CN104217069A - Cavitation forecasting method for mechanical sealing lubricating liquid film - Google Patents

Abstract

The invention discloses a method for forecasting cavitation of a mechanical sealing lubricating liquid film, and particularly relates to a forecasting method for lubricating liquid film cavitation simulation and mass conservation conforming. A reynolds equation is calculated by an SUPG (streamline upwind petrov-galerkin) finite element method, and high-efficiency value iteration is performed. The forecasting method comprises the steps of respectively calculating rigidity matrixes Kp and Ktheta of liquid film pressure p and a liquid film density ratio theta, then introducing uniform comprehensive variables phi of the pressure and the density, modifying an original linear equation, finally calculating an obtained linear supplementary equation by a value iteration technology, and calculating the pressure p and the density theta of the lubricating liquid film at the same time. The high-efficiency and precise forecasting method is supplied to value simulation for the cavitation of the mechanical sealing lubricating liquid film, so that the performance of elements with complicated dynamic pressure slots can be accurately forecast; compared with other methods, the method disclosed by the invention has the advantages of higher calculation speed and higher calculation precision and can meet a requirement on the mass conservation of the liquid film.

Description

A kind of mechanical seal lubricating liquid film cavitation Forecasting Methodology

Technical field

The present invention relates to the mechanical seal field in rotating machinery, particularly relate to a kind of Forecasting Methodology of mechanical seal lubricating liquid film Cavitation Problems.

Background field

Liquid film lubrication mechanical seal is widely used in rotating machinery, for improving load-bearing capacity and the rubbing wear that reduces relative motion interface, often on one or two moving interface, conventionally processes some complicated dynamic pressure grooves.The cavitation effect producing just because of the existence of complicated dynamic pressure groove and Poiseuille flow and Couette flow, makes to be difficult to lubricating liquid film hydrodynamic effect is predicted accurately.Cause the main cause that adopts classic method to solve the failure of lubricated governing equation-Reynolds equation to have three: 1. the cavitation effect under mass conservation condition; 2. under high rotating speed (high Peclet number), the convection current of the ellipse-hyperbolic type [kinetic stability problem that solves bringing that is dominant; 3. the dynamic pressure groove structure of complexity causes the discontinuous and high pressure gradient of thickness of liquid film.

Nowadays,, for the Solve problems of Reynolds equation under cavitation conditions, a lot of mathematical models and numerical algorithm have been proposed.Wherein, JFO cavitation boundary condition is that its corresponding mathematical model of boundary condition of the liquid film mass conservation of widespread use is " ρ-θ " model, and wherein JFO cavitation boundary condition and system of equations become complementary equation.Ausas etc. have proposed a kind of effective relaxation method the solution that meets complementarity condition have been carried out to iteration renewal.Because Finite Element can be set up complicated border, Kumar and Booker and Boedo and Booker adopt finite element method under JFO theoretical condition, cavitation zone to be predicted, to liquid-sheet disintegration with improve border and distinguish.Hajjam and Bonneau have proposed a kind of improved ρ-θ finite element algorithm, and Quan Mo district high-order shape function is simulated lip seal and transient state sliding bearing in cavitation district with low order shape function.Schweizer adopts Lagrange-Eulerian equation to process Reynolds equation and JFO cavitation boundary condition, and provides finite element equation for some the numerical example.But these algorithm ubiquity calculated amount are large, algorithm is realized comparatively complicated shortcoming, affected to a great extent the research and development to lubricating liquid film Cavitation Problems.

Summary of the invention

In order to overcome existing method cavitation boundary condition liquid film leakage rate nonconservation, to thering is the shortcomings such as the performance of liquid lubrication mechanical seal of complicated dynamic pressure groove cannot be carried out Accurate Prediction and calculated amount is large, computational accuracy is low, speed of convergence is slow, the invention provides a kind of mechanical seal lubricating liquid film cavitation Forecasting Methodology, the method has computing velocity faster, higher computational accuracy, can meet the requirement of the liquid film mass conservation.

