A method for calculating ultra-low frequency circular waves generated by underwater moving targets

Abstract

The present invention provides a method for calculating ultra-low frequency circular waves generated by underwater moving targets, comprising the following steps: step 1: analyzing the characteristics of ultra-low frequency circular waves; step 2: the causes of the generation of ultra-low frequency circular waves; step 3: ultra-low frequency circular wave calculation method - multi-level potential method. The present invention provides a new idea for underwater target detection, that is, detecting targets through circular waves, studying the principle and calculation method of ultra-low frequency circular waves generated by underwater moving targets, and providing a theoretical basis for underwater target detection.

Description

Technical Field

[0001] The present invention belongs to the field of underwater target detection, and in particular relates to a method for calculating ultra-low frequency circular waves generated by underwater moving targets. Background Art

[0002] Acoustics and oceanography are closely connected. Acoustic phenomena such as underwater acoustic channels, deep-water scattering layers, and marine biological noise all have their basis in oceanography. The sound field is the only physical field capable of transmitting information over long distances underwater, and therefore plays a vital role in monitoring natural events and conducting communications. However, the two have become somewhat disconnected. Underwater acoustics, introducing homogeneous, isotropic media, invented its own version of the ocean. In classical acoustics, long-range underwater target detection can be neglected compared to the speed and frequency of sound. The extremely low-frequency oscillations of moving objects, scattered vibrations, and fluid flows at very low Mach numbers are all negligible. Noise measurements often reveal fluctuations exceeding 20 decibels, which are explained by Taylor's assumption. Oceanography has neglected mesoscale and small-scale phenomena, focusing instead on large-scale and long-duration ocean phenomena. The two disciplines have pursued separate paths, to the detriment of both. With the development of underwater target acoustic stealth technology, the target detection frequency has shifted to low frequency and ultra-low frequency. Domestic and foreign scholars have gradually shifted their attention to ultra-low frequency detection and started to combine target motion with small and medium-scale phenomena in oceanography. Using medium and small-scale phenomena in oceanography to detect targets has become a new research direction. A possible method is to use a dual-base active sonar installed on a ship to achieve long-range detection. Conventional active detection methods are too complex and power-intensive. Experiments have found that forward scattered acoustic detection can use the method of the external field of the vehicle to solve the detection problem. This patent mainly describes the ultra-low frequency circular wave field generated by underwater moving targets, providing a theoretical basis for ultra-low frequency detection. Summary of the Invention

[0003] The purpose of the present invention is to study the principle and calculation method of ultra-low frequency circular waves generated by underwater moving targets in order to study the ultra-low frequency circular wave field generated by moving targets.

[0004] A method for calculating ultra-low frequency circular waves generated by an underwater moving target comprises the following steps:

[0005] Step 1: Analyze the characteristics of ultra-low frequency circular waves;

[0006] Step 2: The cause of ultra-low frequency circular waves;

[0007] Step 3: Ultra-low frequency circular wave calculation method - multi-level potential method.

[0008] Furthermore, the step 1 is specifically as follows: the Karman vortex shedding frequency of the three-dimensional fluid is given by the Strouhal number:

[0009]

[0010] Where St is the Strouhal number, V and u are the volume and velocity of the target.

[0011] From the perspective of kinetic energy, the Kirchhoff equation describes the translational and rotational motion of the object:

[0012]

[0013] Where u is the translation velocity of the object, ω is the angular velocity of the object, T b , T f is the kinetic energy of the object and fluid, F e is the external force, L e is the torque.

[0014] Furthermore, the step 3 is specifically as follows:

[0015] Step 3.1: Velocity potential

[0016] Assume the target center coordinates are (0,0,d), and define the spherical coordinate system centered at (0,0,d) as:

[0017] x=rsinθcosα=Rcosαy=rsinθsinα=Rsinα,zd=rcosθ

[0018] The total velocity potential in the fluid is Φ, which can be linearly decomposed into:

[0019]

[0020] Among them, μ0 and μ2 represent the vertical and horizontal amplitudes of the sphere, respectively, and φ0 and φ1 are the spatial parts of the velocity potential under the unit amplitude of vertical and horizontal sway. Then the velocity potential φ m The Laplace equation and boundary conditions need to be satisfied:

[0021] Anywhere in the fluid

[0022]

[0023] Surface of a sphere

[0024] Where g is the acceleration due to gravity, n is the unit normal vector of the sphere surface, and n0 and n1 represent the projections of n on the x-axis and z-axis, respectively.

