Direct Numerical Simulation of turbulent boundary layers requires a high spatial resolution specially near the wall to capture the different length scales inherent of turbulence. To avoid the heavy time consuming related to the high spatial resolution, we investigate techniques that completely bypass the wall region by using an artificial boundary condition at a bound of the computational domain. The present work focuses on a POD reconstruction strategy to define specific synthetic boundary conditions capable of fulfilling the correct characteristics and dynamics of the turbulence in both time and space in the near wall region.

In a reference simulation (domain including the wall region), the flow field can be decomposed as the superposition of the POD spatial modes with time-varying amplitudes. Our goal is to predict the boundary conditions on the bounds of a reduced domain that discards the computationally expensive wall region. In the compressible regime, the flow field on the boundaries are determined following the Riemann invariants and, depending on the sign of the characteristic velocities, the solution is either calculated from the solution inside the domain or prescribed by using the boundary conditions. To evaluate the boundary conditions in the reduced domain, the POD time coefficients are built by projecting the flow variables onto the spatial modes. POD bases relative to both primitive and conservative variables have been reviewed. We found that the best result is obtained from conservative variables because this choice enforces conservation of the mass flux. A rescaling procedure has however been used to ensure that the rms of the reconstructed compressible variables match the rms reference values on the reconstructed plane.

A DNS of the complete channel flow has first been performed at Mach=0.5, a wall temperature of 300 K and Re=3000, to compute the POD basis. Reduced channel flow computations have then been conducted with POD-based synthetic boundary conditions implemented at a specific height above the solid walls (z+ = 18). Let us note that, in the reduced channel flow computations, the upper and the lower wall layers have been both discarded up to these altitudes. The mean quantities of the density and the temperature are very well recovered up to a time of 36 time scales (H/U0). Regarding the velocity, the present method is able to well recover the log law in the longitudinal velocity profile. As far as the rescaling procedure is employed, the rms values are also rather well predicted. The largest discrepancies are found on the rms of the wall-normal velocity (w), which is a quantity strongly dominated by the small scales and therefore sensitive to the accuracy of the estimation. The shear stress matches rather well the reference in the middle part of the channel. In the vicinity of the upper and lower bounds, large discrepancies are however recorded. Later on, continuing the simulation up to 54 time scales, the discrepancies with the reference increase for the Reynolds stress components. These discrepancies are attributed to the lack of correlation between conservative variables. A new method of rescaling is under study to solve the problem. Simulations were also run with a synthetic boundary condition prescribed at a higher altitude (i.e. z+=54) that are in good agreement with the reference for shorter integration times(6 time scales). More complete results will be presented and discussed at the conference.