Journal articles: 'Transonic small disturbance equation'

1

Balakrishnan, A. V. "Transonic Small Disturbance Potential Equation." AIAA Journal 42, no. 6 (June 2004): 1081–88. http://dx.doi.org/10.2514/1.5101.

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Lyrintzis, A. S., A. M. Wissink, and A. T. Chronopoulos. "Efficient iterative methods for the transonic small disturbance equation." AIAA Journal 30, no. 10 (October 1992): 2556–58. http://dx.doi.org/10.2514/3.11263.

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3

Čanić, Sunčica, Barbara Lee Keyfitz, and Eun Heui Kim. "Free boundary problems for the unsteady transonic small disturbance equation: Transonic regular reflection." Methods and Applications of Analysis 7, no. 2 (2000): 313–36. http://dx.doi.org/10.4310/maa.2000.v7.n2.a4.

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4

Chern, Mary, and Barbara Lee Keyfitz. "The unsteady transonic small disturbance equation: Data on oblique curves." Discrete and Continuous Dynamical Systems 36, no. 8 (March 2016): 4213–25. http://dx.doi.org/10.3934/dcds.2016.36.4213.

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5

Fishelov, Dalia. "Spectral Methods for the Small Disturbance Equation of Transonic Flows." SIAM Journal on Scientific and Statistical Computing 9, no. 2 (March 1988): 232–51. http://dx.doi.org/10.1137/0909015.

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Lyrintzis, A. S., and Y. Xue. "Acoustics of Transonic Flow Around an Oscillating Flap." Journal of Fluids Engineering 114, no. 2 (June 1, 1992): 240–45. http://dx.doi.org/10.1115/1.2910021.

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Investigation of noise mechanisms due to unsteady transonic flow is important for aircraft noise reduction. In this work, the near-field impulsive noise due to an oscillating flap is simulated numerically. The problem is modeled by the two-dimensional high frequency transonic small disturbance equation (VTRAN2 code). The three types of unsteady shock wave motion have been identified. Two different important disturbances exist in the pressure signal. The disturbances are related to the unsteady motion of the supersonic pocket and fluctuating lift, and drag forces. Pressure wave signatures, noise frequency spectra, and noise directivity are shown.

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7

Goorjian, Peter M., and Robert D. Van Buskirk. "Second-order-accurate spatial differencing for the transonic small-disturbance equation." AIAA Journal 23, no. 11 (November 1985): 1693–99. http://dx.doi.org/10.2514/3.9153.

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8

Batina, John T. "Efficient algorithm for solution of the unsteady transonic small-disturbance equation." Journal of Aircraft 25, no. 7 (July 1988): 598–605. http://dx.doi.org/10.2514/3.45629.

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9

Liu, Ya, Shijun Luo, and Feng Liu. "Multiple solutions and stability of the steady transonic small-disturbance equation." Theoretical and Applied Mechanics Letters 7, no. 5 (September 2017): 292–300. http://dx.doi.org/10.1016/j.taml.2017.09.011.

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10

Čanić, Sunčica, and Barbara Lee Keyfitz. "An Elliptic Problem Arising from the Unsteady Transonic Small Disturbance Equation." Journal of Differential Equations 125, no. 2 (March 1996): 548–74. http://dx.doi.org/10.1006/jdeq.1996.0040.

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11

Whitlow, Woodrow, and David A. Seidel. "Nonreflecting boundary conditions for the complete unsteady transonic small-disturbance equation." AIAA Journal 23, no. 2 (February 1985): 315–17. http://dx.doi.org/10.2514/3.8913.

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12

Keyfitz, Barbara Lee, and Suncica Canic. "Riemann Problems for the Two-Dimensional Unsteady Transonic Small Disturbance Equation." SIAM Journal on Applied Mathematics 58, no. 2 (April 1998): 636–65. http://dx.doi.org/10.1137/s0036139996300.

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13

RUSAK, ZVI, and JANG-CHANG LEE. "Transonic flow of moist air around a thin airfoil with non-equilibrium and homogeneous condensation." Journal of Fluid Mechanics 403 (January 25, 2000): 173–99. http://dx.doi.org/10.1017/s0022112099007053.

