## Abstract

Implementing wind farms in heights of a hilly terrain where wind speed is expected to be large may be viewed as a means to increase wind energy production without occupying fertile lands. Micro sitting of a wind farm in these conditions can gain dramatically from CFD simulation of fluid flow in the ABL above complex topography. However, this issue still poses tough challenges regarding the turbulence model to be used and the way to operate the near wall treatment in the presence eventually of separation. In this work, prediction capacity of RANS turbulence models was studied for a typical hill under the assumption of steady state and incompressible airflow regime in neutral ABL. Two models were analyzed by using *COMSOL Multiphysics* software packages. These included standard

## 1 Introduction

Many wind farms are erected in areas where the site has complex topography. This enables to avoid misleading fertile agricultural lands and to keep windmills away from transport infrastructures. Moreover, there is the advantage of beneficiating from elevated wind speed which permits maximizing wind energy production [1]. However, in comparison with flat ground, airflow characteristics in hilly terrain are more uneven and wind potential is more delicate to estimate [2].

The flow of air occurring in the ABL overlaying a complex topography has been the subject of active research in many areas. Among the subjects that were investigated one finds identifying atmospheric pollution areas, forecasting smoke dispersion in fire events, evaluating wind energy and improving the installation of wind energy structures[3].

In addition to the acceleration experienced by air which increases wind speed in the upstream zone around the summit of hills, the adverse effects resulting from the turbulence caused by terrain and flow separation that takes place in the downstream zone are important features that should be considered in assessing effective wind energy [4].

It was recognized that wind flow pattern in the ABL is widely affected by the presence of upstream obstacles. It depends, among other things, on the mean roughness of the site and curvature of the obstacle [5]. To evaluate the amount of wind energy that is liable to be extracted by a wind turbine, for a given implementation site, the distribution of wind speed over the height of the wind turbine rotor has to be evaluated with adequate accuracy. As in this case the boundary layer turbulence characteristics in terms of skin-friction and shear stress are more complex, advanced Computational Fluid Dynamics (CFD) is required [6].

Carrying out CFD simulation of the ABL has been the subject of many studies: complex terrain [7, 8], pollutant dispersion [9, 10], wind load on turbines buildings [11] and urban environment [12]. In particular many studies have been examined by several research groups for studying the turbulent airflow around hill (i.e., [13, 14, 23, 24, 15–22]). Most of these works have emphasized the severe limitation resulting from the size of the problem yielded by the necessity to discretize the unbounded ABL domain, which increases considerably the demand on computing capacity.

Various turbulence models are used for predicting airflow in the ABL. Most of them are derived based on the Reynolds Averaged Navier-Stokes equations (RANS). This approach assumes the decomposition of an instantaneous quantity into its time-averaged and fluctuating parts. The problem is then governed by a simplified system of mean flow equations, which enables to avoid the difficulties resulting from the Direct Numerical Simulation (DNS) of explicit full time-dependent flow field. The main motivation of this approach is related to the fact that, in most CFD applications, knowing how turbulence affects the mean flow is enough and no real need exists to resolve all the details of the turbulent fluctuations.

Other flow prediction methods include Large Eddy Simulation (LES). This approach is based on time dependent calculations performed on space-filtered equations. Larger eddies are then explicitly calculated. The effect of small eddies on the flow pattern is considered through a refined “subgrid model”. LES approach leads to enhanced representation of turbulence and constitutes an intermediate alternative between RANS based equations and the DNS. However computational cost associated to LES is still very high [25–27].

The RANS based approach leads to the apparition of a nonlinear Reynolds stress term, which requires extra steps in the modeling in order to close the mean flow equations. The closure of the problem was first performed by Boussinesq, who introduced the concept of eddy viscosity. The additional turbulence stresses are then treated by increasing the molecular viscosity with an eddy viscosity. Among the classical closure models following the RANS approach that have been extensively used one finds: standard

Considering hilly terrain, the above mentioned closure models have been used to predict wind flow in the ABL around hills. Due to its robustness, low computational cost, and reasonably accurate predictions, the standard

Recently, Uchida [37] presented the analysis of complex turbulent flow for the case of a 3D hill with high slopes. By conducting a comparison between LES and RANS models, he has shown that the LES, unsteady

