The aim of the present work is to perform Direct Numerical Simulations (DNS) of high-speed turbulent boundary layers over flat plates subjected to favorable and adverse pressure gradients. The computational setup for such problems is particularly challenging due to the intrinsically hyperbolic nature of the compressible Navier–Stokes equations. Pressure gradients can be introduced either by shaping the upper boundary (inviscid or viscous wall) [1] or by prescribing suitable boundary conditions at the top of the computational domain [2]. In the first case, acoustic waves generated by the boundary layer—especially the leading wave originating at the inflow edge—are reflected by the top wall, contaminating the spatially developing flow field. In the second case, the main challenge lies in prescribing proper boundary conditions that ensure a well-posed problem, which is particularly difficult to achieve. In this work, we provide a numerical framework based on the setup of zero-pressure-gradient turbulent boundary layers developing over flat plates, in which we augment the governing equations with tailored forcing terms in the mass, streamwise momentum, and energy equations. These terms are constructed based on the “quasi-1D” compressible Euler equations, such that they replicate the effect of a suitably designed inviscid upper wall, without the associated negative consequences. References [1] Wang, X., Wang, Z., Sun, M., Wang, Q., and Hu, Z. (2019). Direct numerical simulation of a supersonic turbulent boundary layer subject to adverse pressure gradient induced by external successive compression waves. AIP Advances, 9 (8): 085215 [2] Wenzel, C., Gibis, T., Kloker, M., and Rist, U. (2019). Self-similar compressible turbulent boundary layers with pressure gradients. Part 1. Direct numerical simulation and assessment of Morkovin’s hypothesis. Journal of Fluid Mechanics, 880, 239–283