Resistance and Self-Propulsion Numerical Experiments
on TwoTankers at Model and Full Scale
TECNNSCNE UNVER$ffEfT
George D. Tzabiras
Laboratorium 'icorNational Technical University of Athens Scheepshdrornochanca,
.. . MekeIwogZ2 De!ft
1. Introduction
The numerical methods presented in the SSPA-CTH-IEHII. workshop (Larsson et al. 1990) have shown the great )togIess in I)rec!ictillg the tu r.Iutent flow arou ud tite stern of shi1) models, Despite the differences in the employed discretiza.tion schemes and turbulence models, nìaiiy methods have produced reliable results with almost uniform trends. However,, a complete
itunierical solution of the real self-propulsion prol)leIu, incliidiiìg the free-surface deformation
and the I)rOpeller operation. remains a. (Illite complicatQd task. Generally, two approaches
have been employed to solve the viscous free-surface problem. The first solves numerically the Navier-Stokes equations to predic.t the free-surface, e.g. ilirio (1989), while the second decouples the problem and calculates the wave elevation by potential flow methods, e.g. Larsson eL al. (1989). These efforts are still in early stages. For the propeller effect better approximations have been achieved, using the body force approximtíón (Stern ct.al. 1986,, and encouraging
results of engineering interest have been obtained.
The major part ofthe resistance oflarge tankers is due to viscous effects that dominate the
flow. lu these cases a numerical method able to predict the full scale propulsion characteristics l)y neglecting the free surface effec.t would be a valuable tool, since the empirical extrapóla.tion methods used by towing tanks suffer from scale-effect uncertainties. Unfortunately, due to the lack of detailed experimental data for real ships, it is difficult to evaluate the numerical results.
Therefore, grid dependence tests should be performed to verify that the numerical solution is virtually independent of the spatia.l discretization. This self-validation is ¿t least required to
produce reliable results for high Reynolds number flows.
The numerical method of T:abinm' (1984), originally developed to model stern flows, has
been employed to calculate the whole flowfield around the ship. This way, the flow around the
1)0w ¡S calculated by solving both the velocity and tile pressure fields, avoiding the uncértainties
introduced in the thin boundary layer approximations. A successive mesh refinement tech nique
is introduced to perform grid dependence tests as well as to accelerate convergence. This
technique proved very efficient especially for the high Reyrolds calculations where fine grid resolutions are required. An iterative method is followed for the solution of theself-propulsion problem, similar to the one developed for computations around bodlés of revolution (T.abiriis and Garofallidis 1989,. In this procedure the propeller is modelled by a simple actuator disk. The fiowfield is treated as fully turbulent for both model and full scale, and the k-E model is used to simulate the turbulence stresses. Tile wall function method, appealing to be valid within wide ranges at high Reynolds computatiolis, is applied to calculate the near wall flow parameters.
Computations have been carried out for both the tanker models which were used as the test cases in the SSPA workshop to study tile effect of aU- or V- shaped stern, as Dyne (1974) found
experimentally that the stern geometry influences significantly the resistance and propulsion characteristics. The results concern the trends of the integrated resistance coefficients and the variation of the flow variables at the stern and the near wake region of the two models.
'Dept. NA&ME, POB 64070, 15710 Zorafos, Athens, Greece
Op . th7j = + A32cr12 + A23cr13 +
-z---Ox1 (IX1
¡Op
.--'h-; b-
+ puJt
P'UtL:m1'5i:m +da1.,
+ _.. + ____
I Oa2 I ¿)oOx1 h Ox-4 h3 Ox3
lop
C(u3) = --- +pu;.h23 - pu2'u3I32 +
û3 uX3
C( u
+--+--+f
1 ¿)o- i 8013h Ox2 h3 ¿ix3
033) ¡t32 + 2cr23 It23 +
(029 - a33)1t23 + 2a23K32 +
Schiffstechnik Rd. 40 1993 / Ship Technology Research VoI. 40 - 1993 21
2. Grid generation and transport equations
TILe transport equations for the turbulent flow around a ship are solved in a computational doniaiji which consists of subsequent orthogonal curvilinear grids on transverse sections
(Tz-abi ras and Loukaki 1983,). The orthogonal grid generation is based on the conformal mapping technique of von Iicztk and 'I'ack (196.9): tIte coniplex ¡)lane z on the unit circle is mapped to time complex plane z of a transverse section according to the transformation
i =
(i)
u = i
The derivation of (1) is based on the existence, of two symmetry planes, i.e. the waterplaiie and time longitudinal symmmmetry plaime which are intersected by time generated coiítour a.t right angles. However, real ship sections do not. always fulfil this restriction. The front or aft part stern sections usually intersect the waterplane at non-orthogonal angles. The same holds for
time longitudinal plazie when, e.g., a bulbous bow or ste.rn exists. In order to obtain an accurate
a.imalytical representation in these cases, a Karman-Trefftz mapping is performed first (von Ierczek and Stern 1983,), and transformation (1) is applied to the' resulting section.
