2. Unsteady aerodynamic force

This section is focused on the illumination of unsteady aerodynamic forces (motion-induced forces), which result from the wind-structure interaction. Fluctuating deflections of the structure may be excited by the turbulence in oncoming flow, or the wake instability caused by vortex shedding in the structural wake. The unsteady aerodynamic forces result from the modification of the flow as the structure vibrates or changes shape, in other words, the interaction of the wind flow and structure. These forces may lead to instability. The unsteady aerodynamic force is described as two components: the aerodynamic stiffness term that is in-phase with the displacement and the aerodynamic damping term that is out-phase with displacement.

The aerodynamic stiffness is the added stiffness of the air surrounding the structure, which may increase or effectively reduce available structural static stiffness. For a conventional heavy structure, the aerodynamic stiffness is generally insignificant in comparison to the structural stiffness. However, for a long-span light-weight structure, which vibrates more easily in the wind, the aerodynamic stiffness may change the structural response. For instance, if the total static stiffness of the system in wind is reduced to zero, then a divergent instability may be induced.

When a structure is vibrating in the wind, the relative velocity of the structure to the wind flow changes in magnitude and direction. This phenomenon effectively produces an added damping force, referred to as aerodynamic damping. The aerodynamic damping may add to the structural damping to reduce the response of structure, or become negative and increase the response of the structure. The chances of aerodynamic instability are high as the total damping in the system approaches zero.

#### 2.1. Definition of unsteady aerodynamic force

The displacement of structure in the jth mode may be represented by the following equation,

$$Z\_j(\mathbf{s}, t) = \varphi\_j(\mathbf{s})\mathbf{x}\_j(t) \tag{1}$$

where φ<sup>j</sup> and xj are the mode shape and generalized displacement of the jth mode, respectively; and s represents the circumferential coordinate taken along the roof.

Applying a modal analysis to the equation of motion for the roof, we obtain the following equation of motion for the jth generalized displacement,

$$\mathbf{M}\_{\mathbf{s}\_{\uparrow}}\ddot{\mathbf{x}}\_{j}(t) + \mathbf{C}\_{\mathbf{s}\_{\uparrow}}\dot{\mathbf{x}}\_{j}(t) + \mathbf{K}\_{\mathbf{s}\_{\uparrow}}\mathbf{x}\_{j}(t) = F\_{W\_{j}}(t) + F\_{A\_{j}}(t) \tag{2}$$

$$
\ddot{\mathbf{x}}\_{\dot{\jmath}}(t) + 2\xi\_{s\_{\dot{\jmath}}}\omega\_{s\_{\dot{\jmath}}}\dot{\mathbf{x}}\_{\dot{\jmath}}(t) + \omega\_{s\_{\dot{\jmath}}}^2 \mathbf{x}\_{\dot{\jmath}}(t) = F\_{\dot{\jmath}}(t)/M\_{s\_{\dot{\jmath}}}\tag{3}
$$

$$F\_j(t) = F\_{W\_j}(t) + F\_{A\_j}(\mathbf{x}, \dot{\mathbf{x}}, \ddot{\mathbf{x}} \dots) \tag{4}$$

where Msj = generalized mass, ωsj = natural circular frequency, ζsj = critical damping ratio, and Fj = generalized force. FWj represents the fluctuating wind force due to the oncoming flow and wake instability, while FAj represents the unsteady aerodynamic force due to the wind-roof interaction.

In the case of the forced-vibration test, a steady vibration in the first anti-symmetric mode represented by a sine curve is applied to the roof. The unsteady aerodynamic force FAj (here j = 1) can be obtained from Eq. (5) by using the Fourier series at the frequency fm of the forced vibration:

$$F\_{A\_{\rangle}}(t) = F\_{\mathbb{R}\_{\rangle}} \cos 2\pi f\_m t - F\_{I\_{\rangle}} \sin 2\pi f\_m t \tag{5}$$

$$F\_{R\_{\dot{\gamma}}} = \frac{1}{T} \int\_{-T}^{T} F\_{\dot{\gamma}}(t) \cos 2\pi f\_m t \tag{6}$$

$$F\_{I\_j} = \frac{1}{T} \int\_{-T}^{T} F\_j(t) \sin 2\pi f\_m t \tag{7}$$

where FRj and FIj are the in-phase and out-of phase components of the unsteady aerodynamic force, respectively.

