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Rodney O Fox - One of the best experts on this subject based on the ideXlab platform.
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higher order quadrature based moment methods for kinetic equations
Journal of Computational Physics, 2009Co-Authors: Rodney O FoxAbstract:Kinetic equations containing terms for spatial transport, body forces, and particle-particle collisions occur in many applications (e.g., rarefied gases, dilute granular gases, fluid-particle flows). The direct numerical solution of the kinetic equation is usually intractable due to the large number of independent variables. A useful alternative is to reFormulate the problem in terms of the moments of the velocity distribution function. Closure of the moment equations is challenging for flows sufficiently far away from the Maxwellian limit. In previous work, a quadrature-based third-order moment closure was derived for approximating solutions to the kinetic equation for arbitrary Knudsen number. A key component of quadrature-based closures is the moment-inversion algorithm used to find the non-negative weights and velocity abscissas. Here, a robust inversion procedure is proposed for three-component velocity moments up to ninth order. By reconstructing the velocity distribution function, the spatial fluxes in the moment equations are treated using a kinetic-based finite-volume solver. Because the quadrature-based moment method employs the moment transport equations directly instead of a Discretized Form of the kinetic equation, the mass, momentum and energy are conserved for arbitrary Knudsen and Mach numbers. The computational algorithm is tested for the Riemann shock problem and, for increasing Knudsen numbers (i.e. larger deviations from the Maxwellian limit), the accuracy of the moment closure is shown to be determined by the discrete representation of the spatial fluxes.
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a quadrature based third order moment method for dilute gas particle flows
Journal of Computational Physics, 2008Co-Authors: Rodney O FoxAbstract:Dilute gas-particle flows can be described by a kinetic equation containing terms for spatial transport, gravity, fluid drag, and particle-particle collisions. However, the direct numerical solution of the kinetic equation is intractable for most applications due to the large number of independent variables. A useful alternative is to reFormulate the problem in terms of the moments of the velocity distribution function. Closure of the moment equations is challenging for flows away from the equilibrium (Maxwellian) limit. In this work, a quadrature-based third-order moment closure is derived that can be applied to gas-particle flows at any Knudsen number. A key component of quadrature-based closures is the moment-inversion algorithm used to find the weights and abscissas. A robust inversion procedure is proposed for moments up to third order, and tested for three example applications (Riemann shock problem, impinging jets, and vertical channel flow). Extension of the moment-inversion algorithm to fifth (or higher) order is possible, but left to future work. The spatial fluxes in the moment equations are treated using a kinetic description and hence a gradient-diffusion model is not used to close the fluxes. Because the quadrature-based moment method employs the moment transport equations directly instead of a Discretized Form of the Boltzmann equation, the mass, momentum and energy are conserved for arbitrary Knudsen number (including the Euler limit). While developed here for dilute gas-particle flows, quadrature-based moment methods can, in principle, be applied to any application that can be modeled by a kinetic equation (e.g., thermal and non-isothermal flows currently treated using lattice Boltzmann methods), and examples are given from the literature.
Li-shi Luo - One of the best experts on this subject based on the ideXlab platform.
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theory of the lattice boltzmann method from the boltzmann equation to the lattice boltzmann equation
Physical Review E, 1997Co-Authors: Li-shi LuoAbstract:In this paper, the lattice Boltzmann equation is directly derived from the Boltzmann equation. It is shown that the lattice Boltzmann equation is a special Discretized Form of the Boltzmann equation. Various approximations for the discretization of the Boltzmann equation in both time and phase space are discussed in detail. A general procedure to derive the lattice Boltzmann model from the continuous Boltzmann equation is demonstrated explicitly. The lattice Boltzmann models derived include the two-dimensional 6-bit, 7-bit, and 9-bit, and three-dimensional 27-bit models.
Antonis Papachristodoulou - One of the best experts on this subject based on the ideXlab platform.
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a framework for input output analysis of wall bounded shear flows
Journal of Fluid Mechanics, 2019Co-Authors: Mohamadreza Ahmadi, Giorgio Valmorbida, Dennice F Gayme, Antonis PapachristodoulouAbstract:We propose a new framework to evaluate input-output amplification properties of nonlinear models of wall-bounded shear flows, subject to both square integrable and persistent disturbances. We focus on flows that are spatially invariant in one direction and whose base flow can be described by a polynomial, e.g. streamwise constant channel, Couette and pipe flows. Our methodology is based on the notion of dissipation inequalities in control theory and provides a single unified approach to examining flow properties such as energy growth, worst case disturbance amplification, and stability to persistent excitation (i.e., input-to-state stability). It also enables direct analysis of the nonlinear partial differential equation (PDE) rather than of a Discretized Form of the equations, thereby removing the possibility of truncation errors. We demonstrate how to numerically compute the input-output properties of the flow as the solution of a (convex) optimization problem. We apply our theoretical and computational tools to plane Couette, channel and pipe flows. Our results demonstrate that the proposed framework leads to results that are consistent with theoretical and experimental amplification scalings obtained in the literature.
