The spatial averaging theorem revisited

Howes, F. A.; Whitaker, Stephen

doi:10.1016/0009-2509(85)80078-6

Cited by 271 publications

(131 citation statements)

References 12 publications

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“…In general, larger width are useful to reduce fluctuations in quasiperiodic configurations. For periodic geometries, especially for m ⊓ , it does not regularize the fields, contrary to what was suggested in Howes and Whitaker (1985). -Does it mean that studies in the literature using only m ⊓ are obsolete?…”

Section: Resultsmentioning

confidence: 84%

“…4 that the width of the averaging window does not impact the norm of the fluctuations, as was suggested in Howes and Whitaker (1985), but only its phase ( π 2 between p even and odd). In fact, the use of an averaging volume of different shape, for instance a cylinder as proposed in Howes and Whitaker (1985), does not help either, as shown in Quintard and Whitaker (1994b). Therefore, we conclude that the standard averaging operator is incompatible with the usual form of Darcy's law.…”

Section: Hydrostatic Equilibriummentioning

confidence: 70%

“…-Smoothness one of the main problems with the rectangular kernel is that it is discontinuous and may yield derivatives of the average that do not exist everywhere (Howes and Whitaker 1985). In fact, we have g f ⊓ ∈ C 0 c (R n ) (without entering mathematical details, this requires that m ⊓ and χ f g are both in L 2 (R n )) and the derivative is discontinuous at several points, as shown in Fig.…”

Section: Rectangular Kernelmentioning

confidence: 99%

“…The potential for the standard volume average to yield nondifferentiable fields was also raised in a number of papers including Veverka (1981), Mls (1987). Historically, some of the potential solutions proposed in Howes and Whitaker (1985), which consisted in changing the shape of the averaging volume or making it larger, turned out …”

Section: Introductionmentioning

confidence: 99%

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Technical Notes on Volume Averaging in Porous Media I: How to Choose a Spatial Averaging Operator for Periodic and Quasiperiodic Structures

Davit

Quintard

2017

Transp Porous Med

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OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. Abstract This paper is a first of a series aiming at revisiting technical aspects of the volume averaging theory. Here, we discuss the choice of the spatial averaging operator for periodic and quasiperiodic structures. We show that spatial averaging must be defined in terms of a convolution and analyze the properties of a variety of kernels, with a particular focus on the smoothness of average fields, the ability to attenuate geometrical fluctuations, Taylor series expansions, averaging of periodic fields and resilience to perturbations of periodicity. We conclude with a set of recommendations regarding kernels to use in the volume averaging theory.

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Section: Resultsmentioning

confidence: 84%

Section: Hydrostatic Equilibriummentioning

confidence: 70%

Section: Rectangular Kernelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Technical Notes on Volume Averaging in Porous Media I: How to Choose a Spatial Averaging Operator for Periodic and Quasiperiodic Structures

Davit

Quintard

2017

Transp Porous Med

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“…This case of a porous medium surrounded by a Stokes flow has been a topic of active investigation with the main goal of deriving appropriate boundary conditions at the interface (Beavers & Joseph 1967;Beavers, Sparrow & Magnuson 1970;Richardson 1971;Saffman 1971;Taylor 1971;Neale & Nader 1974;Howes & Whitaker 1985;Goyeau et al 2003;Chandesris & Jamet 2006;Valdés-Parada, Goyeau & Ochoa-Tapia 2007;Tlupova & Cortez 2009;Valdés-Parada et al 2009, 2013.…”

Section: Introductionmentioning

confidence: 99%

Boundary integral formulation for flows containing an interface between two porous media

2017

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A system of boundary integral equations is derived for flows in domains composed of a porous medium of permeability k 1 , surrounded by another porous medium of different permeability, k 2 . The incompressible Brinkman equation is used to describe the flow in the porous media. We first apply a boundary integral representation of the Brinkman flow on each side of the dividing interface, and impose continuity of the velocity at the interface to derive the final formulation in terms of the interfacial velocity and surface forces. We discuss relations between the surface stresses based on the additional conditions imposed at the interface that depend on the porosity and permeability of the media and the structural composition of the interface. We present simulated results for test problems and different interface stress conditions. The results show significant sensitivity to the choice of the interface conditions, especially when the permeability is large. Since the Brinkman equation approaches the Stokes equation when the permeability approaches infinity, our boundary integral formulation can also be used to model the flow in sub-categories of Stokes-Stokes and Stokes-Brinkman configurations by considering infinite permeability in the Stokes fluid domain.

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Method of finite difference solutions to the transient bubbly air-water flows

Espinosa-Paredes

Soria

1998

Int. J. Numer. Meth. Fluids

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The spatial averaging theorem revisited

Cited by 271 publications

References 12 publications

Technical Notes on Volume Averaging in Porous Media I: How to Choose a Spatial Averaging Operator for Periodic and Quasiperiodic Structures

Technical Notes on Volume Averaging in Porous Media I: How to Choose a Spatial Averaging Operator for Periodic and Quasiperiodic Structures

Boundary integral formulation for flows containing an interface between two porous media

Method of finite difference solutions to the transient bubbly air-water flows

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