A method for predicting the motion trajectory of an underwater vehicle under the action of internal solitary waves

Abstract

The present invention belongs to the research field of underwater vehicle motion response analysis, and in particular, relates to a method for predicting the motion trajectory of an underwater vehicle under the action of an internal solitary wave. The present invention adopts a time-domain finite displacement motion equation, and for the motion process of an underwater vehicle in the water section, establishes a theoretical model of its related motion response under the strong nonlinear action of an internal solitary wave, predicts the motion trajectory of an underwater vehicle under the action of an internal solitary wave, and quantitatively evaluates the motion response of an underwater vehicle under different conditions such as different positions of the underwater vehicle from the trough of the internal solitary wave, different initial launch depths, and different wave heights of the internal solitary wave, so as to be used for the motion response analysis of an underwater vehicle under the action of an internal solitary wave.

Description

Technical Field

[0001] The present invention belongs to the research field of underwater vehicle motion response analysis, and in particular relates to a method for predicting the motion trajectory of an underwater vehicle under the action of internal solitary waves. Background Art

[0002] In some sea areas, the seawater density stratification phenomenon is significant, the seabed topography changes dramatically, and the flow pattern is changeable, all of which provide natural marine environmental conditions for the generation of internal waves in the sea area. For a vertically launched underwater high-speed moving vehicle, if it is launched at a large depth underwater, the underwater vehicle needs to pass through a variable density layer and encounter internal waves in the horizontal direction, or even large-scale internal solitary waves. Compared with uniform density fluids, the density stratification effect will produce a greater shear effect on the moving body in the lateral and vertical directions, increase the instability of the underwater vehicle's motion posture, and affect the underwater vehicle's exit posture. At present, a large number of engineering software and related theoretical calculation methods for analyzing the motion response characteristics of underwater vehicles under the action of flow before exiting the water have been successfully applied in engineering practice. However, the understanding of the motion response performance of underwater vehicles under the action of internal solitary waves is not very clear. Therefore, it is necessary to carry out the analysis and research of the motion response of underwater vehicles under the action of internal solitary waves. Summary of the invention

[0003] In order to solve the above problems, the present invention proposes a method for predicting the motion trajectory of an underwater vehicle under the action of an internal solitary wave. The present invention adopts the time-domain finite displacement motion equation, and for the motion process of the underwater vehicle in the water section, establishes a theoretical model of its related motion response under the strong nonlinear action of the internal solitary wave, predicts the motion trajectory of the underwater vehicle under the action of the internal solitary wave, and quantitatively evaluates the motion response of the underwater vehicle under different conditions such as different positions of the underwater vehicle from the trough of the internal solitary wave, different initial launch depths, and different wave heights of the internal solitary wave, which is used for the motion response analysis of the underwater vehicle under the action of the internal solitary wave.

[0004] The technical solution of the present invention:

[0005] A method for predicting the motion trajectory of an underwater vehicle under the action of an internal solitary wave, the specific steps are as follows:

[0006] 1. Establish a coordinate system

[0007] Assume that the depth and density of the upper fluid are h1 and ρ1 respectively, the depth and density of the lower fluid are h2 and ρ2 respectively, the total water depth is h, and the subscripts 1 and 2 correspond to the upper and lower layers respectively. Establish a two-dimensional rectangular coordinate system XOY, where the XOY plane coincides with the undisturbed internal interface, the OY axis is directly above the trough, the OX axis points to the propagation direction of the internal solitary wave, and the OY axis is vertically upward. Assume ζ is the displacement of the internal solitary wave interface, propagating along the positive direction of the OX axis at a wave velocity V, as shown in Figure 1As shown. The body-attached coordinate system xoy and the fixed coordinate system XOY are established with the buoyancy center of the fully submerged underwater vehicle as the origin, as shown Figure 2 shown.

[0008] The coordinate transformation relationship between the body coordinate system xoy and the fixed coordinate system XOY is:

[0009] x=X+x d , z=Zh d

[0010] Among them, x d h is the horizontal distance from the trough to the central axis of the underwater vehicle, d It is the vertical distance from the center of the underwater vehicle to the interface between the two layers of fluid.

