2004-ahs Paper

A G ENERALIZED M ODEL FOR T RANSITIONAL B LADE T IP VORTICES J. Gordon Leishman †

Manikandan Ramasamy

Alfred Gessow Rotorcraft Center Department of Aerospace Engineering Glenn L. Martin Institute of Technology University of Maryland College Park, Maryland 20742

Abstract A new mathematical model is proposed to represent the induced velocity of a rotor blade tip vortex at any vortex Reynolds number. A complete description of the tip vortex requires a solution to the Navier-Stokes (N-S) equations, which can only be obtained by reducing these equations under certain assumptions and approximations. Such reductions give a completely laminar vortex model when the Reynolds number is low, or a completely turbulent model at very high vortex Reynolds numbers. However, many rotating-wings operate at conditions where the vortex Reynolds number is in the intermediate (transitional) regime when the vortex is neither fully laminar nor turbulent. This is particularly true at model scale, where most experimental measurements exist. A new analytical model for a transitional vortex has been developed using an eddy viscosity intermittency function in such a way that this function smoothly and continuously models the eddy viscosity variation across the vortex from its inner rotational region into the outer potential flow region. This intermittency function is developed based on Richardson number concept, which brings in the effects of flow rotation on turbulence present inside the vortex boundaries, and is incorporated into the N-S equations governing the development of an axisymmetric vortex flow. A unique aspect of the proposed model is the Reynolds number dependency of the final solutions. Furthermore, because the solutions satisfy the N-S equations, vortex velocity profiles can be solved for any given vortex Reynolds number. The model is shown to correctly reduce to the solution for a laminar (Lamb–Oseen) modelfor very low Reynolds numbers. The proposed model is validated using high resolution tip vortex measurements for a hovering rotor.

Graduate Research Assistant. [email protected] [email protected] Presented at the 60th Annual Forum and Technology Display of the American Helicopter Society International, Baltimore, MD, June 7–11, 2004. c 2004 by M. Ramasamy & J. G. Leishman. Published by the AHS International with permission. † Professor.

Nomenclature a b A AR c C0 CT g l Nb rc r r R Rev Ri t T VIF V1 V1new Vr Vz Vθ αL αI αnew γ γv γ Γ Γv Γ1 δ ζ η η1 ηa η κ µ ν

Empirical constants Rotor disk area, m2 Aspect ratio of the rotor blade Blade chord, m Constant Coefficient of rotor thrust, = T ρAΩ2 R2 Core circulation function Prandtl’s mixing length, m 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 Thrust, N Vortex intermittency function Peak swirl velocity, m/s Peak swirl velocity in new model, m/s Radial velocity of the tip vortex, m/s Axial velocity of the tip vortex, m/s Swirl velocity of the tip vortex, m/s Lamb’s constant, = 1 25643 Iversen’s constant, = 0 01854 New empirical constant, = 0 0655 Reduced circulation, = rVθ , m2 /s Reduced circulation at large distances, m2 /s Non-dimensional circulation, = γ γv Circulation, = 2πrVθ , m2 /s Circulation of the vortex at large distances, m2 /s Circulation at core radius, m2 /s Ratio of apparent to actual viscosity Wake age, deg. Similarity variable, = r2 4γv t Similarity variable at the core radius,= r2c 4γv t Empirical constant Scaled similarity variable, = η αnew 2 Newly developed function, = αnew 2 VIF Dynamic viscosity, kg/m s 1 Kinematic viscosity, = µ ρ, m2 /s

νt νT ρ σ σe ψ Ω

Eddy or tubulent viscosity Total kinematic viscosity, = ν νt Density of the fluid, kg/m3 Shear stress, N/m2 Effective rotor solidity, = Nb c πR Azimuthal position, deg. Rotational speed of the rotor, rad/s

Introduction The proper modeling of rotor blade tip vortices has been a challengetohelicopteraerodynamicistsformanydecades. A better understanding of the blade tip vortices continues to beessentialin predicting theaerodynamic performance, blade airloads and the resulting acoustics of the helicopter rotor. Unlike their fixed wing counterparts, where the tip vortices trail well downstream, rotor tip vortices lie in close proximity to each other for a longer time and may closely interact with other blades. This interaction leads to impulsive aerodynamic loads, resulting in high rotor vibrations and noise production. Furthermore, the rotor wake downwash on the fuselage, tail rotor and/or the empennage can lead to further degradation in overall helicopter performance. In each case a model of the tip vortices is key to enabling adequate predictions of the airloads. Numerous experiments have been conducted in the past in attempts to understand the nature of flow inside a lift generated tip vortex (Refs. 1–9). A detailed summary of the experiments along with the relative capabilities, limitations, uncertainties and precision of the different measurement techniques is given by Martin et al. (Ref. 10). Unlike fixed wing tip vortices, which are predominantly rectilinear in nature, helicopter vortices are curved and lie in close proximity to other vortices, producing mutually induced effects on their development. This makes the accurate measurement of their flow field properties much more difficult. Several mathematical models of vortex flows have also been proposed (Refs. 11–19). However, almost all the proposed models have met only with limited success in rotor analyses such as airloads, vibration and noise prediction. These models are developed either analytically by making various assumptions to the N-S equations, or semi-empirically based on experiments. The most popular among the proposed models are the Lamb– Oseen model (Ref. 11) (which is fully laminar) and the Squire (Ref. 12) and Iverson models (Ref. 18) (which are both fully turbulent). However, most rotating wings operate at conditions where the vortex Reynolds numbers are in the intermediate (transitional) regime, when the flow is neither fully laminar nor turbulent. This is especially true with sub-scale rotors, whichgeneratetip vorticeswith much lower vortex Reynolds numbers. Comprehensive measurements of the complete vortex structure is essential for the development and validation of a generic, high-

fidelity vortex model that can be used in any flight regime. This has been difficultbecause ofthelackofproperinstruments capable of measuring flow fields at the necessary high resolutions, and also the costs and time involved in conducting such experiments. A comparison of vortex Reynolds number between full scale and sub scale rotor will throw some light in the scaling issues involved. The circulation of the tip vortex, Γv is given approximately (Ref. 20) by Γv

kΩRc

CT σ

where k 2 in hover. This implies that the vortex Reynolds number, Rev , is given by Rev

Γv ν

2 ΩR c ν

CT σ

Most vortex flow measurements are performed using subscale rotors that match CT σ and blade aspect ratio, AR R c of the full-scale rotors. The above expression in terms of aspect ratio AR can be written as 2 ΩR R ν AR

Rev

CT σ

The ratio of the votex Reynolds number between modelscale and full-scale is then given by Rev Model scale Rev Full scale

ΩR R Model scale ΩR R Full scale

1 n

The value of n varies with the scaling of the model rotor and the ratio of the tip speed between the model-scale and full-scale rotors. The value of n for some experiments are Experiments HART II test (Ref. 21) McAlister, 2003 (Refs. 22, 23) Ramasamy & Leishman, 2003 (Ref. 24)

ΩR Ratio 1

Scale

n

1/2.5

2.5

2.5

1/3

1/7

21

1

1/3

1/15

45

48

Rev 105 105

104

Table 1: Value of n for various rotor experiments given in Table 1. Clearly this raises issues about scaling effects because the Reynolds number is known to affect both the structure and the viscous growth rate of the tip vortices (e.g., Refs. 12, 13). This dependency is yet to be quantified because of the dearth of high fidelity measurements over a sufficiently wide range of vortex Reynolds numbers. Vortex models derived from model scale experiments cannot necessarily be expected to give valid results when applied to full-scale rotors at full-scale flight conditions.

