A physics-based gap-flow model for potential flow solvers · 2016. 11. 1. · Computational Fluid Dynamics (CFD) solvers permit high-ﬁdelity, viscous simulations of gap-ﬂow, but

A physics-based gap-flow model for potential flow solvers

C.M. Harwood, Y.L. Young n

Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, MI 48109, USA

a r t i c l e i n f o

Article history:Received 21 January 2013Accepted 16 March 2014

Keywords:Potential flowLifting lineCavitationHydrofoilPropellerGap flow

a b s t r a c t

Gap/clearance flows (also known as tip gap flows) affect the hydrodynamic forces, flow structure, andcavitation in both turbomachines and hydrofoils. Computational Fluid Dynamics (CFD) solvers permithigh-fidelity, viscous simulations of gap-flow, but the computational expense often prohibits their usefor fully exploring design spaces. Conventional potential flow solvers, on the other hand, cannot capturethe viscosity-dominated gap-flow dynamics. In the present study, a physics-based gap-flow model ispresented to capture the critical effects of gap-flow using general potential flow solvers. This isaccomplished by re-casting a lift-retention model as a corrected boundary condition. A simple lifting-line formulation is used to demonstrate the applicability of the gap-flow model. Results from the lifting-line analysis, modified by the gap-flow model, are compared with experimental measurements andhigh-fidelity CFD simulations over a range of gap sizes for two confined-wing arrangements withdifferent geometries and flow conditions. The modified lifting-line analysis improves significantly uponthe circulation, downwash, and drag predictions, compared to a standard lifting-line formulationwithout the gap-flow model. A viscosity-corrected expression for tip-vortex strength is proposed. Usingthe viscous correction, qualitatively-correct trends are predicted for vortex strength and tip vortexminimum pressure coefficients across a range of foil aspect-ratios, gap sizes, and angles of attack.

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1. Introduction

Two-dimensional (2-D) foil theory is a long-time fixture ofaero- and hydro-dynamics. Three-dimensional (3-D) potentialtheory for wings and propellers in a free stream is a likewisetried-and-true design tool. The case of a lifting surface with a tipproximal to a confining wall, however, lies between these bound-ing geometries and presents a theoretical gray area. The pressurejump from the pressure-side to the suction-side of a lifting sur-face pushes fluid through this gap, creating complex vortical“gap-flows" and altering pressure distributions on the liftingsurface. A tip leakage vortex (TLV) develops from the gap-flow,with steep pressure gradients and low minimum pressures thatcan cause cavitation to occur with associated noise, erosion,vibration, or other deleterious impacts.

The topic of gap-flow has enjoyed a great deal of research inrecent years – and justifiably so. In any model test of a flexible wing,there exists a measurable gap between the tip and end-wall. Evenmore importantly, ducted turbomachines are designed with appreci-able gap clearances, necessitating the consideration of gap-flowsduring the design stage. Numerical and physical studies have sought

to address this need, but all have met with challenges. Direct physicalexperimentation, unfortunately, is expensive and time consuming,making it difficult to canvass large design-spaces. High-fidelitycomputational fluid dynamics (CFD) solvers are well-suited for thedetailed analysis of the viscous flow in the gap region, but highcomputational-overheads all but prohibit their use in early-stagedesign. Potential methods, on the other hand, represent the otherend of the spectrum of numerical complexity and are many orders ofmagnitude faster than CFD, but standard formulations are not able tocapture the effects of the viscosity-dominated gap-flow. A physics-based model for gap-flow is presented in this paper with the aim ofimproving the accuracy of potential methods while retaining thecomputational efficiency of such solvers.

1.1. Background

The ubiquitous nature of gap-flow has led to a large number ofstudies, generally divisible into three camps: inviscid numericalsimulations, viscous numerical simulations, and physical experi-mentation. This section gives an overview of the work pertinent tothis paper, beginning with inviscid models in Section 1.1.1, viscousstudies in Section 1.1.2, and experimental work in Section 1.1.3.

1.1.1. Inviscid modelsPotential flow formulations frequently use vortex-based solutions

and panel methods to solve Laplace's equation for inviscid, irrotational,

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n Corresponding author. Tel.: þ1 734 647 0249.E-mail addresses: [email protected] (C.M. Harwood),

[email protected] (Y.L. Young).

Please cite this article as: Harwood, C.M., Young, Y.L., A physics-based gap-flow model for potential flow solvers. Ocean Eng. (2014),http://dx.doi.org/10.1016/j.oceaneng.2014.03.025i

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and incompressible flow. One of the simplest such methods is thelifting-line analysis of 3-D wings, first introduced by Prandtl (1918),and generalized by Glauert (1943) for arbitrary planforms and twist-distributions. The derivation reduces a wing to a 1-D domain bycollapsing chordwise variations in vorticity, pressure, and velocity to asingle point on each section, thus representing the wing as a spanwisedistribution of its section properties. In this way, the theory posits thata finite-aspect-ratio wing may be replaced by a spanwise distributionof "bound'' circulation, with zero-circulation boundary conditions ateach free tip.

Sugiyama (1970) performed experimental and numerical studiesof a split rectangular hydrofoil with an RAF-6 section in a flowchannel. The split foil arrangement involved an instrumented foil anda physical “dummy” foil mirrored about the mid-span of the flowchannel to symmetrize the flow, with the plane of symmetrymimicking a free-slip confining wall with an infinite Reynoldsnumber ReWall¼1. The experiments evaluated the effects of gapsize and angle of attack through an extensive test matrix. Sugiyamaalso extended the classic lifting-line analysis of Glauert (1943) to thecase of a confined channel by including a large number of image-foilsto impose symmetry at the walls. His analysis used the classic zero-load boundary condition at the tip. In comparisons with experimentalresults, Sugiyama (1970) found the results of the lifting-line analysisto be qualitatively correct. Quantitative predictions of theoretical lift,however, were much lower than experimental measurements, andthe theoretical predictions of induced drag were much higher thanmeasured values for small gap sizes.

