Differentially Heated Cavity

Why this case matters

Buoyancy is the engine of every thermal wind-engineering flow: warm air rising off a sun-heated facade, a buoyant exhaust plume bending the approach wind, the stratified layering of an urban canopy at night. In nassu these flows are driven by the Boussinesq coupling, where the transported temperature scalar feeds a body force back onto the fluid (see Thermal Extension (Boussinesq)). Before trusting that coupling in a turbulent urban setting you need a case where it is the only physics under test.

The differentially heated square cavity is that case. It is the canonical natural-convection benchmark: a closed square cavity with one hot and one cold vertical wall, no inflow, no turbulence model and no immersed geometry. The only thing that can move the fluid is the buoyancy force generated by the temperature field, so any error in the scalar-to-force path shows up directly in the circulation. The benchmark numerical solution of de Vahl Davis[1] fixes the expected flow at four Rayleigh numbers, giving tabulated wall heat transfer and velocity extrema to compare against. Passing it confirms that the Boussinesq force, the scalar Dirichlet walls and the adiabatic walls all act correctly together, which is the foundation for thermal CWE applications such as heated facades and buoyant plumes.

Physical description

A square cavity of side \(L\) is filled with air. The two vertical walls are held at fixed temperatures, hot on the west wall and cold on the east wall; the two horizontal walls are adiabatic (zero heat flux). The temperature difference sets up a buoyant circulation: fluid rises along the hot wall, crosses under the top, sinks along the cold wall and returns along the floor, forming a single recirculating cell whose strength grows with the Rayleigh number.

In nassu the temperature is a passive scalar with a Boussinesq buoyancy coupling back onto the fluid. The case runs as a thin periodic slab in \(z\) (8 cells) so the quasi-2D benchmark is reproduced while staying on the 3-D D3Q27 fluid and D3Q7 scalar kernels.

Governing equations

The flow is governed by two dimensionless groups. The Rayleigh number measures the strength of buoyancy relative to the diffusive damping of momentum and heat:

(1)\[\mathrm{Ra} = \frac{g\, \beta\, \Delta T\, L^3}{\nu\, \alpha}\]

where \(g\) is gravity, \(\beta\) the thermal expansion coefficient, \(\Delta T\) the hot-cold temperature difference, \(L\) the cavity side, \(\nu\) the kinematic viscosity and \(\alpha\) the thermal diffusivity. The Prandtl number is the ratio of momentum to thermal diffusivity:

(2)\[\mathrm{Pr} = \frac{\nu}{\alpha}\]

The buoyancy enters the fluid as a body force \(F_\alpha = -\rho_0\, \beta\, (\phi - \phi_{\text{ref}})\, g_\alpha\), so the temperature field and the velocity field are coupled at every step.

The validation quantity is the wall heat transfer, expressed as the surface-averaged Nusselt number on the hot wall:

(3)\[\mathrm{Nu}_{\text{avg}} = -\frac{1}{L} \int_0^L \left. \frac{\partial \phi}{\partial x} \right|_{\text{hot wall}} \frac{L}{\Delta T}\, dy\]

which compares the convective wall flux to the purely conductive flux. At low \(\mathrm{Ra}\) the cell is weak and \(\mathrm{Nu}_{\text{avg}} \to 1\) (conduction dominated); as \(\mathrm{Ra}\) rises the boundary layers thin and \(\mathrm{Nu}_{\text{avg}}\) grows.

Simulation setup

The case is run at \(\mathrm{Pr} = 0.71\) (air) over the Rayleigh range \(\mathrm{Ra} = 10^3\) to \(10^6\). The buoyant velocity scale \(U = \sqrt{g\,\beta\,\Delta T\,L}\) and the cavity side \(L\) are fixed, so \(\beta\) and the buoyancy force are identical across the sweep; only the fluid viscosity (through \(\tau\)) and the scalar diffusivity \(D\) change with \(\mathrm{Ra}\).

Parameter	Value
Prandtl number \(\mathrm{Pr}\)	0.71 (air)
Rayleigh number \(\mathrm{Ra}\)	\(10^3,\ 10^4,\ 10^5\) (and \(10^6\) reference target)
Domain \(L_x \times L_y \times L_z\) (lattice units)	\(128 \times 128 \times 8\) (thin periodic \(z\) slab)
Buoyant velocity scale \(U\) (lattice units)	0.1
Thermal expansion \(\beta\)	\(7.8125 \times 10^{-5}\) (\(= U^2 / (\Delta T\, L)\))
Fluid velocity set / operator	D3Q27 / RRBGK
Fluid relaxation time \(\tau\)	1.5232 / 0.8236 / 0.6023 (\(\mathrm{Ra} = 10^3 / 10^4 / 10^5\))
Scalar velocity set / operator	D3Q7 / RRBGK
Scalar diffusivity \(D\) (lattice units)	0.48038 / 0.15191 / 0.04804 (\(= \nu / \mathrm{Pr}\))
Fluid boundary conditions	All four in-plane walls HWBB no-slip; \(z\) periodic
Scalar boundary conditions	West (hot) ScalarRegularizedDirichlet \(\phi_w = 1\); east (cold) ScalarRegularizedDirichlet \(\phi_w = 0\); north / south ScalarHWBB adiabatic (zero flux)
Initial scalar field	linear hot-to-cold ramp \(1 - x / 127\) (shortens the transient)

The Boussinesq force points along \(-y\) (vertical), so a hot parcel (\(\phi > \phi_{\text{ref}} = 0.5\)) feels an upward force. Each Rayleigh number runs to a fixed step budget; the steady state is reached when the wall Nusselt number and the velocity extrema stop drifting.

Reference and acceptance

The reference is the benchmark numerical solution of de Vahl Davis[1], which tabulates the surface-averaged hot-wall Nusselt number and the peak velocities at each Rayleigh number:

\(\mathrm{Ra}\)	\(\mathrm{Nu}_{\text{avg}}\) (de Vahl Davis 1983)
\(10^3\)	1.118
\(10^4\)	2.243
\(10^5\)	4.519
\(10^6\)	8.800

A passing result reproduces the hot-wall \(\mathrm{Nu}_{\text{avg}}\) to within 2-5% of these values across the Rayleigh sweep, and the maximum horizontal and vertical velocities (with their wall-normal locations) to within 3-5% of the benchmark velocity extrema.

Results

The companion notebook compares the hot-wall \(\mathrm{Nu}_{\text{avg}}\) and the velocity extrema against the benchmark values of de Vahl Davis[1] across the Rayleigh sweep.

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[1] (1,2,3)

G. de Vahl Davis. Natural convection of air in a square cavity: a bench-mark numerical solution. International Journal for Numerical Methods in Fluids, 3(3):249–264, 1983. doi:10.1002/fld.1650030305.