TNSRDC 18/08 | DAVID W. TAYLOR NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER Bethesda, Md. 20084 DTNSRDC-78/081 A NOTE ON BLOCKAGE CORRECTION by Kwang June Bai APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED ae a i ay 1 ¢ ~ fo WHO! SHIP PERFORMANCE DEPARTMENT RESEARCH AND DEVELOPMENT REPORT ;LOCKAGE CORRECTION | GC | : ye aa November 1978 | DTNSRDC-78/081 Cs MAJOR DTNSRDC ORGANIZATIONAL COMPONENTS DTNSRDC COMMANDER 00 TECHNICAL DIRECTOR 01 OFFICER-IN-CHARGE OFFICER-IN-CHARGE CARDEROCK +: ANNAPOLIS SYSTEMS DEVELOPMENT DEPARTMENT ,, SHIP PERFORMANCE DEPARTMENT = AVIATION AND SURFACE EFFECTS DEPARTMENT 6 STRUCTURES COMPUTATION, DEPARTMENT | MATHEMATICS AND LOGISTICS DEPARTMENT ‘118 SHIP ACOUSTICS PROPULSION AND DEPARTMENT AUXILIARY SYSTEMS DEPARTMENT 957 SHIP MATERIALS ENGINEERING DEPARTMENT 9, CENTRAL INSTRUMENTATION DEPARTMENT 49 L/WHO!I MB IMMUN MI 0037099 tl | UN Ml 5 o 0301 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) REPORT DOCUMENTATION PAGE Be aS TIONS oil T. REPORT NUMBER 2. GOVT ACCESSION NO.| 3. RECIPIENT'S CATALOG NUMBER i DINSRDC-78/081 4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED A NOTE ON BLOCKAGE CORRECTION 6. PERFORMING ORG. REPORT NUMBER 8. CONTRACT OR GRANT NUMBER(2) 7. AUTHOR(s) Kwang June Bai 10. PROGRAM ELEMENT, PROJECT, TASK AREA & WORK UNIT NUMBERS Task Area ZR O11 0201 Element 61152N 12. REPORT DATE November 1978 13. NUMBER OF PAGES 33 15. SECURITY CLASS. (of this report) UNCLASSIFIED 9. PERFORMING ORGANIZATION NAME AND ADDRESS David W. Taylor Naval Ship Research and Development Center Bethesda, Maryland 20084 11. CONTROLLING OFFICE NAME AND ADDRESS David W. Taylor Naval Ship Research and Development Center Bethesda, Maryland 20084 14. MONITORING AGENCY NAME & ADDORESS/(if different from Controlling Office) DECL ASSIFICATION/ DOWNGRADING SCHEDULE 15a. 16. DISTRIBUTION STATEMENT (of this Report) APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED 17. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report) 18. SUPPLEMENTARY NOTES 19. KEY WORDS (Continue on reverse side if necessary and identify by block number) Blockage correction, Mean speed correction formula, Towing tank experiment, Wind tunnel experiment 20. ABSTRACT (Continue on reverse side if necessary and identify by block number) It has recently been shown that a jump in velocity potential exists between infinite upstream and downstream directions when a body translates uniformly along a channel of finite cross section such as a towing tank or wind tunnel. In this report a new blockage correction formula for body speed is proposed. The speed correction formula due to blockage is (Continued on reverse side) 1 R DD , oon"; 1473 EDITION OF 1 Nov 65 Is OBSOLETE UNCLASSIFIED S/N 0102-LF-014-6601 ——— SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) (Block 20 continued) obtained by dividing computed potential jump by body length, assuming that the body is slender or flat in the direction of motion. The potential jump is expressed explicitly in terms of the effective volume, i.e., the sum of the displaced volume and added mass/density of the submerged body, and the depth Froude number, if a free surface is present. As a test of the present speed correction formula, two cases are considered: (1) the Wigley para- bolic ship model, tested in both a small and a large towing tank, (2) a body of revolution (prolate spheroid) tested in a circular wind tunnel. In each case the mean-speed increment averaged over the entire body surface is computed by a three- dimensional, finite-element method applicable to free-surface flow problems. These are shown to be in good agreement with those obtained by the approximate speed correction formula. At high values of Froude numbers, the main difference in the total resistance coefficients measured in the two towing tanks by Tamura is due primarily to difference in model wave resistance computed for the two tanks by a full-fledged, three-dimensional, finite-element method. Results are also compared to those ob- tained by using the speed correction formula of Lock and Johansen. The present formula renders a better approximation than that of Lock and Johansen when the cross sectional area of a flow tunnel