Submarine Fluid-Structure Interaction Analysis Method Based on Divergence-Free Immersed Boundary Method

Abstract

A fluid-structure interaction analysis method for submarines based on the divergence-free immersed boundary method is disclosed. In this method, the two-dimensional submarine shape and region, as well as the fluid pressure and velocity, are collected. A grid is established based on the flow field of the two-dimensional submarine shape and region. An interleaved Cartesian grid is established in the flow field region, where the fluid pressure p and velocity components u and v are stored in different grids. The interleaved Cartesian grid of the watershed structure is spatially discretized and divergence-free interpolation is performed. The Lagrangian force is diffused into the fluid grid without divergence using the fast Fourier algorithm. The flow field calculation is completed in the time step, and the result is assigned to the next time step to update the flow field and velocity field. The calculation is then performed for the next time step.

Description

Technical Field

[0001] This invention relates to the field of numerical simulation technology for submarine fluid-structure interaction, and in particular to a submarine fluid-structure interaction analysis method based on the divergence-free immersed boundary method. Background Technology

[0002] The ocean is increasingly becoming a domain of focus and competition among global powers, with countries worldwide increasing their investment in the research and development of naval weapons. Submarines, as the world's most important underwater combat platform, are strategic weapons with nuclear deterrent capabilities in naval warfare. Stealth is a key indicator reflecting a submarine's combat and survivability, and a focus of widespread attention in the military communities of various countries. The mechanical noise of a submarine primarily originates from its power source equipment (diesel engines, electric pumps). This power source equipment, acting as a noise source, transmits noise through the floating raft system and piping system to the hull section, causing hull vibration and generating mechanical noise, which then propagates through underwater radiation. Therefore, fluid-structure interaction analysis is an important method for predicting underwater noise in submarines.

[0003] Traditional fluid-structure interaction (FSI) analysis methods require separate mesh generation and calculation for the fluid and structure, with a one-to-one mapping between the nodes of the two types of meshes. This means the fluid mesh must be a body-fitted mesh, making it impossible to achieve interface crossing between the fluid and solid with a single mesh. As the iterative algorithm runs, the mesh needs continuous updates to adapt to the changing coupling effects. When the boundary shape of the object is complex, this method first requires a significant amount of time to generate a high-quality mesh. For moving boundaries, dynamic meshing techniques are also needed for real-time mesh updates, further increasing the computational load and significantly limiting its application scope.

[0004] To overcome the limitations of traditional fluid-structure interaction (FSI) analysis methods, Peskin proposed the immersed boundary method in 1972. It uses a uniform Cartesian mesh over the global solution domain, and its most significant advantage is its ability to perform cross-interface calculations. Fluid mesh generation can cross the fluid domain into the solid domain, enabling automatic tracking of the FSI interface and effectively avoiding the problems of independent mesh generation and updates inherent in traditional methods. Later, Lai and Peskin treated rigid boundaries as elastic boundaries with large stiffness coefficients and allowed for small deformation displacements, studying flow problems at rigid boundaries. In 1993, Goldstein proposed an immersed boundary method based on a spectral method using a feedback force scheme, known as the virtual boundary method. In 2007, Kim proposed the immersed boundary penalty method, aiming to solve elastic boundary problems with mass. Researchers such as Chen Xiaoming from South China University of Technology have conducted research on dynamic boundary problems based on the implicit force immersed boundary method. The improved implicit immersed boundary method not only has the advantage of not requiring real-time mesh updates but also accurately applying no-slip conditions to rigid boundaries. Researchers such as Liu Jianming developed an immersion boundary method for finite volume schemes using unstructured meshes in compressible fluids. Researchers such as Wang Wenquan from Kunming University of Science and Technology presented a fast, direct solution algorithm for the projected immersion boundary method. Researchers such as Wang Liang from Nanjing University of Science and Technology introduced a fluid volume method to identify immersion boundaries, thus providing an adaptive method for immersion boundary methods based on complex static and moving boundaries simulating fluids.

