The study of non-Newtonian fluids has received prodigious attention from the researchers in many branches of science because of the ubiquitous presence of non-Newtonian fluids in nature. Examples include shampoo, toothpaste, ketchup, whipped cream, biological fluids (e.g., blood, saliva, and synovia), industrial fluids (e.g., lubricants, paints, and emulsions), polymer solutions, Oobleck, and body armor, to name a few. Unlike Newtonian fluids, these fluids exhibit variations of viscosity in a nonlinear form with shear-rate change. For shear-thinning (shear-thickening) behavior of non-Newtonian fluids, viscosity decreases (increases) with increasing shear rate. There are many mathematical models [1
] to describe the characteristics of these types of fluid. Numerical simulations of non-Newtonian fluids are more complicated than those of Newtonian fluids because of the constitutive equation, representing non-Newtonian fluid behavior. Many researchers paid attention to the numerical simulation of non-Newtonian fluid flow by considering different constitutive equations and numerical methods such as the finite difference method, finite volume method, finite element or spectral method, and boundary element method. Among these, the work of Bell and Surana [2
] is remarkable. They investigated two dimensional incompressible non-Newtonian power-law fluid flow in a lid-driven square cavity using p
-version least-squares finite element formulation for various values of power-law index n
). Neofytou [3
] studied the 3rd order upwind finite volume method for generalized Newtonian fluid flows in a lid-driven cavity and considered the power-law, Bingham, Casson, and Quemada model and presented the results in graphical form. Rafiee [4
] also studied the modeling of generalized Newtonian lid-driven cavity flow using the smooth particle hydrodynamics (SPH) method. Syrjälä [5
] investigated the fully developed laminar flow of a power-law non-Newtonian fluid in a rectangular duct. Bose and Carrey [6
] studied p
finite element methods for incompressible non-Newtonian flows.
An enormous amount of research has been reported on non-Newtonian power-law fluids using Navies–Stokes (N–S) nonlinear partial differential equations (PDEs) of order two with an employment of the above-mentioned numerical methods. Indeed, the N–S equations associated with the conservation law of mass, momentum, and energy are utilized to describe the flow as a continuous medium at the macroscopic level. All the macroscopic variables, such as velocity, pressure, and temperature, are obtained in a region of interest by solving the N–S equations with the given boundary conditions. Although finding the macroscopic variables from the governing macroscopic equations (i.e., N–S equations) are methodically straightforward; however, researchers need to emphasize the truncation errors that come from the truncation of the Taylor series. Furthermore, the numerical instability arises when solving the discretized algebraic equations for the macroscopic variables at each mesh point, obtained from the nonlinear PDEs.
Compared to the above mentioned numerical method, the lattice Boltzmann method (LBM) is developed based on the fact the fluids are comprised of coarse-grained fictive particles that reside on a regular mesh and perform translation and collision steps entailing overall fluid-like behavior [7
]. Instead of solving macroscopic quantities through N-S equations, LBM uses particle distributions for calculating different properties of the fluid. It is worth being mentioned that the LBM approach with the simple collision model of Bhatnagar–Gross–Krook (BGK) eventually constructs a stable, accurate, and computationally efficient method for simulating complex fluid flows such as non-Newtonian fluid flows, chemical reacting fluid flows, single-phase, multiphase fluid flows, and so on. An efficient strategy was developed for building suitable collision operators and used in a simplified form of the lattice gas Boltzmann equation by Higuera et al. [10
]. They found that the resulting numerical scheme was linearly stable while applied to the computation of the flow in a channel containing a periodic array of obstacles. Succi et al. [11
] did another earlier study on the three-dimensional fluid flow simulation in complex geometry with the low Reynolds number flows. Lavallee et al. [12
] investigated the wall boundary conditions for the lattice Boltzmann simulation, and they reported that the bounce-back reflection was not the only interaction for no-slip boundary conditions and but Knudsen-type interaction was also appropriate. Ziegler [13
] developed a heuristic interpretation of no-slip boundary conditions for lattice Boltzmann and lattice gas simulations. He also suggested an improvement that consists of including the wall nodes in the collision operation.
