CN114818531A - Motion interface tracking numerical dissipation calculation method - Google Patents

Motion interface tracking numerical dissipation calculation method Download PDF

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CN114818531A
CN114818531A CN202210365672.7A CN202210365672A CN114818531A CN 114818531 A CN114818531 A CN 114818531A CN 202210365672 A CN202210365672 A CN 202210365672A CN 114818531 A CN114818531 A CN 114818531A
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吕文朋
张红伟
田震
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Northeastern University China
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Abstract

The invention relates to the technical field of motion interface tracking, and discloses a motion interface tracking numerical dissipation calculation method, which comprises a motion interface tracking algorithm, wherein the motion interface tracking algorithm comprises the following steps: firstly, a VOF algorithm is established on the basis of an MAC (media access control) method, the VOF method is a method capable of processing any free surface, and the basic idea of a model is to solve the volume fraction F of a certain fluid in a computational grid; the invention compares the tracking effect of the VOF (Hirt, Young, FLAIR, FCT, Superbee) method and the Level Set method based on the Eulerian method, the numerical dissipation of the algorithm and the time consumption by reducing and tracking the classic example (shearing flow field) of the motion interface through the Fortran language writing program: (1) the motion interface effect is tracked. The integral average Superbee-VOF method in the VOF methods has a good tracking effect, but the interface is thick, the Levelset method has a best tracking effect, and the interface is thinnest.

Description

Motion interface tracking numerical dissipation calculation method
Technical Field
The invention relates to the technical field of motion interface tracking, in particular to a motion interface tracking numerical dissipation calculation method.
Background
The moving interface refers to an interface between two or more immiscible fluids, and is widely applied to practical engineering such as aerospace, ship water conservancy, fluid machinery and the like, for example, sloshing generated by free surface flow in a liquid rocket fuel tank greatly affects ballistic accuracy, in computational fluid dynamics, the moving interface is used as a special boundary of a flow field and needs to be solved, numerical simulation development is relatively mature so far, but numerical dissipation and accuracy problems exist all the time, and the significance of the research is great.
The motion interface tracking numerical solution can be divided into a gridding method and a non-gridding method, wherein the gridding method is to divide a continuous space into discrete grids and discretely control equations on grid nodes, and the non-gridding method is to solve integral equations or partial differential equation sets with various boundary conditions through a series of randomly distributed nodes. The grid method can be divided into two types, namely a Lagrange method and an Eulerian method, the Lagrange method is also called a satellite method, the change rule of each physical quantity along with the position and the time in the motion process of fluid particles is researched, the motion rule of the whole flow field can be obtained by tracking all the fluid particles, the concept is simple, and the grid method is not suitable for fluids which are severely deformed and mixed; the Eulerian method is also called local method, and expresses fluid physical quantity as a function of space position and time, and the coordinate of the Eulerian method is fixed, so that the Eulerian method can adapt to large deformation of fluid, is dominant in the problem of tracking a moving interface, and has the defect of easy distortion.
The method is characterized in that a Fortran language is used for writing a program, interface evolution tracked by Hirt & Nichols-VOF (hereinafter referred to as Hirt), Young-VOF (hereinafter referred to as Young), Flair-VOF (hereinafter referred to as Flair), FCT-VOF (hereinafter referred to as FCT), integral average Superbee-VOF (hereinafter referred to as Superbee) and Level Set methods under a shear flow condition is calculated, tracking effects, numerical value dissipation conditions and CPU time consumption conditions of the six motion interface tracking algorithms are analyzed, and a better numerical value method is found.
Disclosure of Invention
Technical problem to be solved
Aiming at the defects of the prior art, the invention provides a calculation method for tracking numerical dissipation of a moving interface, which solves the problems in the prior art.
