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07-05 Fluid Dynamics

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    A topic highly related to the finite element method is computational fluid dynamics--CFD--
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    the study of gases and liquids with the help of the computer.
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    For instance, to find the optimum shape of an airfoil or the optimum shape of a car body
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    or to design hydraulic machinery.
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    I've titled this section "Why CFD is hard,"
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    so now let's look into that.
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    If we were looking at a single particle of mass m that is subject to a force F
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    Then the rate of change of the velocity would be proportional to that force.
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    That's one of Newton's laws--force equals mass times acceleration,
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    which is the rate of change of velocity,
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    the derivative of velocity with respect to time.
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    For the fluid, something similar has to happen,
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    but now we're not dealing with a single particle.
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    We're dealing with a virtually infinite amount of particles.
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    What we are working with is not the velocity of the particle.
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    It's the velocity field.
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    For every position in space, we specify the velocity,
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    so the velocity that we specify is the velocity of that particle
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    at that instant of time.
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    Before and after, most probably, this location is going to be occupied
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    by other particles at other times.
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    When we write down Newton's equation for this particle--
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    force equals mass times the derivative of velocity with respect to time--
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    We have to be a little careful.
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    Let's look at what happens after a very short time step.
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    It's mass times 1 over the time step,
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    and now we have to form the difference of the velocity after that time step
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    minus the velocity before that time step.
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    The before part is easy.
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    That's simply our velocity field at the current time and the current location.
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    The tricky thing is the after part.
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    It's the velocity field at the later time--t plus time step.
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    Now we have to take care of the fact that our particle has moved a little.
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    We don't need the velocity field at that later time.
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    At position x it has to be a slightly different position,
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    namely, how far did we advance?
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    We advanced by time step times velocity.
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    Now, this is going to make things ugly.
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    The velocity field of something that includes the velocity field.
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    A function applied to itself.
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    This is what makes things ugly and eventually leads to computational fluid dynamics
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    and eventually leads to computational fluid dynamics being hard.
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    If we do the math right, this becomes the following.
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    First we have to look into the change of the velocity field with time,
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    so we get its partial derivative with respect to time.
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    But then we also have to look into its change with position,
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    which is the partial derivative with respect to x, for instance.
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    The larger the velocity is, the more effect the spatial derivative has.
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    What we get in the end is the x component of the velocity times the partial derivative
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    of the velocity with respect to x.
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    Of course, the same happens with y and z.
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    This is going to be the acceleration,
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    and this is inherently nonlinear.
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    We have a product of a function that we're looking for--the velocity field--with itself.
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    This is going to make solving the differential equation that results from this really hard.
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    Finally, however, even though the resulting equation--
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    the so-called Navier Stokes equation--is going to look pretty complex,
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    it's nothing else but Newton's law applied to the velocity field.
Cím:
07-05 Fluid Dynamics
Team:
Udacity
Projekt:
CS222 - Differential Equations
Duration:
03:21
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