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A new small-disturbance model for a steady transonic flow of moist air with non-equilibrium and homogeneous condensation around a thin airfoil is presented. The model explores the nonlinear interactions among the near-sonic speed of the flow, the small thickness ratio and angle of attack of the airfoil, and the small amount of water vapour in the air. The condensation rate is calculated according to classical nucleation and droplet growth models. The asymptotic analysis gives the similarity parameters that govern the flow problem. Also, the flow field can be described by a non-homogeneous (extended) transonic small-disturbance (TSD) equation coupled with a set of four ordinary differential equations for the calculation of the condensate (or sublimate) mass fraction. An iterative numerical scheme which combines Murman & Cole's (1971) method for the solution of the TSD equation with Simpson's integration rule for the estimation of the condensate mass production is developed. The results show good agreement with available numerical simulations using the inviscid fluid flow equations. The model is used to study the effects of humidity and of energy supply from condensation on the aerodynamic performance of airfoils.

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Cramer, M. S., S. T. Whitlock, and G. M. Tarkenton. "Transonic and Boundary Layer Similarity Laws in Dense Gases." Journal of Fluids Engineering 118, no. 3 (September 1, 1996): 481–85. http://dx.doi.org/10.1115/1.2817783.

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We discuss the validity of similarity and scaling laws for transonic flow and compressible boundary layers when dense gas effects are important. The physical mechanisms for the failure of each class of scaling law are delineated. In the case of transonic flow, a new similitude based on a modified small disturbance equation is presented.

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15

KIM, JONG-YUN, KYUNG-SEOK KIM, SEUNG-JUN LEE, and IN LEE. "TRANSONIC AEROELASTIC ANALYSIS OF AIRCRAFT WINGS CONSIDERING THE BOUNDARY-LAYER EFFECTS." Modern Physics Letters B 23, no. 03 (January 30, 2009): 473–76. http://dx.doi.org/10.1142/s0217984909018680.

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Aerodynamic solver using the transonic small-disturbance (TSD) equation has frequently been used to perform practical aeroelastic analysis for many aircraft models. In the present study, the more accurate aeroelastic analysis solver using the TSD equation was developed by considering the viscous effects of the boundary-layer. The viscous effects were considered using Green's lag-entrainment equations and an inverse boundary-layer method. Through aerodynamic analyses for several aircraft wings, the viscous-inviscid interaction approach could improve the accuracy of the aerodynamic computation using the TSD equation. Finally, the aeroelastic characteristics were investigated using comparisons of the time responses between the inviscid and viscous flows.

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16

Faria, Luiz M., Aslan R. Kasimov, and Rodolfo R. Rosales. "Theory of weakly nonlinear self-sustained detonations." Journal of Fluid Mechanics 784 (November 3, 2015): 163–98. http://dx.doi.org/10.1017/jfm.2015.577.

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We propose a theory of weakly nonlinear multidimensional self-sustained detonations based on asymptotic analysis of the reactive compressible Navier–Stokes equations. We show that these equations can be reduced to a model consisting of a forced unsteady small-disturbance transonic equation and a rate equation for the heat release. In one spatial dimension, the model simplifies to a forced Burgers equation. Through analysis, numerical calculations and comparison with the reactive Euler equations, the model is demonstrated to capture such essential dynamical characteristics of detonations as the steady-state structure, the linear stability spectrum, the period-doubling sequence of bifurcations and chaos in one-dimensional detonations and cellular structures in multidimensional detonations.

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17

TESDALL, ALLEN M., and BARBARA L. KEYFITZ. "A CONTINUOUS, TWO-WAY FREE BOUNDARY IN THE UNSTEADY TRANSONIC SMALL DISTURBANCE EQUATIONS." Journal of Hyperbolic Differential Equations 07, no. 02 (June 2010): 317–38. http://dx.doi.org/10.1142/s0219891610002153.

