Implicit wall modeling method for large eddy simulation based on high-resolution spectral cell method and its application

Abstract

This invention discloses an implicit large eddy wall simulation method based on the high-precision spectral element method. This method constructs an implicit large eddy simulation (SVV-iLES) based on the hp-type spectral element method combined with the spectral viscosity elimination method (SVV), and implements wall simulation within this framework. The method includes the following steps: adjusting the boundary layer according to the actual situation of the turbulent boundary layer to ensure that the corresponding points fall within the logarithmic region; interpolating the element velocity and velocity u within the spectral element method code framework to obtain velocity components parallel to the wall velocity, which are then substituted into the algebraic model; analytically obtaining the wall stress through the algebraic model; and finally, modifying the boundary conditions in real time using the logarithmic rate model to complete the simulation. This significantly reduces the enormous computational burden caused by fine boundary layer meshes in flow problems. This invention also proposes an application of this simulation method to high Reynolds number wall flow scenarios, providing an efficient foundation for accurate flow calculations.

Description

Technical Field

[0001] This invention belongs to the field of large eddy simulation technology, specifically relating to an implicit large eddy wall simulation method based on high-precision spectral element method and its application. Background Technology

[0002] In recent years, with the improvement of computing power, more and more researchers have been studying complex flow structures using high-precision numerical methods. The theoretical basis of Large Eddy Simulation (LES) is the elimination of small-scale eddies in high Reynolds number turbulence, and LES has become an important tool for studying turbulent motion in the last decade or so. The models used in LES are generally explicit subgrid stress models, requiring relatively small grids, which makes calculations cumbersome and data analysis prone to errors. Therefore, related LES methods need further improvement.

[0003] Because the high-precision scheme of the spectral element method (SEM) has low dispersion and dissipation, it can perform more accurate analysis of complex flow structures compared to traditional low-order schemes. To improve the stability of the classical Fourier spectral method, a spectral viscosity elimination (SVV) method was proposed. Combining SVV with the SEM, SVV-iLES was constructed, which has been verified to effectively improve computational stability and accuracy. However, in high Reynolds number flows, coarse boundary layer meshes are difficult to accurately resolve wall stresses, resulting in significant discrepancies between calculated and actual results. Conversely, using finer boundary layer meshes requires enormous computational resources. Therefore, to further improve computational efficiency in high Reynolds number flow applications, we added a wall model to the implicit large eddy simulation (SVV-iLES) based on the SEM. We continue to use the high-precision hp-type SEM with good dispersion, geometric adaptability, and dissipation characteristics as a foundation, while leveraging its strong parallel scalability in large-scale parallel environments. This allows us to model complex turbulent problems and provide reliable numerical results. Summary of the Invention

[0004] The purpose of this invention is to solve the above-mentioned problems by providing an implicit large eddy wall simulation method based on the high-precision spectral element method and its application, so as to facilitate the handling of complex turbulence problems and improve the calculation accuracy and efficiency.

[0005] The technical solution of this invention is: an implicit large eddy wall simulation method based on the high-precision spectral element method. This method is based on the hp-type spectral element method and combines it with the viscosity elimination method SVV to construct SVV-iLES, and realizes wall simulation within its framework. The method is characterized by the following steps:

[0006] Step (1) After preserving the Nth-order mode through the expression of the inviscid Burgers equation, artificial SVV dissipation effect is introduced into the Fourier coefficient mathematical expression of the control equation SVV kernel, thereby defining the SVV operator, introducing it into the NS equation, and then determining the SVV-iLES code framework.

[0007] Step (2) In the turbulent boundary layer, it is divided into a viscous sublayer, a transition layer, a logarithmic layer and an outer boundary layer structure. By setting the height h from the wall in the program, the velocity u at the corresponding position is obtained. Then, it is adjusted according to the actual situation to ensure that the corresponding point falls within the logarithmic region. Under the code framework based on the spectral element method, the unit velocity u(x) and velocity u are interpolated.

[0008] Step (3) Based on the interpolation process in step (2), a velocity component parallel to the wall velocity is obtained.

[0009] Step (4) Substitute the velocity components from step (3) into the algebraic model, obtain the wall stress through Newton's iteration method, and then use the wall stress as the Newman wall boundary condition. At the same time, modify the boundary condition in real time based on the logarithmic law model of the turbulent boundary layer, set the height of the first layer of mesh in the logarithmic law layer, and generally set the value of y+ to above 30.

