CN111400968B - Method for constructing compressible two-phase flow interface condition under curve coordinate system - Google Patents
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Abstract
The invention relates to a method for constructing compressible two-phase flow interface conditions under a curve coordinate system, which is used for popularizing the interface flow numerical simulation technology under a traditional Cartesian coordinate system so as to be suitable for flow field simulation of different fluid interface evolution under a general curve coordinate system. Comprising the following steps: converting a level set equation and a reinitialization equation representing the interface position information into a general curve coordinate system for solving through coordinate transformation; the virtual fluid method reflecting the two-phase flow effect is generalized to the general curve coordinate system for interface condition definition. The advantages are that: the improved level set method under the curve coordinate system is suitable for representing the dynamic position information of the interface under any structural grid; the improved virtual fluid method under the curve coordinate system can be applied to conventional CFD software, so that the CFD software has the capacity of simulating two-phase flow; the interface condition expression form designed by adopting the coordinate transformation method is suitable for solving the compressible two-phase flow interface problem based on curve grids under complex appearance.
Description
Technical Field
The invention belongs to the technical field of computational fluid mechanics, and particularly relates to a method for constructing compressible two-phase flow interface conditions under a curve coordinate system.
Background
The dynamic characteristics generated by the interface movement of the compressible two-phase flow have very important significance for researching a high-speed flow field containing liquid drops, bubbles and elastoplastic wall surfaces. The numerical simulation effect of high confidence is increasingly enhanced, and the method can completely replace some expensive, dangerous and even difficult experiments, greatly reduce the development cost, shorten the development period and play an increasingly important role.
Because of the large differences in physical parameters of the different fluids on either side of the interface, numerical instability can occur if the numerical format simulating a single media fluid is applied directly to simulate a multi-media fluid problem. Especially when highly nonlinear waves such as shock waves act on the interface, non-physical oscillations tend to occur near the interface, even making the calculation difficult. The virtual fluid method treats the interface as a special boundary, builds boundary conditions on the basis of the two-phase flow Riemann problem solution on the virtual flow field near the interface, can avoid unstable values near the interface, and is easy to popularize into multiple dimensions.
However, the above-mentioned technique is mainly applied to cartesian grids, and cannot be used in a general curved coordinate system at present.
Disclosure of Invention
The invention aims to provide a compressible two-phase flow interface condition construction method under a curve coordinate system, which is used for popularizing a level set method and a virtual fluid method under a traditional Cartesian coordinate system based on coordinate transformation, so that the method is suitable for flow field simulation of different fluid interface evolution under a general curve coordinate system and has important application value.
The technical scheme of the invention is as follows:
a method for constructing compressible two-phase flow interface conditions under a curve coordinate system comprises the following steps:
1) Defining virtual fluid areas outside each fluid interface according to the initial value of the symbol distance function phi of each physical grid node; the medium properties of two adjacent fluids are different and the two adjacent fluids are not mutually compatible; the medium attributes are specifically: at least one of the velocity, pressure or density of two adjacent fluids is different; the fluids are compressible fluids; the virtual fluid region contains 3-5 physical grid nodes of another fluid near the interface; the physical grid is obtained by grid division of a calculation flow field area by utilizing grid generation software;
2) Updating the initial value of the symbol distance function phi in the step 1) by using a level set equation under a general curve coordinate system to obtain an updated symbol distance function phi;
3) Solving a re-initialization equation of the symbol distance function under a general curve coordinate system according to the updated symbol distance function phi in the step 2) until a stable solution of the re-initialization equation is obtained, and taking the stable solution as a corrected symbol distance function phi;
4) Updating the fluid interface position information according to the corrected symbol distance function phi in the step 3);
5) According to the virtual fluid area outside the fluid interface in the step 1), for each virtual fluid node P in the virtual fluid area, searching a real fluid node P ' in a real fluid area on the other side of the interface, so that an included angle theta formed by an interface unit normal vector n of the virtual fluid node P and an interface unit normal vector n ' of the real fluid node P ' is minimum; the interface normal direction of each node is determined according to the initial value of the symbol distance function phi in the step 1); wherein θ=arccoss (n·n');
6) Projecting the velocity fields of the virtual fluid node P and the corresponding real fluid node P' in the step 5) to the interface normal direction respectively to obtain the interface normal velocity u of the virtual fluid node P and the real fluid node P n ；
7) The interface normal velocity u of the virtual fluid node P and the real fluid node P' according to step 6) n The pressure and density corresponding to the virtual fluid node P and the real fluid node P' are used for constructing and solving the two-phase flow Riemann problem upward by the interfacial method to obtain the interfacial pressure P I Interface normal speed u n,I And interface two-side density ρ IL ,ρ IR ；
8) According to step 7) the interface pressure p I Interface normal speed u n,I And interface two-side density ρ IL ,ρ IR Defining an interface condition of the virtual fluid node P, i.e. a virtual fluid state;
9) Repeating steps 5) to 8) until the interface conditions of all virtual fluid nodes P of each fluid virtual fluid zone are obtained.
