# Three-Dimensional Unsteady Axisymmetric Viscous Beltrami Vortex Solutions to the Navier–Stokes Equations

## Abstract

**:**

^{−1}], is either constant or nonconstant. In the former case, the well-known exact nondivergent three-dimensional unsteady vortex solutions are obtained by solving the evolution equations for the stream function directly. In the latter case, the consistency conditions are given by nonlinear equations of the stream function, one of which corresponds to the Bragg–Hawthorne equation for steady inviscid flow. Solutions of a novel type are found by numerically solving the nonlinear constraint equation at a fixed time. Time dependence is recovered by taking advantage of the linearity of the evolution equation of the stream function. The proportionality coefficient is found to depend on space and time. A phenomenon of partial restoration of the broken scaling invariance is observed at short distances.

## 1. Introduction

^{−1}], thereby raising the possibility of generating vortices of finite extension in space and/or time. Incidentally, we recall that tilting the vorticity is also accompanied by the breaking of the scaling invariance [8].

^{−1}], thereby violating the scaling invariance of the original Navier–Stokes equations. With a characteristic length scale being defined, it is thus expected that the spatial extension of vortex solution may also be rendered finite. The generalized Beltrami flows [10,38,39] and the extended Beltrami flows [40,41] will share the same features in common.

## 2. Dynamical and Constraint Equations for the Axisymmetric Beltrami Vortex

**ω**given by (4) is divergence-free, too, these conditions lead to another one on c:

_{r}or ω

_{z}may vanish.

## 3. Consistency of Dynamical and Constraint Equations

**ω**, (12) is written as

_{θ}on the rate of temporal change of $\psi $. Noting that $\psi $ is the angular momentum component per unit mass about the symmetry axis, the first term with a negative sign in the large braces of (18) represents the direct resistance on the fluid element. This resistance term, being proportional to c

^{2}and an invariable concomitant of alignment of velocity and vorticity, reveals the hallmark of the Beltrami flow. When $\psi $ is steady and has negligible spatial variations, the direct resistance is balanced with the torque. The second term involves the mixed effects of the rate of strain and vorticity.

^{−1}. Many analytic Beltrami flow solutions are known for constant c.

_{r}and F

_{z}are related to c through (A2) and (A3) in Appendix A. Then, we have the second condition

**F**related by (20) or (21) and (22). Once each component of $v$ is determined as a function of r, z, and t, c is obtained from the Beltrami relation. That this procedure is possible is a peculiarity of the axisymmetric Beltrami flow.

**F**will be present when c is not constant. The details on the relation of nonconstant c and

**F**are elaborated in Appendix B.

## 4. Vortex Solutions

#### 4.1. Constant c

_{θ}= 0. From (13) and (14), c can be constant when $\psi $ is constant or ${v}_{\theta}\propto 1/r$, too. However, this does not satisfy (16).

_{1}is the Bessel function of the first order, α(t) a function of t to be determined later and δ a phase. k and δ are both real or pure imaginary. If we require the solution to be nondivergent in the whole spatial region, k must be real and |k| < c. Single mode solutions of this type frequently appear, irrespective of the symmetry when the nonlinear terms are somehow removed [22,35]. See also [49], wherein the Coriolis force is taken into account. From (23) and (4), it follows that

#### 4.2. Nonconstant c

**F**is nonconservative or H + V is not spatially constant. Unfortunately, we do not know the functional form of $c(\psi ,t)$ or $\varphi (\psi ,t)$, whose temporal evolutions will be governed by the dynamics. However, it may be sensible to set their instantaneous form as an initial condition. We consider a simplest case given by

^{−1}] and [length

^{−1}·velocity

^{−1}], respectively, are spatially constant (but possibly time-dependent). The assumption (26) amounts to keeping the first two terms in the power series expansion of ϕ in $\psi $. With this choice, the scales of the spatial extension and velocity of the solution will be given by A

^{−1}and A/B, respectively. The turn-over time of the vortex rotation will be the order of B/A

^{2}. The representative Reynolds number is given by $\mathrm{Re}=1/\nu B$.

