# On Roof Geometry for Urban Wind Energy Exploitation in High-Rise Buildings

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Turbulence Modeling and Computational Tools Validation

#### 2.1. Governing Equations, Turbulence Modeling and Computational Settings

_{k}and σ

_{ε}(Prandtl numbers); C

_{ε}

_{1}and C

_{ε}

_{2}are closure constants; and P

_{k}is the production of k:

_{t}and the turbulence velocity time scale (T) is:

_{µ}is a model coefficient for the turbulence model and T

_{SKE}= k/ε is adopted for the standard k − ε (SKE) turbulence model. With the aim of reducing the overestimation of k on the impingement wall of bluff bodies, Durbin [17] proposed a bound in this time scale,

_{µ}= 0.0333, C

_{ε}

_{1}= 1.176, C

_{ε}

_{2}= 1.92, σ

_{k}= 1.0, σ

_{ε}= 1.3 and κ = 0.42. These coefficients were obtained by Crespo et al. [20] from the atmospheric measurements of Panofsky and Dutton [21].

^{−}

^{5}for all of the variables’ residuals.

_{U}= 87.5% and HR

_{k}= 75.0% were obtained [18]. Additionally, we carried out a careful grid convergence analysis using three different meshes: fine (9.8 M cells), medium-sized (3.1 M) and coarse (1.7 M) mesh. We have obtained a convergence rate of 2.23 and a grid convergence index of 0.0577 (5.77%). See Toja-Silva et al. [18] for more details about both validation and grid convergence analysis and for a deeper discussion about how the modified Durbin model (Equation (6)) was derived.

#### 2.2. Additional Validation of Numerical Schemes Using a Curved Roof Model in Wind Tunnel

^{−3}and ν = 1.57 × 10

^{−5}m

^{2}·s

^{−1}.

^{2}·s

^{−2}. Notice that we use the logarithmic inlet wind profile for U (according to Richards and Hoxey [25]) instead of the exponential profile reported from the experiment (see Figure 2a). Since values of ε are not reported from the experiment, the entire inlet wind profile (Figure 2c) is defined according to Richards and Hoxey [25]. As is shown below, the validation is successful, and the agreement between the simulation results and the experimental values is reasonably good.

_{i}and EXP

_{i}are the simulation and experimental values and RD and AD are the relative and absolute maximum admissible deviation from the experimental data, respectively. These values are RD = 0.25 and AD = 0.05 and AD = 0.017 for U and k, respectively [18]. The values of the hit rate that allow considering the validation work as successful are HR ≥ 66%.

_{U}= 94.8% and HR

_{k}= 100%. Note that the absolute difference between the values in Figure 5d is very small, although a clear overestimation is observed.

## 3. State-of-the-Art Roof Shapes

^{+}) at the roof shapes simulated (note that the vertical section of spherical and vaulted roofs is the same).

_{0}= 0.01 m and U

_{ref}= 4.4 m·s

^{−1}and z

_{ref}= H = 40 m (H being the height of the building) are considered as reference values in order to calculate U

_{*}by using Equation (9). Since 30 < y

^{+}< 1000, we use standard turbulent wall laws [26,37,38] for the treatment of the near wall regions of the flow.

_{ref}) and the TI threshold height for the state-of-the-art cases at the roof positions described above. The most interesting cases from the wind energy exploitation point of view are at the low-right position in Figure 11d. They show the highest speed-up and lowest turbulence intensity. The most interesting cases are both vaulted and spherical. The vaulted shape generates a higher speed-up and the spherical a lower TI threshold height.

## 4. Influence of the Roof Edge Shape on the Wind Flow

_{ref}= −0.4) at the central-upstream region of the roof. The curved edge has a very positive effect on U on the whole roof, reaching upstream velocities even higher than the free-stream velocity.

_{R}= 0.08), and U shows a speed-up around the upstream edge, while the TI threshold height decreases substantially (12%–16%). Regarding the flow inclination, no significant differences are found with respect to the simple edge case: the flow is skewed on the upstream roof edge around 5° at 0.19 < z/H < 0.31, and below this height, the inclination angle increases until 40° close to the roof edge (z/H < 0.05) in all cases.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

