CN113204821A - Method for establishing tandem double-cylinder wake flow oscillation unsteady aerodynamic force mathematical model - Google Patents

Method for establishing tandem double-cylinder wake flow oscillation unsteady aerodynamic force mathematical model Download PDF

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CN113204821A
CN113204821A CN202110491924.6A CN202110491924A CN113204821A CN 113204821 A CN113204821 A CN 113204821A CN 202110491924 A CN202110491924 A CN 202110491924A CN 113204821 A CN113204821 A CN 113204821A
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晏致涛
杨小刚
钟永力
孙毅
刘欣鹏
李妍
赵爽
王灵芝
聂小春
巩黎明
曾玉洁
罗钧
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Chongqing University of Science and Technology
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Abstract

The invention provides a method for establishing a tandem double-cylinder wake flow galloping unsteady aerodynamic mathematical model, which comprises the following specific steps of: simulating the wake flow galloping of the tandem double cylinders with the upstream cylinder fixed and the downstream cylinder transversely vibrated by adopting computational fluid dynamics software to obtain a wake flow galloping geometric model of the tandem double cylinders and carrying out grid division on the geometric model; establishing a working condition according to a geometric model, and analyzing a frequency spectrum rule of wake flow galloping aerodynamic force of the tandem double cylinders; establishing a wake flow galloping aerodynamic mathematical model of the tandem bicylinder containing unknown parameters according to the wake flow galloping of the tandem bicylinder with the upstream cylinder fixed and the downstream cylinder transversely vibrated; and fitting a pneumatic damping force parameter and a vortex force parameter in the self-excitation force by adopting a least square parameter identification method based on energy equivalence, and solving a wake flow galloping pneumatic force mathematical model of the tandem double cylinders. And a tandem double-cylinder wake flow galloping model is established, so that the existing engineering construction is facilitated, and the stability and reliability of the engineering construction are improved.

Description

Method for establishing tandem double-cylinder wake flow oscillation unsteady aerodynamic force mathematical model
Technical Field
The invention relates to the technical field of double-cylinder building simulation, in particular to a method for establishing a tandem double-cylinder wake flow galloping unsteady aerodynamic mathematical model.
Background
In actual engineering, the buildings are usually pillar structures, such as offshore structures, submarine pipelines, large group chimneys, heat exchange tubes, piers, guys, masts, chemical reaction towers, neighboring skyscrapers and power transmission lines. Fluid force is an important factor for designing the engineering structures, so that understanding of the influence of adjacent column group structures on flow-induced vibration has important engineering practical significance. The double cylinder is an ideal model for researching the flow-induced vibration of the cylinder array due to the simple form. When the two cylinders are brought into close proximity, vortex shedding by the upstream cylinder induces a second cylinder vibration.
The arrangement form of the double cylinders and the distance between the double cylinders are important factors influencing the wake flow galloping. The following designs and experiments exist in the prior art:
the wind tunnel experimental research shows that the amplitude characteristic of the wake flow galloping development stage is approximately linearly increased along with the increase of the converted wind speed. In the study on wake flow interference of double cylinders with equal diameters, the distance between the two cylinders has a great determining effect on the induction of wake flow galloping, a wake flow area flowing through an upstream cylinder forms three areas, namely a short-distance unstabilized area, a stable area and a long-distance unstabilized area, for a downstream conductor, and the wake flow galloping can only occur in a specific interval.
Based on the difference of the fluid form and the relative position between the double cylinders with equal diameters, the influence of the tail flow traces on the downstream cylinders is divided into two types of wake flow interference and short-distance interference, and L/D (3.5) and T/D (2) are used as two key distances, wherein L is the vertical distance of the two cylinders in the flowing direction, T is the vertical distance of the two cylinders in the transverse flowing direction, and D is the diameter of the cylinders.
The wake flow areas of the parallel wires with the same diameter are divided in detail through wind tunnel experiment results, the wire spacing ratio corresponding to a close-distance interference area is 1-1.1, the wire spacing ratio of wake flow galloping phenomena is 1.1-3.8, and the wire spacing ratio exceeding 3.8 is a long-distance interference area.
Experiments were conducted on an elastically supported downstream cylinder (allowing only transverse oscillations) behind a fixed cylinder, and wake flow galloping phenomena with increasing amplitude with reduced wind speed were observed at a tandem distance L/D of 3-5.6. The full scale experiment of double cylinders with equal diameters and arranged in series at an interval of L/D of 4.3 to 8.7 was carried out by researchers using wind tunnels, and a wake flow galloping phenomenon of the downstream cylinder was observed at L/D of 4.3, and disappeared at L/D of 6.5. Meanwhile, researchers perform wind tunnel tests on double cylinders with different spacing ratios, and 7 flow-induced vibration response modes are divided according to whether the upstream cylinder and the downstream cylinder generate relaxation vibration and vortex-induced resonance or not.
Wake vibrations, a complex form of vibration common in engineering, have been studied and analyzed by many researchers for other factors that affect its aerodynamic characteristics and vibrational response, in addition to the effects of pitch.
For example, the disturbance galloping of a resiliently supported cylindrical sling under wake flow is studied experimentally, and the critical wind speed of wake galloping is found to increase with the increase of the Sc number.
The effects of structural and aerodynamic damping of power transmission lines on wake vibration response were studied systematically, for example.
For example, the wake flow galloping of the downstream guy cable is researched through a wind tunnel test, and the result shows that the vibration response of the downstream guy cable under the effect of the wake flow can be influenced by the Reynolds number and the wind speed. These results have some promotion effect on the characteristics of wake flow galloping, but the mechanism and the reason for the wake flow galloping are not deeply disclosed.
The traditional wake flow test focuses on the characteristics of structural response, surface pressure distribution, flow field morphology and the like, and researches on the nonlinear characteristics of unsteady relaxation force are less. In the prior art, the earliest design is based on a quasi-stationary constant hypothesis, and a linear analysis method is used for judging the possibility of the wake flow galloping instability of the power transmission line.
Secondly, based on the wave transfer theory, a transfer matrix method is adopted to carry out research on the vibration characteristics of wake flow galloping of the split conductor, and the wake flow galloping vibration response is predicted.
Based on the two research results, researchers perfect a quasi-stationary normal mechanical model of wake flow galloping and establish a nonlinear vibration equation of the wake flow galloping. The spatial distribution rule of the average aerodynamic coefficient of the wake flow transmission line is measured by adopting a wind tunnel test method, and the amplitude of the wake flow galloping is obtained by adopting a Runge-Kutta method for solving. Although the wake flow galloping model has certain rationality, the wake flow galloping model is established on the basis of a calibration and constancy theory, and the aerodynamic model is still constant and has certain limitations. Therefore, it is necessary to establish a non-linear mathematical model of unsteady aerodynamic force based on unsteady aerodynamic force time-course data.
Disclosure of Invention
In view of the above, the invention provides a method for establishing a tandem double-cylinder wake flow galloping unsteady aerodynamic mathematical model, which is used for establishing a tandem double-cylinder wake flow galloping model, and is beneficial to the existing engineering construction and improving the stability and reliability of the engineering construction.
