calculation of the propulsive e ciency for airships …lars.mec.ua.pt/public/lar...

Calculation of the Propulsive E!ciency forAirships with Stern Thruster !

Thorsten Lutz, Peter Funk, Andreas Jakobi and Siegfried Wagner

Institute for Aerodynamics and Gas Dynamics

University of Stuttgart, Pfa!enwaldring 21

D-70550 Stuttgart, Germany

Tel.: +49 711 685 3409 Fax: +49 711 685 3438

e-mail: {lutz, funk, jakobi, wagner}@iag.uni-stuttgart.de

Abstract A method for the aerodynamic analysis of bodies of revolution with a stern propellerwas developed. The viscous flow about the displacement body is determined by a panel method incombination with an integral boundary-layer procedure. Special emphasis is laid on the iterativeviscous/inviscid coupling. For this purpose the transpiration technique is applied along with anunder-relaxation scheme. The relaxation factor is determined from a local stability analysis of thecoupled governing equations. A vortex sheet model is used to calculate the time-averaged influenceof the propeller with a fully-relaxed slipstream. Both modules are coupled taking viscous interactione!ects into account. The complete tool was applied to analyse airship hulls with stern-mountedthrusters and finally to predict the increased e"ciency of wake propellers.

Nomenclature

a exponent of the power-law profilesA amplitudec chord lengthcd drag coe!cientcD dissipation coe!cientcf skin-friction coe!cientcp pressure coe!cientcP power coe!cient of the propellercT thrust coe!cient of the propellerd distance between body stern and

propeller disc areaD propeller diameter, dragDPropeller propeller-induced hull dragH12 = !1

!2boundary-layer shape factor

H32 = !3!2

boundary-layer shape factorL body length

!Presented at the 14th AIAA Lighter-Than-Air Sys-tems Technical Committee Convention and Exhibition,July 15-19, 2001, Akron, Ohio, USA

n wall normal distance,amplification factor of the TS-waves,propeller rotational speed [rps]

P powerr distanceReL Reynolds number (reference length L)R body radius, propeller radiuss arc lengthS surface areaT propeller thrustTu turbulence factoru local velocity inside the boundary layeru, ! error functionsUe velocity at the boundary-layer edgeU! axial onset flow velocity"v velocity vectorvn transpiration velocityx, y, z cartesian coordinatesx, r, # cylindrical coordinates

1

2 Lutz, Funk, Jakobi & Wagner

$ angle of attack% propeller twist angle at r

R = 0.75& boundary-layer thickness&1 displacement thickness

(axisymmetric definition)&2 momentum thickness

(axisymmetric definition)&3 energy thickness

(axisymmetric definition)&1, &2, &3 integral boundary-layer parameters

(plane definition)' propeller e!ciency'e propulsive e!ciency" velocity potential( propeller advance ratio,

wave lengthµ relaxation factor, doublet strength) wave number* density! source singularity strength# dimensionless transpiration velocity

Subscripts

crit. critical valuee boundary-layer edgemax maximum valueopt optimum valuesep separation locationtra transition location" undisturbed freestream condition

Superscripts

(n) iteration step# converged value

1 Introduction

Fundamental Aerodynamic E"ects ofStern-Mounted Propellers

It is well known that an adequately designed sternpropeller is more e!cient than a propeller withconventional installation. The reason is that thestern-thruster is operating in the viscous wake ofthe airship hull, i.e. in a flow domain with reducedaxial velocity. As discussed in Refs. [7], [10] thislowers the power which is required to obtain a cer-tain thrust. As a consequence, the resulting pro-peller e!ciency ' (based on U!!) is significantlyincreased and may be well above 100%.

On the other hand, the suction e$ect of the pro-peller slipstream accelerates the flow in the tail re-gion of the airship hull which yields an increaseof the pressure drag. For obvious reasons, thepropeller-induced hull drag as well as the e!ciency' of the stern-thruster is higher for hull shapes witha blunt tail.

Detailed investigations on aerodynamic e$ects ofairships with stern-propulsion are hardly reported.An exception represent the wind-tunnel tests per-formed by McLemore [10]. Because of its impor-tance, the main results of these studies shall besummarized. McLemore examined a 1 : 20 scalemodel (L ! 6m) of a complete airship configu-ration with two di$erent stern-mounted propellerswhich were designed specifically for the flow con-ditions occuring in the hull wake.

First of all, McLemore measured the thrust co-e!cient cT , the power coe!cient cP and the ef-ficiency ' in dependency of the advance ratio (.According to the usual definitions these propellercharacteristics are given as:

cT =T

"2 U2

! +R2(1)

cP =P

"2 U3

! +R2(2)

' =T U!

P=

cT

cP(3)

( =U!n · D (4)

In his experiments, the larger propeller turnedout to be very e!cient and shows a maximume!ciency of ' $ 140%. McLemore pointed outthat values of ' above 100% only result becauseaccording to Eq. (3) the e!ciency is defined bythe thrust-to-power ratio multiplied by the undis-turbed freestream velocity U!. Actually, the pro-peller operates in the viscous hull wake where thelocal velocities are well below U!. If the e!ciencywould be defined based on the local onset flow ve-locity, the resulting value of ' would, of course, belower than 100% also for a stern thruster.

In a second test the characteristics were mea-sured when the propeller produced enough thrustto propel the airship, i.e. for the condition that thethrust is equal to the total airship drag includingpropeller-induced drag. The maximum e!ciencydecreased to ' $ 122% which indicates that the de-sign point of the propeller does not exactly matchthe steady-state flight conditions.

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Calculation of the Propulsive E!ciency for Airships with Stern Thruster 3

In a last evaluation the propulsive e!ciency 'e

was determined which is defined as:

'e =(T % DPropeller) · U!

P(5)

This definition accounts for the fact that due tothe propeller-hull interference the thrust requiredto propel the airship is increased by the propeller-induced hull drag DPropeller. This is a more real-istic measure to rate the total e$ectiveness of thepropulsive system1. For steady-state flight condi-tions the measured propulsive e!ciency was about103% compared to a value of only 59% resultingfor a fin-mounted propeller.

Previous Investigations and Overview

In the frame of previous investigations [7] avortex-sheet model to determine the time-averagedvelocity-field induced by a propeller with fully-relaxed slipstream was developed. This propellermodule was coupled to a panel/boundary-layermethod for the analysis of the viscous hull flow.The coupled method enables the correct predictionof the hull drag including propeller e$ects. How-ever, with the implementation [7], the propeller ef-ficiency was underpredicted since only the inviscidonset-flow velocity-field was considered in the eval-uation of the propeller characteristics.

