Computational method for unstable steady solution of airfoil flow based on reduced-order model and gradient optimization

Abstract

The present invention discloses a method for calculating unstable steady-state solutions of airfoil flows based on a reduced-order model and gradient optimization. First, an airfoil geometric model is constructed, a grid is divided, and the grid is imported into a CFD solver for flow field calculation; a flow field snapshot is saved during the CFD solution process; the POD modes of the flow field snapshot are calculated, high-order modes greater than 5 orders are truncated, and a reduced-order model for the remaining POD modes is constructed; then, the residual of the reduced-order model is minimized using gradient optimization to obtain a set of POD modal coefficients that minimize the residual of the reduced-order model, and the POD modal coefficients are back-projected into the physical space to obtain a new flow field; repeated iterations are performed until the flow field converges to obtain an unstable steady-state solution of the airfoil flow. The present invention has the characteristics of high efficiency, robustness, and no dependence on parameters. In addition to airfoils, it can also be used to calculate steady-state solutions of various unstable flows.

Description

Technical Field

[0001] The invention belongs to the technical field of fluid mechanics, and in particular relates to a method for calculating a steady solution of unstable flow of an airfoil. Background Art

[0002] Unstable steady-state solutions are crucial for stability analysis and flow control of fluid systems. They can be used for modal and non-modal analysis in linear stability, and can also be used as design targets for flow control. When there are global unstable modes in the airfoil fluid system, the steady-state solutions of the Navier-Stokes equations cannot be calculated by the time-marching method. Usually, the Newton iteration method (1. Knoll DA Jacobian-free Newton-Krylov methods: a survey of approaches and applications [J]. J. Comput. Phys, 2004, 193: 357-397.) and the selective frequency damping method (1. Espen, Brandt L., Henningson DS, et al. Steady solutions of the Navier-Stokes equations by selective frequency damping [J]. Physics of Fluids, 2006, 18 (6): 357-397.) calculate unstable steady-state solutions. However, the initial value sensitivity of the Newton iteration method and the computational cost for large systems seriously limit its application scope. The selective frequency damping method is a more popular method for calculating unstable steady-state solutions, but this method is overly dependent on parameters. Inappropriate parameters may lead to non-convergence or infeasible computational costs in practice, and when there are multiple unstable modes in the system, this method can hardly obtain unstable steady-state solutions (Casacuberta J, Groot KJ, Tol HJ, et al. Effectiveness and efficiency of selective frequency damping for the computation of unstable steady-state solutions [J]. Journal of Computational Physics, 2018, 375: 481-497.). Therefore, there is still a lack of an efficient, robust and parameter-independent method for calculating unstable steady-state solutions for airfoils. Summary of the invention

[0003] In order to overcome the shortcomings of the prior art, the present invention provides a method for calculating the unstable steady-state solution of airfoil flow based on a reduced-order model and gradient optimization. First, an airfoil geometric model is constructed, a grid is divided, and the grid is imported into a CFD solver for flow field calculation; a flow field snapshot is saved during the CFD solution process; the POD mode of the flow field snapshot is calculated, and high-order modes greater than 5 orders are truncated to construct a reduced-order model for the remaining POD modes; then the residual of the reduced-order model is minimized using gradient optimization to obtain a set of POD modal coefficients that minimize the residual of the reduced-order model, and the POD modal coefficients are back-projected back into the physical space to obtain a new flow field; repeated iterations are performed until the flow field converges to obtain an unstable steady-state solution of the airfoil flow. The present invention has the characteristics of high efficiency, robustness, and independence of parameters. In addition to airfoils, it can also be used to calculate the steady-state solutions of various unstable flows.