Certainly, the capture of the recirculation zone on downstream of the hill following the large separated flow and turbulence production are difficult to simulate numerically within the context of RANS closure models. Some researchers have introduced the idea of adjusting the closure constants of turbulence models in terms of the characteristics of the boundary layer region in neutral atmospheric conditions: Panofsky et al. [38] for

Many software packages are now available to users for CFD fluid flows analysis. They mostly have the benefit of being well accurate and quite handy by providing a genuine user interface [44]. However, monitoring adequately simulations for an original problem is still being far out of reach in many circumstances where the results are sensitive to actual mesh and other convergence criteria, in addition to the model capability. In this study, COMSOL Multiphysics, which is based on the specific Finite Element Method (FEM) [45], is used. Fundamentals settings associated with the FEM are first to be fixed by examining the model convergence in terms of the grid independence test, near wall treatment and the location of the border of the discretized finite domain of calculation.

This paper proposes at first a comparison of the following models standard

The second objective in this article is to carry out a detailed CFD study by comparing it to experimental data of the turbulent airflow over two shapes of hills in terms of velocity profiles during a long of the air domain and to investigate the effect of hills profile and turbulence models on wind speed-up. Therefore, it is significant to notice the impact of a different hill shaped on general flow behavior over complex terrain, which can contribute to finding the optimal turbine placements and to ensuring its reliability and appropriate performance.

This paper presents some new findings that aim at improving simulations of the ABL in complex terrain with hill shape obstacles. A methodology providing wise selection of near wall region treatment as function of the purpose of CFD simulation will be indicated. New closure parameters in the case of

The remainder of the paper is organized as follows: The formulation of the governing equations, constants are given in model formulation and model description are presented briefly in Section 2. The numerical procedure, computational domain, geometric description of the hill, and the appropriate flow parameters are presented in Section 3. The mesh description, the near-wall study of two turbulence closure models and the convergence test are described in Section 4. The results obtained from the numerical simulations and the effect of the topography in the terrain are illustrated in Section 5, the conclusions and discuss the limitations are reported as a final point.

## 2 Governing equations and model description

In RANS (Reynolds averaged Navier-Stokes) simulations in the case of steady incompressible flows, the Navier-Stokes equations in two spatial dimensions yield to solve the nonlinear Reynolds stress term and to close the system of equations [47]. The turbulence models provided in this section use the turbulence viscosity hypothesis concept that was developed by [48]. They correspond to the stress-rate-of-strain relation for Newtonian fluid [49].

### 2.1 Standard $k-\u03f5$ model formulation

Jones and Launder (1972) were the first to introduce the standard

Summary of the model closure coefficients used for the standard

Standard | 0.09 | 1.44 | 1.92 | 1.0 | 1.3 | 0.41 |

Neutral ABL | 0.033 | 1.176 | 1.92 | 1.0 | 1.3 | 0.42 |

### 2.2 Shear stress transport $k-\omega $ model formulation

Menter [55], described the “shear-stress transport”

Summary of the model closure coefficients used for the SST *k-w* turbulence model

Standard | 0.556 | 0.075 | 1.176 | 2 | 0.44 | 0.0828 | 1 | 1.168 | 0.09 | 0.31 |

Neutral ABL | 0.413 | 0.0333 | 1.176 | 2 | 0.20 | 0.0368 | 1 | 1.168 | 0.028 | 0.31 |

Variables and Sources terms for the turbulence models

Constants | ||||

Reynolds filtered variables for (a)-(b) | ||||

(a)Standard | ||||

(b) |

## 3 Numerical setup

### 3.1 Solver details

The FEM has been used to discretize in space the governing equations presented previously. It is obvious that Galerkin formulation can lack stability for the Navier-Stokes equations. For the stationary solver, COMSOL uses Newton's method (Newton-Raphson) to solve the non-linear Navier-Stokes equations [45]. For the simulations of this work, a Direct Linear Solver was used due to them being more robust than iterative solvers. This was also the default choice made by COMSOL for most simulations. The direct solver makes use of Gaussian elimination, or LU factorization, to solve the linearized matrix system. For the simulations with the turbulence model, the variables of turbulence were solved separately from the velocity and pressure by using a segregated solver, and the default solvers chosen by COMSOL were always used. The “PARDISO/Parallel Direct Sparse Solver Interface’’ solver [57] was chosen for all our simulations, for solving large linear systems of equations on shared memory multiprocessors.

