NASA Technical JIT FILE COPY Paper A- 9%08~~ z 3598 DATE DELIVERED ah3h6 A Computational and Experimental Study of Nonlinear Aspects of Induced Drag 1996 Stephen C. Smith, Ames Research Center, Mofsett Field, California National Aeronautics and Space Administration Ames Research Center NASA Technical Paper 3598 . A Computational and Experimental Study of Nonlinear Aspects of Induced Drag February 1996 Stephen C. Smith National Aeronautics and Space Administration TABLE OF CONTENTS . Page ... NOMENCLATURE ...................................................................... Xlll SUMMARY ................................................................................ 1 I 1 INTRODUCTION ...................................................................... 2 ~ 1.1 Motivation and Objectives.. ..................................................... 2 1.2 Historical Background ........................................................... 3 1.3 Present Work.. ................................................................. 7 2 COMPUTATIONAL METHODS FOR PREDICTION OF INDUCED DRAG.. .......... 9 2.1 Linear Potential Flow-Governing Equations .................................... 9 2.2 The High-Order Panel Method-A502 .......................................... 10 2.3 Wake Modeling with Linear Panel Methods.. ................................... 11 2.4 Role of Viscosity in Induced Drag .............................................. 12 2.5 Full-Potential Flow-Governing Equations. ..................................... 13 2.6 Full Potential Flow Solver-nanair ............................................ 14 2.7 Accuracy of Computational Results. ............................................ 15 2.8 Surface Pressure Integration. ................................................... 16 3 FAR-FIELD DRAG COMPUTATION AND THE INFLUENCE OF WAKE MODELING .................................................................... 19 3.1 Far-Field Drag Computation ................................................... 19 3.2 Trefftz-Plane Integration of Computational Results ............................. 26 3.3 Influence of Wake Shape on Far-Field Drag Computation.. ..................... 26 3.4 Nonlinear Considerat ions in the Application of Munk’s Stagger Theorem.. ..... 32 4 APPLICATIONS OF COMPUTATIONAL METHODS FOR INDUCED DRAG PREDICTION ................................................................. 35 4.1 Subsonic Induced Drag Computation ........................................... 35 4.2 Sensitivity to Paneling-Surface Pressure Integration ........................... 38 ... 111 4.3 Sensitivity to Paneling-Trefftz Plane Integration. ............................... 42 4.4 Sensitivity to Angle of Attack.. ................................................ 43 4.5 Conclusions Regarding Computational Prediction of Induced Drag .............. 43 4.6 Modification of Elliptical Wing to Achieve Elliptic Span-Loading ................ 43 4.7 Transonic Induced Drag Computation .......................................... 45 5 INFLUENCE OF COMPRESSIBILITY ON INDUCED DRAG.. ........................ 49 5.1 Relationship Between Circulation and Lift in Transonic Flow ................... 49 5.2 Influence of Mach Number on the Induced Drag of a Swept Wing.. ............. 50 6 COMPUTATIONAL MODELING OF FORCEFREE WAKES .......................... 55 6.1 Initial Studies with Vortex-Lattice Wake Relaxation ............................ 55 6.2 A Hybrid Wake Relaxation Scheme.. ........................................... 56 6.3 Application of Hybrid Wake-Relaxation Scheme ................................ 60 6.4 Error from Neglecting the u-Perturbation Velocity for Force-Free Wakes. ........ 61 6.5 Effect of Streamwise Wake Substitution. ........................................ 63 7 PLANAR WING DESIGNS TO EXPLOIT FAVORABLE WAKE INTERACTION.. .... 65 7.1 Nonplanar Wake from Planar Wing with Force Side-Edge Separation. ........... 66 7.2 Nonplanar Wake from Planar Split-Tip Wing. .................................. 72 8 EXPERIMENTAL STUDIES OF THE ELLIPTICAL AND SPLIT-TIP WING PLANFORMS .......................................................................... 77 8.1 Model Description. ............................................................. 77 8.2 Test Facility ................................................................... 79 8.3 Instrumentation. ............................................................... 80 8.4 Calibrations. ................................................................... 81 8.5 Instrumentation Error in Drag Measurement. ................................... 82 8.6 Data Acquisition and Reduction.. .............................................. 84 8.7 Test Conditions ................................................................ 