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Doesn't exist, it's trying to simplify and approximate the truth. Flow is 5 dimensional, we assume 2 orthogonal axis of spin, and we call the other stuff secondary flow. Flow is locally structured in 3 dimensions, and translated in another 2. Locally structured in x, y, and z planes, and the other 2 dimensions are borrowed from electromagnetism, divergence and curl. Divergence is basically pressure gradient, and curl is basically vorticity.

Aerodynamicists create formulas and approximations to encapsulate all 5 dimensions of airflow. Prandtl's lifting line theory, Martin Kutta's theories regarding circulation, and the subsequent Kutta-Jukowski theorem. Which is really just a description of flow around a cylinder, with an applied Fourier transformation. As we all know Fourier transformations are 2-d projections of more complex phenomena.

The problem is we are pea-brained, it's difficult for us to see the nested donuts in airflow, and the 5 dimensions they encompass. It's bad enough we can't see the wind, but worse our tools for analyzing it are dependent on butchered approximations.

A while ago I saw a new propeller design, and then it hit me like a ton of bricks.

https://www.youtube.com/watch?v=MNnB_50Z20I

This thing is just ribbons of a torus.

https://i.ytimg.com/vi/px5kZn_hPbU/maxresdefault.jpg

The vortex ring is a torus, a non-destructive interference pattern. The vortex ring encapsulates all 5 dimensions of flow. X,Y,Z planes of symmetry, divergence and curl, all in one.

https://youtu.be/NhZ08el9IfQ?t=2743

This stuff applies well to anyone dealing with turning vanes, or strakes to guide flow. Often underestimated or outright missed by CFD is the 3D nature of the flow and secondary flow structures between strakes/compressor rows.

In order to understand fluid dynamics, I felt it would be helpful to share how air behaves under different circumstances. This will be a multi-part series, where I describe air and why it flows the way it does. I will try to keep things simple and easy to understand, with little to no math. Any equations I present will be explained in simple terms that are intuitive.

Air is capable of creating incredible forces, it keeps planes up in the air, it can crush steel drums, it keeps water as a liquid, how is this possible? This is because air has mass, at sea level the pressure is nearly constant and there is a pressure of ~14.7 pounds over every square inch of area. This pressure is pressing on all objects from all sides, as above so below the saying goes. It is this fact that we use and exploit in order to create aerodynamic forces. Air wants to fill everything with itself, and whenever something moves through the air, the physical object removes air for a split second, and as a result air will try to fill that displacement. This is true of fluids in general.

You may have heard the term "nature abhors a vacuum" this is especially true of air. It is the motion of air attempting to fill that vacuum that creates aerodynamic forces. Then following this logic, air moves as a result of the difference in pressure, the default atmospheric pressure, or what is known as "static pressure" will move towards anything that is less than this standard pressure.

Where it starts to get complicated is the path the air takes to fill that difference in pressure. Logic would tell you that the most efficient path between two points is a straight line, so naturally you would think that air follows the path of least resistance, I.E. a straight line. However this is not the case, air rarely if ever flows in a smooth straight line except under the most carefully controlled conditions. By and large, air takes a curved path, that is its preferred path of motion, why does air do this?

The primary reason is that air has mass, and as a result it is subject to inertial and viscous forces. Thus we have the Reynolds number, which accounts for the inertial and viscous forces in a packet of air and serves as a rough estimate of when a flow becomes turbulent. The Reynolds number is estimated as the density of the fluid times the velocity, times the distance the fluid covers, over the viscosity of the fluid. The smaller the Reynolds number the more likely the flow is to be laminar, linear and smooth. Inversely, the greater the Reynolds number the more likely it is to be turbulent and chaotic.

This means that increasing the size of the air packet, or parcel, increasing the velocity of the flow, or its density will give you a greater reynolds number, in other words a flow more likely to be turbulent and chaotic. Inversely the greater the viscosity of air, the lower the reynolds number. Interestingly, the viscosity of air increases with temperature, and its density decreases. Thus temperature has a big influence in the tendency of air to become chaotic and turbulent, however this relationship is not linear.

Another reason why air likes to take curved paths is a result of the speed air moves at, this is because as mentioned previously air has mass, as a mass of air moves, it displaces air, and as a result lowers the static or default pressure of a given volume. Bernouli tried to capture this in his famous equation. As a result of moving or displaced air in a given volume the surrounding air will move to fill that slightly lower pressure from all sides. The easiest way for something to surround something from all sides is a sphere. However, if the air keeps moving, the leading part of the displaced air will have higher pressure as it butts or collides with the standard or static pressure, and a lower pressure on the trailing side. Thus, the tendency of air to fill that low pressure will be curved.

