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Snow accretion prediction on an inclined cable
Cold Regions Science and Technology 157 (2019) 224–234
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Cold Regions Science and Technology journal homepage: www.elsevier.com/locate/coldregions
Snow accretion prediction on an inclined cable Krzysztof Szilder
⁎
T
Aerospace Research Centre, National Research Council, Ottawa, ON K1A 0R6, Canada
ARTICLE INFO
ABSTRACT
Keywords: Snow Freezing rain Snow accretion prediction Bridge cable Cable aerodynamics
We have developed a novel analytical model to predict snow accretions. Our analysis of snow and freezing rain accretions forming on vertical, inclined and horizontal cables revealed that the two most influential parameters are the ratio of windspeed to particle terminal velocity, and cylinder inclination angle. We have also adapted our existing morphogenetic icing model to simulate snow accretions on arbitrarily oriented cylinders/cables. The adapted morphogenetic model can be used to predict complex details of the snow accretion shape, including surface roughness and embedded voids. Detailed accretion structures predicted by the morphogenetic model can be 3D-printed and used to examine the relationship between the amount of snow precipitation and the resulting changes in the aerodynamic characteristics of cables.
1. Introduction During a snowstorm, snow may accrete on the inclined cables of a bridge, such as stay cables on a cable-stayed bridge or on the main cables of a suspension bridge. Over time, snow metamorphosis may occur, allowing the accretion to settle and consolidate. Partial melting and refreezing may also occur. Snow/ice formation on bridge cables can result in ice falling onto pedestrians and traffic, leading to personal injuries and property damage. In severe cases, the snow/ice accretions may lead to bridge closures due to induced cable vibrations. A number of bridges in cold regions have experienced snow/ice shedding and associated vibrations as shown by Roldsgaard et al. (2013), Kleissl and Georgakis (2010), Lai (2009), and Kumpf et al. (2012). With the increasing potential, due to climate change, for freezing rain events to occur, ice accumulation and shedding from bridge cables could become of greater concern in the coming years and decades. The need to understand and predict these phenomena to build durable structures, while ensuring the safety of all bridge users, is the motivation behind this project. For example, the Port Mann cable-stayed bridge in Vancouver, Canada was opened to traffic in 2011. During its first year of service, ice and wet snow built up on the stay cables and ice/snow pieces fell onto the traffic lanes, leading to damage of multiple vehicles, injuries to pedestrians and closure of the bridge for several hours. Recently, in December 2016, a similar situation occurred on the same bridge. The dynamic behaviour of the bridge had to be investigated and a retrofit of the surface of the stay cables has been considered. The Pierre-Laporte Bridge in Québec City, Canada was closed in February 2018 because pieces of ice fell from the cables of this
⁎
suspension bridge. Such events are expected to become more frequent due to the effects of climate change. To better understand ice/snow accretion on cables, a number of predictive tools have been developed. There are several snow accretion models for cables, but they were all developed for power line cables that tend to rotate under load, so that a snow sleeve forms (Kollár et al., 2010; Makkonen and Wichura, 2010; Dalle and Admirat, 2011). Bridge stay cables, however, do not rotate. Moreover, existing models make simple assumptions about the resulting accretion shape, and the resulting shape is often circular or semi-circular. While this assumption may be adequate for estimating gravitational loads, it is insufficient for analyzing the subtle effects of accretion shape on cable aerodynamics. The objective of this research was to extend the ice accretion models developed for freezing rain applications by Szilder (2018) to wet and dry snow conditions. The numerical models developed by Szilder (2018) have enabled the prediction of ice accretion details that could not have been anticipated using existing models. These details are crucial for determining the aerodynamic behaviour of bridge cables and other nonrotating cylindrical objects during precipitation. Using the updated numerical model for snow and ice, accretion shapes can be predicted for any desired environmental conditions and can be 3D-printed, using a technique such as selective laser sintering, to make physical models. The physical models can then be tested in a wind tunnel to investigate the aerodynamic characteristics of the snow covered cables. In this paper, two types of snow accretion models were developed:
• Analytical models that allow an estimate of snow accretion size; and • Numerical models that predict details of snow accretion shapes.
Corresponding author. E-mail address: [email protected].
https://doi.org/10.1016/j.coldregions.2018.11.001 Received 13 July 2018; Received in revised form 18 September 2018; Accepted 1 November 2018 Available online 03 November 2018 0165-232X/ Crown Copyright © 2018 Published by Elsevier B.V. All rights reserved.
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2. Analytical model of snow accretion on a bridge cable
mass on the cylinder, over plus/minus 90° from the stagnation line, gives the snow thickness, h, as a simple measure of the total snow accretion. The expression for cross-sectional area is valid for h ≪ D; therefore, for relatively large snow accretions this simplification might be questionable. Alternatively, it could be assumed that there is no displacement of snowflakes after impingement and that a snow layer of constant thickness grows in a direction towards the impinging straightline precipitation. With the latter assumption, the accretion cross-section can be expressed as the product of accretion thickness and cable diameter. This requires replacement of 0.5π by unity both in Eq. (2) and in all subsequent equations in the paper. Future experimental data may help to identify conditions when the factor 0.5π should be replaced by unity. For now, however, we will retain the factor 0.5π in all relevant expressions in the paper. In addition, snow accretion experimental data may help to identify accretion shapes like those considered in Szilder et al. (2002), where more complex geometrical models were developed for ice accretion on a cable. It should be noted that in all models of snow accretion developed in this paper, it has been assumed that there is no snow shedding. This is equivalent to an assumption that there is complete snow sticking and hence that the accretion efficiency is unity. The non-dimensional ratio of accretion thickness to vertical precipitation is obtained by combining Eqs. (1) and (2):
The first half of the paper presents the development of an analytical model for snow accretion on bridge cables that can be used to understand the role of key physical factors using a simplified, systematic approach. The general equations for snow accretion are presented, followed by several case studies and model simplifications in order to understand the snow accretion process on inclined cables and the parameters that have the most significant impact on snow accretion. The influence of snow precipitation on bridge fence porosity is also evaluated using the developed analytical tools. 2.1. Accretion on a bridge cable in 3D A relationship for impinging freezing rain mass on a bridge cable, with arbitrary orientation in three-dimensional space, was derived in Szilder (2018) assuming straight-line drop trajectories. While a straightline trajectory assumption can be justified for freezing rain, it could be questionable for snow due to its low density and low terminal speed. Consequently, in the present analysis, which applies to snow accretion, trajectories can depart from straight lines, leading to modification of the governing equations derived in Szilder (2018). The total collision efficiency, E, is introduced and defined as the ratio of the mass flux of impinging particles (snow crystals) to the mass flux that would be experienced by the surface if the particles were not deflected by the airflow. Consequently, the impinging mass per unit length of bridge cable, (kg/m), is given by: m=E
PV D
where
P
=
sin cos
and
cos = cos sin
h 2 = PV
(3)
The following six cases have been considered: where: d is the particle diameter. Freezing rain values were adopted from Khvorostyanov and Curry (2002) and a relationship between the snow particle size and its density and terminal velocity was taken from Colli et al. (2016). It was assumed that snow particles are spherical and their physical characteristics, bulk density and terminal velocity are given by a power law as a function of particle diameter.
