RANS simulation of the atmospheric boundary layer over complex terrain with a consistent k-epsilon model

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1 RANS simulation of the atmospheric boundary layer over complex terrain with a consistent k-epsilon model C. Peralta 1), A. Parente 2), M. Balogh 3), and C. Benocci 4) 1) Fraunhofer IWES, Ammerländer Heerstr. 136, Oldenburg, Germany 2) Université Libre de Bruxelles, Brussels, Belgium 3) Department of Fluid Mechanics, Budapest University of Technology and Economics, Budapest, Hungary 4) Von Karman Institute for Fluid Dynamics, Rhode-St-Genèse, Belgium *) presenting author, carlos.peralta@iwes.fraunhofer.de ABSTRACT The most common approach to the computational fluid dynamics (CFD) simulation of the atmospheric boundary layer (ABL) in complex terrain assumes the flow to be incompressible, steady-state and turbulent. Therefore, the problem is set in terms of the Reynold Averaged Navier Stokes equations (RANS), where the turbulent effects are usually modelled by a two-equation (k ɛ) turbulence model. However, it is well known that models of this class present a major drawback when applied to the simulation of the homogeneous neutrally stratified ABL, namely an undesired and unphysical decay of the velocity and turbulent fully-developed profiles specified at the inlet of the computational domain (Blocken et al., 2007). Recently, Parente et al. (2011a) have proposed a modification of the standard k ɛ turbulence model and boundary conditions formulation that achieves an overall consistent treatment of the neutral ABL. This model has been successfully implemented in the open source CFD software OpenFOAM, and tested on the simulation of the flow over complex terrains and hills, at wind tunnel and atmosphere scale (Balogh et al., 2013). In this paper, we test this model on more challenging test cases, namely, in the recent benchmark case of Bolund (Bechmann et al., 2011) and other complex terrain cases. Firm conclusions will be drawn on the capabilities of the present approach and the accuracy and reliability of two-equation turbulence models for complex topography. 1 INTRODUCTION The prediction of wind flow in complex terrain is based on theoretical and experimental techniques. On the experimental side, the first step in order to assess the wind potential for a site is usually based on long- term on-site measurements. It is not easy to determine how long a measuring campaing should last in order to get reliable results, but usually it varies from one to three years. Occasionally, a scaled wind tunnel version of the domain is used, which can be useful in determining the best locations for met mast installations in the real site (Buckingham, 2010). Theoretical techniques are based on empirical, analytical or numerical methods. The most commonly used numerical wind engineering tools use linear solvers, which makes them limited to relatively flat terrains and soft hills. This leads to inaccurate prediction of recirculation and separated flows in more complex terrains. The vast majority of the currently available numerical solvers for the wind flow in complex terrain are based on RANS equations combined with two-equation turbulence models. However, most models lack a consistent formulation of the law of the wall in rough surfaces when used in combination with the Richardson and Hoxey (Richards and Hoxey., 1993) (RH93) inlet conditions for a neutral ABL. The model recently proposed by Parente et al. (2011a) (PGVB11) rederives the k ɛ model in a consistent way and ensures a proper combination of inlet profiles and wall functions based on aerodynamic roughness. This approach has been implemented in open source CFD library OpenFOAM and

