Abstract:
One of the key characteristics of the Wisdom Approach is its ability to integrate point and range characteristics. However, a key parameter set requirement is the involvement of constructs of inherent instability. This poses an issue when attempting to assist in the development of a mathematical model (normally dealing with point estimate characteristics) to show how fluids work with respect to turbulence.
However, this article tries to deal with this specific Millennium Prize problem[1] because the underlying nature of how fluid (and gases) works deal with range estimate characteristics or quantum logic. Given the fact that I am not a mathematician, this article does not attempt to contribute by developing a mathematical model, but rather focuses on using the Wisdom Approach to guide mathematicians towards solving an important question with real life applications that have existed since 1900.
The conceptual model of the Wisdom Approach find that the current two equations used to understand turbulence, the Reynold’s number and the Navier-Stokes equation, are most likely incomplete in the capture of the phenomenon. The equations miss out critical elements, especially those dealing with the path dependent nature of how turbulence can be created, and quantum-oriented elements like variance (or inherent instability).
A conceptual model with stages has implications for mathematicians to utilize multiple equations to address the issue of turbulence, and provides with specific areas to look into in order to finally solve the turbulence problem.
[1] The Millennium Prize Problems are 7 unsolved problems in Mathematics that were stated by the Clay Mathematics Institute on May 24, 2000. A correct solution to any of the problems results in a US$1 million prize being awarded by the institute to the discoverer. This has echoes of a set of 7 problems set by the mathematician David Hilbert in 1900 which were influential of driving the
progress of mathematics in the twentieth century.
[1] The Millennium Prize Problems are 7 unsolved problems in Mathematics that were stated by the Clay Mathematics Institute on May 24, 2000. A correct solution to any of the problems results in a US$1 million prize being awarded by the institute to the discoverer. This has echoes of a set of 7 problems set by the mathematician David Hilbert in 1900 which were influential of driving the progress of mathematics in the twentieth century.
Introduction
Historically, the goal to understand how fluids and gases work have been undertaken by scientists from primarily the fields of physics and mathematics. However, traditional physics and mathematics are steep in point-estimate perception and approaches, which makes it difficult to deal with range estimate constructs with inherent instability. While there have now been significant advances in our understanding of quantum physics, the development of the field of quantum physics still takes a strong “hard science” approach. Even in mathematics, conceptual aspects which deal with theoretical reasoning, such as in the field of “Pure Mathematics” are strongly influenced by the point-estimate nature of mathematics. As such, I am hopeful that the thinking approach I have developed (the Wisdom Approach) will not be tied down by such a perspective as it was built to consider both point and range estimates simultaneously.
The nature of fluids and gases are such that they are inherently unstable. As such, a quantum perspective as taken by the Wisdom Approach is applicable to understand it better. However, the codification of how fluids work typically is represented in a mathematical form that deals with the field of physics. Unfortunately, this writer is not a “hard scientist”[1] and am limited by content expertise to be able to produce a mathematical model to explain how fluids work. However, what I am able to do is to utilize the Wisdom Approach to create a conceptual model that is accurate enough to help mathematicians to solve this Millennium Prize problem, and hopefully be recognized for the contribution.
Since this is an objective oriented article, I will dwell directly into the conceptual solution, with explanation to the logic behind the solution, as well as, to guide any mathematician towards what can be done to solve the problem.
[2] Brian R. Tan is trained in doing research in the social sciences and has a Ph.D. from the University of Washington, Seattle. His doctoral major is in the field of strategy (management), with minors in economics, statistics, psychology and entrepreneurship. The thinking approach he has developed, the Wisdom Approach, utilizes knowledge from all fields, even those outside his training and expertise, highlighting the maxim that “Knowledge is Universal”.
Setting Up the Basis for the Wisdom Approach’s Conceptual Solution
In order to derive the solution, I needed to understand the parameter set of the problem. Starting with a basic idea of how liquids and gases work, from the youtube video [3] that started this search for a solution, I went on to look for information that dealt with drag, turbulence, pressure, fluids and gases. From there, I will utilize two mathematical formulas which are relevant for comparison and contrast in this article. Specifically, these are:
(1) The Reynolds number for turbulence
(2) The Navier-Stokes equation
I will proceed to break down the two mathematical formulas so that they can be used as a benchmark comparison to the Wisdom Approach’s conceptual model.
The Reynolds number for turbulence is described as:
where: V is the flow velocity, D is a characteristic linear dimension, (travelled length of the fluid; hydraulic diameter etc.) ρ fluid density (kg/m3), μ dynamic viscosity (Pa.s), ν kinematic viscosity (m2/s); ν = μ / ρ.
It is noted that the formula has identified the elements of (1) velocity, (2) a combined identification of D which comprises of multiple characteristics (e.g. length, diameter of pipe, etc), (3) density, (4) dynamic viscosity which deals with the interaction between force and internal friction, (5) kinematic viscosity which deals with internal friction.
