2005-ahs-reynolds Number Paper

R EYNOLDS N UMBER BASED B LADE T IP VORTEX M ODEL 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 stratification (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 filament 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 viscosity 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 represent 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 c C0 CT g l

Empirical constants Blade chord, m Constant Coefficient of thrust, = T /ρAΩ2 R2 Core circulation function Prandtl’s mixing length, m

∗ Research

Associate. [email protected] Martin Professor. [email protected] Presented at the 61st Annual Forum and Technology Display of the American Helicopter Society International, Grapevine, TX, c June 1–3, 2005. 2005 by M. Ramasamy & J. G. Leishman. Published by the AHS International with permission. † Minta

Nb rc r r R Rev Ri t T V1 V1new Vr Vz Vθ αL αI αnew γ γv γ Γ Γv Γ1 δ ζ η η1 ηa η κ µ ν νt νT ρ σ σe ψ Ω

Number of blades Core radius of the vortex, m Radial distance, m Non-dimensional radial distance, = r/rc Radius of the blade, m Vortex Reynolds number, = Γv /ν Richardson number Time, s Rotor thrust, N Peak swirl velocity, ms−1 Peak swirl velocity in new model, ms−1 Radial velocity of the tip vortex, ms−1 Axial velocity of the tip vortex, ms−1 Swirl velocity of the tip vortex, ms−1 Lamb’s constant, = 1.25643 Iversen’s constant, = 0.01854 New empirical constant, = 0.0655 Reduced circulation, = rVθ , m2 s−1 Reduced circulation at large distances, m2 s−1 Non-dimensional circulation, = γ/γv Circulation, = 2πrVθ , m2 s−1 Circulation of the vortex at large distances, m2 s−1 Circulation at the core radius, m2 s−1 Ratio of apparent to actual viscosity Wake age, deg. Similarity variable, = r2 /4γvt Similarity variable at the core radius,= rc2 /4γvt Empirical constant Scaled similarity variable, = η/αnew 2 Newly developed function, = αnew 2 VIF Dynamic viscosity, kgm−1 s−1 Kinematic viscosity, = µ/ρ, m2 s−1 Eddy viscosity Total kinematic viscosity, = ν + νt Density, kg/m3 Shear stress, N/m2 Effective rotor solidity, = Nb c/πR Azimuthal position, deg. Rotational speed of the rotor, rad/s

Introduction Understanding the temporal development of helicopter rotor blade tip vortices has been the subject of intensive research for several decades. The motivation is clear, in that a more complete understanding of the structure of the tip vortices is essential for accurately predicting the unsteady airloads on helicopter blades. It has been hypothesized by many investigators that the evolution of rotor tip vortices, such as the core growth and induced velocity distribution, are directly related to the details of the turbulent flow structures present inside the tip vortex core (Refs. 1– 5). A better understanding of these details is critical for helicopters because the tip vortices generated by one blade can interact with following blades, resulting in a problem known as blade-vortex interaction (BVI). This BVI problem results in high unsteady airloads, and is a source of significant noise and vibration levels on helicopter rotors. Because of the continued emphasis on reducing helicopter noise and rotor vibration, higher-fidelity tip vortex models need to be developed to more accurately predict BVI. Eventually, a deeper understanding of how vortices develop and the factors that influence their evolution should help analysts to devise better strategies to alleviate the adverse effects associated with vortex induced airloads. This goal, however, is a longer way off. Most existing vortex models assume either a completely laminar or turbulent interior flow. For example, the classic Lamb–Oseen model (Refs. 6, 7) assumes a completely laminar interior flow, while Squire (Ref. 1) and Iversen (Ref. 2) assume completely turbulent flow for their models. Even though there are measurements to support both laminar and turbulent vortex flow assumptions (Refs. 8, 9), new improvements in measurement instrumentation have allowed for better clarity into the details of the flows inside rotor tip vortices. Flow visualization studies (Refs. 10, 11) and high-resolution flow measurements performed on model scale helicopter rotors (Refs. 10, 12, 13) have confirmed an original hypothesis made by Tung et al. (Ref. 14) that the tip vortex can be classified into three distinct flow regions: an inner laminar region free of all turbulence, a transitional region with eddies of small scale, and an outer turbulent region with larger eddies. The extent of the three regions, however, depends on several factors, including the vortex Reynolds number. Semi-empirical vortex models have been developed in the past that have recognized such a multi-region vortex structure, i.e., Tung model (Ref. 14) and Hoffman & Joubert model (Ref. 15). However, all the aforementioned models have been limited in their application to helicopter rotor problems because they are not general enough to be applied for any vortex Reynolds number, i.e., they do not account for scaling issues. Today, the size scales involved between full-scale helicopter rotors, laboratory or

