R EYNOLDS N UMBER BASED B LADE T IP VORTEX MODEL Manikandan Ramasamy ∗ J. Gordon Leishman † Alfred Gessow Rotorcraft Center Department of Aerospace Engineering Glenn L. Martin Institute of Technology University of Maryland College Park, Maryland 20742 Abstract A mathematical model has been developed to estimate the temporal growth properties of helicopter blade tip vortices at any vortex Reynolds number. One unique feature of the model is that it takes into account rotational stratifi- cation (Richardson’s number) effects on the distribution of turbulent viscosity inside the tip vortices. This model is combined with another model for the effects of fila- ment stretching in predicting the temporal evolution of the vortex. A turbulent growth model solves exactly for the tangential (swirl) velocity starting from the Navier– Stokes equations using an assumed variation in eddy vis- cosity across the vortex core. This variation is a function of the local Richardson’s number, and the final solution becomes dependent on vortex Reynolds number. A more parsimonious functional approximation is given to repre- sent the induced velocity distribution in the tip vortices for practical applications. It is shown that the temporal core growth rate predicted by the new model increases with an increase in vortex Reynolds number, which is consistent with experimental observations. The predictions from the model were validated, wherever possible, with tip vortex measurements from both model- and full-scale rotors. Nomenclature a, b Empirical constants c Blade chord, m C 0 Constant C T Coefficient of thrust, = T /ρAΩ 2 R 2 g Core circulation function l Prandtl’s mixing length, m ∗ Research Associate. [email protected]† Minta Martin Professor. [email protected]Presented at the 61 st Annual Forum and Technology Display of the American Helicopter Society International, Grapevine, TX, June 1–3, 2005. c 2005 by M. Ramasamy & J. G. Leishman. Published by the AHS International with permission. N b Number of blades r c Core radius of the vortex, m r Radial distance, m r Non-dimensional radial distance, = r/r c R Radius of the blade, m Re v Vortex Reynolds number, = Γ v /ν Ri Richardson number t Time, s T Rotor thrust, N V 1 Peak swirl velocity, ms −1 V 1new Peak swirl velocity in new model, ms −1 V r Radial velocity of the tip vortex, ms −1 V z Axial velocity of the tip vortex, ms −1 V θ Swirl velocity of the tip vortex, ms −1 α L Lamb’s constant, = 1.25643 α I Iversen’s constant, = 0.01854 α new New empirical constant, = 0.0655 γ Reduced circulation, = rV θ ,m 2 s −1 γ v Reduced circulation at large distances, m 2 s −1 γ Non-dimensional circulation, = γ/γ v Γ Circulation, = 2πrV θ ,m 2 s −1 Γ v Circulation of the vortex at large distances, m 2 s −1 Γ 1 Circulation at the core radius, m 2 s −1 δ Ratio of apparent to actual viscosity ζ Wake age, deg. η Similarity variable, = r 2 /4γ v t η 1 Similarity variable at the core radius,= r 2 c /4γ v t η a Empirical constant η Scaled similarity variable, = η/α new 2 κ Newly developed function, = α new 2 VIF µ Dynamic viscosity, kgm −1 s −1 ν Kinematic viscosity, = µ /ρ,m 2 s −1 ν t Eddy viscosity ν T Total kinematic viscosity, = ν + ν t ρ Density, kg/m 3 σ Shear stress, N/m 2 σ e Effective rotor solidity, = N b c/πR ψ Azimuthal position, deg. Ω Rotational speed of the rotor, rad/s
Transcript
REYNOLDS NUMBER BASED BLADE
TIP VORTEX MODEL
Manikandan Ramasamy∗ J. Gordon Leishman†
Alfred Gessow Rotorcraft CenterDepartment of Aerospace Engineering
Glenn L. Martin Institute of TechnologyUniversity of Maryland
College Park, Maryland 20742
Abstract
A mathematical model has been developed to estimate thetemporal growth properties of helicopter blade tip vorticesat any vortex Reynolds number. One unique feature ofthe model is that it takes into account rotational stratifi-cation (Richardson’s number) effects on the distributionof turbulent viscosity inside the tip vortices. This modelis combined with another model for the effects of fila-ment stretching in predicting the temporal evolution ofthe vortex. A turbulent growth model solves exactly forthe tangential (swirl) velocity starting from the Navier–Stokes equations using an assumed variation in eddy vis-cosity across the vortex core. This variation is a functionof the local Richardson’s number, and the final solutionbecomes dependent on vortex Reynolds number. A moreparsimonious functional approximation is given to repre-sent the induced velocity distribution in the tip vortices forpractical applications. It is shown that the temporal coregrowth rate predicted by the new model increases with anincrease in vortex Reynolds number, which is consistentwith experimental observations. The predictions from themodel were validated, wherever possible, with tip vortexmeasurements from both model- and full-scale rotors.
