scholarly journals Three-Dimensional Incompressible Flow Solution of an Axial Compressor Using Pseudostream-Functions Formulation

1989 ◽  
Author(s):  
J. Z. Xu ◽  
J. C. Shi ◽  
W. Y. Ni
Keyword(s):  
Flow Field ◽  
Incompressible Flow ◽  
Three Dimensional ◽  
Axial Compressor ◽  
Second Order ◽  
Partial Derivatives ◽  
Analysis And Design ◽  
Characteristic Features ◽  
Derivatives Of ◽  
Convergent Solution

Based on considering the characteristic features of the momentum equations and meeting the requirement of the continuity, two pseudostream-functions are introduced. The principal equation of each pseudostream-function only contains the terms of its own second-order partial derivatives and does not include the second-order partial derivatives of another pseudostream-function. So the relation between the two equations are loosed considerablly and both equations may be solved separately. This property is valueful to obtain the convergent solution of flow field easily. The equations of the pseudostream-functions and the corresponding boundary conditions are given. Two incompressible flow examples show that this method may become a powerful tool in the aerothermodynamic analysis and design of a compressor.

scholarly journals Optimization Design of a 2.5 Stage Highly Loaded Axial Compressor with a Bezier Surface Modeling Method

2020 ◽  
Vol 10 (11) ◽  
pp. 3860
Author(s):  
Song Huang ◽  
Jinxin Cheng ◽  
Chengwu Yang ◽  
Chuangxin Zhou ◽  
Shengfeng Zhao ◽  
...  
Keyword(s):  
Flow Field ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Axial Compressor ◽  
High Dimensions ◽  
Peak Efficiency ◽  
Stall Margin ◽  
Second Stage ◽  
Highly Loaded ◽  
Three Dimensional Space

Due to the complexity of the internal flow field of compressors, the aerodynamic design and optimization of a highly loaded axial compressor with high performance still have three problems, which are rich engineering design experience, high dimensions, and time-consuming calculations. To overcome these three problems, this paper takes an engineering-designed 2.5-stage highly loaded axial flow compressor as an example to introduce the design process and the adopted design philosophies. Then, this paper verifies the numerical method of computational fluid dynamics. A new Bezier surface modeling method for the entire suction surface and pressure surface of blades is developed, and the multi-island genetic algorithm is directly used for further optimization. Only 32 optimization variables are used to optimize the rotors and stators of the compressor, which greatly overcome the problem of high dimensions, time-consuming calculations, and smooth blade surfaces. After optimization, compared with the original compressor, the peak efficiency is still improved by 0.12%, and the stall margin is increased by 2.69%. The increase in peak efficiency is mainly due to the rotors. Compared with the original compressor, for the second-stage rotor, the adiabatic efficiency is improved by about 0.4%, which is mainly due to the decreases of total pressure losses in the range of above 30% of the span height and 10%–30% of the chord length. Besides, for the original compressor, due to deterioration of the flow field near the tip region of the second-stage stator, the large low-speed region eventually evolves from corner separation into corner stall with three-dimensional space spiral backflow. For the optimized compressor, the main reason for the increased stall margin is that the flow field of the second-stage stator with a span height above 50% is improved, and the separation area and three-dimensional space spiral backflow are reduced.

Improved Approach to the Streamline Curvature Method in Turbomachinery

1987 ◽  
Vol 109 (3) ◽  
pp. 213-217 ◽  
Author(s):  
S. Abdallah ◽  
R. E. Henderson
Keyword(s):  
Flow Field ◽  
Velocity Gradient ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Second Order ◽  
Curvilinear Coordinates ◽  
First Order ◽  
Streamline Curvature ◽  
Stators And Rotors ◽  
Streamline Curvature Method

