content_image/308355.jpg content_image/308354.jpg content_image/308353.jpg content_image/308352.jpg content_image/308351.jpg
defined a criterion for minimizing the square of the error as 

<img src='content_image/308355.jpg'>

In order to obtain a local value, varying in time and space in a fairly wide range, for the model constant C , one takes ∂E ∂C and sets it zero to get 

<img src='content_image/308354.jpg'>

A negative C represents the transfer of flow energy from the subgrid-scale eddies to the resolved eddies, which is known as back-scatter and regarded as a desirable attribute of the dynamic model. 

## 2.2 The Werner and Wengle wall layer model 

The Large Eddy Simulation (LES) of turbulent flow over tube-banks is hampered by expensive computational cost incurred when the dynamic and thin near-wall layer is fully resolved. To obviate the computational cost associated with calculating the wall shear stress from the laminar stress- strain relationship that requires the first cell to be put within the range of y + ≈ 1, Werner et al.[37] proposed a simple power-law to replace the law of the wall, in which the velocity profile on a solid wall is given as following,  

<img src='content_image/308353.jpg'>

where A = 8 . 3 and B = 1 / 7. An analytical integration of Eq. (21) results in the following relations for the wall shear stress 

<img src='content_image/308352.jpg'>

where u$_{p}$ is velocity component parallel to the wall and given by: 

<img src='content_image/308351.jpg'>

