Linearized Theory of a Partially Cavitating Plano-Convex Hydrofoil Including the Effects of Camber and Thickness

Transcription

1 Linearized Theory of a Partially Cavitating Plano-Convex Hydrofoil nluding the Effets of Camber and Thikness By R. B. Wade HfORODYNAMlCS LASORA l C. -.ltor NA NSTTUTE OF TECHNOLO(.. ll l. CAUfORt-llA ST1tf T PASA CAUfODAA The linearized treatment of the flow over a partially avitating single hydrofoil having a flat pressure surfae and a irular-ar sution side is presented. The flow is treated as a two-dimensional, steady, invisid flow. Further assumptions made are those of inompressibility and irrotationality. The results obtained are ompared with experiment and generally good orrelation is found for the ranges of validity of the linearization. ntrodution THE investigation of the behavior of hydrofoils under varying degrees of avitation has been the objet of a great deal of interest. Several theoretial approahes have been developed for treating these problems both from a nonlinear and linearized point of view. From experimental observation [1 ), it has been established that the avity flow around a hydrofoil may be divided into three distint regimes: The partially avitating region, the fully avitating region, and an inherently unsteady zone onneting these two flow onfigurations. The partially avitating region is assoiated with flows where the avity length is less than the hord 1 Researh Fellow, Division of Engineering and Applied Siene California nstitute of Tehnology, Pasadena, Calif. 1 Numbers in braket.! designate Referenes at end of paper. Manusript reeived at SNAME Headquarters, November 3 1., length of the body and onsequently the avity terminates on the upper (or sution) surfae of the foil. n the fully avitating region, however, the avity ollapses downstream of the trailing edge and the entire upper surfae of the foil is enlosed within the avity. The problem of fully avitating flows past arbitrary shaped hydrofoils has been thoroughly treated by W u and Wang using nonlinear tehniques [] and by many authors using linearized methods [3]. For the ase of the partially avitating body, however, relatively few results have been published. n this region of flow where the amber and thikness of the profile play a role linearized tehniques are muh more amenable to solution than are nonlinear methods owing to the diffiulty of hoosing a suitable model for representing the flow in this latter ase. Wu [4] has worked out the ase of a partially avitating flat plate using a nonlinear method but all other solutions have been obtained using the linearized tehnique. These solutions inlude that of the Nomenlature a = sale parameter - A, B = onstant.! G lift oeffiient = _L_ pv 1 / hord length omplex funtion. = -- wm H(n H = homogeneous solution = -vr<r - 1) i integral -v= = avitation number = p., - p. pv 1 / l = Poo = avity length pressure at infinity P avity pressure p. al f. At + B rat1n omp ex unt1n = r<r _ l) R = radius u, 11 = veloity omponent.! in x, y-diretions u, v perturbation omponent.! V upstream veloity W(z) w(z) omplex veloity funtions x, y oordinate axes in physial plane z = x + iy omplex physial plane a = angle of attak p density angle subtended by tangent.! to upper foil surfae r = + i 7] at leading and trailing edges transformed omplex plane. 71 oordinate S..l:es in transformed plane Reprinted from the Journol of Ship Reseorh, Vol. 11, No. 1, pp. - 7 JOURNAL OF SHP RESEARCH

