Vibrational relaxation in compressible isotropic turbulence with thermal nonequilibrium

As pioneered by Donzis and Maqui [J. Fluid Mech. 797, 181 (2016)] and Khurshid and Donzis [Phys. Fluids 31, 015103 (2019)], the compressible isotropic turbulence in thermal nonequilibrium is drawing attention in the fluid dynamics community. In the present study, the vibrational rate and the dissipation or production of vibrational energy fluctuation of compressible isotropic turbulence with solenoidal forcing in vibrational nonequilibrium are investigated using numerical simulations. The turbulent Mach number (M-t) is set to be 0.44 and 1.09, and the Taylor Reynolds number (Re-lambda) equals 98.58 approximately. The focus is on the effect of the normalized vibrational relaxation time (K-tau) and characteristic temperature (theta(v)) on statistical features of the vibrational rate and the dissipation or production components of vibrational energy fluctuation. When K-tau is small (less than or similar to 1.0), the average of normalized vibrational rate conditioned on the normalized dilatation is positive in the compression region and negative in the expansion region, enhanced with increase of compression and expansion level, respectively. It indicates that, on average, the energy transfers from translational-rotational to vibrational modes in the compression region and in the inverse direction in the expansion region. The conditional average of the normalized vibrational rate reduces with increase of K-tau, and is insensitive to theta(v). The joint probability density functions (PDFs) and conditional PDFs of the normalized vibrational rate are also analyzed, to reveal the effect of compression and expansion motions on the internal energy exchange between the translational-rotational and vibrational modes. The dissipation or production of vibrational energy fluctuation can result from effects of dilatation, thermal diffusion, and vibrational relaxation. The dissipation component due to thermal diffusion always weakens the vibrational energy fluctuation in both compression and expansion regions for the weakly and highly compressible turbulence, but its effect is insignificant compared to the other two components. For the weakly compressible turbulence, the dissipation or production of vibrational energy fluctuation mainly comes from effects of dilatation and vibrational relaxation when K-tau is small (less than or similar to 1.0), and the vibrational relaxation component loses its significance with increasing of K-tau. For the highly compressible turbulence, both the dilatation and vibrational relaxation effects play an important role in the dissipation and production of vibrational energy fluctuation, and their effects are significantly different from that of the weakly compressible turbulence. The conditional average, joint PDFs, and conditional PDFs of each dissipation or production component are calculated and analyzed in detail to reveal their effects on the vibrational energy fluctuation in different compression and expansion regions. Furthermore, the bulk viscosity effect on the compression or expansion motion in flow field, the vibrational rate, and the dissipation or production components of vibrational energy fluctuation is discussed briefly.