Nomenclature
a, b Empirical constantsc Blade chord, mC0 ConstantCT Coefficient of thrust, = T/ρAΩ2R2
g Core circulation functionl Prandtl’s mixing length, m
νt Eddy viscosityνT Total kinematic viscosity, = ν+νt
ρ Density, kg/m3
σ Shear stress, N/m2
σe Effective rotor solidity, = Nbc/πRψ Azimuthal position, deg.Ω Rotational speed of the rotor, rad/s
Introduction
Understanding the temporal development of helicopterrotor blade tip vortices has been the subject of intensiveresearch for several decades. The motivation is clear, inthat a more complete understanding of the structure of thetip vortices is essential for accurately predicting the un-steady airloads on helicopter blades. It has been hypothe-sized by many investigators that the evolution of rotor tipvortices, such as the core growth and induced velocity dis-tribution, are directly related to the details of the turbulentflow structures present inside the tip vortex core (Refs. 1–5). A better understanding of these details is critical forhelicopters because the tip vortices generated by one bladecan interact with following blades, resulting in a problemknown as blade-vortex interaction (BVI). This BVI prob-lem results in high unsteady airloads, and is a source ofsignificant noise and vibration levels on helicopter rotors.Because of the continued emphasis on reducing helicopternoise and rotor vibration, higher-fidelity tip vortex mod-els need to be developed to more accurately predict BVI.Eventually, a deeper understanding of how vortices de-velop and the factors that influence their evolution shouldhelp analysts to devise better strategies to alleviate the ad-verse effects associated with vortex induced airloads. Thisgoal, however, is a longer way off.
Most existing vortex models assume either a com-pletely laminar or turbulent interior flow. For example,the classic Lamb–Oseen model (Refs. 6, 7) assumes acompletely laminar interior flow, while Squire (Ref. 1)and Iversen (Ref. 2) assume completely turbulent flow fortheir models. Even though there are measurements to sup-port both laminar and turbulent vortex flow assumptions(Refs. 8, 9), new improvements in measurement instru-mentation have allowed for better clarity into the details ofthe flows inside rotor tip vortices. Flow visualization stud-ies (Refs. 10, 11) and high-resolution flow measurementsperformed on model scale helicopter rotors (Refs. 10, 12,13) have confirmed an original hypothesis made by Tunget al. (Ref. 14) that the tip vortex can be classified intothree distinct flow regions: an inner laminar region free ofall turbulence, a transitional region with eddies of smallscale, and an outer turbulent region with larger eddies.The extent of the three regions, however, depends on sev-eral factors, including the vortex Reynolds number.
Semi-empirical vortex models have been developed inthe past that have recognized such a multi-region vortexstructure, i.e., Tung model (Ref. 14) and Hoffman &Joubert model (Ref. 15). However, all the aforementionedmodels have been limited in their application to helicopterrotor problems because they are not general enough tobe applied for any vortex Reynolds number, i.e., they donot account for scaling issues. Today, the size scales in-volved between full-scale helicopter rotors, laboratory or
wind tunnel models, and rotating-wing micro air vehicles(MAVs) means that several orders of magnitude of differ-ence in the tip vortex Reynolds numbers is involved. Untilthe Reynolds number issues are understood and modelled,airload predictions will remain unreliable.
The overall turbulence present inside the tip vortex hasbeen previously hypothesized to change with the geomet-ric scale of the rotor (Ref. 14). Consequently, this af-fects the core growth, peak swirl velocity, and inducedvelocity distribution of the tip vortex. Bearing in mindthat most vortex models are developed empirically frommeasurements made on sub-scale laboratory size rotors,this for that matter raises many questions about the appli-cability of these models to full-scale rotors or to MAVs.Certainly, full-scale rotor tip vortex measurements havebeen made in the past, but are very few in number andcannot be made under the same controlled conditions thatare possible in the laboratory or the wind tunnel. Thereare no vortex measurements that have been made at MAVscale. Boatwright (Ref. 16) made measurements in thewake of a hovering, full-scale OH-23B rotor using hot-wire anemometry, while Cook (Ref. 17) made hot-wiremeasurements using a full-scale S-58 rotor on a hovertower. The paucity of full-scale rotor measurements hasits roots not only from the substantial financial invest-ments but also from the numerous practical difficulties in-volved in making vortex flow measurements at this scale,including the required spatial and temporal fidelity.
As previously alluded to, the inherent difficulties inmaking full-scale measurements in an already compli-cated rotor flow field has caused rotor analysts to performtip vortex experiments mostly on sub-scale model rotors(e.g., Refs. 11, 12, 18–20). Vortex models that have beendeveloped based on these sub-scale measurements havebeen used to try to explain the persistence of rotor tip vor-tices to the relatively old wakes ages that are observed infull-scale rotor tests, but with limited success. This failurecan be attributed, in part, to the neglect of Reynolds num-ber scaling issues while developing a tip vortex model.
