Shock Wave Behavior in Transonic Compressor Noise Generation

1971 ◽  
Vol 93 (4) ◽  
pp. 397-403 ◽  
Author(s):  
M. R. Fink
Keyword(s):  
Shock Wave ◽  
Near Field ◽  
Leading Edge ◽  
Rotation Frequency ◽  
Order Theory ◽  
Shock Strength ◽  
First Order ◽  
Transonic Compressor ◽  
First Order Theory ◽  
Pure Tones

Compressor noise at transonic tip speeds contains strong tones at multiples of shaft rotation frequency as well as harmonics of blade passage frequency. These multiple pure tones or combination tones are caused by rotor blade nonuniformities which result in pronounced irregularities in the shock pattern attached to the rotor. Nonlinear first-order theory, similar to that used in analysis of sonic boom strength, is utilized to determine shock wave decay with upstream distance. In the extreme near field of the rotor, shock strength varies inversely as the square root of upstream distance from the blade leading edge, as with an isolated airfoil. Somewhat further upstream, the expansion region from the neighboring blade in the cascade interacts with the shock so that shock strength varies as the inverse first power of distance. These aerodynamic results are used to infer some characteristics of transonic compressor noise which in turn are compared with experimental results.

On the Unsteady Supersonic Cascade With a Subsonic Leading Edge—An Exact First Order Theory—Part 2

1974 ◽  
Vol 96 (1) ◽  
pp. 23-31 ◽  
Author(s):  
M. Kurosaka
Keyword(s):  
Closed Form ◽  
Stability Boundary ◽  
Pressure Rise ◽  
Leading Edge ◽  
Order Theory ◽  
Back Pressure ◽  
First Order ◽  
First Order Theory ◽  
The Stability ◽  
Stagger Angle

Pursuant to Part 1, the analysis of the aerodynamic forces acting on slowly oscillating airfoils in a supersonic cascade with a subsonic leading edge is presented. First the flow field between adjacent airfoils is determined. In the limit of sonic leading edge, the present results for the velocity potential agree with the sonic limit of Lane’s supersonic leading edge analysis. The requirement of the continuity of pressure in the “train” leads to functional equations for the train velocity; their solutions, obtained in closed form, are found to involve arbitrary constants which are related to the back pressure. The effect of the back pressure on the “train” is discussed in detail. For a cascade with zero pressure rise across it, the train velocity is determined completely and the formulas for lift and moment, accurate to the first order of a frequency parameter, are obtained in closed form. Stability criteria for a single-degree-of-freedom motion are examined. A pure bending motion is found to be stable, but a pure torsional motion becomes unstable under certain circumstances. These results are consistent with analogous oscillations of an isolated airfoil. However, the stability boundary for a typical cascade differs significantly from the case of the isolated airfoil, being strongly influenced by such cascade parameters as solidity, blade-to-blade phase difference, and stagger angle.

On the Unsteady Supersonic Cascade With a Subsonic Leading Edge—An Exact First Order Theory—Part 1

1974 ◽  
Vol 96 (1) ◽  
pp. 13-22 ◽  
Author(s):  
M. Kurosaka
Keyword(s):  
Flow Field ◽  
Steady Flow ◽  
Integral Equations ◽  
Flow Analysis ◽  
Leading Edge ◽  
Order Theory ◽  
Suction Surface ◽  
First Order ◽  
First Order Theory ◽  
Theoretical Predictions

This paper presents, in two parts, the theoretical predictions of the aerodynamic forces acting on slowly oscillating airfoils in a supersonic cascade with a subsonic leading edge. The analysis is based on the assumption of an inviscid, two-dimensional and linearized flow. In the first part of the paper, the flow field ahead of the cascade is considered. An initial value problem is posed and, from the periodicity requirement in the cascade, the problem is reformulated in terms of integral equations. Solution of the integral equations, accurate to the first order of a frequency parameter, are obtained in closed form. In the limit of the steady flow, the unsteady flow analysis yields a mathematical verification of the unique incidence effect. Based on this proof, a simple rule is presented for the airfoil suction surface contour satisfying steady flow requirement ahead of the cascade. The complete aeroelastic problem, including the solution for the flow field between the blades and the trailing interference zone, is treated in Part 2.

scholarly journals On pseudo-finite near-fields which have finite dimension over the centre

1989 ◽  
Vol 32 (3) ◽  
pp. 371-375
Author(s):  
Peter Fuchs
Keyword(s):  
Finite Fields ◽  
Finite Dimension ◽  
Near Field ◽  
Order Theory ◽  
Structure Theory ◽  
First Order ◽  
First Order Theory ◽  
Infinite Model

In [1] J. Ax studied a class of fields with similar properties as finite fields called pseudo-finite fields. One can prove that pseudo-finite fields are precisely the infinite models of the first-order theory of finite fields. Similarly a near-field F is called pseudo-finite if F is an infinite model of the first-order theory of finite near-fields. The structure theory of these near-fields has been initiated by U. Feigner in [5].

scholarly journals Effects of Tip Leakage Flow on the Aerodynamic Performance and Stability of an Axial-Flow Transonic Compressor Stage

Energies ◽  
2021 ◽  
Vol 14 (14) ◽  
pp. 4168
Author(s):  
Botao Zhang ◽  
Xiaochen Mao ◽  
Xiaoxiong Wu ◽  
Bo Liu
Keyword(s):  
Boundary Layer ◽  
Shock Wave ◽  
Axial Flow ◽  
Leading Edge ◽  
Rotating Stall ◽  
Tip Clearance ◽  
Leakage Flow ◽  
Tip Leakage Flow ◽  
Tip Leakage ◽  
Transonic Compressor

To explain the effect of tip leakage flow on the performance of an axial-flow transonic compressor, the compressors with different rotor tip clearances were studied numerically. The results show that as the rotor tip clearance increases, the leakage flow intensity is increased, the shock wave position is moved backward, and the interaction between the tip leakage vortex and shock wave is intensified, while that between the boundary layer and shock wave is weakened. Most of all, the stall mechanisms of the compressors with varying rotor tip clearances are different. The clearance leakage flow is the main cause of the rotating stall under large rotor tip clearance. However, the stall form for the compressor with half of the designed tip clearance is caused by the joint action of the rotor tip stall caused by the leakage flow spillage at the blade leading edge and the whole blade span stall caused by the separation of the boundary layer of the rotor and the stator passage. Within the investigated varied range, when the rotor tip clearance size is half of the design, the compressor performance is improved best, and the peak efficiency and stall margin are increased by 0.2% and 3.5%, respectively.

scholarly journals A first-order theory of Ulm type

Computability ◽  
2019 ◽  
Vol 8 (3-4) ◽  
pp. 347-358
Author(s):  
Matthew Harrison-Trainor
Keyword(s):  
Order Theory ◽  
First Order ◽  
First Order Theory

scholarly journals Quasi-decidability of a Fragment of the First-Order Theory of Real Numbers

2015 ◽  
Vol 57 (2) ◽  
pp. 157-185 ◽  
Author(s):  
Peter Franek ◽  
Stefan Ratschan ◽  
Piotr Zgliczynski
Keyword(s):  
Order Theory ◽  
Real Numbers ◽  
First Order ◽  
First Order Theory
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