Reviewed the common section of the experiment chapter.

doc/report/report_philip/thesisPage__experiments__common.tex

@@ -28,8 +28,8 @@ For this we have developed the \SIBYL{} System, which is described in chapter \r
 		%
 		Thus the new problem is, how we can compute a volume/extension of the markers.
 		
-		We call the volume occupied by a marker its \emph{theoretical} extension.
-		Its main idea is, that a marker has influence and imposes its own state, on its surrounding.
+		We call the volume occupied by a marker its \emph{theoretical extension}.
+		Its main idea is, that a marker has influence and imposes its own state on its surrounding.
 		We assume that inside the theoretical extension, the properties are fully determined by a single marker.
 		
 		There are several different possibility to define such an extension.
@@ -47,11 +47,11 @@ For this we have developed the \SIBYL{} System, which is described in chapter \r
 			A_{m} = \frac{A_{(i,j)}^{C}}{N_{(i,j)}^{(M)}} .			\label{eq:experiments:methodology:calculation:mass:theoExt}
 		\end{align}
 		%
-		This value will vary in time and depends also on the cell the marker is located in.
+		This value will vary with time and depends also on the cell the marker is located in.
 		However all markers within a cell have the same extension.
 		%
 		\\
-		To get the mass of marker $m$ we simply multiply the theoretical extension with its density.
+		To get the mass of marker $m$ we simply multiply its theoretical extension with its density.
 		
 		
 			\paragraph{Uniform Mass}
@@ -73,14 +73,14 @@ For this we have developed the \SIBYL{} System, which is described in chapter \r
 	
 	
 		\subsubsection{Pressure Force Using the Convex Hull}~\label{sec:experiments:methodology:calculation:ConvexHull}
-		Generally, the force exerted by pressure is computed by a surface integral, such as 
+		Generally, force exerted by pressure is computed by a surface integral, such as 
 		%
 		\begin{align}
 			\Vek{F}_{p} = \int_{\partial\mathcal{B}} - \Fkt{p}{\Vek{x}} \cdot \Fkt{\Vek{n}}{\Vek{x}} \udvol{\Vek{x}} .	\label{eq:experiments:methodology:calculation:ConvexHull:pressForceInt}
 		\end{align}
 		%
 		In the above equation, $\partial\mathcal{B}$ is the body's surface, $\Vek{n}$ is an \emph{outward} pointing normal vector and $\Fkt{p}{\Vek{x}}$ is the pressure.
-		The minus is needed for the correct orientation.
+		The minus is needed for the correct direction of the force.
 		%
 		It is quite obvious that an important requirement for this process is, that some kind of surface or at least an approximation thereof, is available.
 		%
@@ -89,16 +89,16 @@ For this we have developed the \SIBYL{} System, which is described in chapter \r
 		However we would like to say, that this method has some problems.
 		
 		In the following we will only consider the markers, which forms the object, for which we would like to compute a surface.
-		Since the markers does not have an extension, we can treat them as points.
-		A further requirement of the method is, that the must be convex, since we only consider balls, this is always the case.
+		Since the markers do not have an extension, we can treat them as points.
+		A further requirement of the method is, that the body must have a convex shape, which is the case for balls, the only geometry we consider.
 		
 		In the current work, we have studied the \emph{convex hull} as an approximation to the body's surface.
 		%
 		\\
 		An intuitive explanation of the convex hull is a rubber band.
-		The markers are represented by nails, that are only partially hammered into a wood plate, such that their heads are still belly out.
+		The markers are represented by nails, that are only partially hammered into a wooden plate, such that their heads are still belly out.
 		Now a rubber band is stretched and put around the markers, such that all nails are enclosed.
-		Then the rubber band is released, it will then start contract itself, it will stop as soon as a nail is encountered.
+		Then the rubber band is released, it will then contract itself, this will stop as soon as a nail is encountered.
 		In the end it will enclose the nails tightly, forming the convex hull.
 		
