A Macroscopic Approach for Simulating Horizontal Convection in a Vegetated Pond

ORIGINAL ARTICLE A Macroscopic Approach for Simulating Horizontal Convection in a Vegetated Pond Vassilios Papaioannou 1 &Panagiotis Prinos 1 Received: 1 June 2020 /Accepted: 24 November 2020/ #The Author(s), under exclusive licence to Springer Nature Switzerland AG part of Springer Nature 2021 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’slawwhereasthe incoming radiation is absorbed from the water column. Three types of vegetation porosities φvarying 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 oneand three-waveband attenuation models •Vegetation effects on the characteristics of horizontal convection •Effect of Grashof number on the motion of horizontal convective currents Keywords Convection .Radiation .Porous media .Vegetation .Currents .Lake https://doi.org/10.1007/s40710-020-00484-x *Vassilios Papaioannou [email protected] Panagiotis Prinos [email protected] 1 Hydraulics Laboratory, Department of Civil Engineering, Aristotle University of Thessaloniki, 54006 Thessaloniki, Greece Published online: 12 January 2021 Environmental Processes (2021) 8:199–218 Content courtesy of Springer Nature, terms of use apply. Rights reserved. 1 Introduction Thermally driven flows are very significant in various geophysical systems. Classical examples of such flows are: (a) the Rayleigh-Benard convection (Grossmann and Lohse 2003; Chilla and Schumacher 2012; Chand et al. 2019); (b) the vertical convection (Ng et al. 2015; Shishkina 2016); and (c) the horizontal convection (Gramberg et al. 2007; Hughes and Griffiths 2008; Griffiths et al. 2013; Gayen et al. 2014; Shishkina and Wagner 2016). In (a) above, the fluid is heated from the bottom and cooled from the top; in (b), heat is supplied through one vertical surface and is removed through another vertical surface; and in (c), heat is supplied and removed through a single surface of a fluid (top or bottom). In nature, vertical convection can be observed between shallow and deep-water regions (Farrow and Patterson 1994; Lei and Patterson 2002a,b; Farrow 2004) while horizontal convection can be observed between the open (non-vegetated) region and the vegetated region of a lake or pond. The horizontal convective currents have attracted the interest of many researchers as they transport nutrients and other chemicals and affect the water quality and the wetland ecology in general. Convective currents also play a catalytic role in sediment transport and flushing of the littoral regions. Such currents are observed in natural lakes (Adams and Wells 1984; Monismith et al. 1990; Nepf and Oldham 1997;Adams et al. 2002; Mariana et al. 2003; Lightbody et al. 2008, among others). Horizontal convective currents are generated when heat is supplied and removed through the same fluid surface. In shallow regions of reservoirs and lakes, the water surface in an open area is affected by the solar radiation, while in an area with emergent vegetation, the water surface is covered by vegetation which blocks the penetration of solar radiation. Patterson (1984) investigated unsteady natural convection in a cavity with internal heating and cooling which is due to non-uniform absorption of radiation. The latter was taken into account through a longitudinally varied internal heat source, which was constant vertically and in time. Trevisan and Bejan (1986) provided a more realistic model for the internal source, which considered the local heating generated by the absorption of radiation through the top surface of a tank. Coates and Patterson (1993,1994) investigated experimentally and numerically horizontal convection in a tank with a radiation model based on Beer’s law. The model included an internal heat source which varied with the water depth in one part of the tank and was equal to zero in the other part of the tank, where there was an opaque layer with no radiation absorption. The above studies proved that, initially, the current velocity increases with t1.5 (t = time) and the acceleration is balanced with the buoyancy-induced pressure gradient (conduction regime). Then, the velocity increases linearly with t0.5 (inertia dominated regime). Finally, the current velocity becomes constant and the absorption of radiation is balanced with energy removal through advection (energy-limited regime). In shallow aquatic systems, emergent vegetation, which completely covers the water surface, is often