Atmospheric Environment

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1 Atmospheric Environment 43 (2009) Contents lists available at ScienceDirect Atmospheric Environment journal homepage: CFD modelling of small particle dispersion: The influence of the turbulence kinetic energy in the atmospheric boundary layer C. Gorlé a,b, *, J. van Beeck b, P. Rambaud b, G. Van Tendeloo a a University of Antwerp, Department of Physics, EMAT, Groenenborgerlaan 171, 2020 Antwerp, Belgium b Von Karman Institute for Fluid Dynamics, Waterloosesteenweg 72, 1640 Sint-Genesius-Rode, Belgium article info abstract Article history: Received 2 July 2008 Received in revised form 26 September 2008 Accepted 26 September 2008 Keywords: Computational Fluid Dynamics (CFD) Numerical simulation Atmospheric Boundary Layer (ABL) Turbulence kinetic energy Particle dispersion When considering the modelling of small particle dispersion in the lower part of the Atmospheric Boundary Layer (ABL) using Reynolds Averaged Navier Stokes simulations, the particle paths depend on the velocity profile and on the turbulence kinetic energy, from which the fluctuating velocity components are derived to predict turbulent dispersion. It is therefore important to correctly reproduce the ABL, both for the velocity profile and the turbulence kinetic energy profile. For RANS simulations with the standard k e model, Richards and Hoxey (1993. Appropriate boundary conditions for computational wind engineering models using the k e turbulence model. Journal of Wind Engineering and Industrial Aerodynamics 46 47, ) proposed a set of boundary conditions which result in horizontally homogeneous profiles. The drawback of this method is that it assumes a constant profile of turbulence kinetic energy, which is not always consistent with field or wind tunnel measurements. Therefore, a method was developed which allows the modelling of a horizontally homogeneous turbulence kinetic energy profile that is varying with height. By comparing simulations performed with the proposed method to simulations performed with the boundary conditions described by Richards and Hoxey (1993. Appropriate boundary conditions for computational wind engineering models using the k e turbulence model. Journal of Wind Engineering and Industrial Aerodynamics 46 47, ), the influence of the turbulence kinetic energy on the dispersion of small particles over flat terrain is quantified. Ó 2008 Elsevier Ltd. All rights reserved. 1. Introduction Because of the increasing interest in urban air quality modelling, the application of Computational Fluid Dynamics (CFD) to study atmospheric dispersion processes in the lower part of the Atmospheric Boundary Layer (ABL) has become an important research subject. Validation is an essential aspect of this research and several comparative studies between CFD and wind tunnel or field measurements have been performed, e.g. Leitl et al. (1997), Meroney et al. (1999), Garcia Sagrado et al. (2002), Chang and Meroney (2003), Dixon et al. (2006), and Wang and McNamara (2007). In all of these publications the intermittent nature of the dispersion process in the wind tunnel and field measurements as opposed to the Reynolds Averaged Navier Stokes (RANS) solution of the CFD simulations is indicated as a reason for the observed discrepancies. These discrepancies could be reduced by either averaging measured values over a sufficiently long period or by performing Large Eddy * Corresponding author. University of Antwerp, Department of Physics, EMAT, Groenenborgerlaan 171, 2020 Antwerp, Belgium. Tel.: þ ; fax: þ address: gorle@vki.ac.be (C. Gorlé). Simulations (LES), which resolve the larger scales of turbulence Schatzmann and Leitl (2002). Other significant sources of error reported are inaccuracies in the boundary conditions for the flow or the pollutant source and the underestimation of the turbulence kinetic energy (Garcia Sagrado et al., 2002; Dixon et al., 2006). It is obvious that a correct reproduction of the ABL from the wind tunnel or field experiments is essential to obtain accurate simulation results for atmospheric dispersion. This is usually achieved by imposing fully developed inlet profiles for the velocity and turbulence characteristics that represent the influence of upstream roughness elements, which are not included in the domain. These inlet profiles should be identical to the measurement conditions and should also be established at the locations of interest further downstream in the computational domain. However, a frequently reported problem is that the inlet profiles are not horizontally homogeneous throughout the domain (Franke et al., 2004, 2007; Blocken et al., 2007a,b; Riddle et al., 2004; Hargreaves and Wright, 2007). The reason for horizontal variations in the profiles is an incompatibility of the inlet profiles with the applied wall functions. As a result the profiles will quickly adapt to the implemented wall functions, leading to unacceptable changes further downstream in the domain. It has been shown that both the velocity and /$ see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi: /j.atmosenv

