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REVIEW 4 major objections 6 minor 61 references

Hybrid LBM-FVM and LBM-MCM Methods for Fluid Flow and Heat Transfer Simulation

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that hybrid LBM-FVM and LBM-MCM schemes, built with two coupling strategies, reproduce benchmark lid-driven cavity flows and natural convection within a few percent, making them valid for multiscale fluid flow and heat…

desk verdict A well-written chapter compilation of the authors' own prior hybrid LBM-FVM/LBM-MCM results—no new result, honest about its weaknesses, but not a standalone research contribution. read the letter →

arxiv 1908.08386 v1 pith:HRJR3EDX submitted 2019-08-18 cs.CE physics.flu-dyn

Zheng Li , Mo Yang , Yuwen Zhang This is my paper · ORCID
classification cs.CEphysics.flu-dyn MSC 76M2865M0865C05
keywords latticeBoltzmannmethodfinitevolumeMonteCarlomultiscalemethodsnaturalconvectionlid-drivencavityNusseltnumbermessagepassinginterfaces
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that mesoscale lattice Boltzmann solvers can be grafted onto macroscale finite-volume and Monte Carlo solvers so that fluid flow and heat transfer are solved together, and that the resulting hybrid schemes are accurate enough for practical use. It develops two coupling strategies: splitting the domain into subdomains and passing velocity, temperature, and density information across the interface, or using LBM for the velocity field while FVM or MCM supplies the temperature. Tested against lid-driven cavity flow at Reynolds numbers $100$–$1000$ and natural convection at Rayleigh numbers $10^4$–$10^6$, the hybrids reproduce benchmark streamlines, temperature fields, and Nusselt numbers, with the largest reported benchmark differences around 4.4%. If the claim holds, complex industrial flows could be decomposed into parts each solved by the cheapest appropriate method without losing accuracy at the seams.

What carries the argument

The load-bearing machinery is the message-passing interface between solvers. For domain-splitting LBM-FVM, the interface is closed with two LBM boundary recipes: the nonequilibrium extrapolation scheme, which sets the unknown boundary density's nonequilibrium part equal to that of a nearby inner node, and the finite-difference velocity-gradient method, which builds the boundary density from the velocity-gradient tensor supplied by FVM. For the coupled natural-convection method, the additional closure is the pressure-gap equality of Eqs. (91)–(93), which converts FVM pressure into the LBM boundary density. In the parameter-splitting hybrids, the working objects are the discrete LBM velocity solution, interpolated onto FVM control-volume faces, and the Monte Carlo random walker whose transition probabilities encode both conduction and convection in the temperature field.

What would settle it

Run the coupled LBM-FVM scheme beyond the tested low-Mach, weakly buoyant range—for example, a taller cavity, a higher Rayleigh number, or a fluid with strong density variation—and check whether the Nusselt number drifts from the benchmark beyond a few percent; alternatively, evaluate both sides of the pressure-gap equality using separate pure LBM and pure FVM solutions on the same overlapping zone and show that the equality fails.

Watch

Extended reading notes

Core claim

The central discovery is that two existing LBM boundary treatments—the nonequilibrium extrapolation scheme and the finite-difference velocity-gradient method—are sufficient to exchange information between LBM and FVM zones, provided the dimensionless lid velocity is made equal in both solvers. For heat transfer, a coupled LBM-FVM scheme reconstructs the LBM boundary density from FVM pressure through an assumed pressure-gap equality, and a second strategy solves the whole velocity field with LBM while FVM or MCM supplies the temperature. In the tested cases pure LBM and pure FVM agree with each other and with benchmark references, and the hybrid results sit at the same accuracy level: the maximum Nusselt number from the hybrid LBM-FVM is within about 3% of the reference in the natural-convection cases, and the LBM-MCM comparison to the square-cavity benchmark reports errors below about 4.4% in maximum and 3.1% in average Nusselt number. The authors conclude that LBM-FVM and LBM-MCM are valid for fluid flow and heat transfer problems.

