Alex.

Check whether the proposed field satisfies the continuity equation, because necessarily du / dx + dv / dy + dw / dz = 0 (where d are partial derivatives) or nabla (V) = 0.

In the case du / dx = 0 dv / dy = 0 and dw / dz = 2z, then nabla (V) is not null!

The velocity vector resonates with our experience of objects translating from one point to another. It makes sense to us: an object from 'here' moves to 'there', and in doing so the velocity vector that is constructed from the motion must point in the direction of travel.

The vorticity has no real-world analogue in our experience. It is a construct as unreal (and as useful!) as a wavefunction. There is therefore no reason for it to conform to our built-in prejudices about the natural world. The simplest formulation has the vorticity at right-angles as a result of the use of the cross product - you're welcome to invent a new vector quantity that points parallel to the local velocity, yet whose magnitude scales with the curl of the field.

But I see no compelling reason to do so.

Are you sure that velocity and vorticity are always perpendicular "by vector analysis"? Consider the following very simple (counter)example:

u1= F(x2), u2=G(x1), u3=H(x3)

where F and G and H are arbitrary functions. The vorticity has components

w1=du3/dx2-du3/dx2=0

w2=du1/dx3-du3/dx1=0

w3=du2/dx1-du1/dx2= G'(x1)-F'(x2)

therefore, unless F and G are identically zero, velocity and vorticity are not parallel!

The scalar (dot) product between velocity and vorticity defines the helicity which plays an important role in many phenomena (it is an invariant of Euler equation).

line 10: I wrote "not parallel" but I meant "not perpendicular", Sorry.

Vorticity (pseudo-vector) is a "local spin" of the vector field (velocity). Describing its direction with cross-product (hence curl of velocity) is mathematically useful trick for doing vector operations with vorticity. It is the same as asking "why angular velocity (or indeed torque or angular momentum) vector is perpendicular to the plane of rotation?" (hence to the tangential velocity/force/momentum vector).

The physical sense of vorticity could be thought as "local angular velocity" of a vector field (though they are different quantities)...

Thank you for nice question (Mustafa) as well as for nice answers (James, Michele and Alexe).

To Michele: intersting example of vector analysis. But, could you give some "practical example" of your functions (F, G and H) anlysis? Thanks in advance.

Keep going please, I am reading. I am wondering why single elbows (n=1 to infinity) in SAME plane cause Dean Vortices - repeatedly . While a switch to a left or right hand turn change the dean vortices to a full body rotation. This discussion is related.

Dear sirs,

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I'm sorry, but I have to disagree frontally of the response by James, the opinion of Michele and Alex's comment, I would say in a direct way that all of these are wrong. The construction of the vorticity vector is extremely real, has a physical sense, the direction of the velocity vector is orthogonal, there is no way to have vorticity vector that is not orthogonal to speed and is not no trick, it is a physical reality.

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To make clear my observations I have to be kind of long, and come back the origins of the concept of vorticity.

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The reason the squareness between the vectors comes from a physical analogy to the concept of average speed between infinitesimal displacements in a continuous medium. The concept was introduced to solids introduced by Cauchy in "Exercices d'analyze et de physique mathematique" Vol ii 1841 302p Barchilier Paris.

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Cauchy introduce the concept of an average rotational any one point as the only way to modify the rotation, since only introducing a torque, which should be orthogonal to the system, this gives a rotation velocities

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Do not forget that in the rotation, not infinitesimal and finite system with mass, is introduced or removed through a torque and this is induced or subtracted by an orthogonal axis (physical reason of the orthogonality of the rotational).

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Do not forget also that an irrotational movement also if no boundary condition that induces rotation (for example, retardation in the boundary layer), causing a torque particles in the fluid, this does not become rotationally, then the ratio of representation is not arbitrary.

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In fluids, Hermann L. F. von Helmholtz in "Uber Intregrale der Hydrodynamischen Gleichungen welche den Wirbelbewegugen entrsprechen. 1858 Journal für die reine und angewandte Mathematik / Zeitschriftenband p25-55 - (On Integrals of the Hydrodynamical Equations, Which Express Vortex-motion), introduced the concept of vorticity, giving himself the name with vorticity and vorticity tube. .

Helmholtz set the properties of vorticity, as shown on page 25 of the cited work.

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I put one of the parts of the work in German, which has a translation into English though I do not have it, this is one section that shows the need for orthogonality (second paragraph) and other properties of the vorticity vector.

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“Die folgende Untersuchung wird nun lehren , dafs in den Fällen , wo ein Geschwindigkeitspotential existirt , die kleinsten Wassertheilchen keine Rotationsbewegungen haben , wohl aber ist wenigstens ein Theil der Wassertheilchen in Rotation begriffen in solchen Fällen , wo kein Geschwindigkeitpotential existirt .