10 And (3) repeating the steps 1) to 9) for constructing the interface condition at the next computing moment according to the interface condition at the current computing moment in the step 9).
The level set equation under the general curve coordinate system in the step 2) is specifically:
φ t +Uφ ξ +Vφ η +Wφ ζ ＝0；
U＝ξ x u+ξ y v+ξ z w；
V＝η x u+η y v+η z w；
W＝ζ x u+ζ y v+ζ z w；
wherein phi is t Representing the partial derivative of the symbol distance function phi with respect to time t, t representing time, U, V, W being the fluid inversion speed of each node; u, v, w is the fluid velocity of each node; (x, y, z) represents the physical grid coordinates of the flow field, and grid division can be performed on the calculated flow field area by using grid generation software to obtain the x, y, z coordinate values of each node; (ζ, η, ζ) represents a calculated grid coordinate obtained by converting the physical grid coordinate from a general curve coordinate system to a cartesian coordinate system; zeta type toy x ,ξ y ,ξ z ,η x ,η y ,η z ,ζ x ,ζ y ,ζ z The measurement coefficient of the calculation grid represents the partial derivative of the calculation grid coordinates xi, eta and zeta to the physical grid coordinates x, y and z, and is usually calculated by using a central difference; phi (phi) ξ 、φ η And phi ζ The partial derivatives of the symbol distance function phi pair calculation grid coordinates xi, eta and zeta can be subjected to differential dispersion according to the selected format precision.
Step 3), the re-initialization equation of the symbol distance function under the general curve coordinate system is specifically:
a＝ξ x η x +ξ y η y +ξ z η z ，b＝ξ x ζ x +ξ y ζ y +ξ z ζ z ，c＝η x ζ x +η y ζ y +η z ζ z ；
wherein, tau represents pseudo time, epsilon is the side length of the physical grid where each node is located; phi (phi) 0 Representing the symbol distance function value prior to reinitialization.
Step 5), the unit normal vector n of the node near the interface under the general curve coordinate system is specifically:
step 6), the normal interface velocity u of the node near the interface under the general curve coordinate system n The method specifically comprises the following steps:
step 8) said step based on the obtained interface pressure p I Interface normal speed u n,I And interface two-side density ρ IL ,ρ IR The method for defining the interface condition of the virtual fluid node P specifically comprises the following steps:
if the sign distance function phi corresponding to the virtual fluid node P is less than 0, then
If the sign distance function phi corresponding to the virtual fluid node P is greater than 0, then
Wherein superscript denotes a virtual fluid. ρ * Representing the density of the virtual fluid, p * Representing the pressure of the virtual fluid; u (u) * 、v * And w * Representing a virtual fluid velocity; l represents a node where the fluid symbol distance function phi is less than 0; r represents a node where the fluid symbol distance function phi is greater than 0; let h=l or R, u H 、v H 、w H And U H 、V H 、W H The fluid velocity and the fluid inversion velocity of the real fluid node P', respectively.
Compared with the prior art, the invention has the advantages and innovations mainly in the following aspects:
1) The invention extends the level set equation and the reinitialization equation under the traditional Cartesian coordinate system to the general curve coordinate system, and is basically suitable for representing the dynamic position information of the interface under any structural grid;
2) The invention extends the flow of constructing interface conditions by the virtual fluid method under the Cartesian coordinate system to the general curve coordinate system, and can be basically applied to conventional CFD software, so that the virtual fluid method has the capacity of simulating two-phase flow;
3) The invention designs an interface condition expression form by adopting a coordinate transformation method, and is suitable for solving the compressible two-phase flow interface problem based on curve grids under complex appearance.