In order to allow a correct prediction of the re-quired power and the propeller e!ciency the vis-cous velocity profile in the propeller disc area hasto be considered. To obtain a realistic boundary-layer profile at the stern of the airship, first of all,a reliable viscous/inviscid coupling of the integralboundary-layer procedure and the panel method isneeded. Therefore, the first extention of the modeldealt with the implementation of an improved iter-ative coupling scheme using the transpiration tech-nique. To stabilize the iteration process an under-relaxation technique is necessary. The approachchosen to determine adequate relaxation factors isbased on a linearized local stability analysis of thecoupled governing equations.

In the subsequent chapter the fundamentals ofthe panel method and the propeller module aresummarized, followed by a more detailed discus-sion of the boundary-layer procedure and espe-cially of the coupling scheme developed. Sec. 4 isdevoted to the application of the enhanced method.1Another appropriate approach would be to just consider

the power (coe!cient) needed to propel the airship at acertain speed, regardless of the thrust required.

First, validation examples with respect to the newviscous/inviscid coupling will be given. Thereafter,the method is applied to calculate the characteris-tics of a propeller operating in the viscous wake ofan airship hull. Comparisons to the experimentsof McLemore will be presented.

2 Calculation of the InviscidOuter-Flow Field

In general, the viscous flowfield about a config-uration can be calculated by numerically solv-ing the (Reynolds-averaged) Navier-Stokes-equations, which is rather time consuming withrespect to mesh-generation and computational ef-fort. This is especially true for a configurationwith rotating propeller because this requires anunsteady analysis. If the Reynolds number ishigh, viscous e$ects are mainly restricted to a thinboundary layer along the body surface, whereasviscosity can be neglected in the remaining exter-nal flowfield. As long as no massive separationoccurs, both flow domains can be calculated sep-arately by solving simplified governing equationsrespectively. Hereby, the inviscid outer-flow solu-tion yields the boundary conditions to perform thesubsequent boundary-layer analysis.

In this chapter the method to determine the in-viscid flow about a displacment body with sternpropeller is summarized whereas in Sec. 3 theboundary-layer procedure and the viscous/inviscidcoupling scheme is discussed in more detail.

2.1 Panel Method

The inviscid flow about the airship hull is calcu-lated using a 3D low-order panel method. Withthis approach an irrotational and isentropic flow-field is considered which allows to introduce a ve-locity potential ". This potential consists of theknown potential ,! of the undisturbed onset flowand the disturbance potential , caused by the sub-merged body.

The continuity equation represents the govern-ing equation of the problem. For incompressiblefluid this equation reads in terms of the distur-bance potential as follows:

%, =-2,

-x2+

-2,

-y2+

-2,

-z2= 0 (6)

Applying Green’s third identity enables to trans-form this Laplace-equation into the following in-

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tegral equation:

, (x, y, z) = % 14+

!!

S

!

rdS

+14+

!!

S

µ"n ·&"

1r

#dS

(7)

Hereby, ! represents a source and µ a doubletsingularity distribution on the solid body surface.For non-lifting cases, as considered in the presentinvestigations, the use of a source distribution issu!cient. To solve the above integral equation,the external Neumann boundary-condition is ap-plied. This condition implies that the velocity nor-mal to the surface has to vanish. Since the velocityis given by the derivative of the potential, this con-dition can be written as:

$"v · "n

%&&surface

='(&") · "n

(&&&surface

= 0 (8)

To enable a numerical solution of Eq. (7) thebody surface is discretized into flat quadrilateralpanels with piecewise constant source strength.The discretisation finally yields a linear equationsystem to solve for the unknown source strengths"!:

)AIC

*· "! = %%%'

RHS

%%%'RHS = %"U! · "n

(9)

)AIC

*is the matrix of the aerodynamic influ-

ence coe!cients which give the induction of onesource panel of unit strength onto the collocationpoint of another panel. The components of thevectors "! and %%%'

RHS represent the correspondingvalues for a particular panel k.

Once the source strengths are known, the invis-cid velocity vector at an arbitrary field point andespecially on the surface can be determined. Ap-plying the Bernoulli-equation finally gives thepressure field.

2.2 Propeller Model

The panel method is coupled to a model which en-ables to calculate the time-averaged influence ofa propeller located on the body axis [7]. In thismodel the propeller disc is represented by a vor-tex sheet whereas the slipstream is modelled by adiscrete number of vortex tubes of variable vortic-ity. Due to this representation, actually, a pro-peller with an infinite number of blades is mod-elled. However, the resulting induced velocity field

0.90 0.95 1.00 1.05 1.10-0.10

-0.05

0.00

0.05

0.10

0.90 0.95 1.00 1.05 1.10-0.10

-0.05

0.00

0.05

0.10

x/L

r/L

Figure 1: Calculated slipstream geometry of a pro-peller located downstream of a displace-ment body [7]

coincides with the time-averaged velocities causedby a propeller with finite number of blades.

The vortex tubes in the slipstream are dis-cretized by truncated concentric cones. Each tubeis terminated by a semi-infinite vortex cylinder.The contracting slipstream geometry is determinedby an iterative relaxation procedure based on thecondition that the slipstream has to be force-free.This implies that the vorticity vector is parallel tothe local velocity vector at each collocation point ofthe slipstream. With this relaxation procedure theself-influence of the propeller and its slipstream aswell as the velocity field induced by a displacementbody are considered.

The influence of the propeller on the hull is con-sidered by a modification of the right-hand-side ofEq. (9). Hereby, the velocity vector induced by thepropeller and its slipstream onto each panel collo-cation point on the hull surface is added to theundisturbed freestream velocity "U!. Thereafter,the source distribution is recalculated to satisfythe updated right-hand-side. This alters the body-induced velocity field which requires an adaptionof the slipstream geometry. This procedure is re-peated until convergency is achieved. An exampleof a calculated fully-relaxed propeller slipstreamincluding propeller-hull interference e$ects is givenin Fig. 1.

The propeller thrust and the required powerfinally are determined by an integration of theforces acting on the bound circulation distribution.These forces are obtained from application of the


Kutta-Joukowsky-law. As a major improve-ment to the model described in Ref. [7] the viscousinstead of the inviscid propeller onset flowfield isconsidered during the calculation of the propellerforces. Furthermore, the viscous drag of the pro-peller blades is estimated based on the drag coe!-cient of a flat blade with fully laminar or fully tur-bulent boundary layer respectively. The flat platedrag-coe!cient depends on the chord Reynoldsnumber Rec(r) at the particular blade section andis given by:

cd|laminar =1.328(

Rec(10)

cd|turbulent =2 · 0.455

log10 (Rec)2.58 (11)

The consideration of the detailed aerodynamiccharacteristics of the blade sections will be imple-mented next.

It is important to mention that with the presentimplementation of the propeller model the boundcirculation distribution on the propeller blades isnot calculated but has to be specified by the user.This distribution might e.g. be taken from a pre-viously performed propeller design, from experi-ments or from the known distribution of ‘ideal’ iso-lated propellers. An extension of the model to the-oretically predict the circulation distribution dur-ing the coupled analysis will be investigated in fu-ture.

