8.1 Characteristics of Turbulent Flows

Turbulent flows are chaotic and unpredictable, with irregular fluctuations in velocity and pressure. They're characterized by enhanced mixing, diffusion, and dissipation compared to laminar flows, leading to increased heat transfer and drag.

Turbulence has a multi-scale structure, with eddies of various sizes interacting and transferring energy. The energy cascade describes how kinetic energy moves from larger to smaller scales, ultimately dissipating as heat at the Kolmogorov microscale.

Turbulent flow features

Irregular fluctuations and enhanced transport

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• Turbulent flows characterized by irregular fluctuations and mixing in fluid motion exhibit chaotic changes in pressure and velocity fields
• Enhanced diffusion, dissipation, and mixing compared to laminar flows lead to increased heat transfer and drag
• High levels of vorticity with vortex stretching play crucial role in energy cascade process
• Intermittency causes intense, localized events to occur sporadically within flow field
  • Examples: sudden bursts of high-velocity fluid, formation of coherent structures (hairpin vortices)

Multi-scale structure and energy cascade

• Multi-scale structure fundamental property with eddies of various sizes interacting and transferring energy
  • Large eddies: contain most of the kinetic energy
  • Small eddies: responsible for viscous dissipation
• Non-linear interactions between different scales lead to complex energy transfer mechanisms
• Energy cascade describes transfer of kinetic energy from larger to smaller scales
  • Ultimately dissipates as heat at Kolmogorov microscale
  • Kolmogorov microscale: smallest scale of turbulent motion (typically ~0.1-1 mm)

Laminar vs Turbulent flow

Flow characteristics and transition

• Laminar flows exhibit smooth, predictable fluid motion with parallel layers sliding past one another
• Turbulent flows display irregular, chaotic motion with significant mixing
• Transition from laminar to turbulent flow occurs at critical Reynolds number
  • Varies depending on specific flow geometry and conditions
  • Example: pipe flow critical Re ≈ 2300, flow over flat plate critical Re ≈ 5 × 10^5
• Velocity profile in laminar flow typically parabolic
• Turbulent flow profiles flatter and more uniform due to increased momentum transfer

Transport properties and visualization

• Turbulent flows have higher momentum and heat transfer rates compared to laminar flows
  • Enhanced mixing and diffusion processes
• Laminar flows governed by viscous forces
• Turbulent flows dominated by inertial forces and exhibit wide range of eddy sizes
• Drag coefficient in turbulent flows generally higher than in laminar flows
  • Increased energy dissipation and pressure drop in fluid systems
• Visualization techniques reveal distinct patterns in laminar and turbulent flows
  • Laminar flow: smooth streamlines (dye injection in water tunnel)
  • Turbulent flow: complex, irregular patterns (smoke visualization in wind tunnel)

Reynolds number for turbulence

Reynolds number fundamentals

• Reynolds number (Re) dimensionless parameter quantifies ratio of inertial forces to viscous forces in fluid flow
  • Key indicator for onset of turbulence
  • $Re = \frac{\rho UL}{\mu}$ where ρ density, U characteristic velocity, L characteristic length, μ dynamic viscosity
• Critical Reynolds number marks transition from laminar to turbulent flow
  • Varies depending on specific flow geometry and boundary conditions
• As Reynolds number increases beyond critical value, flow becomes increasingly turbulent
  • Wider range of eddy sizes and more intense mixing

Reynolds number effects on flow structure

• Influences structure of turbulent boundary layer
  • Affects distribution of mean velocity and turbulent fluctuations near solid boundaries
• High Reynolds number flows exhibit separation of scales between largest and smallest eddies
  • Leads to more developed inertial subrange in energy spectrum
• Plays crucial role in scaling laws and similarity principles for turbulent flows
  • Allows comparison of flows across different scales and conditions
  • Example: use of scaled-down models in wind tunnel testing for aircraft design

Statistical nature of turbulence

Statistical methods and decomposition

• Turbulent flows characterized by random fluctuations in velocity and pressure fields
  • Necessitates statistical methods for description and analysis
• Reynolds decomposition separates flow variables into mean and fluctuating components
  • Forms basis for statistical analysis of turbulence
  • $u(x,t) = \bar{u}(x) + u'(x,t)$ where $\bar{u}$ mean velocity, $u'$ fluctuating component
• Non-Gaussian probability distributions for velocity fluctuations
  • Intermittent events lead to heavy-tailed distributions
• Energy spectrum of turbulent flows follows -5/3 power law in inertial subrange
  • Predicted by Kolmogorov's theory of isotropic turbulence
  • $E(k) \propto k^{-5/3}$ where k wavenumber

Correlation functions and predictability

• Two-point correlation functions and structure functions essential statistical tools
  • Characterize spatial and temporal coherence of turbulent flows
• Closure problem in turbulence modeling arises from non-linear nature of Navier-Stokes equations
  • Leads to infinite hierarchy of statistical moments
• Despite deterministic governing equations, turbulent flows display sensitive dependence on initial conditions
  • Results in chaotic behavior and limited predictability over long time scales
  • Example: weather forecasting accuracy decreases rapidly beyond a few days