[0011] 2. Force analysis of underwater vehicles under the action of internal solitary waves

[0012] The external forces on the underwater vehicle are the gravity of the underwater vehicle itself, the buoyancy of the submerged part, the thrust and fluid dynamics and torque acting on the underwater vehicle. The specific formulas for calculating the various forces are given below.

[0013] (1) Headwind resistance and side resistance

[0014] When an underwater vehicle moves in an internal solitary wave, the fluid dynamics on the head of the underwater vehicle is defined as the headwind resistance, and the fluid dynamics on the side of the underwater vehicle is defined as the side resistance. The underwater vehicle is mainly subject to fluid resistance during the launch process, which is divided into headwind resistance and side resistance. The dimensionless drag coefficients corresponding to headwind resistance and side resistance are defined as Cx and Cy , the formulas for headwind resistance and side resistance are as follows:

[0015] Headwind resistance

[0016] Side resistance

[0017] Where S1=πDL+πD, S2=LD, D is the diameter of the underwater vehicle, and L is the length of the underwater vehicle.

[0018] sign(v1) is the direction of v1. The definition of sign(v1) is as follows:

[0019]

[0020] Where v1 is the speed of the underwater vehicle in the x direction.

[0021] Headwind resistance coefficient C x and the side drag coefficient C yThe specific calculation method is as follows:

[0022] Headwind resistance coefficient C x The friction coefficient C xf and pressure difference resistance coefficient C xp Composition, namely C x =C xf +C xp .

[0023] The friction drag coefficient is directly calculated using the flat plate friction drag coefficient formula:

[0024]

[0025] Pressure difference resistance coefficient C xp The calculation formula of the pressure difference resistance coefficient of the rotating body given by Bapmier is:

[0026]

[0027] Where R e is the Reynolds number; S is the maximum cross-sectional area of ​​the underwater vehicle; Ω is the wetted area of ​​the underwater vehicle; e is the length of the tapered part of the rear section of the underwater vehicle;

[0028] Side drag coefficient C y In the small deflection angle range, it can be expressed as:

[0029]

[0030] Where f is the slenderness ratio of the underwater vehicle, f = L / d, L is the calculated length of the underwater vehicle, and d is the radius of gyration of the cross section of the underwater vehicle; S M It is the maximum longitudinal section area of ​​the underwater vehicle.

[0031] (2) Gravity

[0032] Gravity G is

[0033] G=mg

[0034] Where m is the mass of the underwater vehicle and g is the free fall acceleration.

[0035] (3) Buoyancy

[0036] Buoyancy F B for

[0037] F B =ρV

[0038] Where ρ is the density of seawater and V is the volume of the underwater vehicle. The net buoyancy ΔB can be expressed as

[0039] ΔB=FB -G

[0040] (4) Thrust

[0041] During unpowered launch

[0042] F T =0

[0043] Among them, F T It is the launching thrust of underwater vehicle.

[0044] (5) Internal solitary wave load

[0045] The internal solitary wave force on the axial direction of the underwater vehicle can be calculated by the pressure difference between the upper and lower surfaces, namely the Froude-Krylov force, as shown in the following formula:

[0046] dF p =pnds

[0047] Among them, P is the dynamic pressure in the flow field, S is the upper and lower surface areas of the underwater vehicle, and n is the unit normal vector of the upper and lower surfaces.

[0048] The side surface of the underwater vehicle facing the flow is subject to drag force and inertia force, which are calculated using the Morison formula as shown below:

[0049]

[0050] Where S is the cross-sectional area of ​​the underwater vehicle, D is the equivalent diameter of the underwater vehicle, and V are the instantaneous acceleration and instantaneous horizontal velocity of the water particle perpendicular to the underwater vehicle, respectively.

[0051] The drag coefficient C d and the inertia coefficient C m It is obtained by fitting the experimental results of solitary wave loads in underwater navigation bodies. The fitting formula is as follows:

[0052] C d =173.3×exp(-9.171×10 -3 ×Re)+0.6265

[0053] C m =[95.02-21.97(h1 / h2) 2 ]KC -1.108

[0054] Where Re is the Reynolds number and KC is the KC number.

