The Calculation of Axisymmetric Duct Geometries with Hagen-Poiseuille Flow for an Incompressible Fluid

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In this paper a numerical algorithm is described for solving the boundary value problem associated with axisymmetric, inviscid, incompressible, rotational (and irrotational) flow in order to obtain duct wall shapes from prescribed wall velocity distributions. The governing equations are formulated in terms of the stream function ψ(x,y) and the function φ(x,y) as independent variables where for irrotational flow φ(x,y) can be recognized as the velocity potential function, for rotational flow φ(x,y) ceases being the velocity potential function but does remain orthogonal to the stream lines. A numerical method based on finite differences using an integral formula on a uniform mesh is employed. The technique described is capable of tackling the so-called inverse problem where the velocity wall distributions are prescribed from which the duct wall shape is calculated, as well as the direct problem where the velocity distribution on the duct walls are calculated from  prescribed duct wall shapes. The two different cases as outlined in this paper are in fact boundary value problems with Neumann and Dirichlet boundary conditions respectively. Even though both approaches are discussed, only numerical results for the case of the Dirichlet boundary conditions are given. A downstream condition is prescribed such that cylindrical flow, that is flow which is independent of the axial coordinate, exists.
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Incompressible; Hagen-Poiseuille Flow; Upstream Conditions and Downstream Cylindrical Flow Condition; Adjoint Equation; Green’s Function and Integral Formula

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