This can be explained using Fig. 1, which includesCook’s full-scale rotor tip vortex measurements (Ref. 17),and measurements by Martin et al. (Ref. 10), Ma-halingam et al. (Ref. 18), Ramasamy & Leishman(Ref. 11), McAlister (Ref. 12), and the 40% full-scaleHART II test measurements (Ref. 21). For a helicopterrotor, the tip vortex Reynolds number is given approxi-mately by the result
Rev =2ΩRc
ν
(CT
σ
)(1)
so that matching tip speed ΩR and blade loading coeffi-cient CT /σ leaves only a linear dependence on geometricrotor scaling. From the results in Fig. 1, it is apparent thatthe vortex Reynolds number of the model-scale experi-
100
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106
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108
0 0.2 0.4 0.6 0.8 1 1.2
Ramasamy & Leishman, 2004
Martin & Leishman, 2003
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Cook, 1972
Mahalingam et al., 1998
HART II test
Ratio rotor radius (Model scale/Full scale)
Model scale
Full scale
Micro-air vehicles
Vor
tex
Rey
nold
s nu
mbe
r, R
ev
ω
ω
Swirl velocity
ΓvSwirl velocity
Filament undergoesviscous diffusion Γv
Figure 1: Comparison of the vortex Reynolds numberfor sub-scale and full-scale rotor measurements.
ments is lower by orders of magnitude when comparedwith the full-scale tests, mainly because of geometric scal-ing issues. In the case of the HART II tests, which are40% of full-scale, the measured circulation of the tip vor-tices seems relatively lower than the expected value (theRev values are closer to 105 than to 106, for reasons thatare not yet clear).
Another important, but most often neglected, issue isthe effect of vortex filament strain on the growth prop-erties of tip vortices. Helicopter rotor vortices developin a highly three-dimensional and nonuniform velocityfield, which causes the vortex filaments to undergo ei-ther a stretching or contraction process as they convectin the flow. Stretching of the vortex filaments with posi-tive velocity gradients can intensify the core vorticity andincrease swirl velocities; contracting the vortices in a neg-ative strain field produces the opposite effect. Therefore,neglecting this strain process can alter significantly BVInoise and rotor vibration predictions. The effects of vor-tex filament strain on the development of rotor tip vorticeshave been shown significant within the context of free-vortex rotor wake predictions (Ref. 22).
A blade tip vortex model combining the effects of dif-fusion and strain in predicting vortex evolution was pro-posed by Ananthan et al. (Ref. 22), and was validatedin an experiment by Ramasamy & Leishman (Ref. 11).A Reynolds number dependent transitional vortex modelthat takes into account the effects of rotational stratifica-tion effects (or Richardson number effects) on the tur-bulence present inside the vortex was also developed byRamasamy et al. (Ref. 13), and was validated using avail-able measurements from various sources. The presentwork involves combining and extending these two vor-tex models to develop a comprehensive vortex model thattakes into account both the filament straining issues andthe rotational stratification (Richardson number) effectson the effective turbulent viscosity within the vortex in-terior. The final model is dependent on vortex Reynoldsnumber.
Core Growth Theory
An important aspect of predicting vortex evolution is pre-dicting the temporal growth rate of the vortex core. It isconvenient to quantify the development of the vortex interms of its core size because the peak swirl velocities areobtained at the core boundary. It is widely accepted thatthe growth of the tip vortex core depends on the natureof the flow inside the tip vortices, i.e., whether it is lam-inar or turbulent and to what extent. For example, a tur-bulent flow state increases mixing and so the transfer ofmomentum across the layers of the vortex. This causes thecore to grow as its vorticity spreads radially away from thecore axis. In a laminar flow, momentum transfer is possi-ble only by molecular diffusion. A schematic explainingthe effect of diffusion on the growth properties of the tipvortices is shown in Fig. 2. The vortex core size increaseswith increasing in time and the core vorticity decreases(but total circulation is conserved).
Lamb (Ref. 6) and Oseen (Ref. 7) assumed the flowinside the vortex to be completely laminar and derivedan exact solution to the one-dimensional, incompressibleNavier–Stokes equations. The core growth with time pre-dicted by the Lamb–Oseen model (Refs. 6, 7) is givenby
rc =√
4αLνt (2)
where αL is Lamb’s constant (αL = 1.25643). This result,however, suggests a growth rate that is substantially lowerthan found with experimental measurements – see Fig. 3.Also, the Lamb–Oseen model approaches a singularity attime t = 0 with infinite kinetic energy, which is not phys-ically realistic.
Squire (Ref. 1) and Bhagwat & Leishman (Ref. 23)modified the Lamb–Oseen model by including an eddyviscosity for turbulence that was present inside the tip vor-
Figure 2: Schematic explaining the physics of a vortexfilament undergoing diffusion.
Region 1: Fully laminarRegion 2: TransitionalRegion 3: Fully turbulent
tex. The modified core growth is given by
rc =√
r20 +4ανδt (3)
where r0 is the initial core radius that removes the sin-gularity at t = 0, and δ is the ratio of apparent to actualviscosity, i.e.,
δ =ν+νt
ν= 1+
νt
ν(4)
where νt is the effective turbulent value of viscosity. Thevalues of r0 and δ (or νt ) must be obtained from mea-surements. Squire assumed that the eddy viscosity, whichresults from turbulence, is a function of the kinematic vis-cosity, however, with a different magnitude. Because theprincipal permanent characteristic of a tip vortex is its cir-culation, Squire assumed that the eddy viscosity was pro-portional to the total vortex circulation, i.e.,
δ = 1+a(γv
ν
)= 1+a1
(Γv
ν
)(5)
where a and a1 are empirical constants and the ratio Γv/νis the vortex Reynolds number.
