Entrance Length Calculator
Entrance Length Calculator
How much straight inlet duct you need before the profile is developed — hydrodynamic and thermal, laminar and turbulent, for pipes and ducts — with the spread across the published correlations shown rather than hidden, because it is a disagreement about the word “developed” and not about the physics.
Hydrodynamic and thermal entrance length for a pipe or duct
Water at 20 °C at 2 m/s in a 50 mm bore pipe, with 500 mm of straight inlet duct available and a constant-wall-temperature tube at the 5 per cent criterion
Two entrance lengths, four correlations, and the definition that makes them disagree
Turbulent hydrodynamic: Lh/Dh = 4.4 Re1/6, or 1.359 Re1/4, or 10
Laminar thermal: Lt/Dh = C Re Pr, with C = 0.0335 (constant Tw) or 0.0433 (constant q″) at the 5 per cent criterion
Turbulent thermal: Lt ≈ Lh, nearly independent of Pr
Fully developed: Nu = 3.657 (constant Tw), Nu = 48/11 = 4.364 (constant q″), f·Re = 64 (round pipe)
- L_h
- hydrodynamic entrance length: the distance over which the velocity profile settles. Conventionally measured to where the centreline velocity is within a small tolerance of its developed value, and the tolerance is why three coefficients exist
- L_t
- thermal entrance length: the distance over which the local Nusselt number settles to its developed value. Depends on Pr as well as Re, and on the wall condition
- Re
- UD_h/ν on the bulk mean velocity. Not the centreline velocity — in laminar pipe flow that is exactly twice the mean, and the entrance length is linear in Re
- Pr
- c_pμ/k. It multiplies the laminar thermal entrance length directly, so water at Pr = 7 needs seven times the duct that its velocity profile needs, and engine oil at Pr = 10³ needs a thousand times
- 0.0335 / 0.0433
- the thermal coefficients at the 5 per cent criterion, obtained here by solving for the station at which the local Nu falls within 5 per cent of its developed value. They reproduce the 0.033 and 0.043 usually printed, which identifies what tolerance those printed values correspond to — something the textbooks rarely state
- Nu = 3.657
- the fully developed Nusselt number for constant wall temperature in a round pipe, the first eigenvalue of the Graetz problem. For constant heat flux it is exactly 48/11 = 4.3636, which is analytic
- 4.4 Re^(1/6)
- the turbulent correlation used across this site. It grows very slowly with Re, which is why turbulent entrance lengths are quoted as a flat number of diameters despite depending on Re at all
Worked example
Water at 20 °C at 2 m/s in a 50 mm bore pipe, with 500 mm of straight inlet duct available and a constant-wall-temperature tube at the 5 per cent criterion
ν = 0.0010014/998.21 = 1.003196×10−6 m²/s, so Re = 2 × 0.05 / 1.003196×10−6 = 99,681. That is well past 2300, so the flow is turbulent and the laminar correlations do not apply.
Lh/D = 4.4 Re1/6 = 4.4 × 99,6811/6. 99,6811/6 = 6.8093, so Lh/D = 29.96 diameters and Lh = 29.96 × 0.05 = 1.498 m.
You have 500 mm. 0.5/1.498 = 0.334, so the inlet duct is a third of what the profile needs and the flow arriving at your region of interest is still developing. You need a further 998 mm of straight pipe, or a developed profile imposed at the inlet.
For contrast, the other correlation: 1.359 Re1/4 = 1.359 × 17.7665 = 24.15 diameters = 1.207 m, 19.4 per cent shorter. And the flat rule of thumb gives 10 D = 500 mm, which would have told you the inlet was exactly adequate. That is the difference between a defensible answer and a comfortable one.
If this were laminar at the same velocity — say the same pipe with a fluid 50 times more viscous, Re = 1994 — the entrance length would be 0.06 × 1994 = 120 diameters = 6.0 m. Four times longer at a fiftieth of the Reynolds number. That non-monotonic jump at Re = 2300 is the cliff in the chart below.
Thermal. The flow is turbulent, so Lt ≈ Lh ≈ 30 D and the Prandtl number barely matters. Had it been laminar at Re = 1994 in water at Pr = 7.00, the thermal length at the 5 per cent criterion would be 0.0335 × 1994 × 7.00 = 468 diameters = 23 m — nearly four times the hydrodynamic length, and the reason laminar liquid heat exchangers are almost never thermally developed.