^{2}g

^{−1}ln|g| results from a variation of $U(g)={A}^{2}{(\mathrm{ln}|g|)}^{2}/2$, we may approximate it around the minimum point g = 1 by a harmonic potential $U(g)\approx {A}^{2}{(g-1)}^{2}/2$ and linearize (28) as

#### 4.3. Time Dependence

_{c}reflects the t-dependences of the parameters A, B, and ${\partial}_{r}\psi ({r}_{0})$, too.

_{c}does not depend on t exists for a more general form of $c(\psi ,t)$ is an open question.

## 5. Conclusions and Discussion

^{−1}] generates a characteristic length $l={c}^{-1}$ and a decay time $\tau ={l}^{2}/\nu $ for a single-mode flow. It should be noted that the decay time does not depend on the wavenumber, k, appearing in (23). For air at normal pressure and room temperature, ${\tau}_{\mathrm{air}}\approx 5\times {10}^{4}{(l/\mathrm{m})}^{2}$s. Similarly, for water, we have an estimation ${\tau}_{\mathrm{water}}\approx {10}^{6}{(l/\mathrm{m})}^{2}$s. These periods shall be long enough for a macroscopic Beltrami vortex to persist. We also saw that a nonconstant c will result in a longevity of the vortex when the spectrum function obeys a power law.

_{θ}acts to directly grow the azimuthal component of the flow and induce the meridional components of the motion via the stream function. This is why the vortices undergoing such external forces as the Coriolis force or the Lorentz force are worth studying. The vortex–boundary and vortex–vortex interactions will also play the role. The numerical simulation method is expected to provide powerful tools to elicit information on this issue [27,42,43,44].

**C**, in the present paper, we have examined only the simplest form of the consistency condition for the viscous vortices. Numerous types of consistency conditions remain to be explored.

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Consistency of Constraints with the Navier–Stokes Equations (1) and (3)

## Appendix B. Variable c Generally Implies Time-Dependent External Force

_{r}when H is spatially constant, is required to be nonzero. Since $\psi $ depends on time, so does F

_{r}. In this case, F

_{z}is determined from the constraint (18).