wind tunnel models, and rotating-wing micro air vehicles (MAVs) means that several orders of magnitude of difference in the tip vortex Reynolds numbers is involved. Until the Reynolds number issues are understood and modelled, airload predictions will remain unreliable. The overall turbulence present inside the tip vortex has been previously hypothesized to change with the geometric scale of the rotor (Ref. 14). Consequently, this affects the core growth, peak swirl velocity, and induced velocity distribution of the tip vortex. Bearing in mind that most vortex models are developed empirically from measurements made on sub-scale laboratory size rotors, this for that matter raises many questions about the applicability of these models to full-scale rotors or to MAVs. Certainly, full-scale rotor tip vortex measurements have been made in the past, but are very few in number and cannot be made under the same controlled conditions that are possible in the laboratory or the wind tunnel. There are no vortex measurements that have been made at MAV scale. Boatwright (Ref. 16) made measurements in the wake of a hovering, full-scale OH-23B rotor using hotwire anemometry, while Cook (Ref. 17) made hot-wire measurements using a full-scale S-58 rotor on a hover tower. The paucity of full-scale rotor measurements has its roots not only from the substantial financial investments but also from the numerous practical difficulties involved in making vortex flow measurements at this scale, including the required spatial and temporal fidelity. As previously alluded to, the inherent difficulties in making full-scale measurements in an already complicated rotor flow field has caused rotor analysts to perform tip vortex experiments mostly on sub-scale model rotors (e.g., Refs. 11, 12, 18–20). Vortex models that have been developed based on these sub-scale measurements have been used to try to explain the persistence of rotor tip vortices to the relatively old wakes ages that are observed in full-scale rotor tests, but with limited success. This failure can be attributed, in part, to the neglect of Reynolds number scaling issues while developing a tip vortex model. This can be explained using Fig. 1, which includes Cook’s full-scale rotor tip vortex measurements (Ref. 17), and measurements by Martin et al. (Ref. 10), Mahalingam et al. (Ref. 18), Ramasamy & Leishman (Ref. 11), McAlister (Ref. 12), and the 40% full-scale HART II test measurements (Ref. 21). For a helicopter rotor, the tip vortex Reynolds number is given approximately by the result
 2ΩRc CT Rev = (1) ν σ so that matching tip speed ΩR and blade loading coefficient CT /σ leaves only a linear dependence on geometric rotor scaling. From the results in Fig. 1, it is apparent that the vortex Reynolds number of the model-scale experi-

10

8

10

7

10

6

10

5

10

4

v

Vortex Reynolds number, Re

Core Growth Theory

Full scale

Model scale Ramasamy & Leishman, 2004 Martin & Leishman, 2003 McAlister, 2003 Cook, 1972

Micro-air vehicles

1000

Mahalingam et al., 1998 HART II test

100 0

0.2

0.4

0.6

0.8

1

1.2

Ratio rotor radius (Model scale/Full scale)

Figure 1: Comparison of the vortex Reynolds number for sub-scale and full-scale rotor measurements. ments is lower by orders of magnitude when compared with the full-scale tests, mainly because of geometric scaling issues. In the case of the HART II tests, which are 40% of full-scale, the measured circulation of the tip vortices seems relatively lower than the expected value (the Rev values are closer to 105 than to 106 , for reasons that are not yet clear). Another important, but most often neglected, issue is the effect of vortex filament strain on the growth properties of tip vortices. Helicopter rotor vortices develop in a highly three-dimensional and nonuniform velocity field, which causes the vortex filaments to undergo either a stretching or contraction process as they convect in the flow. Stretching of the vortex filaments with positive velocity gradients can intensify the core vorticity and increase swirl velocities; contracting the vortices in a negative strain field produces the opposite effect. Therefore, neglecting this strain process can alter significantly BVI noise and rotor vibration predictions. The effects of vortex filament strain on the development of rotor tip vortices have been shown significant within the context of freevortex rotor wake predictions (Ref. 22). A blade tip vortex model combining the effects of diffusion and strain in predicting vortex evolution was proposed by Ananthan et al. (Ref. 22), and was validated in an experiment by Ramasamy & Leishman (Ref. 11). A Reynolds number dependent transitional vortex model that takes into account the effects of rotational stratification effects (or Richardson number effects) on the turbulence present inside the vortex was also developed by Ramasamy et al. (Ref. 13), and was validated using available measurements from various sources. The present work involves combining and extending these two vortex models to develop a comprehensive vortex model that takes into account both the filament straining issues and the rotational stratification (Richardson number) effects on the effective turbulent viscosity within the vortex interior. The final model is dependent on vortex Reynolds number.