Drawings
FIG. 1 is a schematic representation of a coordinate transformation of a two-phase flow field interface problem from physical space to computational space;
FIG. 2 is a flow chart of construction of two-phase flow field interface conditions under a general curve coordinate system;
FIG. 3 is a graph showing the distribution of virtual fluid regions and node correspondence in a general curve coordinate system;
FIG. 4 is a grid division of a simulated underwater explosion problem in a general curve coordinate system, and a comparison of a numerical schlieren at a time of 1.0ms with a result in a Cartesian coordinate system;
fig. 5 is a comparison of the density distribution on the 0.2ms and 1.2ms moments x=0 axis of the simulated underwater explosion problem in a general curve coordinate system with the results of the self-grinding procedure, literature in a cartesian coordinate system.
Detailed Description
The research problems of gas-liquid mixing, underwater explosion and the like in the engine combustion chamber all relate to the interface evolution process of bubble/droplet deformation and collapse, shock wave and interface interaction and the like. In general, the calculation area or the outline is complex, and a method of dividing grids by a patch is generally adopted, and the grids are generally irregular and not strictly orthogonal. An interface condition model under a general curve coordinate system needs to be established, and two-phase flow field calculation is carried out by using the interface condition model, so that simulation of interface evolution and instability processes is obtained.
The invention adopts a coordinate transformation method to transform the state of a substance near the interface of a two-phase flow field and a correlation equation from a physical space (x, y, z) to a calculation space (ζ, eta, ζ), and the two-dimensional schematic of the coordinate transformation is shown in figure 1. In a general curve coordinate system, a two-phase flow field interface condition construction flow is shown in fig. 2, and mainly comprises two aspects of interface position information acquisition and updating and interface condition definition. The method comprises the following steps:
1) Defining virtual fluid areas outside each fluid interface according to the initial value of the symbol distance function phi of each physical grid node; the medium properties of two adjacent fluids are different and the two adjacent fluids are not mutually compatible; the medium attributes are specifically: at least one of the velocity, pressure or density of two adjacent fluids is different; the fluids are compressible fluids; the virtual fluid region contains 3-5 physical grid nodes of another fluid near the interface; the physical grid is obtained by grid division of a calculation flow field area by utilizing grid generation software;
2) Updating the initial value of the symbol distance function phi in the step 1) by using a level set equation under a general curve coordinate system to obtain an updated symbol distance function phi;
3) Solving a re-initialization equation of the symbol distance function under a general curve coordinate system according to the updated symbol distance function phi in the step 2) until a stable solution of the re-initialization equation is obtained, and taking the stable solution as a corrected symbol distance function phi;
4) Updating the fluid interface position information according to the corrected symbol distance function phi in the step 3);
5) According to the virtual fluid area outside the fluid interface in the step 1), for each virtual fluid node P in the virtual fluid area, searching a real fluid node P ' in a real fluid area on the other side of the interface, so that an included angle theta formed by an interface unit normal vector n of the virtual fluid node P and an interface unit normal vector n ' of the real fluid node P ' is minimum; the interface normal direction of each node is determined according to the initial value of the symbol distance function phi in the step 1); wherein θ=arccoss (n·n');
6) Projecting the velocity fields of the virtual fluid node P and the corresponding real fluid node P' in the step 5) to the interface normal direction respectively to obtain the interface normal velocity u of the virtual fluid node P and the real fluid node P n ；
7) The interface normal velocity u of the virtual fluid node P and the real fluid node P' according to step 6) n The pressure and density corresponding to the virtual fluid node P and the real fluid node P' are used for constructing and solving the two-phase flow Riemann problem upward by the interfacial method to obtain the interfacial pressure P I Interface normal speed u n,I And interface two-side density ρ IL ,ρ IR ；
8) According to step 7) the interface pressure p I Interface normal speed u n,I And interface two-side density ρ IL ,ρ IR Defining interface conditions of the virtual fluid node P, i.e. virtual fluid states;
9) Repeating steps 5) to 8) until the interface conditions of all virtual fluid nodes P of each fluid virtual fluid area are obtained;
10 And (3) repeating the steps 1) to 9) for constructing the interface condition at the next computing moment according to the interface condition at the current computing moment in the step 9).
