Use of proper orthogonal decomposition and linear stochastic estimation technique to investigate real-time detailed airflows for building ventilation

Abstract

As energy efficiency concerns grow for building heating, ventilation, and air-conditioning systems, more advanced mathematical models and methods may be needed for implementing more comprehensive building environmental monitoring practices. This work investigates the methodology of applying a coupled proper orthogonal decomposition (POD) and linear stochastic estimation (LSE) technique to provide detailed real-time velocity information for a single-room test environment under either mechanical ventilation or wind-driven cross natural ventilation with the support of limited on-site monitoring measurements. In addition, an example application of the proposed POD–LSE methodology with limited measurements to obtain real-time fluctuating wind-driven ventilation rates using a POD–LSE reconstructed velocity profile across a window opening is presented and shows very good agreement with the measured ventilation rates.

Keywords

Proper orthogonal decomposition Linear stochastic estimation Mechanical ventilation Natural ventilation Ventilation rate

Introduction

Commercial and residential buildings currently consume close to 40% of the annual energy used in the US.¹ Heating, ventilation, and air-conditioning (HVAC) systems whose primary function is to maintain appropriately conditioned indoor building environments are responsible for consuming upwards of half of this energy. The building environment is an unsteady system of many variables, some of which change unpredictably in terms of location and time. Monitoring the unsteady building environment is one way to better assess and address thermal comfort and thus affects HVAC system control and energy consumption.

Whether by traditional mechanical ventilation systems or by more complex natural ventilation and mixed-mode ventilation systems, airflow patterns and airflow fields are the major driver behind the indoor building environment and are responsible for the transportation and distribution of contaminants, space air temperatures, ventilation rates, and general airflow organization.^2,3 However, monitoring detailed building environment airflow fields is complex. Instead, HVAC systems are designed and operated based on simplified assumptions, average environmental parameters, and basic airflow concepts.⁴

Constructing and implementing more advanced mathematical models and methods may be needed in order to improve performance and maximize energy efficiency for HVAC systems in the future through better building control strategies. More accurate control over indoor thermal comfort and air quality may need detailed real-time monitoring of indoor airflow fields. Having the capability to acquire spatially and temporally detailed information about the ongoing environmental airflow field changes could further advance HVAC system’s operation, design and delivery strategies, and ultimately energy efficiency. Therefore, monitoring detailed building environmental airflow field information and reporting it to a building automation system or energy management system is an important step.

Numerical simulations such as multi-zone simulations can provide bulk airflow quantities for various cases at a relatively low cost, which is helpful for design and assessment of indoor building ventilation systems including mechanical and natural ventilation.^5–7 However, these simulations rely on accurate measurements for the boundary conditions, which can be difficult to acquire for complex ventilation strategies such as natural ventilation.^8,9

Detailed computational fluid dynamics (CFD) simulations are capable of providing highly resolved spatial and temporal building environmental information.^10–12 Particularly, for natural ventilation, advanced turbulence models such as large eddy simulation (LES) models can be applied to simulate the effect on the building environment and ventilation rates from the variability of wind. Jiang and Chen¹³ studied this effect using an unsteady CFD calculation and LES turbulence model. Wright and Hargreaves¹⁴ used a detached eddy simulation turbulence model to capture unsteady cross ventilation, which was also analysed by LES turbulence models.¹⁵ However, CFD simulations quickly reach the limits of computability when modelling real-world building environments and therefore generally use steady state boundary conditions to examine a single-design condition instead of prescribing monitored boundary conditions fluctuating in both speed and direction.^10,16–18 In addition, simulation results cannot be completely trusted unless validated by corresponding high-quality experimental data.¹⁹ The large computation power required for unsteady real-world simulations make CFD not feasible for real-time building environmental monitoring with current technology.

Detailed experimentation is another methodology used to better understand the indoor building environment and to help validate and improve numerical methods.^{10,12,16,19,20} Tracer gas decay methods are often applied to derive average ventilation rates through an opening. For example, Dascalaki et al.²¹ carried out four single-sided wind-driven natural ventilation experiments using the tracer gas decay methods to show the performance of wind-driven natural ventilation is significantly influenced by the boundary conditions of the wind. Since the wind direction and speed are fluctuating constantly under real conditions, it is important to consider the fluctuating effects in experiments and in numerical simulations, and ultimately considered to improve methods of accurate monitoring.

