A method of constructing a wave number-frequency spectrum of turbulent fluctuating pressure on a ship surface

Abstract

The present application relates to a kind of methods for constructing ship surface turbulent fluctuating pressure wave number-frequency spectrum, comprising the following steps: S1, first to the object of carrying out calculation carries out numerical modeling.S2, using variational multiscale large eddy simulation method carries out the fine numerical simulation of ship flow field;S3, using equidistant sensor line array obtains ship surface turbulent fluctuating pressure time history data;S4, according to Fourier transform method processing array point turbulent fluctuating pressure information, to obtain ship surface turbulent fluctuating pressure wave number-frequency spectrum;S5, finally to the analysis of migrating ridge feature, wherein, migrating ridge slope, it indicates migration speed, that is, the transport speed of turbulent energy.Based on the specific measures of 64 equidistant line array, it can be used on the real surface of ship body that exists pressure gradient, and the time and space distribution of fluctuating pressure can be accurately calculated or measured by the space correlation of 64 elements, and then the ship surface turbulent fluctuating pressure wave number-frequency spectrum is constructed.

Description

Technical Field

[0001] This invention relates to the field of marine hydrodynamics, and in particular to a method for constructing the pressure wavenumber-frequency spectrum of turbulent fluctuations on the surface of a ship. Background Technology

[0002] As is well known, hydrodynamic noise, along with mechanical noise and propeller noise, constitutes the three main noise sources of ships. The intensity of hydrodynamic noise increases rapidly with increasing speed, and its radiated sound power is proportional to the 5th to 6th power of the speed. At low speeds, its contribution to radiated noise is often masked by mechanical and propeller noise. Generally, at higher speeds, the proportion of hydrodynamic noise in radiated noise increases rapidly. With the effective suppression of mechanical and propeller noise in recent years, the role of hydrodynamic noise has become more prominent. Especially when ships are sailing at high speeds, hydrodynamic noise is the most significant noise source, and its effect is substantial and cannot be ignored.

[0003] Hydrodynamic noise is induced by turbulent pulsating pressure generated by flow around the ship's hull surface. This turbulent pulsating pressure constitutes a noise source in itself, and its pulsating nature also acts as an excitation on the hull structure surface, intensifying vibrations and increasing radiated and cabin noise. Therefore, research on turbulent pulsating pressure and its wavenumber-frequency spectrum is of significant engineering practical importance for assessing and suppressing ship hydrodynamic noise.

[0004] Turbulent pulsating pressure is a fundamental issue in turbulence research. Studying the wavenumber-frequency spectrum of turbulent pulsating pressure can reveal the distribution patterns of turbulent energy and the impact of changes in external conditions on this energy distribution, thus guiding practical engineering applications related to turbulent flow. Reducing hydrodynamic noise from ships necessitates reducing turbulent pulsating pressure. Researching the pulsating pressure within the turbulent boundary layer on ship walls contributes to a deeper understanding of the mechanisms of flow-induced noise generation and provides a more comprehensive understanding of turbulent phenomena, thereby advancing theoretical research in the interdisciplinary field of flow-acoustic coupling. Therefore, research on wall-borne turbulent pulsating pressure and its wavenumber-frequency spectrum is of great significance from both practical application and theoretical research perspectives.

[0005] Currently, research on turbulent fluctuating pressure and its wavenumber-frequency spectrum is mainly based on experimental testing. However, with the increasing maturity of numerical calculation methods and the improvement of computational capabilities, some scholars are also conducting numerical calculations of turbulent fluctuating pressure. In the field of numerical calculation, RANS, LES, and DNS methods are frequently studied and discussed. Large eddy simulation (LES) is a numerical simulation method between DNS and RANS. Its computational cost is significantly lower than direct numerical simulation, but much higher than the Reynolds-averaged method. LES was established by Smagorinsky (1963), who proposed the first subgrid vortex model. Its core idea is to directly solve for large-scale motions in turbulence, while using subgrid stress to characterize the influence of unsolvable scales on solvable scales. Smagorinsky's subgrid vortex model has some shortcomings, such as incorrect asymptotic features near the wall, excessive dissipation of large-scale coherent structures, and the inability to assess backscattering effects. To address this deficiency, various other types of subgrid vortex models emerged in the following decades. However, in essence, they were all developed within the framework of Smagorinsky's ideas, remaining largely rooted in the original subgrid vortex model. The most famous of these is the dynamic Smagorinsky subgrid vortex model, which employs two filtering processes and treats the Smagorinsky constant as a function of time and space. While the dynamic Smagorinsky subgrid vortex model has indeed improved the applicability of large eddy simulation methods in many cases, its computational cost has also increased significantly.