where: λ is the overall impingement coefficient, PV is the vertical precipitation (m), D is the cylinder diameter (m), and ρP is the precipitation density (kg/m3), δ is the approach angle, γ is the yaw angle, and ω is the cylinder inclination angle, Fig. 1. The precipitation angle is given by ε = tan−1(U/VT) where U is the horizontal windspeed (m/s) and VT is the terminal speed of the falling particle (m/s). For more details, please refer to Szilder (2018). For simplicity, we have assumed that the impinging precipitation forms an accretion of constant thickness, h (m), on the exposed half of the cable surface: A
E
2.2. Examined precipitation cases
sin cos cos
(1)
m = 0.5 D h
P A
2.3. Collision efficiency in 2D The collision efficiency of the impinging particles for these six cases was calculated assuming 2D potential flow around a circular cylinder. The results are presented as a function of freestream velocity for the six cases in Fig. 2. When particle trajectories are straight lines, the collision efficiency is unity. Departure from straight-line trajectories results in a decreasing collision efficiency. The trajectories and impingement location for the two cases identified by dots in Fig. 2 are shown in Fig. 3. For these freezing rain and dry snow cases, respectively, 92% and 64% of the incoming mass flux impinges on the surface. Fig. 3 also illustrates the definition of the impingement angle α. The dependence of this angle on freestream velocity and particle type and size is shown in Fig. 4. Lower windspeeds and smaller particles are associated with smaller collision efficiencies and smaller impingement angles. Particle trajectory calculations were also performed for a smaller cylinder of 0.01 m diameter and the results are shown in Fig. 5. Smaller
(2)
where: ρA is the accretion density. The above assumption of uniform spreading of the impinging snow
Table 1 Properties of examined precipitation cases.
Freezing rain Wet snow Dry snow
Fig. 1. A sketch of the general geometry for freezing precipitation impinging on a cable. The lines that are parallel are indicated in the schematic. 225
d (mm)
ρΡ (kg/m3)
VT (m/s)
1 5 1 5 1 5
1000 1000 720 144 170 34
4 9 1.4 1.9 0.7 0.9
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Fig. 2. Collision efficiency as a function of freestream velocity for six different particles impacting a cable of 0.20 m diameter.
Fig. 4. Impingement angle as a function of freestream velocity for six different particles impacting a cable of 0.20 m diameter.
Fig. 3. Limiting trajectories for two particles of 1 mm diameter in a freestream airflow of 1 m/s: freezing rain (black) and dry snow (blue). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 5. Collision efficiency as a function of freestream velocity for six different particles impinging on a cable of 0.01 m diameter.
obstacles are associated with smaller trajectory deflections and consequently higher collision efficiencies. Based on these results, assuming a collision efficiency of unity for a 0.01 m diameter cylinder, the errors would be less than 5%, if the freestream velocity exceeds 0.5 m/s.
and the accretion density is the density of ice, making the density ratio 1.1. For snow, we assume that the density of precipitating snow (measured on a horizontal surface) is the same as the density of snow accreting on the cylinder, making the density ratio unity. To the extent that snow density may be influenced by gravity and wind, this assumption may need to be revisited. The collision efficiency is a function of the component of freestream velocity perpendicular to the cylinder axis, which is equal to the horizontal windspeed for this case. When the wind speed is negligible, there is no accretion on a vertical cylinder. To a first approximation, the accretion thickness ratio is a linear function of wind speed. The slope of this linear relationship
2.4. Accretion on a vertical cylinder We begin by examining freezing rain and snow impingement in the simple case of a vertical cylinder, inclination angle 90°. Since λ = tanε = U/VT, Eq. (3) may be simplified to:
h 2 = PV
P A
E (U )
U VT
(4)
For freezing rain, the precipitation density is the density of water 226
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Fig. 6. Accretion thickness ratio as a function of freestream wind speed for six different particles on a vertical 0.20 m diameter cylinder.
Fig. 7. Accretion thickness ratio as a function of wind speed for six different particles on a horizontal 0.20 m diameter cylinder.
increases with wind speed due to enhanced collision efficiency. The slope is determined by the terminal velocity of the particles. The downward trajectories of particles with smaller terminal velocity are more easily deflected by horizontal wind, leading to larger accretions on a vertical cylinder. The results presented in Fig. 6, especially for large accretions, should be treated with caution since the model does not consider any accretion shedding process. One could envisage that all or part of large accretions could eventually be shed due to gravity and aerodynamic forces. To identify those limits, an experimental campaign should be undertaken. However, whatever the shedding limits, they will be reached faster when the accretion thickness ratio is higher, that is, with increasing wind speed.
2.6. Simplified model for vertical and horizontal cylinders with yaw angle of 90° In order to obtain a rough estimate of accretion growth, the analysis could be further simplified. While the collision efficiency can, in some cases, diminish the incoming flux by up to 40% (see Fig. 2), in most conditions this decrease is much smaller, especially for small diameter cylinders (Fig. 5). Consequently, to a first approximation, one may assume a collision efficiency of unity. This assumption eliminates the influence of cylinder diameter on the results. Assuming that the ratio of precipitation density to accretion density is unity eliminates the distinction between freezing rain and snow. Applying these two assumptions, the accretion thickness ratio as a function of wind speed for vertical and horizontal cylinder orientations is shown in Fig. 8. In this very simplified model, the particle terminal velocity and wind speed uniquely determine the accretion growth for both vertical and horizontal cable configurations. The accretion thickness ratio increases with increasing wind speed and decreasing terminal velocity. The difference between vertical and horizontal configurations decreases with increasing wind speed and decreasing terminal velocity.
2.5. Accretion on a horizontal cylinder with yaw angle of 90° We next examine freezing rain and snow impingement in the simple case of a horizontal cylinder whose inclination angle is 0°. Since the approach angle is 90°, = 1/ cos = (U /VT ) 2 + 1 . The free stream speed perpendicular to the cylinder axis is given by the vector sum of the wind speed and terminal velocity. Consequently, Eq. (3) assumes the following form:
h 2 = PV
P A
E ( U 2 + VT 2 )
U VT
2.7. Simplified model for an inclined cylinder with yaw angle of 90° In this section, we examine the influence of cable inclination angle on accretion size using the same simplifying assumptions as in Section 2.6 above. While the general result is given by Eqs. (1–3), we will consider a typical environmental situation where the yaw angle is 90° (precipitation is arriving from a direction perpendicular to the projection of the cylinder axis onto a horizontal surface). For simplicity, we will assume that the collision efficiency is unity. With these assumptions, Eq. (3) may be written in dimensionless form as:
2
+1
(5)
We begin by considering the accretion forming on a horizontal cylinder under calm conditions (U = 0), Fig. 7. For freezing rain with straight vertical trajectories, E = 1 and the accretion thickness ratio is 0.7. For wet and dry snow, the accretion thickness ratio is smaller due to a lower collision efficiency and a density ratio of unity. The accretion thickness ratio reaches a minimum of 0.36 for 1 mm dry snow particles. For a wind speed of approximately 1 m/s, the accretion thickness ratio is around 0.75 for all of the six cases under consideration. For larger values of wind speed, the results diverge, mainly due to different terminal velocities. Interestingly, for large wind speeds, the variation of the accretion thickness ratio with wind speed for a horizontal cylinder is similar to that for a vertical cylinder.
