Diurnal heating and cooling of a sloping water body with vegetated shallow regions

ORIGINAL ARTICLE Diurnal heating and cooling of a sloping water body with vegetated shallow regions Panagiotis Prinos 1 Vassilios Papaioannou 1 Received: 13 June 2023 / Accepted: 18 December 2023 / Published online: 20 January 2024 The Author(s) 2024 Abstract In this work, diurnal heating, by solar radiation, and cooling in a sloping water body with vegetated shallow regions are investigated numerically. The sloping water body consists of a vegetated region with a bottom slope equal to 0.1 and a deep region with a horizontal bottom. The Volume-Averaged Navier-Stokes equations together with the Volume-Averaged Energy equation are solved numerically in the vegetated region. The latter has porosity equal to 0.85 (typical of aquatic plants found in lakes) and a length equal to the total length of the sloping region. At the top free surface, a time-dependent thermal forcing is applied, which is reduced in the vegetated region. Shallow water is considered with a maximum water depth being less than the penetration depth of the solar radiation. The non-vegetated sloping water body is also considered for validation purposes. The results (isotherms, streamlines, exchange flow rate), after ten full thermal forcing cycles, indicate significant vegetation effects on the daytime and night circulation and the exchange flow rates. Keywords Circulation · Cooling · Solar radiation · Natural convection · Vegetation 1 Introduction Natural convection in sloping water bodies (lakes, reservoirs, wetlands) can occur when surface waters are heated or cooled with the same rate [1–5]. The natural convection induced by diurnal heating and cooling in a sloping water body has been investigated by few researchers [6–8]. Field observations [9–11] have shown that the circulation in the littoral region of a lake is not in phase with the thermal forcing, while Farrow and Patterson [6] confirmed the lag of the flow response to the thermal forcing and indicated that such a lag can be up to 12 h. Lei and Patterson [7], based on numerical simulations, found out that there is a distinct time lag in the overflow response to the switches of the thermal forcing and this lag depends on the Grashof number. Dittko et al. [8] &Panagiotis Prinos [email protected] Vassilios Papaioannou [email protected] 1 Hydraulics Laboratory, Department of Civil Engineering, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece 123 Environmental Fluid Mechanics (2024) 24:57–74 https://doi.org/10.1007/s10652-023-09965-7(0123456789().,-volV)(0123456789().,-volV) Content courtesy of Springer Nature, terms of use apply. Rights reserved. used LES to model two lake sidearms which are part of Lake Audrey, NJ USA and Lake Alexandria, SA, Australia. They concluded that the average and maximum volumetric flow rates have a linear relationship with the cubic root of the Rayleigh number for both heating and cooling phases in the two side arms. The vegetation presence in the shallow nearshore region of sloping water bodies and its effect on natural convection induced by cooling or heating have been studied by few researchers [12–17]. The vegetation effects on diurnal heating and cooling have been investigated numerically by Lin and Wu [18] and Ji et al. [19] in a water body of simple triangular cross section. Using asymptotic solutions, they determined the effect of vegetation shading (blockage), during daytime heating and nighttime cooling, on the exchange flow rates and the general circulation in a sloping water body with a very small slope (equal to 10 −5 ). The asymptotic solutions were able to provide reasonable estimates of horizontal velocity and exchange flow rates, however they ignored the non-linear terms of the governing equations, excluded the flow instabilities, and limited the vegetation density. In this work a 2D water body model is used, which consists of a compound cross section. One region with a slope S o and length L s and another region with horizontal bottom (deep region) and length L d is considered. The vegetation, with porosity φ(the ratio of volume occupied by water to the total volume), is present in the sloping region with length L v equal to L s and its effect on the periodic solar radiation and surface cooling is investigated numerically for the first time. A 2D simulation can adequately reproduce the nature of aflow that is approximately 2D in nature. It cannot capture 3D aspects of the flow. More details about the validity of 2D simulations in such studies are discussed by Lei and Patterson [1]. The unsteady 2D Volume-Averaged Navier-Stokes (VANS) equations are solved in conjunction with the unsteady 2D Volume-Averaged equation in the vegetated region, using the Boussinesq approach used by Papaioannou