CN106777770B - The analogy method of hole stream in aqueduct based on finite volume method - Google Patents
The analogy method of hole stream in aqueduct based on finite volume method Download PDFInfo
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Abstract
The invention discloses a kind of analogy method of hole stream in aqueduct based on finite volume method, this method builds the transient flow governing equation containing free gas first；Then the grid system containing free gas, and discrete are established according to finite volume method (FVM, Finite Volume Method)；Then, the flux controlled during pure convection current at unit interface is calculated using the Godunov forms that space-time is second order accuracy；Followed by introducing source item in above-mentioned solution, obtain the explicit FVM Godunov forms of second order of discrete equation solution；Finally, air pocket volume using two methods amendment pressure and is calculated according to the pressure being calculated.In addition, the present invention suppresses the numerical oscillation of falseness by introducing slope limiter.And pressure correcting coefficient C‑apIntroducing, the influence of free gas, hole to pressure is successfully considered, and eliminate the non-genuine pulse near hole stream pressure peak, so as to more accurately simulate the cavitation in water delivery pipeline system.
Description
Technical Field
The invention relates to a finite volume method-based simulation method for cavity flow in a water pipeline, and belongs to the technical field of hydraulic numerical simulation calculation of hydropower stations (pump stations).
Background
When the pressure in the water piping system is lower than the vaporization pressure of the liquid, the liquid may vaporize to form cavitation bubbles. These cavitation bubbles may be uniformly distributed in the conduit, forming a homogeneous flow of cavities; or may be gathered and combined into one or more large cavities, and water column separation occurs; even two phenomena exist in the system at the same time. The generation of cavitation bubbles makes the transient response of the system more complex and dangerous, and many scholars indicate that the column of separation fluid may produce short-term high-pressure pulses above the Joukowsky pressure after remaking, resulting in hydraulic system damage.
At present, for cavitation phenomena in water pipelines, simulation methods thereof are mainly discrete steam-cavity models (DVCM) proposed based on a characteristic line Method (MOC), and discrete gas-cavity models (DGCM) considering a small amount of free gas. The MOC is widely applied in engineering calculation because of its simple calculation and better prediction of the pressure after the first collapse of the cavity. However, in the classical MOC-DVCM model, when a fine grid is adopted, collapse of multiple cavities can cause unrealistic pressure pulses to appear in the calculation result. Although the introduction of free gas (MOC-DGCM) suppressed this pulse to some extent, the computational accuracy did not improve much. Furthermore, the solution of this type of method is limited to first order accuracy due to the limitations of MOC itself. However, the existing method for obtaining a high-order precision solution is too complex in algorithm (e.g., homogeneous continuous model proposed by Chaiko based on FVM method) on one hand, and too long in computation time (e.g., two-dimensional CFD method proposed by Wang Huan, etc.) on the other hand. Zhou Ling and the like are based on a DVCM (dynamic virtual machine) model proposed by a one-dimensional FVM (fuzzy finite element modeling) method, although compared with an MOC-DVCM model, unreal pulses are obviously restrained and second-order accuracy is achieved, the result still fluctuates to a certain degree near the highest pressure. Therefore, further methods to suppress the spurious pulses are still necessary.
Disclosure of Invention
The purpose of the invention is as follows: in order to overcome the defects of false oscillation, low calculation precision and the like in the prior art when the cavity flow in the water pipeline is simulated, the invention provides a simulation method which is simple in algorithm and easy to realize by introducing free gas based on FVM (fuzzy utility model), so that unreal pulses do not appear and a second-order precision solution is obtained when a fine grid is adopted.
The technical scheme is as follows: a finite volume method-based simulation method (FVM-DGCM for short) for hole flow in a water pipeline comprises the following specific steps:
step 1: constructing a transient flow basic control equation containing free gas under an FVM system, and determining a calculation domain, an initial condition and a boundary condition according to an engineering example;
and 2, step: dividing a computational grid according to the FVM, and establishing a discrete equation;
and 3, step 3: calculating flux at the interface of the control unit during pure convection by adopting a Godunov format with time and space both having second-order precision;
and 4, step 4: introducing a source term by a time operator splitting method based on a second-order Runge-Kutta discrete format to obtain a second-order explicit FVM-Godunov format of a discrete equation solution;
and 5: the pressure is corrected and the cavitation volume is calculated based on the calculated pressure.
