The y plus calculator above answers the question every CFD mesh starts with: how thin does the first cell against the wall need to be? The dimensionless wall distance y+ is the yardstick turbulence models use to decide whether they can see the viscous sublayer or whether they have to model it, and getting the first cell into the right y+ band is the difference between a wall shear stress you can trust and one that is quietly wrong.
The tool takes a velocity, a length scale and the fluid properties, computes the Reynolds number, applies a published skin friction correlation, converts that into a wall shear stress and a friction velocity, and inverts the y+ definition for a wall distance. It reports every intermediate step, because each one is an assumption you may want to replace. Arb Digital's Reynolds number calculator covers the first of those steps in more depth, including the transition thresholds for different geometries; this page carries it through to a mesh dimension.
The Two Regimes, and Why the Gap Between Them Matters
A turbulent boundary layer has structure. Right at the wall, viscosity dominates and the velocity rises linearly with distance: this is the viscous sublayer, which occupies roughly y+ below 5. Far from the wall, turbulent mixing dominates and the velocity follows a logarithmic profile: the log-law region, which begins somewhere around y+ of 30 and extends outward. Between them, from about 5 to 30, is the buffer layer, where neither description holds and both are wrong.
That structure gives you two legitimate strategies and one trap.
Resolving the boundary layer means putting cells inside the viscous sublayer so the solver computes the near-wall profile directly. This needs a first cell at y+ of about 1 or below, and it needs enough cells — typically ten to twenty — stacked across the boundary layer at a modest growth ratio, not just one thin cell followed by a jump. It is what low-Reynolds-number models such as k-omega SST are designed for, and it is what you need for accurate drag, heat transfer or separation prediction.
Wall functions take the opposite approach: put the first node out in the log-law region and use the analytical log profile to bridge the gap to the wall. This needs y+ roughly between 30 and 300, and it is dramatically cheaper because you are not resolving a layer that may be microns thick. The price is that you have assumed an equilibrium log-law profile, which stops being true near separation, reattachment, strong pressure gradients or stagnation points.
The trap is landing in the buffer layer, y+ between about 5 and 30, where the first cell is too coarse to resolve and too fine for a standard wall function. Some modern codes blend the two treatments to make this survivable, but a mesh deliberately targeted there is a mesh nobody chose. Aim at one regime or the other. The models these regimes belong to, and the validation cases used to test them, are catalogued in the NASA Turbulence Modeling Resource.
How to Use It
- Pick the flow type. External for a wing, a hull, a vehicle or a plate; internal for a pipe, a duct or a channel. They use different correlations and give different answers.
- Set the fluid. Use a preset for ordinary air or water, or enter your own density and dynamic viscosity for anything else.
- Enter velocity and reference length. Free-stream velocity and distance from the leading edge for external flow; bulk velocity and hydraulic diameter for internal flow.
- Set the target y+. One for a resolved layer, thirty or above for wall functions.
- Check the cell-height convention. The y+ definition refers to the distance of the first computational point from the wall. In a cell-centred finite-volume code that point is the centroid, so the cell you build is twice as tall.
The Formula and the Correlation It Rests On
The dimensionless wall distance is defined as y+ = ρ uτ y / μ, where uτ is the friction velocity, uτ = √(τw / ρ), and τw is the shear stress at the wall. Inverting it for the wall distance gives y = y+ · μ / (ρ uτ).
Everything therefore reduces to estimating τw, and τw = ½ ρ U² Cf where Cf is the skin friction coefficient. That is where a correlation has to be chosen, and this tool names both of the ones it uses.
For external flow it applies the Schlichting flat-plate relation for a turbulent boundary layer, Cf = (2 log10Rex − 0.65)−2.3, which is quoted as valid up to a Reynolds number of about 109. For internal flow it applies the Blasius smooth-pipe relation, Cf = 0.079 Re−1/4, which is established for smooth pipes roughly between Reynolds numbers of 4000 and 105.
