NAVAL SHIP RESEARCH AND OEVELOPMENTENTER
Bethesda, Maryland 20034PROGRESS IN THE ANALYSIS OF THE VISCOUS RESISTANCE OF SURFACE SHIPS
by
Paul S. Granville
Approved for Public Release: Distribution Unlimited
Ship Performance Department
October 1974 SPD 581-01
26 AUG. 1975
Lab. v.
The Naval Ip Riseaich and Development Cmli, Is S 13. !. Navy center for laboratory effort directed at achieving iffipenved sea and air vehicles it aa formed in March 19(17 by
m.rgin the DavId Taylor Model Basin at ('srdemck Maryland nith the MarIne PaiØnee,ine 1.abnrètory it Annipclta. Merylant ,
Neval ship Research and Development Center Bethesda, Md. 20034
* REPORT ORIGINATOR
MAJOR NSRDC ORGANIZATIONAL COMPONENTS
OFFICER-IN-CHARGE CARDE ROCK SYSTEMS DEVELOPMENT DEPARTMENT SHIP PERFORMANCE DEPARTMENT ST Ru CU RE S DEPARTMENT 17 SHIP ACOUSTICS DEPARTMENT 19 MATERIALS DEPARTMENT NSRDC COMMANDER 00 TECHNICAL DIRECTOR01 OFFICER-IN-CHARGE ANNAPOLIS 04 AVIATION AND SURF4CE EFFECTS DEPARTMENT - 16-COMPUTATION AND MATHEMATICS DEPARTMENT 18 PROPULSION AND AUXILIARY SYSTEMS DEP AR H EN I 27 CENTRAL INSTRUMENT ATION DEPARTMENT
SECURITY CLASSIFICATION OF THIS PAGE (When Det. Enfered
DD 1473 EDITION OF I NOV 68 IS OBSOLETE
S/N 0102-014-6601 I
UNCLASSIFIED
SECURITY CLASSIFICATION OFTHIS PAGE (Wiofl Pit. Hnt.r.d)-l
Il%5 UVR% l
AE
BEFORE COMPLETING FORMREAD INSTRUCTIONSI. REPORT NUMBER
SPD-581-01
2. GOVT ACCESSION NO. 3. RECIPIENTS CATALOG NUMBER
4. TITLE (arid Subtitle)
Progress in the Analysis of the Viscous Resistance of Surface Ships
5. TYPE OF REPORT & PERIOD COVERED
Final
6. PERFORMING ORG REPORT NUMBER.
7. AUTHOR(s)
-Paul S. Granville
8. CONTRACTOR GRANT NUMBER(s)
9. PERFORMINGORGANIZATIONNAMEAND ADDRESS -.
Naval Ship Research and Development Center Bethesda, Maryland 20084
tO. PROGRAM ELEMENT. PROJECT. TASK AREA & WORK UNIT NUMBERS
6li53N R 023-01
It. CONTROLLING OFFICE NAME AND ADDRESS
Naval Sea Systems Command Washington, D.C. 20360
12. REPORT DATE
September 1974
13. NUMBEROFPAGES 14. MONITORING AGENCY NAME & ADDRESS(I1 different from Controlling Office)
-IS. SECURITYCLASS(OI this iport)
N one
-IS.. DECL ASSI FICATION/ DOWNGRADING SCHEDULE
16. DISTRIBUTION STATEMENT (of this Report)
-Approved for Public release; Distribution Unlimited
i7 DISTRBT ISTATEMENT (of the ibàtraOt enteiid ln Block 20. If different from Report)
18. SUPPLEMENTARY NOTES
19. KEY WaRPS (Continue on re ide if necessary end Identify by black number)
Viscous resistance; Surface ships
20. ABSTRACT (Continue on reverse aidel necessary and identify by block number) - - - - -- -Progress in the analysis of the viscous resistance of surface ships since
the last ITierican Towing Tank Conference in 1971 is summarized from the avail-able: technical literature. The summary is considered at various theoretical levels when examining the flow field about thehull. Emphasis is placed on-boundary-layer theory. The appendix has a new empirical formula relating form
factor to hull geometry. The form factor was obtained from a boundary-layer calculation of equivalent bodies of revolution and is to be used for the extrapolation of towed modei.resistanceto.fuli_scalecQflditiO.fl,
--LLIiITV CLASSIFICATION OF NIS PAGE(Wh.n Data Ent.,.d
ABSTRACT
Progress in the ana1ysis of the viscous resistance of surface ships since
the last American Towing 'Tank Conference in 1971 is summarized from the available technical literature. The' summary is considered at, various theoretical levels when éxamining'the flow field about the hull. Emphasis is placed on boundary-U.
