https://ntrs.nasa.gov/search.jsp?R=19720004573 2020-03-23T13:40:44+00:00Z th i N ASACONTRACTOR REPORT h o* h v I w U CALCULATION OF COMPRESSIBLE, NONADIABATICBOUNDARY LAYERS IN LAMINAR, TRANSITIONAL AND TURBULENT FLOW BY THE METHOD OF INTEGRAL RELATIONS Prepared by NIELS EN: ENGINEERING AND RESEARCH, INC. Mountain View, Calif. 94040 for Langley Research Center NATIONALAERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. NOVEMBER 1971 I TECH LIBRARY KAFB, NY - -~~ ~ .~ 1. Report No. I 2. Government A-ion No. 3. Recipient's clltalog No. CALCULATION OF COMPRESSIBLE, NONADIABATIC BOUNDARY LAYERS IN 1 November 1971 LAMINAR, TRANSITIONAL AND TURBULZNT FLOW BY W METHOD OF 6. PerformingOrgsnization Code INTEGRALRELATIONS ~" ~ .. .~ . ~~~ - I 7. Author(s) 8. Performing Orgsnization Report No. Gary D. Kuhn NEAR TR 25 . ~ ~~ ~~" - . 10. Work Unit No. 9. MormingOrginization Name and Address 722-03-10-01 Nielsen Engineering and Research, Inc. ~.~ 850 Maude Avenue 11. Contract or Grant No. Mountain View, 94040 .. Calif. NASl-9429 .. " . 13. Type of Report andPeriod Covered 2. SponsoringAgency Name and Address . ContractorReport National Aeronautics and Space Administration 14. SponsoringAgency Code Washington, DC 20546 .- .. .. ... ~ ~~ - -- - __ _~ - r 5. .Supplementary Notes " .. " . ~~~ ~ . 6. Abstract The method of integral relations has been applied to the calculation ofcompressible, nonadiabaticlaminar, transitional, and turbulentboundary-layer characteristics. A two-layer eddy-viscosity model was employed for turbulent flow with transition produced by modulating the eddy viscosity by an intermittency function..--A computqr program was developed to do the calcu- lations for two-dimensional or axisynrmetricconfigurations from low speeds to hypersonicspeeds witharbitrary Prandtl number. The program containsprovisions for tabular input of arbitrary configurationshapes and arbitrarystreamwise pressure, temperature, and Mach number distributions. Options are provided for obtaining initial conditions either from experimentalinformation or from a theoretical similarity solution. The transition region can be described either by an arbitrary distribution of intermittency or by a functionbased on Emnons' probabilitytheory. Correlations were developedfor use in est- imatingthe parameters of thetheoretical intermittency function. Correlations obtained from other sources are used for estimating the transition point. Comparisonswere made between calculated and measured boundary-layer quantities for laminar, transitional, and turbulentflows on flat plates, cones, cone-flares, and a waisted body ofrev- olution.Excellent comparisons were obtainedbetween the present theory and two other theories based on the method of finite differences. The intermittency required to reproduce some experimental heat transfer results in hypersonic flow was found to bequite different from the theoretical'function. It is suggestedthat the simple probability theory of mons may not be valid for representing the intermittency of hyper- sonic transitiohal boundary layers and that the program could be useful as a tool for detailed study of the intermittency of the transition region. 2 18. Distribution Statement Boundary layer Laminarboundary layer Unclassified - Unlimited Transitional boundary layer Turbulent boundary layer ____ I 9. Security Classif. (of this repbrt) 20. Security Classif. (of this page) 21. NO. of pager 22. Rice* Unclassified Unclassified 132 $3.00 For sale by the NationalTechnical Information Service, Springfield, Virginia 22151 . ”. .. .. - FOReWORD The research described herein was sponsored by LangleyResearch Center, NASA, underContract NAS1-9429. Computer facilities were pro- vided at LRC and at Ames ResearchCenter, NASA. The assistanceof the TechnicalMonitor, Mr. S. 2. Pinckney, incoordinating the research effort and providing technical information is gratefully acknowledged. The work was performed under the supervision of Dr. Jack N. Nielsen. iii TABLE OF CONTENTS Page Section -NO. SUMMARY 1 INTRODUCTION 2 LIST OF SYMBOLS 3 THEORETICAL BASIS 7 Assumptions 8 Boundary-Layer Equations €or Compressib1.e Turbulent Flow 8 Transformation of Axisymmetric Boundary-Layer Equations to Almost Two-Dimensional Form 13 Transformation of the Compressible Boundary-Layer Equations 15 Stewartson transformation 16 Dorodnitsyn transformation 19 Derivation of Ordinary Differential Equations 22 Development of integro-differential equations 22 Selection of approximating €unctions 26 Development of ordinary differential equations 29 Initial Conditions 36 Similarity solution 36 Direct €it of known velocity and temperature profiles 38 Known boundary-layer parameters 38 Least squares €it of velocity and temperature profiles 38 CALCULATION OF TRANSITIONAL BOUNDARY