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Vorticity, circulation and Kelvin's circulation theorem
The curl of the velocity as local spin, circulation around loops, and why ideal fluids keep their circulation.
The vorticity is the curl of the velocity, \(\boldsymbol{\omega} = \nabla\times\mathbf{u}\). To see what it measures, split the velocity gradient into symmetric and antisymmetric parts, \(\nabla\mathbf{u} = S + W\) with \(S = \tfrac12(\nabla\mathbf{u} + \nabla\mathbf{u}^T)\) and \(W = \tfrac12(\nabla\mathbf{u} - \nabla\mathbf{u}^T)\). Near a point, \(\mathbf{u}(\mathbf{x}+\mathbf{h}) \approx \mathbf{u}(\mathbf{x}) + S\mathbf{h} + W\mathbf{h}\), and a direct calculation gives \(W\mathbf{h} = \tfrac12\boldsymbol{\omega}\times\mathbf{h}\). So a small fluid element translates, is strained by \(S\), and rotates rigidly with angular velocity \(\boldsymbol{\omega}/2\). For solid-body rotation \(\mathbf{u} = \Omega(-y,x,0)\) the vorticity is \((0,0,2\Omega)\), twice the rotation rate.
Vorticity is not the same as streamlines curving. The shear flow \(\mathbf{u} = (y,0,0)\) has straight streamlines and vorticity \((0,0,-1)\): a small paddle wheel placed in it spins, because the top is pushed harder than the bottom. The point vortex \(\mathbf{u} = \frac{\Gamma}{2\pi r^2}(-y,x)\) has circular streamlines and zero vorticity away from the origin: the paddle wheel goes round the centre without turning about its own axis.
The circulation around a closed curve \(C\) is \(\Gamma = \oint_C \mathbf{u}\cdot d\mathbf{l}\). By Stokes' theorem it is the flux of vorticity through any surface spanning the loop, \(\Gamma = \iint \boldsymbol{\omega}\cdot\mathbf{n}\,dS\). For the point vortex every loop around the origin has circulation \(\Gamma\), even though the curl vanishes on the loop, because the region inside contains the singular centre and is not simply connected.
Kelvin's circulation theorem (W. Thomson, 1869). Suppose the fluid is inviscid, the density is constant (or more generally a function of pressure alone), and the body force is the gradient of a potential. Then the circulation around any closed curve that moves with the fluid is constant in time. The proof is short. For a material loop, \[ \frac{d\Gamma}{dt} = \oint_C \frac{D\mathbf{u}}{Dt}\cdot d\mathbf{l} + \oint_C \mathbf{u}\cdot d\mathbf{u}. \] The second integral is \(\oint d(|\mathbf{u}|^2/2) = 0\). Under the hypotheses, the Euler equations make \(D\mathbf{u}/Dt\) a gradient, \(-\nabla(p/\rho + \Phi)\), and the integral of a gradient round a closed loop is zero.
The consequences are strong. Helmholtz (1858) had shown that vortex lines move with the fluid; Kelvin's theorem implies that a flow which starts irrotational stays irrotational, so vorticity in an ideal fluid can be neither created nor destroyed, only moved and deformed. Real vorticity is born at solid walls through viscosity, which is exactly the hypothesis Kelvin's theorem drops.
Picture it: a smoke ring keeps its circulation as it travels; shrinking its radius would speed it up. Think it: vorticity is the part of the velocity gradient invisible to strain, and circulation is its integrated form; Kelvin's theorem says the flow map of an ideal fluid transports the one-form \(\mathbf{u}\cdot d\mathbf{x}\) up to an exact form.
Nhazi ahụ ejirila · curl of [-y, x, 0]
Nzọụkwụ site n'ụdị
- \mathbf{F} = \left\langle - y,\ x,\ 0 \right\rangle,\quad \nabla\times\mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \partial_x & \partial_y & \partial_z \\ P & Q & R \end{vmatrix}
The curl measures rotation. Expand the determinant one component at a time.
- \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} = 0 - \left(0\right) = 0
The i component: differentiate R with respect to y and Q with respect to z, then subtract.
- \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} = 0 - \left(0\right) = 0
The j component: differentiate P with respect to z and R with respect to x, then subtract.
- \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 1 - \left(-1\right) = 2
The k component: differentiate Q with respect to x and P with respect to y, then subtract.
- \nabla\times\mathbf{F} = \left\langle 0,\ 0,\ 2 \right\rangle
Assemble the three components.
Gosi nzaghachi
Ugbu a ị Họrọ nsogbu, mọọbụ tinye mọọbụ depụta gị. Nzọụkwụ ọbụla, inyogo, azịza ahụ echekwara ruo mgbe ịjụla.
Aịkọn ndị ahụ ejirila ebe a
Tinye ọbụla simbụl maka nkọwa zuru ezu, inyogo, na ihe ọbụla ọbụla na-egosi.
Olee otú: Vorticity, circulation and Kelvin's circulation theorem
- Write the velocity components and compute the curl: (w_y - v_z, u_z - w_x, v_x - u_y).
- For circulation, parametrise the loop, form u . dl, and integrate; or integrate the vorticity over a spanning surface if the region has no holes.
- If the loop encloses a singularity, compute the line integral directly: Stokes' theorem needs a smooth field inside.
- For Kelvin's theorem, check the three hypotheses: no viscosity, barotropic density, conservative body force.
Ajụjụ ndị mmadụ jụrụ
Does a whirlpool have vorticity?
Its core does. Outside the core the flow is close to a point vortex, which has circulation but almost no vorticity, which is why a floating leaf goes round the drain without spinning much about its own axis.
How does an aeroplane wing get circulation if Kelvin says it is conserved?
When the wing starts moving, viscosity at the sharp trailing edge sheds a starting vortex with the opposite circulation. The total around a loop enclosing both stays zero, as Kelvin requires.
What do I need before starting fluid dynamics?
Multivariable calculus (divergence, curl, the divergence and Stokes theorems), linear algebra (symmetric matrices and eigenvalues) and partial differential equations (the heat and Laplace equations). The last lessons also use Sobolev spaces, which are introduced where they are needed.
What are the Navier-Stokes equations in one sentence?
Newton's second law for each particle of a viscous incompressible fluid: acceleration equals the pressure force plus viscous diffusion of momentum, with the constraint that the velocity field is divergence-free.
Why is two-dimensional flow easier than three-dimensional flow?
In 2D the vorticity is a scalar that is only carried and diffused, so its maximum never grows; in 3D vortex lines can be stretched, which amplifies vorticity, and no known bound rules out unlimited growth.
Is this course physics or mathematics?
Both, in order. The first half derives the equations from physical principles and solves classical flows; the second half treats the equations as mathematical objects and studies which of their properties can be proved.
Oge Fluid Dynamics
The continuum hypothesis and fieldsEulerian and Lagrangian descriptions, the material derivativeKinematics: streamlines, pathlines and streaklinesConservation of mass, the continuity equation and incompressibilityThe stream function and two-dimensional incompressible flowThe stress tensor and Cauchy's momentum equationHydrostatics and pressureThe Euler equations and Bernoulli's theoremPotential flow and the Laplace equationViscosity, Newtonian fluids and the Navier-Stokes equationsExact solutions: Couette, Poiseuille and Stokes' first problemDimensional analysis, the Reynolds number and the scaling symmetryStokes flow at low Reynolds numberBoundary layers