1 Introduction

Understanding the movement of air within and across a building provides useful insight into the distribution of various quantities such as temperature, pollutants, aerosols, dust, etc. This is also crucial for determining key metrics such as the thermal comfort and air quality index, which have been shown to affect human health, productivity, and general well-being [1, 2]. In most cases of building flows, this air movement is due to a temperature difference, the direction of which may be seasonal. In other cases, winds may cause the pressure-induced transport,varying with local topography and building design. Any unprotected opening in the fabric of such buildings, e.g., doorway, window, etc., therefore provides a passage for the heat losses and, may also cause an influx of dirt, pollutants, and insects, resulting in degradation of the conditioned indoor air quality of a building. Adopting a solid separator (wooden or metallic door, glass window, etc.) is the traditional approach towards suppressing such undesirable fluid exchange. While they are still a cost-effective solution, physical obstructions often create inconvenience in many commercial and domestic situations that involves significantly higher human or vehicular traffic. Therefore, technical advancements in building flow research have aimed to provide a non-obstructing barrier for traffic that is efficient to inhibit the movement of air, that may carry along with it the energy resulting in heat losses or dust and pollutants.

Air curtain devices (ACD) were originally conceived at the start of the 20th century. However, their adoption has increased significantly only over the last few decades. Fundamentally, ACD is a plane turbulent jet that blows across a doorway opening to suppress the buoyancy-induced movement of air [3]. They are usually driven by a centrifugal blower (in some cases, multiple blowers) and span along the entire width of the doorway. The dynamics of an air curtain flow closely resembles that of a plane turbulent impinging jet due to the presence of a wall (floor) [4]. Closer to the discharge nozzle, the velocity remains nearly constant in the potential core. A self-similar region persists a few jet widths downstream of the nozzle where the profile of the longitudinal velocity component exhibits a Gaussian variation in the lateral direction [5]. This self-similarity is affected closer to the wall, where the velocity decreases to zero. Under isothermal conditions, an air curtain is identical to a jet. As buoyancy is introduced across the two sides of the jet in the form of, for instance, a temperature difference, the resultant stack effect owing to this horizontal density stratification modifies the behavior of the planar jet. In particular, it provides a horizontal forcing that deflects the axis of the jet and could potentially deny its impingement.

The most common configuration of an ACD installation is that of a downward-blowing single air curtain unit as shown in Fig. 1; other configurations, including sideward-blowing, inclined, as well as an upward-blowing ACD, have also been considered in the past. In addition to these, the usage of a twin air curtain with an intermediate exhaust fan and a recirculating ACD has also been documented in the literature [6, 7]. Further possibilities in their variation arise from the operating conditions that include whether the primary air supply is drawn from inside or outside of a building and/or if any additional heating or cooling is done before discharging through the nozzle. An air curtain is said to be neutrally buoyant if its discharge density is the same as that of the density of the surrounding fluid it is being injected into. Whereas the density of a ‘heavy’ air curtain (HAC) is higher than the corresponding surrounding fluid. All these configurations have been developed over the years and are aimed at numerously distinct use cases. Some of them have been adopted extensively in cold room applications and refrigerated display cases to reduce the associated thermal losses [8,9,10], to suppress the contamination transport in building flows that include public buildings such as hospitals, industrial premises, and shopping complexes, etc. [11, 12]. Other applications include prevention of smoke spread in the case of a compartment or tunnel fire [13, 14], to improve the performance of a cooking range hood [15].

Fig. 1
figure 1

An overview of the air curtain installation. Blue and orange colors represent a colder and a warmer fluid respectively

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The experimental study conducted by Hayes and Stoecker [3] considered a vertically blowing air curtain and provided a systematic framework to evaluate the heat transfer characteristics of ACDs. They developed theoretical models, compared them with the experimental measurements, and found the agreement satisfactory. The characterization of the stability of ACDs was done using a dimensionless parameter, deflection modulus (\(D_m\)), which is mathematically defined as the ratio of the momentum flux contained within the ACD at nozzle exit and the transverse forces acting on it due to the stack effect as given below:

$$\begin{aligned} D_m = \frac{\rho _0 a_0 w_0^2}{g H^2 (\rho _d - \rho _l)} \end{aligned}$$
(1)

In this expression, \(\rho _0\), \(\rho _l\), and \(\rho _d\) denote the densities associated with the air curtain, lighter (warm), and dense (cold) fluids, respectively. The width of the ACD jet at the nozzle exit is represented as \(a_0\) while the average discharge velocity is \(w_0\). In Eq. (1), g is the acceleration due to gravity, and H is the height of the unprotected opening. Based on the conservation of momentum in the horizontal direction, they showed that a critical value of \(D_{m, min} = \frac{1}{8}\) is needed to achieve an air curtain that could reach the other side of the doorway if we ignore the jet expansion owing to the entrainment of ambient fluid. If the deflection modulus is smaller than \(D_{m, min}\), it is deflected sideways owing to the stack effect. The concept of deflection modulus has been proved to be robust over the years and is being actively used to characterize air curtain and study their behavior even today.

Another parameter that has been used to characterize the ACDs is sealing effectiveness, E. It compares the efficacy of air curtains in suppressing the buoyancy-induced fluid exchange as compared to the case when it is not installed, and the opening is considered unprotected. Mathematically, E is expressed as:

$$\begin{aligned} E = \frac{q - q_{ACD}}{q} = 1 - \frac{q_{ACD}}{q} \end{aligned}$$
(2)

Here, q and \(q_{ACD}\) are volumetric exchange flow rates associated with a doorway opening in the absence and presence of an air curtain, respectively, and are quantified later in Sect. 3. An open door (OD) situation corresponds to an effectiveness of zero, whereas a perfectly sealed opening is obtained when \(E =\) 1. Past studies have shown that E depends on the deflection modulus, openings in the building fabric such as windows and exhausts, and the operating conditions e.g. jet angle, exit velocity, ambient temperature etc. [3, 8, 16,17,18,19]. \(D_m\) dependence of E has largely been documented for the neutrally buoyant air curtains (NBAC) where \(\rho _0 = \rho _d\) for the configuration shown in Fig. 1a [8, 16, 17, 20, 21]. On the other hand, a heavy air curtain (HAC), which has an operating density larger than the ambient density is a relatively lesser studied problem. In the present study, we consider the case of a HAC installed on the cold (heavy) side with \(\rho _0 > \rho _d\). The primary goal of this work is to compare the performance of a HAC installation with its neutral counterpart and the corresponding \(D_m\) dependence of effectiveness E. In the process, we provide quantitative differences in the velocity dynamics of an air curtain to that of an isothermal jet. In addition, we seek the implications of providing an assistive buoyancy in the indoor space. Specifically, we aim to understand how the infiltration of the ambient fluid of density \(\rho _l\) is neutralized by this assistive buoyancy (proportional to \(\rho _0 - \rho _d\)), when compared to a neutrally buoyant air curtain with \(\rho _0 = \rho _d\).