Other studies have investigated the influence of wind fluctuations on cross natural ventilation with an aim at improving the evaluation accuracy of natural ventilation.^10,22 Particle image velocimetry (PIV) is an increasingly utilized technology providing detailed velocity field information, but relying on expensive systems. More importantly, a PIV system cannot be permanently installed and utilized in an operational building environment for monitoring purposes.

Traditional monitoring technologies have limited spatial and temporal monitoring capabilities and may not provide enough feedback information about building environmental information, especially about airflow fields, for more advanced HVAC system control. Currently, even more sophisticated controllers such as those with proportional–integral–derivative control or model predictive control that addresses building environmental information over varying time horizons are still constrained by the reliance on single-point monitoring measurements as uniform representations.^23–25

This work applies a coupled multi-variant statistical technique in an attempt to obtain detailed air velocities through mathematical reconstruction for a single-room test environment under mechanical ventilation and cross natural ventilation based on limited on-site measurements in real-time. Using experimentation and through development of the methodology to conduct analysis of the coupled proper orthogonal decomposition (POD) and linear stochastic estimation (LSE) technique, this work has demonstrated the feasibility and limitations of the POD–LSE application to different ventilation conditions.

Review of the POD–LSE technique

Proper orthogonal decomposition

Traditional techniques widely used for ventilation study such as multi-zone models, zonal models, and CFD models^5–20 can only provide either bulk flow information but with limited details or comprehensive and detailed indoor environmental information but with a very high cost of computation, and thus do not fit the requirement of providing detailed and real-time ventilation information. There are techniques in related research fields characterizing turbulent behaviour that may provide alternative methods potentially applicable to analysing the dynamics of building ventilation. In particular, POD is a statistical analysis tool that has been used to study many different turbulent flows including shear flows,²⁶ wake dynamics,²⁷ boundary layers,²⁸ flows in an open cavity,²⁹ flow over a backward facing step,³⁰ and more. This technique has also been used to reconstruct temperature and velocity fields of indoor building environments through numerical simulation.^31–33 In addition, POD has been coupled with different stochastic estimation techniques to predict turbulence in wakes²⁷ and the backward facing step problem.³⁰ More recently, POD has been coupled with LSE and successfully applied to a methodology to reconstruct building environmental thermal fields using a limited number of monitoring temperature measurements.³⁴

Theoretically, POD is a multivariate statistical technique that seeks to construct reduced-order models of interdependent variables using a small number of uncorrelated variables, while still retaining the statistical variance of the original data.³⁰ More detailed presentations for the POD technique are available in literature.^30,33–35 POD attempts to project the fluctuating velocity field onto a set of linear orthogonal basis functions φ(x) such that the average variance captured by the projection is greater than any other set of basis functions.

The solution to the optimization yields a set of linear orthogonal Eigen functions henceforth termed the POD basis functions shown in equation (1). The fluctuating velocity field can be thus exactly reconstructed by the infinite summation shown in equation (2)

\int_{Ω} á u' (x, t), u' (x', t) ñ φ (x') d Ω = λ φ (x)

(1)

u' (x, t) = \sum_{i = 1}^{\infty} a_{i} (t) φ_{i} (x)

(2)

where,

$a_{i} (t)$ = the set of temporal coefficients corresponding to the ith POD mode;

$φ_{i} (x)$ = the basis function corresponding to the ith POD mode.

There are at least two methods for solving the POD basis functions and temporal coefficients. The direct method requires the solution to the two-point correlation over the entire flow field for every variable. An alternative method called the snapshot method was first proposed by Sirovich³⁶ and is typically more computationally efficient than the direct method and is utilized in this work.^30,37

A snapshot is the instantaneous fluctuating velocity field at discrete locations over the entire domain, typically captured from experimental measurements or computational modelling such as CFD. Following the method proposed by Sirovich,³⁶ a correlation matrix for a discrete number of field snapshots M is calculated using equation (3) as follows, where u is the u-component of 3D instant velocity measurement:

C_{ij} = \frac{1}{M} \int_{Ω} u' (x, t_{i}) u' (x, t_{j}) d Ω

(3)

Then, the POD basis functions are computed from equation (4), where Einstein notation is continually used to represent tensor order. The temporal coefficients are defined in equation (5).