0.8

θ

πcV / Γ

Non-dimensional swirl velocity, 2

Various attempts (e.g., Refs. 14, 16) have been made to develop avortex modelbased on an analogy with aboundary layer theory. Tung (Ref. 16) made measurements on a sub-scale rotor in hover and proposed a semi-empirical, multi-region modelfor the tip vortex (laminar, transitional and turbulent). However, no attemptwas made to generalize the model to higher (full-scale) vortex Reynolds numbers. Furthermore, such models do not include the effects of flow rotation on the turbulence present inside the vortex. Flow rotation has been hypothesized to have a significant influence on the vortex structure (e.g., Refs. 15, 25, 26) and flow visualization tend to support this hypothesis. The objectives of the present work were to consider the effects of flow rotation on the structure of a tip vortex, and to develop a generalized mathematicalmodelfor rotor tip vortices that is consistent with both flow visualization and flow field measurements. The first approach towards developing such a model for fully turbulent vortices using an analogy with boundary layer theory was made by Hoffman et al. (Ref. 14). Iversen (Ref. 18) later developed a mathematical model for turbulent tip vortices that were a function of vortex Reynolds number. The present work combines and extends both these concepts to the modeling of transitional vortices and provides a more generalized model to predict the vortex flow at any vortex Reynolds number.

0.4

0 Increasing time

-0.4

-0.8 -1 -0.5 0 0.5 1 Non-dimensional distance from the vortex axis, r / c

Figure 1: Swirl velocity distribution for an aging vortex as predicted by the laminar Lamb–Oseen model. where ηL is the similarity variable proposed by Lamb (=r2 4 ν t . For time t 0 the distribution of swirl velocity versus the non-dimensional radial distance is shown in Fig. 1. It is apparent that the peak swirl velocity decreases and the core radius decreases with time, indicating diffusion of vorticity away from the vortex core. However, the Lamb–Oseen model approaches a singularity at time t 0, which is not physically realistic. Moreover, because this model is developed by making a laminar assumption, molecular diffusion is the only source of momentum transport. In this case, the growth of the viscous core with time is given by

Theory The incompressible Navier-Stokes (N-S) equation for one dimensional, axisymmetric vortex in polar coordinates is given by ρ

DVθ Dt

∂ ∂Vθ µT ∂r ∂r

µT

Vθ r

∂Vθ ∂r

2µT r

Vθ r

∂ ∂ γ r νT r ∂r ∂r r2

2νT

∂ γ r ∂r r2

(2)

The exact solutions to this equation can be obtained by assuming that the viscosity has the form νT

ν

νt

(3)

where ν is the kinematic viscosity (a property of the fluid) and νt is the turbulent or “eddy” viscosity. The classic Lamb–Oseen vortex model (Ref. 11) is an exact solution to Eq. 2 under the assumption that νt 0 (i.e., fully laminar flow). The swirl velocity, Vθ , surrounding an isolated vortex filament from this model is Vθ r

Γ 1 2πr

exp

ηL

(4)

(5)

where αL is Lamb’s constant (αL 1 25643). This result, however, is found to be unrealistically slow in light of experimental evidence (e.g., Refs. 10, 24, 23). Equation 4 can also be written in terms of core radius as

(1)

Writing Eq. 1 in terms of local circulation, γ, and kinematic viscosity, νT , results in ∂γ ∂t

4αL νt

rc t

Γv 1 2πrc

Vθ r

e r

αL r2

(6)

where r is the non-dimensional radial distance (r rc . Squire (Ref. 12) modified the vortex core growth obtained from the Lamb–Oseen model by including an eddy viscosity component to account for the effects of turbulence. Because the principal permanent characteristic of a line vortex is its circulation at large distances from the core axis, Squire assumed that the eddy viscosity was proportional to the vortex circulation, Γv . The total viscosity was assumed to be of the form νT

ν

a

Γv 2π

(7)

where a is an empirical constant determined from experiments. It should be noted that the “eddy” component of viscosity was assumed to be constant with r, and so it represents an average eddy viscosity throughout the structure of the vortex. In this case, the modified growth rate of the

v

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

vortex core is given by a simple modification to Eq. 5 as rc ζ

4αL δνζ Ω

(8)

where δ is defined as the ratio of the total to kinematic viscosity, i.e., δ

νT ν

1

a 2π

Γv ν

1

a1

Γv ν

(9)

Notice that the Squire model reduces to the Lamb model when δ 1. The value of a 2π in Eq. 7 (which is normally represented by the coefficient a1 ) was estimated by Bhagwat and Leishman (Ref. 27) to lie between the values of0 00005 and 0 0002 based on nearly all ofthe available experiments on tip vortices conducted over a wide range of Reynolds number. Squire also proposed an effective or virtual origin offset to eliminate the singular nature of the Lamb–Oseen vortex. As a result, Eq. 8 can be written in the form for rotor applications as rc ζ

4αL δν

ζ

ζ0 Ω

r02

4αL δνζ Ω

Correlation ~ ( ζ - ζ0 )-0.5

~ ζ

-0.5

1

0.1

0.01 0.01

Iversen Correlation Ramasamy & Leishman, 2002 Martin & Leishman, 2000 Mahalingam et al., 1998 McAlister, 1996, 2003 Bhagwat & Leishman, 1998 Lamb-Oseen model All fixed-wing data Cook, 1972

0.1

1

10

100

1000

Equivalent downstream distance, ζ Γ v / Ωc2 f (Γv/ν)

Figure2: Vortexcore growth predicted by Squire’s model, (ζ0 30 ).

ζ 4αL δν Ω

10

θ max

0.2

Equivalent peak velocity, V

/c

McAlister, 2003 Martin & Leishman, 2000 Ramasamy & Leishman, 2003

c

Non-dimensional core radius, r

c/ Γ

0.25

(10)

where r0 is the core radius of the tip vortex at time t 0 and ζ0 is the time offset. Figure 2 shows the increase in size of the vortex core with wake age for various assumed values of δ. Clearly, increased values of δ lead to an increased core growth rate. These two modifications have been shown to result in a more physically realistic representation of the vortex growth rate that correlates better with experimental measurements (e.g., Ref. 10, 23, 24). Iversen (Ref. 18) proposed a turbulent vortex model using a variation of Prandtl mixing length theory. In this model, the eddy viscosity was assumed to vary across the radial dimension of the vortex. The mixing length proposed by Iversen increased linearly with the radial distance; this linear dependency was modeled empirically

Figure 3: Iversen-type correlation of peak swirl velocity with equivalent downstream distance for fixed-wing and rotor tip vortex measurements. based on various measurements. The total viscosity variation across the vortex was assumed to vary as νT

ν

l2r

∂ γ ∂r r2

(11)

where l αI r represents the mixing length. The value of αI (Iversen’s constant) was determined by analyzing results from a specific set of experiments (Refs. 1–3) to be 0 01854. Using dimensional analysis, Iversen determined a similarity variable η r2 4γv t which is 2πηL Rev , where ηL is the similarity variable used by Lamb. By performing a similarity transformation on Eq. 2, the circulation distribution ofan isolated linevortex (based on the form of the eddy viscosity variation given in Eq. 11) is given by ∂γ ∂η

