CN103473419A - Processing method of pressure difference in low Reynolds number incompressible flow at bending boundaries - Google Patents

Abstract

The invention relates to a processing method of pressure difference in low Reynolds number incompressible flow at bending boundaries. The method comprises the following steps of S1, determining if the following formula (1) is met or not; if the formula (1) is met, executing step S2, or R0<<x0, L>>R0, Re<1; S2, calculating the volume force FA1 with the formula (2) at the inlet cross section of a pipeline, wherein rao is the fluid density; S3, substituting the formula (2) in step S2 into the formula (3), to obtain the volume force FA2 at any cross section of the pipeline, wherein dA represents surface element. The method can conveniently convert the pressure difference in the low Reynolds number incompressible flow at the bending boundaries into the volume force at any part.

Description

The disposal route of the pressure differential in the low reynolds number incompressible flow on curved boundaries

Technical field

The present invention relates to the hydrodynamics technology field, more particularly, the particularly disposal route of the pressure differential in the low reynolds number incompressible flow on a kind of curved boundaries.

Background technology

Apply smooth particle flux body dynamics (Smoothed Particle Hydrodynamics, be called for short SPH) while simulating the low reynolds number incompressible flow, the pressure gradient that solves Fluid Motion Driven By Moving is very important, because pressure just shows as gradient in the Navier-Stokes equation.At weak compressible SPH(Weakly Compressible SPH, being called for short WCSPH) in algorithm, general pressure is broken down into dynamic pressure and hydrostatic force usually, so the gradient of general pressure also just can obtain by the gradient of these two pressure.

For the WCSPH method, simulation dynamic pressure gradient is simple and direct, and Hydrostatic pressure gradient is counted as a body force usually.Morris has studied the low reynolds number incompressible flow in 1997 with WCSPH, and his test example is that Poiseuille flows and, around post stream, the result of gained and the result of method of finite difference are coincide finely.The colleague of Liu Moubin and Ta has also simulated Poiseuille stream in 2005 by limited particle method, and result is also pretty good.They are converted into body force to Hydrostatic pressure gradient (or hydrostatic force is poor).

The straight low reynolds number incompressible flow for border, this conversion is simple, because in these cases, the Hydrostatic pressure gradient in flow field is a constant, corresponding body force can be simply by the difference of the pressure of entrance and outlet, the length divided by flow field obtains.Yet, for the low reynolds number incompressible flow of border bending, Hydrostatic pressure gradient is inhomogeneous, Hydrostatic pressure gradient everywhere is not constant, how to calculate corresponding body force everywhere and has just become a problem.Therefore, need on a kind of curved boundaries of research the disposal route of pressure differential in the low reynolds number incompressible flow.

Summary of the invention

The object of the invention is to have for prior art the technical matters that can not process the pressure differential in the low reynolds number incompressible flow on curved boundaries, the disposal route of the pressure differential in the low reynolds number incompressible flow on a kind of curved boundaries is provided.

In order to achieve the above object, the technical solution used in the present invention is as follows:

The disposal route of the pressure differential in the low reynolds number incompressible flow on curved boundaries, in the method, the low reynolds number incompressible flow flows in having the pipeline of curved boundary, the described pipeline with curved boundary is pipeline axisymmetric, non-straight and that tube wall is the Gu Bi border, and two ends, the gateway pressure reduction of this pipeline is △ p, wherein △ p=p ₁-p ₂, p ₁pressure, p for porch ₂for the pressure in exit, and adopt the cylindrical coordinate that means respectively axial coordinate and radial coordinate with x and r, the method specifically comprises the following steps,

Step S1, be confirmed whether to meet following formula (1), if meet, performs step S2,

$\mathrm{\&epsiv;}=\frac{\mathrm{\&delta;}}{x_0}<<1,\frac{R_0}{x_0}=O$ Or R ₀<<x ₀, L>>R ₀, R _e<1 (1);

Wherein, δ is wall thickness, x ₀for the axial coordinate of any point, R ₀for the radius at the straight place of pipeline, the length that L is pipe, R _efor Reynolds number;

Step S2, the body force F at the entrance section place of calculating pipeline _a1,

$F_{A1}=\frac{\mathrm{\&Delta;p}}{\mathrm{\&rho;}{\&Integral;}_{-0.5L}^{0.5L}\frac{\underset{A1}{\&Integral;}f(r,x)\mathrm{dA}}{\underset{A2}{\&Integral;}f(r,x)\mathrm{dA}}\mathrm{dx}}---\left(2\right);$

Wherein, ρ is fluid density;

Step S3, by the formula (3) below the formula in step S2 (2) substitution, draw the body force F of any section of pipeline _a2,

$\frac{F_{A2}}{F_{A1}}\&ap;\frac{\underset{A1}{\&Integral;}f(r,x)\mathrm{dA}}{\underset{A2}{\&Integral;}f(r,x)\mathrm{dA}}---\left(3\right);$

Wherein, dA means cell area.

Preferably, described formula (3) is the fluid in pipeline to be met to continuity and the equation of motion carry out magnitude and calculate.

