๐ CFD y+ & First Layer Height
Calculate boundary layer first cell height (Dy1), y+, skin friction (Cf), wall shear stress, and friction velocity for CFD meshes.
โก Fortran 90 Engine
Double Precision (IEEE 754)
โ ISO / ASME Validated
Cfd
๐ Solver Telemetry
โ ACTIVE
๐๏ธ Views
65
โก Solves
54
๐พ Downloads
1,190
๐ฆ Fortran Code
4.4 KB
๐
Released
Jun 2026
โฑ๏ธ Latency
< 1 ms
๐ Boundary Layer & Near-Wall Mesh Structure
Interactive Canvas (Viscous sublayer • Buffer • Log-law • First cell centroid)๐ Flow & Mesh Parameters
Key Formulation:
$$Re_L = \frac{\rho U_\infty L}{\mu} = \frac{U_\infty L}{\nu}$$ $$\tau_w = \frac{1}{2} C_f \rho U_\infty^2 \quad \Longrightarrow \quad u_\tau = \sqrt{\frac{\tau_w}{\rho}}$$ $$y = \frac{y^+ \nu}{u_\tau} \quad \Longrightarrow \quad \Delta y_1 = 2 y \quad \text{(for cell-centered FVM)}$$
$$Re_L = \frac{\rho U_\infty L}{\mu} = \frac{U_\infty L}{\nu}$$ $$\tau_w = \frac{1}{2} C_f \rho U_\infty^2 \quad \Longrightarrow \quad u_\tau = \sqrt{\frac{\tau_w}{\rho}}$$ $$y = \frac{y^+ \nu}{u_\tau} \quad \Longrightarrow \quad \Delta y_1 = 2 y \quad \text{(for cell-centered FVM)}$$
๐ Results & Diagnostics
CFD Mesh Sizing Summary
Recommended First Layer Cell Height ($\Delta y_1$)
4.3753 μm
= 0.0043753 mm | = 4.3753e-6 m (Centroid $y = 2.188\ \mu\text{m}$)
Turbulence Model Diagnostic: Optimal for Low-Re resolved models (SST k-omega, Spalart-Allmaras, LES/DNS).
Reynolds ($Re_L$)2.739e+7
Fully Turbulent
Fully Turbulent
Skin Friction ($C_f$)2.2283e-3
Wall Shear ($\tau_w$)54.59 Pa
Friction Vel ($u_\tau$)6.676 m/s
Viscous Length ($\ell^+$)2.19 μm
BL Thickness ($\delta$)24.1 mm
๐ Universal Law of the Wall ($u^+$ vs $y^+$)
๐ Near-Wall Physical Velocity $u(y)$ [m/s]
Fortran Solver Raw Output:
=============================================================== THERMOFLUIDCALC โ CFD Y+ & FIRST LAYER HEIGHT SOLVER =============================================================== U_inf [m/s] = 2.000000E+02 L_ref [m] = 2.000000E+00 Density rho [kg/m3] = 1.225000E+00 Viscosity mu [Pa.s] = 1.789000E-05 Nu [m2/s] = 1.460408E-05 Target y+ = 1.000000E+00 Cf Model = Schlichting (Prandtl-Schlichting) Cell Definition Mode= 1 --------------------------------------------------------------- Reynolds Number Re_L= 2.738960E+07 Flow Regime = Fully Turbulent Skin Friction Cf = 2.228270E-03 Wall Shear Tau_w[Pa]= 5.459260E+01 Friction Vel u_tau = 6.675732E+00 Viscous Length l+ = 2.187638E-06 Wall Dist y [m] = 2.187638E-06 Cell Height Dy1 [m] = 4.375275E-06 Cell Height Dy1 [mm]= 4.375275E-03 Cell Height Dy1 [um]= 4.375275E+00 --------------------------------------------------------------- BL Thickness delta = 2.408322E-02 Displacement delta* = 3.010403E-03 Momentum theta = 2.341424E-03 Shape Factor H = 1.285714E+00 Viscous Sublayer [m]= 1.093819E-05 Buffer Layer Top [m]= 6.562913E-05 Log-law Region [m] = 