Environ Fluid Mech https://doi.org/10.1007/s10652-018-9580-1 ORIGINAL ARTICLE
Numerical simulation and optimization study of the wind flow through a porous fence Bingbing San1 • Yuanyuan Wang1 • Ye Qiu1
Received: 10 August 2017 / Accepted: 19 January 2018 Springer Science+Business Media B.V., part of Springer Nature 2018
Abstract Three turbulence closure models (RNG k-e, SST k-x and RSM) were used to investigate the flow characteristics around a two-dimensional isolated porous fence. The comparison between the numerical results and the experimental measurements indicated that RSM model shows a better performance than the other two models. The aim of this paper is to accurately and efficiently determine the optimum porosity that attain the best shelter effect of the wind fence in the near wake region (0–4hb) and in the far wake region (4hb–10hb) respectively, where hb is the height of the fence. The gradient algorithm was adopted as the optimization algorithm and the RSM model was used to model turbulent features of the flow. The shelter effect was parameterized by the peak velocity ratio involving velocity and turbulence. The objective was to reduce the peak velocity ratio in the near or far wake region by changing the design variable porosity (/) of the fence, which ranged between 2 and 60%. The results revealed that a porosity of 10.2% was found as the optimum value giving rise to the best shelter effect in the near wake region, and / = 22.1% was determined in the case of the far wake region. In addition, based on the proposed optimization method, it is found that the recirculating bubble behind the fence can only be detected when / \ 29.9%. Keywords CFD Porosity optimization Shelter effect Wind fence
1 Introduction In an attempt to control the wind erosion problem, wind fence has been widely used in many practical applications such as protecting piles from decay in outdoor coal storages and in harbours [7, 17, 35]. It is generally accepted that the solid and porous fences can provide a shelter effect by reducing the wind velocity in the wake region behind the fence & Ye Qiu
[email protected] 1
College of Civil and Transportation Engineering, Hohai University, Nanjing 210098, China
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[6, 34]. Hagen [14] and Dong et al. [10] suggested that the aerodynamics of the wind fence mainly depends on two parameters, including the mean velocity and the turbulence intensity, which are often given as a justification for evaluating the shelter effect of the fence. Different wind velocity reductions and turbulence features can be found in different wake regions, due to the influences of fence characteristics, such as the porosity (defined as the ratio of open to total area of the fence), porosity distribution, the height of the fence and the distance between the neighboring fences [8, 12]. However, for an isolated wind fence, the porosity is considered as the most crucial parameter in influencing the performance of shelter devices [2, 15]. The shelter effect of porous fences has been the subject of many research works that used both experiments and numerical simulations. Raine and Stevenson [29] measured the flow characteristics behind porous fences using a hot wire anemometer, and found that a fence with a porosity of 20% showed the best overall reduction in the leeward mean velocity. Papesch [26] experimentally optimized the fence spacing and porosity to reduce wind damage in shelter systems, and suggested an optimum fence space to height ratio of between 6 and 8 with 40% porosity. Santiago et al. [33] numerically studied the flow characteristics behind fences using three variants of k-e turbulence closure models, including standard k-e, realizable k-e and RNG k-e. The porosity of their investigated fences ranged from 0 to 50%, and an optimum porosity of 35% for sheltering effect of an isolated wind fence was determined. In addition, they concluded that the performance of RNG and realizable k-e models is relatively better than that of the standard k-e model, however, other turbulence closure models such as SST k-x or RSM models should be considered to further improve the simulation accuracy of the peak velocity. Luo et al. [22] carried out quantitative flow-visualization measurements of wake flow behind isolated solid and porous fences based on particle image velocimetry (PIV). The authors reported that the turbulent fluctuation in the wake flows can be effectively reduced with porosities of the wind fence ranging from 20 to 30%. Chen et al. [5] simulated the flow field around a triangular-shaped prism model behind a porous fence by solving Reynolds-average Navier–Stokes (RANS) equations with standard k-e model. They found that a fence with porosity / = 30% seemed to be most effective in reducing the dust emission. Although the researchers had made great progress in the experimental and numerical studies, there are no firm guidelines on the shelter performance of the porous fences. For instance, a solid fence may be best for reducing the wind velocity in the near wake region, however, a fence with a porosity of 10% may provide good shelter characteristics in the far wake region [28]. To the authors’ knowledge, the porosity of the wind fence in the present literatures is discrete ranging between 0 and 60%, and the optimum value was then determined among the studied porosities on the basis of the traditional ‘‘cut and try’’ approach. It is difficult to clearly understand the shelter effects in different wake regions behind the fence with the porosity regarded as a continuous variable, because the wind tunnel test and computational fluid dynamics (CFD) technique tend to be labor-intensive and time-consuming. With the rapid development of computer and CFD technology, a variety of optimization methods have successfully used for aerodynamic optimization problems by a more quick and economical way. Aerodynamic optimization has progressed significantly over the years and has been applied in many fields of engineering. However, studies dealing with the aerodynamic porosity optimization of the wind fence are very limited, which can help to identify the optimum porosities that yield the best shelter performance in different wake regions behind the fence. One relevant study by Du et al. [11] carried out the porosity optimization of fibrous porous materials for the thermal insulating performance by using the simulated annealing method, and proposed that
