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cI3 SE?.
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ARCHIEF
Ref. PAPER 4/2 - SESSION i
lz)
Lab. v ScheepsbouwkunJe
Tech&sche Hogeschoo(
Dell L
SYMPOSIUM ON
"HYDRODYNAMICS OF SHIP AND OFFSHORE PROPULSION SYSTEMS"
HØVIK OUTSIDE OSLO, MARCH 20. 25,1977
"INTERACTION OF AFTERBODY BOUNDARY LAYER AND PROPELLER"
By
Thomas T. Huang and Bruce D. Cox
David W. Taylor Naval Ship Research and Development Center Bethesda, Maryland
ABSTRACT
Measurements of the boundary layer characteristics and pressure distribution on axisymmetric bodies with and without a stern propeller in operation are used to verify new analytical techniques for computing the
hydrodynamic interaction. It is shown that both the in-fluence of the propeller on the body pressure distribution, which causes an increase in resistance (thrust deduction),
and the interaction of the propeller with the stern boundary layer velocity profile can be predicted accurately by
in-viscid flow analyses. A new technique is proposed for cal-culating the effective wake which is an important input in
propeller design. Finally, it is concluded that the primary effect of scale is a change in the nominal wake profile which can be accurately computed using boundary-layer theory
for axlsymmetric bodies.
INTRODUCTION
Many single-screw ship propellers operate inside of thick and.possibiy separated stern boundary layers. The propeller normally actsto accelerate
the flow in the thick boundary layer which results in a decrease of
boundary-layer thickness. The effective inflow velocity distribution experienced by
the propeller depends on the interaction of the propeller and the thick stern boundary layer, and jg different from the nominal velocity
distribu-tion measured in the absence of the propeller., In addition, the upstream
suction of an operating propeller generates higher stern pressure drag and
skin-friction drag. Naval architects ref er to the drag' increase due to
pro-peller suction on the afterbody as thrust deduction. A knowledge of both the effective inflow and the added drag is essential for design of a pro-peller which meets specified propulsion requirements.
In order to focus on the physical nature of the comple,x interaction
between a propeller and a thick stern boundary layer, axisymmetric bodies were chosen for the present investigation. The geometric simplicity of the axisymmetric bodies offers considerable experimental and computational con-venience in studying the fundamental aspects of the interaction. The Laser
Doppler Velocimeter (LDV) was successfully used by Huang et al.,1 to measure
velöclty profiles very close to the propeller. The measured differences
between thé velocity profiles, stern pressure distributions, and stern shear stress distributions with nd without an operating propeller, provide the
necessary clues to the proper understanding of the af.terbody boundary layer and propeller interaction.
New and improved analytical techniques1 have been developed as' a result
of insignt gained from the experiments The first new technique computes
trie effective velocity distribution from the nominal velocity distribution
tneasuredjn the absence of a propeller. he second.improved analytical technique solves for the pressure distribution and velocity profiles over the entire body by considering the interaction between the displacement thicknéss of the boundary layer on the body and the external potential flow
in the absence óf a propeller. The third improved technique uses a lifting-surface representation of the propeller to solve för the thrust deduction añd the difference in pressure distribution with and without the propeller.
A brief derivation of the analytical techniques will be given and còm-parison of theOretical predictions and experimental datá at model Segle will
be made. The application of the present techniques to solve the problem of full-scale propeller/boundary-layer interaction on axisynetric bodies will
be discussed.
PROPELLER AND STERN BOUNDARY-LAYER INTERACTION
The bouñdary-layer velocity distribution in the absence of the
propel-ler will be called the nominal profile. The nominal profile at the
propel-ler plane of a ship model is usually measured by a standard wake rake. With a propeller operating, the flow over the afterbody is. normally accelerated.
The mutual interaction of the propeller and the thick stern boundary layer
results 'in a new resultant velocity profile. An effective velocity profile
is defined to be the resultant inflow velocity profile with propeller in
operation minus trie propeller-induced velocity profile as computed, from lifting-surface theory. The effective profile is an important input to propeller design and is essential for the correct prediction of powering,
cavitation performance, and unsteady forces. It is. therefore essential to
develop reliable and sound theoretical procedures to calculate the effective
profile.
The influence of a stern-mounted propeller on the flow field past
different bodies of revolution and, a flat plate was measured. by Hucho,2'3'4 although no attempt was, made to actually calculate the effective wake
dis-tribution. Investigations related to this subject were made by Wertbrecht,5
6 '.7
'8
flickling, Tsakonas and Jacobs, and Wald. Estimation of effective wake
9 . . lÓ . . '' 11
was made byRaestad, Nagamatsu and Sasajima, and Titof f and Qtlesnov.
The on'lyknown previous effort to theoretically addtess this problem is due to D.M. Nelson* who developed an unpublished computer program fôrcalc4ating
the effective' wake 'from the measured nominal wake and static pressure dis-tribution across the boundary layer. The method to. be presented 'below.
*.
Priv ate counïcation.
requires only the measured nominal wake distribution. Seriouè effort has been made in this work to compare the theoretical wake prèdictions withwake distributions measured by an LDV in the presence of a propeller.
The experimental evidence given in Reference 1. allows one to conclude
that the influence of propellers n stern boundary layers is significant in a region extending two propeller diameters upstream of the propeller. Up-streamof the propeller, the mean circumferential velocity, Ve, Is idén-tically equal to zero on an axlsyetric body both with and without the. propeller in operation. The following assumptions are made to derive a
theoretical approximation of the hydrodynamic interaction between a
propel-ler and a. thick stern boundary layer upstream of the propelpropel-ler: (a) the flow is axisynunetric and the fluid is incompressible; (b) the interaction
of propeller and nominal velocity profile is considered to be inviscid in nature; thus, propeller-induced viscous losses and turbulent Reynolds stresses are neglected; (c) the conventional boundary-layer assumption,
v/ax<3u/ar, is assumed to be valid för the nominal boundary layer .n
the absence of a propeller; and Cd) upstream of the propeller, no energy is added to the fluid by the propeller, and the propeller-induced velocity
field upstream of the propeller Is irrotational.
The vector equat-ion Of steady motion for an inviscid fluid is given by (see, for éxample, Ref erence.12)
-V X (A) = grad H, (1)
where is the fluid velocity, = V X is the vorticity vector, and
H.= p+-. p(.) is the total heàd. For cylindrical polar coordthätes (r,
O, x) w1th= (v,v = O, u), the radial
cOmponent Of Equation (1) maybe writteñ1 as
lau aii\ ru
i x
ri
laHd'Y
xaH
X\_
¡) WT'
where 'Y is the stream function, for an incompressible axisyetric flow
definedby .
i. a'v
u
--x r
r r
Since the flow velocities are increased due to the action of the pro-peller, stream surfaces are shifted closer to the body surface. As shown
in Figure 1, a typical stream surface moves inward from r to r while the velocity is increased from the nominal velocity u to u due to the
inter-X p
action of the propeller with the nominal velocity profile. The resultant
velocity u1 as would be measured in front of the propeller will be called
the apparent total velocity profile. The effective velocity is obtained by substracting the computed propeller-induced velocity from the apparent
total velocity.
