An immersed boundary-lattice Boltzmann method is introduced that can be employed for different thermal and thermo-solutal problems of Newtonian and non-Newtonian fluids. The general macroscopic and mesoscopic equations are presented and discussed. It is shown and proved that the macroscopic equations are satisfied by the proposed lattice Boltzmann equations. This approach removes the limitation of the conventional lattice Boltzmann method in constitutive equations and boundary conditions. To validate the accuracy of the method, it is compared against several cases of complex geometries with curved boundaries for natural convection in enclosures. To demonstrate the ability of this method for the simulation of thermo-solutal flows of non-Newtonian fluids with curved boundaries, double diffusive natural convection of Carreau fluid between a square cylinder and two circular cylinders is investigated and results are reported. Next, double diffusive mixed convection of a Bingham fluid in a cavity with a curved boundary condition is studied.
REFERENCES
1.
M. Y.
Ha
, I.-K.
Kim
, H. S.
Yoon
, and S.
Lee
, “Unsteady fluid flow and temperature fields in a horizontal enclosure with an adiabatic body
,” Phys. Fluids
14
, 3189
(2002
).2.
J. R.
Lee
, M. Y.
Ha
, S.
Balachandar
, H. S.
Yoon
, and S. S.
Lee
, “Natural convection in a horizontal layer of fluid with a periodic array of square cylinders in the interior
,” Phys. Fluids
16
, 1273
(2004
).3.
Z.
Chen
, C.
Shu
, L. M.
Yang
, X.
Zhao
, and N. Y.
Liu
, “Immersed boundary-simplified thermal lattice Boltzmann method for incompressible thermal flows
,” Phys. Fluids
32
, 013605
(2020
).4.
Y.
Kim
and C. S.
Peskin
, “A penalty immersed boundary method for a rigid body in fluid
,” Phys. Fluids
28
, 033603
(2016
).5.
C.-C.
Liao
and C.-A.
Lin
, “Influence of Prandtl number on the instability of natural convection flows within a square enclosure containing an embedded heated cylinder at moderate Rayleigh number
,” Phys. Fluids
27
, 013603
(2015
).6.
H. S.
Yoon
, M. Y.
Ha
, B. S.
Kim
, and D. H.
Yu
, “Effect of the position of a circular cylinder in a square enclosure on natural convection at Rayleigh number of 107
,” Phys. Fluids
21
, 047101
(2009
).7.
R.
Mittal
and G.
Iaccarino
, “Immersed boundary methods
,” Annu. Rev. Fluid Mech.
37
, 239
(2005
).8.
C. S.
Peskin
, “Flow patterns around heart valves: A numerical method
,” J. Comput. Phys.
10
, 252
(1972
).9.
C. S.
Peskin
, “Numerical analysis of blood flow in the heart
,” J. Comput. Phys.
25
, 220
(1977
).10.
M. F.
McCracken
and C. S.
Peskin
, “A vortex method for blood flow through heart valves
,” J. Comput. Phys.
35
, 183
(1980
).11.
C. S.
Peskin
and D. M.
McQueen
, “A three-dimensional computational method for blood flow in the heart I. Immersed elastic fibers in a viscous incompressible fluid
,” J. Comput. Phys.
81
, 372
(1989
).12.
C. S.
Peskin
and D. M.
McQueen
, “A three-dimensional computational method for blood flow in the heart II. Contractile fibers
,” J. Comput. Phys.
82
, 289
(1989
).13.
C. S.
Peskin
and B. F.
Printz
, “Improved volume conservation in the computation of flows with immersed elastic boundaries
,” J. Comput. Phys.
105
, 33
(1989
).14.
D.
Goldstein
, R.
Handler
, and L.
Sirovich
, “Modeling a no-slip flow boundary with an external force field
,” J. Comput. Phys.
105
, 345
(1993
).15.
M.-C.
Lai
and C. S.
