Displacement and Momentum Thickness Calculator

Q: Why is the turbulent shape factor 9/7 exactly?

Because the one-seventh-power profile u/Ue = (y/δ)1/7 gives δ*/δ = 1 − 7/8 = 1/8 and θ/δ = 7/8 − 7/9 = 7/72, and (1/8)/(7/72) = 72/56 = 9/7. It is arithmetic on an assumed profile, not a measurement. A real turbulent flat-plate layer has H between about 1.3 and 1.4 because the 1/7 profile is a little too full near the wall, so treat 9/7 = 1.2857 as a floor rather than an expectation. Neither of the two thickness correlations offered changes it, because they set δ and leave the profile shape alone.

δ*, θ and the shape factor H, with two separate threshold ladders because H = 2.6 is a healthy laminar layer and a turbulent layer on the point of separating — plus the momentum-integral identity C_f/2 = dθ/dx that lets you check your own CFD, and the duct blockage that is the practical reason to compute δ* in the first place.

Last updated 30 Sep 2026

Displacement thickness, momentum thickness and shape factor

station and fluid, or your own two thicknesses → H, the Reynolds numbers, the momentum-integral check and duct blockage

Where the thicknesses come fromThe first two compute δ*, θ and H from the station and the fluid, which is the right route when you are sizing a mesh or checking an order of magnitude. The third takes the two integrals your post-processor gives you and tells you what H means — which is the route to take when you have a real pressure gradient, because no flat-plate relation applies then and the shape factor is the only thing on this page that still does.

Edge velocity U_eThe velocity at the edge of the layer, not far upstream. Read in all three modes: in the third it is used only for the Reynolds numbers Re_θ and Re_δ*, which are what decide whether your H is in the range the separation criteria were measured over.

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

Distance x from the leading edgeRunning length along the surface. Not read when you are entering your own thicknesses.

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.

Turbulent thickness correlationOnly read in turbulent mode, and only to fix δ; the ratios δ*/δ = 1/8 and θ/δ = 7/72 come from the 1/7-power profile itself and are the same either way, so H = 9/7 whichever you pick. The first is the one the y+ page and the thickness page use.

Which regime is your layer inThis decides which set of thresholds H is read against, and it matters enormously: H = 2.6 is a perfectly healthy laminar layer and a turbulent layer about to separate. One ladder of bands for both regimes would be wrong, so there are two. Read only when you enter your own thicknesses; in the first two modes the regime is already known. If you are not sure, the Re_θ printed below is the usual discriminator: a flat-plate layer is not turbulent below Re_θ ≈ 320.

Displacement thickness δ* (mm)∫(1 − u/U_e)dy across the layer. Most post-processors will integrate this for you along a wall-normal line; the two traps are choosing the edge velocity (use the local U_e, not the far field) and truncating the integral before u/U_e has really reached 1. In the first two modes this field shows the computed value.

Momentum thickness θ (mm)∫(u/U_e)(1 − u/U_e)dy. Always smaller than δ*, for every attached profile: the integrand is the same as δ*’s multiplied by u/U_e ≤ 1. If your θ comes out larger than your δ*, the integration is wrong — usually the edge velocity.

Duct hydraulic diameter for the blockage check (mm)Set this to the D_h of the passage if you want the blockage figures below: the displacement thickness on every wall reduces the effective flow area and speeds the core up. Use the hydraulic diameter page for a non-round section. Leave it alone for external flow; the blockage rows are then simply irrelevant rather than wrong.

2.5911Example

A 1 m plate in air at 20 °C and 30 m/s, at the trailing edge, in Blasius laminar mode, with a 50 mm duct for the blockage check

Two integrals, their ratio, and the identity that ties the ratio to the wall shear

δ* = ∫₀^∞(1 − u/U_e) dy · θ = ∫₀^∞(u/U_e)(1 − u/U_e) dy · H = δ*/θ
Momentum integral (zero pressure gradient): C_f/2 = dθ/dx
Blasius: δ* = 1.72078766 x/√Re_x, θ = 0.66411467 x/√Re_x, H = 2.5911 · 1/7-power: δ*/δ = 1/8, θ/δ = 7/72, H = 9/7
Duct blockage: δ*P/A = 4δ*/D_h · U_core/U_bulk = 1/(1 − 4δ*/D_h)