present and can influence the propagation of convective currents (Azza et al. 2006). The vegetated area can be considered as a porous medium with porosity φ ranging from 0.55 (mangroves) to 0.99 (water lily) as presented in Mazda et al. (1997). The most common values in fields are from 0.70 up to 0.90 (Crites et al. 2006). Aquatic vegetation dissipates energy with a production of significant drag which reduces the current velocity. Coates and Ferris (1994) investigated, experimentally, convective currents due to radiation absorption in a cavity with floating vegetation. Oldham and Sturman (2001) studied the emergent vegetation effects on these currents through laboratory and mesocosm experiments. Zhang and Nepf (2009) performed experiments in a tank with open water and aquatic canopy for investigating the thermally driven flow due to surface radiation. They 200 Papaioannou V., Prinos P. Content courtesy of Springer Nature, terms of use apply. Rights reserved. identified two flow regimes: (a) an early regime, where the current velocity increases with t0.5, named inertia; and (b) a constant current velocity, named vegetated drag regime. In this case, the initial regime of the conduction is not important, and the energy-limited regime is unlikely to occur within the typical diurnal heating cycle (12 h or less). Although in most studies the intensity of radiation is assumed to be constant, Lin and Wu (2014) studied the presence of vegetation on the periodically thermal-driven flow and found that: (a) the drag force increases with increasing water depth; and (b) the viscous effects are important in very shallow water. In experiments, arrays of rigid cylinders are often used to mimic vegetation (Prinos et al. 2003;Taninoetal.2005;Jamalietal.2008; Zhang and Nepf 2008,2009). In numerical simulations, two approaches are used which account for the presence of vegetation: (a) the microscopic; and (b) the macroscopic. The former approach is based on three-dimensional (3D) numerical models in which the vegetation is represented by rigid cylinders (Stoesser et al. 2009;Huaietal.2011;TsakiriandPrinos 2016; Shishkina and Horn 2016) and the flow characteristics are determined around individual cylinders. The latter approach is based on the Volume Average Theory (VAT; Whitaker 1999) and the numerical models (2D or 3D) account for vegetation effects through additional resistance terms in the momentum equations. With this approach, a main issue is the estimation of the linear and non-linear resistance terms which can be done with two methods. With the first method, the non-linear resistance force is modeled using a quadratic drag law (Finnigan 2000). The drag force is determined through the drag coefficient CD, which is the main input parameter (Wang et al. 2019). The estimation of CDis difficult, especially for vegetation with complex leaves and foliage. The second method is based on the porous media flow theory, in which the non-linear resistance term is the DarcyForchheimer term (Whitaker 1996; Sidiropoulou et al. 2007; Moutsopoulos et al. 2009). In this case, the resistance terms (linear and non-linear) are based on the vegetation permeability k. Oldham and Sturman (2001) applied this theory for turbulent flow through emergent vegetation, while Lowe et al. (2008) applied a synthesis of porous media and canopy flow approaches for coral communities. Tsakiri et al. (2016) simulated numerically the turbulent gravity currents in aquatic canopies using a k-ε turbulence model accounting for the vegetation effects on turbulence characteristics. In the present study, horizontal convection in a partly vegetated pond is investigated numerically using the macroscopic approach. The horizontal convective currents are generated in an idealized tank where heating, through radiation transfer, takes place through the halfopen water surface. The other half, with vegetation, is not heated since the emergent vegetation is considered to block the radiation. The water in the open region is assumed to be heated with constant heat flux due to solar radiation, and hence, an internal heat source is imposed within the water due to radiation absorption. The internal source is varied within the water depth and with time. As the water temperature in the open area is increased and the water temperature in the vegetated region remains constant, a thermal current is generated at the surface of the tank and travels from the open region to the vegetated