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We formulate a problem for the unsteady transonic small disturbance equations which describes a situation analogous to the reflection of a weak shock off a wedge, with the incident shock replaced by an incident rarefaction. We linearize this problem and solve it exactly, and we compute a numerical solution of the full nonlinear problem. The solution of this problem has several features in common with the solution of the weak shock reflection problem, known as Guderley Mach reflection. In both cases, a rarefaction wave reflects off a sonic line and forms a transonic shock. There is transonic coupling between the supersonic and subsonic regions across the sonic line and shock. In both situations, this sonic line/shock can be considered a free boundary in the formulation of a new type of free boundary problem which has not previously been formulated or analyzed. The free boundary problem that arises in the context of the problem considered here is, however, simpler than the free boundary problem that arises in the weak shock reflection problem.

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18

Opstal, T. M. van. "A Robust Solution Procedure for the Transonic Small-Disturbance Equation in Fluid-Structure Interactions." Open Aerospace Engineering Journal 4, no. 1 (March 22, 2011): 1–10. http://dx.doi.org/10.2174/1874146001104010001.

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19

Rusak, Z. "Novel Similarity Solutions of the Sonic Small-Disturbance Equation with Applications to Airfoil Transonic Aerodynamics." SIAM Journal on Applied Mathematics 54, no. 2 (April 1994): 285–308. http://dx.doi.org/10.1137/s0036139991250050.

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20

Mirkovic, Dragan, and Suncica Canic. "A Numerical Study of Riemann Problems for the Two-Dimensional Unsteady Transonic Small Disturbance Equation." SIAM Journal on Applied Mathematics 58, no. 5 (October 1998): 1365–93. http://dx.doi.org/10.1137/s003613999730884x.

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21

Cavalieri, A. V. G., and P. A. O. Soviero. "Analysis of compressible potential flow over aerofoils using the dual reciprocity method." Aeronautical Journal 116, no. 1178 (April 2012): 391–406. http://dx.doi.org/10.1017/s0001924000005285.

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Abstract The use of the linearised potential model for the analysis of compressible flows is quite widespread, and provides good results for subsonic and supersonic flows. However, the calculation of aerofoils and wings subject to transonic flows requires a non-linear model, such as the transonic small-disturbance (TSD) potential equation. The solution of the problem by a singularity distribution requires singularities over the field, as well as panels on the boundary, characterising the procedure known as field panel method. The present work shows results of calculations of the transonic small-disturbance potential equation for flows without shock waves using the dual reciprocity method (DRM), which permits calculation of integrals only at the boundary of the problem, without the need of field distributions. This approach, compared to the field panel methods, takes considerably less computer time, and shows a significant improvement when compared to results of linear theory without much additional computer time, making this technique adequate to design phases of aircraft. Pressure distribution results show good agreement with other methods found in litterature. The low computational cost of the present method allows us to perform parametric tests and explore the effects of thickness and Mach number on the lift and pitching moment coefficients. A discussion of the physical effect of these parameters on the problem is presented, and the thickness of the aerofoil is shown to increase the lift and change the position of the aerodynamic centre. However, this non-linear effect depends on the precise shape of the thickness distribution.

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22

Gamba, Irene M., Rodolfo R. Rosales, and Esteban G. Tabak. "Constraints on possible singularities for the unsteady transonic small disturbance (UTSD) equations." Communications on Pure and Applied Mathematics 52, no. 6 (June 1999): 763–79. http://dx.doi.org/10.1002/(sici)1097-0312(199906)52:6<763::aid-cpa4>3.0.co;2-3.

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23

WANG, CHUN-WEI, and ZVI RUSAK. "Numerical studies of transonic BZT gas flows around thin airfoils." Journal of Fluid Mechanics 396 (October 10, 1999): 109–41. http://dx.doi.org/10.1017/s0022112099005893.