Technical scheme of the present invention:

A kind of mechanical seal lubricating liquid film cavitation Forecasting Methodology, is characterized in that:

Described Forecasting Methodology be by the related liquid film pressure of mechanical seal lubricating liquid film Cavitation Problems and liquid film density two known variables comprehensively in a partial differential equation, to this equation adopt streamline windward SUPG finite element method process, artificial viscosity is introduced in the convective term in partial differential equation along velocity reversal, obtain its weak integral form, introduce the unified generalized variable of lubricating liquid film pressure and density ratio, form the complementary system of linear equations about generalized variable, adopt numerical value iterative technique to obtain the solution of generalized variable, the pressure distribution and the density ratio that finally obtain end face distribute, the region of density ratio θ <1 is cavitation district, and the region of density ratio θ=1 is liquid film district, thereby realize the processing of Cavitation Problems,

Comprise the steps: in detail

(1) computational fields is carried out to finite element grid division;

(2) calculated rigidity matrix K ^pand K ^θ, when starting condition k=0 is set (k is iterations), F ^k=1: stiffness matrix K ^pand K ^θadopt streamline upstreame scheme to obtain, its calculation expression is respectively

$\left\{\begin{array}{c}{K}_{\mathrm{ij}}^p={\&Integral;}_{\mathrm{\&Omega;}}\frac{h^3}{6\mathrm{\&mu;}}(\frac{\&PartialD;N_i}{\&PartialD;x}\frac{\&PartialD;N_j}{\&PartialD;x}+\frac{{\&PartialD;N}_i}{\&PartialD;y}\frac{{\&PartialD;N}_j}{\&PartialD;y})\mathrm{d\&Omega;}\\ K_{\mathrm{ij}}^{\mathrm{\&theta;}}={\&Integral;}_{\mathrm{\&Omega;}}{\mathrm{hN}}_i(U\frac{\&PartialD;N_j}{\&PartialD;x}+V\frac{{\&PartialD;N}_j}{\&PartialD;y})\mathrm{d\&Omega;}-{\&Integral;}_{\mathrm{\&Omega;}}\frac{1}{2}{\mathrm{\&tau;}}^{\mathrm{SUPG}}h(U\frac{\&PartialD;N_i}{\&PartialD;x}+V\frac{\&PartialD;N_i}{\&PartialD;y})(U-\frac{\&PartialD;\left({\mathrm{hN}}_j\right)}{\&PartialD;x}+V\frac{\&PartialD;\left(h{N}_j\right)}{\&PartialD;y})\mathrm{d\&Omega;}\end{array}\right.$

Wherein, N is finite element shape function, and h is thickness of liquid film, and μ is lubricating fluid kinetic viscosity, and U, V are the component velocity of slidingsurface along x, y coordinate axis, stability factor and h _tat speed U (U _x, U _y) characteristic length of finite element grid in direction; Ω is integral unit.

(3) calculate matrix A=K ^pc-K ^θ(I-C);

Wherein, F is that numerical value is 1 or 0 cavitation instruction vector (1 represents liquid film district, and 0 represents cavitation district); I is unit matrix; C is that a diagonal entry is the sparse matrix that cavitation is indicated vectorial F;

(4) adopt GMRES alternative manner to solve system of linear equations A Φ=K ^θf, obtains unified generalized variable Φ, and supposes that lubricating liquid film be all non-cavitation district, the consistance that adopts block iteration technology innovation and inspection unification variable Φ and cavitation to indicate vectorial F when initial calculation;

(5) adopt following rule to check and upgrade F ^kif: Φ _i<0 and F _i ^k=1, make F _i ^k+1=0; If Φ _i>1 and F _i ^k=0, make F _i ^k+1=1;

(6) if F _i ^k+1-F _i ^k=0, iteration stops, otherwise k=k+1, repeating step (3)～(6);

(7) lubricating liquid film pressure p and density θ solve acquisition by following expression

p＝FΦ

θ＝(1-F)Φ+F

Mechanical seal lubricating liquid film pressure p and density ratio θ have been obtained thus simultaneously; Lubricating liquid film cavitation district and non-cavitation district can be distinguished by the size of density ratio θ: the regions of lubrication of density ratio θ <1 is cavitation district, and the region of density ratio θ=1 is liquid film complete section; The related seals performance of mechanical seal can be calculated and be obtained by acquisition liquid film pressure.

Stiffness matrix K of the present invention ^pand K ^θwith Gauss integration calculate and only need calculate once, do not need to participate in loop iteration calculate, therefore counting yield is higher.

Beneficial effect of the present invention is mainly manifested in:

The present invention adopt streamline windward SUPG finite element method solve Reynolds equation, ensured stability of solution, for the numerical simulation of lubricating liquid film Cavitation Problems provides a kind of efficient accurate computing method.

