Numerical simulation of fluid flow and heat transfer in rotating channels using parallel lattice Boltzmann method
Introduction
Several engineering fluid flow and heat transfer problems involve moving boundaries and rotating fluids. Conventional numerical methods are well established for simulation of wide variety of fluid flow and heat transfer involving moving boundary, slide and rotating domains [1], [2], [3], [4], [5]. From past three decades, Lattice Boltzmann method (LBM) has emerged as new numerical method for simulation of fluid flow and heat transfer problems with stationary boundaries, moving boundaries [6], [7], [8], [9], [10], sliding mesh [11] and rotating boundary [12]. In LBM, dynamics of flow evolve through fictitious particles collision and redistribution on a lattice grid with pre-defined lattice velocities. Under a low Mach number assumption, Chapman-Enskog analysis [13] of LB equation associates moments of equilibrium particles to physical (macroscopic) fluid flow variables, such as density, velocity and temperature, in Navier-Stokes equations. Flows with moving boundaries are modeled via source terms at the boundary, while sliding mesh and rotating fluid flows are modeled via sources terms in the LB equations. However, applications of LBM for solving rotating fluid flows and heat transfer are limited. Following is the literature summary related to application of LBM for solving laminar and turbulent fluid flow problems with moving and rotating wall.
Application of LBM for moving boundary fluid flow problem was presented by Lallemand and Luo [14]. Moving boundary was modeled considering simple bounce-back and interpolation concept at flat walls. Kao and Yang investigated curved moving boundary treatments in the LBM [15]. Hybrid method for solving swirl and rotating flow for Quasi 3D problems was proposed by Huang et al. [16]. This model works for low Mach number axis symmetric flow, and finite difference method was used to solve swirl velocity and temperature field. Chen [17] simulated compositional convection in rotating annulus cavity using LBM. Two sets of boundary conditions namely (a) horizontal temperature and vertical solutal gradient, (b) vertical temperature and horizontal solutal gradient were considered. It was concluded that rotation was suppressing the convection effect, especially for the first set of boundary condition. In another study, Chen [8] proposed a simple LBM model for simulation of heat transfer inside a rotating disc-cylinder domain. This model was based on vorticity-stream function and was shown to be more efficient, stable for simulation of high Grashof number flows.
Cai [12] studied fluid flow and heat transfer over a rotating cylinder for various rotation numbers. They proposed a second order accurate method to deal with moving curved boundaries. Secondary flow behavior in a transient rotating periodic channel was presented by Zhang et al. [9]. Their study found formation of multiple secondary flow vortexes, which were regulated by a ratio of pressure gradient and centrifugal force in short periodic computational domain. Flow in a bifurcated channel resembling draft tube of a turbine with rotating inflow boundary condition was studied by Qing-Dong [12]. This study gives details of particle distribution functions at the various boundaries. Result showed that the rotating flow causes mal distribution in the bifurcated channel. Recently, Zhang et al. [10] proposed a method for simulation of fluid flow in a sliding mesh domain within a local reference frame context. Direct numerical simulation of 2D rotating cylinder, 2D blade motion in cross flow and 3D turbulent flow past propeller were conducted for validating this model. The proposed model was able to compute fluxes accurately across the interface of local reference frame.
A large eddy simulation of agitated turbulent flow in a tank driven by Rushton and pitched blade turbines was presented by Jos Derksen [18]. Smagorinsky subgrid-scale model was used in this study, and it was found that flow field was accurate but with high magnitude of turbulent kinetic energy in the vicinity of impeller. All simulations were conducted on Beowulf cluster. In another study, Lu et al. [19] used nonuniform grid with arbitrary computation domain to solve agitated turbulent flow in a tank driven by Rushton turbine. This method is found to reduce 75 % of simulation time when compared with high resolution uniform grid. Eggels [20] conducted direct numerical simulation and large eddy simulation of agitated turbulent flow in tank. This was one of the earliest studies, which support application of LBM for simulation of turbulent flow. Several authors have studied rotation induced turbulence [21], [22].
From the literature it can be emphasized that application of LBM for simulation of rotating fluid flow and heat transfer is very sparse. However, LBM simulations in the literature considered fully developed flow region at isothermal conditions. Rotating fluid flow and heat transfer in entrance region has many engineering applications such as turbine blade cooling, and electrical winding cooling. Therefore, the objective of this study was to develop an efficient parallel LBM code for simulation of rotating fluid flow and heat transfer using OpenMP library. The parallel LBM model will consider all body forces acting in a non-isothermal rotating fluid flow problem. Two rotation modes namely orthogonal and parallel rotations were considered, in conjunction with stable fluid flow [23] and thermal [24] boundary conditions. Domain decomposition method of data parallelism was adopted.
Section snippets
Methodology
Incompressible LBGK model proposed by He et al. [25] is adopted here. In LBM space is discretized into uniform lattice size of and velocity is discretized into finite number of velocities to form particle distribution functions . The LBGK evolution equation is as follows. is the BGK [26], [27] collision operator which defines particle interaction on lattice sites. represents force term. Flow dynamics evolve through series of
Parallel method
Typically, for stable and accurate LB simulation, lattice nodes should scale with Reynolds number and hence large number of lattice nodes are required for simulation of high Reynolds number flows. In lattice units, Mach number is function of lattice size and lattice time which should be less than 0.3. Actually, LBM simulates pseudo incompressible or weakly compressible fluid flow. To reduce the inherent compressibility effect, lattice size should be proportional to square the order of
Problem description, governing equations and normalization
Fig. 2 shows the schematics of computational domains for orthogonal rotating channel with orientation of lattice stencils, and Fig. 3 shows the schematics of computation domain for parallel rotating channel. The study considers square channel (AR = 1) with length equal to ten hydraulic diameter. The length between rotating axis and the channel inlet is 20 Dh. The Reynolds number based on hydraulic diameter is equal to 100. Walls are assumed to be at constant temperature, and its value is based on
Boundary conditions
Boundary conditions affect the stability and accuracy of LBM simulations. After every time-step, incoming distribution functions at the boundaries are unknown and should be defined based on required physical boundary condition and outgoing distribution functions. There are several methods that are proposed for calculating unknown distribution functions. Zou and He [29] used bounce back of nonequilibrium part of distribution functions to calculate unknown distribution function at the boundaries
Results and discussion
For testing and validation of the parallel LBM code, laminar mixed convection in 3D rotating channels is simulated. Two modes of rotations namely orthogonal and parallel are considered. Parallel simulations are conducted on a workstation equipped with 3.1 GHrtz dual processors 64 GB RAM using OpenMP library. Reynolds number based on hydraulic diameter was kept constant at 100, and walls are kept at constant temperature. Nusselt number definition is based on bulk temperature. Table 1 shows
Conclusion
Incompressible laminar mixed convection in 3D rotating channels was modeled and simulated using parallel LB-BGK method. Reynolds number was held constant at 100. Both density ratio and rotation were set to 0.2. Two rotational modes namely orthogonal and parallel modes were considered. OpenMP libraries was used to parallelize the serial LBM code with domain decomposition method. Predicted velocity and temperature were found to agree well with velocity and temperature obtained from Fluent. From
Conflict of interest
There is no conflict of interest for the following article.
Acknowledgment
The authors acknowledge the support of the Universiti Teknologi Malaysia, Malaysia for carrying out the present research work.
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