ν γv α2I

4η

∂γ ∂η

γ

(12)

where η η α2I . Iversen obtained a similarity solution to the above equation as a function of vortex Reynolds number, i.e., as a function of Γv ν. This unique nature of the similarity solution helped Iversen derive a correlation function that can be used to compare measurements performed at various vortex Reynolds numbers, as shown in Fig. 3. Swirl velocity profiles at different vortex Reynolds numbers can then be obtained using the similarity solution, which isshown in Fig. 4. Itcan be seen that Iversen’s model reduces to the laminar Lamb–Oseen model at very low vortex Reynolds number, as it should. Based on the variable eddy viscosity model given in Eq. 11, Iversen predicted a core growth of a tip vortex that is dependent on the vortex Reynolds number, the result from which is shown in Fig. 5. At very low Reynolds numbers δ 1, which is fully laminar. As the Reynolds number increases above 103 , the value of δ increases until it changes linearly with vortex Reynolds number. Because Squire’s modelpredicts a linear increase

1

/V

θ

1.2

Non-dimensional swirl velocity, V

1 0.8

1

0.6

2 0.4

Lamb-Oseen model Iversen's model, Re V = 200 Iversen's model, Re V = 18000

0.2

Iversen's model, Re V = 1X10

7

3

Lamb-Oseen & Iversen (Re V= 200)

0 0

1 2 3 Non-dimensional distance from core center, r / r

4

c

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

Figure 4: Swirl velocity distribution predicted by Iversen model at various vortex Reynolds numbers.

Figure 6: A representative flow visualization image of a tip vortex emanating from a rotor blade showing three distinct regions (1) Laminar region, (2) Transitional region, (3) Turbulent region.

Constant viscosity (Lamb-Oseen) model Iversen's model

/V

1

100

1.2 Measurements, Re v = 48,000 (Ref. 24)

θ

10 Squire model valid

1 Lamb-Oseen model valid

Transitional regime

0.1 1

10

100 1000 104 105 Vortex Reynolds number, Re

106

107

v

Figure 5: Variation of δ with Reynolds number based on Iversen’s model. of δ at higher Reynolds numbers, it can be concluded that Squire’s model is valid only at high Reynolds numbers (above 105 ). Even though the core growth predicted by Iversen is more physically realistic compared to measurements, it should be kept in mind that the measurements used by Iversen in developing this model are subject to several uncertainty issues such as vortex core wandering effects, and so the value of αI may be unreliable if not corrected for these effects (e.g., see Refs. 28, 29). Because flow field instrumentation has improved since the 1970s more accurate flow measurement techniques have developed with the spatial resolution necessary for measurements of vortex flows. This has resulted in more reliable and higher fidelity measurements (e.g., Refs. 10, 23, 24). New techniques have also been developed to measure and correct for the effects of core wandering (Refs. 28, 29). These measurements, which have been performed over the past few years, have consistently suggested a multi–region vortex structure. For example, flow visualization performed on a tip vortex emanating from a rotating blade is shown in Fig. 6. It can be observed that the flow is laminar near the core, which is marked by a re-

Non-dimensional swirl velocity, V

Apparent to actual viscosity ratio,

δ

1000

1

Lamb-Oseen model Iversen's model at Re v = 48,000

0.8 0.6 0.4 0.2 0 0

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

8

c

Figure 7: Swirl velocity distribution of a tip vortex using Lamb–Oseen and Iversen models compared to measurements. gion where there is no interaction between adjacent fluid layers. This is followed by a transition region that has eddies of different sizes, outside of this there is a more unsteady turbulent region. This multi–region vortex structure conceptdiffers from the above mentioned models in a way thatitis neithercompletely laminar like Lamb–Oseen model nor completely turbulent like Squire’s or Iversen’s model. Recent velocity measurements made using LDV (Refs. 24, 30) also support the idea of a multi–region or transitional structure of the tip vortex, as shown in Fig. 7.

Richardson Number Effects The effects of flow rotation on the development of turbulence present inside the vortex has been hypothesized to play an important role in determining the structure of a vortex (e.g., Refs. 14, 25, 26). Rayleigh’s centrifugal in-

Ri

2Vθ ∂ Vθ r r 2 ∂r

∂Vθ ∂r

2

(13)

which involves the velocity gradients in the vortex flow. Bradshaw derived the numerator and denominator of the above expression in two different frames of reference. This is misleading because this results in a maximum stability at the core radius of the vortex (Ref. 25). Holzapfel (Ref. 26) corrected the definition of Richardson number by taking both the denominatorand numerator in the same inertial frame of reference. The corrected Richardson number proposed by Holzapfel is given by Ri

2Vθ ∂ Vθ r r 2 ∂r

r

∂ Vθ r ∂r

6

10

105

Richardson number, Ri

stability theory (Ref. 31), which uses a buoyancy force concept, suggests that the vortex will never develop turbulence provided that the product of velocity and radial distance increases with the increase in radial coordinate. Few vortex models (e.g., Refs. 15, 17) have been developed that recognise this concept but have achieved limited success. The various properties of the vortex flow predicted using these models, such as its core growth and the distribution of velocity with radial distance, did not correlate well with experimental results. Bradshaw (Ref. 32) developed an analogy between rotating flows and stratified fluids. His analysis is based on the theory that the flow rotation causes the higher speed fluid to prefer the outside of the vortex while conserving angular momentum, even if the density is assumed constant throughout the vortex. Using energy principles, an expression was developed for the local strength of the analogous stratification, expressed as an equivalent gradient Richardson number. This number comes directly from the turbulent kinetic energy (TKE) budget equation (e.g., Refs. 26, 32) and is basically a ratio of the turbulence produced or consumed inside the vortex as a result of buoyancy (centrifugal effects) to the turbulence produced by shearing in the flow. It can also be thought of as the ratio of potential to kinetic energy in a stratified flow. In a swirling flow Bradshaw’s “Richardson number” is given by

Lamb-Oseen model Iversen's model Measurements (Ref. 24) Stratification line

4

10

1000 100 10

1/4

Stratification Line Ri = Re 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 8: Plot of Richardson number with radial coordinate for a vortex flow. below this threshold value. Any turbulence present inside this boundary will be either relaminarized or suppressed; even Kolmogorov size eddies will not be able to penetrate this vortex boundary. The local gradient Richardson number calculated using the measurements from Ramasamy et al. (Ref. 24) are shown in Fig. 8, along with the Lamb–Oseen and Iversen vortex models. The Richardson number variation for both the vortex models (and the measurements) is seen to approach infinity at the center of the vortex. As the radial distance from the center of the vortex increases, the Richardson number quickly reduces in value and goes below the assumed stratification threshold (i.e., for values of Ri above the stratification threshold only laminar flow is possible). In this region, diffusion at a molecular level will be the only means to transport the momentum. This concept helps explain the persistence of tip vortices, in general, to relatively old wake ages (3, 4, or even 5 rotor revolutions). Below the stratification threshold, flow turbulencecan develop. This argumentserves to augmentthe hypothesis of a multi-region vortex that is nearly always laminar inside the core region, which then progressively transitions to turbulence outside the vortex core. This mechanism affects the induced velocity field.