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Numerical studies of two-dimensional, transonic flows of dense gases of retrograde type, known as BZT gases, around thin airfoils are presented. The computations are guided by a recent asymptotic theory of Rusak & Wang (1997). It provides a uniformly valid solution of the flow around the entire airfoil surface which is composed of outer and inner solutions. A new transonic small-disturbance (TSD) equation solver is developed to compute the nonlinear BZT gas flow in the outer region around most of the airfoil. The flow in the inner region near the nose of the airfoil is computed by solving the problem of a sonic flow around a parabola. Numerical results of the composite solutions calculated from the asymptotic formula are compared with the solutions of the Euler equations. The comparison demonstrates that, in the leading order, the TSD solutions of BZT gas flows represent the essence of the flow character around the airfoil as computed from the Euler equations. Furthermore, guided by the asymptotic formula, the computational results demonstrate the similarity rules for transonic flows of BZT gases. There are differences between the self-similar cases that may be related to the error associated with the accuracy of the asymptotic solution. A discussion on the flow patterns around an airfoil at transonic speeds and at various upstream thermodynamic conditions is also presented. The paper provides important guidelines for future studies on this subject.

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24

Rusak, Z. "Transonic flow around the leading edge of a thin airfoil with a parabolic nose." Journal of Fluid Mechanics 248 (March 1993): 1–26. http://dx.doi.org/10.1017/s0022112093000667.

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Transonic potential flow around the leading edge of a thin two-dimensional general airfoil with a parabolic nose is analysed. Asymptotic expansions of the velocity potential function are constructed at a fixed transonic similarity parameter (K) in terms of the thickness ratio of the airfoil in an outer region around the airfoil and in an inner region near the nose. These expansions are matched asymptotically. The outer expansion consists of the transonic small-disturbance theory and it second-order problem, where the leading-edge singularity appears. The inner expansion accounts for the flow around the nose, where a stagnation point exists. Analytical expressions are given for the first terms of the inner and outer asymptotic expansions. A boundary value problem is formulated in the inner region for the solution of a uniform sonic flow about an infinite two-dimensional parabola at zero angle of attack, with a symmetric far-field approximation, and with no circulation around it. The numerical solution of the flow in the inner region results in the symmetric pressure distribution on the parabolic nose. Using the outer small-disturbance solution and the nose solution a uniformly valid pressure distribution on the entire airfoil surface can be derived. In the leading terms, the flow around the nose is symmetric and the stagnation point is located at the leading edge for every transonic Mach number of the oncoming flow and shape and small angle of attack of the airfoil. The pressure distribution on the upper and lower surfaces of the airfoil is symmetric near the edge point, and asymmetric deviations increase and become significant only when the distance from the leading edge of the airfoil increases beyond the inner region. Good agreement is found in the leading-edge region between the present solution and numerical solutions of the full potential-flow equations and the Euler equations.

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25

EVERS, I., and N. PEAKE. "Noise generation by high-frequency gusts interacting with an airfoil in transonic flow." Journal of Fluid Mechanics 411 (May 25, 2000): 91–130. http://dx.doi.org/10.1017/s0022112099008095.

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The method of matched asymptotic expansions is used to describe the sound generated by the interaction between a short-wavelength gust (reduced frequency k, with k [Gt ] 1) and an airfoil with small but non-zero thickness, camber and angle of attack (which are all assumed to be of typical size O(δ), with δ [Lt ] 1) in transonic flow. The mean-flow Mach number is taken to differ from unity by O(δ2/3), so that the steady flow past the airfoil is determined using the transonic small-disturbance equation. The unsteady analysis is based on a linearization of the Euler equations about the mean flow. High-frequency incident vortical and entropic disturbances are considered, and analogous to the subsonic counterpart of this problem, asymptotic regions around the airfoil highlight the mechanisms that produce sound. Notably, the inner region round the leading edge is of size O(k−1), and describes the interaction between the mean-flow gradients and the incident gust and the resulting acoustic waves. We consider the preferred limit in which kδ2/3 = O(1), and calculate the first two terms in the phase of the far-field radiation, while for the directivity we determine the first term (δ = 0), together with all higher-order terms which are at most O(δ2/3) smaller – in fact, this involves no fewer than ten terms, due to the slowly-decaying form of the power series expansion of the steady flow about the leading edge. Particular to transonic flow is the locally subsonic or supersonic region that accounts for the transition between the acoustic field downstream of a source and the possible acoustic field upstream of the source. In the outer region the sound propagation has a geometric-acoustics form and the primary influence of the mean-flow distortion appears in our preferred limit as an O(1) phase term, which is particularly significant in view of the complicated interference between leading- and trailing-edge fields. It is argued that weak mean- flow shocks have an influence on the sound generation that is small relative to the effects of the leading-edge singularity.

