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Numerical Study of Nonlinear Heat Transfer from a Wavy Surface to a High Permeability Medium with Pseudo-Spectral and Smoothed Particle Methods

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Abstract

Motivated by petro-chemical geological systems, we consider the natural convection boundary layer flow from a vertical isothermal wavy surface adjacent to a saturated non-Darcian high permeability porous medium. High permeability is considered to represent geologically sparsely packed porous media. Both Darcian drag and Forchheimer inertial drag terms are included in the velocity boundary layer equation. A high permeability medium is considered. We employ a sinusoidal relation for the wavy surface. Using a set of transformations, the momentum and heat conservation equations are converted from an (x, y) coordinate system to an (\(x, \eta )\) dimensionless system. The two-point boundary value problem is then solved numerically with a pseudo-spectral method based on combining the Bellman–Kalaba quasi linearization method with the Chebyschev spectral collocation technique (SQLM). The SQLM computations are demonstrated to achieve excellent correlation with smoothed particle hydrodynamic (SPH) Lagrangian solutions. We study the effect of Darcy number (Da), Forchheimer number (Fs), amplitude wavelength (A) and Prandtl number (Pr) on the velocity and temperature distributions in the regime. Local Nusselt number is also computed for selected cases. The study finds important applications in petroleum engineering and also energy systems exploiting porous media and undulating (wavy) surface geometry. The SQLM algorithm is shown to be exceptionally robust and achieves fast convergence and excellent accuracy in nonlinear heat transfer simulations.

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Authors and Affiliations

  1. Fluid Mechanics and Propulsion, Mechanical and Aeronautical Engineering, School of Computing, Science and Engineering, UG17 Newton Building, University of Salford, Salford, M5 4WT, UK

    O. Anwar Bég

  2. School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg, 3209, South Africa

    S. S. Motsa

  3. Renewable Energy and Geodynamics, Israfil House, Dickenson Rd., Manchester, M13, England, UK

    T. A. Bég

  4. Spray Research Group, Petroleum and Gas Engineering, School of Computing, Science and Engineering, Room G82 Newton Building, University of Salford, Salford, M5 4WT, UK

    A. J. Abbas & A. Kadir

  5. Applied Mathematics, Mathematics Department, COMSATS, Lahore, Pakistan

    Ayesha Sohail

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  1. O. Anwar Bég
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  2. S. S. Motsa
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  3. T. A. Bég
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Correspondence to O. Anwar Bég.

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Bég, O.A., Motsa, S.S., Bég, T.A. et al. Numerical Study of Nonlinear Heat Transfer from a Wavy Surface to a High Permeability Medium with Pseudo-Spectral and Smoothed Particle Methods. Int. J. Appl. Comput. Math 3, 3593–3613 (2017). https://doi.org/10.1007/s40819-017-0318-4

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