Julian D. Cole

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All published works (42)

Early transonic ideas in the light of later developments

2005-11-13

Julian D. Cole

Optimization of Conical Wings in Hypersonic Flow

1998-12-01

Susan A. Triantafillou , Donald W. Schwendeman , Julian D. Cole

Optimal Hypersonic Conical Wings

1998-11-01

Susan A. Triantafillou , Donald W. Schwendeman , Julian D. Cole

A Blunt-Nosed Thin Body in Hypersonic Flow

1998-04-01

О. С. Рыжов , Julian D. Cole , Norman Malmuth

An inverse problem is posed to simulate the influence of the nose bluntness on the hypersonic inviscid steady flow field around a slender two-dimensional section. In this formulation a bow … An inverse problem is posed to simulate the influence of the nose bluntness on the hypersonic inviscid steady flow field around a slender two-dimensional section. In this formulation a bow shock is given in advance whereas the body contour comes at the end of analysis being identified with a particular streamline. An equation for the shock shape involves two terms in the form of different powers of the distance measured along the direction of oncoming stream. The leading term derives from a classical similar solution for the strong viscous/inviscid interaction regime; a correction to it models bluntness effects at large distances downstream. Matched asymptotic expansions are used to solve the problem within the hypersonic small-disturbance theory. In the outer region bounded by the shock, two sets of ordinary differential equations control the pressure, density, and velocity distributions. The second-order approximation admits of an explicit integral obtainable from a consideration of the finite drag exerted on the blunted nose of a section. The use of the momentum conservation law allows us to predict the power of the exponent of the correction term entering the shock equation. The asymptotic behavior of both first- and second-order approximations is established and employed for providing conditions for a solution in the inner region occupied by a high-entropy layer. Governing equations here are solved, explicitly determining a dependence of the body shape on the correction in the shock representation. A thorough analysis of the Newtonian approach reveals certain limitations inherent in this simplified treatment of steady hypersonic flows.

Transonic Hodograph Design Problems

1998-01-01

Julian D. Cole

On self-similar solutions of Barenblatt's nonlinear filtration equation

1996-04-01

Julian D. Cole , Barbara Wagner

We derive the asymptotic form of the self-similar solutions of the second kind of the Cauchy problem for Barenblatt's nonlinear filtration equation by perturbing the Lie group of the underlying … We derive the asymptotic form of the self-similar solutions of the second kind of the Cauchy problem for Barenblatt's nonlinear filtration equation by perturbing the Lie group of the underlying linear problem. We also show that the decay rate, appearing in the similarity solutions, can be found by a simple inspection of the corresponding energy dissipation law.

Multiple-Scale Expansions for Partial Differential Equations

1996-01-01

J. Kevorkian , Julian D. Cole

Limit Process Expansions for Ordinary Differential Equations

1996-01-01

J. Kevorkian , Julian D. Cole

Book Review Transonic Vortical Gas Flows

1995-04-01

Julian D. Cole

Book Review: Matching of asymptotic expansions of solutions of boundary value problems

1993-10-01

Julian D. Cole

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Shock wave location on a slender transonic body of revolution

1989-11-01

Julian D. Cole , Norman Malmuth

Some Asymptotic Problems of Transonic Flow Theory

1989-01-01

Julian D. Cole , L. Pamela Cook

Transonic wave drag estimation and optimization using the nonlinear area rule

1987-03-01

Norman Malmuth , C. C. Wu , Julian D. Cole

Nonlinear area ruling procedures based on the transonic slender body and lift-dominated theories are described as a means of providing low-cost wave drag estimates and optima for basepoint definition. The … Nonlinear area ruling procedures based on the transonic slender body and lift-dominated theories are described as a means of providing low-cost wave drag estimates and optima for basepoint definition. The computational implementation is capable of accurately predicting drag rise of realistic configurations and shows applicability to moderate supersonic Mach numbers. An analogy between the zero-lift and liftdominated case establishes a basis for sizeable wave drag due to lift reduction through planform and sectional shaping. Results illustrating the potential benefits are shown for a fighter configuration in which a small movement of the maximum thickness location of the equivalent body of revolution with the volume fixed gives a fourfold reduction in zero-lift wave drag. This benefit can be translated into similar reductions in transonic wave drag due to lift.

Slender body theory and optimization procedures for transonic lifting wing bodies

1983-01-10

Norman Malmuth , C. C. Wu , Julian D. Cole

Limit process asymptotic expansion has been applied to develop an unsteady equivalence rule for transonic speeds The rule characterizes the near field of the flow over a transonic body as … Limit process asymptotic expansion has been applied to develop an unsteady equivalence rule for transonic speeds The rule characterizes the near field of the flow over a transonic body as harmonic in cross planes perpendicular to the freestream direction, and the far field as a pulsating nonlinear line source Derived ex pressions for the loading indicate a substantial simplification of the prediction problem for three dimensional transonic unsteady airfoils using the theory In another application of transonic slender body asymptotics, substantial reductions in wave drag have been demonstrated using a parametric inverse method The procedure leads to a nearly shockless equivalent body of revolution for a slender airplane using iteration concepts and elimination of jump discontinuities in the equivalent body surface pressure distributions Multiple constraints such as fixed volume and base area are satisfied in the method by correlation of the equivalent body geometry with features of the smoothed surface pressure distribution

