commit 423971d9631438622bfdda332c2ab2e5f26edb03
parent e647ffaec71a2edc3321762a5453d3f33aef6669
Author: Ivan Gankevich <i.gankevich@spbu.ru>
Date: Sat, 14 Mar 2020 13:52:09 +0300
Verification.
Diffstat:
| main.tex | | | 76 | ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++---------- |
1 file changed, 66 insertions(+), 10 deletions(-)
diff --git a/main.tex b/main.tex
@@ -5,6 +5,8 @@
\usepackage{graphicx}
\usepackage{url}
+\newcommand{\VectorR}[1]{\left[\begin{array}{r}#1\end{array}\right]}
+\newcommand{\VectorL}[1]{\left[\begin{array}{l}#1\end{array}\right]}
\newcommand{\Length}[1]{\big|#1\big|}
\begin{document}
@@ -163,12 +165,7 @@ surface normal \(\vec{n}\) is given by the law of reflection
When we add velocity of incident and reflected air particles we get a vector
that is parallel to the boundary. As we move away from the boundary its impact
on the velocity decays quadratically with the distance. The known analytic
-solution for the potential flow around a cylinder contains similar term:
-\begin{equation*}
- \phi\left(r,\theta\right) = U r \left( 1 + \frac{R^2}{r^2} \right) \cos\theta.
-\end{equation*}
-Here \(r\) and \(\theta\) are polar coordinates, \(R\) is cylinder radius and \(U\)
-is velocity magnitude.
+solution for the potential flow around a cylinder contains similar term (see~sec.~\ref{sec-cylinder}).
\begin{figure}
\centering
@@ -211,7 +208,8 @@ C = -\frac{ \vec\upsilon\cdot\vec{n} }{ \vec\upsilon_r\cdot\vec{n} } = 1
\end{equation*}
and velocity is written simply as
\begin{equation}
-\vec\nabla\phi = \vec\upsilon + \vec\upsilon_r.
+ \label{eq-solution-on-the-boundary}
+ \vec\nabla\phi = \vec\upsilon + \vec\upsilon_r.
\end{equation}
This solution satisfies continuity equation. It gives velocity only at the
@@ -237,7 +235,8 @@ where \(\Length{\cdot}\) is vector length. Plugging the solution into boundary
condition and assuming that neighbouring panels do not affect each other (this allows
removing the integral) gives the same coefficient \(C=1\), but velocity vector is written
differently as
-\begin{equation*}
+\begin{equation}
+\label{eq-solution-near-the-boundary}
\vec\nabla\phi =
\vec\upsilon +
\iint\limits_{a,b\,\in{}A}
@@ -248,7 +247,7 @@ differently as
da\,db;
\qquad
s = 1+\Length{\vec{r}-\vec{S}}^2.
-\end{equation*}
+\end{equation}
Besides the term for reflected air particle velocity that decays quadratically with
the distance to the panel, there is a term that decays quaternary with the distance and
that can be neglected because of this.
@@ -259,9 +258,66 @@ quadratically with the distance to the panel.
\section{Results}
-\subsection{Verification of potential flow around a cylinder}
+\subsection{Verification of the solution on the example of potential flow around a cylinder}
+\label{sec-cylinder}
+
+Potential flow around a cylinder in two dimensions is described by the
+following well-known formula:
+\begin{equation*}
+ \phi\left(r,\theta\right) = U r \left( 1 + \frac{R^2}{r^2} \right) \cos\theta.
+\end{equation*}
+Here \(r\) and \(\theta\) are polar coordinates, \(R\) is cylinder radius and \(U\)
+is \(x\) component of velocity. Cylinder is placed at the origin. To prove that our
+solution on the boundary~\eqref{eq-solution-on-the-boundary} reduces to this solution
+we reduce it to cartesian form using polar coordinate identities
+\begin{equation*}
+ r = \sqrt{x^2+y^2}; \qquad \theta = \arccos{\frac{x}{\sqrt{x^2+y^2}}}.
+\end{equation*}
+Then in cartesian coordinates the solution is written as
+\begin{equation*}
+ \phi(x,y) = U x \left(1 + \frac{R^2}{x^2+y^2}\right)
+\end{equation*}
+and velocity is written as
+\begin{equation}
+ \label{eq-solution-cylinder}
+ \vec\nabla\phi = \VectorL{U \left(R^2 (y^2-x^2) + (x^2+y^2)^2 \right) \\
+ -2 R^2 U x y} / (x^2+y^2)^2.
+\end{equation}
+On the boundary \(x^2+y^2=R^2\) and velocity is written as
+\begin{equation*}
+ \vec\nabla\phi = \frac{2 U}{R^2}\VectorR{y^2 \\ -x y\phantom{^2}}.
+\end{equation*}
+Now if we write surface normal as \(\vec{n}=(x/R,y/R)\) and let
+\(\vec\upsilon=(U,0)\), our solution \eqref{eq-solution-on-the-boundary}
+quite surprisingly reduces to the same expression.
+
+To reduce solution near the boundary~\eqref{eq-solution-near-the-boundary} to
+the solution for potential flow around a cylinder, we let
+\(s=\Length{\vec{r}}^2/\Length{\vec{S}}^2\) (here \(\vec{r}\) is the radius
+vector in cartesian coordinates). Then the solution is written as
+\begin{equation*}
+ \vec\nabla\phi = \vec\upsilon + \frac{1}{s} \vec\upsilon_r
+\end{equation*}
+and reduces to general form of the solution for potential flow around a
+cylinder given in~\eqref{eq-solution-cylinder}.
\section{Discussion}
+
+Solution on the boundary \eqref{eq-solution-on-the-boundary} provides simple
+explanation of areas with the highest and lowest pressure for potential flow
+around a cylinder. At left-most and right-most points on the cylinder boundary
+velocity is nought because incident and reflected particle velocities have
+opposite directions and cancel each other out. At top-most and bottom-most points
+incident and reflected particle velocities have the same direction and
+total velocity is two times larger than the velocity of the flow.
+
+In order to be compatible with the surface of any object, solution near the
+boundary \eqref{eq-solution-near-the-boundary} uses different term \(s\) than
+the solution for potential flow around a cylinder which makes reflected
+velocity term reach maximum value for the point on the boundary, and for the
+point near the boundary the solution includes reflected velocity vectors for
+each panel.
+
\section{Conclusion}
Future work is to include circular motion in the model.