3 Calculation of the ViscousFlow About the Airship Hull

3.1 Boundary-Layer Procedure

Governing Equations and IntegralBoundary-Layer Parameters

For the present investigations a first-order integralboundary-layer procedure was applied to calculatethe viscous e$ects along the hull. The method isbased on the numerical integration of the integralmomentum and energy equations. In a surface-oriented coordinate system (Fig. 2) these equationsread for incompressible axisymmetrical flows as fol-lows:

d&2ds

+ &2

+"H12 + 2

Ue

#dUe

ds+

1R

dR

ds

,=

cf

2(12)

d&3ds

+ &3

+3Ue

dUe

ds+

1R

dR

ds

,= cD (13)

r

U x!

n, v

s, uP

R

Figure 2: Surface-oriented coordinate system usedfor the calculation of axisymmetricboundary layers

These governing equations also hold for &2 )/ Rif the integral parameters according to the follow-ing axisymmetric definitions are presumed:

&1 =!!

0

+1 % u(n)

Ue

,'1 + n.

(dn (14)

&2 =!!

0

u(n)Ue

+1 % u(n)

Ue

, '1 + n.

(dn (15)

&3 =!!

0

u(n)Ue

-

1 %"

u(n)Ue

#2. '

1 + n.(

dn

(16)

with . =

/1

R2%

"1R

dR

ds

#2

(17)

In contrast the integral parameters according tothe plane definition are denoted &1, &2 and &3 with:

&1 =!!

0

+1 % u(n)

Ue

,dn (18)

&2 =!!

0

u(n)Ue

+1 % u(n)

Ue

,dn (19)

&3 =!!

0

u(n)Ue

-

1 %"

u(n)Ue

#2.

dn (20)

The system of ordinary di$erential equations(12), (13) is closed by three algebraic relationsto solve for the five unknown integral parame-ters &1, &2, &3, cf and cD. With the present


method the closure relations proposed by Eppler[1] are used. These closure conditions were derivedfrom Falkner-Skan self-similar profiles for laminarboundary layers whereas empirical relations werecorrelated for turbulent flows. The complete sys-tem of equations is solved for the dependent pa-rameters &2 and &3 based on the velocity distribu-tion along the boundary-layer edge Ue(s) resultingfrom the inviscid outer-flow computation.

Transition Prediction

The location of the laminar to turbulent transitioncan be either specified by the user or predicted bya local empirical criterion [1] or a semi-empiricalen-method. If laminar separation occurs upstreamof transition it is switched to turbulent closure re-lations at the separation point.

In the present investigations the en-method wasapplied to recalculate wind-tunnel experiments fea-turing natural transition. The en-approach takesadvantage of the fact that the transition processin subsonic 2D-flows is typically associated withthe growth and breakdown of Tollmien-Schlichtingwaves (TS-waves). As long as the disturbance am-plitude A is small enough, the amplification canbe predicted by means of linear stability theory.The basic idea of the en-method is that transi-tion may be assumed when the most amplified fre-quency reaches a certain critical amplification fac-tor ncrit. = ln (Acrit./Ainitial). The value of ncrit.

depends on freestream conditions, the receptivitymechanism and on the definition of the transition‘point’. With the present en-approach, spatial dis-turbance growth is considered. To minimize thecomputational e$ort a database method was im-plemented as an alternative to a direct solution ofthe Orr-Somerfeld equation. More details alongwith validation examples are given in Refs. [6], [8]and [9].

Drag Prediction

To determine the viscous drag of the hull the well-known Squire-Young formula is applied. This ap-proach is based on the evaluation of the calculatedmomentum thickness at the stern and accounts forthe skin-friction as well as for the form drag of theboundary layer. If the propeller is switched on, thepropeller-induced pressure drag is calculated froman integration of the pressure distribution alongthe hull surface. The sum of both contributionsrepresents the total hull drag.

3.2 Viscous/Inviscid Coupling

With the theoretical model described so far, theinviscid velocity distribution Ue(s) along the hullsurface is specified as boundary condition for theboundary-layer calculation. This gives the correctresult only for the asymptotic condition of infiniteReynolds number and vanishing boundary-layerthickness. For practical problems with finite val-ues of Re the displacement e$ect of the boundarylayer a$ects the outer flow and thus the distribu-tion Ue(s). This, in turn, has an impact on theboundary-layer development. The displacement ef-fect is most pronounced in the pressure recovery re-gion near the tail. Therefore, an iterative couplingof the boundary layer and the outer-flow computa-tion is needed if realistic boundary-layer propertiesin the stern region of the hull are of interest. Theobjective, hereby, is to modify the inviscid flow cal-culation in such a way that it provides the correctvelocity distribution Ue along the outer edge of theboundary layer under consideration of its displace-ment e$ect. We shall call this equivalent inviscidflow computation [4].

Transpiration Technique

In principle, two di$erent coupling schemes arepossible. With the solid displacement surfacemodel the calculated displacement thickness isadded to the solid hull shape to find the rele-vant contour for the subsequent outer-flow com-putation. The main disadvantage of this approachis that the geometry-dependent aerodynamic in-fluence coe!cients

)AIC

*have to be recalculated

during every iteration step, which is rather time-consuming.

For this reason the surface transpiration modelis usually preferred. With this second approachan additional source-singularity distribution on thesurface of the solid hull shape is introduced to sim-ulate the boundary-layer displacement e$ect. Forthis purpose the right-hand-side of the linear equa-tion system (9) is modified by adding a transpira-tion velocity vn.

At a certain position on the hull the transpira-tion velocity can be derived from an integration ofthe continuity equation across the boundary layerfor both, the real viscous flow and the equivalentinviscid flow. The value of vn can be determinedfrom the fact that the mass flow normal to the wall(caused by vn) in the equivalent inviscid flow hasto be equal to the streamwise rate of change of the

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mass flow deficit in the boundary layer [4]. Forincompressible axisymmetrical flows it follows:

vn =d(Ue · &1)

ds+

Ue · &1R

· dR

ds(21)

During the present investigations it has shownthat in some cases Eq. (21) yields small irreg-ularities of the vn-distribution in the vicinity ofthe stern. An improvement, i.e. a smoother vn-distribution was achieved by introducing the inte-gral momentum equation (12) into Eq. (21) whichfinally gives:

vn = Ue

"d&1ds

+ H12cf

2% H12

d&2ds

%H12&1Ue

dUe

ds% &1

Ue

dUe

ds

#(22)

If the full transpiration velocity as resulting fromEq. (21) or (22) is added to the right-hand-sideof Eq. (9) severe stability problems arise: Nearthe tail region of an axisymmetric body a strongflow deceleration results from the first inviscid flowanalysis. Along with the reducing body radius,an unrealistic strong growth of the displacementthickness is predicted by the subsequent boundary-layer calculation. According to Eq. (21) this yieldsto very high values for vn in the tail region. Asa consequence, the beginning of the pressure re-covery is shifted in upstream direction during thenext iteration loop (Fig. 3) which in turn causesan earlier growth of the boundary layer. Usuallyan unstable or a divergent behaviour of the vis-cous/inviscid coupling process results.