#### 3. Large eddy simulation

Ohkuma et al. [9] investigated the mechanism of aeroelastic instability of long-span flat roofs using a forced vibration test in a wind tunnel. At present, long-span curved roofs are universally constructed. However, there is an insufficient research on unsteady aerodynamic forces on long-span curved roofs, and the characteristics of unsteady aerodynamic forces are not well understood. Therefore, it is necessary to investigate this problem further for proposing more

In this chapter, we investigate the characteristics of unsteady aerodynamic forces acting on long-span curved roofs for improving the wind-resistant design method. The large eddy simulation (LES) is used to discuss the influences of a roof's vibration on the wind pressure and flow field around a vibrating roof. The characteristics of unsteady aerodynamic forces in a wider range of reduced frequency of vibration are also investigated. The results of LES are

This section is focused on the illumination of unsteady aerodynamic forces (motion-induced forces), which result from the wind-structure interaction. Fluctuating deflections of the structure may be excited by the turbulence in oncoming flow, or the wake instability caused by vortex shedding in the structural wake. The unsteady aerodynamic forces result from the modification of the flow as the structure vibrates or changes shape, in other words, the interaction of the wind flow and structure. These forces may lead to instability. The unsteady aerodynamic force is described as two components: the aerodynamic stiffness term that is in-phase with the displacement and the aerodynamic damping term that is out-phase with displacement. The aerodynamic stiffness is the added stiffness of the air surrounding the structure, which may increase or effectively reduce available structural static stiffness. For a conventional heavy structure, the aerodynamic stiffness is generally insignificant in comparison to the structural stiffness. However, for a long-span light-weight structure, which vibrates more easily in the wind, the aerodynamic stiffness may change the structural response. For instance, if the total static stiffness

of the system in wind is reduced to zero, then a divergent instability may be induced.

When a structure is vibrating in the wind, the relative velocity of the structure to the wind flow changes in magnitude and direction. This phenomenon effectively produces an added damping force, referred to as aerodynamic damping. The aerodynamic damping may add to the structural damping to reduce the response of structure, or become negative and increase the response of the structure. The chances of aerodynamic instability are high as the total

The displacement of structure in the jth mode may be represented by the following equation,

where φ<sup>j</sup> and xj are the mode shape and generalized displacement of the jth mode, respectively;

ð Þs xjð Þt (1)

Zjð Þ¼ s; t φ<sup>j</sup>

and s represents the circumferential coordinate taken along the roof.

reasonable methods of response analysis for these roofs.

validated by comparing with the experimental results.

2. Unsteady aerodynamic force

102 Flight Physics - Models, Techniques and Technologies

damping in the system approaches zero.

2.1. Definition of unsteady aerodynamic force

The LES is used to investigate the characteristics of unsteady aerodynamic forces. The influences of a roof's vibration on the wind pressure and flow field around a vibrating roof are also investigated. The simulation is carried out by using a CFD software 'STAR-CD'.

#### 3.1. Computational outline

#### 3.1.1. Computational model

The computational model used in the 'STAR-CD' is shown in Figure 1. In order to investigate the effect of geometric shape on the unsteady aerodynamic force, the rise/span ratio r/L of computational models is assumed to be 0.15, 0.20, and 0.25. The curved roof model is forced to vibrate in the first anti-symmetric mode as shown in Figure 1.

#### 3.1.2. Computational parameters

Table 1 summarizes the computational parameters. In order to discuss the effect of geometric shape on wind-roof interaction, the rise/span ratio is changed from 0.15 to 0.25. The amplitude

Figure 1. Computational model.


