CN109948109A - Unsteady flow in open mesh free particle simulation method containing changes of section - Google Patents
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Abstract
The present invention relates to fluid mechanics technologies to simulate the unsteady flow in open problem containing changes of section to propose a kind of unsteady flow in open mesh free particle simulation method containing changes of section.For this reason, the technical scheme adopted by the present invention is that the unsteady flow in open mesh free particle simulation method containing changes of section, steps are as follows: step 1, initializes the correlated variables and operating parameter of system；Step 2 generates particle information；Step 3 is listed and solves equation and iterate to calculate: step 4 exports result: every calculating for completing a time step just updates as a result, simulating the depth of water, cross-sectional area and the flow velocity of each moment open-channel flow；The circulation of deadline step, exports final result.Present invention is mainly applied to unsteady flow in open mesh free particle simulation occasions.
Description
Technical field
The present invention relates to fluid mechanics technology, a kind of unsteady flow in open containing changes of section is specifically related to without net
Lattice particle simulation method.
Background technique
In simulation Problems of Shallow Water Wave, traditional grid method solves the governing equation and is easy to appear higher-dimension Lagrangian method method
The problem of mesh distortion of middle appearance, so as to cause various numbers such as the unstability of numerical dissipation, numerical value concussion or numerical solution
Value problem.Non-mesh method based on Lagrangian particle model can fully overcome all kinds of numerical value caused by grid method to ask
Topic, calculating process is also more convenient, Lagrangian mesh free particle method can handle with large deformation, move material interface,
The problems such as Free Surface.
Summary of the invention
In order to overcome the deficiencies of the prior art, the present invention is directed to propose a kind of unsteady flow in open mesh free containing changes of section
Particle simulation method, this method solve the shallow water equation under moving coordinate system using Smoothed Particle Hydrodynamics Method,
It has fully considered that traditional grid method brings various numerical problems, has used Lagrangian method to solve under same precision one-dimensional
Governing equation simulates the unsteady flow in open problem containing changes of section.For this reason, the technical scheme adopted by the present invention is that containing section
The unsteady flow in open mesh free particle simulation method of variation, steps are as follows:
Step 1 initializes the correlated variables and operating parameter of system；
Step 2 generates particle information；
Step 3 is listed and solves equation and iterate to calculate:
The one-dimensional shallow water wave equation of solution are as follows:
Wherein, t is the time, and x is position, and A is area of section, and u is water velocity, and H is the depth of water, and g is acceleration of gravity, S0
For channel gradient, SfFor friction term；Lagrange can be converted by equation (1) and (2) according to the individual derivative of moving fluid
Form:
For arbitrary one-dimensional smooth function f (x), integral expression form are as follows:
Wherein δ (x- ξ) is Dirac function, and Ω is the siding-to-siding block length comprising x, in Smoothed Particle Hydrodynamics Method
In, if replacing Dirac function, then the approximate integration expression formula of f (x) with tight branch smooth function W (x- ξ, h) are as follows:
The integral expression of functional derivative are as follows:
And because
[f ' (ξ)] W (x- ξ, h)=[f (ξ) W (x- ξ, h)] '-f (ξ) W ' (x- ξ, h) (8)
So having for compactly supported functions W (x- ξ, h)
In order to convert summation form for integrated form, progress particle is approximate to be obtained:
Wherein N is the particle number in smooth function compacted support, xjFor the position coordinates of j particle, △ xjIt is corresponding for particle
Length is the corresponding area of particle under two-dimensional case, is the corresponding volume of particle, derivative W ' (x-x under three-dimensional situationj, h) and grain
Sub- j is related；
Because
So the particle approximate expression of functional gradient is written as at particle i:
Further, at particle i, the particle approximate expression of open channel cross-sectional area function A is written as:
Wherein VjParticle approximate expression for the corresponding volume of particle, function AH derivative is written as:
It is using formula (14) that equation (4) is discrete, it obtains
Wherein ΠijFor Monaghan type artificial viscosity；
The area and speed of each fluid particles under each time step are calculated separately out by equation (13) and (15), in turn
Solve height and the position of particle；
Step 4 exports result:
1. every calculating for completing time step just update as a result, simulate the depth of water of each moment open-channel flow,
Cross-sectional area and flow velocity；
2. the circulation of deadline step, exports final result.