6.562913E-04 =============================================================== CFD TURBULENCE MODEL RECOMMENDATION: STATUS: Optimal for Low-Re resolved models (SST k-omega, Spalart-Allmaras, LES/DNS). --- LAW OF THE WALL PROFILE --- y_plus u_plus u_phys[m/s] y_phys[m] 1.000000E-01 9.999998E-02 6.675730E-01 2.187638E-07 1.165914E-01 1.165914E-01 7.783330E-01 2.550598E-07 1.359356E-01 1.359356E-01 9.074695E-01 2.973779E-07 1.584893E-01 1.584892E-01 1.058031E+00 3.467172E-07 1.847850E-01 1.847848E-01 1.233574E+00 4.042426E-07 2.154435E-01 2.154431E-01 1.438241E+00 4.713122E-07 2.511886E-01 2.511880E-01 1.676864E+00 5.495097E-07 2.928645E-01 2.928633E-01 1.955077E+00 6.406813E-07 3.414549E-01 3.414528E-01 2.279447E+00 7.469796E-07 3.981072E-01 3.981032E-01 2.657630E+00 8.709142E-07 4.641589E-01 4.641516E-01 3.098551E+00 1.015411E-06 5.411695E-01 5.411559E-01 3.612612E+00 1.183883E-06 6.309573E-01 6.309320E-01 4.211933E+00 1.380306E-06 7.356423E-01 7.355951E-01 4.910635E+00 1.609319E-06 8.576959E-01 8.576078E-01 5.725159E+00 1.876328E-06 1.000000E+00 9.998352E-01 6.674631E+00 2.187638E-06 1.165914E+00 1.165606E+00 7.781270E+00 2.550598E-06 1.359356E+00 1.358776E+00 9.070826E+00 2.973779E-06 1.584893E+00 1.583801E+00 1.057303E+01 3.467172E-06 1.847850E+00 1.845787E+00 1.232198E+01 4.042426E-06 2.154435E+00 2.150525E+00 1.435633E+01 4.713122E-06 2.511886E+00 2.504454E+00 1.671906E+01 5.495097E-06 2.928645E+00 2.914470E+00 1.945622E+01 6.406813E-06 3.414549E+00 3.387457E+00 2.261375E+01 7.469796E-06 3.981072E+00 3.929279E+00 2.623081E+01 8.709142E-06 4.641589E+00 4.542945E+00 3.032748E+01 1.015411E-05 5.411695E+00 5.225782E+00 3.488592E+01 1.183883E-05 6.309573E+00 5.966299E+00 3.982941E+01 1.380306E-05 7.356423E+00 6.742984E+00 4.501435E+01 1.609319E-05 8.576959E+00 7.527886E+00 5.025414E+01 1.876328E-05 1.000000E+01 8.294498E+00 5.537184E+01 2.187638E-05 1.165914E+01 9.024715E+00 6.024657E+01 2.550598E-05 1.359356E+01 9.710534E+00 6.482492E+01 2.973779E-05 1.584893E+01 1.035153E+01 6.910406E+01 3.467172E-05 1.847850E+01 1.095138E+01 7.310850E+01 4.042426E-05 2.154435E+01 1.151530E+01 7.687303E+01 4.713122E-05 2.511886E+01 1.204862E+01 8.043338E+01 5.495097E-05 2.928645E+01 1.255624E+01 8.382211E+01 6.406813E-05 3.414549E+01 1.304237E+01 8.706737E+01 7.469796E-05 3.981072E+01 1.351055E+01 9.019283E+01 8.709142E-05 4.641589E+01 1.396374E+01 9.321819E+01 1.015411E-04 5.411695E+01 1.440438E+01 9.615977E+01 1.183883E-04 6.309573E+01 1.483449E+01 9.903107E+01 1.380306E-04 7.356423E+01 1.525575E+01 1.018433E+02 1.609319E-04 8.576959E+01 1.566957E+01 1.046058E+02 1.876328E-04 1.000000E+02 1.607710E+01 1.073264E+02 2.187638E-04 1.165914E+02 1.647933E+01 1.100116E+02 2.550598E-04 1.359356E+02 1.687707E+01 1.126668E+02 2.973779E-04 1.584893E+02 1.727102E+01 1.152967E+02 3.467172E-04 1.847850E+02 