region. Also, a cold current is generated at the bottom and it travels in the opposite direction. The macroscopic model includes the unsteady two-dimensional (2D) Volume-Averaged Navier-Stokes (VANS) equations and the Volume-Averaged Energy (VAE) equation. Additional resistance terms, based on vegetation porosity and permeability, are included in the VANS equations to account for the effects of vegetation (rigid cylinders). An additional source term is included in the VAE equation to account for the absorption of radiation. This term is 201A Macroscopic Approach for Simulating Horizontal Convection in a... Content courtesy of Springer Nature, terms of use apply. Rights reserved. estimated based on the Beer’s law radiation model. The Boussinesq approximation is also applied since temperature differences are not significant. The commercial CFD code Fluent 15.0.7 (ANSYS Inc. 2013) is used in conjunction with User Defined Functions (UDF) for including the additional terms in the equations. The numerical model focuses on cases with high as well as low vegetation porosity (φ< 0.85) and permeability (k) ranging from 1.56·10−6to 3.31·10−5m2. For cases with low φ, there is no sufficient experimental information due to difficulties in measuring the characteristics of the currents. Also, using the porous media approach, the resistance terms are based on the vegetation permeability k, which can be easily determined for vegetation with a complex structure (stems, foliage, etc.). Conversely, based on canopy flow theory for simulating vegetated flows, the resistance terms are determined through the drag coefficient CD, which is not easily estimated for a medium with a complex structure. The above issues regarding the use of vegetation with high as well as low porosity and the approach based on the vegetation permeability for determining vegetation resistance can be regarded as the innovative elements of this study. In addition, a three-waveband attenuation model is used for the absorption of radiation (Papaioannou and Prinos 2019) in comparison with the simple one-waveband model which has been used mostly in previous studies (Coates and Patterson 1993; Zhang and Nepf 2009; Tsakiri and Prinos 2016). In this work, the effect of the radiation model (one-waveband, three-waveband) on the characteristics of the current (velocity, temperature increase) is investigated. In addition, the value of the bulk absorption coefficient used in both models, and its effect on current characteristics is revealed. The effect of the vegetation porosity φ and subsequently of the permeability, as well as the effect of the Grashof number on the convective current characteristics is shown for vegetation porosity φvarying from 0.75 to 0.97 and respective permeabilities from 1.56·10−6to 3.31·10−5m2. Numerical results are presented together with available experimental data (Zhang and Nepf 2009). In addition, one case without vegetation (φ= 1) is examined and numerical results are compared against empirical relationships of Coates and Patterson (1994). Thecaseofφ= 1 can be considered as one in which there is a thin opaque layer on the water surface (floating vegetation mats). 2 Governing Equations and Numerical Procedure In this section, the VANS and the VAE equations are presented. The additional source terms due to vegetation and solar radiation are analyzed in detail and the determination of characteristic parameters (e.g., vegetation permeability k, attenuation coefficient n0) is described. In addition, the numerical procedure and the case studies are presented. 2.1 VANS and VAE Equations The VANS equations (Eqs. 1and 2), for two-dimensional, unsteady, incompressible flow together with the VAE equation (Eq. 3) are solved. The Boussinesq approximation is also used which treats the density as constant in all equations apart from the buoyancy term of the momentum equation, in which it varies due to temperature difference. The derivation of Eqs. (1), (2) and (3) has been presented previously (e.g., Finnigan 2000;Nikoraetal.2007; 202 Papaioannou V., Prinos P. Content courtesy of Springer Nature, terms of use apply. Rights reserved. Souliotis and Prinos 2008) using VAT. The effect of vegetation on the motion of the convective currents is accounted through additional resistance terms in Eq. (2). ∂ρ ∂tþ∂ρUi hi ðÞ ∂xi¼0ð1Þ ∂ ∂tρUi hiðÞþUj  ∂ ∂xj ρUi hiðÞ¼−∂p hi ∂xiþ∂ ∂xj μ∂Ui hi ∂xjþ∂Uj  ∂xi  þρβ Thi−T0 hiðÞgi hiþFð2Þ ∂ ∂tρCpThi  þUj  ∂ ∂xj ρCpThi  ¼∂ ∂xj κ∂T hi ∂xj  þShð3Þ In the above equations, Ui(m/s) is the fluid velocity in the i direction, p (Pa) is the effective pressure (the difference between static and hydrostatic pressure), T (K) is the fluid temperature, T0(K) is the initial fluid temperature, F (N/m3) is the source term due to vegetation, Sh(W/m3) is the source term due to solar radiation absorption. Also, ρis the fluid density (998.2 kg/m3), μis the fluid dynamic viscosity (0.0009982 kg/(m·s)), βis the thermal expansion coefficient (0.00021 K−1), g is the gravity acceleration (9.81 m/s2), Cpis the specific heat of water (4182 J/ (kg·K)) and κis the thermal conductivity (0.6 W/(m·K)). The symbol < > indicates volume averaging excluding the space occupied by vegetation elements. For a general parameter ψ, the fluid average quantity for spatial averaging is defined as ψhi¼1 Vf ∫ Vf ψdV ð4Þ where Vfis the fluid volume contained in a volume V. The latter consists of a horizontal slab, extensive enough to eliminate plant-to-plant variations but thin enough to preserve the characteristic variation of properties in the vertical. The source term F in Eq. (2) is estimated by Eq. (5) for the current in the vegetated part of the pond: F¼−φν kρUi hi −φ2Cf ffiffiffi k pρUi hi Ui hijj ð5Þ where φis the vegetation porosity, ν(1*10−6m2/s) is the kinematic viscosity of the fluid, k is the intrinsic permeability (or “permeability”,m 2) and Cfis a dimensionless parameter. The source term due to vegetation is derived from the extended Darcy-Forchheimer equation (Whitaker 1996) based on the theory for porous media flow. It includes the laminar resistance force (first term on the right-hand side) as well as the form drag (second term on the right-hand side). These terms have different impacts depending on the prevailing flow regime. The laminar (Darcy) flow regime prevails for Rp< 1 (Rp is the pore Reynolds number) which is defined as Rp=U xd50/ν,withU x(m/s) the horizontal velocity and d50 the mean size of porous material. For 1 < Rp< 10 the boundary layer around the solid elements is more pronounced, while for 10 < Rp< 150, a non-linear relationship is observed between resistance and flow rate. For 150 < Rp<300andR p> 300, an unsteady laminar and a fully turbulent flow regime are observed, respectively. The d50 is defined as 1.5d (d = cylinder diameter) for a porous medium which consists of cylinders (Lowe et al. 2008). For a characteristic horizontal 203A Macroscopic Approach for Simulating Horizontal Convection in a... Content courtesy of Springer Nature, terms of use apply. Rights reserved. velocity Ux= 1 mm/s (Coates and Patterson 1993) and a typical d equal to 6 mm (in the field d may vary between 1 and 10 mm; Leonard and Luther 1995), Rpis approximately equal to 9, and hence, the laminar force is expected to dominate over the form drag. The determination of the permeability k and the dimensionless parameter Cfis as follows. The former can be determined using: (i) empirical relationships (Etminan et al. 2017) and (ii) a three-dimensional microscopic model for simple geometries (Tsakiri and Prinos 2016). Using methodology (i) above, k and Cfcan be determined by Eqs. (6)and(7), respectively (Johnson et al. 2005; van Gent 1995): k¼d2 50φ3 a11−φðÞ 2ð6Þ Cf¼b1 1−φ φffiffiffi k p d50 ð7Þ where a1and b1are empirical coefficients. The empirical coefficients a1and b1take different values, depending on the type and the length scale of the porous medium. For example, van Gent (1995) obtained values of a1between 1000 and 2000 and b1between 0.3 and 1.1. However, most of these values in literature are referred to permeable coastal structures and there is not sufficient information for aquatic canopies. The laminar resistance force depends on the permeability k and thus on a1, while the form drag depends on Cfand thus on b1. Using the UVdrag dominated velocity (m/s) methodology (ii) above, k is determined directly through a three-dimensional microscopic model, while the empirical coefficient Cfis determined from Eq. (7)withb 1equal to 1.1, as proposed by van Gent (1995). Tsakiri and Prinos (2016) have applied such a methodology which shows that k decreases with increasing (1-φ), with the relationship k = 1.60 × 10−7(1 - φ)-1.78 (regression coefficient R2= 0.99). Also, the empirical coefficient a1varies with φand decreases with increasing (1-φ) with the relationship a1= 150 (1 - φ)-0.57 (R2=0.98). 