Hydrodynamics of ship propellers- BRESLIN E POUL, Notas de estudo de Engenharia Naval

Baixe Hydrodynamics of ship propellers- BRESLIN E POUL e outras Notas de estudo em PDF para Engenharia Naval, somente na Docsity! Cambridge Ocean Technology Series 1. Faltinsen: Sea Loads on Ships and Offshore Structures 2. Burcher & Rydil1: Concepts in Submarine Design 3. Breslin & Andersen: Hydrodynamics of Ship Propellers John P. Breslin Professor Emeritus, Department of Ocean Engineering, Stevens Institute of Technology and Poul Andersen Department of Ocean Engineering, The Technical University of Denmark PUBUSHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pin Building, Trumpingron Street, Cambridge CB2 1RP, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, United Kingdom 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia © Cambridge University Press 1994 This book is in copyright. Subject to stamtory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the wrinen permission of Cambridge University Press First published 1994 Reprinted 1996 First paperback edition 1996 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Breslin, John P. Hydrodynamics of ship propellers I John P. Breslin, Poul Andersen. p. cm. - (Cambridge ocean technology series; 3) Includes bibliographical references and index. 1. Propellers. 2. Ships-Hydrodynamics. 1. Andersen, Poul, 1951- . II. Tide. III. Series. VM753.B6S 1993 623.S·73-dc20 93-26511 CIP ISBN 0521413605 hardback ISBN 0 521 574706 paperback Contents Preface xi Notation xiv Abbreviations xxiv 1 Brief review of basic hydrodynamic theory 1 Continuity 1 Equations of motion 2 Velocity fields induced by basic singularities 7 Vorticity 17 2 Properties of distributions of singularities 26 Planar distributions in two dimensions 26 Non-planar and planar distributions in three dimensions 33 3 Kinematic boundary conditions 42 4 Steady flows about thin, symmetrical sections in two dimensions 46 The ogival section 51 The elliptical section 54 Generalization to approximate formulae for families of two-dimensional hydrofoils 57 A brief look at three-dimensional effects 62 5 Pressure distributions and lift on flat and cambered sections at small angles of attack 66 The flat plate 66 Cambered sections 74 6 Design of hydrofoil sections 86 Application of linearized theory 87 Application of non-linear theory 103 7 Real fluid effects and comparisons of theoretically and experimentally determined characteristics 111 Phenomenological aspects of viscous flows 111 Experimental characteristics of wing sections and comparisons with theory 117 vii xii Preface hope that the book in this form will be equally suitable as a text in uni- versity courses, a guide for self-tuition and a reference book in ship-design offices. The subject matter is broadly divided into two parts. In the first, basic hydrodynamics is outlined with comprehensive applications to the con- struction of practical representations of the steady performance of hydro- foils, with and without cavitation, wings and propellers. Here lifting-line theory is described, including propeller design and analysis via computer and pragmatic considerations from actual performance. The last part ad- dresses the unsteady forces on propellers in wakes via lifting-surface theory as well as propeller-induced vibratory forces on simple, nearby boundaries and upon ship hulls. Both non-eavitating and cavitating pro- pellers are treated. In the final chapter a rational procedure for the optim- ization of compound propulsors for increased efficiency is described. Throughout the book, in addition to the theoretical developments, the results of calculations are correlated with experimental findings. Remarks and developments that the reader may wish to skip in his first reading are set in small print. No exercises are provided; to achieve proficiency, the reader, after initial study of the text, should derive the results indepen- dently. An immense pleasure, when writing this book, has been to experience the interest and help from colleagues, institutions and companies all over the world. They generously spent their time answering our questions and supplied us with material, including photographs and figures, with permis- sion to reproduce them in the text. These sources are acknowledged in the figure captions. We are very grateful for this assistance without which this book would have been much more incomplete and less useful. We are particular indebted to Dr. W. van Gent, Maritime Research Institute Netherlands; Professor M. D. Greenberg, University of Delaware; Mr. C.- A. Johnsson, SSPA Maritime Consulting AB; Professor J. E. Kerwin and Dr. S. A. Kinnas, Massachusetts Institute of Technology. Our sincere thanks are also due to Mr. J. H. McCarthy, David Taylor Research Center; Dr. K. Meyne, Ostermann Metallwerke; Dr. W. B. Morgan, David Taylor Research Center; Mr. P. Bak Olesen, A.P. M011er;and Mr. H. Vagi, Mitsui Engineering and Shipbuilding Co., Ltd. for help and suppoit and to Professor R. Eatock Taylor, Oxford University, for his effective proposal of our manuscript to Cambridge University Press. We also wish to express our gratitude to present and former colleagues at the Depart- ment of Ocean Engineering, The Technical University of Denmark. They include Professor Emeritus Sv. Aa. Harvald and Professor P. Terndrup Pedersen who initiated vital parts of the entire process and later together with Professor J. Juncher Jensen, Head of Department, gave us encour- Preface 'xiii agement and support. Invaluable help was provided by the Staff; Ms. L. Flicker typed the lecture-notes version of the manuscript and later ver- sions were typed by Ms. V. Jensen. We acknowledge the financial support of F. L. Smidth & Co. A/S who, on the occasion of their lOOth-year anniversary, sponsored the first au- thor's stay as visiting professor. Later support was provided by The Dan- ish Technical Research Council under their Marine Design Programme. Lyngby, Denmark John P. Breslin October 1992 Poul Andersen Notation The following list of symbols is provided partly as an aid to the reader who wants to use this text as a reference book and read selected chapters. The list contains mainly globally used symbols while many other symbols, including those distinguished by subscript, are defined locally. The nota- tion is not entirely consistent, symbols being used with different defini- tions, however, rarely in the same sections. Practical usage has been given priority. For this reason ITTC notation has only been partly used. The coordinate systems are as follows: For two-dimensional flows the x- axis is horizontal, generally displayed in figures as pointing to the right, with the y-axis vertical and positive upwards. Incoming flow is along the x-axis but opposite in direction. For three-dimensional flows the x-axis is horizontal, with a few exceptions coinciding with the propeller axis and generally displayed in figures as pointing to the right. The y-axis is also horizontal, pointing to port and the z-axis is vertical, pointing upwards. As in the two-dimensional case the incoming flow is along the x-axis but opposite in direction. Moreover, a cylindrical system is used. Its x-axis coincides with that of the cartesian system while the angle is measured from the vertical (z-axis), positive in the direction of rotation of a right- handed propeller. For the two-dimensional case this orientation of axes is in contrast to that used by aerodynamicists (who take the incoming flow along the pos- itive x-axis). However, it is consistent with the three-dimensional defini- tion as well as with the long tradition in naval architecture that the ship is viewed from starboard and the bow consequently is to the right hand. ~~~n ~ xvi Notation Fr Fy, F; fa, fes fe G g g H(x) HÉDA, É) h by, h dg, he Ta(k) Im E Lbk da dr, de ERRO force axial, radial and tangential force components velocity Froude number = >D"— ] length x», y- and z-directed force components camber (maximum) of section axial, radial and tangential partialforce operators Green function acceleration due to gravity acceleration due to gravity vector Heaviside step function Hankel functions of the first and second kind of order n helicoidal coordinate of helicoidal-normal coordinate sys- tem (hn) lengths to leading and trailing edges in helicoidal coor- dinates helicoidal coordinates of cavity leading and trailing edges modified Bessel function of the first kind of order n imaginary part of integral direction vectors in x-, y- and z-directions direction vectors in axial, radial and tangential diree- tions axial, radial and tangential induction factors U advance ratio, free stream = —— ND : . E UN advance ratio, behind ship = A ND Bessel function of the first kind of order n integral —seei, j k Notation Le Lr ja Ma My, Mz z2z EB xvii constant in inverted airfoil equation F force coefficient = NaDã : M moment coefficient = IDE : Fy normalorce coefficient = apa : T thrust coefficient = aiia torque coefficient = um pN?D5 modified Bessel function of the second kind of order n : Pp pressure coefficient. = mê we teduced frequency = 20 —seei,j, k lift Hft due to cavitation ft of wing Lagrange multiplier, or inverse of cavity length moment source strength moment of section about quarter-chord point x-, y- and z-directed moment components source strength, distribution cavity source strength, distribution normal force rate of revolutions = pj 8 xx Vtr) Ve Vi w(t) XI, 7 Xyz XI, Xe Xr Ya(k) yr Notation volume resulting inflow to blade section due to translation and rotation cavity volume on propeller blade critical or (cavitation-) inception speed —seeu, v, w wake fraction — see also u, v, w circumferentially mean wake fraetion at radius r cylindrical coordinates. x-axis positive forward co inciding generally with propeller axis. y measured from vertical, positive in direction of rotation of right-handed propeller. cartesian coordinates for two-dimensional flow. x-axis positive to the right in negative inflow direction. y-axis vertical and upwards. Cartesian coordinates. x-axis positive forward coinciding generally with propelier axis and shown in figures to