δ*: displacement thickness. The distance the wall would have to be moved outwards, in a hypothetical inviscid flow, to pass the same mass flow the real viscous flow passes. It is the blockage the outer flow feels, and it is the only one of the three thicknesses with a direct physical construction
θ: momentum thickness. The thickness of free stream carrying the momentum the boundary layer has lost. Its growth rate IS the wall shear, which is why it is the natural variable for every integral boundary-layer method
H: shape factor δ*/θ. Always greater than 1 for an attached profile. Independent of x, of velocity and of the fluid for any similarity solution, which is what makes it the diagnostic: it reports the shape of the profile with the scale divided out
C_f/2 = dθ/dx: the momentum integral at zero pressure gradient, and the cheapest check there is on a boundary-layer computation: extract θ at two neighbouring stations, difference it, double it, and compare with the wall shear your solver reports. If they disagree by more than a few per cent the streamwise mesh is too coarse
Re_θ: U_eθ/ν. The Reynolds number every separation and transition criterion is actually written in. A flat-plate layer does not sustain turbulence below Re_θ ≈ 320, and correlations for turbulent separation were measured at Re_θ of a few thousand
4δ*/D_h: the fraction of a duct’s area the displacement thickness removes, for a section of hydraulic diameter D_h with the same δ* on every wall. It follows from δ*P/A with D_h = 4A/P, which is exactly why the hydraulic diameter carries its factor of four
H at separation: 4.029 for a laminar layer, from the Falkner–Skan solution at zero wall shear (β = −0.198838), solved here. For a turbulent layer there is no such number: published incipient-separation values span about 1.8 to 2.8

Worked example

A 1 m plate in air at 20 °C and 30 m/s, at the trailing edge, in Blasius laminar mode, with a 50 mm duct for the blockage check
ν = 1.8134×10⁻⁵/1.2041 = 1.50602×10⁻⁵ m²/s, so Re_x = 30/1.50602×10⁻⁵ = 1.992×10⁶ and √Re_x = 1411.38.
δ* = 1.72078766 × 1 / 1411.38 = 1.21922 mm and θ = 0.66411467 × 1 / 1411.38 = 0.47054 mm. Both come from integrating the Blasius equation for this build: δ*'s coefficient is lim(η − f(η)) = 1.72078766 and θ's is 2f″(0) = 0.66411467.
H = 1.21922/0.47054 = 2.5911. Notice that neither the velocity, nor the station, nor the fluid appears in that ratio. Put 3 m/s and 100 mm in instead and δ* and θ both change by a factor of 10, and H does not move.
The momentum-integral check. θ = 0.66411467 √(νx/U), so dθ/dx = 0.33205734 √(ν/(Ux)) = 0.33205734/1411.38 = 2.3527×10⁻⁴. Twice that is 4.7054×10⁻⁴, and the Blasius local C_f is 0.66411467/1411.38 = 4.7054×10⁻⁴. They agree to ten figures, which they must — C_f/2 = dθ/dx is an identity, not a correlation, and reproducing it is how this page's integration was verified.
Blockage. In a 50 mm duct, 4δ*/D_h = 4 × 1.21921/50 = 9.75 per cent of the area gone, so the core runs 1/(1 − 0.0975) = 1.108 times the bulk velocity and the core dynamic pressure is 1.228 times what the bulk velocity would suggest. A 10 per cent velocity error is a 23 per cent dynamic-pressure error, and that is what makes a displacement thickness worth computing in a duct.
For contrast, the turbulent mode at the same station gives δ* = 2.518 mm — twice as much — but δ99 = 20.15 mm against the laminar 3.48 mm, six times as much. A turbulent layer is much thicker and only somewhat more blocking, because its profile is fuller: H falls from 2.59 to 1.29.

Shape factor against pressure gradient, from the Falkner–Skan solution — solved here, not quoted