Table 1. Parameter of CFD simulation.

x<sup>0</sup> of vibration is fixed to 4.0 mm (i.e. x0/L = 1/100). In this study, based on the assumptions of the mean roof height for real structure H\_r = 20 m; the wind speed at mean roof height UH\_r = 20–40 m/s; the natural frequency fs = 0.4–2.5 Hz. We calculated the reduced frequency for real roof f\_r \* = 0.2–2.5, as shown in Table 3. In order to satisfy the similarity principle of real long-span roofs f<sup>m</sup> \* = f\_r \* (f<sup>m</sup> \* = the reduced frequency for model), the forced vibration frequency f<sup>m</sup> should be set at 12.5–156.25 Hz, as shown in Table 2. With regard to the limitation of forced vibration equipment used in the wind tunnel experiment [10], the forced vibration

Table 2. Determination of forced vibration frequency.

frequency cannot be set as large as this. Therefore, it is necessary to use LES to examine the characteristics of unsteady aerodynamic forces in a wider range of reduced frequency. For the LES, we change the forced vibration frequency from 0 to 160 Hz and the range of reduced frequency of vibration is from 0 to 2.5, as shown in Table 1.

#### 3.1.3. Computational domain

Figure 2 shows the computational domain. In this study, the length of span direction equals the span of roof to generate two-dimensional flow that is corresponded with that used in the wind tunnel experiment.

#### 3.1.4. Computational mesh

x<sup>0</sup> of vibration is fixed to 4.0 mm (i.e. x0/L = 1/100). In this study, based on the assumptions of the mean roof height for real structure H\_r = 20 m; the wind speed at mean roof height UH\_r = 20–40 m/s; the natural frequency fs = 0.4–2.5 Hz. We calculated the reduced frequency

Forced vibration frequency (fm) 0–160 Hz (10 Hz increment)

) 0–2.5

Wind speed 5 m/s Forced vibration amplitude 4 mm

Rise/span ratio (r/L) 0.15, 0.20, 0.25

quency f<sup>m</sup> should be set at 12.5–156.25 Hz, as shown in Table 2. With regard to the limitation of forced vibration equipment used in the wind tunnel experiment [10], the forced vibration

*H*\_real

Mean roof height for real structure H\_r = 20 m Mean roof height for model H = 0.08 m Wind speed at mean roof height UH\_r = 20–40 m/s Wind speed at mean roof height UH = 5 m/s Natural frequency fs = 0.4–2.5 Hz Forced vibration frequency fm = 12.5–156.25 Hz

∗ <sup>m</sup> <sup>¼</sup> <sup>f</sup> <sup>m</sup> <sup>H</sup> UH

\* = 0.2–2.5 Reduced frequency for model fm

\* = 0.2–2.5, as shown in Table 3. In order to satisfy the similarity principle of real

*UH*\_model

\* = the reduced frequency for model), the forced vibration fre-

*fm*䠖Forced vibration frequency

䛆 CFD Simulation 䛇

\* =f\_r

\* = 0.2–2.5

*H*\_model

for real roof f\_r

*UH*\_real

f ∗ \_<sup>r</sup> <sup>¼</sup> <sup>f</sup> <sup>s</sup>H\_ <sup>r</sup>

long-span roofs f<sup>m</sup>

Figure 1. Computational model.

104 Flight Physics - Models, Techniques and Technologies

Reduced vibration frequency(fm

Table 1. Parameter of CFD simulation.

\* = f\_r

*f***<sup>s</sup>** 䠖Natural frequency

UH\_ <sup>r</sup> f

䛆 Real Structure 䛇

Table 2. Determination of forced vibration frequency.

Reduced frequency for real roof f\_r

\* (f<sup>m</sup>

\*

In the simulation, various types of mesh arrangements were calculated. We compared the results of LES with those of wind tunnel experiment. And then, the mesh arrangement was selected which leads to the most corresponding results with that of experiment, as shown in Figure 3. The magnitude of minimum mesh is 0.15 <sup>10</sup><sup>3</sup> . And the dynamic mesh is used to simulate the vibration of model.

Figure 2. Computational domain.

Figure 3. Mesh arrangement around roof.

