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Edited Stokes flow and Level Set tutorials (grammar etc.)

2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
3 %
4 % Copyright (c) 2003-2008 by University of Queensland
5 % Earth Systems Science Computational Center (ESSCC)
6 % http://www.uq.edu.au/esscc
7 %
8 % Primary Business: Queensland, Australia
9 % Licensed under the Open Software License version 3.0
10 % http://www.opensource.org/licenses/osl-3.0.php
11 %
12 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
15 \section{Rayleigh-Taylor Instability}
16 \label{LEVELSET CHAP}
18 In this section we will implement the Level Set Method in Escript for tracking the interface between two fluids for Computational Fluid Dynamics (CFD). The method is tested with a Rayleigh-Taylor Instability problem, which is an instability of the interface between two fluids with differing densities. \\
19 Normally in Earth science problems two or more fluids in a system with different properties are of interest. For example, lava dome growth in volcanology, with the contrast of the two mediums as being lava and air. The interface between the two mediums is often referred to as a free surface (free boundary value problem); the problem arises due to the large differences in densities between the lava and air, with their ratio being around 2000, and so the interface between the two fluids move with respect to each other.
20 %and so the lava with the much higher density is able to move independently with respect to the air, and the interface between the two fluids is not constrained.
21 There are a number of numerical techniques to define and track the free surfaces. One of these methods, which is conceptually the simplest, is to construct a Lagrangian grid which moves with the fluid, and so it tracks the free surface. The limitation of this method is that it cannot track surfaces that break apart or intersect. Another limitation is that the elements in the grid can become severely distorted, resulting in numerical instability. The Arbitrary Lagrangian-Eulerian (ALE) method for CFD in moving domains is used to overcome this problem by remeshing, but there is an overhead in computational time, and it results in a loss of accuracy due to the process of mapping the state variables every remesh by interpolation.
23 There is a technique to overcome these limitations called the Level Set Method, for tracking interfaces between two fluids. The advantages of the method is that CFD can be performed on a fixed Cartesian mesh, and therefore problems with remeshing can be avoided. The field equations for calculating variables such as velocity and pressure are solved on the the same mesh. The Level Set Method is based upon the implicit representation of the interface by a continuous function. The function takes the form as a signed distance function, $\phi(x)$, of the interface in a Eulerian coordinate system. For example, the zero isocontour of the unit circle $\phi(x)=x^2 + y^2 -1$ is the set of all points where $\phi(x)=0$. Refer to Figure \ref{UNITCIRCLE}.
24 %
25 \begin{figure}
26 \center
27 \scalebox{0.8}{\includegraphics{figures/unitcircle.eps}}
28 \caption{Implicit representation of the curve $x^2 + y^2 = 1$.}
29 \label{UNITCIRCLE}
30 \end{figure}
31 %
32 The implicit representation can be used to define the interior and exterior of a fluid region. Since the isocontour at $\phi(x)=0$ has been defined as the interface, a point in the domain can be determined if its inside or outside of the interface, by looking at the local sign of $\phi(x)$. For example, a point is inside the interface when $\phi(x)<0$, and outside the interface when $\phi(x)>0$. Parameters values such as density and viscosity can then be defined for two different mediums, depending on which side of the interface they are located.
35 \subsection{Calculation of the Displacement of the Interface}
37 The displacement of the interface at the zero isocontour of $\phi(x)$ is calculated each time-step by using the velocity field. This is achieved my solving the advection equation:
38 %
39 \begin{equation}
40 \frac{\partial \phi}{\partial t} + \vec{v} \cdot \nabla \phi = 0,
41 \label{ADVECTION}
42 \end{equation}
43 %
44 where $\vec{v}$ is the velocity field. The advection equation is solved using a mid-point, which is a two step procedure:
46 Firstly, $\phi^{1/2}$ is calculated solving:
47 %
48 \begin{equation}
49 \frac{\phi^{1/2} - \phi^{-}}{dt/2} + \vec{v} \cdot \nabla \phi^{-} = 0.
