A numerical simulation method for flow around a rotationally symmetric geometry

Abstract

This invention discloses a numerical simulation method for flow around a rotationally symmetric geometry, relating to the field of numerical simulation. The method includes: generating a body-fitted mesh containing a singular axis; setting virtual mesh points outside the singular axis; providing spatial coordinates and flow field variables at the virtual mesh points; calculating the mesh transformation derivatives and Jacobian determinants at the boundary and internal mesh points of the singular axis; providing the mesh transformation derivatives and Jacobian determinants at the virtual mesh points; for viscous flow, providing the first-order derivatives of the flow field variables at the virtual mesh points; discretizing the spatial convection and viscous terms of the governing equations; updating the flow field variables around the rotationally symmetric geometry; calculating the flow field variables on the singular axis; and obtaining the flow field data corresponding to the input data based on all calculation results. This method solves the compatibility problem between the high-precision flux reconstruction finite difference method and the singular axis boundary treatment, reduces numerical errors, and improves the flow field quality at the singular axis boundary when numerically simulating flow around a rotationally symmetric geometry.

Description

Technical Field

[0001] This invention relates to the field of numerical simulation, and more specifically, to a method for numerical simulation of flow around a rotationally symmetric geometry. Background Technology

[0002] A singular axis is a special boundary introduced in computational fluid dynamics (CFD) when dealing with rotationally symmetric geometry. To numerically simulate flow around rotationally symmetric geometries (such as the sharp spherical nose or cylinder of a high-speed aircraft), an axisymmetric body-fitted mesh is typically generated around the axis of symmetry. In this axisymmetric body-fitted mesh, the mesh lines of the surface mesh converge at a point on the axis of symmetry, becoming a singular point. These points, when forming a volume mesh in space, constitute the singular axis. The singular axis appears as a straight line in physical space but is mapped to a plane in computational space. Although mesh points on the singular axis theoretically rotate around the axis, their x, y, and z coordinates remain unchanged; geometrically, this means the area of ​​the singular axis in physical space is zero. When performing coordinate transformations from the physical domain (x, y, and z) to the computational domain (ξ, η, ζ), the inverse of the Jacobian matrix determinant degenerates to zero at the singular axis (J...). -1 = 0). The mathematical singularity introduced by coordinate system transformation itself (the grid derivative tends to zero or infinity) and the inherent smoothness requirement of physical quantities create a contradiction in discrete numerical calculations, thus forming the singular axis problem, which needs to be specially handled in CFD calculations.

[0003] In the finite volume method, the singular axis behaves as a degenerate boundary surface, resulting in zero flux integral, thus eliminating the need to set boundary values. However, in the finite difference method, it is necessary to assign values ​​to the grid points on the singular axis. If the grid contains a singular axis, traditional finite difference calculations typically employ the basic approach of directly setting the numerical flux of the grid points on the singular axis to zero. Then, based on the flow field variable values ​​at adjacent grid nodes inside the singular axis, the flow field variable values ​​at the grid points on the singular axis are obtained by interpolation, circumferential summation, and averaging. This method seems reasonable when there is no angle of attack (AoA). However, when the mesh near the singular axis is sparse, the calculated results often have large errors, with errors concentrating on the singular axis and fluctuations in the calculation. When there is an angle of attack, since the area of ​​the singular axis in physical space is zero, although the actual flow passes through the singular axis, the zero flux treatment of the singular axis means that the variables on the axis do not directly participate in the solution of the flow equations. The lack of direct influence on the flow field on the singular axis may lead to structural distortion or instability in the flow field near the singular axis. Therefore, traditional singular axis boundary treatment methods for finite difference methods face problems of low numerical stability and accuracy.

[0004] Recently, researchers have proposed a hybrid algorithm to solve the problem of calculating singular axes using the finite difference method, building upon traditional approaches. Specifically, a finite difference method based on conserved variable reconstruction is used near the singular axis, while a finite difference method based on flux reconstruction is used in regions away from the singular axis. Conserved variable reconstruction avoids the problem of insufficient information transfer caused by the singular axis area being zero in flux calculations, reflecting the interaction between variables on the singular axis and the surrounding flow field. Because it requires dividing the region for separate conserved variable reconstruction and flux reconstruction, the hybrid algorithm is not only complex to implement, but the region division also relies somewhat on experience. Furthermore, the smoothness of the transition between different reconstruction methods may also exist. Summary of the Invention

[0005] To address the shortcomings of existing finite difference methods for handling singular axis boundaries in numerical simulations of flow around rotationally symmetric geometries, this invention proposes a finite difference method for handling singular axis boundaries based on virtual grid points and employing unified high-precision flux reconstruction across the entire computational domain. This method solves the compatibility problem between high-precision flux reconstruction finite difference methods and singular axis boundary handling, reduces numerical errors, and improves the flow field quality at singular axis boundaries when numerically simulating flow around rotationally symmetric geometries.