Book-Review - Perturbation Methods in Applied Mathematics

1983-01-01

J. Kevorkian , Julian D. Cole , R. W. John

An asymptotic theory of solid tunnel wall interference on transonic airfoils

1982-06-07

Julian D. Cole , Norman Malmuth

Applications to Partial Differential Equations

1981-01-01

J. Kevorkian , Julian D. Cole

Perturbation Methods in Applied Mathematics

1981-01-01

J. Kevorkian , Julian D. Cole

Computational transonic airfoil design in free air and a wind tunnel

1978-01-16

V. S. Shankar Sriram , Norman Malmuth , Julian D. Cole

Modern Developments in Transonic Flow

1975-12-01

Julian D. Cole

A survey is given of transonic small disturbance theory. Basic equations, shock relations, similarity laves, lift and drag integrals are derived., The airfoil boundary value problem is formulated. Finite difference … A survey is given of transonic small disturbance theory. Basic equations, shock relations, similarity laves, lift and drag integrals are derived., The airfoil boundary value problem is formulated. Finite difference methods and computational algorithms are described. Results are compared with other calculation methods and experiments.

Singular perturbation

1975-06-01

Julian D. Cole

Similarity Methods for Differential Equations

1974-01-01

George W. Bluman , Julian D. Cole

Ordinary Differential Equations

1974-01-01

George W. Bluman , Julian D. Cole

Partial Differential Equations

1974-01-01

George W. Bluman , Julian D. Cole

Perturbation Methods in Applied Mathematics

1973-11-01

Julian D. Cole , Lawrence E. Levine

1 Introduction.- 2 Limit Process Expansions Applied to Ordinary Differential Equations.- 3 Multiple-Variable Expansion Procedures.- 4 Applications to Partial Differential Equations.- 5 Examples from Fluid Mechanics.- Author Index. 1 Introduction.- 2 Limit Process Expansions Applied to Ordinary Differential Equations.- 3 Multiple-Variable Expansion Procedures.- 4 Applications to Partial Differential Equations.- 5 Examples from Fluid Mechanics.- Author Index.

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Calculation of plane steady transonic flows

1971-01-01

Earll M. Murman , Julian D. Cole

Transonic small disturbance theory is used to solve for the flow past thin airfoils including cases with imbedded shock waves. The small disturbance equations and similarity rules are presented, and … Transonic small disturbance theory is used to solve for the flow past thin airfoils including cases with imbedded shock waves. The small disturbance equations and similarity rules are presented, and a boundary value problem is formulated for the case of a subsonic freestream Mach number. The governing transonic potential equation is a mixed (elliptic-hyperbolic) differential equation which is solved numerically using a newly developed mixed finite difference system. Separate difference formulas are used in the elliptic and hyperbolic regions to account properly for the local domain of dependence of the differential equation. An analytical solution derived for the far field is used as a boundary condition for the numerical solution. The difference equations are solved with a line relaxation algorithm. Shock waves, if any, and supersonic zones appear naturally during the iterative process. Results are presented for nonlifting circular arc airfoils and a shock free Nieuwland airfoil. Agreement with experiment for the circular arc airfoils, and exact theory for the Nieuwland airfoil is excellent.

Hypersonic similarity solutions for airfoils supporting exponential shock waves

1970-02-01

Jerome Aroesty , Julian D. Cole

Abstract : A study was made of airfoil optimization, using the equations of hypersonic gas dynamics to explore the 'Newtonian chine strip' theory that airfoil concavity enhances the lift-to-drag ratio … Abstract : A study was made of airfoil optimization, using the equations of hypersonic gas dynamics to explore the 'Newtonian chine strip' theory that airfoil concavity enhances the lift-to-drag ratio for a fixed drag penalty. The flow behind concave and convex exponential shock waves is investigated, and the corresponding airfoil surfaces are determined. The calculations show that the optimum lifting surface for fixed drag is only slightly more concave than a flat plate and that the improvement in performance is small. A limit line is shown to exist in the flow field behind convex exponential shock waves, so that is is not possible to construct a convex airfoil that supports an exponential shock wave over its entire length if the nose curvature is too large. (Author)

Perturbation Methods in Applied Mathematics

1970-01-01

Julian D. Cole , George I. Bell

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Perturbation Methods in Applied Mathematics

1969-07-01

Harold Weitzner , Julian D. Cole

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A class of three-dimensional, optimum hypersonic wings.

