1、r c NASA * h cr) E I m I- TECHNICAL REPORT NASA TR I_- c, / I CALCULATION OF NONLINEAR CONICAL FLOWS BY THE METHOD OF LINES i by E. 3. Klunker, Jerry C. South, JY., and Ruby M. Davis Langley Research Center Hamptoiz, Vu. 23365 NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. OCTOBER 1
2、971 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-TECH LIBRARY KAFB, NM 2. Government Accession No. - 1 1. Report No. NASA TR R-374 - 4. Title and Subtitle CALCULATION OF NONLINEAR CONICAL FLOWS BY THE METHOD OF LINES - - - - _- 7. Author() E. B. K
3、lunker, Jerry C. South, Jr., and Ruby M. Davis -. _ - -. _ 9. Performing Organization Name and Address NASA Langley Research Center Hampton, Va. 23365 - _ 2. Sponsoring Agency Name and Address National Aeronautics and Space Administration Washington, D.C. 20546 - -. .- 15. Supplementary Notes 006840
4、3 3. Recipients Catalog No. 5. Report Date 6. Performing Organization Code October 1971 8. Performing Organization Report No. L-7813 IO. Work Unit No. 136 -13 -0 5-0 1 11. Contract or Grant No. 13. Type of Report and Period Covered Technical Report . 14. Sponsoring Agency Code _ - - - -. . _ - 16 Ab
5、stract A computational technique, called the method of lines, is developed for computing the flow field about conical configuratiuns at incidence in a supersonic flow. The method, which makes use of the self-similarity property, is developed for the nonlinear flow equations. The method has proved to
6、 be an efficient and versatile procedure for constructing the numerical solutions to conical flow problems. It has been successful in computing the flow about circular and elliptic cones at conditions where small regions of supersonic cross flow develop and for the conical delta wings where the regi
7、on of supersonic cross flow is extensive. The calcula- tions made for circular and elliptic cones as well as for the compression side of various conical delta wings are in good agreement with experiment except in regions where viscous effects become important. - - _- - . - . - - - - - - 17. Key Word
8、s (Suggested by Author(s) Method of lines Conical flow Entropy layer Delta wings . - -. - -. . . . . - _ 18. Distribution Statement Unclassified - Unlimited 21. NO. of Pages 22. Price 1 80 1 $3.00 20. Security Classif. (of this page) Unclassified -_ _- - I - - . 19. Security Classif. (of this report
9、) Unclassified For sale by the National Technical Information Service, Springfield, Virginia 22151 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-CONTE
10、NTS Page SUMMARY . 1 INTRODUCTION 1 BACKGROUND 3 Nonlinear Conical Methods . 3 Method of Lines . 5 SYMBOLS . 6 METHOD . 9 Conical Coordinates 9 Differential Equations . 10 11 Geometric Parameters . 14 The Method of Lines 15 Symmetry and Boundary Conditions 16 Symmetry conditions . 16 Flow tangency a
11、t surface . 16 Shock-wave conditions 17 Attached shock at wing leading edge . 17 Determination of the Shock Shape . 17 Newton iteration for shock shape 17 Modified Newton iteration procedure 18 Approximate starting shock shapes . 19 Extrapolation to Surface 21 Corrected isentropic surface values . 2
12、1 Computation of the surface entropy . 22 Stability and Error Growth . 22 Force Coefficients . 23 RESULTS AND DISCUSSION . 24 Circular Cone 24 Difficulties at large relative incidence . 25 Extrapolation of surface pressures to large angles of incidence 28 Artificial hump on leeward side . 29 Entropy
13、 Layer and Vortical Singularities . 31 Entropy layer . 32 Vortical singularity lift-off . 34 Transformation to a rectangular region . iii Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Page Elliptic Cone . 36 Cross-flow streamline pattern 38 Effect
14、of number of lines . 41 44 44 Extrapolated surface pressures . 37 Computation history 39 Elliptic cone with large axis ratio 41 Comparison of an elliptic cone computation with other methods . Elliptic cone at yaw Conical Delta Wings . 49 Parabolic-arc cross section . 50 Circular-arc cross section 52
15、 Flat delta wing 53 Wing with reverse curvature . 57 Convergence history . 59 Convergence with increasing N . 59 Variable Line Spacing 61 CONCLUDING REMARKS . 61 APPENDIX A . GEOMETRICAL RELATIONS 63 ArcLength . 63 Direction Cosines 65 Shock Conditions 67 Attached Shock at Wing Leading Edge 69 APPEN
16、DIX C . FORCE AND MOMENT COEFFICIENTS 71 REFERENCES . 74 APPENDIX B . FLOW PROPERTIES BEHIND SHOCK WAVE 67 iv Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-CALCULATION OF NONLINEAR CONICAL FLOWS BY THE METHOD OF LINES By E. B. Klunker, Jerry C. Sou
17、th, Jr., and Ruby M. Davis Langley Research Center SUMMARY A computational technique, called the method of lines, is developed for computing the flow field about conical configurations at incidence in a supersonic flow. which makes use of the self-similarity property, is developed for the nonlinear
