In this PhD thesis, the compressible fluid dynamics of high-speed impact
of a spherical liquid droplet on a rigid substrate is investigated. The impact phenomenon
is characterised by the compression of the liquid adjacent to the target
surface, whereas the rest of the liquid droplet remains unaware of the impact. Initially,
the area of compressed liquid is assumed to be bounded by a shock envelope,
which propagates both laterally and upwards into the bulk of the motionless
liquid. Utilizing a high-resolution axisymmetric solver for the Euler equations, it
is shown that the compressibility of the liquid medium plays a dominant role in
the evolution of the phenomenon. Compression of the liquid in a zone defined by
a shock wave envelope, lateral jetting of very high velocity and expansion waves
in the bulk of the medium are the most important mechanisms identified, simulated
and discussed.
During the first phase of impact, all wave propagation velocities are
smaller than the contact line velocity, thus the shock wave remains attached to the
latter. At a certain point, the radial velocity of the contact line decreases below the
shock velocity and the shock wave overtakes the contact line, starting to travel
along the droplet free surface. The resulting high pressure difference across the
free surface at the contact line region triggers an eruption of intense lateral jetting.
The shock wave propagates along the free surface of the droplet and it is reflected
into the bulk of the liquid as an expansion wave. The development of pressure and
density in the compressed area are numerically calculated using a front tracking
method. The exact position of the shock envelope is computed and both onset and
magnitude of jetting are determined, showing the emergence of liquid jets of very
high velocity (up to 6000 m/s). Computationally obtained jetting times are validated
against analytical predictions. Comparisons of computationally obtained
jetting inception times with analytic results show that agreement improves significantly if the radial motion of the liquid in the compressed area is taken into
account.
An analytical model of the impact process is also developed and compared
to the axisymmetric numerical solution of the inviscid flow equations.
Unlike the traditional linear model - which considers all wave propagation velocities
to be constant and equal to the speed of sound, the developed model predicts
the exact flow state in the compressed region by accommodating the real equation
of state. It is shown that the often employed assumption that the compressed area
is separated from the liquid bulk by a single shock wave attached to the contact
line, breaks down and results in an anomaly. This anomaly emerges substantially
prior to the time when the shock wave departs from the contact line, initiating lateral
liquid jetting. Due to the lack of more sophisticated mathematical models, this
tended to be neglected in most works on high speed droplet impact, even though
it is essential for the proper understanding of the pertinent physics. It is proven that
the presence of a multiple-wave structure (instead of a single shock wave) at the
contact line region resolves the aforementioned anomaly. The occurrence of this
more complex multiple wave structure is also supported by the numerical results.
Based on the developed analytical model, a parametric representation of
the shock envelope surface is established, showing a substantial improvement
with respect to previous linear model, when validated against numerical findings.
In the final part of the thesis, the assumption of a multiple wave structure
which removes the above mentioned anomaly is underpinned with an analytical
proof showing that such a structure is indeed a physically acceptable solution.