3.1. Random fluctuations of leeward–windward pressure difference

The deterministic steady states given by (2.11)–(2.12) are obtained by considering $W$ constant over time. In reality, the wind exhibits a strong variability both in direction and in intensity due to atmospheric turbulence. To assess the effects of this random variability on the system, a stochastic forcing is added to the equations. We follow the procedure presented by Vesipa et al. (Reference Vesipa, Ridolfi and Salizzoni2023), in which zero-mean stochastic fluctuations modelled as an Ornstein–Uhlenbeck process (Uhlenbeck & Ornstein Reference Uhlenbeck and Ornstein1930) are added to the wind velocity. In contrast to Vesipa et al. (Reference Vesipa, Ridolfi and Salizzoni2023), we use the Ornstein–Uhlenbeck process to model fluctuations in the induced pressure difference. This is motivated by the fact that the wind parameter $W$ , which is the source of randomness in the model, is proportional to the pressure difference (see (2.10)). The pressure difference between the windward and the leeward faces of the room is non-dimensionalised by the time-average $\Delta \hat {P}_{w,0}$ and denoted as $\Delta P_w=\Delta \hat {P}_w/\Delta \hat {P}_{w,0}$ , so that the fluctuations $\Delta \hat {P}^{\prime}_{w}=\Delta \hat {P}_{w}-\Delta \hat {P}_{w,0}$ are expressed as $\Delta \hat {P}^{\prime}_w(t)/\Delta \hat {P}_{w,0}$ . As shown in § 4.3, the measured pressure difference is characterised by a Gaussian distribution and exhibits correlation over finite time steps, which justifies the choice of modelling $\Delta P^{\prime}_w(t)$ with a coloured Gaussian noise. Then, the time-dependent fluctuation $\Delta P^{\prime}_w(t)$ is characterised by the following features: (i) a Gaussian probability density function of the realisations, with zero mean and standard deviation $\sigma _{w}$ (which is equal to the coefficient of variation, $C_V=\sigma _{w}$ ); (ii) an exponential autocorrelation, $R_w(\tau ) = C_V^2 \exp [-\tau /\tau _{w}]$ , where $R_w(\tau )$ is the autocorrelation of $\Delta P^{\prime}_w$ and $\tau _{w}$ is the non-dimensional autocorrelation (relaxation) time (normalised as in (2.5)); and (iii) the stationarity of the process ( $C_V$ and $\tau _{w}$ do not change over time). Given these properties, changing the intensity and temporal memory of the fluctuations requires adjusting only the coefficient of variation $C_V$ and the relaxation time $\tau _{w}$ .

Under these assumptions, the pressure fluctuations can be numerically simulated as (Gillespie Reference Gillespie1996)

(3.1) \begin{align} \textrm{d} \Delta P^{\prime}_w(t)= a\Delta P^{\prime}_w(t) \textrm{d} t+b \textrm{d}\varOmega (t), \end{align}

where $\varOmega (t)$ is a standard Brownian motion (Karatzas & Shreve Reference Karatzas and Shreve1991), $a=-\tau _{w}^{-1}$ is the so-called ‘speed of the mean reversion’ and $b=C_V\sqrt {2/\tau _{w}}$ is the so-called ‘diffusion constant’. The diffusion constant $b$ indicates how large is the deviation from the zero mean. Ultimately, the Ornstein–Uhlenbeck process consists of two components: the stochastic term $b\textrm{d}\varOmega (t)$ , which causes random and independent fluctuations, and the deterministic term $a\Delta P^{\prime}_w(t)\textrm{d} t$ , which supports the dampening of the fluctuations and the decay to the mean value.

A realisation of the pressure difference fluctuations $\Delta P^{\prime}_w(t)$ is given by the so-called ‘exact update formula’ (Gillespie Reference Gillespie1996):

(3.2) \begin{align} \Delta P^{\prime}_w(t+\Delta t)=\Delta P^{\prime}_w(t)\exp [-\Delta t/\tau _{w}]+C_V\sqrt {1-\exp [-2\Delta t/\tau _{w}]} \frac {\Delta \varOmega (t)}{\sqrt {\Delta t}}, \end{align}

where $\Delta t$ is the time step of the process and $\Delta \varOmega (t)=\varOmega (t+\Delta t)-\varOmega (t)$ is the increment of the Brownian motion over $\Delta t$ .

Using $\Delta P_w(t)$ , the time series of the wind parameter $W(t)$ is computed by substituting $\Delta P_w(t)$ and $g'_H$ into (2.10):

(3.3) \begin{align} W(t)=\frac {C H^{2/3}\Delta P_{w,0}}{B^{2/3}V^*\rho } \Delta P_w(t). \end{align}

In the case of a constant pressure difference $\Delta P_w(t)=1$ , the wind parameter is also constant and equal to its mean value, $W(t)=W_0 =( {C H^{2/3}}/({B^{2/3}V^*\rho })) \Delta P_{w,0}$ . In the general case of fluctuating wind, (3.3) can then be rewritten as

(3.4) \begin{align} W(t)=W_0 \Delta P_w(t)=W_0\left (1+\Delta P'_w (t)\right ), \end{align}

indicating that wind fluctuations are characterised by two factors: the mean value $W_0$ , about which the parameter fluctuates, and the term $(1+\Delta P^{\prime}_w)$ , which quantifies the fluctuation amplitude.

