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# Line 15  Line 15 
15  \section{Rayleigh-Taylor Instability}  \section{Rayleigh-Taylor Instability}
16  \label{LEVELSET CHAP}  \label{LEVELSET CHAP}
17    
18  In this chapter 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. \\  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.    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.  %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.  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.
# Line 24  There is a technique to overcome these l Line 24  There is a technique to overcome these l
24  %  %
25  \begin{figure}  \begin{figure}
26  \center  \center
27  \scalebox{0.7}{\includegraphics{figures/unitcircle.eps}}  \scalebox{0.8}{\includegraphics{figures/unitcircle.eps}}
28  \caption{Implicit representation of the curve $x^2 + y^2 = 1$.}  \caption{Implicit representation of the curve $x^2 + y^2 = 1$.}
29  \label{UNITCIRCLE}  \label{UNITCIRCLE}
30  \end{figure}  \end{figure}
# Line 86  into Equation (\ref{TAYLOR EXPANSION}) Line 86  into Equation (\ref{TAYLOR EXPANSION})
86  \end{equation}  \end{equation}
87    
88    
89  \subsection{Governing Equations for Fluid Flow}  %\subsection{Governing Equations for Fluid Flow}
90    
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:  %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}  %\begin{equation}
94  \nabla \cdot (\eta(\nabla \vec{v} + \nabla^{T} \vec{v})) - \nabla p = -f,  %\nabla \cdot (\eta(\nabla \vec{v} + \nabla^{T} \vec{v})) - \nabla p = -f,
95  \label{GENERAL NAVIER STOKES}  %\label{GENERAL NAVIER STOKES}
96  \end{equation}  %\end{equation}
97  %  %
98  with the incompressibility condition  %with the incompressibility condition
99  %  %
100  \begin{equation}  %\begin{equation}
101  \nabla \cdot \vec{v} = 0.  %\nabla \cdot \vec{v} = 0.
102  \label{INCOMPRESSIBILITY}  %\label{INCOMPRESSIBILITY}
103  \end{equation}  %\end{equation}
104  %  %
105  where $p$, $\eta$ and $f$ are the pressure, viscosity and body forces, respectively.  %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):  %Alternatively, the Stokes equations can be represented in Einstein summation tensor notation (compact notation):
107  %  %
108  \begin{equation}  %\begin{equation}
109  -(\eta(v\hackscore{i,j} + v\hackscore{j,i})),\hackscore{j} - p,\hackscore{i} = f\hackscore{i},  %-(\eta(v\hackscore{i,j} + v\hackscore{j,i})),\hackscore{j} - p,\hackscore{i} = f\hackscore{i},
110  \label{GENERAL NAVIER STOKES COM}  %\label{GENERAL NAVIER STOKES COM}
111  \end{equation}  %\end{equation}
112  %  %
113  with the incompressibility condition  %with the incompressibility condition
114  %  %
115  \begin{equation}  %\begin{equation}
116  -v\hackscore{i,i} = 0.  %-v\hackscore{i,i} = 0.
117  \label{INCOMPRESSIBILITY COM}  %\label{INCOMPRESSIBILITY COM}
118  \end{equation}  %\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}:  %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}  %\begin{equation}
123  \sigma^{'}\hackscore{ij} = 2\eta D^{'}\hackscore{ij},  %\sigma^{'}\hackscore{ij} = 2\eta D^{'}\hackscore{ij},
124  \label{STRESS}  %\label{STRESS}
125  \end{equation}  %\end{equation}
126  %  %
127  where the deviatoric stretching $D^{'}\hackscore{ij}$ is defined as  %where the deviatoric stretching $D^{'}\hackscore{ij}$ is defined as
128  %  %
129  \begin{equation}  %\begin{equation}
130  D^{'}\hackscore{ij} = D^{'}\hackscore{ij} - \frac{1}{3}D\hackscore{kk}\delta\hackscore{ij}.  %D^{'}\hackscore{ij} = D^{'}\hackscore{ij} - \frac{1}{3}D\hackscore{kk}\delta\hackscore{ij}.
