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# Turbulence and viscous mixing using smoothed particle hydrodynamics

thesis

posted on 08.02.2017, 06:10 authored by Robinson, Martin JeremyThis thesis describes the application of Smoothed Particle Hydrodynamics (SPH) to viscous
and turbulent mixing. It is comprised of two main sections that study two important
classes of mixing flows from each end of the Reynolds Number range. The first section
describes an SPH study of very viscous mixing using a two-dimensional Twin Cam mixer
and the development of numerical tools to study the chaotic mixing within this device.
The second section studies the application of SPH to Direct Numerical Simulations (DNS)
of two-dimensional turbulence in a square box with no-slip boundaries. The primary
focus of this section is to evaluate how well SPH can reproduce the primary characteristics
of 2D wall-bounded turbulence. These characteristics include those near the no-slip
boundaries of the box (e.g. the boundary layer and vortex roll-up) as well as those in the
central turbulent flow (e.g. the inverse energy and direct enstrophy cascades).
Chapter 1 provides an introduction to the two types of mixing flows that are investigated
in this thesis. The chapter covers the motivations behind this research and provides a
summary of the relevent literature.
Chapter 2 gives an overview of the SPH method and the particular formulation used in
this thesis.
Chapter 3 describes 2D SPH simulations of a Twin Cam mixer and compares the results
against experimental data and results from two published Finite Element Method (FEM)
simulations. A methodology for the analysis and quantification of the chaotic mixing is
presented and applied to the Twin Cam mixer. The first half of this methodology is based
on the use of Finite-Time Lyapunov Exponents (FTLE) to visualise the chaotic manifolds
of the flow. The topology of the manifolds describe the stretching and folding actions of
the mixing, and define regions in the flow that are substantially isolated (ie. slow to mix)
from neighbouring regions. A method of calculating the spatial distribution of FTLE directly
from the SPH particle data is presented, which represents a considerable reduction
in computational cost compared to previously published methods. The second half of
the analysis methodology is based on a quantitative measure of mixing. Given a length
scale of interest, this measure calculates the local amount of mixing between two or more
regions. The measure is used to show the differences in both the spatial variation and
total amount of mixing between the important regions in the flow previously identified
by the chaotic manifolds. The chapter finishes by comparing the time scales of mixing
over different length scales in the Twin Cam mixer.
Chapter 4 provides an overview of the current theoretical description of homogeneous
and isotropic 3D and 2D turbulence. It also covers recent numerical and experimental
results for DNS turbulence in periodic and wall-bounded domains. A literature review
of SPH turbulence is given. The chapter ends by discussing the primary motivations
behind this investigation of SPH DNS of two-dimensional wall-bounded turbulence.
Chapter 5 presents the results of ensemble SPH simulations of decaying wall-bounded
2D turbulence at a Reynolds number (Re) of 1500. These are compared against published
results from a pseudospectral code. The qualitative variables of the SPH turbulence evolution
(e.g. kinetic energy decay, angular momentum and average vortex wavenumber)
compare well with the pseudospectral results. However, the production of long-lived
coherent vortices from the boundaries is not seen in the SPH simulations. Subsequent
results show that the boundary layer and vortex roll-up are modelled well by the SPH
method, but excess numerical dissipation prevents the vortex from surviving once it has
detached from the boundary.
Chapter 6 investigates SPH simulations of forced wall-bounded (Re = 1581) and periodic
(Re = 2645) 2D turbulence. As for the decaying case, these results are compared against
published pseudospectral simulations and physical experiments. The SPH simulations
reproduce the direct enstrophy cascade well. The kinetic energy spectrum follows the
expected k-3 scaling in the direct enstrophy range for wavelengths larger than 8 particle
spacings. However, velocity fluctuations at wavelengths less than 8 particle spacings are
responsible for a significant amount of numerical dissipation. This dissipation acts to
weaken the inverse energy cascade and prevents the build-up of energy in the longest
wavelength. Investigations into the statistics of particle pair dispersion show deviations
in the expected scalings at wavelengths less than the forcing scale. These deviations are
consistent with the small-scale velocity fluctuations acting as an additional forcing term
and increasing the rate of mixing at small length scales.
SPH parameter studies show that the turbulence is very sensitive to the SPH sound
speed, with increasing sound speed resulting in a significant increase in numerical dissipation
and a subsequent reduction in the strength of the inverse energy cascade. The
maximum wavelength of the small-scale velocity fluctuations decreases slowly with increasing
resolution. It is estimated that a minimum particle resolution of 3500x3500 is
needed to reduce the minimum wavelength of the velocity fluctuations below the dissipation
length scale of the turbulence (for Re = 1581).
Chapter 7 compares the Cubic Spline and Wendland kernels and their effect on particle
clumping in the forced 2D turbulence simulations. The Cubic Spline is found to generate
significant clumping on a length scale equal to the location of the spline point (typically
chosen to be the smoothing length h). In contrast, the Wendland kernel results in a very
even distribution of particles, which dramatically reduces the numerical dissipation in
the forced turbulence simulations and strengthens the inverse energy cascade. Particle
clumping in SPH simulations is often attributed to the Tensile Instability. However, the
criteria for this instability is unchanged between the Cubic Spline and Wendland kernels
leading to the conclusion that the Tensile Instability is not the cause of the clumping seen
in these simulations.