#### 3.1.5. Computational and boundary conditions

There are mainly three types of CFD approaches, which are used in computational wind engineering (CWE): the Reynolds-averaged Navier-Stokes (RANS), the large eddy simulations (LES), and direct numerical simulation (DNS). Due to the limitation of available computer memory and speed at present, DNS cannot be widely used in CWE for solving complicated practical problems. RANS solves the time-averaged NS equations, and the averaged solution reflects the averaged properties of the turbulent flow. Thus, the time-averaged solution is less trustable in nonstationary flows. On the other hand, LES resolves the scale of motion larger than the gird size and the effect of motion of turbulent eddy smaller than grid scale needs to be modeled. The unsteady motions of large eddy can be explicitly predicted and the accuracy is usually much better than RANS models, since the effects of only small eddy are modeled. Therefore, the LES is adopted in this study.

The governing equations adopted in the present LES method are the spatially filtered continuity and Navier-Stokes equations as follows,

$$\frac{\partial \overline{u}\_i}{\partial x\_i} = 0 \tag{8}$$

$$\frac{\partial \overline{u}\_i}{\partial t} + \frac{\partial \overline{u}\_i \overline{u}\_j}{\partial \mathbf{x}\_j} = -\frac{1}{\rho} \frac{\partial \overline{p}}{\partial \mathbf{x}\_i} + \frac{\partial}{\partial \mathbf{x}\_j} \left( \nu \frac{\partial \overline{u}\_i}{\partial \mathbf{x}\_j} \right) - \frac{\partial \tau\_{ij}}{\partial \mathbf{x}\_i} \tag{9}$$

$$
\pi\_{i\bar{j}} = \overline{u\_i u\_{\bar{j}}} - \overline{u\_i} \overline{u\_{\bar{j}}} \tag{10}
$$

where ui, uj represent flow velocity in i-direction and j-direction. The p, ρ, and υ represent pressure, density, and dynamic viscosity of the fluid, respectively. The (u) denotes application of the spatial filter. The τij is subgrid-scale (SGS) stress, which is parameterized by an eddy viscosity model. The standard Smagorinsky model is adopted to estimate the term of τij, as shown in Eqs. (11)–(13).

$$\tau\_{i\dot{j}} = -2\nu\_{SGS}\overline{S}\_{i\dot{j}} + \frac{1}{3}\delta\_{\dot{j}}R\_{kk} \tag{11}$$

$$\overline{S}\_{i\dot{\jmath}} = \frac{1}{2} \left( \frac{\partial \overline{u}\_i}{\partial \mathbf{x}\_{\dot{\jmath}}} + \frac{\partial \overline{u}\_{\dot{\jmath}}}{\partial \mathbf{x}\_i} \right) \tag{12}$$

$$\nu\_{\rm SGS} = \left(\mathbb{C}\_{\rm s}\overline{\Delta}\right)^{2} \sqrt{2\overline{\mathbb{S}\_{\rm ij}\overline{\mathbb{S}\_{ij}}}} \tag{13}$$

where υSGS is the subgrid-scale turbulent eddy viscosity. Sij is the rate of strain tensor for the resolved scale. Δ means the spatial filter width and Cs is the Smagorinsky constant and is taken as 0.12 (Cs = 0.12). The computational and boundary conditions are summarized in Table 3.


Table 3. Computational and boundary conditions.

#### 3.1.6. Inflow turbulence

3.1.5. Computational and boundary conditions

106 Flight Physics - Models, Techniques and Technologies

Therefore, the LES is adopted in this study.

ity and Navier-Stokes equations as follows,

shown in Eqs. (11)–(13).

∂ui ∂t þ ∂uiuj ∂xj

There are mainly three types of CFD approaches, which are used in computational wind engineering (CWE): the Reynolds-averaged Navier-Stokes (RANS), the large eddy simulations (LES), and direct numerical simulation (DNS). Due to the limitation of available computer memory and speed at present, DNS cannot be widely used in CWE for solving complicated practical problems. RANS solves the time-averaged NS equations, and the averaged solution reflects the averaged properties of the turbulent flow. Thus, the time-averaged solution is less trustable in nonstationary flows. On the other hand, LES resolves the scale of motion larger than the gird size and the effect of motion of turbulent eddy smaller than grid scale needs to be modeled. The unsteady motions of large eddy can be explicitly predicted and the accuracy is usually much better than RANS models, since the effects of only small eddy are modeled.

