Flutter Prediction Based on Fluid-Structural Interactions of Wing with Winglet

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Flutter Prediction Based on Fluid-Structural Interactions of Wing with Winglet

K. M. Sridhar

Assistant Professor of Aero Dept

M.A.M School of Engineering

P. Palanisamy

Dept. of Aero Eng

M.A.M School of Engineering

M. Prasanth

Dept .of Aero Eng

M.A.M School of Engineering

B. Sajahan

Dept. of Aero Eng

M.A.M School of Engineering

Abstract:- Aircraft components are naturally elastic which has its own natural frequency. When the source frequency is equal to objects natural frequency, the object may be tends to vibrate and deform. This may result in flutter. Flutter is an oscillatory instability occurs in airplane wing and control surfaces. The oscillatory motion of fluttering cantilever beam has both flexural and torsional component. The aircraft wing has infinitely many degrees of freedom due to cantilever beam structure. The main focus of this article is to predict the flutter of an aircraft wing with combination of winglet for different mach number and different altitude. Modelling of fluid and structural domain are required to solve FSI phenomena. The wing model has been analysed at certain constant altitude in subsonic range using optimization CFD and FEA tools. The deformation exist in a wing with winglet model caused by dynamic aeroelastic effects. The resulting structural deformation and stress variation corresponding to the flow are fully studied and validated with the help of numerical analysis.

Keywords: Aeroelasticity, deformation, flutter, FSI, stress variation

  1. The aerodynamic forces are solved by using flow field governing equations. The structural displacement are solved by using structural governing equations.

    Designed Initial Flow Field and Structural Models

    Designed Initial Flow Field and Structural Models

    Initial Flow Field model for CFD Residual

    Initial Flow Field model for CFD Residual

    Solving Flow Field Governing Equations for Aerodynamic Forces

    Solving Flow Field Governing Equations for Aerodynamic Forces

    CFD Residuals

    CFD Residuals

    1. INTRODUCTION

      If Converged

      If Converged

      Wing are the sources for lift in an aircraft. Due to the aeroelastic characteristics and stress distribution of wing, it may deform. The aircraft performance may change or decrease due to this

      Initial Structural Model for Structural Residual

      Initial Structural Model for Structural Residual

      deformation. In this article, the flutter predicted interms of deformation and stress distribution over a wing model. There are two ways to calculate fluid structural interaction i.e., strongly coupled fluid structural interaction and partly coupled fluid structural interaction [1].

      1.1 PREDICTING THE EFFECTS OF AEROELASTICITY

      The aeroelastic effect occurring on a wing with winglet model due to the flow separation is considered for present analysis. The aerodynamic force predictions and their influence is done using a three-dimensional Navier- Stroke model with fully coupled iterations to identify the physical phenomena. Once the flow field solutions are converged using CFD tool, then the flexural motion of the wing caused by the influence of aerodynamic forces will be computed using FEA tool. The algorithm for partly coupled fluid structural interaction analysis is presented in figure

      Solving Structural Governing Equations for Structural Displacements

      Solving Structural Governing Equations for Structural Displacements

      Structural Residuals

      Structural Residuals

      If Converged

      If Converged

      Post processing Result

      FIGURE 1: ALGORITHM FOR PARTLY COUPLED FOR FLUID STRUCTURAL INTERACTION ANALYSIS

    2. PROBLEM DESCRIPTION

        1. MODEL DESCRIPTION

          Rectangular wing with 60º cant angle winglet is used for aeroelastic analysis. The airfoil used was a NACA 653218. The wing model were modelled using design software. NACA six series, NACA 653218 is used to model the rectangular wing with elliptical winglet. In NACA 653218 airfoil, the position of minimum pressure is 0.5, design lift coefficient is 0.2, range of lift coefficient is 0.3, maximum thickness is 18% at 39.9% chord and maximum camber is 1.1% at 50% chord. The specification of wing and winglet are shown in table 1 and 2.

          TABLE 1: SPECIFICATION OF WING

          SI.

          NO.

          PARAMETERS OF MODEL

          TYPES AND DIMENSIONS

          1.

          Wing type

          Rectangular wing model

          2.

          Airfoil type

          NACA 653218 airfoil

          3.

          Chord length

          121mm

          4.

          Wing span length

          660mm

          5.

          Semi-span length

          330mm

          FIGURE 2: NACA 653218 AIRFOIL

          TABLE 2: SPECIFICATION OF WINGLET

          SI. NO

          .

          PARAMETERS OF MODEL

          TYPES AND DIMENSIONS

          1.

          Winglet type

          Elliptical winglet

          2.

          Airfoil type

          NACA 653218 airfoil

          3.

          Winglet root chord

          121mm

          4.

          Winglet tip chord

          60.5mm

          5.

          Angled height

          55.1mm

          6.

          Vertical height

          47.7mm

          7.

          Horizontal height

          27.6mm

          8.