[0005] While the classical immersion boundary method boasts wide applicability and ease of implementation, it suffers from a significant drawback: relatively poor volume conservation. Volume conservation refers to the preservation of the volume enclosed by the structure immersed in the fluid within a continuous medium, a direct consequence of the fluid's incompressibility. Even under the simplest quasi-static pressurized membrane, the volume error of the traditional immersion boundary method continuously increases over time.

[0006] The information disclosed in the background section is only intended to enhance the understanding of the background of the present invention, and therefore may contain information that does not constitute prior art known to those skilled in the art. Summary of the Invention

[0007] The purpose of this invention is to provide a submarine fluid-structure interaction analysis method based on the divergence-free immersion boundary method to achieve volume conservation.

[0008] To achieve the above objectives, the present invention provides the following technical solution:

[0009] The present invention provides a submarine fluid-structure interaction analysis method based on the divergence-free immersion boundary method, comprising:

[0010] Step 1: Collect the 2D submarine's shape and region, along with its fluid pressure and velocity. Establish a mesh based on the 2D submarine's shape and the flow field within the region. Create an interleaved Cartesian mesh within the flow field region, storing the fluid pressure p and velocity components u and v in different Cartesian meshes, where u is the velocity component in the x-direction and v is the velocity component in the y-direction. Set the fluid periodicity computation domain to Ω = [0, L]. 2 Ω represents the computational domain; L is the length of the computational domain. The periodic computational domain is uniformly divided into N×N Eulerian grids with a spatial step size h = L / N, where N is the number of grids in a certain direction of the computational domain; for the fluid pressure p, it is stored at the center position of each Eulerian grid, and the center position is x. i，j = ((i+1 / 2)h, (j+1 / 2)h), for the fluid velocity u = (u, v), it is stored at the edge center position of each fluid Eulerian grid, and the position of the fluid velocity in the x-direction is x. i-1 / 2，j = (ih, (j+1 / 2)h), where the position of the fluid velocity in the y-direction is x. i，j-1 / 2 = ((i+1 / 2)h, jh), establish Lagrange coordinates for the submarine hull model;

[0011] Step 2: Set the initial conditions for numerical simulation. Treat the initial shape of the submarine as a high-stiffness elastic boundary and perform a pure fluid steady simulation. The fluid velocity is constant in the flow domain. Update the flow field velocity on the left side of the flow domain to keep it constant as it flows into the flow domain.

[0012] Step 3: Spatial discretization of the staggered Cartesian grid of the watershed structure, wherein a central difference operator is constructed:

[0013]

[0014] Where: e β (β = 1, 2) is a set of orthogonal bases in two-dimensional space, using orthogonal bases along the x and y directions, e1 = (1, 0), e2 = (0, 1); φ(x) is a scalar function; using the above-mentioned partial derivatives with respect to the first order. Central difference operator To define the vector center difference operator D h ,but

[0015] Due to the vector center difference operator For the Laplace operator in the viscosity term, use D h ·D h The Laplace difference operator L can be defined. h :

[0016]

[0017] In the formula: φ(x) is a scalar function; for the finite difference discretization of the convection term, based on the incompressible... The conditions are as follows: spatial discretization is performed using an antisymmetric form, where: Represents the Hamiltonian operator;

[0018]

[0019] In the formula: u is the fluid velocity; u β Represent the velocity components in different directions; discretize them and define them as the antisymmetric difference operator S. β :

[0020]

[0021] In the formula: The average convection velocity, its components are related to u β Defined on the same grid, taking the average value of u as Discrete results of the Navier-Stokes equations for viscous incompressible fluids can be obtained:

[0022]

[0023] D h u = 0;

[0024] In the formula: ρ represents fluid density; p represents fluid pressure; μ represents fluid viscosity coefficient; f FSI Represents fluid force;

[0025] Step 4: To perform divergence-free interpolation, the Poisson equation for a(x) is obtained through Hellholmz decomposition, where a(x) is first set to satisfy the following conditions.

[0026]

[0027] In the formula: u0 is the mean velocity; the curl of the above conditions is processed and the identity property of vectors is used.