In the next part, I'll cover more advanced aspects of airflow, including why vortexes happen, thank you for your time.

Suppose you have a pipe with x amount of air flowing through it. Suppose the flow is nice and laminar throughout the pipe, it's velocity distribution will look something like this.

https://files.catbox.moe/ysa5y8.jpg

The flow near the walls will be slower because the drag from the wall's boundary layer and the moving flow slows it down, while it moves faster near the center of the pipe because the greatest mass of air is moving uniformly carrying its momentum forward with no wall to obstruct it.

Now, what happens if we put a vortex in the middle of that pipe?

https://files.catbox.moe/yk9slc.jpg

The vortex itself is occupying the middle of the pipe, as long as the vortex maintains its integrity, the air cannot pass through the vortex it must go around it. Therefore the vortex is acting like a flow restriction, reducing the cross sectional area of the pipe. The shear stress of the vortex squeezes the air against the walls of the pipe, and reduces the cross sectional area of the pipe. In order to maintain continuity, the flow speeds up, and the pressure in the pipe drops.

https://files.catbox.moe/phr2r0.png
This update has improved the car performance. The update I'm pointing out is the fillet radius at the bottom of the strakes. This builds up the pressure on the leading face of the strakes, and keeps the high pressure from bleeding into the low pressure side. This increases the vorticity of the flow downstream. Everyone else puts the bottom fillet radius in the direction of airflow.

https://files.catbox.moe/1gic06.webp

The Ferrari for instance tried to go with the flow with their strakes.

https://www.sciencedirect.com/science/article/pii/S1000936116300383

Very interesting study that shows that using a winglet on the pressure side of a rotor cascade improves rotor performance, particularly by extending the stall point. In other words, the pressure side winglet creates a higher pressure ratio at a lower rotor speed. This is achieved while having a negligible impact on drag.

"On the contrary, the
pressure-side winglet greatly improves the stall margin and
introduces only a very small penalty in efficiency. At peak efficiency point, there is an efficiency reduction of about 0.27%.
The predicted penalty in rotor isentropic efficiency is due to
the additional surface offered by tip winglet which increases
the additional skin friction loss. Moreover, the pressure-side
winglet causes a slightly higher pressure ratio near the stall
point relative to the reference case..."

"By applying pressure-side winglet, the stall range predicted
by the present work is extended by 33.74%. This shows a significant improvement in the stall range of the compressor
rotor."

This is highlighted in this image.
https://files.catbox.moe/37sing.png

The study further goes on to say

"In the condition with the suction-side winglet
applied, the shock wave/tip leakage vortex interaction is being
intensified which leads to a stronger change in the tip leakage
vortex structure. It is found that the tip leakage twists seriously
and a spiral type breakdown seems to occur at the middle of
the rotor passage. In the case with pressure-side winglet, the
tip leakage vortex trajectory is more inclined in the streamwise
direction. In addition, the distance from the first tip leakage
vortex appearance at the suction surface to the intersection
with the shock is longer than the corresponding distance in
baseline tip case. With the longer distance, the low momentum
core fluid is reenergized as tip leakage vortex mixing with main
flow."

https://files.catbox.moe/k2fytu.png


Not complete proof but evidence that pressure side winglets extend the stall range in a wing cascade, this would be useful in something like a current generation Formula 1 car that uses floor fences in a cascade arrangement. The regulations allow the use of the pressure side winglets as there is a 50mm fillet radius that is allowed on these cascades.


https://files.catbox.moe/ce7igv.jpg

https://www.fia.com/sites/default/files/fia_2023_formula_1_technical_regulations_-_issue_1_-_2022-06-29.pdf

3.5.2 subsection d states
"Once each Floor Fence has been fully defined it is permitted to apply a Fillet at the
boundary between it and the Floor Body, having radius of curvature no greater than
50mm. Such a Fillet would then be considered part of the associated Floor Fence."

As far as I can tell, this means that you can apply a fillet once the floor fence is within the bounds of the actual floor, it makes no mention if the fillet can be applied at the top or bottom of the floor fence.

Is to isolate the motion of the *front knee. There is no one way to do this, but it is very difficult to do.

https://www.youtube.com/shorts/DK9bIkQg4yc

Here you can see a perfect demonstration of the principle in action.

The knee can go up and down, but it cannot wobble side to side, or twist, it must work like a hydraulic jack, only moving up and down.

By loading the front knee with the weight of your torso, you keep the knee from buckling. This allows you to turn the upper torso independently of the lower.

This is in accord with Taoist principles, in order to generate power, there must be a yin yang separation. In this case, the legs are solid, the torso is pliant. That separation delivers great power and knocks the fuck out of the nigger.