lii aimy case, time coefficients a, of ( I ) are calculated by thé proposed iterative procedure of .,,o,1 Ke,czek and Tuck (196.9) where time imumnber N depends on the complexity of the contour. Fcr iiiidship sectiomis, N usually varies between G and 10, while for complex contours, such as a bülhous stern, time reciuired imuimiber of coefficients is drastically increased and ranges between 50, and 70. Omice coefficients a,, im = 1, ..., N are calculated, the orthogonal curvilinear grid can I)C easily created I)' transfoimuimig time corresponding grid omm the circle plane, which is composed of ia.dii and concemitric citcies. 'l'ue distribution of the grid points on the unit circle depends on time clesited arrangement ou the reai contour. If this distribution follows equal arcs on the
circle, the grid nodes on the station contour are concentrated near the two symmetry planes
where high positive curvatures exist, while they are sparse at the concave parts of a U- or
V-shaped section with negative curvature. Sluice at time couicave regiomm of stern sections rapid cimaimges of the flow variables often occur, time above distribution is not always effective with respect to the numerical discretization. Therefore, a combination of two techniques is followed
in time present work. At the regions near the two symmetry planes equal arcs define the points
on the circle, while at the concave parts the points on the circle correspond to equal arcs on the section contour. The latter is achieved by using the inverse mapping of (1). Intermediate
sections and grids caim be easily created after calcuiatimig time corresponding coefficients by
cubic interpolation among the original ship sections. This fast and effective grid generation is ncessary to perform grid dependence tests by successive refinement, as described below.
Fig.i shows an orthogona.l curvilinear grid around a bulbous stern section of a tanker model. A local co-ordinate system for the velocity components can be defined as (uj,u2,u3) where 'u2
and u3 coincide with time gild lutes miorinal al)d along time section contour, while the third
coimmponent u1 is always normai to the section pIane; The corresponding time-averaged
mo-mentum (Reynolds) equations, iii orthogonal curvilinear coordinates are (Tzabira.s, Loukcikis and (Jarofallidis 1990):
+
03
+i O3
+ i ¿la33Ox1 h2 ¿1x. h3 OX3
where h1 ( = 1), h2, h3 are the metrics of the curvilinear system, Kjj the curvatures, p is the pressure, p is the fluid density and C(u) shows the convection terms:
with a = 1.3. The generation term G appearing in the source ternis of k and e transport
equation can be expanded as:
G = 2/L3 {e + 2 + + (e2 + e3 + e3) (10)
The term fo on the right side of equation (2) represents the action on the fluid when it passes through the propeller blades; it is modelled according to the body force approximation (Stern
et al. 1986, Zhang 199O). It is assumed that the propeller acts as an actuator disk with
pie-scribed circulation r(r) which in the present study follows a simple sinusoidal distribution.
3. The finite volume equations
A finite-volume approach (Roache I972,, solves numerically the momentum (2), (3), (4) and the turbulence model (8), (9) transport equations. Each equation is integrated in a control
volume which depends on the variable according to a staggered arrangement. This integration
results in the algebraic form (T:abiins ¡984):
Ap4p = ANN + ASS + AE$E + AWW + AD4D +
+ £.
(li)
where P is the central node of the control volume, Fig.2, and N, S, E, W, D,U the
neigh-bouring nodes along the directions x2,x3 and z1, respectively.
The coefficients A,i =
P, N, S, E, W, D, U are generally functions of and take into account the combined effect
of the convection and diffusion ternis which exist in the original differential equations.
22 Schiffstechnik Bd. 40 - 1993 / Ship Technology Research Vol. 40 - 1993
C(u)
-
-e--- [Oh2h3uiuih2h3 + 0h3u2u1 ± Oh2u3u] (5)Ox1 Ox2
The stress tensor components c are expressed as:
a11
= 2pe1i = 21e
a22 = 2,e22 =
+ u3K23 = 2/.LeC33 = 2 ¿ h3 8xt
+ u2132a12 = ¿.e12 = ¡tt ILa ± h (6)
a13 =
IteEI3 = I +J-[O h3 Or3
a23 =
/2eE23 = 1h2 ax3+ --
h3 a - u31t32 - U91t23]The effective viscosity ¡ in expressions (G) is related to the isotropic turbulent viscosity
according to the assumptions of the k-e turbulence model (Launder and Spa/ding 1974):
Pe = /L + /2 = IL + 0.09pk2/e (7)
0 0k 1
C(k) =
(,Ut--)
0 h3 ¿1k 0 h ¿1kûx + G - (8)
+
-Ox1 Ox1 1z2h3 Ox2
(p.--) + (/L3---)
'2 Ox2 ù13 h3O ii 0e
-
1C()
= Oji h3 ô
Ou h
Oc i e+ 1.44G
-k
l.92
k (9)+
Ox1 aùx1
h2h3 Ox9 a11z2Ox9(--.) +
Ox3 ah3òx3 j
)I
more differential equations:
Schiffstechnik Bd. 40 - 1993 / Ship Technology Research Vol. 40 - 1993 23 In this study, the diffusion ternis are always approximated by central differencing, while two schemes have been used for the modelling of the convection part of A1. The first scheme, related
to the hybrid scheme of Spaldiny (1972), models the convective ternis by upwind or centrai differencing according to the local Peclet numbers. lt is used for the calculation of the Au and AD coefficients, i.e. along the dominant flow direction, as well as in the transverse directions
around the front part of the body where the boundary layet is thin. The second scheine
is applied at the thick boundary layer region around the stern and uses central differences to
calculate the convective part of A1 in the transvérse directions (i.e. AN, A5, AE, Aw). Tzabirzs (1991) shows in numerical tests that this scheine, being more accurate than upwind differences, can be used without convergence or bounduess problems.