The paper is organized as follows. The numerical framework of the employed Reynolds-averaged Navier–Stokes (RANS) and the large-eddy simulation (LES) methods is discussed in Sects. 2.1 and 2.2, respectively. Initial and boundary conditions for these simulations are also discussed therein in Sect. 2. All the experiments that were conducted are elaborated in Sect. 3 with a discussion on estimating the exchange flow rates. Section 4 presents an analytical model based on the conservation equations of mass, momentum, and buoyancy that can be used to estimate the properties associated with an air curtain at a steady state. In Sect. 5, the results from various simulations and experiments are discussed. The general flow field is illustrated in terms of buoyancy distribution in the domain as well as the velocity statistics along the centerline of the ACD jet. Subsequently, the estimates of the sealing effectiveness, E, are compared across different methodologies as a function of the deflection modulus. The implications of the positive buoyancy are discussed in Sect. 5.2.2. The major findings of the present work are summarized in Sect. 6.

2 Numerical simulations

The present study employs two methodologies to simulate the air curtain flows; Reynolds-averaged Navier–Stokes (RANS) and large-eddy simulation (LES). The overarching aim of the numerical simulations is to get the instantaneous fields of velocity and density which are not easily obtainable from experiments. The parameter space for all the different computations is tabulated in Table 1. A two-dimensional (2D) RANS solver is primarily employed to understand the parametric effect in ACD flows, and it is expected that their predictions may not be very accurate due to the various complexities associated with the problem, which will also be demonstrated later. Three-dimensional solvers, e.g., 3D RANS and LES, are used to obtain accurate quantitative estimates of the flow quantities involved. These simulations are conducted in a computational box of size \(L_x = L =\) 2 m, \(L_y = W =\) 0.175 m, \(L_z = H =\) 0.35 m. The size of this geometry is based on the waterbath technique, where the flow is established using water as the working fluid since the ratio of the kinematic viscosity of the two fluids, \(\frac{\nu _{Air}}{\nu _{Water}}\), is approximately 15, therefore, such a simulation requires a smaller domain under similar dynamic conditions [22]. Considering that the origin is taken at the center of the domain in the x-direction (see Fig. 1a), the region under investigation, i.e., the indoor space, spans from \(x = -\frac{L}{2}\) to \(x =\) 0, whereas the ambient fluid is initially filled between \(x =\) 0 to \(x = \frac{L}{2}\). The air curtain has a thickness of \(a_0 =\) 5 mm and is placed between \(x = -a_0\) and \(x =\) 0 within the indoor space. Subsequently, the jet height to width ratio, \(\frac{H}{a_0}\), is set to 70, which is required for ACD applications [17]. Various aspects of the computational setup are shown in Fig. 1, which also presents a cross-sectional view of the typical climatic separation in the presence of an air curtain. In these simulations, the ACD was modeled as a top-hat velocity-inlet boundary with a prescribed uniform velocity and turbulence intensity (see the red patch in Fig. 1a). The side faces at \(x = \pm \frac{L}{2}\), shown as green, were enforced with a zero Neumann boundary condition to represent an outflow boundary. This configuration is generally chosen for the simulations of planar jets and accounts for possible lateral flow exits from both the indoor and outdoor spaces [23, 24]. A no-slip, no-penetration boundary condition is imposed on all the other walls in the domain. The mesh is refined in the central region where the air curtain operates to capture the large gradients in the flow associated with the development of flow instabilities and sharp-density interfaces.

Table 1 Various computations that were conducted for different values of deflection modulus, \(D_m\), and the buoyancy ratio, S, where \(S =\) 0 corresponds to the neutrally buoyant air curtain
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Initially at \(t =\) 0, \(\rho _l\) and \(\rho _d\) are chosen such that it corresponds to \(T_l =\) 45 \(^\circ\)C and \(T_d =\) 25 \(^\circ\)C. The resulting reduced gravity due to this buoyancy difference, \(g^\prime = g \frac{\Delta \rho }{\rho _d}\), before the air curtain operates, is approximately 6.8 cm/s\({}^2\). These temperature levels are good representative values of a typical tropical region in the summer season. Herein, the primary objective of the air curtain installation is to ensure the thermal comfort of building occupants where the indoor temperature could change due to the infiltration of the outdoor warmer fluid. Therefore, the ACD should eliminate the undesirable fluid exchange between the two spaces in order to help prevent any temperature drops in indoors. Furthermore, for the air curtain to assist towards thermal comfort, an assistive buoyancy (\(\rho _0 > \rho _d\)) could potentially help in neutralizing any temperature increment due to fluid infiltration. The strength of this added buoyancy is quantified in the form of a dimensionless parameter, S, that compares the assistive buoyancy in the air curtain to the buoyancy difference across the doorway and can be mathematically written as:

$$\begin{aligned} S = \frac{\rho _0 - \rho _d}{\rho _d - \rho _l} \end{aligned}$$
(3)

The present work reports simulations conducted for the jet temperature, \(T_j\), ranging between 10 \(^\circ\)C and 25 \(^\circ\)C where \(T_j =\) 25 \(^\circ\)C corresponds to a neutrally buoyant air curtain (NBAC). Therefore, in the non-dimensional sense, S varies between 0 for an NBAC installation to approximately 0.39 for an air curtain with \(T_j\)= 10 \(^\circ\)C. The inlet velocity, \(w_0\), is changed to obtain different values of deflection modulus (\(D_m\)). The Reynolds number, Re, based on the air curtain parameters \(\left( Re = \frac{w_0 a_0}{\nu }\right)\) is kept in a similar order (see Table 2) as that in full-scale installations to ensure dynamic similarity. For the heavy air curtains, the corresponding Richardson number, \(Ri = \sqrt{\frac{g^\prime a_0}{w_0^2}}\) is also tabulated that compares the relative magnitude of the added buoyancy (in the form of a reduced gravity) to that of the supplied momentum. It should be noted that Ri is related to the deflection modulus as \(Ri = \left( \frac{a_0}{H}\right) \frac{1}{\sqrt{D_m}}\) and the chosen parameter space for Richardson number closely resemble a full-scale air curtain installation.

Table 2 The values of dimensionless parameters, Reynolds number (Re) and Richardson number (Ri), corresponding to heavy air curtains at various deflection modulus, \(D_m\)
Full size table

In the following subsections, a brief description of the simulation methodologies is presented. The reader may refer to the literature for the complete mathematical framework [25,26,27].

2.1 Reynolds-averaged Navier–Stokes (RANS)

RANS computations are based on the idea of Reynolds decomposition wherein any instantaneous flow parameter, for instance, a velocity component \(u_i\), is partitioned into a mean (\(U_i\)) and fluctuating (\(u_i^\prime\)) part. The following set of equations is solved, which model the behavior of mean velocity and temperature fields under Boussinesq approximation as:

$$\begin{aligned} \frac{\partial {U_i}}{\partial {x_i}}=0 \end{aligned}$$
(4)
$$\begin{aligned} \frac{\partial U_i}{\partial t} + U_j\frac{\partial {U_i}}{\partial {x_j}}=- \frac{1}{\rho _0} \frac{\partial {P}}{\partial {x_i}}+\frac{\partial }{\partial {x}_j} \left[ \nu \frac{\partial {U_i}}{\partial {x}_j}-\overline{{u_i^\prime }{u_j^\prime }}\right] -g [1 - \beta (T - T_0)]\delta _{13} \end{aligned}$$
(5)
$$\begin{aligned} \frac{\partial {{T}}}{\partial {t}}+{U_j}\frac{\partial { {T}}}{\partial {x_j}}=\frac{\partial }{\partial {x}_j} \left[ \kappa \frac{\partial {{T}}}{\partial {x}_j}-\overline{{u_j^\prime }{T^\prime }}\right] , \end{aligned}$$
(6)

