CN109726421B

CN109726421B - Method for acquiring envelope curve of wave force amplitude of cylindrical array based on constructive cancellation

Info

Publication number: CN109726421B
Application number: CN201810784157.6A
Authority: CN
Inventors: 曾晓辉; 于法军; 孙哲
Original assignee: Institute of Mechanics of CAS
Current assignee: Institute of Mechanics of CAS
Priority date: 2018-07-17
Filing date: 2018-07-17
Publication date: 2020-08-14
Anticipated expiration: 2038-07-17
Also published as: CN109726421A

Abstract

The invention provides a method for acquiring envelope curves of wave force amplitude of a cylindrical array based on constructive and destructive, which comprises the steps of establishing a coordinate system for a cylindrical array consisting of a limited number of sitting cylinders arranged in a straight line, determining related parameters, analyzing interference effects of diffracted waves of any cylinder and cylinders upstream and downstream of the cylinder, further obtaining a description model of wave force amplitude curve fluctuation intervals when wave incidence angles are equal to zero and not equal to zero, and a final expression of the horizontal coordinates of peak points and valley points, and then determining the vertical coordinates of the peak points and the valley points so as to obtain the positions of the peak points and the valley points; and connecting the peak points/valley points to obtain the upper/lower envelope curve of the wave force curve in the non-capture area. The invention can provide a reasonable preliminary solution under the condition of avoiding complicated calculation, proves that the fluctuation distance is only related to the total number of the cylinders in the cylinder array, the number of the columns for marking the positions of the cylinders and the wave incidence angle, can reduce the workload on the premise of ensuring the precision and shortens the design and evaluation period.

Description

Method for acquiring envelope curve of wave force amplitude of cylindrical array based on constructive cancellation

Technical Field

The invention relates to the field of ocean engineering, in particular to a method for acquiring an envelope curve of a wave force amplitude of wave force borne by any cylinder in a non-capture region under the action of waves by a cylinder array consisting of a large number of cylinders penetrating through a water surface based on a constructive cancellation theory.

Background

The ocean which accounts for 71 percent of the surface area of the earth contains abundant renewable energy sources such as recoverable resources such as petroleum, natural gas and the like and wind energy, wave energy and the like which can be used by human for a long time. With the increasing demand for energy and resources in economic development, it has become a clear trend to expand living space and seek various materials and energy supplies in the ocean.

Offshore structures as carriers need to be developed no matter marine resource exploitation, offshore space development or actual utilization of marine renewable energy. There is an important class of structures, although the upper structures are different, whose floats/support structures are composed of a plurality of cylinders (i.e., an array of cylinders) that penetrate the water surface. Such as offshore oil platforms, sea-crossing bridges, ultra-large floats, wave-power arrays, and the like. With the continuous expansion of the demand of the economic society for ocean development, the overall size of the ocean structure becomes larger and larger, and the size of the cylindrical array as the floating body/supporting structure of the ocean structure is also increased. The number of cylinders in a cylinder array increases from the first ones to tens, hundreds, and even thousands. A single row of bottomed cylindrical arrays is one of the typical versions of cylindrical arrays. The single row bottomed cylinder array here refers to: the circle center of the circular cross section obtained by the intersection of the horizontal plane and each cylinder is on a straight line, and the circular cross section is continuously communicated with the water surface from the water bottom and penetrates through the water surface to extend upwards to form a cylinder array. The wave force applied to the underwater cylindrical array is a key factor for determining the design scheme of the cylindrical array and ensuring the structural safety, and therefore, the rule that the amplitude of the wave force changes along with the dimensionless wave number needs to be mastered.

The amplitude of the wave force applied to any cylinder in the single-row seated cylinder array shown in fig. 1 fluctuates and fluctuates with the dimensionless wave number. Generally, the amplitude of the wave force experienced by a single column in a single row of a large number (e.g., a number greater than 9) finite array of columns varies with the number of waves with three distinct characteristics: 1) the wave force curve formed by the wave force amplitude changing with the dimensionless wave number has several high peak, the area where these high peak is located is called area I (region I) in the invention; 2) near the region I, the curve has a plurality of gradually-reduced secondary peaks and valleys, the heights of the secondary peaks are all lower than the peaks of the region I, and the fluctuation distance of the wave force curve changes along with the change of dimensionless wave numbers, and the region is called as a region II (region II); 3) outside the two regions mentioned above, there are very regular fluctuations in many places, which are referred to herein as region iii (region iii). The schematic diagram of the three regions is shown in fig. 2.

Region I and region II are related to near-trapping, and these two regions are referred to herein as "trapping-related regions". And region III, which is referred to herein as the "non-capture region". There are a number of studies published internationally and well understood for region I and region II involved in capture. For the non-capture area, the fluctuation rule of the non-capture area is not deeply researched, and a description model for describing the fluctuation distance of the non-capture area is further lacked. The fluctuation distance of the invention refers to: and the distance between the abscissas of two adjacent maximum values (or minimum values) on a wave force curve formed by the wave force amplitude along with the change of the dimensionless wave number. In the present invention, the maximum value point or the minimum value point is also referred to by "peak" or "valley". The non-capture zone fluctuation pitch is described to increase the design level to help extend the fatigue life of the structure at a lower cost. This is because, after a large number of calculations, it is found that the relative difference between the values of adjacent peaks and valleys in the region III of the wave force curve is sometimes large, and can be found only from a limited number of calculations, and can reach a maximum of about 20% in the region III.