The external body force due to the propeller action is included in the source terni S, of the
'u1 - momentum equation (11) as:
fB=CEr(r)
(12)where E is the intersection area of the propeller disk with the corresponding control volumes. It
is uniformly distributed to two successive volumes in the longitudinal direction. The constant C' can be calculated from relation
T =
cJr(r)rdr
(13)for a known value of the thrust T. In (13) rh and r correspond to the hub and the tip radius.
The pressure field is calculated by a SIMPLE-like procedure (Pcitankar and Spalding 1972,,
and an equation similar to (11) is introduced for the pressure correction ¡J
App' = ANPJ + A5p + AEP'E + Ap', + ADP'D + Aup'1, + 5m (14) where
A = D'uE,
A =
A1.'Sn, is the integrated velocity divergence, L is the projected area of the i face of the control volume normal to u1-component, and Du is approximated as (Tzabitis 1992f
Du = D'u° +
ADt4'
(15)In ( 15) A equals to A/A p from the corresponding nionientuin equation (11). The functions
link the pressure gradients to the velocity changes. lii the orthogonal co-ordinate system
under consideration they are simple fuiictiotts of the geometry and A, along the transverse directions x2, s3. It is evident from the integrated form of the momentum equations (3) and (4) that a velocity change is directly related to a pressure change Ap' through
+ Du1Ap' (16)
where i = 2, 3 and Aj! refer to the respective direction. In a staggered grid Ap' is simply the difference between two adjacent nodes. An expression similar to (16) but approximate can
be easily derived foi time niahi directiolL X1 (TzabilTL8 19SÇ). Time applicatioii of relation (15),
which implies an il-step iterative procedure, was proved quite efficient for the calculation of axisymnmetric flows, allowing foi high. under'laxatioii factors and, therefore, leading to high
convergence rates. lu real shuip forms it lias also been applied vitlu success as compared with the SIMPLE 01. PISO (Lssa 1981) algoiithimms. Convergemice can be achieved at stern sections where the two other lllethìO(lS fail. Wlieii steep variations of time pressure occur (such as at the stagnation regioil around the bow), strong underelaxation for the pressure correction is
required to obtain convergence with the solution method which is followed.
4. The boundary conditions
The computational domain is divided into a front and rear region at tile ship, Fig.3. This
subdivision is necessary for high grid resolutions and - almost - grid independent solutions. At
the front part (I), covering the how and a middle region, the houudary layer is thin and fine grid resolution along the normal direction x on the body surface is required by a. differential method which solves siriuiltaneously the velocity and the IressIre fleId nlhis tiue, siiìce the e<ternal boundary has to be placed far enough fi'om the solid surface in ordei to ¡'educe the displacement
effect on the outer flow. Moreover, the high pressure gradients which exist around the bow
region demand a fine discretization along the longitudinalx axis, while relatively coarser grids
can be adopted in z direction due to the rather smooth variation of the transport quantities
girthwise. At the rear part calculation domain (li), which starts amidships and extends in
the wake, the existence of a. thick boundary layer region. the formation of strong longitudinal vortices and the interaction with the near wake are associated with rapid changes of the flow
variables. Therefore, high spatial discretization is required in the three dimensions x1, x2, x3 in order to obtain reliable numerical results.
Boundary conditions have to be specified at each boundary of the sub-regions(I) and (Il) due
to the elliptic character of the discretized transport equations (Il). At the external boundary N of both regions as well as at the inlet boundary (J of region (i) time values of k and r are
set equal to zero, while the velocity components and tIme pressureare assumed to be equal to
the values calculated by the potential flow solution around the actual ship (Hess andSmith 1966). This assumption, being crucial especially for the location of the external boundary,has to be applied in order to avoid excessive miumbers of grid nodes that would be necessary if N wäs located within the undisturbed flow region. However, the shortcoming is that numerical
tests are IIee(led to estimate the location of N, so tlma.t the nimmmiemica.l calculations near the body surface are practically not influenced by the imposed extei'nal conditions. Also, since the potential flow boundary conditions are essential foi' the viscous flow solution, a sufficient surface panel distribution is required to model the flow, particularly at the how region.
