Fractal foundation of Fluid Mechanics

[jehovajah]

Deep within our thought world rotations are embedded as entities that exist spatially. It has taken me a while to apprehend that ineal motion is in fact singularly rare! Vortices do not need to be generated in my opinion, because they exist at all scales. What I am slowly reaching for in this thread is a method of synthesis and analysis that is founded on this Förderung, that is the idea I am promoting.

Claes Johnson shows numerically that vortices form part of the solution for explaining flight. The drag on the wing , the frictional forces of the boundary layer attached to the wing create an instability that results in vortices streaming from the trailing wing edge.

In fluid dynamics these are traditionally called turbulence, but Claes has shown they are initially well formed vortex plumes.

Drag does not create vortices per se, because the idea by Prandtll that boundary layers separate at the crest of the curve for an aerofoil is not physical. Instead the boundary layer separates as it must at the trailing edge creating high potential instabilities that spin out as vortices, and consequently have a low or mean zero potential!

Also, the aerofoil shape creates a venturri effect in which the lower pressure sucks the wing upwards

At supersonic speed however this description alters. The high compression of the boundary layer creates a " solid " like fluid mass which can no longer follow the conyours of the wing . The separation of this boundary layer produces profound instability as a shock wave of sound, pressure and heat and electromagnetic properties promulgates through the surrounding medium. The vortices are do powerful that they form a cone, and the aerofoil is heated and stressed by their powerful dynamic . Drag and lift are compromised as the plane is held aloft by powerful impulse forces generated by vorticular variation. Sometimes thes can reverse the control surfaces generating gravity instead of levity!

These are the kinds of fractal dynamics I want to understand better.

http://claesjohnson.blogspot.co.uk/2012/11/lifting-line-theory-illposed.html

http://secretofflight.wordpress.com/

<a href="http://www.youtube.com/v/5WKU7gG_ApU&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/5WKU7gG_ApU&rel=1&fs=1&hd=1</a>

These kinds of vortex plumes are common in fluid dynamics, but we tend to ignore them in our thinking as Claes points out.

They have a Barycentric connection which Helmholtz seems to have left out of his theorems, as did Kelvin.

Defining Torque as vorticity obscures rotation, because rotation has many axes or barycentres and the vortex Barycentre is the systems rotational one in hich centrifugal and centripetal acceleration balance.

If centrifugal does not balance centripetal acceleration then body deformational forces arise and the shape is deformed, resulting in a new Barycentre and new tangential contact points.

Vortex rotation starts when centripetal and centrifugal body forces balance around the Barycentre which becomes a vortex Barycentre.

Current definitions of torque should not be.confused with rotational vorticity. However torque explains shear deformations of a body and when a body will break into several masses that spiral around the vortex Barycentre.

The notion of rotation as vorticity and the notion of Barycentre allows fluid dynamics to describe fluid element ensembles using a youngs modulus approach or a spring mesh with a damper and a youngs modulus distribution ,

I can now model a fluid dynamic not by Kelvins kinetic theory but by a Barycentric mesh kinetic energy model! Instead of Root mean collision paths we can use Barycentric and tangential contact point analysis to. Odell the fluid element deformation and include spring and damping forces with modulii reflecting the material nature of the fluid.

These spring and damping forces model the electro Thermo and magneto interactions that are evident in matter.

While we cannot leave out collisions ltogether, the nature of most of these are elastic near misses modelled by electro Thermo magneto  considerations, or simply gyroscopic rotations of point mass or Barycentric centres fractally distributed in the volume. Some direct collisions of barycentres must be figured in.

If youngs modulus for a material is exceeded then cracks and complete bodily separation has also to be figured in.

For a gas it is usually assumed that this is the case, but pressure behaviour in the kinetic theory has little to say on this matter.

[youhn]

Just a small addition to this topic.

Benefits of turbulence small-scale-vortices on the slightly bigger scale properties: drag-reduction and self-cleaning

<a href="https://www.youtube.com/v/9nxRO1EhmmM&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/9nxRO1EhmmM&rel=1&fs=1&hd=1</a>


Additional sources:
http://rsta.royalsocietypublishing.org/content/368/1929/4775.full.pdf
https://www.beilstein-journals.org/bjnano/content/pdf/2190-4286-2-9.pdf

[youhn]

In fluid dynamics these are traditionally called turbulence, but Claes has shown they are initially well formed vortex plumes.