Quasi three dimensional blade-to-blade solutions for stators and rotors of turbomachines are obtained using the Streamline Curvature Method (SLCM). The first-order velocity gradient equation of the SLCM, traditionally solved for the velocity field, is reformulated as a second-order elliptic differential equation and employed in tracing the streamtubes throughout the flow field. The equation of continuity is then used to calculate the velocity. The present method has the following advantages. First, it preserves the ellipticity of the flow field in the solution of the second-order velocity gradient equation. Second, it eliminates the need for curve fitting and strong smoothing under-relaxation in the classical SLCM. Third, the prediction of the stagnation streamlines is a straightforward matter which does not complicate the present procedure. Finally, body-fitted curvilinear coordinates (streamlines and orthogonals or quasi-orthogonals) are naturally generated in the method. Numerical solutions are obtained for inviscid incompressible flow in rotating and non-rotating passages and the results are compared with experimental data.

scholarly journals Simpson’s Second-Type Inequalities for Co-Ordinated Convex Functions and Applications for Cubature Formulas

2022 ◽  
Vol 6 (1) ◽  
pp. 33
Author(s):  
Sabah Iftikhar ◽  
Samet Erden ◽  
Muhammad Aamir Ali ◽  
Jamel Baili ◽  
Hijaz Ahmad
Keyword(s):  
Convex Functions ◽  
Cubature Formula ◽  
Applied Mathematics ◽  
Second Order ◽  
Absolute Value ◽  
Partial Derivatives ◽  
Cubature Formulas ◽  
Inequality Theory ◽  
Type Integral ◽  
Derivatives Of

Inequality theory has attracted considerable attention from scientists because it can be used in many fields. In particular, Hermite–Hadamard and Simpson inequalities based on convex functions have become a cornerstone in pure and applied mathematics. We deal with Simpson’s second-type inequalities based on coordinated convex functions in this work. In this paper, we first introduce Simpson’s second-type integral inequalities for two-variable functions whose second-order partial derivatives in modulus are convex on the coordinates. In addition, similar results are acquired by considering that powers of the absolute value of second-order partial derivatives of these two-variable functions are convex on the coordinates. Finally, some applications for Simpson’s 3/8 cubature formula are given.

A Three-Dimensional Numerical Method for Turbomachinery Blading

1992 ◽  
Author(s):  
C. W. Gu ◽  
J. Z. Xu ◽  
J. Y. Du
Keyword(s):  
Boundary Conditions ◽  
Numerical Method ◽  
Stream Function ◽  
Three Dimensional ◽  
Computational Results ◽  
Three Dimensional Flow ◽  
Dimensional Flow ◽  
Stream Functions ◽  
Derivatives Of ◽  
Convergent Solution

By inversing one of the stream functions and their principal equations in a three–dimensional flow the equations with the second–order partial derivatives of both the coordinate and another stream function are derived. The corresponding boundary conditions are easily specified. Based on these equations and the boundary conditions the convergent solution for turbomachinery blading is obtained. The computational results show that the method is simple and effective.

Impact of Sweep on Part Speed Performance of an Axial Compressor Rotor With Circumferential Casing Grooves

2019 ◽  
Author(s):  
Shraman Goswami ◽  
M. Govardhan
Keyword(s):  
Flow Field ◽  
High Performance ◽  
Three Dimensional ◽  
Axial Compressor ◽  
Field Investigation ◽  
Performance Comparison ◽  
Rotor Blades ◽  
Stall Margin ◽  
Compressor Rotor ◽  
Design Speed

Abstract High performance and increased operating range of an axial compressor is obtained by employing three-dimensional design features, such as sweep, as well as shroud casing treatments, such as circumferential casing grooves. A number of different rotor blades with different amounts of sweeps and different sweep starting spans are studied at design speed. Different swept rotors, including zero sweep, are derived from Rotor37 rotor geometry. In the current study the best performing rotor with sweep is analyzed at part speed. The analyses were done for baseline rotor, devoid of any sweep, and with and without circumferential casing grooves. A detailed flow field investigation and performance comparison is presented to understand the changes in flow field at part speed. It is found that that at 100% design speed, stall margin improvement is achived by both sweep and casing grooves, but at 90% speed improvement in stall margin due to sacing groove is very minimal over and above the gain due to sweep. It is also noticed that due to reduced shock loss efficiency is higher at 90% speed than at 100% speed.