Since no energy is added to the upstream fluid by the propeller and since we assume no change of viscous losses due to propeller-induction effects, the total pressure head within the same stream annulus remains constant with (denoted by a subscript p) and without the propeller in operation; thus, upstream of the propeller
1
3Hl(3H\
a
pk5i7)p
or from Equation (2)
The radial velocity with the propeller in operation is assumed equal to
y + y
, where y is the radial velocity without propeller and y is ther pr r pr
circumferential average propeller-induced radial velocity. Furthermore,
5 /9u x
3v\
r 1 r p 3u p 3(vr+v
pr) (3)rI3r
3x1
f 3r pthe normal boundary-layer açprox1.mation avr/ « au/ar is assumed to be
valid for the nominal profile and the propeller-induced velocity field is assumed to be irrotatioñal, which requires
av 3u
Vx
. prpl ax ar
p
where1 is the resultant propeller-induced velocity, and ua is the circumferential-average propeller-induced axial velocity. With these two
approximations, Equation (3) yields a simple relationship for the location of the new stream surface r
p
3u
¡au
3u1 x i i p a
Tta.r
:\.
-
pWithin a. given Stream annulus the. massf low is constant, which at a given
axial location Of x may be written as
d'1'.á ru dr r u dr
:P p:p
Inserting this expresion Into Equation. (4) yelds
u d
xx
n u d(u - u ) (6)p p a
Equations (5) and (6) are theTgoverning eqùatons for the propeller.
and stern bOundary-layer interaction. The nomiñal velocity profile, ui(r).
and the circumferential-average propeller-induced axial velocity profile, u (r ), can be used to obtain the new location of, the stream
ap
urfäce r- p
are equal on the body surface. Far outside of the boundary layer the flow
is uniform without any vorticity. Then we have du O as r - in Equation (6), which implies d(u - u. ) O and u - u constant V , since u is
p a p a .s a
zero and u has to be equal to V as r - Thus, for z - , the effective
p s p
velocity is identically equal to ship speed V. The present results are,
of course, applicable to a propeller operating in open watér where du is
idéntically, equal to zero. Thus, the effective velocity for a propeller operating in open water is identically equal to V; the apparent, total velocity with the propeller in operation is equal to V5 + ua(rp). As can be Seen from EquatIon (6), the necessary condition for the effective pro-file to be different from the nominal propro-file Is'that there must be a ve-locity gradient in the nominal profile (du # O). In other wordS, the pro-. peller can only interact with a nominal profile which. has. non-zero. vorticity. The numerical solutions of the simultaneous Equations (5) and (6). are
straightforward and are given iñ Reference 1.
In current propeller design and performance prediction practice, Only
the measured model nominal velocity profile is available. The full-scale
nominal velocity profile is then estimated from the measured model nominal
profile. In order to ¿ompute the effective velocity profile either for the
model propeller or for the full-scale propeller,, the present theory can be applied in an iterative procedure:.
First, guess an effective velocity' prof:ile from the nominal ve-locity profile. One may start with the guess u(r) =.c, .u(r) where the'
constant c is assumed to be the ratio of the Taylor thrust-identity wake fraction to the .volumetric-mean wáke. fraction. .
The conventional propeller design computer programs13. can.be used
to design a propeller to operate In the estimated effective velocity
pro-file
Lep) wIth a prescribed nondimensional circulation distributionG(r)
of. a propeller of given geometry and rpm, computer programs14'15 are
avail-able to predict G(r) for the estimated values of Ue(tp)
(c.) The circumferential-average própeller-induced áxial velocity,
Ua
at the propeller disk is then calculated by using a field-point velocity program13 which Includes lifting surface corrections. A simpler result,
which may be used as a first. approximation, canbederived from moderately
loaded lifting line theory:
ir'
r'
dG()
___._2.._ p
ir'
ir'
Ir'
d (&_.) tan
p p p
(7)
8
where G r/27rRv5. tan
= (Ue + ua)/(fZrp - u), K is the number of blades,
is the propeller angular velocity, and u is the tangential
in-duced velocity. . .
With u(r) and ua(rp) now estimated, Euations (5) and (6) are
applied to determine u(r). . .
The new effective profile becomes u (r ) u (r) - u (r.).
e p p p
ap
Steps (b) through (e) are repeated until the values of ue(rp) and G(r) converge. .
The final computed nondimensional circulation distribution G(r) for a measured nominal axial velocity profile is shown in Figure2.. It has been found that the values of the effective, axial velocity distribution
Ue(p)/'Vs and G(r) essentially converge o their final values after three
iterations. The results of computations shownin Figure 2 indicate that the ratio of the effective to nominal velocity varies with. radius, having a larger value near the hub of the propeller than near the tip.
PROPELLER-HULL INTERACTION: FRICTIONAL AND PRESSURE-DEFECT COONENTS OF THRUST DEDUCTION
It is well known in naval architecture practice that the delivéréd
of the propeller. Traditionally, the increase in resistance due to
propeller-hull interaction has been. defined in terms of the thrust deduc-tion fracdeduc-tion t,
T=R
Twhere T is the delivered propeller thrust and R is the hull resistance.
The theoretical and experimental literature on thrust deduction is very
2,7,16,17,18,19,20 17
extensive. . Weinbium summarized the earlier contribu-tions of hydrodynamicists to the understanding and formulation of the
phySics. of thrust'deduction. In addition, an extensive list of references
20 on the subject can be found in the paper by Nowacki and Sharma. A
potential-flow computational scheme for calculating thrust deduction by Lagally's theorem was presented by Beveridge.16
An operatitg propeller produces an upstream suction which caúse 'a redúction in pressure on the afterbody thereby increasing the hull pressure
drag.. Simultaneously, this suction also causes an increaseof flow
veloc-ities on the 'afterbo4y and hence, additional frictional drag. Thus, the thrust deduction for a fully-submerged, axisymmetrical, self-propelled bod.y
may be divided into two cmponents,
t = tF + t,
where the pressure-defect compònent, t, is given, by
L/R: t. p 2 C TS r
JR
max pj
(C)
rh/RP 9 (8) 2 CTSand the frictioúa]. componént, tF Is given by wh e r è CTS = p V5TrR
(C), - (C)
2 CTS 2 =[(ci
(CT)b] (£
L/R p Xh/Ri
max p e s lo rT ,IR
o pd(f)
p(thrust loading coefficient based
on ship, speed)
(decrease of pressure coefficient due to propeller)
(increase of local skin friction coefficient due to propeller)
.(9)
tan dr/dx. (slope of body profile)
The pressure cOntribution to rope1lér-hull Intetaction is approxi-mated by the potential-flow approximation. of .C. The propeller-inducéd.
velocity on the hull can be computed from the propeller field-point pro-grarn)3 To determine the change in pressure due to the presènce.of the
21
propeller by the Doùglas-Neumann computer program, hull Sourçe strengths must be adjusted to cancel the normal. velocit.y induced by the propeller,
=
(V) =v
cosc-u sina
where n is theunit outward normal to the surface, v is,. the propeller-induced radialvelocity, 'andua is thé ptoe1ler-propeller-induced axial velocity.
Thus, when the velocity fields of the hull and the propeller are,
superim-posed, the normal velocity must be zeroalong the body contour. The total
pressure coefficient in the presence of the propeller is
(C) = i
-t) =
- (
+++.$)
/
u u u.