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Hall, K. C., W. S. Clark, and C. B. Lorence. "A Linearized Euler Analysis of Unsteady Transonic Flows in Turbomachinery." Journal of Turbomachinery 116, no. 3 (July 1, 1994): 477–88. http://dx.doi.org/10.1115/1.2929437.

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A computational method for efficiently predicting unsteady transonic flows in two-and three-dimensional cascades is presented. The unsteady flow is modeled using a linearized Euler analysis whereby the unsteady flow field is decomposed into a nonlinear mean flow plus a linear harmonically varying unsteady flow. The equations that govern the perturbation flow, the linearized Euler equations, are linear variable coefficient equations. For transonic flows containing shocks, shock capturing is used to model the shock impulse (the unsteady load due to the harmonic motion of the shock). A conservative Lax–Wendroff scheme is used to obtain a set of linearized finite volume equations that describe the harmonic small disturbance behavior of the flow. Conditions under which such a discretization will correctly predict the shock impulse are investigated. Computational results are presented that demonstrate the accuracy and efficiency of the present method as well as the essential role of unsteady shock impulse loads on the flutter stability of fans.

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27

Zhu, N. G., L. Xu, and M. Z. Chen. "Similarity Transformations for Compressor Blading." Journal of Turbomachinery 114, no. 3 (July 1, 1992): 561–68. http://dx.doi.org/10.1115/1.2929180.

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Improving the performance of high-speed axial compressors through low-speed model compressor testing has proved to be economical and effective (Wisler, 1985). The key to this technique is to design low-speed blade profiles that are aerodynamically similar to their high-speed counterparts. The conventional aerodynamic similarity transformation involves the small disturbance potential flow assumption; therefore, its application is severely limited and generally not used in practical design. In this paper, a set of higher order transformation rules are presented that can accommodate large disturbances at transonic speed and are therefore applicable to similar transformations between the high-speed high-pressure compressor and its low-speed model. Local linearization is used in the nonlinear equations and the transformation is obtained in an iterative process. The transformation gives the global blading parameters such as camber, incidence, and solidity as well as the blade profile. Both numerical and experimental validations of the transformation show that the nonlinear similarity transformations do retain satisfactory accuracy for highly loaded blades up to low transonic speeds. Further improvement can be made by only slightly modifying profiles numerically without altering the global similarity parameters.

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28

Bhutani, O. P., and L. Roy-Chowdhury. "Applications of some recent techniques for the exact solutions of the small disturbance potential flow equation of nonequilibrium transonic gas dynamics." Computers & Mathematics with Applications 40, no. 12 (December 2000): 1349–61. http://dx.doi.org/10.1016/s0898-1221(00)00244-3.

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29

TESDALL, ALLEN M. "HIGH RESOLUTION SOLUTIONS FOR THE SUPERSONIC FORMATION OF SHOCKS IN TRANSONIC FLOW." Journal of Hyperbolic Differential Equations 08, no. 03 (September 2011): 485–506. http://dx.doi.org/10.1142/s0219891611002470.

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We present numerical solutions of two problems for the unsteady transonic small disturbance equations whose solutions contain shocks. The first problem is a two-dimensional Riemann problem with initial data corresponding to a slightly supersonic flow hitting the corner of an expanding duct at t = 0. The second problem is a boundary value problem that describes steady transonic flow over an airfoil. In both cases, the solutions contain regions of supersonic and subsonic flow, and an expansion wave interacts with a sonic line to produce a shock. We use high resolution methods, together with local grid refinement, to investigate the nature of the solution in the neighborhood of the point where the shock forms. We find that the shock originates in the supersonic region as originally proposed by Guderley, and very close to, but not at, the sonic line.

