Mustafa Engin Danis

Compressibility transformations are widely used to relate hypersonic zero-pressure-gradient turbulent boundary layers to incompressible reference states, but they are usually assessed only by the collapse of transformed mean velocity profiles alone. Here, a stricter consistency requirement is proposed: the transformed velocity profile is required to agree pointwise with the profile obtained from a fixed incompressible inner–outer model at the same transformed wall-normal coordinate. A second requirement is that the same coordinate should also improve the collapse of semilocal eddy viscosity and turbulent kinetic energy production profiles without modifying their traditional definitions. Using hypersonic direct numerical simulation data under these criteria, existing transformations are shown to incur velocity errors of 1–28%, especially for strongly cooled walls. A new forward transformation is formulated by correcting the semilocal coordinate and adding a corrected wake contribution to the transformed velocity. The proposed transformation satisfies the proposed consistency requirements for a wide range of hypersonic zero-pressure-gradient turbulent boundary-layer cases with 1.2–4.5% velocity errors with minimal calibration. Embedded in an inverse incompressible-to-compressible framework driven by freestream and wall conditions at a prescribed boundary-layer thickness, this model reconstructs key boundary-layer parameters, skin friction, and mean velocity profiles with 2.2–7.5% velocity errors, demonstrating its predictive capability for cold-wall hypersonic turbulent boundary layers.

In this paper, we introduce high-order tensor-train (TT) finite-volume methods for the shallow-water equations (SWEs). We present the implementation of the third-order upwind and the fifth-order upwind and weighted essentially nonoscillatory (WENO) reconstruction schemes in the TT format. It is shown in detail that the linear upwind schemes can be implemented by directly manipulating the TT cores, while the WENO scheme requires the use of TT cross interpolation for the nonlinear reconstruction. In the development of numerical fluxes, we directly compute the flux for the linear SWEs without using TT rounding or cross interpolation. For the nonlinear SWEs where the TT reciprocal of the shallow water layer thickness is needed for fluxes, we develop an approximation algorithm using Taylor series to compute the TT reciprocal. The performance of the TT finite-volume solver with linear and nonlinear reconstruction options is investigated under a physically relevant set of validation problems. In all test cases, the TT finite-volume method maintains the formal high-order accuracy of the corresponding traditional finite-volume method. In terms of speed, the TT solver achieves up to 124 times acceleration of the traditional full-tensor scheme.

This study highlights the importance of satisfying the eddy viscosity equivalence below the logarithmic layer, to deriving accurate compressibility transformations. First, we analyze the ability of known transformations to satisfy the eddy viscosity equivalence and show that the accuracy of these transformations is strongly dependent on this ability. Second, in a step-by-step manner, we devise new transformations that satisfy this hypothesis. An approach based on curve fitting of the incompressible Direct Numerical Simulation data for eddy viscosity profiles below the logarithmic layer provides an extremely accurate transformation, which motivates self-contained methods, making use of mixing length formulas in the inner region. It is shown that the accuracy of existing transformations can be significantly improved by applying these ideas, below the logarithmic layer. Motivated by the effectiveness of the formulations derived from eddy viscosity equivalence, we introduce a new integral transformation based on Reynolds number equivalence between compressible and incompressible flows. This approach is based on defining a new compressible velocity scale, which affects the accuracy of transformations. Several choices for the velocity scale are tested, and in each attempt, it is shown that the eddy viscosity equivalence plays a very important role for the accuracy of compressibility transformations.

A compressibility correction for k−ω models is proposed, and it explicitly takes the effects of high Mach numbers and wall-cooling into account. The correction is validated by a recent set of direc...