Turbulence or turbulent flow can be characterized by which branch of  mathematics?
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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.
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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.
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Real analysis Real analysis (traditionally, the theory of functions of a real variable) is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity, smoothness and related properties of real-valued functions.
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Turbulence (disambiguation) Turbulence is a phenomenon involving the irregular motion of air and fluids, studied in fluid dynamics.
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Applied mathematics Applied mathematics is a branch of mathematics that deals with mathematical methods that find use in science, engineering, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics.
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Mean flow In fluid dynamics, the fluid flow is often decomposed into a mean flow – and deviations from the mean. The averaging can be done either in space or in time, or by ensemble averaging.
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Uriel Frisch Uriel Frisch (born in Agen, in France, on December 10, 1940) is a French mathematical physicist known for his work on fluid dynamics and turbulence.
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Dimension theory In mathematics, dimension theory is a branch of general topology dealing with dimensional invariants of topological spaces.
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Mean curvature flow In the field of differential geometry in mathematics, mean curvature flow is an example of a geometric flow of hypersurfaces in a Riemannian manifold (for example, smooth surfaces in 3-dimensional Euclidean space). Intuitively, a family of surfaces evolves under mean curvature flow if the normal component of the velocity of which a point on the surface moves is given by the mean curvature of the surface. For example, a round sphere evolves under mean curvature flow by shrinking inward uniformly (since the mean curvature vector of a sphere points inward). Except in special cases, the mean curvature flow develops singularities.
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Nonlinear acoustics Nonlinear acoustics (NLA) is a branch of physics and acoustics dealing with sound waves of sufficiently large amplitudes. Large amplitudes require using full systems of governing equations of fluid dynamics (for sound waves in liquids and gases) and elasticity (for sound waves in solids). These equations are generally nonlinear, and their traditional linearization is no longer possible. The solutions of these equations show that, due to the effects of nonlinearity, sound waves are being distorted as they travel.
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Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a "Riemannian metric", i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.
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Pui-Kuen Yeung Pui-Kuen Yeung from the Georgia Institute of Technology, was awarded the status of Fellow in the American Physical Society, after they were nominated by their Division of Fluid Dynamics in 2006, for "insightful contributions to the understanding and modeling of similarity scaling in turbulence and the mixing of passive scalars, especially the study of Lagrangian statistics and dispersion in turbulence through high-resolution simulations addressing Reynolds number and Schmidt number dependencies."
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Riemannian manifold In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space ("M","g") is a real smooth manifold "M" equipped with an inner product formula_1 on the tangent space formula_2 at each point formula_3 that varies smoothly from point to point in the sense that if "X" and "Y" are vector fields on "M", then formula_4 is a smooth function. The family formula_1 of inner products is called a Riemannian metric (tensor). These terms are named after the German mathematician Bernhard Riemann. The study of Riemannian manifolds constitutes the subject called Riemannian geometry.
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Physical mathematics The subject of physical mathematics is concerned with physically motivated mathematics and is different from mathematical physics. The "Journal of Physical Mathematics" is an important journal in the field.
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Bony–Brezis theorem In mathematics, the Bony–Brezis theorem, due to the French mathematicians Jean-Michel Bony and Haïm Brezis, gives necessary and sufficient conditions for a closed subset of a manifold to be invariant under the flow defined by a vector field, namely at each point of the closed set the vector field must have non-positive inner product with any exterior normal vector to the set. A vector is an "exterior normal" at a point of the closed set if there is a real-valued continuously differentiable function maximized locally at the point with that vector as its derivative at the point. If the closed subset is a smooth submanifold with boundary, the condition states that the vector field should not point outside the subset at boundary points. The generalization to non-smooth subsets is important in the theory of partial differential equations.
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Eddy covariance The eddy covariance (also known as eddy correlation and eddy flux) technique is a key atmospheric measurement technique to measure and calculate vertical turbulent fluxes within atmospheric boundary layers. The method analyzes high-frequency wind and scalar atmospheric data series, and yields values of fluxes of these properties. It is a statistical method used in meteorology and other applications (micrometeorology, oceanography, hydrology, agricultural sciences, industrial and regulatory applications, etc.) to determine exchange rates of trace gases over natural ecosystems and agricultural fields, and to quantify gas emissions rates from other land and water areas. It is frequently used to estimate momentum, heat, water vapour, carbon dioxide and methane fluxes.
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Fluid kinematics Fluid kinematics is a field of physics and mechanics concerned with the movement of fluids. Fluids tend to flow easily, which causes a net motion of molecules from one point in space to another point as a function of time. These fluids may be liquids or may be materials with fluid properties, including crowds of people or volumes of grains.
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Helmholtz decomposition In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field; this is known as the Helmholtz decomposition or Helmholtz representation. It is named after Hermann von Helmholtz.
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Airy wave theory In fluid dynamics, Airy wave theory (often referred to as linear wave theory) gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the fluid layer has a uniform mean depth, and that the fluid flow is inviscid, incompressible and irrotational. This theory was first published, in correct form, by George Biddell Airy in the 19th century.
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Nikolay Yegorovich Zhukovsky Prof Nikolay Yegorovich Zhukovsky (Russian: Никола́й Его́рович Жуко́вский ; January 17 [O.S. January 5] 1847  – March 17, 1921) was a Russian scientist, mathematician and engineer, and a founding father of modern aero- and hydrodynamics. Whereas contemporary scientists scoffed at the idea of human flight, Zhukovsky was the first to undertake the study of airflow. He is often called the Father of Russian Aviation.
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Geometry and topology In mathematics, geometry and topology is an umbrella term for the historically distinct disciplines of geometry and topology, as general frameworks allow both disciplines to be manipulated uniformly, most visibly in local to global theorems in Riemannian geometry, and results like the Gauss–Bonnet theorem and Chern–Weil theory.
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Branched manifold In mathematics, a branched manifold is a generalization of a differentiable manifold which may have singularities of very restricted type and admits a well-defined tangent space at each point. A branched "n"-manifold is covered by "n"-dimensional "coordinate charts", each of which involves one or several "branches" homeomorphically projecting into the same differentiable "n"-disk in R. Branched manifolds first appeared in the dynamical systems theory, in connection with one-dimensional hyperbolic attractors constructed by Smale and were formalized by R. F. Williams in a series of papers on expanding attractors. Special cases of low dimensions are known as train tracks ("n" = 1) and branched surfaces ("n" = 2) and play prominent role in the geometry of three-manifolds after Thurston.