The wake structure behind a porous square cylinder is numerically investigated by using both pore... more The wake structure behind a porous square cylinder is numerically investigated by using both pore-scale and macroscopic approaches. The pore-scale simulations (PSSs) concern about the steady flow through and around square arrays of multiple circular cylinders with a wide range of solid fraction. The macroscopic porous media model (PMM) employed is the generalized equation, where the dimensionless permeability Dam is assigned based on the macroscopic permeability Das estimated from PSS via Darcy's law. The connection between pore-scale and macroscopic flow properties is studied in terms of the flow pattern, the geometric parameters, and the occurrence of the recirculating wake behind the array. It is found that the consistency between PSS and PMM is highly dependent on the ratio of Das and Dam. Discussions in terms of the scale analysis of PMM, the discrepancy between Dam and Das, and the effects of stress-jump parameters are also provided.
We quantitatively characterize the metastability in a multi-phase lattice Boltzmann model. The st... more We quantitatively characterize the metastability in a multi-phase lattice Boltzmann model. The structure factor of density fluctuations is theoretically obtained and numerically verified to a high precision, for all simulated wave-vectors and reduced temperatures. The static structure factor is found to consistently diverge as the temperature approaches the critical-point or the density approaches the spinodal line at a sub-critical temperature. Theoretically predicted critical exponents are observed in both cases. Finally, the phase separation in the unstable branch follows the same pattern, i.e. the generation of interfaces with different topology, as observed in molecular dynamics simulations. All results can be independently reproduced through the "idea.deploy" framework https://github.com/lullimat/idea.deploy
The mechanism of recirculating wake formation was examined based on a series of numerical experim... more The mechanism of recirculating wake formation was examined based on a series of numerical experiments on steady flow through and around periodic square arrays of evenly spaced circular cylinders. The Reynolds number of the array Re ranged from 1 to 50, and the solid fraction of the array ϕ was varied from 9.69 × 10−11 to 0.785. The recirculating wake was found to be completely detached from the array under a certain range of ϕ and Re, and varied in size with ϕ and Re. The trailing edge bleeding from the array affects the vortex formation in a way consistent with the entrainment-detrainment mechanism. The combination effect of vorticity accumulation and decay also gives rise to the dissipative vortex. By examining the shear layer formed by the bleeding flow and outer flow, which varies with Re and ϕ, it is concluded that the recirculation mainly results from flow separation at the junction point between the shear layer and the extended centerline of the array due to an adverse pressu...
Journal of Computational and Nonlinear Dynamics, 2015
This paper presents two approaches to the stability analysis of flexible dynamical systems in the... more This paper presents two approaches to the stability analysis of flexible dynamical systems in the time domain. The first is based on the partial Floquet theory and proceeds in three steps. A preprocessing step evaluates optimized signals based on the proper orthogonal decomposition (POD) method. Next, the system stability characteristics are obtained from partial Floquet theory through singular value decomposition (SVD). Finally, a postprocessing step assesses the accuracy of the identified stability characteristics. The Lyapunov characteristic exponent (LCE) theory provides the theoretical background for the second approach. It is shown that the system stability characteristics are related to the LCE closely, for both constant and periodic coefficient systems. For the latter systems, an exponential approximation is proposed to evaluate the transition matrix. Numerical simulations show that the proposed approaches are robust enough to deal with the stability analysis of flexible dyn...
Body-force modelling in thermal compressible flows with the lattice Boltzmann method
Journal of Fluid Mechanics
Body-force modelling in the lattice Boltzmann method (LBM) has been studied extensively in the in... more Body-force modelling in the lattice Boltzmann method (LBM) has been studied extensively in the incompressible limit but rarely discussed for thermal compressible flows. Here we present a systematic approach of incorporating body force in the LBM which is valid for thermal compressible and non-equilibrium flows. In particular, a LBM forcing scheme accurate for the energy equation with second-order time accuracy is given. New and essential in this scheme is the third-moment contribution of the force term. It is shown via Chapman–Enskog analysis that the absence of this contribution causes an erroneous heat flux quadratic in Mach number and linear in temperature variation. The theoretical findings are verified and the necessity of the third-moment contribution is demonstrated by numerical simulations.
