In cylindrical coordinates, pure-strain extensional (stretching) flow along the z-axis is given by: uR = −(γ/2)R and uz = γz, where γ is the strain rate. A vortex aligned with the z-axis has vorticity ω = ωzez = ωo(R)ez at t = 0. What is ωz(R,t) in this flow when the fluid is inviscid? When γ is

Question

In cylindrical coordinates, pure-strain extensional (stretching) flow along the z-axis is given by: [latex]u_{R}=-(\gamma/2)R[/latex] and [latex]u_{z}=\gamma z[/latex], where γ is the strain rate. A vortex aligned with the z-axis has vorticity [latex]\omega=\omega_{z}\mathbf{e}_{z}=\omega_{o}(R)\mathbf{e}_{z}[/latex] at t = 0. What is [latex]\omega_{z}(R,t)[/latex] in this flow when the fluid is inviscid? When γ is positive, does the vorticity at R = 0 strengthen or weaken as t increases? When γ ≠ 0, does the vortex’s circulation change?

Accepted Answer

To determine [latex]\omega_{z}(R,t)[/latex], (5.13) must be solved for the given flow field and initial condition. For the given velocity field with ν = 0, the z-direction component of (5.13) is:

[latex]\frac{D\omega}{Dt}=(\omega\cdot abla)\mathbf{u}+ u abla^{2}\omega[/latex]. (5.13)

[latex]\frac{\partial\omega_{z}}{\partial t}+u_{R}\frac{\partial\omega_{z}}{\partial R}+\frac{u_{\varphi}}{R}\frac{\partial\omega_{z}}{\partial\varphi}+u_{z}\frac{\partial\omega_{z}}{\partial z}=\omega_{z}\frac{\partial u{z}}{\partial z}[/latex].

When [latex]\omega_{z}[/latex] depends only on R and t only, the third and fourth terms on the left side are zero. In addition, [latex]u_{R}=-(\gamma/2)R[/latex] and [latex]\partial u_{z}/\partial z=\gamma[/latex], so this vorticity component equation becomes:

[latex]\frac{\partial\omega_{z}}{\partial t}-\frac{\gamma R}{2}\frac{\partial\omega_{z}}{\partial R}=\gamma\omega_{z}[/latex], or [latex]\frac{\partial\omega_{z}}{\partial t}-\frac{1}{R}\frac{\partial}{\partial R}\left(\frac{1}{2}\gamma R^{2}\omega_{z} ight)=0[/latex].

To solve the second equation, rewrite it in terms of a new dependent function [latex]\phi(R,t)=(1/2)\gamma R^{2}\omega_{z}(R,t)[/latex]:

[latex]\frac{\partial\phi}{\partial t}-\frac{\gamma R}{2}\frac{\partial\phi}{\partial R}=0[/latex],

and assume that there is a trajectory R(t) in R-t space where dR/dt = −(1/2)γR so that the last equation represents the total derivative [latex]d\phi/dt=0[/latex] (see Appendix B.1). Here, the equation for the trajectories R(t) is readily separated and integrated to find: [latex]R(t)=R_{o}\exp\left\{-\gamma t/2 ight\}[/latex] where [latex]R_{o}[/latex] is the radial distance of a trajectory at t = 0.
In this case [latex]\phi[/latex] will be constant along the curves R(t), so the solution for [latex]\phi[/latex] is given by:

[latex]\phi(R(t),t)=\mathrm{const}. =\frac{\gamma R^{2}(t)}{2}\omega_{z}(R,t)=\frac{\gamma R^{2}_{o}}{2}\omega _{o}(R_{o})[/latex],

where the final equality follows from the initial condition. Algebraic rearrangement of the last equality using [latex]R_{o}/R=\exp\left\{+\gamma t/2 ight\}[/latex] produces the final solution for [latex]\omega_{z}[/latex]:

[latex]\omega_{z}(R,t)=e^{+\gamma t}\omega _{o}(Re^{+\gamma t/2})[/latex].

When γ is positive, the vorticity at R = 0 increases exponentially as t increases. The circulation of the vortex at time t, is:

[latex]\Gamma=\int\omega_{z}dA=2\pi\overset{\infty}{\underset{0}{\int}}\omega_{z}(R,t)RdR=2\pi\overset{\infty}{\underset{0}{\int}}e^{+\gamma t}\omega _{o}(Re^{+\gamma t/2})RdR=2\pi\overset{\infty}{\underset{0}{\int}}\omega_{o}(\xi)\xi d\xi[/latex].

where an integration variable substitution ξ = Rexp{+γt/2} has been made, and the final form is the circulation of the vortex at t = 0. Thus, the vortex’s initial circulation is not changed by axial stretching, and this makes sense; axial stretching does apply a torque to fluid elements.
This example illustrates how vortex stretching can concentrate vorticity, and is analogous to the figure skater who starts by moving in a wide circle but ends up in a rapid vertical spin. It also suggests two means for developing a strong vortex. First, start with a large horizontally-spread vorticity-containing fluid mass so that Γ is substantial. And second, allow vertical stretching to take place over a sufficiently long period of time (γt ≫ 1) so that the exponential factors dominate the solution for [latex]\omega_{z}[/latex]. Although both also involve three-dimensional effects, drain vortices and tornadoes are primarily formed by the vorticity-intensification mechanism illustrated in this example.

Question 5.3: In cylindrical coordinates, pure-strain extensional (stretch...

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