Session G29: Compressible Turbulence

Energy spectrum in compressible turbulence

The general scaling of spectra in compressible turbulence is not well understood,
in particular the circumstances under which different theoretical proposals such as  
pseudo-sound and equipartition apply. The evidence supporting different scaling laws seems  
inconsistent across the literature, especially at high turbulent Mach numbers ($M_{t}$).
 From an asymptotic expansion based on two parameters characterizing compressibility (the traditional $M_{t}$ as a measure of scale separation between acoustic and turbulent processes,  and ${u^{'}_{d}}/{u^{'}_{s}}$ the ratio of dilatation to solenoidal rms velocity  
 as a measure of compressibility levels).   We identify the dominant processes in the governing equations and determine the behavior expected for the spectrum.  Using a very large DNS database with different forcing modes and a wide range of governing parameters  
(Taylor microscale Reynolds numbers, $Re_{\lambda}$ up to 170 and turbulent Mach numbers, $M_{t}$ up to 0.8) we support these  predictions and reconcile the seemingly contradictory results in the literature.  We will discuss the potential implications of these findings for compressible scalar mixing.   

The governing parameters in compressible turbulence

Incompressible turbulence is commonly characterized by a single non-dimensional parameter, the Reynolds number ($R_\lambda$). However, when compressible effects are present other parameters are needed to characterize the flow, in part, because of the coupling between hydrodynamics and thermodynamics, and the type of external forcing used to sustain
fluctuations against dissipation. It is common to use the turbulent Mach number $M_t=u'/c$
($u'$ is the rms velocity and $c$ the mean speed of sound) as a measure of compressibility.
However, using a large database of Direct Numerical Simulations of stationary compressible isotropic turbulence forced with both solenoidal and dilatational forcing at a range $R_\lambda$ (38-450) and $\mt$ (0.1-0.6) we show that these two parameters alone fail to describe the state of the flow uniquely. This is illustrated through fluctuations of both
thermodynamic as well as hydrodynamic variables.  We propose a new set of governing parameters which can indeed scale all the data successfully regardless of the levels of dilatations  introduced by the forcing and allow us to distinguish different statistical regimes the flow could be in.  Our proposal is also compared successfully against other results in the literature.

  

Cascades of temperature and entropy fluctuations in three-dimensional compressible turbulence

Cascades of temperature and entropy fluctuations are studied by numerical simulations of stationary three-dimensional compressible turbulence. The -5/3 scaling behavior is identified for the fluctuation spectra of temperature and entropy, with the Obukhov-Corrsin constants close to that of a passive scalar spectrum. The average subgrid-scale (SGS) fluxes of temperature and entropy normalized by the total dissipation rates are close to 1 in the inertial range. The average SGS fluxes conditioned on the filtered velocity divergence exhibit the scale-invariant property in the inertial range. Moreover, strong compression motions enhance the direct SGS fluxes of temperature and entropy from large scales to small scales.   

Experimental Characterization of Inertial Range Statistics in Compressible Turbulence

Compressible turbulence has applications to astrophysical fluid dynamics, high-speed civil transport, and combustion. For compressible subsonic flows, large turbulent fluctuations can generate randomly distributed regions of local vigorous fluid compression.  Measurements are necessary to further develop physical models that quantify the influence of compressibility on turbulent flows. In our experiments, we fabricated nanoscale hot wire probes inspired by Vallikivi et al. (J. Microelectromech. Syst., 2014) using semiconductor manufacturing techniques, and measured turbulence in a specialized pressure vessel filled with sulfur hexafluoride gas. With the use of smaller than conventional hot-wire probes, we resolve inertial range statistics. The dense gas sulfur hexafluoride makes it possible to reach high Mach numbers due to its lower speed of sound compared to air, and high Reynolds numbers since we can lower its kinematic viscosity by an order of magnitude with pressure adjustments. We report measurements of turbulence statistics as well as power spectra for turbulent Mach numbers up to 0.3, and for Taylor microscale Reynolds numbers between 200 and 3700.    

Time-evolution of passive scalar structures in shock-turbulence interaction

A tracking algorithm is introduced to study the time evolution of isosurfaces of any scalar quantity in turbulent flows by relating the geometry of the surface with the underlying flow physics. At every instant in time each educed isosurface is represented as a point in a feature space of parameters that characterize the structures, including geometric and physical information. Correspondences between all structures educed in consecutive time instants are used to construct a graph in which each vertex represents an individual structure at a given time and the edges indicate the found correspondences between the structures over time.
Application of the methodology is presented to study structures of passive scalars (Sc ≈ 1) with well-defined initial shapes and sizes transported in a background turbulent flow (Re_λ≈ 40, M_t ≈ 0.3) passing through a nominally planar shock wave (M ≈ 1.5). Changes of the non-local geometry of the structures across the shock are related to changes of carried physical quantities, such as enstrophy, dissipation rates, and the alignment of strain-rate eigenvectors, vorticity and scalar gradient. Ensemble statistics for structures with common patterns of time evolution will be highlighted.