2 j--z partially avitating flat plate by Aosta [5] whih was also subsequently treated by Geurst [6]. n Geursts paper the formulation is arried out for an arbitrary profile having zero thikness. The ase of a profile inluding thikness effets has not been treated in the literature. The present paper deals with the linearized treatment of a partially avitating flow over a plano-onvex (flat pressure surfae and irular-ar sution side) hydrofoil inluding the effets of amber and profile thikness. The results so obtained are then ompared with the experimental results obtained on suh hydrofoils by several authors, viz., Balhan [7], Meijer [8), and Wade [1]. The purpose of studying suh a profile setion, a member of the arman-trefftz family of airfoils, arises from the extensive use of slight variant from this form in propeller work. Fig. 1 Partially avitating plano-onvex hydrofoil Formulation ofj Problem The hydrofoil is held at an angle of attak, a, to the free-stream veloity, V, as illustrated in Fig. 1. For the present problem it is assumed that a avity forms on the top side of the hydrofoil starting at the leading edge. The avity then terminates on the upper surfae. The angle subtended by the tangents to the irular surfae at the leading and trailing edges, n, is assumed to be small as is the angle of attak, a. These assumptions are in keeping with the linearized theory [3]. With these assumptions it is possible to onsider the veloity field as a perturbation on the free-stream veloity, V, allowing one to write the veloity at any point in the fluid as q = V(1 + u, v) = (u, v) (1) where u, v are the perturbation omponents. Furthermore the equation for the irular-ar sution surfae an be written in the form ely = (1 - x) dx The boundary onditions on the veloity funtion, within the framework of the linearized theory, then beome with the help of Bernoullis equation v = - V a + Vf! (1 - x), for the top wetted surfae v = - V a, for the bottom wetted surfae u = V ( 1 + ) on the avity surfae MARCH 1967 z - PL ANE Fig. y r dl v:-va+v dx Linearized problem in physial z-plane is the avitation number defined as where = P - P pv / P = pressure at infinity P = avity pressure These bow1dary onditions are applied along a slit representing the body in the physial plane, as illustrated in Fig.. From the initial assumptions that the flow is inompressible and irrotational, the funtion (3) W(z) = u - iv (4) is therefore an analyti funtion of the omplex varia ble z. The transformation where r = w 1 (5a) (5b) transforms the slit in the z-plane into the upper half r plane, suh that the entire real axis of the r-plane beomes the surfae of the foil. Furthermore the region < r < 1 beomes the avity surfae. With this transformation the point at infinity in the z-plane is transformed into the point ia in the r-plane as seen in Fig. 3. The relevant boundary onditions are also shown in this figure. i o (z: a> ) v : - Vo u:v(+l v =- Va +Y{! [ r-x( )] t- PLANE Fig. 3 Transform s-plane 1

3 Further onditions to be satisfied are that the veloity at infinity be equal to the free-stream veloity, V, i.e. lim W (z) = V (6) -"" and that the avity-hydrofoil system form a losed body. This requirement may be expressed as r dy = o (7) J body where y represents the ordinates of points on the bodyavity system. Furthermore, at the trailing edge, due to the finite trailing-edge angle of the hydrofoil, the veloity there must behave logarithmially. This replaes the usual utta ondition at the trailing edge. Considering the ftmtion w(z) = W(z) - V ( 1 + ) - iva in the r-plan e, we have: maginary part w = - ro < < Realpartw =O << 1 maginary part w = - VQ [1 - xw] 1 < < ro f we ontinue w(z) analytially through the interval < < 1 into the lower half plane, suh that w(r ) = - w(r ) then the real part of w is an uneven funtion of 7J and the imaginary part of w is an even funtion of 7J. We an thus formulate the following boundary-value problem in the r -plane: w+ + w- = - ro < < w+ - w- = w+ + w- =- i vn [1 - xw 1 < < 1 where the supersripts refer to the value of w(t) as 71 - ±. The problem therefore redues to a Hilbert problem the solution of whih an be found by applying the proedures given in referene [9]. Solution of the Problem Let us first onsider the homogeneous problem H + + H H+- H H + + H - - ro << < < 1 1 <<ro t an be seen that H (t) is ontinuous for < < 1 but A funtion satis has a jump for outside of this range. fying these onditions is H = vnr - 1) where we take the branh uts of H to be along the real axis outside the interval < < 1, and we further require that H,..., r We now onsider the funtion as r - ro a (r) = w(r) H (r) The boundary onditions for this funtion are w+ w - a + - a - = = (w+ + w-) j H + = H + H- w+ w- a + - a- = = (w+ - w-)j H + = H + H - w+ w- a+ - a- = = (w+ + w-)j H + H + H- - ivq [1 Wl vh - 1) - x - ro << << 1 By means of Plemeljs formula, we an express an analyti funtion in the upper half plane by its values along the entire real axis aording to the formula F(z) =!. J "" wherej(x) = j+(x) - j-(x). Aordingly, we obtain j(x) dx 7rZ - "" X - Z a ( ) _ Vn f "" 1 - x dt r - - 1r J1 vh - )C- r) " This solution represents a partiular solution of the problem; the general solution being given by w(r) = - vrr - 1) f "" 1 - x d J1 vh - 1 )(- r) + P (r) vnr - 1) where P (t) is a rational funtion of r whih an have poles only at the points =, 1. At the trailing edge, i.e., the point r = ro in the r-plane, w(r ) an at most behave logarithmially. This behavior, a onsequene of the linearized thikness effet, is already inorporated in the integral part of the solution. Hene P(r) an only be of the form Ar + B rr - 1) where A, B are real onstants. We finally get for our solution JOURNAL OF SHP RESEARCH