(10)
where U is the total surface potential velocity in the presence of the
propeller, which consists of u5, t.he perturbation velocity due to the bare
hull, Usn the tangential perturbation velocity which results from cancelling
the normal propetler-induced velocity, and u ' u 'cos a .+
y
sin ct, thesp a
pr''
propeller-thduced tangential velocity o. the hull. 'The pressure coefficient' of the 5bare
'hull
is given by (Cp)b = i '- (i + u5/V5)2 Since the 'pressure-defect component of thrust deduction is of a potential-flow natue, thevalue of t computed by Equation (8) will be the sameas that' compute4 from
p 16
Lagally's 'theorem in the Beveridge' formuation of interaction of hull-induced velocity on the propeller. By application of Lágaily's theory22 the interaction force arising from a lifting-line (sink disk)
represen-tation of the propeller may be written as
=
f
o u.n, dS '4K=çS
t. ' u'n
Jv
rh/Rfr'
'p
d fr''p
li
f
Equation (il) invOlves integration of the.product of hull-induced axial velocity, uh(rP) and propeller sink strength a(r) over the propeller disk.
Here we have, used Weinblum's17 finding that the propeller sink strength a is equal to -2u at the propeller disk; the lifting-linê computation of u, in Equation (7), has al8o been used The value of t can also be
obtained by integration of the product of propèller-induced axial velocity
U and hull source strength ah over the hull surface dSh,
-$
°h U a- u dS 2 iha
h = -Cf
VV
2 TS j s sTrR Sh pIn Equation (12) the values of U can be computed from thé propeller 13
f ield-ppint computer program using either a line or lifting-surface propeller representation, and the hull source strength ah can be computed from the Douglas potential-flow computer program.21 In Equation
(il), the hull-induced velocity at the propeller can be obtained by the method of Reference 21 and the values of G and tan 81can be obtained from the propeller lifting-line design13 r a performance prediction computer
14,15
program. It is important to note, that lifting-surface theory can be
used to compute Ua 'for Equation (12). It s therefore, possible to examine
the separate contributions of additional propeller characteristics,
includ-ing blade thickness, blade location (skew and rake)' and chordwièe varia-tion in loa4ing,, to the thrust, deducvaria-tion t. ''
The value of tF is computed hereby Equation (9) frOm values of shear
stress coefficients with and without the propeller -in operation, as derived from a boundary-layer computation.
POTENTIAL FLOW/BÖUNDARY-LAYER INTERACTION
Satisfactory predictions of turbulént boundary-layer characteristics can be made for the forward portions of a body by solving the boundary-layer equations, in either integral or differential forms. However, at the ship
stern, thethickness of the boundary layer increases rapidly, mainly due
to the diminishing cross sectional area. The thickness of the stern bound-ary layer usually .exceeds the thickness of the body. Therefore, the
trans-verse curvature of the flow in the boundary layer cannot be approximated by
the transverse curvature of the body and the pressure distribution on the body cannot be solved accurately without considering the displacement
effect of the thick boundary layer. The general approach used here is to perform the initial boundary-layer computation using the potential-flow
21
pressure distribution on the body. The initial flow calculations are then used to modify the geometry of the body and wake, by adding the local displacement thickness as suggested by Lighthlll.24 Potential-flow methods
are then used to compute the pressure distribution around the modified bod and the boundary-layer calculations are repeated using the new pressure
distribution. The basic scheme is continued until the pressure distribu-tions on the body from two successive approximadistribu-tions agree t within a
given error criterion.
25 25 27. 28
The work of Beatty, Cebeci et al., Beveridge, and Myring follow this general approach. Nakayama et al.,29 considered an even more complex model where, allowance is made foi transverse variation of static pressure across the boundary layer and a simple linear profile is assumed for the nominal velocity. A signific.nt difference df the approach of Nakayama et al.29 from the present approach is that the displacement thickness,
which is an integrated effect of the boundary layer, is not uSed to modify
the body. Instead, the potential solution is matched to the boundary-layer
calculation at the edge of the boundary layer and wake.
23
In the present work, the Douglas CS differential boundary-layer method, modified to properly accounted for the effects of transverse curvature, is used to calculate the boundary layer over the axisymmetric
13
body. The integral wake relations given by Granville3°
are used to calcu-late the flow in the wake. Since neither of the two methods properly model
the thick boundary layer in the stern/near-wake region, the calculated dis-placement body is not assumed to be valid in the region 0.95 <X/L<l.05. In this region, a fifth-degree polynòminal is used with the constants de termined by requiring that the thickness, slope and curvature be equal to
those calculated at X/L = 0.95 and 1.05. Should separation occur before
X/L = 0.95, the upstream matching is moved to the separation point. A detailed description of the potential flow/boundary-layer computation
scheme is given in Reference 1.
MODELS AND EXPERIMENTAL METHODS
Three. axisyinmetric afterbodies having afterbody length-diameter ratios
(LAID) of 4.308, 2.247, and l.484were selected for the present
experi-mental investigation (see Figure 3). Afterbody prismatic ratios
were 0.606, 0.526, and 0.416, respectively. As shown in Tablet, each afterbody was coñnected to a parallel middle L and a strèamlined forebody with bow-entrance length/diameter ratio (LEID) of 1.82. The total length of each model was fixed at a constant value of 3.066 m. Other hull par
ticulars of the three models are listed in Table 1.
TABLE t
HULL PARTICULARS FOR AXISY!»NETRIC AFTERBODIES
(L = .3.066 M,L/D l0.97)
14
Model . Afterbody 1 Afterbody? Afterbody3
LM/D 4.85 6.91 7.67
LAID . 4o31 2.25 1.48
pA 0.606 0.526 0.416
p 0.787 0.844 0.862
An existing model propeller whose diameter is 54.5 percent of the maxi-mum body diameter,was located at X/L 0.983 for the experimental investi-gation of propeller and stern boundary-layer interaction. This propeller
was designed for a wake distribution which was different from the wakes of
the three afterbodies. Thus, the iterative procedure described earlier usiñg the propeller/boundary-layer-interaction program and a propeller per-formance computer program was required to calculate the hydrodynamic char-acteristics of the propeller for the given propeller geometry, rpm, and the
specified nominal axial velocity distribution. The measured nominal axial
velocity distributions of the thteè afterbodies are shown in Figure 4. The computed nonditnensional circulations for the propeller operating in the
wakes of the three afterbodies are shown in Figure 5.
The experimental investigation was conducted in the wind tunnel of
the Anechoic Flow Facility of DTNSRDC. The wind tunnel has a 2.438 by 2.438 meters closed-jet test section with a maximum air speed of 61. metérs per second. The model was supported from below by two streamlined Struts located roughly one-third of the model length apart., The disturbances
generated by the supporting strUts were fourìd to 11e within th region below the horizontal center plane. Therefore, all the measurements were made In the vertical center plane along the upper meridian. Each stern
protruded from the wind tunnel closed-jet working section. into the anechoic
chamber (6.4 x 6.4.x 6.4 meters) located upstream o the diffuser. The
propeller was driven by a 9-kIlowatt high-speed motor mounted inside the stern of' the model. Propeller rpm was measured- by a magnetic pickup.
Mea-surëment of the following quantities was made-on each stern without and
then with a propeller operating at one or two advance coefficients.: surface shear stress measured by Preston, tubes, surface pressure distribution
sured by pressure taps, and boundary-layer axial velocity distribution
mea-sured remotely by a Laser Doppler. Velocimeter. The LDV'was located on an
optical bench in the quiescent region of' the anechòic chamber and vas
operated in a dual-beam off-axis backscatter mode Photographs of
the model and optical arrangements are shown in Figuré 6and 7. . The.
focal length of the LDV optics-was 1.5 meters and the total beam angle was 3.72 degrees. The effective probe volume for these optics was roughly eJ-lipsoidal having dimensions of about 0. 5 x 5 , and the probe volumi could
be focused in the stern boundary. layer at distances down to 2 from the full surface.
