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1. Asroacoustr Moise prediction modeling Aeroelastic modeling Fatigue He hability Recommended by Virgilitu Niculae CONSTANTINESCU member of the Romanian Academy Horia DUMITRESCU Florin FRUNZULICA 2 Because the unsteady aerodynamic model is the starting step in aeroelastic and aeroacoustic studies our attention will be concentrated in the next section on the development of the aerodynamic model In the present work we adopted the vortex lattice method VLM because of simplicity and easily program implementation 2 BASIC ASUMPTIONS For the aerodynamic model we have adopted the following assumptions 1 The rotor blades are regarded as thin lifting surface 2 The resultant velocity on the blade is subsonic 3 Compressibility effect is considered by relating a pressure coefficient C with an incompressible pressure coefficient C at a given subsonic Mach number by Prandtl Glauert rule C C 9 J1 M 1 Equation 1 relates an incompressible flow over a given two dimensional profile to a subsonic compressible flow over the same profile 4 The elastic displacements of the blades are negletected the aerodynamic model is calibrated using assumption that the blades are rigid Each blade has fixed the angle of attack and precone angle p 3 VORTEX RING MODEL FOR A THIN LIFTING SURFACE We have choised for the numerical simulation of lifting surfaces the aerodynamic model with vortex rings di2. Leishman J G Flow Visualization of Compressible VortexStructures using Density Gradient Techniques Experiments in Fluids 15 1993 pp 431 442 8 Norman T R Light J S Rotor Tip Vortex Geometry Measurements Using the Wide Field Shadowgraph Technique Journal of the American Helicopter Society Vol 32 No 2 April 1987 pp 40 50 9 Vatistas G H Kozel V Mih W C A Simpler Model for Concentrated Vortices Experiments in Fluids 11 1991 pp 73 76 10 Scully M P Computation of Helicopter Rotor Wake Geometry and Its Influence on Rotor Harmonic Airloads Massachusetts Institute of Technology ASRL TR 178 1 March 1975 11 Bliss D B Quackenbush T R Bilanin A J A New Methodology for Helicopter Free Wake Analysis Presented at the 39th Annual National Forum of the American Helicopter Society St Louis MO May 9 11 1983 12 Lamb Sir Horace Hydrodynamics 6th Edition Cambridge University Press 1932 pp 592 593 668 669 13 Johnson G M Researches on the Propagation and Decay of Vortex Rings ARL 70 0093 June 1970 14 Ogawa A Vortex Flow CRC Series on Fine Particle Science and Technology CRC Press Inc 1993 15 Meyer J R Falabella G An investigation of the experimental aerodynamic loading on a model helicopter rotor blade NACA TN 2953 May 1953 Received December 18 2003
3. THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY Series A i OF THE ROMANIAN ACADEMY Volume 5 Number 3 2004 pp 000 000 A FREE WAKE AERODYNAMIC MODEL FOR HELICOPTER ROTORS Horia DUMITRESCU Florin FRUNZULICA PhD Professor Institute of Statistics and Applied Mathematics Bucharest Romania Lecturer PhD Department of Aerospace Engineering University POLITEHNICA of Bucharest Romania Corresponding author Florin FRUNZULICA E mail ffrunzi yahoo com The accurate computation of the induced velocity on helicopter rotor plane is a prerequisite for the precise evaluation of the angle of attack distribution along the rotor blades which leads to improved air loads prediction for aerodynamics and aeroacustics design and performance analysis The Vortex Lattice Method VLM provides a transparent investigation concerning the role of various physical parameters which influence the aerodynamic problem of rotor downwash calculation This paper presents a method for the calculation of the nonuniform induced downwash of a helicopter rotor using the vortex ring model for a thin lifting surface coupled with a free wake model Key words Helicopter rotor induced velocity Vortex Lattice Method free wake model 1 INTRODUCTION During the last few years considerable research effort has been made with respect to the various aspects of rotors In terms of modeling there is no doubt that significant progress has been achieved during the
4. a viscous core model has been used to avoid such numerical perturbations The experiments that have been conducted to measure the cores of rotary and fixed wing tip vortices have demonstrated that in real flows tip vortices have cores with finite radii 7 8 Because this fact the selection of a finite core vortex model in the numerical schemes must be approached cautiously 3 A free wake aerodynamic model for helicopter rotors Vortex core radius Fa Inner solid body Outer potential rotation flow region Figure 3 Tangential velocity profile Vortex models used in free wake analysis are defined in terms of their tangential velocity profiles Thus the inner viscous region usually consists of a solid body rotation core and the outer region simulates the potential vortex profile fig 3 One series of general velocity profiles for tangential velocity in a two dimensional cross sectional plane rectilinear vortex 7 9 is expressed by the formula 0 E l n 8 2a r r where ris the radial distance from the center of the vortex For n gt the Rankine vortex profile is obtained and for n 1 the Scully velocity profile is recoverd 10 In ref 7 9 the authors have found that for n 2 the model is physically most representative for actual vortex profiles It can be clearly seen that the Rankine velocity profile is discontinous at the boundary between the inner region and the outer region fig 4 while the
