On the K\'arm\'an momentum-integral approach and the Pohlhausen paradox

TL;DR

This paper analyzes the Kármán momentum-integral approach, explains the Pohlhausen paradox, and develops improved velocity profiles that enhance the accuracy of boundary-layer predictions and extend solutions beyond classical limits.

Contribution

It identifies the reasons behind the Pohlhausen polynomial paradox and proposes optimal piecewise velocity profiles to improve boundary-layer analysis accuracy.

Findings

Improved velocity profiles reduce prediction errors by an order of magnitude.

A rational explanation for the Pohlhausen paradox is provided.

New solutions extend boundary-layer models beyond classical limits.

Abstract

This work explores simple relations that follow from the momentum-integral equation absent a pressure gradient. The resulting expressions enable us to relate the boundary-layer characteristics of a velocity profile, $u(y)$, to an assumed flow function and its wall derivative relative to the wall-normal coordinate, $y$. Consequently, disturbance, displacement, and momentum thicknesses, as well as skin friction and drag coefficients, which are typically evaluated and tabulated in classical monographs, can be readily determined for a given profile, $F(\xi)=u/U$. Here $\xi=y/\delta$ denotes the boundary-layer coordinate. These expressions are then employed to provide a rational explanation for the 1921 Pohlhausen polynomial paradox, namely, the reason why a quartic representation of the velocity leads to less accurate predictions of the disturbance, displacement, and momentum thicknesses than using cubic or quadratic polynomials. Not only do we identify the factors underlying this behaviour, we proceed to outline a procedure to overcome its manifestation at any order. This enables us to derive optimal piecewise approximations that do not suffer from the particular limitations affecting Pohlhausen's $F(\xi)=2\xi-2\xi^3 +\xi^4$. For example, our alternative profile, $F(\xi)=(5\xi-3\xi^3+\xi^4)/3$, leads to an order-of-magnitude improvement in precision when incorporated into the K\'arm\'an-Pohlhausen approach in both viscous and thermal analyses. Then noting the significance of the Blasius constant, $\bar{s} \approx 1.630398$, this approach is extended to construct a set of uniformly valid solutions, including $F(\xi)=1-\exp[-\bar{s}\xi(1+\bar{s}\xi/2+\xi^2)]$, which continues to hold beyond the boundary-layer edge as $y\rightarrow\infty$. Given its substantially reduced error, the latter is shown, through comparisons to other models, to be practically equivalent to the Blasius solution.