18、flow equations. The basic idea is to discretize all but one of the independent variables in the partial differential equations so that a coupled system of approximate, simultaneous, ordinary differential-difference equations is obtained. Initial values of these differential- difference equations are
19、 determined from the shock relations after the shock shape is estimated or otherwise specified. The system of equations is integrated numerically and an iterative process is utilized for adjusting the shock shape to satisfy the boundary condition of flow tangency on the body. The method, The method
20、has proved to be an efficient and versatile procedure for constructing the numerical solutions to conical flow problems. the flow about circular and elliptic cones at conditions where small regions of supersonic cross flow develop and for the conical delta wings where the region of supersonic cross
21、flow is extensive. The calculations made for circular and elliptic cones as well as for the compression side of various conical delta wings are in good agreement with experi- ment except in regions where viscous effects become important. It has been successful in computing INTRODUCTION In 1935 Busem
22、ann (ref. 1) introduced the concept of a general conical flow field as one in which the fluid properties are constant along any ray emanating from a common point in the flow. Solutions for such self-similar conical flows are of great importance to the aerodynamicist since (1) significant regions of
23、the flow about many practical con- figurations are conical, or nearly so; (2) conical bodies and wings are the simplest class of three-dimensional shapes and thereby provide “benchmark“ cases for both theoretical and experimental studies in supersonic and hypersonic flow. Although the self -similari
24、ty property of conical flow allows the reduction of the problem from three to two space dimensions, the analyst finds himself confronted with a formidable free-boundary problem for nonlinear partial differential equations of elliptic Provided by IHSNot for ResaleNo reproduction or networking permitt
25、ed without license from IHS-,-,-or mixed type. Hence until the last decade, most conical solutions have been obtained only for the simplest cases or after linearization or other approximations to the equa- tions. However, recent advances in speed and storage of digital computers have spurred the dev
26、elopment of numerical solutions of the full nonlinear equations, to the point where solutions to very general conical flow problems can be obtained in a few minutes. A particularly efficient numerical technique for solving conical flow problems has been reported in references 2 to 4. The method is s
27、emidiscrete, wherein one independent variable is discretized while the other remains as a continuous variable. This method is referred to as the method of lines, to distinguish it from grid or network computations where all independent variables are discretized. The method of lines is “direct“ in th
28、e sense that the body shape is given and is one of the bounding coordinate surfaces; yet, the shock wave is another bounding coordinate surface, and the governing differential equations are solved by integrating inward from the shock. Thus, in that respect, the method is like the inverse methods. Th
29、e technique employed for solving this free boundary problem has three distinguishing features: (1) the coordinate transformation which maps the region between the shock and body onto a rectangle, (2) the solution of the equations by the semidiscrete method of lines, and (3) an iteration procedure fo
30、r satisfying the boundary conditions. None of these features are new; yet when combined, they prove to be an efficient means of solving free boundary problems such as the supersonic blunt-body problem or conical flows. The basic idea of the method of lines is to discretize all but one of the indepen
31、dent variables in the partial differential equations so that a system of approximate, simultaneous, ordinary, differential-difference equations is obtai2ed. Initial values for the system of equations are estimated, or otherwise specified, and the system of equations is integrated numer- ically. An i
32、terative process is utilized to satisfy the boundary conditions; thus, the initial values are subsequently altered and the equations are again integrated. The SUCCCSS of the method of lines as a computational tool hinges upon (1) formulation of the problem in a form which requires relatively few lin
33、es, (2) use of an efficient integration routine that yields good accuracy with relatively large integration steps, and (3) development of an efficient interative process to satisfy the boundary conditions. The first requirement is largely met through the choice of the coordinate system and the secon
