Showing papers on "Method of matched asymptotic expansions published in 1975"

Contributions to the theory of aerodynamic sound, with application to excess jet noise and the theory of the flute

Abstract: This paper describes a reformulation of the Lighthill (1952) theory of aerodynamic sound. A revised approach to the subject is necessary in order to unify the various ad hoc procedures which have been developed for dealing with aerodynamic noise problems since the original appearance of Lighthill's work. First, Powell's (1961 a) concept of vortex sound is difficult to justify convincingly on the basis of Lighthill's acoustic analogy, although it is consistent with model problems which have been treated by the method of matched asymptotic expansions. Second, Candel (1972), Marble (1973) and Morfey (1973) have demonstrated the importance of entropy inhomogeneities, which generate sound when accelerated in a mean flow pressure gradient. This is arguably a more significant source of acoustic radiation in hot subsonic jets than pure jet noise. Third, the analysis of Ffowcs Williams & Howe (1975) of model problems involving the convection of an entropy ‘slug’ in an engine nozzle indicates that the whole question of excess jet noise may be intimately related to the convection of flow inhomogeneities through mean flow pressure gradients. Such problems are difficult to formulate precisely in terms of Lighthill's theory because of the presence of an extensive, non-acoustic, non-uniform mean flow. The convected-entropy source mechanism is actually absent from the alternative Phillips (1960) formulation of the aerodynamic sound problem.In this paper the form of the acoustic propagation operator is established for a non-uniform mean flow in the absence of vortical or entropy-gradient source terms. The natural thermodynamic variable for dealing with such problems is the stagnation enthalpy. This provides a basis for a new acoustic analogy, and it is deduced that the corresponding acoustic source terms are associated solely with regions of the flow where the vorticity vector and entropy-gradient vector are non-vanishing. The theory is illustrated by detailed applications to problems which, in the appropriate limit, justify Powell's theory of vortex sound, and to the important question of noise generation during the unsteady convection of flow inhomogeneities in ducts and past rigid bodies in free space. At low Mach numbers wave propagation is described by a convected wave equation, for which powerful analytical techniques, discussed in the appendix, are available and are exploited.Fluctuating heat sources are examined: a model problem is considered and provides a positive comparison with an alternative analysis undertaken elsewhere. The difficult question of the scattering of a plane sound wave by a cylindrical vortex filament is also discussed, the effect of dissipation at the vortex core being taken into account.Finally an approximate aerodynamic theory of the operation of musical instruments characterized by the flute is described. This involves an investigation of the properties of a vortex shedding mechanism which is coupled in a nonlinear manner to the acoustic oscillations within the instrument. The theory furnishes results which are consistent with the playing technique of the flautist and with simple acoustic measurements undertaken by the author.

Uniform asymptotic expansions in transport theory with small mean free paths, and the diffusion approximation

Abstract: We consider initial−boundary value and boundary value problems for transport equations in inhomogeneous media. We consider the case when the mean free path is small compared to typical lengths in the domain (e.g., the size of a reactor). Employing the boundary layer technique of matched asymptotic expansions, we derive a uniform asymptotic expansion of the solution of the problem. In so doing we find that in the interior of the domain, i.e., away from boundaries and away from the initial line, the leading term of the expansion satisfies a diffusion equation which is the basis of most computational work in reactor design. We also derive boundary conditions appropriate to the diffusion equation. Comparisons with existing results such as the asymptotic and P1 diffusion theories, the PN approximation, and the extrapolated end point condition for these approximations, are made. Finally the uniform validity of our expansions is proved, thus yielding the desired error estimates.

Initial flow field over an impulsively started circular cylinder

Abstract: The initial flow field of an incompressible, viscous fluid around a circular cylinder, set impulsively to move normal to its axis, is studied in detail. The nonlinear vorticity equation is solved by the method of matched asymptotic expansions. Analytic solutions for the stream function in terms of exponential and error functions for the inner flow field, and of circular functions for the outer, are obtained to the third order, from which a uniformly valid composite solution is found. Also presented are the vorticity, pressure, separation point and drag. These quantities agree with the numerical computations of Collins & Dennis. Extended solutions developed by Pade approximants indicate that higher than third-order approximations will yield only minor improvements.

Generation and Decay of Viscous Vortex Rings

Abstract: Decay of a vortex ring in a viscous fluid is discussed by using a solution in the form of an asymptotic expansion for large time t . It is found that the velocity of the vortex ring varies as t -1.5 in the final state of low Reynolds number. The asymptotic expansion is not uniformly valid, and an improvement is made by using the method of matched asymptotic expansions. Generation and development of vortex rings are simulated by numerical integration of the Navier-Stokes equation as an initial and boundary value problem. Time variations of physical quantities such as total energy, impulse and velocity of the vortex rings, etc. are obtained, and have been shown to approach asymptotically to those obtained from the asymptotic expansion. Comparison with experimentally produced vortex rings is also given briefly.