In reality, fluid viscosity impedes flow through the gap, whileinviscid theory permits arbitrarily-large gap-flow velocities tooccur in order to satisfy the zero-load condition at the wing tip.Lakshminarayana and Horlock (1962) introduced the retained liftfraction (KLH) in Lakshminarayana and Horlock (1962) as a non-dimensional value that represents the experimentally-observedretention of circulatory lift at the tip of a confined lifting surface bythe equation,

KLH ¼ClTipCl2D

; ð1Þ

where Cl2D is the ideal lift coefficient given by 2-D theory and ClTipis the sectional lift coefficient at the tip of the foil, determinedexperimentally by using pressure taps at the foil tip. Theyproposed a modified lifting-line model, in which a line of constantbound vorticity is used to represent the foil, discounting 3-Dvortex shedding. Instead, all shed vorticity is present in a TLV, thestrength of which is calculated as

ΓTLVLH ¼ ð1�KLHÞΓ2D; ð2Þwhere Γ2D is the ideal circulation of an airfoil section from 2-Dtheory.

Lewis and Yeung (1977) also studied lift-retention, but under aslightly different definition, given as,

KLY ¼ClWallClTip

¼ ClWallCl2DKLH

: ð3Þ

The quantity CLWall was calculated by installing pressure tapsaround a projected foil profile on the confining wall; integrationof the pressure coefficient around the curve yielded virtual wall-retained components of lift and drag. The experiments drew uponDr. Yeung's experimental testing (Yeung, 1977) of a rectangularairfoil with gap-to-chord ratios ranging from λ¼0 to λ¼0.18,giving a closed-form approximation of the retained lift fraction,

KLY ðλÞ ¼ e�14λ: ð4ÞThe retained lift fractions of Lakshminarayana and Horlock

(1962) and Lewis and Yeung (1977) are compared in Fig. 1. Thestriking agreement between the two correlations is only coinci-dental, considering the different definitions given above, but itsuggests that KLY(λ) is a suitable closed-form approximation to KLH,allowing either one to be represented by a common K :

K ¼ KLH ¼ KLY ðλÞ ¼ e�14λ: ð5ÞKerwin (1989) presented another simple method for approx-

imating viscous gap-effects, based on Bernoulli's obstructiontheory, that has appeared in recent vortex lattice method (VLM)codes by Baltazar et al. (2012) and Gu (2006). The gap is treatedas a porous panel, across which a reduced velocity is permitted.

Nomenclature

a0 section lift curve slope,∂Cl2D∂α

CL;CD 3-D lift, drag coeff., L;D0:5ρU21CSCP pressure coefficient, P�P10:5ρU21CQ orifice coefficientCl;Cd section lift, drag coeff.,

l;d0:5ρU21C

CDi 3-D induced (inviscid) drag coeff.L;D 3-D lift, dragl; d 2-D section lift, dragK ;KLH ;KLY retained lift fractionP local pressureP1 free-stream pressurePv fluid vapor pressureRex Reynolds number (based on length x),

jU1jρxμ

u induced velocity vector, uêxþvêyþwêzU1 free-stream velocity vector, U1êxþV1êyþW1êzU total velocity vector (excluding boundary-layer velo-

city profiles), U1þuUn total velocity vector (including boundary-layer velo-

city profiles)AR aspect ratio, S/CC chord

G gap sizeH test-section width, SþGN number of panels in discretizationS spanT maximum foil thicknessZi locations of nodes for vortex-sheddingnimages number of image foilsΔ influence distanceΓ section circulationΓ2D ideal section circulationΓTLV TLV circulation strengthΨ ; β correlation constantsα geometric angle of attackϵ gap-to-thickness ratio, G/Tλ gap-to-chord ratio, G/CΩ matrix of influence coefficientsω vorticity vector, ∇� Uμ fluid laminar viscosityρ fluid densitys cavitation index, P1 �Pv

0:5ρU21τ thickness-to-chord ratio, T/Cζj location of panel centers

C.M. Harwood, Y.L. Young / Ocean Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎2


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The porous boundary condition is given as

Uj � n̂j ¼ �U1αþCQffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2U1ΓTipj

q; ð6Þ

where Uj is the total velocity vector, n̂j is the unit normal vector,and ΓTipj is the bound vortex strength on the jth chordwise panelat the foil tip. U1 is the axial component of the free streamvelocity, α is the angle of attack, and CQ is the orifice dischargecoefficient. Eq. (6) restricts the induced velocities through the gapto approximate viscous impedence, which results in a modifiedcirculation distribution that, like K , demonstrates a degree of lift-retention at the foil tip. The application is elegant, but is limited bythe orifice coefficient CQ, which is a constant that remainsindependent of the angle of attack, gap size, or Reynolds number.Most studies use a value of CQ¼0.84, recommended by Kerwin(1989).

1.1.2. Viscous modelsTallman and Lakshminarayana (2001) used a compressible

Reynolds-Averaged Navier–Stokes (RANS) code to study the phy-sics of gap-flow in a compressor turbine cascade for two gap sizes.They demarcated regions of fluid inside of the gap on the basis ofvorticity development, and they characterized secondary flows bytheir interaction with flow through the gap. Harwood et al. (2012)presented a similar physical interpretation of subcavitating andcavitating flow around an isolated rectangular hydrofoil inside of acavitation tunnel, simulated using a commercial RANS code and atransport-based cavitation model and validated with experimentaldata from Ducoin et al. (2012). Both groups concluded that the TLVis a dominant secondary flow that depends strongly on the physicsinside of the gap. Viscous mechanisms, as well as inviscid oneswere found to contribute to the TLV development. In addition tothe shedding of the bound circulation near the tip, vorticity isgenerated inside of boundary layers in the gap and convectedinto the TLV. Interference with or enhancement of vorticity-development in the gap by cavitation or end-wall relative motion(such as that for ducted rotating machinery) strongly affect the TLVsize and strength. Tallman and Lakshminarayana (2001) concludedthat the strength of the TLV increases with gap size and, con-versely, that the circulation at the foil tip decreases with theincrease in gap size, consistent with the lift-retention modelsdescribed in Section 1.1.1.

