Session E28: Compressible Turbulence

Effects of shock structure on temperature field in compressible turbulence

Effects of shock structure on temperature in compressible turbulence were investigated. The small-scale shocklets and large-scale shock waves were appeared in the flows driven by solenoidal and compressive forcings, i.e. SFT {\&} CFT, respectively. In SFT the temperature had Kolmogorov spectrum and ramp-cliff structures, while in CFT it obeyed Burgers spectrum and was dominated by large-scale rarefaction and compression. The power-law exponents for the p.d.f. of large negative dilatation were -2.5 in SFT and -3.5 in CFT, approximately corresponded to model results. The isentropic approximation of thermodynamic variables showed that in SFT, the isentropic derivation was reinforced when turbulent Mach number increased. At similar turbulent Mach number, the variables in CFT exhibited more anisentropic. It showed that the transport of temperature was increased by the small-scale viscous dissipation and the large-scale pressure-dilatation. The distribution of positive and negative components of pressure-dilatation confirmed the mechanism of negligible pressure-dilatation at small scales. Further, the positive skewness of p.d.f.s of pressure-dilatation implied that the conversion from kinetic to internal energy by compression was more intense than the opposite process by rarefaction.   

Compressible turbulent channel flow with impedance boundary conditions

We have performed large-eddy simulations of compressible turbulent channel flow at one bulk Reynolds number, Reb $=$ 6900, for bulk Mach numbers Mb $=$ 0.05, 0.2, 0.5, with linear acoustic impedance boundary conditions (IBCs). The IBCs are formulated in the time domain following Fung and Ju (2004) and coupled with a Navier-Stokes solver. The impedance model adopted is a three-parameter Helmholtz oscillator with resonant frequency tuned to the outer layer eddies. The IBC's resistance, R, has been varied in the range, R $=$ 0.01, 0.10, 1.00. Tuned IBCs result in a noticeable drag increase for sufficiently high Mb and/or low R, exceeding 300{\%} for Mb $=$ 0.5 and R $=$ 0.01, and thus represents a promising passive control technique for delaying boundary layer separation and/or enhancing wall heat transfer. Alterations to the turbulent flow structure are confined to the first 15{\%} of the boundary layer thickness where the classical buffer-layer coherent vortical structures are replaced by an array of Kelvin-Helmholtz-like rollers. The non-zero asymptotic value of the Reynolds shear stress gradient at the wall results in the disappearance of the viscous sublayer and very early departure of the mean velocity profiles from the law of the wall.   

Numerical investigation of non-equilibrium effects in hypersonic turbulent boundary layers

Direct numerical simulations of a spatially developing hypersonic boundary layer have been conducted in order to investigate thermal and chemical non-equilibrium effects in a hypersonic turbulent boundary layer. Two different flows, pure oxygen and pure nitrogen flows with specific total enthalpy, $h_{0,O_2} = 9.5017\,MJ/kg$ and $h_{0,N_2} = 19.1116\,MJ/kg$, respectively, have been considered. The boundary edge conditions were obtained from a separate calculation of a flow over a blunt wedge at free-stream Mach numbers $M_{\infty, O_2} = 15$ and $M_{\infty, N_2} = 20$. The inflow conditions were obtained from a simulation of a turbulent boundary layer of a perfect gas. Non-equilibrium effects on turbulence statistics and near-wall turbulence structures were examined by comparing with those obtained in a simulation of the same boundary layer with a perfect-gas assumption.   

Reynolds Stress Anisotropy Effects on the Modeling of Production in Compressible Homogeneous Turbulent Shear Flow

Direct numerical simulations of compressible homogeneous shear flow are performed for a range of gradient Mach numbers and time scale ratios. A pseudo-spectral Fourier collocation method is used to perform the simulations. Compressibility effects associated with larger gradient Mach numbers result in increased anisotropy of the Reynolds stresses and a reduced growth rate of turbulent kinetic energy. However, the time scale ratio also affects Reynolds stress anisotropy. The DNS results seem to indicate that the model coefficient $C_\mu$ in the production term of the turbulent kinetic energy equation is mainly dependent on the time scale ratio for low gradient Mach numbers and collapses to a single function of gradient Mach number for higher gradient Mach numbers. A model for $C_\mu$ is formulated based on this behavior in the DNS. Computations using this model are then compared to the DNS results.   

Fluctuations of thermodynamic variables in compressible isotropic turbulence

A distinguishing feature of compressible turbulence is the appearance of fluctuations of thermodynamic variables. While their importance is well-known in understanding these flows, some of their basic characteristics such as the Reynolds and Mach number dependence are not well understood. We use a large database of Direct Numerical Simulation of stationary compressible isotropic turbulence on up to $2048^3$ grids at Taylor Reynolds numbers up to $450$ and a range of Mach numbers ($M_t \approx 0.1-0.6$) to examine statistical properties of thermodynamic variables. Our focus is on the PDFs and moments of pressure, density and temperature. While results at low $M_t$ are consistent with incompressible results, qualitative changes are observed at higher $M_t$ with a transition around $M_t\sim 0.3$. For example, the PDF of pressure changes from negatively to positively skewed as $M_t$ increases. Similar changes are observed for temperature and density. We suggest that large fluctuations of thermodynamic variables will be log-normal at high $M_t$. We also find that, relative to incompressible turbulence, the correlation between enstrophy and low-pressure regions is weakened at high $M_t$ which can be explained by the dominance of the so-called dilatational pressure.   

The effect of thermal non-equilibrium in decaying turbulence using direct numerical simulations

We study effects of thermal non-equilibrium (TNE), in particular vibrational non-equilibrium, in decaying turbulence using Direct Numerical Simulation (DNS). The exchange mechanism between molecular vibrational and translational energy modes is introduced using the well-known Landau-Teller approximation. A change in the fundamental cascade is observed with dissipation $(\epsilon)$ increasing significantly relative to cases without TNE at time scales of $ O(\tau_{v}) $ where $\tau_v$ is the characteristic relaxation time of vibrational energy. This is also found to depend on the initial degree of TNE $(\Delta E_{v0})$. The relative contributions of energy transfer through classical energy cascade and transfer through TNE exchanges can be represented by a new non-dimensional parameter $S=\frac{\Delta E_{v0}}{\tau_v \epsilon}$. For example, $S$ can be used to understand DNS data, in particular to distinguish different regimes in the interaction. $S$ is also useful to characterize the time at which dissipation peaks as well as its peak value. Results are compared satisfactorily with experimental evidence available. Turbulence is also observed to decelerate the transfer from vibrational to translational mode in flows with initial vibrationally hot states.