1.766177E+01 1.179052E+02 4.042426E-04 2.154435E+02 1.804980E+01 1.204956E+02 4.713122E-04 2.511886E+02 1.843553E+01 1.230707E+02 5.495097E-04 2.928645E+02 1.881933E+01 1.256328E+02 6.406813E-04 3.414549E+02 1.920150E+01 1.281840E+02 7.469796E-04 3.981072E+02 1.958228E+01 1.307261E+02 8.709142E-04 4.641589E+02 1.996191E+01 1.332603E+02 1.015411E-03 5.411695E+02 2.034056E+01 1.357881E+02 1.183883E-03 6.309573E+02 2.071839E+01 1.383104E+02 1.380306E-03 7.356423E+02 2.109554E+01 1.408282E+02 1.609319E-03 8.576959E+02 2.147212E+01 1.433421E+02 1.876328E-03 1.000000E+03 2.184823E+01 1.458529E+02 2.187638E-03 --- END PROFILE ---
๐ Calculation Methodology: CFD y+ Wall Distance & First Layer Sizing
Mathematical Model & Theory
Dimensionless wall distance $y^+$ defines boundary layer grid resolution relative to viscous sublayer thickness ($y^+ < 1-5$ for low-Re resolved models, $30 < y^+ < 300$ for wall functions):
$$y^+ = \frac{\Delta y_1 \cdot u_\tau}{\nu}, \quad u_\tau = \sqrt{\frac{\tau_w}{\rho}}, \quad \tau_w = \frac{1}{2}\rho U_\infty^2 C_f$$
$$C_f = \frac{0.0592}{Re_L^{0.2}}, \quad Re_L = \frac{\rho U_\infty L}{\mu}$$
Assumptions
- Turbulent boundary layer over smooth wall.
- Prandtl friction velocity scaling $u_ au = \sqrt{ au_w/ ho}$.
Academic References
- Schlichting, H., & Gersten, K.: Boundary-Layer Theory, Springer.
- Wilcox, D. C.: Turbulence Modeling for CFD.
Worked Engineering Example
Problem Statement:
Air ($\rho = 1.225\text{ kg/m}^3$, $\mu = 1.789 \times 10^{-5}\text{ Pa}\cdot\text{s}$) flows over an airfoil ($L = 2.0\text{ m}$) at $U_\infty = 50\text{ m/s}$. Find $\Delta y_1$ for target $y^+ = 1.0$.
Step-by-step Solution:
1. $Re_L = 1.225 \times 50 \times 2.0 / (1.789 \times 10^{-5}) \approx 6.847 \times 10^6$.
2. $C_f = 0.0592 \times (6.847 \times 10^6)^{-0.2} \approx 0.002546$, $\tau_w = 0.5 \times 1.225 \times 50^2 \times 0.002546 = 3.899\text{ Pa}$.
3. $u_\tau = \sqrt{3.899 / 1.225} = 1.784\text{ m/s}$, $\Delta y_1 = (1.0 \times 1.789 \times 10^{-5}) / (1.225 \times 1.784) \approx 8.19\ \mu\text{m}$.
Final Result:
Required first cell height is $\Delta y_1 = \mathbf{8.19\ \mu m}$.
Air ($\rho = 1.225\text{ kg/m}^3$, $\mu = 1.789 \times 10^{-5}\text{ Pa}\cdot\text{s}$) flows over an airfoil ($L = 2.0\text{ m}$) at $U_\infty = 50\text{ m/s}$. Find $\Delta y_1$ for target $y^+ = 1.0$.
Step-by-step Solution:
1. $Re_L = 1.225 \times 50 \times 2.0 / (1.789 \times 10^{-5}) \approx 6.847 \times 10^6$.
2. $C_f = 0.0592 \times (6.847 \times 10^6)^{-0.2} \approx 0.002546$, $\tau_w = 0.5 \times 1.225 \times 50^2 \times 0.002546 = 3.899\text{ Pa}$.
3. $u_\tau = \sqrt{3.899 / 1.225} = 1.784\text{ m/s}$, $\Delta y_1 = (1.0 \times 1.789 \times 10^{-5}) / (1.225 \times 1.784) \approx 8.19\ \mu\text{m}$.
Final Result:
Required first cell height is $\Delta y_1 = \mathbf{8.19\ \mu m}$.