Modelling of Generalized Newtonian Lid-Driven Cavity Flow Using an SPH Method

Abstract

In this paper a smoothed particle hydrodynamics (SPH) method is introduced for simulating two-dimensional incompressible non-Newtonian fluid flows, and the nonNewtonian effects in the flow of a fluid which can be modelled by generalized Newtonian constitutive equations are investigated. Two viscoplastic models including Bingham-plastic and power-law models are considered along with the Newtonian model. The governing equations include the conservation of mass and momentum equations in a pseudo-compressible form. The spatial discretization of these equations is achieved by using the SPH method. The temporal discretization algorithm is a fully explicit two-step predictor-corrector scheme. In the prediction step, an intermediate velocity field is obtained using a forward scheme in time without enforcing incompressibility. The correction step consists of solving a pressure Poisson equation to satisfy incompressibility by providing a trade-off between the pressure and density variables. The performance of the proposed scheme is evaluated by studying a benchmark problem including flow of viscoplastic fluids in a lid-driven cavity. Both Newtonian and non-Newtonian cases are investigated and the results are compared with available numerical data. It was shown that in all cases the method is stable and the results are in very good agreement with available data.2000 Mathematics subject classification: primary 76M28; secondary 76A05.

Multiple-relaxation-time lattice Boltzmann model for generalized Newtonian fluid flows

“…From another point of view, the complexity of constitute equations also inspire researchers to develop some new numerical schemes or methods to study non-Newtonian behaviors. For example, Neofytou proposed a 3rd order upwind finite volume method to study GNF (including the Power-Law fluid, Quemada fluid, Bingham plastics, and Casson fluid) flows in a lid-driven cavity, and validated the method through a comparison with numerical data in available literature [3]; we note that the problem of GNF flows in the lid-driven cavity is also studied recently by Rafiee using smoothed particle hydrodynamics (SPH) method [4]. Bell used a p-version least squares finite element formulation (LSFEF) to investigate two-dimensional, incompressible, Power-law fluid flow under isothermal and non-isothermal conditions, and demonstrated the convergence characteristics and accuracy of the formulation through some numerical examples [5].…”

“…The fluid flow in a lid-driven cavity is another classic problem, and has been widely used to validate numerical methods by many authors [3][4][5]37,38] since it has a simple geometry and shows rich vortex phenomena. In this work, this problem is also used to verify the capacity of present MRT model through a comparison between the present numerical results and those reported in some available literature [3,5,37].…”

“…Above brief summary shows that some traditional methods (except for the SPH in Ref. [4]) have been widely used to study GNF flows. However, since the viscosity of GNF is related to the shear rate or the gradient of velocity, we need an appropriate difference scheme to treat the gradient of velocity appeared in viscosity, which not only brings additional difficulties, but also causes the numerical methods to be unstable.…”

Smoothed particle hydrodynamics simulation of non-Newtonian moulding flow

“…Extensive reviews can be found in Monaghan 1992 and2005 [10,11]. Recently, SPH has been applied to simulate non-Newtonian and viscoelastic transient and free surface flows, for example see Ellero and Tanner [17], Fang et al [18], Hossein et al [19], Rafiee et al [20] and Rafiee [21].…”

Numerical approximations for flow of viscoplastic fluids in a lid-driven cavity

“…This benchmark problem has been addressed by some authors [2,22,23,25,28,29]). In the present work, the mechanical model is approximated by multi-field Galerkin least squares method in terms of extra-stress, pressure and velocity.…”