The real secret of turbulence is the sequential , and thus discrete nature of it . It is quite clear that a streamline oscillates after impacting a density and lubricity/ viscosity region. This oscillation is generated along and at the boundary between the different regional substances. While it is easy to overlook it is plain to see that vortices not only dance around each other they oscillate sequentially relative to each other. This sequence is given the name phase which obscures its function. The sequential process is to seed, grow and protect the formation of a regional vortex until it is given birth to by reason of exceeding its dynamic boundary conditions, at the same time as this grows , the protecting vortex is shed by a thinning umbilical cord which is eventually and rapidly drawn into the anti flow around the developing vortex and becomes the seed for another anti vortex to grow and repeat the cycle.
The drive for this cyclical vortex production is the head pressure of the flow, but the inherent vorticity shapes the dynamics through the density and viscosit/ lubricity profiles. In a fluid, rigidity drops off on a sliding scale. Thus a rigid rotational description is misleading. The rotation of a fluid is easily shaped on its boundary. Thus to draw a line of circulation around a vortex is erroneous, while rotation implies a closed loop it does not imply a continuous boundary. In fact the boundary is best described as a collection of arcs. The nature of these arcs could be quite arbitrary, but observation suggests trochoids as good approximations or splines. Thus we may define a circulation boundary as an intersection of trochoidal arcs and the vorticity as the sum of the vorticity of the trochoids.
This then leaves sections of the circulation amenable to interaction, so that a powerful contra trochoid can knock out a circulation spline, but still. Lose the loop. The reduction in the circulation this implies can lead to a vortex core with higher vorticity peeling off and running against the unravelling circulation boundary.
The collapse of the outer boundary allows pressure deformations to seed a contra vortex
The determination of the rotational profile can be modelled by the winding of a piece of elastic. The tension that moves the fluid behaves like a compression or stress front that travels at a fixed speed along a spiral surface rotating out against a spiral surface being dragged in and thus creating another compression or stress front between successive layers of the first. This continues until as many interleaved spiral urfaces as can be supported are achieved and the whole fluid is subject to these cylindrical stress surfaces. The vortex movement is then block like but not rigid. Individual sheets may reach fracture points a and shear away from the elastic like sheet allowing local turbulence to develop. This may grow and lead to a collapse of the sheets dependent on it, or form a trapped bubble that rotates in the vortex, depending on the local density, lubricity and viscosity. These bubbles may be long lasting , intermittent or explosive.
The behaviour of fluids is certainly very interesting and capable of all inconceivable motions! …
Let us consider a fluid flowing in a dielectric in laminar flow
The pressure driving the flow is uniform orthogonally to the flow and appears uniform through the flow.
I place a thin cylinder in the flow and observe drag phenomena around the cylinder
I place another identical cylinder parallel to the first and thereby create a channel flow within the dielectric based flow.
The complex dynamics between the cylinders can be observed in similar general terms to a fluid flow with low viscosity, high lubricity, low density, and low turbulence.
While this is a general description in fluid mechanics it makes several assumptions worth modifying
The flow is laminar, but not free of vorticity or turbulence. I do not assume that laminar means that turbulence Is not regionalised, nor that vorticity is not regionalised into larger scale structures that adopt a lower turbulence and vorticity in a larger scale structure.
Alternatively I could view the uniformity as cascading down into lower scalded regions which therefore maintain regional uniformity until sufficient conditions are met for them to exhibit turbulence fractally or scale free. Similarly for vorticity.
Fluids and bubbles seem to indicate the first condition.
The flow between the 2 cylinders may now partially be analysed using Bernoulli's principle and indeed all the principles of fluid dynamics.
Starting at the blunt ends of the cylinders we consider the boundary layer flow, the velocity gradient exhibited, the vorticity within that velocity gradient and the establishment of subsonic and supersonic regions within this boundary layer, and the strain conditions involved. We also will need to look for a sub luminal and super luminal regions.
