I asked a professional English editor to improve the language of this guidebook. 17th April, 1999 Hajime Yamaguchi. - i -

Transcription

1 This document is downloaded from index-e.html, where the prblg.f program package including FRTRAN77 source codes is available to the public. This document explains what can be performed using this program package. The prblg.f program package is freeware. You are free to use and modify any of the files included in the package. However the copyright belongs to Dr. Haime Yamaguchi Associate Professor Department of Environmental and cean Engineering Graduate School of Engineering University of Tokyo Hongo 7-3-, unkyo-ku, Tokyo , apan Phone: ext.6536 Phone: (After April 3, 999 Fax: Commercial use of the computed results is permitted, but sale of the program is prohibited. This is freeware. Please use this software at your own risk. I am not responsible for any problems that may arise from the use of this package. You may contact me at the above address, though I may not be able to respond to you because of numerous other commitments. 9th an., 999 Haime Yamaguchi Perhaps, you cannot understand the contents of this guidebook without the knowledge of fundamentals of foil section theory and boundary layer. Please study them with textbooks. I asked a professional English editor to improve the language of this guidebook. 7th April, 999 Haime Yamaguchi - i -

2 Index of This Document:. What You Can Do with the prblg.f Program. Displacement ody Concept 3. utline of Computation Procedure 3 4. utline of Potential Flow Computation 4 4. Coordinate System 4 4. Singularity Distribution oundary Conditions Restriction on Foil Shape 7 5. oundary Layer Calculation 8 6. Curve-Fitting of oundary Layer Displacement Thickness 9 7. Hydrodynamic Forces 9 8. Data Input and utput 9. Demonstration of Computed Results 9. Pressure Distribution and oundary Layer Characteristics 9. Hydrodynamic Forces 7. Concluding Remarks 9 References Appendix: Influence Functions due to Surface Singularity Distribution 7 A. Influence Functions for an Isolated Foil in Infinite Flow Field (Case in Fig. 7 A.. Induced Velocity by Point Source or Vortex 7 A.. Induced Velocity by Line Source or Vortex 7 A. Influence Functions for Foils in Cascade (Case 3 in Fig. 9 A.. Infinite Row of Point Sources/Vortices 9 A.. Infinite Row of Line Sources/Vortices 3 A..3 Condition at Upstream Infinity 33 A..4 Influence Functions 34 A.3 Influence Functions for a Foil between Parallel Walls (Case in Fig. 37 A.3. Conugate Complex Velocity 37 A.3. Influence Functions 39 - ii -

3 Revised, April, an., 999 A Computer Program for Predicting the Hydrodynamic Characteristics of Two-Dimensional Foil or Cascade in Steady Flow with oundary Layer Effects Taken into Account - utline of the prblg.f Program - by Haime Yamaguchi Associate Professor Department of Environmental and cean Engineering Graduate School of Engineering University of Tokyo Hongo 7-3-, unkyo-ku, Tokyo , APAN Phone: ext.6536 Phone: (After April 3, 999 Fax: yama@fluidlab.naoe.t.u-tokyo.ac.p. What You Can Do with the prblg.f Program Hydrodynamic characteristics of two-dimentional foils with boundary layer effects taken into account can be calculated for the three conditions shown in Fig., under the following assumptions. Flow is -dimensional and incompressible. Reynolds number is so high that boundary layer approximation is effective. There is no large separation of the boundary layer. Viscous/inviscid flow interaction is taken into account by iterative computation between the outer potential flow and the boundary layer.. Displacement ody Concept When a foil section is placed in a viscous flow at a moderate angle of attack, boundary layers develop on both the upper and lower surfaces of the foil, as shown in Fig.. The whole flow domain out of the boundary layers can be well expressed by potential (inviscid flow approximation. Since velocity loss occurs in boundary layers, the outer potential flow is represented by a potential flow around the displacement body, which is a body of a foil shape plus displacement thicknesses of the boundary layers and their wake. Since the boundary layer characteristics depend on the pressure distribution on the foil surface, or more precisely on the pressure distribution at the outer edge of the boundary layer, interaction between the outer potential flow and boundary layer should be taken into account through iterative calculation. In this manner, lift drop and pressure drag which appear as a result of the pressure distribution change due to the effect of the boundary layer displacement thickness, can be calculated. [Section.., pp , of Cebeci and radshaw, 977] - -

4 ( A foil in a uniform and infinite flow field Uniform Flow ( A foil in uniform flow between parallel upper and lower walls Uniform Flow (If you place a foil close to the lower wall, you can calculate a WIG, Wing In Ground effect. (3 Foils in cascade in uniform flow Uniform Flow Fig. Three flow conditions which can be treated by the "prblg.f" program uter Potential Flow Displacement Thickness of oundary Layer Flow Foil Section "Displacement ody" Fig. "Displacement ody" concept This is a classical problem, and many researchers successfully solved it many years ago [e.g., Powell, 967; Casey, 974]. There is nothing new about the prblg.f program. Compared with the methods developed by the other researchers, this program even has a shortcoming, i.e. the - -

5 problem of trailing edge pressure which will be described in chapter 7. However, this program can treat a foil between parallel walls and foils in cascade, as shown in Fig., by extending the formulations of outer potential flow. Also, favorable agreement between the calculated and experimental results is obtained, as will be shown in chapter 9. Currently, CFD (computational fluid dynamics tools are being rapidly developed for viscous flow computations. It is still very difficult, however, to treat the transition from laminar to turbulent flow, and fully turbulent flow computations are often performed by adopting a turbulence model. However, in usual hydrofoil problems, where the Reynolds number is low, compared, for example, with ship resistance problems, prediction of transition to turbulence is one of the keys for predicting hydrodynamic performance accurately. Also, the location and pressure of the laminar separation or turbulent transition point must be known for sheet cavitation predictions. Traditional boundary layer calculation methods, particularly those adopted in the prblg.f program, contain many empirical formulae and must have limitations due to the database used to develop the empirical formulae. However, the computations are much faster than in the case of usual CFD programs and an empirical transition prediction can also be made. In my opinion, therefore, a program such as prblg.f is still effective and practical for design purposes where repeated predictions are required under the conditions of moderate angles of attack. This program is also effective for practise exercises for university students. Actually, I have usually used this program for such purposes. In short, this program has the following features. <Merits> ( Short computation time of a few to several seconds on a UNIX workstation, Sun SPARC Station 5. ( Good agreement with the experimental data at moderate angles of attack. <Demerits> ( Many empirical equations are used in the boundary layer computation. This might restrict the applicable range of the program. Instead, a short computation time and good agreement with the experiments are realied. ( Since the computation is based on boundary layer calculations, this program cannot follow large separation. This program outputs a warning when a turbulent separation is detected upstream of the 95% chord station. 3. utline of Computation Procedure The outline of the computation procedure is as follows. ( Potential flow computation for a given foil section shape in the beginning or for displacement body during iterations. ( oundary layer computation using surface pressure (velocity distribution obtained from the potential flow computation