is not much larger than the maximum cross section area of the body. UNCLASSIFIED SECURITY CLASSIFICATION OFTHIS PAGE(When Data Entered) TABLE OF CONTENTS Page LUSH OW WGURIES o 6 6 6 6 6 © 00) 6 6 6 OO. 6° 0%) 0 0 00.0 Boo 10 qabat LUGE OW WABHAS 6° 6 616.0 6.0.0 6 0 0006 06 6 6 0 -6.0 686610 iv WMOAWAMIOM 6 6 blo 6 6% 6 06 6 6 O66 G6 6 6 Oe 6 5 6 018 OO O68 Vv INBSTERVNGI SG 6 og ola nd) 6 0 16 SNe ohlG 6 Go a bua Guo! Golo Guin Na il JNO OER USERV AU EIE AD; JON HORIVEMCILOIN, — GiG= os 1G oreo O10 G8 0 6 0 6 9.0 0 G0 1 ile TONMERYOIDOIGIGUON = 6 0 6 0) oo) of OGG oe HG 6646 6°66 6 0"15).6, oFf0 © i BHEOCKAG EH CORRE CMON Mesures simon ireti det) (ru oolon tol otto olen NT oN oe Suro o 3 EXACT MEAN=SPEED! INCREMENT: 2 27. 2 si) cece 2 es 6 wale © Jena e 3 APPROXIMATE MEAN-SPEED INCREMENT . ......-. + © «© » «© « « » 6 INPPAT ICME, 6-6 a 6 GO 00-0 0 6 0 oO 8 8 6 6 6506 66 0 G0 Go 8 THOMAONE AVNSIK JEP IUUIINAE 9G 6 kto ig Go) GN ono 6 6 oN do O86) boo) 0 8 (IONID) GOUININDIL, TGRIIRIGWISE 592655 6 66 094 0 6 6 6 go 160 0-0 8b oOo 0% db COMGLUSIIONS: choo 5 oH 6 GoM B50 co 5 0 6 6 56 6 3 0 6 O55 0 4 oO 6 iL7/ INGRNOILRDEUINNAE 6 6 Ge O66 0 6 GO 6 0 6.50 46505 6 010 0 5 OO © OC 19 UA BANINGHS ¢ 6.a 6 d 56 4 ovGg of0 6 6 6 6-6 6 6 6 6'0 9 OG O10 3 6 21 LIST OF FIGURES 1 - Resistance Coefficients Cos Cs and ce cee in ch Boul OU OM mo os Geta WZ 2 - Corrected Values of (Cp-C ) from Small Tank o, Large Tank x, and ITTC 1957 and ATTC Curves, 3 - Velocity Potential for a Spheroid (a/b=4) in a Wind Tunnel with a Circular Cross Section of Radius Ro ore storie 6 16 4 - Added Mass Coefficient m and Speed Correction Au/U fora Spheroid in) al Circular;Wind Tunnel) - 20.5. 3 3 2% 2 6 « 18 atabag LIST OF TABLES Dimensions of Small, Large, and Extra Large Towalos Werks G 6 6 66 6 05,6 6 6 6 5 0.00 5 0 Wigley Parabolic Model (Tamura Model M1719) .. . Comparisons of Mean-Speed Increment, Computed by Numerical Results and by Present Formula for Bee Ohara ch iis ste ei vorbeedilv eine ta fotancail nolalst vs: pbieleliiest) tony resmitentixenars L Resistance Coefficients of the Wigley Parabolic Model at Two Different Towing Tanks Mieakaik WHI) S 64.0) 6 6 a5060 6.6560 0 6! © 0 6 Frictional Resistance Coefficients Ch and Ga Computed from ITTC (1957) and ATTC Friction Formulas, at a Freshwater Temperature of DOMGeawas. Hecke, Volenere! comer tom tees Comparisons of Mean-Speed Increments on a Spheroid in Wind Tunnel, Computed by a Numerical Method, Approximate Formulas (u_/U=0.0813557 Obtained by Lamb was Used) . . PS hs Nien ee hkl, iv Page 13) 13 18 NOTATION Cross sectional area of towing tank One-half of length of prolate spheroid Beam of ship Radius of maximum cross section of spheroid Block coefficient Frictional resistance coefficient Frictional Resistance Coefficient Total resistance coefficient Wave resistance coefficient Prismatic coefficient Hull fineness parameter Water depth Froude number Ship length Froude number Acceleration of gravity Water depth Potential jump due to blockage Partial form factor Body length; length between perpendiculars Length of waterline Added mass of submerged body Added mass coefficient Ww Radius in cylindrical coordinates R, Frictional resistance Rp Total resistance Wave resistance RY Reynolds number Ro Radius of tunnel wall Sy Wetted surface T Draft of ship U Uniform incoming stream velocity at upstream infinity u Mean speed due to blockage uU Mean speed on the body in unbounded water W Width of towing tank SVs Right-handed rectangular coordinates Au Speed increment Au Mean-speed increment averaged over body v Kinematic viscosity of water fo) Density of water T= (T) T5573) Tangential unit vector co) Total velocity potential oe Total velocity potential in absence of tank (or tunnel) walls 0) Perturbation velocity potential 0) Perturbation velocity potential in absence of tank (or tunnel) wall vi yy V Displaced volume; volume Gradient operator = ‘yt Li TBS gram Task ABSTRACT It has recently been shown that a jump in velocity potential exists between infinite upstream and downstream directions when a body translates unformly along a channel of finite cross section such as a towing tank or wind tunnel. In this report a new block- age correction formula