The Prandtl Plus Scaling Approximation for Turbulent Boundary Layer Flows

TL;DR

This paper critically examines the Prandtl Plus scaling for turbulent boundary layers, demonstrating it lacks theoretical justification due to the absence of whole profile similarity in such flows, supported by simulations and experimental insights.

Contribution

The paper reveals the theoretical limitations of Prandtl Plus scalings and proposes that their applicability is only approximate due to the lack of whole profile similarity in turbulent boundary layers.

Findings

Prandtl Plus scalings do not satisfy flow similarity principles.

Alternative scalings also require whole profile similarity, which is not observed.

Simulations support the theoretical argument against the universal applicability of these scalings.

Abstract

Using the flow governing equation approach to similarity, Weyburne (D. Weyburne, arXiv:1701.02364, 2016) recently showed that for 2-D turbulent boundary layer flows, the Prandtl Plus scalings are NOT, in general, the proper similarity scaling parameters. Based on that failure, Weyburne proposed alternative length and velocity scaling parameters that satisfy the flow governing approach to similarity for external turbulent boundary layer flows. Herein, we show that both the proposed new scalings and the Prandtl Plus scalings have a theoretical underpinning which requires that the boundary layer must show whole profile similarity for the scalings to be applicable. The problem is that wall-bounded turbulent boundary layers are generally acknowledged as not showing whole profile similarity under any circumstances. Thus, applying scaling parameters that subsume whole profile similarity to boundary layers that do not show whole profile similarity means that, at best, the result represents a good approximation. Computer simulation results are used to explore these issues and provides at least preliminary experimental support of the theoretical failures.