# Natural Velocity Decomposition: A Review

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## Abstract

A review of the natural velocity decomposition (NVD) formulation for the analysis of Navier–Stokes flows developed by the authors is presented. The decomposition consists in expressing the velocity field as the sum of two terms, one of which is given as the gradient of a potential \(\varphi \), whereas the other, \(\mathbf{w}\), is rotational and is governed by its own evolution equation. Hence, contrary to all the related approaches such as the Helmholtz decomposition, the NVD does not require the evaluation of the vorticity. The theoretical formulation is presented in detail for incompressible viscous flows and briefly outlined for compressible viscous flows. Of particular significance is the relationship between the present approach for thin vortical layers (almost-potential flows) and that for zero-thickness layers (quasi-potential flows), when this is combined with the transpiration-velocity correction by Lighthill. The methodology may also be a valuable computational tool. To assess this, two problems are addressed. The first consists in the planar flow around a disk, for low Reynolds numbers (separated laminar flows). The computational scheme used is discussed thoroughly. The corresponding numerical results compare favorably with data available in the open literature. These include the separation angle and the length of the recirculation bubble. Also, the flow-field results are in very good agreement with experimental flow visualizations available in the open literature. The second problem that we addressed is the flow due to a jet. This is tackled by the almost-potential flow approximation of the NVD formulation. Again, the results are in good agreement with experimental data available in the open literature. Theoretical innovations (and limitations) of the methodology are discussed, along with its advantages and drawbacks.

## Keywords

Computational fluid dynamics Viscous flows Potential/vortical-field decomposition## Notes

### Acknowledgements

We wish to express our appreciation to the Hariri Center of Boston University, and its Director, Prof. Azer Bestavros, for hosting P.G. and L.M., as well as the Engineering Faculty of Embry-Riddle Aeronautical University and its Dean, Prof. Anastasios Lyrinzis, for hosting M.C. and L.S., in connection with the activity reported here.

