NUMERICAL DEVELOPMENT OF UNSTEADY VORTEX-LATTICE METHOD EXPANSION FOR LEADING-EDGE SHEDDING PREDICTION WITH EXPERIMENTAL VALIDATION Monteiro, T. P. * , Ramesh, K. ** , Silvestre, F. * * Instituto Tecnolø sgico de Aeronautica, Brazil , ** University of Glasgow, UK Keywords: Aerodynamics, Aeroelasticity, Wind Tunnel Abstract With a steady grow in fuel emissions and flight efficiency requirements, airplanes are being de- veloped with higher aspect ratios and lighter materials. This increases overall flexibility, and requires more advanced aerodynamic and structural models. The Unsteady Vortex-Lattice Method (UVLM) has been successfully used for these cases, however one of its shortcommings is it lacks the ability to model leading-edge stall. This work shows the ongoing development of an expanded UVLM, capable of predicting leading- edge vortex shedding and its application to flexi- ble aircraft. The model is validated against exper- imental wind-tunnel tests of a flat plate undergo- ing prescribed movement and flutter limit-cycle oscillations. 1 Introduction Higher aspect ratios and lighter materials seek to reduce fuel emissions and increase flight ef- ficiency by reducing lift to drag ratio and overall weight, respectively, of the aircraft. In particu- lar, High-Altitude Long Endurance (HALE) air- caft tend to have this characteristics as their main engineering feature. A number of modern aircraft, for example the F-16s, have been exposed to flutter conditions, and limit-cycle oscillations (LCOs) where obe- served when nonlinearities on the system, such as geometric, structure and aerodynamic, acted to limit the motion amplitude [3]. This, with the recent trend in design of higher aspect ratios and lighter materials, have created instances where rigid-body and aeroelastic dynamics are coupled and need to be taken into account together. Flutter, in this cases, is usually modeled with linear analysis, however LCOs are by their very nature nonlinear. This inability to model LCOs results in extensive, and expensive, flight testing. These nonlinearities can be of structural or aero- dynamic nature, with structural nonlinearity ris- ing from large deformations, material properties, or loose linkages, whereas aerodynamic nonlin- earities results from compressibility or viscous effects. This work focus on aerodynamic non- linearities[1]. The Unsteady Vortex-Lattice Method (UVLM) is a three-dimensional aerodynamic model based on potential-flow formulation that shows great promise to model nonlinear aero- dynamics due to its ability to model nonplanar wakes. The computational cost of this method sits between linear models such as Doublet- Lattice Method, and high-fidelity ones such as Computation Fluid Dynamics, making it an ideal candidate for HALE aircraft[2]. By dynamically modeling the wake at each time-step, it can account for large, and fast move- ments, however, flow separation is still a limi- tation. In particular, Leading-Edge flow separa- tion has been proven to be major contributor to 1
Transcript
NUMERICAL DEVELOPMENT OF UNSTEADYVORTEX-LATTICE METHOD EXPANSION FOR
With a steady grow in fuel emissions and flightefficiency requirements, airplanes are being de-veloped with higher aspect ratios and lightermaterials. This increases overall flexibility,and requires more advanced aerodynamic andstructural models. The Unsteady Vortex-LatticeMethod (UVLM) has been successfully used forthese cases, however one of its shortcommingsis it lacks the ability to model leading-edge stall.This work shows the ongoing development of anexpanded UVLM, capable of predicting leading-edge vortex shedding and its application to flexi-ble aircraft. The model is validated against exper-imental wind-tunnel tests of a flat plate undergo-ing prescribed movement and flutter limit-cycleoscillations.
1 Introduction
Higher aspect ratios and lighter materials seekto reduce fuel emissions and increase flight ef-ficiency by reducing lift to drag ratio and overallweight, respectively, of the aircraft. In particu-lar, High-Altitude Long Endurance (HALE) air-caft tend to have this characteristics as their mainengineering feature.
