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

(*) Corresponding author

DOI's assignment:
the author of the article can submit here a request for assignment of a DOI number to this resource!
Cost of the service: euros 10,00 (for a DOI)

#### Abstract

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.
Copyright © 2014 Praise Worthy Prize - All rights reserved.

#### Keywords

Incompressible; Hagen-Poiseuille Flow; Upstream Conditions and Downstream Cylindrical Flow Condition; Adjoint Equation; Green’s Function and Integral Formula

PDF

#### References

Abramowitz, M and Stegun, I Handbook of mathematical functions, Dover Publications, Inc, New York 1970.

Batchelor, G,K., An introduction to fluid dynamics, Cambridge University Press, 1973

Cousins, J,M., Special Computational problems associated with axisymmetric flow in Turbomachines. Ph.D thesis (CNAA), 1976.

Curle, N and Davies, H.J., Modern Fluid Dynamics, van Nostrand Reinhold Company, 1971 Chapter 1.

Klier, M., Aerodynamic Design of Annular Ducts, Ph.D thesis (CNAA), 1990 Chapter 1.

Pavlika, V., Vector Field Methods and the Hydrodynamic Design of Annular Ducts, Ph.D thesis, University of North London, Chapter VI, 1995.

Pavlika, V., Vector Field Methods and the Hydrodynamic Design of Annular Ducts, Ph.D thesis, University of North London, Chapter VIII, 1995.

Shapiro, A. H. The Dynamics and Thermodynamics of Compressible Fluid Flow, John Wiley and Sons 1953.

### Refbacks

• There are currently no refbacks.

Please send any question about this web site to info@praiseworthyprize.com
Copyright © 2005-2024 Praise Worthy Prize