Compared with prior art, the invention has the advantages that: the present invention can be converted into pressure differential in the low reynolds number incompressible flow on curved boundaries the body force at any place easily.

The accompanying drawing explanation

Below in conjunction with drawings and Examples, the invention will be further described.

Fig. 1 is the pipe segment structure schematic diagram on rotational symmetry non-straight of the present invention border.

Fig. 2 is the analog result figure of the Poiseuille stream in embodiments of the invention one.

Fig. 3 is the structural representation of the differential expansion pipe stream in embodiments of the invention two.

Fig. 4 is the analog result figure of differential expansion pipe stream on X-axis in embodiments of the invention two.

Fig. 5 is the analog result figure of differential expansion pipe stream on Y-axis in embodiments of the invention two.

Fig. 6 is the structural representation of the tilt flat plate stream in embodiments of the invention three.

Fig. 7 is the analog result figure of tilt flat plate stream on X-axis in embodiments of the invention three.

Embodiment

Below in conjunction with the drawings and specific embodiments, the present invention is described in further detail.

Proposition the object of the invention is to: the straight low Reynolds number flows for border, Hydrostatic pressure gradient is a constant in whole flow field, corresponding body force can be simply by the difference of the pressure of entrance and outlet, the length divided by flow field obtains.Yet, for the flow field of border bending, Hydrostatic pressure gradient everywhere is not constant, body force everywhere can not be simply by the difference of the pressure of entrance and outlet, the length divided by flow field obtains.Therefore, must propose a method and calculate corresponding body force everywhere.

Consult shown in Fig. 1, in order to facilitate the description of this invention, the Reynolds number incompressible flow provided in the present invention flows in having the pipeline of curved boundary, and the pipeline with curved boundary is pipeline axisymmetric, non-straight and that tube wall is the Gu Bi border, two ends, the gateway pressure reduction of given this pipeline is △ p, wherein △ p=p ₁-p ₂, p ₁pressure, p for porch ₂for the pressure in exit, and adopt the cylindrical coordinate that means respectively axial coordinate and radial coordinate with x and r.

Hydrostatic pressure gradient along the X-direction diverse location in this flow field do not equate everywhere, and the body force that how pressure reduction at two ends, flow field is converted into to particle is the SPH simulation to be carried out in this flow field need the problem solved.

The diameter of setting pipeline is very little, and horizontal positioned, and the pressure gradient of the fluid in pipeline between entrance and outlet drives current downflow like this, and the impact of its gravity can be ignored.

Now, disposal route of the present invention is specially:

At first, confirm whether meet following formula (1) in pipeline, when meeting formula (1), can carry out subsequent treatment:

$\mathrm{\&epsiv;}=\frac{\mathrm{\&delta;}}{x_0}<<1,\frac{R_0}{x_0}=O$ Or R ₀<<x ₀, L>>R ₀, R _e<1 (1);

Wherein, δ is wall thickness, x ₀for the axial coordinate of any point, R ₀for managing the radius at straight place, the length that L is pipe, R _efor Reynolds number;

Secondly, continuity that must be satisfied to the pipeline inner fluid and the equation of motion carry out estimation of measure grade, can obtain the body force F at the entrance section place of pipeline herein after the line number of going forward side by side derivation (a kind of calculating derivation method of this mathematical derivation process, repeat no more) _a1body force F with any section of pipeline _a2meet formula (2):

$\frac{F_{A2}}{F_{A1}}\&ap;\frac{\underset{A1}{\&Integral;}f(r,x)\mathrm{dA}}{\underset{A2}{\&Integral;}f(r,x)\mathrm{dA}}---\left(2\right);$

Wherein, dA means cell area;

Like this, from formula (2), can find out, if F _a1known, can obtain at an easy rate in flow field the body force F at x point place arbitrarily _a2; Yet, seldom directly provide F in the known conditions in general flow field _a1, more situation is to the pressure differential between gateway and outlet.

Finally, set up F _a1and the relation between pressure differential △ p, draw the body force F at the entrance section place of pipeline _a1meet formula (3);

$F_{A1}=\frac{\mathrm{\&Delta;p}}{\mathrm{\&rho;}{\&Integral;}_{-0.5L}^{0.5L}\frac{\underset{A1}{\&Integral;}f(r,x)\mathrm{dA}}{\underset{A2}{\&Integral;}f(r,x)\mathrm{dA}}\mathrm{dx}}---\left(3\right);$

Wherein, ρ is fluid density.

By above-mentioned formula (2) and formula (3), and the body force F of any section of pipeline will be drawn in formula (3) substitution formula (2) _a2; The pressure reduction △ p at two ends, flow field can be converted into to the body force of any place's particle, make with the low reynolds number incompressible flow of SPH simulation curved boundaries and can carry out smoothly.

In order to check method of the present invention, remove to simulate three low reynolds number incompressible flows below by it, comprise Poiseuille stream, differential expansion pipe stream and tilt flat plate stream.For relatively, these examples are simulated with the WCSPH method of constant body force equally, and the fluid of simulating is all water, have identical initial density ρ 0=10 ³kg/m ^3;simultaneously, three embodiment have provided Theory Solution for comparing.