Near the body surface, i.e. the bdundary S of Fig.3, the classical wall function method of Launder and Spalding (1974) is applied to model the turbu1nt flow. lt is assumed that the
nn-dimnensiona1 velocity parallel to the wall follows the logarithmic law:
= 1/icln(Ey'4') (17)
with i = 0.42, E = 9.79, u+ = 'u/u1, u =velocity component parallel to the wall, u1 =
r,,. =the wall shear stress, y+ = YRU1/1-', u =fluid kinematic viscosity, YR =normal distance
frbm the wall. Relation (17) together with the values of k and r, derived from the hypothesis
of a nea.r wall one-dimensional turbulent flow, are implicitly introduced in the u1 - and
u3-momentum and turbulence model' equations (2),(4),(8),(9) as the wall boundary conditions. A special treatment is also needed to calculate the convection and diffusion coefficients on the south face of the next to the wall control volume of the u2-cornponent. The value of u2 is calculated to satisfy the integrated continuity equation in the volume which is surrounded by
this face a.nd the wall, while the combined effect of the viscous stresses is taken into account by the projection of the wall shear stress in the x2 direction. The convection: terms on the normal
to tile body faces are calculated by appropriate integration of the logarithmic law (17). The values of the velocity components and the turbulent quantities on the exit planes D1 and D,, of the two sub-regions are assumed to be equal to those of the previous transverse planes, i.e. Neumann conditions are applied. The pressure values on D,1 are calculated by linear extrapolation, while on D, the pressure is estimated by the potentia.l flow solution. The
inlet boundary conditions for ali variables at the inlet plane 1111 of region (11) are calculated
by linear interpolation among the corresponding values obtained after the front part solution
is completed. Fiiially, 011 tile two symmetry planes Neumanì conditions are applied for all variables except for component u which is equal to zero.
5. The solution procedure
The niotnentuni and turbulence model finite volume equations (ii) are solved in both regions following the principles of the partially parabolic algorithm of Praap and Spalding (1975). The
contribution of this aigorithni in tite solution of marine hydrodyitamic problems is of major inìporta.lice, silice a ittarching proced we cali be aJ)plied to solve tite cotiiplete 3D Reynolds equations, resulting in a drastic decrease of the itumber of stored variables. A prerequisite to apply the partially parabolic algorithm is tite existence of a dominant flow direction, which in shipUke bodies can be considered as tile direction that is parallel to the x1 axis.
For a transverse section (constant
x1) the u1, u2, and u3 momentum equations are
solved first. Titeit time pressure field is updated by solving tite system of equations (14) and the velocity components are corrected to satisfy the integrated continuity equation. Next, titek-. equations are solved and the effective viscosities are calculated through (6). The solution
of tite systems of the discretized eqima.tiomLs is pemformiied foi each variable by the application
of a two-step TDMA algorithm (Roache 1972). Tite upstream and downstream values of the variables (including pressure) are known front tile previoùs iteratiolF, that is the terms ADD and Ad! mJ areassumed to be a part of tite source terms of eq.( 11). After this solution, which is
performed 0CC for each variable, the same calculation steps are repeated for the next station etc.. until a sweep of tite calculation dommiaimi is completed. Several sweeps are required to obtain
convergence of tile velocity and pressure fields. Convergence is açhieved whemi the integrated skin friction C1 and pressure C, coefficients satisfy, for 30 successive K sweeps, the following criteria:
Cf
-
C7'I
<0.0002ICf'I
IC- C-'I <
0.0002I(hI
Time integrated C1 and C coefficients are defined as: R
- f)SU'
' - pS(J
where (' is lite velocity at infinity and the skin friction I?. and the pressure R forces are
cdicuiated by:
RF=J1w(S.k) c18,
R=_JPw()ds.
(19)s s
(18)
Ii (19) Sis the integration surface, .flS the tangential vector to the body contour, the normal one amid k the unit vector parallel to s1. The surface pressure value Pw is equal to the calculated omie in the pressure control volume adjacent to tite wall, while tite wall shear stress is estimated
i)y tite wall function method
CIbh'4Kku/2u
TW - P
ln(Ey)
(20)where u is the velocity component parallel to the wall and CD = 0.09.
A successive grid refiuiemnent technique similar to tile one applied for bodies of revolution
(T:abiras 1992), studies the effect of the grid size Oli tite numerical results and accelerates
commvergence at high grid resolutions. Suppose convergence has been achieved with a given grid
(ii. X.,. x) and the solution ° has been obtained. The starting values for a fluer grid
(x , r,, ;v) calL timeim be determined by a two-step limmeam interpolation. First the iuterpoiation
is ittade oit tite oid transverse planes, Fig.4, as
L_
C) O O O0i 1+a2 2+ú3 3+C14
(21)The coefficients c1,a2,a3,a4 are simple functions of the local grid geometries. Then, linear interpolation is applied in the third direction x1 between the L and L + I planes. to calculate
the initial values:
- a
L a L+1- I-'l"p T /J2'rP
This procedure proved to be quite efficient with respect to the achieved convergence rates at high grid resolutions, reducing the required computer costs by more than five times. Grid dependence tests can be performed UI) to the desired grid densities with remarkably lower computational effort.