Why the laminar coefficient is not one number
| Re | 0.05 Re | 0.0567 Re + 0.619/Re (Chen) | 0.06 Re | Spread, largest over smallest |
|---|---|---|---|---|
| 100 | 5.0 D | 5.7 D | 6.0 D | 20 % |
| 500 | 25.0 D | 28.4 D | 30.0 D | 20 % |
| 1000 | 50.0 D | 56.7 D | 60.0 D | 20 % |
| 2000 | 100.0 D | 113.4 D | 120.0 D | 20 % |
| 2300 | 115.0 D | 130.4 D | 138.0 D | 20 % |
The turbulent spread is worse, and the flat rule of thumb is optimistic everywhere
| Re | 4.4 Re^(1/6) | 1.359 Re^(1/4) | Flat 10 D rule | Ratio, first over second |
|---|---|---|---|---|
| 4,000 | 17.5 D | 10.8 D | 10 D | 1.62 |
| 10,000 | 20.4 D | 13.6 D | 10 D | 1.50 |
| 30,000 | 24.5 D | 17.9 D | 10 D | 1.37 |
| 100,000 | 30.0 D | 24.2 D | 10 D | 1.24 |
| 300,000 | 36.0 D | 31.8 D | 10 D | 1.13 |
| 1,000,000 | 44.0 D | 43.0 D | 10 D | 1.02 |
| 10,000,000 | 64.6 D | 76.4 D | 10 D | 0.85 |
Thermal entrance length: what “developed” costs, solved from the Graetz problem
| Criterion on the local Nusselt number | Constant wall temperature, L_t/(D·Re·Pr) | Constant heat flux, L_t/(D·Re·Pr) | Ratio, flux over temperature |
|---|---|---|---|
| Within 5 % of developed | 0.0335 | 0.0433 | 1.29 |
| Within 2 % of developed | 0.0466 | 0.0614 | 1.32 |
| Within 1 % of developed | 0.0570 | 0.0757 | 1.33 |
| Fully developed Nusselt number | 3.657 | 48/11 = 4.3636 | 1.19 |
How long the inlet duct has to be, why the published answers differ by 60 per cent, and the one thing a CFD user gets wrong here
The question is practical and the failure is silent. A CFD user builds a pipe or duct model, puts a uniform velocity inlet at one end, and places the thing they actually want to predict — a bend, an orifice, a heated section, a measurement plane — some distance downstream. If that distance is shorter than the entrance length, the flow arriving at the region of interest is still developing, and the solver will converge cleanly and report a wrong answer with no indication that anything is amiss. The errors are systematic, not random: a developing profile has higher wall shear than a developed one, a steeper pressure gradient, higher heat transfer, and a different response to a bend. Extending the inlet duct is usually the cheapest fix in CFD. Not knowing you needed to is the problem.
The published correlations differ by 20 per cent in laminar flow and 60 per cent in turbulent flow, and that is a disagreement about a word. ‘Developed’ means the profile has stopped changing, which it never exactly does; in practice it means the centreline velocity is within some tolerance of its asymptote, and different authors choose different tolerances without saying so. In laminar flow you meet 0.05 Re, 0.0567 Re and 0.06 Re, which are the same law with three constants. In turbulent flow you meet 4.4 Re1/6, 1.359 Re1/4 and a flat 10 D, which have three different shapes and cross each other. This page prints all of them rather than choosing quietly, and defaults to the ones the Reynolds number page and the pipe friction page already use, so the three pages cannot contradict one another.
The laminar-to-turbulent cliff is the most surprising thing about entrance lengths. A laminar entrance length is proportional to Reynolds number and reaches 138 diameters at Re = 2300. The instant the flow goes turbulent, at the same Reynolds number, it drops to 16.0 diameters. A factor of 8.6 for an infinitesimal change in Reynolds number, because turbulent mixing transports momentum across the pipe far more effectively than viscosity can. This is why turbulent entrance lengths get quoted as a flat number of diameters at all: the Re1/6 dependence is so weak that the answer only moves from 20 to 65 diameters across three decades of Reynolds number. And it is why laminar internal flow is usually modelled with an imposed parabolic inlet rather than a meshed developing region — the developed laminar profile is known exactly, so there is nothing to gain from computing it.
