Thermal Extension (Boussinesq)¶

For thermally driven flows (atmospheric boundary layers, urban heat island, indoor convection), nassu couples the scalar to the fluid through the Boussinesq approximation He et al.[1], Guo et al.[2]: temperature is transported as a passive scalar (see Collision and State Storage) and exerts a buoyancy force on the fluid. Density is treated as constant everywhere except in the buoyancy term, where its variation with temperature drives the flow.

This is the constant-density Boussinesq route - the first and production tier of the thermal modeling hierarchy - and it is the right choice whenever the density variation is small, which covers the great majority of wind-engineering and building-physics problems. When the temperature difference is large enough to vary the density appreciably, the flow leaves this envelope and needs one of the stronger, non-Boussinesq regimes: the weakly-compressible thermal route (the temperature deviation \(\theta\) enters the fluid equilibrium, \(p = \rho c_s^2(1+\theta)\), within \(\mathrm{Ma} < 0.1\)) or the variable-density low-Mach route (density slaved to the equation of state \(\rho = P/(rT)\), for fire-scale density ratios). The page below documents the Boussinesq tier only.

Scope: the DDF Boussinesq (Tier 1) thermal route

The thermal path documented here is the double-distribution-function (DDF) route: temperature rides its own D3Q7 scalar lattice and is solved through the same regularised scalar kernel as any other passive scalar, with the buoyancy force below as the only fluid-side coupling. Everything on this page refers to the DDF-Boussinesq scalar temperature; the two stronger regimes are set out on the thermal modeling hierarchy page.

Buoyancy term¶

The temperature deviation from a reference value adds a body force on the fluid:

\[F^{\text{buoy}}_\alpha = -\rho_0\, \beta\, (T - T_{\text{ref}})\, G_\alpha\]

where \(\beta\) is the thermal expansion coefficient, \(T_{\text{ref}}\) the reference temperature, and \(G_\alpha\) the gravitational acceleration vector (a distinct symbol from the scalar populations \(g_i\)).

Sign convention. \(G_\alpha\) points along gravity (downward), e.g. \(G_\alpha = (0, 0, -|G|)\) when \(z\) is up. With this convention the minus sign in the force is required: a warm parcel (\(T > T_{\text{ref}}\)) then feels \(F^{\text{buoy}}_z = +\rho_0\, \beta\, (T - T_{\text{ref}})\,|G| > 0\), i.e. an upward (buoyant) force. The reference \(T_{\text{ref}}\) is subtracted before the force is formed so that the hydrostatic part is absorbed into the pressure and not driven numerically.

This buoyancy contribution is added to the existing Guo body force in the fluid kernel; it therefore enters the velocity update, the \((1 - \omega/2)(F_\alpha u_\beta + F_\beta u_\alpha)\) collision term, and the IBM half-step force update through the same path as any other body force. The fluid collision operator itself is unchanged, and the equilibrium uses the constant reference density \(\rho_0\).

Thermal scalar equation¶

Temperature satisfies the standard ADE

\[\partial_t T + u_\alpha \partial_\alpha T = \kappa \nabla^2 T + S\]

with thermal diffusivity \(\kappa = \nu / Pr\) (\(Pr\) being the Prandtl number). It is solved with the same scalar transport kernel as any other passive scalar: D3Q7 lattice, regularised collision, macroscopic-only state storage. The coupling is two-way and evaluated once per step: \(T\) from the current scalar field builds the buoyancy force, and the resulting fluid velocity advects \(T\) at the next step.

Dimensionless groups and stability¶

The buoyant regime is governed by the Rayleigh number \(Ra\) and the Prandtl number \(Pr\), with the thermal diffusivity \(\kappa = \nu / Pr\); the thermodynamics dimensionless page defines the full set of thermal groups and derives them.

For numerical stability the buoyancy force per step must stay small in lattice units (a low force Mach number), the same constraint that bounds the velocity Mach number. Strong-\(Ra\) cases are therefore resolved by refining the grid rather than by allowing a larger force per step.