the tight. y-axis positive to port and z-axis vertical and up- wards. position of aerodynamic center leading- and trailing-edge coordinates, corresponding to helicoidal coordinates h; and ht take rake induced by skew Bessel function of the second kind of order n —seex,y andx,y, z camber ordinate argument of associated Legendre function number of propeller blades — see also x, y and x, y, Z complex variable =x+iy Notation w v- Yx &x) bmom 8T, 89 Eq angle of incidence or attack ideal angle of attack induced angle angle of zero lift fluid pitch angle fluid pitch angle, mean inflow iluid pitch angle, local inflow induced pitch angle blade pitch angle circulation Gamma function angle, — see also x, r, 7 U tan A wr H wr n P tanl — 271 strength of vortex distribution (vortex density) blade-position angle, key blade blade-position angle, »th blade gradient operator Laplace operator divergence operator eurl operator Dirac delta-funetion boundary-layer momentum thickness variation of T, Q, etc. parameter, =1forq=0,=2forqjo complex variable — see also € vorticity vector, = £i + nj + (k = ut 2av =2t% Zz =t+ig xxii Notation " n " ideal Hu v ? z Ee, By E o 9, O PN Va Ox, Oy, Ox IR 7 8,6 Pe és by Y cavity ordinate J Kp propeller efficiency =D 27 Kq — see also 6, € ideal efficiency dynamic viscosity blade index number, = z-1.v=0» key blade kinematic viscosity —see 6 6 mass density of water smali surface region strengths of x-, y- and z-directed point dipoles tati P- Pe cavitation number = aU? strength of normaliy directed dipoles p ati .P—Py cavitation number = Tap? nei Do Py cavitation number = Luz 2oUA strengths of x-, y- and z-directed distributed dipoles D—Pv cavitation number = IvioiRP semi-thickness, body, section or blade section velocity potentials cavity velocity potential source velocity potential vortex velocity potential local angle for point on blade relative to blade-position angle %, (Stokes') stream function Notation xxiii Y velocity potential of j-th motion A skew angle [3 angular velocity of propeller = 23N & dimensionless form of a a mean value of à & amplitude of a x, x" dummy variable; dummy-point coordinate in distinction to field-point coordinate x t derivative of function f with respect to argument D , x substantive derivative f Cauchy principal-value integral f finite-part integral (p) az qz-+h amplitude function of p (gZ-th blade frequency) Vector symbols and operators are printed boldfaced; the NACA mean-line designation, a, is the only other symbol so printed. Abbreviations ATTC American Towing Tank Conference CETENA Centro per gli Studi di Tecnica Navale DTMB David Taylor Model Basin - later DTRC DTRC David Taylor Research Center HSVA Hamburgische Schiffbau-Versuchsanstalt INA Institution of Naval Architects - later RINA ITTC International Towing Tank Conference L.E. leading edge MARIN Maritime Research Institute Netherlands MIT Massachusetts Institute of Technology NACA National Advisory Committee for Aeronautics -later NASA NASA National Air and Space Administration NPL National Physical Laboratory PUF Propeller Unsteady Force (MIT computer program) RINA The Royal Institution of Naval Architects SNAME The Society of Naval Architects and Marine Engineers SSPA SSPA Maritime Consulting AB - (SSPA: Statens Skeppsprovningsanstalt) T .E. trailing edge TMB Taylor Model Basin - later DTRC VWS Versuchsanstalt fUr Wasserbau und Schiffbau xxiv 1 Brief Review of Basic Hydrodynamic Theory An extensive, highly mathematical literature exists dealing with fluid- mechanical aspects of ship propellers. Invariably, the mathematical developments are only outlined, impeding easy comprehension even by knowledgeable readers. Our aim is to eluci- date the mathematical theory in much greater detail than is generally available in extant papers. In this context, the first three chapters are provided as aids for those who have not had extensive practice in the ap- plication of classical hydrodynamical theory to flows induced in fluids by the motions of bodies. The fluid of interest is water which is taken to be incompressible and inviscid. Modifications arising from viscosity are de- scribed in a later chapter (Chapter 7) through reference to experimental observations. This review begins with the derivation of the concept of continuity or conservation of mass at all points in sourceless flow and proceeds to the development of the Euler equations of motion. In the restricted but important class of irrotational motions (zero vorticity) Laplace's equation for the velocity potential is obtained. The remainder of this chapter is devoted to derivations of fundamental solutions of Laplace's equation in two and three dimensions. It is emphasized that these first two chapters are necessarily limited in scope, being directed to our needs in subsequent chapters. There are many excellent books which should be consulted for those seeking greater depth and broader description of hydrodynamic theory. Among these we suggest Batchelor (1967), Lamb (1963), Lighthill (1986), Milne-Thomson (1955), and Yih (1988), and Newman (1977) for modern applications. CONTINUITY Consider a general, three-dimensional flow field whose vector velocity is defined by q = iu(x,y,z,t) + jv(x,y,z,t) + kw(x,y,z,t) (Ll) for an incompressible fluid. 6 Brief Review of Basic Hydrodynamic Theory Velocity Fields Induced by Basic Singularities 7 8 Brief Review of Basic Hydrodynamic Theory Velocity Fields Induced by Basic Singularities 9 10 Brief Review of Basic Hydrodynamic Theory Velocity Fields Induced by Basic Singularities 11 16 Brief Review of Basic Hydrodynamic Theory Expanding by the binomial expansion theorem 1 f 1[ 2ex ofe? =0>— 1-|0"" 4 . mrel aera)t A) (1.65) Put this in (1.64), then limg=d= eo tim ! Los, O(e?) — 4TITEDHD——— + Of eso fr Je +2 f2+r pes rr? | Defining the dipole moment strength by E = Me (1.66) we obtain Byx fa =— (1.67) 4m [x2 + 2/2 as the potential of an x-directed dipole at the origin with axis in the positive x-direction. This can be achieved directly by differentiation of the source. To secure the positively directed x-dipole at the origin Place a source at x = x,y = z=0. Then x =0 ô M 1 ga=— | d = | 47 (x — 2 = where the derivative is evaluated at x! = 0 a, ter differentiation. Then a [3 2x -x)(-1) | êrlo2 x- x)2+ 2B2 luco Then formally replacing M by 5, we have (1.67). In general to achieve à dipole whose axis has the direction cosines Ty, Ny, Ma at any point (x',y!2") apply the following operation & z fr à 4 3 N ô ] 1 “a go tia tu lDDW > >————— 4rl O dy de GP + Gy) + (a-29% (1.68) Vorticity where Je = n«5, Ey = ny, E, = 1,5 Dipoles may be distributed on lines and surfaces in the same manner as Sources. VORTICITY We have seen previously that by manipulating the equations of motion (Equations (1.8) to (1.14)), terms arise which, together, are identified as components of a so-called vorticity vector C=G+m+cg the components of which are given by velocity gradients, Le, = ()>(y) ôw dy du Ow Com E 2, 1.69 % & (1.69) ei» O eae 6) > 6) > (8) Here we see that the operations are in eyclic order and the numerators are in acyelic or reverse alphabetic order. To give a more physically based exposition of the rôle of vorticity we can examine the excursions in the velocity components between two neighboring points (x,y,2) and (x + dx, y + &y, 2 + 62). Thus the changes in the velocities over these infinitely small distances are formally du dd éu = gx =6y + —dz ao ta ddr O =Dk+by + —k 1.70) + ao + (1.70) ôw ôw dm du = Se + by + gg dx “dy dz Our previous experience may perhaps suggest the formation of an array or matrix and that some insight might result if this array is made sym- metric, in part, about the diagonal as indicated in (1.70). Taking the first equation in (1.70) and observing the coefficients of &x in the second and third equations we add and subtract terms in the first equation as follows 18 Brief Review of Basic Hydrodynamic Theory > +4-— du du 1 du 1ôw lv 1ôm bt sogoe sei] du= x + = +-— — — + Ox ô 28x 2x 2 dx The vorticity components can be obtained by the following alteration and analogous manipulations of the remaining two equations of (1.70) to give 2 $ A du, Alda 1(du dl Alô de 1[ôr du "= à “Color a) aloe * ax] “ola a “alô dy, pl[ôa E à a J[ôs Da) pel à “Cata niatajto SC 858 1fôs dy 1fôw O de a t =D + Olga, O Ea - & it los 6 nos Sê Thus the motion of a small element is composed of three parts: i. Translation as a whole with components u,v,w. ii. Straining as given by the first three terms in each row of the array. ii, Rotations of the element as given by the last two terms in each equation; the components of angular velocity being €/2, 1/2, efa (termed 'mean rotations' by Cauchy in 1841, (cf. Lamb (1963)). Although we limit our attention to flows in which the vorticity is gen- erally zero we shall find it useful to represent lifting surfaces by isolated vorticity on the body and into the wake. Tt is to be noted that the vorticity in analogy with the velocity field is divergence-free, i.e., dé nd La mi wta 0 (1.72) as may be seen by carrying out this operation on (1.69). To calculate the velocity field qv induced by any given distribution of vorticity in three (or two) dimensions we may proceed as follows: We have seen that the velocity field qv and vorticity vector € are related by Teq=e (1.73) Vorticity 19 Assume that there exists a vector potential By such that VíB=q (1.74) Thus qy is mutually perpendicular to the vector operator Y and the vector potential Bo. Then the field equation for By is got by replacing qv in (1.73) i.e., VaVrB=( (1.75) The vector triple product is equivalent to V(T-By)— WB, =4 and if we can assume for the present that the divergence of B, vanishes everywhere i.e., V-B,=0 (1.76) then the field equation for By is VB, = — Gxys2) (um) which is an inhomogeneous Laplace Equation named after Poisson The solution of this equation is known to be (cf. Milne“Thomson (1955)) B= | Sor) my (1.78) EL R where Rx prg-yprte-? and Syoav ts à volume integral (dV' = dx'dy'dz!) throughout the entire volume in which the vorticity is distributed. This is essentially a volumetric dis- tribution of sources of vector strength density € over all dummy points (xtyt2). With hindsight we can see that (1.78) is a solution of (1.77) because of the previ- q observed behavior of the Laplacian of the source potential 1/R (see Equation 1.57), ie. pasar des) dy =) &e —2) 40 that the $functions upon integration pick out Catx =x, y'=y,2!