Dimensionless groups for thermal flows

The units page shows that an isothermal LBM run is governed by a single dimensionless group, the Reynolds number: match it and the lattice solves the physical problem. A buoyant flow adds a small handful of further groups. This page introduces them and, crucially, shows that the temperature ratio is not one of the numbers you set directly: the Boussinesq non-dimensionalisation absorbs it into a buoyancy velocity scale. Understanding that absorption is what makes the stability page work.

The groups that govern a buoyant flow

Note

A dimensionless group is a ratio of physical quantities whose units cancel, so it takes the same value in any unit system, including lattice units. Two thermal flows are dynamically similar when their governing groups match. The job at setup time is to decide which groups to match and which to deliberately leave unmatched.

Group	Definition	What it compares
Prandtl	\(\mathrm{Pr} = \nu / \kappa\)	momentum diffusivity vs. thermal diffusivity
Grashof	\(\mathrm{Gr} = G\,\beta\,\Delta T\, L^3 / \nu^2\)	buoyancy vs. viscous forces
Rayleigh	\(\mathrm{Ra} = \mathrm{Gr}\,\mathrm{Pr} = G\,\beta\,\Delta T\, L^3 / (\nu\,\kappa)\)	buoyancy driving vs. diffusive damping
Richardson	\(\mathrm{Ri} = \mathrm{Gr}/\mathrm{Re}^2 = G\,\beta\,\Delta T\, L / U^2\)	buoyancy vs. forced inertia
Reynolds	\(\mathrm{Re} = U L / \nu\)	inertia vs. viscous forces

Here \(G = |G_\alpha|\) is the gravitational acceleration, \(\beta\) the thermal expansion coefficient, \(\Delta T\) the characteristic temperature difference, \(L\) a reference length, and \(U\) a reference velocity. The Prandtl number is a property of the fluid (air is \(\mathrm{Pr} \approx 0.71\)) and is matched by setting the scalar diffusivity to \(\kappa = \nu/\mathrm{Pr}\). Rayleigh (or equivalently Grashof) measures how hard buoyancy drives the flow; Richardson measures buoyancy against an imposed forced velocity and decides whether a flow is buoyancy-dominated (\(\mathrm{Ri} \gg 1\)) or shear-dominated (\(\mathrm{Ri} \ll 1\)).

The densimetric (Froude) buoyancy velocity

In a purely buoyancy-driven flow there is no externally imposed velocity to build a Reynolds number from. Instead the flow sets its own speed: the velocity a parcel reaches by falling (or rising) through the buoyancy field over the reference length. Balancing buoyant acceleration \(G\,\beta\,\Delta T\) against inertia over a length \(L\) gives the densimetric or buoyancy (Froude) velocity scale

(1)\[ U_b = \sqrt{G\, \beta\, \Delta T\, L}. \]

This is the natural velocity of the problem. With it, the buoyancy Reynolds number is \(\mathrm{Re}_b = U_b L / \nu = \sqrt{\mathrm{Gr}}\), and the Richardson number built on \(U_b\) is exactly one, confirming that \(U_b\) is the speed at which buoyancy and inertia balance.

The temperature ratio disappears as an independent parameter

Write the temperature deviation as the dimensionless ratio \(\theta = \Delta T / T_0\) (or, in the Boussinesq force, \(\beta\,\Delta T\)). Non-dimensionalise the velocity by \(U_b\) and the temperature by \(\Delta T\), and \(\theta\) no longer appears on its own anywhere in the governing equations: it is absorbed into \(U_b\) (1) and through it into \(\mathrm{Re}_b\) and \(\mathrm{Ra}\). The non-dimensional solution depends only on geometry, \(\mathrm{Pr}\), and \(\mathrm{Ra}\) (or \(\mathrm{Gr}\), \(\mathrm{Re}\)). A run with a hundred-times-smaller temperature ratio is the same non-dimensional flow as long as \(\mathrm{Ra}\), \(\mathrm{Pr}\) and the geometry are preserved.

This is the single most useful fact for setting up a buoyant LBM run, and the stability page is built on it: because \(\theta\) is absorbed, the lattice is free to run at a small, safe temperature ratio while still representing a physically large one, provided the groups that are independent are matched.

A note for buoyancy-velocity scaling

One subtlety follows directly from (1) and is worth stating before it bites. Suppose the physical problem has a temperature ratio \(\lambda\) times larger than the lattice run. The temperature scale then grows like \(\lambda\), but the velocity scale grows only like \(\sqrt{\lambda}\), because \(U_b \propto \sqrt{\beta\,\Delta T}\). Temperature and velocity do not rescale by the same factor. A reconstruction that simply multiplies every field by one common factor is wrong; the stability page returns to this when it reconstructs dimensional fields.

The dimensionless heat-release rate (compartment fire)

When the thermal driving is a heat source rather than a fixed wall temperature, as in the Steckler room-fire case, the relevant group is the dimensionless heat-release rate

(2)\[ Q^*_H = \frac{\dot{Q}}{\rho_\infty\, c_p\, T_\infty\, \sqrt{G\, H}\; H^2} \]

where \(\dot{Q}\) is the heat-release rate, \(H\) a reference height (the doorway height in the Steckler benchmark), and the \(\infty\) subscript denotes the ambient. \(Q^*_H\) measures the fire’s heat output against the buoyant enthalpy flux the opening can carry. It plays the role that \(\mathrm{Ra}\) plays in fixed-temperature convection: it is the group to match between the experiment and the lattice. Because it folds the heat input, the geometry and the buoyancy velocity into one number, matching \(Q^*_H\) (together with \(\mathrm{Pr}\) and the geometry) reproduces the gross compartment exchange even when the temperature ratio and the Reynolds number are deliberately reduced.

What to match, and what to leave

A practical thermal setup matches the groups that govern the physics of interest and knowingly leaves the rest:

• Match \(\mathrm{Pr}\) and the geometry always; match \(\mathrm{Ra}\) (fixed-temperature convection) or \(Q^*_H\) (heat-source / fire problems).

• Leave unmatched the Reynolds (or Grashof) number when the quantity of interest is Reynolds-independent - the gross doorway exchange and neutral-plane height in a compartment fire are set by the buoyant balance, not by the precise turbulence level, so a reduced \(\mathrm{Re}\) is acceptable and keeps the run affordable.

• Never match the raw temperature ratio \(\theta\) as an end in itself: it is emergent from the matched groups, not an input. Trying to force a large \(\theta\) directly is what drives a run supersonic, as the next page shows.

The stability page turns these choices into the concrete lattice knobs and the constraints that keep the run stable.