JP2000081366A - Fluid motion analysis method and storage medium recording fluid motion analysis program - Google Patents

Abstract

PROBLEM TO BE SOLVED: To improve the accuracy in the motion analysis of a non-compression fluid when a fluid is expressed as a group of particles by using a common proportional coefficient where the total becomes zero for the proportional coefficient of a motion equation and that of a non-compression conditional expression. SOLUTION: In a fluid motion analysis method, a non-compression fluid is expressed as a group of a plurality of particles (i). A motion equation including terms that are proportional to a pressure pi of each of the particles (i) and a non-compression conditional expression including terms that are proportional to a speed vi of each of the particles (i) are used. Then, a speed vi' and a pressure pi' of each of the particles (i) at time t+Δt after a time unit width of Δt are obtained from the speed vi and the pressure pi of each of the particles (i) at time (t). A common proportional coefficient where the total reaches zero is used for the proportional coefficient of the motion equation and that of the non-compression conditional expression. The motion equation is expressed by an expression I, while the non-compression conditional expression is expressed by an expression II. Further, the common proportional constant may the one that satisfies an expression III.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a fluid motion analysis method and a fluid motion analysis for expressing a fluid as a group of particles in a fluid motion analysis required for designing an ink jet printer or the like. The present invention relates to a recording medium on which a program is recorded.

[0002]

2. Description of the Related Art In various industrial fields, it is sometimes necessary to predict fluid motion. As a method therefor, there are a method of performing an experiment using a wind tunnel or the like, a method of theoretically solving an equation of fluid motion, and a method of numerically calculating the fluid motion using a digital computer. Among them, the method of calculating using a computer has been particularly widely used due to the recent improvement in the capability of the computer.

For example, an ink jet recording system used in a printer includes a so-called bubble jet system in which bubbles are formed in ink by heating the ink with a heater, and the ink is ejected from a nozzle by this pressure. The so-called piezo jet method is known in which ink is ejected by using the pressure due to the bulk vibration of the ink.However, by grasping the behavior of the ink in the nozzle using the above-described method, further improvement in efficiency and High quality is desired.

Conventionally, as a method of numerically solving a fluid motion, a method of dividing a space into a large number of meshes and discretizing a motion equation, such as a difference method or a finite element method, is often used. Used. On the other hand, in the field of celestial mechanics, a fluid is expressed as a group of particles, S
PH (Smoothed Particle Hydr
odiamics) is frequently used (reference JJ Monaghan: An Intro).
DUCTION TO SPH, COMPUTER P
physics Communications, Vol
ume 48 (1988) 89-96). In SPH, we consider that the mass of the fluid is distributed around the center of the particle according to the function w (r), and the density ρ (r) of the fluid at point r in space is

[0005]

(Equation 7) Consider the model given by Here, m _j is the mass of the j-th particle, r _j is the position vector of the j-th particle, and n is the total number of particles. The function w (r) is called a weight function, and a function as shown in FIG. 3 is often used as a function of the absolute value | r | of the position vector r. As the equation of motion of particles, for example,

[0006]

(Equation 8) An equation of the form Where v _i is the velocity vector of the _ith particle, p _i is the pressure at r _i , ρ _i
Is the density at r _i , and F _i is a force such as an external force or a viscous force acting on the i-th particle. The pressure p _i should be determined simultaneously with the equation relating to energy, but can be determined as a function of the density ρ _{i in} the case of barotropy flow.

The above-mentioned formulation is applied when the fluid has compressibility, and a slight change is required for an incompressible fluid. That is, in the case of an incompressible fluid, the pressure p is

[0008]

(Equation 9) Must be satisfied. Koshizuka et al. Described the equation of motion as a method for treating incompressible fluids.

[0009]

(Equation 10) And the non-compression condition instead of equation (4)

[0010]

[Equation 11] (See JP-A-7-334484).
Here, _ni is the particle number density at r _i , and n _i ⁰ is the particle number density at time 0. In equation (5), instead of W

[0011]

(Equation 12) And rewritten as ρ = ρ _i , _ni = ρ _i / m _i ,

[0012]

(Equation 13) And a similar expression to expression (2) is obtained, but expression (2)
Is not a sum form in which p _i and p _j have ρ _i ² and ρ _j ² in the denominator, but is a feature that they are included in the form of a pressure difference such as p _j −p _i . The equation (6) rather than that temporally constant particle number density, with m _i is temporally constant,

[0013]

[Equation 14] And the momentum of equation (1) and density ρ (r)

[0014]

(Equation 15) Obtained from

[0015]

(Equation 16) Is equivalent to

[0016]

However, in the above conventional example, the total momentum P of the system is reduced by setting the force F such as an external force or a viscous force to zero.

[0017]

[Equation 17] And the total kinetic energy E of the system

[0018]

(Equation 18) Is naturally conserved, that is, the time variation of the total momentum P and the total kinetic energy E of the system

[0019]

[Equation 19] Where equation should be zero is given by equation (2) or equation (8)
In the case where is used, the conservation of the momentum and the kinetic energy is not guaranteed, and there is a possibility that the analysis accuracy is reduced.