φ_{i} (x) = \sum_{k = 1}^{M} ζ_{ik} u' (x, t_{k})

(4)

ζ_{ik} = \frac{ϕ_{k}^{i}}{\sqrt{M \sum_{m = 1}^{M} \sum_{r = 1}^{M} ϕ_{m}^{i} ϕ_{r}^{i} C_{mr}}}

(5)

Here

ϕ_{k}^{i}

represents the kth component of the ith Eigenvector of the correlation matrix C_ij. Finally, the POD temporal coefficients for the ith POD mode at the jth instant in time are computed from equation (6), and the fluctuating velocity field is estimated by the finite summation of POD modes shown in equation (7).

a_{i} (t_{j}) = M \sum_{k = 1}^{M} ζ_{ik} C_{kj}

(6)

\tilde{u}' (x, t) = \sum_{i = 1}^{P} a_{i} (t) φ_{i} (x)

(7)

Linear stochastic estimation

Stochastic estimation was introduced to fluid dynamics by Adrian et al.^38,39 and has since been used widely in fluid dynamics research. There are different types of stochastic techniques, including LSE, quadratic stochastic estimation, kernel ridge regression, and principal component regression. LSE has shown to be a good enough approximation of conditional averages.^38,39

When applied to fluid flows, the purpose of this technique is to estimate the conditional average of a fluctuating velocity vector at a spatial position x + r, given a measured condition occurring at position x as shown in equation (8).

u *' (x + r) = á u' (x + r) | u' (x) ñ

(8)

This can be directly computed by conditionally sampling a velocity field. However, stochastic estimation creates a power series representing the conditional average (equation 9).

u *' (x + r) = A_{ij} (r) u'_{j} (x) + B_{ijk} (r) u'_{j} (x) u'_{k} (x) + \dots

(9)

The coefficients A_ij(r), B_ijk(r),… are found by minimizing the mean square error between the fluctuating velocity and the estimate at position x + r. The LSE technique is determined by truncating the power series at the linear term in equation (9).

POD–LSE complementary technique

The LSE technique can be mathematically coupled with the POD technique to estimate the temporal coefficients based on measured fluctuating velocity events for a few locations. An estimated temporal coefficient $a_{i}^{*}$ is calculated by equation (10), where A_ki is a matrix of LSE coefficients calculated by equation (11) for the ith POD mode at the kth measurement point. The $u'_{n} (x_{k}, t_{j})$ is the new measured velocity event for the ith POD mode at the jth instant in time for position k, and $u' (x_{k}, t_{j})$ represents all the measured fluctuating velocity events at the kth measurement point used in the original POD decomposition. Here the 〈〉 symbol represents time averaging, a_i is the actual temporal coefficient for the ith POD mode calculated from equation (6), and $n_{p}^{*}$ is the number of spatial data points used to estimate temporal coefficients.

a_{i}^{*} (t_{j}) = \sum_{k = 1}^{n * p} A_{ki} u'_{n} (x_{k}, t_{j})

(10)

A_{1 i} á u' (x_{1}, t) u' (x_{1}, t) ñ \dots + A_{n * p 1} á u' (x_{n * p}, t) u' (x_{1}, t) ñ = á a_{1} u' (x_{1}, t) ñ \dots A_{1 i} á u' (x_{1}, t) u' (x_{n * p}, t) ñ \dots + A_{n * p 1} á u' (x_{n * p}, t) u' (x_{n * p}, t) ñ = á a_{i} u' (x_{n * p}, t) ñ

(11)

This technique as described allows a new POD temporal coefficient to be estimated based on an instantaneous velocity measurement or set of measurements. Then the entire velocity field can be estimated using the POD basis functions and the new LSE estimated temporal coefficient in near real-time (so-called reconstruction process), which is shown in equation (12).

u *' (x, t_{j}) = \sum_{i = 1}^{P} a_{i}^{*} (t_{j}) φ_{i} (x)

(12)

Methods

Indoor airflows

Experimental setup and methodology

Two experimental setups were carried out using the same single-room test environment shown in Figure 1, one for mechanical ventilation indoor airflows and one for cross natural ventilation indoor airflows. The electric heat source in the test environment was the data acquisition system and the analogue to digital signal converter with a combined heat output of 100 W. The room is 4.65 m long by 3.66 m wide by 2.44 m high (above a raised floor system) with two double-paned operable windows, one on the south and one on the east-facing facades. Each window opening measured 0.73 m wide by 0.47 m high. The room was equipped with an under-floor air distribution (UFAD) system, and two diffusers were placed at the centre of the room for mechanical ventilation conditions.

