Abstract
A macroscopic approach for simulating horizontal convection in a vegetated pond is presented. The generated convective currents are due to the differential radiation absorption between the two regions of the pond, one with emergent vegetation up to the free surface and one without vegetation. The Volume-Averaged Navier-Stokes (VANS) equations are used for the simulation of the laminar, unsteady, two-dimensional horizontal convection. The vegetation effects on the motion of the currents are taken into account through additional resistance terms based on the vegetation porosity and permeability. The Volume-Averaged Energy (VAE) equation is also solved with an additional source term accounting for the absorption of radiation which is based on a one-waveband and a three-waveband radiation model. The model setup is based on Beer’s law whereas the incoming radiation is absorbed from the water column. Three types of vegetation porosities φ varying from 0.75 to 0.97 are examined in investigating the motion of the convective currents within the vegetated area. The case without vegetation (φ = 1.0) is also examined for assessing the effectiveness of the radiation model. The numerical current velocity and the water temperature increase are presented and compared against available experimental data.
Highlights
• Absorption of radiation based on one- and three-waveband attenuation models
• Vegetation effects on the characteristics of horizontal convection
• Effect of Grashof number on the motion of horizontal convective currents
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Data Availability
All data are available by the authors upon request.
Abbreviations
- a1 :
-
Empirical coefficient (-)
- b1 :
-
Empirical coefficient (-)
- CD :
-
Drag coefficient (-)
- Cf :
-
Dimensionless parameter (-)
- Cp :
-
Specific heat of water (J/(kg·K))
- d:
-
Cylinder diameter (m)
- d50 :
-
Mean size of porous material (m)
- F:
-
Source term (due to vegetation) (N/m3)
- g:
-
Gravity acceleration (m/s2)
- Gr:
-
Grashof number
- h:
-
Water depth (m)
- hI :
-
Intrusion depth (m)
- I0 :
-
Bulk radiation intensity at the surface (W/m2)
- k:
-
Intrinsic permeability (m2)
- L:
-
Total tank length (m)
- Lveg :
-
Length of the vegetated region (m)
- Lop :
-
Length of the open region (m)
- N:
-
Number of bands (-)
- n0 :
-
Bulk attenuation coefficient (1/m)
- P:
-
Effective pressure (Pa)
- R2 :
-
Regression coefficient (-)
- Rp :
-
Pore Reynolds number (-)
- Sh :
-
Source term (W/m3)
- t:
-
Time (s)
- T:
-
Fluid temperature (K)
- tc :
-
Time to the inertia-buoyancy balance (s)
- tΕ :
-
Time to the energy-limited regime (s)
- tv :
-
Time start of the drag dominated flow (s)
- UE :
-
Energy limited velocity scale (m/s)
- Ui :
-
Fluid velocity in the i direction (m/s)
- Ux :
-
Horizontal velocity (m/s)
- Ux,mean :
-
Mean horizontal velocity (m/s)
- UV :
-
Drag dominated velocity (m/s)
- UV,sim :
-
Drag dominated simulation regime (m/s)
- Uvis :
-
Viscous dominated velocity (m/s)
- V:
-
Volume of a horizontal slab (m3)
- Vf :
-
Volume of fluid contained in volume V (m3)
- y:
-
Distance from the tank bottom (m)
- β:
-
Thermal expansion coefficient (1/K)
- ΔΤE :
-
Temperature difference energy -limited regime (K)
- ΔΤV :
-
Temperature difference drag regime (K)
- κ:
-
Thermal conductivity (W/(m·K))
- μ:
-
Fluid dynamic viscosity (kg/(m·s))
- ν:
-
Fluid kinematic viscosity (m2/s)
- ρ:
-
Fluid density (kg/m3)
- φ:
-
Vegetation porosity (-)
- ψ:
-
General parameter (-)
References
Adams EE, Wells SA (1984) Field measurements on side arms of Lake Anna. J Hydraul Eng 110(6):773–793. https://doi.org/10.1061/(ASCE)0733-9429(1984)110:6(773)