Greek | |
---|---|

κ | Von Karman constant (-) |

ε | Turbulence dissipation (m ^{2}/s^{3}) |

ρ | Fluid density (kg/m ^{3}) |

σ_{k} | Kinetic energy Prandtl number (-) |

σ_{ε} | Dissipation Prandtl number (-) |

ν | Kinematic viscosity (m ^{2}/s) |

ν_{t} | Kinematic eddy viscosity (m ^{2}/s) |

Latin | |
---|---|

AD | Absolute maximum admissible deviation from the experimental data |

C | Calculated |

C_{ε}_{1} | Closure constant k − ε model (-) |

C_{ε}_{2} | Closure constant k − ε model (-) |

C_{µ} | Model coefficient turbulence model (-) |

CAD | Computer-aided design |

CFD | Computational fluid dynamics |

DIC | Diagonal incomplete-Cholesky |

DILU | Diagonal incomplete LU |

EU | European Union |

EXP_{i} | Experimental value |

fV | fixed value |

GAMG | Generalized geometric-algebraic multi-grid |

H | Building height (m) |

HAWT | Horizontal axis wind turbine |

HR_{k} | Hit rate for k (%) |

HR_{U} | Hit rate for U (%) |

iP | Inlet profile |

k | Turbulent kinetic energy (m ^{2}/s^{2}) |

LU | Lower upper (factorization) |

M | Millions |

n | Total number of points compared (-) |

PBiCG | Preconditioned bi-conjugate gradient |

P_{k} | Production of k (m ^{2}/s) |

RANS | Reynolds Averaged Navier-Stokes equations |

RD | Relative maximum admissible deviation from the experimental data |

Re | Reynolds number (-) |

SIM_{i} | Simulation value |

S | Modulus of the rate of strain tensor (-) |

SKE | Standard kε turbulence model |

sl | Slip |

sP | Symmetry plane |

STL | Stereolithography |

T | Turbulence velocity time scale (s) |

T_{D} | Turbulence velocity time scale adopted for the Durbin turbulence model (s) |

T_{SKE} | Turbulence velocity time scale adopted for the SKE turbulence model (s) |

TI | Turbulence intensity (-) |

U | Streamwise velocity (m/s) |

U_{∗} | Frictional velocity (m/s) |

$\overline{{{u}^{\prime}}_{i}{{u}^{\prime}}_{j}}$ | Reynolds stresses (m ^{2}/s^{2}) |

U_{ref} | Reference velocity (m/s) |

VAWT | Vertical axis wind turbine |

wF | Wall function |

X_{R} | Recirculation (or reattachment) distance on the roof |

y^{+} | Non-dimensional wall distance (-) |

z | Height (m) |

z_{0} | Roughness height (m) |

z_{ref} | Reference height (m) |

zG | zeroGradient |

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**Figure 1.**Diagram of the wind-tunnel geometry and the axes V1–V3 for the validation of the results. All lengths are in meters.

**Figure 2.**Inlet wind profiles for the validation case: mean streamwise velocity (U) (

**a**), turbulent kinetic energy (k) (

**b**) and turbulent dissipation (ε) (

**c**). The points represent the inlet profiles used in the experiment of Ntinas et al. [24], and the solid lines are the numerical inlets of the simulations.

**Figure 3.**Vertical section of the refined mesh obtained using snappyHexMesh for the validation case. (

**a**) General view mesh, 6.5 M cells; (

**b**) detail of the refinement area.

**Figure 4.**Vertical section at the center of the domain of the U and k fields around the validation building. (

**a**) U field in m·s

^{−1}; (

**b**) k field in m

^{2}·s

^{−1}.

**Figure 5.**Comparison of U and k at the vertical section at the center of the domain for the validation case. (

**a**) U

_{V}

_{1}; (

**b**) U

_{V}

_{2}; (

**c**) U

_{V}

_{3}; (

**d**) k

_{V}

_{3}.

**Figure 6.**Diagram of the computational domain of the base-building (flat roof). All values are in meters.

**Figure 7.**Central vertical section detail of the different roof shapes investigated. (

**a**) Flat roof; (

**b**) shed roof; (

**c**) pitched roof; (

**d**) spherical/vaulted roof.

**Figure 8.**Diagram of the different curved roof shapes investigated. (

**a**) Vaulted roof; (

**b**) spherical roof.

**Figure 9.**Vertical section of the refined mesh obtained using snappyHexMesh. (

**a**) Vertical section of the external mesh (3.3 M cells); (

**b**) general view of the final flat roof mesh (6.7 M cells); (

**c**) detail of the mesh refinement around the flat roof building.

**Figure 10.**Vertical section detail of the refined meshes obtained using snappyHexMesh for the state-of-the-art analysis. (

**a**) Flat roof, 6.7 M cells (y

^{+}≈ 325); (

**b**) shed, 6.4 M cells (y

^{+}≈ 355); (

**c**) pitched, 6.5 M cells (y

^{+}≈ 337); (

**d**) spherical (6.4 M cells, y

^{+}≈ 338)/vaulted (6.6 M cells, y

^{+}≈ 334).

