THE WAKE FRACTION OF A GEOSIM

Transcription

1 THE WAKE FRACTON OF A GEOSM By Z. BENEDEK Department of Hydraulc Machnes Poly techncal Unversty BUdapest (Receved September 1966) Presented by Prof. B. BALOGH a model scale A wetted surface of shp (m 2 ) Ao propeller dsc area (m 2 ) C F d D g h k L p RF T v v A V 4 " o=t D - n vscous resstance coeffcent Symbols used RF cf=--- _1_ov2 A 2 - mean value of the thckness of the boundary layer (m) dameter of screw (m) gravtatonal acceleraton (m sec-2) loss of knetc energy of kg water (m) constant length of waterlne (m) functon of wake fracton vscous resstance of shp (kp) draught (mean value) (m) shpspeed (m sec-) propeller advance speed (m sec-) effectve wake fracton W = v - v VA Q water densty (kp m- 4 sec 2 ) j specfc gravty of water (kp m- 3) The value of the effectve wake fracton s the most mportant factor n the desgn of a propeller. We use a lot of smple approxmatve relatons for precalculaton but the exact value s determnable only by the nvestgaton of the model of the shp and the propeller. However the nvestgaton of the Vctory model famly has ndcated that models made n dfferent szes gve us dfferent values of the wake fracton for the same shpspeed [1]. E.g. the values of the wake fracton are the follow~g n loaded condton of shp (at even keel) at knots shpspeed: model scale W

2 418 Z. BENEDEK The values are dfferent n the same way also at other shpspeeds and n other shp condton (lght condton trmmed by the stern). f the model scale s greater the wake fracton s also greater. We could not say that there s some error of the measure but we must assume that t s a knd of scale effect. Therefore we need a method for the extrapolaton whch gves us a possblty to calculate the wake fracton of the shp from the measured wake fracton of her model. The speed of adv-ance of a propeller can be defned wth ts three components. The potental flow around the shp gves a relatve velocty n the place of the propeller. The local speed of water n the stern wav-e system gv-t's the second component. The thrd component s defned by the local v-eloctes of the boundary layer of the shp hull. The pcture of a potental flow of a perfect flud s determned by the shp form only. But there s a boundary layer around the shp n the v-scous flud and the thckness of the boundary layer s dfferent at the shp and the model therefore the potental flow around the shp and model s also dfferent. But ths dfference s neglgble and we can say that the potental component s the same at the model and at the shp. The second component s a functon of the Froude number and the shp form. Because the Froude numbers of shp and model are equal ths component s also equal f we dsregard the changes of the thckness of the boundary layer. The veloctes of the water n the boundary layer are defned by the Reynolds number and hy the roughness of the shp hull. Thel'efore the thrd component the so-called vscous component s dfferent at the shp and at the model. Thus there s a dfference hetween wakes of the shp and model or of models made n dfferent szes owng to the dfference of the vscous wake component [2]. The water gong along the shp near the hull surface has a loss of ts knetc energy. The loss of the knetc energy of one kg water s equal to the power of the frctonal resstance dvded by the mass of water gong near the surface of the shp durng one second: 1 9A --(CF v- v h = _2 = _1 L_'_C_F v 2 A 2 g d yd-v L (1) where ( s the densty of water y s the specfc grav-ty of water r s the shpspeed A s the ".-etted surface of the shp L s the length of the waterlne of

3 THE WAKE FRACTO.Y OF A GEOSDf 419 the shp d s the mean value of the thckness of the boundary layer. But there s a defned mean value at each shpspeed and propeller advance speed whcb s constant at each shpspeed. Therefore we can wrte the shpspeed: (2) where X s a constant for a shpspeed. W'th ths the frst equaton s: h Lcp X2 (' 2 g d 4. =----- t'-r-t'a)- f we ntroduce the k constant k-~ - X2 then Thus the loss of knetc energy of one kg water: From ths 1 _ v A = ~ + t'a ' v k t'. (5) Thc effectve wake fracton defned by TAYLOR [3]: from ths w = 1- v A VA 1 -- V 2 ---= va W V V therefore =~+~ w 2 2 (6)

4 420 Model scale Shp. speed kt 3 w calculated 'l-- ~;; b Z. BENEDEK Table 1 Loaded condton ' w measured b dfference % o o _ : w calculated Lght condton lo3 w measured ~ dfference 0' " ; ;

5 THE WAKE FRACTOS OF A GEOSM 421 Table 1 contnued Model scale Shpspeed kt 3 w calculated Loaded condton ' w measured dfference 0/ ' ' ' w calculated Lght condton dfrrence 0' ' The value of k contans the mean thckness of the boundary layer (d) the length of shp (L) and the quotent of shpspeed and mean speed (X) so t s the functon of Reynolds number and the roughness of the surface at the same shp form (at geometrcally smlar models so-called "geosms"). The Cp s also the functon of these two thus we may say the k s to be the functon of Cp. n ths way the rght sde of equaton (6) s the functon of the vscous resstance coeffcent: We have calculated the values of k Cp f(' P=-+-= cp) 2 2 C p p=w (7) wth the measured data of the Vctory model famly [1] for all models at dfferent shpspeeds. The calculated values of p are plotted on the Cp n Fgs 1 and 2.