Eqs. (4, 5, and 6) are the mathematical analog of the conservation laws of mass, momentum, and energy. P represents the mean fluid pressure and \(\rho _0\) is the reference density. g is the acceleration due to gravity, and the fluid viscosity, thermal expansion coefficient, and thermal diffusivity are denoted by \(\nu\), \(\beta\), and \(\kappa\), respectively. The inclusion of Reynolds stresses \(\overline{{u_i^\prime }{u_j^\prime }}\) and \(\overline{{u_j^\prime }{T^\prime }}\) is an unavoidable consequence of Reynolds decomposition and poses a challenge to mathematical closure of this system of equations. In the present study, we employ the renormalization group (RNG) k-\(\epsilon\) model [28] to close this system by solving additional equations for the turbulent kinetic energy, k, and the turbulent dissipation rate, \(\epsilon\). These are respectively given as:

$$\begin{aligned} \frac{D k}{D t} = \frac{\partial }{\partial {x_i}} \Bigg [ \left( \nu + \frac{\nu _t}{\sigma _k} \right) \frac{\partial {k}}{\partial {x_i}} \Bigg ] + {\mathcal {P}}_k + {\mathcal {B}}_k - \epsilon \end{aligned}$$
(7)
$$\begin{aligned} \frac{D \epsilon }{D t} = \frac{\partial }{\partial {x_i}} \Bigg [ \left( \nu + \frac{\nu _t}{\sigma _k} \right) \frac{\partial {\epsilon }}{\partial {x_i}} \Bigg ] + \frac{\epsilon }{k}\left( C_{1 \epsilon } {\mathcal {P}}_k + C_{1 \epsilon } (1 - C_{3 \epsilon }) {\mathcal {B}}_k - C^*_{2 \epsilon } \epsilon \right) \end{aligned}$$
(8)

In these equations, the operator\(\frac{D}{Dt}\) symbolizes a material derivative that includes the effect of fluid advection. \(P_k\) represents the rate of production of turbulent kinetic energy due to mean flow and is evaluated as:

$$\begin{aligned} P_k = \nu _t \Bigg [\frac{\partial {U_i}}{\partial {x_j}} + \frac{\partial {U_j}}{\partial {x_i}} \Bigg ] \frac{\partial {U_i}}{\partial {x_j}} \end{aligned}$$
(9)

The turbulent viscosity, \(\nu _t\), in this expression is calculated as \(\nu _t = C_\mu \frac{k^2}{\epsilon }\) where \(C_\mu\) is a model coefficient. The production of turbulence due to buoyancy, \({\mathcal {B}}_k\), is an added term in Eqs. (7 and 8) to account for the effect of stratification and is given as:

$$\begin{aligned} {\mathcal {B}}_k = \beta g \rho _0 \frac{\nu _t}{\sigma _t} \frac{\partial T}{\partial z} \end{aligned}$$
(10)

where \(\sigma _t\) is the turbulent Prandtl number. The constant \(C^*_{2 \epsilon }\) in Eq. (8) is given as:

$$\begin{aligned} C^*_{2 \epsilon } = C_{2 \epsilon } + \frac{C_\mu \eta ^3(1 - \eta /\eta _0)}{1 + \beta \eta ^3} \end{aligned}$$
(11)

Here, the parameter \(\eta\) is estimated as \(Sk/\epsilon\) where \(S = \sqrt{2 S_{ij} S_{ij}}\) and \(S_{ij} = \frac{1}{2} \left( \frac{\partial {U_i}}{\partial {x_j}} + \frac{\partial {U_j}}{\partial {x_i}} \right)\) is the strain rate. The values of coefficients associated with any turbulence model are generally based on empirical and experimental data. For the present work employing the RNG \(k-\epsilon\) model, we have taken these as:

$$\begin{aligned} \left\{ \begin{matrix} C_\mu \\ C_{1 \epsilon }\\ C_{2 \epsilon }\\ \sigma _k\\ \eta _0\\ \beta \end{matrix} \right\} = \left\{ \begin{matrix} 0.0845\\ 1.42\\ 1.68\\ 0.7194\\ 4.38\\ 0.012 \end{matrix} \right\} \end{aligned}$$

The ANSYS Fluent code, which is based on the finite-volume method (FVM), is utilized to solve these equations on two-dimensional and three-dimensional grids. Suitable meshes were chosen after a systematic mesh sensitivity study which is reported in Agrawal et al. [29]. The grids for the 2D and 3D RANS computations have 64,000 and 422,400 points, respectively. For both cases, at least ten cells span across the air curtain width, and the vertical (z) resolution is kept uniform at \(\frac{a_0}{2}\). The spatial derivatives in the governing equations are approximated using a second-order upwind scheme. A second-order implicit method is employed to evaluate the temporal derivative, and the pressure and velocity fields are coupled using the Coupled algorithm. The methodology was validated for the case of a planar jet, and the results are presented in appendix 1.

2.2 Large-eddy simulation (LES)

The RANS approach involves modeling all the scales in a turbulent flow. On the other hand, an LES based-solver calculates the large scales explicitly, which generally play a crucial role in mass and momentum transfer, and models only the impact of small-scale eddies. This is achieved by using a spatial filtering operation to separate the large (resolved) scales from the small (subgrid) ones. The LES solver employed in the present study works on the following set of filtered continuity, momentum, and scalar equations using the Boussinesq approximation, and further assuming a linear relationship between density and the equation of state:

$$\begin{aligned} \frac{\partial {\overline{u_i}}}{\partial {x_i}}=0 \end{aligned}$$
(12)
$$\begin{aligned} \frac{\partial \overline{u_i}}{\partial t} + \overline{u_j}\frac{\partial {\overline{u_i}}}{\partial {x_j}} = - \frac{1}{\rho _0} \frac{\partial {\overline{p}}}{\partial {x_i}} - {{\overline{b}}} \ \delta _{i3} + \frac{\partial }{\partial {x}_j}\left[ \nu \frac{\partial {\overline{u_i}}}{\partial {x}_j}-\tau ^R_{ij}\right] \end{aligned}$$
(13)
$$\begin{aligned} \frac{\partial \overline{b}}{\partial t} + \overline{u_j}\frac{\partial {\overline{b}}}{\partial {x_j}} = \kappa _b \frac{\partial }{\partial {x}_j}\left[ \frac{\partial {\overline{b}}}{\partial {x}_j}-\lambda ^R_{ij}\right] \end{aligned}$$
(14)

where the \(i^{th}\) component of the filtered velocity field is denoted using \({\overline{u_i}}\), \(\kappa _b\) is the scalar diffusivity, \(\overline{p}\) and \(\overline{b} = g {\overline{\rho }}/\rho _0\) correspond to the filtered pressure and buoyancy fields respectively. An eddy-viscosity-based approximation models the anisotropic part of the subgrid stress tensor, \(\tau _{ij}^R = \rho \overline{u_i u_j} - \rho \overline{u}_i \overline{u}_j\), and subgrid buoyancy flux, \(\lambda ^R\), as:

$$\begin{aligned} \tau ^R_{ij} - \frac{1}{3}\tau ^R_{kk}\delta _{ij} = -2 \nu _{sgs} \overline{S}_{ij} \end{aligned}$$
(15)
$$\begin{aligned} \lambda ^R = -2 \kappa _{sgs} \frac{\partial \overline{b}}{\partial x_j} \end{aligned}$$
(16)