A
2
P
h = PV
cos 2
+
U VT
2
(6)
Because Eq. (6) has a simple form, it is easy to identify the influence of all of the essential parameters on the size of snow accretions. A graphical representation of Eq. (6) is depicted in Fig. 9 for three cylinder inclination angles. The results for all inclination angles lie 227
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Using the results developed in the present paper, we can now propose a more accurate approach. Two typical fence designs will be considered: vertical rod and chain link. The porosity of a fence is defined as =1 − AB/AT, where AB is the blocked area and AT is the total area. Using simple geometrical analysis, the porosity can be expressed as: for a vertical rod fence: ϕ = 1 − (D + 2h)/L and for a chain link fence: ϕ = [1 − (D + 2h)/L]2 where D is the fence rod diameter, L is the fence rod spacing, and h is the radial ice thickness. Based on Taylor et al. (2017), the following fence geometry values have been assumed: for vertical rod, D = 0.8 cm and L = 10 cm, and for chain link, D = 0.2 cm and L = 4.9 cm. Radial ice thickness has been estimated using Eq. (6) assuming ω = 90° for a vertical rod fence and ω = 45° for a chain link fence. Assuming a density ratio of unity, the fence porosity for both designs is shown as a function of the vertical precipitation and velocity ratio in Fig. 10. The horizontal black line at 30% porosity depicts the condition when the flow stagnates and no longer passes through the fence. The critical vertical precipitation that results in 30% fence porosity can be readily estimated from Fig. 10. A higher velocity ratio (stronger winds or smaller precipitation particles) produces critical stagnation conditions with less total precipitation, illustrating the important effect of wind in producing blockage. The total precipitation that leads to critical stagnation conditions is approximately three times less for chain link fences than for vertical rod fences.
Fig. 8. Accretion thickness ratio for vertical and horizontal cables as a function of wind speed, for five particle terminal velocities.
2.9. Simplified model for an inclined cylinder with yaw angle of 0° We have also examined the case when the airflow velocity vector is parallel to the vertical projection of the cable axis onto a horizontal surface, with snow impinging on the upper cable surface. In addition, collision efficiency is assumed to be unity. With these assumptions, Eq. (3) after transformation leads to the following expression: A
2
P
h U = sin PV VT
+ cos
(7)
The impinging mass is proportional to the velocity ratio and the slope of this relationship is given by the sine of the inclination angle. For a horizontal cable, airflow velocity does not influence the amount of impinging precipitation, Fig. 11. For calm conditions, the dimensional snow thickness clearly does not depend on yaw angle, as can be seen by comparing Figs. 9 and 11. For vertical cables, the value of yaw angle is irrelevant and snow thickness is proportional to the velocity ratio, as can be seen by comparing Figs. 9 and 11. It should be emphasised that all of the presented results are based on an assumption that all impinging snowflakes adhere to the cable and that there is no snow shedding. In addition, we have assumed that snowflakes approach the cable along straight-line trajectories, leading to a collision efficiency of unity. Both of those assumptions could lead to overestimation of the snow accretion mass. This is especially true for larger accretions where snow is expected to be shed at last partially, especially at high wind speeds. Experimental data will be needed to constrain snow accretion growth, by combining a snow shedding parameterization with the existing model.
Fig. 9. Dimensionless accretion thickness for three different cylinder inclination angles as a function of precipitation particle velocity ratio. Assuming the yaw angle of 90°.
between the vertical and horizontal configuration lines. This figure illustrates the important effect of wind speed in increasing accretion thickness, whatever the cylinder orientation and terminal velocity of the precipitation particles.
3. The morphogenetic snow accretion model
2.8. Influence of snow precipitation on fence porosity
Morphogenetic modelling was originally developed for aerospace in-flight icing applications (Szilder and Lozowski, 2004 and Szilder and Lozowski, 2018). Recently, the model has been modified to predict ice accretion shapes due to freezing rain precipitation (Szilder, 2017 and Szilder, 2018). In this contribution, the morphogenetic model will be further extended for snow accretion applications.
Snow and ice accretion forming on safety fences installed on bridge decks can have a significant effect on the aerodynamic stability of the bridge. The low porosity of snow-covered fences influences the flow around the bridge and in the bridge wake. The assumed relationship between precipitation conditions and decreasing fence porosity in a recent state-of-the art paper is very simplistic (Taylor et al., 2017). 228
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Fig. 10. Decrease of fence porosity due to snow/ice accretion as a function of vertical precipitation for three velocity ratio values. (a) vertical rod design (b) chain link design.
A 3D rectangular lattice defines the accretion domain. The results presented in this paper were obtained for an element size of 2 × 2 × 2 mm and a morphogenetic model domain of 200 × 300 × 400 cells, giving 0.4 × 0.6 × 0.8 m in the horizontal x and y, and vertical z directions, respectively. 3.2. Model result for wind-driven snow We have considered snow accretion on a cylinder of 0.2 m diameter. In all cases, the total vertical precipitation was 5 cm. This is the depth of the snow layer that would have accumulated on a horizontal surface. The morphogenetic model was run for wet and dry snow, with a particle size of 1 mm and two different terminal velocities (see Table 1). We investigated the influence of cable inclination angle, windspeed and wind yaw angle on the snow accretions. We began by examining wet snow accretion under calm conditions. When the snow accretes on a horizontal cylinder, the surface is relatively smooth, Fig. 12a. The maximum snow thickness is almost 5 cm at the top of the cable, Fig. 13a. Due to shadowing effects, the snow surface becomes rough on its edges. When the cylinder is inclined by 45°, the entire snow surface becomes rough for a similar reason, Fig. 12b. The cylinder inclination leads to a smaller impinging flux and so the maximum snow layer thickness is approximately 3 cm. Wet snow accretions were also examined for a windspeed of 2 m/s. Because the terminal velocity of wet snow particles of 1 mm diameter is 1.4 m/s, the particles approach the cylinder at a velocity of 2.4 m/s and the precipitation trajectory angle is 55° from the vertical. When the yaw angle is 0° (the wind blows parallel to the vertical projection of the cable axis onto a horizontal surface), the snow accretion on a horizontal cylinder has a porous structure, Figs. 12c and 13c. This porous structure arises because of strong shadowing effects due to a low particle approach angle of 35°. The horizontal wind enhances the approaching snow mass flux, and this leads to an increase in the snow layer thickness. When the cylinder is inclined at 45°, the snow structure is compact (approach angle of 80°) and the maximum thickness is approximately 8 cm, which is 60% greater than the snow thickness that would have formed on a horizontal surface. In the next two cases, we have assumed that the yaw angle is 30°. For a horizontal cylinder, this results in a particle approach angle of 45°, making the snow accretion more compact than for the case of a yaw angle of 0°, as can be seen by comparing Fig. 13e and Fig. 13c. Snow accreting on an inclined cylinder for a yaw angle of 30° has a compact structure, as shown in Figs. 12f and 13f.
Fig. 11. Dimensionless accretion thickness for three cylinder inclination angles, as a function of precipitation particle velocity ratio, assuming a yaw angle of 0°.
3.1. Morphogenetic model The principal characteristics of morphogenetic modelling applied to snow accretion are described below. A morphogenetic model is a discrete element, random walk model that emulates the motion of individual elements arriving at the accretion surface. For snow accretion, individual model elements may be imagined to be a single snowflake or to consist of an ensemble of snowflakes which undergo identical histories. Keeping this in mind, we will henceforth use the term “element” to describe the particles that comprise the model accretion. Each element has a different history because of the stochastic nature of the model. Element impingement on a cylinder is determined by the distribution of collision efficiency calculated using the tools described in the analytical model section. The element impingement locations are determined randomly according to a specified probability distribution. Once an element impinges on the surface, it moves to the closest “cradle” location (Szilder and Lozowski, 2004). The model is sequential, so that as soon as the final location of a particular element is determined, the behaviour of the next element is considered. 229
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Fig. 12. Wet snow accretion on a cylinder of 0.20 m diameters for a total vertical precipitation of 5 cm. (a) Windspeed 0 m/s, inclination angle 0°. (b) Windspeed 0 m/s, inclination angle 45°. (c) Windspeed 2 m/s, inclination angle 0°, yaw angle 0°. (d) Windspeed 2 m/s, inclination angle 45°, yaw angle 0°. (e) Windspeed 2 m/s, inclination angle 0°, yaw angle 30°. (f) Windspeed 2 m/s, inclination angle 45°, yaw angle 30°.