[0010] In step (5), two algebraic models are used in the analytical wall stress calculation: the traditional logarithmic law expression or the Reichardt law expression. The Reichardt law expression is characterized by achieving a smooth connection between the viscous sublayer and the logarithmic layer.

[0011] Furthermore, in the above-mentioned implicit large eddy wall simulation method based on the high-precision spectral element method, the specific implementation algorithm of SVV in step (1) is as follows:

[0012] The expression for the non-viscosity Burgers equation is:

[0013]

[0014] The corresponding expression for the viscous solution is:

[0015]

[0016] Under the SVV method, preserving the Nth mode, the expression for the solution above is:

[0017]

[0018] The mathematical expression for the Fourier coefficients of the SVV kernel is:

[0019]

[0020]

[0021]

[0022] Expanded in Fourier space using Fourier expansion It only applies to high wavenumbers. It is a mapping operator.

[0023] ★ indicates the convolution symbol; in the initial work, the wavenumber k is defined to be greater than P. cut hour, =1, less than P cut A value of 0 means it has no effect. In future work, Maday will use ∈≈N. -1 .

[0024] The SVV operator expression is:

[0025]

[0026] v svv This represents the effective dissipation intensity when the spectral elements are discrete.

[0027] Furthermore, in the above-mentioned implicit large eddy wall simulation method based on the high-precision spectral element method, in step (2),

[0028] On the one hand, the unit velocity u(x) is determined by the following interpolation formula:

[0029]

[0030] The velocity u(x) is composed of Lagrange interpolation functions as basis functions;

[0031] u i These are the corresponding coefficients of the basis functions;

[0032] φ i (x) is a basis function within the unit.

[0033] On the other hand, the velocity u is obtained through the following interpolation formula:

[0034]

[0035] The computational domain of the velocity u is decomposed into hexahedral elements. During the solution process, the velocity field and pressure field are represented by Np-order polynomials within each element. In the Cartesian coordinate system, when the physical coordinates are linearly related to the corresponding element ID, the physical coordinate parameters are mapped to local coordinates within the element ID. Here, r, s, t, and hi are one-dimensional Lagrange polynomials.

[0036] Furthermore, in the above-mentioned implicit large eddy wall simulation method based on the high-precision spectral element method, in step (3), the expression for the velocity component is:

[0037] u || =ucosθ

[0038] θ is the angle between the flow velocity and the wall.

[0039] Furthermore, in the above-mentioned implicit large eddy wall simulation method based on the high-precision spectral element method, in step (5), the logarithmic law expression is:

[0040]

[0041] Where K = 0.41, B = 5.2;

[0042] The Reichardt law expression:

[0043]

[0044] Where D = 9.8, E = 1 / 11, F = 1 / 3, K = 0.41.

[0045] The present invention relates to an application of a high Reynolds number wall flow scenario based on the implicit large eddy wall simulation method using the high-precision spectral element method, including simulation scenarios of internal flow of aero-engines and flow of underwater vehicles.

[0046] Compared with existing technologies, the technical solution of this invention enables wall simulation within the framework of implicit large eddy simulation (SVV-iLES) based on the spectral element method. This significantly reduces the enormous computational burden caused by the fine boundary layer mesh in high Reynolds number wall flow problems, allowing the method (SVV-iLES) to be applied in large-scale industrial calculations. Moreover, by directly modifying the dissipation of the numerical scheme, the method (SVV-iLES) is equivalent to the subgrid model of the traditional large eddy simulation method. Compared with explicit large eddy simulation, it does not require the establishment of a new wall model, thus reducing truncation errors and various uncertainties in the calculation process. Attached Figure Description

[0047] Figure 1 This is a schematic diagram of the turbulent boundary layer of the flat plate according to the present invention;

[0048] Figure 2 This is a schematic diagram comparing the calculation results of the traditional logarithmic rate based on the present invention with the results of flow velocity and logarithmic rate and DNS from direct numerical simulation.

[0049] Figure 3 This is a schematic diagram comparing the calculation results based on the traditional logarithmic law of this invention with the results of direct numerical simulation of velocity perturbation (root mean square) and DNS.

[0050] Figure 4 This is a schematic diagram comparing the calculation results based on the traditional logarithmic law of the present invention with the results of the average Reynolds stress term and DNS from direct numerical simulation;

[0051] Figure 5 This is a schematic diagram comparing the calculation results based on Reichardt's law of the present invention with the results of direct numerical simulation of flow velocity and logarithmic law and DNS;

[0052] Figure 6 This diagram illustrates a comparison between the calculation results based on Reichardt's law and the results of direct numerical simulation of velocity perturbation (root mean square) and DNS.

