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): <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mfrac> <m:mrow> <m:mi mathvariant="normal">d</m:mi> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> </m:msup> </m:mrow> <m:mrow> <m:mi mathvariant="normal">d</m:mi> <m:msup> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> </m:msup> </m:mrow> </m:mfrac> <m:mo>+</m:mo> <m:msup> <m:mrow> <m:mi>κ</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:msup> <m:mrow> <m:msup> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> </m:msup> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:msup> <m:mrow> <m:mfenced close=")" open="("> <m:mrow> <m:mfrac> <m:mrow> <m:mi mathvariant="normal">d</m:mi> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> </m:msup> </m:mrow> <m:mrow> <m:mi mathvariant="normal">d</m:mi> <m:msup> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> </m:msup> </m:mrow> </m:mfrac> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:math> $\frac{\mathrm{d}{u}^{+}}{\mathrm{d}{y}^{+}}+{\kappa }^{2}{{y}^{+}}^{2}{\left(\frac{\mathrm{d}{u}^{+}}{\mathrm{d}{y}^{+}}\right)}^{2}=1$ 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.