[0025] k0 is the root of the equation that satisfies the following dispersion relation:

[0026] K=k0tanh(k0h)

[0027] Step 3.2: Multi-level expansion method

[0028] Expand the velocity potential to a multipolar potential:

[0029]

[0030] in P n (x) is a Legendre polynomial.

[0031]

[0032] in,

[0033] Step 3.3: Circular Wave

[0034] The velocity generated by the vertical and horizontal swing of the sphere can be expressed as U0 = Re{Ue iwt}k and U1=Re{Ue iwt Where k is the unit vector pointing to the z-axis, i is the unit vector pointing to the x-axis, the potential is expressed as φ0 and φ1, and the Legendre function is P1 0 (cosθ)=cosθ,P1 1 (cosθ)=sinθ, and the boundary conditions for the sway and swing of the sphere are:

[0035]

[0036] According to the multi-level potential constructed above, the radiation potential of the fluid can be expressed as:

[0037]

[0038] The unknown coefficient b is obtained from the boundary conditions and the orthogonal properties of the Legendre function n,m The relationship:

[0039]

[0040] Among them, δ ns is the Kronecker product, if i=j, then δ ns =1, in other cases δ ns =0.

[0041] The displacement ξ of the sphere on the free surface when it oscillates:

[0042] Φ t +gξ=0,z=0

[0043] The beneficial effects of the present invention are:

[0044] This paper provides a new idea for underwater target detection, namely detecting targets through circular waves, and studies the principle and calculation method of ultra-low frequency circular waves generated by underwater moving targets, providing a theoretical basis for underwater target detection. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1 This is a schematic diagram of the energy spectrum of the ocean's natural fluid dynamic field according to the present invention;

[0046] Figure 2 This is a diagram of the vortex shedding phenomenon of an object surrounded by flow in the present invention;

[0047] Figure 3 is the angle of attack diagram of the flow around the target of the present invention;

[0048] Figure 4 This is the target periodic oscillating motion diagram of the present invention;

[0049] Figure 5 This is a schematic diagram of the target environment of the present invention;

[0050] Figure 6 The surface displacement map generated by the sphere at different depths of the present invention;

[0051] Figure 7 This is the surface displacement diagram of the present invention with Ka=0.3, f / a=-2, and t=0. DETAILED DESCRIPTION

[0052] The present invention will be further described below with reference to the accompanying drawings.

[0053] Step 1: Characteristics of ultra-low frequency circular waves.

[0054] The ultra-low frequency field is a hydrodynamic field, not an acoustic field. The basic equations of hydrodynamics and acoustics are derived from the constraints of the fluid mechanics equations. The main derivation of hydrodynamics is that under incompressible conditions, the hydrodynamic pressure satisfies the Laplace equation, and the basic solution is 1 / r, which satisfies It is not a wave, and the pressure field decreases monotonically with distance. Acoustics mainly studies small amplitude motion in static fluids, emphasizing the compressibility of fluids. The sound pressure obeys the wave equation. In the simple harmonic case, the basic solution of the free space Helmholtz equation is satisfy The combination of the time factor and the fundamental solution represents an outward-propagating spherical wave. Here, δ(r) is the Dirac δ function. Comparing the two formulas, we can see that the limit of the wave equation when k0→0 is the hydrodynamic pressure equation. As can be seen from k0=w / c, when the sound wave frequency is very low, that is, when w→0, the sound field can be considered an incompressible flow field. Therefore, in acoustics, there is no ultra-low frequency field. When considering ultra-low frequency fields, the radiation effect of the wave can be ignored, and only the incompressible flow field needs to be considered. For example, the low-frequency "pseudo-sound" measured by a hydrophone in a turbulent region is actually the hydrodynamic field.

[0055] According to the ocean wave spectrum standard, we call the fluctuations between Rossby waves and wind-induced waves Ultra Low Frequency (ULF) waves, which are hydrodynamic waves with an inertial period (IP) close to the potential field. The most sensitive period is 15 minutes, and the period range is 2 minutes to 2 hours. The ULF range of the ocean's natural hydrodynamic field is Figure 1 The winning bid was awarded.