The next set of results was obtained for dry snow particles, assuming a diameter of 1 mm and a terminal velocity of 0.7 m/s. To simplify the comparison between dry and wet snow accretion, the same values of windspeed, inclination and yaw angle have been assumed (compare Fig. 12 with Fig. 14 and Fig. 13 with Fig. 15). Under calm conditions, the dry snow accretion thickness and extent are smaller than for wet snow. This results from the smaller collision
efficiencies, Fig. 2, and impingement extent angles, Fig. 4, for dry snow. The lower terminal velocity of dry snow leads to a smaller particle approach velocity of 2.1 m/s for a windspeed of 2 m/s. Consequently, the precipitation flux has a larger horizontal component and the precipitation fall angle is 70.7° from the vertical. Snow accretions forming on a horizontal cylinder have a highly porous structure, Figs. 14c and 15c. However, for an inclined cylinder, the empty voids inside the snow 230
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Fig. 13. Cross-section in the middle of the snow accretions depicted in Fig. 12. The red point represents the stagnation line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).
matrix disappear, Fig. 15d. The greater snow layer thickness is, paradoxically, due to the decreased terminal velocity, which gives rise to a higher precipitation flux directed towards the cylinder (compare Fig. 15d and Fig. 13d). For a yaw angle of 30°, the dry snow accretion is greater than the comparable wet snow accretion (Figs. 15e and f versus Figs. 13e and f). In addition, the dry snow accretions tend to be more porous and located more on the side of the cylinder.
3.3. Comparison between the morphogenetic and analytical models The cross-sectional area of snow accretion predicted by the morphogenetic model has been compared with predictions by two analytical models. The cross-sectional area A = 0.5π D h, see Eq. (2), is used for comparison and the accretion density is assumed to equal the precipitation density ρA = ρP. The following two analytical models have been considered: 1. General analytical model based on Eq. (3) 231
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Fig. 14. Dry snow accretion for the same conditions as given in the Fig. 12 caption.
A=E
(8)
D PV
derived assuming a yaw angle of 0°. In addition, Eq. (9) was derived assuming a collision efficiency of unity, leading to overprediction of snow accretion especially for low snowflakes terminal velocities. All the morphogenetic model results presented in Section 3.2. have been compared with the two analytical models listed above. The cylinder diameter, D = 20 cm, and vertical precipitation, PV = 5 cm, are the same for all of the examined cases. The snow cross-sectional area depicted in Figs. 13 and 15 has been compared with the analytical model prediction, Fig. 16. For wet snow conditions, the agreement
2. Simplified model for inclined cylinder with yaw angle of 0° based on Eq. (7)
A=
U sin VT
+ cos
D PV
(9)
It should be noted that the conditions for which the morphogenetic model was run include cases with a yaw angle of 30°, but Eq. (9) was 232
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Fig. 15. Cross-section in the middle of the snow accretions depicted in Fig. 14.
between the morphogenetic model and the general analytical model, Eq. (8), is very good for all 6 cases. The largest discrepancy is approximately 10% for cases (c) and (e). This difference results from the variation of snow cross-sectional area along the cable. The agreement between the morphogenetic model and the simplified model defined by Eq. (9) is more pronounced. For calm conditions, when the yaw angle is irrelevant (cases (a) and (b)) and when the yaw angle is 0° (cases (c) and (d)), the simplified model overpredicts due to its collision efficiency of unity assumption. For cases (e) and (f) when the yaw angle is 30°, the comparison with Eq. (9) is questionable since, Eq. (9) was derived assuming a yaw angle of 0°.
Since dry snow is characterized by smaller terminal velocity, the collision efficiency and accreting mass under windless conditions are smaller (cases (a) and (b)) than for wet snow, as predicted by the morphogenetic model. However, when windspeed is non-zero, the horizontal component of snowflake velocity is larger, leading to an increasing impinging snow mass, especially for an inclined cable. The agreement between the morphogenetic model and the general analytical model is overall good, with the biggest discrepancy (about 30%) for porous snow accretions, case (c) (see Fig. 15c). The simplified model given by Eq. (9), however, shows a large discrepancy, due to smaller terminal velocities. 233
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Fig. 16. Snow accretion cross sectional area for 6 wet snow and 6 dry snow conditions, Figs. 12-13 and Figs. 14-15, respectively, as predicted by two analytical models and the morphogenetic model. (a) Wet snow, (b) Dry snow.
4. Conclusions
Acknowledgements
We have developed a novel analytical model to predict snow accretions. The snow impingement characteristics were computed using a particle tracking trajectory code, coupled with 2D potential flow solutions. This simple model allows a quick analysis of snow accretion size as a function of precipitation type and amount, cylinder inclination angle, and wind speed and direction. Our analysis of snow and freezing rain accretions forming on vertical, inclined and horizontal cylinders revealed that the two most influential parameters are the ratio of windspeed to particle terminal velocity, and cylinder inclination angle. We have also demonstrated that the porosity of a chain fence decreases approximately three times faster than the porosity of a vertical rod fence, when exposed to the same environmental conditions. Our model allows examination of other possible geometries for safety fences. We have also adapted our existing morphogenetic icing model in order to simulate snow accretions on arbitrarily oriented cylinders/ cables. The adapted morphogenetic model can be used to predict complex details of the snow accretion shape, including surface roughness and embedded voids. Detailed accretion structures predicted by the morphogenetic model can be 3D-printed and used to examine the relationship between the amount of snow precipitation and the resulting changes in the aerodynamic characteristics of cables. It should be noted that some model assumptions, such as a constant non-fluctuating windspeed and no snow shedding or sliding, may compromise the model predictions. Consequently, some of the present results should be treated with caution, especially when snow impinges on the cylinder sides. Because there is no shedding, the models tend to overpredict the accreted snow mass, which is unrealistic but adds an engineering safety factor. In the future, we plan to perform a model validation exercise using experimental data. It is anticipated that this will lead to the development of a snow shedding module that will improve the accuracy of the snow accretion predictions. In addition, since the developed models relate atmospheric conditions and cable configuration to snow accretion size, it is envisaged that the models can be used to help correlate local climatological snow data with resulting extreme snow accretion events.