131  \label{DEVIATORIC STRETCHING}  %\label{DEVIATORIC STRETCHING}
132  \end{equation}  %\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 $f = -g \rho \delta\hackscore{i3}$.  %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.  %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:  %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}  %\begin{equation}
139  C = \frac{v \delta t}{h}.  %C = \frac{v \delta t}{h}.
140  \label{COURANT}  %\label{COURANT}
141  \end{equation}  %\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}.  %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}.
144    
145    
146  \subsection{Reinitialization of Interface}  \subsection{Reinitialization of Interface}
# Line 199  $\vec{w}$ is the characteristic velocity Line 199  $\vec{w}$ is the characteristic velocity
199  \label{REINITIAL CONVERGE}  \label{REINITIAL CONVERGE}
200  \end{equation}  \end{equation}
201  %  %
202  where $\epsilon$ is the convergence tolerance. Normally, the reinitialization procedure is performed every third time-step of solving the Stokes equation.  where $\epsilon \hackscore{\psi}$ is the convergence tolerance. Normally, the reinitialization procedure is performed every third time-step of solving the Stokes equation.
203    
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:  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  %  %
# Line 212  The accuracy of $\phi$ is only needed wi Line 212  The accuracy of $\phi$ is only needed wi
212  %  %
213  \begin{figure}  \begin{figure}
214  \center  \center
215  \scalebox{0.45}{\includegraphics{figures/LevelSetFlowChart.eps}}  \scalebox{0.5}{\includegraphics{figures/LevelSetFlowChart.eps}}
216  \caption{Flow chart of level set procedure \cite{LIN2005}.}  \caption{Flow chart of Level Set Method procedure \cite{LIN2005}.}
217  \label{LEVELSET FLOWCHART}  \label{LEVELSET FLOWCHART}
218  \end{figure}  \end{figure}
219  %  %
# Line 253  where $\Delta \rho$ is the difference in Line 253  where $\Delta \rho$ is the difference in
253  \end{figure}  \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).  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).
257    
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. The simulation may take a long time to complete on a desktop computer due to the number of elements, so it is recommended to run it on the super computer.    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}  \begin{python}
261    
# Line 304  smooth = alpha*h Line 304  smooth = alpha*h
304  #boundary conditions  #boundary conditions
305  x = mesh.getX()  x = mesh.getX()
306  #left + bottom + right + top  #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] + whereZero(x[1]-l1)*[1.0,1.0]  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]
309    
310  velocity = Vector(0.0, ContinuousFunction(mesh))  velocity = Vector(0.0, ContinuousFunction(mesh))
311  pressure = Scalar(0.0, ContinuousFunction(mesh))  pressure = Scalar(0.0, ContinuousFunction(mesh))
# Line 335  while t_step <= t_step_end: Line 336  while t_step <= t_step_end:
336    #get velocity and pressure of fluid    #get velocity and pressure of fluid
337    Y[1] = -rho*g    Y[1] = -rho*g
338    solution.initialize(fixed_u_mask=b_c,eta=eta,f=Y)    solution.initialize(fixed_u_mask=b_c,eta=eta,f=Y)
339    velocity,pressure=solution.solve(velocity,pressure,max_iter=max_iter,verbose=verbose,useUzawa=useUzawa)    velocity,pressure=solution.solve(velocity,pressure,max_iter=max_iter,verbose=verbose, \
340      useUzawa=useUzawa)
341        
342    #update the interface    #update the interface
343    func = levelset.update_phi(velocity, dt, t_step)      func = levelset.update_phi(velocity, dt, t_step)  
# Line 369  The simulation output is shown in Figure Line 371  The simulation output is shown in Figure
371  \subfigure[t=600]{\label{RT OUTPUT600}\includegraphics[scale=0.252]{figures/RT2D200by200t600.eps}}  \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}}  \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}}  \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$200 elements.}  \caption{Simulation output of Rayleigh-Taylor instability, showing the movement of the interface of the fluids. The contour line represents the interface between the two fluids; the zero contour of the Level Set function. Velocity vectors are displayed showing the flow field. Computational mesh used was 200$\times$200 elements.}
375  \label{RT2D OUTPUT1}  \label{RT2D OUTPUT1}
376  \end{figure}  \end{figure}
377  %  %

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