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30

RUSAK, ZVI, and CHUN-WEI WANG. "Transonic flow of dense gases around an airfoil with a parabolic nose." Journal of Fluid Mechanics 346 (September 10, 1997): 1–21. http://dx.doi.org/10.1017/s0022112097006411.

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Transonic potential flow of dense gases of retrograde type around the leading edge of a thin airfoil with a parabolic nose is studied. The analysis follows the approach of Rusak (1993) for a perfect gas. Asymptotic expansions of the velocity potential function are constructed in terms of the airfoil thickness ratio in an outer region around the airfoil and in an inner region near the nose. The outer expansion consists of the transonic small-disturbance theory for dense gases, where a leading-edge singularity appears. Analytical expressions are given for this singularity by constructing similarity solutions of the governing nonlinear equation. The inner expansion accounts for the flow around the nose, where a stagnation point exists. A boundary value problem is formulated in the inner region for the solution of an oncoming uniform sonic flow with zero values of the fundamental derivative of gasdynamics (Γ=0) and the second nonlinearity parameter (Λ=0) around a parabola at zero angle of attack. The numerical solution of the inner problem results in a symmetric flow around the nose. The outer and inner expansions are matched asymptotically resulting in a uniformly valid solution on the entire airfoil surface. In the leading terms, the flow around the nose is symmetric and the stagnation point is located at the leading edge for every transonic Mach number, and small values of Γ and Λ of the oncoming flow and any shape and small angle of attack of the airfoil. Furthermore, analysis of the inner region in the immediate neighbourhood of the stagnation point reveals that the flow is purely subsonic, approaching critical conditions in the limit of large (scaled) distances, which excludes the formation of shock discontinuities in the nose region.

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ESLER, J. G., O. J. RUMP, and E. R. JOHNSON. "Non-dispersive and weakly dispersive single-layer flow over an axisymmetric obstacle: the equivalent aerofoil formulation." Journal of Fluid Mechanics 574 (February 15, 2007): 209–37. http://dx.doi.org/10.1017/s0022112006003910.

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Non-dispersive and weakly dispersive single-layer flows over axisymmetric obstacles, of non-dimensional height M measured relative to the layer depth, are investigated. The case of transcritical flow, for which the Froude number F of the oncoming flow is close to unity, and that of supercritical flow, for which F > 1, are considered. For transcritical flow, a similarity theory is developed for small obstacle height and, for non-dispersive flow, the problem is shown to be isomorphic to that of the transonic flow of a compressible gas over a thin aerofoil. The non-dimensional drag exerted by the obstacle on the flow takes the form D(Γ) M5/3, where Γ = (F-1)M−2/3 is a transcritical similarity parameter and D is a function which depends on the shape of the ‘equivalent aerofoil’ specific to the obstacle. The theory is verified numerically by comparing results from a shock-capturing shallow-water model with corresponding solutions of the transonic small-disturbance equation, and is found to be generally accurate for M≲0.4 and |Γ| ≲ 1. In weakly dispersive flow the equivalent aerofoil becomes the boundary condition for the Kadomtsev–Petviashvili equation and (multiple) solitary waves replace hydraulic jumps in the resulting flow patterns.For Γ ≳ 1.5 the transcritical similarity theory is found to be inaccurate and, for small M, flow patterns are well described by a supercritical theory, in which the flow is determined by the linear solution near the obstacle. In this regime the drag is shown to be $c_d M^2/(F\sqrt{F^2-1})$, where cd is a constant dependent on the obstacle shape. Away from the obstacle, in non-dispersive flow the far-field behaviour is known to be described by the N-wave theory of Whitham and in dispersive flow by the Korteweg–de Vries equation. In the latter case the number of emergent solitary waves in the wake is shown to be a function of ${\cal A}= 3M/(2\delta^2 \sqrt{F^2-1})$, where δ is the ratio of the undisturbed layer depth to the radial scale of the obstacle.

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32

LEE, SEUNG JUN, DONG-KYUN IM, IN LEE, and JANG-HYUK KWON. "THE WING-BODY AEROELASTIC ANALYSES USING THE INVERSE DESIGN METHOD." Modern Physics Letters B 24, no. 13 (May 30, 2010): 1479–82. http://dx.doi.org/10.1142/s0217984910023918.

