CN117313570A - Power response analysis method for offshore wind power generation structure - Google Patents
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Abstract
A dynamic response analysis method for an offshore wind power generation structure belongs to the technical field of offshore wind power generation. A method for establishing a single pile type fan integral structure power response time domain analysis model taking pile-soil interaction into consideration under the combined action of wind and wave is provided based on a modified phyllin momentum theory, a semi-empirical semi-theory Rainey slender body theory and a p-y curve method, so that coupling calculation of fan aerodynamic force-hydrodynamic force-structure power response-soil resistance is realized; the analysis method is realized based on an aerodynamic module, a hydrodynamic module, a pile-soil interaction module and an offshore wind turbine integral structure power response calculation module, wherein the four modules are mutually coupled, and the interaction between external loads of all parts and the offshore wind turbine can be fully considered. The response analysis method of the offshore wind power generation structure provided by the invention can consider the coupling effect of the wind power generation structure response and the external load of each part under the actions of wind, wave and the like, and the dynamic response calculation result of the whole structure of the wind power generation structure is more accurate, thereby having an important role in comprehensively measuring the safety of the offshore wind power generation structure.
Description
Technical Field
The invention belongs to the technical field of offshore wind power generation, relates to a calculation and analysis method of the power response of an overall structure of an offshore wind turbine under the action of wind, wave and other loads, and particularly relates to a method for analyzing the power response of an offshore wind power generation structure.
Background
Wind energy which is never exhausted is natural, compared with land wind power, offshore wind power generation has the advantages of small occupied area, high wind speed, stable wind direction and the like, and is becoming the focus of development in the field of new energy. The offshore wind energy resource in China is rich, the development benefit is high, and the energy is an economically feasible green energy source. At present, a horizontal shaft fan system is mainly adopted for offshore wind power generation, a wind turbine generator mainly comprises a blade, a hub, a cabin, a tower, an underwater foundation and the like, and the design and construction of a fan foundation with sufficient rigidity and strength are necessary conditions for maintaining stable operation of a fan within the service life and creating economic value.
In summary, the invention establishes an integrated coupling calculation model of aerodynamic force-hydrodynamic force-structural dynamic response-soil resistance of the offshore wind turbine, and provides a research and analysis method of the overall structural dynamic characteristics of the offshore wind turbine, which can provide technical support for the structural design, construction and operation maintenance of the offshore wind turbine (research current situation and development prospect of basic structure of the offshore wind turbine, zhang Ji, he Haihua and Zhang Zhaode).
Disclosure of Invention
Aiming at the problems existing in the prior art, the invention provides a method for analyzing the dynamic response of the integral structure of the offshore wind turbine under the combined action of wind and wave, which can solve the problem of interaction between fluid and the structure of the offshore wind turbine and the seabed foundation, and ensure the accuracy of the dynamic response calculation result of the integral structure of the offshore wind turbine on the basis of greatly improving the calculation efficiency.
In order to achieve the above purpose, the invention adopts the following technical scheme:
the method is realized based on a power response coupling calculation model of the whole structure of the offshore wind turbine, and the calculation model comprises four modules: the system comprises an aerodynamic module, a hydrodynamic module, a pile-soil interaction module and an offshore wind turbine integral structure power response calculation module, wherein the hydrodynamic module is also a wave load calculation module. As can be seen from fig. 1, the three load calculation sub-modules (aerodynamic module, hydrodynamic module, pile-soil interaction module) in the integral coupling calculation model are connected through the offshore wind turbine integral structure dynamic response calculation module. Firstly, calculating wave load, wind load and soil load by using three load calculation sub-modules respectively, converting the wave load, the wind load and the soil load into equivalent node loads in an offshore wind turbine integral structure power response calculation module, then solving the wind turbine integral structure power response by using the offshore wind turbine integral structure power response calculation module to obtain displacement, speed and acceleration at each node, and transmitting the displacement, speed and acceleration to a corresponding load calculation module respectively. Therefore, the model established by the invention is mutually coupled in the time domain, and the interaction between external loads of all parts and the offshore wind turbine structure can be fully considered. The method specifically comprises the following steps:
First step, simulating a nonlinear extreme wave field by establishing a high-order spectrum value wave water tank model
The actual ocean waves are random and nonlinear, and are generally simulated by using irregular waves in numerical analysis. Although in linear theory, irregular waves may be represented as a linear superposition of different wave components; however, when the nonlinearity of the wave is strong, the result of the linear theory simulation cannot meet the precision requirement, and the method needs to establish a complete nonlinearity Gao Jiepu numerical wave water tank model to simulate nonlinear extreme waves.
The Gao Jiepu numerical wave water tank model firstly decomposes the total velocity potential of fluid in the water tank into a periodic velocity potential and an additional velocity potential, utilizes the periodic velocity potential to process periodic boundary conditions, and utilizes the additional velocity potential to process non-periodic wave-making boundary conditions, thereby realizing the wave-making function of the numerical water tank; secondly, in order to improve the calculation efficiency, solving two decomposed velocity potentials by adopting a quasi-spectral method, and respectively expanding the two velocity potentials into a Fourier series form meeting a control equation and corresponding boundary conditions; thirdly, calculating additional speed potential according to the wave-making boundary condition of the water tank at each calculation moment; and finally substituting the obtained additional velocity potential into a nonlinear free water surface boundary condition, performing time integration to obtain a water tank wave surface at the next moment and a periodic velocity potential at the wave surface, thereby constructing a complete high-order spectrum numerical wave water tank model, and repeating the steps until the calculation is finished. The Gao Jiepu numerical wave water tank model constructed by the invention can evolve the target wave calculated by theory into more complex and real nonlinear wave through nonlinear propagation, namely, the model can be used for efficiently and accurately simulating random and strong nonlinear real sea wave, namely, the model can realize a high-order spectrum numerical wave water tank model. The method comprises the following specific steps:
1) Constructing a Gao Jiepu numerical wave water tank model, defining a water tank domain as a calculation domain, and decomposing the total velocity potential of fluid in the calculation domain into two parts, namely a periodic velocity potential and an additional velocity potential, as shown in fig. 2, wherein the specific steps are as follows:
φ＝φ p +φ add (1)
the periodic velocity potential is phi p It needs to meet the free water surface boundary condition, the water bottom boundary condition, the water-proof boundary condition of the left and right ends:
where x represents a horizontal coordinate,representing partial derivatives of x; z represents the vertical coordinate ++>Representing partial derivatives of z; l (L) x The length of the water tank is d, and the water depth of the water tank is d.
