Reynolds Number and Flow Regime Calculator

Re for a round pipe, a rectangular duct, an annulus, an open channel, a flat plate, a cylinder or a sphere — with the hydraulic diameter worked out for you, the characteristic length named on the answer, and regime bands that come from the geometry you picked instead of one remembered number.

Last updated 29 Sep 2026

Reynolds number and flow regime

geometry, velocity and fluid → Re, D_h and the regime

GeometryThis sets the characteristic length AND the regime bands, because they are not separable. A Reynolds number of 3,000 is transitional in a pipe, thoroughly laminar on a flat plate and a turbulent wake behind a cylinder. Naming the length is the single most common omission in a quoted Reynolds number.

Velocity U (m/s)Bulk mean velocity for an internal flow (volumetric flow divided by area, not the centreline value — in a turbulent pipe the centreline is about 1.2 times the mean, in a laminar one exactly 2 times). Free-stream velocity for an external flow.

First dimension (m)Pipe bore D; duct width a; annulus OUTER diameter; open-channel width b; distance x from the leading edge for a plate; diameter D for a cylinder or sphere.

Second dimension (m)Duct height b; annulus INNER diameter; open-channel flow depth y. Ignored for a round pipe, a plate, a cylinder and a sphere.

FluidThe presets are Cengel and Ghajar Table A-9 for air at 1 atm and Table A-3 for water. Choose Other to enter your own; you can give either the kinematic viscosity or the dynamic viscosity with the density.

If you are entering your own fluid, which viscosityOnly read when the fluid is set to Other. Both routes give the same Reynolds number, because Re = ρUL/μ = UL/ν identically. The density is still used to report μ and the mass flow rate either way.

Kinematic viscosity ν (m²/s)Used only when the fluid is Other and the mode above is kinematic. Grey otherwise.

Dynamic viscosity μ (Pa·s)Used only when the fluid is Other and the mode above is dynamic. 1 Pa·s = 1000 cP; water at 20 °C is 1.002×10⁻³ Pa·s, or 1.002 cP. Grey otherwise.

Density ρ (kg/m³)Used only when the fluid is Other. Needed for μ when you entered ν, for ν when you entered μ, and for the mass flow rate. Grey otherwise.

99,602Example

Water at 20 °C flowing at 2 m/s in a 50 mm bore pipe

Reynolds number, and the characteristic length for each geometry

Re = ρUL/μ = UL/ν · D_h = 4A/P · pipe: L = D · rectangular duct: L = 2ab/(a+b) · annulus: L = D_o − D_i · open channel: L = 4by/(b+2y) · plate: L = x · cylinder or sphere: L = D

Re: Reynolds number: the ratio of inertial to viscous forces, ρU²/(μU/L). It is only meaningful once the length L is named, which is why this page prints the length it used
L: characteristic length. The hydraulic diameter for internal flow, the distance from the leading edge for a plate, the diameter for a bluff body. Not interchangeable: the same flow has three different Reynolds numbers on three different lengths
D_h: hydraulic diameter, 4A/P, with A the flow area and P the WETTED perimeter. For a round pipe it reduces to D exactly; for an annulus it reduces to D_o − D_i exactly; for an open channel the free surface is not wetted and is not counted
R: hydraulic radius, A/P, exactly a quarter of the hydraulic diameter. Open-channel hydraulics uses R and quotes the transition thresholds as 500 and 2,000; pipe work uses D_h and quotes 2,300 and 4,000. The factor of four between them is the commonest error in this subject
ν: kinematic viscosity, μ/ρ, m²/s. Either ν or the pair (μ, ρ) will do, and they give identical Reynolds numbers
U: bulk mean velocity for internal flow — volumetric flow over area, not the centreline value. Free-stream velocity for external flow