The specific implementation flow is as follows:
(I) Acquisition and update of interface location information
(I-1) defining a virtual fluid region outside the interface for each fluid based on the value of the symbolic distance function phi. Nodes with a sign of negative value (phi < 0) represent one fluid on the left side of the interface, nodes with a sign of positive value (phi > 0) represent the other fluid on the right side of the interface, and the locations with a sign of zero (phi=0) represent the interface. The virtual fluid area outside the interface generally spans at least two grids of the interface, depending on the numerical format employed.
(I-2) solving a level set equation under a general curve coordinate system:
φ t +Uφ ξ +Vφ η +Wφ ζ ＝0
U＝ξ x u+ξ y v+ξ z w；
V＝η x u+η y v+η z w；
W＝ζ x u+ζ y v+ζ z w；
wherein phi is t Representing the partial derivative of the symbol distance function phi with respect to time t, t representing time, U, V, W being the fluid inversion speed of each node; u, v, w is the fluid velocity of each node; (x, y, z) represents the physical grid coordinates of the flow field, and grid division can be performed on the calculated flow field area by using grid generation software to obtain the x, y, z coordinate values of each node; (ζ, η, ζ) represents a calculated grid coordinate obtained by converting the physical grid coordinate from a general curve coordinate system to a cartesian coordinate system; zeta type toy x ,ξ y ,ξ z ,η x ,η y ,η z ,ζ x ,ζ y ,ζ z The measurement coefficient of the calculation grid represents the partial derivative of the calculation grid coordinates xi, eta and zeta to the physical grid coordinates x, y and z, and is usually calculated by using a central difference; phi (phi) ξ 、φ η And phi ζ The partial derivatives of the symbol distance function phi pair calculation grid coordinates xi, eta and zeta can be subjected to differential dispersion according to the selected format precision. For example, using WENO format to construct left and right discrete derivative values
And->
Then, taking:
to phi η And phi ζ Similar processing is performed.
(I-3) re-initializing the symbol distance function under a general curve coordinate system, namely solving:
a＝ξ x η x +ξ y η y +ξ z η z ，b＝ξ x ζ x +ξ y ζ y +ξ z ζ z ，c＝η x ζ x +η y ζ y +η z ζ z ；
wherein τ represents pseudo time, ε is a small amount, and may correspond to the side length of the physical grid where each node is located; phi (phi) 0 Representing a symbol distance function value prior to reinitialization; phi (phi) ξ 、φ η And phi ζ The partial derivatives of the symbol distance function phi pair calculation grid coordinates xi, eta and zeta can be subjected to differential dispersion according to the selected format precision. For example, using WENO format to construct left and right discrete derivative values
And->
Then take
Wherein,,
to phi η And phi ζ Similar processing is performed.
(I-4) obtaining new time interface position information, the position of phi=0 representing the interface position at the new time.
(II) definition of interface conditions
For each virtual fluid node P of the virtual fluid region near the interface:
(II-1) searching for the real fluid node P ' of the real fluid region on the other side of the interface such that the included angle θ formed by the interface unit normal vector n of the virtual fluid node P and the interface unit normal vector n ' of the real fluid node P ' is minimized, see fig. 3. Where θ=arccose (n·n'), the interfacial unit normal vector is expressed as
(II-2) projecting velocity fields of the virtual fluid node P and the real fluid node P' to interface normal directions, respectively, establishing interface normal speeds u of the virtual fluid node P and the real fluid node P n ，
Wherein the subscript n represents the interface normal.
(II-3) based on the obtained interface normal velocity u of the virtual fluid node P and the real fluid node P n And other fluid states, constructing and solving the interfacial method upward two-phase flow Riemann problem
Obtaining the interface pressure p I Interface normal speed u n,I And interface two-side density ρ IL ,ρ IR . Here, U n ＝[ρ,ρu n ,E n ] T And
the conservation variable and flux in the interfacial normal direction are shown respectively. ρ is the fluid density; p is the fluid pressure; u (u) n Is the fluid normal velocity;
Is the total energy of the fluid in the upward direction, where e is the internal energy per unit mass of the fluid. U (U) n,L And U n,R Is located at interface position n I The two fluid constant states where the material interfaces separate, subscripts L and R denote the fluids on both sides of the interface at the virtual fluid node P and the real fluid node P'. L represents the left side (phi) of the flow field interface<0 node), R represents the flow field interface right side (phi)>Nodes of 0).