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connection profile robustness in a heterogeneous network of piecewise affine fitzhugh nagumo models
Society of Instrument and Control Engineers of Japan, 2011Co-Authors: Kenji Kashima, Antonis Papachristodoulou, Frank AllgowerAbstract:FitzHugh-Nagumo is one of the best known nonlinear neuronal models in mathematical physiology. In particular, spatio-temporal Forms of this model have been used for modeling oscillatory behavior in living organisms such as the behavior of cardiac pacemaker cells. In this paper, we investigate the stability of their spatially Discretized Form, that is, diffusively coupled large-scale networks of piecewisely affine approximated FitzHugh-Nagumo models. Scalable criteria for ultimate boundedness are given taking connection profile (coupling strength, delay length, subsystem heterogeneity) robustness into account. This result is accomplished through a linear multi-agent system perspective for a class of piecewise affine systems.
Oday Ibraheem Abdullah - One of the best experts on this subject based on the ideXlab platform.
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numerical simulation of thermally developing turbulent flow through a cylindrical tube
The International Journal of Advanced Manufacturing Technology, 2019Co-Authors: Ali Belhocine, Oday Ibraheem AbdullahAbstract:A numerical study was conducted using the finite difference technique to examine the mechanism of energy transfer as well as turbulence in the case of fully developed turbulent flow in a circular tube with constant wall temperature and heat flow conditions. The methodology to solve this thermal problem is based on the energy equation, a fluid of constant properties in an axisymmetric and two-dimensional stationary flow. The global equations and the initial and boundary conditions acting on the problem are configured in dimensionless Form in order to predict the characteristics of the turbulent fluid flow inside the tube. Using Thomas’ algorithm, a program in FORTRAN was developed to numerically solve the Discretized Form of the system of equations describing the problem. Finally, using this elaborate program, we were able to simulate the flow characteristics, for changing parameters such as Reynolds, Prandtl, and Peclet numbers along the pipe to obtain the important thermal model. These are discussed in detail in this work. Comparison of the results to published data shows that results are a good match to the published quantities.
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numerical simulation of thermally developing turbulent flow through a cylindrical tube
International Journal on Interactive Design and Manufacturing (ijidem), 2019Co-Authors: Ali Belhocine, Oday Ibraheem AbdullahAbstract:A numerical study was conducted using the finite difference technique to examine the mechanism of energy transfer as well as turbulence in the case of fully developed turbulent flow in a circular tube with constant wall temperature and heat flow conditions. The methodology to solve this thermal problem is based on the energy equation a fluid of constant properties in an axisymmetric and two-dimensional stationary flow. From the mathematical side, a numerical technique for solving the problem of fluid–structure interaction with a fully developed turbulent incompressible Newtonian flow is described. The global equations and the initial and boundary conditions acting on the problem are configured in dimensionless Form in order to predict the characteristics of the turbulent fluid flow inside the tube. Using Thomas’ algorithm, a program in FORTRAN was developed to numerically solve the Discretized Form of the system of equations describing the problem. Finally, using this elaborate program, we were able to simulate the flow characteristics, for changing parameters such as Reynolds, Prandtl and Peclet numbers along the pipe to obtain the important thermal model. These are discussed in detail in this work. Comparison of the results to published data shows that results are a good match to the published quantities.
Mohamadreza Ahmadi - One of the best experts on this subject based on the ideXlab platform.
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a framework for input output analysis of wall bounded shear flows
Journal of Fluid Mechanics, 2019Co-Authors: Mohamadreza Ahmadi, Giorgio Valmorbida, Dennice F Gayme, Antonis PapachristodoulouAbstract:We propose a new framework to evaluate input-output amplification properties of nonlinear models of wall-bounded shear flows, subject to both square integrable and persistent disturbances. We focus on flows that are spatially invariant in one direction and whose base flow can be described by a polynomial, e.g. streamwise constant channel, Couette and pipe flows. Our methodology is based on the notion of dissipation inequalities in control theory and provides a single unified approach to examining flow properties such as energy growth, worst case disturbance amplification, and stability to persistent excitation (i.e., input-to-state stability). It also enables direct analysis of the nonlinear partial differential equation (PDE) rather than of a Discretized Form of the equations, thereby removing the possibility of truncation errors. We demonstrate how to numerically compute the input-output properties of the flow as the solution of a (convex) optimization problem. We apply our theoretical and computational tools to plane Couette, channel and pipe flows. Our results demonstrate that the proposed framework leads to results that are consistent with theoretical and experimental amplification scalings obtained in the literature.