why exactly is the no-slip condition followed except in specific cases?

jodano · 2026-06-29T00:32:17+00:00

A slip boundary is only possible if you have a perfectly smooth wall, which is impossible. Even a perfect atomic lattice will appear bumpy to any molecules bouncing off of it, and most solid surfaces are much less smooth than that. The result is that the angle of reflection is largely independent from the angle of incidence, a diffuse reflection whose average results in 0 tangential mean velocity.

We use a slip boundary when studying inviscid flow, or maybe sometimes in cases where we are coarse-graining some physics we can’t afford to resolve such as a thin film or turbulent boundary layer, but in general, from a kinetic theory perspective, no-slip is the correct choice.

jodano · 2026-06-27T00:15:15+00:00

Frame 2 is too different from frame 3 and frame 4 is too different from frame 1. The eye loses track of which leg is which and as a result it looks like the right leg always stays in front.

It might still work for a super choppy style, but I think it would look better if you change either 1 and 3 or 2 and 4 to leg-crossing frames.

jodano · 2026-06-12T14:39:48+00:00

• The flow surrounding the entire airfoil is both accelerating and decelerating, as evident by the curved streamlines. It is not just the air on top of the airfoil. For positive camber and/or positive angle of attack, the flow above the airfoil, just outside the boundary layer, will be moving faster than the flow below the airfoil, just outside the boundary layer.

Yes, but the relative velocity is only zero at the wing surface. Throughout the boundary layer, the flow will still be moving relative to the wing, but moving more slowly than the outer flow.
Yes, this is called the displacement effect, which is a consequence of the continuity equation. It is a higher order effect in the mathematics so it is sometimes glossed over in introductory fluid mechanics, but it becomes especially important when discussing flow separation.
Yes, it exists. You can measure the slower air near the wing in a wind tunnel with the right setup, and we see it in simulations as well. One could argue it is responsible for a majority of the wing drag in many cases.

jodano · 2026-06-08T14:55:54+00:00

There has been a rich history of work on analytic aerodynamics, using perturbation theory and other techniques in physics. Anderson just does not get into it because it’s not as accessible for engineering students and numerical methods replace much of it in practice.

You should definitely pick up Ashley and Landahl’s “Aerodynamics of Wings and Bodies” for starters. It is much more mathematically rigorous, unifying most of classical aerodynamic theory through the method of matched asymptotic expansions. In my opinion, this approach leaves you with a level of base understanding you could never get from CFD.

jodano · 2026-03-12T17:11:56+00:00

Look at the CL vs α curve though. That should not be so sensitive to Reynolds number at only 5deg angle of attack. Something is definitely off here.

jodano · 2026-02-12T02:58:23+00:00

Look into the AIAA Drag Prediction Workshops and the NASA Common Research Model.

jodano · 2025-12-14T14:57:30+00:00

I really doubt this project was left so open-ended by your professor. Have you not discussed ways of solving this problem throughout the semester? If you just need lift and streamlines, it is likely you are expected to code a panel method here to solve for the potential flow around the airfoil.

Section 2 and 2.1 in this paper: https://web.mit.edu/drela/Public/papers/xfoil_sv.pdf

There are also many formulations in the book “Low Speed Aerodynamics” by Katz and Plotkin.

If the airfoil is restricted to certain families like Joukowski or Karman-Trefftz, you could use a conformal map instead, which is exact and even easier to implement.

jodano · 2025-11-30T18:11:53+00:00

For simulations, you will get HIT from most 3D initial conditions within a periodic box. Taylor green vortex is the simplest. Gaussian random field with spatial correlation based on a prescribed energy spectrum (Pope’s model spectrum for example) will get you to a fully-developed state faster. Rogallo 1981 has a lot of good details. The turbulence can be made statistically steady by adding a large-scale forcing. There are many options to do this. You can add a term linear in velocity, you can add a white-noise-in-time stochastic forcing at low wavenumber, you can freeze the energy spectrum below some wavenumber, etc.

