astern conditions and of the same propeller in open water ... · pdf filein the course of...

Propeller-hull interaction in inland navigation vessel

J. Kulczyk

Department of Mechanical Engineering, Technical University of

Wroclaw, Poland

Abstract

The author presents a computational model of propeller operating in the wakeof inland navigation flat-bottomed vessel. Using this model it is possible tocompute propeller performance as well as propeller-hull interaction coefficients(wake fraction and thrust deduction factor) with limited depth of waterwaytaken into account. The surface distribution technique was applied to solve aflow over nozzle (part of ducted propeller) and potential component of flowbetween hull and waterway bottom. Vortex lattice model was used for screwpropeller or impeller. Simplified Reynolds equation is being solved to determineaxial velocity in flow domain. Results of performed calculations exhibit goodagreement with available results of model tests.

1 Introduction

Screw propeller located at the stern of a hull operates in non-uniform velocityfield that is a result of disturbance caused by presence of a hull in uniformstream. Propeller also induces disturbance velocities which interfere with thehull flow. This mutual influence usually cannot be neglected in analysis of shippropulsion system and is called strong propeller-hull interaction.

Three factors describe that interaction: wake fraction, thrust deduction factorand rotational efficiency of propeller.

The first coefficient includes information about difference between uniformstream and real velocity field in propeller disk. The second describes differencein hull resistance in the case with and without operating propeller. Rotationalefficiency includes proportion between efficiency of propeller operating in

Transactions on the Built Environment vol 11, © 1995 WIT Press, www.witpress.com, ISSN 1743-3509

74 Marine Technology and Transportation

astern conditions and of the same propeller in open water conditions (in uniforminflow). Correct design of ship propulsion system depends on accuracy indetermination of those factors.

For many years model tests have been the only reliable source of informationon hull-propeller interaction. Because model tests are both time-consuming andexpensive their scope is often reduced to one speed and loading condition, andvariation of the above mentioned factors at different operating parameters is nottaken into consideration. For the same reason model tests are not performed forevery ship under design, especially in the case of inland navigation vessels. Eventhough model tests were eventually performed, the results are affected by 'scaleeffect'.

Design of screw propeller consist in determining the parameters of propelleron the base of diagrams for extensively tested series of systematically variedpropellers (e.g. Wageningen B- or Ka-series). The alternative approach iscomposing a propeller of standard airfoil sections. Sometimes, for somereasons, the shape of the original propeller selected from propeller series has tobe modified. For example ducted propellers designed for inland navigationvessels are usually embedded in hull in order to maximise diameter of impeller.The significant difference between geometry of such propeller with incompletenozzle and original propeller is reflected in performance of propulsion system.In the course of design the influence of change in propeller geometry on itsperformance is ignored or included approximately, usually on the basis ofdesigners individual experience.

In this paper the author presents computational model of flow in stern regionof ship with operating ducted propeller. The application of this model isconfined to flat-bottomed hulls and includes limited depth of waterway that isparticularly important in inland navigation.

The well-known vortex lattice model is adopted for screw propeller(impeller). The surface vorticity distribution is applied for nozzle as well as incalculation of potential flow between hull and waterway bottom. Both methodsof flow modelling are valid for ideal fluid. Therefore influence of viscosity istaken into account only by introducing resistance coefficients in calculations ofnet forces on propeller blades and nozzle. Analysis of hull flow is confined tothe region between ship and waterway bottom. Von Karman algorithm forviscous fluid has been applied.

The described method enables:- determination of velocity at arbitrary point in flow domain with and withoutoperating propeller;

- calculation of propeller performance (thrust, torque and efficiency) forarbitrary shape of nozzle section and impeller blade, including hull-propellerinteraction;

- determination of nominal and effective wake fraction as well as thrustdeduction factor.Each element of the considered system (hull, impeller and nozzle) is

calculated separately. Applied iterative algorithm of computation enables to


Marine Technology and Transportation 75

include the influence of one element on the flow over the other.

2 Computational model of ducted propeller

Application of flow modelling with vortex techniques requires the assumptionof ideal fluid. Although compressibility of water is negligible, viscosity playssignificant part in generation of tangent stresses and was included in the form ofairfoil drag coefficients in calculation of total forces.

The vortex lattice model was employed for impeller blades (in detailsdescribed in [1]).