2. by Gauss integration calculated rigidity matrix K ^pand K ^θand only need calculate once, not participate in loop iteration and calculate, only have sparse matrix A in the inner renewal of circulation, therefore greatly reduce calculated amount.

3. it is discontinuous that the finite element mesh that adopted can avoid processing lubricating liquid film thickness, therefore solved the error of calculation problem of bringing thus.

4. than other method, the algorithm that the present invention adopts has advantages of that computing velocity is faster, computational accuracy is higher, can meet mass conservation requirement.

Brief description of the drawings

Fig. 1 is schematic flow sheet of the present invention.

Fig. 2 is the algorithm that adopts of the present invention and the performance parameter comparison diagram of other two kinds of algorithms.

Embodiment

Following examples are used for illustrating this patent, but are not used for limiting the scope of this patent.Now by drawings and Examples, the present invention is described in further detail.

Embodiment 1

The schematic flow sheet of a kind of mechanical seal lubricating liquid film cavitation Forecasting Methodology that Fig. 1 provides for the embodiment of the present invention 1, the present embodiment mainly comprises following steps:

Step 1: computational fields is carried out to finite element grid division;

Step 2: adopt Gauss integration to be calculated as follows integration type on above-mentioned finite element grid, obtain stiffness matrix K after overall package ^pand K ^θ

$\left\{\begin{array}{c}{K}_{\mathrm{ij}}^p={\&Integral;}_{\mathrm{\&Omega;}}\frac{h^3}{6\mathrm{\&mu;}}(\frac{\&PartialD;N_i}{\&PartialD;x}\frac{\&PartialD;N_j}{\&PartialD;x}+\frac{{\&PartialD;N}_i}{\&PartialD;y}\frac{{\&PartialD;N}_j}{\&PartialD;y})\mathrm{d\&Omega;}\\ K_{\mathrm{ij}}^{\mathrm{\&theta;}}={\&Integral;}_{\mathrm{\&Omega;}}{\mathrm{hN}}_i(U\frac{\&PartialD;N_j}{\&PartialD;x}+V\frac{{\&PartialD;N}_j}{\&PartialD;y})\mathrm{d\&Omega;}-{\&Integral;}_{\mathrm{\&Omega;}}\frac{1}{2}{\mathrm{\&tau;}}^{\mathrm{SUPG}}h(U\frac{\&PartialD;N_i}{\&PartialD;x}+V\frac{\&PartialD;N_i}{\&PartialD;y})(U-\frac{\&PartialD;\left({\mathrm{hN}}_j\right)}{\&PartialD;x}+V\frac{\&PartialD;\left(h{N}_j\right)}{\&PartialD;y})\mathrm{d\&Omega;}\end{array}\right.$

Step 3: make k=0, F ^k=1;

Step 4: by following formula compute sparse matrix A:

A＝K ^pC-K ^θ(I-C)

Step 5: solve following linear equation with GMRES alternative manner, obtain unified variable Φ;

AΦ＝K ^θF

Step 6: check and upgrade F ^kif: Φ _i<0 and F _i ^k=1, make F _i ^k+1=0; If Φ _i>1 and F _i ^k=0, make F _i ^k+1=1;

Step 7: if F _i ^k+1-F _i ^k=0, iteration stops, otherwise k=k+1, repeating step (4)～(7);

Step 8: obtain lubricating liquid film pressure p and density θ,

p＝FΦ，θ＝(1-F)Φ+F

Embodiment bis-

This Forecasting Methodology and the method for finite difference (JFO-FDM) that adopts JFO cavitation boundary condition that Fig. 2 provides for the embodiment of the present invention two, the comparison that adopts the Finite Element (REYNOLDS-FEM) of Reynolds cavitation boundary condition, parameter comprises time t, opening force F _o, and the slip Q at internal-and external diameter place _iand Q _o.The leakage rate of the internal-and external diameter that this Forecasting Methodology obtains is almost equal, meets mass conservation requirement; The opening force that obtains is the shortest computing time accurate and the most used, and counting yield is the highest.

Prove by actual computation, the result that the present invention obtains more accurately, reliably, has been verified rationality of the present invention and accuracy.

Content described in this instructions embodiment is only enumerating of way of realization to inventive concept; protection scope of the present invention should not be regarded as only limiting to the concrete form that embodiment states, protection scope of the present invention also and conceive the equivalent technologies means that can expect according to the present invention in those skilled in the art.

