Self-similar turbulent boundary layer in pressure gradient. Four flow regimes

Abstract

Self-similar flows in a turbulent boundary layer when the free-stream velocity is specified as a power, with the exponent m, function of longitudinal coordinate are investigated. The self-similar formulation not only simplifies solving of the problem by reducing the equations of motion to ordinary differential equations but also provides a mean for formulating closure conditions. It is shown that for the class of flows under consideration that depend on three governing parameters the dimensionless mixing length is a function of the normalized distance from the wall and the exponent m in the outer region and a universal function of local Reynolds number in the wall region, the latter corollary being true even when the skin friction vanishes. In calculations this function is set to be independent of pressure gradient, which gives the results very close to experimental data.There exist four different self-similar flow regimes. Each regime is related to its similarity parameter, one of which is the well-known Clauser equilibrium parameter and the other three are established for the first time. In case of adverse pressure gradient when the exponent m lies within certain limits, which depend on Reynolds number, the problem has two solutions with different values of the boundary layer thickness and the skin friction, which points out the possibility of hysteresis in near-separating flow. Separation occurs not at the minimal value of m that corresponds to the strongest adverse pressure gradient but at m = −0.216 − 0.4Rep−1/3 + O(Rep−2/3), where Rep is the Reynolds number based on the longitudinal pressure gradient. The results of the theory are in good agreement with experimental data.