The rules under which the grids are successivelly refined are associated to the rates of
changes of the flow variables. Along x2-direction the mesh varies exponentially with high node
concentration near the solid boundary. The arrangement of grid points along the .x3 direction, i.e. parallel to the section contour, follows the cnforlìal transformation technique as already
described. In the main direction x1, five regions are defluieci, Fig.5. In regions I and II the grid is refined near the fore end of the ship according to an exponential distribution. The
steep negative and positive pressure gradients which occur in these areas together with the transition from a free to a boundary layer flow require high grid densities for a sufficiently accurate numerical solution. The same procedure is followed in the region III, where the grid nodes are concentrated iiear the trailing edge, while iii legion IV, including the frames above the propeller, the transverse sections are equally spaced. Finally, in region V, which extends
from the aft end of the ship to some extend in the wake, again an exponential spacing is used.
The aforementioned solution procedure comprises the first iterative step when the self-propulsion problem is considered. Then the propeller thrust T is assumed to he equal to the
calculated total resistance RT = R + R and the constant C of ¡dation (13) is derived. The
body forces are calculated through (12) and a new numerical solution is obtained for the rear part domain, using the finer grid. The resulting value of RT is again set equal to T and the iterative procedure is followed until the criterion
- R'I <O.001IRI
holds for three successive steps. (i.e. N, N + i and N + 2).
6. The test cases
The two tanker models of the 1990-SSPA-CTH-IIHR workshop were selected as the test cases
of the present work. Tanker i is the original HSVA model having V-shaped stern sections and approximate principal characteristics: LWL = 2.74m, B = 0.40m, T = 0.15m and C'B = 0.84. Tanker 2, designed at SSPA, has the same forebody as the HSVA model but an U-shaped
afterbody and a different overall length, LWL 2.72m. The twö models have the sanie length
between perpendiculars, L = 2664m, which is used in the results as the reference length.
The first part of the computations performed is concerned with the comparison of calculated
results with the existing measurements for Tanker 1 (Wicçjlzardt and Kux 1980,), and for Tanker 2 (Krzaack 1991).
For both models the computations were carried out at a Reynolds number 5 . 10e, based on L. In order to use accurate boundary cou«litious as far as possible, calculations have been performed only for the stern part, the inlet plane placed at x/L = 0.645, where experimental data concerning the integrated boundary layer parameters exist (Hoffmann 197G) (z is the
distance from the front perpendicular). Using the measured boundary layer thickness. the shape
factor and the cross-flow angle girthwise, input velocity profiles can be generated while the k and values are obtained by empirical relations (T:abin-ts 1991). The external boundary on the wa.terpla.ne was located at a constant distance.from the centerline which was equal to six tiimmes
the maximum boundary layer thickness around the first section. Extensive numerica.! tests on (22)
bodies of revolution (Tzubirs 1992,), have indicated that this defiñition of boundary N leads
to almost insensitive resistance calculations with respect to the imposed external condition.
A NlNJNK grid equal to 55 . 45 . 97 was used in each test case, where NI is the number
of nodes in the circumfereiitial x3-diiection, NJ along tite iìormal x2 and NR. the number of traiisverse sections (x1-directioii). The difference iii tue calculation of the stern part resistance coefficients between the above grid and a coarser one of 453077 nodes was below 2%, which is
considered satisfactory with regard to the dependence of the produced results on the grid size. TiLe values of ¿i in the first iiear wall calculation points ranged between 40 and 80 for both
niodels, which is the indicated range foi the application of wall functions.
Potential flow calculations were peifoiiiïed using 2348 quadrilateral eleiiieiits for Tanker i and 2280 for Tanker 2. These numbers of surface panels refer to the one-half of the model,
taking into account the two flow symmetry planes. The sanie panel arrangement has been used
ill all numerical tests. All the compute!' runs were made in a SUN-330 SPARC workstation.
Fig.6 presents calculated and expei'imental isowakes (i.e. u1/(Jo constant lines) at the station a/L = 0.9598 of Tanker 1. The experhitental results show a tendeiicy for sepaxation around the projection of the center of the propeller disk. This is not the case for the numerical results which have the trend to overpredict the u values in the saziie region. An investigation of stern
flow calculations (T:abiiu 1991), shows that this behaviour is due to both the h- turbulence modelling and the wall function appioxiiva.tiou. A remarIab1e iIfll)roVelueflt can be achieved if, instead of tite wall functions, a direct near wall solution is applied. However, in order to compare (1Uite similar models at low and high Reynolds iìumber computations, the wall functioii method
is used in both cases. The comparison of calcuIate(l and measured isowa.kes on the propeller plane, x/L = 0.976 in Fig.7, shows similar trends, namely there is an upstream influence on the formation of the u1 velocity profiles. The numerically predicted isowakes on the propeller
plane. x/L = 0.989 of Tanker 2, Fig.8, are in accordance with the observed results for U-shaped sterns (Dyne 197.), but discrepancies are observed with respect to. the experimental values
which present a local ti1 maximum at a distance of 0.5R from the propeller center (R is the propeller radius). Predictions agree better with measurements downstream at z/L = 1.018, Fig.9.