[0055] The moment generated by the internal solitary wave load on the underwater vehicle is:

[0056] dMW =r B ×dF c =(zz c )(dF d +dF m )

[0057] 3. Motion equations of underwater vehicles under the action of internal solitary waves

[0058] The motion equations of the underwater vehicle with three degrees of freedom under the action of internal solitary waves are:

[0059]

[0060] Where m is the mass of the underwater vehicle, J is the moment of inertia of the underwater vehicle, and λ 11 and λ 22 is the additional mass in the longitudinal and vertical directions, λ 66 is the additional moment of inertia, F1 is the horizontal external load of the underwater vehicle, F2 is the vertical external load of the underwater vehicle, F3 is the external load moment of the underwater vehicle, and the points above the variables represent the derivatives with respect to time.

[0061] Underwater navigation external loads include internal solitary wave dynamic loads F W and its torque M W , resistance F f and its torque M f , buoyancy F B and its torque M B , and the underwater vehicle's own gravity. After obtaining the above-mentioned loads and moments in turn, the external load vector F on the underwater vehicle can be expressed as

[0062] F=F1i+F2K=F W +F B +F f -mgk

[0063] M=F3e3=M W +M B +M f

[0064] Among them, i, j, k are the unit vectors of the fixed coordinate system XOY, e1, e2, e3 are the unit vectors of the body coordinate system xoy, and g is the gravitational acceleration.

[0065] The method for selecting the additional mass is given in detail below.

[0066] When an object moves unconstrained in a fluid, it can not only move in three directions, but also rotate, so all the additional masses constitute a (6×6) additional mass matrix λ.

[0067] First, for an irrotational flow characterized by a velocity potential φ, the kinetic energy T in the fluid region V is given by:

[0068]

[0069] And i takes 1, 2, and 3.

[0070]

[0071] S is the boundary of the fluid region. Now assume that the translational velocity of the underwater vehicle has components U1, U2, and U3 at the origin, and the rotational angular velocity of the underwater vehicle has components U4, U5, and U6. For each component α of the object's motion, the fluid has a corresponding velocity potential. Unit U α The corresponding velocity potential is φ α In this case, the Laplace equation and boundary conditions (no separation and cavitation that produces a free surface) are linear, so the solutions of the equation can be superimposed. Therefore, no matter what kind of motion the underwater vehicle is in, for the general case:

[0072] φ=U α φ α , α=1,2,...,6

[0073] The result is:

[0074]

[0075] The subscripts α and β are integers from 1 to 6, and:

[0076]

[0077] It can be proved from the formula that λ αβ are the components of the additional mass tensor, and λ αβ is symmetrical.

[0078] Kinematic boundary conditions on the surface of an object:

[0079]

[0080] Among them is Velocity vector, is the rotation vector, is the position vector (from the origin to the point on the surface of the object), and is the unit vector of the normal line of the underwater navigation body. Let n4, n5, n6 be defined as

[0081]

[0082] Then the equation can be simplified to:

[0083]

[0084] The value of α ranges from 1 to 6. α The values ​​are all valid, so:

[0085]

[0086] Thus we can get

[0087]

[0088] Because φ α It's U α =1, which means that the additional mass depends only on the shape and orientation of the object and the way it moves, but has nothing to do with the object's linear velocity, angular velocity or acceleration.

[0089] If the underwater vehicle is simplified to a simple cylinder with a diameter of D and a length of l, its additional mass can be simplified. In the fully wet and cavitation-free state, the slender cylinder is an axisymmetric body. Considering the symmetry, only λ is included in the additional mass matrix λ. 11 , 22 , 33 , 55 , 66 , 26 and λ 35 is a non-zero term, and the motion equation of the underwater vehicle with three degrees of freedom under the action of internal solitary waves is a two-dimensional simplified model, so λ 33 , 55 , 35 It is not considered in motion. Using the slender body theory, that is, assuming that there is no interference between the sections of the cylinder, the additional mass of the cylinder is the integral of the additional mass of the section along the longitudinal axis of the cylinder. The additional mass λ of the section is calculated below: ij (x) are as follows:

[0090]

[0091] The total added mass is You can get:

[0092]

[0093] λ 11 It is a small amount compared to the mass of the object, and its approximate formula is as follows:

[0094]

[0095] 4. Prediction of the motion trajectory of underwater vehicles under the action of internal solitary waves

[0096] Assume that the components of the angular velocity and the velocity at the center of buoyancy of the underwater vehicle in the body coordinate system are ω x , v x , v y ; The mass of the underwater vehicle is m, and the moment of inertia of the underwater vehicle about the oz axis is J ZZ .

