On the ROTationality of an inviscid flow: Laplace =/ Euler

Although the title of this blog article sounds too formal, as if it were a scientific publication, it follows the same idea as most of the previous ones, maintaining more or less a simple explanation of some topics of interest in fluid dynamics, within an acceptable extension. Therefore, I will logically explain why the rotationality remains from the incompressible Navier-Stokes equations (i-NSE) to incompressible Euler (i-Euler) ones after its simplification (from viscous fluid to inviscid flow) and how this concept leads to a better understanding of fluid motion from an alternative vorticity-based perspective.

The i-NSE are a set of non-linear partial differential equations (PDEs) that allow approximating the numerical solution for a viscous fluid since they can include all the acting forces such as gravity, pressure, viscous diffusion, and advection (sometimes called convection) terms. Such equations are mainly described in their velocity-pressure (v-p) formulation, however, they can be transformed into their velocity-vorticity (v-ω) form, where the pressure term disappears (see Fig. 2). When the viscous term is removed from such equations, the i-NSE are reduced to the i-Euler equations, which describe an incompressible, inviscid, and rotational flow. Such a rotationality condition is a well-known characteristic of such simplified equations, presented in many formal texts. Even, if someone thinks that artificial intelligence has all the answers, they could try to ask it...  

Fig. 1 Retrieved from "An Album of Fluid Motion" (van Dyke, 1982).

Specifically, the i-NSE in their v-ω form describes the dynamics of a viscous fluid that is inherently rotational (vorticity is not zero everywhere; ω≠0). When the viscous term is dropped, it continues describing the temporal variation of vorticity due to advection (and vortex stretching in the 3D case), which according to such equations of motion is an inviscid phenomenon (-na) per se, since the removed viscous term only directly affects the diffusion of vorticity (in the viscous case, viscosity also affects advection and vortex stretching, but in an indirect way: "at the next time-step").* Thus, the i-Euler equations in their v-ω form describe the evolution of an existing vorticity, despite an inviscid condition. In simpler words, they describe a vortex flow even though the diffusion of vorticity (due to fluid viscosity) is neglected.

Fig. 2 Incompressible Navier-Stokes (momentum) equations in their velocity-vorticity form.

But why can an inviscid flow solution (by the i-Euler equations) describe the evolution of vorticity, which, according to the majority opinion, must be generated only in a viscous medium? The short answer is that viscosity is not necessary for the generation of vorticity. At this point, a historical debate in fluid dynamics is still open. In fact, neither the i-NSE nor the i-Euler equations by themselves describe how vorticity is generated near a surface since such a term is absent in these equations of motion.** However, some theories have been raised in the sense that vorticity can be generated between an inviscid medium and a surface due to a difference in velocities (a gradient) at the flow-solid wall interface. In a previous blog article, I tried to present my point of view on such a debate: A fundamental question in fluid dynamics: vorticity generation

On the other hand, some theories try to justify the opposite idea, I mean a viscous-based vorticity generation mechanism, which first of all seems to be more theoretically complex than the previous inviscid ones, which in any case would be valid only for an attached fluid condition (based on shear stresses), leaving aside more complex fluid dynamics behavior, such as separated fluids. Starting with simple reasoning, I will now try to argue against such viscosity-based theories: viscosity ALWAYS diffuses (spreads/decreases with time) vorticity within a fluid; the former never tends to increase the latter, this is a fact.*** For such a reason, viscosity cannot be a justification for vorticity generation. How can viscosity, which ultimately 'destroys' vorticity, also create it? It just makes no sense! Furthermore, from a microscale perspective, a hypothetical first layer of a viscous fluid must have a zero velocity near the surface to maintain the non-slip condition. So do such irregular particles (e.g., molecules) stay at the same point on the surface but rotate around their axis? This is absurd! According to the zero velocity assumption, such particles should remain completely static (without rotation) trying to slow down the second layer (already diffusing vorticity due to shear stresses!), and so on. Therefore, a viscous fluid is also rotational because such a property arises from its primordial inviscid condition; I mean, a viscous fluid initially behaves as an inviscid flow as described by the corresponding equations of motion.

One of the most common misinterpretations among computational aerodynamicists, above all who work with panel methods, is that inviscidity means per se an irrotational flow. This idea is wrong since the potential flow is only a limiting case of the i-Euler equations (i-Euler with an imposed zero vorticity). According to the classical Potential Flow Theory (PFT), an ideal flow is described as irrotational (vorticity-free), incompressible (divergence-free), and inviscid (viscous-free) which is governed by Laplace's equation, a second-order PDE. Numerical solutions for such an equation are well-known as they define perfect streamlines around arbitrary (any shape) objects through the stream function (ψ). In aerodynamics, it has been successfully applied to obtaining lift for airfoils, wings, and even, complete aircraft configurations, under an attached flow assumption (see Fig. 3). However, such numerical results and flow patterns only appear when a mathematical trick, called the Kutta condition is applied to the rear part of the body, which in an airfoil coincides with the trailing edge. Notice that even in a circular cylinder (to describe the D'Alembert paradox) such a condition is also applied to the rear part, independently of its shape, i.e. it does not have a wedge shape as in the airfoil case. Hence, such a trick is a mathematical generalization and is unrelated to any smooth flow detachment, as some authors attribute it as one of its characteristics. From this general point of view, it can be applied at any point on the body surface, even more than one (for instance, to the top and bottom parts in the cylinder and airfoil cases to approximate more than one flow separation point), as some authors have already proposed.

Fig. 3 Experimental (thus, viscous) vs inviscid (by PFT) results for a NACA 0012 airfoil. Are such "inviscid" results viscous in the low AoA range? Source: (PDF) 2D and 3D Potential Flow Simulation around NACA 0012 Airfoil with Ground Effect

But, what does it imply to maintain an ideal flow perfectly attached to surfaces, as PFT does? This subject has been presented and discussed before in this blog, based on the numerical solution that resembles the typical behavior of viscous fluids at the low Reynolds number (Re) range, while theoretically, the PFT is defined in the opposite limit (infinite Re due to zero viscosity). Besides, it is well-known that high Re (inviscid-dominated) conditions naturally imply turbulent wakes, hence, vorticity as the primordial structure of turbulence. So where does the vorticity go in the PFT description? The answer is that it is confined in an infinitely thin vortex sheet at the surface and then detached downstream (see Fig. 4). One of the few formal texts where such a concept is explicitly explained can be found here (see section 2.2.2): Chapter 2: Aerodynamics model (J.S. Tyll). By following this order of ideas, such a hypothesis can be easily demonstrated by a surface vorticity numerically discretized within the classical PFT (e.g., by a distribution of vorticity or vortex ring elements, in 2D and 3D, respectively), which inherently implies that at the interface (flow-wall) vorticity is present despite the inviscid flow (and a free-slip boundary condition!) assumption.  So, why force vorticity to be attached to the whole surface? Let it free, like your mind, and see what happens; Laplace is not the same as Euler! [1].

Fig. 4 Retrieved from: Unsteady Aerodynamics and Trailing-edge Vortex Sheet of An Airfoil | AIAA SciTech Forum

* Numerically, this is done by a temporal splitting scheme.

** Probably because the generation of vorticity is theoretically an instantaneous mechanism...

*** I am sure that no one here expects a high-viscosity fluid (e.g., honey, oil, or pitch) to flow around an object (typically in the low Reynolds number range) full of vortices!

[1] (PDF) Development of a three-dimensional vortex method for solving detached fluid dynamics