The additional velocity potential is phi add It needs to meet the wave generation of the left end of the water tankBoundary conditions, water bottom boundary conditions, right end watertight sidewall boundary conditions:
wherein u is the horizontal velocity of the target wave input at the inlet of the water tank, and the u value is directly calculated by utilizing a second-order Stokes wave theory according to the target wave parameters, and the specific method is as follows: wave theory and its application Zhili scientific press 2005, chapter two. The target wave parameters comprise wave period, wave height and water depth, and the horizontal speed of the target wave can be obtained by the parameters.
2) Solving for the additional velocity potential phi add Time derivative of additional velocity potential
2.1 Determining an additional velocity potential phi add
φ add In the extended computing domain D add As shown in FIG. 3, the total width L of the extended domain z ＝h add +d, where h add Representing the vertical coordinate of the upper boundary of the extended calculation domain, and d represents the water depth; horizontal velocity of input boundary about centerline z c ＝(h add -d)/2 antisymmetric distribution, so the present invention selects h add =3d, centerline height z c D, guarantee centerline z c Always above the free water surface.
In order to accurately simulate steeper nonlinear waves, the invention provides a wave generation method for inputting a target wave horizontal speed from the bottom z= -d of the water tank to the instantaneous wave surface z=eta. Specific:
the wave surface eta (0, t) at the inlet of the water tank can be directly calculated by the target wave parameters according to the second-order Stokes wave theory (water wave theory and application thereof Zhili. Scientific press 2005, chapter II). In the extended domain D add Is according to phi described in the first step formula (3) add The satisfied boundary conditions are developed into the following form according to the characteristic function:
wherein B is n (t) is the additional velocity potential phi add Amplitude of the nth mode; kappa (kappa) n ＝(2n-1)π/(h add +d) is the extension domain D add N-th eigenmode in vertical direction, h add Representing vertical coordinates at the upper boundary of the extended computational domain, N z Representing the total number of modes in the vertical direction; l (L) x Indicating the length of the sink. Then according to the additional velocity potential phi add Can directly calculate the partial derivative of the X direction of the characteristic expansion of the (E)
Will partial derivativeSubstituting the boundary condition of the incident wave making of the water tank:
at x=0 (6)
Where u (z) represents the horizontal velocity of the target wave input at the inlet of the tank, which is directly calculated from the target wave parameters using the second order Stokes wave theory (Water wave theory and its application Zhili. Scientific Press, 2005, chapter two), using the vertical characteristic function cos [ kappa ] n (z+d)]From the orthogonality of the horizontal velocity u (z) at the wave-making boundary, the coefficient B can be determined n (t) and determining phi add 。
2.2 Determining the time derivative of the additional velocity potential
The time derivative of the additional velocity potentialThe following boundary conditions are satisfied:
at x=0 (7)
Wherein u is t (z) represents the horizontal acceleration of the target wave input at the entrance of the tank, which is directly calculated from the target wave parameters using the second order Stokes wave theory (water wave theory and its application Zhili. Science Press, 2005, chapter two).
Then the velocity potential time derivative is appendedThe characteristic function is developed into the following form:
then the velocity potential time derivative is appendedX-direction partial derivative +.>The resolvable representation is in the form of:
Wherein C is n (t) isAmplitude of the nth mode; partial derivative +.>Substitution (7) using the vertical characteristic function cos [ kappa ] n (z+d)]According to the orthogonality of the horizontal velocity time derivative u at the wave-making boundary t (z) vertical distribution, coefficient C is determined n (t) and further determining->
To avoid initial effects, u (z) and u t (z) need to be multiplied by the buffering function R m (t) the expression is as follows:
wherein T is m For the buffer duration, the invention takes twice the wave characteristic period.
3) Solving the periodic velocity potential phi p
3.1 A) applying a periodic velocity potential function phi at the free water surface of the basin p Represented as being dependent only on horizontal coordinatesForm:
in the formula, t is the current calculation time, and eta is the wave surface height at the time t.
3.2 A completely nonlinear free water surface boundary condition expression for the sink is as follows:
on z=η (12)
On z=η (13)
Wherein g represents a gravitational acceleration; w represents the vertical speed of a water particle at the free water surface;representing partial derivatives over time; v represents the spatial damping parameter of the wave-absorbing region added at the tail end of the water tank, wave reflection can be effectively prevented after the wave-absorbing region is added, and the spatial damping distribution function in the form of cubic polynomials is selected:
wherein x is 0 The starting point of the wave-absorbing region is represented, and the length of the wave-absorbing region is L x -x 0 Then the dimensionless length coefficient μ= (x-x) 0 )/(L x -x 0 ) The method comprises the steps of carrying out a first treatment on the surface of the Alpha is damping strength;
3.3 A) the periodic velocity potential phi p Corresponding water particle vertical velocityThe method is obtained through recursive calculation:
wherein M is the nonlinear reserve total order, W (m) Represents an mth order component, j is an integer;
according to the periodic velocity potential phi described in the first step equation (2) p And expanding the satisfied boundary condition into a Fourier series, and recursively calculating according to the following formula to obtain the composite material:
in the method, in the process of the invention,represents the mth order periodic velocity potential->Amplitude, k of the nth mode n ＝nπ/L x Is the N natural eigenmode of the wave water tank, N x Is the total number of modes in the horizontal direction.
And finally constructing a Gao Jiepu numerical wave water tank model, namely a Gao Jiepu numerical wave water tank model, wherein the numerical wave water tank model is formed by formulas (1), (4), (8), (12), (13) and (16).
