Coherent structure-turbulence interaction studied via a vortex column embedded in fine-scale turbulence

Abstract

The coupling of large-scale coherent structures (CSs) with fine-scale turbulence and its possible relevance to cascade are explored via numerical simulations of a vortex column as an idealized CS embedded in homogeneous, isotropic turbulence. LES of this high Reynolds number (Re) vortex-turbulence interaction supports the turbulence cascade scenario – showing that finer-scale filaments arch over vorticity thread dipoles which themselves wrap around the column, hence the vorticity portrait of the cascade process. LES of a ring, idealizing a thread around a column, embedded in turbulence shows external turbulence and ring-perturbed core fluid stripped from the column are organized into threads acting as an oppositely signed, coaligned pseudo-ring, hence forming a dipole encircling the column. This evolving dipole then reconnects and breaks up, generating new turbulence around the column – a possible scenario for turbulence cascade. Furthermore, contrary to the expected viscous decay within the column due to its no core strain rate, core turbulence intensifies via radial stretching due to the column’s “core dynamics,” further generating new turbulence, now surprisingly within the column. The breakup of the threads is delineated via reconnection of two antiparallel vortices embedded in turbulence. Slender orthogonal filaments form between these vortices, perturbing the collision of the vortices, generating additional threads and new turbulence structures – providing further insight into the cascade process. Hence, studying vortex-turbulence interaction and that of the hierarchy of the resulting threads reveal further steps in the cascade process, seemingly consistent with our vortex cascade scenario of shear flow turbulence.

Appendices

Appendix A: Numerical methods

Simulations are done using a pseudo-spectral method with the Rennich and Lele [30] potential velocity correction to solve the vorticity equations for flows with non-zero circulation, discussed in [31]. High $Re\left(>\mathrm{10,000}\right)$ simulations employ this code modified for Large Eddy Simulations (LES) with a Smagorinsky model for the unresolved velocity field. To approximate turbulence, the Mansour and Wray [24] energy spectrum is applied to random fluctuations, with a given initial amplitude, peak wavenumber, ${\kappa }_{p}$, and high wavenumber spectral slope, $\sigma$. The peak wavenumber controls the dominant length scale. The high wavenumber slope controls how quickly the energy decreases at high wavenumbers, i.e. the energy within finer-scale turbulence. Radial coordinates $\left(r,\theta ,z\right)$ and velocities ($u,v,w)$ are primarily used for discussion, with Cartesian coordinates $\left(x,y,z\right)$ and velocities (${u}_{x},{u}_{y},{u}_{z})$ are used where appropriate; note that ${u}_{z}=w$. The Reynolds number, $Re\equiv {\Gamma }_{C}/\nu$, is a column’s circulation, ${\Gamma }_{C}$, divided by viscosity, $\nu$.

Note that the use of LES in sections 2.2 and 2.3 to capture high $Re$ dynamics of fine scale turbulence raises concern about inverse cascade (energy transfer from the unresolved finer scales into the resolved scales) that could energize and amplify the smallest threads observed. However, such dynamics also should naturally occur in vortex-turbulence interaction, as observed by the inverse cascade responsible for the generation of axial flow within a vortex column [13]. Higher $Re$ DNS simulations of such flows, capturing the complete physics of the thread-filament wrapping scenario detailed in sections 2.1 through 2.3 are needed to settle such concerns, and are out of reach of current computational capacity. The LES inverse cascade is not likely to significantly modify the dynamics as the finer scale energy increases due to inverse cascade could be equivalent to increasing the initial fine scale energy of the simulation.

Appendix B: Vorticity dynamics

As a review, the important vorticity generation mechanisms for perturbations to a vortex column are summarized for an Oseen vortex – an idealized profile of a viscously diffused CS (unless otherwise stated, summarized from [12]).

