Title:

Further assessment of the LET theory

This thesis extends and analyses the Local Energy Transfer (LET) approximation for turbulence. LET is a twopoint twotime second moment closure for the NavierStokes equations, developed using renormalised perturbation theory in an Eulerian coordinate system. Analytical and numerical calculations of LET for the velocity field have been made in previous work. The LET approximation is extended to treat the transport of a passive scalar. The LET equations for passive scalar transport are derived and used in numerical calculations at a range of Reynolds and Prandtl numbers. The evolution in time of the scalar energy, dissipation and transfer spectra is calculated, and these spectra are shown to become selfsimilar under convective or Kolmogorov scaling. The scalar energy, dissipation and transfer spectra at R_{λ im} 40 compare well with experiment. The twotime scalar correlation is calculated and the relevant scaling for the time separation is shown to be convective at small Reynolds number and Kolmogorov (i.e. inertial) at large Reynolds number. The effect of the ratio of the velocity energy spectrum peak wavenumber to the scalar energy peak wavenumber on the thermal to mechanical timescale ratio is compared with experiment. At large Reynolds number the scalar energy spectrum is shown to have a k^{5/3} inertialconvective range at Pr = 0.5, with a value of 1.13 for the ObukhovCorrsin constant β. The scalar energy balance is calculated at several Reynolds numbers and at large Reynolds number shows a clear separation in wavenumber of the production (in fact the energy peak in decaying turbulence) and dissipation ranges. The dependence of the velocityscalar cross derivative skewness on the Reynolds and Prandtl numbers is compared with direct numerical simulation and experiment. The magnitude and the Reynolds number dependence of the skewness is in fair agreement with the simulation, but the Prandtl number dependence is reversed. The Galilean transformation properties of the NavierStokes equations, velocity moment equations, perturbation expansion and LET are investigated. The perturbation expansion (used to derive LET) is shown to be invariant under a Galilean transformation, term by term, thus any truncation will be Galilean invariant. The LET equations are also shown to be Galilean invariant. The concept of Random Galilean Transformation (RGT) is analysed. The RGT was developed by Kraichnan to model the convective effects of the large scales in turbulence. Invariance under a RGT is violated by Eulerian renormalised perturbation theories  this led to the development of quasiLagrangian theories. The RGT is shown to be a change of ensemble rather than a symmetry transformation. This change of ensemble makes the derivation of an Eulerian renormalised perturbation theory impossible as the zeroorder solution is no longer Gaussian and the zeroorder propagator/response function becomes a random variable.