50 \label{MIDPOINT FIST}
51 \end{equation}
52 %
53 Secondly, using $\phi^{1/2}$, $\phi^{+}$ is calculated solving:
54 %
55 \begin{equation}
56 \frac{\phi^{+} - \phi^{-}}{dt} + \vec{v} \cdot \nabla \phi^{1/2} = 0.
58 \end{equation}
59 %
60 This procedure works provided that the discretization of the left-hand side of Equations (\ref{MIDPOINT FIST}) and (\ref{MIDPOINT SECOND}) is a lumped mass matrix. For more details on the mid-point procedure see reference \cite{BOURGOUIN2006}. In certain situations the mid-point procedure has been shown to produce artifacts in the numerical solutions. A more robust procedure is to use the Taylor-Galerkin scheme with the presence of diffusion, which gives more stable solutions. The expression is derived by either inserting Equation (\ref{MIDPOINT FIST}) into Equation (\ref{MIDPOINT SECOND}), or by expanding $\phi$ into a Taylor series:
61 %
62 \begin{equation}
63 \phi^{+} \simeq \phi^{-} + dt\frac{\partial \phi^{-}}{\partial t} + \frac{dt^2}{2}\frac{\partial^{2}\phi^{-}}{\partial t^{2}},
65 \end{equation}
66 %
67 by inserting
68 %
69 \begin{equation}
70 \frac{\partial \phi^{-}}{\partial t} = - \vec{v} \cdot \nabla \phi^{-},
72 \end{equation}
73 %
74 and
75 %
76 \begin{equation}
77 \frac{\partial^{2} \phi^{-}}{\partial t^{2}} = \frac{\partial}{\partial t}(-\vec{v} \cdot \nabla \phi^{-}) = \vec{v}\cdot \nabla (\vec{v}\cdot \nabla \phi^{-}),
78 \label{SECOND ORDER}
79 \end{equation}
80 %
81 into Equation (\ref{TAYLOR EXPANSION})
82 %
83 \begin{equation}
84 \phi^{+} = \phi^{-} - dt\vec{v}\cdot \nabla \phi^{-} + \frac{dt^2}{2}\vec{v}\cdot \nabla (\vec{v}\cdot \nabla \phi^{-}).
86 \end{equation}
89 %\subsection{Governing Equations for Fluid Flow}
91 %The fluid dynamics is governed by the Stokes equations. In geophysical problems the velocity of fluids are low; that is, the inertial forces are small compared with the viscous forces, therefore the inertial terms in the Navier-Stokes equations can be ignored. For a body force $f$ the governing equations are given by:
92 %
93 %\begin{equation}
94 %\nabla \cdot (\eta(\nabla \vec{v} + \nabla^{T} \vec{v})) - \nabla p = -f,
96 %\end{equation}
97 %
98 %with the incompressibility condition
99 %
100 %\begin{equation}
101 %\nabla \cdot \vec{v} = 0.
103 %\end{equation}
104 %
105 %where $p$, $\eta$ and $f$ are the pressure, viscosity and body forces, respectively.
106 %Alternatively, the Stokes equations can be represented in Einstein summation tensor notation (compact notation):
107 %
108 %\begin{equation}
109 %-(\eta(v\hackscore{i,j} + v\hackscore{j,i})),\hackscore{j} - p,\hackscore{i} = f\hackscore{i},
111 %\end{equation}
112 %
113 %with the incompressibility condition
114 %
115 %\begin{equation}
116 %-v\hackscore{i,i} = 0.
118 %\end{equation}
119 %
120 %The subscript comma $i$ denotes the derivative of the function with respect to $x\hackscore{i}$. A linear relationship between the deviatoric stress $\sigma^{'}\hackscore{ij}$ and the stretching $D\hackscore{ij} = \frac{1}{2}(v\hackscore{i,j} + v\hackscore{j,i})$ is defined as \cite{GROSS2006}:
121 %
122 %\begin{equation}
123 %\sigma^{'}\hackscore{ij} = 2\eta D^{'}\hackscore{ij},
124 %\label{STRESS}
125 %\end{equation}
126 %
127 %where the deviatoric stretching $D^{'}\hackscore{ij}$ is defined as
128 %
129 %\begin{equation}
130 %D^{'}\hackscore{ij} = D^{'}\hackscore{ij} - \frac{1}{3}D\hackscore{kk}\delta\hackscore{ij}.