=2 20 Brief Review of Basic Hydrodynamic Theory To demonstrate the requirement which must be met to ensure compli- ance with our assumption (1.76) we can operate on (1.78) with veis =D A xy to get vB,= LF etega)-v Day (1.79) a jy Ca . Now as R involves (x,y,2) and (x!,y',2!) as differences respectively, 1 1 Po=-Vos, v= Then 1 c 1 py D=q SS) dy. vs [É] Rs (1.80) But the last term is zero (see (1.72)) and hence 1 é v.B=-— a a y = lr z ] dv (1.81) which by Gauss! theorem (see Mathematical Compendium, Section 3, Equation (M3.1), p. 505) yields «nº 7 B= IS dSt ; nº is normal to surface S (1,82) 4r]s R Thus V-B, will be zero if the vorticity vector at the bounding surface is normal to the surface i.e., $n = 0. This will always be the case in all applications considered herein. We may now utilize (1.78) and (1.74) to arrive at a formula for the velocity induced by an arbitrary distribution of vorticity. Thus APM = lr av (1.83) which is equivalent to Vorticity 21 = av! (1.84) 1 I Sxy'z)aR v RB The practical application of this is to vorticity distributions along lines and surfaces. In line distributions ( is zero everywhere except along the line and hence the volume integral reduces to à line integral which must de closed or end normal to boundaries. The vorticity vector is everywhere tangent to the curve and is positive in the direction which produces circulation about the line element in the sense of the right-hand rule for rotation or moment vectors. Figure 1.10 Velocity induced in point P by differential line element ds in Q. The local induced velocity at P due to (ds is mutually perpendicular to R and in the direction obtained by the Tight-hand rule obtained by rotating ( into R. The velocity at P must be obtained by integrating completely around the vortex filament. If the vortex extends to infinity then it may be considered to be closed by adding a cireuit at infinite radius which closes the curve but makes no contribution. Thus for vorticity distributed over a closed curve, cf. Figure 1.10 Iva tes) 4 q = | re re as (1.85) 47), R$ This formula is called the Biot-Savart law and was first deduced experi- mentally by Biot and Savart in 1820 (cf. Yih (1988)) as the magnetic vector induced by a steady current flowing in a closed conducting wire. To apply this and recapture the two-dimensional formula deduced earlier consider vorticity of strength I distributed along the 2-axis from —oo to 90. Then ds = de, x =y'=0,(=0i+0j+TkandR=xi+yjs (z-2")k. Then Planar Distributions in Two Dimensions 27 2 Properties of Distributions of Singularities In this chapter we determine the basic behavior of the velocity fields of the various singular solutions of Laplace's equation when they are dis- tributed or "smeared" along lines and surfaces of finite extent. Their properties are particularly important on the lines and surfaces as they will be repeatedly used to generate approximate flow envelopes about thin hydrofoil sections in two dimensions and about slender bodies and lifting surfaces in three space dimensions. PLANAR DISTRIBUTIONS IN TWO DIMENSIONS Source Distributions Source distributions are useful in generating section shapes symmetrical about the long axis. It is therefore important to understand the connec- tion between the source density and the velocity components induced by the entire distribution of sources. From the foregoing, the potential of a line of sources in two dimensions is from (1.35) 28 Properties of Distributions of Singularities Planar Distributions in Two Dimensions 29 30 Properties of Distributions of Singularities Planar Distributions in Two Dimensions 31 36 Properties of Distributions of Singularities Non-Planar and Planar Distributions in Three Dimensions 37 38 Properties of Distributions of Singularities Non-Planar and Planar Distributions in Three Dimensions 39 40 Properties of Distributions of Singularities Non-Planar and Planar Distributions in Three Dimensions 41 4 Steady Flows About Thin, Symmetrical Sections in Two Dimensions Steady Flows About Thin, Symmetrical Sections in 2-D 47 48 Steady Flows About Thin, Symmetrical Sections in 2-D Steady Flows About Thin, Symmetrical Sections in 2-D 49 50 Steady Flows About Thin, Symmetrical Sections in 2-D The Ogival Section 51 56 Steady Flows About Thin, Symmetrical Sections in 2-D Approximate Formulae for Families of 2-D Hydrofoils 57 58 Steady Flows About Thin, Symmetrical Sections in 2-D Thus the curvature imposed by the thickness distribution is a dominant mechanism in producing the minimum pressure. Indeed, when the curva- ture is constant as in the case of the ogive, then 7"(0) = -2i, and as all higher derivatives are zero, we recapture our previous result, Cpmin = - (8/7r)i. However, this analysis is far from complete because it fails to give a correct leading term in the case of an ellipse. We can only conclude that for sections having blunt leading edges the first order theory is inadequate and moreover we cannot expect to capture the effect of the distribution of section slopes by a Taylor expansion about Xm·Indeed the linearized theory for an ellipse as given by the integral in (4.30) suggests strong contributions from the leading and trailing edges by the presence of the weighting factor 1/ ~ 1 - x'2 which becomes square-root singular at x' =±1. Approximate Formulae for Families of 2-D Hydrofoils 59 7 = (1 - xn)l/n Freeman (1942)4 has shown that for n < 2 (less blunt than an ellipse) the minimum pressure occurs at midlength but is always more negative than for the ellipse. For n > 2 (more blunt than an ellipse) the minimum pressure occurs very close to the leading edge and is again more negative than that for an ellipse. These results are displayed in Figure 4.11. Data from measurements made in the NACA (now NASA) windtunnels show the effect of leading-edge curvature and suggests that a curvature slightly greater than that at the nose of an ellipse yields a small improvement over the ellipse. See Figures 4.12 and 4.13. For our purposes, the ellipse will be taken as the best possible (i.e., having the least negative Cpmin). 60 Steady Flows About Thin, Symmetrical Sections in 2-D and B is similarly some definite integral of the distribution of section shape. To test this hypothesis we have plotted results from exact numeri- cal calculations for families of sections as generally given in Abbott & von Doenhoff (1959). A tabulation of the coefficients (from Breslin & Land- weber (1961)) is given Table 4.1 and a graphical comparison is provided in Figure 4.14. It is clear from this figure that the assumed relation (4.39) fits the results from exact evaluations over the range of thickness ratios 0 < t ~ 0.3. The elliptical section is the best but of course is not suitable Approximate Formulae for Families of 2-D Hydrofoils 61 Figure 4.15 Examples of several thickness distributions. From: Breslin, J.P. & Landweber, L. (1961). A manual for calculation of inception of cavitation on two- and three-tiimensional forms. SNAME T&R Bulletin, no. 1-21. New York, N.Y.: SNAME. Copyright © (1961) SNAME. All rights reserved. By courtesy of SNAME, USA. The Flat Plate 67 5 Pressure Distributions and Lift on Flat and Cambered Sections at Small Angles of Attack The non-symmetrical flow generated by flat and cambered laminae at angles of attack is at first modelled by vorticity distributions via classical linearized theory. Here, in contrast to the analysis of symmetrical sec- tions, we encounter integral equations in the determination of the vorti- city density because the local transverse component of flow at anyone point depends upon the integrated or accumulated contributions of all other elements of the distribution. Pressure distributions at non-ideal incidence yield a square-root-type infinity at the leading edge because of the approximations of first order theory. Lighthill's (1951) leading edge correction is applied to give realistic pressure minima at non-ideal angles of incidence. Our interest in pressure minima of sections is due to our concern for cavitation which can occur when the total or absolute pressure is reduced to the vapor pressure of the liquid at the ambient temperature. Since cavitation may cause erosion and noise it should be avoided or at least mitigated which may possibly be done by keeping the minimum pressure above the vapor pressure. This corresponds to maintaining the (negative) minimum-pressure coefficient Cpminhigher than the negative of the cav- itation index. At this point we shall not go deeper into the details of cavitation which is postponed until Chapter 8. Instead we shall continue our theoretical development with flat and cambered sections. THE FLAT PLATE We now seek the pressure distributions and the lift on sections having zero thickness but being cambered and, in general, set at any arbitrary (but small) angle of attack to the free stream, -D. Consider a flat plate at small angle a. Then we might envisage a flow pattern as sketched in Figure 5.1 as a plausible one with stagnation points Sand S', S being on the lower side and S' on the upper side. The flow around the leading edge is strongly curved (very small radius of curvature) resulting in high local velocity and hence (by Bernoulli) develops very low pressure on the upper side, par- ticularly near the leading edge. (In a real fluid the flow about such a 66 68 Pressure Distributions on Flat and Cambered Sections The Flat Plate 69 70 Pressure Distributions on Flat and Cambered Sections The Flat Plate 71 76 Pressure Distributions on Flat and Cambered Sections Cambered Sections 77 78 Pressure Distributions on Flat and Cambered Sections Cambered Sections 79 80 Pressure Distributions on Flat and Cambered Sections Cambered Sections 81 6 Design of Hydrofoil Sections Criteria for the design of blade sections may be selected to include: i. Minimum thickness and chord to meet strength requirements; ii. Sufficient