4 1.8 a=6 1.6 Ct r = O"o 1--- r--- 7o/o 6Yo 1-- 4"/o t-- "/o -... O"o Fig. 4 Lift oeffiient versus avitation number for various thikness ratios at a fixed angle of attak of 6 deg ( r) vnv ( 1) r 1 - x d w - 1r r r - J 1 vh - 1)( - n At+ B + -vt(t - 1) (S) where x an be obtained from the transformation equation (5a) as x = e + a The onstants A and B an be evaluated from the ondition that w (t) = v at r = ia As we will only be onerned with evaluating the foregoing integral at r = ia, we an write this integral as r Ha - e> = J V(( + a) d. r (a - ) + w J l vr- l )(e + a ) For purposes of omputation it is more onvenient to hange the limits of integration to a finite interval. By suessively substituting = 1/ t and t = 1 - x, these integrals redue to t (a t - 1). f 1 t(a t - 1) = Jo (1 + at) dx + ta Jo (1 + at) dx or =/+ ii where / and are only funtions of the parameter a. The integrals are represented in this fashion as it simplifies the numerial integration proess. t is possible to evaluate these integrals in losed form but this leads to a ompliated expression whih is not very useful. The onstants A and B an now be evaluated with the a.id of equation (6). After some manipulation we obtain A = - V ( 1 + a )!. [!f sin 1/t_ + a os!] a and B = - Val(l + a );. [!i + Vf! (1 + a ) ( sin 1/1 + / os 1/1 J 11" os 1/t_- a sin V:.J + Vr! a(l + a )1 ( os 1/1 - / sin 1/1) 11" where!f = 1r + tan- 1/ a. These,_ expressions express the onstants A and B in terms of the parameter a. The first terms in the aforementioned expressions orrespond to the flat-plate solution and the amber and thikness effets are inorporated in the seond terms. The relationship between the avitation number and the avity length l is obtained by applying the losure ondition, equation (7). This ondition may be written in the following form: MARCH

5 1. a Q = 6 CL / { t4 -t. BA LHAN MEJER. f/ "1 )(_ v op 1(. v / L // --:L: 6. / j " --!L". 4. / Y.. e Fig. 5 Ratio of avitation number to twie angle of attak as a funtion of avity length-to-hord ratio for various thikness ratios at a fixed angle of attak of 6 deg Fig. 6 Comparison of theoretial lift oeffiient with experimental results for a 4-perent-thik plano-onvex hydrofoil as a funtion of avitation number for various angles of attak or f dy= f vdx=o J body J body m f W(r) ddz ds = J body where m denotes the imaginary part of the integral. We are furthermore interested in the lift on the body. Within the limitations of the linearized theory this is given by or s CL = - f udx V J body CL = - Re f W(s) dz ds V J body ds where Re denotes the real part of the integral. Sine the body is now the entire real axis in the s-plane, we have to evaluate the integral where 4 = J-" W(s) dzd ds +m S dz a s ds <r - ia)(s + ia) (9) Sine dz _,... 1 asr - ds s 3 there is no ontribution to t he integral by ompleting the ontour by a large irle R and letting R - ro. The value of the integral is then given by the residue of the integrand at the double pole s = ia. Sine the integral traverses the body in a ounter-lokwise fashion in the z-plane whih orresponds to a lokwise sense in the s-plane, we have whih redues to ro = -7Ti[Residue at s = ia] = - 7ra [dwj ds r=ia (1) Carrying out the indiated proedure and separating the result into real and imaginary parts we obtain, after onsiderable algebra, the following expressions for the losure ondition and for the lift oeffiient, C L: 1rA 3.J; ---.,-----,.,. os - Va1(1 + a )1 + 1rB [. 3,Y +. 3,YJ Va1(1 + a)1 SD a SD + JOURNAL OF SHP RESEARCH

