Turbulence Modelling: An inverse problem or a forward dilemma?

Turbulence modelling, and fluid flows in general, represent some of the most complex problems in nature. Although analytical solutions elide all but the simplest of problem statements, the advent of ever more powerful supercomputers has made large-scale problems approachable via numerical predictions. The most widely used approximation is the Reynolds-averaging of the Navier-Stokes equations (Favre-averaging in the case of compressible flows). RANS significantly reduces the computational cost but introduces the need of a closure model for the Reynolds stress terms that arise from the derivation of the time-averaged N-S eqns. The closure problem presents a research challenge often trading quality for performance. More recently, there has been a renewed interest in using Machine Learning techniques to approach physical problems. In this work, we will explore two approaches to the closure problem, an inverse problem formulation (data-driven) and a forward problem formulation (Neural PDE Solver). We will assess both the complexity of implementation and the accuracy of the approximation via Direct Numerical Simulation of compressible, turbulent boundary layer over adiabatic flat plates at Mach numbers of 0.8, 1.6, 2.86 and 5.