6 y Y v Velocity Leading Edge Direction of Surface Discretiation u Angle of Attack, α Foil or Displacement ody θ (* Trailing Edge X x Uniform Flow, U i (* Uniform flow inclination angle, θ, is for the cases of ( and ( in Fig., since the angle of attack is set by inclining the foil. Nonero θ can be given in case (3 in Fig. since the foil row is placed only in the y-direction in the program. In that case, -α is the stagger angle of the cascade and θα is the angle of attack. y Uniform Flow x θ < in this case Fig. 3 Coordinate System and Definition of Parameters (3 Curve fitting of the boundary layer displacement thicknesses on the upper and lower surfaces of the foil. (4 Making-up of a new displacement body by thickening the foil by the curve-fitted displacement thicknesses. (5 Return to ( until convergence. 4. utline of Potential Flow Computation A velocity-based boundary element method is performed for outer potential flow computations. A very simple method, i.e., straight-line-segment-expressed-body and constant singularity distribution, is used. This method is similar to Hess method [Hess, 97], but the formulations were developed by me. 4. Coordinate System As shown in Fig.3, two Cartesian coordinate systems are defined. The (X,Y coordinate system is fixed to the foil and the (x,y coordinate system to the flow. (u,v is the velocity vector - 4 -

7 at an arbitrary point (x,y. The X-axis is the baseline of the body, i.e., usually nose-tail line. The foil chord length should be. The details will be described in chapter 8, Data Input and utput. The surface of the foil or displacement body is discretied into small line segments along the upper edge of the T.E. (trailing edge, the upper surface, the L.E. (leading edge, the lower surface and the lower edge of the T.E. 4. Singularity Distribution Assume that the foil or displacement body surface is discretied into nw segments. The segments are numbered along the direction of discretiation. The (x,y coordinates of end points of the segments are ( L( ( End Points : x,y ; nw In the program, the X-coordinates of the initial end points (initial foil surface discretiation before iteration are given by - X cos π nw and then rotated to give the angle of attack. The X-coordinates of the end points can be given arbitrarily by editing the subroutine wgprf. As shown in Fig. 4, constant source/sink m is distributed on each segment, and constant vortex γ is distributed on whole segments. m m The (nw unknown constants, m ( nw (x,y and γ, are determined from the boundary conditions to be described in the next section. It γ (x nw,y nw m m nw can be said that m represents the shape of the This figure is sometimes misunderstood. body through the tangential flow condition and The arrows denote the positive direction in the definition. f course, m γ gives the lift force through the Kutta condition although all of them interact with one an- can be < (sink and γ is negative in the case of upward lift. Fig. 4 Singularity Distribution other. The sum of the source/sink forms the wake flowing from the trailing edge to infinitely downstream. The (x,y direction velocity (u,v can be written as ( nw u u u( x,y Uicos( θ m f ( x,y γg ( x,y nw v v v( x,y Uisin( θ m f ( x,y g ( x,y γ (3-5 -

8 u u v where f ( x,y, g ( x,y, f ( x,y and g v ( x,y are influence functions whose forms are different in accordance with the computation cases (, ( and (3 in Fig.. They are described in the Appendix (equations (A- - (A-4 for case (, equations (A-39 and (A-54 - (A-6 for case ( and equations (A-39 - (A-45 for case (3. It should be noted that the upper bound of the summation in equation (3 n is nw. The (nwth end point appears in the influence functions. n (xc,yc (xc k,yc k (xc 4.3 oundary Conditions nw,yc nw The following boundary conditions are satisfied at the midpoints (control points of the segments. n nw n k Fig. 5 Normal Vectors Tangential Flow Condition Flow should be tangential to the surface at each control point. Letting the k-th control point be (xc k,yc k ((x k x k /,(y k y k /; k nw, this condition can be expressed as ( u( xc k,yc k,v( xc k,yck ( nx k,nyk u( xc k,yck nxk v( xc k,yck ny k (4 u u v f x,y, g x,y, f x,y ; k Lnw where n k (nx k,ny k is the unit vector normal directed outward with respect to the k-th segment (Fig. 5. nx (y y / (x x (y y k k k k k k ny (x x / (x x (y y k k k k k k k k (5 Substituting (3 into (4, we obtain a set of nw linear equations for the nw unknown constants, m and γ, as follows: nw u v [ f ( xc k,yck nxk f ( xc k,yck nyk] m u v [ g ( xc k, yck nxk g ( xc k, yck nyk] γ Uicos( θ nxk Uisin( θ nyk (6 ; k Lnw Another equation to fix m and γ is obtained from the following Kutta condition

