Abstract
ABSTRACTThe double-averaging methodology is used in this paper for deriving equations for the second-order velocity moments (i.e. turbulent and dispersive stresses) that emerge in the double-averaged momentum equation for incompressible Newtonian flows over mobile boundaries. The starting point in the derivation is the mass and momentum conservation equations for local (at a point) instantaneous variables that are up-scaled by employing temporal and spatial averaging. First, time-averaged conservation equations for mass, momentum, and turbulent stresses for mobile bed conditions are derived. Then, the double-averaged hydrodynamic equations obtained by spatial averaging the time-averaged equations are proposed. The derived second-order equations can serve as a basis for the construction of simplified mathematical and numerical models and for interpretation of experimental and simulation data when bed mobility is present. Potential applications include complex flow situations such as free-surface flows over vegetated or mobile sedimentary beds and flows through tidal and wind turbine arrays.
Highlights
Overland, river, coastal and atmospheric flows frequently exhibit high levels of multi-scale spatial heterogeneity due to geometrically complex and, occasionally, mobile boundaries
In response to this need, the current paper presents the derivation of equations describing the balances of the mean, form-induced, and spatially-averaged turbulent stresses and energies for mobile-boundary flows
T0 θs(xi, t) = φT(xi, t)θ(xi, t) where θ is velocity, pressure or any other hydrodynamic variable, overbar denotes the time-averaging operation, superscript s denotes superficial averaging, xi and t are space and time coordinates, τ is a “local” time coordinate used in the integrand, γ is an indicator of the point occupancy by the fluid (i.e. γ = 1 when the domain is occupied by the fluid and γ = 0 otherwise), Tf refers to the sum of periods during which a point xi is occupied by fluid, and T0 is the total averaging time (Tf ≤ T0)
Summary
River, coastal and atmospheric flows frequently exhibit high levels of multi-scale spatial heterogeneity due to geometrically complex and, occasionally, mobile boundaries. Giménez-Curto & Lera, 1996; Monin & Yaglom, 1971; Raupach & Shaw, 1982) Equations for these fluid stresses, known as second-order equations, represent an important part of the theoretical background underpinning modern studies of turbulent wall-bounded flows. All three types of equations (for mean, turbulent, and form-induced stresses) are important as they underpin any consideration of energy fluxes in turbulent flows over complex boundaries, and are required to provide a proper theoretical foundation for dealing with mobile-boundary flows In response to this need, the current paper presents the derivation of equations describing the balances of the mean, form-induced, and spatially-averaged turbulent stresses and energies for mobile-boundary flows. Appendix 1 provides supplementary details on the conditions and theorems that are necessary for averaging operations employed in the derivation process outlined in Appendix 2
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