Hostname: page-component-848d4c4894-hfldf Total loading time: 0 Render date: 2024-05-21T01:26:40.903Z Has data issue: false hasContentIssue false
  • English
  • Français

The vortex-entrainment sheet in an inviscid fluid: theory and separation at a sharp edge

Published online by Cambridge University Press:  13 March 2019

A. C. DeVoria Open the ORCID record for A. C. DeVoria [Opens in a new window]  and
K. Mohseni Open the ORCID record for K. Mohseni [Opens in a new window]
A. C. DeVoria
Affiliation:
Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611, USA
K. Mohseni*
Affiliation:
Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611, USA Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611, USA
*
†Email address for correspondence: mohseni@ufl.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper a model for viscous boundary and shear layers in three dimensions is introduced and termed a vortex-entrainment sheet. The vorticity in the layer is accounted for by a conventional vortex sheet. The mass and momentum in the layer are represented by a two-dimensional surface having its own internal tangential flow. Namely, the sheet has a mass density per-unit-area making it dynamically distinct from the surrounding outer fluid and allowing the sheet to support a pressure jump. The mechanism of entrainment is represented by a discontinuity in the normal component of the velocity across the sheet. The velocity field induced by the vortex-entrainment sheet is given by a generalized Birkhoff–Rott equation with a complex sheet strength. The model was applied to the case of separation at a sharp edge. No supplementary Kutta condition in the form of a singularity removal is required as the flow remains bounded through an appropriate balance of normal momentum with the pressure jump across the sheet. A pressure jump at the edge results in the generation of new vorticity. The shedding angle is dictated by the normal impulse of the intrinsic flow inside the bound sheets as they merge to form the free sheet. When there is zero entrainment everywhere the model reduces to the conventional vortex sheet with no mass. Consequently, the pressure jump must be zero and the shedding angle must be tangential so that the sheet simply convects off the wedge face. Lastly, the vortex-entrainment sheet model is demonstrated on several example problems.