Generation particle information described in the correlated variables of initialization system and step 2 described in step 1, specifically includes:
1. initializing variable information relevant to problem and operating parameter；
2. generating fluid particles information, particle distribution is initialized in fluid domain, and add initial information；
3. virtual particle information is generated, according to the initial information of the fluid particles setting virtual particle apart from fluid domain boundary 2h.
Further, the initialization correlated variables information and operating parameter are specifically provided that
Computational domain is the one-dimensional space that length is 50m.Dam is located at the center 25m of computational domain, and 0~25m is to have pool
Domain is highly 1m, and it is highly 0 that 25~50m, which is dry bed region,.River bed friction and channel gradient, i.e. s are not considered0=sf=0；
SPH discrete scheme (13) and (15) based on shallow water equation, simulate the Lagrangian particle method of one-dimensional dam-break
Are as follows:
Step 1, initialization.The correlated variables and operating parameter of initialization system, specifically include:
The one-dimensional space that length of field is 50m is calculated, dam is located at the center 25m of computational domain, and 0~25m is to have water area,
Interparticle distance △ x0=0.01m, primary height are 1m, and 25~50m is dry bed region, and the initial velocity of particle is all 0m/
S, quality 10kg, density 1000kg/m3, calculating time step is 0.001s, and the calculating time is 5s, using the length that can polish
Degree is calculated, initial smooth length h0=1.0 △ x, take cubic spline function as kernel function；
Step 2 generates particle information, specifically includes:
It initializes in particle step, collectively generates fluid particles 2500, do not include virtual particle, in the left side cloth of computational domain
Virtual particle is set, guarantees to be able to carry out normal operation, quality, density, the height of virtual particle away from the fluid particles within the scope of the 2h of left end
It is identical with speed as fluid particles.
The area and speed of each fluid particles under each time step are calculated separately out by equation (13) and (15), in turn
Solve height and the position of particle, specific calculating process are as follows:
1. recycling each time step；
2. searching for the neighborhood particle of intended particle after initializing the particle in computational domain, calculated first by equation (13)
Secondly the area of the particle solves its height according to the area and known open channel shape of the particle, the final updating particle
Area and height；
3. calculating the speed of intended particle by equation (15), then the variation of particle position is calculated by speed, finally
Update speed and the position of the particle；
4. determining whether the type of change particle by the calculated particle position of previous step；
5. each fluid particles in cycle calculations domain, repetition do 2. 3. 4. in operation.
The features of the present invention and beneficial effect are:
Non-mesh method based on Lagrangian particle model can fully overcome all kinds of numerical value caused by grid method
Problem, calculating process is also more convenient, and Lagrangian mesh free particle method can be handled with large deformation, mobile material circle
The problems such as face, Free Surface.
Detailed description of the invention:
Fig. 1 program flow diagram.
The one-dimensional dry bed Dam Break Problems physical model of Fig. 2.
(a) of 2s moment particle highly and (b) speed after dam break in Fig. 3 CASE1.
Fig. 4 upstream and downstream depth of water is than the one-dimensional dam-break problem physical model for 2.
(a) of 2s moment particle highly and (b) speed after dam break in Fig. 5 CASE2.
Specific embodiment
The technical scheme is that
A kind of unsteady flow in open mesh free particle simulation method containing changes of section, comprising the following steps:
Step 1 initializes the correlated variables and operating parameter of system；
Step 2 generates particle information；
Step 3 is listed and solves equation and iterate to calculate:
The one-dimensional shallow water wave equation of solution are as follows:
Wherein, t is the time, and x is position, and A is area of section, and u is water velocity, and H is the depth of water, and g is acceleration of gravity, S0
For channel gradient, SfFor friction term.
Lagrangian Form can be converted by equation (1) and (2) according to the individual derivative of moving fluid:
For arbitrary one-dimensional smooth function f (x), integral expression form are as follows:
Wherein δ (x- ξ) is Dirac function, and Ω is the siding-to-siding block length comprising x.In Smoothed Particle Hydrodynamics Method
In, if replacing Dirac function, then the approximate integration expression formula of f (x) with tight branch smooth function W (x- ξ, h) are as follows:
Similarly, the integral expression of functional derivative are as follows:
And because
[f ' (ξ)] W (x- ξ, h)=[f (ξ) W (x- ξ, h)] '-f (ξ) W ' (x- ξ, h) (8)
So having for compactly supported functions W (x- ξ, h)
In order to convert summation form for integrated form, carrying out particle approximation can be obtained:
Wherein N is the particle number in smooth function compacted support, xjFor the position coordinates of j particle, △ xjIt is corresponding for particle
Length (being the corresponding area of particle under two-dimensional case, be the corresponding volume of particle under three-dimensional situation), derivative W ' (x-xj, h) with
Particle j is related.