[0006] Hughes et al. (2000-2001) proposed a novel large eddy simulation (LES) method based on the variational multiscale (VMS) theoretical framework. This method offers significant innovation and advantages compared to traditional LES methods. Based on the variational multiscale (VMS) theoretical framework presented in the literature, and building upon previous exploratory work, this application of the present invention independently derived the complete VMS-LES equations and developed a calculation program. Preliminary calculations were performed on the vortex structure and flat plate boundary layer velocity distribution of the SUBOFF submarine model, verifying the reliability of the calculation method.

[0007] Based on this, the present invention proposes a method for constructing the pressure wavenumber-frequency spectrum of turbulent fluctuations on the ship surface based on variational multi-scale large eddy simulation. Summary of the Invention

[0008] To address the shortcomings of existing production technologies, the applicant provides a method for constructing a turbulent pulsating pressure wavenumber-frequency spectrum on a ship surface. Based on specific measures of a 64-element equally spaced linear array (including array spacing, 64-element linear array spatiotemporal Fourier transform technology, etc.), this method can be used on the actual surface of a ship hull where pressure gradients exist. The spatiotemporal correlation of the 64 elements is sufficient to accurately calculate or measure the temporal and spatial distribution of pulsating pressure, thereby constructing a turbulent pulsating pressure wavenumber-frequency spectrum on the ship surface.

[0009] The technical solution adopted in this invention is as follows: 1. A method for constructing the pressure wavenumber-frequency spectrum of turbulent fluctuations on a ship's surface, characterized by comprising the following steps: S1. First, numerical modeling is carried out on the object to be calculated. During the mesh generation process, the mesh near the 64-element equally spaced sensor linear array and near the hull wall is refined.

[0010] S2. Employ variational multiscale (VMS) large eddy simulation method to conduct detailed numerical simulation of the flow field around the ship; S3. Time history data of turbulent pulsating pressure on the ship surface are obtained by using an equally spaced linear array of sensors. S4. Process the turbulent pulsating pressure information at the array points using the Fourier transform method to obtain the wavenumber-frequency spectrum of the turbulent pulsating pressure on the ship surface. S5. Finally, the characteristics of the migrating ridge are analyzed, including the slope of the migrating ridge. , which represents the migration velocity, that is, the transport velocity of turbulent energy.

[0011] Furthermore, the variational multiscale (VMS) large eddy simulation method is used to conduct fine flow field simulation of the ship, obtaining time history data of turbulent fluctuation pressure on the ship surface. The variational multiscale (VMS) large eddy simulation method includes the variational multiscale formula of the Navier-Stokes equations, scale separation, simulation of the effects of unsolvable scales, specific processing methods for the upwind difference scheme, and time discretization. The variational multiscale formula for the Navier-Stokes equations is as follows:

[0012] in, , P, E, , These represent density, velocity, pressure, viscous stress, total energy, thermal conductivity, and temperature, respectively.

[0013] The viscous stress tensor is:

[0014] in, It is viscous. Always ,in This is the specific heat at constant volume. For gases, it is assumed that density, pressure, and temperature conform to the ideal gas law. Generally speaking, it is advisable to take and It is a constant.

[0015] Furthermore, the spacing of the linear array is required to be 3~5mm, and the Re applicability range is (1~3)×10. 7 .

[0016] Furthermore, the method for obtaining the turbulent pulsating pressure on the ship surface employs specific numerical simulation methods, the requirement of an equally spaced sensor linear array, and a data processing method for the wavenumber-frequency spectrum of the turbulent pulsating pressure.

[0017] Furthermore, the unsteady data of the turbulent pulsating pressure on the ship surface are obtained based on numerical simulation results.

[0018] Furthermore, the method for calculating the wavenumber-frequency spectrum of turbulent fluctuating pressure includes using Fourier analysis to achieve the conversion between the time-space and wavenumber-frequency domains, thereby enabling the study of the statistical characteristics of stationary random signals in the wavenumber-frequency domain. The wavenumber-frequency spectrum of turbulent fluctuating pressure is defined as the Fourier transform of the correlation function of the turbulent fluctuating pressure time-space signal in time and space, and its mathematical expression is: (1.16) In practice, the wavenumber-frequency spectrum of turbulent pulsating pressure is usually obtained by performing a fast Fourier transform (FFT) on the discrete spatiotemporal signal of turbulent pulsating pressure and then ensemble averaging the squares of its amplitude

[11] . At this time, the expression of the wavenumber-frequency spectrum of turbulent pulsating pressure is as follows: (1.17) Among them, symbols Represents the expected value. This represents the pulsating pressure measured by the m-th sensor. The number of time nodes. For the number of sensors, For sensor spacing, For time step, (1.18) (1.19) (1.20) and All are window functions; this paper uses the Hanning window. (1.21) This is the window constant.