camber to generate the design lift; iii. Distribution of thickness and camber to yield the least negative pressure coefficient to avoid or mitigate cavitation; iv. Thickness- and loading-pressure distributions to avoid boundary layer separation with least chord to yield minimum drag consist- ent with requirements i. and iii.; v. Leading and trailing edges to satisfy strength and manufacturing requirements. The first part of this chapter follows from linearized theories developed by aerodynamicists more than 50 years ago, placing emphasis on the use of existing camber and thickness distributions yielding least negative minimum pressure coefficients, Cprninat ideal angle of attack. At non- ideal angles (which always occur in operation in the spatially and tem- porally varying hull wake flows) we are required to seek sections having greatest tolerance to angle deviations and at the same time having nega- tive minimum pressure coefficients exceeding the level that indicates occurence of cavitation. This tolerance depends critically upon the leading edge radius and the forebody shape as well as upon the extent of the flat part of the pressure distribution. Thus we are led to the more recent findings of researchers who have developed profiles having greater toler- ance to angle of attack. When cavitation is unavoidable the latest ap- proach is to use blunter leading edges to generate shorter, more stable cavities thereby avoiding "cloud" cavitation which causes highly deleteri- ous erosion or pitting of the blades. The older procedures are treated next under the heading of Applications of Linearized Theory and the modern developments are described in the section entitled Application of Non- Linear Theory, p. 103 and sequel. 86 Application of Linearized Theory 87 Since propeller blade sections operate in real water, cavitation and friction must be taken into account in an actual design. The authors have chosen, however, to highlight the application of the theories developed in the preceding chapters without letting the real fluid effects obscure the development. Such effects are pursued later in Chapter 7 (Real Fluid Effects) and Chapter 8 (Cavitation), but are dealt with briefly in the present chapter in the form of comments in the text. This applies in particular to the section on Application of Non-Linear Theory since such applications have been devised to include more real fluid effects to provide superior designs. Readers who find this order awkward may prefer to read Chapters 7 and 8 prior to reading this chapter. 88 Design of Hydrofoil Sections Application of Linearized Theory 89 90 Design of Hydrofoil Sections Application of Linearized Theory 91 96 Design of Hydrofoil Sections which are the relations imposed by the kinematic conditions found earlier when we considered the symmetrical and asymmetrical flows about sec- tions separately (d. Equations (5.34) and (4.14)). This is not surprising as we have neglected the axial components of the perturbation velocities due to thickness, camber and angle of attack in the kinematic condition which if retained would couple the source and vorticity densities through the inclusion of terms which are of second order except in the neighborhood of the leading and trailing edges. It is important to realize that the bound- ary or kinematic condition is linear in all the velocity components and in the slopes Yf' and r' 10. This is not true of the full Bernoulli equation. We must not think of the first order kinematic condition as a linearization since the exact kinematic condition is always linear in u, v, U, Yf' and r'. It is a first order approximation where terms of order of the square of the thickness and camber ratios and their products are neglected. Indeed, as we have seen, the approximations are not uniformly valid and give rise to unrealistic results at the leading edge (due to loading and thickness) and at the trailing edge (due to thickness). As the pressure equation has been linearized, to obtain the combined pressures due to loading and thickness we can add their separate contribu- tions to get, at ideal angles of attack, on the upper side Application of Linearized The0T'1J 97 98 Design of Hydrofoil Sections Application of Linearized Theory 99 100 Design of Hydrofoil Sections Application of Linearized Theory 101 106 Design of Hydrofoil Sections ments. In actual constructions, the leading-€dge radii are some 10 to 20 times those of the NACA tables. This requires modification of a substan- tial portion of the forebody. We have seen in Chapter 4, p. 59, that sec- tions blunter than an ellipse produce sharp minimum pressures hard upon the leading edge at ideal incidence and that the minimum pressure at non-ideal incidence is dominated by the leading-ooge contribution which varies inversely to the leading-€dge radius of curvature. It is clear that to accommodate the large excursion in incidence angles in the wakes of full form ships it is necessary to employ sections with larger leading-ooge radii and hence maximum thickness locations much closer to the nose. This and other factors lead Johnsson (ibid.) to design a new class of sections composed of Application of N01/;-Linear Theory 107 While a less negative Cpminis obtained from the joining of two semi- ellipses giving the maximum thickness at Cm = 0.0375c from the leading edge its value of ~. 70 is more than 10 times that of a complete ellipse for which Cpmin = -2(tjc) = ~.0692! All of these surely giving low cavitation inception speeds at ideal incidence! Johnsson's measurements with a model propeller (built with this type of sections having radial distributions of position of maximum thicknesses as shown in Figure 6.14) 108 Design of Hydrofoil Sections Application of Non-Linear Theory 109 110 Design of Hydrofoil Sections Our own interpretation of the superior performance of Johnsson's sec- tional design relies on two mechanisms: i. The rising pressure downstream of his minimum Cp produces short cavities as compared to the flat pressure distribution over an extensive portion of the chord provided by the NACA-16 sections. ii. The far blunter leading edge dramatically reduces cavity length and volume. These mechanisms are revealed by the theories elaborated in Chapter 8. Similar studies of blunter bladEHection design have been underway in Japan, see for example Yamaguchi et al. (1988). The subject of blade section design is well summarized in the ITTC Propulsor Committee Report, ITTC (1990b), which should be consulted for further information and references. 7 Real Fluid Effects and Comparisons of Theoretically and Experimentally Determined Characteristics We have this far completely neglected the fact that all fluids possess viscosity. This property gives rise to tangential frictional forces at the boundaries of a moving fluid and to dissipation within the fluid as the "lumps" of fluid shear against one another. The regions where viscosity significantly alters the flow from that given by inviscid irrotational theory are confined to narrow or thin domains termed boundary layers along the surfaces moving through the fluid or along those held fixed in an onset flow. The tangential component of the relative velocity is zero at the surface held fixed in a moving stream and for the moving body in still fluid all particles on the moving boundary adhere to the body. The resulting detailed motions in the thin shearing layer are compli- cated, passing from the laminar state in the extreme forebody through a transitional regime (due to basic instability of laminar flow) to a chaotic state referred to as turbulent. We do not calculate these flows. In what follows we show that viscous effects are a function of a dimen- sionless grouping of factors known as the Reynolds number and review the significant influences of viscosity in terms of the magnitude of this num- ber upon the properties of foils as determined by measurements in wind- tunnels at low subsonic speeds. PHENOMENOLOGICAL ASPECTS OF VISCOUS FLOWS The equations of motion for an incompressible but viscous fluid can be derived in the same way as for a non-viscous fluid, d. Chapter 1, p. 3 and sequel, but now with inclusion of terms to account for the viscous shear stresses. The assumption that the fluid is Newtonian yields a Si~ relationship between stresses and rates of deformation, d. for example Ih (1988). The equations of motion or Namer-Stokes Equations for an incom- pressible, Newtonian fluid are then, in vector notation, and in the presence of gravity 116 Theoretically and Experimentally Determined Characteristics Experimental Characteristics of Wing Sections 117 As turbulence is achieved in ship-hull and propeller-blade boundary layers hard upon the leading edge, the non-scaled persistence of laminar boundary layers on ship and propeller models downstream of their leading edges is a basic cause of lack of flow similarity which affects drag and lift or thrust at high angles of attack. For this reason trip wires and studs are used on hull models and fine leading-€dge roughness is frequently applied to model propellers to artificially induce early transition to turbulence thereby securing a more stable boundary layer. The foregoing overview of the phenomenological aspect of viscous flows about sections enables us to interpret, in a qualitative sense, the experi- mental characteristics obtained from wind tunnel measurements of airfoil sections of interest to us. We may now turn to an inspection of selected experimental results. EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS12 AND COMPARISONS WITH THEORY To overcome the deficiencies of earlier tests of wing models of finite aspect ratio (whose history predated the successful flight of aircraft) the U.S. National Committee for Aeronautics (NACA) built a two--dimen- sional, low turbulence wind tunnel. This enabled measurements of lift, drag, moments and pressure distributions to be made at elevated Re using models of 0.61 m (2 ft.) chord length which completely spanned the 0.91 m (3 ft.) width of the test section. Lift was measured by integration of pressures arising from reactions on the floor and ceiling of the tunnel. Drag was obtained from wake survey measurements (momentum defect) and pitching moments directly by a balance. Usual tests were made over a Reynolds number range of 3-9.106 and at Mach numbers (the ratio of the velocity of the flow to the velocity of sound) less than 0.17. Free stream turbulence intensities the order of a few hundredths of one per cent of the speed were maintained. With this understanding of the model test conditions we may turn to an inspection of the significant characteristics and their comparison with theory. ~ Reactions due to the hydrodynamic pressures and shearing stresses on a foil section can be specified by two force components, perpendicular and parallel to the direction of the ambient flow, (the lift and drag respective- ly) and by a moment perpendicular to their plane (the pitching moment). These forces and moments are primarily a function of the angle of attack and camber and secondarily of the thickness distribution. We may now examine several significant characteristics. 