A method for calculating adiabatic wall temperature suitable for non-orthogonal grid

Abstract

The present application relates to the field of computational fluid dynamics, and especially relates to a kind of adiabatic wall temperature calculation method suitable for non-orthogonal grid.The method is based on the high-order accuracy discrete expression of temperature in wall normal gradient, and wall temperature is solved by coupling iteration method.Firstly, construct the grid of body structure and calculate grid derivative and grid transformation jacobian;Subsequently, high-order finite difference discrete solution is carried out to flow control equation in the calculation coordinate system;At wall, according to the discrete equation of temperature normal gradient of adiabatic boundary condition, and wall temperature is solved by iteration method.The method strictly satisfies the temperature normal gradient of adiabatic wall to be zero in theory, is not dependent on grid orthogonality, is suitable for high-precision numerical calculation under the condition of orthogonal and non-orthogonal grid, and can effectively improve the stability and accuracy of adiabatic wall temperature calculation.

Description

Technical Field

[0001] This invention relates to the field of computational fluid dynamics, and more particularly to a method for calculating the temperature of adiabatic walls applicable to non-orthogonal grids. Background Technology

[0002] The Navier-Stokes equations (NS equations) are a set of governing equations for fluid dynamics, consisting of the mass conservation equation, momentum conservation equation, and energy conservation equation. These equations describe the evolution of flow field variables such as density, velocity, pressure, and temperature. Computational Fluid Dynamics (CFD) obtains approximate solutions to complex flow problems by numerically discretizing these governing equations. In practical engineering applications, when fluid flows over a solid wall, heat exchange occurs between the fluid and the wall due to fluid viscosity and convective heat transfer, resulting in wall temperature or heat flux conditions at the boundary.

[0003] Accurate calculation of wall temperature is crucial for engineering design. When wall temperature exceeds the design tolerance, it can lead to serious consequences. For example, in aero-engines, turbine blades operate in a high-temperature exhaust environment for extended periods, and their safe operation depends on precise control of wall temperature. During atmospheric reentry, the cabin surface generates significant heat through intense friction with the air; the accuracy of wall temperature prediction directly impacts the design of thermal protection structures and flight safety. Therefore, high-precision calculation of wall temperature is one of the key issues in CFD numerical simulations.

[0004] However, existing methods for handling wall temperature are mostly based on virtual point assignment strategies. When the mesh near the wall is non-orthogonal, severely oblique, or has poor smoothness, the numerical solution process becomes highly sensitive to mesh quality, leading to non-physical oscillations or even convergence failure, severely impacting computational stability and result reliability. Therefore, it is necessary to propose a wall temperature calculation method applicable to non-orthogonal mesh conditions to improve the stability and accuracy of numerical calculations. Summary of the Invention

[0005] This invention proposes a method for calculating the temperature of adiabatic walls applicable to non-orthogonal meshes. This method is based on a fully high-order precision discretization of the temperature normal gradient of the adiabatic wall and employs a coupled iterative approach to calculate the wall temperature. Specifically, it includes the following steps: Step 1: Generate computational mesh: Generate a body-fitted structured mesh around the geometry to be simulated, and achieve coverage of the flow field region through multiple connected structured mesh blocks; Step 2: Calculate mesh feature information: Under the finite difference discretization framework, establish the coordinate transformation relationship between the computation coordinate system and the rectangular coordinate system, transform the flow control equations in the rectangular coordinate system to the computation coordinate system, and calculate the mesh derivative and mesh transformation Jacobian. The mesh derivative and mesh transformation Jacobian are discretized using a symmetric conservation form. Among them, the rectangular coordinate system The flow control equations are as follows: , Let be the flow field variables to be determined. , and All are about The function; Calculate coordinate system With rectangular coordinate system The coordinate transformation relationship between them is as follows: , Calculate coordinate system The flow control equations are as follows: , in: , In static grid In this context, the expression for the grid derivative is: , Where the subscripts denote partial derivatives, Representing coordinates For calculating coordinates Partial derivatives in direction; The expression for the Jacobian of the mesh transformation is: .

