Stream Function Solution of Transonic Flow Along S2 Streamsurface of Axial Turbomachines

1986 ◽  
Vol 108 (1) ◽  
pp. 138-143 ◽  
Author(s):  
Xiaolu Zhao
Keyword(s):  
Stream Function ◽  
Transonic Flow ◽  
Fluid Velocity ◽  
Measurement Data ◽  
Curvilinear Coordinates ◽  
Design Problems ◽  
Artificial Compressibility ◽  
Function Solution ◽  
Compressor Rotor ◽  
Design Speed

Based on Wu’s general equations of 3-D turbomachine flow, expressed with respect to nonorthogonal curvilinear coordinates, the conservative stream-function formulations of transonic flow along S2 streamsurface have been discussed. The problem of mixed flow can be solved by the use of the artificial compressibility method, and the passage shock on the S2 streamsurface can be captured. The distribution of the fluid velocity from hub to shroud can be obtained directly by integrating the velocity gradient equation, after the principal equation has been solved, so that the difficulty of the nonuniqueness of density-mass flux relation is avoided. The density is determined after the velocity has been obtained. Two computer programs have been coded; one can be used to compute the hybrid or design problems, the other is suitable to compute the analysis problem. The former has been used to compute the transonic flow field along a mean S2 streamsurface in the DFLVR compressor rotor at design speed. The numerical results agree well with L2F measurement data.

scholarly journals Transonic Flow Along Arbitrary Stream Filament of Revolution Solved by Separate Computations With Shock Fitting

1986 ◽  
Author(s):  
Wang Baoguo ◽  
Hua Yaonan ◽  
Huang Xiaoyan ◽  
Wu Chung-Hua
Keyword(s):  
Flow Field ◽  
Stream Function ◽  
Transonic Flow ◽  
Matrix Method ◽  
Axial Compressor ◽  
Iteration Scheme ◽  
Curvilinear Coordinates ◽  
Algebraic Equations ◽  
Compressor Rotor ◽  
Orthogonal Curvilinear Coordinates

The transonic flow field in a cascade of blades lying on an S1 stream surface of revolution is solved by separate computations in the supersonic and the transonic region. The characteristics method is used to solve the supersonic flow upstream of the passage shock and the direct matrix method is used to solve the transonic flow downstream of the passage shock. The transonic stream-function equation in weak conservative form was discretized with respect to general non-orthogonal curvilinear coordinates. Using the artificial density technique and a new iteration scheme between the stream function and the density, the set of algebraic equations was solved by the direct matrix method. A computer program has been developed and is applied to compute the flow field on several S1 stream surfaces of revolution for the DFVLR transonic axial compressor rotor. It is found that the thickness of the S1 stream filament and the variation of entropy along the streamlines have strong influence on calculation. The calculated result agrees with the experimental data fairly well.

Numerical Solution of Transonic Stream Function Equation on S1 Stream Surface in Cascade

1986 ◽  
Author(s):  
Hua Yaonan ◽  
Wu Wenquan
Keyword(s):  
Stream Function ◽  
Transonic Flow ◽  
Momentum Equation ◽  
Curvilinear Coordinates ◽  
Algebraic Equations ◽  
Central Difference ◽  
Linear Algebraic Equations ◽  
Difference Formula ◽  
Grid Nodes ◽  
Transonic Cascade

A method is presented in this paper for calculating transonic flow field in turbomachinery cascades. With respect to non-orthogoanl curvilinear coordinates, the stream function equation governing fluid flow was established. Using the Artificial Compressibility Method, the discretization of the partial differential equation was carried out by use of the standard central difference formula. The set of linear algebraic equations obtained is solved by means of the Direct Matrix Method. In order to overcome the non-uniqueness of density in transonic flow in the stream function method, the velocities at grid nodes are first obtained by integrating the momentum equation and then the densities are determined from the energy equation. Application of this method to some transonic cascade-flow with supersonic or subsonic inlet velocity shows that the solution obtained is in fair agreement with experimental data.

3D Transonic Potential Flow Computation in an Axial-Flow Compressor Rotor by an Approximate Factorization Scheme

1986 ◽  
Vol 108 (1) ◽  
pp. 112-117
Author(s):  
Jialin Zhang
Keyword(s):  
Flow Field ◽  
Transonic Flow ◽  
Axial Flow ◽  
Curvilinear Coordinates ◽  
Full Potential ◽  
Approximate Factorization ◽  
Factorization Scheme ◽  
Compressor Rotor ◽  
Fully Implicit ◽  
Flow Compressor

A conservative full-potential equation of 3D transonic flow in a turbomachine has been derived with the tensor method and expressed with respect to nonorthogonal curvilinear coordinates, and a fully implicit approximate factorization scheme to calculate the flow field has been developed in this paper. The new algorithm has been used to compute the 3D transonic flow field within an axial-flow single-stage compressor rotor tested by DFVLR. Comparisons between the computed flow field and the DFVLR data have been made. Results demonstrate that fast convergence can be achieved by the presented algorithm and that the agreement with the measurements obtained with an advanced laser velocimeter is quite good.