2 674 C. Gorlé et al. / Atmospheric Environment 43 (2009) turbulence profiles of the approach flow affect the results for the downstream velocity, turbulence and concentration (Castro and Robins, 1977; Saathoff et al., 1998; Gao and Chow, 2005). Consequently, horizontal variations in the inlet ABL profiles will influence the downstream prediction of these quantities. For RANS simulations with the standard k e model several solutions have been proposed which result in a horizontally homogeneous velocity profile. The most frequently used method is based on the boundary conditions proposed by Richards and Hoxey (1993). They suggested profiles for velocity, turbulence kinetic energy and turbulence dissipation rate in combination with a logarithmic wall function. When using their inlet profiles without the proposed wall function, one will not obtain a horizontally homogeneous velocity profile, so either the correct wall function has to be implemented in the code (Hargreaves and Wright, 2007), or the constants in the existing rough wall functions have to be adapted to fit with the inlet conditions (Blocken et al., 2007b). A drawback of these methods is that the proposed inlet boundary condition for the turbulence kinetic energy is a constant value over the full height of the lower ABL, which might not correspond to the profiles measured in the atmosphere or generated in a wind tunnel. Although the influence hereof on the velocity field is probably limited, the influence on the concentration field could be more important, since turbulence will enhance dispersion. The goal of this study is to propose a method that allows the correct reproduction of the ABL profiles of velocity and turbulence kinetic energy as measured in wind tunnel tests, using RANS simulations with the standard k e turbulence model. The simulations will be performed using Fluent The influence of the proposed correction on the dispersion of tracer particles in the ABL over flat terrain will be determined. 2. Characteristics of modelled ABL The ABL considered in this study was taken from the CEDVAL database. The profile was generated in the BLASIUS wind tunnel at the Meteorological Institute of the University of Hamburg (Donat, 1995). Using modified Standen Spires (Standen, 1972) and a uniform LEGO-roughness an ABL was modelled at a scale of 1:200. Profiles of mean velocity and turbulence were obtained from two-component LDV measurements in one horizontal and one vertical plane. They are reported in terms of three mean velocity components U (longitudinal), V (lateral), W (vertical) [m s 1 ], three turbulence intensities TI u,ti v,ti w [%] and two Reynolds stresses Cu 0 v 0 D and Cu 0 w 0 D [m 2 s 2 ]. All profiles are shown in Fig. 1. For simulations with the k e turbulence model, the boundary conditions for the turbulence should preferably be described in Fig. 2. k profile calculated for ABL from CEDVAL database. terms of the turbulence kinetic energy k and the turbulence dissipation rate e. The reported turbulence intensity profiles should thus be converted in profiles for k and e. In the present study only the u- component of the measured turbulence, Cu 0 D ¼ TI u U=100, is used for determination of the k-profile: k ¼ 3 2 hu0 i 2 (1) The resulting profile for k is shown in Fig. 2. Using Eq. (1), a rather high value of turbulence kinetic energy is achieved, which will clearly demonstrate the problem related to the horizontal homogeneity of the profile. As the k e model is an isotropic turbulence model, the resulting velocity fluctuations extracted by the model for the determination of the turbulent dispersion will be correct for the u-component, but will be overpredicted for the v- and w-components. The profile for e will be determined according to Eq. (8). 3. ABL modelling using RANS with the standard k e turbulence model The simulations presented in this section were performed on a 2D domain of 4 m length and 1 m height (model scale), with a grid of cells. A gradient was applied in the vertical direction to have the centre point of the wall adjacent cell at a height of m. Richards and Hoxey (1993) outline the following conditions for modelling the ABL in steady incompressible two-dimensional flow using the standard k e model: Fig. 1. ABL profiles from CEDVAL database.

