[granzer]

Why do vortices undergo stretching, i.e. what causes vortices to stretch?

[LuckyTran]

Nothing but the conservation laws. It falls right out when you write down (i.e derive) the vorticity transport equation.

Let us presume that there is a 3D flowfield with some vorticity in it, which implies some form of velocity gradient. Velocity gradients means (there are differences in velocity) and different layers of fluid will tend to move at different speeds. However, angular momentum must be conserved. The conservation of angular momentum as the fluid in different speed-layers move automatically yields the longitudinal vortex stretching.

[granzer]

Originally Posted by LuckyTran

Nothing but the conservation laws. It falls right out when you write down (i.e derive) the vorticity transport equation.

Let us presume that there is a 3D flowfield with some vorticity in it, which implies some form of velocity gradient. Velocity gradients means (there are differences in velocity) and different layers of fluid will tend to move at different speeds. However, angular momentum must be conserved. The conservation of angular momentum as the fluid in different speed-layers move automatically yields the longitudinal vortex stretching.

Thank you. Busy how does product of vorticity along a direction and velocity gradient in that same direction give change of vorticity? I get that (positive) velocity gradient along a stream line causes static pressure to decrease and a elongate in (assume incompressible) fluid to elongate. This elongation causes (due to conservation of angular momentum) vorticity to increase. But i am not able to intutivey understand how the product of vorticity and velocity gradient is giving rate of change of vorticity. In a way velocity gradient can be seen as change in length per unit length per unit time...but how multiplying this to vorticity gives rate of change.of vorticity is lost on me.

[sbaffini]

Have you seen this?

https://en.m.wikipedia.org/wiki/Vortex_stretching

I think it is pretty clear about it

[FMDenaro]

Originally Posted by granzer

Thank you. Busy how does product of vorticity along a direction and velocity gradient in that same direction give change of vorticity? I get that (positive) velocity gradient along a stream line causes static pressure to decrease and a elongate in (assume incompressible) fluid to elongate. This elongation causes (due to conservation of angular momentum) vorticity to increase. But i am not able to intutivey understand how the product of vorticity and velocity gradient is giving rate of change of vorticity. In a way velocity gradient can be seen as change in length per unit length per unit time...but how multiplying this to vorticity gives rate of change.of vorticity is lost on me.

First, the stretching of vorticity does not mean directly that a vortices is deformed.
If you read the fluid dynamics texbook of Chorin you understand you need to think about a vortical tube, that is a tube having the vorticity vector field tangent to the surface. Then you have the conservation of vorticity as same as the conservation of mass. Action of the stretching of vorticity on the vortical tube can be identified.

Note: the stretching term can be analysed in term of matrix operator on a vector. You have the matrix in the velocity gradient acting on the vector that is the vorticity. Express in terms of eigenvectors problem.

[granzer]

Originally Posted by sbaffini

Have you seen this?

https://en.m.wikipedia.org/wiki/Vortex_stretching

I think it is pretty clear about it

Thank you and yes I did have a look at it before but it did not clear my doubt. Also, the article is very basic. The dot product of vorticity vector with velocity gradient tensor, the so called vortex stretching term of vorticity equation, has vortex stretching components and votex turning components. The wiki article only mentions about the stretching even thought they have written the 'vortex stretching term'.

[granzer]

Originally Posted by FMDenaro

First, the stretching of vorticity does not mean directly that a vortices is deformed.
If you read the fluid dynamics texbook of Chorin you understand you need to think about a vortical tube, that is a tube having the vorticity vector field tangent to the surface. Then you have the conservation of vorticity as same as the conservation of mass. Action of the stretching of vorticity on the vortical tube can be identified.

Note: the stretching term can be analysed in term of matrix operator on a vector. You have the matrix in the velocity gradient acting on the vector that is the vorticity. Express in terms of eigenvectors problem.

Ok...maybe the way I am thinking about vorticity is wrong...when I think of vorticity o am thinking of a infinitesimal 'vorticity' element spinning inside each point of the vortex tube(with direction along vortex lines,both inside and along the boundary of the tube, all pointed in same direction)and as it's diameter reduce s (due to stretching) it's spinning increases due to conservation of angular momentum. Thank you for referring the book...I will look into it.

[FMDenaro]

Originally Posted by granzer

Ok...maybe the way I am thinking about vorticity is wrong...when I think of vorticity o am thinking of a infinitesimal 'vorticity' element spinning inside each point of the vortex tube(with direction along vortex lines,both inside and along the boundary of the tube, all pointed in same direction)and as it's diameter reduce s (due to stretching) it's spinning increases due to conservation of angular momentum. Thank you for referring the book...I will look into it.

You have to think that a vortical tube is the envelope of the vorticity field, therefore the stretching of the vorticity affects the vortical tube. This is the link you missed. Not only conservation of angular moment but also the fact that the vorticity field is divergence-free is useful.

[granzer]

Originally Posted by FMDenaro

You have to think that a vortical tube is the envelope of the vorticity field, therefore the stretching of the vorticity affects the vortical tube. This is the link you missed. Not only conservation of angular moment but also the fact that the vorticity field is divergence-free is useful.

Wow thank you! So, if there is viscosity then the vorticity tube expands due to viscous diffusion of vorticity(and die out due to dissipation)? And in an inviscid fluid ( I am assuming a initial vorticity already present)the vorticity tube will just contract due to stretching ( and also twist and turn due to off diagonal velocity gradient terms? Side question: can you please tell me if the off diagonal terms of the velocity gradient still be present if there is no viscosity? ) and this stretching will go on forever?

[FMDenaro]

Originally Posted by granzer

Wow thank you! So, if there is viscosity then the vorticity tube expands due to viscous diffusion of vorticity(and die out due to dissipation)? And in an inviscid fluid ( I am assuming a initial vorticity already present)the vorticity tube will just contract due to stretching ( and also twist and turn due to off diagonal velocity gradient terms? Side question: can you please tell me if the off diagonal terms of the velocity gradient still be present if there is no viscosity? ) and this stretching will go on forever?

Assume a vortical tube of characteristic diameter d. Now only if d is comparable to the Taylor microscale the viscosity starts to have effects until d is of the order of the Kolmogorov lenght when the vortical structures disappear. Only in this range the energy dissipation takes place.
You understand that the stretching is responsible of deformation of the structures in an inviscid-like manner until d is small enough. That means you have an inviscid energy transfer also in viscous flows.