For very low vortex Reynolds numbers, the value of δapproaches 1, so it reduces to the laminar Lamb–Oseenmodel. Higher values of δ correspond to an increasedlevel of turbulence inside the vortex. This would resultin increased vortex core growth rate, as shown in Fig. 3.It should, however, be noted that δ is not a function of r,which means that the turbulence present inside the tip vor-tex is implied to be independent of radial location. Eventhough, the Squire vortex model differs from the laminarLamb–Oseen model by including (on average) the effectsof turbulence on the core growth properties, the swirl ve-locity distribution predicted by both models are the same.This result is also independent of vortex Reynolds num-ber or other scaling. Yet this is not consistent with mea-surements and this deficiency with the modeling was onemotivation for the present work.
Estimating the value of a1 (and, hence, δ) from Eq. 5largely depends upon the way in which the eddy viscos-ity is assumed to vary radially across the entire tip vor-tex region from its core axis into the outer far-field region.For example, a1 approaches zero for laminar flow assump-tions, such as in Lamb–Oseen model. While Squire as-sumed uniform eddy viscosity, Iversen (Ref. 24) hypoth-esized that the eddy viscosity varies linearly with the ra-dial distance. This different assumption affects the coregrowth. Bhagwat & Leishman (Ref. 23) suggested thatthe average value of a1 lies within the fairly broad rangefrom 0.0004 to 0.00005 based on a summary of all avail-able measurements. However, determining a more exactvalue for a1 is required for applications that must pre-dict accurately rotor airloads, helicopter vibrations, and
Figure 3: Vortex core growth predicted by Squire’smodel (In this case the results that ζ0 ≈ 30).
Figure 4: A representative flow visualization imageof a tip vortex emanating from a rotor blade showingthree distinct flow regions: (1) a fully laminar region,(2) a transitionally turbulent region, and (3) a fully tur-bulent region.
rotor noise. This point has been addressed previously byRamasamy & Leishman (Ref. 13).
There are also observations from both velocity fieldmeasurements (Refs. 10, 13–15) and flow visualization(Refs. 10, 13) suggesting that flow structures in the vor-tex are not consistent with either the fully laminar or fullyturbulent assumption. It is now known that a rotor tip vor-tex is made of three main regions: an inner laminar re-gion that rotates like a solid body, a transitional flow re-gion, and a more fully turbulent outer region. One suchexample is shown in Fig. 4. This means the eddy viscos-ity is nearly zero near the vortex core axis and approachesa value equivalent to the fully turbulent region far awayfrom the center of the tip vortex. Between these regions is
Filament is strainedor "stretched"
ω
Swirl velocity
ωSwirl velocity
Sl
S
l + ∆l
Γv
Γv
a transitional region with eddies of different length scales.Ramasamy & Leishman (Ref. 13) developed a transi-tional vortex model that is a function of vortex Reynoldsnumber based on the premise of this multi-region vortexconcept. The predictions of the model were later validatedwith measurements, with good agreement.
Methodology
A vortex Reynolds number dependent core growth modelfor rotor tip vortices can be developed that takes into ac-count the effects of vortex filament strain and flow rotationeffects on turbulence (or the eddy viscosity) present insidethe tip vortices. This is obtained by combining two ex-perimentally validated individual vortex models (a strainmodel and a transitional flow model).
Vortex Strain Model
A vortex strain model was developed by Ananthan et al.(Ref. 22), which quantifies the effects of vortex filamentstretching on the core properties of rotor tip vortices. Con-servation of fluid mass and momentum were used in de-riving this model. A filament undergoing pure diffusionresults in increased core size per unit length (for the samefilament circulation) with increase in time, as shown pre-viously in Fig. 3. However, the strain model suggests thata tip vortex filament must result in a reduced core size andan increased core vorticity when it is subjected to a pos-itive stretching in a flow with positive velocity gradients,as shown by the schematic in Fig. 5. Conservation of massand momentum leads to an expression for the vortex coregrowth, which is given by
rc =
√r0
2 +4ανδ∫ ζ
ζ0
(1+ ε)−1dζ (6)
where r0 is the initial core radius, ζ is the (wake) age ofthe filament, and ε(ζ) is the instantaneous filament strainas given by
ε =∆ll
(7)
Applying zero strain rate will reduce the strain model(Eq. 6) into a diffusion based Squire-like core growthmodel, as given previously in Eq. 3.
To validate this model, measurements were made in thewake of a hovering rotor in the presence of a ground plane.This resulted in very high velocity gradients being pro-duced to strain the vortices. Flow visualization of the tipvortex developments allowed the strain field to be mea-sured. These high velocity gradients strained the vortexfilaments allowing the core properties of tip vortices to be
Figure 5: Schematic explaining the physics of a vortexfilament undergoing positive filament strain.
measured in a known strain field (Ref. 11). The measure-ments were found to correlate well with the core growthpredicted by the strain model.
Transitional Vortex Model
A transitional rotor tip vortex model was developed byRamasamy & Leishman (Ref. 13) and takes into accountthe swirling flow (rotation) effects on the turbulent struc-ture of tip vortices. This model was developed usingan eddy viscosity function in such a way that the func-tion smoothly and continuously models the eddy viscosityvariation across the vortex from its inner laminar region tothe outer turbulent flow region.