Peskin
, “An immersed boundary method with formal second-order accuracy and reduced numerical viscosity
,” J. Comput. Phys.
160
, 705
(2000
).16.
L.
Zhu
and C. S.
Peskin
, “Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method
,” J. Comput. Phys.
179
, 452
(2002
).17.
Y.
Kim
and C. S.
Peskin
, “Penalty immersed boundary method for an elastic boundary with mass
,” Phys. Fluids
19
, 053103
(2007
).18.
L.
Zhu
, G.
He
, S.
Wang
, L.
Miller
, X.
Zhang
, Q.
You
, and S.
Fang
, “An immersed boundary method based on the lattice Boltzmann approach in three dimensions, with application
,” Comput. Appl. Math.
61
, 3506
(2011
).19.
F.-B.
Tian
, H.
Luo
, L.
Zhu
, J. C.
Liao
, and X.-Y.
Lu
, “An efficient immersed boundary-lattice Boltzmann method for the hydrodynamic interaction of elastic filaments
,” J. Comput. Phys.
230
, 7266
(2011
).20.
L.
Zhu
, X.
Yu
, N.
Liu
, Y.
Cheng
, and X.
Lu
, “A deformable plate interacting with a non-Newtonian fluid in three dimensions
,” Phys. Fluids
29
, 083101
(2017
).21.
W.
Tong
, Y.
Yang
, and S.
Wang
, “Characterizing three-dimensional features of vortex surfaces in the flow past a finite plate
,” Phys. Fluids
32
, 011903
(2020
).22.
N.
Tofany
, Y. M.
Low
, C.-H.
Lee
, and Y.-M.
Chiew
, “Two-phase flow simulation of scour beneath a vibrating pipeline during the tunnel erosion stage
,” Phys. Fluids
31
, 113302
(2019
).23.
J.
Ryu
, J.
Yang
, S. G.
Park
, and H. J.
Sung
, “Phase-mediated locomotion of two self-propelled flexible plates in a tandem arrangement
,” Phys. Fluids
32
, 041901
(2020
).24.
D.
Zhang
, G.
Pan
, L.
Chao
, and Y.
Zhang
, “Effects of Reynolds number and thickness on an undulatory self-propelled foil
,” Phys. Fluids
30
, 071902
(2018
).25.
J. W.
Park
, J.
Ryu
, and H. J.
Sung
, “Effects of the shape of an inverted flag on its flapping dynamics
,” Phys. Fluids
31
, 021904
(2019
).26.
R.
Subburaj
, P.
Khandelwal
, and S.
Vengadesan
, “Numerical study of flow past an elliptic cylinder near a free surface
,” Phys. Fluids
30
, 103603
(2018
).27.
J.
Yuan
, A. A.
Mishra
, G.
Brereton
, G.
Iaccarino
, and M.
Vartdal
, “Single-point structure tensors in turbulent channel flows with smooth and wavy walls
,” Phys. Fluids
31
, 125115
(2019
).28.
A. S.
Bharadwaj
and S.
Ghosh
, “Numerical investigation of lift enhancement in flapping hover flight
,” Phys. Fluids
32
, 051901
(2020
).29.
S.
Wang
, J.
Ryu
, G.
He
, F.
Qin
, and H. J.
Sung
, “A self-propelled flexible plate with a Navier slip surface
,” Phys. Fluids
32
, 021906
(2020
).30.
F.
Liu
, G.
Liu
, and C.
Shu
, “Fluid–structure interaction simulation based on immersed boundary-lattice Boltzmann flux solver and absolute nodal coordinate formula
,” Phys. Fluids
32
, 047109
(2020
).31.
J. R.
Pacheco
, A.
Pacheco-Vega
, T.
Rodić
, and R. E.
Peck
, “Numerical simulations of heat transfer and fluid flow problems using an immersed-boundary finite-volume method on non-staggered grids
,” Numer. Heat Transfer, Part B
48
, 1
(2005
).32.
