A Space-Time Accurate Finite-State Inflow Model for

A Space-Time Accurate Finite-State Inflow Model for Aeroelastic Applications Massimo Gennaretti Full Professor Roma Tre University Roma, Italy

Riccardo Gori Felice Cardito PhD student PhD student Roma Tre University Roma Tre University Roma, Italy Roma, Italy Giovanni Bernardini Assistant Professor Roma Tre University Roma, Italy

Jacopo Serafini Research Fellow Roma Tre University Roma, Italy

Dynamic wake inflow modelling is one of the main issues in the development of efficient and high-fidelity simulation tools for design of new generation rotorcraft. The aim of this paper is the development of a LTI, space-accurate, finitestate model for the prediction of dynamic wake inflow perturbations of helicopter rotors in arbitrary steady flight, extracted from high-fidelity aerodynamics solvers. The proposed dynamic wake inflow model, coupled with sectional aerodynamic load theories, may be conveniently applied for aeroelastic and aero-servo-elastic analysis of rotors with the introduction of a limited number of additional states, thus turning out to be suitable for real-time predictions. The outcomes of numerical investigations concerning a helicopter rotor in hovering and forward flight conditions are shown to demonstrate the capability of the introduced model to capture with a good level of accuracy the time evolution of the wake inflow radial distribution on blades of rotors subject to arbitrary perturbations.

NOTATION

α,c α,s λ jα,0 , λ j,m , λ j,m

wake inflow parameters

λ

vector of wake inflow parameters radial basis functions

A1 , A0 , A, B, C

rational-matrix approximation matrices

c

blade chord

N/2 φ j0 , φ jc , φ js , φ j

α,c α,s H α,0 jk , H jk,m , H jk,m

inflow parameters transfer functions

ω

perturbation frequency

H

inflow parameters transfer matrix

Ω

rotor angular velocity

Nb

number of blades

Nm

number of multi-harmonic components

N/2 Nr0 , Nrs , Nrs , Nr

number of radial functions

qr

kinematic perturbation

q

vector of kinematic perturbations

R

rotor radius

(r, ψ)

polar coordinates on rotor disc

r

wake inflow dynamics states

s

Laplace-domain variable

v

approximated wake inflow

vi

wake inflow evaluated on i-th blade

v0 , vc , vs , vN/2

wake inflow multiblade coefficients

θ0 , θc , θs

blade pitch controls

θtw

blade twist

µ

advance ratio

N/2 λ j0 , λ jc , λ js , λ j

wake inflow components

INTRODUCTION Wake inflow is one of the main issues in helicopter aeroelasticity and flight mechanics analysis in that, altering the blade radial distribution of the effective angle of attack, it deeply affects main rotor aerodynamic responses. While the use of aerodynamics tools based on three-dimensional, unsteady formulations would represent a definitive solution, their computational cost is still too high for design oriented applications, thus making the sectional-aerodynamics/wake-inflow combination a convenient solution process for aeroelastic, flight dynamics and flight control system analyses. Finite-state dynamic inflow modelling is an effective solution strategy to include the wake unsteady effects in rotorcraft simulation tools. Most of the developed finite-space inflow models are based on the closed form acceleration-potential solution of the fluid dynamics over a disk. Widely-used tools are the Pitt-Peters (Refs. 1, 2) and the Peters-He (Ref. 3) models, along with their extended versions taking into account wake distortion effects (see, for instance, (Refs. 4, 5)). Based on simplifying assumptions for the flow field solution, they are computationally very efficient. The Pitt-Peters model is suited for low-frequency flight dynamics simulations, whereas the

Presented at the AHS 72nd Annual Forum, West Palm Beach, c 2016 by the AmeriFlorida, May 17–19, 2016. Copyright can Helicopter Society International, Inc. All rights reserved. 1

Peters-He one, with the introduction of more accurate approximation forms of azimuthal and radial inflow distributions, is applicable to high-frequency aeroelastic problems.

