Skin friction prediction for attached flows based on two-dimensional inviscid solutions

Boundary layer theory and its analytical methods for skin friction coefficients provide an important basis for aerodynamic analysis.

However, classical analytical formulas are mostly limited to flat-plate flows.

High-fidelity numerical simulations are not only computationally expensive but also yield predictions that are highly sensitive to physical models, numerical schemes, and grid resolution.

To overcome these limitations, symbolic AI opens a new pathway to discover novel laws of complex physical systems from data.

Using limited data from surface solutions of the Euler equations and the skin friction coefficient from viscous flows over airfoils, we employ symbolic regression to progressively discover a generalizable, interpretable analytical formula chain for fast skin friction prediction in subsonic and supersonic attached flows.

From the perspective of physical mechanisms, the discovered analytical expression chain reveals scaling laws for skin friction at different Mach numbers: the basic form captures the logarithmic decay of skin friction along the streamwise direction in the turbulent boundary layer; the inclusion of a pressure coefficient correction term quantifies the effect of surface pressure variation; and the Mach number correction term evolves with flow regimes, transitioning from the compressibility correction term in subsonic regimes to the thermodynamic effects term in supersonic and hypersonic regimes.

This knowledge chain exhibits a unified structure across different Mach numbers, and omitting the correction terms under certain conditions recovers classical theoretical forms, further demonstrating its physical consistency.

Validation against typical geometries shows that this analytical formula chain achieves a low average integrated skin friction drag prediction error, with good generalization capability across different freestream conditions and geometric shapes.