7.1 Characteristics of turbulent flows

Turbulent flows are chaotic and irregular, characterized by rapid velocity fluctuations and enhanced mixing. They differ from laminar flows, which have smooth, parallel layers. Understanding turbulence is crucial for engineering applications like combustion and aerodynamics.

Turbulent flows exhibit increased friction and energy dissipation compared to laminar flows. The transition between flow regimes depends on factors like fluid velocity and viscosity, often characterized by the Reynolds number. Turbulent boundary layers have distinct structures and velocity profiles.

Turbulent vs laminar flow

• Turbulent and laminar flows are two distinct flow regimes in fluid dynamics, characterized by different flow patterns and behaviors
• Laminar flow exhibits smooth, parallel layers of fluid with no mixing between layers, while turbulent flow is characterized by chaotic and irregular motion with enhanced mixing
• The transition between laminar and turbulent flow depends on factors such as fluid velocity, viscosity, and geometry of the flow domain

Characteristics of turbulent flows

Highly irregular velocity fluctuations

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• Turbulent flows exhibit rapid and seemingly random fluctuations in velocity magnitude and direction
• These fluctuations occur in all three spatial dimensions and vary with time
• Velocity fluctuations are superimposed on the mean flow velocity, resulting in a highly complex and chaotic flow field
• The presence of irregular velocity fluctuations distinguishes turbulent flows from laminar flows, which have smooth and predictable velocity profiles

Chaotic and random motion

• Turbulent flows are characterized by chaotic and apparently random motion of fluid particles
• The motion of fluid particles in turbulence is highly sensitive to initial conditions, meaning that small perturbations can lead to drastically different particle trajectories over time
• The chaotic nature of turbulence makes it challenging to predict the exact motion of individual fluid particles
• The random motion in turbulence is often described using statistical tools, such as probability density functions and correlation functions

Enhanced mixing and diffusion

• Turbulent flows exhibit significantly enhanced mixing and diffusion compared to laminar flows
• The chaotic motion of fluid particles in turbulence promotes rapid mixing of momentum, heat, and mass across the flow domain
• Turbulent mixing is crucial in many engineering applications, such as combustion, chemical reactions, and heat transfer
• The enhanced diffusion in turbulence leads to increased rates of mass, momentum, and energy transport compared to molecular diffusion in laminar flows

Increased friction and energy dissipation

• Turbulent flows experience higher friction and energy dissipation compared to laminar flows
• The irregular velocity fluctuations in turbulence generate additional shear stresses and viscous dissipation
• The increased friction in turbulent flows leads to higher pressure drops and energy losses in pipes, channels, and other flow systems
• Energy dissipation in turbulence occurs through the cascade of energy from large-scale eddies to smaller scales, where viscous dissipation ultimately converts kinetic energy into heat

Transition from laminar to turbulent

Critical Reynolds number

• The transition from laminar to turbulent flow is often characterized by a critical Reynolds number ($Re_{cr}$)
• The Reynolds number is a dimensionless quantity that represents the ratio of inertial forces to viscous forces in a flow
• For pipe flow, the critical Reynolds number is approximately 2300, above which the flow becomes turbulent
• The critical Reynolds number varies depending on the flow geometry and can be affected by factors such as surface roughness and flow disturbances

Factors affecting transition

• Several factors influence the transition from laminar to turbulent flow
• Increasing the fluid velocity or decreasing the fluid viscosity promotes the transition to turbulence by increasing the Reynolds number
• Surface roughness and flow disturbances (vibrations, obstacles) can trigger the transition to turbulence at lower Reynolds numbers
• Pressure gradients and flow curvature can also affect the transition process
• Understanding the factors affecting transition is important for controlling and manipulating flow regimes in engineering applications

Turbulent boundary layers

Velocity profile in turbulent flows

• The velocity profile in turbulent boundary layers differs from that in laminar boundary layers
• In turbulent boundary layers, the velocity profile is more uniform in the outer region due to enhanced mixing
• The velocity gradient near the wall is steeper in turbulent flows, resulting in a thinner viscous sublayer
• The logarithmic law of the wall describes the mean velocity profile in the overlap region of turbulent boundary layers

Turbulent boundary layer structure

• Turbulent boundary layers consist of distinct regions with different flow characteristics
• The inner region (viscous sublayer and buffer layer) is dominated by viscous effects and has a steep velocity gradient
• The outer region (log-law region and wake region) is dominated by turbulent mixing and has a more uniform velocity profile
• The overlap region (log-law region) exhibits a logarithmic velocity profile and serves as a transition between the inner and outer regions

Boundary layer separation in turbulence

• Boundary layer separation occurs when the flow detaches from the surface due to adverse pressure gradients or geometric discontinuities
• In turbulent flows, boundary layer separation is delayed compared to laminar flows due to the higher momentum transfer in the boundary layer
• Turbulent boundary layers are more resistant to separation because of the enhanced mixing and energy transfer from the outer region to the near-wall region
• Flow control techniques, such as surface roughness or active flow control, can be used to manipulate turbulent boundary layers and prevent or delay separation

Turbulence scales and energy cascade

Energy-containing eddies

• Turbulent flows contain a wide range of eddy sizes, from large-scale eddies to small-scale eddies
• The largest eddies, known as energy-containing eddies, are responsible for most of the turbulent kinetic energy
• Energy-containing eddies extract energy from the mean flow through a process called vortex stretching
• The size of energy-containing eddies is typically comparable to the flow domain dimensions (integral length scale)

