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 ($\Delta y_1$)
37.672 μm
= 0.037672 mm | = 3.7672e-5 m (Centroid $y = 18.84\ \mu\text{m}$)
Turbulence Model Diagnostic: Optimal for Low-Re resolved models (SST k-omega, Spalart-Allmaras, LES/DNS).
Reynolds ($Re_L$)1.541e+5
Laminar
Laminar
Skin Friction ($C_f$)5.3433e-3
Wall Shear ($\tau_w$)0.7364 Pa
Friction Vel ($u_\tau$)0.7753 m/s
Viscous Length ($\ell^+$)18.8 μm
BL Thickness ($\delta$)1.91 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] = 1.500000E+01 L_ref [m] = 1.500000E-01 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= 1.540665E+05 Flow Regime = Laminar Skin Friction Cf = 5.343319E-03 Wall Shear Tau_w[Pa]= 7.363762E-01 Friction Vel u_tau = 7.753215E-01 Viscous Length l+ = 1.883616E-05 Wall Dist y [m] = 1.883616E-05 Cell Height Dy1 [m] = 3.767233E-05 Cell Height Dy1 [mm]= 3.767233E-02 Cell Height Dy1 [um]= 3.767233E+01 --------------------------------------------------------------- BL Thickness delta = 1.910764E-03 Displacement delta* = 6.573029E-04 Momentum theta = 2.537495E-04 Shape Factor H = 2.590361E+00 Viscous Sublayer [m]= 9.418081E-05 Buffer Layer Top [m]= 5.650849E-04 Log-law Region [m] = 5.650849E-03 =============================================================== 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 7.753214E-02 1.883616E-06 1.165914E-01 1.165914E-01 9.039583E-02 2.196135E-06 1.359356E-01 1.359356E-01 1.053938E-01 2.560506E-06 1.584893E-01 1.584892E-01 1.228801E-01 2.985331E-06 1.847850E-01 1.847848E-01 1.432676E-01 3.480640E-06 2.154435E-01 2.154431E-01 1.670377E-01 4.058128E-06 2.511886E-01 2.511880E-01 1.947515E-01 4.731430E-06 2.928645E-01 2.928633E-01 2.270632E-01 5.516443E-06 3.414549E-01 3.414528E-01 2.647357E-01 6.431700E-06 3.981072E-01 3.981032E-01 3.086580E-01 7.498811E-06 4.641589E-01 4.641516E-01 3.598667E-01 8.742972E-06 5.411695E-01 5.411559E-01 4.195698E-01 1.019356E-05 6.309573E-01 6.309320E-01 4.891752E-01 1.188482E-05 7.356423E-01 7.355951E-01 5.703227E-01 1.385668E-05 8.576959E-01 8.576078E-01 6.649217E-01 1.615570E-05 1.000000E+00 9.998352E-01 7.751937E-01 1.883616E-05 1.165914E+00 1.165606E+00 9.037191E-01 2.196135E-05 1.359356E+00 1.358776E+00 1.053488E+00 2.560506E-05 1.584893E+00 1.583801E+00 1.227955E+00 2.985331E-05 1.847850E+00 1.845787E+00 1.431078E+00 3.480640E-05 2.154435E+00 2.150525E+00 1.667348E+00 4.058128E-05 2.511886E+00 2.504454E+00 1.941757E+00 4.731430E-05 2.928645E+00 2.914470E+00 2.259651E+00 5.516443E-05 3.414549E+00 3.387457E+00 2.626368E+00 6.431700E-05 3.981072E+00 3.929279E+00 3.046455E+00 7.498811E-05 4.641589E+00 4.542945E+00 3.522243E+00 8.742972E-05 5.411695E+00 5.225782E+00 4.051661E+00 1.019356E-04 6.309573E+00 5.966299E+00 4.625800E+00 1.188482E-04 7.356423E+00 6.742984E+00 5.227980E+00 1.385668E-04 8.576959E+00 7.527886E+00 5.836532E+00 1.615570E-04 1.000000E+01 8.294498E+00 6.430903E+00 1.883616E-04 1.165914E+01 9.024715E+00 6.997055E+00 2.196135E-04 1.359356E+01 9.710534E+00 7.528786E+00 2.560506E-04 1.584893E+01 1.035153E+01 8.025766E+00 2.985331E-04 1.847850E+01 1.095138E+01 8.490843E+00 3.480640E-04 2.154435E+01 1.151530E+01 8.928057E+00 4.058128E-04 2.511886E+01 1.204862E+01 9.341557E+00 4.731430E-04 2.928645E+01 1.255624E+01 9.735126E+00 5.516443E-04 3.414549E+01 1.304237E+01 1.011203E+01 6.431700E-04 3.981072E+01 1.351055E+01 1.047502E+01 7.498811E-04 4.641589E+01 1.396374E+01 1.082639E+01 8.742972E-04 5.411695E+01 1.440438E+01 1.116802E+01 1.019356E-03 6.309573E+01 1.483449E+01 1.150150E+01 1.188482E-03 7.356423E+01 1.525575E+01 1.182811E+01 1.385668E-03 8.576959E+01 1.566957E+01 1.214895E+01 1.615570E-03 1.000000E+02 1.607710E+01 1.246492E+01 1.883616E-03 1.165914E+02 1.647933E+01 1.277678E+01 2.196135E-03 1.359356E+02 1.687707E+01 1.308516E+01 2.560506E-03 1.584893E+02 1.727102E+01 1.339060E+01 2.985331E-03 1.847850E+02 1.766177E+01 1.369355E+01 3.480640E-03 2.154435E+02 1.804980E+01 1.399440E+01 4.058128E-03 2.511886E+02 1.843553E+01 1.429347E+01 4.731430E-03 2.928645E+02 1.881933E+01 1.459103E+01 5.516443E-03 3.414549E+02 1.920150E+01 1.488733E+01 6.431700E-03 3.981072E+02 1.958228E+01 1.518256E+01 7.498811E-03 4.641589E+02 1.996191E+01 1.547689E+01 8.742972E-03 5.411695E+02 2.034056E+01 1.577047E+01 1.019356E-02 6.309573E+02 2.071839E+01 1.606341E+01 1.188482E-02 7.356423E+02 2.109554E+01 1.635583E+01 1.385668E-02 8.576959E+02 2.147212E+01 1.664780E+01 1.615570E-02 1.000000E+03 2.184823E+01 1.693941E+01 1.883616E-02 --- 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 $\Delta y_1 = \mathbf{8.19\ \mu m}$.