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Johnson JT, Madadi M, Ladiges DR, Shi Y, Hughes BD, Sader JE. Variational solution to the lattice Boltzmann method for Couette flow. Phys Rev E 2024; 109:055305. [PMID: 38907478 DOI: 10.1103/physreve.109.055305] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/04/2023] [Accepted: 04/16/2024] [Indexed: 06/24/2024]
Abstract
Literature studies of the lattice Boltzmann method (LBM) demonstrate hydrodynamics beyond the continuum limit. This includes exact analytical solutions to the LBM, for the bulk velocity and shear stress of Couette flow under diffuse reflection at the walls through the solution of equivalent moment equations. We prove that the bulk velocity and shear stress of Couette flow with Maxwell-type boundary conditions at the walls, as specified by two-dimensional isothermal lattice Boltzmann models, are inherently linear in Mach number. Our finding enables a systematic variational approach to be formulated that exhibits superior computational efficiency than the previously reported moment method. Specifically, the number of partial differential equations (PDEs) in the variational method grows linearly with quadrature order while the number of moment method PDEs grows quadratically. The variational method directly yields a system of linear PDEs that provide exact analytical solutions to the LBM bulk velocity field and shear stress for Couette flow with Maxwell-type boundary conditions. It is anticipated that this variational approach will find utility in calculating analytical solutions for novel lattice Boltzmann quadrature schemes and other flows.
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Wang L, Yin X. Apparent permeability of flow through periodic arrays of spheres with first-order slip. POWDER TECHNOL 2017. [DOI: 10.1016/j.powtec.2017.01.072] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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Feuchter C, Schleifenbaum W. High-order lattice Boltzmann models for wall-bounded flows at finite Knudsen numbers. Phys Rev E 2016; 94:013304. [PMID: 27575233 DOI: 10.1103/physreve.94.013304] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/06/2015] [Indexed: 06/06/2023]
Abstract
We analyze a large number of high-order discrete velocity models for solving the Boltzmann-Bhatnagar-Gross-Krook equation for finite Knudsen number flows. Using the Chapman-Enskog formalism, we prove for isothermal flows a relation identifying the resolved flow regimes for low Mach numbers. Although high-order lattice Boltzmann models recover flow regimes beyond the Navier-Stokes level, we observe for several models significant deviations from reference results. We found this to be caused by their inability to recover the Maxwell boundary condition exactly. By using supplementary conditions for the gas-surface interaction it is shown how to systematically generate discrete velocity models of any order with the inherent ability to fulfill the diffuse Maxwell boundary condition accurately. Both high-order quadratures and an exact representation of the boundary condition turn out to be crucial for achieving reliable results. For Poiseuille flow, we can reproduce the mass flow and slip velocity up to the Knudsen number of 1. Moreover, for small Knudsen numbers, the Knudsen layer behavior is recovered.
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Affiliation(s)
- C Feuchter
- Department of Mechanical Engineering and Materials Science, Aalen University, Beethovenstrasse 1, D-73430 Aalen, Germany
| | - W Schleifenbaum
- Department of Mechanical Engineering and Materials Science, Aalen University, Beethovenstrasse 1, D-73430 Aalen, Germany
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Ambruş VE, Sofonea V. Implementation of diffuse-reflection boundary conditions using lattice Boltzmann models based on half-space Gauss-Laguerre quadratures. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:041301. [PMID: 24827186 DOI: 10.1103/physreve.89.041301] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/21/2013] [Indexed: 06/03/2023]
Abstract
The Gauss-Laguerre quadrature method is used on the Cartesian semiaxes in the momentum space to construct a family of lattice Boltzmann models. When all quadrature orders Qx, Qy, Qz equal N+1, the Laguerre lattice Boltzmann model LLB(Qx,Qy,Qz) exactly recovers all moments up to order N of the Maxwell-Boltzmann equilibrium distribution function f(eq), calculated over any Cartesian octant of the three-dimensional momentum space. Results of Couette flow simulations at Kn=0.1, 0.5, 1.0 and in the ballistic regime are reported. Specific microfluidic effects (velocity slip, temperature jump, longitudinal heat flux) are well captured up to Kn=0.5, as demonstrated by comparison to direct simulation Monte Carlo results. Excellent agreement with analytic results is obtained in the ballistic regime.
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Affiliation(s)
- Victor Eugen Ambruş
- Center for Fundamental and Advanced Technical Research, Romanian Academy, Bd. Mihai Viteazul 24, 300223 Timişoara, Romania
| | - Victor Sofonea
- Center for Fundamental and Advanced Technical Research, Romanian Academy, Bd. Mihai Viteazul 24, 300223 Timişoara, Romania
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Ambruş VE, Sofonea V. High-order thermal lattice Boltzmann models derived by means of Gauss quadrature in the spherical coordinate system. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:016708. [PMID: 23005564 DOI: 10.1103/physreve.86.016708] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/14/2012] [Indexed: 06/01/2023]
Abstract
We use the spherical coordinate system in the momentum space and an appropriate discretization procedure to derive a hierarchy of lattice Boltzmann (LB) models with variable temperature. The separation of the integrals in the momentum space into angular and radial parts allows us to compute the moments of the equilibrium distribution function by means of Gauss-Legendre and Gauss-Laguerre quadratures, as well as to find the elements of the discrete momentum set for each LB model in the hierarchy. The capability of the high-order models in this hierarchy to capture specific effects in microfluidics is investigated through a computer simulation of Couette flow by using the Shakhov collision term to get the right value of the Prandtl number.