The thermal entrance length is longer than the hydrodynamic one in every liquid, and it is usually the one that binds. In laminar flow it carries a factor of Pr: water at Pr = 7.00 needs seven times the duct its velocity profile needs, and engine oil at Pr around 103 needs a thousand times. This is why laminar liquid heat exchangers are essentially never thermally developed, and why using a developed Nusselt number of 3.657 or 4.364 for one will badly understate the heat transfer — the local Nusselt number in the developing region is several times its developed value. The wall condition matters too, and by a definite amount: constant heat flux takes about 30 per cent longer than constant wall temperature, because it is approaching a higher developed Nusselt number (48/11 = 4.364 against 3.657). In turbulent flow the Prandtl number dependence largely disappears and the two entrance lengths are comparable.
What to do about it in a model. Three options, in increasing order of effort and accuracy. Extend the inlet duct by the length printed above — simple, always defensible, and for turbulent flow at 20 to 40 diameters usually cheap enough. Impose the developed profile at the inlet — free for laminar flow where the profile is analytic, and reasonable for turbulent flow from a power law or a log law, though it will not give you the right turbulence quantities. Or recycle: map the outlet profile back to the inlet, or run a short translationally periodic section as a precursor and impose its solution. The third is the only one that gets the turbulence right as well as the mean profile, and the mean profile is not the slow part: the Reynolds stresses take tens of diameters longer than the mean velocity to settle. Whichever you pick, say so in the report, because the geometry alone does not tell a reader what was assumed. For a non-round section, get Dh from the hydraulic diameter page first, and for the boundary-layer view of the same thing — the displacement thickness growing until the two walls see each other — see the shape-factor page.
Frequently asked questions
Which laminar coefficient should I use, 0.05 Re or 0.06 Re?
Either, as long as you say which. They are the same law with different tolerances built in, and the 20 per cent between them is smaller than the uncertainty in most things you would do with the answer. If you want the best-founded number, Chen’s 0.0567 Re + 0.619/Re is a fit to a numerical solution of the developing flow rather than a rule of thumb, and it sits between the two textbook values. This site defaults to 0.06 Re because that is what the Reynolds number and pipe friction pages already use, and having three pages agree is worth more than the third figure.
Why is the turbulent entrance length so much shorter than the laminar one?
Because turbulent mixing moves momentum across the pipe far more effectively than viscosity does, so the wall’s influence reaches the centreline much sooner. In laminar flow the information travels by molecular diffusion and the time it takes scales with D²/ν, which is why the length scales with Re. In turbulent flow it travels by eddies whose size scales with the pipe itself, so the length is a nearly fixed number of diameters. At Re = 2300 the two differ by a factor of 8.6, which produces the discontinuity you can see in the chart above. The discontinuity is real in the sense that the two regimes genuinely behave differently; what is not real is the sharpness, because between 2300 and 4000 the flow is intermittent and neither correlation applies.
Is 10 diameters enough?
Almost never, for CFD. Ten diameters is a flow-meter installation rule, where a few per cent of profile distortion is acceptable and where the meter has been calibrated with a known amount of it. The two real turbulent correlations give 20 and 14 diameters at Re = 104 and 30 and 24 at 105, so 10 D is between a half and a third of what the profile actually needs, and in laminar flow it is nowhere at all. If you want a single number to remember for turbulent CFD, use 40 diameters; it covers the whole range up to Re = 106 on the more conservative correlation.
Does the mean velocity profile being developed mean the turbulence is developed?
No, and this is the limitation none of the correlations state. The mean profile settles first; the Reynolds stresses, the turbulent kinetic energy distribution and the near-wall structure continue to adjust for a further 20 to 50 diameters, depending on what the inlet turbulence looked like. If what you are predicting depends on turbulence rather than on the mean profile — mixing, a scalar variance, heat transfer, noise, particle dispersion — then the entrance length here is a lower bound and you should either double it or use a periodic precursor. The turbulence inlet page is about how much this matters at the inlet in the first place.
Where do 0.033 and 0.043 for the thermal length come from?
