Reconstruction of an unsteady flow from incomplete boundary data: Part I. Theory

B. Tomas Johansson

Research output: Contribution to journalArticle

Abstract

Purpose – To propose and investigate a stable numerical procedure for the reconstruction of the velocity of a viscous incompressible fluid flow in linear hydrodynamics from knowledge of the velocity and fluid stress force given on a part of the boundary of a bounded domain.
Design/methodology/approach – Earlier works have involved the similar problem but for stationary case (time-independent fluid flow). Extending these ideas a procedure is proposed and investigated also for the time-dependent case.
Findings – The paper finds a novel variation method for the Cauchy problem. It proves convergence and also proposes a new boundary element method.
Research limitations/implications – The fluid flow domain is limited to annular domains; this restriction can be removed undertaking analyses in appropriate weighted spaces to incorporate singularities that can occur on general bounded domains. Future work involves numerical investigations and also to consider Oseen type flow. A challenging problem is to consider non-linear Navier-Stokes equation.
Practical implications – Fluid flow problems where data are known only on a part of the boundary occur in a range of engineering situations such as colloidal suspension and swimming of microorganisms. For example, the solution domain can be the region between to spheres where only the outer sphere is accessible for measurements.
Originality/value – A novel variational method for the Cauchy problem is proposed which preserves the unsteady Stokes operator, convergence is proved and using recent for the fundamental solution for unsteady Stokes system, a new boundary element method for this system is also proposed.
Original languageEnglish
Pages (from-to)53-63
Number of pages11
JournalInternational Journal of Numerical Methods for Heat and Fluid Flow
Volume19
Issue number1
DOIs
Publication statusPublished - 2009

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Unsteady Flow
Unsteady flow
Fluid Flow
Flow of fluids
Boundary element method
Boundary Elements
Bounded Domain
Cauchy Problem
Annular Domains
Stokes Operator
Stokes System
Colloidal Suspensions
Microorganisms
Weighted Spaces
Numerical Procedure
Incompressible Flow
Fundamental Solution
Numerical Investigation
Viscous Fluid
Variational Methods

Keywords

  • differential equations
  • flow
  • fluid phenomena

Cite this

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abstract = "Purpose – To propose and investigate a stable numerical procedure for the reconstruction of the velocity of a viscous incompressible fluid flow in linear hydrodynamics from knowledge of the velocity and fluid stress force given on a part of the boundary of a bounded domain. Design/methodology/approach – Earlier works have involved the similar problem but for stationary case (time-independent fluid flow). Extending these ideas a procedure is proposed and investigated also for the time-dependent case. Findings – The paper finds a novel variation method for the Cauchy problem. It proves convergence and also proposes a new boundary element method. Research limitations/implications – The fluid flow domain is limited to annular domains; this restriction can be removed undertaking analyses in appropriate weighted spaces to incorporate singularities that can occur on general bounded domains. Future work involves numerical investigations and also to consider Oseen type flow. A challenging problem is to consider non-linear Navier-Stokes equation. Practical implications – Fluid flow problems where data are known only on a part of the boundary occur in a range of engineering situations such as colloidal suspension and swimming of microorganisms. For example, the solution domain can be the region between to spheres where only the outer sphere is accessible for measurements. Originality/value – A novel variational method for the Cauchy problem is proposed which preserves the unsteady Stokes operator, convergence is proved and using recent for the fundamental solution for unsteady Stokes system, a new boundary element method for this system is also proposed.",
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Reconstruction of an unsteady flow from incomplete boundary data : Part I. Theory. / Johansson, B. Tomas.

In: International Journal of Numerical Methods for Heat and Fluid Flow, Vol. 19, No. 1, 2009, p. 53-63.

Research output: Contribution to journalArticle

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