Extension of generalized forced convective heat transfer coefficient expressions for isolated buildings taking into account oblique wind directions

The surface-averaged forced Convective Heat Transfer Coefficient (CHTCavg) at a windward building facade is influenced by the complex interaction between a wide range of parameters. Existing CHTC expressions, however, consider the impact of these parameters either incompletely or not at all. Earlier studies have shown that this shortcoming can lead to significant errors in Building Energy Simulations. In this paper, therefore, the combined impacts of wind speed (U10), building height (H) and width (W), and wind direction (θ) on the CHTCavg for the windward facade of buildings are systematically investigated. High-resolution CFD simulations of wind flow and forced convective heat transfer, validated with wind-tunnel measurements, are performed for 64 building geometries (10m≤H and W≤ 80m), 8 wind directions (0°≤ θ≤ 78.75°) and 4 reference wind speeds (1 m/ s≤U10≤ 4m/s). The 3D steady RANS equations with the realizable k-ε turbulence model and the low-Re number Wolfshtein model are used. The results show that for a given building geometry and U10, the CHTCavg decreases as θ increases from 0° to 78.75°. The maximum reduction of about 42% occurs for the building with H=8W=80m. In addition, for a given θ and U10, by increasing H, the CHTCavg increases, while increasing W has the opposite impact on the CHTCavg. Finally, a new generalized CHTC expression is presented as a function of U10, H, W and θ, and its accuracy is confirmed by detailed in-sample and out-of-sample evaluations.


Introduction
Research and practice in urban physics, building energy and building component durability require the knowledge of the exterior forced convective heat transfer (CHT) at the building facades. For building energy studies, for example, the convective heat transfer coefficient (CHTC) is typically used to model the CHT between buildings and their environment as part of the calculation of the heating and cooling demand of buildings and to assess the energy performance of the building envelope. Knowledge of the CHTC is also important for modeling of heat waves in urban areas, the urban heat island effect, indoor and outdoor heat stress, etc., which are topics that are strongly related to the grand societal challenges energy, health and climate [1,2].
Earlier CFD studies have shown that the impact of building geometry on the forced CHTC can be very large and to some extent counter-intuitive [4,6]. For example, for a 10 m wide windward facade, as the building height (H) increases from 10 m to 80 m, the forced surfaceaveraged CHTC (CHTC avg ) on the windward facade increases by about 20% [4]. For H = 10 m, however, increasing the building width from 10 to 80 m has the opposite impact on the forced surface-averaged CHTC, which decreases by more than 33% [4]. The first trend can be explained by the increase of wind speed with height in the atmospheric boundary layer. The second trend is attributed to the so-called windblocking effect that refers to the upstream wind deceleration due to the blockage by the building [12,13].
Previous experimental and numerical studies have indicated that the forced CHTC changes as a function of Reynolds number. For example, a power-law dependence was found between the CHTC at the surfaces of wall-mounted rectangular prisms and the Reynolds number. For windward building facades, several power-law correlations between the CHTC avg and U 10 have been provided (e.g., = CHTC 5. 15U 10 0.81 [8], = CHTC 4.6U 10 0.89 [9], = CHTC 5.01U 10 0.85 [14]), where U 10 is the mean wind speed in the undisturbed flow at a height of 10 m above the ground.
The impact of wind direction on the CHTC was investigated in previous CFD studies (e.g. Refs. [8,9]). In each of these studies, only one specific low-rise isolated building was investigated while the simultaneous impact of building geometry was not taken into account. It was shown that the surface-averaged CHTC at the windward building facades varies substantially as a function of the wind direction. For example, Blocken et al. [9] showed that for a low-rise cubic building (10 m × 10 m × 10 m) and for U 10 = 3 m, by increasing the wind direction from 0°to 78.75°, the CHTC avg reduces by more than 25%.
