CFD Domain Size and Blockage Ratio Calculator
CFD Domain Size and Blockage Ratio Calculator
How far the inlet, outlet and side boundaries of an external-aerodynamics domain have to sit, and the blockage ratio that results — with the Maskell correction so that a confined domain gives you an error estimate instead of a scolding. Works the same arithmetic for a wind-tunnel model, and treats the downstream length separately, because wake decay sets it and it is nearly always the longest dimension.
Domain size, blockage ratio and the error it costs
A 4.5 m car, 2.2 m² frontal area, expected drag coefficient 0.32, inlet 3 body lengths upstream, outlet 7 downstream, side boundaries 3 from the centreline, top boundary 3 above, ground plane at the bottom, Maskell ε = 0.96, three-dimensional
Blockage, the Maskell correction, and the three independent boundary criteria
- B
- blockage ratio. Body frontal area over domain cross-sectional area in three dimensions; projected height over domain height in two, because a 2-D body has no area
- ε
- Maskell blockage factor. 0.96 is Maskell’s own value for sharp-edged bluff bodies and is also used for sub-critical circular cylinders; a published fit for rectangular prisms gives ε = 1.11 − 0.14(B/D). The correction is explicit in the MEASURED coefficient, which is why it needs no iteration
- ε·C_D·B
- the relative overprediction of drag. Note what it depends on: the PRODUCT of drag coefficient and blockage, so a single blockage threshold is only a threshold at one drag coefficient
- Δp/q
- the static-pressure difference the axial momentum balance requires between inlet and outlet when the side boundaries carry no shear. Derived below; it equals C_D·B exactly, and it does not depend on how far away the outlet is
- a
- equivalent body radius, √(A/π) in three dimensions and half the projected height in two. It is what makes the potential-decay criterion usable on a real shape
- u′/U
- velocity perturbation from the body’s own potential field at distance d. Exact for a sphere and a circular cylinder, and the right order for anything else. The exponents differ, and that difference is why 2-D domains are habitually too small
- Δu_max
- centreline velocity deficit in the far wake. Self-similar decay, so the constant cancels in any ratio: to halve it you multiply the downstream distance by 2^(3/2) = 2.83 in three dimensions and by 4 in two
Worked example
A 4.5 m car, 2.2 m² frontal area, expected drag coefficient 0.32, inlet 3 body lengths upstream, outlet 7 downstream, side boundaries 3 from the centreline, top boundary 3 above, ground plane at the bottom, Maskell ε = 0.96, three-dimensional
The cross-section. The domain is 2 × 3 × 4.5 = 27 m wide and, with a ground plane, 3 × 4.5 = 13.5 m high, so C = 364.5 m². Blockage B = 2.2/364.5 = 0.604%. Comfortably inside every published guideline.
Now the number that actually matters, which is not the blockage ratio. Maskell: the drag overprediction is ε·C_D·B = 0.96 × 0.32 × 0.00604 = 0.186%. A corrected drag coefficient of 0.3194 against a measured 0.32. Negligible, and you can say so with a figure rather than a rule of thumb.
INVERT IT, because this is what a reader with a confined domain needs. At this drag coefficient the blockage ratio that would cost 1% of drag is 1/(0.96 × 0.32) = 3.26%, and 3% of drag needs 9.77%. You have three to five times more room than the 1% rule of thumb implies — because that rule is the 1% contour for a body of drag coefficient 1.0, and this car is a third of that. A flat disc at C_D = 1.17 hits 1% of drag error at 0.89% blockage. One blockage threshold cannot serve both.
The outlet boundary condition, from an exact momentum balance rather than a correlation. Take the whole domain as a control volume. The side boundaries are slip or symmetry, so they carry no shear, and the drag on the body must therefore appear as a static-pressure difference between inlet and outlet: Δp·C = D = q·C_D·A, so Δp/q = C_D·B = 0.32 × 0.604% = 0.193%. Pinning the outlet at free-stream static pressure mis-specifies it by that much. Note what is NOT in that expression: the downstream distance. Moving the outlet further away does not reduce this error at all — only a wider domain does.
