Eigenmodes of lined flow ducts with rigid splices

2011 ◽  
Vol 690 ◽  
pp. 399-425 ◽  
Author(s):  
E. J. Brambley ◽  
A. M. J. Davis ◽  
N. Peake
Keyword(s):  
Recurrence Relation ◽  
Fourier Coefficients ◽  
Acoustic Pressure ◽  
Asymptotic Expression ◽  
Numerical Procedure ◽  
Mean Flow ◽  
Exact Analytic Expression ◽  
Analytic Expression ◽  
Comparable Magnitude ◽  
Cylindrical Duct

AbstractThis paper presents an analytic expression for the acoustic eigenmodes of a cylindrical lined duct with rigid axially running splices in the presence of flow. The cylindrical duct is considered to be uniformly lined except for two symmetrically positioned axially running rigid liner splices. An exact analytic expression for the acoustic pressure eigenmodes is given in terms of an azimuthal Fourier sum, with the Fourier coefficients given by a recurrence relation. Since this expression is derived using a Green’s function method, the completeness of the expansion is guaranteed. A numerical procedure is described for solving this recurrence relation, which is found to converge exponentially with respect to number of Fourier terms used and is in practice quick to compute; this is then used to give several numerical examples for both uniform and sheared mean flow. An asymptotic expression is derived to directly calculate the pressure eigenmodes for thin splices. This asymptotic expression is shown to be quantitatively accurate for ducts with very thin splices of less than 1 % unlined area and qualitatively helpful for thicker splices of the order of 6 % unlined area. A thin splice is in some cases shown to increase the damping of certain acoustic modes. The influences of thin splices and thin boundary layers are compared and found to be of comparable magnitude for the parameters considered. Trapped modes at the splices are also identified and investigated.

scholarly journals A PRECISE DESCRIPTION OF THE p-ADIC VALUATION OF THE NUMBER OF ALTERNATING SIGN MATRICES

2011 ◽  
Vol 07 (01) ◽  
pp. 57-69
Author(s):  
CLEMENS HEUBERGER ◽  
HELMUT PRODINGER
Keyword(s):  
Fourier Coefficients ◽  
Counting Function ◽  
Precise Description ◽  
Alternating Sign Matrices ◽  
Sum Of Digits ◽  
Sum Of Digits Function ◽  
Adic Valuation ◽  
Exact Analytic Expression ◽  
Analytic Expression

Following Sun and Moll ([4]), we study vp(T(N)), the p-adic valuation of the counting function of the alternating sign matrices. We find an exact analytic expression for it that exhibits the fluctuating behavior, by means of Fourier coefficients. The method is the Mellin–Perron technique, which is familiar in the analysis of the sum-of-digits function and related quantities.

scholarly journals ENERGY EXTRACTION FROM GRAVITATIONAL COLLAPSE TO STATIC BLACK HOLES

2003 ◽  
Vol 12 (01) ◽  
pp. 121-127 ◽  
Author(s):  
REMO RUFFINI ◽  
LUCA VITAGLIANO
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Extraction Process ◽  
Maximum Efficiency ◽  
Energy Extraction ◽  
Mass Energy ◽  
Classical Level ◽  
Static Black Hole ◽  
Exact Analytic Expression ◽  
Analytic Expression

The mass-energy formula of black holes implies that up to 50% of the energy can be extracted from a static black hole. Such a result is reexamined using the recently established analytic formulas for the collapse of a shell and the expression for the irreducible mass of a static black hole. It is shown that the efficiency of energy extraction process during the formation of the black hole is linked in an essential way to the gravitational binding energy, the formation of the horizon and the reduction of the kinetic energy of implosion. Here a maximum efficiency of 50% in the extraction of the mass energy is shown to be generally attainable in the collapse of a spherically symmetric shell: surprisingly this result holds as well in the two limiting cases of the Schwarzschild and extreme Reissner–Nordström space–times. Moreover, the analytic expression recently found for the implosion of a spherical shell to an already formed black hole leads to a new exact analytic expression for the energy extraction which results in an efficiency strictly less than 100% for any physical implementable process. There appears to be no incompatibility between General Relativity and Thermodynamics at this classical level.