In order to calculate the flow over a nozzle the surface vorticity distributiontechnique (which falls among boundary element methods) was adopted. Thistechnique was successfully applied in calculations of pressure distribution onaxisymmetrical annular airfoils in uniform flow [2]. The present model isextended to cope with flow over non-axisymmetrical nozzle in arbitrary three-dimensional flow field. Detailed description of that model was included in Ref

[5].

Rigid body in flow field is substituted with region of motionless fluidseparated from the ambient flow by vortex sheet of infinitesimal thickness Inorder to keep fluid inside the separated region in rest the appropriate boundarycondition must be imposed. In this case the Dirichlet boundary condition of zerotangent velocity on the inside surface of vortex sheet has been applied. Thecondition can be written in the form of two integral equations:

K«.-.«2-)+iny,(»,..«2.Ki2A,A>,>:. +v,,, =o2 4O (i)

- 2(«,-.«2j + JJy2(«,..«2.K«'A,AA>2. +K,. =0

where K[^ and K^ are the influence coefficients and are equal to tangent

component of velocity induced at the considered point on the boundary

(%]m,%2m) by the element of vortex sheet of unit strength, located at position

(%in,%2n); , and F, are tangent components of velocity in undisturbed flow

at considered point, in %, and %% direction respectively; y, and y ~, are strengths

of vorticity components in curvilinear two-dimensional coordinate systemcoincident with boundary (Fig.l). First terms in the above equations account forthe velocity jump on the boundary (being a vortex sheet) at the considered point

denoted (u ,u )

According to the Helmholtz theorem on vorticity conservation bothcomponents of surface vorticity are related to each other. Hence the strength ofone component can be expressed in terms of the other and only one of



equations (1) suffices to solve the problem.In order to solve one of equations (1) numerically it has to be discretized and

written in the form of set of algebraic equations. The method of spatialdiscretization of the boundary is shown in Fig. 1.

Figure 1. Distribution of vortices on nozzle surface.

Influence of small quadrilateral patch of dimensions As A/ and vorticity

strength /-,, on the flow field is approximated by disturbance introduced by

element of vortex line of length A/ and strength F, = y^As. In the following

the vorticity y -, will be referred to as bound vorticity (denoted y ) and,

consequently, corresponding vortices F^ as bound vortices (F ). The bound

vortices in the case of axisymmetrical nozzle compose annular vortices. Thedescribed substitution results in concentration of the remaining component of

surface vorticity ( y, called trailing vorticity y ) in vortex lines F, orthogonal



to bound vortices and referred to as trailing vortices (F .). Analogously

vorticity in wake of the nozzle is concentrated in vortex filaments referred to as

free vortices (F ). In the case of isolated nozzle in uniform flow the free

vortices have a shape of semi-infinite rays attached to trailing edge of thenozzle. When a nozzle surrounds an operating impeller the wake of a nozzle isrepresented by the system of helical vortices coaxial with impeller. Strength ofeach trailing and free vortex is a linear function of strengths of bound vortices.Each bound vortex is 'included' in trailing and free vortices lying downstreamthat vortex. There is assumed that free vortices have constant strength fromtrailing edge to infinity. In the case of axi symmetrical nozzle in axi symmetricalvelocity field the trailing, and consequently, free vortices vanish in the absenceof circumferential gradient in strength of bound vorticity.

The Dirichlet boundary condition applied to control point located in midspanof bound vortex reads:

where:

r*7 B t, ~ — ~ --- strength of bound vorticity,

ajb

— y g -velocity jump across vortex sheet at considered control point,

r^ -strength of bound vortex,

ij -indices of control point.a,b -indices of bound vortices,2N -number of annular vortices composed of bound vortices,M -number of bound vortices composing each annular vortex,

AT; jw . -coupling coefficient equivalent to tangent velocity induced in control

point (ij) by unit bound vortex F^ and all trailing and free vortices

related to that bound vortex,

YI -tangent component of undisturbed (outer) velocity vector at control

point (ij),

As^ -length of a patch of vorticity.

Writing the above boundary condition for all control points on nozzle surfaceone obtains a set of 2NxM linear equations with the same number of unknown



values of /g^ .

According to the vorticity conservation theorem the strength of trailing

vortices F^ is related to strengths of bound vortices located upstream the

considered trailing vortex.

where N indicates the bound vortex closest to leading edge. The strength oftrailing vorticity is:

where:

r. -distance of control point from the axis of a nozzle,

A0 -angular span of each bound vortex.The coupling coefficients are calculated according to the Biot-Savart law.