1.1.3. Experimental studiesIn addition to the experimental studies by Lakshminarayana

and Horlock (1962) and Sugiyama (1970), as described in Section1.1.1, Farrell and Billet (1994) performed extensive testing of anaxial-flow pump to study TLV cavitation. Lift was directly mea-sured by instrumenting the blade tip, and laser velocimetry was

used to measure the TLV size and circulation. They suggested thatEq. (2) underestimates the TLV strength because it neglects thespanwise gradient of bound circulation. They recommended acorrected expression,

ΓTLVFB ¼ ð1�KLY ÞΓ2Dþ0:18Γ2D; ð7Þwhere Γ2D was defined at the tip section of the rotor blade whichagreed well with experimental data. In addition, they used aRankine vortex model to relate ΓTLV to the minimum vortexpressure coefficient, which occurs at the interface between thevortex viscous core and surrounding irrotational vortex sheath:

CPMin ¼ �2ΓTLVFϵ

2PrcU1

� �2; ð8Þ

where rc is the viscous core radius, given as

rc ¼ ð1�e�6ϵÞrc0 ; ð9Þ

rc0 ¼ βC Re�1=7C : ð10Þϵ¼G/T is the gap-to-thickness ratio, ReC is the foil chord-basedReynolds number, and a value of β¼0.36 was used by Farrell andBillet (1994) (also adopted in the present study). Eq. (10) invokesthe hypothesis of McCormick (1962) that the viscous core radiusdepends upon the pressure-side boundary-layer thickness for afoil in a free stream. Eq. (9) gives the reduced core radius in thepresence of a confining wall. Farrell and Billet (1994) used theabove equations to predict the cavitation inception index over arange of gap sizes and flow coefficients, finding that the modelingapproach worked well for gap-to-thickness ratios commonlyfound in turbomachinery ðϵ� 0:1Þ, while it underpredicted theinception index at larger gap-to-thickness ratios and overpre-dicted the index at smaller ratios.

Miorini et al. (2010) studied the flow through the rotor-casing clearance of the optically-matched waterjet facility at JohnsHopkins University. Stereoscopic particle image velocimetryrevealed a boundary layer on the blade tip inside the gap, whichdetached from the suction side and formed a shear layer that fedthe TLV. Also inside the gap, the boundary layer on the casing-wallresulted in a detached counter-rotating vortex. The flows observedin the experiment corroborate the numerical results of Tallmanand Lakshminarayana (2001) and Harwood et al. (2012), despitethe lack of relative motion between the foil tip and end-wall in thenumerical simulations.

Ducoin et al. (2012) observed significant gap and vortexcavitation while conducting physical experiments with a flexiblecantilevered foil in the IRENAV cavitation tunnel at the FrenchNaval Academy. A cavitating vortex core clearly indicated thepresence of a strong TLV. The CFD simulations in the present workuse the geometry of Ducoin et al. (2012). Harwood et al. (2012)previously validated their RANS results with experimental resultsfor the same geometry from the French Naval Academy.

1.2. Objectives

It is clear that a great deal of time and effort has been dedicatedto the study of gap-flow. However, its numerical treatmentremains nebulous. The objectives of this work are

� Develop a physics-based gap-flow model for potential flowsolvers based on the concept of lift-retention, and derive thenecessary constants by assimilating data from RANS simula-tions and experiments.

� Validate the gap-flow model using results from physicalexperiments and high-fidelity RANS simulations.

� Use the gap-flow model to quantify the influence of gapsize, angle of attack, and aspect ratio on the foil circulation

0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1

λ, G/C

KLH

, KLY

Lakshminarayana and Horlock, 1962 (KLH)Lewis and Yeung, 1977 (KLY)

e−14λ

Fig. 1. Retained lift fraction as a function of gap-to-chord ratio.

C.M. Harwood, Y.L. Young / Ocean Engineering ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3



distribution, induced velocities, TLV strength, TLV cavitationinception index, and hydrodynamic force coefficients.

2. Methodology

Section 2.1 describes a modified lifting-line model with thegap-flow correction. Section 2.1.1 presents the numerical set-up ofthe CFD simulations performed by the authors.

2.1. Derivation of a lifting-line model

Consider a foil or wing of span S and chord C, with a small gapof size G, as shown in Fig. 2(a). The Kutta–Joukowski theorem,

lðZÞ ¼ΓðZÞUðZÞρ ð11Þmay be applied in a section-plane Z of the wing to yieldthe sectional lift-per-unit-span, l(Z), as a function of the boundcirculation strength in that plane, Γ(Z), and the chordwise com-ponent of the inflow velocity, U(Z). Bound circulation is a functionof the section's effective angle of attack. According to Helmholtz'ssecond theorem, the change in bound vorticity from one section tothe next must be accompanied by the shedding of a trailing vortexfilament; the aggregate of the spanwise circulation gradient is atrailing sheet of vorticity, which in turn induces downwash inneighboring section-planes, reducing the effective angle of attackin those sections. This sets up the fundamental problem, whereinthe interplay of circulation and downwash must be resolved.

