5.5 Flow separation

Flow separation is a critical phenomenon in fluid dynamics where fluid detaches from a surface, causing turbulence and drag. This topic explores the types, causes, and effects of flow separation, including boundary layer separation, wake formation, and vortex shedding.

Understanding flow separation is crucial for engineers designing aircraft, vehicles, and structures. The notes cover prediction methods, control techniques, and applications in various fields. Experimental and mathematical approaches for studying flow separation are also discussed.

Types of flow separation

• Flow separation occurs when a fluid flow detaches from a solid surface, resulting in a region of reversed flow and increased turbulence
• The three main types of flow separation are boundary layer separation, wake formation, and vortex shedding, each with distinct characteristics and impacts on fluid dynamics

Boundary layer separation

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• Occurs when the boundary layer, the thin layer of fluid adjacent to a surface, separates from the surface due to adverse pressure gradients or geometric irregularities
• Characterized by a reversal of flow direction near the surface, leading to the formation of recirculation zones and increased drag
• Boundary layer separation can be laminar or turbulent, depending on the Reynolds number and surface conditions
• Examples include flow over a curved surface (airfoil) and flow in a diverging channel (diffuser)

Wake formation

• Happens downstream of a body immersed in a fluid flow, where the separated boundary layers from opposite sides of the body merge to form a wake
• Wakes are characterized by a low-pressure region, reduced velocity, and increased turbulence compared to the surrounding flow
• The size and shape of the wake depend on the body's geometry and the flow conditions (Reynolds number, angle of attack)
• Examples include flow behind a circular cylinder, a sphere, or a bluff body (vehicle, building)

Vortex shedding

• An unsteady flow phenomenon that occurs when alternating vortices are shed from the sides of a body in a periodic manner
• Caused by the interaction between the separated boundary layers and the surrounding flow, leading to the formation of a von Kármán vortex street
• The frequency of vortex shedding depends on the body's size, shape, and the flow velocity, characterized by the Strouhal number
• Examples include flow around a circular cylinder, a square prism, or a flag pole

Causes of flow separation

Adverse pressure gradients

• Flow separation often occurs when the fluid encounters an adverse pressure gradient, where the pressure increases in the flow direction
• Adverse pressure gradients cause the boundary layer to decelerate and eventually reverse direction, leading to separation
• The severity of the adverse pressure gradient determines the location and extent of flow separation
• Examples include flow over the rear portion of an airfoil, flow in a diffuser, or flow over a backward-facing step

Viscous effects near walls

• The viscosity of the fluid plays a crucial role in flow separation, especially near solid boundaries where the velocity gradients are high
• The no-slip condition at the wall leads to the formation of a boundary layer, where viscous effects are dominant
• As the boundary layer grows and encounters an adverse pressure gradient, the near-wall fluid particles lose momentum and are more susceptible to separation
• Examples include flow over a flat plate with a sudden change in surface roughness or flow in a pipe with a sharp bend

Geometric factors

• The geometry of the body or the flow domain can significantly influence the occurrence and location of flow separation
• Sharp corners, edges, or abrupt changes in the surface contour can cause the boundary layer to separate prematurely
• Streamlined shapes, such as airfoils or teardrop-shaped bodies, are designed to minimize flow separation and reduce drag
• Examples include flow around a square cylinder (sharp corners) compared to a circular cylinder, or flow over a smooth sphere versus a golf ball (dimpled surface)

Effects of flow separation

Increased drag

• Flow separation leads to a significant increase in pressure drag, also known as form drag, due to the formation of a low-pressure region behind the body
• The separated flow region, or wake, is characterized by increased turbulence and mixing, which further contributes to the drag force
• The magnitude of the drag increase depends on the size and shape of the separated flow region, which is influenced by the body's geometry and flow conditions
• Examples include the high drag experienced by bluff bodies (vehicles, buildings) compared to streamlined shapes (airfoils, submarines)

Reduced lift

• In the case of lifting bodies, such as aircraft wings or turbine blades, flow separation can lead to a substantial reduction in lift force
• Separation on the upper surface of an airfoil causes a loss of suction and a decrease in the pressure difference between the upper and lower surfaces, resulting in reduced lift
• The onset of flow separation limits the maximum lift coefficient and determines the stall angle of an airfoil
• Examples include aircraft wing stall during takeoff or landing, or the performance degradation of wind turbine blades at high angles of attack

Unsteady flow patterns

• Flow separation often leads to unsteady and turbulent flow patterns downstream of the body, characterized by vortex shedding, oscillations, and increased mixing
• The unsteady nature of separated flows can cause structural vibrations, acoustic noise, and increased fatigue loading on the body
• The interaction between the separated flow and the body can lead to complex flow phenomena, such as galloping, flutter, or lock-in
• Examples include the vibration of power transmission lines due to vortex shedding, or the flapping of a flag in the wind