### 3.2 Computational air domain

In the piece of work, the hill is appointed as Hill3 according to which a/H ratio and its corresponding maximum hill slope is 26°. The extended distance of the computational domain is

### 3.3 Boundary conditions

The set of the fluid dynamics and turbulence governing equations mentioned in section 2 gives a full model for the description of turbulent flows. Therefore, it can be difficult to solve these equations due to the existence of the non-linear convective term presented in equation (1), which makes an extensive variety of length and time scales. Additionally, the hypotheses employed to derive the high Reynolds number model are not valid near walls in which the tangential wind speed disappears. A Suitable description of the boundary conditions is required. The different boundary conditions were fixed as velocity inlet and pressure outlet for the two side surfaces of the computational field, while the other surfaces (wall, hill, and top) have been identified as no-slip conditions.

In the context of ABL numerical simulations, completely developed inlet profiles for velocity and turbulent variables are usually imposed on the inlet of the computational air domain. The average velocity profile, turbulent kinetic energy, and dissipation rate under neutral ABL conditions proposed by [59] are used in this work and it has been mentioned in Table 4.

Summary of boundary conditions

B.C /Turbulence models | Inlet | Outlet | Wall | Top |

No-slip (Wall- Function) | Open boundary | |||

No-slip (enhanced wall treatment) | Open Boundary |

The wall function and enhanced wall treatment have been applied to specify the ground boundary conditions for the standard

## 4 Mesh description

### 4.1 Mesh generation and analysis

The mesh for the two-dimensional model has been created using the definition of the triangle elements size for the completely computational domain and by determining the mesh distribution (quadratic element) close to the wall (Fig. 4) by fixing the number of layers in the boundary layer properties.

The mesh characteristics for the two turbulence models studied in this work, based on the thickness of the first layer *h* for many types of meshes are presented in Table 5. The successive ratio employed for all the meshes is 1.2. This makes it possible to analyze the behavior of the different turbulence models and treatments approximately the walls according to the different precision regions of resolution as determined by the wall lift-off

Computational performance of the various turbulence models

Turbulence models | Type of the wall- treatment | Type of Mesh | h | Number of elements | Number of degrees of freedom | Solution time (s) | Distance from the wall |

High Reynolds | M1 Coarse | 0.0577H | 4,485 | 49,819 | 362 | ||

M2 Normal | 0.0425H | 7,364 | 78,265 | 506 | |||

M3 Fine | 0.0303H | 12,238 | 122,989 | 770 | |||

M4Extra Fine | 0.0212H | 18,487 | 179,906 | 1,292 | |||

Low Reynolds | M1 Coarse | 3.035e-3H | 4,485 | 49,317 | 1,021 | ||

M2 Normal | 1.821e-3H | 7,364 | 77,591 | 1,463 | |||

M3 Fine | 7.286e-4H | 12,238 | 122,179 | 2,171 | |||

M4 Extra Fine | 6.071e-4H | 18,487 | 178,936 | 2,964 |

The time of the numerical simulation is toughly influenced by mesh characteristics to calculate the most significant variables. However, the mesh must be refined to produce reliable results that are independent of the grid size, but at the same time, an unnecessary fine grid will greatly increase the simulation time, which is one of the most important tasks for engineering problems to make time limitations and to achieve the fast and reliable solution. However, reducing the computational costs is an essential task. The parameters indicating computational performance are presented in Table 5 below as solution time until convergence.

The application of the wall law at the first mesh cell of the computational grid yields to use a specific strategy of the dimensionless wall-lift off distance

In COMSOL Multiphysics code, the wall law is gotten by the namely scalable wall function, whose

The surface resolution should be approximately 11.06 on the walls surrounding the fluid. If the value is higher, it means that the mesh is relatively coarse and that the precision can be compromised.

Another technique to flow simulation is to control the dimensionless center named

Therefore, fixing the distance from the wall in the computational domain, the refinement is relatively natural. However, importantly, the distance used to calculate the wall law remains constant, which yields to decreasing a numerical error in FEM.

