On the Accuracy of Compressibility Transformations

TL;DR

This paper investigates the importance of eddy viscosity equivalence in compressibility transformations and introduces new, more accurate transformations based on this principle and Reynolds number equivalence.

Contribution

It develops new compressibility transformations that satisfy eddy viscosity equivalence, significantly improving accuracy below the logarithmic layer.

Findings

Existing transformations' accuracy depends on eddy viscosity equivalence

New transformations based on eddy viscosity fitting are highly accurate

Reynolds number-based integral transformation enhances compressibility modeling

Abstract

This study highlights the importance of satisfying the eddy viscosity equivalence below the logarithmic layer, to deriving accurate compressibility transformations. First, we analyze the ability of known transformations to satisfy the eddy viscosity equivalence and show that the accuracy of these transformation is strongly dependent on this ability. Secondly, in a step-by-step manner, we devise new transformations that satisfy this hypothesis. An approach based on curve fitting of the incompressible Direct Numerical Simulation data for eddy viscosity profiles below the logarithmic layer provides an extremely accurate transformation. That motivates self-contained methods, making use of mixing length formulas, in the inner region. It is shown that the accuracy of existing transformations can be significantly improved by applying these ideas, below the logarithmic layer. Motivated by the effectiveness of the formulations derived from eddy viscosity equivalence, we introduce a new integral transformation based on Reynolds number equivalence between compressible and incompressible flow. This approach is based on defining a new compressible velocity scale, which affects the accuracy of transformations. Several choices for the velocity scale are tested, and in each attempt, it is shown that the eddy viscosity equivalence plays a very important role for the accuracy of compressibility transformations.