CN110569541A - Pipeline water hammer analysis method - Google Patents
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Abstract
The invention relates to the technical field of hydrodynamics, and provides a new method for analyzing a pipeline water hammer, which solves a water hammer equation under a mobile coordinate system and realizes simulation of a water hammer phenomenon in a pipeline. Meanwhile, on the premise of meeting the numerical precision, the problem of pipeline water hammer can be simulated more conveniently. Therefore, the technical scheme adopted by the invention is that the pipeline water hammer analysis method comprises the following steps: step one, initialization: initializing related variable, particle, upstream and downstream virtual particle information; step two, listing solution equations and carrying out iterative calculation: and step three, outputting the result. The invention is mainly applied to the fluid mechanics engineering design occasion.
Description
Technical Field
The invention relates to the technical field of hydromechanics, in particular to a new method for analyzing a pipeline water hammer.
background
The pipeline water hammer problem can be solved by using a rigid column theory, a characteristic line method or some grid method when the elasticity of water is not considered, however, in most cases, the elasticity of the water body must be considered at times for solving the problem more accurately, and the characteristic line method or the grid method generally tracks the movement of an interface, i.e. a complex calculation problem is generated, and the calculation precision is reduced along with the accumulation of interpolation errors. When the method of the density summation type non-grid particles is used for solving the problem of the pipeline water hammer, the elasticity of water can be fully considered, the calculation process is more convenient, and the problem of the pipeline water hammer can be better simulated in practical application.
Disclosure of Invention
in order to overcome the defects of the prior art, the invention aims to provide a new method for analyzing the pipeline water hammer, which solves the water hammer equation under a mobile coordinate system and realizes the simulation of the water hammer phenomenon in the pipeline. Meanwhile, on the premise of meeting the numerical precision, the problem of pipeline water hammer can be simulated more conveniently. Therefore, the technical scheme adopted by the invention is that the pipeline water hammer analysis method comprises the following steps:
step one, initialization: initializing related variable, particle, upstream and downstream virtual particle information;
step two, listing solution equations and carrying out iterative calculation:
Solving the water hammer equation as follows:
where P is pressure, V is the velocity of the fluid, ρ is the density of the fluid, c is the acoustic velocity, g is the gravitational acceleration, θ is the pipe inclination, D is the pipe inside diameter, λ is the coefficient of friction t and x represents the time and distance distribution along the pipe;
when solving for pressure in the water hammer using density summation, the governing equation is:
ρi＝∑mjWij (4)
Pi-P0＝c2(ρi-ρ0) (5)
where ρ is0is given a reference pressure P0fluid density of [ rho ]iDenotes the density of i particles, mjDenotes the mass of j particles, Pidenotes the pressure at the i particle, W denotes the smoothing function;
The upstream boundary condition is
where P (0, t) represents a pressure at 0 on the pipeline at any time t, PRrepresenting reservoir pressure, V (0, t) representing the velocity at 0 on the pipeline at any time t;
The downstream boundary condition is
V(L,t)＝0 (8)
The above equation indicates that at any time t, the velocity at the pipe location L is 0;
in the method of smooth particle hydrodynamics, the integral form of an arbitrary function f (x) is expressed as:
wherein δ (x-x') is a dirac function, Ω is an integral volume containing x, and the specific expression is as follows:
If the Dirac function is replaced by a smooth function W (x-x', h), the integral of f (x) is expressed as:
where h is the smoothing length of the smoothing function, in the derivation of the SPH (smooth particle hydrodynamics) method, the kernel approximation operator is generally labeled with an angular bracket, i.e., the integral of the function is expressed as:
<f(x)>＝∫Ωf(x')W(x-x',h)dx' (12)
to get an approximation of the functional derivative, we replace f (x) with f' (x), i.e. the functional derivative is approximated as:
and because:
[f'(x')]W(x-x',h)＝[f(x')W(x-x',h)]'-f(x')W'(x-x',h) (14)
Therefore:
converting the integral form into a summation form, and performing particle approximation to obtain:
wherein N is the number of particles in the smooth function tight branch domain,The derivative W' (x-x) for the corresponding length of the particlejh) is associated with particle j; and because:
Therefore:
And because:
therefore, the particle approximation of the function at particle i is written as:
discretizing the control equations (3), (4) and (5) by using a smooth particle fluid dynamics method to obtain:
ρi＝∑mjWij (21)
Pi-P0＝c2(ρi-ρ0) (23)
Wherein, ViDenotes the velocity at i particle, Wij,xrepresenting the first derivative, Π, of the smoothing function with respect to xijfor the artificial viscosity item, the specific expression of the Monaghan type viscosity item is as follows:
wherein the content of the first and second substances,
in the formula, c and V respectively represent artificial sound velocity and velocity vector of particles, alpha and beta are constants, and a Monaghan type viscosity term is the most common viscosity term in smooth particle fluid dynamics;
calculating the pressure and speed information of each particle at different moments according to a control equation;
Step three, outputting a result:
1) Ending the particle circulation, storing and outputting an intermediate result;
2) And ending the time loop and outputting a final result.