Under-Relaxation Scheme and Determinationof the Relaxation Factor

To obtain a stable behaviour of the coupling pro-cess, only a part / of the theoretical update of vn

may be added to the right-hand-side of Eq. (9), i.e.an under-relaxation scheme has to be introduced.Usually the relaxation factor / is chosen empiri-cally. For the present investigations a more sophis-ticated approach was applied which is based on alocal linear stability analysis of the coupled pro-cess as proposed by Le Balleur [3] (see also Lock[4]). Following the derivation of an ‘optimum’ re-laxation factor for plane flows as given by Lock [4]an analogous theory will now be derived for ax-isymmetrical boundary layers along a body withlocally cylindrical shape.

x/L

c p

0 0.25 0.5 0.75 1

-0.5

0

0.5

1

Inviscid solutionCoupled viscous/inviscid analysis(1 iteration, no relaxation)

Figure 3: Unstable behaviour of the vis-cous/inviscid coupling process withoutrelaxation, ‘Lotte’ hull, ReL = 16 · 106,xtra/L = 0.05

First of all, the transpiration velocity vn ac-cording to Eq. (21) is nondimensionalized with theouter-flow velocity Ue:

# =vn

Ue(23)

The quantities #(n), v(n)n and U (n)

e shall denotethe flow properties at the n-th iteration whereas0#, 0vn and 0Ue represent the correct final values af-ter convergency of the iteration process. To definethe discrepancy from the converged solution thefollowing error functions are introduced:

!(n) = #(n) % 0# (24)

u(n) =U (n)

e % 0Ue

0Ue

(25)

We assume that the rate of change of any flowproperty is negligibly small compared to the vari-ation of the above error-functions. Then, a locallylinearized analysis of the behaviour of u and ! canbe performed. To describe the equivalent inviscidflow a potential function is introduced which sat-isfies the Laplace-equation in case of incompress-ible fluids. In cylindrical coordinates this equationreads as follows:

"xx + "rr +1R

"r = 0 (26)


If, furthermore, the crude assumption is metthat the analysed body has locally a cylindricalshape2, the outer-edge velocity is oriented in axialand the transpiration velocity in radial directionwhich allows to determine the error-functions asfollows:

u =-"-x

= "x (27)

! =-"-r

= "r (28)

The error-functions u and ! are coupled by theLaplace-equation (26) and, on the other hand, bythe boundary-layer equations via Eq. (21).

Eq. (26) is now represented by a superpositionof harmonic waves, i.e. a Fourier-expansion is in-troduced in order to investigate the amplificationof particular waves of the errors in downstream di-rection. It can easily be shown that a solution ofEq. (26) for a wave number ) = 2#

$ is given by:

"(x, r) = A · ei%x · e"&(r"R) = A · f(x) · g(r)

(29)

with: . =1R +

11

R2 + 4)2

2and i =

(%1

(30)

For the partial derivatives it follows:

u = A · i) ei%x · g(r) = A · i)f(x) · g(r) (31)

ux =-u

-x= %A · )2f(x) · g(r) (32)

! = A · f(x) · gr(r) (33)

Combining Eqs. (32) and (33) yields:

ux = %)2!g(r)gr(r)

=)2!

.(34)

Now, the dependency of the nondimensionalizedtranspiration velocity # = vn

Ueon the velocity gra-

dient dUeds resulting from the viscous analysis shall

be derived. For this purpose the gradient d!1ds is

eliminated in Eq. (21) by inserting the integral

2An analogous theory was develeoped for a locally coni-cal body shape. Both approaches yield almost identicalresults with respect to the calculated relaxation factor,see Fig. 6. For simplicity only the cylindrical approachis reported in the present paper.

boundary-layer equations (12) and (13). Rearrang-ing the resulting expression finally yields:

# =vn

Ue=

dUe

ds

&2Ue

B + C (35)

with B =+

H12 % 3H32dH12

dH32(36)

% (H12 + 2)"

H12 % H32dH12

dH32

#,

and C ="

H12 % H32dH12

dH32

# "cf

2% &2

R

dR

ds

#

+dH12

dH32

"cD % &3

R

dR

ds

#+

&1R

dR

ds(37)

The derivative of the shape factor dH12dH32

can bedetermined from the closure relations used [1] inanalytical form.

Introducing the error-functions (24) and (25)into Eq. (35), taking into account that the deriva-tives of the flow properties are assumed to be smalland that dUe

ds = dUedx for the flow direction consid-

ered, it follows:

! =du

dx· &2 B = B &2 · ux (38)

During the viscous/inviscid coupling the tran-spiration velocity for the next iteration (n + 1) iscalculated based on the velocity gradient of the n-th iteration. Thus:

!(n+1) = B &2 · u(n)x (39)

Inserting Eq. (34) as resulting from the evalua-tion of the Laplace-equation into (39) shows howthe error ! behaves from one iteration of the vis-cous/inviscid coupling to the next:

!(n+1) = µ !(n) (40)

with µ =B&2)2

.(41)

with µ representing an amplification factor. Notethat the error in the transpiration velocity is am-plified for |µ| > 1. In this case it can be expectedthat the viscous/inviscid coupling will show an un-stable behaviour and that an under-relaxation hasto be introduced to stabilize the process.

Now a relation to determine a suitable relaxationfactor / shall be derived. Rewriting the error-function ! for iteration step (n) and (n + 1) and


solving for the converged solution 0# allows to elim-inate ! if Eq. (40) is inserted:

0# = #(n) % !(n)

and 0# = #(n+1) % !(n+1) = #(n+1) % µ!(n)

* 0# =µ

µ % 1#(n) % 1

µ % 1#(n+1) (42)

This equation states that the converged valueof the transpiration velocity can, in principle, bedetermined from the values #(n) and #(n+1) oftwo subsequent iterations. Because of the simplify-ing assumptions introduced during the derivation,Eq. (42) can only be expected to give an estimate.Nevertheless, it might be used to obtain an ‘opti-mum’ relaxation factor /. Rearranging Eq. (42)yields:

0# = #(n) + /'#(n+1) % #(n)

((43)

with / =1

1 % µ=

11 % B!2%2

&

(44)

In the above equation no statement about thewave number ) was made so far. To specify thevalue of ) it should be mentioned that with a nu-merical method at least two nodes within one wave-length are needed to resolve a harmonic wave. Forthe panel method this means that the highest wavenumber )max which can result for the error u isdetermined by the distance %s between the neigh-bouring panel collocation points. With

)max $ +

%s(45)

a reasonable minimum for the ‘optimum’ relax-ation factor is given by:

/opt =1

1 % B!2&

$#

!s

%2 (46)

This equation can be applied locally at eachpanel collocation point along the hull surface to ob-tain a distribution for the relaxation factor /opt(s).It was used in the present investigations to stabilizethe viscous/inviscid coupling process and yieldsmost often good convergence characteristics. Nev-ertheless, in some particular cases, the couplingprocess tends to become unstable after conver-gency seemed to have been achieved. In those casesthe relaxation factor was attenuated.