The governing equations adopted in the present LES method are the spatially filtered continu-

where ui, uj represent flow velocity in i-direction and j-direction. The p, ρ, and υ represent pressure, density, and dynamic viscosity of the fluid, respectively. The (u) denotes application of the spatial filter. The τij is subgrid-scale (SGS) stress, which is parameterized by an eddy viscosity model. The standard Smagorinsky model is adopted to estimate the term of τij, as

> 1 3

2SijSij q

τij ¼ �2νSGSSij þ

∂ui ∂xj þ ∂uj ∂xi � �

<sup>ν</sup>SGS <sup>¼</sup> Cs<sup>Δ</sup> � �<sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffi

where υSGS is the subgrid-scale turbulent eddy viscosity. Sij is the rate of strain tensor for the resolved scale. Δ means the spatial filter width and Cs is the Smagorinsky constant and is taken as 0.12 (Cs = 0.12). The computational and boundary conditions are summarized in Table 3.

Sij <sup>¼</sup> <sup>1</sup> 2 ¼ 0 (8)

δijRkk (11)

(9)

(12)

(13)

� <sup>∂</sup>τij ∂xi

τij ¼ uiuj � uiuj (10)

∂ui ∂xi

¼ � <sup>1</sup> ρ ∂p ∂xi þ ∂ ∂xj ν ∂ui ∂xj � �

As is known, the flow around a structure is strongly affected by the flow turbulence. Therefore, the proper generation of the inflow turbulence for the LES is essential in the determination of wind loads on structures. At present, several techniques have been developed. In general, there are three kinds of inflow turbulence generation methods. The first approach is to store the time history of velocity fluctuations obtained from a preliminary LES computation. Nozu and Tamura [11] employed the interpolation method with the periodic boundary condition to simulate a fully developed turbulent boundary layer and tried to change the turbulent characteristics by using roughness blocks. Another approach is to numerically simulate the turbulent flow in auxiliary computational domains (often called a driver region set at the upstream region of a main computational domain). Lund et al. [12] proposed the method to generate turbulent inflow data for the LES of a spatially developing boundary layer. Kataoka and Mizuno [13] simplified Lund's method by assuming that the boundary layer thickness is constant within the driver region, and only the fluctuating part of velocity is recycled in the streamwise direction. Nozawa and Tamura [14, 15] discussed the potential of large eddy simulation for predicting turbulence characteristics in a spatially developed turbulent boundary layer over a rough ground surface and improved Lund's method. The third approach is to use artificial numerical models to generate inflow turbulence statistically [16–19].

In this study, we use a preliminary LES to simulate inflow turbulence and store the time history of velocity fluctuations. Figure 4 shows a schematic illustration of the domain of the preliminary computation. In the domain, the roughness blocks with heights 3, 5, and 8 cm are distributed on the ground to generate turbulence. The computational and boundary conditions are summarized in Table 4.

Figure 4. Preliminary computational domain.


Table 4. Computational and boundary conditions.

The profiles of the mean wind speed and turbulent intensity at the inlet of the computational domain are shown in Figure 5(a). The longitudinal velocity spectrum at a height of H = 90 mm is shown in Figure 5(b). In both figures, the results of wind tunnel flow are also plotted for comparative purposes. It can be seen that the inflow turbulence used in the LES is generally in good agreement with that used in the wind tunnel experiment.

Figure 5. Comparison of inflow turbulence between wind tunnel experiment and LES; (a) profiles of mean wind speed (U(z)) and turbulence intensity (Iu); (b) longitudinal velocity spectrum (H = 90 mm).