Claims (1)

1. a mechanical seal lubricating liquid film cavitation Forecasting Methodology, is characterized in that:

Described Forecasting Methodology be by the related liquid film pressure of mechanical seal lubricating liquid film Cavitation Problems and liquid film density two known variables comprehensively in a partial differential equation, to this equation adopt streamline windward SUPG finite element method process, artificial viscosity is introduced in the convective term in partial differential equation along velocity reversal, obtain its weak integral form, introduce the unified generalized variable of lubricating liquid film pressure and density ratio, form the complementary system of linear equations about generalized variable, adopt numerical value iterative technique to obtain the solution of generalized variable, the pressure distribution and the density ratio that finally obtain end face distribute, the region of density ratio θ <1 is cavitation district, and the region of density ratio θ=1 is liquid film district, thereby realize the processing of Cavitation Problems,

Comprise the steps: in detail

(1) computational fields is carried out to finite element grid division;

(2) calculated rigidity matrix K ^pand K ^θ, when starting condition k=0 is set (k is iterations), F ^k=1: stiffness matrix K ^pand K ^θadopt streamline upstreame scheme to obtain, its calculation expression is respectively

$\left\{\begin{array}{c}{K}_{\mathrm{ij}}^p={\&Integral;}_{\mathrm{\&Omega;}}\frac{h^3}{6\mathrm{\&mu;}}(\frac{\&PartialD;N_i}{\&PartialD;x}\frac{\&PartialD;N_j}{\&PartialD;x}+\frac{{\&PartialD;N}_i}{\&PartialD;y}\frac{{\&PartialD;N}_j}{\&PartialD;y})\mathrm{d\&Omega;}\\ K_{\mathrm{ij}}^{\mathrm{\&theta;}}={\&Integral;}_{\mathrm{\&Omega;}}{\mathrm{hN}}_i(U\frac{\&PartialD;N_j}{\&PartialD;x}+V\frac{{\&PartialD;N}_j}{\&PartialD;y})\mathrm{d\&Omega;}-{\&Integral;}_{\mathrm{\&Omega;}}\frac{1}{2}{\mathrm{\&tau;}}^{\mathrm{SUPG}}h(U\frac{\&PartialD;N_i}{\&PartialD;x}+V\frac{\&PartialD;N_i}{\&PartialD;y})(U-\frac{\&PartialD;\left({\mathrm{hN}}_j\right)}{\&PartialD;x}+V\frac{\&PartialD;\left(h{N}_j\right)}{\&PartialD;y})\mathrm{d\&Omega;}\end{array}\right.$

Wherein, N is finite element shape function, and h is thickness of liquid film, and μ is lubricating fluid kinetic viscosity, and U, V are the component velocity of slidingsurface along x, y coordinate axis, stability factor and h _tat speed U (U _x, U _y) characteristic length of finite element grid in direction, Ω is integral unit;

(3) calculate matrix A=K ^pc-K ^θ(I-C);

Wherein, F is that numerical value is 1 or 0 cavitation instruction vector (1 represents liquid film district, and 0 represents cavitation district); I is unit matrix; C is that a diagonal entry is the sparse matrix that cavitation is indicated vectorial F;

(4) adopt GMRES alternative manner to solve system of linear equations A Φ=K ^θf, obtains unified generalized variable Φ, and supposes that lubricating liquid film be all non-cavitation district, the consistance that adopts block iteration technology innovation and inspection unification variable Φ and cavitation to indicate vectorial F when initial calculation;

(5) adopt following rule to check and upgrade F ^kif: Φ _i<0 and F _i ^k=1, make F _i ^k+1=0; If Φ _i>1 and F _i ^k=0, make F _i ^k+1=1;

(6) if F _i ^k+1-F _i ^k=0, iteration stops, otherwise k=k+1, repeating step (3)～(6);

(7) lubricating liquid film pressure p and density θ solve acquisition by following expression:

p＝FΦ

θ＝(1-F)Φ+F

Mechanical seal lubricating liquid film pressure p and density ratio θ have been obtained thus simultaneously; Lubricating liquid film cavitation district and non-cavitation district can be distinguished by the size of density ratio θ: the regions of lubrication of density ratio θ <1 is cavitation district, and the region of density ratio θ=1 is liquid film complete section; The related seals performance of mechanical seal can be calculated and be obtained by acquisition liquid film pressure.