CN112613249B - Labyrinth seal three-dimensional small-gap turbulent flow field solving method - Google Patents

Labyrinth seal three-dimensional small-gap turbulent flow field solving method Download PDF

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CN112613249B
CN112613249B CN202011575201.6A CN202011575201A CN112613249B CN 112613249 B CN112613249 B CN 112613249B CN 202011575201 A CN202011575201 A CN 202011575201A CN 112613249 B CN112613249 B CN 112613249B
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曹小建
刘建林
李玉坤
李家亮
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China University of Petroleum East China
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Abstract

The invention discloses a labyrinth seal three-dimensional small-gap turbulent flow field solving method, and particularly relates to the field of three-dimensional turbulent flow field calculation. According to an actual labyrinth seal structure, MATLAB software is utilized to establish a labyrinth seal model and place the labyrinth seal model in a three-dimensional coordinate system, radial averaging processing is carried out on the flow velocity and pressure of interstitial fluid, a fluid continuous equation of the interstitial fluid is determined based on the integral flow theory, an interstitial fluid three-control model is established, fluid control equation sets of the interstitial fluid in each control body are respectively determined, then the eccentric whirling condition of a rotating shaft is combined, each control body fluid control equation set is decomposed into an equation set consisting of a zero-order concentric whirling equation and a first-order eccentric whirling equation based on a perturbation method, a zero-order velocity field and a first-order velocity field of each control body along the circumferential direction of the rotating shaft are obtained after solving, and an annular seal model interstitial fluid velocity field is determined. The invention greatly reduces the solving difficulty of the fluid control equation set of the three-dimensional turbulent flow and provides a theoretical basis for solving the complex flow field.

Description

Labyrinth seal three-dimensional small-gap turbulent flow field solving method
Technical Field
The invention relates to the field of three-dimensional turbulent flow field calculation, in particular to a labyrinth seal three-dimensional small-gap turbulent flow field solving method.
Background
The sealing system is widely applied to turbomachines such as a gas turbine, a pump and a compressor, the clearance circulation is commonly used in the sealing system, a transmission shaft penetrates through the inside and outside of equipment and has a clearance with the equipment, so that a medium in the equipment leaks outwards through the clearance, and labyrinth sealing is mostly adopted at the present stage to prevent the medium inside the equipment from leaking.
The labyrinth seal clearance circular flow is driven by pressure difference to flow from an outlet to an inlet, the clearance circular flow continuously flows in the labyrinth seal structure, fluid in the labyrinth seal clearance flows into an expansion cavity to form a vortex, pressure energy, kinetic energy and internal energy of the vortex are converted with each other, three-dimensional turbulence is formed in the labyrinth seal clearance, and the three-dimensional turbulence flow field is extremely complex, so that the solution is very difficult.
Disclosure of Invention
The invention aims to solve the problems and provides a method for solving a three-dimensional small-clearance turbulent flow field of a labyrinth seal, which reduces the difficulty in solving the three-dimensional turbulent flow field in the labyrinth seal structure, overcomes the defect that the circulating flow field of the labyrinth seal clearance cannot be solved in the prior art, and realizes the solution of the complex three-dimensional turbulent flow field.
The invention specifically adopts the following technical scheme:
a labyrinth seal three-dimensional small-gap turbulent flow field solving method specifically comprises the following steps:
step 1, according to an actual labyrinth seal structure, establishing a labyrinth seal model consisting of a labyrinth seal ring, a rotating shaft and a gap fluid by using MATLAB software, wherein a groove body is arranged on the labyrinth seal ring, the structural parameters and boundary conditions of the labyrinth seal model are set, a three-dimensional rectangular coordinate system is established by taking the axis of the rotating shaft at the gap fluid inlet of the labyrinth seal ring as an origin of coordinates, and the x-axis direction of the three-dimensional rectangular coordinate system is the radial direction of the rotating shaft and the z-axis direction of the rotating shaft is the axial direction of the rotating shaft;
step 2, in a three-dimensional rectangular coordinate system, carrying out radial averaging processing on the flow velocity and the pressure of the fluid in the clearance of the labyrinth seal model, wherein the calculation formula is as follows:
Figure BDA0002863382470000011
in the formula (I), the compound is shown in the specification,
Figure BDA0002863382470000012
representing the flow rate of interstitial fluid along the y-axis,
Figure BDA0002863382470000013
representing the flow rate of interstitial fluid along the x-axis,
Figure BDA0002863382470000014
indicating interstitial fluid along the z-axisThe flow rate of the liquid is controlled by the flow rate,
Figure BDA0002863382470000021
representing the fluid pressure of the interstitial fluid; h represents the distance from the wall surface of the rotating shaft to the inner wall of the annular sealing ring; u represents the flow velocity of the gap fluid after radial averaging along the y-axis, V represents the flow velocity of the gap fluid after radial averaging along the x-axis, W represents the flow velocity of the gap fluid after radial averaging along the z-axis, and P represents the fluid pressure of the gap fluid after radial averaging;
based on the integral flow theory, combining the flow speed and the pressure of the clearance fluid after radial averaging to determine a fluid continuous equation of the clearance fluid in the labyrinth seal model, as shown in formula (2):
Figure BDA0002863382470000022
where t represents time and ρ represents the density of the interstitial fluid;
according to the fluid continuous equation of the gap fluid, momentum equations of the gap fluid along the z-axis and the y-axis are respectively obtained, and the momentum equation of the gap fluid along the z-axis is shown as the formula (3):
Figure BDA0002863382470000023
in the formula (I), the compound is shown in the specification,
Figure BDA0002863382470000024
representing the shear stress of the surface of the rotating shaft along the z-axis;
the momentum equation of the interstitial fluid along the y-axis is shown in equation (4):
Figure BDA0002863382470000025
in the formula (I), the compound is shown in the specification,
Figure BDA0002863382470000026
representing the shear stress of the surface of the spindle along the y-axis;
based on a fluid continuous equation of the gap fluid, simulating the flow condition of the gap fluid by using a labyrinth seal model, and determining the fluid properties of the gap fluid at different positions of the labyrinth seal model;
step 3, dividing the gap fluid in the labyrinth seal model into a control body I, a control body II and a control body III according to the flow attributes of the gap fluid at different positions in the labyrinth seal model, defining the gap fluid between the labyrinth seal ring non-groove body section and the rotating shaft as the control body I, the gap fluid between the labyrinth seal ring groove body inlet and the rotating shaft as the control body II and the gap fluid in the labyrinth seal ring groove body as the control body III, establishing a gap fluid three-control-body model, setting the flow attributes of the gap fluid in each control body, and determining a fluid control equation set of the gap fluid in each control body by combining an integral flow theory based on the gap fluid three-control-body model;
the fluid control equation of the fluid in the internal gap of the control body I is as follows:
Figure BDA0002863382470000031
wherein the content of the first and second substances,
Figure BDA0002863382470000032
in the formula, R represents the radius of the annular sealing ring, and theta represents the included angle between the gap fluid and the xz plane of the three-dimensional rectangular coordinate system; hIThe distance from the wall surface of the rotary shaft to the inner wall surface of the annular sealing ring in the control body I is represented; u shapeΙRepresents the flow velocity of the gap fluid along the y-axis after radial averaging in the control body I, WIRepresents the flow velocity of the gap fluid along the z-axis after radial averaging in the control body I, PIThe fluid pressure of the gap fluid after radial averaging in the control body I is represented; ω represents the rotational speed of the shaft; μ represents gap fluid flow viscosity, ms、mrDenotes a first wall surface friction coefficient, ms＝mr；ns、nrDenotes a second wall surface friction coefficient, ns＝nr；