Therefore, an object of the present invention is to improve the accuracy of motion analysis of an incompressible fluid when the fluid is represented as a group of particles.

[0021]

In order to achieve the above object, a fluid motion analysis method of the present invention expresses an incompressible fluid as a group of a plurality of particles i, and is proportional to the pressure p _i of each particle i. Using a motion equation including a term and an incompressible conditional expression including a term proportional to the velocity v _i of each particle i, the time step size is obtained from the velocity v _i and the pressure p _{i of} each particle i at time t. a fluid motion analysis method for analyzing a fluid movement of the incompressible fluid by determining the velocity v _i 'and the pressure p _i' of each particle i at time t + Delta] t after Delta] t has elapsed, the proportional coefficients of the equation of motion and It is characterized in that a common proportional coefficient whose sum is zero is used as the proportional coefficient of the non-compression conditional expression.

In the fluid motion analysis method of the present invention as described above, a common proportional coefficient is used for the equation of motion and the incompressible conditional expression, and the sum of the common proportional coefficients is zero. When coefficients other than the proportional coefficient and external force are zero, the momentum and kinetic energy of the system are preserved.

The equation of motion is

[0024]

(Equation 20) And the non-compression conditional expression is

[0025]

(Equation 21) And the common proportionality factor is

[0026]

(Equation 22) May be satisfied.

The recording medium on which the fluid motion analysis program of the present invention is recorded has a motion equation that expresses an incompressible fluid as a group of a plurality of particles i and includes a term proportional to the pressure p _i of each particle i. by using the non-compressed condition includes a term proportional to the velocity v _i of each particle i, the velocity v _i and the pressure from p _i, the time step size Δt of the at time t each particle i
A recording medium recording a fluid motion analysis program for analyzing a fluid movement of the incompressible fluid by determining the velocity v _i 'and the pressure p _i' of each particle i at time t + Delta] t after the lapse of the motion equation And a common proportional coefficient having a total sum of zero is used as the proportional coefficient and the proportional coefficient of the non-compression conditional expression.

The equation of motion is

[0029]

(Equation 23) And the non-compression conditional expression is

[0030]

(Equation 24) And a common proportionality factor is

[0031]

(Equation 25) May be satisfied.

[0032]

DESCRIPTION OF THE PREFERRED EMBODIMENTS (First Embodiment) First, the principle of a fluid motion analysis method for analyzing fluid motion according to the present embodiment will be described.

In the present embodiment, the conservation of momentum and kinetic energy is guaranteed by using a common proportionality coefficient for the kinetic equation and the incompressible conditional equation of the system. That is, as the equation of motion

[0034]

(Equation 26) Is used. Also, as the non-compression conditional expression,

[0035]

[Equation 27] , And as a coefficient

[0036]

[Equation 28]Use one that satisfies Where a_ijIs the equation of motion and
A proportionality coefficient vector common to the non-compression conditional expressions, b_ijIs viscous
Coefficient representing the effect, m_iIs the mass of particle i, v_i, V _jIs grain
Child i, velocity of particle j, c_iIs a term representing the effect of external force,
λ_jIs the pressure p of the particle j_jAnd density ρ_jIs a function of

The time change of the total momentum P of the system expressed by the equation (12) when the external force c _i = 0 is used by using the coefficients satisfying the conditions shown in the above equations (18) and (19). When calculating,

[0038]

(Equation 29) Becomes When the external force c _i and the viscosity coefficient b _ij are both 0, the time change of the total kinetic energy E of the system expressed by the equation (13) is calculated as follows.

[0039]

[Equation 30] Becomes The above calculations show that when the external force c _i is 0, the momentum is conserved, and when the viscosity coefficient b _ij is 0, the kinetic energy is also conserved.

Next, the basic flow of the fluid motion analysis using the coefficients satisfying the above conditions will be described.

FIG. 1 schematically shows the configuration of a fluid analysis apparatus used in this embodiment.

An input unit 1 for inputting conditions and the like, a calculation unit 2 for performing calculations according to the present embodiment, and an output unit 3 for outputting analysis results.

The input unit 1 is composed of a keyboard, a mouse and the like. The calculation unit 2 includes a CPU 4 and a memory 5 and can read the recorded content recorded on a recording medium 9. The output unit 3 includes a CRT 6 for displaying an analysis result and a printing unit 7 capable of outputting by printing. Also,
The analysis result can be stored in the analysis result storage unit 8 such as a hard disk.

The recording medium 9 stores a fluid motion analysis program for executing the above-described analysis, and the calculation unit 2 executes calculations according to the read program. In addition,
The input unit 1, the calculation unit 2, and the output unit 3 may be in the form of a workstation or a personal computer.

Next, FIG. 2 shows a flowchart of the analysis method of the present invention when the calculation unit 2 executes the fluid motion analysis program recorded on the recording medium 9 shown in FIG.