Further, in step 1, under the FVM system, a transient flow basic control equation containing free gas is constructed, and it is assumed on the basis of the classical water hammer problem: (a) The content of the introduced free gas is determined and follows the isothermal change rule of the ideal gas; (b) The free gas is concentrated in the center of the control unit, and the rest part is still pure water; (c) In order to calculate the cavitation volume, each control unit is equally divided into two small control bodies; (d) The pressure and flow in each aliquot are the solution to the pure water hammer problem under FVM, and the influence of free gas and cavities on pressure is considered by introducing a pressure correction coefficient; (e) The amount of change in cavitation volume can be calculated from the amount of change in flow or pressure within the aliquot.
Further, in step 1, the basic differential equation of the classical water hammer is:
wherein, along the pipeline, the distance x and the time t are independent variables; h (x, t) is the piezometric tube head; v (x, t) is the average cross-sectional velocity; g is the acceleration of gravity; a is the wave velocity; f is the Darcy-Weisbaha friction coefficient; d is the pipe diameter.
Further, in step 1, the free gas follows the ideal gas isothermal variation law:
wherein T is temperature, M g For free gas mass, R g Is a constant of the gas and is,is the volume of the gas-water mixture,as a standard condition(ii) a gas void fraction;is the absolute value of the gas pressure.
Further, in step 1, the continuity equation for the cavitation volume is:
wherein,is the cavitation volume, Q out And Q in Respectively the flow on the upstream and downstream sides of the cavitation volume.
Further, in step 2, the spatial domain x is discretized into N s Control units of length 2 deltax and time domain t discrete at intervals deltat, the free gas being concentrated in the centre of each control unit. For the jth control unit, the serial numbers of the upstream and downstream interfaces are defined as j-1/2 and j +1/2 respectively.
Further advance toStep 2, in order to consider the influence of free gas on transient flow and calculate the cavitation volume, an equivalent grid system is constructed: each control unit j is equally divided into 2 small control volumes i, i +1 in the spatial domain, and the time domain is the same as the original grid. The length of each small control body is delta x, and the grid number of the equivalent grid system is N = N s . For the ith small control body, the serial numbers of the upstream and downstream interfaces are defined as i-1/2 and i +1/2 respectively.
Further, in step 2, on the basis of the FVM grid system containing free gas, the step of establishing the discrete equation is:
(a) The riemann problem of differential equations (1) (2) can be approximated as the riemann problem of a linear hyperbolic system with constant coefficients:
wherein Is the average value of V, which is a constant.
(b) Integrating equation (5) over small control volume i (from interface i-1/2 to i + 1/2) and time period Δ t (from t to t + Δ t) to obtain a discrete equation for flow variable u:
wherein the superscripts n and n +1 represent t and t + Δ t time steps, respectively.Is the average value of u over the entire control volume; f is the flux at the interface.
Further, in step 3, the step of calculating the flux at the interface when pure convection is:
(a) Calculating flux at internal control unit interface
According to the Godunov method, when pure convection is carried out, for any control body i (1 < -i and N), the flux at the interface i +1/2 is as follows:
wherein,the average value of u on the left side and the right side of the interface i +1/2 respectively in n time steps.Andthe calculation method of (2) determines the calculation accuracy. In the invention, the spatial and temporal second-order precision is obtained by introducing a MUSCL (monomer Upstream-centered Scheme for consistency rows) Handock format, and a MINMOD limiter is introduced to ensure that false oscillation does not occur in the solution.
(b) Calculating flux at boundary control unit interface
To obtain second-order accuracy, two virtual controllers I are constructed on the upstream side of the start controller 1 and the downstream side of the end controller N -1 、I 0 And I N+1 、I N+2 And assume that the flow information at the virtual volume is consistent with the boundary. The boundary Riemann problem can thus be solved, and the corresponding Godunov flux f 1/2 And f N+1/2 The calculation can also be done like an internal unit.
Furthermore, in step 3, the convection term of pure convection should satisfy the CFL (Courant-Friedrichs-Lewy criterion) condition.
Further, in step 4, after the source term is introduced, the second-order FVM-Godunov format of the transient flow basic differential equation solution containing the free gas is as follows:
wherein,in the time step of n +1, the flux of a flow variable u is controlled by a control unit i when the flow is pure convection;the flux after the first update using time-splitting.
