Small Disturbance Theory Research Articles

Inviscid separated flows of finite extent

Two-dimensional, inviscid, incompressible flow is considered when the flow region contains a separation bubble of finite length. Within the separation bubble a slender-eddy approximation is employed, whilst outside it small disturbance theory is used to solve the potential-flow equations. The solution is completed by matching the pressure across the vortex sheet that divides the two regions of flow. Solutions are presented for the flow past smooth indentations in an otherwise plane boundary.

Vector Monte Carlo methods for computing disturbances and derivatives with respect to parameters

A triangular system of integral equations is obtained, which, under certain conditions, are satisfied by the derivatives with respect to the parameters of the solution of an integral equation of the 2nd kind. On this basis, some algorithms of the Monte Carlo method are constructed and studied. The corresponding estimates of the first derivatives are in fact the same as the familiar scalar estimates, but the matrix approach greatly simplifies the study of the variances and extends the sufficient condition for them to be finite. Similar results are obtained for the parametric disturbances of the solution of an integral equation. Examples are given in which the method is used in the theory of radiation transport, where it turns out that, in certain cases, the new algorithm for the estimation of the small disturbance differs from the earlier algorithm based on the theory of small disturbances for the transport equation.

Cone-derived waveriders with longitudinal curvature

Lifting-body waverider configurations with longitudinal curvature are derived by means of known flowfields past ogival bodies. The flowfields are generated by means of perturbations of the supersonic flow past a circular cone. Approximate analytical solutions for the shock-layer flowfield are found within the framework of hypersonic small-disturbance theory. Perturbations due to longitudinal curvature produce reduced drag, increased lift-to-drag ratios, and provide design advantages for a more efficient volume distribution related to packaging of guidance, propulsion, and payload units. Nomenclature Ab =base area of waverider Cp = pressure coefficient D = drag / =body perturbation function Gm = shock perturbation function for power law g = shock perturbation function Kd =M00d, hypersonic similarity parameter; d in radians L =lift t = length of basic cone MO, =freestream Mach number m = perturbation index factor Q -Poo^oo/2

Flow instabilities in transonic small-disturbance theory

The dynamics of unsteady transonic small disturbance flows about two-dimensional airfoils is examined, with emphasis on the behavior in the region where the steady state flow is nonunique. It is shown that nonuniqueness results from an extremely long time scale instability which occurs in a finite Mach number and angle of attack range. The similarity scaling rules for the instability are presented and the possibility of similar behavior in the Euler equations is discussed.

Transonic small-disturbance theory for dusty gases

Transonic small-disturbance theory for dusty gases

The effects of gusts on the fluctuating airloads of airfoils in transonic flow

Unsteady interactions of distributed and sharp-edged gusts with a stationary airfoil have been analyzed in two-dimensional transonic flow.A simple method of introducing such disturbances has been numerically implemented within the framework of unsteady, transonic small-disturbance theory. Representative solutions for various airfoils subjected to chordwise and transverse gusts show that the strength and unsteady motion of the shock wave on the airfoil significantly affect the flowfield development and, consequently, the dynamic airloads. Also a study was made of the reductions in the unsteady airloads that can be achieved by the proper active control motion of a trailing-edge flap, and a simple gust-alleviation strategy was developed. However, the chordwise pressure distributions associated with gusts are very different from those produced by trailing-edge flap oscillations. Consequently, the fluctuating lift and the unsteady pitching moments cannot both be eliminated simultaneously.

Theoretical study of the transonic flow past wedge profiles with detached shock waves

This report presents a numerical study of the two-dimensional, steady and inviscid flow of a perfect gas past wedge profiles with detached shock waves for free-stream Mach numbers between 1.05 and 1.44.Utilizing the hodograph transformation, a boundary-value problem for the mixed flow is formulated in the hodograph plane and subsequently solved by a finite-difference scheme, iterating respectively between the elliptic and hyperbolic regions. The use of boundary-fitted coordinates and a graded lattice failed to achieve satisfactory results, owing to either convergence difficulties or numerical inaccuracies introduced through the coordinate generator. On the other hand, Cartesian coordinates presented no convergence problems and yielded accurate results.After transforming the solution back to the physical plane, the flow field between the shock wave and the limiting Mach wave is determined. A comparison of the results with experiments and other theories shows a good agreement both for the pressure distribution on the wedge and for the shock stand-off distance.Also of special interest is a quantitative comparison of the results with the small-disturbance theory (solution of the Tricomi equation) and the limiting case of the free-stream Mach number 1.Moreover, an example of the flow past round-nosed wedges, with and without an angle of attack, is treated using the inverse method.