[0019] Furthermore, the core program for performing a velocity Fourier transform on the turbulent pulsating pressure time history data of a 64-element equally spaced sensor linear array includes:

[0020] The beneficial effects of this invention are as follows: The variational multiscale large eddy simulation method proposed in this invention has the following innovations compared with the widely used traditional large eddy simulation method based on filtering: First, traditional large eddy simulation (LES) methods employ grid filtering, which, limited by the number of grid cells, leads to increasingly higher computational costs and lower computational efficiency. Variational multiscale LES, however, uses variational projection instead of filtering for the Navier-Stokes equations—a significant innovation. While traditional LES methods effectively filter periodic boundary conditions, filtering introduces complex mathematical problems for wall turbulence. In ship hydrodynamics, wall turbulence is often the primary concern, and variational projection avoids these issues. Second, the variational multiscale large eddy simulation method separates the flow scales a priori, or deductively, meaning that the separation is performed before the numerical simulation begins. Different flow scales have strict governing equations, while the traditional large eddy simulation method only has governing equations for large-scale flows and approximates for small-scale flows. Third, in the variational multiscale large eddy simulation method, the unsolvable scale only affects the smallest solvable scale. That is to say, the unsolvable scale term only exists in the smallest solvable scale equation, and does not contain unsolvable scale terms in the larger solvable scale equation. Fourth, in the variational multiscale large eddy simulation method, energy is extracted from the smallest solvable scale, thus exhibiting good asymptotic properties near the wall and having minimal impact on dissipation at larger solvable scales. Fifth, the governing equations in variational multiscale large eddy simulation (LES) are based on original variables, and unsolvable scales only affect the smallest solvable scale. In contrast, the governing equations in traditional LES are based on filtered variables, and unsolvable scales affect all solvable scales. Traditional LES first filters the Navier-Stokes equations, decomposing the flow variables into filtered and fluctuating components (subgrid tensors), posing the challenge of simulating subgrid scales. In variational multiscale LES, filtering is unnecessary; the governing equations are expressed using original variables, thus eliminating the problem of simulating fluctuating variables.

[0021] The method proposed in this application for constructing the pressure wavenumber-frequency spectrum of turbulent fluctuations on the ship surface using a 64-element equally spaced sensor linear array also has the following innovative features: 1. The variational multi-scale large eddy simulation method proposed in this application has the following technical advantages over the currently widely used traditional large eddy simulation methods based on filtering: (1) Variational multiscale large eddy simulation provides a more solid mathematical foundation for turbulence simulation; (2) Variational multiscale large eddy simulation method can be easily extended to complex geometric bodies such as ships. The method itself does not have mathematical problems such as commutativity and homogeneity. These problems need to be dealt with when using traditional large eddy simulation spatial filtering, but they are difficult to solve. (3) In traditional large eddy simulation, the model of constant coefficient Smagorinsky type acts on all scales being simulated. In fact, it is only proven to be effective when it acts on small scales, especially for wall turbulence. The variational multiscale large eddy simulation method handles this in this way. (4) Compared with traditional large eddy simulation methods, especially dynamic subgrid models, variational multiscale large eddy simulation methods simulate more rigorous and concise equations, and the calculation program is easier to solve in parallel, thus the computational efficiency is very high, significantly higher than that of traditional large eddy simulation methods.

[0022] 2. The 64-element equally spaced pressure sensor linear array technology proposed in this application has the following technical advantages over the 48-element equally spaced pressure sensor linear array technology currently possessed by the United States: (1) In the 64-element equally spaced pressure sensor linear array proposed in this application, the spacing between each two pressure sensors is 1 cm, which makes it easier to perform spatiotemporal correlation and more accurately reflect the spatiotemporal characteristics of turbulent pulsating pressure. In contrast, in the 48-element equally spaced pressure sensor linear array in the United States, the spacing between each two pressure sensors is 2.54 cm, which is obviously insufficient in spatial resolution.

[0023] (2) After the 64-element equally spaced pressure sensor linear array obtains the turbulent pulsating pressure, it is necessary to perform spatiotemporal Fourier transform technology. Since the number of sensors increases to 64, the transformation matrix increases significantly, requiring a special matrix load recoding method. This method is different from the American method and significantly improves the calculation accuracy and efficiency. Attached Figure Description

[0024] Figure 1 This is a flowchart of the method of using the present invention.

[0025] Figure 2 This refers to the 64-element equally spaced linear array in this invention.

[0026] Figure 3 This is a diagram illustrating the process of mesh agglomeration into a large mesh in the two-dimensional case of this invention.