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optimization methods showed great potential in complicated industrial and engineering applications. The intention of the current paper is to propose an aerodynamic optimization method with applications in porous fence design. The shelter performance of an isolated fence was parameterized by the peak velocity ratio, which takes into account both the mean velocity and the turbulent fluctuation. The porosity of the fence was considered as the design variable that mainly determines the shelter effect, and the gradient algorithm was applied to aerodynamic porosity optimization problems. In the first part of this paper, different turbulence closure models (RNG k-e, SST k-x and RSM) are employed to analyze the flow characteristics behind the porous fence, and then evaluated by wind tunnel experiments [3, 33]. The RSM model was consequently chosen to simulate the turbulent flow field. In the second part, the implementation of aerodynamic optimization using gradient algorithm was carried out with porosity varying between 2 and 60%, which aim at obtaining the best shelter performance in near wake region (0–4hb) and in far wake region (4hb–10hb), respectively. Finally, the proposed optimization approach was also adopted to accurately and efficiently determine the value of porosity at which the recirculating bubble behind the fence disappears.
2 Numerical methods 2.1 Governing equations The commercial CFD software, FLUENT was used to simulate the flow field around an isolated porous fence. Flow simulation was carried out using a 2-D double precision, steady-state and segregated solver. It is assumed that the viscous incompressible flow satisfies the continuity equation (Eq. 1) and Reynolds-averaged Navier–Stokes (RANS) momentum equations (Eq. 2), and the governing equations can be written as follows: oui ¼0 oxi
ð1Þ
oui oui 1 op o l oui 0 0 ¼ þ ui uj þ uj q oxi oxj q oxj ot oxj
ð2Þ
where ui is the fluid velocity in the ith direction, xi is the ith coordinate, t is the time, q is the density of the air and l is the dynamic viscosity. The Reynolds stress can be expressed as follows: 2 oui ouj 0 0 ð3Þ þ ui uj ¼ sij ¼ kdij þ mt 3 oxj oxi where mt = Cl k2/e denotes the turbulence kinematic viscosity; k is the turbulence kinetic energy and e is the kinetic energy dissipation rate; dij is Kronecker delta function; Cl = 0.09 is model constant. Three different turbulence closure models (RNG k-e, SST k-x and RSM) were used in the simulations. The complete formulation of the RNG k-e turbulence model is given as follows:
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ok ok 1 o ok Gk Gb YM SK þ ¼ ðak leff Þ þ ui þ e þ ot oxi q oxj oxj q q q q
ð4Þ
oe oe oe 1 o e e2 Cl g3 ð1 g=g0 Þ e2 þ ui þ C1e ðGk þ C3e Gb Þ C2e ¼ ae leff ot oxi q oxj oxj kq 1 þ bg3 k k Se þ q ð5Þ where leff is the effective turbulent dynamic viscosity; aj and ae are the inverse of the Prandtl numbers for k and e, respectively; Gk and Gb are the turbulence kinetic energy productions caused by the mean velocity gradients and buoyancy, respectively; YM represents the impact of compressible turbulent fluctuation expansion to the overall dissipation rate; SK and Se are user-defined source terms; g = Sk/e, where S is the scalar measure of the deformation tensor; C1e, C2e, C3e, g0 and b are model constants (C1e = 1.42; C2e = 1.68; g0 = 4.38; b = 0.012). More details on the RNG model have been given in Kim and Baik [19]. The shear stress transport (SST k-x) model comprises two equations: one for k (Eq. 6) and another for x, the specific turbulent dissipation rate (Eq. 7): ok ok o ok P~k þ uj þ b kx ¼ lk ð6Þ ot oxj oxj oxj q ox ox o ox Px 1 1 okox þ uj bx2 þ 2ð1 F1 Þ ¼ ðl Þþ ot oxj oxj x oxj rx;2 x oxi oxi q
ð7Þ
where P~k is the effective rate of production of k; Px is the rate of production of x; lk and lx are the effective viscosities; F1 is the blending function; b, b* and rx,2 are model constants (b = 3/40; b* = 9/100; rx,2 = 1/0.856). More details on the underlying equations in this method can be found in Rocha et al. [31]. The Reynolds-Stress model (RSM) was also adopted and the transport equations for the Reynolds stresses are given as follows: o 0 0 o quk u0i u0j ¼ Dij þ Pij þ /ij þ eij ð8Þ qui uj þ ot oxk " # ou0i u0j o lt ou0i u0j ð9Þ Dij ¼ þl oxk rk oxk oxk ouj oui Pij ¼ q u0i u0k þ u0j u0k oxk oxk /ij ¼
ou0i ou0j þ oxj oxi
ou0 ou0j eij ¼ 2l i oxk oxk
ð10Þ
ð11Þ
ð12Þ
where Dij denotes the turbulent and molecular diffusions; Pij, /ij and eij, are, respectively, the stress production, the pressure strain and the dissipation rate. Equations of k and e used
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in RSM model are associated with a second-order model, which can provide good results in the trailing and exit regions of the jet. Further details on the RSM model are provided in the literature [32].