[0006] The symmetric conservation expression for the grid derivative is: , The symmetric conservation expression for the Jacobian of the grid transformation is: .

[0007] Step 3: Discretization and solution of flow control equations: In the computational coordinate system, the flow control equations are discretized using a high-order finite difference scheme, and the half-nodal flux is calculated using the numerical flux method to solve for the flow field variables. The discretization is performed using a high-order finite difference scheme, and the discretization expression is as follows: , Among them, the numerical flux of the half node for: , In this invention, the numerical flux of the half-node Several commonly used numerical flux calculation methods can be employed, among which the Roe flux calculation method, which is frequently used, can be described as follows: , in, and The flow field variables for the semi-nodes are calculated using third-order linear explicit interpolation, and the calculation formula is as follows: .

[0008] Step 4: Calculation of adiabatic wall temperature: Based on the adiabatic boundary conditions of the wall, a discrete expression for the temperature gradient along the normal direction of the wall is established, and the wall temperature is solved by a coupled iterative method until the preset convergence condition is met.

[0009] The wall insulation boundary conditions are as follows: , in For temperature, Let represent the unit normal vector of the solid wall, where , and Cartesian coordinate system The unit coordinate basis vector, , and They are the unit normal vectors exist , and The projection values ​​in the direction, and satisfying the normalization condition. .

[0010] A discrete expression for the temperature gradient along the wall normal is established, specifically as follows: According to the mathematical definition of the directional derivative: , Based on the aforementioned coordinate transformation relationship, the spatial derivative of temperature in the calculation coordinate system The following expression is: , The last term in the above equation is the geometric conservation law error term, which has a value of zero when using the grid derivative symmetric conservation calculation form. Therefore, we can further obtain: , Similarly, we have: , The expression for the temperature gradient along the wall normal is: , Consider the adiabatic boundary conditions of the wall: , Without loss of generality, we take the computation coordinate system. If the downward direction is the normal direction of the wall mesh, then: , The temperature gradient at the solid boundary is discretized using a high-order precision difference scheme: , After some organization, we get: , The above equation does not explicitly give the temperature values ​​on all wall surfaces; the right side of the equation... direction and The spatial derivative terms of the directions implicitly contain the temperature of the wall, and all are discretized using the central difference scheme of the interior points. The remaining temperature terms on the right side of the equation are non-boundary interior field temperature values, which are all considered known values ​​during the boundary treatment process.

[0011] This invention uses an iterative solution based on the above-mentioned temperature relationship of the adiabatic wall. The expression for the iterative solution process is as follows: , Among them, the initial value is assigned to the wall temperature. The wall temperature value is calculated based on the above expression. And calculate the temperature difference. Then the wall temperature value Substituting the above expression into the calculation yields... This cycle continues until the temperature difference is reached. Until it is small enough or the maximum number of iterations is reached.

[0012] This allows us to obtain the wall temperature that accurately meets the insulation requirements. The wall flow field data calculated in the above steps is then output according to the post-processing requirements to facilitate the extraction and application of flow field characteristic data in the post-processing process.

[0013] The adiabatic wall temperature calculation method proposed in this invention theoretically guarantees the temperature normal gradient condition of the wall, regardless of whether the grid near the wall is orthogonal or not. It is fully applicable to the accurate calculation of wall temperature for both orthogonal and non-orthogonal grids, breaking the limitation of traditional wall temperature processing methods that are not applicable to non-orthogonal grids.

[0014] The method for calculating the temperature of an insulating wall surface proposed in this invention has the following beneficial effects: (1) This invention is based on the gradient expression of temperature in the normal direction of the wall, which theoretically strictly satisfies the adiabatic boundary condition, that is, the normal gradient of the wall temperature is zero, and solves the wall temperature through a coupled iterative method. This method does not depend on the orthogonality of the mesh near the wall and can be applied to the calculation of wall temperature under both orthogonal and non-orthogonal mesh conditions. This overcomes the problem of poor applicability of traditional wall temperature processing methods under non-orthogonal mesh conditions and improves the versatility and stability of the method. (2) The present invention directly calculates the wall temperature based on the temperature gradient information obtained in the flow solution process. Compared with the traditional one-dimensional assignment method based on virtual points, it avoids the error propagation problem caused by artificially constructing virtual points, and can better maintain the high-order precision of the numerical discreteness at the wall, improve the accuracy of the wall temperature calculation, so that the obtained flow field and wall information are more in line with the physical reality. Attached Figure Description

[0015] To more clearly illustrate the technical solution of the present invention, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other embodiments can be obtained based on these drawings without creative effort.

