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Computational analysis of wind driven natural ventilation in buildings
Energy and Buildings 38 (2006) 491–501 www.elsevier.com/locate/enbuild
Computational analysis of wind driven natural ventilation in buildings G. Evola a,*, V. Popov b a
Department of Mechanical and Industrial Engineering, University of Catania, Viale Andrea Doria 6, Catania 95125, Italy b Wessex Institute of Technology, Ashurst Lodge, Ashurst, Southampton SO40 7AA, UK Received 31 March 2005; received in revised form 20 August 2005; accepted 23 August 2005
Abstract The design of natural ventilation in buildings is often performed by means of computational fluid dynamics (CFD) techniques, whose application is gaining popularity. In the present study, Reynolds averaged Navier–Stokes equation (RANS) approach is applied to wind driven natural ventilation in a cubic building. Two different models are considered, namely the two-equation k–e model and the Renormalization Group (RNG) theory. The velocity and pressure distribution inside and around the building are determined, as well as the ventilation rate, for three different configurations: cross ventilation, single-sided ventilation with an opening on the windward wall and single-sided ventilation with an opening on the leeward wall. The numerical results are compared with experimental data, showing a good agreement, particularly when using RNG. The discrepancy in the determination of the ventilation rate is reasonable and the flow distribution inside the building is properly described when RNG model is used. However, the k–e model fails to determine the correct velocity components near the horizontal surfaces. According to these results, the RNG model can be considered a useful tool for the study of wind driven natural ventilation, especially for the assessment of the ventilation rate and of the air distribution inside a room. # 2005 Elsevier B.V. All rights reserved. Keywords: Single-sided ventilation; Cross ventilation; Computational fluid dynamics; RANS modelling; Ventilation rate
1. Introduction Mechanical ventilation in buildings is nowadays a common practice, due to the need to provide thermal comfort and good indoor air quality in enclosed spaces. The energy consumption related to the operation of heating, ventilation and airconditioning systems (HVAC) is considerable: according to recent studies, nearly 68% of the total energy use in service and residential buildings is attributable to HVAC systems [1]. On the other hand, natural ventilation replaces indoor air with fresh outdoor air without any energy consumption, and it also helps to overcome common health problems related to an insufficient maintenance of HVAC systems. For these reasons, great attention is being focused on the design of natural ventilation in buildings. The correct design of a naturally ventilated building is a challenging task, due to the complexity of the physical phenomena involved. In particular, the air flow through an opening, either purpose provided or adventitious, depends on * Corresponding author. Tel.: +39 0957382421; fax: +39 0957382496. E-mail address: [email protected] (G. Evola). 0378-7788/$ – see front matter # 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.enbuild.2005.08.008
the pressure difference between the sides of the opening, as well as on the resistance opposed to the air flow by the opening itself; the latter is a function of opening shape and dimension. The pressure difference is produced by wind and buoyancy forces, several studies have been performed in order to better understand the interaction between these two driving forces. Allocca et al. [2] showed that, under certain circumstances, the wind effect might not be beneficial, as it may reduce the ventilation rate provided by buoyancy forces alone. In this work, the attention is focused on wind driven natural ventilation. In order to obtain reliable information concerning the air flow and the pressure distribution around and inside a naturally ventilated building, full-scale measurements can be performed [3,4]. However, wind tunnel tests on small-scale models are usually preferred, as they allow the control of wind speed and direction as well as the study of different configurations [5,6]. An alternative approach is represented by computational fluid dynamics (CFD), based on the numerical solution of the set of governing equations which describe the flow field. CFD-based programs are widespread and provide a detailed description of the air flow. In recent years, their application has
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become more and more popular thanks to the increase of computational power and to the improvements in turbulence modelling [7]. Among the available models, Reynolds averaged Navier–Stokes equation (RANS) and large-eddy simulation (LES) are the most common. RANS is widely used by most of the commercial CFD software. It is based on the time-averaging of the flow field governing equations; this means that all of the unsteadiness is considered as part of the turbulence and is averaged out. On the other hand, LES is based on the space filtering of the turbulent structures. The fluid flow is separated into large eddies and small eddies; the first ones are solved by the governing equations, while the last ones are approximated and described by suitable subgrid-scale models. Jiang et al. [8] applied LES to wind driven ventilation; they determined pressure and velocity distribution inside and around a scale cubic building model, showing a good agreement between numerical and experimental results. LES models have also been successfully applied to cross natural ventilation in a group of apartments [9]. Despite their reliability, LES models are time-consuming and for this reason they are seldom used, unless an extreme precision is required. On the contrary, RANS models require less computing time, but sometimes they lack the due precision. As an example, Jiang and Chen [10] compared LES and RANS performance in wind driven natural ventilation, concluding that RANS might not be appropriate for the determination of the ventilation rate. A comparison between the results provided by RANS and LES models in natural ventilation problems is also available in [11]. Due to the large use of RANS models, an extensive research is needed to further test their reliability and improve their application to natural ventilation problems. Previous studies concerning RANS modelling of wind driven natural ventilation in a building immersed in an atmospheric boundary layer can be found in literature [12], but they were based on a twodimensional approach and little attention was paid to the determination of the ventilation rate. The present investigation will be focused on the application of three-dimensional RANS modelling on wind driven natural ventilation in buildings. Detailed information concerning air flow inside and pressure distribution around a scale model building will be provided, and compared to the experimental measurements performed by Jiang et al. [8]. Furthermore, the ventilation rate determined through the simulation will be compared with results provided by LES models and empirical methods.
where the openings are located on a single fac¸ade. As a consequence, in the present study three different cases are considered: Case 1: single-sided ventilation with an opening on the windward wall; Case 2: single-sided ventilation with an opening on the leeward wall; Case 3: cross ventilation with openings on both windward and leeward walls. In order to allow a comparison with the experimental results available in literature, the simulations are performed on a building-like model similar to the one used in [8], whose dimensions are 250 mm 250 mm 250 mm. Two different building models are used: the first one is provided with a doorlike opening sized 84 mm 125 mm (width height), and it is used to study single-sided wind driven ventilation (Cases 1 and 2). The second building model presents two openings of the same dimension on opposite sides, in order to simulate cross ventilation (Case 3). The thickness of the walls is 6 mm in both building models. Fig. 1 shows a schematic view of the building model with an opening in the windward wall. Furthermore, a logarithmic wind profile upwind of the building is considered. The dimension of the computational domain, displayed in Fig. 2, is chosen large enough in order not to disturb the air flow around the building. 2.2. Numerical method Computational fluid dynamics is based on the resolution of the governing equations which describe the flow field in the computational domain, namely the continuity equation for mass transfer, the Navier–Stokes equation for momentum transfer and the thermal energy equation for heat transfer. In the present analysis, buoyancy effects are neglected; hence, the flow is considered isothermal and the last equation is not used.