Turbulence is not equal to chaos or badly-shaped vortices. It has to do with things happening at smaller scales than the scale which is used to evaluate the fluid system. In computational fluid dynamics it takes to much computing power to model all those small vortices, which of course eat up a little of the total energy and convert it to heat. Instead of modelling these all smaller-scaled and transient vortices, the loss of energy is accounted for in a turbulence-model. Most of these are both space-averaged and time-averaged. This makes it possible to carry out steady-state fluid dynamic simulations. To obtain a more physical faithfull results, a very fine mesh, no turbulence models at all (just the navier-stokes equations) and a transient approach is needed.

Source:
http://www.cfd-online.com/Wiki/Turbulence_modeling
http://www.innovative-cfd.com/turbulence-model.html

[jehovajah]

Thanks Youhn.
The initial start of the plume I think is a vortex, but very quickly it becomes a complex structure I thought might be turbulent because of all the vortices implied . Frankly at this stage I do not know how to describe it, but claes computation gives well defined vortex plumes , which may be physically observable. The examples I had in mind always described it as turbulent, but then they admit it was too complex to describe it by any other name. The fractal structure was observed however.

[jehovajah]

The importance of the Barycentre AND the tangential contact points in a description of space behaviour under strain reminds me of the strain Ellipsoid!

A search is intriguing as it hints at this being used in slowly deforming rock , but as yet I have not found a good enough reference to mine!

However this link
http://www.researchgate.net/publication/22681376_Perceiving_the_centroid_of_configurations_on_a_rolling_wheel/file/e0b4951a510855345d.pdf
Shows how our perception is naturally attuned to these kind of relationships..

For me the use of the Barycentric calculus of Grassmann epithet his tangential contact points could be linked to the ideas of the strain Ellipsoid. This then gives fluids a representation by this structure which simplifies regional computation of fluid behaviours, potentially.

http://journals.ametsoc.org/doi/full/10.1175/1520-0469(1998)055%3C3358:SALRNF%3E2.0.CO;2

A discussion in the   Second topic heading  refers.

[hermann]

But in practice ... what would higher dimensional rotation mean for (an engineer like) me?

Hallo Youhn,

When we speak of rotation we always have a rotation axis. You can think of this rotation axis as a vector. (Right hand rule)
In three dimensional space this vector has three components in higher dimensions it has more components.
One component for each dimension

Here is a short over view on the rotation of an vector (not the vector of the rotation axis)

A three dimensional vector can be written in the following form:



A four dimensional vector can be written in the following form:



No problem to write down even higher dimensions:



In mathematics you can perform rotations with the help of special matrixes called rotation matrixes.
Let be the rotation matrix, be the original vector and the rotated vector:



In this form a rotation can be seen as a simple matrix multiplication. (No archmetrics with sin, tan, cos etc.)
Even in higher dimensions!!!
That's the trick.

In three dimensions you have a 3x3 matrix for rotations. In four dimensions you have a 4x4 matrix, in higher dimensions a nxn matrix.
This kind of matrix operations (rotations) are called ortogonal transformations in mathematics. If you allow complex numbers as components of the vector and the matrix this is called a unitary transformation.

It is also possible to derive the rotation axsis from a rotation matrix and vise versa. The vector on the rotation achses is allso called the "Eigenvector" of this rotation. A more abstract formulation of Eigenvector is used in Quantenmechanics.

[jehovajah]

This lecture on stress strain fields relates to the topic.
<a href="http://www.youtube.com/v/r8KzP7G7Uks&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/r8KzP7G7Uks&rel=1&fs=1&hd=1</a>

In this case density of materials allows stresses to act more slowly in the materials showing the strain fields. For less dense materials stresses act more quickly and so the strain fields propagate more quickly to catastrophic collapse or permanent , liquid deformation.  In fluids the strain field propagates almost instantaneously, catastrophically collapses almost as quickly and fragments are dispersed by the environmental stressors..

Within the body of any fluid the environmental stressors are likely to be hydrostatically arranged ( in a gravitational field) or symmetrically arranged by electro Thermo magneto complex energy rotations. Thus stress introduced internally should create strain propagation that leads to catastrophic collapse , but those products are held together and are  dispersed by the environmental stressors or flow currents. This would be the basis of vortex shedding?

[youhn]

In this case density of materials allows stresses to act more slowly in the materials showing the strain fields.

What do you mean when you write about "stresses act more slowly" ... ?

For less dense materials stresses act more quickly and so the strain fields propagate more quickly to catastrophic collapse or permanent , liquid deformation.