6 4~2 A lnear functon s defned by the plotted ponts n both the loaded and the lght condton of the shp: p = a cp b (8) - 7f. f. VCTOR' ~/ loaded ccndtfo.l o (+a=50) 9 A 0 0 (oq =30)..." ~~. + (oa =23J (+Q=) 8 L- ~ ~~ f er Fg. 1 tff. w la VCTORY lght condton 9 8 Fg The values of the constants a and b determned by the mean valu(~s wake fractons and vscous resstance of the models n loaded condton (at even keel) a = 0.77 b = n lght condton (trmmed by the stern) a = 1.43 b = of

7 THE WAKE FRACTON OF A GEOSM 423 The most mportant geometrcal data of the shp are the follown g: n loaded condton n lght condton length of the waterlne L (m) draught (mean value) T (m) wetted surface of shp A (m 2 ) dameter of screw D (m) dsc area of screw Ao (m 2 ) A relatons: Ao L T T D We can wrte the values of the constants a and b wth a good approxmaton: L n the loaded condton n the lght condton a =.3 T a=.3--- T A D Ao T A b = DA o = "J 0.77 b = = a = = "-/ b = = 0.00'7 ~ The calculated values of the wake fracton accordng to equatons (7) and (8) wth the above-mentoned values of a and b are gven n the table

8 424 Z.BENEDEK The measured values of the wake fractons and the dfference of' the measured and calculated wake fractons n the percentage of the measured values are also gven. The dfferences (the errors) are the followng n the loaded condton: n 33.3% of the cases the errors are below +1% n 29.2% of the cases the errors are between ±1-2% n.7% of the cases the errors are between J..2-3% n 20.8% of the cases the errors are over ±3% The mean value of the errors s 2.1%. f we dsregard the extremely hgh errors the mean value s 1.3%. The errors are ahout % only at the model made n scale 40. We can assume that there s some error of measurement n the consequence of the low Reynolds number (lg Re = ). n the lght condton of the shp the errors are the followng: n 33.3% of the cases the errors are below : 1% n 31.3% of the cases the errors are between +1-2% n.5% of the cases the errors are between ±2-3% n 22.9% of the cases the errors are over ±3% The mean value of errors s 2.7% but apart from the extremely hgh errors the mean value s only 1.1 %. The errors are about % only at the model made n scale 50. The results of the nvestgaton of the rough model made n scale 6 are not gven n the mentoned paper [1]. Therefore t would not he possble to control ths method n the feld of the hgher roughness of ths motorboat (CF = = ). f we consder the extremely hgh errors of the mentoned models we can obtan the followng concluson accordng to the nvestgatons of the two Vctory famles at - knots shpspeed: 1. We can wrte the effectve wake fracton of a shp as the functon of the vscous resstance coeffcent wth a good approxmaton (see equatons (7) and (8)): 1 b -=a+- (9) w Cp 2. The gravtatonal component of the wake fracton s neglgble because the dfferences of the measured and calculated values of the wake fracton are about 2-3% at dfferent shpspeeds. 3. The mentoned low values of errors prove that the potental component of wake fracton gves very low dfferences at dfferent Reynolds numbers and dfferent roughnesses.

9 THE WA.KE FRA.CTO\- OF A GEOS[y[ But the presence of the potental and gravtatonal component s demonstrated besde the frctonal component. n Fgs 1 and 2 we can see that the lne determned by the measured ponts of one of the models has a bgger :;lope than the lne of the p. = f (c F)' We cannot obtan more exact nformaton n ths queston. The results of further nvestgatons are doubtful f we take nto consderaton that the dfferences of the values of effectve wake fractons are very low at dfferent shpspeeds (we can assume that the errors of the test measurement have the same values) and the propeller has a dfferent Reynolds number n open water and behnd-condton at the tests. 5. f we make the model experments of any shp wth two models n dfferent szes or wth one model but wth two dfferent roughnesses and we calculate the values of Cp p=w n the two mentoned cases by the mean values of the measured wake fractons and vscous resstance coeffcent we can obtan the straght of p = f (c p). The effectve wake fracton s determnable wth ths extrapolator for the dfferent roughenesses of the shp. f we nvestgate one model wth two dfferent roughnesses we could obtan the tests of the rougher model nstead of the usual overload tests. Summary The scale effect of the dfferent measured data of shps were nvestgated by means of the results of geometrcally smlar models (geosm). The values of the wake fracton of models made at dfferent scales are very dfferent. But t s possble to wrte the wake fracton as a smple functon of the vscous resstance coeffcent wth a good approxmaton. The wake fracton of the shp s determnable wthout scale effect by means of ths functon from the measured data of ~wo models of shps. The method gves a possblty for the determnaton of the wake fracton of shp wth dfferent roughness too. References 1. LAP A. J. W. and VAN MANEN J. D.: Scale effect experments on Vctory shps and models (Parts and V). Transacton of the Royal nst. of Naval Archtects GROTHEUS-SPORK H.: On geosm tests for the research vessel Meteor and a tanker. Transacton of the nst. of :Marne Engneers B.UOGH B.-VK..\R T.: Haj6k elmelete. Akadema Kad Zoltan BENEDEK Budapest X. Sztoczek u Hungary 6 Perodc. Polytcchuca M_ X/4_