132 %\end{equation}
133 %
134 %where $\delta\hackscore{ij}$ is the Kronecker $\delta$-symbol, which is a matrix with ones for its diagonal entries ($i = j$) and zeros for the remaining entries ($i \neq j$). The body force $f$ in Equation (\ref{GENERAL NAVIER STOKES COM}) is the gravity acting in the $x\hackscore{3}$ direction and is given as $f = -g \rho \delta\hackscore{i3}$.
135 %The Stokes equations is a saddle point problem, and can be solved using a Uzawa scheme. A class called StokesProblemCartesian in Escript can be used to solve for velocity and pressure.
136 %In order to keep numerical stability, the time-step size needs to be below a certain value, known as the Courant number. The Courant number is defined as:
137 %
138 %\begin{equation}
139 %C = \frac{v \delta t}{h}.
140 %\label{COURANT}
141 %\end{equation}
142 %
143 %where $\delta t$, $v$, and $h$ are the time-step, velocity, and the width of an element in the mesh, respectively. The velocity $v$ may be chosen as the maximum velocity in the domain. In this problem the Courant number is taken to be 0.4 \cite{BOURGOUIN2006}.
146 \subsection{Reinitialization of Interface}
148 As the computation of the distance function progresses, it becomes distorted, and so it needs to be updated in order to stay regular \cite{SUSSMAN1994}. This process is known as the reinitialization procedure. The aim is to iteratively find a solution to the reinitialization equation:
149 %
150 \begin{equation}
151 \frac{\partial \psi}{\partial \tau} + sign(\phi)(1 - \nabla \psi) = 0.
153 \end{equation}
154 %
155 where $\psi$ shares the same level set with $\phi$, $\tau$ is pseudo time, and $sign(\phi)$ is the smoothed sign function. This equation is solved to meet the definition of the level set function, $\lvert \nabla \psi \rvert = 1$; the normalization condition. Equation (\ref{REINITIALISATION}) can be rewritten in similar form to the advection equation:
156 %
157 \begin{equation}
158 \frac{\partial \psi}{\partial \tau} + \vec{w} \cdot \nabla \psi = sign(\phi),
160 \end{equation}
161 %
162 where
163 %
164 \begin{equation}
165 \vec{w} = sign(\phi)\frac{\nabla \psi}{|\nabla \psi|}.
167 \end{equation}
168 %
169 $\vec{w}$ is the characteristic velocity pointing outward from the free surface. Equation (\ref{REINITIALISATION2}) can be solved by a similar technique to what was used in the advection step; either by the mid-point technique \cite{BOURGOUIN2006} or the Taylor-Galerkin procedure. For the mid-point technique, the reinitialization technique algorithm is:
171 1. Calculate
172 %
173 \begin{equation}
174 \vec{w} = sign(\phi)\frac{\nabla \psi}{|\nabla \psi|},
176 \end{equation}
177 %
179 2. Calculate $\psi^{1/2}$ solving
180 %
181 \begin{equation}
182 \frac{\psi^{1/2} - \psi^{-}}{d\tau/2} + \vec{w} \cdot \nabla \psi^{-}= sign(\phi),
184 \end{equation}
185 %
187 3. using $\psi^{1/2}$, calculate $\psi^{+}$ solving
188 %
189 \begin{equation}
190 \frac{\psi^{+} - \psi^{-}}{d\tau} + \vec{w} \cdot \nabla \psi^{1/2}= sign(\phi),
192 \end{equation}
193 %
195 4. if the convergence criterion has not been met, go back to step 2. Convergence is declared if
196 %
197 \begin{equation}
198 ||\nabla \psi \hackscore{\infty}| - 1| < \epsilon \hackscore{\psi}.