[0027] Figure 4 The coarse-scale component in this invention Distribution characteristics (one-dimensional illustration).

[0028] Figure 5 Interpolation point in this invention and Location diagram. Detailed Implementation

[0029] The specific embodiments of the present invention will now be described with reference to the accompanying drawings.

[0030] like Figure 1 As shown, numerical modeling of the computational object is first performed. During mesh generation, the mesh near the 64-element equally spaced sensor linear array and near the hull wall is refined. The linear array spacing is required to be 3~5mm, and the Re value is applicable to a range of (1~3)×10⁻⁶. 7 Then, a detailed numerical simulation of the flow field around the ship was conducted using the established variational multiscale (VMS) large eddy simulation method. Time history data of turbulent fluctuation pressure on the ship surface were obtained using an equally spaced linear array of sensors. The turbulent fluctuation pressure information at the array points was processed using the Fourier transform method to obtain the wavenumber-frequency spectrum of the turbulent fluctuation pressure on the ship surface. Finally, the characteristics of the migrating ridge were analyzed. The slope of the migrating ridge was determined. , which represents the migration velocity, that is, the transport velocity of turbulent energy.

[0031] The following provides a detailed introduction to the theory and formula derivation of the variational multiscale (VMS) large eddy simulation method and Fourier transform method involved in this invention.

[0032] Based on existing fundamental theories, this invention derives and studies various theories, and finally combines them to form the variational multiscale (VMS) large eddy simulation method.

[0033] 1. Large eddy simulation method based on variational multiscale (VMS) 1.7 Variational Multiscale Formula for the Navier-Stokes Equation The conserved form of the Navier-Stokes equations, including the energy equation, is as follows:

[0034] in, , P, E, , These represent density, velocity, pressure, viscous stress, total energy, thermal conductivity, and temperature, respectively.

[0035] The viscous stress tensor is:

[0036] in, It is viscous. Always ,in This is the specific heat at constant volume. For gases, it is assumed that density, pressure, and temperature conform to the ideal gas law. Generally speaking, it is advisable to take and It is a constant.

[0037] Several key processes in the study are described below: (a) Bounded flow region It can be discretized by unstructured tetrahedral mesh, which greatly reduces the difficulty of mesh generation. Based on this, a dual mesh can be obtained by defining mesh cells and control volumes. (b) Discretize the flux using the finite volume method (FV), and define the variables within each control volume. Treat it as a constant. ,in For nodes Corresponding control body The characteristic function on, For variables This is a constant value on the control volume. This first-order approximation can be achieved using the MUSCL method. [1、2] It is transformed into a semi-discrete form of the sixth derivative, which will be introduced in later chapters.

[0038] (c) Dissipative fluxes are discretized using the finite element method (FE), and the flow variables are... It is approximated by using continuous piecewise linear functions. ,in For nodes Related Shaped functions (linear functions) express At the node The value on.

[0039] The discretization method used in this paper is called the finite volume / finite element hybrid method (hybrid FV / FE). The advantage of using the hybrid method is that, for both convection and diffusion operators, a step-by-step technique is employed, using the most suitable method for each. In the first step, an upwind discretization scheme is used for the convection operator, and in the second step, a lumped-mass finite element method is used for the diffusion operator. It should be noted that the basic methods used in deriving the variational multiscale large eddy simulation method in this paper can be directly applied to the complete finite element method or the complete finite difference method; that is, the basic derivation in this paper is not limited by the discretization method.

[0040] Discretizing the Navier-Stokes equations using the hybrid FV / FE method yields a system of equations in the following form: (1.1) in, superscript This indicates transpose.

[0041] in, Indicates branches The boundary, Represents the outward normal vector of the boundary.

[0042] make and They represent the characteristic functions respectively. The space formed and Shape function The space constitutes the composition. To separate the coarse and fine scales, the above space is decomposed as follows: (1.2) (1.3) The superscript line in this article ( () indicates a coarse scale, and the superscript () indicates a coarse scale. ) indicates a finer scale. Similarly, relative to formulas (1.2) and (1.3), Also perform the above decomposition: (1.4) When the decomposition methods represented by (1.2) and (1.3) are applied to the system of equations (1.1), the following conventions must be followed. Only when Multiply or hour, Only then does it need to be decomposed into coarse and fine scales. If Multiply or At that time, decomposition is unnecessary, in order to avoid producing [products / destructive processes]. and There is no need to simulate higher-order small terms. Therefore, the problem represented by the system of equations (1.1) is transformed into the following two subproblems: (1.5) (1.6) in,

[0043] (1.7)

[0044]

[0045]

[0046]

[0047] (1.8)

[0048]

[0049]

[0050]

[0051]

[0052] (1.9) Generally speaking, ,so .