**Figure 11.**Comparison of U, k and TI for the state-of-the-art cases at the most advantageous vertical axis for each case: upstream edge for the flat roof, downstream edge for the shed roof and center of the roof for the rest of the shapes. (

**a**) Comparison of U; (

**b**) comparison of k; (

**c**) comparison of TI below the limit of TI < 0.15; (

**d**) comparison of the speed-up (U/U

_{ref}) and the TI threshold height.

**Figure 12.**Comparison of U and TI fields on the roof for the state-of-the-art cases: sharp roofs. (

**a**) U field flat roof; (

**b**) TI field flat roof; (

**c**) U field pitched roof; (

**d**) TI field pitched roof; (

**e**) U field shed roof; (

**f**) TI field shed roof.

**Figure 13.**Comparison of U and TI fields on the roof for the state-of-the-art cases: curved roofs. (

**a**) U field vaulted roof; (

**b**) TI field vaulted roof; (

**c**) U field spherical Abohela; (

**d**) TI field spherical Abohela.

**Figure 14.**Examples of the different roof edges tested. (

**a**) Simple edge; (

**b**) curved edge; (

**c**) railing; (

**d**) cantilever.

**Figure 15.**Central vertical section detail of the different roof edge shapes investigated. (

**a**) Simple edge; (

**b**) curved edge; (

**c**) railing; (

**d**) cantilever.

**Figure 16.**Vertical section detail of the refined meshes obtained using snappyHexMesh for the different roof edge shapes investigated. (

**a**) Simple edge, 6.7 M cells (y

^{+}≈ 325); (

**b**) curved edge, 6.7 M cells (y

^{+}≈ 329); (

**c**) railing, 6.9 M cells (y

^{+}≈ 314); (

**d**) cantilever, 6.9 M cells (y

^{+}≈ 297).

**Figure 17.**Diagram of the axes V1–V4 on the base-building at the central vertical plane of the domain, for the comparison of roof edge shapes.

**Figure 19.**Comparison of k between the different roof edge shapes investigated at the vertical section at the center of the domain.

**Figure 20.**Comparison of TI below the limit of TI < 0.15 between the different roof edge shapes investigated at the vertical section at the center of the domain.

**Figure 21.**Comparison of U and TI fields on the roof for the different roof shapes investigated. (

**a**) U field cantilever; (

**b**) TI field cantilever; (

**c**) U field railing; (

**d**) TI field railing; (

**e**) U field curved edge; (

**f**) TI field curved edge.

**Table 1.**Boundary conditions imposed at each boundary of the domain. Nomenclature: C = calculated; fV = fixed value; iP = inlet profile; sl = slip; sP = symmetry plane; wF = wall function; zG = zero gradient.

U | k | ε | ν_{t} | p | |
---|---|---|---|---|---|

Inlet | iP | iP | iP | C | zG |

Outlet | zG | zG | zG | C | fV zero |

Ground | fV zero | kqRwF | epsilon wF | nutkrough wF | zG |

Building | fV zero | kqR wF | epsilon wF | nutk wF | zG |

Sky | sl | sl | sl | C | zG |

Sides | sP | sP | sP | sP | sP |

© 2015 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Toja-Silva, F.; Peralta, C.; Lopez-Garcia, O.; Navarro, J.; Cruz, I.
On Roof Geometry for Urban Wind Energy Exploitation in High-Rise Buildings. *Computation* **2015**, *3*, 299-325.
https://doi.org/10.3390/computation3020299

**AMA Style**

Toja-Silva F, Peralta C, Lopez-Garcia O, Navarro J, Cruz I.
On Roof Geometry for Urban Wind Energy Exploitation in High-Rise Buildings. *Computation*. 2015; 3(2):299-325.
https://doi.org/10.3390/computation3020299

**Chicago/Turabian Style**

Toja-Silva, Francisco, Carlos Peralta, Oscar Lopez-Garcia, Jorge Navarro, and Ignacio Cruz.
2015. "On Roof Geometry for Urban Wind Energy Exploitation in High-Rise Buildings" *Computation* 3, no. 2: 299-325.
https://doi.org/10.3390/computation3020299