and Prinos [20]. The averaging of the Navier-Stokes equation and the energy equation is based on the method of Volume Averaging according to Whitaker [21]. In the non-vegetated region these equations are simplified to the classical NS and Energy equations. Initially, the non-vegetated sloping water body is considered for validation purposes since numerical results of Lei and Patterson [7] are available. Also, it is used for comparing the results with those of the vegetated cases and identifying the vegetation effects. The water body with vegetation in the sloping region is considered next. During the daytime heating due to solar radiation, a time-dependent thermal forcing is applied at the top free surface with a peak flux I o in the deep region and I v , equal to φI o , in the vegetated region (the top of the emergent vegetation blocks 15% of the incoming radiation). During the nighttime cooling, heat loss through the free surface occurs with peak flux I o in the deep region and I v , equal to φI o , in the vegetated region. Also, for investigating the effect of the vegetation shading, the case with complete blocking of incoming heating and cooling by the top of vegetation is considered (I v = 0.0). This is an extreme case, however, in the field, Lövstedt & Bengtsson [15] found that Phragmites australis vegetation, with vertical straws and horizontal leafs, was able to reduce the heating 85% in comparison with that in the open lake. Special emphasis is given to the vegetation effects (a) on the exchange flow rates during heating and cooling for water bodies with shallow waters (h is less than the penetration depth of the solar radiation (h\η -1 ), (b) on the time lag of the flow response to the switch between daytime radiative heating and nighttime surface cooling. Also, the effect of vegetation shading on the flow response during heating and cooling is considered. 123 58 Environmental Fluid Mechanics (2024) 24:57–74 Content courtesy of Springer Nature, terms of use apply. Rights reserved. 2 Model specification and numerical procedure 2.1 Model specification A two-dimensional (2D) model of the sloping water body is considered (Fig. 1). Its cross section is compound with two distinct regions, (1) a sloping region with length L s = 1.0 m, maximum flow depth h=0.1 m, slope S o (=h/L s ) equal to 0.1 and vegetation length L v equal to L s , (2) a region of length L d and uniform water depth (horizontal bottom) with no vegetation. The total length is L=2.0 m. The model dimensions are those used by Lei and Patterson [7] for comparison purposes. The adopted Cartesian coordinate system starts at the left bottom point of the model (Fig. 1). The (a) geometry is the non-vegetated case with peak radiation flux I o equal to 50 W/m 2 . The absorption coefficient ηis set equal to 6.2 m −1 (the dimensionless absorption coefficient, ηh, is equal to 0.62). The maximum water depth h is less that the penetration depth of the solar radiation (h\η −1 ) and hence shallow water body is considered. It is used in this work for validation purposes and computational results are compared against those of Lei and Patterson [7]. The vegetation porosity φis equal to 0.85 and the peak flux I v is either equal to 0.85 I o , or 0.0. The slope S o is kept constant and equal to 0.1 which is considered as a large bottom slope. The Rayleigh number Ra, based on peak radiation flux I o and maximum flow depth h, is set equal to 1.21×10 8 (Ra = gbIoh4 qoCpvk2,g= acceleration due to gravity, β=thermal expansion coefficient, ρ ο =initial water density, C P = specific heat of water, v=kinematic viscosity, k=thermal diffusivity). The same Ra number has been used by Lei &Patterson [7] for both experiments and numerical simulation. Two types of thermal forcing are developed for simulating the diurnal cycle. They are represented by the sinusoidal function I surf =I o sin(2πt/P) which is shown in Fig. 2. The flux at the water surface is I surf , the peak flux is I o , t is the time and P is the period of the thermal forcing (14,000 s). During the first half of every forcing cycle (shaded areas of Fig. 2) solar radiation enters the water body from the free surface and hence I surf is positive. It is absorbed by the water according to the Beer’s law and adopting a single-band model, the radiation flux I at a given distance from the free surface is given as I=I surf ×e (h−y)η . During the second half of the forcing cycle there is a heat loss through the free surface and hence I surf is negative. The present diurnal model ensures that the total heat absorbed by the water body during daytime is in balance with the total heat released by the water body during the night cooling. The parameters associated with the flow dynamics are (a) the Fig. 1 Water body Geometry with and without vegetation 123 Environmental Fluid Mechanics (2024) 24:57–74 59 Content courtesy of Springer Nature, terms of use apply. Rights reserved. Prandtl number Pr (=7.07) and (b) the Rayleigh number Ra (=1.21×10 8 ). Heat exchange between water and vegetation is neglected. 