Load-bearing premise

In the coupled LBM-FVM method, the density handed from FVM to LBM at the artificial boundary is reconstructed from the assumption that the pressure gap between the two solvers at that boundary equals the pressure gap between their zone averages; the text notes that even a small pressure mismatch produced clear error, so this hand-made equality is the premise the coupled method most depends on.

Editorial extensions

If this is right

  • The nonequilibrium extrapolation scheme is the better interface closure at low Reynolds numbers, while the finite-difference velocity-gradient method is the better one at high Reynolds numbers.
  • Coupled LBM-FVM reproduces natural-convection fields and wall Nusselt numbers for Rayleigh numbers $10^4$–$10^6$ with either vertical or horizontal domain splits, so geometric placement of the interface does not by itself control accuracy.
  • A parameter-splitting LBM-FVM route, with LBM for velocity and FVM for temperature, matches benchmark maximum Nusselt numbers within about 3% in the tested cases.
  • Combining LBM velocity with a Monte Carlo temperature solver matches the square-cavity benchmark Nusselt numbers with reported errors below about 4.4%, making the statistical temperature solver usable despite its rough isotherms.
  • The total time of any hybrid is controlled by the slower LBM zone, so the payoff is not speed in these benchmarks but the ability to apply each solver where it is strongest.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The pressure-gap equality that converts FVM pressure into LBM density is the most likely point of failure outside the tested range; a direct check is to compute both sides from separate pure solver solutions on the same overlap zone.
  • The statistical roughness visible in the Monte Carlo isotherms suggests that adding variance reduction or smoothing to the temperature solver could improve the LBM-MCM scheme without changing its coupling logic.
  • The benchmarks are all laminar two-dimensional cavities, so a natural extension would be to test the same interface recipes for turbulent or three-dimensional flows, where mesh requirements and time-step matching change the balance between solvers.
  • If the central claim holds, the practical payoff is largest for problems that combine complex-geometry regions, where LBM excels, with extended or conjugate heat-transfer regions, where FVM or MCM is cheaper.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The manuscript, framed as a book chapter, develops two families of multiscale methods for fluid flow and heat transfer: hybrid LBM-FVM methods (with two subdomain-coupling variants and one parameter-splitting variant) and a combined LBM-MCM method. In the subdomain-coupling LBM-FVM approach, the domain is split into LBM and FVM zones with message passing at artificial boundaries; in the parameter-splitting approaches, LBM provides the velocity field while FVM or MCM provides the temperature field. The methods are tested on lid-driven cavity flow (Re = 100, 400, 1000) and natural convection in a square cavity (Ra = 10^4, 10^5, 10^6, Pr = 0.71), with comparisons to Ghia et al. (1982), de Vahl Davis (1983), and Hortmann et al. (1990). The reported maximum Nusselt number errors for LBM-MCM are 1.0% and 4.4% for the two Rayleigh numbers, and the LBM-FVM variants agree with benchmark solutions within roughly 3% in the tabulated cases. The conclusion states that 'LBM-FVM and LBM-MCM proposed in this chapter are valid to solve fluid flow and heat transfer problems.'

Significance. If the central claim holds, the chapter provides a useful engineering validation of several hybrid meso-macro strategies, particularly the less common LBM-MCM combination and the parameter-splitting LBM-FVM method. The main strengths are the use of external benchmark solutions for the core validations (Ghia et al. for lid-driven flow, de Vahl Davis and Hortmann et al. for natural convection), the inclusion of quantitative Nusselt number comparisons, and the testing of two different domain decompositions for the subdomain-coupling method. These features give the numerical results independent grounding rather than relying only on fitted parameters. However, the validity conclusion rests on a small set of benchmark cases, a single grid resolution for each method, and one hand-made pressure-to-density closure whose justification is only qualitative, so the breadth of the claim exceeds what the evidence supports.