The inlet, which has a clean criterion of its own. The equivalent body radius is a = √(2.2/π) = 0.837 m. A sphere’s potential field perturbs the axial velocity by (a/d)³, so at d = 13.5 m the inlet sees 0.024% of the free stream — three body lengths is generous here. A 0.1% perturbation would need only d = 10a = 8.37 m, or 1.86 body lengths. In two dimensions the same 0.1% needs d = 31.6a: the same tolerance costs three times the distance, which is why 2-D domains built from 3-D habits are too small.
The outlet distance, which is a wake question and not a blockage question. Seven body lengths is 31.5 m, or 18.8 equivalent diameters. A self-similar axisymmetric wake’s centreline deficit decays as x⁻²ᐟ³, so relative to 5 diameters the deficit here is (18.8/5)^(−2/3) = 0.41 of what it was. The ratio needs no experimental constant because the constant cancels. To halve the deficit again you would need 2^(3/2) = 2.83 times the distance: 89 m, or 19.8 body lengths. In two dimensions the wake decays as x⁻¹ᐟ² and halving it takes four times the distance.
What that says about the guidance. COST Action 732 asks for 15 body heights downstream and the AIJ guidelines for 10; both ask for 5 upstream and 5 laterally, with a maximum blockage of 3%. The upstream figure is conservative by the potential-decay criterion — 2 body lengths is enough in three dimensions for 0.1%. The downstream figure is not conservative at all, because it is buying uniformity at the outlet plane and that is the slowest-decaying thing in the domain. Spend the boundary distance downstream and the cells sideways.
The finished box is 49.5 × 27 × 13.5 m, 18 043 m³. Take that to the mesh cell count estimator — and note that widening the domain to halve the blockage costs only large background cells, which is the cheapest volume in the entire mesh.
Drag overprediction from blockage, ε = 0.96 — the error is the product, not the blockage
| Blockage ratio | Cd 0.3 (car) | Cd 0.5 (van) | Cd 0.8 | Cd 1.0 | Cd 1.2 (building, cylinder) | Cd 2.0 |
|---|---|---|---|---|---|---|
| 0.5% | 0.14% | 0.24% | 0.38% | 0.48% | 0.58% | 0.96% |
| 1% | 0.29% | 0.48% | 0.77% | 0.96% | 1.15% | 1.92% |
| 2% | 0.58% | 0.96% | 1.54% | 1.92% | 2.30% | 3.84% |
| 3% | 0.86% | 1.44% | 2.30% | 2.88% | 3.46% | 5.76% |
| 5% | 1.44% | 2.40% | 3.84% | 4.80% | 5.76% | 9.60% |
| 10% | 2.88% | 4.80% | 7.68% | 9.60% | 11.52% | 19.20% |
Published domain-size guidance, and what each figure is actually buying
| Source | Upstream | Downstream | Lateral | Max blockage | What sets it |
|---|---|---|---|---|---|
| Franke et al., COST Action 732 (2007) | 5H | 15H | 5H | 3% | Urban and building flows; the downstream figure is set by wake redevelopment before the outlet |
| Tominaga et al., AIJ guidelines (2008) | 5H | 10H | 5H | 3% | Same class of problem, a shorter outlet distance accepted |
| Potential-decay criterion, derived here (3-D) | 10a for 0.1%, which is 1.9L here | not applicable | set by blockage | not applicable | The body’s own potential field, (a/d)³. Shows the 5H upstream figure is conservative in three dimensions |
| Potential-decay criterion, derived here (2-D) | 31.6a for 0.1%, which is 5.3L here | not applicable | set by blockage | not applicable | Decay is only (a/d)², so three times the distance for the same tolerance |
| Momentum balance, derived here | not applicable | does not help | the only lever | Cd·B sets Δp/q | The mean outlet pressure error is Cd·B whatever the outlet distance — width is the only fix |
Three independent criteria, three different boundaries
| Boundary | What constrains it | How it scales | Symptom when it is too close |
|---|---|---|---|
| Inlet | The body’s own potential field must have decayed below your tolerance | (a/d)³ in 3-D, (a/d)² in 2-D | The imposed uniform inlet profile fights the real non-uniform flow; total pressure at the inlet is not what you set |
| Side and top | Blockage: the walls accelerate the flow past the body and confine the wake | Error goes as Cd × B; B is a cross-section ratio | Drag over-predicted, base pressure raised, separation moved, wake asymmetry suppressed or locked to the walls |
| Outlet | Wake non-uniformity at a plane where you are imposing a uniform pressure | Peak deficit as x⁻²ᐟ³ in 3-D, x⁻¹ᐟ² in 2-D. The MEAN pressure error is Cd × B and does not decay at all | Drag that drifts as you move the outlet; reversed flow through a pressure outlet; convergence stalling |
Why one blockage threshold cannot work, and what to do when you cannot widen the domain
Blockage ratio is the right number to compute and the wrong number to judge. The quantity that has a universal threshold is not the blockage ratio B but the product of B and the drag coefficient, because that product is what the error is proportional to. Maskell’s correction for a body in a closed working section is C_D/C_D,corrected = 1 + ε·C_D·B, so the relative overprediction of drag is ε·C_D·B with ε close to one. Put numbers in it and the familiar guidance falls into place: 1% blockage on a body of drag coefficient 1.0 costs about 1% of drag, and 5% blockage costs about 5%. The rule of thumb is exactly right — in one column of the table. A modern car at drag coefficient 0.32 could run at 3.3% blockage for the same 1% error, and a flat disc at 1.17 hits 1% at 0.89%. If you take one thing from this page, take that the threshold you need is a value of C_D × B, and that the blockage ratio at which your own body crosses 1% is printed under the answer.
