CN113688554A - Flow field calculation method combining discontinuous finite element and immersion boundary method - Google Patents
Flow field calculation method combining discontinuous finite element and immersion boundary method Download PDFInfo
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Abstract
The invention discloses a flow field calculation method combining an interrupted finite element and an immersed boundary method, which comprises the steps of S1, generating right-angle grid units for the whole flow field; s2, constructing an anisotropic shape function in the right-angle grid unit; s3, initializing a flow field; and S4, performing time-marching circulation on the flow field by using an immersion boundary method based on the constructed anisotropic shape function to obtain flow field data and realize flow field analysis. The method can greatly improve the flow numerical simulation efficiency of various complex geometric areas in the practical application of engineering, and has better application prospect.
Description
Technical Field
The invention belongs to the technical field of fluid mechanics, and particularly relates to a flow field calculation method combining an interrupted finite element method and an immersion boundary method.
Background
In recent years, immersion Boundary methods (Immersed Boundary methods) in the field of Computational Fluid Dynamics (CFD) have attracted much attention in the academic and application fields, and in the immersion Boundary methods, rectangular grids are used for the whole flow field calculation, and a finite difference Method discrete control Method is matched. In the vicinity of the solid wall surface boundary, aiming at the intersection condition of the solid surface and the rectangular grid, the immersion boundary method applies boundary conditions by adopting an extrapolation method or an equivalent external force method. The step of generating the grid is omitted, and the problem of coordinate conversion from a physical space to a calculation space does not need to be processed in the calculation, so that the calculation efficiency can be greatly improved, and the method becomes a hotspot of research in the field of CFD at present.
As shown in fig. 1, the direct mesh of the submerged boundary method cannot resolve the extremely thin boundary layer of the surface of an object in high reynolds number flow, since it is not possible to avoid normal encryption alone in a skin mesh. If the thin boundary layer is directly distinguished by encrypting the isotropic right-angle grid, which is equivalent to performing unnecessary encryption on the flow direction and the spreading direction while encrypting the normal grid, the grid number is increased sharply, the calculation amount is huge, and the numerical simulation can not be completed under reasonable calculation resources (calculation power, memory and time consumption). Therefore, the immersion boundary method is only used for simulating low Reynolds number flow initially, because the boundary layer of the flow is thick and can be resolved by a thicker right-angle grid, as shown in FIG. 2, and the high Reynolds number attached boundary layer is simulated later by combining the wall function technology, because the wall function technology abandons the direct resolution of the internal structure of the boundary layer by the grid and adopts an approximate model to estimate the influence of the boundary layer indirectly, thereby effectively reducing the requirement on the size of the grid. However, this approach actually circumvents the difficulties, and the immersion boundary method still fails in the convection field analysis.
Disclosure of Invention
Aiming at the defects in the prior art, the flow field calculation method combining the discontinuous finite element and the immersion boundary method solves the problem that the flow of high Reynolds number wall surfaces is difficult to flow respectively in the process of performing numerical simulation on the boundary layer structure by using the conventional boundary entering method.
In order to achieve the purpose of the invention, the invention adopts the technical scheme that: a flow field calculation method combining an interrupted finite element method and an immersion boundary method comprises the following steps:
s1, generating right-angle grid units for the whole flow field;
s2, constructing shape functions of all units in the flow field based on the generated right-angle grid units;
s3, initializing a flow field;
and S4, performing time advancing circulation on each unit of the flow field by using an immersion boundary method based on the shape function of each unit of the structure to obtain flow field calculation data.
Further, in step S1, in the process of generating the rectangular grid cells, the encryption process is performed near the wall surface.