2.2 Porous media model In numerical simulations, the flow through a permeable wind fence is modeled in accordance with Darcy’s law. That is, the fence is considered as a thin porous media by the addition of a momentum source term to the standard fluid flow equations. This source term contributes to the pressure gradient in the porous material, creating a pressure drop through it. The pressure change caused by a porous fence in the incompressible flow can be written as Dp = 1/2krqv2, where v is the normal velocity component to the fence surface; kr is the (dimensionless) resistance coefficient or pressure loss coefficient. For modeling convenience, a viscous porous media flow through a fence can be estimated by means of porosity (/). Thus, the resistance coefficient itself can be defined as a function of fence porosity, and there are also a few empirical formulas available: kr = 0.5(3/(2/) - 1)2 [16]; ` kr = 1.04(1 - /2)//2 [30]. In this study, the fence resistance coefficient provided by Reynolds, necessary for model simulations, is adopted.
2.3 Numerical validation by the Bradley and Mulhearn’s fence Before optimal analysis of a porous fence, it is necessary to validate the numerical method and the possible applicability of the turbulence models (RNG k-e, SST k-x and RSM) used in this paper. Bradley and Mulhearn [3], denoted B&M hereafter, has experimentally measured the flow through a very long, isolated, porous fence located on a uniform ground. Their experimental data have been used to evaluate the performance of different turbulence models in several numerical studies [20, 37]. Here, we also chose B&M data as a benchmark. Note that, only the simulated mean velocity (temporal average velocity) was compared with the test data in Sect. 2.3, while in Sect. 4, the peak velocity ratio defined by the mean velocity and turbulence will be analyzed. Parameters for the porous fence in B&M experiment are height hb = 1.2 m, roughness length z0 = 0.002 (hb/z0 = 600) and resistance coefficient kr = 4. Two-dimensional fence flows were simulated using three turbulence closure models mentioned above. The height of the computational domain ranged 0 B z/hb B 41, which will lead the blockage ratio less than 3%. In the horizontal direction, the domain spanned from x/hb = - 10 to 80 with the fence placed at x/hb = 0. To consider both accuracy and efficiency for numerical calculations, Santiago et al. [33] investigated the grid size effects on the simulation results of B&M fence. They suggested that the maximum spatial resolution near the fence is Dx/hb = 0.125 and Dz/hb = 0.125, which is enough to obtain accurate grid-independent simulations of the wind fence flows. Thus, a resolution grid with Dx/hb = 0.125 and Dz/hb = 0.125 was used close to the fence, and the computational domain was discretized into 22,260 grid cells by using structural grid technology. Standard wall functions were used at the ground and in the case of a solid fence. Porous fence was represented by the porous media model. Zero normal velocity and gradients were imposed at the inlet boundary of the computational domain, and the flow has been considered fully developed (outflow) at the outlet boundary. The SIMPLE algorithm (semiimplicit method for pressure linked equations) is chosen as scheme to couple pressure and
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velocity, which has been widely used for atmospheric flow simulations, such as flow around porous fences or street canyons [1, 36]. The convergence criteria of the numerical solution for the flow field were that all the residuals of the governing equations were less than 10-6. It should be noted that all the CFD simulations in present study were conducted in a neutral atmosphere. The inlet velocity profile was obtained from the B&M experiment, and corresponded to a logarithmic profile: u z ð13Þ u ¼ ln z0 j where u* is the friction velocity (u* = 0.3 m/s); j is von Karman’s constant (j = 0.4). Figure 1 shows the inlet velocity profile normalized with the mean velocity of upstream flow at height z = 4 m (u04). Based on the assumption of an equilibrium boundary layer, the inlet profiles of the turbulence kinetic energy kin and its dissipation rate ein can be expressed as follows: u2 ffi ¼ 0:3m2 s2 kin ¼ pffiffiffiffiffi Cl
ð14Þ
2=3 ein ¼ Cl3=4 kin =jz
ð15Þ
For comparison convenience, the computed streamwise velocity is also normalized by the upstream mean velocity at z = 4 m (u04). Normalized velocities at fixed heights of z/ hb = 0.38 and 1.88 are shown in Fig. 2, and the vertical profiles of u/u04 at fixed position of x/hb = 4.2 are shown in Fig. 3. In general, the simulation data based on the three turbulence closure models are qualitative in good agreement with those obtained from the B&M experiment. It can be observed from Fig. 2 that, at z/hb = 0.38 and 1.88, the differences among the three simulated cases are not substantial and all shows an overall good performance under the same numerical method conditions (grid, domain and etc.). However, in the case of x/hb = 4.2 (see Fig. 3), the numerical results with z/hb \ 1.5 of RSM model are slightly more accurate than the RNG k-e and SST k-x models.
