  We investigate the modes of deformation of an initially spherical bubble
immersed in a homogeneous and isotropic turbulent background flow. We perform
direct numerical simulations of the two-phase incompressible Navier-Stokes
equations, considering a low-density bubble in the high density turbulent flow
at various Weber number (the ratio of turbulent and surface tension forces)
using the air-water density ratio. We discuss a theoretical framework for the
bubble deformation in a turbulent flow using a spherical harmonic
decomposition. We propose, for each mode of bubble deformation, a forcing term
given by the statistics of velocity and pressure fluctuations, evaluated on a
sphere of the same radius. This approach formally relates the bubble
deformation and the background turbulent velocity fluctuations, in the limit of
small deformations. The growth of the total surface deformation and of each
individual mode is computed from the direct numerical simulations using an
appropriate Voronoi decomposition of the bubble surface. We show that two
successive temporal regimes occur: the first regime corresponds to deformations
driven only by inertial forces, with the interface deformation growing linearly
in time, in agreement with the model predictions, whereas the second regime
results from a balance between inertial forces and surface tension. The
transition time between the two regimes is given by the period of the first
Rayleigh mode of bubble oscillation. We discuss how our approach can be used to
relate the bubble lifetime to the turbulence statistics and eventually show
that at high Weber number, bubble lifetime can be deduced from the statistics
of turbulent fluctuations at the bubble scale.