where \(\overline{S}_{ij}\) represents the strain rate associated with the filtered velocity field. The subgrid-scale Prandtl number \(Pr_{sgs} = \nu _{sgs}/\kappa _{sgs}\) has been taken as unity for all the reported LES simulations. The above set of equations is solved using the in-house code Stratified Ocean Model with Adaptive Refinement (SOMAR). Various kinds of stratified flows, including tidal flows, low mode wave scattering, gravity currents, internal and solitary waves propagation, etc., have been studied in the past using SOMAR [30,31,32,33,34]. The numerical discretization in this solver is based on second-order finite differences for the advection, viscous and diffusion terms, whereas time advancement is done using the RK3 scheme. The grid employed in the present LES simulations comprises a base level containing \(N_x \times N_y \times N_z =\) 1536 \(\times\) 32 \(\times\) 128 points with a finer level overlaid in the central region having a refinement factor of 4, 2 and 4 in the x, y and z directions respectively. The spatial extent of the finer level is kept between \(x =\) \(-\)0.05 m to \(x =\) 0.05 m to accurately resolve the shear layers in the planar air curtain jet.

3 Experiments

The experiments are conducted in a glass tank of dimensions 2 m \(\times\) 0.185 m \(\times\) 0.4 m, a schematic of which is shown in Fig. 2. While an initial temperature difference is used in the simulations to create the desired buoyancy effect across the doorway, we dissolve table salt in freshwater to manipulate the fluid density in the experimental system. Using dissolved substances such as salt eliminates the need to create a thermally insulated system and, thus, greatly aids the ease with which the stratified flow measurements may be conducted and the flow may be visualized. The effect of different scalar diffusivity (\(\kappa _{T} \approx\) 0.14 mm\(^2\)/s, \(\kappa _{S} \approx\) 0.0013 mm\(^2\)/s) has shown to produce similar transport phenomenon associated with the macroscopic scales in buoyancy-driven flows [35,36,37]. While this may not be always true for full-scale measurements, this simplification is valid for similitude experiments such as those conducted in the present study [38]. Therefore, heat (temperature) and salt can be interchangeably used as stratifying agents when the macroscopic motions are concerned in a laboratory waterbath.

Fig. 2
figure 2

Schematic of the experimental setup at the instant before removing the lock-gate. The curtain fluid is brine solution for the NBAC and aqueous solution of sugar for HAC

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The tank was then divided into two equal sections using a perspex sheet (lock gate, denoted as LG) that spanned across the entire width of the facility and, thus, represented a doorway. The two halves were simultaneously filled using freshwater and saltwater, maintained at a density of \(\rho _l\) and \(\rho _d\), respectively. Any minor fluid leakage across the lock gate due to hydrostatic pressure difference was carefully monitored during the filling process. In a separate test, this leakage was estimated to be within 5 ml/min, which corresponds to less than 0.5% of the total fluid volume at either side of the lock gate. The water level at the beginning of an experiment was kept at \(H =\) 0.22 m. Generally, the fluid level on the saltwater side was kept approximately 5 mm lower than the freshwater side to allow the establishment of ACD, which is described now.

The air curtain was installed on the saltwater side next to the lock gate, as shown in Fig. 2. It was fabricated from a cylindrical pipe of length 182 mm, which was closed at its ends. A fine sponge wrapped in a flexible steel wire mesh was fitted inside this cylinder to facilitate a uniform distribution of the air curtain fluid. 32 pinholes of size \(\phi _p =\) 1.1 mm, as shown in the top left side of Fig. 2, were uniformly drilled using a vertical milling machine along the length of the cylinder. In any experiment, the air curtain was installed such that these holes were just submerged in saltwater. The resultant round water jets from these holes usually merge a few diameters downstream of the cylindrical pipe and form a planar jet required for the sealing [39]. The effective planar jet width can be estimated as the ratio of the total area over which the flow occurs to the cylinder length and is equal to 0.17 mm in the present experiments.

The fluid supply for the ACD is taken from a tank (denoted as ST) using a centrifugal pump whose delivery head is regulated through a control valve, V1. In the case of NBAC, salt is dissolved in the supply tank such that \(\rho _0 = \rho _d\) and \(\Gamma _0 = \Gamma _d\) where \(\Gamma\) denotes the electrical conductivity of the solution. Whereas for HAC, we dissolve excess sugar in the curtain fluid such that \(\rho _0 > \rho _d\) for establishing the extra buoyancy. However, the conductivity, in this case, is related as \(\Gamma _0 < \Gamma _d\). The reason behind using sugar instead of salt for creating a heavy air curtain will be discussed later. The flow rate through the air curtain nozzle is measured using a turbine flowmeter (TFM) with an accuracy of ± 3%. Another control valve, V2, was provided to fine-tune the flow rate to a predetermined value which was kept constant at \(Q_0 =\) 6 LPM across all the experiments  (see Table 3).

Table 3 Various experiments conducted in the present study
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After filling the tank to the desired level, fluids on the two sides are stirred to ensure a homogeneous medium. Six samples are then collected from each half of the tank at different spatial locations. The density and conductivity of these samples are measured using the Anton Paar DMA 1001 density meter (accuracy: ± 0.0001 gm/cc) and Thermo Scientific A222 conductivity meter (accuracy: ± 0.5%). The random error in these measurements is within ± 0.0002 gm/cc and ± 0.5%, respectively. The estimates of the salt concentration, c (in gm/L), are derived from the measured conductivity through a calibration that was performed for a known amount of salt concentrations ranging between 0 to 80 gm/L. The air curtain is then switched on, and the flow is allowed an establishment time (usually of the order of \(t_0 =\) 10 s), after which the flow rate through the ACD becomes nearly constant. The initial difference of 5 mm in the fluid levels is provided for this adjustment such that at the time of removing the lock-gate, the free surfaces on both sides are at the same level. Upon the establishment of the air curtain, the lock gate is rapidly removed for the commencement of the fluid exchange. The ACD continues to operate for \(t_{ex} =\) 45 s, during which a high speed camera, described below, records the entire exchange process. The lock gate is then reinstalled firmly, followed by switching off the air curtain and closing of valve V2 to avoid any possible return flow that may occur due to gravity-induced siphon. Subsequently, the fluid in each half of the tank undergoes the homogenization process as described above. At the end of the experiment, samples are taken from the indoor region to measure \(\rho _d^n\) and \(\Gamma ^n_d\) that provides an estimate of the exchange that occurred during the time span of \(t_{ex}\) seconds, where the superscript n represents the new values of the corresponding property.

To visualize the bulk structure of air curtains, a colored dye (laundry brightener) was injected through a 0.8 mm needle placed near the jet exit. The dye injection was gravity driven from a small supply tank (shown as DIT), and its flow rate was regulated using a control valve, V3. The needle head was carefully positioned, and the valve was adjusted such that a dye parcel loses its momentum completely after reaching underneath the air curtain. In most experiments, the dye injection starts at \(t = t_0\) after removal of the lock-gate and finishes at \(t = t_0 + t_{ex}\) just before the LG is reinstalled. In some experiments, another dye was premixed with the outdoor fluid (freshwater) to observe the infiltration that occurs during a door opening cycle. The resultant flow is imaged using a Canon EOS 90D DSLR camera with an adjustable focal length (18–135 mm) lens at a sampling rate of 100 Hz.