12 Partly abstracted from Abbott & von Doenhoff, (1959), by courtesy of Dover Publications, USA. 118 Theoretically and ExperimentaUy Determined Characteristics Experimental Characteristics of Wing Sections 119 120 Theoretically and Experimentally Determined Characteristics Experimental Characteristics of Wing Sections 121 The variation of CL with angle and with zero and standard roughness for four sections at Re = 9.106 is given in Figure 7.10. Here we see that the older section NACA 23012 (zero camber, 12 per cent thick) exhibits abrupt stall at 18 degrees. It also shows a small negative angle of zero lift which since it is a symmetrical section should be zero according to theory. The dependence of drag coefficient on the operating CL is also shown in Figure 7.10. Here, in smooth condition, it is important to note the low drag obtained at and to either side of the design C L .= 0040, for NACA- 64:0415, the s~alled drag bucket, and the steep risel in CD below CL = 0.2 and above CL = 0.6. It is also to be noted that for moments taken about the aerodynamic center, the moment coefficients are independent of CL· The listed values of the position, xac/a, of the aerodynamic center (about which point the moment is essentially constant) are quite close to the theoretical value of 0.50 (an aspect we did not examine in our theoretical development but it is of structural significance). The drag or resistance of a section is made up of the tangential stress arising from skin friction and a form drag from the lack of full pressure recovery (attained in inviscid flow) over the afterbody. The skin-friction drag is comparable to that of a flat plate of the same length in that the shearing stresses in the region of the forebody where the pressure is falling (negative pressure gradient) are greater than that of a flat plate (both having turbulent or both having laminar flows) and in the afterbody the 126 Theoretically and Experimentally Determined Characteristics - -- -. _0Pressure DIstributIons A single comparison of calculated and measured pressure coefficient distributions on a sectibn is provided in Figure 7.14 showing excellent agreement on both lower and upper sides, especially in the region of most negative Cpo This comparison is based on a superposition-{)f-velocities procedure described in Abbott & von Doenhoff, (1959) (pp. 75 - 79) and cannot be regarded as representative of all theoretical processes which include non-linear and precise representation of the geometry. Nonetheless other correlations between measured and calculated pressure distributions have shown comparable agreement so the reader may rest assured that two-dimensional theoretical estimates can be used with confidence and that they will generally be conservative when applied to propeller blade sections when the angle of flow incidence is known to be accurate. Both the unsteady and three-dimensional effects present on ship propeller sections serve to reduce the negativeness of Cpmin as determined from quasi-steady, two-dimensional values as deduced from the theory pre- sented herein. This is in keeping with the traditional conservatism of the naval architect! Experimental Characteristics of Wing Sections 127 In summary, the foregoing overview of experimental characteristics reveals that in a comfortable range about the design lift coefficient, the lift associated characteristics are very well predicted by theory (with the exception of the rectangular (a = 1.0) camber loading). As inviscid theory predicts zero drag and no stalling characteristics we must employ empiri- cisms for these aspects. Thus relatively simple theory is found to be of great utility in the understanding of the performance of blade sections and in the pragmatic design of propellers. We may now turn with increased confidence to the prediction of condi- tion of cavitation inception and thereafter to the theory of cavitating hydrofoils. 8 Cavitation Here, following a brief account of early observations of the effects of cav- itation on ship propellers, we present methods of estimation of conditions at inception of cavitation followed by an outline of the development of linearized theory of cavitating sections. Application of this theory is made to partially cavitating sections, employing the rarely used method of coupled integral equations. The chapter concludes with important correc- tions to linear theory and a brief consideration of unsteady cavitation. HISTORlCAL OVERVIEW Cavitation or vaporization of a fluid is a phase change observed in high speed flows wherein the local absolute pressure in the liquid reaches the vicinity of the vapor pressure at the ambient temperature. This phenom- enon is of vital importance because of the damage (pitting and erosion) of metal surfaces produced by vapor bubble collapse and degradation of per- formance of lifting surfaces with extensive cavitation. It is also a source of high-frequency noise and hence of paramount interest in connection with acoustic detection of ships and submarines. Both "sheet" and "bubble" forms of cavitation are shown in Figure 8.1. One of the earliest observations of the effects of extensive cavitation on marine propellers was made by Osborne Reynolds (1873) when investiga- ting the causes of the "racing" or over-speeding of propellers. The first fully recorded account of cavitation effects on a ship was given by Barna- by (1897) in connection with the operation of the British destroyer Daring in 1894. About that time, Sir Charles Parsons (inventor of the steam tur- bine) obtained very disappointing results from the initial trials (1894) of his vessel Turbinia, fitted with a single, two-bladed propeller 0.75 m in diameter. He concluded from the trials of the Daring, that the limiting thrust because of the formation of large cavities corresponded to an aver- age pressure on the blades of 77.6 kN/m2 (11.25 Ibs/in2). After experi- mentation with three tandem propellers on a single shaft, he finally fitted the Turbinia with three shafts, each with three tandem propellers of 0.46 m (1.50 ft) diameter and blade-area ratio of 0.60. His vessel then achieved the very remarkable speed of 32.75 knots at 1491 kW (2000 hp) and later was said to have reached 34 knots as detailed by Burrill (1951). Historical Overview 129 130 Cavitation Since the turn of this century an enormous literature has grown dealing with the physics of cavitation and the damage to pumps, propellers, hy- drofoils etc. as well as the effect on their hydrodynamic performance. It will be far beyond the limits of this book to give even a review of this literature. Instead we refer interested readers to the limited number of books on the subject where also such reviews of the literature can be found. They give comprehensive treatments of the phenomena of cavitation such as its formation, the dynamics of bubbles, their collapse and the erosion. Knapp, Dailey &. Hammitt (1970) also describe the effects on flow over hydrofoils and treat cavitation scaling which is of importance in connection with model testing of hydrofoils and propellers. We also mention Young (1989), and Isay (1981) who includes cavitation on hydrofoils and propellers. PREDICTION OF CAVITATION INCEPTION To predict the inception of cavitation we are interested in finding the conditions, in particular the position on the body, where the local pressure drops to the vapor pressure. The vapor pressure of a liquid is a fundamen- tal characteristic (analogous to density, surface tension, viscosity etc.) which depends on the temperature. Volatile liquids such as benzine have high vapor pressures relative to water for which at lOoC (50°F) the vapor pressure Pv = 1227.1 N/m2 (0.178 Ibs/in2). We may immediately note that this vapor pressure at sea temperature is very small compared to the atmospheric pressure which is 101325 N/m2 (14.696 Ibs/in2). The vari- ation of Pv with temperature for water is given in Table 8.1. Prediction of Cavitation Inception 131 Although cavitation according to theory should take place when the pressure drops to the vapor pressure it has been observed to occur at pressures above and below the vapor pressure depending on the amount and distribution of nuclei or particles to which minute pockets of undis- solved gas or air are attached. These act as interfaces on which vaporiz- ation or boiling initiates. Indeed, Harvey, the great researcher of human blood, showed long ago that distilled water requires an enormous negative pressure, some -60 atmospheres before rupture and vaporization ensued. However in our application to propellers, sea water is profused with undis- solved air and we may take the condition for the onset of cavitation to occur where the local total pressure is close to the vapor pressure. In this connection it is important to realize that in model test facilities such as variable pressure tunnels and towing tanks it is necessary to ensure a sufficient supply of nuclei; otherwise cavitation inception and its subsequent extent will not comport to full scale. There are many other factors which affect the scaling of such test results, cf. for instance Knapp, Daily & Hammitt (1970). In flowing liquids the tendency of the flow to cavitate is indicated by the so-called cavitation index or vapor cavitation number which (by the ITTC Standard Symbols (1976)) is expressed by 136 Cavitation As a convenience in evaluating (8.19) the nomograph in Figure 8.4 is provided. One need only connect the depth to the value avi = [Conin| with a straight line and its intercept on the speed scale is the answer. For comparative purposes the cavitation inception speeds for a variety of two-dimensional and axially symmetric forms are graphed in Figure 8.6 for the forms displayed in Figure 8.5. Two features should be noted. Firstly, axially symmetric forms have much higher speeds of inception than two-dimensional sections because the flow is “relieved” by three- dimensionality. Secondly, decrease in cavitation speed is more rapid with decrease in fineness ratio (length/diameter) for axially symmetric forms than for two-dimensional sections of increasing thickness ratio. The minimum pressure coefficient for spheroids at zero angle of attack has been found by Breslin & Landweber (1961) to be expressed simply by 3 t where d is the diameter and £ the length. Thus we see that Cpmin varies more sharply with "thickness ratio" d/f than two-dimensional forms. The variation of minimum pressure with nose radius for various bodies of revolution calculated from axially symmetric source-sink distributions are shown in Figure 8.7. Again it is seen that forms blunter than ellipses of revolution yield relatively large negative Cpmin- 4 (ap? d 0<7<02 (8.20) Comin = 1 a l T [— “eo 3 | ln cido speet 4 ra | naçie ess 2 s ET] 1 aii Lo] o É Back tsuction) side bubble [TT dA + a 4 Imlfaco fprescura Lo = I ni o F HA | LI 4 0 02 04 DO 08 10 12 14 16 18 20 22 24 26 28 30 Comin Figure 8.3 Variation of Cpmin with angle of attack and cavitation types to be expected if [Cpmin| < Gy. Adapted from Brockett, T. (1966). Minimum Pressure Envelopes for Modified NACA-66 Sections with NACA à = 0.8 Camber and BuSkips Tape 1 ond Type JT Sections. Hydromechanies Laboratory, Research and Development Report 1790. Washington, D.C.: David Taylor Model Basin — Department of the Navy. By courtesy of David Taylor Research Center, USA Prediction of Cavitation Inception This chart applies to sea water at 15º C To estimate the speed of incipient covitation for q body, toy a ruler on the given h and g, ; Covitation index Mr ad + he resultant cavitotion speed + As the intercept an the Y, scale Depth h in meters below water surface NT [pre [o in m/s 20 30; 10 sd Speed af incipient covitation v, 137 Figure 8.4 Nomograph showing the relationshi , ip of depth tati index, and speed of incipient cavitation for any Dodpo ereenee, cavitation 138 Cavitation BODIES OF REVOLUTION Ellipsoid or elliptical cylinder 4/d=5 Akron airship model : British airship model C-Class aírship model CYLINDERS TMB Form c/t=60 TMB Form c/t=2.5 Joukowski girfoil c/t=25 NACA symmetrical airtoil 0012 CD) <= 00 > 0/t:8.33 NACA symmetrical airfoil 0010 c/t=10.0 Parabola y=1-0,00444 x? c/=15 | ! s66NS Oval x27=430701-y21) c/1:80 Oval x!º =251611-418) 0/t=60 2 =36(1-y2 = Parabola Ellipse Eltipse x/=36(1-y?) c/t=60 veO Os Pp eABl-y) "612 €1=654 Porabx ; ola : D(1-92) y=1-00123 x semi-cirete c/t-50 c/t=25 Figure 8.5 Sections of cylinders and bodies of revolution for which cavitation speeds were calculated. Figure 1 from Landweber, L. (1951). The Asialy Symmetric Potential Fiow about Elongated Bodies of Revolution. Report No. 761. Washington, D.C.: Navy Department, The David W. Taylor Model Basin. By courtesy of David Taylor Research Center, USA. Prediction of Cavitation Inception 139 55 Y f m/s Ellipsoids 50) Lj df | / Hg Elliptical Cylinders q 3] / l nes ylinders o 2 Tr “ x XÁ A / Lito k 1 Joukowsky ” ” Symmetrical Airfoils o 0 2 4 6 8 Do no ou % 10,04 Figure 8.6 Calculated cavitation speeds for various bodies of revolution and cylinders. die lower case letters in circles refer to the body shapes shown in Figure Figure 2 from Landweber, L. (1951). The Azialy Symmetric Potenti Fiow abost Elongated Bodies of Revolution. Report No: 761. Wahingione D.C.: Navy Department, The David W. Taylor Model Basin. ! By courtesy of David Taylor Research Center, USA. 140 Cavitation CAVITATING SECTIONS The dominant feature of steady high-speed flows of inviscid fluids about blunt forms, as for example, displayed in Figure 8.8, is the development of free-stream surfaces or streamlines. Along these the pressure is the con- stant vapor pressure and hence, by Bernoulli's equation, the magnitude of the tangential velocity is also constant. For the pressure in the cavity equal to the ambient total pressure, the tangential velocity component along the stream surface is equal to the speed of the incident flow far up- stream, yielding an open cavity extending (theoretically) to infinity down- stream. Mathematical solutions of such flows about two-dimensional forms were derived more than 120 years ago by Helmholtz (1868) and Kirchhoff (1868) using complex-variable theory and mapping procedures, see Milne- Thomson (1955) for a detailed account. An excellent summary of the ex- tensive literature about cavity flows has been given by Wu (1972) who has made basic contributions. However, as Tulin (1964) pointed out, there were no important applications of free-streamline theory to flows about forms of engineering interest for more than 60 years until Betz & Peter- sohn (1931) explained the stalled operation of pumps and compressors by extension of the theory to an infinite cascade of flat foils. Cavitating Sections 141 Figure 8.8 Flow past a flat plate with a cavity at ambient pressure corresponding to a cavitation number of zero. During World War II the impetus of naval problems lead to the devel- opment of approximate theories for cavities of finite length by assuming the existence of re-entrant flow at the aft end of the cavity and alterna- tively by use of a flat plate to permit the cavity to close and the pressure to rise on the flat plate to the stagnation pressure. The assumption was that for cavities of sufficient length, the flow at their termini has little effect on the flow about the body. This allowed the use of mapping pro- cedures to cavities of finite length. Posdunin (1944, 1945) pioneered research on propellers designed to operate with large trailing cavities which he referred to as "supercavitating" propellers. This appellation has been in common use thereafter for all flows with cavities which extend beyond the trailing edge of the body. The highly significant application of theory to supercavitating flows of interest to nav·al architects began in the early 1950's by the dramatic breakthroughs made by Tulin (1953) at DTRC who introduced lineariz- ation of the equations of motion. As an aeronautical engineer fully famil- iar with the application of linearized theory to thin wing sections, Tulin quickly perceived that the same procedure (with some modifications) could be applied to cavity flows generated by thin wedges and hydrofoils. In contrast, and prior to that time most professors of naval architecture as well as mathematicians regarded the non-linearity of the Bernoulli equation as sacrosant and were uninformed of the wide use of linear theory by aerodynamicists. Tulin's first analysis in 1953, clearly displayed the adaption of small perturbation, aerodynamic-type procedures to the flow about slender, blunt-based strut sections (with port-starboard symmetry) with trailing cavities of finite length at small cavitation indices. His result for the limiting case of (J = 0 for which the cavity is theoretically open and of infinite length agreed with the exact results of Kirchhoff for wedges of small apex angles. He next demonstrated (with Burkart), Tulin & Burkart 146 Cavitation a era — dx! = —g(x) s;MSxSay=o0 x 32 8.29 where (629) g(x) = a(oU + 2u0(x,0.)) (8.30) and a 1 edson =| E jasx£a (831) 27 x — x 2 a uo(x,0,) is the horizontal component of the velocity induced along the cavity by the section thickness and the non-avitating vortex distribution. We can see that the cavity is "driven'" by the right side of (8.29) which is composed of the cavitation number, the speed U and the horizontal induction of the fully wetted flow, u9 on the cavity. We now have three integral equations involving four unknown functions, “ot Yca Mç and the cavity length £e = a-aç for a given cavitation index q and foil geometry. The fourth relationship is given by enforcing require- ment v. above. This condition for closure implies | melx)dx = O (8.32) It is expeditious to fix the cavity length and solve for the required cavita- tion number. We can solve (8.28) for Yca in terms of Ye, using that result in (8.27). This last equation can then bc inverted to give %s às an opera- tion on me which is used in (8.29) to obtain an integral equation for mc alone. For the inversion of (8.28) we use the gencral form in Appendix A (p. 484 and sequel) in which the function h(x) is now the integral of %of in (8.28). The solution is given in (A.20) p. 489, with ay = —a and à» = àç. The constant K in (4.20) is found by imposing the condition vi. that Ycal—a) is finite (the same procedure as explained on p. 71 for the flat plate). This yields ax [O fam [O e dxr [ÉS Td (8.33) ao ax | (e) fc) —a ae Partially Cavitating Hydrofoils 147 Expansion of the inner kernel in partial fractions gives Ea) 63) = xx a (e) (eme xx and upon interchange of the order of integration, (8.33) becomes 1 far [O ve [ Ta ma = — de Je dx io nm jaç-—x xx! a x! lua! ac —a 1 BLxL a (8.35) The x'-integrals are evaluated by (MB8.21), p. 525 noting that x is inside the integration limits and x! is outside, giving a 1 ja +x x'-ae Tot ti Feal 7 fax | o eo dx';-aLx< ac (8.36) aç Insertion of this into (8.27), noting that x then becomes x'!, yields a 1 ae nm a 1 1 TS qe sl] que [2X x'—ãe es(x!) | x=xt u 1 to Not dx r ac ax! (ext) (x x!) a = a e =q mx); de<xLa We again expand in partial fractions, interchange orders of integration and evaluate the x'-integrals by use of (M8.22), p. 525 selectively to obtain ax! x! a X'—de Jor — dx = q mex, , do Em (8.37) Now h(x) in Appendix A is the right-hand