[0028] D h ×(D h ×u)=D h (D h ·u)-L h u,

[0029] The resulting equation is -L h a = D h ×(u-u0)=D h ×u, a structure defined on the mesh is obtained by solving the Fast Fourier Transform algorithm. The vector A(X) defined on the Lagrange grid points is obtained by applying the regularized δ function interpolation scheme to a(x), thus from The defined Lagrange velocity field is automatically non-divergent, thus yielding... In the formula: U is the velocity of the Lagrange boundary point obtained by interpolation, x is the Eulerian coordinate, and X is the coordinate of the Lagrange boundary point; δ h For regularization functions;

[0030] Step 5: Perform divergence-free force diffusion based on the energy conservation between Lagrange and Euler variables, characterized as... In the formula: f FSI For fluid force; F FSI For boundary forces; For the defined grid; U m and F FSI，m Let M be the velocity and fluid-structure interaction force at the m-th Lagrange point; M be the total number of Lagrange points; Δs be the grid spacing; and the discrete vector identity be used. The above formula can be converted to

[0031]

[0032] in,

[0033] Using divergence-free velocity interpolation and supplementary conditions:

[0034] all In the formula: It is an arbitrary scalar field. The fluid-structure interaction force f acting on the Euler grid is obtained. FSI Poisson's equation for (x):

[0035]

[0036] Similarly, the divergence-free diffusion of the Lagrangian force into the fluid mesh is solved using the Fast Fourier Transform algorithm;

[0037] Step 6: Complete the flow field calculation within the time step. This yields the velocity field information, pressure field information, fluid force on the model, and deformation of the model within one time step. Assign these values ​​to the next time step as initial values, update the flow field and velocity field, and perform calculations for the next time step.

[0038] In the above technical solution, the submarine fluid-structure interaction analysis method based on the divergence-free immersed boundary method provided by the present invention has the following beneficial effects: The divergence-free immersed boundary method of the present invention solves the problems of discontinuous velocity field and volume conservation crossover caused by the use of local grid points in the traditional immersed boundary method through traditional velocity interpolation at the boundary Lagrange points; Compared with the area error of the traditional scheme, the area error of the divergence-free high-stiffness immersed boundary method of the present invention decreases continuously as the grid division of the boundary is continuously refined; The staggered grid method of the present invention avoids non-physical oscillation of the pressure field and provides a more stable and accurate fluid pressure field. Attached Figure Description

[0039] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments recorded in this invention. For those skilled in the art, other drawings can be obtained based on these drawings.

[0040] Figure 1 This is a schematic diagram of variable storage using the interlaced grid used in this invention;

[0041] Figure 2 This is a flowchart of the divergence-free velocity interpolation method of the present invention;

[0042] Figure 3 This is a flowchart of the divergence-free force diffusion of the present invention;

[0043] Figures 4(a) to 4(b) This invention is used to calculate the flow around a stationary single cylinder at a Reynolds number Re = 100 and h s = Normalized area error variation of h;

[0044] Figure 5 This is the pressure distribution cloud map of a stationary flow around a single cylinder at a Reynolds number Re = 100, according to the present invention. Detailed Implementation

[0045] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below in conjunction with the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0046] Therefore, the following applies to the appendix Figures 1 to 5The detailed description of the embodiments of the present invention provided herein is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

[0047] It should be noted that similar labels and letters in the following figures indicate similar items. Therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures.

[0048] In the description of this invention, it should be understood that the terms "center," "longitudinal," "lateral," "length," "width," "thickness," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," "outer," "clockwise," and "counterclockwise," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on this invention.

[0049] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature. In the description of this invention, "a plurality of" means two or more, unless otherwise explicitly specified.

[0050] In this invention, unless otherwise explicitly specified and limited, the terms "installation," "connection," "linking," and "fixing," etc., should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral part; they can refer to a direct connection or an indirect connection through an intermediate medium; they can refer to the internal communication of two components or the interaction between two components. Those skilled in the art can understand the specific meaning of the above terms in this invention according to the specific circumstances.

