Consider a vortex approaching a wall as shown in attached figure. In the second case vortex is near to the wall compared to the first case. So, does the force experienced by the wall is more or less in second case compared to the first one?
Extending this to a lid-driven cavity problem, if I change the cavity shape to make the vortex to bring closer to the top plate, does the top plate experiences more force?
Hi Gaddam,
As you move away from the core of the vortex the velocity diminish, since continuity equation must be satisfied, and hence the velocity gradient induced by the vortex at the wall is smaller when you are further away than when your are closer from the wall. That is, the shear stress exerted by the vortex (or per unit spanwise length in the 2D case) at one point of the wall is tau=mu*du/dy, which you can integrate over the total length of your wall or plate. Intuitively, if your place your vortex at infinity you would expect no perturbation in the fluid --at rest-- near the wall, hence 0 exerted force.
Also note that if you are in 2D you should expect a couple of vortices so circulation sums exactly to zero, and if you are in 3D to a loop vortex. Somewhere in your sketch should appear another vortex. In addition, as you get closer to the wall (at the viscous sublayer: y+
Dear Gaddam,
A vortex core is a low pressure region. In your figure (assuming same vortex in both images) the wall in second case will definitely feel sucked more strongly towards the vortex as the low pressure core in this case is closer to wall and will induce more negative cp on the wall.
Extending from that, in case of the driven cavity flow if you push the vortex closer to the lid on top, the downward force on the lid (suction) will increase.
In fact this concept of near-wall vortex creating stronger suction (more negative cp) on the wall is well utilized in delta-wings in fighter aircraft (e.g. F-16) to boost lift force (see attached figure taken from www.sciencedirect.com).
The insects generate lift using the theory of LEV (Leading Edge Vortex theory) which is nothing but something similar to your figure where a leading edge vortex remains attached to the flapping insect wing surface and the insect is able to generate very efficient lift force. (see attached figure taken from jeb.biologist.org).
Rajeev's answer only give practical examples of the problem posed, but does not answer the original problem posed.
Dear all,
Thank you for you insight. I have something in my mind, please correct me if I am wrong.
Consider a vortex in a closed cavity (say, lid-driven cavity). If a vortex is a low pressure zone, then a high pressure zone is associated with its surroundings. After all, a vortex is a distribution of pressure in a closed cavity (in a lid-driven cavity). In that case, if a vortex moves closer to the lid, it carries a high pressure zone along with it. In that case, does it push the lid away from the cavity? Or consider the lid to be a elastic membrane. Then the membrane will be pulled or pushed towards/away from the cavity?
Thank you.
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