(vc_ergun_pressure_drop)=

# Ergun packed-bed pressure drop (Darcy -> Forchheimer)

## Why this case matters

Nassu's volumetric porous region applies a momentum sink with a linear (Darcy)
coefficient `porous_alpha` and a quadratic (Forchheimer / canopy) coefficient
`porous_beta`. The two earlier porous cases each exercise only part of this: case
`02_porous_pipe_flow` validates the linear Darcy term against the analytic
Brinkman Bessel profile (`porous_beta = 0`), and case `01_flow_through_trees`
uses `porous_beta` on a canopy geometry with no clean analytic target. This case
is the **isolated pressure-drop sweep** that activates both terms together and
drives the flow across the linear -> quadratic regime transition, validating the
combined sink against the Ergun packed-bed correlation {footcite:t}`ergun1952`.

It also carries a multiblock variant whose porous region straddles a refinement
interface, guarding the issue #704 fix that restored the porous body force to
the block-border population reconstruction.

## The Ergun balance

A packed bed of spheres of diameter `d_p` and voidage `eps` (void fraction)
develops a streamwise pressure gradient that the Ergun correlation splits into a
viscous (Darcy) and an inertial (Forchheimer) contribution:

```{math}
---
label: vc_ergun_correlation
---
\frac{dP}{dx} = 150\,\frac{(1-\varepsilon)^2}{\varepsilon^3}\,\frac{\mu\,u}{d_p^2} \;+\; 1.75\,\frac{(1-\varepsilon)}{\varepsilon^3}\,\frac{\rho\,u^2}{d_p}
```

with `u` the superficial velocity. In Nassu a volumetric region adds the body
force $F_\alpha = -\alpha\,u_\alpha - \beta\,|u|\,u_\alpha$ to the Guo source
term on every region node. In a fully-developed, gradient-free porous-filled duct
driven by a constant streamwise body force `G`, the viscous and advective terms
vanish and the steady momentum balance is purely local:

```{math}
---
label: vc_ergun_balance
---
G = \alpha\,u + \beta\,u^2
```

This is {eq}`vc_ergun_correlation` with $dP/dx = \rho\,G$ ($\rho = 1$) once
the lattice coefficients are identified ($\mu = \rho\nu$, $\rho = 1$):

```{math}
---
label: vc_ergun_coeffs
---
\alpha = 150\,\frac{(1-\varepsilon)^2}{\varepsilon^3}\,\frac{\nu}{d_p^2}, \qquad \beta = 1.75\,\frac{(1-\varepsilon)}{\varepsilon^3}\,\frac{1}{d_p}
```

The two terms are equal at the **transition velocity** $u^* = \alpha/\beta$,
where the Forchheimer number $Fo = \beta u/\alpha = 1$. Sweeping the drive `G`
moves `u` from the Darcy-dominated regime (`Fo << 1`) through the transition to
the Forchheimer-dominated regime (`Fo >> 1`).

## Physical -> lattice mapping

Everything in the YAML is lattice units; the physical interpretation is recovered
in the notebook from the parsed config.

| Parameter      | Lattice value | Note                                        |
| -------------- | ------------- | ------------------------------------------- |
| `cs^2`         | 1/3           | speed of sound squared                      |
| `tau`          | 0.56          | relaxation time                             |
| `nu`           | 0.02          | `cs^2 (tau - 1/2)`                          |
| `eps`          | 0.4           | voidage (typical dense bed)                 |
| `d_p`          | 50            | particle diameter (lattice nodes)           |
| `porous_alpha` | 6.75e-3       | `150 (1-eps)^2/eps^3 nu / d_p^2`            |
| `porous_beta`  | 0.328125      | `1.75 (1-eps)/eps^3 / d_p`                  |
| `u*`           | 0.02057       | transition velocity `alpha/beta` (`Fo = 1`) |

## `ergunGate` - the pressure-drop sweep

### Setup

A `32^3` triply-periodic box, whole domain porous (D3Q27, RRBGK). A constant
streamwise body force `G` drives the flow; with no gradients the steady state is
the exact Ergun balance {eq}`vc_ergun_balance`. The drive is swept via `!unroll`
over `F.x`, with `porous_alpha` / `porous_beta` baked once and shared across the
sweep (v1 bakes one coefficient pair per kernel).

### Velocity / Reynolds sweep

Each `G` is $\alpha u + \beta u^2$ solved for the target `u`; `sim_id` follows
the table order.

| `sim_id` | `u_target` | `Fo`  | `Ma`   | `G = F.x`   | regime                |
| -------- | ---------- | ----- | ------ | ----------- | --------------------- |
| 0        | 0.004      | 0.194 | 0.0069 | 3.225e-05   | Darcy-dominated       |
| 1        | 0.010      | 0.486 | 0.0173 | 1.00312e-04 | linear, near-onset    |
| 2        | 0.020      | 0.972 | 0.0346 | 2.6625e-04  | transition (`Fo ~ 1`) |
| 3        | 0.035      | 1.701 | 0.0606 | 6.38203e-04 | Forchheimer-leaning   |
| 4        | 0.050      | 2.431 | 0.0866 | 1.15781e-03 | Forchheimer-dominated |

All `Ma < 0.1` and `tau in (0.5, 2)`.

### Validation metrics

For each `sim_id` the steady superficial velocity `u` is the domain-mean
streamwise velocity at the last snapshot. The pair `(u, dP/dx = G)` must land on
the analytic Ergun curve {eq}`vc_ergun_correlation` to within **~3%** across the
whole sweep, with the curvature of `dP/dx(u)` (the quadratic Forchheimer term)
resolved at the high-`Fo` end. A point probe records the exponential approach to
the terminal velocity.

## `ergunRefinedSlab` - multiblock interface guard

### Setup

The same Ergun box driven at the transition point (`u_target = 0.02`, `Fo ~ 1`,
so both terms are active), but with an inner slab in `x` (`x in [8, 24]`) refined
to level 1. The porous region straddles two C2F / F2C refinement interfaces.

### Validation metrics

This is the regression guard for issue #704 (the porous force was omitted from
the block-border population reconstruction, producing a per-block sawtooth at
interfaces). Because the Ergun balance is purely local and level-independent, the
converged streamwise velocity must be **spatially uniform** - the same value on
the coarse and fine blocks, with no sawtooth at `x = 8` or `x = 24`, and matching
the `sim_id 2` value of the gate sweep within ~1%.

```{note}
The notebook comparison and the committed figures require a GPU run; results are
pending. The configs parse and are physically well-posed.
```

```{footbibliography}
```

```{toctree}
---
hidden:
---
03_ergun_pressure_drop.ipynb
```