2.1.1. Governing equationsGlauert (1943) presents the following governing equations

relating the induced downwash velocity, v(Z), and bound circula-tion, Γ(Z):

vðZÞ ¼Z S0

�∂ΓðζÞ∂ζ

4πðζ�ZÞ ∂ζ; ð12Þ

and

ΓðZÞ ¼ CðZÞ2

a0ðZÞðUðZÞaðZÞ�vðZÞÞþCl0 ðZÞUðZÞ� �

; ð13Þ

where S is the span, C is t.he chord length, a0 is the slope of the 2-Dlift curve, α is the local geometric angle of attack, and Cl0 is the 2-D

lift coefficient at α¼01. All of the quantities except span are seento vary with the spanwise coordinate Z only, so the wing is reducedto a 1-D distribution of section properties, called a lifting-line. Eqs.(12) and (13) may be combined into the fundamental integralequation for the circulation distribution, which must be satisfied atall points on the lifting-line:

Z S0

�∂ΓðζÞ∂ζ

4pðζ�ZÞ ∂ζþ2ΓðZÞ

a0ðZÞCðZÞ¼UðZÞ aðZÞþCl0 ðZÞ

a0ðZÞ

� �: ð14Þ

2.1.2. Physical boundary conditionsThe Z component of velocity at the root-wall (Z¼0) and the

end-wall (Z¼H¼SþG) is zero, signifying a symmetry condition oneach plane. To satisfy this boundary condition, a mirror “image”foil must be reflected about Z¼0, and the composite foil and imagemust be repeated a sufficient number of times in each directionto effectively symmetrize the flow. The present study adoptsSugiyama's approach (Sugiyama, 1970) by using nimages¼20, wherenimages is the number of instances of the foil and its root-reflectedimage. Eq. (12) can be re-expressed to include the images:

vðZÞ ¼ZFoil

�∂Γ∂ζ

4πðζ�ZÞ ∂ζþ ∑nimages

k ¼ �nimages

ZImagek

�∂Γ∂ζ

4πðζ�ZÞ ∂ζ: ð15Þ

The circulation distribution is identical on each image, with theexception that the gradient assumes the opposite sign on thereflection about the root-wall and all repetitions thereof.

2.1.3. Numerical solution methodThe lifting-line is first discretized into N�1 spanwise panels,

with the Nth virtual panel spanning the gap, as shown in Fig. 2(b).Eq. (14) is satisfied at the panel-centers located at ζj. Vortexshedding is assumed to occur from the nodes located at Zi, where

ζj ¼ 12 ðZjþZjþ1Þ; j¼ 1;2;…;N

Γ j and vj are the bound circulation strengths and induceddownwash velocities at the panel centers, respectively, for j¼1,2,…, N�1. ΓN is the circulation at the foil tip (node point ZN), and isthe key to the gap-flow model.

By imposing Eq. (14) at the center of each panel, a second-ordernumerical algorithm can be created. Derivative terms in theintegrand of Eq. (12) may be replaced with central-differencingat all nodes except the tip, and with a second-order backwards-Euler approximation at the tip node. The integral itself can beapproximated by using closed Newton–Cotes quadrature. Afterfurther algebraic manipulations, Eq. (12) may be expressed in thecompact discrete form,

vj ¼ ∑N

i ¼ 1Ωj;iΓi; j¼ 1;2;…;N ð16Þ

where Ωj;i is defined as the influence coefficient of the boundcirculation at panel-center j on the downwash velocity at panel-center i, including the influence of respective panels located on thewing images.

The discrete form of Glauert's second governing equation(Eq. (13)) may be re-ordered as

vjþ2ΓjCja0j

¼UjajþUjCl0;ja0j

; j¼ 1;2;…;N � 1 ð17Þ

and combined with Eq. (16) to yield the linear discrete form ofEq. (14):

XNi ¼ 1

Ωj;iΓiþ2ΓjCja0j

¼ UjajþUjCl0;ja0j

; j¼ 1;2;…;N � 1 ð18ÞFig. 2. Illustration of 1-D domain for lifting-line analysis. (a) Schematic illustrationof foil. (b) Foil discretization scheme.




The system is, of course, under-constrained as a result of the(N�1)�N coefficient matrix. In this work, ΓN is assumed to beknown, which allows it to be moved to the right hand side ofEq. (18) to yield the final linear discrete form of the governingintegral equation:

XN�1i ¼ 1

Ωj;iΓiþ2Γjcja0j

¼ UjajþUjCl0;ja0j

� ΓNΩj;N ; j¼ 1;2;…;N � 1 ð19Þ

The circulation at the foil tip (ΓN) is determined from the gap-to-chord ratio (λ) by applying Eq. (5) for the retained lift fractionK :

ΓN ¼ KΓ2D ¼ K ða0NaNþCl0;N ÞUNCN2; ð20Þ

where Γ2D is again defined at the tip section of the hydrofoil.Substituting Eq. (20) into (19) recasts the retained lift fraction as amodified boundary condition, and the remaining N�1 unknowncirculation values corresponding to the physical foil may befound by solving the resultant linear system. Solving the samelinear system with a homogeneous boundary condition (ΓN¼0)mimics the zero-load condition used in classic lifting-line analyses(Glauert, 1943). In the subsequent discussion, results for the“corrected” lifting line analysis indicate the use of the lift-retention model (Eq. 20), while “uncorrected” lifting-lifting lineresults are obtained with the classic zero-tip-load assumption(ΓN¼0).