A number of modern aircraft, for example theF-16s, have been exposed to flutter conditions,and limit-cycle oscillations (LCOs) where obe-served when nonlinearities on the system, such
as geometric, structure and aerodynamic, actedto limit the motion amplitude [3]. This, with therecent trend in design of higher aspect ratios andlighter materials, have created instances whererigid-body and aeroelastic dynamics are coupledand need to be taken into account together.
Flutter, in this cases, is usually modeled withlinear analysis, however LCOs are by their verynature nonlinear. This inability to model LCOsresults in extensive, and expensive, flight testing.These nonlinearities can be of structural or aero-dynamic nature, with structural nonlinearity ris-ing from large deformations, material properties,or loose linkages, whereas aerodynamic nonlin-earities results from compressibility or viscouseffects. This work focus on aerodynamic non-linearities[1].
The Unsteady Vortex-Lattice Method(UVLM) is a three-dimensional aerodynamicmodel based on potential-flow formulation thatshows great promise to model nonlinear aero-dynamics due to its ability to model nonplanarwakes. The computational cost of this methodsits between linear models such as Doublet-Lattice Method, and high-fidelity ones such asComputation Fluid Dynamics, making it an idealcandidate for HALE aircraft[2].
By dynamically modeling the wake at eachtime-step, it can account for large, and fast move-ments, however, flow separation is still a limi-tation. In particular, Leading-Edge flow separa-tion has been proven to be major contributor to
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MONTEIRO, T. P. , RAMESH, K. , SILVESTRE, F.
LCOs[3].Ramesh et al[1], has developed a two-
dimensional method that, by using leading-edgesuction as a modulator, can predict leading-edge flow separation and shed leading-edge vor-tices (LEV) into the flow, effectively modelingLCOs on an airfoil, this method was called Lesp-Modulated Discrete Vortex Method (LDVM).This method is currently being expanded to 3Dcases using a strip theory approach, with promis-ing results[3].
An UVLM implementation that combinesLDVM’s leading-edge vortexes and UVLM wasproposed by Hirato [4]. With special attentionto the numerical accuracy and stability, this workproved that it was possible to add LDVM’s LESPmodulation to UVLM with good results.
This work expands the findings Hirato,proposing a matrix-based way to calculate theLeading-Edge Vortex Modulator in UVLM. Thiswork also seeks to validate LESP-Modulationagainst wind tunnel test results for prescribedflexible body movement and coupled struc-tural/aerodynamic flutter. LDVM results werealso used to validate the new model.
2 Model Development
2.1 Aerodynamic Model
The current UVLM model was based on Katz andPlotkin [4] implementation, with an expansion tocalculate the LESP for each chordwise strip fol-lowing the two-dimensional panel method pre-sented in the previous section. The implementa-tion was done in the MATLAB environment, withextensive use of C code generation technology tospeed up simulation time.
Katz and Plotkin[4] model calculates thebound vortex distribution by a system of equa-tions (1), where the right-hand side (RHS) is thenormal velocity contribution from the developedwake and the wing kinematics, and the ai j coef-ficients are the calculated as the influence of unitbound vortexes rings in each other.
a11 a12 · · · a1ma21 a22 · · · a2m
...... . . . ...
am1 am2 · · · amm
Γ1Γ2...
Γm
=
RHS1RHS2
...RHSm
(1)
Where,
m = MpanelRows ∗NpanelCollumns (2)
From the bound vortex distribution, it is pos-sible to calculate the pressure distribution at eachpanel. Equation 2 calculates the force vector ateach panel, with the third element of the vectorrepresenting the Lift force.
Where [U(t),V (t),W (t)] is the kinematiccontribution, [uW ,vW ,wW ] is the wake contribu-tion, τi and τ j are the tangent unit vector chord-wise and spanwise, respectively. ∆ci j and ∆bi jare the panel sizes, chordwise and spanwise re-spectively. Finally, ρ is the air density, ∆Si j isthe panel area and ni j is the normal vector at thepanel. Lift and Moment coefficients were cal-culated for each spanwise location by combin-ing the forces and moments of every chordwisepanel. Equations 3 and 4 show how the lift andmoment coefficient calculation for each spanwiselocation.