3.2. Wind fluctuations cause noise-induced phenomena

To illustrate some effects of wind fluctuations on naturally ventilated systems, numerical simulations are performed. First the venting parameter $V^*$ and the wind characteristics (the mean wind parameter $W_0$ and the pressure difference fluctuation properties $C_V$ and $\tau _{w}$ ) are set accordingly to realistic values provided by Ma et al. (Reference Ma, Fouladirad and Grall2018). Then, the non-dimensional pressure difference $\Delta P_w(t)$ is computed according to (3.2). At $t=0$ , the fluctuation $\Delta P^{\prime}_w(t=0)$ is set equal to 0. The time step of the simulation is $\Delta t= \tau _{w}/50$ , so that the temporal resolution is finer than the relaxation time of the Ornstein–Uhlenbeck process. The random variables $\Delta \varOmega _i(t) /\sqrt {\Delta t}$ ( $i = 1, \ldots ,N$ , where $N$ is the total number of time steps of the Ornstein–Uhlenbeck process) required for the generation of the stochastic term of the Ornstein–Uhlenbeck process are obtained by using a standard Matlab routine (randn). The time series of $W(t)$ is then obtained by substituting $\Delta P_w(t)$ in (3.4).

To solve the system (2.8), the buoyancy-dominated steady values of regime $\mathcal{A}$ (obtained for $W_0$ ) are chosen as initial conditions, i.e.

(3.5) \begin{align} h(t=0)=h_0,\quad g'(t=0)=g'_0. \end{align}

Figure 3.

The deterministic steady states (a) $h$ and (b) $g'$ under constant opposing wind conditions (black lines), along with their mean values under fluctuating wind (grey lines). For the deterministic case, the solid lines represent the stable solutions, while the dashed lines the unstable solutions. For the stochastic case, each value of $h$ and $g'$ corresponds to the mean value $W=W_0$ for which it is obtained. The coefficient of variation $C_V$ varies between $0.1$ (top line) and $1.5$ (bottom line). The vertical black line marks the critical value $W_{\textit{crit}}=1$ . The venting parameter is fixed at $V^*=6.96$ (consistent with the experimental set-up; see § 5.1) in all cases. $(c{-}f)$ Time series of $h$ and $g'$ normalised by their deterministic equilibria $h_0,g'_0$ , with a mean wind parameter (c,d) $W_0=0.86$ and (e,f) $W_0=1.15$ . The corresponding mean values are highlighted by markers in $(a{,} b)$ .

Figure 3 presents the mean values of $h$ and $g'$ as functions of $W$ (figure 3 $a{,}b$ ), alongside time series of $h$ and $g'$ normalised by their deterministic equilibria $h_0$ and $g'_0$ , respectively (figure 3 $c{-}f$ ). In figure 3( $a{,}b$ ), the $x$ -axis value $W$ corresponds to the mean of the fluctuating $W(t)$ used to obtain $h(t)$ and $g'(t)$ . The deterministic equilibria are shown for reference as black lines. The venting parameter is fixed at $V^*=6.96$ (this value is consistent with the experimental set-up; see § 5.1). Results are reported for various coefficients of variation $C_V$ of the pressure difference. The correlation time $\tau _{w} = 0.005$ is kept constant (three orders of magnitude smaller than the characteristic filling box time scale, consistent with field data (Ma et al. Reference Ma, Fouladirad and Grall2018)). It is not varied across the simulations since Vesipa et al. (Reference Vesipa, Ridolfi and Salizzoni2023) showed that the system is relatively insensitive to changes in correlation time within the same order of magnitude. Furthermore, changing the order of magnitude would fall outside the scope of this work as we intend to investigate the effect of wind fluctuations rather than long-term, highly autocorrelated transients. In figure 3( $c{-}f$ ), normalising by $h_0$ and $g'_0$ emphasises the effect of noise on the ventilation dynamics: deviations from unity indicate a shift of the mean behaviour. Normalising by $h_0$ and $g'_0$ also facilitates comparison of the effects of fluctuating wind with varying mean strengths $W_0$ and coefficients of variation $C_V$ .

Figure 3 reveals the occurrence of a noise-induced phenomenon when the system undergoes stochastic wind fluctuations: the variables fluctuate and their statistical equilibrium departs from the deterministic equilibrium. Five key features arise, highlighted by both the spaces $W{-}h$ and $W{-}g'$ and the time series. First, the mean values of the interface height $h$ and the reduced gravity $g'$ are significantly lower than the steady states attained under constant wind. This reduction is evident in the spaces $W{-}h$ and $W{-}g'$ (figure 3 $a{,}b$ ), where the grey lines deviate downward from the black curves. In the time series (figure 3 $c{-}f$ ), the reduction is quantified by the distance of the interface height and the reduced gravity normalised by $h_0$ and $g'_0$ from unity. Notably, the decrease is larger for $h$ than for $g'$ .

Second, the deviation from the deterministic equilibrium increases as $W_0$ grows. Figure 3( $a{,}b$ ) shows that these deviations intensify significantly as $W_0$ approaches $W_{\textit{crit}}$ . The time series in figure 3( $e{,}f$ ) reveal a larger gap from unity compared with those in figure 3( $c{,}d$ ).

Third, higher values of $C_V$ affect more the system, as they induce a larger deviation: the average of $h$ and $g'$ undergoes a greater reduction. In the spaces $W{-}h$ and $W{-}g'$ , mean values corresponding to larger $C_V$ always lie below those of lower $C_V$ . Figure 3( $c{-}f$ ) shows that the deviations of $h$ and $g'$ increase with $C_V$ .