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Flutter phenomenon is one of the most dangerous problems in aeroelasticity. When it occurs, the aircraft structure can fail in a few second. In recent aeroelastic research, computational fluid dynamics (CFD) techniques become important means to predict the aeroelastic unstable responses accurately. Among various flow equations like Navier-Stokes, Euler, full potential and so forth, the transonic small disturbance (TSD) theory is widely recognized as one of the most efficient theories. However, the small disturbance assumption limits the applicable range of the TSD theory to the thin wings. For a missile which usually has small aspect ratio wings, the influence of body aerodynamics on the wing surface may be significant. Thus, the flutter stability including the body effect should be verified. In this research an inverse design method is used to complement the aerodynamic deficiency derived from the fuselage. MGM (modified Garabedian-McFadden) inverse design method is used to optimize the aerodynamic field of a full aircraft model. Furthermore, the present TSD aeroelastic analyses do not require the grid regeneration process. The MGM inverse design method converges faster than other conventional aerodynamic theories. Consequently, the inverse designed aeroelastic analyses show that the flutter stability has been lowered by the body effect.

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33

ZAKHARIAN, A. R., M. BRIO, J. K. HUNTER, and G. M. WEBB. "The von Neumann paradox in weak shock reflection." Journal of Fluid Mechanics 422 (November 3, 2000): 193–205. http://dx.doi.org/10.1017/s0022112000001609.

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We present a numerical solution of the Euler equations of gas dynamics for a weak-shock Mach reflection in a half-space. In our numerical solutions, the incident, reflected, and Mach shocks meet at a triple point, and there is a supersonic patch behind the triple point, as proposed by Guderley. A theoretical analysis supports the existence of an expansion fan at the triple point, in addition to the three shocks. This solution is in complete agreement with the numerical solution of the unsteady transonic small-disturbance equations obtained by Hunter & Brio (2000), which provides an asymptotic description of a weak-shock Mach reflection. The supersonic patch is extremely small, and this work is the first time it has been resolved in a numerical solution of the Euler equations. The numerical solution uses six levels of grid refinement around the triple point. A delicate combination of numerical techniques is required to minimize both the effects of numerical diffusion and the generation of numerical oscillations at grid interfaces and shocks.

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34

Kim, Eun Heui. "Higher regularity for free boundary and tangential derivative problems and global solutions to regular reflections for the unsteady transonic small disturbance (UTSD) equations." Nonlinear Analysis: Theory, Methods & Applications 64, no. 4 (February 2006): 844–55. http://dx.doi.org/10.1016/j.na.2005.05.046.

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35

Zeigler, Fredrick J., and Donald A. Drew. "Transonic small-disturbance theory for dusty gases." AIAA Journal 23, no. 6 (June 1985): 958–60. http://dx.doi.org/10.2514/3.9015.

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36

Williams, Marc H., Samuel R. Bland, and John W. Edwards. "Flow instabilities in transonic small-disturbance theory." AIAA Journal 23, no. 10 (October 1985): 1491–96. http://dx.doi.org/10.2514/3.9115.

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37

Rusak, Z., J. C. Lee, and J. J. Choi. "A small-disturbance model of transonic combustion." Combustion Theory and Modelling 12, no. 1 (December 18, 2007): 93–113. http://dx.doi.org/10.1080/13647830701449417.

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38

Luo, Shijun, and Lixia Wang. "Shock wave in transonic small-disturbance flow." Computer Methods in Applied Mechanics and Engineering 167, no. 1-2 (December 1998): 101–8. http://dx.doi.org/10.1016/s0045-7825(98)00112-1.

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39

Evers, I. "Gust-Shock Interaction in Transonic Small-Disturbance Flow." AIAA Journal 39, no. 1 (January 2001): 29–36. http://dx.doi.org/10.2514/2.1298.

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40

Evers, I. "Gust-shock interaction in transonic small-disturbance flow." AIAA Journal 39 (January 2001): 29–36. http://dx.doi.org/10.2514/3.14693.

