The lift force on a spherical bubble in a viscous linear shear flow
Authors: LEGENDRE DOMINIQUE , MAGNAUDET JACQUES 
Published: July 2002

The three-dimensional flow around a spherical bubble moving steadily
in a viscous
linear shear flow is studied numerically by solving the full Navier–Stokes
equations.
The bubble surface is assumed to be clean so that the outer flow obeys
a zero-shear-stress
condition and does not induce any rotation of the bubble. The main goal
of the
present study is to provide a complete description of the lift force experienced
by the
bubble and of the mechanisms responsible for this force over a wide range
of Reynolds
number (0.1[les ]Re[les ]500, Re being based on
the bubble diameter) and shear rate (0[les ]Sr[les ]1, Sr

being the ratio between the velocity difference across the bubble and the

relative velocity). For that purpose the structure of the flow field, the
influence of the
Reynolds number on the streamwise vorticity field and the distribution
of the
tangential velocities at the surface of the bubble are first studied in
detail. It is shown
that the latter distribution which plays a central role in the production
of the lift force
is dramatically dependent on viscous effects. The numerical results concerning
the lift
coefficient reveal very different behaviours at low and high Reynolds numbers.
These
two asymptotic regimes shed light on the respective roles played by the
vorticity
produced at the bubble surface and by that contained in the undisturbed
flow. At low
Reynolds number it is found that the lift coefficient depends strongly
on both the
Reynolds number and the shear rate. In contrast, for moderate to high Reynolds

numbers these dependences are found to be very weak. The numerical values
obtained
for the lift coefficient agree very well with available asymptotic results
in the low- and
high-Reynolds-number limits. The range of validity of these asymptotic
solutions is
specified by varying the characteristic parameters of the problem and examining
the
corresponding evolution of the lift coefficient. The numerical results
are also used for
obtaining empirical correlations useful for practical calculations at finite
Reynolds
number. The transient behaviour of the lift force is then examined. It
is found that,
starting from the undisturbed flow, the value of the lift force at short
time differs from
its steady value, even when the Reynolds number is high, because the vorticity
field
needs a finite time to reach its steady distribution. This finding is confirmed
by an
analytical derivation of the initial value of the lift coefficient in an
inviscid shear flow.
Finally, a specific investigation of the evolution of the lift and drag
coefficients with the
shear rate at high Reynolds number is carried out. It is found that when
the shear rate
becomes large, i.e. Sr=O(1), a small but
consistent decrease of the lift coefficient
occurs while a very significant increase of the drag coefficient, essentially
produced by
the modifications of the pressure distribution, is observed. Some of the
foregoing
results are used to show that the well-known equality between the added
mass
coefficient and the lift coefficient holds only in the limit of weak shears
and nearly
steady flows.