An 'inviscid' boundary layer! This is still a bug (Part 3)

Go to the first part: An 'inviscid' boundary layer! Is this a bug?! (Part 1)

Approximately a year ago, I published the initial parts of a series detailing a significant visualization anomaly encountered within the Results module of ANSYS Workbench. I recently tested the newest ANSYS Student version (2025 R2) with the same 2D test case, and unfortunately, the issue persists despite having been reported in the official forum: a modeled boundary layer (BL) appears in the visualization for an inviscid flow simulation (over both an airfoil and a 3D wing; see Fig. 1). It is crucial to emphasize that this visual error does not appear in the ANSYS Fluent solver module. Since the solver correctly recognizes the inviscid condition (μ=0), it does not impose a no-slip condition at the wall, and thus, no boundary layer is formed in the flow solution itself. 

As those with a fundamental knowledge of fluid dynamics understand this is not a personal interpretation error, I believe this bug is primarily a visualization flaw, not a serious computational error. I estimate it could be resolved with minimal changes to the source code, likely related to how it renders data near wall boundaries. While the spurious visual presence of the BL is confusing, the more significant issue lies in the discrepancy in the plotted velocity magnitude between the two modules, as explained in the previous parts of this series. This disparity, derived from a single simulation result file, strongly suggests an interpolation error is occurring. The Results module appears to be improperly applying or interpolating velocity constraints at the wall, leading to both a visual misrepresentation and a quantitative inaccuracy in the velocity (and pressure) field.

Fig. 1 A modeled boundary layer still appears for a 3D inviscid simulation (with a free-slip boundary condition; Results module).

ANSYS Fluent is undoubtedly a standard in the world of Computational Fluid Dynamics (CFD) simulation, used extensively across both industry and academia. My intention in this series is not to point out its weaknesses, especially when the solver itself helps validate the core hypothesis of my doctoral research: that an hypothetical inviscid flow past a body is not described by the Potential Flow Theory (PFT). This concept directly confronts the majority opinion held by most theoretical and numerical researchers in aerodynamics and fluid dynamics globally [1]. The conventional wisdom often fails to distinguish between two distinct idealizations: 

1. Inviscid flow described by the Euler Equations and, 
2. Potential flow described by the Laplace Equation. 

The error lies in equating the Euler solution to the limit condition of a viscous fluid (Reynolds number tending to infinity) and assuming zero vorticity (irrotationality) a priori. This discussion has been presented from various perspectives in previous articles on this blog. Now, I intend to take the argument one step further: I am looking to provide a numerical demonstration that the three-dimensional Euler equations allow vorticity generation in a non-viscous medium.

Fig. 2 A closed wing CAD model based on a NACA 0006 airfoil.

To accomplish the numerical task described above, a closed wing (a duct) was modeled using the same airfoil profile from the 2D case: the NACA 0006. The specific geometry choices were made to isolate the phenomenon of interest:

Mitigating 3D complexities: The closed wing design was chosen to circumvent the need to model the complex, three-dimensional effects of wingtip vortices, simplifying the approximation of the numerical solution. Furthermore, the trailing edge was rounded to avoid the geometric singularities associated with a perfectly sharp edge.

Ensuring free-stream conditions: The duct's diameter was set to two chord lengths. This size was selected to minimize perpendicular interaction between the duct surfaces and the flow around the wing, ensuring the conditions near the airfoil closely resemble a free-stream condition. 

The control volume was modeled by subdividing the domain into six subdomains based on two concentric cylinders (see Fig. 3). This strategy aimed for maximum control during the meshing process and facilitated the generation of a high-quality, structured-type mesh where feasible. The resulting meshing strategy balanced quality and the constraints of the academic license:

Far-field mesh: High-quality hexahedral elements were used far from the body.

Near-field mesh: The region around the airfoil utilized tetrahedrons, which were then automatically converted to polyhedrons by Fluent (improving convergence and resolution).

Geometric Adaptation: An inflation layer was included despite modeling an inviscid case. Its purpose here is not to model the boundary layer, but to maintain a better adapted mesh to the geometry, which is crucial given the 500,000 element limit imposed by the academic version.

Fig. 3 Meshing of the control volume (around 460K elements and 259K nodes).

For this 3D simulation, consistent with the previous 2D analysis, the inviscid flow solver based on the Euler equations was selected. By definition, the fluid viscosity is zero, and a free-slip condition is inherently imposed at all solid walls. The simulation utilized standard boundary conditions:

• The inlet BC was set to a free-stream velocity of 10 m/s (α=0 deg.).
• The outlet BC used a default zero gauge pressure (0 Pa).
• The outermost external BC was defined as symmetry to minimize external influence.

All other solver settings were left at their default values, including the coupled pressure-velocity scheme and least squares cell based spatial discretization. The solution was initialized using the hybrid initialization method. The simulation was monitored by tracking the scaled residuals and the drag coefficient (CD). The convergence criterion was strict: the scaled residuals for continuity and all three velocity components had to fall below 1e-6. All four variables achieved this criterion after just 224 iterations, demonstrating rapid and stable convergence (see Fig. 4). The monitored CD stabilized at a final value of 0.0065. This non-zero drag value, calculated under the assumption of zero viscosity, provides the first significant numerical evidence supporting the hypothesis that the three-dimensional Euler equations inherently admit a force generation mechanism, directly contradicting the predictions of classical PFT.