Specially, it should be noted that D would captures aspects of external friction and time. D is regarded as a proxy capture for these two elements.
A quick analysis of the Reynolds number highlights its limited usage because of its content specificity (study pertaining to fluid action in pipes). It considers 4 elements and an interaction between 2 elements as identified by the Wisdom Approach:
(1) Velocity / Speed – captured directly
(2) Density / Viscosity – capture directly
(3) Friction – internal friction captured by kinematic viscosity; external friction proxied by tube characteristics under D.
(4) Time – proxied by the interaction between velocity and D (specifically the length of pipe)
(5) Interaction between force and internal friction – captured by dynamic viscosity
Indirectly, the Reynold’s number seems to proxy of the energy state of the scenario, whereby a number in excess of 4000 (this is a judgment call based on judgment and empirical evidence) is deemed to signal turbulence. Reynold’s number can go up to very large numbers (e.g. 100,000) whereby the patterns of turbulence changes.
While the Navier-Stokes equation is described as:
Where:
v>0 is the kinematic viscosity,
f (x, t) is the external volumetric force,
{\displaystyle \nabla } is the gradient operator,
is the Laplacian operator,
Within the Navier-Stokes equation, pressure and velocity is also considered. There is also a consideration using a 3-dimensional vector field and it follows the conservation of mass principle. An analysis of the Navier-Stokes equation identifies
(1) Velocity / Speed – captured directly
(2) Friction – internal friction captured by kinematic viscosity, overall friction captured by pressure, friction towards external objects captured by external volumetric force
(3) Direction – captured by gradient, divergence of gradient as represented by the Laplace operator, and measured using 3-dimensional vector force.
(4) Time – captured directly
(5) Viscosity – captured directly
It can broadly be seen that the Navier-Stokes equation is a more complex and fine-grained version of the Reynold’s number, in that (1) it captures more elements than the Reynolds number (5 compared to 4), (2) The new element of directionality is captured in a complex manner, using a 3-dimensional vector field, (3) the element of friction is captured in a more comprehensive manner, (4) time is integrated into the equation directly, and not captured as a proxy. In addition, there is no context limitation to the usage of this equation. However, there are no direct capture of any interaction effects (the Reynold’s number capture one interaction effect in the form of dynamic viscosity).
[2] Brian R. Tan is trained in doing research in the social sciences and has a Ph.D. from the University of Washington, Seattle. His doctoral major is in the field of strategy (management), with minors in economics, statistics, psychology and entrepreneurship. The thinking approach he has developed, the Wisdom Approach, utilizes knowledge from all fields, even those outside his training and expertise, highlighting the maxim that “Knowledge is Universal”.
[3] https://www.facebook.com/veritasium/videos/932098520753904
The Conceptual Solution by the Wisdom Approach
To build the conceptual model that would serve as a basis to guide the formulation of a mathematical formula to explain turbulence, I visualized the process of turbulence using several examples, including the creation of turbulence by running my hand through a tank of water, the traveling of an object through the air (stone, golf ball, airplane), and visualization of what would create a storm system. These visualizations were meant to generate different variations of turbulence generation to gain some benefits of triangulation and generalization of thought.
From these examples, I identified all possible elements that would be relevant. Some of the elements identified actually framed the way to conceptualize the solution in terms of structure, rather than the solution itself. Let me explain.
I had realized that turbulence was a path dependent process that occurred over time, given a base state of non-turbulence. This is in contrast with the Reynold’s number and the Navier-Stokes equation which seems to capture turbulence from a relatively more static frame of reference. The 2 equations try to incorporate dynamic aspects (e.g. time) directly into explaining turbulence, which increases complexity of the equation and might not represent the phenomenon to be understood well enough or easily enough.
Suggestion: explore the possibility of breaking it down into several simpler equations that can more simply explain a specific aspect of the path-dependent nature of turbulence. I utilize the concept of “the rules of the game” as described by institutional theory, the economics version, that is linked to game theory.
Thus, the elements of “path dependency” and “time” have influenced the structure of the solution.
With this in mind, these are the elements which I had identified as meaningful and relevant to addressing the phenomenon of turbulence:
Elements:
(1) Velocity / speed
(2) Viscosity / Density
(3) Friction
(4) Variance
(5) Direction
(6) Synchronicity
Key Interactive Effects identified:
(7) Interaction between speed and viscosity – energy value
(8) Interaction between friction and variance – the friction differential
(9) Interaction between direction and synchronicity – internal equilibrium
Additional consideration identified:
(10) External equilibrium – for comparison with an external force (e.g. another system – think of hot vs cold air or hot and cold water or saline and fresh water)
(11) The rules of the game – understanding the parameter set WITHIN and BETWEEN different bodies (e.g. hot vs cold water)
(12) Interaction between external equilibrium and the rules of the game – to answer questions on how turbulent bodies might interact with an external force
Broad assumption – the conservation of mass, implying a closed system
(13) Externality – to consider bypassing the conservation of mass assumption by considering leakages and injections of mass or energy. This might be a critical element to consider given it would normally require an injection of energy or an obstacle to create turbulence from a base state.