side of (8.37) and the unknown function is Yet 4(x'-ac)/(a+x!). Applying the general inversion formula in Appendix A, taking the result which gives regularity at x = ac, we find a 1 ja+x mia ac de! (8.38) 148 Cavitation Use of (8.38) in (8.29) yields the final integral equation for me a ax ax! Dall ax qadx ac The forms in the kernel suggest the substitutions = = p= [E = (Rx (8.40) a+x ax! With these transformations (8.39) (quite miraculously!) transforms to Metx) dx = — g(x) (8.39) xx! ale) (8.41) Cortina) (rel o (Bem) AI) where s = [(a-ac)/(a+ac) = J&/(e-to) ; e = 2a is the chord, fe the cavity length. This equation is of the form | Mt) de = G(2) z 0 with M(2') = 2! me(z)/(1+2'2). We may now use the general inversion formula (A.20) from Appendix À to solve for M(z) and hence for me(z) to obtain Gleyde! + K| (842) Me(z) = — lt? E (sa) 0 12312 [es za! where K is the arbitrary constant which must be determined by imposing a side condition somewhat analogous to the Kutta condition. The behaviour of the flow at a sharp leading edge was examined by Geurst (1959) who analyzed the tocal velocity there from the exact solu- tion of Helmholtz for a cavitating plate. He found the singularity in the vertical component at the leading edge to vary as (a-x) 1% for x + a. As z= Item) /(2-+8), this corresponds to 1/fe. We find K in a similar way to that ir Chapter 5, p. 71 by requiring lim 2!mdz) = 0 yielding 0 Partially Cavitating Hydrofoils 149 8 — K-] ê 47 and with this (8.42) becomes upon substitution for G and z s 2 z t XD, mola) = — A 4 de (o0) olruo(a0.) (8.43) 2m/z(s-2) 1l+ 2! z-a o This gives the quarter root singularity at x = a and à square-root singu- larity at the cavity end x = ac, both of which are integrable. Gla)da! Cavitation Number — Cavity Length Relation The cavitation number, q, can be found as a function of the cavity length, te = a — ac by evaluation of Equation (8.32) using the transformation (8.40) and the source density (8.43) and solving for o. This relation ig found to be Port |sa! “1 1 dat — uo(2!,0,))) — — | l+at2 do! o(e”0+) o 2-2! 1422 (s-z de 2 U sq Z [81 qt [sm dz =—— —— —— |—— dg! o ldsaz Jozat lero doar (8.44) The z- and z*- integrals can be evaluated by application of relevant inte- grals (M8.23)-(M8.32) at the end of Section 8, Table of Airfoil Integrais of the Mathematical Compendium. This gives 42 g=— Val ) 1/4 8 1H1-%, te [a (e lt) E ole Ode (8.45) (1=0)1/4 (14+22)2 o=-— where ?e is the cavity length in fraction of the chord. Evaluating for the case of a flat plate for which m = O and from (5.29) uo(x0,) =— 7/2 = — Va Aa+9)/(0-0) = —Ua/z, we find (with assist- ance from the Table of Airfoil Integrals) the simple result a afã : — 55 —— 8.46) “Mt han (549) 150 Cavitation This result can be shown to agree with that of Acosta (1955) and Geurst (1959). We observe that, in general, the cavity length is an implicit func- tion of q and a functional of the horizontal component of the flow induced by the camber, thickness and angle of attack of the fully wetted section. A graph of (8.46) given in Figure 8.11 shows that two cavity lengths are given for a single value of the parameter al q in the range al q < 0.0962 and none for larger values as found by Acosta (ibid.). Observations of tests of two-<limensional sections in a water tunnel by Wade & Acosta (1966) confirmed that stable short cavities exist for al q < 0.10, giving good agreement with the theory. For 0.10 < alq < 0.20, the flow was found to give cavity oscillations having low reduced frequencies welU in the range 0.10 to 0.20. For al q > 0.20 quite steady, supercavitating flow ensued. Hence the theory is useful only for cavityIchord length, lc < 0.75 or al q < 0.10. Unrealistic lift and cavity cross-sectional areas are obtained for lc > 0.75. Modification of Linear Theory 151 which for lc = 0 (q ...,00) gives agreement with the fully wetted result. However as lc"" 1, CL ...,00 ! Again we see that the theory applies only to short cavities. It is to be noted in Figure 8.12 that the lift coefficient is composed of the fully wetted linear variation with a plus a non-linear increment which grows rapidly with alq to give Cd27r = 1.4 at the largest value for which the theory may be valid. The increase in lift may be thought of as due to the effective curvature or camber provided by the cavity. The fact that cavity length and area are highly non-linear in al q means that we cannot identify contributions of individual harmonics of the in- flow velocity to propellers as cavity area etc. depends on the total inflow angle a to the sections. MODIFICATION OF LINEAR THEORY Linear theory is, of course, approximate because of the several simplifica- tions made to secure tractable mathematics. Primary motivation to im- prove prediction of cavity size has arisen from the need to predict fluc- tuating pressures from intermittent sheet cavitation on propeller blades. Numerical experience with representations of three-dimensional unsteady cavities on blades revealed that the extent and volume of the cavities were overly large in comparison with ship and model observations. Tulin & Hsu (1977) were the first to show by an ingenious modification of the exact flow about hydrofoils that the effect of forebody thickness is to markedly reduce cavity length and sectional area. This follows from the fact that linear theory gives a poor prediction of the horizontal velocity of the non-cavitating section uo(x,O+)at and near the leading edge. We have in Chapters 5 and 6 and the beginning of this chapter seen that the Lighthill correction is essential to give predictions of inception of cavita- tion. A most adroit correction to linearized cavity flow analogous to that of Lighthill for the non-cavitating case has been developed by Kinnas (1985, 1991). Although Kinnas' procedure involves only the leading-€dge 156 Cavitation SUPERCAVITATING SECTIONS At gufficientiy low values of the cavitation number (about one-half the o at inception) sheet cavitation on the back of sections extends beyond the trailing edge. These flows are of interest to designers of high-speed hydro- foils and propellers for operation at speeds in excess of 40 knots. As linear theory for supercavitating sections such as shown schematically in Figure 8.19 is analogous to that exhibited for partial cavitation we omit details, giving formulas only for the flat plate which are equivalent to those obtained by Geurst (1960) but are expressed in physical parameters. y Apdx cavity ! P Figure 8.19 Schematic of idealized supercavitating section (cavity thickness exag- gerated). The relation of the dimensionless cavity length e to a/o for the super- cavitating flat plate is extremely simple, being te=1+ E (8.52) Ss Hence for o > 0, 2ç > 00, giving an open cavity. For o + 00, to= 1 which is an unrealistic result to be discussed later. The lift coefficient is Me Wte- (ic o1] a (8.53) L or 2 c= ze(t + (Za/o)?) [1 + Qa/0) - 2/0) (8.54) 2 This subsumes Tulin's (1956) result for o = O as seen next. For o => 0, using the binomial expansion of (8.54) we find 2x slol=5a (8.55) 2 1º o 2a] Supercavitating Sections 157 Hence for a flat plate with a long cavity the lift rate dCy/da = 1/2 which is one fourth of that of non-cavitating sections (ef. (5.48) p. 77). This says that supercavitating propellers will produce much less vibratory forces from the non-uniform flow in the hull wake. For o + 00, Cj > 00 which is again à non-physical result because the theory becomes invalid as the end of the cavity approaches the trailing edge of the foil. Tulin (1979) has Pointed out that the cavity lengths for 0.096 < afo < 0.20 as shown in Figure 8.20 given by the linearized steady theory for partial and supercavitation have not been observed ex- perimentally because of flow instability. He developed à qualitative un- steady theory for partial cavitation on thin foils of finite aspect ratio at reduced frequences we/U of order unity which explains the oecurence of stable flows for C,/270 < 0.096 and instability for C/279 > 0.096 as well as flow hysteresis. te te | ] [ Steady super- 14-— — -Branches not observed [2 covitoting flow | experimentally p dvd dA : 10 7 SN Limit cycle 08-—— — ++ oscillations in Ti covity length 06 Se :015 / 1? | 0 T | 0.2 t DaS62 0 4 0 040 015 020 025 030 0 05 a/g Pigure 8.20 Cavity length as function of inflow angle/cavitation number for flat Plate. Figure 9 from Tulin, M.P. & Hsu, CC. (1980). New applications of cavity low theory. In Proc. Thiricenth Symp. on Naval Hydrodynamics, ed. T. Inui, pp. 107-31. Tokyo: The Shipbuilding Research Associatioi of Japan. By courtesy of Office of Naval Research, USA. Cavity-caused drag is simply the horizontal component of the local pressure jump integrated over the length of the section, i.e. D= [ “aofs) e dx (8.56) —a 158 Cavitation since there is no leading-edge suction force with leading-edge cavitation. For the flat plate, dy-/dx = o yielding an aft directed force or Cp = 00y, giving a lift-drag ratio of 1/a. However, for a foil with a curved lower side, forward components can be obtained over part of the surface, as indicated in Figure 8.19. This gives the possibility of designing Super- cavitating sections of high lift-drag ratios. Figure 8.21 displays designs of supercavitating sections having largê lift-drag ratio as developed by Posdunin (1945), by Tulin, by Newton & Rader (1961) and at Hydro- nautics Inc.. Those labeled TMB-Tulin and Hydronautics were designed by the nse of linearized theory by Tulin & Burkart (1955) and by Johnson (1957). Tulin & Burkart (ibid.) elegantly showed that the flow at o = O about supercavitating sections of chord c is equivalent to that Flow direction about aerodynamic sections of + chord length equal to Te, and de- duced the equivalence relations between the coeificients C= Cy: Cp= C'/8r THB “Tulin (8.57) Posdunin (1945) LS Cu Newton & Rader (1961) where Cy and Cp are the hydro- foil coefficients and Cy and O are the moment (about the lead- Hydronutics ing edge) and lift coefficient re- spectively of the related airfoil. As a consequence of the equiv- Figure 8.21 Supercavitating propeller : - profiles. alences, Tulin pointed out that qussa