[0006] To achieve the above-mentioned objective, this invention provides a numerical simulation method for flow around a rotationally symmetric geometry, the method comprising:

[0007] Step 1: Based on mesh generation software, generate a body-fitted mesh containing singular axes required for numerical simulation of flow around a rotationally symmetric geometry;

[0008] Step 2: Set virtual mesh points outside the singular axes of the body-fitted mesh;

[0009] Step 3: Based on the spatial coordinates of the internal grid points of the body-fitted structure mesh, provide the spatial coordinates of the virtual grid points;

[0010] Step 4: Based on the spatial coordinate values ​​of virtual grid points, singular axis boundary grid points, and internal grid points, calculate the grid transformation derivatives and Jacobian determinants of singular axis boundary grid points and internal grid points, which are used to discretize the spatial convection derivatives and spatial viscous derivatives of the control equations;

[0011] Step 5: Based on the mesh transformation derivative and Jacobian determinant at the internal mesh points of the body-fitted structure mesh, the mesh transformation derivative and Jacobian determinant at the virtual mesh points are used to discretize the spatial convection derivative and spatial viscosity derivative terms of the control equations;

[0012] Step 6: Based on the flow field variable values ​​at the internal grid points of the body-fitted structure mesh, provide the flow field variable values ​​at the virtual grid points;

[0013] Step 7: For the viscous flow problem around a rotationally symmetric geometry, based on the flow field variable values ​​at the boundary grid points of the singular axis, the internal grid points, and the virtual grid points, calculate the first derivative values ​​of the flow field variables at the boundary grid points of the singular axis and the internal grid points, and provide the first derivative values ​​of the flow field variables at the virtual grid points for discretizing the spatial viscous derivative terms of the control equations.

[0014] Step 8: Based on the mesh transformation derivatives, Jacobian determinants, flow field variable values, and first-order derivative values ​​of the flow field variables on virtual mesh points, singular axis boundary mesh points, and internal mesh points, discretize the spatial convection derivative term and spatial viscous derivative term of the control equations;

[0015] Step 9: Based on the spatial convection derivative and spatial viscous derivative of the discretized control equations, update the flow field variable values ​​around the rotationally symmetric geometry iteratively through time-progression.

[0016] Step 10: Based on the updated flow field variable values ​​around the rotationally symmetric geometry, calculate the flow field variable values ​​on the singular axis of the rotationally symmetric geometry;

[0017] Step 11: Determine whether the total number of calculation time steps or the calculation convergence condition has been met. If not, return to step 6. If met, end the simulation and obtain the flow field data of the rotationally symmetric geometry corresponding to the input data based on all calculation results.

[0018] The principle of this method is as follows: by setting virtual grid points outside the singular axis boundary of the body-fitted mesh of rotationally symmetric geometry, and reasonably setting the grid coordinates, grid transformation derivatives, flow field variables and their first derivatives on the virtual grid points, the spatial derivative terms of the control equations are discretized in the entire computational domain, including the singular axis, using a unified high-precision flux reconstruction difference method. This solves the compatibility problem between the high-precision flux reconstruction finite difference method and the singular axis boundary treatment of the body-fitted mesh.

[0019] Preferably, the rotationally symmetric geometry is the spherical head or cylindrical body of an aircraft or vehicle.

[0020] Preferably, step 1 specifically includes:

[0021] Based on mesh generation software, a body-fitted mesh containing singular axes is generated for numerical simulation of flow around a rotationally symmetric geometry. The physical coordinates of the body-fitted mesh are (x, y, z), with the singular axes conventionally falling on the x-axis. The computational coordinates of the body-fitted mesh are (ξ, η, ζ), where the ξ direction corresponds to the normal mesh line and is denoted by the index i, i=I. maxThe surface represents the object, the η direction corresponds to the flow grid line and is denoted by the index j, j=1 represents the singular axis, the ζ direction corresponds to the circumferential grid line and is denoted by the index k; the number of grid points on the normal grid line is I. max The number of grid points flowing along the grid line is J. max The number of grid points on the circumferential grid line is K. max K max For odd values, the grid points are evenly distributed along the circumferential grid lines.

[0022] Preferably, step 2 specifically includes:

[0023] The grid points on the singular axis are expanded circumferentially in the computational space to form a plane j=1. For each grid point on this plane, several layers of virtual grid points are set in the negative direction of index j (i.e., the direction in which index j decreases).

[0024] Preferably, the spatial coordinates and flow field variables at the virtual grid points are obtained as follows:

[0025] The spatial coordinates and flow field variables at virtual grid points are obtained as follows:

[0026] The spatial coordinates and flow field variables of the current virtual grid point are set based on the internal grid points of the body-fitted structure mesh. The specific correspondence is as follows:

[0027] ;

[0028] Among them, flow field variables , , , and These represent density, the velocity components in the x, y, and z directions, and pressure, respectively; jg is the index of the virtual grid point in the η direction, and jt and kt are the indices of the internal grid points of the corresponding virtual grid point in the η and ζ directions, respectively. This represents the number of layers in the virtual grid.