## Appendix C. Extracting the Gaussian Factor and the Spectrum Function from the Solution (i) in Figure 1

## References

- Burgers, J.M. A Mathematical Model Illustrating the Theory of Turbulence. In Advances in Applied Mechanics; Von Mises, R., Von Kármán, T., Eds.; Elsevier: Amsterdam, The Netherlands, 1948; Volume 1, pp. 171–199. [Google Scholar]
- Rott, N. On the Viscous Core of a Line Vortex. J. Appl. Math. Phys. (ZAMP)
**1958**, 9, 543–553. [Google Scholar] [CrossRef] - Sullivan, R.D. A Two-Cell Vortex Solution of the Navier-Stokes Equations. J. Aerosp. Sci.
**1959**, 26, 767–768. [Google Scholar] [CrossRef] - Bellamy-Knights, P.G. An Unsteady Two-Cell Vortex Solution of the Navier—Stokes Equations. J. Fluid Mech.
**1970**, 41, 673–687. [Google Scholar] [CrossRef] - Bellamy-Knights, P.G. Unsteady Multicellular Viscous Vortices. J. Fluid Mech.
**1971**, 50, 1–16. [Google Scholar] [CrossRef] - Craik, A.D.D. Exact Vortex Solutions of the Navier–Stokes Equations with Axisymmetric Strain and Suction or Injection. J. Fluid Mech.
**2009**, 626, 291–306. [Google Scholar] [CrossRef] - Weinbaum, S.; O’Brien, V. Exact Navier-Stokes Solutions Including Swirl and Cross Flow. Phys. Fluids
**1967**, 10, 1438–1447. [Google Scholar] [CrossRef] - Gibbon, J.D.; Fokas, A.S.; Doering, C.R. Dynamically Stretched Vortices as Solutions of the 3D Navier–Stokes Equations. Phys. D Nonlinear Phenom.
**1999**, 132, 497–510. [Google Scholar] [CrossRef] [Green Version] - Takahashi, K. Three-Dimensional Unsteady Axisymmetric Vortex Solutions to the Bellamy-Knights Equation and the Distribution of Boundary Conditions. AIP Adv.
**2022**, 12, 085324. [Google Scholar] [CrossRef] - Drazin, P.G.; Riley, N. The Navier-Stokes Equations: A Classification of Flows and Exact Solutions; Cambridge University Press: Cambridge, UK, 2006. [Google Scholar]
- Trkal, V. A Note on the Hydrodynamics of Viscous Fluids. Czechoslov. J. Phys.
**1994**, 44, 97–106. [Google Scholar] [CrossRef] - Gromeka, I.S. Some Cases of the Motion of an Incompressible Fluid; Collected Works; Academy of Sciences: Moscow, USSR, 1952. (In Russian) [Google Scholar]
- Wood, V.T.; Brown, R.A. Simulated Tornadic Vortex Signatures of Tornado-Like Vortices Having One- and Two-Celled Structures. J. Appl. Meteorol. Climatol.
**2011**, 50, 2338–2342. [Google Scholar] [CrossRef] - Baker, C. Some Musings on Tornado Vortex Models. Available online: https://profchrisbaker.com/2020/06/30/some-musings-on-tornado-vortex-models/ (accessed on 30 July 2023).
- Barber, T.A.; Majdalani, J. On the Beltramian Motion of the Bidirectional Vortex in a Conical Cyclone. J. Fluid Mech.
**2017**, 828, 708–732. [Google Scholar] [CrossRef] [Green Version] - Maicke, B.; Majdalani, J. On the Compressible Bidirectional Vortex. Part 1: A Bragg-Hawthorne Stream Function Formulation. In Proceedings of the 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Nashville, TN, USA, 9–12 January 2012; Aerospace Sciences Meetings. American Institute of Aeronautics and Astronautics: Reston, VA, USA, 2012. [Google Scholar]
- Williams, L.L.; Majdalani, J. Exact Beltramian Solutions for Hemispherically Bounded Cyclonic Flowfields. Phys. Fluids
**2021**, 33, 093601. [Google Scholar] [CrossRef] - Arnold, V.I.; Arnold, V.I. Sur La Topologie des Écoulements Stationnaires des Fluides Parfaits; Givental, A.B., Khesin, B.A., Varchenko, A.N., Vassiliev, V.A., Viro, O.Y., Eds.; Springer: Berlin/Heidelberg, Germany, 1965; pp. 15–18. [Google Scholar]
- Dombre, T.; Frisch, U.; Greene, J.M.; Hénon, M.; Mehr, A.; Soward, A.M. Chaotic Streamlines in the ABC Flows. J. Fluid Mech.
**1986**, 167, 353. [Google Scholar] [CrossRef] - Bouya, I.; Dormy, E. Revisiting the ABC Flow Dynamo. Phys. Fluids
**2013**, 25, 037103. [Google Scholar] [CrossRef] [Green Version] - Chakraborty, P.; Roy, A.; Chakraborty, S. Topology and Transport in Generalized Helical Flows. Phys. Fluids
**2021**, 33, 117106. [Google Scholar] [CrossRef] - Bělík, P.; Su, X.; Dokken, D.P.; Scholz, K.; Shvartsman, M.M. On the Axisymmetric Steady Incompressible Beltrami Flows. OJFD
**2020**, 10, 208–238. [Google Scholar] [CrossRef] - Yi, T. Some General Solutions for Linear Bragg–Hawthorne Equation. Phys. Fluids
**2021**, 33, 077113. [Google Scholar] [CrossRef] - Viúdez, A. Multipolar Spherical and Cylindrical Vortices. J. Fluid Mech.