Fig. 4 Scaled residual plots. A smooth convergence!

The solution plotted directly in Fluent (see Fig. 5) is consistent with the physically expected outcome for an inviscid flow. Since no boundary layer (BL) should exist, the velocity at the wall is not zero; instead, the minimum velocity value recorded around the leading edge is approximately 4 m/s, which is an intermediate, non-zero value. The maximum velocity observed is 11.24 m/s, slightly exceeding the free-stream velocity of 10 m/s due to acceleration over the airfoil. As noted earlier, a critical inconsistency arises when comparing these results to the Results module visualization. In the Results module, the maximum velocity recorded on the same plane is 10.89 m/s, a value slightly lower than the 11.24 m/s observed directly in the Fluent solver. The minimum velocity values are also dramatically mismatched: the Results module incorrectly plots a minimum velocity of 0 m/s near the surface (see Fig. 1), as it incorrectly models a viscous BL with a no-slip condition within the first element layer. These differences strongly indicate a possible interpolation error in the visualization module, confirming that the post-processor is misrepresenting the quantitative velocity field derived from the core simulation data.

Fig. 5 Velocity magnitude contours on the vertical plane and on the closed wing's surface (visualized in Fluent).

The pressure field visualized in the Results module appears consistent and well-resolved, showing no abrupt or unphysical changes in the vicinity of the streamlined body (see Fig. 6). However, these pressure values are also subject to closer review, as discrepancies exist between the quantitative values reported in the Fluent solver and those displayed in the Results module visualizations (up to 17 Pa for the suction values).

Fig. 6 Static pressure field on and around the "duct" (visualized in the Results module).

After demonstrating that the simulation process described here is both well-implemented and meets the criteria for high quality and convergence, it is time to present the primary objective of this investigation: the numerical evidence that vorticity can be generated around a surface (under a free-slip condition!) in a non-viscous medium, not only in the bidimensional case but also in the three-dimensional one (see Fig. 7). As mentioned earlier, this result directly challenges the conventional conception that an inviscid flow is inherently irrotational and must be accurately described by PFT—the "dry water" model famously critiqued by von Neumann*. These findings refute the common assumption that any trailing edge vortex is exclusively caused by the "pressure difference" between the upper and the lower part of an airfoil or wing. In aerodynamics, this simulation suggests that the mechanism for vorticity generation is inherent in the three-dimensional Euler equations themselves, independent of viscosity. 

Fig. 7 Contours of vorticity (curl of the velocity) in the z-direction. Vorticity has relatively high values but not close to zero! (3D simulation).

While I acknowledge the visualization issues in the Results module, I must use it to display the vorticity field, whereas the Fluent solver itself is currently unsuitable for direct plotting of the vorticity vector components. The core reason is that the native inviscid (Euler) solution does not provide the necessary velocity derivatives required to customize and plot the corresponding vorticity vector component (∂v/dx - ∂u/dy) directly within the Fluent interface. To numerically demonstrate vorticity generation and access the required velocity derivatives, I must rely on a workaround: using the laminar model (Navier-Stokes) and manually setting the fluid viscosity to zero with a free-slip condition at the walls. In the 2D case, this modified laminar model yielded results very similar to the true inviscid model, including the crucial existence of vorticity, thereby allowing me to compute and visualize vorticity. Unfortunately, applying this modified laminar model to the current 3D mesh has proven challenging. The continuity residual has yet to converge below 1e-4, indicating numerical instability due to lack of the damping effect of the fluid viscosity or difficulty associated with the pseudo-inviscid formulation on this specific mesh.** Therefore, for the present 3D visualization, the Results module remains the only available tool for graphically inspecting the existence of the vorticity field. Will the vorticity still be displayed when the visualization error is corrected?**

As emphasized from the beginning of this series, I am fully aware that a computer simulation cannot, on its own, prove a physical law, especially when the numerical method is simply treated as a "black box". However, the numerical evidence presented here serves a critical purpose: it opens the door to alternative theoretical frameworks regarding the generation of circulation and force in fluid dynamics [2]. This evidence suggests that conventional assumptions—specifically regarding the nature of inviscid flow—may need reconsideration, encouraging a deeper look into the underlying mathematical physics rather than dismissing the results based on established but incomplete paradigms.

* John von Neumann's perspective on PFT was famously dismissive and cynical, encapsulating his view on the limitations of mathematical rigor when divorced from physical reality, especially in the context of fluid dynamics. His most famous line regarding PFT is: "Mathematicians study 'dry water'."

** If you are interested in trying to solve this challenge on your own, you can download the duct's geometry here: Closed wing (step file)

The style of this text has been improved by IA (Google Gemini Flash).

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

[2] [2506.18719] The Full Nonlinear Vortex Tube-Vorton Method: the post-stall condition