A Theory of Rotating Stall of Multistage Axial Compressors: Part II—Finite Disturbances

A small-disturbance theory of rotating stall in axial compressors is extended to finite amplitude, assuming the compressor characteristic is a parabola over the range of the disturbance. An exact solution is found which requires the operating point to be at the minimum or maximum of the parabola. If the characteristic is flat in a deep-stall regime, the previous harmonic solution applies with neither reverse flow or “unstalling.” If the characteristic is concave upward in deep stall, the disturbance has a skewed shape, steeper at the stall-zone trailing edge as experiment shows. Propagation speed is only slightly affected by this nonlinearity. Near stall inception, negative curvature in combination with multiple stall zones can limit the nonlinear oscillation, in the manner of “progressive” stall. If, as seems likely, lag at stall inception is negative (opposite to inertia), propagation speed exceeds 1/2 wheel speed, as experiments suggest.

Hypersonic large-deflection similitude for quasi-wedges and quasi-cones

SummaryGhosh's large-deflection hypersonic similitude and piston theory for plane flows have been extended to axisymmetric flows: a new piston motion in conico-annular space as an extension to Sedov's axisymmetric piston has been proposed and a set of ordinary differential equation for 1-D unsteady motion obtained. Wedge and cone flow solutions are obtained without the constant-density assumption of the earlier theory; the pressure ratio is expressed as a closed form function of the similarity parameter whereas the previous constant density theory gave an iterative procedure. The present theory unifies small disturbance theory and constant density theory.

Stability of ideal incompressible flow with constant vorticity in an elliptic cylinder

STABILITY OF IDEAL INCO)~RESSIBLE FLOW WITH C~ISTANT VORTICITY IN AN ELLIPTIC CYLINDER V. A. Vladimirov UDC 532.5+532.516 A large number of papers are devoted to the study of the stability of rotating flows. Significant progress has been made in understanding the particular idealized case, viz., the stability of ideal fluid flows with circular streamlines (see, e.g., [1-7]). A study of this problem made it possible to establish the existence of two fundamental mechanisms of instability in rotating flows, viz, "centrifugal" and "shear." An understanding of these instability mechanisms became the basis for modeling a whole series of transition and turbulence phenomena [2, 4, 7-12]). At the same time actual flows are frequently only approximately circular. Hence the question arises as to the influence of small deviations from circular geometry on the flow stability. This problem was studied in [13-16]. In [13- 14] experimental and theoretical investigations were carried out on the stability of rotating fluid inside an elliptic cylinder with a small eccentricity. Instability leading to inflec- tions of the axis of rotation was experimentally observed. In order to theoretically study this instability, a model based on Galerkin method using two well-chosen base functions was suggested. The existence of instability in such a formulation appears, at the first glance, surprising since it is known that rigid body rotation has a large stability margin [4]. The problem of the stability of linear vortex in a potential flow which is qualitatively close to [13-14] was investigated in [15-16]. The vortex core was assumed to be subject to small deformation so that the shape of its cross-section is close to an ellipse with a small eccen- tricity. Computations using small disturbance theory also showed the existence of instabil- ity associated with the inflexion of the axis of rotation. As in [13-14] and also in [15- 16], theoretical investigation is limited to the study of flow stability with respect to dis- turbances of a particular type within the framework of linear theory. The stability of flow [13-14] inside an elliptic cylinder is the simplest among a large class of problems associat -~ ed with stability of deformed fluid rotation and deserves detailed study. In the present paper results are given for the stability of this flow with respect to the general form of disturbances. Small disturbance theory is used in terms of the small parameter s. According to computations, the flow is always unstable, even to the first-order approximation in ~, with respect to three-dimensional disturbances with wavelength 2~/k along the axis of rota- tion. The corresponding wave numbers k continuously fill even number of segments of width of the order e. At the center of each segment (points ko) the growth rate of disturbances is a maximum. The values of ko correspond to conditions for first order singularity of the problem. As regards the physical aspect of instability, we note that its mechanism is simi- lar to the known "resonant interactions" [17] in the complex case when the plane waves are not the solutions to the linear problem. The result obtained can be interpreted as a varia- tion of Hasselman's [18] statement on the instability of finite amplitude wave in the pres- ence of corresponding "resonance triad~ Here, resonance conditions take the form of already mentioned singularity conditions. The deformation of the rotational flow by the elliptic walls plays the role of the 'first" wave splitting into two 'resonant" waves. The Galerkin method is not used in this paper but the complete linearized equations of motion are solved. Hence the results presented here generalize [13-14] for the type of disturbances considered in these studies. There is a qualitative agreement here with the conclusions of [13-14]. i. Let us formulate the problem. Consider an elliptic cylinder whose surface is defined by the following equation in cylindrical coordinate system (r, 0, z)

