Turbulence Inlet Conditions Calculator
Turbulence Inlet Conditions Calculator
Turbulence intensity, length scale, k, ε, ω and the eddy-viscosity ratio from one velocity and one length — with νt/ν checked against the band ANSYS Fluent actually names, because that is the number that tells you whether the inlet you just typed is sane.
Turbulence inlet conditions
Air at 20 °C entering a 100 mm duct at 10 m/s, fully developed, with the standard 0.07·D_h length scale
The six quantities, and the one constant that ties them together
- I
- turbulence intensity, u′/U, the ratio of the RMS velocity fluctuation to the mean velocity. Entered as a percentage everywhere in this trade and used as a fraction in every formula
- ℓ
- turbulent length scale in the MIXING-LENGTH definition, ℓ = Cμ^(3/4)k^(3/2)/ε. This is what Fluent and OpenFOAM mean by the words. CFX means k^(3/2)/ε, which is 6.09 times larger, and the classical mixing length Cμ·k^(3/2)/ε is 1.83 times smaller
- k
- turbulent kinetic energy per unit mass, m²/s². The factor 1.5 is the isotropy assumption: k = ½(u′² + v′² + w′²) becomes 1.5u′² when the three components are equal
- ε
- turbulent dissipation rate, m²/s³. What the k–ε family asks for
- ω
- specific dissipation rate, 1/s — dissipation per unit turbulent kinetic energy. What the k–ω and SST families ask for. ω = ε/(Cμ·k) exactly, so the two model families are never asking for different physics
- Cμ
- 0.09 in every standard two-equation model. It is the constant in νt = Cμ k²/ε and it is fitted to the log layer, where production balances dissipation and −u′v′ = Cμ^(1/2)k
- νt/ν
- eddy-viscosity ratio, identical to μt/μ. The one number that says whether the whole set is sane, because it is the only combination that has an expected magnitude
Worked example
Air at 20 °C entering a 100 mm duct at 10 m/s, fully developed, with the standard 0.07·D_h length scale
Reynolds number first, because the intensity depends on it: Re_Dh = U·D_h/ν = 10 × 0.1 / 1.516×10⁻⁵ = 65,963
Intensity from the duct correlation: I = 0.16 × 65,963^(−1/8) = 4.00 per cent. That is the number everyone half-remembers as “about 5 per cent for a pipe”, and it is only about 5 per cent near Re = 10⁴ — at Re = 10⁶ it is 2.8 per cent
Length scale: ℓ = 0.07 × 0.1 = 7 mm. The 0.07 is the maximum mixing length in fully developed turbulent pipe flow, not a tuning factor
Turbulent kinetic energy: u′ = 0.03997 × 10 = 0.3997 m/s, so k = 1.5 × 0.3997² = 0.2396 m²/s²
Dissipation: ε = Cμ^(3/4)·k^(3/2)/ℓ = 0.09^0.75 × 0.2396^1.5 / 0.007 = 2.753 m²/s³. And ω = ε/(Cμ·k) = 2.753/(0.09 × 0.2396) = 127.7 s⁻¹. Check it the other way: k^0.5/(Cμ^0.25·ℓ) = 0.4895/(0.5477 × 0.007) = 127.7. The two routes must agree, and if yours do not, one of the exponents on Cμ is wrong
NOW THE ONLY NUMBER THAT CAN BE SANITY-CHECKED. νt = Cμ·k²/ε = 0.09 × 0.2396²/2.753 = 1.877×10⁻³ m²/s, so νt/ν = 1.877×10⁻³/1.516×10⁻⁵ = 123.8. Fluent puts this ratio at 100 to 1000 for fully developed duct flow, and 123.8 sits just inside that
And here is the check worth doing once, by hand, so you trust the page. Substitute the two correlations into νt = Cμ^(1/4)k^(1/2)ℓ and everything cancels except the Reynolds number: νt/ν = Cμ^(1/4)·√1.5·0.16·0.07·Re^(7/8) = 0.007513·Re^(7/8). That reaches 100 at Re = 5.2×10⁴ and 1000 at Re = 7.2×10⁵. Fluent’s “100 to 1000 for high-Reynolds fully-developed duct flows” and the two correlations are therefore the same statement, arrived at independently — which is the strongest evidence available that neither is mistyped