An important aspect of predicting vortex evolution is predicting the temporal growth rate of the vortex core. It is convenient to quantify the development of the vortex in terms of its core size because the peak swirl velocities are obtained at the core boundary. It is widely accepted that the growth of the tip vortex core depends on the nature of the flow inside the tip vortices, i.e., whether it is laminar or turbulent and to what extent. For example, a turbulent flow state increases mixing and so the transfer of momentum across the layers of the vortex. This causes the core to grow as its vorticity spreads radially away from the core axis. In a laminar flow, momentum transfer is possible only by molecular diffusion. A schematic explaining the effect of diffusion on the growth properties of the tip vortices is shown in Fig. 2. The vortex core size increases with increasing in time and the core vorticity decreases (but total circulation is conserved). Lamb (Ref. 6) and Oseen (Ref. 7) assumed the flow inside the vortex to be completely laminar and derived an exact solution to the one-dimensional, incompressible Navier–Stokes equations. The core growth with time predicted by the Lamb–Oseen model (Refs. 6, 7) is given by  rc = 4αL νt (2) where αL is Lamb’s constant (αL = 1.25643). This result, however, suggests a growth rate that is substantially lower than found with experimental measurements – see Fig. 3. Also, the Lamb–Oseen model approaches a singularity at time t = 0 with infinite kinetic energy, which is not physically realistic. Squire (Ref. 1) and Bhagwat & Leishman (Ref. 23) modified the Lamb–Oseen model by including an eddy viscosity for turbulence that was present inside the tip vorSwirl velocity

ω

Γv

Filament undergoes
viscous diffusion

Γv

ω Swirl velocity

Figure 2: Schematic explaining the physics of a vortex filament undergoing diffusion.

(3)

where r0 is the initial core radius that removes the singularity at t = 0, and δ is the ratio of apparent to actual viscosity, i.e., δ=

ν + νt νt = 1+ ν ν

(4)

where νt is the effective turbulent value of viscosity. The values of r0 and δ (or νt ) must be obtained from measurements. Squire assumed that the eddy viscosity, which results from turbulence, is a function of the kinematic viscosity, however, with a different magnitude. Because the principal permanent characteristic of a tip vortex is its circulation, Squire assumed that the eddy viscosity was proportional to the total vortex circulation, i.e.,
 γ  Γv v δ = 1+a = 1 + a1 (5) ν ν where a and a1 are empirical constants and the ratio Γv /ν is the vortex Reynolds number. For very low vortex Reynolds numbers, the value of δ approaches 1, so it reduces to the laminar Lamb–Oseen model. Higher values of δ correspond to an increased level of turbulence inside the vortex. This would result in 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 vortex is implied to be independent of radial location. Even though, the Squire vortex model differs from the laminar Lamb–Oseen model by including (on average) the effects of turbulence on the core growth properties, the swirl velocity distribution predicted by both models are the same. This result is also independent of vortex Reynolds number or other scaling. Yet this is not consistent with measurements and this deficiency with the modeling was one motivation for the present work. Estimating the value of a1 (and, hence, δ) from Eq. 5 largely depends upon the way in which the eddy viscosity is assumed to vary radially across the entire tip vortex region from its core axis into the outer far-field region. For example, a1 approaches zero for laminar flow assumptions, such as in Lamb–Oseen model. While Squire assumed uniform eddy viscosity, Iversen (Ref. 24) hypothesized that the eddy viscosity varies linearly with the radial distance. This different assumption affects the core growth. Bhagwat & Leishman (Ref. 23) suggested that the average value of a1 lies within the fairly broad range from 0.0004 to 0.00005 based on a summary of all available measurements. However, determining a more exact value for a1 is required for applications that must predict accurately rotor airloads, helicopter vibrations, and

McAlister, 2003 Martin et al., 2001 Lamb–Oseen model Squire model, δ = 2 Squire model, δ = 8 Squire model, δ = 16 Ramasamy & Leishman, 2004 Cook, 1972

0.25 Non-dimensional core radius, rc/ c

tex. The modified core growth is given by  rc = r02 + 4ανδt

0.2

δ = 16

δ=8

0.15 0.1

δ=2 δ = 1 (Lamb-Oseen)

0.05 0 0

180

360 540 720 Wake age, ζ (deg)

900

Figure 3: Vortex core growth predicted by Squire’s model (In this case the results that ζ0 ≈ 30◦ ).

1 2 3

Region 1: Fully laminar
Region 2: Transitional
Region 3: Fully turbulent

Figure 4: A representative flow visualization image of a tip vortex emanating from a rotor blade showing three distinct flow regions: (1) a fully laminar region, (2) a transitionally turbulent region, and (3) a fully turbulent region. rotor noise. This point has been addressed previously by Ramasamy & Leishman (Ref. 13). There are also observations from both velocity field measurements (Refs. 10, 13–15) and flow visualization (Refs. 10, 13) suggesting that flow structures in the vortex are not consistent with either the fully laminar or fully turbulent assumption. It is now known that a rotor tip vortex is made of three main regions: an inner laminar region that rotates like a solid body, a transitional flow region, and a more fully turbulent outer region. One such example is shown in Fig. 4. This means the eddy viscosity is nearly zero near the vortex core axis and approaches a value equivalent to the fully turbulent region far away from the center of the tip vortex. Between these regions is

a transitional region with eddies of different length scales. Ramasamy & Leishman (Ref. 13) developed a transitional vortex model that is a function of vortex Reynolds number based on the premise of this multi-region vortex concept. The predictions of the model were later validated with measurements, with good agreement.