The study was funded by Infrastructure Canada as part of the Climate-Resilient Buildings and Core Public Infrastructure project. References Colli, M., Lanza, L.G., Rasmussen, R., Theriault, J.M., 2016. The collection efficiency of shielded and unshielded precipitation gauges. Part II: modeling particle trajectories. J. Hydrometeorol. 17, 245–255. Dalle, B., Admirat, P., 2011. Wet snow accretion on overhead lines with French report of experience. Cold Reg. Sci. Technol. 65, 43–51. Khvorostyanov, V.I., Curry, J.A., 2002. Terminal velocities of droplets and crystals: power laws with continuous parameters. J. Atmos. Sci. 59, 1872–1884. Kleissl, K., Georgakis, C.T., 2010. Bridge ice accretion and de- and anti-icing systems: a review. In: The 7th International Cable Supported Bridge Operators' Conference, China. Kollár, L.E., Olqma, O., Farzaneh, M., 2010. Natural wet-snow shedding from overhead cables. Cold Reg. Sci. Technol. 60, 40–50. Kumpf, J., Helmicki, A., Nims, D.K., Hunt, V., Agrawal, S., 2012. Automated ice inference and monitoring on the Veterans' glass city Skyway bridge. Journal of Bridge Engineering, ASCE Vol. 17, 975–978. Lai, T., 2009. Critical analysis of the Shoots bridge, central span of the Second Severn Crossing. In: Proceedings of Bridge Engineering 2 Conference 2009, April 2009. University of Bath, Bath, UK. Makkonen, L., Wichura, B., 2010. Simulating wet snow loads on power line cables by a simple model. Cold Reg. Sci. Technol. 61, 73–81. Roldsgaard, J.H., Kiremidjian, A., Georgakis, C.T., Faber, M.H., 2013. Preliminary probabilistic prediction of ice/snow accretion on stay cables based on meteorological variables. In: Safety, Reliability, Risk and Life-Cycle Performance of Structures and Infrastructures, ISBN 978-1-138-00086-5. Szilder, K., 2017. Numerical modeling of ice accretion shape due to freezing precipitation on engineering structures. In: 7th European and African Conference on Wind Engineering, EACWE. 2017. pp. 5 64. Szilder, K., 2018. Theoretical and experimental study of ice accretion due to freezing rain on an inclined cylinder. Cold Reg. Sci. Technol. 150, 25–34. Szilder, K., Lozowski, E.P., 2004. Novel two-dimensional modeling approach for aircraft icing. J. Aircr. 41 (4), 854–861. Szilder, K., Lozowski, E.P., 2018. Comparing experimental ice accretions on a swept wing with 3D morphogenetic simulations. J. Aircr. https://doi.org/10.2514/1.C034879. Szilder, K., Lozowski, E.P., Reuter, G., 2002. Analytical modelling of ice load for a family of dry ice accretion shapes. Int. J. Offshore Polar Eng. 12 (2), 178–183. Taylor, Z., Stoyanoff, S., Dallaire, P.O., Kelly, D., Irwin, P., 2017. Aerodynamics of longspan bridges: susceptibility to snow and ice accretion, ASCE. J. Struct. Eng. 143 (7) 04017039.
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Snow accretion prediction on an inclined cable Krzysztof Szilder
⁎
T
Aerospace Research Centre, National Research Council, Ottawa, ON K1A 0R6, Canada
ARTICLE INFO
ABSTRACT
Keywords: Snow Freezing rain Snow accretion prediction Bridge cable Cable aerodynamics
We have developed a novel analytical model to predict snow accretions. Our analysis of snow and freezing rain accretions forming on vertical, inclined and horizontal cables revealed that the two most influential parameters are the ratio of windspeed to particle terminal velocity, and cylinder inclination angle. We have also adapted our existing morphogenetic icing model to simulate snow accretions on arbitrarily oriented cylinders/cables. The adapted morphogenetic model can be used to predict complex details of the snow accretion shape, including surface roughness and embedded voids. Detailed accretion structures predicted by the morphogenetic model can be 3D-printed and used to examine the relationship between the amount of snow precipitation and the resulting changes in the aerodynamic characteristics of cables.
1. Introduction During a snowstorm, snow may accrete on the inclined cables of a bridge, such as stay cables on a cable-stayed bridge or on the main cables of a suspension bridge. Over time, snow metamorphosis may occur, allowing the accretion to settle and consolidate. Partial melting and refreezing may also occur. Snow/ice formation on bridge cables can result in ice falling onto pedestrians and traffic, leading to personal injuries and property damage. In severe cases, the snow/ice accretions may lead to bridge closures due to induced cable vibrations. A number of bridges in cold regions have experienced snow/ice shedding and associated vibrations as shown by Roldsgaard et al. (2013), Kleissl and Georgakis (2010), Lai (2009), and Kumpf et al. (2012). With the increasing potential, due to climate change, for freezing rain events to occur, ice accumulation and shedding from bridge cables could become of greater concern in the coming years and decades. The need to understand and predict these phenomena to build durable structures, while ensuring the safety of all bridge users, is the motivation behind this project. For example, the Port Mann cable-stayed bridge in Vancouver, Canada was opened to traffic in 2011. During its first year of service, ice and wet snow built up on the stay cables and ice/snow pieces fell onto the traffic lanes, leading to damage of multiple vehicles, injuries to pedestrians and closure of the bridge for several hours. Recently, in December 2016, a similar situation occurred on the same bridge. The dynamic behaviour of the bridge had to be investigated and a retrofit of the surface of the stay cables has been considered. The Pierre-Laporte Bridge in Québec City, Canada was closed in February 2018 because pieces of ice fell from the cables of this
⁎
suspension bridge. Such events are expected to become more frequent due to the effects of climate change. To better understand ice/snow accretion on cables, a number of predictive tools have been developed. There are several snow accretion models for cables, but they were all developed for power line cables that tend to rotate under load, so that a snow sleeve forms (Kollár et al., 2010; Makkonen and Wichura, 2010; Dalle and Admirat, 2011). Bridge stay cables, however, do not rotate. Moreover, existing models make simple assumptions about the resulting accretion shape, and the resulting shape is often circular or semi-circular. While this assumption may be adequate for estimating gravitational loads, it is insufficient for analyzing the subtle effects of accretion shape on cable aerodynamics. The objective of this research was to extend the ice accretion models developed for freezing rain applications by Szilder (2018) to wet and dry snow conditions. The numerical models developed by Szilder (2018) have enabled the prediction of ice accretion details that could not have been anticipated using existing models. These details are crucial for determining the aerodynamic behaviour of bridge cables and other nonrotating cylindrical objects during precipitation. Using the updated numerical model for snow and ice, accretion shapes can be predicted for any desired environmental conditions and can be 3D-printed, using a technique such as selective laser sintering, to make physical models. The physical models can then be tested in a wind tunnel to investigate the aerodynamic characteristics of the snow covered cables. In this paper, two types of snow accretion models were developed:
• Analytical models that allow an estimate of snow accretion size; and • Numerical models that predict details of snow accretion shapes.
Corresponding author. E-mail address: [email protected].
https://doi.org/10.1016/j.coldregions.2018.11.001 Received 13 July 2018; Received in revised form 18 September 2018; Accepted 1 November 2018 Available online 03 November 2018 0165-232X/ Crown Copyright © 2018 Published by Elsevier B.V. All rights reserved.
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2. Analytical model of snow accretion on a bridge cable
mass on the cylinder, over plus/minus 90° from the stagnation line, gives the snow thickness, h, as a simple measure of the total snow accretion. The expression for cross-sectional area is valid for h ≪ D; therefore, for relatively large snow accretions this simplification might be questionable. Alternatively, it could be assumed that there is no displacement of snowflakes after impingement and that a snow layer of constant thickness grows in a direction towards the impinging straightline precipitation. With the latter assumption, the accretion cross-section can be expressed as the product of accretion thickness and cable diameter. This requires replacement of 0.5π by unity both in Eq. (2) and in all subsequent equations in the paper. Future experimental data may help to identify conditions when the factor 0.5π should be replaced by unity. For now, however, we will retain the factor 0.5π in all relevant expressions in the paper. In addition, snow accretion experimental data may help to identify accretion shapes like those considered in Szilder et al. (2002), where more complex geometrical models were developed for ice accretion on a cable. It should be noted that in all models of snow accretion developed in this paper, it has been assumed that there is no snow shedding. This is equivalent to an assumption that there is complete snow sticking and hence that the accretion efficiency is unity. The non-dimensional ratio of accretion thickness to vertical precipitation is obtained by combining Eqs. (1) and (2):
The first half of the paper presents the development of an analytical model for snow accretion on bridge cables that can be used to understand the role of key physical factors using a simplified, systematic approach. The general equations for snow accretion are presented, followed by several case studies and model simplifications in order to understand the snow accretion process on inclined cables and the parameters that have the most significant impact on snow accretion. The influence of snow precipitation on bridge fence porosity is also evaluated using the developed analytical tools. 2.1. Accretion on a bridge cable in 3D A relationship for impinging freezing rain mass on a bridge cable, with arbitrary orientation in three-dimensional space, was derived in Szilder (2018) assuming straight-line drop trajectories. While a straightline trajectory assumption can be justified for freezing rain, it could be questionable for snow due to its low density and low terminal speed. Consequently, in the present analysis, which applies to snow accretion, trajectories can depart from straight lines, leading to modification of the governing equations derived in Szilder (2018). The total collision efficiency, E, is introduced and defined as the ratio of the mass flux of impinging particles (snow crystals) to the mass flux that would be experienced by the surface if the particles were not deflected by the airflow. Consequently, the impinging mass per unit length of bridge cable, (kg/m), is given by: m=E
PV D
where
P
=
sin cos
and
cos = cos sin
h 2 = PV
(3)
The following six cases have been considered: where: d is the particle diameter. Freezing rain values were adopted from Khvorostyanov and Curry (2002) and a relationship between the snow particle size and its density and terminal velocity was taken from Colli et al. (2016). It was assumed that snow particles are spherical and their physical characteristics, bulk density and terminal velocity are given by a power law as a function of particle diameter.