In the mechanical ventilation experimentation, six Omni-direction anemometer probes from Dantec Dynamics A/S (Denmark) were placed throughout test environment to measure air velocities. Figure 2 shows the layout out of the Omni-directional probes for the mechanical ventilation experiment. All Omni-directional probes were placed 1.0 m above the raised floor, except for the probe located at the supply diffuser that was placed at raised floor level.

Eight anemometer probes were used to measure air velocities for the natural ventilation indoor airflow experiments. Two, three axis (3D) gold-plated hot wire probes from Dantec Dynamics A/S (Denmark) were used to gather three component velocity data at the centre of each of the two openings. The 3D probes were used to verify cross natural ventilation conditions since the experiments were exposed to uncontrollable natural wind. In addition, the other six Omni-directional probes were placed throughout the test environment according to the layout shown in Figure 3.

All anemometer probes sampled air velocities at 2 Hz during the mechanical and natural ventilation experiments. When converting the voltage data to velocity data for Omni-directional probes, local temperature, pressure, and humidity measurements were used. Calibrated velocities for the 3D probes were obtained using a fourth-order linearization function regression for each wire from a 30-point speed calibration.

Summary of POD–LSE methodology

There are a number of steps involved in the POD–LSE methodology process. A summary description given below describes the entire methodology used for each experimental analysis. All computational analysis in this study was completed using MATLAB coding.

Step 1: Experimental data were acquired using the experimental setups discussed above for hours of mechanical and natural ventilation experiments.

Step 2: For POD, 1000 instantaneous velocity field snapshots measured from the probe locations for each experiment (see Figures 2 and 3) were decomposed into basis functions and temporal coefficients (see equation (4)).

Step 3: Then a matrix of LSE coefficients was calculated using the velocity field snapshots from only two probes and the POD temporal coefficients (see equation (14)). This step allowed the reconstruction (steps 4 and 5) to be completed using only two probes.

Step 4: An LSE estimated temporal coefficient was calculated from a new instantaneous velocity measurement at the two probe locations.

Step 5: Finally, the velocity field was reconstructed for all probe locations specific to each experiment (see Figures 2 and 3) using the LSE estimated temporal coefficient (step 4) and the original POD basis functions (step 2). In other words, the velocity field for all probes was reconstructed using limited on-site measurements from only two probes (for mechanical ventilation). The reconstruction was done for the entire experimental data set.

Natural ventilation airflows across window opening

Experimental setup

Eight anemometer probes were used to measure air velocities in the experiment to study the natural ventilation airflow across a window opening. Two 3D probes were installed in the centre of the two window openings to gather three component velocity data, while other six Omni probes were deployed in the south-facing window opening as shown in Figure 4. Anemometer probes were sampled at a rate of 2 Hz.

Results

Mechanical ventilation indoor airflow POD–LSE

The UFAD system was turned on and off during experimentation to maintain the thermostat temperature within the dead band of the temperature setting for cooling conditions. Therefore, two mean velocities were calculated, corresponding to either the on or off mode of the UFAD, for each probe location when constructing fluctuating velocity data sets (see equation (1)). POD decomposition was done on the velocity measurements from the six Omni probes using 1000 data snapshots to ensure that multiple UFAD system cycles were covered. Figure 5 shows the variance captured by the first 10 POD modes. Over 98% of the fluctuating velocity variance, or turbulent kinetic energy, from all six Omni-directional probe measurements were captured by the first POD mode. POD modes 2 and 3 captured less than 1.5% of the turbulent kinetic energy. This result demonstrated the construction of a significantly reduced-order POD model of the fluctuating velocity field for the single-room test environment under mechanical ventilation. Therefore, two POD modes could be used to complete the reconstruction of the entire velocity field in the study.

