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\newtheorem{proposition}{\bf Proposition}
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\newtheorem{project}{\bf Proj{\`e}t}
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\title{Master International de M\'ecanique, 2018-19}
\author{ F. Hecht}
\date{ Version 1.}
\begin{document}
\maketitle
\begin{abstract}
Pick a project, report your choice to F.H.. to make sure that no one else has selected the same before you. Project reports are to be handed in .pdf format at the time of the exam. There will be a small bonus for those who use \LaTeX ~to make their report.
\end{abstract}
\section*{List of CFD Projects using freefem++}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection*{Project 1**: Multiple wings}
%
%The geometry uses two wing profiles NACA0012 as in the example 9.3 in the freefem++ documentation
%made of 2 wing profiles parallel about the center of a big circle. Use Equation set 1 (irrotational flow) and compute the flow
%\\
%Q1: without incidence angle ($\psi = y$ on the big circle, $\psi=0$ on both wings.
%\\
%Q2: with incidence ($\psi=y\cos\alpha+x\sin\alpha$ on the big circle and $\psi=a$ on the lower wing and $\psi=b$ on the upper one). Adjust a,b so as to satisfy the Joukowski conditions.
%\\
%Q3: Replace equation set 1 (irrotational flow) by equation set 2 (Navier-Stokes). Compare both solutions
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Project 2***: Swimming of micro-organisms}
Complete exercise of Stokes flow; the object swim is a Naca012 wing
with a force movement of the with the following deformation $ [x,y] -> [x, y+x^2*\beta] $
with $\beta = 0.2*sin(t)$,
the freefem++ code to build these king meshes:
{\small \bFF
@real beta =0.1;
@real x0 = 0, y0=0, Cx =0.2, Cy=0,theta=0; //30*pi/180.;
@func @bool MoveWing(@real cx,@real cy,@real ct,@real st,@real x0,@real y0)
{
@real X=x-cx,Y=y-cy; // save x,y;
x = X*ct + Y*st + cx + x0;
y = -X*st + Y*ct + cy + x0;
@return 1;
}
@func Naca012=0.17735*sqrt(x)-0.075597*x
- 0.212836*(x^2)+0.17363*(x^3)-0.06254*(x^4);
@border C(t=0,2*pi) { x=0.5+5.*cos(t); y=5.*sin(t);label=1}
@border Splus(t=0,1) { x = t; y = beta*t^2 + Naca012;
MoveWing(Cx,Cy,cos(theta),sin(theta),x0,y0) ; label=2;}
@border Sminus(t=1,0){ x = t; y= beta*t^2 - Naca012;
MoveWing(Cx,Cy,cos(theta),sin(theta),x0,y0) ; label=2;}
@plot(Splus(70)+Sminus(70),wait=1);
@for(real cc = -0.3; cc<0.3; cc += 0.01)
{
verbosity=0;
beta = cc;
theta = cc*100*pi/180.;// angle
plot(Splus(70)+Sminus(70),wait=0);
}
\eFF}
\begin{enumerate}
\item[Q0] Compute the force on the Objet, and the barycenter of the wing
\item[Q1] Make a plot of velocity components and rotation angle versus time.
Choose a velocity of the two line, such that the objet move.
\item[Q2] Make a film of the swimming
\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Project 3: Comparison between inviscid and viscous flow}
Q1. Use Geometry 1 (ellipse in a rectangle) with $\alpha=20^o$ and Equation set 2 (Navier-Stokes). Then compute the flow at Reynolds number 50.
\\
Q2. Then on the same geometry use Equation set 1 (irrotational flow) and try to find the constant value for $\psi$ on the ellipse that makes the flow nearest to the result of Question 1.
\\
Q3. Compute the lift of the ellipse by using eq. set 2 (Navier-Stokes) at Reynolds number $50,100,200,+\infty$.
Use the lift to compute the appropriate constant for $\psi$ on the ellipse and plot the results for these values.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection*{Project 4: Lift, drag and temperature}
%
%With Geometry 1 (ellipse in a rectangle) and equation set 2 (Navier-Stokes), compute the lift and the drag of the ellipse as a function of $a$ for a fixed value of $b$,
%\\
%Q1. first for a Reynolds number where the flow is stationary
%\\
%Q2. then as a function of time at Reynolds number 200.