3 C. Gorlé et al. / Atmospheric Environment 43 (2009) The vertical velocity is zero. The pressure is constant. The shear stress, s 0 is constant: ðm l þ m t Þ du dz ¼ s 0 ¼ ru 2 * (2) where m l and m t are the laminar and turbulent viscosities, respectively, r is the density and u * is the boundary layer friction velocity. m t is related to k and e using r and a constant C m as follows: m t ¼ rc m k 2 e The turbulence kinetic energy and dissipation rate satisfy their conservation equations which are given by: v mt vk vu 2 þ m vz s k vz t re ¼ 0 (4) vz v mt ve vu 2 e þ C vz s e vz 1 m t vz k C 2r e2 k ¼ 0 (5) where s k is the turbulence kinetic energy Prandtl number, s e the turbulence dissipation Prandtl number and C 1 and C 2 are constants. If the profiles defined for U, k and e satisfy Eqs. (2 5), a horizontally homogeneous ABL should be obtained. In the following sections different combinations of inlet profiles and their effect on the horizontal homogeneity of the ABL are discussed Constant inlet value for k The approach described by Richards and Hoxey (1993) consists of defining the following inlet boundary conditions for U: UðzÞ ¼ u * z þ z0 ln k z 0 where z 0 is the ABL roughness height and k the von Karman constant. A good log law fit for the velocity profile shown in Fig. 1 is found for a model scale roughness of m and a friction velocity of m s 1 for k ¼ (3) (6) For k and e the inlet conditions should be given by: kðzþ ¼ eðzþ ¼ p u2* ffiffiffiffiffi (7) C m u 3 * kðz þ z 0 Þ These inlet conditions imply equilibrium of turbulence production and dissipation: qffiffiffiffiffi eðzþ ¼ C m kðzþ du dz At the ground boundary, the retarding shear stress s g ¼ ru 2 *g should be determined with the local friction velocity equal to: u * g ¼ (8) (9) ku g ln z g þ z 0 =z0 (10) Combining these boundary conditions with the following relation for the constants of the turbulence model provides a solution to the standard k e model: qffiffiffiffiffi k 2 ¼ðC 2 C 1 Þs e C m (11) When using a commercial CFD code, however, the wall boundary condition is usually not as prescribed above. Blocken et al. (2007b) provide a solution for this by deriving a relationship which brings the rough wall functions in equilibrium with the inlet profiles. For Fluent this relation is given by: k s;abl ¼ 9:793z 0 C s (12) where k s is the roughness height as defined in Fluent for the rough wall function and C s is a constant required for the wall function. k s should be smaller than the height of the centre point of the wall adjacent cell and was consequently set to m. The resulting value for C s is The Fluent Manual states that C s should be smaller than or equal to 1. This requirement, however, results in a first cell height of minimum z 0, which is too large to obtain a good resolution. Therefore, the requirement of limiting C s to 1 is ignored and a higher value is defined through a User Defined Function. The inlet conditions for the CFD simulations for k and e as described by Eqs. (7) and (8) are plotted in Fig. 3. For comparison the value of k derived from the wind tunnel measurements is also Fig. 3. Comparison of inlet profiles from wind tunnel measurements and those from Richards and Hoxey (1993) and the profiles at the domain outlet.

4 676 C. Gorlé et al. / Atmospheric Environment 43 (2009) shown. The plot shows that the turbulence kinetic energy is underpredicted when using Eq. (7) with the standard value for C m of The profiles obtained for U, k and e at the outlet are almost identical to the inlet conditions. Identical results were obtained, with a grid refined by 2 in all directions. This refinement does require to decrease k s and increase C s by a factor of 2. These results show that the approach described by Blocken et al. (2007b) results in homogeneous profiles throughout the domain. The inlet condition for k as described by Eq. (7) does however underestimate the values derived from the wind tunnel tests Measured inlet profile for k In Franke et al. (2007) it is recommended to use measured values for the turbulence quantities, if available, as inlet boundary conditions. Therefore a simulation was performed to verify the homogeneity of the ABL when using the measured values for k, combined with the boundary conditions for the velocity, the turbulence dissipation and the wall as described before. The resulting profiles at the in- and outlet are shown in Fig. 4.It is observed that the new inlet values for k are not maintained through the domain. In addition the higher value of k seems to slightly disturb the horizontal homogeneity of the velocity profile. The development of the inlet profiles through the domain can be explained by the fact that the new profile for k in combination with the given profiles for U and e does not satisfy Eqs. (2) (5). The profiles will then develop to values that do provide a solution, which in this case results in underestimation of the turbulence kinetic energy Correction of profile for e and turbulence model constants The results from Fig. 4 show that, if the inlet profiles should be horizontally homogeneous through the domain, profiles for k and e have to be found that result in an approximate solution