(II) technical scheme
In order to achieve the purpose, the invention provides the following technical scheme: a method for calculating a dissipation of a kinematic interface tracking value, comprising a kinematic interface tracking algorithm, the kinematic interface tracking algorithm comprising:
firstly, VOF algorithm
The VOF method is established on the basis of the MAC method and is a method capable of processing any free surface, and the basic idea of the model is to solve the volume fraction F of a certain fluid in a computational grid;
let the computational domain unit grid volume be σ i，j Defining a scalar function f (x, y, t) to be 1 if there is a target fluid at the (x, y) location, and 0 otherwise. Defining a volume fraction F of fluid for an arbitrary grid cell i,j Comprises the following steps:
Figure BDA0003585653340000021
assuming that the flow field is not compressible, F is derived from time to obtain a VOF equation:
Figure BDA0003585653340000022
(1) the Hirt & Nichols-VOF solving method comprises the steps of regarding a moving interface as a local single-value function Y (x) or X (y), estimating the absolute values of the slope dY/dX and dX/dY of the interface in each grid, determining the interface to be horizontal or vertical, and then determining the position and the direction of the interface by combining the volume fraction of fluid, wherein the method is simple, but lays a foundation for the later interface reconstruction;
(2) the Youngs-VOF solving method is different from the Hirt method that the interface is represented by a horizontal vertical line segment, the Youngs method approximately represents a motion interface by an inclined straight line segment in a single grid, and a calculation domain is divided into a grid filled with a main fluid (F ═ 1) and a secondary fluid (F ═ 0) and an interface grid (0)<F<1) From grid boundary velocity values (u) t 、u r 、u b 、u l ) The mass flow (f) through the grid boundary is determined t 、f r 、f b 、f l )；
When the trellis state F is 1, the mass flow through each boundary is:
Figure BDA0003585653340000031
Figure BDA0003585653340000032
Figure BDA0003585653340000033
Figure BDA0003585653340000034
when the trellis state F is 0, the mass flow through each boundary is: f. of t，b，l，r ＝0；
When the grid state is more than 0 and less than 1, calculating the interface normal vector for the interface grid
Figure BDA0003585653340000035
Figure BDA0003585653340000036
Figure BDA0003585653340000037
Included angle α of the interface with the x-axis:
Figure BDA0003585653340000038
the form of the fluid existing on the interface can be approximately divided into 16 cases, which are simplified into 4 cases through inversion and symmetry, and taking the first type as an example, when the first type is 0 < F < 1, the mass flow calculation method comprises the following steps:
Figure BDA0003585653340000039
Figure BDA00035856533400000310
Figure BDA0003585653340000041
Figure BDA0003585653340000042
the calculation of the grid four-side mass flow F is completed, and the F value is updated through F to obtain the volume fraction of the fluid in the whole field at the next moment:
F i,j t+δt ＝F i,j t -(f t +f b +f l +f r )/(δxδy) (7)
(3) the method for solving the Flair-VOF is different from a Youngs method that an interface is represented by inclined straight-line segments on a single grid, and the Flair method constructs a straight-line segment on the boundary of two adjacent grids to be used as an approximate interface;
(4) the FCT-VOF solving method is different from the three geometric solving methods, and the VOF partial differential equation is solved to obtain the volume fraction of the fluid;
(5) the method for solving the integral average type Superbee-VOF is characterized in that a VOF equation is integrated in a control volume according to time and space:
Figure BDA0003585653340000043
defining integral average, and constructing a solving function into a piecewise linear function:
Figure BDA0003585653340000044
the slope of the interface is
Figure BDA0003585653340000045
Introducing a Superbee limiter;
Figure BDA0003585653340000046
the slope is limited so that the format remains monotonic, having a TVD format,
Figure BDA0003585653340000047
the volume fraction of the fluid at the next time is further determined by the following equation:
Figure BDA0003585653340000051
②, Level Set method
On the calculation domain sigma, a motion interface gamma (t) is defined, namely a Level Set function
Figure BDA0003585653340000052
Zero iso-surface of (d):
Figure BDA0003585653340000053
for example, a circular area may be defined as:
Figure BDA0003585653340000054
the Level Set function satisfies the continuity equation, and a Level Set equation can be obtained:
Figure BDA0003585653340000055
and (3) solving a Level Set equation, wherein a three-order TVD Runge Kutta method is adopted for time dispersion, and a five-order weno method is adopted for space dispersion. Several steps later due to numerical dissipation
Figure BDA0003585653340000056
Will not be able to preserve the properties of the distance function, reconstruct the distance function
Figure BDA0003585653340000057
Referred to as re-initialization. This can be achieved by solving the following stable solution:
Figure BDA0003585653340000058
in the formula
Figure BDA0003585653340000059
For the Level Set function before reinitialization, ε is a small quantity chosen to avoid the denominator being zero.