[0029] Preferably, the method for calculating the derivative of the mesh transformation from physical space to computational space in step 4 is as follows:

[0030] ;

[0031] Where J is the Jacobian determinant of the coordinate transformation. , , , , , , , and These are the calculation of spatial coordinates ( , , The derivative of the mesh transformation in the physical space along the (x, y, z) direction. For partial derivative operators, , , , , , , , and These are the physical space coordinates (x, y, z) in the computational space (x, y, z). , , The derivative of the grid transformation in the direction of )

[0032] Specifically, at the boundary grid points of the singular axis, let:

[0033] ;

[0034] in, , , , , , .

[0035] Preferably, step 5 specifically involves determining the mesh transformation derivative and Jacobian determinant at the virtual mesh point according to the following formula:

[0036] ;

[0037] in, , , , , , , , and Derivatives of grid transformation , , , , , , , and The value at the virtual grid point (i, jg, k). , , , , , , , and Derivatives of grid transformation , , , , , , , and The value at the internal grid point (i, jt, k). Let $\mathbf{i}$ be the value of the Jacobian determinant at the virtual grid point (i, jg, k). The value of the Jacobian determinant at the internal grid point (i, jt, k).

[0038] Preferably, in step 7, if the flow in the viscous flow problem around a rotationally symmetric geometry is axisymmetric, then the first derivative values ​​of the flow field variables at the virtual grid points are obtained according to the following formula:

[0039] ;

[0040] in, and Let be the values ​​of the first derivative of the velocity component u in the x-direction at the virtual grid point (i, jg, k) and the internal grid point (i, jt, k), respectively. and Let be the values ​​of the first derivative of the velocity component u in the y-direction at the virtual grid point (i, jg, k) and the internal grid point (i, jt, k), respectively. and Let be the values ​​of the first derivative of the velocity component u in the z-direction at the virtual grid point (i, jg, k) and the internal grid point (i, jt, k), respectively. and Let be the values ​​of the first derivative of the velocity component v in the x-direction at the virtual grid point (i, jg, k) and the internal grid point (i, jt, k), respectively. and Let be the values ​​of the first derivative of the velocity component v in the y-direction at the virtual grid point (i, jg, k) and the internal grid point (i, jt, k), respectively. and Let be the values ​​of the first derivative of the velocity component v in the z-direction at the virtual grid point (i, jg, k) and the internal grid point (i, jt, k), respectively. and Let be the values ​​of the first derivative of the velocity component w in the x-direction at the virtual grid point (i, jg, k) and the internal grid point (i, jt, k), respectively. and Let be the values ​​of the first derivative of the velocity component w in the y-direction at the virtual grid point (i, jg, k) and the internal grid point (i, jt, k), respectively. and Let be the values ​​of the first derivative of the velocity component w in the z-direction at the virtual grid point (i, jg, k) and the internal grid point (i, jt, k), respectively. and Let be the values ​​of the first derivative of temperature t in the x-direction at the virtual grid point (i, jg, k) and the internal grid point (i, jt, k), respectively. and Let be the values ​​of the first derivative of temperature t in the y-direction at the virtual grid point (i, jg, k) and the internal grid point (i, jt, k), respectively. and Let be the values ​​of the first derivative of temperature t in the z-direction at virtual grid point (i, jg, k) and internal grid point (i, jt, k), respectively.

[0041] In step 7, if the flow in the viscous flow problem around a rotationally symmetric geometry is non-axisymmetric, then the first derivative of the flow field variables at the virtual grid points is calculated using the one-sided difference method based on the flow field variable values ​​at the virtual grid points and the internal grid points.

[0042] Preferably, step 8 includes: based on the grid transformation derivatives, Jacobian determinants, flow field variable values, and first-order derivative values ​​of the flow field variables at virtual grid points, singular axis boundary grid points, and internal grid points, the spatial convection derivative terms of the control equations are discretized at the singular axis boundary and its adjacent region using the flux reconstruction difference method consistent with the method used for internal grid points, and the spatial viscous derivative terms of the control equations are discretized using the central difference method consistent with the method used for internal grid points, to obtain the spatial derivative terms of the discretized control equations.

[0043] Preferably, step 10 specifically includes:

[0044] Based on the updated flow field variable values ​​around the rotationally symmetric geometry, the flow field variable values ​​on the singular axis are interpolated along the j direction for the index position (i, k); the interpolated flow field variable values ​​are then averaged along the circumferential direction as the flow field variable values ​​on the singular axis of the rotationally symmetric geometry.

