Haar wavelet numerical simulation method for hypersonic flow

Abstract

The application provides a Haar wavelet numerical simulation method for hypersonic flow, comprising: obtaining a calculation domain, boundary conditions and a flow field conservation variable vector at a current time; in the calculation domain, based on Haar wavelet base functions and the boundary conditions, determining a Haar matrix, a first integral matrix after correction of the Haar wavelet base functions and a second integral matrix after correction; using the boundary conditions and the second integral matrix after correction, solving a coefficient vector of the corresponding Haar matrix; according to the first integral matrix after correction and the coefficient vector of the Haar matrix, determining a first derivative; according to the Haar matrix and the coefficient vector of the Haar matrix, calculating a second derivative; based on the first derivative and the second derivative, determining a spatial discrete operator; using a target time advancing method, combining the spatial discrete operator, and updating the hypersonic flow numerical value. Through the application, the stability and reliability of the hypersonic flow simulation are improved.

Description

Technical Field

[0001] This invention belongs to the field of computational fluid dynamics numerical simulation technology, specifically relating to a Haar wavelet numerical simulation method for hypersonic flow. Background Technology

[0002] Hypersonic flow is of significant research value in aerospace vehicle design and reentry processes. Its flow field often exhibits multi-scale characteristics such as strong shock waves, expansion waves, shear layers, and boundary layers. The shock wave thickness is typically much smaller than the reference scale, and the flow field physical quantities undergo drastic abrupt changes at the shock wave location, exhibiting discontinuous characteristics. Therefore, numerical simulations of hypersonic flow must accurately capture the shock wave location while maintaining overall accuracy; otherwise, numerical instability or distorted results may occur.

[0003] Existing finite difference and finite volume methods typically require the introduction of artificial viscosity or filters to maintain stability when dealing with shock waves. This inevitably leads to numerical dissipation and reduces solution accuracy. While spectral methods and spectral element methods offer high accuracy in smooth regions, they suffer from the Gibbs phenomenon when encountering discontinuities. This phenomenon manifests as non-physical oscillations near the shock wave, affecting convergence and stability. Figure 1 As shown, the first derivative (a) and second derivative (b) of the Heaviside step function are presented. The blue line represents the result of the HWM method, and the red line represents the result of the Chebyshev method. It can be clearly seen from the figure that the Chebyshev method produces severe Gibbs oscillations at discontinuous locations. In addition, most existing wavelet methods rely on the assumption of solution smoothness, which still produces oscillations when dealing with strong discontinuities in hypersonic flows, lacking stability and universality.

[0004] Therefore, existing methods cannot simultaneously guarantee computational efficiency while maintaining high accuracy and stability, especially in hypersonic flow problems dominated by strong shock waves. There is an urgent need for a numerical simulation method that can accurately capture shock waves while avoiding numerical oscillations. Summary of the Invention

[0005] In view of this, the purpose of this invention is to provide a Haar wavelet numerical simulation method for hypersonic flow to meet the need to improve the stability and reliability of hypersonic flow simulation.

[0006] To achieve the above objectives, the present invention provides the following technical solution: According to a first aspect, the present invention provides a Haar wavelet numerical simulation method for hypersonic flow, comprising: obtaining a computational domain, boundary conditions, and the current flow field conserved variable vector; within the computational domain, determining the Haar matrix, the corrected first integral matrix of the Haar wavelet basis functions, and the corrected second integral matrix based on the Haar wavelet basis functions and boundary conditions; solving for the coefficient vector of the corresponding Haar matrix using the boundary conditions and the corrected second integral matrix; determining the first derivative of the flow field conserved variable vector in space based on the corrected first integral matrix and the coefficient vector of the Haar matrix; calculating the second derivative of the flow field conserved variable vector in space based on the Haar matrix and the coefficient vector of the Haar matrix; determining a spatial discrete operator to describe the spatial changes of the flow field based on the first and second derivatives; and updating the hypersonic flow numerical values ​​using a target time-progression method combined with the spatial discrete operator.