2.2 Governing equations For the numerical simulation, the unsteady VANS equations are solved [Eqs. (1) and (2)] in conjunction with the unsteady VAE equation [Eq. (3)] in the vegetated region, using the Boussinesq approach. The derivation of these equations is based on the method of Volume Averaging used by Whitaker [21] and is presented in detail by [22–25], among others. In the non-vegetated region these equations are simplified to the classical NS and Energy equations. These equations are shown below: oUi hi f oxi ¼0ð1Þ oUi hi f otþUj  foUi hi oxj f ¼1 qo op hi oxi f þo oxj voUi hi f oxj þoUj  oxi f !"# þgbT hi fTo hi f  þFi ð2Þ oThi ot f þUj  foThi oxj f ¼o oxj koThi oxj f !"# þ\Sh[fð3Þ where U i = velocity in the direction i, p=pressure, T, T o =fluid temperature and initial temperature respectively, F i = resistance term due to vegetation and S h = heat source term determining the absorption of the solar radiation by the water body during daytime heating (It is set to zero during nighttime cooling). The symbols\[and \[ f indicate the superficial and the intrinsic volume average of a parameter A (scalar, vector, or tensor) respectively which are defined as: Fig. 2 Periodic thermal forcing at the free surface of the water body 123 60 Environmental Fluid Mechanics (2024) 24:57–74 Content courtesy of Springer Nature, terms of use apply. Rights reserved. A hi ¼1 VZ Vf AdV and A hi f¼1 VfZ Vf AdV ð4Þ where V f is 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 dimension. The term F i , due to vegetation, is given by the Darcy-Forchheimer equation which includes linear and non-linear forces using porous media characteristics (e.g., vegetation permeability k p ) Fi¼uv kp Ui hi fu2Cf ffiffiffiffiffi kp pUi hi fUi hi f ð5Þ where φ=vegetation porosity, k p = vegetation permeability (m 2 ) and C f = dimensionless empirical coefficient. The permeability k p and the dimensionless coefficient C f are determined by the following Eqs. (6) and (7)[26,27]. kp¼d2 50u3 a1ð1uÞ2ð6Þ Cf¼b1 1u uffiffiffiffiffi kp p d50 ð7Þ where d 50 is the grain diameter, a 1 and b 1 are empirical coefficients. When the porous medium consists of cylinders then d 50 is equal to 1.5 d (d is the diameter of the cylinders, Lowe et al. [26]). The empirical coefficients a 1 and b 1 take different values, depending on the type and the length scale of the porous medium. For example, Losada et al. [28] obtained values of a 1 between 1000 and 2000 and b 1 between 0.3 and 1.1. For φ=0.85, d 50 =9.0 mm, α 1 =1000 and b 1 =1.1 the values for C f and k p are calculated as: C f = 0.484, k p = 5.04×10 −6 m 2 . The term S h is given by the following: Sh¼ Iosin 2pt P  gegðhyÞ;if sin 2pt P  0 0;if sin 2pt P  \0 8 > > < > > : ð8Þ The boundary conditions are defined as follows: a) The left tip on Fig. 1is cut off at x=0.016 m for avoiding a singularity and the side wall is assumed to be rigid non-slip and adiabatic. b) The side wall in the deep region is also assumed to be rigid non-slip and adiabatic. c) The bottom of the water body is also rigid, non-slip and the following thermal flux is applied, assuming that the radiation absorbed by the boundary is conducted back into the fluid, Lei & Patterson [29]. 123 Environmental Fluid Mechanics (2024) 24:57–74 61 Content courtesy of Springer Nature, terms of use apply. Rights reserved. oT on¼ 1 kqoCp uIosin 2pt P  egðhyÞ;if sin 2pt P  0 0;if sin 2pt P  \0 8 > > < > > : ð9Þ where n is the direction normal to the bottom boundary. d) The water surface is assumed to be stress free and the applied thermal boundary condition is given by: oT oy¼ 0;if sin 2pt P  0 1 kqoCp uIosin 2pt P  ;if sin 2pt P  \0 8 > > < > > : ð10Þ Initially, the water body has a temperature T o = 293.15 K and is motionless. 2.3 Numerical scheme and mesh dependency tests The Fluent 15.0.7 CFD code is applied for the numerical computations using a control volume technique. The extra source terms F i and S h are introduced to the initial equations using User Defined Functions (UDF) based on C++code. 