Does vorticity "induces" velocity?

Eddies, vortices, whirlpools, tornadoes, and hurricanes; all of these fluidic structures have the same thing in common: they all rotate around a point by a vectorial quantity (which may vary with time). This is where the term "vorticity" comes in. Mathematically, vorticity is defined as the curl of the velocity: 𝜔(r, t)=∇×𝑢(r, t). Since the velocity field depends on its position (r) and time (t), vorticity also does. Then, vorticity within a vortex can be explained as a measure of the rotation of a flow element as it moves along a closed (circular) streamline about an axis, maintaining a circumferential translation. In an ideal (incompressible, irrotational and inviscid) vortex, all flow elements remain without local rotation, where they are still irrotational (𝜔=0 everywhere, except along the axis).

Fig. 1 Spiral vorticity. Unlike solid-body rotation, the tangential velocity decreases with increasing distance from the vortex center. Source: https://64.media.tumblr.com/3d03d0003b6ad142b69af8d310811934/tumblr_pv4tui7H5x1y9iby6o2_540.gif

There is another term associated with vorticity that is more abstract due to its mathematical (rather than physical) nature: "circulation" (Γ). This term always appears in connection with the generation of lift theory, for example in the Kutta-Zhukovskiy theorem, where the lift (L) is simply obtained by the flow density (ρ) times the magnitude of the free-stream velocity (U) times the circulation: L=ρUΓ. This simple definition allows to obtain a fundamental measure in aerodynamics (e.g., lift on a two-dimensional flat plate). Now, I will not go deeper into the analysis of what circulation represents, since the goal of this text is to go in another sense. However, keep in mind that circulation, like vorticity, can also be represented by a vectorial quantity (α)*. In the three-dimensional case, vorticity can be defined as circulation density, since both terms are related by a fluid volume (α= 𝜔V; α is sometimes denoted as Γ).

Fig. 2 Black points: Solid-body rotation; dark blue: large-viscosity fluid; light blue: small-viscosity fluid ➝ inviscid flow.

Returning to the two-dimensional case to simplify the explanation, according to the theory, a point (ideal) vortex can induce (or cause) a tangential velocity (v_t) at a certain distance (r) and it depends on the magnitude of its circulation: v_t=Γ/2πr. From such an equation, it is clear that the tangential velocity decreases as the evaluation point moves away from the vortex center (see Fig. 3b). Note that this phenomenon is exactly the opposite of what occurs in a solid-body (or infinitely viscous fluid) rotation, where the tangential velocity increases with distance: v_t=Ωr, where Ω is the angular velocity (ω/2; see Fig. 3a). In a solid-body rotation, the circumferential translation of a particle remains about the axis, but includes a local rotation (it is a rotational behavior)**.

Fig. 3 Tangential velocity vs radius for a) solid-body rotation and b) an ideal bidimensional vortex. 

Now it is time to get to the point of this publication: Does a vortex induce velocity at a distance? It is true that "induction" is a term often used in electromagnetics, where the concepts of Potential Flow Theory (including the Biot-Savart law; see below), the basis of current theoretical aerodynamics, apply. From a broader perspective, however, the word "induction" should not be treated as exclusive of certain fields of knowledge, since it means "caused by" or "provoked by". Since it is clear that this post is not about the origin of the words or their etymology, I will now try to justify why the term "induced velocity" should be correct, or at least accepted in the most broadest sense (disconected from electromagnetics), in the context of the Circulation Theory of Lift and vortex methods, i.e. in aerodynamics.

In the three-dimensional case, and by analogy from electromagnetics to aerodynamics, the Biot-Savart law states that a vortex filament (i.e., a wire) with constant circulation (i.e., electric current) causes a vectorial velocity (i.e., magnetic field) at a fixed point at a distance (now such a distance is defined by a series of position vectors: r_0, r_1, and r_2):

Fig. 4 Induced velocity at a point (P) caused by a vortex filament (A-B) with circulation Γ. 