[0016] Figure 1 This is a diagram illustrating the division of the calculation area into equal parts.

[0017] Figure 2 This is a schematic diagram showing the alternating distribution of solution points and flux points.

[0018] Figure 3 (a) is the orthogonal grid cylindrical turbulence. Temperature contour map near the wall surface. Figure 3 (b) is a cylindrical turbulence with a 45-degree oblique grid. Temperature contour map near the wall. Detailed Implementation

[0019] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0020] It should be noted that the terms "first," "second," etc., in the specification, claims, and accompanying drawings of this invention are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments of the invention described herein can be implemented in orders other than those illustrated or described herein. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover a non-exclusive inclusion; for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0021] Specifically, the method for calculating the temperature of adiabatic walls using non-orthogonal grids proposed in this invention includes the following steps: Step 1: Generate the computational mesh: Generate a body-fitted mesh around the geometry to be simulated, and cover the flow field region using multiple connected mesh blocks. The multiple body-fitted mesh blocks are mainly to facilitate subsequent numerical simulation of the flow problem using a high-order finite difference numerical scheme.

[0022] Step 2: Calculate mesh feature information: Under the finite difference discretization framework, establish the coordinate transformation relationship between the computation coordinate system and the rectangular coordinate system, transform the flow control equations in the rectangular coordinate system to the computation coordinate system, and calculate the mesh derivative and mesh transformation Jacobian.

[0023] To improve the adaptability of high-order precision finite difference schemes to the simulation of complex configuration turbulence, the detailed discretization of the grid derivative and grid transformation Jacobian under the finite difference scheme is as follows: Among them, the rectangular coordinate system The flow control equations are as follows: , Let be the flow field variables to be determined. , and All are about The function; When performing finite difference discretization on multiple body-fitted mesh structures, it is necessary to transform them to a computational coordinate system and establish the computational coordinate system. With rectangular coordinate system A one-to-one coordinate transformation relationship exists between them, where the coordinate system is calculated. With rectangular coordinate system The coordinate transformation relationship between them is as follows: , Calculate coordinate system The flow control equations are as follows: , in: , In static grid In this context, the expression for the grid derivative is: , Where the subscripts denote partial derivatives, Representing coordinates For calculating coordinates Partial derivatives in direction; The expression for the Jacobian of the mesh transformation is: .

[0024] To strictly satisfy the geometric conservation law while accurately reflecting the geometric properties of the computational grid, this invention adopts the following symmetric conservation calculation form for the grid derivative: , Meanwhile, the symmetric conservation calculation form of the grid transformation Jacobian is adopted as follows: .

[0025] Step 3: Discretization and solution of flow control equations: In the computational coordinate system, the flow control equations are discretized using a high-order finite difference scheme, and the half-nodal flux is calculated using the numerical flux method to solve for the flow field variables. Since the finite difference method based on structured grids always simulates multidimensional physical flow problems by solving them in a dimension-by-dimensional discrete manner, the three directional derivatives in the computation coordinate system are... , and The discrete processes are independent and the algorithms are similar, therefore this invention only uses... This paper uses the calculation of a flow control equation as an example to illustrate the high-order accuracy finite difference discretization method in the computation coordinate system.

[0026] The discrete region in the computational coordinate system is divided into Equal division, corresponding structural mesh in Distributed in directions Grid points, such as Figure 1 As shown in the figure. The solution points and flux points satisfy an alternating distribution pattern, as follows: Figure 2 As shown.

[0027] It is particularly noteworthy that the boundary points of the computational domain contain both solution points and flux points.