N.
Zhang
, Z. C.
Zheng
, and S.
Eckels
, “Study of heat-transfer on the surface of a circular cylinder in flow using an immersed-boundary method
,” Int. J. Heat Fluid Flow
29
, 1558
(2008
).33.
B. S.
Kim
, D. S.
Lee
, M. Y.
Ha
, and H. S.
Yoon
, “A numerical study of natural convection in a square enclosure with a circular cylinder at different vertical locations
,” Int. J. Heat Mass Transfer
51
, 1888
(2008
).34.
Z.
Wang
, J.
Fan
, K.
Luo
, and K.
Cen
, “Immersed boundary method for the simulation of flows with heat transfer
,” Int. J. Heat Mass Transfer
52
, 4510
(2009
).35.
H. K.
Jeong
, H. S.
Yoon
, M. Y.
Ha
, and M.
Tsutahara
, “An immersed boundary-thermal lattice Boltzmann method using an equilibrium internal energy density approach for the simulation of flows with heat transfer
,” J. Comput. Phys.
229
, 2526
(2010
).36.
W. W.
Ren
, C.
Shu
, J.
Wu
, and W. M.
Yang
, “Boundary condition-enforced immersed boundary method for thermal flow problems with Dirichlet temperature condition and its applications
,” Fluids
57
, 40
(2012
).37.
C.-C.
Liao
and C.-A.
Lin
, “Simulations of natural and forced convection flows with moving embedded object using immersed boundary method
,” Comput. Methods Appl. Mech. Eng.
213-216
, 58
(2012
).38.
C.-C.
Liao
and C.-A.
Lin
, “Influences of a confined elliptic cylinder at different aspect ratios and inclinations on the laminar natural and mixed convection flows
,” Int. J. Heat Mass Transfer
55
, 6638
(2012
).39.
W.
Ren
, C.
Shu
, and W.
Yang
, “An efficient immersed boundary method for thermal flow problems with heat flux boundary conditions
,” Int. J. Heat Mass Transfer
64
, 694
(2013
).40.
F.
Ilinca
and J.-F.
Hétu
, “Immersed boundary solution of natural convection in a square cavity with an enclosed rosette-shaped hot cylinder
,” Numer. Heat Transfer, Part A
65
, 1154
(2014
).41.
C.
Choi
, S.
Jeong
, M. Y.
Ha
, and H. S.
Yoon
, “Effect of a circular cylinder’s location on natural convection in a rhombus enclosure
,” Int. J. Heat Mass Transfer
77
, 60
(2014
).42.
Y.
Hu
, D.
Li
, S.
Shu
, and X.
Niu
, “Study of multiple steady solutions for the 2D natural convection in a concentric horizontal annulus with a constant heat flux wall using immersed boundary-lattice Boltzmann method
,” Int. J. Heat Mass Transfer
81
, 591
(2015
).43.
Y.
Wang
, C.
Shu
, and L. M.
Yang
, “Boundary condition-enforced immersed boundary-lattice Boltzmann flux solver for thermal flows with Neumann boundary conditions
,” J. Comput. Phys.
306
, 237
(2016
).44.
J.
Wu
, Y.
Cheng
, and L. A.
Miller
, “An iterative source correction based immersed boundary-lattice Boltzmann method for thermal flow simulations
,” Int. J. Heat Mass Transfer
115
, 450
(2017
).45.
S. G.
Park
, C. B.
Chang
, B.
Kim
, and H. J.
Sung
, “Simulation of fluid-flexible body interaction with heat transfer
,” Int. J. Heat Mass Transfer
110
, 20
(2017
).46.
J. B.
Lee
, S. G.
Park
, and H. J.
Sung
, “Heat transfer enhancement by asymmetrically clamped flexible flags in a channel flow
,” Int. J. Heat Mass Transfer
116
, 1003
(2018
).47.