Maskell’s factor is about one, and that is not a coincidence. Take the whole domain as a control volume and balance axial momentum. The inlet and outlet carry the pressure and the momentum flux; the side boundaries, if they are slip or symmetry planes, carry no shear at all. The drag on the body therefore has to appear as a static-pressure difference between the two ends: Δp·C = D = q·C_D·A, so Δp/q = C_D·B exactly, with no empirical constant anywhere in it. That is the same group Maskell’s correction is built on, which is why his suggested ε = 0.96 sits where it does. It also tells you something the correction does not: if you pin a pressure outlet at free-stream static pressure, you have mis-specified it by C_D·B of dynamic pressure, and no amount of downstream distance reduces that, because the downstream distance does not appear in the expression. Only a larger cross-section does.
The inlet distance has a clean criterion that almost nobody uses. A body in otherwise uniform flow perturbs the velocity ahead of it through its own potential field, and for a sphere that perturbation is exactly (a/r)³ of the free stream on the axis, where a is the radius. Take the equivalent radius a = √(A/π) of your real body and the criterion becomes arithmetic: 0.1% perturbation at the inlet needs d = 10a, 1% needs d = 4.64a. On the worked example that is 8.4 m, under two body lengths. The published guidance of five body heights is therefore generous for a three-dimensional body. For a two-dimensional one it is not, because a cylinder’s disturbance decays only as (a/r)²: 0.1% now needs d = 31.6a, more than three times as far. Two-dimensional domains are habitually built too small, and this is why — the numbers get carried over from three-dimensional experience where the exponent is larger.
The downstream length is a wake problem and deserves its own argument. It is nearly always the longest dimension in the domain, and the reason is that wake decay is the slowest process in the flow. In the self-similar far wake the centreline velocity deficit falls as x⁻²ᐟ³ for an axisymmetric body and only x⁻¹ᐟ² for a two-dimensional one. Because those are similarity laws, ratios of them are exact and free of any experimental constant: to halve the centreline deficit you must multiply the downstream distance by 2^(3/2) = 2.83 in three dimensions and by 4 in two. That is a brutal exchange rate, and it is why guidance asks for 10 to 15 characteristic lengths downstream while asking for only 5 upstream. What the distance buys is uniformity ACROSS the outlet plane, so that a constant-pressure boundary condition is a reasonable thing to impose; it does not buy a smaller mean error, which as shown above is fixed by blockage. The practical test is the one everybody already knows: move the outlet and see whether the drag moves. If it does, the outlet was inside the wake.
The wind-tunnel case is the same arithmetic read backwards. “Is my tunnel model too big” is “what is B” with the working-section area in the denominator, and the answer is the same correction. Two differences are worth naming. A real tunnel has boundary layers on its walls, so the effective cross-section is smaller than the geometric one and the blockage is slightly worse than it looks — a displacement thickness of a few millimetres on each wall of a small tunnel is a real effect. And an open-jet tunnel behaves oppositely: the jet can expand, so the correction changes sign and is roughly a fifth of the closed-section value. The simple one-dimensional continuity estimate printed below, (1 − B)⁻² − 1 on dynamic pressure, is the crude upper bound for the closed case — it assumes the entire cross-section accelerates uniformly, which potential flow says it does not, so it over-reads. It is included because it is the calculation most people do in their heads, and because seeing it sit above the Maskell figure is the clearest way to understand what Maskell added.