[0053] Figure 7 This is a schematic diagram comparing the calculation results based on Reichardt's law of the present invention with the results of the average Reynolds stress term of direct numerical simulation and DNS;

[0054] Figure 8 This is a cloud map simulating the flow of an underwater vehicle according to the present invention;

[0055] Figure 9 This is a high Reynolds number airfoil flow simulation cloud diagram of the present invention;

[0056] Figure 10 This is a flow effect cloud diagram of the instantaneous flow field in the flat plate channel flow of the present invention;

[0057] Figure 11 This is a flow effect contour map of the average flow field in the flat plate channel flow of the present invention.

[0058] Figure 12 This is a diagram illustrating the spanwise velocity of the flat plate channel flow at a Reynolds number of 550 (located in the z = 1.57 plane) according to the present invention.

[0059] Figure 13 This is a diagram illustrating the spanwise velocity of the flat plate channel flow at a Reynolds number of 950 (located in the z = 1.57 plane) according to the present invention.

[0060] Figure 14 This is a diagram illustrating the spanwise velocity of the flat plate channel flow at a Reynolds number of 2000 (located in the z = 1.57 plane) according to the present invention.

[0061] Figure 15 This is a diagram illustrating the spanwise velocity of the flat plate channel flow mesh 2 (located in the z=1.57 plane) of the present invention. Detailed Implementation

[0062] The technical solution of the present invention will be described below with reference to the accompanying drawings to make it easier to understand and master. To make the objectives, technical solutions, and advantages of the present invention clearer, the present invention will be further described in detail below through specific embodiments, but this should not be construed as limiting the scope of the above-mentioned subject matter of the present invention to the following embodiments. Specific implementation examples:

[0064] like Figure 1 As shown, this paper illustrates an implicit large eddy wall simulation method based on the high-precision spectral element method. It utilizes the hp-type spectral element method and combines it with the viscous elimination method (SVV) to construct SVV-iLES, achieving wall simulation within this framework. Previous techniques set the height of the first mesh layer within the viscous sublayer, essentially setting the y+ value to around 1. However, when applied to high Reynolds number wall flow calculations, including airfoil flow and aerodynamic noise calculations, the computational load is enormous without a wall model. This invention, based on a turbulent boundary layer logarithmic rate model, modifies boundary conditions in real-time using an algebraic model to ensure computational accuracy. By increasing the height of the first mesh layer, computational accuracy is maintained, significantly improving computational efficiency.

[0065] The specific steps are as follows:

[0066] Step (1): After preserving the Nth-order mode through the expression of the inviscid Burgers equation, an artificial SVV dissipation effect is introduced into the mathematical expression of the Fourier coefficients of the SVV kernel of the governing equation, thereby defining the SVV operator, introducing it into the NS equation, and then determining the SVV-iLES code framework.

[0067] The expression for the non-viscosity Burgers equation is as follows:

[0068]

[0069] The corresponding expression for the viscous solution is:

[0070]

[0071] Under the SVV method, preserving the Nth mode, the expression for the solution above is:

[0072]

[0073] The mathematical expression for the Fourier coefficients of the SVV kernel is:

[0074]

[0075]

[0076]

[0077] Expanded in Fourier space using Fourier expansion It only applies to high wavenumbers. It is a mapping operator.

[0078] ★ indicates the convolution symbol; in the initial work, the wavenumber k is defined to be greater than P. cut hour, =1, less than P cut A value of 0 means it has no effect. In future work, Maday will use ∈≈N. -1 ;

[0079] The SVV operator expression is:

[0080]

[0081] v svv This represents the effective dissipation intensity when the spectral elements are discrete.

[0082] Step (2): In the turbulent boundary layer, it is divided into a viscous sublayer, a transition layer, a logarithmic layer, and an outer boundary layer structure. By setting the height h from the wall in the program, the velocity u at the corresponding position is obtained. Then, it is adjusted according to the actual situation of the turbulent boundary layer to ensure that the corresponding point falls within the logarithmic region. Under the code framework based on the spectral element method, the unit velocity u(x) and velocity u are interpolated respectively.

[0083] The unit velocity u(x) is obtained through the following interpolation formula:

[0084]

[0085] The velocity u(x) is composed of Lagrange interpolation functions as basis functions;

[0086] u i These are the corresponding coefficients of the basis functions;

[0087] φ i (x) is a basis function within the unit.