Fig. 1 Vertical profile of the mean velocity at the upper boundary
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3 Optimization methods 3.1 Porosity optimization model Porosity is one of the most important parameters that determine the shelter performance of an isolated wind fence. Optimum porosity exists in the wind fence design, which provides the best shelter performance in a given shelter region. An aerodynamic porosity optimization problem can be treated as a single-objective optimization problem. This problem usually consists of objective functions, design variables, a flow solver that discussed in Sect. 4.1 and a numerical optimization method. It can be expressed as follows: ( Min : Fð/Þ ð16Þ s:t: a\/\b where F(/) is the objective function; / is the design variable; a and b are the upper and lower limits of the design variable. The principal effect of the wind fence is to reduce the wind velocity, which involves two components: mean velocity and turbulent fluctuation. Thus, in this work we intend to evaluate the shelter performance of porous fences by using just one parameter, namely the
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peak velocity ratio Ru(x, z), which was proposed by Santiago et al. [33], and the form of 1/ Ru(x, z) was also used by Frank and Ruck [13] as a fence protection factor. Ru ðx; zÞ ¼
U ðx; zÞ þ rðx; zÞ Uref þ rref
ð17Þ
where x and z are the horizontal and vertical coordinates, respectively; U(x, z) is the mean velocity; r(x, z) is the velocity standard deviation and defined as r(x, z) = U(x, z) Iu(x, z), where Iu(x, z) is the turbulence intensity. U(x, z) ? r(x, z) denotes the peak velocity, which takes both the mean value and the turbulent fluctuation into account. Subscript ‘‘ref’’ represents the reference values at z/hb = 6.67 upstream from the wind fence. Actually, the peak velocity in the shelter region is compared with that in the undisturbed upstream region. It is noteworthy that, both the mean velocity and turbulent fluctuations will increase remarkably with increasing fence height when z/hb [ 1 [18, 25]). Accordingly, statistics of shelter performance parameters (mean velocity, turbulent fluctuation, peak velocity, etc.) at z/hb B 1 (shelter region of interest) will be evaluated and analyzed through CFD simulations. In the porosity optimization model, the objective function F(/) is minimizing the total values of Ru(x, z) at z/hb B 1 in different wake regions of the wind fence, and is therefore given by: Min : F ð/Þ ¼
N X 8 X
Ru ðxn ; zm ; /Þ
ð18Þ
n¼1 m¼1
where Ru(xn, zm; /) indicates, in the case of porosity /, the peak velocity ratio at fixed position of x = xn and z = zm (n = 1, …, N and N is the number of the horizontal grid nodes in the wake region (x [ 0); z ranges between 0 and hb, m = 1, …, 8 and Dzm = 0.125hb).
