The occurrences and happenings at Shalbourne Soaring Society. A gliding club near Andover, Newbury and Hungerford.

Sunday (9th June) confusion

The weather guru wasnt sure and had us all confused over what advection was - its simple really  \frac{1}{2} \mathbf{u} \cdot \nabla \mathbf{u} + \frac{1}{2} \nabla (\mathbf{u} \mathbf{u}) = \nabla \left( \frac{\|\mathbf{u}\|^2}{2} \right) + \left( \nabla \times \mathbf{u} \right) \times \mathbf{u} + \frac{1}{2} \mathbf{u} (\nabla \cdot \mathbf{u})
durr!

Anyway despite that we went flying using the standard technique of throwing gliders skyward until they stuck. Variously different pilots added velco , pritstick(TM) , treacle  and drying weetabix to the upper surfaces of their wings until Phil and Carol tried a Cumulus cloud  - this seemed to do the trick! After that most people got a soaring flight, though those practicing circuits also got a few of those...

In addition a guy came to watch for a while:  the chap used to be a member of the club and was in the Pirat syndicate but then he moved to Canada in 1986. You may remember him – his name is Colin Aldridge.
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4 comments:

  1. Anonymous12 June 2013 at 08:04

    I can't believe you actually put the equation for advection on this page, especially with the risk that our members will try to simplify the answer, which I am not sure is the answer. As you are aware this form also makes visible that the skew symmetric operator introduces error when the velocity field diverges. However rather than use your equation let us assume p:R n ×(0,∞)→R is a scalar concentration, u∈R n is the underlying continuous velocity field, x∈R n are the physical coordinates, t is time.

    A typical advection (or transport) equation in absence of diffusion is given by:


    ∂p(x,t) ∂t +∇⋅(p(x,t)u(x))=0


    Further assume velocity field is divergence free. In some circumstances (i.e. when the scalar is passive), the underlying velocity field is not affected by the flow of the scalar. Hence, here u(x) is constant, and the scalar trajectories correspond with the streamlines of the velocity field, which are given as solution of:


    dx dt =u(x)


    Now if we add diffusion, we get:
    ∂p(x,t) ∂t +∇⋅(p(x,t)u(x))=K∇ 2 p(x,t)


    But this can be written as :


    ∂p(x,t) ∂t +∇⋅[p(x,t)[u(x)−K∇p(x,t) p(x,t) ]]=0


    we have made the "effective" velocity field v dependent on scalar p .

    where v(x,t)=u(x)−K∇p(x,t) p(x,t) plays the role of the new 'velocity' field.

    Now p(x,t) represents the scalar transport (advection without diffusion) under this modified velocity field.

    My question is if this approach has been explored in studying the original advection-diffusion equations in the literature. I feel there is additional insight to be gained by this change of viewpoint, although I don't have concrete proof of that.

    What I mean is that now we can study the dynamical system:


    dx dt =v(x),
    and apply techniques from dynamical systems to study fixed points etc. Obviously, the dependence of v on p complicates things here, but the approach seems to me as something useful to pursue.
    Chris.

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  2. Anonymous13 June 2013 at 07:14

    Couldn't have put it better myself :-)

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  3. Anonymous14 June 2013 at 15:24

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  4. Anonymous15 June 2013 at 17:48

    I remember Colin Aldridge - didn't he invent advection? Or was it velcro?

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