Because of the complexity involved in obtaining accurate values of the CHTC, a large number of empirical and semi-empirical expressions have been established using on-site (full-scale) measurements (e.g. the Mobile Window Thermal Test (MoWiTT) model [15], and the model by Liu and Harris [16]) or wind-tunnel (reduced-scale) measurements (e.g. the McAdams [17] and CIBS (Chartered Institute of Building Services) models [18]) as well as Computational Fluid Dynamics (CFD) simulations (e.g. the models by Emmel [8] and Montazeri & Blocken [6]). Many of these expressions are implemented in Building Energy Simulation (BES) programs [19,20]. However, the accuracy and reliability of these expressions are of concern because: • Many of these expressions are based on invalid assumptions: an example is that the CHTC is often assumed constant across a building facade and that a measurement resulting from a single point is considered to be valid for the whole facade. Further, some expressions have been derived based on wind-tunnel measurements over a flat plate. For example, the McAdams [17] and CIBS models [18] that are widely used in BES tools have been derived based on wind-tunnel experiments for a vertical square copper plate (0.5 × 0.5 m 2 ) in a uniform air flow parallel to the plate [21]. The flow structure around buildings, however, is more complex than the one over flat plates.
• Generally, the influence of important parameters such as wind direction, surrounding buildings, and surface roughness is taken into account either incompletely, or not at all. In addition, a vast majority of these expressions do not contain the complex interaction between these parameters. An example is the correlation proposed in the CIBS Guide [18]: CHTC = 4.1U loc +5.8, where U loc = 2/3U R . In this correlation, U loc is the wind speed at a certain distance from the building facade and at a certain height from the ground, and U R is the wind speed at a certain height from the roof surface.
• Many expressions are case-specific, i.e., they were derived for specific building characteristics, and under specific meteorological conditions (wind speed and wind direction) and they are therefore strictly only applicable to the conditions for which they were derived. For example, the model by Liu & Harris [16] is based on fullscale measurements for a one-storey building in a rural area that is partially sheltered by a nearby building. The model provides three CHTC expressions based on U loc (CHTC = 6.31U loc +3.32), U R (CHTC = 2.08U R +2.97), and U 10 (CHTC = 1.53U 10 +1.43), which were measured 0.5 m away from the wall surface, 1 m above the roof and 10 m above the ground level, respectively. Since a copper sheet (smooth surface) was used to measure the CHTC at the facades of this one-storey building, it is expected that this model is accurate only for low-rise buildings with smooth surfaces.
The accuracy of these empirical or semi-empirical expressions, socalled secondary sources, depends on availability and accuracy of the above-mentioned so-called primary sources, i.e., on-site or wind-tunnel experiments, and CFD simulations. Table 1 presents an overview of the wind-tunnel experiments and CFD simulations of convective heat transfer performed on wall-mounted rectangular prisms. It can be observed that: • Most of the available high-resolution wind-tunnel data of CHTC are based on measurements at relatively low Reynolds numbers (10 3 -10 4 ) [22][23][24][25], which limits the applicability of the available data for building applications.
• CFD simulations have been used to investigate CHTC at realistic Reynolds numbers for building applications (Re ∼ 10 6 -10 7 ) as, unlike wind-tunnel testing, CFD does not suffer from potentially incompatible similarity requirements because simulations can be conducted at full scale [26,27]. The impact of the reference wind speed (or Re), building geometry and urban surroundings have been investigated. However, not all studies have resolved the flow in the thin viscous sublayer at the building surfaces that constitutes the largest resistance to surface convective heat transfer, as this requires very small cells in the near wall region (typically a few ten or hundreds of micrometer).
• The combined effects of different parameters on CHTC have been taken into account in only a few studies. Natarajan and Chyu [23], Emmel et al. [8], and Blocken et al. [9] investigated the combined effect of Re and wind direction and presented CHTC expressions as a function of Re for different approaching wind directions. Montazeri et al. [4] and Montazeri and Blocken [6] investigated the combined impact of Re and building dimensions. The results of the latter studies led to new generalized expressions for the surface-averaged CHTC for different building surfaces as a function of reference wind speed, and height and width of the windward facade.
Earlier studies have shown that the use of existing CHTC expressions may lead to incorrect results in building energy simulations (BES). For example, using different expressions has revealed deviations of more than 30% in the yearly cooling energy demand of a simple low-rise building [20]. For high-rise buildings, the deviations could go up to 42% [28]. The above-mentioned shortcomings of the existing CHTC expressions and these of their underlying primary data sources indicate the need for more research efforts in terms of wind-tunnel testing, CFD simulations and the establishment of new and more generally applicable CHTC expressions. This paper is intended to provide a further step in that direction.