[0045] One or more technical solutions provided by this invention have at least the following technical effects or advantages:

[0046] This invention introduces virtual grid points outside the singular axis boundary of a body-fitted mesh for a rotationally symmetric geometry. By reasonably setting the grid coordinates, grid transformation derivatives, flow field variables, and their first derivatives on the virtual grid points, and using the same flux reconstruction method as for the internal grid points to discretize the spatial terms at the singular axis boundary and its adjacent region, the influence and contribution of the flow field variables on the other side of the singular axis to the local flow field are taken into account. This achieves singular axis boundary processing of the body-fitted mesh based on the high-precision flux reconstruction finite difference method, reducing numerical errors and improving the flow field quality at the singular axis boundary when numerically simulating flow around a rotationally symmetric geometry. Attached Figure Description

[0047] The accompanying drawings, which are provided to further illustrate embodiments of the invention and constitute a part of this invention, are not intended to limit the scope of the invention.

[0048] Figure 1 This is a flowchart illustrating a numerical simulation method for flow around a rotationally symmetric geometry.

[0049] Figure 2 A schematic diagram of the coordinate axes and indices of a body-fitted structure mesh with a singular axis for a supersonic ball head;

[0050] Figure 3 This is a schematic diagram of the wall pressure distribution in the calculation results of the inviscid flow around the supersonic ball head in Case 1.

[0051] Figure 4 This is a schematic diagram of the pressure distribution on the symmetry plane in the calculation results of the inviscid flow around the supersonic ball head in Case 1.

[0052] Figure 5 This is a schematic diagram of the wall pressure distribution in the calculation results of the viscous flow around the supersonic ball head in Case 2.

[0053] Figure 6 This is a schematic diagram of the pressure distribution on the symmetry plane in the calculation results of the viscous flow around the supersonic ball head in Case 2. Detailed Implementation

[0054] To better understand the above-mentioned objectives, features, and advantages of the present invention, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be noted that, where there is no conflict, the embodiments of the present invention and the features thereof can be combined with each other.

[0055] Many specific details are set forth in the following description in order to provide a full understanding of the invention. However, the invention may also be practiced in other ways different from those described herein, and therefore the scope of protection of the invention is not limited to the specific embodiments disclosed below.

[0056] Example 1;

[0057] Please refer to Figure 1 , Figure 1 This is a flowchart illustrating a numerical simulation method for flow around a rotationally symmetric geometric body. Embodiment 1 of the present invention provides a numerical simulation method for flow around a rotationally symmetric geometric body, the method comprising:

[0058] Step 1: Based on mesh generation software, generate a body-fitted mesh containing singular axes required for numerical simulation of flow around a rotationally symmetric geometry;

[0059] Step 2: Set virtual mesh points outside the singular axes of the body-fitted mesh;

[0060] Step 3: Based on the spatial coordinates of the internal grid points of the body-fitted structure mesh, provide the spatial coordinates of the virtual grid points;

[0061] Step 4: Based on the spatial coordinate values ​​of virtual grid points, singular axis boundary grid points, and internal grid points, calculate the grid transformation derivatives and Jacobian determinants of singular axis boundary grid points and internal grid points, which are used to discretize the spatial convection derivatives and spatial viscous derivatives of the control equations;

[0062] Step 5: Based on the mesh transformation derivative and Jacobian determinant at the internal mesh points of the body-fitted structure mesh, the mesh transformation derivative and Jacobian determinant at the virtual mesh points are used to discretize the spatial convection derivative and spatial viscosity derivative terms of the control equations;

[0063] Step 6: Based on the flow field variable values ​​at the internal grid points of the body-fitted structure mesh, provide the flow field variable values ​​at the virtual grid points;

[0064] Step 7: For the viscous flow problem around a rotationally symmetric geometry, based on the flow field variable values ​​at the boundary grid points of the singular axis, the internal grid points, and the virtual grid points, calculate the first derivative values ​​of the flow field variables at the boundary grid points of the singular axis and the internal grid points, and provide the first derivative values ​​of the flow field variables at the virtual grid points for discretizing the spatial viscous derivative terms of the control equations.

[0065] Step 8: Based on the mesh transformation derivatives, Jacobian determinants, flow field variable values, and first-order derivative values ​​of the flow field variables on virtual mesh points, singular axis boundary mesh points, and internal mesh points, discretize the spatial convection derivative term and spatial viscous derivative term of the control equations;

[0066] Step 9: Based on the spatial convection derivative and spatial viscous derivative of the discretized control equations, update the flow field variable values ​​around the rotationally symmetric geometry iteratively through time-progression.

[0067] Step 10: Based on the updated flow field variable values ​​around the rotationally symmetric geometry, calculate the flow field variable values ​​on the singular axis of the rotationally symmetric geometry;

[0068] Step 11: Determine whether the total number of calculation time steps or the calculation convergence condition has been met. If not, return to step 6. If met, end the simulation and obtain the flow field data of the rotationally symmetric geometry corresponding to the input data based on all calculation results.