[0007] According to a second aspect, embodiments of the present invention provide an electronic device, the device comprising: a memory, a processor, and a computer program stored in the memory and executable on the processor, the processor performing the steps of the Haar wavelet numerical simulation method for hypersonic flow described in the first aspect or any embodiment of the first aspect.

[0008] This embodiment provides a Haar wavelet numerical simulation method for hypersonic flows. Through a workflow of "location points - integral matrix - Haar coefficient solution - derivative calculation - time progression," it achieves high-precision numerical simulation of hypersonic flows. The piecewise nature of the Haar wavelet ensures stability and accuracy in regions with strong discontinuities such as shock waves, while the introduction of the modified integral matrix allows boundary conditions to be naturally integrated into the numerical scheme, avoiding the additional boundary processing required in traditional methods. This makes the entire simulation process more compact and efficient in both mathematical structure and physical implementation.

[0009] Furthermore, the method proposed in this embodiment fully utilizes the piecewise constant characteristics and tight support of Haar wavelets, enabling numerically stable capture of strong discontinuities such as shock waves and contact discontinuities. Compared to traditional spectral methods, this embodiment eliminates the Gibbs phenomenon at discontinuities naturally without the need for additional limiters or artificial viscosity, ensuring the physical consistency and numerical stability of the calculation results. Simultaneously, this method exhibits excellent resolution in strong discontinuity regions, accurately characterizing shock wave structures, thereby significantly improving the stability and reliability of hypersonic flow simulations.

[0010] Other advantages, objectives, and features of the invention will be set forth in the following description and will be apparent to those skilled in the art in some respects, or may be learned by practice of the invention. The objectives and other advantages of the invention can be realized and obtained through the following description. Attached Figure Description

[0011] To make the objectives, technical solutions, and beneficial effects of this invention clearer, the following figures are provided for illustration: Figure 1 The experimental results of the HWM method and Chebyshev method in the background art of this invention are shown in the figure. Figure 2 This is a flowchart illustrating a specific example of a Haar wavelet numerical simulation method for hypersonic flow in this invention. Figure 3 This is a schematic diagram of the structural composition of a Haar wavelet numerical simulation method for hypersonic flow according to the present invention. Figure 4 The figures show the experimental results of the first three Haar wavelets when the resolution level J=7 in this invention; Figure 5 This is a schematic block diagram of a specific example of an electronic device in an embodiment of the present invention. Detailed Implementation

[0012] The technical solution of the present invention will now be clearly and completely described with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. 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.

[0013] In the description of this invention, it should be noted that, unless otherwise explicitly specified and limited, the terms "installation," "connection," and "linking" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; they can also refer to the internal connection of two components; and they can refer to a wireless connection or a wired connection. Those skilled in the art can understand the specific meaning of the above terms in this invention based on the specific circumstances.

[0014] Furthermore, the technical features involved in the different embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0015] This invention provides a Haar wavelet numerical simulation method for hypersonic flows, such as... Figure 2As shown, it includes: S101, obtain the computational domain, boundary conditions, and the current flow field conservation variable vector; S102, within the computational domain, based on the Haar wavelet basis functions and boundary conditions, determine the Haar matrix, the first integral matrix after the correction of the Haar wavelet basis functions, and the second integral matrix. S103, using the boundary conditions and the corrected quadratic integral matrix, solve for the coefficient vector of the corresponding Haar matrix; S104. Based on the corrected first integral matrix and the coefficient vector of the Haar matrix, determine the first derivative of the flow field conserved variable vector in space; based on the Haar matrix and the coefficient vector of the Haar matrix, calculate the second derivative of the flow field conserved variable vector in space. S105, based on the first and second derivatives, determines the spatial discrete operator used to describe the spatial changes of the flow field; S106 employs a target time advancement method combined with spatial discretization operators to update hypersonic flow values.