Assume that there is a vortex column with azimuthal velocity, $V$, and a perturbation consisting of radial $\left({\omega }_{r}\right)$, azimuthal $\left({\omega }_{\theta }\right)$, and axial $\left({\omega }_{z}\right)$ vorticity; the column also has axial vorticity associated with it and, in general, the total axial vorticity (column plus perturbation) will be denoted as ${\omega }_{z}$. The first dynamic of interest is ${\partial }_{t}{\omega }_{\theta }\approx {\omega }_{r}{S}_{r\theta }$, generation of ${\omega }_{\theta }$ due to ${\omega }_{r}$ being tilted by the column’s strain rate, ${S}_{r\theta }=r{\partial }_{r}\left(V/r\right)$ (figure 16a); generation of ${\omega }_{\theta }$ is referred to as “vortex line coiling.” The generated ${\omega }_{\theta }$ is oppositely signed to ${\omega }_{r}$ as ${S}_{r\theta }$ is negative. This vorticity tilting also causes negative Reynolds shear stress, $\overline{uv }<0$, which is necessary for production of turbulent kinetic energy. As the Oseen vortex is normal mode stable [32], ${\omega }_{\theta }$ generation is eventually halted; this occurs due to the meridional velocity associated with ${\omega }_{\theta }$ tilting ${\omega }_{z}$ into ${\omega }_{r}$ (figure 16b). For the Oseen vortex, the column’s ${\omega }_{z}$ extends to infinity (though rapidly decaying), so the tilted ${\omega }_{z}$ tends to be associated with the column, and hence positive. Taking the meridional velocity gradients associated with $-{\omega }_{\theta }$, in particular $\partial u/\partial z$ (figure 16b), the tilting of $+{\omega }_{z}$ generates $-{\omega }_{r}$; thus, the initial ${\omega }_{r}$ is reduced over time. When ${\omega }_{r}$ reaches zero, generation of ${\omega }_{\theta }$ ends. Note that if $-{\omega }_{z}$ (associated with decreasing circulation, hence unstable due to Rayleigh’s circulation criterion) is tilted, this tilting increases the magnitude of ${\omega }_{r}$ (discussed further in [23]). Concurrent with the reduction of ${\omega }_{r}$, there is also tilting of existing ${\omega }_{r}$ into ${\omega }_{z}$ (figure 16c); this tilting generates $+{\omega }_{z}$, increasing the existing ${\omega }_{z}$, which, combined with the reduction in ${\omega }_{r}$, means the vortex lines in the meridional plane become more aligned in the axial direction. Thus, any perturbation vorticity outside a vortex column is tilted and stretched azimuthally, forming a vorticity thread, before the generation of vorticity is limited by tilting of column ${\omega }_{z}$.

Figure 16

Generation of $\left(a\right)\, {\omega }_{\theta }$ via mean straining due to the column’s $V\left(r\right)$, $\left(b\right)\, {\omega }_{r}$ by tilting of ${\omega }_{z}$ due to $\partial u/\partial z$ associated with ${\omega }_{\theta }$ in $\left(a\right)$, and $\left(c\right)\, {\omega }_{z}$ by tilting of ${\omega }_{z}$ due to $\partial w/\partial r$ associated with ${\omega }_{\theta }$ in $\left(a\right)$.

Continued tilting of the column ${\omega }_{z}$ reverses the sign of ${\omega }_{r}$, causing generation of ${\omega }_{\theta }$ that reduces the previously generated ${\omega }_{\theta }$, eventually reducing ${\omega }_{\theta }$ to zero and reversing it (termed uncoiling and opposite coiling). Thus, these mechanisms also describe vortex core dynamics (MH94) – the repeated coiling, uncoiling, and opposite coiling of vortex lines due to axial variation of the vortex core radius. Note that even though stretching by the column’s strain causes linear in time ${\omega }_{\theta }$ generation (which cannot be represented by an exponentially growing Fourier mode, hence transient growth), the inherent self-limitation of the filament stretching by core dynamics means a finite amount of vorticity (or energy) growth during this process. As the generation of ${\omega }_{\theta }$ due to strain rate cannot occur within the vortex core, as the strain rate is approximately zero, the generation of ${\omega }_{\theta }$ (i.e. vortex line coiling) is due to differential rotation of the vortex surface as the core radius changes (discussed in MH94 and [26]). In the axisymmetric case, the vortex surface is also a surface of constant circulation (radial and axial derivatives of circulation are ${\omega }_{z}$ and ${\omega }_{r}$, respectively; MH94), changing the radius of the surface also changes the azimuthal velocity of the surface, causing different points along the axis to rotate at different velocities. Thus, a vortex line with no azimuthal component is tilted into the azimuthal direction. Similarly, the meridional velocity of a thread outside a vortex induces a variation in the core size, naturally causing core dynamics (sketched in figure 1b).