**2022**, 936, A13. [Google Scholar] [CrossRef] - Chandrasekhar, S. On force-free magnetic fields. Proc. Natl. Acad. Sci. USA
**1956**, 42, 1–5. [Google Scholar] [CrossRef] - Chandrasekhar, S.; Woltjer, L. On force-free magnetic fields. Proc. Natl. Acad. Sci. USA
**1958**, 44, 285–289. [Google Scholar] [CrossRef] - Pelz, R.B.; Yakhot, V.; Orszag, S.A.; Shtilman, L.; Levich, E. Velocity-Vorticity Patterns in Turbulent Flow. Phys. Rev. Lett.
**1985**, 54, 2505–2508. [Google Scholar] [CrossRef] [PubMed] - Golubkin, V.N.; Sizykh, G.B. Some General Properties of Plane-Parallel Viscous Flows. Fluid Dyn.
**1987**, 22, 479–481. [Google Scholar] [CrossRef] - Freedman, M.H. A Note on Topology and Magnetic Energy in Incompressible Perfectly Conducting Fluids. J. Fluid Mech.
**1988**, 194, 549–551. [Google Scholar] [CrossRef] - Moffatt, H.K. Helicity and Singular Structures in Fluid Dynamics. Proc. Natl. Acad. Sci. USA
**2014**, 111, 3663–3670. [Google Scholar] [CrossRef] [PubMed] - Davies-Jones, R. Can a Descending Rain Curtain in a Supercell Instigate Tornadogenesis Barotropically? J. Atmos. Sci.
**2008**, 65, 2469–2497. [Google Scholar] [CrossRef] [Green Version] - Yoshida, Z.; Mahajan, S.M.; Ohsaki, S.; Iqbal, M.; Shatashvili, N. Beltrami Fields in Plasmas: High-Confinement Mode Boundary Layers and High Beta Equilibria. Phys. Plasmas
**2001**, 8, 2125–2131. [Google Scholar] [CrossRef] - Bhattacharjee, C. Beltrami–Bernoulli Equilibria in Weakly Rotating Self-Gravitating Fluid. J. Plasma Phys.
**2022**, 88, 175880101. [Google Scholar] [CrossRef] - Morgulis, A.; Yudovich, V.I.; Zaslavsky, G.M. Compressible Helical Flows. Comm. Pure Appl. Math.
**1995**, 48, 571–582. [Google Scholar] [CrossRef] - Berker, R. Integration des Équations du Mouvement d’un Fluide Visqueux Incompressible; Handbuch der Physik, Bd VIII/2, Encyclopedia of Physics; Springer: Berlin, Germany, 1963. [Google Scholar]
- Ershkov, S.V. Non-Stationary Helical Flows for Incompressible 3D Navier–Stokes Equations. Appl. Math. Comput.
**2016**, 274, 611–614. [Google Scholar] [CrossRef] - Dierkes, D.; Cheviakov, A.; Oberlack, M. New Similarity Reductions and Exact Solutions for Helically Symmetric Viscous Flows. Phys. Fluids
**2020**, 32, 053604. [Google Scholar] [CrossRef] - Wang, C.Y. Exact Solutions of the Unsteady Navier-Stokes Equations. Appl. Mech. Rev.
**1989**, 42, S269–S282. [Google Scholar] [CrossRef] - Joseph, S.P. Polynomial Solutions and Other Exact Solutions of Axisymmetric Generalized Beltrami Flows. Acta Mech.
**2018**, 229, 2737–2750. [Google Scholar] [CrossRef] - Wang, C.Y. Exact Solutions of the Steady-State Navier-Stokes Equations. Annu. Rev. Fluid Mech.
**1991**, 23, 159–177. [Google Scholar] [CrossRef] - Dyck, N.J.; Straatman, A.G. Exact Solutions to the Three-Dimensional Navier–Stokes Equations Using the Extended Beltrami Method. J. Appl. Mech.
**2020**, 87, 011004. [Google Scholar] [CrossRef] - Holm, D.D.; Kerr, R. Transient Vortex Events in the Initial Value Problem for Turbulence. Phys. Rev. Lett.
**2002**, 88, 244501. [Google Scholar] [CrossRef] [Green Version] - Choi, Y.; Kim, B.-G.; Lee, C. Alignment of Velocity and Vorticity and the Intermittent Distribution of Helicity in Isotropic Turbulence. Phys. Rev. E
**2009**, 80, 017301. [Google Scholar] [CrossRef] - Jacobitz, F.G.; Schneider, K.; Bos, W.J.T.; Farge, M. On Helical Multiscale Characterization of Homogeneous Turbulence. J. Turbul.
**2012**, 13, N35. [Google Scholar] [CrossRef] [Green Version] - Constantin, P.; Majda, A. The Beltrami Spectrum for Incompressible Fluid Flows. Commun. Math. Phys.
**1988**, 115, 435–456. [Google Scholar] [CrossRef] - Ershkov, S.V.; Shamin, R.V.; Giniyatullin, A.R. On a New Type of Non-Stationary Helical Flows for Incompressible 3D Navier-Stokes Equations. J. King Saud Univ. Sci.
**2020**, 32, 459–467. [Google Scholar] [CrossRef] - Hicks, W.M. Researches in Vortex Motion. Part III. On Spiral or Gyrostatic Vortex Aggregates. [Abstract]. Proc. R. Soc. Lond.
**1897**, 62, 332–338. [Google Scholar] - Bragg, S.L.; Hawthorne, W.R. Some Exact Solutions of the Flow Through Annular Cascade Actuator Discs. J. Aeronaut. Sci.
**1950**, 17, 243–249. [Google Scholar] [CrossRef] - Gledzer, E.B.; Makarov, A.L. A Class of Steady Axisymmetric Incompressible Flows. Fluid Dyn.
**1991**, 25, 832–838. [Google Scholar] [CrossRef] - Taylor, G.I. LXXV. On the Decay of Vortices in a Viscous Fluid. Lond. Edinb. Dublin Philos. Mag. J. Sci.
**1923**, 46, 671–674. [Google Scholar] [CrossRef]