Work the default through. Air at 1.225 kg/m³ and 1.789 × 10−5 Pa·s, at 20 m/s over 1 m, gives Re = 1.225 × 20 × 1 / 1.789e−5 = 1,369,480. Then log10Re = 6.1365, so Cf = (12.273 − 0.65)−2.3 = 11.623−2.3 = 0.0035462. Wall shear stress is 0.5 × 1.225 × 400 × 0.0035462 = 0.8688 Pa, friction velocity is √(0.8688/1.225) = 0.8422 m/s, and the wall distance for y+ = 1 is 1.789e−5 / (1.225 × 0.8422) = 1.734 × 10−5 m, about 17.3 micrometres. The full cell, centroid at that distance, is about 34.7 micrometres tall. That number is the reason wall-resolved external aerodynamics meshes are expensive.
This Sizes a Cell. It Does Not Validate a Mesh.
The most important limitation on the page. The estimate above comes from a flat-plate or smooth-pipe correlation applied to your reference length and velocity. Your actual geometry has curvature, pressure gradients, corners, wakes, separation and probably a stagnation point, and the real wall shear stress varies enormously across it. Near a leading edge or a stagnation point the local shear is far higher than the correlation predicts, so the same cell height yields a much larger y+ there; in a separated region the shear collapses towards zero and y+ falls with it.
What that means in practice is that the real y+ distribution is only known after the solution has converged. Every serious CFD workflow includes the same step: run the case, plot y+ over the wall surfaces, and check that the distribution actually lies in the band the turbulence model wanted. If it does not, remesh and run again. This page produces a starting point for that loop, not a result from it.
A related point about the reference length. For external flow the Reynolds number rises along the surface, so the friction coefficient falls and the required cell height grows with distance from the leading edge. If you size on the full body length you will be too coarse at the front. Sizing on a length near the leading edge, or simply on the smallest length of interest, gives a more conservative mesh. Some meshing tools do this automatically per-surface; many do not.
One Thin Cell Is Not a Resolved Boundary Layer
Hitting y+ of 1 with the first cell is necessary and nowhere near sufficient. A low-Reynolds-number turbulence model needs the whole near-wall region resolved, which in practice means something like ten to twenty cells within the boundary layer thickness, growing at a ratio around 1.1 to 1.2 so the transition into the bulk mesh is smooth.
A mesh with one micron-thin cell followed by a jump straight into millimetre cells has a wonderful y+ and a badly under-resolved boundary layer, and it will produce a plausible-looking but inaccurate wall shear stress. Aspect ratio matters too: extremely elongated cells at the wall are normal and acceptable, but a sudden change in growth ratio is not, and it shows up as a kink in the velocity profile.
The other half of the same discipline is knowing how thick the boundary layer is in the first place, so you know how many cells you need to fill it. For a turbulent flat-plate layer the thickness is roughly 0.37 x Rex−1/5, which for the default case gives about 22 mm at one metre — more than a thousand times the first cell height. That ratio is what sets the growth-ratio arithmetic.
Where the Fluid Properties Come In
The wall distance is directly proportional to the dynamic viscosity and inversely proportional to the density, so property errors propagate straight into the mesh. A water case run with room-temperature properties when the loop actually runs at 60 °C is using a viscosity more than twice too high, and will produce a first cell more than twice too thick.
Both presets here are ordinary reference conditions and both are easy to replace. For water at another temperature, the water viscosity calculator evaluates the reference formulation directly, and the water density calculator gives the matching density. For air, the air viscosity calculator applies Sutherland's law and the air density calculator handles pressure and temperature. If your properties arrive in poise or stokes, the viscosity converter rescales them into SI before you enter them here.
Arb Digital builds free tools like this one because genuinely useful pages earn attention. If you want calculators, tools or technical content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Treating the estimate as the answer — the real y+ distribution is a property of the converged solution and has to be read back off it, surface by surface.