CL j =∑∆F(3)i j
q∞S1(4)
CM j =∑L∗ (xpvt
j − xΓi j)
q∞cS1(5)
Where the xpvtj is the aerodynamic pivot posi-
tion for the spanwise position j, xΓi j is the vortex
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Numerical Development of Unsteady Vortex-Lattice Method Expansion for Leading-Edge SheddingPrediction with Experimental Validation
ring position at the panel, and q∞ is the dynamicpressure.
From the model developed by Hirato, Y[4]we can calculate an approximation for A0 fromthe leading-edge panel circulation. This approxi-mation is empirically determined and will be ap-plied to the current model. It is currently underdevelopment a more robust, and efficient, way ofcalculating the A0 is under development to reusesome of the assets already calculated by UVLM.Equation 3 gives the A0 for each strip in the wing.
A0(t,y j) =
1.13Γ1(t,y j)
U∞c[cos−1(1− 2∆xc )+ sin(cos−1(1− 2∆x
c ))](6)
This approach to the solution proved wellsuited for panel based methods, not requiring afiner mesh, and providing a good approximationfor the A0.
2.2 Structural Model
Two structural models were used to verify theproposed UVLM changes against the experi-ments: The first model, presents a controlled firstbending mode deflection, allowing for the mod-eling of the prescribed results. The second wasa coupled structural model to allow the model-ing of flutter condition. In both cases, integrationwas made around changes to the local grid usedin UVLM, with the elastic displacements happen-ing around the local axis.
2.2.1 Prescribed Motion Structural Model
For the prescribed motion model, BisplinhoffR.[6] provides a good approximation for the firstmode shape of a cantilever beam. This type ofmodel is considered an uncoupled model: Posi-tions and deformations are defined as a vector intime and aerodynamic forces and moments arecalculated without acting on the structure. Equa-tion 4 gives the vertical displacement of a pointin the wing span.
h(y) = D[(sin(
√wna b)− sinh(
√wna b)
cosh(√wn
a b)+ cos(√wn
a b))
(sinh(√
wn
ay)− sin(
√wn
ay))
+(cosh(√
wn
ay)− cos(
√wn
ay))] (7)
Where,
a =
√EIm
(8)
In equation 5, E is the elastic coefficient, I thewing Inertia and m is the mass. The parameter Dfrom Equation 4 can be obtained by applying theequation o a known condition, in this case, themaximum measured wing tip displacement.
2.2.2 Flutter Structural Model
For flutter prediction and simulation, a coupledmodel is required. A modal superposition ap-proach can be used to great effect in calcu-lating the deformations in the wing and cou-pling it to the aerodynamic forces calculated byUVLM.Writing the physical coordinates as equa-tion 6, it is possible to express the deformationsas a combinations of modal shapes. These can betruncated to allow for a order reduction and fastexecuting code.
q = φη (9)
Where φ is the eigenvector matrix represent-ing the modal shapes and η is the generalizeddisplacements. The equations of motion can berewritten as:
ηmx1 +2εmxmωnmxmηmx1 +ω2nmxm
ηmx1 =
µ−1mxmQmx1 (10)
With m being the number of modes usedat the truncation, ε being the diagonal matrixof modal damping, ωn the diagonal matrix ofmodal natural frequencies, µ the diagonal matrix
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MONTEIRO, T. P. , RAMESH, K. , SILVESTRE, F.
of modal masses and Q the vector of generalizedforces.
The generalized forces Q can be written asa function of CL and CM, both calculated by theUVLM model. Equation 8 shows generalizedforces calculation.
Q = φq∞
−CLc∆yCMc2∆y
0
(11)
3 Wind Tunnel Experiments
Initial experiments were developed to verifyaerodynamic nonlinearities caused by leadingedge vortex formation in a flat plate. Two basetests were developed: the first test representinga prescribed movement with leading-edge vor-tex influence, and a the second test represent-ing a flutter condition with Limit-Cycle Oscilla-tions[2].