LAYERS 40 Prediction of the Transition Point 41 Turbulent Eddy-Viscosity Model 45 Modulation of the Eddy Viscosity in the Transition Region 50 Correlation €or the Extentof the Transition Zone 52 DETERMINATION OF ORDER OF APPROXIMATION FOR THE ENTHALPY FUNCTION 54 DESCRIPTION OF COMPUTER PROGRAM 58 V Page Section -No. EVALUATION OF THE METHOD 61 Incompressible Transitional Flow on a Flat Plate 61 Compressible Transitional Flow on a Sharp Flat Plate 62 Transitional Flowon a Sharp Cone 64 Adverse Pressure Gradient Flow on a Cone-Flare 65 Comparison with a Finite-Difference Method 66 Calculations with Specified Initial Conditions 68 Effect of Transverse Curvature in Laminar Flow 71 Turbulent Flow with Transverse Curvature 71 CONCLUDING REMARKS 73 REFERENCES 77 FIGURES 1 THROUGH20 a3 vi CALCULATION OF COMPRESSIBLE, NONADIABATIC BOUNDARY LAYERS IN LAMINAR, TRANSITIONALAND TURBULENT FLCW BY THE METHOD OF INTEGRAL RELATIONS By Gary D. Kuhn Nielsen Engineering& Research, Inc. SUMMARY The method of integral relations has been applied to the calculation of compressible, nonadiabatic laminar, transitional, and turbulent boundary-layer characteristics. A two-layer. eddy-viscosity model was employed for turbulent flow with transition produced by modulating the eddy viscosity by an intermittency function. A computer program was developed to do the calculations for two-dimensional or axisymmetric configurations from low speeds to hypersonic speeds with arbitrary Prandtl number. The program contains provisions for tabular input of arbitrary configuration shapes and arbitrary streamwise pressure, tempera- ture, and Mach number distributions. Options are provided for obtaining initial conditions either from experimental information or from a theoretical similarity solution. The transition region can be described either by an arbitrary distri- bution of intermittency or by a function basedmons' on probability theory. Correlations were developed for use in estimating the parameters of the theoretical intermittency function. Correlations obtained from other sources are used for estimating the transition point. Comparisons were made between calculated and measured boundary-layer quantities for laminar, transitional, and turbulent flows on flat plates, cones, cone-flares, and a waisted body of revolution. Excellent compari- sons were obtained between the present theory and two other theories based on the method of finite differences. The intermittency required to reproduce some experimental heat transfer results in hypersonic flow was found to be quite different from the theoretical function. It is suggested that the simple probability theory ofmons may notbe valid for representing the intermittencyof hypersonic transitional boundary layers and that the program could be useful as a tool for detailed study of the intermittency of the transition region. INTRODUCTION Many empiricalmethods are available for estimating the properties of laminar and turbulentboundary layers. With the development of sophisticated electronic computers, many computerprograms have also been developedfor calculating the solutions of the boundary-layer equations in thelaminar and turbulentregions. At the same time, much researchhas been done on the nature of the boundary-layer flow in the transition regionbetween laminar and turbulentflow. Even now, however,the traLlsi- tionregion is notcompletely understood. The methodsof computing the boundarylayer in the transition region usually employ some empirical variationof the growth of the turbulent eddy viscosity, itself an empirical quantity, along with a prediction of the starting point of the transitionregion developed from still other empirical correlations. At thepresent time, there are two mathematicaltechniques for computingthe details of the boundary layer. Both techniques use the boundary-layerequations together with an empirical eddy-viscosity model forthe turbulent region. The techniquesdiffer essentially in how they solvethe boundary-layer equations. In the first method,an example of which is described. in reference 1, the solution is soughtby the method of finitedifferences. In the second method,described in references 2, 3, and 4, thesolution is bythe method ofintegral relations, in which the solution is represented to anydegree of accuracy by partial sums of infinite series. In the work described in this report, the method of integral relations is adapted to developmentof a generalizedprogram for use as anengineer- ing tool for the calculation of boundary-layers on arbitrary two- dimensionalor axisymmetric bodies, with arbitrary flow conditions. Themethod ofintegral relations was appliedin references 2, 3, and 4 to the calculation of laminar and turbulent,nonadiabatic, compressibleboundary layers. In