          Cant angle

          60

          Using the specification which is tabulated in table I and II, the wing with winglet model is designed using modelling software. The designed wing model is shown in figure 3.

          FIGURE 3: DESIGNED RECTANGULAR WING WITH ELLIPTICAL WINGLET MODEL IN 3D VIEW

          A C-Domain control volume is used for flow field analysis. The wing with winglet model including the C-Domain Control Volume is depicted in figure 4. The control volume and the wing with winglet model are subtracted with each other. The total wing with winglet model area is immersed inside the control volume and one face of the wing model is attached with the side face of the control volume.

        2. GRID GENERATION

          A fine tetrahedron grid is generated for both the control volume and wing model as shown in figure 5. Tetrahedron grid is preferred for 3-D solid structures. Totally 178484 nodes and 703930 elements are generated for the control volume which is used for flow field analysis as shown in figure 4.

          FIGURE 4: FINE GRID VIEW OF CONTROL VOLUME WITH THE WING STRUCTURE

          Totally 199675 nodes and 126597 elements are generated for wing which is used for structural analysis as shown in figure 5.

          FIGURE 5: FINE GRID VIEW OF WING STRUCTURE

        3. BOUNDARY CONDITION

      In the control volume the face upright to the leading edge is considered as velocity inlet, the face just opposite to the velocity inlet is taken as pressure outlet and the remaining four faces of the

      control volume are considered as symmetry faces. The upper and lower faces of the wing model are taken as fluid solid interface faces. The numerical

      values of the boundary conditions are given as the International Standard Atmospheric (ISA) properties at 0km to 11km altitude which values are taken from reference 4. The velocity range is considered from 0.05 Mach to 0.75 Mac.

      One face of the wing model attached to one face of the control volume and the symmetry face is converted into a fixed support for the structural analysis. The imported pressure load from the fluid flow solver is applied on the fluid- solid interface faces. Aluminium alloy is the material used on wing model design and its properties are shown in table 3.

      TABLE 3: PROPERTIES OF MATERIAL

      N

      o

      Variables

      Properties of material

      1

      Material

      Aluminium alloy

      2

      Youngs Modulus

      71GPa

      3

      Density

      2770 kg/m3

      4

      Bulk Modulus

      69.608GPa

      5

      Poissons Ratio

      0.33

    3. RESULTS AND DISCUSSIONS

        1. FUNDAMENTAL VIBRATION ANALYSIS

          The fundamental vibrational analysis of wing model carried out using modal analysis software package. The vector contour of Mode Shape 4 is shown in figure 6.

          TABLE 4: FIRST TEN VIBRATIONAL ANALYSIS DATA

          SI. NO.

          MODE SHAPE

          FREQUENCY (Hz)

          1.

          1

          106.84

          2.

          2

          513.6

          3.

          3

          651.08

          4.

          4

          807.35

          5.

          5

          1752.8

          6.

          6

          2385.2

          7.

          7

          2700.7

          8.

          8

          3063.00

          9.

          9

          3455.6

          10.

          10

          4066.9

          FIGURE 6: VECTOR CONTOUR OF MODE SHAPE 4

          The main objective of modal analysis is to determine the dynamic characteristics of aircraft wing such as natural frequency, deformation and mode shapes. First ten vibrational analysis data are shown in table 4.

        2. FLOW FIELD ANALYSIS

          The flow field analysis are carried out for wing with winglet model at various mach numbers from 0.05 to 0.75 and at different altitude from 0km to 11km using Computational fluid dynamics tool.

          Mach Numbe r

          Maximum Static Pressure (Pa)

          Altitud e

          h = 0km

          Altitud e

          h= 1km

          Altitud e

          h= 2km

          Altitud e

          h= 3km

          Altitud e

          h= 4km

          Altitud e

          h= 5km

          0.05

          1131.0

          3

          991.61

          9

          877.40

          786.34

          3

          676.68

          597.55

          0.10

          4580.0

          9

          4052.6

          7

          3584.2

          3155.8

          2859.2

          2484.8

          0.15

          10352.

          30

          9170.0

          4

          8116.2

          7158.8

          6349.3

          5611.4

          0.20

          18263.

          50

          16188.

          00

          14487

          12643

          11196

          9775.5

          0.25

          28542.

          70

          25145.

          60

          22275

          21226

          17320

          15133

          0.30

          40454.

          20

          35901.

          10

          32087

          28096

          24820

          21883

          0.35

          54589.

          90

          48530.

          60

          42925

          38245

          33521

          29294

          0.40

          71616.

          20

          62898.

          00

          55683

          49153

          43495

          37979

          0.45

          90245.

          10

          79145.

          30

          70611

          62252

          54688

          47717

          0.50

          111435

          .0

          97680.

