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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.

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It is well known that the correlation between the Smagorinsky model and the subgrid scale stress is low, while the model based on the scale similarity assumption has considerably higher correlation. However, the scale similarity model by itself was found to be insufficiently dissipative. Therefore, the model is usually used together with the Smagorinsky model. Model coefficients are commonly computed using the two-parameter dynamic procedure. Nevertheless, the dynamic two-parameter mixed model still does not work well for wall bounded flows, since the model predicts a high value of the wall shear stress. In this study, we propose a modification to the two-parameter dynamic procedure for wall bounded flows, which removes that defect: the Smagorinsky parameter, C S ,i s computed exactly the same way as in the dynamic Smagorinsky model, then the other parameter, C L , is computed dynamically as C S is known. This ensures that the mixed model provides proper wall shear stress and mean velocity profile. Computational tests are done for turbulent channel flow where the Reynolds numbers based on the channel half-width and wall friction velocity are 395 and 1400. To remove the ambiguity regarding the accuracy of the finite difference scheme, we use high ͑up to 12th͒ order accurate fully conservative finite difference schemes in a staggered grid system.

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 main objective of this study is to clarify the effect of thermal wall boundary conditions on turbulence statistics and structures in a compressible turbulent flow. This work is an extension of Morinishi et al. (J. Fluid Mech. vol. 502, 2004, p. 273), who performed DNS of compressible turbulent channel flow between adiabatic and isothermal walls at Mach number $M\,{=}\,1.5$ (Case 2). We address the question of whether the modification of turbulence statistics is attributable to the effect of the adiabatic wall boundary condition or the effect of the increase of wall temperature caused by the adiabatic wall boundary condition. New DNS of the compressible turbulent channel flow between isothermal walls with the wall temperature difference at the Mach number $M=1.5$ (Case 1) and DNS of the corresponding incompressible turbulent flow with passive scalar transport (Case I) are performed. The present study shows that the mean temperature profile near the high-temperature wall for Case 1 has an additional maximum due to the friction work, while such an additional maximum does not appear for Cases 2 and I. The additional maximum leads to a corresponding near-wall maximum of temperature fluctuations. We find the direction of energy transfer due to pressure work near the adiabatic wall for Case 2 being opposite to that near the isothermal wall to be due to the effect of the high-temperature wall, not to the effect of the adiabatic wall. These findings are explained by using the budgets of internal energy and temperature variance transport equations.

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