A Reynolds shear stress model for constant-pressure boundary layers

Closure models for the Reynolds shear stress in wall-bounded flows are mostly based on various turbulence quantities like the turbulent kinetic energy (k), its dissipation rate (ϵ), the Reynolds stress ( v r 2) in the wall-normal direction, the mean flow velocity gradient (S), the normalized strain rate ( S k / ϵ ), etc.; vr is the intensity of the fluctuating velocity component in the wall-normal direction (y). Close to the wall, ( u ′ v ′ ¯ ) is known to vary with y approximately as ∼ y 3. With an emphasis on this near-wall feature, a closure model is proposed here in terms of k, ( v r / u r ), and a weighted mean strain rate ( S k v r + / ϵ ); ur is the intensity of the fluctuating velocity component in the streamwise direction. The choice of ur in the present proposal is in view of the fact that, although the energy associated with u r 2 in a simple flow, for example, is distributed to v r 2 by fluctuating pressure, it has never been used in past closure models. In terms of the known approximate variations of k , v r , u r, etc., with y in the near-wall region, the present model suggests ( u ′ v ′ ¯ ) ∼ y 3.25 and so nearly retains its known approximate variation ∼ y 3 (for small y). It is suggested here that v r + in the weighted strain rate damps the strain rate ( S k / ϵ ) significantly in the near-wall region and plays a similar role in Durbin's model [“Near-wall turbulence closure modeling without damping functions,” Theor. Comput. Fluid Dyn. 3, 1 (1991)]. It is also reported here that v r ≈ k 1 / 2 and u r ≈ ( ϵ / S ) 1 / 2 in the outer region, and these velocity scales are associated with ( S k / ϵ ) ∼ constant in this region; alternatively, in terms of these velocity scales, the normalized strain rate ( S k / ϵ ) is the ratio of the Reynolds stresses ( v r 2 / u r 2 ) in the outer region.