2.2 The Solar Radiation Model The solar radiation model is based on the Beer’s Law model stating that an incoming radiation is absorbed vertically through a water column. The model is introduced to the EnergyTemperature equation through an additional source term. The source term Shis an internal heating source, which represents the absorption of radiation by the water. It is added in the VAE equation (Eq. 3) only in the open region of the pond. In the vegetated region, the water does not absorb any radiation due to the cover of the free surface by vegetation. This internal heat source develops horizontal temperature gradients between the open water and the water in the region with vegetation, and these gradients induce circulation. For the extra source terms in the momentum and the energy equations, User Defined Functions (UDF) are used. Based on the Beer’s Law, the radiation intensity decreases with increasing water depth and the source term is given as: Sh¼1 ρ⋅CP ∑ N i¼1 niIie−nih−yðÞ ð8Þ where N is the number of bands, Iiis the surface radiation intensity within waveband i (W/m2), ηiis the attenuation coefficient within waveband i (m−1), h is the water depth and y is the 204 Papaioannou V., Prinos P. Content courtesy of Springer Nature, terms of use apply. Rights reserved. distance from the bottom of the tank (m). In order to compute the Iiintensities, the blackbody radiation distribution must be taken into account. The lamps used in past experiments (Coates and Patterson 1993; Lei and Patterson 2002a; Zhang and Nepf 2009)weremostlyof3200K color temperature generating a surface radiation heat flux Io, which was divided into i intensities based on the distribution of the spectral radiance as given in Fig. 1. As the color temperature decreases the spectral distribution moves towards the infrared region of the spectrum. The attenuation coefficient for the water depends on the wavelength of the radiation and the water turbidity. For a solar pond of 1 m depth, Rabl and Nielsen (1975) developed a model with N= 4 to quantify the variation of the solar radiation intensity. They concluded that the attenuation of the radiation through a water column cannot be described with a single exponential, because different wavelengths differ widely in their attenuation coefficients. Coates and Patterson (1993) developed a model with N= 3 using temperature measurements in a 300-mm water column and found that it reproduced the observed data accurately. Hattori et al. (2014) divided the spectrum of the attenuation coefficient using N= 50 with equal radiation intensity and found that the variation between the solution with N=50 and that with N= 3 is marginal. However, in limnological applications, the attenuation coefficient is usually characterized by a single bulk attenuation coefficient (Farrow and Patterson 1994; Lei and Patterson 2002a,b;ZhangandNepf2009). AmodelwithN= 3 is used in this work, based on Hattori et al. (2014) where the ni/n0 coefficient ratio was numerically derived using a least square fit of a three-wave band model to aN= 50 bands of equal intensity distribution. The percentage of the incoming radiation Io divided in the three-waveband model, as well as the ratio ni/n are presented in Table 1,where n0is the characteristic bulk attenuation coefficient. The spectrum of the attenuation coefficient for the water, which was used by Hattori et al. (2014), was obtained from the data of Hale and Querry (1973), shown in Fig. 2.Thesmallest attenuation coefficient is found in the ultra-violet and visible range, while longer waves are Fig. 1 Spectral radiance distribution for the color temperature of 3200 K 205A Macroscopic Approach for Simulating Horizontal Convection in a... Content courtesy of Springer Nature, terms of use apply. Rights reserved. increasingly attenuated. The infrared band of the spectrum is most easily absorbed in water and most of the energy associated with the infrared region is absorbed near the water surface. The attenuation of light decreases with decreasing wavelength and reaches a minimum attenuation for blue (Hale and Querry 1973; Rabl and Nielsen 1975), and then increases again in the ultraviolet (UV) region. About 53% of the total light energy is transformed into heat and absorbed in the first meter of water (Wetzel 2001). 