JFM classification

Mathematical Foundations: General fluid mechanics Vortex Flows: Vortex dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 866 , 10 May 2019 , pp. 660 - 688
Copyright
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alben, S. 2008 Optimal flexibility of a flapping appendage in an inviscid fluid. J. Fluid Mech. 614, 355–380.10.1017/S0022112008003297Google Scholar
Alben, S. 2009 Simulating the dynamics of flexible bodies and vortex sheets. J. Comput. Phys. 228 (7), 2587–2603.10.1016/j.jcp.2008.12.020Google Scholar
Aris, R. 1962 Vectors, Tensors and the Basic Equations of Fluid Mechanics. Prentice-Hall.Google Scholar
Baker, G. R., Meiron, D. I. & Orszag, S. A. 1982 Generalized vortex methods for free-surface flow problems. J. Fluid Mech. 123, 477–501.10.1017/S0022112082003164Google Scholar
Basu, B. C. & Hancock, G. J. 1978 The unsteady motion a two-dimensional aerofoil in incompressible inviscid flow. J. Fluid Mech. 87, 159–178.10.1017/S0022112078002980Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Bhatia, H., Norgard, G., Pascucci, V. & Bremer, P.-T. 2013 The Helmholtz-Hodge decomposition – A survey. IEEE Trans. Vis. Comput. Graphics 19 (8), 1386–1404.10.1109/TVCG.2012.316Google Scholar
Clements, R. R. 1973 An inviscid model of two-dimensional vortex shedding. J. Fluid Mech. 57 (2), 321–336.10.1017/S0022112073001187Google Scholar
Cortelezzi, L. & Leonard, A. 1993 Point vortex model of the unsteady separated flow past a semi-infinite plate with transverse motion. Fluid Dyn. Res. 11 (6), 263–295.10.1016/0169-5983(93)90013-ZGoogle Scholar
DeVoria, A. C. & Mohseni, K. 2018 Vortex sheet roll-up revisited. J. Fluid Mech. 855, 299–321.10.1017/jfm.2018.663Google Scholar
DeVoria, A. C. & Ringuette, M. J. 2013 On the flow generated on the leeward face of a rotating flat plate. Exp. Fluids 54 (4), 1–14.10.1007/s00348-013-1495-5Google Scholar
Giesing, J. P. 1969 Vorticity and Kutta condition for unsteady multienergy flows. Trans. ASME J. Appl. Mech. 36 (3), 608–613.10.1115/1.3564724Google Scholar
Greenberg, M. D. 1998 Advanced Engineering Mathematics, 2nd edn. Prentice Hall.Google Scholar
Haroldsen, D. J. & Meiron, D. I. 1998 Numerical calculation of three-dimensional interfacial potential flows using the point vortex method. SIAM J. Sci. Comput. 20 (2), 648–683.10.1137/S1064827596302060Google Scholar
Howe, M. S. 2007 Hydrodynamics and sound. Cambridge University Press.Google Scholar
Jackson, J. D. 1998 Classical Electrodynamics, 3rd edn. Wiley.Google Scholar
Jones, M. A. 2003 The separated flow of an inviscid fluid around a moving flat plate. J. Fluid Mech. 496, 405–441.10.1017/S0022112003006645Google Scholar
Katz, J. 1981 A discrete vortex method for the non-steady separated flow over an airfoil. J. Fluid Mech. 102, 315–328.10.1017/S0022112081002668Google Scholar
Kellogg, O. D. 1929 Foundations of Potential Theory, 1st edn. Springer.10.1007/978-3-642-90850-7Google Scholar
Koochesfahani, M. 1989 Vortical patterns in the wake of an oscillating airfoil. AIAA J. 27 (9), 1200–1205.10.2514/3.10246Google Scholar
Lamb, H. 1945 Hydrodynamics. Dover.Google Scholar
Leonard, A. 1980 Vortex methods for flow simulation. J. Comput. Phys. 37 (3), 289–335.10.1016/0021-9991(80)90040-6Google Scholar
Lighthill, M. J. 1963 Introduction: Boundary layer theory. In Laminar Boundary Theory, pp. 46–113. Oxford University Press.Google Scholar
Liu, L. Q., Zhu, J. Y. & Wu, J. Z. 2015 Lift and drag in two-dimensional steady viscous and compressible flow. J. Fluid Mech. 784, 304–341.10.1017/jfm.2015.584Google Scholar
Maskell, E. C.1971 On the Kutta-Joukowski condition in two-dimensional unsteady flow. Unpublished note, Royal Aircraft Establishment, Farnborough, England.Google Scholar
Michelin, S. & Llewellyn Smith, S. G. 2009 An unsteady point vortex method for coupled fluid-solid problems. Theor. Comput. Fluid Dyn. 23 (2), 127–153.10.1007/s00162-009-0096-7Google Scholar
Muskhelishvili, N. I. 1946 Singular Integral Equations, 1st edn. P. Noordhoof Ltd.Google Scholar
Nitsche, M. & Krasny, R. 1994 A numerical study of vortex ring formation at the edge of a circular tube. J. Fluid Mech. 276, 139–161.10.1017/S0022112094002508Google Scholar
Phillips, H. B. 1959 Vector Analysis, 1st edn. Wiley; 18th printing.Google Scholar
Poling, D. R. & Telionis, D. P. 1986 The response of airfoils to periodic disturbances: The unsteady Kutta condition. AIAA J. 24 (2), 193–199.10.2514/3.9244Google Scholar
Pozrikidis, C. 2000 Theoretical and computation aspects of the self-induced motion of three-dimensional vortex sheets. J. Fluid Mech. 425, 335–366.10.1017/S0022112000002202Google Scholar
Pullin, D. I. 1978 The large-scale structure of unsteady self-similar rolled-up vortex sheets. J. Fluid Mech. 88 (3), 401–430.10.1017/S0022112078002189Google Scholar
Rosenhead, L. 1931 The formation of vortices from a surface of discontinuity. Proc. R. Soc. Lond. A 134 (823), 170–192.10.1098/rspa.1931.0189Google Scholar
Rott, N. 1956 Diffraction of a weak shock with vortex generation. J. Fluid Mech. 1, 111–128.10.1017/S0022112056000081Google Scholar
Saffman, P. G. 1981 Dynamics of vorticity. J. Fluid Mech. 106, 49–58.10.1017/S0022112081001511Google Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Sarpkaya, T. 1975 An inviscid model of two-dimensional vortex shedding for transient and asymptotically steady separated flow over an inclined plate. J. Fluid Mech. 68, 109–128.10.1017/S0022112075000717Google Scholar
Schlichting, H. 1955 Boundary Layer Theory, 2nd edn. McGraw-Hill.Google Scholar
Scriven, L. E. 1960 Dynamics of a fluid interface equation of motion for newtonian surface fluids. Chem. Engng Sci. 12 (2), 98–108.10.1016/0009-2509(60)87003-0Google Scholar
Sears, W. R. 1956 Some recent developments in airfoil theory. J. Aeronaut. Sci. 23 (5), 490–499.10.2514/8.3588Google Scholar
Slattery, J. C., Sagis, L. & Oh, E. S. 2007 Interfacial Transport Phenomena, 2nd edn. Springer.Google Scholar
Stakgold, I. 1968 Boundary Value Problems of Mathematical Physics: Volume 2, 1st edn. Society for Industrial and Applied Mathematics.Google Scholar
Wang, C. & Eldredge, J. D. 2013 Low-order phenomenological modeling of leading-edge vortex formation. Theor. Comput. Fluid Dyn. 27 (5), 577–598.10.1007/s00162-012-0279-5Google Scholar
Westwater, F. L.1935 The rolling up of a surface of discontinuity behind an aerofoil of finite span. Report R&M 1692. Aeronautical Research Council.Google Scholar
Wu, J.-Z., Ma, H.-Y. & Zhou, M.-D. 2006 Vorticity and vortex dynamics. Springer.10.1007/978-3-540-29028-5Google Scholar
Xia, X. & Mohseni, K. 2013 Lift evaluation of a two-dimensional pitching flat plate. Phys. Fluids 25 (9), 091901.10.1063/1.4819878Google Scholar
Xia, X. & Mohseni, K. 2017 Unsteady aerodynamics and vortex-sheet formation of a two-dimensional airfoil. J. Fluid Mech. 830, 439–478.10.1017/jfm.2017.513Google Scholar