2. Methods 2.1. Problem description When dealing with wind driven natural ventilation, two different ventilation patterns can be identified, namely cross ventilation and single-sided ventilation. The first one is characterized by the presence of two or more openings on opposite sides of the building, and it can provide ventilation rates larger than those obtained by single-sided ventilation,
Fig. 1. Description of the building model in single-sided ventilation with an opening in the windward wall (Case 1); all the dimensions are in mm.
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Table 1 Empirical constants used for the formulation of k–e model 0.09 1.44 1.92 1.0 1.3
Cm C1 C2 sk se
constants used in the previous relations are provided. These values may be not suitable for low Re number flows, or in some local domains of high Re number flows where the damping effect of the surfaces is prominent; in these cases, some corrections would be required. Concerning this problem, Renormalization Group (RNG) model [13] is gaining popularity for modelling indoor air movement; the difference between RNG and k–e models mainly results in the substitution of the constant C1 in Eq. (5) with the following expression: Fig. 2. Dimensions of the computational domain.
For a steady incompressible flow and according to RANS approach, time-averaged continuity and Navier–Stokes equations can be, respectively, written as: @ ðrUi Þ ¼ 0 @xi
g ðr r0 Þ (2) where Uj (j = 1–3) are the components of the velocity vector, g the gravity acceleration, dj = 1 if j = 3 and 0 otherwise and m is the dynamic viscosity. The term mt is called turbulent viscosity and is introduced in RANS modelling to describe the timeaveraged effects of the turbulent stress. According to the widely used two-equation k–e model, mt can be expressed as follows: k2 e
(3)
where Cm is a constant and k and e are the turbulent kinetic energy and its dissipation rate, respectively. In order to close the model two other equations are required, namely the transport equations for k and e: @ @ ðrUi kÞ ¼ @xi @xi
@ @ ðrUi eÞ ¼ @xi @xi
m mþ t sk
m mþ t se
@k þ Pk re @xi
(4)
@e e e2 þ C1 Pk C2 r @xi k k
(5)
In Eqs. (4) and (5), Pk is the production rate of turbulent kinetic energy, which depends on the turbulent viscosity and the velocity distribution. In Table 1, the values of all the empirical
hð1 h=h0 Þ 1 þ bh3
(6)
where C10 1:44, h0 4.4, b 0.015 and
(1)
@ @p @ @U j @Ui ðrUi U j Þ ¼ þ þ ðm þ mt Þ dj @xi @x j @xi @xi @x j
mt ¼ Cm r
C1 ¼ C10
h¼
k e
rffiffiffiffiffi Pk m
(7)
Through Eqs. (6) and (7), the production term in Eq. (5) is modified in such a way that it properly accounts for the larger dissipation rate experienced in the laminar regions near solid surfaces. Detailed information about the application of RNG model is available in [14]. 2.3. Numerical solution and boundary conditions A CFD software based on the finite volume method has been used to solve the set of equations provided by both RANS models introduced in the previous section. A rectangular Cartesian grid with 676,000 control volumes is used; the size of the smallest control volumes, placed close to the walls and the ground is 0.005 m and a stretching factor of 1.1 is adopted. A sensitivity analysis of the numerical scheme on the grid refinement has been performed by using a finer grid (900,000 volumes), but the change in the results was negligible. As far as the boundary conditions are concerned, at the upwind boundary a logarithmic profile of the streamwise velocity component U is applied: U0 h UðhÞ ¼ ln (8) ho k where h is the distance from the ground, k = 0.41 is the Von Karman’s constant, U0 = 1.068 m/s and h0 = 0.005 m have been determined through the best fit of the experimental data provided in [8]. The velocity components along vertical and spanwise direction are equalled to zero.
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In addition, as the distribution of k and e on the inlet boundary is not known, the following relations are used [15]: 3 k ¼ ðUavg Ti Þ2 2
e ¼ Cm3=4
k3=2 lt
(9)
where Uavg is the average flow velocity, Ti the turbulence intensity and lt is the turbulence length scale. According to the literature, Ti = 4% [16–18] and lt = 0.4 m [19] have been used. It is to be underlined that the results showed a sensitivity to Ti much larger than to lt. At the outlet of the computational domain, a constant pressure is assumed, while the gradients of all the dependent variables are assumed to vanish. Furthermore, a co-located variable arrangement is employed, together with the algorithm proposed by Rhie and Chow to avoid checker boarding [20]. The coupling between pressure and velocity distribution is obtained using SIMPLE procedure [21]. The second order linear upwind difference scheme has been used, after an accurate analysis of the errors produced by numerical diffusion when using the first order upwind scheme. The Gauss–Seidel point-implicit method is used to solve the finite-volume linear system, while the convergence criterion is based on the reduction of all the scaled solution residuals under the threshold of 103.
3. Results and discussion 3.1. Velocity distribution The main purpose of the present study is to investigate the reliability of the RANS scheme when modelling wind driven natural ventilation in buildings. For this reason, the comparison between numerical and experimental results has been limited to the zone of the computational domain closer to the building. In Fig. 3, five vertical lines in the middle section of the building are identified (from A to E), in which experimental data are available; the determination of the velocity in the points along the lines ranging from B to D allows a full description of the flow inside the building, while locations A and E have been chosen in order to describe the flow pattern close to the openings. The velocity components U and V have been determined, along streamwise and vertical direction, respectively. As far as the third velocity component is concerned
Fig. 3. Locations along which experimental and numerical results have been compared.
