diff --git a/doc/report/report_philip/thesisPage__finalConclOutlook.tex b/doc/report/report_philip/thesisPage__finalConclOutlook.tex
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--- a/doc/report/report_philip/thesisPage__finalConclOutlook.tex
+++ b/doc/report/report_philip/thesisPage__finalConclOutlook.tex
@@ -2,95 +2,141 @@
% This chapter we make the final conclusion
\chapter{Final Conclusion and Outlook}~\label{chap:finConcOutlook}
-In this chapter we will discuss the results that we have seen in the previous chapter in a larger context.
-% I weiss es braucht noch etwas mehr hier.
+In this chapter we will draw a final conclusion about the present work.
+In the previous chapter we have already discussed the experiments, but we focused on experiment at a time.
+However in this chapter we will put the results into a larger context.
+%
+\\
+In addition, we will also discuss new aspects for further investigations.
\section{Final Conclusion}~\label{chap:finConcOutlook:finCinc}
- In chapter \ref{chap:experiments} we have discussed the results of several experiments.
- Despite their simple nature, they were testing some of fundamental principles of the universe.
- %
- \\
- Before, we have discussed them individually, but we would like to summarize and put them into a larger context.
-
+ Despite their simple nature, we were able to gain valuable insights from our experiments.
+ We will now summarize them and put them in a larger context.
- We looked at flows, that were generated by moving an object through a fluid.
- There we have learned, that flow patterns observed in different flows, but at the same Reynolds number, looks the same.
- And as we have expected it, flows at different Reynolds numbers looks different.
- In these experiments another important observation was made, flows at low Reynolds numbers looked laminar.
- However at higher numbers we saw, that patterns started to looks chaotic and vortexes were formed.
- This phenomena is known as turbulence and also observed in nature, however in $2D$ there are some differences.
+ In this work we have studied the patterns generated by an object pulled through a media.
+ While this sounds rather simple and trivial, it has important technological applications, which makes it a good test case.
+ %
+ We learned that these patterns look similar for different flows, if they have same Reynolds number.
+ Further we saw, that if the Reynolds numbers are different, the flows are different as well.
+ %
+ \\
+ Form these experiments we also learned, that flows at low Reynolds numbers behave laminar.
+ But if the Reynolds number is increased, patterns start to look chaotic and flow vortexes were formed.
+ This is a phenomena that is known as turbulence\footnote
+ {
+ We have discussed this already in section \ref{sec:experiments:constBall:Similarities:turb} on page \pageref{sec:experiments:constBall:Similarities:turb}, but we would like to repeat ourself here again.
+ %
+ \\
+ Turbulence is an inherent $3$ dimensional problem.
+ It exists in $2$ dimensions but its internal working is \emph{entirely} different, from the three dimensional case.
+ }.
%
\\
- To conform that this effect is caused by inertia and not some other effects we have redone the experiments, but ignoring inertial effects.
- The observed flow was laminar, which indicates that the chaotic behaviour is an inertial effect.
+ Nearly impossible in reality, but very simple inside a computer is to repeat these experiments, while \emph{neglecting} all inertial effects.
+ %
+ These experiments looked rather differently from before, most importantly, for all systems the flows stayed laminar.
+ From our results we concluded, that the method is indeed able to capture inertial effects.
- We also have studied the interaction between an object and the media.
- We did this by measuring the added mass, which describe the portion of fluid that must be accelerated by the body.
- In these experiments we learned, that this coupling works and the correct portion of fluid is accelerated.
+ Before the velocity of the objects was constant, while they moved through the media.
+ However if velocity varies with time, we expect to observe additional effects.
+ %
+ For example, if the body accelerates, the media will oppose it.
+ A certain fraction of the media, which is known as added mass, must be accelerated as well.
+ From our experiments we learned that the added mass is simulated correctly.
%
\\
- However this experiments also revealed, that it is not easy to define a ``surface'' for an object.
+ But there is a different effect that we expect to be present.
+ The fluid opposes the ball's acceleration by generating a pressure on its surface.
%
- Being able to define a surface and perform calculations on it is important.
- For example a surface is needed for calculating fiction or handling interaction with other bodies.
-
-
- Most of the time, we used the Jenny--Meyer integrator.
- However some experiments were conducted several times, each time with a different integrator.
+ We tried to measure this force directly by reconstructing its surface and integrating pressure over it.
+ However we had great problems and the results are not very accurate.
+ But we believe that the main problems lie with the unfit surface reconstruction and grid artefacts, which perturbed the signals.
%
- A final verdict is difficult, but we think that most of them are able to capture the most important aspects.