2 validated for the Askervein hill benchmark case. A comparison with the commercial solver Fluent was also performed by Balogh et al. (2013). In this investigation, we test the consistent approach of PGVB11 in two more challenging complex-terrain benchmark cases: the Bolund hill (Berg et al., 2011) and the Alaiz mock-up wind tunnel test case (Cabezón et al., 2007). In the first case, we compare our results with the experimental data for the main (270 ) wind direction, as well as previous simulations done with the standard k ɛ model available in OpenFOAM (Peralta et al., 2014). For the Alaiz data set we compare our results with the data available from the wind tunnel experiments The paper is organized as follows. In section 2 we describe briefly the consistent k ɛ model used in this investigation, as well as the details of its implementation in OpenFOAM. In section 3 we discuss the numerical solution of the RANS equations in complex terrain and present the two case studies considered in this investigation. We discuss the results of the simulations in Section 4 and state some conclusions in section 5. 2 CONSISTENT k ɛ MODEL Using the standard k ɛ model transport equations for turbulent kinetic energy k (TKE) and turbulent dissipation ɛ (Launder and Spalding., 1974), one can derive the RH93 inlet profiles for a homogeneous ABL when assuming constant pressure and shear stress, and the crosswise and normal components of the velocity vanishing. These widely used profiles can be written as u = u κ ln ( z + z0 z 0 ), k = u2 u 3, ε = Cµ κ(z + z o ), (1) where u is the friction velocity, z 0 the surface roughness length and κ the von Karman constant. These expressions automatically satisfy a 1D version of the k ɛ equations for a homogenous neutral ABL if the turbulent Prandtl number of the dissipation rate is given by σ ɛ = κ/ C µ (C ɛ2 C ɛ1 ), where κ is von Karman s constant. Although these profiles satisfy such a 1D model, a constant TKE does not agree well with experiments (Hargreaves and Wright, 2007). Modifications to the k and ɛ inlet profiles of RH93 have been proposed by Yang et al. (2009), with k varying with height and new fitting constants determined from experiments. Additionally, Gorlé et al (Gorlé et al., 2010b) introduced a modification to C µ and σ ɛ which allow an approximate solution to the 1D system of equations using the TKE profile suggested by Yang et al. Later on Parente et al. (2010) introduced two new source terms in the transport equations for k and ɛ in oder to guarantee their exact solution. These terms have the form S k = u [ κ (z + z 0 ) k ] u 4 (C ɛ2 C ɛ1 ) C µ, S ɛ = σ k z z (z + z 0 ) 2 1 κ 2 σ. (2) ɛ The constant C µ is redefined as a function depending on the distance to the wall z, C µ = u 4 /k 3 (z), where a new, hight dependent expression for k is found by assuming local equilibrium between turbulence production and dissipation k(z) = A ln(z + z 0 ) + B, (3) where A and B are constants determined by fitting the above equation to a measured profile of k. Recently Balogh and Parente (2014) developed a 4-parameter version of the expression above, with a modified S k source term.

3 2.1 Wall function treatment The consistent approach described above also includes a modification of the common wall functions used in the literature, which suffer from an inconsistency with the RH93 inlet profiles (see Blocken et al. (2007) for a discussion on typical remedial measures). In the approach used in this paper, the production of turbulent kinetic energy at the wall is computed at a location indicated by the first cell center z p, displaced by the aerodynamic roughness z 0. The resulting wall function constants depend directly on the roughness length, while preserving the same form of the universal law of the wall (Balogh et al., 2013). Velocity and turbulent dissipation rate are fixed at z p, following RH93, but unlike RH93 the production of turbulent kinetic energy at the wall is not integrated over the first cell height but computed at z p + z 0. This avoids the typical peak in turbulent kinetic energy observed by many authors (Hargreaves and Wright, 2007). Additionally, the formulation offers the possibility of selecting a smooth or rough wall according to the roughness characteristics of the surface (Balogh et al., 2013; Parente et al., 2011a). 2.2 Building influence area (BIA) filter The approach described in the previous sections was successfully tested in the simulation of unperturbed ABLs but turned out to be inadequate for the simulation of separated regions (Parente et al., 2011b). This problem was solved using the Building Influence Area (BIA) filter described in Gorlé et al. (2010a). This allows a gradual transition or blending of the turbulence model parameters from their values in an unperturbed ABL to values adequate for regions disturbed by obstacles. The blending of the standard k ɛ and the new formulation is achieved by changing the C µ, S k and S ɛ terms using a function of power N (Balogh, 2014) C bld µ = C std µ + (C hom µ C std µ ) f N b (U err), (4) S bld k = S k f N b (U err), S bld ɛ = S ɛ f N b (U err), (5) where the superscripts std, hom and bld denote the standard, homogeneous and blended forms of the corresponding terms. The blending function f b depends on the difference between the local and homogeneous values of the velocity, calculated using U err = U U hom / U hom, where the blending function is defined as f b (U err ) = [ ] (min 2 sin max(uerr U tr, 0), 1 π π ). (6) 1 U tr 2 Small deviations from the velocity homogeneous profiles are present near the ground, which originate from the unrealistic profiles of k. For this reason, a threshold U err = 0.05 is chosen. 2.3 OpenFOAM implementation The consistent approach described in the previous sections was implemented in OpenFOAM The following modifications of the k ɛ model are implemented as runtime selectable options: Standard (STD) form of the production term for turbulent kinetic energy: P std k = ν t S 2, where S is the modulus of the rate of strain tensor. Launder and Kato (1993) modification (KL) to the production term for turbulent kinetic energy: Pk KL = ν t S Ω, where Ω is the vorticity. This modification was intended to reduce the tendency of the STD model to over-predict the turbulent production in regions with strong acceleration or deceleration.