An intermittency function was developed based on aRichardson number concept, which also brings in the ef-fects of swirling flow (rotation) on the turbulence presentinside the vortex boundaries. The Richardson number isdefined as the ratio of turbulence produced or consumedas a result of centrifugal forces to the turbulence producedby shear. Bradshaw (Ref. 25) derived an expression forthe Richardson number based on an analogy between ro-tational and stratified flows. This expression, which waslater modified by Holzapfel (Ref. 26), is given by
Ri =(
2Vθ
r2
∂(Vθr)∂r
)/(r
∂(Vθ/r)∂r
)2
(8)
and involves the velocity gradients in the vortex flow.Cotel & Breidenthal (Ref. 27) and Cotel (Ref. 28)
suggested that the tip vortex will not develop or sustainany turbulence until the local gradient Richardson num-ber, Ri, falls below a critical value (or stratification thresh-old). Based on experiments, Cotel et al. (Ref. 29) deter-mined that the critical value of Richardson number to beRi = Re1/4
v . This would mean that at any radial location
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MeasurementsStratification line
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hard
son
num
ber,
Ri
Non-dimensional distance from core center, r / rc
Stratification line, Ri = ReV
1/4
0
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Intermittency function
Vor
tex
inte
rmitt
ency
func
tion,
VIF
Non-dimensional distance from core center, r / rc
1
2
3
Core radius
Figure 6: Plot of Richardson number with radial coor-dinate for a vortex flow.
the vortex will not be able to develop or sustain any turbu-lence if the local value of Ri stays above the stratification
threshold of Re1/4v .
Figure 6 shows the variation of Ri for the previouslymentioned Squire and Iversen models along with the mea-surements made by Ramasamy & Leishman (Ref. 11). Itis evident that there exists a multi-region vortex structurewith laminar flow until a particular distance from the cen-ter of the vortex where the Richardson number is alwaysabove the threshold value. This is followed by a transitionflow region and then an outer turbulent region on movingfar away from the vortex core axis. This concept is clearlyconsistent with flow visualization (Fig. 4).
Using this Richardson number concept, Ramasamy &Leishman (Ref. 13) developed a generalized eddy viscos-ity function to represent the variation of eddy viscosityacross the tip vortex. The expression for eddy viscosity,which was derived using an analogy based on boundarylayer theory, is given mathematically by
νt = VIF αnew2(
r2 ∂∂r
(Γr2
) )(9)
where αnew is an empirical constant (found empirically).VIF is called the vortex intermittency function, which isdefined by
VIF =12
[1+ erf
(b
[√ηη1
−ηa
])](10)
where b and ηa are empirical constants, η is the similarityvariable and
√η/η1 is equivalent to the ratio (r/rc)2.
The variation of the vortex intermittency function (VIF)with respect to the radial location of tip vortices is shownin Fig. 7. It can be observed that near the center of thevortex the VIF approaches zero. This results in zero eddyviscosity, as assumed by the laminar Lamb–Oseen model
Figure 7: Eddy viscosity intermittency function acrossthe vortex: (1) laminar flow region, (2) transitionalflow region, and (3) fully turbulent region.
based on Eq. 9. Far away from the vortex core axis, thevalue of VIF approaches one, resulting in the value ofeddy viscosity equivalent to the eddy viscosity for a com-pletely turbulent flow.
This expression for eddy viscosity was then incorpo-rated into the momentum equation governing the develop-ment of an axisymmetric vortex flow, i.e.,
∂γ∂t
= r∂∂r
[νT r
∂∂r
(γr2
)]+2νT
∂∂r
(γr2
)(11)
This results in a similarity solution for the circulation dis-tribution that is a function of vortex Reynolds number, i.e.,
− ∂γ∂η
=[
νγ0
1
α2i
+4VIF2 · |X|]
∂2γ
∂η2 +2|X |X
η∂
∂ηVIF
(12)where
X =1γ0
[η
∂γ∂η
− γ
]
The sequence of steps involved in deriving the solutionfor this equation, along with the various assumptions, isgiven in Ref. 13. It can be understood from Eq. 12 thatthe model reduces to the laminar Lamb–Oseen model orto a completely turbulent model for values of the VIF ap-proaching 0 and 1, respectively. Also, for very low vor-tex Reynolds numbers, the model approaches the laminarLamb–Oseen model for any value of the VIF. The threeempirical constants involved in the model were derivedusing vortex flow measurements from various availablesources.
Solving Eq. 12 numerically using a Runge-Kuttascheme showed that the circulation and induced veloc-ity distribution of tip vortices predicted by the transitionalmodel correlated extremely well with experimental mea-surements. Examples are shown in Figs. 8 and 9 in terms
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MeasurementsLamb–Oseen modelIversen's modelTransitional modelTung model
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-dim
ensi
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circ
ulat
ion,
Γ /
Γv
Non-dimensional distance from core center, r / rc
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-dim
ensi
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irl v
eloc
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θ / V
1
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Effe
ctiv
e vi
scos
ity c
oeffi
cien
t, δ
Vortex Reynolds number, Rev
Model scale Full scale
Figure 8: Predicted ratio of circulation to circulationat large distance, Rev = 48,000, (1) laminar region, (2)transitional region and, (3) turbulent region.