What are strain fields? When you say liquid deformation, do you mean plastic deformation (as in opposite to elastic deformation) ... ? More keywords could be Young's modulus and creep (which adds some viscosity to solids like PTFE materials show very much, but also all kinds of steel at elevated temperatures).

In fluids the strain field propagates almost instantaneously, catastrophically collapse almost as quickly and fragments are dispersed by the environmental stressors..

This sounds like the reverse of the statement above, which was about dense (solid?) bodies. If I understand correctly, then I agree. Forces acting upon fluids generate almost instantenious plastic (non-elastic) deformation, while dissipating all internal stresses rather quick. Forces acting upon solids lead to very slow (almost non-existing) plastic deformation (called creep), while elastic deformation happens almost instantenious together with the internal stresses.

Within the body of any fluid the environmental stressors are likely to be hydrostatic ally arranged ( in a gravitational field) or symmetrically arranged by electro Thermo magneto complex energy rotations.

What do you mean by "hydrostatically arranged" ? Same question for the symmetrically arrangement.

Thus stress introduced internally should create strain propagation that leads to catastrophic collapse , but those products are held together and are  dispersed by the environmental stressors or flow currents. This would be the basis of vortex shedding?

Yes, internal stresses lead to strain (propagation). But what do you mean by the catastrophic collapse? Collapse of the fluid? The shape of the fluid? The relation between stress and strain ... ?  

I'll scan the lecture later this evening. Might answer some ofthe questions.

[youhn]

I think you mentioned density as a key word, but probably in another topic. But since the whole universe can be seen as a fluid on the bigger scales of time;

Titel: If the Universe is expanding, why are galaxies still merging?
source: https://medium.com/starts-with-a-bang/26c0f36ddc89

The Greek already knew; Panta Rhei.

[jehovajah]

Youhn I apologise for my lazy editing and correction policy! Typos abound and I only correct what I spot or if I am reviewing a post to answer a question.

I am not a materials engineer and barely study above A level physics if I can help it! Youngs modulus I understand , plastic deformation in rigid materials I understand intuitively from pictures and graphs.

In a solid the elastic limit is reached and breached when the rigid material does not return to its dimensions. The plastic deformation of a solid or creep is like bonds slowly snapping gradually exposing the undersurface material. Slow snapping is where a crack appears in the surface and propagates inwards to the centre. Meanwhile the exposed faces deform forming a kind of bottle neck. I think I used liquid deformation to mean plastic deformation, and I apologise for any confusion.

Again catastrophic collapse is my term to describe when the material separates into distinct fragments with their own surface tensions. When this occurs they are surrounded by the environmental material
And interact with it as a fragment.

Strain fields are my terminology for fluctuating regions of strain. The fluctuation, from one point of view may look like progressive density variations. From another it may be a standing density variation, a region suddenly increases or decreases in density.

Density is a fundamental Archimedian concept measured by displacement of a fluid which is weighed and the proportion used as a definition of density. Because of this I used hydrostatic to describe one kind of density variation. This is akin to gravitational compaction of matter.

The propagation of strain fields creates tensile stress variations. Sometimes these stress variations are slow especially larger amplitude strain , giving rise to large stresses in that region . But some strain propagation is associated with fast minuscule amplitude propagation. The stresses that these strain fields initiate are minuscule. These fast propagations are typically in denser materials.
<a href="http://www.youtube.com/v/-wgci_qDAy8&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/-wgci_qDAy8&rel=1&fs=1&hd=1</a>

[jehovajah]

Certain Hugh speed events reveal the fundamental fluidity of matter.

<a href="http://www.youtube.com/v/QfDoQwIAaXg&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/QfDoQwIAaXg&rel=1&fs=1&hd=1</a>

What is still harder to see is the fluidity of light, but at a trillion frames per second even that is revealed.

The notion of continuity has to be changed. Continuity has to be a signal that we have reached the limit of our ability to distinguish!

For example, much of our mathematical modelling provides us only with frames showing actual positions as calculated or differences as calculated. This means, for a rotating object or physical region it would appear that it rotated along a continuous arc without physically passing through the space in between calculated frames.

It is only when we run the frames together that we notice in the "film" effects that we unconsciously " fill" in the causal pathway for. This causal pathway is what we experience as physicality.

Our formalisms or mathematical models can not tell us the full physical picture. We have to intuit that ourselves. In so doing the notion of cause arises in the sense of continuity. However, we can also consider cause in the sense of contiguity! Thus we experience the notion of first or most primitive cause.