- But from the results we have presented in this work, we think that the cell centred velocity point integrator\footnote{Also known as ``standard'', which is described in section \ref{sec:discretization:step6:CCSTD} on page \pageref{sec:discretization:step6:CCSTD}.} should not be used.
+ \\
+ However from a visual inspection of the pressure distribution, we concluded that inertia is indeed able to generate a \emph{plausible} pressure on the body's surface.
+ On top of that, we were not able to find a similar distribution in systems that ignored inertial effects.
+
+
+ Most of the time the Jenny--Meyer integrator was used, but some experiments were simulated using different integrators.
+ %
+ A final verdict is difficult, but most of them seam to be able to capture the most important aspects.
+ However from the results we have presented in this work, we think that the cell centred velocity point integrator\footnote{Also known as ``standard'', which is described in section \ref{sec:discretization:step6:CCSTD} on page \pageref{sec:discretization:step6:CCSTD}.} should not be used.
%
We also observed problems with the Jenny--Meyer integrator, but they are mostly related to some internal checks, that are too restrictive and conservative.
- Most of the experiments focused on linear movements and effects that are related to them.
- However we have considered rotational movements as well, but not as extensive as linear movements.
+ Most of our experiments focused on linear movement and effects related to it.
+ However we have considered some aspects of rotational movements as well.
%
- Our results are not as conclusive as they are for linear movements.
+ But our results were not as conclusive as they were for linear movements.
%
\\
Our the data suggests that it is very likely the method is indeed able to handle rotational movements, maybe even as well as linear ones.
Nevertheless we suggest more investigations on such processes.
- We started this thesis with the question, how well a code, written for geodynamical simulations, is able to handle conditions that are radical different?
+ We started this thesis with a question about, how well a code, written for geodynamical simulations, is able to handle conditions that are radical different?
%
In this thesis we did not find a catastrophic or inherent problem in such an endeavour.
But we are hesitating to unconditionally answer it with \emph{yes}.
- Instead we conclude that the method might be suitable for it, but more research is needed for i to conform it.
+ Instead we conclude that the method might be suitable for it, but more research is needed to conform it.
% END: Final Conclusion
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Outlook and Further Work}~\label{chap:finConcOutlook:outlook}
- When we have discussed the experiments in the last chapter, we also mentioned points, that needs further investigation.
- We would like to summarize them here and extend them.
+ When we have discussed the experiments in the last chapter, we also mentioned aspects which need further investigations.
+ In this section however, we would like to discuss points, which are present in all systems.
+ Further we will also points towards completely new directions for investigations.
\paragraph{Rotational Effects}~\label{chap:finConcOutlook:outlook:rot}
- As we have pointed out before, most of the problems we had during the project and most of the unanswered questions, are related to rotational movement, we studied them in section \ref{sec:experiments:angMom} on page \pageref{sec:experiments:angMom}.
+ We mentioned it before, most of the experiments we conducted for the present work were concerned with linear movements.
+ However most of the problems as well as unanswered questions, are related to rotational movements.
+ Such movements were studied in section \ref{sec:experiments:angMom} on page \pageref{sec:experiments:angMom}.
%
- After some tuning, mostly lowering the time step, we were able to obtain results that were at least to some degree consistent.
+ \\
+ Most of our problems were because initially we were unable to understand our data.
+ In order to obtain data, that we were able to understand and deemed consistent, we had to tune, primarily lowering the time step, some system parameters.
+
- While it is important to understand the conservation of momentum, we suggest put it on hold, at least for the time being.
+ While it is important to study if angular momentum is conserved, we suggest put it on hold, at least for the time being.
%
- In order to verify if the method has indeed a \emph{fundamental} problem with rotational effects, we should first focus on a different experiment.
- Which represents the same problem domain, but is easer to understand\footnote{The \RotDisc{} scenario is quite easy, but it is hard to analyse, because there are a lot of potential factors, that can result in a decay of momentum.}.
+ In order to verify if the method has a \emph{fundamental} problem with rotational effects, we suggest to focus on a new experiment.
%
- As we have pointed out before, we suggests to implement a Taylor--Couette flow.
+ We recommend to study a Taylor--Couette flow.
%
\\
- Since we have to ensure that the disc is rotating at approximately the same speed\footnote{We suggests to use a special kind of material model, as we have done it for the \NoBC{} models.}, slowing down is not much of an issue, because the disc is powered.
+ It belongs to the same problem domain, but is easer to understand\footnote
+ {
+ The \RotDisc{} scenario is quite easy, but it is hard to analyse, because there are a lot of potential factors, that influnce the decay of momentum.
+ }.
%
- Viscous effects are no problem, but \emph{needed} to transmit the rotation to other parts of the domain.
