Buoyancy-driven natural ventilation: the role of thermal stratification and its impact on model accuracy

. Since the invention of mechanical ventilation systems, natural ventilation has been deemed inferior compared to active systems for ventilation of buildings. The recent COVID-19 pandemic and raising awareness of climate change issues have rekindled the interests in natural ventilation as a sustainable method for ventilation and pollutant removal. Modelling natural ventilation is challenging due to uncontrollable outdoor conditions. Simple models such as the well-mixed air model assume uniform indoor air temperature. However, thermal stratification can induce significant temperature differences in the vertical direction, thereby violating the well-mixed assumption. This study evaluates the performance of the well-mixed model, the two-layer stratification model, and computational fluid dynamics (CFD) models in predicting the indoor air temperature under buoyancy-driven displacement ventilation. Compared to experimental measurements, the well-mixed model significantly overpredicts the indoor air temperature without thermal stratification since it assumes a uniform indoor air temperature. The two-layer stratification model overpredicts the upper layer air temperature and underpredicts the lower layer air temperature. The CFD models can capture the trend of the thermal stratification of a gradual increase in temperature with height. However, the CFD models underpredict the indoor air temperature, possibly due to errors introduced by the assumption of adiabatic indoor surfaces. Since simplified models cannot resolve thermal stratification, high-fidelity models, such as CFD models, should be used to model natural ventilation. Experimental studies of natural ventilation should include measurements of the thermal stratification, as well as the temperatures or heat fluxes on the indoor surfaces so the results can be used to develop and evaluate numerical models.


Introduction
Natural ventilation has been adopted for space cooling (or heating) until the invention of heating, ventilation, and air conditioning (HVAC) systems which allows for full control of the indoor environment. Since then, natural ventilation has been shunned by designers and planners due to its complexity and uncertainties from uncontrollable ambient environments [1]. However, with the growing awareness of the energy crisis and the impacts of climate change, low-energy and net-zeroenergy buildings have been gaining popularity. A large fraction of energy consumption in the building sector is contributed by space cooling and heating [2]. Therefore, adopting passive or natural ventilation has a huge potential to reduce energy consumption in the building sector.
Natural ventilation is induced by the pressure difference between indoor and outdoor. Wind-driven ventilation is induced by pressure difference generated by outdoor wind, while buoyancy-driven ventilation is driven by pressure difference generated by density difference in indoor and outdoor air of different temperatures. This paper focuses on buoyancy-driven ventilation under calm outdoor conditions, i.e., in the absence of outdoor wind or very weak outdoor wind. More specifically, we consider displacement ventilation * Corresponding author: lupwai@nus.edu.sg with more than one opening. In addition, we consider only cooling by natural ventilation, where the outdoor air has a lower temperature than the indoor air, which is often the case.
In a buoyancy-driven ventilated building, cold outdoor air enters through opening(s) at the lower level, while warm indoor air exits through opening(s) at the higher level. This induces a thermal stratification, where indoor air temperature increases with height. In other words, the indoor air temperature is not uniform. However, conventional building envelope models assume a uniform indoor temperature [3]. Though the zonal models can divide the indoor space into multiple zones, thus enabling spatial temperature variation, this can be achieved by a coarse-grid computational fluid dynamics (CFD) model at a similar computational cost [4]. This paper aims to quantify the errors of ignoring thermal stratification in buoyancy-driven natural ventilation flows. Two simplified models, namely the well-mixed model and the two-layer stratification model, as well as CFD models will be compared with experimental data to evaluate their performance in predicting indoor air temperature in a full-scale building under buoyancy-driven displacement ventilation. Section 2 describes the methodology and models, Section 3 provides the results and discussion, Section 4 concludes the study.

Methodology
Three methods to model buoyancy-driven ventilation are briefly described in this section. The well-mixed model is the simplest model by assuming a uniform indoor air temperature. The two-layer stratification model assumes two layers of indoor air: the less dense upper layer is warmer while the denser lower layer is cooler. The computational fluid dynamics (CFD) model solves for the velocity and temperature fields in the entire indoor volume and can therefore resolve the spatial variability of temperature variation in the indoor air. A major drawback of CFD models is that they are computationally expensive.

