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.

Quasi-Three-Dimensional and Full Three-Dimensional Rotational Flow Calculations in Turbomachines

1985 ◽  
Vol 107 (2) ◽  
pp. 277-284 ◽  
Author(s):  
Qinghuan Wang ◽  
Genxing Zhu ◽  
Chung-Hua Wu
Keyword(s):  
Flow Field ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Relaxation Method ◽  
Computing System ◽  
Curvilinear Coordinates ◽  
Rotational Flow ◽  
Three Dimensional Flow ◽  
Dimensional Solution ◽  
Dimensional Flow

Progress in the development of quasi-three-dimensional and full three-dimensional numerical solutions for steady subsonic rotational flow through turbomachines is presented. An iterative calculation between the flow on a mean hub-to-tip S2 stream surface and a number of blade-to-blade S1 stream surfaces gives the quasi-three-dimensional solution, which is very easily extended to give full three-dimensional solution by merely calculating a few more S2 surface flows and relaxing the restriction that S1 surfaces are surfaces of revolution. A new S2–S1 iteration scheme has been developed and employed in the present code. The governing equations on the S1 and S2 surfaces are expressed in terms of general nonorthogonal curvilinear coordinates so that they are body-fitted without any coordinate transformation and are solved by either matrix method or line-relaxation method. An automatic computing system is used, which first computes the quasi-three-dimensional flow for blade design and then computes the full three-dimensional flow for the blade row just designed. The results obtained by applying this computing system to the design and determination of full three-dimensional flow field of a two-stage axial compressor and a high subsonic compressor stator are obtained and shows clearly the amount of the twist of the general S1 surfaces and the difference in the flow field between the quasi-three-dimensional and full three-dimensional solutions.

scholarly journals Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Darae Jeong ◽  
Yibao Li ◽  
Chaeyoung Lee ◽  
Junxiang Yang ◽  
Yongho Choi ◽  
...  
Keyword(s):  
Parabolic Equations ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
Second Order ◽  
Order Convergence ◽  
Numerical Convergence ◽  
Verification Method ◽  
First Order ◽  
Convergence Test ◽  
Second Order Convergence

In this paper, we propose a verification method for the convergence rates of the numerical solutions for parabolic equations. Specifically, we consider the numerical convergence rates of the heat equation, the Allen–Cahn equation, and the Cahn–Hilliard equation. Convergence test results show that if we refine the spatial and temporal steps at the same time, then we have the second-order convergence rate for the second-order scheme. However, in the case of the first-order in time and the second-order in space scheme, we may have the first-order or the second-order convergence rates depending on starting spatial and temporal step sizes. Therefore, for a rigorous numerical convergence test, we need to perform the spatial and the temporal convergence tests separately.

scholarly journals A Partial Lagrangian Approach to Mathematical Models of Epidemiology

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
R. Naz ◽  
I. Naeem ◽  
F. M. Mahomed
Keyword(s):  
Mathematical Models ◽  
Exact Solutions ◽  
Numerical Solutions ◽  
First Integrals ◽  
Order Equation ◽  
Second Order ◽  
Lagrangian Approach ◽  
Hiv Model ◽  
First Order ◽  
Partial Lagrangian

This paper analyzes the first integrals and exact solutions of mathematical models of epidemiology via the partial Lagrangian approach by replacing the three first-order nonlinear ordinary differential equations by an equivalent system containing one second-order equation and a first-order equation. The partial Lagrangian approach is then utilized for the second-order ODE to construct the first integrals of the underlying system. We investigate the SIR and HIV models. We obtain two first integrals for the SIR model with and without demographic growth. For the HIV model without demography, five first integrals are established and two first integrals are deduced for the HIV model with demography. Then we utilize the derived first integrals to construct exact solutions to the models under investigation. The dynamic properties of these models are studied too. Numerical solutions are derived for SIR models by finite difference method and are compared with exact solutions.

Flow Analysis in Francis Water Turbines

1967 ◽  
Vol 89 (3) ◽  
pp. 445-451
Author(s):  
W. Jansen
Keyword(s):  
Flow Field ◽  
Flow Analysis ◽  
Analytical Techniques ◽  
Small Drop ◽  
Streamline Curvature ◽  
Underlying Theory ◽  
Upper Limits ◽  
Water Turbines ◽  
Flow Reversals ◽  
Streamline Curvature Method

It is the intent of this study to investigate the flow field in Francis water turbines operated at various guidevane settings with the use of analytical techniques. The flow fluid is analyzed with a streamline-curvature method and the underlying theory is discussed briefly. With the analysis, certain lower and upper limits for the guidevane settings can be defined. Within these limits only a small drop-off in efficiency at the extreme settings can be expected. Outside these limits severe cavitation and flow reversals are shown to take place. In the Appendix, it is shown how the analysis can be used to evaluate the effects of different passage and blade geometry on the flow field.

