Field inversion for transitional flows using continuous adjoint methods

Transition modeling represents one of the key challenges in computational fluid dynamics. While numerical efforts were traditionally devoted to either improving Reynolds-averaged Navier–Stokes-based turbulence modeling or developing scale-resolving simulations, cautious attention has been recently given to field inversion and machine learning techniques. This paper discusses an updated development of field inversion model for transitional flows based on k–ω shear stress transport model using the continuous adjoint approach, instead of the typical discrete adjoint method. The original model is modified by multiplying the production term of the turbulent kinetic energy equation by a spatially varying discrepancy function η(x). The adjoint equations and the relevant boundary conditions are specifically derived and integrated in OpenFOAM. The present model is validated using two zero pressure-gradient and four non-zero pressure-gradient from flow-over-flat-plate T3-series test cases. The gradient descent method is employed in the optimization process to minimize the discrepancy in the calculated shear stress. The inferred solution indicates a smooth transition to turbulence at the reported critical Reynolds numbers. The optimized model significantly improves the predictions of skin-friction coefficients, originally incorporated in the objective function. To demonstrate the usefulness of the present approach, the investigation is further extended to determine both velocity and shear Reynolds-stress profiles, which to our knowledge has not been reported before. Furthermore, a reduction in the percentage error from 50.2% to 7.3% is well observed in the predicted boundary layer thickness considering the laminar regime in the T3C5 test case.