In these regions we need to study the choking and blocking regions as the contribute to the disposition of vorticity in the boundary layer, the growth of the vorticity layer and it's separation from the cylinders to shape the laminar flow.
The wake of this disturbance, it's spread and oscillation and the influence of the pressure in the laminar flow between the cylinders and external to the parallel system.
By looking at the behaviours of supersonic flow I would hope to gain sight into super luminal flow and how the shock pressures' vorticity propagate in the channel flow.
The turbulence in a supersonic flow is streamline in appearance, and is accompanied by shock pressure waves. But the supersonic flow is determined by the Venturi shape, and diminishes as soon as it widens, allowing a growth in the stream turbulence profile, which can be later blocked by another Venturi.. Jet flow out of the Venturi is accompanied by back rotating vorticular turbulence.
All these regions need to be examined and their combined effect understood, plus the generation of heat , light and sound.
The reference frames for fluid mechanics have to consist of at least 3: two external and at least one local to a region. The axes of the frames ( including polar axes) can be used as axes of " rotation": that is moments , vorticity, torques twistors, twistorques can all be described within this system of three frames of reference. The local frame is therefore not a rigid link to the external frame and so fluid regions of pressure can be coordinated and understood in terms of their local dynamics, while having an underpinning transformation relationship to every other reference frame via an external one. To achieve this it is customary to choose a local reference frame that has its origin at a centre of symmetry, but this is not a necessary practice.
The ability to resolve a general rotation in a given frame to axial rotation components is very useful , but it must be properly understood. For example: the thre rotational axes generating a rotation on a point given as lengths along the axes cannot be used independently of that " translational " reference frame coordinate. But as soon as a combined rotation is described the translational reference point is changed and thus has to be updated, thus a full understanding of Newtonian motion depends on this fundamental iterative/ recursive process.
It is always possible to describe a displacement as a mere change in the translational reference frame, but this fundamentally leads to anomalies in observation. The usual assumption is a straight line translation no matter how infinitesimal. This assumption leaves out vorticity that is angular rotation, and causes consternation in the empirical data when " anomalies" of orientation and position occur! As complicated as it may be, which is why Newton reduced his formulation to point masses etc, the actual motions of an extended body( ie having a local reference frame) have to be included. Today's computational power makes this essential standard a pragmatic reality. Thus we see the rise of games with "real physics" included in the behaviours of moving objects in 3d simulations.
This throws a revealing light on the hunt for a mathematical models of rotation. Rodrigues found the rotational matrices based on 1/2 angles before Hamilton discovered Quaternion rotation. The merest difference in result apparently lead to Gibbs denouncing Quaternions in America and Kelvin in Europe, the Quaternions always gave double the expected rotation, due to the naively of those using the tool. Instead of realising they needed to half the input angle they blamed Quaternions for all sorts of Ills! When Rodrigues equations were eventually rediscovered they could see that it explicitly included a half angle process in the formulation! Nevertheless quaternion notation was cumbersome and difficult to work with due to its overwhelming appearance, something Hamilton worked on in his final work "The Elements". However it was a good few decades before the notation we use today was developed, making it easier to write but obscure to compute.
We have traded off the ease of notation against the rituals of computation. Fortunately computers can perform these routines easily for us, making any method of computation and combination now a breeze. As a consequence, Hamilton's Quaternions have proved themselves superior in many applications requiring rotation to be included as well as perspective.
In a fluid motion is not relative to a frame of reference but to a system of spherical pressure centres that also move relative to each other. Thus a fractal distribution of local reference frames is posited with a dynamic relationship between each other and the external reference frame.
Such systems are now computationally possible.
Finally in such a system it is possible to have 2 contra streams of flow which pass by each other with little proposed interaction( high lubricity low viscosity, low turbulence). This requires an "anti "matter proposition. I consider this in a later post, but here would draw attention to reflection properties of fluids, so far not discussed, and refraction and diffraction. The important, but not intuitive notion is that separate stream lines can exist side by side , particularly when fluidsin each stream have different density characteristics.