Visualization of Flow Patterns Induced by an Impinging Jet Issuing from a Circular Planform

1/N monotonically. The values of CL at the end of cycles 6-12 were 0.1867 (6), 0.2010 (7), 0.2072 (8), 0.2118 (9), 0.2148 (10), 0.2169 (11), and 0.2181 (12). Hence, the conclusion is that at these flow conditions the symmetric steady flow is unstable and that a slight perturbation will cause the flow to evolve to a nonsymmetric stable state. For this calculation the initial shock waves were located at 77% of the chord from the leading edge and the local Mach number just ahead of the shock was M= 1.172. During the course of the airfoil motion, the shock waves did not approach closer to the trailing edge than 92% chord location and the maximum local Mach number did not exceed 1.184. Therefore, the flow was consistent with the approximations of the unsteady transonic, small-disturbance theory used in LTRAN2. Also, the shock waves did not reach sufficiently close to the trailing edge that possible interference with the imposition of the Kutta condition might explain the anomalous flow. Multiple solutions have also been found for the transonic full potential steady flow, equation by Steinhoff and Jameson. Steinhoff and I will be investigating these anomalous computed flows further with the use of more complete governing equations. The motivation behind this effort is to determine whether the shock wave instability in these calculations also occurs in the Euler equations. If such an instability is found, it could provide an inviscid mechanism for invoking the buffet phenomenon, which also occurs over a narrow range of Mach numbers in the transonic regime.

Optimization of waverider configurations generated from axisymmetricconical flows

Analytic results from inviscid hypersonic small-disturbance theory for axisymmetric conical flow are used to generate waverider configurations valid for all values of Kb ^A/^5, where d is the semi vertex angle of the basic cone. By means of the calculus of variations, those configurations are sought that yield the maximum lift-todrag ratio subject to suitable constraints, such as fixed lift, fixed volume, fixed base area, fixed planform area, and so on. The inviscid analysis accounts for wave drag only, but the effects of viscous drag are discussed. For the fixed-lift constraint, two types of optimized configurations are found: types with pointed noses, and types with round noses with sharp leading edges. The results are analytic in nature and particularly suitable for studying the various tradeoffs that are involved in missile design. Comparison of the results with other types of lifting bodies suggests that properly selected cone-derived waveriders are among the best producers of large liftto-drag ratios.

Note on the Axisymmetric Sonic Jet

The axisymmetric sonic jet is considered in transonic small disturbance theory. A new hodograph equation is derived from which it can be deduced that there is a local similarity solution describing the final state. From this similarity solution it follows that the uniform sonic state is reached at a finite distance from the orifice.

Embedded flow characteristics of sharp-edged rectangular wings

A exploratory investigation into the construction of semiempirical, analytic expressions for steady, symmetric flow characteristics on thin rectangular wings at transonic speed is presented. Early contributions from the small-disturbance transonic flow theory are used as a guide. These solutions for two-dimensional wings with rombic sections, extending over most of the supersonic portion of the transonic region, are represented by analytic expressions obtained through modification of corresponding expressions prepared for subsonic flow. The result is applied on a wing with an aspect ratio of 2 and three thickness-to-chord ratios. Experiments for comparison are not available.

Hypersonic Flow Past an Axisymmetric Body with Small Longitudinal Curvature

The problem of steady supersonic flow past an axisymmetric body with small longitudinal curvature is solved by means of a perturbation technique. Assuming a slender body within the framework of hypersonic smalldisturbance theory, the resulting differential equations are solved explicitly in closed form using the weak polar crossflow approximation. The perturbation quantities are expressed as series expansions involving powers of the radial distance from the nose of the body and functions of a single angular variable 0. The various flowfield quantities including the surface pressure coefficient have been calculated explicitly in terms of the series. The present analytical results compare favorably with available experimental data and Van Dyke's numerical calculations for hypersonic flow past an ogival-shaped slender body.