Predicting flow separation

Boundary layer theory

• Boundary layer theory provides a framework for analyzing the flow behavior near solid surfaces and predicting the onset of separation
• The boundary layer equations, derived from the Navier-Stokes equations, describe the velocity and pressure distribution within the boundary layer
• Analytical solutions, such as the Blasius solution for laminar flow over a flat plate, can be used to estimate the boundary layer thickness and skin friction coefficient
• Numerical methods, such as the Thwaites' method or the Kármán-Pohlhausen method, can be employed to solve the boundary layer equations for more complex geometries and flow conditions

Pressure gradient analysis

• Analyzing the pressure gradient along the surface of a body can provide insights into the likelihood and location of flow separation
• Adverse pressure gradients, where the pressure increases in the flow direction, are indicative of potential flow separation
• The pressure gradient can be obtained from experimental measurements (pressure taps, pressure-sensitive paint) or computational fluid dynamics (CFD) simulations
• Examples include the pressure distribution over an airfoil at different angles of attack, or the pressure gradient in a diffuser with varying area ratios

Computational fluid dynamics (CFD)

• CFD simulations solve the governing equations of fluid motion (Navier-Stokes equations) numerically to predict the flow behavior, including separation
• CFD models can capture the complex interaction between the boundary layer and the surrounding flow, providing detailed information on velocity, pressure, and turbulence fields
• Turbulence modeling is crucial for accurately predicting flow separation, with models ranging from Reynolds-Averaged Navier-Stokes (RANS) to Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS)
• Examples include the CFD analysis of flow over an aircraft wing, a wind turbine blade, or a car body to optimize the design for minimum separation and drag

Controlling flow separation

Streamlining surfaces

• Streamlining involves shaping the body or surface to minimize the adverse pressure gradient and delay flow separation
• Streamlined shapes, such as airfoils, are designed with a smooth and gradual change in surface contour to maintain attached flow over a wide range of operating conditions
• Proper streamlining can significantly reduce drag and improve the performance of vehicles, aircraft, and other fluid-dynamic systems
• Examples include the design of aircraft wings (supercritical airfoils), high-speed trains (nose cones), and submarines (teardrop shape)

Active flow control methods

• Active flow control involves the use of external energy input to manipulate the boundary layer and prevent or delay flow separation
• Techniques such as boundary layer suction, blowing, or synthetic jets can be used to energize the near-wall flow and overcome adverse pressure gradients
• Active flow control can be adaptive, with the control input adjusted based on real-time flow measurements or feedback
• Examples include the use of micro-jets on aircraft wings to prevent stall, or the application of plasma actuators to control flow separation on turbine blades

Passive flow control devices

• Passive flow control relies on geometric modifications or the use of fixed devices to alter the flow field and suppress separation
• Vortex generators, such as small fins or protrusions, can be placed on the surface to create streamwise vortices that energize the boundary layer and delay separation
• Riblets, which are small grooves aligned with the flow direction, can be used to reduce the skin friction drag and improve the resistance to separation
• Examples include the use of vortex generators on aircraft wings (stall strips) or wind turbine blades, and the application of riblets on the hulls of ships or the surfaces of pipelines

Applications of flow separation

Aircraft wing stall

• Stall occurs when an aircraft wing exceeds its critical angle of attack, leading to flow separation and a sudden loss of lift
• The onset of stall is characterized by the separation of the boundary layer on the upper surface of the wing, resulting in a dramatic increase in drag and a loss of control
• Understanding and predicting stall is crucial for ensuring safe and efficient aircraft operation, particularly during takeoff, landing, and maneuvering
• Examples include the use of stall warning systems, such as stick shakers or angle-of-attack indicators, and the design of high-lift devices (slats, flaps) to delay the onset of stall

Bluff body aerodynamics

• Bluff bodies, such as vehicles, buildings, and bridges, are characterized by their non-streamlined shape and the presence of large separated flow regions
• The aerodynamics of bluff bodies is dominated by the formation of wakes, vortex shedding, and unsteady flow patterns, which contribute to high drag forces and structural loading
• Understanding and controlling flow separation around bluff bodies is essential for improving their aerodynamic performance, stability, and safety
• Examples include the design of low-drag vehicle shapes (cars, trucks), the optimization of building facades for wind loading, and the mitigation of vortex-induced vibrations in bridges or tall structures

Turbomachinery performance

• Flow separation can significantly impact the performance and efficiency of turbomachinery, such as compressors, turbines, and pumps
• In compressors, flow separation on the blade surfaces can lead to stall, surge, and reduced pressure rise, limiting the operating range and stability of the machine
• In turbines, flow separation can cause a reduction in the power output and efficiency, as well as increased blade loading and vibration
• Examples include the design of advanced compressor blade profiles (controlled diffusion airfoils) to minimize separation, and the use of active flow control techniques (blowing, suction) to improve the performance of gas turbine engines

Experimental techniques

Flow visualization methods

• Flow visualization techniques are used to qualitatively observe and analyze the flow patterns, including separation, in experimental settings
• Surface flow visualization methods, such as oil flow or tufts, can reveal the surface streamline patterns and the location of separation and reattachment points
• Off-body flow visualization methods, such as smoke or dye injection, can provide insights into the three-dimensional flow structures and the development of wakes and vortices
• Examples include the use of smoke generators in wind tunnel tests to visualize the flow over an aircraft model, or the application of hydrogen bubble technique to study the flow separation in a water channel