### Compliance with Ethical Standards

### Conflict of interest

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

## Supplementary material

## References

- 1.Attaway, D.C.: The boundary element method for the diffusion equation: a feasibility study. In: Morino, L., Piva, R. (eds.) Boundary Integral Methods—Theory and Applications. Springer, Berlin (1991)Google Scholar
- 2.Beauchamp, P.: A potential-vorticity decomposition for the boundary integral equation analysis of viscous flows, Ph. D. Thesis, Graduate School, Division of Engineering and Applied Science, Boston University, Boston, MA (1990)Google Scholar
- 3.Bykhovskiy, E.B., Smirnov, N.V.: On orthogonal expansions of the space of vector functions which are square-summable over a given domain and the vector analysis operators. Trudy Mat. Inst. Steklove, vol. 59, No. 5, pp. 5–36, Academy of Sciences USSR Press (1960) (In Russian; English translation available as NASA TM-77051, 1983)Google Scholar
- 4.Carrier, F.G., Krook, M., Pearson, C.E.: Functions of a Complex Variable; Theory and Technique. McGraw-Hill, New York (1966)zbMATHGoogle Scholar
- 5.Casciola, C.M., Piva, R.: A Lagrangian approach for vorticity intensification in swirling rings. Comput. Mech.
**21**, 276–282 (1998)CrossRefzbMATHGoogle Scholar - 6.Chen, Y., Maki, K.J.: A velocity decomposition approach for three-dimensional unsteady flow. Eur. J. Mech. B/Fluids
**62**, 94–108 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Chorin, A.J., Marsden, J.E.: A Mathematical Introduction to Fluid Mechanics, 2nd edn. Springer, New York (1990)CrossRefzbMATHGoogle Scholar
- 8.Coderoni, M.: A Novel Decomposition of the Velocity Field—Computational Issues and Applications, Tesi di Laurea Magistrale in Ingegneria Aeronautica. Università Roma Tre, Roma (2014)Google Scholar
- 9.Cossu, C., Morino, L.: A vorticity-only formulation and a low-order asymptotic expansion solution near Hopf bifurcation. Comput. Mech.
**20**, 229–241 (1997)CrossRefzbMATHGoogle Scholar - 10.Dennis, S.C.R., Chang, G.Z.: Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100. J. Fluid Mech.
**42**, 471–489 (1970)CrossRefzbMATHGoogle Scholar - 11.Douglas Jr., J.: Alternating direction methods for three space variables. Numer. Math.
**4**, 41–63 (1962)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Douglas Jr., J., Gunn, J.E.: A general formulation of alternating direction methods. Numer. Math.
**6**, 418–453 (1964)CrossRefGoogle Scholar - 13.Ffowcs Williams, J.E., Hawkings, D.L.: Sound generated by turbulence and surfaces in arbitrary motion. Philos. Trans. R. Soc. A
**264**, 321–342 (1969)CrossRefzbMATHGoogle Scholar - 14.Farassat, F., Myers, M.K.: Extension of Kirchhoff’s formula to radiation from moving surfaces. J. Sound Vib.
**123**(3), 451–460 (1988)MathSciNetCrossRefzbMATHGoogle Scholar - 15.Fornberg, B.: A numerical study of steady viscous flow past a circular cylinder. J. Fluid Mech.
**98**, 819–855 (1980)CrossRefzbMATHGoogle Scholar - 16.Gennaretti, M., Salvatore, F., Morino, L.: Forces and moments in incompressible quasi-potential flows. J. Fluids Struct.
**10**, 281–303 (1996)CrossRefGoogle Scholar - 17.Gradassi, P.: Methodologies for the analysis of vortex flows in fluid dynamics and aeroacoustics, Doctoral Thesis, Università Roma Tre, Scuola Dottorale di Ingegneria, Sezione di Ingegneria Meccanica e Industriale (2013)Google Scholar
- 18.Grove, A.S., Shair, F.H., Petersen, E.E., Acrivos, A.: An experimental investigation of the steady separated flow past a circular cylinder. J. Fluid Mech.
**19**, 60–80 (1964)CrossRefzbMATHGoogle Scholar - 19.Guggenheim, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Biforcations of Vector Fields. Springer, Berlin (1983)Google Scholar
- 20.Hirsch, M.W., Smale, S.: Differential Equation, Dynamical Systems, and Linear Algebra. Academic Press, Boston (1974)zbMATHGoogle Scholar
- 21.Kellog, O.D.: Foundations of Potential Theory. Fredrick Ugar, New York (1929). (Also available as Dover, New York, NY, 1953)CrossRefGoogle Scholar
- 22.Kress, R.: Linear Integral Equations. Springer, New York (1989)CrossRefzbMATHGoogle Scholar
- 23.Lemmerman, L.A., Sonnad, V.R.: Three-dimensional viscous-inviscid coupling using surface transpiration. J. Aircr.
