Thermodynamics and the LBM

The LBM chapter builds the solver from kinetic gas theory: it starts from the Boltzmann equation, tracks a distribution function and its velocity moments, and recovers the Navier-Stokes equations as a by-product. That chapter is, in effect, an introduction to the mechanical side of the fluid: mass and momentum. This chapter does the same job for the thermal side. It introduces the thermodynamic concepts a buoyant or heated simulation rests on, shows how they relate to the energy balance and back to Navier-Stokes, and ends on the practical question every thermal run faces: how to choose and balance the dimensionless groups so the lattice stays stable.

Where the kinetic chapter starts from molecules, this one starts from the first law of thermodynamics and the energy equation. The two pictures meet: in kinetic theory the temperature is the second velocity moment of the distribution function (the spread of molecular velocities about the mean), just as density and momentum are the zeroth and first moments. So a thermal LBM is not a bolt-on; it is the next moment of the same mesoscopic object.

A hierarchy of thermal models

Temperature couples back to the fluid through two physically distinct mechanisms: a buoyancy body force, and a thermodynamic coupling of density, temperature and pressure through an equation of state. How strongly the density responds to temperature decides which mechanisms a simulation needs, and that orders the thermal capability into three nested regimes of increasing density variation.

The production thermal route is a passive scalar plus Boussinesq buoyancy

For the atmospheric, urban and indoor-convection flows Nassu targets, temperature is transported as a passive scalar on its own double-distribution-function (DDF) lattice and feeds back to the fluid through a single Boussinesq buoyancy force. The full compressible energy equation is not solved. This is the route documented in the scalar-transport thermal page; the present chapter supplies its thermodynamic foundation and its stability rules.

  1. Constant-density Boussinesq (production). Density is held constant everywhere except in the buoyancy term, where its temperature dependence drives the flow. The fluid solver is unchanged; the temperature rides the scalar transport machinery and exerts a body force. This is valid when temperature differences are moderate (small density variation) and speeds are far below the speed of sound, which covers the great majority of wind-engineering and building-physics problems.

  2. Weakly-compressible thermal. The temperature deviation \(\theta\) enters the fluid equilibrium itself, promoting the lattice equation of state to \(p = \rho c_s^2(1+\theta)\) while the density is still the streamed \(\sum_i f_i\). This extends Boussinesq to moderate density variation within \(\mathrm{Ma} < 0.1\); the thermal equilibrium moments are derived on the compressible LBM page.

  3. Variable-density low-Mach. Density is slaved to temperature through the equation of state \(\rho = P/(rT)\), with a thermodynamic/hydrodynamic pressure split and an energy equation. This is the regime of strongly buoyant, fire-scale flows, the only one that reaches large density ratios.

The thermal modeling hierarchy page lays out the three routes, their validity envelopes and how they relate. This chapter is written for the first, production route, and is explicit wherever the stronger two differ.

How to read this chapter

The pages below form one short narrative, from the energy balance to the practical balancing of dimensionless numbers:

  1. The thermal modeling hierarchy separates the buoyancy force from the thermodynamic equation-of-state coupling and sets out the three nested regimes - constant-density Boussinesq, weakly-compressible thermal, and variable-density low-Mach - with the validity envelope and the density treatment of each, and how to choose between them.

  2. From thermodynamics to the energy equation starts at the first law, builds internal energy and enthalpy, writes the compressible energy equation alongside continuity and momentum, and then performs the low-Mach and Boussinesq reductions that justify treating temperature as an advected scalar with a buoyancy back-coupling.

  3. The double-distribution energy backend covers transporting the conserved energy on its own lattice distribution: the continuity identity that makes the material derivative exactly conservative, the conserved variable and its machine-precision conservation, the advection-diffusion equilibrium, the deviation and variable-density correction terms that are kept rather than dropped, and the stencil-order argument for why energy needs its own distribution.

  4. Dimensionless groups for thermal flows introduces the groups that govern a buoyant flow - Prandtl, Grashof, Rayleigh, Richardson, the densimetric (Froude) buoyancy scaling and the heat-release rate \(Q^*\) - and shows how the Boussinesq non-dimensionalisation absorbs the temperature ratio into a buoyancy velocity scale.

  5. Balancing the dimensionless values and keeping the LBM stable is the practical centrepiece: the force Mach number, why the buoyancy force per step must stay small, the lattice-gravity lever that lets a run match the physical \(Q^*\) without going supersonic, why the temperature ratio is emergent rather than dialed, the temperature-versus-velocity scaling subtlety, and how to reconstruct dimensional fields by similarity afterwards.

For the implementation side - the DDF collision, the buoyancy force in the kernel, the thermal boundary conditions - see the scalar transport chapter. For the unit system and the mechanical Reynolds/Mach constraints this chapter builds on, see Physical and Lattice Units.