2

New Vortex Model

(14)

It should be noted thatthis form ofthe Richardson number has strain rate in the denominator instead of plane shear. Cotel (Refs. 25, 33) used this stratification concept and has determined a threshold value for the Richardson number that is a function of vortex Reynolds number. Their analysis is based on a non-dimensional parameter called a “Persistence Parameter,” which is defined as the ratio of rotational to translational speed of the vortex. If the persistenceparameter ishigh, which isthe case formostwing generated tip vortices (including rotorcraft), the threshold value of the Richardson number was found to be Rev 1 4 . This means that the vortex will be laminar up to a radial distance where the local gradient Richardson number falls

The newvortex modeldeveloped here recognizesthe foregoing described effects of flow rotation on the turbulence developments within the vortex core. The incompressible Navier-Stokes (N-S) equation for a one-dimensional, axisymmetric vortex in cylindrical coordinates is given by DVθ Dt

∂ ∂Vθ νT ∂r ∂r

νT

Vθ r

2νT r

∂Vθ ∂r

Vθ r

(15)

An assumption of one dimensional, axisymmetric vortex results in the axial and the radial velocities of the vortex along with any variation in the axial and radial direction to be zero, i.e., Vz Vr 0

and

∂ ∂θ This implies that DVθ Dt becomes ∂ ∂t

γ r

∂ 0 ∂z ∂Vθ ∂t and, therefore, Eq. 15

∂ ∂ γ γ νT νT 2 ∂r ∂r r r 2νT ∂ γ γ r ∂r r r2

(16)

where γ is given by the product Vθ r. But we have that r

∂ γ ∂r r2

and also

r ∂ γ ∂r r

1 ∂γ r2 ∂r γ r2

1 ∂γ r ∂r

2γ r3

1 ∂γ r 2 ∂r

2γ r

2γ r2

(17)

(18)

r

∂ ∂ γ νT r ∂r ∂r r 2

2νT r

∂ γ ∂r r2

(19)

This is made using an assumption that the flow inside the vortex is analogous to the time dependent flow of an infinite line vortex. A classical way (Ref. 34) of writing the total viscosity νT is νT ν νt ν l 2 σ (20) where l κr is the mixing length and σ is the time average ofthe shearstress inside the flow ofthe tip vortex. Assuming a value of zero for κ would result in constant viscosity model. A constant value for κ (i.e., using Iversen’s value of κ αI 0 01854) will result in a completely turbulent flow model. In the new model, κ is considered as a function of r and is varied in a manner analogous to the intermittency function used in boundary layer theory (e.g., Refs 35, 36). Assuming the flow in the vortex to be self-similar, a similarity variable η is obtained from a dimensional analysis that is given by η

r2 4γv t

(21)

which is in agreement with Iversen’s model. The next step is to get Eq. 19 in terms of the similarity variable η. Let Eq. 19 be written as ∂γ ∂t

r

∂ ∂ γ νT r ∂r ∂r r 2

2νT r

∂ γ ∂r r2

Step 2: Substituting νT reducing it.

ν into the RHS of Eq. 22 and

Step 3: Substituting νT ducing it.

l 2 σ into RHS of Eq. 22 and re-

Step 4: Finally, combining allthe three steps to get a nondimensionalized circulation distribution in terms of the similarity variable η, which is given by ∂γ ∂η Part1

ν 1 γv α2I

4κ2 X α2I

Part2

Part3

∂2 γ ∂η2

1 2 X X ∂κ2 (23) η ∂η α2I Part4

Here, γ γ γv and η η αI . From Eq. 23itcanbe noted that the result reduces to a constant viscosity model when the value of κ approaches zero and reduces to Iversen’s model when the value of κ approaches αI . Notice that Eq. 23 has four parts. 2

Substituting these lattertwo equations into Eq. 16, the distribution of circulation for a tip vortex with a variable effective viscosity is given by ∂γ ∂t

Step 1: Similarity transformation of the LHS of Eq. 22, i.e., transforming the term ∂γ ∂t

(22)

RHS

Transforming the above expression in terms of the similarity variable η is carried out in four steps, the details of which given are in the appendix.

Part 1: The variation of circulation, γ, with respect to the similarity variable η. Part 2: The variation in circulation that results from the assumption of constant viscosity. Part 3: The circulation variation as a result ofthe assumption of variable eddy viscosity. Part 4: A result of the “transitional” eddy viscosity variation that is yet to be modeled. The new model would be complete by defining a function for κ. As explained in the previous sections, the tip vortex can be assumed to be made of three regions: an inner laminar region, a transitional region, followed by a turbulent region. Therefore, the variation of κ should represent the variation of the eddy viscosity over all of these three regions. It can be observed from Fig. 8 that in the laminar region (i.e., until a particular distance from the vortex core where the Richardson’s number falls below the threshold value)the vortex cannotdevelop orsustainany turbulence. This will result in essentially zero eddy viscosity. The developed function for κ should also be zero until that radial distance is reached in this region. As the radial distance increases, the vortex flowbecomes transitional, andfinally the flow becomes completely turbulent at large distances. This would mean that the proposed function of κ should also startincreasing and reach a value equivalent to that of a completely turbulent vortex at large radial distances. To satisfy all the above mentioned conditions, a new value of κ is defined as κ

αI

VIF

(24)

Vortex intermittency function, VIF

1

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 / r

3

c

Figure 9: Eddy viscosity intermittency function across the vortex: (1) Laminar region, (2) Transitional region, (3) Turbulent region Vortex intermittency function, VIF

1

3

0.8

is known to vary linearly with the normal distance. Also, the stress is usually assumed constant and is equal to the value of stress at its surface. Similarly, in a vortex flow the turbulent fluctuations are suppressed by flow rotation effects (stratification) and, therefore, the eddy viscosity that results from turbulence is negligible. Also, the tangential inertia of this “eye” or “equivalent viscous sub-layer” of the vortex (represented by Region 1 in Fig. 9 where the VIF 0) is very high when compared with the Reynolds stress because of the high tangential velocity gradients. Therefore, the hypothesis by Hoffman et al. (Ref. 14) that the circulation distribution varies logarithmically with radial distance is not valid inside this region. In the new model the VIF forces the eddy viscosity to be zero in this region, resulting in laminar flow near the vortex core axis. Furthermore, it should be noticed that the mean swirl velocity increases linearly with distance (solid body rotation) similar to that of the boundary layer. By writing the expression for total viscosity as

Intermittency function

0.6

νT

2

0.4

∂ ∂r

Γ r2

ν

κ2 r 2

ν

VIF αnew2 r2

Core radius

∂ ∂r

Γ r2

(26)

0.2 1

0

1 3 5 7 Non-dimensional distance from core center, η / η

9 1

Figure 10: Eddy viscosity intermittency function across the vortex: (1) Laminar region, (2) Transitional region, (3) Turbulent region where αI is Iversen’s constant and VIF is a vortex intermittency function as given by VIF

1 1 2

erf b

η η1

ηa

(25)

where η1 represents the value of the similarity variable at the pointwhere the peak swirlvelocity is maximum, b and ηa are empirical constants. The coefficient b represents the rate at which the transition from laminar to turbulent flow occurs, and ηa representsthe value atwhich the value of VIF is 0 5. A typical variation of the VIF with respect to nondimensional radial distance r rc and η η1 is shown in Figs. 9 and 10, respectively. A value of VIF 0 corresponds to κ 0 and VIF 1 corresponds to κ αI , i.e., values of κ that would result in a completely laminar and turbulent values for eddy viscosity, respectively. The new expression for the VIF is derived in a manner analogous to the intermittency function in boundary layer theory (e.g., 35, 36). Near the wall in a boundary layer (viscous sub-layer) the turbulent fluctuations are damped by the presence ofa solid surface. Here, the mean velocity

it is apparent from Fig. 9 that the VIF (and hence the eddy viscosity) is zero near the center of the vortex. Here, αnew is a new empirical constant yet to be determined. The necessity and procedure to determine the value of αnew is explained in the next section. Region 2 represents a transition region where the flow transitions from a laminar to a turbulent flow. Here, the tangential inertia is much smaller than the Reynolds stress because of the low velocity gradient. The Richardson number in this region falls below the threshold value (Ri Rev 1 4 ), and any turbulence present cannot be suppressed by stratification– see Fig. 20. As a result both types of shear: viscous (molecular) shear and turbulent (eddy) shear are equally important here. Region 3 represents the potential flow regime (analogous to the outer region ofthe boundary layer), where the circulation remains constant. Here, the eddy viscosity plays a significant role and is more dominant than the molecular viscosity. The variation of the eddy viscosity in all the aforementioned three regions is modeled in a continuous manner using the vortex intermittency function: an inner laminar region when the value of VIF is approximately zero, a transition region when 0 VIF 1, and then turbulent region when VIF approaches unity. Upon substituting the expression for the VIF (Eq. 25) into Eq. 23 and replacing αI by αnew we get ∂γ ∂η