4) The constructed high-order spectrum value wave water tank model is used for efficiently and accurately simulating random and strong nonlinear real sea waves, and as shown in fig. 4, the method comprises the following steps:
4.1): dividing grids of the water tank in the horizontal direction and the vertical direction through the constructed numerical wave water tank model, and giving initial conditions of the water tank. Setting the free surface potential of the numerical wave water tank model at all positions of initial time And the free surface Gao Cheng are as follows:
η=0, at t=0 (17)
4.2): at the current time t, solving according to the water tank wave-making boundary condition formulas (6) and (7) to obtain an additional speed potential phi add Andthe method specifically comprises the following substeps:
4.2.1 According to the target wave parameters, calculating the wave surface eta, the water particle velocity u and the acceleration u of the target wave at the inlet of the water tank by utilizing the second-order Stokes wave theory t ；
4.2.2 According to the boundary conditions (6) and (7) of the wave generation of the water tank, calculating the additional velocity potential phi in the expansion domain add Time derivative of additional velocity potential
4.2.3 At the time of obtaining the additional velocity potential phi add On the basis of the characteristic expansion coefficient, solving the additional velocity potential phi at the free water surface by adopting an analytic method add Spatial derivative of (2)
4.3): calculating the periodic velocity potential phi p And the spatial derivative thereof and the spatial derivative of the wave surface eta, and calculates the velocity and the acceleration of water particles in the flow field, comprising the following steps:
4.3.1 Known at the current time)Performing recurrence calculation +.>Finally, the sum is added up to obtain the +.>
4.3.2 Respectively calculating the periodic velocity potential at the wave surface and the horizontal derivative of the wave surfaceA periodic velocity potential phi p The vertical guide number W of (a);
4.3.3 At the calculated periodic velocity potential phi p Additional velocity potential phi add On the basis of the above, the space and time derivatives of the two are further calculated by an analysis method and summed, so that the speed and the acceleration of the water particles at any position in the flow field can be obtained.
4.4): substituting the physical quantity calculated in the steps 4.2) and 4.3) into the completely nonlinear free water surface conditional expression (12) and (13) as a forcing term to perform time integration to obtain the free surface potential at the next momentAnd a free surface Gao Cheng. Judging whether the calculation time is over or not, if not, repeating the steps 4.2) to 4.4) repeatedly, and entering the calculation of the next time t+delta t;
4.5): when the wave simulation ending time is reached, gao Jiepu numerical wave water tank model calculation is ended, and the whole process numerical simulation of the numerical wave water tank on nonlinear wave generation and propagation can be realized.
The invention adopts a high-order spectrum numerical wave water tank model to generate the focused wave to simulate the nonlinear extreme waves on the sea, as shown in figure 2, the inlet speed of the nonlinear numerical wave water tank is calculated according to the simulated target wave parameters by adopting a second-order Stokes wave theory, and each wave component of the focused wave is set at the time t=t 0 Spatial position x =x 0 The elevation of the wave surface reaches the maximum. The wave components are distributed at equal intervals in the frequency range of the focused wave, and the wave amplitude is determined by the wave spectrum according to the NewWave model:
Wherein A is i A linear focusing amplitude corresponding to the ith wave component; n is the total number of wave components; a is the total focus amplitude; s (f) i ) The spectrum value corresponding to the ith wave component; s (f) n ) The spectral value corresponding to the nth wave component. Wherein S (f) is the spectrum of waves, and the invention adopts a JONSWAP spectrum:
wherein H is 1/3 Is the effective wave height; t (T) p Is the period of the spectrum peak; f (f) p Is the spectral peak frequency; gamma is the spectral peak rise factor of the spectrum,
the second-order sum frequency and difference frequency components of the focused wave can only be calculated by the traditional irregular wave second-order Stokes wave theory (water wave theory and application thereof. Zhili. Scientific press. 2005. Second chapter), but the higher-order spectrum numerical water tank model established by the invention can simulate higher-order components in strong nonlinear focused waves, the calculation result is more accurate, and the real extreme waves at sea can be simulated.
The second step, a wave load calculation module is constructed, and the wave load of the extreme waves simulated in the first step on the fan structure is calculated according to the Rainey slender body theory at each moment, and the wave load calculation module is divided into two parts: line wave load f Rainey (t, z) contribution part and water surface concentrationForce (F) x ) surface Contribution portions. The method comprises the following steps:
(1) Calculating the line wave load f Rainey (t, z) contribution part
The invention adopts Rainey slender body theory to directly calculate the wave force of extreme wave acting on any height of the center line of the pile foundation, namely the line wave load f of the fan structure in unit length Rainey (t,z)：
Wherein ρ is the sea water density; u,The horizontal velocity and the acceleration of water particles at the z height of the central axis of the pile in the incident wave field are respectively; w (w) z Representing the partial derivative of the vertical velocity of the water particle in the z direction. X, & gt>Respectively representing the displacement, the speed and the acceleration of the fan structure at the z-height; for the incidence situation of the focused wave, the focused wave simulated by the high-order spectrum value wave water tank model is used for representing the extreme wave, the wave field in the water tank is restored at each time step, and the water quality point speed and the acceleration of the fan structure in the flow field are calculated. C (C) m Is an additional inertial force coefficient or mass coefficient; c (C) D Is a drag coefficient or a velocity force coefficient; a=pi D 2 And/4 is the cross-sectional area, and D is the pile diameter. Notably, the Rainey elongate body theory adopted in the invention additionally considers the axial divergence term +.>
The line wave load f Rainey The wave load of the (t, z) contribution part is calculated by the formula (22) Rainey (t, z) integrating vertically along the fan structure from the sea bottom z= -d to the instantaneous wave surface z=η (t) expressed as follows:
Wherein z represents a vertical coordinate; d represents the water depth; η (t) represents the wave surface at the fan axis at the instant of the next calculation; f (F) x Represents f Rainey (t, z) horizontal forces of waves acting on the fan structure; m is M y Representing the overturning moment of the wave acting on the fan structure at the mud-surrounding surface.
(2) Water surface concentrated force contribution part
Said force being concentrated by the water surface (F x ) surface The wave load calculation formula of the contribution part is as follows:
wherein (M) y ) surface Represents the water surface concentration force (F) x ) surface Overturning moment acting on the fan structure around the mud surface;
and (3) adding the two loads calculated in the formulas (23) and (24) together to obtain the total wave load of the wave acting on the fan structure at each moment.
Thirdly, constructing an aerodynamic load calculation module, and calculating wind load on a fan structure according to a phyllin momentum theory at each moment, wherein the wind load calculation module comprises the following specific steps:
according to the blade parameters of the upper structure of the fan, the aerodynamic load of the fan is calculated by utilizing the classical phyllin momentum theory. According to the theory, the thrust and torque calculated by the momentum theory and the phyllin theory are equal, axial and tangential induction factors a and a' are obtained by iterative solution, as shown in fig. 5, the horizontal thrust T born by the whole fan blade is obtained by radial integration along the blade:
Wherein V is 0 Is wind speed ρ air Is of air density, R hub Is the radius of the hub, R is the radius of the fan blade, FIs a loss factor.