2.1.4. Near-wall treatmentThe low-velocity regions near the walls have been neglected in

the analysis thus far, replaced by the irrotational velocity field U tosustain Helmholtz's second theorem. Under the assumption that astrip-theory-type correction is valid in the small region near thewall, one can use the Kutta–Joukowski theorem (Eq. (11)) to showthat circulation varies linearly with local velocity. The vector ofbound-circulation strengths, corrected for the boundary-layervelocity profiles, is given as

ΓModj ¼ΓjUnjUj

; j¼ 1;2;…N ð21Þ

where Unj is the local velocity at ζj, including viscous velocitydeficits. A physical interpretation is that bound circulation is lostto viscous dissipation in the boundary layer near the wall, and isnot shed downstream. Unj may be found by using any closed-formapproximation for the velocity profile near the wall, such as apower-law model. The relationship between lift coefficient andcirculation becomes

Clj ¼ΓModj2UnjU2j Cj

: ð22Þ

2.1.5. Extension to other potential flow solversThe gap-flow correction was derived for a 1-D lifting-line,

but a similar correction can be generally applied to 2-D and 3-Dpotential solvers. For panel solvers, the circulation boundarycondition (ΓN) will itself correspond to a chordwise sum ofthe unknown vortex strengths at the tip. In such a case, anassumption must be made about the chordwise load distribution.The approach of Rains (1954), who assumed a triangular vorticitydistribution, is suggested as a good first approximation.

2.2. High-Fidelity RANS simulation

The commercial code ANSYS CFX was used to solve the incom-pressible, steady RANS equations for sub-cavitating flow around arectangular cantilevered hydrofoil inside of a flow channel with

eight different gap sizes, given in Table 1. The simulated domainand geometry were based on the experiments of Ducoin et al.(2012), who used a foil with a span (S) of 191 mm and a fixed gapsize of 1 mm. Fig. 3 is a schematic depiction of the computationaldomain and boundary conditions. A constant velocity inlet wasspecified in conjunction with a static-pressure outlet. Zero-slipconditions were specified on all solid boundaries except the rootwall, where a symmetry condition was imposed.

For each gap size, an unstructured mesh of between fivemillion and eight million elements was created, with structuredlayers near the foil pressure and suction faces, foil tip, andconfining wall to resolve boundary-layer profiles. The structuredboundary-layer meshes were characterized by Y þ � 2 to capturethe viscous sublayer via the two-equation k-ω turbulence model.The flow over the suction surface of a NACA-66 profile is almostentirely turbulent at an angle of attack of 81 (Ducoin and Young,2013), so a transition model was deemed unnecessary. As shownin Fig. 4, non-uniform refinements were performed at the foilleading and trailing edges, the foil wake, the TLV trajectory, andthe gap itself. 50–60 prismatic structured elements were specifiedacross the gap for each gap-size. The fine gap-mesh is necessary toresolve the exceedingly-high velocity gradients inside of the gapand to capture interactions between the vortex and boundarylayers on the foil tip and confining wall. Mesh refinements studies

Table 1Geometries and flow conditions for two validation cases: Case A, with experimentaldata from Sugiyama (1970) and Case B, with present CFD results, which have beenvalidated against the experimental results of Ducoin et al. (2012).

Geometry Case A (Sugiyama, 1970) Case B (Ducoin et al., 2012)Validation Data Experiment CFD

C(Chord) 70 mm 150 mmS (Span) 150 mm - G 192 mm - GAR (Aspect ratio at G¼0) 2.143 1.28τ (T/C ratio) 10% 12%Foil section RAF-6 NACA-66-312G (Gap size) 0.01, 1, 5, 10 mm 0.5, 1, 2, 3, 4, 6, 12, 24 mmα (Angle of attack) 31, 61 81U1 30 m/s 5 m/s

ReWall ¼ U1ρðXFoil�XInlet Þμ 1 3:29� 106ReC 1:84� 106 7:5� 105

Fig. 3. Domain and boundary conditions for RANS simulations (Case B).

Fig. 4. Surface mesh showing refinement regions (Case B).




were performed by Harwood et al. (2012); converged results for afixed gap size of 1 mm in subcavitating and cavitating conditionswere then validated against experimental and previous numericalresults from Ducoin et al. (2010, 2012), showing good agreementwith respect to force coefficients, flow structures, and pressuredistributions. The authors have assumed sufficient validity in thesimulations to merit the extension to other gap sizes.

3. Results and discussion

To validate the gap-flow model for the discrete lifting-lineapproach, results are compared with the RANS simulationsdescribed above for a NACA-66 hydrofoil (Case B) and withexperimental measurements (Sugiyama, 1970) for an RAF-6 airfoil(Case A). Uncorrected lifting-line results, which use the zero-tip-loading assumption, but include a large number of image-foils(Sugiyama, 1970), are also included for comparison. The geome-tries and flow characteristics of the test cases are given in Table 1.

Sections 3.1 and 3.2 present validation results for local andglobal flow quantities, respectively. Section 3.2.2 gives a viscosity-corrected model for TLV circulation strength. Section 3.2.3 sum-marizes the effects of gap size on incipient vortex cavitation, usingthe gap-flow model and viscosity-corrected TLV strength.

3.1. Local effects of gap-flow

Local effects are defined as the influence of the gap onspanwise-varying quantities, such as circulation, induced down-wash, and sectional lift and drag coefficients.

3.1.1. 2-D circulationFig. 5(a) compares the bound circulation distributions pre-

dicted by the lifting-line analysis with and without the gap-flowmodel (respectively denoted as Mod. LL and Uncorr. LL) to theexperimental results of Sugiyama (1970) for Case A, denoted inthe figure as (Exp.). Two gap-to-thickness ratios are shown(ϵ¼G/T¼0.143 and ϵ¼1.43). The physically-symmetric split-foilarrangement used by Sugiyama (1970) in the experimental setupplaced the tip in a uniform velocity field, so no corrections aremade for a boundary layer near the tip in the lifting-line analysis(i.e. Unj =Uj ¼ 1 in Eq. (21)). For the larger gap size (ϵ¼1.43), bothlifting-line solutions closely match the experimental data. How-ever, as the gap size decreases, the zero tip-circulation constrainton the uncorrected solution causes an under-prediction of circula-tion everywhere along the span. The gap-flow model improvesmarkedly upon the lifting-line prediction, particularly near the tipfor cases with small gap sizes, where lift-retention is mostinfluential. A local increase in sectional lift near the tip is seen inthe experimental data, a consequence of the TLV, which induceslow pressures on the suction side of the foil. This is a non-circulatory component of lift, but is difficult to separate from thecirculatory component. The result is a slight discrepancy at the tipbecause the lifting-line analysis only resolves the circulatorycomponent of lift.