for the turbulence model when combined with the logarithmic profile for U. If equilibrium between turbulence dissipation and production is imposed, the profile for e can be determined from Eq. (9). Yang et al. (2007) already derived that substituting this relation for e in the conservation equation for k, results in a solution for k of the following form: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kðzþ ¼ Alnðz þ z 0 ÞþB (13) where A and B are constants that can be determined by fitting the equations to the measured profile of k. For the profile under consideration A ¼ 0.11 and B ¼ According to Eq. (9), e will then be given by: pffiffiffiffiffi C mu* qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eðzþ ¼ Alnðz þ z kðz þ z 0 Þ 0 ÞþB (14) Using these relations, the conservation equation for k is satisfied, but Eqs. (2) and (5) are not. Therefore, simply imposing Eqs. (13) and (14) as inlet boundary condition will not provide a correct solution and will most likely still result in a development of the ABL profiles inside the domain. The first remaining equation to be satisfied is Eq. (2). Substituting Eqs. (13) and (14), the relation to be fulfilled is derived as: C m ¼ u 4 * Alnðz þ z 0 ÞþB (15) As it is not possible to impose a value for C m that varies with height, the value for C m in the wall adjacent cell will be used. For the given profile this results in C m ¼ The second remaining equation is the conservation equation for e. Again substituting Eqs. (13) and (14), it can be derived that the following relation should be satisfied for s e : k 2 A=2 þ kðzþ 2 s e ðzþ ¼ u 2 * ðc 2 C 1 ÞkðzÞ½1 Eðz þ z 0 ÞŠ where E is a function given by: E ¼ 1 e B=A p ffiffiffi p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ Erf kðzþ 2 =A q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 kðzþ 2 =A (16) (16a) For the present values of k(z), B and A, an approximate solution is obtained when assuming E ¼ 0, which considerably simplifies the equation. The resulting value for s e also varies with height and can be imposed in Fluent through a User Defined Function. Fig. 5 shows the resulting profiles for k, e and s e with a value of C m equal to The profiles were used for a simulation in combination with the logarithmic profile for U (Eq. (6)). k s and C s were identical to the values used before (Eq. (12)). The profiles obtained for U, k and e at the outlet are shown in Fig. 6. The profile for U is perfectly homogeneous, while the profiles Fig. 4. Comparison of inlet profiles from wind tunnel measurements with the profiles at the domain outlet.

5 C. Gorlé et al. / Atmospheric Environment 43 (2009) for k and e still show a small deviation. This is explained by the fact that the provided inlet profiles are not an exact solution for the constant value of C m. When comparing the results to Fig. 4 it is however obvious that a much better reproduction of the turbulence kinetic energy is obtained. 4. Influence of the turbulence kinetic energy profile on small particle dispersion The influence of the different profiles of turbulence kinetic energy on the dispersion of spherical particles with a 1 mm diameter was investigated by performing simulations for dispersion in an ABL over flat terrain. The same problem has been solved with a simple Gaussian model for comparison with the CFD results Computational domain and boundary conditions The domain represents a rectangular wind tunnel test section with dimensions m as shown in Fig. 7. A circular source with a diameter of 1 cm is located at the ground, 1.2 m behind the domain inlet. Air is entering the domain through this source with a velocity of 1 m s 1 and 20,000 particles are released from random positions within the source at the same velocity. The reference wind speed of the incoming boundary layer is 5.61 m s 1 at 0.4 m height. A symmetric boundary condition is applied at half the width of the domain. The grid resolution is 0.5 mm at the source and increases gradually at locations further away from the source. The total amount of cells is 340,000. The centrepoint of the wall adjacent cell, z 1, is at a height of m, resulting in z þ values just above 30. The value of k s was set to m. The grid dependency of the solution was checked by obtaining a solution with a grid refined by 2 in all directions within the region of interest. The results presented did not show significant changes when performing this refinement. The two sets of boundary conditions used will be referred to as Type 1 (low k) and Type 2 (high k) and are presented in Table 1. The values used for fitting the measured profiles of U and k to the equations are u * ¼ m s 1, z 0 ¼ m, A ¼ 0.11 and B ¼ Particle dispersion modelling The particle dispersion is modelled using a discrete phase model, which determines the particle trajectory by integrating the force balance of the particle written in a Lagrangian reference frame. In the present analysis the only forces considered are the inertia and the drag force, hence the force balance can be written as: Fig. 5. Inlet profiles for k and e and corresponding values for s e. Fig. 6. Comparison of inlet profiles from Eqs. (6, 13, 14) with the profiles at the domain outlet.