Discussion of the Parameters A"ecting theRelaxation Factor

Since B (Eq. (36)) is always negative, at least forthe closure relations used in the present boundary-layer method, Eq. (46) shows that smaller relax-ation factors are needed (compare discussion ofLock [4]):

. . . for higher &2-values, i.e. for lower Reynolds-numbers

. . . for smaller values of %s, i.e. for finer panelmeshes

. . . for higher absolute values of B

The evaluation of the shape factor relations usedshows that the module of B becomes larger as H32

decreases, see Fig. 4. Therefore, small values of /are required in the vicinity of separation.

H32

H12 B

1.4 1.5 1.6 1.7 1.8 1.9 21

1.5

2

2.5

3

3.5

4

-100

-80

-60

-40

-20

0

H12B

Figure 4: Shape factor relation for turbulent2D boundary-layers according to Ep-pler [1] and resulting behaviour of theparameter B

Application Example of the Coupling Scheme

As an example the analysis of the ‘Lotte’ hull givenin Fig. 3 was repeated using the present relax-ation scheme. The pressure distribution obtainedafter 20 iterations along with the inviscid resultis depicted in Fig. 5. No tendency to an unsta-ble behaviour could be observed and the solutionwas practically converged after 20 iteration cycles.Fig. 6 shows the distribution of the ‘optimum’ re-laxation factor after the first iteration step. It canbe seen that in the nose region almost no under-relaxation is needed whereas very small values of /result near the tail for the reasons discussed above.For comparison, the /-distribution which would re-sult if a locally conical body shape is assumed dur-ing the derivation of /opt, is also depicted in Fig. 5.


x/L

c p

0 0.25 0.5 0.75 1

-0.5

0

0.5

1

Inviscid solutionCoupled viscous/inviscid analysis(100 iterations, with relaxation)

Figure 5: Viscous/inviscid coupling with relax-ation, Lotte hull, ReL = 16 · 106,xtra/L = 0.05

s/L

"op

t

0 0.25 0.5 0.75 10

0.2

0.4

0.6

0.8

1

locally cylindrical shapelocally conical shape

Lotte, ReL=16 x106, fully turbulent

Figure 6: Distribution of the ‘optimum’ relaxationfactor /opt according to Eq. (46), Lottehull, ReL = 16 · 106, fully turbulentboundary layer, first iteration

Since the curves practically coincide the ap-proach described in the present paper is regardedto be su!cient and was therefore used for all sub-sequent calculations.

3.3 Derivation of the Boundary-LayerProfiles from Integral Parameters

With an integral boundary-layer procedure no in-formation about the detailed velocity distribu-tion across the boundary layer is provided. Toobtain the viscous onset-flow velocity field forthe stern propeller, the boundary-layer profileuUe

(r, L), therefore, has to be determined from thecalculated axisymmetric integral parameters like&1(L), &2(L) or &3(L).

Representation of Turbulent Boundary-LayerProfiles

For plane flows several families of velocity pro-files are well established like the Falkner-Skanself-similar profiles for laminar boundary layers.For turbulent flows some analytical expressions areknown. A rather simple representation is given bythe power-law profiles:

u

Ue=

'n

&

(a(47)

Swa$ord and Whitfield [12] proposed an analyt-ical expression to describe incompressible attachedor separated turbulent boundary layers. This ap-proach is more physically motivated and takes thezonal character of turbulent boundary layers intoconsideration:

u

Ue=

u'Ue

sign(cf )arctan (0.09y+)

0.09(48)

+"

1 % u'Ue

+sign(cf )0.18

#tanh

12

2

a

"n

&2

#b3

with u' = Ue ·4&&&

cf

2

&&&

(49)

and y+ =u'n

)= Re!2 ·

u'Ue

· n

&2

(50)

Notice that the momentum thickness &2 accord-ing to the plane defintion (19) has to be used whenevaluating Eq. (48). The parameters a, b are con-stant across the boundary-layer profile but are de-pendent on its shape and thus have to be deter-mined for each profile considered. For this pur-pose, ansatz (48) is introduced into the definitions(14), (15) and (16) to solve for &1, &2, &3 by nu-merical integration. The corresponding shape fac-tors H12 and H32 have to be equal to the specifiedvalues. These compatibility equations enable thedetermination of the parameters a, b by solvinga nonlinear equation system, e.g. using a dampedNewton-Raphson method.

The profile families discussed might also be usedto represent the velocity distribution across ax-isymmetric boundary layers as long as its thicknessis moderate.

Now it shall be described how to obtain the ex-ponent a of the power-law profiles and the rele-vant boundary-layer thickness & from given val-ues &1 and &2 as resulting from the axisymmet-ric boundary-layer calculation. First of all, ansatz


(47) is introduced into Eqs. (14) and (15) whichyields:

&1&

= 1 % 1a + 1

+. &

2% . &

a + 2(51)

&2&

=1

a + 1% 1

2a + 1+ . &

"1

a + 2% 1

2a + 2

#

(52)

Since !1/!!2/! = !1

!2= H12 which is known from the

boundary-layer calculation and . from the geomet-ric properties of the analyzed body, Eqs. (51) and(52) yield a first equation to determine a and &:

H12 + 1a + 1

+. & (H12 + 1)

a + 2

% H12

2a + 1% H12 . &

2a + 2% 2 + . &

2= 0

(53)

Rearranging Eq. (51) gives the second equation re-quired:

& =

/(a + 2)2

.2(a + 1)2+

2(a + 2). a

· &1 % a + 2.(a + 1)

(54)

The coupled equations (53) and (54) can be solvediteratively using a Newton method.

With the known values of a and & the velocitydistribution u

Ue(n) across the boundary layer can

finally be obtained from ansatz (47).To give an example, Fig. 7 shows the approx-

imation of measured boundary-layer profiles [11]by the profile families discussed. The power-lawprofiles were determined from measured values of&1 and &2 whereas Re!2 and cf are required addi-tionally to derive the Swa$ord profiles. Both ap-proaches are in good agreement to the actual veloc-ity distributions even though very thick boundarylayers near the stern of an axisymmetric body wereconsidered. It is, therefore, expected that withthe present post-processing approach the propelleronset-flow velocity field is determined to su!cientaccuracy as long as the coupled viscous/inviscidcalculation yields reliable results for the integralboundary-layer parameters. For the present inves-tigations the power-law approach is used to obtainthe boundary-layer profile at the stern of the hull.