Particle image velocimetry (PIV)

• PIV is a non-intrusive optical measurement technique that provides instantaneous velocity fields in a fluid flow
• Small tracer particles are seeded into the flow, and their positions are recorded using high-speed cameras at two closely spaced time instances
• The displacement of the particles between the two images is used to calculate the velocity vectors, yielding a high-resolution velocity field
• PIV can be used to study the detailed flow structures and turbulence characteristics in separated flows, such as wakes, vortices, and recirculation zones
• Examples include the PIV measurements of the flow around a circular cylinder, an airfoil at high angles of attack, or in the wake of a wind turbine

Hot-wire anemometry

• Hot-wire anemometry is a point-measurement technique that uses a thin wire heated by an electric current to measure the local fluid velocity
• The wire is exposed to the flow, and the heat transfer from the wire to the fluid depends on the velocity, allowing for high-frequency velocity measurements
• Hot-wire probes can be single, double, or triple-wired, enabling the measurement of one, two, or three velocity components, respectively
• Hot-wire anemometry is particularly useful for studying the turbulence characteristics and the unsteady nature of separated flows, providing information on velocity fluctuations, spectra, and correlations
• Examples include the use of hot-wire probes to measure the velocity profiles in a turbulent boundary layer, the wake of a bluff body, or the flow in a turbomachinery passage

Mathematical modeling

Navier-Stokes equations

• The Navier-Stokes equations are the fundamental governing equations of fluid motion, describing the conservation of mass, momentum, and energy in a fluid flow

• The equations are a set of coupled, non-linear partial differential equations that relate the velocity, pressure, density, and temperature fields in a fluid

• In their general form, the Navier-Stokes equations are:

  • Conservation of mass (continuity equation): $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$

  • Conservation of momentum: $\rho \left( \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = -\nabla p + \mu \nabla^2 \mathbf{u} + \rho \mathbf{g}$

  • Conservation of energy: $\rho c_p \left( \frac{\partial T}{\partial t} + \mathbf{u} \cdot \nabla T \right) = k \nabla^2 T + \Phi$

  where $\rho$ is the density, $\mathbf{u}$ is the velocity vector, $p$ is the pressure, $\mu$ is the dynamic viscosity, $\mathbf{g}$ is the gravitational acceleration, $c_p$ is the specific heat capacity, $T$ is the temperature, $k$ is the thermal conductivity, and $\Phi$ is the viscous dissipation term.

• Solving the Navier-Stokes equations analytically is challenging due to their non-linearity and coupling, and is only possible for simple flow cases with many assumptions and simplifications

• Numerical methods, such as finite difference, finite volume, or finite element methods, are commonly used to solve the Navier-Stokes equations for complex flow problems, including those involving flow separation

Boundary layer equations

• The boundary layer equations are a simplified form of the Navier-Stokes equations that describe the flow behavior in the thin layer adjacent to a solid surface

• The equations are derived by applying the boundary layer approximations, which assume that the flow is predominantly parallel to the surface, the pressure gradient normal to the surface is negligible, and the viscous effects are confined to a thin region near the wall

• The two-dimensional steady boundary layer equations for incompressible flow are:

  • Continuity equation: $\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$

  • Momentum equation: $u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = -\frac{1}{\rho} \frac{dP}{dx} + \nu \frac{\partial^2 u}{\partial y^2}$

  where $u$ and $v$ are the velocity components in the $x$ and $y$ directions, respectively, $P$ is the pressure, and $\nu$ is the kinematic viscosity.

• The boundary layer equations are parabolic in nature and can be solved using various analytical and numerical methods, such as the Blasius solution, the Falkner-Skan solution, or the Keller box method

• The boundary layer equations provide valuable insights into the flow behavior near surfaces, including the growth of the boundary layer, the skin friction distribution, and the onset of flow separation

Turbulence modeling approaches

• Turbulence modeling is essential for accurately predicting flow separation and its consequences, as turbulence plays a crucial role in the dynamics of separated flows
• Direct Numerical Simulation (DNS) involves solving the Navier-Stokes equations without any turbulence modeling, resolving all the spatial and temporal scales of turbulence
  • DNS is computationally expensive and limited to low Reynolds number flows and simple geometries
• Large Eddy Simulation (LES) is a technique that resolves the large-scale turbulent motions and models the smaller scales using a subgrid-scale model
  • LES provides a more accurate representation of turbulence than RANS models but is still computationally demanding
• Reynolds-Averaged Navier-Stokes (RANS) models are based on the Reynolds decomposition, where the flow variables are split into mean and fluctuating components
  • RANS models solve for the mean flow quantities and model the effects of turbulence using various closure models, such as the $k-\epsilon$, $k-\omega$, or Reynolds stress models
  • RANS models are computationally efficient and widely used in engineering applications but may have limitations in accurately predicting flow separation and unsteady phenomena
• Hybrid RANS-LES methods, such as Detached Eddy Simulation (DES) or Delayed Detached Eddy Simulation (DDES), combine RANS modeling near the walls with LES in the separated regions, providing a balance between accuracy an