When the evidence is not enough: on "inviscid attached" flows

I remember when some years ago I found an interesting and nice visualization on YouTube called "Inviscid Flow ovrer an airfoil", where some kind of green flow (or fluid) passes an aerodynamic/hydrodynamic shape with incidence (at a high angle of attack; AoA). Most aeronautical engineers know that under normal operating conditions (at a large Reynolds number; Re), fluid detaches near the leading edge, forming a chaotic wake downstream, leading to stall since for this particular case, the AoA is about 22 degrees (I made the measurement by a print screen in Paint and based on pixels; sorry for the lack of scientific rigor). Now the question is: is that title correct? And the answer is: Absolutely not!

Video 1: Viscous fluid over an airfoil (Re~1). Source: Inviscid Flow ovrer an airfoil

Despite the flow in such a visualization, or even better, the fluid seems to follow more or less perfectly defined streamlines as it flows around the airfoil, similar to the shown by the Potential Flow Theory (PFT) during the fundamentals of fluid dynamics or aerodynamics course, which, supposedly, models inviscid flows, free of any vorticity (laminar patterns; see Fig. 1). Since it is clear that such a visualization comes from an experimental test (it is not a computer-based simulation or animation) it should correspond to a superfluid (e.g. helium-4) past an airfoil. However, as far as I know, no single experiment has been performed or reported for an inviscid fluid past an object. In fact, last year Prof. Haithem Taha's research group (University of California-Irvine; UCI) earned one million dollars in funds for testing with superfluids and lifting surfaces to determine the role of viscosity in lift, which is still an open question on fundamentals of fluid dynamics.

Fig. 1 'Inviscid' flow past an airfoil by the current Potential Flow Theory. Source: Fluid flow patterns - Nerdy Mechanical

Now, I will describe what I saw in the previous video. It is clearly a viscous-dominated case, close to a unitary Reynolds number (Re~1), where shear stresses are present, sharply deforming the fluid tracers, above all, close to the leading edge. But what do I want to probe by denying the title of a nice random video on YouTube? The answer comes past the half-video around the trailing edge. The viscous fluid tries to round it, going from the lower to the upper surface, however, a fluid separation phenomenon better known as the Kutta condition*, in the scope of the PFT, happens. In fact, such a detached fluid is clearly an inviscid manifestation (no more shear stresses lead to detachment due to the upperside fluid flow), while the attached condition is inherently viscous. Such concepts are completely opposite to the popular belief, including top researchers in fluid dynamics but among all, in aerodynamics, some of them ensuring that an inviscid flow is theoretically defined by perfect streamlines according to Laplace's equation (irrotational flow) instead of Euler's one (rotational flow). They overlook that originally, the latter is the general case to describe mathematically an inviscid flow, while the former is only a particularity (for an infinite Reynolds number condition) of such a set of equations of motion, as is presented in a previous post: On the ROTationality of an inviscid flow: Laplace /= Euler 

Fig. 2 Starting vortex formation from an impulsively started wing in an incompressible flow. Contours distinguish the circulation about the wing and the opposite signed circulation about the starting vortex. Source: AA200_Ch_11_Two-dimensional airfoil theory_Cantwell

From the previous description, it is clear that viscosity (but not inviscidity) is the cause of fluid turning around the trailing edge (it is not an inertial effect but viscous one)**. In fact, in a super-viscous hypothetical condition (where no separation is allowed) the fluid will continue flowing or circulating along the upper side of the airfoil, in a counter-clockwise direction, as is shown in the corresponding part to the airfoil (C_I) in Fig. 2. In the scope of the PFT, one of the simplest definitions for circulation is: Circulation is the line integral of the velocity field around a closed contour. It measures how much the fluid is rotating within a region enclosed by a contour line by summing the velocity components along the contour path. According to such a definition, and by analogy, in the super-viscous hypothetical case (without any fluid detachment), circulation should be a viscous manifestation but not an inviscid one. So, does the PFT define a viscous condition? Yes, it does. The current interpretation of the PFT does not describe an inviscid flow but the inviscid behavior of a viscous fluid (due to an attached fluid assumption), which behaves as an inviscid fluid (no shear and irrotational) away from the surfaces. Such a viscous attribution does not come from the mathematics behind the PFT, but from imposing an attached circulation to the surface (and a parallel alignment of streamlines/tangential velocity near the surface). However, a truly inviscid flow can be modeled [1] and solved [2] by PFT concepts, leaving circulation detached without restrictions from the whole surface, depending only on the developed vorticity-velocity field.

The previous inviscid flow-viscous fluid-inviscid fluid 'complexity' is perfectly explained during the first 16 minutes of Lecture 28 by Prof. S. Chakraborty (IIT Kharagpur; see video 2). Related to this, I have selected three phrases that help to understand the general meaning of such a lecture: 

1. "So, I want to iterate repeatedly that inviscid does not literally mean that viscosity of the fluid that is under consideration is zero".
Here it is clear that Prof. Chakraborty is referring to an inviscid behavior within a viscous fluid (away from a surface) described by the PFT.

2. "In the outer region [outside the boundary layer] the flow is inviscid and irrotational in this example because an irrotational flow was already imposed on it".
As described by Laplace's equation, PFT has already been manipulated by imposing irrotationality (zero vorticity) to almost the whole domain, except at the surface. Such an exception is justified since a detached circulation-vorticity exists as an advected wake behind the body (see Fig. 2); a perfectly irrotational condition should not have any detached wake, similar to the super-viscous hypothetical condition described before. By the way, What does such a hypothetical case look like? Something like this:
Fig. 3 Pure clockwise circulation around an airfoil. Source:  3  Airfoils and Airflow (Dr. John S. Denker)

3. "So, you have to be careful about the use of inviscid and irrotational flow. It is not a question of whether all the domain can be treated as that, but even a part of the domain if it can be treated as that; it helps in simplifying the governing equations considerably".
Inviscid does not mean irrotational per se; inviscid plus ROTational flow exists and is defined by the general case (Euler's equations). The current PFT describes a simplification; hence, it is only applicable for a certain specific operating condition (i.e., an attached fluid assumption under a large Re); more complex phenomena, such as fluid separation, cannot be addressed by just performing some corrections or by simplified models; Physics must be respected at all costs.

Video 2: Lecture 28: Potential Flow (Prof. Suman Chakraborty; IIT Kharagpur, India).

* It happens even though it is not a perfectly sharp (wedge) trailing edge.
** Viscosity, via the Reynolds number, defines the fluid flow pattern, but not the "pressure difference" between the upper and lower sides; the pressure field is a consequence of fluid motion. Subject to perform measurements, and for this particular case, the fluid at the bottom reaches the trailing edge earlier (close to the surface), implying a lower pressure (the marker is stretched towards the TE: acceleration) than at the top (the marker is squeezed towards the TE: deceleration). Therefore, the "pressure difference" hypothesis would not be applicable to justify the fluid turning.

[1] The Full Multi-wake Vortex Lattice Method: a detached flow model based on Potential Flow Theory | Advances in Aerodynamics | Full Text
[2] The Full Non-linear Vortex Tube-Vorton Method: the pre-stall condition | Advances in Aerodynamics | Full Text

"If they can imagine a perfect inviscid flow, why could not I hypothesize a super-viscous fluid?"

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