[0069] The following is a detailed explanation of this method:

[0070] This invention addresses the numerical simulation of flow around rotationally symmetric geometries. It proposes a finite difference method for handling singular axis boundaries based on virtual grid points and employing unified high-precision flux reconstruction across the entire computational domain. The method is implemented in the following 11 steps:

[0071] Step 1: Using mesh generation software such as Pointwise, generate a body-fitted mesh containing singular axes required for numerical simulations of flow around rotationally symmetric geometry. Specific requirements are as follows: Figure 2 (Taking a spherical flow mesh as an example) The coordinates in physical space are (x, y, z), and the singular axis is conventionally defined to lie on the x-axis; the coordinates in computational space are (ξ, η, ζ), and ξ is conventionally defined to correspond to the normal mesh line and denoted by the index i, where i = I. max The surface represents the object, η corresponds to the flow direction grid line and is denoted by the index j, where j=1 represents the singular axis, and ζ corresponds to the circumferential grid line and is denoted by the index k. The number of grid points on the normal grid line is I. max , i.e. i=1, …, I max The number of grid points flowing along the grid line is J. max That is, j=1,…,J max The number of grid points on the circumferential grid line is K. max That is, k=1, …, K max Here, it is required that the grid points on the circumferential grid lines be uniformly distributed, and that the number of circumferential grid points K is [value missing]. max It should be an odd number.

[0072] Step 2: Set up virtual mesh points outside the singular axes of the rotationally symmetric geometry's body-fitted mesh. The mesh points on the singular axes are expanded circumferentially in the computational space to form a plane with index j=1. For each mesh point on the plane, set up several layers of virtual mesh points in the negative direction of index j. The number of virtual mesh point layers is determined based on the accuracy of the finite difference method. For the fifth-order accurate finite difference WENO method, the number of virtual mesh point layers is set to Nghost=3.

[0073] Step 3: Given the spatial coordinates of virtual mesh points outside the singular axis of the rotationally symmetric geometry, these coordinates are used to calculate the derivative of the mesh transformation. The spatial coordinates of the current virtual mesh points are set based on the mesh coordinates of the mesh points inside the body-fitted mesh, with the specific correspondence as follows:

[0074] (1)

[0075] Where jg is the index of the virtual grid point in the η direction, and jt and kt are the indices of the internal grid points of the corresponding virtual grid point in the η and ζ directions, respectively. This represents the number of layers in the virtual grid.

[0076] Step 4: Calculate the mesh transformation derivatives and Jacobian determinants at the boundary and interior mesh points of the singular axis of the rotationally symmetric geometry. These are used to discretize the spatial convection and viscous derivative terms of the governing equations. Based on the virtual mesh points and the spatial coordinates at the boundary and interior mesh points of the singular axis, a consistent high-precision central difference method (such as fourth- or sixth-order central difference) is used to calculate the mesh transformation derivatives and Jacobian determinants at the boundary and interior mesh points of the singular axis.

[0077] The method for calculating the derivative of the mesh transformation from physical space to computational space is as follows:

[0078] (2)

[0079] Where J is the Jacobian determinant of the coordinate transformation. , , , , , , , and It is the derivative of the corresponding grid transformation.

[0080] It is important to note that at grid points on the singular axis boundaries, the derivative of the grid transformation has the following expression: , , , , , and In actual numerical simulations, to avoid calculating the reciprocal of zero, at the boundary grid points of the singular axis, let:

[0081] (3)

[0082] Step 5: Given the mesh transformation derivatives and Jacobian determinants at virtual mesh points outside the singular axis of the rotationally symmetric geometry, use them to discretize the space convection derivative and viscous derivative terms of the governing equations. Based on the mesh transformation derivatives and Jacobian determinants at the internal mesh points, determine the mesh transformation derivatives and Jacobian determinants at the virtual mesh points according to the following correspondence:

[0083] (4)

[0084] Step 6: Given the flow field variable values ​​at virtual mesh points outside the singular axis of the rotationally symmetric geometry. Set the flow field variable values ​​at the current virtual mesh point based on the flow field variable values ​​at the internal mesh points, with the specific correspondence as follows:

[0085] (5)

[0086] Among them, flow field variables , , , and These are density, the velocity component in the x-direction, the velocity component in the y-direction, the velocity component in the z-direction, and pressure.

[0087] Step 7: For the viscous flow problem around a rotationally symmetric geometry, calculate the first derivative values ​​of the flow field variables at the singular axis boundary grid points and internal grid points, based on the flow field variable values ​​at the singular axis boundary grid points, internal grid points, and virtual grid points. Also, provide the first derivative values ​​of the flow field variables at the virtual grid points for discretizing the spatial viscous derivative terms of the governing equations. For viscous flow problems, the construction of the viscous flux in the viscous terms of the governing equations requires the first derivatives of the flow field variables. If the flow is axisymmetric, the first derivative values ​​of the flow field variables at the virtual grid points are given according to the following relationship:

[0088] (6)

[0089] If the flow is non-axisymmetric, the above relationship does not hold. The first derivative of the flow field variables at virtual grid points can be calculated using high-precision one-sided difference methods (such as fourth- or sixth-order precision one-sided difference) based on the flow field variable values ​​at virtual grid points and internal grid points.

