.. _vc_flow_over_stat_sphere:

===========================
Flow Over Stationary Sphere
===========================

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

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 :math:`C_{p}` around the sphere surface is calculated from the local pressure :math:`p` as:

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

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

where :math:`F_{\mathrm{D}}` is the drag force exerted by the fluid on sphere surface, and :math:`A_{\perp}` is solid body area perpendicular to the flow direction, which in the present case is the area of a circle of diameter :math:`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 :math:`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 :math:`F_{\mathrm{D}}` can be found from the summation of spread forces in the flow direction.

.. toctree::
   :maxdepth: -1
   :hidden:

   Flow Over Stationary Sphere <07.1_flow_over_sphere.ipynb>
   Flow Over Stationary Sphere - Wall model <07.2_flow_over_sphere_wm.ipynb>

.. footbibliography::