major comments (4)
  1. [Section 4.1.2, Eqs. (91)-(93)] The load-bearing pressure closure is not derived or independently tested. Equation (91) assumes that the difference between the LBM boundary pressure and the average LBM pressure equals the difference between the FVM boundary pressure and the average FVM pressure, based only on the observation that pressure-gradient differences 'are not evident' in the tested cases. The manuscript itself states that a small LBM-FVM pressure difference produced clear error, so accuracy is sensitive to this closure. Because the same natural-convection configurations were used to motivate the closure and then to validate the coupled method, the validation is partly circular. The authors should either derive Eq. (91) from the discrete momentum balances of the two schemes, or test it on cases outside the square-cavity Boussinesq family (e.g., different geometries, Rayleigh numbers, message-zone widths) before concluding that the coupled method is generally valid.
  2. [Sections 4.2, 5.2, and 6.3] The validity claim is supported by results at a single grid resolution: 160 x 160 grids are used for the coupled LBM-FVM method, and the LBM-MCM method appears to use only one grid as well, with no grid-refinement or convergence study anywhere in the chapter. Without evidence that the reported errors are not accidental at this particular resolution, the statement that the methods 'are valid to solve fluid flow and heat transfer problems' is under-supported. A grid-dependence study (at least two additional resolutions) for at least one case per method is needed to establish that the hybrid interfaces and interpolations converge.
  3. [Section 4.1.2, Eq. (96)] The artificial-boundary density is further approximated by the neighboring inner-node density through Eq. (96), after being reconstructed from FVM pressure through Eq. (93). This second approximation is introduced without justification or error analysis. Since the boundary density enters the nonequilibrium extrapolation scheme used for the velocity and temperature fields, errors in Eq. (96) propagate into the interior solution. The authors should quantify the effect of this approximation, for example by comparing results with and without it, or by testing an alternative interpolation for the boundary density.
  4. [Section 6.3.2, Table 3] The reported maximum Nusselt number error rises to 4.4% for Ra = 10^5 in the LBM-MCM method, yet the text describes the agreement as 'good' and the conclusion treats the method as validated. The 4.4% error may be acceptable for engineering purposes, but the authors should state an explicit accuracy criterion and discuss why this error level supports the general validity claim. The same applies to the qualitative Nusselt-number comparisons in Section 4.2, where no quantitative error table is provided for the coupled LBM-FVM method.
minor comments (6)
  1. [General] Several equations are garbled or contain rendering errors (e.g., Eq. (4) for the D2Q9 discrete velocities, Eqs. (6)-(8) for the equilibrium distribution, and the non-dimensional equations in Section 3.2.2). The notation is often inconsistent (for example, V is used both for the velocity vector and for the non-dimensional vertical velocity). These issues make it difficult to reproduce the methods without consulting the authors' earlier papers.
  2. [Section 4.2, Figures 19-20] The caption of Figure 20 reads 'Nusselt numbers at Ra=10^4' while the text says Figures 19 and 20 show temperature fields and streamlines; the figure content appears to be isotherms or streamlines, not Nusselt numbers. Please correct the caption and the corresponding text reference.
  3. [Sections 5 and 6, Tables 1-3] The table numbering is confusing: Section 5 introduces Tables 1 and 2 for the hybrid LBM-FVM Nusselt number comparison, while Section 6 refers to 'Table 1' and 'Table 3' when presenting LBM-MCM results. Renumber the tables consistently.
  4. [Section 6.1] The ordering of Eqs. (98) and (99) and the surrounding text is confusing: the sentence introducing convection effects appears before Eq. (99), but the conduction probabilities are presented in Eq. (98) with convection modifications only later. Please reorganize the derivation and state explicitly which equation applies to conduction and which to convection.
  5. [References] Reference [46] (de Vahl Davis) lists page numbers as '249-164', which is clearly a typo for 249-264. Also, some references to the authors' own prior work ([36], [38], [42], [45]) overlap heavily with the content of this chapter; this should be acknowledged or flagged as a compilation of prior results.
  6. [Conclusion] The final sentence makes a very broad claim that LBM-FVM and LBM-MCM are 'valid to solve fluid flow and heat transfer problems', but the tested cases are limited to lid-driven flow and natural convection in square enclosures. A more cautious statement that the methods are valid for the tested benchmark problems would better match the evidence.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: validation is against external benchmarks; repeated self-citations are not load-bearing.