[0004] The technical solution adopted by the present invention to solve the technical problem includes the following steps:

[0005] Step 1: Build the airfoil geometry model, divide the mesh, and import the mesh into the CFD solver for flow field calculation;

[0006] Step 2: Save the flow field snapshot during the CFD solution process; for the finite difference based CFD solver, the flow field snapshot includes the values ​​of the solution variables and the derivatives of each order of the discretized NS equation; for the finite volume CFD solver, the flow field snapshot includes the grid center value, the left and right values ​​of the face center, the face center value and the face center derivative value;

[0007] Step 3: Calculate the POD modes of the flow field snapshot, truncate the high-order modes greater than 5, and construct a reduced-order model for the remaining POD modes;

[0008] Step 4: Use gradient optimization to minimize the residual of the reduced-order model to obtain a set of POD modal coefficients that minimize the residual of the reduced-order model, and back-project the POD modal coefficients back to the physical space to obtain a new flow field;

[0009] Step 5: Use the new flow field obtained in step 4 as the initial field of the pseudo-time iteration step, and repeat steps 2 to 4 until the flow field converges, that is, the unstable steady-state solution of the airfoil flow is obtained.

[0010] Preferably, the method for calculating the POD mode of the flow field snapshot in step 4 is as follows:

[0011] Center value U m The snapshot matrix X composed of POD analysis is used to obtain its POD mode Φ m , and obtain the transformation matrix T between the snapshot and the modality, where the matrix T satisfies Φ m =X m T, calculate the center value U through matrix T mp , face-centered derivative Face-centered left value Face-centered right value The POD modality:

[0012]

[0013] Preferably, the method for constructing the reduced-order model R(ξ) about the residual POD modal coefficients is as follows:

[0014]

[0015]

[0016] Where m is the grid number, N is the total number of grids, and V m represents the grid volume of grid m, ξ represents the modal coefficient, which is the optimization variable; n mp represents the surface normal direction of the pth face of mesh m, ΔS mp Indicates area, They are the face-centered inviscid flux and viscous flux respectively. A horizontal bar above the variable indicates the snapshot mean of the corresponding variable.

[0017] The beneficial effects of the present invention are as follows:

[0018] The present invention has the characteristics of high efficiency, robustness and independence of parameters, and can be used to calculate steady solutions of various unstable flows in addition to airfoils. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Figure 1 It is a schematic diagram of the process of the present invention.

[0020] Figure 2 This is the airfoil calculation grid of S809 in the embodiment of the present invention.

[0021] Figure 3 The figure is a comparison of the residual convergence process between the airfoil initialization method of S809 in the embodiment of the present invention and the residual convergence process of the present invention. DETAILED DESCRIPTION

[0022] The present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0023] The present invention is a CFD solution technology that combines the time-marching method, the reduced-order model and the gradient optimization, which can obtain steady-state solutions for various typical unstable flows in engineering and is the basis for various flow stability analyses in engineering. The method collects flow field snapshots during the pseudo-marching process of the CFD solver to construct a reduced-order model, and then uses gradient optimization to minimize the residual of the reduced-order model, and finally re-substitutes the optimized flow field into the CFD solver for calculation. This method combines the efficiency and robustness of the time-marching method with the convergence of the gradient optimization method, and only adds a small amount of additional calculations.

[0024] like Figure 1 As shown, a method for calculating the unstable steady solution of airfoil flow based on a reduced-order model and gradient optimization includes the following steps:

[0025] Step 1: For a three-dimensional wing, construct an airfoil geometry model, divide the mesh, and import the mesh into a CFD solver for flow field calculation;

[0026] Step 2: Save the flow field snapshot during the CFD solution process; for the finite difference based CFD solver, the flow field snapshot includes the values ​​of the solution variables and the derivatives of each order of the discretized NS equation; for the finite volume CFD solver, the flow field snapshot includes the grid center value, the face center left value and right value (for calculating the inviscid flux), the face center value and the face center derivative value (for calculating the viscous flux);

[0027] Step 3: Calculate the POD modes of the flow field snapshot, truncate the high-order modes greater than 5, and construct a reduced-order model for the remaining POD modal coefficients;

[0028] Step 4: Use gradient optimization to minimize the residual of the reduced-order model to obtain a set of POD modal coefficients that minimize the residual of the reduced-order model, and back-project the POD modal coefficients back to the physical space to obtain a new flow field;

[0029] Step 5: Use the new flow field obtained in step 4 as the initial field of the pseudo-time iteration step, and repeat steps 2 to 4 until the flow field converges, that is, the unstable steady-state solution of the airfoil flow is obtained.