Fourth, the deviation of $h$ and $g'$ from their deterministic steady states occurs in an orderly way: the lower the noise (the lower the value of $C_V$ ), the higher the value of $W_0$ at which $h$ and $g'$ start deviating (figure 3 $a,b$ ). The reduced gravity diverges from the deterministic equilibrium at higher $W_0$ compared with $h$ . This is due to the lesser sensitivity of $g'$ , already observed. For $h$ , deviations at lower noise intensities are characterised by shallower slopes, while higher noise intensities lead to steeper deviations. For $g'$ , a similar pattern holds as long as the system remains within the basin of attraction associated with stable forward flow: lower $C_V$ leads to more gradual changes, and higher $C_V$ to steeper ones. However, a different behaviour emerges during the transition from the basin of attraction of the fixed point of regime $\mathcal{A}$ to that associated with stable reverse flow. For $C_V\lt 0.8$ , this transition is abrupt, occurring over a narrow range of $W_0$ and marked by a very steep trend. On the contrary, for higher $C_V$ , this transition is softer and the trend smoother.

Lastly, large noise intensities cause the system to move to the basin of attraction associated with stable reverse flow, although the initial state is the equilibrium of regime $\mathcal{A}$ (see the red time series in figure 3 $e{,}f$ ). As $C_V$ increases, the mean value $W_0$ at which this transition occurs reduces (see the grey lines in figure 3 $a{,}b$ ).

4.1. Experimental set-up and measurements

Experiments are conducted in a recirculating wind tunnel of the Laboratoire de Mécanique des Fluides et d’Acoustique (LMFA) at the Ecole Centrale de Lyon. The wind tunnel has a test section that is $8$ m long, $1$ m high and $0.7$ m wide. Figure 4 shows the wind tunnel, including a zoomed view of the test section and a photograph taken during the experiments. The zoom highlights the experimental set-up within the test section. A transparent Plexiglas box inside the test section of the wind tunnel reproduces the room. The box is placed on a shelf at 0.37 m from the ground in order to avoid the interferences with the 0.02 m thick boundary layer developing at the wind tunnel. The interior of the box, measuring 29.5 cm long, 15 cm wide and 25 cm high, has the same geometry as that used by Coomaraswamy & Caulfield (Reference Coomaraswamy and Caulfield2011). The density difference is created by injecting carbon dioxide. Since the density of carbon dioxide $\hat {\rho }_{\text{CO}_2}$ is greater than that of air $\hat {\rho }_a$ ( $\hat {\rho }_{\text{CO}_2} \approx 1.5 \hat {\rho }_a$ ), the experimental set-up is inverted (see figure 4). Consequently, the top of the box corresponds to the floor of the room, while the bottom represents the ceiling. Carbon dioxide is injected through a circular nozzle with a diameter of 0.9 cm, located at the centre of the box’s upper face. This generates a turbulent plume directed downward, causing carbon dioxide to accumulate in a stratified layer, near the bottom of the box, whose interface is located at height $\hat {h}$ below the top of the box. The opening on the windward facade is cut near the bottom, while the opening on the leeward facade is cut near the top. Each opening consists of two holes, each with a diameter of 2 cm, positioned at the same height but near opposite edges. The openings are positioned sufficiently far from the plume entry point into the $\hat {\rho }_{\text{CO}_2}$ layer to minimise direct disturbance from rapid inflows or outflows, so that the flow primarily affects only the buoyant-layer dynamics.

Figure 4.

Schematic diagram of the recirculating wind tunnel at the LMFA at the Ecole Centrale de Lyon with a zoom on the experimental set-up inside the test section. The shaded area inside the box represents the carbon dioxide seeded with oil droplets. The sketch is accompanied by a photo of the box taken during the experiments. In the photo, the plume is not visible because the laser sheet is near the face of the box and does not intersect the plume.

To make the buoyant layer and the interface visible and measurable, the carbon dioxide flow passes through an oil seeder before reaching the box nozzle. In this way, the carbon dioxide entering the box carries also a small amount of nebulised oil that allows the buoyant layer to be visualised by means of a laser sheet. The latter is in a plane further from the plume, to ensure stable and reliable results as the entry of the plume into the buoyant layer is expected to generate some localised fluctuations. Experiments are filmed by a camera whose frequency is 25 frames per second, and video processing allows the elevation of the interface to be continuously measured (see Appendix A). Not including the plume in the field of view enlarges the visualisation window of the interface, facilitating the computation of its height.

In addition to carbon dioxide, a gas tracer – namely ethane ( $\text{C}_2\text{H}_6$ ) – is injected through the nozzle to infer the concentration and compute the reduced gravity of the buoyant layer. Ethane concentration is measured by a Cambustion HFR400 flame ionisation detector (FID). Applying the procedure developed in Vidali et al. (Reference Vidali, Marro, Correia, Gostiaux, Jallais, Houssin, Vyazmina and Salizzoni2022, Reference Vidali, Marro, Gostiaux, Houssin-Agbomson, Vyazmina and Salizzoni2025) and knowing the ratio between the concentration of carbon dioxide and ethane at the source, the computation of the density of the buoyant layer is performed (further details are provided in Appendix B). The FID has a single probe, so that the concentration measurement is punctual and provides a localised estimate of the density. The FID probe is placed close to the edge between the top face and the leeward face of the box. In this way, any direct effects due to the entrainment of the plume or inflows through the opening are avoided. Some preliminary FID measurements at different heights within the buoyant layer yielded comparable results, so that the location of the probe can be considered representative of the bulk. It is worth mentioning that the presence of oil seeding particles required for visualising the interface has a slight effect on the mean ethane concentration detected by the FID. However, Marro et al. (Reference Marro, Gamel, Méjean, Correia, Soulhac and Salizzoni2020) show that the uncertainty in the measurements is lower than $5\,\%$ .