A vortex Reynolds number dependent core growth model for rotor tip vortices can be developed that takes into account the effects of vortex filament strain and flow rotation effects on turbulence (or the eddy viscosity) present inside the tip vortices. This is obtained by combining two experimentally validated individual vortex models (a strain model 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 filament stretching on the core properties of rotor tip vortices. Conservation of fluid mass and momentum were used in deriving this model. A filament undergoing pure diffusion results in increased core size per unit length (for the same filament circulation) with increase in time, as shown previously in Fig. 3. However, the strain model suggests that a tip vortex filament must result in a reduced core size and an increased core vorticity when it is subjected to a positive stretching in a flow with positive velocity gradients, as shown by the schematic in Fig. 5. Conservation of mass and momentum leads to an expression for the vortex core growth, which is given by rc =

r0 2 + 4ανδ

 ζ ζ0

(1 + ε)−1 dζ

Swirl velocity

ω

l S

Γv

Filament is strained or "stretched"

Methodology



Γv

(6)

where r0 is the initial core radius, ζ is the (wake) age of the filament, and ε(ζ) is the instantaneous filament strain as given by ∆l ε= (7) l Applying zero strain rate will reduce the strain model (Eq. 6) into a diffusion based Squire-like core growth model, as given previously in Eq. 3. To validate this model, measurements were made in the wake of a hovering rotor in the presence of a ground plane. This resulted in very high velocity gradients being produced to strain the vortices. Flow visualization of the tip vortex developments allowed the strain field to be measured. These high velocity gradients strained the vortex filaments allowing the core properties of tip vortices to be

l + ∆l S

ω

Swirl velocity

Figure 5: Schematic explaining the physics of a vortex filament undergoing positive filament strain. measured in a known strain field (Ref. 11). The measurements were found to correlate well with the core growth predicted by the strain model.

Transitional Vortex Model A transitional rotor tip vortex model was developed by Ramasamy & Leishman (Ref. 13) and takes into account the swirling flow (rotation) effects on the turbulent structure of tip vortices. This model was developed using an eddy viscosity function in such a way that the function smoothly and continuously models the eddy viscosity variation across the vortex from its inner laminar region to the outer turbulent flow region. An intermittency function was developed based on a Richardson number concept, which also brings in the effects of swirling flow (rotation) on the turbulence present inside the vortex boundaries. The Richardson number is defined as the ratio of turbulence produced or consumed as a result of centrifugal forces to the turbulence produced by shear. Bradshaw (Ref. 25) derived an expression for the Richardson number based on an analogy between rotational and stratified flows. This expression, which was later modified by Holzapfel (Ref. 26), is given by
Ri =

2Vθ ∂(Vθ r) r2 ∂r




∂ (Vθ /r) 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 sustain any turbulence until the local gradient Richardson number, Ri, falls below a critical value (or stratification threshold). Based on experiments, Cotel et al. (Ref. 29) determined that the critical value of Richardson number to be 1/4 Ri = Rev . This would mean that at any radial location

10 6

Richardson number, Ri

10

Lamb–Oseen model Iversen's model

4

Measurements Stratification line

1000 100 10

Stratification line, Ri = Re

1/4

V

1 0.1 0.01 0

0.5 1 1.5 2 2.5 Non-dimensional distance from core center, r / r c

Figure 6: Plot of Richardson number with radial coordinate for a vortex flow. the vortex will not be able to develop or sustain any turbulence if the local value of Ri stays above the stratification 1/4 threshold of Rev . Figure 6 shows the variation of Ri for the previously mentioned Squire and Iversen models along with the measurements made by Ramasamy & Leishman (Ref. 11). It is evident that there exists a multi-region vortex structure with laminar flow until a particular distance from the center of the vortex where the Richardson number is always above the threshold value. This is followed by a transition flow region and then an outer turbulent region on moving far away from the vortex core axis. This concept is clearly consistent with flow visualization (Fig. 4). Using this Richardson number concept, Ramasamy & Leishman (Ref. 13) developed a generalized eddy viscosity function to represent the variation of eddy viscosity across the tip vortex. The expression for eddy viscosity, which was derived using an analogy based on boundary layer theory, is given mathematically by

 ∂ Γ νt = VIF αnew 2 r2 (9) ∂r r2 where αnew is an empirical constant (found empirically). VIF is called the vortex intermittency function, which is defined by

  1 η VIF = (10) − ηa 1 + erf b 2 η1 where b and η a are empirical constants, η is the similarity variable 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 shown in Fig. 7. It can be observed that near the center of the vortex the VIF approaches zero. This results in zero eddy viscosity, as assumed by the laminar Lamb–Oseen model

