Pulsating flow through porous media

Pulsating Flow Through Porous Media

Michele Iervolino, Marcello Manna and Andrea Vacca

Abstract The present work investigates the response of a porous media to an un-steady forcing resulting from the superposition of an harmonic component to a meanone. The analysis is carried out both in terms of global parameters and local fieldsobtained processing data from Direct Numerical Simulation of the Navier–Stokesequations at pore level performed with a spectrally accurate multi–domain algo-rithm.

1 Introduction

The transport of fluid through porous media is encountered in several groundwater,marine, chemical and mechanical engineering applications. In many circumstancesthe relevant global flow parameters may be predicted through the Darcy’s law, i.e.assuming steady flow conditions and negligible inertial effects. However, in someof the above applications both unsteadiness and considerable inertia may be en-countered. While many efforts have been devoted to the description of the purelyoscillatory flow, i.e. with zero mean pressure gradient [8, 5, 3, 4, 1, 2], the pulsatingones have been far less investigated. Starting from experimental evidences and ac-counting for the acceleration of the fluid in the global momentum equation, severalcorrections to the Darcy–Forchheimer’s law have been proposed [12, 13, 14]. Thefirst attempt to extend the analysis also at the pore level is due to Graham & Hig-don [6] who considered two–dimensional periodic media consisting of constrictedchannels or cylinder arrays with high solid volume fraction (φ ). In [6] it has been

Michele Iervolino and Andrea VaccaDipartimento di Ingegneria Civile, Seconda Universita di Napoli, Via Roma, 29, 81031 Aversa(Ce), Italy, e-mail: [email protected], [email protected]

Marcello MannaDipartimento di Ingegneria Meccanica per l’Energetica, Universita di Napoli ”Federico II”, ViaClaudio, 21, 80125 Napoli, Italy, e-mail: [email protected]

1

2 Iervolino, Manna and Vacca

shown that an oscillatory forcing larger than the mean one may produce a reduc-tion of the time–averaged flow rate up to 40% of the steady state value. The presentpaper analyses the pulsating flow field in a two-dimensional porous medium whosebase element consists of square cylinders with moderate solid volume fraction. Theanalysis is carried out by solving the unsteady Navier-Stokes equations at the porescale.

2 Flow problem and computational setup

The problem under investigation is the pulsating flow through a periodic array ofl × l square cylinders spaced by 2h, (l = 3h, φ = 0.36, see figure 1a) driven by aharmonically time varying pressure gradient ∆ p/L:

∆ pL

=∆ pL

+∆ posc

Lcos(ω t) . (1)

The time mean value ∆ p/L, the amplitude ∆ posc/L, and the pulsation ω = 2π/T ,T being the dimensional oscillating period, have been chosen to span at least oneorder of magnitude of the governing dimensionless parameters. Among the possibletriplets of the relevant dimensionless parameters we shall use the following ones:

Re =∆ pL

h3ρµ2 , Reosc =

∆ posc

Lh3ρµ2 , χ =

hδ

, (2)

in which µ and ρ are the fluid viscosity and density and δ =√

2µ/ρω is the Stokeslayer thickness. Table 1 summarizes the investigated space parameters.

Table 1 Run matrix

Re 32 64 128

Reosc 32 64 96 128 32 64 128 32 128

χmin 0.50 0.32 0.50χmax 12.94 12.94 12.94

The analysis is carried out solving at the pore scale the unsteady 2–D Navier–Stokes equations with the spectral algorithm developed and validated in [7]. Thecomputational domain Ω , shown with dashed line in figure 1a, consists of a squareregion with dimensions 10h×10h. No-slip boundary conditions are imposed at thesolid walls, while periodicity is enforced on the boundaries of Ω . Viscous termsare discretised implicitly (Crank-Nicholson); the convective operator and the sourceterm (1) are instead treated with the explicit Adams-Bashforth scheme. The timediscretization is carried out with the second order projection scheme of Van Kan

Pulsating Flow Through Porous Media 3

[10]. The three resulting elliptic equations, two for the velocity components and onefor the pressure, are solved by a weak Legendre multi–domain algorithm.

The computational domain has been discretized with 77 patched sub–domains,each of which with 15 Legendre modes. Resolution adequacy was assessed througha grid convergence study, carried out increasing the number of the Legendre modesup to 19× 19. The flow variables have been made dimensionless assuming L = hand U = h2∆ p/(Lµ) as length and velocity characteristic scales, respectively. Inde-pendently on the values of the triplet Re,Reosc,χ , all the computations, after an ini-tial transient, reached a periodic behavior with dimensionless period T = TL /U .Under the above conditions, based on the local velocity field u, the spatially averagefluid velocity (or seepage velocity) U and its time mean value U have been evaluatedrespectively as:

U(t) =1

Ω f

∫

Ω f

|u(t)| dΩ , U =1T

∫ T

0U(t) dt (3)

where Ω f = (1−φ)Ω represents the volume occupied by the fluid.Steady–state preliminary computations were carried out in order to define the up-

per bound of applicability of the Darcy’s law, i.e. negligibility of the inertial effects,along with the limiting pressure gradient above which steady flow conditions cannottake place (Re ∼ 300). Figure 1b shows the computed dimensionless steady seep-age velocity Ust versus the Reynolds number, i.e. Ust = f (Re). The independenceof Ust on the Reynolds number confirms the validity of the Darcy’s law up to Re ∼ 1[9, 11, 15]. Based on the steady results, the quasi–steady approximation U qst of theperiod–averaged seepage velocity has been computed as:

Uqst =1T

∫ T

0f (Re+Reosccos(ωt))dt (4)

2 h

l

∆p

a)

10-1 100 101 102

0.01

0.02

0.03

0.04

Ust

Re

b)

Fig. 1 Sketch of the geometry of the porous medium (a); seepage velocity Vs steady pressuregradient (b).

