(vc_flow_over_stat_sphere)=

# Flow Over Stationary Sphere

## Why this case matters

The flow over a stationary sphere is the canonical bluff-body benchmark and the first real test of the immersed boundary method on a curved, three-dimensional surface. Its geometry is simple but its physics is rich: as the Reynolds number {eq}`vc_reynolds_sphere` increases the wake passes through steady, transitional and turbulent regimes, so a single configuration exercises separation, pressure recovery and drag across a wide range. Because the surface pressure coefficient and the drag coefficient {eq}`vc_drag_coefficient` are extensively documented, the IBM force integration can be checked quantitatively against decades of experiment {footcite:t}`Achenbach1972,Sakamoto1990-112,Taneda1956-11,Rodriguez2011` and simulation {footcite:t}`Constantinescu2003,Constantinescu2004-16`. For computational wind engineering it is the proving ground for the IBM that later carries arbitrary building geometry.

The problem of a flow around a sphere is described as a open flow around a sphere to which a uniform velocity is found at regions sufficiently far from the sphere surface. As illustrated below:

```{figure} /_static/img/solver/validation/cases/flow_over_sphere.svg
---
width: 70%
align: center
---
```

The behavior of this flow will be dependent on the flow Reynolds number, which for in this case is defined as:

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

Due to its simple geometry and great amount of investigation in the literature, the open flow around a stationary sphere can be seen as a fundamental validation case. The experimental studies used as main reference are those from {footcite:t}`Achenbach1972,Sakamoto1990-112,Taneda1956-11,Rodriguez2011`; and the numerical are {footcite:t}`Constantinescu2003,Constantinescu2004-16`.

The pressure coefficient $C_{p}$ around the sphere surface is calculated from the local pressure $p$ as:

```{math}
---
label: vc_pressure_coefficient
---
C_{p}=\frac{p-p_{\infty}}{\frac{1}{2}\rho_{\infty}U_{\infty}^{2}}
```

where $p_{\infty}$, $\rho_{\infty}$, and $U_{\infty}$ are the free stream fluid's pressure, density, and velocity, respectively.

Some examples of pressure coefficient profiles taken from experimental data which can be found in the cited literature for various Reynolds numbers is shown below:

```{figure} /_static/img/solver/validation/cases/cp_sphere.svg
---
width: 100%
align: center
---
```

Another parameter frequently used for validation of flow around a sphere is the drag coefficient, which is Reynolds dependent and is calculated as:

```{math}
---
label: vc_drag_coefficient
---
C_{\mathrm{D}}=\frac{F_{\mathrm{D}}}{\frac{1}{2}\rho_{\infty}A_{\perp}U_{\infty}^{2}}
```

where $F_{\mathrm{D}}$ is the drag force exerted by the fluid on sphere surface, and $A_{\perp}$ is solid body area perpendicular to the flow direction, which in the present case is the area of a circle of diameter $d$. Through experimental and numerical experiments, some correlations were well-established for certain Reynolds number ranges, such as those from {footcite:t}`Concha1979-349,Schiller1933`. The curve of $C_{\mathrm{D}}$ as a function of the Reynolds number is shown below:

```{figure} /_static/img/solver/validation/cases/drag_coefficient.svg
---
width: 70%
align: center
---
```

```{note}
With an IBM simulation the value of $F_{\mathrm{D}}$ can be found from the summation of spread forces in the flow direction.
```

```{toctree}
---
hidden:
---
01.1_flow_over_sphere.ipynb
01.2_flow_over_sphere_wm.ipynb
```

```{footbibliography}
```