layer the9ry.' The" appendix ha&' a new empirical formula relating form factor to hull geometry. The form factor was obtained from a .bounary-layer calculation
of equivalent bodies of revolution and 'is 'to be used for the extrapo1ationof'' towed' model resistance tofui1sca1e conditions..
PROGRESS IN THE ANALYSIS OF THE VISCOUS RESISTANCE OF SURFACE SHIPS
(Presented to 17th American Towing TankCoñference, Pasadena, Cal. - June 1974)
Progress in the analysis' of the viscous' resistance of surface ships is reviewed ,since the 1971 meeting of the American Towing Tank Conference)' See
also the summary by Wieghardt2 'to the 1972 International Towing Tank Conference.
Progrés.s in the analysis of the viscous resistance of surface ships may be examined. at different theoretical levels when considering the flow about the hull. 'T'hese levels depend on the nature of the analytical, models describing the flow about a'sh.i'p which is advancing along a free surface. The flaw may. be,
laminar and/or tubulent and wavy at the free surface.. Resistance to the flow
arises from the pressure and shearing stres distributiofls over the hul1 surface.
Level 1 Navier-Stokes Equations and the Free Surfce Condition .
,
Level 1 may be considered the most' theoretical level and p'rbbab1. the least developed. Here the complete flow field 'is to be solved for a ship producing waves on a free surface. The resulting total résistance including' wavemaking' and
viscous components is the summation of the pressure and shearing stress over the
hull surface:in the axial direction. What is requiredis three-dimensiona'1
solution ofthe Navier-Stokes equations for both laminar and turbulent floW
with a free-surface boundary conditionof constant pressure.
The Nayier-Stokes equations represent Newton's second law of motion and
Newton's linear law of viscosity man Eulerian continuum system.. Mathematically
the Navier-Stokes equati.ons are non-linear partial differential equations .of
1Granvil.le, P.S., "Progress in the Analysis of the Viscous Reispnce of Surface
Vessels," Naval Ship Research and Development Center, Ship Performance Tech
Note 204 (Aug 1971).. . ' ' '
.
2Wieghardt, K., "Viscous Resistance," 13 th International Towing Tank Conference, Report of Resistance Committee,. Appendix 4, pp. 4-14 (Sept 1972).
the elliptic kind. The free-surface conditiOn satisfying Bernoulli's equation
is also non-linear,
There are currently determined efforts to develop numérlcal niethOds. to
solve, the Navier-Stokes 'equationfor laminar flow, Emphasis ison simple
geometries and unsteady separated flow3'4 with results so far limited to low
Reynolds numbers. Ship geometries and the free-surfac cond1toñ are nowhere.
in sight. .
There seemsto beno perceptible progress onhoW tosolve the
Navier-Stokes equations for turbulent flow.. Current theories pfturbuience assume
that turbulent motion does obey the Navier-Stokes 'equations .nstantaneously.
However the irregular fiuctuationpropertieS of turbulence. such as. the distribution of amplitude, frequency and phase seem torqu1re hypotheses
additional to those implied In the Navier-Stokes équatibns. Level II - Reynolds Equations and. Free Surface Condltin
For turbulent flow the Navi'er-Stokes eauat.ons may be.averaged over a small time interval to yield the Navier-Stokes equations for:the mean velocity and
mean pressure, now.tern,ed the Reynolds equations. However.the'non-liflear.nature of the Navier-Stokes equations results in"additionai apparent stresses called
Reynolds or'turbül.ent stresses. The. solution of the Reynolds eauatinsreauires
then additional analytical 'relationsfor the turbulent.stresses.. Since there
is no general theory for the turbulent stresses, recourse: has to be made' to approximate methods.. Because turbuient.shearing stress Is the most. important,
it has received the most attention.. Historically Boussinesq by analogy, to..
laminar viscosity introduced the eddy viscosity, the value of which unfortunately
varies with the flow field unlike lan.inar yiscoslt,y. Prandtl by analogy to,
the mean free path of gases introduced the mi:xlng length which also, varies with the flow field. More recently differential, equations or rate equations
have been introduced for the turbulent shear stress involving various empirical
coefficients. Launder and Spaldl.ng5 present a recent summary some of the
methods. . . , .