### 4. Results and discussion

#### 4.1. Comparison with wind tunnel experiment

In order to validate the LES computation, the distributions of the mean wind pressure coefficient Cp\_mean and fluctuating wind pressure coefficient Cp\_RMS along the centerline of the vibrating roof is compared with those obtained from the wind tunnel experiment. Figures 6 and 7 show the results, in which the results for the frequencies of 0, 10, and 15 Hz are plotted. It can be seen that there is generally a good agreement between the LES and the wind tunnel experiment. In Figure 6, the difference is somewhat larger near the rooftop; the LES values are approximately 10% larger in magnitude than the experimental ones. This difference may be due to a difference in surface roughness of the roof between the LES and the wind tunnel experiment. In Figure 7, when the fm = 0 Hz, the value of Cp\_RMS for the LES is larger than that for the wind tunnel test. That maybe because that the turbulence intensity of inflow turbulence used in the LES is slightly larger than that used in the wind tunnel test (see Figure 5).

#### 4.2. Distribution of wind pressure on the roof

The profiles of the mean wind speed and turbulent intensity at the inlet of the computational domain are shown in Figure 5(a). The longitudinal velocity spectrum at a height of H = 90 mm is shown in Figure 5(b). In both figures, the results of wind tunnel flow are also plotted for comparative purposes. It can be seen that the inflow turbulence used in the LES is generally in

Upper boundary Zero normal velocity and zero normal gradients of other variables

good agreement with that used in the wind tunnel experiment.

Inlet boundary Cyclic boundary condition

Side boundary Cyclic boundary conditions Outlet boundary Cyclic boundary conditions

Diffusion schemes Centered difference scheme Time differential schemes First order Euler implicit

Floor and surfaces of roughness blocks No-slip condition Convection schemes MARS method

Numerical algorithm PISO algorithm Time step Δt = 2.0E�04 s

Table 4. Computational and boundary conditions.

Figure 4. Preliminary computational domain.

108 Flight Physics - Models, Techniques and Technologies

The distributions of mean and rms fluctuating wind pressure coefficients for various forcedvibration frequencies are shown in Figure 8. It can be seen that the mean wind pressure

Figure 6. Comparisons for the distribution of the mean wind pressure coefficients along the centerline between LES and wind tunnel experiment.

Figure 7. Comparisons for the distribution of the fluctuating wind pressure coefficients along the centerline between LES and wind tunnel experiment.

coefficients Cp\_mean near the rooftop increase in magnitude and the rms fluctuating wind pressure coefficients Cp\_rms generally increase, as the forced-vibration frequency increases. Furthermore, the variation is significant near the position of the greatest forced-vibration amplitude. These results indicate that the wind pressure field around the vibrating roof is strongly influenced by the vibration of the roof.

LES of Unsteady Aerodynamic Forces on a Long-Span Curved Roof http://dx.doi.org/10.5772/intechopen.70880 111

Figure 8. Variation of mean and fluctuating wind pressure coefficients with forced vibration frequency (r/L = 0.15).

Figure 9 shows the variations of mean and rms fluctuating wind pressure coefficients with the rise/span ratio. It can be seen that the Cp\_mean changes from negative to positive at the leading edge of the roof as the rise/span ratio increases. Furthermore, the negative peak value increases in magnitude with an increase in rise/span ratio. The value of Cp\_rms increases with an increase in rise/span ratio at the middle part of the roof. However, the effect of r/L on Cp\_rms is less significant than on Cp\_mean.

#### 4.3. Discussion flow field around the roof

coefficients Cp\_mean near the rooftop increase in magnitude and the rms fluctuating wind pressure coefficients Cp\_rms generally increase, as the forced-vibration frequency increases. Furthermore, the variation is significant near the position of the greatest forced-vibration amplitude. These results indicate that the wind pressure field around the vibrating roof is

Figure 7. Comparisons for the distribution of the fluctuating wind pressure coefficients along the centerline between LES

Figure 6. Comparisons for the distribution of the mean wind pressure coefficients along the centerline between LES and

strongly influenced by the vibration of the roof.

wind tunnel experiment.

110 Flight Physics - Models, Techniques and Technologies

and wind tunnel experiment.

The roof configurations at several steps (phases) during one period of vibration are shown in Figure 10. The deformation of the windward side is upward and becomes the greatest at step 2; and that of the leeward side is upward and becomes the greatest at step 4.

Figure 9. Variation of mean and fluctuating wind pressure coefficient with rise/span ratio (fm = 10 Hz).