The new artificial viscosity constructed by Riemann decomposition is used for replacing the artificial viscosity item, and the method is realized as follows:
The pressure term P in equation (22)iand PjUsing Riemann's solutionInstead, we get a governing equation that replaces the artificial viscosity term in traditional SPH with a new artificial viscosity constructed by the Riemann solution:
wherein the content of the first and second substances,
Calculating pressure and velocity information for each particle at different times according to a discrete form of a control equation:
According to a control equation, calculating the pressure and speed information of each particle at different moments, wherein the specific calculation process is as follows:
1) Cycling the time variable;
2) circulating the particles;
3) obtaining pressure information of the fluid particles according to the control equation in the discrete form for the initialized particles, and updating the pressure information of the fluid particles;
4) Calculating and updating the pressure information of the upstream virtual particle according to the equations (6) and (7);
5) Calculating and updating velocity information of the fluid particles according to the control equation in the discrete form;
6) respectively updating the speeds of the upstream virtual particles and the downstream virtual particles according to the speeds of the upstream boundary fluid particles and the downstream boundary fluid particles;
7) and updating the particle position, the pressure of the particle, the upstream and downstream virtual particle speeds and the pressure information according to the velocity of the particle.
the invention has the characteristics and beneficial effects that:
the invention provides a new method for analyzing a pipeline water hammer, which solves the water hammer pressure through a density summation method, solves the water hammer speed through an SPH method and further solves a water hammer equation under a mobile coordinate system. Meanwhile, aiming at the artificial viscosity item in the SPH method, the invention utilizes the new artificial viscosity constructed by Riemann solution to solve the water hammer equation, and the new viscosity does not need to manually adjust the coefficient of the viscosity. The influence of weak compressibility of water can be fully considered, and the problem of pipeline water hammer can be simulated more conveniently on the premise of meeting numerical precision.
description of the drawings:
FIG. 1 is a diagram of a physical model of a pipeline water hammer;
FIG. 2 is a computational flow diagram for solving the pipeline water hammer problem;
FIG. 3 is a graph of pressure results obtained with the new method of solving for pipeline water hammer.
FIG. 4 is a graph of pressure results for solving for water hammer with a new artificial viscosity term.
Detailed Description
The technical problem to be solved by the invention is to provide a new method for analyzing the pipeline water hammer, the method solves the water hammer pressure through a density summation method, solves the water hammer speed through an SPH (smooth particle hydrodynamics) method, and further solves the water hammer equation under a mobile coordinate system. The simulation of the water hammer phenomenon in the pipeline is realized. Meanwhile, aiming at the artificial viscosity item in the SPH method, the invention solves the water hammer equation by using the new artificial viscosity constructed by Riemann solution. The influence of weak compressibility of water can be fully considered, and the problem of pipeline water hammer can be simulated more conveniently on the premise of meeting numerical precision. In order to solve the technical problems, the technical scheme of the invention is as follows:
a new method of analyzing a pipeline water hammer, comprising the steps of:
initializing related variables, particles and upstream and downstream virtual particle information;
step two, listing solution equations and carrying out iterative calculation:
Solving the water hammer equation as follows:
Where P is pressure, V is the velocity of the fluid, ρ is the density of the fluid, c is the acoustic velocity, g is the gravitational acceleration, θ is the pipe inclination, D is the pipe inside diameter, λ is the coefficient of friction t and x represents the time and distance distribution along the pipe;
When solving for pressure in the water hammer using density summation, the governing equation is:
ρi＝∑mjWij (4)
Pi-P0＝c2(ρi-ρ0) (5)
Where ρ is0is given a reference pressure P0fluid density of [ rho ]idenotes the density of i particles, mjDenotes the mass of j particles, Pidenotes the pressure at the i particle, W denotes the smoothing function;
the upstream boundary condition is
where P (0, t) represents a pressure at 0 on the pipeline at any time t, PRrepresents the pressure of the reservoir, V (0, t) represents the speed at 0 on the pipeline at any time t;