In this paper, new high-resolution CFD simulations of wind flow and forced convective heat transfer are performed, and a new generalized CHTC expression is presented that yields the surface-averaged forced CHTC for the windward facade of buildings as a function of four parameters: wind speed, building height, building width and wind direction.
It should be noted that in BES (building energy simulation) and BE-HAM (building envelope heat, air and moisture transfer) tools, generally three categories of CHTC models (expressions) are implemented with respect to surface orientation regarding the wind direction, i.e., CHTC models for windward facades, leeward facades and roofs. The focus in the present paper is on windward building facades because previous experimental and numerical studies indicated that the highest surface-averaged CHTC for wall-mounted rectangular prisms is obtained for the windward surface (e.g. Ref. [37]). The lowest CHTC is found for the leeward surfaces where the CHTC is rather insensitive to the wind direction and the building geometry [6,23]. In addition, in BE-HAM programs, the windward facade is the one that is wetted by winddriven rain and for which knowledge of CHTC and of the related CMTC (convective moisture transfer coefficients) is of particular importance [38][39][40].
The results of this study will bring a better understanding of the interaction between the CHTC and its influencing parameters wind speed, building height, building width and wind direction at the windward facades of buildings. In addition, this study will lead to a new generalized CHTC expression for windward building facades, intending to support more accurate BES and BE-HAM simulations.
The outline of the paper is as follows. The wind-tunnel experiments of surface temperature at the surfaces of a wall-mounted cube by Meinders et al. [24] and the validation study are briefly presented in Section 2. Section 3 describes the computational settings and parameters for CFD simulations. Section 4 presents the CFD results, the new CHTC expression and in-sample and out-of-sample evaluations of the new expression. A discussion on the limitations of the study and the main conclusions are presented in Sections 5 and 6.

CFD validation study
In this study, the wind-tunnel measurements by Meinders et al. [24,37] are used for the CFD validation. Because this validation study has been published as a separate paper [4], it is only briefly presented here.
Meinders et al. [24,37] determined the exterior convective heat transfer at the surfaces of a cube (H c = 0.015 m) in developing turbulent channel flow. The channel had dimensions width × height = 0.6 × 0.05 m 2 . Experiments were conducted for several Reynolds numbers (based on H c ) ranging from 2000 to 5000. Laser-Doppler anemometry was used to measure the approach-flow turbulent boundary layer mean velocity and turbulent kinetic energy, while the external surface temperature distribution of the cube surface was measured with infrared thermography.
For the validation study, Re = 4440 is considered. The 3D steady RANS equations are solved with the commercial CFD code Ansys/ Fluent 12.1 [41]. The realizable k-ε model (Rk-ε) [42] is used in combination with the low-Re number Wolfshtein model [43] for closure. The SIMPLE algorithm is used for pressure-velocity coupling, pressure interpolation is second order, and second-order discretization schemes are used for both the convection terms and the viscous terms of the governing equations. All other settings and parameters are reported in the previous validation study [4]. Fig. 1 compares the wind-tunnel results and CFD results of surface temperature along lines in a vertical and horizontal midplane on the windward surface of the cube. The general agreement is good. In this case, the average deviation along the vertical and horizontal lines is about 1.7 and 2.4%, respectively. This is also in line with the LES results by Iousef et al. [44] and Hu et al. [45]. Nevertheless, some overestimations can be clearly seen close to the ground. The main reason for these overestimations is not clear. But it could be due to additional heat loss through the base wall in the experiment, which is not taken into account in the simulations [4,44]. Another reason could be related to an incorrect prediction of the size and shape of the standing vortex due to the upstream longitudinal gradients in the approach-flow profiles [46,47]. Less good agreements are found for the other surfaces (not shown in Fig. 1) that are attributed to the well-documented inaccurate flow field prediction by steady RANS downstream of the windward facade [26,27,44,[48][49][50]. Given the good agreements between CFD and wind-tunnel experiments for the windward surface and the fact that the focus of this study is on the windward facade of buildings, the computational parameters, and settings of the validation study are also used for the CFD simulations for the full-scale buildings.