9 Kutta Condition t There are a few different ways to satisfy the Kutta condition. Here, a simple criterion that the pressures above and below the trailing edge are equal, (xc nw,yc nw (xc,yc T.E. is taken. From the ernoulli equation, this criterion t nw turns into the velocity criterion, i.e., the absolute Fig. 6 Unit Tangential Vectors at T.E. values of the velocities above and below the trailing edge are equal. Letting the unit vectors tangential to the st and nw-th segments be t (tx,ty and t nw (tx nw,ty nw (Fig. 6, this condition can be written as vi. where ( u( xc,yc,v( xc,yc ( tx,ty ( u( xc nw,yc nw,v( xc nw,ycnw ( tx nw,tynw (7 u( xc,yc tx v( xc,yc ty u( xc nw,ycnw txnw v( xc nw,ycnw tynw (8 tx (x x / (x x (y y ty (y y / (x x (y y tx (x x / (x x (y y nw nw nw nw nw ty (y y / (x x (y y nw nw nw nw nw nw nw nw nw (9 Substituting (3 into (8 yields nw [ ( ( ( nw nw nw ( nw nw nw] u v u v f xc, yc tx f xc, yc ty f xc, yc tx f xc, yc ty m u v u v [ g xc, yc tx g xc, yc ty g xc nw, ycnw txnw g xc nw, ycnw tynw] γ U cos( θ ( tx tx U sin( θ ( ty ty i nw i nw ( Equations (6 and ( form a set of nw linear equations by which we can obtain the nw unknown constants, m and γ. nce we obtain these unknown constants, we can calculate the velocity at each control point using equation (3, and the pressure there using the ernoulli equation. 4.4 Restriction on Foil Shape Since this method is based on source/sink distribution, a foil with a very thin part leads to significant numerical error. For example, a flat plate foil with ero thickness can never be calculated by this method. Even in the case of a thick foil, one should be careful when comput

10 ing for a foil with not only ero trailing edge thickness but also ero trailing edge angle, because thickness near the trailing edge becomes very thin. If the pressure distribution close to the trailing edge becomes unusual, modify the thickness distribution near the trailing edge. 5. oundary Layer Calculation The so-called integral methods based on the momentum integral equation of the boundary layer are adopted to save computation time and simplify the programming. Please refer to Cebeci and radshaw [977] for details on the calculation methods adopted in this program. Laminar oundary Layer and Its Separation Thwaites method modified by Curle and Skan, pp.8- of Cebeci and radshaw[977]. oundary layer calculation starts from a control point which is the closest to the stagnation point. Natural Transition to Turbulence Equation by Cebeci and Smith, pp of Cebeci and radshaw[977]. Turbulent oundary Layer Head s method modified by Cebeci, pp.9-94 of Cebeci and radshaw[977]. It is assumed at the starting point of the turbulent boundary layer that momentum thickness is continuous to the end of the laminar boundary layer and the the shape factor H (displacement thickness/(momentum thickness is.4, which is a typical value of the flat plate boundary layer. The turbulent boundary layer starts at the natural transition point or at the end of the laminar separation bubble. It is assumed that the turbulent separation occurs at H. because it actually occurs at H You can easily change this value by editing one line in the program. Length of Laminar Separation ubble At the beginning of the iteration, the boundary layer is calculated for pressure distribution with no boundary layer effect. If a negative pressure peak at the leading edge appears at that particular angle of attack, the adverse pressure gradient there is higher than that of the expected convergence. This too high adverse pressure gradient sometimes causes immediate turbulent separation after the leading edge laminar separation, disallowing any further iteration. Iumida [984] found from Gaster s measurements [Gaster, 967] that the length of the laminar separation bubble is approximately 5 times the momentum thickness at the laminar separation point, and he started turbulent boundary layer calculation from the end of the laminar separation bubble, which is 5 times the momentum thickness downstream of the laminar separation point. The same countermeasure is adopted in my program if laminar separation occurs upstream of X.. No boundary layer development is assumed in the laminar separation bubble. If the laminar separation occurs downstream of X., the length of the laminar separation bubble is assumed to be and turbulent boundary layer calculation starts immediately

11 6. Curve-Fitting of oundary Layer Displacement Thickness During iteration, the program generates a displacement body by adding the boundary layer displacement thickness to the foil surface. However, displacement thickness is discontinuous at the starting point of the turbulent boundary layer, as described in the preceding chapter. Also, small undulation in the computed displacement thickness disturbs the convergence. Thus, the computed displacement thickness is curve-fitted by the following equation using the least squares method: ( L ( 6/7 3 7 δ s b s b s b3s b4s b7s where δ fitted displacement thickness, s arc-length along the foil surface from the stagnation point, b -b 7 unknown constants to be determined by the least squares. The first and second terms on the right-hand side of equation ( correspond to laminar and turbulent boundary layers on a flat plate, respectively. When turbulent separation occurs, no further boundary layer calculation can be executed downstream of the turbulent separation point. To perform iteration calculation anyway, boundary layer displacement thickness is linearly extrapolated downstream of the turbulent separation point, keeping the same inclination as that at the separation point. 7. Hydrodynamic Forces y integrating the pressure and friction along the foil surface (not the surface of the displacement body, we can calculate the lift coefficient CL, moment coefficient CM and drag coefficient CD. A simple trapeoidal formula is used in numerical integration. In the case of a foil with an open trailing edge (nonero trailing edge thickness, pressure there is assumed to be equal to that at adacent control points where the Kutta condition is satisfied (Fig. 7, and no friction is taken into account. Drag Coefficient Cp Cp Pressure Coefficient T.E. (xc,yc (xc nw,yc nw Cp TE Cp (Cp nw Cp nw Fig. 7 Pressure at pen Trailing Edge Drag coefficient is an important parameter for discussing the foil performance. It is difficult, however, to obtain the drag coefficient precisely because it is a second-order small value. Numerical errors due to pressure computation and numerical integration can become of the same order as the drag coefficient. It is expected that this problem would be improved by using higher-order singularity distribution and surface discretiation [Hess, 973] In addition, the present program tends to predict a lower drag coefficient than the actual - 9 -