Because
So the particle approximate expression of functional gradient is writeable at particle i are as follows:
Further, at particle i, the particle approximate expression of open channel cross-sectional area function A is writeable are as follows:
Wherein VjFor the corresponding volume of particle.The particle approximate expression of function AH derivative is writeable are as follows:
It is using formula (14) that equation (4) is discrete, it can obtain
Wherein ΠijFor Monaghan type artificial viscosity.
The area and speed of each fluid particles under each time step can be calculated separately out by equation (13) and (15),
And then solve height and the position of particle.Specific calculating process are as follows:
1. recycling each time step；
2. searching for the neighborhood particle of intended particle after initializing the particle in computational domain, calculated first by equation (13)
Secondly the area of the particle solves its height according to the area and known open channel shape of the particle, the final updating particle
Area and height.
3. calculating the speed of intended particle by equation (15), then the variation of particle position is calculated by speed, finally
Update speed and the position of the particle.
4. determining whether the type of change particle by the calculated particle position of previous step.
5. each fluid particles in cycle calculations domain, repetition do 2. 3. 4. in operation.
Step 4 exports result:
1. every calculating for completing time step just update as a result, simulate the depth of water of each moment open-channel flow,
Cross-sectional area and flow velocity；
2. the circulation of deadline step, exports final result.
Generation particle information described in the correlated variables of initialization system and step 2 described in step 1, specifically includes:
1. initializing variable information relevant to problem and operating parameter；
2. generating fluid particles information, particle distribution is initialized in fluid domain, and add initial information；
3. virtual particle information is generated, according to the initial information of the fluid particles setting virtual particle apart from fluid domain boundary 2h.
Further, in the above scheme, the initializing variable information and operating parameter are specifically provided that
CASE 1: one-dimensional dry bed Dam Break Problems
The physical model of this experiment problem of modelling is shown in that Fig. 2, computational domain are the one-dimensional spaces that length is 50m.Dam is located at meter
At the center 25m for calculating domain, it is highly 1m that 0~25m, which is to have water area, and it is highly 0 that 25~50m, which is dry bed region,.Do not consider river
Bottom friction and channel gradient, i.e. s0=sf=0.
SPH discrete scheme (13) and (15) based on shallow water equation, simulate the Lagrangian particle method of one-dimensional dam-break
Are as follows:
Step 1, initialization.The correlated variables and operating parameter of initialization system, specifically include:
The one-dimensional space that length of field is 50m is calculated, dam is located at the center 25m of computational domain, and 0~25m is to have water area,
Interparticle distance △ x0=0.01m, primary height are 1m, and 25~50m is dry bed region.The initial velocity of particle is all 0m/
S, quality 10kg, density 1000kg/m3, calculating time step is 0.001s, and the calculating time is 5s.In this experiment, it adopts
It is calculated with variable smoothing distance, initial smooth length h0=1.0 △ x.Take cubic spline function as kernel function.
Step 2 generates particle information, specifically includes:
It initializes in particle step, fluid particles 2500 (not including virtual particle) is collectively generated, in the left side of computational domain
It arranges virtual particle, guarantees to be able to carry out normal operation, the quality of virtual particle, density, height away from the fluid particles within the scope of the 2h of left end
Degree and speed are identical as fluid particles.
Step 3 is listed and solves equation and iterate to calculate:
The one-dimensional shallow water wave equation of solution are as follows:
Wherein, t is the time, and x is position, and A is area of section, and u is water velocity, and H is the depth of water, and g is acceleration of gravity, S0
For channel gradient, SfFor friction term.
Lagrangian Form can be converted by equation (1) and (2) according to the individual derivative of moving fluid:
For arbitrary one-dimensional smooth function f (x), integral expression form are as follows:
Wherein δ (x- ξ) is Dirac function, and Ω is the siding-to-siding block length comprising x.In Smoothed Particle Hydrodynamics Method
In, if replacing Dirac function, then the approximate integration expression formula of f (x) with tight branch smooth function W (x- ξ, h) are as follows:
Similarly, the integral expression of functional derivative are as follows:
And because
[f ' (ξ)] W (x- ξ, h)=[f (ξ) W (x- ξ, h)] '-f (ξ) W ' (x- ξ, h) (8)
So having for compactly supported functions W (x- ξ, h)
In order to convert summation form for integrated form, carrying out particle approximation can be obtained:
Wherein N is the particle number in smooth function compacted support, xjFor the position coordinates of j particle, △ xjIt is corresponding for particle
Length (being the corresponding area of particle under two-dimensional case, be the corresponding volume of particle under three-dimensional situation), derivative W ' (x-xj, h) with
Particle j is related.