From the Graetz problem: the temperature field developing in an already-parabolic velocity profile. The local Nusselt number approaches 3.657 for constant wall temperature and 48/11 for constant heat flux, and the entrance length is the station at which it gets close enough. Solving for the station at which Nu is within 5 per cent gives 0.0335 and 0.0433 — which are the printed 0.033 and 0.043. So the usual printed values are the 5 per cent criterion, which almost nobody says. Ask for 1 per cent instead and the coefficients become 0.0570 and 0.0757, 70 per cent longer. That is what makes a thermal entrance length a range.
My duct is a rectangle. Can I really just use the hydraulic diameter?
It is the standard approximation and it is what the correlations expect, but it is an approximation. A rectangular duct does not develop the same way a round pipe of the same Dh does: the corners are slow, the mid-wall regions are fast, and secondary flows develop in the corners that have no analogue in a pipe. A reasonable proxy for the size of the error is how far the laminar friction constant f·Re departs from the round-pipe 64 for your aspect ratio — 62.2 at 2:1, 72.9 at 4:1, 82.3 at 8:1, all computed on the hydraulic diameter page. Expect the same order of error here, and more at high aspect ratio.
Should I mesh the developing region or impose a profile?
Impose a profile if you know it and the developing region is not what you are predicting. For laminar flow that is a clear win: the developed profile is exactly parabolic in a round pipe and available in closed form for other sections, so meshing a hundred diameters of pipe to rediscover it is wasted effort. For turbulent flow it is less clear, because an imposed power-law or log-law profile comes with guessed turbulence quantities that then have to relax anyway. The best answer for turbulent flow is a periodic or mapped inlet, which costs a short extra domain and gets both the mean profile and the turbulence right. The worst answer is a uniform inlet close to the region of interest, which is also the most common.
Is the entrance length affected by the inlet geometry?
Yes, and none of these correlations know about it. They assume a uniform velocity entering a straight duct. A sharp-edged inlet produces a separation and a vena contracta that lengthen the process; a bell-mouth shortens it; a bend or a valve upstream can leave a swirl or an asymmetry that persists for a hundred diameters and more, far longer than the profile itself takes to develop. Swirl decay in particular is slow: a single elbow can leave measurable swirl 50 diameters downstream. If your real installation has fittings upstream, the entrance length here is the optimistic case — see the minor loss page for what those fittings cost in pressure, which is a different question from what they cost in profile.
Related calculators
References
- R.-Y. Chen (1973), Flow in the entrance region at low Reynolds numbers, J. Fluids Eng. 95, 153–158, for Lh/D = 0.0567 Re + 0.619/Re. Cited as an equation, not reproduced as a table; it is the only one of the three laminar options that is a fit to a solution of the developing-flow problem rather than a rule of thumb, and it lands between the two textbook constants.
- Zhi-qing Wang (1982) for Lh/D = 4.4 Re1/6, and M. S. Bhatti and R. K. Shah (1987) for 1.359 Re1/4. The first is the correlation already in use on the Reynolds number page and the pipe friction page, which is why it is the default here.
- R. K. Shah and A. L. London, Laminar Flow Forced Convection in Ducts, for the Graetz-problem eigen-solutions. No table is reproduced. The thermal coefficients on this page were obtained by solving for the station at which the local Nusselt number falls within 5, 2 and 1 per cent of its developed value, using Shah’s local-Nu representations, and the 5 per cent results 0.0335 and 0.0433 reproduce the widely printed 0.033 and 0.043 — which is both the verification and the finding, because it identifies the tolerance those printed numbers silently assume.
- The fully developed Nusselt numbers are the first eigenvalue of the Graetz problem, 3.6568, for constant wall temperature, and the exact 48/11 = 4.36364 for constant heat flux. The second is analytic and was rederived rather than quoted; the first is checked against it by the requirement that the constant-flux value exceed it, which it does by 19.3 per cent.
- K. Avila et al. (2011), The onset of turbulence in pipe flow, Science 333, 192–196, and the 2300 transition threshold used on the Reynolds number page. This page switches correlations at the same 2300 rather than at a different number, so the regime label on one page and the arithmetic on the other cannot disagree.
- NIST Chemistry WebBook (US Government) for water density and viscosity at 0.101325 MPa, and the US Standard Atmosphere 1976 for air via Sutherland’s formula. Prandtl numbers were computed as cpμ/k from those same viscosities: 7.00 for water at 20 °C and 0.712 for air, both matching the standard published values, which confirms the property set is self-consistent.
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/