The above pioneer development of the lattice Boltzmann method is based on the single-relaxation-time (SRT) or Bhatnagar–Gross–Krook (BGK) approach. Later on, it was found that the GGK-LBM has some limitations due to the numerical instability for the higher Reynolds number fluid flows and fixed Prandtl number (
) problem. To overcome these defects d’Humieres [14
] developed a multiple-relaxation-time (MRT)-based lattice Boltzmann method. Later on, Lallemand and Luo [15
] and d’Humieres [16
] found in their studies that MRT-LBM is much more stable than the BGK-LBM because the different relaxation parameters change the optimal numerical stability. Due to the superiority of The MRT-LBM over BGk-LBM, many researchers applied MRT-LBM to simulate the fluid flow and convective heat transfer, such as [17
In recent decades, the LBM has become attractive for numerical simulations in computational fluid dynamics because of its parallelism in the algorithm, simplicity of programming, high accuracy, lesser computational time, and simplicity for modeling complex geometrical flow problems [22
]. Nevertheless, very few investigations are related to non-Newtonian fluids that have been conducted in the lattice Boltzmann simulations. Fillips and Roberts [27
] analyzed the non-Newtonian flows using the lattice Boltzmann simulation. A lattice Boltzmann simulation of non-Newtonian flows past confined cylinders has been done by Nejat et al. [28
]. Chai et al. [29
] used the multiple-relaxation-time (MRT) lattice Boltzmann method for investigating the generalized Newtonian flows. Mendu and Das [30
] investigated the non-Newtonian cavity flow driven by two facing lids using the LBM. Immersed boundary SRT or BGK-LBM has been applied to investigate the non-Newtonian flows over a heated cylinder by Deloueci et al. [31
]. Li et al. [32
] studied power-law fluid flows in the two-dimensional square cavity using the multiple-relaxation-time lattice Boltzmann method.
A massively scalable implementation approach of the 2D and 3D lattice Boltzmann method for CPU/GPU heterogeneous clusters has been studied by Riesinger et al. [33
]. They used massage passing interface (MPI) programming for the multi-core Xeon E5-2680v2 CPU and two different NVIDIA GPUs: Tesla K20x and Tesla P100, for the heterogeneous GPU clusters. They found that Tesla P100 showed the best computational performance. Geier and Schönherr [34
] implemented the esoteric twist algorithms for the lattice Boltzmann method. Esoteric Twist is a thread-safe in-place streaming method that combines streaming and collision and requires only a single data set, which is particularly suitable for the implementation of the lattice Boltzmann method on graphics processing units.
Franco et al. [35
] investigated the 2D lid-driven cavity flow simulation using GPU-CUDA with a high-order finite difference scheme for the Navier–Stokes equations with a maximum
= 10,000. They used NVIDIA GeForce GTX-570 card and found that for the lid-driven cavity flow, GPU computing performed well for the larger grid sizes, and a similar performance had been found by Molla et at. [36
]. GPU accelerated simulations of the three-dimensional flow of power-law fluids in a lid-driven cubic cavity based on the Navier–Stokes equations have been studied by Jin et al. [37
]. They used NVIDIA Tesla k20x GPU to simulate the laminar flow for the shear-thinning (
) and shear-thickening (
) along with Newtonian fluid (
) for maximum
. Recently, Rahim et al. [38
] have done the BGK-based lattice Boltzmann simulations of natural convection and heat transfer from multiple heated blocks using the Tesla k40 GPU and found that GPU computing is much faster than the single-core central process unit (CPU) computing.
The literature mentioned above for non-Newtonian fluid flow employed the traditional Ostwald-de Waele power-law (PL) model [39
], to characterize the non-Newtonian fluid behavior. Indeed, the PL model is the most deliberately used in all branches of rheology because of its straightforward description of non-Newtonian fluids [1
]. However, the model contains several limitations in its description. Firstly, an unphysical prediction of infinite viscosity arises at the zero shear rate. Therefore, a singularity problem arises at the zero shear rate in the case of shear-thinning fluids. Secondly, the non-Newtonian viscosity for the shear-thickening fluids obtained by the traditional power-law model is substantially large at an infinite shear rate. Therefore, a modification has been adopted in the literature considering that all time-independent type non-Newtonian fluids behave Newtonian at a very low and high shear rates [42
]. The amendment is considered according to an experimental result conducted by Boger [43
], who first reported a lower and upper non-Newtonian regime for the pseudo-plastic fluids against applied shear-stress. Indeed, pseudo-plastic fluids sustain a constant viscosity at very low and high shear rates, and that viscosity decreases when the shear rate is increasing within a range. In this approach, Yao and Molla [42
] proposed a modified power-law viscosity model for the external boundary layer studies, which is free from the singularity that arises in the traditional Ostwald-de Waele power-law model. A lower and upper threshold of shear rates has been imposed in the original power-law model in the modification. The viscosity is predicted as Newtonian before and after the threshold limit, according to the report of Boger [43