ν 1 γv αnew 2

2X X ∂ VIF η ∂η (27) Equation 27 requires the first derivative of the VIF with 4VIF2 X

∂2 γ ∂η2

respect to η, i.e., ∂ VIF ∂η

Upon substituting this in Eq. 30 gives

1 ∂ 1 2 ∂η

η η1

erf b

1 exp π

b

ηa

ν 1 γv αnew 2

ηa

∂2 γ ∂η2

4 VIF2 X

1 2X X exp π η

b

b 2

2

η η1

2

ηa

(28)

Determination of αnew It should be noted that Eq. 28 has three empirical constants, αnew, b, ηa . Iversen determined the value of αI from various measurements that are at large equivalent downstream distances. The reason for doing this was to ensure a more uniform turbulent decay of the trailing vortices. Also, it should be kept in mind that the measurements used by Iversen were performed with fixed wings, including results at very high vortex Reynolds numbers. Even though this model predicted a higher core growth than the laminar Lamb–Oseen model, the distribution of swirl velocity, Vθ , predicted by the model did not correlate well with experimental measurements (Refs. 10, 24, 22, 23). An approach similar to that of Iversen is taken using new measurements to determine a value for αI . Let the new value be called αnew. Following Iversen let

At the core radius, Vθ V1 and r at the core radius is given by V1 c γv But

c rc

γ

r2c Ω 4γv ψ

rc2 4γv t

η1

constant

This would result in c rc

ν γv

(31)

Γv Ωc

C0 g

Γv ν

(32)

B

Γv Rev

VIF αnew2 r2

∂ ∂r

Γ r2

(33)

it is apparent that the total viscosity will remain constant at very high Reynolds number. This will result in the core circulation function g (which is a function of vortex Reynolds number) and hence, the peak swirl velocity (γ η1 2 ) to asymptote to a constant value at higher vortex Reynolds numbers. This implies that the value of C0 is 1 641 if the asymptotic value of g is assumed to approach unity at large vortex Reynolds numbers. The value of αnew was determined, however, by using the asymptotic value of γ η1 2 ( 0 539) obtained from Iversen’s model (as shown in the Fig. 11) to assure that the new model will approach the Iversen’s model when the VIF approaches the value of unity. By writing Eq. 32 as Vc Γv

ψ c

Γv Ωc

γ

1 2

η

1 2

1 2αnew 2π

1 2

(34)

then αnew can be written as (29)

rc . Therefore, Eq. 29

r2c ν 4γv t γv

γ η1

where part A in Eq. 32 represents the vortex velocity scaling parameter, B represents the distance scaling paramenter that is also called an “equivalent downstream distance,” C0 is a constant as yet to be determined, and g is the core circulation function. Table 2 shows the average values of the quantity on the left-hand side ofEq. 32 versus equivalentdownstream distances for a set of rotor experiments. The average value of all of data was found to be 1 641. By re-writting Eq 26 in terms of vortex Reynolds number as νT

c ν γ η r γv

ψ c

V1 c Γv A

b 2 ηη1

1 2

1 2

1 ηη1

Solving this expression numerically using a Runge-Kutta schemes provides the required distribution of circulation for a given vortex Reynolds number.

Vθ c γv

1 4η1

or

Substituting the above expression into Eq. 27 gives ∂γ ∂η

Ωc γv

c ψ

2

η η1

2

V1 c γv

(30)

αnew

γ η

1 2

Iversen

4

2 π 1 641

0 0655

(35)

Because this value is determined by averaging various measurements that were performed over a range of vortex Reynolds numbers, the velocity profile corresponding to the solution of Eq. 28 for any vortex Reynolds number can be predicted.

Discussion Swirl Velocity and Circulation

c2 Ω 4η1 γv ψ

1 2

A numerical solution to the circulation distribution given by Eq. 28 can be obtained using Runge-Kutta integration

Experiments Martin & Leishman, 2000 Bhagwat & Leishman, 1998 Mahalingam et al., 1998 McAlister, 1996 McAlister, 2003 Ramasamy & Leishman, 2003

Equivalent downstream distance ψ c Γv Ω c 0.5–8.9 0.19–10 0.41–3.48 0.06–0.4 0.02–0.6 0.4–28.6

LHS of Eq. 32 V1 c Γv ψ c Γv Ωc 2.197 1.9362 1.4386 0.9957 0.741 2.5264

1 2

Table 2: Values of equivalent peak swirl velocity from various rotorcraft experiments.

Non-dimensional circulation,

1

0.1 Lamb-Oseen model Iversen's model

0.01 4

1

6

100 10 10 Vortex Reynolds number, Rev

0.6 Measurements (Ref. 24) Lamb-Oseen model Iversen's model New model Tung model

0.4 0.2 0 0

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

12

c

Figure 13: Ratio of circulation to circulation ar large distance, Rev 48 000, (1) Laminar region (2) Transitional region (3) Turbulent region.

1

1.2

/V

1

0.8

8

θ

/V

3

2

1

10

Figure 11: Variation of peak swirl velocity versus Reynolds number for Lamb–Oseen and Iversen’s model.

1 0.8 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

1.2

θ

Measurements (Ref.24) New model Lamb-Oseen model Iversen's model

8

c

Figure12: Swirlvelocity distribution using new modelfor Rev 48 000, (1) Laminar region (2) Transitional region (3) Turbulent region. schemes, also satisfying the boundary conditions γ 0 0 and γ ∞ 1. The resulting swirl velocity distribution versusnon-dimensionalradialdistance isshown in Fig. 12 along with the measurements obtained from Ramasamy et al. (Ref. 24) for Rev 48 000. This figure also includes the velocity distribution obtained from the laminar Lamb– Oseen model and the completely turbulent Iversen model for comparison. It is apparent that the new model predicts the velocity much better than either of the other two models because the new model is able to demarcate the three different regions of vortex.

Non-dimensional swirl velocity, V

Non-dimensional swirl velocity, V

1

v

Γ /Γ

Equivalent peak swirl velocity,

γ/η

1/2

1.2

1 0.8

All results overlap here

0.6 0.4

Lamb-Oseen model Iversen's model New model

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: Predicted swirl velocity profiles at Rev

200.