Figure 9: Predicted swirl velocity distribution pre-dicted by the transitional vortex model for Rev =48,000, (1) laminar region, (2) transitional region and,(3) turbulent region.
of the circulation distribution and the swirl velocity pro-files, respectively. The agreement of the new model withthe measurements is clearly better than for either a modeldeveloped on laminar flow assumptions or a model basedon fully turbulent flow assumptions.
New Vortex Model
The development of the new vortex model, which is ob-tained by combining the strain and transitional modelspreviously discussed, will be complete by: (i) formulat-ing an expression for δ in Eq. 6 using the temporal growthpredictions from the transitional vortex model, and (ii) de-riving a more convenient algebraic expression to representthe swirl velocity distribution.
The expression for δ based on the transitional vortexmodel is given by
δ =Revα2
new
2παL
[Γv Lamb
V1new
]2
(13)
Figure 10: Variation of δ with vortex Reynolds num-ber based on new core growth model.
where αnew is a “new” empirical constant estimated basedon measurements from various available sources that werelisted in Ref. 13, and V1 is the peak swirl velocity pre-dicted by the transitional model.
The ratio δ predicted by the transitional vortex model isplotted against vortex Reynolds number in Fig. 10. It canbe observed that the Lamb–Oseen model predicts a con-stant core growth independent of Reynolds number (be-cause of the inherent laminar flow assumption), while theIversen model and the transitional model predicts an in-creased growth rate for an increase in vortex Reynoldsnumber. It should, however, be noted that the Iversen coregrowth model shows a much higher core growth rate com-pared with measurements or with the transitional vortexmodel. This is because the Iversen model assumes that theeddy viscosity inside the vortex is fully turbulent from thevortex core axis to the outer potential region. The valueof δ predicted by the transitional model correlates wellwith measurements both in sub-scale experiments (wherethe vortex Reynolds number is lower) as well as full-scaletests (that have vortex Reynolds numbers at least an orderof magnitude higher).
Comparing the value of δ predicted by the transitionalmodel with the value of δ based on the uniform eddy vis-cosity model proposed by Squire (as given in Eq. 5) en-ables the determination of a unique value for the constanta1, which is 6.5× 10−5 – see Fig. 11. With this uniquevalue for a1, substituting the expression for δ from Eq. 5
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a1 = 6x10-5
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ctiv
e vi
scos
ity c
oeffi
cent
, δ
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a1
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-dim
ensi
onal
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irl v
eloc
ity, V
θ /
V1
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All results overlap
here
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irl v
eloc
ity, V
θ /
V1
Non-dimensional distance from core center, r / rc
Transitional modeland
curve fit
Figure 11: Comparison of δ between the transitionalvortex model and Squire’s model to determine the con-stant a1.
into Eq. 6 results in the core growth model
rc =
√r0
2 +4αν(1+a1Rev)∫ ζ
ζ0
(1+ ε)−1dζ (14)
Because this model is a function of vortex Reynolds num-ber, the core growth properties for a tip vortex that devel-ops in time (i.e., with age) through any strain field ε(ζ) atany vortex Reynolds number, Rev, can now be determined.
Swirl Velocity Distribution
The induced velocity profile as a function of the Reynoldsnumber can be obtained from the circulation distributionpredicted by the vortex model, as given by Eq. 12. Thedifficulty in using this model arises from the relativelyinconvenient mathematical form of the final result andthe need for its numerical solution. The computationalcost involved in repeatedly solving a differential equa-tion for the velocity profile is significant within the con-text of routine velocity field evaluations in comprehensiverotor codes. Therefore, a more approximate solution wassought.
Algebraic vortex models have gained increasing popu-larity over the past few years because of their extremelylow computational cost (such as for inclusion within free-vortex wake models) and good fidelity when compared tomeasurements, at least at a single Reynolds number, e.g.,the Vatistas vortex model (Refs. 30, 31). To have a rel-atively parsimonious mathematical function to representthe velocity profile, an expression of the form
Vθ =Γv
2πr
[1−
3
∑n=1
an exp(−bnr)
](15)
has been used to approximate the results given by the vor-tex model in Eq. 12. Here, an and bn are constants that
Figure 12: Numerical prediction of swirl velocity ver-sus expeonential approximation for Rev = 1×102.
Figure 13: Numerical prediction of swirl velocity ver-sus expeonential approximation for Rev = 48×103.
are determined by fitting this expression to the numeri-cally predicted velocity profiles. Because the swirl veloc-ity profile changes with vortex Reynolds number, differentvalues of an, and bn are obtained as Rev varies. The curve-fit was made in such a way that
3
∑n=1
an = 1 (16)
so as to satisfy the boundary condition that the swirl ve-locity is zero at the vortex core axis. Care was also takento make sure that the predicted swirl velocity based on thecurve fit asymptotes to zero at large values of r.