Further, the step S2 is specifically:
s21, in the rectangular grid cells, marking the rectangular cells near the wall surface as near-wall cells, and defining a xi-eta coordinate system of the near-wall cells according to the direction of the local wall surface;
wherein xi is a wall surface tangential direction, and eta is a wall surface normal direction;
s22, respectively selecting polynomials of corresponding times in xi and eta directions in the near wall unit, constructing corresponding anisotropic shape functions, and forming an approximate space corresponding to a near wall unit solution, namely an approximate space of a discontinuous finite element;
and simultaneously, adopting an isotropic shape function of an x-y space in a non-near-wall unit to complete the shape function construction of all units in the flow field.
Further, in step S21, the ξ - η coordinate system of the near-wall unit is determined specifically as follows:
in the rectangular grid unit, for the rectangular unit of the coarse rectangular grid in the x-y space, a xi-eta coordinate system is set at the centroid of the unit, and the xi-eta is obtained by arbitrarily rotating the x-y coordinate system of the coordinate system.
Further, in step S22, when the gradient of the tangential velocity field in the vicinity of the wall surface in the wall surface normal direction is larger than the gradient of the wall surface tangential direction in the viscous flow, the degree of the η -direction corresponding polynomial is larger than the degree of the ξ -direction corresponding polynomial.
Further, the step S22 is specifically:
establishing a local coordinate system xi-eta by taking the unit centroid as an origin (0,0), and setting the distribution of the flow field variable u (xi, eta) in the unit to be approximately obtained by the linear combination of corresponding shape functions, namely:
in the formula, B(m，n)(xi, eta) is a shape function, the highest degree of xi direction is n times, the highest degree of eta direction is m times, the polynomial of xi is p times, the polynomial of eta is q times,
coefficients that are shape functions;
the degree of freedom and the shape function of the (p +1) × (q +1) term pair are obtained based on u (xi, η) and are used as an approximate space of the near wall unit solution.
Further, the step S4 is specifically:
s41, applying physical boundary conditions;
s42, based on the applied physical boundary conditions, using the immersion boundary method to correct the freedom degree of the wall surface cut in the right-angle grid unit;
s43, calculating residual errors of discontinuous finite elements of the units according to the shape functions of the constructed units;
s44, performing time advancing by adopting a set time dispersion mode, and updating the degree of freedom of solution variables in the units according to the residual errors corresponding to each unit;
wherein each cell comprises a near-wall cell and a non-near-wall cell;
s45, calculating current flow field data according to the degree of freedom of solution variables in each current unit and outputting the data;
s46, repeating the steps S41-S45 until the set time step length is reached;
and S47, completing time advancing circulation, storing output flow field data and completing flow field calculation.
Further, the flow field data in the step S45 includes an aerodynamic coefficient, a fluid density, a fluid pressure, and a fluid temperature.
The invention has the beneficial effects that:
the method can greatly improve the flow numerical simulation efficiency of various complex geometric areas in practical engineering application, has good application prospect and is embodied in the following points:
(1) the degree of freedom configuration of the anisotropic shape function discontinuous finite element adopted in the invention does not depend on the shape of the grid, and has better flexibility;
(2) in the invention, the high Reynolds number boundary layer of the irregular wall surface can be really distinguished only by adopting the right-angle grids;
(3) the invention adopts the right-angle grid, which can greatly save the manpower, time and computing resources required by grid generation and can easily realize the step of grid automatic generation in CFD.
Drawings
Fig. 1 is a schematic diagram of a cylindrical skin grid structure grid (left) and a cylindrical cartesian grid (right) in the background art provided by the present invention.
Fig. 2 is a schematic diagram illustrating the boundary layer resolving power comparison between the conformal grid (left) and the rectangular grid (right) of the NACA0012 airfoil structure in the background art provided by the present invention.
FIG. 3 is a flow chart of a flow field calculation method combining discontinuous finite element and dipping boundary methods according to the present invention.
FIG. 4 is a schematic view of an anisotropic flow field under a coarse grid of right angles provided by the present invention.
FIG. 5 is a schematic diagram of a ξ - η coordinate system of a rectangular unit provided by the present invention.
Fig. 6 is a schematic diagram of a unit cut by a wall surface according to the present invention.