Figure 13 shows the distribution of the ratio of local circulation Γ to large radius circulation Γv versus nondimensional radial distance. By plotting the measurements from Ref. 24 it is evident that the new model predicts the non-dimensional circulation distribution much better than the constant viscosity model (Lamb) or the variable viscosity model (Iversen), or the multi-region vortex model (Tung). Figures 14 through 18 show the predicted distribution of swirl velocities at four different Reynolds numbers of 200 25 000 48 000 75 000 and 105 respectively. All the figures include the swirl velocity distribution from

1

/V

0.6 0.4 0.2 0 0

5 10 Non-dimensional distance from core center, r / r

15

1

Non-dimensional swirl velocity, V

1

Non-dimensional swirl velocity, V

0.8

1.2

θ

Lamb-Oseen model Iversen's model New model

θ

/V

1

0.8 0.6 0.4

Lamb-Oseen model Iversen's model New model

0.2 0 0

c

/V Non-dimensional swirl velocity, V

Lamb-Oseen model Iversen's model New model

0.6 0.4 0.2 0 0

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

10

c

1

1 Lamb-Oseen model Iversen's model New model

θ

/V

Figure 16: Predicted swirl velocity profiles at Rev 48 000.

0.8 0.6 0.4 0.2 0 0

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

c

Figure 17: Predicted swirl velocity profiles at Rev 75 000. the laminar Lamb–Oseen model and the turbulent Iversen model for comparison. It can be observed from Fig. 14 that the predicted swirl velocity distribution from the new model (as well as Iversen model) lie on the constant viscosity results, indicating that all three models behave like the fully laminar model at low Reynolds numbers. As the Reynolds number increases, however, the eddy viscosity increases and the completely turbulent Iversen model shows a different profile from the laminar profile. The

1

Lamb-Oseen model

1.2

ReV = 200

θ

1

Non-dimensional swirl velocity, V

θ

/V

1

Figure 18: Predicted swirl velocity profiles at Rev 105 .

1 0.8

4

c

Figure 15: Predicted swirl velocity profiles at Rev 25 000.

Non-dimensional swirl velocity, V

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

ReV = 25 X 10 3

1

ReV = 48 X 10 3 ReV = 75 X 10 3

0.8

ReV = 1 X 10 5

0.6 0.4

Increasing Re v

0.2 0 0

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

6

c

Figure 19: Predicted swirl velocity profiles at various vortex Reynolds number. new model which includes flow rotation effects and suppresses the turbulence near the vortex core axis, behaves like a laminar model near the core axis. As the radial distance increases, it slowly transitions to the completely turbulent flow model. The swirl velocity profiles obtained from the new model for a range of vortex Reynolds numbersissummerized in Fig. 19. Itis apparentthatthevelocity profile changes with increasing vortex Reynolds number, and becomes closer to a more fully turbulent profile at high vortex Reynolds numbers. The Richardson number predicted by the new model along with the measurements from Ref. 24 is shown in Fig. 20. Again, this plot includes results from the Lamb– Oseen model and Iversen’s model for comparison. It is apparent that the new model behaves exactly the same way as that of the measurements by being laminar until a particular radial distance from the center of the vortex is reached (where the Richardson number is above the stratification threshold).

Peak Swirl Velocity and Core Radius The variation of non-dimensional peak swirl velocity obtained from the numerical solution of Eq. 23 versus Reynolds number is shown in Fig. 21. Values from var-

6

10

1000

ReV = 1x10 4 1

104

Richardson number, Ri

ReV = 1x10 0 - 1x10 3

Lamb-Oseen model Iversen's model Measurements (Ref. 24) New model Stratification line

100 10

Stratification Line Ri = Re 1/4

1 0.1 0.01 0

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

3

c

Figure 20: Variation of Richardson number for various models with non-dimensional radial distance, Rev 48 000.

1/2

γ/η Equivalent peak swirl velocity,

ReV = 1x10 5

1

ReV = 1x10 6 McAlister, 2003 Martin & Leishman, 2000 Ramasamy & Leishman, 2003

0.8 0.6 0.4 0.2 0 0

180

360 540 720 Wake age, ζ (deg)

900

Figure 22: Variation of peak swirl velocity with wake age for various vortex Reynolds numbers.

Lamb-Oseen model Iversen's model New model Ramasamy & Leishman, 2003 Martin & Leishman, 2000 Mahalingam et al., 1998 McAlister, 1996, 2003 Bhagwat & Leishman, 1996 Cook, 1972

10

Non-dimensional peak swirl velocity, V

105

which the peak swirl velocity reduces with wake age increases. Similarly, the vortex core size increases rapidly with wake age for high vortex Reynolds numbers. This is important because a small change in the strength or the size ofthevortex can havesignificantimpacton predicting the unsteady airloads and propagated noise of helicopter rotors.

1

Core Growth 0.1 Model scale

Full scale

0.01 1

100 104 106 Vortex Reynolds number, Re

108

v

Figure 21: Variation of peak swirl velocity with vortex Reynolds number. ious experiments are included in the figure. It can be observed that the new model predicts the peak swirl velocity better than either of the Lamb–Oseen and Iversen models. Because this new model is developed based on the Iversen model, both the new model and the completely turbulent Iversen model shows the tendency to asymptote to a constant value at high vortex Reynolds numbers. The asymptotic value, however, is different in both models because ofthedifference in theway the eddy viscosity varies between these two models. It should be noticed that the peak swirl velocity V1 for large Reynolds number is proportional to Γ1 1 2 t 1 2 , which is independent of Reynolds number. The variation of peak swirl velocity and core radius with wake age for various vortex Reynolds numbers is shown in Figs. 22 and 23 respectively. It can be observed that as the vortex Reynolds number increases the rate at

The ratio of apparentto actualviscosity for the new model can be used to determine the growth rate of the vortex core. The similarity variable η at the core radius is given by rc 2 η1 4αnew2 γv t i.e., rc 2

4γv αnew2 η1 t 4

4 Rev ν αnew2 η1 t 2π

Rev αnew2 η1 νt 2π

(36)

From the Squire model we have rc 2

4αL δνt

where αL is Lamb’s constant (αL ing Eqs. 36 and 37 we get αL δ

(37) 1 25643). By compar2

Rev αnew 2 ΓvLamb 2π V1new

where V1new is the peak swirl velocity obtained from the new model at a given vortex Reynolds number. This means that δ

Rev αnew2 ΓvLamb 2παL V1new

2

(38)

5

10 3

δ

0

ReV = 1X10 - 1x10

Effective viscosity coefficent,

ReV = 1x10 4 ReV = 1x10 5

ReV = 1x10 6 McAlister, 2003 Martin & Leishman, 2000

Non-dimensional core radius, r

c

/c

0.5

Ramasamy & Leishman, 2003

0.4 0.3 0.2

a1 = 6x10

-5

Lamb-Oseen model

1000

a

1

100 10 1 0.1 100 104 106 Vortex Reynolds number, Re

1

0.1

108

v

Figure 25: Variation of δ with vortex Reynolds number to determine a1 .

0 0

180

360 540 720 Wake age, ζ (deg)

900

Figure23: Variation ofcoreradius of thevortexwith wake age for various vortex Reynolds number.

4

δ

10 Effective viscosity coefficient,

New model

104

1000

Lamb-Oseen model Iversen's model New model Ramasamy & Leishman, 2003 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

1

a 2π

Γv ν

1

a1

Γv ν

Conclusions

10 1

1000

δ

for various vortex Reynolds number against the δ variation predicted by the new model (as shown in Fig. 25) the constant a1 (=a 2π) was determined to be 6 10 5 . This value lies within the range suggested by Bhagwat & Leishman (Ref. 27), as shown in Fig. 26. The unique determination of a1 will help rotorcraft analysts in various applications better determine growth rate ofthe vortex in a physically correctmanner and, therefore, predictunsteady airloads and noise to a higher level of fidelity.