%\\
%Q3. Compute the temperature in the fluid when the ellipse is at 400K and the inflow and slip wall at 300K.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection*{Project 5: Swimming at high Reynolds number}
%
%(\emph{Here you need a better computer. Those of Atrium are not sufficient.})
%
%Design a snake by taking a section of a sine-wave ($y=a\sin(x+\alpha t)$ rotated $\beta(t)$ and translated $\vec v(t)$) and giving it some thickness. Place it at the center of a rectangle. The snake is able to slide on the sine-wave used to design it. Hence its shape is always the same but its head is not at the same place on the sine-wave. In other words for a snake of known length swimming with known amplitude $a$ the snake position is given by $\alpha,\beta,\vec v$ and the position of the head on the sine-wave (one parameter).
%\\
%Q1: Use the strategy of lesson 3 (Stokes swimming) for this animal swimming in a long rectangle. First use Stokes equations.
%\\
%Q2 Then switch to Navier-Stokes and increase the Reynolds (decrease $\nu$).
%\\
%Q3 Plot the speed of the head versus time and the angle of the sine-wave versus time.
\subsection*{Project 6*: mixing}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
With Geometry 2 (Y-shape tube)
\\
Q1: Compute the flow with Equation set 1 (irrotational flow), parabolic inflow at the top of both branches of the Y, no slip condition on the sides and free outflow at the bottom.
\\
Q2: Assume that on one branch the liquid entering the pipe is blue and on the other branch red. Using an equation for the transport of a chemical compute and display the mixing.
\\
Q3: Same with Stokes flow
\\
Q4: Compare the FEM-Characteristics with another numerical scheme for the transport equation (you may if you wish add a small diffusion term).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Project 8: Buoyancy**}
(\emph{Here you need a better computer. Those of Atrium are not sufficient.})
The geometry is the unit square with Equation set 2 (Navier-Stokes) coupled with the temperature
\[
\p_t\theta + u\n\theta -\n\cdot(\kappa\n\theta)=\frac\nu 2|\n u+\n u^T|^2 .
\]
With the Boussinesq approximation a force $f $ is added to the right hand side of Equation set 2 (Navier-Stokes):$ f_2= -C\theta$.
\\
Q1: assuming free slip for the velocity at the 4 walls of the cavity, and room temperature everywhere except at the vertical wall where it is $100^o$ on right and $0^o$on the left, and
compute the flow.
\\
Q2: Same with no slip conditions at the wall
\\
Q3: Change you numerical scheme for the temperature equation and redo the calculations, with full implicit schema and using Newton methods
to solve the no-linearity problem.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Project 9***: Richtmayer instabilities}
The domain is the unit square, the initial state is the heavy fluid on top of the light fluid with an interface of equation $y=\sin(a x)$.
\\
Q1. Let the flow evolve in time and show the results.
\\
Q2 Same but with mesh adapted to $\rho$.
\\
Q3. Study the influence of $a$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Project 10***: Kelvin-Helmohltz instabilities}
The geometry is a rectangle $(0,L)\times(-H,H)$. The inflow conditions are $u.n=-1$ when $y>0$ and zero otherwise. There is free slip on the lateral walls and free outflow.
The fluid velocity at time zero is $u_2=0$, $u_1=0$ if $y<0$ $u_1=1$ otherwise
\\l
Q1: Compute the flow; if it is stationary perturb it and see how the instabilities grow.
\\
Q2: Use a stream function - vorticity formulation to compute the same.
\\
Q3: Redo Q1 till time T then continue but with periodic conditions between the inflow and the outflow.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection*{Project 11: Convection schemes}
%
%Try various convection schemes on the Navier-Stokes equation on the Geometry 1 (ellipse in a rectangle) with a=b (circle).
%\\
%Q1: The FEM-Characteristics
%\\
%Q2: a Diconstinuous-Galerkin scheme
%\\
%Q3: An SUPG method
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection*{Project 12***: RANS k-epsilon}
%
%The Geometry is a flat plate, i.e. a rectangle $(0,L)\times(-H,H)$ split by a fat segment
%$(L,2L)\times(-\epsilon,\epsilon)$.
%Equation set 2 (Navier-Stokes) is complemented with the k-epsilon model (see notes)
%\\
%Q1: Try Equation set 2 (Navier-Stokes) without turbulence model and $u=0,$ on the flat plate, uniform inflow, slip condition on the lateral walls and free output flow.