First cause or most primitive cause occurs as an experience where the observed or calculated behaviour has no continuous physical interpretation. Thus my knowledge of an event is initial and I assume that all motion is prior to this initial knowledge state. Then as my knowledge tracks the changes in each frame I intuit a physical continuous cause.for motion( say, Newtonian motive). In this intuitive ascription, concepts like density and reaction and interaction are appended.

Now when a motion becomes apparent, at some stage in the middle of the " film" sequence, it will have no prior frames to intuit its cause in a continuous model. This is when I intuitively shift to a contiguous model of cause, and intuit a first cause at the frame or between the frames where the change is first noticed..

Contiguous causality really allows for the mysterious and mythical explanation of causality to have its justifiable place.  Between empirical observations and intuited continuous causality we must allow for contiguous causality, and that means unobserved causes which reveal themselves only by a change in observed motions.

No matter how rigorous we become continuity and contiguity support both myth and fact making in science, so called.

Contiguous causality inspires us to search more carefully and to have a broader view of causality. So for example a change in motive behaviour may accompany a change in local density, and this may be sufficient to simply explain observations. But as Maxwell Observed, we cannot rule out the little daemon in the fabric of space!

It is therefore a personal choice, outside of empirical observation and deduction and induction whether whether you rule out or rule in some none observable reality , apprehended o ly by its effects on what is observed. But certainly, to develop a mythical world reality on the basis of empirical data is not supportable by empirical data. Empirical data used in this way serves only as a metaphorical or analogous description. Thus if one accepts this implied causality one is free to creatively mythologise it, but with no moral authority or empirical one.

That being said, one being free to believe as one wishes does impact on what one ultimately observes, and recognising that makes any notion of ultimate truth a miasma of coercive opinions without ultimate justification.

To claim the scientific method as the source of ultimate truth is as misleading for scientists as it is for those wo support their religion by it. Absolute anything is absolutely misleading! Lol!

It is possible to live approximately, always learning, always skeptical, but always striving to improve to the benefit of the majority who want it, or a minority who do not reject the change.

So back to the fluid nature of physicality: contiguous causality is associated with rotational motion and rotational motive/ energy., almost by definition

[jehovajah]

I had a singular question in my mind!
How could normal forces contribute to tangential force in a viscous medium?
I was watching the following video in which the viscous properties of water are apparent.

<a href="http://www.youtube.com/v/i8nZ7B1h5Mw&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/i8nZ7B1h5Mw&rel=1&fs=1&hd=1</a>

It is clear that a distribution of velocities and thus forces shape the evolving structure, but always a core flows into a rotating ball that breaks away from the supporting and transmitting " rest" of a plume.
So how does laminar flow become rotational!

The answer is all flow is rotational. Thus when constrained to be laminar by some energy constraint it represents a suppression of a curvature.

Newton recognised this dynamic behaviour by establishing it within his inertial reference frame.

Most confuse Cartesian coordinates with the inertial reference frame. In fact the advanced reference frames respect curvilinear coordinate systems in dynamic rotational motion. This sounds and is complex, but Newton simplified it to a combination of Cartesian and polar coordinates combined! In particular his centripetal and centrifugal axes are radial And are combined with a tangential axes and a circular arc axes curving from the point of tangency.

Thus any tangential force also has a curving arc force associated with the point of tangency.

When a tangential force contacts a resistive surface, 2 forces are generated that are equally opposed: an anti tangent and the anti tangent arc! This arc is responsible for the notion that normal forces can contribute to tangential forces! By altering the centripetal centrifugal arc or polar reference frame a tangential force is generated instantaneously. In fact it is not a " tangential" force but a tendency for the centre to move!

<a href="http://www.youtube.com/v/51-6QCJTAjU&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/51-6QCJTAjU&rel=1&fs=1&hd=1</a>

This tendency is also measured by viscosity and the Reynolds number. It is a measure of how likely a centre of rotation is to move in a direction or a preferred direction.

Viscosity is a measure of how proportionally connected normal and tangential forces on a surface are. The Reynolds number therefor measures this tangential point arc force and thus is a measure of force curvature at the point of tangency . It is clearly related to the substance surface boundary as well as the fluid substance the surface is moving through.

Instead of viewing the normal and tangent axes as the reference frame for curvature it is better to view them as resolved axes of curvature, that is generated by the curvature! In this regard an orthogonal reference frame is generated by a curved surface using tangential line segments. Obtaining any 2 non parallel tangents to a point on a curved surface allows a normal to be generated.

However the movement of the surface is not describable by a single normal or tangent. It is better described by spherically rotating elements.