Therefore, in the actual calculation of the hydrodynamic force, if the step size of the abscissa is not small enough, the error of the wave force calculation result in the region III may reach 20% or even more. For the "one-time" strength failure problem caused by extreme loads, this may not be much affected because the amplitude of the wave force at the peak of the trapping region is much higher than that of the non-trapping region, and the relatively small error of about 20% of the amplitude of the wave force in the non-trapping region does not affect the "one-time" failure of the structure. However, for fatigue failure due to cyclic loading, the above-mentioned wave force calculation error may have a significant negative impact, since the calculation of fatigue life requires accounting for the combined contribution of the wave force in a certain frequency range (rather than just considering the corresponding maximum value at the near-trapping frequency of the trapping region, as in the case of intensity analysis). This is because, when analyzing the linear time invariant system fatigue life, the spectral density function of the alternating stress response is equal to the input ocean wave spectral density multiplied by the square of the system transfer function digital-to-analog. The natural frequency of the elastic mode of the conventional marine structure is far higher than the wave frequency, so that the transfer function of the amplitude of the alternating stress amplitude can be obtained by multiplying the transfer function of the amplitude of the wave force shown in fig. 2 by a certain coefficient. If the wave force transfer function is calculated with a large error due to an improper step selection, the alternating stress magnitude transfer function also has a large error, and the squared error becomes larger (e.g., if the transfer function has a modulus error of 10%, the squared error increases to 20%, and if the transfer function has a modulus error of 20%, the squared error increases to 36%). Therefore, inaccurate alternating stress response results can be obtained, and the accuracy of fatigue life evaluation is further influenced. Considering that the design of the cylindrical array can make the near-tracking frequency of the system avoid the frequency band with larger energy of the sea waves, the alternating stress of the non-capture area can occupy a large part in the contribution to the fatigue damage. Thus, accurate calculation of the wave force in the non-capture zone as shown in fig. 2 is of great significance for accurate assessment of fatigue life.

In summary, the precondition for efficiently and accurately acquiring the wave force of the non-capture area is to grasp the wave force curve fluctuation characteristics of the non-capture area and obtain a description model capable of accurately predicting the wave force curve fluctuation distance of the non-capture area in advance. For a non-capture area occupying most of the wave force curve, namely an area III (the area has practical significance for evaluating the fatigue life of a structure), the fluctuation characteristics of the non-capture area are still lack of deep knowledge, and a description model of the fluctuation distance of the wave force curve formed by the wave force amplitude along with the dimensionless wave number change in the non-capture area is not used as a basis for efficient evaluation and design.

At present, when the design and evaluation of a relevant actual engineering structure are carried out based on a wave force curve, in order to avoid missing peak and valley points of the wave force curve, a large amount of complex calculations are usually required to be carried out on the wave force curve, and the problems of overlarge time cost, high calculation cost and the like are caused. When the scheme is evaluated, selected or initially designed, the method causes low efficiency, greatly increases cost and consumes time, and is very uneconomical. At this time, if the upper and lower limits of the wave force curve fluctuation can be accurately and quickly mastered, a reasonable preliminary solution can be provided under the condition of avoiding complicated calculation, so that a method for quickly and accurately obtaining the peak point and the valley point of the wave force curve is urgently needed, and the problems are solved.

Disclosure of Invention

The invention aims to provide a method for acquiring an envelope curve of a wave force amplitude curve of a wave force applied to any cylinder along the direction of a line connecting centers of circles in the horizontal section of a cylinder array in a non-capture area by using a constructive cancellation theory under the action of waves in the cylinder array consisting of a large number of cylinders with the same diameter penetrating through a water surface and arranged in a straight line.

Particularly, the invention provides a method for acquiring envelope curves of wave force amplitudes of a cylindrical array based on constructive cancellation, which comprises the following steps:

step 100, taking a region where a plurality of high-rise peaks in a wave force curve formed by wave force amplitude values changing along with dimensionless wave numbers as a region I, taking a region where secondary peaks and valleys which are lower than the high-rise peaks and have curve fluctuation distances changing along with dimensionless wave numbers near the high-rise peaks are as a region II, and taking the wave force curve excluding the region I and the region II as a region III as an analysis object region;

step 200, establishing a cylinder array coordinate system consisting of a plurality of same seated cylinders arranged in a straight line, determining related parameters, and converting a dimensionless wave number into a ratio of a distance between two adjacent cylinders to a wavelength, so that a wave force curve of each cylinder has a peak point when the diffracted waves of the cylinders generate constructive interference and a wave force curve has a valley point when the diffracted waves of the cylinders generate destructive interference; for the condition that the wave incident angle is equal to zero, analyzing the wave path difference of two paths of incident wave transmitted to any cylinder for diffraction and incident wave transmitted to the last cylinder at the downstream of the cylinder for diffraction, and solving a preliminary expression of the abscissa of any peak point and a preliminary expression of the abscissa of any valley point in a wave force curve area III;

step 300, at the upstream of any cylinder, summing the upstream-transmitted left-transmitted diffraction waves generated by the cylinder and the downstream-transmitted left-transmitted diffraction waves generated by each cylinder, simplifying the sum by using a hankel function to obtain specific positions of two equivalent cylinders which are equivalent to the current cylinder array and different from the current cylinder in position for the fluctuation distance problem, and substituting the position peak of the equivalent cylinder farther away from the cylinder into the preliminary expression of the abscissa of the valley point and the preliminary expression of the abscissa of the valley point for correction to obtain the final expression of the abscissa of any peak point and the final expression of the abscissa of any valley point in the wave force curve area III when the wave incident angle is equal to zero;

step 400, for the condition that the wave incident angle is not equal to zero, by using the same method as above, first analyzing the difference between the two paths of the incident wave transmitted to any cylinder for diffraction and the difference between the two paths of the incident wave transmitted to the first cylinder at the array end part at the upstream of the cylinder for diffraction, then analyzing the difference between the two paths of the incident wave transmitted to any cylinder for diffraction and the difference between the two paths of the incident wave transmitted to the last cylinder at the downstream of the cylinder for diffraction, and thus obtaining a first preliminary expression of the abscissa of any peak point and a first preliminary expression of the abscissa of any valley point of any cylindrical wave force curve in the area III due to the action of the cylinder diffracted wave at the upstream of the cylinder, and a second preliminary expression of the abscissa of any peak point and a second preliminary expression of the abscissa of any valley point due to the action of the cylinder diffracted wave at the downstream of the cylinder;

step 500, firstly, at the downstream of any one cylinder, summing the downstream-transmitted diffraction wave generated by the cylinder and the downstream-transmitted diffraction wave generated by each cylinder at the upstream of the cylinder, and correcting by using the same method, so as to obtain a final expression I of the abscissa of any peak point and a final expression I of the abscissa of any valley point when the wave incident angle is not equal to zero; then, at the upstream of any cylinder, summing the upstream-transmitted diffraction waves generated by the cylinder and the downstream-transmitted diffraction waves generated by each cylinder, and continuously correcting to obtain a final expression II of the abscissa of any peak point and a final expression II of the abscissa of valley point;