5. it is possible to replace the exact formula for the induced velocity by the vortex ring the Biot Savart formula DV gptxdrF T p 3 F n r ar se cae ame do 11 with simplified formula V l A e P 7 12 Arr 1 12 where A n and represent the panel area the normal to panel and the distance between collocation point and the weight center of the panel The formula 12 is valid for 2 3 4 d where d is a reference length of the panel 5 COMPUTATIONAL PROCEDURE The numerical analysis closely follows the concepts presented in the previous section In all cases variables are non dimensionalized to provide economy in utilization of the computer codes Velocities are normalized with RQ distances with R areas with R time with 1 Q and circulation with R Q Force torque and power are non dimensionalized as indicates in eq 20 The general procedure requires that calculations be made at small time increments until a periodic solution is obtained Initially there is no wake structure and it is only as the wake develops sufficiently that a periodic solution is obtained Based on experience it appears that the wake must be propagating three or four revolutions for a periodic solution to be achieved The computation is initialized by setting the bound vorticity in each blade panel to zero The perturbation velocities at each blade panel are then calculated based on all of the vortex filaments
6. last ten years This progress was mainly based on the existing experience of other engineering fields As regards the design of helicopter rotors a number of methods for structural aerodynamic an aeroacoustic analysis used in aeronautical engineering have been applied However in many cases only simply models were used This is true especially as far as Complete Design Tools are concerned Of course in order to devise a flexible and practical design tool simple models constitute to some extent an obligatory choice However the choice of simple models means that only approximate results can be obtained Therefore we need to determine their limitations On the other hand during the last few years within the numerous activities a number of more elaborate models has been developed In most cases the corresponding work was related to the analysis of only one aspect of the whole design problem of rotor e g advanced aerodynamic models were applied to rigid rotors and for steady inflow conditions Though we cannot expect that the advanced models will be applied to practical design problems at least in the near future we can use them in order to validate and improve the simple models The collection of the a fore mentioned problems defines the setting of a complete design tool summarized in the following diagram Inflow Condition o tead Drimamic a ve Wake modelling Aerodyraruc rodel a a Drmarmie output pre dictions Loading predictor
7. over the same panel it is possible to replace the exact formula for the induced velocity by the vortex ring the Biot Savart formula VT erxdF T f3 n n V f 5 lt 3 do 6 4qa r 4r with simplified formula 7 where A n and 7 represent the panel area the normal to panel and the distance between collocation point and the weight center of the panel The formula 7 is valid for 7 2 3 4 d where d is a reference length of the panel 4 VORTEX WAKE MODELLING It is commonly agreed that the key to an accurate calculation of the rotor aerodynamic behaviour is the correct modeling of the rotor wake There are two main approaches to the problem of wake modeling 1 The first method is known as the prescribed wake or rigid wake According to this method the geometry of the wake is known a priori which implies that the velocity field or rather an approximation to it has been assumed Once the wake geometry has been prescribed the corresponding induced velocity and circulation distributions along the blade can be calculated The geometry of the wake is determined by using different kinds of assumptions while in most of the cases these assumptions are based on experimental evidence 2 4 2 The second method is the free wake analysis 5 6 In this method an initial geometry of the vortex wake is assumed This wake is regarded as being composed of a large number of discrete vortex elements and these elements are allo
8. values n 1 and n 2 assure a smooth continous transition Figure 4 Non dimensional tangential velocity profiles Another observation is necessary the induced velocities depend of the vortex core radius In the far wake it is possible that distance between the vortex and a collocation point can be very small and can result over distorted local wake structure and also lead to oscilations in the evolving wake In ref 11 the authors propose a model with larger tip core radius for older vortex elements in this way reducing the intensity of local interaction between the filaments The model uses the Lamb Osen vortex profile 12 2 1 exp 9 207 7 4v t vy 7 t where the exponential term represents the viscous decay of the vortex with time due to the kinematic viscosity V of the fluid The variation of radius core r in time can be determinate by calculating the maxima of eq 9 Thus the variation of r can be written as 13 14 Horia DUMITRESCU Florin FRUNZULICA 6 r 1 12 4v t 10 where is an experimentally determined decay coefficient The purpose of this coefficient is to increase the rate at which the vortex core grows in time In this stage of the program development the equations 9 and 10 are not yet implemented Induced velocity at far distance At far distance from panel using equivalence between the vortex ring model and the doublet with uniform intensity over the same panel