3.2 Optimization algorithms For the porosity optimization problem, the optimization algorithms will find the values of the porosity that optimize the objective function while satisfying the constraints. The strategy proposed in this work for solving the porosity optimization problem entails the use of gradient algorithm, which is in general computationally faster than non-gradient based methods [23]. According to the gradient algorithm, the aim of porosity optimization is to lead the gradient of the objective function converge to zero as fast as possible, and consequently obtain the global optimal solution. The basic idea is to select the negative gradient direction of the objective function as the search direction of each iteration step, and gradually approach the minimum function value. The iterative algorithm of the gradient based method (steepest-descent method) can be written as follows: ð19Þ /ðtþ1Þ ¼ /ðtÞ þ at rðtÞ F ð/Þ equivalent to DðtÞ / ¼ at rðtÞ F ð/Þ where t is the iteration step; at is the search step size solved by the golden section method; r F(/) is the derivative of F(/). Equation 19 can be approximated by the Taylor expansion of F(/) and expressed as follows: 2 F ðtþ1Þ ð/Þ ¼ F ðtÞ ð/Þ þ rðtÞ F ð/Þ DðtÞ / ¼ F ðtÞ ð/Þ at rðtÞ F ð/Þ F ðtÞ ð/Þ ð20Þ It follows directly from Eq. 20 that, the value of F(/) gradually decreases as the iteration proceeds. Thus, the optimum results can be achieved by the minimizing the objective
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function (Min: F(/)) through repeatedly modifying the design variable /. The gradient algorithm is frequently characterized by fast convergence rate, because only the first derivative of the objective function needs to be calculated. However, the convergence rate becomes relatively slow in the later period. The convergence criterion is defined as: ðtþ1Þ F ð/Þ F ðtÞ ð/Þ \s ð21Þ where s is the convergence tolerance and the procedure terminates when s B 1910-4. In summary, the basic process for porosity optimization consists of the following steps: assuming an initial porosity value /(0) and calculating the initial objective function (F(0)(/)); ` searching for the optimum porosity by using gradient algorithm (Eq. 19); ´ checking the convergence criterion, if not satisfy and ˆ repeating the second and third steps until convergence. In this study, CFD simulations are adopted for modeling the turbulent flow around the fence, and therefore obtaining the optimum porosity. A general approach for the aerodynamic porosity optimization using CFD is shown in Fig. 4.
4 Results and discussion 4.1 Simulations of the Santiago’s wind fence A sample of two-dimensional isolated wind fence, provided by Santiago et al. [33] (denoted hereafter as SAN) is selected as the aerodynamic optimization example. For the Fig. 4 Flow chart of the aerodynamic porosity optimization utilizing CFD
Start Assumption of initial variable value Yes (porosity ( )) Solve objective function value based on CFD numerical simulation Variable (the value of ) update by gradient algorithm Obtain the new value of objective function
No
Convergence check
Yes End
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SAN’s experiments to be reported, the fence with a height (hb) of 0.12 m and a thickness of 0.01 m was located on a flat surface in a wind tunnel. The inflow velocity profile was fitted to a logarithmic profile with z0 = 2.4 9 10-14 m and u* = 0.3 m/s. The Reynolds number based on the fence height was about 1.9 9 105. As a matter of fact, the choice of the turbulence model has a significant influence on the accuracy and efficiency of the optimization. Thus, CFD simulations were firstly carried out to represent the SAN wind tunnel experiment. The performances of three different turbulence closure models (RNG k-e, SST k-x and RSM) will be evaluated with the same numerical setups (gird, domain, boundary conditions, etc.), and further to determine the turbulence model for optimization procedure. In the streamwise direction, the computational domain ranges from x/hb = - 50 to 100 with the fence located at x/hb = 0. The height of the domain is 30hb with a blockage ratio approximate 3.3%. The structural mesh is used: near the fence the grid size is 0.0075 m 9 0.0075 m (Dx/hb = 0.0625 and Dz/hb = 0.0625), and it is exponentially increased away from the fence. The boundary conditions (wall function at ground, outflow at the outlet boundary) are the same as used in Sect. 2.3. The inlet velocity profile is calculated by Eq. 13 to simulate the experimental data. The turbulence kinetic energy and dissipation rate are computed in the same way as for B&M case. The pressure and velocity fields were decoupled by using the SIMPLE method for pressure-linked equations. Figures 5 and 6 present the peak velocity ratio profiles obtained from the simulations and SAN wind tunnel measurements of fences with porosity / = 0 (solid fence) and 35%,
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Fig. 5 Peak velocity ratio profiles behind the solid fence with porosity / = 0 at several vertical positions (z/hb = 0.25, 0.5, 0.75, 1, 4), obtained from a SAN experimental data and different turbulence models: b RNG k-e model; c SST k-x model; d RSM model
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Fig. 6 Peak velocity ratio profiles behind the porous fence with porosity / = 35% at several vertical positions (z/hb = 0.25, 0.5, 0.75, 1, 4), obtained from a SAN experimental data and different turbulence models: b RNG k-e model; c SST k-x model; d RSM model
respectively. It has been shown that the simulation results of the three different turbulence models against the experimental data are not particularly good. However, the performance of RSM turbulence model is in general acceptable, especially for Ru(x, z) values at z/ hb \ 1. Results of SST k-x are worse than those of the other two turbulence models. For the RNG model case of porosity / = 35% (Fig. 6b), lower Ru(x, z) value is generally observed because RNG k-e produces lower values of turbulence kinetic energy than other turbulence models. It is commonly accepted that, RSM turbulence model provides better prediction of bluff-body flow characteristics such as velocity profiles and recirculating regions in comparison with RNG k-e and SST k-x models [38]. As a result, the RSM model is employed to model turbulent features of the flow in the later optimization procedure. It can be observed from Fig. 5a that, in the region near the fence (0 \ x/hb B 4), a solid fence provides lower values of peak velocity ratio than those in the far wake region (4 \ x/ hb B 10). In contrast to this, as shown in Fig. 6a, the value of Ru(x, z) obviously increases with increasing porosity in the near wake region, however, the porous fence shows better shelter performance at z/hb \ 1 in the far wake region. It must be noted that, a solid or porous fence with low porosity may be best for protecting the near wake region, while a fence with high porosity may provide good shelter effect in the far wake region [28]. Thus, the optimum porosities with the best shelter effects of a wind fence in different wake regions will be accurately determined based on the proposed optimization methods.
