Closed form solution of plane turbulent boundary flow – solution process, validity conditions and extension

Abstract The Prandtl mixing length plane turbulent boundary flow is governed by the nonlinear ordinary differential equation (ODE): d u + d y + + κ 2 y + 2 d u + d y + 2 = 1 $\frac{\mathrm{d}{u}^{+}}{\mathrm{d}{y}^{+}}+{\kappa }^{2}{{y}^{+}}^{2}{\left(\frac{\mathrm{d}{u}^{+}}{\mathrm{d}{y}^{+}}\right)}^{2}=1$ with boundary condition y + = 0 : u + = 0. Although van Driest first derived the closed-form solution for this ODE in 1956 (E. R. van Driest, “On turbulent flow near a wall,” J. Aeronaut. Sci. , vol. 23, no. 11, pp. 1007–1011, 1956), but the lack of detailed solution process led to its long-term neglect in subsequent fluid mechanics textbooks and research. As a result, the problem of finding the solution has remained an open question in the community. In this work, we independently rederive the solution and provide a complete, step-by-step solution process. Furthermore, we explicitly clarify the validity conditions of the solution (i.e., smooth wall, zero pressure gradient, fully developed turbulent flow, incompressible flow) and propose a modified analytical extension that conforms well to direct numerical simulation (DNS) results. The proposed modification incorporating van Driest-type damping into the solution framework is physically motivated, and this extension is semi-empirical, thus should be interpreted as a model enhancement/refinement rather than an exact mathematical solution. The present work not only supplements the missing solution details of van Driest’s solution but also improves its applicability.