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Vortex sheet
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<h2 id="diffusion-of-a-vortex-sheet">Diffusion of a vortex sheet</h2> <p>Once a vortex sheet, it will diffuse due to viscous action. Consider a planar unidirectional flow at t = 0 {\displaystyle t=0} , </p> u = { + U , for y > 0 − U , for y < 0 {\displaystyle u={\begin{cases}+U,&{\text{for }}y>0\\-U,&{\text{for }}y<0\end{cases}}} <p>implying the presence of a vortex sheet at y = 0 {\displaystyle y=0} . The velocity discontinuity smooths out according to<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> u ( y , t ) = U 2 π ν t [ ∫ 0 ∞ e − ( s − y ) 2 / 4 ν t d s − ∫ 0 ∞ e − ( s + y ) 2 / 4 ν t d s ] . {\displaystyle u(y,t)={\frac {U}{2{\sqrt {\pi \nu t}}}}\left[\int _{0}^{\infty }e^{-(s-y)^{2}/4\nu t}\mathrm {d} s-\int _{0}^{\infty }e^{-(s+y)^{2}/4\nu t}\mathrm {d} s\right].} <p>where ν {\displaystyle \nu } is the <a href="/facts/Kinematic_viscosity/QLVfrdCz">kinematic viscosity</a>. The only non-zero vorticity component is in the z {\displaystyle z} direction, given by </p> ω z = − U π ν t e − y 2 / 4 ν t {\displaystyle \omega _{z}=-{\frac {U}{\sqrt {\pi \nu t}}}e^{-y^{2}/4\nu t}} . <h2 id="vortex-sheet-with-periodic-boundaries">Vortex sheet with periodic boundaries</h2> <p>A flat vortex sheet with periodic boundaries in the streamwise direction can be used to model a temporal free shear layer at high Reynolds number. Let us assume that the interval between the periodic boundaries is of length 1 {\displaystyle 1} . Then the equation of motion of the vortex sheet reduces to </p><p> ∂ z ∗ ∂ t = − ı 2 ∫ 0 1 cot π ( z ( Γ , t ) − z ( Γ ′ , t ) ) d Γ ′ {\displaystyle {\frac {\partial z^{*}}{\partial t}}=-{\frac {\imath }{2}}\int \limits _{0}^{1}\cot \pi (z(\Gamma ,t)-z(\Gamma ',t))\;d\Gamma '} </p> <p>Note that the integral in the above equation is a Cauchy principal value integral. The initial condition for a flat vortex sheet with constant strength is z ( Γ , 0 ) = Γ {\displaystyle z(\Gamma ,0)=\Gamma } . The flat vortex sheet is an equilibrium solution. However, it is unstable to infinitesimal periodic disturbances of the form ∑ k = − ∞ ∞ A k e ı 2 π k Γ {\displaystyle \sum _{k=-\infty }^{\infty }A_{k}\mathrm {e} ^{\imath 2\pi k\Gamma }} . Linear theory shows that the Fourier coefficient A k {\displaystyle A_{k}} grows exponentially at a rate proportional to k {\displaystyle k} . That is, higher the wavenumber of a Fourier mode, the faster it grows. However, a linear theory cannot be extended much beyond the initial state. If nonlinear interactions are taken into account, asymptotic analysis suggests that for large k {\displaystyle k} and finite t < t c {\displaystyle t<t_{c}} , where t c {\displaystyle t_{c}} is a critical value, the Fourier coefficient A k {\displaystyle A_{k}} decays exponentially. The vortex sheet solution is expected to lose analyticity at the critical time. See Moore (1979), and Meiron, Baker and Orszag (1983). </p><p>The vortex sheet solution as given by the Birkoff-Rott equation cannot go beyond the critical time. The spontaneous loss of analyticity in a vortex sheet is a consequence of mathematical modeling since a real fluid with viscosity, however small, will never develop singularity. Viscosity acts a smoothing or regularization parameter in a real fluid. There have been extensive studies on a vortex sheet, most of them by discrete or point vortex approximation, with or without desingularization. Using a point vortex approximation and delta-regularization Krasny (1986) obtained a smooth roll-up of a vortex sheet into a double branched spiral. Since point vortices are inherently chaotic, a Fourier filter is necessary to control the growth of round-off errors. Continuous approximation of a vortex sheet by vortex panels with arc wise diffusion of circulation density also shows that the sheet rolls-up into a double branched spiral. </p><p>In many engineering and physical applications the growth of a temporal free shear layer is of interest. The thickness of a free shear layer is usually measured by momentum thickness, which is defined as </p><p> θ = ∫ y = − ∞ ∞ ( 1 4 − ( ⟨ u ⟩ 2 U ) 2 ) d y {\displaystyle \theta =\int \limits _{y=-\infty }^{\infty }\left({\frac {1}{4}}-\left({\frac {\left\langle u\right\rangle }{2U}}\right)^{2}\right)\mathrm {d} y} </p><p>where ⟨ u ⟩ = 1 L ∫ 0 L u ( x , y , t ) d x {\displaystyle \left\langle u\right\rangle ={\frac {1}{L}}\int _{0}^{L}\ u(x,y,t)dx} and U {\displaystyle U} is the freestream velocity. Momentum thickness has the dimension of length and the non-dimensional momentum thickness is given by θ N D = θ / L {\displaystyle \theta _{ND}=\theta /L} . Momentum thickness can be used to measure the thickness of a vortex layer. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Vortex_ring/N5IpqxrV">Vortex ring</a></li> <li><a href="/facts/Burgers_vortex/2rnkeFCd">Burgers vortex sheet</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>McGraw-Hill Dictionary of Scientific and Technical Terms Retrieved July 2012 <a href="http://www.answers.com/topic/vortex-sheet#ixzz21j0Kc9pU" target="_blank">http://www.answers.com/topic/vortex-sheet#ixzz21j0Kc9pU</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Drazin, P. G., & Riley, N. (2006). The Navier-Stokes equations: a classification of flows and exact solutions (No. 334). Cambridge University Press. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>