7.5 Turbulent boundary layers

Turbulent boundary layers are complex fluid regions near solid surfaces, characterized by chaotic motion and rapid mixing. They play a crucial role in many engineering applications, affecting drag, heat transfer, and flow separation.

Understanding turbulent boundary layers is essential for optimizing fluid systems. This topic explores their structure, governing equations, and control methods, providing insights into how engineers can manipulate these layers to improve performance in various fields.

Turbulent vs laminar flow

• Turbulent flow characterized by chaotic, irregular motion with rapid mixing and fluctuations in velocity, pressure, and temperature
• Laminar flow characterized by smooth, parallel layers of fluid with no mixing between layers
• Turbulent flow occurs at high Reynolds numbers while laminar flow occurs at low Reynolds numbers

Boundary layer theory

• Describes the thin layer of fluid near a solid surface where viscous effects are significant
• Boundary layer develops due to the no-slip condition at the surface, causing velocity to increase from zero to the freestream value

Boundary layer thickness

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• Defined as the distance from the surface where the velocity reaches 99% of the freestream value
• Increases with distance from the leading edge of the surface
• Affected by factors such as Reynolds number, pressure gradient, and surface roughness

Boundary layer separation

• Occurs when the boundary layer detaches from the surface due to adverse pressure gradient
• Leads to increased drag, reduced lift, and flow instability
• Can be delayed or prevented by using flow control techniques such as suction or vortex generators

Turbulent boundary layer structure

• Consists of distinct regions with different flow characteristics and scaling laws
• Dominated by turbulent mixing and momentum transfer across the layer

Inner vs outer region

• Inner region close to the wall where viscous effects are dominant and velocity scales with wall units
• Outer region further from the wall where turbulent mixing dominates and velocity scales with boundary layer thickness
• Overlap region in between where both scaling laws apply

Viscous sublayer

• Thin layer closest to the wall where viscous stresses dominate and velocity profile is linear
• Velocity scales with wall units $u^+ = y^+$
• Typically extends up to $y^+ \approx 5$

Buffer layer

• Transition region between the viscous sublayer and the logarithmic layer
• Velocity profile deviates from the linear and logarithmic laws
• Extends from $y^+ \approx 5$ to $y^+ \approx 30$

Logarithmic layer

• Region where the velocity profile follows a logarithmic law $u^+ = \frac{1}{\kappa} \ln y^+ + B$
• $\kappa$ is the von Karman constant $(\approx 0.41)$ and $B$ is a constant $(\approx 5.2)$
• Extends from $y^+ \approx 30$ to $y/\delta \approx 0.2$

Overlap region

• Region where the inner and outer scaling laws overlap
• Velocity profile follows a power law $u/U_\infty = (y/\delta)^\alpha$
• $\alpha$ is a constant that depends on the pressure gradient and Reynolds number

Turbulent boundary layer equations

• Governing equations for turbulent boundary layers derived from the Navier-Stokes equations
• Require modeling of the turbulent stresses to close the system of equations

Reynolds-averaged Navier-Stokes equations

• Obtained by decomposing the velocity and pressure fields into mean and fluctuating components
• Result in additional terms called Reynolds stresses that represent the effect of turbulent mixing
• Require modeling to close the system of equations

Closure problem

• Arises because there are more unknowns than equations in the Reynolds-averaged Navier-Stokes equations
• Requires modeling of the Reynolds stresses using turbulence models such as eddy viscosity models or Reynolds stress models
• Choice of turbulence model depends on the flow configuration and desired accuracy

Turbulent boundary layer parameters

• Various dimensionless parameters that affect the structure and behavior of turbulent boundary layers
• Used to characterize the flow and compare results from different experiments and simulations

Reynolds number effects

• Turbulent boundary layers become thicker and more energetic as the Reynolds number increases
• Viscous sublayer and buffer layer become thinner relative to the boundary layer thickness
• Turbulent mixing and momentum transfer increase with Reynolds number

Pressure gradient effects

• Favorable pressure gradient (decreasing pressure in the flow direction) leads to a thinner boundary layer and delayed separation
• Adverse pressure gradient (increasing pressure in the flow direction) leads to a thicker boundary layer and earlier separation
• Zero pressure gradient corresponds to a boundary layer with constant thickness

Surface roughness effects

• Roughness elements on the surface can trigger transition to turbulence and increase turbulent mixing
• Roughness increases the skin friction drag and heat transfer at the surface
• Effect of roughness depends on the size and shape of the roughness elements relative to the boundary layer thickness

Turbulent boundary layer measurements

• Experimental techniques used to measure velocity, pressure, and temperature in turbulent boundary layers
• Provide data for validating turbulence models and studying the physics of turbulent flows

Hot-wire anemometry

• Measures velocity by sensing the change in resistance of a thin wire exposed to the flow
• Provides high temporal resolution but limited spatial resolution
• Requires careful calibration and correction for temperature effects

Particle image velocimetry

• Measures velocity by tracking the displacement of tracer particles in the flow
• Provides high spatial resolution but limited temporal resolution
• Requires optical access to the flow and careful seeding of tracer particles

Turbulent boundary layer control

• Techniques used to manipulate the structure and behavior of turbulent boundary layers
• Aim to reduce drag, enhance heat transfer, or delay separation

Passive vs active control

• Passive control uses fixed geometric modifications such as riblets or vortex generators
• Active control uses dynamic actuation such as suction, blowing, or wall motion
• Active control can adapt to changing flow conditions but requires energy input

Riblets

• Streamwise grooves on the surface that reduce turbulent mixing and skin friction drag
• Work by restricting the spanwise motion of near-wall vortices
• Optimal riblet size and spacing depend on the boundary layer thickness and Reynolds number

Polymer additives

• Long-chain polymer molecules added to the fluid that reduce turbulent drag
• Work by suppressing near-wall turbulence and increasing the viscous sublayer thickness
• Effective at low concentrations but can degrade over time and cause environmental issues

Suction

• Removal of fluid from the boundary layer through porous walls or slots
• Reduces the boundary layer thickness and delays separation
• Requires careful design of the suction system to avoid flow disturbances

Turbulent boundary layer applications

• Practical applications where understanding and controlling turbulent boundary layers is important
• Span a wide range of fields including aerospace, automotive, and energy systems

Aerodynamic drag reduction

• Reducing turbulent skin friction drag on aircraft, cars, and ships
• Achieved through techniques such as riblets, polymer additives, and suction
• Can lead to significant fuel savings and emission reductions

Heat transfer enhancement

• Increasing turbulent heat transfer in heat exchangers, cooling systems, and combustion chambers
• Achieved through techniques such as surface roughness, vortex generators, and jet impingement
• Can lead to more compact and efficient heat transfer devices

Flow-induced noise reduction

• Reducing turbulent noise generated by flow over surfaces such as aircraft wings, wind turbine blades, and submarine hulls
• Achieved through techniques such as porous surfaces, serrations, and active flow control
• Can lead to quieter and more environmentally friendly transportation systems