Note that the tilting of ${\omega }_{r}$ and ${\omega }_{z}$ (figure 16b, c) are due to the velocity gradients associated with $\omega_{\theta }$, not due to $\omega_{\theta }$ itself. Hence, it is possible for $\omega_{\theta }$ in filament outside the column to non-locally tilt vorticity within the column’s core. In this way, strong core fluctuations can be excited through specific resonance conditions. For example, a helical filament outside the core excites helical Kelvin waves; when the angular velocity of the filament matches the oscillation frequency of the Kelvin wave, the filament continually excites the core [12]. Even without resonance, turbulence excites significant core fluctuations [14]. Similarly, vortex rings (an idealization of azimuthally wrapped vortex filaments) excite axisymmetric core waves via tilting of the column’s $\omega_{z}$ (discussed in [16] and sketched in figure 1b). Such non-local interactions between external threads and the vortex core play an important role in the coupling between the column and turbulence.

Appendix C: Solution to coupled ${\varvec{\omega}}_{{\varvec{r}}}$ and ${\varvec{\omega}}_{{\varvec{\theta}}}$ equations

From equations (3) and (4) from section 2.1, the inviscid axisymmetric $\omega_{r}$ and $\omega_{\theta }$ evolution equations:

$$\frac{{\partial \omega_{r} }}{\partial t} \approx \omega_{\theta } \omega_{z} \;{\text{and}}\;\frac{{\partial \omega_{\theta } }}{\partial t} \approx - \omega_{r} \omega_{z} ,$$

an analytical solution can be determined. Taking the time derivative of the $\omega_{r}$ equation, equation (3),

$$ \frac{{\partial^{2} \omega_{r} }}{{\partial t^{2} }} \approx \frac{{\partial \omega_{\theta } }}{\partial t}\omega_{z} , $$

(14)

the $\partial \omega_{\theta } /\partial t$ equation, equation (4), can be substituted into equation (14), giving an equation only in terms of $\omega_{r} ,\omega_{z}$ and $t$. By treating $\omega_{z}$ as a constant, this equation,

$$ \frac{{\partial^{2} \omega_{r} }}{{\partial t^{2} }} \approx - \omega_{r} \omega_{z}^{2} , $$

(15)

has oscillating in time solutions, as

$$\omega_{r} \left( t \right) = A\sin \left( {\omega_{z} t} \right) + B\cos \left( {\omega_{z} t} \right).$$

(16)

Given that the vorticity field must have some initial $\omega_{r}$ to generate the thread, we let $B = \omega_{r}^{0}$, and, as discussed in Appendix B, $\omega_{r}$ decreases due to tilting of $\omega_{z}$ reducing it, there is no growth of $\omega_{r}$, which leads to the choice of $A = 0$. Note that, in tilting in shear flow $U\left( y \right)$, an $\omega_{y}$ vortex filament, initially perpendicular to the flow direction, is tilted over time in the flow direction $x$. For this configuration, the $\omega_{y}$ vorticity remains unchanged while tilting generating $\omega_{x}$. In contrast, for the current scenario of a filament outside a vortex column, the tilting of the column’s $\omega_{z}$ by the meridional flow of the filament’s $\omega_{\theta }$ inherently reduces the original $\omega_{r}$ of the filament, always causing it to be reduced to zero. Note that this neglects the generation of $\omega_{r}$ by other sources, e.g. non-axisymmetric tilting of $\omega_{\theta }$ into $\omega_{r}$ due to helical ($m = 1$) perturbations [13]. Then, by integrating equation (4) with equation (16), $\omega_{\theta }$ has the form:

$$ \omega_{\theta } = \omega_{\theta }^{0} \sin \left( {\omega_{z} t} \right), $$

(17)

where $\omega_{\theta }^{0}$ is the peak azimuthal vorticity resulting from the column straining $\omega_{r}$, detailed in section 2.1.