[0090] Step 8: Based on the grid transformation derivatives, Jacobian determinants, flow field variable values, and first-order derivative values ​​of the flow field variables at virtual grid points, singular axis boundary grid points, and internal grid points, discretize the spatial convection derivative and viscous derivative terms of the control equations. At the singular axis boundary and its adjacent region, a high-precision flux reconstruction difference method (such as the fifth-order precision finite difference WENO method) consistent with the method used for internal grid points is used to discretize the convection derivative term, and a high-precision central difference method (such as the fourth-order or sixth-order central difference method) consistent with the method used for internal grid points is used to discretize the viscous derivative term, thus obtaining the spatial derivative term of the discretized control equations. The fifth-order precision finite difference WENO method can be found in the references; this embodiment of the invention will not elaborate further.

[0091] 1. GS Jiang, C.-W. Shu, Efficient implementation of weighted ENOschemes, J. Comput. Phys. 126 (1996) 202–228.

[0092] 2. R. Borges, M. Carmona, B. Costa, WS Don, An improved weightedessentially non-oscillatory scheme for hyperbolic conservation laws, J.Comput. Phys. 227 (2008) 3101–3211.

[0093] Step 9: Based on the spatial convection derivative and viscous derivative terms of the discretized control equations, update the flow field variable values ​​around the rotationally symmetric geometry iteratively through time-progression.

[0094] Step 10: Based on the updated flow field variable values, the flow field variable values ​​on the singular axis of the rotationally symmetric geometry are obtained by interpolation and then averaging.

[0095] Specifically, the following is a method: First, for a certain index position (i, k), interpolation is performed along the j direction to obtain the flow field variables on the singular axis (j=1). Different k positions may interpolate different flow field variables on the axis. Second, the obtained values ​​are averaged along the circumferential direction (traversing all points where k takes values) and then used as the flow field variable values ​​on the axis. Taking variable u as an example, there are two interpolation methods: (1) Zero gradient condition: For inviscid flow, second-order difference discretization is used, and for viscous flow, third-order difference discretization is used. (2) Equal rate of change, that is, the second derivative of the flow direction at the second point outside the singular axis is zero, specifically realized as u i =2ui+1 - u i+2 .

[0096] Step 11: Determine whether the total number of calculation time steps or the calculation convergence condition has been met. If not, return to step 6. If met, end the simulation and obtain the flow field data of the rotationally symmetric geometry corresponding to the input data based on all calculation results.

[0097] This invention solves the compatibility problem between the high-precision flux reconstruction finite difference method and the handling of the singular axis boundary of the body-fitted mesh by setting virtual mesh points outside the singular axis boundary of the body-fitted mesh and reasonably setting the mesh coordinates, mesh transformation derivatives, flow field variables and their first derivatives on the virtual mesh points. It uses a unified high-precision flux reconstruction difference method to discretize the spatial derivative terms of the control equations in the entire computational domain, including the singular axis.

[0098] Example 2;

[0099] Based on Example 1, Example 2 will be described and illustrated with specific implementation cases.

[0100] Implementation Case 1 is: Supersonic flow around a non-sticky ball head;

[0101] The calculation conditions for supersonic inviscid spherical head flow are as follows: incoming Mach number M = 3.0, angle of attack zero. The spherical head surface is a slip boundary, with supersonic inlet and outlet conditions. The circumferential grid has 31 uniformly distributed points; the normal grid has 101 points; and the flow-directed grid has 61 points. The governing equations are the Euler equations in a general curvilinear coordinate system. The convection term spatial discretization uses the fifth-order accurate finite-difference WENO method based on flux reconstruction, and the time propagation uses the third-order accurate TVD Runge-Kutta method. Figures 3-4 It is the calculation result, including the pressure distribution on the wall and the pressure distribution on the symmetry plane.

[0102] Implementation Case 2 is: Supersonic viscous ball head flow;

[0103] Calculation conditions for supersonic viscous flow around a ball head: laminar flow, angle of attack zero, incoming Mach number M=3.0, Reynolds number based on ball head radius Re=1×10⁻⁶. 5 The spherical head surface is subjected to no-slip conditions, and the surface pressure is obtained from the zero pressure gradient along the normal direction of the surface. Supersonic inlet and outlet conditions are implemented. The circumferential grid has 31 points, uniformly distributed; the normal grid has 101 points; and the flow-directed grid has 61 points. The governing equations are the Navier-Stokes equations in a general curvilinear coordinate system. The spatial discretization of the convection term uses the fifth-order accurate finite-difference WENO method based on flux reconstruction, the spatial discretization of the viscous term uses the sixth-order accurate central difference method, and the time-propagation uses the third-order accurate TVD Runge-Kutta method. Figures 5-6 It is the calculation result, including the pressure distribution on the wall and the pressure distribution on the symmetry plane.