34、d can be satisfied with any of a number of integration schemes such as a fourth-order Runge-Kutta method. The computational time and the utility of the method depend to a large part on the itera- tive process. The present paper expands upon the material in reference 4; further details and refinement
35、s of the method are presented, together with numerous applications to a variety of conical flow problems. Comparisons of the present calculations with other theories and experiment are given for circular and elliptic cones, and conical delta wings. 2 Provided by IHSNot for ResaleNo reproduction or n
36、etworking permitted without license from IHS-,-,-The stream velocity vector lies in a plane of symmetry for all the configurations; how- ever, this restriction is not a limitation of the method itself. ical aspects of conical flow are touched on, such as the inviscid entropy layer with the attendant
37、 steep gradients adjacent to the surface, and the nodal-type singularities of the cross -flow streamline patterns. Some of the more theoret- To aid the reader, numerous headings and subsections are employed. A separate section “Background“ is included which cites most of the recent work in nonlinear
38、 coni- cal flow theory, in particular, the related work in the U.S.S.R. which seems to have gone relatively unnoticed. BACKGROUND In this section, a review of past work in nonlinear supersonic conical flow theory is given. No effort has been made to consider the large body of literature which con- c
39、erns linearized conical flow theory. Nonlinear Conical Methods The earliest treatment of nonaxisymmetric conical flow was given by Stone (ref. 5) together with the numerical computations carried out under the direction of Kopal (ref. 6), where the flow about circular cones at small incidence was con
40、structed as a perturbation about the axisymmetric nonlinear Taylor-Maccoll solution (ref. 7). ognized the singularities of these conical flows, which were not accounted for in the Stone solution, and discussed the general features of the streamlines. The many analyt- ical papers published since (ref
41、s. 9 to 11 and many other papers referenced in these works) have been concerned largely with the construction of solutions to conical flows by means of matched asymptotic expansions. These papers have concentrated primarily on the theoretical development; consequently, there has been relatively litt
42、le computational work presented. Ferri (ref. 8) rec- Two basic approaches are available for the numerical development of exact non- linear conical solutions : 1 (1) Distance -asymptotic methods, where some initial distribution of the flow vari- ables and shock-wave shape is used near the apex as ini
43、tial values for continuing the calculation downstream by some three -dimensional computation scheme. The calculation proceeds until conical similarity conditions are sufficiently satisfied. .- - - _- - - 1Exact in the sense that the only approximation made is the reduction of the gov- erning partial
44、 differential equations to ordinary differential equations or algebraic equa- tions by using finite-difference expressions for the derivatives with respect to one or more of the coordinates. 3 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-(2) Metho
45、ds which invoke the conical self-similarity and thereby reduce to two the number of independent variables are referred to simply as “conical“ methods for brevity. Both general approaches have their merits. The distance-asymptotic techniques develop the solution as a well-posed initial-boundary probl
46、em for equations of hyperbolic type, and convergence is “almost“ guaranteed from both physical and theoretical consid- erations. However, to achieve a satisfactory solution in many problems where a fine mesh is needed, these methods require a large amount of computer storage and time. The conical me
47、thods reduce the problem to one in two dimensions but in the more diffi- cult form of a free boundary problem for equations of elliptic or mixed type. In fact, many of the conical methods are similar to methods used for solving the blunt-body problem. The methods of references 12 to 20 are examples
48、of the distance-asymptotic method. References 12, 13, and 14 considered circular cones at angle of attack, refer- ences 15 and 16 included cones of elliptic cross section, and reference 17 presents cal- culations for the compression side of conical delta wings with the shock wave attached not only a
49、t the apex, but also along the swept leading edges. More recently, a three- dimensional characteristic method has been developed to compute the flow about some delta-wing configurations, also with an attached leading-edge shock (ref. 18). ences 19 and 20, a method is presented which is essentially a distance-asymptotic method, and whic
