I would like to evaluate the strength of vortex in lid driven cavity flow. i am simulating the cases in Fluent. Although stream function (m^2/s)indicates the strength of vortex, I want to know vortex strength in Joules. How is it possible?
Here is my take based on dimensional analysis:
Vortex is a part of continuum that has a tendency for spinning motion indicated by a non-zero vector quantity called vorticity it possesses. Vorticity vector is given as the curl of the velocity field and its direction is given by the right hand rule.
Another quantity called circulation which is line integral of velocity along a closed path is basically the strength of vortex. Circulation along a closed curve in a flow field is related to vorticity through Stokes Equation i. e. Circulation along a curve enclosing an infinitesimal area having normal vector n is equal to the dot product of vorticity vector and the area vector.
So Circulation = Vorticity · Area and has dimensions of L2T-1. If you know the mass flow rate in the flow (having dimensions of MT-1 ). The product of mass flow rate and circulation = Energy (ML2T-2).
So if you have a flow with mass flow rate of X kg/s in a flow with circulation of Y m2/s, then the vortical flow has a strength equivalent to XY kgm2s-2 = XY Joules.
(Hope the dimensional analysis is accurate)
As pointed out by Aziz, measuring vortex strength in terms of energy is not obvious nor sufficiently unique. A standard approach would be to measure it in terms of second invariants of velocity gradient, usually referred to as Q.
This quantity tells you how much the rotational part (the antisymmetric) part of your velocity gradient tensor is dominating over the strain part (the symmetric part).
To do that, it compares the norms of these tensor: if Q>0, rotation is dominating over strain and vice-versa.
I attach the derivation of Q for a 2D flow. As you can see, it only involves first order derivatives of the flow field. In the second attachment, you can see a practical example of using Q to determine (and locate) large scale recirculation structures.
Best Regards,
Miguel
Conference Paper Quantitative flow visualization of confinement-driven instab...
Here is my take based on dimensional analysis:
Vortex is a part of continuum that has a tendency for spinning motion indicated by a non-zero vector quantity called vorticity it possesses. Vorticity vector is given as the curl of the velocity field and its direction is given by the right hand rule.
Another quantity called circulation which is line integral of velocity along a closed path is basically the strength of vortex. Circulation along a closed curve in a flow field is related to vorticity through Stokes Equation i. e. Circulation along a curve enclosing an infinitesimal area having normal vector n is equal to the dot product of vorticity vector and the area vector.
So Circulation = Vorticity · Area and has dimensions of L2T-1. If you know the mass flow rate in the flow (having dimensions of MT-1 ). The product of mass flow rate and circulation = Energy (ML2T-2).
So if you have a flow with mass flow rate of X kg/s in a flow with circulation of Y m2/s, then the vortical flow has a strength equivalent to XY kgm2s-2 = XY Joules.
(Hope the dimensional analysis is accurate)
Dear Rajeev,
Detecting vortices from vorticity is not a good method.
Take, for example, a simple 2d flow having no vertical component and a parabolic horizontal component. Let's say u=a*Y^2. If you take the curl, it easy to see that you will have a linearly growing function of the Y. High vorticity does not necessarily means vortex; it might well mean high "cross" gradients.
On the other hand you can have "vortices" even if the curl is zero. That's the very well known case of irrotational vortex, where the velocity decreases as 1/r, with r the distance from the origin: sufficiently far from the origin, the curl is zero.
Using the circulation can make things more complex as you need to define the path, and that will surely not be easy nor unique.
I suggest to have a look at this paper
Miguel
Article An objective definition of a vortex
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