full rationale

The central claims are that the proposed LBM-FVM and LBM-MCM hybrid schemes are valid for fluid flow and heat transfer. These claims are supported by comparisons with external reference solutions: Ghia et al. [35] for the lid-driven cavity in Section 3, Hortmann et al. [41] for the Nusselt-number benchmarks in Section 5, and Davis [46] for the LBM-MCM natural-convection cases in Section 6. The coupled method in Section 4 is checked against the authors' own pure LBM and pure FVM results, but those two pure methods are standard, are themselves shown against literature benchmarks elsewhere in the chapter, and are not fitted to the hybrid result. The pressure-density closure in Eqs. (91)-(93) is an ad hoc assumption and is the weakest load-bearing premise, as the skeptic notes, but it is not fitted to the benchmark output; the observed small pressure difference motivated the closure, and the same test cases are then used for validation. That is a robustness/generalization concern, not circularity by construction. The paper does contain many self-citations, including refs. [28], [30]-[32], [36], [38], [39], [42], and [45], and several figure captions reference the authors' own prior papers, but none of these citations is used to justify the validity claim in place of an external benchmark or independently checkable computation. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors, and no known result is merely relabeled as new. Thus the derivation chain is self-contained with respect to its external validation, and the only caveat is the prevalence of non-load-bearing self-citations.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central validation rests on standard LBM and FVM theory, but several ad hoc closures are load-bearing: the nonequilibrium extrapolation at Eq. (60), the pressure reconstruction at Eq. (91), and the unstated random walker statistics in MCM. No new physical entities are introduced.

free parameters (3)
  • Message passing zone width = 3 shared grids (3 x 160)
    Section 3.2.2 says 'Three shared grids are selected after testing'; no systematic convergence study is shown, and this width controls interface accuracy and computational cost.
  • MCM random walker count N = Not stated
    Section 6.1 defines temperature as the average over N random walkers in Eq. (102), but N and the convergence criteria are never reported.
  • Lid velocity to sound speed ratio = 0.1
    Eqs. (39)-(40) set the lid velocity to 0.1 times the sound speed so that LBM and SIMPLE dimensionless velocities match; this is a convenience choice, not a physical constraint.
assumptions (5)
  • domain assumption The Boussinesq approximation applies to natural convection in a square cavity.
    Introduced in Section 4.1.1 before Eqs. (73)-(76) and used for all natural convection validations.
  • domain assumption The spurious term in the LBM momentum equation can be neglected at low Mach number, so LBM and FVM solve effectively the same macroscopic equations.
    Section 2.3 justifies coupling by dropping the density-gradient term; this approximation must hold in the coupled zone.
  • domain assumption Nonequilibrium extrapolation scheme: the nonequilibrium part of the density distribution at the boundary equals that at the adjacent inner node.
    Eq. (60) closes the LBM artificial boundary in Section 3.2.2; the paper itself notes accuracy degrades where velocity gradients are large.
  • ad hoc to paper Pressure relation Eq. (91): Lp - p = Lp - Sp, used to reconstruct LBM boundary density from FVM pressure.
    Section 4.1.2 states 'it is reasonable to assume' this relation; the author also notes that small pressure differences caused evident error, making this a fragile hand-made closure.
  • domain assumption MCM transition probabilities in Eqs. (98)-(100) remain positive and sum to 1 on the chosen grid and velocity field.
    Random walk Monte Carlo requires valid probabilities; no check of positivity or normalization is reported for the convection cases.

