On the Kármán momentum-integral approach and the Pohlhausen paradox

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(ξ) = u/U. Here, ξ = y/δ 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 behavior but also 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 = 2ξ − 2ξ3 + ξ4. For example, our alternative profile, F = (5ξ − 3ξ3 + ξ4)/3, leads to an order-of-magnitude improvement in precision when incorporated into the Kármán–Pohlhausen approach in both viscous and thermal analyses. Then, noting the significance of the Blasius constant, s¯≈1.630 398, this approach is extended to construct a set of uniformly valid solutions, including F=1−exp[−s¯ξ(1+12s¯ξ+ξ2)], which continues to hold beyond the boundary-layer edge as y → ∞. Given its substantially reduced error, the latter is shown, through comparisons to other models, to be practically equivalent to the Blasius solution.