In the present paper, we derive exact solutions for the helically invariant Navier–Stokes equations. The approach is based on an invariant solution ansatz emerging from the Galilean group in helical coordinates, which leads to linear functions in the helical coordinate ξ = az + bφ for the two helical velocity components uξ and uη. The variables z and φ are the usual cylinder coordinates. Starting from this approach, we derive a new equation for the radial velocity component ur in the helical frame, for which we found two special solutions. Moreover, we present an exact linearization of the Navier–Stokes equations by seeking exact solutions in the form of Beltrami flows. Using separation of variables, we found exponentially decaying time-dependent solutions, which consist of trigonometric functions in the helical coordinate ξ and of confluent Heun-type functions in the radial direction.
REFERENCES
1.
L. J.
Vermeer
, J. N.
Sørensen
, and A.
Crespo
, “Wind turbine wake aerodynamics
,” Prog. Aerosp. Sci.
39
, 467
–510
(2003
).2.
A.
Mitchell
, S.
Morton
, and J.
Forsythe
, “Wind turbine wake aerodynamics
,” Report No. ADA425027, Air Force Academy Colorado Springs, Department of Aeronautics
, 1997
.3.
O. I.
Bogoyavlenskij
, “Helically symmetric astrophysical jets
,” Phys. Rev. E
62
, 8616
–8627
(2000
).4.
D. D.
Schnack
, E. J.
Caramana
, and R. A.
Nebel
, “Three-dimensional magnetohydrodynamic studies of the reversed-field pinch
,” Phys. Fluids
28
, 321
–333
(1985
).5.
J. L.
Johnson
, C. R.
Oberman
, R. M.
Kulsrud
, and E. A.
Frieman
, “Some stable hydromagnetic equilibria
,” Phys. Fluids
1
, 281
–296
(1958
).6.
T.
Sarpkaya
, “On stationary and travelling vortex breakdowns
,” J. Fluid Mech.
45
, 545
–559
(1971
).7.
S. L.
Bragg
and W. R.
Hawthorne
, “Some exact solutions of the flow through annular cascade actuator discs
,” J. Aeronaut. Sci.
17
, 243
–249
(1950
).8.
H.
Grad
and H.
Rubin
, “Hydromagnetic equilibria and force-free fields
,” J. Nucl. Energy
7
, 284
–285
(1958
).9.
V. D.
Shafranov
, “Propagation of an electromagnetic field in a medium with spatial dispersion
,” Sov. Phys. JETP
7
, 1019
–1029
(1958
).10.
O. I.
Bogoyavlenskij
, “Exact helically symmetric plasma equilibria
,” Lett. Math. Phys.
51
, 235
–247
(2000
).11.
A. F.
Cheviakov
and O. I.
Bogoyavlenskij
, “Exact anisotropic MHD equilibria
,” J. Phys. A: Math. Gen.
37
, 7593
(2004
).12.
C. Y.
Wang
, “On the low-Reynolds-number flow in a helical pipe
,” J. Fluid Mech.
108
, 185
–194
(1981
).13.
M.
Germano
, “On the effect of torsion on a helical pipe flow
,” J. Fluid Mech.
125
, 1
–8
(1982
).14.
M.
Germano
, “The Dean equations extended to a helical pipe flow
,” J. Fluid Mech.
203
, 289
–305
(1989
).15.
E. R.
Tuttle
, “Laminar flow in twisted pipes
,” J. Fluid Mech.
219
, 545
–570
(1990
).16.
L.
Zabielski
and A. J.
Mestel
, “Steady flow in a helically symmetric pipe
,” J. Fluid Mech.
370
, 297
–320
(1998
).17.
S. V.
Ershkov
, R. V.
Shamin
, and A. R.
Giniyatullin
, “On a new type of non-stationary helical flows for incompressible 3D Navier-Stokes equations
,” J. King Saud Univ., Sci.
32
(1
), 459
–467
(2020
).18.
S. V.
Ershkov
, “Non-stationary helical flows for incompressible 3D Navier–Stokes equations
,” Appl. Math. Comput.
274
, 611
–614
(2016
).19.
I.
Delbende
, M.
Rossi
, and O.
Daube
, “DNS of flows with helical symmetry
,” Theor. Comput. Fluid Dyn.
26
, 141
–160
(2012
).20.
D. G.
Dritschel
, “Generalized helical Beltrami flows in hydrodynamics and magnetohydrodynamics
,” J. Fluid Mech.
222
, 525
–541
(1991
).21.
M.
Jamil
and C.
Fetecau
, “Helical flows of Maxwell fluid between coaxial cylinders with given shear stresses on the boundary
,” Nonlinear Anal.: Real World Appl.
11
, 4302
–4311
(2010
).22.
J.-Z.
Wu
, H.-Y.
Ma
, and M.-D.
Zhou
, Vorticity and Vortex Dynamics
(Springer Science & Business Media
, 2007
).23.
N. H.
Ibragimov
, CRC Handbook of Lie Group Analysis of Differential Equations
(CRC Press
, 1995
), Vol. 2.24.
V. K.
Andreev
, O. V.
Kaptsov
, V. V.