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Cite this review

Pith. "Pith review of Hybrid LBM-FVM and LBM-MCM Methods for Fluid Flow and Heat Transfer Simulation." pith.science (2026). https://pith.science/paper/HRJR3EDX

@misc{pith2026190808386,
  author       = {Pith},
  title        = {Pith review of: Hybrid LBM-FVM and LBM-MCM Methods for Fluid Flow and Heat Transfer Simulation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HRJR3EDX}},
  note         = {Machine review of arXiv:1908.08386}
}
read the original abstract

The fluid flow and heat transfer problems encountered in industry applications span into different scales and there are different numerical methods for different scales problems. It is not possible to use single scale method to solve problems involving multiple scales. Multiscale methods are needed to solve problems involving multiple scales. In this chapter, meso-macro-multiscale methods are developed by combining various single scale numerical methods, including lattice Boltzmann method (LBM), finite volume method (FVM) and Monte Carlo method (MCM). Macroscale methods include FVM, while LBM and MCM belongs to mesoscale methods. Two strategies exist in combing these numerical methods. For the first one, the whole domain is divided into multiple subdomains and different domains use various numerical methods. Message passing among subdomains decides the accuracy of this type of multiscale numerical method. For the second one, various parameters are solved with different numerical methods. These two types of multiscale methods are both discussed in this chapter.

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Reference graph

Works this paper leans on

61 extracted references · 61 canonical work pages

  1. [1]

    Assume the velocity on x = 2l

    work page

  2. [2]

    LATTICE BOLTZMANN METHOD Lattice Boltzmann method is a promising mesoscale method for fluid flow and heat transfer simulation. Instead of solving mass, velocity and energy conservation equations as traditional CFD methods, LBM reaches macroscale parameter using statistical behaviors of particles as shown in Fig. 1, which represent large amounts of fluid m...

    work page

  3. [3]

    The key point of the hybrid method is to pass the information on the interface between LBM and FVM

    TWO SCHEMES FOR HYBRID LATTICE BOLTZMANN and FINITE VOLUME METHODS Two schemes for hybrid LBM-FVM method are proposed in this Section 3. The key point of the hybrid method is to pass the information on the interface between LBM and FVM. It is difficult to transfer velocity obtained from FVM into node population that is needed in LBM. Nonequilibrium extrap...

    work page

  4. [4]

    2 Combined LBM and FVM method for pure fluid flow problems 3.2.1 Problem Statement Lid driven flow is used to test the method to combine LBM and FVM. No-s lip boundary conditions are applied to this 2-D problem, and the flow is driven by a constant lid velocity 0u on the top of the square cavity while the velocities on all other boundaries are zero shown ...

    work page

  5. [5]

    Use SIMPLE to solve the velocity in the SIMPLE zone

    work page

  6. [6]

    Transfer the information on x = 1l from the SIMPLE zone to the LBM zone

    work page

  7. [7]

    Use the LBM to solve the velocity in the LBM zone

    work page

  8. [8]

    Transfer the information on 2l from LBM zone to SIMPLE zone

    work page

Show all 61 references
  1. [9]

    Go back to step 2 until the velocity is converged. 3.3 Two schemes results comparison (a) SIMPLE (b) LBM (c) Hybrid method 1 (d) Hybrid method 2 Figure 11 Streamlines at Re = 100 [36] Hybrid LBM-FVM and LBM-MCM Methods for Fluid Flow and Heat Transfer Simulation 15 15 The lid ...

  2. [10]

    O’Connell, P

    S. O’Connell, P. Thompson, Molecular dynamics-continuum hybrid computations: A tool for studying complex fluid flows, Physical Review E, 52, 1995, 5792–5795

    1995

  3. [11]

    Heat transfer plays an important role in many multiscale problems

    A COUPLED LATTICE BOLTZMANN AND FINITE VOLUME METHOD In Section 3, first strategy to fulfill LBM-FVM multiscale method for fluid flow simulation is discussed. Heat transfer plays an important role in many multiscale problems. A coupled LBM-FVM method for fluid flow and heat tr...

  4. [12]

    Temperature fields and streamlines obtained from pure LBM and pure FVM agree with each other well as shown in Fig

    It is obvious that natural convection has dominated the heat transfer process and there is a stream line vertex near the center of the cavity. Temperature fields and streamlines obtained from pure LBM and pure FVM agree with each other well as shown in Fig. 19 and 20. In addit...

