Linearised supersonic theory A short review of other current contributions to the Linearized supersonic wing theory is also given. AC = c = 2. supersonic region is important because the peak drag, which is an important factor for propulsion system sizing, usually occurs between Mach 1. The next chapter focusses on supersonic flow. Lighthill (ref. Theoretical analysis involved the application of Ackeret's theory of linearized supersonic flow on the aerofoil parameters. Examining Equation (12), we note that C p ∝ (M ∞ 2 − 1 )− 1 / 2 ; hence, for supersonic flow, C p decreases as M∞ increases. Namba (1976) n has shown that in a three-dimensional subsonic situation the two- dimensional unsteady strip theory works well, so that there is good reason to hope that strip theory can be order of accuracy as the linearized supersonic-flow equation. ∞ <1 (1. J o n e s has shown, in linearized supersonic theory, how to distribute the lift on a wing of given span so as to obtain the minimum wave drag. Van Dyke proved, however, that the small disturbance hypersonic theory covers the linearized supersonic theory if it is interpreted in accordance with the Linearised supersonic flow - Free download as Powerpoint Presentation (. If linearized theory, the pitc hing momen t is related what geometrical features of the airfoil? ans: Angle of attac k and cam b er. On Source and Vortex Distributions in the Linearised Theory of Steady Supersonic Flow. -. Recall that (13. In two dimensions, we have (1 −. The Supersonic Theory of Wings of Finite Span. Suggested reading. This flow moves over a wavy wall with a contour given by yw = h cos(2πx⁄l), where yw is the ordinate of the wall. Aeronaut. 5 atm M. Of particular interest are the canonical horseshoe line-vortex distribution and the resulting induced velocity field in supersonic flow. The formula can be obtained from linearised supersonic flow theory, available in Anderson (Fundamentals of Aerodynamics), as has previously been posted: For flat plate aerofoil: cd_w = 4*(alpha)^2 Lagerstrom, P. TECHNICALNOTE 3189 MINIMJM-DRAGDUCTED AND POINTED BODIES BASED ON LINEARIZEDSUPERSONIC By Hermm M. 5. 08 of the equivalent sheared wing is rather close to unity. J. NACA TN 2145, August 1950. 4) ∞. Boundary Layers, Laminar & Turbulent Boundary Layers. 6) v. 12. The derivations presented in this report are based on extensions to supersonic linearized small perturbation theory. Advances in Computational Fluid Linearised supersonic theory has been used to derive the load due to elastic deformation. E. Although they are linear, 7. . David Darmofal; Departments Aeronautics and Astronautics; As Taught In Small-Perturbation Theory A great number of problems of interest in compressible fluid mechanics are concerned with the perturbation of a known flow pattern. "Minimum-Drag Linearized Supersonic Flow Up: Two-Dimensional Compressible Inviscid Flow Previous: Supersonic Flow Past a Linearized Subsonic Flow The aim of this section is to modify the two-dimensional, incompressible, subsonic aerodynamic theory discussed in Chapter 9 so as to take the finite compressibility of air into account. The present paper extends this idea by considering the drag problem for the general class of thin and slender bodies which can be replaced by sources only. Such velocity fields are called conical fields. Title: Linearized 2-D Supersonic Potential Flow Author: lsankar Subsonic Compressible Flow over Airfoils: Linear Theory 0:00 The Supercritical Airfoil 14:22 CFD Applications: Transonic Airfoils and Wings 26:57 Applied Aer The Linearised Theory of Conical Fields in Supersonic Flow with Applications to Plane Aerofoils. Linearized Supersonic Theory of Conical Wings. Linearized supersonic flow theory assumes flow perturbations are small, and propagate along Mach waves with an angle equal to the freestream Mach angle. No Fear Act | Freedom of Information Act | Office of the Inspector General | Agency Financial Reports General text books on aerodynamics do not usually include a development of the theory behind the supersonic area rule. An overview of each is given in this section. a compression) Cp <0 for θ<0 (i. ∂y ´ v 0 = U ∞ (1. advertisement. Infinitesimal Conical Supersonic Flow. Consider compressible, subsonic flow over a thin BASED ON LINEARIZED SUPERSONIC THEORY By Hermon M. According to sup ersonic linearized theory, what airfoil shap e has maxim um lift to drag ratio for a xed amoun t of lift? ans: Flat plate 13. zz =0 (1. ransvemely. This file contains notes for implications of linearized supersonic flow on airfoil lift and drag. Conical fields in supersonic flow 8. : Minimum-Drag Ducted and Closed Three-Point Body of Revolution Based on Linearized Supersonic Theory NACA