A wall function turbulent flow calculation method of immersed boundary method

Abstract

The application discloses a wall function turbulent flow calculation method of an immersed boundary method, relates to the field of computational fluid dynamics, and can improve the calculation efficiency of the FIVER immersed boundary method for solving high Reynolds number turbulent flow problems. The application comprises the following steps: combining a Spalart-Allmaras wall function and the FIVER immersed boundary method to establish a turbulent flow calculation method based on a wall function; calculating the tangential velocity and modified eddy viscosity of the wall near the grid by using the wall function; and determining relevant variables required for calculating the flux near the wall. The application reduces the number of wall near the grid required by the FIVER immersed boundary method for calculating turbulent flow problems under high Reynolds number, and improves the ability of the FIVER immersed boundary method for calculating turbulent flow problems under high Reynolds number.

Description

Technical Field

[0001] This invention relates to computational fluid dynamics numerical simulation techniques, and in particular to a method for calculating wall function turbulence based on the immersed boundary method. Background Technology

[0002] In nature and engineering, many problems involve fluid-structure interaction (FSI) under large deformations, such as the flight of birds or insects, or the deployment of parachutes. In these problems, the object undergoes significant structural deformation, causing changes in the surrounding flow field; conversely, these changes affect the object's deformation, resulting in a complex FSI process. Simulating FSI problems has long been a classic and challenging problem in computational science. Traditional body-fitted mesh methods can represent object surfaces well, but when dealing with FSI problems, dynamic meshing is needed to modify the flow field mesh to maintain mesh quality. When an object undergoes large deformations, the flow field mesh quality often drops sharply, even exhibiting negative volume, leading to computational failure. The immersed boundary method is another flow field solution method. Its characteristic is a fixed spatial mesh; even with large deformations, the spatial mesh remains stationary, thus avoiding mesh quality degradation or negative volume issues. It is an effective method for handling large deformation FSI problems and facilitates automatic mesh generation. However, the cost of the immersion boundary method is that the treatment of the wall boundary is more complex than that of the body mesh method, requiring special methods, and the calculation of the integral at the boundary is also more complex than that of the body mesh method.

[0003] Flow problems in nature and engineering are generally high Reynolds number problems. For these problems, traditional body-fitted meshing methods typically use mesh elements with extremely high aspect ratios to accurately capture the boundary layer effect. However, the immersion boundary method generally uses isotropic Cartesian meshes for calculation. Therefore, to accurately capture the boundary layer, a huge number of meshes are required near the object surface, which greatly affects the application of the immersion boundary method in high Reynolds number flow problems. Summary of the Invention

[0004] To address the issue of the enormous number of grids required for calculating turbulence near the object surface using the immersed boundary method at high Reynolds numbers, this invention provides a compressible flow immersed boundary method turbulence calculation method based on Spalart-Allmaras wall functions. The aim is to reduce the number of grids required for turbulence calculations near the object surface by using wall functions, thereby improving the ability of the immersed boundary method to handle turbulent flows.

[0005] To achieve the above objectives, the technical solution provided by this invention is: a method for calculating wall function turbulence using the immersion boundary method, comprising:

[0006] Step 1: Establish the main governing equations, which are the Favre-averaged Navier-Stokes equations.

[0007] Step 2: Establish the turbulence model equations, which are the Spalart-Allmaras turbulence model equations; Step 3: Based on the main governing equations, spatial discretization is performed using the lattice-based finite volume method in the region far from the wall, while the FIVER immersion boundary method is used in the region near the wall, establishing the following one-and-a-half Riemann problem:

[0008]

[0009] in, For the flow field grid points near the object surface, along the local normal direction of the object surface The flow field state, ρ is density, and ξ is a one-dimensional local normal direction coordinate pointing inwards from the object, i.e., along... direction, It is the velocity of the object, Δt n Indicates the current time step;

[0010] Step 4: Calculate the flow field state at the wall based on the exact solution of the one-dimensional half-Riemann problem, thereby obtaining the flux at the wall;

[0011] Step 5: Based on the turbulence model method, spatial discretization is performed in the region far from the wall. In the region near the wall, the density, normal velocity and total energy are calculated using the FIVER immersion boundary method. In particular, the improved wall function is used to calculate the tangential velocity and eddy viscosity coefficient.

