2.4 Vorticity and circulation

Vorticity and circulation are key concepts in fluid dynamics that describe rotational motion in fluids. They help us understand complex flow behaviors, from the lift on airplane wings to the swirling patterns in turbulent flows.

These concepts are crucial for analyzing fluid motion and its effects on objects. By studying vorticity and circulation, we can predict and explain phenomena like vortex shedding, boundary layer separation, and energy transfer in turbulent flows.

Definition of vorticity

• Vorticity is a fundamental concept in fluid dynamics that quantifies the local rotation of a fluid element
• It plays a crucial role in understanding the behavior of fluids, especially in situations involving swirling or rotating flows
• Vorticity is a vector quantity that describes the curl of the velocity field, indicating the direction and magnitude of rotation

Mathematical representation

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• Mathematically, vorticity is defined as the curl of the velocity field: $\vec{\omega} = \nabla \times \vec{u}$
  • $\vec{\omega}$ represents the vorticity vector
  • $\vec{u}$ represents the velocity field
• In Cartesian coordinates, the vorticity components are given by:
  • $\omega_x = \frac{\partial w}{\partial y} - \frac{\partial v}{\partial z}$
  • $\omega_y = \frac{\partial u}{\partial z} - \frac{\partial w}{\partial x}$
  • $\omega_z = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}$
• The magnitude of vorticity is given by $|\vec{\omega}| = \sqrt{\omega_x^2 + \omega_y^2 + \omega_z^2}$

Physical interpretation

• Physically, vorticity represents the angular velocity of a fluid element as it moves through the flow field
• It measures the rate at which a fluid element rotates about its own axis, independent of the overall motion of the fluid
• Vorticity is often associated with the presence of vortices or eddies in the flow, such as those observed in the wake of a bluff body (cylinder) or in a swirling flow (tornado)
• High vorticity regions indicate areas of intense rotation, while low vorticity regions correspond to more irrotational or potential flow

Vorticity vs. rotation

• Vorticity and rotation are closely related concepts in fluid dynamics, but they have some key differences

Similarities

• Both vorticity and rotation describe the rotational motion of a fluid element
• They are vector quantities that have a magnitude and direction
• Vorticity and rotation are important in understanding the behavior of fluids in various applications, such as aerodynamics, turbomachinery, and geophysical flows

Key differences

• Rotation refers to the angular velocity of a fluid element relative to a fixed reference frame, while vorticity is the angular velocity of a fluid element relative to its own axis
• Rotation can be caused by external forces or boundary conditions (rotating cylinder), whereas vorticity is an intrinsic property of the flow field
• Vorticity is a local quantity that varies from point to point in the flow, while rotation can be uniform throughout the fluid domain
• In a rotating reference frame (rotating tank), the fluid may have a non-zero rotation but zero vorticity if the flow is irrotational

Vorticity equation

• The vorticity equation is a fundamental equation in fluid dynamics that describes the evolution of vorticity in a flow field

Derivation

• The vorticity equation can be derived by taking the curl of the Navier-Stokes equations, which govern the motion of fluids
• Starting with the Navier-Stokes equations in vector form: $\frac{\partial \vec{u}}{\partial t} + (\vec{u} \cdot \nabla)\vec{u} = -\frac{1}{\rho}\nabla p + \nu \nabla^2 \vec{u}$
• Taking the curl of both sides and using the vector identity $\nabla \times (\vec{u} \cdot \nabla)\vec{u} = (\vec{u} \cdot \nabla)\vec{\omega} - (\vec{\omega} \cdot \nabla)\vec{u} + \vec{\omega}(\nabla \cdot \vec{u})$, we obtain the vorticity equation: $\frac{\partial \vec{\omega}}{\partial t} + (\vec{u} \cdot \nabla)\vec{\omega} = (\vec{\omega} \cdot \nabla)\vec{u} + \nu \nabla^2 \vec{\omega}$

Terms and their meanings

• $\frac{\partial \vec{\omega}}{\partial t}$: The local rate of change of vorticity
• $(\vec{u} \cdot \nabla)\vec{\omega}$: The advection of vorticity by the velocity field, representing the transport of vorticity by the fluid motion
• $(\vec{\omega} \cdot \nabla)\vec{u}$: The vortex stretching term, which describes the intensification or weakening of vorticity due to the stretching or compression of vortex lines
• $\nu \nabla^2 \vec{\omega}$: The diffusion of vorticity due to viscous effects, which tends to smooth out vorticity gradients
• The vorticity equation shows that vorticity can be generated, transported, stretched, and diffused in a flow field, leading to complex vortical structures and dynamics

Vortex lines and tubes

• Vortex lines and tubes are geometric constructs used to visualize and analyze the vorticity field in a fluid flow