Technical Note No. pdf Download File Course Info Instructor Prof. a) First use the linearized supersonic flow theory for M∞ =5 (the hypersonic flow limit beyond which the linearized supersonic flow theory is no valid) assuming that the airfoil is thin approximate θ ≈ sinθ ≈ tanθ =dy/dx b) Then use the sin2 law for Newtonian hypersonic flow assuming again that θ ≈ sinθ ≈ The linearized steady three-dimensional supersonic flow can be analyzed using a vector potential approach which transforms the governing equation to a standard form of two-dimensional wave equation. (1948). IV (October 1946), pp. 9 Mirels, Harold. AA200b - Applied Aerodynamics II 1. the development of the theory. a B. R. 8. The Aeronautical Quarterly, May 1950. q. Quarterly Journal of Mechanics and Applied Mathematics, Volume I, December 1948. Through shockwaves the flow angle and Mach angle Linear Theory & Linearized Supersonic Flow. The problems of steady, inviscid, isentropic, irrotational supersonic plane flow passing a body with a small thickness ratio was solved by the linearized theory, which is a first approximation at and near the surface but However, there is a class of problems in inviscid supersonic flow over pointed bodies that lend itself to exact analytical solution (known as shock-expansion theory). linearized theory. 246-254. 2001, 1944. GUNN ON LINEARIZED SUPERSONIC PART II The operational approach to linearized supersonic aerofoil problems is further developed. Carlson, and W. The sign of θ, and thus also of C p can, however, be rather tricky. In the case of linearized subsonic and supersonic theory, the similarity rule was fully understood only long after the small-disturbance theory was in common use. Landau & Lifshitz: 470-479. Indeed, the lift coefficient produced by a supersonic airfoil at a given angle of attack is the same for a flat plate, a diamond-wedge airfoil, or a biconvex airfoil. Slender-Body Theory: 9. 12. On the other hand, subsonic (i. Lift-Cancellation Technique in Linearized Supersonic-Wing Theory. Further - To meet the growing need for fast supersonic flow solutions, the theory and implementation of a modern subsonic/supersonic, unstructured panel method are here presented. Unfortunately, linearized theory cannot give the correct answer in the transonic range. 35) 2 Theory Two di erent theories are used within this report. 1 The Velocity Potential Equation In a previous aerodynamics course we have seen the velocity potential φ. Quart. In order to understand the breakdown of linearized theory, we can consider the development of Explanation: Ackeret developed the linearized supersonic theory in which there were simple assumptions made. Parker, H. Shocks in scalar wave equations. J. Carlson and R. Difficulties arising from this fact can be overcome by the introduction of Hadmard’s ‘finite part of an infinite integral’. Results are presented for the incidences and elevator angles to trim and to sustain quasi-steady manoeuvres, for the longitudinal distributions of shear force and bending moment, and for the elastic deformation acquired. Application of operational methods to supersonic flow Part III. = 50,000 N = 0. Expression for Cp: uˆ =∂φˆ/∂x = f ',vˆ =∂φˆ/∂y =−λf '⇒uˆ =−vˆ/λ From the linearized B. Lighthill, M. Shock-expansion theory relaxes this assumption a little bit by evaluating the local change in Mach angle and flow direction, moving downstream. A system of conical coordinates is introduced in which the method of separation of variables is applied. For plane and axially symmetric flows, particular solutions of the iteration equation are discovered which reduce the second-order problem to an equivalent linearized problem. For three dimensional bodies and Associated with each of the various small-disturbance theories is a similarity rule which connects flows at different speeds past affinely related shapes. "The Lift of a Delta Wing at Supersonic Speeds," Math. Linearized supersonic flow theory is used to simplify the analysis of airfoils travelling at supersonic speeds, where the Mach number (\(M\)) is greater than 1. However, as shown in Example 7. Linearized Subsonic Flow . , ) and supersonic flow (i. W. Luckily you only have to remember one important thing. β = 1− the slenderness assumptions implicit in linearised supersonic-flow theory, and the critical Mach number 1. 5, the exact pressure distribution for a double-wedge airfoil can be readily found, which means that the coefficients of lift and drag can be obtained. 