1969-02-01

Julian D. Cole , TS-FOU ZIEN

A limiting case of hypersonic flow is considered in which Mm —> °o • the flow deflections are small so that hypersonic small-disturbance theory applies. Within this framework there are … A limiting case of hypersonic flow is considered in which Mm —> °o • the flow deflections are small so that hypersonic small-disturbance theory applies. Within this framework there are various, known, exact solutions for flow past axisymmetric bodies. These flows are those for which the shock shape follows a power law rs '~ x*. The idea used in this paper is to construct the compression side of a lifting wing from the known streamlines in the flow behind the power-law shock wave. By considering families of such wings an optimum problem is considered, namely, to find the wing with given lift which produces a minimum wave resistance. The optimum problem is solved by variatioiial methods. Numerical results are are obtained for a range of n from ^ to 10, with y = 1.4.

Asymptotic Behavior of Certain Nonlinear Boundary-Value Problems

1968-11-01

José Canosa , Julian D. Cole

The asymptotic properties of a class of nonlinear boundary-value problems are studied. For large values of a parameter, the differential equation is of the singular-perturbation type, and its solution is … The asymptotic properties of a class of nonlinear boundary-value problems are studied. For large values of a parameter, the differential equation is of the singular-perturbation type, and its solution is constructed by means of matched asymptotic expansions. In two special cases, very simple approximate analytic solutions are obtained, and their accuracy is illustrated by showing their good agreement with the exact numerical solution of the problem.

The blowhard problem—Inviscid flows with surface injection

1968-07-01

Julian D. Cole , Jerry Aroesty

A class of three dimensional optimum hypersonic wings

1968-01-22

Julian D. Cole , Tse-Fou Zien

SINGULAR PERTURBATION METHODS AND NONLINEAR BOUNDARY VALUE PROBLEMS.

1968-01-01

José Canosa , Julian D. Cole

Erratum: "Expansion of a Finite Mass of Gas Into Vacuum"

1965-11-01

Carl Greifinger , Julian D. Cole

Expansion of a finite mass of gas into vacuum

1965-06-01

Carl Greifinger , Julian D. Cole

Abstract : A summary is presented of research on the problem of the expansion into vacuum of a finite mass of gas initially at rest and in a uniform state, … Abstract : A summary is presented of research on the problem of the expansion into vacuum of a finite mass of gas initially at rest and in a uniform state, assuming a perfect and inviscid gas. The cases of plane, cylindrical, and spherical flow are considered.

The Theory of Laminar Boundary Layers in Compressible Fluids. By K. S<scp>TEWARTSON</scp>. Oxford University Press, 1964. 191 pp. £3. 3s.

1964-12-01

Julian D. Cole

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Uniformly Valid Asymptotic Approximations for Certain Non-Linear Differential Equations

1963-01-01

Julian D. Cole , J. Kevorkian

Similarity and Dimensional Methods in Mechanics

1961-03-01

L. I. Sedov , Morris D. Friedman , Maurice Holt +1 more

Similiarity and Dimensional Methods in Mechanics, 10th Edition is an English language translation of this classic volume examining the general theory of dimensions of physical quantities, the theory of mechanical … Similiarity and Dimensional Methods in Mechanics, 10th Edition is an English language translation of this classic volume examining the general theory of dimensions of physical quantities, the theory of mechanical and physical similarity, and the theory of modeling. Several examples illustrate the use of the theories of similarity and dimensions for establishing fundamental mechanical regularities in aviation, explosions, and astrophysics, as well as in the hydrodynamics of ships. Other interesting areas covered include the general theory of automodel motions of continuum media, the theory of propagation of explosion waves in gases, the theory of one-dimensional nonestablished motion in gases, the fundamentals of the gas-dynamics theory of atom-bomb explosion in the atmosphere and the theory of averaging of gaseous flows in channels. Aspects of modeling include the dimensionless characteristics of compressor operation, the theories of engine thrust, and efficiency of an ideal propeller for subsonic and supersonic speeds. Similiarity and Dimensional Methods in Mechanics, 10th Edition is an ideal volume for researchers and students involved in physics and mechanics.

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Experiments on propagation of weak disturbances in stationary supersonic nozzle flow of chemically reacting gas mixtures

1961-01-01

Peter P. Wegener , Julian D. Cole

On a quasi-linear parabolic equation occurring in aerodynamics

1951-10-01

Julian D. Cole

On the transonic flow past thin airfoils

1947-06-01

Julian D. Cole

Common Coauthors

Coauthor	Papers Together
J. Kevorkian	6
Norman Malmuth	6
George W. Bluman	3
C. C. Wu	2
José Canosa	2
Susan A. Triantafillou	2
Donald W. Schwendeman	2
Carl Greifinger	2
L. I. Sedov	1
Barbara Wagner	1
О. С. Рыжов	1
Earll M. Murman	1
George I. Bell	1
Jerry Aroesty	1
TS-FOU ZIEN	1
Tse-Fou Zien	1
Jerome Aroesty	1
R. W. John	1
Maurice Holt	1
L. Pamela Cook	1
Lawrence E. Levine	1
Harold Weitzner	1
Morris D. Friedman	1
V. S. Shankar Sriram	1
Peter P. Wegener	1

