Turbulence in supersonic channel flow is studied using direct numerical simulation. The ability of outer and inner scalings to collapse profiles of turbulent stresses onto their incompressible counterparts is investigated. Such collapse is adequate with outer scaling when sufficiently far from the wall, but not with inner scaling. Compressibility effects on the turbulent stresses, their anisotropy, and their balance equations are identified. A reduction in the near-wall pressure-strain, found responsible for the changed Reynolds-stress profiles, is explained using a Green's-function-based analysis of the pressure field.

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Direct numerical simulation of flow past a sphere in a stratified fluid is carried out at a subcritical Reynolds number of 3700 and $Fr=U_{\infty }/ND=1,2$ and 3 to understand the dynamics of moderately stratified flows with $Fr=O(1)$. Here, $U_{\infty }$ is the free stream velocity, $N$ is the background buoyancy frequency and $D$ is the sphere diameter. The unstratified flow past the sphere consists of a separated shear layer that transitions to turbulence, a recirculation zone and a wake with a mean centreline deficit velocity, $U_{0}$, that decreases with downstream distance as a power law. With increasing stratification, the separated shear layer plunges inward vertically and its roll up is inhibited, the recirculation zone is shortened and the mean wake decays at a slower rate of $U_{0}\propto (x_{1}/D)^{-0.25}$ in the non-equilibrium (NEQ) region. The transition from the near wake where $U_{0}$ has a decay rate similar to the unstratified case to the NEQ regime occurs as an oscillatory modulation by a steady lee wave pattern with a period of $t=2\unicode[STIX]{x03C0}/N$ that leads to accelerated $U_{0}$ between $Nt=\unicode[STIX]{x03C0}$ and approximately $Nt=2\unicode[STIX]{x03C0}$. Far downstream, the wake is dominated by coherent horizontal motions. The acceleration of $U_{0}$ by the lee wave and the lower turbulence production in the NEQ regime, thereby less loss to turbulence, prolongs the lifetime of the wake relative to its unstratified counterpart. The intensity, temporal spectra and structure of turbulent fluctuations in the wake are assessed. Buoyancy induces significant anisotropy among the velocity components and between their vertical and horizontal profiles. Consequently, the near wake ($x_{1}/D<10$) exhibits significant differences in turbulence profiles relative to its unstratified counterpart. Spectra of vertical velocity show a discrete peak in the near wake that is maintained further downstream. The turbulent kinetic energy (TKE) balance is computed and contributions from pressure transport and buoyancy are found to become increasingly important as stratification increases. The findings of this investigation will be helpful in designing accurate initial conditions for the temporally evolving model of stratified wakes.

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