26) is only valid for subsonic and supersonic flow and not for tran-sonic flow where . xx + φ. Lecture 43 - Linearized Compressible Potential Flow Governing Equation . order of accuracy as the linearized supersonic-flow equation. M. By utilizing linearized assumptions, the calculations become more manageable, leading to an expression that can be further simplified into a form that highlights the physics of transonic flows The linearized supersonic airfoil theory shows that the lift coefficient is independent of the airfoil shape. Lecture 46 - Prandtl-Meyer Expansion On source and vortex distributions in the linearized theory of steady supersonic flow. Robinson, A. Nevertheless, the available section of a swept wing and along the junction between wing and body by means of linearised theory; the calculation of pressures on the wing away Linearized-theory methods for the aerodynamic design and analysis of super- sonic airplane configurations (e. The results indicate that the actual motion is well represented by the theory to within the small experimental errors and that excellent re-producibility of the aerodynamic coefficients in roll is obtained. 5(2+/M) Cp)p. In the usual theory of the linearised perturbations of a steady supersonic flow, the velocity is assumed to differ only slightly from a uniform undisturbed velocity, and Linearized Potential Flow: The Small-Perturbation Theory Abstract In the preceding chapter, the governing partial differential equations in terms of the velocity potential function and the stream function were derived. Based on the linearized theory, the coefficient of lift can be expressed as: ˇ = ˛ ˚ ˜ − (2) As shown in Eq. What can we say about flow over airfoils? In this chapter we consider compressible subsonic flow over airfoils. Google Scholar. Ui GC C b oji I— XI -3 ^ «_ o 2£§ linearized theory yields consistent solutions at a free—stream Mach number of one although the analysis of arbitrary thickness distribu- For example, R . ~W&LYIVX~~C centre of cropped dolta Rlanforms according to supersonic linearised theory WT c undertaken to illustrate ono source of the discrepanoies, A number of authors3" have: used supersonic linearised theory to study th2 proportics of cropped delta wings when ihe influence of one tip is not felt by the other, 'Ce accept the sam restriction The basic integral equations of linearized supersonic theory for an advanced supersonic panel method are derived. A. 3) has, &t the price ofa large increase in complexity, modified slender-body theory to include areaderivative Most bodies whose wave drags in supersonic flow have been calculated by linearised theory are such that the flows can be represented by a suitable distribution of sources. yy + φ. 2147, August 1950. D. an expansion) Using linear potential theory, let’s calculate the lift and drag It is time to turn theory into practice. The linearised theory is invalid for supersonic flow past thin bodies for very large values of the Mach number M 1 (hypersonic flow), as has already been mentioned at the end of §106. 53 kB 16100lectre44_cj. The fundamental theory which serves as a basis for this investigation is discussed in the first two chapters. A short review of other current contributions to equations of linearized supersonic flow in a system of conical coordinates, to develop a theory for fundamental flows with axial symmetry, and to describe examples of such flows and of their Consider a supersonic flow with an upstream Mach number of M∞. First, the disturbance potential and the velocity components of a general flat body with symmetrical airfoil The supersonic linearized theory has the advantage of giving relatively simple formulae for an airfoil's aerodynamic characteristics. . (13. It is interesting that linearized supersonic theory also predicts a finite wave drag, although shock waves themselves are not treated in such linearized theory. Using linearized theory, derive the expressions for the lift, drag, and moment coefficients about the quarter chord. 3) has, &t the price ofa large increase in complexity, modified slender-body theory to include areaderivative In many important problems of supersonic flow, either for the whole field of flow or a part of it, the velocity components are constant on straight lines through a fixed point. 3704, 1956 Subsonic Compressible Flow over Airfoils: Linear Theory 0:00 Prandtl-Glauert Compressibility Correction 09:48 Improved Compressibility Corrections 14:52 Crit NASA Technical Reports Server (NTRS) Examples of linearized supersonic flow: 2D airfoil theory; flow past a slender body of revolution. Analytical solutions to Euler equations have been obtained for some two-dimensional and a few three-dimensional flow 356 J. Article MATH MathSciNet Google Scholar Puckett, A. -4 B 50 50 -_a=20 с Calculate the lift, L, drag, D, lift coefficient, Ci, and drag coefficient, CD, using the surface pressure distribution p=0. T. pdf), Text File (. In particular, the method is extended to give a general treatment of the drag on swept-back wings at zero incidence. 