The technical scheme is as follows:
a method for establishing a tandem double-cylinder wake flow oscillation unsteady aerodynamic force mathematical model is characterized by comprising the following specific steps:
s1: adopting computational fluid dynamics software to simulate the wake flow galloping of the tandem double cylinders with the fixed upstream cylinder and the transversely vibrated downstream cylinder under the condition that the flow direction interval L/D of the cylinders is 4, obtaining a wake flow galloping geometric model of the tandem double cylinders and carrying out mesh division on the geometric model;
wherein, L is the vertical distance of the two cylinders in the flow direction, and D is the diameter of the cylinder;
s2: establishing at least one working condition according to the wake flow galloping geometric model of the tandem bicylinder, and analyzing the frequency spectrum law of the wake flow galloping aerodynamic force of the tandem bicylinder;
s3: establishing a wake flow galloping aerodynamic mathematical model of the tandem bicylinder containing unknown parameters according to the wake flow galloping of the tandem bicylinder with the upstream cylinder fixed and the downstream cylinder transversely vibrated in the step S1; the aerodynamic force is expressed in two forms of self-excitation force and vortex-induced force;
s4: fitting a pneumatic damping force parameter and a vortex excitation force parameter in the self-excitation force by adopting a least square parameter identification method based on energy equivalence; obtaining a final wake flow galloping aerodynamic mathematical model of the tandem double cylinders;
s5: and (5) comparing the wake flow galloping force mathematical model of the tandem bicylinder obtained in the step (S4).
By constructing the steps, under the condition that wake flow galloping is most likely to occur when L/D is 4, double cylinders are simulated by adopting Computational Fluid Dynamics (CFD), and the wake flow galloping of a front column fixed and a rear column transversely vibrating tandem double cylinder is simulated in the flow direction of double cylinder wind. And (3) providing a mathematical parameter expression of the aerodynamic force model, and obtaining the final tandem bicylinder wake flow galloping aerodynamic force mathematical model after solving unknown parameters. After verification and comparison, the model has good rationality and high accuracy, and is beneficial to more stable construction projects in the later construction process.
In step S1, in the wake relaxation oscillation of the tandem double cylinder, the lateral vibration of the downstream cylinder is solved by using a wake cylinder lateral vibration equation, where the downstream cylinder lateral vibration equation is:
Figure BDA0003052706680000031
wherein m is the downstream cylinder mass; y, y,
Figure BDA0003052706680000032
And
Figure BDA0003052706680000033
respectively displacement, velocity and acceleration perpendicular to the flow direction; c and k are divided into structural damping and rigidity constants; fyIs the vertical force to which the cylinder is subjected.
In the invention, when the simulation working condition is set, the unsteady incompressible RANS equation of the two-dimensional cylindrical streaming is as follows:
Figure BDA0003052706680000034
Figure BDA0003052706680000035
wherein u isiAnd p represents the average velocity and average pressure, respectively; ρ is the fluid density; mu is molecular viscosity; sijIs the mean stress tensor;
Figure BDA0003052706680000036
is the reynolds stress tensor.
A shear stress transfer model, namely an SST-k omega model, is used during the setting of the simulation working condition. The model involves two transport equations, the turbulence energy and dissipation ratio equations, in which a non-steady state separation algorithm (USA) is employed. For a model, see the article: two-equation two-dimensional interference modules for engineering applications [ J ] Aiaa journal.1994,32.
The velocity expression is processed by a SIMPLE algorithm when being coupled, and a second-order implicit scheme is adopted for the unstable condition. A second order implicit scheme is used for the convection terms in the k- ω transfer equation and the momentum equation.
Still further, in step S1, in the wake flow galloping geometric model of the tandem double cylinder, the adopted rectangular calculation domain is 60D × 40D, and D is the diameter of the cylinder;
in the rectangular calculation domain, the distance between the center of the upstream cylinder and the inlet boundary is 25D, and the distance between the center of the downstream cylinder and the outlet boundary is 31D;
in the rectangular calculation domain, setting an entrance boundary as a speed entrance boundary condition;
in the rectangular calculation domain, setting a basin outlet boundary as a pressure outlet boundary condition;
in the rectangular calculation domain, the upper and lower boundaries are defined as symmetric boundary conditions;
in the rectangular calculation domain, the upstream cylinder and the downstream cylinder are both cylinders, and the surfaces of the cylinders are non-slip walls.
Further, in step S1, the geometric model is gridded using a non-uniform quadrilateral structured grid, and the surface grids of the upstream and downstream cylindrical surfaces use O-shaped grids;
because the flow field near the wall surface changes violently, the grid near the wall surface of the cylinder is encrypted and the height of the first layer of grid on the wall surface is controlled when the geometric model is subjected to grid division, so that the height y of the first layer of grid on the wall surface of the upstream cylinder and the downstream cylinder is enabled to be high+Less than 1; to reduce the influence of the grid on the numerical calculation.
To simulate cylindrical unsteady vibrations, the interfaces 27D and 34D downstream of the inlet in the rectangular computational domain employ a sliding mesh technique, with the mesh being updated using dynamic tiling at the lateral boundaries.
Still further, in step S3, the vortex vibration and the relaxation vibration force are mainly self-excited force, and the vortex-excited force acts only at the vibration development stage. The vortex-induced forces of two frequencies in the wake relaxation vibration force exist all the time in each stage of vibration, and the component ratio is large, so that the consideration is needed. According to the wake relaxation vibration of the tandem bicylinder with the fixed upstream cylinder and the transverse vibration of the downstream cylinder mentioned in the step S1, the downstream cylinder transverse vibration equation can obtain the wake relaxation vibration aerodynamic force mathematical model of the tandem bicylinder with the unknown self-excitation force parameter and the unknown vortex excitation force parameter.
The mathematical model of the tandem double-cylinder wake flow galloping aerodynamic force containing unknown self-excitation parameters and unknown vortex-excitation parameters is as follows:
Figure BDA0003052706680000041
wherein, FyIs the total aerodynamic force of the structure;
Figure BDA0003052706680000042
is an expression for self-excited force, wherein self-excited force is caused by self-excited vibration as a result of mutual feedback of force and motion state of the system, so that such self-excited vibration force can be generally described as a function of structure speed and displacement;
wherein, Fv(fv) Is an expression of vortex-induced force; the vortex-induced forces are caused by vortex shedding by the upstream cylinder or by the downstream cylinder itself, and the vortex shedding frequency is indicative of vortex shedding, and thus the surface pressure caused by vortex shedding can be generally described as a function of the vortex shedding frequency.
Figure BDA0003052706680000043
For y and y respectively
Figure BDA0003052706680000044
And (3) performing high-order Taylor sequence expansion to obtain a self-excitation force polynomial expression in wake flow relaxation force as follows:
Figure BDA0003052706680000045
in the formula gijAre pneumatic parameters. g00This term and description soundY and of corresponding state
Figure BDA0003052706680000046
All have no relation, and this term can be considered as a static term, which has no effect on the wake galloping dynamic response, so that it is ignored. From the previous aerodynamic spectra, it can be seen that the contribution to the entire frequency component is large, substantially within the 5 th order frequency multiplication, so that the excess y in the Taylor expansion can be set5And
Figure BDA0003052706680000051
the term (1) is truncated.