Writing strengths of trailing and free vortices in terms of relevant boundvortices (equation (3)) reduces number of unknowns by two and is equivalent toneglecting one of equations (1). As usually in the case of lifting bodies withairfoil-like sections the satisfaction of Kutta condition at trailing edge isrequired. The condition is introduced directly into the obtained equations byassuming that strengths of two bound vortices closest to the trailing edge andlying on opposite sides of the section have the same absolute values batopposite signs.

The theorem of zero circulation along the curve enclosing no vortex line isutilised in calculation of coefficients coupling opposite vortices (located at thesame distance from leading edge but on opposite sides of the airfoil). Theseparticular coupling coefficients are calculated according to the followingformula:

where: /,#=l,2,...,Mand nJ=\,2,...,2N.Pressure distribution on the nozzle is related to tangent velocity (and hence

to surface vorticity) by Bernoulli equation:

vj

Axial force (thrust or resistance according to sense of a vector) is obtainedby integration of pressure distribution:



1 . M2N-\To,=--pV;-L Zc -Ar;,,.A0,,.;;,, (7)

£ j-\ i=\

where V \ denotes speed of a nozzle in relation to the ambient fluid,

kr.j =—\r.^j -r_,^j-an increment of radial ordinate along the contour of

airfoil section.The final solution is obtained on way of iterative coupling of two solutions.

After calculation of nozzle or impeller flow the disturbance in velocity field iscomputed and subsequently taken into account in calculation of the otherpropeller component. Convergence of the process is tested by comparison ofnozzle, impeller or total thrust obtained in two consecutive iterations.

Because the flow over impeller blade in non-axisymmetrical velocity field isunsteady it was necessary to calculate the specified blade at several angularpositions and results are averaged circumferentially. At each position the flow isconsidered steady.

The described above model of flow over a nozzle along with the commonvortex lattice model applied for impeller blades has been successfully applied incalculation of open water characteristics of ducted propellers [5].

3 Flow between hull and waterway bottom

3.1 Viscous flowThe method is based on the von Karman algorithm for turbulent flow betweentwo parallel plates. Original model was extended in order to take into accountthe varying distance between solid boundaries. In the case of present applicationthe upper boundary corresponds to hull surface and the lower boundary - to flatwaterway bottom. The shape of a hull is described by the following function(seeFig.2):

4r(*) -area of transverse section (read from sectional area curve),

B -ship breadth,h -waterway depth.

Starting from the simplified Reynolds equation applied to flow domaon oneobtains the following equation describing the distribution of axial velocity [6]:

V = 2C (9)



where:V -ship speed,z -distance from waterway bottom 0<z<z(x).

B/2

P.P

Figure 2. Shape of a frame at propeller disk .

The value of coefficient denoted C is obtained from continuity equation andBernoulli equation applied to the considered configuration, and fromassumption that pressure gradient is a sum of gradients in viscous and potentialflows. The appropriate formula reads:

c = -5

where:

fO/

[*<*>]

-pressure gradient in potential flow,

(10)

q^ -non-dimensional pressure gradient in viscous flow.

The value of qf is determined according to the ITTC formula for frictionalresistance. The pressure gradient in potential flow is calculated using the modeldescribed in next section.

In Cartesian coordinates with origin in the middle of propeller disk (Fig.2)the axial component of velocity with respect to actual shape of a frame z(y) canbe calculated from the formula:

00



where:

V^ (z>y) -axial component of velocity induced by propeller,

T -ship draught.

If one assumes zero velocity induced by propeller (K, (z,y) = 0) the obtained

velocity field corresponds to condition with no operating propeller. Then thefollowing formula for local value of nominal wake fraction can be applied:

This way the circumferential and radial distribution of the wake fraction inpropeller disk can be calculated.

3.2 Potential flowPotential flow in the region between hull and waterway bottom is calculatedwith application of the surface vorticity distribution technique in similar way asin the case of a nozzle. It is assumed that wave pattern around the ship has noinfluence on the flow. Therefore free water surface is substituted with rigid flatplate. In Cartesian coordinates shown in Fig. 3 boundary condition (1) has aform:

- r , , y . ] ] y y K , y - d x . d y = -v,(x,,y,) (is)

where:

y j, -strength of vorticity,

Kf,(x,y) -coupling coefficient,

F, (*,,>>,) -tangent component of undisturbed velocity at point (%,,}%)

(including velocity induced by propeller).Continuous distribution of vorticity on the hull and rigid water surface after

discretization is substituted with a systems of orthogonal vortex line elements.Equation (13) written at any control point on this surface takes the followingform:

ij -indices of control point (/=l,2,3,...;y=l,2,3,...),/,& -indices of vortex element (7=1,2,3,...; 6=1,2,3,...).