f⁮ + ff″ + β(1 − f′²) = 0 was integrated for this page by fourth-order Runge–Kutta, shooting on f″(0) to f′(∞) = 1. The β = 0 row is the check: it reproduces the Blasius H = 2.5911 to seven figures, and multiplying its two integral columns by √2 gives 1.72078766 and 0.66411467 — the Blasius coefficients on x/√Re_x — to eight figures. The two integral columns are in the Falkner–Skan similarity variable, which differs from the Blasius one by exactly √2, so multiply them by √2 before comparing them with anything quoted on x/√Re_x. The RATIO H needs no conversion. The last row is separation: H = 4.029 at β = −0.198838, with the wall shear exactly zero. Notice how little θ moves across the whole family and how much δ* does — that is why H, and not either thickness alone, is the separation diagnostic.
Hartree beta	Physical meaning	f″(0), relative to Blasius	Displacement thickness, Falkner–Skan scaling	Momentum thickness, Falkner–Skan scaling	H
+0.5	Strongly accelerating, U ∝ x^(1/3)	1.976	0.80455	0.35027	2.2969
+0.3	Accelerating, U ∝ x^(3/17)	1.650	0.91099	0.38574	2.3617
0	Flat plate, Blasius	1.000	1.21678	0.46960	2.5911
−0.10	Mildly decelerating	0.680	1.44270	0.51504	2.8011
−0.15	Decelerating	0.461	1.64697	0.54518	3.0209
−0.18	Strongly decelerating	0.274	1.87158	0.56771	3.2967
−0.19	Close to separation	0.182	2.00676	0.57652	3.4808
−0.198838	Separation, zero wall shear	0.000	2.35884	0.58543	4.0292

The three thicknesses and the shape factor, for the two flat-plate profile families

The laminar column is exact: the coefficients come from integrating the Blasius equation, and the last two rows agree identically because C_f/2 = dθ/dx holds exactly for a similarity solution. The turbulent column is a correlation closed with an assumed profile, and the last two rows differ by **1.25 per cent** — the published pair (0.16, 0.027) does not quite satisfy the momentum integral, which demands (0.16, 0.026667) or (0.162, 0.027). That gap is far inside the scatter of the data, and it is reported here rather than silently corrected, because 0.16 and 0.027 are what the literature and the y+ page use. The one-fifth-power pair, δ/x = 0.37 Re^−1/5 with C_f = 0.0576 Re^−1/5, satisfies it exactly.
Quantity	Laminar, Blasius	Turbulent, one-seventh power	Ratio, turbulent over laminar
δ99	4.9100 x/√Re_x	0.16 x Re_x^(−1/7)	depends on Re — 5.8× at Re = 2 × 10⁶
δ*	1.72078766 x/√Re_x	δ/8	2.07× at Re = 2 × 10⁶
θ	0.66411467 x/√Re_x	7δ/72	4.16× at Re = 2 × 10⁶
δ*/δ99	0.350467	0.125 exactly	0.357
θ/δ99	0.135258	0.0972222 exactly	0.719
H = δ*/θ	2.5911002	9/7 = 1.2857143	0.496
Local C_f	0.66411467 Re_x^(−1/2)	0.027 Re_x^(−1/7) (White)	—
C_f implied by 2dθ/dx	0.66411467 Re_x^(−1/2)	0.026667 Re_x^(−1/7)	—

What a displacement thickness costs in a duct

A displacement thickness δ* on every wall of a section with hydraulic diameter D_h removes δ*P/A = 4δ*/D_h of the flow area, because D_h = 4A/P by definition. The core must speed up by the reciprocal of what is left, and dynamic pressure goes as the square, so a blockage of 10 per cent is a 23 per cent error in anything computed from a nominal bulk velocity head. This is the practical reason a CFD engineer computes δ* at all, and it is also the honest signal that a duct’s inlet region has stopped behaving like two independent flat plates — at which point the entrance length page is the right tool and this one is not.
Blockage, 4 times displacement thickness over D_h	Effective area left	Core velocity multiplier	Core dynamic pressure multiplier	Reading
1 %	99 %	1.0101	1.0203	Negligible; inside the noise of most measurements
2 %	98 %	1.0204	1.0412	Detectable in a careful pressure-drop measurement
5 %	95 %	1.0526	1.1080	Matters: 11 % on a velocity head
10 %	90 %	1.1111	1.2346	The flat-plate assumption is finished; opposite walls interact
20 %	80 %	1.2500	1.5625	Well on the way to fully developed flow
25 %	75 %	1.3333	1.7778	The two walls’ layers have merged; there is no core left to correct

What δ* and θ actually are, why H needs two threshold ladders, and the one identity that will catch a bad boundary-layer mesh

δ* and θ are the two thicknesses that mean something, and δ99 is the one that does not. A boundary layer has no edge, so δ99 depends entirely on where you decide to put it; δ* and θ are integrals across the whole profile and need no edge at all, only an edge VELOCITY. δ* has a physical construction: it is how far out you would have to move the wall, in an equivalent inviscid flow, to pass the same mass. θ has a dynamic one: it is the thickness of free stream carrying the momentum the layer has lost, and its growth rate is the wall shear. Between them they are what every integral boundary-layer method, every panel-method viscous coupling and every blockage correction is built from.