12 Flow Decreasing Pressure due to Contraction Additional Drag Contraction due to oundary Layers on Walls Fig. 8 Additional Drag due to Pressure Gradient Induced by oundary Layer Development along Channel Walls value, because of the rough expression of the wake. As shown in Fig., the viscous wake from the trailing edge contracts downstream. To express this more precisely, we should distribute sinks along the wake as Powell did [967]. However, this is not included in this program, resulting in slightly higher pressure near the trailing edge because acceleration due to sinks in the wake is not taken into account. This leads to slightly lower drag coefficient. Error due to this is enlarged for a foil with an open trailing edge. Even a negative pressure drag is often calculated. As another indicator for drag, this program outputs another drag coefficient using Squire and Young s empirical formula modified by Eppler and Somers [98, pp.3], where the drag coefficient is calculated from boundary layer characteristic values at the trailing edge or turbulent separation point. This drag coefficient is usually more reliable than that obtained from numerical integration. It should be noted, however, that Squire and Young s formula was developed for an isolated foil in an infinite flow field. In the case of a foil between parallel walls, you might encounter another problem due to a slight difference in the flow conditions between the computation and experiment. If you perform a foil experiment in a straight channel, boundary layers develop on the top, bottom and side walls, as shown in Fig. 8. Contraction due to the displacement effect of these boundary layers reduces the pressure along the flow because flow is accelerated downstream. Thus, the foil surface pressure in the aft part is reduced, resulting in additional drag. This phenomenon is not taken into account in the present program. This additional drag is usually very small, but you must take it into consideration if you are predicting the foil performance in a very narrow channel. Hopefully, the channel used in the experiment is designed with a very slight diffusion, with boundary layer development on the walls taken into account! 8. Data Input and utput The data input procedure is simplified as much as possible. I hope it is clear how to input data into the program from the comment lines in the program source file and sample data files. Figure 9 is attached to aid understanding. I hope no additional explanation is necessary for the output. To ensure understanding, the definition of geometrical mean velocity which appears in the case of cascade computation is shown in Fig.. - -

13 y Y (X, Y coordinate system is fixed to the foil to represent the foil shape. Leading Edge Angle of Attack, α "sycnt" y-coordinate of the rotation center Foil Trailing Edge X x θ (* "xcnt" X-coordinate of the rotation center to incline the foil by the angle α. Uniform Flow, U i (* Not required to input θ except for cascade computation. θ is automatically set equal to. "xcnt" and "sycnt" determine the foil position in the flow field. Case ( in Fig. y Uniform Flow "b" x (Centerline of the Channel Case (3 in Fig. y x Uniform Flow "b" θ < in the Case of This Figure Fig. 9 Input Data - -

14 V mean Geometrical Mean Velocity Vector (V in V out / V in Velocity Vector Infinitely Upstream U i V out Velocity Vector Infinitely Downsteam V mean V in V out Fig. Geometrical Mean Velocity in case of Cascade 9. Demonstration of Computed Results I have been using this program and its previous versions for about 5 years as a base for my own research mainly on cavitation around hydrofoils, in addition to performing numerous experiments. Therefore, I have had sufficient opportunity to compare the computed results with the experimental data, and I have not observed any maor discrepancy between them because I know the features of this computation procedure well and can evaluate the results properly. In demonstrating the accuracy of predictions by the present program in this document, I mainly use the experimental data obtained by organisations other than my laboratory, since some people view comparison between calculated and experimental results obtained by the same organisation with suspicion. A recent comparison to our tunnel experiment is described by Yamaguchi et al. [999]. 9. Pressure Distribution and oundary Layer Characteristics rebner and agley [95] measured pressure distribution and boundary layer characteristics in detail using a symmetrical foil section, R.A.E., whose thickness-to-chord ratio is %. Powell [967] developed a theoretical computation method based on the displacement body concept and showed very good agreement between his calculation and the results of rebner & agley s experiment. Similar comparisons are shown here for the present program. Figures and show foil and displacement body shape and pressure distribution for angles of attack of 4.9 deg. and 8.8 deg., respectively. The Reynolds number based on the uniform flow velocity and foil chord length is.6 6. Thin lines show the foil shape and pressure distribution with no boundary layer effect as the starting point of the iteration. Thick lines show the results of convergence, i.e., the displacement body shape and corresponding pressure distri- - -

15 Exp. by rebner and agley, 95 Cal. by "prblg.f" with No oundary Layer Effect Cal. by "prblg.f" with oundary Layer Effect Pressure Coefficient, Cp Wing Profile R.A.E. Re.6 x ^6 Angle of Attack 4.9 deg. Pressure Distribution.6 Y Chord Station, X X : Y : R.A.E. Wing Profile "Displacement ody" Profile X Fig. Foil Shape and Pressure Distribution of R.A.E. Foil Section; α 4.9 deg., Re

16 Exp. by rebner and agley, 95 Cal. by "prblg.f" with No oundary Layer Effect Cal. by "prblg.f" with oundary Layer Effect Wing Profile R.A.E. Re.6 x ^6 Angle of Attack 8.8 deg. Pressure Coefficient, Cp Pressure Distribution -.5 Y Chord Station, X X : Y : R.A.E. Wing Profile "Displacement ody" Profile X Fig. Foil Shape and Pressure Distribution of R.A.E. Foil Section; α 8.8 deg., Re

17 Displacement Thickness Momentum Thickness Exp. by rebner and agley; Upper Surface Exp. by rebner and agley; Lower Surface Cal. by "prblg.f"; Upper Surface Cal. by "prblg.f"; Lower Surface Wing Profile R.A.E. Re.6 x ^6 Angle of Attack 4.9 deg. Displacement and momentum thicknesses are nondimensionalied by the foil chord length Chord Station, X Natural Transition Point Experiment: X. (Upper Surface X.85 (Lower Surface Calculation: X.87 (Upper Surface X.78 (Lower Surface Chord Station, X Shape Factor, H Chord Station, X Fig. 3 oundary Layer Characteristics of R.A.E. Foil Section; α 4.9 deg., Re

18 Displacement Thickness Momentum Thickness Chord Station, X Exp. by rebner and agley; Upper Surface Exp. by rebner and agley; Lower Surface Cal. by "prblg.f"; Upper Surface Cal. by "prblg.f"; Lower Surface Wing Profile R.A.E. Re.6 x ^6 Angle of Attack 8.8 deg. Displacement and momentum thicknesses are nondimensionalied by the foil chord length Chord Station, X Experiment: Upper Surface: Natural Transition(? at X. Lower Surface: No Transition up to the T.E. Calculation: Upper Surface: Laminar Separation at X.5 and Turbulent oundary Layer Starts at X. Lower Surface: Laminar Separation at X.937 and Turbulent oundary Layer Starts at X.945 Shape Factor, H Chord Station, X Fig. 4 oundary Layer Characteristics of R.A.E. Foil Section; α 8.8 deg., Re