To evaluate the effect of an adverse wind, the box is submitted to the forcing action of a steady flow (having residual turbulence intensity of approximately $\hat {\sigma }_v/\hat {v}_{w,0}\sim 0.01$ ) within the wind tunnel. To perturb the flow and reproduce the wind fluctuations, a bluff body is placed upwind of the box at the height of the box centre. It is a rectangular cylinder with a section $8\,\textrm{cm}\times 4$ cm, spanning the whole width of the test section. When the incoming steady flow passes the bluff body, vortex-shedding occurs downstream, producing a turbulent wake whose intensity gradually decays moving farther downwind. The turbulence intensity and the pressure fluctuations at the box openings depend therefore on the distance $\hat {d}$ from the bluff body. The pressure difference $\Delta \hat {P}_w$ across the box is detected between the centre of the windward and the leeward faces. The mean pressure difference is measured using a Furness 501 manometer, which has a resolution of $0.01$ Pa; the fluctuations are detected by means of a Kulite pressure scanner with an acquisition frequency of 1 kHz (Kulite 2016).

Table 1 summarises all the ventilation experiments performed. For each configuration the mean velocity $\hat {v}_{w,0}$ of the incoming flow, the mean pressure difference $\Delta \hat {P}_{w,0}$ between the faces of the box, and the distance $\hat {d}$ of the bluff body from the box are displayed. The distance $\hat {d}=\infty$ corresponds to the configuration without the bluff body. The different configurations enumerated differ in terms of the distance $\hat {d}$ . Since each of these configurations consists of several runs at different wind speed, a range is reported for the mean wind velocity $\hat {v}_{w,0}$ and the mean pressure difference $\Delta \hat {P}_{w,0}$ (see Appendix C for all the runs). The acquisition time is 120 s.

Table 1.

List of ventilation box configurations investigated during the experimental campaign. For each configuration, the mean wind velocity, the mean pressure difference between the faces of the box, the distance $\hat {d}$ of the bluff body from the box and the Reynolds number (defined in § 4.2) are displayed. The distance $\hat {d}=\infty$ corresponds to the absence of the bluff body.

For each run, the flow of carbon dioxide seeded with oil is $\hat {q}_{\text{CO}_2}=1.19$ l min⁻¹, mixed to an ethane flux of about $\hat {q}_{\text{C}_2\text{H}_6}=0.01$ ml min⁻¹. For each configuration – characterised by the distance $\hat {d}$ of the bluff body from the box – the first run performed is made for null wind velocity and $W=0$ . Then, the wind speed is gradually increased. This procedure allows the system to remain in the stratified regime up to higher wind velocities than those that would trigger a transition to the mixed regime under a sudden increase in $\hat {v}_{w,0}$ . Once the well-mixed regime is reached, some other runs are made for higher wind speed. The ranges of mean wind velocity, Reynolds number and mean pressure difference reported in table 1 exclude the first run with $\hat {v}_{w,0} = \Delta \hat {P}_{w,0} = 0$ , in order to report the range of strictly positive values.

4.2. Flow similarity

In the existing literature, similar experiments have been mostly performed in water flumes, where buoyancy release is reproduced using saline solutions (Hunt & Linden Reference Hunt and Linden2005; Bower et al. Reference Bower, Caulfield, Fitzgerald and Woods2008; Coomaraswamy & Caulfield Reference Coomaraswamy and Caulfield2011; Mott & Woods Reference Mott and Woods2011; Mott & Woods Reference Mott and Woods2012; Partridge & Linden Reference Partridge and Linden2013). In these experiments, the Reynolds number is sufficiently large to ensure that the flow is independent of viscosity, while density differences remain sufficiently small for the Boussinesq approximation to hold. These conditions are generally considered sufficient to guarantee dynamic similarity with full-scale flows. Furthermore, although the Schmidt number for salt in water ( $Sc \sim 10^3$ ) differs significantly from the Prandtl number for air ( $Pr \sim 0.7$ ), this discrepancy is considered negligible because the Péclet number is sufficiently large for convective transport to dominate over molecular diffusion. In the present study, experiments are instead performed using CO $_2$ emissions in air. Notably, this set-up implies slightly lower Reynolds and Péclet numbers compared with water flume studies, and introduces a density ratio which, in principle, exceeds the limit of the Boussinesq approximation. Nevertheless, as discussed below, the flow conditions remain consistent with those observed in water flume experiments. To discuss this, we first focus on the internal flow within the box and subsequently on the external flow.

