8 - Free Streamline Flows
Published online by Cambridge University Press: 05 October 2013
Summary
Introduction
In this chapter we briefly survey the extensive literature on fully developed cavity flows and the methods used for their solution. The terms “free streamline flow” or “free surface flow” are used for those situations that involve a “free” surface whose location is initially unknown and must be found as a part of the solution. In the context of some of the multiphase flow literature, they would be referred to as separated flows. In the introduction to Chapter 6 we described the two asymptotic states of a multiphase flow, homogeneous and separated flow. Chapter 6 described some of the homogeneous flow methods and their application to cavitating flows; this chapter presents the other approach. However, we shall not use the term separated flow in this context because of the obvious confusion with the accepted, fluid mechanical use of the term.
Fully developed cavity flows constitute one subset of free surface flows, and this survey is intended to provide information on some of the basic properties of these flows as well as the methods that have been used to generate analytical solutions of them. A number of excellent reviews of free streamline methods can be found in the literature, including those of Birkkoff and Zarantonello (1957), Parkin (1959), Gilbarg (1960), Woods (1961), Gurevich (1961), Sedov (1966), and Wu (1969,1972). Here we shall follow the simple and elegant treatment of Wu (1969,1972).
- Type
- Chapter
- Information
- Cavitation and Bubble Dynamics , pp. 206 - 246Publisher: Cambridge University PressPrint publication year: 2013
References
(1970). Boundary layer separation at a free streamline. Part 1. Two-dimensional flow. J. Fluid Mech., 44, 211–225.Google Scholar
(1975). The effects of capillarity on free streamline separation. J. Fluid Mech., 70, 333–352.Google Scholar
(1955). A note on partial cavitation of flat plate hydrofoils. Calif. Inst. of Tech. Hydro. Lab. Rep. E-19.9.
(1960). Cavitating flow past a cascade of circular arc hydrofoils. Calif. Inst. of Tech. Hydro. Lab. Rep. E-79.2.
and (1959). Remarks on cavitation in turbomachines. Calif. Inst. of Tech. Eng. Div. Rep. No. 79. 3
and (1971). Experimental investigation of non-steady forces on hydrofoils oscillating in heave. Proc. IUTAM Symp. on non-steady flow of water at high speeds, Leningrad, USSR, 95–104.
(1975). Viscous effects on the position of cavitation separation from smooth bodies. J. Fluid Mech., 68, 779–799.Google Scholar
(1953). Abrupt and smooth separation in plane and axisymmetric flow. Memo. Arm. Res. Est., G.B., No.22/63.
and (1953). Axisymmetric cavity flow. Rep. Res. Est., G.B., No. 12/53.
and (1954). Axisymmetric cavity flow about ellipsoids. Proc. Joint Admiralty-U.S.Navy Meeting on Hydroballistics.
and (1931). Application of the theory of free jets. NACA TM No. 667.
and (1957). Jets, wakes, and cavities. Academic Press.
(1969a). A numerical solution of axisymmetric cavity flows. J. Fluid Mech., 37, 671–688.Google Scholar
(1969b). Some viscous and other real fluid effects in fully developed cavity flows. In Cavitation State of Knowledge (eds: ), ASME, N.Y.
(1970). Some cavitation experiments with dilute polymer solutions. J. Fluid Mech., 44, 51–63.Google Scholar
and (1973). Theoretical, quasistatic analysis of cavitation compliance in turbopumps. J. Spacecraft and Rockets, 10, No. 3, 175–180.Google Scholar
, , and (1980). On the leading edge flutter of cavitating hydrofoils. J. Ship Res., 24, No. 3, 135–146.Google Scholar
(1911). Les surfaces de glissement de Helmholtz et la résistance des fluides. Ann. Chim. Phys., 23, 145–230.Google Scholar
(1948). Introduction to complex variables and applications. McGraw-Hill Book Company.
and (1957). Two-dimensional, steady, cavity flow about slender bodies in channels of finite width. ASME J. Appl. Mech., 24, 170–176.Google Scholar
, , and (1957). Wall effects in cavitating hydrofoil flow. J. Ship Res., 1, No. 3, 31–39.Google Scholar
, , and (1993). Navier-Stokes analysis of 2-D cavity flows. ASME Cavitation and Multiphase Flow Forum, FED-153, 149–155.Google Scholar
(1966). On the linear theory of cascades of supercavitating hydrofoils. U.K. Nat. Eng. Lab. Rep. No. 218.
(1946). Hydrodynamical theory of two-dimensional flow with cavitation. Dokl. Akad. Nauk. SSSR, 51, 267–270.Google Scholar
and (1948). Water tunnel investigations of steady state cavities. David Taylor Model Basin Rep. No. 668.
