A Green function approach for the investigation of the incompressible flow past an oscillatory thin hydrofoil including floor effects
© Răpeanu and Carabineanu; licensee Springer. 2014
Received: 21 November 2013
Accepted: 4 April 2014
Published: 8 May 2014
In the framework of the small perturbations theory, we study the incompressible inviscid flow of a uniform stream past an oscillatory/undulatory thin hydrofoil including floor effects. A Green function is used to deduce the integral equation for the jump of the pressure past the foil. The integral equation is numerically solved and the average drag coefficient is calculated. For some wings there appears a propulsive force and this force increases when the hydrofoil is close to the floor.
MSC: Primary 76B10; secondary 65R20; 45H99; 31A10.
Keywordsflexible hydrofoil hypersingular integral equation floor effect thrust
In the present paper we study the small-amplitude oscillatory/undulatory motion of an incompressible fluid past a thin flexible plate which performs prescribed oscillations in the presence of a wall (floor) with which it is parallel in its undisturbed state, and relative to which it is moving with constant speed. We shall limit the analysis to bodies large enough so that the Reynolds number is large. As is stated by Eloy et al. in , when the flexible surface has a typical speed of several body lengths per second, the flow can be considered irrotational, meaning that the flow vorticity is concentrated in thin boundary layers adjacent to the body surface and in a thin wake (vortex sheet) behind the body. Since the effects of viscosity manifest inside the thin boundary layers, we may treat the fluid as inviscid in the rest of the flow domain. Recalling Lagrange-Cauchy’s theorem which states that if the flow is potential in a certain configuration, it remains potential in every configuration arising from the initial one, we deduce that the theory of the unsteady motion of lifting wings as well as the theory of potential flow can be successfully utilized (see the papers of Carabineanu [2–6], Dowell and Hall , Dragoş , Homentcovschi [9, 10], Lighthill , Street , Taylor , Wu [14, 15], Watkins et al. ).
The periodic motion of a flexible foil is oscillatory if the foil or parts of it remain rigid during the motion. The undulatory motion involves a traveling wave down the foil (Street ). As we know from aerodynamics and hydrodynamics studies (Dragoş and Carabineanu [17, 18], Dragoş et al. ), the hydrodynamic coefficients of a hydrofoil are influenced by the presence of the floor. The aim of the paper is to predict the drag or the thrust enhancement generated by the presence of the floor. We employ like in [2–6, 8, 9] the linearized Euler equations for the incompressible flow. For taking into account the floor effect we use the Green function of the Laplacean for the Neumann problem in the half-space. We use the integral representation for the harmonic functions and the slipping condition to obtain an integral equation for the jump of the pressure over the hydrofoil. In order to discretize the integral equation, we split the kernel of the equation into several kernels for which we provide appropriate approximation formulas depending on the type of singularity of the kernel. Assuming that the hydrofoil is subjected to harmonic oscillations, we simplify the integral equation making it independent of time. By solving the discretized integral equation we calculate the jump of the pressure over the wing.
After obtaining the pressure field, the average drag is calculate by performing a numerical integration. We study an example of undulatory motion of the flexible thin delta wing. When the frequency surpasses a critical value, the drag becomes negative i.e. it appears a propulsive force. We notice that the distance between the wing and the floor influences the drag and the thrust.
The statement of the problem
where p is the pressure and is the constant density of the fluid. The aim of the present paper is to use the boundary conditions (7), (9), and the partial differential equations (10) for obtaining an integral equation for the jump of the pressure across . In order to ensure the uniqueness of the solution we shall impose a certain periodic in time behavior to the unknowns.
The Green function. The integral equation of the problem
In (27) may have complex values. By convention means the real part of . For the sake of simplicity, we shall calculate (as is usual in the oscillatory hydrofoil theory) complex values for the jump of pressure and then we shall consider the real part.
We discretize the hypersingular integral equation (32) in order to solve it numerically. In Appendix A we split the kernel into several kernels and describe the type of singularity for each one. In Appendix B, depending on the kind of singularity, we deliver appropriate approximation formulas. In order to ensure the uniqueness of the solution, we impose a certain behavior of the pressure jump in the vicinity of the leading edge.
The propulsive force. Numerical results. Floor effects
Appendix 1: The singularities of the kernel of the integral equation
into several kernels in order to show the kind of singularities we are dealing with and to find afterwards the most convenient approximation formulas.
The kernels and have strong singularities of order . The kernel has a polar singularity. The kernels and have integrable logarithmic singularities. Taking into account the series expansions of , , , , , we may easily prove that the kernels , , , , , have no singularity and they are continuous functions. , , are also continuous functions. We notice that for .
