Dependence of the Smagorinsky-Lilly's constant on inertia, wind stress, and bed roughness for large eddy simulations

Equations governing large eddy simulations are usually closed by incorporating with the Smagorinsky-Lilly's turbulence model of eddy viscosity. The model contains a so-called filtering length and a Sinagorinsky-Lilly constant that changes among different researchers. The variation range of the constant is wide and its value is usually determined in a sense of guessing. Since the constant is closely related to the magnitude of eddy viscosity, hence to our numerical solutions eventually, setting a more precise and determinate procedure for prescribing the constant seems to be worthy it. The constant, is first estimated in use of the properties of fluid flow within the inertia subrange. Then, along with a general derivation, the explicit closed-form expression for the constant is presented for steady uniform flows. It is found that, with the analogy between the filtering technique and Reynolds average, C-SL may not necessarily be constant but proportional to the Manning n and water depth. Other than the determination of C-SL, the vertical flow velocity profile in an infinitely long wide rectangular channel without spiral flow motion is obtained through the use of the Smagorinsky-Lilly's turbulence closure model. It is shown analytically that the velocity profile in unsteady open channel flow can be expressed as a function of an integration function J(n)(z) that accounts for wind stress and inertia terms. With the velocity profile, effects of inertia terms, wind stress, and channel bed roughness on C-SL are deeply explored in response to the dependence of C-SL on J(n)(z).