where: λ is the overall impingement coefficient, PV is the vertical precipitation (m), D is the cylinder diameter (m), and ρP is the precipitation density (kg/m3), δ is the approach angle, γ is the yaw angle, and ω is the cylinder inclination angle, Fig. 1. The precipitation angle is given by ε = tan−1(U/VT) where U is the horizontal windspeed (m/s) and VT is the terminal speed of the falling particle (m/s). For more details, please refer to Szilder (2018). For simplicity, we have assumed that the impinging precipitation forms an accretion of constant thickness, h (m), on the exposed half of the cable surface: A
E
2.2. Examined precipitation cases
sin cos cos
(1)
m = 0.5 D h
P A
2.3. Collision efficiency in 2D The collision efficiency of the impinging particles for these six cases was calculated assuming 2D potential flow around a circular cylinder. The results are presented as a function of freestream velocity for the six cases in Fig. 2. When particle trajectories are straight lines, the collision efficiency is unity. Departure from straight-line trajectories results in a decreasing collision efficiency. The trajectories and impingement location for the two cases identified by dots in Fig. 2 are shown in Fig. 3. For these freezing rain and dry snow cases, respectively, 92% and 64% of the incoming mass flux impinges on the surface. Fig. 3 also illustrates the definition of the impingement angle α. The dependence of this angle on freestream velocity and particle type and size is shown in Fig. 4. Lower windspeeds and smaller particles are associated with smaller collision efficiencies and smaller impingement angles. Particle trajectory calculations were also performed for a smaller cylinder of 0.01 m diameter and the results are shown in Fig. 5. Smaller
(2)
where: ρA is the accretion density. The above assumption of uniform spreading of the impinging snow
Table 1 Properties of examined precipitation cases.
Freezing rain Wet snow Dry snow
Fig. 1. A sketch of the general geometry for freezing precipitation impinging on a cable. The lines that are parallel are indicated in the schematic. 225
d (mm)
ρΡ (kg/m3)
VT (m/s)
1 5 1 5 1 5
1000 1000 720 144 170 34
4 9 1.4 1.9 0.7 0.9
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Fig. 2. Collision efficiency as a function of freestream velocity for six different particles impacting a cable of 0.20 m diameter.
Fig. 4. Impingement angle as a function of freestream velocity for six different particles impacting a cable of 0.20 m diameter.
Fig. 3. Limiting trajectories for two particles of 1 mm diameter in a freestream airflow of 1 m/s: freezing rain (black) and dry snow (blue). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 5. Collision efficiency as a function of freestream velocity for six different particles impinging on a cable of 0.01 m diameter.
obstacles are associated with smaller trajectory deflections and consequently higher collision efficiencies. Based on these results, assuming a collision efficiency of unity for a 0.01 m diameter cylinder, the errors would be less than 5%, if the freestream velocity exceeds 0.5 m/s.
and the accretion density is the density of ice, making the density ratio 1.1. For snow, we assume that the density of precipitating snow (measured on a horizontal surface) is the same as the density of snow accreting on the cylinder, making the density ratio unity. To the extent that snow density may be influenced by gravity and wind, this assumption may need to be revisited. The collision efficiency is a function of the component of freestream velocity perpendicular to the cylinder axis, which is equal to the horizontal windspeed for this case. When the wind speed is negligible, there is no accretion on a vertical cylinder. To a first approximation, the accretion thickness ratio is a linear function of wind speed. The slope of this linear relationship
2.4. Accretion on a vertical cylinder We begin by examining freezing rain and snow impingement in the simple case of a vertical cylinder, inclination angle 90°. Since λ = tanε = U/VT, Eq. (3) may be simplified to:
h 2 = PV
P A
E (U )
U VT
(4)
For freezing rain, the precipitation density is the density of water 226
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Fig. 6. Accretion thickness ratio as a function of freestream wind speed for six different particles on a vertical 0.20 m diameter cylinder.
Fig. 7. Accretion thickness ratio as a function of wind speed for six different particles on a horizontal 0.20 m diameter cylinder.
increases with wind speed due to enhanced collision efficiency. The slope is determined by the terminal velocity of the particles. The downward trajectories of particles with smaller terminal velocity are more easily deflected by horizontal wind, leading to larger accretions on a vertical cylinder. The results presented in Fig. 6, especially for large accretions, should be treated with caution since the model does not consider any accretion shedding process. One could envisage that all or part of large accretions could eventually be shed due to gravity and aerodynamic forces. To identify those limits, an experimental campaign should be undertaken. However, whatever the shedding limits, they will be reached faster when the accretion thickness ratio is higher, that is, with increasing wind speed.
2.6. Simplified model for vertical and horizontal cylinders with yaw angle of 90° In order to obtain a rough estimate of accretion growth, the analysis could be further simplified. While the collision efficiency can, in some cases, diminish the incoming flux by up to 40% (see Fig. 2), in most conditions this decrease is much smaller, especially for small diameter cylinders (Fig. 5). Consequently, to a first approximation, one may assume a collision efficiency of unity. This assumption eliminates the influence of cylinder diameter on the results. Assuming that the ratio of precipitation density to accretion density is unity eliminates the distinction between freezing rain and snow. Applying these two assumptions, the accretion thickness ratio as a function of wind speed for vertical and horizontal cylinder orientations is shown in Fig. 8. In this very simplified model, the particle terminal velocity and wind speed uniquely determine the accretion growth for both vertical and horizontal cable configurations. The accretion thickness ratio increases with increasing wind speed and decreasing terminal velocity. The difference between vertical and horizontal configurations decreases with increasing wind speed and decreasing terminal velocity.
2.5. Accretion on a horizontal cylinder with yaw angle of 90° We next examine freezing rain and snow impingement in the simple case of a horizontal cylinder whose inclination angle is 0°. Since the approach angle is 90°, = 1/ cos = (U /VT ) 2 + 1 . The free stream speed perpendicular to the cylinder axis is given by the vector sum of the wind speed and terminal velocity. Consequently, Eq. (3) assumes the following form:
h 2 = PV
P A
E ( U 2 + VT 2 )
U VT
2.7. Simplified model for an inclined cylinder with yaw angle of 90° In this section, we examine the influence of cable inclination angle on accretion size using the same simplifying assumptions as in Section 2.6 above. While the general result is given by Eqs. (1–3), we will consider a typical environmental situation where the yaw angle is 90° (precipitation is arriving from a direction perpendicular to the projection of the cylinder axis onto a horizontal surface). For simplicity, we will assume that the collision efficiency is unity. With these assumptions, Eq. (3) may be written in dimensionless form as:
2
+1
(5)
We begin by considering the accretion forming on a horizontal cylinder under calm conditions (U = 0), Fig. 7. For freezing rain with straight vertical trajectories, E = 1 and the accretion thickness ratio is 0.7. For wet and dry snow, the accretion thickness ratio is smaller due to a lower collision efficiency and a density ratio of unity. The accretion thickness ratio reaches a minimum of 0.36 for 1 mm dry snow particles. For a wind speed of approximately 1 m/s, the accretion thickness ratio is around 0.75 for all of the six cases under consideration. For larger values of wind speed, the results diverge, mainly due to different terminal velocities. Interestingly, for large wind speeds, the variation of the accretion thickness ratio with wind speed for a horizontal cylinder is similar to that for a vertical cylinder.