Fourth, constructing a pile-soil interaction module, and calculating the soil load on the fan structure by using a p-y curve method at each moment, wherein the concrete steps are as follows:
the offshore wind turbine structure is inserted into a sand foundation to a certain depth for fixation, the wind turbine structure can generate lateral deflection under the action of wind and wave loads, one part of lateral load is born by the pile body, and the other part of load is transmitted to the soil body through the pile foundation.
The invention calculates the soil load by using a p-y curve method, and the functional relation between the soil resistance p and the pile body lateral displacement y on the unit pile length established for the sandy foundation is as follows:
wherein A is s As a coefficient related to the load type, it is determined by the following equation:
k is initial foundation counterforce modulus and is related to the internal friction angle of the soil body; z is the depth of soil body and is positive from the mud surface downwards; p (P) u The ultimate bearing capacity of the soil body at any depth Z is represented by the following formula:
wherein D is the diameter of a single pile foundation; gamma is the effective volume weight of the soil; z is Z r ＝((C 3 -C 2 )D)/C 1 Is the critical depth; coefficient C 1 、C 2 And C 3 As a function of the internal friction angle phi' of the soil body, is determined by the following equation:
FIG. 6 shows a schematic soil resistance calculation, and the expression for the total resistance of the fan structure obtained by integrating the soil resistance in the foundation along the vertical direction is as follows:
Wherein L is p The length of the pile foundation soil-entering part is the water depth.
Fifthly, constructing a fan overall structure power response finite element solving model, and calculating the power response of the offshore fan overall structure at each moment, wherein the power response is specifically as follows:
5.1 Establishing a dynamic response finite element solving model of the integral structure of the offshore wind turbine
The offshore wind turbine is vertical in structure under static state, and has a length of l and a linear density of pile foundationThe elastic modulus of the fan material is E, the cross-sectional moment of inertia is I, the cross-sectional bending stiffness is EI, and the fan blade, the cabin and other upper structures are regarded as concentrated masses. Because of the small aspect ratio of the blower structure, the finite element method is applied to the Euler-Bernoulli beam unit (finite element method [ M ]]Wangcheng Beijing, university of Qinghai Press, 2003.) discrete blower structures, a blower structure line elastic finite element model was built. The distribution parameter system satisfies the following motion differential equation:
wherein X is,Respectively representing the displacement, the speed and the acceleration of the fan structure at the z height, c is the structural damping, and f (t, z) is the wave load born by the unit length of the fan structure. The top and the bottom of the offshore wind turbine are unconstrained, and the boundary conditions are as follows:
the offshore wind turbine structure is discrete by adopting standard four-degree-of-freedom beam units, and each node in the units has two degrees of freedom (node displacement and rotation angle) as shown in fig. 7. The weighting function is taken as an interpolation function, and an integral equation of a weighting allowance is established and solved by using Galerkin finite element method (Wangcheng. Finite element method basic principle and numerical method [ M ]. Beijing: university of Qinghai Press, 1997.). Integrating each unit can obtain a unit mass array and a unit stiffness array:
Converting the wave load f (t, z) received by the unit length of the fan into a unit equivalent node load, wherein the conversion process is shown in fig. 8, and assembling the unit matrix into an overall matrix to obtain a time-domain differential equation of the power response of the overall structure of the fan system, namely, the time-domain differential equation is as follows:
wherein X is,Column vectors consisting of node displacement, speed and acceleration in the finite element model; m is M G 、C G And K G Respectively an integral mass, damping and stiffness matrix, a structural damping matrix C G The Rayleigh damping matrix is adopted, namely, a damping matrix is obtained by linearly superposing a stiffness matrix and a mass matrix:
C G ＝αM G +βK G (35)
wherein the damping coefficient alpha=2ζω 1 ω 2 /(ω 1 +ω 2 )，β＝2ξ/(ω 1 +ω 2 ) ζ is the damping ratio, ω 1 And omega 2 Is the first two-order vibration frequency of the fan structure. F (F) total Converting the wave load, the aerodynamic load and the soil load obtained by the calculation in the second step, the third step and the fourth step into equivalent node loads of a fan finite element model at each moment, wherein the conversion process is as shown in figure 8And the assembled load column vectors are shown.
5.2 Numerical solution of the dynamic response of the overall structure of the offshore wind turbine by a stepwise integration method
The basic idea of the gradual integration method is that a time domain dispersion of a time domain differential equation (34) of the dynamic response of the integral structure of the fan is converted into a time differential format, and the response quantity at the moment t is gradually calculated by adopting a high-precision numerical integration method according to the response quantity (displacement, speed and acceleration) of t-deltat at the previous moment and the structural load at the moment t (note: because the fan structure response is relatively small in water particle motion and wind speed, the calculation of wave load and wind load and the solution of the dynamic response of the integral structure of the fan adopt a weak coupling form, namely the dynamic response quantity of the fan structure in the wave load calculation formula still adopts the response result of t-deltat at the previous moment). The invention adopts the classical NewMark method (NewMark N M.A method of computation for structural dynamics [ J ]. Journal of Engineering Mechanics-ASCE,1959,85 (3): 67-94) to complete time integration, namely, displacement, speed and acceleration at each node at the moment t can be solved, and the displacement, the speed and the acceleration are transmitted to a corresponding load calculation module, and the calculation is repeated until the calculation is finished.
5.3 Outputting the power response of the whole structure of the fan at each moment, analyzing the result to find out the extreme value response and the corresponding time, and providing an efficient and accurate analysis means for the actual engineering design of the offshore wind turbine.
The innovation point of the invention is analyzed:
the invention provides and establishes the integrated coupling calculation model of aerodynamic force-hydrodynamic force-structural dynamic response-soil resistance of the offshore wind turbine, and realizes the numerical analysis of the dynamic response of the integral structure of the offshore wind turbine, thereby providing a numerical calculation method for the actual engineering design of the single-pile offshore wind turbine.
The invention has the beneficial effects that:
(1) The algorithm provided by the invention can fully consider the coupling effect of the vibration and pneumatic action of the offshore wind turbine tower under the combined action of wind, wave and the like, and can accurately calculate the dynamic response of the whole structure of the wind turbine;
(2) The offshore wind turbine dynamic response full-coupling calculation model developed by the invention has the characteristics of high calculation efficiency and stability, can fully consider the interaction of extreme waves of nonlinear wave-wave interaction and a single pile type wind turbine, and provides an effective analysis means for the structural high-frequency resonance phenomenon of the offshore wind turbine with strong nonlinearity and transient property.
