6 C. Jin, I. Potts, D. C. Swailes, M. W. Reeks 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 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: 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 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] M$_{ij}$ = ˜ Δ $^{2}$| ˜ S | ˜ S$_{ij}$ − Δ 2 ˜ | S | S$_{ij}$ (17) L$_{ij}$ − 1 3 L$_{kk}$ δ$_{ij}$ = − 2 CM$_{ij}$ (16) L$_{ij}$ = T$_{ij}$ − ˜ τ$_{ij}$ (15) T$_{ij}$ = ˜ u$_{i}$u$_{j}$ − ˜ u$_{i}$$_{˜}$ u$_{j}$ (14) ∂$_{˜}$ u$_{i}$ ∂t + ∂$_{˜}$ u$_{i}$$_{˜}$ u$_{j}$ ∂x$_{j}$ = − 1 ρ ∂$_{˜}$ p ∂x$_{i}$ + ∂ ∂x$_{j}$ ( ν ∂$_{˜}$ u$_{i}$ ∂x$_{j}$ ) − ∂ T$_{ij}$ ∂x$_{j}$ (13)