[0104] Although preferred embodiments of the invention have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including both the preferred embodiments and all changes and modifications falling within the scope of the invention.

[0105] Obviously, those skilled in the art can make various modifications and variations to this invention without departing from its spirit and scope. Therefore, if these modifications and variations fall within the scope of the claims of this invention and their equivalents, this invention also intends to include these modifications and variations.

Claims

1. A numerical simulation method for flow around a rotationally symmetric geometry, characterized in that, The method includes: Step 1: Based on mesh generation software, generate a body-fitted mesh containing singular axes required for numerical simulation of flow around a rotationally symmetric geometry; Step 2: Set virtual mesh points outside the singular axes of the body-fitted mesh; Step 3: Based on the spatial coordinates of the internal grid points of the body-fitted structure mesh, provide the spatial coordinates of the virtual grid points; Step 4: Based on the spatial coordinate values ​​of virtual grid points, singular axis boundary grid points, and internal grid points, calculate the grid transformation derivatives and Jacobian determinants of singular axis boundary grid points and internal grid points, which are used to discretize the spatial convection derivatives and spatial viscous derivatives of the control equations; Step 5: Based on the mesh transformation derivative and Jacobian determinant at the internal mesh points of the body-fitted structure mesh, the mesh transformation derivative and Jacobian determinant at the virtual mesh points are used to discretize the spatial convection derivative and spatial viscosity derivative terms of the control equations; Step 6: Based on the flow field variable values ​​at the internal grid points of the body-fitted structure mesh, provide the flow field variable values ​​at the virtual grid points; Step 7: For the viscous flow problem around a rotationally symmetric geometry, based on the flow field variable values ​​at the boundary grid points of the singular axis, the internal grid points, and the virtual grid points, calculate the first derivative values ​​of the flow field variables at the boundary grid points of the singular axis and the internal grid points, and provide the first derivative values ​​of the flow field variables at the virtual grid points for discretizing the spatial viscous derivative terms of the control equations. Step 8: Based on the mesh transformation derivatives, Jacobian determinants, flow field variable values, and first-order derivative values ​​of the flow field variables on virtual mesh points, singular axis boundary mesh points, and internal mesh points, discretize the spatial convection derivative term and spatial viscous derivative term of the control equations; Step 9: Based on the spatial convection derivative and spatial viscous derivative of the discretized control equations, update the flow field variable values ​​around the rotationally symmetric geometry iteratively through time-progression. Step 10: Based on the updated flow field variable values ​​around the rotationally symmetric geometry, calculate the flow field variable values ​​on the singular axis of the rotationally symmetric geometry; Step 11: Determine whether the total number of calculation time steps or the calculation convergence condition has been met. If not, return to step 6. If met, end the simulation and obtain the flow field data of the rotationally symmetric geometry corresponding to the input data based on all calculation results.

2. The numerical simulation method for flow around a rotationally symmetric geometry according to claim 1, characterized in that, Rotationally symmetric geometry consists of a spherical head and a cylindrical body for an aircraft or spacecraft.

3. The numerical simulation method for flow around a rotationally symmetric geometry according to claim 1, characterized in that, Step 1 specifically includes: Based on mesh generation software, a body-fitted mesh containing singular axes is generated for numerical simulation of flow around a rotationally symmetric geometry. The physical coordinates of the body-fitted mesh are (x, y, z), with the singular axes conventionally falling on the x-axis. The computational coordinates of the body-fitted mesh are (ξ, η, ζ), where the ξ direction corresponds to the normal mesh line and is denoted by the index i, i=I. max The surface represents the object, the η direction corresponds to the flow grid line and is denoted by the index j, j=1 represents the singular axis, the ζ direction corresponds to the circumferential grid line and is denoted by the index k; the number of grid points on the normal grid line is I. max The number of grid points flowing along the grid line is J. max The number of grid points on the circumferential grid line is K. max K max For odd values, the grid points are evenly distributed along the circumferential grid lines.

4. The numerical simulation method for flow around a rotationally symmetric geometry according to claim 3, characterized in that, Step 2 specifically includes: The grid points on the singular axis are expanded circumferentially in the computational space to form a plane j=1. For each grid point on this plane, several layers of virtual grid points are set in the negative direction of index j.

5. The numerical simulation method for flow around a rotationally symmetric geometry according to claim 3, characterized in that, The spatial coordinates and flow field variables at virtual grid points are obtained as follows: The spatial coordinates and flow field variables of the current virtual grid point are set based on the internal grid points of the body-fitted structure mesh. The specific correspondence is as follows: ； Among them, flow field variables and These are density, the velocity component in the x-direction, the velocity component in the y-direction, the velocity component in the z-direction, and pressure; jg It is the index of the virtual grid point in the η direction. jt and kt These are the indices of the internal grid points of the corresponding virtual grid points in the η and ζ directions, respectively. This represents the number of layers in the virtual grid.

