Jiaquan Deng

A Boltzmann based model for open channel flows

A finite volume, Boltzmann Bhatnagar-Gross-Krook (BGK) numerical model for one- and two-dimensional unsteady open channel flows is formulated and applied. The BGK scheme satisfies the entropy condition and thus prevents unphysical shocks. In addition, the van Leer limiter and the collision term ensure that the BGK scheme admits oscillation-free solutions only. The accuracy and efficiency of the BGK scheme are demonstrated through the following examples: (i) strong shock waves, (ii) extreme expansion waves, (iii) a combination of strong shock waves and extreme expansion waves, and (iv) one-and two-dimensional dam break problems. These test cases are performed for a variety of Courant numbers (C r ), with the only condition being C r < 1. All the computational results are free of spurious oscillations and unphysical shocks (i.e., expansion shocks)

A Boltzmann-based mesoscopic model for contaminant transport in flow systems

Abstract The objective of this paper is to demonstrate the formulation of a numerical model for mass transport based on the Bhatnagar–Gross–Krook (BGK) Boltzmann equation. To this end, the classical chemical transport equation is derived as the zeroth moment of the BGK Boltzmann differential equation. The relationship between the mass transport equation and the BGK Boltzmann equation allows an alternative approach to numerical modeling of mass transport, wherein mass fluxes are formulated indirectly from the zeroth moment of a difference model for the BGK Boltzmann equation rather than directly from the transport equation. In particular, a second-order numerical solution for the transport equation based on the discrete BGK Boltzmann equation is developed. The numerical discretization of the first-order BGK Boltzmann differential equation is straightforward and leads to diffusion effects being accounted for algebraically rather than through a second-order Fickian term. The resultant model satisfies the entropy condition, thus preventing the emergence of non-physically realizable solutions including oscillations in the vicinity of the front. Integration of the BGK Boltzmann difference equation into the particle velocity space provides the mass fluxes from the control volume and thus the difference equation for mass concentration. The difference model is a local approximation and thus may be easily included in a parallel model or in accounting for complex geometry. Numerical tests for a range of advection–diffusion transport problems, including one- and two-dimensional pure advection transport and advection–diffusion transport show the accuracy of the proposed model in comparison to analytical solutions and solutions obtained by other schemes.

A Boltzmann-based finite volume algorithm for surface water flows on cells of arbitrary shapes

An explicit two–dimensional conservative finite volume model for shallow water equations is formulated and tested. The algorithm for the mass and momentum fluxes at the control surface of the finite volume is obtained from the solution of the Bhatnagar–Gross–Krook (BGK) Boltzmann equation. Unlike classical methods, BGK schemes do not require an ad–hoc splitting of advection and diffusion. The BGK scheme is second order in both time and space. The formulation of the BGK algorithm is performed for a cell of arbitrary irregular shape, but the test cases are conducted using a structured grid of quadrilateral cells. Two approximate Riemann solvers, the HLLC scheme and the two–stage Hancock–HLLC scheme, where HLL stands for Harten, Lax and van Leer and C stands for contact discontinuity, are also considered. The second–order accuracy of HLL and Hancock–HLLC schemes is obtained by MUSCL approach, where MUSCL is the acronym for Monotone Upstream–centered Schemes for Conservation Laws. The data reconstruction for all three schemes is carried out by theVan Leer limiter. The test cases involve strong shocks and expansion waves. The accuracy of the schemes are measured using an absolute error norm and a waviness error norm. The HLLC scheme is highly oscillatory for Courant number larger than 0.5, while the BGK and the Hancock–HLLC schemes are applicable for Courant numbers as high as 1.0. For a fixed value of the central processing unit (CPU) time, the absolute error of the Hancock–HLLC is slightly smaller than that of the BGK while the waviness error of the BGK is quite close to that of Hancock–HLLC. This is because (i) the Hancock–HLLC is a two–step method while the BGK is a single–step method (i.e., the Hancock–HLLC requires storage of intermediate variables, but the BGK does not), and (ii) the Hancock–HLLC schemes requires larger number of grid points than the BGK scheme for the same level of accuracy. For example, to achieve an absolute error of 0.01, the BGK requires about 600 grid points while the Hancock–HLLC requires about 800 grid points. Both the BGK and Hancock–HLLC schemes have similar convergence properties. Unlike exact or approximate Riemann solvers, BGK fluxes accounts for both waves and diffusion. The ability of the BGK scheme to model diffusion is illustrated using a viscous flow problem. Excellent agreement between the analytical and computed viscous flow solution is found. Although the BGK and Hancock–HLLC schemes perform similarly for hyperbolic problems, BGK schemes have the added advantage of being able to solve hyperbolic–parabolic problems without the need for an ad–hoc operator splitting. This is important given that the artificial splitting of advection and diffusion is known to cause artificial widening in shear layers and introduces artificial transient in regions with sharp gradients. Such problems arise when the splitting operation fails to faithfully represent the correct coupling between the physics of advection and the physics of waves.

A Numerical Scheme For Free-surface Flow Modeling Based On Kinetic Theory

Although most of numerical schemes based on shallow water equations work well for continuous surface flows, they are not suitable for flows containing vertical motions, such as the dam break problem, expansion and shock waves problems because they are based on the governing equations which have no effective mechanism to account for the vertical motion. To overcome this shortcoming, the authors of this paper apply the Boltzmann equation of gas kinetic theory to the surface water flow modeling and develop a set of equations governing surface water flows which incorporate the energy losses at the vertical motion. The test examples show that the Boltzmann equation based scheme has high resolution and high accuracy in modeling the discontinuities flows. In addition, Boltzmann based scheme is highly accurate in modeling the smooth surface flows.