Parameter of CFD simulation.
Abstract
The present paper discusses the unsteady aerodynamic forces on long-span curved roofs by using large eddy simulation (LES). The forced vibration test in a turbulent boundary layer is simulated. The models are force vibrated in the first anti-symmetric mode to investigate 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. A comparison between the wind tunnel experiment and the LES indicates that the LES can be used effectively to evaluate the unsteady aerodynamic force.
Keywords
- large eddy simulation
- unsteady aerodynamics force
- long-span curved roof
- forced vibration test
1. Introduction
Wind-structure interaction is a critical consideration in the design of many structures in civil engineering, especially for structures being flexible and light, such as long-span bridges, high-rise buildings, and long-span roofs. Such structures are generally vulnerable to the dynamic wind actions because of low damping and frequency [1, 2, 3, 4]. The wind-structure interaction induces unsteady aerodynamic forces, or motion-induced wind forces, which may affect the wind-induced response significantly [5, 6]. Therefore, the unsteady aerodynamic force is an important consideration in the design of long-span vaulted roofs. Uematsu and Uchiyama [7] conducted a series of wind tunnel tests using elastic models of a one-way type of suspended roof. The mechanism of the wind-induced vibrations and the effect of wind-roof interaction on the dynamic response were discussed. Daw and Davenport [8] carried out a forced vibration test on a semi-circular roof to investigate the dependence of unsteady aerodynamic forces on the turbulence intensity, wind speed, vibration amplitude, and geometric details of the roof. Ohkuma
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 validated by comparing with the experimental results.
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
where
Applying a modal analysis to the equation of motion for the roof, we obtain the following equation of motion for the
where
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
where
3. Large eddy simulation
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
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
Wind speed | 5 m/s |
Forced vibration amplitude | 4 mm |
Rise/span ratio ( |
0.15, 0.20, 0.25 |
Forced vibration frequency ( |
0–160 Hz (10 Hz increment) |
Reduced vibration frequency( |
0–2.5 |
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
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 × 10^{−3}. And the dynamic mesh is used to simulate the vibration of model.
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,
where
where
Computational domain | 9.75L(x) × 0.6L(y) × 2.5L(z) |
Inlet boundary | Inflow turbulence is generated in preliminary computational domain |
Upper boundary | Zero normal velocity and zero normal gradients of other variables |
Side boundary | Cyclic boundary conditions |
Outlet boundary | Zero normal gradients of all variables |
Floor and model surfaces | No-slip condition |
Grid discretization | 260(x) × 24(y) ×64(z)=(399, 360) |
Convection schemes | Second-order centered difference scheme |
Time differential schemes | Euler implicit |
Numerical algorithm | PISO algorithm |
Time | T = 4 s, Δt = 2.0E−04 s (Courant Number: 9.1E−02) |
3.1.6. Inflow turbulence
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
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 .
Inlet boundary | Cyclic boundary condition |
Upper boundary | Zero normal velocity and zero normal gradients of other variables |
Side boundary | Cyclic boundary conditions |
Outlet boundary | Cyclic boundary conditions |
Floor and surfaces of roughness blocks | No-slip condition |
Convection schemes | MARS method |
Diffusion schemes | Centered difference scheme |
Time differential schemes | First order Euler implicit |
Numerical algorithm | PISO algorithm |
Time step | Δt = 2.0E−04 s |
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
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
4.2. Distribution of wind pressure on the roof
The distributions of mean and rms fluctuating wind pressure coefficients for various forced-vibration frequencies are shown in
Figure 8
. It can be seen that the mean wind pressure coefficients
Figure 9
shows the variations of mean and rms fluctuating wind pressure coefficients with the rise/span ratio. It can be seen that the
4.3. Discussion flow field around the roof
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 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 (
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
4.4. Evaluation of unsteady aerodynamic forces
In this study, we use aerodynamic stiffness coefficient
where
The generalized force
where
Figure 13
shows the aerodynamic stiffness and damping coefficients,
The distribution of aerodynamic stiffness and damping coefficients
5. Concluding remarks
The unsteady aerodynamic forces acting on the long-span curved roof have been investigated based on a numerical simulation (LES). A forced vibration test was carried out. First, the influence of a roof’s vibration on the wind pressure was investigated. It is found that the wind pressure on a vibrating roof is strongly influenced by the roof’s vibration. Furthermore, the flow field around a vibrating roof was also investigated. It is found that the vibration of the roof may restrain the separation of a vortex near the trailing edge of the roof. Finally, the characteristics of unsteady aerodynamic force acting on a long-span vaulted roof were evaluated. Both the wind tunnel experiment and CFD simulation show similar results for the variation of aerodynamic stiffness and damping coefficients
Therefore, it is necessary to consider the effects of unsteady aerodynamic forces in the wind-resistant design of long-span curved roof with lightweight and low stiffness for evaluating the response of the roof more reasonably.
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