Worked example

Water at 20 °C flowing at 2 m/s in a 50 mm bore pipe
The characteristic length for a round pipe is the bore, and the hydraulic diameter confirms it: D_h = 4A/P = 4(π × 0.05²/4)/(π × 0.05) = 0.05 m exactly. This is worth doing once, because it is the only geometry where 4A/P gives you back what you started with
Re = U·D_h/ν = 2 × 0.05/1.004×10⁻⁶ = 99,602. Firmly turbulent — 43 times the transition value
The actionable inverse: the velocity at which this pipe would reach Re = 2,300 is 2,300 × 1.004×10⁻⁶/0.05 = 0.046 m/s. You would have to slow the water to 4.6 cm/s to make it laminar, which is why laminar water in a 50 mm pipe is a laboratory condition and not an industrial one
NOW THE SAME FLOW IN A RECTANGULAR DUCT, which is where the characteristic length starts to matter. A 200 × 100 mm duct has D_h = 2ab/(a+b) = 2 × 0.2 × 0.1/0.3 = 0.1333 m, so at the same 2 m/s the Reynolds number is 265,600 — nearly three times the pipe figure. Using the duct height, or the diagonal, or the width alone would have given 199,200, 445,400 or 398,400. All four numbers are arithmetic and only one is the Reynolds number
AND THE ANNULUS, which is the one people get wrong. A 100 mm pipe with a 60 mm tube down the middle: D_h = 4A/P = 4 × π(0.1² − 0.06²)/4 ÷ π(0.1 + 0.06) = 0.04 m, which is exactly D_o − D_i. Not the gap of 20 mm, and not the outer diameter. The closed form falls out of 4A/P with no approximation, so there is nothing to remember beyond that
AND THE OPEN CHANNEL, where the convention changes under you. A 1 m wide channel running 300 mm deep: the wetted perimeter is the bed plus two sides, 1 + 2(0.3) = 1.6 m, and the free surface is NOT part of it. D_h = 4 × 0.3/1.6 = 0.75 m, so at 2 m/s Re_Dh = 1,494,000. But hydraulics texts quote this on the hydraulic radius, R = A/P = 0.1875 m, giving Re_R = 373,500 and thresholds of 500 and 2,000 instead of 2,000 and 8,000. Same flow, same regime, a factor of four in the number
The regime bands are not one number either. For the pipe, 2,040 is where sustained turbulence first becomes possible (Avila and co-workers, 2011, measured to ±10), 2,300 is where an ordinary pipe is observed to transition, and 4,000 is where it is reliably fully turbulent. On a flat plate the same question is answered by a band from 10⁵ to 3×10⁶ depending on roughness and free-stream turbulence, with 5×10⁵ the conventional value for a commercial surface. For a cylinder there are seven distinct regimes and the first drag crisis alone is placed between 1×10⁵ and 3×10⁵ by different sources. Anything that gives you one sharp transition number for all geometries is wrong about at least six of them
Two by-products worth having, and a caution about the first of them. The Blasius smooth-wall factor gives 0.3164 × 99,602^(−0.25) = 0.01781, while the Petukhov form (0.79·ln Re − 1.64)^(−2) gives 0.01801. Iterating Colebrook for a smooth wall gives 0.01800, so Petukhov is right to 0.01 per cent here and Blasius is 1.1 per cent low; by Re = 10⁶ Blasius is 14 per cent low and by 10⁷ it is 31 per cent low. This page prints Blasius for familiarity and uses Petukhov for everything downstream of it. So uτ = U√(f/8) = 2√(0.01801/8) = 0.0949 m/s, which is the number the y+ calculation needs, and the first cell height for y+ = 1 is ν/uτ = 10.6 μm. The turbulent entrance length is 4.4·Re^(1/6)·D_h = 1.50 m, or 30 diameters, so a tapping closer than that to the inlet is not measuring developed flow

Hydraulic diameter: one definition, seven sections

4A/P is the only thing to remember; every closed form in the last column falls out of it. Two traps are worth naming. The annulus hydraulic diameter is D_o − D_i, which is TWICE the radial gap, and using the gap halves the Reynolds number. And in an open channel the free surface is not a wetted boundary, so it is not in P — counting it would give a smaller D_h and a smaller Reynolds number for no physical reason.
Section	Flow area A	Wetted perimeter P	D_h = 4A/P	Closed form
Round pipe, bore D	πD²/4	πD	D	D_h = D exactly
Rectangular duct a × b	ab	2(a+b)	2ab/(a+b)	the harmonic mean of a and b
Square duct, side a	a²	4a	a	D_h = a
Concentric annulus D_o, D_i	π(D_o² − D_i²)/4	π(D_o + D_i)	D_o − D_i	twice the gap, not once
Parallel plates, gap h, wide	bh	2b	2h	twice the gap
Open channel, width b, depth y	by	b + 2y	4by/(b+2y)	free surface NOT wetted
Equilateral triangle, side a	√3a²/4	3a	a/√3	0.577a