The fluid control equation of the fluid in the internal clearance of the control body II is as follows:
Figure BDA0002863382470000033
wherein the content of the first and second substances,
Figure BDA0002863382470000034
Uj＝0.42UШ+0.58UII
Figure BDA0002863382470000035
Figure BDA0002863382470000036
Figure BDA0002863382470000037
Figure BDA0002863382470000038
Figure BDA0002863382470000039
in the formula, HIIThe distance from the wall surface of the rotary shaft to the inner wall surface of the annular sealing ring in the control body II is represented; u shapeIIRepresents the flow velocity of the gap fluid along the y-axis after radial averaging in the control body II, WIIRepresents the flow velocity of the gap fluid along the z-axis after radial averaging in the control body II, PIIRepresenting the fluid pressure, U, of the radially averaged interstitial fluid in the control body IIШRepresents the flow velocity of the gap fluid along the y-axis after radial averaging in the control body III, WIIIRepresents the flow velocity of the gap fluid along the z-axis after radial averaging in the control body II, VIIPresentation controlIn the body II, the flow velocity V of the gap fluid along the x-axis after radial averagingIIIRepresenting the flow velocity of the gap fluid along the x axis after radial averaging in the control body III; alpha represents the boundary gradient of the control body II and the control body III, and L represents the sealing total length of the annular sealing ring; u shapejThe flow velocity of the gap circulation at the junction of the control body II and the control body III along the y-axis is shown, WjRepresenting the flow velocity of the gap circulation at the junction of the control body II and the control body III along the z-axis; vintRepresents the rate of gap fluid flow from control body II into control body III; c. C1、c4Is a coefficient, betaz、βθRepresents a wall shear force parameter, betaz＝βθ；βvA calculated parameter representing the eddy velocity of the interstitial fluid along the z-axis;
the fluid control equation of the fluid in the internal gap of the control body III is as follows:
Figure BDA0002863382470000041
wherein the content of the first and second substances,
Figure BDA0002863382470000042
Figure BDA0002863382470000043
in the formula, HIIIThe distance from the wall surface of the rotary shaft to the inner wall surface of the annular sealing ring in the control body III is represented; u shapeIIIRepresenting the flow velocity of the gap fluid along the y axis after radial averaging in the control body III; tau isgθRepresenting the component of the shearing force of the inner wall of the groove body of the annular sealing ring along the z axis;
step 4, simulating the eccentric whirl condition of the rotating shaft by using a labyrinth seal model, combining the eccentricity of the rotating shaft in the labyrinth seal model, decomposing the fluid control equation set of each control body based on a perturbation method, and respectively representing a pressure field, a temperature field, a speed field along the y axis and a speed field along the z axis in each control body as perturbation forms and bringing the perturbation forms into the fluid control equation set by taking the eccentricity of the rotating shaft as a shooting amount, so that the fluid control equation set of each control body is decomposed into an equation set consisting of a zero-order concentric whirl equation and a first-order eccentric whirl equation, and a fluid control equation set consisting of a zero-order concentric whirl equation and a first-order eccentric whirl equation of each control body is obtained;
the fluid control equation set of the control body I is decomposed into:
Figure BDA0002863382470000051
wherein epsilon represents the eccentricity of the rotating shaft; p is a radical ofIRepresenting the pressure field, p, of the control body II0Pressure field, p, calculated by a system of equations representing the zeroth order of the control volume II1Expressing the pressure field calculated by a first order equation set of the control body I; u. ofIRepresenting the velocity field, u, of the interstitial fluid in the control body I along the y-axisI0Representing the velocity field u of the interstitial fluid along the y-axis calculated by the zeroth order equation system of the control body II1Representing the velocity field of the gap fluid along the y axis calculated by a first order equation set of the control body I; w is aIRepresenting the velocity field of the interstitial fluid in the control body I along the z-axis, wI0The velocity field of the interstitial fluid along the z-axis, w, calculated by a zeroth order equation system representing the control volume II1Representing the velocity field of the gap fluid along the z axis calculated by a first order equation set of the control body I; t isIDenotes the temperature field, T, of the control body II0Temperature field, T, calculated by a zeroth order equation set representing the control volume II1Representing the temperature field calculated by a first order equation set of the control body I;
the fluid control equation set for control volume II is decomposed as:
Figure BDA0002863382470000052
in the formula, pIIDenotes the pressure field, p, of the control body IIII0Representing the pressure field, p, calculated by a system of zeroth order equations of the control volume IIII1Expressing the pressure field calculated by a first order equation set of the control body II; u. ofIIRepresenting the velocity field, u, of the interstitial fluid in the control body II along the y-axisII0Interstitial fluid calculated by zeroth order equation system expressing control body IIVelocity field along the y-axis, uII1The velocity field of the clearance fluid along the y axis calculated by a first order equation set of the control body II is represented; w is aIIRepresenting the velocity field of the interstitial fluid in the control volume II along the z-axis, wII0Representing the velocity field of the interstitial fluid along the z-axis, w, calculated by a zeroth order equation set of the control body IIII1Representing the velocity field of the gap fluid along the z axis calculated by a first order equation set of the control body II; t isIIDenotes the temperature field, T, of the control body IIII0Temperature field, T, calculated by a zeroth order equation set representing the control volume IIII1The temperature field calculated by a first order equation set of the control body II is expressed;
the fluid control equation set for control volume III is decomposed as:
Figure BDA0002863382470000053
in the formula uIIIRepresenting the velocity field, u, of the interstitial fluid in the control volume III along the y-axisIII0Representing the velocity field u of the interstitial fluid along the y axis calculated by a zeroth order equation system of a control body IIIIII1Representing the velocity field of the gap fluid along the y axis calculated by a first order equation set of the control body III; t isIIIDenotes the temperature field, T, of the control body IIIIII0Temperature field, T, calculated by a zeroth order equation set representing the control volume IIIIII1Representing the temperature field calculated by a first order equation set of the control body III;
step 5, solving a zero-order concentric vortex equation and a first-order eccentric vortex equation in each control body fluid control equation set, solving the zero-order equation set in each control body fluid control equation set by utilizing a Newton-Raphoson iteration method to obtain a zero-order velocity field of the clearance fluid along the y axis in all the control bodies of the labyrinth seal model, expanding the first-order equation set in each control body fluid control equation set in a simple harmonic manner according to the eccentric simple harmonic vortex condition and the periodic characteristics of a rotating shaft of the labyrinth seal model along the y axis, and calculating to obtain a first-order pressure field, a first-order temperature field and a first-order velocity field along the y axis of the clearance fluid in all the control bodies of the labyrinth seal model;
and 6, combining the zero-order velocity field and the first-order velocity field of each control body along the y axis according to the zero-order velocity field and the first-order velocity field of each control body along the y axis of the labyrinth seal model to obtain the velocity field of the clearance fluid of the labyrinth seal model.