Inertial subrange and Kolmogorov scale

• The inertial subrange is a range of intermediate eddy sizes in which energy is transferred from larger eddies to smaller eddies without significant dissipation
• In the inertial subrange, the energy spectrum follows a -5/3 power law, known as Kolmogorov's five-thirds law
• The smallest eddies, known as Kolmogorov-scale eddies, are responsible for the viscous dissipation of turbulent kinetic energy
• The Kolmogorov length scale ($\eta$) represents the size of the smallest eddies and is determined by the viscosity and dissipation rate of the flow

Energy dissipation at small scales

• Energy dissipation in turbulence occurs primarily at the smallest scales (Kolmogorov scales) through viscous dissipation
• The rate of energy dissipation is determined by the viscosity and the velocity gradients at the small scales
• The energy cascade process transfers energy from larger eddies to smaller eddies, ultimately leading to dissipation at the Kolmogorov scales
• The dissipation rate is an important parameter in turbulence modeling and is often assumed to be equal to the production rate of turbulent kinetic energy in equilibrium turbulence

Statistical description of turbulence

Turbulent velocity fluctuations

• Turbulent flows are characterized by random velocity fluctuations superimposed on the mean flow velocity
• The velocity fluctuations are often decomposed into mean and fluctuating components using Reynolds decomposition
• The mean velocity represents the time-averaged velocity, while the fluctuating component represents the turbulent fluctuations
• The statistical properties of turbulent velocity fluctuations, such as the root-mean-square (RMS) velocity and the turbulence intensity, are used to quantify the level of turbulence

Probability density functions

• Probability density functions (PDFs) are used to describe the statistical distribution of turbulent velocity fluctuations
• PDFs provide information about the likelihood of observing a particular velocity value at a given point in the flow
• The shape of the PDF can reveal important characteristics of the turbulence, such as the presence of intermittency or non-Gaussian behavior
• Joint PDFs can be used to describe the correlation between velocity components or other flow variables

Turbulence intensity and length scales

• Turbulence intensity is a measure of the level of turbulence in a flow, defined as the ratio of the RMS velocity fluctuations to the mean velocity
• Higher turbulence intensities indicate stronger turbulent fluctuations relative to the mean flow
• Turbulent length scales, such as the integral length scale and the Taylor microscale, characterize the size of turbulent eddies
• The integral length scale represents the size of the largest eddies, while the Taylor microscale represents the intermediate scales at which viscous dissipation becomes significant

Turbulent flow modeling approaches

Direct Numerical Simulation (DNS)

• Direct Numerical Simulation (DNS) is a computational approach that solves the Navier-Stokes equations without any turbulence modeling assumptions
• DNS resolves all spatial and temporal scales of turbulence, from the largest eddies to the Kolmogorov scales
• DNS requires extremely fine spatial and temporal resolution, making it computationally expensive and limited to low Reynolds number flows
• DNS is mainly used for fundamental research and validation of turbulence models, rather than for practical engineering applications

Large Eddy Simulation (LES)

• Large Eddy Simulation (LES) is a computational approach that resolves the large-scale turbulent eddies and models the effects of smaller scales
• In LES, the Navier-Stokes equations are filtered to remove the small-scale eddies, and a subgrid-scale (SGS) model is used to represent their effects on the resolved scales
• LES captures the unsteady and three-dimensional nature of turbulent flows, providing more accurate results than RANS models
• LES is less computationally expensive than DNS but still requires significant computational resources, especially for high Reynolds number flows

Reynolds-Averaged Navier-Stokes (RANS) models

• Reynolds-Averaged Navier-Stokes (RANS) models are the most widely used approach for turbulent flow simulations in engineering applications
• RANS models solve the time-averaged Navier-Stokes equations and model the effects of turbulence using additional transport equations
• The most common RANS models are the k-epsilon and k-omega models, which solve transport equations for the turbulent kinetic energy (k) and either the dissipation rate (epsilon) or the specific dissipation rate (omega)
• RANS models are computationally efficient and provide reasonable predictions for many engineering flows, but they have limitations in capturing unsteady and separated flows

Turbulence in engineering applications

Turbulent flow in pipes and channels

• Turbulent flow is commonly encountered in pipes and channels in various engineering applications, such as fluid transportation and heat exchangers
• The presence of turbulence in pipes and channels leads to increased friction losses and pressure drop compared to laminar flow
• Turbulent flow in pipes and channels is often characterized by the Darcy-Weisbach friction factor, which relates the pressure drop to the flow velocity and pipe roughness
• Turbulent flow in pipes and channels can be influenced by factors such as wall roughness, flow obstructions, and changes in cross-sectional area

Turbulence in aerodynamics and wind engineering

• Turbulence plays a crucial role in aerodynamics and wind engineering applications, such as aircraft design, wind turbines, and buildings
• In aerodynamics, turbulence affects the lift, drag, and stability of aircraft, and it can lead to phenomena such as flow separation and stall
• Wind turbines operate in the atmospheric boundary layer, which is highly turbulent and can impact the performance and loads on the turbine blades
• In wind engineering, turbulence influences the wind loads on buildings and structures, as well as the dispersion of pollutants and heat in urban environments

Turbulent mixing in combustion and chemical processes

• Turbulent mixing is essential in combustion and chemical processes, as it enhances the mixing of reactants and promotes efficient reactions
• In combustion systems, such as internal combustion engines and gas turbines, turbulence improves fuel-air mixing and flame stabilization
• Turbulent mixing in chemical reactors increases the contact between reactants and catalyst surfaces, leading to higher reaction rates and improved product yields
• The design and optimization of combustion and chemical systems often involve the control and manipulation of turbulent mixing to achieve desired performance characteristics