          00

          86310

          76143

          67308

          58993

          0.55

          134176

          .0

          118918

          .00

          104160

          91713

          81113

          71197

          0.60

          158012

          .0

          139805

          .00

          123350

          108730

          96147

          84253

          0.65

          184838

          .0

          163612

          .00

          144400

          127130

          112480

          98507

          0.70

          213681

          .0

          189334

          .00

          167110

          147150

          130190

          113120

          0.75

          244942

          .0

          218313

          .00

          191430

          168480

          148990

          129460

          TABLE 5: MAXIMUM STATIC PRESSURE VS MACH NUMBERS FOR 0KM TO 5KM ALTITUDE

          The maximum static pressure variation

          increases while mach number increases and decreases while altitude (h) increases as shown in figure 7. The minimum static pressure variation against mach number is shown in figure 9. The static pressure variation over the wing at 0.2 and 0.6 Mach number at 4km altitude is shown in figure 8 and 10 as a vector contour.

          From this vector representation, it is evident that the maximum static pressure variation occurs at wing leading edge and minimum static pressure variation occurs at wing trailing edge. The pressure occurred at winglet region is less while compare with wing region as shown in figure 10 and 11. The computed maximum static pressure value against mach number is shown in table 5 and 6. The computed minimum static pressure value against mach number is shown in table 7 and 8.

          Mach Numb er

          Maximum Static Pressure (Pa)

          Altitud e

          h = 6km

          Altitud e

          h= 7km

          Altitud e

          h= 8km

          Altitud e

          h= 9km

          Altitud e

          h= 10km

          Altitud e

          h= 11km

          0.05

          517.94

          450.05

          388.53

          336.45

          288.36

          246.66

          0.10

          2115.9

          1886.3

          1596.9

          1377.9

          1181.8

          1010.4

          0.15

          4901.4

          4184.2

          3699.9

          3132.0

          2691.8

          2304.2

          0.20

          8652.0

          7478.2

          6474.1

          5591.6

          4811.5

          4119.6

          0.25

          13287

          11586

          10069

          8707.5

          7495.5

          6446.0

          0.30

          19142

          16536

          14499

          12439

          10716

          9201.8

          0.35

          25851

          22528

          19561

          16773

          14461

          12415

          0.40

          33221

          28962

          25152

          21932

          18747

          16086

          0.45

          42089

          p>36382

          31878

          27349

          23549

          20210

          0.50

          51233

          44634

          39077

          33534

          28902

          24780

          0.55

          62263

          54274

          47145

          40489

          34763

          29835

          0.60

          73634

          64187

          55770

          48243

          41302

          35430

          0.65

          85994

          74917

          65049

          56270

          48500

          41354

          0.70

          98941

          86484

          75075

          64951

          55714

          47769

          0.75

          11369

          0

          98684

          85829

          74046

          63750

          54629

          TABLE 6: MAXIMUM STATIC PRESSURE VS MACH NUMBERS FOR 6KM TO 11KM ALTITUDE

          FIGURE 7: MAXIMUM PRESSURE VARIATION ABOUT MACH NUMBER AT 0KM TO 11KM ALTITUDE

          FIGURE 8: COMPUTED STATIC PRESSURE VARIATION AT 0.2 MACH NUMBER AT 4KM ALTITUDE

          ach Numbe

          r

          Minimum Static Pressure (Pa)

          h = 0km

          h= 1km

          h= 2km

          h= 3km

          h= 4km

          h= 5km

          0.05

          2.6261

          6

          2.3985

          5

          2.1787

          0

          1.8895

          1.7551

          1.5694

          0.10

          10.692

          0

          9.6941

          5

          8.5818

          6.6773

          7.3119

          5.8982

          0.15

          19.453

          2

          17.520

          8

          15.991

          15.36

          15.459

          12.880

          0.20

          37.349

          4

          35.743

          1

          26.447

          27.627

          27.179

          19.406

          0.25

          52.998

          5

          49.072

          0

          44.845

          45.723

          45.415

          29.455

          0.30

          68.678

          0

          63.391

          6

          58.942

          49.057

          62.412

          45.306

          0.35

          88.147

          2

          89.423

          1

          75.806

          69.473

          80.699

          48.374

          0.40

          127.96

          1

          99.825

          9

          93.983

          88.543

          102.55

          60.474

          0.45

          111.22

          3

          128.20

          8

          116.56

          102.04

          128.24

          87.783

          0.50

          207.65

          6

          150.89

          0

          122.6

          132.40

          146.72

          108.91

          0.55

          236.42

          6

          233.57

          3

          206.31

          116.83

          140.44

          126.55

          0.60

          179.73

          4

          177.51

          6

          221.45

          211.13

          158.78

          141.10

          0.65

          237.93

          4

          195.09

          5

          175.22

          231.85

          135.80

          124.74

          0.70

          310.69

          4

          208.15

          3

          246.60

          175.93

          252.72

          210.99

          0.75

          371.29

          7

          314.84

          4

          218.78

          249.19

          264.24

          246.61

          TABLE 7: MINIMUM STATIC PRESSURE VS MACH NUMBERS FOR 0KM TO 5KM ALTITUDE

          Mach Numbe r

          Minimum Static Pressure (Pa)