The consequence nobody states: the eddy-viscosity ratio for a duct is not a fixed band, it grows as Re^(7/8). At Re = 10⁴ it is 24; at Re = 10⁷ it is 10,000. A guide that tells you 10 to 100 for internal flow is describing Re below about 5×10⁴ and nothing else
What the duct correlation actually produces, across the Reynolds numbers people run
| Re_Dh | I = 0.16·Re^(−1/8) | νt/ν with ℓ = 0.07·D_h | Is this inside Fluent’s 100 to 1000? |
|---|---|---|---|
| 1×10³ | 6.75 % | 3.2 | no — far below |
| 5×10³ | 5.52 % | 12.9 | no — below |
| 1×10⁴ | 5.06 % | 23.8 | no — below |
| 5×10⁴ | 4.14 % | 97.2 | just below |
| 1×10⁵ | 3.79 % | 178 | yes |
| 5×10⁵ | 3.10 % | 728 | yes |
| 1×10⁶ | 2.85 % | 1,336 | just above |
| 5×10⁶ | 2.33 % | 5,463 | no — above |
| 1×10⁷ | 2.13 % | 10,019 | no — far above |
Three definitions of ‘turbulent length scale’, and what each code means by the words
| Definition | Formula | Relative size | ℓ for this page’s worked example | Who uses it |
|---|---|---|---|---|
| Classical mixing length | Cμ·k^(3/2)/ε | 1.00 | 3.834 mm | textbook mixing-length theory |
| Mixing-length (turbulence) scale | Cμ^(3/4)·k^(3/2)/ε | 1.826 | 7.000 mm | ANSYS Fluent, OpenFOAM (turbulentMixingLength… inlets) |
| Dimensional (eddy) length scale | k^(3/2)/ε | 11.11 | 42.60 mm | ANSYS CFX eddy length scale |
Which pair your solver is asking for
| Solver and menu | Asks for | Take from this page |
|---|---|---|
| Fluent, k–ε — Intensity and Hydraulic Diameter | I (%) and D_h | enter your own numbers; Fluent applies the same 0.16·Re^(−1/8) and 0.07·D_h internally |
| Fluent, k–ε — K and Epsilon | k and ε | k and ε above |
| Fluent, k–ω / SST — K and Omega | k and ω | k and ω above |
| Fluent — Intensity and Viscosity Ratio | I (%) and μt/μ | the intensity and the headline figure |
| OpenFOAM — turbulentIntensityKineticEnergyInlet on k | intensity as a fraction | I above, divided by 100 |
| OpenFOAM — turbulentMixingLengthDissipationRateInlet on epsilon | mixingLength | ℓ above, mixing-length definition |
| OpenFOAM — turbulentMixingLengthFrequencyInlet on omega | mixingLength | ℓ above, same definition |
| Star-CCM+ — Turbulence Specification: Intensity + Viscosity Ratio | I and μt/μ | the intensity and the headline figure |
| CFX — Turbulence Intensity and Eddy Length Scale | I and the eddy length scale | the intensity, and the DIMENSIONAL ℓ above, not the mixing-length one |
Where these numbers come from, and the one of them that can be checked
Every inlet in every turbulent simulation needs two numbers, and only two. Give the solver a velocity scale for the turbulence and a length scale for it, and everything else on this page is algebra. The velocity scale is the intensity, I = u′/U; the length scale is ℓ. From those, k = 1.5(UI)² by the isotropy assumption, ε = Cμ^(3/4)k^(3/2)/ℓ by dimensional analysis with the equilibrium constant, ω = ε/(Cμ k) by definition, and νt = Cμ k²/ε by the eddy-viscosity hypothesis. The reason solvers offer four different input pairs is not that they need different information; it is that different people arrive holding different halves of the same two numbers.