for body speed is proposed. The speed correction formula due to blockage is obtained by dividing com- puted potential jump by body length, assuming that the body is slender or flat in the direction of motion. The potential jump is expressed explicitly in terms of the effective volume, i.e., the sum of the displaced volume and added mass/density of the submerged body, and the depth Froude number, if a free surface is present. As a test of the present speed correction formula, two cases are considered: (1) the Wigley parabolic ship model, tested in both a small and a large towing tank, (2) a body of revolution (prolate spheroid) tested in a circular wind tunnel. In each case the mean-speed increment averaged over the entire body surface is computed by a three-dimensional, finite-element method applicable to free-surface flow problems. These are shown to be in good agreement with those obtained by the approximate speed correction formula. At high values of Froude numbers, the main difference in the total resistance coefficients measured in the two towing tanks by Tamura is due primarily to difference in model wave resistance computed for the two tanks by a full- fledged, three-dimensional, finite-element method. Results are also compared to those obtained by using the speed correction formula of Lock and Johansen. The present formula renders a better approximation than that of Lock and Johansen when the cross sectional area of a flow tunnel is not much larger than the maximum cross section area of the body. ADMINISTRATIVE INFORMATION This work was authorized and funded by the Independent Research Pro- at the David W. Taylor Naval Ship Research and Development Center, Area ZR O11 0201, Element 61152N. INTRODUCTION Many authors have investigated blockage effect and proposed approxi- mate blockage formulas to account for towing tank or wind tunnel bound- aries. 1=9* The first approximation concerning towing tank blockage effects date back more than four decades. Owing to the difficulty encountered in computing flow separation, wake flow, free-surface effects, etc., the exact *A complete listing of references is given on page 21. 1 magnitude of the blockage effect on fluid force acting on a body is too complicated to analyze by purely theoretical means. However, these diffi- culties did not stop engineers from attempting to make simple engineering approximations of the blockage problem. For engineering purposes, compu- tation of a mean-speed increment on a body due to blockage effects has been the main focus of interest in order to make a blockage correction to frictional drag. In the computation, the incremented change in frictional drag due to blockage is determined directly from the computed incremental increase of mean speed over the body surface caused by flow blockage. Two basic inviscid flow-theory approaches have been previously em- ployed. The first approach is based on the so-called one-dimensional, mean-flow theory, using the Kreitner equation, which was first obtained by eretener from the Bernoulli and the mass continuity equations under the assumption that velocity is uniform in each cross sectional plane. To name a few, epee and Kim? used this approach. The second approach is based on successive reflection of images in the walls of a rectangular tank or simpler axisymmetric singularities in case of axisymmetric flows. In this approach, the velocity potential of the flow inside a specified tank boundary can be computed exactly in principle; usually, the potential is represented by a series expansion, and only the first few terms are >" and Landweber and Naleayemae have used the computed. Oniigeraa,© Tamura, latter approach. In all, there exist about a dozen formulas proposed for blockage corrections and each is somewhat different from the other. Some formulas introduce empirical correction faeroren whereas others claim to be based on analytical derivations. Some formulas are proposed to be used only for frictional resistance corrections, whereas other formulas are used for total resistance corrections. An extensive review of the subject has been made by Gross and Watanabenc In the present preliminary study, skepticism is exercised about proposals in