List of cases
The simulations for establishment of the expression are performed for 64 different building geometries, 8 wind directions, and 4 reference wind speeds. In addition, simulations are performed for out-of-sample evaluation of the expressions for additional building geometries, wind directions, and reference wind speeds. In total 625 simulations are performed. The simulations can be classified into three groups ( Table 2) Table 1 Overview of studies on forced convective heat transfer at wall-mounted rectangular prisms based on wind tunnel or CFD.  geometries with dimensions within the range of original data (Group 2) and 4 building geometries beyond the range of the original data. In this case, 6 reference wind directions (θ = 0°, 17°, 28°, 39°, 51°and 62°) and 4 reference wind speeds (U 10 = 1, 1.5, 2.5 and 3.5 m/s) are considered.

Computational domain and gird
The dimensions of the computational domain are chosen based on the best practice guidelines by Tominaga et al. [51] and Franke et al. [52]. In this case, the upstream and downstream domain lengths are 5H and 15H, respectively. The height of the domain is 6H. As shown in the literature, a properly sized computational domain is of high importance for accurate CFD simulations of various applications, e.g. Refs. [53,54]. A high-resolution hybrid grid with prismatic and hexahedral cells is generated using the surface-grid extrusion technique [55], resulting in 1,911,316 cells (Fig. 2). The distance from the center point of the walladjacent cell to the wall, y p , is about 400 μm, which leads to a maximum y* value below 5 for all building geometries, reference wind speeds, and wind directions.

Computational settings and parameters
At the inlet of the domain, neutral atmospheric boundary layer inflow profiles of mean wind speed U (m/s), turbulent kinetic energy k (m 2 /s 2 ) and turbulence dissipation rate ε (m 2 /s 3 ) are imposed: The buildings are assumed to be situated on a large grass-covered terrain with an aerodynamic roughness length z 0 = 0.03 m [56]. The reference wind speed at 10 m height, U 10 , ranges from 1 to 4 m/s, resulting in building Re (based on the building height H) ranging from 0.7 × 10 6 to 8.5 × 10 6 . Note that the use of the relatively low reference wind speeds is to avoid a prohibitively high total number of computational cells and the need for excessive computational resources, because the thickness of the viscous sublayer at the building surfaces decreases with increasing Re. For all simulations, the longitudinal turbulence intensity I u , that is imposed at the inlet ranges from 20% at ground level with exponential decay to 5% at gradient height. The turbulent kinetic energy k is calculated from U and I u using Eq. (2) and assuming that the standard deviations of the turbulent fluctuations in the three directions u v w . The building and ground surfaces are considered smooth no-slip walls. Zero static pressure is applied at the outlet plane. Symmetry conditions (zero normal velocity and zero gradients) are applied at the top and lateral sides of the domain. The thermal boundary conditions are a uniform inlet air temperature of 10°C and a fixed surface temperature of 30°C for the building surfaces. In this study, only forced convection is considered and buoyancy effect is assumed negligible. Therefore, the CHTC is independent of the given temperature difference between the building surfaces and the air. It should be noted that there are different ways to define the CHTC using different definitions of the reference temperature. In this study, all CHTC values are determined based on the upstream (approach-flow) uniform temperature, at a position where it is unaffected by the presence of the building. The adiabatic boundary condition is used for the ground surface. Radiation and natural convection are not taken into account. The same solver settings as in the validation study (Section 2.2) are used. The 3D steady RANS equations are solved with the realizable k-ε turbulence model and the low-Re number Wolfshtein model. In the one-equation model of Wolfshtein, the turbulent viscosity, μ t , and dissipation, ε, in the viscosity-affected near-wall region are computed from Eqs. (4) and (5): where ρ is the density and k is the turbulent kinetic energy. The length scales l μ and l ε are given by Eqs. (6) and (7): where y is the normal distance from the wall at the cell centers and the

Relationship between CHTC and reference wind speed