Indeed, introducing over the rotor disc the hub-fixed polar coordinate system, (r, ψ), for a four-bladed rotor, the perturbation wake inflow, v, is represented through the following extension of the form applied in the Pitt-Peters model (Refs. 1,2)

The main objective of this paper is to present a methodology to extract radial/azimuthal-accurate, finite-space LTI models of rotor dynamic inflow perturbations from highfidelity aerodynamics solvers. It is an extension of the LTI, finite-state modelling approach recently presented by the authors (Refs. 6,7), suitable for helicopter flight dynamics applications, and closely related to the recent work in (Ref. 8). The model parameters are identified by a multi-step technique similar to that recently introduced in (Ref. 6). Considering rotors in arbitrary steady flight, the transfer functions relating perturbations of the system state variables to wake inflow are determined by the analysis of an appropriate sequence of timemarching aerodynamic responses, while a rational-matrix approximation algorithm is applied for their finite-state approximation (see also (Refs. 9, 10), for applications of similar approaches to aerodynamic and aeroelastic loads). The wake inflow responses are evaluated through simulations provided by a Boundary Element Method (BEM) tool for potential-flow solutions, suited for rotors in arbitrary motion (Ref. 11): it is capable of accurate simulations taking into account free-wake and aerodynamic interference effects in multi-body configurations (like coaxial rotors or rotor-fuselage systems), as well as severe blade-vortex interactions.

v(r, ψi ,t) = v0 (r,t) + vc (r,t) cos ψi + vs (r,t) sin(ψi )+ + vN/2 (r,t) (−1)i

(1)

where ψi denotes the azimuth position of the i-th blade, whereas v0 , vc , vs and vN/2 are, respectively, the instantaneous multiblade collective, cyclic and differential inflow coefficients at a given radial position, r. For vi denoting the wake inflow perturbation evaluated by the high-fidelity aerodynamic tool on the i-th rotor blade, applying the separation of variables technique, and choosing suitable sets of linearly independent radial basis functions, φ jα (r), the multiblade wake inflow coefficients are expressed in the following form v0 (r,t) =

1 Nb ∑ vi (r,t) = Nb i=1

Nr0

∑ λ j0 (t) φ j0 (r)

j=1

Nb

2 vc (r,t) = ∑ vi (r,t) cos ψi = Nb i=1 2 Nb vs (r,t) = ∑ vi (r,t) sin ψi = Nb i=1

Considering a helicopter rotor in hovering and advancing flight conditions, the numerical investigation presents the identified transfer functions, the assessment of the accuracy of their rational approximations, and the validation of the proposed dynamic inflow model by comparison with wake inflow directly calculated by the time-marching BEM solver. In particular, the attention is focused on the capability of capturing accurately radial distribution variations and multi-harmonic components arising in forward flight.

Nrc

∑ λ jc (t) φ jc (r)

j=1

(2)

Nrs

∑ λ js (t) φ js (r)

j=1

N/2

Nr 1 Nb N/2 N/2 vN/2 (r,t) = vi (r,t) (−1)i = ∑ λ j (t) φ j (r) ∑ Nb i=1 j=1

where Nb denotes the number of rotor blades, Nrα is the number of functions used to define the α-coefficient radial distribution, with λ jα representing the corresponding component onto the basis function φ jα . The differential component appears only for rotors with an even number of blades, whereas higher-harmonic multiblade cyclic components are included for Nb > 4. It is worth noting that, neglecting the differential and (if present) the higher-harmonic cyclic components, for Nr0 = Nrc = Nrs = 1, φ10 = 1, and φ1c = φ1s = r, the proposed inflow distribution coincides with that of the Pitt-Peters model (Refs. 1, 2).

FINITE-STATE INFLOW MODEL EXTRACTION The main steps in the development of the proposed finite-state wake inflow model extraction from arbitrary high-fidelity aerodynamic solvers are: definition of a suited approximation form of the inflow distribution; LTI representation of the model parameters identified from responses of the aerodynamic solver; finite-state approximation of the parameters dynamics, and hence of the inflow model.

Extraction of Model Parameters LTI Dynamics The methodology proposed for the extraction of the dynamic wake inflow model from a high-fidelity aerodynamic solver starts with the identification of the transfer functions relating the perturbations of the system state variables with the components of the inflow multiblade coefficients.

The way these steps are implemented in the proposed methodology is described in the following.

To this purpose, following an approach similar to that used in (Refs. 6, 7), first, a high-fidelity aerodynamic tool is applied to evaluate the blade wake-inflow perturbations, vi (r,t), corresponding to harmonic perturbations of the fixed-frame variables, qk , associated to the degrees of freedom and control variables of interest (like, for instance, blade deformation multiblade coordinates, hub motion variables, blade pitch control variables).