speed correction formulas that can be used to correct the total resistance which include the wave resistance in water of finite depth with sidewalls. Herein is proposed a new speed correction formula to be used only for frictional resistance. The wave resistance which has been computed for any towing tank and/or model conditions by using the localized finite-element method previously developed by the em gnerey It seems to be impossible to make blockage corrections to total resistance by using only a single-speed correction formula, even though such formulas have been proposed in the past. The approach used to derive a blockage correction formula herein is different from the two inviscid flow-theory approaches described pre- viously. Derivation of a mean-speed correction formula in this report is based on the potential jump occurring in the three-dimensional flow in a towing tank or wind tunnel. To test the new speed correction formula, numerical computations for a full-fledged, three-dimensional wave resist- ance problem were made. The numerical mean-speed increment on a spheroid was computed exactly for a circular wind tunnel and compared with the results obtained by the new formula; results obtained agree reasonably well with exact numerical results. BLOCKAGE CORRECTION EXACT MEAN-SPEED INCREMENT Steady uniform flow past a ship fixed in a channel has been con- sidered; see sae? The coordinate system is right handed and rectangular. Under the usual assumptions, steady uniform flow may be described by a total velocity potential ® defined by ® (x,y,z) = Ux + > (x,y,z) (1) where » is the perturbation-velocity potential in a channel of finite cross section. Similarly the total velocity potential e. (x,y,z) = Ux + Oe (x,y,z) (2) is defined to describe the flow about the same body in an unbounded fluid, i.e., in the absence of channel boundaries. The fluid speed on a body surface in general increases due to the blockage effect when compared with that of unbounded fluid. However, the speed increment on the body surface is not uniform over the entire surface. For example, the forward stag- nation point of an axisymmetric body remains the same whether in an un- bounded fluid or in a wind tunnel of circular cross section. Nevertheless, a mean speed correction has been traditionally employed for the blockage correction mainly due to its simplicity. To describe a mean-speed: incre- ment, speed increment due to blockage locally on the body surface is defined as Ay Au V (aon) O (3) ay V (6-5) ° > where T = (T, 57 T3) is a unit tangential vector on the body surface; T Die 1 is the component along the x-axis, i.e., the longitudinal direction, and To and T3 are, respectively, the normal and tangential components in the cross sectional plane of the body. Then the "exact'' mean speed increment averaged over the entire submerged body surface is given by Agee - {J V (6-9,) * T ds (4) S (e) oO where 55 is the wetted surface area, and Tt is specified. One natural way of specifying Tt would be as the unit potential flow streamline vector on the body. However, streamlines on a body in bounded and unbounded flows, described by © and oR; respectively, do not coincide in general, except in the special case of an axisymmetric body in a flow facility of circular cross section. In the case of a ship hull, if T= (1,0,0), and 55 = 2° L * T under the assumption that the ship is thin, Equation (4) can be reduced to Au=u-u (5) where 0 2c ar |¢ (3 -¥2) - @(-F>¥°) Je -T 0 uy “An : T { E (4 .y,0) oC (- ¥ .y,0] oy -T where L and T are the ship length and draft, respectively. In Equation (5), the draft T is assumed to be uniform from the bow at x =- L/2 to the stern at x = L/2; the centerplane of the ship is on z = 0. Similarly, for a slender axisymmetric body of revolution in a wind tunnel of circular cross section, the mean-speed increment averaged over the body surface is given by fe) where L — eT bei oy =0 hell) L See ISTE R=0 (6) ta 1 x= a =0 Ws saa 3 E x =- 3 > R=0 where 2 and the peripheral length along a body meridian is approximated by the body length, assuming that the body is slender. APPROXIMATE MEAN-SPEED INCREMENT In this subsection, the method of obtaining an approximate speed correction