In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along with the flow. It is an important quantity in the dynamical theory of fluids and provides a convenient framework for understanding a variety of complex flow phenomena, such as the formation and motion of vortex rings.Mathematically, the vorticity
ω
→
{\displaystyle {\vec {\omega }}}
is the curl of the flow velocity
u
→
{\displaystyle {\vec {u}}}
:
ω
→
≡
∇
×
u
→
,
{\displaystyle {\vec {\omega }}\equiv \nabla \times {\vec {u}}\,,}
where
∇
{\displaystyle \nabla }
is the del operator. Conceptually,
ω
→
{\displaystyle {\vec {\omega }}}
could be determined by marking parts of a continuum in a small neighborhood of the point in question, and watching their relative displacements as they move along the flow. The vorticity
ω
→
{\displaystyle {\vec {\omega }}}
would be twice the mean angular velocity vector of those particles relative to their center of mass, oriented according to the right-hand rule.
In a two-dimensional flow,
ω
→
{\displaystyle {\vec {\omega }}}
is always perpendicular to the plane of the flow, and can therefore be considered a scalar field.
I'm learning Meteorology, and using the book Atmospheric Science by Wallace and Hobbs. We're discussing the kinematics of the winds (fluids). I shall post some images to say what I don't understand. This is how they define their natural coordinate system
I know and understand the concepts of...
The classical free surface profile for the solitary wave for irrotational and incompressible fluids for small amplitude and long wavelength is the classical Korteweg-deVries(KdV) equation given by:\frac{\partial\eta}{\partial t}+\frac{\partial \eta}{\partial x}+\eta\frac{\partial\eta}{\partial...
This is not homework. I'm studying fluid mechanics/dynamics in the heart/blood vessels and I just want to understand this, so I can have a better appreciation for it's clinical relevance. I'm more of biology/biochem type of person so this has been a bit of challenge. I have basic physics course...
Can someone check if my answer is correct please?
Question:
If liquid contained within a finite closed circular cylinder rotates about the axis k of the cylinder prove that the equation of continuity and boundary conditions are satisfied by u = ΩxR where Ω = Ωk is the constant angular velocity...
Homework Statement
The compressible inviscid vorticity convection equation:
$$\frac{D(\frac{w}{\rho})}{Dt}=(\frac{w}{\rho})\cdot \nabla U + \frac{1}{\rho}\nabla P \times \nabla (\frac{1}{\rho})$$
differs from the incompressible version in two important ways :
1) The convected quantity is...
Homework Statement
The vorticity vector ##\vec{\omega} = \text{curl}\ \vec{v}##, defined as usual by ##\omega^{2}=i_{{\vec{\omega}}}\text{vol}^{3}##, is ##\textit{not}## usually invariant since the flow need not conserve the volume form.
The mass form, ##\rho\ \text{vol}^{3}##, however...
There are several unreasonable assumptions in the formulation of the Prandlt Mixing Theory. However, it works reasonably well for simple 1-D flow. An attempt to explain why is given by Davidson in his book 'Turbulence', on page 122 - 124, section 4.1.4.
It's stated that it still works because...
Hi I am trying to understand what forces are at work to slow a liquid flowing in a circular pipe
As an example the above torus pipe fully filled with an incompressible low viscosity liquid. This is rotated until the fluid achieves solid body rotation and then the pipe is suddenly stopped...
I would like to reproduce results from a much older code to test a new one. I only have the old code's results in the form of plots, not data, but I need data. The older code solves the unsteady vorticity transport equation in 2D with a constant kinematic viscosity term. I'm interested in 2-D...
So mathematically I understand that divergence of curl of something is zero.
However, talking specifically about vorticity, this is what it seems to imply to me:
When there is vorticity in a fluid, the tiny particles spin around their own axes, so a net circulation is formed around the surface...
Hi guys. Let me just say at the outset that I know very little fluid mechanics but I keep coming back to the same issue over and over in a general relativity related problem so I figured I'd just ask the fluid mechanics question here.
In countless places the interpretation of the vorticity...
Hi asking for help
how to calculate vorticity using fortran
∇×c ⃗ where both nabla and c are three dimension vector, and c is the wind speed with three component
The quantity ##\vec{\omega} = \vec{\nabla} \times \vec{f}## is called vorticity and is the measure of the local circulation of the vector field ##\vec{f}##.
So, given the same vector field ##\vec{f}##, is possible measure the local flux by ##\vec{\nabla} \cdot \vec{f}##. This quantity has...
If invisid flow starts with no vorticity then no vorticity will be produced. This can be understood
intuitively: we note that of the three types of force that can act on a cubic fluid
element, the pressure, body forces, and viscous forces, only the viscous shear forces are
able to give rotary...
