On conditional scalar increment and joint velocity–scalar increment statistics

Abstract

Conditional velocity and scalar increment statistics are usually studied in the context of Kolmogorov's refined similarity hypotheses and are considered universal (quasi-Gaussian) for inertial-range separations. In such analyses the locally averaged energy and scalar dissipation rates are used as conditioning variables. Recent studies have shown that certain local turbulence structures can be captured when the local scalar variance ⟨ϕ″2⟩r and the local kinetic energy kr are used as the conditioning variables. We study the conditional increments using these conditioning variables, which also provide the local turbulence scales. Experimental data obtained in the fully developed region of an axisymmetric turbulent jet are used to compute the statistics. The conditional scalar increment probability density function (PDF) conditional on ⟨ϕ″2⟩r is found to be close to Gaussian for ⟨ϕ″2⟩r small compared with its mean and is sub-Gaussian and bimodal for large ⟨ϕ″2⟩r, and therefore is not universal. We find that the different shapes of the conditional PDFs are related to the instantaneous degree of non-equilibrium (production larger than dissipation) of the local scalar. There is further evidence of this from the conditional PDF conditional on both ⟨ϕ″2⟩r and χr, which is largely a function of ⟨ϕ″2⟩r/χr, a measure of the degree of non-equilibrium. The velocity-scalar increment joint PDF is close to joint Gaussian and quad-modal for equilibrium and non-equilibrium local velocity and scalar, respectively. The latter shape is associated with a combination of the ramp–cliff and plane strain structures. Kolmogorov's refined similarity hypotheses also predict a dependence of the conditional PDF on the degree of non-equilibrium. Therefore, the quasi-Gaussian (joint) PDF, previously observed in the context of Kolmogorov's refined similarity hypotheses, is only one of the conditional PDF shapes of inertial range turbulence. The present study suggests that such analyses provide a connection between statistical and structural descriptions of inertial-range turbulence.