H needs two ladders, and this is the single most important thing on the page. A laminar Blasius layer has H = 2.5911 and is in perfect health. A turbulent layer at H = 2.4 is on the point of separating. Both statements are standard, and put side by side they mean that a shape factor read against the wrong regime’s thresholds gives the opposite of the right answer. This page therefore carries two sets of bands and picks between them from the regime, and when you enter your own δ* and θ it asks you which regime you are in rather than guessing. The laminar ladder is anchored on numbers that can be derived: the Falkner–Skan family gives 2.297 at β = +0.5 (strongly accelerated), 2.5911 at β = 0, and 4.029 at β = −0.198838, where the wall shear is exactly zero. Those were obtained by solving f⁮ + ff″ + β(1 − f′²) = 0 for this build, and the β = 0 case reproduces Blasius to seven figures, which is the check. The turbulent ladder cannot be anchored that way, because turbulent separation has no exact solution: published incipient-separation shape factors span roughly 1.8 to 2.8 depending on the criterion and on Re_θ. The page uses 2.4 as an indicative threshold and says plainly that it is the middle of a wide band.

C_f/2 = dθ/dx is the cheapest boundary-layer check in existence, and almost nobody runs it. At zero pressure gradient the momentum integral reduces to that identity: the momentum thickness grows at exactly half the skin-friction coefficient. It is not a correlation, it is a consequence of conservation of momentum, so it holds for any profile at zero pressure gradient, laminar or turbulent, at any Reynolds number. To use it, extract θ at two neighbouring stations from your solution, difference it, double it, and compare with the wall shear the solver reports at the midpoint. If the two disagree by more than a few per cent, your streamwise mesh is too coarse or your wall treatment is inconsistent with your near-wall mesh, and no amount of refining normal to the wall will fix it. This page prints both sides of the identity for the flat-plate profiles so you can see what agreement looks like: in the laminar mode they match to ten figures, because they must.

The practical reason to compute δ* is blockage. In a duct, the displacement thickness on every wall removes 4δ*/D_h of the flow area — the factor of four is exactly why the hydraulic diameter is defined as 4A/P, and the hydraulic diameter page has more on that. Ten per cent blockage means the core runs 11 per cent faster than the bulk velocity and its dynamic pressure is 23 per cent higher, which is enough to invalidate a flow-meter calibration, a pressure-drop prediction or a comparison against a nominal inlet velocity. It is also the signal that the two walls have started to see each other and that the flow is on its way to being developed, at which point the entrance length page is the right tool.

Where these relations stop applying. Everything computed from a station here assumes a flat plate at zero pressure gradient. With a pressure gradient, δ* and θ still mean exactly what they meant — they are definitions — but their flat-plate values do not apply and neither does H = 2.5911 or 9/7. That is the case the third mode is for: give the page the two integrals from your own solution and it will tell you what the shape factor says about the layer’s health, which is information a flat-plate correlation cannot give you at all. The skin friction page has the local and average wall stress, and the thickness page has δ99, the thermal layer and the transition band.

Frequently asked questions

Which is bigger, δ* or θ?

δ*, always, for any attached profile. The two integrands are (1 − u/U_e) and (u/U_e)(1 − u/U_e), and the second is the first multiplied by u/U_e, which is at most 1. So H = δ*/θ > 1 with no exceptions, and if your post-processor gives you H below 1 something is wrong with the integration rather than with the flow. The commonest cause is an edge velocity taken too low, which makes u/U_e exceed 1 over part of the profile and contributes negatively to δ*.

Why does H not change when I change the velocity or the station?

Because both flat-plate modes use a similarity profile, and a similarity profile has one shape at every station. δ* and θ each scale with x/√Re_x in laminar flow, so their ratio is a pure number: 2.5911. In the turbulent mode both scale with δ, so the ratio is 9/7 exactly. That invariance is the reason H is useful. It is also the reason H is the right thing to compare between a CFD run and an experiment — it does not care whether the two got the same Reynolds number, only whether they got the same profile shape.

Is H = 2.4 really the turbulent separation threshold?

It is an indicative value in the middle of a wide band, and the page says so rather than pretending otherwise. Published criteria for incipient turbulent separation run from about 1.8 to 2.8: Sandborn and Kline put it near 2.7 at moderate Re_θ and lower at high Re_θ, other criteria give 2.0 or 2.2, and the disagreement is partly about what ‘incipient’ means. The thing that actually defines separation is zero wall shear stress, and that is what to look at. H is the early warning, and what matters about it is the trend along the surface rather than the value at one station.