Vortex intermittency function, VIF

1

10 5

3

0.8 Intermittency function

0.6 2 0.4 Core radius

0.2 1 0 1 2 Non-dimensional distance from core center, r / rc

3

Figure 7: Eddy viscosity intermittency function across the vortex: (1) laminar flow region, (2) transitional flow region, and (3) fully turbulent region. based on Eq. 9. Far away from the vortex core axis, the value of VIF approaches one, resulting in the value of eddy viscosity equivalent to the eddy viscosity for a completely turbulent flow. This expression for eddy viscosity was then incorporated into the momentum equation governing the development of an axisymmetric vortex flow, i.e., 

 ∂ ∂ γ ∂ γ ∂γ + 2ν (11) =r νT r T ∂t ∂r ∂r r2 ∂r r2 This results in a similarity solution for the circulation distribution that is a function of vortex Reynolds number, i.e.,  2  ∂γ ∂ γ 2|X|X ∂ ν 1 2 − + 4VIF · |X| + = VIF ∂η γ0 α2i η ∂η ∂η2 (12) where   ∂γ 1 η −γ X= γ0 ∂η The sequence of steps involved in deriving the solution for this equation, along with the various assumptions, is given in Ref. 13. It can be understood from Eq. 12 that the model reduces to the laminar Lamb–Oseen model or to a completely turbulent model for values of the VIF approaching 0 and 1, respectively. Also, for very low vortex Reynolds numbers, the model approaches the laminar Lamb–Oseen model for any value of the VIF. The three empirical constants involved in the model were derived using vortex flow measurements from various available sources. Solving Eq. 12 numerically using a Runge-Kutta scheme showed that the circulation and induced velocity distribution of tip vortices predicted by the transitional model correlated extremely well with experimental measurements. Examples are shown in Figs. 8 and 9 in terms

1

3

2

1 0.8 0.6 Measurements Lamb–Oseen model Iversen's model Transitional model Tung model

0.4 0.2

4

0 0

2 4 6 8 10 Non-dimensional distance from core center, r / r

12

c

Non-dimensional swirl velocity, Vθ / V1

Figure 8: Predicted ratio of circulation to circulation at large distance, Rev = 48, 000, (1) laminar region, (2) transitional region and, (3) turbulent region. 1.2 1

10

1000 100 10 1 0.1 1000

Measurements Transitional model Lamb–Oseen model

0.8

Effective viscosity coefficient, δ

Non-dimensional circulation, Γ / Γ v

1.2

Lamb–Oseen model Iversen's model Transitional model Ramasamy & Leishman, 2004 Martin & Leishman, 2000 McAlister, 2003 McAlister, 1996 Cook, 1972 Cliffone & Orloff, 1975 Jacob et al., 1996 Mahalingam & Komerath, 1998 Bhagwat & Leishman, 1998 Govindaraju & Saffman, 1971 Jacob et al., 1995 Kraft, 1955 McCormick, Tangler & Sherrib, 1963 Rose & Dee, 1963 Corsiglia et al., 1973 Baker et al., 1974 Dosanjh et al., 1964

Model scale 4

Full scale 5

6

10 10 10 Vortex Reynolds number, Re

7

10

v

Figure 10: Variation of δ with vortex Reynolds number based on new core growth model.

Iversen's model

0.6 0.4 0.2 1

2

3

0 0

1 2 3 4 5 6 7 Non-dimensional distance from core center, r / r

8 c

Figure 9: Predicted swirl velocity distribution predicted 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 profiles, respectively. The agreement of the new model with the measurements is clearly better than for either a model developed on laminar flow assumptions or a model based on fully turbulent flow assumptions.

New Vortex Model The development of the new vortex model, which is obtained by combining the strain and transitional models previously discussed, will be complete by: (i) formulating an expression for δ in Eq. 6 using the temporal growth predictions from the transitional vortex model, and (ii) deriving a more convenient algebraic expression to represent the swirl velocity distribution. The expression for δ based on the transitional vortex model is given by   Rev α2new Γv Lamb 2 δ= (13) 2παL V1new

where αnew is a “new” empirical constant estimated based on measurements from various available sources that were listed in Ref. 13, and V1 is the peak swirl velocity predicted by the transitional model. The ratio δ predicted by the transitional vortex model is plotted against vortex Reynolds number in Fig. 10. It can be observed that the Lamb–Oseen model predicts a constant core growth independent of Reynolds number (because of the inherent laminar flow assumption), while the Iversen model and the transitional model predicts an increased growth rate for an increase in vortex Reynolds number. It should, however, be noted that the Iversen core growth model shows a much higher core growth rate compared with measurements or with the transitional vortex model. This is because the Iversen model assumes that the eddy viscosity inside the vortex is fully turbulent from the vortex core axis to the outer potential region. The value of δ predicted by the transitional model correlates well with measurements both in sub-scale experiments (where the vortex Reynolds number is lower) as well as full-scale tests (that have vortex Reynolds numbers at least an order of magnitude higher). Comparing the value of δ predicted by the transitional model with the value of δ based on the uniform eddy viscosity model proposed by Squire (as given in Eq. 5) enables the determination of a unique value for the constant a1 , which is 6.5 × 10−5 – see Fig. 11. With this unique value for a1 , substituting the expression for δ from Eq. 5

