content_image/296923.jpg content_image/296922.jpg content_image/296921.jpg content_image/296920.jpg content_image/296919.jpg
major drawback of the model used in LES is that there is an inherent inability to represent a wide range of turbulent flows with a single value of the model coefficient C$_{ν}$ . Given that the turbulent flow over tube-banks in the present study is fully three-dimensional, the standard Smagorinsky SGS model is not used here to compute the coefficient C$_{ν}$ . 

Germano et al.[35] proposed a new procedure to dynamically compute the model coefficient C$_{ν}$ based on the information obtained from the resolved large scales of motion. The new procedure employes another coarser filter ˜ Δ (test filter) whose width is greater than that of the default grid filter. Applying the test filter to the filtered Navier-Stokes equations, one obtains the following equations 

<img src='content_image/296923.jpg'>

where the tilde denotes the test-filtered quantities. T$_{ij}$ represents the subgrid scale stress tensor from the resolved large scales of motion and is given by 

The quantities given in (6) and (14) are related by the Germano identity: 

<img src='content_image/296922.jpg'>

<img src='content_image/296921.jpg'>

which represents the resolved turbulent stress tensor from the SGS tensor between the test and grid filters, T$_{ij}$ and τ$_{ij}$ . Applying the same Smagorinsky model to T$_{ij}$ and τ$_{ij}$ , the anisitropic parts of L$_{ij}$ can be written as 

where 

<img src='content_image/296920.jpg'>

<img src='content_image/296919.jpg'>

One hence obtains the value of C from (17) that is solved on the test filter level and then apply it to Eq. (12). The model value of C is obtained via a least squares approach proposed by Lilly[36], since Eq. (17) is an overdetermined system of equations for the unknown variable C . Lilly[36] 