%\\
%Q2: Same as Q1 but with the equation set " (RANS) and $u=0,\p_n k=0$ on the flat plate. Compare with Blasius's solution.
%\\
%Q3: Same as Q2 but with the $k-epsilon$ model (see notes)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Project 13**: Euler versus Navier-Stokes}
Geometry 1 (ellipse in a rectangle) but with the ellipse replaced by a wing profile NACA0012 (equation is given in example 3.9 of freefem++ documentation).
\\
Q1: compute the lift by using Equation set 1 (irrotational flow).
\\
Q2 : Compute the flow using Equation set 2 (Navier-Stokes)
\\
Q3: Set $\nu=0$ in Q2 and see if you can find the same solution as in Q1?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Project 14**: A pseudo rotating geometry}
The geometry is an ellipse centered at the origin of radius $a,b$ and the long axis is horizontal; the fluid is bounded by a circle centered at the center of the ellipse and of radius $R>>a$. Use equation set 2 (Navier-Stokes). with boundary condition on the large circle $u=(cos\alpha, \sin\alpha)^T$.
\\
Q1: Compute the flow and the resulting force $\d f$ on the ellipsoid elementary area $\d S$.
\\
Q2: Change $\alpha$ by writing Newton's law on the ellipse discretized by an explicit time scheme.
\\
Q3: Check whether the fact that we are using a frame of reference attached to the ellipse requires a Coriolis term added to the Navier-Stokes equation and do the time dependent calculation of the flow in the frame of reference where the ellipse is fixed but with visualization in the frame of reference where the ellipse rotates,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Project 15***: A rotating Geometry}
With Geometry 1 (ellipse in a rectangle) we will let the ellipse $E$ rotate freely around its center. It is the same project as Project 11 but we do not change the reference frame. Since this project is harder than 11 you may team with 11.
\\
Q1: Compute the flow using Equation set 2 (Navier-Stokes) and compute the resulting force $\d f$ on the ellipsoid elementary area $\d S$.
\\
Q2: Change $\alpha$ by writing Newton's law on the ellipse discretized by an explicit time scheme.
\\
Q3: Include in a time loop Q1,Q2 and a remeshing step to allow for the geometry change.
\subsection*{Project 16: A compressible flow}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Use Geometry 3 (nozzle with a bump) with slip conditions on the top and bottom boundaries, uniform inflow $u_\infty$ normal velocity and free outflow.
\\
Q1: Compute the flow using Equation set 1 (irrotational flow).
\\
Q2: Same but with a potential function $\phi$ instead of $\psi$ (equation set 0).
\\
Q3: Use the transonic equation $\n\cdot(1-|\n\phi|^2)^\beta\n\phi=0$ (see chapter 2 in the notes) and compute the flow by a fixed point method:
\\
\centerline{Loop on m: $\n\cdot(1-|\n\phi^m|^2)^\beta\n\phi^{m+1}=0$}
\\
Start with $u_\infty$ small and enlarge it till a shock appears.
%\subsection*{Project 14b: Acoustics}
%
%
%Q1: Starting from the equations of compressible flow, show that under certain circumstances one can solve
%\[
% \p_t\rho + \n\cdot u =0,~~~~\p_t u + \n p=0,~~~ \frac{p}{\rho}=\hbox{constant}
%\]
%Q2: Propose a scheme and implement it with freefem++ for Geometry 3 (nozzle with a bump).
%\\
%Q3: Compare with the classical wave equation for acoustics
\subsection*{ Project 17*: Numerical Methods}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Geometry 1 (ellipse in a rectangle):
Q1: Solve the Navier-Stokes equation with P2/P1
Q2: Compare with P1b/P1 and also with the projection scheme of NSproject.edp
Q3: Don't use characeteristic-Galerkin but use Newton on the time independent problem
%\subsection*{ Project 18***: Inertia}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%Q1: Solve the Navier Stokes equation when the outer circle rotate at angular speed $\omega$ and the inner circle is fixed
%a the center but can be rotating.
%
%Q2: Study convergence with respect to mesh size
%
%Q3: The hole is now a heavy disk of mass m which is free to rotate around its center. The motion of the outer circle will stir fluid motion which will force the inner center to rotate according to Newton's law.