(1962). Thin airfoil theory applied to hydrofoils with a single finite cavity and arbitrary free-streamline detachment. J. Fluid Mech., 12, 227–240.Google Scholar
(1964). Choked flow about vented or cavitating hydrofoils. ASME J. Basic Eng., 86, 561–568.Google Scholar
and (1927). On the flow of air behind an inclined flat plate of infinite span. Proc. Roy. Soc., London, Series A, 116, 170–197.Google Scholar
and (1985). Attached cavitation and the boundary layer: experimental investigation and numerical treatment. J. Fluid Mech., 154, 63–90.Google Scholar
and (1973). A note on the calculation of supercavitating hydrofoils with rounded noses. ASME J. Fluids Eng., 95, 222–228.Google Scholar
(1974). Supercavitating linear cascades with rounded noses. ASME J. Basic Eng., SeriesD, 96, 35–42.Google Scholar
(1975b). Three-dimensional theory on supercavitating hydrofoils near a free surface. J. Fluid Mech., 71, 339–359.Google Scholar
(1956). The mathematical theory of three-dimensional cavities and jets. Bull. Amer. Math. Soc., 62, 219–235.Google Scholar
(1949). A generalization of the Schwarz-Christoffel transformation. Proc. U.S. Nat. Acad. Sci., 35, 609–612.Google Scholar
(1960). Jets and cavities. In Handbuch der Physik, Springer-Verlag, 9, 311–445.
(1961). Theory of jets in ideal fluids. Academic Press, N.Y. (1965).
(1968). Aerodynamic loading on a two-dimensional airfoil during dynamic stall. AIAA J., 6, 1927–1934.Google Scholar
(1868). Über diskontinuierliche Flüssigkeitsbewegungen. Monatsber. Akad. Wiss., Berlin, 23, 215–228.Google Scholar
(1972). On flow past a supercavitating cascade of cambered blades. ASME J. Basic Eng., SeriesD, 94, 163–168.Google Scholar
and (1954). Water tunnel experiments on spheres in cavity flow. Calif. Inst. of Tech. Hydro. Lab. Rep. No.E-24.9.
(1961). Theoretical and experimental investigation of supercavitating hydro-foils operating near the free surface. NASA TR R-93.
(1890). I. A modification of Kirchhoff's method of determining a two-dimensional motion of a fluid given a constant velocity along an unknown streamline. II. Determination of the motion of a fluid for any condition given on a streamline. Mat. Sbornik (Rec. Math.), 15, 121–278.Google Scholar
(1967). An extension of the Woods theory for unsteady cavity flows. ASME J. Basic Eng., 89, 798–806.Google Scholar
and (1990). Non-linear analysis of the flow around partially or super-cavitating hydrofoils on a potential based panel method. Proc. IABEM-90 Symp. Int. Assoc. for Boundary Element Methods, Rome, 289–300.
(1946). Cavitation with finite cavitation numbers. Admiralty Res. Lab. Rep. R1/H/36.
(1951). The axially symmetric potential flow about elongated bodies of revolution. David Taylor Model Basin Report No. 761.
(1971). Supercavitating hydrofoil of finite span. Proc. IUTAM Symp. on Non-steady Flow of Water at High Speeds, Leningrad, 277–298.
and (1988). Another approach in modelling cavitating flows. J. Fluid Mech., 195, 557–580.Google Scholar
(1962). Unsteady lift and moment on fully cavitating hydrofoils at zero cavitation number. J. Ship Res., 6, No. 1, 15–25.Google Scholar
(1956). Supercavitating flow past bodies with finite leading edge thickness. David Taylor Model Basin Rep. No. 1081.
(1970). Lifting line theory of supercavitating hydrofoil of finite span. ZAMM, 50, 645–653.Google Scholar
and (1971). Linearized potential flow models for hydrofoils in supercavitating flows. ASME J. Basic Eng., 93, Series D, 550–564.Google Scholar
(1961). Cavitation tests on hydrofoils designed for accelerating flow cascade: Report 1. ASME J. Basic Eng., 83, Series D, 637–647.Google Scholar
(1964). Cavitation tests on hydrofoils designed for accelerating flow cascade: Report 3. ASME J. Basic Eng., 86, Series D, 543–555.Google Scholar
(1958). Experiments on circular-arc and flat plate hydrofoils. J. Ship Res., 1, 34–56.Google Scholar
(1959). Linearized theory of cavity flow in two-dimensions. The RAND Corp. (Calif.) Rep. No. P-1745.
(1962). Numerical data on hydrofoil reponse to non-steady motions at zero cavitation number. J. Ship Res., 6, No. 1, 40–42.Google Scholar
(1871). On the mathematical theory of stream lines, especially those with four foci and upwards. Phil. Trans. Roy. Soc., 267–306.
