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.
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
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