What to do when the domain genuinely cannot be widened. Sometimes the walls are the problem — you are simulating a tunnel test, or a vehicle in a road tunnel, or a channel. Then model them. Keep the walls, give them the right boundary condition, and compare like with like: a confined simulation against a confined experiment is a valid validation, and applying a blockage correction to both is not. When the walls are only an artefact of a domain you cannot afford to enlarge, apply the Maskell figure, quote it explicitly, and be clear about what it does not fix. It corrects an integrated force. It does not correct the pressure distribution, the separation location, or the wake asymmetry that gives a bluff body its side force — all of which are suppressed by confinement and none of which a single scalar correction can restore. And check the cost of the alternative before accepting the correction: the cells you add by widening a domain are the largest cells in the mesh, and the mesh cell count estimator will usually show that doubling the width costs far less than the refinement box you already drew around the body.
Frequently asked questions
Is 5% blockage really the limit?
There is no single limit, and that is the point of the table above. The published maximum in the two main wind-engineering guidelines — COST Action 732 and the AIJ guidelines — is 3%, not 5%; 5% appears widely in the cylinder and vehicle literature. Both are blind to the drag coefficient, which is what the error is proportional to. At a drag coefficient of 1.0, 5% blockage costs about 4.8% of drag; at 0.32 it costs 1.5%; at 1.17 it costs 5.6%. Compute ε·C_D·B, decide what error you can live with, and read the blockage ratio off from that.
Does making the domain longer downstream reduce blockage?
No. Blockage is a cross-section ratio, so the streamwise length does not enter it at all — and neither does it enter the mean static-pressure error at the outlet, which is C_D·B by an exact momentum balance. Downstream length buys you uniformity across the outlet plane, so that imposing a constant pressure there is defensible. Those are different problems with different fixes: widen the domain to reduce blockage, lengthen it to make the outlet condition honest.
Why does the Maskell correction not need iterating?
Because it is written in terms of the MEASURED coefficient: C_D,corrected = C_D/(1 + ε·C_D·B). Everything on the right is known. Some presentations write the same relation with the corrected coefficient inside the bracket, and that version is implicit and does need iterating — but it is not the form Maskell’s own derivation gives and the difference is second order in C_D·B, which is a fraction of a per cent at any blockage you should be running at. The explicit form is used here and it is exact in its own terms.
How far upstream does the inlet really need to be?
Far enough that the body’s potential field has decayed below the tolerance you care about, which is a calculation rather than a habit. For a three-dimensional body the perturbation is (a/d)³ with a = √(A/π), so 0.1% needs d = 10a and 1% needs d = 4.64a. On a car that is under two body lengths for 0.1%. For a two-dimensional body it is (a/d)² and 0.1% needs d = 31.6a — three times as far for the same tolerance. If the inlet has to be closer than the criterion allows, use a velocity inlet rather than a total-pressure inlet, because imposing a velocity on a flow that is genuinely decelerating is a smaller lie than imposing a total pressure on it.
My drag changes when I move the outlet. What is wrong?
The outlet is inside the wake. A constant-static-pressure outlet forces a uniform pressure across a plane that physically has a strong transverse variation, and the reaction to that shows up on the body. The diagnostic is in the numbers here: if the outlet is fewer than about eight equivalent diameters downstream the wake is not self-similar yet. Move it out until the drag stops moving — remembering the exchange rate, that halving the remaining deficit costs 2.83 times the distance in three dimensions — or use a pressure-outlet formulation with a radial-equilibrium or averaged-pressure option so that only the MEAN is imposed. And check that no reversed flow is entering through the outlet, which is the same symptom with a louder alarm.
How does a symmetry plane change the blockage ratio?
It does not, if you are consistent. A symmetry plane on the body centreline halves the frontal area and halves the domain width, so the ratio is unchanged — which is the right answer, because the physics is unchanged. The mistake to avoid is halving one and not the other, which moves the blockage ratio by a factor of two in whichever direction you got it wrong. It does halve the cell count, which is the reason to use it, and it does forbid any asymmetric wake mode, which is the reason not to on a bluff body where the wake bistability is part of the answer.
