The present study deals with the generalization of a macroscopic turbulence model in porous media using a capillary model. The additional source terms associated with the production and dissipation of turbulent kinetic energy due to the presence of solid matrix are calculated using the capillary model. The present model does not require any prior pore scale simulation of turbulent flow in a specific porous geometry in order to close the macroscopic turbulence equations. Validation of the results in packed beds, periodic arrangement of square cylinders, synthetic foams, and longitudinal flows such as pipes, channels, and rod bundles against available data in the literature reveals the ability of the present model in predicting turbulent flow characteristics in different types of porous media. Transition to the fully turbulent regime in porous media and different approaches to treat this phenomenon are also discussed in the present study. Finally, the general model is modified so that it can be applied to lower Reynolds numbers below the range of fully turbulent regime in porous media.

References

References
1.
De Lemos
,
M. J.
,
2012
,
Turbulence in Porous Media: Modeling and Applications
,
Elsevier
,
London
.
2.
Ljung
,
A.-L.
,
Frishfelds
,
V.
,
Lundström
,
T. S.
, and
Marjavaara
,
B. D.
,
2012
, “
Discrete and Continuous Modeling of Heat and Mass Transport in Drying of a Bed of Iron Ore Pellets
,”
Drying Technol.
,
30
(
7
), pp.
760
773
.
3.
Antohe
,
B.
, and
Lage
,
J.
,
1997
, “
A General Two-Equation Macroscopic Turbulence Model for Incompressible Flow in Porous Media
,”
Int. J. Heat Mass Transfer
,
40
(
13
), pp.
3013
3024
.
4.
Masuoka
,
T.
, and
Takatsu
,
Y.
,
1996
, “
Turbulence Model for Flow Through Porous Media
,”
Int. J. Heat Mass Transfer
,
39
(
13
), pp.
2803
2809
.
5.
Nakayama
,
A.
, and
Kuwahara
,
F.
,
1999
, “
A Macroscopic Turbulence Model for Flow in a Porous Medium
,”
ASME J. Fluids Eng.
,
121
(
2
), pp.
427
433
.
6.
Nakayama
,
A.
, and
Kuwahara
,
F.
,
2008
, “
A General Macroscopic Turbulence Model for Flows in Packed Beds, Channels, Pipes, and Rod Bundles
,”
ASME J. Fluids Eng.
,
130
(
10
), p.
101205
.
7.
Pedras
,
M. H.
, and
de Lemos
,
M. J.
,
2001
, “
Macroscopic Turbulence Modeling for Incompressible Flow Through Undeformable Porous Media
,”
Int. J. Heat Mass Transfer
,
44
(
6
), pp.
1081
1093
.
8.
Nield
,
D.
,
2001
, “
Alternative Models of Turbulence in a Porous Medium, and Related Matters
,”
ASME J. Fluids Eng.
,
123
(
4
), pp.
928
934
.
9.
Guo
,
B.
,
Yu
,
A.
,
Wright
,
B.
, and
Zulli
,
P.
,
2006
, “
Simulation of Turbulent Flow in a Packed Bed
,”
Chem. Eng. Technol.
,
29
(
5
), pp.
596
603
.
10.
Takeda
,
K.
,
1994
, “
Mathematical Modelling of Pulverised Coal Combustion in a Blast Furnace
,” Ph.D. thesis, Imperial College London, London.
11.
Bey
,
O.
, and
Eigenberger
,
G.
,
1997
, “
Fluid Flow Through Catalyst Filled Tubes
,”
Chem. Eng. Sci.
,
52
(
8
), pp.
1365
1376
.
12.
Alvarez
,
G.
,
Bournet
,
P.-E.
, and
Flick
,
D.
,
2003
, “
Two-Dimensional Simulation of Turbulent Flow and Transfer Through Stacked Spheres
,”
Int. J. Heat Mass Transfer
,
46
(
13
), pp.
2459
2469
.
13.
Alvarez
,
G.
, and
Flick
,
D.
,
2007
, “
Modelling Turbulent Flow and Heat Transfer Using Macro-Porous Media Approach Used to Predict Cooling Kinetics of Stack of Food Products
,”
J. Food Eng.
,
80
(
2
), pp.
391
401
.
14.
Chandesris
,
M.
,
Serre
,
G.
, and
Sagaut
,
P.
,
2006
, “
A Macroscopic Turbulence Model for Flow in Porous Media Suited for Channel, Pipe and Rod Bundle Flows
,”
Int. J. Heat Mass Transfer
,
49
(
15
), pp.
2739
2750
.
15.
Hoffmann
,
M. R.
,
2004
, “
Application of a Simple Space-Time Averaged Porous Media Model to Flow in Densely Vegetated Channels
,”
J. Porous Media
,
7
(
3
), pp.
183
191
.
16.
Jouybari
,
N. F.
,
Maerefat
,
M.
, and
Nimvari
,
M. E.
,
2015
, “
A Macroscopic Turbulence Model for Reacting Flow in Porous Media
,”
Transp. Porous Media
,
106
(
2
), pp.