9. g the presented procedure Since 1s based on a flexible coupling between different modules it becomes a useful tool for the aerodynamic study and optimization of helicopter rotors The presented computational methodology is under further development regarding more sophisticated models for blade aerodynamic and dynamic characteristics as well as optimization techniques for wake geometry calculations REFERENCES 1 KATZ J PLOTKIN A Low Speed AerodynamicsCambridge University Press 2001 2 Miller R H On the Computation of Airloads Acting on Rotor Blades in Forward Flight Journal of the American Helicopter Society Vol 7 No 2 April1962 pp 56 66 3 Landgrebe A J An Analytical and Experimental Investigation of Helicopter Rotor Performance and Wake Geometry Characteristics UsAAM RDL TR 71 24 June 1971 4 Egolf T A Landgrebe A J Helicopter Rotor Wake Geometry and its Influence in Forward Flight Vol I Generalized Wake Geometry and Wake Efect on Rotor Airloads and Performance NASA CR3726 Oct 1983 5 Quackenbush T R Bliss D B Wachspress D A Computational Analysis of Hover Performance Using a New Free Wake Method Presented at the Second International Conference on Rotorcraft Basic Research College Park MD Feb 16 18 1988 6 Miller W O Bliss D B Direct Periodic Solutions of Rotor Free Wake Calculations Journal of the American Helicopter Society Vol 38 No 2 April 1993 pp 53 60 7 Bagai A
10. h situation it is useful to express the quantities in terms of relative values as the difference between upper and lower surface values The net pressure coefficient assumes the form 2 2 V V 2 d oP a6 scu en lle ie el E ii where u corresponds the upper surface and corresponds the lower surface of the wing The pressure difference across the wing surface is given by Ap p P AR AR sa 15 P PiP P 2 2 ar ar 15 The velocity V could be broken up on the wing s surface according to two directions T a direction tangent chordwise to the wing s surface and T a direction tangent spanwise to the wing s surface dP Y lij SE 9P glij P ae 16 OT 2 Ab Ue gt OT 2 ZAG c6 99 where the sign corresponds to the upper surface and the sign corresponds to the lower surface and Ac and Ab are the panel lengths in the ith and jth directions The 7 and T are defined to be on the i J panel For a vortex ring the bond between the variation of the potential function by time and the Ta n E 17 variation of the circulation by time is 2 Hence the pressure difference between the upper and the lower surface is given by u u JE v t v j w t w t k i j Ap de l j moe pS oe 2s oI re a ee Ac Ab dt The force acting upon the i f j element is obtained AF Ap AS A 19 The projection of these forces
11. in the flow using eq 2 These velocities will initially be equal to zero since no wake structure exists and since the bound vorticity has been set equal to zero The bound vorticity is then calculated for each panel using eq 3 and the last calculated value of the induced velocity This process is repeated in order to correct the predicted values of the induced velocities and bound velocities This simple predictor corector method appear to be adequate since the bound vortex rings do not cause large perturbation velocities at nearby blade panels due to their relative geometric orientation The next step is the calculation of blade element performance torque and power output These values are output at this point along with induced velocities at each blade element The next major step is to recalculate the position of all wake vortex elements using a simple convection equation dr dt V Time is incremented and a new set of shed vortices are created If a revolution has been completed the rotor performance for the revolution is output The process is repeated for the desired number of revolution of the rotor The pressure coefficient in a point of interest on the blade surface can be expressed as 7 A free wake aerodynamic model for helicopter rotors 2 p p V 2 d ae ee e eam 13 P pve 2 ta V t US For thin bodies the vorticity inside the bound surface is proportional to the velocity jump across that surface In suc
12. in the reference system attached to the rotor axis allows calculation of thrust and torque of the rotor The thrust torque and power coefficients are defined as nondimensional parameters of the rotor as follows T O P ea Oo __ _ _______ pn R QO Q pr R QO P pr R Q 20 T Horia DUMITRESCU Florin FRUNZULICA 8 6 RESULTS Figure 5 shows the wake development in two situation a ascending flight with velocity V z 1 75m s and b forward flight with V 42m s 8 tilt axis of rotor In case a the neglect of development a large core radius for older vortex elements conducts at distorted local wake structure The beginning of rotor motion was set impulsively Because the time for a complete wake development is too large the process was stopped after five revolutions In the future we intend to split the wake in two regions the near region where the wake is treated numerically complete and a far region where the wake is treated numerically simplified i l nee N b i Figure 5 The wake development a ascending flight and b forward flight The rotor has two blades with following characteristics 3 5 m blade tip radius 0 3 m hub radius 3 conical angle The each blade is modeled with a rigid flat plate with 4 geometric angle of attack and its are fixed at 35 of blade chord elastic axis of blade For the case test number two we consider a two blades rotor B 2 in hover wi