step 600, taking the smaller of the abscissa in the final expression I and the final expression II as the final expression of the abscissa of the arbitrary peak point and the final expression of the abscissa of the arbitrary valley point when the wave incident angle is not equal to zero, and synthesizing the final expressions of the wave incident angle when the wave incident angle is equal to zero and not equal to zero to obtain the final expression of the abscissa of the arbitrary peak point and the final expression of the abscissa of the arbitrary valley point in the region III of the arbitrary cylindrical wave force curve;

step 700, obtaining the abscissa of any peak point and valley point according to the final expression of the abscissa, so as to obtain the corresponding wave number, solving a linear equation set to obtain an unknown diffraction coefficient in the expression of the velocity potential, further obtaining the wave force borne by any cylinder, dimensionless and modular the wave force, and obtaining the ordinate of the wave force curve at any peak point and valley point in the region III;

step 800, determining the position of each peak point and valley point according to the obtained abscissa of any peak point and valley point and the obtained ordinate of any peak point and valley point, and connecting the peak points to obtain an upper envelope line of the wave force curve in the region III; by connecting these valleys, the lower envelope of the wave force curve in region III is obtained.

In one embodiment of the present invention, the acquisition method according to claim 1,

in the cylindrical array coordinate system, an included angle between a plane incident wave propagation direction and the positive direction of an x axis in the cylindrical array global coordinate system is called a wave incident angle, and the building of the global coordinate system enables the wave incident angle to be smaller than or equal to 90 degrees; k is the serial number of any cylinder in the cylinder array, and the increasing direction of the serial number k is consistent with the positive direction of the x axis in the whole coordinate system of the cylinder array.

In an embodiment of the present invention, the preliminary expression of the abscissa of the arbitrary peak point and the preliminary expression of the abscissa of the arbitrary valley point given in the step 200 may be obtained, and the difference between the abscissas of the two adjacent peak points or the two adjacent valley points is equal, so that the preliminary expression of the fluctuation distance of the arbitrary cylindrical wave force curve in the region III may be further obtained, and the preliminary expression of the abscissa of the arbitrary peak point and the preliminary expression of the abscissa of the arbitrary valley point in the step 200 are obtained as follows:

the dimensionless wave number Kd/π can be rewritten as: kd/pi ═ R/λ;

first peak point R in wave force curve_p(1)And valley point R_v(1)The corresponding pillar spacing is represented by the following formula:

2(N-k)R_p(1)＝λ

where K is the wavenumber, R ═ 2d is the column spacing of adjacent cylinders, λ is the wavelength, in the subscripts, p represents the peak point, v represents the valley point, (1) represents the first peak or valley;

let the column pitch corresponding to any s-th and s + 1-th peak points in the region III be R_p(s)And R_p(s+1)，R_p(s+1)＝R_p(s)+R_pIf the peak occurs under the condition that constructive interference occurs, the wave path difference corresponding to the adjacent s-th and s + 1-th peak should be the wavelength which is s times and s +1 times respectively, and thus the preliminary expression of the difference between the abscissas of the adjacent peaks is obtained as follows:

the valley points occur under the condition that destructive interference occurs, and the column pitch R corresponds to the s-th and s + 1-th valley points_v(s)And R_v(s+1)＝R_v(s)+R_vThe corresponding path differences are equal to 2s-1 times and 2s +1 times the wavelength, respectively, from which a preliminary expression for the difference between the abscissas of adjacent valleys is derived:

from the above derivation, if the differences between the abscissa of the adjacent peak points and the abscissa of the adjacent valley points are equal, the preliminary fluctuation distance expression is:

n is the total number of cylinders in the cylinder array; at this time, the preliminary expression of the abscissa of any peak point and the preliminary expression of the abscissa of any valley point in the wave force curve region III are as follows:

in one embodiment of the present invention, the process of obtaining the final expression of the wave pitch when the wave incident angle is equal to zero, the final expression of the abscissa of the arbitrary peak point, and the final expression of the abscissa of the arbitrary valley point is as follows:

at any k column upstream | x_kL position (x)_k<0) Summing the diffraction potentials of the k column and the columns downstream of the k column to obtain:

wherein,

A_nis the coefficient on column 1, i is an imaginary unit, ω is the wave circular frequency, t is time, n is an integer, Z_n＝J_n′(Ka)/H_n' (Ka), K is the wave number, J_nIs a Bessel function of the first kind, H_nIs a first type of hankel function, a is the cylinder radius;

the method for simplifying the Hankel function by using an asymptotic expression comprises the following steps:

wherein,

the second term and the third term of the above formula are equivalent to the superposition effect of two equivalent cylindrical left waves with the wave path difference of 2(R/2) and 2(N +1/2-k) R with the k-pillar left wave, and the two equivalent cylinders are positioned at R/2 and (N +1/2-k) R at the downstream of the k-pillar;

and correcting the preliminary fluctuation distance expression by using the position of the equivalent cylinder farther away from the k column to obtain a final expression of the fluctuation distance in the wave force curve area III:

and obtaining the final expression of the abscissa of any peak point and the final expression of the abscissa of any valley point in the corrected wave force curve area III as follows:

in one embodiment of the present invention, for the case that the wave incident angle is not equal to zero, the preliminary expression one for the abscissa of any peak point and the preliminary expression one for the abscissa of any valley point are obtained as follows:

wherein, the k column is any one column in the column array, β is the wave incident angle, and when β ≠ 0, the column spacing corresponding to the random s-th and s + 1-th peak points and valley points in the region III is