Transition is a band in every geometry, and the band is different in each

The single numbers in the second column are what gets quoted and they are useful as anchors. The bands in the third column are what actually happens, and the difference matters most where the band is widest: on a flat plate it spans more than an order of magnitude, so a drag prediction that assumes a fully turbulent boundary layer from the leading edge and one that assumes transition at 5×10⁵ can differ by tens of per cent on the same geometry.
Geometry	Conventional single number	The honest band	What moves it
Pipe or duct, internal	2,300	2,040 to about 4,000, and laminar flow has been held to 10⁵ in a very careful rig	entry geometry, roughness, vibration, upstream fittings
Open channel, on R = A/P	500	500 to 2,000 transitional; 2,000 and 8,000 on D_h	the same, plus the free surface
Flat plate, Re_x	5×10⁵	about 10⁵ to 3×10⁶	free-stream turbulence, surface roughness, pressure gradient, trips
Cylinder — shedding starts	47	46 to 49	aspect ratio, end conditions, blockage
Cylinder — wake goes 3-D	200	190 to 300	spanwise length, free-stream turbulence
Cylinder — drag crisis	3×10⁵	1×10⁵ to 3×10⁵ for the onset, ending near 3.5×10⁶	surface roughness above all
Sphere — shedding starts	270	about 270	little
Sphere — drag crisis	3×10⁵	3×10⁵ to 4×10⁵ smooth; far lower if rough	roughness and free-stream turbulence

The same flow, five characteristic lengths

A factor of 2.2 between the largest and smallest of these, all computed from the same duct and the same velocity. This is the single commonest mistake in a quoted Reynolds number, and it is invisible in the number itself — which is why every Reynolds number in a report should be written with its length, as Re_Dh or Re_x or Re_D, and never bare.
Length used for a 200 × 100 mm duct at 2 m/s in water at 20 °C	Value	Reynolds number	Is it the Reynolds number?
Hydraulic diameter, 2ab/(a+b)	0.1333 m	265,600	yes
Duct width a	0.2 m	398,400	no
Duct height b	0.1 m	199,200	no
Diagonal	0.2236 m	445,400	no
Equivalent-area circle diameter	0.1596 m	317,900	no, and it is a plausible-looking wrong answer

Naming the length, and where the regime boundaries really are

A Reynolds number without its characteristic length is not a number, it is a rumour. Re = UL/ν, and L is a choice. For the 200 × 100 mm duct in the table above, five defensible-sounding choices of L give Reynolds numbers spanning a factor of 2.2, and nothing in the figure itself tells you which was used. The convention that makes reported Reynolds numbers comparable is the hydraulic diameter, D_h = 4A/P, with A the flow area and P the wetted perimeter — and the reason it is 4A/P rather than A/P is precisely so that it collapses to the bore for a round pipe, which is where the familiar thresholds were measured. Write it as Re_Dh, Re_x or Re_D in anything anyone else will read.

The hydraulic diameter has two traps and both are common. The first is the annulus: D_h = D_o − D_i, which is twice the radial gap, and it comes out of 4A/P with no approximation — π(D_o² − D_i²)/4 over π(D_o + D_i)/4 is exactly D_o − D_i. Using the gap instead halves the Reynolds number and can move a real case across the transition band. The second is the open channel: the free surface is not a wall, so it is not part of the wetted perimeter. A 1 m channel running 300 mm deep has P = 1.6 m, not 1.9 m.

And in an open channel the threshold numbers change with the convention. Hydraulic engineering has always used the hydraulic radius R = A/P rather than the hydraulic diameter, and quotes laminar flow below Re_R = 500, transitional to 2,000 and turbulent above. Because D_h = 4R, the identical thresholds are 2,000 and 8,000 on the hydraulic-diameter basis. This page computes everything on D_h so that all seven geometries are directly comparable and prints Re_R beside it, but if you take a threshold from a hydraulics text and apply it to a hydraulic-diameter Reynolds number you will be out by a factor of four, in the direction that makes turbulent flow look laminar.