Definitions

• A vortex line is a curve that is tangent to the vorticity vector at every point along its length
  • Mathematically, a vortex line is defined by the equation $\frac{dx}{\omega_x} = \frac{dy}{\omega_y} = \frac{dz}{\omega_z}$
• A vortex tube is a bundle of vortex lines that form a tubular structure
  • The cross-sectional area of a vortex tube is inversely proportional to the magnitude of vorticity, as required by the conservation of circulation

Properties

• Vortex lines cannot start or end within the fluid; they must either form closed loops or extend to the boundaries of the domain
• Vortex lines cannot cross each other, as this would imply a discontinuity in the vorticity field
• The strength of a vortex tube, defined as the circulation around its cross-section, remains constant along its length (Kelvin's circulation theorem)
• Vortex tubes move with the fluid, and their evolution is governed by the vorticity equation
• In inviscid flows, vortex lines and tubes are material lines, meaning they are composed of the same fluid particles over time

Helmholtz's vortex theorems

• Helmholtz's vortex theorems are a set of fundamental principles that describe the behavior of vorticity in inviscid, barotropic flows

Kelvin's circulation theorem

• Kelvin's circulation theorem states that the circulation around a closed loop moving with the fluid remains constant over time
• Mathematically, $\frac{D}{Dt} \oint_C \vec{u} \cdot d\vec{l} = 0$, where $C$ is a closed loop moving with the fluid
• This theorem implies that vorticity cannot be created or destroyed in an inviscid, barotropic flow, and that the strength of a vortex tube remains constant along its length

Vortex tube evolution

• Helmholtz's second theorem states that a vortex tube moves with the fluid and retains its strength, even as it is stretched or deformed by the flow
• This means that the circulation around a vortex tube remains constant, and the vortex lines within the tube are material lines that move with the fluid particles

Vortex lines and material lines

• Helmholtz's third theorem asserts that in an inviscid, barotropic flow, vortex lines are material lines
• This implies that fluid particles that initially lie on a vortex line will remain on that vortex line as the flow evolves, and the vorticity of each fluid particle is conserved
• The connection between vortex lines and material lines highlights the Lagrangian nature of vorticity in inviscid flows

Circulation

• Circulation is a scalar quantity that measures the macroscopic rotation of a fluid along a closed curve

Definition and properties

• Circulation is defined as the line integral of the velocity field along a closed curve $C$: $\Gamma = \oint_C \vec{u} \cdot d\vec{l}$
• It quantifies the net amount of rotation along the curve, with counterclockwise rotation being positive and clockwise rotation being negative
• Circulation is a global quantity that depends on the choice of the closed curve, unlike vorticity, which is a local quantity
• According to Stokes' theorem, the circulation around a closed curve is equal to the flux of vorticity through any surface bounded by that curve: $\Gamma = \int_S \vec{\omega} \cdot d\vec{A}$

Circulation vs. vorticity

• While circulation and vorticity are related concepts, they have some key differences:
  • Circulation is a scalar quantity, while vorticity is a vector quantity
  • Circulation is a global measure of rotation, while vorticity is a local measure of rotation
  • Circulation depends on the choice of the closed curve, while vorticity is independent of any specific path
• In inviscid flows, the circulation around a closed material curve remains constant (Kelvin's circulation theorem), while the vorticity of individual fluid particles is conserved
• The connection between circulation and vorticity is given by Stokes' theorem, which relates the circulation around a closed curve to the flux of vorticity through a surface bounded by that curve

Kutta-Joukowski theorem

• The Kutta-Joukowski theorem is a fundamental principle in aerodynamics that relates the lift generated by an airfoil to the circulation around it

Lift generation

• The Kutta-Joukowski theorem states that the lift force per unit span acting on an airfoil is equal to the product of the fluid density, the freestream velocity, and the circulation around the airfoil: $L' = \rho_{\infty} U_{\infty} \Gamma$
• The circulation around the airfoil is established by the Kutta condition, which requires the flow to leave the trailing edge smoothly, with finite velocity
• The presence of circulation around the airfoil leads to a pressure difference between the upper and lower surfaces, resulting in lift generation

Circulation around airfoils

• The circulation around an airfoil can be calculated using the Kutta-Joukowski theorem, given the lift force and the freestream conditions
• In potential flow theory, the circulation around an airfoil can be modeled using vortex elements, such as point vortices or vortex panels
• The circulation distribution along the airfoil is determined by enforcing the Kutta condition and the no-penetration boundary condition on the airfoil surface
• The circulation around an airfoil varies with the angle of attack, with higher angles of attack generally resulting in increased circulation and lift, up to the point of stall

Potential flow

• Potential flow is a simplified model of fluid flow that assumes the flow is inviscid, irrotational, and incompressible

Irrotational vs. rotational flow

• In potential flow, the vorticity is assumed to be zero everywhere in the fluid domain, making the flow irrotational
• Irrotational flows are characterized by the existence of a velocity potential $\phi$, such that the velocity field can be expressed as the gradient of the potential: $\vec{u} = \nabla \phi$
• In contrast, rotational flows have non-zero vorticity and cannot be described by a velocity potential alone
• Many real flows exhibit a combination of irrotational and rotational regions, with vorticity being generated at boundaries and advected into the flow