First, parameters such as initial conditions, boundary conditions, and calculation conditions of the system necessary for calculation are read (step 1).
1). Next, based on the parameters read in step 1, variables such as the particle position, density, pressure, and velocity are initialized (step 12). Then, while updating the time t by Δt, the calculation is repeated until the end time is reached (step 13). If the end time has not been reached, calculation of external force (step 14), calculation of viscous force (step 15),
In each step of the pressure calculation (step 16), c _i ,

[0047]

[Outside 1] Was calculated, v _i the value v _i 'in t + Delta] t of the calculated explicit. However, v _i '
Does not generally satisfy the incompressibility condition, so that it is modified with the pressure p _i ′ at t + Δt (step 1
8, 19), and iterative calculation is performed until the non-compression condition is satisfied (step 17). If the non-compression condition is satisfied, the time t
Is advanced by Δt (step 20), the calculation result is output (step 21), and the calculation is repeated until the end time is reached.

This calculation is performed similarly to the difference scheme for the incompressible fluid. For example, when a first-order forward difference is applied to dv _i / dt, and a value p ′ at t + Δt is used as the pressure p, Expression (2) shown in the conventional example becomes

[0049]

(Equation 31) Becomes Here, values at time t are used for values such as ρ, ω, and F. The uncompressed condition is

[0050]

(Equation 32) And it is sufficient. in this case,

[0051]

[Equation 33] It is. Note that the non-compression conditional expression (23) is simply expressed as

[0052]

(Equation 34) However, in this case, kinetic energy is not strictly preserved due to a discretization error with respect to time, and therefore, it is preferable to use Expression (23).

As described above, by using the common coefficient a _ij for the kinetic equation and the incompressible conditional expression, and by making the sum of the common coefficients a _ij equal to zero, the condition that the momentum and the kinetic energy should be preserved , The momentum and the kinetic energy are preserved, so that the accuracy of the motion analysis of the incompressible fluid when the fluid is represented as a group of particles can be improved. (Second Embodiment) In this embodiment, the equation of motion (22) of the first embodiment is used as the equation of motion.

[0054]

(Equation 35) And, as the non-compression condition,

[0055]

[Equation 36] Is used. in this case,

[0056]

(37) It is.

Note that the calculation steps and the structure of the fluid analysis device are the same as those in the first embodiment, and a description thereof will be omitted.

In this embodiment, as in the first embodiment, the momentum and the kinetic energy are preserved.
Therefore, the accuracy of the motion analysis of the incompressible fluid when the fluid is expressed as a group of particles can be improved.

[0059]

As described above, the fluid motion analysis method of the present invention uses a common proportional coefficient for the equation of motion and the incompressible conditional expression, and the sum of the common proportional coefficients is zero. When the coefficients other than the common proportional coefficient and the external force are zero, the momentum and the kinetic energy of the system are conserved. Thereby, the accuracy of the motion analysis of the incompressible fluid when the fluid is represented as a group of particles can be improved.

[Brief description of the drawings]

FIG. 1 shows a schematic configuration of an analyzer according to the present invention.

FIG. 2 shows a flowchart of the present invention.

FIG. 3 shows a representative weight function.

[Explanation of symbols]

 DESCRIPTION OF SYMBOLS 1 Input part 2 Calculation part 3 Output part 4 CRT 5 Printing part 6 Analysis result storage part 7 CPU 8 Memory 9 Recording medium

Claims (8)

[Claims] 1. An equation of motion that expresses an incompressible fluid as a group of a plurality of particles i, including a term proportional to the pressure p _i of each particle i and a non-motion equation including a term proportional to the velocity v _i of each particle i. by using the compression condition, from the velocity v _i and the pressure p _i of each particle i at time t, velocity v _i of each particle i at time t + Delta] t after the step size Delta] t time elapsed 'and pressure p _i' Is a fluid motion analysis method for analyzing the fluid motion of the incompressible fluid by calculating the proportional coefficient of the equation of motion and the proportional coefficient of the incompressible conditional expression, a common proportional coefficient that the sum is zero A fluid motion analysis method characterized by being used. 2. The equation of motion is: The fluid motion analysis method according to claim 1, wherein 3. The non-compression conditional expression is as follows: The fluid motion analysis method according to claim 2, wherein 4. The common proportionality factor is: The fluid motion analysis method according to any one of claims 2 and 3, wherein 5. An equation of motion that expresses an incompressible fluid as a group of particles i, including a term proportional to the pressure p _i of each particle i and a non-linear equation including a term proportional to the velocity v _i of each particle i. by using the compression condition, from the velocity v _i and the pressure p _i of each particle i at time t, velocity v _i of each particle i at time t + Delta] t after the step size Delta] t time elapsed 'and pressure p _i' Is a recording medium that records a fluid motion analysis program for analyzing the fluid motion of the incompressible fluid, wherein the sum of the proportional coefficient of the equation of motion and the proportional coefficient of the incompressible conditional expression becomes zero. A recording medium on which a fluid motion analysis program is recorded, wherein a common proportionality coefficient is used. 6. The equation of motion is: A recording medium on which the fluid motion analysis program according to claim 5 is recorded. 7. The non-compression conditional expression is: A recording medium on which the fluid motion analysis program according to claim 6 is recorded. 8. The common proportionality factor is: A recording medium on which the fluid motion analysis program according to claim 6 is satisfied.