Pompa çarkları içindeki akışın sayısal çözümlenmesi

Abstract

Numerical solution is done on a finite mesh and incoming and outgoing waves cross the boundary of the mesh. In order to have well-posed problem appropriate boundary conditions must be specified. The numerical treatment of boundary conditions along the boundaries in the physical domain is one of the problem in solving the Euler equations. Their improper implementation can result inaccurate results. A major problem involved in solving incompressible Euler equations comes from the lack of a pressure term in the continuity equation. The techniques used for the solution of the incompressible equations can be classified as follows (LAKSHIMINARAYANA, 1990). Pressure Based Methods : In this technique the assumed pressure is updated using an auxiliary equation for pressure. And this equation is solved together with momentum equations in an uncoupled manner. The equations are iterated until a divergence free flow field is satisfied. This procedure usually requires a relaxation scheme. Convergence slows down because of the uncoupled approach. Fractional Step Method : This method is used for the time dependent computations of the incompressible Euler equations. The time evaluation is approximated by several steps. The common application of this method is done by two steps. The first step is to solve for an auxiliary velocity field using the momentum equation in which the pressure-gradient term can be computed from the pressure in the previous time step. In the second step, the pressure is computed which will map the auxiliary velocity onto a divergence-free velocity field. Pseudo-compressibility Method : Recent advances and most of the studies in computational fluid dynamics have been made in compressible flow equations. Therefore many algorithms has been developed for compressible equations. Instead of using Poisson equation for pressure artificial compressibility approach suggested by CHORIN (1967) is used. In this formulation which is also used in this study, the continuity equation is modified by adding a time derivative of the pressure term. This is done to preserve the hyperbolic character of the equation. Together with the unsteady momentum equations, this forms a hyperbolic type of time-dependent equations. NUMERICAL ANALYSIS OF THE FLOW E The numerical analysis of the flow in radial pump impeller is not studied properly. There are a few studies in the literature. In this study it is attempted to solve steady flow field in radial pump impeller, by: using time-dependent incompressible Euler equations with the assumption of pseudo-compressibility. The flow models used in this study as follows. : ' Two Dimensional Incompressible Flow : The two dimensional incompressible time- dependent Euler equations written in conservative form has been used. This section has been done to test the limits of pseudo-compressibility approach by using computer calculations. Several two-dimensional test cases where experimental results are available has been solved numerically.. : ? ' ' The governing equations by using a Cartesian coordinate system are as follows.

