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2D fault systems are now working. 3D still needs work.

 1 lgraham 2191 2 \section{Stokes Flow} 3 \label{STOKES FLOW CHAP} 4 5 lgraham 2192 In this section we will look at Computational Fluid Dynamics (CFD) to simulate the flow of fluid under the influence of gravity. The StokesProblemCartesian class will be used to calculate the velocity and pressure of the fluid. 6 lgraham 2191 The fluid dynamics is governed by the Stokes equation. 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: 7 % 8 \begin{equation} 9 \nabla \cdot (\eta(\nabla \vec{v} + \nabla^{T} \vec{v})) - \nabla p = -f, 10 \label{GENERAL NAVIER STOKES} 11 \end{equation} 12 % 13 with the incompressibility condition 14 % 15 \begin{equation} 16 \nabla \cdot \vec{v} = 0. 17 \label{INCOMPRESSIBILITY} 18 \end{equation} 19 % 20 where $p$, $\eta$ and $f$ are the pressure, viscosity and body forces, respectively. 21 Alternatively, the Stokes equations can be represented in Einstein summation tensor notation (compact notation): 22 % 23 \begin{equation} 24 -(\eta(v\hackscore{i,j} + v\hackscore{j,i})),\hackscore{j} - p,\hackscore{i} = f\hackscore{i}, 25 \label{GENERAL NAVIER STOKES COM} 26 \end{equation} 27 % 28 with the incompressibility condition 29 % 30 \begin{equation} 31 -v\hackscore{i,i} = 0. 32 \label{INCOMPRESSIBILITY COM} 33 \end{equation} 34 % 35 The subscript comma $i$ denotes the derivative of the function with respect to $x\hackscore{i}$. 36 %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}: 37 % 38 %\begin{equation} 39 %\sigma^{'}\hackscore{ij} = 2\eta D^{'}\hackscore{ij}, 40 %\label{STRESS} 41 %\end{equation} 42 % 43 %where the deviatoric stretching $D^{'}\hackscore{ij}$ is defined as 44 % 45 %\begin{equation} 46 %D^{'}\hackscore{ij} = D^{'}\hackscore{ij} - \frac{1}{3}D\hackscore{kk}\delta\hackscore{ij}. 47 %\label{DEVIATORIC STRETCHING} 48 %\end{equation} 49 % 50 %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 ($i \neq j$). 51 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 $f = -g \rho \delta\hackscore{i3}$. 52 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; more detail on the class can be view in Chapter \ref{MODELS CHAPTER}. 53 In order to keep numerical stability, the time-step size needs to be kept below a certain value, to satisfy the Courant condition. The Courant number is defined as: 54 % 55 \begin{equation} 56 C = \frac{v \delta t}{h}. 57 \label{COURANT} 58 \end{equation} 59 % 60 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 time-step size was calculated for a Courant number of 0.4. 61 62 lgraham 2193 The following PYTHON script is the setup for the Stokes flow simulation, and is available in the example directory as 'fluid.py'. It starts off by importing the classes, such as the StokesProblemCartesian class, for solving the Stokes equation and the incompressibility condition for velocity and pressure. Physical constants are defined for the viscosity and density of the fluid, along with the acceleration due to gravity. Solver settings are set for the maximum iterations and tolerance; the default solver used is PCG. The mesh is defined as a rectangle, to represent the body of fluid. The gravitational force is calculated base on the fluid density and the acceleration due to gravity. The boundary conditions are set for a slip condition at the base of the mesh; fluid movement in the x-direction is free, but fixed in the y-direction. An instance of the StokesProblemCartesian is defined for the given computational mesh, and the solver tolerance set. Inside the while loop, the boundary conditions, viscosity and body force are initialized. The Stokes equation is then solved for velocity and pressure. The time-step size is calculated base on the Courant condition, to ensure stable solutions. The nodes in the mesh are then displaced based on the current velocity and time-step size, to move the body of fluid. The output for the simulation of velocity and pressure is then save to file for visualization. 