To quantify the amount of fluid exchange during the air curtain operation, we employ the mass conservation in the indoor space, similar to that adopted by Frank and Linden [18], Jha et al. [20], which yields:

$$\begin{aligned} \rho _d^n (V + V_{ACD}) = \rho _d V - \rho _d V_{ex} + \rho _l V_{ex} + \rho _0 V_{ACD} \end{aligned}$$
(17)

Here, the left-hand side represents the total mass in the dense (indoor) fluid region at the end of an experiment. The initial volume, V, of the indoor space is taken as \(\frac{LWH}{2}\) and \(V_{ACD}\) accounts for the volume of the air curtain fluid added to the indoor space and is calculated as \(V_{ACD} = 0.5Q_0t_0 + \beta Q_0 t_{ex}\). In this expression, \(0.5Q_0t_0\) approximates the volume of fluid added to the indoor space during the start-up phase assuming a linear variation of flow rate (in time) from 0 to \(Q_0\). In the second term, \(\beta\) accounts for the spillage of the air curtain inside the dense fluid half after impingement on the floor of the tank and is taken as 0.5 in the subsequent calculations. This value of \(\beta\) was also chosen in the experimental studies conducted by other researchers [11, 17, 18]. In addition, our numerical simulations also confirm that the air curtain jet divides nearly equally between the two halves after impingement. The first term on the right-hand side of Eq. (17) is the initial mass of the region under consideration, whereas the second and third terms represents the mass transfer across the doorway owing to fluid exchange. The corresponding volume exchange is represented as \(V_{ex}\). The last term signifies the mass added due to the air curtain operation. For a neutrally buoyant air curtain, \(\rho _0 = \rho _d\), and the corresponding exchange flow rate across the air curtain device, \(q_{ACD}\), can be estimate as:

$$\begin{aligned} q_{ACD} = \frac{V_{ex}}{t_{ex}} = \left( \frac{\rho _d - \rho _d^n}{\rho _d - \rho _l} \right) \left( \frac{V + V_{ACD}}{t_{ex}} \right) \end{aligned}$$
(18)

For a heavy air curtain, since \(\rho _0 > \rho _d\), the final density in the indoor space, \(\rho _d^n\), can change either due to infiltration of the ambient fluid of density \(\rho _l\) or due to the addition of the air curtain of density \(\rho _0\). It should be noted that while the infiltration from the ambient space results in a reduction of the mean fluid density within the indoor region, the air curtain device, owing to the added buoyancy, increases the mean density. To quantify the overall fluid exchange, a mass conservation equation similar to Eq. (17) can still be formulated for a HAC, however, it cannot explicitly differentiate between the two aforementioned contributions and, therefore, only provides an apparent estimate of the exchange flow based on the mean density, \(\rho _d^n\), within the indoor space. Consider a scenario where a contaminant is present in the ambient fluid, and some exchange takes place between the indoor and outdoor space. It is possible that the reduction in the fluid density in the indoor space due to this exchange may be counterbalanced by the air curtain device and thus, the effectiveness based on Eqs. (2 and 18) may not reflect the sealing capabilities of an ACD associated with the transport of such contaminants. Therefore, one needs to segregate the infiltration effect of the ambient fluid from the air curtain contribution to \(\rho _d^n\). This can be achieved by introducing a different tracer (sugar) for the air curtain device which distinguishes between the ambient and the ACD fluids. A conservation equation for the “salt mass", as was first proposed by Frank and Linden [18], can now be written for the dense fluid half as:

$$\begin{aligned} c_d^n (V + V_{ACD}) = c_d V - c_d V_{ex} + c_l V_{ex} + c_0 V_{ACD} \end{aligned}$$
(19)

The physical meaning of individual terms in Eq. (19) is identical to that of Eq. (18), where the total mass is replaced using the salt mass, which can be inferred from the measured salt concentration c. An algebraic rearrangement of Eq. (19) leads to:

$$\begin{aligned} q_{ACD}^\prime = \frac{V_{ex}}{t_{ex}} = \left( \frac{c_d - c_d^n}{c_d - c_l} \right) \frac{V}{t_{ex}} + \left( \frac{c_0 - c_d^n}{c_d - c_l} \right) \frac{V_{ACD}}{t_{ex}} \end{aligned}$$
(20)

For a neutrally buoyant air curtain, \(c_0 = c_d\) and \(\rho _0 = \rho _d\), and thus Eqs. (18 and 20) are identical. Whereas for a heavy air curtain, \(c_0 = c_l < c_d\) and \(\rho _0> \rho _d > \rho _l\), therefore, the exchange flow rate associated with the salt mass transfer in Eq. (20) may be different than that of Eq. (18). In the context of building flows, measuring the density \(\rho _d^n\) and the associated \(q_{ACD}\) could be a possible surrogate for the mean fluid temperature within the indoor region and thus, can be a good indicator of thermal comfort for building occupants. On the other hand, the corresponding measurements of \(q_{ACD}^\prime\) could act as a proxy for fluid infiltration from the ambient space. For the same reasons, the effectiveness based on the measure of exchange flow in Eq. (20) may be different from E (Eq. 2) and can be calculated as \(E^\prime = 1 - \frac{q_{ACD}^\prime }{q^\prime }\). Both of these measures are reported in Sect. 5.2.

4 Analytical description

In this section, we present an analytical model to describe the spatial evolution of an air curtain jet. A domain similar to that of our simulations, with two fluids at densities, \(\rho _d\) and \(\rho _l\), is considered, as shown in Fig. 3a. An ACD with an initial density, \(\rho _0\), and a uniform velocity, \(w_0\), is provided at the interface between the two fluids. The small density differences in the system imply that the Boussinesq approximation is applicable. We also consider that the jet width at the nozzle exit, \(a_0\), is much smaller compared to the span of the domain, thereby enabling a two-dimensional formulation of the analytical model, which is valid for present experiments. In addition, the jet is considered to be uniform along the spanwise (y) direction, so that the dynamical quantities could be evaluated per unit length of the plane buoyant jet.

Consider a rectangular control volume ABCD as shown in Fig. 3b, where the air curtain is having an arbitrary geometric configuration defined by the width, a(z), and an inclination angle \(\theta (z)\). The analytical model, outlined next, is based on the Newtonian conservation equations applied to this Eulerian control volume. The framework presented below for the analytical development is similar to that of Frank and Linden [18], however in the present work, the sense of buoyancy flux is reversed owing to different practical motivation. In addition, we have also utilised the developed model for studying the vertical evolution of mean velocity of the air curtain and its deflection as discussed later.