[0054] and, ,so .

[0055] However, in order to simulate only the effect of unsolvable scales on the smallest solvable scale, we still use replace ,use replace For air, this effect is negligible; for water, these issues are completely irrelevant.

[0056] The system of equations (1.5) can be written as:

[0057]

[0058] (1.10)

[0060]

[0061] (1.11) Equation set (1.10) controls the coarse scale, and equation set (1.11) controls the fine scale. These two sets of equations are coupled, and the mechanism for separating these scales will be introduced below.

[0063] 1.8 Scale Separation According to reference [3], the definition is as follows: for exist The projection on the surface.

[0064] (1.12) (1.13) Define a simple and efficient projection operator This is the main purpose of this section.

[0065] Mesh agglomeration is a key aspect of this method. For any given tetrahedral mesh, a dual mesh defined by mesh elements and control volumes can be obtained. Such dual meshes can be fused into large meshes through the mesh agglomeration process given in reference [4]. The specific agglomeration operations and related theories are quite extensive, so please refer to reference [4] for details, which will not be elaborated here. Figure 3 The mesh condensation process in the two-dimensional case is given. The mesh condensation process in reference [4] has the following characteristics: 1. The mesh around a certain corner point is merged into one in a certain form; 2. The coarsened mesh is no longer the classic finite element (triangle, tetrahedron) form, but a more general finite volume partition form; 3. Spatial approximation is given at each level.

[0066] After aggregating the mesh into a larger mesh, it is necessary to define the coarse-scale component in the dissipation term. Its meaning is Spatial average on a large grid.

[0067] (1.14) in, (1.15) For the surrounding corner point Grid cells, Indicate its volume, , Indicates the presence of grid cells A large grid. The specific meaning is explained as follows: Consider including corner points All triangular meshes (2D) are divided into six smaller triangles using the three medians of each triangle, with all triangles surrounding the corner points. Small triangles combined together are The same principle applies to three dimensions.

[0068] According to equations (1.14) and (1.15), we can... The expression is as follows: (1.16) (1.17) Given Shape function After considering the characteristics, the coarse-scale component will then be... An approximation is made using a continuous function, which is a uniformly invariant constant within each cell contained in a large grid, and linearly distributed within adjacent cells of two large grids. Therefore, It can be viewed as a piecewise constant function with linear connectivity. See Figure 4 .

[0069] (1.18) The averaging process described above can also be used to define the coarse-scale component in the convection term. Among them, use This can be replaced. Therefore, for the convection term, The definition is as follows: (1.19) For dissipation terms, The definition is as follows: (1.20) In both cases, All satisfied ,therefore This is the projection operator.

[0070] That's it. as well as The two definitions of decomposition, in which, or Represented by function or To approximate the coarse-scale space, or Represented by function or To approximate the space at fine scales. If Given the total number of nodes in a grid, If the total number of large grids after agglomeration processing is , then or The dimension is , or The dimension is .

[0071] 1.9 Simulation of the Influence of Unsolvable Scales The turbulent energy transfer process can be described by an "energy level grid," where energy is transferred from large eddies to smaller eddies, level by level, until it is dissipated as heat and eventually disappears. Therefore, it can be assumed that energy transfer occurs, in most cases, between adjacent scales: from larger solvable scales to smaller solvable scales, and from smaller solvable scales to unsolvable scales. Consequently, it can be assumed that, since unsolvable scales are far from larger solvable scales, the influence of unsolvable scales on larger solvable scales is negligible compared to the influence of unsolvable scales on smaller solvable scales.

[0072] In equation (1.11), the integral term This can be compared to the Reynolds stress term in the classical large eddy simulation method, therefore it can be... This can be considered as a fine-scale Reynolds stress term. Similarly, the following equation can be... It is considered as a fine-scale Reynolds enthalpy flow rate.

[0073] Fine-scale Reynolds stresses cause rapid pulsations at the fine scale, which are ultimately dissipated by molecular viscosity. The effect of this stress tensor cannot be accurately captured during discretization unless all scales are solved, as in DNS, but this defeats the purpose of constructing a VMS-LES. For this reason, its indirect effect on energy dissipation is addressed through a generalization of the Smagorinsky eddy viscosity model, which includes only the fine scale and considers only its governing equations. Furthermore, it should be noted that the integral term... Adding only to the fine-scale momentum equation in equation system (1.11), using the following relationship,

[0074] , , Indicates the local grid scale. In this report, Set per unit The cube root of the volume (1.21) Similarly, the fine-scale Reynolds entropy flow rate is first simulated by analogy to the Smagorinsky eddy viscosity model, but only at the fine scale; this is... Then this term is added to the fine-scale energy equation in equation system (11). For isobaric specific heat, Let be the Prandtl number at the subgrid scale, assumed to be a constant.