- Confusing the wall distance with the cell height — in a cell-centred code the first node sits at the centroid, so the cell you build is twice the distance the y+ definition refers to.
- Landing in the buffer layer — between roughly y+ 5 and 30 the first cell is too coarse to resolve the sublayer and too fine for a standard wall function.
- Meeting y+ with a single thin cell — a resolved boundary layer needs ten to twenty cells across it at a controlled growth ratio, not one thin cell and a jump.
- Sizing on the full body length — friction is highest near the leading edge, so a mesh sized on the whole length is too coarse where it matters most.
Related Free Tools From Arb Digital
The Reynolds number this page starts from is covered in more depth, with the transition thresholds for different geometries, by the Reynolds number calculator. For fluid properties, use the water viscosity calculator, the air viscosity calculator, the water density calculator and the air density calculator, with the viscosity converter for unit changes. On the internal-flow side, the friction factor calculator and the pipe flow calculator take the same Reynolds number to a pressure drop, and for external flow the drag force calculator covers the force the surface actually feels. Everything Arb Digital publishes is indexed on the free online tools hub. Boundary layer theory itself is developed in MIT OpenCourseWare's 2.25 Advanced Fluid Mechanics and applied to wings in 16.100 Aerodynamics.
Frequently Asked Questions
It is a dimensionless wall distance, defined as the density times the friction velocity times the distance from the wall, divided by the dynamic viscosity. It measures where a point sits within the structure of a turbulent boundary layer. Turbulence models make different assumptions in different parts of that structure, so the y+ of the first cell decides whether the model is being used the way it was designed to be used.
Around one or below if you intend to resolve the viscous sublayer with a low-Reynolds-number model, and roughly thirty to three hundred if you intend to use standard wall functions and bridge the near-wall region with the log law. The band between about five and thirty is the buffer layer, where the first cell is too coarse to resolve and too fine for a wall function, and it should be avoided.
Usually not. The y+ definition refers to the distance of the first computational point from the wall, and in a cell-centred finite-volume code that point is the cell centroid. The cell you actually create is therefore twice that distance tall. Node-centred codes place the point on the cell face instead. Getting this convention wrong halves or doubles the near-wall mesh, which is exactly the size of error that matters here.
For external flow it uses the Schlichting flat-plate relation for turbulent skin friction, quoted as valid to a Reynolds number of about ten to the ninth. For internal flow it uses the Blasius smooth-pipe relation, established for smooth pipes between Reynolds numbers of roughly four thousand and one hundred thousand. Both are named on the page because the estimate depends entirely on which one is applied.
Because the correlation describes a flat plate or a smooth straight pipe, and your geometry has curvature, pressure gradients, corners and probably separation. The local wall shear stress varies across the surface, rising sharply near leading edges and stagnation points and collapsing in separated regions, so the same cell height produces very different y+ values in different places. Only the converged solution knows the real distribution.
No. It means the first cell is in the right place. Resolving the layer also requires enough cells stacked across its thickness, typically ten to twenty, at a growth ratio around 1.1 to 1.2 so the mesh blends smoothly into the bulk. One very thin cell followed by a jump into coarse cells satisfies the y+ check and still under-resolves the profile, producing a wall shear stress that looks plausible and is not.
The distance from the leading edge to the location you care about. Skin friction falls as that distance grows, so the required cell height increases along the surface. Sizing on the full body length gives a mesh that is too coarse near the front, where the friction is highest. Using a shorter length, or the smallest length of interest, produces a more conservative mesh at the cost of more cells.
Directly. The wall distance is proportional to the dynamic viscosity and inversely proportional to the density, so a property error goes straight into the mesh. Water viscosity more than halves between twenty and sixty degrees Celsius, so a hot loop meshed with room-temperature properties gets a first cell more than twice as thick as it needed, and the y+ it reports back after solving will be correspondingly higher.
This page produces a mesh sizing estimate from published correlations for idealised geometries. The y+ actually achieved is a property of the converged solution and must be verified on it before any result is relied upon.