2.3 Numerical Procedure The Fluent 15.0.7 CFD code is applied for the numerical computations using a control volume technique. The computational domain is divided into finite control volumes on which the governing equations are integrated. The Gambit program is used for the mesh generation. The segregated solution method is used and the velocity-pressure coupling is achieved with the SIMPLE algorithm. The PRESTO scheme is used for discretizing the pressure equation (continuity equation), while the Second Order Upwind scheme is used for the momentum and the energy equations (ANSYS Inc. 2013). The extra source terms F and Share introduced to the initial equations using User Defined Functions (UDF) based on C++ code. The estimated maximum horizontal velocity does not exceed 1 mm/s (Zhang and Nepf 2009), the Reynolds number is equal to 100 (Re = Uxh/ν), and hence, the generated convective currents are laminar. The time step is set equal to 0.1 s since the simulations are unsteady. Table 1 Three wave band model characteristics Band Wavelength (nm) Ii/I0ni/n0 1 < 900 0.252 0.24 2900–1200 0.194 5.25 3 > 1200 0.544 110.0 Fig. 2 Spectrum of attenuation coefficient for water 206 Papaioannou V., Prinos P. Content courtesy of Springer Nature, terms of use apply. Rights reserved. The macroscopic model is used in a two-dimensional rectangular tank (Fig. 3). Initially the velocity components in x and y directions are equal to zero and the temperature is equal to T0 in both the open and the vegetated domains. No-slip wall boundary conditions are applied at the bottom and the side walls of the tank as well as at the top surface. The temperature at the side walls remains constant and equal to the initial temperature (adiabatic condition), while at the bottom wall and the top surface ∂T ∂y¼0 is set. The grid is two-dimensional with orthogonal cells. For grid independent results, two meshes are considered for the case with the porosity φ= 0.97. The first has cell dimensions Δx=Δy = 0.001 m (6·104cells) and the second Δx=Δy = 0.0005 m (24·104cells), respectively. In Fig. 4, the velocity Ux, at distance x = −0.09 m, is shown for the two meshes at t = 300 s and 600 s. The difference between the results is less than 1.0%, and thus, the mesh with 6·104cells is used for all the cases studied. 2.4 Study Cases The simulation of the convective currents is based on the experimental model presented by Zhang and Nepf (2009), and hence, numerical results are compared with experimental measurements. The numerical tank has a total length L = 0.6 m and height equal to the water depth h which varies between 0.1 m and 0.2 m (Fig. 3). The tank has a vegetated area (Lveg = 0.3 m) and an area without vegetation (Lop = 0.3 m). The above dimensions are the same with those of the experimental tank. Initially, the tank contains water with density ρ= 998.2 kg/m3 and constant temperature T0= 294.55 K. At t = 0 s, a radiation is applied at the water surface and the water of the open region absorbs the incoming radiation. The surface radiation intensity I0is equal to 157 W/m2, while the bulk attenuation coefficient is found equal to n0=6.65m −1based on temperature measurements taken within the water column (Zhang and Nepf 2009). The surface radiation intensity remains constant during the simulation. The vegetation is simulated as rigid emergent cylinders with diameter d = 0.006 m. The vegetation porosity φvaries from 0.75 to 1.0 (no vegetation) for investigating the vegetation effects on the convective currents. The magnitude of the resistance drag increases with decreasing porosity φand permeability k, and hence, the velocity of the convective currents decreases. Using the method described in Hattori et al. (2014), the incoming radiation I0is split in three-wave bands with non-dimensional intensities (Ii/I0) equal to 0.544, 0.194 and 0.252 and attenuation coefficients (ni/n0) equal to 110, 5.25 and 0.24, respectively. All cases, both vegetated and non-vegetated, have a constant Prandtl number equal to 6.96. Fig. 3 Tank geometry, initial and boundary conditions 207A Macroscopic Approach for Simulating Horizontal Convection in a... Content courtesy of Springer Nature, terms of use apply. Rights reserved. g⋅β⋅Ιο⋅hI2 ρο⋅CP⋅ν  0:5 . The intrusion depth hIis derived from the numerical runs. It is shown that the velocity ratio decreases with increasing vegetative drag to viscous stress ratio. As the latter decreases (decreasing vegetative drag) the velocity ratio increases, and hence, the mean current velocity tends to the viscous–dominated velocity. The same tendency is also observed in the experimental measurements of Zhang and Nepf (2009). 