Abstract:
The main objective of this thesis is to study the instability and evaluate the likely mechanisms of
transition to turbulence in divergent and small-scale (as discussed below) pipe flows. It was first
demonstrated by Reynolds (1883) that pipe flow becomes turbulent at a particular value of a nondimensional
parameter which now bears his name. In recent times, this problem has received a lot
of renewed attention [Faisst & Eckhardt (2003); Hof et al. (2003); Peixinho & Mullin (2006)]. With
many demonstrations [see e.g. Hof et al. (2004)] that pipe flow may be maintained laminar up to high
Reynolds numbers (of the order of hundred thousand), an understanding of the effect of variations
in geometry and flow conditions are increasingly relevant.
Fully developed laminar flow (Hagen-Poiseuille flow) through a straight pipe is linearly stable
at any Reynolds number, Re. In this case, nonlinear and transient growth mechanisms drive the
transition to turbulence. However Hagen-Poiseuille flow is attained only when the pipe wall is straight
and smooth and the pipe is long enough. The length required increases with Reynolds number.
There are many variations from these conditions where linear instability can play a significant role
in transition to turbulence. Some of these situations are addressed in this thesis. They are, flow
through (i) a divergent pipe, (ii) a variety of diverging-converging pipes with constant average radius,
(iii) diverging pipes/channels with velocity slip at the wall, and (iv) the entry region of a straight pipe.
The instabilities of these spatially developing laminar flows is shown to be fundamentally different
from flows that do not vary downstream. We have also studied separation in diverging channels and
pipes.
It is not in general possible to derive analytical solutions for the laminar flows described above.
We obtain the mean flow by solving the steady two-dimensional/axisymmetric Navier-Stokes equations
exactly. For the accuracy desired, the computational time required for solving the elliptic equations
is very large. A full-multigrid algorithm (FMG) is used to accelerate the convergence. The code
is parallelised at the National Aerospace Laboratories, Bangalore. The FMG speeds up the solution
by a factor of hundred as compared to many traditional algorithms like Gauss-Seidel and Jacobi
iteration technique, and the parallel code (using an eight processor machine) gives a superlinear
speed-up of 11 times over a single processor.
(i) The laminar flow through a divergent pipe is shown to be linearly unstable at any angle of divergence
a, with the instability critical Reynolds number tending to infinity as the angle of divergence
goes to zero. At small a (< 1 ) the instability is determined by a parameter S (x) aRe describing
the basic flow profile, and the mechanism is inviscid. The flow is linearly unstable to the swirl mode
for S > 10. The instability critical Reynolds numbers are surprisingly low, e.g. about 150 for a
divergence of 3 , which would suggest a role for such instabilities in the transition to turbulence.
For small angles of divergence and high Reynolds numbers, an axisymmetric Jeffery-Hamel
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equation (AJH) is derived to describe the mean flow. At larger angles of divergence (1 or greater)
the axisymmetric Navier-Stokes equations are solved directly. The partial differential equations for
non-parallel stability are solved as an extended eigenvalue problem by a novel technique. (ii) We
then study the effect of local asymmetric convergence/divergence on laminar flow through a pipe
of constant average radius. The main finding is that the instability behaviour can be changed dramatically
by reversing the direction of flow. This is offered as a possible mechanism that could be
operating in small-scale flows, due to the presence of wall roughness.
(iii) Fluid dynamics and the role of the walls at small-scale can be very different from that
at large scales. We make a minor foray into this regime, by considering the effect of wall slip at
Knudsen numbers less than 0:1. Recent studies indicate that a velocity slip at the wall dramatically
stabilizes the linear mode in a plane two-dimensional channel, but has very little effect on the algebraic
transient growth of disturbances [Lauga & Cossu (2005)]. At microscales, apart from slip, local
divergences and convergences of the wall are frequently encountered, here we focus on the effect
of divergence. Whereas transient growth is more important in a plane channel, at wall divergences
of less than a degree, it is linear instability, taking place two orders of magnitude lower in Reynolds
number than in a plane channel, which is dominant. Unlike in a plane channel, the effect of velocity
slip at the wall is to reduce stability. Transient growth is shown to be an insignificant player in the
process of transition to turbulence.
(iv) The laminar velocity profile through a circular pipe is parabolic once the flow is fully developed.
However, the distance le required to reach this fully-developed state can be very long,
and scales linearly with the Reynolds number, Re, roughly as le=R Re=20, where R is the pipe
radius. Therefore high Reynolds number laminar flow through a pipe of limited length may never
reach a parabolic state. We show that in such circumstances linear stability can play an important
role in transition. We solve for the basic flow exactly, and conduct a non-parallel stability analysis, to
show that flow can be linearly unstable even at a Reynolds number of 1000. In contrast to what is
expected of a boundary layer type flow, disturbance growth is higher in the core region. Our results
are consistent with that of experiment. Earlier theoretical studies predicted critical Reynolds number
an order of magnitude higher than that observed in experiment and were in serious disagreement
with each other.
(v) Some other studies which are in the preliminary stage are described at the end of the
thesis. These include the study of pulsatile flow through a straight pipe. Prescribing time periodic
velocity profiles at the inlet we have solved the Navier-Stokes equation directly using a full-multigrid
algorithm on a parallel machine Venkatesh et al. (2005). We also studied separated flows in the
diverging channel/pipe. At higher angles of divergence, we study flow separation in channels and
pipes. The size, location and shape of the separated region for divergent angle varying from 0 to 90
degree are discussed in the thesis. In future, we are planning to conduct full non-parallel stability
analysis e.g. global stability analysis of such flows without the approximations made in this thesis.