Detailed Description
The following description of the embodiments of the present invention is provided to facilitate the understanding of the present invention by those skilled in the art, but it should be understood that the present invention is not limited to the scope of the embodiments, and it will be apparent to those skilled in the art that various changes may be made without departing from the spirit and scope of the invention as defined and defined in the appended claims, and all matters produced by the invention using the inventive concept are protected.
As shown in fig. 3, a flow field calculation method combining discontinuous finite element and dipping boundary method includes the following steps:
s1, generating right-angle grid units for the whole flow field;
s2, constructing shape functions of all units in the flow field based on the generated right-angle grid units;
s3, initializing a flow field;
and S4, performing time advancing circulation on each unit of the flow field by using an immersion boundary method based on the shape function of each unit of the structure to obtain flow field calculation data.
In step S1 of the present embodiment, in the process of generating the rectangular grid cells, an encryption process is performed near the wall surface to improve the approximation to the irregular wall surface.
In step S2 of this embodiment, an anisotropic shape function discontinuous finite element is constructed, and decoupling of the degree of freedom configuration and the mesh is achieved, that is, the degree of freedom configuration no longer depends on the shape, size, and other factors of the mesh.
The degrees of freedom in the conventional immersion boundary method are arranged at the vertices of the rectangular meshes or the cell centers, and are tightly bound to the mesh shapes, so that since the meshes cannot decompose the wall geometry, the degrees of freedom defined by the coarse rectangular meshes cannot match the wall geometry to the boundary layer anisotropy resolution form. The invention combines the anisotropic shape function discontinuous finite element in the immersion boundary method, and can distinguish the irregular wall surface shape and the thin boundary layer in the coarse right-angle grid.
The discontinuous finite element of the anisotropic function in this embodiment is obtained by sampling any anisotropic shape function and improving on the basis of the taylor unfolding discontinuous finite element method, and based on this, the step S2 specifically includes:
s21, in the rectangular grid cells, marking the rectangular cells near the wall surface as near-wall cells, and defining a xi-eta coordinate system of the near-wall cells according to the direction of the local wall surface;
wherein xi is a wall surface tangential direction, and eta is a wall surface normal direction;
s22, respectively selecting polynomials of corresponding times in xi and eta directions in the near wall unit, constructing corresponding anisotropic shape functions, and forming an approximate space corresponding to a near wall unit solution, namely an approximate space of a discontinuous finite element;
and simultaneously, adopting an isotropic shape function of an x-y space in a non-near-wall unit to complete the shape function construction of all units in the flow field.
In step S21, the method for determining the ξ - η coordinate system of the near-wall unit specifically includes:
in the rectangular grid unit, for the rectangular unit of the coarse rectangular grid in the x-y space, a xi-eta coordinate system is set at the centroid of the unit, and the xi-eta is obtained by arbitrarily rotating the x-y coordinate system of the coordinate system.
As the shape function of the near-wall unit is based on the xi-eta coordinate system, the shape function and the corresponding freedom degree can be matched with the designated direction at will, and the decoupling of the freedom degree configuration and the rectangular grid is realized.
The step S22 is specifically:
establishing a local coordinate system xi-eta by taking the unit centroid as an origin (0,0), and setting the distribution of the flow field variable u (xi, eta) in the unit to be approximately obtained by the linear combination of corresponding shape functions, namely:
in the formula, B(m，n)(xi, eta) is a shape function, the highest degree of xi direction is n times, the highest degree of eta direction is m times, the polynomial of xi is p times, the polynomial of eta is q times,
coefficients that are shape functions;
obtaining the degree of freedom and the anisotropic shape function of (p +1) × (q +1) term pairing based on u (xi, eta), taking the degree of freedom and the anisotropic shape function as an approximate space of a near-wall unit solution, and performing spatial dispersion on a flow field control equation; since p and q are not necessarily the same, the approximation capability of the approximation space of these shape functions to form the near-wall cell solution is different in both ξ and η directions, i.e., anisotropic for the ξ - η space, in order to better resolve the drastic changes in the normal direction within the wall bounding layer.