[0075] On the other hand, the energy equations that appear at fine scales This term has a small effect at high Reynolds numbers and low Mach numbers, and therefore does not need to be simulated.

[0076] Using the simulation method described above, equations (1.10) and (1.11) will be transformed into the following system.

[0077]

[0078] (1.22) Note that when dealing with laminar flow, the equations in equation (1.22) , Therefore, in the Smagorinsky subgrid vortex model... , , , ,in This represents the Reynolds stress tensor in the usual sense. This represents Smagorinsky eddy viscosity.

[0080] 1.10 Specific processing method of the windward difference scheme As mentioned earlier, the convection term is semi-discretized using the FV method. This involves discretizing Roe... [5] The proposed method is similar to MUSCL. [1、2] The methods are used together, in the MUSCL method, by using the 6th derivative. [6] Stability is achieved through numerical dissipation. At high frequencies, this dissipation exhibits a very significant local effect, which reduces the mutual interference between it and sub-grid-scale simulations. [7] This improved numerical dissipation method is an important component of the final numerical algorithm for LES because it reconciles the contradiction between artificial dissipation and simulated dissipation.

[0081] make Representing corner points Let be the set of adjacent nodes. (1.23) Convection Flow

[0082] Through numerical flow To approximate the process, where , (1.24) in, For real coefficients, introduce This is to directly control the windward term and its corresponding numerical dissipation. The windward term is given by the following formula: (1.25) in, Represents the Roe matrix [5] State parameters and for At the border Reconstructed values ​​on, Indicates that the center is at the node. and The common boundary of the two grids.

[0083]

[0084]

[0085] gradient Calculated by the following formula, (1.26) in Indicates the upwind grid gradient in (see) Figure 5 ), Indicates downwind grid gradient in For unit The nodal gradient, For unit The nodal gradient, Represents the central gradient. , and Corner point In the grid and The foot of the perpendicular on the corresponding plane (see Figure 5 ), For point The gradient at a given point can be calculated by interpolating the gradients at the three surrounding nodes.

[0087] gradient Do the same treatment.

[0088] To obtain a stable windward difference term, the windward difference parameters are set as follows, please refer to reference [6]: , ,

[0089] Generally speaking, , The choice of is quite complex. It should be chosen to be as small as possible to reduce the mutual interference between artificial dissipation and simulated dissipation; at the same time, it should be as large as possible to produce the minimum numerical dissipation that can eliminate non-physical pressure oscillations.

[0090] 1.11 Time Discreteness After semi-discretization, the simulation system (1.22) can be applied to each control volume. Write it in the following form, (1.27) in, The combination of semi-discrete convection flows is represented by equations (1.24)-(1.26)). (1.28) This represents the combination of semi-discrete laminar diffusion fluxes. (1.29) (1.30) and It can be done using ordinary shape functions. To express (see equations (1.17) and (1.18)), all All items can be evaluated using standard finite element analysis, in which the combination relationship between the mesh and the larger mesh must be considered. A loop must be performed on the tetrahedral mesh, for each tetrahedron... The large grid containing its corner points needs to be addressed one by one. If it is a tetrahedron... Corner Belongs to large grid Then tetrahedron Participate in each and every part of the large grid nodes Connected The calculation of the term (see formula (1.30)).

[0091] Explicit time integration can be performed directly on (1.27). This paper adopts the following fourth-order Runge-Kutta algorithm.

[0092]

[0093] in, .

[0094] Implicit integration of (1.27) generally requires the Jacobian matrix to be considered. , as well as Linearization is performed. Constructing a second-order approximation of these three Jacobian matrices is a very complex but very useful task (see reference [8]). Loss correction techniques can also be used. [9] In this technique, a linearization method based on a first-order approximation is used to construct each Jacobian matrix, or the Newton-Krylov method can be used.

[10] In this method, the Jacobian matrix is ​​approximated using a second-order finite difference technique based on flow rate, and the synthesized linear equations are solved at each time step. A simpler computational approach involves employing a semi-discrete method, where the turbulence dissipation term is solved at each time step. Both must be evaluated, and (1.27) is discretized using the following formula. (1.31) 2. Calculation method of turbulent pulsating pressure wavenumber-frequency spectrum Fourier analysis is used to achieve the transformation between the time-space and wavenumber-frequency domains, thereby enabling the study of the statistical properties of stationary random signals in the wavenumber-frequency domain. The wavenumber-frequency spectrum of turbulent fluctuating pressure is defined as the Fourier transform of the correlation function of the turbulent fluctuating pressure time-space signal in time and space, and its mathematical expression is: (1.16) In practice, the wavenumber-frequency spectrum of turbulent fluctuating pressure is usually obtained by performing a Fast Fourier Transform (FFT) on the discrete-time signal of turbulent fluctuating pressure and then taking an ensemble average of the squares of its amplitude.