Inflow Representation In order to represent the complex wake inflow field over a rotor blade with a level of accuracy suitable for (high-frequency) aeroelastic applications, it is expressed in a non-rotating frame in terms of spanwise-varying multiblade variables. 2

In this context, it is important to observe that, except for axi-symmetric hovering operating conditions, the timeperiodic nature of rotor aerodynamics yields multi-harmonic responses to single-harmonic inputs, with multiplicity related to the number of blades (when fixed-frame and/or multiblade input/output variables are considered, as in the present case). For instance, for small-perturbation harmonic cyclic inputs of frequency ω, non-zero harmonic components of the corresponding cyclic outputs appear at the frequencies ω, mNb Ω − ω, and mNb Ω + ω, for m = 1, 2, ..., Nm , with Nm related to the maximum order of periodicity of the aerodynamic operator. Considering a four-bladed rotor in forward flight at advance ratio µ = 0.2, and wake inflow spatial distribution of Pitt-Peters-model type (namely, Nr0 = Nrc = Nrs = 1, φ10 = 1, and φ1c = φ1s = r), this is confirmed by the result in Figure 1 that presents the spectrum of the λ1s response to a lowfrequency harmonic longitudinal cyclic pitch. Indeed, this figure shows that it is characterized by a single tonal peak at the very-low input frequency, and by two couples of tonal peaks, one around the 4/rev harmonic and one (smaller) around the 8/rev one.

coefficient Fourier series of fundamental frequency Nb Ω λ jα (t) = λ jα,0 (t) i (3) Nm h α,c α,s + ∑ λ j,m (t) cos(m Nb Ωt) + λ j,m (t) sin(m Nb Ωt) m=1

and Nm

λ jα (t) =

∑

n α,c λ j,m (t) cos[(1/2 + m) Nb Ωt)]

m=0

+

(4)

o

α,s λ j,m (t) sin[(1/2 + m) Nb Ωt]

with Eq. 3 applied to collective/cyclic responses to fixedframe/collective/cyclic inputs and differential responses to differential inputs, whereas Eq. 4 is valid for collective/cyclic responses to differential inputs and differential responses to fixed-frame/collective/cyclic inputs. Indeed, considering, for instance, input/output variables related to Eq. 3, for a given harmonic input qk at frequency ω, the corresponding harmonic components of λ jα at the same frequency coincide with those of λ jα,0 , whereas from its harmonic components at the frequencies (m Nb Ω − ω) and (m Nb Ω + ω) it is possible to determine the harmonic compoα,c α,s nents of λ j,m and λ j,m at frequency ω (similar considerations may be applied to the input/output relations associated to Eq. 4). The harmonic components at frequency ω of the inflow paα,c α,s rameters, λ jα,0 , λ j,m and λ j,m , suitably combined with the input harmonic component, provide the complex values, at this α,c α,s frequency, of the transfer functions H α,0 jk , H jk,m , H jk,m relating them to the input, qk . Finite-State Approximation The transfer functions between inputs and the inflow parameters introduced in Eqs. (3) and (4) represent LTI dynamics, thus allowing the description of a LTP operator through combination of outputs of LTI sub-operators (this approach is equivalent to that outlined in (Ref. 12), widely applied for LTP system identification).

Fig. 1. Spectrum of λ1s response to harmonic longitudinal cyclic pitch.

In order to obtain a finite-state description of the wake inflow dynamics, the LTI dynamics of the parameters introduced in its space-time representation (collected in the vector λ ) is conveniently represented through the following transfermatrix form

However, it is important to observe that a different input/output frequency correlation occurs if the input is of differential type and the output is of collective/cyclic type, or the input is of fixed-frame/collective/cyclic type and the output is of differential type. Indeed, in this case, the response to small-perturbation harmonic cyclic inputs of frequency ω, has non-zero harmonic components at the frequencies (1/2 + m)Nb Ω − ω, and (1/2 + m)Nb Ω + ω, for m = 0, 1, ..., Nm .

λ˜ = H(s) q˜

(5)

where s is the Laplace-domain variable, the matrix H collects all the transfer functions extracted from the aerodynamic responses, whereas q denotes the vector of the inputs of interest (expressed through fixed-frame/multiblade variables). Considering the frequency domain of interest of the problem the inflow model is identified for, the process of evaluation of matrix H is repeated for a discrete number of frequencies within that range, so as to get an adequate sampling

Therefore, it is convenient to express each inflow component, λ jα , through one of the following two time-dependent 3

of it. Then, the rational-matrix approximation (RMA) of the following form H (s) ≈ s A1 + A0 + C [s I − A]−1 B

(6)

that provides the best fitting of the sampled H-matrix values is determined through a least-square technique (Refs. 9, 13). A1 , A0 , B and C are real, fully populated matrices, whereas A is a square block-diagonal matrix containing the poles of the approximated transfer functions. Finally, transforming into time domain the combination of Eqs. 5 and 6 provides the following LTI, finite-state model for the wake inflow parameters λ =A1 q˙ + A0 q + C r r˙ =A r + B q

(7)

where r is the vector of the additional states representing wake inflow dynamics. The above LTI differential model relating rotor kinematics to inflow parameters, combined with Eqs. (3) and (4) provides a LTP operator for the wake inflow components. Applying these components in Eq. 2 and coupling with Eq. 1 yields a space-time accurate LTP prediction model of the wake inflow on the rotor disc, suitable for aeroelastic (and flight dynamics) applications.

