Turbulent Channel Flow with Passive Scalar

Validates the scalar advection-diffusion module under fully-developed wall-bounded turbulence. The fluid is the same turbulent plane channel as Turbulent Channel Flow at \(\mathrm{Re}_\tau \approx 180\); a passive scalar \(\phi\) is added on top, with Dirichlet walls at \(y = 0\) (\(\phi_w = 0\)) and \(y = L_y\) (\(\phi_w = 1\)). The case is the canonical heated-channel benchmark and the first exercise of the scalar regularised wall BCs in a turbulent regime.

The fluid setup follows the case 04 multilevel configuration (same physics, BC topology, hot-start artefact and near-wall refinement strategy) at a reduced box: \(\delta = 40\) lattice units and a \(6\delta \times 2\delta \times 3\delta\) domain matching Kawamura’s own \(6.4 \times 2 \times 3.2\) aspect ratio. This keeps the total cost in the same budget class as the Taylor-Green 3-D case while leaving the box dimensions (\(L_x^+ = 1080\), \(L_z^+ = 540\)) well above the minimal-flow-unit thresholds, so the buffer-layer streak dynamics and the scalar statistics are unconstrained. The fluid component of the exact case-04 configuration has been validated against Kim et al.[1]; the channel here re-validates at the smaller box as part of this case.

Setup

• Geometry. Plane channel of size \(L_x \times L_y \times L_z = 240 \times 80 \times 120\) in lattice units. Wall normal is \(y\); periodic in \(x, z\).

• Fluid. D3Q27 RR-BGK with \(\tau = 0.503333\) (\(u^* = 0.005\), \(\nu = u^* \delta / \mathrm{Re}_\tau\)), body force \(F_x = u^{*2}/\delta = 6.25 \times 10^{-7}\). Two static refinement slabs at level 1 cover the near-wall regions \(0 \le y < 8\) and \(72 \le y < 80\), giving \(\Delta y^+ \approx 2.25\) at the finest level (covering \(y^+ \le 36\)). Walls at \(y = 0\) and \(y = L_y\) use RegularizedHWBB. Hot-started from the same artefact turbulent_channel.vtm as case 04, so the case begins in a statistically stationary state.

• Scalar. D3Q7 RR-BGK with molecular diffusivity \(D = 1.565 \times 10^{-3}\) in lattice units, giving a molecular Prandtl number \(\mathrm{Pr} = \nu / D = 0.71\) (air; the most-cited Kawamura curve). The case runs without LES (quasi-DNS, matching the case 04 channel family), so the \(\mathrm{Sc}_t\) subgrid coupling is inactive. Walls at \(y = 0\) and \(y = L_y\) use ScalarRegularizedDirichlet at \(\phi_w = 0\) and \(\phi_w = 1\) respectively. Initial scalar field is the laminar steady-state linear ramp \(\phi(y) = y / L_y\), which puts the initial condition close to the eventual time-averaged mean so the statistics window can stay short.

Governing equation

The passive scalar field \(\phi(\mathbf{x}, t)\) satisfies the advection-diffusion equation

(1)\[\frac{\partial \phi}{\partial t} + u_\alpha \frac{\partial \phi}{\partial x_\alpha} = D\, \frac{\partial^2 \phi}{\partial x_\alpha \partial x_\alpha}\]

with no source term. Under LES the molecular diffusivity is augmented by a subgrid contribution \(\nu_\mathrm{SGS} / \mathrm{Sc}_t\), so the effective diffusivity used by the lattice Boltzmann scalar operator is

(2)\[D_\mathrm{eff} = D + \frac{\nu_\mathrm{SGS}}{\mathrm{Sc}_t}.\]

Reynolds-averaging (1) in the streamwise and spanwise directions gives the statistically stationary balance between molecular diffusion and the turbulent scalar flux:

(3)\[0 = D\, \frac{d^2 \langle \phi \rangle}{dy^2} - \frac{d \langle v' \phi' \rangle}{dy}.\]

Integrating (3) once gives the total scalar flux (molecular plus turbulent)

\[q_\mathrm{tot}(y) = -D\, \frac{d \langle \phi \rangle}{dy} + \langle v' \phi' \rangle = \mathrm{const.}\]

which is the canonical Kawamura balance. In wall units \(y^+ = y u^* / \nu\) with friction scalar \(\phi^* = q_w / (\rho c_p u^*)\), the mean profile follows the usual three-layer structure (viscous sublayer linear, log layer with Karman constant \(\kappa_\phi\), and a wake correction at the core).

Reference

The reference data is the Kawamura et al.[2] [3] DNS database for the heated turbulent plane channel, which provides:

• the mean scalar profile \(\langle \phi \rangle^+(y^+)\);

• the scalar fluctuation RMS \(\langle \phi'^2 \rangle^{+}(y^+)\);

• the turbulent scalar flux \(\langle u' \phi' \rangle^{+}(y^+)\) and \(\langle v' \phi' \rangle^{+}(y^+)\);

at multiple \(\mathrm{Re}_\tau\) and \(\mathrm{Pr}\) values. For \(\mathrm{Re}_\tau = 180,\ \mathrm{Pr} = 0.71\) (the configuration of this case) the Kawamura database remains the canonical validation target in the LBM and LES literature.

