# Pressure Coefficient

The **Pressure Coefficient**, $c_p$, is a dimensionless quantity that provides a **generalized representation** of the pressure distribution on a surface, or body, exposed to a fluid flow.
It allows us to assess how the local pressure at a specific point differs from the surrounding free-stream pressure, **accounting for the dynamic pressure** of the fluid flow.

## Definition

The pressure coefficient is a **dimensionless form of the pressure signal**.
It is obtained by the following expression:

$$
c_{p}(t) = \frac{p(t) - p_{\infty}(t)}{q}
$$

By definition, the pressure coefficient is a local property for each triangle of the mesh.

## Use Case

It is used primarily for **analysis and interpretation** of the measured data.

It should always be generated, since it is the first analysis step. 
It is a fundamental property of the pressure normalization, and **it is used to calculate the other coefficients**.
However, it is not the final result to be delivered to clients.

## Artifacts

The Cp template declares:

1. **Body pressure** (`kind: surface`): per-timestep pressure on every
   mesh triangle.
2. **Reference probe** (`kind: points`, optional): atmospheric reference
   pressure signal. If omitted, the reference pressure is taken as 0.

and a pipeline that subtracts the reference (`sub`), divides by the
dynamic pressure (`scale` by `1 / q`), and reduces to per-element
`statistics`. The outputs are a `cp` time series and a `cp` statistics
data source, each writable to an XDMF+H5 pair (ParaView-readable).

## Usage

Run the shipped template from the command line:

```bash
cfdmod run fixtures/tests/pressure/templates/cp.yaml
```

or drive it from Python over any storage backend:

```python
from cfdmod import load_template, run_template, XdmfH5Storage

template = load_template("cp.yaml")
bindings = run_template(template, storage=XdmfH5Storage(root="."))
cp_t = bindings["cp_t"]          # SurfaceDataSource, one row per triangle
```

The [calculate_cp.ipynb](calculate_cp.ipynb) notebook walks through this
template step by step; a fuller end-to-end version (Cp / Cf / Cm / Ce over
a container pack) lives at `examples/container_pack/process_container_pack.ipynb` in
the repository.

## Data format

:::{note}
For more information about the normalized time scale ($t^*$), check the [Time Normalization section](./time_normalization.md)
:::

```{list-table} $c_p(t)$
:widths: 20 20 20 20 20
:header-rows: 1

* - time_idx/point_idx
  - Normalized time ($t^*$)
  - 0
  - 1
  - 2
* - 0
  - 0.0
  - 1.25
  - 1.15
  - 1.32
* - 0
  - 1.0
  - 1.1
  - 1.5
  - 1.13
```

```{list-table} $c_p (stats)$
:widths: 20 20 20 20 20
:header-rows: 1

* - scalar
  - 0
  - 1
  - 2
  - 3
* - min
  - -1.25
  - -0.9
  - -1.1
  - -0.2
* - max
  - 1.15
  - 0.95
  - 1.13
  - 0.19
* - mean
  - 0.83
  - 0.9
  - 0.5
  - 0.13
* - rms
  - 0.26
  - 0.25
  - 0.13
  - 0.19
* - skewness
  - 1.15
  - -0.95
  - 1.13
  - 0.19
```

```{toctree}
:maxdepth: -1
:hidden:

Transform cp <calculate_cp.ipynb>
```