vortex stretching
 

Hello, I want to know what is vortex stretching why it is 3D what we mean by turbulence scale what is skin fraction thanks

Re: vortex stretching
 

Hello Miss Saba18,

I guess you are looking into a 3D incompressible flow, where the vorticity (per unit mass - vortencity) is conserved (the flow is e.g. polytropic, isothermal or barotropic - i.e. no baroclinic instability to generate vorticity). In that case because of the conservation of vortencity, as a 3D vortex (looking like a tornado for example) is elongated it gets thinner (like stretching). This is similar to conservation of angular momentum in physics of rigid bodies. The rotational velocity also increases as it gets thinner. You can have 2D vortices (in a plane, e.g. in the atmosphere of Jupiter) and they are streched due to the differential rotation and strong winds. Then the vortices, instead of being circular, take an oval form - I believe that's not what you are interested in.

I do not know what skin fraction is, I suspect it has to do with boundary layer size... Or with the outer edge of a vortex?.. I'll leave here the place for someone else to help you.

Cheers, Patrick

Re: vortex stretching
 

hello I read that vortex stretching is a 3D phenomena that is why turbulence is 3D. If we have vortices in 2D so why not vortex stretching is 2D. What is baroclinic instability, you use word vorticity per unit mass, its mean we may measure vorticity. What is the measurment of vorticity, dia of vortex or number of vortex.

Re: vortex stretching
 

Basically vortex stretching is the term on the right hand side of the vorticity equation

Dw/Dt = w.grad u, w= curl(u)

which allows, in 3D since in 2D the right-hand-side is zero implying pointwise conservation of vorticity, the vorticity to change along a particle path (providing it's not zero!). If you now consider a vortex tube, the circulation about which is conserved as it is convected by the fluid, then the vortex stretching term describes the stretching/compression of the vortex tube. The reason why vortex-stretching is important is that if a vortex tube is stretched it's cross-sectional area reduces and so the vorticity of the tube must increase in order to preserve the circulation; i.e. w.dA is constant. - This is believed to be on of the mechanism by which solutions of the Euler equations can fail to exist at a finite time.

Skin friction is the surface stress; if n is the unit normal to the surface, nu.D is the stress-tensor and s is a unit vector in the tangent plane of the surface then nu.(Dn).s is the component of skin friction in the s-direction.

Re: vortex stretching
 

From a turbulence point of view, the term of vortex stretching is sometimes used when explaining the production of turbulence by the mean velocity gradient. It describes exactly what Patrick explains but at turbulence scale.

Turbulence scale is the length scale of turbulence. It is not far from the energy containing eddies radius.

At last "skin friction" = "wall friction" and u+ is the velocity skin friction.

Sylvain

Re: vortex stretching
 

Thanks Ananda,

for your detailed explanation. I am not sure John Chien would have liked it... but i am sure what you wrote is correct, that is, in it very own context.

Cheers, Patrick

Re: vortex stretching
 

As Tom wrote, the vorticity is the curl of the velocity and quantifies the 'amount of rotation' of the flow. When you take the curl of the flow equations you get an equation for the voticity (again see Tom's answer) and on the right hand side of the equation for the vorticity there is a source term that can generate vorticity. This source term is composed of the gradient of the pressure curl the gradient of the density times some other quantities. If the gradient of the pressure and the gradient of the density are parallel (aligned) then this term vanishes (because the curl of two vectors parallel is zero). The gradient of the pressure defines the lines of constant pressure, and the same for the density. So this term is zero when the lines of constant pressure are aligned to the lines of constant density, and the flow is called 'baropropic' (this is the case in isothermal flow or more generally polytropic flow, or when the pressure is just a function of the density - barotropic flows). When this is not the case the source term on the right hand side of the equations for the vorticity is non zero and can generate vorticity. Such a flow is called baroclinic and it is subject to the baroclinic instability, namely the generation of vorticity by the source term. A baroclinic flow is one where the lines of constant pressure (isobars) are not parallel to the lines of constant density.

The quantity that remains constant in a barotropic flow is the volume integral of the vorticity over the entire domain. Yes, you can measure invidual vortices by checking their vorticity, curl v, not the opposite.

Cheers, Patrick

Re: vortex stretching
 

Patrick,

strictly speaking the vortex stretching term is the w.grad u term where w is the vorticity (along with the additional contribution w.div u for compressible flow) and is only active in 3D flows. The term you are referring to is the baroclinic forcing term which can, as you've pointed out, be a source of vorticity (although the Ertel Potential vorticity will be still conserved) even in 2D flows. These two mechanisms for the production of vorticity are distinct and should not be confused.

To understand why the w.grad u term is called the vortex stretching term consider a cylindrical tube of fluid. Assume that initially the vorticity points along the axis of the cylinder and that the axial component of velocity is increasing in this direction. Then the vortex stretching term will tend to stretch the cylinder with the consequence (conservation of circulation) that the vorticity of the cylinder is amplified. It's this amplification of vorticity that is important in turbulent flow.

Tom.

Re: vortex stretching
 

what is the physical defination of circulation, and what is vortex tube. Is there any difference between vorticity (Curl of velocity), rotational flow ( rotation of fluid partical about an axis perpendicular to rotation of flow), vortex and vortices.

Re: vortex stretching
 

what is the physical defination of circulation, and what is vortex tube. Is there any difference between vorticity (Curl of velocity), rotational flow ( rotation of fluid partical about an axis perpendicular to rotation of flow), vortex and vortices.

Re: vortex stretching
 

I think you need to read a book on fluid dynamics (such as Elementary fluid dynamics by D.Acheson)!

Circulation is the line integral of the vorticity about a simple closed circuit within the fluid.

A vortex tube is like a streamtube but instead of the surface of the tube being made up of streamlines it's made up of vortex lines. (A vortex line is a line in the fluid which is parallel to the local vorticity vector).

Vorticity shouldn't be confused with rotation (it is not angular momentum! since an infinitessimal body may rotate in an irrotational flow - if the body in question is a sphere it's angular momentum will however be half the vorticity),

Tom.