(II-4) based on the obtained interface pressure p I Interface normal speed u n,I And interface two-side density ρ IL ,ρ IR Defining interface conditions (i.e., virtual fluid states) for the virtual fluid node P:
if the sign distance function phi corresponding to the virtual fluid node P is less than 0, i.e., the virtual fluid node P is located to the left of the interface, then:
if the sign distance function phi corresponding to the virtual fluid node P is greater than 0, i.e., the virtual fluid node P is located to the right of the interface, then:
here, superscript * Representing a virtual fluid. ρ * Representing the density of the virtual fluid, p * Representing the pressure of the virtual fluid; u (u) * 、v * And w * Representing a virtual fluid velocity; l represents a node with the fluid symbol distance function phi smaller than 0, namely the node is positioned at the left side of the interface, and R represents a node with the fluid symbol distance function phi larger than 0, namely the node is positioned at the right side of the interface; let h=l or R, u H 、v H 、w H And U H 、V H 、W H The fluid velocity and the fluid inversion velocity of the real fluid node P', respectively.
Figures 4 and 5 show the numerical results of solving the underwater explosion problem in a curvilinear coordinate system using the technique of the present invention. In the initial condition, the explosive gas is p=1GPa, ρ=1250 kg/m 3 Water p=100 kPa, ρ=1000 kg/m 3 Air was p=100 kPa, ρ=1 kg/m 3 Are all in a static state. The state equation uses the rigid gas state equation p= (N-1) ρe-NB, b=0 for gas n=1.4, b=600 MPa for water n=4.4. Fig. 4 compares the meshing, 1.0ms moment numerical schlieren with the results of the self-grinding procedure in cartesian coordinates, and fig. 5 compares the 0.2ms and 1.2ms moment x=0 on-axis density distribution results with the results of the self-grinding procedure, literature (Hu et al, j. Comp. Phys.228 (17) (2009) 6572-6589) in cartesian coordinates. It can be seen that although the grid division and topology are completely different from the cartesian coordinate system under the curved coordinate system, the calculation result is well matched with the calculation result under the cartesian coordinate system in the self-grinding procedure and literature.
Aiming at the problem that the prior art cannot be used under a general curve coordinate system, the invention aims to apply a virtual fluid method to the general curve coordinate system, and realize the numerical simulation of the compressible two-phase flow under any structural grid by constructing a definition mode of the interface condition of the compressible two-phase flow under the general curve coordinate system through coordinate transformation. The method can solve the practical problem with complex appearance and has important application value.
The foregoing is merely illustrative of the best embodiments of the present invention, and the present invention is not limited thereto, but any changes or substitutions easily contemplated by those skilled in the art within the scope of the present invention should be construed as falling within the scope of the present invention.
The present invention is not described in detail as being well known to those skilled in the art.