  5. [13]

    The whole domain is divided into two subdomains and these subdomain are solved with LBM and FVM respectively

    A HYBRID LATTICE BOLTZMANN AND FINITE VOLUME METHOD In Sections 3 and 4, LBM-FVM multiscale method was proposed to solve fluid flow and heat transfer problems. The whole domain is divided into two subdomains and these subdomain are solved with LBM and FVM respectively. This st...

  6. [14]

    Monte Carlo method has valid advantages in solving some heat transfer problems

    A COMBINED LATTICE BOLTZMANN AND MONTE CARLO METHOD To take advantages of both LBM and FVM, a LBM-FVM hybrid method was proposed and verified in Section 5. Monte Carlo method has valid advantages in solving some heat transfer problems. LBM-MCM combined method is proposed for f...

    1983

  7. [15]

    Latt, Hydrodynamic Limit of Lattice Boltzma nn Equations, PhD Thesis, University of Geneva, Geneva, 2007

    J. Latt, Hydrodynamic Limit of Lattice Boltzma nn Equations, PhD Thesis, University of Geneva, Geneva, 2007

    2007

  8. [16]

    Abraham, Dynamically spanning the length scales from the quantum to the continuum, International Journal of Modern Physics C, 11, 2000, 1135-1148

    F. Abraham, Dynamically spanning the length scales from the quantum to the continuum, International Journal of Modern Physics C, 11, 2000, 1135-1148

    2000

  9. [17]

    Grest, K

    G. Grest, K. Kremer, Molecular dynamics simulation for polymers in the presence of a heat bath, Physical Review A, 33, 1986, 3628–3631

    1986

  10. [18]

    Pomeau, 20 Years of Lattice Dynamics: A Person al view, International Journal of Modern Physics C, 18(4), 2007, 437-446

    Y. Pomeau, 20 Years of Lattice Dynamics: A Person al view, International Journal of Modern Physics C, 18(4), 2007, 437-446

    2007

  11. [19]

    Anderson, Computational fluid dynamics, McGraw-Hill Inc

    J. Anderson, Computational fluid dynamics, McGraw-Hill Inc. New York, 1995

    1995

  12. [20]

    Versteeg, W

    H. Versteeg, W. Malalasekra, An introduction to computational fluid dynamics: the finite volume method approach, Prentice Hall, New Jersey, 1996

    1996

  13. [21]

    W. Q. Tao, Numerical Heat transfer, Xi’an Jiaotong University Press, Xi’an, China, 2001

    2001

  14. [22]

    Kandhai, D

    D. Kandhai, D. Vidal, A. Hoekstra, Lattice-Boltzmann and finite element simulations of fluid flow in a SMRX static mixer reactor, International Journal for Numerical Methods in Fluids, 31, 1999, 1019- 1033

    1999

  15. [23]

    X. Nie, S. Chen, A continuum and molecular dynamics hybrid method for micro- and nano-fluid flow, Journal of Fluid Mechanics, 500, 2004, 55-64

    2004

  16. [24]

    Flekkoy, G

    E. Flekkoy, G. Wagner, J. Feder, Hybrid model for combined particle and continuum dynamics, Europhysics Letters, 52, 2000, 271–276

    2000

  17. [25]

    Mezrhab, M

    A. Mezrhab, M. Bouzidi, P. Lallemand, Hybrid lattice-Boltzmann finite-difference simulation of convective flows, Computers & Fluids, 33, 2004, 623–641

    2004

  18. [26]

    Dupuis, E

    A. Dupuis, E. Kotsalis, P. Koumoutsakos, Coupling lattice Boltzmann and molecular dynamics models for dense fluids, Physical Review E, 75, 2007, 046704

    2007

  19. [27]

    Werder, J

    T. Werder, J. Walther, P. Koumout, Hybrid atomistic-continuum method for the simulation of dense fluid flows, Journal of Computational Physics, 205(1), 2005, 373-390

    2005

  20. [28]

    Fedosov, G

    D. Fedosov, G. Karniadakis, Interfacing atom istic–mesoscopic–continuum flow regimes, Journal of Computational Physics, 228(4), 2009, 1157–1171

    2009

  21. [29]