Figure 10. Roof configurations at several steps.

Figure 11 shows representative flow fields around a stationary or vibrating roof at a frequency of 10 or 20 Hz. It can be seen that the wind speed increases near the roof regardless of the roof's vibration. In the case of a stationary roof (fm = 0 Hz), the flow separates near the 3/4 position of the roof from the leading edge. On the other hand, in the case of a vibrating roof, the separated vortex seems smaller than that in the stationary roof case, which may be due to the vibration of the roof that restrains the separation of the vortex. In addition, the separated position at the

Figure 11. Flow fields around the roof for various forced vibration frequencies (r/L = 0.15); (a) 0 Hz; (b) 10 Hz; (c) 20 Hz.

rear of roof changes with the vibration of roof. The separated position is relatively forward when the roof is vibrated in step 1 to step 3, because the deformation at the windward side of roof makes the flow separated in advance. On the other hand, the separated position is relatively backward when the roof is vibrated in step 3 to step 5, as the result that the deformation at the leeward side of roof restrains the flow separated.

Figure 12 shows the effect of the rise/span ratio on the flow field around a vibrating roof at a forced vibration frequency of 20 Hz. It can be seen that the wind speed near the rooftop becomes higher, generating larger suction as the rise/span ratio increases. Therefore, the negative peak value of Cp\_mean increases with an increase in rise/span ratio (see Figure 8). Furthermore, as the rise/span ratio increases, the vortex at the rearward of roof becomes larger. Flow fields around the roof for various rise/span ratios (fm = 20 Hz); (a) r/L = 0.15; (b) r/L = 0.20; (c) r/L = 0.25.

#### 4.4. Evaluation of unsteady aerodynamic forces

Figure 11 shows representative flow fields around a stationary or vibrating roof at a frequency of 10 or 20 Hz. It can be seen that the wind speed increases near the roof regardless of the roof's vibration. In the case of a stationary roof (fm = 0 Hz), the flow separates near the 3/4 position of the roof from the leading edge. On the other hand, in the case of a vibrating roof, the separated vortex seems smaller than that in the stationary roof case, which may be due to the vibration of the roof that restrains the separation of the vortex. In addition, the separated position at the

Figure 9. Variation of mean and fluctuating wind pressure coefficient with rise/span ratio (fm = 10 Hz).

Figure 10. Roof configurations at several steps.

112 Flight Physics - Models, Techniques and Technologies

In this study, we use aerodynamic stiffness coefficient aKj and aerodynamic damping coefficient aCj to investigate the characteristics of unsteady aerodynamic forces acting on a vibrating long-span curved roof, which are given by the following equations [8]:

Figure 12. Flow fields around the roof for various rise/span ratios (fm = 20 Hz); (a) r/L = 0.15; (b) r/L = 0.20; (c) r/L = 0.25.

$$a\_{kj} = \frac{F\_{Rj}(f\_m)}{q\_H A\_s(\mathbf{x}\_0/\mathcal{L})} = \frac{1}{q\_H A\_s(\mathbf{x}\_0/\mathcal{L})} \frac{1}{T} \int\_{-T}^{T} F\_j(t) \cos 2\pi f\_m t \tag{14}$$

$$a\_{\mathbb{C}\bar{\jmath}} = \frac{F\_{\bar{\jmath}}\left(f\_m\right)}{q\_H A\_s(\mathbf{x}\_0/\mathcal{L})} = \frac{1}{q\_H A\_s(\mathbf{x}\_0/\mathcal{L})} \frac{1}{T} \int\_{-T}^{T} F\_{\bar{\jmath}}(t) \sin 2\pi f\_m t \tag{15}$$

where FRj is the in-phase component with the generalized displacement represented as the aerodynamic stiffness term, FIj is the in-phase component with velocity represented as the aerodynamic damping term, qH = velocity pressure at the mean roof height H, As = roof area, x<sup>0</sup> = forced vibration amplitude, L = span of the roof, T = vibration period, fm = forced vibration frequency, and fm \* = reduced frequency of vibration, defined by fmH/UH, with UH being the mean wind speed at the mean roof height H.