4 (a) (b) Figure 1: (a) Computational grid for the simulation of Bolund hill, showing the ground, and one side and the outlet patch. The direction of the flow (270 ) is indicated by the measurement line B (Berg et al., 2011). (b) Computational grid used for the simulations of the Alaiz wind tunnel, showing refinement regions around the hilly terrain, as well as one of the sides and the outlet patch. The measurement positions R1, R2 and P1-P7 are also shown (Buckingham, 2010). Yap (1987) correction for separated flows, aimed at correcting TKE prediction in separated regions through an additive source term in the equation. This modification is usually used together with the KL modification (referred to as KLY), as suggested by Launder (Balogh et al., 2013). 3 CASE STUDIES IN COMPLEX TERRAIN We study two complex terrain benchmark cases, one in real terrain (Bolund) and a wind tunnel mockup version of the Alaiz hill. We solve the steady state RANS equations for the wind field, using OpenFOAM s simplefoam RANS solver, with the consistent k model described in section 2 for calculating the turbulent viscosity. In all the simulations presented in this paper, a structured grid is generated with the new library terrainblockmesher (Schmidt et al., 2012), based on OpenFOAM s blockmesh native mesher. The mesher was used in a previous investigation of the Askervein and Bolund data sets using OpenFOAM s standard k model (Peralta et al., 2014). We typically used 2-5 million cells meshes in this investigation. We tested three different combinations of the k models: the comprehensive approach, including all the modifications to the k PGVB11 model (C) as described in section 2.3, with blending filter (CB) and with KLY modification and blending filter (CBKLY). An exponent N=3 was used In all simulations with the sinusoidal blending function. Due to lack of space, we present only the combinations that produced results closest to the experimental data. 3.1 Simulations using the Bolund hill dataset The Bolund hill, a classical bechmark test case for CFD solvers for complex terrain, is located north of DTU Wind Energy (Denmark). It is a steep escarped island of about 130 m lenth, 12 m height and 75 m width (Berg et al., 2011). The island and surrounding sea are characterized by roughness lengths z0 = m and z0 = m, respectively. A measuring campaign was performed in 2007 and 2009.

5 (a) (b) Figure 2: Speedup (a) and TKE (b) versus position along line B at 2 m height. The black squares are the experimental data. The blue curve corresponds to the STD k ɛ model and the red curve to the comprehensive model of Parente with blending (CB). Case Speedup HR (%) T KE/Ure 2 f FB Bolund (STD k ɛ ) Bolund (CB k ɛ ) Alaiz (STD k ɛ ) Alaiz (CBKLY k ɛ ) Alaiz (C4 k ɛ ) Table 1: Hit rates (HR) and fractional bias (FB) of the speedup and TKE for the Bolund and Alaiz test cases. We performed steady state numerical simulations for the 270 wind direction. The computational domain is shown in Fig. 1a. The peak height of the hill is 12 m. The dimensions of the domain are m 3 in the streamwise, spanwise and vertical directions respectively. We use the expressions (1)as inlet profiles, with T KE/u 2 = 5.8 and u = 0.4 m s 1 (Bechmann et al., 2011). The outlet boundary is defined as a pressure outlet and the lateral boundaries as zero gradient. The top boundary is defined by fixing the inlet values of wind speed, TKE and ɛ at the top height. We calculate the speedup factor at 2 m height, defined as (U U 0 )/U 0, where U is the horizontal velocity at a height of 2 m above the local terrain along line B and U 0 is a reference velocity at the M0 mast position. Additionally, we calculate the normalized TKE, defined as (T KE T KE re f )/U 2 0, where T KE re f is a reference TKE at the M0 mast position (Peralta et al., 2014). In Fig. 3 we present a comparison between the experimental data (black squares), the simulations performed with the STD k ɛ model presented in Peralta et al. (2014) (blue curve) and a simulation using the comprehensive approach of Parente with blending filter (CB, red curve). 3.2 Numerical simulations using the Alaiz wind tunnel data set The Alaiz test site, located south of Pamplona (Navarra, Spain), is a blind test case developed within the frame of the European WAUDIT project (Cabezón et al., 2007). It is approximately 1050 m in