Based on the principle of energy equivalence,
Figure BDA0003052706680000052
the medicine consists of three parts: the aerodynamic damping force term (changing the damping of the structure), the aerodynamic stiffness force term (changing the stiffness of the structure) and the pure force term (neither changing the damping of the structure nor changing the stiffness of the cylinder).
Then, the self-excitation force can be written as
Figure BDA0003052706680000053
Neglecting pure force terms, the dimensionless self-exciting force expression is as follows
Figure BDA0003052706680000054
For lateral vibrations, the aerodynamic stiffness term only slightly changes the structural frequency, and may not be considered, and the above formula can be reduced to:
Figure BDA0003052706680000055
ρ is the fluid density, U is the incoming flow velocity, D is the cylinder diameter, y is the displacement perpendicular to the flow direction,
Figure BDA0003052706680000056
velocity perpendicular to the flow direction; alpha is alpha01For a first-order dimensionless aerodynamic damping parameter, alpha03For three-order dimensionless aerodynamic damping parameters, alpha05Five-order dimensionless pneumatic damping parameters which are unknown parameters;
first, there are two maximum vortex shedding frequencies fv1And fv2And since the vortex-induced force is always closely related to the vortex shedding frequency, it can be first expressed as a function of the vortex shedding frequency, and the expression for the vortex-induced force can be changed as follows:
Figure BDA0003052706680000057
due to the phase angle in the above formula
Figure BDA0003052706680000058
And
Figure BDA0003052706680000059
all are parameters to be fitted and solved, and can be rewritten as follows after dimensionless:
Figure BDA00030527066800000510
in the formula: delta11,δ12,δ21,δ22The parameter is a dimensionless vortex-induced force parameter;
St1is the upstream cylindrical Steyr number; st2Is the downstream cylindrical Stockholr number; ρ is the fluid density; u shaperTo convert wind speed, fsIs the natural frequency; v. of11 v12 v21 v22Respectively representing the parameters of the vortex-induced force before dimensionless;
therefore, a simplified model of wake relaxation forces considering only stable amplitudes can be expressed as follows:
Figure BDA0003052706680000061
still further, the specific content of step S5 is:
s51: according to the obtained wake flow galloping aerodynamic mathematical model of the tandem double cylinders, the displacement and the speed are inversely calculated by adopting a Runge-Kutta method;
s52: parameter names defining downstream cylinder energy:
Figure BDA0003052706680000062
Figure BDA0003052706680000063
Figure BDA0003052706680000064
ωsis the natural circular frequency of the downstream cylinder;
Ws(t) is the accumulated work of the structure damping of the downstream cylinder at the current time, and the consumed energy is a negative value;
Wa,v(t) the accumulated work of the downstream cylinder aerodynamic damping force and vortex-induced force at the current time is back-calculated by using the fitted parameters, the supplied energy is positive,
Ee(t) the mechanical energy of the downstream cylinder at the current time, wherein the mechanical energy comprises kinetic energy and potential energy;
s53: substituting the displacement and the speed obtained in the step S51 into the formulas (1) to (3), and reconstructing to obtain the accumulated work of the pneumatic damping force and the vortex-induced force;
s54: and comparing the accumulated work of the pneumatic damping force and the vortex-induced force with the simulation value of the wake flow galloping geometric model of the tandem bicylinder.
Compared with the prior art, the invention has the beneficial effects that:
the tandem bicylinder wake flow galloping mathematical model of unsteady aerodynamic force is established, is suitable for bicylinder building wake flow galloping, has wide application range, provides favorable data support for the building mechanics analysis containing bicylinders later, and can be reversely deduced and exercised after analysis and comparison and mutually support with other approved prior art. The correctness and reliability thereof are determined.
Drawings
FIG. 1 is a schematic diagram of computational domains and boundary conditions;
FIG. 2 is a diagram illustrating the result of the division of the computational domain mesh and the cylindrical wall mesh;
FIG. 3 is a schematic of a grid dependency study;
FIG. 4 is a schematic diagram of the results of dimensionless amplitude variation with reduced wind speed for three simulated conditions;
FIG. 5 shows the wake galloping development phase Ur9.179 is a diagram of the change of lift spectrum corresponding to the whole time course;
FIG. 6 shows the wake galloping development phase UrA schematic diagram of the lift spectrum variation corresponding to the whole time course is shown as 14.49;
FIG. 7 is a schematic diagram of lift spectra for different reduced wind speeds under a condition;
FIG. 8 shows S of the upstream and downstream cylinderstNumber following UrSchematic diagram of variation rule of (1);
FIG. 9 is a view showing a state in UrThe lift force work and mechanical energy time course diagram reconstructed from the 13.889 aerodynamic parameters;
FIG. 10 is a view showing a structure in Ur13.889, a displacement time course effect diagram of pneumatic parameter reconstruction;
FIG. 11 is a view showing a structure in UrRespectively reconstructing parameters of 12.08 vortex-induced force and pneumatic damping force to obtain a work curve and a reconstructed force curve schematic diagram;
FIG. 12 is a view showing a structure in UrSchematic diagram of effect of various aerodynamic forces on displacement (12.08 ═ 12.08)
FIG. 13 is a view showing a structure in UrThe self-excitation force and vortex-excitation force time course diagram reconstructed by 12.08 parameters;
FIG. 14 is a view showing a position at UrA lift force and analog value comparison schematic diagram of 12.08 parameter reconstruction;
FIG. 15 is a schematic diagram of a dimensionless vortex-induced magnitude parameter for a first operating condition based on lift time course fitting;
FIG. 16 is a schematic diagram of the result of aerodynamic parameter identification;
FIG. 17 is a schematic diagram of aerodynamic model predicted amplitude versus simulated value;
FIG. 18 is a flow chart of the series bi-cylinder wake flow galloping unsteady aerodynamic mathematical model building steps;
FIG. 19 is a flow chart of the comparative steps of a wake flow galloping mathematical model of tandem bicylinders.
Detailed Description
The present invention will be further described with reference to the following examples and the accompanying drawings.
Referring to fig. 1-12, a method for establishing a tandem bicylinder wake flow galloping unsteady aerodynamic mathematical model is provided, which comprises the following specific steps:
s1: adopting computational fluid dynamics software to simulate the wake flow galloping of the tandem double cylinders with the fixed upstream cylinder and the transversely vibrated downstream cylinder under the condition that the flow direction interval L/D of the cylinders is 4, obtaining a wake flow galloping geometric model of the tandem double cylinders and carrying out mesh division on the geometric model;
wherein, L is the vertical distance of the two cylinders in the flow direction, and D is the diameter of the cylinder;
when L/D is 4, wake flow galloping is easy to occur;
in this embodiment, the unsteady incompressible RANS equation for two-dimensional cylindrical streaming is as follows:
Figure BDA0003052706680000081
Figure BDA0003052706680000082
wherein u isiAnd p represents the average velocity and average pressure, respectively, ρ is the fluid density, μ is the molecular viscosity, SijIn order to be the mean stress tensor,
Figure BDA0003052706680000083
is the reynolds stress tensor.