19 bution. It is seen that the pressure distribution with the boundary layer effect agrees very well with the measured one. A slight discrepancy at the trailing edge is due to the rough wake treatment in the computation, as described in chapter 7. As may already be known to you, the lift coefficient is almost equal to the area surrounded by the pressure distribution, and lift in real flow is a little lower than that obtained from full potential flow calculation with no boundary layer effect taken into account. The present comparison clearly reveals the reason for this. That is because the camber and angle of attack of the displacement body are reduced by a thicker boundary layer on the upper surface attributed to adverse (positive pressure gradient on the upper surface. Early transition to turbulence on the upper surface, which is seen in Figs. 3 and 4, enhances this effect. Figures 3 and 4 compare the boundary layer development along the upper and lower surfaces of the foil for angles of attack of 4.9 deg. and 8.8 deg., respectively. Comparisons of the displacement thickness, momentum thickness and shape factor H (displacement thickness/ (momentum thickness are shown. Although the present program predicts a slightly early transition to turbulence compared to the experiment, computed boundary layer characteristics agree very well with the experiment. It is considered that the discrepancy of the lower surface shape factor is due to measurement error, since the lower surface boundary layer is laminar and very thin. I think that the calculated shape factor is more probable, taking into consideration that H of the flat plate laminar boundary layer is.6. At the angle of attack of 8.8 deg., the computation shows that laminar separation with reattachment occurs at the upper surface leading edge. rebner and agley [959] did not clearly describe whether the natural transition at X. of their measurement is that due to laminar separation or not. I think, however, that laminar separation occurs also in real flow, on the basis of my experience, since the Reynolds number is moderate and a sharp negative pressure peak appears at the leading edge. 9. Hydrodynamic Forces Extensive foil series tests were performed at NACA, USA [Abbott and van Doenhoff, 959]. Their results are very well-known and widely used all over the world. Comparisons on lift, drag and moment coefficients are shown here for a symmetrical NACA66 - section and a nonsymmetrical NACA66 - section. oth foils have a thickness-to-chord ratio of %. Data and programs for computing these foils are contained in the set of the prblg.f package. NACA6 foil series are one of the laminar foil series which were developed to keep a wide laminar flow area at low angles of attack in order to reduce drag by suppressing boundary layer development. NACA6 series were designed theoretically and have a ero trailing edge angle as well as ero trailing edge thickness, resulting in very small thickness in the vicinity of the trailing edge. As described in section 4.4, the present potential flow calculation method has a problem for such a foil. Thus, thickness distribution near the trailing edge was slightly modified. Figure 5 ( shows the NACA66 - foil shape and its modification. The original thickness distribution was replaced with a straight line from the X.7 position to the trailing edge, main

20 ..5 X : Y : riginal Shape Modified Shape (after X.7 Y X ( NACA66 - X : Y :.5 Y X ( NACA66 - Fig. 5 riginal and Modified Shapes of NACA66 - and NACA66 - Foils - 8 -

21 taining the ero trailing edge thickness. The NACA66 - foil shown in Fig. 5 ( is a combination of the NACA66 - thickness distribution with the NACA camber line of a. and cl i.. The details are described in the comment lines of the program naca66.f. First, comparison on a symmetric NACA66 - foil is described. The Marine Propeller Cavitation Tunnel at the University of Tokyo has a Foil Test Section, whose working section is 6mm high, 5mm wide and,mm long [Applied Fluid Engineering Laboratory, 998]. A stainless steel foil model of NACA66 - with 5mm chord and span lengths was tested at this section. The test method is the same as that described by Yamaguchi et al. [999]. The experiment was performed at the Reynolds number Re.5 6. The results computed by the prblg.f program are compared with the results of this test and NACA s data at Re Figure 6 shows the variation of the lift coefficient with angle of attack. The yellow parts of the calculated lines denote that turbulent separation occurs upstream of X.95. Although the calculations agree well with the experiments, stall characteristics cannot be predicted by the present program. This is because the present program is based on boundary layer calculation. Figure 7 compares the variation of the drag coefficient. The drop of the drag coefficient at low angles of attack is characteristic of a laminar foil. The present program expresses this phenomenon well. It also well expresses the fact that a higher Reynolds number gives lower drag coefficient. Figure 8 shows the variation of the lift/drag ratio. Good agreement is also seen between the calculation and the experiment. Small peaks at the angles of attack of about ± deg. are due to the characteristics of laminar foil. The present calculation expresses these peaks to some extent, although NACA s data show the peaks more clearly. Figure 9 illustrates the variation of the moment coefficient around the /4 chord position. Since this foil is symmetric, the moment coefficient obtained from a thin wing theory is. Although a slight discrepancy between the calculation and experiment is observed, I think this figure should be observed from the viewpoint that both calculated and experimental moment coefficients are almost. NACA s data are rather unreasonable since they do not show symmetric variation with respect to the origin. Similar comparison is made for a nonsymmetric foil section, NACA66 -, as shown in Figs. and. Similar good agreement between the calculation and the experiment can be seen, including the effect of the Reynolds number.. Concluding Remarks The characteristics of the computer program prblg.f have been described. This program can compute the hydrodynamic characteristics of an isolated foil, a foil between parallel walls or foils in cascade, in -dimensional incompressible uniform flow field. An iterative computation between the outer potential flow and boundary layer on the foil surface is executed. This technique is a classical one as compared to the recent CFD technique. However, good results can be obtained in a much shorter computation time at moderate angles of attack. This program package is freeware; anyone can use and modify it. However, the copyright - 9 -