Preferably, in step 1, the parameters of the labyrinth seal structure include the radius of the annular seal ring, the radius of the rotating shaft, the eccentricity of the rotating shaft, the rotating speed of the rotating shaft, the concentric whirling gap, the number of the groove bodies, the depth of the groove bodies, the width of the groove bodies, the distance between the groove bodies and the total length of the annular seal ring; the boundary conditions of the labyrinth seal structure comprise an inlet loss coefficient, an outlet pressure recovery coefficient, an outlet static pressure, a gap fluid flow rate and contact temperatures of the gap fluid with the surface of the rotating shaft and the surface of the annular seal ring of the labyrinth seal structure.
Preferably, in step 3, the gap fluid in each control body is set to be a newtonian incompressible fluid, the gap fluid in the control body II is set to be a through fluid, the gap fluid in the control body III is a vortex fluid, a certain boundary gradient exists at the intersection of the control body II and the control body III, the pressure values of the gap fluids in the control body II and the control body III are the same, and there is no mass exchange between the gap fluids at the intersection of the control body II and the control body III.
Preferably, the gap fluid in the control body III is in a single vortex, and the flow velocity of the gap fluid in the control body III along the x axis and the z axis is zero.
Preferably, in the step 5, solving a zeroth order equation set in each control body fluid control equation set based on a Newton-raphson iterative method specifically includes the following steps:
step 5.1, dispersing the labyrinth seal model gap fluid flow field into a plurality of tiny units, collecting the flow velocity of each unit gap fluid along the z axis to form a vector x, and then forming a vector f by using the y axis motion equation of the control body where each unit is located, as shown in formula (14):
Figure BDA0002863382470000061
in the formula, N1Representing the total number of discrete units of interstitial fluid located within the control body I, N2The total number of the gap fluid discrete units in the control body II is shown;
step 5.2, setting a zero-order flow velocity initial value x of the clearance fluid in the labyrinth seal model along the y axis0And the calculation accuracy εe；
And 5.3, substituting the zero-order flow velocity value of the gap fluid along the y axis into an iterative equation, and calculating to obtain the updated zero-order flow velocity value of the gap fluid along the y axis, wherein the iterative equation is shown as the formula (15):
Figure BDA0002863382470000071
wherein k represents the number of iterations, and k is 0 in the first iteration; x is the number ofkRepresenting the zeroth order flow velocity of the interstitial fluid along the y-axis, x for the first iterationk＝x0；xk+1Representing the zeroth order flow velocity value of the gap fluid along the y-axis after updating;
and (3) obtaining the zero-order flow velocity value of the updated gap fluid along the y axis by calculating an iterative equation:
Figure BDA0002863382470000072
step 5.4, if max | x is satisfiedk+1-xk|≤εeThen let xk＝xk+1Returning to the step 5.3 to continue iterative computation; if not satisfying max | xk+1-xk|≤εeIf so, the iterative computation is ended;
step 5.5, outputting a zero-order flow velocity value x of the gap fluid along the y axis obtained by iterative computationk+1And determining the zero-order velocity field of the clearance fluid along the y-axis in all control bodies of the labyrinth seal model.
Preferably, in step 5, the first order equation set in each control body fluid control equation set is expanded in a simple harmonic form, and the result is as follows:
the first order system of equations for control volume I is expanded as:
Figure BDA0002863382470000073
the first order system of equations for control volume II is expanded as:
Figure BDA0002863382470000074
the first order system of equations for control volume III is expanded as:
Figure BDA0002863382470000075
in the formula, pI1Pressure field, w, representing the first order equation of the control body II1Representing the velocity field, u, of the first order equation of the control volume I along the z-axisI1Representing the velocity field of the first order equation of the control body I along the y-axis, TI1Temperature field along y-axis, p, representing first order equation of control volume II1s、wI1s、uI1sAnd TI1sFor controlling the coefficient of the sine term, p, in the simple harmonic form of the first order equation of the body II1c、wI1c、uI1cAnd TI1cIs a cosine term coefficient in a simple harmonic form of a first order equation of a control body I;
pII1pressure field, w, representing the first order equation of the control body IIII1Representing the velocity field along the z-axis of the first order equation of the control volume II, uII1Representing the velocity field of the first order equation of the control body II along the y-axis, TII1Temperature field along y-axis, p, representing first order equation of control body IIII1s、wII1s、uII1sAnd TII1sFor controlling the coefficient of the sinusoidal term in the harmonic form of the body II first order equation, pII1c、wII1c、uII1cAnd TII1cIs the cosine term coefficient in the simple harmonic form of the first order equation of the control body II;
uШ1representing the velocity field, T, of the first order equation of control body III along the y-axisШ1Temperature field along y-axis, u, representing first order equation of control volume IIIШ1sAnd TШ1sFor controlling the sine term coefficient, u, in the harmonic form of the first order equation of the body IIIШ1cAnd TШ1cIs the cosine term coefficient in the harmonic form of the first order equation of the control body III.
The invention has the following beneficial effects:
the invention is based on the integral flow theory, combines the characteristic of small radial thickness of the clearance fluid in the labyrinth seal structure, simplifies the three-dimensional turbulence in the labyrinth seal structure into two-dimensional flow, the method simplifies the solution problem of the gap fluid flow field while maintaining the property of the gap fluid flow field, establishes a gap fluid three-control-body model, decomposes a fluid control equation set of the labyrinth seal structure into an equation set consisting of a zero-order concentric vortex equation and a first-order eccentric vortex equation based on a perturbation method by utilizing the eccentric vortex characteristic of a rotating shaft of the labyrinth seal structure, calculates the internal flow field of the labyrinth seal structure, greatly reduces the solution difficulty of the fluid control equation set, provides a theoretical basis for solving a complex flow field, meanwhile, the invention can also study the flow fields of the labyrinth seal models with different groove body sizes by adjusting the structural parameters of the labyrinth seal models, thereby being beneficial to guiding the design of the labyrinth seal structure.
Drawings
FIG. 1 is a schematic view of a labyrinth seal.
FIG. 2 is a schematic view of a labyrinth seal model.
Fig. 3 is a schematic diagram of a model of a gap fluid three-control body.