The intensity is the easy one and the length scale is the one that is guessed. For a fully developed duct there is an honest correlation, I = 0.16·Re_Dh^(−1/8), which comes out of the Blasius friction law and is good to about a per cent over the usual range. For anything else the intensity is a measurement or a judgement: 1 to 5 per cent downstream of a grid, well under 1 per cent in a good tunnel, 5 to 20 per cent inside a compressor. The length scale has one honest correlation too — ℓ = 0.07·D_h for a developed duct, where 0.07 is the peak mixing length in fully developed pipe flow — and one more for a boundary layer arriving at the inlet, ℓ = 0.4·δ99. Outside those two cases it is a guess, and because νt is linear in ℓ while it only grows as √k, a length scale guessed a factor of three too large is an eddy viscosity three times too large.
The eddy-viscosity ratio is the only quantity on the page with an expected value, which makes it the only one you can check. k has no expected value; it is whatever U² and I² make it. ε has no expected value either. But νt/ν — identical to μt/μ, because the density cancels — is a pure number with a known magnitude for each kind of flow, and it is the quantity the solver will actually use. That is why this page puts it in the headline and everything else below it. The ANSYS Fluent user guide is explicit about both ends: “at the free-stream boundaries of most external flows, μt/μ is fairly small. Typically, the turbulence parameters are set so that 1 < μt/μ < 10”, and in “high-Reynolds-number boundary layers, shear layers, and fully-developed duct flows” it is “on the order of 100 to 1000”.
That internal band is not a constant, and this is the correction worth carrying away. Put the two duct correlations into νt = Cμ^(1/4)k^(1/2)ℓ and the velocity and the diameter both cancel: νt/ν = Cμ^(1/4)·√1.5·0.16·0.07·Re^(7/8), which is 0.007513·Re^(7/8). The ratio therefore grows almost linearly with Reynolds number. It is 24 at Re = 10⁴, 178 at Re = 10⁵, 1,336 at Re = 10⁶ and about 10,000 at Re = 10⁷ — all of them correct. Fluent’s “100 to 1000” corresponds exactly to Re_Dh between 5.2×10⁴ and 7.2×10⁵, which is what “high-Reynolds-number duct flow” meant when that sentence was written. So a ratio of 10⁴ is a red flag on an aerofoil and entirely right on a large water main. Judge it against the reference figure this page prints for your own Reynolds number, not against a remembered band.
The word ‘length scale’ means three different lengths depending on which manual you are holding. Fluent and OpenFOAM mean Cμ^(3/4)k^(3/2)/ε. CFX’s eddy length scale means k^(3/2)/ε, which is 6.09 times larger. The classical mixing length is Cμ·k^(3/2)/ε, 1.83 times smaller than Fluent’s. Nothing in any of the three interfaces tells you which you are typing into, and there is no dimensional check that can catch it because all three are lengths. All three are printed under the answer here for exactly that reason.
Two last cautions that cost people days. First, the inlet condition decays: turbulence set at a boundary five diameters upstream of the thing you care about will not be the turbulence that arrives there, and for a low-turbulence external case the decay between the inlet plane and the model is usually larger than the difference between any two sensible inlet values. Set the inlet so that the intensity is right AT THE MODEL. Second, ℓ must be smaller than the passage: a length scale larger than the hydraulic diameter asks the model for eddies bigger than the duct that contains them, which is not a physical boundary condition and will show up as an inlet region that behaves like a reservoir.
Frequently asked questions
Which of these six numbers does my solver actually want?
Fluent’s k–ε menu will take intensity and hydraulic diameter, intensity and length scale, intensity and viscosity ratio, or k and ε outright; its k–ω and SST menus take k and ω. OpenFOAM uses turbulentIntensityKineticEnergyInlet on k (intensity as a FRACTION, not a percentage), turbulentMixingLengthDissipationRateInlet on epsilon and turbulentMixingLengthFrequencyInlet on omega, both of which take a mixing length. Star-CCM+ defaults to intensity plus turbulent viscosity ratio. CFX asks for intensity and an eddy length scale, which is a different definition of length scale — see the table above. All of them are the same two pieces of information in different clothing.