100

0.1

By plotting δ from the Squire’s eddy viscosity model as given by Eq. 9, i.e.,

Model scale 4

Full scale 5

6

10 10 10 Vortex Reynolds number, Re

7

10

v

Figure 24: Variation of δ with vortex Reynolds number. Aplotshowing thevariationof δ with Reynoldsnumber for the new model along with Lamb–Oseen and Iversen models (with rotating wing measurements) is shown in Fig. 24. Clearly the completely turbulent Iversen model overpredicts the core growth. Also, the laminar Lamb– Oseen model under predicts the core growth because molecular diffusion is the only source of momentum transport. However, the new model predicts the core growth betterthaneitherofthe Lamb–Oseen and Iversen’s models. It can also be observed that at low vortex Reynolds numbers, both the new model and the Iversen’s model approach the laminar solution (δ 1).

A new generalized vortex model was developed from the N-S equations that includes the effects of flow rotation on turbulence present inside the tip vortex. The model is developed using an intermittency function based on the concept of local Richardson number. The function smoothly and continuously models the eddy viscosity variation across the vortex and accounts for the effects of flow rotation on turbulence development. The vortex cannot develop or sustain any turbulence until the local Richardson number falls below a threshold value. This results in a multi-region vortex structure: an inner laminar region, a transitional region, and an outer turbulentregion. The new model is able to demarcate these three distinct regions of the vortex clearly and is consistant with flow visualization and velocity field measurements. It is shown thatthe vortex velocity profile predicted by the new model is initially laminar until a particular radial distance and then slowly transitions to a turbulent vortex. The following conclusions have been drawn from this work: 1. The swirl velocity and circulation profiles predicted by the new model correlates well with experimental

eddy viscosity variation of the new model to develop a simple model for the growth rate of the vortex core. As a result, the value of the constant a1 that was used in the Squire model was determined and was found to lie within the range suggested by Bhagwat & Leishman.

Laminar trend New model Corsiglia et al., 1973 Cliffone & Orloff, 1975 Rose & Dee, 1963 McCormick, Tangler & Sherrib, 1963 Kraft, 1955 Jacob et al., 1996 Jacob et al., 1995 Govindaraju & Saffman, 1971 Bhagwat & Leishman, 1998 Mahalingam & Komerath, 1998 Cook, 1972 McAlister, 1996 Baker et al.,1974 Dosanjh et al., 1964 Martin & Leishman, 2000 Ramasamy & Leishman, 2003

0.01

Acknowledgments This research was supported, in part, by the National Rotorcraft Technology Center under Grant NCC 2944. The authorswishtothank Mr.Karthikeyan Duraiswamyfor his valuable suggestions in this work.

0.001

References

Range suggested by Bhagwat & Leishman, 2002

1

Coefficient, a

0.0001

New model

1

Kraft, C. C., “Flight Measurements of The Velocity Distribution and Persistence of the Trailing Vortices of an Airplane,” NACA TN 3377, 1955.

10-5

2

Model scale

-6

McCormick, B. W., Tangler, J. L., and Sherrieb, H. E., “Structure of Trailing Vortices,” Journal of Aircraft, Vol. 5, No. 3, July 1968, pp. 260–267.

Full scale

10

1000

4

10

5

10

6

10

7

10

Vortex Reynolds number, Rev

Figure 26: Effective viscosity parameter, a1 , as a function of vortex Reynolds number. measurements, and much better than existing models.

3

Rose, R., and Dee, W. F., “Aircraft Vortex Wake and Their Effectson Aircraft,” AeronauticalResearch Council Report No. CP-795, 1965. 4

Cook, C.V., “TheStructure oftheRotorBladeTip Vortex,” Paper 3, Aerodynamics of Rotary Wings, AGARD CP-111, September 13–15, 1972. 5

2. The Reynolds number dependency of the model helps understand trends shown in measurements that are made both at sub-scale models and at full-scale. The swirl velocity and circulation profiles obtained from the new model changes with an increase in vortex Reynolds number. It is shown that the model reduces to a constant viscosity laminar Lamb model at very low Reynolds number. 3. It was found that the increase in vortex Reynolds number corresponds to an increase in turbulence present inside the vortex, and would result in an increased growth rate of the vortex core. This would also lead to a more rapid reduction of peak swirl velocity with increasing time. 4. The new model also predicts that at large Reynolds numbers, the peak swirl velocity V1 is proportional to Γ1 1 2 t 1 2 and is independent of Reynolds number. This is consistent with Iversen’s model and also with the various measurements that are available. 5. An equivalent average eddy viscosity distribution suggested by Squire was matched with the assumed

Govindraju, S. P., and Saffman, P. G., “Flow in a TurbulentTrailing Vortex,”Physics ofFluids , Vol. 14, No. 10, October 1971, pp. 2074–2080. 6

Corsiglia, V. R., Schwind, R. G., and Chigier, N. A., “Rapid Scanning, Three Dimensional Hot Wire Anemometer Surveys of Wing-Tip Vortices,” NASA CR2180, 1973. 7

Baker, G. R., Barker, S. J., Bofah, K. K., and Saffman, P. G., “Laser Anemometer Measurements of Trailing Vortices in Water,” Journal of Fluid Mechanics, Vol. 65, 1974, pp. 325–336. 8

Dosanjh, D. S., Gasparek, E. P., and Eskinazi, S., “Decayof aViscousTrailing Vortex,” TheAeronautical Quarterly, Vol. 3, No. 3, 1962, pp. 167–188. 9

Ciffone, D. L., and Orloff, K. L., “Far-Field WakeVortex Characteristics of Wings,” Journal of Aircraft, Vol. 12, No. 5, May 1975, pp. 464–470. 10

Martin, P. B., Pugliese, G., and Leishman, J. G., “High Resolution Trailing Vortex Measurements in the Wake of a Hovering Rotor,” American Helicopter Society 57th Annual National Forum, Washington, DC, May 9–11, 2001.

11 Lamb,

H., Hydrodynamics, 6th ed., Cambridge University Press, Cambridge, 1932, pp. 592–593. 12

Squire, H. B., “The Growth of a Vortex In Turbulent Flow,” Aeronautical Quarterly, Vol. 16, August 1965, pp. 302–306. 13 Owen,

P. R., “The Decay of a Turbulent Trailing Vortex,” The Aeronautical Quarterly, Vol. 10, February 1969, pp. 69–78. 14 Hoffman,

E. R., and Joubert, P. N., “Turbulent Line Vortices,” Journal of Fluid Mechanics, Vol. 16, 1963, pp. 395–411. 15

Donaldson, C. D., “Calculation of Turbulent Shear Flows for Atmospheric and Vortex Motions,” AIAA Journal, Vol. 10, No. 1, January 1972, pp. 4–12. 16 Tung,

C., Pucci, S. L., Caradonna, F. X., and Morse, H. A., “The Structure of Trailing Vortices Generated by Model Helicopter Rotor Blades,” NASA TM 81316, 1981. 17

Baldwin, B. S., Chigier, N. A., and Sheaffer, Y. S., “Decay of Far-Flowfield in Trailing Vortices,” AIAA Journal, Vol. 11, No. 12, December 1973, pp. 1601–1602. 18 Iversen,

J. D., “Correlation of Turbulent Trailing Vortex Decay Data,” Journal of Aircraft, Vol. 13, No. 5, May 1976, pp. 338–342. 19 Phillips, W. R. C.,

“The Turbulent Trailing Vortex During Roll-Up,” JournalofFluid Mechanics, Vol. 105, 1981, pp. 451–467. 20

Leishman, J. G., Principles of Helicopter Aerodynamics, Cambridge University Press, New York, 2000, Chap. 3, pp. 99–100. 21 Yu.,

Y. H., and Tung, C., “The HART-II Test: Rotor Wakes and Aeroacoustics with Higher Harmonic Pitch Control (HHC) Inputs,” Proceedings of the American Helicopter Society 58th Annual National Forum, Montr´eal, Canada, June 11–13 2002. 22 McAlister,

K. W., “Measurements in the Near Wake of a Hovering Rotor,” AIAA Paper 96-1958, Proceedings of 27th AIAA Fluid Dynamic Conference, New Orleans, June 18–20, 1996. 23

McAlister, K., “Rotor Wake Development During the First Rotor Revolution,” Proceedings of the American Helicopter Society 59th Annual National Forum, Phoenix, Arizona, May 6–8 2003.