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 QUICK scheme is used for the momentum and the energy equations [30]. The time step is set equal to 0.1 s. Such a time step is selected, in conjunction with the Δx (the grid size in the x direction) and the computed velocities, for satisfying the Courant-Friedrichs-Lewy (CFL ¼uDt=Dx\1) criterion. The maximum CFL number in the simulations was 0.52. The total simulation time is 10P. Three different grids were tested for examining the solution dependence on the grid spacing. The maximum number of grid points (in the x and y directions) was 1000×100= 1×10 5 , 4000×200=8×10 5 and 8000×400=3.2×10 6 . The grid size in the y direction was smaller in the vicinity of the top and bottom boundaries while it was uniform in the x direction. The smallest cell dimensions (Δx×Δy), adjacent to the top and bottom boundary, were 1.0 mm × 0.16 mm, 0.5 mm × 0.08 and 0.25 mm × 0.04 mm for the three meshes while the biggest ones, in the deep section of h=0.1 m, were 1.0 mm × 1.4 mm, 0.5 mm × 0.7 mm and 0.25 mm × 0.35 mm. The mesh with 3.2×10 6 was finally used which gave better structured plumes (more complete eddying motion in the plumes) in both the cooling and heating phases. Detailed results about the grid-sensitivity analysis are presented by Papaioannou and Prinos [17]. 3 Analysis of results Numerical results (temperatures, streamlines, exchange flow rates) for all cases are presented and analysed in the following paragraphs. Initially the sloping water body with no vegetation is presented and analysed which is used for validation purposes as well as for comparison with the cases with vegetation. 123 62 Environmental Fluid Mechanics (2024) 24:57–74 Content courtesy of Springer Nature, terms of use apply. Rights reserved. Figure 3shows the temperature difference ΔΤ (=T −T o ), made dimensionless with I o h/κ,at six characteristic times of the last forcing cycle. The first three (t/P=0.0, 0.1 and 0.5) correspond to the daytime heating process while the last three (t/P=0.7, 0.9 and 0.98) to the night cooling. At the beginning of the last cycle (t/P=0.0) the isotherms show that gravity currents, formed during the previous cooling phase, continue to flow downward along the sloping bottom and the horizontal bed. At t/P=0.1 the temperature of the water body increases due to the absorption of radiation. The increase is higher in the shallow region of the water body and less in the deeper part. The above findings are supported by the streamlines (ψ/ρ ο L s k) shown in Fig. 4at the same times (ψis defined as mass discharge (kg/s)). Initially, at t/P=0.0 the streamlines show a large-scale overturning circulation with multiple cores which has been developed during the preceding cooling phase as it can also be seen at t/P=0.9 and 0.98 of the last cycle. At t/ P=0.1 a cellular flow structure is shown in the deep region of the waterbody which is due to the rising thermals while a large recirculation is observed in the sloping region of the water body, due to the development of a horizontal temperature difference between the shallow and the less shallow waters of the sloping region. This large recirculation, with clockwise rotation, is further developed during the heating phase which extends through the whole water body (t/P=0.5). During the cooling phase the growing thermal boundary layer at the free surface and the instabilities in the form of sinking plumes break the general circulation, developed previously, and at t/P=0.7 a cellular flow structure is formed in the main part of the deep region. Also, the gravity currents, Fig. 3 Contours of (ΔT/(I o h/κ)) at various stages for a diurnal thermal forcing (water body with no vegetation) 123 Environmental Fluid Mechanics (2024) 24:57–74 63 Content courtesy of Springer Nature, terms of use apply. Rights reserved. developed in the nearshore sloping region, result in a recirculation pattern which flushes the whole sloping region and part of the deep section. As the gravity currents continue to travel into the deep region this recirculation expands further covering the whole water body (t/P= 0.9 and 0.98). The above findings during the heating and cooling phase are in agreement with those found by Lei and Patterson [7] for shallow waters (Fig. 3of Lei and Patterson [7]). Τhe variation of the averaged exchange flow rate Q(made dimensionless with k) with time t/Pis presented in Fig. 5for the last five cycles in conjunction with the corresponding thermal forcing. Q is determined by integrating the flow rate Q(x), at a given xlocation, along the horizontal direction x. Hence, Q¼1 LZ L 0 QðxÞdx where QðxÞ¼1 2Z h 0 u jj dy ð11Þ Figure 5shows that, in the beginning of the daytime heating, the thermal forcing increases continuously, however the Q/k decreases, due to the residual night time circulation from the preceding cooling phase. This decrease continues up to approximately t/P=0.06 for each cycle. After t/P=0.06, Q starts to increase in phase with the increasing thermal forcing. Fluctuations of Q during the heating phase are apparent, especially around the heating peaks which are due to the instabilities associated with the bottom heating. When the radiative heating becomes sufficiently weak and the instabilities are weak Q decreases, even in the beginning of the cooling phase. There is also a time lag