Detailed analysis of the measurement accuracies has not been made. Hwever, the standard deviations of the measured data were estimated from
repeat runs. The standard deviations of the measured static wall pressure
and shear Stress were less than 5 percent of their mean values and the Stan-dard deviation of the measured velocities was less than 2 percent of the
free-stream velocity.
COMPARISON 0F EXPERIMENTAL AND THEORETICAL RESULTS
Figure 8 shows the measured and computed valúes of pressure coefficient for After-bodies 1, ., and 3 at a length Reynolds number R = 5..9 x i,o6.
The transition is fixed at X/L = 0.015 which corresponds to the virtual. origin of turbulence of a 0.61-nun diameter trip wire located at X/L = 0.05. The results show that agreement is very good for Afterbody 1, fairly good. for Afterbody 2, and relatively poor for Afterbody 3. The measured and
computed local shear stress distributions CT are shown for After-body 1 in
Figure 9, for Afterbody 2 in Figure 10, and for Afterbody 3 in Figure 11. The agreement between measured and computed shear stresses IS good for
Afterbodies 1 and 2. For Afterbody 3, good agreement exists up to the
point of separation (X/L 0.92). After separation, the present boundary-layer computation method. breaksdown and no calculated results are
pre-sented in Figure 11 after separation. Qualitatively, the measured shear Stress takes small negative values in the separation bubble and becomes
-positive again after flow reattachment at a value f X/L àf about Ó.97.
The disagreement between measured.and computed values of,C for After-body 3 in Figure 8 was caused by the presence of shoulder separation at X/L 0,92.
The effect of the propeller on the stern pressure distribution was
com-puted by the potential-f làw approximation. As can be séen from Equation (10),
the total body surface velocity In the presence of the propeller is the sum of the free-stream velocity, the bare-hull perturbation velocity
(in-cluding the displacement effect) , the perturbation velocity as the result
of cancelliñg the propeller-induced velocity normal to the hull, and the propeller-induced velocity tàngentia1 to the hull. The computed
boundary-layer characteristics in the presence of the propeller aré based on this
surface velocity.
The measured and computed values of
CT with the propeller in operation
are shown in Figures 9 through 11. As expected, the propeller accelerates the flow at the stern resulting in an increase of shear stress. However, this increase of C is limited to the region X/L >0.90. Nô effect Is noted
at distañces larger than two propeller diameters (2%) upstream of the
pro-peller. As shown in Figure 11, the suction of the propeller does not change
the point of boundary-layer separation on Afterbody 3. The distance between the propeller plane and the. point of separation is 1.3 propeller diameters where the influence of the propeller is not strong. enough to alter the sep-aration location.
Figures 12 through 14 show the measured and computed pressure ,distri-butions with and without the propeller in operation. Again, the effect of
the propeller òn pressüre distribution is felt up to a distance of 2D
up-stream of the propeller. As can be seen from Figures 12 and 13, the mea-sured values of (C) (Cp)b and Ê = (Cp)b - (C) are in good agreement
with the computed values for Afterbodies i and 2. Figure 14 for Afterbody 3 shows that the measured values of i C are smaller than the computed values
of ¿ C aft of the separation point X/L = 0.92. These smaller measured
valuesof may be caused by the cushion effect o the separationbubble.
However, the measured values of C are larger than the computed values of
C upstream of the separation point, reflecting the pòssibie contraction
of the separation streamline due to the propeller. ' The values of t, obtaiqed by integration of the measured values of
C according to Equation (8) are given in Figures 12 through 13. The
good agreement between the measured and computed values of t shown in Table 2 for Afterbodies 1, 2, and 3 shows that the pressure. component of
TABLE 2
COMPARISON OF COMPUTED AND MEASURED THRUST DEDUCTION FRACTÏONS
thrust deductiOn can be well predicted by the potential-flow analysis propeller/hull interaction approximation. However, if there exists flow
separation at the stern, the potential-flow interaction approximátion for
t should be used with caution. Although the computed values of t, for
Afterbody 3 agrees well with the value' ôf t obtained by the integration of the measured values of C , the measured and computed values of C do
p p
not agree very well in the vicinity Of the separation point (Figure. 14).
23
The CS boundary-layer computation can be performed with the pres-sure distribution (C) as modified by the propeller. I The measured and
computed. local shear strEss distributions and the values of tF computed by integration of measured values of (using Equation (9)) over the three
afterbodies are shown in Figures 9 through 11. The measured values of t and tF for the three afterbodies at R = 5.9 x 10 aresuarized in.
- n
Table 3. . .
The agreement between the computed and measured pressure components of thrust duduction. t is satisfactory. The measured ratio t
It
is found toFp
be less than 5 percent for the three afterbodies at J = 1.25 and .J 1.07. Therefore, it is sufficient to estimate the value of thrust deduction fraction, t, from its preSsuré component t.
18 Computed t P Measured t p Measured .t F Measured
t.t+t
pF
Measuredt/t
Fp
Afterbody l,J1.25 0.068 0.07 0.0024 0.0724 3.4% Afterbody 2, JE1.25 0.129 0.143 0.0023 0.1453 1.6% Afterbody 2, J=l.07 0.126 0.140 0.0026 0.1426 1.9% Afterbody 3, Jul.25 0.106 0.109 0 0.109 0 Afterbody 3, J1.07 0.103 0.103 0 .0.103 0.TABLE 3
MEASURED AND COMPUTED VALUES OF THRUST DEDUCTION FOR AN APPENDED SERIES 58 BODY (NSRDC MODEL 4620, PROPELLER 3638)
The computation procedure developed for calculating t can be applied to arbitrary bodies. As an illustrative example, the potential f low pro-peller/hull InteractIon aialyses have, been applied to a Séries 58.appended
body. The details of afterbody (NSRDC Model 4620), appendages, and propel-ler (NSRDC Model 3638) are given by Beveridge)6 The results of the
mimer-16' ical calculation are shown in Table 3. In Equation (il) Beveridge.. set
a
= -
22 where is the induced velocity at the lifting line... However, thep a ...a'
:'
:
present metho4 properly used, ap = - 2Ua where is thé. circumférential-mean propeller indücèd axial velOcity at the sink disk. It is interesting to note that the lifting surfacé cortectións. (in. this case primarily thickness
19 o '
METHD
. THRUST DEDUCTIONMeasured (Resistance and Self-Propulsion . t. + tf
Experiments)
.
0150
LiftIng Body Pressure Integration, Surface Equation (10)
t 0.135
p
Method
. /
(for Ua) Legally Theorem, Equation (12)
- .
t 0.133
Computed '
--Lifting Legally Théorem, Equations (11) t = 0.141
Line or (12) ..
Method
Beveridge,16 Equation (11) with t = 0 159
'Disk)
a
=-2tz
,p 'a ,
since this propeller has no rake) yield a 5-percent -lower válue of t
corn-pared with the lifting-line method. The measured value of the total thrust deduction, t = t + tF obtained from towing tank resistance and
self-propulsion experiments is also shown in Table 3 for comparison. Although the value of tF was not measured or computed, one may estimate
tF 0.05 t.