Simulation parameters

Parameter	Value
Friction Reynolds number \(\mathrm{Re}_\tau\)	\(\approx 180\)
Molecular Prandtl number \(\mathrm{Pr}\)	0.71 (air)
LES / \(\mathrm{Sc}_t\)	none (quasi-DNS; the Sc_t subgrid coupling is inactive)
Domain \(L_x \times L_y \times L_z\) (lattice units)	240 x 80 x 120 (\(6\delta \times 2\delta \times 3\delta\), \(\delta = 40\))
Refinement	Two static slabs at level 1 in \(0 \le y < 8\) and \(72 \le y < 80\)
\(\Delta y^+\) (finest level)	\(\approx 2.25\)
Fluid velocity set	D3Q27
Fluid collision operator	RRBGK
Fluid relaxation time \(\tau\)	0.503333
Streamwise body force \(F_x\) (lattice units)	6.25e-7
Scalar velocity set	D3Q7
Scalar collision operator	RRBGK
Scalar diffusivity \(D\) (lattice units)	1.565e-3 (= \(\nu / \mathrm{Pr}\))
Scalar boundary conditions	South wall (\(y = 0\)): ScalarRegularizedDirichlet \(\phi_w = 0\). North wall (\(y = L_y\)): ScalarRegularizedDirichlet \(\phi_w = 1\). \(x, z\) periodic.
Initial scalar field	\(\phi(y) = y / L_y\) (laminar steady-state ramp)
Fluid initialisation	turbulent_channel.vtm (hot-start from case 04)
Number of steps	150000 (statistics from step 50000 onwards, \(\approx 12.5\) eddy turnovers at \(\mathrm{ETT} = \delta / u^* = 8000\) steps)

Validation metrics

All Reynolds-averaged scalar quantities are reconstructed in the notebook from a single source: the channel_profile.mid_channel plane time series. The plane is normal to the spanwise axis \(z\) and spans the full streamwise / wall-normal \((x, y)\) domain, sampled every 100 steps from step 50000 to 150000. Because the flow is homogeneous in \(x\), each wall-normal node averages over every \((x, t)\) sample (1000 snapshots x 240 streamwise nodes), and the two walls are folded onto the half-channel by parity under the reflection \(y \to L_y - y\) combined with \(\phi \to 1 - \phi\):

1. Mean scalar profile \(\langle \phi \rangle^+(y^+)\), the sample mean of scalar_phi.

2. Scalar RMS profile \(\langle \phi'^2 \rangle^{+}(y^+)\), from \(\langle \phi^2 \rangle - \langle \phi \rangle^2\).

3. Turbulent scalar fluxes \(\langle u' \phi' \rangle^{+}(y^+)\) and \(\langle v' \phi' \rangle^{+}(y^+)\), formed as the covariances \(\overline{u_\alpha \phi} - \overline{u_\alpha}\,\overline{\phi}\) over the same samples. A single plane therefore yields the cross-correlation fluxes that an auto-variance statistics kernel cannot.

Wall units are computed from the friction velocity \(u^*\) (fixed analytically by the driving body force, \(u^* = \sqrt{F_x \delta}\)) and the friction scalar \(\phi^* = q_w / (\rho c_p u^*)\), where the wall flux \(q_w\) is read off the near-wall gradient of \(\langle \phi \rangle\).

A passing result reproduces the linear viscous sublayer \(\langle \phi \rangle^+ = \mathrm{Pr}\, y^+\) close to the wall, matches the slope of the log layer and the position / amplitude of the scalar RMS peak, and stays within \(\sim 5\%\) of the Kawamura \(\langle v' \phi' \rangle^{+}\) profile across the channel core.

Note

The notebook runs end to end today but skips the scalar panels until the scalar BC reconstruction kernels move past the TODO stubs: the configuration parses and dispatches through the BC pipeline, but the per-BC g_i reconstruction is currently a no-op, so the scalar field is non-finite and only the flow field is valid. Once the kernels produce a finite scalar and the case is re-run, the scalar profiles populate automatically.

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[1]

John Kim, Parviz Moin, and Robert Moser. Turbulence statistics in fully developed channel flow at low reynolds number. Journal of Fluid Mechanics, 177:133–166, 1987. doi:10.1017/S0022112087000892.

[2]

Hiroshi Kawamura, Kouichi Ohsaka, Hiroyuki Abe, and Kiyoshi Yamamoto. Dns of turbulent heat transfer in channel flow with low to medium-high Prandtl number fluid. International Journal of Heat and Fluid Flow, 19(5):482–491, 1998. doi:10.1016/S0142-727X(98)10026-7.

[3]

Hiroshi Kawamura, Hiroyuki Abe, and Yuichi Matsuo. Dns of turbulent heat transfer in channel flow with respect to Reynolds and Prandtl number effects. International Journal of Heat and Fluid Flow, 20(3):196–207, 1999. doi:10.1016/S0142-727X(99)00014-4.