Commonly Cited References

Introduction to Hypersonic Flow

1962-12-01

G. G. Chernyĭ , Raymond J. Seeger

First Page First Page

Perturbation Methods in Applied Mathematics

1981-01-01

J. Kevorkian , Julian D. Cole

On the Differential Equations of the Simplest Boundary-Layer Problems

1942-04-01

Hermann Weyl

Computation of Transonic Flows past Lifting Airfogs and Slender Bodies

1972-07-01

J. A. Krupp , Earll M. Murman

Solutions of the transonic small disturbance equation are presented for flow past thin lifting airfoils and slender bodies with M^ < 1, including cases with imbedded shock waves. The results … Solutions of the transonic small disturbance equation are presented for flow past thin lifting airfoils and slender bodies with M^ < 1, including cases with imbedded shock waves. The results are obtained numerically using a mixed finite-difference relaxation method previously reported by the authors. Results are presented for four lifting airfoils at various angles of attack and are compared with shock free theory and experimental data. For the slender body case, comparisons with experiments are given for five geometries both with and without aft stings. The results are also compared with approximate theory. Discussion is given on the treatment of the boundary conditions, computing times and accuracies, and ranges of applicability of the small disturbance theory.

Applications of Hypersonic Small-Disturbance Theory

1954-03-01

Milton D. Van Dyke

The role of coordinate systems in boundary-layer theory

1954-03-01

Saul Kaplun

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Numerical Solution of Partial Differential Equations.

1966-12-01

G. W. Hedstrom , G. D. Smith

Ordinary and Delay Differential Equations

1977-01-01

R. D. Driver

The partial differential equation ut + uux = μxx

1950-09-01

Eberhard Hopf

Hypersonic Flow Theory

1964-04-01

Robert E. Wilson , W Hayes , Ronald F. Probstein

Asymptotic Solutions of Initial Value Problems for Nonlinear Partial Differential Equations

1970-06-01

Joseph B. Keller , Stanley Kogelman

Previous article Next article Asymptotic Solutions of Initial Value Problems for Nonlinear Partial Differential EquationsJoseph B. Keller and Stanley KogelmanJoseph B. Keller and Stanley Kogelmanhttps://doi.org/10.1137/0118067PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail … Previous article Next article Asymptotic Solutions of Initial Value Problems for Nonlinear Partial Differential EquationsJoseph B. Keller and Stanley KogelmanJoseph B. Keller and Stanley Kogelmanhttps://doi.org/10.1137/0118067PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Julian D. Cole, Perturbation methods in applied mathematics, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1968vi+260 MR0246537 0162.12602 Google Scholar[2] Edward A. Frieman, On a new method in the theory of irreversible processes, J. Mathematical Phys., 4 (1963), 410–418 10.1063/1.1703968 MR0160580 0124.45704 CrossrefISIGoogle Scholar[3] Joseph B. Keller and , Lu Ting, Periodic vibrations of systems governed by nonlinear partial differential equations, Comm. Pure Appl. Math., 19 (1966), 371–420 MR0205520 0284.35004 CrossrefISIGoogle Scholar[4] Martin H. Millman and , Joseph B. Keller, Perturbation theory of nonlinear boundary-value problems, J. Mathematical Phys., 10 (1969), 342–361 10.1063/1.1664849 MR0237867 0169.12702 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails On perturbation theory and its application in solving ordinary differential equations using the asymptotic expansion methodJournal of Physics: Conference Series, Vol. 1963, No. 1 | 1 Jul 2021 Cross Ref Causal properties of nonlinear gravitational waves in modified gravityPhysical Review D, Vol. 96, No. 6 | 21 September 2017 Cross Ref Introduction to Bifurcation and StabilityIntroduction to Perturbation Methods | 16 October 2012 Cross Ref Introduction to Asymptotic ApproximationsIntroduction to Perturbation Methods | 16 October 2012 Cross Ref The Method of HomogenizationIntroduction to Perturbation Methods | 16 October 2012 Cross Ref Matched Asymptotic ExpansionsIntroduction to Perturbation Methods | 16 October 2012 Cross Ref The WKB and Related MethodsIntroduction to Perturbation Methods | 16 October 2012 Cross Ref Multiple ScalesIntroduction to Perturbation Methods | 16 October 2012 Cross Ref Accuracy Improvement of the Method of Multiple Scales for Nonlinear Vibration Analyses of Continuous Systems with Quadratic and Cubic NonlinearitiesMathematical Problems in Engineering, Vol. 2010 | 1 Jan 2010 Cross Ref Asymptotic analysis of a perturbation problemJournal of Computational and Applied Mathematics, Vol. 190, No. 1-2 | 1 Jun 2006 Cross Ref ASYMPTOTIC APPROACH FOR NON-LINEAR PERIODICAL VIBRATIONS OF CONTINUOUS STRUCTURESJournal of Sound and Vibration, Vol. 249, No. 3 | 1 Jan 2002 Cross Ref Vibrational control of a non-linear elastic panelInternational Journal of Non-Linear Mechanics, Vol. 36, No. 4 | 1 Jun 2001 Cross Ref ON INTERACTIONS OF OSCILLATION MODES FOR A WEAKLY NON-LINEAR UNDAMPED ELASTIC BEAM WITH AN EXTERNAL FORCEJournal of Sound and Vibration, Vol. 235, No. 2 | 1 Aug 2000 Cross Ref An Asymptotic Theory for a Weakly Nonlinear Beam Equation with a Quadratic PerturbationG. 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Functions of a Complex Variable: Theory and Technique.