After the symbols representing the column spacing are added with the superscript u, the symbols are represented asThe corresponding amount caused by the action of the traveling right wave of each column;

obtaining a first preliminary expression of the abscissa of the arbitrary peak point and a first preliminary expression of the abscissa of the arbitrary valley point:

and a preliminary fluctuation pitch expression one:

correcting the three formulas to obtain a final expression I of the abscissa of any peak point, a final expression I of the abscissa of any valley point and a final fluctuation distance expression I;

in one embodiment of the present invention, for the case that the wave incident angle is not equal to zero, the preliminary expression two of the abscissa of the arbitrary peak point and the preliminary expression two of the abscissa of the arbitrary valley point, and the preliminary fluctuation pitch expression two are obtained as follows:

wherein,

the sign of equal representing the column spacing is increased and the superscript l represents the corresponding quantity caused by the left wave propagation effect of each column downstream;

then, a preliminary expression two for an arbitrary peak abscissa, a preliminary expression two for an arbitrary valley abscissa, and a preliminary fluctuation pitch expression two are obtained:

correcting the three formulas to obtain a final expression II of the abscissa of any peak point and a final expression II of the abscissa of any valley point when the wave incident angle is not equal to zero, and a final fluctuation interval expression II:

in one embodiment of the present invention, at a wave incidence angle equal to zero, the wave spacing of any cylindrical wave force curve in region III is:

when the wave incident angle is not equal to zero, the wave distance of any cylindrical wave force curve in the region III is as follows:

in one embodiment of the present invention, the process of obtaining the abscissa of each of the peak points is as follows:

for the case of a wave incident angle equal to zero, the final expression in terms of the column pitch-wavelength ratio for the abscissa of any peak in the wave force curve region III is:

for the case where the wave incidence angle is not equal to zero, combining the final expression one for the abscissa of the arbitrary peak point and the final expression two for the abscissa of the arbitrary peak point can obtain the final expression for the abscissa of the arbitrary peak point expressed in terms of the column pitch-wavelength ratio:

the process of obtaining the abscissa of each valley point is as follows:

for the case of a wave incident angle equal to zero, the final expression in the column pitch-wavelength ratio for the abscissa of any valley point in the wave force curve region III is:

for the case where the wave incidence angle is not equal to zero, combining the final expression one for the abscissa of any valley point and the final expression two for the abscissa of any valley point yields the final expression for the abscissa of any valley point in terms of the ratio of the column pitch to the wavelength:

in one embodiment of the present invention, the ordinate of each of the peak point and the valley point is obtained as follows:

according to the space factor phi (r) of the velocity potential near any k column in the water wave diffraction problem of the bottomed cylinder array_k,θ_k) The formula:

wherein,

for the diffraction coefficient, k is the number of any column in the column array, and the increasing direction of the number k is consistent with the positive direction of the x axis, (r)_k,θ_k) Polar coordinate of a local cylindrical coordinate system passing through the k-pillar axis for the vertical axis Z-axis, Z_n＝J_n′(Ka)/H_n' (Ka), K is the wave number, a is the cylinder radius, J_nIs a Bessel function of the first kind, H_nIs a first type of hank function, n is an integer;

solving a linear equation system of the diffraction coefficient in the velocity potential expression as follows:

wherein β is the angle between the plane incident wave propagation direction and the positive direction of the x axis in the cylindrical array global coordinate system (wave incident angle), and the global coordinate system is established to make the wave incident angle β not more than pi/2, R_jkIs the kth columnThe distance from the child to the jth pillar, i in imaginary units, m in integer α_jkIs the angle of orientation from the kth post to the jth post, I_kThe phase factor of the incident wave at the kth pillar;

substituting the wave number K corresponding to the peak point and the valley point into the equation to obtain the diffraction coefficients of the peak point and the valley point under the wave number

Value of (2) is the diffraction coefficient

Substituting the following formula

The wave force F along the connection line of the centers of circles in the horizontal section of the cylindrical array on any cylinder k under the wave number corresponding to any peak point and valley point in the region III of the wave force curve can be obtained^k(ii) a Wherein rho is the density of water, g is the acceleration of gravity, A is the amplitude of incident waves, and h is the water depth;

carrying out non-dimensionalization on the wave force shown by the formula by using the wave force borne by the cylinders with the same geometric dimension under the same environmental condition, and obtaining the dimensionless wave force of any kth column in the cylinder array under the wave numbers corresponding to any peak point and valley point in the region III of a wave force curve as follows:

the dimensionless wave force amplitude is obtained by taking the mode, and the dimensionless wave force amplitude is the ordinate of the wave force curve at any peak point and valley point in the area III.

The present invention provides such recognition and understanding: the wave distance of a non-capture area (area III) in a wave force curve formed by the wave force amplitude value changing along with the dimensionless wave number does not change along with the dimensionless wave number, the wave distance is only related to the total number of cylinders in the cylinder array, the number of columns for marking the positions of the cylinders and the wave incidence angle, and the formula provided by the invention can be used for accurately predicting.

The invention can deepen understanding and cognition on the fluctuation characteristic of a non-capture area (area III) in a wave force curve formed by the wave force amplitude changing along with dimensionless wave numbers, and provides a method for determining the upper envelope line and the lower envelope line in the area III of the wave force curve of the cylindrical array. Therefore, when the scheme is evaluated, selected or initially designed, the peak-valley points of the wave force curve can be prevented from being missed, and a large number of other data points which are not important for the stages of the actual engineering can be avoided, so that the calculation amount can be greatly reduced, the cost is saved, the efficiency is improved, and the technical support is provided for finally improving the evaluation and design level.

In addition, the invention also provides a prediction formula of the fluctuation distance in the non-capture area which can be directly quoted so as to reduce the actual calculation amount.

Drawings

FIG. 1 is a schematic view of a cylindrical array of cylinders of the same diameter aligned in a line in accordance with one embodiment of the present invention;

fig. 2 is a schematic diagram of three regions, namely, a wave force curve formed by the amplitude of the wave force of a single-row seated-bottom column group with the total number N of columns being 17, the column number k being 9, the wave incidence angle β being 0 and the diameter-column spacing ratio a/d being 1/4, along with the dimensionless wave number, and a capture-related Region (Region i and Region II) and a non-capture Region (Region III);

fig. 3 is an image of the amplitude of the wave force varying with the column number at a dimensionless wave number corresponding to the near-drawing peak for a single-row array of bottomed cylinders with total number of cylinders N301, column number k 151, wave incidence angle β 0, and diameter-column spacing ratio a/d 1/2;

fig. 4 is an image of the amplitude of the wave force at a dimensionless wave number corresponding to the first valley point to the left of the near-drawing peak for a single-row bottom-seated cylinder array with total number of cylinders N301, column number k 151, wave incidence angle β 0, and diameter-column spacing ratio a/d 1/2, as a function of column number;

fig. 5 is an image of the amplitude of the wave force at a dimensionless wave number corresponding to the first peak point to the left of the near-drawing peak for a single-row bottom-seated cylinder array with total number of cylinders N301, column number k 151, wave incidence angle β 0, and diameter-column pitch ratio a/d 1/2, as a function of the column number;

