Effects of Rayleigh number on thermal siphons in Triangular Water Bodies Vassilios PAPAIOANNOU¹, Panagiotis PRINOS² ¹ Senior Researcher, Information Technologies Institute/ CERTH, Thessaloniki, Greece, email:
[email protected] ² Emeritus Professor, Dept. of Civil Eng./ AUTh, Thessaloniki, Greece, email:
[email protected] https://sciforum.net/event/ECWS-9 INTRODUCTION & AIM CONCLUSION FUTURE WORK / REFERENCES METHOD Natural convection in enclosed or semi-enclosed water bodies (e.g., lakes, reservoirs) drives vertical and horizontal heat and momentum exchange. Surface cooling creates density differences between shallow and deep regions, generating thermal siphons [1], [2]. In triangular or sloping-bottom basins, nearshore areas cool faster, triggering downslope gravity currents and upwelling in deeper zones, affecting mixing, stratification, and heat distribution [3]. Most studies focus on low to moderate Rayleigh numbers (Ra) [4], while high-Ra turbulent flows remain less explored. This study aims to: 1. Examine thermal siphon formation under surface cooling in triangular basins at high Ra. 2. Compare Large Eddy Simulation (LES) with WALE subgrid modeling to 2D DNS results. 3. Analyze temperature and flow fields to assess interactions between downslope currents and convective plumes. Future work will extend simulations to 3D, explore LES-RANS hybrids, and investigate effects of uneven cooling, wind, and seasonal changes, as well as extreme Ra and long-term cooling to better understand turbulent thermal siphons. [1] Mao, Y., Lei, C., & Patterson, J. C. (2010). Unsteady near-shore natural convection induced by surface cooling. Journal of Fluid Mechanics, 642,213–233. https://doi.org/10.1017/S0022112009991765 [2] Papaioannou, V., & Prinos, P. (2023). Natural convection due to surface cooling in sloping water bodies with vegetation. Journal of Hydraulic Research, 61,382–395. https://doi.org/10.1080/00221686.2023.2222094 [3] Ulloa, H. N., Ramón, C. L., Doda, T., Wüest, A., & Bouffard, D. (2022). Development of overturning circulation in sloping waterbodies due to surface cooling. Journal of Fluid Mechanics, 930, A18.https://doi.org/10.1017/jfm.2021.883 [4] Doda, T., Ulloa, H. N., Ramón, C. L., Wüest, A., & Bouffard, D. (2023). Penetrative convection modifies the dynamics of downslope gravity currents. Geophysical Research Letters, 50. https://doi.org/10.1029/2022GL100633 The numerical models for simulating thermal siphons in water bodies, particularly at high Rayleigh numbers (turbulent natural convection), fall into two categories: 1. Large Eddy Simulation (LES) models (2D), which require significant computational resources, and 2. Reynolds-Averaged Navier–Stokes (RANS) models, coupled with a turbulence model to account for turbulence effects. In this study, Large Eddy Simulation (LES) is employed to investigate the quasi-steady state behavior of thermal siphons induced by surface cooling at high Ra numbers, using the WALE model to account for subgrid-scale turbulence and compares results with 2D DNS. Simulations use a time step Δt = 0.1 s (CFL condition) over 30,000 s. Boundary conditions (Fig. 1) are rigid, non-slip, adiabatic sides and bottom (∂𝑇/∂𝑛=0), a stress-free water surface with an applied thermal flux (−∂𝑇/∂𝑦=B0/(gβk), and an initially motionless water body at 293.15 K. Figure 1. Characteristic flow regions (left) and conceptual model (right) •The surface buoyancy outflow, B0(m2/s3)is defined as B0=gβI0/ρ0Cp,where g: gravitational acceleration [m/s2], β: thermal expansion coefficient [1/K]: surface cooling flux I0[W/m2], ρ0: fluid density [kg/m3], Cp: water specific heat capacity at constant pressure [J/(kg⋅K)]. •The Rayleigh number, Ra is defined as Ra =𝐵0ℎ4/(𝜈𝑘2), where: 𝜈: kinematic viscosity [m2/s], h: maximum water depth [m], k: thermal diffusivity [m2/s]. •The Prandtl number, 𝑃𝑟 is Pr =𝜈/𝑘= 7.07 for water. •Characteristic time scales were estimated following Ulloa et al. [3] and adapted for triangular water bodies for: ▪the onset of thermal instabilities at the free surface, τB≈ √657.5 √(ν/B0).It decreases from 1813.1 s up to 57.3 s with increasing Ra from 1010 up to 1013 ▪the time for plumes to reach the bottom, τRB ≈h2/3/B01/3.It decreases from 1710.0 s up to 171.0 s with increasing Ra number ▪the quasi-steady state, τSS (≈2L2/3/B01/3 ,L= total body length). It also decreases from 15873.7 s up to 1587.4 s with increasing Ra number. Figure 5 shows the large circulation pattern for the highest Ra number. Both DNS and LES capture the persistent large-scale circulation, though DNS reveals transient small-scale vortices mostly absent in LES. Figure 2. Temperature Ratio DNS (upper) & LES (lower) (Ra = 10¹¹, t/τss = 1.07) Figure 3. Stream-function DNS (upper) & LES (lower) (Ra = 10¹¹, t/τss = 1.07) Figure 4. Temperature ratio DNS (upper) & LES (lower) (Ra = 10¹³, t/τss = 5.04) Figure 5. Stream-function DNS (upper) & LES (lower) (Ra = 10¹³, t/τss = 5.04) For Ra =10¹¹ at t/τss = 1.07 the DNS field exhibits sharp thermal plumes and fine-scale temperature fluctuations, capturing the small-scale turbulent structures. In contrast, the LES smooths these gradients due to its subgrid-scale modeling, which filters out the smallest scales of motion. Despite these differences, both DNS and LES accurately reproduce the gravity current responsible for flushing the bottom layer. With increasing Ra (Ra =10¹3), Both cases exhibit similar large-scale flow structures and thermal patterns, indicating comparable overall mixing behavior. The DNS figure shows slightly cooler regions and sharper spatial gradients, reflected by the presence of more blue-toned areas. RESULTS & DISCUSSION The effect of Ra number on temperature and stream-function for the two highest Ra numbers studied is shown in the following figures. Figure 2 shows the temperature for Ra=1011. The DNS temperature field shows sharp plumes while the LES smooths gradients and underpredicts the peak by ~6%, slightly damping small-scale heat transfer. The DNS stream-function (Fig. 3) shows intricate vortices and secondary eddies, highlighting intense high-Ra turbulence. LES captures circulation, but misses smaller-scale features, with peak stream-function ψ₍DNS₎(=ψ/ρ0qc) equal to 2.88 while is equal 2.82 for LES. The temperature Τ/Τ0(Fig. 4) fields at t/τss = 5.04 show that the DNS exhibits finer thermal structures and larger temperature variations, while the LES appears smoother and warmer. The LES field spans a similar range (0.983≤𝑇/𝑇0≤1.0) with that of DNS (0.982≤𝑇/𝑇0≤1.0).