Four different mesh sizes are indicated in Table 5 and schematized in Fig. 2 below to investigate the concept of sensitivity of dimensionless distance from the wall in flow domain as a mean if identifying the suitable near wall treatment. For these types of simulation, it is recommended to check the wall lift-off in viscous unit

### 4.2 Convergence criteria and grid independence study

When the successive results of the CFD simulation do not change significantly by adding further iterations, the solution converges. All the simulations presented in this part have converged on a solution whose relative error is less than 10^{−4}. On the other hand, it should be noted that a variation in the damping factor for each variable of speed and turbulence has been achieved until reaching convergence.

A grid independence study was investigated to confirm that the results should be independent during the additional refinements. This piece of work was based on simulations executed the average velocity takes as the convergence criteria.

Figure 3 showed the comparison of the numerical computation of the mean velocity for standard

At the second level of the mesh (7,364 elements), it shows no further changes for the average velocity obtained by using the

Finally, these numerical computation lead to the conclusion that it is more adequate to ensure results on the two turbulence models by choosing the mesh with 12,238 elements, independently of the mesh size.

### 4.3 Mesh scheme

A free triangular mesh was used inside the flow stream away from the walls although boundary layer was added close to the walls. Since the velocity distribution changes immediately normal to the wall, near to the boundary, and slightly in the tangential direction of the wall. To achieve sufficiently small wall lift-off, boundary layer meshes are needed. They can be generated automatically in COMSOL. The boundary layers mesh is smaller than the different places in whole domain, which was made to contain quadrilateral elements strongly packed in the direction normal to the wall and thinly in the tangential direction (see Fig. 4). Additionally, the all-sharp edges and contours of flow separation require the mesh refinements. It is therefore a suitable way to use denser meshes, which implied smaller elements just in the most sensitive areas, where unstructured meshes have been used in this work for the large spatial variation of the examined fields.

The Finer Mesh (M3) has been created using the parameters in Table 6. This is the mesh used for all simulations in this work.

Property of the mesh shown in Fig. 6

Configuration | Domain | Values |

Mesh size | Air | Max element size (Block A):0.072m Min element size (Block C): 0.0032m |

Corner Refinement | Min. angle between the limits: 240deg Element size scale factor: 0.35 | |

Boundary layers | Number of layers: 10 Growth rate: 1.2 |

## 5 Flow analysis over a steep shaped hill 3

### 5.1 Effect of turbulence models on wind speed profiles: Validation with wind tunnel data

The simulation results of the two turbulence models for airflow over a steep shaped hill (H3) were performed to their comparison with the wind tunnel data from [46]. The streamwise at vertical plane y = 0m by two turbulence models including standard

Therefore, this figure presents the comparison of expected results for the vertical profiles of the horizontal velocity component at x/H = −3 (upwind), x/H = 0 (summit), x/H = 3 (downwind) of the hill with the experiment RUSHIL data. As indicated from the profiles distributions, the correspondence between the measured data forecasted by the standard

Additionally, it is noteworthy that the x-velocity profiles for two turbulence models at x/H = −3 near the inlet boundary agree quite well to those measured in the wind tunnel testing as illustrated in Fig. 5 (a). However, the predicted velocity with these models tends to give different results at locations behind the steep slope of the hill. It was found that the acceleration over the crest of the 2D hill x/H = 0 obtained by CFD modelling Fig. 5 (b) in the case of the

On the other hand, the horizontal velocity profile calculated by using the

It should be noted that the standard

Measured and predicted separation, reattachment and recirculation lengths behind hill

Turbulence models | Source | Separation point | Reattachment Point | Recirculation length |

standard | Present work | 0.85H | 3.41H | 2.56H |

Present work | 0.51H | 5.46H | 4.95H | |

Present work | 1.28H | 6.41H | 5.13H | |

« Modified » standard | Fluent [58] | ------- | 4.1H | ------ |

LES | [26] | ------- | 5.75H | ------ |

Wind tunnel experiment | [46] | 0.5H | 6.5H | 6H |

Generally, the differences predicted among the standard

### 5.2 CFD results and discussion

The velocity and pressure contours plots resulting of the two-dimensional computational domain under neutral ABL condition are presented in Fig. 7 and Fig. 8. It is examined that the isovalue of the velocity contour for the models predicts a deceleration of the flow immediately upstream from the hill, followed by an acceleration of the windward slope with a maximum speed at the top, then a negative flow in the separation region. Downstream, along the slope, all these phenomena predict a vortex with a negative eddy direction. Another observation is the convergence in the fastest upper levels of the flow fields for the