It is evident that the present methods cai. be used with confidence t
pre-dict the potential flow propeller/hull interaction for arbitrary bòdiês
and propeller conf:igurations.
The mean axial velocity profiles in the boundary layers of the three
afterbodies with and without a propeller n operation vere measured by a Laser Doppler Velocinieter. The boundary-layer profiles were also computed
13
by the Douglas CS differential boundary-layer computer. program. The
pres-sure distributions calculated from the present potential flow/boundary layer iteration scheme were used to compute t-he profiles for the bare-hull.
The pressure. distributions calculated from the potentIal f làw propeller!
hull interaction approximation (Equation (10)) were used to compute. the
pro-files with the propeller in operation. Figures 15 through 17 show the
measured and computed mean axial velocity profiles immédiately upstream.of
the propeller (X/L = 0.977). The "present" approximations shown in
Fig-ures 15 through 17 for the profiles with a propeller in operation are the results computed from the inviscid propeller/stern boundary layer inter-action approximation, using the measure nominai velocity profiles as input.
As shown in Reference 1, the agreement between the measured and com-puted nominal profiles in the absence of a propeller ls.excellent for all
afterbodies in the region X'/L < 0.90. For Afterbody 1, which has a mild
adverse pressure gradient, the present potential flow/boundary-layer interaction scheme does provide a very good approximation of the stern f low (even t X/L = 0.977 as shown in 'Figure 15).
However, for Afterbody 2, which has a strong adverse pressure gradient, the measured velocities are
progressively slower than the computed velocities as the end of the stetn
is approached (Figure 16). Since the boundary-layer computation Is term-inated when flow separation is encountered, no computed results are pre-sented in Figure 17. .
The measured and computed axial apparent total velocity profiles u(r) with the propeller in operation are shown in Figures 15 through 17. Good
agreement between the measured and computed values of u is found for all
of the cases. The measured difference in axial velocities obtained by the LDV with and without the propeller in operation, u/V, and the computed propeller axial velocities, Ua/V are shown in Figure 18. The present
approximation for tu/V is in excellent agreelent with the measured data.
It is important to note that the propeller plane is et X/L 0.983 in all
cases. The measurement stations vere located at a distance of 0.12 D from
13 p
the propeller. The propeller field-point velocity program was used to calculate the circumferential-mean propeller-induced axial velocities Ua at the positions where the nominal and apparent total velocity profiles were measured by the LDV. The calculated radial distribution of circulation for
the propeller operating in the wakeS of the three different afterbodies which is used in the field-point program, has been shown in Figuré 5.
Re-call that the efective velocity prof:ile Ue is the apparent total velocity
profile minus the propeller-induced velocity profile Ua (Figure 18).
The good agreement between the measured and computed values òf shown in
Figures 15 through 18 suggests that the present inviscld approximation can be used with confidence to calculate the effective velocity profile for a propeller on an axisyetric body from the measured nominal velòcity
pro-file u and propeller-induced velocity profile ua.
..
SCALE EFFECT ON NOMINAL WAKE DISTRIBUTION
Ship models are tested at Reynolds numbers which are two orders of magnitude or more smaller than Reynolds numbers of full-scale ships. Thus,
the nominal velocities for a full-scale ship can be expected to be greater
than those measured at model scale. Most towiñg tanks have developed their own empirical apprpaches to account for the scale effect on the wake. Some
empirical methods have been presented at the ITT meetings, and a compréhen-sive review of varlous.approaches has been given by Dyne.31'32
The empirical approach proposed by Sasajima and Tanaka33 was the first attempt to extrapolate the wake distribution measured on a modèl to the
full-scale ship. The method assuméd that the magnitude of the frictional
component of. the wake varied linearly with the flat-plate frictional
coef-f icient. However, even on a fiat plate, the turbulent boundary layer does not have the necessary similarity property required for this assumption to
be valid. Furthermore, the intrinsic nature of a thick boundary layer on a ship stern cannot be properly represented by the flat-plate approximation. The thickness of the boundary layer increases rapidly at the stern as the
cross-sectional area decreases. Since the thickness of the stern boundary layer usually exceeds the radius of the body, the transverse curvature of the boundary layer flow cannot be realistically represented by the
trans-verse curvature of the body. The differences between predicted, flat-plate.
4nd axisymmetric stern boundary-layer velocity profiles on Afterbody i are
illustrated iñ Figure 19. The Douglas CS23. differential method was used to
compute the flat-plate boundary layer without considering the displacement
effect. It can be Seen from Figure 19 that the boundary-layer thickness at an axisymmétricstern. is much larger than the flat-piateboundary layer
at the same length Reynolds number.
In the following we digress briefly to .outline a derivation of the
.33
simplified wake-scaling formula used by Sasajirna and Tanaka. It is well known34 that two-dimensional laminar boundary layers possess a
similarity characteristic; for a flat plate, the longitudinal, velocities
take the following similarity form
'X'ff...Y/X\f\.ffz1\
s'.'\/Vx/V1
\6/Yc
F'
where x is the verse distance efficient (i.e characteristic (13)longitudinal distance. from the leading edge, y is the trans-6 is the boundary-layer thickneàs, and is the drag co-., the 1957 TTC line). There is no corresponding similarity
for two-dimensional turbulent boundary layers. ven for
a
turbulent,boundary layer alonga
flat plate, the well-known laws Of the wall and wake cannot be brought into the form35However, for
a
given vElue of Reynolds number, the vélocity profile of a two-dimensional flat-plate turbulent boundary layer can be crudelyapprox-imated by a power law
i
u n
v
t6)
s
where n is a function of Reynolds number. In this case, the total
resistance coefficient becomes35
which implies that & is proportional to for a fixed
value
of n.There-fore, a similarity form of the solution for the - power velocity profile
when n is fixed may be approximated by
= (ï
X
u\ u
n .6V/ V
\ó/ (n+l)(n+2) x. (14) (15) (16)If the velocity profilé ls.known (either from meäsurement' or computation)
at a given Reynolds number, the velocity pröfLle can be obtained for other
Reynolds numbers .f n' is assumEd to have a fixed valüe. For a given value
of u/V, Equation. (16) states that the coordinate of the velocIty ptofile
isshifted by ' . .
(.
X
swhere the subscripts m and s denote the values corresponding to the. model and full-scale ship, respectively.
Formula. (17) vas f.irst proposed by Sasajima and Tanaka.33 The value
of can be considered to include the smooth flat-plate- resistance
co-efficient and a roughness allowance
AF.
However, the. value of the power in reality changes with Reynolds number. The well-known + - power lawis only approximately valid for io6 < R < Figure 20 shóws the
theo-retically computed and empirically calcúlated scale effect on a flat plate
velocity profile. Again, the Douglas CS23 differential method was used to compute the axial velocity profiles at two length Reynolds numbers. The flat-plate émplrical method was used to calculate the profile for R =
9
- 6
-s
3 X
10: from the profile at a-lower Reynolds number R = . 6 X 10 via
Equation (17). If thé flat-plate empirical methods were adequate, theñ
Curve ii should be a' good approximation to Curve Iwhich is'not the
caé
as shown in Figure'20. The discrepancy is quité large nEar theplae.
-' As for the flat. plate,.if»the axisytnmetric nominal velocityprofile at the propeller disk is assumed to follow a'* power law, Equation (14),
can be related to' boundary-layer thickness by
2
/''u\ 'u
2.
ô I11__-.L_
Id(L\
n ô S (x)J
\.