%
\subsection*{ Project 19*: Flow Through Porous Media}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The geometry is a rectangle $(0,L)\times(0,H)$. The equations are
\begin{eqnarray}&&
\n ({\bf K}\n p) = 0,~~ u=K\n p,~~~\p_t c + u\cdot\n c -\nu\n(K\n c) = f
\end{eqnarray}
where $p$ is the hydrostatic pressure, $c$ the concentration of the pollutant, $u$ the velocity of the flow, ${\bf K}$ the permeability tensor of the medium, $\nu$ the diffusivity of the pollutant and $f$ the pollutant source.
The medium is an horizontal layer of clay in $(0,L)\times(0,H/3)$ with an horizontal layer of limestone over it.
First we suppose that ${\bf K}$ is the identity matrix times a scalar $K$ which in the clay is 1000 and $K$ in the limestone is 1; $\nu=1.0e-3$. $p$ on the boundary is $x+y$, which $\p c/\p n$ is imposed on the boundary at all times. The source term is a Dirac at $x=L/2, y=H/4$, constant in time.
Q1: Use P1 elements for both u and c and compute the evolution of c. Compute the integral of c in the domain and plot it as a function of time.
Q2: Use a discontinuous Galerkin method to improve convervativity
Q3: Now ${\bf K}$ is a diagonal matrix with $K_{11}=10 K$ and $K_{22}=K$. Redo Q1 and Q2.
%\subsection*{ Project 20***: Shallow water equations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%We shall use Geometry 1 (ellipse in a rectangle) and the ellipsoid represents an island in the river. The equations are
%\begin{eqnarray}&&
%\p_t (h u) + u\cdot\n(h u) + g\n h^2 -\nu\Delta u = 0,
%\cr&&
%\p_t h + \n\cdot(h u)=0~~ (\Leftrightarrow ~~\p_t h^2 + \n\cdot( h^2 u)+ h^2\n\cdot u=0)
%\end{eqnarray}
%where $h$ is the water depth, u the velocity and p the pressure.
%
%Q1: Use P1 elements for $v=hu,H=h^2$ and approximate $\Delta u\approx \n\cdot(h^{-1}\n v)$. Compare Characteristic-Galerkin and semi-Newton
%
%Q2 Reformulate the problem in terms of $u, h$ and use P1 elements for these variables and compare (with semi-Newton only)
%
%Q3: Identify a regime with a shock and compute it.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection*{ Project 21: Free Boundary}
%\emph{(Interesting but not easy)}\\
%We wish to model the flow coming out of a tap. So we consider a geometry made by a square with an entry a 2/3 of the height on the left vertical wall and the 3 other boundaries are free flow (zero constraint).
%The equations are Navier-Stokes. The initial flow is parabolic in an horizontal rectangle which extends the entry. Due to gravity the jet will curve down; the boundary conditions on the free boundaries of the jet are
%$u.n=0,~ -pn+\nu\p_n u=-g e^2$ where $e^2=(0,1)^T$.
%
%Q1: Compute the flow when $g=0$ only in the jet. The mesh should be moved so as to match all the boundary conditions on the jet.
%
%Q2 Same but now $g$ grows slowly with time up to 9.81.
%
%Q3: Use the equation set 5 with $\rho=1$ in the water and $\rho=0.01$ in air and compute in the entire square.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{ Project 21**: Free Boundary}
The problem is to find the Free Boundary of the river when a wake enter on the river.
To simplify the coordinate is $x,y$ ( y corresponding to the z axis with gravity).
the domain is a kind of rectangle with up free surface.
The equation are Stokes or Navier-Stokes, the 3 left, down, right boundary condition are $u=0$
and on the top, $u=$given and $\sigma.n =$ given .
Use the free surface equation equation of FreeFem++ as a good starting point.
\subsection*{ Project 22: Turbulence**}
The geometry is a backward step: $(0,a)\times(c,d)\cup(a,b)\times(0,d)$. The flow is modeled by the set 3 (RANS equations). Compute a turbulent flow with uniform inflow conditions and free outflow conditions.
\\
Q1: Begin with Navier-Stokes
\\
Q2: Then add the equation for $k$.
\\
Q3 Find on the web or in the literature a similar test case (but not the same turbulence model) and compare.
