(vc_heated_cylinder)=

# Heated Cylinder in Cross-Flow

## Why this case matters

Most thermal CWE geometries are curved: a heated chimney, a warm cable, a sun-baked cylindrical tank. On a Cartesian lattice these surfaces are represented by a voxelized band that staircases the curve, and the scalar Dirichlet condition (a fixed surface temperature) must be enforced on that band. The flat-wall heated-channel case ({ref}`vc_turb_channel_passive_scalar`) validates the scalar wall BC on a grid-aligned surface; this case is the curved counterpart, where the wall heat flux must be recovered from a staircased boundary under forced convection.

The heated cylinder in cross-flow is the textbook forced-convection benchmark. A circular cylinder held at a fixed surface temperature sheds heat into the oncoming flow, and the surface-averaged Nusselt number as a function of Reynolds number is captured by the widely used Churchill-Bernstein correlation {footcite:t}`churchill1977correlating`. Comparing nassu against that correlation across two grid resolutions exposes how the staircased curved Dirichlet wall converges, and tying it to the known Strouhal number confirms the coupled thermal-and-shedding dynamics are right. This is the validation gate for the curved voxel-Dirichlet scalar BC that every thermal urban geometry relies on.

## Physical description

A circular cylinder of diameter $d$ sits in a uniform cross-flow at freestream velocity $U_\infty$. The cylinder surface is held at a fixed elevated temperature (scalar $\phi_w = 1$) while the freestream is clean ($\phi = 0$). Heat is carried away by the boundary layer that develops over the front of the cylinder, then mixed into the wake behind it.

The flow regime depends on the Reynolds number. At $\mathrm{Re} = 40$ the wake is steady with a symmetric recirculation bubble; by $\mathrm{Re} = 100$ the wake is unsteady, shedding a laminar Von Karman vortex street that periodically detaches the thermal plume. The cylinder is voxelized on a single uniform grid level (no IBM, no refinement) so that the curved surface band is the only wall treatment under test, carrying both the no-slip velocity condition and the heated scalar Dirichlet condition.

## Governing equations

The Reynolds number on the diameter sets the flow regime:

```{math}
---
label: vc_hc_reynolds
---
\mathrm{Re} = \frac{U_\infty\, d}{\nu}
```

The validation quantity is the surface-averaged Nusselt number on the diameter, $\mathrm{Nu}_D$, compared with the Churchill-Bernstein correlation:

```{math}
---
label: vc_hc_churchill
---
\mathrm{Nu}_D = 0.3 +
\frac{0.62\, \mathrm{Re}^{1/2}\, \mathrm{Pr}^{1/3}}
{\left[1 + (0.4 / \mathrm{Pr})^{2/3}\right]^{1/4}}
\left[1 + \left(\frac{\mathrm{Re}}{282000}\right)^{5/8}\right]^{4/5}
```

valid for $\mathrm{Re}\,\mathrm{Pr} > 0.2$ over a wide range of Reynolds and Prandtl numbers. The thermal field is a passive scalar coupled to the flow only through advection and diffusion (buoyancy off), so the heat transfer is purely forced convection. See {doc}`/theory/scalar_transport/boundary_conditions` for the scalar wall BC and {doc}`/theory/scalar_transport/index` for the advection-diffusion scheme.

## Simulation setup

The primary sweep is laminar and 2-D (a thin periodic span). Two Reynolds numbers ($\mathrm{Re} = 40$, steady; $\mathrm{Re} = 100$, shedding) are each run at two lattice diameters ($d = 40$ and $d = 80$) so the staircase convergence of the wall flux can be reported. The lattice freestream is fixed at $U_\infty = 0.05$ ($\mathrm{Ma} = 0.087$) and the viscosity follows from the target Reynolds number; the scalar diffusivity follows from $\mathrm{Pr} = 0.71$ (air).

| Parameter                                      | Value                                                                                                                   |
| ---------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------- |
| Prandtl number $\mathrm{Pr}$                   | 0.71 (air)                                                                                                              |
| Reynolds number $\mathrm{Re}$                  | 40 (steady), 100 (shedding)                                                                                             |
| Lattice diameter $d$                           | 40 and 80 (two resolutions per Reynolds number)                                                                         |
| Freestream velocity $U_\infty$ (lattice units) | 0.05 ($\mathrm{Ma} = 0.087$)                                                                                            |
| Domain                                         | $20\,d$ streamwise, $12\,d$ cross-stream, thin periodic span; cylinder centred $6\,d$ from the inlet                    |
| Fluid velocity set / operator                  | D3Q27 / RRBGK                                                                                                           |
| Fluid relaxation time $\tau$                   | 0.65 / 0.80 ($\mathrm{Re} = 40$, $d = 40 / 80$); 0.56 / 0.62 ($\mathrm{Re} = 100$, $d = 40 / 80$)                       |
| Scalar velocity set / operator                 | D3Q7 / RRBGK                                                                                                            |
| Scalar diffusivity $D$ (lattice units)         | $\nu / \mathrm{Pr}$ per variant (0.07042 / 0.14085 / 0.02817 / 0.05634)                                                 |
| Cylinder wall treatment                        | voxelized band: `RegularizedHWBB` no-slip plus `ScalarRegularizedDirichlet` $\phi_w = 1$ (heated)                       |
| Fluid boundary conditions                      | west `UniformFlow` inlet, east / top / bottom `RegularizedNeumannOutlet`, span periodic                                 |
| Scalar boundary conditions                     | west `ScalarRegularizedDirichlet` $\phi_w = 0$ (clean inflow), east / top / bottom `ScalarRegularizedNeumann` zero-flux |
| Initial scalar field                           | 0 everywhere; the heated band injects the thermal layer                                                                 |

```{note}
A $\mathrm{Re} = 3900$ 3-D LES variant (turbulent wake, Smagorinsky subgrid model with a turbulent-Schmidt scalar coupling $\mathrm{Sc}_t$) extends this case beyond the laminar primary sweep. It requires a full 3-D span, an active subgrid model and a much longer averaging window.
```

## Reference and acceptance

The reference is the Churchill-Bernstein correlation {footcite:t}`churchill1977correlating` in {eq}`vc_hc_churchill`, which at $\mathrm{Pr} = 0.71$ gives $\mathrm{Nu}_D \approx 3.4$ at $\mathrm{Re} = 40$ and $\approx 5.2$ at $\mathrm{Re} = 100$.

Because the curved Dirichlet wall is staircased onto the lattice, the surface-averaged wall heat flux converges at **first order** in $d / \Delta x$; the two-resolution pair at each Reynolds number is run precisely to expose that trend. A passing result reproduces $\mathrm{Nu}_D$ to within roughly 10-15% of the correlation at both resolutions (the correlation itself carries 10-20% scatter), with the convergence improving from the coarse to the fine diameter and the local $\mathrm{Nu}(\theta)$ qualitatively correct (a peak at the front stagnation point, a minimum near separation). As a sanity check on the unsteady coupling, the $\mathrm{Re} = 100$ shedding gives a Strouhal number $\mathrm{St} \approx 0.16\text{--}0.17$, rising toward $\mathrm{St} \approx 0.21$ at the higher Reynolds numbers of the turbulent regime.

## Results

The companion notebook compares $\mathrm{Nu}_D$ against the Churchill-Bernstein correlation at both lattice resolutions and reports the $\mathrm{Re} = 100$ Strouhal number as a check on the unsteady coupling.

```{toctree}
---
hidden:
---
02_heated_cylinder.ipynb
```

```{footbibliography}
```