(1945). The physical laws governing the cavitation bubbles produced behind solids of revolution in a fluid flow. Kaiser Wilhelm Inst. Hyd. Res., Gottingen, TPA3/TIB.
and (1950). Rotationally symmetric source-sink bodies with predominantly constant pressure distributions. Arm. Res. Est. Trans. No. 1/50.
(1920). On steady fluid motion with a free surface. Proc. London Math. Soc., 19, 206–215.Google Scholar
(1954). A new hodograph for free streamline theory. NACA TN 3168.
and (1948). Cavitation and pressure distribution: headforms at zero angles of yaw. Bull. St. Univ. Iowa, Eng., No. 32.
(1966). Plane problems in hydrodynamics and aerodynamics (in Russian). Izdat. “Nauka”, Moscow.
(1959). Experimental studies of supercavitating flow about simple two-dimensional bodies in a jet. J. Fluid Mech., 5, 337–354.Google Scholar
(1967). Linearized theory of non-stationary cascades at fully stalled or supercavitating conditions. Zeitschrift fur Angewandte Mathematik und Mechanik, 8, 531–542.Google Scholar
and (1946). Fluid motions characterized by free streamlines. Phil. Trans., 240, 117–161.Google Scholar
(1948). Relaxation methods in mathematical physics. Oxford Univ. Press.
and (1962). Cavitation in turbopumps—Part 1. ASME J. Basic Eng., Series D, 84, 326–338.Google Scholar
(1970). Discontinuous flows and free streamline solutions for axisymmetric bodies at zero and small angles of attack. NASA TN D-5634.
and (1958). Finite cavity cascade flow. Proc. 3rd U.S. Nat. Cong. of Appl. Math., 837–845.
(1958) A general linearized theory for cavitating hydrofoils in nonsteady flow. Proc. 2nd ONR Symp. on Naval Hydrodynamics, 559–579.
(1916). Über die Kontraktion kreisförmiger Flüssigkeits-strahlen. Z. Math. Phys., 64, 34–61.Google Scholar
(1953). Steady two-dimensional cavity flows about slender bodies. David Taylor Model Basin Rep. 834.
(1964). Supercavitating flows—small perturbation theory. J. Ship Res., 7, No. 3, 16–37.Google Scholar
(1978). A partially cavitated hydrofoil of finite span. ASME J. Fluids Eng., 100, No. 3, 353–354.Google Scholar
(1987). The surface singularity method applied to partially cavitating hydrofoils. J. Ship Res., 31, No. 2, 107–124.Google Scholar
(1989). The surface singularity or boundary integral method applied to supercavitating hydrofoils. J. Ship Res., 33, No. 1, 16–20.Google Scholar
(1914). Sur la validité des solutions de certains problèmes d'hydrodynamique. J. Math. Pures Appl.(6), 10, 231–290.Google Scholar
(1967). Linearized theory of a partially cavitating cascade of flat plate hydrofoils. Appl. Sci. Res., 17, 169–188.Google Scholar
and (1966). Experimental observations on the flow past a planoconvex hydrofoil. ASME J. Basic Eng., 88, 273–283.Google Scholar
and (1967). Investigation of cavitating cascades. ASME J. Basic Eng., Series D, 89, 693–706.Google Scholar
and (1965). General formulation of a perturbation theory for unsteady cavity flows. ASME J. Basic Eng., 87, 1006–1010.Google Scholar
(1957). Aerodynamic forces on an oscillating aerofoil fitted with a spoiler. Proc. Roy. Soc. London, Series A, 239, 328–337.Google Scholar
(1951). A new relaxation treatment of flow with axial symmetry. Quart. J. Mech. Appl. Math., 4, 358–370.Google Scholar
(1961). The theory of subsonic plane flow. Cambridge Univ. Press.
and (1966). The theory of cascade of cavitating hydrofoils. Quart. J. Mech. Appl. Math., 19, 387–402.Google Scholar
(1956). A free streamline theory for two-dimensional fully cavitated hydrofoils. J. Math. Phys., 35, 236–265.Google Scholar
(1957). A linearized theory for nonsteady cavity flows. Calif. Inst. of Tech. Eng. Div. Rep. No. 85–6.
(1962). A wake model for free streamline flow theory, Part 1. Fully and partially developed wake flows and cavity flows past an oblique flat plate. J. Fluid Mech., 13, 161–181.Google Scholar
(1969). Cavity flow analysis; a review of the state of knowledge. In Cavitation State of Knowledge (eds: , ), ASME, N.Y.
and (1964a). A wake model for free streamline flow theory, Part 2. Cavity flows past obstacles of arbitrary profile. J. Fluid Mech., 18, 65–93.Google Scholar
and (1964b). An approximate numerical scheme for the theory of cavity flows past obstacles of arbitrary profile. ASME J. Basic Eng., 86, 556–560.Google Scholar
, , and (1971). Cavity-flow wall effect and correction rules. J. Fluid Mech., 49, 223–256.Google Scholar