Well-mixed model
Under the well-mixed assumption, the indoor air temperature is uniform and takes a single value. The flow rate Q across an opening is expressed in Equation where Cd is the discharge coefficient, often assumed to be a constant where Cd = 0.6 [5]. A is the area of the opening, β is the thermal expansion coefficient, ΔT is the indoor-outdoor temperature difference, g is the gravitational constant, H is a vertical length scale. For two openings at different heights, H is the height difference between the openings (not the height of a room or a building). Under steady state and assuming no heat loss through the building envelope, indoor heat generation rate balances heat removal rate by natural ventilation, i.e., Egen = Env where E represents energy rate, subscripts gen and nv represent generation and natural ventilation. Since Env = ρCpQΔT, where ρ is density and Cp is specific heat capacity, ΔT can be solved by Equation (2): (2)

Two-layer stratification model
A simple deviation from the well-mixed assumption is the two-layer stratification model. An example of the two-layer stratification model is shown in Fig. 1 obtained from a water bath experiment [6]. Linden et al. [7] and Linden [8] formulated an analytical solution for displacement ventilation, predicting that a room with a constant heat source (i.e., source of buoyancy) will develop a stable, two-layer stratification flow when steady state is reached. The bottom layer has cold air with the same temperature as the outdoor air, while the top layer has a higher temperature. The height of the interface h, normalized by H, is determined by the effective opening area A * following Equation ( , where α is the entrainment constant for a thermal plume [9]. Hunt & Linden [10] estimated α ≈ 0.117, giving C ≈ 0.142. It is interesting to note from Equation (3) that ξ depends only on geometrical parameters but not on the strength of the buoyant source. The analytical solution has a good agreement with reduced-scale experiments, as shown in Fig. 2 [7].  [6] showing two stable layers of water of different temperatures formed by a thermal plume. Tu is the temperature of the upper layer temperature while Tl is the temperature of the lower layer. Once h is known, the air temperature of the upper layer, Tu, can be calculated. The reduced gravity is given in Equation (4): where Tl is the air temperature of the lower layer, which is known (equals the outdoor temperature). The strength of the buoyancy source, = , determines g' [6] following Equation (5): The only unknown in Equation (4) and Equation (5) is Tu, and therefore Tu can be solved.

Computational fluid dynamics model
Computational Fluid Dynamics (CFD) models explicitly resolve the geometry of a building and its openings. CFD models are computationally expensive, but they solve and provide the velocity and temperature fields in the entire indoor volume. Large eddy simulation (LES) is generally more accurate but computationally intensive, while Reynolds-Averaged Navier-Stokes (RANS) simulations are computationally efficient but less accurate, especially for non-isothermal flows [11]. CFD models also have the advantage of modeling full-scale buildings with full controls of the boundary conditions. However, CFD models need to be validated with experiments. Ideally, full-scale CFD models should be validated with full-scale experiments, although many experiments were conducted at a reduced scale. We chose the full-scale experiment conducted by Yang et al. [12] in our study for three reasons. First, it has a simple geometry with only two openings, as shown in Fig. 3. Second, it measured seven locations in the vertical direction so the thermal stratification can be estimated. Third, the measurements were collected over a long period to achieve a quasisteady state, so the results can be directly compared with the (steady) well-mixed model and the two-layer stratification model. Fig. 3. The geometry of the test room in Yang et al. [12]. A heat source of 100 W is placed in the middle of the room (not shown).
All CFD simulations are run with ANSYS Fluent 2021 R1. The geometry follows that in Fig. 3 in the experiment [12]. The model is meshed with ANSYS Meshing with 220k hexahedral grids, as shown in Fig.  4. The mesh is refined near the vents (openings) and near the heater to more accurately resolve the flows across the openings and the buoyant flow induced by the heater. All grids are hexahedral with zero skewness, ensuring a high-quality structured mesh. The maximum grid expansion ratio is 1.2.
In this study, both steady RANS and LES are run to compare their results with experiment. The initial and boundary conditions are derived to resemble the experimental conditions [12]. Both RANS and LES are initialized with a uniform initial temperature of 26.3 °C and zero velocities. The heater is modeled as a volumetric heat source releasing heat at 100 W in the middle of the domain (see Fig. 4). The no-slip boundary condition is applied to all room surfaces. The heat conduction through the room surfaces was not measured in the experiment, so we run two RANS simulations, one with an adiabatic thermal boundary condition for all room surfaces, the other with a constant temperature (equals the initial indoor air temperature) boundary condition. Due to its high computational resources, the LES is run with only the adiabatic room surface boundary condition. The lower opening is a pressure inlet with an inlet temperature of 22.1 °C. The upper opening is a pressure outlet.
The temperature difference in this study is small (< 10 °C) so Boussinesq approximation is employed to model the effect of buoyancy. The Coupled Scheme with the Pseudo Time Method is used for pressurevelocity coupling. The k-ω SST turbulence closure scheme is used for the RANS model, while the Wall-Adapting Local Eddy-Viscosity (WALE) Model is used as the subgrid-scale model in the LES. Second order differencing/upwind schemes are used for spatial discretization. For the LES, the Bounded Second Order Implicit Scheme is used for time discretization. The time step size is allowed to vary such that the Courant Number < 1.0 to ensure stability. This results in a time step size of 0.01 s to 0.03 s. The experiment shows that the measured temperatures reach a quasi-steady state after about 2 hours, so the LES is run for two hours (simulation time) as a ramp up period. The simulation is then continued for another 30 minutes simulation time, with time-averaging turned on. The averaging period of 30 minutes is verified to be sufficient, as further averaging for another 30 minutes does not change the results. The RANS takes about 4 CPU-hours to complete, while the LES takes about 3000 CPU-hours.