FIG. 4 is a flow chart for decomposing a set of fluid control equations based on perturbation.
Detailed Description
The following description of the embodiments of the present invention will be made with reference to the accompanying drawings:
a labyrinth seal three-dimensional small-gap turbulent flow field solving method specifically comprises the following steps:
step 1, according to an actual labyrinth seal structure, as shown in fig. 1, a labyrinth seal model composed of a labyrinth seal ring, a rotating shaft and a gap fluid is established by using MATLAB software, a groove body is arranged on the labyrinth seal ring, as shown in fig. 2, the labyrinth seal model is arranged in a three-dimensional rectangular coordinate system, the origin of coordinates of the three-dimensional rectangular coordinate system is the position of the axis of the rotating shaft at the inlet of the gap fluid of the labyrinth seal ring, the central axis of the rotating shaft is used as the z axis, the radial direction of the rotating shaft is used as the x axis, and the x axis, the y axis and the z axis are perpendicular in pairs;
setting structural parameters of a labyrinth seal model, wherein the structural parameters of the labyrinth seal model are shown in a table 1; setting boundary conditions of a labyrinth seal model, wherein the inlet and outlet loss coefficient of the labyrinth seal model is 0.05, the outlet pressure recovery coefficient is 0.95, the outlet static pressure is 0Pa, the flow speed of gap fluid entering the labyrinth seal ring is 5m/s, the contact temperature of the gap fluid and the surface of the rotating shaft is 20 ℃, and the contact temperature of the gap fluid and the surface of the annular seal ring is 90 ℃.
TABLE 1 labyrinth seal model structural parameters
Figure BDA0002863382470000091
Step 2, because the thickness of the clearance fluid in the labyrinth seal ring in the x-axis direction is very small, the radial averaging processing is performed on the flow speed and the pressure of the clearance fluid of the labyrinth seal model in a three-dimensional rectangular coordinate system by taking the thickness of the clearance fluid in the x-axis as a whole, that is, the average value in the direction is used for replacing parameter values for the flow speed and the pressure of the clearance fluid of the labyrinth seal model in the x-axis direction, and the calculation formula is as follows:
Figure BDA0002863382470000092
in the formula (I), the compound is shown in the specification,
Figure BDA0002863382470000093
representing the flow rate of interstitial fluid along the y-axis,
Figure BDA0002863382470000094
representing the flow rate of interstitial fluid along the x-axis,
Figure BDA0002863382470000095
representing the flow velocity of the interstitial fluid along the z-axis,
Figure BDA0002863382470000096
representing the fluid pressure of the interstitial fluid;h represents the distance from the wall surface of the rotating shaft to the inner wall of the annular sealing ring; u represents the flow velocity of the gap fluid after radial averaging along the y-axis, V represents the flow velocity of the gap fluid after radial averaging along the x-axis, W represents the flow velocity of the gap fluid after radial averaging along the z-axis, and P represents the fluid pressure of the gap fluid after radial averaging;
based on the integral flow theory, converting the three-dimensional turbulence of the gap fluid in the labyrinth seal model into two-dimensional flow by combining the flow speed and the pressure of the gap fluid after radial averaging, wherein the converted two-dimensional flow still keeps the original three-dimensional turbulence characteristic of the gap fluid, and determining a fluid continuous equation of the gap fluid in the labyrinth seal model, as shown in formula (2):
Figure BDA0002863382470000101
where t represents time and ρ represents the density of the interstitial fluid;
according to the fluid continuous equation of the gap fluid, momentum equations of the gap fluid along the z-axis and the y-axis are respectively obtained, and the momentum equation of the gap fluid along the z-axis is shown as the formula (3):
Figure BDA0002863382470000102
in the formula (I), the compound is shown in the specification,
Figure BDA0002863382470000103
representing the shear stress of the surface of the rotating shaft along the z-axis;
the momentum equation of the interstitial fluid along the y-axis is shown in equation (4):
Figure BDA0002863382470000104
in the formula (I), the compound is shown in the specification,
Figure BDA0002863382470000105
representing the shear stress of the surface of the spindle along the y-axis;
meanwhile, the gap fluid is subjected to radial averaging processing, so that the gap fluid has no momentum equation in the x-axis direction;
and (3) simulating the flow condition of the gap fluid by using the labyrinth seal model based on a fluid continuous equation of the gap fluid, and determining the fluid property of the gap fluid at different positions of the labyrinth seal model.
Step 3, according to the flow attributes of the gap fluid at different positions in the labyrinth seal model, dividing the gap fluid in the labyrinth seal model into a control body I, a control body II and a control body III, and establishing a gap fluid three-control-body model, as shown in fig. 3, wherein the gap fluid between the labyrinth seal groove-free section and the rotating shaft is the control body I, the gap fluid between the labyrinth seal groove body inlet and the rotating shaft is the control body II, the gap fluid in the labyrinth seal groove body is the control body III, and the boundary gradient at the intersection of the control body II and the control body III is 0.008; the gap fluid in each control body is set to be Newtonian incompressible fluid, the gap fluid in the control body II is set to be fluid, the gap fluid in the control body III is a single vortex, the pressure change in the inner part of each groove body of the annular sealing ring is ignored, the pressure value of the gap fluid in the control body II is the same as that of the gap fluid in the control body III, and no mass exchange exists between the gap fluids at the intersection of the control body II and the control body III.