Following the results obtained before [2],both tests were expanded with a velocity sen-sor positioned to measure the trailing edge ve-locity during the prescribed motion and flutteroscillations. More advanced signal conditionerswere also used to mitigate inacuracies in the forcemeasurement.
Figure 01 shows the prescribed test setup,composed by a flat plate with a PZT actuatormounted on top of a balance device in front of awind-tunnel, the DSpace data acquisition system,and the laser velocity sensor.
Figure 01: Prescribed Test Setup
3.1 Prescribed Movement Tests
The prescribed movement test used the flat plate,positioned against a steady free-flow of 10 m/s atseveral angles of attack. Measurements were per-formed for at a steady-state static situation andwith the flat plate excited by PZT actuators at-tached to its root. These PZTs, when activatedclose to the plate’s natural frequency for the firstbending mode, generated a high amplitude bend-ing motion, that combined with the freestream,produces high apparent angle of attack. Figure02 shows a schematic of the test.
Figure 02: Experimental Test Schematic
These tests were executed by achieving the10 m/s freestream velocity, and then measuringthe static, steady-state, condition, followed by thedynamic prescribed movement. For this article.Results from three angles of attack will be used toverify the UVLM model calibration to expectedLEV formation.
3.2 Flutter Condition Tests
The Flutter condition test was designed to visual-ize Leading-Edge Vortices influence when Limit-Cycle Oscillations are achieved. Several flatplates, with a ballast, where tested. The ballastwas positioned to lower the flutter velocity andallow it to occur inside the tunnels operationalspeed. Figure 03 shows the flutter test setup.
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Numerical Development of Unsteady Vortex-Lattice Method Expansion for Leading-Edge SheddingPrediction with Experimental Validation
Figure 03: Flutter Test Setup
Tests consisted of increasing the tunnel ve-locity until flutter was perceived. The velocitywas them kept until Limit-Cycle Oscillations aresustained.
A more flexible wing was eventually selectedfor the flutter tests, this was intended to reducesimulation time by requiring a lower freestreamvelocity to achieve flutter. This wing has the di-mensions of 0.4m for span, 0.027m chord and thesame 0.8mm for thickness.
During the tests, it was observed couplingbetween the wing’s and balance’s modes of vi-bration. Different wings presented different cou-plings, and, although lift readings presented somedistortions, trailing edge velocity was success-fully captured by the measurement apparatus.
4 Results
Static Tests were run at the wind tunnel to ver-ify the correlation between lead edge vortexesand high angles of attack. For the Aluminumflat plate, a angle of attack sweep was performedfrom 0 to 11 degrees. Figure 04, show that after6 degrees of angle of attack a nonlinear responseis observed.
Figure 04: Static angle of attack test sweep
By running the static test with UVLM andLDVM for a seven degree angle of attack, it is
possible to verify that this condition should in-deed have Lead-Edge Vortex generation. The ref-erence value of 0.1 for LEV was defined in [1]for flat plates. Figure 05, shows good relation-ship between the UVLM and LDVM results forA0 estimation.
Figure 05: LESP Static Test
Two important aspects are important to no-tice:
1. Due to UVLM’s 3D implementation, thereis a reduced circulation, and therefore A0at the wing tip. This explains the differentvalues between each strip, with the stripsclose to the tip showing reduced values.
2. UVLM A0 calculation, as of implemented,is an approximation based on previoustests[4].
When analyzing the prescribed motion re-sults, with a initial angle of attack of 7 degrees, itis possible to notice that UVLM doesn’t achievethe CL values measured at aerodynamic balance(figure 06 - right). This is consistent with theresults obtained by Ramesh, et al [3] that alsofound greater values from the experiments, withthe mainreason being that we have inertial forcesthat are captured by the balance and are not easilymodeled.
Figure 06: Prescribed Test Results
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MONTEIRO, T. P. , RAMESH, K. , SILVESTRE, F.