[0088] The velocity u is obtained through the following interpolation formula:

[0089]

[0090] The computational domain of the velocity u is decomposed into hexahedral elements. During the solution process, the velocity field and pressure field are represented by Np-order polynomials within each element. In the Cartesian coordinate system, when the physical coordinates are linearly related to the corresponding element ID, the physical coordinate parameters are mapped to local coordinates within the element ID. Here, r, s, t, and hi are one-dimensional Lagrange polynomials.

[0091] Step (3): Based on the interpolation process in step (2), the velocity component parallel to the wall velocity is obtained, and the expression of the velocity component is:

[0092] u || =ucosθ

[0093] θ is the angle between the flow velocity and the wall.

[0094] Step (4): Substitute the velocity components from step (3) into the algebraic model, obtain the wall stress through Newton's iteration method, and then use the wall stress as the Newman wall boundary condition. At the same time, modify the boundary condition in real time based on the logarithmic law model of the turbulent boundary layer, set the height of the first layer of mesh below the logarithmic law layer, and set the value of y+ above 30.

[0095] Step (5): In the analytical calculation of wall stress, the traditional logarithmic law expression or the Reichardt law expression can be used. The characteristic of using the Reichardt law expression is that it achieves a smooth connection between the viscous sublayer and the logarithmic law layer.

[0096] The logarithmic expression:

[0097]

[0098] In the formula, K = 0.41 and B = 5.2.

[0099] The Reichardt law expression:

[0100]

[0101] In the formula, D = 9.8, E = 1 / 11, F = 1 / 3, and K = 0.41.

[0102] The algebraic model in step (5) is verified, and the structure is obtained in the form of a line graph. The calculation parameters and corresponding parameter values ​​required for the verification are selected as follows: see the data in Table 1.

[0103]

[0104] Table 1: Verification Parameter Table

[0105] On the one hand, such as Figures 2-4As shown in the figure, the flow-average velocity of the implicit large eddy simulation based on the wall model satisfies the logarithmic law well and is basically consistent with the results of direct numerical simulation (DNS). Furthermore, a comparison of the mean Reynolds stress term reveals that the results of the implicit large eddy simulation based on the wall model become increasingly close to the results of direct numerical simulation (DNS) as the value of y+ increases. In conclusion, by verifying the traditional algebraic model, it is shown that the calculation results of the implicit large eddy simulation based on the wall model are fast and accurate.

[0106] On the other hand, such as Figures 5-7 As shown in the figure, the flow-average velocity of the wall model based on Reichardt's law can also satisfy the logarithmic law, which is basically consistent with the results of direct numerical simulation (DNS). At the same time, it is consistent with the calculation results of the algebraic model based on the traditional logarithmic law equation. Since the velocity coordinate points are selected in the logarithmic layer, the calculation results of the two in the logarithmic layer are not much different, and they also have good applicability.

[0107] Specifically, the implicit large eddy wall simulation method based on the high-precision spectral element method described above can be applied to high Reynolds number wall flow scenarios.

[0108] More specifically, such as Figure 8 As shown, when performing flow simulation of underwater vehicles, the wall model constructed by the implicit large eddy wall simulation method based on the high-precision spectral element method can effectively reduce the wall mesh height in the flow calculation of underwater vehicles, significantly reduce the consumption of computing resources, and efficiently improve the computing efficiency.

[0109] More specifically, such as Figure 9 As shown, when performing flow simulation inside an aero-engine, the wall model constructed using this implicit large eddy wall simulation method based on the high-precision spectral element method can effectively reduce the height of the first layer mesh and significantly reduce computational consumption.

[0110] More specifically, such as Figures 10-11 As shown, when performing flow simulation in a flat plate channel, the instantaneous flow field and the average flow field are displayed respectively; moreover, as Figures 12-15 As shown, when located in the z=1.57 plane, the spanwise velocity effect of different Reynolds numbers 550, 950, 2000 and Mesh2 for flat plate channel flow gradually stabilizes with the increase of Reynolds number, effectively reducing the mesh height of the flat plate channel flow wall and improving computational efficiency.

[0111] In this invention, the key technology is the wall simulation achieved within the framework of Implicit Large Eddy Simulation (SVV-iLES) based on the spectral element method. As described above, compared with existing technologies, this invention significantly reduces the enormous computational burden caused by the fine boundary layer mesh in high Reynolds number wall flow problems by achieving wall simulation within the framework of Implicit Large Eddy Simulation (SVV-iLES) based on the spectral element method. This allows the method (SVV-iLES) to be applied in large-scale industrial calculations. Furthermore, this method (SVV-iLES) directly modifies the dissipation of the numerical scheme to achieve the subgrid model of traditional large eddy simulation methods, i.e., it uses an algebraic model to modify the boundary conditions in real time. While increasing the height of the first layer mesh, it ensures computational accuracy. Compared with explicit large eddy simulation, it does not require the establishment of a new wall model, reducing truncation errors and various uncertainties in the calculation process.