Solution of Transonic S1 Surface Flow by Successively Reversing the Direction of Integration of the Stream Function Equation

1985 ◽  
Vol 107 (2) ◽  
pp. 317-322 ◽  
Author(s):  
Zhengming Wang
Keyword(s):  
Stream Function ◽  
Integration Method ◽  
Surface Flow ◽  
Curvilinear Coordinates ◽  
Governing Equations ◽  
Rapid Convergence ◽  
Surface Flows ◽  
Stream Surface ◽  
Artificial Compressibility ◽  
Supersonic Inlet

In the solution of the stream function equation on an S1 relative stream surface with transonic velocities, the occurrence of two values of the density is avoided by using a method of combining simple iteration with an integration method. In this method, the direction of integration is successively reversed, i.e., the starting line for the integration is varied from iteration to iteration. The governing equations are therefore satisfied as fully as possible during each iteration, and the procedure leads to rapid convergence. The method uses nonorthogonal curvilinear coordinates and artificial compressibility. The technique can be used to calculate transonic S1 surface flows, with either subsonic or supersonic inlet velocities. Example calculations indicate that the method is very effective.

Lee waves in three-dimensional stratified flow

1984 ◽  
Vol 148 ◽  
pp. 97-108 ◽  
Author(s):  
G. S. Janowitz
Keyword(s):  
Stream Function ◽  
Displacement Field ◽  
Three Dimensional ◽  
Frequency Dispersion ◽  
Vertical Displacement ◽  
Function Solution ◽  
Dispersion Surface ◽  
Phase Calculation ◽  
Lee Waves ◽  
The Green Function

The effect of a shallow isolated topography on a linearly stratified, three-dimensional, initially uniform flow in the x-direction is considered. The Green-function solution for the velocity disturbance due to this topography, which is equivalent to that due to a dipole at the origin, is shown to be without swirl, i.e. the velocity disturbance lies strictly in planes passing through the x-axis. Thus this disturbance can be described in terms of a stream function. The asymptotic forms of the wavelike portion of the stream function and the vertical displacement field are obtained. The latter is in agreement with the limited versions due to Crapper (1959). The Gaussian curvature of the zero-frequency dispersion surface is obtained analytically as a step in the stationary-phase calculation. The model is extended to determine the vertical displacement field for an arbitrary shallow topography far downstream. For topographies that are even functions of x and y it is shown that the details of the topography affect the displacement field only in the vicinity of the x-axis. Elsewhere, the amplitude of the displacement is proportional to the net volume of the topography.

Studying the Blood Plasma Flow past a Red Blood Cell with the Mathematical Method of Kelvin's Transformation

2014 ◽  
Vol 2 (1) ◽  
pp. 57-66
Author(s):  
Maria Hadjinicolaou ◽  
Eleftherios Protopapas
Keyword(s):  
Blood Cell ◽  
Blood Plasma ◽  
Stream Function ◽  
Plasma Flow ◽  
Red Blood Cell ◽  
Fluid Velocity ◽  
Prolate Spheroid ◽  
Mathematical Tool ◽  
Flow Past ◽  
Analytical Expressions

A mathematical tool, namely the Kelvin transformation, has been employed in order to derive analytical expressions for important hydrodynamic quantities, aiming to the understanding and to the study of the blood plasma flow past a Red Blood Cell (RBC). These quantities are the fluid velocity, the drag force exerted on a cell and the drag coefficient. They are obtained by employing the stream function ? which describes the Stokes flow past a fixed cell. The RBC, being a biconcave disk, has been modelled as an inverted prolate spheroid. The stream function is given as a series expansion in terms of Gegenbauer functions, which converge fast. Therefore we employ only the first term of the series in order to derive simple and ready to use analytical expressions. These expressions are important in medicine, for studying, for example the transportation of oxygen, or the drug delivery to solid tumors.

Computation of Potential Flow on S2 Stream Surface for a Transonic Axial-Flow Compressor Rotor

1985 ◽  
Vol 107 (2) ◽  
pp. 323-328 ◽  
Author(s):  
Pan-Ming Lu¨ ◽  
Chung-Hua Wu
Keyword(s):  
Potential Function ◽  
Axial Compressor ◽  
Axial Flow ◽  
Full Potential ◽  
Stream Surface ◽  
Compressor Rotor ◽  
Transonic Axial Compressor ◽  
Design Speed ◽  
Flow Compressor ◽  
Good Agreement

A set of conservative full potential function equations governing the fluid flow along a given S2 streamsurface in a transonic axial compressor rotor was obtained. By the use of artificial density and a potential function/density iteration, this set of equations can be solved, and the passage shock on the S2 streamsurface can be captured. A computer program for this analysis problem has been developed and used to compute the flow field along a mean S2 streamsurface in the DFVLR transonic axial compressor rotor. A comparison of computed results with DFVLR L2F measurement at 100 percent design speed shows fairly good agreement.

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