From the identification of the elements, its core interactive effects and major considerations of structure and assumptions, the following conceptual solution have been developed [4]. It should be noted that the conceptual solution is made within the context of an energy consideration.
Figure 1: Wisdom Approach Conceptual Model
All 6 elements (1 to 6) and 3 interactive effects (7 to 9) are represented in the conceptual model above. It is done using the conservation of mass assumption, meaning that the externality (13) is excluded. Likewise, additional considerations (10 to 12) are not included in the conceptual model. However, 10 to 13 do represent important considerations that can be considered upon further development of the model. For now, a more concise conceptual model is presented.
The interaction of Velocity (Ve) and Viscosity (Vi) produces the base Energy (E) value, and energy considerations need to be considered throughout the model. The elements of time and path dependency are captured through the structure of the model. This means that the model includes 2 additional elements (path dependency and time) but in a way that influences structure. Time can probably be incorporated into the model through mathematical formulation later on. For now, the construct of cancellation rate consists of the elements of friction, direction, synchronicity per unit of time.
Identified interaction and relationship effects are captured with the tagged information beside the identified elements. For example, under the element of Variance (Var), the relationship with turbulence is seen to be positive (implying the higher the level of variance or internal instability within the fluid, the higher the level of turbulence). On the other hand, the lower the level of Variance (Var), the higher the Directionality, Synchronization and the interaction of Direction and Synchronization.
Interaction effects and relationships can be notoriously problematic in identification for conceptual models (which is done before empirical results are used), and given my own lack of context knowledge, I apologize if certain relationships are misidentified.
From the conceptual model, it can be seen that turbulence would generally be classified based on a certain energy floor, that has characteristics driven by direction of flows and synchronicity, while taking into account the cancellation rate. The drivers for these are the 4 elements to the left, capturing Velocity, Viscosity, Friction and Variance.
[4] It should be noted that while it followed the sequence of actions described, there were some reiterations in a cyclical manner, especially with regards to the identification of the key interaction effects.
Conclusions using the Conceptual Solution by the Wisdom Approach
The prevailing equations used to understand turbulence in fluids seem to be underspecified. There are critical elements that have not been considered.
a. E.g. the element of variance (Var), which captures the inherent instability of the fluid (e.g. hot water near boiling point have greater inherent instability or variance, which would lead to greater levels of turbulence; Likewise, mixing of hot and cold water or air will also create greater variance within the fluid or gaseous bodies, thus increasing turbulence).
b. Direction was considered in the Navier-Stokes equation, with the absence of synchronicity, and the corresponding interaction between direction and synchronicity. Synchronicity has a time element inbuilt with it, and is critical in understanding turbulence. For example, Newton’s 3rd law of motion states that “when two bodies interact, they apply forces to one another that are equal in magnitude and opposite in direction”. This must happen at the same time and be synchronized in nature. When it is not in a synchronized manner, in a chaotic and complex scenario, the results cannot be easily predicted.
c. The Reynold’s number uses proxies that would make it difficult to be more broadly applied and a proxy capture also can lead to inaccuracies of results.
d. There might be model under-specificity because both equations do not take into consideration the path dependent nature of turbulence and how that interacts with time, and different rules of the game. When considered holistically, understanding turbulence might require more than 1 equation.
e. Both the Reynold’s number and the Navier-Stokes equation do not seem to capture key interaction effects that would have meaningful and relevant impact on turbulence. For example, they do not capture the friction x variance interaction (by virtue of the non-consideration of variance) or the notion of a cancellation rate, which would be meaningful when trying to understand specific patterns of turbulence. I’ve have included a picture below, whereby cancellation rate, together with “additional considerations”10 to 13 would help with understanding patterns 4 and 5 in the picture below.
Picture 1: Patterns of Turbulence[5]
f. There seems to be relatively little consideration on range-estimate characteristics. i.e. using a quantum perspective. The Navier-Stokes equation seems to do a better job at this, by taking in some aspect of quantum understanding (for example, with its consideration of a 3-dimensional vector field). However, all in all, the reliance on point-estimate calculations being forced fed into an equation, with insufficient consideration on broad parameters and structure might have prevented a solution from being found. Essentially, you cannot accurately predict a scenario with strong range estimate characteristics using a point-estimate solution.
It is the hope of this author that the Wisdom Approach, with its characteristics of integrating and manipulating point and range estimate characteristics, together with its understanding of identification of elementals and its corresponding elemental fingerprint, will provide with direction and guidance towards the solving of a problem that would normally be considered to be outside its parameter set of application. However, with thoughtful manipulation of point and range estimate manipulation and integration, I am happy that a conceptual model can be proposed, and I hope it will prove useful for the Mathematicians and Physicists, and later on, for the engineers.
[5] Source: http://labman.phys.utk.edu/phys221core/modules/m8/turbulence.html
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