[0051] In this invention, unless otherwise explicitly specified and limited, "above" or "below" the second feature can include direct contact between the first and second features, or contact between the first and second features through another feature between them. Furthermore, "above," "over," and "on top" of the second feature includes the first feature directly above or diagonally above the second feature, or simply indicates that the first feature is at a higher horizontal level than the second feature. "Below," "below," and "under" the second feature includes the first feature directly below or diagonally below the second feature, or simply indicates that the first feature is at a lower horizontal level than the second feature.

[0052] To enable those skilled in the art to better understand the technical solution of the present invention, the present invention will be further described in detail below with reference to the accompanying drawings. Figures 1 to 5 As shown, a submarine fluid-structure interaction analysis method based on the divergence-free immersion boundary method includes,

[0053] This example provides a submarine fluid-structure interaction analysis method based on the divergence-free immersion boundary method, specifically including:

[0054] Step 1: A mesh is established for the two-dimensional submarine's shape and the regional flow field. An interleaved Cartesian mesh is created for the flow field region, where the pressure p and velocity components u and v are stored on different mesh systems. For the two-dimensional case, the periodic computation domain for the fluid is set to Ω = [0, L]. 2 The computational domain is uniformly divided into N×N Eulerian grids with a spatial step size h = L / N, and the variables of the staggered grids are stored as follows: Figure 1 As shown. For the fluid pressure p, it is stored at the center of each Euler grid cell, with the center position being x. i，j = ((i+1 / 2)h, (j+1 / 2)h), denoted by C. For the fluid velocity u = (u, v), it is stored at the edge center of each fluid Eulerian grid, and the position of the fluid velocity in the x-direction is x. i-1 / 2，j = (ih, (j+1 / 2)h), where the position of the fluid velocity in the y-direction is x. i，j-1 / 2= ((i+1 / 2)h, jh). Lagrange coordinates are established for the submarine hull model. In one embodiment, a two-dimensional contour is established by measuring the scale of the model; Lagrange boundary points are set on the two-dimensional contour at certain intervals, thus completing the establishment of the two-dimensional submarine shape. The fluid region is set to be larger than the submarine model size, and a fluid domain grid is set with a certain grid width to store flow field information in a matrix. The pressure matrix of each point includes the coordinates and pressure magnitude of that point; similarly, the velocity matrix is ​​also included; throughout the region, the pressure information grid is spread out with a certain grid size. While the principle of the velocity information grid is the same, the grid points do not coincide with the pressure information grid points, hence the name "interlaced". These two information grids together form an Eulerian grid. The Lagrange boundary points are established around the two-dimensional submarine contour, that is, the Lagrange boundary points are used to represent the boundaries where the submarine "immerses" in the entire two-dimensional fluid domain.

[0055] Step 2: Set the initial conditions for the numerical simulation based on the computational state of the problem. Treat the initial shape of the submarine as a high-stiffness elastic boundary and perform a pure fluid steady-state simulation. The fluid velocity is set to constant within the flow domain, and the flow field velocity on the left side of the flow domain is updated to maintain a constant inflow into the flow domain. This is the step of setting initial conditions; the boundary stiffness coefficient is set to the stiffness corresponding to the submarine hull material. In the force diffusion step of step 5, the corresponding model of the high-stiffness elastic boundary is used to calculate the boundary forces.

[0056] Step 3: Spatial discretization of the staggered grid of the watershed structure, including constructing the central difference operator.

[0057]

[0058] Gradient Discrete Operator Laplace difference operator:

[0059]

[0060] For the finite difference discretization of the convection term, based on the incompressibility Conditions, spatial discretization is performed using an antisymmetric form.

[0061]

[0062] Discrete results of the Navier-Stokes equations for viscous incompressible fluids can be obtained:

[0063]

[0064] D h .u=0

[0065] Step 4: Perform divergence-free interpolation. First, obtain the Poisson equation for a(x) through Hellholmz decomposition. Then, establish an edge-centered staggered grid a(x) using the Fast Fourier Transform algorithm. Finally, apply a regularized δ-function interpolation scheme to obtain the vector A(X) defined on the Lagrange grid points, thus... The defined Lagrange velocity field is automatically non-divergent, thus yielding...