3Roache, P.J., "Computatna.1 Fluid Dynamics," Hermosa Publishers, Albuquerque, New Mexico (1972).
4Marshal1, G. and E. van Spiegel, "On the Numerical Treatment-of, the Na.vier-Stokes, Equations for an Incompressible Fluid," Journal of Engineering Mathematics, Vol. 7, No. 2, pp. 173-88 (April 1973).
5Launder, B.E. and D.B. Spalding, "Lectures in Mathematical Models .of Turbulence,"
3
Application to ship flows at this theoretical level seems to be practically nonexistent.
Level 1.11 - General Boundary Layer
A great simplification in analyzing the flow field about a ship results from applying the boundary-layer concept. Physically it is recoqnized that there is a region of flow next to the hull where the viscous effects of the fluid
predominate. In fact the turbulent flow next to the hull is cuite visible to an observer. The exaggeratiQn of..the,. viscous effects next to the hull is due nainly tO the no-slip condition on the hull.
A boundary layer or the flow region next to the hull is defined, inside of
which the flow is viscous and outside of which the flow is inviscid. The viscous flow in the boundary layer originates at a stagnation point on the bow and
increases progressively in thickness downstream. In additionthe viscous flow.. starts out laminar and, usually undergoes transition to turbulent flow farther:.
downstream. The inviscid flowoutside the boundary layer provides the pressure
field which controls the growth of th.e boundary layer and phenomena such as
separation.
Analyticallythe inviscid flow outside the boundary layer is treated as a
potential flow which is naturally a great simplication. Inside the boundary layer the Navier-Stokes eauations may be applied to the laminar flow and the Reynolds eauations to the turbulent flow. A great simplication is to use the -pressure.fleld outside the boundary layer as a boundary condition to the viscous flow inside.
The general mathematical problems of the flow about a ship more or less at this theoretical level is treated by Landweber.6 The exact boUndary,
conditions on the hull surface and on the free surface are included in general mathematical statements.
The. boundary-layer concept also provides the rationale for separatjnq the resistance of a ship into wavemaking resistance and viscous resistance. The wavemaking resistance arises mostly from the potential field outside the boundary layer which for ships is strongly affected by. the waves produced on the free surface. The viscous resistance conies rnostly from the boundary-layer flow which ends up as the viscous wake far downstream.
6Landweber, L. "ContribUtiOns on Some Current Problems of Ship Reistance" in "International Jubilee Meeting, 40th Anniversary of Netherlands Ship Model Basin 1972,' Netherlands Shi'p 'Model Basin, Wageningen 1973.
Since the boundary-layer flow 1 ariven by the inviscid pressure field outside the boundary layer, the anaytcal solution o the boundary layer
flow requires as an input the analytical solution of the inviscid wavemaking flow about the
ship0
In other words tne wavemaking resistance has to be knowh analytically before the boundary-layer can be calculated analytically0 Progress in the calculation of viscous resistance at this theoretical level is dependent then on progress in the calculatiOn of wavemaking resistance Level IV - Boundary Layer Equations and Free Surface ConaitionAt this level the boundary layer concept still remains However, by an order of magnitude analysis for th comparatively thin boundary layer, the Navier-Stokes equations and the Reynolds eauations become simplitied by discarding tenns considered negligible0 The results are the boundary layer equations of motion which ae now parabolic partial differential equations
instead of the more difficult elliptic partial differential eauations
The ship boundary.layers are three-dimensional in geometry with, however, one great simplication: symmetry about the center fore-and-aft plane0 In general, there has been continual development In the solution
of,three-dimensional boundary-layer equations and this will be reviewed separately, The presence of the free surface and its waves provides a great difficulty in obtaining the pressure field and even th upper extent of the boundary
layer at the free surface0 In fact, knowing the pressure. field means knowing the wavemaking resistance a subject whose analysis is still being aeveloped.
