Shock and Vibration

Volume 2018, Article ID 6839062, 8 pages

https://doi.org/10.1155/2018/6839062

## A New Erosion Model for Wind-Induced Structural Vibrations

^{1}School of Civil Engineering, Hubei Polytechnic University, Huangshi, Hubei 435003, China^{2}Department of Civil Engineering, Tsinghua University, Beijing 100084, China^{3}Geodätisches Institut, Leibniz Universität Hannover, 30167 Hannover, Germany

Correspondence should be addressed to Yi Zhang; moc.361@78iy_gnahz

Received 16 March 2018; Revised 7 June 2018; Accepted 17 July 2018; Published 10 September 2018

Academic Editor: Aly Mousaad Aly

Copyright © 2018 Keqin Yan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In recent years, computational fluid dynamics (CFD) method has been widely utilized in simulating wind-induced snow drifting. In the simulating process, the erosion flux is the main controlling factor which can be calculated by the product of erosion coefficient and the differences between the flow stress and threshold stress. The erosion coefficient is often adopted as an empirical constant which is believed not to change with time and space. However, in reality, we do need to consider the influences of snow diameter, density, and wind speed on the erosion coefficient. In this technical note, a function of air density, sow particle density, snow particle radius, and snow particle strength bond is proposed for the erosion coefficient. Based on an experiment study, the effects of these parameters in erosion coefficient is analyzed and discussed. The probability distribution and value range of erosion coefficient are also presented in this technical note. The applicability of this approach is also demonstrated in a numerical study for predicting the snow distributions around a cube structure. The randomness of the structural vibrations is studied with details.

#### 1. Introduction

In heavy snow areas, wind-induced snow drifting causes unbalanced snowdrift around buildings/on roofs. It is not only difficult to remove but also causes trouble for vehicles and pedestrians. Roof collapse occurs for unbalanced snow distribution.

Wind-induced snow drifting belongs two-phase flow. There are many parameters affecting the result of this phenomenon: wind speed, friction velocity, threshold friction velocity, snow particle radius, density, and cohesion. At present, there are four kinds of research methods for this phenomenon: theoretical analysis, field investigation, wind tunnel (water flume) experiment, and numerical simulation. Field investigation can obtain first hand data, and it is the basis of all other methods. But the field investigation is usually constrained by natural conditions, is time-consuming, and can only obtain the result under certain conditions. Therefore, it is not easy to reveal the inherent law of this phenomenon. Wind tunnel test can change parameters and can make up the shortage of field measurement to some extent. But only a few wind tunnels can carry out this kind of test. Furthermore, similarity criterion is difficult to satisfy due to reduced model. Many former researches have been done on the topic of wind-induced snow drifting through either field experiment [1–8] or numerical simulation [9–13]. Most researchers tried to combine theoretical analysis with empirical formulas from results of field investigations and use it in the numerical simulations [5, 9–13]. In an early works of the authors [14], a new method is developed to measure the air velocity profile surrounding an existing building structure considering snow effects. The snow particle size and its distribution are considered in plotting the velocity profile. In this method, the experiment was conducted based on a simple wind tunnel powered by a fan in the lab. The influences from the field such as the potential damages that might be caused to the equipment due to snow particles are not considered. Different snow size effects are also not catered. Obviously, the input from the field results has a significant influence on the reliability of the analysis. Among these, the modeling of erosion flux is one of the most dominant factors in numerical analysis. Plenty of studies on determining the erosion flux have been carried out in recent years [15]. In these pioneering works, the erosion flux is found to be dependent on the difference between the friction flow stress and threshold friction stress. The erosion flux can be calculated based on the product of a coefficient and the stress difference. The determination of the value of is based on field experiments. For instance, Schmidt [16] had once carried out experimental tests in a wind tunnel to investigate behavior of drifting spherical glass microbeads, which has a diameter of 350 *μ*m and density of 2.5 g/cm^{3}. From the experiment, it was found that when flow friction velocity is , has a value around . It is also reported that the value of coefficient increases as the particle diameter decreases. For instance, Anderson [17] carried out investigations with mineral particles in size of sand and found out that the value of should be in orders of . Therefore, the particle size is an influential factor in determining the value of . In fact, there are more factors that need to be considered in the estimate of . The authors believe that it depends on snow conditions.

Since 1990s, computational fluid dynamics (CFD) theory has been brought into the simulation of snow drifting. Similar to the model of Andersen [16], Naaim [11] suggested that the erosion flux of snow can be computed by the product of two factors. The first factor is the difference between the square of flow friction velocity and the square of threshold friction velocity. The second factor is a coefficient , or so-called erosion coefficient , which usually takes a value of . Because of its simplicity, many research works have adopted this concept and taken this value for the erosion coefficient in their CFD simulations [18]. Very limited studies have been carried out to discuss about the value range of this coefficient. And studies on which factors can affect this coefficient are also lacking. Thus, a comprehensive study on the attributes for an erosion coefficient is quite demanding.