Results and discussion
In the experiment of Yang et al. [12], the outdoor temperature is 22.1 °C, A * = 0.040 m 2 , Egen = 100 W, B = 0.0027 m 4 s -3 , and H = 1.62 m (note that H is the height difference between the lower and upper openings, not the room height, which is 2.6 m).

Results from well-mixed model
From Fig. 3, the upper opening area of the test room in Yang et al. [12] is 0.042 m 2 , smaller than the lower opening area of 0.072 m 2 . From Section 2.1, Equation (2) gives ΔT = 5.9 °C for A = 0.042 m 2 and ΔT = 4.1 °C for A = 0.072 m 2 . The outdoor temperature is 22.1 °C so the well-mixed indoor air temperature is between 26.2 °C and 28.0 °C. Since both openings are relatively small compared to the length scale of the room (the room height is 2.6 m), the opening with a smaller area (i.e., the upper opening) has a larger limiting effect on the flow rate. Therefore, the well-mixed indoor air temperature should be closer to 28.0 °C, the upper limit.

Results from two-layer stratification model
For the two-layer stratification model, Equation (3) gives ξ = 0.52 and hence h = 0.85 m under a steady state. Substituting h into Equation (5) gives g' = 0.18 ms -2 and substituting g' into Equation (4) gives Tu − Tl = 5.5 °C. Since Tl = 22.1 °C (same as the outdoor temperature), Tu = 27.6 °C. Note that ξ = h/H, where H = 1.62 m is the height difference between the upper and lower openings, not the height of the room (2.6 m as shown in Fig. 3). The vertical space between the upper opening and the ceiling (between 1.62 m and 2.6 m) is filled with warm air of temperature Tu, and this space does not affect h. Even if the ceiling height is much taller, say 5 m, h remains unchanged unless H is varied. Therefore, to increase h or to maximize the flow rate of natural ventilation, the upper opening should be placed as high as possible to maximize H.  Fig. 5(a) plots the temperature contour on the plane between the heater and a wall (see Fig. 4 for the location of the plane), while Fig. 5(b) plots the vertical temperature profile on the dashed line in Fig. 5(a). This plane and line are selected because experimental data is available [12] and the comparison of the results to experiment will be discussed in Section 3.4. The temperature contour in Fig. 5(a) indicates thermal stratification, where the nearground temperature is about 22 °C while the near-ceiling temperature is about 25 °C. The dotted line in Fig. 5(b) indicates the height of the upper opening, H. As discussed in Section 3.2, the space between the upper opening and the ceiling (height 1.62 m to 2.6 m) does not affect the flow: it simply acts as an additional space for the accumulation of warm air. Therefore, the temperature is constant at about 25 °C in this space. We will focus on the temperature profile between the ground and H (height 0 m to 1.62 m). The most striking feature of the temperature contour and profile in Fig. 5 is that the thermal stratification is continuous and varies almost linearly, except near H. This (nearly) linear thermal stratification indicates that the indoor air is neither uniform in temperature (the well-mixed assumption) nor having two distinct temperatures (two-layer stratification model). The timeaveraged results from LES is similar to that from RANS and is not shown here. The next section will compare the results from all models (including LES) with the experimental data. First, the measured temperatures in the experiment (the open symbols) shows that the air is not well-mixed but is thermally stratified, where the near-ground temperature is about 22 °C while the temperature at height 1.6 m is about 26 °C. The experimental data also shows that the temperature increases gradually with height, instead of having a layer of warm air above another layer of cool air.