Based on a gap fluid three-control-body model, combined with an overall flow theory, because the flow speed of gap fluid along the x axis is 0 under the flow theory, the control body II has no x-axis momentum equation, meanwhile, because the gap fluid in the control body III is a single vortex, the flow speeds of the gap fluid in the control body III along the z axis and the x axis are both zero, and therefore a fluid control equation set of the gap fluid in each control body is determined as follows:
the fluid control equation of the fluid in the internal gap of the control body I is as follows:
Figure BDA0002863382470000111
wherein the content of the first and second substances,
Figure BDA0002863382470000112
in the formula, R represents the radius of the annular sealing ring, and theta represents the included angle between the gap fluid and the xz plane of the three-dimensional rectangular coordinate system; hIThe distance from the wall surface of the rotary shaft to the inner wall surface of the annular sealing ring in the control body I is represented; u shapeΙRepresents the flow velocity of the gap fluid along the y-axis after radial averaging in the control body I, WIRepresents the flow velocity of the gap fluid along the z-axis after radial averaging in the control body I, PIThe fluid pressure of the gap fluid after radial averaging in the control body I is represented; ω represents the rotational speed of the shaft; μ represents gap fluid flow viscosity, ms、mrRepresenting a first wall friction coefficient; n iss、nrRepresents a second wall surface friction coefficient;
the fluid control equation of the fluid in the internal gap of the control body II is as follows:
Figure BDA0002863382470000113
wherein the content of the first and second substances,
Figure BDA0002863382470000114
Uj＝0.42UШ+0.58UII
Figure BDA0002863382470000115
Figure BDA0002863382470000116
Figure BDA0002863382470000117
Figure BDA0002863382470000118
Figure BDA0002863382470000119
in the formula, HIIThe distance from the wall surface of the rotary shaft to the inner wall surface of the annular sealing ring in the control body II is represented; u shapeIIRepresents the flow velocity of the gap fluid along the y-axis after radial averaging in the control body II, WIIRepresents the flow velocity of the gap fluid along the z-axis after radial averaging in the control body II, PIIRepresenting the fluid pressure, U, of the radially averaged interstitial fluid in the control body IIШRepresents the flow velocity of the gap fluid along the y-axis after radial averaging in the control body III, WIIIRepresents the flow velocity of the gap fluid along the z-axis after radial averaging in the control body II, VIIRepresents the flow velocity of the gap fluid along the x-axis after radial averaging in the control body II, VIIIRepresenting the flow velocity of the gap fluid along the x axis after radial averaging in the control body III; alpha represents the boundary gradient of the control body II and the control body III, and L represents the sealing total length of the annular sealing ring; u shapejRepresents the flow velocity of the clearance circulation at the junction of the control body II and the control body III along the y axis, WjRepresenting the flow velocity of the gap circulation at the junction of the control body II and the control body III along the z-axis; vintRepresents the rate of gap fluid flow from control body II into control body III; c. C1、c4Is a coefficient, betaz、βθRepresents a wall shear force parameter, betaz＝βθ＝0.275；βvA calculation parameter representing an eddy current velocity of the gap fluid along the z-axis;
the fluid control equation of the fluid in the internal gap of the control body III is as follows:
Figure BDA0002863382470000121
wherein the content of the first and second substances,
Figure BDA0002863382470000122
Figure BDA0002863382470000123
in the formula, HIIIThe distance from the wall surface of the rotary shaft to the inner wall surface of the annular sealing ring in the control body III is represented; u shapeIIIRepresenting the flow velocity of the gap fluid along the y axis after radial averaging in the control body III; tau isgθThe component of the shearing force of the inner wall of the groove body of the annular sealing ring along the z-axis is shown.
Step 4, simulating the eccentric whirl condition of the rotating shaft by using a labyrinth seal model, combining the eccentricity of the rotating shaft in the labyrinth seal model, decomposing the fluid control equation set of each control body based on a perturbation method, and respectively representing a pressure field, a temperature field, a speed field along a y axis and a speed field along a z axis in each control body as perturbation forms and bringing the perturbation forms into the fluid control equation set by taking the eccentricity of the rotating shaft as a perturbation amount as shown in fig. 4, so that the fluid control equation set of each control body is decomposed into an equation set consisting of a zero-order concentric whirl equation and a first-order eccentric whirl equation, and a fluid control equation set consisting of a zero-order concentric whirl equation and a first-order eccentric whirl equation of each control body is obtained;
the fluid control equation set of the control body I is decomposed into:
Figure BDA0002863382470000131
wherein epsilon represents the eccentricity of the rotating shaft; p is a radical ofIRepresenting the pressure field, p, of the control body II0Pressure field, p, calculated by a system of equations representing the zeroth order of the control volume II1Expressing the pressure field calculated by a first order equation set of the control body I; u. ofIRepresenting the velocity field, u, of the interstitial fluid in the control body I along the y-axisI0Representing the velocity field u of the interstitial fluid along the y-axis calculated by the zeroth order equation system of the control body II1Representing the velocity field of the gap fluid along the y axis calculated by a first order equation set of the control body I; w is aIRepresenting the velocity field of the interstitial fluid in the control body I along the z-axis, wI0The velocity field of the interstitial fluid along the z-axis, w, calculated by a zeroth order equation system representing the control volume II1Representing the velocity field of the gap fluid along the z axis calculated by a first order equation set of the control body I; t isIDenotes the temperature field, T, of the control body II0Presentation controlTemperature field, T, calculated by system I zeroth order equation setI1Representing the temperature field calculated by a first order equation set of the control body I;
the fluid control equation set for control volume II is decomposed as:
Figure BDA0002863382470000132
in the formula, pIIDenotes the pressure field, p, of the control body IIII0Representing the pressure field, p, calculated by a system of zeroth order equations of the control volume IIII1Expressing the pressure field calculated by a first order equation set of the control body II; u. ofIIRepresenting the velocity field, u, of the interstitial fluid in the control body II along the y-axisII0The velocity field u of the interstitial fluid along the y axis, which is calculated by a zeroth order equation system of the control body IIII1Representing the velocity field of the gap fluid along the y axis calculated by a first order equation set of the control body II; w is aIIRepresenting the velocity field of the interstitial fluid in the control volume II along the z-axis, wII0Representing the velocity field of the interstitial fluid along the z-axis, w, calculated by a zeroth order equation set of the control body IIII1Representing the velocity field of the gap fluid along the z axis calculated by a first order equation set of the control body II; t isIIDenotes the temperature field, T, of the control body IIII0Temperature field, T, calculated by a zeroth order equation set representing the control volume IIII1The temperature field calculated by a first order equation set of the control body II is expressed;
the fluid control equation set for control volume III is decomposed as:
Figure BDA0002863382470000133
in the formula uIIIRepresenting the velocity field, u, of the interstitial fluid in the control volume III along the y-axisIII0Representing the velocity field u of the interstitial fluid along the y axis calculated by a zeroth order equation system of a control body IIIIII1Representing the velocity field of the gap fluid along the y axis calculated by a first order equation set of the control body III; t isIIIDenotes the temperature field, T, of the control body IIIIII0Temperature field, T, representing the calculation of zeroth order equation set of control volume IIIIII1Representing the temperature field calculated by the first order equation set of the control body III.