**16**, 353–358 (1979)CrossRefGoogle Scholar - 24.Lighthill, M.J.: On sound generated aerodynamically; I. General theory. Proc. R. Soc.
**A11**, 564–587 (1952)MathSciNetzbMATHGoogle Scholar - 25.Lighthill, M.J.: On displacement thickness. J. Fluid Mech.
**4**, 383–392 (1958)MathSciNetCrossRefzbMATHGoogle Scholar - 26.Lyrintzis, A.S.: Review: the use of Kirchhoff’s method in computational aeroacoustics. ASME J. Fluids Eng.
**116**(4), 665–676 (1994)CrossRefGoogle Scholar - 27.Meleshko, V.V., Gourjii, A.A., Krasnopolskaya, T.S.: Vortex rings: history and state-of-the-art. J. Math. Sci.
**173**(4), 1–37 (2011)MathSciNetzbMATHGoogle Scholar - 28.Morino, L.: A general theory of unsteady compressible potential aerodynamics, NASA CR-2464, NASA, Washington, DC (1974)Google Scholar
- 29.Morino, L.: Helmholtz decomposition revisited: vorticity generation and trailing edge condition, Part 1: incompressible flows. Comput. Mech.
**1**, 65–90 (1986)CrossRefzbMATHGoogle Scholar - 30.Morino, L.: Material contravariant components: vorticity transport and vortex theorems. AIAA J.
**24**, 526–528 (1986)MathSciNetCrossRefGoogle Scholar - 31.Morino, L.: Helmholtz and Poincaré potential-vorticity decompositions for the analysis of unsteady compressible viscous flows. In: Banerjee, P.K., Morino, L. (eds.) Boundary Element Methods in Nonlinear Fluid Dynamics, vol. 6 of Developments in Boundary Element Methods, pp. 1–54. Elsevier Applied Science, London (1990)Google Scholar
- 32.Morino, L.: Boundary integral equations in aerodynamics. Appl. Mech. Rev.
**46**, 445–466 (1993)CrossRefGoogle Scholar - 33.Morino, L.: Is there a difference between aeroacoustics and aerodynamics? An aeroelastician’s viewpoint. AIAA J.
**41**, 1209–1223 (2003)CrossRefGoogle Scholar - 34.Morino, L.: Boundary elements in primitive variables and FWH equation revisited—viscosity effect. In: 10th AIAA/CEAS Aeroacoustics Conference, Manchester, UK, AIAA Paper 2004-2890 (2004)Google Scholar
- 35.Morino, L.: From primitive-variable boundary-integral formulation to FWH equation. In: 12th AIAA/CEAS Aeroacoustics Conference, Cambridge, MA, AIAA Paper 2006-2485 (2006)Google Scholar
- 36.Morino, L.: A primitive-variable boundary integral formulation unifying aeroacoustics and aerodynamics, and a natural velocity decomposition for vortical fields. Int. J. Aeroacoust.
**10**, 295–400 (2011)CrossRefGoogle Scholar - 37.Morino, L., Bharadvaj, B.K.: A unified approach for the potential and viscous free-wake analysis of helicopter rotors. Vertica
**12**, 147–154 (1988)Google Scholar - 38.Morino, L., Beauchamp, P.P.: A potential-vorticity decomposition for the analysis of viscous flows. In: Tanaka, M., Cruse, T.A. (eds.) Boundary Element Methods in Applied Mechanics. Pergamon Press, Oxford (1988)Google Scholar
- 39.Morino, L., Bernardini, G.: Singularities in discretized BIE’s for Laplace’s equation; trailing-edge conditions in aerodynamics. In: Bonnet, M., Sändig, A.-M., Wendland, W.L. (eds.) Mathematical Aspects of Boundary Element Methods, pp. 240–251. Chapman & Hall, London (2000)Google Scholar
- 40.Morino, L., Bernardini, G.: On the vorticity generated sound for moving surfaces. Comput. Mech.
**28**, 311–316 (2002)CrossRefzbMATHGoogle Scholar - 41.Morino, L., Bernardini, G., Caputi-Gennaro, G.: A vorticity formulation for computational aeroynamic and aeroelastic analyses of viscous flows. J. Fluids Struct.
**25**, 1282–1298 (2009)CrossRefGoogle Scholar - 42.Morino, L., Bernardini, G., Gennaretti, M.: A boundary element method for the aerodynamic analysis of aircraft in arbitrary motions. Comput. Mech.
**32**, 301–311 (2003)CrossRefzbMATHGoogle Scholar - 43.Morino, L., Corbelli, A., Tseng, K.: A primitive variable boundary element formulation for the Euler equations in aeroacoustics of rotors and propellers. In: 5th AIAA/CEAS Aeroacoustics Conference, Bellevue, WA, AIAA Paper 99-0399 (1999)Google Scholar
- 44.Morino, L., Gennaretti, M.: Boundary integral equation methods for aerodynamics. In: Atluri, S.N. (ed.) Computational Nonlinear Mechanics in Aerospace Engineering, Progress in Aeronautics and Astronautics, vol. 146, pp. 279–321. American Institute of Aeronautics and Astronautics, AIAA, Reston, VA (1992)Google Scholar