A matrix of LSE coefficients was calculated from the POD temporal coefficients and the velocities measured by the two probes that produced 1000 snapshot data. The fluctuating velocities were then reconstructed for the entire duration of the experiment by keeping the original POD basis functions and LSE coefficients, but using instantaneous (real-time) velocity measurements at probe locations 5 and 6, located at the thermostat and supply diffusers (see Figure 2), to estimate new temporal coefficients and complete the reconstruction. Figure 6 compares the measured velocities and POD–LSE reconstructed velocities measured at probe locations 1 and 2, which were estimated using the real-time velocity measurements at probe locations 5 and 6.

A root mean square error (RMSE) and a normalized root mean square error (NRMSE) were calculated between the measured and POD–LSE reconstructed velocities for all six Omni-directional probe locations using equations (13) and (14).

{RMSE}_{i} = \sqrt{\sum_{t = 1}^{T} (u' (x_{i}, t) - \tilde{u} *' (x_{i}, t)) / T}

(13)

{NRMSE}_{i} = \frac{{RMSE}_{i}}{{\bar{U}}_{s}}

(14)

Here

{\bar{U}}_{s}

is the average velocity measured at the supply diffuser when the UFAD is running, which corresponds to measurement at probe location 6 and T represents the experiment duration. The RMSE is presented to provide an error with velocity unit and because the calculated NRMSE is small. The RMSE and the NRMSE for all probes is summarized in Table 1.

Table 1.

Summary of RMSE and NRMSE for POD–LSE reconstruction for mechanical ventilation experiment.

	Probe 1	Probe 2	Probe 3	Probe 4	Probe 5 (thermostat)	Probe 6 (supply)
RMSE (m/s)	0.03	0.001	0.04	0.02	0.00	0.00
NRMSE	0.0037	0.0017	0.0048	0.0024	0.0000	0.0000

RMSE: root mean square error; NRMSE: normalized root mean square error.

The coupled POD–LSE methodology demonstrates great potential for reconstructing detailed dynamic velocity fields for mechanical ventilation in near real-time depending on the frequency of real-time measurements. POD enables simulation of a significantly reduced-order model for an unsteady velocity field under the tested UFAD mechanical ventilation condition. Measured velocity fluctuations at all probe locations were highly correlated, which shows why one dominant POD mode exists. Over 99% of the turbulent kinetic energy was captured by the first two POD modes. This allowed for the fluctuating velocity field at all six Omni-directional probe locations to be reconstructed using only two measurements in real-time to a high level of accuracy.

Natural ventilation indoor airflow POD–LSE

POD decomposition was carried out on 1000 snapshots of the measured data from the natural ventilation indoor airflow experimental setup (see Figure 3). Figure 7 shows the fluctuating velocity variance from the eight probes captured by the first 10 POD modes.

The first POD mode captured nearly 65% of the fluctuating velocity field’s variance. Over 95% of the fluctuating velocity field’s variance was captured by the first six POD modes. The POD decomposition for the natural ventilation indoor airflow experiment did not construct a significantly reduced-order mode of the velocity field. Eight probes were used to measure air velocities of the indoor airflows for the single room under wind-driven natural ventilation. However, POD would simulate a model with at least six measurable modes. For this type of system, a limited number of monitoring measurements was not able to completely reconstruct the measured variance. This is because POD temporal coefficients were uncorrelated events, and the LSE technique was used to correlate temporal coefficients with specific monitoring measurements. Therefore, more monitoring measurements would be needed to reconstruct the entire natural ventilation indoor airflow field instead of only two seen with the mechanical ventilation experiment. The proposed methodology may not be feasible in a practical sense for analysing and reconstructing the indoor airflow velocity field for the single-room test environment under cross natural ventilation conditions due to the necessary number of monitoring probes.

Natural ventilation window airflow POD–LSE

POD decomposition was carried out using 1000 snapshots of the measured data from the natural ventilation rate experiment (see Figure 4) where seven probes were located in the south-facing window opening. POD would reduce the seven probe signals into two or three dominant POD modes. Figure 8 shows the fluctuating velocity field’s variance captured by each mode. The matrix of LSE coefficients was calculated from the constructed POD temporal coefficients and measured velocities of the 1000 snapshots.