A. A.
Delouei
, M.
Nazari
, M. H.
Kayhani
, and S.
Succi
, “Non-Newtonian unconfined flow and heat transfer over a heated cylinder using the direct-forcing immersed boundary-thermal lattice Boltzmann method
,” Phys. Rev. E
89
, 053312
(2014
).48.
P.
Lee
, B. E.
Griffith
, and C. S.
Peskin
, “The immersed boundary method for advection-electrodiffusion with implicit timestepping and local mesh refinement
,” J. Comput. Phys.
229
, 5208
(2010
).49.
X.
Gong
, Z.
Gong
, and H.
Huang
, “An immersed boundary method for mass transfer across permeable moving interfaces
,” J. Comput. Phys.
278
, 148
(2014
).50.
S.
Succi
, The Lattice Boltzmann Equation: For Fluid Dynamics and Beyond
(Oxford University Press
, 2001
).51.
S.
Succi
, “Mesoscopic modeling of slip motion at fluid-solid interfaces with heterogeneous catalysis
,” Phys. Rev. Lett.
89
, 064502
(2002
).52.
A. A.
Mohamad
, Lattice Boltzmann Method: Fundamentals and Engineering Applications with Computer Codes
(Springer
, 2011
).53.
Z.
Chen
, C.
Shu
, and D.
Tan
, “Highly accurate simplified lattice Boltzmann method
,” Phys. Fluids
30
, 103605
(2018
).54.
Z.
Chen
, C.
Shu
, and D.
Tan
, “Three-dimensional simplified and unconditionally stable lattice Boltzmann method for incompressible isothermal and thermal flows
,” Phys. Fluids
29
, 053601
(2017
).55.
Z.
Chen
, C.
Shu
, and D.
Tan
, “Immersed boundary-simplified lattice Boltzmann method for incompressible viscous flows
,” Phys. Fluids
30
, 053601
(2018
).56.
Z.
Chen
, C.
Shu
, and L. Q.
Zhang
, “A simplified axisymmetric lattice Boltzmann method for incompressible swirling and rotating flows
,” Phys. Fluids
31
, 023605
(2019
).57.
H. Z.
Yuan
, Y.
Wang
, and C.
Shu
, “An adaptive mesh refinement-multiphase lattice Boltzmann flux solver for simulation of complex binary fluid flows
,” Phys. Fluids
29
, 123604
(2017
).58.
H.
Sajjadi
, M.
Salmanzadeh
, G.
Ahmadi
, and S.
Jafari
, “Turbulent indoor airflow simulation using hybrid LES/RANS model utilizing lattice Boltzmann method
,” Comput. Fluids
150
, 66
(2017
).59.
H.
Sajjadi
, A. A.
Delouei
, M.
Izadi
, and R.
Mohebbi
, “Investigation of MHD natural convection in a porous media by double MRT lattice Boltzmann method utilizing MWCNT–Fe3O4/water hybrid nanofluid
,” Int. J. Heat Mass Transfer
132
, 1087
(2019
).60.
H.
Sajjadi
, A. A.
Delouei
, M.
Atashafrooz
, and M.
Sheikholeslami
, “Double MRT lattice Boltzmann simulation of 3-D MHD natural convection in a cubic cavity with sinusoidal temperature distribution utilizing nanofluid
,” Int. J. Heat Mass Transfer
126
, 489
(2018
).61.
S. C.
Fu
, R. M. C.
So
, and W. W. F.
Leung
, “A discrete flux scheme for aerodynamic and hydrodynamic flows
,” Comput. Phys. Commun.
9
, 1257
(2011
).62.
S. C.
Fu
, R. M. C.
So
, and W. W. F.
Leung
, “Linearized-Boltzmann-type-equation-based finite difference method for thermal incompressible flow
,” Comput. Fluids
69
, 67
(2012
).63.