[0056] When conducting dual-base forward scattering detection of long-range targets, the ultra-low frequency circular wave field caused by the moving target will produce ultra-low frequency modulation on the sound wave, providing a basis for ultra-low frequency detection of far-field targets.

[0057] In early experiments, there are two explanations for the fluctuations in the ultra-low-frequency signal generated by the fluid flow. One explanation is that it is due to the internal waves excited by the wake vortex, whose frequency is determined by the Vaisala frequency and the Kariolis inertial frequency. The second explanation is related to the low-intensity shock wave formed in front of the blunt object when the object flows around it. The frequency is related to the vortex formation frequency. Because the experiments were conducted too early and the theory was not perfect, the three-dimensional vortex formation frequency is given as an approximation. Where u is the target velocity and L is the target characteristic size. With the continuous improvement of the theory, the Karman vortex shedding frequency of the three-dimensional fluid is given by the Strouhal number:

[0058]

[0059] Among them, St is the Strouhal number, V and u are the volume and velocity of the target. Based on the literature, this paper gives the following third explanation.

[0060] When a moving object flows through an incompressible ideal fluid, it is subject to a pressure differential force caused by a pressure gradient. If the fluid and solid are treated as a single dynamic system, the pressure on the object's surface is equal to the internal pressure of the system, so measurement is unnecessary. The object's motion causes the fluid to move. Once the object stops moving, the fluid also stops moving. Therefore, the motion of the object in the fluid is examined from the perspective of kinetic energy. The Kirchhoff equations describe the object's translational and rotational motion:

[0061]

[0062] Where u is the translation velocity of the object, ω is the angular velocity of the object, T b , T f is the kinetic energy of the object and fluid, F e is the external force, L e The left side represents the resultant force and torque on the target, and the other items on the right side are the force and torque of the fluid on the object.

[0063] From the above formula, we can see that for a uniform non-rotating object, the fluid reaction force will decrease to zero (Dalbert's paradox), and the torque is The effect of fluid pressure is just a torque, and if u and When parallel, that is, when the target's velocity u is along one of the target's three principal axes, the torque becomes zero. For an ellipsoid, there are three mutually perpendicular stable directions of motion. The shortest axis is the most stable direction of motion. If the direction of motion deviates slightly, the torque tends to reduce this deviation. The longest axis is the least stable direction of motion, and the torque tends to amplify this deviation. This provides a theoretical explanation for some common motion instabilities. Rockets and torpedoes move in the direction of their longest axis, which causes them to wobble.

[0064] With the discovery of the d'Alembert fallacy and the Karman vortex street phenomenon, attention gradually shifted to the problem of flow around viscous fluids. When a target flows around a cylinder, vortices are generated due to the viscosity of the fluid. The pressure gradient provides the centripetal force that propels the vortices into circular motion. At high Reynolds numbers, the vortices periodically detach from the target surface in an asymmetric manner. Numerous experimental and numerical results have demonstrated that in the problem of flow around a cylinder, the forces acting on the target are closely related to the development of the surrounding vortex field, particularly the vortex separation and shedding processes. Therefore, in this process, the velocity gradient and viscosity induce viscous forces, which provide the force that forces the axially stable moving object off its axis. Under the combined action of pressure differential and viscous forces, the target experiences not only resistance or drag in the direction of fluid flow, also known as normal force, but also lift perpendicular to the flow, also known as lateral force. A target designed for uniform linear motion will experience vortex oscillations due to the combined effects of pressure differential and viscous drag. This means that the actual motion trajectory becomes a periodic oscillation. The oscillation frequency is twice the vortex shedding frequency.

[0065] The target oscillation excites surface wave modes on the free surface. The initial domain of the surface wave propagating above the moving object is called a circular wave, whose lateral dimension (radius) is equal to the half-gravity wave length. The following mainly explains the reasons why the target motion produces oscillations and the calculation method and simulation of the circular waves caused by the oscillation.

[0066] Step 2: The cause of ultra-low frequency circular waves

[0067] In viscous fluids, the force acting on a moving target consists of two parts: the pressure difference force related to the pressure gradient and the viscous force related to the fluid viscosity and velocity gradient.