Step 5, solving a zero-order concentric vortex equation and a first-order eccentric vortex equation in each control body fluid control equation set, wherein the zero-order concentric vortex equation of each control body is not related to the flow time t of the gap fluid and the circumferential coordinate theta (the angle of the gap fluid moving along the circumferential direction of the rotating shaft), and the first-order eccentric vortex equation of each control body is related to the flow time t of the gap fluid, the circumferential coordinate theta (the angle of the gap fluid moving along the circumferential direction of the rotating shaft) and the axial coordinate (z-axis coordinate) of the rotating shaft;
solving a zeroth order equation set in each control body fluid control equation set by utilizing a Newton-Raphoson iterative method, and determining a zeroth order velocity field of the clearance fluid along the y axis in all control bodies of the labyrinth seal model, wherein the specific process comprises the following steps:
step 5.1, dispersing the labyrinth seal model gap fluid flow field into a plurality of tiny units, collecting the flow velocity of each unit gap fluid along the z axis to form a vector x, and then forming a vector f by using the y axis motion equation of the control body where each unit is located, as shown in formula (14):
Figure BDA0002863382470000141
in the formula, N1Representing the total number of discrete units of interstitial fluid located within the control body I, N2The total number of the gap fluid discrete units in the control body II is shown;
step 5.2, setting a zero-order flow velocity initial value x of the clearance fluid in the labyrinth seal model along the y axis0And the calculation accuracy εe＝1e-4；
And 5.3, substituting the zero-order flow velocity value of the gap fluid along the y axis into an iterative equation, and calculating to obtain the updated zero-order flow velocity value of the gap fluid along the y axis, wherein the iterative equation is shown as the formula (15):
Figure BDA0002863382470000142
wherein k represents the number of iterations, and k is 0 in the first iteration; x is the number ofkBetween representationsZero order flow velocity value of gap fluid along y axis, x at first iterationk＝x0；xk+1Representing the zeroth order flow velocity value of the gap fluid along the y-axis after updating;
and (3) obtaining the zero-order flow velocity value of the updated gap fluid along the y axis by calculating an iterative equation:
Figure BDA0002863382470000143
step 5.4, if max | x is satisfiedk+1-xkIf | is less than or equal to 1e-4, let xk＝xk+1Returning to the step 5.3 to continue iterative computation; if not satisfying max | xk+1-xkIf the | is less than or equal to 1e-4, finishing the iterative computation;
step 5.5, outputting a zero-order flow velocity value x of the gap fluid obtained by iterative computation along the y axisk+1And determining the zero-order velocity field of the clearance fluid along the y-axis in all control bodies of the labyrinth seal model.
And then according to the eccentric simple harmonic vortex condition and the periodic characteristics of the rotating shaft of the labyrinth seal model along the y axis, a first order equation set in each control body fluid control equation set is expanded in a simple harmonic mode, and the expansion result is as follows:
the first order system of equations for control volume I is expanded as:
Figure BDA0002863382470000151
the first order system of equations for control volume II is expanded as:
Figure BDA0002863382470000152
the first order system of equations for control volume III expands as:
Figure BDA0002863382470000153
in the formula, pI1Presentation controlPressure field of first order equation of body I, wI1Representing the velocity field, u, of the first order equation of the control volume I along the z-axisI1Representing the velocity field of the first order equation of the control body I along the y-axis, TI1Temperature field along y-axis, p, representing first order equation of control volume II1s、wI1s、uI1sAnd TI1sFor controlling the coefficient of the sine term, p, in the simple harmonic form of the first order equation of the body II1c、wI1c、uI1cAnd TI1cIs a cosine term coefficient in a simple harmonic form of a first order equation of a control body I;
pII1pressure field, w, representing the first order equation of the control body IIII1Representing the velocity field, u, of the first order equation of the control volume II along the z-axisII1Representing the velocity field of the first order equation of the control body II along the y-axis, TII1Temperature field along y-axis, p, representing first order equation of control body IIII1s、wII1s、uII1sAnd TII1sFor controlling the coefficient of the sinusoidal term in the harmonic form of the body II first order equation, pII1c、wII1c、uII1cAnd TII1cIs the cosine term coefficient in the simple harmonic form of the first order equation of the control body II;
uШ1representing the velocity field, T, of the first order equation of control body III along the y-axisШ1Temperature field along y-axis, u, representing first order equation of control volume IIIШ1sAnd TШ1sFor controlling the coefficient of the sinusoidal term, u, in the harmonic form of the first order equation of the body IIIШ1cAnd TШ1cIs the cosine term coefficient in the simple harmonic form of the first order equation of the control body III;
and calculating to obtain a first-order pressure field, a first-order temperature field and a first-order speed field along the y axis of the clearance fluid in all control bodies of the labyrinth seal model.
And 6, combining the zero-order velocity field and the first-order velocity field of each control body along the y axis according to the zero-order velocity field and the first-order velocity field of each control body along the y axis of the labyrinth seal model to obtain the velocity field of the clearance fluid of the labyrinth seal model.
It is to be understood that the above description is not intended to limit the present invention, and the present invention is not limited to the above examples, and those skilled in the art may make modifications, alterations, additions or substitutions within the spirit and scope of the present invention.

Claims (6)

1. A labyrinth seal three-dimensional small-gap turbulent flow field solving method is characterized by comprising the following steps:
step 1, according to an actual labyrinth seal structure, establishing a labyrinth seal model consisting of a labyrinth seal ring, a rotating shaft and a gap fluid by using MATLAB software, wherein a groove body is arranged on the labyrinth seal ring, the structural parameters and boundary conditions of the labyrinth seal model are set, a three-dimensional rectangular coordinate system is established by taking the axis of the rotating shaft at the gap fluid inlet of the labyrinth seal ring as an origin of coordinates, and the x-axis direction of the three-dimensional rectangular coordinate system is the radial direction of the rotating shaft and the z-axis direction of the rotating shaft is the axial direction of the rotating shaft;
step 2, in a three-dimensional rectangular coordinate system, carrying out radial averaging processing on the flow velocity and the pressure of the fluid in the clearance of the labyrinth seal model, wherein the calculation formula is as follows:
Figure FDA0002863382460000011
in the formula (I), the compound is shown in the specification,
Figure FDA0002863382460000012
representing the flow rate of interstitial fluid along the y-axis,
Figure FDA0002863382460000013
representing the flow rate of interstitial fluid along the x-axis,
Figure FDA0002863382460000014
representing the flow velocity of the interstitial fluid along the z-axis,
Figure FDA0002863382460000015
representing the fluid pressure of the interstitial fluid; h represents the distance from the wall surface of the rotating shaft to the inner wall of the annular sealing ring; u represents the flow velocity of the gap fluid after radial averaging along the y-axis, V represents the flow velocity of the gap fluid after radial averaging along the x-axis, and W represents the gap fluid after radial averagingThe flow rate of the interstitial fluid along the z-axis, P representing the fluid pressure of the radially averaged interstitial fluid;
based on the integral flow theory, combining the flow speed and the pressure of the clearance fluid after radial averaging to determine a fluid continuous equation of the clearance fluid in the labyrinth seal model, as shown in formula (2):
Figure FDA0002863382460000016
where t represents time and ρ represents the density of the interstitial fluid;
according to the fluid continuous equation of the gap fluid, momentum equations of the gap fluid along the z-axis and the y-axis are respectively obtained, and the momentum equation of the gap fluid along the z-axis is shown as the formula (3):
Figure FDA0002863382460000017
in the formula (I), the compound is shown in the specification,
Figure FDA0002863382460000018
representing the shear stress of the spindle surface along the z-axis;
the momentum equation of the interstitial fluid along the y-axis is shown in equation (4):
Figure FDA0002863382460000019
in the formula (I), the compound is shown in the specification,
Figure FDA00028633824600000110
representing the shear stress of the surface of the spindle along the y-axis;
based on a fluid continuous equation of the gap fluid, simulating the flow condition of the gap fluid by using a labyrinth seal model, and determining the fluid properties of the gap fluid at different positions of the labyrinth seal model;