Acoustic energy harvesting from vortex-induced tonal sound in a baffled pipe

2011 ◽  
Vol 225 (8) ◽  
pp. 1847-1850 ◽  
Author(s):  
R Hernandez ◽  
S Jung ◽  
K I Matveev
Keyword(s):  
Acoustic Pressure ◽  
Electrical Energy ◽  
High Amplitude ◽  
Acoustic Mode ◽  
Acoustic Energy ◽  
Electric Load ◽  
Mean Flow ◽  
Acoustic Resonators ◽  
Acoustic Energy Harvesting ◽  
Mean Flow Velocity

Energy of high-amplitude sound that often appears in acoustic resonators with mean flow can be harnessed and converted into electricity for powering sensors and other devices. In this study, tests were conducted in a simple setup consisting of a pipe with a pair of baffles and a piezoelement. Tonal sound, corresponding to the second acoustic mode of the resonator, was excited due to vortex shedding/impinging on baffles in the presence of mean flow. Generated sound energy was partially converted into electrical energy by a piezoelement. About 0.55 mW of electric power was produced on a resistive electric load at acoustic pressure amplitudes in the pipe about 170 Pa and mean flow velocity 2.6 m/s.

The acoustic field of sources in shear flow with application to jet noise: convective amplification

1977 ◽  
Vol 79 (1) ◽  
pp. 33-47 ◽  
Author(s):  
T. F. Balsa
Keyword(s):  
Stress Tensor ◽  
Acoustic Field ◽  
Acoustic Pressure ◽  
Time Derivative ◽  
Jet Noise ◽  
Mean Flow ◽  
Retarded Time ◽  
High Frequencies ◽  
Classic Theory ◽  
Derivatives Of

Lighthill, in his elegant and classic theory of jet noise, showed that the far-field acoustic pressure of noise generated by turbulence is proportional to the integral over the jet volume of the second time derivative of the Lighthill stress tensor, the integrand being evaluated at a retarded time. The purpose of this paper is to generalize the above results to include the effects of mean flow (velocity and temperature) surrounding the source of sound. It is shown quite generally that the integrand is now a certain functional of the Lighthill stress tensor evaluated at a retarded time. More important, however, at low and high frequencies this functional assumes an extremely simple form, so that the acoustic field can once more be given by integrals of the time derivatives of the Lighthill tensor. Both the self- and the shear-noise contributions to the pressure are evaluated.

A Viscous-Inviscid Interaction Procedure—Part 2: Application to Turbulent Flow Over a Rearward-Facing Step

1986 ◽  
Vol 108 (1) ◽  
pp. 71-75 ◽  
Author(s):  
O. K. Kwon ◽  
R. H. Pletcher
Keyword(s):  
Turbulent Flow ◽  
Turbulence Modeling ◽  
Numerical Procedure ◽  
Length Scale ◽  
Two Dimensional ◽  
Reattachment Length ◽  
Mean Flow ◽  
Channel Expansion ◽  
Good Agreement

The viscous-inviscid interaction numerical procedure described in Part 1 is used to predict steady, two-dimensional turbulent flow over a rearward-facing step. The accuracy of predictions is observed to be quite sensitive to the specification of length scale in the turbulence modeling. The best results are observed when the length scale is specified algebraically downstream of the step using parameters characteristic of the step geometry. Predictions of mean flow quantities and reattachment length are shown to be in generally good agreement with measurements obtained over a range of channel expansion ratios.

scholarly journals Analysis of Aerodynamically Induced Whirling Forces in Axial Flow Compressors

2000 ◽  
Vol 122 (4) ◽  
pp. 761-768 ◽  
Author(s):  
Z. S. Spakovszky
Keyword(s):  
System Analysis ◽  
Axial Flow ◽  
Tip Clearance ◽  
Design Tools ◽  
Mean Flow ◽  
Blade Loading ◽  
Force Prediction ◽  
Engine Design ◽  
Comparable Magnitude ◽  
Axial Flow Compressors

A new analytical model to predict the aerodynamic forces in axial flow compressors due to asymmetric tip-clearance is introduced. The model captures the effects of tip-clearance induced distortion (i.e., forced shaft whirl), unsteady momentum-induced tangential blade forces, and pressure-induced forces on the spool. Pressure forces are shown to lag the tip-clearance asymmetry, resulting in a tangential (i.e., whirl-inducing) force due to spool pressure. This force can be of comparable magnitude to the classical Alford force. Prediction and elucidation of the Alford force is also presented. In particular, a new parameter denoted as the blade loading indicator is deduced. This parameter depends only on stage geometry and mean flow and determines the direction of whirl tendency due to tangential blade loading forces in both compressors and turbines. All findings are suitable for incorporation into an overall dynamic system analysis and integration into existing engine design tools. [S0889-504X(00)01604-4]