(spanwise component), it has been neglected as it is supposed to be zero, due to the symmetry of the domain. The plots in Figs. 4–6 show the vertical distribution of experimental and numerical velocity for the three ventilation pattern evaluated; numerical results have been obtained through both RNG and k–e model. The plots refer to the scaled values of U and V, where the scaling factor Uref = 12 m/s is the maximum streamwise velocity measured in the experimental tests. As can be seen from Figs. 4 and 5, there is a good agreement between experimental and numerical values of the streamwise component U when dealing with single-sided ventilation (Cases 1 and 2); some discrepancies can be noticed in the zones over the building roof and close to the leeward wall (Y > 0.25, Sections from B to E), where both RANS models tend to underestimate the measured U-velocity profile. This peculiarity is in full agreement with the results obtained in other studies concerning the wind distribution around a bluff body ([18,22]), which illustrated the inaccuracy of RANS models in the separation region above and behind the building. Similar conclusions can be drawn with reference to the vertical component V, even if the discrepancies displayed in the region above the roof appear larger than for streamwise component U. Nevertheless, the description of the flow inside the building and close to the openings, which is the main purpose in natural ventilation design, may be considered satisfying. In addition, the better performance of RNG model with respect to k–e model can be underlined. The difference is evident in the regions close to horizontal surfaces; as an example, in Fig. 4, Sections B to D, RNG results closely describe the variation of the streamwise component near the floor (Y < 0.05), while an important discrepancy is shown by k–e results. This is due to the ability of RNG model to account for the damping effect of the solid surfaces. A further evidence of this superiority of RNG model in this kind of problems may be provided by Fig. 7, which refers to the centre of the building (Section C) in single-sided windward ventilation (Case 1). The vertical profile of turbulent kinetic energy dissipation rate e is shown: when approaching the floor, its increase is expected to be outstanding, but only the results provided by RNG reveal this steep variation. As far as cross ventilation is concerned, some inaccuracy can be detected, especially in the determination of the V-velocity profile; the distribution of the streamwise component is described with a good accuracy, even if RNG results show a slight discrepancy in the bottom–middle region of the enclosed space (0.05 < Y < 0.15), where the results provided by k–e are better. It is worth mentioning that the velocity distribution obtained by Jiang et al. [8] using LES approach is better than that provided by RANS models for cross ventilation, while the results are comparable when dealing with single-sided ventilation; in this case LES is slightly more precise only in the description of the separation region above the roof, but the importance of this aspect is minor when the aim of the analysis is the study of the ventilation inside the building rather than the flow around it. Finally, Fig. 8 illustrates the flow distribution inside the building for the three configurations considered in this analysis, obtained using the RNG model. The distribution provided by k–
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Fig. 4. Velocity distribution for single-sided, windward ventilation (Case 1). Solid line: RNG model; dashed line: k–e model; dots: experimental values. (a) U/Uref; (b) V/Uref.
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Fig. 5. Velocity distribution for single-sided, leeward ventilation (Case 2). Solid line: RNG model; dashed line: k–e model; dots: experimental values. (a) U/Uref; (b) V/Uref.
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Fig. 6. Velocity distribution for cross ventilation (Case 3). Solid line: RNG model; dashed line: k–e model; dots: experimental values. (a) U/Uref; (b) V/Uref.
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where Cd is the discharge coefficient of the openings, DCp the difference between the mean pressure coefficients associated to the opposite openings and Aeff (m2) is the effective area; the latter is defined by: 1 1 1 ¼ 2 þ 2 2 Aeff Ain Aout
Fig. 7. Vertical profile of turbulent kinetic energy dissipation rate e (Case 1, Section C). Solid line: RNG model; dashed line: k–e model.
e model is not shown, as it is almost the same. Due to the good agreement with experimental data of streamwise and vertical velocity components previously determined, the plots of Fig. 8 can be considered accurate enough, especially those referring to single-sided ventilation. It is interesting to notice that leeward ventilation seems to produce an air movement inside the enclosed space more significant than windward ventilation, especially in the region close to the opening. This point will be further analysed in the following section. 3.2. Ventilation rate The design calculation of the air flow rate provided by natural ventilation is usually accomplished by means of simple relationships. For single-sided wind driven natural ventilation an empirical model is available [23], according to which the ventilation rate can be calculated as: QSS ¼ 0:025 A Ubh
(10)
where A (m2) is the opening area and Ubh (m/s) is the reference wind velocity, measured at the building height. The wind profile adopted in this study provides Ubh = 10.2 m/s, then the expected ventilation rate is QSS = 2.67 103 m3/s. For cross ventilation, the following semi-empirical model can be used, derived from the application of the Bernoulli equation: qffiffiffiffiffiffiffiffiffi QCR ¼ Cd Aeff Ubh DCp (11)
(12)
In the present analysis, Aeff = 7.425 103 m2; we can also reasonably assume Cd = 0.61, which is the typical value of the discharge coefficient for a sharp opening [24–26], and DCp = 0.9, considering typical values of Cp = 0.7 and 0.2 for windward and leeward faces, respectively [27,28]. Hence, the expected ventilation rate for cross ventilation is QCR = 4.38 102 m3/s. Table 2 compares the ventilation rate provided by the simulations with the expected values previously defined; the values obtained by LES approach in [8] are also reported. As a direct consequence of the better performance of RNG model in the determination of the velocity profiles, RNG-based ventilation rates present better agreement with the expected values than k–e results. The biggest deviation between expected and RNG results is measured in Case 3 (cross ventilation) and is approximately 10%; this discrepancy reduces to only 2% in Case 2 (leeward single-sided ventilation). According to these figures, RNG model may be considered a reliable tool for the determination of wind driven ventilation rate; on the contrary, k–e model is quite inaccurate, and only for cross ventilation its performance gets close to RNG model, the difference between the two RANS models reducing to 6%. Anyway, for a correct interpretation of these data it is important to remember that the expected ventilation rates have not been obtained through experimental measurements, but they are given by empirical relationships which have by now shown a good accuracy, but which may be affected by some limitations. As an example, Eq. (10) does not take into account the wind direction, providing the same value for windward and leeward ventilation rate. In fact, Heiselberg and co-workers [29] performed full-scale measurements for different wind velocities and directions on a building equipped with a single opening: on average, their results showed a good agreement with Eq. (10), but a significant variation of the ventilation rate with the wind direction was also displayed, resulting in a smaller air exchange rate when the opening was placed in the leeward face rather than in the windward one. On the contrary, Melaragno [30] pointed out that the mean air velocity inside the building could be smaller when the opening is on the windward face. The results of the present work, namely those shown in Fig. 8 and Table 2, obtained using the RNG model, suggest that the air movement, and hence the ventilation rate, is slightly higher in Case 2 (leeward face). Moreover, it should be underlined that LES results are those which best approach the expected values, the only exception being represented by leeward single-sided ventilation, whose large ventilation rate (4.8 103) should anyway be interpreted in the light of the previous considerations.
