Heated Channel Thermal Entrance (Graetz Problem) Validation · Boreas FFD

Last updated: 2026-11-04 11 min read

This thermal-entrance (Graetz) problem has fully developed laminar flow in a plane channel. Heating begins at one location on an isothermal upper wall while the lower wall is insulated. Bulk temperature approaches wall temperature and local Nusselt number falls toward the fully developed limit Nu = 4.861 (Shah and London, 1978). It checks Boreas's isothermal-wall heat transfer, formula-defined inlet velocity, heat balance, and temperature-advection convergence. Results are from Boreas v0.10.0 on GPU; CPU was used only to diagnose the coarsest mesh.

The benchmark is a high-resolution solution of the two-dimensional steady energy equation with the same domain and boundary conditions. Its downstream Nusselt number agrees with the classical Graetz limit, Nu = 4.861. Errors below compare two numerical solutions; they are not discrepancies from an experiment.

Model and boundary conditions

heated-duct-vardan-dunn1997 model and boundary conditions

Fully developed laminar flow enters at 20 °C. The upper wall is held at 40 °C from x = 0.1 m onward; the lower wall and upstream upper wall are insulated. Schematic; not to scale.

Reference solution

For prescribed velocity u(y) = 6U_m(y/a)(1 − y/a), the steady energy equation is:

u(y)\,\frac{\partial T}{\partial x} = \alpha \left(\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2}\right)

The reference solves this equation with the Boreas domain and boundary conditions: inlet-face T_in, zero-gradient outlet, upper-wall T_w for x ≥ X0, and insulated lower wall. Cell-centered finite volumes and central differences on 1920 × 160 and 3840 × 320 meshes are Richardson-extrapolated; their maximum difference is 5.4e-3 K. A classical Graetz solution without axial conduction independently gives downstream Nu_Dh = 4.8607 (published 4.861). Axial conduction raises dimensionless bulk temperature θ by about 1.5e-3, so the two-dimensional solution including it is the proper Boreas reference.

Bulk temperature and interval-average Nusselt number are defined below, with D_h = 2a. The same interval estimator is applied to the reference solution.

\theta = \frac{T_b - T_{in}}{T_w - T_{in}}, \qquad \mathrm{Nu}_{Dh} = \frac{U_m\, a\, D_h}{\alpha\,(x_2 - x_1)} \ln\!\frac{T_w - T_{b,1}}{T_w - T_{b,2}}

Simulation setup

Item	Boreas setting
Channel	Gap a = 0.05 m, length L = 1.2 m, two-dimensional slab
Inlet	Fully developed u = 6U_m(y/a)(1 − y/a) prescribed by a spatial formula; U_m = 0.03 m/s; T_in = 20 °C
Walls	Upper wall at 40 °C for x ≥ 0.1 m (first 0.1 m insulated), lower wall insulated; pressure outlet
Air	ρ = 1.2 kg/m³, c_p = 1005 J/(kg·K), ν = 1.5e-5 m²/s, α = 2.376e-5 m²/s; Re_Dh = 200, Pe_Dh = 126
Mesh	Uniform h = a/N, N = 8, 16, 32, 64 (up to 295,000 cells)
Time	dt = 0.25 s, 400 s total; residence time 40 s, transverse diffusion time a²/α = 105 s
Advection	Semi-Lagrangian and upwind
Comparison	Bulk temperature at planes spaced 0.05 m apart (RMS θ error), interval-average Nu_Dh, outlet heat balance

Bulk temperature and Nusselt number

Boreas bulk temperature along the channel versus the two-dimensional reference

Upwind bulk temperatures on four meshes (markers), the two-dimensional reference (solid line), and Graetz without axial conduction (gray dashed line). θ rises from the heating start at x = 0.1 m.

Boreas interval-average Nusselt number versus reference

Nu_Dh falls from a high entrance value toward 4.861. The first heated interval, x = 0.10–0.15 m, contains the heating-start singularity and has larger coarse-mesh differences.

Mesh convergence

Bulk-temperature error under mesh refinement

RMS θ error on heated-section planes. Upwind follows the first-order dashed guide. Semi-Lagrangian ceases improving beyond N = 16 with dt fixed at 0.25 s.

N	Upwind θ RMS	Upwind Nu error, x ≥ 0.5 m [%]	Upwind heat imbalance [%]	Semi θ RMS	Semi Nu error [%]	Semi heat imbalance [%]
8	1.73e-3	−1.02	0.00	1.81e-3	−1.16	−0.08
16	9.64e-4	−0.30	0.01	9.90e-4	−0.53	−0.11
32	5.22e-4	−0.10	0.01	1.13e-3	−0.38	−0.11
64	2.21e-4	+0.01	−0.04	1.48e-3	−0.37	−0.10

Halving h reduces upwind error by a factor of 1.8–2.4, consistent with first-order streamwise upwind differencing. At N = 64, downstream Nu is 4.863 versus reference 4.861. Conservative discretization closes the heat balance within 0.04%.

Transient history

Bulk-temperature histories on three planes for N = 64

Bulk temperatures at x = 0.25, 0.6, and 1.15 m (solid upwind, dashed semi-Lagrangian). They settle within 50 s and remain steady through 400 s.

Quantitative results

Metric	Value
Downstream Nu_Dh, upwind N = 64 (reference 4.861)	4.863
Downstream Nu error, upwind N = 64 [%]	+0.01
θ RMS error, upwind N = 8 → 64	1.73e-3 → 2.21e-4
Upwind heat-balance imbalance [%]	≤ 0.04
Upwind mesh convergence order	1

Interpretation and limits

Semi-Lagrangian convergence stalls at high CFL. At dt = 0.25 s and N = 64, maximum CFL is about 14 and θ RMS stays near 1e-3 as the mesh is refined. Reducing dt to 0.05 s (CFL 2.9) on the same deck gives θ RMS 1.25e-4, better than upwind. A mesh ladder alone cannot establish semi-Lagrangian accuracy when CFL grows.
v0.10.0 outlet temperature-boundary defect. Specifying a reverse-flow temperature on a pressure outlet also applies temperature Dirichlet to outward-flow faces in v0.10.0, conducting heat out of the outlet. In the initial run, outlet mass-average temperature was 3 K below the preceding plane and heat imbalance reached 25%. All results here omit that key; the defect was fixed in v0.10.1.
Inlet/heated-wall corner. Starting heating at x = 0 lets conductive heat leave backward through the inlet face without entering the heat-balance monitor. Its imbalance grew from 1.5% to 4.3% with refinement. Heating therefore starts at x = 0.1 m.