If one considers the typical flat-bottomed hull of inland navigation cargovessel it appears that vortex model of hull surface can be limited to the bottomof stern part of a hull. The performed test calculations confirmed no essentialinfluence of that simplification on results. On the other hand the described



simplification reduces substantially number of unknowns and consequently timeoccupied by calculations.

1 2

is const8/2

Figure 3. Model of potential flow.

Inflow velocity at the transverse section limiting the considered flow regionis assumed to vary linearly from Vps at symmetry plane to the value of sailingspeed at ship side [4]. The velocity at plane of symmetry is calculated accordingto formula:

h

i-T,15,

Equation (14) does not include the influence of waterway bottom on theflow. In order to take into account the limited depth of waterway the bottom iscovered with vorticity in the same way as hull. Boundary condition can bewritten at control points on waterway bottom in the form similar to (14).However, if one assumes that waterway bottom is a plane parallel to watersurface and moving with the speed of stream rate the vorticity on it can beexpressed in terms of hull vorticity and vice versa. The equation (14) can thenbe rewritten including the influence of waterway bottom on the flow:

_J_

where:i

J

DjKi

06)

-index of control point on the hull (/=l,2,.-index of vortex element on waterway bottom (j=l

-coupling coefficient describing tangent velocity induced at i-th control



point on a hull by j-th vortex element lying on waterway bottom,

KK.D- -coupling coefficient describing tangent velocity induced at position of

j-th vortex element on waterway bottom by i-th vortex element lyingon a hull.

Relation between the strength of vorticity on waterway bottom and thestrength of vorticity on a hull reads:

*KD (17)

Set of linear equations (16) written for all control points on a hull can besolved with inlet velocity distribution (15) entered in the RHS vector. Vortexlines representing vorticity on a hull are aligned with outlines of frames andapproximated with series of straight line elements called vortex elements. In thecase of twin screw vessels only one side of hull is considered and the influenceof the other, symmetrical in relation to plane of symmetry, is included byapplication of 'mirror image' technique. The acceptable results have beenobtained with total number of 180 vortex elements on a hull (Mx7V=180) and 70vortex elements on waterway bottom (KxL=10).

Velocity distribution in flow domain for given geometry of boundaries anddetermined strength of vortices can be calculated with application of the Biot-Savart law. The mean value of net induced velocity at specified transversesection x=const. (see Fig.2) equals:

'W% , ,

. I f z -dzh-B-A^ ,

Hence factor of axial pressure gradient in potential flow can be calculated:

, OH

If resultant velocities V (z,y) used in calculation of mean value include

components induced by operating propeller then the effective wake fraction canbe calculated from equations (11) and (12). In that case the axial component of

velocity induced by propeller (V^ (z,y)) is omitted in equation (11).

4 Propeller-hull interaction - algorithm of computations

The outlined computational model of ducted propeller and described in moredetails model of viscous flow between hull and waterway bottom can becoupled in order to solve a complex task of determination of wake fraction andthrust deduction factor. Comprehensive algorithm of computations is presentedin Fig.4.



CALCULATION OF NOMINAL WAKEIN PROPELLER DISK

CALCULATION OF PROPELLER FLOW(SCREW OR DUCTED PROPELLER)

CALCULATION OF INFLUENCEOF OPERATING PROPELLER ON HULL FLOW

CALCULATION OF EFFECTIVE WAKEIN PROPELLER DISK

CALCULATION OF PROPELLER PERFORMANCE

Figure 4. Flow chart of calculations of propeller-hull interaction.