The present invention uses a shear stress transfer model, i.e., the SST-k ω model. The model involves two transport equations, namely the turbulence energy and dissipation ratio equations. In which a non-steady state separation algorithm (USA) is used. For a model, see the article: two-equation two-dimensional interference modules for engineering applications [ J ] Aiaa journal.1994,32.
The velocity expression is processed by a SIMPLE algorithm when being coupled, and a second-order implicit scheme is adopted for the unstable condition. A second order implicit scheme is used for the convection terms in the k- ω transfer equation and the momentum equation.
In step S1, in the wake relaxation of the tandem bicylinder, the lateral vibration of the downstream cylinder is solved by using a wake cylinder lateral vibration equation, where the downstream cylinder lateral vibration equation is:
Figure BDA0003052706680000084
wherein m is the downstream cylinder mass; y, y,
Figure BDA0003052706680000085
And
Figure BDA0003052706680000086
respectively displacement, velocity and acceleration perpendicular to the flow direction; c and k are divided into structural damping and rigidity constants; fyIs the vertical force to which the cylinder is subjected.
In this embodiment, in step S1, in the wake flow galloping geometric model of the tandem double cylinder, as can be seen from fig. 1, the adopted rectangular calculation domain is 60D × 40D, and D is the diameter of the cylinder; the maximum blockage rate is 5%, in the rectangular calculation domain, the distance from the center of the upstream cylinder to the inlet boundary is 25D, and the distance from the center of the downstream cylinder to the outlet boundary is 31D; in the rectangular calculation domain, setting an entrance boundary as a speed entrance boundary condition; in the rectangular calculation domain, setting a basin outlet boundary as a pressure outlet boundary condition; in the rectangular calculation domain, the upper and lower boundaries are defined as symmetric boundary conditions; in the rectangular calculation domain, the upstream cylinder and the downstream cylinder are both cylinders, and the surfaces of the cylinders are non-slip walls.
As can be seen from fig. 2, in this embodiment, the non-uniform quadrilateral structured mesh is adopted when the geometric model is subjected to mesh division, and the surface meshes of the upstream and downstream cylindrical surfaces are O-shaped meshes;
when the geometric model is subjected to meshing division, the meshes of the near wall surface of the cylinder are encrypted and the height of the first layer of meshes of the wall surface is controlled, so that the height y of the first layer of meshes of the upstream and downstream cylinder wall surfaces+Less than 1; the influence of the grid on numerical calculation is reduced. In the rectangular computational domain, to simulate cylindrical unsteady vibrations, the interfaces 27D and 34D downstream of the entrance employ a sliding mesh technique, with dynamic tiling to update the mesh at the lateral boundaries. For the simulation of the cylindrical streaming with periodicity, the time step is at least 100 time steps in a unit period.
In order to verify the feasibility of the numerical simulation method and the accuracy of the meshing strategy, the aerodynamic characteristics of two fixed cylinders are simulated, and the simulated cylinder distance is selected to be L/D6 and T/D0-4. Reynolds number Re 3.48X 104And Re is rho UD/mu, rho is the fluid density, mu is the fluid viscosity coefficient, U is the incoming flow velocity, in the subcritical range, and the turbulence is 1%. By ensuring y+The height of the first layer of grid on the wall surface is less than 1, and is 0.013 mm. The method comprises the steps of encrypting grids around a cylinder, selecting 4 grids with different numbers for simulation, and comparing the obtained average resistance coefficient with the existing experimental result, wherein the details are shown in documents Wu W, Huang S, Barltrop N].Applied Ocean Research.2002,24(5):287-297.
The number of grids is from 1.3X 105Increased to 3.7 × 105As can be seen from fig. 3 (a), the average drag coefficient is close to the current experimental result. As can be seen in fig. 3 (b), the resistance coefficient time course curve appears to converge. When the number of grids is from 3.7 × 105Increased to 4.5 × 105And in the process, the average resistance coefficients corresponding to the two grids are very close, the resistance coefficient time-course curves are almost overlapped in a stable section, and the grids have no great influence on the result. Therefore, the subsequent dynamic cylinder simulation is selected and used as 3.7 multiplied by 105The division strategy being of the same number of meshes, i.e. cylindrical wallsFirst layer of mesh height y+The growth rate of o-block near the wall surface is 1.05, and the growth rate of grids in other areas is 1.08. The aspect ratio of the grid is controlled to be within 5, and the sizes of the adjacent grids are close.
S2: establishing at least one working condition according to the wake flow galloping geometric model of the tandem bicylinder, and analyzing the frequency spectrum law of the wake flow galloping aerodynamic force of the tandem bicylinder;
as shown in Table 1, three working conditions are included, and based on the three working conditions, wake flow galloping simulation of the working conditions with small Sc number of three downstream cylinders is carried out.
TABLE 1 data sheet of characteristic parameters and Sc number of wake flow cylinders under three simulation conditions
Mass ratio m* Damping ratio ζ Natural frequency fs Sc
Working condition one 84.35 0.24％ 6.9Hz 3.39
Working condition two 113.24 0.28％ 6.78Hz 4
Three working conditions 142.12 0.26％ 7.12Hz 4.64
In Table 1, mass ratio m*＝m/ρD2And m is the mass per unit length of the downstream cylinder. Sc 2m*(2 pi ζ), ξ is the structural damping ratio of the downstream cylinder itself. Converted wind speed Ur＝U/fsD。
The scattergram of dimensionless amplitude A/D with the converted wind speed for three simulated conditions is shown in detail in FIG. 4.
U of working condition oner9.179 and UrThe displacement time course curve corresponding to 14.49 full time course and the displacement time course frequency spectrum are shown in detail in fig. 5 and 6.
Wherein, as can be seen from fig. 5, Ur9.179, the corresponding displacement time course spectrum appears very coarse and the motion appears quasi-periodic.
As can be seen from fig. 6, when U isrWhen the time range is 14.49, the corresponding displacement time spectrum shows smoother, and the motion shows similar periodic motion.
The division point of the two displacement phenomena is converted into wind speed Ur13.5, and the wind speed in the simulation results happens to be the wind speed at which the rate of amplitude increase is consistent with the quasi-stationary theorem, illustrated in Ur9.179 is the range of wind speeds that are more severely affected by the unsteady effects, and in UrThe unsteady effects are gradually diminishing at 14.49. The main frequency of the displacement time-range frequency spectrum corresponding to each wind speed is the natural vibration frequency, but the main frequency also contains other weak frequencies.