The CFD simulations from Group 1 ( Table 2) are used to obtain the relationship between the forced surface-averaged CHTC (CHTC avg ) and the reference wind speed U 10 . Fig. 3 shows the results of the CFD simulations for the windward facade for θ = 0°and for different building geometries: (i) buildings with H = 10 m and W ranging from 10 to 80 m (Fig. 3a), (ii) buildings with W = 10 m and H ranging from 10 to 30 m  Table 2 Geometry of buildings and values of reference wind speed and wind direction.    ( Fig. 3b), and buildings with W = H ranging from 10 to 30 m (Fig. 3c). As pointed out by Montazeri and Blocken [6], based on fitting with power-law functions, the fit with power-law exponent 0.84 yields the best performance for all building geometries (R 2 = 0.9829 to 0.9994). This is also indicated in Fig. 3d-f, where the parameter CHTC avg /U 10 0.84 is plotted versus U 10 . It appears that for a given building geometry, the ratio CHTC avg /U 10 0.84 is independent of U 10 for wind directions perpendicular to the windward facade. Fig. 4a-c indicates profiles of the CHTC avg at the windward facade of the buildings with H = 10 m and W = 40 m (Fig. 4a), H = 40 m and W = 10 m (Fig. 4b) and H = W = 10 m (Fig. 4c) as a function of U 10 for different reference wind directions. The ratio CHTC avg /U 10 0.84 for these three geometries is presented in Fig. 4d-f. It can be seen that for a given building geometry, the ratio CHTC avg /U 10 0.84 is nearly independent of U 10 for these different reference wind directions. This indicates that the establishment of the relationship between CHTC avg , building height, and width and reference wind direction can be performed for only a single value of U 10 . Therefore, this approach is adopted in the next subsection. Afterwards, out-ofsample evaluations will be performed to validate this approach.

New CHTC expression
To establish the relationship between the surface-averaged CHTC (CHTC avg ), building height (H), building width (W) and wind direction (θ), the simulations from Group 2 ( and 16% for θ = 0°, 22.5°, 45°and 67.5°, respectively. Note that increasing the width of the buildings results in a more pronounced wind-blocking effect upstream of the building, which reduces the local wind speeds near the windward facade. This also increases the time that the air is in contact with the facade, which reduces the temperature difference between the air and the surface especially for the top and edges of the facade. These phenomena lead to a  • For a given building geometry (constant height and width of the windward facade), the ratio CHTC avg /U 10 0.84 reduces as the reference wind direction increases. This is because the local and surfaceaveraged CHTC at the surfaces of a bluff body are highly dependent on the immediate flow structure around the obstacle [4,9,57,58] that substantially varies by changing the wind direction [59]. This is in line with the wind-tunnel measurements of Natarajan and Chyu [23], though in those measurements the boundary layer thickness was relatively thin, i.e., about 1/4 to 1/3 of the cube height.
In this study, a polynomial is used to fit the CFD data in four dimensions (CHTC/U 10 , H, W and θ) and to yield the CHTC expression as a function of the four parameters (U 10 , H, W, θ). This polynomial can subsequently be used to obtain values for the CHTC by interpolation or by extrapolation. However, as pointed out by Montazeri and Blocken [6], the use of polynomials beyond the original data should be treated with caution because of the rather large extrapolation errors that can occur with polynomials. In the present study, the following steps are taken to improve the performance of the polynomial beyond the original data: • For a given value of wind direction and building width, the CFD results for different building heights, Group 2 in Table 2 (10 m ≤ H ≤ 80 m), are used to generate additional data for 90 m ≤ H ≤ 150 m using two-dimensional fitting. A sensitivity analysis shows that the power-law functions yield the best performance to describe the correlation between CHTC avg /U 10 0.84 and H. For example, for θ = 0°and W = 10 m, the fit with power-law exponent 0.085 yields the best performance. Therefore, the power-law function CHTC avg /U 10 0.84 = 5.05H 0.085 is implemented to obtain data beyond the original data for buildings with H = 90, 100, 110, 120, 130, 140 and 150 m.
• The abovementioned procedure is used to obtain additional data for different building heights (90 m ≤ H ≤ 150 m) for all wind directions and building widths mentioned in Group 2 of Table 2.
• For a given value of wind direction and building height, the CFD results for different building widths (10 m ≤ W ≤ 80 m) are used to generate additional data for 90 m ≤ W ≤ 150 m using two-dimensional fitting. In this case, the logarithmic functions are found to be superior to fit the CFD data.
