Question: Turbulence or turbulent flow can be characterized by which branch of  mathematics?
Context: 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. 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. 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. 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. 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. 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. vascular murmur" ( ; from French, "noise"), or vascular murmur"', is the abnormal sound generated by turbulent flow of blood in an artery due to either an area of partial obstruction; or a localized high rate of blood flow through an unobstructed artery.  The bruit may be heard ("auscultated") by pressing a stethoscope to the skin over the turbulent flow and listening.  Most bruits occur only in systole, so the bruit is intermittent and its frequency dependent on the heart rate.  Anything increasing the blood flow velocity such as fever, anemia, or hyperthyroidism, can increase the amplitude of the bruit. 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. 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. 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.
Answer: Chaos theory