6 (a) (b) Figure 3: (a)speedup and (b) TKE versus position along the line of met masts R1-P7 at m height for the Alaiz wind tunnel mockup. The black squares are the experimental data. The blue curve corresponds to the STD k ɛ model and the red curve to the comprehensive model of Parente with blending (CB k ɛ ), and the green curve to the comprehensive model of Parente without blending (C k ɛ ). A 2D cut of the hilly terrain is also shown as solid line, indicating the locations of the measuring points R1-P7. height and extends over about 4 km long with a main orientation East-West. A mock-up version of the hill was studied in the wind tunnel of the von Karman institute for Fluid Mechanics (VKI). The mockup model represents 1/5357 scaled version of the real rectangular domain. Hot-wire anemometers and PIV measurements were performed at the reference positions R1 and R2 and measurement points P1-P7 (see Figure 2b). Simulations were done in a 3D domain aligned with the measurement plane (indicated by the line R1-P7 in Fig. 2b), in order to compare with the experimental study described in Croonenborghs (2010). An empty fetch region of 3 m is placed before the terrain, representing a rough wall. The dimensions of the domain are m 3 in the streamwise (indicated by the measuring points), spanwise and vertical directions respectively. Figure 3 shows a comparison between the experimental data (black squares), a simulation performed with the STD k ɛ model (blue curve) and a simulation using the comprehensive approach of Parente with blending and the KLY modification (CBKLY, red curve). Unlike the Bolund case, where the inlet TKE is a constant value, the vertical profile from the Alaiz wind tunnel test case follows a more complex function. For this reason, we additionally tested the 4-parameter TKE developed by Balogh and Parente (2014) using the comprehensive approach (C4, green curve). A 2D cut of the hilly terrain is also shown, indicating the locations of the measuring points R1-P7. The experimental line follows a real scale height of 90 m, corresponding to m in model scale. In this case the normalized TKE is defined as TKE/U 2 re f, where the reference velocity U re f was measured at point R2 at 0.5 m above the model surface (Croonenborghs, 2010). For the normalized TKE only the points R1, P4 and P6 are available from the published data. 4 DISCUSSION We calculated the hit rate for the speedup, defined as the fraction of N measurement locations where the CFD results are within a 25 % interval of the measurement data (Balogh, 2014). For the normalized TKE, given the lack of data for the Alaiz case, we calculated instead the fractional bias FB = 2(Ō

7 P)/(Ō + P), where Ō and P are the means of observed and predicted values. A positive (negative) value indicates an over (under) estimation. Results are presented in table 1. For the Bolund benchmark, the discrepancies in the speedup are similar for both approaches (STD k ɛ and CB). The CB approach gives better results downstream of the hill, with a hit rate of 50 % (cf, 33.3 % for the STD k ɛ ). Both approaches show an overestimation, but the CB approach produced a peak value closest to the experimental data, with a 36 % smaller FB, as shown in Table 1. For the Alaiz wind tunnel benchmark, the CBKLY approach gives a speedup with a slightly higher hit rate (44 %) than the STD k ɛ (33.3 %). We additionally tested the comprehensive approach with a 4-parameter TKE (C4, green curve), which gave a similar result as the CBKLY. The flow is more closely reproduced by the C4 k ɛ approach (green curve) on the leeward side of the hilly terrain between the observation points R2 and P1. The flow in this region is more influenced by the topography and consequently more three-dimensional. The inlet profile will probably have a stronger influence in this region. The C4 approach better predicts the flow in the leeward side of the hill than the CBKLY approach, but it overpredicts the speedup on the windward side along met masts P3-P7. For the normalized TKE, both the CBKLY and C4 approaches show an underestimation, while the STD approach shows an overestimation of the TKE of similar magnitude (see FB values in table 1). 5 CONCLUSIONS We performed additional validations of the consistent k ɛ PGVB11 model in two complex terrain benchmark cases. The implementation was recently done in OpenFOAM, and validated for a 2D empty fetch and Askervein hill (Balogh et al., 2013). The results agree well with the experimental data, in particular for the Bolund data set, for which a marked improvement in TKE prediction was observed. For the Alaiz wind tunnel data, the agreement is less clear. The 3D simulations with the C4 approach show an improvement in the prediction of the speedup in the leeside of the hill with respect to the simulations using the STD approach, while the CBKLY approach gave a better prediction in the windward side. Overall, the hit rates of speedup are higher for the PGVB11 model than with the STD model. For the TKE the improvement is more marginal. The results obtained in this investigation will be used as a guide for performing real scale simulations of the Alaiz hill benchmark case. Acknowledgements We acknowledge the computer time provided by the Facility for Large-scale COmputations in Wind Energy Research (FLOW) at the University of Oldenburg. We thank Prof. J. van Beeck and Ms S. Buckingham, from the von Karman Institute for Fluid Dynamics, for providing experimental data and the cloud of points for generating the Alaiz wind tunnel topography. References M. Balogh. Numerical simulation of atmospheric flows using general purpose CFD solvers. PhD thesis, Budapest University of Technology and Economics, Budapest, Hungary, M. Balogh and A. Parente. In Sixth International Symposium on Computational Wind Engineering (CWE2014), M. Balogh, A. Parente,, and C. Benocci. RANS simulation of ABL flow over complex terrains applying an Enhanced k-e model and wall function formulation: Implementation and comparison for fluent and OpenFOAM. J. Wind Eng. Ind. Aerodyn., 104: , 2013.

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