Result comparison
Second, the profiles obtained from the well-mixed model (blue dash-dot lines) indicate the lower and upper limits of the well-mixed air temperatures at 26.2 °C and 28.0 °C (see Section 3.1). The experiment shows that the highest measured temperature is below the lower limit predicted by the well-mixed model. In other words, the well-mixed model significantly overpredicts the air temperature.
Third, the profile obtained from the two-layer stratification model (red solid line) shows that the model underpredicts the temperature of the lower layer, which is expected, as the temperature of the lower layer is https://doi.org/10.1051/e3sconf/202339602038 IAQVEC2023 equals to the outdoor temperature. The temperature of the upper layer is overpredicted. The limitation of the two-layer stratification model is that it assumes the temperature in each layer to be uniform, therefore it cannot accurately predict the gradual increase of temperature with height as observed in the experiment. Fourth, the profile obtained from the steady RANS CFD simulation (solid black line), which is the same profile in Fig. 5(b), shows that the thermal stratification trend is well captured but the temperature is consistently underpredicted. The profile obtained from LES (black dotted line) agrees better with the experiment but still underpredicts the temperature. These two profiles are obtained from the CFD simulations with adiabatic room surfaces, i.e., there is no heat transfer from the room surfaces to the air. In the experiments, the initial condition was achieved by conditioning the room to reach a uniform temperature using ceiling fans and airconditioning systems before the start of each experiment [12]. The room surface temperature was not measured. However, it is reasonable to assume that the room surface temperature has reached an equilibrium with the indoor air temperature during the initial conditioning of the room (else, the indoor air temperature will continue to vary until an equilibrium is reached). The room surface temperature will drop when the natural ventilation starts. Since the surface temperature is not measured, the heat transfer from room surfaces to air cannot be calculated, but we can estimate an upper limit of this heat transfer by assuming that the room surface temperature equals the initial air temperature. Another RANS simulation is run with a constant temperature boundary condition on all room surfaces (as opposed to the adiabatic boundary condition in the other simulations). The profile from this RANS simulation is plotted as the dash line in Fig. 6. It over-predicts the air temperature, as expected, since this simulation assumes the maximum heat transfer from room surfaces to the indoor air. Nevertheless, both two RANS simulations (adiabatic boundary condition and constant surface temperature boundary condition) provide a range of air temperature that encompasses the experimental data. This also suggests that the heat transfer from room surfaces is not negligible in the experiment.

Discussion
The experiment on buoyancy-driven displacement ventilation across two openings show thermal stratification with a gradual increase of temperature with height [12]. Simple models, such as the well-mixed model and the two-layer stratification model, fail to reproduce the thermal stratification observed in the experiment. The CFD simulations capture the trend of the thermal stratification well, but the temperature cannot be accurately predicted unless the boundary condition of the room surfaces is known. Our previous works measuring both indoor air temperature and surface temperatures show that CFD can indeed predict the air temperature accurately with well-defined boundary conditions [13,14]. More importantly, thermal stratification is observed in a real operational building with complex geometry under buoyancy-driven displacement ventilation. For example, Fig. 7 shows that the thermal stratification is stable and is maintained after three hours of natural ventilation in a three-story building. Each floor has a localized thermal stratification, as shown by the layers of warm air below the ceiling of each floor, and an overall thermal stratification across the atrium. Natural ventilation is inherently a complex process and therefore care should be taken in modeling it. Using highly simplified models or assumptions may introduce significant errors. In addition, we also hope that future experimental studies of natural ventilation to not only measure the air temperature, but also the surface temperatures or heat fluxes on the indoor surfaces. Such an experimental dataset is valuable for model development and evaluation to study natural ventilation [15].

Conclusion
Buoyancy-driven displacement ventilation is studied using three different models, namely the well-mixed model, the two-layer stratification model, and CFD models. The experimental measurement in a full-scale room by Yang et al. [12] is used to compare the performance of the models. The experiment shows thermal stratification in the indoor air, where the air temperature gradually increases from the ground. The key findings are: • The well-mixed model assumes a uniform indoor air temperature and therefore cannot predict the thermal stratification. The air temperature is significantly overpredicted. • The two-layer stratification model underpredicts the air temperature in the lower layer and overpredicts the air temperature in the upper layer. The trend of thermal stratification cannot be captured. • CFD simulations can capture the trend of thermal stratification but underpredicts the air temperature. This could be due to the assumption of adiabatic room surfaces, while in the experiments, the heat transfer from room surfaces may not be negligible. LES performs slightly better than RANS in this study, but with a computational cost three orders of magnitude higher than that of RANS. Based on the above findings, we conclude that the simplified models, namely the well-mixed model and the two-layer stratification model, cannot accurately predict buoyancy-driven displacement ventilation flows as they cannot resolve the thermal stratification in such flows. Thermal stratification plays an important role in both modeling and applications of natural ventilation. High-fidelity models, such as CFD models, should be used. Experimental studies of natural ventilation in buildings should include measurements of the thermal stratification, as well as the temperatures or heat fluxes on the indoor surfaces so the results can be used to develop and evaluate numerical models.