Energy-Based Volumetric LBM

• Energy-Based Volumetric LBM is a computational approach that enforces conservation of mass, momentum, and energy to accurately simulate thermohydrodynamic flows.
• It leverages Gauss–Hermite quadrature and finite-volume discretization to address complex geometries, phase changes, and conjugate heat transfer challenges.
• Validated against benchmark tests, the method demonstrates high fidelity in simulating compressible, multiphase, and particulate thermal flows.

The energy-based volumetric lattice Boltzmann method (LBM) encompasses a class of schemes for the direct numerical simulation of thermohydrodynamic flows where energy conservation is enforced at the discrete level and macroscopic density, momentum, and energy equations are rigorously recovered. These methods achieve volumetric (finite-volume or cell-centered) discretization of the Boltzmann equation, generalize the velocity set using Gauss–Hermite quadrature, and incorporate flexible treatments for complex geometries, phase change, conjugate heat transfer, and heterogeneous media. Energy-based formulations are particularly relevant for compressible, high–Mach-number, and multi-component flows, as well as for systems involving solid–liquid or solid–fluid boundaries where heat transfer and jump conditions are prominent. The approach has been formalized for general compressible thermal flows (Sbragaglia et al., 2010), extended to phase-change phenomena in porous or mushy media (Liu et al., 2016), and adapted for conjugate thermal particulate suspensions (Zhang et al., 2024).

1. Theoretical Foundations and Core Equations

The volumetric LBM advances the classical discrete-velocity Boltzmann framework by enforcing exact conservation of density, momentum, and energy through Hermite polynomial expansion and Gauss–Hermite quadrature selection of the velocity set. The general single-relaxation-time (BGK) kinetic equation for discrete populations $f_l$ at velocity $\mathbf c_l$ takes the form:

$$\partial_t f_l(\mathbf x,t) + \mathbf c_l \cdot \nabla f_l = -\frac{1}{\tau}(f_l - f_l^{(eq)}) + \frac{1}{\tau_g}\frac{f_l^{(eq)}}{\rho T^2}\ \mathbf \Pi : (\mathbf c_l - \mathbf u)(\mathbf c_l - \mathbf u)$$

where $\tau$ and $\tau_g$ control viscosity and thermal diffusivity, and $\mathbf \Pi$ encodes the non-equilibrium stress tensor. The equilibrium $f_l^{(eq)}$ is computed via Hermite expansion to chosen order $N$,

$$f_l^{(eq)} = \omega_l \sum_{n=0}^N \frac{1}{n!} \mathbf a_0^{(n)}(\rho,\mathbf u, T) \mathcal H_l^{(n)}$$

with explicit constraints on mass, momentum, and energy moments.

The finite-volume discretization integrates conservation laws over control volumes, yielding time updates:

$$f_l(\mathbf x, t + \Delta t) = f_l(\mathbf x, t) + \Delta t \left[\Omega_l(\mathbf x, t) - \mathbf c_l \cdot \nabla f_l(\mathbf x, t)\right]$$

The numerical fluxes and collision terms are evaluated with higher-order finite-volume stencils, ensuring accuracy near boundaries and at interfaces (Sbragaglia et al., 2010).

2. Velocity Space Discretization and Gauss–Hermite Quadrature

Accurate recovery of energy transport and higher-order non-equilibrium effects demands velocity spaces with isotropy at required moment ranks. The velocity set $\{\mathbf c_l\}$ and quadrature weights $\{\omega_l\}$ are chosen as the nodes and weights of $D$-dimensional Gauss–Hermite quadrature:

$$\int w(\mathbf c)\mathcal H^{(n)}(\mathbf c)\mathcal H^{(m)}(\mathbf c)\,d\mathbf c \approx \sum_{l=0}^{Q-1} \omega_l \mathcal H_l^{(n)}\mathcal H_l^{(m)} = \delta_{nm} n!$$

Typical choices include off-grid $D2Q21$, $D2Q28$ lattices for 2D simulations, guaranteeing isotropy up to rank 8 or above. This enables accurate thermodynamic and compressibility effects in flows where the velocity distribution deviates significantly from local equilibrium (Sbragaglia et al., 2010).

3. Volumetric Coupling at Solid–Fluid Interfaces, Phase Fronts, and Multi-Physics Extensions

Energy-based volumetric LBM applies naturally to multi-phase and conjugate heat transfer systems via blending schemes and source-term reformulation.

• In phase-change systems, as in solid-liquid transitions in porous media, volumetric LBM tracks the local liquid fraction $f_\ell$ (PCM fraction), blending the collision operator between physical states. The phase interface is implicitly captured via the enthalpy field $H_f$, eliminating the need for explicit front tracking (Liu et al., 2016).
• For conjugate heat transfer, the method introduces per-cell solid fraction $f_s(\mathbf x, t) \in [0,1]$, allowing the same update equations in both solid, fluid, and interface regions. Velocities and thermophysical coefficients (e.g., $c_v$, $\lambda$) are locally interpolated,

$$c_v = (1-f_s) c_{v,f} + f_s c_{v,s}, \quad \lambda = (1-f_s)\lambda_f + f_s\lambda_s$$

Volumetric blending ensures strict no-slip and continuity of temperature and flux at the interface (Zhang et al., 2024).

