scholarly journals A Stream Function Relaxation Method for Solving Transonic S1 Stream Surface With Pre-Determination of the Density

1985 ◽  
Author(s):  
Ge Manchu
Keyword(s):  
Difference Scheme ◽  
Stream Function ◽  
Flow Theory ◽  
Relaxation Method ◽  
Curvilinear Coordinates ◽  
Transonic Flows ◽  
Stream Surface ◽  
Numerical Examples ◽  
Orthogonal Curvilinear Coordinates

On the basis of Prof. Wu’s 3-D flow theory (ref.1, 2, 3, 4, 5), a general streamfunction equation in non-orthogonal curvilinear coordinates is developed. The equation can be used to calculate subsonic or transonic flows on S1 or S2 stream surfaces of turbomachinery. In this paper streamlines coordinates and a mixed difference scheme are adopted in solving the stream function equation. A procedure for pre-determination of the density is developed and used to determine the unique-value of density from the known value of the stream function. Numerical examples are given.

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.

scholarly journals A high-precision algorithm for axisymmetric flow

1995 ◽  
Vol 1 (1) ◽  
pp. 11-25
Author(s):  
A. Gokhman ◽  
D. Gokhman
Keyword(s):  
Velocity Field ◽  
Boundary Conditions ◽  
Stream Function ◽  
Potential Flow ◽  
Axisymmetric Flow ◽  
Curvilinear Coordinates ◽  
Shape Functions ◽  
Quadrilateral Elements ◽  
Orthogonal Curvilinear Coordinates ◽  
Double Integrals

We present a new algorithm for highly accurate computation of axisymmetric potential flow. The principal feature of the algorithm is the use of orthogonal curvilinear coordinates. These coordinates are used to write down the equations and to specify quadrilateral elements following the boundary. In particular, boundary conditions for the Stokes' stream-function are satisfied exactly. The velocity field is determined by differentiating the stream-function. We avoid the use of quadratures in the evaluation of Galerkin integrals, and instead use splining of the boundaries of elements to take the double integrals of the shape functions in closed form. This is very accurate and not time consuming.

scholarly journals Analytical solution for the stationary model of pollutant propagation in an aquatic medium

2019 ◽  
Vol 14 (2) ◽  
pp. 1
Author(s):  
Cíntia Ourique Monticelli ◽  
Jorge Rodolfo Zabadal ◽  
Daniela Muller Quevedo ◽  
Carlos Augusto Nascimento
Keyword(s):  
Stream Function ◽  
Concentration Distribution ◽  
Pollutant Dispersion ◽  
Curvilinear Coordinates ◽  
Qualitative Behavior ◽  
Inviscid Flows ◽  
Advection Diffusion Equation ◽  
Thermotolerant Coliforms ◽  
Orthogonal Curvilinear Coordinates ◽  
Pollutant Propagation

This work presents a new analytical approach for solving pollutant dispersion problems along irregular-shaped water bodies. In this approach, the advection-diffusion equation is expressed in terms of orthogonal curvilinear coordinates, defined by the velocity potential and the corresponding stream function for inviscid flows. The boundary condition rewritten in terms of these new coordinates is reduced to a classical third kind one, i.e., the derivative of the concentration distribution with respect to the stream function is proportional to the numerical value of the local pollutant concentration. The solution obtained from the proposed formulation was employed to simulate pollutant dispersion (thermotolerant coliforms) along the Pampa Creek, a tributary of the Sinos River at the outskirts of Novo Hamburgo city, South Region of Brazil. The results obtained reproduce the qualitative behavior of the expected concentration distribution.

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.

Difference Schemes for Nonlinear BVPS on the Semiaxis

2007 ◽  
Vol 7 (1) ◽  
pp. 25-47 ◽  
Author(s):  
I.P. Gavrilyuk ◽  
M. Hermann ◽  
M.V. Kutniv ◽  
V.L. Makarov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Boundary Value Problem ◽  
Difference Scheme ◽  
Adaptive Algorithm ◽  
Order Differential Equation ◽  
Constructive Method ◽  
Numerical Examples ◽  
Exact Difference ◽  
The Given

Abstract The scalar boundary value problem (BVP) for a nonlinear second order differential equation on the semiaxis is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme whose solution coincides with the projection of the exact solution of the given differential equation onto the underlying grid. A constructive method is proposed to derive from the EDS a so-called truncated difference scheme (n-TDS) of rank n, where n is a freely selectable natural number. The n-TDS is the basis for a new adaptive algorithm which has all the advantages known from the modern IVP-solvers. Numerical examples are given which illustrate the theorems presented in the paper and demonstrate the reliability of the new algorithm.

XVI.—On Triple Systems of Surfaces and Non-Orthogonal Curvilinear Coordinates

1927 ◽  
Vol 46 ◽  
pp. 194-205 ◽  
Author(s):  
C. E. Weatherburn
Keyword(s):  
Curvilinear Coordinates ◽  
Triple Systems ◽  
Orthogonal Curvilinear Coordinates ◽  
Orthogonal Systems ◽  
Considerable Detail

The properties of “triply orthogonal” systems of surfaces have been examined by various writers and in considerable detail; but those of triple systems generally have not hitherto received the same attention. It is the purpose of this paper to discuss non-orthogonal systems, and to investigate formulæ in terms of the “oblique” curvilinear coordinates u, v, w which such a system determines.

Contracting ducts of finite length

1951 ◽  
Vol 2 (4) ◽  
pp. 254-271 ◽  
Author(s):  
L. G. Whitehead ◽  
L. Y. Wu ◽  
M. H. L. Waters
Keyword(s):  
Stream Function ◽  
Finite Length ◽  
Velocity Potential ◽  
Three Dimensional ◽  
Relaxation Method ◽  
Two Dimensional ◽  
Dimensional Flow ◽  
Hodograph Plane ◽  
Two Dimensional Flow ◽  
Working Section

SummmaryA method of design is given for wind tunnel contractions for two-dimensional flow and for flow with axial symmetry. The two-dimensional designs are based on a boundary chosen in the hodograph plane for which the flow is found by the method of images. The three-dimensional method uses the velocity potential and the stream function of the two-dimensional flow as independent variables and the equation for the three-dimensional stream function is solved approximately. The accuracy of the approximate method is checked by comparison with a solution obtained by Southwell's relaxation method.In both the two and the three-dimensional designs the curved wall is of finite length with parallel sections upstream and downstream. The effects of the parallel parts of the channel on the rise of pressure near the wall at the start of the contraction and on the velocity distribution across the working section can therefore be estimated.

scholarly journals Study on Optimal Placement and Reasonable Number of Viscoelastic Dampers by Improved Weight Coefficient Method

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Ji-ting Qu ◽  
Hong-nan Li
Keyword(s):  
Mathematical Model ◽  
Weight Coefficient ◽  
Optimal Placement ◽  
Optimal Method ◽  
Analogy Method ◽  
Numerical Examples ◽  
Viscoelastic Dampers ◽  
Reasonable Number ◽  
Coefficient Method

A new optimal method is presented by combining the weight coefficient with the theory of force analogy method. Firstly, a new mathematical model of location index is proposed, which deals with the determination of a reasonable number of dampers according to values of the location index. Secondly, the optimal locations of dampers are given. It can be specific from stories to spans. Numerical examples are illustrated to verify the effectiveness and feasibility of the proposed mathematical model and optimal method. At last, several significant conclusions are given based on numerical results.

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