(1)
Zhou Yi-Ren Chen Guo-Ping
Nanjing Hydraulic Research Institute, Nanjing, 210029 China
Abstract: In most cases, the wave characteristics around a large circular cylinder were studied to calculate wave forces on the piles. In this paper, by analyzing wave characteristics with Mac Camy and Fuchs’ diffraction theory, two relationships associated with the research of scour problem around a large circular cylinder are obtained. They will provide an essential theoretical basis for the study of scour around a large circular cylinder.
Keywords: wave characteristics, large cylinder, scour
A review of the previously published literature [1, 2, 3, 4] on the scour adjacent to an obstacle under wave action show that there are usually three different scour types according to the relative size of the structure, namely breakwater type, large cylinder type and pile type. For breakwater type the maximum scour depth is irrelative to the size of structure and is deeper than that of other types in same wave and sediment condition. For large cylinder type the maximum scour depth usually occurs in the front of the cylinder like breakwater type. However, for pile type the maximum scour depth may appear by the sides of a cylinder as in flow conditions. From the experiment’s results [5, 6], the criterion to divide three types were given with the relative diameter D/L, i.e.: As D/L ³ 0.5, in breakwater type; As 0.5>D/L³0.15, in large cylinder type; As D/L<0.15, in pile type.
The scour of bed particles adjacent to an obstacle begins when the velocities and accelerations of the water particles cause hydrodynamic forces sufficient to overcome gravity and cause the bed particles to move. Because the factors influencing the scour are very complex, the first step should be to analyze the wave characteristics around a cylinder when we study the scour around it. From the preceding discussion, it is concluded that the wave characteristics around a large circular cylinder are influenced directly by the relative diameter D/L. It is very important to analyze the relationship between wave characteristics and D/L for the purpose of studying the local scour around a cylinder.
When a progressive wave is incident on a vertical circular cylinder, it will be reflected backward along the surface of cylinder. The incident wave and the reflect wave, propagating in different directions, are superimposed, making the wave field around the cylinder changed greatly.
Mac Camy and Fuchs (1954)[7] have presented a diffraction theory in which the wave characteristics around a large circular cylinder could be well described by making use of the linear velocity potential f.
The Grid system used in wave field calculation is illustrated in Fig.1. In this system, based on the linearized, small- amplitude wave theory with the assumption of irrotational incompressible fluid, the complex velocity potential fi of incident wave can be written as
(1)
in which
，
，t is time.
Fig.1 Grid system
Transforming the grid system to the cylindrical coordinate (r, q, z) with the center of the cylinder as origin of coordinates, the Eq.1 can be altered as
(2)
in which
is the first kind of Bassel
function of order n.
The velocity potential fD of dispersion wave, which satisfies 3-dimensional Laplace equation and reflection condition of the cylinder face, can be obtained by using the boundary condition of the cylinder face, i.e.:
(3)
Combining fi and fD, and substituting the cylinder boundary condition: r=a（where “a ” is radius of the cylinder）into it, the total velocity potential f of the wave field can be expressed as
(4)
in which
is the first kind of Hankel
function of order n, expressed as
(5)
is the second kind of Bassel
function of order n.
is derivative of the first kind of Bassel function of order n;
is derivative of the first kind of
Hankel function of order n; By using Eq.5, the wave
field around the circular cylinder can be calculated.
Fig. 2 Comparison of the ratio of wave height between calculation and experiment
Saito, Sato and Shibayama（1988）[6] have measured wave fields above scoured bed around a large circular cylinder in their experiment. The comparison of the ratio of wave height between calculation and experiment is shown in Fig.2. From Fig.2, it can be seen that the calculation values have a general agreement with experiment ones in almost all regions, except of some area behind the cylinder. So it can be said that the diffraction theory presented by MacCamy – Fuchs may be used to analyze the wave characteristics in this study.
Wave characteristics in the front of a large cylinder
Because the maximum scour depth usually occurs in the front of a large circular cylinder, it is very important to analyze the wave characteristics there. Due to the superimposition between incident waves and reflect waves, the partial standing waves are produced in the front of a large cylinder, in which the real wave height is higher than incident one, but is lower than that in the front of a breakwater. The ratio of wave height calculated by MacCamy– Fuchs’ diffraction theory around a large circular cylinder in different wave periods and diameters is shown in Fig.3.
Fig.3 Ratio of wave height around a large circular cylinder in different wave periods and diameters
In Fig.3, it can be seen that there is a reflection area in the front of the cylinder because the incident waves are obstructed by it. The greater relative cylinder diameter is, the higher the maximum ratio of wave height is. As D =100cm, the maximum ratio of wave height in the front of cylinder gets to the value up to 1.8, when the wave velocity can be calculated by using standing wave theory.
Fig.4 The maximum ratio of wave height versus the relative diameter D/L
In the other hand, for same diameter cylinder,
different wavelength may also induce the change of wave height in the front of
the cylinder. The longer the wavelength is, the lower the ratio of wave height
is. Spotting the calculated data of the maximum ratio of wave height versus the
relative diameter D/L in Fig.4, we can find that the maximum ratio of wave
height raises with the increase of D/L. And as D/L>0.5, the value of
approximates to 2, when the waves
in front of a cylinder display standing wave characteristics. By using a
hyperbolic function, the relation curve can be presented as
(6)
in which
is the maximum ratio of wave height, D is diameter of circular cylinder, L is
incident wavelength. By using Eq.6, the real maximum wave height in the front of
the cylinder, which influencing scour depth directly, can be calculated.