Pukhnachev
, and A. A.
Rodionov
, Applications of Group-Theoretical Methods in Hydrodynamics
(Springer Science & Business Media
, 1998
), Vol. 450.25.
S. V.
Meleshko
and V. V.
Pukhnachev
, “One class of partially invariant solutions of the Navier-Stokes equations
,” J. Appl. Mech. Tech. Phys.
40
, 208
–216
(1999
).26.
O.
Bogoyavlenskij
, “Infinite families of exact periodic solutions to the Navier-Stokes equations
,” Moscow Math. J.
3
, 263
–272
(2003
).27.
V. V.
Pukhnachev
, “Symmetries in the Navier–Stokes equations
,” Usp. Mekh.
4
, 6
–76
(2006
).28.
G. W.
Bluman
, A. F.
Cheviakov
, and S. C.
Anco
, Applications of Symmetry Methods to Partial Differential Equations
(Springer
, 2010
).29.
M.
Kumar
and R.
Kumar
, “On some new exact solutions of incompressible steady state Navier-Stokes equations
,” Meccanica
49
, 335
–345
(2014
).30.
H. K.
Moffatt
, “Helicity and singular structures in fluid dynamics
,” Proc. Natl. Acad. Sci. U. S. A.
111
, 3663
–3670
(2014
).31.
O.
Kelbin
, A. F.
Cheviakov
, and M.
Oberlack
, “New conservation laws of helically symmetric, plane and rotationally symmetric viscous and inviscid flows
,” J. Fluid Mech.
721
, 340
–366
(2013
).32.
D.
Dierkes
and M.
Oberlack
, “Euler and Navier-Stokes equations in a new time-dependent helically symmetric system: Derivation of the fundamental system and new conservation laws
,” J. Fluid Mech.
818
, 344
–365
(2017
).33.
P. G.
Drazin
and N.
Riley
, The Navier-Stokes Equations: A Classification of Flows and Exact Solutions
(Cambridge University Press
, 2006
), Vol. 334.34.
C. R.
Ethier
and D. A.
Steinman
, “Exact fully 3D Navier-Stokes solutions for benchmarking
,” Int. J. Numer. Methods Fluids
19
, 369
–375
(1994
).35.
S. V.
Alekseenko
, P. A.
Kuibin
, V. L.
Okulov
, and S. I.
Shtork
, “Helical vortices in swirl flow
,” J. Fluid Mech.
382
, 195
–243
(1999
).36.
P. M.
Gresho
, “Incompressible fluid dynamics: Some fundamental formulation issues
,” Annu. Rev. Fluid Mech.
23
, 413
–453
(1991
).37.
P.
Constantin
and A.
Majda
, “The Beltrami spectrum for incompressible fluid flows
,” Commun. Math. Phys.
115
, 435
–456
(1988
).38.
O. I.
Bogoyavlenskij
, “Exact unsteady solutions to the Navier–Stokes and viscous MHD equations
,” Phys. Lett. A
307
, 281
–286
(2003
).39.
G. I.
Taylor
, “LXXV. On the decay of vortices in a viscous fluid
,” London, Edinburgh Dublin Philos. Mag. J. Sci.
46
, 671
–674
(1923
).40.
41.
G.
Kristensson
, Second Order Differential Equations: Special Functions and Their Classification
(Springer Science & Business Media
, 2010
).42.
43.
A. M.
Ishkhanyan
and A. E.
Grigoryan
, “Fifteen classes of solutions of the quantum two-state problem in terms of the confluent Heun function
,” J. Phys. A: Math. Theor.
47
, 465205
(2014
).44.
L. J.
El-Jaick
and B. D. B.
Figueiredo
, “Solutions for confluent and double-confluent Heun equations
,” J. Math. Phys.
49
, 083508
(2008
).45.
M.
Hortaçsu
, “Heun functions and their uses in physics
,” in Mathematical Physics
(World Scientific
, 2013
), pp. 23
–39
.46.
O. V.
Motygin
, “On numerical evaluation of the Heun functions
,” in 2015 Days on Diffraction (DD)
(IEEE
, 2015
), pp. 1
–6
.47.
P. A.
Clarkson
and M. D.
Kruskal
, “New similarity reductions of the Boussinesq equation
,” J. Math. Phys.
30
, 2201
–2213
(1989
).48.
K.
Kunii
, T.
Ishida
, Y.
Duguet
, and T.
Tsukahara
, “Laminar-turbulent coexistence in annular Couette flow
,” J. Fluid Mech.
879
, 579
–603
(2019
).49.
M.
Ali
and M.
Abid
, “Self-similar behaviour of a rotor wake vortex core
,” J. Fluid Mech.
740
, R1
(2014
).50.
C.
Selçuk
, I.
Delbende
, and M.
Rossi
, “Helical vortices: Quasiequilibrium states and their time evolution
,” Phys. Rev. Fluids
2
, 084701
(2017
).51.
Maplesoft
, Maple User Manual, Maplesoft, A Division of Waterloo Maple, Inc.
, Waterloo, Ontario
, 2017
.© 2020 Author(s).
2020
Author(s)
You do not currently have access to this content.
