Vorticity dynamics of dilute two-way-coupled particle-laden mixing layers
Authors: MEIBURG E. , WALLNER E. , PAGELLA A. , RIAZ A. , HÄRTEL C. , NECKER F. 
Published: July 2002

The two-way coupling mechanisms in particle-laden mixing layers are investigated,
with and without particle settling, and with an emphasis on the resulting modifications
to the fluid vorticity field. The governing equations are interpreted with respect to
the production and cancellation of vorticity. These mechanisms are shown to be
related to the misalignment of the concentration gradient and the slip velocity, as
well as to the difference in fluid and particle vorticities. Preliminary insight into the
physics is obtained from an analysis of the unidirectional base flow. For this model
problem, the conditions are established under which the particle velocity remains
a single-valued function of space for all times. The resulting simplified set of two-way-coupled equations governing the vorticity of the fluid and particulate phases,
respectively, is solved numerically. The formation of a decaying travelling wave
solution is demonstrated over a wide range of parameters. Interestingly, the downward
propagation of the fluid vorticity field is not accomplished through convection, but
rather by the production and loss of vorticity on opposite sides of the mixing layer.
For moderate settling velocities, the simulation results reveal an optimal coupling
mechanism between the fluid and particle vorticities at intermediate values of the
mass loading parameter. For large settling velocities and intermediate mass loadings,
more than one local maximum is seen to evolve in the vorticity field. A scaling law
for the downward propagation rates of the vorticity fronts is derived.

Two-dimensional particle-laden mixing layers are investigated by means of a mixed
Lagrangian–Eulerian approach which is based on the vorticity variable. For uniformly
seeded mixing layers, the simulations confirm some of the features observed by
Druzhinin (1995b) for the model problem of a two-way-coupled particle-laden Stuart
vortex, as well as by Dimas & Kiger (1998) in a linear stability analysis. For small
values of the Stokes number, a mild destabilization of the mixing layer is observed.
At moderate and large Stokes numbers, on the other hand, the transport of vorticity
from the braids into the core of the evolving Kelvin–Helmholtz vortices is seen to
be slowed by the two-way coupling effects. As a result, the particle ejection from the
vortex cores is weakened. For constant mass loadings, the two-way coupling effects
are strongest at intermediate Stokes number values. For moderately large Stokes
numbers, the formation of two bands of high particle concentration is observed in
the braids, which reflects the multi-valued nature of the particle velocity field. For
mixing layers in which only one stream is seeded, the particle concentration gradient
across the mixing layer leads to strong vorticity production and loss, which results
in an effective net motion of the vortex in the flow direction of the seeded stream.
Under particle settling, the vortex propagates downward as well. For the parameter
range explored here, its settling velocity agrees well with the scaling law derived from
the unidirectional flow analysis.