Theoretical Investigation of a Passive Scalar such as Temperature in Isotropic Turbulence.
Abstract
A passive scalar is any quantity convected by a fluid which does not affect the dynamics of that fluid. Examples are temperature, salinity, strongly diffusive magnetic fields, and the concentration of chemical reactants. Temperature and salinity can be considered passive only when buoyancy is neglected. While eventually we want to study convection and the transport properties of scalars, the objective of this thesis is to understand the statistical properties of an isotropic scalar field in turbulence. This is the simplest problem in turbulence beyond studying the Navier Stokes equations alone. It was originally believed that the dynamics of a passive scalar would be dominated by the same integral scales which control the velocity field. Experiments have shown that this is not always true. In particular, the decay rate of temperature fluctuations depends on the initial temperature scale size. Chapters I to III use a simple dynamical model of the spectrum and phenomenological arguments to resolve these problems. We show that the observed dependence on initial scale size is in fact a transient and derive a formula which describes the crossover to asymptotic scalar decay. Unfortunately, these methods cannot describe intermittency, which is the observation that the velocity and temperature fluctuations occur in bursts. In Chapter IV we develop a threedimensional spectral code for the velocity and a passive scalar and use it to calculate the higher order statistics which describe intermittency. To maintain steady state turbulence ands collect statistics, we force the equations. We find that the scalar derivative flatness increases much faster with Reynolds number than the velocity flatness. This agrees with experiments and implies that the scalar field is more intermittent than the velocity field. We also find a strong correlation between the rate of strain and the scalar derivative, but an anticorrelation between the vorticity and the scalar derivative. In the last chapter, we study decaying turbulence with our code. These simulations are all at very low Reynolds numbers. We compare our results to spectral closures and discover that the singletime assumptions used in the test field model lead to errors at low Reynolds numbers. In this regime we find that the direct interaction approximation is more accurate. Next we discuss our decay rates, which are faster than the phenomenological predictions. This is due to the absence of an inertial range at low Reynolds numbers. We make a detailed comparison of one and three dimensional length scales and conclude that L(,1) = E(,1)(k(,p))/E is the best experimental equivalent of the threedimensional length. Finally, we begin to analyze how a coherent, anisotropic scalar field returns to isotropy. We find that the rate of return depends on the wavenumber of the initial mode. These initial conditions are similar to those in heated screen experiments. By comparing simulation and experiment, we conclude that the experiments are isotropic and suggest a method to test this.
 Publication:

Ph.D. Thesis
 Pub Date:
 1981
 Bibcode:
 1981PhDT........88K
 Keywords:

 Physics: Fluid and Plasma