Lower Bounds for Sonic Bangs in the Far Field

This is a calculation of the chordwise lift distribution, total lift, and moment on a two-dimensional wing striking a sharp-edged gust a t supersonic speed. A direct solution is established by considering a distribution of sources in a fluid at rest. An alternate method using Busemann's conical flow is also shown to be applicable. The time history of the total lift and mid-chord moment is discussed. I t is shown tha t the total lift increases with time and reaches a maximum that corresponds to the steadystate phase of the flow. The mid-chord moment goes through a maximum independent of the Mach Number if the latter value is larger than 4/7r, while this maximum can become infinite for a range of Mach Numbers between 4/TT and 1. (1) INTRODUCTION CONSIDERABLE ATTENTION has been given lately to nonstationary flow problems of wings flying at supersonic speeds. Most of the work, however, has been concerned with the aerodynamic forces on an oscillating airfoil from the standpoint of flutter analysis. The problem of the wing hitting a sharp-edged gust is of a different nature and turns out to be actually much simpler than the oscillating airfoil problems. It is shown in section 2 that it may be treated by a distribution of sources of a simple type along the chord and that the pressure distribution may be derived by elementary methods. The procedure does not introduce a moving fluid but considers a fluid at rest in which nonstationary sources are distributed in a layer of variable extent. This point of view, which is closer to acoustics than to aerodynamics, is somewhat novel and seems to present advantages of simplicity and closeness to physical reality in certain categories of problems. The pressure distribution derived by this method is applied to the calculation of the time history of lift and moment on the wing in section 3. Particular attention is given to the value of the mid-chord moment;.-which starts from zero, rises to a maximum,,and goes back to zero. The value of this maximum and related 4ata is evaluated in section 4. These results are of particular interest to the designer. The derivation of the pressure as given in section 1 is only one of the methods that may be used in this problem. As an independent check and as an illustration of the application of Busemann's method of conical flow to a nonstationary problem, an alternate derivation is given for the pressure distribution in section 4. Received August 15, 1948. * Member of Cornell Aeronautical Laboratory consulting staff. Also Professor of Applied Physical Sciences, Brown University. In a paper by Schwarz procedures used in oscillating airfoil theory are extended to the problem of a wing striking a sharp-edged gust at supersonic speed. Results for an oscillating down-wash lead to the gust problem by a Fourier integral representation. This method constitutes a considerable detour and introduces intermediate results of a transcendental nature which are actually not needed and are more complicated than the result. I t may be verified that the expression derived in the present paper for the pressure distribution is equivalent to that derived by Schwarz. He does not, however, discuss the physical aspects of the problem or derive expressions for lift and moment. (2) DERIVATION OF THE PRESSURE DISTRIBUTION The wing of chord / enters a uniform gust of upward velocity i/0 at the supersonic velocity V (Fig. 1). The velocity component normal to the wing must remain zero, and this condition is equivalent to the generation of a velocity normal to the wing which cancels the gust velocity (Fig. 2). This may also be considered as a reflection of the gust on the wing. Because the velocity is supersonic, the pressure distribution on. one side does not influence the pressure on the other, and therefore we need only consider the bottom side. The pressure distribution on top will be the same except for a reversal of sign. For the same reason the pressure distribution is not influenced by the trailing edge, and

The assumptions of the thin airfoil theory are found to provide certain necessary conditions for the minimum drag of airfoils having a given total lift, a given maximum thickness, or a given volume. The conditions are applicable to steady or unsteady motions and to subsonic or supersonic speeds without restriction on the planform. The computation of drag and the statement of the conditions for minimum drag depend on the consideration of a combined flow field, which is obtained by superimposing the disturbance velocities in forward and reversed motions. If the planform of the airfoil and its total lift are given, it is found that, for minimum drag, the lift must be distributed in such a way that the downwash in the combined field is constant over the entire planform. If the planform is given and the thickness of the airfoil is required to contain a specified volume, then the thickness must be distributed over the planform in such a way that the pressure gradient of the combined field in the direction of flight is constant at all points of the wing.