[0030] The flow field can then be applied to various engineering models for stability analysis or flow control applications. Specific embodiment:

[0032] Implementation example description: Using the S809 airfoil turbulence example, the calculation grid is as follows Figure 2 As shown, the total number of object surface grids is 200, the total number of grid cells is 76356, and the steady flow field is solved. The calculation state is selected as the incoming flow Mach number Ma = 0.2, the angle of attack α = 15.2°, and the Reynolds number Re = 2e6. The solver based on the finite volume method is used, the time advancement format uses the implicit Gauss-Seidel format, the inviscid flux uses the Roe format, and the viscous flux uses the central format.

[0033] (1) During the pseudo-time advancement process, a flow field snapshot (including the grid center value U) is stored every 10 iteration steps. m , face center value U mp , face-centered derivative Face-centered left value Face-centered right value ), take 5 flow field snapshots at equal intervals, that is, call the reduced-order model every 50 steps.

[0034] (2) Perform POD analysis on the snapshot matrix X composed of grid center values ​​to obtain its POD mode Φ m , and obtain the transformation matrix T between the snapshot and the modality, where the matrix T satisfies Φ m =X m T. The POD modes of the face-centered value, face-centered derivative, face-centered left value, and face-centered right value are calculated using the matrix T.

[0035]

[0036] (3) Use POD modes to construct a reduced-order model:

[0037]

[0038]

[0039] in, They are the face-centered inviscid flux and the viscous flux, respectively. They can be solved using different flux formats, but must be consistent with the CFD solver. Gradient optimization is used to minimize the residual of the reduced-order model. The reduced-order model is implemented in python, the derivatives are calculated using automatic differentiation in pytorch, and the gradient optimization uses the AdamW optimizer.

[0040] (4) Substitute the optimized flow field with lower residual into the CFD solver as the initial value, and repeat steps (1) to (4) until convergence.

[0041] Figure 3 The figure is a comparison result of the convergence process of the residual value of the flow field and the original iterative method. It is obvious that the original iterative method cannot be calculated to converge, while the method of the present invention can quickly calculate and converge to obtain the unstable steady-state solution of the flow.

Claims

1. A method for calculating the steady-state solution of airfoil flow instability based on reduced-order model and gradient optimization, characterized in that: The steps include: Step 1: Build the airfoil geometry model, divide the mesh, and import the mesh into the CFD solver for flow field calculation; Step 2: Save the flow field snapshot during the CFD solution process; for the finite difference based CFD solver, the flow field snapshot includes the values ​​of the solution variables and the derivatives of each order of the discretized NS equation; for the finite volume CFD solver, the flow field snapshot includes the grid center value, the left and right values ​​of the face center, the face center value and the face center derivative value; Step 3: Calculate the POD mode of the flow field snapshot, truncate the high-order modes greater than 5, and construct a reduced-order model for the remaining POD modes; the method for calculating the POD mode of the flow field snapshot is as follows: Center value U m The snapshot matrix X composed of POD analysis is used to obtain its POD mode Φ m , and obtain the transformation matrix T between the snapshot and the modality, where the matrix T satisfies Φ m =X m T, calculate the center value U through matrix T mp , face-centered derivative Face-centered left value Face-centered right value The POD modality: The method for constructing the reduced-order model R(ξ) about the residual POD modal coefficients is as follows: Where m is the grid number, n is the total number of grids, V m represents the grid volume of grid m, ξ represents the modal coefficient, which is the optimization variable; n mp represents the surface normal direction of the pth face of mesh m, ΔS mp Indicates area, They are the face-centered inviscid flux and the viscous flux, respectively. A horizontal bar above the variable indicates the snapshot mean of the corresponding variable; Step 4: Use gradient optimization to minimize the residual of the reduced-order model to obtain a set of POD modal coefficients that minimize the residual of the reduced-order model, and back-project the POD modal coefficients back to the physical space to obtain a new flow field; Step 5: Use the new flow field obtained in step 4 as the initial field of the pseudo-time iteration step, and repeat steps 2 to 4 until the flow field converges, that is, the unstable steady-state solution of the airfoil flow is obtained.

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