\subsection*{Project 23: Generation of a axisymmetric droplet very viscous}
The problem is a times depend Stokes in moving domain in green defined by a
level set function. You have forces at the top to push the fluid , no sleep boundary condition on
boundary mold, on free surface just surface tension proportional to the radius of curvature.
\begin{figure}[hpt]
\begin{center}
\includegraphics[height=5cm]{droplet.eps}
\end{center}
\caption{\label{fig:domain}Domain $\Omega$ et $\Omega_t$ }
\end{figure}
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection*{Project 23: bifurcations in a flow}
%
%The geometry is a disk with a hole, representing the perpendicular cross section of a pipe containing a rotating rod: the pipe is the inside of the unit circle, the rod is a hole centered at (0,1/2) and with radius R<1/2. The flow is incompressible modeled b the Navier-Stokes equations.
%\\
%Q1: Choose a low Reynolds number to obtain a stationary solution with no eddy.
%\\
%Q2: Increase the Re to obtain a stationary flow with eddies
%\\
%Q3: Plot the drag on the pipe versus Re.
%\\
%Q4: Try the RANS model (set 3) on this problem.
\section*{Definitions}
\paragraph{Geometry 1 (ellipse in a rectangle)} An ellipse in a long rectangle $(0,L)\times(0,H)$. The ellipse is centered at (L/5,H/2) and has a long radius a and a short radius b and the long axis makes an angle $\alpha$ with the horizontal axis of the reference frame.
\paragraph{Geometry 2 (Y-shape tube) } A bifurcating tube shaped like the letter Y. The parameters are the lengths a and b of the 3 portions of the tubes, the diameter d and the angle $\alpha$
between the 2 branches of the Y.
\paragraph{Geometry 3 (nozzle with a bump)} A long rectangle $(0,L)\times(0,H)$ where the flat bottom boundary has a bump shaped as a portion of circle, $x(t)=\frac{L}3+R*cos(\beta t),~y(t)=-R+a +R\sin(\beta t), t\in(t_1,t_2)$; naturally $t_1,t_2$ needs to be calculated as functions of $R,\beta$ so that the boundary is closed and simply connected.
\paragraph{Equation set 0: potential flow}
The flow is assumed irrotational and inviscid so that $-\Delta\phi=0$.
The flow is $u=\nabla\phi, p=p_0-\frac12|u|^2$; it is parallel to the walls so $\partial_n\phi=0$ on the boundaries where there is a slip condition $u\cdot n=0$. Typical inflow/outflow condition is $\phi$ constant .
\paragraph{Equation set 1 (irrotational flow)}
The velocity of the flow and the pressure are $u=\n\times\psi, p=p_0-\frac12|u|^2$ .
The flow is assumed irrotational and inviscid so that $-\Delta\psi=0$.
The flow is parallel to the walls so $\psi$ is constant on the boundaries where there is a slip condition $u\cdot n=0$. The inflow is assumed parabolic and compatible with the slip conditions. The outflow condition is $\p_n\psi=0$.
\paragraph{Equation set 2 (Navier-Stokes): Navier-Stokes flow}
Navier-Stokes flow with slip conditions on the walls (i.e., $u.n=0,~\partial_n(u.s)=0$, s being the tangent, n the normal), parabolic inflow and free outflow (i.e. $-\nu\partial_n u +p n =0$).
\paragraph{Equation set 3: RANS model}
A one equation RANS turbulence model to compute the turbulent viscosity in the Reynolds averaged equations
by $\nu_T=\nu + c_1\sqrt{k}$ where $k$ is the turbulent kinetic energy :
\begin{eqnarray}&&
\p_t u + u\n u -\n\cdot(\nu_T(\n u+ (\n u)^T)) + \n p =0,~~ \n\cdot u = 0
\cr&&
\p_t k + u\n k - c_1\n(\sqrt{k}\n k) = c_1\sqrt{k}\frac12|\n u + \n u^T|^2
\end{eqnarray}
with $\p_n k =0$ on walls and at the outflow boundaries and initial and inflow conditions given.
For the velocity no slip conditions on the walls and parabolic inflow and free outflow (i.e. $-\nu_T\partial_n u +p n =0$).
\paragraph{Equation set 5: piecewise incompressible viscous flow}
Incompressible Navier-Stokes flow with multiple regions of different densities is modeled by
\begin{eqnarray}&&
\p_t(\rho u) +\n\cdot(\rho u\otimes u) +\n p -\mu\Delta u =0
\cr &&
\p_t\rho+\n\cdot(\rho u)=0,~~~\n\cdot u=0
\end{eqnarray}
$\rho$ constant in each region
%$$\includegraphics[width=10cm]{sketch2}$$
\end{document}