The downstream boundary condition is
V(L,t)＝0 (8)
the above equation indicates that at any time t, the velocity at the pipe location L is 0;
In the method of smooth particle hydrodynamics, the integral form of an arbitrary function f (x) is expressed as:
wherein δ (x-x') is a dirac function, Ω is an integral volume containing x, and the specific expression is as follows:
if the Dirac function is replaced by a smooth function W (x-x', h), the integral of f (x) is expressed as:
where h is the smoothing length of the smoothing function, in the derivation of the SPH (smooth particle hydrodynamics) method, the kernel approximation operator is generally labeled with an angular bracket, i.e., the integral of the function is expressed as:
<f(x)>＝∫Ωf(x')W(x-x',h)dx' (12)
to get an approximation of the functional derivative, we replace f (x) with f' (x), i.e. the functional derivative is approximated as:
and because:
[f'(x')]W(x-x',h)＝[f(x')W(x-x',h)]'-f(x')W'(x-x',h) (14)
therefore:
converting the integral form into a summation form, and performing particle approximation to obtain:
Wherein N is the number of particles in the smooth function tight branch domain,the derivative W' (x-x) is the length (area for particles in two dimensions and volume for particles in three dimensions) for a particlejAnd h) is associated with particle j.
and because:
Therefore:
and because:
therefore, the particle approximation of the function at particle i can be written as:
The control equations (3), (4), (5) are discretized by a smooth particle fluid dynamics method to obtain:
ρi＝∑mjWij (21)
Pi-P0＝c2(ρi-ρ0) (23)
Wherein, ViDenotes the velocity at i particle, Wij,xrepresenting the first derivative, Π, of the smoothing function with respect to xijfor the artificial viscosity item, the specific expression of the Monaghan type viscosity item is as follows:
wherein the content of the first and second substances,
in the formula, c and V respectively represent the artificial sound velocity and the velocity vector of the particles, alpha and beta are constants and are usually about 1.0, but the values of the c and the beta are properly adjusted in different fluid states, a Monaghan type viscosity term is the most common viscosity term in smooth particle fluid dynamics, kinetic energy can be converted into heat energy, dissipation generated by shock waves is provided, and non-physical penetration when the particles are close to each other can be solved.
When the problem is solved by using the artificial viscosity, the coefficient of the artificial viscosity needs to be adjusted manually, which brings inconvenience to the experiment. The present invention replaces the manual sticky term in conventional SPH with a new manual sticky constructed by Riemann's solution to this problem. The concrete implementation is as follows:
will be given in equation (22)Pressure term P ofiAnd Pjusing Riemann's solutionInstead, we get a governing equation that replaces the artificial viscosity term in traditional SPH with a new artificial viscosity constructed by the Riemann solution:
wherein the content of the first and second substances,
calculating pressure and velocity information for each particle at different times according to a discrete form of a control equation:
the specific calculation process is as follows:
1) Cycling the time variable;
2) Circulating the particles;
3) Obtaining pressure information of the fluid particles according to the control equation in the discrete form for the initialized particles, and updating the pressure information of the fluid particles;
4) Calculating and updating the pressure information of the upstream virtual particle according to the equations (6) and (7);
5) calculating and updating velocity information of the fluid particles according to the control equation in the discrete form;
6) Respectively updating the speeds of the upstream virtual particles and the downstream virtual particles according to the speeds of the upstream boundary fluid particles and the downstream boundary fluid particles;
7) updating the particle position, the particle pressure, the upstream and downstream virtual particle speeds and pressure information according to the particle speed;
Step three, outputting a result:
1) Ending the particle circulation, storing and outputting an intermediate result;
2) and ending the time loop and outputting a final result.
further, in the above scheme, the initialization of the relevant variable, particle, upstream and downstream virtual particle information is performed in step one. The method specifically comprises the following steps:
1) initializing variable information related to the problem;
2) initializing fluid particle information, uniformly distributing particles in a fluid domain, and adding initial information;
3) initializing virtual particle information, respectively distributing two layers of virtual particles on the upstream and downstream boundaries of the fluid, and adding initial information according to boundary conditions.