• The abovementioned procedure is used to obtain additional data for different building widths (90 m ≤ W ≤ 150 m) for all wind directions and building heights mentioned in Group 2 of Table 2. • All original data obtained by CFD simulations (Group 2 in Table 2 A fourth-order polynomial equation including cross-terms (i.e., terms involving the product of the independent variables) is used to express the ratio CHTC avg /U 10 0.84 as a function of H, W and θ with a good approximation (R 2 = 0.9919). Table 3 presents the new CHTC expression as a function of U 10 , H, W and θ. Using the coefficients with 2 decimal digits instead of 8 provides a deviation of about 0.21%. Using the coefficients with 3 decimal digits instead of 8 provides a deviation of only 0.010%. Therefore, only 3 decimal digits are retained in the table. The new expression can be easily implemented and used in numerical BES and BE-HAM programs.

In-sample and out-of-sample evaluation of new CHTC expression
In-sample and out-of-sample evaluations are performed to evaluate the accuracy of the new CHTC expression, presented in Table 3. The terms "in-sample" and "out-of-sample" evaluations refer to the assessment of the accuracy of the new CHTC expression within and beyond the range of the original CFD data, respectively. Fig. 9 illustrates the results of the in-sample evaluation in which the CHTC values obtained by the new expression are compared with the values directly obtained from the CFD simulations from Group 2 in Table 2. A very good agreement is observed with the high values of R 2 ranging from 0.9579 (θ = 78.75°) to 0.9981 (θ = 0°). Out-of-sample evaluations for combinations of the variables U 10 , H, W and θ are performed by comparing with the CFD results (Group 3 in Table 2). Figs. 10 and 11 show that for all combinations the resulting values of surface-averaged CHTC are well predicted by the new expression. In this case, the maximum deviation is about 7%, which occurs for the building with H = 15 m and W = 65 m at θ = 62°for U = 2.5 m/s. The average deviation for all cases is about 4%. The related coefficient of determination is R 2 = 0.9980. Note that for the building geometries outside the range of those for which the CFD simulations were performed, the maximum deviation is 1.6% for the building with H = 60 m and W = 90 m (Fig. 11). The in-sample and out-of-sample evaluations provide confidence in the new expressions for an accurate prediction of the CHTC for different buildings.  Table 3) and five commonly used CHTC expressions in BES tools with the values directly obtained from the CFD simulations. The results are presented for three different building geometries as a function of the reference wind speed and wind direction. The EmmelVertical [8], Nusselt-Jürges [21], McAdams [17], Mitchell [60] and TARP [61] expressions are considered. The calculations are based the implementations of these model in the BES tool EnergyPlus. It should be noted that EmmelVertcial is the only model in which wind direction is considered as a parameter. The TARP and Mitchell models implicitly consider building geometry [20], while Nusselt-Jürges and McAdams only include wind speed to describe the CHTC. It can be seen that the new expression (Table 3) provides a good agreement with the CFD results for different values of reference wind speed and wind direction. For the building with H = W = 10 m, for example, the deviation between the CFD results and the new expression results is about 0.8% for θ = 0°and U 10 = 1 m/s. This is about 2.8%, 1.4% and 0.3% for θ = 22.5°, 45°and 67.5°, respectively. For the other expressions, however, the deviations can be very large. For the building with H = W = 10 m, the use of the Nusselt-Jürges expression, for example, leads to a deviation of about 48% for θ = 0°and U 10 = 1 m/s. This deviation increases by increasing the wind direction and goes up to 93% for θ = 67.5°.

New expression vs other CHTC models
In addition, the CHTC avg values obtained by the new expression (Table 3) are compared with the values given by the existing generalized CHTC expression that contains the reference wind speed, the width and the height of the windward building facade as parameters [6]. The evaluation is performed for the 64 building geometries presented in Table 2 (Group 2, 10 m ≤ H ≤ 80 m and 10 m ≤ W ≤ 80 m) and for θ = 0°. The results show that the maximum and the average deviation between the CHTC values obtained by the two expressions are about 3.7% and 1.4%, respectively.