Nonlinear magnetohydrodynamics by Galerkin approximation computation
ABSTRACT A fully spectral numerical code is used to explore the properties of voltage-driven diss... more ABSTRACT A fully spectral numerical code is used to explore the properties of voltage-driven dissipative magnetofluids inside a periodic cylinder with circular cross-section. The trial functions are orthonormal eigenfunctions of the curl (Chandrasekhar-Kendall functions). Transitions are observed from axisymmetric resistive equilibria without flow to helically-deformed laminar states with flow, and between pairs of helical laminar states with different pairs of poloidal and toroidal m and n numbers. States of minimum energy dissipation rate seem to be preferred. At high values of the pinch ratio, fully-developed magnetohydrodynamic turbulence is observed.
It was shown previously [X. Shan, Phys. Rev. E, 81, 036702, (2010)] that the lattice weights of t... more It was shown previously [X. Shan, Phys. Rev. E, 81, 036702, (2010)] that the lattice weights of the lattice Boltzmann method are the solutions to an optimization problem subjecting to a set of linear equality and inequality constraints. Here, a solution strategy is developed and complete solutions for such optimization problem are obtained in one-, two-, and three-dimensions up to 9th order. In addition to recovering the wellknown lattices, a number of high-order lattices are also discovered for the first time. These lattices can be used to construct high-order lattice Boltzmann schemes for nonequilibrium flow simulation.
In nonideal gas lattice Boltzmann ͑LB͒ models, obtaining the correct form of the pressure tensor ... more In nonideal gas lattice Boltzmann ͑LB͒ models, obtaining the correct form of the pressure tensor is essential in determining many of the statistical mechanical properties such as the surface tension and the density profile. Here we outline a general approach for calculating the pressure tensor in LB models with interactions beyond nearest neighbors. The statistical mechanical properties calculated from such a pressure tensor are shown to agree very well with those measured from numerical experiments. Comparisons with alternative theories are also made.
Multicomponent lattice Boltzmann model from continuum kinetic theory
Physical Review E, 2010
We derive from the continuum kinetic theory a multicomponent lattice Boltzmann model with intermo... more We derive from the continuum kinetic theory a multicomponent lattice Boltzmann model with intermolecular interaction. The resulting model is found to be consistent with the model previously derived from a lattice-gas cellular automaton [X. Shan and H. Chen, Phys. Rev. E 47, 1815 (1993)] but applies in a much broader domain. A number of important insights are gained from the kinetic theory perspective. First, it is shown that even in the isothermal case, the energy equipartition principle dictates the form of the equilibrium distribution function. Second, thermal diffusion is shown to exist and the corresponding diffusivities are given in terms of macroscopic parameters. Third, the ordinary diffusion is shown to satisfy the Maxwell-Stefan equation at the ideal-gas limit.
Accuracy of high-order lattice Boltzmann method for non-equilibrium gas flow
Journal of Fluid Mechanics
Abstract The lattice Boltzmann method (LBM), which was originally designed for near-incompressibl... more Abstract The lattice Boltzmann method (LBM), which was originally designed for near-incompressible Navier–Stokes flows, has been extended to rarefied gas flows with high-order quadrature in recent years. Although the ability of the high-order LBM to capture rarefaction effects has been demonstrated by many authors, its accuracy and efficiency are often undermined by numerical dissipation introduced by the off-lattice abscissas in Gauss–Hermite quadrature. Here, using the spontaneous Rayleigh–Brillouin scattering problem as the benchmark, we assess the accuracy and efficiency of the high-order LBM with on-lattice quadrature rules up to 39th order. The numerical error comprises two parts, one due to the rarefaction effect and the other due to temporal-spatial discretization, and we find that the former depends not only on the number of discrete velocities, but also on their distribution in velocity space. With a quadrature of 29th order, the error between the LBM and the discrete velocity method is found to be below 1 % up to $Kn=2.0$. Compared with a finite-volume Bhatnagar–Gross–Krook solver using Gauss–Hermite quadrature, the on-lattice LBM has a numerical dissipation several orders of magnitude lower, and achieves the same accuracy with fewer discrete velocities.