[0066] Step 5: Perform divergence-free force diffusion based on the energy conservation between Lagrange and Euler variables, which can be characterized as follows: The above equation can be transformed using the discrete vector identity. And through divergence-free velocity interpolation and supplementary conditions all

[0067] The fluid-structure interaction force f acting on the Euler grid can be obtained. FSI Poisson equation for (x) Similarly, the divergence-free diffusion of the Lagrangian force into the fluid mesh is solved using the Fast Fourier Transform algorithm.

[0068] Step 6: Complete the flow field calculation in the time step, assign it to the next time step, update the flow field and velocity field, and perform the calculation for the next time step.

[0069] In one embodiment, the method includes,

[0070] Step 1: Construct a geometric model of the object's shape in the flow problem;

[0071] Step 2: Divide the fluid computation region in the model from Step 1 into an equidistant Cartesian grid. Unlike the traditional immersion boundary method, this is a staggered grid, where the pressure p and velocity components u and v are stored on different grid systems. For the two-dimensional case, the periodic computation domain of the fluid is set to Ω = [0, L]. 2 The computational domain is uniformly divided into N×N Eulerian grids with a spatial step size h = L / N, and the variables of the staggered grids are stored as follows: Figures 3 to 1 As shown. For the fluid pressure p, it is stored at the center of each Euler grid cell, with the center position being x. i，j = ((i+1 / 2)h, (j+1 / 2)h), denoted by C. For the fluid velocity u = (u, v), it is stored at the edge center of each fluid Eulerian grid, and the position of the fluid velocity in the x-direction is x. i-1 / 2，j = (ih, (j+1 / 2)h), where the position of the fluid velocity in the y-direction is x. i，j-1 / 2 =((i+1 / 2)h,jh).

[0072] Step 3: Discretize the space according to the staggered grid, including defining the first-order partial derivatives. Central difference operator:

[0073]

[0074] In the formula, φ(x) is a scalar function, {e1, e2} is a set of orthogonal bases, and the formula is used... To define the gradient discrete operator D h It is easy to know For the Laplace operator that appears in the viscous term, use D h .D h Then the Laplace difference operator L h Defined as:

[0075]

[0076] Finite difference discretization of the convection term, based on incompressibility Conditions, spatial discretization is performed using an antisymmetric form.

[0077]

[0078] The spatial discretization of the Navier-Stokes equations for viscous incompressible fluids is then:

[0079]

[0080] D h .u=0

[0081] Step 4: Establish a vector a(x) centered on the edge on the staggered grid. Use traditional velocity interpolation to obtain the vector A(X) at the Lagrange grid points, thus obtaining the divergence-free interpolated velocity U(X). Create a vector a(x) defined on the Eulerian grid points according to the Hellheims decomposition, where a(x) satisfies the following conditions:

[0082]

[0083] Where u0 is the mean velocity, expressed as:

[0084]

[0085] Solving for a(x), we obtain the Poisson equation:

[0086] -L h a = D h ×(u-u0)=D h ×u

[0087] After solving using the Fast Fourier Transform algorithm, the vector A(X) defined at the Lagrange grid points is obtained by interpolating a(x) at the Eulerian grid points to the Lagrange grid points of the immersion boundary using the regularized δ function:

[0088]

[0089] Step 5: Perform divergence-free force diffusion based on the energy conservation relationship between the Lagrange and Euler variables. According to the energy conservation relationship between the Lagrange and Euler variables:

[0090] (u, SF) x =(S * u, F) X

[0091] Discrete can be represented as:

[0092]

[0093] Using the properties of a(x) defined in step 4 and the divergence-free velocity interpolation, it can be rewritten as follows:

[0094]

[0095]

[0096] Apply discrete curl treatment and apply gauge condition D. h ·f FSI =0 yields Poisson's equation for fluid-structure interaction.

[0097]

[0098] The divergence-free diffusion of the Lagrange force can be achieved using the Fast Fourier Transform algorithm.

[0099] Step 6: Assign the obtained relevant flow field values ​​to the next time step, update the flow field, and iterate for the next time step;

[0100] As a preferred technical solution, a Lagrange coordinate system is established for rigid small deformation boundaries to characterize the boundary points;

[0101] As a preferred technical solution, in step 2, an interleaved Cartesian grid is used to establish the flow domain. For the fluid pressure p, it is stored at the center of each Eulerian grid cell, and for the fluid velocity u = (u, v), it is stored at the center of the edge of each fluid Eulerian grid.