5

1

4

10

Non-dimensional swirl velocity, V θ / V

Effective viscosity coefficent, δ

10

New model a = 6x10

-5

1

1000

Lamb–Oseen model

a1 100 10 1 0.1 1

4

6

100 10 10 Vortex Reynolds number, Re

8

10

1.2 1 0.8

All results overlap here

0.6 0.4

Transitional Model Lamb–Oseen model Curve fit

0.2 0 0

v

into Eq. 6 results in the core growth model  rc =

r0 2 + 4αν(1 + a1 Rev )

 ζ ζ0

(1 + ε)−1 dζ

(14)

Because this model is a function of vortex Reynolds number, the core growth properties for a tip vortex that develops in time (i.e., with age) through any strain field ε(ζ) at any vortex Reynolds number, Rev , can now be determined.

Swirl Velocity Distribution The induced velocity profile as a function of the Reynolds number can be obtained from the circulation distribution predicted by the vortex model, as given by Eq. 12. The difficulty in using this model arises from the relatively inconvenient mathematical form of the final result and the need for its numerical solution. The computational cost involved in repeatedly solving a differential equation for the velocity profile is significant within the context of routine velocity field evaluations in comprehensive rotor codes. Therefore, a more approximate solution was sought. Algebraic vortex models have gained increasing popularity over the past few years because of their extremely low computational cost (such as for inclusion within freevortex wake models) and good fidelity when compared to measurements, at least at a single Reynolds number, e.g., the Vatistas vortex model (Refs. 30, 31). To have a relatively parsimonious mathematical function to represent the velocity profile, an expression of the form

3 Γv Vθ = (15) 1 − ∑ an exp(−bn r) 2πr n=1 has been used to approximate the results given by the vortex model in Eq. 12. Here, an and bn are constants that

4 c

1

Figure 12: Numerical prediction of swirl velocity versus expeonential approximation for Rev = 1 × 102 . Non-dimensional swirl velocity, V θ / V

Figure 11: Comparison of δ between the transitional vortex model and Squire’s model to determine the constant a1 .

0.5 1 1.5 2 2.5 3 3.5 Non-dimensional distance from core center, r / r

1.2 1 Transitional model and curve fit

0.8 0.6 0.4

Transitional model Lamb–Oseen model

0.2

Curve fit

0 0

0.5 1 1.5 2 2.5 3 3.5 Non-dimensional distance from core center, r / r

4 c

Figure 13: Numerical prediction of swirl velocity versus expeonential approximation for Rev = 48 × 103 . are determined by fitting this expression to the numerically predicted velocity profiles. Because the swirl velocity profile changes with vortex Reynolds number, different values of an , and bn are obtained as Rev varies. The curvefit was made in such a way that 3

∑ an = 1

(16)

n=1

so as to satisfy the boundary condition that the swirl velocity is zero at the vortex core axis. Care was also taken to make sure that the predicted swirl velocity based on the curve fit asymptotes to zero at large values of r. The value of the constants that are obtained for various vortex Reynolds number are given in Table 1. It can be observed that at low vortex Reynolds numbers a1 = 1 and an = 0 for n = 1 and b1 = αL = 1.25643 and bn = 0 for n = 1. This confirms that the expression for estimating the swirl velocity distribution reduces to the laminar Lamb–Oseen model for low values of Rev . Figures 12, 13 and 14 show the results for the swirl velocity that were obtained at three different vortex Reynolds numbers. Using this parsimonious mathematical form in representing the induced velocity field of the

Rev 1 100 1000 10, 000 2.5 × 104 4.8 × 104 7.5 × 104 1 × 105 2.5 × 105 5.0 × 105 7.5 × 105 1 × 106

a1 1.0000 1.0000 1.0000 0.8247 0.5933 0.4602 0.3574 0.3021 0.1838 0.1386 0.1011 0.0792

b1 1.256 1.2515 1.2328 1.2073 1.3480 1.3660 1.3995 1.4219 1.4563 1.4285 1.4462 1.4716

a2 0.0000 0.0000 0.0000 0.1753 0.2678 0.3800 0.4840 0.5448 0.6854 0.7432 0.7995 0.8352

b2 0.0000 0.0000 0.0000 0.0263 0.01870 0.01380 0.01300 0.0122 0.0083 0.0058 0.0048 0.0042

b3 0.0000 0.0000 0.0000 0.0000 0.2070 0.1674 0.1636 0.1624 0.1412 0.1144 0.1078 0.1077

th

1/7 scale model, Re = 8X10

1.2

4

v

th

1/15 scale model, Re = 1.7X10

5

v

1

Lamb-Oseen model 5

S-58 Full-scale, Re = 8 X 10 v

0.8

Ramasamy & Leishman, 2004 McAlister, 1996 McAlister, 2003 Cook, 1972 Mahalingam et al., 1998

0.6 0.4

Transitional model Lamb–Oseen model Curve fit

0.2 0 0

0.5 1 1.5 2 2.5 3 3.5 Non-dimensional distance from core center, r / r

4 c

Figure 14: Numerical prediction of swirl velocity versus expeonential approximation for Rev = 1 × 106 . tip vortices is not only computationally inexpensive, but also takes into account the required change in the form of Vθ with Rev , which will be necessary for accurate aeroacoustic predictions. Values of the coefficients for intermediate Reynolds numbers can be found through interpolation.