The course of calculations starts with computation of velocity distribution inpropeller disk (nominal wake in absence of operating propeller). The obtainedvelocity field constitutes the undisturbed flow in the subsequent propellercalculation. After the solution of propeller flow is completed the velocitiesinduced by propeller on hull and waterway bottom are calculated. In turn theviscous flow between hull and waterway bottom is solved again, this time withrespect to disturbances due to operating propeller. Obtained velocitydistribution in propeller disk can serve to compute the first approximation ofwake fraction and as data in the second approximation of propeller flow. Thedescribed steps, starting from propeller flow calculation, are then repeated untilconvergence of mean value of effective wake fraction from two consecutiveloops is gained. Finally the performance of propeller is calculated (thrust,torque, efficiency). Additionally the resultant velocities in flow field can be

calculated according to formula (11) this time, however, with velocity V^ (z,y)

induced by propeller taken into account.The calculations proceed in the same way for screw propeller as well as for

ducted propeller. In the casse of nozzle embedded in a hull it is considered asincomplete annular airfoil. No additional discretization of vortex surface in thevicinity of nozzle-hull junction is then introduced.

Some manipulations were necessary in calculation of velocity induced atcontrol points on a hull by propeller and nozzle wakes consisting of helicalvortex filaments. First of all the deformation of the vortices had to be



introduced when they approached ship skin. Next additional spatial distributionof vorticity was applied in places where vortex line passed very close to controlpoint and excessive induced velocities were expected.

It takes about 2 hours to perform complete calculations on microcomputer(IBM PC486/66) for the case with ducted propeller.

Effective wake fraction can be also determined according to standardprocedure usually applied to model test results, with assumption of thrust ortorque identity. To follow this way open water characteristics of propeller haveto be additionally calculated.

Thrust deduction factor can be, according to the proposal presented in [7],calculated using the following formula:

f = l-— (20)

where:

7^ -thrust of an isolated propeller,

T -thrust of the same propeller operating in astern conditions.

The value of 7^ is read from open water characteristics with assumption that

diameter and revolutions are equal to those of propeller in astern conditions andthat speed of isolated propeller corresponds to ship speed.

5 Results of calculations, concluding remarks

Test calculations have been performed for several typical inland navigationcargo vessels for which model test results were available. There were:- twin screw motor cargo vessel [8],- a train of twin screw pushboat and four EUROPE II -type dumb bargesarranged in two rows [9],

- a train of three-screw pushboat and six EUROPE II -type dumb bargesarranged in two rows [10].

Main particulars of those ships are collected in Table 1. The calculations wereperformed for the same operating conditions as in model tests. The author alsocarried out a series of additional calculations with alternative propellers, atdifferent waterway depth, propeller loading and number and arrangement ofbarges in train. There was also an isolated pushboat considered.

Sample results are collected in Table 2. Calculated distributions of velocitywith and without operating propeller are presented in Figures 5 and 6.

In the case of motor cargo vessel the computed wake fraction and thrustdeduction factor were applied in prediction of top sailing speed at two differentoperating conditions (different waterway depths H=3.5m, H=5m, andconsequently different resistance curves). In Fig.7 the calculated prognosis iscompared to one obtained from model test.

The results confirm the strong dependence of mean wake fraction from



waterway depth, arrangement of barges in train and position of propeller axis.Enlargement of waterway depth, rearrangement of barges so that total length ofa train becomes reduced, or enlargement of distance of propeller axis from shipskin results in decrement of nominal wake fraction. Changes in ship draught atconstant waterway depth do not cause such univocal changes in wake fraction.Anyway trends are similar in both nominal and effective fractions.

Table 1. Main particulars

Lc[m]B [m]T[m]Lwi [m]CB[-\D|m]P/D [-]AE/AO [-]zh[m]Model scale

Motor carvessel6.80.757

0.2 0.2246.604 6.760.874 0.874

0.1200.650.564

0.4 and OJ12.5

;o

0.2566.7680.876

!8

2-screwpushboat2.18750.8750.10932.11880.6220.131251.0520.714

0.312516

3 -screwpushbot2.18750.93440.106252.11880.64260.131251.0520.714

0.312516

Dumb barge

4.78130.0781

0.1754.5650.947

----

0.1874.5870.946

----

0.24.6110.945

----

0.312516

Table 2. Results of calculations and model test

h[m]T[m]T [m] (barge)n [1/s]F[m/s]

w«Test \VT

Wzpt

™nCal- WTcula- H>*tions Wzp

tkT

Motor cargo vessel

0.40.2-

26.221.3360.2760.270.090.2450.1590.1140.1190.0640.1750.145

0.24-

26.11.2610.2540.230.060.270.1380.1060.1050.0440.1330.152

0.256-

26.01.1710.2320.200.010.270.0980.0430.052-0.0150.0600.154

0.280.2-

25.81.10.4380.260.110.2920.3030.2040.2210.1320.1860.190

2-screw pushboat

0.31250.1093

0.17516.070.88750.4380.3180.0720.2000.40850.21100.23560.10730.11070.3711

0.216.1330.8350.4710.3240.0900.2100.46260.24400.25600.11280.10660.3828

3 -screw pushboatmiddleprop.

