"The flow of an incompressible fluid past a cylinder is one of the first mathematical models that a student of fluid dynamics encounters. This flow is an excellent vehicle for the study of concepts that will be encountered numerous times in mathematical physics, such as vector fields, coordinate transformations, and most important, the physical interpretation of mathematical results." [1]
Mathematical Solution
PD Image Colors: pressure field. Red is high and blue is low. Velocity vectors.
PD Image Close-up view of one quadrant of the flow. Colors: pressure field. Red is high and blue is low. Velocity vectors.
Pressure field (colors), streamfunction (black), velocity potential (white).
A cylinder (or disk) of radius
is placed in two-dimensional, incompressible, inviscid flow.
The goal is to find the steady velocity vector
and pressure
in a plane, subject to the condition that
far from the cylinder the velocity vector is
![{\displaystyle {\vec {V}}=U{\widehat {i}}+0{\widehat {j}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d6727f65589c236c2942c133ff08c59c1f00364)
and at the boundary of the cylinder
![{\displaystyle {\vec {V}}\cdot {\widehat {n}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93e702c397410f280a2c7868c01c64d5b2a10546)
where
is vector normal to the cylinder surface. The upstream flow is uniform and has no vorticity. The flow is inviscid, incompressible and has constant mass density
. The flow therefore remains without vorticity, or is said to be irrotational, with
everywhere. Being irrotational, there must exist a velocity potential
:
![{\displaystyle {\vec {V}}=\nabla \phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a30e2ae390dfa0954b7f8c1b653f8c6c7a0a1b9e)
Being incompressible,
, so
must satisify Laplace's equation:
![{\displaystyle \nabla ^{2}\phi =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/046b1f213d422442e4710eb5d4927f67d58953e2)
The solution for
is obtained most easily in polar coordinates <matth>r</math> and
, related to conventional Cartesian coordinates by
and
. In polar coordinates, Laplace's equation is:
![{\displaystyle {\frac {\partial ^{2}\phi }{\partial r^{2}}}+{\frac {1}{r}}{\frac {\partial \phi }{\partial r}}+{\frac {1}{r^{2}}}{\frac {\partial ^{2}\phi }{\partial \theta ^{2}}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a242c51cb2fa596502072e5cba51d7f17c8706fe)
The solution that satisfies the boundary conditions is
![{\displaystyle \phi (r,\theta )=U\left(r+{\frac {R^{2}}{r}}\right)\cos \theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea94ca3eeb30fbbd6f47828cb265073e6cc04f77)
The velocity components in polar coordinates are obtained from the components of
in polar coordinates:
![{\displaystyle V_{r}={\frac {\partial \phi }{\partial r}}=U\left(1-{\frac {R^{2}}{r^{2}}}\right)\cos \theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2006be98e794bb45ecac2ada27c19bc67ee7526b)
and
![{\displaystyle V_{\theta }={\frac {1}{r}}{\frac {\partial \phi }{\partial \theta }}=-U\left(1+{\frac {R^{2}}{r^{2}}}\right)\sin \theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/65e37d47a5c163533677128978d94b6b8c085593)
Being invisicid and irrotational, Bernoulli's equation allows the solution for pressure field to be obtained directly form the velocity field:
![{\displaystyle p={\frac {1}{2}}\rho \left(U^{2}-V^{2}\right)+p_{\infty }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b0f9dc7dfd635138e60f487a1c0ed1c73439460)
where the constants
and
appear to that
far from the cylinder, where
.
Using
,
![{\displaystyle p={\frac {1}{2}}\rho U^{2}\left(2{\frac {R^{2}}{r^{2}}}\cos(2\theta )-{\frac {R^{4}}{r^{4}}}\right)+p_{\infty }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b851b19045d007abbd2a2143809c7485b8e75047)
In the figures, the colorized field referred to as "pressure" is a plot of
![{\displaystyle 2{\frac {p-p_{\infty }}{\rho U^{2}}}=2{\frac {R^{2}}{r^{2}}}\cos(2\theta )-{\frac {R^{4}}{r^{4}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7480e98c62ea0fb6c8673e9a5f3caf1f1702e414)
On the surface of the cylinder, or
, pressure varies from a maximum of 1 (red color) at the stagnation points at
and
to a minimum of -3 (purple) in the limb of the cylinder at
and
. Likewise,
varies from V=0 at the stagnation points to
on the sides, in the low pressure.
References