[0068]

[0069] Where p is the pressure distribution in the ambient fluid, σ ijis the viscous stress tensor.

[0070] The first summation is obtained in an incompressible ideal fluid. Assume that the target moves under the action of an external force, the fluid motion caused by it is potential, and the velocity potential satisfies the Laplace equation Select a motion coordinate system with the origin at the center of the target and r as the distance to the origin. At a distance from the object, the general solution of the Laplace equation is:

[0071]

[0072] Among them, a and A are independent of coordinates. Vector A is determined by the target motion speed and actual shape. It can be solved by solving the target surface boundary conditions and flow conditions. and Velocity potential The speed given is Because the flow rate of an incompressible fluid through any closed surface is zero, the total flow rate through a sphere with a radius of R is ρ(a / r)4πr 2 =4πρa, so a=0. Ignoring higher-order terms when solving for the velocity at a distance, we obtain:

[0073]

[0074] The speed is:

[0075]

[0076] The kinetic energy of the fluid is:

[0077]

[0078] Among them, V f , V s For the fluid and target volumes. Introduce an additional mass tensor m ij and the inertia coefficient k ij , the fluid kinetic energy can be rewritten as:

[0079]

[0080] From the relationship between the total momentum and total energy of the fluid dE = u·dP, the components of momentum P are obtained as follows:

[0081] P i =m ij u j =k ij ρV s u j (7)

[0082] The pressure difference force of the fluid on the moving target is the derivative of the total momentum of the fluid, and the fluid resistance is obtained as:

[0083]

[0084] The force f is the derivative of the total momentum of the system with respect to time. The total momentum is equal to the sum of the momentum of the object mu (m is the mass of the object) and the momentum of the fluid:

[0085]

[0086] This is the equation of motion for an object immersed in an ideal fluid. For a sphere, A = 0.5R 3 u, then the diagonal components of all adjacent mass tensors are the same m xx =m yy =m zz , so k ij =(1 / 2)δ ij , where δ ij is the unit Kronk tensor.

[0087] In addition to the pressure resistance, the actual underwater flow target is also subject to viscous resistance. For the second viscous resistance F in formula (1), i 2 , for small values ​​of Reynolds number Re=ul / v Re<1, the quadratic term of flow velocity in the Wiener-Stokes equation is neglected and the Stokes formula F2=6πRηu is valid, where l is the target length and v is the fluid viscosity, As the Reynolds number increases, Stokes' law is no longer valid and the quadratic term of velocity must be considered, resulting in a linear equation for fluid velocity:

[0088]

[0089] The Osen equation solution in the literature directly gives a modified expression:

[0090]

[0091] It can be seen that as the Reynolds number increases, the viscous drag no longer depends significantly on the value of viscosity ν, and the force is related to ρu 2 S is proportional to the cross-section of the object. Sometimes, for smooth objects, the viscous drag is proportional to V when Re>>1. 2 / 3 Rather than S, as mentioned above, where V is the volume of the object, then:

[0092]

[0093] The constant const depends only on the shape of the object. Usually we consider the resistance factor C, which is defined as:

[0094]

[0095] The drag factor C = const, which depends only on the shape of the object. For example, the expression for static drag is F 2x =C(ρu 2 / 2)V 2 / 3 However, at large Reynolds numbers, an asymmetric Karman vortex shedding phenomenon will occur on the target tail surface when the target moves. The two-dimensional Karman vortex shedding phenomenon is as follows: Figure 2 shown

[0096] Two-dimensional vortex shedding frequency f B The 3D vortex shedding frequency f has been given above. The vortex shedding viscosity force provides the power to stabilize the target moving axially and deviate from the axis, resulting in an attack angle α and a deviation angle β of the target trajectory relative to the flow direction, as shown in Figure 3 Shown is the angle of attack α resulting from the target motion.

[0097] When α (α≤10°) and β (β≤10 ° ) is very small, except for the resistance F 2x , and also produces a large lift F 2y (vertical) and F 2z (lateral):

[0098] F 2x =C x (ρu 2 / 2)V 2 / 3

[0099] F 2y =C y (ρu 2 / 2)V 2 / 3

[0100] F 2z =C z (ρu 2 / 2)V 2 / 3 (14)

[0101] Due to the viscosity of the fluid, at large Reynolds numbers, the moving target will produce periodic oscillation motion under the action of external forces and fluids. Although the actual target will correct this oscillation, even so, it cannot change the natural periodicity of its translational motion. Figure 4 Shown is the oscillatory motion of the target in two dimensions.

