Data-knowledge-driven semi-empirical model augmentation method for nonlinear vortex-induced vibration

Abstract

Vortex-induced vibration is a typical nonlinear fluid–structure interaction phenomenon. Significant challenges to high-precision prediction by the prevalent methods rely on three complex nonlinear dynamic behaviors: nonlinear evolution (NE), vibration peak deviating from the resonance (PD), and nonlinear hysteresis. Although the semi-empirical model is a theoretical and efficient manner, it is difficult to accurately predict the above nonlinear phenomena due to the incomplete mathematical expressions and uncertain parameters. In this paper, a data-knowledge-driven (DKD) augmentation method is proposed to modify the typical wake oscillator model. A comprehensive analysis is first conducted for the effect of the potential aerodynamic damping terms. Motivated by the above analysis, a delay damping term is proposed which contributes to the NE and PD phenomenon by affecting the growth rate of the flow frequency and triggers the mode transition of the coupled system. With these physical understandings, a new model architecture is constructed by combining the delay damping and the Rayleigh damping. Besides, experimental data are utilized to identify the empirical parameters of the model by the ensemble Kalman filter data assimilation technique. The results indicate that the DKD model (marked as Van-Delay-Rayleigh) can accurately compute these nonlinear behaviors for 25 different cylinders. Compared with the original model, the prediction accuracy of the DKD model is improved by 2–5 times. It also shows the generalization capability with various mass-damping parameters, which can reduce the number of wind tunnel tests by 70%.

Appendices

Appendix

A Wake oscillator model of the 2DOF circular cylinder

The basic wake oscillator model of the 2DOF circle is described in this section and the detailed formulae can be referred to [50]. The main mathematical expression and the dimensionless variable are the same as in Sect. 2.1. However, the dimensionless time is expressed by $t=\omega _{sy} T$, where $\omega _{sy}$ is the natural structure frequency of the cross-flow direction and $\omega _{sx}$ is the natural structure frequency of the in-line direction. The nonlinear coupled equations of in-line displacement ($x =X /{D}$), cross-flow displacement (y), fluctuating drag ($p=2C_{D}/C_{D0}$) and lift (q) are obtained:

$$\begin{aligned}{} & {} \ddot{x}+\lambda _{x}\dot{x}+f^{*2}(x+\alpha _{x} x^{3}+\beta _{x}xy^{2})\nonumber \\{} & {} \quad =M_{D}\varOmega ^{2}p-2\pi M_{L}\varOmega ^{2}(q\dot{y}/U_{r}) \end{aligned}$$

(18)

$$\begin{aligned}{} & {} \ddot{p}+2\varepsilon _{x}\varOmega ({p}^2-1)\dot{p}+4\varOmega ^2{p}=B_{3}\ddot{x} \end{aligned}$$

(19)

$$\begin{aligned}{} & {} \ddot{y}+\lambda _{y}\dot{y}+y+\alpha _{y}{y}^3+\beta _{y}{y}{x}^2\nonumber \\{} & {} \quad =M_{L}\varOmega ^{2}q+2\pi M_{D}\varOmega ^{2}(p\dot{y}/U_{r}) \end{aligned}$$

(20)

$$\begin{aligned}{} & {} \ddot{q}+2\varepsilon _{y}\varOmega ({q}^2-1)\dot{q}+4\varOmega ^2{q}=B_{3}\ddot{y} \end{aligned}$$

(21)

where $f^{*}=\omega _{sx}/\omega _{sy}$ means the ratio of structural frequencies of two freedoms, $M_{L}$ and $M_{D}$ are the system mass parameters and ax are the geometrical parameters. $\varOmega =S_{t}U_{r}$. According to the Srinil [50], $f^{*}=1$, $a_{x}=a_{y}=b_{x}=b_{y}=0.7$ and $B_{3}=12$.

$$\begin{aligned} M_{D}= & {} C_{D0}/(16\pi ^{2}S_{t}^{2}\mu _{x}) \end{aligned}$$

(22)

$$\begin{aligned} M_{L}= & {} C_{L0}/(16\pi ^{2}S_{t}^{2}\mu _{y}) \end{aligned}$$

(23)

$$\begin{aligned} \lambda _{x}= & {} 2\xi _{x}f^{*}+\gamma \varOmega /\mu _{x} \end{aligned}$$

(24)

$$\begin{aligned} \lambda _{y}= & {} 2\xi _{y}+\gamma \varOmega /\mu _{y} \end{aligned}$$

(25)

$C_{L0}$ and $C_{D0}$ are the associated lift and drag coefficients of a stationary (empirically $C_{L0}=0.3$ and $C_{D0}=0.2$). $\mu _{y}=\mu _{x}$, $\xi _{x}=\xi _{y}$ and $\gamma =0.8$.

B Prediction of the VIV

This part shows more detailed results for the VIV predictions by the new model. Figure 28 displays the results of the rectangular with a 4:1 section. The Van der Pol model and the Van der Pol-Rayleigh model are difficult to consider the amplitude-varied damping in the results, which produces a much larger amplitude than the actual vibration. Figure 29 shows the response of the cylinder with a 2:1 section. Due to the ignorance of the strongly unsteady effects, the results predicted by the quasi-steady theory in this paper and the reference [40] deviate greatly from the actual values. But the DKD model proposed by the augmentation method can successfully capture the PD and the NE phenomena. Moreover, the hysteresis phenomenon is represented by the red dashed line. Furthermore, the effectiveness of the Van-Delay-Rayleigh model is demonstrated by the VIV of the 2DOF circular cylinder in Fig. 30. All three nonlinear phenomena calculated by the Van-Delay-Rayleigh model are in agreement with the experimental data.