355
381
.
17.
Nimvari
,
M.
,
Maerefat
,
M.
,
Jouybari
,
N.
, and
El-Hossaini
,
M.
,
2013
, “
Numerical Simulation of Turbulent Reacting Flow in Porous Media Using Two Macroscopic Turbulence Models
,”
Comput. Fluids
,
88
, pp.
232
240
.
18.
Pinson
,
F.
,
Grégoire
,
O.
, and
Simonin
,
O.
,
2006
, “
k–ε Macro-Scale Modeling of Turbulence Based on a Two Scale Analysis in Porous Media
,”
Int. J. Heat Fluid Flow
,
27
(
5
), pp.
955
966
.
19.
Comiti
,
J.
, and
Renaud
,
M.
,
1989
, “
A New Model for Determining Mean Structure Parameters of Fixed Beds From Pressure Drop Measurements: Application to Beds Packed With Parallelepipedal Particles
,”
Chem. Eng. Sci.
,
44
(
7
), pp.
1539
1545
.
20.
Mickley
,
H.
,
Smith
,
K.
, and
Korchak
,
E.
,
1965
, “
Fluid Flow in Packed Beds
,”
Chem. Eng. Sci.
,
20
(
3
), pp.
237
246
.
21.
Jolls
,
K.
, and
Hanratty
,
T.
,
1966
, “
Transition to Turbulence for Flow Through a Dumped Bed of Spheres
,”
Chem. Eng. Sci.
,
21
(
12
), pp.
1185
1190
.
22.
Van der Merwe
,
D.
, and
Gauvin
,
W.
,
1971
, “
Velocity and Turbulence Measurements of Air Flow Through a Packed Bed
,”
AIChE J.
,
17
(
3
), pp.
519
528
.
23.
Dybbs
,
A.
, and
Edwards
,
R.
,
1984
, “
A New Look at Porous Media Fluid Mechanics—Darcy to Turbulent
,”
Fundamentals of Transport Phenomena in Porous Media
,
J.
Bear
and
M. Y.
Corapcioglu
, eds.,
Martinus Nijhoff
,
The Hague, The Netherlands
, pp.
199
254
.
24.
Latifi
,
M.
,
Midoux
,
N.
,
Storck
,
A.
, and
Gence
,
J.
,
1989
, “
The Use of Micro-Electrodes in the Study of the Flow Regimes in a Packed Bed Reactor With Single Phase Liquid Flow
,”
Chem. Eng. Sci.
,
44
(
11
), pp.
2501
2508
.
25.
Rode
,
S.
,
Midoux
,
N.
,
Latifi
,
M.
,
Storck
,
A.
, and
Saatdjian
,
E.
,
1994
, “
Hydrodynamics of Liquid Flow in Packed Beds: An Experimental Study Using Electrochemical Shear Rate Sensors
,”
Chem. Eng. Sci.
,
49
(
6
), pp.
889
900
.
26.
Hall
,
M.
, and
Hiatt
,
J.
,
1996
, “
Measurements of Pore Scale Flows Within and Exiting Ceramic Foams
,”
Exp. Fluids
,
20
(
6
), pp.
433
440
.
27.
Seguin
,
D.
,
Montillet
,
A.
,
Comiti
,
J.
, and
Huet
,
F.
,
1998
, “
Experimental Characterization of Flow Regimes in Various Porous Media—II: Transition to Turbulent Regime
,”
Chem. Eng. Sci.
,
53
(
22
), pp.
3897
3909
.
28.
Horton
,
N.
, and
Pokrajac
,
D.
,
2009
, “
Onset of Turbulence in a Regular Porous Medium: An Experimental Study
,”
Phys. Fluids
,
21
(
4
), p.
045104
.
29.
Bağcı
,
Ö.
,
Dukhan
,
N.
, and
Özdemir
,
M.
,
2014
, “
Flow Regimes in Packed Beds of Spheres From Pre-Darcy to Turbulent
,”
Transp. Porous Media
,
104
(
3
), pp.
501
520
.
30.
Hellstrom
,
J. G. I.
,
Jonsson
,
P. J. P.
, and
Lundstrom
,
T. S.
,
2010
, “
Laminar and Turbulent Flowthrough an Array of Cylinders
,”
J. Porous Media
,
13
(
12
), pp.
1073
1085
.
31.
Khayamyan
,
S.
, and
Lundström
,
T. S.
,
2015
, “
Interaction Between the Flow in Two Nearby Pores Within a Porous Material During Transitional and Turbulent Flow
,”
J. Appl. Fluid Mech.
,
8
(
2
), pp.
281
290
.http://jafmonline.net/JournalArchive/download?file_ID=35754&issue_ID=221
32.
Khayamyan
,
S.
,
Lundström
,
T. S.
, and
Gustavsson
,
L. H.
,
2014
, “
Experimental Investigation of Transitional Flow in Porous Media With Usage of a Pore Doublet Model
,”
Transp. Porous Media
,
101
(
2
), pp.
