An 'inviscid' boundary layer! Is this a bug?! (Part 1)

In my opinion, one of the worst ways to justify a physical hypothesis is to perform simulations, especially when they are subject to modeling and approximation rather than the direct solution of equations, which could also be limited by simplifications and assumptions. However, this time I will make use of a commercial tool that is accepted with enough credibility in the field of Computational Fluid Dynamics (CFD) to justify that vorticity can be generated between the interaction of a solid body and an inviscid medium, following the line of previous articles in this blog. To accomplish this task, I have tried to reduce all the steps involved to a minimum, keeping only the essential ones (e.g. I avoided using a viscous model, put the viscosity to zero, and manually set a free-slip wall)*.

A two-dimensional simulation in one of the most popular CFD software (based on the finite volume method; FVM), consisting of an inviscid flow past a NACA 0006 airfoil at zero degrees of angle of attack (AoA), was performed. Such an airfoil was chosen to be compatible with Potential Flow Theory (PFT), which theoretically can only be solved under an attached flow assumption (small angle of attack and thin bodies) in the high (theoretically infinite) Reynolds number (Re) range. I will avoid going into the details of the pre-processing since such a step does not have any particularities: importing points that define the airfoil, generating a C-shape domain, designing a good quality mesh (without additional viscous computations for y+) for a fast solution on a conventional laptop (32K quad elements; see Fig. 2 for convergence plots). The complete project (.wbpj file and folder) can be downloaded here: Ansys Workbench project (version: 2024 R2).

Fig. 1 An inviscid boundary layer near the surface. It this a software bug?! (in Results module).

Within the solver (Fluent), an inviscid flow simulation is selected, air for the fluid material (only density is available since it is inviscid; left by default), the boundary conditions (BCs) are defined as velocity inlet (50 m/s for a Re~3E6 in an equivalent viscous case), a pressure-outlet (gauge pressure: zero pascals) for the remaining outer boundary, and a stationary wall (without any available options due to the inviscid model) for the airfoil surface. Related to this, section 9.5.1 (Setting Up an Inviscid Flow Model) of the Fluent user manual says: "Walls are assumed to be slip surfaces (the velocity is not zero, unlike viscous flows) and therefore have a tangential velocity computed based on the solution of the governing equations". But why does a velocity gradient (similar to a viscous boundary layer; BL) appear near the airfoil surface?! (see Fig. 1). Maybe an inviscid flow should not model a free-slip boundary condition "naturally"? Or is this a bug, or worse, a numerical trick to force convergence for one of the simplest fluid dynamics simulations?

Fig. 2 Residuals of continuity and components of velocity (smooth convergence until millionths; 1E-6).

To try to answer the previous question I will refer to section 1.7.1 (Euler equations) of Fluent's theory guide: "For inviscid flows, Ansys Fluent solves the Euler equations. The mass conservation equation is the same as for a laminar flow, but the momentum and energy conservation equations are reduced due to the absence of molecular diffusion". Unfortunately, such a theory guide omits information for the laminar model, however from Fig. 1 is possible to infer that such inviscid BL is being modeled since its gradients are more or less within the first layer mesh (notice that plotted wall shear is zero along the entire surface). Thus, since velocity (and pressure) gradients are present, does exist vorticity near the solid surface?! Yes, according to this numerical result, an inviscid flow is associated with detached vorticity (hence DRAG)! (see Fig. 3). But is this type of flow the same as that described by the PFT? NO, in a strict sense. In the case of the Euler equations, these describe an inviscid and rotational flow (it can be incompressible or compressible), whereas from the ideal flow definition, the latter is inviscid and "irrotational" (except at an infinitely thin BL due to an infinite Re condition)**. In this sense, the PFT describes a particular limit condition of the Euler equations, where all quantities are transported instantaneously, and due to its attached flow assumption (all vorticity is attached to the surface) is not able to solve flow separation; therefore a generalization of the PFT should allow to solve a detached flow condition, leaving vorticity to advect, stretch (in a 3D case) and diffuse freely, without any assumption [1].

Fig. 3 Vorticity around the upper and lower surface near the trailing edge: a "laminar BL" (with vorticity!).

Then should there be a truly inviscid and irrotational flow? The answer is quite abstract as well as hypothetical (and probably physically impossible). In the first instance, no shear stresses between flow and surface (truly free-slip BC) will mean no momentum transfer between both media, hence neither resultant forces nor pressures. However, this idea may be debatable due to the acting normal stresses...Anyway, at this point, my original hypothesis, from a practical (and real) point of view remains: an inviscid flow, in the context of PFT, is capable of generating, thus detaching vorticity from the whole surface.

What results do you get from your CFD software or code? Why does the continuity equation not converge when the flow symmetry is barely broken (e.g. AoA=3 degrees)?

* I also ran two simulations with the laminar model, setting the viscosity to zero and selecting non-slip and free-slip walls. In the first case, the velocity field is quite similar to the inviscid model (with some kind of BL around the airfoil); in the second case, there is no BL. In both cases, the numerical values for drag are quite similar: the no-slip case is greater than the free-slip case by 3 drag counts or 3 ten-thousandths. The laminar free-slip results are practically the same as for the inviscid model (supposedly "free-slip" but appearing non-slip): Cd=0.0021. Besides, in both cases a vorticity field is present!

** This subject has been explained and discussed before.

[1] The Full Non-linear Vortex Tube-Vorton Method: the pre-stall condition | Advances in Aerodynamics | Full Text

Go to the second part: An 'inviscid' boundary layer! Is this a bug?! (Part 2)