Pipe transition is three numbers, not one. 2,300 is the value in every textbook and it is where an ordinary pipe with an ordinary entry is observed to transition. It is not where the physics changes. Avila and co-workers settled that in 2011: by measuring how long an artificially created turbulent puff survives against how often it splits, and finding the Reynolds number where those two rates cross, they located the critical point for SUSTAINED turbulence at 2,040 ± 10. Below that, turbulence you inject decays; above it, turbulence you inject persists and spreads. Above about 4,000 the flow is reliably fully turbulent. And in the other direction the upper limit is not a limit at all — laminar pipe flow has been maintained past Re = 10⁵ in rigs built to eliminate every disturbance, because transition in a pipe is not an instability of the laminar solution but a response to finite disturbances. For CFD, the practical consequence is that the 2,040 to 4,000 band is where a steady RANS answer should not be trusted and where the choice of model decides the answer.

Flat-plate transition is a band spanning more than a decade, and quoting 5×10⁵ alone is a defect. Linear stability puts the first unstable Tollmien–Schlichting wave at Re_x of about 9.1×10⁴, so nothing transitions below roughly 10⁵. 5×10⁵ is the value for a typical commercial surface in ordinary conditions. A carefully polished plate in a low-turbulence tunnel stays laminar to about 3×10⁶. Free-stream turbulence, roughness, an adverse pressure gradient, a trip and vibration all move it forward; a favourable pressure gradient moves it back. Because this is a LOCAL Reynolds number, the regime also changes along the plate, so the number that matters for a drag estimate is the distance at which Re_x reaches your assumed transition value, which this page prints.

Bluff bodies have regimes, not a transition. A cylinder passes through seven distinguishable states: unseparated creeping flow, a steady symmetric vortex pair, a clean two-dimensional laminar vortex street from Re ≈ 47, a three-dimensional wake transition through roughly 190 to 300, a long subcritical plateau with St ≈ 0.2 and C_D ≈ 1.2, the drag crisis where C_D collapses to about 0.3, and a transcritical regime above roughly 3.5×10⁶ where regular shedding returns and C_D recovers to about 0.7. The boundaries above the first drag crisis are experimental and roughness-dependent, and published sources genuinely disagree: the onset of the drag crisis is placed anywhere from 1×10⁵ to 3×10⁵. A sphere has its own sequence, with separation appearing near Re = 20, axisymmetry lost near 210, shedding from about 270, a long C_D plateau near 0.4 to 0.5, and its own drag crisis at 3×10⁵ to 4×10⁵ when smooth. The practical CFD consequences are concrete: two-dimensional cylinder simulations are valid below about Re = 190 and not above; no RANS model predicts a drag crisis; and a wind-tunnel test in the subcritical band tells you little about a full-scale structure in the transcritical one unless the model is roughened to force the same regime.

What to do with the answer. If it is comfortably laminar, turn the turbulence model off — leaving one on will manufacture eddy viscosity the flow does not have. If it is comfortably turbulent, the Reynolds number tells you the near-wall mesh you need, and the friction velocity printed beside the answer is the input to that. If it lands in a transitional band, the honest answer is a range and a transition-sensitive model, not a number. And whichever it is, carry the length with it.

Frequently asked questions

Which length should I use for a non-round duct?

The hydraulic diameter, D_h = 4A/P, where A is the flow area and P the wetted perimeter. For a rectangular duct a × b that is 2ab/(a+b); for a square duct it is the side; for a concentric annulus it is D_o − D_i; for a wide gap between parallel plates it is twice the gap. The reason for the factor of four is that it makes D_h equal to D for a round pipe, so the familiar thresholds of 2,300 and 4,000 carry over. Not the diagonal, not the width, and not the diameter of a circle of equal area — all three are used by mistake and all three are wrong.

Is the pipe transition at 2,300 or 2,040?

Both, and they answer different questions. 2,300 is where an ordinary pipe with an ordinary entry is observed to transition, and it is the right number for engineering work. 2,040 ± 10 is where sustained turbulence first becomes possible at all, measured by Avila and co-workers in 2011 from the competition between puff decay and puff splitting: below it, injected turbulence dies; above it, it persists. And neither is an upper bound — laminar flow has been held past Re = 10⁵ in rigs designed to remove every disturbance, because pipe transition is triggered by finite disturbances rather than by an instability of the laminar solution.

