ALBERT

Description: A potential function/stream function formulation is introduced for the solution of the fully 3-D inverse potential ‘target pressure’ problem. In the companion paper (Part 1) it is seen that the general 3-D inverse problem is ill-posed but accepts as a particular solution elementary streamtubes with orthogonal cross-section. Under this simplification, a novel set of flow equations was derived and discussed. The purpose of the present paper is to present the computational techniques used for the numerical integration of the flow and geometry equations proposed in Part 1. The governing flow equations are discretized with centred finite difference schemes on a staggered grid and solved in their linearized form using the preconditioned GMRES algorithm. The geometry equations which form a set of first-order o.d.e.s are integrated numerically using a second-order-accurate space marching scheme. The resulting computational algorithm is applied to a double turning duct and a 3-D converging-diverging nozzle ‘reproduction’ test case. © 1995, Cambridge University Press. All rights reserved.

Description: An inverse potential methodology is introduced for the solution of the fully 3-D target pressure problem. The method is based on a potential function/stream function formulation, where the physical space is mapped onto a computational one via a body-fitted coordinate transformation. A potential function and two stream vectors are used as the independent natural coordinates, whilst the velocity magnitude, the aspect ratio and the skew angle of the elementary streamtube cross-section are assumed to be the dependent ones. A novel procedure based on differential geometry and generalized tensor analysis arguments is employed to formulate the method. The governing differential equations are derived by requiring the curvature tensor of the flat 3-D physical Eucledian space, expressed in terms of the curvilinear natural coordinates, to be zero. The resulting equations are discussed and investigated with particular emphasis on the existence and uniqueness of their solution. The general 3-D inverse potential problem, with ‘target pressure’ boundary conditions only, seems to be ill-posed accepting multiple solutions. This multiplicity is alleviated by considering elementary streamtubes with orthogonal cross-sections. The assumption of orthogonal stream surfaces reduces the number of dependent variables by one, simplifying the governing equations to an elliptic p.d.e. for the velocity magnitude and to a second-order o.d.e. for the streamtube aspect ratio. The solution of these two equations provides the flow field. Geometry is determined independently by integrating Frenet equations along the natural coordinate lines, after the flow field has been calculated. The numerical implementation as well as validation test cases for the proposed inverse methodology are presented in the companion paper (Paper 2). © 1995, Cambridge University Press. All rights reserved.

Description: A new inverse inviscid method suitable for the design of rotating blade sections lying on an arbitrary axisymmetric stream-surface with varying streamtube width is presented. The geometry of the axisymmetric stream-surface and the streamtube width variation with meridional distance, the number of blades, the inlet flow conditions, the rotational speed and the suction and pressure side velocity distributions as functions of the normalized arc-length are given. The flow is considered irrotational in the absolute frame of reference and compressible. The output of the computation is the blade section that satisfies the above data. The method solves the flow equations on a (phi 1, psi) potential function-streamfunction plane for the velocity modulus, W and the flow angle beta; the blade section shape can then be obtained as part of the physical plane geometry by integrating the flow angle distribution along streamlines. The (phi 1, psi) plane is defined so that the monotonic behavior of the potential function is guaranteed, even in cases with high peripheral velocities. The method is validated on a rotating turbine case and used to design new blades. To obtain a closed blade, a set of closure conditions were developed and referred.