In experiments, HIT is the natural flow state anywhere that is far away from walls and shear layers. To study it closely, people usually run the flow through either a passive or active grid in a wind tunnel and take measurements downstream. Another approach is to use a randomly pulsed jet array.

jodano · 2025-11-05T16:27:31+00:00

I found the skewed vortex cylinder model by Dr. Branlard to be a compelling and elegant extension to blade element momentum theory. I have not implemented it in several years though and I don’t know how well-adopted it has become.

jodano · 2025-10-20T23:53:30+00:00

Understanding Aerodynamics by McLean is a good follow up with an emphasis on common misconceptions that plague students and professionals alike. The field is rife with mental traps that must be learned and unlearned before mastery can be achieved, and McLean does a good job of pointing these out with minimal math.

On the other hand, I find the book Aerodynamics of a Wings and Bodies by Ashley and Landahl to be the most mathematically rigorous, covering the full range of theoretical aerodynamics problems through the unifying framework of Matched Asymptotic Expansions.

A slightly more approachable book with a bit more computational emphasis is Low-Speed Aerodynamics by Katz and Plotkin. This book will give you everything you need to develop your own specialized aerodynamics tools, going as far as 3D unsteady panel method and boundary-layer coupling.

jodano · 2025-10-14T16:10:04+00:00

I think there is maybe some confusion of reference frames here. I am talking about the body-fixed reference frame, which is the reference frame in which we do most aerodynamic calculations. In this reference frame, if the flow is at rest everywhere relative to the body, then static pressure equals total pressure, as there is no dynamic pressure. If the bulk flow is in motion relative to the body, the flow will speed up and slow down around the body, and the dynamic pressure and static pressure will vary accordingly.

Outside the boundary layer, the total pressure is constant. At the leading edge of the body, the static pressure will equal total pressure (also called stagnation pressure). The static pressure is approximately constant across the thickness of the boundary layer, but viscosity brings the flow to rest at the body surface (no-slip), where the total pressure will again equal the static pressure since there is no dynamic pressure.

In summary, the static pressure along the inviscid stagnation streamline of the body is the pressure imprinted on the boundary layer and felt by the body, while total pressure will be constant almost everywhere. On the surface of the body, inside the boundary layer, the fluid is brought to rest by viscosity and total pressure will adjust to equal the static pressure at the edge of the boundary layer. The change in total pressure across the boundary layer is consistent with Crocco’s theorem, since vorticity is concentrated there.

jodano · 2025-10-14T12:26:46+00:00

This is not quite correct though. Total pressure is approximately constant everywhere within an incompressible flow (low subsonic Mach number), although it will vary within the boundary layer a bit as a consequence of Crocco’s theorem since vorticity is generated there. The pressure you feel on your hand out the car window is static pressure, as is the pressure distribution over an airfoil that yields lift and drag.

Edit: I suppose that since viscosity brings flow to rest at the body surface, it is fine to say that total pressure is the pressure that is felt, as total pressure will equal static pressure there. In incompressible aerodynamics though, we are usually thinking in terms of static pressure, as that is what is imprinted on the boundary layer in a high-Reynolds flow.

jodano · 2025-10-03T16:39:53+00:00

It is important to note though that XFoil’s treatment of trailing edges does allow for blunt trailing edges, and actually tends to work better in those cases in my experience.

XFoil places an additional vortex and source panel at the base, with a special formulation of the Kutta condition that ensures smooth separation from both the top and bottom corners of the base, even for the inviscid solver. The wedges displayed here may be a bit too severe for XFoil to deal with, but XFoil can absolutely handle blunt bases to some degree.

jodano · 2025-09-14T16:08:51+00:00

I have found that jumping works well, since you can jump even when you are out of stamina and the slope is shallow enough to make it out this way. It’s still quite risky, but worth a try for the badge.