Drawings
FIG. 1 is a schematic diagram of the power response coupling calculation of the overall structure of an offshore wind turbine.
FIG. 2 is a schematic diagram of the operation of a digital wave water tank and a fan.
FIG. 3 is a extended computational domain D add Schematic diagram.
FIG. 4 is a flow chart of the numerical wave trough calculation.
FIG. 5 is a schematic diagram of wind load calculation for an offshore wind turbine structure.
FIG. 6 is a schematic representation of the calculation of lateral soil load by the p-y curve method.
FIG. 7 is a schematic diagram of a finite element model unit of the overall structure of the offshore wind turbine.
And (3) converting the external load into a finite element model equivalent node load schematic diagram. FIG. 8 (a) is the conversion of line load to equivalent node load; fig. 8 (b) is the conversion of concentrated force loading into equivalent node loading.
Fig. 9 is a plot of offshore wind turbine horizontal load duration.
FIG. 10 is a plot of fan tower top horizontal displacement versus time calculated for different hydrodynamic models.
FIG. 11 is a graph of the horizontal displacement duration of the top of a wind turbine tower for different incident wave fields.
In the figure, x 0 Is the starting point coordinate of a wave-absorbing region of the water tank, L x The length of the water tank is; l (L) z Calculate the total width of the domain for the water tank extension, D add To expand the computational domain, h add Calculating the boundary coordinates on the domain for continuation, z c The vertical coordinates of the axes in the domain are calculated for the continuation.
Detailed Description
The following describes the embodiments of the present invention further with reference to the drawings and technical schemes.
(1) At present time t, simulating extreme waves by using nonlinear extreme wave generation module
The invention utilizes the wave water tank with high-order spectrum value to simulate the focused wave field to represent actual extreme waves, and sets the length of the wave water tank to be 10 times of characteristic wavelength (namely 10L) p ) Wherein the length of the damping area at the tail end of the water tank is 4L p . Each focused wave group consists of 25 wave components, each wave component being at 0.5f p <f<1.5f p Equidistant distribution in the range, setting focus position x f ＝1.0L p Time of focus t f ＝8.0T p . 2T at the beginning of the computation p And a buffer function is applied in a time period, so that the influence of transient clutter in the water tank is reduced.
(2) At the current time t, calculating the wave load on the fan by utilizing a wave load calculating module (a hydrodynamic module)
The invention adopts the Rainey slender body theory of half theory and half experience to calculate the wave load acting on the fan and sets an additional inertia force coefficient C m Coefficient of drag force c=1.0 D =1.0. The pile foundation of the NREL 5-MW reference fan is simplified into a constant-section cylinder, the radius R=3.0m, the pile length l=110.0m above the mud surface, and the linear densityPile body bending stiffness ei=1.037×10 12 Nm 2 。
(3) At the current time t, calculating the wind load on the fan by using a wind load calculation module (aerodynamic module)
The wind load is calculated by utilizing a phyllin momentum theory according to structural parameters (including blade length, single blade mass, position of blade mass center and the like) of the NREL 5-MW fan blade. The upper structure of the fan mainly comprises a hub, a cabin and blades, wherein the height of the hub relative to the hydrostatic surface is 90.0M, the mass of the hub is 56780kg, the mass of the cabin is 240000kg, the mass of each blade is 17740kg, the length of each fan blade is 61.5M, and the upper cabin and the upper blade structure are simplified into a concentrated mass M at the free end of the pile top Top =350,000 kg. Each fan blade is radially divided into 17 phylloxera, and wind load on each phylloxera is sequentially calculated and radially integrated along the bladeAnd obtaining the total wind load on the fan.
(4) At the current time t, calculating the soil load on the fan by using a soil load calculating module (pile-soil interaction module)
According to the invention, the interaction between the fan and the foundation is calculated according to a p-y curve method. The NREL 5-MW fan pile foundation is buried into the soil for fixing at 36.0m, and the pile bottom is free and unconstrained. The foundation is three layers of sandy soil, and detailed soil layer distribution and soil parameters are shown in table 1. And (3) calculating the soil resistance of the fan pile foundation per unit length according to the soil layer parameters and the lateral displacement of the pile foundation, and integrating along the vertical direction of the pile foundation to obtain the total soil load of the fan.
TABLE 1 soil layer distribution and soil parameters
(5) At the current time t, calculating and outputting the power response of the whole structure of the fan by using a power response calculation module of the whole structure of the offshore wind turbine
The invention establishes a power response finite element solving model of the integral structure of the offshore wind turbine, calculates the load on the wind turbine by using three load modules of wind, wave and soil respectively, converts the load into an equivalent node load in the finite element model of the integral structure of the wind turbine, adopts a classical NewMark method (NewMark N M.A method of computation for structural dynamics [ J ]. Journal of Engineering Mechanics-ASCE,1959,85 (3): 67-94) to carry out time integration, and gradually calculates the response (displacement, speed and acceleration) at the moment t by adopting a high-precision numerical integration method according to the response (displacement, speed and acceleration) of the previous moment t-deltat and the structural load at the moment t, and outputs the power response of the integral structure of the wind turbine at the moment t;
(4) Judging whether the calculation time is over, if not, repeating the steps (2) to (4) repeatedly, and entering the calculation of the next time t+delta t;
(5) And when the calculation end time of the power response of the integral structure of the fan is reached, the calculation and the solving of the power response of the integral structure of the fan can be realized, and the result is analyzed to find the extremum response and the corresponding time, so that an efficient and accurate analysis means is provided for the actual engineering design of the offshore fan.
Examples
(1) Wind load and soil load calculation and analysis of offshore wind turbine
The wind speed of the normal wind is 11.4m/s, the rotating speed of the wind wheel is 12.1rpm, the wind load is modulated by adopting a buffer function, the wind load is slowly increased to the maximum value, and the buffer time is set to be 50.0s. The horizontal force duration curve on the fan is shown in fig. 9, and as can be seen from the graph, the wind load and the soil load are equal in size and opposite in direction and always keep balanced, so that the accuracy of a fan wind and soil load calculation model is verified.
(2) Power response calculation and analysis for integral structure of offshore wind turbine
The wind speed of the normal wind is 11.4m/s, the rotating speed of the wind wheel is 12.1rpm, the water depth d=20m, the focusing amplitude A=1.80 m, and the spectral peak period T p The fan generates lateral vibration under the combined action of wind and wave, and the structural damping ratio xi=0.01.