Fig. 3
figure 3

Control volume chosen for the formulation of the analytical model

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First, we write the equation for the conservation of volume flux. Physically, it implies that the total fluid volume leaving the face EC, \(Q (z + dz)\), must equal to that entering through AF, Q(z) and the volume being entrained through faces AE and FC, \(Q_{E}\). To quantify the fluid entrainment, we use the model proposed by Morton, Taylor and Turner [40], according to which the entrainment velocity \(u_{E}\) is taken to be proportional to a characteristic velocity scale of the flow with the proportionality constant being the entrainment coefficient \(\alpha _{E}\). For the case of jets, this characteristic velocity is usually taken to be the centerline velocity, \(w_s (z)\). It is to be noted that the fluid entrainment, owing to the tilt of the air curtain, takes place along the interfaces AE and FC, which may not necessarily be vertical. Therefore, to accommodate for the slant angle \(\theta\), we consider the flux along \(ds = dz \sqrt{1 + \tan ^2\theta }\). The resultant volume conservation equation therefore becomes:

$$\begin{aligned} \frac{d}{d z}(a w_s) = \alpha _{E} w_s \sqrt{1+\tan ^2 \theta } \end{aligned}$$
(21)

The next set of conservation equations are derived from the two components of the total momentum flux, \(m = aw_s^2\). We first consider the horizontal component of the total momentum flux, \(m_x = aw_s^2 sin\theta\), where the only force affecting the fluid momentum is due to the ambient stack effect resulting from the horizontal density stratification. For an isothermal jet, this force will be zero. In the context of building flows, it should be highlighted that this forcing is zero at the neutral height where the outside and inside fluid pressures are same. A representative stack profile is also shown in Fig. 3a. It has previously been shown that the location of neutral height has a significant effect on the dynamics of fluid exchange that occurs in buildings [17]. The horizontal momentum flux equation, therefore, under the Boussinesq approximation, can be written as:

$$\begin{aligned} \frac{d}{d z}\left( a w_s^2 \sin \theta \right) =-g^{\prime }\left( z-\frac{H}{2}\right) \end{aligned}$$
(22)

where z is measured from the nozzle exit. On the other hand, in the vertical direction, the corresponding momentum flux \(m_z = aw_s^2 cos\theta\) could change due to two effects. First, a net pressure difference that acts between the faces BE and FD of the control volume. Here, the length element dx can be expressed as \(dz \ tan \theta\) to estimate the component of this pressure force acting in the vertical direction. The second contribution to the vertical momentum is owing to the buoyancy force, \(g(\rho - \rho _d)\), that the air curtain experiences as it penetrates in the ambient fluid. Therefore, the resulting equation for the momentum flux conservation in z-direction assumes the following form:

$$\begin{aligned} \frac{d}{d z}\left( a w_s^2 \cos \theta \right) =g^{\prime }\left( z-\frac{H}{2}\right) \tan \theta -\frac{a g}{\cos \theta }\left( \frac{\rho }{\rho _d}-1\right) \end{aligned}$$
(23)

Lastly, considering the initial density difference at the nozzle exit, \(\rho - \rho _d\), and invoking the conservation of the associated buoyancy flux per unit jet length, \(aw_sb\), yields:

$$\begin{aligned} \frac{d}{d z}\left( a w_s g\left( \frac{\rho }{\rho _d}-1\right) \right) =0 \end{aligned}$$
(24)

To simplify the equation set (21)–(24), we note that the characteristic velocity and length scales, \(w_s\) and a, can be expressed as m/Q and \(Q^2/m\) respectively where \(m = \sqrt{m_x^2 + m_z^2}\) with \(tan \theta = m_x/m_z\). This yields the following equations which are subsequently solved in MATLAB to obtain the spatial evolution of the air curtain jet:

$$\begin{aligned} \begin{aligned}&\frac{d}{d z}(Q)=\frac{\alpha _{E}}{m_z} \frac{m_x^2+m_z^2}{Q} \\&\frac{d}{d z}\left( m_x\right) =-g^{\prime }\left( z-\frac{H}{2}\right) \\&\frac{d}{d z}\left( m_z\right) =+g^{\prime }\left( z-\frac{H}{2}\right) \frac{m_x}{m_z}-\frac{Q^2 b}{m_z} \\&\frac{d}{d z}(Q b)=0 \end{aligned} \end{aligned}$$
(25)

For a vertically blowing ACD installation with an exit velocity \(w_0\) and density \(\rho _0\), the initial conditions for the variables describing equation set 25 are:

$$\begin{aligned} \begin{aligned}&Q(0) = Q_0 / W \\&m_x(0) = 0 \\&m_z(0) = a_0 w_0^2 \\&b(0) = g\left( \rho _0 / \rho _d - 1\right) \end{aligned} \end{aligned}$$
(26)

5 Results

In this section, we would first present a qualitative description of the buoyancy (temperature) field in the domain obtained from the experiments and simulations. This allows demarcation of an ACD installation into an unstable or a stable regime. Subsequently, we calculate the performance of neutrally buoyant and heavy air curtains in the form of sealing effectiveness using different methodologies and discuss the implications of an assisting buoyancy towards better climatic separation. In the last part of this section, a quantitative analysis comparing the velocity statistics of an air curtain jet with an isothermal jet is presented.

Fig. 4
figure 4

Comparison between the experimental flow visualization and the pseudocolor buoyancy contours of the LES simulations at different deflection modulus

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5.1 Structure of the air curtain flow

Figure 4 presents a comparison of the dye visualization obtained from the experiments with that of the buoyancy distribution predicted by the LES simulations. In experiments, a green dye was premixed with the dense fluid, whereas the laundry brightener (blue) was used to mark the curtain fluid of density \(\rho _0\). The resultant flow field for two values of deflection modulus, \(D_m\), is shown in Fig. 4a and c respectively. At small values of \(D_m\), the ACD fails to reach the opposite end of the doorway and is deflected sideways due to the disruptive buoyancy difference across it. As a result, there is infiltration of the ambient fluid (notice the trajectory of the air curtain) accompanied by leakage from the indoor region. This regime is, therefore, commonly referred to as an unstable (or breakthrough). The buoyancy distribution obtained from the LES simulation of an unstable ACD is shown in Fig. 4b. Here, it should be pointed out that while the blue color corresponds to an equivalent density of \(\frac{\rho _d + \rho _l}{2}\), it is a reasonably good marker for the air curtain fluid that separates the indoor and outer regions. From simulations, the data is extracted from the center plane (\(y = L_y/2\)) to illustrate the characteristic features associated with the air curtain flows. The presence of overturning vortices at the interface between the light (white) and the dense (green) fluid is attributed to shear instability. Note that these features are not evident in experimental images since the camera sensor inevitably performs a spatial averaging across the width/span of the tank during the recording process. Overall, a good qualitative agreement can be observed between experiments and simulations.

When the momentum flux of the air curtain is increased while keeping the stack effect the same, i.e., as \(D_m\) increases, the jet possesses sufficient strength to reach the other end. As a result, a near-vertical trajectory is established, which is also predicted by the numerical computations as shown in Fig. 4d. This results in an aerodynamic barrier that significantly suppresses the pure buoyancy-induced exchange that occurs at small \(D_m\). However, as the jet travels from the nozzle exit to the bottom wall, some amount of fluid is entrained into the jet leading to a slight dilution. In addition to this, upon impingement of the ACD on the floor, the jet is divided into two parts that travel to the indoor and outdoor spaces. While the flow physics of this impingement region is not properly understood, it is likely that the high turbulence in this region results in some fluid exchange across the air curtain, leading to an effectiveness smaller than unity.

5.2 Air curtain effectiveness

In this section, we first quantify the sealing effectiveness corresponding to the neutrally buoyant air curtains and validate our results with the existing literature. Subsequently, we present estimates of E for heavy air curtains and compare them with their neutral counterparts to assess the influence of the assisting buoyancy.