As a preliminary step, we perform tests in which the buoyancy flux imposed at the source is varied by changing the CO $_2$ flow rate. These tests are used to assess the influence of the inner Reynolds number and of possible non-Boussinesq effects on the internal flow dynamics. The inner Reynolds number is defined as $ Re_{inn}=\sqrt {\hat {g}_H'\hat {H}}\,\hat {H}/\hat {\nu }_k$ , i.e. adopting $\sqrt {\hat {g}_H'\hat {H}}$ as the characteristic velocity (Linden Reference Linden1999), where $\hat {\nu }_k$ is the kinematic viscosity of the working fluid. In our ventilation experiments, $Re_{inn}\simeq 3.6\times 10^3$ , while in the preliminary tests $Re_{inn}$ spans the range $2\times 10^3{-}4\times 10^3$ . The resulting interface heights remain nearly constant, demonstrating that the inner dynamics has a negligible dependence on the Reynolds number across this range, consistent with previous literature. By extension, this result also indicates a negligible dependence on the Péclet number $Pe=\sqrt {\hat {g}'\hat {H}}\,\hat {H}/\hat {\kappa }$ , where $\hat {\kappa }$ is the molecular diffusivity (Linden Reference Linden1999), since both numbers scale identically with the buoyant velocity $\sqrt {\hat {g}_H'\hat {H}}$ . In our experiments, $Pe\simeq 3.4\times 10^3$ , which is sufficiently large to ensure that molecular diffusion is negligible compared with convection. Note that our set-up yields a Schmidt number $Sc\simeq 1$ , ensuring proximity to the ratio of viscous to diffusive effects encountered in full-scale cases. Apart from confirming the negligible dependence on $Re_{inn}$ , these preliminary experiments suggest that the flow does not exhibit dynamical effects associated with density variations exceeding the validity range of the Boussinesq approximation. This is consistent with recent findings by Salizzoni et al. (Reference Salizzoni, Vaux, Creyssels, Craske and van Reeuwijk2024), who showed that the near-field entrainment coefficient of a buoyant release does not exhibit any specific dependence on the density ratio.

Regarding the external flow, similarity conditions depend solely on the external Reynolds number, which can be conveniently defined as $Re=\hat {v}_{w,0} \hat {L}/\hat {\nu }_k$ , where $\hat {L}=0.04$ m is the reference length of the bluff body perturbing the flow and $\hat {\nu }_k=1.5\times 10^{-5}$ m² s⁻¹ is the kinematic viscosity of air at $20\,^\circ$ C. To verify this similarity, two sets of measurements are performed. Firstly, we measured vertical profiles of the turbulence intensity $\hat {\sigma }_v/\hat {v}_{w,0}$ in the wake of the bluff body using a single-probe Dantec Dynamics hot-wire anemometer (Jorgensen Reference Jorgensen2002). These measurements show that the profiles of $\hat {\sigma }_v/\hat {v}_{w,0}$ do not exhibit any dependence on the incoming wind velocity (see table 1), and therefore on $Re$ . Secondly, we focus on the pressure difference across the box, both in the absence of the bluff body upstream (steady incident flow) and within its wake. In the whole set of experiments, the mean pressure difference fully rescales with the square of the velocity ( $\Delta \hat {P}_{w,0}\sim ({1}/{2})\hat {\rho }\hat {v}_{w,0}^2$ ), confirming that viscous forces have a negligible effect on the flow. All these results demonstrate the independence of the Reynolds number of both first- and second-order velocity statistics, as well as of the time-averaged pressure difference. This is consistent with the findings of Saha, Muralidhar & Biswas (Reference Saha, Muralidhar and Biswas2000), who showed that the Reynolds number (whose definition is consistent with ours) has a negligible effect on the flow past a square cylinder for $Re\gt 600$ , which is the lower bound of our experimental range. Based on this, we assume that both the variance and the correlation time of the pressure signals – which could not be directly verified across the entire experimental range due to sensor limitations at low pressure signals – can also be considered Reynolds number-independent. This implies that the coefficient of variation $C_V$ is constant and independent of $\hat {v}_w$ , and that the correlation time $\hat {\tau }_w$ rescales with $\hat {v}_w$ .

4.3. Characterisation of the pressure difference fluctuations

Table 2 shows the statistic of pressure difference $\Delta \hat {P}_{s}$ (subscript ‘ $s$ ’ stands for ‘scanner’) signals measured with the pressure scanner, for all the considered distances of the bluff body from the box. These include the mean value $\Delta \hat {P}_{s,0}$ , the coefficient of variations $C_V$ , the skewness, the kurtosis and the correlation time. In figure 5, the probability distribution functions of signals normalised by the mean $\Delta \hat {P}_{s,0}$ (histogram plots) in three configurations are shown (for the bluff body at 500 mm and at 1500 mm and in the case without it). The thick lines display Gaussian distributions having standard deviations equal to those of the signals (and mean 1).

Table 2.

Mean, coefficient of variation, skewness, kurtosis and correlation time of the pressure difference $\Delta \hat {P}_s$ signals.

Figure 5.

Probability density functions of the normalised pressure difference $\Delta \hat {P}_s/\Delta \hat {P}_{s,0}$ between the windward and the leeward faces of the box. Three different configurations are displayed, corresponding to three different distances of the bluff body from the box. The histogram plots come from the scanner signals; the thick lines are Gaussian distributions with mean 1 and standard deviations matching those of the signals (colours indicate the same standard deviation).

In each configuration, the Gaussian distribution approximates the distribution of the pressure difference (all samples pass the Kolmogorov–Smirnov test at the $5\,\%$ significance level). The statistical moments and the shape of the probability distribution functions in figure 5 confirm this: the skewness takes a value close to zero and the kurtosis is approximately 3 for all the configurations. As expected, $C_V$ decreases as the distance of the bluff body from the box increases. Thus, the closer the obstacle, the greater the magnitude of the fluctuations.