Based on the obtained anisotropic shape function discontinuous finite element, in a coarse right-angle grid of the immersion boundary method, the anisotropy characteristics of the irregular wall surface shape and the wall surface boundary layer in the right-angle grid can be distinguished through flexible configuration freedom.
As shown in fig. 4, in step S22 described above, when the gradient of the tangential velocity field in the vicinity of the wall surface in the wall surface normal direction is larger than the gradient of the wall surface tangential direction in the viscous flow, the degree of the η -direction corresponding polynomial expression is larger than the degree of the ξ -direction corresponding polynomial expression. When the variable has larger change in the tangential direction, the tangential polynomial degree can also be properly increased, in short, the polynomial degrees in different directions are self-adaptively adjusted according to the local flow field change to form an anisotropic shape function.
Specifically, the flow field near the wall surface has a larger gradient in the normal direction of the wall surface and a less variable flow field in the tangential direction. In order to analyze the flow field, enough degrees of freedom need to be arranged in the wall surface normal direction, for a right-angle unit close to the wall surface, the wall surface tangential direction and the wall surface normal direction can be selected as a shape function space, for a right-angle unit close to the wall surface, and the wall surface tangential direction and the wall surface normal direction can be selected as a xi-eta coordinate system of the shape function space. Meanwhile, in order to distinguish the drastic change of the flow field near the wall surface in the normal direction (eta direction), a higher-order polynomial can be adopted for the shape function in the eta direction, and a lower-order polynomial can be adopted for the shape function along the flow direction (xi direction) (as shown in fig. 5). In this way, the identification of the wall surface geometric characteristics and the analysis of the anisotropic boundary layer under the coarse right-angle grid can be realized.
Step S4 of this embodiment specifically includes:
s41, applying physical boundary conditions;
s42, based on the applied physical boundary conditions, using the immersion boundary method to correct the freedom degree of the wall surface cut in the right-angle grid unit;
as shown in fig. 6, for a cell cut by a wall surface (in the figure, a dark color is a flow field area, and a light color is a solid area), the value of the degree of freedom is corrected, so that the value of the flow area is almost unchanged, and at the same time, a function is ensured to meet boundary conditions, thereby avoiding boundary application in the sense of an immersion boundary method is completed, and specific embodiments refer to boundary measures of a conventional immersion method, such as cut-cell method, ghost-cell method, grid-less method, and the like;
s43, calculating residual errors of discontinuous finite elements of the units according to the shape functions of the constructed units;
s44, performing time advancing by adopting a set time dispersion mode, and updating the degree of freedom of solution variables in the units according to the residual errors corresponding to each unit;
wherein each cell comprises a near-wall cell and a non-near-wall cell;
s45, calculating current flow field data according to the degree of freedom of solution variables in each current unit and outputting the data;
s46, repeating the steps S41-S45 until the set time step length is reached;
and S47, completing time advancing circulation, storing output flow field data and completing flow field calculation.
The flow field data in the above step S45 includes an aerodynamic coefficient, a fluid density, a fluid pressure, and a fluid temperature.
In the flow field calculation process, the key point of distinguishing the high Reynolds number boundary layer is that enough degrees of freedom are arranged along the normal direction of the wall surface, so that the step length of the normal close-fit radius layer grid is very small, and the step lengths in other directions can be larger, so that the thin boundary layer with high anisotropy can be economically and effectively distinguished.
The invention provides a novel method for realizing flow field analysis by combining an anisotropic shape function and an immersion boundary method, which has the capability of realizing more freedom degree in the wall surface normal direction in a thicker right-angle grid so as to ensure that a high Reynolds number thin boundary layer is distinguished in a thick right-angle grid, and simultaneously reserves the advantages of simple generation and easy realization of automation of the right-angle grid in the immersion boundary method.