5.2.1 Neutrally buoyant ACDs

A neutrally buoyant air curtain comprises a jet having the same density as that of the space it is discharging into, i.e., \(\rho _0 = \rho _d\). The effectiveness can be calculated using the primitive definition introduced in Eq. (2) which compares the fluid exchange in the presence of an ACD with the case when the doorway is unprotected. Herein, the exchange flow rates are estimated using Eq. (18), which takes into account the density of the indoor fluid at any time instant t. It is to be noted that the effectiveness would also vary with time, and in this paper, we present the values of ACD effectiveness at a quasi-steady state when the exchange flow rates are nearly constant. The variation of the estimated effectiveness as a function of deflection modulus is shown in Fig. 5 for all the methodologies and are compared with the experimental values of [20] who conducted measurements in a facility of similar size with similar operating conditions in the same range of \(D_m\). An open door situation corresponds to \(D_m =\) 0.

Fig. 5
figure 5

E vs. \(D_m\) curve obtained from the experiments and compared with the measurements of Jha et al. [20] and the LES data

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Firstly, the qualitative variation is represented accurately using both numerical and experimental techniques. The sealing effectiveness assumes low values for unstable ACDs (\(D_m \lessapprox {\mathcal {O}} (0.1)\) for which the air curtain breakthrough and allows a certain amount of fluid exchange to occur. As the deflection modulus is increased, the values of E correspond to a typical stable configuration for which the \(E_{max}\) are approximate of the order 0.8, i.e., only 20% of the original exchange flow remains across the air curtain. Upon an increment in the \(D_m\) beyond this point, a gradual reduction in the sealing effectiveness is observed, which is likely due to stronger turbulence in the impingement region that results in mass transfer.

Second, it can be seen that the effectiveness predicted by the RANS-based numerical simulations differs considerably from the experimental measurements. Generally, we observe that for 2D computations, the values of E are over-predicted beyond a \(D_m\) of 0.2, whereas the 3D framework results in an under-prediction. A trend line for the stable ACDs is also shown in Fig. 5 that shows the difference in magnitude from the experimental measurements, which is denoted using the black dashed line. Upon careful observation of the buoyancy distribution in various simulations, it is seen that the two-dimensional computations fail to accurately reproduce the physical phenomenons of vortex stretching, due to which the overturning motions do not cascade into smaller scales. Therefore, the unavailability of a third direction leads to an overall lower fluid exchange in the presence of an air curtain resulting in a higher effectiveness. Whereas in the corresponding LES data, we observe a very good collapse with the experimental estimates, i.e., within the experimental uncertainty limits (\(\approx 5 \%\)). This is not surprising since the LES explicitly calculates the motion of large eddies, which are largely responsible for the bulk exchange associated with air curtain flows. The effect of the modeled (subgrid) scales, as it appears, does not significantly affect the estimated value of q.

Finally, as evident from Fig. 5, the present experiments also conform very well to that of the measurements of Jha et al. [20] and the LES data. This also serves as a validation for the data acquisition and analysis techniques for our measurements. Using the same process, we then alter the operating density of the air curtain to create a HAC such that \(\rho _0 > \rho _d\) and the corresponding effectiveness estimates are discussed next.

5.2.2 Influence of assistive buoyancy

It is expected that the added buoyancy (\(B_a \propto (\rho _d - \rho _0)\)) would suppress the effects caused due to the infiltrating exchange flow (\(B_i \propto (\rho _d - \rho _l)\)). Overall, the indoor fluid density \(\rho _d^n (t)\), at the end of an experiment (or simulation), would be a manifestation of the competing effect of \(B_a\) and \(B_i\). Therefore, any assisting buoyancy, i.e., \(S>\) 0, is likely to result in higher effectiveness than its neutral counterpart (\(S =\) 0).

The parametric variation between E and S was studied using the 2D RANS simulations due to their lower computational cost and is shown in Fig. 6a. It can be seen that the effectiveness increases as the positive buoyancy (S) increases, irrespective of the value of \(D_m\). The change in the sealing effectiveness relative to a neutrally buoyant air curtain, \(\frac{\Delta E}{E_{NBAC}}\), was observed to be around 10 - 20% for \(S =\) 0.3, which is also investigated using LES and experiments and is discussed later. It is also possible that for a large value of S (\(\rho _0 \gg \rho _d\)), the effectiveness could exceed unity, implying that the temperature within the indoor space would be lower than \(T_d =\) 25 \(^\circ\)C at the end of the simulation. However, in the present computations, the maximum value of S was taken to be 0.39, which did not exhibit this behavior for \(D_m \le\) 1.5. As established in the last section, 2D RANS simulations often over-predict the values of E associated with air curtain flows, and the same is observed for the case of HAC, as discussed next.

Fig. 6
figure 6

Estimates of the sealing effectiveness of heavy air curtain from various methods

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The estimates of the sealing effectiveness obtained using the experiments, and that of the LES computations are plotted in Fig. 6b. Through both methodologies, a positive correlation is observed between E and S; however, a minimal increment (of \({\mathcal {O}} (5\%)\)) is observed in the heavy air curtains as compared to their neutral counterpart. Equivalently, it implies that the fluid infiltration associated with a HAC installation is 25% lesser as compared to a neutrally buoyant ACD. Frank and Linden [18] computed the effectiveness of buoyant air curtains (with equivalent \(\rho _0 < \rho _d\)) thus making them negatively buoyant. They showed experimentally that the effectiveness, E, reduces with increasing the extent of opposing buoyancy i.e. a lighter air curtain is less effective. The findings of the present study are, therefore, in line with the previous work.

In addition, the usage of two scalars (salt and sugar) in the experiments allowed an estimation of the exchange flow based on the infiltration effect as discussed in Sect. 3 using Eq. (20). While the conventional definition, E, provides a quantitative measure accounting for the thermal comfort of building occupants in this context, the alternate definition based on the infiltration, \(E^\prime\), focuses on the effect of contamination transport, therefore, providing a metric for indoor air quality. Our measurements suggest a difference of less than 2% in the calculated values of E and \(E^\prime\), thereby indicating a good coupling between the effect of an added buoyancy on the thermal comfort as well as the indoor air quality of building space. A similar procedure could be adopted in the numerical simulations to differentiate between the two measures of effectiveness by introducing passive scalars, which could be a potential problem for future research.

5.3 Velocity statistics

The buoyancy difference across an air curtain is a significant contributor towards influencing the flow dynamics when compared to an isothermal jet. The case of an isothermal jet is a widely studied problem [5, 23, 41] and, therefore, will now be used to compare the velocity statistics at varying deflection modulus for the HAC case with \(S =\) 0.3. At this junction, it shall be reiterated that an ACD with smaller \(D_m\) corresponds to a higher buoyancy difference across it, whereas an ACD approaches an isothermal jet as \(D_m \rightarrow \infty\).

