# From Modelling Turbulence to General Systems Modelling

## Abstract

**:**

## 1. Introduction

“The revolution developed from people’s attempt to understand disorder—or apparent disorder—in nature, including turbulence in fluids, the erratic flows of epidemics, and the arrhythmic writhing of a heart in the moments before death. These ideas have begun to be applied within the social realm, so now there is use of chaos and complexity theory by social theorists, economists, and people looking at therapies and therapeutic communities.”Antony Bryant [5]

## 2. The New Age of Data Collection and Its Implications for Modelling Complex Systems

## 3. Turbulence as the Beginning of Complexity

#### 3.1. Solving Turbulence

#### 3.2. Turbulent Reacting Flows

#### 3.3. Turbulent Combustion Models

_{i}, conditioned on a given value of the mixture fraction Z (see Klimenko and Bilger [18] for the complete form of the CMC equation). Here, ${N}_{Z}=\langle {(\nabla Z)}^{2}|Z\rangle $ is the conditional scalar dissipation,

**u**is velocity, and W is the source term. The models associated with the stationary frame of reference are referred to as Eulerian, while models connected to moving fluid are called Lagrangian. The conditional and unconditional (quasi-laminar) models specified above are Eulerian. The conditional models are intermediate in their complexity and accuracy between quasi-laminar models and Lagrangian PDF models considered below.

_{1}, Y

_{2}, … Here,

**w**

^{(k)}(t) denotes the independent Wiener process. The Lagrangian PDF methods were analysed in the seminal work of Pope [19]. The PDF models also allow for Eulerian implementations called “stochastic fields” [20]. The synergy of the PDF and conditional methods resulted in the MMC (multiple mapping conditioning) approach, which involves adding stochastic equations for the so-called reference variables [21,22],

**x**= (x

_{1}, x

_{2}, x

_{3}) but also on the reference variables x

_{1}, x

_{2}, … MMC models are often implemented combining Eulerian simulations of dynamic properties and sparse Lagrangian simulations of reactive components. Note that using Markov families of larger dimension due to additional (i.e., reference) stochastic variables such as those in Equation (5) allows us to represent a wider spectrum of effects.

#### 3.4. Transplantation of Models

## 4. Classes of Systems Models

#### 4.1. Historical Classification

#### 4.2. Conceptual Classification

**Average models**- ○
- System dynamics
- ○
- Other average models (AI, neural networks, …)

**Agent-based models (Monte Carlo and particle methods)**- ○
- Global (homogeneous)
- ○
- Eulerian
- ○
- Lagrangian
- ○
- Combined (Eulerian–Lagrangian)

**Modified and hybrid models**- ○
- Conditional, multiscale, multilevel, etc.

## 5. Major Features of Complex Systems and Models

#### 5.1. Modelling Multiscale Processes

^{−5}to 10

^{−1}s for turbulence and from 10

^{−9}to 1 s for chemical kinetics [17]. Interactions of different scales are one of the major problems that turbulent combustion models need to deal with. For example, the Flamelet model [11,29] is very effective in dealing with fast localised reactions interacting with slower and larger turbulence, while MMC implements PDF treatment of smaller scales combined with conditional modelling at larger scales [22].

#### 5.2. Conserved Properties and Information

_{f}transits to the burned state y

_{b}when and only when it receives a temperature boost from another burned particle [4].

_{f}= 0 and y

_{b}= 1; in combustion, such a variable y is conventionally called the reaction progress variable. The possibility of extinction is not considered in this model. If we have many particles in a uniform container, then

^{−ct})

^{−1}.

#### 5.3. Emergence of Chaotic Order

**y**

^{(k)},

**y**

^{(l)}) = P(

**y**

^{(k)})P(

**y**

^{(l)}), where P(

**y**

^{(k)}) is probability of particle k having properties

**y**

^{(k)}. This hypothesis imposes severe constraints on the complexity of the system, restricting system behaviour to basic thermodynamics-like randomness and prohibiting hierarchal multiscale dependencies. Further research into the particle systems indicates that dependencies violating particle chaos emerge under some conditions [31]. This generally is not desirable in conventional combustion simulations but can be instrumental in simulating complex systemic effects.

#### 5.4. Emergence of Intransitivity

#### 5.5. Complex Topologies, Networks and Emergence of the Small World

**p**

^{(n)}= (p

_{1}, p

_{2}, …, p

_{k})

^{(n)}of particle location at nodes 1, 2, …, k for the timestep n. Here,

**T**is the stochastic matrix (positive elements summing up to unity for each column), which specifies Markov transition probabilities [41].

_{r}in a graph grows exponentially n

_{r}~ exp(cr) with the distance r from a selected central node [42]. This is in contrast with n

_{r}~ r

^{2}for a localised grid on a two-dimensional surface. This small-world effect results in an exponentially fast propagation of epidemic, making the modern interconnected world more capable of and more susceptible to the fast propagation of information and viruses.