itati il of mini- Figure 13 from Tutin, M. (1964). he supercavitatins foil o! m Supercavitating flows — small pertuebo- mum drag for a fixed Cy is given So ton theory. Journal of Ship Research, gs , 7, no. 3, 16-37. by the shape cf the related airíoil hdi é (1964) SNAME. AI rights i its li t its d. as di ao eentratod ao By couriesy of SNAME, USA. cL = =C (8.58) M a1gU2et L giving from (8.57) [€1/Cp)om = 85/01 (8.59) Unsteady Cavitation 159 This ideal upper bound cannot be attained in practice for as he found, the flow in the physical plane has negative cavity thickness in the region of the hydrofoil. It is mandatory to account for the relative shape of foil and cavity and, of course, to provide for sufficient foil thickness for strength. Auslaender (1962) has taken this geometry into account, yielding remark- ably high lift-drag ratios. Application of supercavitating sections to propellers must take into ac- count the effects of the thick, trailing cavity shed by the preceding blade. Early ignorance of this proximity of a “free surface" has, in one case, resulted in a 50 per cent underdevelopment of design thrust! An approxi- mate theory has been outlined by Tulin (1964) to correct for the presence of a nearby cavity on sectional performance. Nowadays, computer-effected representations in three dimensions include this interference effect. Use of supercavitating sections for propellers- is necessary for operation at low cavitation numbers according to a criterion devised by Tachmindji & Morgan (1958) as indicated in Figure 8.22. 8 Un Marginal a Not practical Not practical for super- " for covitating propellers 04 conventional or super- Pa] 02 caviating o propellers Practical for super - 01 1 covitating propellers [= 008 : ! 006 : 00 | 0 02 04 06 08 10 12 14 k 16 Figure 8.22 Areas for practical use of supercavitating propellers. Figure 1 from Tachmindiji, AJ. & Morgan, W.B. (1058). The design and estimated performance of a series of supercavitating propellers. In Proc Second Symp. on Naval Hydrodymamics, ed. R.D. Cooper, pp. 4809-532. Washington, D.G.: Office of Naval Research — Department of the Navy. By courtesy of Office of Naval Research, USA. UNSTEADY CAVITATION Most ship screws do not cavitate continuously as do those on very high speed craft but are prone to intermittent cavitation, generally in the case of centerline propellers, in the angular region from 10 to 2 o'clock. This could be modelled in two dimensions by considering the case of the sec- 160 Cavitation tion moving at constant speed through a stationary cyclic variation in cross flow. This is known as a gust problem and is here treated in detail in Chapter 17 for the non-cavitating section. For the cavitating sections in such a travelling transverse gust the varying cavity is modelled by a source distribution which varies with time. This yields the untenable result that at large distance the perturbation pressure becomes infinite, This obtains because the potential for the source distribution at large distance R takes the form é Lá “obe2) Ink de 1d (8.60) => | xt) nRdt=— . 2d), 27 Ot ac where R = 4x? + y2, n is the time-dependent cavity ordinate and Ac(t) is the cavity area. We can neglect the convective part of the pressure (cf. Equation (1.30); a justification of this is given on p. 411) and the dominant pressure is then InR (8.61) Hence p becomes infinite as & > 00, unless RAç/ôt2 = 0, which in turn requires 94c/ôt to be a constant, This is the consequence of the incom- pressibility of the fluid and the absence of any pressure-relieving bound- aries in this two-dimensional boundless domain. In actuality there is always a free water surface and none of the bodies of interest are two-dimensional in character. Introduction of the water surface with a boundary condition appropriate to high Froude number (Appendix F) (consistent with low o) would provide a velocity potential from the sources and their negative image in the water surface on y = 0, cf. Figure 8.23 d2 + x2 + 2(yd R2 d+ x2 —2yd + x'x) R2 ó sã]. fe) =— — atx!st) Im 27 de Jo (8.62) where d is the depth to the line source below the water surface which as R + 00 becomes Unsteady Cavitation = 1 de vd 2 IRA yd Ê 7xôt Rº ep Tã? Rº >0 asR>500 (8.63) This representation (together with vertical dipoles over the chord and onto the "wake" and their negative images) would be amenable to solu- tion by computer as there is no inversion formula which applies to the coupled integral equations. (Surprisingly, this problem has not been solved to the authors! knowledge). Intermittent cavitation on propeller blades is, in any event, very three- dimensional and must be represented b; ee-dii j j ity di ms y three-dimensional singul: - tributions which yield conver, gularity dis gent pressure fields. This topic is i i Chapa a pic is outlined in 166 Actuator Disc Theory HEAVILY LOADED DISC In all textbooks of naval architecture the flow around an actuator disc is described by a partly non-linear theory. We shall try to obtain additional insight by avoiding this simplification as this allows us to deal with a disc of heavy loading. We follow the elegant theory by Wu (1962) (a landmark paper) with some alternatives and with greater exposition of mathematical details. In the previous paragraphs of this chapter all velocities were total veloc- ities. Since in our applications we generally consider propellers in an axial inflow of velocity -U we shall distinguish between the total velocity and the perturbation velocity like we have done with wing sections in the previous chapters. We shall then use Ua for the perturbation velocity and u! for the total velocity ul = Ua - U. Since the inflow contains neither swirl nor contraction, Ur and Ut represent both total and perturbation velocities. As a start we make a further, general modification of (9.8) by defining the velocity vector Heavily Loaded Disc 167 168 Actuator Disc Theory The equation of motion is then in vector notation (9.18) Figure 9.4 Actuator disc. The disc of radius R is located in the plane x = 0 in a uniform inflow —U, see Figure 9.4. It applies axial, radial! and tangential forces to the fluid having the components Fa, Fr, Fe, cf (9.13) with x! = 0. These forces only apply on the disc, The disc is rotating with angular velocity vo We now impose that the flow is stationary and that it is axisymmetric so that ui = uitsr) Ur = u(xr) wu, = u(xr) and that u, = O upstream of the disc and outside the race but is non-zero in the race because of the swirl imposed by the tangential blade forces on the fluid. This will be seen later. . For axisymmetric flow we can employ the Stokes' stream function w, cÊ. Milne-Thomson (1955). The axial and radial velocities are then given by ed 1% (9.19) uf=-— rõr Up = We see that 4 is constructed to satisfy the continuity equation (9.7) EE 5] 5] [9] +0=0 rôx óxlrôr] al rôx) rlrôx Heavily Loaded Disc 169 The stream function is constant along stream tubes (Milne-Thomson (ibid.)). For the uniform stream the stream function is r2 W=-Uy (9.20) or lôb ua g=>""" =D =— rôr r2ã ie., far upstream of the disc. The kinematics of the flow is completely specified by | and the tan- gential velocity component u, which we must now find. The geometry of the flow in any meridional plane is sketched in Figure 9.5. Trace of bounding stream surface Actuator disc rotating at angular velocity w A Vw x Pigure 9.5 Upper half of meridional plane. As the disc is heavily loaded the equations of motion cannot be linear- ized. The only simplification in (9.18) is that the derivative with Tespect to time vanishes because of stationary flow. With the force terms given in (9.13), (9.18) is valid for an infinitesimal force element. To obtain the influence from the entire disc we must integrate. For the axial component R 27 r rio) = | ár l PO ade = do pray o 0 r = [o Ap(r)&s) O<r<R (631) o R<r whcre the longitudinal coordinate x has been substituted by s which is the curvilinear coordinate along a trace of stream surface, cf. Figure 9.6. For the other force components we replace n; with the relevant compo- nent of the normal. Note that F4 is still a force density or force per unit 170 Actuator Disc Theory volume (Ap = force/unit area, 6 = 1/unit length) for which reason we have kept the ' on F). Note also that n is not the geometrical normal to the disc since this normal is axially directed. But n defines the direction of the forces and we must therefore keep all components. The force can now be written as r= o Aptr)é(s) 0O<r<R 9.22 0 R<r (9.22) From (9.22) we see that outside the disc the force term is everywhere zero leaving 1 Ui 4+- a) =qx( (9.23) p 2 Taking the scalar products pl a vr+odj=a-(gx0)=0 p 2 cferjel-car0=0 since the vector q x 4 is orthogonal to both q and q. Therefore the sur- faces p 1 —+- q2 = constant , 2 outside the blade row (the disc) coincide with the axisymmetric stream and vortex tubes, the latter being generally in the race of the propeller. (In the case of incident flow with shear or vorticity there will also be vortex tubes in the entire region of the ship hull wake). Wu (1962) has pointed out that the relations of dynamical characteristics can be most clearly put in evidence when expressed in a system of curvilinear coordinates (s,$) in any meridional plane, where s is along a trace of a stream surface where 4) is constant and q varies along the normal. To determine the relationships between the differential operators we Heavily Loaded Disc 171 use Equation (3.6) or similar relations from the differential geometry of surfaces to calculate the components of the normal np = — tda (dbfóx)2+( dj dr)? aw/ dr (9.24) nç= [o/)Teoaap by po of (9.19) we also have (as we should since Vl) = c ig a stream ne, (9.25) where Vo= Juz + uZ, the velocity along the stream surface. Using the chain rule and from geometrical considerations (Figure 9.6) à ko &d a ua aa. ê x à or à ô ô ..D 0,00 Mm md TT sing — + cosô ô ô 8 — + sing — cos: mind The derivatives can now be expressed by the velocities using (9.25) ô 1 ê ô =— vit ua] & Vl é a 1 2 ô (9.26) —=-luft--u dr Vl Cx Noting that 64/6n! = rVs and 8 ma 1.8 dp dp dar iva then from (9.26) da f,ô 8 ap vila “a (927