The choice of. coordinate systems for ship boundary layers is considered by MIloh and Patei70 The usual use of streamhne coordinates means a
different coordinate system for each different Froude number
with
itsdifferent wave pattern0 Mlloh and Patel advocate a fixed orthogonal coordinate system with axial stations as one of the coordinates; the whole system then being independent of Froude number- in general, coordinate systems for three-dimensional boundary layers are based on the given body surface as a reference
surface it should be. noted that streamline coordinates are. physically meaning-fül and are also required by integral methods for solving boundary layer
equations0
7Miloh, T0. and V0C- Patel, "Orthogonal Coordnate Systems for Three-Dimensional
Boundary Layers, with Particular Reference to Ship Forms," Journal of Ship Research,
Vol0
17, No 1, pp. 50-58 (Mar 1973).As an input to boundary layercalculations, Adee8 solves the potential
flow around a ship hull using a linearized free-stream boundary condition. Sources are distributed on the hull surface and the results are given in terms
of streamlines. Adee9usesan integral method, the entrainment method of
Cumptsy and Head, to perform boundary layercalculations.on a ship hull.
Since at this theoretical level, the wavemaking resistance is obtained. from the calculated pressure field and the viscous resistanceisobtained from
the boundary-layer calculation, the analytic result is a calculated total resistance.
Level V. - Froüde MethOd and Boundary Layer on Double Model
The levels..to follow involve model tests at full-scale Froude numbers to take care.of wavemaking resistance. The viscous resistance is then to be
calculated by some other means.
The inability to calculate the resistanceof ship led long ago to the use
of reduced-scale models. In the universally used Froude method, the model is towed at the full-scale Froude number to deduce the wavernaking resistance, leaving to calculation the determination of the change In viscous resistance
due to thechange inRe,'nplds number. from model sc1è to. full scale. A recent analysis by Granville1° argues that even in considering variousinteractions
between wavemaking and viscous resistances, the changein viscous resistancedue
to Reynolds number can be accomodated by the change in viscous resistance of
th&ship at zero Froude number. ThecaseOf zero Froude number Is the same
as that of adeeply-submerged double model. The zero .Froude number case is
relatively simple since itinvolves the determination of the potential pressure
distribution without the great complications ofthe free surface. There are;
numerical methods available as programs for potential flow calculations involving
source distributions'.
There is also the simpler butlessaccuratepotential-flow slender body method of TUck and von. Kerczek..38Adee, B.H., Calculation of the StreamlinesAbout a Ship Assuming a Linearized gFree-Surface Boundary. Condition,"JoUr.of Ship Res., Vol. 17, No.3, pp.140-6
Adee, B.H., "Boundary Layers on Ships," Ph.D. Thesis, Univ. ofCalifornia
10(Berkey) 1972.
Granville, P S , "A Modified Froude Method for Determining Full-Scale Resistance
of Surface Ships from Towed Models," Naval Ship R&D Center Report 4201 (July 73), Vito appear in Journal of Ship Research.
Hess, J L and A M 0 Smith, "Calculation of Potential Flow about Arbitrary Bodies," in Progress in Aeronautical Sciences, Vol. 8, Pergamon Press, New York.
,d,(l966)..
"Dawson, C.W. and J.S. Dean, 'The XYZ Potential Flow Program," Naval Ship Research 1and Development Center .Report 3892 (June 1972).
Tuck, E.O.and C. vonKerczek, "StreamlInes and Pressure Distribution on Arbitrary Ship Hulls at Zero Fróude No.", Journal of Ship Research, Vol. 12, No.. 3 (Sept 68).
The boundary-layer calculation on the double model is that of a three.-dimensional method adapted tohull geometries0 Huang.andvon-..Kerczek14 and
von Kerczek15 use the entrainment method of Cumptsy and Head for. the boundary-layer, calculations and the slender body method of Tuck andvon Kerczek for the.
potentlal-flowcalculations. A similar calculation has besn performed by
Hatano et'al'6' and by Hotta17.
There St:ill rémainS a. number of important prbblems-in'the calculation of
the viscOus resistance of double models from the boundary-layer development.