## 6. Concluding Remarks

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Mayer-Schonberger, V.; Cukier, K. Big Data: A Revolution that Will Transform How We Live, Work, and Think; Mariner Books: Boston, MA, USA, 2014. [Google Scholar]
- Pope, S.B. Turbulent Flows; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Klimenko, A.Y. The convergence of combustion models and compliance with the Kolmogorov scaling of turbulence. Phys. Fluids
**2021**, 33, 25112. [Google Scholar] [CrossRef] - Klimenko, A.Y.; Pope, S.B. Propagation speed of combustion and invasion waves in stochastic simulations with competitive mixing. Combust. Theory Model.
**2012**, 16, 679–714. [Google Scholar] [CrossRef] - Bryant, A. Liquid modernity, complexity and turbulence. Theory Cult. Soc.
**2007**, 24, 127–135. [Google Scholar] [CrossRef] - Eisenhauer, W.I. Big problems with big data. ISE Ind. Syst. Eng. Work
**2016**, 48, 22. [Google Scholar] - Kaiser, B. Targeted; HarperCollins Publishers: New York, NY, USA, 2019. [Google Scholar]
- Cummings, D. The Campaign, Physics and Data Science. 2016. Available online: https://dominiccummings.com/2016/10/29/on-the-referendum-20-the-campaign-physics-and-data-science-vote-leaves-voter-intention-collection-system-vics-now-available-for-all/ (accessed on 17 May 2022).
- Heinz, S. Statistical Mechanics of Turbulent Flows; Springer: Berlin/Heidelberg, Germany, 2003. [Google Scholar]
- Frisch, U. Turbulence: The legacy of A.N. Kolmogorov; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Kuznetsov, V.R.; Sabelnikov, V.A. Turbulence and Combustion, 2nd ed.; Hemisphere: New York, NY, USA, 1990. [Google Scholar]
- Klimenko, A.Y. Complex competitive systems and competitive thermodynamics. Phil. Trans. R. Soc. A
**2013**, 371, 20120244. [Google Scholar] [CrossRef] - Ge, Y.; Cleary, M.J.; Klimenko, A.Y. A comparative study of sandia flame series (d–f) using sparse-lagrangian mmc modelling. Proc. Combust. Inst.
**2013**, 34, 1325–1332. [Google Scholar] [CrossRef] - Williams, F.A. Combustion Theory, 2nd ed.; Addison-Wesley, Reading: Boston, MA, USA, 1985. [Google Scholar]
- Libby, P.A.; Williams, F.A. Turbulent Reactive Flows; Springer: Berlin/Heidelberg, Germany, 1980. [Google Scholar]
- Fox, R. Computational Models for Turbulent Reacting Flows; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Warnatz, J.; Maas, U.; Dibble, R.W. Combustion: Physical and Chemical Fundamentals, Modelling and Simulation, Experiments, Pollutant Formation; Springer: Berlin/Heidelberg, Germany, 1996. [Google Scholar]
- Klimenko, A.Y.; Bilger, R.W. Conditional moment closure for turbulent combustion. Prog. Moment Energy Combust. Sci.
**1999**, 25, 595–687. [Google Scholar] [CrossRef] - Pope, S.B. Pdf methods for turbulent reactive flows. Prog. Energy Combust. Sci.
**1985**, 11, 119–192. [Google Scholar] [CrossRef] - Readshaw, T.; Ding, T.; Rigopoulos, S.; Jones, W.P. Modeling of turbulent flames with the large eddy simulation; probability density function (pdf) approach, stochastic fields, and artificial neural networks. Phys. Fluids
**2021**, 33, 035154. [Google Scholar] [CrossRef] - Klimenko, A.Y.; Pope, S.B. A model for turbulent reactive flows based on multiple mapping conditioning. Phys. Fluids
**2003**, 15, 1907–1925. [Google Scholar] [CrossRef] [Green Version] - Sundaram, B.; Klimenko, A.Y.; Cleary, M.J.; Ge, Y. A direct approach to generalised multiple mapping conditioning for selected turbulent diffusion flame cases. Combust. Theory Model.
**2016**, 20, 735–764. [Google Scholar] [CrossRef] - Sawford, B. Micro-mixing modelling of scalar fluctuations for plumes in homogeneous turbulence. Flow Turbul. Combust.
**2004**, 72, 133–160. [Google Scholar] [CrossRef] - Rogallo, R.S.; Moin, P. Ten questions concerning the large-eddy simulation of turbulent flows. Annu. Rev. Fluid Mech.
**1984**, 16, 99–137. [Google Scholar] [CrossRef] - Gell-Mann, M. The Quark and the Jaguar; W.H. Freemen and Company: New York, NY, USA, 1994. [Google Scholar]
- Gilbert, G.N.; Troitzsch, K.G. Simulation for the Social Scientist, 2nd ed.; Open University Press: Maidenhead, UK, 2005. [Google Scholar]
- Forrester, J.W. System dynamics-the next fifty years. Syst. Dyn. Rev.
**2007**, 23, 359–370. [Google Scholar] [CrossRef] - Clarke, K.C. Cellular Automata and Agent-Based Models; Springer: Berlin/Heidelberg, Germany, 2014; pp. 1217–1233. [Google Scholar]
- Peters, N. Turbulent Combustion; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Sterman, J.D. Business Dynamics: Systems Thinking and Modeling for a Complex World; Irwin/McGraw-Hill: Boston, MA, USA, 2000. [Google Scholar]
- Klimenko, A.Y. Lagrangian particles with mixing. I. simulating scalar transport. Phys. Fluids
**2009**, 21, 065101. [Google Scholar] [CrossRef] - Klimenko, A.Y. Lagrangian particles with mixing. II. sparse-lagrangian methods in application for turbulent reacting flows. Phys. Fluids
**2009**, 21, 065102. [Google Scholar] [CrossRef] - Gyftopoulos, E.P.; Beretta, G.P. Thermodynamics. Foundations and Applications; Macmillan: New York, NY, USA, 1991. [Google Scholar]
- Klimenko, A.Y. Complexity and intransitivity in technological development. J. Syst. Sci. Syst. Eng.
**2014**, 23, 128–152. [Google Scholar] [CrossRef] [Green Version] - Zhabotinsky, A. Belousov-Zhabotinsky reaction. Scholarpedia
**2007**, 2, 1435. [Google Scholar] [CrossRef] - Poddiakov, A.; Valsiner, J. Intransitivity cycles and their transformations: How dynamically adapting systems function. In Qualitative Mathematics for the Social Sciences: Mathematical Models for Research on Cultural Dynamics; Rudolph, L., Ed.; Routledge Abingdon: New York, NY, USA, 2013; pp. 343–391. [Google Scholar]
- Klimenko, A.Y.; Klimenko, D.A. The Evolution of Technology and Emergence of the Knowledge Society; Glasstree Academic Publishing: Morrisville, NC, USA, 2019. [Google Scholar]
- Klimenko, A.Y.; Saulov, D.N.; Massarotto, P.; Rudolph, V. Conditional model for sorption in porous media with fractal properties. Transp. Porous Media
**2012**, 92, 745–765. [Google Scholar] [CrossRef] - Klimenko, A.Y.; Abdel-Jawad, M.M. Conditional methods for continuum reacting flows in porous media. Proc. Combust. Inst.
**2007**, 31, 2107–2115. [Google Scholar] [CrossRef] - Klimenko, D.A.; Hooman, K.; Klimenko, A.Y. Evaluating transport in irregular pore networks. Phys. Rev. E
**2012**, 86, 011112. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bouet, V.; Klimenko, A.Y. Graph clustering in industrial networks. IMA J. Appl. Math.
**2019**, 84, 1177–1202. [Google Scholar] [CrossRef] - O’Sullivan, D. Spatial Network Analysis; Springer: Berlin/Heidelberg, Germany, 2014; pp. 1252–1273. [Google Scholar]
- Mensah, P.; Katerere, D.; Hachigonta, S.; Roodt, A. Systems Analysis Approach for Complex Global Challenges; Springer: Cham, Switzerland, 2018. [Google Scholar]

