Turbulent Flow

Turbulent Flow

When analyzing the effects of turbulence in CFD models, it is important to take into account certain parameters that influence its calculation.

Y+ parameter

The behavior of the flow near the wall is a complicated phenomenon and to distinguish the different regions near the wall the concept of wall y+ has been formulated. Thus y+ is a dimensionless quantity, and is distance from the wall measured in terms of viscous lengths.
If we intend to resolve the effects near the wall i.e., in the viscous sub layer then the size of the mesh size should be small and dense enough near the wall so that almost all the effects are captured. But in some cases, if the wall effects are negligible then there is option of including semi-empirical formulae to bridge between the viscosity affected region and fully turbulent region and in this case the mesh need not to be dense or small near the wall i.e., coarse mesh would work.
For near wall modelling it is well-known that the mesh size should be small enough, however then the question follows is how small? Thus, here comes the concept of y+, and based on this value the first cell height can be calculated. The near wall region is meshed using the calculated first cell height value with gradual growth in the mesh so that the effects are captured and avoiding overall heavy mesh count.
In a flow bounded by a wall, different scales and physical processes are dominant in the inner portion near the wall, and the outer portion approaching the free stream. These layers are typically known as the inner and outer layers. Considering the flow over a smooth flat plate the boundary layer can be distinguished into two types namely laminar boundary layer and turbulent boundary layer. Since we are dealing with the turbulent boundary layer let us not get into the laminar boundary layer. Typical boundary layer structure over a flat plate is shown below. In between the laminar and turbulent boundary layer there lies a transition region. Typically for flow over a flat plate the transition usually occurs around Re ≈ 5 · 10⁵.
Flat plate boundary layer
In turbulent boundary layer region flow near the wall has been analyzed in terms of three layers:
Viscous layer (y+ < 5)
Also called linear layer, it collects the surface friction forces. A very small cell size next to the surface is necessary if viscous forces are to be calculated. In applications where these forces are negligible these cells can be larger by reaching the logarithmic layer.
Buffer layer (5 < y+ < 30)
It is a transition layer that does not conform to linear or logarithmic behavior.
Logarithmic layer (30 < y+ < 120)
A layer in which the mean velocity of a turbulent flow at a given point is proportional to the logarithm of the distance from that point to the surface. It is a good approximation for the rest of the boundary layer profile.
Chart y+ vs U+

Y+ calculation

The centroid of the cell adjacent to the surface of a body, the first cell of the mesh, is given by:

y_p= \Large \frac {y^+  \mu} {u_{\tau}  \rho}

Therefore, the height of the first cell will be:

y_H= 2  y^+

The calculation of y+ is an iterative process so it must be assumed in the first instance. The software will calculate this y+, if we have deviated, we will have to calculate the new cell with the new y+ given by the software. If the calculation of the viscous forces is important in our application, as for example in airfoils, we will have to take an initial y+ below 5, in the viscous layer. If, on the other hand, the viscous forces are not significant, we can place ourselves in the logarithmic layer with a y+ above 30 and below 120.
ρ is the density and µ is the dynamic viscosity of the fluid, both properties that can be considered constant.

The friction velocity ( u_{\tau} ) is calculated as:

u_{\tau} = \sqrt{ \Large \frac {\tau_{\omega}} {\rho}}

Where wall shear stress ( \tau_{\omega} ) depends on the skin friction coefficient ( C_f):

\tau_{\omega} = \Large \frac {1}{2}\normalsize \rho  U^2  C_f

Where U is the free stream velocity of the fluid and the skin friction coefficient ( C_f ) must be estimated since we do not know it, so we will take the flat plate one in turbulent regime.

C_f = (2 \log_{10} (Re) - 0.65)^{-2.3}

Other approaches are possible:

For internal flows: C_f= 0.079 Re^{-0.25}

For external flows: C_f= 0.058 Re^{-0.2}

Where Re is the Reynolds number:

Re = \Large \frac {\rho U L} {\mu}

With L equal to the characteristic length.
Once the size of the first cell size is defined, we will calculate y+ until it matches the one that we had assumed in an iterative process.

Turbulence coefficients

Turbulent kinetic energy (k):

k = \Large \frac {3} {2} \normalsize U^2 I^2

Where U is the free stream velocity and I is the turbulence intensity.

If the maximum free stream velocity is defined as U_{max} , then the turbulence intensity (I) is calculated as:

I (\%) = \Large \frac {U_{max}-U} {U} \normalsize \cdot 100 = (G-1)\cdot 100

Being G the gust factor defined as:

U_{max} = G \cdot U

Dissipation ratio (ϵ):

  \epsilon = C_{\mu} \Large \frac {\rho k^2}{\mu} \normalsize \beta^{-1}

Where C_{\mu} = 0.09 in the methods k-ϵ and k-ω, and β is the turbulent viscosity ratio.

\beta = \Large \frac{\mu_t}{\mu}

Specific dissipation ratio (ω):

  \omega = \Large \frac {\rho k}{\mu} \normalsize \beta^{-1}

The following tables show I and beta values according to experience.
For internal flows:
TurbulenceReI (%) β
Very low<30000.05-10.1-0.2
Very high>50000100
For external flows:

1< \beta < 10