Since the decomposition yields three dominate POD modes, a correlation was calculated to see if any probe was highly correlated with a particular POD mode. This step was not discussed in the mechanical ventilation experiment because the fluctuating velocity measurements of each probe for studying the mechanical ventilation were well correlated and one dominant POD mode would exist (see Figure 5). Table 2 provides a summary of the correlation coefficients for each probe between the velocity data and the temporal coefficients for the first three POD modes under natural ventilation airflow across window opening experimentation.

Table 2.

Correlation coefficient summary for three dominant POD modes and the measured fluctuating velocities of seven probes used for studying natural ventilation.

	Probe 1	Probe 2	Probe 3	Probe 4	Probe 5	Probe 6	Probe 7
Mode 1	0.98	0.72	0.86	0.96	0.88	0.99	0.86
Mode 2	−0.13	−0.49	−0.50	0.27	−0.47	0.01	0.45
Mode 3	−0.11	0.48	0.11	0.05	0.04	−0.11	0.18

The correlation calculation reveals that certain fluctuating velocities were correlated with temporal coefficients for certain POD modes. This correlation allows the use of a single probe to estimate a specific temporal coefficient through the LSE technique (see equations (13) and (14)) as opposed to using all seven probes. Two probes, probes 6 and 3, were chosen because of their greatest correlation with the first two dominant POD modes and thus were used to recalculate LSE coefficients for the three dominant POD modes from the original 1000 data snapshots, and then to reconstruct the entire fluctuating velocity field for the entire experiment.

Using the first three dominant POD modes, the fluctuating velocities were then reconstructed for the entire duration of the experiment by keeping the original POD basis functions and recalculated LSE coefficients, but using real-time velocity measurements from probe locations 3 and 6 beyond the initial data subset (i.e. the 1000 snapshots) to estimate new temporal coefficients. This calculation process was accomplished for hours of velocity data beyond the initial 1000 snapshots. Figure 9 shows a portion of the reconstructed fluctuating velocities beyond the initial data subset used in the decomposition, specifically for probe locations 5 and 6, where probe location 5 represents near real-time velocity reconstruction and probe location 6 represents real-time measurements (i.e. the velocity probe data always used).

As shown in equation (15), the NRMSE was calculated to compare the reconstructed fluctuating velocity field using real-time measurements from two probe locations against the measured fluctuating velocity field. The RMSE was normalized by the maximum measured velocity fluctuation for each probe since the mean of the fluctuating quantity is approaching zero. The NRMSE values for fluctuating velocity observed over hours of data reconstruction were less than 7% for each probe location when only two probes were used to estimate new temporal coefficients.

NRMSE = \frac{\sqrt{\sum_{t = 1}^{T} (u' (x, t) - u *' (x, t)) 2 / T)}}{max (u' (x, t))}

(15)

Therefore, a potential application of the POD–LSE technique for our study of natural ventilation was to obtain a spatially resolved fluctuating velocity field across an opening using a small number of probes (e.g. two probes) to make real-time measurements. This could provide a more accurate, real-time evaluation of ventilation rates for wind-driven natural ventilation conditions and thus could offer a better understanding of airflows and contaminant transportation in perimeter zones for a naturally ventilated building.

Real-time ventilation rates from POD–LSE

An important realization from the statistical analysis was that even the fluctuating velocity signals measured in a single window opening, with a relatively small area, were not consistently correlated with one another.⁴⁰ Therefore, two or three dominant POD modes were observed in the natural ventilation airflow across window opening experiment, and thus two or three probes measuring real-time velocities would be needed for an accurate reconstruction.

A potential application of the POD–LSE methodology for natural ventilation would be to provide real-time ventilation rates. For comparison, the ventilation rates were calculated using two methods. First, all seven probe velocity measurements in the window opening were used, and the opening area was divided equally into seven sub-areas where the actual airspeed across the sub-area was assumed uniform and represented by the probe measured velocity. Figure 10 shows the window opening subdivided into the seven sub-areas.

