Nonlinear two-dimensional free surface solutions of flow exiting a pipe and impacting a wedge

Alex Doak, Jean Marc Vanden-Broeck

Research output: Contribution to journalArticlepeer-review

3 Citations (SciVal)


This paper concerns the flow of fluid exiting a two-dimensional pipe and impacting an infinite wedge. Where the flow leaves the pipe there is a free surface between the fluid and a passive gas. The model is a generalisation of both plane bubbles and flow impacting a flat plate. In the absence of gravity and surface tension, an exact free streamline solution is derived. We also construct two numerical schemes to compute solutions with the inclusion of surface tension and gravity. The first method involves mapping the flow to the lower half-plane, where an integral equation concerning only boundary values is derived. This integral equation is solved numerically. The second method involves conformally mapping the flow domain onto a unit disc in the s-plane. The unknowns are then expressed as a power series in s. The series is truncated, and the coefficients are solved numerically. The boundary integral method has the additional advantage that it allows for solutions with waves in the far-field, as discussed later. Good agreement between the two numerical methods and the exact free streamline solution provides a check on the numerical schemes.

Original languageEnglish
Article number8
JournalJournal of Engineering Mathematics
Issue number1
Publication statusPublished - 28 Jan 2021

Bibliographical note

Funding Information:
Jean-Marc Vanden-Broeck was supported in part by EPSRC under Grant EP/NO18559/1.

Publisher Copyright:
© 2021, The Author(s).

Copyright 2021 Elsevier B.V., All rights reserved.


  • Free surface flows
  • Gravity–capillary flows
  • Numerical methods

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)


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