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4.2 Optimal design of isolated porous fences In mathematical terms, the pressure loss coefficient kr used in porous media model (Sect. 2.2) tends to be infinite when the porosity is close to zero. In order to improve the efficiency of the optimization, lower and upper bounds are considered to be imposed on the design variable. A prior investigation (not shown in this paper) has been demonstrated that, the fence with a porosity of / = 2% has almost the same flow characteristics as the solid fence. Additionally, Lee and Park [21] had noted that the shelter effect of porous fences gradually decreases as / [ 40%, and can be neglected when / = 65%. Accordingly, the design variable / is determined ranges from 2 to 60% in the optimization procedure.
4.2.1 Optimum porosity in the near wake region (0 \ x/hb B 4) For the single-objective optimization problems, a unique global optimal solution exists. To ensure this, two initial porosities (/ = 2 and 60%), corresponding to the lower and upper bounds of the design variable, are used simultaneously to minimize the objective functions. The objective function Fnear(/) is defined in the near region of the fence (0 \ x/hb B 4) at heights less than hb (z/hb \ 1), and the convergence criterion (s B 1910-4) is satisfied at the 10th iteration step. The optimal results indicate that a fence with / = 10.2% is best for protecting the near wake region (0 \ x/hb B 4). Figure 7 shows the numerical peak velocity ratio profiles behind a porous fence with porosity / = 10.2% by using RSM turbulence model. It can be observed that values of Ru (x, z) are between 0 and 0.2 in the region of x/hb B 4, and are almost constant (close to 0.1) from x/hb = 1 to x/hb = 3. Comparing with the cases of the solid fence and porous fence with / = 35%, as shown in Figs. 5 and 6, it is evident that a fence with / = 10.2% provides a better sheltered region in the range of 0 \ x/hb B 4, however, its protection is not so good at larger distances. The shelter effects of wind fences with porosity / = 0, 10.2 and 35% can be more clearly illustrated by comparing the contours of the mean velocity distributions, as shown in Fig. 8. In addition, streamline contours around the fence are also shown in the insets in Fig. 8. The mean velocity behind the solid and porous fences is reduced to different extents. For porosities of / = 0 and 10.2% (Fig. 8a, b), the mean velocity near the fence (x/hb B 4) is effectively reduced and a better shelter performance is observed for the case Fig. 7 Peak velocity ratio profiles behind the fence with porosity / = 10.2% at z/ hb = 0.25, 0.5, 0.75, 1
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Fig. 8 Contours of the mean velocity distributions around the wind fence: a porosity / = 0, solid fence; b porosity / = 10.2%; c porosity / = 35%. Streamline contours around the fence are shown in the insets
of / = 10.2%. However, recirculating regions (with reverse or negative streamwise velocity) develop behind the fences, which result in poor shelter characteristics in the far wake. For porosity case of / = 35% (Fig. 8c), the shelter effect in the near wake is not so effective, but the protection far from the fence (x/hb [ 4) is better than close to it. Park and Lee [27] found that a fence with porosity larger than 30% is useful for reducing the mean velocity in the far wake without the formation of a recirculating bubble behind the fence. Turbulence kinetic energy is caused mechanically by the generated recirculating bubbles that arise behind the fences. Contours of the turbulence kinetic energy distributions for different fence porosities are shown in Fig. 9. High values of turbulence kinetic energy generated by the presence of the fence can be observed at heights between hb and 3hb. For fences with porosities / = 0 and 10.2% (Fig. 9a, b), the distributions of turbulence kinetic energy (k) are similar and low values can be found near the fence (x/hb B 4) where the turbulent velocity is very small. For the fence with a porosity of / = 35% (Fig. 9c), most of the turbulence kinetic energy is located in a relatively small region just behind the fence and insignificant values of k can be seen in the other downstream areas. Note that, the center of high k area is approximately above the reattachment region (x/hb [ 10, z/hb [ 1) when the fence porosity / \ 20% [10], and we should not evaluate the shelter performance at heights z/hb \ 1 by just the turbulence kinetic energy (or turbulent fluctuation). Hence, it is reasonable to balance the reduction of mean velocity with change of turbulent fluctuation and shelter distance in this study.