6 o a < t = 4t lj,, 1: /l /J a.. /,1 t tl t/..._ 4 _,/// :- " ME JER o.a 1. Fig. 7 Some avity-length measurements ompared with the theory for a 4-perent-thik plano-onvex hydrofoil at various angles of anak g CL.. a o.a.6.4 f 7,.. f/. lk. y a s!q v. / fv / // v/6 f /6 r--- L /) 4 lg( v z WADE a -. a 6 1 o Fig. 8 Comparison of theoretial lift oeffiient with experimental results for a 7-perent-thik plano-onvex hydrofoil as a funtion of avitation number for various angles of anak! and where 1 )/ [11 ( sin f + a os!) 4 a 1 + a -! ( os - a sin ) J +, 1 ( = 4 i ( t ta(at - 1) dx 4 Jo (1 + at)3 From the e results the graphs shown in Figs. 4-1 were obtained. The omputations were arried out on a BM 794 omputer. The range of values investigated was for angles of attak from to 1 deg and values of Q orrespondjng to thikness ratios of from to 1 perent. Disussion Figs. 4 and 5 show t he effet of profile ontour on the performane of the hydrofoil under partially avitating onditions. Only urves for one angle of attak, a = 6 deg, are shown. t is een that the effets of amber and thikness are to inrease lift at any given ayjtation number with a orresponding inrease in avity length. n the linearized theory the validity of the results is usually re trited to a ertain range of ayjtation numbers due to the type of avity losure used. For the present ase this range of validity holds for values of avity length-to-hord ratio less than about.75, the same value as for the flat-plate ase [3, 4]. This is apparent from Fig. 5 where it is seen that the avitation number reahes a minimum value at this point. Values outside t his range would give rise to the possibility of two avity lengths for any given ayjtation number. t should be noted that the foregoing alulations are MARCH

7 a u l t a a = WADE 4. 6 a to : / - / 4 /, /7 - e - / A L Fig. 9 Some avity-length measurements ompared with theor y for a 7-per ent-thik plano-onvex hydrofoil at various angles of attak a <.3 : jl 1 1!,. 7t. CORRECTED TH EORY fy : -,... t-,,.,, 6.,, _ _ e -..A _... _ to WADE e to Fig. 1 Comparison of experimental avity-length m easurements with orreted theory for a 7-perent-thik hydrofoil at various angles of attak based on the assumption that the avity springs from the leading edge at all times. Under ertain onfigurations of angle of attak and thikness ratio this assumption may not be physially possible, as brought out by the findings of experimental investigation [1], vhere at lower angles of attak a avity starts downstream of the leading edge at approximately the point of maximum thikness. This should therefore be kept in mind when applying the aforementioned results. t may be noted that the fully wetted results obtained from this linearized theory for a hydrofoil of 7 perent thikness ratio give a zero lift angle of attak of -4 deg with a orresponding lift oeffiient at zero angle of attak of.438. This ompares with values of -4 deg 1 min and.479 obtained by onformal mapping tehniques, and with -4 deg and.38 obtained from experiment [1]. n Figs. 6 to 1 a omparison of the theoretial results is shown with points obtained from various experimental investigations. Fig. 6 illustrates the lift oeffiients for a 4-perent-thik plano-onvex hydrofoil as obtained by Balhan and Meijer. t will be seen that good agreement is found between experiment and theory. Fig. 7 ompares some avity-length measurements obtained by Meijer with those predited by the present theory. Here the agreement is not as good. The disrepany in this ase may be partly due to the 6 diffiulty in measuring avity lengths under these irumstanes owing to a ertain arbitrariness in interpreting where the avity ends. However, a more likely explanation is the fat that in Meijers experiments the avitation numbers are based on vapor pressure and not on measured avity pressure. t is well known that the avitation number is dereased if measured avity pressure is used in determining this parameter. This lowering would t herefore tend to improve the orrelation between experiment and theory. n Fig. 8 omparison is made between experiment and theory for a 7-perent-thik hydrofoil. Here it will be seen that the experimental results are between to 3 perent lower than the theoretial values-the larger value orresponding to the largest angle of attak. This disrepany is probably due to the deterioration of the linearization at these higher thiknesses in onjuntion with the inreasing angle of attak. The orresponding avity lengths are shown in Fig. 9 where again a similar disrepany exists. t is knmvn from airfoil experiments that if the theoretial lift is arbitrarily adjusted to the experimental value, t he theoretial pressure distribution on the foil agrees on the whole with the experimental one. This artifie ahieves two purposes. First, it endeavors to some extent to aount for real fluid effets and seond, it affords a means of heking whether the experimental JOURNAL OF SHP RESEARCH