[0097] From the analysis of steps 1-3, we can get the kinematic equations of the underwater vehicle in the body coordinate system as follows:

[0098]

[0099] Combined angular velocity ω z and attitude angle change rate The relationship between

[0100]

[0101] The simplified equation of motion is

[0102]

[0103] in,

[0104] A=[(mx c +λ 26 )(G.L oG cosθ+F y L or sinθ-M W -mx c v x ω x )-(J zt +λ 66 )(ΔBcosθ+F y sinθ-mv x ω z )]

[0105] B=[(mx c +λ 26 )(ΔBcosθ+F y sinθ-F c -mv x ω z )-(m+λ 22 )(G.L oG cosθ+F y ·L or sinθ-mx c v x ω z )]

[0106] Through the simplified motion equation shown in the above formula, the motion trajectory of the underwater vehicle under the action of internal solitary waves can be predicted.

[0107] The beneficial effects of the present invention are as follows: the present invention adopts the time-domain finite displacement motion equation to analyze the force of an underwater vehicle under the action of an internal solitary wave, obtains the loads and moments of various forces respectively, establishes its three-degree-of-freedom motion equation under the strong nonlinear action of an internal solitary wave, predicts the motion trajectory of the underwater vehicle under the action of an internal solitary wave, and can be used for the motion response analysis of the underwater vehicle under the action of an internal solitary wave. BRIEF DESCRIPTION OF THE DRAWINGS

[0108] Figure 1 It is a schematic diagram of solitary waves in two layers of fluid and their coordinate system;

[0109] Figure 2 It is the force analysis diagram of underwater navigation body;

[0110] Figure 3 It is a schematic diagram of the motion trajectory of the underwater vehicle at different launch positions. DETAILED DESCRIPTION

[0111] The specific implementation of the present invention is further described below in conjunction with the accompanying drawings and technical solutions.

[0112] In this embodiment, the method of the present invention is used to predict the motion trajectory of the underwater vehicle at different positions away from the trough of the internal isolated wave. The initial depth is 30m, the internal wave height is 50m, and the displacement and deflection angle of the underwater vehicle at the initial positions X0=0m, X1=100m, X2=200m, X3=300m, X4=400m, and X5=500m are shown in Table 1.

[0113] The initial position in the table is the initial coordinate of the center of the underwater vehicle in the coordinate system XOY. It can be seen from the table that when the initial position is at X0 = 0, that is, the launch position of the underwater vehicle is directly above the trough of the internal solitary wave, the displacement and deflection of the underwater vehicle relative to other positions are the largest, because the fluid velocity near this position is the largest, and the internal solitary wave has a greater effect, causing the motion trajectory of the underwater vehicle to deviate from the original trajectory. Figure 3 The same result can be observed. At a position far away from the internal solitary wave (X5=500m), the underwater vehicle moves approximately vertically upward; the closer to the internal solitary wave, the more obvious the deviation of the motion trajectory of the underwater vehicle.

[0114] Table 1 Displacement and deflection angle of underwater vehicle when launched at different distances from the trough

[0115]