The correlation time $\hat {\tau }_s$ reported in table 2 is obtained by computing the autocorrelation function for increasing time lags; $\hat {\tau }_s$ corresponds to the integral of the autocorrelogram from lag zero to the lag where the autocorrelation coefficient is equal to $ \exp [-1]$ (Tritton Reference Tritton1988). As shown in table 2, it is almost the same for all the configurations in which the bluff body is present; without the bluff body, it is up to two orders of magnitude smaller.

Given the flow similarity discussed in § 4.3, the coefficient of variation of pressure difference is invariant for each configuration, while the correlation time scales with the wind velocity (and thus with the square root of the pressure difference). The non-dimensional correlation time rescaled to the range of the ventilation experiments (i.e. the measurements of $h$ and $g'$ ) takes values between $0.009$ and $0.04$ . These values are in agreement with the correlation time used in the simulations shown in § 2.

The analysis of the pressure difference time series obtained with the pressure scanner highlights that the use of the Ornstein–Uhlenbeck process modelling the pressure difference is appropriate and well justified: the signals of pressure difference have a Gaussian distribution and are time-correlated. The noise can be therefore modelled with a coloured Gaussian random process.

Figure 6.

Experimental visualisations extracted from videos. (a,b) Frames come from the case without the obstacle ( $C_V=0.07$ ) for $W_0=0.5$ . The white dashed line refers to the top of the box at the laser sheet plane, while the withe solid line corresponds to the mean value of the run. The interface height is denoted as $h_0$ as this case is obtained with a uniform incoming flow field. (c–f) Frames are taken at the same wind strength $W_0=0.5$ but higher noise $C_V=0.29$ . The red solid line refers to the mean $h$ for this case. For reference, $h_0$ is also shown.

5.1. Model parameter fitting

To compare the theoretical and the experimental results, it is necessary to align the model with the physical characteristics of the experimental set-up. This requires adjusting parameters of difficult estimation through data fitting, to account for dissipative head losses of the box in the set-up and idealisations used in the mathematical model. In particular, the plots in figure 8 are obtained setting two parameters: the virtual origin correction $z_{v}$ of the plume and the venting parameter $V^*$ .

The virtual origin correction $z_{v}$ is required to settle a discrepancy between the experimental set-up, in which buoyancy is provided by a steady circular source of dense fluid (having a finite diameter, volume and momentum fluxes), and the theoretical model that assumes an ideal point source of pure buoyancy (i.e. implying zero volume and momentum fluxes). To reconcile this discrepancy, Morton (Reference Morton1959) demonstrated that the flow and buoyancy fluxes from a finite-diameter source are equivalent to those from a point source located at elevation $z=z_v$ . Alternative approaches to evaluate $z_v$ were subsequently proposed by Caulfield (Reference Caulfield1991), Caulfield & Woods (Reference Caulfield and Woods1995), Hunt & Kaye (Reference Hunt and Kaye2001) and Van Den Bremer & Hunt (Reference Van Den Bremer and Hunt2010).

Since existing solutions in the literature are comparable and closely aligned, we treated $z_v$ as a free parameter to fit the experimental values of $h$ and $g'$ with the theoretical model and then verified its consistency with reported values. The red markers in figure 8 represent data obtained under constant external wind (i.e. without the bluff body upstream), so we expect them to lie on the black curve. In the $g' {-} h$ phase diagram (figure 8 $c$ ), the deterministic equilibrium line for the deterministic case satisfies $g' = h^{-5/3}$ . This relation is independent of parameters like the venting parameter $V^*$ , implying the red markers should fall on it after applying the virtual origin correction. We selected $z_v = 0.04$ (corresponding to $\hat {z}_v = 1$ cm in dimensional terms), consistent with literature corrections and the value used by Coomaraswamy & Caulfield (Reference Coomaraswamy and Caulfield2011) for the same geometry. Notably, this adjustment also accurately captures the behaviour in the $W_0 {-} h$ plane (figure 8 $a$ ).

The venting parameter $V^*$ can be instead directly computed from (2.7) as a function of the total effective opening area $\hat {A}^*$ , the coefficient $C$ – which takes into account the ‘top-hat’ entrainment coefficient for a plume $\alpha \simeq 0.1$ – and the height of the box $\hat {H}$ . For the calculation of $\hat {A}^*$ , whose definition is given in (2.1), we consider equal discharge coefficient $C_d$ for the high- and low-level openings, as they have the same size and shape. Using the conventional value $C_d=0.6$ (Hunt & Linden Reference Hunt and Linden2005), the venting parameter reads $6.55$ , in agreement with that used by Coomaraswamy & Caulfield (Reference Coomaraswamy and Caulfield2011) (the difference is about $7.7\,\%$ ). Since both the ‘top-hat’ entrainment coefficient $\alpha$ for a plume and the discharge coefficient $C_d$ involve some uncertainty, the value of $V^*$ is chosen to improve the agreement between experimental results and the equilibrium values of $h$ and $g'$ as $W$ varies. In this way, we obtain $V^*=6.96$ , which deviates only slightly (by 16 %) from the value predicted theoretically.