Fig. 5(b) compares circulation distributions predicted by theRANS simulations and the lifting-line analysis (Mod. LL andUncorr. LL) for Case B. A turbulent boundary layer is presenton the wall, so the circulation distribution was corrected withEq. (21), using a 1/7th power-law velocity profile to compute Unj .Again, the discrepancy between the uncorrected lifting-line solu-tion and the CFD results increases with decreasing gap size, whilethe corrected model (Mod. LL) maintains a good qualitativeagreement, with only a slight over-prediction of bound circulationnear the root. As in the preceding figure, the RANS simulationscapture some non-circulatory lift augmentation near the tip.

3.1.2. 2-D downwash velocitiesFig. 6(a) compares the lifting-line predictions of induced down-

wash velocity (v) with the experimental results of Sugiyama(1970) for Case A. The same comparison is made with CFD resultsin Fig. 6(b) for Case B. The uncorrected lifting-line predictionsexhibit excessive downwash velocities at small gap sizes. Con-versely, good agreement outside of the immediate tip-region isshown between the modified lifting-line analysis and the resultsof both experiments (Fig. 6(a)) and CFD simulations (Fig. 6(b)). Eq.(13) is used to extract the downwash from experimental measure-ments and CFD results, so the non-circulatory lift at the tippropagates into an artificial reduction in induced downwash. Thus,the discrepancy near the tip may be attributed to the samephenomenon mentioned in Section 3.1.1.

3.2. Global effects of gap-flow

The gap-flow model's utility as a design tool is more-directlymeasured by predictions of global (integral) quantities such as 3-Dlift and drag (Section 3.2.1), vortex strength (3.2.2), and minimumvortex pressures (3.2.3) as they vary with gap size.

3.2.1. Hydrodynamic forcesThe 3-D lift coefficient (CL) is plotted in Fig. 7 as a function of

the gap-to-thickness ratio (ϵ). Included in the figure are theexperimental data of Sugiyama (1970) at two angles of attack ina uniform flow (Case A), and results of RANS simulations at asingle angle of attack with a turbulent end-wall boundary layer(Case B). Uncorrected and modified lifting-line results are included

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

Z/S

Γ/Γ 2

D

α=6°Rewall=∞

Mod. LL (Present Study), ε=0.143Uncorr. LL (Present Study), ε=0.143Exp [Sugiyama, 1970], ε=0.143Mod. LL, ε=1.43Uncorr. LL, ε=1.43Exp, ε=1.43

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

Z/S

Γ/Γ 2

D

α=8°Rewall=3.29e+006

Mod. LL (Present Study), ε=0.056Uncorr. LL (Present Study), ε=0.056CFD (Present Study), ε=0.056Mod. LL, ε=0.222Uncorr. LL, ε=0.222CFD, ε=0.222Mod LL, ε=0.667Uncorr. LL, ε=0.667CFD, ε=0.667

Fig. 5. Validation of normalized bound circulation ðΓ=Γ2DÞ distributions alongnormalized span (Z/S) for varying gap-to-thickness ratios (ϵ). (a) Comparison withresults of Sugiyama (Case A). (b) Comparison with results of RANS simulations(Case B).




for all cases, with respective boundary-layer corrections (Eq. (21)).The underestimation of circulation by the uncorrected lifting-lineanalysis is brought to bear, with similarly under-predicted valuesof CL for small gap sizes. The gap-flow model improves themodified lifting-line solutions greatly and agrees very well withSugiyama's experimental results (Case A). In comparison with theCFD results (Case B), the modified lifting-line solution also agreeswell, while the uncorrected lifting-line drastically underpredictslift for small non-zero gap sizes. The effect of the smaller aspectratio of Case B is a steeper slope, due to a greater proportion of liftbeing lost to 3-D effects.

The 3-D induced drag coefficients ðCDi Þ are plotted from thesame sources as functions of ϵ in Fig. 8. The overestimation of

sectional downwash by the uncorrected lifting-line approach inFig. 6 leads to an overly-high integral quantity of CDi throughoutthe range shown. On the other hand, the modified lifting-lineresults agree very well with those from Sugiyama's experiments(Case A) and with high-fidelity CFD simulations (Case B). Theslopes of the curves increase as the angle of attack is increased andas the aspect ratio is reduced (Case B), indicating that the gap moststrongly affects heavily-loaded lifting surfaces with low aspect-ratios.

3.2.2. Vortex strengthAs previously mentioned, Tallman and Lakshminarayana (2001)

and Harwood et al. (2012) concluded that TLV development isinfluenced by viscous stresses in the gap. Strong negative helicitydevelops in the thin boundary layer attached to the foil tip, whichis then convected out of the gap, combining with 3-D vortexshedding to form the rotational TLV core and surrounding irrota-tional region. Fig. 9 depicts profile views of the streamlines passingthrough the gap and section views through the gap for Case B only.Projections of the velocity vectors are shown, and the cross-section planes are shaded by the X-component of vorticity, definedas

ωx ¼ ð∇� U�Þ � êx;where êx is a unit vector in the X-direction. Gap sizes of ϵ¼0.0278(G¼0.5 mm) and ϵ¼0.333 (G¼6 mm) are shown in Fig. 9a and b,respectively. The smaller gap size represents a strongly-confinedflow; the larger gap represents a case where confinement is lessdominant, but the effect of the end-wall is still present.