85 8.8 Saniplirig Duration for Data Acquisition ........................................ 86 8.9 Estimated Error in Induced Drag from Periodic Rolling Moment.. .............. 88 iv 8.10 Stream Angle Correction ...................................................... 91 8.11 Buoyancy ..................................................................... 92 lw 8.12 Flow Visualization ............................................................. 93 8.13 Experiment.al Results Before Wall Corrections .................................. 93 8.14 Wall Corrections .............................................................. 96 8.15 Experimental Results After Wall Corrections ................................... 99 8.16 Estimation of Viscous Drag .................................................... 99 8.17 Span Efficiency............................................................... 106 8.18 A Wake Survey Method for Experimental Drag Decomposition ................ 106 9 SUMMARY AND CONCLUSIONS ..................................................... 109 9.1 Summary ..................................................................... 109 9.2 Conclusions ................................................................... 109 APPENDIX 2ND ORDER ACCURATE APPROXIMATION OF THE COMPRESSIBLE BERNOULLI EQUATION ................................................... .. 115 REFERENCES ........................................................................... 117 V LIST OF TABLES Table Page 1 Span efficiency versus Mach number for the Xt = 1.00 crescent wing. a = 5" . 47 2 Variation of span efficiency with Mach number for swept wing. cr = 2" ....... 53 3 Comparison of pressure-integrated lift and lift based on circulation .......... 53 4 Span efficiency of Xt = 1.00 crescent wing. C, = 0.30 ........................ 61 5 Span efficiency vs. force-free wake length for the Xt = 1.00 crescent wing .... 64 6 Span efficiency vs. force-free wake length for the Xt = 0.25 elliptical wing .... 64 7 Span efficiency of Planform A compared with the Xt = 1 .0 crescent wing ..... 70 8 Chord distribution of Planform B ........................................... 70 9 Span efficiency of Planform B ............................................... 71 10 Predicted span efficiency of the splibtip wing ................................ 76 11 Task mk . I11 balance capacity ............................................... 80 12 Results of drag decomposition from experimental wake surveys (ref.24) ..... 107 vii LIST OF FIGURES I- Figure Page ~. 1 Lanchester’s concept of trailing vortex wake ...................................... 3 2 Prandtl’s lifting line model ....................................................... 4 3 Morino boundary conditio11 and Kutta condition for panel method ............... 11 4 Cartesian grid for Tranair ....................................................... 14 5 Effect of leading edge panel density on pressure integration ...................... 16 6 Control volume for momentum conservation ..................................... 20 7 U-perturbation produced when wake is not aligned with freestream .............. 23 8 Additional bounding surfaces on shock discontinuity ............................. 25 9 Survey points used to determine wake properties in Trefftz plane ................ 27 10 Nonphysical wake shape with zero drag ......................................... 27 11 Intermediate partition in control volume ........................................ 29 12 Near-field and wake control volumes ............................................ 29 13 Wake substitution modifies induced velocities ................................... 30 14 Near-field portion of force-free wake remains after substitution .................. 31 15 Planar wing producing a nonplanar wake ........................................ 33 16 Two elliptical wing planforms ................................................... 36 17 Family of wings studied by Van Dam (ref . 3) .................................... 37 18 Span Efficiency versus Xt predicted by VanDam (ref . 3) ......................... 37 19 Typical surface panel model .................................................... 39 20 Effect of spanwise panel density on span efficiency computed by surface pressure integration ........................................................ 40 21 Spanwise lift distribution for elliptical and crescent wings ........................ 41 22 Spanwise drag dish-ibution for elliptical and crescent wings ...................... 41 ix 23 Effect of spanwise panel density on span efficiency computed by Trefftz-plane integration .................................................... 42 24 Effect of angle of attack on span efficiency computed by surface pressure integration .......................................................