Further, in step 4, the riemann problem solved by the present invention ignores the convection term, so the caudalus number satisfies the CFL condition.
Further, in step 5, it is determined which method to correct the pressure and calculate the cavitation volume based on the calculated pressure: if the pressure obtained in the step 4 is lower than the liquid vaporization pressure, correcting the pressure by adopting a method I and calculating the cavitation volume; otherwise, calculating the cavitation volume and correcting the pressure by adopting a method II; two methods for correcting the pressure and calculating the cavitation volume are:
(a) Method I, pressure above vaporization pressure:
to any control unit j (1)<j<N s ) And the pressure is calculated according to a second-order FVM-Godunov format solution calculated by two corresponding equal division control bodies i and i +1 in the equivalent grid system:
wherein,which represents the pressure of the control unit j,respectively representing the pressure of the control bodies i and i + 1;
substituting equation (9) into equation (3) allows calculation of the cavitation volume:
where ρ is l Represents the liquid density; z (j) represents the elevation of the location at control unit j; h v Indicating the fluid vaporization pressure.
In order to take into account the influence of the free gas on the partial control bodies, the pressure in the partial control bodies needs to be corrected. For this purpose, a pressure correction factor C is introduced -ap To obtain a corrected pressure in the aliquot control:
wherein, C is more than or equal to 0 -ap ≤1：C -ap =0 means that the pressure in both aliquots is corrected according to the cavitation pressure; c -ap By =1 is meant that the pressure within the two aliquots is not corrected, i.e. the effect of cavitation is neglected.
(2) Process II, pressure equal to or less than the vaporization pressure:
once the pressure within any one aliquot is equal to or less than the liquid vaporization pressure, the liquid is considered to vaporize forming vapor pockets. At this time, the air pocket volume is calculated from the air pocket volume continuity equation (4)). In the discrete format of FVM, the expression is as follows:
wherein,the flow rates of the upper and lower sides of the cavity in the control unit j are respectively;indicating the gas cavity volume at time step n within the control unit j.
After considering the influence of the cavity on the pressure in the equal-division control body, the pressure in the equal-division control body is corrected into:
has the beneficial effects that: compared with the prior art, the invention has the following advantages:
(1) The FVM-DGCM model provided by the invention successfully overcomes the difficult problems of a one-dimensional FVM method in the aspects of considering free gas, tracking cavitation and predicting development-collapse of the gas, obtains second-order precision, and is simple and easy to realize as MOC methods; (2) Coefficient of pressure correction C -ap The introduction of (2) successfully considers the influence of free gas and holes on pressure, and is the key for realizing a DGCM model by adopting FVM; (3) Coefficient of pressure correction C -ap The reasonable value of the method ensures that no artificial pulse peak value appears in the result regardless of a sparse grid or a fine grid; (4) In the limit case C -ap In the case of =1, the model ignores the influence of free gas and can be considered as FVM-DVCM, so that the model is more adaptive; (5) The model can also simulate the transient flow containing gas without generating a cavity, and the application range of the model is wider.
Drawings
FIG. 1 is a basic flow diagram of the present invention;
FIG. 2 is a diagram of an initial grid discretization system under the FVM of the embodiment;
FIG. 3 is a diagram of an equivalent grid discretization system under the FVM of the embodiment;
FIG. 4 shows the pressure correction coefficient C of the embodiment -ap Pressure profile at the end of the valve when = 1.0;
FIG. 5 shows the pressure correction coefficient C of the embodiment -ap Pressure profile at the end of the valve when = 0.9;
FIG. 6 shows the pressure correction coefficient C of the example -ap Pressure profile at the end of the valve when = 0.5;
in fig. 2: 1-free gas.
Detailed Description
The present invention is further illustrated by the following examples, which are intended to be purely exemplary and are not intended to limit the scope of the invention, as various equivalent modifications of the invention will occur to those skilled in the art upon reading the present disclosure and fall within the scope of the appended claims.
The method for simulating the cavity flow in the water conveying pipeline based on the finite volume method comprises the following steps of: 1. constructing a transient flow basic control equation containing free gas under an FVM system, and determining a calculation domain, an initial condition and a boundary condition according to an engineering example; 2. dividing a computational grid according to the FVM, and establishing a discrete equation; 3. calculating flux at the interface of the control unit during pure convection by adopting a Godunov format with time and space both having second-order precision; 4. introducing a source term by a time operator splitting method based on a second-order Runge-Kutta discrete format to obtain a second-order explicit FVM-Godunov format of a discrete equation solution; 5. the pressure is corrected and the cavitation volume is calculated based on the calculated pressure. Figure 1 shows a basic flow diagram of the present invention.