(III) advantageous effects
The invention provides a method for calculating the dissipation of a tracking numerical value of a motion interface, which has the following beneficial effects:
the invention compares the tracking effect of the VOF (Hirt, Young, FLAIR, FCT, Superbee) method and the Level Set method based on the Eulerian method, the numerical dissipation of the algorithm and the time consumption by reducing and tracking the classic example (shearing flow field) of the motion interface through the Fortran language writing program:
(1) the motion interface effect is tracked. The integral average Superbee-VOF method in the VOF methods has a good tracking effect, but the interface is thick, the Level Set method has a best tracking effect, and the interface is thinnest.
(2) Numerical dissipation. Firstly, with the advance of calculation time, the numerical value dissipation of the Superbee method in the VOF method is small, the numerical value dissipation rate of the Levelset method is minimum, but the numerical value dissipation is greatly increased when the operation time is increased. Secondly, when the time step changes, the Superbee method always keeps good conservation, while the Level Set method is greatly influenced by the time step, has good numerical value conservation when the time step is 0.01s, and increases the numerical value dissipation rate when the time step is increased or decreased. When the space step length changes, the numerical value dissipation of the Superbee, Hirt, Flair and Level Set methods is aggravated along with the increase of the space step length, and the numerical value dissipation is lightened on the contrary because grids of the Youngs and FCT methods are reduced, the calculated amount is reduced; too large a spatial step (Δ t 0.1m) leads to computational divergence.
(3) The calculation time of the Level Set method is far more than that of the VOF method.
Drawings
FIG. 1 is a schematic representation of four interface types in the present invention;
FIG. 2 is a table showing the share of the interface type of FIG. 1 in each side length according to the present invention;
FIG. 3 is a schematic diagram illustrating interface tracking results of a simulated shear flow according to six interface tracking methods of the present invention;
FIG. 4 is a schematic diagram of the trend of numerical dissipation ratios with calculation time according to six interface tracking methods of the present invention;
FIG. 5 is a schematic diagram showing the trend of the numerical dissipation ratios of six interface tracking methods of the present invention along with the time step;
FIG. 6 is a schematic diagram of the trend of numerical dissipation ratios along with the change of spatial step length in six interface tracking methods according to the present invention;
FIG. 7 is a schematic diagram of CPU operation time of six interface tracking methods according to the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The invention provides a technical scheme that: a method for calculating a dissipation of a kinematic interface tracking value, comprising a kinematic interface tracking algorithm, the kinematic interface tracking algorithm comprising:
firstly, VOF algorithm
The VOF method is established on the basis of the MAC method and is a method capable of processing any free surface, and the basic idea of the model is to solve the volume fraction F of a certain fluid in a computational grid;
setting a computing DomainCell grid volume of σ i，j Defining a scalar function f (x, y, t) to be 1 if there is a target fluid at the (x, y) location, and 0 otherwise. Defining a volume fraction F of fluid for an arbitrary grid cell i,j Comprises the following steps:
Figure BDA0003585653340000071
assuming that the flow field is not compressible, F is derived from time to obtain a VOF equation:
Figure BDA0003585653340000072
(1) the Hirt & Nichols-VOF solving method comprises the steps of regarding a moving interface as a local single-value function Y (x) or X (y), estimating the absolute values of the slope dY/dX and dX/dY of the interface in each grid, determining the interface to be horizontal or vertical, and then determining the position and the direction of the interface by combining the volume fraction of fluid, wherein the method is simple, but lays a foundation for the later interface reconstruction;
(2) the Youngs-VOF solving method is different from the Hirt method that the interface is represented by a horizontal vertical line segment, the Youngs method approximately represents a motion interface by an inclined straight line segment in a single grid, and a calculation domain is divided into a grid filled with a main fluid (F ═ 1) and a secondary fluid (F ═ 0) and an interface grid (0)<F<1) From grid boundary velocity values (u) t 、u r 、u b 、u l ) The mass flow (f) through the grid boundary is determined t 、f r 、f b 、f l )；
When the trellis state F is 1, the mass flow through each boundary is:
Figure BDA0003585653340000081
Figure BDA0003585653340000082
Figure BDA0003585653340000083
Figure BDA0003585653340000084