[0012] As a preferred technical solution, step 5 further includes:

[0013] Step 51: Determine the internal points of the flow field based on the flux near the wall and the normal of the nearest wall;

[0014] Step 52: Modify the attenuation function in the Spalart-Allmaras turbulence model equations using the distance from the point inside the flow field to the wall as the cutoff distance;

[0015] Step 53: Substitute the tangential velocity value of this point relative to the wall into the wall function to determine the friction velocity of the wall;

[0016] Step 54: Substitute the friction velocity of the wall and the distance from the wall at the flux calculation location into the wall function to obtain its tangential velocity, and obtain the eddy viscosity coefficient from the friction velocity of the wall and the distance from the wall.

[0017] Step 55: Calculate the flux at the flux calculation location.

[0018] As a preferred technical solution, the flow field state value of the point inside the flow field is obtained by linear interpolation of the element to which it belongs.

[0019] As a preferred technical solution, the distance from the point inside the flow field to the wall is three times the size of the grid near the wall.

[0020] As a preferred technical solution, the spatial grid in step 3 uses tetrahedral elements and is discretized using the finite volume method with grid format. The flow field state is stored on the nodes of the grid elements, and the spatial grid does not contain the object surface grid. Only when the spatial grid is generated, the local grid is refined in the area near the wall to accurately capture the flow state near the object.

[0021] As a preferred technical solution, in the turbulence model equation, the convection term is calculated using a first-order upwind scheme in the region far from the object surface, and the dissipation term is calculated using a central difference scheme. In the region near the wall, within the cutoff distance, the Spalart-Allmaras turbulence model equation uses a modified decay function.

[0022] As a preferred technical solution, the friction velocity in step S53 is solved by Newton's iteration method. The advantages of this invention compared to the prior art are:

[0023] (1) This invention achieves numerical simulation of high Reynolds number turbulence problems by combining Spalart-Allmaras wall function and FIVER immersion boundary method, thereby reducing the number of boundary layer grids required for turbulence calculation;

[0024] (2) This invention is based on the compressible Navier-Stokes equations and can handle high-speed flow problems;

[0025] (3) The present invention has no restrictions on the shape of the object or whether it is closed, and has a wide range of applications. Attached Figure Description

[0026] Figure 1 This is a schematic diagram of the area near the wall provided in an embodiment of the present invention. Detailed Implementation

[0027] The embodiments of the present invention will be described in further detail below with reference to the accompanying drawings.

[0028] This embodiment provides a compressible flow immersion boundary method for turbulence calculation based on the Spalart-Allmaras wall function. This method reduces the number of meshes near the wall required for immersion boundary method to calculate turbulence problems at high Reynolds numbers, and improves the ability of immersion boundary method to calculate turbulence problems at high Reynolds numbers.

[0029] To address the aforementioned issues, this embodiment provides a method for calculating wall function turbulence using the immersed boundary method, specifically including the following steps:

[0030] Step 1: Establish the main control equations. In this embodiment, the main control equations are the Favre-averaged Navier-Stokes equations, expressed as follows:

[0031]

[0032] Where W represents the conserved variables (density, momentum, and total energy), and Ω represents the volume (three-dimensional) or area (two-dimensional) of the control volume. For the control volume boundary, F c F represents the flux. v This represents the viscous flux. When considering turbulent flow, it is expressed as the Favre-averaged Navier-Stokes equations, where the flow field variables are average values.

[0033] Step 2: Establish the turbulence model equations. In this embodiment, the Spalart-Allmaras turbulence model equations are adopted.