[0016] For example, the core structure of this embodiment consists of a wavelet basis function construction module, an integral matrix and correction matrix solving module, a spatial derivative calculation module, and a time propagation module. The entire method expands the flow field variables within the computational domain by placing points and utilizes the piecewise constant characteristics of the Haar wavelet to accurately capture strong discontinuities. Its structural composition consists of... Figure 3 As shown.

[0017] In this embodiment, the computational domain differs for different flow field calculation scenarios, including the flow field around a hypersonic vehicle and the flow field in the thermal protection region of a reentry vehicle. This embodiment defines the computational domain as the spatial range used to solve for hypersonic flow field variables during numerical simulation, and it can be manually defined. For example, when simulating the flow field around a hypersonic vehicle, the computational domain can be defined as the region covering the surface of the vehicle.

[0018] After obtaining the computational domain, boundary conditions, and the current flow field conservation variable vector, within the computational domain, based on the Haar wavelet basis functions and boundary conditions, the Haar matrix, the first integral matrix after correction of the Haar wavelet basis functions, and the second integral matrix are determined. This includes: determining the location of the placement points within the computational domain based on a pre-defined resolution level; determining the Haar wavelet basis functions based on the location of the placement points within the computational domain; constructing the Haar matrix, the first integral matrix, and the second integral matrix of the Haar wavelet basis functions based on the Haar wavelet basis functions; and correcting the first integral matrix and the second integral matrix according to the boundary conditions to obtain the corrected first integral matrix and the second integral matrix.

[0019] Specifically, in terms of the spatially discrete structural composition, firstly, a standard interval [0,1] is selected. There are 1 configuration point, the location of which is given by the following formula: (1) in, , The resolution level determines the calculation accuracy. This value is a pre-set calculation parameter. Figure 4 These are the first three Haar wavelets at resolution level J=7. Based on these location points, Haar wavelet basis functions are constructed. The integral forms of H, P, and Q are obtained, which are the basic building blocks of this method. It should be noted that the process of obtaining H, P, and Q is prior art known to those skilled in the art and will not be elaborated upon here.

[0020] In terms of working principle, the conserved variables of the flow field are determined, and the vector of conserved variables of the flow field is defined. S : S (2) in, For fluid density, , , w are respectively x, y, z Momentum in the three Cartesian coordinate directions, E is the total energy per unit volume (including internal and kinetic energy). Velocity components of the flow field. u, v, w It is possible u = / , v = / , w = / Extraction. In addition to the conserved variable form, the flow field can also be characterized by primitive variables (velocity, pressure, temperature), which can be selected by those skilled in the art as needed.

[0021] Next, the flow field conserved variable vector S is projected into the Haar basis function space to establish the antiderivative (flow field variable function) and Haar coefficients. The relationship between the two functions. The second derivative of the function at the locus of points can be expressed as: (3) in, f The vector represents the conserved variables of the flow field to be solved. S For ease of expression, a component in equation (2) is referred to as a flow field variable function in this embodiment, for example: fIt can be taken as the fluid density. It can also be taken as x Directional momentum density , y Directional momentum density , z Directional momentum density Or, the total energy density E, where H is the Haar matrix. x Represents standardized computational space coordinates. This is the Haar coefficient vector.

[0022] By integrating twice, we can obtain the conserved variables of the flow field to be solved with respect to the coordinates in the computation space. x The reconstructed form of the distribution function: (4) in, f(x) Represents the conserved variables of the flow field to be solved with respect to spatial coordinates. x The distribution function, also known as the flow field variable function, is determined by the boundary conditions. A and B are integration constants. Determine that Q represents the quadratic integral matrix, and then correct the quadratic integral matrix according to the boundary conditions to obtain the corrected quadratic integral matrix. : (5) Simultaneously, the first-order integral matrix P can be corrected according to the boundary conditions to obtain the corrected first-order integral matrix. : (6) Then, using the boundary conditions and the corrected quadratic integral matrix... Solve for the coefficient vector a of the corresponding Haar matrix: (7) After obtaining the coefficients, the first and second derivatives can be recovered.