5.2. Behaviour of measured time-averages of $h$ and $g'$

We present here the key findings of our experimental campaign, namely the time-averaged behaviour of the interface height $h$ and the reduced gravity $g'$ . The different runs of the ventilation experiment were conducted under varying wind conditions, characterised by different mean wind velocities and turbulence intensities. Once the wind conditions were set, the measurements were performed after some time allowing the system to reach dynamical equilibrium, so that the time-averaging did not include the transient from one condition to another. The corresponding results, shown as markers in figure 8, are in remarkable agreement with the theoretical findings, in the cases of both constant and fluctuating wind. The ‘constant-wind case’ is depicted by the red markers, obtained for a constant incoming flow field (i.e. without the bluff body upstream of the box). These red markers lie on the deterministic equilibria depicted by black thick lines. The interface height is slightly lower than the black line for $W_0\geqslant W_{\textit{crit}}$ . This deviation arises from the presence of wind fluctuations ( $C_V$ is slightly larger than zero) that starts affecting the system when the wind parameter exceeds its threshold value. Indeed, a perfectly fluctuation-free ‘constant-wind case’ is unachievable, since even under nominally constant conditions, slight fluctuations occur due to the presence of the box itself.

In the case of strong fluctuations induced by the bluff body, the laboratory experiments exhibit all the key features of the noise-induced phenomenon observed in the numerical simulations presented in § 3.2. First, the time-averages of $h$ and $g'$ are systematically lower than the deterministic equilibrium of regime $\mathcal{A}$ for constant wind (thick black line), although the stratification holds and the system is in the basin of attraction associated with stable forward flow. For a given value of $W_0$ , all the markers are lower than the red ones. In agreement with the numerical results, this behaviour is more evident for the interface height $h$ (figure 8 $a$ ) rather than for the reduced gravity $g'$ (figure 8 $b$ ).

Second, the deviation increases with $W_0$ . While all markers align with the black curve at low $W_0$ , deviations become apparent as $W_0$ grows, with both $h$ and $g'$ decreasing more significantly at higher $W_0$ .

Third, increasing $C_V$ has a stronger impact on the system. For a fixed $W_0$ , as the markers deviate from the constant-wind equilibrium, the green markers are consistently lower than the blue, which in turn are lower than the magenta and light-blue markers. Regarding $h$ , the black markers ( $C_V=0.15$ ) lie between the red markers and the others across all $W_0$ . Since reduced gravity $g'$ is less influenced by noise, this trend is less pronounced.

In the fourth place, as the noise level (expressed by $C_V$ ) increases, the markers of $h$ and $g'$ deviate from the deterministic equilibrium values at progressively lower values of $W_0$ . The green markers ( $C_V=0.29$ ) representing $h$ start deviating for $W_0\simeq 0.2$ , the blue and magenta ones ( $C_V=0.23,0.21$ ) for $W_0\simeq 0.25$ , the light-blue ones ( $C_V=0.17$ ) for $W_0\simeq 0.3$ , while the black ones ( $C_V=0.15$ ) when $W_0 \simeq 0.4$ . It is noteworthy that the reduced gravity maintains a mean value comparable with the constant-wind equilibrium for larger $W_0$ compared with the interface height: the green markers overlap the red ones until $W_0\simeq 0.3$ , while the blue, magenta and light-blue ones deviate for $W_0\simeq 0.65$ . As already observed for the numerical results of $h$ , larger noise intensities correspond to steeper deviations, while lower noise intensities correspond to shallower ones. The same pattern is observed for the reduced gravity as long as the system remains in the basin of attraction of the fixed point of regime $\mathcal{A}$ : the light-blue markers maintain a constant slope until $W_0\simeq 0.75$ , while the magenta and blue markers change incline for $W_0\gt 0.6$ . However, during the transition to the basin of attraction associated with stable reverse flow, a different behaviour emerges observed also in the simulations. For $C_V\leqslant 0.21$ (red, light-blue and magenta markers) this transition is abrupt: the markers jump from one regime to the other. For larger $C_V$ (blue and green markers), this transition is softer and the markers exhibit a more gradual trend.

Lastly, for larger $C_V$ , the system reaches the well-mixed regime $\mathcal{C}$ for lower $W_0$ . It is noteworthy that for $C_V\gt 0.15$ , the transition to regime $\mathcal{C}$ ( $h=0$ ; see figure 8 $a$ ) occurs for $W_0\lt W_{\textit{crit}}$ . The corresponding values of reduced gravity $g'$ are represented by the lowest markers in figure 8 $b$ for $W_0\lt W_{\textit{crit}}$ . As in the constant-wind case no equilibrium of regime $\mathcal{C}$ exists for $W\lt W_{\textit{crit}}$ ; these markers do not lie on any theoretical black curve.

The noise-induced phenomenon is evident also in the phase space $g' {-} h$ . In figure 8( $c$ ), the paired values of $h$ and $g'$ (obtained simultaneously) are displayed. Figure 8( $c$ ) reports all the data points shown in figure 8( $a{,}b$ ). We recall that the black curve represents the equilibria in the case of constant wind, and so the red markers lie on the deterministic equilibrium line. The others overlap it just until $h\simeq 0.35$ and $g'\simeq 0.9$ . Then, they descend opening in a fan-shaped way. The effects on the interface height are much greater. The deviation from the black curve occurs mainly in the vertical direction: for comparable values of $g'$ , the interface height takes value lower and lower.