Computation of the Flow Field of Transonic Axial Compressor by Steady Arbitrary Lagrangian Eulerian Formulation

2008 ◽  
Author(s):  
Adel Ghenaiet ◽  
Nouredine Djeghri
Keyword(s):  
Flow Field ◽  
Three Dimensional ◽  
Axial Compressor ◽  
Arbitrary Lagrangian Eulerian ◽  
Viscous Effects ◽  
Stability Range ◽  
Transonic Compressor ◽  
Ale Formulation ◽  
Transonic Axial Compressor ◽  
Residual Smoothing

This paper presents a multi-block solver dealing with an inviscid three dimensional compressible flow through a transonic compressor blading. For efficient computations of the 3D time dependant Euler equations, this solver that we have developed has been cast within a stationary ALE ‘Arbitrary Lagrangian Eulerian’. The main contribution of this paper is by consolidating this ALE formulation, to alleviate the shortcomings linked to rotation effects and the mixed relative subsonic–supersonic inlet flow conditions, which are now simply implemented through an absolute subsonic flow velocity. The finite volume based solver is using the central differencing scheme known as JST (Jameson-Schmidt-Turkel). The explicit multistage Runge-Kutta algorithm is used as a pseudo time marching to the steady-state, coupled with two convergence accelerating techniques; the variable local time-stepping and the implicit residual smoothing procedure. The adaptive implicit residual smoothing has extended the stability range of this explicit scheme, and proved to be successful in accelerating the rate of convergence. This code is currently being extended to include viscous effects, where fluxes are discretized based on Green’s theorem. To support this solver, an H type grid generator based on algebraic and elliptic methods has been developed. The segmentation of the complete domain into smaller blocks has provided full topological and geometrical flexibilities. The code was used to compute the flow field of a transonic axial compressor NASA rotor 37, and comparisons between the calculations and some available experimental data under the design speed and part speed, show qualitatively good agreement.

Measurement of the Three-Dimensional Tip Region Flow Field in an Axial Compressor

1993 ◽  
Vol 115 (3) ◽  
pp. 468-476 ◽  
Author(s):  
R. C. Stauter
Keyword(s):  
Flow Field ◽  
Three Dimensional ◽  
Axial Compressor ◽  
Time Resolved ◽  
Tip Leakage Vortex ◽  
Tip Leakage ◽  
Rotating Blade ◽  
Flow Phenomena ◽  
Component Velocity ◽  
And Behavior

A two-color, five-beam LDV system has been configured to make simultaneous three-component velocity measurements of the flow field in a two-stage axial compressor model. The system has been used to make time-resolved measurements both between compressor blade rows and within the rotating blade passages in an axial compressor. The data show the nature and behavior of the complex, three-dimensional flow phenomena present in the tip region of a compressor as they convect downstream. In particular, the nature of the tip leakage vortex is apparent, being manifested by high blockage as well as the expected vortical motion. The data indicate that the radial flows associated with the tip leakage vortex begin to decrease while within the rotor passage, and that they temporarily increase aft of the passage.

scholarly journals Second Order Perturbations of Elliptic Elements with Respect to the Initial Ones

1999 ◽  
Vol 172 ◽  
pp. 453-454
Author(s):  
F.J. Marco Castillo ◽  
M.J. Martínez Usó ◽  
J.A. López Ortí
Keyword(s):  
Quadratic Form ◽  
Second Order ◽  
Initial Instant ◽  
Orbital Elements ◽  
Partial Derivatives ◽  
Orbital Parameters ◽  
The Matrix ◽  
The Individual ◽  
Derivatives Of

AbstractThe following paper is devoted to the theoretical exposition of the obtention of second order perturbations of elliptic elements and is a follow-up of previous papers (Marco et al., 1996; Marco et al., 1997) where the hypothesis was made that the matrix of the partial derivatives of the orbital elements with respect to the initial ones is the identity matrix at the initial instant only. So, we must compute them through the integration of Lagrange planetary equations and their partial derivatives.Such developments have been applied to the individual corrections of orbits together with the correction of the reference system through the minimization of a quadratic form obtained from the linearized residual. In this state two new targets emerged: 1.To be sure that the most suitable quadratic form was to be considered.2.To provide a wider vision of the behavior of the different orbital parameters in time.Both aims may be accomplished through the consideration of the second order partial derivatives of the elliptic orbital elements with respect to the initial ones.

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