Kolmogorov's theory of turbulence revolutionized our understanding of fluid dynamics. It explains how energy cascades from large to small eddies in turbulent flows, introducing key scales that characterize these complex systems.

The theory provides a statistical framework for describing turbulence, including the famous five-thirds law for energy spectra. While it has limitations, Kolmogorov's work remains fundamental to modern fluid dynamics and its applications.

Kolmogorov's theory of turbulence

• Groundbreaking work by Soviet mathematician Andrey Kolmogorov in the 1940s laid the foundation for modern understanding of turbulence
• Provides a statistical description of turbulence based on the concept of energy cascade and similarity hypotheses
• Introduces important length, time, and velocity scales that characterize turbulent flows

Energy cascade in turbulent flows

• Turbulent flows exhibit a hierarchical structure of eddies, ranging from large to small scales
• Energy is transferred from larger eddies to smaller eddies through a process called energy cascade
• Large eddies break down into smaller eddies, which in turn break down into even smaller eddies, and so on
• Energy is dissipated into heat at the smallest scales by viscous forces

Kolmogorov length scale

• Represents the smallest eddy size in a turbulent flow, below which viscous dissipation dominates
• Denoted by $\eta$ and defined as $\eta = (\nu^3/\epsilon)^{1/4}$, where $\nu$ is the kinematic viscosity and $\epsilon$ is the energy dissipation rate
• Eddies smaller than the Kolmogorov length scale are isotropic and homogeneous

Kolmogorov time scale

• Characteristic time scale associated with the smallest eddies in a turbulent flow
• Defined as $\tau_\eta = (\nu/\epsilon)^{1/2}$
• Represents the turnover time of the smallest eddies

Kolmogorov velocity scale

• Characteristic velocity scale of the smallest eddies in a turbulent flow
• Defined as $v_\eta = (\nu\epsilon)^{1/4}$
• Velocity fluctuations at scales smaller than the Kolmogorov length scale are determined by the viscosity and energy dissipation rate

Kolmogorov's first similarity hypothesis

• Also known as the local isotropy hypothesis
• States that at sufficiently high Reynolds numbers, the small-scale turbulent motions are statistically isotropic
• Implies that the statistics of small-scale turbulence are universal and independent of the large-scale flow geometry

Kolmogorov's second similarity hypothesis

• Relates the statistics of velocity increments in the inertial subrange to the energy dissipation rate
• States that in the inertial subrange, the statistics of velocity increments depend only on the energy dissipation rate and the separation distance
• Forms the basis for the famous five-thirds law

Kolmogorov's five-thirds law

• Describes the scaling of the energy spectrum in the inertial subrange of turbulent flows
• States that the energy spectrum $E(k)$ in the inertial subrange follows a power law with an exponent of -5/3
• Mathematically expressed as $E(k) = C\epsilon^{2/3}k^{-5/3}$, where $C$ is the Kolmogorov constant and $k$ is the wavenumber

Energy spectrum in inertial subrange

• The inertial subrange is a range of scales in turbulent flows where energy is transferred from larger to smaller scales without significant dissipation
• In the inertial subrange, the energy spectrum follows the five-thirds law
• The existence of the inertial subrange is a key prediction of Kolmogorov's theory

Dissipation range of energy spectrum

• The dissipation range is the range of scales smaller than the Kolmogorov length scale, where viscous dissipation dominates
• In the dissipation range, the energy spectrum deviates from the five-thirds law and decays rapidly
• The shape of the energy spectrum in the dissipation range is not universal and depends on the specific flow conditions

Universal equilibrium range

• Refers to the range of scales in turbulent flows where the energy spectrum is determined by the energy dissipation rate and viscosity alone
• Includes both the inertial subrange and the dissipation range
• In the universal equilibrium range, the statistics of turbulence are expected to be universal and independent of the large-scale flow geometry

Limitations of Kolmogorov's theory

• Assumes homogeneous and isotropic turbulence, which may not hold in all flow situations
• Does not account for intermittency effects, which lead to deviations from the predicted scaling laws
• Limited to high Reynolds number flows, where there is a clear separation between the large and small scales

Intermittency in turbulent flows

• Refers to the spatial and temporal fluctuations in the energy dissipation rate
• Leads to deviations from the predicted scaling laws, particularly in the dissipation range
• Requires more advanced statistical descriptions, such as the refined similarity hypotheses, to capture the effects of intermittency

Refined similarity hypotheses

• Proposed by Kolmogorov and Oboukhov to account for intermittency effects in turbulent flows
• Introduce a local energy dissipation rate, which varies in space and time
• Modify the scaling laws for velocity increments and energy spectrum to include the effects of intermittency
• Provide a more accurate description of turbulence statistics, particularly in the dissipation range

Experimental validation of Kolmogorov's theory

• Numerous experimental studies have been conducted to test the predictions of Kolmogorov's theory
• Measurements of velocity fluctuations and energy spectra in various turbulent flows (e.g., grid turbulence, pipe flow, atmospheric turbulence) have provided support for the five-thirds law and the existence of the inertial subrange
• Deviations from the predicted scaling laws have been observed, particularly in the dissipation range, due to intermittency effects

Applications of Kolmogorov's theory

• Turbulence modeling: Kolmogorov's theory provides a framework for developing subgrid-scale models in Large Eddy Simulations (LES) and turbulence closure models in Reynolds-Averaged Navier-Stokes (RANS) simulations
• Turbulent mixing: Understanding the energy cascade and the scales of turbulence is crucial for predicting mixing rates and scalar transport in turbulent flows (e.g., chemical reactions, pollutant dispersion)
• Aerodynamics and hydrodynamics: Kolmogorov's theory is used to estimate the dissipation of turbulent kinetic energy and the drag force in various engineering applications (e.g., aircraft design, wind turbines, ship hulls)
• Atmospheric and oceanic turbulence: The concepts of energy cascade and similarity hypotheses are applied to study turbulence in the atmosphere and oceans, which plays a crucial role in weather prediction and climate modeling