Velocity potential

• The velocity potential $\phi$ is a scalar function that fully describes the velocity field in an irrotational flow
• It is related to the velocity components by: $u = \frac{\partial \phi}{\partial x}$, $v = \frac{\partial \phi}{\partial y}$, $w = \frac{\partial \phi}{\partial z}$
• The velocity potential satisfies Laplace's equation, $\nabla^2 \phi = 0$, which is a linear partial differential equation
• The linearity of Laplace's equation allows for the superposition of elementary potential flow solutions (uniform flow, source, sink, doublet) to construct more complex flow fields
• The velocity potential provides a convenient framework for analyzing irrotational flows and has been widely used in aerodynamics, hydrodynamics, and other branches of fluid mechanics

Vorticity in viscous flows

• In real fluids, viscosity plays a crucial role in the generation, diffusion, and dissipation of vorticity

Diffusion of vorticity

• Viscous effects lead to the diffusion of vorticity, causing vorticity gradients to smooth out over time
• The diffusion of vorticity is governed by the viscous term in the vorticity equation, $\nu \nabla^2 \vec{\omega}$
• The diffusion of vorticity is a dissipative process that leads to the decay of vortical structures and the eventual homogenization of the vorticity field
• The rate of vorticity diffusion depends on the kinematic viscosity of the fluid, with higher viscosity leading to faster diffusion

Vorticity generation at boundaries

• In viscous flows, vorticity is generated at solid boundaries due to the no-slip condition, which requires the fluid velocity to match the velocity of the boundary
• The presence of velocity gradients near the boundary leads to the production of vorticity, as described by the vorticity equation
• The generated vorticity diffuses away from the boundary and is advected into the flow by the velocity field
• Boundary layers, which are thin regions of high velocity gradients near solid surfaces, are a primary source of vorticity in viscous flows
• The interaction between the boundary-generated vorticity and the outer flow can lead to complex vortical structures, such as separation bubbles, vortex shedding, and turbulent boundary layers

Vortex shedding

• Vortex shedding is a phenomenon that occurs when a fluid flows past a bluff body, resulting in the periodic formation and detachment of vortices in the wake

Mechanism

• As the fluid flows past a bluff body (cylinder), boundary layers develop on the surface due to viscous effects
• The adverse pressure gradient behind the body causes the boundary layers to separate, leading to the formation of shear layers
• The shear layers roll up into vortices, which are shed alternately from the upper and lower surfaces of the body
• The shedding of vortices creates an oscillating flow pattern in the wake, known as the von Kármán vortex street

von Kármán vortex street

• The von Kármán vortex street is a stable, periodic arrangement of vortices in the wake of a bluff body, named after Theodore von Kármán
• It consists of two staggered rows of vortices with opposite circulation, which are shed alternately from the upper and lower surfaces of the body
• The vortex street is characterized by a specific spacing ratio between the vortices, known as the Strouhal number, which depends on the Reynolds number of the flow
• The formation of the von Kármán vortex street is associated with unsteady lift and drag forces on the body, as well as acoustic noise and structural vibrations
• Vortex shedding and the von Kármán vortex street are important phenomena in many engineering applications, such as flow around buildings, bridges, and offshore structures, as well as in the design of heat exchangers and musical instruments (Aeolian harp)

Vorticity in turbulence

• Turbulent flows are characterized by the presence of a wide range of scales of motion, from large eddies to small dissipative scales, and are strongly influenced by vorticity dynamics

Vortex stretching

• Vortex stretching is a key mechanism in the dynamics of turbulent flows, responsible for the transfer of energy from large scales to small scales
• In three-dimensional turbulence, vortex lines can be stretched by the velocity gradients, leading to the intensification of vorticity
• The vortex stretching term in the vorticity equation, $(\vec{\omega} \cdot \nabla)\vec{u}$, describes the amplification of vorticity due to the stretching of vortex lines
• Vortex stretching is a fundamental process in the energy cascade, where energy is transferred from large eddies to smaller eddies, until it is dissipated by viscosity at the Kolmogorov scale

Enstrophy and energy dissipation

• Enstrophy is a scalar quantity that measures the total amount of vorticity in a flow, defined as the integral of the square of the vorticity magnitude: $\varepsilon = \int_V |\vec{\omega}|^2 dV$
• In turbulent flows, enstrophy is closely related to the dissipation of kinetic energy by viscosity
• The enstrophy equation, derived from the vorticity equation, describes the evolution of enstrophy in a flow and includes terms for enstrophy production, dissipation, and transport
• The dissipation of enstrophy is proportional to the dissipation of kinetic energy, with the proportionality constant being the kinematic viscosity: $\varepsilon_{\omega} = \nu \varepsilon$