The wake structure behind a porous square cylinder is numerically investigated by using both pore... more The wake structure behind a porous square cylinder is numerically investigated by using both pore-scale and macroscopic approaches. The pore-scale simulations (PSSs) concern about the steady flow through and around square arrays of multiple circular cylinders with a wide range of solid fraction. The macroscopic porous media model (PMM) employed is the generalized equation, where the dimensionless permeability Dam is assigned based on the macroscopic permeability Das estimated from PSS via Darcy's law. The connection between pore-scale and macroscopic flow properties is studied in terms of the flow pattern, the geometric parameters, and the occurrence of the recirculating wake behind the array. It is found that the consistency between PSS and PMM is highly dependent on the ratio of Das and Dam. Discussions in terms of the scale analysis of PMM, the discrepancy between Dam and Das, and the effects of stress-jump parameters are also provided.
We quantitatively characterize the metastability in a multi-phase lattice Boltzmann model. The st... more We quantitatively characterize the metastability in a multi-phase lattice Boltzmann model. The structure factor of density fluctuations is theoretically obtained and numerically verified to a high precision, for all simulated wave-vectors and reduced temperatures. The static structure factor is found to consistently diverge as the temperature approaches the critical-point or the density approaches the spinodal line at a sub-critical temperature. Theoretically predicted critical exponents are observed in both cases. Finally, the phase separation in the unstable branch follows the same pattern, i.e. the generation of interfaces with different topology, as observed in molecular dynamics simulations. All results can be independently reproduced through the "idea.deploy" framework https://github.com/lullimat/idea.deploy
The mechanism of recirculating wake formation was examined based on a series of numerical experim... more The mechanism of recirculating wake formation was examined based on a series of numerical experiments on steady flow through and around periodic square arrays of evenly spaced circular cylinders. The Reynolds number of the array Re ranged from 1 to 50, and the solid fraction of the array ϕ was varied from 9.69 × 10−11 to 0.785. The recirculating wake was found to be completely detached from the array under a certain range of ϕ and Re, and varied in size with ϕ and Re. The trailing edge bleeding from the array affects the vortex formation in a way consistent with the entrainment-detrainment mechanism. The combination effect of vorticity accumulation and decay also gives rise to the dissipative vortex. By examining the shear layer formed by the bleeding flow and outer flow, which varies with Re and ϕ, it is concluded that the recirculation mainly results from flow separation at the junction point between the shear layer and the extended centerline of the array due to an adverse pressu...
Journal of Computational and Nonlinear Dynamics, 2015
This paper presents two approaches to the stability analysis of flexible dynamical systems in the... more This paper presents two approaches to the stability analysis of flexible dynamical systems in the time domain. The first is based on the partial Floquet theory and proceeds in three steps. A preprocessing step evaluates optimized signals based on the proper orthogonal decomposition (POD) method. Next, the system stability characteristics are obtained from partial Floquet theory through singular value decomposition (SVD). Finally, a postprocessing step assesses the accuracy of the identified stability characteristics. The Lyapunov characteristic exponent (LCE) theory provides the theoretical background for the second approach. It is shown that the system stability characteristics are related to the LCE closely, for both constant and periodic coefficient systems. For the latter systems, an exponential approximation is proposed to evaluate the transition matrix. Numerical simulations show that the proposed approaches are robust enough to deal with the stability analysis of flexible dyn...
Body-force modelling in thermal compressible flows with the lattice Boltzmann method
Journal of Fluid Mechanics
Body-force modelling in the lattice Boltzmann method (LBM) has been studied extensively in the in... more Body-force modelling in the lattice Boltzmann method (LBM) has been studied extensively in the incompressible limit but rarely discussed for thermal compressible flows. Here we present a systematic approach of incorporating body force in the LBM which is valid for thermal compressible and non-equilibrium flows. In particular, a LBM forcing scheme accurate for the energy equation with second-order time accuracy is given. New and essential in this scheme is the third-moment contribution of the force term. It is shown via Chapman–Enskog analysis that the absence of this contribution causes an erroneous heat flux quadratic in Mach number and linear in temperature variation. The theoretical findings are verified and the necessity of the third-moment contribution is demonstrated by numerical simulations.