          h = 6km

          h= 7km

          h= 8km

          h= 9km

          h= 10km

          h= 11km

          0.05

          1.4678

          1.2543

          1.0382

          0.9288

          6

          0.9004

          6

          0.83003

          0.10

          4.9476

          4.7553

          4.7218

          4.0647

          0

          3.4839

          0

          2.97210

          0.15

          11.776

          10.285

          9.3470

          8.1781

          0

          7.2082

          0

          6.24590

          0.20

          22.837

          18.727

          14.422

          12.683

          11.254

          0

          10.1430

          0.25

          25.80

          4

          26.00

          2

          22.55

          3

          20.06

          1

          18.89

          40

          14.68

          60

          0.30

          35.75

          8

          31.88

          7

          36.30

          2

          31.57

          6

          28.48

          60

          22.53

          00

          0.35

          48.12

          4

          40.80

          9

          36.35

          5

          38.44

          4

          37.64

          80

          32.41

          90

          0.40

          53.41

          0

          53.97

          2

          55.57

          6

          40.63

          1

          43.09

          80

          40.65

          20

          0.45

          80.55

          6

          58.98

          2

          53.64

          6

          65.95

          5

          58.31

          70

          47.35

          40

          0.50

          78.18

          6

          69.41

          9

          64.57

          9

          77.36

          5

          68.73

          40

          60.39

          10

          0.55

          117.8

          6

          100.3

          0

          79.36

          5

          74.84

          4

          81.76

          50

          70.63

          00

          0.60

          129.6

          6

          115.0

          8

          96.85

          3

          78.61

          6

          86.48

          70

          82.95

          10

          0.65

          145.6

          5

          129.8

          1

          108.8

          6

          88.21

          3

          79.80

          70

          98.30

          50

          0.70

          123.2

          6

          135.8

          3

          117.3

          2

          96.14

          8

          126.6

          7

          110.4

          3

          0.75

          184.6

          3

          126.0

          6

          127.7

          8

          123.2

          5

          143.6

          9

          122.2

          3

          Mach Numbe r

          Minimum Static Pressure (Pa)

          h = 6km

          h= 7km

          h= 8km

          h= 9km

          h= 10km

          h= 11km

          0.05

          1.4678

          1.2543

          1.0382

          0.9288

          6

          0.9004

          6

          0.83003

          0.10

          4.9476

          4.7553

          4.7218

          4.0647

          0

          3.4839

          0

          2.97210

          0.15

          11.776

          10.285

          9.3470

          8.1781

          0

          7.2082

          0

          6.24590

          0.20

          22.837

          18.727

          14.422

          12.683

          11.254

          0

          10.1430

          0.25

          25.80

          4

          26.00

          2

          22.55

          3

          20.06

          1

          18.89

          40

          14.68

          60

          0.30

          35.75

          8

          31.88

          7

          36.30

          2

          31.57

          6

          28.48

          60

          22.53

          00

          0.35

          48.12

          4

          40.80

          9

          36.35

          5

          38.44

          4

          37.64

          80

          32.41

          90

          0.40

          53.41

          0

          53.97

          2

          55.57

          6

          40.63

          1

          43.09

          80

          40.65

          20

          0.45

          80.55

          6

          58.98

          2

          53.64

          6

          65.95

          5

          58.31

          70

          47.35

          40

          0.50

          78.18

          6

          69.41

          9

          64.57

          9

          77.36

          5

          68.73

          40

          60.39

          10

          0.55

          117.8

          6

          100.3

          0

          79.36

          5

          74.84

          4

          81.76

          50

          70.63

          00

          0.60

          129.6

          6

          115.0

          8

          96.85

          3

          78.61

          6

          86.48

          70

          82.95

          10

          0.65

          145.6

          5

          129.8

          1

          108.8

          6

          88.21

          3

          79.80

          70

          98.30

          50

          0.70

          123.2

          6

          135.8

          3

          117.3

          2

          96.14

          8

          126.6

          7

          110.4

          3

          0.75

          184.6

          3

          126.0

          6

          127.7

          8

          123.2

          5

          143.6

          9

          122.2

          3

          TABLE 8: MINIMUM STATIC PRESSURE VS MACH NUMBERS FOR 6KM TO 11KM ALTITUDE

          FIGURE 9: MINIMUM PRESSURE VARIATION ABOUT MACH NUMBER AT 0KM TO 11KM ALTITUDE

          0.70

          352100

          00

          314100

          00

          277720

          00

          245680

          00

          214900

          00

          1892000

          0

          0.75

          404980

          00

          363700

          00

          319710

          00

          281900

          00

          248230

          00

          2182600

          0

          0.70

          352100

          00

          314100

          00

          277720

          00

          245680

          00

          214900

          00

          1892000

          0

          0.75

          404980

          00

          363700

          00

          319710

          00

          281900

          00

          248230

          00

          2182600

          0

          FIGURE 10: COMPUTED STATIC PRESSURE VARIATION AT 0.6 MACH NUMBER AT 4KM ALTITUDE

        3. STRUCTURAL ANALYSIS

      The total deformation and the equivalent Von- Mises stress distributions of the wing with winglet model are computed using Finite Element Analysis tool. The computed pressure load imported over wing with winglet model as shown in figure 8 and 10.

