Skip to main content
SpringerLink
Log in

Oscillatory Darcy Flow in Porous Media

  • Published:
Transport in Porous Media Aims and scope Submit manuscript

Abstract

We investigate the flow in a porous medium subjected to an oscillatory (sinusoidal) pressure gradient. Direct numerical simulation (DNS) has been performed to benchmark the analytical solutions of the unsteady Darcy equation with two different expressions of the time scale: one given by a consistent volume averaging of the Navier–Stokes equation with a steady-state closure for the flow-resistance term, another given by volume averaging of the kinetic energy equation with a closure for the dissipation rate. For small and medium frequencies, the analytical solutions with the time scale obtained by the energy approach compare well with the DNS results in terms of amplitude and phase lag. For large dimensionless frequencies (\(\omega \tau \gtrsim 10\)), we observe a slightly smaller damping of the amplitude than predicted by the unsteady Darcy equation with the low-frequency time scale. This can be explained by a change in the velocity fields towards a potential flow solution. Note that at those high frequencies, the flow amplitudes remain below 1 % of those of the steady-state flows. Our DNSs, however, indicate that the time scale predicted by the steady-state closure for the flow-resistance term is too small. In general, this study supports the use of the unsteady form of Darcy’s equation with constant coefficients to solve time-periodic Darcy flow, provided the proper time scale has been found.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  • Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1967)

    Google Scholar 

  • Breuer, M., Peller, N., Rapp, C., Manhart, M.: Flow over periodic hills—numerical and experimental study over a wide range of Reynolds numbers. Comput. Fluids 38(2), 433–457 (2009)

    Article  Google Scholar 

  • Breugem, W.P., Boersma, B.J., Uittenbogaard, R.E.: The influence of wall permeability on turbulent channel flow. J. Fluid Mech. 562, 35–72 (2006)

    Article  Google Scholar 

  • Burcharth, H.F., Andersen, O.H.: On the one-dimensional steady and unsteady porous flow equations. Coast. Eng. 24, 233–257 (1995)

    Article  Google Scholar 

  • Fan, J., Wang, L.: Analytical theory of bioheat transport. J. Appl. Phys. 109(104), 702 (2011)

    Google Scholar 

  • Gu, Z., Wang, H.: Gravity waves over porous bottoms. Coast. Eng. 15(5), 497–524 (1991)

    Article  Google Scholar 

  • Habibi, K., Mosahebi, A., Shokouhmand, H.: Heat transfer characteristics of reciprocating flows in channels partially filled with porous medium. Transp. Porous Media 89(2), 139–153 (2011)

    Article  Google Scholar 

  • Hall, K.R., Smith, G.M., Turcke, D.J.: Comparison of oscillatory and stationary flow through porous media. Coast. Eng. 24, 217–232 (1995)

    Article  Google Scholar 

  • Hill, A.A., Straughan, B.: Poiseuille flow in a fluid overlying a porous medium. J. Fluid Mech. 603, 137–149 (2008)

    Article  Google Scholar 

  • Hokpunna, A., Manhart, M.: Compact fourth-order finite volume method for numerical solutions of Navier–Stokes equations on staggered grids. J. Comput. Phys. 229, 7545–7570 (2010)

    Article  Google Scholar 

  • Kuznetsov, A.V., Nield, D.A.: Forced convection with laminar pulsating flow in a saturated porous channel or tube. Transp. Porous Media 65, 505–523 (2006). doi:10.1007/s11242-006-6791-6

    Article  Google Scholar 

  • Laushey, L.M., Popat, L.V.: Darcy’s law during unsteady flow. In: Tison LJ (ed) Ground water: general assembly of Bern (Sep. 25–Oct. 7, 1967), International Union of Geodesy and Geophysics (IUGG) and International Association of Scientific Hydrology (IASH), vol. 77, pp. 284–299 (1968)

  • Loudon, C., Tordesillas, A.: The use of the dimensionless Womersley number to characterize the unsteady nature of internal flow. J. Theor. Biol. 191, 63–78 (1998)

    Article  Google Scholar 

  • Lowe, R.J., Koseff, J.R., Monismith, S.G.: Oscillatory flow through submerged canopies: 1. Velocity structure. J. Geophys. Res. 110(C10), 016 (2005)

    Google Scholar 

  • Lowe, R.J., Shavit, U., Falter, J.L., Koseff, J.R., Monismith, S.G.: Modeling flow in coral communities with and without waves: a synthesis of porous media and canopy flow approaches. Limnol. Oceanogr. 53, 2668–2680 (2008)

    Article  Google Scholar 

  • Manhart, M.: A zonal grid algorithm for DNS of turbulent boundary layers. Comput. Fluids 33(3), 435–461 (2004)

    Article  Google Scholar 

  • Peller, N.: Numerische simulation turbulenter Strömungen mit Immersed Boundaries. Ph.D. thesis, Technische Universität München (2010)

  • Peller, N., Le Duc, A., Tremblay, F., Manhart, M.: High-order stable interpolations for immersed boundary methods. Int. J. Numer. Methods Fluids 52, 1175–1193 (2006)

    Article  Google Scholar 

  • Rajagopal, K.R.: On a hierarchy of approximate models for flows of incompressible fluids through porous solids. Math. Models Methods Appl. Sci. 17(2), 215–252 (2007)

    Article  Google Scholar 

  • Sollitt, C.K., Cross, R.H.: Wave transmission through permeable breakwaters. In: Proceedings of 13th Coastal Engineering Conference, ASCE, vol. 3, pp. 1827–1846 (1972)

  • Tilton, N., Cortelezzi, L.: Linear stability analysis of pressure-driven flows in channels with porous walls. J. Fluid Mech. 604, 411–445 (2008)

    Article  Google Scholar 

  • Wang, C.Y.: The starting flow in ducts filled with a Darcy–Brinkman medium. Transp. Porous Media 75, 55–62 (2008). doi:10.1007/s11242-008-9210-3

    Article  Google Scholar 

  • Whitaker, S.: Flow in porous media I: a theoretical derivation of Darcy’s law. Transp. Porous Media 1(1), 3–25 (1986)

    Article  Google Scholar 

  • Whitaker, S.: The Forchheimer equation: a theoretical development. Transp. Porous Media 25(1), 27–61 (1996)

    Article  Google Scholar 

  • Williamson, J.H.: Low-storage Runge–Kutta schemes. J. Comput. Phys. 35(1), 48–56 (1980)

    Article  Google Scholar 

  • Womersley, J.R.: Method for the calculation of velocities, rate of flow and viscous drag in arteries when the pressure gradient is known. J. Physiol. 127, 553–563 (1955)

    Article  Google Scholar 

  • Zhu, T., Waluga, C., Wohlmuth, B., Manhart, M.: A study of the time constant in unsteady porous media flow using direct numerical simulation. Transp. Porous Media 114, 161–179 (2014)

    Article  Google Scholar 

Download references

Acknowledgments

Financial support from the International Graduate School of Science and Engineering (IGSSE) of the Technische Universität München for research training group 6.03 is gratefully acknowledged. Our special thanks go to Christian Waluga and Barbara Wohlmuth for numerous fruitful discussions.

Author information

Authors and Affiliations

  1. Technische Universität München, Fachgebiet Hydromechanik, Arcisstr. 21, 80333, Munich, Germany

    Tao Zhu & Michael Manhart

Authors
  1. Tao Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Manhart
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Manhart.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhu, T., Manhart, M. Oscillatory Darcy Flow in Porous Media. Transp Porous Med 111, 521–539 (2016). https://doi.org/10.1007/s11242-015-0609-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-015-0609-3

Keywords

Search

Navigation