sideprop.

ductedprop.(side)

0.31250.106250.187515.20.873

0.6250.3900.064

0.6280.6570.5290.2530.2680.440

0.5200.460-0.029

0.6290.5430.3890.1240.2390.424

0.520

-0.413

0.6290.5430.5140.180.3440.434

- nominal wake fraction- effective wake fraction based on thrust identity- effective wake fraction (calculated)

p- wake fraction with operating propeller



P[rad]

Figure 5. Circumferential distribution of nominal wake fraction (twin screwpushboat).

Figure 6. Circumferential distribution of wake fraction in front of operatingpropeller (twin screw pushboat).


Marine Technology and Transportation

X Calculated

400

200

10012 14 16 18 V[km/h]

Figure 7. Predicted sailing speed of motor cargo vessel.

The type of propeller and its loading have strong influence on effective wake.The magnitude of effective wake fraction is higher in the case of ductedpropeller in relation to screw propeller. Increment in propeller loading as well asin pitch-diameter ratio results in flow acceleration and corresponding decrementin effective wake fraction.

The presented method takes into account the influence of both radial and



circumferential variations of fluid velocity on propeller performance. Results ofcalculations reveal that in the case of screw propeller this variation has littleeffect whereas in the case of ducted propeller radial variation of wake fractionmay become significant. Evident rise of total propeller thrust was observedwhen the fraction values were redistributed so that values at blade tip wereincreased at unchanged mean value. This gain was mainly due to higher thrustof a nozzle. That example shows that increment in propeller efficiency can beachieved by appropriate control over velocity distribution in ship wake.

References:

1. Szantyr, J A method to determine pressure distribution on ship propellerblade operating in a nonuniform velocity field using the model that accountsfor unsteady hydrodynamic processes, Ph.D. Thesis translation,DTTVSRDC/Tra -JJ 1982

2. Lewis, R.I. & Ryan, P.G. Surface vorticity theory for axisymmetricpotential flow past annular aerofils and bodies of revolution with applicationto ducted propellers an cowls, JMES, vol. 14, No 4, 1972

3. Glover, E.J. & Ryan, P.G. A ducted propeller design method: A newapproach using surface vorticity distribution techniques and lifting linetheory, Trans. RINA, vol. 114, 1972.

4. Kulczyk, J. Numerical modelling of hydrodynamic interactions in propulsionsystem of inland navigation vessel, Prace Naukowe Instytutu Konstrukcji iEksploatacji Maszyn Politechniki Wroclawskiej, Seria: Monografie, No 17,Wroclaw 1992, (in polish).

5. Kulczyk, J. & Tabaczek, T Computational model for calculation of ductedpropeller performance in entire forward operating condition range, FourthInternational Symposium on Practical Design of Ships and Mobile Units,PRADS, Varna, Bulgaria, 1989, Vol. 3.

6. Kulczyk, J. Analyze und Berechnung der Nachstromziffer fur Binnenschiffe,Schiffstechnik, B.28, Heft 3, 1981

7. Jarzyna, H Szantyr, J. & Koronowicz, T. Screw Propeller Design, ZeszytyNaukowe Instytutu Maszyn Przeplywowych PAN, No 290/1171/89, Gdansk,1989.

8. Luthra, G Untersuchung der Nachstromverteilung an einem 2- Schrauben-Binnengutermotorschiff. Versuchsanstalt fur Binnenschiffbau E.V.,Duisburg, Bericht Nr 788.

9. Luthra, G. Untersuchung der Nachstomverteilung eines im Verbandschiebenden Schubboots in Pontonform mit einer zwecks Verbesserungzum Propeller geanderten Zwierumpf- Unterwasserformgebung, Versuchs-anstalt fur Binnenschiffbau E. V., Duisburg, Bericht Nr 702.

10. Luthra, G. Untersuchung der Nachstromverhaltnisse an Drei- und VierSchrauben-Schubbooten, Versuchsanstalt fur Binnenschiffbau E.V.,Duisburg, Bericht Nr 919.


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