[0028] The flow control equations in the computational coordinate system are discretized using a high-order finite difference scheme, and the discretized expression is: , And the difference scheme near the left boundary is: , The difference scheme near the right boundary can be obtained directly by symmetry.

[0029] Among them, the numerical flux of the half node for: , In this invention, the numerical flux of the half-node Several commonly used numerical flux calculation methods can be employed, among which the Roe flux calculation method, which is frequently used, can be described as follows: , Among them, the flow field variables of the half-node and The calculation formula is obtained from third-order linear explicit interpolation: .

[0030] Step 4: Calculation of adiabatic wall temperature: Based on the adiabatic boundary conditions of the wall, a discrete expression for the temperature gradient along the normal direction of the wall is established, and the wall temperature is solved by a coupled iterative method until the preset convergence condition is met.

[0031] The thermal insulation conditions of a solid wall surface can generally be described as follows: , in For temperature, Let represent the unit normal vector of the solid wall, where , and Cartesian coordinate system The unit coordinate basis vector, , and They are the unit normal vectors exist , and The projection values ​​in the direction, and satisfying the normalization condition. .

[0032] According to the mathematical definition of the directional derivative:

[0033] According to the mathematical definition of the directional derivative: , Based on the aforementioned coordinate transformation relationship, the spatial derivative of the temperature in the calculation coordinate system... The following can be expressed as: , The last term in the above equation is the geometric conservation law error term. When the grid derivative symmetric conservation calculation form described above is used, its value is zero, and we can further obtain: , Similarly, we have: , Furthermore, the temperature gradient along the wall normal can be expressed as: , Consider the adiabatic boundary conditions of the wall: , Without loss of generality, we take the computation coordinate system. If the downward direction is the normal direction of the wall mesh, then: , The temperature gradient at the solid boundary is discretized using a high-order precision difference scheme: , After some organization, we get: , The above equation does not explicitly give the temperature values ​​on all wall surfaces; the right side of the equation... direction and The spatial derivative terms of the directions implicitly contain the temperature of the wall, and all are discretized using the central difference scheme of the interior points. The remaining temperature terms on the right side of the equation are non-boundary interior field temperature values, which are all considered known values ​​during the boundary treatment process.

[0034] In actual calculations, this invention uses the above-mentioned temperature relationship of the adiabatic wall to perform iterative solutions. The specific iterative solution process can be described as follows: , That is, assigning an initial value to the wall temperature. The wall temperature value is calculated based on the above relationship. And calculate the temperature difference. Then the wall temperature value Substituting the above expression into the calculation yields... This cycle continues until the temperature difference is reached. The number of iterations continues until the value is sufficiently small or the maximum number of iterations is reached. Based on this, the wall temperature that accurately meets the adiabatic requirements can be obtained.

[0035] The wall flow field data obtained from the above steps are output according to the post-processing requirements to facilitate the extraction and application of flow field feature data in the post-processing process.

[0036] This patented method for calculating the temperature of adiabatic walls using non-orthogonal meshes calculates the wall temperature based on the temperature normal gradient through a coupled iterative approach. By strictly adhering to the expression for the temperature normal gradient and employing a coupled iterative method, the wall temperature is theoretically guaranteed to be strictly zero. This method is applicable to both orthogonal and non-orthogonal meshes, overcoming the limitation of traditional temperature processing methods that are not suitable for non-orthogonal wall meshes.

[0037] It should be noted that, for those skilled in the art, the technical features in the above embodiments can be freely combined, and the resulting technical solutions also belong to the embodiments disclosed in this invention.

[0038] Furthermore, without departing from the principles of this invention, several improvements and modifications can be made to this invention, and these improvements and modifications also fall within the protection scope of the claims of this invention.

