[Paper Review] Implementation of the force term in half-range lattice Boltzmann models

Victor E. Ambruş, Victor Sofonea+2 more|arXiv (Cornell University)|Aug 10, 2017

Lattice Boltzmann Simulation Studies73 references3 citations

TL;DR

This paper proposes a distribution theory-based method to implement the force term in half-range lattice Boltzmann models for rarefied gas flows with wall-normal forces, overcoming mass non-conservation issues in existing approaches. By treating the momentum-space gradient of a discontinuous distribution function via distribution theory, the method enables accurate simulation of force-driven flows with diffuse-reflective boundaries, validated through gravitational Poiseuille flow with full-range and half-range comparisons showing improved accuracy in rarefied regimes.

ABSTRACT

In the frame of the Boltzmann equation, wall-bounded flows of rarefied gases require the implementation of boundary conditions at the kinetic level. Such boundary conditions induce a discontinuity in the distribution function with respect to the component of the momentum which is normal to the boundary. Expanding the distribution function with respect to half-range polynomials allows this discontinuity to be captured. The implementation of this concept has been reported in the literature only for force-free flows. In the case of general forces which can have non-zero components in the direction perpendicular to the walls, the implementation of the force term requires taking the momentum space gradient of a discontinuous function. Our proposed method deals with this difficulty by employing the theory of distributions. We validate our procedure by considering the simple one-dimensional flow between diffuse-reflective walls of equal or different temperatures driven by the constant gravitational force. For this flow, a comparison between the results obtained with the full-range and the half-range Gauss-Hermite LB models is also presented.

Motivation & Objective

• To address the challenge of implementing non-parallel external forces in half-range lattice Boltzmann models where the force has a component normal to the wall.
• To resolve zeroth-order errors (e.g., mass non-conservation) arising when the force is not strictly parallel to the boundary in existing half-range models.
• To extend the applicability of half-range Gauss-Hermite quadratures to force-driven flows by properly handling the momentum-space gradient of a discontinuous distribution function.
• To validate the method using a one-dimensional gravitational Poiseuille flow between diffuse-reflective walls of different temperatures, comparing results with full-range models.

Proposed method

• The method employs distribution theory to compute the momentum-space gradient of a discontinuous distribution function, which arises due to wall-induced hemispheric discontinuities in the velocity space.
• It derives a modified expansion for the derivative of the weighted half-range Hermite polynomial product, enabling consistent projection of the force term in the lattice Boltzmann framework.
• The approach uses a recursive relation for half-range Hermite polynomials and derives coefficients for the momentum derivative via integration by parts and orthogonality properties.
• The force term is incorporated by expressing the gradient of the distribution function in terms of half-range basis functions, ensuring consistency with the kinetic boundary conditions.
• The method is implemented using a two-stream quadrature scheme, with separate Gauss-Hermite nodes for positive and negative momentum directions.
• The kernel matrices for the half-range model are constructed to handle the non-smooth distribution function, enabling accurate computation of fluxes and moments.

Experimental results

Research questions

• RQ1 How can the momentum-space gradient of a discontinuous distribution function be consistently computed in the presence of a non-parallel external force in half-range lattice Boltzmann models?
• RQ2 What modifications are required in the standard lattice Boltzmann force term formulation to preserve mass conservation when the force has a component normal to the wall?
• RQ3 Can the half-range Gauss-Hermite quadrature framework be extended to force-driven flows while maintaining accuracy in rarefied regimes?
• RQ4 How does the performance of the half-range model compare to the full-range model in simulating gravitational Poiseuille flow with diffuse-reflective walls?
• RQ5 What is the impact of wall temperature differences on the accuracy and stability of the half-range model in force-driven rarefied flows?

Key findings

• The proposed method successfully eliminates zeroth-order errors (e.g., mass non-conservation) in half-range lattice Boltzmann simulations when the external force has a component normal to the wall.
• The method enables accurate simulation of gravitational Poiseuille flow between diffuse-reflective walls of different temperatures, with results matching reference solutions from the full-range model.
• The half-range model shows significant accuracy gains over the full-range model in rarefied regimes (high Knudsen number), particularly in capturing velocity slip and temperature jump effects.
• The distribution theory-based approach allows consistent projection of the force term onto half-range basis functions, even when the distribution function is discontinuous in momentum space.
• The derived kernel matrices for the half-range model ensure proper handling of the momentum-space gradient, enabling stable and accurate simulations.
• Validation against the full-range model confirms that the half-range approach maintains high accuracy while reducing computational cost in rarefied flow regimes.