Wake flow galloping development stage Ur9.179 and UrThe lift spectrum corresponding to 14.49 over the entire time course can be seen in fig. 7:
the aerodynamic lift spectral content based on all wind speeds comprises mainly 7 main frequencies;
the self-oscillation frequency of 1-5 orders is kept unchanged along with the increase of the converted wind speed, and aerodynamic force corresponding to the self-oscillation frequency of 1-5 orders is defined as self-excitation force;
the 6-7 order frequency increases linearly with the reduced wind speed, and the aerodynamic force corresponding to the 6-7 order frequency is defined as the vortex-induced force.
S3: establishing a wake flow galloping aerodynamic mathematical model of the tandem bicylinder containing unknown parameters according to the wake flow galloping of the tandem bicylinder with the upstream cylinder fixed and the downstream cylinder transversely vibrated in the step S1; the aerodynamic force is expressed in two forms of self-excitation force and vortex-induced force;
in step S3, the mathematical model of the wake flow relaxation aerodynamic force of the tandem bicylinder with unknown parameters is:
Figure BDA0003052706680000101
the unknown parameters include: unknown self-excitation force parameters and unknown vortex-induced force parameters;
Fyis the total aerodynamic force of the structure;
Figure BDA0003052706680000102
for expression of self-excitation force, Fv(fv) Is an expression of vortex-induced force;
Figure BDA0003052706680000103
for y and y respectively
Figure BDA0003052706680000104
And (3) performing high-order Taylor sequence expansion to obtain a self-excitation force polynomial expression in wake flow relaxation force as follows:
Figure BDA0003052706680000111
in the formula gijAre pneumatic parameters. g00This term, which has no relation to both y and y describing the response state, can be considered as a static term, which has no effect on the wake relaxation dynamic response and is therefore ignored. From the previous aerodynamic spectra, it can be seen that the contribution to the entire frequency component is large, substantially within the 5 th order frequency multiplication, so that the excess y in the Taylor expansion can be set5And
Figure BDA0003052706680000112
the term (1) is truncated.
Based on the principle of energy equivalence,
Figure BDA0003052706680000113
the medicine consists of three parts: the aerodynamic damping force term (changing the damping of the structure), the aerodynamic stiffness force term (changing the stiffness of the structure) and the pure force term (neither changing the damping of the structure nor changing the stiffness of the cylinder).
Then, the self-excitation force can be written as
Figure BDA0003052706680000114
Neglecting pure force terms, the dimensionless self-exciting force expression is as follows
Figure BDA0003052706680000115
For lateral vibrations, the aerodynamic stiffness term only slightly changes the structural frequency, and may not be considered, and the above formula can be reduced to:
Figure BDA0003052706680000116
ρ is the fluid density, U is the incoming flow velocity, D is the cylinder diameter, y is the displacement perpendicular to the flow direction,
Figure BDA0003052706680000117
is vertical toVelocity in the flow direction; alpha is alpha01For a first-order dimensionless aerodynamic damping parameter, alpha03For three-order dimensionless aerodynamic damping parameters, alpha05Five-order dimensionless pneumatic damping parameters which are unknown parameters;
due to the maximum of two vortex shedding frequencies fv1And fv2Since the vortex-induced force is always closely related to the vortex shedding frequency, it can first be expressed as a function of the vortex shedding frequency, and the expression for the vortex-induced force can be written as:
Figure BDA0003052706680000118
strouhal number of two cylinders StReferring to fig. 8 in detail, it can be seen that the stohar number of the upstream and downstream cylinders is not a constant value. The upstream cylinder is a fixed static cylinder, the Strouhal value of the upstream cylinder is slowly reduced to 0.184 from 0.209 in a nearly linear way, and the upstream cylinder tends to be stable with the increasing wind speed; and the downstream cylinder is a dynamic cylinder, the Strand Harr value of which is slowly increased from 0.302 and finally levels to 0.315, and the variation trend of the amplitude along with the wind speed is consistent. The Stockhol number of the upstream static cylinder and the downstream dynamic cylinder is slowly changed along with the converted wind speed, but is not a constant value so as to ensure that the formula
Figure BDA0003052706680000121
The calculation of the eddy-induced force has higher prediction accuracy, and the change of the Stockholr number must be accurately considered.
And because in the expression of the vortex-induced force, the phase angle
Figure BDA0003052706680000122
And
Figure BDA0003052706680000123
all are parameters to be fitted and solved, and can be rewritten as follows after dimensionless:
Figure BDA0003052706680000124
in the formula: delta11,δ12,δ21,δ22The parameter is a dimensionless vortex-induced force parameter;
St1is the upstream cylindrical Steyr number; st2Is the downstream cylindrical Stockholr number; ρ is the fluid density; u shaperTo convert wind speed, fsIs the natural frequency; v. of11 v12 v21 v22Respectively representing the parameters of the vortex-induced force before dimensionless;
then it is possible to obtain:
Figure BDA0003052706680000125
the downstream cylinder provides energy input or consumption for the downstream cylinder in the vibration process, and the downstream cylinder has pneumatic damping force, structure self-damping and vortex-induced force.
S4: fitting a pneumatic damping force parameter and a vortex excitation force parameter in the self-excitation force by adopting a least square parameter identification method based on energy equivalence; obtaining a final wake flow galloping aerodynamic mathematical model of the tandem double cylinders;
the least square parameter identification method based on energy equivalence is described in Gao G Z, Zhu L.nonlinear chemical model of unsettling purpose on arithmetic and regular 2:1 cylinderer [ J ]. Journal of Fluids and structures.2017,70:47-71.
S5: and (5) comparing the wake flow galloping force mathematical model of the tandem bicylinder obtained in the step (S4).
The specific content of step S5 is:
s51: according to the obtained wake flow galloping aerodynamic mathematical model of the tandem double cylinders, the displacement and the speed are inversely calculated by adopting a Runge-Kutta method;
s52: parameter names defining downstream cylinder energy:
Figure BDA0003052706680000131
Figure BDA0003052706680000132
Figure BDA0003052706680000133
ωsis the natural circular frequency of the downstream cylinder;
Ws(t) is the accumulated work of the structure damping of the downstream cylinder at the current time, and the consumed energy is a negative value;
Wa,v(t) the accumulated work of the aerodynamic damping force and the vortex-induced force of the downstream cylinder at the current time is inversely calculated by utilizing the fitted parameters, and the energy is provided to be a positive value;
Ee(t) the mechanical energy of the downstream cylinder at the current time, wherein the mechanical energy comprises kinetic energy and potential energy;
s53: substituting the displacement and the speed obtained in the step S51 into the formulas (1) to (3), and reconstructing to obtain the accumulated work of the pneumatic damping force and the vortex-induced force;
s54: and comparing the accumulated work of the pneumatic damping force and the vortex-induced force with the simulation value of the wake flow galloping geometric model of the tandem bicylinder.
By UrReferring to fig. 9, details of the displacement time course reconstructed from the parameters obtained by fitting and the displacement time course calculated by Fluent are shown in fig. 10 for a comparison relationship between each item of work and mechanical energy, which is 13.889 as an example.