8 data are self-onsistent. This approah was utilized here. The theoretial lift oeffiient was adjusted to the experimental value for the same avity lengths and the orresponding theoretial avitation number was altered aordingly. These adjusted avitation numbers are shown plotted in Fig. 1. t is seen that the orreted theoreti<ial values are in good agreement with the experimental points. The..: limiting ase when the thikness of the hydrofoil is "teo redues in the limit to that of the performane of a partially avitating flat plate. Conlusion n onlusion it may be stated that the linearized theory presented predits with suffiient auray the performane of a partially avitating plano-onvex hydrofoil having perentage thiknesses up to 5 perent. For larger thiknesses the linearization breaks down and for 7 perent thikness ratios overestimates the lift oeffiients by to 3 perent, for angles of attak up to 1 deg. The limiting ases of a flat plate and a fully wetted plano-onvex hydrofoil are retrieved by taking the appropriate limits. Aknowledgments The author wishes to thank Dr. A. J. Aosta for his onstant interest and many helpful disussions. This work was supported by the Department of the Navy under Contrat Nonr (4). Referenes 1 R. B. Wade and A. J. Aosta, "Experimental Observations on the Flow Past a Plano-Convex Hydrofoil," Trans. ASME, Journal of Basi Engineering, Paper presented June 7, 1965, Applied Mehanis/ Fluids Engineering Conferene, Washington, D. C. T. Yao-Tsu Wu and D. P. Wang, "A Wake Model for Free Streamline Flow Theory, Part, Cavity Flows Past Obstales of Arbitrary Profile," J ournal of Fluid Mehanis, vol. 18, part 1, 1964, pp B. R. Parkin, "Linearized Theory of Cavity Flow in Two Din1ensions," Rand Corporation Report P-1745, T. Yao-Tsu Wu, "A Wake Model for Free-Streamline Flow Theory, Part 1, Fully and Partially Developed Wake Flows and Cavity Flows Past an Oblique Flat Plate," J ournal of Fluid Mehanis, vol. 13, part, 196, pp A. J. Aosta, "A Note on Partial Cavitation of Flat Plate Hydrofoils," California nstitute of Tehnology Hydrodynamis Laboratory Report No. E-19.9, J. A. Geurst, "Linearized Theory for Partially Cavitated Hydrofoils," nternational Shipbuilding Progress, vol. 6, no. 6, J. Balban, "Metingen aan Enige bij Sheepshroenen Gebruikelijke Profielen in Vlokke Stroming met en Zonder Cavitie," Ned. Sheepsbouwkundig Proefstation te Wageningen, M. C. 1lleijer, "Some Experiments on Partially Cavitating Hydrofoils," nternational Shipbuilding Progress, vol. 6, no. 6, N.. u skbelisbvili, Singular ntegral Equations, P. Noordboff, Limited, Groningen, Holland, MARCH