A
2
P
h = PV
cos 2
+
U VT
2
(6)
Because Eq. (6) has a simple form, it is easy to identify the influence of all of the essential parameters on the size of snow accretions. A graphical representation of Eq. (6) is depicted in Fig. 9 for three cylinder inclination angles. The results for all inclination angles lie 227
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Using the results developed in the present paper, we can now propose a more accurate approach. Two typical fence designs will be considered: vertical rod and chain link. The porosity of a fence is defined as =1 − AB/AT, where AB is the blocked area and AT is the total area. Using simple geometrical analysis, the porosity can be expressed as: for a vertical rod fence: ϕ = 1 − (D + 2h)/L and for a chain link fence: ϕ = [1 − (D + 2h)/L]2 where D is the fence rod diameter, L is the fence rod spacing, and h is the radial ice thickness. Based on Taylor et al. (2017), the following fence geometry values have been assumed: for vertical rod, D = 0.8 cm and L = 10 cm, and for chain link, D = 0.2 cm and L = 4.9 cm. Radial ice thickness has been estimated using Eq. (6) assuming ω = 90° for a vertical rod fence and ω = 45° for a chain link fence. Assuming a density ratio of unity, the fence porosity for both designs is shown as a function of the vertical precipitation and velocity ratio in Fig. 10. The horizontal black line at 30% porosity depicts the condition when the flow stagnates and no longer passes through the fence. The critical vertical precipitation that results in 30% fence porosity can be readily estimated from Fig. 10. A higher velocity ratio (stronger winds or smaller precipitation particles) produces critical stagnation conditions with less total precipitation, illustrating the important effect of wind in producing blockage. The total precipitation that leads to critical stagnation conditions is approximately three times less for chain link fences than for vertical rod fences.
Fig. 8. Accretion thickness ratio for vertical and horizontal cables as a function of wind speed, for five particle terminal velocities.
2.9. Simplified model for an inclined cylinder with yaw angle of 0° We have also examined the case when the airflow velocity vector is parallel to the vertical projection of the cable axis onto a horizontal surface, with snow impinging on the upper cable surface. In addition, collision efficiency is assumed to be unity. With these assumptions, Eq. (3) after transformation leads to the following expression: A
2
P
h U = sin PV VT
+ cos
(7)
The impinging mass is proportional to the velocity ratio and the slope of this relationship is given by the sine of the inclination angle. For a horizontal cable, airflow velocity does not influence the amount of impinging precipitation, Fig. 11. For calm conditions, the dimensional snow thickness clearly does not depend on yaw angle, as can be seen by comparing Figs. 9 and 11. For vertical cables, the value of yaw angle is irrelevant and snow thickness is proportional to the velocity ratio, as can be seen by comparing Figs. 9 and 11. It should be emphasised that all of the presented results are based on an assumption that all impinging snowflakes adhere to the cable and that there is no snow shedding. In addition, we have assumed that snowflakes approach the cable along straight-line trajectories, leading to a collision efficiency of unity. Both of those assumptions could lead to overestimation of the snow accretion mass. This is especially true for larger accretions where snow is expected to be shed at last partially, especially at high wind speeds. Experimental data will be needed to constrain snow accretion growth, by combining a snow shedding parameterization with the existing model.
Fig. 9. Dimensionless accretion thickness for three different cylinder inclination angles as a function of precipitation particle velocity ratio. Assuming the yaw angle of 90°.
between the vertical and horizontal configuration lines. This figure illustrates the important effect of wind speed in increasing accretion thickness, whatever the cylinder orientation and terminal velocity of the precipitation particles.
3. The morphogenetic snow accretion model
2.8. Influence of snow precipitation on fence porosity
Morphogenetic modelling was originally developed for aerospace in-flight icing applications (Szilder and Lozowski, 2004 and Szilder and Lozowski, 2018). Recently, the model has been modified to predict ice accretion shapes due to freezing rain precipitation (Szilder, 2017 and Szilder, 2018). In this contribution, the morphogenetic model will be further extended for snow accretion applications.
Snow and ice accretion forming on safety fences installed on bridge decks can have a significant effect on the aerodynamic stability of the bridge. The low porosity of snow-covered fences influences the flow around the bridge and in the bridge wake. The assumed relationship between precipitation conditions and decreasing fence porosity in a recent state-of-the art paper is very simplistic (Taylor et al., 2017). 228
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Fig. 10. Decrease of fence porosity due to snow/ice accretion as a function of vertical precipitation for three velocity ratio values. (a) vertical rod design (b) chain link design.
A 3D rectangular lattice defines the accretion domain. The results presented in this paper were obtained for an element size of 2 × 2 × 2 mm and a morphogenetic model domain of 200 × 300 × 400 cells, giving 0.4 × 0.6 × 0.8 m in the horizontal x and y, and vertical z directions, respectively. 3.2. Model result for wind-driven snow We have considered snow accretion on a cylinder of 0.2 m diameter. In all cases, the total vertical precipitation was 5 cm. This is the depth of the snow layer that would have accumulated on a horizontal surface. The morphogenetic model was run for wet and dry snow, with a particle size of 1 mm and two different terminal velocities (see Table 1). We investigated the influence of cable inclination angle, windspeed and wind yaw angle on the snow accretions. We began by examining wet snow accretion under calm conditions. When the snow accretes on a horizontal cylinder, the surface is relatively smooth, Fig. 12a. The maximum snow thickness is almost 5 cm at the top of the cable, Fig. 13a. Due to shadowing effects, the snow surface becomes rough on its edges. When the cylinder is inclined by 45°, the entire snow surface becomes rough for a similar reason, Fig. 12b. The cylinder inclination leads to a smaller impinging flux and so the maximum snow layer thickness is approximately 3 cm. Wet snow accretions were also examined for a windspeed of 2 m/s. Because the terminal velocity of wet snow particles of 1 mm diameter is 1.4 m/s, the particles approach the cylinder at a velocity of 2.4 m/s and the precipitation trajectory angle is 55° from the vertical. When the yaw angle is 0° (the wind blows parallel to the vertical projection of the cable axis onto a horizontal surface), the snow accretion on a horizontal cylinder has a porous structure, Figs. 12c and 13c. This porous structure arises because of strong shadowing effects due to a low particle approach angle of 35°. The horizontal wind enhances the approaching snow mass flux, and this leads to an increase in the snow layer thickness. When the cylinder is inclined at 45°, the snow structure is compact (approach angle of 80°) and the maximum thickness is approximately 8 cm, which is 60% greater than the snow thickness that would have formed on a horizontal surface. In the next two cases, we have assumed that the yaw angle is 30°. For a horizontal cylinder, this results in a particle approach angle of 45°, making the snow accretion more compact than for the case of a yaw angle of 0°, as can be seen by comparing Fig. 13e and Fig. 13c. Snow accreting on an inclined cylinder for a yaw angle of 30° has a compact structure, as shown in Figs. 12f and 13f.