 		Computing the convex hull is a task that is often performed in computational geometry or video games.
@@ -106,7 +106,7 @@ For this we have developed the \SIBYL{} System, which is described in chapter \r
 		%
 		\\
 		Regardless of the algorithm used for computing the hull, the result is always a polygon.
-		Performing a surface integral on it is particular easy, since all elements are straight lines.
+		Performing a surface integral on such an object, is particular easy, since all elements are straight lines.
 		
 		
 			\paragraph{Problem}
@@ -114,7 +114,8 @@ For this we have developed the \SIBYL{} System, which is described in chapter \r
 			Depending on the situation it is either ``not bad'' or ``good try'' to ``plain wrong.''
 			
 			We think that the main problem is, that the convex hull is in it self, not a very good approximation of the surface.
-			Obviously it is the tightest possible surface, however it is very reasonable to assume that the extension of a body is a bit larger.
+			Obviously it is the tightest possible surface.
+			However it is very reasonable to assume that the extension of a body is a bit larger.
 			What further complicate things is, that due to the finite resolution, the boundary of an object is blurt out.
 			%
 			Or to say it differently, the convex hull is most likely \emph{inside} the object itself.
@@ -123,7 +124,7 @@ For this we have developed the \SIBYL{} System, which is described in chapter \r
 			We also have tried to increase the number of markers, but were not able to see much improvement.
 			
 			However we have not investigated the topic in much more detail.
-			But some data, that explains the problem in more details can be found in section \ref{sec:experiments:pBallSin} on page \pageref{sec:experiments:pBallSin}.
+			But some discussion on the topic can be found in section \ref{sec:experiments:pBallSin} on page \pageref{sec:experiments:pBallSin}.
 			
 			% END:		Problem
 			%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -135,8 +136,8 @@ For this we have developed the \SIBYL{} System, which is described in chapter \r
 		\subsubsection{Momentum}~\label{sec:experiments:methodology:Momentum}
 		The momentum of an object is defined as the velocity at which the object travels, multiplied with its mass.
 		%
-		We have marker centric view, so we used the markers to compute the momentum of an body.
-		As we have seen above, we can define \emph{a} mass for each marker, since they all have \emph{a} velocity\footnote
+		We have a marker centric view, so we used the markers to compute the momentum of an body.
+		As we have seen above, we can define \emph{a} mass for each marker, since they have \emph{a} velocity\footnote
 			{
 				See section \ref{sec:experiments:methodology:Velocity} to see what ``velocity'' actually is.
 			}
@@ -152,10 +153,10 @@ For this we have developed the \SIBYL{} System, which is described in chapter \r
 		
 			
 			\paragraph{Uniform Mass}
-			We have pointed out, that the marker mass fluctuates to some extend, this means that the momentum has the same problem, when the mass is based on the theoretical extension.
+			We have pointed out, that the marker mass fluctuates to some extend, this means that the momentum has the same problems, when the mass is based on the theoretical extension.
 			As we have said above, we can assume that the markers have uniform mass, which would also allow to drop mass completely, since it is only a scaling constant.
 			%
-			However this can only be done, as long as we do not compare the momentum of different objects.
+			However this can only be done, as long as we do not compare momentum of different objects.
 						
 			% END:		Uniform mass
 			%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -188,7 +189,7 @@ For this we have developed the \SIBYL{} System, which is described in chapter \r
 	
 	\subsection{Velocity}~\label{sec:experiments:methodology:Velocity}
 	As we have pointed out in chapter \ref{chap:discretization}, markers have two different velocities.
-	Once, they have an intrinsic velocity, which serves as some kind of memory and is used to implement inertia.
+	First, they have an intrinsic velocity, which serves as some kind of memory and is used to implement inertia.
 	Secondly they have a velocity, which is actually used to advect/move them through the domain, we called it the \emph{feel} velocity.
 	%
 	\\
@@ -196,20 +197,20 @@ For this we have developed the \SIBYL{} System, which is described in chapter \r
 	