For example the wall shearing stress distribution is calculated over the hull. What is needed is a method for calculating the viscous resistance from the wake
far downstream. This also includes the contribution from the viscous pressure
resistance. There is 'alsd the problem of calculating the separation resistance for ships with full sterns, and proper modeling Of cross-flow in the boundary
layer with making a small crossflow approximation. Level VI - Equivalent Body of Revolution
For the deterrninatic'n of viscous resistance, the double model maybe
approimated be an equivalent body of revolution. The cakuiatfon.'df'the''iscous
résistance of.bodies of revolution from the boundary-layer development is. relatively simple. The fit of a body of revolution to a ship hull is fairly 'clóse"wlth the greatest discrepancy at the bow and' .stern.
'I:n order to accommodate the boundary layer development as closely as possible, G'ranviile1° defines a dual equivalent body of revolution. For the. pressure'distthutiOn, an equivalent bodyof revolution is givenby the
cross-sectional -area -of the hull. For the surface area, an equivalent, body of
revolution is given by the perimeter of the hull. . Both are used in.the
boundary-layer cal'cülatiqn; the first indirectly through thepressure d.lstrlbt,ion.
14Huang, T T
andC H
van Kerczek, "Shear Stress and Pressure Distribution on a Surface Ship Model:. Theory and Experiment" presented to 9th SymposiumNaval Hydrodynamics, Paris, August1972.
von Kerczek, C , "Calculation of the Turbulent Boundary Layer on a Ship Hull
atZero Froude Number," Journal Of Ship Research,Vol. 17, No.2., pp. 106-120
l6Hatano, S., M. Nakato, T.. Hotta,1973). and.S. Matsui, "Calculation of Ship Frictional Resistance by Three-Dimensional Boundary Layer Theory," Journal of Society 17of Naval Architects of Japan, Vol. 130,, pp. 1-10 .(Dec 1971).
Hotta, T., "A. Method for Calculating Three-Dimensional Turbulent Boundary Layers by Using Streamline Co-ordinates," 'Japanese Journal, of Applied Physics,
Vol. 12, No. 6, pp. 908-15' (June 1973.). .
In general, there have been improvements in the calculation of boundary
layers on bodies of revolution. Cebeci et al18 use a differential method with an eddy viscosity model for the shearing stress Nakayama and Patel19
use the integral entrainment method of Head, together with a consideration
of thethi.k'b'oiindáry läyeron the tail;
Level VII - Form Factor fromEquivalént BodyOf Revo1utôn
Granviile19 obtains a form factor fromthe boundary layer calculation of
the equivalent body of revolution with the dual definition. The form factor
is used in Landweber's hypothesis relating the viscous resistance of the ship
to thatof an, equivalent flat plate with allowance for aconstant separation
resistance.coefficient. Granville calculates this fonn.factor which is independent..of Reynolds number by using a power-law similarity law for the ,turbulent boundary layer. Sincethe Froude method only requires the change
in viscous resistance from model tO full-scale, it Is notnecessary to evaluate
the constant separation coefficient which drops. out. In the Appendix these calculated form factors .are related to overall hull geometrical factors.
Level VIII FormFactor for Equivalent Flat Plate
In the Hughes method, the viscous resistance.at zero Froudenumber is
proportionalto the equivalent flat plate.resistance, the constantof
proportionality being the form factor The form factor is obtained empirically
from the total resistance measurements at a close-to-zero Froude number Owing to difficulties instimulating transitiont low Reynolds numberandthe p,or'
precision of dynamometers at low towing speeds, the determination of the Hughes'
factOr. is quite uncertain. Investigators have correlated the Hughes form factor ii'th Simple geometrical coefficients of the hull.
Gross.and Watanbe20glvea
summary of recentefforts in this regard.
Level IX -. Equivalent Flat Plate . . .
This is the traditional Froude niethodwhere the change in.viscous resistance
from model to full-scale is made equal to the change in flat plate resistance
The equivalent flat plate.is one with the ship length and an area equal to
that.ofthe wetted hull surface.
'8Cebeci, T , G J Mosinkis and A M 0 Smith, "Calculation of Viscous Drag in Incompressible Flows," Journal of Aircraft, Vol. 9, No. 10, .p. 691-2
19(Oct 1972). . .