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Fig. 8. Air flow distribution inside the building (RNG results).
3.3. Pressure Table 3 shows the values of the mean pressure coefficient Cp on all of the external surfaces of the building, determined by:
Cp ¼
P P0 0:5 r U
2
(13)
and averaging out the values obtained for each of the three configurations investigated. In Eq. (13), P0 (Pa) is the static
pressure in the undisturbed flow and U (m/s) is the free stream velocity calculated at building height. The values reported in literature refer to a low-rise, square building in open flat country. As far as the roof is concerned, the two figures displayed refer to the front and the rear side of a sloping roof with a pitch <108, the value for a plan roof can be reasonably assumed to lie between these values. The results of RNG simulations show a good agreement with the standard values provided in literature, with a discrepancy ranging from 2% for left side wall to 17% for leeward face; this suggests a good confidence of RNG in determining the mean
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Table 2 Empirical and numerical values of the ventilation rate (m3/s) Single-sided windward ventilation (Case 1) RNG model k–e model LES [8] Empirical methods
3
Single-sided leeward ventilation (Case 2) 3
4.8 102 5.08 102 4.65 102 4.38 102
2.62 10 1.48 103 4.8 103 2.68 103
2.4 10 3.75 103 2.7 103 2.68 103
Cross ventilation (Case 3)
Table 3 Mean pressure coefficients for the surfaces of the cubic building Windward face
Leeward face
Right wall
Left wall
Roof
RNG model k–e model
0.566 0.766
0.229 0.125
0.478 0.361
0.484 0.362
0.524 0.363
Literature [27,28] [31]
0.7 0.6
0.2 0.35
0.5 0.45
0.5 0.45
0.8 to 0.4 –
pressure distribution around a building when wind driven natural ventilation is concerned. 4. Conclusions In the present study, Reynolds averaged Navier–Stokes approach has been applied to wind driven natural ventilation in a cubic building; three different configurations were considered: single-sided ventilation with an opening on the windward wall, single-sided ventilation with an opening on the leeward wall and cross ventilation. The simulations were performed using both standard two-equation k–e model and Renormalization Group theory. As a result, the velocity distribution inside and around the building was determined, as well as the ventilation rate and the mean pressure coefficients on all of the building surfaces; all the results were then compared to experimental data provided in literature and well established empirical expressions for the prediction of the ventilation rate. The numerical results obtained using RNG method show a good agreement with the experimental data, especially for single-sided ventilation. The profiles of streamwise and vertical velocity component are accurate, and the discrepancy between calculated and expected ventilation rate lies under 10%. On the contrary, the results suggest the difficulty of k–e model in describing the flow close to the surfaces, where the damping effect is relevant, thus determining a noticeable inaccuracy in the prediction of the ventilation rates. Despite the greater precision, the computation effort experienced with RNG model is only slightly more intensive than k–e model. It is also worth mentioning that positioning the opening on the leeward side rather than on the windward side, when dealing with single-sided ventilation, results in a stronger air movement inside the building; hence, in a larger ventilation rate. However, in literature different opinions have been expressed about this point, thus raising the need of further experimental investigations. Other authors have previously dealt with wind driven natural ventilation by means of large-eddy simulation, which is a more
precise but more time-consuming method; the difference between RNG and LES results is not outstanding, at least for the description of the flow inside the building. According to these findings, RNG model can be considered a useful tool for the study of air flow inside and around a building when dealing with wind driven natural ventilation, especially if a reasonable accuracy is required but not an extreme precision. On the contrary, the use of LES may be preferable in wind engineering application and advanced fluid dynamics. References [1] M. Orme, Estimates of the energy impact of ventilation and associated financial expenditures, Energy and Buildings 33 (2001) 199–205. [2] C. Allocca, Q. Chen, L.R. Glicksman, Design analysis of single-sided natural ventilation, Energy and Buildings 35 (2003) 785–795. [3] C.J. Koinakis, Combined thermal and natural ventilation modelling for long-term energy assessment: validation with experimental measurements, Energy and Buildings 37 (2005) 311–323. [4] M.P. Straw, C.J. Baker, A.P. Robertson, Experimental measurements and computations of the wind-induced ventilation of a cubic structure, Journal of Wind Engineering and Industrial Aerodynamics 88 (2000) 213–230. [5] M.M. Eftekhari, L.D. Marjanovic, D.J. Pinnock, Air flow distribution in and around a single-sided naturally ventilated room, Building and Environment 38 (2003) 389–397. [6] N.H. Wong, S. Heryanto, The study of active stack effect to enhance natural ventilation using wind tunnel and computational fluid dynamics (CFD) simulations, Energy and Buildings 36 (2004) 668–678. [7] Q. Chen, Using computational tools to factor wind into architectural environmental design, Energy and Buildings 36 (2004) 1197–1209. [8] Y. Jiang, D. Alexander, H. Jenkins, R. Arthur, Q. Chen, Natural ventilation in buildings: measurements in a wind tunnel and numerical simulation with large-eddy simulation, Journal of Wind Engineering and Industrial Aerodynamics 91 (2003) 331–353. [9] Y. Jiang, Q. Chen, Effect of fluctuating wind direction on cross natural ventilation in buildings from large eddy simulation, Building and Environment 37 (2002) 379–386. [10] Y. Jiang, Q. Chen, Study of natural ventilation in buildings by large eddy simulation, Journal of Wind Engineering and Industrial Aerodynamics 89 (2001) 1155–1178. [11] S. Murakami, Overview of turbulence models applied in CWE-1997, Journal of Wind Engineering and Industrial Aerodynamics 74–76 (1998) 1–24.