An error-probability upper-bound is developed for coherent phase-shift keying (CPSK) considering the combined effects of Gaussian noise, intersymbol interference (ISI), and several peak-limited circular-symmetric (CS) components such as adjacent-channel interference (ACI) and cochannel interference (CCI). In an earlier paper [1], an upper bound was introduced, based on the maximizing probability density function concept, for CPSK with Gaussian noise and a single CS interference. The natural extension of this type of bound to multiple mixed interferences-such as are encountered in practice-is given in this sequel. This multiple interferer bound, which is computationally simple to use, requires only the peak and mean-square values of the several interferences involved, and is optimal in the sense that no other bound, based solely on the same information, will be lower than it. Through examples we draw comparisons with previously available results, which show that the multiple-interference limited-peak bound, when applied to composite interference situations, can be an effective tool for system design and analysis.

Let q 1 > q 2 > q 3 q_1>q_2>q_3 be odd primes and N = q 1 q 2 q 3 N=q_1q_2q_3 . Put \[ d = gcd ( q 1 − 1 , q 2 − 1 , q 3 − 1 ) and h i = q i − 1 d , i = 1 , 2 , 3. d=\gcd (q_1-1,q_2-1,q_3-1) \text { and } h_i=\tfrac {q_i-1}d, \;i=1,2,3. \] Then we call d d the kernel , the triple ( h 1 , h 2 , h 3 ) (h_1,h_2,h_3) the signature , and H = h 1 h 2 h 3 H=h_1h_2h_3 the height of N N , respectively. We call N N a C 3 C_3 -number if it is a Carmichael number with each prime factor q i ≡ 3 mod 4 q_i\equiv 3\mod 4 . If N N is a C 3 C_3 -number and a strong pseudoprime to the t t bases b i b_i for 1 ≤ i ≤ t 1\leq i\leq t , we call N N a C 3 C_3 -spsp ( b 1 , b 2 , … , b t ) (b_1, b_2,\dots ,b_t) . Since C 3 C_3 -numbers have probability of error 1 / 4 1/4 (the upper bound of that for the Rabin-Miller test), they often serve as the exact values or upper bounds of ψ m \psi _m (the smallest strong pseudoprime to all the first m m prime bases). If we know the exact value of ψ m \psi _m , we will have, for integers n > ψ m n>\psi _m , a deterministic efficient primality testing algorithm which is easy to implement. In this paper, we first describe an algorithm for finding C 3 C_3 -spsp(2)’s, to a given limit, with heights bounded. There are in total 21978 21978 C 3 C_3 -spsp ( 2 ) (2) ’s > 10 24 >10^{24} with heights > 10 9 >10^9 . We then give an overview of the 21978 C 3 C_3 - spsp(2)’s and tabulate 54 54 of them, which are C 3 C_3 -spsp’s to the first 8 8 prime bases up to 19 19 ; three numbers are spsp’s to the first 11 prime bases up to 31. No C 3 C_3 -spsp’s > 10 24 >10^{24} to the first 12 12 prime bases with heights > 10 9 >10^9 were found. We conjecture that there exist no C 3 C_3 -spsp’s > 10 24 >10^{24} to the first 12 12 prime bases with heights ≥ 10 9 \geq 10^9 and so that ψ 12 a m p ; = 3186 65857 83403 11511 67461 (24 digits) a m p ; = 399165290221 ⋅ 798330580441 , \begin{equation*} \begin {split} \psi _{12}&= 3186\; 65857\; 83403\; 11511\; 67461\; \text {(24 digits)} \\ &=399165290221\cdot 798330580441, \end{split}\end{equation*} which was found by the author in an earlier paper. We give reasons to support the conjecture. The main idea of our method for finding those 21978 21978 C 3 C_3 -spsp ( 2 ) (2) ’s is that we loop on candidates of signatures and kernels with heights bounded, subject those candidates N = q 1 q 2 q 3 N=q_1q_2q_3 of C 3 C_3 -spsp ( 2 ) (2) ’s and their prime factors q 1 , q 2 , q 3 q_1,q_2,q_3 to Miller’s tests, and obtain the desired numbers. At last we speed our algorithm for finding larger C 3 C_3 -spsp’s, say up to 10 50 10^{50} , with a given signature to more prime bases. Comparisons of effectiveness with Arnault’s and our previous methods for finding C 3 C_3 -strong pseudoprimes to the first several prime bases are given.