[0023] After calculating a and Under the premise of [condition], the first derivative of the flow field conservation variable vector in space is determined based on the corrected first integral matrix and the coefficient vector of the Haar matrix, as shown in the following formula: (8) Next, based on the Haar matrix and its coefficient vector, the second derivative of the flow field conservation variable vector in space is calculated. The calculation of the second derivative is directly given by equation (3). In this way, the antiderivative and derivatives of each order can be obtained simultaneously without adding extra boundary treatment, realizing the unified calculation of the convection and viscous terms in the governing equations.

[0024] Based on the above derivation, for the three-dimensional compressible Navier-Stokes equations, in practical applications, to distinguish it from the above derivation process, the computational space used in actual use... Alternative (x,y,z) And the above process uses f This represents the flow field variable function, which in the actual process... f What it actually means is S To express each specific flow field conservation variable in the vector, the following processing is performed on each variable in the S vector, hereinafter referred to as... S Replace the above f。

[0025] In computational space Construct the corresponding modified quadratic integral matrix in each of the three directions. and the Haar matrix The Haar coefficient vector and first and second derivatives are obtained by repeating the above derivation process. Conserved variables are... The first and second derivatives in the direction are as follows: ; (9) in, The vector of conserved flow field variables is represented in space. First derivative in the direction, In order to be in The corresponding Haar coefficient vector in the direction, These are the left and right boundary values. Indicates in The first-order integral matrix after directional correction; The vector of conserved flow field variables is represented in space. Second derivative in the direction, In order to be in Haar matrix in the direction, In order to be in The coefficient vector of the Haar matrix in the direction.

[0026] right The direction is treated in the same way, and the result is transformed into physical space through coordinate transformation to complete the calculation of the three-dimensional derivative. The above formula (9) corresponds to formula (8) and formula (3) in the above derivation process, respectively.

[0027] In the time-progression part, the hypersonic flow values ​​at the next time step are determined based on the third-order Runge-Kutta method and combined with spatial discretization operators. The time interval between the current time step and the next time step satisfies the CFL stability condition, specifically: A third-order Runge-Kutta method is adopted as the core structure. Let... Let be the time step, and let be the spatial discrete operator. The update process at each time step is as follows: , (10) .

[0028] in, Indicates the time step. This represents the first intermediate step of Runge-Kutta. This represents the second intermediate step of the Runge-Kutta. This represents the conserved flow field variables at the nth time step. Let represent the updated solution at time step (n+1). This indicates that a fixed spatial discrete operator acts at time step n. superior, This indicates that a fixed spatial discrete operator is applied in the first intermediate step. superior, This indicates that the fixed spatial discrete operator acts on the second intermediate step. .

[0029] The third-order Runge-Kutta method completes time advancement through three sub-steps: the first step calculates the intermediate steps. The second step is to calculate the intermediate steps. The third step yields the updated conserved variables. After completing these three steps, the evolution from the nth time step to the (n+1)th time step is achieved. L itself is a fixed spatial discrete operator, which only acts on S at different times.

[0030] The time step satisfies the CFL stability condition: (11) in, The spatial step size is in three directions. u,v,w ) represents the fluid velocity in three spatial directions, | u |+ c The characteristic velocity in the x-direction (the sum of convection velocity and the speed of sound). The characteristic velocity in the y-direction is represented. The characteristic velocity in the z-direction is represented by CFL, where CFL is the stability coefficient and c is the local speed of sound.

[0031] This embodiment provides a Haar wavelet numerical simulation method for hypersonic flows. Through a workflow of "location points - integral matrix - Haar coefficient solution - derivative calculation - time progression," it achieves high-precision numerical simulation of hypersonic flows. The piecewise nature of the Haar wavelet ensures stability and accuracy in regions with strong discontinuities such as shock waves, while the introduction of the modified integral matrix allows boundary conditions to be naturally integrated into the numerical scheme, avoiding the additional boundary processing required in traditional methods. This makes the entire simulation process more compact and efficient in both mathematical structure and physical implementation.