The technical solution of the present invention will be further described in detail with reference to the accompanying drawings and examples.
In the embodiment, in order to verify and analyze the simulation effect of the FVM-DGCM simulated hole flow provided by the invention, a water column separation experimental device system designed and established by Simpson in 1986 is selected for verifying the effectiveness of the method. The whole system consists of an upstream pressure water tank, a copper pipe, a downstream ball valve and a downstream pressure water tank. The total length of the copper pipe is 36m, the inner diameter is 19.05mm, and the slope of the pipe is 1/36. The water hammer wave speed is 1280m/s. The cavitation flow transient is caused by the sudden closure of the downstream ball valve. The experimental working condition parameters are as follows: the initial flow rate is 0.332m/s, the static pressure head of the upstream reservoir is 23.41m, the on-way head loss is 0.33m, the water temperature is 24.4 ℃, and the valve closing time is 0.022s.
The method comprises the following specific steps:
step 1: and under an FVM system, constructing a transient flow basic control equation containing free gas, and determining a calculation domain, initial conditions and boundary conditions according to the engineering example.
(a) Establishing a fundamental governing equation
Transient elementary differential equation:
wherein, along the pipeline, the distance x and the time t are independent variables; h (x, t) is the piezometer tube head; v (x, t) is the average cross-sectional velocity; g is the acceleration of gravity; a is the wave velocity; f is the Darcy-Weisbaha friction coefficient; d is the pipe diameter.
Free gas isothermal change state equation:
wherein T is temperature, M g For free gas mass, R g Is a constant of the gas, and is,is the volume of the gas-water mixture,as a standard conditionThe porosity of the gas in the following range,is the absolute value of the gas pressure. This embodiment takes down the MOC-DGCM proposed value
Continuity equation for air pocket volume:
wherein,is the volume of the air pocket, Q out And Q in Respectively the flow on the upstream and downstream sides of the cavitation volume.
(b) Determining a computational domain, initial conditions and boundary conditions
Calculating a domain: a pipeline from the water outlet of the upstream reservoir to the valve;
initial conditions were as follows: when the ball valve is fully opened, the initial flow rate is 0.332m/s, and the initial pressure of each node is obtained by subtracting the corresponding steady-state friction loss from the static pressure head of the upstream reservoir;
the boundary conditions are as follows: at the inlet of the pipeline, the reservoir provides a constant pressure boundary which is equal to the static pressure water head of the upstream reservoir; the downstream ball valve closes quickly causing a cavitation flow transient.
Step 2: and dividing the computational grid according to the FVM, and establishing a discrete equation.
Discretizing spatial domain x into N s The length of each control unit is 2 delta x, the time domain t is discrete to control units with the interval delta t, and the free gas is concentrated at the center of each control unit. For the jth control unit, the serial numbers of the upstream interface and the downstream interface of the jth control unit are respectively defined as j-1/2 and j +1/2. Fig. 2 is a diagram of an initial grid discrete system based on the FVM in this embodiment.
To account for the effect of free gas on transient flows and calculate cavitation volume, an equivalent grid system was constructed (see fig. 3): each control unit j is equally divided into 2 small control volumes i, i +1 in the spatial domain, and the time domain is the same as the original grid.The length of each small control body is delta x, and the grid number of the equivalent grid system is N = N s . For the ith small control body, the serial numbers of the upstream and downstream interfaces are defined as i-1/2 and i +1/2 respectively.
On the basis of the FVM grid system containing free gas, the specific steps of establishing a discrete equation are as follows:
(a) The riemann problem of differential equations (1) (2) can be approximated as the riemann problem of a linear hyperbolic system with constant coefficients:
wherein Is the average value of V, which is a constant.
(b) Integrating equation (5) over small control volume i (from interface i-1/2 to i + 1/2) and time period Δ t (from t to t + Δ t) to obtain a discrete equation for flow variable u:
wherein the superscripts n and n +1 represent t and t + Δ t time steps, respectively.Is the average value of u over the entire control volume; f is the flux at the cell interface.
