Question: Turbulence or turbulent flow can be characterized by which branch of  mathematics?

full_context: Document 1: Mathematical and theoretical biology: Mathematical and theoretical biology is an interdisciplinary scientific research field with a range of applications. The field is sometimes called mathematical biology or biomathematics to stress the mathematical side, or theoretical biology to stress the biological side. Theoretical biology focuses more on the development of theoretical principles for biology while mathematical biology focuses on the use of mathematical tools to study biological systems, even though the two terms are sometimes interchanged. Mathematical biology aims at the mathematical representation, treatment and modeling of biological processes, using techniques and tools of applied mathematics. It has both theoretical and practical applications in biological, biomedical and biotechnology research. Describing systems in a quantitative manner means their behavior can be better simulated, and hence properties can be predicted that might not be evident to the experimenter. This requires precise mathematical models.
Document 2: Churn turbulent flow: Churn turbulent flow is a two-phase gas/liquid flow regime characterized by a highly-agitated flow where gas bubbles are sufficient in numbers to both interact with each other and, while interacting, coalesce to form larger distorted bubbles with unique shapes and behaviors in the system. This flow regime is created when there is a large gas fraction in a system with a high gas and low liquid velocity. It is an important flow regime to understand and model because of its predictive value in nuclear reactor vessel boiling flow.
Document 3: Transition modeling: Transition modeling is the use of a model to predict the change from laminar and turbulence flows in fluids and their respective effects on the overall solution. The complexity and lack of understanding of the underlining physics of the problems makes simulating the interaction between laminar and turbulent flow to be difficult and very case specific. Transition does have the wide range of turbulence options available for most CFD applications for the following reasons. Transition involves a wide range of scales where the energy and momentum transfer are strong influenced by inertial or non-linear effects that are unique to the simulation. Transition also occurs using different ways, (such as natural or bypass) where modelling all options are difficult. Most CFD code uses RANS where averaging eliminates linear disturbance.
Document 4: Taylor microscale: The Taylor microscale, which is sometimes called the turbulence length scale, is a length scale used to characterize a turbulent fluid flow. This microscale is named after Geoffrey Ingram Taylor. The Taylor microscale is the intermediate length scale at which fluid viscosity significantly affects the dynamics of turbulent eddies in the flow. This length scale is traditionally applied to turbulent flow which can be characterized by a Kolmogorov spectrum of velocity fluctuations. In such a flow, length scales which are larger than the Taylor microscale are not strongly affected by viscosity. These larger length scales in the flow are generally referred to as the inertial range. Below the Taylor microscale the turbulent motions are subject to strong viscous forces and kinetic energy is dissipated into heat. These shorter length scale motions are generally termed the dissipation range.
Document 5: Turbulence kinetic energy: In fluid dynamics, turbulence kinetic energy (TKE) is the mean kinetic energy per unit mass associated with eddies in turbulent flow. Physically, the turbulence kinetic energy is characterised by measured root-mean-square (RMS) velocity fluctuations.
Document 6: Turbulence: Turbulence or turbulent flow is a flow regime in fluid dynamics characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow regime, which occurs when a fluid flows in parallel layers, with no disruption between those layers.
Document 7: Chaos theory: Chaos theory is a branch of mathematics focused on the behavior of dynamical systems that are highly sensitive to initial conditions. 'Chaos' is an interdisciplinary theory stating that within the apparent randomness of chaotic complex systems, there are underlying patterns, constant feedback loops, repetition, self-similarity, fractals, self-organization, and reliance on programming at the initial point known as "sensitive dependence on initial conditions". The butterfly effect describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state, e.g. a butterfly flapping its wings in Brazil can cause a tornado in Texas.
Document 8: Jorge Luis Borges and mathematics: Jorge Luis Borges and mathematics concerns several modern mathematical concepts found in certain essays and short stories of Argentinian author Jorge Luis Borges (1899-1986), including concepts such as set theory, recursion, chaos theory, and infinite sequences, although Borges' strongest links to mathematics are through Georg Cantor's theory of infinite sets, outlined in "The Doctrine of Cycles" ("La doctrina de los ciclos"). Some of Borges' most popular works such as "The Library of Babel" ("La Biblioteca de Babel"), "The Garden of Forking Paths" ("El Jardín de Senderos que se Bifurcan"), "The Aleph" ("El Aleph"), an allusion to Cantor's use of the Hebrew letter aleph (formula_1) to denote cardinality of transfinite sets. and "The Approach to Al-Mu'tasim" ("El acercamiento a Almotásim") illustrate his use of mathematics.
