Stationary and time-dependent numerical approximation of the lid-driven cavity problem for power-law fluid flows at high Reynolds numbers using a stabilized finite element formulation of the VMS type

Abstract

In this work, a variational multiscale (VMS) nite element formulation is used to approximate numerically the natural convection in a dierentially heated square cavity problem for Newtonian and power-law uids. The Newtonian case has been extensively studied by many authors over the years for air, with a Prandtl number (Pr) of 0.71. For this case there is an agreement about the uid dynamics related to the transition between steady and time dependent behavior, that initiates via a Hopf bifurcation in a well dened Rayleigh number range (1.7 • 10 8 < Ra < 1.93 • 10 8). To our knowledge, this time dependent behavior has not been properly dened for other Prandtl numbers. On the other hand, in the non-Newtonian case, results for Ra > 10 6 are scarce and no time dependent solutions have been reported. Regarding to the Prandtl number, the studies are generally focused for Pr > 1.0 both in the Newtonian and non-Newtonian cases. The presented VMS formulation is validated using existing results, and is used to study high convective cases, to determine the ow conditions at which it becomes time dependent, and to establish new benchmark solutions for non-Newtonian uid ows at dierent Pr and power law indexes. Obtained results have good agreement with those previously reported in the specic literature, and new data related to the oscillatory behavior is found in both Newtonian and non-Newtonian cases. The ow dynamics and its time dependent behavior show strong dependence on Ra, Pr and the power law index n, and the critical Rayleigh number (Ra c) where the rst Hopf bifurcation occurs is determined for dierent values of n in the range 0.01 < Pr < 1, 000. The non-Newtonian inuence of the uid is later checked in a 3D model of a simplied heat exchanger where the capability of pseudoplastic uids for energy transport is compared with the Newtonian case.

The Finite Element Numerical Investigation of Free Surface Newtonian and Non-Newtonian Fluid Flows in the Rectangular Tanks

“…e velocity boundary layer of dilatant fluid is wider than the other two fluids, which is mainly because its viscosity tends to be bigger with the increase in shear rate. ese findings are identical with the reports in the literature [26]. e CPU time is 8580 s for this simulation on Mesh2.…”

“…According to the rheological theory, the viscosity of non-Newtonian fluid will be affected by the velocity gradient. We take the power-law model [26,27] to describe a nonlinear relation between shear stress and the rate of deformation. e strain rate tensor is denoted as d � (∇u + (∇u) T )/2, and its magnitude is c �…”

Analysis of a stabilised finite element method for power-law fluids

“…Finite element approximations of problems with power-law rheology have been extensively studied, including stabilised (or variational-multiscale) methods (cf. [10,1], for example) and local discontinuous Galerkin methods (see, [27], for example). The relevant literature is vast and it is beyond the scope of this work to provide an exhaustive survey of the various contributions; the interested reader may wish to consult [29], for example.…”

Experimental and Numerical Analysis of Chaotic Advection as an Efficient Approach to Maximize Homogeneous Laminar Mixing in a Batch Mixer

“…The performance of the mixer is such that, by creating a perturbation in the flow, the flow path was changed and caused chaotic trajectories of fluid particles. Chaotic mixers can be categorized based on the type of perturbations caused by the flow into two main categories: (1) passive mixer (Aguirre et al, 2018, Grosso et al, 2018, Jung et al, 2018, Luan et al, 2018, Mizuno and Funakoshi, 2002, Mizuno and Funakoshi, 2004, Pacheco et al, 2006, Xu et al, 2016, which creates chaotic flow with a simple geometric perturbation and without input of energy, and (2) the active mixer (Jegatheeswaran et al, 2018, Tohidi et al, 2015, Wünsch and Böhme, 2000, which requires input energy to cause chaos in the flow.…”