Because
So the particle approximate expression of functional gradient is writeable at particle i are as follows:
Further, at particle i, the particle approximate expression of open channel cross-sectional area function A is writeable are as follows:
Wherein VjFor the corresponding volume of particle.The particle approximate expression of function AH derivative is writeable are as follows:
It is using formula (14) that equation (4) is discrete, it can obtain
Wherein ΠijFor Monaghan type artificial viscosity.
The area and speed of each fluid particles under each time step can be calculated separately out by equation (13) and (15),
And then solve height and the position of particle.Specific calculating process are as follows:
1. recycling each time step；
2. searching for the neighborhood particle of intended particle after initializing the particle in computational domain, calculated first by equation (13)
Secondly the area of the particle solves its height according to the area and known open channel shape of the particle, the final updating particle
Area and height.
3. calculating the speed of intended particle by equation (15), then the variation of particle position is calculated by speed, finally
Update speed and the position of the particle.
4. determining whether the type of change particle by the calculated particle position of previous step.
5. each fluid particles in cycle calculations domain, repetition do 2. 3. 4. in operation.
Step 4 exports result:
1. every calculating for completing time step just update as a result, simulate the depth of water of each moment open-channel flow,
Cross-sectional area and flow velocity；
2. the circulation of deadline step, exports final result.
CASE 2: the upstream and downstream depth of water is than the one-dimensional dam-break problem physical model for 2
The physical model of this experiment problem of modelling is shown in that Fig. 4, computational domain are the one-dimensional spaces that length is 50m.Dam is located at meter
At the center 25m for calculating domain, it is highly 1m that 0~25m, which is high water level region, and it is highly 0.5m that 25~50m, which is low water level region,.No
Consider river bed friction and channel gradient, i.e. s0=sf=0.
The present invention is described in detail with reference to the accompanying drawing.
SPH discrete scheme (13) and (15) based on shallow water equation, the simulation upstream and downstream depth of water is than the one-dimensional dam-break drawing for 2
Ge Lang particle method are as follows:
Initialization: step 1 initializes the correlated variables and operating parameter of system, specifically includes:
Experiment simulation is to calculate the one-dimensional space that length of field is 50m, and dam is located at the center 25m of computational domain, 0~
25m is high water level region, interparticle distance △ x0=0.01m, primary height are 1m, 25~50m low water level region, between particle
Away from △ x1=0.02m, primary height are 0.5m.The initial velocity of particle is all 0, quality 10kg, density 1000kg/
m3, calculating time step is 0.001s, and the calculating time is 5s.In this experiment, using the smooth length h=1.4 △ x of fixation1。
Step 2 generates particle information, specifically includes:
3750 particles of initial distribution fluid particles (not including virtual particle) are arranged virtual particle in the two sides of computational domain, are protected
Card is able to carry out normal operation away from the fluid particles in the 2h length of left and right ends, quality, density, height and the speed of virtual particle with
Fluid particles are identical.
Step 3 is the same as CASE 1.
Although above in conjunction with figure, invention has been described, and the invention is not limited to above-mentioned specific embodiment parties
Formula, the above mentioned embodiment is only schematical, rather than restrictive, and those skilled in the art are in this hair
Under bright enlightenment, without deviating from the spirit of the invention, many variations can also be made, these belong to guarantor of the invention
Within shield.