Nakayama, A. & V.C. Patel, "Calculation of the Viscous Resistanceof Bodies Of Revolution," University of Iowa, Institute of Hydtaulic Research, Report 151
20(Oct 73). .
Gross, A. and K. Watanabe, "FormFactor," 13th International Towing Tank Conference, Report.of Performance Committee, Appendix. 4,.pp. 4-14(Sept 72).
There has been almost o recent work on flat plate resistance pe Se.
Wieghardt21 shows the difference in measured flat-plate resistance
coefficients over the yars for selected Reynolds numbers0
Three-Dimensional Turbulent Boundary Layers
The general characteristics of three-dimensional turbulent 'boundary layers over curved bodies are discussed by Eichelbrenner22 and also by Nash and Patel23. Orthogonal curvilinear coordinates are used to describe the boundary-layer flow
field0 Turbulent boundary-layer equations are the reduced Reynolds equations of
motion where in terms considered negligible have been dropped0 An elliptic system
of partial differential equations has been reduced to a simpler parabolic system.
However, the principal difficulty in solving the Reynolds equations remains,
namely, a lack of relations for the Reynolds stresses0 These have to be separately supplied by other considerations, mostly empirical.
In general, there have been two approaches to solving the boundary-layer
equations: the differential approach to solve the equations directly as partial differential equations and the integral approach where the equations are
integrated forijally over the boundary layer and then solved as ordinary differential equations.
Both differential and integral methods are used in two-dimensional flows
and have been adapted to three-dimensional conditions0 In the differential methods the Reynolds shearing stress has to be known to obtain any solution.
Cebeci and his associates24'25'26 use an eddy viscosity model which is considered
21Wieghardt, K0, "Zur Grenzschicht am Schiff," in "Beitrage zur Stromungsmechanik, Insbesondere zur Grenzs chi chttheori e," Deutsche Luftund Raumfahrt Forschungs -22bericht 72-27, pp. 391-7 (1972),
Eichelbrenner, E.A., "Three-Dimensional Boundary Layers" in "Annual Review of Fluid Mechanics,
Vol0
5," M. van Dyke, W0G0 Vincenti, and J.V. Wehausen, eds.,2 Annual Reviews
Inc0,
Palo Alto, California (l973)
3Nash, J.F. and V.E, Patel, "Three-Dimensional Turbulent Boundary Layers," SBC 2ATechnical Books, Atlanta, Ga0 (l972)
Ceheci, T.,G0J. Môsinskis, and K0 Kaups, "A General Method for 'Calculating
Lncompressible 'Lamtnar 'and TurtYulent Boundary Layers 1. Swept. Ln.f.inite Cylinders and Small Cross Flow," Douglas Aircraft Co. Report 2MDC J5694 (Nov l972)
Cebeci, T., K. Kaups, G.J. Mosinskis and J0A0 Rehn, "Some Problems of the
Calculation of Three-Dimensional Boundary-Layer Flows on General Configuration," 26NASA CR-2285 (July 1973).
Cebeci, T., "A General Method for Calculating Three-Dimensional Laminar and Turbulent Boundary Layers0
IL
Three-Dimensional Flows in CartesianCoordinates," Douglas Aircraft Co0 Report MDC J6517 (Mar 1974).
as a scalar quantity for three-dimensional flows. Bradshaw27 developed a rate equation for shearing stress from the turbulence energy equation which was
later extended to the cross-flow shearing stress defined for Cartesian
coordinates. Launder and Spalding5 sumarize the various analytical models
for the Reynolds stress. In general, the differential methods require extensive
numerical analysis for programming.28
Wheeler and Johnston29 compare various three-dimensi onal differential
methods for given measured flows,
Integral methods require streamline coordinates in order that a
two-dimensional method be used in the streamline direction and an additional
velocity model for the cross-flow direction, Granville30 presents a group of two-dimensional integral methods based on a two-parameter velocity law and quasi-equilibrium, conditions.
Kux31 defines various three-dimensional integral parameters.
Drag Reduction
The reduction of hydrodynamic skin friction by an air film next to the
hull surface is based on the principle that the wall shearing stress depends
on the density, other factors being equal. Unfortunately, the lesser density
of air than that of water also leads to buoyancy instabilities in air films. .The:s.urfacetension between air and water also promotes the formation of
bubbles if the air is present in small amounts. Therefore, air lubrication
can only be applied to flat bottoms where the air cannot rise and the air must
be present in a substantial amount to prevent air bubbles.