The objective of this study was to investigate the value range of erosion coefficient and to derive an expression of erosion coefficient in terms of the dominant factors. Realizing this, the paper is organized as follows. After the introduction, Section 2 will first discuss the dominant factors that are influential to erosion coefficient. Based on these factors, an expression of erosion coefficient is derived. After this, the value range of erosion coefficient is investigated in Section 3. The domain of the dominant factors is considered in this evaluation. The applicability of the developed approach is further demonstrated in a case study carried out in Section 4. The final conclusions drawn from this study are summarized in Section 5.

#### 2. Derivation of the Formula for Erosion Coefficient

As mentioned in many of the literature, coefficient , which characterizes the bond strength of snow particles, has the same physical meaning as erosion coefficient in the CFD method [19]. Herein, a short discussion on the relationship between these two coefficients is provided as follows.

In most literature on the topic of wind-induced snow drifting, the snow surface erosion flux is calculated as follows [20]:where is the erosion coefficient, is the friction velocity, and is the threshold friction velocity of snow particles. In most former works, the value of is suggested to be 0.0005 or 0.0007.

Later on, Anderson analyzed the characteristics of surface force and suggested that drifted snow particle number should be calculated as follows [9, 17]:where is a coefficient characterized by the bond strength between snow particles which usually takes a value of , represents the surface shear stress caused by flow, and indicates the threshold shear stress. By combining Equations (1) and (2), the relationship between and can be revealed.

The relationship between surface shear stress and surface friction velocity can be described by the following equation:where is the air density (the value usually adopts a value of for normal air).

By substituting Equation (3) into Equation (2), we can obtain the following equation for computing the snow particle number:

Once the snow particle number is estimated, it can then be used to calculate the erosion flux. Snow surface erosion flux is the drifted snow particle quantity in unit area per unit time. Since drifted snow contains snow particles with different radius, can be expressed as a function of the particle sizes as the following equation:where is the snow particle density, is the snow particle volume, is the snow particle diameter, is the snow particle radius, and represents the snow particle (). For simplicity, threshold friction velocity is assumed to be the constant for all snow particles. Thus, Equation (5) can be further revised as follows:

Therefore, by substituting Equation (6) into Equation (1), the equation for computing the erosion coefficient can be derived as follows:

As can be observed from Equation (7), erosion coefficient is a function of snow particle radius and density, coefficient , and air density. It is a much more complicated factor which should be quite random depending on snow particle properties. In the following, the influence of different parameters on the erosion coefficient in Equation (7) will be elucidated below.

#### 3. Value Range of Erosion Coefficient

##### 3.1. Randomness of Snow Particle Radius

Wind-induced snow drifting contains snow particles with different radius. Budd [3] analyzed the particle radius distribution through field investigation. He suggested that the drifted snow particle radius obeys two-parameter gamma distribution and gave out the distribution of snow radius along the height Schmidt et al. [21, 22]. A specific formula for this distribution function is given as follows Schmidt [21]:where represents the probability density function of snow particle radius, is the shape parameter of gamma distribution which is proportional to height from snow surface, is the scale parameter of gamma distribution, and is a gamma distribution.

It can be seen that and are related to average snow radius and height . The relationship among these parameters can be further described by the following equations:

It is easy to see from the equation that average snow particle radius will decrease as the height *z* from snow surface increases.

##### 3.2. Randomness of Erosion Coefficient

From the above analysis, we can see the distribution of particle size is highly depending on the height from snow surface. Snow particle radius varies inversely with . Tabler’s work indicated that when the 10 m height wind speed is less than 12 m/s, saltation is predominant transportation [23]. Bagnold [24] suggested that the height should be within 0.1 meter. Therefore, based on these concerns, in the following part, we only investigate the value range of at three selected heights, namely, *z* = 0.02 m, 0.05 m, and 0.1 m.

To determine the value range of , the calculation is split into four steps. Firstly, based on the surface height value (0.02 m, 0.05 m, and 0.1 m), the values of and are calculated from Equations (10)–(12). Second, based on the values of and , the gamma distribution function for the snow particle radius is constructed. Thirdly, substitute the distribution function for snow particle radius into Equation (7). By utilizing the kernel smoothing density estimation, probability density distribution function of can be directly obtained (Figure 1(a)). Finally, the cumulative distribution function for erosion coefficient can be derived based on the density function (Figure 1(b)). The comparison between the investigated erosion coefficients at different heights and the reported results in the literature is also illustrated in Figure 1. The estimated statistical parameter values in Figure 1 are recorded in Table 1.