6. The numerical simulation method for flow around a rotationally symmetric geometry according to claim 3, characterized in that, The method for calculating the derivative of the mesh transformation from physical space to computational space in step 4 is as follows: ； Where J is the Jacobian determinant of the coordinate transformation. These are the calculation of spatial coordinates. The derivative of the mesh transformation in the (x, y, z) direction in physical space. For partial derivative operators, These are the physical space coordinates (x, y, z) in the computational space (x, y, z). The derivative of the grid transformation in the direction of ) Specifically, at the boundary grid points of the singular axis, let: ； in, , .

7. The numerical simulation method for flow around a rotationally symmetric geometry according to claim 5, characterized in that, Step 5 specifically involves determining the grid transformation derivative and Jacobian determinant at the virtual grid points according to the following formula: ； in, , , , , , , , and Derivatives of grid transformation and The value at the virtual grid point (i, jg, k). , , , , , , , and Derivatives of grid transformation and The value at the internal grid point (i, jt, k). Jacobian determinant in virtual grid points The value on, The value of the Jacobian determinant at the internal grid point (i, jt, k).

8. The numerical simulation method for flow around a rotationally symmetric geometry according to claim 5, characterized in that, In step 7, if the flow in the viscous flow problem around a rotationally symmetric geometry is axisymmetric, then the first derivative values ​​of the flow field variables at the virtual grid points are obtained according to the following formula: ； in, and Let be the values ​​of the first derivative of the velocity component u in the x-direction at the virtual grid point (i, jg, k) and the internal grid point (i, jt, k), respectively. and Let be the values ​​of the first derivative of the velocity component u in the y-direction at the virtual grid point (i, jg, k) and the internal grid point (i, jt, k), respectively. and Let be the values ​​of the first derivative of the velocity component u in the z-direction at the virtual grid point (i, jg, k) and the internal grid point (i, jt, k), respectively. and Let be the values ​​of the first derivative of the velocity component v in the x-direction at the virtual grid point (i, jg, k) and the internal grid point (i, jt, k), respectively. and Let be the values ​​of the first derivative of the velocity component v in the y-direction at the virtual grid point (i, jg, k) and the internal grid point (i, jt, k), respectively. and Let be the values ​​of the first derivative of the velocity component v in the z-direction at the virtual grid point (i, jg, k) and the internal grid point (i, jt, k), respectively. and Let be the values ​​of the first derivative of the velocity component w in the x-direction at the virtual grid point (i, jg, k) and the internal grid point (i, jt, k), respectively. and Let be the values ​​of the first derivative of the velocity component w in the y-direction at the virtual grid point (i, jg, k) and the internal grid point (i, jt, k), respectively. and Let be the values ​​of the first derivative of the velocity component w in the z-direction at the virtual grid point (i, jg, k) and the internal grid point (i, jt, k), respectively. and Let be the values ​​of the first derivative of temperature t in the x-direction at the virtual grid point (i, jg, k) and the internal grid point (i, jt, k), respectively. and Let be the values ​​of the first derivative of temperature t in the y-direction at the virtual grid point (i, jg, k) and the internal grid point (i, jt, k), respectively. and Let be the values ​​of the first derivative of temperature t in the z-direction at virtual grid point (i, jg, k) and internal grid point (i, jt, k), respectively. In step 7, if the flow in the viscous flow problem around a rotationally symmetric geometry is non-axisymmetric, then the first derivative of the flow field variables at the virtual grid points is calculated using the one-sided difference method based on the flow field variable values ​​at the virtual grid points and the internal grid points.

9. The numerical simulation method for flow around a rotationally symmetric geometry according to claim 1, characterized in that, Step 8 includes: based on the grid transformation derivatives, Jacobian determinants, flow field variable values, and first-order derivative values ​​of the flow field variables at virtual grid points, singular axis boundary grid points, and internal grid points, discretizing the spatial convection derivative and viscous derivative terms of the control equations; at the singular axis boundary and its adjacent region, discretizing the convection derivative terms of the control equations using the flux reconstruction difference method consistent with the method used for internal grid points; and discretizing the viscous derivative terms of the control equations using the central difference method consistent with the method used for internal grid points, to obtain the spatial derivative terms of the discretized control equations.

10. The numerical simulation method for flow around a rotationally symmetric geometry according to claim 3, characterized in that, Step 10 specifically includes: Based on the updated flow field variable values ​​around the rotationally symmetric geometry, the flow field variable values ​​on the singular axis are interpolated along the j direction for the index position (i, K); the interpolated flow field variables are averaged along the circumferential direction as the flow field variable values ​​on the singular axis of the rotationally symmetric geometry.

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