[0102] As a preferred technical solution, in step 3, spatial discretization is performed based on the staggered grid from step 2 to obtain the discrete NS equations based on the staggered grid;

[0103] As a preferred technical solution, in step 4, a divergence-free velocity interpolation method is used to establish a vector a(x) centered on the edge, and the vector A(X) on the Lagrange grid points is obtained by using traditional velocity interpolation, thereby obtaining the divergence-free interpolated velocity U(X);

[0104] As a preferred technical solution, in step 5, divergence-free force diffusion is performed based on the energy conservation relationship between the Lagrange variables and the Euler variables.

[0105] A calculation example of flow around a cylinder that can achieve the effects of this invention:

[0106] Table 1 Comparison of drag coefficient, lift coefficient, and Strouhal number when Re = 100

[0107]

[0108] Table 2 Comparison of drag coefficient, lift coefficient, and Strouhal number when Re = 185

[0109]

[0110] Table 3 Comparison of drag coefficient, lift coefficient and Strouhal number when Re = 200

[0111]

[0112] The sources cited in the table are as follows:

[0113] [1]Yang X, Zhang

[0114] [2]Wu X-d,Liu H-p,Chen F.Numerical investigation of flowcharacteristics around two side-by-side cylinders by immersed boundary-lattice Boltzmann flux solver[J].Journal of Zhejiang University-SCIENCE A,2018,19(5):384-398.

[0115] [3]Constant E,Favier J,Meldi M,et al.An immersed boundary method inOpenFOAM:verification and validation[J].Computers& fluids,2017,157:55-72.

[0116] [4]Pinelli A,Naqavi I,Piomelli U,et al.Immersed-boundary methods forgeneral finite-difference and finite-volume Navier-Stokes solvers[J].Journalof Computational Physics,2010,229(24):9073-9091.

[0117] [5]Choi J-I,Oberoi RC,Edwards JR,et al.An immersed boundary methodfor complex incompressible flows[J].Journal of Computational Physics,2007,224(2):757-784.

[0118] [6]Supradeepan K, Roy A. Characterization and analysis of flow over twoside by side cylinders for different gaps at low Reynolds number: a numerical approach[J]. Physics of Fluids, 2014, 26(6): 063602.

[0119] Tables 1, 2, and 3 present the numerical analysis results of the flow around a stationary single cylinder at Reynolds numbers of 100, 185, and 200, respectively. C′ represents the mean of the drag coefficient. L and (C) L ) rms ... L and (C) L ) rms The rigid boundary immersion method shows some discrepancy with the literature (rigid boundary), while the results obtained by using the divergence-free rigid boundary immersion method are in good agreement with the literature results. This indicates that, compared with traditional velocity interpolation and force diffusion, divergence-free velocity interpolation and force diffusion can obtain more accurate results in numerical simulation of rigid boundary problems.

[0120] Finally, it should be noted that the described embodiments are only some embodiments of this application, not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of this application without creative effort are within the scope of protection of this application.

[0121] The foregoing has only described certain exemplary embodiments of the present invention by way of illustration. Undoubtedly, those skilled in the art can modify the described embodiments in various ways without departing from the spirit and scope of the present invention. Therefore, the foregoing drawings and descriptions are illustrative in nature and should not be construed as limiting the scope of protection of the claims of the present invention.