[0032] Furthermore, the method proposed in this embodiment fully utilizes the piecewise constant characteristics and tight support of Haar wavelets, enabling numerically stable capture of strong discontinuities such as shock waves and contact discontinuities. Compared to traditional spectral methods, this embodiment eliminates the Gibbs phenomenon at discontinuities naturally without the need for additional limiters or artificial viscosity, ensuring the physical consistency and numerical stability of the calculation results. Simultaneously, this method exhibits excellent resolution in strong discontinuity regions, accurately characterizing shock wave structures, thereby significantly improving the stability and reliability of hypersonic flow simulations.

[0033] This application also provides an electronic device, such as... Figure 5 As shown, processor 501 and memory 502 are connected via a bus or other means.

[0034] Processor 501 can be a central processing unit (CPU). Processor 501 can also be other general-purpose processors, digital signal processors (DSPs), application-specific integrated circuits (ASICs), field-programmable gate arrays (FPGAs), or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, or combinations of the above types of chips.

[0035] The memory 502, as a non-transitory computer-readable storage medium, can be used to store non-transitory software programs, non-transitory computer-executable programs, and modules, such as the program instructions / modules corresponding to the Haar wavelet numerical simulation method for hypersonic flow in this embodiment of the invention. The processor executes various functional applications and data processing by running the non-transitory software programs, instructions, and modules stored in the memory.

[0036] Memory 502 may include a program storage area and a data storage area. The program storage area may store the operating system and applications required for at least one function; the data storage area may store data created by the processor, etc. Furthermore, the memory may include high-speed random access memory and non-transitory memory, such as at least one disk storage device, flash memory device, or other non-transitory solid-state storage device. In some embodiments, memory 502 may optionally include memory remotely located relative to the processor, which can be connected to the processor via a network. Examples of such networks include, but are not limited to, the Internet, corporate intranets, local area networks, mobile communication networks, and combinations thereof.

[0037] The one or more modules are stored in the memory 502, and when executed by the processor 501, they perform actions such as... Figure 1 The Haar wavelet numerical simulation method for hypersonic flow shown in the embodiment.

[0038] For specific details regarding the aforementioned electronic devices, please refer to the relevant documentation. Figure 2 The relevant descriptions and effects in the illustrated embodiments are for understanding purposes only and will not be repeated here.

[0039] This embodiment also provides a computer storage medium storing computer-executable instructions that can execute the Haar wavelet numerical simulation method for hypersonic flow in any of the above method embodiments. The storage medium can be a magnetic disk, optical disk, read-only memory (ROM), random access memory (RAM), flash memory, hard disk drive (HDD), or solid-state drive (SSD), etc.; the storage medium may also include combinations of the above types of memory.

[0040] Finally, it should be noted that the above preferred embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail through the above preferred embodiments, those skilled in the art should understand that various changes can be made to it in form and detail without departing from the scope defined by the claims of the present invention.

Claims

1. A Haar wavelet numerical simulation method for hypersonic flow, characterized in that, include: Obtain the computational domain, boundary conditions, and the vector of conserved flow field variables at the current moment; Within the computational domain, based on the Haar wavelet basis functions and boundary conditions, the Haar matrix, the first integral matrix after the correction of the Haar wavelet basis functions, and the second integral matrix are determined. Using the boundary conditions and the corrected quadratic integral matrix, solve for the coefficient vector of the corresponding Haar matrix; Based on the corrected first integral matrix and the coefficient vector of the Haar matrix, determine the first derivative of the flow field conserved variable vector in space; based on the Haar matrix and the coefficient vector of the Haar matrix, calculate the second derivative of the flow field conserved variable vector in space. Based on the first and second derivatives, a spatial discrete operator is determined to describe the spatial variation of the flow field; A target time-progression method combined with spatial discretization operators is used to update the hypersonic flow values.