Further, in the above scheme, specific parameters of the initialization variable information, the particle information, and the upstream and downstream virtual particle information are set as follows: the diameter D of the pipeline is 0.797m, the length L of the pipeline is 20m, the initial flow of the reservoir is 0.5m3/s, and the pressure P of the reservoirR1MPa, density rho of 1000kg/m3, gravity acceleration g of 9.8m/s2, sound wave speed c of 1250.7m/s, initial speed of 1.002m/s and total particle mass of 100. Initial conditions of the water column are V (x,0) ═ 0 and P (x,0) ═ PR(ii) a In the numerical simulation, 205 particles were initially uniformly distributed, including the dummy particles at both upstream and downstream ends. The calculation time step is 0.0001s, and the total calculation time is 5 s.
further, in the above scheme, the specific parameters of the initialized fluid particle information are set as follows: the pipeline is uniformly distributed with 201 fluid particles, and the fluid particle information is that V (x,0) is 0 and P (x,0) is PR。
Further, in the above scheme, the initialization virtual particle information parameter is specifically set as follows: two virtual particles at the upstream and downstream boundaries, the upstream virtual particle pressure being PRthe downstream virtual particle pressure is equal to the pressure of the fluid particle closest to the downstream end, the upstream virtual particle initial velocity is 1.002m/s, and the downstream virtual particle adopts a mirror image particle method, and the initial velocity is-1.002 m/s.
the present invention is described in further detail below with reference to the attached drawings.
As shown in FIG. 1, the diameter D of the test line of the present invention was 0.797m, the length L of the test line was 20m, the initial flow rate of the reservoir was 0.5m3/s, and the pressure P of the reservoir wasR1MPa, density rho of 1000kg/m3, gravity acceleration g of 9.8m/s2, sound wave speed c of 1250.7m/s, initial speed of 1.002m/s and total particle mass of 100. Initial conditions of the water column are V (x,0) ═ 0 and P (x,0) ═ PR(ii) a In the numerical simulation, 205 particles were initially uniformly distributed, including the dummy particles at both upstream and downstream ends. The calculation time step is 0.0001s, and the total calculation time is 5 s.
the new method for analyzing the pipeline water hammer comprises the following steps:
Initializing related variables, particles and upstream and downstream virtual particle information; the method specifically comprises the following steps:
1) initializing variable information related to the problem: the diameter D of the pipeline is 0.797m, the length L of the pipeline is 20m, the initial flow of the reservoir is 0.5m3/s, and the pressure P of the reservoirR1MPa, density rho of 1000kg/m3, gravity acceleration g of 9.8m/s2, sound wave speed c of 1250.7m/s, initial speed of 1.002m/s and total particle mass of 100. Initial conditions of the water column are V (x,0) ═ 0 and P (x,0) ═ PR(ii) a In the numerical simulation, 205 particles (including the imaginary particles at each end) were initially uniformly distributed, the calculation time step was 0.0001s, and the total calculation time was 5 s.
2) initializing fluid particle information, uniformly distributing particles in a fluid domain, and adding initial information: initializing fluid particles, uniformly distributing 201 fluid particles in the pipeline, and obtaining fluid particle information of V (x,0) ═ 0 and P (x,0) ═ PR。
3) initializing virtual particle information, respectively distributing two layers of virtual particles on the upstream and downstream boundaries of the fluid, and adding the initial information according to boundary conditions: two virtual particles at the upstream and downstream boundaries, the upstream virtual particle pressure being PRThe downstream virtual particle pressure is equal to the pressure of the fluid particle closest to the downstream end, the upstream virtual particle initial velocity is 1.002m/s, and the downstream virtual particle adopts a mirror image particle method, and the initial velocity is-1.002 m/s.