4Conclusions Horizontal convective currents are numerically investigated in a vegetated pond. The convective currents are produced due to the differential absorption of solar radiation between the vegetated and the non-vegetated regions of the pond. A two-dimensional macroscopic model, based on the porous media theory, is applied for the simulation of the convective currents. The vegetation effects are considered through extra terms, which are based on the vegetation porosity and permeability, and include both the laminar resistance force and the form drag. Numerical macroscopic results are compared against available experimental measurements (Zhang and Nepf 2009). The following aspects provide new insight to the problem studied: (a) the use of vegetation with high as well as low porosity; (b) the approach used for determining vegetation resistance, based on vegetation permeability; and (c) the three-waveband attenuation model used for the absorption of radiation. The results, after upscaling, can be used effectively in the field (lakes and reservoirs) where convective flows affect significantly the transport of nutrients and other chemicals. The following conclusions can be derived from the previous analysis: (a) The effect of the bulk attenuation coefficient n0on the current characteristics (for typical values ranging from 6.65 to 10 m−1) is more significant for the one-waveband model Fig. 13 Variation of mean current velocity with vegetation permeability 214 Papaioannou V., Prinos P. Content courtesy of Springer Nature, terms of use apply. Rights reserved. rather than for the three-waveband model. The effect is more pronounced on the temperature increase, due to the direct effect of the source term dealing with the absorption of radiation, rather than on the velocity characteristics which are affected by other forces (pressure and drag) as well. (b) The effect of the radiation model (one-waveband, three-waveband) on the current characteristics is also important since each model heats differently the water column. The effect is more significant on the temperatures simulated by each model since the water temperature is affected directly by the incoming radiation and the way the long wave radiation is absorbed. In contrast, the velocities are less affected by the temperature difference since other forces are also important (pressure and drag). (c) The effect of the Grashof number on the dimensionless temperature increase and the dimensionless mean velocity is apparent even for the small range of values considered (from 7.75·106to 1.24·108). Both radiation models present the same behavior. The threewaveband model generates smaller temperature differences than those of the onewaveband model. Such differences are very similar with those observed in the experiments (Zhang and Nepf 2009), indicating the effectiveness of the three-waveband model to correctly simulate the absorption of the long wave radiation. (d) The mean current velocity decreases with increasing vegetation porosity φ(permeability k). When the ratio of vegetative drag to viscous stress tends to one, the mean current velocity tends to the viscous-dominated velocity scale. Computed and experimental velocities (Zhang and Nepf 2009) show the same tendency. (e) The present macroscopic model, which is not time-consuming, is found to be appropriate for the simulation of horizontal convection in vegetated ponds. A three-waveband attenuation model used for the absorption of radiation is shown to perform better in predicting the temperature increase due to the motion of the horizontal convective currents. 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. Author’s Contributions 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. Data Availability All data are available by the authors upon request. Compliance with Ethical Standards All the ethical standards were followed. Conflict of interest On behalf of all authors, the corresponding author states that there is no conflict of interest. Code Availability The C++ code used in the Fluent’s UDF is available by the authors upon request. Nomenclature 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, 215A Macroscopic Approach for Simulating Horizontal Convection in a... Content courtesy of Springer Nature, terms of use apply. Rights reserved. 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,Lengthoftheopenregion(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. 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