    Christensen, S

    A. Christensen, S. Graham, Multiscale lattice Boltzmann modeling of phonon transport in crystalline semiconductor material, Numerical Heat Transfer, Part B: Fundamentals, 57, 2010, 89-109

    2010

  22. [30]

    Z. Li, M. Yang, Y. Zhang, Numerical Simulation of Melting Problems Using the Lattice Boltzmann Method with the Interfacial Tracking Method, Numerical Heat Transfer, Part A: Applications, 68(11), 2015, 1175-1197

    2015

  23. [31]

    J. Latt, B. Chopard, Straight velocity boundaries in the lattice Boltzmann method, Physical Review E, 77, 2008, 056703

    2008

  24. [32]

    Mondal, S

    B. Mondal, S. Mishra, Lattice Boltzmann method applied to the solution of the energy equations of the transient conduction and radiation problems on non-un iform lattices, International Journal of Heat and Mass Transfer, 51, 2008, 68–82

    2008

  25. [33]

    Joshi, A

    H. Joshi, A. Agarwal, B. Puranik, C. Shu, A. Agrawal, A hybrid FVM-LBM method for single and multi- fluid compressible flow problems, International Journal for Numerical Methods in Fluids, 62, 2010, 403– 427

    2010

  26. [34]

    Mondal, S

    B. Mondal, S. Mishra, The lattice Boltzmann method and the finite volume method applied to conduction– radiation problems with heat flux boundary conditions, International Journal for Numerical Methods in Engineering, 78(2), 2009, 172-195

    2009

  27. [35]

    for Reynolds number equals 400 and 1000. Nonequilibrium extrapolation scheme in hybrid method 1 obtain the unknown density distribution by assuming the nonequilibrium part of the density distribution equals to that on the nearby inner nodes. So the bigger the velocity gr adien...

  28. [36]

    H. Luan, H. Xu, L. Chen, Coupling of finite volume method and thermal lattice Boltzmann method and its application to natural convection, International Journal for Numerical Methods in Fluids, 70, 2012, 200-221

    2012

  29. [37]

    H. Luan, H. Xu, L. Chen, Evaluation of the coupling scheme of FVM and LBM for fluid flows around complex geometries, International Journal of Heat and Mass Transfer, 54, 2011, 1975-1985. Hybrid LBM-FVM and LBM-MCM Methods for Fluid Flow and Heat Transfer Simulation 39 39

    2011

  30. [38]

    L. Chen, Y. He, Q. Kang, W. Tao, Coupled numerical approach combining finite volume and lattice Boltzmann methods for multi-scale multi-physicochemical, Journal of Computational Physics, 255(15), 2013, 83-105

    2013

  31. [39]

    Z. Guo, B. Shi, C. Zheng, A coupled lattice BGK model for the Boussinesq equations, International Journal for Numerical Methods in Fluids, 39, 2002, 325–342

    2002

  32. [40]

    X. Nie, X. Shan, H. Chen, A lattice-Boltzmann/ finite-difference hybrid simulation of transonic flow, 47th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, Florida. 2009, 139

    2009

  33. [41]

    G. Peng, H. Xi, C. Duncan, Finite volume scheme for the lattice Boltzmann method on unstructured meshes, Physical Review E, 59, 1999, 4675-4682

    1999

  34. [42]

    Mishra, H

    S. Mishra, H. Roy, Solving transient conduction and radiation heat transfer problems using the lattice Boltzmann method and the finite volume method, Journal of Computational Physics, 23, 2007, 89-107

    2007

  35. [43]

    Z. Li, M. Yang, Y. Zhang, A hybrid lattice Boltzmann and finite volume method for melting with natural convection, Numerical Heat Transfer, Part B: Fundamentals, 66(4), 2014, 307-325

    2014

  36. [44]

    Howell, The Monte Carlo method in radiative he at transfer, Journal of Heat Transfer, 120(3), 1998, 547-560

    J. Howell, The Monte Carlo method in radiative he at transfer, Journal of Heat Transfer, 120(3), 1998, 547-560

    1998

  37. [45]