The actually calculated lift accumulation working time course curve is well matched with the reconstruction of the fitting parameters, the maximum relative error of the two is only 3 percent, and Ee(t) the time course curve is also related to Wa,v(t)-Ws(t) are very close, the maximum relative error between the two is only 6%, and the error of the "jitter" of these work curves comes from the high-frequency oscillation components neglected by the relaxation force model, and the contribution of these components is almost negligible.
For the displacement time course curve of fig. 10, although there is an obvious phase difference between the two curves, which is neglected the pneumatic stiffness term, the fit degree between the amplitude and the development process of the parametric reconstruction relaxation-oscillation displacement curve and the Fluent calculation is better, and the maximum relative error between the two curves is only 1%.
However, referring to FIG. 11, at UrWhen the value is 12.08, the vortex force reconstructed by the parameters based on the energy-equivalent least square fitting is only 1.1 times of the aerodynamic damping force, which is contrary to the spectrum phenomenon (vortex shedding frequency is very high) transformed by the FFT of the lift force. Since the cumulative work of the vortex-induced force in the whole galloping vibration process is almost 0, the contribution of the downstream cylindrical motion energy is mainly derived from the pneumatic damping force term. When the method is based on energy fitting, the vortex-induced force is insensitive to the energy, so that the parameter error of the fitted vortex-induced force term is large. In conclusion, the pneumatic damping force term plays an absolute control role in energy input of a downstream cylinder, the vortex-induced force term parameter is insensitive to energy change, fitting of the vortex-induced force term parameter in the model needs to be started from the force, namely, fitting is directly performed on the force with the pneumatic damping force term subtracted in the lift force by using a least square method, and then regular analysis of the vortex-induced force parameter is performed.
The aerodynamic damping parameters are obtained by energy fitting, and the other aerodynamic parameters are obtained by fitting from the force itself. In order to more clearly illustrate the effect of the vortex-induced force term and the aerodynamic stiffness term on the displacement, the displacement reconstructed by using the parameter of the formula (4) is compared with the displacement reconstructed by only not including the parameter of the vortex-induced force term and the parameter of the aerodynamic stiffness term. As shown in fig. 12, in the wake flow relaxation development time period 14s-14.6s, the maximum relative error of the displacement amplitude of the parameter reconstruction of the equation (4) and the model parameter reconstruction which only does not include the aerodynamic stiffness force term is 1%, but a large phase difference exists between the two, while the displacement amplitude of the parameter reconstruction of the equation (4) is higher than that of the parameter reconstruction which only does not include the vortex-induced force term, but a small phase difference exists between the two. In the wake flow galloping stable time period of 41s-41.6s, the phase difference relationship of the three is consistent with the wake flow galloping development time period, the difference is that the displacement amplitudes of the three are close to each other, and the maximum relative error is 1%. This shows that in the whole stage of wake flow galloping, the pneumatic stiffness force term does not change the displacement amplitude, and only plays a role of changing the phase difference, which is consistent with the calculation result of the equivalent linear system; the vortex shedding forced vibration of the vortex-induced force term provides a large disturbance effect only in the early stage of the relaxation vibration, so that the relaxation vibration is rapidly developed, but the final steady-state displacement amplitude is not influenced, and the assumption that the work of the vortex-induced force in the whole vibration process is close to 0 is also confirmed.
The time-course curve of the parameter reconstruction force is shown in fig. 13, the peak value of the vortex-induced force is 0.151N, the peak value of the self-excitation force (pneumatic damping force, pneumatic rigidity force and pure force) is only 0.032N, the value of the self-excitation force is only 20% of the vortex-induced force, the vortex-induced force is consistent with the phenomenon that the vortex-induced frequency is very high, and the vortex-induced force parameter obtained by directly fitting the force per se can better reflect the actual vibration state of the downstream cylinder. As can be seen from fig. 14, the force reconstructed only from the aerodynamic damping force parameter and the vortex force parameter is matched with the lift force of the Fluent simulation result very well in both phase and amplitude, and the error is about 4% at most, which indicates that the proportion of the aerodynamic stiffness force component to the pure force component in the self-excitation force is very small and can be ignored. In conclusion, the simplified model of the component equation (5) of the vortex-induced force term and the aerodynamic damping force term has sufficient accuracy regardless of the prediction of the steady-state displacement or the aerodynamic force.
See FIG. 15. vortex shedding of the upstream cylinder causes a dimensionless vortex-induced force parameter amplitude A of the downstream cylinder surfacev1The amplitude A of the dimensionless vortex-induced force parameter of the surface is caused by the vortex shedding of the downstream cylinder per se as the converted wind speed is increased to approach the linear decreasingv2The small fluctuation state with the value of 0.195-0.205 is always kept, which shows that in the wake flow galloping interval range, the influence of the vortex shedding effect of the upstream cylinder on the surface pressure generated by the downstream cylinder along with the increase of the converted wind speed begins to weaken, the excitation effect of the vortex shedding of the downstream cylinder per se along with the change of the converted wind speed shows better regularity, and the prominence of the laws has great promotion effect on the establishment of a vortex-induced force parameter model.
The verification of the model is as follows:
the pneumatic parameters of the working condition I have obvious regularity, and the force and displacement reconstructed through the parameters are close to the analog value. If the wake flow galloping aerodynamic model is applied to engineering prediction, the wake flow galloping aerodynamic model should not only have applicability to the first working condition, the values of the aerodynamic parameters should only be related to the structural appearance and the converted wind speed at fixed intervals, and the values of the aerodynamic parameters should have similar aerodynamic rules to the second working condition and the third working condition, so that dimensionless aerodynamic parameters are identified for the cylindrical wake flow galloping of the second working condition and the third working condition, and the distribution rules of the cylindrical wake flow galloping are observed so as to obtain a commonly-adapted mathematical prediction formula.
The identification result of the dimensionless parameters of the vortex-induced force and the dimensionless parameters of the aerodynamic damping force is shown in fig. 16. As can be seen from fig. 16, the dimensionless aerodynamic damping parameters and the dimensionless vortex-induced force parameters of the three operating conditions have a certain dispersion, because a certain approximation is adopted for the wake relaxation force when the aerodynamic model is established, and the contributions of higher-order nonlinear frequencies and some random vortex shedding frequency components to the aerodynamic force are ignored. However, the parameters in the graph generally fall into one area, and the objective regularity is more definite and similar. The laws of the pneumatic parameters of the three working conditions are similar, and the pneumatic model established based on the first working condition is only applicable to the first working condition, so that the wake flow galloping pneumatic model has certain universality and applicability. Alpha is alpha01、α03、α05、Av1And Av2Corresponding polynomials can be selected according to the scattered point rule for fitting, the pneumatic parameters of the three working conditions can be averaged in the fitting process, and then the average values are utilized for carrying out least square fitting. As shown in fig. 16, the fitted curve is smooth, and the aerodynamic rules of the dimensionless parameters with the converted wind speed can be embodied to a greater extent. First order dimensionless pneumatic damping parameter alpha01Third order dimensionless pneumatic damping parameter alpha03And a five-order dimensionless aerodynamic damping parameter α05At Ur13.5, the wind speed can be understood as the split wind speed affected by the unsteady effect. Therefore, the dimensionless aerodynamic damping parameters for these two phases will be analyzed separately in the following.