200 \end{equation}
201 %
202 where $\epsilon \hackscore{\psi}$ is the convergence tolerance. Normally, the reinitialization procedure is performed every third time-step of solving the Stokes equation.
204 The mid-point technique works provided that the left-hand side of Equations (\ref{REINITIAL MIDPOINT2}) and (\ref{REINITIAL MIDPOINT3}) is a lumped mass matrix. Alternatively, for a one-step procedure, the reinitialization equation can be given by:
205 %
206 \begin{equation}
207 \psi^{+} = \psi^{-} - \tau \vec{w} \cdot \nabla \psi^{-} + \frac{d \tau^{2}}{2} \vec{w} \cdot \nabla(\vec{w} \cdot \nabla \psi^{-}).
209 \end{equation}
210 %
211 The accuracy of $\phi$ is only needed within the transition zone; and so it can be calculated in a narrow band between the interface of the fluids.
212 %
213 \begin{figure}
214 \center
215 \scalebox{0.5}{\includegraphics{figures/LevelSetFlowChart.eps}}
216 \caption{Flow chart of Level Set Method procedure \cite{LIN2005}.}
218 \end{figure}
219 %
220 When the distance function, $\phi$, is calculated, the physical parameters, density and viscosity, are updated using the sign of $\phi$. The jump in material properties between two fluids, such as air and water can be extreme, and so the transition of the properties from one medium to another is smoothed. The region of the interface is assumed to be of finite thickness of $\alpha h$, where $h$ is the size of the elements in the computational mesh and $\alpha$ is a smoothing parameter. The parameters are updated by the following expression:
221 %
222 \begin{equation}
223 P =
224 \left \{ \begin{array}{l}
225 P\hackscore{1} \hspace{5cm} where \ \ \psi < - \alpha h \\
226 P\hackscore{2} \hspace{5cm} where \ \ \psi > \alpha h \\
227 (P\hackscore{2} - P\hackscore{1}) \psi/2\alpha h + (P\hackscore{1} + P\hackscore{2})/2 \ \ \ \ \ \ where \ \ |\psi| < \alpha h.
228 \end{array}
229 \right.
231 \end{equation}
232 %
233 where the subscripts $1$ and $2$ denote the different fluids. The procedure of the level set calculation is shown in Figure \ref{LEVELSET FLOWCHART}.
234 Further work is needed in the reinitialization procedure, as it has been shown that it is prone to mass loss and inconsistent positioning of the interface \cite{SUCKALE2008}.
236 \subsection{Benchmark Problem}
238 The Rayleigh-Taylor instability problem is used as a benchmark to validate CFD implementations \cite{VANKEKEN1997}. Figure \ref{RT2DSETUP} shows the setup of the problem. A rectangular domain with two different fluids is considered, with the greater density fluid on the top and the lighter density fluid on the bottom. The viscosities of the two fluids are equal (isoviscous). An initial perturbation is given to the interface of $\phi=0.02cos(\frac{\pi x}{\lambda}) + 0.2$. The aspect ratio $\lambda = L/H = 0.9142$ is chosen such that it gives the greatest disturbance of the fluids. The fluid properties is chosen such that the compositional Rayleigh number is equal to one:
239 %
240 \begin{equation}
241 R\hackscore{b} = \frac{\Delta \rho H^{3}}{\kappa \eta} = 1.
243 \end{equation}
244 %
245 where $\Delta \rho$ is the difference in density between the two fluids, $\eta$ is the viscosity and $\kappa$ is the thermal diffusivity; arbitrarily taken equal to 1 for a ``non thermal'' case.
246 %
247 %
248 \begin{figure}
249 \center
250 \scalebox{0.7}{\includegraphics{figures/RT2Dsetup.eps}}
251 \caption{Parameters, initial interface and boundary conditions for the Rayleigh-Taylor instability problem. The interface is defined as $\phi=0.02cos(\frac{\pi x}{\lambda}) + 0.2$. The fluids have been assigned different densities and equal viscosity (isoviscous) \cite{BOURGOUIN2006}.}
252 \label{RT2DSETUP}
253 \end{figure}
254 %
255 %
256 The following PYTHON code is for the Rayleigh-Taylor instability problem, which is available in the example directory as 'RT2D.py'. This script uses the 'StokesProblemCartesian' class for solving the Stokes equation, along with the incompressibility condition. A class called 'LevelSet' is also used, which performs the advection and reinitialization procedures to track the movement of the interface of the fluids. The details and use of these classes are described in Chapter \ref{MODELS CHAPTER} (Models Chapter).