Why does my open-channel Reynolds number look four times too big?

Because you are almost certainly comparing a hydraulic-diameter Reynolds number with a hydraulic-radius threshold. Hydraulics uses R = A/P and quotes laminar below 500 and turbulent above 2,000; D_h = 4R, so on the hydraulic-diameter basis those same thresholds are 2,000 and 8,000. This page computes on D_h for comparability with the other geometries and prints Re_R alongside. Both numbers describe the same flow and the same regime.

Can I use one transition Reynolds number for everything?

No, and that is the single reason this page asks for the geometry before anything else. 2,300 for internal flow, about 5×10⁵ for a flat plate, 47 for the onset of shedding behind a cylinder: these are not variations on a theme, they are different physical events measured on different lengths. A Reynolds number of 3,000 is transitional in a pipe, solidly laminar on a flat plate, and a turbulent wake behind a cylinder — all at once, for the same fluid at the same speed.

Should I turn the turbulence model off for a laminar case?

Yes. A turbulence model applied to a laminar flow will generate eddy viscosity from the mean shear whether or not the flow is turbulent, and the result is an overpredicted pressure drop and an overly flat velocity profile. If the case is laminar, solve it laminar — and if it is a transitional case that you want to resolve rather than model, that needs a transition model or a scale-resolving run, not a standard two-equation model.

My cylinder case is at Re = 500. Can I run it in 2-D?

Not reliably. The cylinder wake loses spanwise coherence through roughly Re = 190 to 300, so above about 190 a two-dimensional simulation cannot let the shedding decorrelate along the span and overpredicts the fluctuating lift, sometimes by a factor of two, while getting the Strouhal number roughly right. Below Re ≈ 190 the wake genuinely is two-dimensional and a 2-D transient laminar run is the correct and cheap choice — which is why that regime is the standard validation case.

Where does the entrance length come from, and why two formulae?

For laminar flow the hydrodynamic entry length is L/D = 0.05 to 0.06 times Re, depending on which text you take it from — the exact Langhaar solution sits at about 0.0575 Re, so both coefficients bracket it. For turbulent flow this page uses L/D = 4.4·Re^(1/6), which gives 20 to 30 diameters over the range Re = 10⁴ to 10⁵ and is where the familiar rule of thumb of about 10 to 30 diameters comes from. The consequence for CFD is a real one: if your inlet is closer than the entrance length to the region of interest and you specify a flat velocity profile, you are simulating the development and not the developed flow.

Which friction factor does the y+ figure use, and why are two printed?

The friction velocity and the y+ = 1 cell height use the Petukhov form, f = (0.79·ln Re − 1.64)^(−2), because it is explicit and stays within about 1 per cent of a converged Colebrook solution from Re = 10⁴ to 10⁸. The Blasius form 0.3164·Re^(−0.25) is printed beside it because it is the one everybody knows, but it is only good to about Re = 10⁵: it runs 6 per cent low at 3×10⁵, 14 per cent low at 10⁶ and 31 per cent low at 10⁷. Since uτ goes as √f, a 14 per cent error in f is a 7 per cent error in the first-cell height, which is enough to miss a carefully targeted y+ of 1. Both are SMOOTH-wall values; with real roughness use Colebrook or Haaland with your own roughness height, and note that the laminar branch 64/Re is the round-pipe value — a square duct is 56.9/Re and parallel plates 96/Re.

Should I use kinematic or dynamic viscosity?

Whichever you have. Re = ρUL/μ = UL/ν identically, because ν = μ/ρ. Kinematic is more convenient for the Reynolds number and for y+, dynamic is what a datasheet usually quotes and what you need for a stress. The pitfall is units, not physics: dynamic viscosity is often given in centipoise (1 cP = 10⁻³ Pa·s) and kinematic in centistokes (1 cSt = 10⁻⁶ m²/s), and a factor of a thousand in either direction is the commonest way to get a Reynolds number that is three orders of magnitude out.

The velocity — mean or maximum?

Bulk mean for internal flow: volumetric flow rate divided by the flow area. Not the centreline velocity, which in a laminar pipe is exactly twice the mean and in a turbulent one about 1.2 times it, so using it would raise the Reynolds number by 20 per cent to 100 per cent. For an external flow it is the free-stream velocity, undisturbed and upstream of the body.