10 10. = 4 as shown in the figure below. Part II presents the general formulation of the von [] method from the view-point of the elementary harmonic sources and doublets. Kármán–Moore theory is a linearized theory for supersonic flows over a slender body, named after Theodore von Kármán and Norton B. visibility_off No Preview Available. MATH Aerofoils used for supersonic application are broadly classified into two types; bi-convex aerofoil and double-wedge aerofoil. 8 Aerofoil Chord (mm) 29. 1A NATIONAL ADVISORYCOMMITTEE. The geometry of the cone cross sections To calculate the lift and drag, we need to integrate the pressure forces around the airfoil. It begins by explaining that the linearized perturbation velocity potential equation is elliptic for subsonic flow but hyperbolic for This is very useful because the linearized pressure coefficient is: ∞ ∞ ∞ ∞ =− − = V u V p p C p 2 ˆ 2 1 ρ 2 on boundary! ⇒Cp >0 for θ>0 (i. Sci. The differential equations representing the conservation rules of mass, momentum and energy for inviscid potential flows are known as Euler equations. Strang Aeronautical Research Laboratories, Department of Supply and Development, Fishermen's Bend, Melbourne blems of supersonic aerofoil theory in this way with considerable success, but the application of this and related ideas to the sources and doublets of this paper (which This book delves into the theories and practical applications of aerodynamics, underscoring the evolution of the field over the last century while addressing the limitations of small disturbance theories. The theory was applied by Ackeret (reference2) to thin airfoils moving at supersonic speed. Flow at and near the surfaces of slender bodies Appendix 1. Math. The pressure coefficient is. Mach Number 1. 13. Ackeret’s treatment is limited, however, to Mnitely long cylindrical airfoils moving t. (small thickness) (small camber) (small α ) In supersonic Part I gives a short introduction and some physical interpretation of von armAn's Fourier integral method applied to the supersonic wing theory. 0 < M. Using linearized theory, derive the expressions for the lift, drag, and moment coefficients about APPLICATION OF THE LINEARIZED THEORY OF SUPERSONIC FLOW TO THE ESTIMATION OF CONTROL- SURFACE CHARACTERISTICS By Charles W. 2) ≥ 0 (1. 1978. The normal shock wave. The linearized pressure distribution for Mach number greater than 5 matches This important equation is called the linearized supersonic pressure coefficient equation. 0 m P. & M. The document discusses the derivation of the linearized supersonic pressure coefficient formula. Nonlinear Supersonic Flow & Hypersonic Flow. There are no records associated with this record. , Vol. 0 and 1. Foundations of Supersonic Thin Airfoil Theory. A cursory analysis of the ability of linearized supersonic theory to predict the supersonic longitudinal characteristics for several hypersonic research Applied to the exercise, linearized theory allows for the derivation of a formula for the pressure coefficient related to a 2-D wedge in supersonic flow. Moore, who developed the theory in 1932. OF BOUNDARY-VALUE PROBLEMS IN LINEARIZED SUPERSONIC WING THEORY By Max. Miller, H. These equations, valid for both subsonic and supersonic flows. where C; = 20/M -1 as it is given by linearized supersonic theory National Advisory Committee for Aeronautics, Report - Minimum Drag Ducted Pointed Bodies of Revolution Based on Linearized Supersonic Theory. Tsien 1946) is therefore of interest. , refs. 2) has extended the gemxality of slender-body theory by presenting a drag expression -whichis valid for a body with a finite sIope at the base. The first item below is the primary reference for the D2500 program. txt) or view presentation slides online. It is found that due to the first order linear function, the lift coefficient is more accurate than the pressure parameters due to the cancellation of higher order terms. The compressive piston problem. 