And step 3: and calculating the flux at the interface of the control unit in pure convection by adopting a Godunov format with space and time both having second-order precision.
Further, step 3 comprises the following substeps:
step 3.1: flux at the internal control unit interface when calculating pure convection
According to the Godunov method, in pure convection, for any control body i (1 yarn i and yarn n), the flux at the interface i +1/2 is:
wherein,the average value of u on the left side and the right side of the interface i +1/2 respectively in n time steps.Andthe calculation method of (2) determines the calculation accuracy. In the invention, the spatial and temporal second-order precision is obtained by introducing a MUSCL (monomer Upstream-centered Scheme for maintaining Laws) Hancock format, and a MINMOD limiter is introduced to ensure that no false oscillation occurs in the solution.
Step 3.2: boundary control unit interface flux when calculating pure convection
To obtain second-order accuracy, two virtual controllers I are constructed on the upstream side of the start controller 1 and the downstream side of the end controller N -1 、I 0 And I N+1 、I N+2 And assume that the flow information at the virtual volume is consistent with the boundary. So that the boundary Riemann problem can be solved, and the corresponding Godunov flux f 1/2 And f N+1/2 The calculation can also be done like an internal unit.
In this embodiment, two boundary conditions are included:
(a) Constant water level boundary of upstream reservoir:
at the upstream boundary, the riemann invariants associated with the negative feature line are:
wherein H 1/2 ＝H res ，H res Is the upstream reservoir static pressure water head.
Thereby can be pushedVariable u 1/2 (t)＝(H 1/2 ,V 1/2 ). Virtual unit I adjacent to the inlet of the duct according to the assumptions at the virtual unit -1 And I 0 The corresponding values of (a) are:
(b) Downstream valve boundary:
at the downstream boundary, the riemann invariants associated with the positive feature line are:
the downstream boundary conditions are: valve closing time T c And (4) internal closing. The relationship between head and flow rate is:
V N+1/2 ＝0,t<T c (12)
wherein H D Is the outlet pressure head, ξ v As loss coefficient at valve, variable u N+1/2 (t)＝(H N+1/2 ,V N+1/2 ). Proximity I according to assumptions at the virtual cell N Virtual unit I of N+1 And I N+2 The corresponding values of (a) are:
note that in step 3, the convection term should satisfy the CFL condition (Courant-Friedrichs-Lewy criterion):
C r ≤1 (14)
wherein, C r Is the Korotkang number.
And 4, step 4: and introducing a source term by a time operator splitting method based on a second-order Runge-Kutta discrete format to obtain a second-order explicit FVM-Godunov format of a discrete equation solution.
The specific implementation process is as follows:
(a) Pure convection:
(b) Update with the source term multiplied by Δ t/2:
(c) Update again with the source term multiplied by Δ t:
in addition, the Riemann problem involved in the present invention ignores the convection term, so the Korotkoff number satisfies the CFL condition.
And 5: and determining which method to correct the pressure and calculate the cavitation volume according to the calculated pressure: if the pressure obtained in the step 4 is lower than the liquid vaporization pressure, correcting the pressure by adopting a method I and calculating the cavitation volume; otherwise, method II is used to calculate the cavitation volume and correct the pressure.
The two methods are respectively as follows:
(a) Method I, pressure above vaporization pressure:
for any control unit j (1)<j<N s ) And the pressure is calculated according to a second-order FVM-Godunov format solution calculated by two corresponding equal division control bodies i and i +1 in the equivalent grid system:
wherein,which represents the pressure of the control unit j,respectively representing the pressure of the control bodies i and i + 1;
substituting equation (9) into equation (3) allows calculation of the cavitation volume:
where ρ is l Represents the liquid density; z (j) represents the elevation of the location at control unit j; h v Indicating the fluid vaporization pressure.
In order to take into account the influence of the free gas on the partial control bodies, the pressure in the partial control bodies needs to be corrected. For this purpose, a pressure correction factor C is introduced -ap To obtain a corrected pressure in the aliquot control:
wherein, C is more than or equal to 0 -ap ≤1：C -ap =0 means that the pressure in both aliquots is corrected according to the cavitation pressure; c -ap =1 means that the pressure in the two aliquots is not corrected, i.e. the effect of cavitation is neglected.