R. R.
Huilgol
and G. H. R.
Kefayati
, “From mesoscopic models to continuum mechanics: Newtonian and non-Newtonian fluids
,” J. Non-Newtonian Fluid Mech.
233
, 146
(2016
).64.
R. R.
Huilgol
and G. H. R.
Kefayati
, “A particle distribution function approach to the equations of continuum mechanics in Cartesian, cylindrical and spherical coordinates: Newtonian and non-Newtonian fluids
,” J. Non-Newtonian Fluid Mech.
251
, 119
(2018
).65.
G. H. R.
Kefayati
, H.
Tang
, and A.
Chan
, “Immersed boundary-finite difference lattice Boltzmann method through fluid-structure interaction for viscoplastic fluids
,” J. Fluid Struct.
83
, 238
(2018
).66.
P. L.
Quere
, “Accurate solutions to the square thermally driven cavity at high Rayleigh number
,” Comput. Fluids
20
, 29
(1991
).67.
P.
Zhang
, X.
Zhang
, J.
Deng
, and L.
Song
, “A numerical study of natural convection in an inclined square enclosure with an elliptic cylinder using variational multiscale element free Galerkin method
,” Int. J. Heat Mass Transfer
99
, 721
(2016
).68.
Y. G.
Park
, M. Y.
Ha
, C.
Choi
, and J.
Park
, “Natural convection in a square enclosure with two inner circular cylinders positioned at different vertical locations
,” Int. J. Heat Mass Transfer
77
, 501
(2014
).69.
G. S.
Mun
, Y. G.
Park
, H. S.
Yoon
, M.
Kim
, and M.
Yeong Ha
, “Natural convection in a cold enclosure with four hot inner cylinders based on diamond arrays (Part-I: Effect of horizontal and vertical equal distance of inner cylinders)
,” Int. J. Heat Mass Transfer
111
, 755
(2017
).70.
Y.
Wang
, C.
Shu
, and C. J.
Teo
, “Thermal lattice Boltzmann flux solver and its application for simulation of incompressible thermal flows
,” Comput. Fluids
94
, 98
(2014
).71.
C.
Shu
, “Application of differential quadrature method to simulate natural convection in a concentric annulus
,” Int. J. Numer. Methods Fluids
30
, 977
(1999
).72.
T. H.
Kuehn
and R. J.
Goldstein
, “An experimental and theoretical study of natural convection in the annulus between horizontal concentric cylinders
,” J. Fluid Mech.
74
, 695
(1976
).73.
Z.
Chen
, C.
Shu
, and D.
Tan
, “A simplified thermal lattice Boltzmann method without evolution of distribution functions
,” Int. J. Heat Mass Transfer
105
, 741
(2017
).74.
I.
Sezai
and A. A.
Mohamad
, “Double diffusive convection in a cubic enclosure with opposing temperature and concentration gradients
,” Phys. Fluids
12
, 2210
(2000
).75.
A. A.
Mohamad
, R.
Bennacer
, and J.
Azaiez
, “Double diffusion natural convection in a rectangular enclosure filled with binary fluid saturated porous media: The effect of lateral aspect ratio
,” Phys. Fluids
16
, 184
(2004
).76.
P. J.
Carreau
, “Rheological equations from molecular network theories
,” Trans. Soc. Rheol.
16
, 99
(1972
).77.
78.
R. R.
Huilgol
and G. H. R.
Kefayati
, “Natural convection problem in a Bingham fluid using the operator-splitting method
,” J. Non-Newtonian Fluid Mech.
220
, 22
(2015
).79.
G. H. R.
Kefayati
and R. R.
Huilgol
, “Lattice Boltzmann method for simulation of mixed convection of a Bingham fluid in a lid-driven cavity
,” Int. J. Heat Mass Transfer
103
, 725
(2016
).© 2020 Author(s).
2020
Author(s)
You do not currently have access to this content.