 	Our motivation\footnote
 	{
-		We would like to make a small remark concerning it.
+		We would like to make a small remark concerning this \emph{choice} itself.
 		In the early phase of the evaluation phase, we have looked at the intrinsic velocity as well, but only for small experiments, that are not shown here.
 		The \emph{decision} itself, to use the feel velocity, was made \emph{before} we started to analyse data at a larger scale.
 		This means, we have not chosen the feel velocity because the results satisfied us the most, instead it was some \emph{random} decision, that was \emph{not} revisited for consistency.
 	}
 	for using the feel velocity instead of the intrinsic velocity was, because at the very end, the ball travels at that speed.
-	Thus if an object \emph{hits} something, it will collide with that velocity.
+	Thus if an object \emph{hits} something, it will collide at that velocity.
 	
 	We have not studied the difference between intrinsic and feel velocity.
 	Some of our earliest work suggests, that the difference is not very large. 
 	
 	
 		\subsubsection{Grid Velocities}~\label{sec:experiments:methodology:Velocity:gridVel}
-		One could also argue that, instead of deriving quantities from markers, they should be derived from the grid.
+		One could also argue that, instead of deriving quantities from markers, they should be derived from the grid, since the equations are solved there.
 		However there are several problems with this approach.
 		
 		As an example, consider the computation of momentum of an \emph{object}.
@@ -222,7 +223,7 @@ For this we have developed the \SIBYL{} System, which is described in chapter \r
 		
 		There is also a second problem, which velocity should be used?
 		The intrinsic velocity, that was interpolated to the grid or the one that was computed by the solver?
-		A third possibility would be to use the interpolate feel velocity to the grid.
+		A third possibility would be to use the interpolate feel velocity.
 
 		% END:		Grid velocity
 		%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -238,7 +239,7 @@ For this we have developed the \SIBYL{} System, which is described in chapter \r
 	The concept is actually quite simple.
 	Assume a body inside a frictionless fluid, further there is no gravity or other forces.
 	At the beginning the body travels at a constant speed through the fluid.
-	Because there is no friction between body and media, the body will continue to travel with its current velocity for all times and never stop.
+	Because there is no friction between body and media, the body will continue to travel with its current velocity for all times and never stops.
 	%
 	\\
 	Now assume, the body is accelerated, \emph{i.e} its momentum is increased.
@@ -252,14 +253,14 @@ For this we have developed the \SIBYL{} System, which is described in chapter \r
 	The force is generated by the inertia of the moving fluid particles and manifests itself as a pressure increase at the bow.
 	%
 	Since we impose a \emph{velocity} on the body, the force that is needed to accelerate it, must be larger, such that the resistance of the fluid is compensated.
-	This force will accelerate the fluid.
+	This additional force will be used to accelerate the fluid.
 	
 	The question now is ``what portion of the fluid has to be accelerated?''
 	It turns out that this question is of great technological importance, with applications ranging from naval architecture to aeronautics.
 	%
 	The portion of fluid can be seen as an added or virtual mass to the body, generally we can define it as 
 	\begin{align}
-		\hat{m} = f \, \rho_{F} \, V_{K} .		\label{eq:experiments:linMom:linMod:addedMass}
+		\hat{m} = f \cdot \rho_{F} \, V_{K} .		\label{eq:experiments:linMom:linMod:addedMass}
 	\end{align}
 	%
 	Where $\rho_{F}$ is the density of the fluid and $V_{K}$ is the volume\footnote
@@ -270,11 +271,11 @@ For this we have developed the \SIBYL{} System, which is described in chapter \r
 	of the body.
 	%
 	$f$ is a factor that \emph{solely} depends on the body's geometry.
-	In the case of a cylinder we can use potential theory to determine the value of $f$.
+	In the case of a cylinder we can use potential theory to determine its value.
 	After tedious, but straight forward calculations we find $f \equiv 1$.
 	%
 	\\
-	Thus the added mass is exactly the mass that is displaced by the body.
+	Thus the added mass is exactly the mass that is displaced by the cylinder.
 
 	
 	% END:		Virtual Mass