Computational Transonic Inverse Procedure for Wing Design

A computational transonic inverse procedure for three-dimensional wings in which shapes are determined supporting prescribed pressures is presented. The method is based on modified small disturbance (MSD) theory and can handle wing design in the presence of a fuselage. A consistent analysis-inverse differencing is implemented at the wing slit grid points to ensure recovery of specified pressures in the analysis mode. Formation of an open or a fishtail trailing edge is avoided by a systematic alteration of the velocity potential in front of the leading edge of span stations under inverse mode, until closure is achieved. To lend support to the numerical procedure, an analogous incompressible two-dimensional problem is studied analytically. As an illustration of the usefulness and versatility of the method, the development of a laminar flow control (LFC) wing from a given base wing geometry is presented along with the analysis verification.

Linear Acoustic Model for the Exhaust Duct of a Pulsed Chemical Laser

The photo-initiated reaction in a typical pulsed chemical laser produces transient disturbances in the medium with peak pressure magnitudes on the order of 10 atmospheres and temperatures of up to several thousands of degrees Kelvin. The quality of the emitted beam is influenced by the chemical and optical homogeneity of the medium at the time of initiation; it is thus necessary to attentuate the reaction-generated disturbances in the short time between laser pulses. Typical requirements are reduction of disturbance magnitudes to the range of 10-3 to 10-5 atmospheres in times of 10-2 sec. While the initial disturbance is decidedly nonlinear, the acceptable perturbation levels are within the limits of traditional small disturbance theory, and complete analyses of the fluid dynamics of the clearing process must span the entire range of magnitudes. This paper deals with one end of that range, and describes a computational model, based on linear acoustic theory, that has been used to gain insight into the effects of conventional acoustic attenuation devices on laser cavity clearing times. The model developed is for one-dimensional wave propagation in channels with nonuniform geometry, nonuniform wall acoustic-impedance properties and nonuniform medium properties. The method of inner and outer asympotic expansions has been used to develop a transfer matrix computational model that produces a reaction-cavity reflection coefficient spectrum. Results are shown which illustrate the effects of the axial location of a slug of heated gas on the reflection coefficient spectrum of an acoustically-treated exhaust duct. The results suggest that the presence of temperature nonuniformities in the exhaust duct impede the rapid attenuation of disturbances from the reaction cavity. Pulsed laser devices are under development for applications requiring high power and high energy output. The major problems associated with pulsed chemical lasers involve reagent flow technology; the large amplitude, violent transient flow that results from a pulse must be cleared from the reaction

Transonic small-disturbance theory for lightly loaded cascades

Analytical solutions are derived for a class of two-dimension al transonic cascade flows. In the particular limiting case studied, the airfoils are considered as lightly loaded; a similarity parameter is defined for this case. Second-order asymptotic solutions are derived for the nearly one-dimensional flow in regions directly between successive airfoils and for the periodic flow ahead of and behind the cascade. For high subsonic Mach numbers, composite solutions are obtained by a suitable joining of these results with solutions derived for thin regions containing the leading and trailing edges. Surface pressures and lines of constant Mach number are calculated for circular-arc airfoils in some specific examples. For low supersonic speeds a far-field solution, which allows calculation of the decay of shock waves at large distances, is also derived.

Newtonian flow theory for slender bodies in a dusty gas

Hypersonic small-disturbance theory is extended to consider the problem of dusty-gas flow past thin two-dimensional bodies. The mass fraction of suspended particles is assumed to be sufficiently large that the two-way interaction between particle phase and gas phase must be considered. The system of eight governing equations is further reduced by considering the Newtonian approximation γ → 1 andM∞→ ∞. The Newtonian theory up to second order is studied and the equations are solved for the case of a thin wedge at zero angle of attack. Expressions for the streamlines, dust-particle paths, shock-wave location and all flow variables are obtained. It is seen that the presence of the dust increases the pressure along the wedge surface and tends to bend the shock wave towards the body surface. Other effects of the interaction of the two phases are also discussed.

Transonic flow calculations over two-dimensional canard-wing systems

As a prototype for the three-dimensional interaction problem, the transonic interference flowfields over twodimensional canard-wing systems are computed using transonic small disturbance theory. In the calculation, the two airfoils comprising the lifting biplane system are placed in separate computational planes with an overlapped region across which information from one airfoil to the other is transferred at the end of each relaxation cycle. Results showing the favorable interference in overall lift are presented and compared with linear theory calculations. Far field expressions for solid, slotted, and free jet wind tunnel wall cases that correspond to the canard-wing arrangement are also described.