According to the invention, the classical Morisen equation and the Rainey slender body theory are adopted to calculate wave load, and FIG. 10 shows the comparison condition of the fan tower top displacement duration curves calculated by the classical Morisen equation and the Rainey slender body theory, so that the fan tower top displacement calculated by the Rainey slender body theory is slightly larger than the corresponding result of the Morisen equation, the two are well matched integrally, and the accuracy of the hydrodynamic force calculation model is verified.
Then, the influence of linear and nonlinear incident focusing waves on the dynamic response of the whole structure of the fan is compared and analyzed, as shown in fig. 11, the displacement peak value corresponding to the linear incident focusing waves is the smallest, and the consistency is poorer than that of the displacement result corresponding to the nonlinear incident focusing waves; in addition, the peak value of tower top displacement corresponding to the numerical wave water tank simulated incident focused wave is maximum, and the dynamic response nonlinearity of the whole structure of the fan is stronger. From fig. 11, it can be observed that the focused wave main peak passes the offshore wind turbine and then bursts a high frequency resonance response for a number of characteristic wave periods, indicating that the high frequency wave load excites the high frequency resonance response of the offshore wind turbine structure as the high frequency load frequency approaches the natural frequency of the pile foundation.
The examples described above represent only embodiments of the invention and are not to be understood as limiting the scope of the patent of the invention, it being pointed out that several variants and modifications may be made by those skilled in the art without departing from the concept of the invention, which fall within the scope of protection of the invention.
Claims (7)
1. The analysis method is characterized in that the analysis method is realized based on a marine fan integral structure dynamic response coupling calculation model, the calculation model comprises an aerodynamic module, a hydrodynamic module, a pile-soil interaction module and a marine fan integral structure dynamic response calculation module, wherein the hydrodynamic module is also a wave load calculation module, and three load calculation sub-modules are connected through the marine fan integral structure dynamic response calculation module; firstly, calculating wave load, wind load and soil load by using three load calculation sub-modules respectively, converting the wave load, the wind load and the soil load into equivalent node load in a power response calculation module of the whole structure of the offshore wind turbine, and then, solving the power response of the whole structure of the offshore wind turbine by using the power response calculation module of the whole structure of the offshore wind turbine to obtain displacement, speed and acceleration at each node, and respectively transmitting the displacement, the speed and the acceleration to a corresponding load calculation module so as to further realize analysis of the power response of the whole structure of the offshore wind turbine.
2. The method for analyzing the dynamic response of the offshore wind power generation structure according to claim 1, which is characterized by comprising the following steps:
first step, simulating a nonlinear extreme wave field by establishing a high-order spectrum value wave water tank model
The Gao Jiepu numerical wave water tank model firstly decomposes the total velocity potential of fluid in the water tank into a periodic velocity potential and an additional velocity potential, utilizes the periodic velocity potential to process periodic boundary conditions, and utilizes the additional velocity potential to process non-periodic wave-making boundary conditions, thereby realizing the wave-making function of the numerical water tank; secondly, solving two decomposed velocity potentials by adopting a quasi-spectrum method, and respectively expanding the two velocity potentials into Fourier series forms meeting a control equation and corresponding boundary conditions; thirdly, calculating additional speed potential according to the wave-making boundary condition of the water tank at each calculation moment; finally, substituting the obtained additional velocity potential into a nonlinear free water surface boundary condition, performing time integration to obtain a water tank wave surface at the next moment and a periodic velocity potential at the wave surface, constructing a complete high-order spectrum numerical wave water tank model, and repeating all the way until the end;
the high-order spectrum value wave water tank model is adopted to efficiently and accurately simulate the real sea wave with randomness and strong nonlinearity;
The second step, a wave load calculation module is constructed, and the wave load of the extreme waves simulated in the first step on the fan structure is calculated according to the Rainey slender body theory at each moment, and the wave load calculation module is divided into two parts: line wave load f Rainey (t, z) contribution part, and water surface concentration force (F x ) surface A contribution portion;
thirdly, constructing an aerodynamic load calculation module, and calculating wind load on a fan structure according to a phyllin momentum theory at each moment;
fourthly, constructing a pile-soil interaction module, and calculating the soil load on the fan structure by using a p-y curve method at each moment;
fifthly, constructing a fan overall structure power response finite element solving model, and calculating the power response of the offshore fan overall structure at each moment, wherein the power response is specifically as follows:
5.1 Establishing a dynamic response finite element solving model of the integral structure of the offshore wind turbine
The offshore wind turbine is vertical in structure under static state, and has a length of l and a linear density of pile foundationThe elastic modulus of the fan material is E, the cross section moment of inertia is I, the cross section bending rigidity is EI, and the upper structures such as the fan blades, the engine room and the like are regarded as concentrated masses; because the transverse-longitudinal ratio of the fan structure is small, a finite element method is adopted to apply an Euler-Bernoulli beam unit discrete fan structure, and an elastic finite element model of a fan structure line is established; The distribution parameter system satisfies the following motion differential equation:
wherein X is,Respectively representing displacement, speed and acceleration of the fan structure at the z-height, c is structural damping, and f (t, z) is wave load born by the fan structure in unit length; the top and the bottom of the offshore wind turbine are unconstrained, and the boundary conditions are as follows:
the offshore wind turbine structure adopts standard four-degree-of-freedom beam units for dispersion, and each node in the units has two degrees of freedom; taking the weight function as an interpolation function by adopting a Galerkin finite element method, establishing an integral equation of the weighted allowance and solving; integrating each unit can obtain a unit mass array and a unit stiffness array:
converting wave load f (t, z) born by unit length of the fan into unit equivalent node load, and assembling the unit matrix into an integral matrix to obtain a time-domain differential equation of the dynamic response of the integral structure of the fan system, wherein the time-domain differential equation is as follows:
wherein X is,Column vectors consisting of node displacement, speed and acceleration in the finite element model; m is M G 、C G And K G Respectively an integral mass, damping and stiffness matrix, a structural damping matrix C G The Rayleigh damping matrix is adopted, namely, a damping matrix is obtained by linearly superposing a stiffness matrix and a mass matrix:
C G ＝αM G +βK G (35)
Wherein the damping coefficient alpha=2ζω 1 ω 2 /(ω 1 +ω 2 )，β＝2ξ/(ω 1 +ω 2 ) ζ is the damping ratio, ω 1 And omega 2 The front two-order vibration frequency of the fan structure; f (F) total Converting the wave load, the aerodynamic load and the soil load obtained by calculation in the second step, the third step and the fourth step into equivalent node loads of a fan finite element model at each moment, and assembling to form a load column vector;
5.2 Numerical solution of the dynamic response of the overall structure of the offshore wind turbine by a stepwise integration method
The dynamic response time-domain differential equation (34) of the integral structure of the fan is dispersed in the time domain and is converted into a time-difference format, and the response quantity at the time t is gradually calculated by adopting a high-precision numerical integration method according to the response quantity at the time t-delta t at the previous time and the structural load at the time t; the time integration is completed, namely the displacement, the speed and the acceleration of each node at the moment t can be solved, and are transmitted to the corresponding load calculation module, and the process is repeated until the process is finished;
5.3 Outputting the power response of the whole structure of the fan at each moment, analyzing the result to find out the extreme value response and the corresponding time, and providing an efficient and accurate analysis means for the actual engineering design of the offshore wind turbine.