Comparison of the longitudinal component of vorticity is of special interest, since the
lattem is related to the forinatioti of tite longitudinal vortices which affect tite propeller
perfor-¡nance, Fig.i0. The trends for the longitudinal vortex formation are the same, but calculations show a more intense vorticity aiea which is almost the same as the one predicted by the method
of Pat1 i al. (1990,). The measureliments in tite aiea itear tite hull are rather scarce and, there-fore, detailed comparisons can not be ¡nade. The calculated results in the near wake region which are compared to the experimental contours at x/L = 1.005 in Figs. lia and lib, show
an improvement.
'l'lie accurate prediction of wall shear stresses and pressures is important because they define tite ship resistaitce. Unfortunately, IIOffifl(Lflfl (1976) provides only few experimental values for
Tanker 1. Fig.i2 compares calculated aiid measured local C. and values around the after-most measuring station at x/L = 0.942. Tile predictiois agree well with measurements. Local differences are of minor importance with respect to the integrated results.
6.1 Resistance calculations at model and full scale
Two sets of whole-field computations have been performed for each model at Reynolds numbers of 5 . 10's (model) and 2 . i0 (full scale). Time computational domain of tite front
part. region I of Fig.3 started at x/L = 0.075 and extended up to x/L = 0.6. Time external
boundary (same for both models) was specifled on tite waterplane to be at least. six times tile loca.1 boundary layem thickness. The latter is calculated by tile flat-plate empirical formula
(Schlichting 1968). The aft-part coruputatinal domain was between x/L = 0.55 and
x/L =
1.24. After the calculations the values of y+ at. the center of the adjacent to the wall control volumes ranged between 30 a.nd 120 for the low Reynolds and between 400 and 6000 for the high Reynolds number. Extensive numerica! experiments on bodies of revolution or ship forms (T:abiras and Loukakis 1989, Ju and Palet 1991, Tabiras 1992), have shown that the
increasing 9+ values can be reliably tiSed as the Reynolds number increases. It is quite easy, e.g.
to show that for the 1/7 power law profiles of a fiat, plate the u'ange 35 < y < 110 at Re=5 106
c.orrespoll(lS to the range 2500 < j < 7000 at. R.e=2 . l0. when the saine non-dimensional distances from the wall are considered. The highest values of y appear at the a.ftermost part of the body, i.e. region IV in Fig.4, while y ranges between 400 and 2500 at tIte front and the
midship region.
Table 1: Grid dependence test for the front part
Table i shows the dependence of the integrated resistance coefficients on the grid size at the front part. giving results for both Reynolds numbers. The interpolation technique followed becomes very advantageous at the high grid (lelisities. For the low Reynolds number the (' value is insensitive for the last two grids, while C, becomes almost independent at grid densities above 35 35 92. The corresponding values at the high Reynolds number show that C1 remains
practically uninfluenced for all the tested grid sizes, while C is drastically more sensitive to
grid changes and converges to a limiting value when the number of the nodes is increased. The pressure force at the low Reynolds number results in a higher C value of 0.72V i0 than the corresponding 0.47 . iO at the high Reynolds number test, concluding that the viscous calculation of the pressure field at the front part is sensitive to the Reynolds scaling. Therefore, the. estiina.tioii of the pressure force on the bow by a potential flow solution is questionable. As expected, the C, value at the high Reynolds number is closer to the potential flow value, which was estimated to 0.15 . iO over the sanie surface area.. Tables 2 and 3 present grid dependence tests for the stern parts of Tanker i and Tanker 2.
Table 2: Grid dependence test for the stern Part of TANKER 1
Table 3: Grid dependence test for the stern par.t of TANKER. 2
28 SchifÍtechnik Bd. 40 - 1993 / Ship Technology Research Vol. 40 - 1993
R.e=5 I0 R.e=2 . i0 NINJ.NK grid density IT sweeps ¿' .
loa Ç
.iO NINJNK ITC .
iOG .
iO 15.15.37 195 3.69 1.97202051
313 1.68 1.89 25 .25 . 68 250 3.81 1.18 35 .35 . 81 163 1.68 0.87 35.35 .92 147 3.82 0.75 4550 106 155 1.68 0.48 45 .4.5 . 92 95 .3.82 0.72 45 . 65 106 160 1.68 0.47 Re=5 10 R.e=2 l0 NINJ.NK IT C1 . i0 C i0 NINJNK IT C1 . i0 Cr151542
183 3.18 2.76 20.20.47 291 1.48 2.79 30 . :30 . 77 150 3.19 1.69 35 . :35 . 77 101 1.49 1.66454597
66 3.21 1.62455097
118 1.50 1.42 45.ß5.97 60 1.50 1.38 Re=.5 106 R.e=2 10NI.NJNK IT C1 . iO
Ç .
iü NINJNK IT C1 . 103G .
i5 15.42
291 3.24 3.53 20.20.47 396 1.52 3.1130.3077 166 3.25 2.15
353577
178 1.51 1.8345.45.97 53 3.2.5 2.00
455097
182 1.52 1.46The improvement of the convergence rates, by the grid refinement technique, is impressive for the higher grid resolutions which were used at the stern. The obtained values present the same trends as at the bow solution, that is the C1 values converge faster than G as tile grids
become finer.