Core Growth The core growth predicted by the new model for a range of vortex Reynolds numbers is shown in Fig. 15. This includes vortex Reynolds numbers that correspond to both full- and model-scale rotors. Vortex measurements from various available sources at different vortex Reynolds number are shown. It can be observed that the core size predicted by the new model correlates well with the measurements that are made at different vortex Reynolds numbers. An increase in the vortex Reynolds number corresponds to an increase in the amount of turbulence present inside the tip vortices and this, therefore, increases the growth rate of the vortex cores.

Non-dimensional core radius, rc / c

Non-dimensional swirl velocity, V θ / V

1

Table 1: Values of the coefficients used in the curve fit to the numerical predictions of swirl velocity as a function of vortex Reynolds number.

0.4

0.3

0.2

0.1

0 0

180

360 540 Wake age, ζ (deg)

720

Figure 15: Core growth predicted by the new vortex model at different vortex Reynolds numbers. It can be observed that for full-scale flight conditions the vortex core size grows up to 25% of the blade chord within 360◦ of wake age but only up to 35% by two rotor revolutions. Because the new vortex model has its basis in the Iversen model, a logarithmic growth rate is found at older wake ages. Again, it is seen that the model reduces to the laminar Lamb–Oseen model at very low values of vortex Reynolds number (Rev < 1000). The new model slightly overpredicts the tip vortex core growth measured by Cook (Ref. 17) using the full-scale Sikorsky S-58 rotor. This difference can, at least in part, be attributed to the measurement uncertainties involved in making a fullscale measurement in the open air environment; for example, the problem of vortex wandering must clearly be an

Combining Diffusion & Stretching Model The growth of the vortex core, as shown in Eq. 14, depends not only on the vortex Reynolds number but also on the local velocity gradient experienced by the vortex filaments as they age in the flow. These local velocity gradients can stretch or squeeze the vortex filaments, changing the vorticity (but not the net circulation) and so producing a different induced velocity field. This is especially important in the vortex ring state when the helicopter rotor operates in forward flight, or in ground effect, where velocity gradients are typically higher. Ananthan et al. (Ref. 22) used a free vortex-wake method to show that the magnitude of the velocity gradients when the rotor operates in forward flight are large enough to have significant effects on growth properties of tip vortices. This suggests that any vortex measurements that are made in forward flight must be corrected for the effects of strain if the intent is to isolate the competing mechanisms of stretching and diffusion from each other. The ability to do this would seem to be a prerequisite to validate any kind of blade tip vortex model. Non-dimensional core radius, r c/c

0.1 Rectilinear vortex, Vε = 0

0.08

Positive strain, V = 0.25 ε

Negative strain, V = -0.25 ε

0.06 0.04 0.02

The importance and sensititivity of vortex filament strain on the growth properties of tip vortices can be illustrated with reference to the viscous development of a rectilinear vortex at a particular vortex Reynolds number. The effects on the filaments were examined as a function of prescribed strain rate defined by Vε = dε/dt. Figure 16 shows the growth rate predicted by the model given by Eq. 14 for a rectilinear vortex filament operating at Rev = 5 × 105 . This is a representative value of most model-scale rotor measurements. This figure also includes the core growth predicted by the model for a vortex filament undergoing a constant filament strain rate, both positive and negative. It can be observed that an application of positive filament strain rate results in a reduced core growth rate. Generally, these results show that the effects of stretching counter the effects of diffusion with a notable reduction in the core size. On the other hand, a negative filament strain results in an increased core growth rate, showing that the effects of strain enhances the effects of diffusion. This would mean that under the action of certain velocity gradients the vortex would exhibit an increased core growth rate even if the vortex Reynolds number is small. The core growth predicted by the model at different vortex Reynolds numbers when the vortex filament undergoes a constant positive strain rate is shown in Fig. 17. The core growth predicted by the model in the absence of strain for the same type of vortex Reynolds numbers is shown in Fig. 18. It can be observed that the core growth changes both with the sign and magnitude of the strain on the vortex filament and also with vortex Reynolds number. Even if the magnitude of diffusion of the vortex is increased through an increase in turbulence in the core (or an increase in vortex Reynolds number), the vortex filament Re = 100 v