[11] At this point, the expression for the turbulent fluctuation pressure wavenumber-frequency spectrum is as follows: (1.17) Among them, symbols Represents the expected value. This represents the pulsating pressure measured by the m-th sensor. The number of time nodes. For the number of sensors, For sensor spacing, For time step, (1.18) (1.19) (1.20) and All are window functions; this paper uses the Hanning window. (1.21) This is the window constant.

[0095] The following is the core program for performing a velocity Fourier transform on the turbulent pulsating pressure time history data of a 64-element equally spaced sensor linear array:

[0096] In summary, the proposed method for constructing the pressure wavenumber-frequency spectrum of turbulent fluctuations on the ship surface is highly innovative and advanced. Its innovation is primarily reflected in the variational multi-scale large eddy simulation method and the 64-element equally spaced sensor linear array employed.

[0097] The variational multiscale large eddy simulation method proposed in this invention has the following innovations compared with the widely used traditional large eddy simulation method based on filtering: First, traditional large eddy simulation (LES) methods employ grid filtering, which, limited by the number of grid cells, leads to increasingly higher computational costs and lower computational efficiency. Variational multiscale LES, however, uses variational projection instead of filtering for the Navier-Stokes equations—a significant innovation. While traditional LES methods effectively filter periodic boundary conditions, filtering introduces complex mathematical problems for wall turbulence. In ship hydrodynamics, wall turbulence is often the primary concern, and variational projection avoids these issues. Second, the variational multiscale large eddy simulation method separates the flow scales a priori, or deductively, meaning that the separation is performed before the numerical simulation begins. Different flow scales have strict governing equations, while the traditional large eddy simulation method only has governing equations for large-scale flows and approximates for small-scale flows. Third, in the variational multiscale large eddy simulation method, the unsolvable scale only affects the smallest solvable scale. That is to say, the unsolvable scale term only exists in the smallest solvable scale equation, and does not contain unsolvable scale terms in the larger solvable scale equation. Fourth, in the variational multiscale large eddy simulation method, energy is extracted from the smallest solvable scale, thus exhibiting good asymptotic properties near the wall and having minimal impact on dissipation at larger solvable scales. Fifth, the governing equations in variational multiscale large eddy simulation (LES) are based on original variables, and unsolvable scales only affect the smallest solvable scale. In contrast, the governing equations in traditional LES are based on filtered variables, and unsolvable scales affect all solvable scales. Traditional LES first filters the Navier-Stokes equations, decomposing the flow variables into filtered and fluctuating components (subgrid tensors), posing the challenge of simulating subgrid scales. In variational multiscale LES, filtering is unnecessary; the governing equations are expressed using original variables, thus eliminating the problem of simulating fluctuating variables.

[0098] The method proposed in this application for constructing the pressure wavenumber-frequency spectrum of turbulent fluctuations on the ship surface using a 64-element equally spaced sensor linear array has the following innovations: Currently, researchers in the field of ship hydrodynamics, both domestically and internationally, generally use single-point pressure sensors to calculate or measure the turbulent pulsating pressure on the surface of ships in water. Only the United States possesses the technology to measure and construct the pulsating pressure wavenumber-frequency spectrum of a flat plate using a 48-element equally spaced pressure sensor linear array in a silencing tank. This technology is mainly used for mechanistic studies and is geared towards simple bodies without pressure gradients, such as flat plates; it cannot yet be applied to real ship surfaces. The specific measures proposed in this application based on a 64-element equally spaced linear array (including array spacing and 64-element spatial-temporal Fourier transform technology) can be used on real ship hull surfaces with pressure gradients. The 64-element spatial-temporal correlation is sufficient to accurately calculate or measure the temporal and spatial distribution of pulsating pressure, thereby constructing the turbulent pulsating pressure wavenumber-frequency spectrum of the ship surface.

[0099] III. Advantages of the Invention 1. The variational multi-scale large eddy simulation method proposed in this application has the following technical advantages over the currently widely used traditional large eddy simulation methods based on filtering: (1) Variational multiscale large eddy simulation provides a more solid mathematical foundation for turbulence simulation; (2) Variational multiscale large eddy simulation method can be easily extended to complex geometric bodies such as ships. The method itself does not have mathematical problems such as commutativity and homogeneity. These problems need to be dealt with when using traditional large eddy simulation spatial filtering, but they are difficult to solve. (3) In traditional large eddy simulation, the model of constant coefficient Smagorinsky type acts on all scales being simulated. In fact, it is only proven to be effective when it acts on small scales, especially for wall turbulence. The variational multiscale large eddy simulation method handles this in this way. (4) Compared with traditional large eddy simulation methods, especially dynamic subgrid models, variational multiscale large eddy simulation methods simulate more rigorous and concise equations, and the calculation program is easier to solve in parallel, thus the computational efficiency is very high, significantly higher than that of traditional large eddy simulation methods.