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Wirbellinien nenne ich Linien , welche durch die Flüssigkeitsmasse so gezogen sind , dass ihre Richtung überall mit der Richtung der augenblicklichen Rotationsaxe der in ihnen liegenden Wassertheilchen zusammentrifft .

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Wirbelfäden nenne ich Theile der Wassermasse , welche man dadurch aus ihr herausschneidet , dass man durch alle Puncte des Harfangs eines endlich kleinen Flachenelements die entsprechenden Wirbellinien construirt .

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1 ) kein Wassertheilchen in Rotation kommt , welches nicht von Anfang an in Rotation begriffen ist .

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2 ) Die Wassertheilchen , welche zu irgend einer Zeit derselben linie angehören , auch indem sie sich fortbewegen , immer zu derselben Wirbellinie gehörig bleiben.

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3 ) Dafs das Product aus dem Querschnitte und der Rotationsgeschwindigkeit eines unendlich dünnen Wirbelfadens längs der ganzen Länge des Fadens constant ist , und auch bei der Fortbewegung des Fadens denselben Werth behält. Die Wirbelfäden müssen deshalb innerhalb der Flüssigkeit in sich zurücklaufen, oder können nur an ihren Grenzen endigen.”

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Anteriorly Sir William Thomson (Lord Kelvin ou Baron Kelvin of Largs ) introduces the same principle in "On the Theories of the Internal Friction of Fluids in Motion, and of the Equilibrium and Motion of Elastic Solids 1845 Camb. Phil. Trans viii paper I 80, but this work of Lord Kelvin is not clearly explained the concept of vortex.

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Later on, quoting the work of Helmholtz, Lord Kelvin publishes specific text on the subject in "On vortex motion" Edin. Trans XXV 1869 papers, iv. 13., Previous observations where the same may be observed.

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The only thing in the arbitrary concept of vorticity vector is the meaning of it, which was adopted to make an incomplete analogy with the work of Maxwell.

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If one works with ElectroMagnetic (EM) phenomena, one quickly becomes familiar with the idea of curls or vorticity.

Most of the EM phenomena work at 90 degrees to each other (e.g. the magnetic field and induced currents in induction heating).

In the real world there are also many examples in fluid mechanics (e.g. vortexes from aircraft wings) and in mechanics as already pointed out by others, i.e. where rotation is generated by gradients in orthogonal fields.

At its simplest, I think about pushing on a vertical cylinder with my fingers. If I push equally on the left and the right side, the cylinder will move sideways. If I push preferentially on one side it will begin to rotate about its axis. This shows that not only do you need an orthogonal applied force, but you need a gradient in that force to generate a rotation. In a homogeneous velocity field you can generate translation, but not rotation.

So in induction heating, for example, a homogeneous field will not generate heating. This is an issue with 'thin' work pieces and slowly changing magnetic fields. Considering a work piece inserted into a simple helical wound coil, if the gradient of the axial magnetic field in the radial direction is negligible, negligible current is induced. The current J =∇XH (ignoring displacement currents). Current in the phi direction thus being the result of the gradient of H in the radial direction.

Taking the curl of the applied vector, analyses the correct gradient(s) and mathematically delivers the direction of the axis of rotation in the same operation. For an applied field with X, Y and Z components, each of these will of course add to the vorticity and the direction of the resulting rotation will not be parallel to any particular axis.

Things start to get confusing and somewhat complicated in time varying fields, where complex calculus is then required.

Rogerio,

I don't think anyone doubts the "existence" of vorticity, which you seem trying to prove. As for "there is no way for vorticity to be not perpendicular" and the example of Michele:

If I construct following velocity field (I use Matlab):

i=0.8:0.1:1;

x1=i;x2=i;x3=i; %coordinates

[X1,X2,X3]=meshgrid(x1,x2,x3); %coordinates 3D

%velocity components depend on coordinates as in Michele example

U=X2.^2; V=X1.^3; W=X3.^2;

%take curl of this velocity field

[CURLX, CURLY, CURLZ]=curl(X1,X2,X3,U,V,W);

%plot velocity

quiver3(X1,X2,X3,U,V,W,3);

hold on

%plot vorticity

quiver3(X1,X2,X3,CURLX,CURLY,CURLZ,3);

And the result is: (blue is velocity, green is curl of velocity field).

One can see these vectors are not perpendicular... but still curl points to the direction of "local spin"

Alex.

Check whether the proposed field satisfies the continuity equation, because necessarily du / dx + dv / dy + dw / dz = 0 (where d are partial derivatives) or nabla (V) = 0.

In the case du / dx = 0 dv / dy = 0 and dw / dz = 2z, then nabla (V) is not null!