We also can verify that the experiment isbound to have LEV generated by all strips (figure06 - left). This would also increase the lift forcegenerated at the strips and could account for thedifference in the results.
Simulating flutter presented an extra diffi-cult in requiring longer simulations to verify thevibration development through time. However,greater velocity and using more flexible wings,with smaller chords, impacts the time step neededto achieve a stable simulation on UVLM.
For the more flexible flat plate with a ballastin a 5mm offset, flutter was obtained with a 10m/sfreestream velocity, and one degree angle of at-tack (figure 07).
Figure 07: Flutter Test Results
It is possible to verify, from the coupled sim-ulation results, that initial deformation is domi-nated by the first elastic mode (η1), with the sec-ond and third modes (η2 and η3 respectively),that will eventually develop into the LCO, ini-tially with smaller magnitude, but with clear in-crease in magnitude through time (figure 08).
Figure 08: Generalized Modes Displacements
When analyzing the dynamics of the wing tiptogether with the A0 (figure 09), we can see onthe left graphic that the angle of attack starts afast harmonic movement, with the plunge, on the,middle figure, presenting a higher amplitude andlower frequency behavior. From the right graphic
it is possible to verify that the wing tip is stillunder the LESP threshold, however with a clearupward trend, it is expected from test results andlinear simulations that the movement slowly in-creases in amplitude until LEVs are shed and thedynamics migrate to the LCO condition.
Figure 09: Wing Tip Strip Dynamics
Unfortunately, longer simulations were notpossible due to the slow simulation time, thathinder current model applications to initial ver-ifications. Future developments will apply paral-lel computing capabilities to speed-up simulationand allow for longer simulations. This will be es-sential to allow for LEV generation.
5 Conclusions and Next Steps
Static tests showed a correlation between LESPbased LEV generation and angle of attack. Fur-ther investigations should be performed to verifyif LEV generation will be enough to reproducethe nonlinear behavior resulting from static-stallcondition.
Uncoupled models were used to validate andverify A0 calculation in UVLM, with good corre-lation between 2D and 3D models. Comparisonswith wind tunnel results show that LEV genera-tion should be expected and can happen unevenlyspanwise.
Coupled models show wing deformation’s in-fluence on A0 and should be able to represent flut-ter scenarios. Future LEV implementation is nec-essary however to achieve LCO.
However, faster models will be necessaryto achieve feasible simulation time and perfor-mance. The test results currently available shouldbe useful in calibrating and validating LEV gen-eration.
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Numerical Development of Unsteady Vortex-Lattice Method Expansion for Leading-Edge SheddingPrediction with Experimental Validation
5.1 References
References
[1] Ramesh K, et al. Discrete-vortex method withnovel shedding criterion for unsteady airfoilflows with intermittent leading-edge vortexshedding. Journal of Fluid Mechanics, 751,500âAS538, 2014.
[2] Murua J, et al. Applications of the UnsteadyVortex-Lattice Method in Aircraft Aeroelastic-ity and Flight Dynamics. Progress in AerospaceScience, DOI: 10.1016/j.paerosci.2012.06.001,2012.
[3] Ramesh K, et al. Experimental And NumericalInvestigations of Post-Flutter Limit Cycle Os-cillations on a Cantilevered Flat Plate. Interna-tional Forum on Aeroelasticity and StructuralDynamics, 2017.
[4] Hirato Y. Leading-Edge-Vortex Formation onFinite Wings in Unsteady Flow. North CarolinaState University, 2016.
[5] Katz J. Discrete vortex method for the non-steady separated flow over an airfoil. Journal ofFluid Mechanics, 102(1), 315âAS328, 1981.
[6] Katz P, Plotkin A. Low-Speed Aerodynamics.Cambridge University Press.
[7] Bisplinhoff R, et al. Aeroelasticity. Dover Pub-lications INC.
6 Acknowledgements
This work was conducted during a scholar-ship supported by Brazilian Federal Agency forSupport and Evaluation of Graduate Education(CAPES) within the Ministry of Education ofBrazil at Instituto TecnolÃsgico de Aeronautica.
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