Discussion
In this study, a fourth-order polynomial equation including crossterms (i.e., terms involving the product of the independent variables) is used to develop the new CHTC expression. The choice of the polynomial is based on a sensitivity analysis in which polynomials of different orders were evaluated. Six polynomials were considered: second, third  Fig. 9. (a-h) In-sample evaluation of the forced CHTC expression in and fourth-order polynomials including and excluding cross-terms. The in-sample accuracy of the polynomials was evaluated by comparing the coefficients of determination (R 2 ). The results show that a fourth-order polynomial including cross-terms yields the best performance. The coefficients of determination for the second, third and fourth-order polynomials excluding cross-terms are 0.9171, 0.9270, and 0.9286, respectively. These are 0.9691, 0.9847 and 0.9919 for the polynomials including cross-terms, respectively. The good performance of four-order polynomials was also observed in a previous study [6].
The new expression contains wind speed, building height and width, and wind direction as separate parameters. The use of H and W as separate parameters, instead of non-dimensional parameters such as building aspect ratio (H/W), is because of the complex individual impact of H and W on the CHTC avg for different wind directions. For example, for buildings with H = W (H/W = 1) and at θ = 0°, increasing the building height and width from 10 m to 40 m yields a 10% reduction in the CHTC avg ⁄(U 10 0.84 ) on the windward facade. For higher values of H and W, however, the opposite trend is found. As H and W increase from 50 m to 80 m, the ratio CHTC avg ⁄(U 10 0.84 ) increases by 3%. Similar trends can also be seen for the other wind directions. While this study has provided several new insights, and presented a new generalized CHTC expression for more accurate BES and BE-HAM simulations, it is also important to mention the limitations of this study: • The validation study is performed for a wall-mounted cubic obstacle at a relatively low Reynolds number (Re = 4440). This is due to the lack of available high-resolution on-site or wind-tunnel data at realistic Reynolds numbers for building applications (Re ∼ 10 6 -10 7 ). It should be noted that the essential flow features such as impingement, the horse-shoe vortex, etc., and the level of flow complexity at both aforementioned regimes of Re are relatively similar. Therefore, it can reasonably be assumed that the given combination of computational parameters and settings that provided accurate simulation results in the validation study, will also provide accurate results for the full-scale buildings. Nevertheless, given the relatively low Reynolds number in the measurements, it is likely that the boundary layer over the windward surface of the cube is laminar, while the boundary layers over building surfaces are mostly turbulent. Future high-resolution experimental work is, therefore, required at sufficiently high Reynolds numbers to enable further validation of CFD simulations.
• No grid-sensitivity analysis is performed for the CFD simulations of  Table 3 Table 3 for buildings with dimensions outside the range of the CFD data for θ = 0°and U 10 = 1 m/s. the full-scale buildings. Nevertheless, special attention was paid to generating a high-resolution and high-quality computational grid with a minimum near-wall cell size of 400 μm (y* < 5) to resolve the entire boundary layer, including the viscous sublayer and the buffer layer, which dominate the convective surface resistance. In addition, the grid was made according to best practice guidelines for grid generation [2,51,52,62,63].
• In this study, the simulations are performed for perfectly smooth building surfaces, which is generally an implicit assumption of low-Re number modeling when the geometry of roughness features is not modeled explicitly. Note that previous experimental studies have shown the importance of surface roughness (small-scale roughness) on the exterior CHTC [64,65]. In addition, other studies show the importance of large-scale roughness (building facades and roofs with protrusions or recessions) on local flow patterns [66][67][68][69][70][71][72]. Given the fact that building details, such as balconies, can lead to a strong change in the flow pattern around the building, this roughness scale can also considerably affect the CHTC distribution on building facades, and future CHTC studies should also focus on facades with this type of roughness.
• This study is limited to isolated buildings and is not valid for buildings surrounded by other buildings of similar height.
Nevertheless, in many European cities, high-rise buildings are generally positioned among low-rise buildings, and not present in groups/clusters, as in more modern cities on the American continent. Therefore, they can, to some extent, be represented by an isolated building in this study. Note that considering the impact of building surroundings requires an extensive parametric study in which the individual and the combined effects of urban geometry parameters (e.g. frontal area density, plan area density and building height variations), wind speed, wind direction, and building geometry are investigated. Given the complexity of wind flow in urban areas, more accurate results should be pursued using transient simulations with LES. However, LES is computationally expensive especially for extensive parametric studies such as the present study in which numerous simulations are performed.