1967-02-01

Ernest C. Schlesinger , G. F. Carrier , Max Krook +1 more

Preface Erratum 1. Complex numbers and their elementary properties 2. Analytic functions 3. Contour integration 4. Conformal mapping 5. Special functions 6. Asymptotic methods 7. Transform methods 8. Special techniques … Preface Erratum 1. Complex numbers and their elementary properties 2. Analytic functions 3. Contour integration 4. Conformal mapping 5. Special functions 6. Asymptotic methods 7. Transform methods 8. Special techniques Index.

NUMERICAL DESIGN OF TRANSONIC AIRFOILS**The work presented in this paper is supported by the AEC Computing and Applied Mathematics Center, Courant Institute of Mathematical Sciences, New York University, under Contract AT(30-1)-1480 with the U. S. Atomic Energy Commission.

1971-01-01

P. R. Garabedian , David G. Korn

Non-linear wave propagation solutions by Fourier transform perturbation

1981-01-01

S. C. Chikwendu

New approach to the asymptotic theory of nonlinear oscillations and wave-propagation

1975-03-01

Wiktor Eckhaus

Asymptotic Solutions of Some Weakly Nonlinear Elliptic Equations

1976-09-01

S. C. Chikwendu

Multiple-scale perturbation solutions are obtained for a class of nonlinear second order elliptic Dirichlet boundary value problems in two independent variables, on a semi-infinite strip and on a square. Complex … Multiple-scale perturbation solutions are obtained for a class of nonlinear second order elliptic Dirichlet boundary value problems in two independent variables, on a semi-infinite strip and on a square. Complex characteristics are used, and only nonlinearities that involve functions of a first derivative of the dependent variable are covered. The order one approximation is obtained from the solution of a pair of nonlinear ordinary differential equations. The method is especially of interest when boundary conditions are specified on an unbounded domain.

Hypersonic Flow Theory

1959-12-01

L. Howarth

By W. D. Hayes and R. F. Probstein New York and London: Academic Press Inc. Pp. xiv + 464. Price 92s. By W. D. Hayes and R. F. Probstein New York and London: Academic Press Inc. Pp. xiv + 464. Price 92s.

Asymptotic solutions of nonlinear wave equations using the methods of averaging and two-timing

1977-07-01

R. W. Lardner

The method of averaging and the two-time procedure are outlined for a class of hyperbolic second-order partial differential equations with small nonlinearities. It is shown that they both lead to … The method of averaging and the two-time procedure are outlined for a class of hyperbolic second-order partial differential equations with small nonlinearities. It is shown that they both lead to the same integro-differential equation for the lowest-order approximation to the solution. For the special case of the nonlinear wave equation, this lowest-order solution consists of the superposition of two modulated travelling waves, and separate integro-differential equations are derived for the amplitudes of these two waves. As an example, the wave equation with Van der Pol type of nonlinearity is considered.

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Partial differential equations: analytical solution techniques

1990-12-01

J. Kevorkian

1. The Diffusion Equation.- 2. Laplace's Equation.- 3. The Wave Equation.- 4. Linear Second-Order Equations with Two Independent Variables.- 5. The Scalar Quasilinear First-Order Equation.- 6. Nonlinear First-Order Equations.- 7. … 1. The Diffusion Equation.- 2. Laplace's Equation.- 3. The Wave Equation.- 4. Linear Second-Order Equations with Two Independent Variables.- 5. The Scalar Quasilinear First-Order Equation.- 6. Nonlinear First-Order Equations.- 7. Quasilinear Hyperbolic Systems.- 8. Approximate Solutions by Perturbation Methods.- A.1. Review of Green's Function for ODEs Using the Dirac Delta Function.- A.2. Review of Fourier and Laplace Transforms.- A.3. Review of Asymptotic Expansions.- References.

On a quasi-linear parabolic equation occurring in aerodynamics

1951-10-01

Julian D. Cole

A general approach to linear and non-linear dispersive waves using a Lagrangian

1965-06-01

G. B. Whitham

The basic property of equations describing dispersive waves is the existence of solutions representing uniform wave trains. In this paper a general theory is given for non-uniform wave trains whose … The basic property of equations describing dispersive waves is the existence of solutions representing uniform wave trains. In this paper a general theory is given for non-uniform wave trains whose amplitude, wave-number, etc., vary slowly in space and time, the length and time scales of the variation in amplitude, wave-number, etc., being large compared to the wavelength and period. Dispersive equations may be derived from a variational principle with appropriate Lagrangian, and the whole theory is developed in terms of the Lagrangian. Boussinesq's equations for long water waves are used as a typical example in presenting the theory.

