Inner–Outer Layer Matching for Improved Compressibility Transformations

We investigate the role of eddy viscosity equivalence below the logarithmic layer in achieving accurate compressibility transformations. Existing methods are evaluated for their ability to satisfy this condition, revealing a strong correlation between transformation accuracy and adherence to eddy viscosity equivalence. Building on this insight, we develop new transformations through a stepwise approach. A highly accurate formulation is obtained by fitting eddy viscosity profiles from incompressible Direct Numerical Simulation data in the inner region, motivating self-contained models based on mixing length formulations. Applying these ideas below the logarithmic layer significantly improves the performance of conventional transformations. We further propose an integral transformation based on Reynolds number equivalence between compressible and incompressible flows. This framework introduces a modified compressible velocity scale, and several variants are assessed. Finally, we study the effects of the outer layer by implementing a transformation that combines Van Driest damping in the inner region with Cole's wake function in the outer region. Across all cases, the results consistently underscore the importance of eddy viscosity equivalence in improving transformation accuracy.