step 3, dividing the gap fluid in the labyrinth seal model into a control body I, a control body II and a control body III according to the flow attributes of the gap fluid at different positions in the labyrinth seal model, defining the gap fluid between the labyrinth seal ring non-groove body section and the rotating shaft as the control body I, the gap fluid between the labyrinth seal ring groove body inlet and the rotating shaft as the control body II and the gap fluid in the labyrinth seal ring groove body as the control body III, establishing a gap fluid three-control-body model, setting the flow attributes of the gap fluid in each control body, and determining a fluid control equation set of the gap fluid in each control body by combining an integral flow theory based on the gap fluid three-control-body model;
the fluid control equation of the fluid in the internal gap of the control body I is as follows:
Figure FDA0002863382460000021
wherein the content of the first and second substances,
Figure FDA0002863382460000022
in the formula, R represents the radius of the annular sealing ring, and theta represents the included angle between the gap fluid and the xz plane of the three-dimensional rectangular coordinate system; hIThe distance from the wall surface of the rotary shaft to the inner wall surface of the annular sealing ring in the control body I is represented; u shapeΙRepresents the flow velocity of the gap fluid along the y-axis after radial averaging in the control body I, WIRepresents the flow velocity along the z-axis, P, of the gap fluid after radial averaging in the control body IIThe fluid pressure of the gap fluid after radial averaging in the control body I is represented; ω represents the rotational speed of the shaft; μ represents gap fluid flow viscosity, ms、mrDenotes a first wall surface friction coefficient, ms＝mr；ns、nrDenotes a second wall surface friction coefficient, ns＝nr；
The fluid control equation of the fluid in the internal gap of the control body II is as follows:
Figure FDA0002863382460000023
wherein the content of the first and second substances,
Figure FDA0002863382460000031
in the formula, HIIThe distance from the wall surface of the rotary shaft to the inner wall surface of the annular sealing ring in the control body II is represented; u shapeIIRepresents the flow velocity of the gap fluid along the y-axis after radial averaging in the control body II, WIIRepresents the flow velocity of the gap fluid along the z-axis after radial averaging in the control body II, PIIRepresenting the fluid pressure, U, of the radially averaged interstitial fluid in the control body IIIIIRepresents the flow velocity of the gap fluid along the y-axis after radial averaging in the control body III, WIIIRepresents the flow velocity of the gap fluid along the z-axis after radial averaging in the control body II, VIIRepresents the flow velocity of the gap fluid along the x-axis after radial averaging in the control body II, VIIIRepresenting the flow velocity of the gap fluid along the x axis after radial averaging in the control body III; alpha represents the boundary gradient of the control body II and the control body III, and L represents the sealing total length of the annular sealing ring; u shapejRepresents the flow velocity of the clearance circulation at the junction of the control body II and the control body III along the y axis, WjRepresenting the flow velocity of the gap circulation at the junction of the control body II and the control body III along the z-axis; vintRepresents the rate of gap fluid flow from control body II into control body III; c. C1、c4Is a coefficient, betaz、βθRepresents a wall shear force parameter, betaz＝βθ；βvA calculation parameter representing an eddy current velocity of the gap fluid along the z-axis;
the fluid control equation of the fluid in the internal gap of the control body III is as follows:
Figure FDA0002863382460000032
wherein the content of the first and second substances,
Figure FDA0002863382460000033
in the formula, HIIIThe distance from the wall surface of the rotary shaft to the inner wall surface of the annular sealing ring in the control body III is represented; u shapeIIIRepresenting the flow velocity of the gap fluid along the y axis after radial averaging in the control body III; tau isgθRepresenting the component of the shearing force of the inner wall of the groove body of the annular sealing ring along the z axis;
step 4, simulating the eccentric whirl condition of the rotating shaft by using a labyrinth seal model, combining the eccentricity of the rotating shaft in the labyrinth seal model, decomposing the fluid control equation set of each control body based on a perturbation method, and respectively representing a pressure field, a temperature field, a speed field along the y axis and a speed field along the z axis in each control body as perturbation forms and bringing the perturbation forms into the fluid control equation set by taking the eccentricity of the rotating shaft as a shooting amount, so that the fluid control equation set of each control body is decomposed into an equation set consisting of a zero-order concentric whirl equation and a first-order eccentric whirl equation, and a fluid control equation set consisting of a zero-order concentric whirl equation and a first-order eccentric whirl equation of each control body is obtained;
the fluid control equation set of the control body I is decomposed into:
Figure FDA0002863382460000041
wherein epsilon represents the eccentricity of the rotating shaft; p is a radical ofIRepresenting the pressure field, p, of the control body II0Representing the pressure field, p, calculated by a control body I zeroth order equation setI1Expressing the pressure field calculated by a first order equation set of the control body I; u. ofIRepresenting the velocity field, u, of the interstitial fluid in the control body I along the y-axisI0Representing the velocity field u of the interstitial fluid along the y-axis calculated by the zeroth order equation system of the control body II1Representing the velocity field of the gap fluid along the y axis calculated by a first order equation set of the control body I; w is aIRepresenting the velocity field of the interstitial fluid in the control body I along the z-axis, wI0The velocity field of the interstitial fluid along the z-axis, w, calculated by a zeroth order equation system representing the control volume II1Representing the velocity field of the gap fluid along the z axis calculated by a first order equation set of the control body I; t isIDenotes the temperature field, T, of the control body II0Temperature field, T, calculated by a zeroth order equation set representing the control volume II1Representing the temperature field calculated by a first order equation set of the control body I;
the fluid control equation set for control volume II is decomposed as:
Figure FDA0002863382460000042
in the formula, pIIDenotes the pressure field, p, of the control body IIII0Representing the pressure field, p, calculated by a system of zeroth order equations of the control volume IIII1Expressing the pressure field calculated by a first order equation set of the control body II; u. ofIIRepresenting the velocity field, u, of the interstitial fluid in the control body II along the y-axisII0Representing the velocity field u of the interstitial fluid along the y axis calculated by a zeroth order equation system of a control body IIII1Representing the velocity field of the gap fluid along the y axis calculated by a first order equation set of the control body II; w is aIIRepresenting the velocity field of the interstitial fluid in the control volume II along the z-axis, wII0Representing the velocity field of the interstitial fluid along the z-axis, w, calculated by a zeroth order equation set of the control body IIII1Representing the velocity field of the gap fluid along the z axis calculated by a first order equation set of the control body II; t isIIDenotes the temperature field, T, of the control body IIII0Temperature field, T, calculated by a zeroth order equation set representing the control volume IIII1The temperature field calculated by a first order equation set of the control body II is expressed;
the fluid control equation set for control volume III is decomposed as:
Figure FDA0002863382460000051
in the formula uIIIRepresenting the velocity field, u, of the interstitial fluid in the control volume III along the y-axisIII0Representing the velocity field u of the interstitial fluid along the y axis calculated by a zeroth order equation system of a control body IIIIII1Representing the velocity field of the gap fluid along the y axis calculated by a first order equation set of the control body III; t isIIIDenotes the temperature field, T, of the control body IIIIII0Temperature field, T, calculated by a zeroth order equation set representing the control volume IIIIII1Representing a set of first order equations for control volume IIIA calculated temperature field;
step 5, solving a zero-order concentric vortex equation and a first-order eccentric vortex equation in each control body fluid control equation set, solving the zero-order equation set in each control body fluid control equation set by utilizing a Newton-Raphoson iteration method to obtain a zero-order velocity field of the clearance fluid along the y axis in all the control bodies of the labyrinth seal model, expanding the first-order equation set in each control body fluid control equation set in a simple harmonic manner according to the eccentric simple harmonic vortex condition and the periodic characteristics of a rotating shaft of the labyrinth seal model along the y axis, and calculating to obtain a first-order pressure field, a first-order temperature field and a first-order velocity field along the y axis of the clearance fluid in all the control bodies of the labyrinth seal model;
and 6, combining the zero-order velocity field and the first-order velocity field of each control body along the y axis according to the zero-order velocity field and the first-order velocity field of each control body along the y axis of the labyrinth seal model to obtain the velocity field of the clearance fluid of the labyrinth seal model.