4 Iervolino, Manna and Vacca

3 Results

The effects of the pulsating forcing onto the period-averaged seepage velocity isenlightened in figure 2a, for the Re = 32 case. Four values of the amplitude of theoscillating pressure gradient have been considered, namely Reosc = 32,64,96,128,so that the ratio between the steady component and the unsteady one varies between1 and 4. Similar results have been obtained also for the Re = 64 and Re = 128 cases(results not shown) herein.

Independently on the Reosc value, the oscillatory forcing reduces the period-averaged seepage velocity respect to the steady value. The larger reductions occurfor the smaller values of χ . Moreover this effect grows with the amplitude of theoscillation (for Reosc = 32: U/Ust = 0.92; Reosc = 128: U/Ust = 0.56). Increasingthe oscillation frequency at a constant Reosc value, the ratio U/Ust gradually growsand tends to the unity in the high frequency regime (for χ > 4). Figure 2b depictsthe temporal power spectrum of U for the Re = 32 and Reosc = 32 case, for differ-ent values of the χ parameter. In the high frequency regime the power associated tothe fundamental frequency largely overwhelms all the other ones, while in the lowfrequency regime a fuller spectrum has been found.

The main features of the flow field at pore level are shown in figures 3-5. Figure3 reports the period–averaged velocities u for the Re = 32 and Reosc = 32 case, withχ = 12.94 (figure 3a) and χ = 0.50 (figure 3b). The corresponding oscillating ve-locity field u(τ) = u(τ)−u, at the phases τ = 1/8T,3/8T,5/8T,7/8T , are reportedin figures 4 and 5, respectively. In the high frequency regime (see figure 4) the oscil-lating flow field is characterized by a plug flow between the grains, with high vortic-ity values concentrated only at the neighborhood of the solid boundaries. Moreover,the oscillating flow field shows strong similarities with the one computed neglectingthe non–linear terms, i.e. Stokes flow, whereas the mean field resembles the corre-

1 100.50

0.75

1.00

r = 3r = 2.5r = 2r = 1.5

Reosc= 32Reosc= 64Reosc= 96Reosc= 128

χ1 100.50

0.75

1.00a)

__ U/U

st

f/fT

Φ

100 101 10210-11

10-9

10-7

10-5

10-3

10-1

chi0.5

chi1

chi1.42

chi3.54

chi5.01

chi12.94

χ = 0.50χ = 1.00χ = 1.42χ = 3.54χ = 5.01χ = 12.94

b)

Fig. 2 Mean seepage velocity as a function of the relative Stokes layer (Figure 2a): dashed linesdenote the quasi-steady approximation (U = U qst ). Figure 2b depicts seepage velocity power spec-tra made dimensionless with U ; fT = 1/T denotes the fundamental frequency of the oscillation.

Pulsating Flow Through Porous Media 5

sponding steady solution of Navier-Stokes equations forced with ∆ p/L. In contrast,in the low frequency regime (χ < 1, see figures 3b and 5) the period–averaged flowfield strongly differs from the corresponding steady one. The oscillating componentconsiderably deviates from the Stokes solution, exhibiting large separation zonesin the deceleration phase. The results of figure 2 can be therefore explained on thebasis of the flow fields at pore level. In the high frequency regime, the unsteadyterm dominates over the convective ones and the flow is mainly governed by theStokes equations. Indeed in this regime the modifications of period–averaged seep-age velocity caused by the oscillating pressure gradient vanishes. In contrast, in thelow frequency regime the non–linear term cannot be neglected, determining a stronginteraction between the steady and the oscillating components, with a noticeable en-ergy transfer toward the smaller temporal scales, characterised by frequencies oneorder of magnitude higher than the fundamental one.

4 Conclusion

Pulsating flows through a porous medium whose base element consists of squarecylinders have been investigated processing the two–dimensional field data obtainedby a Direct Numerical Simulation of Navier–Stokes equations. A spectrally accuratemulti–domain solver has employed. Both high and low frequency regimes have beencharacterized, analyzing the steady and phase locked mean velocities. Power spectraof the seepage velocity shows that the transition from the low to the high frequencyregimes occurs smoothly.

0.06

0.00

a)

0.16

0.00

b)

Fig. 3 Shaded plot of total velocity for period–averaged flow field with streamlines superposed(Re = 32 and Reosc = 32): a) χ = 12.94, b) χ = 0.50

6 Iervolino, Manna and Vacca

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Fig. 4 Velocity vector plot of the oscillating component u(τ) (Re = 32, Reosc = 32, χ = 12.94): a)τ = 1/8T ; b) τ = 3/8T ; c) τ = 5/8T ; d) τ = 7/8T

Pulsating Flow Through Porous Media 7

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500.1

a)

500.1

b)

500.1

c)

500.1

d)

Fig. 5 Velocity vector plot of the oscillating component u(τ) (Re = 32, Reosc = 32, χ = 0.50): a)τ = 1/8T ; b) τ = 3/8T ; c) τ = 5/8T ; d) τ = 7/8T