Related calculators

References

K. Avila, D. Moxey, A. de Lozar, M. Avila, D. Barkley and B. Hof, The Onset of Turbulence in Pipe Flow, Science 333 (2011), 192–196. The source for the critical point Re = 2040 ± 10 for sustained turbulence in pipe flow, with Re defined on the mean velocity and the pipe diameter. Read directly rather than quoted second hand, because the number is routinely reported as a replacement for 2,300 when it answers a different question.
Y. A. Çengel and J. M. Cimbala, Fluid Mechanics: Fundamentals and Applications, and F. M. White, Fluid Mechanics: the conventional pipe bands (laminar below about 2,300, transitional to about 4,000, turbulent above), the hydraulic diameter 4A/P, the laminar entrance length coefficient of 0.05 to 0.06 and the Blasius smooth-wall friction factor 0.3164 Re^−1/4. Cited, not reproduced.
University of Iowa ME:5160 lecture notes, Turbulent Boundary Layer: “For typical commercial surfaces transition occurs at Re_trx = 5×10⁵. However, one can delay the transition to Re_trx = 3×10⁶ with care in polishing the wall”, with a critical value of 10⁵ below which transition is not observed. The source for the flat-plate band used here rather than a single number.
Purdue ME 30800 course reading, Pipe Flows: turbulent hydrodynamic entry length L/D = 4.4 Re^1/6, giving 20 to 30 diameters over Re = 10⁴ to 10⁵, and laminar L/D = 0.06 Re. Both used on this page.
KTH Mechanics, Flow regimes for a circular cylinder, and Thermopedia, Crossflow over circular cylinders. Two independent regime tables, used together precisely because they disagree on the boundaries — creeping flow to 3–5 against below 1, the pure Kármán street ending at 150–300 against 150, and the critical regime beginning at 1–1.3×10⁵ against 1.5×10⁵. The bands on this page take the middle of the published range and the notes state the spread rather than hiding it.
J. R. Southard, Introduction to Fluid Motions and Sediment Transport, chapter 3, and the standard drag-crisis literature (Achenbach; Roshko) for the sphere sequence: Stokes flow below Re = 1, separation from about Re = 20, loss of axisymmetry near 210, shedding from about 270, a C_D plateau near 0.4 to 0.5, and the drag crisis at 3 to 4×10⁵ for a smooth sphere with C_D falling to roughly 0.1.
V. T. Chow, Open-Channel Hydraulics, and standard open-channel teaching material: laminar below Re_R = 500, transitional 500 to 2,000 and turbulent above, quoted on the HYDRAULIC RADIUS R = A/P. Converted here to the hydraulic-diameter basis (2,000 and 8,000) and both bases printed, because mixing the two is a factor-of-four error in the direction that makes turbulent flow look laminar.
B. S. Petukhov’s explicit friction factor for smooth tubes, f = (0.79 ln Re − 1.64)⁻², as printed with the Gnielinski correlation for 3×10³ < Re_D < 5×10⁶. Checked here against a converged Colebrook iteration for a smooth wall at Re = 4×10³, 10⁴, 3×10⁴, 10⁵, 3×10⁵, 10⁶, 10⁷ and 10⁸: Petukhov agrees within 1.2 % above Re ≈ 1.7×10⁴, but runs high at the bottom of its range — +3.84 % at Re = 4×10³ and +1.93 % at 10⁴, so treat it as indicative in the first decade above transition. Blasius 0.3164 Re^−1/4, by contrast, is 2.5 % high at 10⁴ and 14, 31 and 47 % LOW at 10⁶, 10⁷ and 10⁸. Blasius is reported for familiarity; the friction velocity and the y+ = 1 height use Petukhov.
Y. A. Çengel and A. J. Ghajar, Heat and Mass Transfer, Table A–9 (air at 1 atm: ν = 1.470, 1.516 and 1.562 × 10⁻⁵ m²/s at 15, 20 and 25 °C) and Table A–3 (saturated water: μ = 1.138 and 1.002 × 10⁻³ Pa·s with ρ = 999.1 and 998.0 kg/m³ at 15 and 20 °C, hence ν = 1.139 and 1.004 × 10⁻⁶ m²/s). Every preset on this page is μ/ρ recomputed from those two tables rather than taken from a kinematic-viscosity column.

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