Table 4: Resistance characteristics at model and full scale
'rlie iiìtc'grated overall resistance coefficients C'i, Gp and CT = C1 + G are presented in Table 4 for the two models. The total resistance coefficient CT of the U-shaped Tanker 2 is about 3.5% higher than the corresponding Cr of Tanker I at model scale. Tile difference is teduced to 0.8% at full scale. \hitli respect to tite ITTC-57 friction Une which calculates the
titodel aitci full scale C1 coefficients as 3.397 . l0 p 1.407 . l0 , the predicted friction coefficients of Table 4 aie 6% higher at the model Reynolds and 14% higher at the full scale
Reynolds number. If the ATTC friction formula is adopted, the difference at the low Reynolds is almost 9.5%, i.e. the predicted C'1 curve is roughly parallel to the ATTC-friction line. To test
the validity of the two classical extrapolation methods (Froude hypothesis and the form factor
approach) it. is assumed that the low R.eyitolds calculated values represent an experiment at
model scale. The residual resistance coefficients ('R, derived by subtracting the ITTC values foi C1 from the total coefficient C',', are equal to l.308 I0 for Tanker i resp. 1.472rn i0 for Tanker 2. 'l'lien the full scale CT values are calculated as C = CFH ± CR by time Froude
method or C = (1 + f) . Cpy by the form factor, where CFH is tlìe ITTC C'I value at the
high Reynolds number and f the form factor which is defined as CR/C'i at model scale. Table 5 gives tile predicted values for C. Supposing that the numerical method values are correct,
the Fronde itiethod overpiedicts tite resistance by 9.6% resp. 15% while the use of form factors underpreclicts by almost 20%.
Table 5: Validation of extrapolation methods
Tite last row of Table 4 shows tite calculated nominal wake fraction i - w, i.e. the flow
rate through the entrance of tite propeller disk, non-dimensionalized by U0. Time Reynolds iLurilber affects significantly tile results. Tite wake fraction is increased by almost 30% at the fitti scale. Titis difference is explained by tite isowake contours of F'igs.13 and 14 at the propeller
planes a/L = 0.976 and x/L = 0.989 of the two models, corresponding to time high Reynolds
computations. The isowakes at tite full scale are quitedifferent from those of time model scale,
Figs.7a and 8a. tite former being smoother and concentrated near the longitudinal symmetry platie. in Figs. [5 and 16 t he longitudinal vorticity contours, corresponding to the low and
high Reynolds calculations, are also plotted for tite 1)ropeller' plane of Tanker 2. The positive
values of vorticity at full scale are concentrated closer to tite surface and the symmetry plane,
Schiffstechnik Bd. 40 - 1993 / Ship Technology Research Vol. 40 - 1993 29
TANKER' TANKER2
Re=5 106 Re=2 iQ R.e=5' ]Ø6 Re=2 iO9 C1 . 1O 3.597 1.613 3.612 1.620 ('p iO 1.107 0.864 1.257 0.876 CT . i0 4.705 2.477 4.869 2.497
i - w
0.549 0.709 0.538 0.7 15 TANKER.1 TANKER2 Error Error calculated CT . iO3 2.477 0 2.497 0 Froude Cr' iO 2.715 +9.6% 2.879 +15% Foim factor C'' . iü 1.948 -21% 2.016 -19%References
DYNE, G. (1974), A study of the scale effect on wake, propeller cavitation and vibratory pressure at hull of two tanker models, SNAME Trans.
HESS, iL. nd SMITH, A.M.O. (190G), Calculation of potential flow about arbitrary bodies, Prog. Aeronaut.. Sci. 8
HINO T. (1989), Computation of a free surface flow aroiiiìd an advancing ship by the Navier-Stokes eqiiat.ioiis, 5th lot.. Conf. N uni. Ship Ilydrocl. . ¡1 iroshiina
HOFFMANN, H.P. (197G), Investigation of the three,diiuensioiial turbulent. boundary layer on a double model of a 5h1) In a wiil tunnel, (in German), IIS-Rep.343, Univ.Haiììburg
ISSA, R.J.(1981), Numerical methods for two- and three-dimensional viscous flows, V.1<1. Lecture Series
5
JU, S. and PATEL, VC. (1991), Stern flows at full-scale Reynolds numbers, J. Ship Res. 35
von KER.CZEN, C.. and TUCK, E.U. ( 1969), Th represeiit.ation of ship hulls by cotìformal flapping techniques, J. Ship Res. 19
von KERCZEK, C. and STERN, F. (1983), The representation of ship hulls conformal mapping fune-tions: fractional maps, J.Ship Res.. 27
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LAUNDER, BE. and SPALDING, D.B. (1974), TIme numerical computation of turbulent flows, J. Comput. Methods in App!. Medi. and Eng. 3.
