An extensive direct numerical simulation database over a wide range of Reynolds and Schmidt numbers is used to examine the Schmidt number dependence of the structure function of passive scalars and the applicability of the so-called Yaglom's relation in isotropic turbulence with a uniform mean scalar gradient. For the moderate Reynolds numbers available, the limited range of scales in scalar fields of very low Schmidt numbers (as low as 1/2048) is seen to lead to weaker intermittency, and weaker alignment between velocity gradients and principal strain rates. Strong departures from both Obukhov-Corrsin scaling for second-order structure functions and Yaglom's relation for the mixed velocity-scalar third-order structure function are observed. Evaluation of different terms in the scalar structure function budget equation assuming statistical stationarity in time shows that, if the Schmidt number is very low, at intermediate scales production and diffusion terms (instead of advection) are major contributors in the balance against dissipation.
We are indebted to Professor T. Gotoh for his generous help with the numerical and mathematical considerations in Sec. II of this paper, and to Professor K. R. Sreenivasan for his constant encouragement. We gratefully knowledge support from the National Science Foundation (NSF), via NSF Grant Nos. CBET-1139037 and OCI-0749223.
The computations in this paper used supercomputer resources provided by the National Energy Research Scientific Computing Center (NERSC, at Lawrence Berkeley National Laboratories), National Center for Computational Sciences (NCCS, at Oak Ridge National Laboratory), the Texas Advanced Computing Center (TACC, at the University of Texas at Austin), and the National Center for Supercomputing Applications (NCSA, under the Blue Waters project at the University of Illinois at Urbana-Champaign). Resources at NCCS were provided by INCITE and Director's Discretionary allocations programs at the Oak Ridge Leadership Computing Facility, which is supported by the Office of Science of the (U.S.) Department of Energy (DOE) under Contract No. DE-AC05-00OR22725.
I. INTRODUCTION II. COMPUTATION OF SCALAR STRUCTURE FUNCTION BALANCE A. Advective transport B. Production by mean gradient C. Molecular transport and dissipation D. Statistical stationarity III. SIMULATION DATABASE IV. STATISTICS OF SCALAR INCREMENTS AND GRADIENTS A. Scaling of scalar increments and dissipation B. Amplification by alignment with principal strain rates V. YAGLOM'S RELATION AND SCALAR STRUCTURE FUNCTION BUDGET VI. CONCLUSIONS