Fig. 2. Transfer function λ10 vs θ0 in hover. • sample.

RMA;

Fig. 3. Transfer function λ1s vs θc in hover. • sample.

RMA;

Eliminating the multi-harmonic cyclic terms from Eqs. 3 and (4), the wake inflow components operator would becomes of LTI type, like that of the wake inflow model extraction approach introduced in (Refs. 6, 7), for low-frequency flight dynamics applications. The inclusion of the multi-harmonic cyclic terms improves the accuracy of the predicted wake inflow (particularly, the higher frequency content).

NUMERICAL RESULTS In this section, the proposed space-time accurate wake inflow model is verified and validated. The aerodynamic predictions used to extract the inflow model are obtained by an unsteady, potential-flow, BEM tool for rotorcraft, extensively validated in the past by some of the authors (Refs. 11, 14). Considering a rotor with four blades, radius R = 4.91 m, constant chord c = 0.27 m, twist angle θtw = −8◦ from root to tip, and angular speed Ω = 44.4 rad/s, the proposed wake inflow model is assessed for both hovering condition and forward flight at µ = 0.2. Wake inflow transfer functions and their rational approximations are examined, along with the capability of the resulting finite-state model to predict wake inflow radial distribution and time evolution due to arbitrary blade pitch control perturbations.

Dividing the blade span into a number of finite segments, the basis functions used in Eq. 2 are such to provide a linear distribution of inflow coefficients within each segment, assuring continuity at the edges of them.

The rotor wake in the aerodynamic BEM solver is assumed to have a prescribed helicoidal shape that, in forward flight, coincides with the surface swept by the trailing edges, whereas in hovering has a spiral length given by the mean trimmed inflow. In addition, all the results presented in the following concern nondimensional wake inflow obtained by division with the factor ΩR.

Hovering condition In this flight condition the aerodynamic operator is of timeconstant nature, and thereby multi-harmonic inflow parameters are not introduced. 4

First, the excellent quality of the RMA applied to the transfer functions samples extracted from the BEM solver is shown in Figures 2 and 3, which present the frequency behaviour of λ10 vs θ0 and λ1s vs θc , respectively (note that, in this case, λ10 ≡ λ10,0 and λ1s ≡ λ1s,0 ).

distributions of the percentage prediction error (related to the sectional inflow peak) obtained by the present formulation and by the linear Pitt-Peters-like model of (Ref. 6). The improvement of the inflow prediction provided by the more accurate radial description is considerable throughout the blade span.

A similar high level of accuracy is observed for all of the transfer functions involved in matrix H for the hovering case. The evaluated RMA introduces 17 pairs of complex conjugate poles corresponding to 34 additional states: however, note that these are preliminary results aiming at verifying and validating the proposed methodology; a significant decrease of the number of RMA poles can be obtained by a suitable trade-off between accuracy and computational efficiency of the resulting model. Next, considering a collective pitch perturbation, θ0 , having the form of a chirp signal with frequency sweeping the domain [0, 0.3/rev] and maximum amplitude equal to 1◦ , the corresponding wake inflow perturbations predicted by the proposed methodology are correlated both with those directly computed by the BEM solver and with simulations of the inflow model of (Ref. 6), based on the Pitt-Peters-like linear spatial approximation form (linear PP). This comparison is shown in Figure 4 for the wake inflow evaluated at three blade sections, r/R = {0.7, 0.85, 0.95}. These results demonstrate Fig. 5. Spanwise error distribution for hovering condition. present; linear.

Forward Flight Regarding the advancing flight condition at µ = 0.2, first, Figures 6 and 7 present, two examples of transfer functions, rec,c spectively λ10,0 vs θ0 and λ2,1 vs θs . Akin to the hovering case examined, these figures demonstrate that the applied RMA is of excellent accuracy; this level of accuracy is obtained for all of the transfer functions in matrix H introducing 17 pairs of complex conjugate poles (note that Nm = 2 is considered). Concerning the high number of additional states introduced, observations similar to those discussed for the hovering case can be made: they could be significantly reduced by a suitable trade-off between accuracy and computational efficiency (not considered in theses preliminary results). Fig. 4. Wake inflow predictions for hovering condition, at r/R = 0.7 (top), r/R = 0.85, r/R = 0.95 (bottom). BEM; present; linear PP.