The impact of the larger gap is immediately evident as a larger,more coherent TLV than in the case of the smaller gap size. For therange of gap sizes shown, ωx is highly polarized in the gap, withstrong negative and positive components developing in theboundary layers near the foil tip and end wall, respectively. Thepositive sense of rotation is clockwise in both figures. In the case ofthe larger gap, flow separation occurs at the corner between thepressure surface and the tip, at which point the separated floweither reattaches (for X=Cr0:5) or joins the TLV without reattach-ing to the tip ðX=C40:5Þ. The important observation is thatnegative vorticity is developed inside of the gap by viscousmechanisms, which then contributes to the TLV formation. By itsinviscid nature, the lifting-line approach cannot capture thisphenomenon directly, and neither Eq. (2) nor (7) include a termto represent viscosity-generated vorticity in the TLV circulationstrength. The present authors posit that a correction similar to Eq.(7) may be used to model the viscous contribution. It stands toreason that this component of vorticity is a function of the flow

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.05

0

0.05

0.1

0.15

0.2

Z/S

v/U

∞

α=6°Rewall=∞

Mod. LL (Present Study), ε=0.143Uncorr. LL (Present Study), ε=0.143Exp [Sugiyama, 1970], ε=0.143Mod. LL, ε=1.43Uncorr. LL, ε=1.43Exp, ε=1.43

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.05

0

0.05

0.1

0.15

0.2

Z/S

α=8°Rewall=3.29e+006

Mod. LL (Present Study), ε=0.056Uncorr. LL (Present Study), ε=0.056CFD (Present Study), ε=0.056Mod. LL, ε=0.222Uncorr. LL, ε=0.222CFD, ε=0.222Mod LL, ε=0.667Uncorr. LL, ε=0.667CFD, ε=0.667

v/U

∞

Fig. 6. Validation of normalized downwash velocity ðv=U1Þ distributions alongnormalized span (Z/S) for varying gap-to-thickness ratios (ϵ). (a) Comparison withresults of Sugiyama (Case A). (b) Comparison with results of RANS simulations(Case B).

0 0.5 1 1.50.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

ε, G/T

CL

Mod. LL (Present Study], Case A, α=3°

Uncorr. LL, Case A, α=3°

Exp [Sugiyama, 1970], Case A, α=3°

Mod. LL, Case A, α=6°


Exp, Case A, α=6°

Mod. LL, Case B, α=8°

Uncorr. LL, Case B, α=8°

CFD (Present Study), Case B, α=8°

Rewall=∞

Rewall=3.289e+006

Fig. 7. Variation of 3-D lift coefficient (CL) with gap-to-thickness ratio (ϵ).

0 0.5 1 1.50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

ε, G/T

C Di

Mod. LL (Present Study], Case A, α=3°


Exp [Sugiyama, 1970], Case A, α=3°

Mod. LL, Case A, α=6°


Exp, Case A, α=6°

Mod. LL, Case B, α=8°

Uncorr. LL, Case B, α=8°

CFD (Present Study), Case B, α=8°

Rewall=∞Rewall=3.289e+006

Fig. 8. Variation of 3-D induced drag coefficient ðCDi Þwith gap-to-thicknessratio (ϵ).




velocity through the gap and thus a function of the gap size.Furthermore, the viscous component should vanish for very largegaps, where the effects of confinement disappear and inviscidmechanisms (shedding of bound circulation) dominate vortexformation, and it should vanish at zero gap size, where the flowis 2-D. The authors propose a viscosity-corrected model of theform,

ΓTLVHY ¼ ðΓ1 � ΓNÞþΓ2De�Ψϵϵ; ð23Þwhere Ψ ¼ 2:6 was found to yield a good correlation between theCFD and modified lifting-line results. This model indirectlyaccounts for the effect of aspect ratio by including the root-circulation in the first term, while the viscous correction (secondterm) vanishes for gaps of either zero or infinite size.

Fig. 10 compares the TLV strength predicted by the modifiedlifting-line analysis combined with Eq. (23) to the values obtainedfrom the CFD simulations for Case B only. As described by Oweisand Ceccio (2005), the strength of the leakage vortex varies alongthe streamwise coordinate; it increases in strength as it is fed bythe leakage flow, and decreases again past the foil trailing edge asa result of viscous dissipation. This “feeding” of vorticity isexhibited in Fig. 9(b). The TLV circulation was determined fromthe CFD results for each gap size by integrating only the negativecomponent (CCW component) of ωx over a transverse section ofthe vortex at the streamwise position of the trailing edge. Alsoshown in Fig. 10 are the TLV strengths calculated from the resultsof the modified lifting-line analysis, using Eq. (2) and (7), as well asthe laser-velocimetry measurements of Farrell and Billet (1994).

Unsurprisingly, the modified lifting-line model, in conjunctionwith Eq. (23), shows the shed vortex strength increasing with gapsize, corroborating the numerical results of the present CFDsimulations and those of Tallman and Lakshminarayana (2001).

The exception is for very large gap sizes, where the CFD resultsindicate a reduction in TLV strength. Physically, this may arisebecause the fixed tunnel width dictates a reduction in aspect ratiowith increasing gap size, causing a weaker vortex to be shed. Theviscous correction of Eq. (23) provides a better correlation withCFD results than either Eq. (7) or (2). More scatter is present in theexperimental data of Farrell and Billet (1994) for an axial-flowpump, but the trend is still captured by Eq. (23).