A second ventilation rate was calculated using the POD–LSE reconstructed velocities from two probes (Probes 6 and 3). Figure 11 compares the two calculations for the same portion of the experiment beyond the initial data subset that was used for POD decomposition as shown in Figure 9.

The resulting ventilation rate calculation based on the detailed seven point measurements was estimated at 21.1 Air Changes per Hour (ACH), while the ventilation rate calculated based on the POD–LSE reconstructed velocities was estimated at 20.6 ACH over the duration of the experiment. The POD–LSE methodology would enable more detailed spatial velocity data to be acquired indefinitely with limited probes. In this case, the two probes had provided the spatial detail of seven.

Therefore, for a naturally ventilated space, the POD–LSE technique can be implemented to obtain high spatially resolved fluctuating airflow velocities across a window opening using a few (e.g. two) real-time monitoring measurements and the methodology is given as follows:

Step 1: First, obtain a set of data through on-site measurements with a prescribed number of probes. The number of probes used in the initial data collection depends on the desired spatial resolution of the velocity field across the opening.

Step 2: Then, decompose the data into POD basis functions and temporal coefficients.

Step 3: Conduct correlation analysis between individual probe data and the calculated temporal coefficients to determine where strong correlations exist between probe measurement locations and POD mode temporal coefficients.

Step 4: Next, calculate the matrix of LSE coefficients from the POD temporal coefficients and with those probes that have high correlations to the dominant POD modes, generally two to three probes for a regular size of window.

Step 5: Finally, continue monitor the real-time velocity data using the highly correlated probes that kept remained in the opening. Estimate the new temporal coefficients from these real-time data, using the initial matrix of LSE coefficients and to reconstruct the entire velocity field in real-time.

Conclusions

This work has developed a methodology utilizing a coupled POD–LSE multivariate statistical technique to reconstruct velocities for a single-room test environment under mechanical UFAD ventilation system and under wind-driven cross natural ventilation based on a limited number of real-time on-site monitoring measurements. The POD–LSE technique with limited measurements would provide real-time detailed information of mechanically ventilated airflow fields that would improve understanding of indoor airflows, thermal environments, and contaminant transportation and distribution. The POD–LSE technique could reconstruct real-time fluctuating velocities for the single-room under mechanical ventilation to within 0.04 m/s of measured fluctuating velocities for a number of locations around the space. The POD–LSE technique is capable of reconstructing velocities based on monitoring measurements from only two velocity probes, for instance, one at the supply diffuser and one at the thermostat.

The POD–LSE technique application may be extended to other types of mechanical ventilations, such as mixing ventilation, if there is a roughly steady correlation between the indoor airflows. In contrast, the POD–LSE technique was not able to create a reduced-order model of the indoor airflow system for the single-room test environment under cross natural ventilation conditions. This result could be influenced by the limited number of probes used in the initial data collection phase of the methodology and with the random speed fluctuations. Ultimately good fluctuating wind speed correlations were not constructed between the probes spaced throughout the indoor environment resulting in the inability to create a reduced-order model. This could thus limit the capabilities of the proposed methodology for reconstructing and monitoring detailed indoor airflows under natural ventilation. In addition, a critical difference between natural ventilation and mechanical ventilation is that natural ventilation is continuously affected by external random variations, such as outside fluctuating wind speeds, especially for the cross natural ventilation studied here.

The POD–LSE technique is capable of creating a reduced-order model of velocities in an open window for the single-room test environment under cross natural ventilation conditions. The proposed methodology is capable of reconstructing real-time wind-driven cross natural ventilation velocities across a window opening based on only real-time velocity measurements from two monitoring probes. The proposed methodology would give reliable unsteady boundary conditions that could be used with other numerical models to more accurately predict real-time ventilation rates and contaminants introduced through open fenestration in a naturally ventilated building.

Due to the restriction of current experimental instrument availability, this study used Omni-directional probes to measure indoor airflows although two 3D probes were installed at the windows to monitor cross natural ventilation. Future work will consider using more 3D anemometers such as 3D ultrasonic anemometers to provide 3D in-depth velocity information on indoor airflows, which will enhance the validation accuracy of the proposed POD–LSE technique.

Footnotes

Authors’ contribution

All authors contributed equally in the preparation of this manuscript.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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