An excellent closed-form approximate solution to the Blasius equation

Abstract An approximate analytical solution for the laminar boundary layer on a flat plate governed by the Blasius equation is presented. Although the numerical solution of the Blasius equation is well established, no closed-form analytical solution has yet been found. In this study, the approximate analytical solutions are developed by extending Lin's (1999) exponential formulation of the dimensionless streamwise velocity to include quadratic and cubic exponents. Both proposed forms satisfy the boundary conditions at the wall and the free-stream, and the corresponding analytical expressions for the dimensionless stream function, shear stress, and cross-stream velocity are derived. Comparison with the numerical Blasius solution shows that the quadratic form provides a reasonable approximation, while the cubic form exhibits excellent agreement over the entire range of the similarity variable η which is defined as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>η</m:mi> <m:mo>=</m:mo> <m:mi>y</m:mi> <m:msqrt> <m:mrow> <m:msub> <m:mrow> <m:mi>U</m:mi> </m:mrow> <m:mrow> <m:mi>∞</m:mi> </m:mrow> </m:msub> <m:mo>/</m:mo> <m:mfenced close=")" open="("> <m:mrow> <m:mi>ν</m:mi> <m:mi>x</m:mi> </m:mrow> </m:mfenced> </m:mrow> </m:msqrt> </m:math> $\eta =y\sqrt{{U}_{\infty }/\left(\nu x\right)}$ , where x and y are the wall-parallel and wall-normal coordinates respectively, U ∞ is the free-stream velocity and ν is the kinematic viscosity. The maximum residual of the cubic solution is on the order of 10 −3 at the wall, indicating high accuracy. The proposed cubic analytical solution thus provides a compact and explicit expression that closely represents the Blasius solution, offering a practical alternative for engineering applications requiring analytical evaluation of boundary-layer quantities.