How do I use the momentum-integral identity on my own CFD output?

Pick a flat portion of your wall with negligible pressure gradient. Extract θ on wall-normal lines at two stations a short distance Δx apart — far enough for the difference to be bigger than your integration noise, close enough that C_f has barely changed. Compute 2(θ₂ − θ₁)/Δx and compare it with the C_f the solver reports midway between them. Agreement within a few per cent means the streamwise resolution and the wall treatment are consistent. A systematic shortfall in dθ/dx usually means the streamwise mesh is too coarse; a systematic excess often means the boundary-layer integration is being truncated too close to the wall.

What edge velocity should I use for the integration?

The local one, at the edge of the layer on the wall-normal line you are integrating along — not the far-field or inlet velocity. On anything with curvature that distinction is large: near an aerofoil suction peak the local edge velocity can be 1.5 times the free stream, and using the free-stream value instead will give you a δ* that is wrong by tens of per cent and an H that is wrong in a way you cannot easily spot. The usual practical definition is the velocity at the point where the total pressure stops changing, or where the vorticity falls below a threshold; both are better than picking a distance.

Does the blockage calculation apply to an external flow?

No, and the duct field is there to be ignored in that case. The 4δ*/D_h form assumes a closed section of hydraulic diameter D_h with the same displacement thickness on every wall, which is the inlet of a pipe or a duct. For an external flow the analogous concept is the displacement body — the outer flow behaves as though the surface were displaced outwards by δ* — and its practical consequence is a change in the effective camber and thickness of an aerofoil rather than a velocity increase in a core.

Why is the turbulent shape factor 9/7 exactly?

Because the one-seventh-power profile u/U_e = (y/δ)^1/7 gives δ*/δ = 1 − 7/8 = 1/8 and θ/δ = 7/8 − 7/9 = 7/72, and (1/8)/(7/72) = 72/56 = 9/7. It is arithmetic on an assumed profile, not a measurement. A real turbulent flat-plate layer has H between about 1.3 and 1.4 because the 1/7 profile is a little too full near the wall, so treat 9/7 = 1.2857 as a floor rather than an expectation. Neither of the two thickness correlations offered changes it, because they set δ and leave the profile shape alone.

Related calculators

References

H. Blasius (1908), Grenzschichten in Flüssigkeiten mit kleiner Reibung. f⁮ + ½ff″ = 0 was integrated for this build by fourth-order Runge–Kutta at Δη = 10⁻⁵, shooting on f″(0) to f′(∞) = 1: f″(0) = 0.33205734, δ*√Re_x/x = lim(η − f(η)) = 1.72078766, θ√Re_x/x = 0.66411467, H = 2.5911002.
V. M. Falkner and S. W. Skan (1931), Some approximate solutions of the boundary layer equations, and D. R. Hartree (1937). The separation shape factor on this page was solved, not quoted: f⁮ + ff″ + β(1 − f′²) = 0 integrated at each β, giving f″(0) = 0 at β = −0.198838 with H = 4.0294. The β = 0 case of the same solver returns the Blasius H = 2.5911 to seven figures, which is the identity that validates it.
The momentum integral C_f/2 = dθ/dx at zero pressure gradient, from von Kármán (1921). It is used on this page as a check rather than a formula: in the laminar mode the two sides agree to ten figures by construction, and the 1.25 per cent gap in the turbulent mode is what exposes the internal inconsistency of the published (0.16, 0.027) correlation pair.
F. K. Sandborn and S. J. Kline (1961), Flow models in boundary-layer stall inception, J. Basic Eng. 83, 317–327, and the wider literature on turbulent separation criteria, for the statement that published incipient-separation shape factors span roughly 1.8 to 2.8 and depend on Re_θ. No table is reproduced; the page gives the band and says it is indicative, which is the honest summary of a literature that does not agree with itself.
H. Schlichting and K. Gersten, Boundary-Layer Theory, 9th edition, and F. M. White, Viscous Fluid Flow, cited for the one-seventh-power profile and the turbulent thickness correlations. The ratios δ*/δ = 1/8, θ/δ = 7/72 and H = 9/7 were recomputed from the profile here, and are exact arithmetic on it.
US Standard Atmosphere 1976 (NOAA/NASA/USAF) and the NIST Chemistry WebBook for the fluid properties, both US Government works. The values are identical to those used on the other pages in this section, so a Reynolds number is the same number wherever you compute it on this site.

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/