**Figure 1.**Numerical solutions of (27) under the conditions ${\partial}_{z}\psi ={\partial}_{z}^{2}\psi =0$. Ordinate is ${v}_{\theta}=\psi /r$. Three curves, (i), (ii), and (iii), are the solutions for three sets of the parameters A, B, and ${\partial}_{r}\psi ({r}_{0})$, as designated in the figure. Each $\psi ({r}_{0})$ is chosen so that the straight extrapolation of the curve directs the origin. Integrations were performed in a region ${r}_{0}=0.002\le r\le 8$.

**Figure 2.**The (ordinate)-dependence of $4a/{A}^{2}$ (abscissa), where $a(z,t)\approx \psi /{r}^{2}$ near the symmetry axis. Boundary condition is $a(0,t)=0$.

**Figure 3.**Upper panel: The numerical solution (i) in Figure 1 is shown by red circles and red thick curve. Solid black curve is the result of fitting to the numerical solution (i) by Equation (A9) in Appendix C. Lower panel: Hankel transform $\tilde{v}$ (red solid curve), Gaussian factor $\varpi exp(-\gamma {q}^{2})$ with $\varpi =0.28$ and $\gamma =0.048$ (green broken line) and $f=\tilde{v}(q,0)exp(\gamma {q}^{2})$ (blue dotted curve).

**Figure 4.**Temporal variation due to (15) with ${F}_{\theta}=0$ of the solution (i) for ${v}_{\theta}$ in Figure 1. Integration was performed up to the tenth zero of ${J}_{1}$.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Takahashi, K.
Three-Dimensional Unsteady Axisymmetric Viscous Beltrami Vortex Solutions to the Navier–Stokes Equations. *J* **2023**, *6*, 460-476.
https://doi.org/10.3390/j6030030

**AMA Style**

Takahashi K.
Three-Dimensional Unsteady Axisymmetric Viscous Beltrami Vortex Solutions to the Navier–Stokes Equations. *J*. 2023; 6(3):460-476.
https://doi.org/10.3390/j6030030

**Chicago/Turabian Style**

Takahashi, Koichi.
2023. "Three-Dimensional Unsteady Axisymmetric Viscous Beltrami Vortex Solutions to the Navier–Stokes Equations" *J* 6, no. 3: 460-476.
https://doi.org/10.3390/j6030030