3.2.3. Vortex cavitation inceptionFig. 11 shows the effect of gap size on the predicted tip vortex

cavitation index (si). Eq. (8) is used to predict the minimum vortexpressure coefficient, which is equal to the negative of the incipientcavitation number. It should be noted that Eq. (23) is used tocalculate ΓTLV in Eq. (8). It is immensely expensive to captureincipient vortex cavitation using CFD because of the fine discreti-zations needed on time and space; hence, no RANS results are

Fig. 9. Case B results; streamlines, velocity vectors and vorticity contours at various sections through gap, showing process of TLV roll-up and the viscous contribution tovorticity development. (a) G ¼ 0.5 mm; ϵ ¼ 0.0278. (b) G ¼ 6 mm; ϵ ¼ 0.333.

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

ε, G/T

Γ TLV

/Γ2D

Mod. LL, Eq. 23, Case BMod. LL, Eq. 2, Case BMod. LL, Eq. 7, Case BCFD (Present Study), Case BFarrell and Billet [1994], Experiment (Rotor)

Fig. 10. Variation of normalized tip leakage vortex (TLV) strength ðΓTLV=Γ2DÞ withgap-to-thickness ratio (ϵ).




available for this case. The trend in Fig. 11 shows that tip vortexcavitation incepts earlier (si increases) as the gap-to-thicknessratio (ϵ) increases and as the angle of attack (α) increases. Theeffect of Case B's smaller aspect ratio is to compress the respectivecurve along the ϵ axis, signifying that the low aspect ratio causes astronger dependence upon the gap size. As ϵ increases, all of thecurves reach physically-realistic values. Local minima in thecavitation inception number, which suggest the existence of anoptimal gap size for cavitation delay, appear near values ofϵ� 0:05. Farrell and Billet (1994) noted the existence of suchminima, predicted by Eq. (7) to occur around ϵ¼0.2 for rotors.

4. Conclusions and recommendations

As was mentioned in the introduction, standard potential flowformulations are appropriate for the limiting cases of zero-gap(2-D) and infinite gap (3-D). The results of the unmodified lifting-line analysis corroborate this by trending towards the correctresults at very small and very large gap sizes. For intermediate gapsizes found in most turbomachines and flexible-wing experi-ments, however, classical potential approaches are inadequate.A simple gap-flow model, based upon the experimental findings ofLakshminarayana and Horlock (1962) and Lewis and Yeung (1977),has been applied as a modified boundary condition in a lifting-lineanalysis, with the following results:

� The predicted distributions of bound circulation, downwash,and induced drag have been validated against a combination ofexperimental and high-fidelity CFD results for two distinctgeometries with and without wall boundary layers. Global liftand drag are also validated against both CFD and experimentaldata.

� The lifting-line analysis with a gap-flow model is shown togreatly improve predictions of circulation and downwashvelocity distributions, as well as global lift and drag coefficients,compared to an uncorrected lifting-line analysis. The uncor-rected lifting-line analysis included a large number of imagefoils to account for wall effects, and used the classic zero-loading condition at the wing tip.

� An additional viscosity-correction term was found to improvethe model predictions of the TLV strength when compared tohigh-fidelity CFD data and to previous experimental measure-ments by accounting for the viscous contribution to the TLVformation.

� Realistic values of the incipient TLV cavitation number areyielded by a Rankine vortex model used in concert with theviscosity-corrected TLV model.

� The reduction in circulatory lift or increase in induced-dragcaused by the gap-flow was found to be proportionally more-severe for foils with low aspect ratios or high angles of attack.

Similarly, the incipient TLV cavitation index was found todepend upon aspect ratio, angle of attack, and gap size.

� The computational expense of the RANS simulations was manyorders of magnitude, O(106), greater than that of the modifiedlifting-line analysis.

The proposed physics-based gap-flow correction can be appliedto other potential flow solvers to create efficient and robustphysics-based design and optimization tools for hydrofoils, pro-pellers, turbines, compressors, waterjets, or any other ductedturbomachines. The authors suggest, as further development, thata TLV model, such as the concentrated tip-vortex method (CTVM)of Lewis and Yeung (1977), be added to capture the non-circulatory lift near the tip and to more-accurately predict therolled-up TLV circulation strength. The authors also intend toapply the gap-flow model to a 3-D coupled boundary elementmethod–finite element method (BEM–FEM) presented by Young(2008) to analyze the transient hydroelastic response of ductedturbomachinery.

Acknowledgments

Special acknowledgement is owed to Dr. Antoine Ducoin for hisassistance in preparing the RANS simulations. The authors are alsograteful to the Office of Naval Research (ONR) and Dr. Ki-Han Kim(program manager) for their financial support through Grant no.N00014-09-1-1204. This work was also supported in part by theNational Research Foundation of Korea (NRF) grant funded by theKorean government (MEST) through the GCRC-SOP Grant no.2012-0004783.

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0 0.5 1 1.50

1

2

3

4

5

6

7

ε, G/T

σ i

Case A, α=3°

Case A, α=6°

Case B, α=8°

Fig. 11. Variation of incipient vortex cavitation index with gap-to-thickness ratio.



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A physics-based gap-flow model for potential flow solversIntroductionBackgroundInviscid modelsViscous modelsExperimental studies

Objectives

MethodologyDerivation of a lifting-line modelGoverning equationsPhysical boundary conditionsNumerical solution methodNear-wall treatmentExtension to other potential flow solvers

High-Fidelity RANS simulation

Results and discussionLocal effects of gap-flow2-D circulation2-D downwash velocities

Global effects of gap-flowHydrodynamic forcesVortex strengthVortex cavitation inception

Conclusions and recommendationsAcknowledgmentsReferences

Documents

A physics-based gap-flow model for potential flow solvers · 2016. 11. 1. · Computational Fluid Dynamics (CFD) solvers permit high-ﬁdelity, viscous simulations of gap-ﬂow, but