Is Cμ really 0.09, and does it change between models?
0.09 is the standard value in the standard k–ε model, the RNG and realizable variants use it in the same places for the boundary-condition relations, and the k–ω family uses β* = 0.09 in exactly the role Cμ plays here, which is why ω = ε/(Cμ k) works across the lot. Its origin is the log layer: where production balances dissipation, the shear stress satisfies −u′v′ = Cμ^(1/2) k, and measured log layers give −u′v′/k ≈ 0.3, so Cμ ≈ 0.09. The realizable model makes Cμ a variable in the interior of the flow, but the inlet relations on this page still use 0.09.
My eddy-viscosity ratio comes out at 10,000. Is that wrong?
Only if your Reynolds number is modest. For a fully developed duct the ratio grows as Re^(7/8): 0.007513·Re^(7/8) with the standard correlations. At Re_Dh = 10⁷ that is about 10,000 and it is correct. At Re_Dh = 10⁵ the same reading would mean your length scale is roughly 56 times too large. The page prints the fully developed reference value for your own Reynolds number beside the answer precisely so you can tell the two cases apart — a fixed band cannot.
What intensity should I use for a wind tunnel or an external aerodynamics case?
Whatever the tunnel measures, usually 0.05 to 0.5 per cent, and for a genuinely quiet tunnel below 0.1 per cent. Then choose the length scale to put νt/ν between 1 and 10, which is the range Fluent names for free-stream boundaries; the page prints the two lengths that land it at 1 and at 10. The bigger trap is decay: k falls between the inlet plane and the model, and with a long upstream domain the free stream that reaches the body can be an order of magnitude quieter than the one you specified. If you are using a transition model, this is not a detail — it is the whole answer.
Why is my ω enormous — a hundred or more per second?
Because ω has units of 1/s and is the inverse of a turbulent time scale, so a small length scale makes it large. ω = √k/(Cμ^(1/4)ℓ), and with a 7 mm length scale and k of 0.24 m²/s² it is 128 s⁻¹. Values of 10 to 10⁵ s⁻¹ are all routine. There is nothing to fix unless the length scale itself is wrong: check the eddy-viscosity ratio instead, which is the quantity with a known magnitude.
Where does 0.16·Re^(−1/8) actually come from?
From the Blasius friction law and one measured number. Blasius gives the Darcy friction factor of a smooth pipe as 0.3164·Re^(−1/4), so the friction velocity is uτ/U = √(f/8) = 0.1989·Re^(−1/8). The measured axial turbulence intensity at the centre of a fully developed pipe is u′/uτ ≈ 0.80. Multiply the two: u′/U = 0.80 × 0.1989 × Re^(−1/8) = 0.159·Re^(−1/8), which is 0.16 to two figures, and the reconstruction tracks the published correlation to better than 0.6 per cent from Re = 10⁴ to 10⁶. The −1/8 exponent is not empirical either: it is half the −1/4 in Blasius, because intensity goes as √f. One consequence follows immediately — the correlation inherits Blasius’s range, so above about Re = 10⁵ it is extrapolating and reads slightly low.
Where does the 0.07 in 0.07·D_h come from?
From Nikuradse’s measured mixing-length distribution in fully developed turbulent pipe flow, which peaks at the centreline at l_m/R = 0.14. Since R = D/2 that is l_m/D = 0.07, so the coefficient is a measurement of the largest mixing length a pipe contains rather than a tuning constant — which is what the Fluent documentation says, and it can be checked independently of Fluent. Near the wall the same distribution reduces to l_m = κy with κ ≈ 0.41. The practical reading: an inlet length scale of about 7 per cent of the passage is right, and one approaching the passage size itself is not physical.
Does the density matter?
Not for anything on this page. k, ε, ω and νt are all per unit mass or kinematic, and the eddy-viscosity ratio νt/ν equals μt/μ exactly, because the density cancels top and bottom. You need the density to turn νt into μt if your solver reports the dynamic value, and for nothing else. Only the kinematic viscosity is asked for here.