FIG. 6 is a schematic diagram of wave constructive/destructive interference; wherein, the fluctuation curve is a variation curve of the wave force amplitude of the single-row seated-bottom cylindrical array with the dimensionless wave number, the number k of the column is 21, the wave incidence angle beta is 0, and the diameter-column spacing ratio a/d is 1/4;

fig. 7 is a graph of the variation of the wave force amplitude fluctuation distance measurements with dimensionless wavenumbers for different column numbers k, for a total number N of columns 101, a wave incidence angle β 0, a diameter-column distance ratio a/d 1/4;

FIG. 8 shows the measured value of the wave pitch in region III for the amplitude of the wave force experienced by the kth column at the total number of columns N11, 21,51,101, the wave incidence angle β 0, and the diametric column pitch ratio a/d 1/4

And theoretical calculated value

The image is changed with the column number.

Detailed Description

Before elaborating on the details of the present invention, a method for determining the location and extent of the regions (region I and region II) of interest captured by the wave force curve, which is a combination of the results of the prior art studies and our analysis, is described. (regions I and II are the previously described capture-related regions having significantly higher and deeper peaks and valleys than region III. in addition, these two regions are further characterized by a change in the undulation pitch with a change in dimensionless wave number)

1) Region I (Capture related region)

There is a lot of literature on the frequency of the tapped mode for an infinitely long array of cylinders or a single cylinder placed on the centerline of a water bath, and these results allow estimation of the near-tapping wavenumber for a finite number of arrays of bottomed cylinders, i.e. the location of region I can be obtained. Specifically, according to the ratio a/d of the cylinder diameter to the cylinder spacing (2a is the cylinder diameter, and 2d is the distance between the adjacent cylinder axes), the wave number corresponding to the trailing mode known in the literature is searched, and the region I in the capture relevant region of the finite-length cylinder array can be obtained by searching and calculating the peak in the vicinity of the wave number. For some cases where the a/d literature does not show the corresponding wavenumber of the traced mode, 1/| [20(N-K) +10] can be used as an initial calculation step (N is the total number of columns in the column array and K is the number of columns identifying the column position), and searching for peaks near an integer multiple of Kd/pi of 0.5 (K is the wavenumber) can obtain region I in the region of interest for capturing of a column array of finite length. For a finite long single row cylindrical array, as the diameter-to-cylinder spacing ratio a/d decreases, the peak of region I also moves to the right. By comparing the result with the result corresponding to the close a/d, the calculation range of the region I can be further narrowed. For the wave number value corresponding to the obtained peak, a relation graph of the wave force amplitude and the column number is drawn, if a complete half-wave form can be presented, as shown in fig. 3, the maximum hydrodynamic force acts on the middle column, and the peak position is accurate enough. If not, the encryption step size can be continued to find a more accurate peak point.

2) Region II (Another Capture related region)

The secondary peaks and valleys in the vicinity of the wave force curve region I constitute a region II in which the wave pitch of the curve changes with the dimensionless wave number. At present, the literature researches on some secondary peaks and valleys on the left side of the peak of the region I in the limited long cylindrical array wave force curve. Studies have shown that these secondary peaks, troughs are related to the infinite length cylinder array Rayleigh-Bloch wave problem and the tracked modes with multiple cylinders laterally arranged in the water bath. In particular, for the middle column of the column array composed of N single-row seated columns, the abscissa (dimensionless wave number) of the positions of the secondary peak and the secondary valley left to the peak point of the wave force curve strictly corresponds to the abscissa (dimensionless wave number) of the peak position of the middle column wave force curve in the column array with the number of columns being N/2, N/3, N/4 …, and is specifically as follows:

the abscissa of the peak position of the middle column wave force curve of the single-row bottom-sitting cylindrical array with the number of cylinders being N/2 corresponds to the abscissa of the first valley point position on the left side of the peak of the middle column wave force curve of the single-row bottom-sitting cylindrical array with the number of cylinders being N, the relation graph of the wave force amplitude value of the cylindrical array with the number of cylinders being N under the dimensionless wave number corresponding to the abscissa of the valley point position and the number of the cylinders presents a form of two half waves, as shown in fig. 4, the wave force amplitude value corresponding to the highest peak of the two half waves is equal to the wave force amplitude value of the middle column of the cylindrical array with the number of cylinders being N/2 under the same wave number.

The abscissa of the peak position of the middle column wave force curve in the single-row seated bottom cylinder array with the number of cylinders being N/3 corresponds to the abscissa of the first peak position on the left side of the peak of the middle column wave force curve in the cylinder array with the number of cylinders being N, the relation graph of the wave force amplitude of the cylinder array with the number of cylinders being N under the dimensionless wave number corresponding to the abscissa of the peak position and the number of the cylinders presents a form of three half-waves, as shown in fig. 5, the wave force amplitude corresponding to the highest peak of the three half-waves is equal to the wave force amplitude of the middle column in the cylinder array with the number of cylinders being N/3 under the same wave number.

The number of cylinders is N/4, N/5 …, etc. is similar to that described above, and so on. Generally, when N/N _i10 hours (n)_iIs a natural number), the effect of near-drawing is already rather weak, and the number of cylinders can be (N/N) at this time_iThat is) 10 as the left boundary of the region II of the cylindrical array wave force curve composed of N cylinders.

We found by computational analysis that region II is affected differently by near-bridging for different diameter-to-column spacing ratios a/d. The larger a/d, the larger the range of influence of near-tracking. For example, for a/d equal to 0.25, the number of cylinders N/N_iThe dimensionless wavenumber corresponding to the peak position of the wave force of 20 arrays can be used as the left limit of region II, and for a/d equal to 0.5, this left limit will last until the number of cylinders N/N _i5 array wave force spike position correspondencesDimensionless wave number of (a). For the case of any kth column in a single row of the seated cylinder cluster array, this can be determined with reference to the ranges of the above-mentioned middle columns.