63 lgraham 2191 % 64 \begin{python} 65 from esys.escript import * 66 import esys.finley 67 from esys.escript.linearPDEs import LinearPDE 68 from esys.escript.models import StokesProblemCartesian 69 70 #physical constants 71 eta=1.0 72 rho=100.0 73 g=10.0 74 75 #solver settings 76 tolerance=1.0e-4 77 max_iter=200 78 t_end=50 79 t=0.0 80 time=0 81 gross 2580 verbose=True 82 lgraham 2191 83 #define mesh 84 H=2.0 85 L=1.0 86 W=1.0 87 mesh = esys.finley.Rectangle(l0=L, l1=H, order=2, n0=20, n1=20) 88 coordinates = mesh.getX() 89 90 #gravitational force 91 Y=Vector(0.0, Function(mesh)) 92 Y=-rho*g 93 94 #element spacing 95 h=Lsup(mesh.getSize()) 96 97 #boundary conditions for slip at base 98 boundary_cond=whereZero(coordinates)*[0.0,1.0] 99 100 #velocity and pressure vectors 101 velocity=Vector(0.0, ContinuousFunction(mesh)) 102 pressure=Scalar(0.0, ContinuousFunction(mesh)) 103 104 #Stokes Cartesian 105 solution=StokesProblemCartesian(mesh) 106 solution.setTolerance(tolerance) 107 108 while t <= t_end: 109 110 print " ----- Time step = %s -----"%( t ) 111 print "Time = %s seconds"%( time ) 112 113 solution.initialize(fixed_u_mask=boundary_cond,eta=eta,f=Y) 114 lgraham 2193 velocity,pressure=solution.solve(velocity,pressure,max_iter=max_iter, \ 115 gross 2580 verbose=verbose) 116 lgraham 2191 117 print "Max velocity =", Lsup(velocity), "m/s" 118 119 #Courant condition 120 dt=0.4*h/(Lsup(velocity)) 121 print "dt", dt 122 123 #displace the mesh 124 displacement = velocity * dt 125 coordinates = mesh.getX() 126 mesh.setX(coordinates + displacement) 127 128 time += dt 129 130 vel_mag = length(velocity) 131 132 #save velocity and pressure output 133 saveVTK("vel.%2.2i.vtu"%(t),vel=vel_mag,vec=velocity,pressure=pressure) 134 t = t+1.0 135 136 \end{python} 137 lgraham 2193 % 138 lgraham 2191 The results from the simulation can be viewed with \mayavi, by executing the following command: 139 % 140 \begin{python} 141 mayavi -d vel.00.vtu -m SurfaceMap 142 \end{python} 143 % 144 lgraham 2193 Colour coded scalar maps and velocity flow fields can be viewed by selecting them in the menu. The time-steps can be swept through to view a movie of the simulation. 145 lgraham 2192 Figures \ref{FLUID OUTPUT1} and \ref{FLUID OUTPUT2} shows the simulation output. Velocity vectors and a colour map for pressure are shown. As the time progresses the body of fluid falls under the influence of gravity. 146 lgraham 2193 % 147 gross 2654 \begin{figure}[ht] 148 lgraham 2191 \center 149 jfenwick 2335 \subfigure[t=1]{\label{FLOW OUTPUT 01}\includegraphics[scale=0.25]{figures/stokes-fluid-t01}} 150 \subfigure[t=20]{\label{FLOW OUTPUT 10}\includegraphics[scale=0.25]{figures/stokes-fluid-t10}} 151 \subfigure[t=30]{\label{FLOW OUTPUT 20}\includegraphics[scale=0.25]{figures/stokes-fluid-t20}} 152 \includegraphics[scale=0.25]{figures/stokes-fluid-colorbar} 153 lgraham 2191 \caption{Simulation output for Stokes flow. Fluid body starts off as a rectangular shape, then progresses downwards under the influence of gravity. Color coded distribution represents the scalar values for pressure. Velocity vectors are displayed at each node in the mesh to show the flow field. Computational mesh used was 20$\times$20 elements.} 154 \label{FLUID OUTPUT1} 155 \end{figure} 156 lgraham 2193 % 157 gross 2654 \begin{figure}[ht] 158 lgraham 2191 \center 159 jfenwick 2335 \subfigure[t=40]{\label{FLOW OUTPUT 30}\includegraphics[scale=0.25]{figures/stokes-fluid-t30}} 160 \subfigure[t=50]{\label{FLOW OUTPUT 40}\includegraphics[scale=0.25]{figures/stokes-fluid-t40}} 161 \subfigure[t=60]{\label{FLOW OUTPUT 50}\includegraphics[scale=0.25]{figures/stokes-fluid-t50}} 162 \includegraphics[scale=0.25]{figures/stokes-fluid-colorbar} 163 lgraham 2191 \caption{Simulation output for Stokes flow.} 164 \label{FLUID OUTPUT2} 165 \end{figure} 166 lgraham 2193 % 167 gross 2371 The view used here to track the fluid is the Lagrangian view, since the mesh moves with the fluid. One of the disadvantages of using the Lagrangian view is that the elements in the mesh become severely distorted after a period of time and introduce solver errors. To get around this limitation the Level Set Method can be used, with the Eulerian point of view for a fixed mesh. 168 %The Level Set Method is discussed in Section \ref{LEVELSET CHAP}.