That was my first "feeling" of Michele's example - can´t be a real example. Thank you, Rogerio, for reminding of basic Fluid Mechanic laws.

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Dear Štĕpán.

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The problem is that nowadays people are launched in research without studying the basic principles of Fluid Mechanics. They think that taking a velocity field is simply put values in u, v and w function of the coordinates. Forget to check whether or not this represents a field of velocities which satisfy at least more basic equation of fluid mechanics!

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To not fall into errors as seen, the texts of the late nineteenth and early twentieth century beginning with the concept of potential function, and this is both simple verification of continuity as the calculation of the vorticity vector.

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Today it is easy to represent three-dimensional fields (as done above) a cluster of functions in x, y and z, but no one should NEVER verify a definition by a numerical example, because the pictures can be deceiving too.

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Dear Rogerio,

completely agree with you. The CFD is a powerful as well as DANGEROUS tool if one not validate the results on real problems.

Rogerio,

thanks for your nice reminder of Fluid Mechanics fundamentals.

Indeed, my example had divergence, but that was not the point Michele was making.

If you want to restrict vorticity to be valid only for divergence-free flows (it that so?), here you can have nice divergence-free field with non-zero helicity :

U=-y, V=x; W=const. div(U,V,W)=0 (I hope you agree on that)

Vorticity will have only Z-component (I hope you agree on that). But velocity everywhere also has Z-component (constant).

So velocity vector isn't perpendicular to the vorticity in this case either, even the field is divergence-free.

Shall I restrain myself of adding a picture ...? No...:)

If a field has any component in the direction of its curl, the two obviously cannot be perpendicular.

Vorticity is not necessarily perpendicular to velocity in a three-dimensional flow field (as shown by Alex). However, you cannot construct any two-dimensional fields where they are not perpendicular.

I am very glad that I asked this question. It has been very educating. Scientific discussion is very powerfull tool of learning. Thanks to everyone who contributes with a question or an answer.

Alex.

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Actually we're both wrong and we are right. We did not realize we were talking about two things, I based on physical principles and thou in mathematical operations. Gone unnoticed that we are discussing in R ³, so the physical concept of vorticity which is employed in R ² have to be modified to be able to correlate this spinning vortex stretching and energy dissipation.

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What you did creating a constant term speed was simply a translation of the axis system, for example, if the speed of translation in Z were A or 2A your example would double the vorticity, contradicting the principle of Galilean relativity where laws physical must be described independent of a rectilinear and uniform motion. The math is right but the principle is completely wrong. In your example, over a two-dimensional irrotational field was added a constant and uniform speed, and this led to aggregation from zero to a finite value to rotatability of the flow.

Rogerio,

with all respect, but "... if the speed of translation in Z were A or 2A your example would double the vorticity, contradicting the principle of Galilean relativity" -- IT WON'T , AS Z-COMPONENT OF CURL (THE ONLY COMPONENT OF VORTICITY) DOES NOT DEPEND ON z-component of velocity W...

I wanted to show that the particular field (1st or 2nd or any other) is not too much important, what important is simply "when helicity of the field is not zero, vorticity and velocity are not-perpendicular" . Indeed when a solenoidal field is convected without diffusion (like in my last example) there is a helicity.

There are plentiful "real" examples of such flows: "..Taylor-Gortler vortices, leading edge and trailing vortices shed from wings and slender bodies, streamwise vortices in boundary layers and free shear flows, Langmuir circulations in the ocean, and analogous structures in the atmosphere--tornados and rotating storms.." (H.K.Moffat. Annu. Rev. Fluid Mech. 1992. 24 : 281-312) ...finally Beltrami fields where vorticity and velocity are actually colinear.

Thanks for discussion!

...the link

Article Helicity in Laminar and Turbulent Flow

They may be parallel as in the Arnold-Beltrammy-Childress flow.

Curl of a vector with three components in the direction 3 orthogonal spatial direction is NOT ALWAYS perpendicular to the vector. This can be easily verified by calculating curl for the vector A = \hat{X} A_x(x, y, z)+ hat{y} A_y(x, y, z)+ hat{z} A_zx, y, z) and taking dot product of the vector A with Curl A - which will be non-zero in general

the reason is    

 u.curl u = det(grad,u,u) = - det(grad,u,u)

because the determinant is antisymmetric  formally => u.curl u = 0

for 3d  vector A,B,C  by definition you have

(A x B) .  C = det(A,B,C)

and

curl A = grad x A  , with grad = (d_1, d_2,d_3) vector operator

u. curl u = \sum_ {i,j,k) \in S_3  signe (i,j,k) d_i u_j,u_k  

             = -= \sum_ {i,j,k) \in S_3 ( -signe (i,k,j) ) d_i u_k,u_j

             => = 0.  

where S_3 is set of permutation of order 3.