The cylindrical array in the present invention refers to a cylindrical array in which a large number of cylinders of the same diameter penetrating the water surface are arranged in a straight line (i.e., the centers of the respective circles in the horizontal section of the cylindrical array are on a straight line). The wave force in the invention refers to the wave force applied to any cylinder along the direction of the connection line of the centers of circles in the horizontal section of the cylinder array. The fluctuation distance in the invention refers to the distance between the abscissa of two adjacent maximum value points (or minimum value points) on a wave force curve formed by the wave force amplitude changing along with dimensionless wave numbers. In the present invention, the maximum point or minimum point is also described by "peak" or "valley".

As shown in fig. 2 and fig. 6, the present invention provides a method for obtaining an envelope curve of wave force amplitude of a cylindrical array based on constructive cancellation, comprising the following steps:

region III (region III) is called the non-capture region, region i (region i) and region ii (region ii) are called capture-related regions, and in the non-capture region (region III), the wave force curve has a very regular wave phenomenon.

wherein, the above-mentioned wave path difference is in direct proportion to the distance between two adjacent cylinders, make the wave path difference equal to s times and s +1 times wavelength (s is any natural number) respectively, can solve the preliminary expression of the abscissa of any peak point and preliminary expression of the difference between the abscissas of adjacent peak points, make the above-mentioned wave path difference equal to 2s-1 times and 2s +1 half wavelength respectively, can solve the preliminary expression of the abscissa of any valley point and preliminary expression of the difference between the abscissas of adjacent valley points; further, it can be known that the differences between the abscissas of two adjacent peaks or two adjacent valleys are equal, so that a preliminary fluctuation distance expression of any cylindrical wave force curve in the region III can be obtained.

Here, the number of cylinders in the cylindrical array is usually greater than 9, and the abscissa interval of adjacent maxima or adjacent minima of the wave force amplitude curve is a fixed value in region III, does not change with dimensionless wave frequency, is only related to the total number N of cylinders in the array, the number k of the cylinder identifying the position of the cylinder, and the wave incidence angle β, and can be predicted very accurately with a simple formula.

In a cylindrical array coordinate system, an included angle between a plane incident wave propagation direction and the positive direction of an x axis in a cylindrical array integral coordinate system is called a wave incident angle, and the integral coordinate system is established to enable the wave incident angle to be smaller than or equal to 90 degrees; k is the serial number of any cylinder in the cylinder array, and the increasing direction of the serial number k is consistent with the positive direction of the x axis in the whole coordinate system of the cylinder array.

The preliminary expression of the fluctuation distance, the preliminary expression of the abscissa of any peak point and the preliminary expression of the abscissa of any valley point are obtained as follows:

the dimensionless wavenumber (i.e. the wave force curve abscissa) Kd/π can be rewritten as: kd/pi ═ R/λ; thus, a change in the amplitude of the wave force with dimensionless wavenumber can also be considered as a change in the amplitude of the wave force with the column pitch-to-wavelength ratio. When the diffracted wave of the j column is transmitted to the vicinity of the k column to generate constructive (destructive) interference with the diffracted wave of the k column, the amplitude of the wave force on the k column obtains a peak (valley)) The value is obtained. R_p(1)And R_v(1)Respectively showing the column spacing corresponding to the first peak point and the valley point in the wave force curve and the first peak point R in the wave force curve_p(1)And valley point R_v(1)The corresponding pillar spacing is represented by the following formula:

2(N-k)R_p(1)＝λ

where K is the wavenumber, R ═ 2d is the column pitch of adjacent cylinders, λ is the wavelength, and in the subscripts, p represents the peak point, v represents the valley point, and (1) represents the first peak or valley.

Let the column pitch corresponding to any s-th and s + 1-th peak points in the region III be R_p(s)And R_p(s+1)，R_p(s+1)＝R_p(s)+R_pIf the peak occurs under the condition of constructive interference, the wave path difference corresponding to the adjacent s-th and s + 1-th peak should be s times and s +1 times of wavelength, respectively, so that the wave path difference between the N-column and the k-column left-transmitted diffracted wave (hereinafter referred to as "left-transmitted wave") should satisfy:

2(N-k)R_p(s)＝sλ

2(N-k)(R_p(s)+R_p)＝(s+1)λ

the above two subtraction equations obtain the initial expression of the difference between the abscissas of adjacent peaks:

the valley points occur under the condition that destructive interference occurs, and the column pitch R corresponds to the s-th and s + 1-th valley points_v(s)And R_v(s+1)＝R_v(s)+R_vThe corresponding path differences are equal to 2s-1 times and 2s +1 times the wavelength, respectively, from which the following derivation is possible:

the preliminary expression for obtaining the difference between the abscissas of adjacent valley points is:

n is the total number of cylinders in the cylinder array; at this time, the preliminary expression of the abscissa of any peak point and the preliminary expression of the abscissa of any valley point in the wave force curve region III are:

the process of obtaining the final expression of the wave distance in the wave force curve area III when the wave incident angle is equal to zero, the final expression of the abscissa of any peak point and the final expression of the abscissa of any valley point is as follows:

at any k column upstream | x_kL position (x)_k<0) Summing the diffraction potentials of the k-pillars and the pillars downstream of the k-pillars (which is equivalent to the left wave summation described above) yields:

wherein for a long row of columns, the columns in the middle area far away from the two ends are provided with

the method for simplifying the formula by taking an asymptotic expression from the Hankel function comprises the following steps:

the constants + -pi/2 of the second and third terms in the equation do not contribute to the problem of the undulation pitch discussed below; wherein,

the above formula is actually the superposition of three left-transmitted plane waves, the first term is k-column left-transmitted wave, the second term and the third term are the sum of the left-transmitted waves of N-k columns from k +1 column to N column, the second term and the third term are equivalent to the superposition effect of two equivalent cylindrical left-transmitted waves with the wave path difference of 2(R/2) and 2(N +1/2-k) R with the k-column left-transmitted wave, and the two equivalent cylinders are positioned at R/2 and (N +1/2-k) R at the downstream of the k-column;

similarly, when the incident angle is equal to zero, the final expression of the abscissa of any peak point and the final expression of the abscissa of any valley point in the corrected wave force curve region III are as follows:

the case where the incident angle is not zero (β ≠ 0) is more complicated than the case where the incident angle is zero, and at this time, the difference in the path length between the right diffracted wave of each column upstream of the k-column and the right diffracted wave of the k-column (hereinafter simply referred to as "right-transmitted wave") is different from the case where β ═ 0 is used, and when β ≠ 0, the arbitrary inter-column points corresponding to the s-th and s + 1-th peaks and valleys in Region III are definedDistance is

Since it is necessary to consider the case where the right wave propagation of the upstream columns and the left wave propagation of the downstream columns, respectively, hereinafter, the above symbols indicating the column pitch indicate the corresponding quantity caused by the right wave propagation of the upstream columns when the upper corner mark "u" is added, and indicate the corresponding quantity caused by the left wave propagation of the downstream columns when the upper corner mark "l" is added.