when the trellis state F is 0, the mass flow through each boundary is: f. of t，b，l，r ＝0；
When the grid state is more than 0 and less than 1, calculating the interface normal vector for the interface grid
Figure BDA0003585653340000085
Figure BDA0003585653340000086
Figure BDA0003585653340000087
Included angle α of the interface with the x-axis:
Figure BDA0003585653340000088
the form of the fluid existing on the interface can be approximately divided into 16 cases, which are simplified into 4 cases through inversion and symmetry, as shown in fig. 1, the calculation of the side length share of the main fluid on each side of the unit grid is shown in fig. 2, taking the first type as an example, when the first type is 0 < F < 1, the mass flow calculation method is as follows:
Figure BDA0003585653340000089
Figure BDA0003585653340000091
Figure BDA0003585653340000092
Figure BDA0003585653340000093
the calculation of the grid four-side mass flow F is completed, and the F value is updated through F to obtain the volume fraction of the fluid in the whole field at the next moment:
F i,j t+δt ＝F i,j t -(f t +f b +f l +f r )/(δxδy) (7)
(3) the method for solving the Flair-VOF is different from a Youngs method that an interface is represented by inclined straight-line segments on a single grid, and the Flair method constructs a straight-line segment on the boundary of two adjacent grids to be used as an approximate interface;
(4) the FCT-VOF solving method is different from the three geometric solving methods, and the VOF partial differential equation is solved to obtain the volume fraction of the fluid;
(5) the method for solving the integral average type Superbee-VOF is characterized in that a VOF equation is integrated in a control volume according to time and space:
Figure BDA0003585653340000094
defining integral average, and constructing a solving function into a piecewise linear function:
Figure BDA0003585653340000095
the slope of the interface is
Figure BDA0003585653340000096
Introducing a Superbee limiter;
Figure BDA0003585653340000101
the slope is limited so that the format remains monotonic, having a TVD format,
Figure BDA0003585653340000102
the volume fraction of the fluid at the next time is further determined by the following equation:
Figure BDA0003585653340000103
②, Level Set method
On the calculation domain sigma, a motion interface gamma (t) is defined, namely a Level Set function
Figure BDA0003585653340000104
Zero iso-surface of (d):
Figure BDA0003585653340000105
for example, a circular area may be defined as:
Figure BDA0003585653340000106
the Level Set function satisfies the continuity equation, and a Level Set equation can be obtained:
Figure BDA0003585653340000107
and (3) solving a Level Set equation, wherein a three-order TVD Runge Kutta method is adopted for time dispersion, and a five-order weno method is adopted for space dispersion. Due to numerical dissipation, after several steps
Figure BDA0003585653340000108
Will not be able to preserve the properties of the distance function, reconstruct the distance function
Figure BDA0003585653340000109
Is called reinitiationAnd (5) carrying out initialization. This can be achieved by solving the following stable solution:
Figure BDA00035856533400001010
in the formula
Figure BDA0003585653340000111
For the Level Set function before reinitialization, ε is a small quantity chosen to avoid the denominator being zero.
Numerical dissipation analysis
1. Numerical verification
A two-dimensional calculation area of 1m multiplied by 1m is defined, a circular inner area with the circle center located at (0.5,0.3) and the radius of 0.2m is defined as a main fluid, and other areas are defined as secondary fluids. The grid is divided into 100 × 100, the space step size Δ x is 0.01m, and the time step size Δ t is 0.001 s. The initial velocity field is given as:
Figure BDA0003585653340000112
after a given time t, the velocity field reverses, as:
Figure BDA0003585653340000113
and continuing to calculate the time t, wherein the time t is taken as 2 s. The interface tracking effect of the six interface tracking methods simulating shear flow is shown in fig. 3.
The Hirt method represents a moving interface in an interface grid by a horizontal or vertical straight line, so that the interface is jagged and the tracking effect is the worst; the Youngs method is characterized in that a motion interface grid is represented by an interface normal vector sum and an included angle between the interface normal vector sum and an x axis, the tracking effect is smooth, the interface is thin, and cavities are easy to appear in the motion interface grid; the integration average type Superbee-VOF method introduces a Superbee limiter to make the format monotonous, has a TVD format, has better tracking effect and thicker interface; the Flair-VOF method has a thicker interface and a poorer tracking effect; the FCT-VOF method has the advantages that the interior of the FCT-VOF method is easy to generate holes, the interface is thin, and the tracking effect is poor; the Level Set method solves the high-precision discrete format adopted by the equation, the tracking effect is best, and the interface is thinnest.