Adams C, Boar R, Hubble D, Gikungu M, Harper D, Hickley P, Tarras-Wahlberg N (2002) The dynamics and ecology of exotic tropical species in floating plant mats: Lake Naivasha, Kenya. Hydrobiologia 488(1–3):115–122. https://doi.org/10.1023/A:1023322430005
ANSYS Inc. (2013) ANSYS Fluent 15.0 user’s guide. ANSYS Inc., Canonsburg
Azza N, Denny P, van de Koppel J, Kansiime F (2006) Floating mats: their occurrence and influence on shoreline distribution of emergent vegetation. Fresh Bio 51(7):1286–1297. https://doi.org/10.1111/j.1365-2427.2006.01565.x
Chand K, Sharma M, Vishnu VT, De A (2019) Statistics of coherent structures in two-dimensional turbulent Rayleigh-Benard convection. Phys Fluids 31:115112. https://doi.org/10.1063/1.5125758
Chilla F, Schumacher J (2012) New perspectives in turbulent Rayleigh-Benard convection. Eur Phys J E 35(7):58. https://doi.org/10.1140/epje/i2012-12058-1
Coates M, Ferris J (1994) The radiatively driven natural convection beneath a floating plant layer. Limnol Oceanogr 39(5):1186–1194. https://doi.org/10.4319/lo.1994.39.5.1186
Coates MJ, Patterson JC (1993) Unsteady natural convection in a cavity with non-uniform absorption of radiation. J Fluid Mech 256:133–161. https://doi.org/10.1017/S0022112093002745
Coates MJ, Patterson JC (1994) Numerical simulations of the natural convection in a cavity with nonuniform internal sources. Int J Heat and Fluid Flow 15(3):218–225. https://doi.org/10.1016/0142-727X(94)90041-8
Crites RW, Reed SC, Middlebrooks EJ (2006) Natural wastewater treatment systems. CRC/Taylor & Francis, Boca Raton
Etminan V, Lowe RJ, Ghisalberti M (2017) A new model for predicting the drag exerted by vegetation canopies. Water Resour Res 53:3179–3196. https://doi.org/10.1002/2016WR020090
Farrow DE (2004) Periodically forced natural convection over slowly varying topography. J Fluid Mech 508:1–21. https://doi.org/10.1017/S002211200400847X
Farrow DE, Patterson JC (1994) The daytime circulation and temperature structure in a reservoir sidearm. Int J Heat Mass Transf 37(13):1957–1968. https://doi.org/10.1016/0017-9310(94)90335-2
Finnigan J (2000) Turbulence in plant canopies. Annu Rev Fluid Mech 32:519–571. https://doi.org/10.1146/annurev.fluid.32.1.519
Gayen B, Griffiths RW, Hughes GO (2014) Stability transitions and turbulence in horizontal convection. J Fluid Mech 751:698–724. https://doi.org/10.1017/jfm.2014.302
Gramberg HIJ, Howell PD, Ockendon J (2007) Convection by a horizontal thermal gradient. J Fluid Mech 586:41–57. https://doi.org/10.1017/S0022112007006635
Griffiths RW, Hughes GO, Gayen B (2013) Horizontal convection dynamics: insights from transient adjustment. J Fluid Mech 726:559–595. https://doi.org/10.1017/jfm.2013.244
Grossmann S, Lohse D (2003) On geometry effects in Rayleigh-Bénard convection. J Fluid Mech 486:105–114. https://doi.org/10.1017/S0022112003004270
Hale GM, Querry MR (1973) Optical constants of water in the 200-nm to 200-μm wavelength region. Appl Opt 12:555–563. https://doi.org/10.1364/AO.12.000555
Hattori T, Patterson JC, Lei C (2014) Study of unsteady natural convection induced by absorption of radiation based on a three-waveband attenuation model. J Phys Conf Ser 530:12036. https://doi.org/10.1088/1742-6596/530/1/012036
Huai WX, Wu ZL, Qian ZD, Geng C (2011) Large eddy simulation of open channel flows with non-submerged vegetation. J Hydrodyn 23(2):258–264. https://doi.org/10.1016/S1001-6058(10)60111-4
Hughes GO, Griffiths RW (2008) Horizontal convection. Annu Rev Fluid Mech 40:185–208. https://doi.org/10.1146/annurev.fluid.40.111406.102148
Jamali M, Zhang X, Nepf HM (2008) Exchange flow between a canopy and open water. J Fluid Mech 611:237–254. https://doi.org/10.1017/S0022112008002796