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PATEL, V.C., CIIEN, [IC. and JU, S. (1990), Comput.ationsof ship stern and wake flow and compam-¡sous with experiment., J. Ship Res. :34
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TZABIRAS, G.D. amid GAROFALLIDIS, D.A. (1989), Prediction of the resistance characteristics of an axisymmetric body With a propeller miiodel, 4th mt. Symp. Practical Design of Ships, Varna
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Arb or
TZABIRAS, G.D. (1991), A ituinerical study of the turbulent flow around the stern of ship models, hit. J. Nuni. Meth. in Fluids 13
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Fig. 1. Grid on a transverse section
I
N
D11
II
Fig. 3. Subdivision of the calculation
domain
Fig. 5. Definition of interpolation regions
Fig. 2. Control volume
Fig. 4. Grid interpolation on a transverse section
Schiffstechnik Bd. 40 - 1993 / Ship Technology Research Vol. 40 - 1993 33 Dt
ut'
Fig. 6. Calculated (left) and experimental (right) isowakes at X/L = 0.9598 (Tanker 1: Re 5. 106)
Fig. 7. Calculated (left) and experimental (right) isowakes at X/L = 0.976 (Tanker 1: Re = 5 106)
Fig. 8. Calculated (left) and experimental (right) isowakes at X/L = 0.989 (Tanker 2:
Re = 5. 106)
Fig. 9. Caiculated (left) and experimental (right) isowakes at X/L = 1.018 (Tanker 2:
Re = 5. 106)
Fig. 10. Calculated (left) and experimental (right) iso-voi'ticity contours at X/L = 0.9598 (Tanker 1: Re = 5. 106)
.9
Fig. il. Calculated (left) and experimental (right) iso-vorticity contours at X/L = 1.005 (Tanker 1: Re = 5 106)
0 0080
0.0060
0.0040
0.0020
Keel Girth(%)
Fig. 13. Calculated isowake contours at
X/L = 0.976 (Tanker 1: Re = 2 i0)
80.00 '0000
Fig. 15 (left). Calculated iso-vorticity contours at X/L = 0.989 (Tanker 2:
Re = 5 106)
Fig. 16 (right). Calculated iso-vorticity contours at X/L = 0.989 (Tanker 2: Re = 2. 10g) O f000 0 0009 - '000 Keel Grth(%) -J3000 I I 0.00 2000 4000 60.00 80.00 '00.00
Fig. 12. Distribution of Cj (left) and (J,, (right) at X/L = 0.942 (Tanker i)
Fig. 14. Calculated isowake contours at X/L = 0.989 (Tanker 2: Re = 2 10e)
Fig. 17. Integrated pressure force for Tanker 2
36 Schiffstechnik Bd. 40 - 1993 / Ship Technology Research VoI. 40 - 1993
00 0.0000
Fig. 18. Local skin friction coefficient for Tanker 2
Fig. 20. Calculated isowakes a.t X/L = 0.976
with propulsor (Tanker 1: Re = 5 10G)
200 1.00 -0.00 0.00
y2/R
Re=5* 106 Re=2* 0.50 1.00Fig. 19. Velocity profiles with propulsor in operation
Fig. 21. Calculated isowakes at X/L = 0.976 with propulsor (Tanker 1: Re = 2 10e)
Fig. 22. Calculated isowakes at X/L = 0.989 Fig. 23. Calculated isowakes at X/L 0.989 with propulsor (Tanker 2: Re = 5 . 10) with propulsor (Tanker 2: Re
= 2
jQY) Schiffstechnik Bd. 40 - 1993 / Ship Technology Research Vol. 40 - 1993 370.00 40
0.0020
0.60
0.10
Fig. 24. Calculated iso-vorticity contours at X/L = 0.989 with propulsor (Tanker 2:
Re = 5.106)
0.40
0.80
Fig. 25. Calculated iso-vorticity contours at
X/L 0.989 with propulsor (Tanker 2:
Re = 2 10e) 0.0000 0.60 0.70 Re= 5.106 - without propulsor - with propulsor
Fig. 28. G1 distribution at stern part
(Tanker 2) u H 0.90 1 .00 Re=5. 106
-
without propulsor with propulsorFig. 2G. Ç distribution at stern part
(Tanker 2)
Fig. 27. Ç distribution at stern part
(Tanker 2)
x/L
X/L
38 Schiffscechnik Bd. 40 - 1993 / Ship Technology Research Vol. 40 - 1993
1.20
'1.10