Document 9: Mathematical model: A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, biology, Earth science, meteorology) and engineering disciplines (such as computer science, artificial intelligence), as well as in the social sciences (such as economics, psychology, sociology, political science). Physicists, engineers, statisticians, operations research analysts, and economists use mathematical models most extensively . A model may help to explain a system and to study the effects of different components, and to make predictions about behaviour.
Document 10: Quantum chaos: Quantum chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is: "What is the relationship between quantum mechanics and classical chaos?" The correspondence principle states that classical mechanics is the classical limit of quantum mechanics. If this is true, then there must be quantum mechanisms underlying classical chaos (although this may not be a fruitful way of examining classical chaos). If quantum mechanics does not demonstrate an exponential sensitivity to initial conditions, how can exponential sensitivity to initial conditions arise in classical chaos, which must be the correspondence principle limit of quantum mechanics? In seeking to address the basic question of quantum chaos, several approaches have been employed:
Document 11: Reynolds number: The Reynolds number (Re ) is an important dimensionless quantity in fluid mechanics used to help predict flow patterns in different fluid flow situations. It has wide applications, ranging from liquid flow in a pipe to the passage of air over an aircraft wing. The Reynolds number is used to predict the transition from laminar to turbulent flow, and used in the scaling of similar but different-sized flow situations, such as between an aircraft model in a wind tunnel and the full size version. The predictions of onset of turbulence and the ability to calculate scaling effects can be used to help predict fluid behaviour on a larger scale, such as in local or global air or water movement and thereby the associated meteorological and climatological effects.
Document 12: Turbulator: A turbulator is a device that turns a laminar flow into a turbulent flow. Turbulent flow can be desired on parts of the surface of an aircraft wing (airfoil) or in industrial applications such as heat exchangers and the mixing of fluids. The term “turbulator” is applied to a variety of applications and is used as a derivative of the word turbulent. However, the word has no commonly accepted technical or scientific meaning. As such, it has been approved as a trademark in the U.S. and other countries in conjunction with machine parts used within rotating drums, sterilizers, heat transfer ovens, mixing and pelletizing machines, and air destratification fans for horticultural and agricultural uses, among others.
Document 13: Chaos Theory (film): Chaos Theory is a 2008 comedy-drama film starring Ryan Reynolds, Emily Mortimer, and Stuart Townsend. The film was directed by Marcos Siega, written by Daniel Taplitz and Kathy Gori, and was shot in Vancouver, British Columbia, Canada.
Document 14: Mortal Chaos: Mortal Chaos is a series of novels for teenage and young adult readers which was published by Oxford University Press in 2012. Written by Matt Dickinson, the series explores the world of chaos theory and the butterfly effect. Previous published works by Matt Dickinson have been "The Death Zone" (Random House 1997), "High Risk", (Random House 1999) and "Black Ice" (Random House 2001). Mortal Chaos is inspired by the work of Edward Lorenz and Ray Bradbury's 1952 story "A Sound of Thunder", in which the concept of sensitive dependence on initial conditions is applied to time travel. Each novel in the series starts with the beating of a butterfly's wings and follows parallel (and simultaneous) stories set in different parts of the globe as a disaster unfolds. The films "The Butterfly Effect", "Babel" and "Crash" employed a similar concept. The series is commissioned by OUP's children's division and will be edited by Liz Cross with managing editor Kathy Webb.
Document 15: K-epsilon turbulence model: K-epsilon (k-ε) turbulence model is the most common model used in Computational Fluid Dynamics (CFD) to simulate mean flow characteristics for turbulent flow conditions. It is a two equation model which gives a general description of turbulence by means of two transport equations (PDEs). The original impetus for the K-epsilon model was to improve the mixing-length model, as well as to find an alternative to algebraically prescribing turbulent length scales in moderate to high complexity flows.
ground_truth: Chaos theory