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turb-kinetic-energy: 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
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Fig. 9 Contours of the turbulence kinetic energy distributions around the wind fence: a porosity / = 0, solid fence; b porosity / = 10.2%; c porosity / = 35%
4.2.2 Optimum porosity in the far wake region (4 \ x/hb B 10) The objective function Ffar(/) is defined in the far region behind the fence (4 \ x/hb B 10) at heights of z/hb \ 1. The convergence criteria is satisfied at the 10th iteration step and a global solution with the optimum porosity / = 22.1% is determined by the objective function Ffar(/). Figure 10 shows the peak velocity ratio profiles behind the fence with porosity / = 22.1% in a range from x/hb = 0 to 10. Worse protection for heights z/hb C 1 [higher values of Ru (x, z)] is observed when x/hb increases. However, at heights z/hb \ 1 (especially the case z/hb = 0.75), the protection far from the fence is generally better than close to it, since Ru (x, z) has low values between 0 and 0.1 for x/hb = 4 and 10. Moreover, in comparison with the solid and porous fence with / = 35% (Figs. 5, 6), a porosity / = 22.1% is found to be the best value, relative to the far wake region (4 \ x/hb B 10), to provide an effective shelter effect. Figure 11 shows the contours of the mean velocity distributions around the fence with porosity / = 22.1%. The corresponding streamline contour is shown in the inset in this figure. It is seen that porous fence with / = 22.1% yields a more effective reduction in the mean velocity at larger distances (4 \ x/hb B 10) than fences with porosities / = 0 and 10.2% (Fig. 8a, b). For porosities of / B 22.1%, recirculating regions develop behind the fences, which shift and shrink in the downstream direction with fence porosity increasing up to 22.1%. This phenomenon can be explained by the influence of the flow passing directly through the fence (namely bleed flow), that resulting in the region of reverse
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Fig. 11 Contours of the mean velocity distributions around the fence with porosity / = 22.1%. The inset in the figure shows the streamline contour around the fence
streamwise velocity only occurs in the far wake region as the porosity increases. Furthermore, compared to that of the fence with / = 35% as shown in Fig. 8c, it is observed that the fence with / = 22.1% shows an overall large reduction in the mean velocity in the entire wake region (0 \ x/hb B 10). Dong et al. [9] found that the bleed flow will be dominant in the wake when the porosity is larger than 20%, and the greater the fence porosity, the faster the mean velocity profiles recover. Figure 12 shows the contours of the turbulence kinetic energy distributions for the fence with porosity / = 22.1%. The results of Fig. 12 combined with Fig. 9 reveal that, the
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Fig. 12 Contours of the turbulence kinetic energy distributions around the fence with porosity / = 22.1%
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turbulence kinetic energy at heights z/hb B 1 decreases significantly for wind fences with higher porosities when / C 22.1%. Although the turbulence kinetic energy in the case of / = 22.1% (see Fig. 12) are larger than the / = 35% case (see Fig. 9c), the mean velocity in the range of 4 \ x/hb B 10 and z/hb B 1 are greatly reduced for the fence with optimum porosity / = 22.1%. This indicates that the velocity reduction, rather than turbulence fields, is relatively important in determining the shelter efficiency of fences in the far wake region.