5.3. Quantitative comparison with model outputs

Figure 8 also shows the results of the numerical simulations (grey lines), performed for different noise intensities $C_V$ , spanning from 0.1 (top curve) to 1.5 (bottom curve). These results, already discussed in § 3.2, are qualitatively very similar to those observed in the experimental results. However, the system’s response in the experiments shows some quantitative discrepancies compared with the numerical results. In particular, we identify three main differences. In the first place, the model consistently underestimates the effect of wind fluctuations. This requires using a higher noise intensity (i.e. larger $C_V$ ) in the simulations to achieve closer agreement with the experimental results. Figure 8( $a$ ) shows that constant incoming wind (see red markers) is the only case in which numerical and experimental results align well for comparable value of coefficient of variation: $C_V=0.07$ in the experiments and $C_V=0.1$ in the simulation. For other cases, involving obstacle-induced fluctuating wind, the simulations require a significantly higher $C_V$ to reproduce the experimental trends: the experimental values of $h$ obtained for $C_V=0.15$ fall within the range of numerical results corresponding to $C_V$ between $0.2$ and $0.7$ ; the markers corresponding to coefficients of variation of pressure difference between $0.17$ and $0.23$ (light-blue, magenta and blue) match the simulation curves for $C_V$ between $0.6$ and $1.4$ ; finally, the green markers ( $C_V=0.29$ ) are in agreement with the curve attained for $C_V=1.5$ .

In the second place, the measured values of reduced gravity (figure 8 $b$ ) are slightly higher than the theoretical curves. The red markers (obtained for a constant incoming wind) lie above the deterministic equilibrium of regime $\mathcal{A}$ (black curve). The other markers overlap the red ones until the noise starts affecting the system; then, they descend but remain higher than the grey curves until regime $\mathcal{C}$ is achieved.

Lastly, the experimental points that reach regime $\mathcal{C}$ lie on the deterministic equilibrium (see the markers $h=0$ in figure 8 $a$ and those in figure 8 $b$ lying on the black curve), while the numerical results take larger values. This feature is evident especially for the reduced gravity (see in figure 8 $b$ the grey lines, which are arranged one above the other based on the value of $C_V$ ). Notably, regime $\mathcal{C}$ occurs for $W \lt W_{{crit}}$ when $C_V \gt 0.15$ in the experiments, but only when $C_V \gt 0.7$ in the simulations.

The first discrepancy between the numerical and experimental results – i.e. the need to set a higher noise level in the simulations to better match the experimental results – can be ascribed to the placement of the probe for the pressure measurements, taken between the centres of the box’s windward and leeward faces. Bartoli & Ricciardelli (Reference Bartoli and Ricciardelli2010) showed that – conversely to the pressure fluctuations on the windward face of a box which are strongly correlated to the turbulence of the incoming flow – those on the leeward face are almost uncorrelated with it. This suggests that the pressure distribution on the leeward facade is primarily dominated by the vortex-shedding dynamics behind the box and, therefore, the wake-induced dynamics results in greater pressure fluctuations close to the edges than at the centre of the face (Amin & Ahuja Reference Amin and Ahuja2013; da Silva, Sumner & Bergstrom Reference da Silva, Sumner and Bergstrom2024). For this reason, the magnitude of noise measured at the centre of the leeward face is lower than that experienced at the opening, which is close to the top edge and is likely to be affected by these wake-induced fluctuations. As a result, the actual pressure difference fluctuations between the openings are expected to be larger than those measured at the face centres.

The second discrepancy – i.e. the deviation of the experimental reduced gravity $g'$ from the model in regime $\mathcal{A}$ – can be ascribed to the employment of nebulised oil droplets and to the perfect-mixing assumption. As mentioned in § 4, oil seeding particles have a slight effect on the mean concentration detected by the FID, although the uncertainty of the measures remains within $5\,\%$ . Moreover, the model assumes instantaneous, uniform heating of the buoyant layer, but in reality mixing occurs over a finite time scale and heterogeneity is present. Since the density measurements are punctual, they may not capture the full density structure but they are expected to provide a reasonable approximation of the bulk.

Lastly, the third discrepancy – namely the inability of the simulations to reach the deterministic equilibrium of regime $\mathcal{C}$ , which is instead attained in the experiments – may be attributed to the inherent limitations of the mathematical formulation of the system dynamics: the model is based on the characteristic convection time of the plume, so that it does not capture phenomena at faster time scales. It assumes that the flow is in a quasi-steady state, so that hydrostatic equilibrium can be applied to the interior of the room. Perturbing the model to investigate the effects of wind fluctuations entails rapid and continuous changes of flow, which were not considered in the development of the model, as it does not include any inertia term. In real cases (and in the experiments), the system has some inertia and its own response time. If the system is in the well-mixed state $\mathcal{C}$ , the overall wind force is overcoming the overall buoyancy force, the cool air entering through the high-level opening mixes with the buoyant fluid inside and the stratification vanishes. In this context, the reduced gravity of the fluid is much lower than that of the buoyant layer in the stratified regime. In the case of wind fluctuations making the buoyancy force instantaneously stronger than the wind force, the numerical system responds immediately. It results in an increment of the interface height and in a reduction of the reduced gravity. As the the flow reverses again, the system moves back to regime $\mathcal{C}$ . In reality, and in the experiments, the formation of a stratification requires a sustained density gradient. Such a gradient can develop only if the outflow of mixed air through the windward opening – and the corresponding inflow of cool air through the leeward opening – persists for a sufficient duration. If the forward flow does not last long enough, no increase in reduced gravity or stratification occurs.

These reasons explain the quantitative discrepancy between the markers and the grey lines in figure 8( $a{,}b$ ). On the one hand, the numerical results (grey lines) take values larger than the deterministic equilibrium of regime $\mathcal{C}$ because at any large- $W$ fluctuations the system comes back to the stratified state $\mathcal{A}$ . On the other hand, the experimental results (markers) lie on the black curve because the $W$ fluctuations do not last enough to make the system switch back to regime $\mathcal{A}$ .