Calculation of plane steady transonic flows

1971-01-01

Earll M. Murman , Julian D. Cole

On free perturbations in hypersonic laminar flow behind a profile

1984-01-01

S. V. Manuĭlovich

Measurement of Rate Constants of Fast Reactions in a Supersonic Nozzle

1958-04-01

Peter P. Wegener

First Page First Page

Invariance Group of the Equation$\partial {\bf u} / \partial t = - {\bf u} \cdot \nabla {\bf u}$

1973-05-01

Gerald Rosen , G. W. Ullrich

It is shown that the primordial equation of hydrodynamics is invariant under a 24-parameter Lie group of space-time coordinate transformations. It is shown that the primordial equation of hydrodynamics is invariant under a 24-parameter Lie group of space-time coordinate transformations.

Self-similar motions of a gas with shock waves, spreading according to a power law into a gas at rest

1959-01-01

G. L. Grodzovskii , N.L. Krashchennikova

Weakly Nonlinear Waves for a Class of Linearly Unstable Hyperbolic Conservation Laws with Source Terms

1995-04-01

J. Kevorkian , Jun Yu , L. Wang

We consider a pair of general hyperbolic conservation laws with source terms, and focus on the class of problems that are unstable in the linearized sense. We derive evolution equations … We consider a pair of general hyperbolic conservation laws with source terms, and focus on the class of problems that are unstable in the linearized sense. We derive evolution equations governing the leading approximation of the nonlinear solution using multiple scale expansions. We then analyze these evolution equations to determine conditions under which linearly unstable disturbances equilibrate. In particular, we show that for certain parameter values periodic initial disturbances evolve into travelling waves consisting of piecewise continuous profiles joined by shocks. We also exhibit a novel bifurcation process whereby the wave number of the travelling wave increases a unit amount as a parameter value in the evolution equation is doubled. Numerical solutions are provided throughout.

The Theory of Laminar Boundary Layers in Compressible Fluids

1964-06-01

K. Stewartson , L. Talbot

Anomalous dimensions and the renormalization group in a nonlinear diffusion process

1990-03-19

Nigel Goldenfeld , Olivier Martin , Y. Oono +1 more

We present a renormalization-group (RG) approach to the nonlinear diffusion process ${\mathrm{\ensuremath{\partial}}}_{\mathit{t}}$u=D ${\mathrm{\ensuremath{\partial}}}_{\mathit{x}}^{2}$u, with D=1/2 for ${\mathrm{\ensuremath{\partial}}}_{\mathit{x}}^{2}$u>0 and D=(1+\ensuremath{\epsilon})/2 for ${\mathrm{\ensuremath{\partial}}}_{\mathit{x}}^{2}$u0, which describes the pressure during the filtration of an … We present a renormalization-group (RG) approach to the nonlinear diffusion process ${\mathrm{\ensuremath{\partial}}}_{\mathit{t}}$u=D ${\mathrm{\ensuremath{\partial}}}_{\mathit{x}}^{2}$u, with D=1/2 for ${\mathrm{\ensuremath{\partial}}}_{\mathit{x}}^{2}$u>0 and D=(1+\ensuremath{\epsilon})/2 for ${\mathrm{\ensuremath{\partial}}}_{\mathit{x}}^{2}$u0, which describes the pressure during the filtration of an elastic fluid in an elastoplastic porous medium. Our approach recovers Barenblatt's long-time result that, for a localized initial pressure distribution, u(x,t)\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}(\mathrm{\ensuremath{\alpha}}+1/2)}$f(x/ \ensuremath{\surd}t, \ensuremath{\epsilon}), where f is a scaling function and \ensuremath{\alpha}=\ensuremath{\epsilon}(2\ensuremath{\pi}e${)}^{1/2}$+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$) is an anomalous dimension, which we compute perturbatively using the RG. This is the first application of the RG to a nonlinear partial differential equation in the absence of noise.

Exakte Lösungen der Differentialgleichungen einer adiabatischen Gasströmung

1940-01-01

Friedrich Ringleb

Studies in Non-Linear Stability Theory

1965-01-01

Wiktor Eckhaus

Shock-wave strength for separation of a laminar boundary layer at transonic speeds

1971-06-01

A. F. Messiter , A. Feo , R. E. Melnik

Theorie der Transformationsgruppen I

1880-12-01

Sophus Lie

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Perturbation Methods in Applied Mathematics

1973-11-01

Julian D. Cole , Lawrence E. Levine

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Similarity Analyses of Boundary Value Problems in Engineering

1965-07-01

Garrett Birkhoff , Arthur G. Hansen

Similarity analyses of boundary value problems in engineering , Similarity analyses of boundary value problems in engineering , مرکز فناوری اطلاعات و اطلاع رسانی کشاورزی Similarity analyses of boundary value problems in engineering , Similarity analyses of boundary value problems in engineering , مرکز فناوری اطلاعات و اطلاع رسانی کشاورزی