]. Within the lower and upper threshold limits, the viscosity is measured using the Ostwald-de Waele power-law model. Therefore, the introduced modified power-law model becomes free from singularities in the boundary-layer formulations. Molla and Yao [42
] have investigated several Prandtl boundary-layer studies with a modified power-law model. However, the present modified power-law model for the lattice Boltzmann method is different due to the presence of governing parameter
The present study intended to employ the modified power-law viscosity model to investigate the non-Newtonian fluid flow in a complex benchmark problem; the lattice Boltzmann method (LBM) conducted the numerical simulation. In this approach, the lid-driven cavity flow, channel flow, and backward-facing step (BFS) flow are utilized for the present study as the computational fluid dynamics (CFD) community considered these models a benchmark problem examining the accuracy and efficiency of any numerical method. The backward-facing step (BFS) flows are essential for their wide range of engineering applications such as engine flows, separation flow behind a vehicle (cars and boat), heat transfer systems, inlet tunnel flow of engine or inside a condenser/combustor, and even the flow around buildings [45
]. The BFS model is one representative model that involves essential flow features such as flow separation, vortex evolution, re-circulation, and re-attachment. Many experimental and numerical investigations are carried out using these physical domains since they are considered classical problems in fluid dynamics. For example, Lee and Mateescu [46
] experimentally and numerically investigated the two-dimensional backward-facing step flow. Their study’s working fluid was air, and they have considered the Renolds number
, and the channel’s expansion ratio is 1.17 and 2.0. Armaly et al. [47
] also conducted an experimental and theoretical investigation of backward-facing step flow considering air. They presented results for laminar, transitional, and turbulent airflow in a Reynolds-number range of
. For high Reynolds numbers, Erturk [48
] studied two-dimensional steady incompressible flow over a backward-facing step using a very efficient finite difference method and presented the results both in graphical and tabular forms. Xiong and Zhang [49
] developed a two-dimensional lattice Boltzmann model for uniform channel flow. Zhou et al. [50
] studied the lattice Boltzmann method (LBM) for open channel flow. They discussed the power, potential, applicability, and accuracy of the LBM in many open-channel flow simulations. Wu and Shao [51
] investigated numerical simulation of steady two-dimensional incompressible lid-driven cavity flows (
= 100–500) using a multi-relaxation-time (MRT) model in the parallel lattice Boltzmann Bhatnager–Gross–Krook method (LBGK). Recently, Aditya and Kenneth [52
] investigated the backward-facing step flow using a numerical procedure that involves the coupling of the LBM and the vorticity-stream function method (VSM). They found that the coupled LBM-VSM solver is superior to a single solver for the 2D incompressible BFS flow for
while maintaining the same accuracy. The literature that mentioned and not mentioned here regarding the BFS flow mainly focused on the Newtonian fluid, and an extensive literature review can be found in Chen et al. [45
]. Furthermore, Ameur and Menni [53
] recently investigated a three-dimensional BFS flow for the case of shear-thinning fluids (carboxymethylcellulose) where the power-law index
and the Reynolds number has been varied within the range of
12,000. However, to the best of the author’s knowledge, no study has considered a comprehensive investigation of the non-Newtonian power-law fluids in a backward-facing step flow with the lattice Boltzmann simulation approach.
Recently, several studies based on the finite difference and finite volume methods for the Naiver–Stokes and lattice Boltzmann method with non-Newtonian fluids have been done by the different authors. Shupti et al. [54
] investigated the pulsatile non-Newtonian fluid flows in a model aneurysm with an oscillating wall. A lattice Boltzmann simulation of non-Newtonian power-law fluid flows in a bifurcated channel with low Reynolds number has been studied by Siddikk et al. [55
]. Thohura et al. [56
] have investigated the fluid flows’ numerical simulation of convective heat transfer for the non-Newtonian fluids based the Naiver–Stokes that are solved by the finite volume method.
On top of the above literature review, the present study has been structured as follows: A modified power-law viscosity (MPL) model is proposed for the lattice Boltzmann method. In this approach, the Reynolds number has been found to be free from the power-law index . For the lattice Boltzmann method, a new parameter associated with the time unit is introduced into the power-law model that governs the non-Newtonian fluid flows. Moreover, in the modified power-law model, there is no singularity for which the infinitely large viscosity can generate in the simulation while the fluid shear rate approaches zero. The MPL model was validated for the well-known benchmark problems and then applied for fluid flows through a backward-facing step. The present simulations have been done using the state-of-the-art GPU parallel computing using the CUDA C platform.