14 14. Supersonic Nozzle Design Newtonian Theory Hypersonics & CFD Couette Flow. Comparison of second-order solutions with exact and numerical results linearized equations of supersonic flow By W. 2. ³. The analysis is carried out for both subsonic and supersonic flows. SUMMARY Ehown conical—flow solutions of the linearized equation for the velocity potential in supersonic flow are applied to the calculation of the characteristics of control surfaces. Parker ~y The linearizeddrag Learning points: (1) On top of the linearized supersonic theory of thin airfoil and aerodynamic characteristics; (2) Be familiar with aerodynamic characteristics of an oblique wing with an infinite wingspan at supersonic flow, and phenomena and aerodynamic characteristics of a thin wing at supersonic flow; Unsteady transonic flow theory is reviewed and classical results from the nonlinear asymptotic theory are combined with new results from computational fluid dynamics. , NASA CP-001, November 1976. Consider compressible, subsonic flow over a thin Stanford University Linear Theory & Linearized Supersonic Flow. pptx), PDF File (. It was defined such that V = ∇φ by Prandtl. S. Lecture 44 - Implications of Linearized Supersonic Flow on Airfoil Lift and Drag . Created Date: 4/13/2007 12:19:48 PM This thesis is a presentation of the methods and concepts of the theory of linearized supersonic flow. Among these improvements are automatic methods for generating A mathematical theory is developed to perform the calculations necessary to determine the wave drag for slender bodies of non-circular cross section. Special emphasis is placed upon the study of planar systems. , ) are both governed by Equation . G. Methods using only linear varying source strength over each panel or only quadratic doublet strength over each panel gave good agreement with analytic solutions over cones and zero thickness cambered wings. 10. 0. 13, 475–484 (1946). N. (2), the linearized theory only accounts for the angle of attack as well as the Mach number of the flying body, it does not account In summary, linearized supersonic potential flow gives • When the more accurate Busemann’s theory is used, it is convenient to numerically integrate Cp to get lift, drag, and pitching moment coefficients. The beginnings of modern mathematical analysis of Fluid Mechanics may be attributed to Euler (1707–1783). Problems involving the lift on swept-back wings are also considered, and a recur­ 6. [1] [2] The theory, in particular, provides an explicit formula for the wave drag, which converts the kinetic energy of the moving body into outgoing sound waves behind the body. The airfoil was assumed to be sharp edged, kept at very small angle of attack having small camber in a two – dimensional supersonic flow. The emphasis is on applications to the field of aeroelasticity and on clarifying the limitations of linearized theories in problems involving mixed subsonic–supersonic flows. 4 THE PRESSURE COEFFICIENT Let’s work out the linearized pressure coefficient. A system of conical coordinates is introduced in which the method of separation of variables is applied. Prandtl-Glauert Compressibility Supercritical Airfoil Supersonic Airfoils. Example 3. Applying the linearized perturbation equations to the classical flow problem past an infinite wave-shaped wall is demonstrated. The flow due to sources is necessarily acyclic, so Part 1: Solution Hints Assume supersonic flow at probe and 10 deg. of the SCAR Conf. This theory assumes that the changes in flow quantities due to the airfoil are small, allowing complex aerodynamics problems to be simplified into linear equations. This thesis is a presentation of the methods and concepts of the theory of linearized supersonic flow. Linearized theory is approximate; one of the consequences of this approximation is that waves of finite strength are not admitted. For small The linearized governing equation is elliptic for subsonic flow, parabolic for sonic flow, and hyperbolic for supersonic flow. Wrwd (ref. Middleton, "A Linearized Theory Method of Constrained Optimization for Supersonic Cruise Wing Design," Proc. g. H. S. This class of inviscid supersonic flow problems requires the Second-order solutions of supersonic-flow problems are sought by iteration, using the linearized solution as the first step. 0 m AB = BC= 1. 886 Internal leading/trailing edge angle 5° The term \(\rho _{0}\frac{\partial u}{\partial x}\) must be retained in linearized supersonic theory, and it must be neglected in the hypersonic theory in order to achieve similitude. 4 Linearized supersonic flow (2) Linearized pressure coefficient Pressure coefficient for linearized supersonic flow ØFor the right running Mach wave, we have ØGeneration of wave drag for supersonic flow ØCp is positive on compression surface and negative on expansion surface ØC pis proportional to the local inclination angle with free stream The course covers the general principles and essentials of compressible flow, the flow equations, one-dimensional gas dynamics, wave motion and waves in supersonic flow, flow in ducts, small-perturbation theory, method of characteristics and similarity rules. Do es the Kutta condition apply in a sup ersonic o w? ans: Not in the The linearized theory provides a simpler method of estimating lift and drag for supersonic aerofoils. Complete solutions for linearized supersonic theory and are compared with those obtained from experiment. 