333
348
.
33.
Lage
,
J.
,
1998
, “
The Fundamental Theory of Flow Through Permeable Media From Darcy to Turbulence
,”
Transport Phenomena in Porous Media
,
D. B.
Ingham
and
I.
Pop
, eds.,
Pergamon
,
Oxford, UK
, pp.
1
30
.
34.
Seguin
,
D.
,
Montillet
,
A.
, and
Comiti
,
J.
,
1998
, “
Experimental Characterisation of Flow Regimes in Various Porous Media—I: Limit of Laminar Flow Regime
,”
Chem. Eng. Sci.
,
53
(
21
), pp.
3751
3761
.
35.
Weissberg
,
H. L.
,
1963
, “
Effective Diffusion Coefficient in Porous Media
,”
J. Appl. Phys.
,
34
(
9
), pp.
2636
2639
.
36.
Liu
,
S.
, and
Masliyah
,
J. H.
,
2005
, “
Dispersion in Porous Media
,”
Handbook of Porous Media
,
K.
Vafai
, ed.,
CRC Press
,
Boca Raton, FL
, pp.
81
140
.
37.
Ghanbarian
,
B.
,
Hunt
,
A. G.
,
Ewing
,
R. P.
, and
Sahimi
,
M.
,
2013
, “
Tortuosity in Porous Media: A Critical Review
,”
Soil Sci. Soc. Am. J.
,
77
(
5
), pp.
1461
1477
.
38.
Ergun
,
S.
,
1952
, “
Fluid Flow Through Packed Columns
,”
Chem. Eng. Prog.
,
48
(2), pp.
89
94
.http://dns2.asia.edu.tw/~ysho/YSHO-English/2000%20Engineering/PDF/Che%20Eng%20Pro48,%2089.pdf
39.
Mauret
,
E.
, and
Renaud
,
M.
,
1997
, “
Transport Phenomena in Multi-Particle Systems—I. Limits of Applicability of Capillary Model in High Voidage Beds-Application to Fixed Beds of Fibers and Fluidized Beds of Spheres
,”
Chem. Eng. Sci.
,
52
(
11
), pp.
1807
1817
.
40.
Barrande
,
M.
,
Bouchet
,
R.
, and
Denoyel
,
R.
,
2007
, “
Tortuosity of Porous Particles
,”
Anal. Chem.
,
79
(
23
), pp.
9115
9121
.
41.
Gunn
,
D.
,
1993
, “
On Axial Dispersion in Fixed Beds
,”
Chem. Eng. Process.: Process Intensif.
,
32
(
6
), pp.
333
338
.
42.
Sabiri
,
N.
,
Montillet
,
A.
, and
Comiti
,
J.
,
1997
, “
Pressure Drops of Non-Newtonian Purely Viscous Fluid Flow Through Synthetic Foams
,”
Chem. Eng. Commun.
,
156
(
1
), pp.
59
74
.
43.
Liu
,
S.
, and
Masliyah
,
J. H.
,
1996
, “
Single Fluid Flow in Porous Media
,”
Chem. Eng. Commun.
,
148–150
(
1
), pp.
653
732
.
44.
Kuwahara
,
F.
,
Yamane
,
T.
, and
Nakayama
,
A.
,
2006
, “
Large Eddy Simulation of Turbulent Flow in Porous Media
,”
Int. Commun. Heat Mass Transfer
,
33
(
4
), pp.
411
418
.
45.
Jouybari
,
N. F.
,
Lundstrom
,
T. S.
,
Hellstrom
,
J. G. I.
,
Maerefat
,
M.
, and
Nimvari
,
M. E.
,
2016
, “
Numerical Computation of Macroscopic Turbulent Quantities in a Porous Medium: An Extension to a Macroscopic Turbulence Model
,”
J. Porous Media
,
19
(
6
), pp.
497
513
.
46.
Kim
,
J.-H.
,
Ochoa
,
J. A.
, and
Whitaker
,
S.
,
1987
, “
Diffusion in Anisotropic Porous Media
,”
Transp. Porous Media
,
2
(
4
), pp.
327
356
.
47.
Ryan
,
D.
,
Carbonell
,
R.
, and
Whitaker
,
S.
,
1981
, “
A Theory of Diffusion and Reaction in Porous Media
,”
AIChE J.
,
71
, pp.
46
62
.
48.
Comte-Bellot
,
G.
, and
Craya
,
A.
,
1965
,
Ecoulement Turbulent Entre Deux Parois Paralleles
, Vol.
419
,
Publications Scientifiques Techniques du Ministere de l'Air
,
Paris, France
.
49.
Perry
,
A.
,
Henbest
,
S.
, and
Chong
,
M.
,
1986
, “
A Theoretical and Experimental Study of Wall Turbulence
,”
J. Fluid Mech.
,
165
, pp.
163
199
.
50.
Bear
,
J.
,
2013
,
Dynamics of Fluids in Porous Media
,
Dover Publications
,
Mineola, NY
.
You do not currently have access to this content.