Step two, listing solution equations and performing iterative computation:
solving the water hammer equation as follows:
Where P is pressure, V is the velocity of the fluid, ρ is the density of the fluid, c is the acoustic velocity, g is the gravitational acceleration, θ is the pipe inclination, D is the pipe inside diameter, λ is the coefficient of friction t and x represents the time and distance distribution along the pipe;
When solving for pressure in the water hammer using density summation, the governing equation is:
ρi＝∑mjWij (4)
Pi-P0＝c2(ρi-ρ0) (5)
where ρ is0Is given a reference pressure P0Fluid density of [ rho ]iDenotes the density of i particles, mjdenotes the mass of j particles, PiDenotes the pressure at the i particle, W denotes the smoothing function;
the upstream boundary condition is
where P (0, t) represents a pressure at 0 on the pipeline at any time t, PRRepresents the pressure of the reservoir, V (0, t) represents the speed at 0 on the pipeline at any time t;
the downstream boundary condition is
V(L,t)＝0 (8)
the above equation indicates that at any time t, the velocity at the pipe location L is 0;
in the method of smooth particle hydrodynamics, the integral form of an arbitrary function f (x) is expressed as:
wherein δ (x-x') is a dirac function, Ω is an integral volume containing x, and the specific expression is as follows:
if the Dirac function is replaced by a smooth function W (x-x', h), the integral of f (x) is expressed as:
where h is the smoothing length of the smoothing function, in the derivation of the SPH (smooth particle hydrodynamics) method, the kernel approximation operator is generally labeled with an angular bracket, i.e., the integral of the function is expressed as:
<f(x)>＝∫Ωf(x')W(x-x',h)dx' (12)
to get an approximation of the functional derivative, we replace f (x) with f' (x), i.e. the functional derivative is approximated as:
and because:
[f'(x')]W(x-x',h)＝[f(x')W(x-x',h)]'-f(x')W'(x-x',h) (14)
therefore:
converting the integral form into a summation form, and performing particle approximation to obtain:
Wherein N is the number of particles in the smooth function tight branch domain,the derivative W' (x-x) is the length (area for particles in two dimensions and volume for particles in three dimensions) for a particlejand h) is associated with particle j.
and because:
Therefore:
and because:
therefore, the particle approximation of the function at particle i can be written as:
the control equations (3), (4), (5) are discretized by a smooth particle fluid dynamics method to obtain:
ρi＝∑mjWij (21)
Pi-P0＝c2(ρi-ρ0) (23)
wherein, ViDenotes the velocity at i particle, Wij,xrepresenting the first derivative, Π, of the smoothing function with respect to xijFor the artificial viscosity item, the specific expression of the Monaghan type viscosity item is as follows:
wherein the content of the first and second substances,
In the formula, c and V respectively represent the artificial sound velocity and the velocity vector of the particles, alpha and beta are constants and are usually about 1.0, but the values of the c and the beta are properly adjusted in different fluid states, a Monaghan type viscosity term is the most common viscosity term in smooth particle fluid dynamics, kinetic energy can be converted into heat energy, dissipation generated by shock waves is provided, and non-physical penetration when the particles are close to each other can be solved.
When the problem is solved by using the artificial viscosity, the coefficient of the artificial viscosity needs to be adjusted manually, which brings inconvenience to the experiment. The present invention replaces the manual sticky term in conventional SPH with a new manual sticky constructed by Riemann's solution to this problem. The concrete implementation is as follows:
The pressure term P in equation (22)iand Pjusing Riemann's solutionInstead, we get a governing equation that replaces the artificial viscosity term in traditional SPH with a new artificial viscosity constructed by the Riemann solution:
wherein the content of the first and second substances,
calculating pressure and velocity information for each particle at different times according to a discrete form of a control equation:
The specific calculation process is as follows:
3) cycling the time variable;
4) Circulating the particles;
3) obtaining pressure information of the fluid particles according to the control equation in the discrete form for the initialized particles, and updating the pressure information of the fluid particles;
4) Calculating and updating the pressure information of the upstream virtual particle according to the equations (6) and (7);
5) calculating and updating velocity information of the fluid particles according to the control equation in the discrete form;
6) respectively updating the speeds of the upstream virtual particles and the downstream virtual particles according to the speeds of the upstream boundary fluid particles and the downstream boundary fluid particles;
7) Updating the particle position, the particle pressure, the upstream and downstream virtual particle speeds and pressure information according to the particle speed;
step three, outputting a result:
1) ending the particle circulation, storing and outputting an intermediate result;
2) And ending the time loop and outputting a final result.
although the present invention has been described with reference to the accompanying drawings, the present invention is not limited to the above-mentioned embodiments, which are only one kind (schematic) and not restrictive, and those skilled in the art can make many modifications without departing from the spirit and scope of the present invention.