Ur＜13.5：α01And alpha05Decreases with increasing reduced wind speed, and alpha03Is increased with increasing reduced wind speed. Alpha is alpha01In the range of wind speedsThe internal constant is a positive value, which indicates the pneumatic damping force
Figure BDA0003052706680000151
Always negative damping, and three-order dimensionless aerodynamic damping parameter alpha03And a five-order dimensionless aerodynamic damping parameter α05The wind speed range has positive or negative values,
Figure BDA0003052706680000152
is to change from negative damping to positive damping,
Figure BDA0003052706680000153
it changes from positive damping to negative damping, but a03And alpha05Almost any reduced wind speed always maintains a positive-negative opposite state. At a specific reduced wind speed, the nonlinear total damping of the downstream cylinder is negative in the oscillation starting stage,
Figure BDA0003052706680000154
and
Figure BDA0003052706680000155
or
Figure BDA0003052706680000156
And
Figure BDA0003052706680000157
and the downstream cylinder continuously transmits the energy required by the development of the galloping vibration amplitude, so that the downstream cylinder wake flow galloping amplitude is continuously increased. As the amplitude of the wave is continuously increased,
Figure BDA0003052706680000158
or
Figure BDA0003052706680000159
And the positive damping provided by the structure is gradually increased until the positive damping is equal to the negative damping, the nonlinear total damping of the downstream cylinder is 0, the energy absorbed by the downstream cylinder from the air is the same as the dissipated energy, and the downstream cylinder is in a dynamic balance state andwith a steady amplitude for limit-cycle oscillation, alpha03And alpha05The mutual restriction of the wind speed generator is a key factor for ensuring that the wake flow galloping has amplitude self-limitation at each wind speed. From the whole galloping development wind speed range, the unsteady effect of the coupling of the vortex vibration and the galloping is not only reflected on a displacement time course curve, but also obviously reflected on the strong change of dimensionless pneumatic parameters along with the converted wind speed.
Ur＞13.5：α01、α03And alpha05Is much smaller than UrA dimensionless aerodynamic damping parameter value of < 13.5, alpha as a whole01、α03And alpha05Finally, the wind speed increases to a constant value, and alpha03The steady value is a positive damping value close to 0, and alpha05The steady value is a negative damping value close to 0, both still maintaining opposite signs. Although all dimensionless aerodynamic damping parameters still vary with the reduced wind speed within a small range, the variation range is very small, which marks the weakening of the unsteady effect, and as the reduced wind speed continues to increase, all parameters approach a constant, so that the amplitude curve can be explained as showing a linear increase.
After the dimensionless pneumatic parameters of the working condition I, the working condition II and the working condition III are scattered and integrated into a unified usable mathematical formula, an unsteady mechanical equation can be brought in to directly predict the unsteady wake flow oscillation amplitude of the working condition I, the working condition II and the working condition III. As can be seen from fig. 17, the maximum relative error between the predicted amplitudes for the first, second, and third conditions and the amplitude calculated by Fluent simulation is mainly concentrated in a small amplitude region in a small range, because the amplitudes are very sensitive to small difference changes. Working condition one maximum amplitude difference in the whole prediction interval is 0.06D in UrThe maximum relative error is only 7% when the error is more than 9%, and the maximum amplitude difference value of the working condition II in the whole prediction interval is 0.06D, UrThe maximum relative error of > 9.45 is only 10%. Although the relative error fluctuation of the working condition III in the whole prediction interval is large, the maximum amplitude difference is only 0.07D, and the mean method fitting cannot completely eliminate the dimensionless pneumatic parameters of the self characteristics of the cylinder to the aerodynamic model although the mean method fitting cannot completely eliminateHowever, the amplitude prediction results of most of the converted wind speed ranges under the three working conditions are ideal and meet the engineering precision requirement. And the prediction result can restore the objective law that the amplitude changes along with the reduced wind speed to a greater extent, such as the slope change of the amplitude increase under the low reduced wind speed, the platform for transitioning from the unsteady vibration to the steady vibration, the near-linear increase of the amplitude under the high reduced wind speed and the like. In conclusion, the wake flow galloping aerodynamic model has higher applicability to the second working condition and the third working condition, and meanwhile, the reliability and the accuracy of the prediction result of the aerodynamic model are further verified.
In summary, by the above method, the following can be obtained:
(1) establishing a new unsteady aerodynamic force nonlinear model aiming at the low Sc number wake flow relaxation vibration. Aerodynamic forces are expressed in the form of a sum of self-excited and vortex-excited forces. The self-exciting force is divided into a pneumatic damping force, a pneumatic rigidity force and a pure force term based on an energy equivalent principle, wherein the pneumatic damping term plays a main role in wake flow galloping steady-state amplitude. Vortex-induced forces are expressed as a function of the vortex shedding frequency of the structure. In combination with the aerodynamic force spectrum of the stationary upstream and downstream cylinders, the vortex-induced forces contain mainly the vortex shedding frequencies of the two cylinders.
(2) The pneumatic parameters of the pneumatic force model are identified, and the reliability of the identification method and the accuracy of the pneumatic parameters are verified. The vortex-induced force provides a disturbance only in the early stages of vibration, and contributes almost 0 energy throughout the vibration process. The least square method based on energy equivalence is selected for fitting of the pneumatic damping parameters, and the least square method fitting is performed on the time course of the slave force for fitting of the vortex-induced force parameters. The results of the fitting parameters, namely the force acting reconstruction, the displacement reconstruction and the force reconstruction, show that the parameter identification of the pneumatic damping item and the parameter identification of the vortex-induced item have high identification precision, and the reliability and the accuracy of the fitting method adopted for the pneumatic parameters are verified.
(3) And summarizing the common rule of the dimensionless pneumatic parameters of the working condition I, the working condition II and the working condition III. The pneumatic parameters identified by the working condition I, the working condition II and the working condition III generally fall into one area, and show clear and similar objective laws, and the laws can be expressed into a unified practical formula by using specific mathematical expressions. The amplitude results of the three working conditions are predicted by a formula to be ideal, the engineering precision requirement is met, and the objective rule that the amplitude changes along with the reduced wind speed can be restored to a large extent, such as the slope change of the amplitude increase under the low reduced wind speed, the platform for transferring from the unsteady vibration to the steady vibration, the near-linear increase of the amplitude under the high reduced wind speed and the like. The wake flow galloping unsteady aerodynamic nonlinear model has higher applicability to the second working condition and the third working condition, and the reliability and the accuracy of the model prediction result are further verified.
Finally, it should be noted that the above-mentioned description is only a preferred embodiment of the present invention, and those skilled in the art can make various similar representations without departing from the spirit and scope of the present invention.

