Direct Numerical Simulation and Theory of a Wall-Bounded Flow with Zero Skin Friction

We study turbulent plane Couette-Poiseuille (CP) flows in which the conditions (relative wall velocity ΔU _w ≡ 2U _w , pressure gradient dP/dx and viscosity ν) are adjusted to produce zero mean skin friction on one of the walls, denoted by APG for adverse pressure gradient. The other wall, FPG for favorable pressure gradient, provides the friction velocity u_τ , and h is the half-height of the channel. This leads to a one-dimensional family of flows of varying Reynolds number Re ≡ U _w h/ν. We apply three codes, and cover three Reynolds numbers stepping by a factor of 2 each time. The agreement between codes is very good, and the Reynolds-number range is sizable. The theoretical questions revolve around Reynolds-number independence in both the core region (free of local viscous effects) and the two wall regions. The core region follows Townsend's hypothesis of universal behavior for the velocity and shear stress, when they are normalized with u_τ and h; universality is not observed for all the Reynolds stresses, any more than it is in Poiseuille flow or boundary layers. The behavior at very high Re is unknown. The FPG wall region obeys the classical law of the wall, again for velocity and shear stress, but could suggest a low value for the Karman constant κ, possibly near 0.37. For the APG wall region, Stratford conjectured universal behavior when normalized with the pressure gradient, leading to a square-root law for the velocity. The literature, also covering other flows with zero skin friction, is ambiguous. Our results are very consistent with both of Stratford's conjectures, suggesting that at least in this idealized flow geometry the theory is successful like it was for the classical law of the wall. We appear to know the constants of the law within a 10% bracket. On the other hand, again that does not extend to Reynolds stresses other than the shear stress, but these stresses are passive in the momentum equation.