I'm trying to plot the vorticity field for lid-driven cavity flow. Because I'm looking at 2D flow, the vorticity is plotted as a scalar field The flow field is clockwise, and I would expect the majority of the resulting vorticity field to be negative (right hand rule). However, it seems as...
Homework Statement
A tornado:
of d = 2r = 0.9 km
r = radius
vortic ity 9 rad/s at altitude 0.8 km
What is the velocity at ground level given 2*pi*r = 95 m at that level.
If someone could just guide me on what the relevant equations are. I am revisiting the background material but any...
Hi all,
Am reading a few papers for a Uni case study about structures in Turbulent Boundary Layers over a Flat Plate (water), particularly low-speed streaks.
I'm confused over what mechanism causes the span wise variations in velocity that seems to cause low speed streaks.
Would...
Hello,
I am trying to calculate the potential vorticity (PV) in units (PVU) of thousands of different parcels of air on varying spatial and temporal scales. Variables available are as follows;
parameter Unit Format Precision Default value :
1 date yyyymmdd I8.8 0 99999999
2 time...
Hello all:
I'm reading an old paper (from the 1980s) where someone solved the vorticity equation to get streamlines for flow. This is probably a silly question, but I'd be interested to know why he may have used this method instead of solving for the x and y (it's a 2-D set-up) components of...
Homework Statement
Consider 2-D flow motion (u,v) = (y,-x). Calculate the vorticity field of the flow.
Homework Equations
\omega = vx - uy
The Attempt at a Solution
I calculated \omega = -2, so using the vorticity equation I get zero. So I guess what I am asking is what exactly am...
There is a discussion of expansion, shear, and vorticity in Wald (p. 217) and in Hawking and Ellis (p. 82). My motivation for comparing them was that although Wald's treatment is more concise, Wald doesn't define the expansion tensor, only the volume expansion.
Wald starts off by restricting to...
I've recently been studying aircraft trailing vortices and have been reading various papers regarding vortices. However, my aerodynamics is very rusty, I've been confused about the vorticity of these vortices.
In Fundamentals of Aerodynamics (Anderson), Ch 3, Vortex flow is shown to be...
Homework Statement
Consider an imaginary circular disc, of radius R, whose arbitrary orientation is described by the unit vector, \vec {n} , perpendicular to the plane of the disc. Define the component, in the direction \vec {n} , of the angular velocity, \vec {\Omega} , at a point in the...
Hello,
I have a parallelepiped domain, divided in nodes. Each node has a velocity vector (3D). I want to evaluate the vorticity. Starting from the definition of vorticity (e.g. w_x=\partialu_z/\partialy-\partialu_y/\partialz) I calculated the vorticity in a fortran progem like...
Homework Statement
Consider an imaginary circular disc, of radius R, whose arbitrary orientation is described by the unit vector, \vec {n} , perpendicular to the plane of the disc. Define the component, in the direction \vec {n} , of the angular velocity, \vec {\Omega} , at a point in the...
I'm solving a 2D forced convection problem by using finite difference method. It involves solving the vorticity-stream function equations. I met some problem when I'm trying to solve the vorticity equation, which can be stated as follows:
How to deal with the vorticity boundary condition? after...
Homework Statement
For a fluid flow of velocity u and vorticity w=∆ x u, show that:
(u. ∆ )u=-u x w + ∆(1/2|u|²)
Sorry the triangles should be the other way up!
Homework Equations
∆(u.v)=(u.∆)v + (v.∆)u +u x (∆ x v) + v x (∆ x u )
The Attempt at a Solution
I need to...
Homework Statement
Viscous flow between two rigid plates in which a lower rigid boundary y=0 is suddenly moved with speed U, which an upper rigid boundary to the fluid, y=h, is held at rest.
Homework Equations
\mathbf{u}=(u(y,t),0,0)
\frac{\partial u}{\partial t} = \nu \frac{\partial^2...
This is what I understand about Kelvin's Circulation Theorem
1)for inviscid (where the viscous forces are much LESS than inertial forces) AND uniform density flow, the circulation is conserved.
2)This implies (by some arduos vector calculus manipulations) that the vorticity of each fluid...
Hi all.
In Fluid dynamics, is vorticity equiavlent to angular momentum?
It seems that vorticity is twice of the rate of rotation of a fluid element at a point, and angular momentum is the density times the rate of rotation of a fluid element, so they just differ by some constant?
But it is...
I am given, in each of three cases, an angular velocity \Omega(r) and am told to assume no axial (z) velocity i.e., u_z = 0. I am asked to
(1) find the velocity field in cartesian coordinates
(2) find the vorticity distribution in threee cases.
(1) As setup, the problem asks me to...