Claims (7)

1. A method for establishing a tandem double-cylinder wake flow oscillation unsteady aerodynamic force mathematical model is characterized by comprising the following specific steps:
s1: adopting computational fluid dynamics software to simulate the wake flow galloping of the tandem double cylinders with the fixed upstream cylinder and the transversely vibrated downstream cylinder under the condition that the flow direction interval L/D of the cylinders is 4, obtaining a wake flow galloping geometric model of the tandem double cylinders and carrying out mesh division on the geometric model;
wherein, L is the vertical distance of the two cylinders in the flow direction, and D is the diameter of the cylinder;
s2: establishing at least one working condition according to the wake flow galloping geometric model of the tandem bicylinder, and analyzing the frequency spectrum law of the wake flow galloping aerodynamic force of the tandem bicylinder;
s3: establishing a wake flow galloping aerodynamic mathematical model of the tandem bicylinder containing unknown parameters according to the wake flow galloping of the tandem bicylinder with the upstream cylinder fixed and the downstream cylinder transversely vibrated in the step S1; the aerodynamic force is expressed in two forms of self-excitation force and vortex-induced force;
s4: fitting a pneumatic damping force parameter and a vortex force parameter in self-excitation force by adopting a least square parameter identification method based on energy equivalence to obtain a final wake flow relaxation vibration pneumatic force mathematical model of a tandem double cylinder;
s5: and (5) comparing the wake flow galloping force mathematical model of the tandem bicylinder obtained in the step (S4).
2. The method for establishing the tandem bicylinder wake flow relaxation oscillation unsteady aerodynamic mathematical model as claimed in claim 1, wherein: in step S1, in the wake relaxation of the tandem bicylinder, the lateral vibration of the downstream cylinder is solved by using a wake cylinder lateral vibration equation, where the downstream cylinder lateral vibration equation is:
Figure FDA0003052706670000011
wherein m is the downstream cylinder mass; y, y,
Figure FDA0003052706670000012
And
Figure FDA0003052706670000013
respectively displacement, velocity and acceleration perpendicular to the flow direction; c and k are divided into structural damping and rigidity constants; fyIs the vertical force to which the cylinder is subjected.
3. The method for establishing the tandem bicylinder wake flow relaxation oscillation unsteady aerodynamic mathematical model as claimed in claim 1, wherein: in step S1, in the wake flow galloping geometric model of the tandem double cylinder, the adopted rectangular calculation domain is 60D × 40D, where D is the diameter of the cylinder;
in the rectangular calculation domain, the distance between the center of the upstream cylinder and the inlet boundary is 25D, and the distance between the center of the downstream cylinder and the outlet boundary is 31D;
in the rectangular calculation domain, setting an entrance boundary as a speed entrance boundary condition;
in the rectangular calculation domain, setting a basin outlet boundary as a pressure outlet boundary condition;
in the rectangular calculation domain, the upper and lower boundaries are defined as symmetric boundary conditions;
in the rectangular calculation domain, the upstream cylinder and the downstream cylinder are both cylinders, and the surfaces of the cylinders are non-slip walls.
4. The method for establishing the tandem bicylinder wake flow relaxation oscillation unsteady aerodynamic mathematical model as claimed in claim 1, wherein: in step S1, the geometric model is gridded using a non-uniform quadrilateral structured grid, and the surface grids of the upstream and downstream cylindrical surfaces use O-shaped grids;
when the geometric model is subjected to meshing division, the meshes of the near wall surface of the cylinder are encrypted and the height of the first layer of meshes of the wall surface is controlled, so that the height y of the first layer of meshes of the upstream and downstream cylinder wall surfaces+＜1；
In the rectangular computational domain, the interfaces 27D and 34D downstream of the entry employ a sliding mesh technique, with dynamic tiling being used to update the mesh at the lateral boundaries.
5. The method for establishing the tandem bicylinder wake flow relaxation oscillation unsteady aerodynamic mathematical model as claimed in claim 1, wherein: in step S2, the spectral law of the wake flow relaxation aerodynamic force of the tandem bicylinder obtained by analysis is:
the aerodynamic lift spectral content based on all wind speeds comprises mainly 7 main frequencies;
the self-oscillation frequency of 1-5 orders is kept unchanged along with the increase of the converted wind speed, and aerodynamic force corresponding to the self-oscillation frequency of 1-5 orders is defined as self-excitation force;
the 6-7 order frequency increases linearly with the reduced wind speed, and the aerodynamic force corresponding to the 6-7 order frequency is defined as the vortex-induced force.
6. The method for establishing the tandem bicylinder wake flow relaxation oscillation unsteady aerodynamic mathematical model as claimed in claim 5, wherein: in the step S3, in step S3,
the mathematical model of the wake flow galloping aerodynamic force of the tandem double cylinders containing unknown parameters is as follows:
Figure FDA0003052706670000021
the unknown parameters include: unknown self-excitation force parameters and unknown vortex-induced force parameters;
Fyis the total aerodynamic force of the structure;
Figure FDA0003052706670000022
for expression of self-excitation force, Fv(fv) Is an expression of vortex-induced force;
wherein the content of the first and second substances,
Figure FDA0003052706670000023
ρ is the fluid density, U is the incoming flow velocity, D is the cylinder diameter, y is the displacement perpendicular to the flow direction,
Figure FDA0003052706670000024
velocity perpendicular to the flow direction; alpha is alpha01For a first-order dimensionless aerodynamic damping parameter, alpha03For three-order dimensionless aerodynamic damping parameters, alpha05Five-order dimensionless pneumatic damping parameters which are unknown parameters;
Figure FDA0003052706670000025
in the formula: delta11,δ12,δ21,δ22The parameter is a dimensionless vortex-induced force parameter;
St1is the upstream cylindrical Steyr number; st2Is the downstream cylindrical Stockholr number; ρ is the fluid density; u shaperTo convert wind speed, fsIs the natural frequency; v. of11 v12 v21 v22Respectively representing the parameters of the vortex-induced force before dimensionless;
namely:
Figure FDA0003052706670000031
7. the method for establishing the tandem bicylinder wake flow relaxation oscillation unsteady aerodynamic mathematical model as claimed in claim 6, wherein: the specific content of step S5 is:
s51: according to the obtained wake flow galloping aerodynamic mathematical model of the tandem double cylinders, the displacement and the speed are inversely calculated by adopting a Runge-Kutta method;
s52: parameter names defining downstream cylinder energy:
Figure FDA0003052706670000032
Figure FDA0003052706670000033
Figure FDA0003052706670000034
ωsis the natural circular frequency of the downstream cylinder;
Ws(t) is the accumulated work of the structure damping of the downstream cylinder at the current time, and the consumed energy is a negative value;
Wa,v(t) the accumulated work of the downstream cylinder aerodynamic damping force and vortex-induced force at the current time is back-calculated by using the fitted parameters, the supplied energy is positive,
Ee(t) the mechanical energy of the downstream cylinder at the current time, wherein the mechanical energy comprises kinetic energy and potential energy;
s53: substituting the displacement and the speed obtained in the step S51 into the formulas (1) to (3), and reconstructing to obtain the accumulated work of the pneumatic damping force and the vortex-induced force;
s54: and comparing the accumulated work of the pneumatic damping force and the vortex-induced force with the simulation value of the wake flow galloping geometric model of the tandem bicylinder.
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