For the condition that the wave incident angle is not equal to zero, firstly, analyzing the wave path difference between the right propagation diffracted wave propagating downstream from the first cylinder at the end part of the array positioned at the upstream of the k column and the right propagation diffracted wave of the k column to enable the right propagation diffracted wave and the right propagation diffracted wave of the k column to meet the condition of generating constructive or destructive interference, and obtaining a primary fluctuation interval expression I, wherein the process is as follows:

After the symbols representing the column spacing are increased by the upper corner mark u, corresponding quantities caused by the right wave propagation action of each column at the upstream are represented; obtaining a first preliminary fluctuation interval expression:

when β ≠ 0, the first preliminary expression of the abscissa of the s-th peak point and the first preliminary expression of the abscissa of the s-th valley point in Region III are:

and then analyzing the wave path difference between the left transmitted diffracted wave which is propagated upstream by the last cylinder positioned at the downstream of the k column and the left transmitted diffracted wave of the k column to ensure that the wave path difference meets the condition of generating constructive or destructive interference, and obtaining a preliminary wave pitch expression II, wherein the process is as follows:

wherein,

the sign of equal representing the column spacing is increased and the superscript l represents the corresponding quantity caused by the left wave propagation effect of each column downstream; obtaining a preliminary fluctuation distance expression II as follows:

when β ≠ 0, the preliminary expression two of the abscissa of the s-th peak point and the preliminary expression two of the abscissa of the s-th valley point in Region III are:

step 500, firstly, at the downstream of any one cylinder, summing the downstream-transmitted diffraction wave generated by the cylinder and the downstream-transmitted diffraction wave generated by each cylinder at the upstream of the cylinder, and correcting by using the same method, so as to obtain a final expression I of the abscissa of any peak point and a final expression I of the abscissa of any valley point when the wave incident angle is not equal to zero; then, at the upstream of any cylinder, summing the upstream-transmitted diffraction waves generated by the cylinder and the left-transmitted diffraction waves generated by each downstream cylinder of the cylinder, and continuously correcting to obtain a final expression II of the abscissa of any peak point and an expression II of the abscissa of valley point;

similar to the β ═ 0 process, downstream | x of the k column_kL position (x)_k>0) The sum of the diffraction potentials of the k column and the columns upstream of the k column is

Wherein,

in this equation, the second term and the third term represent the sum of the right waves of the k-1 columns from the 1 st column to the k-1 column. The effect of these two terms can be seen as the effect of two columns located upstream of the kth column at R/2 and (k-1/2) R, so that the preliminary undulation pitch expression one

Obtaining a final fluctuation space expression I when the incident angle is not equal to zero after correction

And the final expression I of the abscissa of the arbitrary peak point and the final expression I of the abscissa of the arbitrary valley point in the state are as follows:

at k column upstream | x_kL position (x)_k<0) The sum of the diffraction potentials of the k column and the columns downstream of the k column is:

wherein,

in the formula, the second term and the third term represent the sum of the left waves of the N-k columns from the k +1 th column to the N column, and the effects of the two terms can be regarded as R/2 and (N +1/2-k) located downstream of the k columnThe effect of left-hand wave propagation of the two cylinders at R, so that the preliminary spacing expression two when the wave incidence angle is not equal to zero

And obtaining a final fluctuation distance expression II after correction:

and the final expression II of the abscissa of the arbitrary peak point and the final expression II of the abscissa of the arbitrary valley point in the state are as follows:

the final expression I of the abscissa of any valley point and the final expression II of the abscissa of any valley point respectively reflect the influence of the upstream pillar and the downstream pillar on the abscissa of the valley point,

combining these two equations yields the final expression for the abscissa of any valley point expressed as the ratio of the column spacing to the wavelength:

the final expression I of the abscissa of the arbitrary peak point and the final expression II of the abscissa of the arbitrary peak point respectively reflect the influence of the upstream column and the downstream column on the abscissa of the peak point,

combining these two equations yields the final expression for the abscissa of any peak in terms of the ratio of column spacing to wavelength:

the vertical coordinates of each peak and valley are obtained as follows:

calculating the space factor phi (r) of the velocity potential near any k column in the water wave diffraction problem of the bottomed cylinder array_k,θ_k) The formula of (1) is:

wherein,

wherein the diffraction coefficient

The (unknown coefficients) are determined by the following linear equation of diffraction coefficients:

wherein β is the angle between the plane incident wave propagation direction and the positive direction of the x axis in the cylindrical array global coordinate system (wave incident angle), and the global coordinate system is established to make the wave incident angle β not more than pi/2, R_jkIs the distance from the kth post to the jth post, I is an imaginary unit, m is an integer, α jk is the angle of orientation from the kth post to the jth post, I_kThe phase factor of the incident wave at the kth pillar;

according to the fact that the abscissa R/lambda is equal to the dimensionless wave number Kd/pi, the final expression of the abscissa of the arbitrary valley point and the final expression of the abscissa of the arbitrary peak point also give the abscissas of the peak point and the valley point expressed by the dimensionless wave number, and therefore the wave numbers K corresponding to the peak point and the valley point can be obtained.

The wave number K corresponding to the peak point and the valley point is substituted into the linear equation of the diffraction coefficient to obtain the diffraction coefficient under the condition of the wave number corresponding to the peak point and the valley point

Will have a value of

Substituting the following formula

The wave force F along the connection line of the centers of circles in the horizontal section of the cylindrical array on any cylinder k under the wave number corresponding to any peak point and valley point in the region III of the wave force curve can be obtained^k(ii) a Wherein rho is the density of water, g is the acceleration of gravity, A is the amplitude of the incident wave, and h is the water depth.

