In this paper, the effects of adiabatic and isothermal conditions on the statistics in compressible turbulent channel flow are investigated using direct numerical simulation (DNS). DNS of two compressible turbulent channel flows (Cases 1 and 2) are performed using a mixed Fourier Galerkin and B-spline collocation method. Case 1 is compressible turbulent channel flow between isothermal walls, which corresponds to DNS performed by Coleman et al. (1995). Case 2 is the flow between adiabatic and isothermal walls. The flow of Case 2 can be a very useful framework for the present objective, since it is the simplest turbulent channel flow with an adiabatic wall and provides ideal information for modelling the compressible turbulent flow near the adiabatic wall. Note that compressible turbulent channel flow between adiabatic walls is not stationary if there is no sink of heat. In Cases 1 and 2, the Mach number based on the bulk velocity and sound speed at the isothermal wall is 1.5, and the Reynolds number based on the bulk density, bulk velocity, channel half-width, and viscosity at the isothermal wall is 3000.To compare compressible and incompressible turbulent flows, DNS of two incompressible turbulent channel flows with passive scalar transport (Cases A and B) are performed using a mixed Fourier Galerkin and Chebyshev tau method. The wall boundary conditions of Cases A and B correspond to those of Cases 1 and 2, respectively. Case A corresponds to the DNS of Kim & Moin (1989). In Cases A and B, the Reynolds number based on the friction velocity, the channel half-width, and the kinematic viscosity is 150.The mean velocity and temperature near adiabatic and isothermal walls for compressible turbulent channel flow can be explained using the non-dimensional heat flux and the friction Mach number. It is found that Morkovin's hypothesis is not applicable to the near-wall asymptotic behaviour of the wall-normal turbulence intensity even if the variable property effect is taken into account. The mechanism of the energy transfers among the internal energy, mean and turbulent kinetic energiesis investigated, and the difference between the energy transfers near isothermal and adiabatic walls is revealed. Morkovin's hypothesis is not applicable to the correlation coeffcient between velocity and temperature fluctuations near the adiabatic wall.

Theoretical and experimental studies have been performed on fully developed twodimensional turbulent channel flows in the low Reynolds number range that are subjected to system rotation. The turbulence is affected by the Coriolis force and the low Reynolds number simultaneously. Using dimensional analysis, the relevant parameters of this flow are found to be Reynolds number Re* = u*D/v (u* is the friction velocity, D the channel half-width) and Ωv/u2* (Ω is the angular velocity of the channel) for the inner region, and Re* and ΩD/u* for the core region. Employing these parameters, changes of skin friction coefficients and velocity profiles compared to nonrotating flow can be reasonably well understood. A Coriolis region where the Coriolis force effect predominates is shown to exist in addition to conventional regions such as viscous and buffer regions. A flow regime diagram that indicates ranges of these regions as a function of Re* and |Ω|v/u2* is given from which the overall flow structure in a rotating channel can be obtained.Experiments have been made in the range of 56 ≤ Re* ≤ 310 and -0.0057 ≤ Ωv/u2* ≤ 0.0030 (these values correspond to Re = 2UmD/v from 1700 to 10000 and rotation number R0 = 2|Ω|D/Um up to 0.055; Um is bulk mean velocity). The characteristic features of velocity profiles and the variation of skin friction coefficients are discussed in relation to the theoretical considerations.

The critical Taylor number, phenomena accompanying the transition to turbulence, and the cellular structure of Taylor–Görtler vortex in the flow between two concentric spheres, of which the inner one is rotating and the outer is stationary, are investigated using three kinds of flow-visualization technique. The critical Taylor number generally increases with the ratio β of clearance to inner-sphere radius. For β [les ] 0.08, the critical Taylor number in spherical Couette flow is smaller than in circular Couette flow, but vice versa for β > 0.08. A pair of toroidal Taylor–Görtler vortices occurs first around the equator at the critical Reynolds number Rec (or critical Taylor number Tc). More Taylor–Görtler vortices are added with increasing Reynolds number Re. After reaching the maximum number of vortex cells, as Re is increased, the number of vortex cells decreases along with the various transition phenomena of Taylor–Görtler vortex flow, and the vortex finally disappears for very large Re, where the turbulent basic flow is developed. The instability mode of Taylor–Görtler vortex flow depends on both β and Re. The vortex flows encountered as Re is increased are toroidal, spiral, wavy, oscillating (quasiperiodic), chaotic and turbulent Taylor–Görtler vortex flows. Fourteen different flow regimes can be observed through the transition from the laminar basic flow to the turbulent basic flow. The number of toroidal and/or spiral cells and the location of toroidal and spiral cells are discussed as a means to clarify the spatial organization of the vortex. Toroidal cells are stationary. However, spiral cells move in relation to the rotating inner sphere, but in the reverse direction of its rotation and at about half its speed. The spiral vortices number about six, and the spiral angle is 2–10°.

The dynamics of the anisotropy of the Reynolds stress tensor and its behavior in decaying homogeneous turbulence subjected to system rotation are investigated in this study. Theoretical analysis shows that the anisotropy can be split into two parts: polarization and directional anisotropies. The former can be further separated into a linear part and a nonlinear part. The corresponding linear solution of the polarization anisotropy is derived in this paper. This solution is found to be equivalent to the linear solution of the anisotropy. While proposing a method to introduce the polarization anisotropy into an isotropic turbulence, direct numerical simulation (DNS) of the rotating turbulence with or without the initial anisotropy is carried out. The linear solution of the anisotropy agrees very well with the DNS result, showing that the evolution of the polarization anisotropy is mainly dominated by the linear effect of the system rotation. With an immediate rotation rate, the coupling effect between the system rotation and nonlinear interactions causes an energy transfer from the region near the pole to the region near the equator in wave space. This type of transfer causes an anisotropic distribution of the kinetic energy between the pole and equator, which relates closely to the directional anisotropy and the two-dimensionalization. In addition, we find that the presence of the initial polarization anisotropy does not affect the evolution of the directional anisotropy, while the presence of the initial directional anisotropy greatly influences the evolution of the polarization anisotropy.

The fundamental frequencies of velocity fluctuations caused by disturbances (vortices and waves) in the laminar–turbulent transition of spherical Couette flow between two concentric spheres with only the inner sphere rotating are investigated by simultaneous and flow-visualization measurements for two small ratios, 0.06 and 0.024, of the clearance to inner-sphere radius for which spiral Taylor–Görtler (TG) vortices and traveling waves on TG vortices play an important role. As the Reynolds number Re is increased for both clearance ratios, first toroidal TG vortices occur at the equator, but the flow is steady. Second, spiral TG vortices (fundamental frequency fSi) occur next to the toroidal ones, so that the flow becomes singly periodic. Third, in addition to these spiral TG vortices, new spiral TG vortices occur between them at the different colatitude θ, so that the pair number of spiral TG vortices around the spherical annulus differs by θ. Therefore, plural fundamental frequencies fSi (i=1,2,…) of spiral TG vortices are detected, corresponding to the different pair numbers. However, the flow is not quasi-periodic, because ratios of the different frequencies fSi are in rational. Fourth, traveling waves (fundamental frequency fW) occur on TG vortices, so that the flow becomes quasi-periodic (fSi and fW). Fifth, the plural frequencies fSi change to a broadband frequency fB, which indicates the onset of chaos. Sixth, fW disappears, so that the flow becomes chaotic spiral TG vortex-flow with only fB. The rotation frequency (wave speed) of traveling waves is larger than that of spiral TG vortices, and the value of fB exists about between the two rotation frequencies.