Since no shocks are expected in centrifugal pump impeller the second order dissipation terms is not used. To verify the three dimensional flow assumption, flow in a 90° elbow which has rectangular cross section has been solved. Free vortex distribution obtained from numerical calculations has been compared with theoretical calculations. Secondly, the impeller where experiments has been done (MINER, 1988) has been solved. Lastly, an industrial pump impeller used to verify the code. The results obtained from numerical calculations has been interpreted. An experimental performance curve given by the company has been compared with the performance curve using numerical procedure. The results are couraging. As a result, using the computer programs described above the characteristics of the radial pump impellers can be predicted satisfactorily. This helps to the designer to improve the efficiency and performance with the reduced cost and time. xx Numerical solution is done on a finite mesh and incoming and outgoing waves cross the boundary of the mesh. In order to have well-posed problem appropriate boundary conditions must be specified. The numerical treatment of boundary conditions along the boundaries in the physical domain is one of the problem in solving the Euler equations. Their improper implementation can result inaccurate results. A major problem involved in solving incompressible Euler equations comes from the lack of a pressure term in the continuity equation. The techniques used for the solution of the incompressible equations can be classified as follows (LAKSHIMINARAYANA, 1990). Pressure Based Methods : In this technique the assumed pressure is updated using an auxiliary equation for pressure. And this equation is solved together with momentum equations in an uncoupled manner. The equations are iterated until a divergence free flow field is satisfied. This procedure usually requires a relaxation scheme. Convergence slows down because of the uncoupled approach. Fractional Step Method : This method is used for the time dependent computations of the incompressible Euler equations. The time evaluation is approximated by several steps. The common application of this method is done by two steps. The first step is to solve for an auxiliary velocity field using the momentum equation in which the pressure-gradient term can be computed from the pressure in the previous time step. In the second step, the pressure is computed which will map the auxiliary velocity onto a divergence-free velocity field. Pseudo-compressibility Method : Recent advances and most of the studies in computational fluid dynamics have been made in compressible flow equations. Therefore many algorithms has been developed for compressible equations. Instead of using Poisson equation for pressure artificial compressibility approach suggested by CHORIN (1967) is used. In this formulation which is also used in this study, the continuity equation is modified by adding a time derivative of the pressure term. This is done to preserve the hyperbolic character of the equation. Together with the unsteady momentum equations, this forms a hyperbolic type of time-dependent equations. NUMERICAL ANALYSIS OF THE FLOW E The numerical analysis of the flow in radial pump impeller is not studied properly. There are a few studies in the literature. In this study it is attempted to solve steady flow field in radial pump impeller, by: using time-dependent incompressible Euler equations with the assumption of pseudo-compressibility. The flow models used in this study as follows. : ' Two Dimensional Incompressible Flow : The two dimensional incompressible time- dependent Euler equations written in conservative form has been used. This section has been done to test the limits of pseudo-compressibility approach by using computer calculations. Several two-dimensional test cases where experimental results are available has been solved numerically.. : ? ' ' The governing equations by using a Cartesian coordinate system are as follows. xvi where experimental measurements done by LIU (1994) has been solved and compared with experimental results. Lastly the impeller, obtained from industry, having the shape of twisted blades solved numerically. It can be said that the code developed for quasi three dimensional flow can predict the performance of the impeller accurately without consuming time. Three Dimensional Flow : In this part of this study full geometry of the impeller is considered. Then the flow pattern from hub to shroud also can be investigated. Secondary flows and flow separation especially at suction side of the shroud in radial impellers can be predicted. The foil 3-D of Euler system of equations written in a conservative form are used. The equations are written in a cylindrical system of coordinates (r, 8, z), fixed to the blade row which rotates at a constant angular velocity around z axis. The unknown variables, which has to be solved, are the components of relative velocities (Wr, We, Wz) and (p) static pressure. Energy equation is not included as explained before. To achieve the steady state solution time marching technique is applied. Pseudo-compressibility technique, further developed by RIZZI and ERIKSSON (1985) has been applied to continuity equation to preserve the hyperbolicity. The governing equations are shown below. 5Ü H/;\ 1 d,^s d (r\ r n - + irf + (g) + - h +b = 0 at rdr^ ' rö0VÖ' dz^ ' (4) u = f = Semi-discretization is applied to governing equations. Finite volume technique and fourth order Runge-Kutta time stepping scheme is applied to the equations like previous parts. Since the solution is performed between two blades, periodic boundary conditions are applied in outlet and inlet part of the impeller channel. In order to preserve the conservative form of the numerical scheme the artificial dissipation terms are introduced by adding dissipative fluxes to the integral form of the Euler equations. xix Since no shocks are expected in centrifugal pump impeller the second order dissipation terms is not used. To verify the three dimensional flow assumption, flow in a 90° elbow which has rectangular cross section has been solved. Free vortex distribution obtained from numerical calculations has been compared with theoretical calculations. Secondly, the impeller where experiments has been done (MINER, 1988) has been solved. Lastly, an industrial pump impeller used to verify the code. The results obtained from numerical calculations has been interpreted. An experimental performance curve given by the company has been compared with the performance curve using numerical procedure. The results are couraging. As a result, using the computer programs described above the characteristics of the radial pump impellers can be predicted satisfactorily. This helps to the designer to improve the efficiency and performance with the reduced cost and time. xx where experimental measurements done by LIU (1994) has been solved and compared with experimental results. Lastly the impeller, obtained from industry, having the shape of twisted blades solved numerically. It can be said that the code developed for quasi three dimensional flow can predict the performance of the impeller accurately without consuming time. Three Dimensional Flow : In this part of this study full geometry of the impeller is considered. Then the flow pattern from hub to shroud also can be investigated. Secondary flows and flow separation especially at suction side of the shroud in radial impellers can be predicted. The foil 3-D of Euler system of equations written in a conservative form are used. The equations are written in a cylindrical system of coordinates (r, 8, z), fixed to the blade row which rotates at a constant angular velocity around z axis. The unknown variables, which has to be solved, are the components of relative velocities (Wr, We, Wz) and (p) static pressure. Energy equation is not included as explained before. To achieve the steady state solution time marching technique is applied. Pseudo-compressibility technique, further developed by RIZZI and ERIKSSON (1985) has been applied to continuity equation to preserve the hyperbolicity. The governing equations are shown below. 5Ü H/;\ 1 d,^s d (r\ r n - + irf + (g) + - h +b = 0 at rdr^ ' rö0VÖ' dz^ ' (4) u = f = Semi-discretization is applied to governing equations. Finite volume technique and fourth order Runge-Kutta time stepping scheme is applied to the equations like previous parts. Since the solution is performed between two blades, periodic boundary conditions are applied in outlet and inlet part of the impeller channel. In order to preserve the conservative form of the numerical scheme the artificial dissipation terms are introduced by adding dissipative fluxes to the integral form of the Euler equations. xix Since no shocks are expected in centrifugal pump impeller the second order dissipation terms is not used. To verify the three dimensional flow assumption, flow in a 90° elbow which has rectangular cross section has been solved. Free vortex distribution obtained from numerical calculations has been compared with theoretical calculations. Secondly, the impeller where experiments has been done (MINER, 1988) has been solved. Lastly, an industrial pump impeller used to verify the code. The results obtained from numerical calculations has been interpreted. An experimental performance curve given by the company has been compared with the performance curve using numerical procedure. The results are couraging. As a result, using the computer programs described above the characteristics of the radial pump impellers can be predicted satisfactorily. This helps to the designer to improve the efficiency and performance with the reduced cost and time.