Claims

1. A method for predicting the motion trajectory of an underwater vehicle under the action of internal solitary waves, characterized in that: The specific steps are as follows: Step 1. Establish a coordinate system Assume that the depth and density of the upper fluid are h1 and ρ1 respectively, the depth and density of the lower fluid are h2 and ρ2 respectively, the total water depth is h, and the subscripts 1 and 2 correspond to the upper and lower layers respectively; establish a two-dimensional rectangular coordinate system XOY, in which the XOY plane coincides with the undisturbed internal interface, the OY axis is directly above the wave trough, the OX axis points to the propagation direction of the internal solitary wave, and the OY axis is vertically upward and positive; let ζ be the displacement of the internal solitary wave interface, propagating along the positive direction of the OX axis at a wave speed V; and establish a body-attached coordinate system xoy and a fixed coordinate system XOY with the buoyancy center of the fully immersed underwater vehicle as the origin; The coordinate transformation relationship between the body coordinate system xoy and the fixed coordinate system XOY is: x=X+x d ,z=Z-h d Among them, x d h is the horizontal distance from the trough to the central axis of the underwater vehicle, d It is the vertical distance from the center of the underwater vehicle to the interface between the two layers of fluid; Step 2. Force analysis of underwater vehicle under the action of internal solitary waves (1) Headwind resistance and side resistance When an underwater vehicle moves in an internal solitary wave, the fluid dynamics on the head of the underwater vehicle is defined as the headwind resistance, and the fluid dynamics on the side of the underwater vehicle is defined as the side resistance. The underwater vehicle is mainly subjected to fluid resistance during the launch process, which is divided into headwind resistance and side resistance. The dimensionless drag coefficients corresponding to the headwind resistance and side resistance are defined as Cx and Cy. The formulas for headwind resistance and side resistance are as follows: Headwind resistance Side resistance Where S1=πDL+πD, S2=LD, D is the diameter of the underwater vehicle, and L is the length of the underwater vehicle; sign(v1) is the direction of v1. The definition of sign(v1) is as follows: Where v1 is the speed of the underwater vehicle in the x direction; Headwind resistance coefficient C x and the side drag coefficient C y The specific calculation method is as follows: Headwind resistance coefficient C x The friction coefficient C xf and pressure difference resistance coefficient C xp Composition, namely C x =C xf +C xp ; The friction drag coefficient is directly calculated using the flat plate friction drag coefficient formula: Pressure difference resistance coefficient C xp The calculation formula of the pressure difference resistance coefficient of the rotating body given by Bapmier is: Where R e is the Reynolds number; S is the maximum cross-sectional area of ​​the underwater vehicle; Ω is the wetted area of ​​the underwater vehicle; e is the length of the tapered part of the rear section of the underwater vehicle; Side drag coefficient C y In the small deflection angle range, it is expressed as: Where f is the slenderness ratio of the underwater vehicle, f = L / d, L is the length of the underwater vehicle, d is the radius of gyration of the cross section of the underwater vehicle; S M is the maximum longitudinal section area of ​​the underwater vehicle; (2) Gravity Gravity G is G=mg Where m is the mass of the underwater vehicle, g is the free fall acceleration; (3) Buoyancy Buoyancy F B for F B =ρM Where ρ is the density of seawater, M is the volume of the underwater vehicle; the net buoyancy ΔB is recorded as ΔB=F B -G (4) Thrust During unpowered launch F T ＝0 where F T It is the launching thrust of underwater vehicles; (5) Internal solitary wave load The internal solitary wave force on the axial direction of the underwater vehicle is calculated using the pressure difference force between the upper and lower surfaces, namely the Froude-Krylov force, as shown in the following formula: dF p =pnds Among them, p is the dynamic pressure in the flow field, s is the upper and lower surface areas of the underwater vehicle, and n is the unit normal vector of the upper and lower surfaces; The side surface of the underwater vehicle facing the flow is subject to drag force and inertia force, which are calculated using the Morison formula as shown below: Where S is the cross-sectional area of ​​the underwater vehicle, d is the equivalent diameter of the underwater vehicle, and V are the instantaneous acceleration and instantaneous horizontal velocity of the water particle perpendicular to the underwater vehicle, respectively; The drag coefficient C d and the inertia coefficient C m It is obtained by fitting the experimental results of solitary wave loads in underwater navigation bodies. The fitting formula is as follows: C d =173.3×exp(-9.171×10 -3 ×Re)+0.6265 C m =[95.02-21.97(h1 / h2) 2 ]KC -1.108 Where Re is the Reynolds number, KC for KC number; The moment generated by the internal solitary wave load on the