4.2.3 Porosity with the disappearance of recirculating bubble As is well known, for fences with low porosities, the recirculating bubble detaches from the fence, and it moves downstream and becomes smaller as the fence porosity increases. However, this bubble (or reverse flow) could no longer be detected for fences with high porosities [24, 28]. From this viewpoint, the optimization method proposed in this paper is used to determine the accurate value of the fence porosity, at which the recirculating bubble disappears (with no reverse streamwise velocity component). The optimization target is to minimize all the reverse streamwise velocity components to achieve zero in a wide wake region of 0 \ x/hb B 16 and 0 \ z/hb B 4, namely, only the positive streamwise velocity remains. The corresponding objective function can be defined as sum|U(t) x (xn, zm; /)|, where porosity / is the design variable; Ux(xn, zm; /) is the mean reverse or negative streamwise velocity at the fixed position (xn, zm) for the fence with porosity /; sum|| is the sum of all absolute values of reverse streamwise velocity components. In addition, the same optimization framework based on the gradient algorithm is utilized as shown in Fig. 4, and the search process will terminate when the value of sum|U(t) x (xn, zm; / )| is found to be zero. The initial value of the design variable is / = 2%, and the value of sum|U(t) x (xn, zm; /)| gradually converges to zero at the 30th iteration step (corresponding porosity / = 29.9%). Figure 13 shows the profiles of peak velocity ratio behind the fence with porosity / = 29.9%. It is found the fence with / = 29.9% seems to have a similar behavior as the / = 35% case (Fig. 6d), but with lower values of Ru (x, z) in general. Besides, the Ru (x, z) values at z/hb = 0.25 and 0.5 are between 0 and 0.1 for x/hb = 6–10, producing a better protection for larger distances than the previous cases. However, for z/hb C 0.75, the
Fig. 13 Peak velocity ratio profiles behind the fence with porosity / = 29.9% at z/ hb = 0.25, 0.5, 0.75, 1
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Fig. 14 Contours of the mean velocity distributions around the fence with porosity / = 29.9%. The inset in the figure shows the streamline contour around the fence
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Fig. 15 Contours of the turbulence kinetic energy distributions around the fence with porosity / = 29.9%
shelter effect decreases obviously with increasing z/hb as compared with the / = 22.1% case (Fig. 10). Figures 14 and 15 show the contours of the mean velocity and turbulence kinetic energy distributions for the fence with porosity / = 29.9%. It is shown that the variations of the mean velocity and turbulence kinetic energy (respectively) are similar as the fence with porosity / = 35% (Figs. 8c, 9c). In the case of / = 29.9%, as shown by the streamline contour in the inset in Fig. 14, no reverse streamwise velocity component is detected in the wake of the porous fence, indicating the disappearance of the recirculating bubble. This is quite coincident with the result of Castro [4], who mentioned that the recirculation bubble behind the porous fence could not be detected above a porosity of 30%.
5 Conclusions In this study, an aerodynamic porosity optimization method is proposed to accurately and efficiently determine the optimum fence porosity, which can provide the best shelter in any optional wake region. At the first beginning, CFD simulations based on the Navier–Stokes equations using three turbulence closure models (RNG k-e, SST k-x and RSM) were utilized for the prediction of wind flow through an isolated porous fence. The shelter effect of the wind fence was evaluated in terms of the peak velocity ratio, with consideration of both mean velocity and turbulent fluctuation. The results showed that the simulated distributions of mean velocity from the three turbulence models are in good agreement with
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the wind tunnel data by Bradley and Mulhearn [3]. When simulating the flow characteristics of the experimental fence by Santiago et al. [33], relatively large differences in the peak velocity ratio distributions are found among the numerical results of the three turbulence models, however, the RSM model has a relatively good performance. The gradient algorithm was used as the optimization algorithm and the RSM turbulence model was chosen to improve the accuracy of the optimal results. The goal was to determine an optimum porosity to minimize the total values of the peak velocity ratio at heights less than hb, in a given shelter region. In this way, designers could optimally control the fence porosity to obtain the best shelter effect in different wake regions more efficiently. The applicability of the proposed strategy was investigated on two case studies of the near wake and far wake regions, respectively. The results have shown that the best fence porosity in the near wake region (0 \ x/hb B 4) is / = 10.2%, and the fence with porosity / = 22.1% provides the best shelter in the far wake region (4 \ x/hb B 10), as confirmed by the numerical results. In addition, the proposed method is also capable of searching the exact porosity (/ = 29.9%) at which the recirculating bubble behind the fence disappears. In future studies, the effect of porosity on the size of a recirculation region behind the fence will be taken into account in the aerodynamic porosity optimization. Acknowledgements This study was financially supported by the National Natural Science Foundation of China under Grant No. 51578211, the Natural Science Foundation of Jiangsu Province under Grant No. BK20170880, the China Postdoctoral Science Foundation under Grant No. 2016M591754, and the Fundamental Research Funds for the Central Universities under Grant No. 2015B29514.
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