+ Since we have to ensure that the disc is rotating at approximately the same speed, slowing down is not much of an issue.
+ %
+ Further viscous effects are no problem, but \emph{needed} to transmit the rotation to other parts of the domain.
This also lowers the influence of ghost viscosity, since we can simply ignore the ``near wall'' behaviour and just look at the undisturbed fluid.
%
- And most importantly, after some time, the system will reach a known steady state solution and maintain it.
+ And most importantly, after some time, the system will reach a known steady state solution and maintain it\footnote
+ {
+ Actually we have done it already, the system is mentioned as \texttt{rotTC} in the appendix, see section \ref{sec:appendix:Corello:Ini:SetUp:TYPE} on page \pageref{sec:appendix:Corello:Ini:SetUp:TYPE}.
+ It is quite similar to the usual \RotDisc{} setting, but there is an outer cylinder at rest to confine some fluid.
+ The inner disc is kept rotating by manipulating its intrinsic velocity, this is done by a special material model.
+ %
+ \\
+ Due to time constrains we were not able to fully review the data we have gathered on it nor to include it into this report.
+ However our primarily analysis suggests, that the simulations indeed reach and maintain the analytical steady state.
+ %
+ We measured the radial velocity profiles and found that the curves look very similar to the expected one.
+ %
+ But there are a few problems.
+ For example the inner disc seams to have a larger radius, while the cavity of the outer cylinder is smaller than it should be.
+ But we think that this is a grid effect.
+ }.
% END: Rot
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -101,18 +147,18 @@ In this chapter we will discuss the results that we have seen in the previous ch
But energy is also a very important quantity, that needs to be conserved by the method as well.
%
\\
- However if we start to consider energy, we should also to enable the energy equation, which we have ignored in this work.
+ However if we start to study energy, we should also think about enabling the energy equation, which we have ignored in this work.
% END: Energy
%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Different Velocity}~\label{chap:finConcOutlook:outlook:diffVel}
- \emph{A} velocity is needed to compute momentum and kinetic energy.
+ \emph{A} velocity is needed to compute momentum, kinetic energy and other quantities.
For internal consistency we have decided to only use the feel velocity defined on markers.
- But,as we have pointed out in section \ref{sec:experiments:methodology:Velocity} on page \pageref{sec:experiments:methodology:Velocity}, this is not the only option we had.
+ But, as we have mentioned it in section \ref{sec:experiments:methodology:Velocity} on page \pageref{sec:experiments:methodology:Velocity}, this was not our only option.
%
- We have seen that the results obtained by this velocity, meet most of our exaptations.
+ We have seen that results obtained by using this velocity, met most of our exaptations.
However we think that it is mandatory, to also study the results when different velocities are used.
% END: diff vel
@@ -126,7 +172,7 @@ In this chapter we will discuss the results that we have seen in the previous ch
\\
Such tests are interesting for many reasons.
For example the method has to conserve both linear \emph{and} angular momentum at the same time.
- Further such a body is subject to the Magnus effect, \cite{kundu_book}, which is well understood and can be used as a very interesting test case.
+ Further such a body is subject to the Magnus effect, \cite{kundu_book}, which is well understood and can be used as a very interesting future test case.
% END: Combined effects
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -155,14 +201,30 @@ In this chapter we will discuss the results that we have seen in the previous ch
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
- \paragraph{Quantitative Analysis}~\label{chap:finConcOutlook:outlook:quantitative}
- In this work we focused on the \emph{qualitative} behaviour of the systems.
- This is fine for the beginning, but we think that at least some degree of quantitative analysis is needed.
+ \paragraph{Fluid-Fluid}~\label{chap:finConcOutlook:outlook:FluidFluid}
+ The systems we have studied in this work, can be summarized as ``rigid body moves through a fluid.''
+ While this is fine on its own, we should also start considering systems involving two fluids instead.
%
- This will also allow us to determine the influence of lost that is unphysical and are caused by numerical effects.
+ For example such problems are covered in \cite{LeVeque}.
- % END: Quantitaive
+ % END: Fluid-Fluid
%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+ \paragraph{Different Geometry}~\label{chap:finConcOutlook:outlook:diffGeo}
+ In the current work we have studied rigid objects inside a media, in all cases this object was a ball\footnote
+ {
+ It is actually a cylinder, but for convenience, we call it a ball.
+ }.
+ We did this because it offered us a particular simple geometry.
+ %
+ However we suggest to simulate other objects including objects without symmetries, such that forces are not cancelled by it.
+
+ % END: Different geomtery
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+
% END: Outlook
%%%%%%%%%%%%%%%%%%%%%%%