Should I use the continuity estimate or the Maskell one?
Maskell, for anything with a separated wake, which is what “bluff body” means. The one-dimensional continuity estimate, (1 − B)⁻² − 1 on dynamic pressure, assumes the whole cross-section accelerates uniformly past the body. Potential flow says it does not — the disturbance decays away from the body — so continuity over-reads, and both figures are printed here so you can see by how much. It is still worth having, because it is the calculation people do in their heads and because it is a genuine upper bound: whatever the real solid-blockage effect is, it is smaller than that.
What about a 2-D case? The numbers look much worse.
They are worse, on all three criteria at once, and this is the single most useful thing to know about 2-D domain sizing. The potential disturbance decays as the square of distance rather than the cube, so the inlet needs about three times the distance for the same perturbation. The wake deficit decays as x⁻¹ᐟ² rather than x⁻²ᐟ³, so halving it costs four times the distance rather than 2.83. And blockage is a height ratio rather than an area ratio, so a body occupying a tenth of the domain height is 10% blockage in 2-D where the same proportion in each direction would be 1% in 3-D. Domain sizes carried over from 3-D experience are therefore far too small, and 2-D cylinder studies in the literature show exactly this: drag coefficients that vary by tens of per cent with domain width.
Related calculators
References
- E. C. Maskell, A Theory of the Blockage Effects on Bluff Bodies and Stalled Wings in a Closed Wind Tunnel, Aeronautical Research Council R&M 3400 / RAE Report Aero 2685 (1963). The origin of the correction used here. The form applied is CD/CD,corr = 1 + εCD(S/C), explicit in the measured coefficient; Maskell’s own suggested value for sharp-edged bluff bodies is ε = 0.96. Cited, not reproduced.
- Verification of ε: the value 0.96 appears independently in the sub-critical circular-cylinder blockage literature and in a Chalmers University thesis appendix that states Maskell’s own suggestion directly. A published fit for rectangular prisms gives ε = 1.11 − 0.14(B/D) over 0.75 < B/D < 3, so 0.83 to 1.01. The figure of 2.5 that circulates for "three-dimensional bluff bodies" could not be traced to Maskell or to any primary source and is not used here.
- J. Franke, A. Hellsten, H. Schlünzen and B. Carissimo (eds), Best Practice Guideline for the CFD Simulation of Flows in the Urban Environment, COST Action 732 (2007). Source of the 5H upstream, 15H downstream, 5H lateral domain and the 3% maximum blockage ratio quoted in the table. Cited, not reproduced.
- Y. Tominaga et al., AIJ guidelines for practical applications of CFD to pedestrian wind environment around buildings, Journal of Wind Engineering and Industrial Aerodynamics 96 (2008). Source of the 5H / 10H / 5H domain and the same 3% blockage limit. Cited, not reproduced.
- Derived for this page rather than taken from a source: the outlet pressure error. Taking the whole domain as a control volume with slip or symmetry side boundaries, axial momentum requires Δp·C = D = q·CD·A, so Δp/q = CD·B with no empirical constant and no dependence on the downstream distance. This is why Maskell’s ε comes out near unity, and it is the reason a domain that is lengthened rather than widened does not fix a drifting drag coefficient.
- Derived for this page: the inlet criterion. For a sphere of radius a in uniform flow the axial velocity perturbation is exactly U(a/r)³, and for a circular cylinder it is U(a/r)², from the standard doublet solutions. With an equivalent radius a = √(A/π), a 0.1% inlet perturbation therefore needs 10a in three dimensions and 31.6a in two — the factor of three between them being the reason two-dimensional domains are habitually undersized.
- H. Tennekes and J. L. Lumley, A First Course in Turbulence, and A. A. Townsend, The Structure of Turbulent Shear Flow, for the self-similar far-wake decay exponents: centreline velocity deficit as x−2/3 for an axisymmetric wake and x−1/2 for a plane wake. Cited for the exponents only. Because they are similarity laws, the RATIO reported on this page carries no experimental constant and is exact within similarity: halving the deficit costs a factor 23/2 = 2.83 in distance in three dimensions and 4 in two.
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/