      Mach Number

      Equivalent Von – Mises Maximum Stress (Pa)

      h = 0km

      h= 1km

      h= 2km

      h= 3km

      h= 4km

      h= 5km

      0.05

      147600

      122500

      107150

      98979

      83749

      72039

      0.10

      585780

      515680

      453330

      394440

      385790

      328490

      0.15

      138360

      0

      120980

      0

      106170

      0

      931680

      913050

      769740

      0.20

      262910

      0

      229790

      0

      213500

      0

      178600

      0

      165540

      0

      1341000

      0.25

      435730

      0

      372340

      0

      323330

      0

      323290

      0

      263110

      0

      2184500

      0.30

      617220

      0

      543240

      0

      496680

      0

      418050

      0

      381220

      0

      3357300

      0.35

      849550

      0

      748730

      0

      661280

      0

      601110

      0

      524330

      0

      4432400

      0.40

      114590

      00

      994810

      0

      874980

      0

      767210

      0

      689370

      0

      5876500

      0.45

      145520

      00

      126300

      00

      114340

      00

      100790

      00

      876360

      0

      7529700

      0.50

      179900

      00

      156270

      00

      138230

      00

      121870

      00

      108560

      00

      9616700

      0.55

      218980

      00

      194540

      00

      168740

      00

      148320

      00

      131980

      00

      1171300

      0

      0.60

      255810

      00

      227380

      00

      201340

      00

      178410

      00

      157450

      00

      1399800

      0

      0.65

      302790

      00

      268710

      00

      237720

      00

      210000

      00

      186390

      00

      1649000

      0

      Mach Number

      Equivalent Von – Mises Maximum Stress (Pa)

      h = 0km

      h= 1km

      h= 2km

      h= 3km

      h= 4km

      h= 5km

      0.05

      147600

      122500

      107150

      98979

      83749

      72039

      0.10

      585780

      515680

      453330

      394440

      385790

      328490

      0.15

      138360

      0

      120980

      0

      106170

      0

      931680

      913050

      769740

      0.20

      262910

      0

      229790

      0

      213500

      0

      178600

      0

      165540

      0

      1341000

      0.25

      435730

      0

      372340

      0

      323330

      0

      323290

      0

      263110

      0

      2184500

      0.30

      617220

      0

      543240

      0

      496680

      0

      418050

      0

      381220

      0

      3357300

      0.35

      849550

      0

      748730

      0

      661280

      0

      601110

      0

      524330

      0

      4432400

      0.40

      114590

      00

      994810

      0

      874980

      0

      767210

      0

      689370

      0

      5876500

      0.45

      145520

      00

      126300

      00

      114340

      00

      100790

      00

      876360

      0

      7529700

      0.50

      179900

      00

      156270

      00

      138230

      00

      121870

      00

      108560

      00

      9616700

      0.55

      218980

      00

      194540

      00

      168740

      00

      148320

      00

      131980

      00

      1171300

      0

      0.60

      255810

      00

      227380

      00

      201340

      00

      178410

      00

      157450

      00

      1399800

      0

      0.65

      302790

      00

      268710

      00

      237720

      00

      210000

      00

      186390

      00

      1649000

      0

      TABLE 9: EQUIVALENT VON MISES MAXIMUM STRESS VS MACH NUMBERS FOR 0KM TO 5KM ALTITUDE

      The equivalent Von – Mises stress variation caused by the pressure load acting on the wing at the inlet velocity of

      0.2 and 0.6 Mach at 4km altitude is illustrated in figure 12 and 114. The maximum stress variations are tabulated which is shown in table 9 and 10 and minimum stress variations are tabulated which is shown in table 11 and 12. The maximum and minimum equivalent Von Mises Stress variation increase while Mach number increases and also decreases while altitude increases as shown in figure 11 and 13.

