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Y+ Calculator

Y+ Formula

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(dimensionless)
(dimensionless)

1. What is the Y+ Calculator?

Definition: This calculator computes the wall distance (\( y \)) required to achieve a target dimensionless wall distance (\( y^+ \)) in a turbulent boundary layer flow. It also calculates the Reynolds number (\( Re_x \)), skin friction coefficient (\( C_f \)), wall shear stress (\( \tau_w \)), and friction velocity (\( u_* \)) based on the freestream velocity (\( U_f \)), fluid density (\( \rho \)), dynamic viscosity (\( \mu \)), and length of the boundary layer (\( L \)).

Purpose: It is used in computational fluid dynamics (CFD) and aerodynamics to determine the appropriate mesh size near a wall for accurate turbulence modeling, particularly in wall-bounded flows.

2. How Does the Calculator Work?

The calculator uses the following formulas:

Formulas: \[ Re_x = \frac{\rho \cdot U_f \cdot L}{\mu} \] Skin Friction Coefficient (\( C_f \)):

  • Prandtl (1927): \( C_f = 0.074 \cdot Re_x^{-0.2} \)
  • Granville (1977): \( C_f = 0.0776 \cdot [\log_{10}(Re_x) - 1.88]^{-2} + 60 \cdot Re_x^{-1} \)
  • Schlichting: \( C_f = [2 \cdot \log_{10}(Re_x) - 0.65]^{-2.3} \)
  • Kempf-Karman (1951): \( C_f = 0.055 \cdot Re_x^{-0.182} \)
  • Schultz-Grunov (1940): \( C_f = 0.427 \cdot [\log_{10}(Re_x) - 0.407]^{-2.64} \)
\[ \tau_w = C_f \cdot \frac{1}{2} \cdot \rho \cdot U_f^2 \] \[ u_* = \sqrt{\frac{\tau_w}{\rho}} \] \[ y^+ = \frac{y \cdot \rho \cdot u_*}{\mu} \] Rearranged to solve for \( y \): \[ y = \frac{y^+ \cdot \mu}{\rho \cdot u_*} \] where:
  • \( Re_x \): Reynolds number (dimensionless)
  • \( C_f \): Skin friction coefficient (dimensionless)
  • \( \tau_w \): Wall shear stress (Pa, bar, psi, at, atm, Torr, hPa, kPa, lb/ft²)
  • \( u_* \): Friction velocity (m/s, km/h, mph, knots, km/s, mi/s, mi/min, km/min, ft/s, ft/min)
  • \( y \): Wall distance (m, cm, mm, in, ft, yd)
  • \( U_f \): Freestream velocity (m/s, km/h, mph, knots, km/s, mi/s, mi/min, km/min, ft/s, ft/min)
  • \( \rho \): Fluid density (kg/m³, lb/ft³)
  • \( \mu \): Dynamic viscosity (Pa·s, cP)
  • \( L \): Length of boundary layer (m, cm, mm, in, ft, yd)
  • \( y^+ \): Dimensionless wall distance (dimensionless)

Unit Conversions:

  • Freestream Velocity (\( U_f \)) and Friction Velocity (\( u_* \)):
    • 1 m/s = 1 m/s
    • 1 km/h = 0.277778 m/s
    • 1 mph = 0.44704 m/s
    • 1 knot = 0.514444 m/s
    • 1 km/s = 1000 m/s
    • 1 mi/s = 1609.34 m/s
    • 1 mi/min = 26.8224 m/s
    • 1 km/min = 16.6667 m/s
    • 1 ft/s = 0.3048 m/s
    • 1 ft/min = 0.00508 m/s
  • Density (\( \rho \)):
    • 1 kg/m³ = 1 kg/m³
    • 1 lb/ft³ = 16.0185 kg/m³
  • Dynamic Viscosity (\( \mu \)):
    • 1 Pa·s = 1 Pa·s
    • 1 cP = 0.001 Pa·s
  • Length (\( L \), \( y \)):
    • 1 m = 1 m
    • 1 cm = 0.01 m
    • 1 mm = 0.001 m
    • 1 in = 0.0254 m
    • 1 ft = 0.3048 m
    • 1 yd = 0.9144 m
  • Wall Shear Stress (\( \tau_w \)):
    • 1 Pa = 1 Pa
    • 1 bar = 100000 Pa
    • 1 psi = 6894.76 Pa
    • 1 at = 98066.5 Pa
    • 1 atm = 101325 Pa
    • 1 Torr = 133.322 Pa
    • 1 hPa = 100 Pa
    • 1 kPa = 1000 Pa
    • 1 lb/ft² = 47.8803 Pa