25 Cotel,

A. J., and Breidenthal, R. E., “Turbulence Inside a Vortex,” Physics of Fluids, Vol. 11, No. 10, 1999, pp. 3026–3029. 26 Holzapfel, A. Hofbauer, T., Gerz, T., and Schumann, U., “Aircraft Wake Vortex Evolution and Decay in Idealized and Real Environtments: Methodologies, Benefits and Limitations,” Proceedings of the Euromech Colloquium, 2001. 27 Bhagwat,

M. J., and Leishman, J. G., “Generalized Viscous Vortex Core Models for Application to FreeVortex Wake and Aeroacoustic Calculations,” Proceedings of the 58th Annual Forum of the American Helicopter Society International, Montr´eal Cananda, June 11– 13 2002. 28 Devenport,

W. J., Rife, M. C., Liapis, S. I., and Follin, G. J., “The Structure and Developmentofa Wing-Tip Vortex,” Journal of Fluid Mechanics, Vol. 312, 1996, pp. 67– 106. 29 Leishman,

J. G., “On the Aperiodicity of Helicopter Rotor Wakes,” Experiments in Fluids, Vol. 25, 1998, pp. 352–361. 30

Martin, P., and Leishman, J. G., “Trailing Vortex Measurements in the Wake of a Hovering Rotor with Various Tip Shapes,” Proceedings of the 58th Annual Forum of the American Helicopter Society International, Montr´eal Canada, June 11–13, 2002. 31 Rayleigh,

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“The Analogy Between Streamline Curvature and Buyoyancy in Turbulent Shear Flows,” Journal of Fluid Mechanics, Vol. 36, pp. 177–191. 33

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and Townsend, A. A., “The Nature of Turbulent Motion at Large Wake Numbers,” Proceedings of the Royal Society, London, A199, 1949, pp. 238–255.

24

Ramasamy, M., and Leishman, J. G., “The Interdependence of Straining and Viscous Diffusion Effetcs on Vorticity in Rotor Flow Fields,” Proceedings of the American Helicopter Society 59th Annual National Forum, Phoenix, Arizona, May 6–8 2003.

Appendix The following are the sequence of steps in deriving the circulation distribution given in Eq. 23

Step - 1

Step - 3

To get ∂γ ∂t in terms of η we have

Let νT κ2 r2 σ based on the assumed value of eddy viscosity. Consequently the RHS of Eq. 22 becomes

∂γ ∂t

∂γ ∂η ∂η ∂t

(A1) RHS

r

Upon differentiating Eq. 21 with respect to t we get ∂η ∂t

r2 4γv

t2

(A2)

∂γ ∂η

η t

r

ν in the RHS of Eq. 22 to get ∂ ∂ γ νr ∂r ∂r r2

2ν r

∂ γ ∂r r2

(A4)

r

∂ ∂r

ν ∂γ r ∂r

2ν ∂γ r ∂r

2γ r

r

∂ γ ∂r r2

(A12)

σ where X gives

∂γ η ∂η

RHS

To write Eq. A5 in terms of η, we need both ∂γ ∂r and ∂2 γ ∂r2 . The first derivative of γ can be obtained using ∂γ ∂η ∂η ∂r

(A6)

RHS ∂η ∂r

∂2 γ ∂η2

2

∂2 η ∂r2

∂γ ∂η

(A7)

The first and second derivatives of η with respect to r can be obtained by differentiating Eq. 21 with respectto r, i.e., ∂η ∂r ∂2 η ∂r2

2r 4γv t ∂ ∂r

2η r

2η r

(A8)

2η r2

(A9)

Upon substituting the derivatives of η into Eqs. A6 and A7, the resulting first and second derivatives of γ are in turn applied in Eq. A5 to get RHS

η ∂2 γ ν γv t ∂η2

(A10)

∂ σσ ∂r

4κ2 σ σ

(A13)

1 ∂γ r ∂r

2γ r2

2 X r2

(A14)

γ. Substituting Eq. A14 into Eq. A13 4 ∂ X X r κ2 r2 ∂r 4 4κ2 2 X X r

κ2 r 3

∂ ∂r

4 XX r4 (A15)

Expanding the middle term in the above expression and canceling like terms results in

Differentiating again results in ∂2 γ ∂r 2

κ2 r

A similarity transformation of the above expression requires the transformation of σ and its firstderivative. Simplifying Eq. A12 results in

2γ r

using the relation from Eq. 17. Further simplification results in ∂2 γ 1 ∂γ RHS ν (A5) ∂r2 r ∂r

∂γ ∂r

∂ 2 κ ∂r

σ σr

r2

RHS

The above equation can be written as

and

Vθ r

Eq. A11 can be rewritten in terms of shear stress as

Substitute νT

RHS

∂Vθ ∂r

σ

(A3)

Step - 2

RHS

(A11)

Because the shear stress inside a one dimensional axisymmetric vortex is given by

Substituting Eq. A2 into Eq. A1 gives ∂γ ∂t

∂ γ ∂r r 2 ∂ γ r ∂r r 2

κ2 r 2 σ r

2 κ2 r 2 σ

η t

1

∂ ∂r

4 ∂ X Xr κ2 r2 ∂r

4κ2 ∂ XX r ∂r

Part1

Part2

(A16)

Notice that the above expression has two parts. A constant value of κ will eliminate the first term and result in a solution for an Iversen type of eddy viscosity variation. This would mean that the first term in the above expression is the additional term in the new model as a result of having κ as the variable rather than a constant as assumed by Iversen. Equation A16 can be written in terms of similarity variable η as RHS

2X X ∂ 2 κ γv t ∂η

4κ2 η ∂2 γ X γv t ∂η2

(A17)

Step - 4 The next step is to combine all the previous three steps to get the circulation distribution in terms of the similarity

variable η. To get this circulation equation, we combine Eqs. A3, A10 and A17, which results in ∂γ ∂η

η t

ν

η ∂2 γ 4κ2 η ∂2 γ X γv t ∂η2 γv t ∂η2 2 X Xη 1 ∂ 2 κ (A18) γv t η ∂η

Letting γ γ γv and cancelling like terms, the above expression becomes ∂γ ∂η

ν γv

∂2 γ ∂η2

4 κ2 X

2X X ∂ 2 κ η ∂η

(A19)

where X is X γv . Let η ηα2I , where the value of αI is obtained from experimental measurements, the above equation takes the form 1 α2I

∂γ ∂η

1 α4I 1 α4I

ν γv

4κ2 X

∂2 γ ∂η2

2X X ∂ 2 κ η ∂η

∂γ Here, it should be noted that X is given by η ∂η Further simplification results in

∂γ ∂η

ν 1 γv α2I

4 κ2 X α2I

which is the required result.

∂2 γ ∂η2

(A20) γ .

1 2 X X ∂κ2 (A21) η ∂η α2I