On the theory of hypersonic gas flow with a power-law shock wave

1960-01-01

В. В. Сычев

Flow of a sonic gas stream past a body of revolution

1964-01-01

S. V. Fal'kovich , Ilya Chernov

Matched-conic approximation to the two fixed force-center problem

1963-03-01

P. A. Lagerstrom , J. Kevorkian

Planar motion of a particle of negligible mass from the neighborhood of a gravitational center (the "earth") of mass 1 - to the neighborhood of a second center (the "moon") … Planar motion of a particle of negligible mass from the neighborhood of a gravitational center (the "earth") of mass 1 - to the neighborhood of a second center (the "moon") of mass is studied by asymp- totic methods for the case 1. The calculations are carried out for the case of two fixed centers. It is pointed out, however, that the methods used are also applicable to the case of the two centers rotating around their center of mass, that is, to the limiting case of the restricted three-body problem for which the second mass is much smaller than the first. A uniformly valid solution describing the passage from the earth to the moon and the motion in the neighborhood of the moon is obtained. Each part of the motion is in the first approximation a Keplerian conic relative to the earth and moon, respectively. However, these conics cannot be matched directly: In order to determine the second part, as well as the subsequent motion, it is necessary to compute a correction of order,o to the first part. This statement is equally true for the restricted three-body problem.

Similarity and Dimensional Methods in Mechanics

1961-03-01

L. I. Sedov , Morris D. Friedman , Maurice Holt +1 more

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Conservative Full Potential, Implicit Marching Scheme for Supersonic Flows

1982-11-01

Vijaya Shankar

Non-equilibrium flow over a wavy wall

1959-11-01

Walter G. Vincenti

A small-disturbance solution is obtained for the steady two-dimensional flow over a sinusoidal wall of an inviscid gas in vibrational or chemical non-equilibrium. The results are based on a single, … A small-disturbance solution is obtained for the steady two-dimensional flow over a sinusoidal wall of an inviscid gas in vibrational or chemical non-equilibrium. The results are based on a single, linear, third-order partial differential equation, which plays the same role here as does the Prandtl–Glauert equation in equilibrium flow. The solution is valid throughout the range from subsonic to supersonic speeds and for all values of the rate parameter from equilibrium to frozen flow (in both of which limits it reduces to Ackert's classical solution of the Prandtl–Glauert equation). The results illustrate in simple fashion some of the properties of non-equilibrium flow, such as the occurrence of pressure drag at subsonic speeds and the absence of the discontinuous phenomena that characterize the Prandtl–Glauert theory when the flow changes from subsonic to supersonic.

Supersonic Nozzle Flow with a Reacting Gas Mixture

1959-05-01

Peter P. Wegener

An investigation has been made of the stationary supersonic nozzle flow of a reacting gas mixture of nitrogen tetroxide and dioxide carried at low concentration in nitrogen. A judicious choice … An investigation has been made of the stationary supersonic nozzle flow of a reacting gas mixture of nitrogen tetroxide and dioxide carried at low concentration in nitrogen. A judicious choice of supply conditions and nozzle geometry made it possible to produce flows either in chemical equilibrium or in states between this and the frozen flow. The flow could be fully determined from pressure and area measurements and, by applying an equation for the reaction mechanism, rate constants of recombination could be found. The results of these measurements agreed with those obtained from independent optical absorption measurements, and the third-order rate constant for recombination was found to be 3 × 1014 cm6 mole−2 sec−1 for 210 &lt; T &lt; 330°K. Finally, experiments with stationary oblique shocks led to a determination of the dissociation rate constant.

Matched Asymptotic Expansions: Ideas and Techniques

1989-07-01

P. A. Lagerstrom , Carl M. Bender

I. Introduction: General Concepts of Singular Perturbation Theory.- II. Layer-type Problems. Ordinary Differential Equations.- III. Layer-type Problems. Partial Differential Equations.- List of References. I. Introduction: General Concepts of Singular Perturbation Theory.- II. Layer-type Problems. Ordinary Differential Equations.- III. Layer-type Problems. Partial Differential Equations.- List of References.

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Eigenvalues for the Blast Wave

1970-02-01

K. Stewartson , Bruce Thompson

Asymptotic expansions can be formed for hypersonic flows past blunt bodies using the blast wave as a leading approximation. Theoretical studies of such expansions have revealed that they are not … Asymptotic expansions can be formed for hypersonic flows past blunt bodies using the blast wave as a leading approximation. Theoretical studies of such expansions have revealed that they are not valid on streamlines which pass close to the body at any stage, but become valid in the neighborhood of the shock wave at large distances downstream of the nose. In order to describe the whole flow field at such distances, an expansion appropriate to the neighborhood of the body is necessary, and when a matching is attempted between the two expansions, irreconcilable logarithmic terms arise unless a certain infinite set of integrals related to entropy all vanish. However, if all these integrals vanish, then the outer expansions of the forms so far proposed are quite independent of any conditions occurring in the inner layer near the nose. In a previous paper the authors advanced the alternative hypothesis that there might exist sets of undiscovered eigenfunctions associated with the blast wave expansion through which the initial conditions near the nose could affect the ultimate flow far downstream. For convenience the problem is studied here in its unsteady analog, and it is shown that there exist two infinite sets of eigenvalues, one real, the other complex. The smallest eigenvalues are computed in the limiting cases γ → 1 + 0 and γ → ∞, and also for an ideal monatomic gas with γ = 53

