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Gil Forsyth committed Feb 12, 2015
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60 changes: 39 additions & 21 deletions lessons/05_Lesson05_InfiniteRowOfVortices.ipynb
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"This time, you'll be on your own creating the code for this flow!\n",
"The objective of this assignement is to visualize the streamlines around an infinite row of vortices. First, you will consider the case of a finite number of vortices, obtained by simple superposition. By adding more and more vortices, you should be able to see how the flow pattern approaches that of an infinite row of vortices. But there will always be some differences (pay attention to what these may be).\n",
"\n",
"The objective of this assignment is to visualize the streamlines induced by an infinite row of vortices.\n",
"Because the computer does not enable the passage to the limit of an infinite number, we first consider the case of a finite number of vortices.\n",
"It's possible to derive an analytical expression for the infinite case, and the derivation is provided below. With this analytical expression, you can visualize the streamlines for the infinite case. Observe and think: how are the streamlines different from one case to the other?\n",
"\n",
"However, it is possible to derive an analytical expression for the infinite case.\n",
"\n",
"Are the streamlines different from one case to the other?\n",
"\n",
"\n",
"In this notebook, their is no Python code... not yet! Your job is to read the theory, to think how you could implement it in an efficient manner and finally to code it and visualize the results."
"In this notebook, there is no Python code. Your job is to study the theory (and follow the mathematics on your own handwritten notes), to think how you could implement it in an efficient manner and finally to code it and visualize the results."
]
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{
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"As seen in a previous lesson, a vortex of strength $\\Gamma$ has a stream-function\n",
"You might not suspect it, but the vortex has a very important role in classical aerodynamic theory. You'll discover some of its uses in this assignment.\n",
"\n",
"First, a little review of the basics. As seen in a previous lesson, a vortex of strength $\\Gamma$ has a stream-function:\n",
"\n",
"$$\\psi\\left(r,\\theta\\right) = \\frac{\\Gamma}{2\\pi}\\ln r$$\n",
"\n",
"and a velocity potential\n",
"\n",
"$$\\phi\\left(r,\\theta\\right) = -\\frac{\\Gamma}{2\\pi}\\theta$$\n",
"\n",
"From this we can derive the velocity components in a polar coordinate system\n",
"We can now derive the velocity components in a polar coordinate system, as follows:\n",
"\n",
"$$u_r\\left(r,\\theta\\right) = 0$$\n",
"\n",
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"In a Cartesian system, the velocity components at $\\left(x,y\\right)$ induced by a vortex, of strength $\\Gamma$ located at $\\left(x_\\text{vortex},y_\\text{vortex}\\right)$, are given by\n",
"In a Cartesian coordinate system, the velocity components at $\\left(x,y\\right)$ around a vortex of strength $\\Gamma$ located at $\\left(x_\\text{vortex},y_\\text{vortex}\\right)$, are given by\n",
"\n",
"$$u\\left(x,y\\right) = +\\frac{\\Gamma}{2\\pi}\\frac{y-y_\\text{vortex}}{(x-x_\\text{vortex})^2+(y-y_\\text{vortex})^2}$$\n",
"\n",
"$$v\\left(x,y\\right) = -\\frac{\\Gamma}{2\\pi}\\frac{x-x_\\text{vortex}}{(x-x_\\text{vortex})^2+(y-y_\\text{vortex})^2}$$\n",
"\n",
"and the stream-function is\n",
"and the stream-function is written as\n",
"\n",
"$$\\psi\\left(x,y\\right) = \\frac{\\Gamma}{4\\pi}\\ln\\left((x-x_\\text{vortex})^2+(y-y_\\text{vortex})^2\\right)$$"
]
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"Adding many vortices"
"Superposition of many vortices"
]
},
{
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"\n",
"where $\\left(x_i,y_i\\right)$ are the Cartesian coordinates of the $i^{\\text{th}}$ vortex.\n",
"\n",
"(In the next section, we'll show you how to derive a closed-form expression for the *infinite* sum of vortices. But hang on.)\n",
"Here is a diagram of the situation:\n",
"\n",
"\n",
"<center>![image](files/resources/infiniteRowVorticesSketch.png)</center>\n",
"\n",
"<center>![image](files/resources/infiniteRowVorticesSketch.png)</center>"
"\n",
"(In the next section, we'll show you how to derive a closed-form expression for the *infinite* sum of vortices. But hang on.)"
]
},
{
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"source": [
"##### Task:\n",
"\n",
"#### Compute the velocity field and plot the streamlines of a row of vortices"
"#### Compute the velocity field and plot the streamlines of a row of vortices."
]
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"You will place a $N$ vortices aligned on the horizontal axis and visualize the flow pattern created. Do the following:\n",
"You will place $N$ vortices aligned on the horizontal axis and visualize the flow pattern. Do the following:\n",
"\n",
"* Using the equations derived above, compute the velocity components of each vortex on a mesh grid.\n",
"\n",
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"source": [
"Now that we have derived the functions for the velocity components, implement them in a code cell and plot the streamlines. \n",
"\n",
"Can you see any difference with the previous task where the number of vortices was finite?"
"*Can you notice the differences with the previous case where the number of vortices was finite?*\n",
"\n",
"Play around with your plotting settings for the *finite* row of vortices, until you can get a plot that looks more like the infinite case. When can you say the finite case is a good approximation of the infinite case?"
]
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"#####Think"
]
},
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"Notice that the streamline pattern is parallel to the *vortex sheet*: there is no net normal flow. Therefore, the sheet can model a solid surface in potential flow. We end up with a slip velocity at such an interface: how is this consistent with potential flow? "
]
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