Claims (4)
1. The method for constructing the compressible two-phase flow interface condition under the curve coordinate system is characterized by comprising the following steps:
1) Defining virtual fluid areas outside each fluid interface according to the initial value of the symbol distance function phi of each physical grid node; the medium properties of two adjacent fluids are different and the two adjacent fluids are not mutually compatible; the medium attributes are specifically: at least one of the velocity, pressure or density of two adjacent fluids is different; the fluids are compressible fluids; the virtual fluid region contains 3-5 physical grid nodes of another fluid near the interface; the physical grid is obtained by grid division of a calculation flow field area by utilizing grid generation software;
2) Updating the initial value of the symbol distance function phi in the step 1) by using a level set equation under a general curve coordinate system to obtain an updated symbol distance function phi;
3) Solving a re-initialization equation of the symbol distance function under a general curve coordinate system according to the updated symbol distance function phi in the step 2) until a stable solution of the re-initialization equation is obtained, and taking the stable solution as a corrected symbol distance function phi;
4) Updating the fluid interface position information according to the corrected symbol distance function phi in the step 3);
5) According to the virtual fluid area outside the fluid interface in the step 1), for each virtual fluid node P in the virtual fluid area, searching a real fluid node P ' in a real fluid area on the other side of the interface, so that an included angle theta formed by an interface unit normal vector n of the virtual fluid node P and an interface unit normal vector n ' of the real fluid node P ' is minimum; the interface normal direction of each node is determined according to the initial value of the symbol distance function phi in the step 1); wherein θ=arccoss (n·n');
6) Projecting the velocity fields of the virtual fluid node P and the real fluid node P' in the step 5) to the interface normal direction respectively to obtain the interface normal direction velocity u of the virtual fluid node P and the real fluid node P n ；
7) The interface normal velocity u of the virtual fluid node P and the real fluid node P' according to step 6) n The pressure and density corresponding to the virtual fluid node P and the real fluid node P' are used for constructing and solving the two-phase flow Riemann problem upward by the interfacial method to obtain the interfacial pressure P I Interface normal speed u n,I And interface two-side density ρ IL ,ρ IR ；
8) According to step 7) the interface pressure p I Interface normal speed u n,I And interface two-side density ρ IL ,ρ IR Defining interface conditions of the virtual fluid node P;
9) Repeating steps 5) to 8) until the interface conditions of all virtual fluid nodes P of each fluid virtual fluid area are obtained;
10 According to the interface condition in the step 9) is the interface condition at the current computing moment, repeating the steps 1) to 9) for constructing the interface condition at the next computing moment;
the level set equation under the general curve coordinate system in the step 2) is specifically:
φ t +Uφ ξ +Vφ η +Wφ ζ ＝0；
U＝ξ x u+ξ y v+ξ z w；
V＝η x u+η y v+η z w；
W＝ζ x u+ζ y v+ζ z w；
wherein phi is t Representing the partial derivative of the symbol distance function phi with respect to time t, t representing time, U, V, W being the fluid inversion speed of each node; u, v, w is the fluid velocity of each node; (x, y, z) represents the physical grid coordinates of the flow field; (ζ, η, ζ) represents a calculated grid coordinate obtained by converting the physical grid coordinate from a general curve coordinate system to a cartesian coordinate system; zeta type toy x ,ξ y ,ξ z ,η x ,η y ,η z ,ζ x ,ζ y ,ζ z The measurement coefficient of the calculation grid represents the partial derivative of the calculation grid coordinates zeta, eta and zeta to the physical grid coordinates x, y and z; phi (phi) ξ 、φ η And phi ζ Representing the partial derivative of the symbol distance function phi with respect to the calculated grid coordinates ζ, eta, ζ;
step 3), the re-initialization equation of the symbol distance function under the general curve coordinate system is specifically:
a＝ξ x η x +ξ y η y +ξ z η z ，b＝ξ x ζ x +ξ y ζ y +ξ z ζ z ，c＝η x ζ x +η y ζ y +η z ζ z ；
wherein, tau represents pseudo time, epsilon is the side length of the physical grid where each node is located; phi (phi) 0 Representing the symbol distance function value prior to reinitialization.
2. The method for constructing a compressible two-phase flow interface condition under a curved coordinate system according to claim 1, wherein in step 5), a unit normal vector n of a node near the interface under the general curved coordinate system is specifically:
3. the method for constructing compressible two-phase flow interface conditions in a curved coordinate system according to claim 1, wherein step 6) is performed with respect to an interface normal velocity u of a node near the interface in the general curved coordinate system n The method specifically comprises the following steps:
4. a method of constructing a compressible two-phase flow interface condition in a curvilinear coordinate system according to any one of claims 1-3, wherein step 8) is based on the obtained interface pressure p I Interface normal speed u n,I And interface two-side density ρ IL ,ρ IR The method for defining the interface condition of the virtual fluid node P specifically comprises the following steps:
if the sign distance function phi corresponding to the virtual fluid node P is less than 0, then:
if the sign distance function phi corresponding to the virtual fluid node P is greater than 0, then:
wherein ρ is * Representing the density of the virtual fluid, p * Representing the pressure of the virtual fluid; u (u) * 、v * And w * Representing a virtual fluid velocity; l represents the fluid symbol distanceNodes with a function phi less than 0, R represents nodes with a fluid sign distance function phi greater than 0; let h=l or R, u H 、v H 、w H And U H 、V H 、W H The fluid velocity and the fluid inversion velocity of the real fluid node P', respectively.
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