Fig. 11. Dimensionless accretion thickness for three cylinder inclination angles, as a function of precipitation particle velocity ratio, assuming a yaw angle of 0°.
3.1. Morphogenetic model The principal characteristics of morphogenetic modelling applied to snow accretion are described below. A morphogenetic model is a discrete element, random walk model that emulates the motion of individual elements arriving at the accretion surface. For snow accretion, individual model elements may be imagined to be a single snowflake or to consist of an ensemble of snowflakes which undergo identical histories. Keeping this in mind, we will henceforth use the term “element” to describe the particles that comprise the model accretion. Each element has a different history because of the stochastic nature of the model. Element impingement on a cylinder is determined by the distribution of collision efficiency calculated using the tools described in the analytical model section. The element impingement locations are determined randomly according to a specified probability distribution. Once an element impinges on the surface, it moves to the closest “cradle” location (Szilder and Lozowski, 2004). The model is sequential, so that as soon as the final location of a particular element is determined, the behaviour of the next element is considered. 229
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Fig. 12. Wet snow accretion on a cylinder of 0.20 m diameters for a total vertical precipitation of 5 cm. (a) Windspeed 0 m/s, inclination angle 0°. (b) Windspeed 0 m/s, inclination angle 45°. (c) Windspeed 2 m/s, inclination angle 0°, yaw angle 0°. (d) Windspeed 2 m/s, inclination angle 45°, yaw angle 0°. (e) Windspeed 2 m/s, inclination angle 0°, yaw angle 30°. (f) Windspeed 2 m/s, inclination angle 45°, yaw angle 30°.
The next set of results was obtained for dry snow particles, assuming a diameter of 1 mm and a terminal velocity of 0.7 m/s. To simplify the comparison between dry and wet snow accretion, the same values of windspeed, inclination and yaw angle have been assumed (compare Fig. 12 with Fig. 14 and Fig. 13 with Fig. 15). Under calm conditions, the dry snow accretion thickness and extent are smaller than for wet snow. This results from the smaller collision
efficiencies, Fig. 2, and impingement extent angles, Fig. 4, for dry snow. The lower terminal velocity of dry snow leads to a smaller particle approach velocity of 2.1 m/s for a windspeed of 2 m/s. Consequently, the precipitation flux has a larger horizontal component and the precipitation fall angle is 70.7° from the vertical. Snow accretions forming on a horizontal cylinder have a highly porous structure, Figs. 14c and 15c. However, for an inclined cylinder, the empty voids inside the snow 230
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Fig. 13. Cross-section in the middle of the snow accretions depicted in Fig. 12. The red point represents the stagnation line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).
matrix disappear, Fig. 15d. The greater snow layer thickness is, paradoxically, due to the decreased terminal velocity, which gives rise to a higher precipitation flux directed towards the cylinder (compare Fig. 15d and Fig. 13d). For a yaw angle of 30°, the dry snow accretion is greater than the comparable wet snow accretion (Figs. 15e and f versus Figs. 13e and f). In addition, the dry snow accretions tend to be more porous and located more on the side of the cylinder.
3.3. Comparison between the morphogenetic and analytical models The cross-sectional area of snow accretion predicted by the morphogenetic model has been compared with predictions by two analytical models. The cross-sectional area A = 0.5π D h, see Eq. (2), is used for comparison and the accretion density is assumed to equal the precipitation density ρA = ρP. The following two analytical models have been considered: 1. General analytical model based on Eq. (3) 231
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Fig. 14. Dry snow accretion for the same conditions as given in the Fig. 12 caption.
A=E
(8)
D PV
derived assuming a yaw angle of 0°. In addition, Eq. (9) was derived assuming a collision efficiency of unity, leading to overprediction of snow accretion especially for low snowflakes terminal velocities. All the morphogenetic model results presented in Section 3.2. have been compared with the two analytical models listed above. The cylinder diameter, D = 20 cm, and vertical precipitation, PV = 5 cm, are the same for all of the examined cases. The snow cross-sectional area depicted in Figs. 13 and 15 has been compared with the analytical model prediction, Fig. 16. For wet snow conditions, the agreement
2. Simplified model for inclined cylinder with yaw angle of 0° based on Eq. (7)
A=
U sin VT
+ cos
D PV
(9)
It should be noted that the conditions for which the morphogenetic model was run include cases with a yaw angle of 30°, but Eq. (9) was 232
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Fig. 15. Cross-section in the middle of the snow accretions depicted in Fig. 14.
between the morphogenetic model and the general analytical model, Eq. (8), is very good for all 6 cases. The largest discrepancy is approximately 10% for cases (c) and (e). This difference results from the variation of snow cross-sectional area along the cable. The agreement between the morphogenetic model and the simplified model defined by Eq. (9) is more pronounced. For calm conditions, when the yaw angle is irrelevant (cases (a) and (b)) and when the yaw angle is 0° (cases (c) and (d)), the simplified model overpredicts due to its collision efficiency of unity assumption. For cases (e) and (f) when the yaw angle is 30°, the comparison with Eq. (9) is questionable since, Eq. (9) was derived assuming a yaw angle of 0°.
Since dry snow is characterized by smaller terminal velocity, the collision efficiency and accreting mass under windless conditions are smaller (cases (a) and (b)) than for wet snow, as predicted by the morphogenetic model. However, when windspeed is non-zero, the horizontal component of snowflake velocity is larger, leading to an increasing impinging snow mass, especially for an inclined cable. The agreement between the morphogenetic model and the general analytical model is overall good, with the biggest discrepancy (about 30%) for porous snow accretions, case (c) (see Fig. 15c). The simplified model given by Eq. (9), however, shows a large discrepancy, due to smaller terminal velocities. 233
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Fig. 16. Snow accretion cross sectional area for 6 wet snow and 6 dry snow conditions, Figs. 12-13 and Figs. 14-15, respectively, as predicted by two analytical models and the morphogenetic model. (a) Wet snow, (b) Dry snow.
4. Conclusions
Acknowledgements
We have developed a novel analytical model to predict snow accretions. The snow impingement characteristics were computed using a particle tracking trajectory code, coupled with 2D potential flow solutions. This simple model allows a quick analysis of snow accretion size as a function of precipitation type and amount, cylinder inclination angle, and wind speed and direction. Our analysis of snow and freezing rain accretions forming on vertical, inclined and horizontal cylinders revealed that the two most influential parameters are the ratio of windspeed to particle terminal velocity, and cylinder inclination angle. We have also demonstrated that the porosity of a chain fence decreases approximately three times faster than the porosity of a vertical rod fence, when exposed to the same environmental conditions. Our model allows examination of other possible geometries for safety fences. We have also adapted our existing morphogenetic icing model in order to simulate snow accretions on arbitrarily oriented cylinders/ cables. The adapted morphogenetic model can be used to predict complex details of the snow accretion shape, including surface roughness and embedded voids. Detailed accretion structures predicted by the morphogenetic model can be 3D-printed and used to examine the relationship between the amount of snow precipitation and the resulting changes in the aerodynamic characteristics of cables. It should be noted that some model assumptions, such as a constant non-fluctuating windspeed and no snow shedding or sliding, may compromise the model predictions. Consequently, some of the present results should be treated with caution, especially when snow impinges on the cylinder sides. Because there is no shedding, the models tend to overpredict the accreted snow mass, which is unrealistic but adds an engineering safety factor. In the future, we plan to perform a model validation exercise using experimental data. It is anticipated that this will lead to the development of a snow shedding module that will improve the accuracy of the snow accretion predictions. In addition, since the developed models relate atmospheric conditions and cable configuration to snow accretion size, it is envisaged that the models can be used to help correlate local climatological snow data with resulting extreme snow accretion events.
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