Numerical simulations are employed to investigate the turbulent kinetic energy (TKE) budgets in turbulent channel flows with pressure gradients and separation. Incompressible, highly resolved large eddy simulations are performed for Re_τ = 170 and 615 to investigate the flow developing along a convergent-divergent channel. The aim of this work is to analyze the TKE budgets both in physical and Fourier spaces to characterize the important scales in the individual processes in such turbulent flows. The study is performed for different positions along the channel where favorable and adverse pressure gradients are present. Proper orthogonal decomposition is employed to understand the role of the most energetic structures in the TKE budgets. Results indicate that such structures account for most of the turbulent effects present in the flow, except for the transport term. A spectral TKE equation in Fourier space is developed for flows with one homogeneous direction to characterize the turbulent processes as a function of the wavelength in the channel spanwise direction. The results show that viscous effects occur at the same range of wavelengths for which production is found and that TKE is transported to the near-wall region, being dissipated by large spanwise scale motion. They also show that favorable pressure gradients change the distribution of processes along the spanwise wavelengths. In the adverse pressure gradient region, TKE is transported both toward the wall and toward the center of the channel, where it is balanced by the advection term.

This paper presents the effects of some convective flux computation schemes on boundary layer and shocked flow solutions. Second-order accurate centered and upwind convective flux computation schemes are discussed. The centered Jameson scheme, plus explicitly added artificial dissipation terms are considered. Three artificial dissipation models, namely a scalar and a matrix version of a switched model, and the CUSP scheme are available. Some implementation options regarding these methods are proposed and addressed in the paper. For the upwind option, the Roe flux-difference splitting scheme is used. The CUSP and Roe schemes require property reconstructions to achieve second-order accuracy in space. A multidimensional limited MUSCL interpolation method is used to perform property reconstruction. Extended multidimensional limiter formulation and implementation are here proposed and verified. Theoretical flow solutions are used in order to provide a representative testbed for the current study. It is observed that explicitly added artificial dissipation terms of the centered scheme may nonphysically modify the numerical solution, whereas upwind schemes seem to better represent the flow structure.