P. Chaviaropoulos

An inverse inviscid method for the design of quasi-three-dimensional turbomachinery cascades

The calculation of the blade shape, when the desired velocity distribution is imposed, has been the object of numerous investigations in the past. The object of this paper is to present a new method suitable for the design of turbomachinery stator and rotor blade sections, lying on an arbitrary axisymmetric stream-surface with varying streamtube width. The flow is considered irrotational in the absolute frame of reference and compressible. The given data are the streamtube geometry, the number of blades, the inlet flow conditions and the suction and pressure side velocity distributions as functions of the normalized arc-length. The output of the computation is the blade shape that satisfies the above data. The method solves an elliptic type partial differential equation for the velocity modulus with Dirichlet and periodic type boundary conditions on the (potential function, stream function)-plane ([Phi], [Psi]). The flow angle field is subsequently calculated solving an ordinary differential equation along the iso-[Phi] or iso-[Psi] lines. The blade coordinates are, finally, compared by numerical integration. A set of closure conditions has been developed and discussed in the paper. The method is validated on several test cases and a discussion is held concerning its application and limitations.

On the 3-D inverse potential target pressure problem. Part 1. Theoretical aspects and method formulation

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 illposed 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).

Compressible flow airfoil design using natural coordinates

Abstract An irrotational inviscid compressible inverse design method for two-dimensional airfoil profiles is described. The potential (φ) and streamfunction (ψ) are used as the independent natural coordinates. The physical space on which the boundaries of the airfoil are sought, is mapped onto the (φ, ψ) space via a body-fitted coordinate transformation. A novel procedure based on differential geometry arguments is employed to derive the governing equations for the inverse problem, by requiring the curvature of the flat 2-D Euclidean space to be zero. An auxiliary coordinate transformation permits the definition of C-type computational grids on the (φ, ψ) plane resulting in a more accurate description of the leading edge region. Geometry is determined by integrating Frenet equations along the grid lines. A two-parameter iterative scheme has been incorporated in the design procedure in order to assure closure of the trailing edge. To validate the method, inverse calculation results are compared with direct, ‘reproduction’, calculation results. The design procedure of a new airfoil shape is also presented.

On the 3-D inverse potential target pressure problem. Part 2. Numerical aspects and application to duct design

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 0.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.

A 3-D Inverse Methodology Applied to the Design of Axisymmetric Ducts

An inverse inviscid potential fully 3-D methodology is introduced. 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. Potential function and two orthogonal stream functions (vectors) are used as independent variables, whilst the velocity magnitude, the aspect ratio of the elementary streamtubes cross-section and density are the dependent ones. Differential geometry and generalized tensor analysis arguments are employed both to formulate the method and derive the governing differential equations. A reduced version of the general 3-D methodology is applied to the design of axisymmetric ducts. The main advantage of the proposed approach is that the flow and geometry solutions are entirely decoupled. The geometry is computed independently after the flow field has been determined. Results are compared to analytical and numerical solutions of a direct method.Copyright

Novel scalar-vector potential formulation for three-dimensional, inviscid, rotational flow problems

A computational method for the calculation of steady, strongly rotational, inviscid, subsonic flowfields in three-dimensional ducts is presented. The method is based on the decomposition of the velocity vector into a potential and rotational part through the Helmholtz theorem. The computational algorithm requires the solution of elliptic-type equations for the scalar and vector potentials, where a completely novel approach for solving the vector potential equation has been adopted. The new formulation has better physical meaning than our previous one and has the great advantage of decoupling the vector potential boundary conditions. The use of very fast elliptic solvers, based on preconditioned minimization techniques, leads to very economic computations. The transport equations are handled in their Lagrangian form.

Computation of Rotational Transonic Flows Using a Decomposition Method

Any vector field may be decomposed, in a unique way, into an irrotational and a rotational part, if appropriate boundary conditions are imposed to the scalar and vector potentials introduced by the above decomposition. In the present work, the transformation is applied to the mass flux vector, in order to calculate two-dimensional, steady, rotational, transonic flows in arbitrarily shaped ducts and plane cascades. The whole procedure is discussed from an analytical and a numerical point of view, while finite difference-finite volume schemes are used to derive numerical results.

Single-Pass Method for the Solution of the 2-D & 3-D Inverse Potential Problem

A method for the solution of the 2-D and 3-D inverse potential problem is presented. Potential and streamfunction are introduced in order to map the physical space onto a computational one via a body-fitted coordinate transformation. The velocity magnitude, the aspect ratio and the cross-section angle of the elementary streamtubes are the dependent variables. A novel procedure based on differential geometry and generalized tensor analysis arguments is employed to formulate the method and derive the governing differential equations. The assumption of orthogonal streamsurfaces reduces the number of dependent variables by one, simplifying the governing equations to an elliptic p.d.e. for the velocity magnitude and 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 of the grid lines, after the flow field has been determined. The 2-D case is treated as a particular case of the general 3-D one. The method has been applied to both 2-D and 3-D reproduction cases of the present workshop with very satisfactory results.

Arbitrary Blade Section Design Based on Viscous Considerations. Blade Optimization

A blade design and optimization procedure is presented in this work, which is based on viscous flow considerations. This procedure concerns the design of optimum rotating arbitrary compressible high subsonic compressor and turbine blade shapes. It takes into account the effects of wall curvature and Coriolis force on turbulence, while it allows the variation of stream surface radius, along which the blade shape is placed, as well as streamtube width, with meridional distance. In order to establish the inverse part of the viscous optimization procedure, aspects such as laminar stability, transition, optimum deceleration and, more generally, the behaviour of compressible attached and separated shear layers are discussed. A plane on which all the general properties of the compressible laminar and turbulent shear layers appear, is constructed and the generation of optimum shear layers for the critical side of the blade shape is established. The complete optimization (design) procedure is then described and discussed, while various designs realized by the present procedure are presented at the end of this paper.

A Potential Prediction of Three-Dimensional Incompressible Flows Through Turbomachinery Blade Rows

A method is presented for the numerical solution of incompressible three-dimensional flows in turbomachinery blade rows. The flow is assumed to be irrotational in the absolute frame of reference.