In Reynolds averaging, fluctuating quantity is represented as the sum of the averaged value and the fluctuation of the same quantity.

Mathematically it can be represented as.This is the Unsteady Reynolds Average Navier Stokes equation.There are two differences between Unsteady Reynolds Average Navier Stokes(URANS) equation and Navier Stoke(NS) equation :1.The URANS equation is based on the quantities which are averaged and in the case of NS equations the quantities are instantaneous in nature.2.

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The URANS equation has an additional term, this term is called the Reynolds Stress.Reynolds averaging causes the ltering of the resting scales which are big enough but are not universal. Due to this, the URANS model is not applicable universally.

We use K- model to solve the flow which is not laminar but is turbulent in nature. The epsilon model is given belowThe K SST Model is a two-equation eddy-viscosity model which is widely used. The shear stress transport formulation combines the K and K, because of which it can be used till the wall where there is low Reynolds number and results of this model are very accurate.The SST model switches between the two models based on the type of flow it encounters. For example, in the case of the free stream, it uses K- because the K model has a problem of being too sensitive to turbulence properties of the free stream which causes loss of accuracy. The K-! SST model gives very good accuracy for adverse pressure gradients and separating flow.

The equation for kinematic eddy viscosity:Grid resolution near the wall:Optimum resolution of the grid is important, the grid near the wall should have high resolution.Near the walls viscosity effects are signicant so we need high resolution near the wall becausethere will be the formation of boundary layers near the wall due to which there will be a change invelocity gradient. If the resolution is not high then we will not be able to capture the changein gradient accurately and there will be approximations which are not accurate leading to higherrors in the solution.The other factor on which mesh resolution depends is the value of y+. y+ is the dimensionlesswall distance.

Mathematically, y+ can be shown as :ut is the friction velocity at the nearest wall; y signies the distance to the nearest wall represents the kinematic viscosity.The stress in wall turbulence can be considered as the sum of the laminar and turbulent stresses: