Hybrid simulation of thunderstorm outflows and wind-excited response of structures

Abstract

Starting from the records detected by the monitoring network of the European Projects “Wind and Ports” and “Wind, Ports and Sea”, this paper proposes a novel strategy for simulating the wind velocity field of thunderstorm outflows. A model of the wind field along a vertical axis is first proposed. Its different ingredients and the inherent sources of randomness are then separated into five groups and a hybrid technique for the simulation of thunderstorm outflows is formulated. This technique is applied to generate artificial time-histories of the aerodynamic wind loading on three real slender vertical test structures whose dynamic response is evaluated by means of a time domain integration of the equations of motion. The results are analysed in a probabilistic frame aiming to inspect the distribution of the maximum value of the response, the role of the aerodynamic admittance, the relevance of the resonant part of the response, and the contribution of higher vibration modes in parallel with the classic analysis of the response of structures to synoptic extra-tropical cyclones. The conclusions draw some prospects on the joint calibration of the response spectrum technique and the time domain simulations.

Appendices

Appendix 1: Approximations involved by the simulation algorithm

Figures 17 and 18 clarify the typical approximations involved by the simulations. The dashed lines in the four schemes (a–d) of Fig. 17 show the slowly-varying mean wind velocity $\bar{v}$ of one of the 93 thunderstorm records examined. The solid lines in schemes (a–c) show the slowly-varying mean wind velocity of three of the 1000 simulated records derived from the real one. The solid line in scheme (d) shows the slowly-varying mean wind velocity as averaged over the ensemble of 1000 simulated records.

Fig. 17

Slowly-varying mean wind velocity of one of the 93 thunderstorm records examined (dashed lines) compared with those of three simulated records (a–c) and the ensemble average of 1000 simulations (d) (solid lines)

Figure 18 shows analogous diagrams for the slowly-varying turbulence intensity $I_{v}$. The simulation of $\bar{v}$ is characterised by very good approximations. The simulation of $I_{v}$ involves slight underestimations especially in correspondence of the maximum values of the turbulence intensity.

Fig. 18

Slowly-varying turbulence intensity of one of the 93 thunderstorm records examined (dashed lines) compared with those of three simulated records (a–c) and the ensemble average of 1000 simulations (d) (solid lines)

Appendix 2: Dynamic alongwind response to synoptic winds

The dynamic alongwind response of a slender vertical structure to a synoptic extra-tropical cyclone is evaluated by means of the method described in [63]. Thus, only the contribution of the first mode of vibration is taken into account. Besides, the mean value and the standard deviation of the maximum displacement are given by:

$$\mu_{{x_{\hbox{max} } }} \left( z \right) = \gamma \left\{ {1 + 2\left[ {\sqrt {2\ln \left( {\nu_{x} \Delta T} \right)} + \frac{0.5772}{{\sqrt {2\ln \left( {\nu_{x} \Delta T} \right)} }}} \right]I_{v} \left( H \right)\sqrt {Q^{2} + R^{2} } } \right\}\psi_{1} \left( z \right)$$

(19)

$$\sigma_{{x_{\hbox{max} } }} \left( z \right) = \gamma \left\{ {2\left[ {\frac{\pi }{\sqrt 6 }\frac{1}{{\sqrt {2\ln \left( {\nu_{x} \Delta T} \right)} }}} \right]I_{v} \left( H \right)\sqrt {Q^{2} + R^{2} } } \right\}\psi_{1} \left( z \right)$$

(20)

where γ is a constant factor, I _v is the (synoptic) turbulence intensity [60], ν _x and ν _Q are the expected frequencies of the displacement and of its quasi-static part, respectively, Q ² and R ² are the quasi-static and the resonant response coefficients, respectively. These quantities are defined by:

$$\nu_{x} = \sqrt {\frac{{\nu_{Q}^{2} Q^{2} + n_{1}^{2} R^{2} }}{{Q^{2} + R^{2} }}} ;\,\nu_{Q} = \frac{{\bar{v}(0.6H)}}{{L_{v} (0.6H)}}\frac{1}{{\sqrt {31.25\tilde{\tau }^{1.44} + 0.74\tilde{H}^{0.64} + 5.41\tilde{\tau }^{0.93} \tilde{H}^{0.71} } }}$$

(21)

$$Q^{2} = \left( {\frac{2\beta + \zeta + 1}{\beta + \zeta + 1}} \right)^{2} \frac{1}{{1 + 0.30{\kern 1pt} \tilde{H}^{0.63} }};\,R^{2} = \left( {\frac{2\beta + \zeta + 1}{\beta + \zeta + 1}} \right)^{2} \frac{\pi }{{4\xi_{1} }}\frac{{6.868\tilde{n}_{1} }}{{\left( {1 + 10.302\tilde{n}_{1} } \right)^{5/3} }}C\left\{ {\tilde{n}_{1} \tilde{H}} \right\}$$

(22)

$$\tilde{\tau } = \frac{{\tau \bar{v}(0.6H)}}{{L_{v} (0.6H)}} ;\quad \tilde{H} = \frac{{k{\kern 1pt} C_{v} H}}{{L_{v} (0.6H)}};\quad \tilde{n}_{1} = \frac{{n_{1} L_{v} (0.6H)}}{{\bar{v}(0.6H)}};\quad k = \frac{1}{2}\exp \left\{ { - 0.27\left( {\beta + \zeta } \right)} \right\}$$

(23)

$$C\left\{ \eta \right\} = \frac{1}{\eta } - \frac{1}{{2\eta^{2} }}\left( {1 - e^{ - 2\eta } } \right)\;\text{;}\quad C\left\{ 0 \right\} = 1$$

(24)

where $\bar{v}$ is the (synoptic) mean wind velocity, $\beta = 1/\left[ {\ln \left( {H/2z_{0} } \right)} \right]$ is the exponent of the power law that best fits the logarithmic profile of $\bar{v}$ [73], $L_{v}$ is the (synoptic) integral length scale of the turbulence, C _v is the exponential decay coefficient [60].

Indicative analyses of the tests structures described in Sect. 2 are here carried out in correspondence of the following parameters: z ₀ = 0.01, 0.1, 1 m, $L_{v} \left( z \right) = \bar{L}\left( {z/\bar{z}} \right)^{\nu }$, being $\bar{L} = 300\;\text{m}$, $\bar{z} = 200\;\text{m}$, $\nu = 0.67 + 0.05\ln \left( {z_{0} } \right)$ (z ₀ in m), C _v  = 10, v _ref  = 27 m/s, v _ref being the mean wind velocity at z = 10 m height in an open flat homogeneous terrain with z ₀ = 0.05 m. The quasi-static part of the displacement can be evaluated assuming R = 0.

Table 12 shows the cov of the maximum displacement at the structure top, pointing out that this quantity increases on increasing the roughness length whereas it does not seem so much influenced by the structure type. In any case the cov of $x_{\hbox{max} } \left( H \right)$ is no greater than 0.1 and its value reduces in the order of 0.06-0.08 for smooth terrains. Thus, as stated by Davenport [66] the spread of the random variable $x_{\hbox{max} } \left( H \right)$ is very limited and it may be reasonably identified with its mean value $\mu_{{x_{\hbox{max} } }} \left( H \right)$.

Table 12 Coefficient of variation of the maximum displacement at the structure top (synoptic extra-tropical cyclone, first mode of vibration)

Table 13 shows the amplification factor of the three test structures pointing out values of A that strongly depend on the structure type whereas they are weakly influenced by the roughness length. On average A = 2.0–2.3 for Structure 1, A = 1.5–1.6 for Structure 2, A = 1.2–1.3 for Structure 3. The large values of A for Structure 1 are mainly due to the limited damping coefficient and to the limited height; this second aspect makes very low the reduction effects due to the non-contemporaneity of the local pressures and thus the role of the aerodynamic admittance.

Table 13 Amplification factor (synoptic extra-tropical cyclone, first mode of vibration)