Claims (8)
1. A flow field calculation method combining an interrupted finite element method and an immersed boundary method is characterized by comprising the following steps:
s1, generating right-angle grid units for the whole flow field;
s2, constructing shape functions of all units in the flow field based on the generated right-angle grid units;
s3, initializing a flow field;
and S4, performing time advancing circulation on each unit of the flow field by using an immersion boundary method based on the shape function of each unit of the structure to obtain flow field calculation data.
2. The method for calculating a flow field according to claim 1, wherein in step S1, the method includes performing encryption processing near the wall surface during the process of generating the rectangular grid cells.
3. The method for calculating a flow field by combining discontinuous finite element and dipping boundary method according to claim 1, wherein the step S2 is specifically as follows:
s21, in the rectangular grid cells, marking the rectangular cells near the wall surface as near-wall cells, and defining a xi-eta coordinate system of the near-wall cells according to the direction of the local wall surface;
wherein xi is a wall surface tangential direction, and eta is a wall surface normal direction;
s22, respectively selecting polynomials of corresponding times in xi and eta directions in the near wall unit, constructing corresponding anisotropic shape functions, and forming an approximate space corresponding to a near wall unit solution, namely an approximate space of a discontinuous finite element;
and simultaneously, adopting an isotropic shape function of an x-y space in a non-near-wall unit to complete the shape function construction of all units in the flow field.
4. A flow field calculation method incorporating discontinuous finite element and dipping boundary method as claimed in claim 3, wherein in step S21, the xi- η coordinate system of the near wall element is determined by:
in the rectangular grid unit, for the rectangular unit of the coarse rectangular grid in the x-y space, a xi-eta coordinate system is set at the centroid of the unit, and the xi-eta is obtained by arbitrarily rotating the x-y coordinate system of the coordinate system.
5. The flow field calculation method combining the discontinuous finite element method and the dipping boundary method according to claim 3, wherein in the step S22, when the gradient of the tangential velocity field near the wall surface in the normal direction of the wall surface is larger than that in the tangential direction of the wall surface under the viscous flow condition, the degree of the η -direction corresponding polynomial is larger than that in the ξ -direction corresponding polynomial.
6. The method for calculating a flow field by combining an interrupted finite element method and an immersed boundary method according to claim 4, wherein the step S22 is specifically as follows:
establishing a local coordinate system xi-eta by taking the unit centroid as an origin (0,0), and setting the distribution of the flow field variable u (xi, eta) in the unit to be approximately obtained by the linear combination of corresponding shape functions, namely:
in the formula, B(m,n)(xi, eta) is a shape function, the highest degree of xi direction is n times, the highest degree of eta direction is m times, the polynomial of xi is p times, the polynomial of eta is q times,
coefficients that are shape functions;
the degree of freedom and the shape function of the (p +1) × (q +1) term pair are obtained based on u (xi, η) and are used as an approximate space of the near wall unit solution.
7. The method for calculating a flow field by combining discontinuous finite element and dipping boundary method according to claim 5, wherein the step S4 is specifically as follows:
s41, applying physical boundary conditions;
s42, based on the applied physical boundary conditions, using the immersion boundary method to correct the freedom degree of the wall surface cut in the right-angle grid unit;
s43, calculating residual errors of discontinuous finite elements of the units according to the shape functions of the constructed units;
s44, performing time advancing by adopting a set time dispersion mode, and updating the degree of freedom of solution variables in the units according to the residual errors corresponding to each unit;
wherein each cell comprises a near-wall cell and a non-near-wall cell;
s45, calculating current flow field data according to the degree of freedom of solution variables in each current unit and outputting the data;
s46, repeating the steps S41-S45 until the set time step length is reached;
and S47, completing time advancing circulation, storing output flow field data and completing flow field calculation.
8. The method of claim 7, wherein the flow field data in step S45 includes an aerodynamic coefficient, a fluid density, a fluid pressure, and a fluid temperature.
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