Johnson HK, Karambas TV, Avgeris I, Zanuttigh B, Gonzalez-Marco D, Caceres I (2005) Modelling of waves and currents around submerged breakwaters. Coast Eng 52:949–969. https://doi.org/10.1016/j.coastaleng.2005.09.011
Kirk JT (1994) Light and photosynthesis in aquatic ecosystems. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511623370
Lei C, Patterson JC (2002a) Natural convection in a reservoir sidearm subject to solar radiation: experimental observations. Exp Fluids 552:207–220. https://doi.org/10.1007/s00348-001-0402-7
Lei C, Patterson JC (2002b) Unsteady natural convection in a triangular enclosure induced by absorption of radiation. J Fluid Mech 460:181–209. https://doi.org/10.1017/S0022112008005077
Leonard LA, Luther ME (1995) Flow hydrodynamics in tidal marsh canopies. Limnol Oceanogr 40:1474–1484. https://doi.org/10.4319/lo.1995.40.8.1474
Lightbody A, Avener M, Nepf HM (2008) Observations of short-circuiting flow paths within a constructed treatment wetland in Augusta, Georgia, USA. Limnol Oceanogr 53(3):1040–1053. https://doi.org/10.4319/lo.2008.53.3.1040
Lin YT, Wu HC (2014) The role of rooted emergent vegetation on periodically thermal-driven flow over a sloping bottom. Environ Fluid Mech 14:1303–1334. https://doi.org/10.1007/s10652-014-9336-5
Lowe RJ, Shavit U, Falter JL, Koseff JR, Monismith G (2008) Modeling flow in coral communities with and without waves: a synthesis of porous media and canopy flow approaches. Limnol Oceanogr 53(6):2668–2680. https://doi.org/10.2307/40058354
Mariana M, Mazzeo N, Moss B, Godrigues-Gallego L (2003) The structuring role of free floating versus submerged plants in a subtropical shallow lake. Aquat Ecol 37:377–391. https://doi.org/10.1023/B:AECO.0000007041.57843.0b
Mazda Y, Wolanksi E, King B, Sase A, Ohtsuka D, Magi M (1997) Drag forces due to vegetation in mangrove swamps. Mangrove Salt Marshes 1:193–199. https://doi.org/10.1023/A:1009949411068
Monismith S, Imberger J, Morison ML (1990) Convective motions in the sidearm of a small reservoir. Limnol Oceanogr 35:1676–1702. https://doi.org/10.4319/lo.1990.35.8.1676
Moutsopoulos KN, Papaspyros INE, Tsihrintzis VA (2009) Experimental investigation of inertial flow processes in porous media. J Hydrol 374(3–4):242–254. https://doi.org/10.1016/j.jhydrol.2009.06.015
Nepf HM, Oldham CE (1997) Exchange dynamics of a shallow contaminated wetland. Aquat Sci 59:193–213. https://doi.org/10.1007/BF02523273
Ng CS, Ooi A, Lohse D, Chung D (2015) Vertical natural convection: application of the unifying theory of thermal convection. J Fluid Mech 764:349–361. https://doi.org/10.1017/jfm.2014.712
Nikora VI, McEwan IK, McLean SR, Coleman SE, Pokrajac D, Walters R (2007) Double-averaging concept for rough-bed open-channel and overland flows: theoretical background. J Hydraul Eng 133(8):873–883. https://doi.org/10.1007/s10652-012-9265-0
Oldham CE, Sturman JJ (2001) The effect of emergent vegetation on convective flushing in shallow wetlands: scaling and experiment. Limnol Oceanogr 46(6):1486–1493. https://doi.org/10.4319/lo.2001.46.6.1486
Papaioannou V, Prinos P (2019) Numerical simulation of horizontal convective currents in a lake with macrophytes. Proc 14th Conference of Hellenic Hydrotechnical Association, Volos, Greece, 16-17 May, Volos, Greece (in Greek)
Patterson JC (1984) Unsteady natural convection in a cavity with internal heating and cooling. J Fluid Mech 140:135–151. https://doi.org/10.1017/S0022112084000549
Prinos P, Sofialidis D, Keramaris E (2003) Turbulent flow over and within a porous bed. J Hydraul Eng 129(9):720–734. https://doi.org/10.1061/(ASCE)0733-9429(2003)129:9(720)
Rabl A, Nielsen C (1975) Solar ponds for space heating. Sol Energy 17:1–12. https://doi.org/10.1016/0038-092X(75)90011-0