Application of an Improved Streamline Curvature Approach to a Modern, Two-Stage Transonic Fan: Comparison With Data and CFD

2002 ◽  
Author(s):  
Keith M. Boyer ◽  
Walter F. O’Brien
Keyword(s):  
Three Dimensional ◽  
Operating Conditions ◽  
Navier Stokes ◽  
Model Parameters ◽  
Two Stage ◽  
Streamline Curvature ◽  
Wide Range ◽  
Improved Model ◽  
Transonic Fan ◽  
Streamline Curvature Method

A streamline curvature method with improvements to key loss models is applied to a two-stage, low aspect ratio, transonic fan with design tip relative Mach number of approximately 1.65. Central to the improvements is the incorporation of a physics-based shock model. The attempt here is to capture the effects of key flow phenomena relative to the off-design performance of the fan. A quantitative analysis regarding solution sensitivities to model parameters that influence the key phenomena over a wide range of operating conditions is presented. Predictions are compared to performance determined from overall and interstage measurements, as well as from a three-dimensional, steady, Reynolds-averaged Navier-Stokes method applied across the first rotor. Overall and spanwise comparisons demonstrate that the improved model gives reasonable performance trending and generally accurate results. The method can be used to provide boundary conditions to higher-order solvers, or implemented within novel approaches using the streamline curvature method to explore complex engine-inlet integration issues, such as time-variant distortion.

scholarly journals A procedure for the calculation of flow through axisymmetric ducts

1983 ◽  
Vol 2 (2) ◽  
pp. 92-103
Author(s):  
P. S. Heyns
Keyword(s):  
Flow Field ◽  
Flow Direction ◽  
Governing Equations ◽  
Streamline Curvature ◽  
Flow Through ◽  
Streamline Curvature Method

A procedure for the calculation of flow through axisymmetric ducts as are typically found in turbomachines, is presented. The procedure is based on a streamline curvature method with the governing equations formulated along quasi-orthogonals in the flow field. This formulation allows the procedure to be used for segments of a duct wherein the flow direction is predominantly radial. It is assumed that the flow on specific stream surfaces is isentropic, but normal entropy gradients may exist because of processes which took place upstream of the duct.

The Consistent Second-Order Wave Theory and Its Application to a Submerged Spheroid

1970 ◽  
Vol 14 (01) ◽  
pp. 23-50
Author(s):  
Young H. Chey
Keyword(s):  
Free Surface ◽  
Solid Wall ◽  
Three Dimensional ◽  
Wave Theory ◽  
Order Theory ◽  
Wave Resistance ◽  
Second Order ◽  
Surface Phenomena ◽  
First Order ◽  
Theoretical Results

Because of the recognized inadequacy of first-order linearized surface-wave theory, the author has developed, for a three-dimensional body, a new second-order theory which provides a better description of free-surface phenomena. The new theory more accurately satisfies the kinematic boundary condition on the solid wall, and takes into account the nonlinearity of the condition at the free surface. The author applies the new theory to a submerged spheroid, to calculate wave resistance. Experiments were conducted to verify the theory, and their results are compared with the theoretical results. The comparison indicates that the use of the new theory leads to more accurate prediction of wave resistance.

An Analysis Method for Multistage Transonic Turbines With Coolant Mass Flow Addition

1998 ◽  
Vol 120 (4) ◽  
pp. 744-752 ◽  
Author(s):  
F. Mildner ◽  
H. E. Gallus
Keyword(s):  
Flow Field ◽  
Mass Flow ◽  
Three Dimensional ◽  
Iterative Solution ◽  
Curvilinear Coordinates ◽  
Main Stream ◽  
Viscous Effects ◽  
Air Mixing ◽  
Volume Method ◽  
Cooling Air

The subject of this paper is a numerical method for the calculation of the transonic flow field of multistage turbines, taking high coolant flow into account. To reduce the processing time, a throughflow method based on the principels of Wu is used for the hub-to-tip calculation. The flow field is obtained by an iterative solution between a three-dimensional inviscid hyperbolic time-dependent algorithm with an implicit finite volume method for the blade-to-blade calculations using C-meshes and a single representative meridional S2m-streamsurface. Along the S2m-plane with respect to nonorthogonal curvilinear coordinates, the stream function equation governing fluid flow is established. The cooling air inflow inside the blade passage forbids the assumption of a constant mass flow along the main stream direction. To consider the change of the aerodynamic and thermodynamic behavior, a cooling air model was developed and implemented in the algorithm, which allows the mixing of radially arbitrarily distributed cooling air in the trailing edge section of each blade row. The viscous effects and the influence of cooling air mixing are considered by the use of selected loss correlations for profile, tip leakage, secondary flow and mixing losses in the S2m-plane in terms of entropy. The method is applied to the four-stage high-temperature gas turbine Siemens KWU V84.3. The numerical results obtained are in good agreement with the experimental data.

The Relative Merits of an Inviscid Euler 3-D and Quasi-3-D Analysis for the Design of Transonic Rotors

1988 ◽  
Author(s):  
D. P. Miller ◽  
A. C. Bryans
Keyword(s):  
Flow Field ◽  
Numerical Solutions ◽  
Axisymmetric Flow ◽  
Three Dimensional ◽  
Viscous Effects ◽  
Blade Loading ◽  
Rotor Design ◽  
Accurate Indication ◽  
Through Flow ◽  
Transonic Rotor

It is the purpose of this paper to examine the flow fields in an advanced modern transonic rotor design using both axisymmetric and three dimensional techniques. Also, to determine the deviation of the axisymmetric flow from three-dimensional flow field and whether this seriously affects the results. Inviscid Euler solvers are now widely used to analyze transonic flows through turbomachines giving a reasonably accurate indication of the flow field in blade passages. Although viscous effects are important, the inviscid analysis provides significant knowledge of the flow field which is essential to transonic design. The blade-to-blade loading and work distributions are determined quite realistically by the 3-D and quasi-3-D inviscid analyses. Through-flow and blade-to-blade inviscid solutions are presented for a highly loaded transonic rotor. Numerical solutions for various transonic rotor designs operating at peak efficiency are also compared with test data.

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