underwater vehicle is: dM W =r B ×dF c =(z-z c )(dF d +dF m ) Step 3. Motion equations of underwater vehicles under the action of internal solitary waves The motion equations of the underwater vehicle with three degrees of freedom under the action of internal solitary waves are: Where m is the mass of the underwater vehicle, J is the moment of inertia of the underwater vehicle, and λ 11 and λ 22 is the additional mass in the longitudinal and vertical directions, λ 66 is the additional moment of inertia, F1 is the horizontal external load of the underwater vehicle, F2 is the vertical external load of the underwater vehicle, F3 is the external load moment of the underwater vehicle, and the points above the variables represent the derivatives with respect to time; Underwater navigation external loads include internal solitary wave dynamic loads F W and its torque M W , resistance F f and its torque M f , buoyancy F B and its torque M B , and the underwater vehicle's own gravity; after obtaining the above-mentioned loads and moments in turn, the external load vector F on the underwater vehicle is expressed as: F=F1i+F2K=F W +F B +F f -mgk M=F3e3=M W +M B +M f Among them, i, j, k are the unit vectors of the fixed coordinate system XOY, e1, e2, e3 are the unit vectors of the body coordinate system xoy, and g is the gravitational acceleration; The method for selecting additional mass is as follows: When an object moves unconstrained in a fluid, it can not only move in three directions, but also rotate, so all the additional masses constitute a (6×6) additional mass matrix λ; First, for an irrotational flow characterized by a velocity potential φ, the kinetic energy T in the fluid region N is given by: And i takes 1, 2, 3; C is the boundary of the fluid region. Now assume that the translational velocity of the underwater vehicle has components U1, U2, and U3 at the origin, and the rotational angular velocity of the underwater vehicle has components U4, U5, and U6; for each component α of the object's motion, the fluid has a corresponding velocity potential; the unit U α The corresponding velocity potential is φ α Indicates; at this time, the Laplace equation and boundary conditions are linear, so the solutions of the equation can be superimposed; therefore, no matter what kind of motion the underwater vehicle is in, for the general case: φ=U α f α ,α=1,2,...,6 The result is: The subscripts α and β are integers from 1 to 6, and: It can be proved from the formula that λ αβ are the components of the additional mass tensor, and λ αβ It is symmetrical; Kinematic boundary conditions on the surface of an object: Among them is Velocity vector, is the rotation vector, is the position vector, and is the unit vector of the normal line of the underwater navigation body; let n4, n5, n6 be defined as Then the equation is simplified to: The value of α ranges from 1 to 6; since for all U α The values ​​are all valid, so: Thus we get: Because φ α It's U α =1, so the added mass depends only on the shape and orientation of the object, and the way the object moves, and has nothing to do with the object's linear velocity, angular velocity or acceleration; If the underwater vehicle is simplified to a simple cylinder with a diameter of D and a length of l, its additional mass can be simplified. In the fully wet and cavitation-free state, the slender cylinder is an axisymmetric body. Considering the symmetry, only λ is included in the additional mass matrix λ. 11 , 22 , 33 , 55 , 66 , 26 and λ 35 is a non-zero term, and the motion equation of the underwater vehicle with three degrees of freedom under the action of internal solitary waves is a two-dimensional simplified model, so λ 33 , 55 , 35 It is not considered in motion; the slender body theory is adopted, that is, it is assumed that there is no interference between the sections of the cylinder, and the additional mass of the cylinder is the integral of the additional mass of the section along the longitudinal axis of the cylinder; the additional mass λ of the section is calculated ij (x) are as follows: The total added mass is get: λ 11 It is a small amount compared to the mass of the object, and its approximate formula is as follows: Step 4. Prediction of the motion trajectory of underwater vehicles under the action of internal solitary waves Assume that the components of the angular velocity and the velocity at the center of buoyancy of the underwater vehicle in the body coordinate system are ω x , v x , v y ; The mass of the underwater vehicle is m, and the moment of inertia of the underwater vehicle about the oz axis is J ZZ ; From the analysis of steps 1-3, the kinematic equations of the underwater vehicle in the body coordinate system are obtained as follows: Combined angular velocity ω z and attitude angle change rate The relationship between The simplified equation of motion is: in, A=[(mx c +λ 26 )(G·L oG cosθ+F y L or sinθ-M W -mx c v x oh x )-(J zt +λ 66 )(ΔBcosθ+F y sinθ-mv x oh z )] B=[(mx c +λ 26 )(ΔBcosθ+F y sinθ-F c -mv x oh z )-(m+λ 22 )(G·L oG cosθ+F y ·L or sinθ-mx c v x oh z )] By simplifying the motion equation, the motion trajectory of the underwater vehicle under the action of internal solitary waves is predicted.

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