Shishkina O (2016) Momentum and heat transport scaling in laminar vertical convection. Phys Rev E 93(5):051102. https://doi.org/10.1103/PhysRevE.93.051102
Shishkina O, Horn S (2016) Thermal convection in inclined cylindrical containers. J Fluid Mech 790:R3. https://doi.org/10.1017/jfm.2016.55
Shishkina O, Wagner S (2016) Prandtl-number dependence of heat transport in laminar horizontal convection. Phys Rev Lett 116:024302. https://doi.org/10.1103/PhysRevLett.116.024302
Sidiropoulou MG, Moutsopoulos KN, Tsihrintzis VA (2007) Determination of Forchheimer equation coefficients a and b. Hydrol Process 21:534–554. https://doi.org/10.1002/hyp.6264
Souliotis D, Prinos P (2008) Turbulence in vegetated flows: volume-average analysis and modeling aspects. Acta Geophys 56(3):894–917. https://doi.org/10.2478/s11600-008-0027-9
Stoesser T, Salvador GU, Rodi W, Diplas P (2009) Large eddy simulation turbulent flow through submerged vegetation. Transp Porous Media 78(3):347–365. https://doi.org/10.1007/s11242-009-9371-8
Tanino Y, Nepf HM, Kulis PS (2005) Gravity currents in aquatic canopies. Water Resour Res 41:W12402. https://doi.org/10.1029/2005WR004216
Trevisan OV, Bejan A (1986) Convection driven by the non-uniform absorption of thermal radiation at the free surface of a stagnant pool. Numer Heat Transfer 10(5):483–506. https://doi.org/10.1080/10407788608913530
Tsakiri M, Prinos P (2016) Microscopic numerical simulation of convective currents in aquatic canopies. Procedia Engineering 162(C):611–618. https://doi.org/10.1016/j.proeng.2016.11.107
Tsakiri M, Prinos P, Koftis T (2016) Numerical simulation of turbulent exchange flow in aquatic canopies. J Hydraul Res 54(2):131–144. https://doi.org/10.1080/00221686.2016.1141803
Van Gent MRA (1995) Wave interaction with permeable coastal structures. Dissertation, Delft University of Technology
Wang W, Huai W, Li S, Wang P, Wang Y, Zhang J (2019) Analytical solutions of velocity profile in flow through submerged vegetation with variable frontal width. J Hydrol 578:124088. https://doi.org/10.1016/j.jhydrol.2019.124088
Wetzel RG (2001) Light in inland water. Limnology, 3rd edn. Academic, San Diego, pp 49–69
Whitaker S (1996) The Forchheimer equation: a theoretical development. Transp Porous Media 25:27–61. https://doi.org/10.1007/BF00141261
Whitaker S (1999) The method of volume averaging. Theory and applications of transport in porous media, vol 13. Springer, Dordrecht
Zhang X, Nepf HM (2008) Density driven exchange flow between open water and an aquatic canopy. Water Resour Res 44(8):W08417. https://doi.org/10.1029/2007WR006676
Zhang X, Nepf HM (2009) Thermally driven exchange flow between open water and aquatic canopy. J Fluid Mech 632:227–243. https://doi.org/10.1017/S0022112009006491
Acknowledgments
Part of this work has been presented in Greek in the 14th Conference of the Hellenic Hydrotechnical Association, Volos, Greece, 16–17 May 2019.
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Prinos Panagiotis contributed to the conception and design of the study. Papaioannou Vassilios conducted simulation runs, data collection and analysis. The first draft of the manuscript was written by Prinos Panagiotis, and all authors reviewed early drafts of the manuscript. All authors read and approved the final manuscript.
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Papaioannou, V., Prinos, P. A Macroscopic Approach for Simulating Horizontal Convection in a Vegetated Pond. Environ. Process. 8, 199–218 (2021). https://doi.org/10.1007/s40710-020-00484-x
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DOI: https://doi.org/10.1007/s40710-020-00484-x