Embodiment mono-

The present embodiment adopts Poiseuille stream, is about to two infinitely-great flat boards and is placed on respectively coordinate y=-0.5YL and y=0.5YL place, and wherein, YL is two distances between flat board.Its initial static fluid is the pressure reduction corresponding to entrance and outlet by the body force F(that is parallel to X-axis) drive, finally will reach a steady state (SS).

In the present embodiment, YL=10 ^-3m, flow field length is d=5 * 10 ^-4m, hydrostatic force is poor is △ p=10 ^-4n/m ², as shown in Figure 2, and as can be seen from Figure 2, method of the present invention is consistent with the result with the simulation of constant body force for the result of through type (1), formula (2) and formula (3) simulation, and all with Theory Solution, coincide well.

Embodiment bis-

The present embodiment adopts differential expansion pipe stream, as shown in Figure 3, can consider one section axisymmetric blood vessel, and its pressure differential fluid flows in pipeline, and pressure reduction is △ p=p ₁-p ₂=1.939006287 * 10 ^-3n/m ², the radius of pipeline is the function about position x, can be expressed as (4) formula:

$\frac{R}{R_0}=\left\{\begin{array}{cc}1& \left|x\right|>X_0\\ 1+\frac{\mathrm{\&delta;}}{{2R}_0}(1+\cos \frac{\mathrm{\&pi;x}}{X_0})& \left|x\right|\&le;X_0\end{array}\right.,---\left(4\right);$

Wherein, R ₀=0.5 * 10 ^-3m, δ=0.2R ₀, L=6 * 10 ^-3m, X ₀=2 * 10 ^-3m, ε=δ/X ₀=0.05.

Analog result, X-axis speed and Y-axis speed are respectively as shown in Figure 4 and Figure 5.And can find out from Fig. 4 and Fig. 5, adopt the analog result of method of the present invention and Theory Solution to coincide well, but adopt the analog result of constant body force not coincide with Theory Solution.

Embodiment tri-

The present embodiment adopts tilt flat plate stream, as shown in Figure 6, in the present embodiment, d _bC=4mm, 2l ₁=0.5mm, α=3.503 °, and the hydrostatic force between B and C poor be △ p=1.21665968 * 10 ^-3n/m ².

Adopt method of the present invention, along the analog result of the horizontal velocity of the fluid particles on X-axis as shown in Figure 7; As can be seen from Figure 7, by the analog result of method of the present invention, with Theory Solution, conform to again, but the simulation in constant body force situation can not conform to.

Although described by reference to the accompanying drawings embodiments of the present invention; but the patent owner can make various distortion or modification within the scope of the appended claims; as long as be no more than the described protection domain of claim of the present invention, all should be within protection scope of the present invention.

Claims (2)

1. the disposal route of the pressure differential in the low reynolds number incompressible flow on a curved boundaries, it is characterized in that: in the method, the low reynolds number incompressible flow flows in having the pipeline of curved boundary, the described pipeline with curved boundary is pipeline axisymmetric, non-straight and that tube wall is the Gu Bi border, and two ends, the gateway pressure reduction of this pipeline is △ p, wherein △ p=p ₁-p ₂, p ₁pressure, p for porch ₂for the pressure in exit, and adopt the cylindrical coordinate that means respectively axial coordinate and radial coordinate with x and r, the method specifically comprises the following steps,

Step S1, be confirmed whether to meet following formula (1), if meet, performs step S2,

$\mathrm{\&epsiv;}=\frac{\mathrm{\&delta;}}{x_0}<<1,\frac{R_0}{x_0}=O$ Or R ₀<<x ₀, L>>R ₀, R _e<1 (1);

Wherein, δ is wall thickness, x ₀for the axial coordinate of any point, R ₀for the radius at the straight place of pipeline, the length that L is pipe, R _efor Reynolds number;

Step S2, the body force F at the entrance section place of calculating pipeline _a1,

$F_{A1}=\frac{\mathrm{\&Delta;p}}{\mathrm{\&rho;}{\&Integral;}_{-0.5L}^{0.5L}\frac{\underset{A1}{\&Integral;}f(r,x)\mathrm{dA}}{\underset{A2}{\&Integral;}f(r,x)\mathrm{dA}}\mathrm{dx}}---\left(2\right);$

Wherein, ρ is fluid density;

Step S3, by the formula (3) below the formula in step S2 (2) substitution, draw the body force F of any section of pipeline _a2,

$\frac{F_{A2}}{F_{A1}}\&ap;\frac{\underset{A1}{\&Integral;}f(r,x)\mathrm{dA}}{\underset{A2}{\&Integral;}f(r,x)\mathrm{dA}}---\left(3\right);$

Wherein, dA means cell area.

2. the disposal route of the pressure differential in the low reynolds number incompressible flow on curved boundaries according to claim 1 is characterized in that: described formula (3) is the fluid in pipeline to be met to continuity and the equation of motion carry out magnitude and calculate.

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