[0100] 2. The 64-element equally spaced pressure sensor linear array technology proposed in this application has the following technical advantages over the 48-element equally spaced pressure sensor linear array technology currently possessed by the United States: (1) In the 64-element equally spaced pressure sensor linear array proposed in this application, the spacing between each two pressure sensors is 1 cm, which makes it easier to perform spatiotemporal correlation and more accurately reflect the spatiotemporal characteristics of turbulent pulsating pressure. In contrast, in the 48-element equally spaced pressure sensor linear array in the United States, the spacing between each two pressure sensors is 2.54 cm, which is obviously insufficient in spatial resolution.

[0101] (2) After the 64-element equally spaced pressure sensor linear array obtains the turbulent pulsating pressure, it is necessary to perform spatiotemporal Fourier transform technology. Since the number of sensors increases to 64, the transformation matrix increases significantly, requiring a special matrix load recoding method. This method is different from the American method and significantly improves the calculation accuracy and efficiency.

[0102] The above description is an explanation of the present invention and not a limitation thereof. The scope of the present invention is defined by the claims. Within the scope of protection of the present invention, any form of modification may be made.

Claims

1. A method for constructing the turbulent pulsed pressure wavenumber-frequency spectrum of a ship's surface, characterized in that, Includes the following steps: S1. First, numerical modeling is carried out on the object to be calculated. During the mesh generation process, the mesh near the 64-element equally spaced sensor linear array and near the hull wall is refined. S2. Employ variational multi-scale large eddy simulation (LES) to conduct detailed numerical simulations of the flow field around a ship. S3. Time history data of turbulent pulsating pressure on the ship surface are obtained by using an equally spaced linear array of sensors. S4. Process the turbulent pulsating pressure information at the array points using the Fourier transform method to obtain the wavenumber-frequency spectrum of the turbulent pulsating pressure on the ship surface. S5. Finally, the characteristics of the migrating ridge are analyzed, including the slope of the migrating ridge. , which represents the migration velocity, that is, the transport velocity of turbulent energy; the variational multi-scale large eddy simulation method is used to carry out fine flow field simulation of ships and obtain time history data of turbulent pulsating pressure on the ship surface; The variational multiscale large eddy simulation method includes the variational multiscale formula of the Navier-Stokes equations, the scale separation method, the simulation of the effects of unsolvable scales, the specific processing method of the upwind difference scheme, and time discretization. The variational multiscale formula for the Navier-Stokes equations is as follows: in, , P, E, , These represent density, velocity, pressure, viscous stress, total energy, thermal conductivity, and temperature, respectively. The viscous stress tensor is: in, It is viscous. Always ,in The specific heat is taken at constant volume; for gases, it is assumed that density, pressure, and temperature conform to the ideal gas law. ;Pick and The value is constant; the spacing of the linear array is required to be 3~5mm, and the applicable range of Re is (1~3)×10. 7 The method for obtaining the turbulent pulsating pressure on the ship surface employs specific numerical simulation methods, the requirement of an equidistant sensor linear array, and a data processing method for the wavenumber-frequency spectrum of the turbulent pulsating pressure. The unsteady data of the turbulent pulsating pressure on the ship surface are obtained based on numerical simulation results. The calculation method for the wavenumber-frequency spectrum of the turbulent pulsating pressure includes using Fourier analysis to achieve the conversion between the time-space and wavenumber-frequency domains, thereby enabling the study of the statistical characteristics of stationary random signals in the wavenumber-frequency domain. The wavenumber-frequency spectrum of the turbulent pulsating pressure is defined as the Fourier transform of the correlation function of the turbulent pulsating pressure time-space signal in time and space, and its mathematical expression is: （1.16） In practice, the wavenumber-frequency spectrum *l* of turbulent fluctuating pressure is obtained by performing a Fast Fourier Transform on the discrete-time signal of turbulent fluctuating pressure and then ensemble averaging the squares of its amplitudes. The expression for the wavenumber-frequency spectrum of turbulent fluctuating pressure is as follows: （1.17） Among them, symbols Represents the expected value. This represents the pulsating pressure measured by the m-th sensor. For the number of time nodes, For the number of sensors, For sensor spacing, For time step, （1.18） （1.19） （1.20） and All are window functions; this paper uses the Hanning window. （1.21） This is the window constant.

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