Large deflection of a supercavitating hydrofoil

LARGE DEFLECTION OF A SUPERCAVITATING HYDROFOIL

Yuri AntipovDepartment of Mathematics Louisiana State UniversityBaton Rouge, Louisiana

Singapore, August 16, 2012

OUTLINE

1. A supercavitating curvilinear elastic hydrofoil: Tulin’s type model

2. Solution for a thin circular elastic hydrofoil3. Nonlinear model on large deflection of an elastic foil 4. Viscous effects: a boundary layer model

A SUPERCAVITATING ELASTIC HYDROFOIL

TULIN’S SINGLE-SPIRAL-VORTEX CLOSURE MODEL

POTENTIAL THEORY MODEL

ELASTIC DEFORMATION MODEL: SHELL THEORY

LARGE DEFLECTION OF A BEAM: BARTEN-BISSHOPP-DRUCKER MODEL

GENERALIZATION OF THE BARTEN-BISSHOPP-DRUCKER MODEL

ARBITRARY LOAD AND RIGIDITY (CONT.)

ARBITRARY LOAD AND RIGIDITY (CONT.)

NON-LINEARITY OF THE COUPLED FLUID-STRUCTURE INTERACTION PROBLEM

A RIGID POLYGONAL SUPERCAVITATING HYDROFOIL

NUMERICAL RESULTS FOR A RIGID POLYGONAL FOIL ZEMLYANOVA & ANTIPOV (SIAM J APPL MATH, 2012)

METHOD OF SUCCESSIVE APPROXIMATIONS

DISPLACEMENTS, PRESSURE, FOIL PROFILE

VISCOUS EFFECTS: BOUNDARY LAYER MODEL

KARMAN-POHLHAUSEN METHOD

KARMAN-POHLHAUSEN METHOD (CONT.)

BOUNDARY LAYER ON THE CAVITY

CONCLUSIONS• The Tulin single-spiral-vortex model has been employed to

describe supercavitating flow past an elastic hydrofoil• The nonlinear equation of large deflection of an elastic

beam (‘elastica’) has been solved exactly in terms of elliptic functions

• The method of conformal mappings and the Riemann-Hilbert formalism have been used to solve the cavitation problem in closed form

• The fluid-structure interaction problem has been solved by the method of successive approximations

• The Prandtl boundary layer equations and the Karman-Pohlhausen method have been applied to derive a nonlinear first-order ODE for the shearing stress on the foil. On the cavity boundary, the shearing stress has been found explicitly