formula is given, based on the potential jump discussed earlier by Ballo Define K as a jump in the velocity potential $ given in Equation (1) between the infinite upstream and downstream directions. The potential jump K is given by integrating the speed increment along a line in the fluid from a point infinitely far upstream to a point infinitely far downstream. Numerical solutions for practical ship forms at sub- critical speeds in towing tanks and for slender bodies of revolution in wind tunnels indicate that most of the potential jump occurs along the body length. This finding, observed in numerical solutions, will be used as the basis for obtaining the present approximate formula for the speed correction. It is possible to prove this empirical finding by showing that the values of the potential at the upstream and downstream stagnation points are approximately equal to the corresponding asymptotic values of the potential in the simple case of axisymmetric flow. However, the proof will not be discussed here. Thus, the mean speed increment u due to blockage is approximated by (7) Cc G i] lA In a recent simple analysis,* the expressions for the potential jump K in terms of the effective volume and the depth Froude number Fy in three dimensions with a free surface were _ (# +m'/o) U 2. WH (1-F,, ) K (8) *A more detailed analysis in general cases has been submitted in a paper to the Journal of Fluid Mechanics (1978). where ¥ = displaced volume 0 = density of water = towing speed U W = tank depth H = water depth m = m' (F.,) = added mass in the longitudinal direction F., = U/vgH In the derivation of Equation (8), it is assumed that the waterplane area of the ship hull is so thin that a line integral term is neglected, and the body boundary condition is satisfied exactly on the body surface. From Equations (7) and (8) is obtained Asi) SP cel y/o U 2 NG KGa) (9) where A = WH is the cross sectional area of the tank. It is of interest to note that when the value of g approaches infinity, Fu approaches zero, and Equation (9) reduces to the case of a wind tunnel, where Au id Mest: m'/0o ‘um AL (10) It is also of interest to note that when the body boundary condition is linearized, i.e., satisfied on the body centerplane, Equations (9) and (10) further reduce to Ta eee eo AL (1-F, ) in the presence of a free surface and Aaa a ' UATE ao") in the absence of a free surface. APPLICATIONS TOWING TANK EXPERIMENT To test the new blockage correction formula, three sets of compu- tations were first made for the same model in three different towing tanks. The first two tanks had the dimensions given by Tamercaea es see Table l. The third tank was approximately four times greater in cross sectional area than the large tank listed in Table 1, i.e., W = 24 m and H=12m. The specific ship model considered was the Wigley parabolic model (Model M1719 in Tamura), and the equation of the hull surface was given by 2 N i] I+ hol| 2 tee | 1-(2) (11) where L/B = 10, and T/L = 0.0625. The geometric particulars of the models have been given in Table 2. In the computations, the ship hull boundary condition was linearized; thus, speed correction formula (Equation (9')) was used. To test the present mean-speed correction formula, computations were also made from Equation (5) the exact mean-speed increment averaged over the hull surface from the local velocities obtained by the finite-element methods In computing the value of u, from Equation (5), the numerical result for the extra large tank was used in place of the perturbation potential for un- bounded water Oe because the effect of the tank wall and the bottom was found to be negligibly small. Comparisons between the "exact" and approximate mean-speed increments are given in Table 3. Agreement is reasonably good. It should be noted in Table 3 that the exact mean speed averaged on the hull surface us defined by Equation (5), is not only nonzero but also independent of Froude number. It should also be noted TABLE 1 - DIMENSIONS OF SMALL, LARGE, AND EXTRA LARGE Extra Large Tank 24 TOWING TANKS"? 