Boundary element method for internal axisymmetric flow

Alexander Gokhman,Dmitry Gokhman

doi:10.1155/s1024123x9900099x

Alexander Gokhman, Dmitry Gokhman

Open Access

https://doi.org/10.1155/s1024123x9900099x

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Abstract

We present an accurate fast method for the computation of potential internal axisymmetric flow based on the boundary element technique. We prove that the computed velocity field asymptotically satisfies reasonable boundary conditions at infinity for various types of inlet/exit. Computation of internal axisymmetric potential flow is an essential ingredient in the three-dimensional problem of computation of velocity fields in turbomachines. We include the results of a practical application of the method to the computation of flow in turbomachines of Kaplan and Francis types.

Highlights

The intrinsic three dimensional problem for internal potential incompressible flows past objects in axisymmetric passages, such as flows past blades of turbomachines, can be solved by superposition of several flows: (a) axisymmetric flow in the passage; (b) constant whirl flow in the passage; (c) flow induced by vortex filaments on the blades; (d) flow induced by distributed sources on the boundary that compensates for the nonzero normal components of velocity at the boundary of flows (a), (b), and (c)
The use of curvilinear coordinates and closed form integration was shown to provide significantly higher accuracy than comparable ordinary Cartesian finite element schemes and can be applied to viscous laminar or turbulent flows using a generalization of the Galerkin method formulated in [2]
These expressions are used with various values of the coefficients a and b

Summary

Introduction

The finite element scheme was formulated using curvilinear coordinates that follow the boundary and closed form integration was used to compute the Galerkin integrals. The use of curvilinear coordinates and closed form integration was shown to provide significantly higher accuracy than comparable ordinary Cartesian finite element schemes and can be applied to viscous laminar or turbulent flows using a generalization of the Galerkin method formulated in [2]. In [1] a boundary element method was formulated for internal potential axisymmetric incompressible flow in a passage with a cylindrical draft tube and applied to the simple problem of external flow past a sphere. The boundary elements and the vorticity distribution were assumed to be piecewise linear and the boundary condition that the normal velocity is zero (vn = 0) was imposed at the midpoints of the elements. It has been used to obtain curvilinear coordinates for the method in [3]

Formulation of the Problem

Integral equation for vorticity on the boundary

Induced velocity

Boundary conditions and the integral equation for vorticity

A priori conditions on vorticity

Computational approach

Boundary elements not incident to the test point

A pair of boundary elements incident to the test point

Improper integrals at the inlet and exit

Application