VJ VS, ô
\ 6 J . (2n+1) (2n+2) S (x) owhere S(x) is the surface area. The approximated similarity form of the
i ,
-solution for the axisymmet-ric power velocity profile is. then given by
24
(17)
(18)
Here, ve have neglected the variation of. static pressure and the normal
stress across the boundary layer.
Figure 21 shows the theoretically
coin-puted and empirically calculated scale effect on the nominal axial velocity
profile at the propeller plane of Afterbody 1.
The potential f.low/
boundary-layer interaction program' was used to compute the axial velocity
profile at two length Reynolds numbers.
The flat-plate and axisytninetric
methods were used to calculate the profile for R
= 3 X
from the
profile at the lower Reynolds númber R
6 Xio6.
It is obvious from
Figure 21 that the flat-plate empirical method is not suitable for
calcu-lating the scale effect on the axisyinmetric thick-stern boundary layer.
Although the profile computed by the axisytnmetric empirical method
(Equa-tion (19)) is closer to the theoretically computed profile, it is still
not a4equate to accurately compute the scale effect on the nominal velocity
profile.
CONCLUDING REMARKS
In this paper we have described recent experimeñtal and theoretical
studies of the classical propeller-body interaction problem.
Acomprè
hensive set of boundary layer measurements is presented for thrée
axisym-metric afterbodies, obtained with and without an.operating stern propeller.
The results ate compared with 'calculations based on new theoretical analyses
and 'provide valuable insights into the manner in which the propeller mf
lu-ences the flow over the afterbody and interacts with a thick stern boundary
layer.
Comparison with experimental results shOws that the potential flow!
boundary-layer interaction program computes, accurate values of 'pressure,
shear stress, and velocity profiles f pr the fine stern.
Less satisfactory
ptedictions are obtained for the somewhat fuller stern, while only fair
predictions ate obtained 'for the fullest stern having shoulder sepáration.
In all cases the agreément is excellent
ver the forward 90-percent of the
bodies.
These results suggest that the modeling of the wake should be
modi-f ied. to, treat cases omodi-f separated modi-flow and that the present boundary-layer
equations should be further improved for the fuller stens.
It is shown that in most cases, the inviscid propeller/stern boundary-layer interaction theory predictS very well the measured total velocity
profile with the propeller in operation. The ratio of the computed
effec-tive and nominal velocities Is found to be largest at the própeller hub
and decreases toward the propeller tip. This is cOntrary to methods which
assume that. the effective uke profile is a constant multiple of the
nominal wake profile. Since the radial diStributiop of. effective wake
plays an important role in propeller propulsion and cavitation performance,
it is recommended that the present t.heory be incorporated into design
procedures for wke adapted propellers.
It is also shown that the added body resistance (thrust deduction) arises almost entirely from the pressure defect on the stern. Moreover, the distribution of this pressure defect and the thrust deduction are pre-dicted very well from potential flow coñsiderations alone. For this reason, both the thrust deduction and propeller/boundary-layer interaction, which
is essentiallyinviscid in. nature, should be independent of scale. The primary scale effect is manifested by a change in the nominal velocity profile, which in the case of axisymmetric bodies, cañ be computed. from boundary-layer theory as described in this paper.
ACKNOWLEDG1ENT
The Initial experimental work reported herein was funded uñder the David W. Taylor Naval Ship R&D Center's Independent Exploratory
Develop-ment Program, EleDevelop-ment Number 62766M. All of the subsequent investigations
were funded under the Naval Materials Command Direct Laboratory Funding,
Element Number 62543N.
The authors wish to express their gratitude to Mr. JustjnH. McCarthy
for his continuous.support and technical guidance. Dr. William.B.. Morgan is thanked for this advice in applying propeller theories. The authors would also. like to thank their co-workers, Mr. N. Santelli, Dr. H.T. Wang, Ms. N.C. Groves,. Dr. A.G. Hansen, and Mr. Thomas A. LaFone for their co-operation and contribution. The staffof the Center's Anechoic Flow
Facility are also thanked for their effective experimental support.
REFERENCES
Huang, T.T., H.T. Wang, N. Santelli, N.C. Groves," Propeller! Stern/Boundary Layer Interaction on Axisyminetric Bodies: Theory and Experiment," David W. Taylor Naval Ship Research and
Development Center Report 76-0113 (Dec 1976).
Hucho, W.-!!., "iiber den Einfluss einer Heckschraube auf die
Druckverteilung und die Grenzschicht eines Rotationskrpers
-Teil II: Untersuchungen bei hoheren Schubbelastungsgraden," Institut FUr Stromungsmechanik der Technischen Hochschule
Branunschweig, Bericht 64/45 (1965).
Hucho, W.-H., "iber den Zusammenhang zwischen Normalsog,
Rei-bungssog und dem Nachstrom bei der Strdmung um Rotation-skrper," Schiff und Hafen, Heft 10, pp. 689-693 (1968).
Hucho, W.-H., "Untersuchungen aber den Einfluss einer Heckschraube auf die Druckverteilung und die Grenzschicht Schiffsähnlicher
Krper," Ingenieur-Archiv Vol. XXXVII, pp. 288-303 (1969). Wertbrecht, H.M., "Vom Sog, ein Versuch Seiner Berechnung," Jahrbuch Schiffbautechnische Gesellschaft, Vol. 42, pp. 147-204
(1941).
Hickling, R., "Propellers in the Wake of an Axisymmetric Body," Transactions of the Royal Institute of Naval Architects, Vol. 99,
pp. 601-617 (1957).
Tsakonas, S. and Jacobs, W.R., "Potential and Viscous Parts of the Thrust Deduction and Wake Fraction for an Ellipsoid of Revolution," Journal of Ship Research, Vol. 4, No. 2, pp. 1-16
(1960).
Wald, Q., "Performance of a Propeller in a Wake and the
Inter-actionof Propeller and Hull," Journal of Ship Research,.Vol. No. 1, pp. 1-8 (1965).
Raestad, A.E.,"Estimation of Marine Propeller's Induced Effects on the Hull Wake-Scale Effect on the Hull Wake Field," Det
norske Ventas Report No. 72-3-M, Chapter 1 (1972).
Nágamatsu, T. and Sasajima, T., "Effect of Propeller Suction on Wake," Journal of the Society of Naval Architects of Japan,
Vôl. 137, pp. 58-63 (1975).
il. Titoff, LA. and Otlesnov, Yu, P., "SOme Aspects .of
Propeller-hull Interftction," Swedish-SOviet Propeller Symposium, M9scow
(1975).
Thwaites., B., Incompressible Aerodynamics, Chapter. XI, OxfOrd
University Press '(1960)..
Kerwin, J..E. and Leopold, L., "A Design Theory for Subcavitating
Theory," Transactions of the Society of Naval Architects and Marine Engineers, Vol. 72,pp. 294-335 (1964).
Cummings; D.E., "Núinerical Prediction of Propeller Charaèteristics,"
JOurnal of'Ship Research, Vol. 17, Part 3, pp. 12-18 (1973). Tsaö, S.S.-K., "Documentation of Program for the Analysis of
Per-formance and Spindle Torque f Controllable Pitch Propellers,"
MIT, Department of Ocean Engineéring Report No. 75-8 (May 1975)
Beveridge, J.L., "Analytical Prediction of Thrust Dèduction for Submersibles and Surface Ships," Journal of Ship Research, Vol. 13, No. 4, pp. 258-271 (1969).