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[21] S.V. Patankar, D.B. Spalding, A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows, International Journal of Heat and Mass Transfer 15 (1972) 1787–1806. [22] R. Martinuzzi, C. Tropea, The flow around surface-mounted, prismatic obstacles placed in a fully developed channel flow, Journal of Fluids Engineering 115 (1993) 85–93. [23] BS 5925, Code of Practice for Design of Buildings: Ventilation Principles and Designing for Natural Ventilation, British Standards Institution, London, 1980. [24] D. Etheridge, M. Sandberg, Building Ventilation: Theory and Measurements, Wiley, Chichester, UK, 1996. [25] H. Awbi, Ventilation of Buildings, Spon Press, London, UK, 2003. [26] ASHRAE Handbook of Fundamentals, American Society of Heating, Refrigeration and Air-Conditioning Engineers, Atlanta, 2001 (Chapter 26). [27] M.W. Liddament, A Guide to Energy Efficient Ventilation, in: Air Infiltration and Ventilation Centre, Coventry, UK, 1996. [28] F. Allard, Ventilation in Buildings: A Design Handbook, James & James Ltd., London, UK, 1998. [29] T.S. Larsen, P. Heiselberg, T. Sawachi, Analysis and design of single-sided natural ventilation, in: Proceedings of the Fourth International Symposium on HVAC, Beijing, China, October, (2003. [30] M. Melaragno, Wind in Architectural and Environmental Design, Van Nostrand Reinhold, New York, 1982. [31] M.V. Swami, S. Chandra, Correlations for pressure distribution of buildings and calculation of natural-ventilation airflow, ASHRAE Transactions 94 (1988) 243–266.
Computational analysis of wind driven natural ventilation in buildings G. Evola a,*, V. Popov b a
Department of Mechanical and Industrial Engineering, University of Catania, Viale Andrea Doria 6, Catania 95125, Italy b Wessex Institute of Technology, Ashurst Lodge, Ashurst, Southampton SO40 7AA, UK Received 31 March 2005; received in revised form 20 August 2005; accepted 23 August 2005
Abstract The design of natural ventilation in buildings is often performed by means of computational fluid dynamics (CFD) techniques, whose application is gaining popularity. In the present study, Reynolds averaged Navier–Stokes equation (RANS) approach is applied to wind driven natural ventilation in a cubic building. Two different models are considered, namely the two-equation k–e model and the Renormalization Group (RNG) theory. The velocity and pressure distribution inside and around the building are determined, as well as the ventilation rate, for three different configurations: cross ventilation, single-sided ventilation with an opening on the windward wall and single-sided ventilation with an opening on the leeward wall. The numerical results are compared with experimental data, showing a good agreement, particularly when using RNG. The discrepancy in the determination of the ventilation rate is reasonable and the flow distribution inside the building is properly described when RNG model is used. However, the k–e model fails to determine the correct velocity components near the horizontal surfaces. According to these results, the RNG model can be considered a useful tool for the study of wind driven natural ventilation, especially for the assessment of the ventilation rate and of the air distribution inside a room. # 2005 Elsevier B.V. All rights reserved. Keywords: Single-sided ventilation; Cross ventilation; Computational fluid dynamics; RANS modelling; Ventilation rate
1. Introduction Mechanical ventilation in buildings is nowadays a common practice, due to the need to provide thermal comfort and good indoor air quality in enclosed spaces. The energy consumption related to the operation of heating, ventilation and airconditioning systems (HVAC) is considerable: according to recent studies, nearly 68% of the total energy use in service and residential buildings is attributable to HVAC systems [1]. On the other hand, natural ventilation replaces indoor air with fresh outdoor air without any energy consumption, and it also helps to overcome common health problems related to an insufficient maintenance of HVAC systems. For these reasons, great attention is being focused on the design of natural ventilation in buildings. The correct design of a naturally ventilated building is a challenging task, due to the complexity of the physical phenomena involved. In particular, the air flow through an opening, either purpose provided or adventitious, depends on * Corresponding author. Tel.: +39 0957382421; fax: +39 0957382496. E-mail address: [email protected] (G. Evola). 0378-7788/$ – see front matter # 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.enbuild.2005.08.008
the pressure difference between the sides of the opening, as well as on the resistance opposed to the air flow by the opening itself; the latter is a function of opening shape and dimension. The pressure difference is produced by wind and buoyancy forces, several studies have been performed in order to better understand the interaction between these two driving forces. Allocca et al. [2] showed that, under certain circumstances, the wind effect might not be beneficial, as it may reduce the ventilation rate provided by buoyancy forces alone. In this work, the attention is focused on wind driven natural ventilation. In order to obtain reliable information concerning the air flow and the pressure distribution around and inside a naturally ventilated building, full-scale measurements can be performed [3,4]. However, wind tunnel tests on small-scale models are usually preferred, as they allow the control of wind speed and direction as well as the study of different configurations [5,6]. An alternative approach is represented by computational fluid dynamics (CFD), based on the numerical solution of the set of governing equations which describe the flow field. CFD-based programs are widespread and provide a detailed description of the air flow. In recent years, their application has
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become more and more popular thanks to the increase of computational power and to the improvements in turbulence modelling [7]. Among the available models, Reynolds averaged Navier–Stokes equation (RANS) and large-eddy simulation (LES) are the most common. RANS is widely used by most of the commercial CFD software. It is based on the time-averaging of the flow field governing equations; this means that all of the unsteadiness is considered as part of the turbulence and is averaged out. On the other hand, LES is based on the space filtering of the turbulent structures. The fluid flow is separated into large eddies and small eddies; the first ones are solved by the governing equations, while the last ones are approximated and described by suitable subgrid-scale models. Jiang et al. [8] applied LES to wind driven ventilation; they determined pressure and velocity distribution inside and around a scale cubic building model, showing a good agreement between numerical and experimental results. LES models have also been successfully applied to cross natural ventilation in a group of apartments [9]. Despite their reliability, LES models are time-consuming and for this reason they are seldom used, unless an extreme precision is required. On the contrary, RANS models require less computing time, but sometimes they lack the due precision. As an example, Jiang and Chen [10] compared LES and RANS performance in wind driven natural ventilation, concluding that RANS might not be appropriate for the determination of the ventilation rate. A comparison between the results provided by RANS and LES models in natural ventilation problems is also available in [11]. Due to the large use of RANS models, an extensive research is needed to further test their reliability and improve their application to natural ventilation problems. Previous studies concerning RANS modelling of wind driven natural ventilation in a building immersed in an atmospheric boundary layer can be found in literature [12], but they were based on a twodimensional approach and little attention was paid to the determination of the ventilation rate. The present investigation will be focused on the application of three-dimensional RANS modelling on wind driven natural ventilation in buildings. Detailed information concerning air flow inside and pressure distribution around a scale model building will be provided, and compared to the experimental measurements performed by Jiang et al. [8]. Furthermore, the ventilation rate determined through the simulation will be compared with results provided by LES models and empirical methods.
