# Y+ Calculator (First Cell Height for CFD)

URL: https://speedycalculator.com/calculators/physics/yplus/
Last updated: 2026-10-30

> The y+ calculator estimates the wall distance of the first mesh cell needed to achieve a target y+ in a CFD simulation, from the free-stream velocity, fluid properties, reference length and a skin-friction correlation.

## Inputs

- Free-stream velocity U (m/s, km/h, mph, ft/s, knots)
- Density ρ (kg/m³, g/cm³, g/mL, lb/ft³, lb/in³)
- Dynamic viscosity μ (Pa·s, mPa·s)
- Reference length L (m, cm, mm, km, in, ft, yd, mi)
- Desired y⁺
- Skin friction correlation

## Results

- Reynolds number
- Skin friction coefficient
- Wall shear stress
- Friction velocity u*
- First cell height

## Formula

```
Re = ρUL/μ,  C_f from correlation,  τ_w = ½C_f·ρU²,  u* = √(τ_w/ρ),  y = y⁺·μ/(ρ·u*)
```

## Explanation

CFD turbulence models need the first grid cell at the right distance from walls: y⁺ ≈ 1 to resolve the viscous sublayer, or 30–300 for wall functions. This calculator estimates that height before you build the mesh.

### How to use the Y+ Calculator

- **Free-stream velocity U**
- **Density ρ**
- **Dynamic viscosity μ**
- **Reference length L**
- **Desired y⁺**
- **Skin friction correlation**

Results update as you type, every field has its own unit menu, and everything is calculated in your browser.

### How it works

Enter the free-stream velocity, density, viscosity, reference length and target y⁺, and choose a skin-friction correlation.

### Inputs explained

FieldWhat to enterFree-stream velocity UUnits: m/s, km/h, mph, ft/s, knotsDensity ρUnits: kg/m³, g/cm³, g/mL, lb/ft³, lb/in³Dynamic viscosity μUnits: Pa·s, mPa·sReference length LUnits: m, cm, mm, km, in, ft …Desired y⁺A numberSkin friction correlationChoose one: Schlichting, Prandtl (1/7 power), Granville, Schultz-Grunov

### What each result means

- **Reynolds number** = [Density ρ]×[Free-stream velocity U]×[Reference length L]÷[Dynamic viscosity μ]
- **Wall shear stress** = 0.5×[Skin friction coefficient]×[Density ρ]×[Free-stream velocity U]^2
- **Friction velocity u*** = sqrt([Wall shear stress]÷[Density ρ])
- **First cell height** = [Desired y⁺]×[Dynamic viscosity μ]÷([Density ρ]×[Friction velocity u×])

### Example values

How the main result changes when **Free-stream velocity U** changes and everything else stays at the example values above:

Free-stream velocity U (m/s)First cell height (µm)1032.97271522.79232017.52722514.29123012.0932409.2881

## Worked example

Inputs — Free-stream velocity U: 20 m/s; Density ρ: 1.225 kg/m³; Dynamic viscosity μ: 1.81e-05 Pa·s; Reference length L: 1 m; Desired y⁺: 1; Skin friction correlation: Schlichting.

Results — Reynolds number = 1,353,591.16022; Skin friction coefficient = 0.00355; Wall shear stress = 0.87055 Pa; Friction velocity u* = 0.843 m/s; **First cell height = 17.5272 µm**.

## Key terms

- **y⁺**: Dimensionless wall distance, y·u*/ν.

## FAQ

**What y⁺ should I aim for?**
About 1 for low-Re models like k-ω SST; 30–300 with wall functions.

**Is the cell height the centroid or the face?**
Many codes use the cell centroid – check your solver.

**Why does the correlation matter?**
Different flat-plate fits give slightly different C_f.

**Should I check y⁺ after solving?**
Yes – this is only an estimate.


## Sources

- [Fluid Mechanics (F. M. White) – reference text](https://www.mheducation.com/highered/product/fluid-mechanics-white-chul/M9781260258318.html)
- [The Engineering ToolBox](https://www.engineeringtoolbox.com/)