[0112] The technical solution, working process, and implementation effects of the present invention have been described in detail above. It should be noted that the described examples are only typical examples of the present invention. In addition, the present invention may have many other specific implementation methods. All technical solutions formed by equivalent substitution or equivalent transformation fall within the scope of protection claimed by the present invention.

Claims

1. An implicit large eddy simulation method based on the high-precision spectral element method, which constructs the implicit large eddy simulation method SVV-iLES based on the hp-type spectral element method combined with the spectral viscosity elimination method SVV, and adds a wall simulation method within its framework, characterized in that... The method includes the following steps: Step S1: After preserving the Nth-order mode through the expression of the inviscid Burgers equation, an artificial SVV dissipation effect is introduced into the Fourier coefficient mathematical expression of the SVV kernel of the control equation, thereby defining the SVV operator, introducing it into the NS equation, and then determining the SVV-iLES code framework. Step S2: In the turbulent boundary layer, it is divided into a viscous sublayer, a transition layer, a logarithmic layer, and an outer boundary layer structure. By setting the height h from the wall in the program, the velocity u at the corresponding location is obtained. Then, it is adjusted according to the actual situation of the turbulent boundary layer to ensure that the corresponding point falls within the logarithmic region. Under the code framework based on the spectral element method, the element velocity is... Interpolate the velocity u; Step S3: Based on the interpolation processing in step S2, obtain the velocity component parallel to the wall velocity; Step S4: Substitute the velocity components from step S3 into the algebraic model, obtain the wall stress through Newton's iteration method, and then use the wall stress as the Newman wall boundary condition. At the same time, modify the boundary condition in real time based on the logarithmic law model of turbulent boundary layer, set the height of the first grid layer below the logarithmic law layer, and set the value of y+ above 30. Step S5: In the analytical calculation of wall stress, two algebraic models are used, namely the traditional logarithmic law expression or the Reichardt law expression.

2. The implicit large eddy wall simulation method based on high-precision spectral element method according to claim 1, characterized in that: In step S1, under the corresponding initial and boundary conditions, the expression for the inviscid Burgers equation is: The corresponding viscous solution expression is: Under the SVV method, preserving the Nth mode, the expression for the solution above is: The mathematical expression for the Fourier coefficients of the SVV kernel is: , , In Fourier space, it is expanded using Fourier methods. It only applies to high wavenumbers. It is a mapping operator. Represents the convolution symbol; in the initial work, the wavenumber k is defined to be greater than... P cut hour, =1, less than A value of 0 means it has no effect. In future work, Maday will use... .

3. The implicit large eddy wall simulation method based on high-precision spectral element method according to claim 2, characterized in that: In step S1, the SVV operator expression is: In the formula, This represents the effective dissipation intensity when the spectral elements are discrete.

4. The implicit large eddy wall simulation method based on high-precision spectral element method according to claim 1, characterized in that: In step S2, the unit velocity It is obtained using the following interpolation formula: Among them, speed It is composed of Lagrange interpolation functions as basis functions; u i These are the corresponding coefficients of the basis functions; These are the basis functions within the unit.

5. The implicit large eddy wall simulation method based on high-precision spectral element method according to claim 4, characterized in that: In step S2, the velocity u is obtained using the following interpolation formula: The computational domain of the velocity u is decomposed into hexahedral elements. During the solution process, the velocity field and pressure field are represented by Np-order polynomials within each element. In the Cartesian coordinate system, when the physical coordinates are linearly related to the corresponding element ID, the physical coordinate parameters are mapped to local coordinates within the element ID. Here, r, s, t, and hi are one-dimensional Lagrange polynomials.

6. The implicit large eddy wall simulation method based on high-precision spectral element method according to claim 1, characterized in that: In step S3, the expression for the velocity component is: θ It is the angle between the flow velocity and the wall.

7. The implicit large eddy wall simulation method based on high-precision spectral element method according to claim 1, characterized in that: In step S5, the expression for the logarithmic rate is, ,in K =0.41, B =5.2; The Reichardt law expression is, Where D=9.8, E=1 / 11, F=1 / 3, K=0.41.

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