where the openings are located on a single fac¸ade. As a consequence, in the present study three different cases are considered: Case 1: single-sided ventilation with an opening on the windward wall; Case 2: single-sided ventilation with an opening on the leeward wall; Case 3: cross ventilation with openings on both windward and leeward walls. In order to allow a comparison with the experimental results available in literature, the simulations are performed on a building-like model similar to the one used in [8], whose dimensions are 250 mm 250 mm 250 mm. Two different building models are used: the first one is provided with a doorlike opening sized 84 mm 125 mm (width height), and it is used to study single-sided wind driven ventilation (Cases 1 and 2). The second building model presents two openings of the same dimension on opposite sides, in order to simulate cross ventilation (Case 3). The thickness of the walls is 6 mm in both building models. Fig. 1 shows a schematic view of the building model with an opening in the windward wall. Furthermore, a logarithmic wind profile upwind of the building is considered. The dimension of the computational domain, displayed in Fig. 2, is chosen large enough in order not to disturb the air flow around the building. 2.2. Numerical method Computational fluid dynamics is based on the resolution of the governing equations which describe the flow field in the computational domain, namely the continuity equation for mass transfer, the Navier–Stokes equation for momentum transfer and the thermal energy equation for heat transfer. In the present analysis, buoyancy effects are neglected; hence, the flow is considered isothermal and the last equation is not used.
2. Methods 2.1. Problem description When dealing with wind driven natural ventilation, two different ventilation patterns can be identified, namely cross ventilation and single-sided ventilation. The first one is characterized by the presence of two or more openings on opposite sides of the building, and it can provide ventilation rates larger than those obtained by single-sided ventilation,
Fig. 1. Description of the building model in single-sided ventilation with an opening in the windward wall (Case 1); all the dimensions are in mm.
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Table 1 Empirical constants used for the formulation of k–e model 0.09 1.44 1.92 1.0 1.3
Cm C1 C2 sk se
constants used in the previous relations are provided. These values may be not suitable for low Re number flows, or in some local domains of high Re number flows where the damping effect of the surfaces is prominent; in these cases, some corrections would be required. Concerning this problem, Renormalization Group (RNG) model [13] is gaining popularity for modelling indoor air movement; the difference between RNG and k–e models mainly results in the substitution of the constant C1 in Eq. (5) with the following expression: Fig. 2. Dimensions of the computational domain.
For a steady incompressible flow and according to RANS approach, time-averaged continuity and Navier–Stokes equations can be, respectively, written as: @ ðrUi Þ ¼ 0 @xi
g ðr r0 Þ (2) where Uj (j = 1–3) are the components of the velocity vector, g the gravity acceleration, dj = 1 if j = 3 and 0 otherwise and m is the dynamic viscosity. The term mt is called turbulent viscosity and is introduced in RANS modelling to describe the timeaveraged effects of the turbulent stress. According to the widely used two-equation k–e model, mt can be expressed as follows: k2 e
(3)
where Cm is a constant and k and e are the turbulent kinetic energy and its dissipation rate, respectively. In order to close the model two other equations are required, namely the transport equations for k and e: @ @ ðrUi kÞ ¼ @xi @xi
@ @ ðrUi eÞ ¼ @xi @xi
m mþ t sk
m mþ t se
@k þ Pk re @xi
(4)
@e e e2 þ C1 Pk C2 r @xi k k
(5)
In Eqs. (4) and (5), Pk is the production rate of turbulent kinetic energy, which depends on the turbulent viscosity and the velocity distribution. In Table 1, the values of all the empirical
hð1 h=h0 Þ 1 þ bh3
(6)
where C10 1:44, h0 4.4, b 0.015 and
(1)
@ @p @ @U j @Ui ðrUi U j Þ ¼ þ þ ðm þ mt Þ dj @xi @x j @xi @xi @x j
mt ¼ Cm r
C1 ¼ C10
h¼
k e
rffiffiffiffiffi Pk m
(7)
Through Eqs. (6) and (7), the production term in Eq. (5) is modified in such a way that it properly accounts for the larger dissipation rate experienced in the laminar regions near solid surfaces. Detailed information about the application of RNG model is available in [14]. 2.3. Numerical solution and boundary conditions A CFD software based on the finite volume method has been used to solve the set of equations provided by both RANS models introduced in the previous section. A rectangular Cartesian grid with 676,000 control volumes is used; the size of the smallest control volumes, placed close to the walls and the ground is 0.005 m and a stretching factor of 1.1 is adopted. A sensitivity analysis of the numerical scheme on the grid refinement has been performed by using a finer grid (900,000 volumes), but the change in the results was negligible. As far as the boundary conditions are concerned, at the upwind boundary a logarithmic profile of the streamwise velocity component U is applied: U0 h UðhÞ ¼ ln (8) ho k where h is the distance from the ground, k = 0.41 is the Von Karman’s constant, U0 = 1.068 m/s and h0 = 0.005 m have been determined through the best fit of the experimental data provided in [8]. The velocity components along vertical and spanwise direction are equalled to zero.
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In addition, as the distribution of k and e on the inlet boundary is not known, the following relations are used [15]: 3 k ¼ ðUavg Ti Þ2 2
e ¼ Cm3=4
k3=2 lt
(9)
where Uavg is the average flow velocity, Ti the turbulence intensity and lt is the turbulence length scale. According to the literature, Ti = 4% [16–18] and lt = 0.4 m [19] have been used. It is to be underlined that the results showed a sensitivity to Ti much larger than to lt. At the outlet of the computational domain, a constant pressure is assumed, while the gradients of all the dependent variables are assumed to vanish. Furthermore, a co-located variable arrangement is employed, together with the algorithm proposed by Rhie and Chow to avoid checker boarding [20]. The coupling between pressure and velocity distribution is obtained using SIMPLE procedure [21]. The second order linear upwind difference scheme has been used, after an accurate analysis of the errors produced by numerical diffusion when using the first order upwind scheme. The Gauss–Seidel point-implicit method is used to solve the finite-volume linear system, while the convergence criterion is based on the reduction of all the scaled solution residuals under the threshold of 103.
3. Results and discussion 3.1. Velocity distribution The main purpose of the present study is to investigate the reliability of the RANS scheme when modelling wind driven natural ventilation in buildings. For this reason, the comparison between numerical and experimental results has been limited to the zone of the computational domain closer to the building. In Fig. 3, five vertical lines in the middle section of the building are identified (from A to E), in which experimental data are available; the determination of the velocity in the points along the lines ranging from B to D allows a full description of the flow inside the building, while locations A and E have been chosen in order to describe the flow pattern close to the openings. The velocity components U and V have been determined, along streamwise and vertical direction, respectively. As far as the third velocity component is concerned
Fig. 3. Locations along which experimental and numerical results have been compared.
