This paper investigates the dynamics of velocity gradients for a spatially developing flow
generated by a single square element of a fractal square grid at low inlet Reynolds number
through direct numerical simulation. This square grid-element is also the fundamental
block of a classical grid. The flow along the grid-element centreline is initially irrotational
and becomes turbulent further downstream due to the lateral excursions of vortical
turbulent wakes from the grid-element bars. We study the generation and evolution of
the symmetric and anti-symmetric parts of the velocity gradient tensor for this spatially
developing flow using the transport equations of mean strain-product and mean enstrophy
respectively. The choice of low inlet Reynolds number allows for fine spatial resolution
and long simulations, both of which are conducive in balancing the budget equations of
the above quantities. The budget analysis is carried out along the grid-element centreline
and the bar centreline. The former is observed to consist of two subregions: one in the
immediate lee of the grid-element which is dominated by irrotational strain, and one
further downstream where both strain and vorticity coexist. In the demarcation area
between these two subregions, where the turbulence is inhomogeneous and developing,
the energy spectrum exhibits the best
−
5
/
3 power law slope. This is the same location
where the experiments at much higher inlet Reynolds number show a well defined
−
5
/
3 spectrum over more than a decade of frequencies. Yet, the Q-R diagram remains
undeveloped in the near grid-element region, and both the intermediate and extensive
strain-rate eigenvectors align with the vorticity vector. Along the grid-element centreline,
the strain is the first velocity gradient quantity generated by the action of pressure
Hessian. This strain is then transported downstream by fluctuations and strain self-
amplification is activated a little later. Further downstream, vorticity from the bar wakes
is brought towards the grid-element centreline, and, through the interaction with strain,
leads to the production of enstrophy. The strain-rate tensor has a statistically axial
stretching form in the production region, but a statistically biaxial stretching form in
the decay region. The usual signatures of velocity gradients such as the shape of Q-R
diagrams and the alignment of vorticity vector with the intermediate eigenvector are
detected only in the decay region even though the local Reynolds number (based on the
Taylor length scale) is only between 30 and 40.