The instantaneous velocity fields are first spatially averaged in the spanwise direction within \(y =\) 0.04 m (0.25W) and \(y =\) 0.13 m (0.75W) to exclude the wall effects that are present at \(y =\) 0 and \(y = W\). Then, a temporal averaging operator is applied during the time period when the exchange flow rate exhibits a quasi-steady behavior. It should be mentioned at this point that a steady-state solution for the present ACD configuration is not possible, as the density difference across the air curtain would asymptotically approach a value of zero. Mathematically, this averaging can be written as:

$$\begin{aligned} \overline{w} (x, z) = \frac{\int _{T_1}^{T_2} \int _{Y_1}^{Y_2} w(x, y, z, t) \ dy \ dt}{(T_2 - T_1)(Y_2 - Y_1)} \end{aligned}$$

To estimate the velocity statistics along the ACD centerline, we identify the location of the centerline, \(x_{CL} (z)\), based on the maxima in vertical velocity component at any horizontal cross-section, \(\overline{w} (x)\). To include the possibility that this velocity maxima could occur between two grid points, we fit a second-order polynomial around the centerline location (\(x_{CL} (z)\)). This process is repeated for all the grid points in the z direction to obtain \(\overline{w}_{CL} (z)\). In a similar manner, the corresponding RMS velocity field is obtained along the ACD centerline as:

$$\begin{aligned} \overline{w^\prime } (x, z) = \sqrt{\frac{\int _{t = T_1}^{t = T_2} \int _{y = Y_1}^{y = Y_2} [w(x, y, z, t) - \overline{w} (x, z)]^2 \ dy \ dt}{(T_2 - T_1)(Y_2 - Y_1)}} \end{aligned}$$

These statistics are then normalised using the exit velocity, \(w_0\), which is a common practice for jet studies, and are shown in Fig. 7.

Fig. 7
figure 7

Comparison of the mean centerline and RMS vertical velocity obtained from the LES simulation at different \(D_m\) to quantify the differences between an air curtain and an isothermal jet

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It can be observed from Fig. 7a that the mean centerline velocity for the isothermal jet obtained using the present simulations agrees very well with the analytical solution [42] in the self-similar region which spans roughly between \(z =\) 0.2H and \(z =\) 0.8H. With the introduction of a buoyancy difference across the jet, \(\overline{w}_{CL} (z)\) was found to be lower than its isothermal value in the self-similar region. This reduction was observed to be larger as \(D_m\) decreases and, therefore, can be attributed to the transverse stack effect, which was shown to deflect the ACD sideways providing it with a horizontal momentum. In fact, for the unstable ACD at \(D_m =\) 0.05, the decay rate of the centerline velocity, \(\frac{d \overline{w_{CL}}}{dz}\), near the jet exit, was found to be maximum, likely due to a complete sideward deflection. Whereas for the stable air curtain installation, while the decay rate was larger as compared to an isothermal jet, it was observed to be less sensitive to \(D_m\) as shown in Fig. 7a. Our preliminary analysis suggests that the entrainment of ambient fluid is larger at smaller values of \(D_m\), which provides a pathway for the ACD deceleration along the centerline. It can also be noted that the \(D_m =\) 0.1 case matches quite well with that of higher \(D_m\) cases suggesting the possibility that the stability of the heavier air curtain is established at a value slightly lower than the theoretical estimate of \(D_{m,min} = \frac{1}{8}\) for the neutrally buoyant air curtain. This is due to the added buoyancy provided in the ACD which allow a deeper penetration of the curtain fluid into its ambient.

On the other hand, Fig. 7b depicts an interesting behavior for the RMS vertical velocity. Similar to the isothermal jet, two peaks are obtained in the distribution of \(\overline{w^\prime }\) at \(z \approx\) 0.3H and \(z \approx\) 0.95H. Physically, the first peak corresponds to the location of maximum turbulence intensity after the initial development of the planar jet, where the individual shear layers across the jet merge. In contrast, the second peak is due to the presence of the impingement region, where the local turbulence intensity is observed to be larger. Overall, there is a good agreement between these numerical results with the measured estimates of Khayrullina et al. [4]. It is found that the magnitude of the second peak associated with the air curtains matches very well with that of an isothermal jet, suggesting that the fluctuating field behaves very similar near the wall, and the buoyancy has negligible effects. In comparison, the normalized RMS value corresponding to the first peak for ACDs is higher than an isothermal jet and does not exhibit any significant \(D_m\) dependence either.

5.3.1 Comments on the analytical model

The system of differential Eq. (25) presented in Sect. 4 are solved numerically using MATLAB. We use the explicit Runge–Kutta method to obtain the spatial evolution of the air curtain jet, having the initial conditions outlined in Eq. (26). In the following, we compare the predictions of the analytical model with the LES data.

Figure 8a and b show the variation of the centerline velocity and the curtain deflection for the case of \(D_m =\) 0.4. To obtain the path traversed by the ACD in the vertical direction, we first calculate \(\theta\) as \(tan^{-1} \left( \frac{m_x}{m_z}\right)\) and use the expression \(dx = dz \ tan(\theta\)) to quantify x(z). The comparison of the air curtain deflection between the analytical model and the LES data seems reasonably good. The jet exhibits a minor deflection towards the outdoor space due to horizontal buoyancy \(g(\rho _d - \rho _l)\). Beyond \(z/H \approx\) 0.75, the plane impinging jets behave differently than a free jet [4], and this region is shaded in Fig. 8. In contrast, significant discrepancies exist in the centerline velocity predicted by the analytical model. Most notably, this formulation does not account for the no-penetration boundary condition that should be imposed at \(z = H\). The initial decay rate also seems to be overestimated by the model. Therefore, it can be formally concluded that while the analytical formulation predicts the trajectory of an air curtain reasonably well, it may not provide good estimates of the dynamical quantities associated with it.

Fig. 8
figure 8

Comparison between the predictions of the analytical model and that of simulation (LES) data corresponding to a stable air curtain at \(D_m =\) 0.4

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6 Conclusions

Numerical simulations and experiments were conducted to quantify the sealing effectiveness, E, of neutrally buoyant and heavy air curtains. This measure of E determines the level of thermal comfort within a building which could be affected by the infiltration of outdoor fluid through any opening in the building fabric. 2D RANS, 3D RANS and LES computations were performed to understand the influence of deflection modulus, \(D_m = \frac{\rho _0 a_0 w_0^2}{g H^2 \Delta \rho }\), and the density ratio, \(S = \frac{\rho _0 - \rho _d}{\rho _d - \rho _l}\) on the resultant exchange flow. On the other hand, dye visualization and bulk density/conductivity measurements were facilitated to provide experimental measure of the general structure of air curtain flows and their corresponding sealing effectiveness.

We compared the effectiveness of neutrally buoyant air curtains (NBAC) obtained from the LES simulations with the literature data and found a satisfactory agreement. Whereas some discrepancies were observed when the RANS solver was employed, likely due to inherent modeling of various flow scales. In addition, a good match of sealing effectiveness was also observed with the present experiments. Upon the introduction of an assisting buoyancy, it was seen that the sealing capabilities increased irrespective of the methodology adopted. This implies that the heavy air curtains positively impact the thermal comfort of building occupants at all values of \(D_m\). As compared to the case of neutrally buoyant ACDs, an increase of approximately 15% and 5% in E was estimated using 2D RANS and LES simulations for stable air curtains (\(D_m \ge {\mathcal {O}}\)(0.1)). The experimental data for the HAC also conformed with the LES estimates, which confirmed the bias associated with 2D RANS computations. Therefore, the infiltration associated with the NBACs could be suppressed by another 25% with the introduction of an assistive buoyancy.

The velocity statistics of heavy air curtains at different values of \(D_m\) were also compared to that of an isothermal jet to study the underlying effects of the various density gradients prominent in such flows. The resulting mean centerline velocity exhibited significant decay as a result of buoyancy difference across the jet. It was found that this decay was much larger for unstable air curtains where the ACD would fail to impinge on the floor and was deflected sideways. Whereas, the onset of stability in the heavier ACDs was observed near \(D_m \approx\) 0.1, which is smaller than NBAC measured in experiments and theoretical values.