22 belongs to Dr. Haime Yamaguchi, University of Tokyo. The contact address is given on the front page of this document. Any comments or questions are welcome, but I may not be able to respond to you because of numerous other commitments. I thank all the students who operated this program. Their comments and feedback were useful for improving the program. References Abbott, I. H. and von Doenhoff, A. E. 959, Theory of Wing Sections, Dover Publications, New York, USA. Applied Fluid Engineering Laboratory, 998, Marine Propeller Cavitation Tunnel, rebner, M. A. and agley,. A., 95, Pressure and oundary-layer Measurements on a Two-Dimensional Wing at Low Speed, A.R.C. R.&M. No p. Casey, M. V., 974, The Inception of Attached Cavitation from Laminar Separation ubbles on Hydrofoils, Proc. Conf. Cavitation, IMechE, Edinburgh, pp.9-6. Cebeci, T. and radshaw, P., 977, Momentum Transfer in oundary Layers, Series in Thermal and Fluids Engineering, Hemisphere Publishing Corporation, McGraw-Hill ook Company. Eppler, R. and Somers, D. M., 98, A Computer Program for the Design and Analysis of Low-Speed Airfoils, NASA Technical Memorandum 8, 77p. FRTRAN program list and 6 figures. Gaster, M., 967, The Structure and ehaviour of Separation ubbles, A.R.C. R.&M. No.3595 Hess,. L., 97, Calculation of Potential Flow about Arbitrary Three Dimensional Lifting odies, Douglas Report MDC Hess,. L, 973, Higher rder Numerical Solution of the Integral Equation for the Two- Dimensional Neumann Problem, Computer Meth. in Appl. Mech. and Eng., Vol.. Iumida, Y., 984, Reply to the Discussion by Ito, M. for the paper "Study on Propeller Design Method in Cavitating Flow (st Report" by Iumida, Y.,. Society of Naval Architects of apan, Vol.55, 984, pp.395 (in apanese. Powell,.., 967, The Calculation of the Pressure Distribution on a Thick Cambered Aerofoil at Subsonic Speeds Including the Effects of the oundary Layer, A.R.C. C.P. No.5, p 8 figures. Yamaguchi, H., Kato, H., Maeda, M. and Toyoda, M., 999, High Performance Foil Sections with Delayed Cavitation Inception, Proc. International Symposium on Cavitation Inception, 3rd ASME/SME oint Fluids Engineering Conference, ASME, San Fransisco, p. (to appear. - -

23 .5 Exp at UT Re.5 x ^6 Cal by "prblg" Cal by "prblg" (Large Separation Lift Coefficient, CL Angle of Attack (deg. ( Comparison with the Results of the Experiment at the University of Tokyo.5 Exp at NACA Re 3. x ^6 Cal by "prblg" Cal by "prblg" (Large Separation Lift Coefficient, CL Angle of Attack (deg. ( Comparison with the Results of the Experiment at NACA Fig. 6 Comparison of Lift Coefficient of NACA66 - Foil - -

24 ..8.6 Drag Coefficient, CD Exp at UT Cal by "prblg" Re.5 x ^6 Cal by "prblg" (Large Separation ( Comparison with the Results of the Experiment at the University of Tokyo Angle of Attach (deg..8.6 Drag Coefficient, CD Exp at NACA Cal by "prblg" Re 3. x ^6 Cal by "prblg" (Large Separation ( Comparison with the Results of the Experiment at NACA Angle of Attach (deg. Fig. 7 Comparison of Drag Coefficient of NACA66 - Foil - -

25 8 Lift/Drag Ratio Exp at UT Re.5 x ^6 Cal by "prblg" Cal by "prblg" (Large Separation Angle of Attack (deg. ( Comparison with the Results of the Experiment at the University of Tokyo Lift/Drag Ratio Angle of Attack (deg. Exp at NACA Cal by "prblg" Re 3. x ^6 Cal by "prblg" (Large Separation ( Comparison with the Results of the Experiment at NACA Fig. 8 Comparison of Lift/Drag Ratio of NACA66 - Foil - 3 -

26 Moment Coefficient, CM-/4C Moment Center /4 Chord Position Positive in Anticlockwise Direction.. -. Exp at UT Cal by "prblg" Re.5 x ^6 Cal by "prblg" (Large Separation Angle of Attack (deg. ( Comparison with the Results of the Experiment at the University of Tokyo Moment Coefficient, CM-/4C Moment Center /4 Chord Position Positive in Anticlockwise Direction Exp at NACA Cal by "prblg" Re 3. x ^6 Cal by "prblg" (Large Separation Angle of Attack (deg. ( Comparison with the Results of the Experiment at NACA Fig. 9 Comparison of Moment Coefficient of NACA66 - Foil - 4 -

27 Lift Coefficient, CL Exp by NACA (Re3x^6 Exp by NACA (Re9x^6 Cal by "prblg" (Re3x^6 Cal by "prblg" (Re9x^6 Cal by "prblg" (Large Separation Angle of Attack (deg. ( Comparison of Lift Coefficient..8 Drag Coefficient, CD Angle of Attack (deg. ( Comparison of Drag Coefficient Fig. Comparison of Lift and Drag Coefficients of NACA66 - Foil - 5 -

28 8 6 Lift/Drag Ratio Exp by NACA (Re3x^6 Exp by NACA (Re9x^6 Cal by "prblg" (Re3x^6 Cal by "prblg" (Re9x^6 Cal by "prblg" (Large Separation Moment Coefficient, CM-/4C Moment Center /4 Chord Position Positive in Anticlockwise Direction Angle of Attack (deg. ( Comparison of Lift/Drag Ratio Angle of Attack (deg. ( Comparison of Moment Coefficient Fig. Comparison of Lift/Drag Ratio and Moment Coefficient of NACA66 - Foil - 6 -