Lifting-line theory for an unsteady wing as a singular perturbation problem

Abstract: A linearized theory, which treats unsteady motions of a wing of large aspect ratio at variable forward speed in an inviscid incompressible fluid, is developed, using the method of matched asymptotic expansions. The wing geometry and motions are specified; and the time-dependent lift and moment are obtained. Long-time asymptotic behaviour of an initial-value harmonic motion is presented, as are the short-time solutions of a wing starting from rest, with constant acceleration and with impulsive acceleration to constant speed. Some attention is given to flapping flight. Results are presented in quadrature form for a general class of unsteady wing motions.

Theory of non-steady state droplet combustion

Abstract: The method of matched asymptotic expansions is applied to the analysis of non-steady state combustion of liquid fuel droplets. Asymptotic solutions are derived for inner and outer regions representing the droplet vicinity and far field, respectively. The expansions are based on a small parameter identified as the ratio of the droplet radius to the diffusion field radius. The outer region is dominated by unsteady diffusion; convection is a higher-order effect. Viewed from the outer region, the droplet appears to be a point source of mass and sink for heat. The inner region is in a quasi-steady state and is characterized by a balance between convection and diffusion. The mass burning rate is governed by the inner-region solution; to lowest order the quasi-steady value is recovered. It is shown that perturbations due to the unsteady outer region increase the burning rate and reduce the droplet lifetime. Flame trajectories calculated from uniformly valid composite expansions show that (i) the flame moves away from the droplet then back towards it, (ii) the ratio of flame-to-droplet radius increases continually, and (iii) flame extinction occurs after the droplet vanishes. These results contradict the quasi-steady state theory of droplet combustion, but they agree with observed droplet behavior. Theoretical predictions exhibit good agreement with experimental data.

Uniform asymptotic solutions for potential flow about a slender body of revolution

Abstract: The general problem of potential flow past a slender body of revolution is considered. The flow incident on the body is described by an arbitrary potential function and hence the results presented here extend those obtained by Handels-man & Keller (1967 α). The part of the potential due to the presence of the body is represented as a superposition of potentials due to point singularities (sources, dipoles and higher-order singularities) distributed along a segment of the axis of the body inside the body. The boundary condition on the body leads to a linear integral equation for the density of the singularities. The complete uniform asymptotic expansion of the solution of this equation, as well as the extent of the distribution, is obtained using the method of Handelsman & Keller. The special case of transverse incident flow is considered in detail. Complete expansions for the dipole moment of the distribution and the virtual mass of the body are obtained. Some general comments on the method of Handelsman & Keller are given, and may be useful to others wishing to use their method.

Asymptotic analysis of stochastic models in population genetics

Abstract: Stochastic models in population genetics, which lead to diffusion equations, are considered. A method for obtaining asymptotic expansions of the solutions of these equations is presented. The expansions are valid for t / N small, where t is the time in generations and N is the population size. The method permits the analysis of models including selection, mutation, migration, etc. The case of two alleles at one locus is considered. Formulas and numerical results are presented. In the few special cases for which exact solutions are known, comparison shows that the asymptotic solution is good for values of t from zero to 2 N generations. The method can be applied to cases of more than two alleles at one locus and to linkage of genes at different loci.

Acoustic radiation from surfaces oscillating at large amplitude and small Mach number

Abstract: The purpose of this paper is to demonstrate that at small values of the Mach number a harmonically rich radiated field will result from the simple harmonic (sinusoidal) motion of an oscillating body when the amplitude of oscillation is large. An asymptotic solution of the equations of inviscid compressible flow is obtained which is valid in the small Mach number limit. To a given order of approximation, the effects of fluid nonlinearity (nonlinearity in the equations of motion and the equation of state) do not enter, and the harmonic distortion is due entirely to large‐amplitude boundary motion. In higher approximations, where both the effects of boundary motion and fluid nonlinearity enter, the former effect can still control the amplitude of certain harmonics of the radiated acoustic field. The results are obtained by the method of matched asymptotic expansions, which is ideally suited for the purpose of distinguishing the harmonic distortion due to large displacements from that due to fluid nonlinearit...

On rotationally symmetric flow above an infinite rotating disk

Abstract: The similarity equations for rotationally symmetric flow above an infinite counter–rotating disk are investigated both numerically and analytically. Numerical solutions are found when α, the ratio of the disk's angular speed to that of the rigidly rotating fluid far from it, is greater than −0.68795. It is deduced that there exists a critical value αcr, of α above which finite solutions are possible. The value of α and the limiting structure as α → αcr are found using the method of matched asymptotic expansions. The flow structure is found to consist of a thin viscous wall region above which lies a thick inviscid layer and yet another viscous transition layer. Furthermore, this structure is not unique: there can be any number of thick inviscid layers, each separated from the next by a viscous transition layer, before the outer boundary conditions on the solution are satisfied. However, comparison with the numerical solutions indicates that a single inviscid layer is preferred.