17. Weinbium, G.P., "The Thrust Deduction," Joürnal Of American
Society of Naval Engineers, Vol. 63, pp.. 363-380 (1951).
18. Amtsberg, H., "Investigatións on the Interaction between Hull and Propeller of odies of Revolution," (in German), Jàhrbuch der Schiffbautechnischen Gesselschaft, Vol. 54, pp. ll7-52
(1960). (David Taylor Model Basin Translation 309 (1965)).
Nowacki, H., "Potential Wake and Thrust Deduction Calcultions for Ship-Like Bodies," Jahrbuch der Schiffbautechnjschen
Gesselschaft, Vol. 57, pp. 330-363 (1963)..
Nowacki, H. and Sharma, S.D., "Surface Effect in Hull Propelle.r Intéraction," The Ninth Office of Nàval Research Symposium on Naval Hydrodynamics (Aug 1972), Paris, Ftance, ACR-203, U.S.
Government Printing Office, Vol. 2, pp. 1845-1961 (1972).
Hess, J.L. and Smith, A.M.O., "Calculation of Potential Flow about
Arbitrary Bodies," Progress in Aeronautical Science.,. Vol. 8, Pérgamon Press, New York (1966).
Curnmins, W.E., "The Force, and Moment on a.. Body in a Time-Varying Potential Flow,! Journal of Ship Research, VOl. 1, pp. 7-18(1957). Cebeci, T. and Smith, A.M.O., Analysis of Turbulent Boundary
Layer, Academic Press, New York., (1974).
Lighthiil, M.J.,. "On Displacement Thickness," Journal of Fluid
Mechanics, Vol. 4, Part 4, pp. 383-392 (1958).
25 Beatty, T D , "A Theoretical Method for the Analysis and Design
of Axisyimnetric Bodies," National Aeronautics and Space Admin-istration CR-2498 (1975).
Cebecï,,T., MosinsIis,G.J., and Smith, A.M.0., "CaÎculation of Viscous Drag and Turbulent Boundary-Layer Separation on Two-Dimensional and Axisymmetric Bodies in Incompressible FlowS,"
Douglas Aircraft Report MDC J0973-.0l (1970).
Beveridge, J.L., "Pressure Distribütion on Towed and Propelled Streamlined Bodies of Revolution at Deep Submergence," David
Taylor Model Basin Report 1665 (1966).
Myring, D.F., "The Profile Drag of Bodies of Revolution, in
Sub-sonic Axisymmetric Flow," Royal Aircraft Establishment Techniçal
Report 72234 (1972).
Nakáyama, A., Patel, V.Ç., and Landweber,L., "Flow Interaction
Near the Tail of a. Bòdy of Revolution: Part 1 - Flow Exterior to Boundary Layer and Wake, Part 2 -Iterative solution for Flow Within and Exterior to Boundary Layer and Wake,' Journal of Fluids Engineering, Transactions of the American Söciety of
Mechani'al Engineers, Vol. 98, Series 1, No. 3, pp. S31-49'(1976). Granville, P.S., "The Calculation of the Viscous Drag of Bödies of Revolution," David Taylor Model Basin Report 849 (1953).
3l. Dyne, G., "On the Scale Effect on Wake and Thrust Dedùction,"
Proceedings, Thirteenth International Towing Tank Conference., Berlin, Hamburg, Wést Germany, Appendix, 6, Report of Perfòrmance Committee, Vpl. 1, 1972.
32. Dyne,, G., "A Study of the Scale Effect on Wake, PröpeUer
Cavita-tion and Vibratory Pressure at Hull of Two Tanker Mo4els," Transactions, The Society of Naval Architects and Marine
Engineers, Vol. 82, pp. l62.l85 (1974)'.
Sasajima,H. and I. Tanaka,, "On the Estimation of Wake of Ships,' Proceedings, Eleventh International Towing Tank Conference,
Tokyo, Japan, Appendix X, Performance Session, pp. 1,40-143 (1966).
Landau, L.D. and E..M. Lifshitz, Fluid Méçhanics, First Edition,' Pergamon Press, London, pp. 145-151 (1959).
Schlichting, H., Boundary-Layer Theory, Sixth Edition,
Chapter XIX, McGraw-Hill Book Co., New York (1968).
Region of Propeller lñfluence Extends to About
2 Dp Upstream
of
the Propeller
Stream Surface without
the Presence of Propeller
Stream Surface with
Propeller in Operation
Du-.-_-...
-.
_o.-9
.-i
t
rF-Ag.
IDefinition Sketch for Propeller
Stern Boundary Layer
Interaction
04
0.004 0.008 0.01 2 0.016 0.020
oil
,/
\\
Computed Measured Nominal/
Circulation G byVelocity, .A
Propeller Inverse UaI
Program .1.
FComputed by Prpeiler
/
FieidPoint,Program
Propeller:/
/
CTS T =0.371 1I
/
/
/
/
'N.
4 1.25 .f
/
uer(r/Rp).
2lrRp V5/
I,
.,,,'
/
N%Up,
V-
s/
,
Computed by the Computer Program.
/
,,
0.2
Ò4
O'5
I I.L
Ux/Vs,LetVs,Ot Up/Vs
Figure 2Computed Nondimensional Circulation and Effective and Apparent
Axial Velocity Profiles after FoUr Iterations for a Typical Propeller
Operated ¡n.a Measured Nominal: Axidl Velocity Profile of a Typical
Axisymmetricul Body
0.6
0.7
0.8
0.9
1.00.9
08
0.7
0.6
r
Rp
0.5
0.3
.20.05
0.04
0.03
0.02
0.0I.
0.7
Afterbody
I-Afterbody 2
Afterbody 3
.0.8
X/L
Agure 3 The Three Axisymmetric Afterbodies
0.9
1.0
0.9
.0.8
0.7
//
/
MEASURED NOMINAL AXIAL
/
/
VELOCITY PROFILES AT THE'
0.6
PLANE OF PROPELLER:
/ /
AFTERBO'Y
I ./
/
/
05
AFTERBODY 2
AFTER800Y 3
0.2
0.1
I
'h
I J I II
I 0.10.2
0.3
0.4
0.5
0.6
0.7
0.8
ux
.vs
Figure 4Measured Nominal Axial Velocity Profiles for the
0.I
- .-.
%% Afterbody ICTS:O.371, JI.25
.CTS:O.420,
J:I.25T'
CTS : 0.637, J: 107
R s»i,'
.1'
'
\/
//
Afterbody 3
///
i)SN,.
.CT$:O.428,J:I.25
__
J..0.647, J=I.07
io
»00.004: 0.008
0.012
0.016
0.020
0.024
0.028
0.032
r(r/p)
.ir
Fig. 5 Nondimeñsional Circulation for the Pröpefler Operated in
the Wakes of Three Aftérbodies
1.0
0.9
0.8
0.7
0.6
r
0.5
0.4
0.3
0.2
(I)
ARGON
LASER
BAC KSCAT TER
MODULE
IL'
PHOTO
DETECTORS
(3)
'(4)
FOCUSING
LENSE
SPLITTER
Fig. 6 Off -Axis Dual
-
Beam Backscatter Optics
Fig. 7 - Photographic View of Afterbody
2 and Optical ßrrangement of Laser
0.3
0.2
0.I
_0i
-_0.2
_0.3
_O.4
_0.5
0.75
Afterbody
Afterbody
Afterbody
0---.