      Mach Numbe r

      Equivalent Von – Mises Maximum Stress (Pa)

      h = 6km

      h= 7km

      h= 8km

      h= 9km

      h= 10km

      h= 11km

      0.05

      60944

      52277

      44329

      37964

      32008

      26838

      0.10

      260750

      248790

      195270

      167090

      142910

      121680

      0.15

      671150

      535960

      502440

      395620

      337310

      288480

      0.20

      124420

      0

      994660

      854680

      733060

      626300

      533630

      0.25

      188260

      0

      163000

      0

      140310

      0

      120660

      0

      102900

      0

      864720

      0.30

      292410

      0

      241240

      0

      218550

      0

      178730

      0

      153080

      0

      129740

      0

      0.35

      402570

      0

      349600

      0

      301870

      0

      248750

      0

      212990

      0

      181170

      0

      0.40

      511280

      0

      443110

      0

      382500

      0

      343540

      0

      282100

      0

      240460

      0

      0.45

      675280

      0

      567500

      0

      507890

      0

      421460

      0

      361940

      0

      308630

      0

      0.50

      816150

      0

      708430

      0

      630260

      0

      526590

      0

      450610

      0

      385100

      0

      0.55

      102060

      00

      885490

      0

      765760

      0

      641660

      0

      550290

      0

      468650

      0

      0.60

      121960

      00

      105930

      00

      917540

      0

      791380

      0

      659100

      0

      561940

      0

      0.65

      143920

      00

      125190

      00

      108530

      00

      928340

      0

      800230

      0

      665850

      0

      0.70

      164350

      00

      145790

      00

      126380

      00

      108710

      00

      910380

      0

      776800

      0

      0.75

      193410

      00

      164700

      00

      145470

      00

      122500

      00

      105260

      00

      896070

      0

      TABLE 10: EQUIVALENT VON – MISES MAXIMUM STRESS VS MACH NUMBERS FOR 6KM TO 11KM ALTITUDE

      0.35

      6574.6

      5767.0

      5093

      4766.3

      4014.8

      3362.7

      0.40

      8877.8

      7758.5

      6824.4

      5952.8

      5170.9

      4517.4

      0.45

      11186.

      0

      9864.1

      8705.2

      7655.7

      6455.0

      5850.8

      0.50

      13813.

      0

      12203

      10831

      9598.3

      7911.1

      7257.9

      0.55

      16657.

      0

      14692

      13155

      11635

      9552.2

      8679.2

      0.60

      19720.

      0

      17531

      15576

      13815

      11375

      10243

      0.65

      23185.

      0

      20585

      18266

      16157

      13292

      11976

      0.70

      26979.

      0

      23935

      21195

      18713

      15368

      14503

      0.75

      30940.

      0

      26964

      24305

      21456

      17712

      16607

      0.35

      6574.6

      5767.0

      5093

      4766.3

      4014.8

      3362.7

      0.40

      8877.8

      7758.5

      6824.4

      5952.8

      5170.9

      4517.4

      0.45

      11186.

      0

      9864.1

      8705.2

      7655.7

      6455.0

      5850.8

      0.50

      13813.

      0

      12203

      10831

      9598.3

      7911.1

      7257.9

      0.55

      16657.

      0

      14692

      13155

      11635

      9552.2

      0.60

      19720.

      0

      17531

      15576

      13815

      11375

      10243

      0.65

      23185.

      0

      20585

      18266

      16157

      13292

      11976

      0.70

      26979.

      0

      23935

      21195

      18713

      15368

      14503

      0.75

      30940.

      0

      26964

      24305

      21456

      17712

      16607

      FIGURE 11: MAXIMUM STRESS VARIATION ABOUT MACH NUMBER AT 0KM TO 11KM ALTITUDE

      Mach Numbe r

      Equivalent Von – Mises Minimum Stress (Pa)

      h = 0km

      h= 1km

      h= 2km

      h= 3km

      h= 4km

      h= 5km

      0.05

      90.017

      69.613

      58.953

      61.813

      44.476

      34.107

      0.10

      372.77

      326.01

      283.57

      243.30

      262.79

      212.76

      0.15

      939.64

      805.10

      699.28

      606.32

      677.10

      522.19

      0.20

      1926.6

      1659.4

      1642.5

      1248.2

      1299.8

      902.08

      0.25

      3465.1

      2792.6

      2393.8

      2541.8

      2115.0

      1552.3

      0.30

      4743.6

      4145.6

      3974

      3150.1

      3007.6

      2642.1

      Mach Numbe r

      Equivalent Von – Mises Minimum Stress (Pa)

      h = 0km

      h= 1km

      h= 2km

      h= 3km

      h= 4km

      h= 5km

      0.05

      90.017

      69.613

      58.953

      61.813

      44.476

      34.107

      0.10

      372.77

      326.01

      283.57

      243.30

      262.79

      212.76

      0.15

      939.64

      805.10

      699.28

      606.32

      677.10

      522.19

      0.20

      1926.6

      1659.4

      1642.5

      1248.2

      1299.8

      902.08

      0.25

      3465.1

      2792.6

      2393.8

      2541.8

      2115.0

      1552.3

      0.30

      4743.6

      4145.6

      3974

      3150.1

      3007.6

      2642.1

      FIGURE 12: COMPUTED EQUIVALENT VON – MISES STRESS VARIATION AT 0.2 MACH NUMBER AT 4KM ALTITUDE

      TABLE 11: EQUIVALENT VON MISES MINIMUM STRESS VS MACH NUMBERS FOR 0KM TO 5KM ALTITUDE

      MAC H NUMB ER

      EQUIVALENT VON – MISES MINIMUM STRESS (PA)

