Skip to main content
Log in

Analytical series solutions for three-dimensional supercritical flow over topography

  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

Abstract

An analytical series solution method for three-dimensional, supercritical flow over topography is presented. Steady, nonlinear solutions are calculated for a single layer of inviscid, constant-density fluid that flows irrotationally over an obstacle that varies significantly in the x-, y- and z-directions. Accurate series solutions for the free surface and a series of stream tubes throughout the flow region are calculated to demonstrate the three-dimensional properties of the problem. These solutions provide valuable insight into the three-dimensional interactions between the fluid and obstacle which is impossible to gain from any two-dimensional model. The model is described by a Laplacian free-boundary problem with fully nonlinear boundary conditions. The solution method consists of iteratively updating the location of the free surface (on top of the fluid) using a cost function which is derived from the Bernoulli equation. Root-mean-square errors in the boundary conditions are used as convergence criteria and a measure of the accuracy of the solution. This method has been used to solve the two-dimensional version of this problem in the past. Here, we detail the extensions required for three-dimensional flow.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Higgins PJ, Read WW, Belward SR (2006) A series solution method for free boundary problems arising from flow over topography. J Eng Math 54: 345–358

    Article  MathSciNet  MATH  Google Scholar 

  2. Baines PG (1995) Topographic effects in stratified flows. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  3. Long RR (1953) Some aspects of the flow of stratified fluids, I. A theoretical investigation. Tellus 5: 42–58

    Article  MathSciNet  ADS  Google Scholar 

  4. King AC, Bloor MTG (1989) A semi-inverse method for free surface flow over a submerged body. Q J Mech Appl Math 42: 183–202

    Article  MathSciNet  MATH  Google Scholar 

  5. Belward SR, Forbes LK (1995) Interfacial waves and hydraulic falls: some applications to atmospheric flows in the lee of mountains. J Eng Math 29: 161–179

    Article  MathSciNet  MATH  Google Scholar 

  6. Forbes LK (1989) An algorithm for 3-dimensional free-surface problems in hydrodynamics. J Comput Phys 82: 330–347

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Parau E, Vanden-Broeck JM (2002) Nonlinear two- and three-dimensional free surface flows due to moving disturbances. Eur J Mech B 21: 643–656

    Article  MathSciNet  MATH  Google Scholar 

  8. Dias F, Bridges TJ (2006) The numerical computation of freely propagating time-dependent irrotational water waves. Fluid Dyn Res 38: 803–830

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Kruger S, Rutschmann P (2006) Modeling 3D supercritical flow with extended shallow-water approach. J Hydraul Eng 132: 916–926

    Article  Google Scholar 

  10. Read WW (1993) Series solutions for Laplace’s equation with nonhomogeneous mixed boundary conditions and irregular boundaries. Math Comput Model 17: 9–19

    Article  MathSciNet  MATH  Google Scholar 

  11. Read WW, Belward SR, Higgins PJ, Sneddon GE (2005) Series solutions for seepage in three dimensional aquifers. ANZIAM J 46: C1126–C1140

    MathSciNet  Google Scholar 

  12. Vanden-Broeck J-M (2010) Gravity-capillary free-surface flows. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  13. Trefethen LN (2000) Spectral methods in MATLAB. SIAM, Philadelphia

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to P. J. Higgins.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Higgins, P.J., Read, W.W. & Belward, S.R. Analytical series solutions for three-dimensional supercritical flow over topography. J Eng Math 77, 39–49 (2012). https://doi.org/10.1007/s10665-012-9543-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10665-012-9543-3

Keywords

Navigation