Remarks on the Equilibrium Turbulent Boundary Layer

Abstract

Two similarity laws are known for the mean-velocity profile in a turbulent boundary layer with constant pressure. These are Prandtl's law of the wall and Karman's momentum-defect law. The first law has recently been generalized empirically to flows with arbitrary pressure gradient by Ludwieg and Tillmann, and the second law to a certain class of equilibrium flows by F. Clauser. In the present paper it is shown that the pressure distribution corresponding to a given equilibrium flow cam be computed by assuming that a certain parameter D = (τ_w/q)dq/dτ_w is constant, where q and τ_w are the dynamic pressure in the free stream and the shearing stress at the wall, respectively. The hypothesis D = constant is suggested by a study of the integrated continuity equation and is supported by a rigorous analogy between the class of equilibrium flows defined by Clauser and the class of laminar flows studied by Falkner and Skan. The hypothesis D = constant is also verified using experimental data for several equilibrium turbulent flows and is interpreted physically from a kinematic point of view. Two hypothetical limiting cases of equilibrium flow are described. At one extreme is the boundary Layer in a sink flow, with a completely logarithmic mean-velocity profile outside the sublayer. At the other extreme is a continuously separating boundary layer in a dimensionless pressure gradient (x/q)dq/dx approximately twice that for the corresponding laminar flow. Typical shearing-stress profiles are computed for several equilibrium turbulent flows, including the two limiting cases.