(spanwise component), it has been neglected as it is supposed to be zero, due to the symmetry of the domain. The plots in Figs. 4–6 show the vertical distribution of experimental and numerical velocity for the three ventilation pattern evaluated; numerical results have been obtained through both RNG and k–e model. The plots refer to the scaled values of U and V, where the scaling factor Uref = 12 m/s is the maximum streamwise velocity measured in the experimental tests. As can be seen from Figs. 4 and 5, there is a good agreement between experimental and numerical values of the streamwise component U when dealing with single-sided ventilation (Cases 1 and 2); some discrepancies can be noticed in the zones over the building roof and close to the leeward wall (Y > 0.25, Sections from B to E), where both RANS models tend to underestimate the measured U-velocity profile. This peculiarity is in full agreement with the results obtained in other studies concerning the wind distribution around a bluff body ([18,22]), which illustrated the inaccuracy of RANS models in the separation region above and behind the building. Similar conclusions can be drawn with reference to the vertical component V, even if the discrepancies displayed in the region above the roof appear larger than for streamwise component U. Nevertheless, the description of the flow inside the building and close to the openings, which is the main purpose in natural ventilation design, may be considered satisfying. In addition, the better performance of RNG model with respect to k–e model can be underlined. The difference is evident in the regions close to horizontal surfaces; as an example, in Fig. 4, Sections B to D, RNG results closely describe the variation of the streamwise component near the floor (Y < 0.05), while an important discrepancy is shown by k–e results. This is due to the ability of RNG model to account for the damping effect of the solid surfaces. A further evidence of this superiority of RNG model in this kind of problems may be provided by Fig. 7, which refers to the centre of the building (Section C) in single-sided windward ventilation (Case 1). The vertical profile of turbulent kinetic energy dissipation rate e is shown: when approaching the floor, its increase is expected to be outstanding, but only the results provided by RNG reveal this steep variation. As far as cross ventilation is concerned, some inaccuracy can be detected, especially in the determination of the V-velocity profile; the distribution of the streamwise component is described with a good accuracy, even if RNG results show a slight discrepancy in the bottom–middle region of the enclosed space (0.05 < Y < 0.15), where the results provided by k–e are better. It is worth mentioning that the velocity distribution obtained by Jiang et al. [8] using LES approach is better than that provided by RANS models for cross ventilation, while the results are comparable when dealing with single-sided ventilation; in this case LES is slightly more precise only in the description of the separation region above the roof, but the importance of this aspect is minor when the aim of the analysis is the study of the ventilation inside the building rather than the flow around it. Finally, Fig. 8 illustrates the flow distribution inside the building for the three configurations considered in this analysis, obtained using the RNG model. The distribution provided by k–
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Fig. 4. Velocity distribution for single-sided, windward ventilation (Case 1). Solid line: RNG model; dashed line: k–e model; dots: experimental values. (a) U/Uref; (b) V/Uref.
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Fig. 5. Velocity distribution for single-sided, leeward ventilation (Case 2). Solid line: RNG model; dashed line: k–e model; dots: experimental values. (a) U/Uref; (b) V/Uref.
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Fig. 6. Velocity distribution for cross ventilation (Case 3). Solid line: RNG model; dashed line: k–e model; dots: experimental values. (a) U/Uref; (b) V/Uref.
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where Cd is the discharge coefficient of the openings, DCp the difference between the mean pressure coefficients associated to the opposite openings and Aeff (m2) is the effective area; the latter is defined by: 1 1 1 ¼ 2 þ 2 2 Aeff Ain Aout
Fig. 7. Vertical profile of turbulent kinetic energy dissipation rate e (Case 1, Section C). Solid line: RNG model; dashed line: k–e model.
e model is not shown, as it is almost the same. Due to the good agreement with experimental data of streamwise and vertical velocity components previously determined, the plots of Fig. 8 can be considered accurate enough, especially those referring to single-sided ventilation. It is interesting to notice that leeward ventilation seems to produce an air movement inside the enclosed space more significant than windward ventilation, especially in the region close to the opening. This point will be further analysed in the following section. 3.2. Ventilation rate The design calculation of the air flow rate provided by natural ventilation is usually accomplished by means of simple relationships. For single-sided wind driven natural ventilation an empirical model is available [23], according to which the ventilation rate can be calculated as: QSS ¼ 0:025 A Ubh
(10)
where A (m2) is the opening area and Ubh (m/s) is the reference wind velocity, measured at the building height. The wind profile adopted in this study provides Ubh = 10.2 m/s, then the expected ventilation rate is QSS = 2.67 103 m3/s. For cross ventilation, the following semi-empirical model can be used, derived from the application of the Bernoulli equation: qffiffiffiffiffiffiffiffiffi QCR ¼ Cd Aeff Ubh DCp (11)
(12)
In the present analysis, Aeff = 7.425 103 m2; we can also reasonably assume Cd = 0.61, which is the typical value of the discharge coefficient for a sharp opening [24–26], and DCp = 0.9, considering typical values of Cp = 0.7 and 0.2 for windward and leeward faces, respectively [27,28]. Hence, the expected ventilation rate for cross ventilation is QCR = 4.38 102 m3/s. Table 2 compares the ventilation rate provided by the simulations with the expected values previously defined; the values obtained by LES approach in [8] are also reported. As a direct consequence of the better performance of RNG model in the determination of the velocity profiles, RNG-based ventilation rates present better agreement with the expected values than k–e results. The biggest deviation between expected and RNG results is measured in Case 3 (cross ventilation) and is approximately 10%; this discrepancy reduces to only 2% in Case 2 (leeward single-sided ventilation). According to these figures, RNG model may be considered a reliable tool for the determination of wind driven ventilation rate; on the contrary, k–e model is quite inaccurate, and only for cross ventilation its performance gets close to RNG model, the difference between the two RANS models reducing to 6%. Anyway, for a correct interpretation of these data it is important to remember that the expected ventilation rates have not been obtained through experimental measurements, but they are given by empirical relationships which have by now shown a good accuracy, but which may be affected by some limitations. As an example, Eq. (10) does not take into account the wind direction, providing the same value for windward and leeward ventilation rate. In fact, Heiselberg and co-workers [29] performed full-scale measurements for different wind velocities and directions on a building equipped with a single opening: on average, their results showed a good agreement with Eq. (10), but a significant variation of the ventilation rate with the wind direction was also displayed, resulting in a smaller air exchange rate when the opening was placed in the leeward face rather than in the windward one. On the contrary, Melaragno [30] pointed out that the mean air velocity inside the building could be smaller when the opening is on the windward face. The results of the present work, namely those shown in Fig. 8 and Table 2, obtained using the RNG model, suggest that the air movement, and hence the ventilation rate, is slightly higher in Case 2 (leeward face). Moreover, it should be underlined that LES results are those which best approach the expected values, the only exception being represented by leeward single-sided ventilation, whose large ventilation rate (4.8 103) should anyway be interpreted in the light of the previous considerations.