Claims

1. A method for calculating the temperature of an adiabatic wall surface applicable to non-orthogonal grids, characterized in that, Includes the following steps: Step 1: Generate computational mesh: Generate a body-fitted structured mesh around the geometry to be simulated, and achieve coverage of the flow field region through multiple connected structured mesh blocks; Step 2: Calculate mesh feature information: Under the finite difference discretization framework, establish the coordinate transformation relationship between the computation coordinate system and the rectangular coordinate system, transform the flow control equations in the rectangular coordinate system to the computation coordinate system, and calculate the mesh derivative and mesh transformation Jacobian. The mesh derivative and mesh transformation Jacobian are discretized using a symmetric conservation form. Among them, rectangular coordinate system The flow control equations are as follows: , Let be the flow field variables to be determined. , and All are about The function; This indicates taking the derivative with respect to the variable; Calculate coordinate system With rectangular coordinate system The coordinate transformation relationship between them is as follows: ， Calculate coordinate system The flow control equations are as follows: ， in: ， Step 3: Discretization and solution of flow control equations: In the computational coordinate system, the flow control equations are discretized using a high-order finite difference scheme, and the half-nodal flux is calculated using the numerical flux method to solve for the flow field variables. Step 4: Calculation of adiabatic wall temperature: Based on the adiabatic boundary conditions of the wall, a discrete expression for the temperature gradient along the normal direction of the wall is established, and the wall temperature is solved by a coupled iterative method until the preset convergence condition is met.

2. The method for calculating the temperature of an adiabatic wall surface applicable to non-orthogonal grids according to claim 1, characterized in that, In step 2, The expression for the grid derivative is: ， Where the subscripts denote partial derivatives, Representing coordinates For calculating coordinates Partial derivatives in direction; The expression for the Jacobian of the mesh transformation is: 。 3. The method for calculating the temperature of an adiabatic wall surface applicable to non-orthogonal grids according to claim 2, characterized in that, In step 2, The symmetric conservation expression for the grid derivative is: ， The symmetric conservation expression for the Jacobian of the grid transformation is: 。 4. The method for calculating the temperature of an adiabatic wall surface applicable to non-orthogonal grids according to claim 1, characterized in that, In step 3, the discrete expression of the higher-order finite difference scheme. for: ， in, and The flow field variables for the semi-nodes are calculated using third-order linear explicit interpolation, and the calculation formula is as follows: 。 5. The method for calculating the temperature of an adiabatic wall surface applicable to non-orthogonal grids according to claim 4, characterized in that, Numerical flux of half nodes When using the Roe flux calculation method, the expression is: 。 6. The method for calculating the temperature of an adiabatic wall surface applicable to non-orthogonal grids according to claim 1, characterized in that, In step 4, the wall insulation boundary conditions are as follows: ,in For temperature, Denotes the unit normal vector of the solid wall, where , and Cartesian coordinate system The unit coordinate basis vector, , and They are the unit normal vectors exist , and The projection values ​​in the direction, and satisfying the normalization condition. .

7. The method for calculating the temperature of an adiabatic wall surface applicable to non-orthogonal grids according to claim 1, characterized in that, In step 4, a discrete expression for the temperature gradient along the wall normal is established, specifically as follows: According to the mathematical definition of the directional derivative: , Based on the aforementioned coordinate transformation relationship, the spatial derivative of temperature in the calculation coordinate system The following expression is: , in, The error term of the geometric conservation law has a value of zero when calculated using the grid derivative symmetric conservation method, which can then be further obtained as follows: ， The expression for the temperature gradient along the wall normal is: .

8. The method for calculating the temperature of an adiabatic wall surface applicable to non-orthogonal grids according to claim 1, characterized in that, In step 4, consider the adiabatic boundary conditions of the wall: ， Without loss of generality, we take the computation coordinate system. If the downward direction is the normal direction of the wall mesh, then: 。 9. The method for calculating the temperature of an adiabatic wall surface applicable to non-orthogonal grids according to claim 1, characterized in that, In step 4, the iterative solution is performed based on the temperature relationship of the adiabatic wall. The expression for the iterative solution process is as follows: ， Among them, the initial value is assigned to the wall temperature. The wall temperature value is calculated based on the above expression. And calculate the temperature difference. Then the wall temperature value Substituting the above expression into further calculations yields... This cycle continues until the temperature difference is reached. Until it is small enough or the maximum number of iterations is reached.

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