Calculate y+ – Estimate initial cell height

Jan 16, 2012   //   by Mark Bell   //   Aerospace Engineering  //  No Comments

Knowing how to calculate y+ is very important when using turbulence modelling.  To obtain an initial cell height above the wall (aerofoil surface, for example), one can calculate an approximate height using the following method.  The following assumptions are made,

  • The gas is ideal
  • A zero-pressure gradient is assumed
The non-dimensional wall distance, y+ is given by,

(1)   \begin{equation*} y^+ = \frac{V_\tau y}{\nu} \end{equation*}

where V_\tau is the friction velocity, y is the initial height above the wall and \nu is the kinematic viscosity.  For a required y+, one can estimate the initial cell height.

For a given Mach number, M_\infty,

(2)   \begin{equation*} V_\infty = M a = M \sqrt{\gamma R T} \end{equation*}

Before calculating the Reynolds number, the fluid viscosity can be corrected for temperature using Sutherland’s Law (Eqn 3),

(3)   \begin{equation*} \mu = \mu_0 \left[\frac{T}{T_0}\right]^{\frac{3}{2}} \left[\frac{T_0 + S_u}{T + S_u}\right] \end{equation*}

where, \mu_0 = 1.716E-05 kg/ms, T_0 = 273.11 K and S_u = 110.56 K.  T is the static temperature of the freestream flow.  The Reynolds number can be calculated from Eqn 4, where \rho is the freestream air density (obtainable from the equation of state, Eqn 5), V is the freestream velocity, c is a reference length and \mu is the dynamic viscosity of the fluid.

(4)   \begin{equation*} Re = \frac{\rho V c}{\mu} \end{equation*}

(5)   \begin{equation*} p = \rho R T \end{equation*}

An estimation of the skin-friction coefficient can be made using the equation for a flat plate with a turbulent boundary layer,

(6)   \begin{equation*} C_F = 0.074 Re^{-0.2} \end{equation*}

Given knowledge of the skin friction coefficient, the dimensional friction velocity may be obtained from,

(7)   \begin{equation*} V_\tau = V_\infty \sqrt{C_F}{2} \end{equation*}

The final step of the equation is to re-arrange Eqn 1 for y.

About the author

Mark Bell wrote 18 articles on this blog.

Mark is a Research Fellow at Queens University Belfast were his specialism is in numerical aerodynamic modelling using CFD. He writes here about aerospace engineering, as well as his other side interests in web design/development and internet marketing.

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