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41

Batina, John T. "Unsteady transonic small-disturbance theory including entropy and vorticity effects." Journal of Aircraft 26, no. 6 (June 1989): 531–38. http://dx.doi.org/10.2514/3.45799.

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42

Gibbons, Michael D., and John T. Batina. "Supersonic far-field boundary conditions for transonic small-disturbance theory." Journal of Aircraft 27, no. 9 (September 1990): 764–70. http://dx.doi.org/10.2514/3.45936.

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43

Lv, Ping, and Yanbo Hu. "Sonic-supersonic solutions for a reactive transonic small disturbance model." Journal of Mathematical Analysis and Applications 504, no. 1 (December 2021): 125380. http://dx.doi.org/10.1016/j.jmaa.2021.125380.

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44

Giddings, Thomas E., Jacob Fish, and Zvi Rusak. "A stabilized finite element formulation for the transonic small-disturbance system." International Journal for Numerical Methods in Engineering 50, no. 9 (2001): 2069–91. http://dx.doi.org/10.1002/nme.110.

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45

Ly, Eddie, Daniel Norrison, and Alexabder Robert Barrett. "Simulation of transonic flows using quad-core OpenMP Euler, flux modified transonic small disturbance, and Fluent codes." ANZIAM Journal 51 (April 30, 2010): 155. http://dx.doi.org/10.21914/anziamj.v51i0.2658.

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46

Bennett, Robert M., John T. Batina, and Herbert J. Cunningham. "Wing-flutter calculations with the CAP-TSD unsteady transonic small-disturbance program." Journal of Aircraft 26, no. 9 (September 1989): 876–82. http://dx.doi.org/10.2514/3.45854.

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47

Silva, Walter A., and Robert M. Bennett. "Application of transonic small disturbance theory to the active flexible wing model." Journal of Aircraft 32, no. 1 (January 1995): 16–22. http://dx.doi.org/10.2514/3.46678.

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48

Kluwick, A., and E. A. Cox. "Steady small-disturbance transonic dense gas flow past two-dimensional compression/expansion ramps." Journal of Fluid Mechanics 848 (June 13, 2018): 756–87. http://dx.doi.org/10.1017/jfm.2018.368.

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Abstract:

The behaviour of steady transonic dense gas flow is essentially governed by two non-dimensional parameters characterising the magnitude and sign of the fundamental derivative of gas dynamics ($\unicode[STIX]{x1D6E4}$) and its derivative with respect to the density at constant entropy ($\unicode[STIX]{x1D6EC}$) in the small-disturbance limit. The resulting response to external forcing is surprisingly rich and studied in detail for the canonical problem of two-dimensional flow past compression/expansion ramps.

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49

van der Vooren, J. "The downstream flow of a transonic aircraft: a small disturbance, high Reynolds number analysis." Aerospace Science and Technology 12, no. 6 (September 2008): 457–68. http://dx.doi.org/10.1016/j.ast.2007.11.001.

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GIDDINGS, THOMAS E., ZVI RUSAK, and JACOB FISH. "A transonic small-disturbance model for the propagation of weak shock waves in heterogeneous gases." Journal of Fluid Mechanics 429 (February 25, 2001): 255–80. http://dx.doi.org/10.1017/s0022112000002779.

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The interaction of weak shock waves with small heterogeneities in gaseous media is studied. It is first shown that various linear theories proposed for this problem lead to pathological breakdowns or singularities in the solution near the wavefront and necessarily fail to describe this interaction. Then, a nonlinear small-disturbance model is developed. The nonlinear theory is uniformly valid and accounts for the competition between the near-sonic speed of the wavefront and the small variations of vorticity and sound speed in the heterogeneous media. This model is an extension of the transonic small-disturbance problem, with additional terms accounting for slight variations in the media. The model is used to analyse the propagation of the sonic-boom shock wave through the turbulent atmospheric boundary layer. It is found that, in this instance, the nonlinear model accounts for the observed behaviour. Various deterministic examples of interaction phenomena demonstrate good agreement with available experimental data and explain the main observed phenomena in Crow (1969).

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Journal articles on the topic 'Transonic small disturbance equation'

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