Determine the relationship between thermal and momentum roughness functions

Determine whether the thermal roughness function ΔΘ⁺ exhibits a linear mapping to the momentum roughness function Δu₁⁺ for compressible turbulent boundary layers over sinusoidal roughness, and establish the precise dependence of ΔΘ⁺ on the wall-to-recovery temperature ratio T_w/T_r.

Background

The paper analyzes thermal roughness functions for Mach 2.5 turbulent boundary layers over three-dimensional sinusoidal roughness at three wall-to-recovery temperature ratios. Using the Liang et al. temperature transformation and the Griffin et al. velocity transformation, the authors report a tentative linear regression between the thermal and momentum roughness functions, but emphasize that the available dataset is too limited to establish whether this relationship is genuinely linear.

The authors also observe an approximately linear dependence of the thermal roughness function on T_w/T_r. Because the cold-wall temperature transformations contain singularities and the reported data include only three rough-wall thermal conditions, additional simulations or measurements are needed to determine both relationships reliably.

References

From the limited dataset it is not possible to conclude whether the thermal roughness functions exhibit linear mapping to the momentum roughness functions. Additionally, the thermal roughness function exhibits an approximately linear dependence on $T_w/T_r$, specifically $\Delta \Theta+ = 3.47\frac{T_w}{T_r} + 3.91$, with coefficient of determination $R2=0.972$. However, additional data points are necessary to confirm the exact relationship.

— Influence of wall thermal boundary condition on mean-flow characteristics of a Mach 2.5 fully rough turbulent boundary layer  (2609.09341 - Braga et al., 8 Sep 2026) in Section 6, subsection “Inner Layer and Roughness Function Analysis”