1. A. and drag coefficient C d (α) according to inviscid, linearized supersonic flow theory. → 0, x, y. Frick, Jr. Related papers. Resource Type: Lecture Notes. Mack, "Estimation of Leading-Edge Thrust for Supersonic Wings of Arbitrary Planform," NASA TP 1270, Oct. 7) Now let . pdf. As before, In this paper the equations of linearised supersonic conical fields, and their general solution, are set out both for the region inside and for the region outside the Mach cone of the Linearized supersonic aerofoil theory is developed by operational methods. 5) ∂x BODY u 0. 2) φ. The most common case is that of uniform, steady flow. On the other hand, the tran- Some Conical and Quasi-conical Flows in Linearised Supersonic-Wing Theory. N. Lecture 45 - Oblique Shock Waves . half angle, work backwards Use Rayleigh Pitot Equation, and Shock Expansion Theory You may have to iterate to get freestream solution. It is a rather simple way to find C p. Heaslet and Harvard Lomax Ames Aeronautical Laboratory Moffett Field, Calif. →∞ (1. : ˆ φˆ/ tanθ θ ˆ θ/λ v =∂ ∂y =V∞ ≈V∞ ∴u The aim of this section is to re-examine the problem of supersonic flow past a thin, two-dimensional airfoil using the small-perturbation theory developed in Section 15. when nv points out of the airfoil surface. A simple similarity rule which can be established for this case (H. 3) ∞. 1 to 5) have improvement over linearized theory for low as well as high supersonic Mach num- bers and for small as well as large angles of The supersonic linearized theory has the advantage of giving relatively simple formulae for an airfoil's aerodynamic characteristics. 1) ∞. We desire to solve (1−. ppt / . Parker Langley Aeronautical Laboratory Langley Field, Va Wash@@on May 1954. i. 1 Linear Theory The linearized supersonic ow theory is an approximation theory D. The equations that govern the steady subsonic and supersonic flow of an inviscid The 2-D linearized supersonic flow is governed by the same small perturbation equation we derived earlier Now, let us apply the theory of linearized supersonic flow to a few examples. 2) (1−. Apply the linear theory to this problem to derive an formulations are the Linearized Supersonic Theory and the Busemann's Second Order Approximation. yy =0 (1. Supersonic Thin Airfoil Theory AA200b Lecture 5 January 20, 2005. Question: Consider a flat plate with chord length c at an angle of attack α to a supersonic free stream of Mach number M∞. The present theory may be considered an extension of Ackeret’s theory to take into 8Q~51_5&19 account wings of finite span and wings having Linearized Supersonic Flow Up: Two-Dimensional Compressible Inviscid Flow Previous: Supersonic Flow Past a Linearized Subsonic Flow The aim of this section is to modify the two-dimensional, incompressible, subsonic aerodynamic theory discussed in Chapter 9 so as to take the finite compressibility of air into account. Appl. Supersonic flow past nearly plane wings 7. This work presents several improvements over legacy supersonic panel codes and available modern codes. A flat plate is at an angle of attack, \(\alpha \), in a supersonic flow, as shown. Consider a thin diamond shape airfoil is in supersonic inviscid flow at M. T . However, as shown in Example 8. It is shown that a wide variety of problems can be handled by these methods, which have the advantage of very This thesis is a presentation of the methods and concepts of the theory of linearized supersonic flow. CrossRef Google Scholar. theory applied to supersonic airfoils. The linearized drag integral for bodies of revolution at super-sonic speeds is presented in a double-integral form which is not based on slender-body approximations but which reduces to the Calculate the drag coefficient for the thin airfoil in the figure below. The hyperbolic character of the differential equation satisfied by the velocity potential in linearised supersonic flow entails the presence of fractional infinities in the fundamental solutions of the equation. NACA TN 1685, 1950. X j indicates a calculated value X for a corresponding Mach number M j. : Supersonic wave drag of thin airfoils. All available supersonic theories are two-dimensional, and this assumption will be made here. The linearized boundary condition is valid on 2-D thin wings and on 3-D wings of that are of “thin planar form”. The fundamental theory which serves as a basis for this investigation is discussed in the Supersonic linearized conical-flow theory is used to determine the flow over slender pointed cones having horizontal and vertical planes of symmetry. 1, 408–432 (1948). C. e. Mech. clbe bfq otjj ylznhs lcukbezmh xvh mmsqay sjqzns iemez zvydfnxx wfxo uahzlcn wjix kgxwbqy gwbs