2. Numerical dissipation analysis
Taking the initial condition and the boundary condition as 2.1 below, the initial total-field main fluid mass is recorded as m0, the velocity field reversal is not considered for the moment, after the time t under the action of the initial velocity field, the total-field main fluid mass is recorded as m, and the numerical dissipation ratio is expressed as n ═ m (m-m) 0 )/m 0 ,％。
The influence of the calculation time length on the numerical dissipation is researched, wherein the calculation time length is 2s and 10s, and the result is shown in FIG. 4.
Along with the advance of the calculation time, the numerical dissipation rates of the six interface tracking methods are increased, within 1s, the Level Set, Superbee and Hirt methods are good in conservative property, the Flair and young methods are poor, the FCT method is the worst, the numerical dissipation rate reaches 20%, and the calculation is distorted. After 2s, the numerical dissipation rate of the Level Set method is greatly increased and becomes the highest dissipation rate quickly. All calculation methods show mass loss in the early stage of calculation, after a certain time of calculation, the Hirt, Young and Level Set methods are converted into mass increase, and the Superbee, Flair and FCT methods continue to show mass loss.
The effect of the time step on the numerical dissipation was explored and the results are shown in fig. 5.
When the time step length is changed, the Superbee method always keeps better conservative property, the Hirt method is second, the Youngs method and the Flair method are slightly worse, and the FCT method is worst. The VOF method numerical value conservation is slightly reduced along with the increase of the time step length; the Level Set method is greatly influenced by the time step, has better numerical value conservation when the time step is 0.01s, and increases the numerical value dissipation rate when the time step is increased or decreased.
The influence of the space step on the numerical dissipation is explored, and the change result of the space step (the grid number n multiplied by n) is shown in fig. 6.
When the space step length changes, the numerical dissipation of the Superbee, Hirt, Flair and Level Set methods is aggravated along with the increase of the space step length, and the numerical dissipation is lightened on the contrary because grids of the Youngs and FCT methods are reduced, the calculated amount is reduced; too large a spatial step (Δ t 0.1m) leads to computational divergence.
The CPU operation time of different interface tracking algorithms is explored, and the result is shown in fig. 7.
In the numerical calculation, the operation time of the CPU greatly affects the judgment of whether the model is practical. The calculation time is respectively Youngs, Hirt, Flair, Superbee, FCT and Level Set from short to long, and the time consumption of the Level Set method is far more than that of the VOF method because the VOF method adopts a first-order precision format on time dispersion and the Level Set method adopts a high-precision third-order TVD Rune-Kutta format.
It is noted that, herein, relational terms such as first and second, and the like may be used solely to distinguish one entity or action from another entity or action without necessarily requiring or implying any actual such relationship or order between such entities or actions. Also, the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus.
Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that changes, modifications, substitutions and alterations can be made in these embodiments without departing from the principles and spirit of the invention, the scope of which is defined in the appended claims and their equivalents.