      h = 6km

      h= 7km

      h= 8km

      h= 9km

      h= 10km

      h= 11km

      0.05

      27.483

      22.632

      18.121

      14.617

      11.356

      9.1446

      0.10

      163.91

      162.27

      128.53

      109.14

      87.708

      70.593

      0.15

      449.57

      339.42

      333.94

      251.65

      216.07

      188.50

      0.20

      892.36

      634.10

      552.60

      471.25

      400.90

      341.09

      0.25

      1324.2

      1124.6

      949.73

      803.48

      677.60

      562.41

      0.30

      2287.1

      1745.1

      1650.9

      1244.9

      1047.7

      869.08

      0.35

      3183.7

      2750.2

      2335.0

      1795.4

      1509.3

      1260.6

      0.40

      3910.1

      3368.3

      2887.9

      2682.4

      2061.8

      1723.7

      0.45

      5247.4

      4359.5

      3999.8

      3196.0

      2693.8

      2259.0

      0.50

      6355.7

      5490.8

      4878.5

      4030.6

      3428.2

      2875.5

      0.55

      7614.3

      6660.3

      5814.3

      4954.6

      4218.3

      3564.4

      0.60

      8966.0

      7836.6

      6837.9

      5928.1

      5098.5

      4314.1

      0.65

      10434

      9109.6

      7932.3

      6903.3

      5981.9

      5135.6

      0.70

      12699

      10478

      9123.0

      7930.6

      7081.4

      6016.9

      0.75

      13797

      12679

      10392

      9535.6

      8207.3

      6958.9

      TABLE 12: EQUIVALENT VON MISES MINIMUM STRESS VS MACH NUMBERS FOR 6KM TO 11KM ALTITUDE

      FIGURE 13: MINIMUM STRESS VARIATION ABOUT MACH NUMBER AT 0KM TO 11KM ALTITUDE

      70

      600

      20

      100

      500

      800

      0.55

      0.8762

      60

      0.7785

      000

      0.6734

      10

      0.5921

      500

      0.5305

      500

      0.4687

      300

      0.60

      1.0190

      00

      0.9056

      800

      0.8024

      30

      0.7118

      100

      0.6327

      900

      0.5601

      200

      0.65

      1.2065

      00

      1.0704

      000

      0.9474

      60

      0.8371

      200

      0.7485

      000

      0.6596

      900

      0.70

      1.4042

      00

      1.2523

      000

      1.1071

      00

      0.9790

      500

      0.8641

      300

      0.7546

      700

      0.75

      1.6157

      00

      1.4564

      000

      1.2744

      00

      1.1237

      000

      0.9972

      000

      0.8700

      300

      0.8024

      30

      70

      600

      20

      100

      500

      800

      0.55

      0.8762

      60

      0.7785

      000

      0.6734

      10

      0.5921

      500

      0.5305

      500

      0.4687

      300

      0.60

      1.0190

      00

      0.9056

      800

      0.7118

      100

      0.6327

      900

      0.5601

      200

      0.65

      1.2065

      00

      1.0704

      000

      0.9474

      60

      0.8371

      200

      0.7485

      000

      0.6596

      900

      0.70

      1.4042

      00

      1.2523

      000

      1.1071

      00

      0.9790

      500

      0.8641

      300

      0.7546

      700

      0.75

      1.6157

      00

      1.4564

      000

      1.2744

      00

      1.1237

      000

      0.9972

      000

      0.8700

      300

      FIGURE 14: COMPUTED EQUIVALENT VON – MISES STRESS VARIATION AT 0.6 MACH NUMBER AT 4KM ALTITUDE

      The total deformation distribution produced on the wing because of the pressure load acting on it (0.2 and 0.6 Mach number) is shown in figure 16 and 17. The total deformation value increases gradually from the wing root to winglet tip. The maximum deformation occurs at trailing edge wing tip to winglet tip. The wing for various velocity inlet conditions at different altitude, deformations values are computed as illustrated and tabulated in figure 15 and table 13 and 14.