6 678 C. Gorlé et al. / Atmospheric Environment 43 (2009) crossing time is defined as t cross ¼ sln½1 ðl e =sju U p jþš where s is the particle relaxation time and L e is the eddy length scale. Concentrations at specified points inside the domain are determined by calculating the time that each particle spends inside a small cell volume surrounding the point of interest. Using the total time spent by all particles in the cell, the non-dimensional concentration C * ¼ C local /C source is given by: C cell ¼ Q P t i;cell NV cell (19) Fig. 7. Sketch of simulation domain. where Q is the source strength (source flow rate), N is the total number of particles tracked through the domain and V cell is the volume of the cell. The resulting values for the concentrations presented in the following sections have been verified to be independent of the choice of N and V cell. du p ¼ 18m dt d 2 p r U Up pc c (17) where d p is the particle diameter, r p the particle density (1550 kg m 3 ) and C c is the Cunningham correction factor, which is equal to 1.17 for particles of 1 mm in air at standard conditions. The turbulent dispersion is accounted for by integrating the trajectory equations for individual particles using the instantaneous fluid velocity U þ u 0 (t) along the particle path. The instantaneous fluid velocity is determined using a stochastic method, i.e. the random walk model (Fluent Inc., 2006). In this method the fluctuating velocity components are discrete piecewise constant functions of time given by: qffiffiffiffiffiffiffiffiffiffi u 0 ¼ z 2k=3 (18) z is a normally distributed random number between 1 and 1, kept constant over an interval of time which indicates the time that a particle is assumed to interact with an eddy. This value is given by the smaller value of the characteristic lifetime of the eddies and the particle eddy crossing time. The characteristic life time is given by s e ¼ T L log(r) where r is a random number between 0 and 1 and T L is the integral time scale given by T L zcðk=eþ. C is a constant which was determined from C ¼ 4=3C 0 with C 0 ¼ C 0N = ð1 þ 7:5 C0N 2 ÞRe 1:64 l. In this equation C 0N is approximately equal to 6 (Oesterlé, 2006). The p Taylor-scale Reynolds number was determined from Re l ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð20=3þre L with Re L ¼ k 2 =en (Pope, 2008). The resulting value for T L is 0.28 for boundary conditions Type 1 and 0.23 for boundary conditions Type 2. The particle eddy Table 1 Boundary conditions and model coefficients for Type 1 and Type 2 (u * and z 0 from fitting the velocity profile; A and B from fitting the k profile; k, C 1 and C 2 are constants (0.4187, 1.44 and 1.92, respectively); z 1 is the centre point of the first cell). Inlet profiles u k e Type 1 (low k) Type 2 (high k) u * ln z þ z 0 u * ln z þ z 0 k z 0 k z 0 u 2* pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi Alnðz þ z 0 ÞþB C m u 3 * kðz þ z 0 Þ Turbulence model C m 0.09 Coefficients s e 1.3 Wall function k s z 1 z 1 pffiffiffiffiffi C mu* kðz þ z 0 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Alnðz þ z 0 ÞþB u 4 * Alnðz 1 þ z 0 ÞþB A=2 þ kðzþ 2 ðc 2 C 1 Þ u2 * k 2kðzÞ Coefficients C s 9:793z 0 k s 9:793z 0 k s 4.3. Gaussian dispersion model The Gaussian dispersion model is governed by the following equation when considering the dispersion along the plume centerline (y ¼ 0): " " # " ## Q ðh z Cðx;z;HÞ¼ exp d Þ 2 ðhþz 2ps y s z u 2s 2 þexp d Þ 2 z 2s 2 z (20) where Q is the source strength, u is the velocity and should be a constant value instead of a profile as in the CFD calculations. H is the effective source height and should include the influence of the plume rise due to buoyancy and momentum. The value for H was determined from the results of the CFD simulations as m. The value for u was set to 2 m s 1, which is the free stream velocity at this height. s y and s z are the dispersion coefficients, whose values increase with increasing turbulence. They may be determined using s y ¼ ax 0:093 and s d z ¼ bx c. Three different stability classes are d considered: C (slightly unstable), D (neutral) and E (slightly stable). The values used for the constants a, b and c depend on the stability class and are given by: a C ¼ 0.2, b C ¼ 0.111, c C ¼ 0.911, a D ¼ 0.130, b D ¼ 0.105, c D ¼ 0.827, a E ¼ 0.098, b E ¼ and c E ¼ (Owen Harrop, 2002) Results The contours for flow velocity and turbulence kinetic energy on the symmetry plane obtained from the CFD simulations are shown in Fig. 8 for both types of boundary conditions. The plots show that there is no significant difference in the velocity, while the difference in turbulence kinetic energy is around 40% at