The value of the constants that are obtained for var-ious vortex Reynolds number are given in Table 1. Itcan be observed that at low vortex Reynolds numbersa1 = 1 and an = 0 for n = 1 and b1 = αL = 1.25643and bn = 0 for n = 1. This confirms that the expressionfor estimating the swirl velocity distribution reduces to thelaminar Lamb–Oseen model for low values of Rev.
Figures 12, 13 and 14 show the results for the swirlvelocity that were obtained at three different vortexReynolds numbers. Using this parsimonious mathemat-ical form in representing the induced velocity field of the
Table 1: Values of the coefficients used in the curve fit to the numerical predictions of swirl velocity as a function ofvortex Reynolds number.
Figure 14: Numerical prediction of swirl velocity ver-sus expeonential approximation for Rev = 1×106.
tip vortices is not only computationally inexpensive, butalso takes into account the required change in the form ofVθ with Rev, which will be necessary for accurate aeroa-coustic predictions. Values of the coefficients for interme-diate Reynolds numbers can be found through interpola-tion.
Core Growth
The core growth predicted by the new model for a rangeof vortex Reynolds numbers is shown in Fig. 15. This in-cludes vortex Reynolds numbers that correspond to bothfull- and model-scale rotors. Vortex measurements fromvarious available sources at different vortex Reynoldsnumber are shown. It can be observed that the core sizepredicted by the new model correlates well with the mea-surements that are made at different vortex Reynolds num-bers. An increase in the vortex Reynolds number corre-sponds to an increase in the amount of turbulence presentinside the tip vortices and this, therefore, increases thegrowth rate of the vortex cores.
Figure 15: Core growth predicted by the new vortexmodel at different vortex Reynolds numbers.
It can be observed that for full-scale flight conditionsthe vortex core size grows up to 25% of the blade chordwithin 360 of wake age but only up to 35% by two rotorrevolutions. Because the new vortex model has its basisin the Iversen model, a logarithmic growth rate is found atolder wake ages. Again, it is seen that the model reducesto the laminar Lamb–Oseen model at very low values ofvortex Reynolds number (Rev < 1000). The new modelslightly overpredicts the tip vortex core growth measuredby Cook (Ref. 17) using the full-scale Sikorsky S-58rotor. This difference can, at least in part, be attributed tothe measurement uncertainties involved in making a full-scale measurement in the open air environment; for exam-ple, the problem of vortex wandering must clearly be an
0
0.02
0.04
0.06
0.08
0.1
Rectilinear vortex, Vε = 0
Positive strain, Vε = 0.25
Negative strain, Vε = -0.25
0 360 720
Non
-dim
ensi
onal
cor
e ra
dius
, rc/c
Wake age, ζ (deg.)
0
0.01
0.02
0.03
0.04
0.05
0.06
Rev = 100
Rev = 1 X 104
Rev = 1 X 105
Rev = 1 X 106
0 360 720
Non
-dim
ensi
onal
cor
e ra
dius
, rc/c
Wake age, ζ (deg.)
issue as it is in the laboratory (Refs. 32, 33). The measure-ments of Boatright (Ref. 16) were noted to exhibit sub-stantial scatter, most likely because of this problem andare not shown here. The measurements of Mahalingam etal. (Ref. 18) suggest a somewhat higher core growth rateat young wake ages than that might be expected based onthe vortex Reynolds number. This could be because themeasurements were made using a rotor operating in for-ward flight; as previously alluded to in this case the vortexfilaments experience velocity gradients in the rotor wake,which can strain or squeeze filaments and can affect themeasured growth properties of the tip vortices. The im-portance of vortex filament strain and its interdependenceon diffusion (and vortex Reynolds number) is explained inthe next section.
Combining Diffusion & Stretching Model
The growth of the vortex core, as shown in Eq. 14, de-pends not only on the vortex Reynolds number but also onthe local velocity gradient experienced by the vortex fila-ments as they age in the flow. These local velocity gradi-ents can stretch or squeeze the vortex filaments, changingthe vorticity (but not the net circulation) and so producinga different induced velocity field. This is especially impor-tant in the vortex ring state when the helicopter rotor oper-ates in forward flight, or in ground effect, where velocitygradients are typically higher. Ananthan et al. (Ref. 22)used a free vortex-wake method to show that the magni-tude of the velocity gradients when the rotor operates inforward flight are large enough to have significant effectson growth properties of tip vortices. This suggests thatany vortex measurements that are made in forward flightmust be corrected for the effects of strain if the intent is toisolate the competing mechanisms of stretching and diffu-sion from each other. The ability to do this would seem tobe a prerequisite to validate any kind of blade tip vortexmodel.
Figure 16: Variation of core growth predicted by thenew model for different values of uniform strain ratesat Rev = 5×105.
The importance and sensititivity of vortex filamentstrain on the growth properties of tip vortices can be il-lustrated with reference to the viscous development of arectilinear vortex at a particular vortex Reynolds num-ber. The effects on the filaments were examined as afunction of prescribed strain rate defined by Vε = dε/dt.Figure 16 shows the growth rate predicted by the modelgiven by Eq. 14 for a rectilinear vortex filament operat-ing at Rev = 5 × 105. This is a representative value ofmost model-scale rotor measurements. This figure also in-cludes the core growth predicted by the model for a vortexfilament undergoing a constant filament strain rate, bothpositive and negative. It can be observed that an appli-cation of positive filament strain rate results in a reducedcore growth rate. Generally, these results show that theeffects of stretching counter the effects of diffusion witha notable reduction in the core size. On the other hand, anegative filament strain results in an increased core growthrate, showing that the effects of strain enhances the effectsof diffusion. This would mean that under the action ofcertain velocity gradients the vortex would exhibit an in-creased core growth rate even if the vortex Reynolds num-ber is small.