3. The method for analyzing the dynamic response of the offshore wind power generation structure according to claim 2, wherein the first step comprises the following steps:
1) Constructing a Gao Jiepu numerical wave water tank model, defining a water tank domain as a calculation domain, and decomposing the total velocity potential of fluid in the calculation domain into two parts, namely a periodic velocity potential and an additional velocity potential, specifically:
φ＝φ p +φ add (1)
the periodic velocity potential is phi p It needs to meet the free water surface boundary condition, the water bottom boundary condition, the water-proof boundary condition of the left and right ends:
where x represents a horizontal coordinate,representing partial derivatives of x; z represents the vertical coordinate ++>Representing partial derivatives of z; l (L) x The length of the water tank is d, and the water depth of the water tank is d;
the additional velocity potential is phi add The method needs to meet the wave-making boundary condition of the left end of the water tank, the water bottom boundary condition and the watertight side wall boundary condition of the right end:
where u is the horizontal velocity of the target wave input at the entrance of the flume;
2) Solving for the additional velocity potential phi add Time derivative of additional velocity potential
2.1 Determining an additional velocity potential phi add
φ add In the extended computing domain D add Solving in, selecting h add =3d, centerline height z c D, guarantee centerline z c Always higher than the free water surface; in order to accurately simulate steeper nonlinear waves, a wave generation method of inputting a target wave horizontal speed from z= -d at the bottom of a water tank to an instantaneous wave surface z=η; specific:
The wave surface eta (0, t) at the inlet of the water tank is directly calculated by the target wave parameters; in the extended domain D add Phi in the first step of formula (3) add The satisfied boundary conditions are developed into the following form according to the characteristic function:
wherein B is n (t) is the additional velocity potential phi add Amplitude of the nth mode; kappa (kappa) n ＝(2n-1)π/(h add +d) is the extension domain D add N-th eigenmode in vertical direction, h add Representing vertical coordinates at the upper boundary of the extended computational domain, N z Representing the total number of modes in the vertical direction; l (L) x Indicating the length of the water tank; then according to the additional velocity potential phi add Can directly calculate the partial derivative of the X direction of the characteristic expansion of the (E)
Will partial derivativeSubstituting the boundary condition of the incident wave making of the water tank:
where u (z) represents the horizontal velocity of the target wave input at the inlet of the trough, using the vertical directionCharacteristic function cos [ kappa ] n (z+d)]From the orthogonality of the horizontal velocity u (z) at the wave-making boundary, the coefficient B can be determined n (t) and determining phi add ；
2.2 Determining the time derivative of the additional velocity potential
The time derivative of the additional velocity potentialThe following boundary conditions are satisfied:
wherein u is t (z) represents the horizontal acceleration of the target wave input at the entrance of the flume;
then the velocity potential time derivative is appendedThe characteristic function is developed into the following form:
Then the velocity potential time derivative is appendedX-direction partial derivative +.>The resolvable representation is in the form of:
wherein C is n (t) isAmplitude of the nth mode; partial derivative +.>Substitution (7) using the vertical characteristic function cos [ kappa ] n (z+d)]According to the orthogonality of the horizontal velocity time derivative u at the wave-making boundary t (z) vertical distribution, coefficient C is determined n (t) and further determining->
3) Solving the periodic velocity potential phi p
3.1 A) applying a periodic velocity potential function phi at the free water surface of the basin p Represented as being dependent only on horizontal coordinatesForm:
wherein t is the current calculation time, and eta is the wave surface height at the time t;
3.2 A completely nonlinear free water surface boundary condition expression for the sink is as follows:
wherein g represents a gravitational acceleration; w represents the vertical direction of water particles at the free water surfaceA speed;representing partial derivatives over time; v represents the spatial damping parameter of the wave-absorbing region added at the tail end of the water tank, and wave reflection can be effectively prevented after the wave-absorbing region is added;
3.3 A) the periodic velocity potential phi p Corresponding water particle vertical velocityThe method is obtained through recursive calculation:
wherein M is the nonlinear reserve total order, W (m) Represents an mth order component, j is an integer;
according to the periodic velocity potential phi described in the first step equation (2) p And expanding the satisfied boundary condition into a Fourier series, and recursively calculating according to the following formula to obtain the composite material:
In the method, in the process of the invention,represents the mth order periodic velocity potential->Amplitude, k of the nth mode n ＝nπ/L x Is the N natural eigenmode of the wave water tank, N x The total mode number in the horizontal direction;
4) The constructed high-order spectrum value wave water tank model is used for efficiently and accurately simulating random and strong nonlinear real sea waves, and comprises the following steps of:
4.1): by passing throughDividing a built numerical wave water tank model into grids of the water tank in the horizontal direction and the vertical direction, and giving initial conditions of the water tank; setting the free surface potential of the numerical wave water tank model at all positions of initial timeAnd the free surface Gao Cheng are as follows:
4.2): at the current time t, solving according to the water tank wave-making boundary condition formulas (6) and (7) to obtain an additional speed potential phi add Andthe method specifically comprises the following substeps:
4.2.1 According to the target wave parameters, calculating the wave surface eta, the water particle velocity u and the acceleration u of the target wave at the inlet of the water tank by utilizing the second-order Stokes wave theory t ；
4.2.2 According to the boundary conditions (6) and (7) of the wave generation of the water tank, calculating the additional velocity potential phi in the expansion domain add Time derivative of additional velocity potential
4.2.3 At the time of obtaining the additional velocity potential phi add On the basis of the characteristic expansion coefficient, solving the additional velocity potential phi at the free water surface by adopting an analytic method add Spatial derivative of (2)
4.3): calculating the periodic velocity potential phi p And the spatial derivative thereof and the spatial derivative of the wave surface eta, and calculates the velocity and the acceleration of water particles in the flow field, comprising the following steps:
4.3.1 Known at the current time)Performing recurrence calculation +.>Finally, the sum is added up to obtain the +.>
4.3.2 Respectively calculating the periodic velocity potential at the wave surface and the horizontal derivative of the wave surfaceA periodic velocity potential phi p The vertical guide number W of (a);
4.3.3 At the calculated periodic velocity potential phi p Additional velocity potential phi add On the basis of the above, the space and time derivatives of the two are further calculated by an analysis method and summed, so that the speed and the acceleration of water particles at any position in the flow field can be obtained;
4.4): substituting the physical quantity calculated in the steps 4.2) and 4.3) into the completely nonlinear free water surface conditional expression (12) and (13) as a forcing term to perform time integration to obtain the free surface potential at the next momentAnd a free surface Gao Cheng; judging whether the calculation time is over or not, if not, repeating the steps 4.2) to 4.4) repeatedly, and entering the calculation of the next time t+delta t;
4.5): when the wave simulation ending time is reached, gao Jiepu numerical wave water tank model calculation is ended, and the whole process numerical simulation of the numerical wave water tank on nonlinear wave generation and propagation can be realized.