The generalized force Fj may be described in terms of the external and internal pressures pe and pi, as shown in Eq. 16,

$$F\_{\dot{\!\!\!/}}(t) = \int\_{0}^{R\_{\text{s}}} \left[ p\_{e\_{\dot{\!\!/}}}(s, t) - p\_{i\_{\dot{\!\!/}}}(s, t) \right] \wp\_{\dot{\!\!/}}(s) ds \tag{16}$$

where Rs = total length of the vaulted roof. Internal pressure pi is ignored in the present study, because the first anti-symmetric mode under consideration causes no change of internal volume. The model's vibration mode almost corresponded with the asymmetric sine mode, as shown in Eq. 17.

$$\varphi\_j(\mathbf{s}) = \sin 2\pi \frac{\mathbf{s}}{R\_s} \tag{17}$$

Figure 13 shows the aerodynamic stiffness and damping coefficients, aK and aC, obtained from the LES and the wind tunnel experiment, plotted as a function of the reduced frequency of vibration fm \* (fm \* = fmH/UH). The wind tunnel experiment was carried out in a limited range of fm \* , while the LES was conducted over a wider range of fm \* . It can be seen that the LES results are consistent with those of the wind tunnel experiment, which indicates that the LES model can be used for investigating the characteristics of unsteady aerodynamic forces. The aerodynamic stiffness coefficient aK is generally positive and increases with an increase in fm \* , which decreases the total stiffness of the system. On the other hand, the aerodynamic damping coefficient aC is negative and increases in magnitude with an increase in fm \* , resulting in an increase in the total damping of the system.

The distribution of aerodynamic stiffness and damping coefficients aK and aC with fm \* for various rise/span ratios is shown in Figure 14. It can be seen that the values of aK for r/L = 0.15, 0.20, and 0.25 are generally consistent with each other when fm \* <0.4. However, when fm \* >0.4, the value of aK decreases with an increase in the rise/span ratio. Regarding the value of aC, the results for various rise/span ratios are generally similar to each other. This figure indicates that the value of aK is influenced by the rise/span ratio of a long-span vaulted roof. However, the effect of the rise/span ratio on the value of aC is small.

aKj <sup>¼</sup> FRj <sup>f</sup> <sup>m</sup>

114 Flight Physics - Models, Techniques and Technologies

aCj <sup>¼</sup> FIj <sup>f</sup> <sup>m</sup>

mean wind speed at the mean roof height H.

frequency, and fm

pi, as shown in Eq. 16,

� � qHAsð Þ <sup>x</sup>0=<sup>L</sup> <sup>¼</sup> <sup>1</sup>

� � qHAsð Þ <sup>x</sup>0=<sup>L</sup> <sup>¼</sup> <sup>1</sup>

FjðÞ¼ t

ðRs 0 pej

qHAsð Þ x0=L

Figure 12. Flow fields around the roof for various rise/span ratios (fm = 20 Hz); (a) r/L = 0.15; (b) r/L = 0.20; (c) r/L = 0.25.

qHAsð Þ x0=L

where FRj is the in-phase component with the generalized displacement represented as the aerodynamic stiffness term, FIj is the in-phase component with velocity represented as the aerodynamic damping term, qH = velocity pressure at the mean roof height H, As = roof area, x<sup>0</sup> = forced vibration amplitude, L = span of the roof, T = vibration period, fm = forced vibration

The generalized force Fj may be described in terms of the external and internal pressures pe and

ð Þ� s; t pij

h i

1 T ðT �T

1 T ðT �T

\* = reduced frequency of vibration, defined by fmH/UH, with UH being the

ð Þ s; t

φj

Fjð Þt cos 2πf <sup>m</sup>t (14)

Fjð Þt sin 2πf <sup>m</sup>t (15)

ð Þs ds (16)

Figure 13. Comparisons of the LES and the wind tunnel experiment for the aerodynamic stiffness coefficient aK and aerodynamic damping coefficient aC versus reduced frequency of vibration fm \* .

Figure 14. Aerodynamic stiffness and damping coefficients versus fm \* for different rise/span ratio r/L.