Re = 1 X 10

4

Re = 1 X 10

5

Re = 1 X 10

6

v

Non-dimensional core radius, r c/c

issue as it is in the laboratory (Refs. 32, 33). The measurements of Boatright (Ref. 16) were noted to exhibit substantial scatter, most likely because of this problem and are not shown here. The measurements of Mahalingam et al. (Ref. 18) suggest a somewhat higher core growth rate at young wake ages than that might be expected based on the vortex Reynolds number. This could be because the measurements were made using a rotor operating in forward flight; as previously alluded to in this case the vortex filaments experience velocity gradients in the rotor wake, which can strain or squeeze filaments and can affect the measured growth properties of the tip vortices. The importance of vortex filament strain and its interdependence on diffusion (and vortex Reynolds number) is explained in the next section.

v

0.06

v

0.05 0.04 0.03 0.02 0.01 0

0 0

360 720 Wake age, ζ (deg.)

Figure 16: Variation of core growth predicted by the new model for different values of uniform strain rates at Rev = 5 × 105 .

0

360 720 Wake age, ζ (deg.)

Figure 17: Variation of core growth predicted by the new model for a uniform strain rate vε = 0.25 at different vortex Reynolds numbers.

ported to be significant. An increased or decreased value of positive filament strain corresponds to reduced or increased core size, respectively, for the same wake age. This change in core size can significantly alter the wake geometry and the induced velocities inside the rotor wake. The implications are that this will also affect predictions of BVI airloads on the rotor and also BVI induced noise.

Re = 100 v

Re = 1 X 10

4

Non-dimensional core radius, r c/c

v

0.06

Re = 1 X 10

5

Re = 1 X 10

6

v

v

0.05 0.04 0.03

Conclusions

0.02 0.01 0 0

360 720 Wake age, ζ (deg.)

Figure 18: Variation of core growth predicted by the new model for a rectilinear vortex, vε = 0.0, at different vortex Reynolds numbers. might show reduced core growth and increased vorticity (with higher peak swirl velocity values) if the vortex filament undergoes high positive strain. Overall, these results show the highly interdependent nature of filament strain and diffusion (or vortex Reynolds number) on the growth of tip vortices. For rotors operating near ground, the magnitude of velocity gradients will be of the same order of magnitude as that of forward flight. Strain values measured using a model scale rotor operating in the presence of a solid boundary (Ref. 13) were applied to the core growth model given by Eq. 14, the results of which are shown in Fig. 19. It can be observed that the model correlates well with the measurements for a value of δ that represents the core growth without the effects of vortex filament strain. Also, the core growth predicted by the new model deviates from the simple diffusion based Squire core growth model at older wake ages where the vortex filament strains were reNon-dimensional core radius, rc/ c

0.25 Lamb-Oseen model Squire model, δ = 8 Martin & Leishman, 2001 Ramasamy & Leishman, 2003 New model with measured strain rate New model with reduced strain rate New model with increased strain rate

0.2 0.15

δ=8

0.1 δ = 1 (laminar)

0.05 0 0

180

360 540 720 Wake age, ζ (deg.)

900

Figure 19: Core growth predicted by the new model for measured values of strain rates. Rev = 48, 000.

A vortex model in terms of vortex Reynolds number was successfully developed and validated using available rotor tip vortex measurements. The model was developed by combining two experimentally validated individual vortex models that take into account the effects of filament stretching and rotational flow (Richardson number) effects on the turbulence present inside the tip vortices in predicting their temporal evolution. An useful exponential series approximation was also developed to represent the swirl velocity distribution predicted by the transitional vortex model. The following conclusions have been derived from the work: 1. Vortex filament strain and diffusion efffects are interdependent processes. Tip vortices will show an increased growth rate even when the vortex Reynolds number is small if the filament undergoes significant (negative) strain. Similarly, the tip vortices will exhibit reduced growth rate despite being at higher Reynolds numbers if they experience large positive strain. 2. The core growth rate predicted by the model for a rectilinear vortex filament increases with an increase in vortex Reynolds number (for Rev > 1, 000). Increasing the vortex Reynolds number increases the turbulence present inside the tip vortices and, hence, gives an increased core growth rate. At lower Rev , however, the core growth rate reduces to the laminar Lamb–Oseen model by being independent of vortex Reynolds number but with a finite core radius at time t = 0. 3. An exponential series approximation emulated the properties of the swirl velocity distribution predicted by the transitional vortex model at a much lower computational cost. This provides a useful approach for incorporating vortex Reynolds number effects into the vortex model for use in a variety of aeroacoustic applications. 4. The effect of strain on growth properties of a tip vortex at a given vortex Reynolds number was found to favor, balance, or counter the effects of diffusion

based on the nature and magnitude of the stretching. Vortex measurements that are made in velocity gradients must be corrected for the effects of vortex filament strain if the results are to be interpreted in correct manner for the development of better vortex models.

Acknowledgments This research was supported, in part, by the National Rotorcraft Technology Center under Grant NCC 2944.

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