Next, similarly the hovering case analysis, the inflow perturbation generated by a chirp-type collective pitch, θ0 , has been examined (in this case the signal frequency sweeps the domain [0, 3/rev]). For the blade sections r/R = {0.7, 0.85, 0.95}, Figures 8-10 present the comparison of the wake inflow directly computed by the BEM solver with predictions from the proposed formulation, with and without inclusion of multi-harmonic components. In addition, it shows also the inflow provided by the model employing a Pitt-Peterslike linear radial distribution, again with and without inclusion of multi-harmonic components.

the capability of the present approach to capture with good accuracy the radial distribution of the wake inflow that, especially at the tip of the blade, shows significant gradients. A more complete view of the relevant enhancements introduced by the radial approximation expressed by the series expansions in Eq. 2 is given in Figure 5. It presents the spanwise 5

improvement with respect to the simpler models already observed in the hovering case. This is also demonstrated by the corresponding spanwise distributions of the percentage error predictions (defined as for the hovering analysis) shown in Figure 11. It highlights the low increase of accuracy produced by inclusion of higher harmonic terms in the linear radial approximation model, as well as the beneficial effects of more accurate radial approximations combined with inclusion of multi-harmonic inflow variables.

c,c Fig. 6. Transfer function λ10,0 vs θ0 and λ2,1 in forward flight. RMA; • sample.

Fig. 8. Wake inflow predictions for forward flight at r/R = 0.7. BEM; present with multi-harmonic terms; present without multi-harmonic terms; linear PP with multi-harmonic terms; linear PP without multi-harmonic terms.

CONCLUSIONS From the numerical assessment of the proposed methodology for extraction of space-time accurate finite-state wake inflow models from high-fidelity aerodynamic solvers, the following conclusion are drawn:

c,c Fig. 7. Transfer function λ2,1 vs θs in forward flight. RMA; • sample.

• for rotors in hovering condition, the radial variation of the wake inflow is captured with good accuracy by introduction of four basis functions per inflow coefficient;

It is interesting to note that, the linear model is not significantly improved by the introduction of multi-harmonic terms (these seem to cancel out by the assumption of linear radial approximation, maybe due to the phase shift of higher harmonic signal at different radial position). Furthermore, it is evident that the application of a more accurate radial distribution approximation provides enhanced predictions, which are further improved by inclusion of multi-harmonic terms. The prediction provided by the proposed inflow model is of good quality throughout the whole blade span, confirming the

• in hovering conditions, in the absence of multi-harmonic outputs due to the time-constant nature of the aerodynamic operator, the inflow dynamics is satisfactorily described by introducing 34 states (note that, these are preliminary results: a suitable trade-off between accuracy and computational efficiency of the model would reduce the number of the additional states); 6

Fig. 9. Wake inflow predictions for forward flight at r/R = 0.85. BEM; present with multi-harmonic terms; present without multi-harmonic terms; linear PP with multi-harmonic terms; linear PP without multi-harmonic terms.

Fig. 11. Spanwise error distribution for forward flight condition. present with multi-harmonic terms; present without multi-harmonic terms; linear PP with multi-harmonic terms; linear PP without multi-harmonic terms. tion; • in forward flight, a more accurate radial approximation provides significant enhancement of the quality of the simulated inflow, which is further improved by inclusion of the multi-harmonic terms; • the number of inflow states introduced by the present model in case of forward flight condition is equal to 34, whether or not the multi-harmonic contributions are included: these are representative of the wake vorticity unsteady effects and affects dynamics of all of the considered inflow parameters (also in this case, a suitable trade-off between accuracy and computational efficiency would reduce the number of additional states required). The preliminary results presented demonstrate the capability of the proposed finite-state modelling to capture multiharmonic inflow terms deriving from the time-periodic nature of the aerodynamic operator in forward flight condition, and hence to predict with good space and time accuracy the wake inflow on advancing and hovering rotors. It seems to be an inflow simulation tool suitable for rotor aeroelastic stability and control applications.

Fig. 10. Wake inflow predictions for forward flight at r/R = 0.95. BEM; present with multi-harmonic terms; present without multi-harmonic terms; linear PP with multi-harmonic terms; linear PP without multi-harmonic terms.

Author contact: M.Gennaretti, [email protected].

• in forward flight, when a linear space approximation form along the rotor disc is applied, the introduction of the inflow parameters associated to the multi-harmonic outputs scarcely affects the quality of the inflow simula-

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