29 Appendix: Influence Functions due to Surface Singularity Distribution Influence functions in equation (3, f u ( x,y, g u ( x,y, f x,y v ( v and g x,y, are derived here. A. Influence Functions for an Isolated Foil in Infinite Flow Field (Case in Fig. A.. Induced Velocity by Point Source or Vortex Assume that a point source M or a point vortex Γ is placed y at x iy in the Cartesian coordinate system (x,y as v wu-iv shown in Fig.A. Here i is the imaginary unit. Letting u and v be the x- and y-direction velocities, respectively, the complex conugate velocity w( at an arbitrary point xiy xiy u is M M Γ w ( u iv π (A- x iy for a point source, and ( w u iv for a point vortex. Γ πi (A- x Fig.A Point Source and Point Vortex in Infinite Flow Field A.. Induced Velocity by Line Source or Vortex Then, let us consider the case where uniform source distribution m is placed along the segment, as shown in Fig.A. ased on equation (A-, we can obtain the complex conugate velocity w( at an arbitrary point as follows: ( w r m dr (A-3 π t where r distance between and, r distance between and t, and t is a point on the segment. Here, we can write r e iarg (A-4 y xiy u m uniform source distribution along -. m r t re arg( - arg( - v wu-iv Fig.A Line Source Distribution x r t t e i arg t i dr e arg dt e t e i arg iarg (A-5 (A-6 (A-7-7 -

30 Using the relations given in equations (A-4 - (A-7, equation (A-3 can be written as ( w m dt e iarg( m t π π (A-8 For the case of uniform vortex distiribution γ along, the following equation can be obtained in a similar manner. ( w γ i e iarg( dt t γ π πi (A-9 Thus, the complex conugate velocity induced by line sources and vortices shown in Fig.4 is nw w u iv m nw ( ( ( γ i (A- π π y dividing equation (A- into real and imaginary parts, we obtain the influence fuctions in equation (3 as f u ( x, y ( x x π ( x x ( y y ( x x ( y y ( x x ( y y ( y y ( y y ( x x ( x x y y arctan ( x x ( x x ( y y y y (A- f v ( x, y ( y y π ( x x ( y y ( x x ( y y ( x x ( y y ( y y ( x x ( x x y y x x arctan ( x x ( x x ( y y y y (A- g u nw ( x, y y y π ( x x ( y y ( x x ( y y ( x x ( y y ( x x ( y y ( x x ( x x y y arctan ( x x ( x x ( y y y y (A-3-8 -

31 Mirror Image Mirror Image Mirror Image Mirror Image Uniform Flow Top Wall ottom Wall Top Wall Uniform Flow riginal Foil ottom Wall Uniform Flow Top Wall ottom Wall Mirror Image Mirror Image Mirror Image Mirror Image Mirror Images due to Top and ottom Walls Positive Mirror Images and riginal Foil Negative Mirror Images Fig.A3 Mirror Images due to Top and ottom Walls Which can be Expressed by Two Cascades g v nw ( x, y x x π ( x x ( y y ( x x ( y y ( x x ( y y ( y y ( y y ( x x ( x x y y arctan ( x x ( x x ( y y y y (A-4 A. Influence Functions for Foils in Cascade (Case 3 in Fig. As shown on the left in Fig.A3, a foil between parallel walls (Case ( in Fig. can be expressed by infinite mirror images due to the top and bottom walls. Since a foil y between parallel walls can be expressed by adding two cascades, as shown in the middle and on the right in Fig.A3, Γ the influence functions of foils in cascade (Case (3 in M ib Fig. are shown here, describing Case ( in section A.3. b M A.. Infinite Row of Point Sources/Vortices Assume that the point sources M and vortices Γ are placed in an infinite row with the period ib, as shown in Fig.A4. The complex coordinates of the sources and vortices are, ±ib, ±ib, ±3ib. In this case, the conugate complex velocity at an arbitrary point is Γ x iy M Γ -ib Fig.A4 Infinite Row of Point Sources/Vortices x - 9 -

32 M w ( u iv Γ i π π ibn n M Γ LL π πi ib ib ib ib M Γ π πi b b 3b LL M Γ π πi ( n ( nb M Γ π πi π b n b n π M Γ π b bi b b π n π b n π M Γ coth b bi b (A-5 The final transformation of equation (A-5 is an application of the power series expansion of the function coth. Equation (A-5 can also be obtained by a conformal mapping procedure. As shown in Fig. A5, let us consider the mapping function which converts the upper half of the ζ-plane into the i b Im i b of the -plane. Using the Schwar-Christoffel transformation, ξ- band area ( axis is folded at ζ: vi., d Aζ dζ A( ζ C (A-6 (A-7-3 -

33 y M xiy ζ exp b η ζξiη b x M M ξ -plane b ( ζ π ζ-plane Fig. A5 Correspondance of Flow between -Plane and ζ-plane where A and C are unknown constants. The point of C corresponds to ζ, and - corresponds to ζ. From (A-7, we obtain ζ exp C A Since (A-8 is a periodic function with the period of πia, πia ib vi., b A π Setting the point of in correspondance to ζ, C Substituting (A-9 and (A- into (A-7 and (A-8, b ( ζ π (A-8 (A-9 (A- (A- π ζ exp b When a source M exists at in a band area ( (A- i b Im i b of the -plane, the flow veloc- ity at - must be M b. Thus the flow volume rate at - is M. Since - and in the -plane correspond to ζ and ζ, respectively, the flow in the ζ-plane is expressed by a sink M at ζ and a source M at ζ. Thus, the complex velocity potential is - 3 -

34 M M M / / Φ M ( ζ ( ζ ( ζ ( ζ ζ π 4π π Substituting (A- into (A-3, (A-3 Φ M ( M ( b b M π ( b exp exp (A-4 π π Thus, the conugate complex velocity is w M dφm ( u iv d π cosh ( M b b π ( b M coth b b (A-5 When a point vortex Γ is placed at in the -plane, the flow in the ζ-plane is expressed by a vortex Γ at ζ and another vortex Γ at ζ. The complex velocity potential is Φ Γ Γ ( ζ ( ζ πi 4πi (A-6 Γ ζ Thus, the conugate complex velocity is w Γ dφγ Γ ( u iv coth d bi b Equations (A-5 and (A-7 agree with equation (A-5. (A-7 A.. Infinite Row of Line Sources/Vortices Now, let us consider the infinite row of line sources/ vortices on the basis of point sources/vortices. As shown in Fig. A6, uniform source distribution m and uniform vortex distribution γ are placed on the segment, and the same flow is repeated with a period of ib. Using equation (A-5 and a similar method to that in section A.., the conugate complex velocity at is expressed by w ( uiv y ib ib m γ ib ib m γ r b bi coth b t dr (A-8 Using the same variable transformations as those in section A.., -ib x -ib Fig. A6 Infinite Row of Line Sources/Vortices - 3 -