The wave force F shown by the formula is subjected to the wave force by cylinders with the same geometric dimension under the same environmental condition^kPerforming dimensionless transformation to obtain the dimensionless wave force of any kth column in the cylinder group array under the wave numbers corresponding to any peak point and valley point in the region III of the wave force curve:

the dimensionless wave force amplitude is obtained by taking the mode, which is the ordinate of the wave force curve at any peak and valley point in the region III.

So far, the abscissa and the ordinate of the wave force curve at any peak point and valley point in the region III are obtained, and the valley points are connected to obtain a lower envelope curve of the wave force curve in the region III; by connecting these peaks, the upper envelope of the wave force curve in region III is obtained.

Furthermore, at a wave incidence angle equal to zero, the final expression for the wave spacing in region III for any cylindrical wave force curve is:

when the wave incidence angle is not equal to zero, the final expression of the wave spacing of any cylindrical wave force curve in the region III is as follows:

FIG. 7 shows the wave force amplitude measurements for a wave pitch ratio of 1/4 with a total number of columns N of 101, a wave incidence angle β of 0, and a diameter column pitch ratio a/d of 1/4

Curve as a function of dimensionless wavenumber. It can be seen that the fluctuation pitch is constant over a large range of wave numbers, this region is region III, the regions where the fluctuation pitch rapidly decreases are regions I and II, the asymptotes and the values in the graph are theoretical predicted values calculated using the final expression of the present invention, and the results are very consistent.

Fig. 8 shows measured values of the wave pitch in the region III of the wave force received by the kth column of the single-row bottomed cylinder cluster array when N is 11, N is 21, N is 51, N is 101, the wave incident angle β is 0, and the diameter column pitch ratio a/d is 1/4

And describing model expression calculations

Comparison of (1). Through comparison, the predicted value of the description model expression is well matched with the actual calculated value.

It will be appreciated by those skilled in the art that while a number of exemplary embodiments of the invention have been shown and described in detail herein, many other variations or modifications can be made, which are consistent with the principles of this invention, and which are directly determined or derived from the disclosure herein, without departing from the spirit and scope of the invention. Accordingly, the scope of the invention should be understood and interpreted to cover all such other variations or modifications.

Claims

1. The method for acquiring the envelope curve of the wave force amplitude of the cylindrical array based on constructive cancellation is characterized by comprising the following steps of:

step 100, taking a region where a plurality of high-rise peaks in a wave force curve formed by wave force amplitude values changing along with dimensionless wave numbers as a region I, taking a region where secondary peaks and valleys which are lower than the high-rise peaks and have curve fluctuation distances changing along with dimensionless wave numbers near the high-rise peaks are as a region II, and taking a wave force curve outside the region I and the region II as a region III and taking the wave force curve as an analysis object region;

2. The acquisition method according to claim 1,

3. The acquisition method according to claim 1,

the preliminary expression of the abscissa of any peak point and the preliminary expression of the abscissa of any valley point given in the step 200 can be obtained, and the differences between the abscissas of two adjacent peak points or two adjacent valley points are equal, so that a preliminary fluctuation distance expression of any cylindrical wave force curve in the region III can be further obtained, and the preliminary fluctuation distance expression and the preliminary expressions of the abscissa of any peak point and the abscissa of any valley point in the step 200 are obtained as follows:

the dimensionless wave number Kd/π can be rewritten as: kd/pi ═ R/λ;

the column spacing R corresponding to the first peak point and the valley point in the wave force curve respectively_p(1)And R_v(1)Represented by the formula:

2(N-k)R_p(1)＝λ

4. the acquisition method according to claim 3,

the process of obtaining the final expression of the wave incident angle equal to the zero time fluctuation distance, the final expression of the abscissa of any peak point and the final expression of the abscissa of any valley point is as follows:

at any k column upstream | x_kI where x_k<0, summing the diffraction potentials of the k columns and the columns downstream of the k columns to obtain:

wherein,

A_nis the coefficient on column 1, i is an imaginary unit, ω is the wave circular frequency, t is time, n is an integer, Z_n＝J′_n(Ka)/H′_n(Ka), K is the wave number, J_nIs a Bessel function of the first kind, H_nIs a first type of hankel function, a is the cylinder radius;

wherein,

5. the acquisition method according to claim 4,

for the condition that the wave incidence angle is not equal to zero, the first preliminary expression of the abscissa of any peak point and the first preliminary expression of the abscissa of any valley point are obtained as follows:

After the symbols representing the column spacing are increased by the upper corner mark u, corresponding quantities caused by the right wave propagation action of each column at the upstream are represented;

and a preliminary fluctuation pitch expression one:

6. the acquisition method according to claim 5,

for the condition that the wave incidence angle is not equal to zero, a second preliminary expression of the abscissa of any peak point and a second preliminary expression of the abscissa of any valley point, and a second preliminary fluctuation distance expression are obtained as follows:

wherein,

7. the acquisition method according to claim 6,

when the wave incident angle is equal to zero, the wave distance of any cylindrical wave force curve in the region III is as follows:

8. the acquisition method according to claim 6,

the process of obtaining the abscissa of each peak point is as follows:

the process of obtaining the abscissa of each valley point is as follows:

9. the acquisition method according to claim 8,

the vertical coordinate of each peak point and each valley point is obtained in the following manner:

wherein,

k is the weave of any cylinder in the cylinder array for the diffraction coefficientThe increasing direction of the number k is consistent with the positive direction of the x axis, (r)_k,θ_k) Polar coordinate of a local cylindrical coordinate system passing through the k-pillar axis for the vertical axis Z-axis, Z_n＝J′_n(Ka)/H′_n(Ka), K is the wave number, a is the radius of the cylinder, J_nIs a Bessel function of the first kind, H_nIs a first type of hank function, n is an integer;

β is the angle between the plane incident wave propagation direction and the positive direction of the x axis in the cylindrical array global coordinate system, namely the wave incident angle, and the global coordinate system is established to make the wave incident angle β not more than pi/2, R_jkIs the distance from the kth pillar to the jth pillar, i is in imaginary units, m is an integer α_jkIs the angle of orientation from the kth post to the jth post, I_kThe phase factor of the incident wave at the kth pillar;

Value of (2) is the diffraction coefficient

Substituting the following formula