Claims (2)
1. a pipeline water hammer analysis method is characterized by comprising the following steps:
step one, initialization: initializing related variable, particle, upstream and downstream virtual particle information;
step two, listing solution equations and carrying out iterative calculation:
solving the water hammer equation as follows:
where P is pressure, V is the velocity of the fluid, ρ is the density of the fluid, c is the acoustic velocity, g is the gravitational acceleration, θ is the pipe inclination, D is the pipe inside diameter, λ is the coefficient of friction t and x represents the time and distance distribution along the pipe;
When solving for pressure in the water hammer using density summation, the governing equation is:
ρi＝∑mjWij (4)
Pi-P0＝c2(ρi-ρ0) (5)
Where ρ is0Is given a reference pressure P0fluid density of [ rho ]iDenotes the density of i particles, mjdenotes the mass of j particles, Pidenotes the pressure at the i particle, W denotes the smoothing function;
The upstream boundary condition is
Where P (0, t) represents a pressure at 0 on the pipeline at any time t, PRRepresenting reservoir pressure, V (0, t) representing the velocity at 0 on the pipeline at any time t;
the downstream boundary condition is
V(L,t)＝0 (8)
the above equation indicates that at any time t, the velocity at the pipe location L is 0;
in the method of smooth particle hydrodynamics, the integral form of an arbitrary function f (x) is expressed as:
wherein δ (x-x') is a dirac function, Ω is an integral volume containing x, and the specific expression is as follows:
if the Dirac function is replaced by a smooth function W (x-x', h), the integral of f (x) is expressed as:
where h is the smoothing length of the smoothing function, in the derivation of the smooth particle hydrodynamics SPH method, the kernel approximation operator is generally marked with an angle bracket, i.e. the integral of the function is expressed as:
<f(x)>＝∫Ωf(x')W(x-x',h)dx' (12)
to get an approximation of the functional derivative, we replace f (x) with f' (x), i.e. the functional derivative is approximated as:
and because:
[f'(x')]W(x-x',h)＝[f(x')W(x-x',h)]'-f(x')W'(x-x',h) (14)
therefore:
converting the integral form into a summation form, and performing particle approximation to obtain:
wherein N is the number of particles in the smooth function tight branch domain,The derivative W' (x-x) for the corresponding length of the particlejh) is associated with particle j; and because:
therefore:
and because:
Therefore, the particle approximation of the function at particle i is written as:
discretizing the control equations (3), (4) and (5) by using a smooth particle fluid dynamics method to obtain:
ρi＝∑mjWij (21)
Pi-P0＝c2(ρi-ρ0) (23)
Wherein, Videnotes the velocity at i particle, Wij,xrepresenting the first derivative, Π, of the smoothing function with respect to xijFor the artificial viscosity item, the specific expression of the Monaghan type viscosity item is as follows:
Wherein the content of the first and second substances,
in the formula, c and V respectively represent artificial sound velocity and velocity vector of particles, alpha and beta are constants, and a Monaghan type viscosity term is the most common viscosity term in smooth particle fluid dynamics;
calculating the pressure and speed information of each particle at different moments according to a control equation;
Step three, outputting a result:
1) ending the particle circulation, storing and outputting an intermediate result;
2) and ending the time loop and outputting a final result.
2. the method for analyzing the pipeline water hammer as claimed in claim 1, wherein the new artificial viscosity constructed by Riemann's Riemann solution is used to replace the artificial viscosity term, and the method is implemented as follows:
The pressure term P in equation (22)iand Pjusing Riemann's solutioninstead, we get a governing equation that replaces the artificial viscosity term in traditional SPH with a new artificial viscosity constructed by the Riemann solution:
Wherein the content of the first and second substances,
Calculating pressure and velocity information for each particle at different times according to a discrete form of a control equation:
According to a control equation, calculating the pressure and speed information of each particle at different moments, wherein the specific calculation process is as follows:
1) cycling the time variable;
2) circulating the particles;
3) Obtaining pressure information of the fluid particles according to the control equation in the discrete form for the initialized particles, and updating the pressure information of the fluid particles;
4) calculating and updating the pressure information of the upstream virtual particle according to the equations (6) and (7);
5) Calculating and updating velocity information of the fluid particles according to the control equation in the discrete form;
6) respectively updating the speeds of the upstream virtual particles and the downstream virtual particles according to the speeds of the upstream boundary fluid particles and the downstream boundary fluid particles;
7) And updating the particle position, the pressure of the particle, the upstream and downstream virtual particle speeds and the pressure information according to the velocity of the particle.
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Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
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| CN111709197A (en) * | 2020-06-17 | 2020-09-25 | 福州大学 | SPH inflow boundary processing method based on Riemann invariant |
| CN116663146B (en) * | 2023-05-30 | 2023-11-17 | 西安理工大学 | Calculation method of non-circular pipeline on-way resistance |
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