• In the present study, only forced convection is considered. Earlier studies have shown that buoyancy can be important in general, but especially in urban areas, where pavement and building facades are heated by the solar radiation, and wind speeds are significantly reduced because of wind sheltering [1]. The presence of the buoyancy driven flows can significantly change the structure of the flow around buildings and strengthen the intensity of the circulation within street canyons, resulting in significant changes in the CHTC

Summary and conclusions
In this study, the combined impacts of the reference wind speed (U 10 ), the height (H) and the width (W) of the windward building facade, and the reference wind direction (θ) on the surface-averaged forced CHTC (CHTC avg ) for the windward facade of buildings are investigated and a new generalized CHTC expression as a function of these parameters is presented. CFD simulations of wind flow and forced convective heat transfer are performed for 70 different building geometries, 8 wind directions (0°≤ θ ≤ 78.75°) and 4 reference wind speed values (1 m/s ≤ U 10 ≤ 4 m/s). In total 625 simulations are performed. The 3D RANS equations with the realizable k-ε turbulence model and the low-Re number Wolfshtein model are used. The simulations are based on an extensive validation study with previously published wind-tunnel measurements. The following conclusions are made: (a) The combined effects of U 10 , H, W and θ on CHTC avg • For a given building geometry (constant height and width of the windward facade) and U 10 , the surface-averaged CHTC avg reduces as the reference wind direction increases. This is because the local and surface-averaged CHTC at the surfaces of a bluff body are highly dependent on the immediate flow structure around the obstacle that substantially varies by changing the wind direction.
• For a given wind direction and U 10 , by increasing the building height, the CHTC avg increases. Increasing the building width, however, has the opposite impact on the CHTC avg .
• For a given U 10  • The first group of simulations is performed to establish the expressions of forced surface-averaged CHTC as a function of reference wind speed U 10 (or Re). The results show that for a given building geometry, the ratio CHTC avg /U 10 0.84 is nearly independent of U 10 for all reference wind directions. This indicates that the establishment of the relationship between CHTC, building height, and width and wind direction should only be performed for a single value of U 10 , which saved time and computational cost.
• The second group of simulations is performed to establish the expression of forced surface-averaged CHTC as a function of wind speed, height and width of windward facade and wind direction.
The results show that the ratio CHTC avg /U 10 0.84 for the windward facades can be presented with high accuracy by a polynomial with windward building facade height and width and wind direction as parameters.
• The third group of simulations is performed to evaluate the out-ofsample accuracy of the expressions, indicating similarly high coefficients of determination as for the in-sample evaluation. The new CHTC expression is an analytical formula as a function of U 10 , windward facade height, and building width, and wind direction that can easily be implemented in BES (building energy simulation) and BE-HAM (building envelope heat-air-moisture transfer) programs.
• The CHTC avg values obtained by the new CHTC expression and five commonly used CHTC expressions in BES tools are compared with the values directly obtained from the CFD simulations. The new expression provides a good agreement with the CFD results for different values of reference wind speed and wind direction. For the building with H = W = 10 m, for example, the deviation between the CFD results and the new expression results is about 0.8% for θ = 0°and U 10 = 1 m/s. This is about 2.8%, 1.4% and 0.3% for θ = 22.5°, 45°and 67.5°, respectively. For the other expressions, however, the deviations can be very large. For the building with H = W = 10 m, the use of the Nusselt-Jürges expression, for example, leads to a deviation of about 48% for θ = 0°and U 10 = 1 m/s. This deviation increases by increasing the wind direction and goes up to 93% for θ = 67.5°.
Future research on this topic will consist of: • Evaluating the combined impacts of building geometry, wind speed and wind direction on the forced CHTC for facades other than the windward facade.
• Investigating the impact of different (small scale and large scale) surface roughness on the forced CHTC.
• Investigating the combined impacts of building geometry, wind speed, wind directions and urban surroundings on the mixed (forced and natural) CHTC, and establishing new expressions.