Expected Maximum Block Size in Critical Random Graphs

2019 ◽  
Vol 28 (4) ◽  
pp. 638-655
Author(s):  
V. Rasendrahasina ◽  
A. Rasoanaivo ◽  
V. Ravelomanana
Keyword(s):  
Real Number ◽  
Random Graphs ◽  
Function Theory ◽  
Block Size ◽  
Maximum Size ◽  
Induced Subgraphs ◽  
Symbolic Method ◽  
Exact Analytic Expression ◽  
Analytic Expression ◽  
Image Position

AbstractLet G(n,M) be a uniform random graph with n vertices and M edges. Let ${\wp_{n,m}}$ be the maximum block size of G(n,M), that is, the maximum size of its maximal 2-connected induced subgraphs. We determine the expectation of ${\wp_{n,m}}$ near the critical point M = n/2. When n − 2M ≫ n2/3, we find a constant c1 such that $$c_1 = \lim_{n \rightarrow \infty} \left({1 - \frac{2M}{n}} \right) \,\E({\wp_{n,m}}).$$ Inside the window of transition of G(n,M) with M = (n/2)(1 + λn−1/3), where λ is any real number, we find an exact analytic expression for $$c_2(\lambda) = \lim_{n \rightarrow \infty} \frac{\E{\left({\wp_{n,{{(n/2)}({1+\lambda n^{-1/3}})}}}\right)}}{n^{1/3}}.$$ This study relies on the symbolic method and analytic tools from generating function theory, which enable us to describe the evolution of $n^{-1/3}\,\E{\left({\wp_{n,{{(n/2)}({1+\lambda n^{-1/3}})}}}\right)}$ as a function of λ.

A Simplified Method to Predict the Steady Cyclic Stress State of Creeping Structures

2001 ◽  
Vol 69 (2) ◽  
pp. 148-153 ◽  
Author(s):  
K. V. Spiliopoulos
Keyword(s):  
Fourier Coefficients ◽  
Numerical Procedure ◽  
Cyclic Stress ◽  
Cyclic Loads ◽  
The Finite Element Method ◽  
Time Points ◽  
State Of Stress ◽  
Short Period ◽  
Simplified Method

Simplified methods have been developed to find the long-term cyclic state of stress for structures that exhibit inelastic creep and are subjected to a short period cyclic loading. In the present work a new simplified method is presented which may be applied to cyclic loads having any period. The method is based on decomposing the residual stress in Fourier series. The various Fourier coefficients are computed, in an iterative way, by satisfying equilibrium and compatibility at a few time points inside the cycle. The whole numerical procedure is formulated within the finite element method and examples of various structures are presented.

Direct Calculation of Sound Radiated From Bodies in Nonuniform Flows

1993 ◽  
Vol 115 (4) ◽  
pp. 573-579 ◽  
Author(s):  
H. M. Atassi ◽  
J. Fang ◽  
S. Patrick
Keyword(s):  
Acoustic Pressure ◽  
Three Dimensional ◽  
Sound Field ◽  
Direct Calculation ◽  
Significant Rise ◽  
Mean Flow ◽  
Unsteady Pressure ◽  
Potential Part ◽  
The Mean ◽  
Radiated Sound

Sound radiated from a single airfoil and a cascade of airfoils in three-dimensional gusts is directly calculated. Euler’s equations are linearized about the mean flow of the airfoil or cascade. The velocity field is split into a vortical part and a potential part. The latter is governed by a single nonconstant-coefficient convective wave equation. For a single airfoil, the radiated sound is calculated using Kirchhoff’s method from the mid field of the unsteady pressure obtained through the unsteady aerodynamic solver. The results indicate the importance of the contribution of the quadrupole effects to the sound field. For a cascade of airfoils, the acoustic pressure is directly obtained by solving the partial differential equation. The results show that, as the maximum Mach number on the blade surface nears unity, there is a significant rise in the local unsteady pressure, and also a significant increase in the upstream acoustic pressure.

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