13. ion points one obtains a linear algebraic system of equations with the unknowns I k 1 M xN This system can be solved using an appropriate algorithm The vortices generated at the trailing edge of each blade maintain their intensity constant in time and are transported with a velocity equal to the local velocity of the air flow such that the theorem of Kelvin will be satisfied DY _ ol Dt ot The shape of the wake is given by the condition V0 4 V V P 0 4 1 e the wake is force free The time step should be chosen such that the vortex generated at the leading edge should not be propagated over a distance greater than the smallest dimension of the panel shape at the trailing edge in order to avoid the degeneration of the propagated vortex rings especially at the blade tips The wake geometry is complete fig 2a But towards the blade tip the vortices which spring from trailing edge roll in space around a curve line generate by the blade tip In this case is numerically efficient to use a new model for wake at blade tip which it replace the rollup vortices with a curvilinear vortex fig 2b reler ine haird Hire shod eferenece ine biada b inboard varie shet Figure 2 a Complete wake development and b simplified wake model at blade tip Horia DUMITRESCU Florin FRUNZULICA 4 At far distance from panel using equivalence between the vortex ring model and the dublet with uniform intensity
14. stributed over the median surface of the blades fig 1 The leading segment of the vortex ring is placed at the panel quarter chord line and the collocation point is at the center of the panel s three quarter chord line This choice is justified by the fact that the velocity induced by a distribution of vortices is the same with the one given by a vortex placed at 1 4 chord line with the calculating point considered at 3 4 of the chord line valid in the case of flat plate 1 A blade LE collocation point 5 normal vector LZS t blade TE Ip S bound vortices r SQ A AN O PETES free wake vortices Xh A Figure 1 Vortex ring model for a thin lifting surface X Y Zp blade coordinate system TE trailing edge LE leading edge W wake 3 A free wake aerodynamic model for helicopter rotors The local velocity V for each collocation point is a combination of the self induced velocity V ina p the kinematic velocity V p t due the motion of the blade and the wake induced velocity V p E Cus 00 E E C lor O j l i Ped M w Vp Vap t Vaa P Vyp VRP t x lt ll M II l The intensity 1 of the vortex rings is found imposing the tangential flow condition on the wing surface M w V tT ti V O i 3 lL J M II E Car T j l i l j M II l Applying this relation or every collocation point k MxN collocat
15. th characteristics solidity B c mR 0 0637 tip Mach number RQ a 0 19 pitch angle B 8 and constant chord distribution The each blade is modeled with a thin lifting surface rigid flat plate discretized in 7 points in chord direction and 13 points in radial direction Figure 6 shows computed and experimental nondimensional lift distribution along blade The numerical results obtained by VLM C 0 00391 lie closer to the experimental data 15 C obtained for an axial wake development equal with 1 6 R related at rotor plane 0 00380 because of more realistic modeling of the free wake development These results are xXp yiR Figure 6 Nondimensional lift distribution along the blade y R nondimensional radius 9 A free wake aerodynamic model for helicopter rotors 7 CONCLUSIONS The primary objective of this paper has been to describe the implementation of a computational procedure for helicopter rotor downwash calculations which utilizes the Vortex Lattice Method for free wake modelling The formulation of rotor wake discretization and the resulting induced downwash prediction were analyzed The major features of the rotor blade lifting surface aerodynamic model have been outlined Several aspects of rotor wake simulation such as tip vortex roll up process and vortex core modeling included in the presented methodology were discussed in some detail An integrated computer code has been developed applyin
16. wed to convect in the velocity field they create Provided the numerical method employed is convergent the vortex elements will move until they take up positions which are consistent with the velocity field As might be expected the computer requirements for such calculations are prodigious which makes this kind of analysis very expensive This is the reason why some investigators have modeled the rotor wake with two or three different regions near intermediate and far wake At each region the computations are done in a way which is appropriate to that region This approach causes a reduction in the computational effort The present free wake model is based on the time marching method where motion begins from an impulsive start vith the subsequent generation of a vortex wake modeled by a equence of discrete shed at equal time intervals Thus for steady state motion the force and moment responses are asymptotically achieved Vortex model It is know that the Biot Savart law for induced velocity produces a singularity whenr 0 Also the induced velocities at points very close to the vortex will lead at unrealistically large induced velocities These numerically aspects can create non physical perturbations to the local angle of attack unusually large wake induced velocities at blade tip with effect over tip vortex geometry large distortion of wake etc A direct consequence is the instability of the solution In the present work
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