the location of interest. The resulting differences in the particle tracks can therefore be attributed to the difference in the fluctuating part of the flow velocity, determined from the turbulence kinetic energy. The influence of this difference in turbulence kinetic energy on the particle tracks is shown in Fig. 9. This plot shows the y- and z- position of the particles when they cross a plane perpendicular to the flow direction at several distances behind the source. A distinctly larger dispersion of the particles is observed when using the boundary conditions and model constants that result in a higher turbulence kinetic energy. In Fig. 10 the resulting non-dimensional concentration values on planes perpendicular to the flow direction at 0.02 and 0.05 m behind the source are plotted. The plots show that a 40% lower turbulence kinetic energy results in a 90% higher value for the maximum value of C *. Fig. 11 shows a plot of the values on the symmetry plane (y ¼ 0.65 m) at 0.05 and 0.1 m behind the source. For comparison

7 C. Gorlé et al. / Atmospheric Environment 43 (2009) Fig. 8. Comparison of velocity and turbulence kinetic energy contours at symmetry plane for boundary conditions Type 1 and Type 2. Fig. 9. Particle positions on three planes downstream of the source. Fig. 10. Non-dimensional concentrations on two planes located downstream of the source, perpendicular to the flow direction.

8 680 C. Gorlé et al. / Atmospheric Environment 43 (2009) Fig. 11. Non-dimensional concentrations on the symmetry plane at two locations downstream of the source. the results from the Gaussian dispersion model are included. The plot shows that the difference between the two CFD simulations is approximately equivalent to the difference between stability classes C (slightly unstable) and E (slightly stable). The CFD simulations show a lower concentration for the high turbulence kinetic energy below m at 0.05 m behind the source and below m at 0.1 m behind the source. Above these heights, the concentration obtained for the higher turbulence is higher due to the increase in the dispersion of the particles. This is the major difference with the results from the Gaussian model, which show a lower concentration for the lower stability classes over the full height. This difference can be explained by considering the formula used for s y and s z. s y depends largely on the stability class, while the variation in s z is limited. Consequently, the dispersion is mainly increasing in the y-direction, leading to lower concentrations over the full height at the centerline, while the distribution over the height is less affected. 5. Discussion The method proposed involves the modification of two constants in the standard k e turbulence model. The value of C m is changed from 0.09 to and the value of s e is changed from a constant value of 1.3 to values varying between 2.2 and 3.0 for the case studied. The significant decrease of C m was already proposed by Richards and Hoxey (1993). They obtained a value of for the modelling of the atmospheric boundary layer at Silsoe. According to Eq. (11), they then increased the turbulent dissipation Prandtl number s e to 3.2. They suggested that a possible cause for the deviation of C m is the significant low frequency contribution to the velocity fluctuations, which contribute little or nothing to the Reynolds stress. The Reynolds stresses are related to the turbulent viscosity by ru 0 w 0 ¼ m t ðdu=dzþ. Substituting Eq. (3) in this relation one obtains u 0 w 0 ¼ C m ðk 2 =eþðdu=dzþ. Consequently, if u 0 w 0 is small compared to k, the value of C m should be decreased. The increase in the value of s e implies that the diffusion of e is decreased. By imposing higher values for s e at the wall the diffusion away from the wall of the value of e defined near the wall is limited. The observation that C m should be decreased has some implications for the modelling of dispersion with the convection diffusion equation. When using this method, the dispersion will be governed by the parameter m t /Sc t, where Sc t is the turbulent Schmidt number. By decreasing C m and imposing higher values for k at the inlet boundary, there will be no net increase in m t. Consequently, when using the proposed boundary conditions and modifications of the constants in the turbulence model, no enhanced dispersion will be observed when using the convection diffusion equation. In several studies (e.g. Di Sabatino et al., 2007; Tominaga and Stathopoulos, 2007; Riddle et al., 2004) it is suggested that for the modelling of dispersion in urban areas the turbulent Schmidt number should be lower than the standard value of 0.7. If