.Such air lubrication has been applied to barge-like ships where the air
ts:..main±ai'ned in "large bottom cornpaTtnlents.32'33 A 20-percent reduction in res.istance has been achieved on such a model,
27Bradshaw., P., "Calculation of Three-Dimensional Turbulent Boundary Layers," 28Journal of Fluid Mechanics, Vol. 46, Pt. 3, pp. 417-45 (April 1971).
Klinksiek, W.F. and F.J. Pierce, "A Finite Difference Solution of the Two, and Three-Dimensional Incompressible Turbulent Boundary Layer Equations," Journal of Fluids Engineering, Trans. ASME, Vol. 95, Series 1, No. 3,
445-8 (Sept 1973).
Wheeler.,, A.J.'...'and'J.P.'Johnsto,' "An Assessment' of Three-Dimensiotal Turbulent 'Boundary. ..L'ayerP:redi'cttQn Methods," Journal of Fluids Engineering, Trans.
3ASME, Vol. 95, Series 1, No. 3, pp. 415-21 (Sept 1973).
vGranville, P.S., 'Integral Methods for Turbu'lent Boundary Layers in Pressure 31Gradients,"'.Journal of Ship Research, Vol. 16, No. 3, pp. 191-204 (Sept 1972).
Kux, J..,. "integra]e 'Grenzschichtparameter Definitlonén, Deutungen,
3,Diffentialgleichungen," Schiffstechnik, Vol. 20, No.102, pp 74-6 (Nov 1973). 'Anon, "Applying the Air-Cushion Principle to Very Large Vessels," Naval 33Architect, No. 2, p. 38 (April 1972).
Courouble, M. "Recherche sur une Technique de Rduction de la Resistance a la Marche ds Navires Lents par Lames d'Air,"Bulietin de 1"Association
Technique' Maritime et Aronautlque No. 71, pp. 461-79 (1971).
The hydrodynarnic aspects of the reduction in turbulent skin friction by
polymer additives are well understood in terms of the boundary layer or pipe
flow similarity laws. A so-called negative roughness analogy seems to be
operative. The polymer additive also leads to a maximum drag reduction for
a thin boundary layer controlled by an interactive layer similarity law.
Granville34 reviews the current status of these hydrodynamic aspects and presents logarithmic resistance laws for flat plates. and for rotating disks35
including. the case of maximum drag reduction.36
Experimental Investigations
Experimental investigations on the viscous resistanceof ships may be
roughly divided into two types. The first type concerns the overall aspects
determined directly by a wake survey or indirectly by a wave-cut. The second
type involves the boundary layer in local measurements of the pressure, shearing stress and velocity profile. Of course local measurements can be
integrated over the hull surface to give the overall resistance. Most of the work has been done at model scale which still leaves the problem of scale effect.
Paffett37 gives a discussion of.the resolution of total resistance into
various components while Granville1° presents a component analysis in terms of dependence on Reynolds number and Froude number including their interaction.
Tsai and Landweber38 show from wake surveys the sinuous .variation of
viscous resistance with Froude number. Conn39 calculates the variation of the
frictional resistance from the wall shearing stress with Froude number by
considering the hydrostatic effect of the wave profile in producing undulations in the controlling velocity fieldoutsidé of the boundary layer.
34Granville, P.S., "Hydrodynamic Aspects of Drag Reduction with Additivies," 3MarThe Tecimology, Vol.
10, No.
pp...284-g2
(.Ju1y 1973).raviiie, P.S..., :"The Resisting Torque. and Turbulent Boundary Layer of
Rotating Disks with Smooth and with Rough Surfaces in Ordinary Fluids and in Drag-Reducing Polymer Solutions," Journal of Ship Research, Vol. 17,
4, pp. 181-195 (Dec 1973).
Granville, P.S., "Maximum Drag Reduction for a Flat Plate in Polymer 37Solution," Journal of Hydronautics, Vol. 6, No. 1, pp. 58-9 (Jan 1972).
Paffett, J.A.H., "The Components of Resistance," 13th International Towing Tank Conference, Report of Resistance Committee, Appendix 2, pp. 4-14 30(Sept 1972).