2 Width in meters Mean Water Depth in meters 182) TABLE 2 - WIGLEY PARABOLIC MODEL (TAMURA MODEL M1719) Length between Perpendiculars in meters 8.000 Length of Waterline in meters 7.984 Beam in meters 0.800 Draft in meters 0.500 Volume in meters 1.422 Wetted Surface in meters 9.408 Block Coefficient 0.4453 Prismatic Coefficient 0.6680 TABLE 3 — COMPARISONS OF MEAN-SPEED INCREMENT, COMPUTED BY NUMERICAL RESULTS AND BY PRESENT FORMULA FOR = * FS 0.4 Exact Numerical Results Equation (9') Tank a J i Lad a u/U u/U Au/U = (u-u)/U Au/U Small} 0.017198 | 0.030514 0.0133 0.0128 Large | 0.017198 | 0.019425 0.0022 0.0029 *Results of extra large tank were used to compute Uo as discussed in text. that the free surface effect on the velocity profile on the body surface would be significantly dependent upon whether the hull is in a shallow towing tank or in unbounded water. The present study indicates that the approximate speed correction formula satisfactorily treats the seemingly complicated free-surface effect on the mean-speed increment on the body. The total resistance coefficient C determined experimentally by EY Tamura, and the wave resistance coefficients Cx computed by the finite element method, are given in Table 4. In presenting our results, the total resistance coefficient Co and the wave resistance coefficient Ci are defined as 2s Dee 2/3 C= /e (12) ch ai eee Ww w/ 2 where Ro and R are, respectively, the total and wave resistances. The frictional resistance coefficients, Cy and cu are defined by CP as u SS S (13) AGE Pa peaiobny 2 (Ole e2 22/3 eae URE sais 7/38 ¥ where Sy is the model wetted surface area. The model length Froude number F. and Reynolds number RO are defined by I; FL = U/¥gL (14) Re v/VOL 10 where V is the kinematic viscosity of water. Here the Reynolds number R is obtained by assuming that the freshwater temperature in two towing tanks was 20 C. Table 4 results are given in Figure 1; the wave resistance computed for the extra large tank was taken to be the same as for unbounded water, already mentioned. In Figure 1, hull wave resistance in the large tank is very close to that for the extra large tank. Thus, the blockage effect on wave resistance is very small for the large tank. Also, the main difference in the total resistance coefficients Cy measured in the small and large towing tanks is due primarily to the difference in the model wave resistance computed for the two tanks. Table 4 gives the speed corrections computed from Equation (9') along with the corrected values of (C,-C_). The corrected value of (C.-C) is given by (C,-C ) (u/ (uU+Au)). Table 5 gives the frictional resistance co- efficients Ch and Co» computed from International Towing Tank Conference (ITTC) (1957) and American Towing Tank Conference (ATTC) friction formulas. In the present study, it is assumed that the total resistance less the computed theoretical wave resistance is approximately equal to the frictional resistance, since the ship hull is thin and smooth, i.e., form drag is assumed to be negligibly small. If we make use of the @renailile correlation of partial form factor k. with hull-fineness parameter = C. ¥(B/L) (2T/L) for the Wigley parabolic model with Fh = 0.5, we find that k = 0.04; i.e., form drag is estimated to be only 4 percent of the frictional drag and a still lower percentage of the total drag. Accord- ingly, speed correction Equation (9') was applied to the resistance component co = Co = ce to correct for blockage effect. Results given in Table 5 are shown in Figure 2. In Figure 2, the corrected values of (C.-C) are lower than the values of CE given by ITTC and ATTC friction formulations, indicating negative hull form drag, which is not acceptable. In other words, if the form drag coefficient and other corrections had been added to the values of ITTC and ATTC friction coefficients, this discrepancy would be even larger. The discrepancy seems to have been caused by computed values of the wave resistance being too large. a LEGEND: SMALL TANK ———=; LARGE TANK re) EXTRA LARGE TANK RESISTANCE COEFFICIENTS 0.325 0.35 0.375 0.40 F, = U/JoL Figure 1 - Resistance Coefficients, Cos Cis and 12 C F TABLE 4 - RESISTANCE COEFFICIENTS OF THE WIGLEY PARABOLIC MODEL AT TWO DIFFERENT TOWING TANKS (MODEL M1719) Cc C a Ww (C.