Wave characteritics at sides of a large cylinder
When incident waves progress around a cylinder, water particles will oscillate along the surface of both cylinder sides, where the maximum horizontal velocity at the bottom is greater than that in no cylinder case. For the purpose of studying scour by the sides of the cylinder, it is very important to expect where the amplitude of bottom horizontal velocity is the greatest. The amplitude of bottom horizontal velocity calculated by MacCamy– Fuchs’ diffraction theory around a large circular cylinder in different wave periods and diameters is shown in Fig.5.
Fig.5 Amplitude of bottom horizontal velocity around a large circular cylinder in different wave periods and diameters
From Fig.5, it can be seen that the maximum amplitude of bottom velocity by the sides is always smaller than that in the front of cylinder as D<100cm. The site of the maximum bottom velocity amplitude in the condition of diameter D=100cm and period T=1.0s is apparently upper than that in T=2.0s. However, the place of the maximum velocity amplitude in D=45cm and T=1.0s is hinder than that in T=2.0. From this all, it is clearly that the place of the maximum velocity amplitude by the sides not only relates to the wave period but the cylinder diameter.
Defining q as the angle between the fiducial line
opposite to wave propagating direction and the line linking the cylinder center
and the maximum velocity spot by the side of cylinder, the calculated data of
the maximum velocity angle q versus the relative diameter D/L is
spotted in Fig.6. From Fig.6, we can find that the calculated data is clearly
divided into two parts: as D/L<0.5, the relationship nearly a line;
as D/L>0.5, the value of q varying little in a range of 750 ~
850. The diagram indicates that the maximum velocity site by the
sides of a cylinder will move downwards with the increase of relative diameter
D/L, but when D/L>0.5, the size of the cylinder almost has no influence over
the maximum velocity site. It is to say that a large cylinder can be looked upon
as a breakwater when D/L>0.5, which also verified the experimental results
presented by Agarie[5] and Saito[6]. Because we chiefly
pay attention to the large cylinder structure (0.15<D/L<0.5) in this
study, the relation curve function of the left side of the diagram can be
presented as
Fig. 6 The maximum velocity angle q versus the relative diameter D/L
(7)
By using Eq.7, the maximum velocity site by the sides of the cylinder, which influencing scour topography directly, can be calculated.
From above analysis, it is clear that wave field around a circular cylinder will change greatly comparing with that in no obstacle case. It can be concluded that the wave field can be approximately divided into three kinds of areas, namely standing wave area in the front of the cylinder, incident wave + diffracted wave area by the sides of the cylinder and sheltered area at the back of the cylinder. The wave characteristics in different areas are not alike. Because the wave height in sheltered area is relative low, it is not analyzed in this study.
The two relationships presented in this study are very important in researching the local scour problems around a large cylinder. Because the deepest scour usually occurs in the front of the large cylinder, Eq.6 can provide real dynamic condition for the empirical formula expecting the maximum scour depth around a large cylinder. Eq.7 can expect the maximum bottom velocity by the sides of a large cylinder, where the sediment moves most violently, which can help us to study sediment movement, especially in study of scour around a large cylinder under wave + flow action.
It should be noticed that we could find that the maximum bottom velocity by the side of cylinder sometimes is greater than that in the front of cylinder in this study, meanwhile the deepest scour appears in the front of the cylinder, which has been verified by experiment [8]. It indicates that the maximum bottom velocity is not the sole dynamic parameter for scour in different areas. So the further study on the wave characteristic around a large circular cylinder should be enhanced.
References
[1] S.Sato, N.Tanaka and I.-Irie, Study on scouring at the foot of coastal structures, Proc. of the 11th Coastal Eng. Conf., ASCE, Vol.1, 1968.
[2] S. L. Xie, Scouring patterns in front of vertical breakwaters and their influences on the stability of the foundations of the breakwaters, Coastal Eng. Group, Dep. of Civil Engineering, Delft University of Technology, Delft, pp. 61, 1981.
[3] D. R. Wells and R. M. Sorensen, Scour around a circular cylinder due to wave motion, Proc. of the 12th Coastal Eng. Conf., ASCE, Vol.2, 1970.
[4] Rukai Wang, J. B. Herbich. Combined current and wave-produced scour around a single pile. Texas A﹠M University, COE Report, No. 269, 1983.3.
[5] T. Agarie, H. Katsui, Scour around a large circular cylinder, Proc. of the 32th Japanese Conference on Coastal Eng., JSCE, 1985 (in Japanese).
[6] H. Saito, S. Sato, and T. Shibayama, Study on the wave field around a large circular column, Proc. of the 35th Japanese Conference on Coastal Eng., JSCE, 1988 (in Japanese).
[7] R. C. MacCamy, R. A. Fuchs, Wave forces on piles (A diffraction theory), Tech. Memo., BEB, No.69, 1954.
[8] Y. R. Zhou, G. P. Chen, Local scour around a large circular cylinder under irregular wave action, Nanjing Hydraulic Research Institute, No.2029, 2000.7 (in Chinese).