U/Ue

n/L

0 0.25 0.5 0.75 10

0.01

0.02

0.03

0.04

0.05

0.06

x/L = 0.90H12 = 1.529

Experiment PatelSwafford profilePower-law profile

U/Ue

n/L

0 0.25 0.5 0.75 10

0.01

0.02

0.03

0.04

0.05

0.06

x/L = 0.93H12 = 1.635


U/Ue

n/L

0 0.25 0.5 0.75 10

0.01

0.02

0.03

0.04

0.05

0.06

x/L = 0.96H12 = 1.957


U/Ue

n/L

0 0.25 0.5 0.75 10

0.01

0.02

0.03

0.04

0.05

0.06

x/L = 0.99H12 = 2.231


Figure 7: Approximation of boundary-layer pro-files based on measured integral param-eters, experiment by Patel et al. [11]

Convertion of Integral Boundary-LayerParameters from Axisymmetric to PlaneDefinition

Besides the approximation of turbulent boundary-layer properties, the power-law ansatz enables asimple convertion of the axisymmetric integral pa-rameters &1, &2, &3 into the properties &1, &2, &3 ac-cording to the plane definition. Inserting Eq. (47)into definitions (18) to (20) yields:

&1 ="

1 % 1a + 1

#· & (55)

&2 ="

1a + 1

% 12a + 1

#· & (56)

&3 ="

1a + 1

% 13a + 1

#· & (57)


x/L

# 1/L

,#1

/L

0.7 0.8 0.9 10

0.01

0.02

0.03

#1/L experiment#1/L convertion#1/L experiment

$$$

x/L

# 2/L

,#2

/L

0.7 0.8 0.9 10

0.01

0.02

0.03

#2/L experiment#2/L convertion#2/L experiment

$

$$

x/L

H12

,H12

0.7 0.8 0.9 11

1.5

2

2.5

H12 experimentH12 convertionH12 experiment

$

$$

Figure 8: Convertion of displacement thickness &1,momentum thickness &2 and shape factorH12 from axisymmetric to plane defini-tion using the power-law approach

Fig. 8 is to demonstrate the capability of thepower-law approach to be used for convertion ofthe integral boundary-layer parameters. Based onmeasured values [11] of &1 and &2 Eqs. (53) to (56)were applied to theoretically determine the corre-sponding values &1, &2 and H12. The obtained dis-tributions almost coincide with the experimentalresults.

4 Results and Discussion

4.1 Viscous Flow about the Hullwithout Propeller Influence

Detailed experiments on axisymmetric bodies withstern propellers are hardly available in literatureopen to the public. Therefore, the present the-oretical model is at first validated for the barehull with respect to the boundary-layer calcula-tion and the viscous/inviscid coupling scheme de-scribed. For this purpose low Reynolds-numberexperiments were selected because for these condi-tions the boundary layer in the tail region is ratherthick and viscous/inviscid interaction e$ects aremost pronounced. Thus, limitations of the first-order boundary-layer procedure and the couplingscheme become more obvious than for the highReynolds-numbers of real airships.

As very first example, Fig. 9 shows the calcu-lated and the measured pressure distribution fora 1 : 20 scale model of the remote-controlled air-ship ‘Lotte’ built at the University of Stuttgart.The experiments were performed in the ‘MediumWind-Tunnel’ of the IAG with a turbulence lat-tice being mounted in some distance upstream ofthe model. The lattice increases the turbulencefactor to Tu $ 0.06 which yields an almost fullyturbulent flow about the entire hull like it is truefor full-scale airships. The wind-tunnel model wassupported by a tailboom which was also consideredin the calculation.

It is obvious that in the tail region the invis-cid pressure distribution strongly deviates from themeasured distribution due to the neglection of theboundary-layer displacement e$ect. The presentcoupled viscous/inviscid analysis is in good agree-ment with the experiments even though the theo-retical result shows a slightly stronger pressure re-covery. However, it should be noticed that the cal-culation yields the surface pressure for the equiva-lent inviscid flow. This pressure di$ers from thatof the real viscous flow if, due to curvature e$ects,the pressure is not constant across the boundarylayer [4]. This is expected to be true in the tailregion of the analyzed body at least for the lowReynolds numbers considered.

The boundary-layer experiments performed byPatel et al. [11] were chosen as second example.The examined shape is a 1:6 ellipsoid with a conicaltail piece (Fig. 10) which yields a strong thickeningof the boundary layer in this area. This test caserepresent a tough challenge for the viscous/inviscid


x [m]

c p

0 0.25 0.5 0.75 1

-0.5

0

0.5

1

Inviscid solutionCoupled viscous/inviscid analysisExperiment IAG

Figure 9: Surface pressure distribution about amodel of the ‘Lotte’-hull with tail-boom, experiment by Lowag & Funk [5],ReL = 1.3 · 106, $ = 0#, fully turbulent

x/L

c p

0 0.25 0.5 0.75 1

-0.5

0

0.5

1

Inviscid solutionCoupled viscous/inviscid analysisExperiment Patel et al.

Figure 10: Surface pressure distribution alonga body of revolution, experimentby Patel et al. [11], ReL = 1.26 · 106,forced transition at xtra/L = 0.05

coupling since it was especially designed to inves-tigate strong interaction e$ects between boundarylayer and external flow.

The strong interaction causes a significant re-duction of the measured pressure recovery in com-parison to the inviscid solution, see Fig. 10. Thecoupled viscous/inviscid analysis yields a correct

x/L

# 1/L

0.6 0.7 0.8 0.9 10

0.01

0.02

0.03

0.04

Without boundary-layer couplingCoupled viscous/inviscid analysisExperiment Patel et al.

x/L

H12

0.6 0.7 0.8 0.9 11

1.5

2

2.5

3

Without boundary-layer couplingCoupled viscous/inviscid analysisExperiment Patel et al.

x/L

c f

0.6 0.7 0.8 0.9 10

0.001

0.002

0.003

0.004

0.005

Without boundary-layer couplingCoupled viscous/inviscid analysisExperiment Patel et al. (Clauser plot)

Figure 11: Distributions of the displacement thick-ness &1, the shape factor H12 andthe skin-friction coe!cient cf alongthe body of revolution examined byPatel [11], ReL = 1.26 · 106, $ = 0#,forced transition at xtra/L = 0.05

tendency but again, the recompression is overpre-dicted.

Fig. 11 gives the results of calculated and mea-sured boundary-layer parameters. The coupledanalysis shows a rather good agreement with theexperiments even in the vicinity of the pointed tail.Notice, that the calculation without coupling givessignificant di$erent results for the shape factor dis-tribution. If the displacement e$ect is neglected asteep increase of H12 in the tail region along withturbulent separation at xsep/L $ 0.98 can be ob-served. On the other hand, the coupled analysis aswell as the experiment give substantial lower H12-values and no separation occurs.