35 w m γ b bi ( ( b t coth dt m γ b bi cosh b t dt b t (A-9 y letting π θ b t b, thereby dt θ, π w m γ b bi ( b π π b π b cosh θ dθ θ m γ π πi π b [ ( θ π b m γ π πi b b (A-3 A..3 Condition at Upstream Infinity Equation (A-3 is the base for influence functions. However, the uniform flow condition is given infinitely upstream, so that no disturbance is desired infinitely upstream. Let us observe the behaviour of (A-3 infinitely upstream. m w u iv γ π πi b π exp exp b lim ( b π exp exp b m γ π πi exp b lim exp b m γ π πi lim exp b

36 m γ π πi b Thus, m γ i b b (A-3 u v m b γ b (A-3 It is reasonable to obtain the result that horiontal and vertical velocities are induced by the source and vortex, respectively. The flow field with no velocity infinitely upstream can be obtained by subtracting (A-3 from (A-3. w ( u iv m b π b b γ i i b π b b (A-33 A..4 Influence Functions Equation (A-33 is the base for influence functions for Case (3 in Fig.. Here, we need to express the velocity (u,v using real variables. Since x iy and x iy, ( x x y y ( x x i y y ( x x i y y ( x x ( y y (A-34 n the other hand,

37 b π b exp π exp b b π π exp exp b b (A-35 π π exp ( exp b b π π exp exp ( π b x x i b y y b x x i b y y exp cos exp sin b x x b y y i b x x b y y π exp cos exp sin b x x b y y i b x x b y y π cos π cosh π sin π b x x b y y i b x x b y y cos cosh sin b x x b y y b x x b y y tan i b y y arctan tanh b x x (A-36 Substituting (A-36 into (A-35, b π b cos cosh sin b x x b y y b x x b y y cos cosh b x x b y y b x x sin b y y

38 i b y y π tan b y y tan arctan arctan b x x π tanh tanh b x x (A-37 The imaginary part of the right-hand term of equation (A-37 is tan tan b y y b y y tanh tanh arctan b x x b x x tan tan b y y b y y tanh π tanh b x x b x x tan tanh tan tanh arctan b y y b x x b y y b x x tanh tanh tan π tan b x x b x x b y y b y y sin cosh cos b x x b y y b x x b y y sin cosh π cos arctan b x x b y y b x x b y y cos cos b x x b x x b y y b y y cosh cosh sin b x x b x x b y y sin π b y y (A-38 Letting r x x y y (A-39 R ( x, y

39 π π π cos cosh sin b x x b y y b x x b y y π π π cos cosh b x x ( b y y b x x sin b y y (A-4 I ( x y, π π π sin cosh cos b x x b y y b x x b y y π π π ( sin cosh arctan b x x b y y b x x cos b y y cos cos b x x b x x b y y b y y cosh cosh sin b x x b x x π ( b y y b y y sin (A-4 we can obtain the influence functions for the Case (3 in Fig. as follows: f f g g r x x R x, y y y I x, y ( x, y b πr u x x I x, y y y R x, y ( x, y πr v ( x, y π ( x x I ( x, y( y y R ( x, y nw u r v nw ( x, y r b π nw ( x x R( x, y ( y y I( x, y r (A-4 (A-43 (A-44 (A-45 A.3 Influence Functions for a Foil between Parallel Walls (Case in Fig. A.3. Conugate Complex Velocity As shown on the left in Fig. A7, a line source/vortex between parallel walls is expressed by infinite mirror images due to top and bottom walls. Here, x iy x iy (A

40 y y y xiy x-iy ib ib ib ib Top Wall b ib ib m γ ib x ib m γ x ib ib ib x - ib ottom Wall - ib - ib -ib -ib -ib -ib Line Source/Vortex and Its Mirror Images Positive Images and Itself Negative Images Fig. A7 Line Source/Vortex between Parallel Walls and x iy x iy (A-47 This flow field consists of two infinite rows of line sources/vortices shown in the middle and on the right in Fig. A7. The middle flow field is characteried by ( and, ±ib, ±4ib, ±6ib, ±8ib,. The right flow field is characteried by ( and ±ib, ±3ib, ±5ib, ±7ib,. Referring to equation (A-3, the middle flow field is expressed by w m γ π πi ( b b (A-48 and the right flow field by

41 w m γ π πi ( ( ( b ib b ib m γ π πi b i π b i π m γ π πi cosh b cosh b (A-49 y adding (A-48 and (A-49, and taking the upstream infinity condition into account, the conugate complex velocity around a line source/vortex between parallel walls is w ( uiv m b π b b π cosh b cosh b γ i π b b cosh b cosh b (A-5 A.3. Influence Functions Since b and b ( b can be easily expressed using real variables in the same manner as that described in section A..4, the other terms in (A-5 are expressed here using real variables

42 ( x iy x iy ( x iy x iy ( x x i y y ( x x ( y y (A-5 cosh π cosh b ( b x x i y y ( ( ( cosh cos sin b x x b y y i b x x b y y (A-5 Thus, cosh b π cosh b ( cosh ( cos sin b x x b y y b x x b y y ( π cosh cos b x x b y y b x x sin b y y ( i arctan tanh π b x x tan π b y y arctan tanh π b x x tan π b y y ( cosh ( cos sin b x x b y y b x x b y y ( π cosh cos b x x b y y b x x sin b y y ( b x x π b y y π b x x π b y y sin cosh cos ( i b x x π sin b y y π cosh b x x arctan cos b y y ( cosh ( cos cosh cos b x x b y y b x x b y y sin ( b x x b y y sin b x x b y y (A-53