258 The script starts off by importing the necessary classes. The physical properties of the two fluids are defined, such as density and viscosity. Acceleration due to gravity is taken as 10.0 $ms^{-2}$. Solver settings are set for solving the Stokes problem, with the number of time-steps, solver tolerance, maximum solver iterations, and the option to use the Uzawa scheme or not; the default solver is the PCG solver. A regular mesh is defined with 200$\times$200 elements. Level set parameters are set for the reinitialization procedure, such as the convergence tolerance, number of reinitialization steps, the frequency of the reinitialization, for example, every third time-step, and the smoothing parameter to smooth the physical properties across the interface. A no-slip boundary condition is set for the top and bottom of the domain, while on the left and right-hand sides there is a slip condition. The initial interface between the two fluids is defined as in Figure \ref{RT2DSETUP}. Instances of the StokesProblemCartesian and LevelSet class are created. The iteration throughout the time-steps involves the update of the physical parameters of the fluids; the initialization of the boundary conditions, viscosity, and body forces; the solving of the Stokes problem for velocity and pressure; then the level set procedure. The output of the level set function, velocity and pressure is saved to file. The time-step size is selected based on the Courant condition. Due to the number of elements in the computational mesh, the simulation may take a long time to complete on a desktop computer, so it is recommended to run it on the super computer. At present, the fine mesh is required to capture the details of the fluid motion and for numerical stability.
259 %
260 \begin{python}
262 from esys.escript import *
263 import esys.finley
264 from esys.escript.models import StokesProblemCartesian
265 from esys.finley import finley
266 from esys.finley import Rectangle
267 from LevelSet import *
269 #physical properties
270 rho1 = 1000 #fluid density on bottom
271 rho2 = 1010 #fluid density on top
272 eta1 = 100.0 #fluid viscosity on bottom
273 eta2 = 100.0 #fluid viscosity on top
274 g=10.0
276 #solver settings
277 dt = 0.001
278 t_step = 0
279 t_step_end = 2000
280 TOL = 1.0e-5
281 max_iter=400
282 verbose=True
283 useUzawa=True
285 #define mesh
286 l0=0.9142
287 l1=1.0
288 n0=200
289 n1=200
291 mesh=Rectangle(l0=l0, l1=l1, order=2, n0=n0, n1=n1)
292 #get mesh dimensions
293 numDim = mesh.getDim()
294 #get element size
295 h = Lsup(mesh.getSize())
297 #level set parameters
298 tolerance = 1.0e-6
299 reinit_max = 30
300 reinit_each = 3
301 alpha = 1
302 smooth = alpha*h
304 #boundary conditions
305 x = mesh.getX()
306 #left + bottom + right + top
307 b_c = whereZero(x[0])*[1.0,0.0] + whereZero(x[1])*[1.0,1.0] + whereZero(x[0]-l0)*[1.0,0.0] \
308 + whereZero(x[1]-l1)*[1.0,1.0]
310 velocity = Vector(0.0, ContinuousFunction(mesh))
311 pressure = Scalar(0.0, ContinuousFunction(mesh))
312 Y = Vector(0.0,Function(mesh))
314 #define initial interface between fluids
315 xx = mesh.getX()[0]
316 yy = mesh.getX()[1]
317 func = Scalar(0.0, ContinuousFunction(mesh))
318 h_interface = Scalar(0.0, ContinuousFunction(mesh))
319 h_interface = h_interface + (0.02*cos(math.pi*xx/l0) + 0.2)
320 func = yy - h_interface
321 func_new = func.interpolate(ReducedSolution(mesh))
323 #Stokes Cartesian
324 solution=StokesProblemCartesian(mesh,debug=True)
325 solution.setTolerance(TOL)
326 solution.setSubProblemTolerance(TOL**2)
328 #level set
329 levelset = LevelSet(mesh, func_new, reinit_max, reinit_each, tolerance, smooth)