between the cooling increase Fig. 4 Streamlines (ψ/ρ ο hk) at various stages for a diurnal thermal forcing (water body with no vegetation). The dashed and solid lines show clockwise and anticlockwise rotation respectively 123 64 Environmental Fluid Mechanics (2024) 24:57–74 Content courtesy of Springer Nature, terms of use apply. Rights reserved. and the exchange flow increase while thermal instabilities are present for most of the cooling phase. After the time of maximum cooling (e.g. t/P=0.75) the exchange flow rate continues to increase slightly and then starts to decrease. The strength of the circulation in the heating phase and the associated exchange flow rate is approximately of the same order as that in the cooling phase. Computed results are found in satisfactory agreement with those of Lei and Patterson [7]. The mean Q/k of the present simulations is 192.3 while that of Lei and Patterson [7] is 217.8 (11.7% difference). In the following paragraphs the vegetation effects on the flow development and flow regimes are investigated for the water body with a vegetated sloping region. Figures 6and 7show the isotherms and the streamlines respectively for vegetation in the sloping region, with I v = 0.85I o (the vegetation blocks 15% of the incoming heating and cooling). At t/P=0.0 the isotherms show that the gravity currents, formed during the previous cooling phase (t/P=0.98), move very slowly, due to the lower horizontal temperature gradient developed in this case, and remain in the sloping region even at t/P=0.1. The rest of the water body is heated continuously with hot rising plumes developed at the bottom of the deep region (instabilities) due to the re-emission of energy from the bottom wall to the water. At t/P=0.50 the shallow region is heated faster than the deep region but not as fast as the shallow non-vegetated region (Fig. 3), due to I v =φΙ ο . In the cooling phase (t/P=0.70) cold sinking plumes are developed along the top surface of the water body, however, the horizontal temperature gradient and the developed gravity currents are weak and move slowly in the sloping region for the whole cooling phase (t/P=0.90 and 0.98) and for the subsequent heating phase. Most of the water body has mean temperatures above the initial ones (positive ΔT) for the full cycle. Fig. 5 Calculated horizontal exchange flow rate over one the last five cycles (Ra=1.21×10 8 ) 123 Environmental Fluid Mechanics (2024) 24:57–74 65 Content courtesy of Springer Nature, terms of use apply. Rights reserved. (c) For the water body with a vegetation which completely blocks the surface heating and cooling the significant difference in radiation flux and temperature between the vegetated and the non-vegetated free surface results in alternate large-scale circulation patterns with night and early day (t/P=0.90, 0.97 and 0.0) convective currents along the free surface and late day (t/P=0.50) gravity currents along the bottom with a largescale overturning circulation. (d) The variation of the exchange flow rate, the maximum, minimum and average temperature of the water body with vegetation in the sloping region indicates that the exchange flow rate is lower (42.2%) for a vegetated sloping region with I v =φΙ ο than that with no vegetation while the variation of temperature (ΔT max ,ΔT min and ΔT avg ) is not significant for I v = 0.0 in comparison with that of the other two cases. (e) The velocity profiles at two locations within the sloping region (x/h=4.0 and 10.0) indicated the decrease of velocity due to vegetation resistance at x/h=4.0, while, at the vegetated/non-vegetated interface (x/h=10.0), profiles, of low velocities, with threeor fourlayers are apparent due to the counter-rotating cells developed there. Acknowledgements Results presented in this work have been produced using the Aristotle University of Thessaloniki (AUTh) High Performance Computing Infrastructure and Recourses. Fig. 12 Diurnal variation of velocity u/(k/h) at (a) x/h=4.0, (b) x/h=10.0, for a water body with and without vegetation 123 72 Environmental Fluid Mechanics (2024) 24:57–74 Content courtesy of Springer Nature, terms of use apply. Rights reserved. Author contributions Professor PP wrote the abstract, introduction and conclusions sessions. PhD candidate PV prepared Figs. 1,2,3,4,5,6,7,8,9,10,11 and 12 and wrote the numerical procedure and the analysis of the results. Both authors reviewed the manuscript. Funding Open access funding provided by HEAL-Link Greece. The authors did not receive support from any organization for the submitted work. Data availability Data can be provided upon request. Declarations Conflict of interest The authors declare no competing interests. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. 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