Basic Concepts Underlying Singular Perturbation Techniques

1972-01-01

P. A. Lagerstrom , Richard G. Casten

In many singular perturbation problems multiple scales are used. For instance, one may use both the coordinate x and the coordinate $x^ * = \varepsilon ^{ - 1} x$. In … In many singular perturbation problems multiple scales are used. For instance, one may use both the coordinate x and the coordinate $x^ * = \varepsilon ^{ - 1} x$. In a secular-type problem x and $x^ * $ are used simultaneously. This paper discusses layer-type problems in which $x^ * $ is used in a thin layer and x outside this layer. Assume one seeks approximations to a function $f(x,\varepsilon )$, uniformly valid to some order in $\varepsilon$ for x in a closed interval D. In layer-type problems one uses (at least) two expansions (called inner and outer) neither of which is uniformly valid but whose domains of validity together cover the interval D. To define “domain of validity” one needs to consider intervals whose endpoints depend on $\varepsilon $. In the construction of the inner and outer expansions, constants and functions of e occur which are determined by comparison of the two expansions “matching.” The comparison is possible only in the domain of overlap of their regions of validity. Once overlap is established, matching is easily carried out. Heuristic ideas for determining domains of validity of approximations by a study of the corresponding equations are illustrated with the aid of model equations. It is shown that formally small terms in an equation may have large integrated effects. The study of this is of central importance for understanding layer-type problems. It is emphasized that considering the expansions as the result of applying limit processes can lead to serious errors and, in any case, hides the nature of the expansions.

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On the theory of hypersonic flow past plane and axially symmetric bluff bodies

1956-10-01

N. C. Freeman

The ‘Newtonian-plus-centrifugal’ approximate solution (Busemann (1933) and Ivey (1948)) for hypersonic flow past plane and axially symmetric bluff bodies in gases with the ratio of the specific heats λ constant … The ‘Newtonian-plus-centrifugal’ approximate solution (Busemann (1933) and Ivey (1948)) for hypersonic flow past plane and axially symmetric bluff bodies in gases with the ratio of the specific heats λ constant and equal to unity is rederived using ‘boundary layer’ techniques together with the von Mises variables x and ψ. A method of successive approximations then gives a closer approximation to this solution for ε (λ − 1)/(λ + 1) small and the free-strea Mach number infinite. Formulae for the streamlines, shock shape and pressure distribution are determined to this approximation. These formulae are valid for any plane or axially symmetric shape, giving the ‘stand-off’ distance of the shock wave from the body as ½εlog(4|3ε) and ε times the nose radius of curvature for plane and axially-symmetric flows respectively. Particular results are computed for a number of special shapes. For certain shapes, the theory has a singular point where the first approximation to the pressure vanishes (θ = 60° for a sphere). Actually, the theory is not applicable where the pressure becomes too small. The corresponding theory for gases of general thermodynamic properties is deduced, the approximation being valid provided the total energy of the gas is large compared with the energy contained in the translational modes of the gas molecules.

Numerical Solution of Partial Differential Equations

1974-01-01

I. Rabuška

Hypersonic weak-interaction similarity solutions for flow past a flat plate

1967-08-11

William B. Bush , Arthur K. Cross

The hypersonic weak-interaction regime for the flow of a viscous, heat-conducting compressible fluid past a flat plate is analysed using the Navier-Stokes equations as a basis. The fluid is assumed … The hypersonic weak-interaction regime for the flow of a viscous, heat-conducting compressible fluid past a flat plate is analysed using the Navier-Stokes equations as a basis. The fluid is assumed to be a perfect gas having constant specific heats, a constant Prandtl number, σ, of order unity, and a viscosity coefficient varying as a power, ω, of the absolute temperature. Limiting forms of solutions are studied for the free-stream Mach number, M , the free-stream Reynolds number (based on the plate length), R L , and the reciprocal of the weak-interaction parameter, (ξ*) −1 = [Fscr ]( M , R L , ω, σ), greater than order unity. By means of matched asymptotic expansions, it is shown that, for (1 − ω) > 0, the zone between the shock wave and the plate is composed of four distinct regions for which similarity exists. The behaviour of the flow in these four regions is analysed.

Analysis of transonic airfoils

1971-11-01

P. R. Garabedian , David G. Korn

Characteristics of the equations of motion of a reacting gas

1958-07-01

L. J. F. Broer

The equations of motion for a chemically reacting gas in the absence of viscosity and heat conduction are set up. It is shown that the characteristic speed defined by this … The equations of motion for a chemically reacting gas in the absence of viscosity and heat conduction are set up. It is shown that the characteristic speed defined by this set of equations is the high-frequency limit of the phase velocity of sound waves as long as the reaction rate is finite. At infinite reaction rate (chemical equilibrium) the characteristics suddenly change to the lowfrequency sound speed. The nature of this transition is discussed in connection with a recent paper of Resler (1957).