Appendix B: The discretization of the integral equation
- Eloy C, Doaré O, Duchemin L, Scouveiller L: A unified introduction to fluid mechanics of flying or swimming at high Reynolds number. Exp. Mech. 2009. 10.1007/s11340-009-9289-7
- Carabineanu A: Incompressible flow past oscillatory wings of low aspect ratio by the integral equations method. Int. J. Numer. Methods Eng. 1999, 45: 1187-1201. 10.1002/(SICI)1097-0207(19990730)45:9<1187::AID-NME625>3.0.CO;2-ZMathSciNetView Article
- Carabineanu A: Self-propulsion of oscillating wings in incompressible flow. Int. J. Numer. Methods Fluids 2008, 56: 1-21. 10.1002/fld.1513MathSciNetView Article
- Carabineanu A: Self-propulsion of aquatic animals by undulatory or oscillatory motion. Series on Mathematical Modelling of Environmental and Life Sciences Problems. In Proc. 3rd Workshop on Math. Modeling of Environmental and Life Sciences. Edited by: Stelian I, Marinoschi G. Ed. Acad. Romane, Bucharest; 2004:131-145. Constanta, May 2004
- Carabineanu A: Self-propulsion of an oscillatory wing including ground effects. Series on Mathematical Modelling of Environmental and Life Sciences Problems. In Proc. 4th Workshop on Math. Modeling of Environmental and Life Sciences. Edited by: Stelian I, Marinoschi G, Popa C. Ed. Acad. Romane, Bucharest; 2004:39-54. Constanta, September 2005
- Carabineanu A: Self-propulsion of an oscillatory wing including tunnel effects. An. Univ. Bucur., Mat. 2006, LV: 5-26.MathSciNet
- Dowell EH, Hall KC: Modeling of fluid-structure interaction. Annu. Rev. Fluid Mech. 2001, 33: 445-490. 10.1146/annurev.fluid.33.1.445View Article
- Dragoş L: The theory of oscillating thick wings in subsonic flow. Lifting line theory. Acta Mech. 1985, 54: 221-238. 10.1007/BF01184848View Article
- Homentcovschi D: Theory of the lifting surface in unsteady motion in an inviscid fluid. Acta Mech. 1977, 27: 205-216. 10.1007/BF01180086View Article
- Homentcovschi D: Determination of the vortex associated with the motion of a body in an ideal fluid. Mech. Res. Commun. 1976, 3(3):191-195. 10.1016/0093-6413(76)90009-4MathSciNetView Article
- Lighthill J: Mathematical Biofluiddynamics. SIAM, Philadelphia; 1987.
- Street, A: Preliminary Finite Element Modeling of a Piezometric Actuated Marin Propulsion Fish. Thesis, Department of Mechanical Engineering, Rochester Institute of Technology, Rocester, New York (2006)
- Taylor GI: Analysis of the swimming of long and narrow animals. Proc. R. Soc. Lond. Ser. A 1952, 214(117):158-183.View Article
- Wu TY: Mathematical biofluiddynamics and mechanophysiology of fish locomotion. Math. Methods Appl. Sci. 2001, 24: 1541-1564. 10.1002/mma.218MathSciNetView Article
- Wu TY: On theoretical modeling of aquatic and aerial animal locomotion. 38. In Advances in Applied Mechanics. Edited by: Giessen E, Wu TY. Elsevier, Amsterdam; 2002:291-353.
- Watkins, CE, Runyan, HL, Woolston, DS: On the Kernel function of the integral equation relating the lift and downwash distributions of oscillating finite wings in subsonic flow. NACA T.R. 1234 (1955)
- Dragoş L, Carabineanu A: A numerical solution for the equation of the lifting surface in ground effects. Commun. Numer. Methods Eng. 2002, 18: 177-187. 10.1002/cnm.481View Article
- Dragoş L, Carabineanu A: The supersonic flow past a thin profile including ground and tunnel effects. Z. Angew. Math. Mech. 2002, 82: 649-652. 10.1002/1521-4001(200209)82:9<649::AID-ZAMM649>3.0.CO;2-2View Article
- Dragoş L, Carabineanu A, Dumitrache R: A numerical solution for the equation of the lifting line including ground and tunnel effects. Proc. Rom. Acad., Ser. A: Math. Phys. Tech. Sci. Inf. Sci. 2008, 9(3):1-7.
- Ditkine V, Proudnikov A: Transformations intégrales et calcul opérationnel. Mir, Moscow; 1978.
- Fox C: A generalisation of the Cauchy principal value. Can. J. Math. 1957, 9: 110-115. 10.4153/CJM-1957-015-1View Article
- Dumitrescu DF: Three methods for solving Prandtl’s equation. Z. Angew. Math. Mech. 1996, 76(6):1-4.
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