[0034] Step 3: For the main governing equations, spatial discretization is performed using the lattice-based finite volume method in the region far from the wall. Flux calculations are performed using the Roe scheme, achieving second-order accuracy through linear reconstruction. Viscous flux calculations are performed using the central difference method, where the gradients of the flow field variables are obtained using the Green-Gauss method. In the region near the wall, the FIVER (Finite Volume method with Exact two-material Riemann Problems) immersion boundary method is used to handle the wall boundary conditions, establishing the following one-and-a-half-dimensional Riemann problem:

[0035]

[0036] in, For the flow field grid points near the object surface, along the local normal direction of the object surface The flow field state, ρ is density, and ξ is a one-dimensional local normal direction coordinate pointing inwards from the object, i.e., along... direction, It is the velocity of the object, Δt n Indicates the current time step.

[0037] Step 4: Calculate the flow field state at the wall based on the exact solution of the one-dimensional half-Riemann problem, thereby obtaining the flux at the wall.

[0038] In this embodiment, the exact solution to the one-dimensional semi-Riemann problem is divided into two cases: shock wave and rarefaction wave. The flow field state (density, normal velocity, and total energy) at the wall is then calculated from the exact solution. For inviscid or laminar flow problems, the flux at the wall is calculated using the Roe scheme. Figure 1 The curve represents the boundary of the object. Point I is an internal point of the flow field, and point J is an internal point of the object. Solve the one-dimensional half-Riemann problem at the midpoint C of edge IJ to obtain the density, normal velocity and total energy at point C.

[0039] Step 5: For the Spalart-Allmaras turbulence model equations, discretize the Spalart-Allmaras turbulence model equations in the region far from the wall using a method similar to the mainstream equations. In the region near the wall, obtain the density, normal velocity, and total energy using the FIVER immersed boundary method. Calculate the tangential velocity and eddy viscosity coefficient in the region near the wall using an improved wall function. The steps are as follows:

[0040] a. A point inside the flow field is determined by the flux calculation location near the wall and the normal to the nearest wall. The flow field state value at this point is obtained by linear interpolation from the element containing it, such as... Figure 1 Point P, as shown, is obtained from point C along the normal direction of the nearest wall. The distance from point P to the wall is three times the mesh size near the wall. The flow field state at point P is calculated by linear interpolation from the element containing point P.

[0041] b. Using the distance from a point inside the flow field to the wall as the cutoff distance, modify the attenuation function f in the Spalart-Allmaras turbulence model equations. v1 That is, when the distance from a point inside the flow field to the wall is less than the distance from point P to the wall, the attenuation function is...

[0042]

[0043] Where, r d =d P / d, d P Let P be the distance from point P to the wall, d be the distance from a point inside the flow field to the wall, and c be the distance from point P to the wall. v1 =0.71, c is the vortex coefficient, and ν is the kinematic viscosity coefficient.

[0044] c. Spalart-Allmaras wall functions are

[0045]

[0046] In this embodiment, a1=8.14822158, a2=-6.92870938, b1=7.46008761, b2=7.46814579, c1=2.54967735, c2=1.33016516, c3=

[0047] 3.59945911,c4=3.63975319,u + =u / u τ y + =u τ d / ν, u τ The velocity is the wall friction velocity.

[0048] For tangential velocity, the following relationship holds:

[0049] u t =u τ f(y + (5)

[0050] The function f is expressed as (3). The tangential velocity at point P, expressed as (5), is used to determine the wall friction velocity u by Newton's iteration method. τ ;

[0051] d. Put u τ Substituting the distance between point C and the wall into equation (5), we obtain the tangential velocity and eddy viscosity at point C.

[0052]

[0053] e. Using the flow field state at point C as the right state, calculate the flux at point C, thereby realizing the wall boundary conditions of the mainstream equation and the Spalart-Allmaras turbulence model equation.