U/Ue

n/L

0 0.25 0.5 0.75 10

0.01

0.02

0.03

0.04

0.05

0.06

x/L = 0.90

Experiment PatelPower-law profilePower-law profile(with coupling)

U/Ue

n/L

0 0.25 0.5 0.75 10

0.01

0.02

0.03

0.04

0.05

0.06

x/L = 0.93


U/Ue

n/L

0 0.25 0.5 0.75 10

0.01

0.02

0.03

0.04

0.05

0.06

x/L = 0.96


U/Ue

n/L

0 0.25 0.5 0.75 10

0.01

0.02

0.03

0.04

0.05

0.06

x/L = 0.99


Figure 12: Predicted and measured boundary-layer profiles for the body of rev-olution examined by Patel [11],ReL = 1.26 · 106, $ = 0#, forced transi-tion at xtra/L = 0.05

Fig. 12 finally shows the comparison of mea-sured boundary-layer profiles and power-law pro-files determined from the calculated integral pa-rameters (compare Sec. 3.3). The velocity pro-files as resulting from a coupled analysis yield apretty good match to the real profiles except for themost downstream measuring location x/L = 0.99.At the stern the inner region of the boundarylayer could not satisfactory represented by the one-parameter power-law approach and, furthermore,the boundary-layer thickness is slightly overpre-dicted. Nevertheless, it becomes obvious that aviscous/inviscid coupling is indispensable to pre-dict reasonable boundary-layer properties in thetail region.

Because for real airships the Reynolds-numberis much higher than for the test cases considered,the displacement e$ect is less pronounced and itcan be expected that the present model enablesthe calculation of the viscous flowfield to su!cientaccuracy for the intended investigations on sternpropellers.

4.2 E!ciency of Stern-MountedPropellers

Because it is instructive for the study of thepropeller-hull interference e$ects, calculation re-sults for the ‘Lotte’ airship with stern propelleras reported in Ref. [7] are depicted once more andcompared to results obtained with the present en-hanced theory (Fig. 13). The full-scale airship oflength L = 16m was examined for a flight speed ofU! = 15m/s which corresponds to a Reynolds-number of ReL = 16 · 106. The hull boundary-layer was assumed to be fully turbulent. To enablea comparison to previous results, the bound cir-culation distribution derived from the experimentsof Hucho [2] was specified even though the actualLotte propeller shows a di$erent design. Duringthe calculation the propeller rotational speed wasiterated until the obtained thrust was equal to theviscous plus the propeller-induced hull drag.

In Ref. [7] the influence of the distance betweenpropeller and stern on thrust coe!cient cT , powercoe!cient cP and e!ciency ' was investigated.With these earlier calculations only the inviscidflow field was considered during the analysis ofthe propeller characteristics. Due to the propeller-induced hull drag the thrust required for steadyflight increases when the propeller approaches thestern, see Fig. 13. As discussed in Ref. [7] the pro-peller e!ciency increases accordingly because theonset-flow velocity of the propeller decreases in thevicinity of the displacement body. Altogether, therequired power which is the relevant quantity torate the propulsive e!ciency remains almost con-stant if the viscous hull wake is not considered.

With the extended calculation method describedin Chapter 3 the influence of the viscous hull wakeon the characteristics of a stern-mounted propellercan be quantified. The obtained results are givenin Fig. 13 by symbols. It is obvious that therequired power is significantly reduced comparedto the previous results whilst the thrust is al-most una$ected. The propeller e!ciency (based onU!) increases accordingly and is well above 100%(' $ 1.1 assuming a fully laminar and ' $ 1.3 for


cT, turbulent bladescP, turbulent blades%, turbulent bladescT, laminar bladescP, laminar blades%, laminar blades

d/R

c T,c P

%

0 0.5 1 1.5 20.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

cTcP%

Inviscid propelleronset flow

Viscous propelleronset flow

Figure 13: Influence of viscous e$ects on thrustresp. power coe!cient and e!ciency ofa stern propeller (viscous results ac-cording to present theory, inviscid re-sults as given in Ref. [7]), Lotte hull,ReL = 16 · 106, $ = 0#, forced transi-tion at xtra/L = 0.05

a fully turbulent blade boundary-layer). It shouldbe mentioned that these values strongly dependon the specified bound circulation distribution andare, therefore, only valid for the particular con-sidered distribution. Nevertheless it becomes obvi-ous that the gain in e!ciency overcompensates thepropeller-induced drag and that an adequately de-signed stern propeller is a very favourable propul-sion device for airships.

As another example, an attempt was made to re-calculate the experiments on an airship model withstern-propeller performed by McLemore [10]. Forthis purpose the shape of the airship hull depictedin report [10] was at first digitized and slightlysmoothed. This contour along with the inviscidpressure distribution is given in Fig. 14. With thepresent calculations the bare hull at zero incidencewas analyzed whereas the experiments were per-formed for the complete configuration with fins andgondola.

Since the actual bound circulation distributionof the stern propeller is not reported in Ref. [10],again, the qualitative distribution resulting fromexperiments of Hucho [2] was specified. The ab-solute level of the circulation strength was scaledsuch that a thrust coe!cient of cT = 0.04 wasobtained for the same advance ratio ( $ 1.14 asin the experiments of McLemore for a blade twist

x/L

c p

0 0.25 0.5 0.75 1

-0.5

0

0.5

1

Figure 14: Inviscid pressure distribution alongan airship hull investigated byMcLemore [10], $ = 0#

of %$

rR = 0.75

%= 20#. The resulting circulation

distribution was specified for the whole range ofthe propeller rotational speed examined. This isof course a crude simplification. The experiments[10] were performed in the Langley full-scale wind-tunnel at U! = 94 ft/s corresponding to a hullReynolds-number of ReL = 11.9 · 106. For thisfacility, a turbulence factor of Tu $ 0.003 is re-ported. According to a correlation proposed by vanIngen [13] the e$ect of such a level of freestreamturbulence on transition can be simulated by spec-ifying a critical amplification factor of ncrit. $ 6.8for the en criterion. This value of ncrit. was usedin the analysis of the viscous hull flow.

Fig. 15 shows the calculated propeller e!ciencyvs. the advance ratio in comparison to measuredcharacteristics obtained for two di$erent bladetwist angles %. In the calculations the viscous in-teraction between hull and propeller and a fully-relaxing propeller slipstream was considered. Twocurves are given, one as resulting for fully lami-nar and one for fully turbulent flow about the pro-peller blades. A direct comparison to the experi-ments is first of all admissible only for the advanceratio which serves to scale the bound circulationdistribution (( = 1.14). For this value of ( themeasured propeller e!ciency lies right in betweenthe predicted curves. But also for other rotationalspeeds the behaviour is determined quite reason-able even though a constant circulation distribu-tion is assumed.


