Abstract
Synthetic turbulence refers to stochastic fields having characteristics of real hydro-
dynamic turbulent flows, which has been useful in the modelling and simulation of
turbulence, and for further understanding fundamental properties of turbulent motion. Synthetic turbulence aims to construct the field variables (such as velocity
distributions) by simpler processes to reproduce characteristic features of turbulent
fluctuations with a reduced computational cost in comparison with a formal numerical solution of the Navier-Stokes equations. A new approach of synthetic turbulence
has been recently proposed, which showed that realistic synthetic isotropic turbulent
fields could be generated using the Multi-scale turnover Lagrangian map (MTLM).
The initial focus of this thesis is on studying the MTLM synthetic fields using the
filtering approach. This approach, which has not been pursued so far, sheds new
light on the potential applications of the synthetic fields in large eddy simulations
and subgrid-scale (SGS) modelling. Our investigation includes SGS stresses, and
SGS dissipations and related statistics, SGS scalar variance, and its relations with
other quantities (such as the filtered molecular scalar dissipation). It is well-known
that, even if a synthetic field had reproduced faithfully the multi-fractal statistics, it
may not be able to produce the energy flux across the energy spectrum. Therefore,
from the LES and/or SGS modelling perspective, many questions remain unclear,
such as the PDF of the SGS dissipation, the amount of back-scattering, among
others. They are addressed in this work. It demonstrates that using the MTLM is
able to build a synthetic SGS model with a number of good features which many
current SGS models (including those for the scalar flux) do not have. We also show
that it has advantages in representing the filtered molecular scalar dissipation. In
addition, we generalize the formulation of MTLM to include the effects of a mean
scalar gradient on the scalar field. Our numerical tests provide the necessary proof
that the effects of the mean gradient can be captured by MTLM. Furthermore, we
investigate the effects of the input spectra on the statistics of the MTLM fields.
We study the effects of the shape of the spectra by using truncated spectra and a
model spectra (the Kovasznay spectra) as the input. The additional case, and the
additional quantities we examine, have shedded light on how to apply the MTLM
technique in simulations, as well as the robustness of the technique.
The Constrained MTLM is a new technique generalizing the MTLM procedure to
generate anisotropic synthetic turbulence in order to model inhomogeneous turbu-
lence by using the adjoint formulation. Li and Rosales [107] derived the optimality
system corresponding to the MTLM map and applied this method to synthesize
two Kolmogorov flows. In this thesis, we derive a new optimality system to generate anisotropic synthetic turbulence according to the CMTLM approach in order
to include the effects of solid wall boundaries, which were not taken into account in
the last study. We consider the difference introduced by the solid wall, under the
impermeable boundary conditions, where the normal components velocity field are
zero, while the tangent components may be non-zero. To accomplish this task, we
have modified the CMTLM procedure to generate a reflectionally symmetric synthetic field which serves as a model of the velocity field in a fully developed channel
flow. That the MTLM procedure preserves the reflectional symmetries is proved, the
adjoint optimality system with reflectional symmetry are derived.
We aim to obtain accurate turbulent statistics, and compare our results with computed and experimental results.
CMTLM procedure formulates MTLM procedure as an optimization problem with
the initial Gaussian random field as the control and some known velocity field as the
target. Thus, with the purpose to quantify the contributions of the adjoint operator
in the modelling process, the effects of the control variable on the cost function
gradient and the corresponding adjoint field is examined. Contours of the mean of
the gradients of the cost functions and adjoint fields for three cases with data taken
from synthetics CMTLM Kolmogorov flows and from CMTLM synthetics velocity
field generated with DNS data as the target are computed.
Finally, in order to define a new SGS model to simulate interactions between different
length scales in turbulence, we will combine DNS data with Constrained MTLM
method. Three data sets are truncated from DNS data with different degrees of
resolution, filtered with the cutoff filter with large filter scale, which are then used
as target fields to synthesize three CMTLM fields. The CMTLM fields are merged
with these target fields. Data from the merged fields are used to predict the SGS
quantities, and are compared with exact SGS quantities which have been computed
from DNS field. In addition, the statistical geometry between the SGS and filtered
quantities for real and predicted data are also investigated.