2. The labyrinth seal three-dimensional small-gap turbulent flow field solving method as claimed in claim 1, wherein in step 1, labyrinth seal structure parameters include annular seal ring radius, rotating shaft eccentricity, rotating shaft rotating speed, concentric whirling gap, groove body number, groove body depth, groove body width, groove body interval and annular seal ring total length; the boundary conditions of the labyrinth seal structure comprise an inlet loss coefficient, an outlet pressure recovery coefficient, an outlet static pressure, a gap fluid flow rate and contact temperatures of the gap fluid with the surface of the rotating shaft and the surface of the annular seal ring of the labyrinth seal structure.
3. The method for solving the three-dimensional small-gap turbulent flow field of the labyrinth seal according to claim 1, wherein in step 3, the gap fluid in each control body is set to be a newtonian incompressible fluid, the gap fluid in the control body II is set to be a through fluid, the gap fluid in the control body III is a vortex fluid, a certain boundary gradient exists at the intersection of the control body II and the control body III, the pressure values of the gap fluids in the control body II and the control body III are the same, and no mass exchange exists between the gap fluids at the intersection of the control body II and the control body III.
4. The method for solving the three-dimensional small-gap turbulent flow field of the labyrinth seal as recited in claim 3, wherein the gap fluid in the control body III is a single vortex, and the flow velocity of the gap fluid in the control body III along the x-axis and the z-axis is zero.
5. The method for solving the labyrinth seal three-dimensional small-gap turbulent flow field according to claim 1, wherein in the step 5, a zeroth order equation set in each control body fluid control equation set is solved based on a Newton-Raphoson iterative method, and the method specifically comprises the following steps:
step 5.1, dispersing the labyrinth seal model gap fluid flow field into a plurality of tiny units, collecting the flow velocity of each unit gap fluid along the z axis to form a vector x, and then forming a vector f by using the y axis motion equation of the control body where each unit is located, as shown in formula (14):
Figure FDA0002863382460000061
in the formula, N1Representing the total number of discrete units of interstitial fluid located within the control body I, N2The total number of the interstitial fluid discrete units positioned in the control body II is represented;
step 5.2, setting a zero-order flow velocity initial value x of the clearance fluid in the labyrinth seal model along the y axis0And the calculation accuracy εe；
And 5.3, substituting the zero-order flow velocity value of the gap fluid along the y axis into an iterative equation, and calculating to obtain the updated zero-order flow velocity value of the gap fluid along the y axis, wherein the iterative equation is shown as the formula (15):
Figure FDA0002863382460000062
wherein k represents the number of iterations, and k is 0 in the first iteration; x is the number ofkRepresenting the zero order flow velocity of interstitial fluid along the y-axisValue, first iteration xk＝x0；xk+1Representing the zeroth order flow velocity value of the gap fluid along the y-axis after updating;
and (3) obtaining the zero-order flow velocity value of the updated gap fluid along the y axis by calculating an iterative equation:
Figure FDA0002863382460000063
step 5.4, if max | x is satisfiedk+1-xk|≤εeThen let xk＝xk+1Returning to the step 5.3 to continue iterative computation; if not satisfying max | xk+1-xk|≤εeIf so, the iterative computation is ended;
step 5.5, outputting a zero-order flow velocity value x of the gap fluid obtained by iterative computation along the y axisk+1And determining the zero-order velocity field of the clearance fluid along the y-axis in all control bodies of the labyrinth seal model.
6. The method for solving the three-dimensional small-gap turbulent flow field of the labyrinth seal as claimed in claim 1, wherein in the step 5, the first-order equation set in each control body fluid control equation set is developed in a simple harmonic form, and the result is as follows:
the first order system of equations for control volume I is expanded as:
Figure FDA0002863382460000064
the first order system of equations for control volume II is expanded as:
Figure FDA0002863382460000071
the first order system of equations for control volume III expands as:
Figure FDA0002863382460000072
in the formula, pI1Pressure field, w, representing the first order equation of the control body II1Representing the velocity field, u, of the first order equation of the control volume I along the z-axisI1Representing the velocity field of the first order equation of the control body I along the y-axis, TI1Temperature field along y-axis, p, representing first order equation of control volume II1s、wI1s、uI1sAnd TI1sFor controlling the coefficient of the sine term, p, in the simple harmonic form of the first order equation of the body II1c、wI1c、uI1cAnd TI1cIs a cosine term coefficient in a simple harmonic form of a first order equation of a control body I;
pII1pressure field, w, representing the first order equation of the control body IIII1Representing the velocity field, u, of the first order equation of the control volume II along the z-axisII1Representing the velocity field of the first order equation of the control body II along the y-axis, TII1Temperature field along y-axis, p, representing first order equation of control body IIII1s、wII1s、uII1sAnd TII1sFor controlling the coefficient of the sinusoidal term in the harmonic form of the body II first order equation, pII1c、wII1c、uII1cAnd TII1cIs the cosine term coefficient in the simple harmonic form of the first order equation of the control body II;
uШ1representing the velocity field, T, of the first order equation of control body III along the y-axisШ1Temperature field along y-axis, u, representing first order equation of control volume IIIШ1sAnd TШ1sFor controlling the coefficient of the sinusoidal term, u, in the harmonic form of the first order equation of the body IIIШ1cAnd TШ1cIs the cosine term coefficient in the harmonic form of the first order equation of the control body III.
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汽轮机组迷宫密封泄漏量及气动特性三维CFD数值模拟研究;韩强等;《电站系统工程》;20130915(第05期);全文 *
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