-C) (FL) Experiment Numerical by Tamura by Bai Corrected Corrected SO OO SHO) CO © © Values in both tanks are converted to the case without blockage. TABLE 5 - FRICTIONAL RESISTANCE COEFFICIENTS Ch AND cm COMPUTED FROM ITTC (1957) AND ATTC FRICTION FORMULAS, AT A FRESHWATER TEMPERATURE OF 20 C DEAN; ALE) 3y7/ ATTC F R L n A A 0.003565 0.003444 0.003516 0.003400 0.003470 0.003360 0.003429 0.003323 13 A ITTC 1957 Ce ° Ee ° wees . 0.325 0.35 0.375 0.40 0.425 FL =U VoL Figure 2 - Corrected Values of (Cp-C_) from Small Tank o, Large Tank x, and ITTC 1957 and ATTC Curves C. 14 In the numerical computation of wave resistance by the finite-element method, 44 nodes on the ship hull surface, i.e., on the centerplane, and 1496 nodes for the entire fluid domain were taken. One may expect more refined results by reducing the size of finite elements. To treat low values of Froude number accurately, smaller and more elements are necessary. WIND TUNNEL EXPERIMENT As a second example, the blockage effect was considered for a wind tunnel having a uniform circular cross section of radius Ro: The specific body geometry considered was a prolate spheroid with its meridian profile given by silt (15) for the special case when a/b = 4. The potential flow for the axisymmetric boundary configurations con- sidered herein could have been computed by the conventional method of integral equations; i.e., the axial source and doublet distributions or the vortex sheet on the surface, etc., as discussed in Landweber. However, the velocity potential has been computed by the finite-element method. Computations have been made for seven values of Ro/b salen 25) 655 25 Sy45 54 nach IS) eulilsatopev y/o a, \eiovexal R/b = 15 was computed, the effect of the tunnel wall on the body surface was negligibly small as if the body were moving in an infinite fluid. The value of u,/U defined in Equation (5), computed by using the result of Ri/b = 15, was 0.08185, whereas that computed by using the exact analytic result for the unbounded Water, -Laec, Ri/b =o, given in item was 0.08156. The computed velocity potential > is shown in Figure 3 for three values of Ri/b = 1.25, 1.5, and 15. To illuminate the assumption made to obtain the present approximate mean-speed correction, Figure 3 shows straight lines drawn from the origin to the asymptotic values of K/2 at the 15 ¢/Ub LEGEND: ON BODY — — — AT R/b = 1.25 —.—— LINEAR POTENTIAL VARIATION ASSUMED IN THE PRESENT SPEED CORRECTION FORMULA (THE SLOPE IS THE SPEED CORRECTION) Figure 3 - Velocity Potential for a Spheroid (a/b=4) in a Wind Tunnel with a Circular Cross Section of Radius Ro 16 the downstream stagnation point x = L/2. The slope of each straight line is equal to the speed correction defined by Equation (7). Owing to the skew symmetry of the potential with respect to x = 0, the result for the upstream half-body can be obtained from the downstream potential shown in Figure 3. The velocity potential increases monotonically from a value slightly lower than - K/2 at the upstream stagnation point to a value slightly higher than K/2 at the downstream stagnation point on the body surface. However, the potentials at R = 1.25b approach monotonically the asymptotic values at both ends for Ri/b S 1525 aml R,/b St 355 In Table 6 the approximate mean speed correction given by Equation (10) is compared with the exact mean-speed correction computed from Equation (6). Table 6 also gives the speed correction obtained by the Lock and Johansen formula, which is given in Ponee as = 3 ta = 2,501 iE ) (16) When Ri/b < 3, our approximate results show better agreement with the exact numerical results than with those of Lock and Johansen. In Figure 4 computed values of the added mass coefficient and the mean speed correction Au/U are shown as a function of b/R.- In Figure 4, note that for b/R, > 0.765, the contribution of the added mass to the speed correction in Equation (10) is more dominant than the contribution of the displaced volume, i.e., m=m'/p¥ > 1. This finding indicates that a crude blockage correction, based on only the local cross sectional area of the body using one-dimensional theory, cannot always give a good approximation of the mean-speed correction when the added mass coefficient is not small. CONCLUSIONS In the present study a new mean-speed formula for corrections caused by blockage is proposed. The approximate formula is tested by comparing Ly) m AND Au/U TABLE 6 — COMPARISONS OF MEAN-SPEED INCREMENTS ON A SPHEROID IN A WIND TUNNEL, COMPUTED BY A NUMERICAL METHOD, APPROXIMATE FORMULAS (u ,/U=0. 