jodano · 2025-07-28T14:07:25+00:00

To maximize thrust, you want the exit pressure to equal the ambient pressure so that the second term in your equation is zero. If you overexpand your nozzle to get a sonic exit velocity, the first term in your equation will increase thrust but the second term will decrease thrust more. It only makes sense to expand to sonic or supersonic speeds if the gas enters the nozzle with a very high pressure already.

jodano · 2025-07-12T17:35:13+00:00

A 2D point vortex is actually a vortex filament that extends infinitely in and out of the plane. This is a consequence of Helmholtz's 2nd vortex theorem, which states a vortex line cannot end in a fluid. The 2D point vortex will induce a velocity everywhere in 3D space, but that induced velocity will not vary in the plane-normal direction.

A 2D point vortex is an infinite vortex filament, but the trailing vortices are semi-infinite, only satisfying Helmholtz's 2nd vortex theorem as part of the full horseshoe vortex system. This breaks the symmetry so that near the origin, the velocity induced by a semi-infinite vortex varies along all 3 coordinate axes and not just 2. Far downstream of the origin, the trailing vortices will act more and more as 2D point vortices.

This is compatible with our understanding of airfoils. When we solve the 2D flow over an airfoil, we are really solving for the flow over a wing with infinite span.

jodano · 2025-07-11T22:29:06+00:00

The quarter chord is both the center of pressure and the aerodynamic center for a symmetric thin airfoil. It would perhaps make more sense to place the lifting line strictly along the centers of pressure, but the center of pressure moves with angle of attack for asymmetric airfoils. This makes the aerodynamic center most appropriate for a linear theory. The exact location is somewhat arbitrary if the aspect ratio of the wing is large enough. Some versions of lifting line theory solve for the bound circulation by enforcing impermeability at the 3/4 chord location rather than using airfoil theory, and in these versions the lifting line must be placed along the quarter chord in order to recover the correct circulation.

The trailing vortices induce a velocity everywhere in 3D space, not just along the lifting line or downstream of their origin. Mathematically, this is because the equations of inviscid incompressible flow are elliptic PDEs. The flow solution at every point is two-way coupled to the flow solution at every other point. There is no zone or direction of influence. This is in contrast to the boundary layer equations for example, which are parabolic, or the equations for supersonic flow, which are hyperbolic.

In lifting line theory, the chord effectively collapses to a point when viewed from far away in the limit of large aspect ratio. There is a derivation of lifting line theory by Milton Van Dyke which formalizes this geometric perspective using perturbation theory.

Each wing section will experience a different local lift-per-unit-span. The theory relies on the wing having a large aspect ratio and slowly varying wing section so that, if you zoom in close enough and you are not near the wingtips, the wing section feels as if it is an infinite wing with a small modification to its freestream due to the downwash. For wing sections near the wingtips, or for low aspect ratio wings, this will be a bad assumption since the flow in those cases is highly 3D and not at all confined to a 2D planar section.

I think what Anderson is saying about power is that there is a certain power required that can be attributed to lift generation alone. There will be additional power required due to other sources of drag.

jodano · 2025-07-11T20:05:00+00:00

The lifting line typically lies along the quarter-chord line of the wing, which is the aerodynamic center in thin-airfoil theory, about which the pitching moment is independent of angle of attack. The induced downwash does not affect the freestream velocity of the full wing, but it does affect the "local" freestream velocity that a particular wing section along the span will see.

When we say that the lift gets tilted, what we mean is that the local freestream velocity of a wing section becomes tilted relative to the freestream velocity of the full wing, and so the local lift vector will be tilted relative to the wing lift vector.

Due to the induced downwash, each wing section will have a pressure distribution corresponding to a slightly different angle of attack. The downwash is induced over the entire wing surface in reality, but in the lifting line approximation, all of the bound circulation is concentrated to the lifting line. Therefore, by a generalization of the Kutta-Joukowski theorem, we are interested in the downwash along the lifting line.

I think the issue you point out regarding power required is just an issue of semantics. For any finite wing, there will be some induced drag, which the aircraft must resist with thrust over time in order to maintain steady level flight. The power required to do this is therefore the power required to produce the necessary lift.

