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 DUw=2Uw and pressure gradient dP=dx) 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 , and h is the half-height of the channel. This leads to a onedimensional family of flows of varying Reynolds number Re=Uh/nu. We apply three codes, and cover three Reynolds numbers stepping by a factor of 2 each time. The agreement between codes is very good. The theoretical questions involve 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 and h; universality is not observed for all the Reynolds stresses. The FPG wall region obeys the classical law of the wall, again for velocity and shear stress, but suggesting a low value for the Karman constant k, between 0.36 and 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, is ambiguous. Our results are very consistent with both of Stranford’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, and we know the constants of the law within a 10% bracket. On the other hand, again that does not extend to all the Reynolds stresses.