Nonlinear magnetohydrodynamics by Galerkin approximation computation
ABSTRACT A fully spectral numerical code is used to explore the properties of voltage-driven diss... more ABSTRACT A fully spectral numerical code is used to explore the properties of voltage-driven dissipative magnetofluids inside a periodic cylinder with circular cross-section. The trial functions are orthonormal eigenfunctions of the curl (Chandrasekhar-Kendall functions). Transitions are observed from axisymmetric resistive equilibria without flow to helically-deformed laminar states with flow, and between pairs of helical laminar states with different pairs of poloidal and toroidal m and n numbers. States of minimum energy dissipation rate seem to be preferred. At high values of the pinch ratio, fully-developed magnetohydrodynamic turbulence is observed.
It was shown previously [X. Shan, Phys. Rev. E, 81, 036702, (2010)] that the lattice weights of t... more It was shown previously [X. Shan, Phys. Rev. E, 81, 036702, (2010)] that the lattice weights of the lattice Boltzmann method are the solutions to an optimization problem subjecting to a set of linear equality and inequality constraints. Here, a solution strategy is developed and complete solutions for such optimization problem are obtained in one-, two-, and three-dimensions up to 9th order. In addition to recovering the wellknown lattices, a number of high-order lattices are also discovered for the first time. These lattices can be used to construct high-order lattice Boltzmann schemes for nonequilibrium flow simulation.
In nonideal gas lattice Boltzmann ͑LB͒ models, obtaining the correct form of the pressure tensor ... more In nonideal gas lattice Boltzmann ͑LB͒ models, obtaining the correct form of the pressure tensor is essential in determining many of the statistical mechanical properties such as the surface tension and the density profile. Here we outline a general approach for calculating the pressure tensor in LB models with interactions beyond nearest neighbors. The statistical mechanical properties calculated from such a pressure tensor are shown to agree very well with those measured from numerical experiments. Comparisons with alternative theories are also made.
Multicomponent lattice Boltzmann model from continuum kinetic theory
Physical Review E, 2010
We derive from the continuum kinetic theory a multicomponent lattice Boltzmann model with intermo... more We derive from the continuum kinetic theory a multicomponent lattice Boltzmann model with intermolecular interaction. The resulting model is found to be consistent with the model previously derived from a lattice-gas cellular automaton [X. Shan and H. Chen, Phys. Rev. E 47, 1815 (1993)] but applies in a much broader domain. A number of important insights are gained from the kinetic theory perspective. First, it is shown that even in the isothermal case, the energy equipartition principle dictates the form of the equilibrium distribution function. Second, thermal diffusion is shown to exist and the corresponding diffusivities are given in terms of macroscopic parameters. Third, the ordinary diffusion is shown to satisfy the Maxwell-Stefan equation at the ideal-gas limit.
Accuracy of high-order lattice Boltzmann method for non-equilibrium gas flow
Journal of Fluid Mechanics
Abstract The lattice Boltzmann method (LBM), which was originally designed for near-incompressibl... more Abstract The lattice Boltzmann method (LBM), which was originally designed for near-incompressible Navier–Stokes flows, has been extended to rarefied gas flows with high-order quadrature in recent years. Although the ability of the high-order LBM to capture rarefaction effects has been demonstrated by many authors, its accuracy and efficiency are often undermined by numerical dissipation introduced by the off-lattice abscissas in Gauss–Hermite quadrature. Here, using the spontaneous Rayleigh–Brillouin scattering problem as the benchmark, we assess the accuracy and efficiency of the high-order LBM with on-lattice quadrature rules up to 39th order. The numerical error comprises two parts, one due to the rarefaction effect and the other due to temporal-spatial discretization, and we find that the former depends not only on the number of discrete velocities, but also on their distribution in velocity space. With a quadrature of 29th order, the error between the LBM and the discrete velocity method is found to be below 1 % up to $Kn=2.0$. Compared with a finite-volume Bhatnagar–Gross–Krook solver using Gauss–Hermite quadrature, the on-lattice LBM has a numerical dissipation several orders of magnitude lower, and achieves the same accuracy with fewer discrete velocities.
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Papers by Xiaowen Shan