The pressure distribution obtained by two turbulence models used are shown in Fig. 8 to envisage the structure of the flow in the stream for the 2D case. It can be represented in the form of pressure level curves the entire computational domain with the two RANS models used the general flow characteristics and the overall forecast of upstream, summit and downstream are similar, although some differences can be observed. In the case of a boundary layer near to the hill, the relative pressure of

However, two turbulence models create a low-pressure zone at the top of the hill when the flow returns to the surface; the relative pressure of

## 6 Influence of 2D hill slopes shape with different turbulence models

The implementation of wind turbines in an optimal position with high wind speeds is considered a challenge and important during onshore wind siting to achieve extra performance of the power output and the effectiveness of wind farms.

The flow around complex terrain has been the subject of numerous field measurements and experimental wind-tunnel operations. The variations in the flow conditions due to changes in the wind directions is the most important point leading to the study of more idealized flow situations with simplified terrain geometries. The investigation of the flow over the bound escarpment in Denmark by Lange et al. (2006) and the study of the flow over a generalized isolated hill by using different inflow conditions presented by [62] can ensure and give more details for these views.

The flow development over hills is largely dependent on their steepness, where the flow separation disappears with lower steep hill, the comportment of the logarithmic boundary layer is similar to that observed on flat terrain [63]. On the other hand, steeper sloped construct the separation surfaces on the trailing edge of a hill [64] has shown that this phenomenon can be deflected in some cases towards the ground caused by the lee waves induced by hills.

The acceleration on the top of the hill leads to increased production of energy in wind farm but not always positively. However, it also induces negative effects due to the presence of several features like the increasing levels of turbulence and wind shear. For purposes to benefit the acceleration of wind speeds, to enable accurate predictions of the flow and at the same time to eliminate the negative effects, the knowledge of the fundamental mechanisms governing flow over hilly terrain, topographical features of wind farms located in complex terrain are necessary [65]. Therefore, it is significant to notice the impact of different hill shapes on general flow behavior over complex terrain.

Figure 9 presents the shape of sloped hills used in this piece of the work, they are defined by the parametric equation specified in section 3.2. The slopes called here by Hill3 and Hill5 as function to their a/H ratios and according to their maximum hill slopes are 26° and 16°, respectively as shown in Table 8 and Fig. 9.

Representation of hill characteristics

Name | Slope | ||||

RUSHIL H3 | 26° | ||||

RUSHIL H5 | 16° |

### 6.1 Effect of turbulence models and hills shape on velocity profiles: Comparison of experiments data with CFD results

The comparison of the CFD simulation and experience measurements results are presented in Fig. 10 below using x-velocity as an essential parameter. The longitudinal velocity over the long measurement field for the two slopes H3 and H5 under two turbulence models

The results calculated by these two models are in agreement in most situations, except for differences on the downstream side of the hill. These curves include their comparison with the experimental measurements in the six positions. As we can see, at the measurement positions x/H=−3 and −1.5, upstream from the hill, the speed profiles are similar for the slope H3 (Fig. 10 (top)) except in the case at x/H = −1.5, a slight difference is marked for the

### 6.2 Simulation results and discussion: Effect of turbulence models and hills shape on speed-up factor profiles

The speed-up ΔS on the top of two hill slopes (H3 and H5) obtained by 2D CFD modeling is presented in Fig. 11 for the two turbulence models

On the higher slope

For the case of the

For the case of the height y = 5.13H and more precisely, moving away from the hill for the two slopes H3 and H5 for the

The comparison of velocity contours by considering isolated hills in 2D are shown in Fig. 12. The development of the flow observed on such geometries depends largely on the presence or absence of flow separation.

The results indicated that the hill with a very gentle slope (Hill5) would cause a slight disturbance in the vertical flow, as illustrated in Fig. 12 (lines 2 & 4).