(b) Process II, pressure equal to or less than the vaporization pressure:
once the pressure within any one of the aliquots is equal to or less than the liquid vaporization pressure, the liquid is considered to be vaporized to form a vapor cavity. At this time, the air pocket volume is calculated from the air pocket volume continuity equation (4)). In the discrete format of FVM, the expression is as follows:
wherein,the flow rates of the upper and lower sides of the cavity in the control unit j are respectively;indicating the cavitation volume at the nth time step in control unit j. .
After considering the effect of the cavity on the partial control body pressure, the partial control body pressure is corrected to be:
after the calculation is programmed by the method, the calculation result of the FVM-DGCM is compared with experimental data. Fig. 4 to 6 show the mesh number Ns = {32,256} and the pressure correction coefficient C in this embodiment, respectively -ap Pressure profile at valve when =1.0,0.9,0.5. It can be seen that the pressure correction coefficient C -ap The value of (a) has a great influence on the non-real pressure pulse in the FVM-DGCM result. Reasonable selection of C -ap The value of (2) can effectively inhibit the unreal pressure pulse no matter how many grids are, but too low value can cause too large pressure attenuation. Therefore, the specific gravity among the unreal pressure pulse, the pressure amplitude and the time response must be balanced and considered, and C which is closest to the experimental result and has the least false oscillation is selected -ap The value is obtained. Taken together, in this example, C -ap =0.9 and under a fine grid, the results are optimal. Aiming at the phenomenon of the cavity flow in the water pipeline, compared with an MOC method and an FVM-DVCM, the invention providesThe FVM-DGCM method can obtain more accurate and stable results under a fine grid.
Claims (3)
1. A simulation method of hole flow in a water conveying pipeline based on a finite volume method is characterized in that a Godunov format of a second-order finite volume method is adopted to simulate the hole flow phenomenon in the water conveying pipeline system, and the method comprises the following specific steps:
step 1: constructing a transient flow basic control equation containing free gas under an FVM system, and determining a calculation domain, an initial condition and a boundary condition according to an engineering example;
step 2: dividing a computational grid according to the FVM, and establishing a discrete equation;
and step 3: calculating flux at the interface of the control unit during pure convection by adopting a Godunov format with time and space both having second-order precision;
and 4, step 4: introducing a source term by a time splitting method based on a second-order Runge-Kutta discrete format to obtain a second-order explicit FVM-Godunov format of a discrete equation solution;
and 5: correcting the pressure according to the calculated pressure and calculating the cavitation volume;
in the step 1, the constructed transient flow basic control equation containing the free gas comprises the following steps:
(1) Basic differential equation of water hammer:
wherein, along the pipeline, the distance x and the time t are independent variables; h (x, t) is the piezometer tube head; v (x, t) is the average cross-sectional velocity; g is the acceleration of gravity; a is the wave velocity; f is the darcy-weisbaha friction coefficient; d is the pipe diameter;
(2) Free gas isothermal change state equation:
wherein T is temperature, M g For free gas mass, R g Is a constant of the gas and is,is the volume of the gas-water mixture,is a standard condition(ii) a gas void fraction;is the absolute value of the gas pressure;
(3) Continuity equation for air pocket volume:
wherein,is the volume of the air pocket, Q out And Q in Flow rates on the upstream and downstream sides of the cavitation volume, respectively;
in step 2, the method for establishing the computational grid containing the free gas under the FVM method comprises the following steps:
(a) Establishing an initial grid:
discretizing spatial domain x into N s The control units are 2 delta x in length and discrete in a time domain t into intervals delta t, and free gas is concentrated in the center of each control unit; defining the serial numbers of the upstream and downstream interfaces of the jth control unit as j-1/2 and j +1/2 respectively;
(b) Establishing an equivalent grid:
each control unit j is arranged in spaceThe time domain is the same as the original grid; the length of each small control body is delta x, and the grid number of the equivalent grid system is N = N s (ii) a For the ith small control body, the serial numbers of the upstream and downstream interfaces are respectively defined as i-1/2 and i +1/2;
in step 2, on the basis of the FVM grid system containing free gas, the steps of establishing a discrete equation are as follows:
(a) The riemann problem of differential equations (1) (2) can be approximated as the riemann problem of a linear hyperbolic system with constant coefficients:
wherein Is the average value of V, which is a constant;
(b) Integrating equation (5) over small control volume i (from interface i-1/2 to i + 1/2) and time period Δ t (from t to t + Δ t) to obtain a discrete equation for flow variable u:
wherein, superscripts n and n +1 represent t and t + Δ t time steps, respectively;is the average value of u over the entire control volume; f is the flux at the interface;
in step 3, the step of calculating the flux at the interface during pure convection is as follows:
(a) Calculating flux at internal control unit interface
According to the Godunov method, in pure convection, for any control body i (1 yarn i and yarn n), the flux at the interface i +1/2 is:
wherein,the average value of u to the left side and the right side of the interface i +1/2 respectively in the time step of n;andthe calculation method of (2) determines the calculation accuracy; second-order precision in space and time is obtained by introducing a MUSCL Hancock format, and meanwhile, a MINMOD limiter is introduced to ensure that false oscillation does not occur in the solution;
(b) Calculating flux at boundary control unit interface
To obtain second-order accuracy, two virtual controllers I are constructed on the upstream side of the start controller 1 and the downstream side of the end controller N -1 、I 0 And I N+1 、I N+2 And assuming that the flow information at the virtual volume is consistent with the boundary; the boundary Riemann problem can thus be solved, and the corresponding Godunov flux f 1/2 And f N+1/2 The calculation can also be done like an internal unit;
in step 5, according to the calculated pressure, determining which method to correct the pressure and calculating the cavitation volume: if the pressure obtained in the step 4 is lower than the liquid vaporization pressure, correcting the pressure by adopting a method I and calculating the cavitation volume; otherwise, calculating the cavitation volume and correcting the pressure by adopting a method II; two methods for correcting the pressure and calculating the cavitation volume are:
(a) Method I, pressure above vaporization pressure:
to any control unit j (1)<j<N s ) And the pressure of the two-order FVM-Goduno is calculated according to the two corresponding equal control bodies i and i +1 in the equivalent grid systemv-format solution calculation:
wherein,which represents the pressure of the control unit j,respectively representing the pressure of the control bodies i and i + 1;
substituting equation (9) into equation (3) allows calculation of the cavitation volume:
wherein ρ l Represents the liquid density; z (j) represents the elevation of the location at control unit j; h v Representing the fluid vaporization pressure;
in order to consider the influence of free gas on the equal-component control bodies, the pressure in the equal-component control bodies needs to be corrected; for this purpose, a pressure correction factor C is introduced -ap To obtain a corrected pressure in the aliquot control:
wherein, C is more than or equal to 0 -ap ≤1：C -ap =0 means that the pressure in both aliquots is corrected according to the cavitation pressure; c -ap =1 means that the pressure in the two aliquots is not corrected, i.e. the effect of cavitation is neglected;
(b) Process II, pressure equal to or less than the vaporization pressure:
once the pressure within any one of the aliquots is equal to or less than the liquid vaporization pressure, the liquid is considered to vaporize forming a vapor cavity; at this time, the air pocket volume is calculated from the air pocket volume continuity equation (4)); in the discrete format of FVM, the expression is as follows:
wherein,the flow rates of the upstream side and the downstream side of the cavity in the control unit j are respectively;represents the cavitation volume at time step n within control unit j;
after considering the influence of the cavity on the pressure in the equal-division control body, the pressure in the equal-division control body is corrected into:
2. the finite volume method-based simulation method for cavity flow in a water pipeline according to claim 1, wherein in step 1, a transient flow basic control equation containing free gas is constructed under an FVM system, and it is assumed that: (a) The content of the introduced free gas is determined and follows the isothermal change rule of the ideal gas; (b) The free gas is concentrated in the center of the control unit, and the rest part is still pure water; (c) In order to calculate the air cavity volume, each control unit is equally divided into two small control bodies; (d) The pressure and flow in each aliquot are the solution to the pure water hammer problem under FVM, and the influence of free gas on pressure is considered by introducing a pressure correction coefficient; (e) The amount of change in cavitation volume can be calculated from the amount of change in flow or pressure within the aliquot.
3. The finite volume method-based simulation method of the cavity flow in the water conveying pipeline according to claim 1, wherein in the step 4, after the source term is introduced, the second-order FVM-Godunov format of the solution of the fundamental differential equation of the transient flow containing the free gas is as follows:
wherein,in the time step of n +1, the flux of a flow variable u is controlled by a control unit i when the flow is pure convection;the flux after the first update using time-splitting.
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