In an earlier paper the authors calculated the main invariant of a tame polynomial over a valued field in terms of the simple invariants associated with a strict system of polynomial extensions that contained that polynomial. In this note we give upper and lower bounds in terms of such invariants for the main invariant of any defectless polynomial. We also determine precisely the polynomials for which the upper bound is the main invariant; this class strictly contains the set of tame polynomials. A class of examples with the same upper and with the same lower bound for the main invariant is given whose main invariants form a dense subset of the interval between the two bounds. A second class of polynomials is given whose strict systems have arbitrarily long length and whose main invariant is the lower bound. A basic tool is a formula for the main invariant which itself gives an algorithm for computing the main invariants of the polynomials in any strict system; in particular, simple formulas are given for the main invariants of some very special types of defectless polynomials including generalized Schönemann polynomials. The Krasner constants of defectless polynomials are also studied.

Let e: 1 -> N - G - K -> 1 be an extension of a finite cyclic group N by a finite cyclic group K .Using homological perturbation theory, we introduce the beginning of a free resolution of the integers Z over the group ring ZG of G in such a way that the resolution reflects the structure of G as an extension of N by K, and we use this resolution to compute the additive structure of the integral cohomology of G in many cases.We proceed by first establishing a number of special cases, thereafter constructing suitable cohomology classes thereby obtaining a lower bound, then computing characteristic classes introduced in an earlier paper, and, finally, exploiting these classes, obtaining upper bounds for the cohomology via the integral cohomology spectral sequence of the extension e.The calculation is then completed by comparing the two bounds.

Improving bounds on link failure tolerance of the star graph

This paper documents aeroelastic results from a new integrating product that allows cross-over between simulation of aerodynamics, dynamic stability and control, and structures in the current version, as well as propulsion and store separation in later versions. The Kestrel software product is an integrating product written in modular form with a Python infrastructure to allow growth to additional capabilities as needed. The Kestrel software product has been designed from the ground up to address fixed-wing aircraft in flight regimes ranging from subsonic through supersonic flight, including maneuvers, multi-aircraft configurations, and operational conditions. Kestrel v2.0 has nine simulation capabilities, static rigid body aircraft, rigid body maneuvering aircraft, aeroelastic aircraft, aircraft with control surface motion, predictive 6DoF aircraft motion, and combinations of the previous capabilities. Aeroelastic results for two wings and a generic fighter configuration are presented with comparison to experiment and other researcher’s data.

In this paper, wing-body configurations for a next generation Supersonic Transport are designed by means of Multiobjectiv e Evolutionary Algorithms. SST wing-body configurations are designed to reduce the aerodynamic drag and the sonic boom for supersonic flight. To lower the sonic boom strength, the present objective function is to satisfy the equivalent area distribution for low sonic boom proposed by Darden. Wing and fuselage is defined by 131 design variables and optimized at the same time. Structured multiblock grids around SST wing-body configuration are generated automatically and an Euler solver is used to evaluate the aerodynamic performance of SST wing-body configuration. Compromised solutions are found as Pareto solutions. Although they have a variety of fuselage configurations, all of them have a similar wing planform due to the imposed constraints. The present results imply that a lifting surface should be distributed innovatively to match Darden’s distribution for low boom.

Marketing surveys of traditional business aircraft operators have shown that there is significant interest customer interest in the development of a supersonic business jet. One requirement for such an aircraft is that it be capable of unrestricted supersonic flight over land. This requirement has focused considerable attention on sonic boom minimization for a class of 100,000 pound, Mach 1.6 to 2.0 cruise airplanes. Configuring the design with sufficient low boom characteristics while at the same time including enough cabin volume has proven difficult. Gulfstream has developed a Quiet Spike concept that significantly improves the sonic boom signature while at the same time allowing for additional fuselage volume. The multi stage extendable nose spike projects out of, and retracts into, the forward fuselage of the airplane. Wind tunnel tests of supersonic equivalent area distributions have validated the near field aerodynamics of the spike concept. CFD analysis of a wing/body configured with a multi stage spike have extended the analysis beyond idealized equivalent area distributions and shown a great potential for tailoring the shape of the sonic boom ground signature while simultaneously allowing greater flexibility in the fuselage design.