& = U! / (n D)

%

0.5 1 1.50

0.5

1

1.5

2Present theory, fully turbulent propeller bladePresent theory, fully laminar propeller bladeExperiment McLemore

' = 15o

' = 20o

Figure 15: Calculated and measured e!ciencyof a stern propeller, experiment byMcLemore [10], ReL(hull) = 11.9 · 106,natural transition (ncrit.(hull) = 6.8),$ = 0#

& = U! / (n D)

%

0.5 1 1.50

0.5

1

1.5

2

Stern propellerIsolated propeller(fully laminar blades)

& = U! / (n D)

%

0.5 1 1.50

0.5

1

1.5

2Stern propellerIsolated propeller(fully turbulent blades)

Figure 16: Calculated e!ciency of a propellerlocated at the stern of the McLemorehull compared to an isolated propeller,upper picture: fully laminar blades,lower picture: fully turbulent blades

With decreasing ( (increasing rotational speedat constant freestream velocity U!) the viscousdrag of the propeller blades becomes more domi-nant. This causes a reduction of the e!ciency '. Ifthe advance ratio is increased the e$ective angle ofattack of the real propeller blades is decreased untilthe zero-lift incidence is reached and the propellerwill not produce thrust anymore. A steep decreaseof the e!ciency can be observed in this (-range.Because in the analysis the same finite circulationdistribution is specified for every ( regardless of theactual blade incidence this sharp drop does not re-sult from the present calculation. The theoreticalcurves represent the result that would be achievedfor a particular advance ratio if the twist distribu-tion is adopted such that the intended circulationdistribution is realized. As expected the e!ciencyincreases continously with the advance ratio in thiscase.

Finally, in Fig. 16 the characteristics obtainedfor the stern propeller are compared to calculatede!ciency curves which result for the isolated pro-peller in the absence of the hull. Identical circula-tion distributions are assumed in both cases. Theresults clearly show that the e!ciency (based onU!) is significantly increased if the propeller is lo-cated in the viscous wake of an upstream displace-ment body.

5 Summary

A tool for the calculation of the time-averagedflowfield about a displacement body with a stern-mounted propeller was developed. The methodtakes viscous interaction e$ects into account andenables, e.g., to determine the propeller-induceddrag and to predict the characteristics of the wakepropeller. The method was, at first, validatedwith respect to viscous flow calculation aboutthe bare hull. It has shown that the coupledpanel/boundary-layer procedure yields reasonableresults even in the vicinity of the stern where astrong growth of the boundary layer can be ob-served.

Thereafter, the complete tool has been used toanalyse di$erent airship hulls propelled by a sternthruster. It has proven that the e!ciency can besignificantly increased if the propeller is located in-side the viscous hull wake. E!ciencies well above100% (based on U!) found in experiments werealso predicted by the present model. Taking thepropeller-induced hull drag into account it was


demonstrated that the power required to propelthe airship is decreasing for this favourable pro-peller installation. This, however, requires thatthe propeller is designed specifically for the realflow conditions in the propeller disc area since theblade incidence is significantly a$ected by the hullwake.

The present calculation method lacks from thefact that the bound circulation of the propellerblades is not determined during the analysis buthas to be specified by the user. Furthermore, thedetailed aerodynamic characteristics of the bladesections are not considered yet. Ongoing workdeals with the extension of the model both re-spects.

Acknowledgement

The present investigations were partly performedduring a research stay of the first author at the Me-chanical Engineering Laboratory, Tsukuba, Japanunder grant of the Japan Industrial Technology As-sociation. The authors gratefully acknowledge forthis opportunity and for the support and the kindhospitality of the host Dr. M. Onda (MEL).

References

[1] R. Eppler: Airfoil Design and Data.Springer Verlag Berlin Heidelberg New York,1990. ISBN 3-540-52505-X.

[2] W.-H. Hucho: Untersuchungen uberden Einfluß einer Heckschraube auf dieDruckverteilung und die Grenzschichtschi!sahnlicher Korper. Dissertation, Tech-nische Hochschule Braunschweig, Fakultatfur Maschinenwesen, January 1967.

[3] J. C. LeBalleur: Couplage visqueuxnon-visqueux: analyse du probleme inculantdecollements et ondes de choc. La RechercheAerospatiale, No. 6, 1977.

[4] R. C. Lock: Prediction of the Drag of Wingsat Subsonic Speeds by Viscous/Inviscid Inter-action Techniques. Aircraft Drag Predictionand Reduction, AGARD R-723, 1985.

[5] M. Lowag: Druckverteilungsmessungen amModell des Solarluftschi!s Lotte. Studienar-beit, Institut fur Aerodynamik und Gasdy-namik, Universitat Stuttgart, October 1999.

[6] Th. Lutz: Berechnung und Optimierungsubsonisch umstromter Profile und Rota-tionskorper. Dissertation, Institut furAerodynamik und Gasdynamik, UniversitatStuttgart, 2000. VDI Fortschritt-Berichte,Reihe 7: Stromungstechnik, Nr. 378, ISBN 3-18-337807-8.

[7] Th. Lutz, D. Leinhos and S. Wagner:Theoretical Investigations of the Flowfield ofAirships with a Stern Propeller. Proceedingsof the International Airship Convention andExhibition, 1996. Bedford, Great Britain, 5–7July 1996.

[8] Th. Lutz and S. Wagner: Drag Reduc-tion and Shape Optimization of Airship Bod-ies. Journal of Aircraft, Vol. 35, No. 3, S. 345–351, May–June 1998. AIAA Paper 97-1483.

[9] Th. Lutz and S. Wagner: NumericalShape Optimization of Natural Laminar FlowBodies. In: Proceedings of the 21st ICASCongress, 13–18 September 1998, Melbourne,Australia, ICAS/AIAA, 1998. AIAA Paper98-31525, ICAS-98-2.9.4.

[10] H. Clyde McLemore: Wind Tunnel Testsof a 1/20-Scale Airship Model with Stern Pro-pellers. TN D-1026, NASA, 1962.

[11] V. C. Patel, A. Nakayama andR. Damian: Measurements in the thickaxisymmetric turbulent boundary layer nearthe tail of a body of revolution. J. FluidMech., Vol. 63, No. 2, S. 345–367, 1974.

[12] T. W. Swafford: Analytical Approxima-tion of Two-Dimensional Separated Turbu-lent Boundary-Layer Velocity Profiles. AIAAJournal, Vol. 21, No. 6, S. 923–925, 1983.

[13] J. L. van Ingen and L. M. M. Boer-mans: Research on Laminar Separation Bub-bles at Delft University of Technology in Re-lation to Low Reynolds Number Airfoil Aero-dynamics. Proceedings of the Conference onLow Reynolds Number Airfoil Aerodynam-ics, UNDAS-CP-77B123, University of NotreDame, Indiana, USA, 1985, S. 89–124.

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