The core growth predicted by the model at different vor-tex Reynolds numbers when the vortex filament undergoesa constant positive strain rate is shown in Fig. 17. The coregrowth predicted by the model in the absence of strain forthe same type of vortex Reynolds numbers is shown inFig. 18. It can be observed that the core growth changesboth with the sign and magnitude of the strain on the vor-tex filament and also with vortex Reynolds number. Evenif the magnitude of diffusion of the vortex is increasedthrough an increase in turbulence in the core (or an in-crease in vortex Reynolds number), the vortex filament
Figure 17: Variation of core growth predicted by thenew model for a uniform strain rate vε = 0.25 at dif-ferent vortex Reynolds numbers.
0
0.01
0.02
0.03
0.04
0.05
0.06
Rev = 100
Rev = 1 X 104
Rev = 1 X 105
Rev = 1 X 106
0 360 720
Non
-dim
ensi
onal
cor
e ra
dius
, rc/c
Wake age, ζ (deg.)
0
0.05
0.1
0.15
0.2
0.25
0 180 360 540 720 900
Lamb-Oseen modelSquire model, δ = 8Martin & Leishman, 2001Ramasamy & Leishman, 2003New model with measured strain rateNew model with reduced strain rateNew model with increased strain rate
Non
-dim
ensi
onal
cor
e ra
dius
, rc/ c
Wake age, ζ (deg.)
δ = 1 (laminar)
δ = 8
Figure 18: Variation of core growth predicted by thenew model for a rectilinear vortex, vε = 0.0, at differentvortex Reynolds numbers.
might show reduced core growth and increased vorticity(with higher peak swirl velocity values) if the vortex fila-ment undergoes high positive strain. Overall, these resultsshow the highly interdependent nature of filament strainand diffusion (or vortex Reynolds number) on the growthof tip vortices.
For rotors operating near ground, the magnitude of ve-locity gradients will be of the same order of magnitudeas that of forward flight. Strain values measured usinga model scale rotor operating in the presence of a solidboundary (Ref. 13) were applied to the core growth modelgiven by Eq. 14, the results of which are shown in Fig. 19.It can be observed that the model correlates well with themeasurements for a value of δ that represents the coregrowth without the effects of vortex filament strain. Also,the core growth predicted by the new model deviates fromthe simple diffusion based Squire core growth model atolder wake ages where the vortex filament strains were re-
Figure 19: Core growth predicted by the new modelfor measured values of strain rates. Rev = 48,000.
ported to be significant. An increased or decreased valueof positive filament strain corresponds to reduced or in-creased core size, respectively, for the same wake age.This change in core size can significantly alter the wakegeometry and the induced velocities inside the rotor wake.The implications are that this will also affect predictionsof BVI airloads on the rotor and also BVI induced noise.
Conclusions
A vortex model in terms of vortex Reynolds number wassuccessfully developed and validated using available rotortip vortex measurements. The model was developed bycombining two experimentally validated individual vor-tex models that take into account the effects of filamentstretching and rotational flow (Richardson number) effectson the turbulence present inside the tip vortices in predict-ing their temporal evolution. An useful exponential seriesapproximation was also developed to represent the swirlvelocity distribution predicted by the transitional vortexmodel. The following conclusions have been derived fromthe work:
1. Vortex filament strain and diffusion efffects are inter-dependent processes. Tip vortices will show an in-creased growth rate even when the vortex Reynoldsnumber is small if the filament undergoes signifi-cant (negative) strain. Similarly, the tip vortices willexhibit reduced growth rate despite being at higherReynolds numbers if they experience large positivestrain.
2. The core growth rate predicted by the model for arectilinear vortex filament increases with an increasein vortex Reynolds number (for Rev > 1,000). In-creasing the vortex Reynolds number increases theturbulence present inside the tip vortices and, hence,gives an increased core growth rate. At lower Rev,however, the core growth rate reduces to the laminarLamb–Oseen model by being independent of vortexReynolds number but with a finite core radius at timet = 0.
3. An exponential series approximation emulated theproperties of the swirl velocity distribution predictedby the transitional vortex model at a much lowercomputational cost. This provides a useful approachfor incorporating vortex Reynolds number effectsinto the vortex model for use in a variety of aeroa-coustic applications.
4. The effect of strain on growth properties of a tip vor-tex at a given vortex Reynolds number was foundto favor, balance, or counter the effects of diffusion
based on the nature and magnitude of the stretch-ing. Vortex measurements that are made in velocitygradients must be corrected for the effects of vortexfilament strain if the results are to be interpreted incorrect manner for the development of better vortexmodels.
Acknowledgments
This research was supported, in part, by the National Ro-torcraft Technology Center under Grant NCC 2944.
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