The airflow generally takes the maximum speed near the top (Fig. 12) and decelerates on the leeward side of the hill. The boundary layer developing on the hill will differ significantly from the turbulent logarithmic boundary layer typically found on a flat plat.

As we can see in Fig. 12 (lines 1 & 3), for a steeper hill (Hill 3), an adverse pressure gradient can lead to the separation of the flow and create a wake area on the downstream side of the hill. The separate flow region of the hill is characterized by lower flow velocities than upstream from the slopes of the hill and by more intense mixing due to the turbulent irregular vortices. A reverse flow region may also form behind the hill due to the recirculation of the flow. The shape of the hill and its local surface roughness influence the location of the flow separation. Furthermore, it is noted that the

By moving the flow down the hill, the blocking effect will gradually decrease and disappear completely when the flow reaches the top. As can be seen in Fig. 12, the effect of acceleration increases continuously and reaches its maximum at the summit, more precisely for the steep slope H3. In this area, the boundary layer is disturbed and a thin layer of shear is formed.

In Fig. 13, the comparison of the turbulent kinetic energy (TKE) predictions generated by the leeward separation of the two hills are briefly illustrated. These contours show similar behavior of velocity for the two models used in this section. Otherwise, this figure indicates the degree of the concentration of the TKE in the shear layer, which separates the recirculation zone from the outer layer.

Simulation results by using the standard ^{2}/s^{2}. This means that it can be considered as a most dissipative model and includes a large length scale. On the other hand, it confirms the length of the recirculation zone presented in section 5.1. However, the ^{2}/s^{2} as shown in Fig. 13 (line 3). In the same way, it implies that this model is less dissipative in comparison by the

In general, the hill (Hill 5) is more sensitive than the hill (Hill 3) due to the lower slope and therefore the fact that the flow is at the threshold of separation. This means that small changes in the upstream boundary layer can cause separation and result a completely different flow upstream and downstream of the separation. On the other hand, the

## 7 Conclusions

The paper set out to perform the two-dimensional computation of airflow over hill in complex terrain by using the commercial code COMSOL Multiphysics. The high-resolution steady-state RANS simulations are used to study the turbulence boundary layer flows over hilly terrain, also, to assess the performance of two turbulence schemes, more especially: standard

The study suggests that the prediction of mean velocity is quite good at most locations and the expected reattachment length is certainly closer to the measurements than some previous studies. Therefore, the

The results indicate overall compatible behavior for two models despite an acceptable deviation from the experimental set of data. Regarding some discrepancies between the predicted streamwise velocity profile and experimental data for three position x/H = −3, x/H = 0, x/H = 3, the standard

The results of CFD simulation in the case study of the impact of hilly configurations on two different slopes have briefly been presented in this work. It should be remarked that modeling complex flow regions, such as recirculation zones, the

Next, we have shifted our attention to the case of the hill slope effect on the mean velocity and turbulence. The main findings of this study can be summarized as follows:

- Hill slopes are an important element for the flow characteristics affect. The steep slope, the separation of the flow due to the high adverse pressure gradient will considerably decrease the average speed, which can provide a negative effect on the energy potential of wind turbines.
- In the crest of the hill, it shows that the shape of the slope has an important effect on the prediction of speed acceleration (speed-up) at the top of the hill. It remarks that the average speed highly increases for the steep slope compared by a gentle slope, which can lead to an extractable maximum power rate of the wind turbine.
- The velocity profile at the foot of the hill is still higher for the low slope due to the small effect of the separation than the high slope. Therefore, it can produce a positive effect on wind speed in this region.

## Nomenclature

Half-length | |

Pressure coefficient | |

Depth of the boundary layer | |

Height of the hill | |

Turbulent kinetic energy | |

Reynolds number | |

Sources terms | |

Time | |

Velocity component | |

Free stream velocity | |

Friction velocity | |

Cartesian coordinate | |

Roughness length | |

Von Karman's constant | |

variables | |

Dynamic viscosity and turbulent viscosity | |

Turbulent kinematic viscosity | |

Effective diffusion coefficient | |

Air density | |

Distance from the wall | |

Wall normal coordinate |

### Abbreviation

ABL | Atmospheric Boundary Layer |

CFD | Computational Fluid Dynamics |

FEM | Finite Element Method |

RANS | Reynolds Averaged Navier-Stokes |

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