    Z. Li, M. Yang, Y. Zhang, A hybrid lattice Boltzmann and Monte Carlo method for natural convection simulation, International Journal for Multiscale Computational Engineering, 13(4), 2015, 297-309

    2015

  38. [46]

    Z. Li, M. Yang, Y. Zhang, Lattice Boltzmann Method Simulation of 3-D Natural Convection with Double MRT Model, International Journal of Heat and Mass Transfer, 94, 2016, 222-238

    2016

  39. [47]

    Z. Li, M. Yang, Y. Zhang, Double MRT Thermal Lattice Boltzmann Method for Simulating Natural Convection of Low Prandtl Number Fluids, International Journal of Numerical Methods for Heat and Fluid Flow, 26(6), 2016

    2016

  40. [48]

    Huber, A

    C. Huber, A. Parmigiani, B. Chopard, M. Manga, O. Bachmann, Lattice Boltzmann model for melting with natural convection, International Journal of Heat and Fluid Flow, 29, 2008, 1469-1480

    2008

  41. [49]

    S. V. Patankar, Numerical Heat Transfer and Fluid Flow, McGraw-Hill Book Co, 1980

    1980

  42. [50]

    U. Ghia, K. Ghia, C. Shin, High-Re solutions for incompressible flow using the Navier–Stokes equations and a multigrid method, Journal of computational physics, 48, 1982, 387-411

    1982

  43. [51]

    Z. Li, M. Yang, Y. Zhang, Hybrid lattice and finite volume methods for fluid flow problem, International Journal for Multiscale Computational Engineering, 12(3), 2014, 177-192

    2014

  44. [52]

    Q. Zou, S. Hou, S. Chen, G. Doolen, An improved incompressible lattice Boltzmann model for time- independent flows, Journal of Statistical Physics; 81(35), 1995, 35-48

    1995

  45. [53]

    Z. Li, M. Yang, Y. Zhang, A coupled lattice Boltzmann and finite volume method for natural convection simulation, International Journal of Heat and Mass Transfer, 70, 2014, 864-874

    2014

  46. [54]

    Z. Li, M. Yang, Z. Wang, Y. Zhang, A corrected method to solve convective-diffusion equation based on SIMPLE, Journal of Engineering Thermophysics, 33(9), 2012, 1563-1566

    2012

  47. [55]

    Mohamad, A

    A. Mohamad, A. Kuzmin, A critical evaluation of force term in lattice Boltzmann method, natural convection problem, International Journal of Heat and Mass Transfer, 53, 2010, 990-99

    2010

  48. [56]

    Hortmann, M

    M. Hortmann, M. Peric, G. Scheuerer, Finite volume multigrid prediction of laminar natural convection: Benchmark solutions, International Journal for Numer Methods in Fluids, 11, 1990, 189-207

    1990

  49. [57]

    Z. Li, M. Yang, Y. Zhang, Hybrid lattice Boltzmann and finite volume method for Natural Convection, Journal of Thermophysics and Heat Transfer, 28(1), 2014, 68-77

    2014

  50. [58]

    Minkowycz, E

    W. Minkowycz, E. Sparrow, J. Murthy, Handbo ok of Numerical Hear Transfer (Second Edition), John Wiley & Sons, INC., New Jersey, 2006

    2006

  51. [59]

    Faghri, Y

    A. Faghri, Y. Zhang, J. Howell, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO., 2010

    2010

  52. [61]

    Davis, Natural convection of air in a square cavity: A benchmark numerical solution, International Journal Numer Methods in Fluids, 3, 1983, 249-164

    G. Davis, Natural convection of air in a square cavity: A benchmark numerical solution, International Journal Numer Methods in Fluids, 3, 1983, 249-164

    1983

  53. [106]

    Thus, the hybrid LBM-FVM is reliable for the natural convection simulation

    The hybrid method also has a good accuracy for th e Nusselt number when comparing with the reference and the two pure methods results. Thus, the hybrid LBM-FVM is reliable for the natural convection simulation. This method is designed to take advantages of both LBM and FVM. It...

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Reviewed August 14, 2026 · model on record in the stance chip above.