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Fig. 8. Air flow distribution inside the building (RNG results).
3.3. Pressure Table 3 shows the values of the mean pressure coefficient Cp on all of the external surfaces of the building, determined by:
Cp ¼
P P0 0:5 r U
2
(13)
and averaging out the values obtained for each of the three configurations investigated. In Eq. (13), P0 (Pa) is the static
pressure in the undisturbed flow and U (m/s) is the free stream velocity calculated at building height. The values reported in literature refer to a low-rise, square building in open flat country. As far as the roof is concerned, the two figures displayed refer to the front and the rear side of a sloping roof with a pitch <108, the value for a plan roof can be reasonably assumed to lie between these values. The results of RNG simulations show a good agreement with the standard values provided in literature, with a discrepancy ranging from 2% for left side wall to 17% for leeward face; this suggests a good confidence of RNG in determining the mean
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Table 2 Empirical and numerical values of the ventilation rate (m3/s) Single-sided windward ventilation (Case 1) RNG model k–e model LES [8] Empirical methods
3
Single-sided leeward ventilation (Case 2) 3
4.8 102 5.08 102 4.65 102 4.38 102
2.62 10 1.48 103 4.8 103 2.68 103
2.4 10 3.75 103 2.7 103 2.68 103
Cross ventilation (Case 3)
Table 3 Mean pressure coefficients for the surfaces of the cubic building Windward face
Leeward face
Right wall
Left wall
Roof
RNG model k–e model
0.566 0.766
0.229 0.125
0.478 0.361
0.484 0.362
0.524 0.363
Literature [27,28] [31]
0.7 0.6
0.2 0.35
0.5 0.45
0.5 0.45
0.8 to 0.4 –
pressure distribution around a building when wind driven natural ventilation is concerned. 4. Conclusions In the present study, Reynolds averaged Navier–Stokes approach has been applied to wind driven natural ventilation in a cubic building; three different configurations were considered: single-sided ventilation with an opening on the windward wall, single-sided ventilation with an opening on the leeward wall and cross ventilation. The simulations were performed using both standard two-equation k–e model and Renormalization Group theory. As a result, the velocity distribution inside and around the building was determined, as well as the ventilation rate and the mean pressure coefficients on all of the building surfaces; all the results were then compared to experimental data provided in literature and well established empirical expressions for the prediction of the ventilation rate. The numerical results obtained using RNG method show a good agreement with the experimental data, especially for single-sided ventilation. The profiles of streamwise and vertical velocity component are accurate, and the discrepancy between calculated and expected ventilation rate lies under 10%. On the contrary, the results suggest the difficulty of k–e model in describing the flow close to the surfaces, where the damping effect is relevant, thus determining a noticeable inaccuracy in the prediction of the ventilation rates. Despite the greater precision, the computation effort experienced with RNG model is only slightly more intensive than k–e model. It is also worth mentioning that positioning the opening on the leeward side rather than on the windward side, when dealing with single-sided ventilation, results in a stronger air movement inside the building; hence, in a larger ventilation rate. However, in literature different opinions have been expressed about this point, thus raising the need of further experimental investigations. Other authors have previously dealt with wind driven natural ventilation by means of large-eddy simulation, which is a more
precise but more time-consuming method; the difference between RNG and LES results is not outstanding, at least for the description of the flow inside the building. According to these findings, RNG model can be considered a useful tool for the study of air flow inside and around a building when dealing with wind driven natural ventilation, especially if a reasonable accuracy is required but not an extreme precision. On the contrary, the use of LES may be preferable in wind engineering application and advanced fluid dynamics. References [1] M. Orme, Estimates of the energy impact of ventilation and associated financial expenditures, Energy and Buildings 33 (2001) 199–205. [2] C. Allocca, Q. Chen, L.R. Glicksman, Design analysis of single-sided natural ventilation, Energy and Buildings 35 (2003) 785–795. [3] C.J. Koinakis, Combined thermal and natural ventilation modelling for long-term energy assessment: validation with experimental measurements, Energy and Buildings 37 (2005) 311–323. [4] M.P. Straw, C.J. Baker, A.P. Robertson, Experimental measurements and computations of the wind-induced ventilation of a cubic structure, Journal of Wind Engineering and Industrial Aerodynamics 88 (2000) 213–230. [5] M.M. Eftekhari, L.D. Marjanovic, D.J. Pinnock, Air flow distribution in and around a single-sided naturally ventilated room, Building and Environment 38 (2003) 389–397. [6] N.H. Wong, S. Heryanto, The study of active stack effect to enhance natural ventilation using wind tunnel and computational fluid dynamics (CFD) simulations, Energy and Buildings 36 (2004) 668–678. [7] Q. Chen, Using computational tools to factor wind into architectural environmental design, Energy and Buildings 36 (2004) 1197–1209. [8] Y. Jiang, D. Alexander, H. Jenkins, R. Arthur, Q. Chen, Natural ventilation in buildings: measurements in a wind tunnel and numerical simulation with large-eddy simulation, Journal of Wind Engineering and Industrial Aerodynamics 91 (2003) 331–353. [9] Y. Jiang, Q. Chen, Effect of fluctuating wind direction on cross natural ventilation in buildings from large eddy simulation, Building and Environment 37 (2002) 379–386. [10] Y. Jiang, Q. Chen, Study of natural ventilation in buildings by large eddy simulation, Journal of Wind Engineering and Industrial Aerodynamics 89 (2001) 1155–1178. [11] S. Murakami, Overview of turbulence models applied in CWE-1997, Journal of Wind Engineering and Industrial Aerodynamics 74–76 (1998) 1–24.
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