  The triple decomposition of a velocity gradient tensor is studied with direct
numerical simulations of homogeneous isotropic turbulence, where the velocity
gradient tensor is decomposed into three components representing an
irrotational straining motion. Strength of these motions can be quantified with
the decomposed components. A procedure of the triple decomposition is proposed
for three-dimensional flows, where the decomposition is applied in a basic
reference frame identified by examining a finite number of reference frames
obtained by three sequential rotational transformations of a Cartesian
coordinate. Even though more than one basic reference frame may be available
for the triple decomposition, the results of the decomposition depend little on
the choice of basic reference frame. In homogeneous isotropic turbulence,
regions with strong rigid-body rotations or straining motions are highly
intermittent in space, while most flow regions exhibit moderately strong
shearing motions in the absence of straining motions and rigid-body rotations.
The shear tensor is also used for detecting intense shear layers.