331 while t_step <= t_step_end:
332 #update density and viscosity
333 rho = levelset.update_parameter(rho1, rho2)
334 eta = levelset.update_parameter(eta1, eta2)
336 #get velocity and pressure of fluid
337 Y[1] = -rho*g
338 solution.initialize(fixed_u_mask=b_c,eta=eta,f=Y)
339 velocity,pressure=solution.solve(velocity,pressure,max_iter=max_iter,verbose=verbose, \
340 useUzawa=useUzawa)
342 #update the interface
343 func = levelset.update_phi(velocity, dt, t_step)
345 print "##########################################################"
346 print "time step:", t_step, " completed with dt:", dt
347 print "Velocity: min =", inf(velocity), "max =", Lsup(velocity)
348 print "##########################################################"
350 #save interface, velocity and pressure
351 saveVTK("phi2D.%2.4i.vtu"%t_step,interface=func,velocity=velocity,pressure=pressure)
352 #Courant condition
353 dt = 0.4*Lsup(mesh.getSize())/Lsup(velocity)
354 t_step += 1
356 \end{python}
357 %
358 %
359 The results from the simulation can be viewed by visualization software such as \textit{visIt}. If the software is installed, it can be opened by simply executing the following command:
360 %
361 \begin{python}
362 visit
363 \end{python}
364 %
365 In the visIt main window, vtk/vtu files can be opened from the File menu; contours and vectors can then be displayed by selecting them from the Plots menu and pressing the Draw button. A movie of the simulation can be watched by pressing the Play button. The graphics are displayed in the Vis window. For more information on \textit{visIt} see the website \cite{VisIt}.
367 The simulation output is shown in Figures \ref{RT2D OUTPUT1} and \ref{RT2D OUTPUT1} showing the progression of the interface of the two fluids. A diapir can be seen rising on the left-hand side of the domain, and then later on, a second one rises on the right-hand side.
368 \begin{figure}
369 \center
370 \subfigure[t=300]{\label{RT OUTPUT300}\includegraphics[scale=0.252]{figures/RT2D200by200t300.eps}}
371 \subfigure[t=600]{\label{RT OUTPUT600}\includegraphics[scale=0.252]{figures/RT2D200by200t600.eps}}
372 \subfigure[t=900]{\label{RT OUTPUT900}\includegraphics[scale=0.252]{figures/RT2D200by200t900.eps}}
373 \subfigure[t=1200]{\label{RT OUTPUT1200}\includegraphics[scale=0.252]{figures/RT2D200by200t1200.eps}}
374 \caption{Simulation output of Rayleigh-Taylor instability, showing the movement of the interface of the fluids. The contour line represents the interface between the two fluids; the zero contour of the Level Set function. Velocity vectors are displayed showing the flow field. Computational mesh used was 200$\times$200 elements.}
375 \label{RT2D OUTPUT1}
376 \end{figure}
377 %
378 \begin{figure}
379 \center
380 \subfigure[t=1500]{\label{RT OUTPUT1500}\includegraphics[scale=0.252]{figures/RT2D200by200t1500.eps}}
381 \subfigure[t=1800]{\label{RT OUTPUT1800}\includegraphics[scale=0.252]{figures/RT2D200by200t1800.eps}}
382 \caption{Simulation output of Rayleigh-Taylor instability.}
383 \label{RT2D OUTPUT2}
384 \end{figure}
385 %
386 %
387 %The Level Set Method can be applied to many areas of science, for example simulating subduction zones in geophysics, motion of bubbles, and flame propagation. Its also used in image processing. However, the Level Set Method does have limitations. The level set function can still become irregular after reinitialisation, leading to artifacts in the simulations, requiring more thought into the implementation of the reinitialisation step.
388 %

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