Claims (2)
1. a kind of unsteady flow in open mesh free particle simulation method containing changes of section, characterized in that steps are as follows:
Step 1 initializes the correlated variables and operating parameter of system；
Step 2 generates particle information；
Step 3 is listed and solves equation and iterate to calculate:
The one-dimensional shallow water wave equation of solution are as follows:
Wherein, t is the time, and x is position, and A is area of section, and u is water velocity, and H is the depth of water, and g is acceleration of gravity, S0For river
The bed gradient, SfFor friction term；Lagrangian Form can be converted by equation (1) and (2) according to the individual derivative of moving fluid:
For arbitrary one-dimensional smooth function f (x), integral expression form are as follows:
Wherein δ (x- ξ) is Dirac function, and Ω is the siding-to-siding block length comprising x, in Smoothed Particle Hydrodynamics Method, if
Dirac function is replaced with tight branch smooth function W (x- ξ, h), then the approximate integration expression formula of f (x) are as follows:
The integral expression of functional derivative are as follows:
And because
[f ' (ξ)] W (x- ξ, h)=[f (ξ) W (x- ξ, h)] '-f (ξ) W ' (x- ξ, h) (8)
So having for compactly supported functions W (x- ξ, h)
In order to convert summation form for integrated form, progress particle is approximate to be obtained:
Wherein N is the particle number in smooth function compacted support, xjFor the position coordinates of j particle, △ xjFor the corresponding length of particle
It spends, is the corresponding area of particle under two-dimensional case, be the corresponding volume of particle, derivative W ' (x-x under three-dimensional situationj, h) and particle
J is related；
Because
So the particle approximate expression of functional gradient is written as at particle i:
Further, at particle i, the particle approximate expression of open channel cross-sectional area function A is written as:
Wherein VjParticle approximate expression for the corresponding volume of particle, function AH derivative is written as:
It is using formula (14) that equation (4) is discrete, it obtains
Wherein ΠijFor Monaghan type artificial viscosity；
The area and speed of each fluid particles under each time step are calculated separately out by equation (13) and (15), and then are solved
The height of particle and position out；
Step 4 exports result:
1. every calculating for completing a time step just updates as a result, being the depth of water for simulating each moment open-channel flow, transversal
Area and flow velocity；
2. the circulation of deadline step, exports final result.
Generation particle information described in the correlated variables of initialization system and step 2 described in step 1, specifically includes:
1. initializing variable information relevant to problem and operating parameter；
2. generating fluid particles information, particle distribution is initialized in fluid domain, and add initial information；
3. virtual particle information is generated, according to the initial information of the fluid particles setting virtual particle apart from fluid domain boundary 2h.
2. the unsteady flow in open mesh free particle simulation method containing changes of section as described in claim 1, characterized in that into
One step, the initialization correlated variables information and operating parameter are specifically provided that
Computational domain is the one-dimensional space that length is 50m, and dam is located at the center 25m of computational domain, and 0~25m is to have water area, high
Degree is 1m, and it is highly 0 that 25~50m, which is dry bed region, does not consider river bed friction and channel gradient, i.e. s0=sf=0；
SPH discrete scheme (13) and (15) based on shallow water equation, simulate the Lagrangian particle method of one-dimensional dam-break are as follows:
Step 1, initialization, initializes the correlated variables and operating parameter of system, specifically includes:
The one-dimensional space that length of field is 50m is calculated, dam is located at the center 25m of computational domain, and 0~25m is to have water area, particle
Spacing △ x0=0.01m, primary height are 1m, and 25~50m is dry bed region, and the initial velocity of particle is all 0m/s, matter
Amount is 10kg, density 1000kg/m3, calculating time step be 0.001s, the calculatings time be 5s, using variable smoothing distance into
Row calculates, initial smooth length h0=1.0 △ x, take cubic spline function as kernel function；
Step 2 generates particle information, specifically includes:
It initializes in particle step, collectively generates fluid particles 2500, do not include virtual particle, arranged in the left side of computational domain empty
Particle guarantees to be able to carry out normal operation, quality, density, height and the speed of virtual particle away from the fluid particles within the scope of the 2h of left end
It spends identical as fluid particles；
The area and speed of each fluid particles under each time step are calculated separately out by equation (13) and (15), and then are solved
The height of particle and position out, specific calculating process are as follows:
1. recycling each time step；
2. searching for the neighborhood particle of intended particle after initializing the particle in computational domain, the grain is calculated by equation (13) first
Secondly the area of son solves its height, the area of the final updating particle according to the area and known open channel shape of the particle
And height；
3. calculating the speed of intended particle by equation (15), then the variation of particle position, final updating are calculated by speed
The speed of the particle and position；
4. determining whether the type of change particle by the calculated particle position of previous step；
5. each fluid particles in cycle calculations domain, repetition do 2. 3. 4. in operation.
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| CN114357907A (en) * | 2022-01-07 | 2022-04-15 | 中国空气动力研究与发展中心计算空气动力研究所 | Parallel method suitable for Lagrange type particle numerical simulation |
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