t = 1.12 δ. For water at 20 °C, Pr = 7.00 and the thermal layer is 0.52 δ. The intuition that the thermal layer is always thinner comes from working with liquids.

Which transition Reynolds number should I choose?

For a CFD run, the one your turbulence model is actually going to use — which for a standard k-ω SST or k-ε run with no transition model is the last option, turbulent from the leading edge. Choose a real transition Reynolds number only if you are running a transition model (γ-Re_θ, k-k_L-ω, an e^N method) or comparing against an experiment whose tunnel turbulence you know. If you are sizing a mesh, the conservative choice is the tripped case, because it gives the thinnest layer near the leading edge and therefore the smallest cells.

Does any of this apply to an aerofoil, a blade or a hull?

As an order-of-magnitude estimate at a station, yes, and that is how it is normally used. As a prediction, no: every relation on this page assumes zero pressure gradient. An adverse gradient thickens the layer, raises the shape factor and can separate it; a favourable gradient thins it and delays transition. The practical route is to take the local edge velocity and the running length from the stagnation point, use this page to size the first cell and the prism stack, and let the solver find the real thickness. What this page is genuinely good for is catching a mesh that is an order of magnitude wrong before you spend a night on it.

What is the Schmidt number field for?

Species transport. If you are solving a scalar — humidity, a tracer, a reacting species — the concentration boundary layer has its own thickness, governed by Sc = ν/D_AB exactly as the thermal layer is governed by Pr. For water vapour in air Sc ≈ 0.6 and the concentration layer is slightly thicker than the velocity layer; for a dissolved gas in water Sc is several hundred and the concentration layer is a small fraction of δ, which is a severe mesh requirement that people discover after the run. The Lewis number printed above, Sc/Pr, tells you whether heat or species is the binding constraint.

How accurate is the turbulent branch?

Take it as ±10 per cent at best in the range it was fitted, and worse outside it. The two correlations offered here agree to 5 per cent near Re_x = 10⁶, cross near 3×10⁶, and are 29 per cent apart at 10⁹. That spread is not a defect in either fit; it is how well a flat-plate turbulent thickness is known, and it is larger than the spread in skin friction because δ depends on the edge definition as well as on the wall shear. Skin friction, which is an integral of the wall shear rather than a position, is known much better: the three common local correlations agree to 0.44 per cent at Re_x = 10⁶.

Related calculators

References

H. Blasius (1908), Grenzschichten in Flüssigkeiten mit kleiner Reibung, Z. Math. Phys. 56, 1–37. Not quoted from a table. The equation f⁮ + ½ff″ = 0 was integrated for this page by fourth-order Runge–Kutta at Δη = 10⁻⁵, shooting on f″(0) to f′(∞) = 1, giving f″(0) = 0.33205734, η₉₉ = 4.909990, δ*√Re_x/x = 1.72078766 and θ√Re_x/x = 0.66411467.
H. Schlichting and K. Gersten, Boundary-Layer Theory, 9th edition. Cited for the turbulent correlations and the transition discussion; no table from it is reproduced here. Every turbulent number on this page is either a named closed form or was recomputed from the 1/7-power profile.
F. M. White, Viscous Fluid Flow and Fluid Mechanics, for δ/x = 0.16 Re_x^−1/7 and its partner C_f = 0.027 Re_x^−1/7. The 1.25 per cent inconsistency between that pair, reported in the third table above, was found by applying the momentum integral to them; it is stated here rather than silently corrected, because the pair is what the literature uses.
L. Prandtl (1927) for the one-fifth-power results δ/x = 0.37 Re_x^−1/5 and C_f = 0.0576 Re_x^−1/5. That pair satisfies C_f/2 = dθ/dx exactly for a 1/7-power profile, which is how it was verified here.
US Standard Atmosphere 1976 (NOAA/NASA/USAF) for Sutherland’s viscosity formula, and the NIST Chemistry WebBook for water density and viscosity at 0.101325 MPa. Both are US Government works, so the values are reproduced here rather than merely cited. Water Prandtl numbers were computed as c_pμ/k from those viscosities and gave 7.00 at 20 °C, matching the published value and confirming the property set is internally consistent.
K. Avila, D. Moxey, A. de Lozar, M. Avila, D. Barkley and B. Hof (2011), The onset of turbulence in pipe flow, Science 333, 192–196. Cited because the Reynolds number page uses its 2040 for pipes; it is the model for treating a transition point as a measured band rather than a constant, which is what this page does for the flat plate.

Setup guidance, not validation. Correlations have ranges of validity and cell-count estimates are order-of-magnitude. A converged simulation is not a correct one. Full disclaimer at calcengines.com/disclaimer/