Claims

1. A submarine fluid-structure interaction analysis method based on the divergence-free immersion boundary method, characterized in that, It includes the following steps: Step 1: Collect the 2D submarine's shape and region, along with its fluid pressure and velocity. Establish a mesh based on the 2D submarine's shape and the flow field within the region. Create an interleaved Cartesian mesh within the flow field region, storing the fluid pressure p and velocity components u and v in different Cartesian meshes, where u is the velocity component in the x-direction and v is the velocity component in the y-direction. Set the fluid periodicity computation domain to Ω = [0, L]. 2 Where Ω represents the computational domain; L is the length of the computational domain; the periodic computational domain is uniformly divided into N×N Eulerian grids with a spatial step size h = L / N, where N is the number of grids in a certain direction of the computational domain. For the fluid pressure p, it is stored at the center position of each Eulerian grid, and the center position is x. i，j = ((i+1 / 2)h, (j+1 / 2)h), for the fluid velocity u = (u, v), it is stored at the edge center position of each fluid Eulerian grid, and the position of the fluid velocity in the x-direction is x. i-1 / 2，j = (ih, (j+1 / 2)h), where the position of the fluid velocity in the y-direction is x. i，j-1 / 2 = ((i+1 / 2)h, jh), establish Lagrange coordinates for the submarine hull model; Step 2: Set the initial conditions for numerical simulation. Treat the initial shape of the submarine as a high-stiffness elastic boundary and perform a pure fluid steady simulation. The fluid velocity is constant in the flow domain. Update the flow field velocity on the left side of the flow domain to keep it constant as it flows into the flow domain. Step 3: Spatial discretization of the staggered Cartesian grid of the watershed structure, wherein a central difference operator is constructed: Where: e β (β = 1, 2) is a set of orthogonal bases in two-dimensional space, using orthogonal bases along the x and y directions, e1 = (1, 0), e2 = (0, 1); φ(x) is a scalar function; using the first-order partial derivatives. Central difference operator To define the vector center difference operator D h ,but Due to the vector center difference operator For the Laplace operator in the viscosity term, use D h ·D h The Laplace difference operator L can be defined. h : In the formula: φ(x) is a scalar function; for the finite difference discretization of the convection term, based on the incompressible... The conditions are as follows: spatial discretization is performed using an antisymmetric form, where: Represents the Hamiltonian operator; In the formula: u is the fluid velocity; u β Represent the velocity components in different directions; discretize them and define them as the antisymmetric difference operator S. β : In the formula: The average convection velocity, its components are related to u β Defined on the same grid, taking the average value of u as Discrete results of the Navier-Stokes equations for viscous incompressible fluids can be obtained: D h ·u=0; In the formula: ρ represents fluid density; p represents fluid pressure; μ represents fluid viscosity coefficient; f FSI Represents fluid force; Step 4: To perform divergence-free interpolation, the Poisson equation for a(x) is obtained through Hellholmz decomposition, where a(x) is first set to satisfy the following conditions. In the formula: u0 is the mean velocity. The above conditions are processed by curl and the identity property of vectors is used. D h ×(D h ×u)=D h (D h ·u)-L h u, The resulting equation is -L h a = D h ×(u-u0)=D h ×u, a structure defined on the mesh is obtained by solving the Fast Fourier Transform algorithm. The vector A(X) defined on the Lagrange grid points is obtained by applying the regularized δ function interpolation scheme to a(x), thus from The defined Lagrange velocity field is automatically non-divergent, thus yielding... In the formula: U is the velocity of the Lagrange boundary point obtained by interpolation, x is the Eulerian coordinate, and X is the coordinate of the Lagrange boundary point; δ h For regularization functions; Step 5: Perform divergence-free force diffusion based on the energy conservation between Lagrange and Euler variables, characterized as... In the formula: f FSI For fluid force; F FSI For boundary forces; For the defined grid; U m and F FSI，m Let M be the velocity and fluid-structure interaction force at the m-th Lagrange point; M be the total number of Lagrange points; Δs be the grid spacing; and the discrete vector identity be used. The above formula can be converted to in, Using divergence-free velocity interpolation and supplementary conditions: all In the formula: It is an arbitrary scalar field. The fluid-structure interaction force f acting on the Euler grid is obtained. FSI Poisson's equation for (x): Similarly, the divergence-free diffusion of the Lagrangian force into the fluid mesh is solved using the Fast Fourier Transform algorithm; Step 6: Complete the flow field calculation within the time step. This yields the velocity field information, pressure field information, fluid force on the model, and deformation of the model within one time step. Assign these values ​​to the next time step as initial values, update the flow field and velocity field, and perform calculations for the next time step.

Images