one considers that the dispersion will be governed by all spatial and temporal scales of the velocity fluctuations, this observation is consistent with the explanation given by Richards and Hoxey (1993) for the lower value of C m : if the velocity fluctuations have significant low frequency contributions, which do not contribute to the Reynolds stresses and therefore not to the value of m t, the value of Sc t should be reduced to obtain a higher value for the dispersion. The effect of changing the constants on the simulation of flows around building configurations should be further investigated. When performing such validation studies it is suggested to determine the boundary conditions using a value for k based on the three fluctuating velocity components: k ¼ 1/2(u 02 þ v 02 þ w 02 ). This should result in a more physically correct value for k. The resulting velocity fluctuations will then be underpredicted in the u-direction and slightly overpredicted in the v- and w-directions. In an attempt to compensate for the isotropy of the turbulence model one could consider the introduction of constant factors in the calculation of the different components of the velocity fluctuations. These could be determined according to the ratio between the magnitude of the three components of the velocity fluctuations that would be expected in the ABL under consideration. The constant C used in the determination of the Lagrangian time scale T L has an important effect on the resulting values for the concentration. The values for C that were determined with the formulas from Oesterlé (2006) and Pope (2008) are between the range of and 0.6 that is given by Oesterlé (2006). They can be considered an educated guess, but are by no means exact values. The default value used in Fluent is When using this value, there is an increase in the maximum concentration of 12 18% compared to, respectively, boundary conditions Type 2 with T L ¼ 0.23 and boundary conditions Type 1 with T L ¼ This demonstrates the importance of verifying the sensitivity of the results to the value of C, since the constant cannot be determined exactly. 6. Conclusions The study presented aimed at proposing a method to correctly model the turbulence kinetic energy in the atmospheric boundary layer and to determine the influence of the turbulence kinetic energy on the dispersion of small particles. The proposed method is based on the boundary conditions that were defined by Richards and Hoxey (1993) combined with the correction of the rough wall function implemented in the CFD code as described by Blocken et al. (2007b). The drawback of this method is the inlet boundary condition for the turbulence kinetic energy, which has to be constant over the height of the ABL.

9 C. Gorlé et al. / Atmospheric Environment 43 (2009) Therefore, the inlet boundary conditions for the turbulence kinetic energy and the turbulence dissipation rate were allowed to vary with height according to the relations derived by Yang et al. (2007). In addition, it was derived that the constants C m and s e, used by the turbulence model, should be determined according to Eq. (15) (using the value in the wall adjacent cell) and Eqs. (16 and 16a). When using these relations, the proposed boundary conditions approximately satisfy the conditions that should be fulfilled to model a horizontally homogeneous boundary layer, both for the velocity and the turbulence kinetic energy. It was shown that the influence of the turbulence kinetic energy on the dispersion of small particles is significant by considering the dispersion of 20,000 particles from a circular source over flat terrain. A comparison was made between a simulation using the correction of the rough wall function to obtain a horizontally homogeneous velocity profile, combined with the constant value for the turbulence kinetic energy and the standard values for the constants in the turbulence model and a simulation that also included the correction of the rough wall function, but combined with a profile for the turbulence kinetic energy varying with height and the modified constants for the turbulence model. This comparison showed that a 40% lower turbulence kinetic energy results in a 90% higher prediction of the maximum concentration close to the source. This difference is comparable to the difference that is obtained between the results obtained from a Gaussian dispersion model for stability classes C and E. Acknowledgements This work was supported by IWT Vlaanderen, the Institute for the Promotion of Innovation by Science and Technology in Flanders, through the SBO project NanoSoc. 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