DNS of compressible turbulent boundary layers: pressure-gradient influence and self-similarity

Direct numerical simulations (DNS) of spatially-evolving compressible turbulent boundary layers have been performed for the zero-pressure-gradient (ZPG) case and various pressure-gradient (PG) cases, both for sub- and supersonic Mach numbers. Whereas it was the main objective of the ZPG cases to provide data with a maximum of reliability to improve the current state of available DNS data, the PG cases are aimed to achieve self-similarity in streamwise direction, the first time for the compressible regime. By using the PG data sets for its validation, a self-similarity theory has been derived for the compressible regime, which only has been available for the incompressible regime so far. The ZPG study is presented for a fine-meshed range of Mach numbers from 0.3 to 2.5. All data are compared to the literature database where significant data scatter can be observed. The wake region of the mean velocity profile is observed to scale much better with the momentum-thickness Reynolds number calculated with the far-field viscosity than with the wall viscosity. The time-averaged velocity fluctuations, density-scaled according to Morkovin’s hypothesis, are found to be noticeably influenced by compressibility effects in the inner layer and in the wake region. As a basis for the PG study, the conditions for self-similarity have been theoretically derived for the compressible regime. It is shown that self-similarity can be obtained for the most important terms of the Favre-averaged momentum and energy equations in the outer layer. Indicating the state of self-similarity for the compressible regime, a compressible pressure-gradient parameter has been formulated. The PG study has been performed for inflow Mach numbers of 0.5 and 2.0, both for favorable and adverse PG (FPG, APG) distributions. Representing the fundamental assumption for both van-Driest’s and Morkovin’s compressibility scaling, the total shear-stress distributions near the wall are not constant for PG cases anymore, but seem to be almost unaffected by compressibility. Hence, van-Driest’s transformation can be assumed to be still approximately valid. Since compressibility and PG effects have been found to be almost uncoupled for the turbulent shear stress, also the spirit of Morkovin’s hypothesis is expected to still hold for the PG cases investigated.