• In solid–liquid phase change with local thermal non-equilibrium (LTNE), enthalpy-based multiple–relaxation-time (MRT) LBM is formulated as a set of coupled distribution functions for flow, PCM, and metal foam temperatures, enforcing non-iterative phase-tracking (Liu et al., 2016).

4. Boundary Conditions and Interface Treatments

Volumetric LBM benefits from consistent construction of physical boundary conditions. For compressible and thermal flows, diffuse-reflection and Dirichlet (zero-slip, zero-temperature-jump) conditions are implemented:

• Diffuse-reflection kernel updates outgoing distributions at wall nodes by proportional redistribution of incoming populations, enforcing exact net zero mass and reproducing slip and temperature-jump as required by kinetic theory.
• Dirichlet conditions are enforced at wall nodes by Newton–Raphson iteration to obtain corrections ensuring prescribed wall velocity and temperature, with the full Jacobian detailed for robustness (Sbragaglia et al., 2010).

In the context of particulate flows and conjugate transfer, solid-fraction blending guarantees exact no-slip and continuity of flux, while special difference schemes and blending constraints (e.g., for $\beta_1=-4$ in MRT source terms) prevent spurious interfacial spikes (Zhang et al., 2024).

5. Grid Refinement, Adaptivity, and Computational Aspects

The finite-volume (volumetric) approach permits arbitrary mesh adaptivity and nonuniform spacing. Grid clustering near boundaries or interfaces, parametrized (e.g., via a $\beta$-parameterized hyperbolic tangent), sharply reduces boundary-layer and interface errors without incurring instability (Sbragaglia et al., 2010):

$$y_j = \frac{L_y}{2} \left(1+\frac{\tanh[\beta(1-2(j-1)/(N_y-1))]}{\tanh\beta}\right)$$

Adaptive spacing enables accurate resolution of boundary layers, thermal jumps, and complex interface dynamics (e.g., in Rayleigh–Bénard convection or solid–liquid front propagation). In particulate and phase–change flows, this framework supports efficient load balancing and scalability, as only streaming is globally coupled—all other updates remain strictly local (Zhang et al., 2024).

6. Benchmark Tests, Accuracy, and Limitations

Energy-based volumetric LBM is validated across a range of canonical and applied problems:

• For compressible thermal flows, decay rates (shear and thermal), shock-tube solutions (Sod test), slip/jump at boundaries, and Rayleigh–Bénard convection structures closely agree with analytical, Euler, and high-order finite-difference solutions, typically within a few percent error (Sbragaglia et al., 2010).
• Solid-liquid phase change simulations (natural convection melting, conductive solidification) match phase front and temperature evolution from reference finite-volume and finite-difference methods to within 1–3% (Liu et al., 2016).
• In conjugate particulate systems, both single-body sedimentation and large-scale suspensions reproduce known regimes and temperature averages, maintaining second-order accuracy and numerical stability under severe packing and property jumps (Zhang et al., 2024).

The table below summarizes computational properties from select studies:

Method Variant & Lattice	Overhead vs. BGK-LBM	Superior Attributes
Energy-based TVLBM (Sbragaglia et al., 2010)	2–3× per timestep	Adaptive grids, robust boundaries
Enthalpy-based MRT (Liu et al., 2016)	6× faster vs. iterative FVM	Explicit phase change, sharp interface
Conjugate particulate (Zhang et al., 2024)	Comparable to standard LB	Single-domain, parallel-friendly

Principal limitations include increased memory requirements for large velocity sets and restrictions on timestep for ultra-low viscosities (CFL-like criteria). Prospective improvements encompass MRT and entropic collision models, real-gas and multi-phase extensions, and dynamically adaptive mesh refinement (Sbragaglia et al., 2010).

7. Connections to Broader Research and Outlook

Energy-based volumetric LBM provides a unified and physically consistent platform for simulating a diverse array of non-isothermal flow phenomena. Its robust handling of interface physics, adaptivity, and local conservation links it to contemporary developments in enthalpy methods, multi-relaxation-time schemes, and single-domain approaches for particle-resolved and porous media simulation (Sbragaglia et al., 2010, Liu et al., 2016, Zhang et al., 2024). The approach is especially pertinent for applications in heat exchanger design, latent-heat storage, conjugate transport in particulate flows, and advanced multi-physics coupled systems.

A plausible implication is that further methodological refinement—particularly incorporation of deformation, phase transitions, and higher–order nonlinear collision operators—will continue to expand the scope and accuracy of energy-based volumetric LBM in computational fluid and thermal science.