Asymptotic Expansions of Singularly Perturbed Quasi-Linear Optimal Systems

Abstract: A class of two-point boundary value problems (TPBVP’s) which arise in fixed final time free endpoint optimal control problems is considered. An asymptotic power series solution of the TPBVP is constructed with respect to a parameter whose perturbation changes the differential order of the problem. Based on a stability hypothesis, the proof of asymptotic correctness is accomplished through a successive approximation scheme.

The Thin-Hydrofoil Thickness Problem Including Leading-Edge Corrections

Abstract: The method of Keldysh and Lavrentiev is used to study the potential flow of a uniform stream past a submerged thin symmetric hydrofoil. An expression for the surface speed is obtained and the method of matched asymptotic expansions is used to develop corrections at the round leading edge. The pressure, lift, drag, and pitching moment are presented for the Joukowski hydrofoil with emphasis on the variation with thickness, Froude number, and chord-to-depth ratio.

On Two Methods of Solution for a Singularly Perturbed Linear State Regulator Problem

Abstract: The asymptotic solution of a singularly perturbed linear state regulator problem with quadratic cost has been considered by Yackel and Wilde, O’Malley, and Kung, among others. Hamilton–Jacobi theory converts the problem to a singularly perturbed two-point boundary value problem. This problem can be solved directly or by solving a terminal value problem for the Riccati gains and, then, an initial value problem for the states. For a $2 \times 2$ time-invariant system, it is shown that both methods require the same hypotheses.

Asymptotic expansions in the perturbed two-body problem with application to systems with variable mass

Abstract: The main theorems of the theory of averaging are formulated for slowly varying standard systems and we show that it is possible to extend the class of perturbation problems where averaging might be used.

The Subcritical Spatially Homogeneous Explosion; Initiation to Completion

Abstract: In this paper a complete solution (initiation to completion) for the sub-critical spatially homogeneous reaction has been developed. Complete reaction consumption is included. The solution is in two parts: (1) the rapid temperature rise (2) the subsequent decay back to initial conditions. The two parts are joined using the method of matched asymptotic expansions. The complete temperature-time history is displayed graphically, to show the dependence of the temperature variation and fuel consumption on reaction order, heat loss parameter and the thermodynamic properties of the system.

On the asymptotic behavior of solutions of certain non-autonomous differential equations

Abstract: (1.2) x+a(t)f(x, Λ, x)x+b(t)g(x9 x)+c(t)h(x) = p(t, x, i, x) where a(t), b(t)y c(t) are positive continuously differentiate and /, g, h, p are continuous real-valued functions depending only on the arguments shown, and the dots indicate the differentiation with respect to t. The asymptotic property of solutions of third order differential equations has received a considerable amount of attention during the past two decades, particularly when (1.2) is autonomous. Many of these results are summarized in [11]. A few authors have studied non-autonomous third order differential equations. K. E. Swick [13] considered the following equations

Newton’s Method Techniques for Singular Perturbations

Abstract: In this paper, existence and uniqueness of solutions to several classes of singularly perturbed initial value and boundary value problems are proven by use of Newton’s method. Asymptotic expansions of solutions to some of these problems are developed and analyzed, and proof of their asymptotic correctness is shown based on Newton’s method. As a by-product some numerical algorithms for these solutions are obtained.

Higher approximations to the free convection flow from a heated vertical flat plate

Abstract: This paper is concerned with the problem of obtaining higher approximations for the free convection from a heated vertical flat plate to that represented by the well known solution of Schmidt and Beckmann. For large Grashof number, the perturbation problem is a singular one and the method of matched asymptotic expansions is used to construct inner and outer expansions for the velocity and temperature distributions. The small perturbation parametere is the inverse of the fourth root of the Grashof number and the expansions are shown to involve only integral powers ofe. The first three terms in the expansion are calculated and numerical results are presented for the velocity, temperature, skin friction and heat transfer. The agreement with experiment is found to be excellent, and the theory fully explains the discrepancies which exist between boundary layer theory and experiment.

Oscillatory and asymptotic properties of differential equations with retarded arguments

Abstract: We give here some results on oscillatory and asymptotic behavior for differential equations with retarded arguments of arbvitrary order. Lemma 1 establishes a comparison priniple from which we derive the oscillatory anbd asymptotic behavior of the solutions by considering simple ordinary differential equations of the form Y (n)+g(t) Y α=0 whose solutions have known behavior. Several known results for diferential equations with retarded arguments and (cf.[1], [6], [8]and[12])are particular cases of ours. Moreover, even in the case of ordinary differential equations, our results appear as generalizations of other ones (cf.[2-4]and[9-11]).

Analysis of transonic flow about lifting wing-body configurations

Abstract: An analytical solution was obtained for the perturbation velocity potential for transonic flow about lifting wing-body configurations with order-one span-length ratios and small reduced-span-length ratios and equivalent-thickness-length ratios. The analysis is performed with the method of matched asymptotic expansions. The angles of attack which are considered are small but are large enough to insure that the effects of lift in the region far from the configuration are either dominant or comparable with the effects of thickness. The modification to the equivalence rule which accounts for these lift effects is determined. An analysis of transonic flow about lifting wings with large aspect ratios is also presented.