Large-scale structures in compressible turbulence and what they can tell us about dilatational motions

The solenoidal nature of the velocity field in incompressible turbulence imposes strong kinematic constraints on the structure of statistical objects describing the flow. Some of these constraints are relaxed in compressible turbulence due to the existence of dilatational motions. Thus, classical statistical relations derived from these constraints and widely used in incompressible flows (e.g. ratios between longitudinal and transverse integral scales or moments of velocity gradients), are modified for compressible flows. Because these dilatational motions affect the flow, it is important to understand their effect on statistics through those constraints and to assess whether those new relations can provide information on compressiblity effects. We show that this is indeed the case and present some new relations. Using Helmholtz decomposition we show that constraints on both solenoidal and dilatational components redefine the way common statistics should be interpreted. For example, the ratio of longitudinal to transverse integral length scales contains information about isotropy (as in incompressible flows) but also on the amount of dilatation providing a direct way of quantifying ``compressibility'' from easy-to-measure correlations. Furthermore, component correlation functions appear to exhibit universal behavior with their combined effect providing information also about dilatational effects. We support our results with a large database of well resolved DNS.