0813557 OBTAINED BY LAMB WAS USED) Present Formula (Equation (6)) Lock and Johansen (Equation (16)) 1.25 i aes | b/R,, Figure 4 - Added Mass Coefficient m and Speed Correction Au/U for a Spheroid in a Circular Wind Tunnel 18 it with an exact numerical mean-speed correction, computed by the finite- element method for both a towing tank experiment and a wind tunnel experi- ment. The two predictions are shown to be in good agreement for both facilities. It is shown that the effect of added mass coefficient on the speed correction of a body is very significant as the blockage effect in- creases. It is also found that the main difference in the total resistance coefficient measured in a large and a small towing tank is due primarily to the difference in the model wave resistances computed for the two tanks. Further investigation is necessary to take into account other blockage corrections due to viscous effects such as flow separation and wake dis- placement thickness effects. ACKNOWLEDGMENT The author would like to thank Mr. Justin McCarthy for suggesting this problem and for his encouragement during the work. 19 [ hee te Ae tea (yer wba « Sauer ien omernn a ry nt : ie = an u rn 7 eeu 1 a vig , REFERENCES 1. Tamura, K., "Study of the Blockage Correction," Journal of the Society of Naval Architects of Japan, Japan, Vol. 131, pp. 17-28 (1972). 2. Tamura, K., "Blockage Correction," The 14th International Towing Tank Conference, Ottawa, Canada, pp. 173-182 (1975). 3. Kreitner, J., "Uber den Schiffswiderstand auf beschranktem Wasser," Zeitschrift Werft, Reederei, Hafen (1934). 4. Hughes, G., "Tank Boundary Effects on Model Resistance," Transactions of the Institution of Naval Architects, Vol. 103, pp. 421-440 (1961). 5. Kim, H.C., "Blockage Correction in a Ship Model Towing Tank," The University of Michigan Report 04542, Ann Arbor (1963). 6. Ogiwara, S., “Calculations of Blockage Effects,'’ The 14th International Towing Tank Conference, Ottawa, Canada, Vol. 3, pp. 163-172 (1975). 7. Landweber, L. and A. Nakayama, "Effect of Tank Walls on Ship- Model Resistance," The 14th International Towing Tank Conference, Ottawa, Canada, Vol. 3, pp. 62-91 (1975). 8. Pope, A., "Wind Tunnel Testing,'' John Wiley and Sons, Inc., New York, p. 319 (1947). 9. Gross, A. and K. Watanabe, "On Blockage Correction," The 13th International Towing Tank Conference, Appendix 3, Berlin/Hamburg, Germany (1972). 10. Bai, K.J., "A Localized Finite-Element Method for Steady, Two- Dimensional Free-Surface Flow Problems,'' Proceedings of the First International Conference on Numerical Ship Hydrodynamics, Edited by J.W. Schot and N. Salvesen, pp. 209-229 (20-22 Oct 1975). Zi 11. Bai, K.J., "A Localized Finite-Element Method for Steady Three- Dimensional Free-Surface Flow Problems," Proceedings of the Second International Conference on Numerical Ship Hydrodynamics, held at the University of California, Berkeley, Calif. (1977). 12. Bai, K.J., "A Localized Finite-Element Method for Two-Dimensional Steady Potential Flows with a Free Surface," Journal of Ship Research, Vol. 22), No. 4 @978). 13. Granville, P.S., "Partial Form Factors form Equivalent Bodies of Revolution for the Froude Method of Predicting Ship Resistance," Paper 9, Proceedings of the First Ship Technology and Research Symposiun, Society of Naval Architects and Marine Engineers, pp. 9-1-9-13 (1975). 14. Landweber, L., "“Axisymmetric Potential Flow in a Circular Tube," Journal of Hydronautics, Vol. 8, No. 4, pp. 137-145 (1974). 15. Lamb, H., "Hydrodynamics," Sixth Edition, Cambridge University Press, Great Britain, p. 738 (1932). 22 Copies INITIAL DISTRIBUTION WES CHONR/438 Cooper NRL 1 Code 2027 1 Code 2627 ONR/Boston ONR/ Chicago ONR/Pasadena ONR/San Francisco NORDA NOO/Lib (Naval Oceanographic Office) USNA 1 Tech Lib 1 Nav Sys Eng Dept 1 Jewell 1 Bhattacheryya 1 Calisal NAVPGSCOL 1 Library 1 Garrison NADC NOSC, San Diego 1 Library 1 Lang 1 Higdon NCSC/712 D. 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