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Affiliation(s)
- Victor Eugen Ambruş
- Department of Physics, West University of Timişoara, Boulevard Vasile Pârvan 4, Timişoara R-300223, Romania.
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de Izarra L, Rouet JL, Izrar B. High-order lattice Boltzmann models for gas flow for a wide range of Knudsen numbers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:066705. [PMID: 22304218 DOI: 10.1103/physreve.84.066705] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/06/2010] [Revised: 05/21/2011] [Indexed: 05/31/2023]
Abstract
The lattice Boltzmann methods (LBMs) have successfully been applied to microscale flows in the hydrodynamic regime, such as flows of liquids in porous media. However, the LBM in its standard formulation does not produce correct results beyond the hydrodynamic regime, i.e., for slip and transitional ones. Following the work of Shan and He [Phys. Rev. Lett. 80, 65 (1998)], we propose to extend the LBM to these regimes in which nonequilibrium effects are obvious and require us to include a larger number of distribution-function moments.
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Affiliation(s)
- Léonard de Izarra
- Université d'Orléans, CNRS/INSU Institut des Sciences de la Terre d'Orléans, UMR 6113, Campus Géosciences 1A, rue de la Férollerie, F-45071 Orléans cedex 2, France.
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Meng J, Zhang Y. Gauss-Hermite quadratures and accuracy of lattice Boltzmann models for nonequilibrium gas flows. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:036704. [PMID: 21517622 DOI: 10.1103/physreve.83.036704] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/18/2010] [Revised: 11/20/2010] [Indexed: 05/30/2023]
Abstract
Recently, kinetic theory-based lattice Boltzmann (LB) models have been developed to model nonequilibrium gas flows. Depending on the order of quadratures, a hierarchy of LB models can be constructed which we have previously shown to capture rarefaction effects in the standing-shear wave problems. Here, we further examine the capability of high-order LB models in modeling nonequilibrium flows considering gas and surface interactions and their effect on the bulk flow. The Maxwellian gas and surface interaction model, which has been commonly used in other kinetic methods including the direct simulation Monte Carlo method, is used in the LB simulations. In general, the LB models with high-order Gauss-Hermite quadratures can capture flow characteristics in the Knudsen layer and higher order quadratures give more accurate prediction. However, for the Gauss-Hermite quadratures, the present simulation results show that the LB models with the quadratures obtained from the even-order Hermite polynomials perform significantly better than those from the odd-order polynomials. This may be attributed to the zero-velocity component in the odd-order discrete set, which does not participate in wall and gas collisions, and thus underestimates the wall effect.
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Affiliation(s)
- Jianping Meng
- Department of Mechanical Engineering, University of Strathclyde, Glasgow G11XJ, United Kingdom.
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Hund SJ, Antaki JF. An extended convection diffusion model for red blood cell-enhanced transport of thrombocytes and leukocytes. Phys Med Biol 2009; 54:6415-35. [PMID: 19809124 DOI: 10.1088/0031-9155/54/20/024] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
Abstract
Transport phenomena of platelets and white blood cells (WBCs) are fundamental to the processes of vascular disease and thrombosis. Unfortunately, the dilute volume occupied by these cells is not amenable to fluid-continuum modeling, and yet the cell count is large enough that modeling each individual cell is impractical for most applications. The most feasible option is to treat them as dilute species governed by convection and diffusion; however, this is further complicated by the role of the red blood cell (RBC) phase on the transport of these cells. We therefore propose an extended convection-diffusion (ECD) model based on the diffusive balance of a fictitious field potential, Psi, that accounts for the gradients of both the dilute phase and the local hematocrit. The ECD model was applied to the flow of blood in a tube and between parallel plates in which a profile for the RBC concentration field was imposed and the resulting platelet concentration field predicted. Compared to prevailing enhanced-diffusion models that dispersed the platelet concentration field, the ECD model was able to simulate a near-wall platelet excess, as observed experimentally. The extension of the ECD model depends only on the ability to prescribe the hematocrit distribution, and therefore may be applied to a wide variety of geometries to investigate platelet-mediated vascular disease and device-related thrombosis.
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Affiliation(s)
- S J Hund
- Carnegie Mellon University, 700 Technology Dr., CMRI/PTC 4218, Pittsburgh, PA 15219, USA.
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Kim SH, Pitsch H, Boyd ID. Lattice Boltzmann modeling of multicomponent diffusion in narrow channels. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:016702. [PMID: 19257161 DOI: 10.1103/physreve.79.016702] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/01/2008] [Revised: 12/08/2008] [Indexed: 05/27/2023]
Abstract
We investigate lattice Boltzmann (LB) modeling of multicomponent diffusion for finite Knudsen numbers. Analytic solutions for binary diffusion in narrow channels, where both molecular and Knudsen diffusion are of importance, are obtained for the standard and higher-order LB methods and validated against the results from the direct simulation Monte Carlo (DSMC) method. The LB methods are shown to reproduce the diffusion slip phenomena. In the DSMC method, while fluid particles are diffusely reflected on a wall, significant component slip and a kinetic boundary layer are observed. It is shown that a higher-order LB method accurately captures the characteristics observed in the DSMC method.
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Affiliation(s)
- Seung Hyun Kim
- Department of Mechanical Engineering, Stanford University, California 94305-3035, USA.
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