Steps:

  • Enter the freestream velocity (\( U_f \)) in m/s, km/h, mph, knots, km/s, mi/s, mi/min, km/min, ft/s, or ft/min (default is 10 m/s, step size 0.00001).
  • Enter the fluid density (\( \rho \)) in kg/m³ or lb/ft³ (default is 1.225 kg/m³ for air at 15°C and sea level, step size 0.00001).
  • Enter the dynamic viscosity (\( \mu \)) in Pa·s or cP (default is 0.0000181 Pa·s for air at 15°C, step size 0.00001).
  • Enter the length of the boundary layer (\( L \)) in m, cm, mm, in, ft, or yd (default is 1 m, step size 0.00001).
  • Enter the target dimensionless distance (\( y^+ \)) (default is 30, step size 0.00001).
  • Select the skin friction coefficient method (Prandtl, Granville, Schlichting, Kempf-Karman, or Schultz-Grunov).
  • Convert all inputs to SI units (m/s for velocities, kg/m³ for density, Pa·s for viscosity, m for lengths).
  • Calculate the Reynolds number (\( Re_x \)) using \( Re_x = \frac{\rho \cdot U_f \cdot L}{\mu} \).
  • Calculate the skin friction coefficient (\( C_f \)) based on the selected method.
  • Calculate the wall shear stress (\( \tau_w \)) using \( \tau_w = C_f \cdot \frac{1}{2} \cdot \rho \cdot U_f^2 \).
  • Calculate the friction velocity (\( u_* \)) using \( u_* = \sqrt{\frac{\tau_w}{\rho}} \).
  • Calculate the wall distance (\( y \)) using \( y = \frac{y^+ \cdot \mu}{\rho \cdot u_*} \).
  • Convert the results to the selected units and display them, using scientific notation if the absolute value is less than 0.001, otherwise rounded to 4 decimal places.

3. Importance of Y+ Calculation

Calculating \( y^+ \) and related parameters is crucial for:

  • CFD Simulations: Determining the appropriate mesh size near walls to accurately resolve the boundary layer in turbulence models.
  • Aerodynamic Design: Optimizing the design of airfoils, vehicles, and structures by understanding near-wall flow behavior.
  • Engineering Applications: Ensuring accurate predictions of drag, lift, and heat transfer in wall-bounded flows.

4. Using the Calculator

Examples:

  • Example 1: Calculate the wall distance for an airflow over a plate with a freestream velocity of 10 m/s, density of 1.225 kg/m³, dynamic viscosity of 0.0000181 Pa·s, boundary layer length of 1 m, and target \( y^+ = 30 \), using the Prandtl method, with shear stress in Pa, friction velocity in m/s, and wall distance in mm:
    • Enter \( U_f = 10 \) m/s.
    • Enter \( \rho = 1.225 \) kg/m³.
    • Enter \( \mu = 0.0000181 \) Pa·s.
    • Enter \( L = 1 \) m.
    • Enter \( y^+ = 30 \).
    • Select method: Prandtl.
    • Reynolds number: \( Re_x = \frac{1.225 \times 10 \times 1}{0.0000181} = 676795.58 \).
    • Skin friction coefficient: \( C_f = 0.074 \times (676795.58)^{-0.2} = 0.00389 \).
    • Wall shear stress: \( \tau_w = 0.00389 \times \frac{1}{2} \times 1.225 \times 10^2 = 0.2384 \, \text{Pa} \).
    • Friction velocity: \( u_* = \sqrt{\frac{0.2384}{1.225}} = 0.4412 \, \text{m/s} \).
    • Wall distance: \( y = \frac{30 \times 0.0000181}{1.225 \times 0.4412} = 0.001005 \, \text{m} \).
    • Convert to mm: \( y = 0.001005 \times 1000 = 1.005 \).
    • Result: \( Re_x = 676795.5801 \), \( C_f = 0.0039 \), \( \tau_w = 0.2384 \, \text{Pa} \), \( u_* = 0.4412 \, \text{m/s} \), \( y = 1.0050 \, \text{mm} \).
  • Example 2: Calculate the wall distance for a flow with a freestream velocity of 30 ft/s, density of 0.00001 kg/m³, dynamic viscosity of 0.0000181 Pa·s, boundary layer length of 1 cm, and target \( y^+ = 1 \), using the Schultz-Grunov method, with shear stress in psi, friction velocity in ft/min, and wall distance in in:
    • Enter \( U_f = 30 \) ft/s.
    • Convert to m/s: \( 30 \times 0.3048 = 9.144 \, \text{m/s} \).
    • Enter \( \rho = 0.00001 \) kg/m³.
    • Enter \( \mu = 0.0000181 \) Pa·s.
    • Enter \( L = 1 \) cm, convert to m: \( 1 \times 0.01 = 0.01 \, \text{m} \).
    • Enter \( y^+ = 1 \).
    • Select method: Schultz-Grunov.
    • Reynolds number: \( Re_x = \frac{0.00001 \times 9.144 \times 0.01}{0.0000181} = 0.0505 \).
    • Skin friction coefficient: \( C_f = 0.427 \times [\log_{10}(0.0505) - 0.407]^{-2.64} \), \( \log_{10}(0.0505) = -1.2967 \), \( C_f = 0.427 \times (-1.7037)^{-2.64} = 0.0044 \).
    • Wall shear stress: \( \tau_w = 0.0044 \times \frac{1}{2} \times 0.00001 \times (9.144)^2 = 1.8383 \times 10^{-6} \, \text{Pa} \).
    • Convert to psi: \( 1.8383 \times 10^{-6} \times 0.000145038 = 2.6669 \times 10^{-10} \).
    • Friction velocity: \( u_* = \sqrt{\frac{1.8383 \times 10^{-6}}{0.00001}} = 0.0136 \, \text{m/s} \).
    • Convert to ft/min: \( 0.0136 \times 196.85 = 2.6741 \).
    • Wall distance: \( y = \frac{1 \times 0.0000181}{0.00001 \times 0.0136} = 0.1334 \, \text{m} \).
    • Convert to in: \( 0.1334 \times 39.3701 = 5.2520 \).
    • Result: \( Re_x = 0.0505 \), \( C_f = 0.0044 \), \( \tau_w = 2.6669 \times 10^{-10} \, \text{psi} \), \( u_* = 2.6741 \, \text{ft/min} \), \( y = 5.2520 \, \text{in} \).

5. Frequently Asked Questions (FAQ)

Q: What is \( y^+ \)?
A: \( y^+ \) is the dimensionless wall distance, used in CFD to determine the mesh size near a wall for accurate turbulence modeling. It represents the ratio of the wall distance to the viscous length scale.

Q: Why are there multiple methods for calculating the skin friction coefficient?
A: Different researchers (Prandtl, Granville, Schlichting, Kempf-Karman, Schultz-Grunov) developed empirical approximations for \( C_f \) based on experimental data, each valid under specific conditions and Reynolds number ranges.

Q: What is the significance of the Reynolds number in this calculation?
A: The Reynolds number (\( Re_x \)) indicates the flow regime (laminar or turbulent) and is used to calculate the skin friction coefficient, which affects the wall shear stress and friction velocity.

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