ThE Reynolds number gives a kind of average curvature for these spherical elements, but what an instantaneous surface looks like  is an aggregate of these curved elements, their curves of intersection and their interaction with light spherical rotations.
Normals and tangents enable a surface plot of triangles and parallelograms to give a rough idea of the spherical elements aggregate surface structure.

[jehovajah]

This is a related thread.

http://www.fractalforums.com/new-theories-and-research/revisiting-the-riemann-sphere-(again)/

[jehovajah]

From Roger Bagula's fractal news email.

http://www.sciencedaily.com/releases/2014/03/140321095023.htm?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+sciencedaily+%28Latest+Science+News+--+ScienceDaily%29

And Escher patterning

http://phys.org/news/2011-06-exploring-tessellations-escher.html

[jehovajah]

Rather than make analogies to liquids, I am making the not unreasonable assumption that liquids and gases( classical fluids) are analogous to the electro Thermo magneto complexes which form fractally distributed regions described by us subjectively( or by me at any rate) as SpaceMatter or plasma, and universally as simply space.

Thus the viscous properties of space exhibit themselves to our perception as overall inertial frame effects. The distinction between inertia nd viscosity disappears at the universal level. Inertia is a local phenomenon of viscosity of space.

Viscosity is of empirical experience, non uniform. This aria tion in viscosity also is apprehended as density. Again density is a local variation in viscosity.

The aspects of space described by the Newtonian Fluid motive, roughly equivalent to modern notions of energy, are the behaviour of this viscous medium in mediating acceleration within itself and it's fractal regional parts. Rotation is fundamentally necessary to provide sufficient ground for regionality and thus variation.

I see that the viscosity of a region , under rotation, creates a contiguous divide, in certain cases the divide may store a returning motive called relaxation motive . This relaxation motive is also called surface tension.
Density variation due to viscosity reveals itself in boundary construction. Boundaries do not form static ally, they represent a dynamic interface where rotational motives creat active and reactive forces through resolvable accelerations. Newtons inertial reference frame concepts enable us to resolve these accelerations into normal and tangential accelerations, however trochoidal the accelerations actually are.

Why resolve into these 2 orientations?

Well in our simplicity we only want to know if 2 identified regions are going to remain together or come apart, And if they are rotating! A net zero acceleration means the regions will maintain relative positions. A net zero rotational acceleration means regions will retain relative orientation, any thing else is likely to be too complicated to explain.

Gravitational behaviours is one of those non zero situations. While we focus on gravity we completely ignore levity which also occurs and is not uncommon in space. However our notions of density obscure these distinct force situations, and are compounded by our willingness to generalise local knowledge to a universal scale!

The viscosity of space and of plasmas seen astronomically reveals a fractal distribution at all scales of this variable property. The attendant energetic properties of a viscous medium can be gathered under a tripartite description electro Thermo magneto complex. Of these Thermo magneto is the scientific ground based on the work of Gilbert who examined more closely and more empirically the notions of the ancient observers and philosophers.

We may safely remark that the amazing viscous behaviour around magnets has been in want of anaccommodating notion which links it naturally to chemical and mechanical behaviours through the empirically evident vortex phenomena. That electric magnetism was allowed to obscure the philosophy is just one of those all too common mis steps men have made in the furtherance of their egos, statuses and social reputations, and latterly financial gain.

While the study of this viscosity in space has progressed through the study of its effects in fluids, it is not to be overthrown as an abstracted notion, for all notions are abstracted. Rather it is a pragmatic choice, for the doorway to ideal or formal thinking is in the control of the pragmatists.

It is the pragmatist who surveying the empirical data, elects from the many choices the most practical compromise. This compromise is therefore ideal! It may only ever be approximate to any empirical experience, but it's use is in freeing the artisan to move on to active construction.

Thus as Newton observed, mechanics gives rise to geometry which in turn perfects Mrchanics in an endlessly iterative process.

In this regard fractal geometry serves a fundamental purpose. It teaches those who will learn that complexity can be arrived at from simplicity, and that having pragmatically chosen an ideal form, it is not to remain static, but to be dynamically employed to improve the accuracy of the developing model.

Our forebears spent great amounts of thought and practice in ascertaining these ideal forms. We now must apply them iteratively to create better fractal models of our empirical experience.

What should hearten us to do so is that as we have developed better magnifying tools we have found these ideal forms existing at the very smallest scales. The Socratic platonic game of which is real, the form or the constructed form has its answer: both are real, and we are connected to our universe at levels we do not realise!