The oscillation of the wake can " flick" fluid elements into adjacent streams , making large scale vorticularity possible , especially if an anti streamline exists.
There is empirical evidence that such a structural system exists between 2 capacitor plates.
There are many substances, and some are substances within substances. We have known this as a culture and as individuals for a very long time. But we have also made up myths and facts about these substances. The most enduring have been those that we call chemical.
In days gone by our ancestors named the things we know earth and sky. And in the earth they found minerals and water living plants and living animates. In the sky the found lights and lightnings air and winds clouds and smoke , rain and dust. From the earth erupted fire and brimstone, from the sky fell fire and lightning and thunder rolled . And our ancestors reacted by doing whatever it took to survive!
Some responded by gaining wisdom, and passing that wisdom on to future generations. That wisdom was recorded as myth and legend, ritual and magic trial and error and success and failure. Culture and tradition encapsulated it all, war and attrition honed it into weapons of attack and defence commerce partitioned it into processes and wealth, each contingency adding to, taking away, shaping our wisdom and knowledge, creating mysteries and mysticisms to shore up power and advantage.
There are many substances, and many are found within others. There is no end to what is within. As we found more and more within alchemists recorded the substances and the processes to extract that which was within. These were explained as earth, water, air and fire. By these and adding biological sacrifices gold silver iron copper were extracted, but much was given up as smoke.
Today our chemical knowledge is huge and systematic. We forget that we have developed from the simple observation that there are many substance and some are within others. There are more substances within others than we think, but our periodic table has systematised our naming of the substances, it has quantified certain substances and it has organised them into a synthetic structure that enables chemists to make predictions, but it has not changed the fundamental relationship between the elements, that some are within other compounds which are within other substances which are within the earth, the water, the wind and the fire!
We have a powerful technology, a powerful systematic knowledge, but these facts are modern myths. We should not let our modern mythologies obscure the essential mysticism which they contain. The mystery which our minds ponder through philosophy has the potential to reshape our myth making and fact making. As powerful , utilitarian and reliable as our present systems are, yet there is more we are not told , much we do not ponder which would transform our apprehension, reshape the configurations of power, liberate the economic slaves and usher in a utopian egalitarianism, when we recognise the way symbiosis and antibiotics and dominance really works for good and or ill.
At the moment we all could die at any instant. Some want to survive even that eventuality, so resources that benefit the few are established, but even these plans are useless against our inherent frailties. Our sensible course has bern known and taught for millennia. The Koheleth tells us to enjoy the now. The Kohelth says be moderate in all things. The shamans say live in harmony with every other animate and process in the system, but live in hope.
There is one wisdom that is millennia old, that is substances have substances within them; the grosser ones have finer ones and so on ad infinitum! Mandelbrot tacked this scale free fractal structure at a time when iterative machines could use the recursive equations that other ages had had to make explicit, even though they new they were implicit. Suddenly he was able to make wonderful "geometrical" patterns based on these recursions. The words he used to try to get past the tendency to big up recursion as some mysterious process serve to illustrate the rhetorical skill of Kairos. He said that "a fundamental process of fractal geometry is based on almost self similarity". Almost self similarity is a description of recursive iteration and recursive or implicit equations. Such equations are not nor ever can be identities.
So their are many substances in fluid mechanics, but the recursive formulations of quantitative measure are the same up to homemorphism. That means: we use different constants, but the same recursive formulations.
There is a fluid that flows within a dielectric, some call it electric some call it magnetic but this fluid is in all and of all. I awoke and realised that the magnetic fluid seeps out of the electric fluid. The more the electric fluid is stressed the more the magnetic fluid seeps out.
There are many substances and many are within other substances, but the electric substance is like air, like a wind in all other substances. And within this electric substance is another substance called magnetic. This too is like air or wind and it seeps out of the electric substance when it is pressured and in motion.
Triboelectric describes electric fluid that is pressured out of substances. Tribomagnetic describes magnetic fluid that is pressured out of electric fluid. By " rubbing" 2 electric fluids together magnetic fluid is squeezed out.