-s.---
0/1
o/
\
il
N
Measured Computed Io
2
3D
Afterbody 3
Location of Flow ASeparation on
R= 5.9x10
At terbody 3UI
s Afterbody 3o
/ II
I
/
/
/
D/
I
-,
"h/
'h/
,Afterbody 2w"
o0.80
0.85
0.90
0.95
loo
X
L
Fig. 8 - Comparison of Measured and
Computed Stern
Pressure
Distributions for the Three Afterbodies
o
cP
00025
xl, 2frCr
)dx?:
R2 TS P-'X00035
0.0002
- 0.0001
0.0030 -.
L0.0
Q Measurements0'
06
0.7
0.8
0.9
1.0X/L
Fig. 9 Measured and Corriputed Skin - Friction Distribution
on Afterbody I With and Without Propeller in
Operation (R
5.9 x
Q6)
maxo
o
- 0.0004
0.0003
i.9o
I-L)
ONo Propeller
0.0020 -
Propeller J = 1.250.0015
Boundary-Layer Computations No Propeller0.0010 -
- - - - Propeller J:l.25
0.0005
0.0045
PropellerJ:125
Measured Derived tF0.0024
0 0040
Fropeller CTS 0.371o
I I i06
0.7
0.8
0.9
1.0X/L
Fig. IO Measured and Computed Skin -
Friction Distribution
on Afterbody 2 With and Without
Propeller in
Operation (R=5.9 x
¡Q8)
I-.L)
O.00O80.0007
0.0006
0.0003
0.0001
0.0
-0.0004
0.0002
i.?0
0.0045
0.0040
0.0035
0.0030
0.0025
0.0020
0.0015
0.0010
0.0005
PropellerJ:I 25
J 1.07 Measured CT0 A Derived 'F0.0023
0.0026
Propeller CTS0.42
0.637
2jxh
I
QI
I
I
,
II
£ -2r(C)dx
C R Xmax -Measurementso
No Propeller
-
Propeller J 1.25 Propeller J 1.07 Boundary-Layer ComputationsNo Propiier
o
Fo
o
- - - - Propeller J:125
Propeller J:l.07
0.0045'
0.0040
0.0035
00030
0 002.5
0.0020
0.0015
00010
0.0005
o
Measured
Computed
o
Propel 1er Has No Measurable
Ef fèct on C1.
Flow Separation
Predicted
Measured
o9i8
L "'
.tiL
I'
I0.6
0.7
0.8
.0.9
1.0X/L
.Fig.:II_ Measured and Çomputed Skin-Friction
DistributIons
a
o
o
5.9.8.8
Potèntiol Flow0.5
. Exper'mentsNo Propellei With Propeller R x 10_6
J:125
Q4
o
e
5.9£
8.8
Computed0.2
0-0.6
0.7
0.8
0.9
I.0
I. IX/L
Fig. I2
Measured and ÇómputedPressure Distribution
on
fterbody
IWith and Without Propeller
in.
Q
o
_04
0.15L 0.10
Q-Qo
»
4 4I__Computationst
/
0.6
0.7
0.8
0.9
X/L
1.0Fig. 13 Measured and Computed Pressure Distribution on
Afterbody 2
With and Without Propeller in
Operation
0.4
0.2
Ex per imentsJ:107
R x
5.98.8
No Propeller With Propeller
J:125
o
0.15
0.10
Q.o
.0 Q.o
%.;0.05
a. C.,il
J=l.25 J:l.07
s
s
L
No Dota. Computation zio-E ,iperimentsJ:125
J:i.07
O5
5.9
--
8.8
Potential Flow0.428 0.646
CTS0.109 0.103
tp
rrh
= 2j r(áC)d
P
C.RJ
mazI
I
I
I
I
I
I
I
I
!
,1.
0,8
I -Ij
0.6
0.7
0.8
0.9
1.0X/L
Fig. 14 Measured and Computed Pressure Distribution on
Afterbody 3
With and Without Propeller in
1.0
0.9
0.8
0.7
2
0.6
0.5
0.4
0.3
,
-
Propeller J'l.25
r,
'0.136
rmaxMeasurement
B-L CalculationNo Propeller
O
oA
ux
vs
Present Approximation
Propeller J
1.254/
/
I
I
'Iup
vs0.2
0.1o
0
0.102 0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0 1.1UI
Up
Fig. 15 Measured and Computed Axial Velocity Profiles at
0.977 of Afterbody I
With and Without Propeller ¡n
Operation (R
5.9 x 10)
1.0
0.9
0.8
0.7
XD.
E
0.4
0.3
0.2
0.I
o
0
0.10.2
0.3
0.4
0.5
0.6 07
0.8
0.9
1.0-.vs.vs..
Fig. 16 - Measured and Computed Axial Velocity Profiles at
= 0.977 of At terbody 2 With and Without Propeller in
Operation (Rn.
5.9 X 106),
.Measurement 8-L Calculation
No Propeller
Q
Propeller J'l.25
A
Propeller Jl.O7
O
Present Approximation
(ix
vs
J1.25
IO?
o0 J 1.25 1.2 /i0
io
0i
/1.25
J- 1.07
o
vs o Ofiji
o
no
A
I Io
0
o ,;
od
o
j
Up vso
0
0.10.2
0.3
0.4
0.5
O.0.7
0.8
0.9
1.0Vs
Vs
1.2Fig. 17 Measured Axial Velocity Profile at
'0.977 of
4f terbody 3 With and Without Propeller
in
Operation (R
= 5.9 x
106)
Mesurernen?
0.9
NProeIier
O
Propeller
'l.2
A
Propeler J-07
o
0.7
No B-L Calculation After Separation
*06
o.
Present Approximation
0.5
0.4
03
0.2
r
;-.
-0.136
'maz o.1.2
1.00.8
0.6
0.4
0.2
nax0.04
Propeller
J'l.25
R = 5.9 x
tt.
't
'
u
u-u
\
?
/!..Measurement
-,
Xp-Present Approxlmatl9n,
Ue Up U.av""v)
.UxAx
o'...
.&?
Ro
rmax
/
.loO
-- ,..
. Computed UaPropeller Induced
-vs i -:.
.L
.Ö8
0:12
0.16
ö0
'0.24
U0 -Vsor-ç,--
sFig 18 Measured and Computed Axial Velocity Increase at
X/L = 0.977 on Afterbody 2 Dué to' the Suction of
Propeller'
' . ..0.050
0.045
0.Ò40
0.035
0.2
0.3
0.4
0.5
0.6
O7
r0.8
0.9
1.0ux
vs . . iFigure 19Fundamental Difference Between the Flat-elate and
0.020
0.018
0.016
0.014
0.012
-J)
0.010
0.008
0.006
0.004
0.002
-TRANSITION AT X/L 0.010
FLAT-PLATE EMPIRICAL METHOD
(II)
FOR R3xI09
FROM THE PROFILE OF
R
- 6x106
mBOUNDARYLAYER COMPUTATION
R_6xIO6
(I)
R'3xIO
0.2
0.3
0.4
0.5
ux
vs
Figure 20Theoretically Computed and Empirically Calculated
Scale Effect on Axial Velocity Profile on Flat-Plate.
0.7
0.6
0.5
0.3
0.2
0.I
o
BOUNDARY-LAYER COMPUTATION
-
6x106
R-
3x109
ns-.----.----EMPIRICAL METHODS FOR
R
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