2. The Haar wavelet numerical simulation method for hypersonic flow according to claim 1, characterized in that, Within the computational domain, based on the Haar wavelet basis functions and boundary conditions, the Haar matrix, the corrected first-order integral matrix, and the corrected second-order integral matrix are determined, including: Based on a pre-defined resolution level, the location of the configuration point within the computational domain is determined; Determine the Haar wavelet basis functions based on the location of the locus of points within the computational domain; Based on the Haar wavelet basis functions, construct the Haar matrix, the first integral matrix of the Haar wavelet basis functions, and the second integral matrix; The first and second integral matrices are corrected according to the boundary conditions to obtain the corrected first and second integral matrices.

3. The Haar wavelet numerical simulation method for hypersonic flow according to claim 1, characterized in that, The hypersonic flow numerical values ​​are updated using a target time-progression method combined with spatial discretization operators, including: Based on the third-order Runge-Kutta method and combined with spatial discretization operators, the hypersonic flow values ​​at the next time step are determined, where the time interval between the current time step and the next time step satisfies the CFL stability condition.

4. The Haar wavelet numerical simulation method for hypersonic flow according to claim 2, characterized in that, Based on a pre-defined resolution level, the location of placement points within the computational domain is determined, including: ； in, Indicates the first The location of each configuration point , For resolution levels, Indicates the number of configuration points.

5. The Haar wavelet numerical simulation method for hypersonic flow according to claim 1, characterized in that, Using the boundary conditions and the corrected quadratic integral matrix, solve for the coefficient vector of the corresponding Haar matrix, including: ； Where a is the coefficient vector of the corresponding Haar matrix. This is the corrected quadratic integral matrix. f Let be the conserved variables of the flow field to be solved, and be any component of the vector of conserved flow field variables. For boundary conditions, x Represents standardized computational space coordinates.

6. The Haar wavelet numerical simulation method for hypersonic flow according to claim 1, characterized in that, Based on the corrected first-order integral matrix and the coefficient vector of the Haar matrix, determine the first-order derivative in space of the flow field conservation variable vector, including: ； in, The vector of conserved flow field variables is represented in space. First derivative in the direction, In order to be in The corresponding Haar coefficient vector in the direction, These are the left and right boundary values. Indicates in The first-order integral matrix after directional correction.

7. The Haar wavelet numerical simulation method for hypersonic flow according to claim 1, characterized in that, Based on the Haar matrix and its coefficient vector, calculate the second derivative in space of the flow field conservation variable vector, including: ; in, The vector of conserved flow field variables is represented in space. Second derivative in the direction, In order to be in Haar matrix in the direction, In order to be in The coefficient vector of the Haar matrix in the direction.

8. The Haar wavelet numerical simulation method for hypersonic flow according to claim 3, characterized in that, Based on the third-order Runge-Kutta method, combined with spatial discretization operators, the hypersonic flow values ​​at the next time step are determined, including: ， ， ； in, Indicates the time step. This represents the first intermediate step of Runge-Kutta. This represents the second intermediate step of Runge-Kutta. This represents the conserved flow field variables at the nth time step. Let represent the updated solution at time step (n+1). This indicates that a fixed spatial discrete operator acts at time step n. superior, This indicates that a fixed spatial discrete operator is applied in the first intermediate step. superior, This indicates that the fixed spatial discrete operator acts on the second intermediate step. .

9. The Haar wavelet numerical simulation method for hypersonic flow according to claim 8, characterized in that, The time interval between the current moment and the next moment satisfies the CFL stability conditions, including: in, The spatial step size is in three directions, ( u,v,w ) represents the fluid velocity in three spatial directions, | u |+ c express x Characteristic velocity in the direction, express y Characteristic velocity in the direction, express z Characteristic velocity in the direction, CFL is the stability coefficient, and c is the local speed of sound.

10. An electronic device, the device comprising: A memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that the processor performs the steps of the Haar wavelet numerical simulation method for hypersonic flow as described in any one of claims 1-9.

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