- 45.Morino, L., Gradassi, P.: Vorticity generated sound and jet noise. In: 19th AIAA/CEAS Aeroacoustics Conference, Berlin, Germany, AIAA Paper 2013-2089 (2013)Google Scholar
- 46.Morino, L., Gradassi, P.: From aerodynamics towards aeroacoustics: a novel natural velocity decomposition for the Navier–Stokes equations. Int. J. Aeroacoust.
**14**, 161–192 (2015)CrossRefGoogle Scholar - 47.Morino, L., Salvatore, F., Gennaretti, M.: A new velocity decomposition for viscous flows: Lighthill’s equivalent-source method revisited. Comput. Methods Appl. Mech. Eng.
**173**, 317–336 (1999)MathSciNetCrossRefzbMATHGoogle Scholar - 48.Morino, L., Tseng, K.: A general integral formulation for unsteady compressible potential flows with applications to airplanes and rotors. In: Banerjee, P.K., Morino, L. (eds.) Boundary Element Methods in Nonlinear Fluid Dynamics, vol. 6 of Developments in Boundary Element Methods, pp. 183–246. Elsevier Applied Science, London (1990)Google Scholar
- 49.Morse, P.M., Feshbach, H.: Methods of Theoretical Physics, vol. I and II. McGraw-Hill, New York (1953)Google Scholar
- 50.Noack, B.R., Eckelmann, H.: A low-dimensional Galerkin method for the three-dimensional flow around a circular cylinder. Phys. Fluids
**6**, 124–143 (1994)CrossRefzbMATHGoogle Scholar - 51.Pierce, A.D.: Acoustics. Acoustical Society of America, New York (1989)Google Scholar
- 52.Piva, R., Morino, L.: Vector Green’s function method for unsteady Navier–Stokes equations. Meccanica
**22**, 76–85 (1987)CrossRefzbMATHGoogle Scholar - 53.Piva, R., Morino, L.: A boundary integral formulation in primitive variable for unsteady viscous flows. In: Banerjee, P.K., Morino, L. (eds.) Boundary Element Methods in Nonlinear Fluid Dynamics, vol. 6 of Developments in Boundary Element Methods, pp. 117–150. Elsevier Applied Science, London (1990)Google Scholar
- 54.Romano, G.P.: The effect of boundary conditions by the side of the nozzle of a low Reynolds number jet. Exp. Fluids
**33**, 323–333 (2002)CrossRefGoogle Scholar - 55.Rosemurgy, W.J., Beck, R.J., Maki, K.J.: A velocity decomposition formulation for 2D steady incompressible lifting problems. Eur. J. Mech. B/Fluids
**58**, 70–84 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - 56.Serrin, J.: Mathematical principles of classical fluid mechanics. In: Flügge, S. (ed.) Encyclopedia of Physics, vol. VIII/1, pp. 125–263. Springer, Berlin (1959)Google Scholar
- 57.Simone, L.: Modeling of incompressible viscous flows. Tesi di Laurea Magistrale in Ingegneria Aeronautica, Università Roma Tre, Roma (2015)Google Scholar
- 58.Taneda, S.: Experimental investigation of the wake behind cylinders and plates at low Reynolds numbers. J. Phys. Soc. Jpn.
**11**, 302–307 (1956)CrossRefGoogle Scholar - 59.Thom, A.: Flow past circular cylinders at low speeds. Proc. R. Soc. A
**141**, 651–669 (1933)CrossRefzbMATHGoogle Scholar - 60.Thomson, W.: The translatory velocity of a circular vortex ring. Philos. Mag. Ser. 4
**34**, 511–512 (1867)Google Scholar - 61.Truesdell, C.: The Kinematics of Vorticity. Indiana University Press, Bloomington (1954). (Also available as Dover, New York, NY, 2018)zbMATHGoogle Scholar
- 62.Van Dyke, M.: An Album of Fluid Motion. Parabolic Press, Stanford (1982)Google Scholar
- 63.Widnall, S.E.: The structure and dynamics of vortex filaments. Annu. Rev. Fluid Mech.
**7**, 141–165 (1975)CrossRefzbMATHGoogle Scholar - 64.Widnall, S.E., Sullivan, J.P.: On the stability of vortex ring. Proc. R. Soc. Lond. A
**332**, 335–353 (1973)CrossRefzbMATHGoogle Scholar - 65.Widnall, S.E., Tsai, C.-Y.: The instability of the thin vortex ring of constant vorticity. Proc. R. Soc. Lond. A
**287**, 273–305 (1977)MathSciNetzbMATHGoogle Scholar - 66.Wu, J.C.: Numerical boundary conditions for viscous flow problems. AIAA J.
**14**, 432–441 (1976)Google Scholar - 67.Wu, J.C.: Recent advances in solution methods for unstead viscous flows. In: Banerjee, P.K., Morino, L. (eds.) Boundary Element Methods in Nonlinear Fluid Dynamics, Vol. 6 of Developments in Boundary Element Methods, pp. 1–54. Elsevier Applied Science, London (1990)Google Scholar
- 68.Wu, J.C.: Elements of Vorticity Aerodynamics. Springer, Berlin (2017)zbMATHGoogle Scholar
- 69.Zdravcovich, M.M.: Flow Around Circular Cylinders. Oxford University Press, Oxford (1997)Google Scholar