Can I just leave the solver defaults alone?
For an internal flow at a reasonable Reynolds number, often yes — the answer a few diameters downstream is set by the walls and not by the inlet, and this is why so many duct cases are insensitive to it. For an external case, a transition model, a short domain, a free shear layer or anything where the inlet is close to the region of interest, no. The test is cheap: run the case twice with the length scale a factor of three apart and see whether anything you care about moves. If it does, the inlet condition is part of your answer and needs to be justified rather than defaulted.
What length scale should I use downstream of a grid or screen?
Start at the order of the mesh or hole pitch — the openings set the size of the eddies that are shed — and then check where νt/ν lands rather than trusting the coefficient, because the integral scale grows with distance downstream while the intensity decays. Grid turbulence genuinely produces large eddy-viscosity ratios: 3 per cent intensity at 10 m/s with a 1 cm scale gives a ratio of about 130 in air, which looks like duct turbulence because in this respect it is. That is why this page judges the grid case against the internal band and not the free-stream one.
Related calculators
References
- ANSYS FLUENT User’s Guide, Determining Turbulence Parameters (Turbulence Intensity; Turbulence Length Scale and Hydraulic Diameter; Turbulent Viscosity Ratio). The source for I = 0.16 ReDh−1/8, for ℓ = 0.07 L with the note that “the factor of 0.07 is based on the maximum value of the mixing length in fully-developed turbulent pipe flow”, for ε = Cμ3/4k3/2/ℓ and ω = k1/2/(Cμ1/4ℓ), and for both viscosity-ratio statements quoted on this page: “1 < μt/μ < 10” at free-stream boundaries and “on the order of 100 to 1000” in fully-developed duct flows. Cited, not reproduced.
- OpenFOAM, turbulentMixingLengthDissipationRateInlet and turbulentMixingLengthFrequencyInlet boundary conditions, source documentation. Confirm εp = Cμ0.75k1.5/L and ωp = k0.5/(Cμ0.25L) with a default Cμ of 0.09 — the same mixing-length definition Fluent uses, checked against the code documentation rather than assumed.
- CFD-Online Wiki, Turbulence length scale. The source for the three competing definitions and their ratios: classical Cμk3/2/ε, mixing-length Cμ3/4k3/2/ε (Fluent, OpenFOAM) and dimensional k3/2/ε (CFX), together with the boundary-layer rules of thumb 0.22 δ99 and 0.4 δ99 for the classical and mixing-length definitions. The ratios 1.826 and 11.11 are recomputed here from Cμ = 0.09 and agree exactly.
- Y. A. Çengel and A. J. Ghajar, Heat and Mass Transfer, Table A–9 (properties of air at 1 atm) and Table A–3 (saturated water). The source for the kinematic viscosities offered: air 1.470, 1.516 and 1.562 × 10−5 m2/s at 15, 20 and 25 °C; 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, giving ν = 1.139 and 1.004 × 10−6 m2/s.
- Independent derivation of both coefficients, performed for this page rather than taken on trust. (1) The intensity correlation is the Blasius friction law plus one measured quantity: uτ/U = √(f/8) with f = 0.3164 Re−1/4 gives 0.1989 Re−1/8, and the measured centreline value u′/uτ ≈ 0.80 for fully developed pipe flow gives 0.159 Re−1/8 — within 0.6 % of the published 0.16 Re−1/8 at Re = 104, 5×104, 105 and 106. (2) The 0.07 is Nikuradse’s measured mixing length in a pipe, which peaks at lm/R = 0.14 at the centreline and hence lm/D = 0.07 (CFD Direct, CFD General Principles — Mixing length).
- Independent numerical check performed for this page. Substituting I = 0.16 Re−1/8 and ℓ = 0.07 Dh into νt = Cμ1/4k1/2ℓ gives νt/ν = 0.0075132 Re7/8, which crosses 100 at Re = 5.17×104 and 1000 at Re = 7.18×105. Fluent’s stated band for fully-developed duct flow and the two correlations are therefore mutually consistent, which is the check that establishes no coefficient on this page has been mistyped.
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