[0102] The additional force generated during the periodic change of direction is mainly provided by the lift. For a real underwater object moving in the x,y plane, the Kirchhoff equation ignores the rotation of the object in the form of the target moving along the x and y axes:

[0103]

[0104] We regard the underwater target as a non-buoyant body, and the mass m=ρV,F e is the external force, F ex is the traction force, F ey is the lateral force. For a small angle of attack α, the traction force is approximately equal to the resistance force, and the lateral force is proportional to the angle of attack. Let the periodically changing angle of attack be α = α0exp(-iΩt). According to formula (15), the target motion speed is:

[0105] u x =const+O(α 2 )

[0106]

[0107] When α0<<1, the velocity u of the non-buoyant target y Simplified to

[0108]

[0109] Similarly, we can get the velocity u about the deviation angle β=β0exp(-iΩt) z The expression:

[0110]

[0111] Step 3: Ultra-low frequency circular wave calculation method - multi-level potential method

[0112] Step 3.1: Velocity potential

[0113] Consider an ideal incompressible and irrotational fluid of finite depth h. A sphere of radius a is hidden within the fluid at a distance d from the free surface, with the surface S being the sphere. It oscillates with small amplitude in both horizontal and vertical motions at a circular frequency ω. A rectangular coordinate system oxz is established with the oz axis pointing vertically upward and the ox axis and free surface coinciding, as shown in the following example: Figure 5 shown.

[0114] Assume the target center coordinates are (0,0,d), and define the spherical coordinate system centered at (0,0,d) as:

[0115] x=r sinθcosα=R cosα y=r sinθsinα=R sinα,zd=r cosθ (19)

[0116] The total velocity potential in the fluid is Φ, which can be linearly decomposed into:

[0117]

[0118] Among them, μ0 and μ2 represent the vertical and horizontal amplitudes of the sphere, respectively, and φ0 and φ1 are the spatial parts of the velocity potential under the unit amplitude of vertical and horizontal sway. Then the velocity potential φ mThe Laplace equation and boundary conditions need to be satisfied:

[0119] Anywhere in the fluid (21)

[0120]

[0121] Surface of the sphere (24) where g is the acceleration due to gravity, n is the unit normal vector of the sphere surface, and n0 and n1 represent the projections of n on the x-axis and z-axis respectively.

[0122] k0 is the root of the equation that satisfies the following dispersion relation:

[0123] K=k0tanh(k0h) (25)

[0124] Step 3.2: Multi-level expansion method

[0125] The multipole potential is a singular solution of the Laplace equation that satisfies the free surface and bottom boundary conditions, similar to the output wave away from the singular point, which in this case will be the center of the sphere. For the velocity potential that meets the previous conditions, the series solution of the velocity potential can be established according to the separation of variables method, expanding the velocity potential into a multipole potential:

[0126]

[0127] in P n (x) is a Legendre polynomial.

[0128]

[0129] in, Step 3.3: Circular Wave

[0130] Whether the sphere is in heaving or swaying motion, it will excite waves with wave number k0 on the free surface. The multipolar potential manifests as a cylindrical wave at a horizontal distance from the singularity, and as a circular wave on the free surface, with a lateral dimension (radius) equal to the half-gravity wave length. Next, we use the multipolar potential constructed above and combine it with the target surface boundary conditions to calculate the circular waves generated by the target oscillation.

[0131] The velocity generated by the vertical and horizontal swing of the sphere can be expressed as U0 = Re{Ue iwt}k and U1=Re{Ue iwt Where k is the unit vector pointing to the z-axis, i is the unit vector pointing to the x-axis, the potential is expressed as φ0 and φ1, and the Legendre function is P1 0 (cosθ)=cosθ,P1 1(cosθ)=sinθ, and the boundary conditions (24) for the sway and swing of the sphere are:

[0132]

[0133] According to the multi-level potential constructed above, the radiation potential of the fluid can be expressed as:

[0134]

[0135] The unknown coefficient b is obtained from the boundary condition (28) and the orthogonality property of the Legendre function n,m The relationship:

[0136]

[0137] Among them, δ ns is the Kronecker product, if i=j, then δ ns =1, in other cases δ ns =0.