**Figure 1.**Examples of coherent structure in turbulent flows. Left: experimental laser-induced fluorescence image of a submerged turbulent jet (Fukushima and Westerwee, Wikipedia). Centre: Kármán vortex street induced by wind flowing around the Juan Fernández Islands (NASA, Wikipedia). Right: MMC simulation of Sandia Flame F [13].

Models for Turbulent Reacting Flows | Models for General Complex Systems |
---|---|

Average and quasi-laminar models, plug-flow reactor | System dynamics and other models dealing with direct emulation of overall performance of the system |

PDF Monte Carlo models, Largangian particle implementations, mixing | Agent-based models with interaction between moving agents; particles are called agents |

Eulerian implementation of stochastic simulations (e.g., stochastic fields) | Stationary agents and/or cellular automata, where agents do not move and usually represented by stationary cells |

Conditional models and conditional/PDF models | Elements of conditional methods are used occasionally but the methodology is not well developed for general systems |

LES and similar models with direct simulation of large scales and modelling small scales | Reproducing large scales in conjunctions with a simplified treatment of processes at small-scales is promising, especially in conjunction with conditional models |

DNS or complete simulation of all (from large-scale to small-scale) features | Modelling of all details is usually impossible for general complex systems |

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Klimenko, A.Y.
From Modelling Turbulence to General Systems Modelling. *AppliedMath* **2022**, *2*, 247-260.
https://doi.org/10.3390/appliedmath2020015

**AMA Style**

Klimenko AY.
From Modelling Turbulence to General Systems Modelling. *AppliedMath*. 2022; 2(2):247-260.
https://doi.org/10.3390/appliedmath2020015

**Chicago/Turabian Style**

Klimenko, Alexander Y.
2022. "From Modelling Turbulence to General Systems Modelling" *AppliedMath* 2, no. 2: 247-260.
https://doi.org/10.3390/appliedmath2020015