4. A method of analyzing the dynamic response of an offshore wind turbine structure according to claim 3, wherein:
and step 2.2) In order to avoid the initial effect, u (z) and u t (z) need to be multiplied by the buffering function R m (t) the expression is as follows:
wherein T is m The invention takes twice wave characteristic period for buffering time length;
in the step 3.2), a spatial damping distribution function in the form of a cubic polynomial is selected:
wherein x is 0 The starting point of the wave-absorbing region is represented, and the length of the wave-absorbing region is L x -x 0 Then the dimensionless length coefficient μ= (x-x) 0 )/(L x -x 0 ) The method comprises the steps of carrying out a first treatment on the surface of the Alpha is damping strength.
5. The method for analyzing the dynamic response of the offshore wind power generation structure according to claim 2, wherein the second step comprises the following steps:
(1) Calculating the line wave load f Rainey (t, z) contribution part
Directly calculating wave force of extreme waves acting on any height of the center line of the pile foundation by adopting Rainey slender body theory, namely line wave load f of fan structure on unit length Rainey (t,z)：
Wherein ρ is the sea water density; u,The horizontal velocity and the acceleration of water particles at the z height of the central axis of the pile in the incident wave field are respectively; w (w) z Representing the partial derivative of the vertical velocity of the water particle in the z direction; x, & gt >Respectively representing the displacement, the speed and the acceleration of the fan structure at the z-height; for the incidence situation of the focused wave, the focused wave simulated by using a Gao Jiepu numerical wave water tank model represents extreme waves, and the wave field in the water tank is restored in each time step, and the water quality point speed and the acceleration of a fan structure in the flow field are calculated; c (C) m Is an additional inertial force coefficient or mass coefficient; c (C) D Is a drag coefficient or a velocity force coefficient; a=pi D 2 4 is the cross-sectional area, D is the pile diameter;
the line wave load f Rainey The wave load of the (t, z) contribution part is calculated by the formula (22) Rainey (t, z) integrating vertically along the fan structure from the sea bottom z= -d to the instantaneous wave surface z=η (t) expressed as follows:
wherein z represents a vertical coordinate; d represents the water depth; η (t) represents the wave surface at the fan axis at the instant of the next calculation; f (F) x Represents f Rainey (t, z) horizontal forces of waves acting on the fan structure; m is M y Representing the overturning moment of the wave acting on the fan structure at the mud-surrounding surface;
(2) Water surface concentrated force contribution part
Said force being concentrated by the water surface (F x ) surface The wave load calculation formula of the contribution part is as follows:
wherein (M) y ) surface Represents the water surface concentration force (F) x ) surface Overturning moment acting on the fan structure around the mud surface;
And (3) adding the two loads calculated in the formulas (23) and (24) together to obtain the total wave load of the wave acting on the fan structure at each moment.
6. The method for analyzing the dynamic response of the offshore wind power generation structure according to claim 2, wherein the third step comprises the following steps: calculating the aerodynamic load of the fan by utilizing a classical phyllin momentum theory according to the blade parameters of the upper structure of the fan; according to the principle that the thrust and torque calculated by the momentum theory and the phyllin theory are equal, the axial and tangential induction factors a and a' are obtained by iterative solution, and the horizontal thrust T born by the whole fan blade is obtained by radial integration along the blade:
wherein V is 0 Is wind speed ρ air Is of air density, R hub The hub radius is R, the radius of the fan blade, and F is a loss factor.
7. The method for analyzing the dynamic response of the offshore wind power generation structure according to claim 2, wherein the fourth step comprises the following steps: the offshore wind turbine structure is inserted into a sand foundation for fixation, the wind turbine structure generates lateral deflection deformation under the action of wind and wave load, one part of lateral load is born by the pile body, and the other part of load is transmitted to the soil body through the pile foundation; the soil load is calculated by using a p-y curve method, and the functional relation between the soil resistance p and the lateral displacement y of the pile body on the unit pile length established for the sandy foundation is as follows:
Wherein A is s As a coefficient related to the load type, it is determined by the following equation:
k is initial foundation counterforce modulus and is related to the internal friction angle of the soil body; z is the depth of soil body and is positive from the mud surface downwards; p (P) u The ultimate bearing capacity of the soil body at any depth Z is represented by the following formula:
wherein D is the diameter of a single pile foundation; gamma is the effective volume weight of the soil; z is Z r ＝((C 3 -C 2 )D)/C 1 Is the critical depth; coefficient C 1 、C 2 And C 3 As a function of the internal friction angle phi' of the soil body, is determined by the following equation:
the expression for the total resistance on the structure of the wind turbine can be obtained by integrating the soil resistance in the foundation along the vertical direction as follows:
wherein L is p The length of the pile foundation soil-entering part is the water depth.
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