Claims (1)

1. A dissipation calculation method for a tracking numerical value of a moving interface comprises a moving interface tracking algorithm, and is characterized in that: the motion interface tracking algorithm comprises:
firstly, VOF algorithm
The VOF method is established on the basis of the MAC method and is a method capable of processing any free surface, and the basic idea of the model is to solve the volume fraction F of a certain fluid in a computational grid;
let the computational domain unit grid volume be σ i，j Defining a scalar function F (x, y, t) with a target fluid at the (x, y) location as F (x, y, t) 1, otherwise as F (x, y, t) 0, defining a volume fraction F of fluid for any grid cell i,j Comprises the following steps:
Figure FDA0003585653330000011
assuming that the flow field is not compressible, F is derived from time to obtain a VOF equation:
Figure FDA0003585653330000012
(1) the Hirt & Nichols-VOF solving method comprises the steps of regarding a moving interface as a local single-value function Y (x) or X (y), estimating the absolute values of the slope dY/dX and dX/dY of the interface in each grid, determining the interface to be horizontal or vertical, and then determining the position and the direction of the interface by combining the volume fraction of fluid, wherein the method is simple, but lays a foundation for the later interface reconstruction;
(2) the Youngs-VOF solving method is different from the Hirt method that the interface is represented by a horizontal vertical line segment, the Youngs method approximately represents a motion interface by an inclined straight line segment in a single grid, and a calculation domain is divided into a grid filled with a main fluid (F ═ 1) and a secondary fluid (F ═ 0) and an interface grid (0)<F<1) From grid boundary velocity values (u) t 、u r 、u b 、u l ) The mass flow (f) through the grid boundary is determined t 、f r 、f b 、f l )；
When the trellis state F is 1, the mass flow through each boundary is:
Figure FDA0003585653330000013
Figure FDA0003585653330000021
Figure FDA0003585653330000022
Figure FDA0003585653330000023
when the trellis state F is 0, the mass flow through each boundary is: f. of t，b，l，r ＝0；
When the grid state is more than 0 and less than 1, calculating the interface normal vector for the interface grid
Figure FDA0003585653330000024
Figure FDA0003585653330000025
Figure FDA0003585653330000026
Included angle α of the interface with the x-axis:
Figure FDA0003585653330000027
the form of the fluid existing on the interface can be approximately divided into 16 cases, which are simplified into 4 cases through inversion and symmetry, and taking the first type as an example, when the first type is 0 < F < 1, the mass flow calculation method comprises the following steps:
Figure FDA0003585653330000028
Figure FDA0003585653330000029
Figure FDA00035856533300000210
Figure FDA0003585653330000031
the calculation of the grid four-side mass flow F is completed, and the F value is updated through F to obtain the volume fraction of the fluid in the whole field at the next moment:
Figure FDA0003585653330000032
(3) the method for solving the Flair-VOF is different from a Youngs method that an interface is represented by inclined straight-line segments on a single grid, and the Flair method constructs a straight-line segment on the boundary of two adjacent grids to be used as an approximate interface;
(4) the FCT-VOF solving method is different from the three geometric solving methods, and the VOF partial differential equation is solved to obtain the volume fraction of the fluid;
(5) the method for solving the integral average type Superbee-VOF is characterized in that a VOF equation is integrated in a control volume according to time and space:
Figure FDA0003585653330000033
defining integral average, and constructing a solving function into a piecewise linear function:
Figure FDA0003585653330000034
the slope of the interface is
Figure FDA0003585653330000035
Introducing a Superbee limiter;
Figure FDA0003585653330000036
the slope is limited so that the format remains monotonic, having a TVD format,
Figure FDA0003585653330000037
the fluid volume fraction at the next time is then determined by the following equation:
Figure FDA0003585653330000041
②, Level Set method
On the calculation domain sigma, a motion interface gamma (t) is defined, namely a Level Set function
Figure FDA0003585653330000042
Zero iso-surface of (d):
Figure FDA0003585653330000043
for example, a circular area may be defined as:
Figure FDA0003585653330000044
the Level Set function satisfies the continuity equation, and a Level Set equation can be obtained:
Figure FDA0003585653330000045
solving a Level Set equation, adopting a three-order TVD (transient voltage divider) Runge Kutta method for time dispersion and adopting a five-order weno method for space dispersion, and solving a Level Set equation after several steps due to numerical dissipation
Figure FDA0003585653330000046
Will not be able to preserve the properties of the distance function, reconstruct the distance function
Figure FDA0003585653330000047
Referred to as reinitialization, can be achieved by solving for a stable solution of:
Figure FDA0003585653330000048
in the formula
Figure FDA0003585653330000049
For the LevelSet function before reinitialization, ε is a small quantity chosen to avoid the denominator being zero.
CN202210365672.7A 2022-04-07 2022-04-07 Motion interface tracking numerical dissipation calculation method Pending CN114818531A (en)

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Cited By (2)

* Cited by examiner, † Cited by third party
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CN115344902A (en) * 2022-10-18 2022-11-15 中国空气动力研究与发展中心计算空气动力研究所 Free surface reconstruction method, device, equipment and medium
CN115344902B (en) * 2022-10-18 2023-03-10 中国空气动力研究与发展中心计算空气动力研究所 Free surface reconstruction method, device, equipment and medium

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