[0054] The advantages of this invention over the prior art are:

[0055] (1) This invention achieves numerical simulation of high Reynolds number turbulence problems by combining Spalart-Allmaras wall function and FIVER immersion boundary method, which reduces the number of boundary layer grids required for turbulence calculation; (2) This invention is based on compressible Navier-Stokes equations and can handle high-speed flow problems;

[0056] (3) The present invention has no restrictions on the shape of the object or whether it is closed, and has a wide range of applications.

[0057] The above are merely preferred embodiments of the present invention. The scope of protection of the present invention is not limited to the above embodiments. All technical solutions falling within the scope of the present invention's concept are within the scope of protection of the present invention. It should be noted that for those skilled in the art, any improvements and modifications made without departing from the principles of the present invention should be considered within the scope of protection of the present invention.

Claims

1. A method for calculating wall function turbulence using the immersion boundary method, characterized in that, include: Step 1: Establish the main governing equations, which are the Favre-averaged Navier-Stokes equations; Step 2: Establish the turbulence model equations, wherein the turbulence model is the Spalart-Allmaras turbulence model equations; Step 3: Based on the aforementioned governing equations, spatial discretization is performed using the lattice-based finite volume method in the region far from the wall, while the FIVER immersion boundary method is used in the region near the wall, establishing the following one-dimensional semi-Riemann problem: （1）； in, For the flow field grid points near the object surface, along the local normal direction of the object surface The flow field state, , It's density. It is a one-dimensional local normal direction coordinate, pointing inwards from the object, that is, along... direction, It is the speed of an object's motion. Indicates the current time step; Step 4: Calculate the flow field state at the wall based on the exact solution of the one-dimensional half-Riemann problem, thereby obtaining the flux at the wall; Step 5: Based on the turbulence model equations, spatial discretization is performed in the region far from the wall. In the region near the wall, the density, normal velocity, and total energy are calculated using the FIVER immersion boundary method. The tangential velocity and eddy viscosity coefficient are calculated using the improved wall function. Step 5 further includes: Step 51: Determine the internal points of the flow field based on the flux near the wall and the normal of the nearest wall; Step 52: Modify the attenuation function in the Spalart-Allmaras turbulence model equations using the distance from the point inside the flow field to the wall as the cutoff distance; Step 53: Substitute the tangential velocity values ​​of the points inside the flow field relative to the wall into the wall function to determine the friction velocity of the wall. Step 54: Substitute the friction velocity of the wall and the distance from the wall at the flux calculation location into the wall function to obtain its tangential velocity, and obtain the eddy viscosity coefficient from the friction velocity of the wall and the distance from the wall. Step 55: Calculate the flux at the flux calculation location.

2. The method for calculating turbulence using wall functions according to claim 1, characterized in that, The flow field state value of a point inside the flow field is obtained by linear interpolation of the element it belongs to.

3. The method for calculating wall function turbulence according to claim 1 or 2, characterized in that, The distance from a point inside the flow field to the wall is three times the size of the grid near the wall.

4. The method for calculating turbulence using wall functions according to claim 1, characterized in that, The spatial mesh in step 3 uses tetrahedral elements and is discretized using the finite volume method with a grid format. The flow field state is stored on the nodes of the grid elements. The spatial mesh does not contain the object surface mesh. Only when the spatial mesh is generated, the local mesh is refined in the area near the wall to accurately capture the flow state near the object.

5. The method for calculating turbulence using wall functions according to claim 1, characterized in that, In the turbulence model equations, the convection term is calculated using a first-order upwind scheme in the region far from the surface, and the dissipation term is calculated using a central difference scheme. In the region near the wall, within the cutoff distance, the Spalart-Allmaras turbulence model equations use a modified decay function.

6. The method for calculating turbulence using wall functions according to claim 1, characterized in that, The friction velocity in step 53 is solved by Newton's iteration method.

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