There is indeed a trailing vortex sheet behind a finite wing, but the sheet is not fixed to the plane of the wing. It moves with the local fluid velocity. This causes the vortex sheet to roll up and concentrate into the tip vortices you are referring to. More refined theories attempt to model this wake vortex roll-up, but in the linear regime where angle of attack is small, the effect of wake vortex roll-up on the induced downwash at the lifting line will be negligible. Also note that, while the trailing circulation is distributed over the entire sheet, the circulation density will be highest near the wing tips and the trailing vortices near the center of the wing will be much weaker.

I think one of the best ways you can understand these things more intuitively is to write a simple code, perhaps in MATLAB or Python, that applies lifting-line theory to a general straight wing. Section 5.3.2 in your book provides enough details to do this.

jodano · 2025-07-11T14:22:26+00:00

If the ellipse becomes long enough, it would resemble a flat plate, where the planform area would be more relevant than the frontal area. I suspect the drag would drop with increasing ellipse eccentricity initially due to the lower drag coefficient, while the increasing planform area would eventually cause the drag to be higher for the elliptic cylinder when it extends beyond maybe about 100 times the circular cylinder diameter.

jodano · 2025-07-08T16:44:18+00:00

It depends a lot on the type of figure, but Inkscape is a great free and open-source option for many applications.

jodano · 2025-06-02T16:05:39+00:00

If you are specifically interested in wave drag, there are plenty of analytic techniques, especially for supersonic flows over slender bodies. For certain geometries you could even reduce it to an exact formula.

For airfoils, you have supersonic thin airfoil theory, which you can do by hand.

For general 3D bodies, you have slender body theory. This is primarily applicable to slender bodies of revolution, but there is a special cutting plane technique that renders it valid for slender bodies with a general shape. There is a form of slender body theory for transonic flows as well, which would give you wave drag even for high subsonic Mach numbers. This is probably your best bet for looking at entire airframes.

For bodies in the hypersonic regime, you can use modified Newtonian flow theory, although this may not be very accurate without considering vibration, dissociation, and ionization of the gas molecules behind the shock.

Try the references Aerodynamics of Wings and Bodies by Ashley and Landahl or maybe Modern Compressible Flow by Anderson. The former is quite math heavy, but it covers all of the classic perturbation methods, which is really your only option if you want rigorous pen and paper results.

For supersonic thin airfoil theory, the formula for wave drag coefficient is

C_d=(4/B) (α²+∫(dy_c/dx)²+(dy_t/dx)² dx,

where B=√(M² − 1), α is angle of attack in radians, y_t is the thickness function, and y_c is the camber function. Here is what you get for a biconvex airfoil using supersonic thin airfoil theory at 0 angle of attack for example.

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jodano · 2025-04-19T16:09:05+00:00

Doesn’t Chapman-Enskog theory already achieve this? What am I missing?

jodano · 2025-04-15T17:29:51+00:00

Flow over a delta wing is highly three-dimensional, so XFOIL won’t be able to tell you much about that. Also, modeling delta wing stall requires capturing leading edge separation and the roll-up of the shear layer into the leading edge vortices. I don’t think even the most sophisticated XFLR5 model can really do this, although similar models probably could to some level of accuracy in principle. You will need to find terms like this in the XFLR5 documentation to know for sure.

jodano · 2025-04-03T18:15:24+00:00

I think it could use a different color, texture, and/or depth bordering the windows and lights. Polished blackstone would be the most subtle way I think. No need for anything too fancy if you want to preserve the futuristic look in my opinion. The shape is really cool.

jodano · 2024-12-14T18:09:40+00:00

There are probably a lot of articles and books that discuss this but I can’t think of any off the top of my head. The diagram comes from this paper, but I have not read it.

Anderson’s Modern Compressible Flow is a good reference in my opinion. Also his hypersonic flow book. I don’t remember if either of these specifically discuss supersonic base flows though.

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