      TABLE 13: TOTAL DEFORMATION VS MACH NUMBERS FOR 0KM TO 5KM ALTITUDE

      Mach Number

      Total Deformation (mm)

      h = 6km

      h= 7km

      h= 8km

      h= 9km

      h= 10km

      h= 11km

      0.05

      0.00225

      83

      0.00192

      87

      0.00162

      88

      0.00139

      51

      0.00116

      97

      0.00097

      594

      0.10

      0.00988

      58

      0.00959

      86

      0.00739

      32

      0.00631

      34

      0.00538

      87

      0.00458

      060

      0.15

      0.02616

      30

      0.02059

      00

      0.01955

      70

      0.01514

      90

      0.01289

      30

      0.01102

      000

      0.20

      0.04893

      50

      0.03856

      70

      0.03307

      90

      0.02833

      70

      0.02418

      00

      0.02057

      100

      0.25

      0.07380

      20

      0.06383

      50

      0.05487

      00

      0.04713

      40

      0.04014

      10

      0.03360

      700

      0.30

      0.11610

      00

      0.09496

      10

      0.08661

      40

      0.07020

      40

      0.06007

      50

      0.05081

      300

      0.35

      0.16025

      00

      0.13909

      00

      0.12000

      00

      0.09811

      80

      0.08392

      70

      0.07130

      400

      0.40

      0.20279

      00

      0.17559

      00

      0.15143

      00

      0.13681

      00

      0.11151

      00

      0.09495

      000

      0.45

      0.26971

      00

      0.22551

      00

      0.20266

      00

      0.16716

      00

      0.14347

      00

      0.12222

      000

      0.50

      0.32535

      00

      0.28223

      00

      0.25167

      00

      0.20941

      00

      0.17900

      00

      0.15289

      000

      0.55

      0.40836

      00

      0.35428

      00

      0.30627

      00

      0.25576

      00

      0.21906

      00

      0.18633

      000

      0.60

      0.48804

      00

      0.42383

      00

      0.36700

      00

      0.31645

      00

      0.26299

      00

      0.22396

      000

      0.65

      0.57586

      00

      0.50080

      00

      0.43396

      00

      0.37146

      00

      0.31982

      00

      0.26562

      000

      0.70

      0.65648

      00

      0.58326

      00

      0.50541

      00

      0.43469

      00

      0.36341

      00

      0.31002

      000

      0.75

      0.77335

      00

      0.65810

      00

      0.58193

      00

      0.48940

      00

      0.42025

      00

      0.35764

      000

      TABLE 14: TOTAL DEFORMATION VS MACH NUMBERS FOR 6KM TO 11KM ALTITUDE

      Mach Numbe r

      Total Deformation (mm)

      h = 0km

      h= 1km

      h= 2km

      h= 3km

      h= 4km

      h= 5km

      0.05

      0.0056

      263

      0.0045

      788

      0.0039

      96

      0.0037

      463

      0.0031

      535

      0.0026

      795

      0.10

      0.0224

      03

      0.0196

      940

      0.0172

      87

      0.0150

      140

      0.0150

      930

      0.0126

      910

      0.15

      0.0535

      40

      0.0466

      990

      0.0409

      48

      0.0359

      070

      0.0360

      840

      0.0300

      380

      0.20

      0.1031

      50

      0.0899

      320

      0.0843

      01

      0.0698

      390

      0.0657

      160

      0.0522

      480

      0.25

      0.1730

      50

      0.1468

      700

      0.1271

      60

      0.1283

      000

      0.1048

      500

      0.0857

      310

      0.30

      0.2446

      10

      0.2149

      800

      0.1977

      00

      0.1650

      300

      0.1522

      700

      0.1334

      000

      0.35

      0.3374

      80

      0.2972

      100

      0.2623

      70

      0.2397

      300

      0.2098

      600

      0.1753

      900

      0.40

      0.4581

      90

      0.3961

      400

      0.3482

      20

      0.3049

      900

      0.2763

      500

      0.2332

      800

      0.45

      0.5820

      10

      0.5033

      600

      0.4572

      00

      0.4030

      500

      0.3530

      000

      0.2997

      800

      0.50

      0.7202

      0.6231

      0.5516

      0.4866

      0.4362

      0.3844

      FIGURE 15: TOTAL DEFORMATION ABOUT MACH NUMBER AT 0KM TO 11KM ALTITUDE

      FIGURE 16: TOTAL DEFORMATION CONTOUR AT

      0.2 MACH NUMBER AT 4KM ALTITUDE

      FIGURE 17: TOTAL DEFORMATION CONTOUR AT

      0.6 MACH NUMBER AT 4KM ALTITUDE

    4. CONCLUSION

The considered rectangular wing with winglet model is kept at 0º Angle of Attack throughout the analysis. Therefore, the pressure distributions obtained for the subsonic Mach numbers from 0.05 to 0.75 at different altitude are verified with the available historical data. The results are fully agreed with the data exist in the NACA report and verified with the fifteen inlet velocity conditions. From the results and contours, it is identified that the non-linear aeroelastic effects in the both incompressible and compressible subsonic velocities are negligible. The methodology used in this article can be implemented for Taper, Swept back and Delta wings with different wingtips with high subsonic Mach numbers to study the aeroelastic nature of such designs.

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  3. H. Haddadpour, R.D. Firouz-Abadi (2006), Evaluation of quasi-steady aerodynamic modeling for flutter prediction of aircraft wings in incompressible flow, Elsevier. Thin-Walled Structures 44 (2006) 931936.

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