[0138] The displacement ξ of the sphere on the free surface when it oscillates is determined by expression (31):

[0139] Φ t +gξ=0, z=0 (31)

[0140] The above describes the specific implementation of each part of the invention content, and the following analyzes the simulation example.

[0141] For infinite-dimensional linear relationships, the truncation method can be used to solve them. In numerical calculations, the relationship is truncated at n=N, and the N×N equation group is used to calculate the coefficient b n,m , the multi-level expansion formula (29) of the radiation potential can converge quickly. After a large number of calculations, it is shown that when N = 4, the influence of the truncation term is already very small. Therefore, in the following numerical calculations, N = 4 is taken.

[0142] The amplitude of the circular wave on the free surface excited by the oscillating motion of the sphere varies with the dimensionless frequency Ka. For this purpose, h / a = 10 and d / a = -2, -3, -4. The results are as follows Figure 6 As shown in the figure, it can be seen that with the increase of d / a, the amplitude of the surface wave mode fluctuation excited by the spherical oscillation motion on the free surface increases in both oscillation motion modes. Figure 7 The circular waves caused by the heave and sway motions of the target under the conditions of Ka=0.3, d / a=-2, t=0 are given. It can be seen that the circular waves generated by the heave motion of the sphere are symmetrical, while the circular waves generated by the sway motion are antisymmetric.

[0143] The foregoing description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Those skilled in the art will readily appreciate that various modifications and variations of the present invention are possible. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of the present invention shall be included within the scope of protection of the present invention.

Claims

1. A method for calculating ultra-low frequency circular waves generated by underwater moving targets, characterized in that: The following steps are involved: Step 1: Analyze the characteristics of ultra-low frequency circular waves; The Karman shedding frequency of a three-dimensional fluid is given by the Strouhal number: Among them, St is the Strouhal number, V and u are the volume and velocity of the target; From the perspective of kinetic energy, the Kirchhoff equation describes the translational and rotational motion of an object: Where u is the translation velocity of the object, ω is the rotation angular velocity of the object, T b , T f is the kinetic energy of the object and the fluid, F e is the external force, L e is the torque; Step 2: The cause of ultra-low frequency circular wave; Step 3: Ultra-low frequency circular wave calculation method - multi-level potential method; Step 3.1: Velocity potential Assume the target center coordinates are (0,0,d), and define the spherical coordinate system centered at (0,0,d) as: x=rsinθcosα=Rcosαy=rsinθsinα=Rsinα,zd=rcosθ The total velocity potential in the fluid is Φ, which decomposes linearly into: Among them, μ0 and μ2 represent the heave and sway amplitudes of the sphere, respectively, φ0 and φ1 are the spatial parts of the velocity potential under the unit amplitude of heave and sway, respectively; then the velocity potential φ m The Laplace equation and boundary conditions need to be satisfied: Anywhere in the fluid Surface of a sphere Where g is the gravitational acceleration, n is the unit normal vector of the sphere surface, and n0 n1 represent the projection of n on the x-axis and z-axis respectively; k0 is the root of the equation that satisfies the following dispersion relation: K = k0tanh(k0h) Step 3.2: Multi-level expansion method Expand the velocity potential to a multipolar potential: in P n (x) is a Legendre polynomial; in, Step 3.3: Circular Wave The velocity generated by the vertical and horizontal swing of the sphere is expressed as U0 = Re {Ue iwt }k and U1=Re{Ue iwt }i where k is the unit vector pointing to the z-axis, i is the unit vector pointing to the x-axis, the potential is represented by φ0 and φ1, and the Legendre function is P1 0 (cosθ)=cosθ,P1 1 (cosθ)=sinθ, and the boundary conditions for the sway and heave of the sphere are: According to the multi-level potential constructed above, the radiation potential of the fluid is expressed as: The unknown coefficient b is obtained from the boundary conditions and the orthogonal properties of the Legendre function n,m The relationship is: Among them, δ ns is the Kronecker product. If i=j, then δ ns =1, in other cases δ ns =0; The displacement ξ of the sphere on the free surface when it oscillates: F t +gξ=0,z=0.

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