As we know, such a law is the basis of the Lifting Line Theory (LLT) and the Vortex Lattice Method (VLM), which have proven to be reliable over several decades for calculating the aerodynamic characteristics of simplified geometries (i.e., airfoils, wings, and aircraft) under specific operating conditions (e.g., attached flow assumption), especially during the preliminary design phase. For such a fact, the application and usefulness of this law cannot be denied, if not in fluid dynamics in general, at least in computational aerodynamics. But what is the physical justification that allows a vortex to induce a velocity at a distance? or, in other words, why does a leaf of a tree move when a tornado is approaching? 

At this point, we must remember that vorticity (hence the Biot-Savart law) is defined within the framework of the PFT. So, such a theory is established for an assumed irrotational, incompressible and inviscid (without friction between circumferential layers around an axis) continuum medium, where flow properties are instantaneously transported far away due to the infinite Reynolds number condition. Note that the velocity induced by an ideal vortex is never zero! (v_t=Γ/2πr is an asymptotic function; see Fig. 3b), so ideally a perfect vortex has no physical boundary. But why relate electromagnetics and aerodynamics if the induction phenomenon is so different? In the first case, it takes place even in vacuum (e.g., in space), while it is quite obvious that a fluid flow cannot exist in such a state due to the absence of matter (and its transport). Perhaps electromagnetics and aerodynamics are not related at all? In addition, we must remember that sources and sinks, which naturally exist in electromagnetics, have no physical meaning in fluid dynamics and aerodynamics (i.e., the flow does not appear or disappear abruptly), although they are often used to model an "ideal" flow past an object (see Fig. 5). This last fact seems to strengthen those who argue against the use of "induced velocity" in aerodynamics. Perhaps it is time to separate aerodynamics from electromagnetics, maintaining only one common law between the two fields.

Fig. 5 Rankine oval by a source and a sink in the scope of the Potential Flow Theory. Source: http://www-mdp.eng.cam.ac.uk

So, returning to Fig. 2, it is quite obvious that the frictional contact between the particles of the solid body (or a veeery viscous fluid!) is the mechanism that allows to induce a linearly increasing tangential velocity from the center to the edge of the solid body/fluid. So what happens in the hypothetical case of an inviscid vortex? Although there is no shear (nor local rotation) between the circumferential layers, the inviscid condition does not prevent (biased) collisions between the flow particles (see Fig. 7), which do not necessarily occur exactly along a streamline (where all particles hypothetically have the same tangential velocity), since each flow particle has a volume (see intermolecular collision theory and Fig. 6). Finally, such a massive collision phenomenon will dramatically reduce the tangential velocity in the inner layers (theoretically infinite along the axis), while in the outer layers such a phenomenon is asymptotically smooth (see Fig. 3b).

 

Fig. 6 Intermolecular collision scheme, where fluid molecules are represented by spheres.
Source: Molecular Collisions (youtube.com)

Fig. 7 Simplified scheme for particle collisions with different tangential velocities (arrows) and crossing trajectories due to their curved paths.

This article, like the previous ones, does not pretend to show absolute truths about the understanding of a complex physical phenomenon, especially by oversimplified explanations, where some fine-grained aspects are left aside (I do not yet have enough knowledge to go deeper***). However, it may help to make an effort to think about the problem from a different perspective; in this case, from a very high viscosity fluid to an inviscid one, where it is assumed that some physical interaction exists within the medium to transport a variable (velocity). In previous cases, such a way of thinking (inverse logic) seems to be successfully applied [1].

Is there another way to 'transport' tangential velocity from the vortex center without the existence of physical particle interaction (collisions)? Perhaps electromagnetically at molecular level? 

*in 3D, both vorticity and circulation vectors must be aligned to maintain a divergence-free vorticity field.

**similar to the moon's orbit around the earth, where the same side is always seen from this latter.

***However, I do know mesoscale principles (based on streaming and collision processes) behind the Lattice-Boltzmann Method (LBM), an intermediate level between micro and macro scales.

[1] https://librepenzzzador.blogspot.com/2023/10/a-four-question-quiz-on-fundamentals-of.html