'Tsai, C.E. and L.LalTdweber, 11Tatal and Viscous Resistance of Four Series-6O,Mode1s-," 13th .rnternationa1 Towing Tank Conference, Materials for
39Reports, 4-14 (Sept 1972) p. 29.
Conn, J., "Note on the Undulations in Cv/Speed Curves," 13th International Towing Tank Conference, Materials for Reports, 4-14, (Sept 1972) p. 19.
For a Victory Ship model, Townsin4° finds that the sum of the viscous
resistance from a wake survey and the wavemaking resistance from a wave
pattern survey agree with the measured total resistance except at excessive
speeds. The undulation in viscous resistance with Froude number was noted. Townsin41 also investigates the added resistance due to wave-breaking at the
bow.,....,At.M.gh,speedstwO:setS.Ofbreakiflg waves were observed, Surveys of
the..side 1obes of. the"wake :due to, wave-breaking .Th.dicatad added resistance
which improved the agreement with the measured' total resistance.
Huang and von Kerczek14 measure the wall shearing stress and pressure
dis.tr4'butionon a towed Series 60 model0 Hot films and Preston tubes are used. tmeasure:both Tth.e magnitude and direction of 'the wall shearing stress. Gadd42 surveys the boundary layer of a model of an ore carrier for comparison
with previous full-scale measurements0 Matheson and Joubert43 measured the boundary layer of a double model of tanker in a wind tunnel, Full-scale
measurements of the boundary layer of a fishing training ship have been
obtained by a Japanese team.44
Voliheim and Nestler45 measured the effect of a bulbous bow on the viscous
resistance of a model of a bulk carrier.
Pope and Bellhouse46 describe the development of hot films to measure the
wall shearing stress of towed models.
40Townsin, R.L., "The Viscous Drag of a 'Victory' Model; Results from Wake and Wake Pattern Measurements," Royal Institution of Naval Architects, Supplimentary 41Papers., Vol. 113, p. 307-21 (T97l).
Townsin..R.L., "Ship Reststance Components Revealed by Wake Momentum and Wave '.PatternMeasurements in the Presence of Breaking Ship Waves," Transactions,
North East Coast Institution of Engineers and Shipbuilders, Vol. 8., p. 23,
42ec 1972).
Gadd, G.E., "A Comparison of some Model and Full-Scale Hull Boundary Layer Measurements," Transactions North East Coast Institution of Engineers and
4Shipbuilders, Vol. 90, No. 2, pp.. 51-8 (Dec 1973).
"Matheson, N., and P.N. Joubert, "Experimental Detenninationof the Components of Resistance of a Small O.8OCB Tanker Model,tm Journal of Ship Research, Vol. 44l7 No. 3, pp. 162-80, (Sept - 1973).
Irnamoto1, H. et..al., "Measurements of Boundary Layers of Ships," Reports of Research Institute for Applied Mechanics, Kyushu University, Vol. 19, No. 63, 45pp. 125-186 (Oct 1971).
Vollheim, R. and W. Nestler, "Messung der Wellenentwicklung und der Reibungsverluste bei dem Modell eines MassengutfracterS, CB = 0.80, mit Bugwulstvarianten," Schiffbauforschung, Part 1, Vol. 10, No. 1/2, pp. 70-89 A(l97l); Part II, Vol., 10, No. 5/6, pp. 187-95 (1971).
9°Pope, R.J. and B.J. Bellhouse, "Measurement of Local Skin Friction on Ship Models Using Thin-Film Probes," Transactons, Royal Institution of Naval Architects, Supplementary Papers, Vol. 113, pp. 401-9 (1971).
APPENDIX - Form Factor for Equivalent Body of Revolution
The extrapolation of the resistance of towed models to full-scale
conditions at Level VII based on an equivalent body of revolution'0 may
be greatly facilitated if the calculated form factor is correlated with
simple hull geometrical parameters. For the cases treated by Granville10
a close correlation is given by
where
k1 = form factor frcm equivalent body of revolution
CB = Block Coefficient
B = Beam of ship
H = Draft of ship
L = Length of ship
The correlation should be considered tentative at this stage of
development and used mainly as a check on the boundary-layer calculations.
k.1 =
67.7
12
