An improved hybrid flow simulation method considering large multi-shaft pipelines

Abstract

The present invention discloses an improved mixed-flow simulation method considering large multi-shaft pipelines, proposes an improved mixed-flow model, combines the two-phase component method of water bodies with discrete gas, and solves it by combining the random selection method and the Godunov-type finite volume method. Considering the applicability of the actual multi-shaft layout and large-scale engineering cases, it can improve the accuracy, stability and practical applicability of mixed-flow simulation, accurately and quantitatively characterize the dynamic pressure characteristics of the water-air two-phase flow in mixed-flow simulation, and detailedly reveal the influence effects of multi-shaft layout and scale effect on mixed-flow simulation, which is of great significance for the mixed-flow simulation and hydraulic safety guidance of actual engineering considering the complex layout of large multi-shafts.

Description

Technical Field

[0001] The present invention relates to the technical field of numerical simulation calculation of hydraulics in hydropower stations (pumping stations), and particularly relates to an improved mixed-flow simulation method considering large multi-vertical shaft pipelines. Background Art

[0002] In recent years, in large rainwater drainage systems and water diversion and regulation projects, due to heavy rainfall or a large inflow from upstream, rapid flow conversions often occur. The characteristics of this flow regime are the initial compression of long-distance air bags and subsequent irregular intermittent flow regime transitions. Due to the requirements of actual engineering layouts, these pipeline systems involve multi-vertical shaft gas discharge problems. These complex flows pose a great threat to the safe operation of the pipeline system, so it is very necessary to develop corresponding mixed-flow models.

[0003] In current research, one-dimensional and three-dimensional numerical methods for mixed-flow simulation have been developed. Considering that large pipelines require a large number of grid cells and long simulation times, the high computational requirements of three-dimensional methods make them impractical. In contrast, one-dimensional methods are more suitable for large pipelines, including the rigid column method, shock capturing method, and shock fitting method. In the one-dimensional shock capturing method, the two-component pressure method (TPA) effectively solves the mixed-flow simulation problem without tracking the air-water interface and is widely used. Therefore, this model is also applied to the water part of the present invention.

[0004] In addition, to solve the problem of trapped air bags in mixed flows, the two-component pressure method needs to be combined with an air model. The air model is based on the ideal gas law and mainly includes the uniform air head model and the discrete air model. Traditional methods: 1. mainly the uniform air head model, simply assuming that the air pressure in a single air bag is uniform, cannot accurately consider the problem of trapped air bags in it; 2. in addition, traditional methods ignore the mutual influence between actual multi-vertical shafts, cannot consider the staged characteristics of gas discharge during the mixed-flow process, and have poor characterization ability for the transient coupling effects of the water-gas two-phase pressure, density, and flow velocity of the gas-liquid two-phase mixed flow; 3. ignore the size effect of large-distance air bags between large-scale prototypes and models, and lack quantitative research and verification on the applicability of the water-gas two-phase mixed-flow model in actual large projects; 4. traditional solution methods have inherent defects of low resolution and poor conservation for shock capturing and wave speed adjustment of multiphase and transient flows.

[0005] Therefore, the present invention aims to solve these gaps by establishing an improved mixed-flow model. This model combines the two-component pressure method for water part simulation and the discrete Euler method for gas part. The numerical solution is the Godunov-type finite volume method, combined with the random choice method (RCM) to improve numerical stability. The applicability in large air bags is further emphasized, and the problems of multi-vertical shafts and size effects are analyzed. Summary of the Invention

[0006] The purpose of this section is to outline some aspects of the embodiments of the present invention and briefly introduce some preferred embodiments. Some simplifications or omissions may be made in this section, as well as in the abstract and title of the present application, to avoid obscuring the purpose of this section, the abstract, and the title. However, such simplifications or omissions shall not be used to limit the scope of the present invention.

[0007] In view of the existing problems in the gas-liquid two-phase transient flow of the water delivery system, such as solving the problems of dynamic coupling of gas and fluid transient parameters, multi-vertical shaft gas discharge, system size effect, and low reproduction accuracy and poor stability of traditional models, the present invention is proposed.

[0008] Therefore, the technical problems solved by the present invention are as follows: improving the mixed flow model, combining the two-phase component method of water body and discrete gas, and using the random selection method and the Godunov-type finite volume method to solve them. Considering the applicability of the actual multi-vertical shaft layout and large-scale engineering cases, the accuracy, stability, and practical applicability of the mixed flow simulation can be improved, the dynamic pressure characteristics of the water-gas two-phase flow in the mixed flow simulation can be accurately quantified, and the influence effects of the multi-vertical shaft layout and scale effect on the mixed flow simulation can be detailedly revealed.

[0009] To solve the above technical problems, the present invention provides the following technical solutions: an improved mixed flow simulation method considering large multi-vertical shaft pipelines, including dynamically coupling gas and fluid transient parameters, quantitatively analyzing multi-vertical shaft gas discharge and system size effect, and improving the reproduction accuracy and stability of the model. The specific steps are as follows: S1: Construct a two-phase component model of the water body and calculate the mass and momentum fluxes of the pipeline main body; S2: Dynamically couple the discrete gas model and combine the discrete gas Euler method; S3: Introduce the random selection method and the Godunov-type finite volume method to achieve accurate and stable solution of the model; S4: Process the calculation results and verify the prediction results of the proposed model with experimental data; S5: Quantitatively consider the influence effects of multi-vertical shaft gas discharge and system scale.

[0010] Among them, a two-phase component simulation model is established for the water body part:

[0011]

[0012] In the formula:

[0013]

[0014]

[0015]

[0016]

[0017] Among them, U is the vector of conserved variables; F is the vector of conserved variable fluxes; S is the vector of source terms; x is the distance along the pipe axis; t is the time; A is the flow cross-sectional area; Q is the flow rate; g is the acceleration due to gravity; h u is the distance between the centroid of the cross-section and the free surface; h p is the additional head; h air is the pressurized air head; θ = π - arccos[(y - D / 2)D / 2]; D is the pipe diameter; A pipe is the cross-sectional area; a is the wave speed in the pressurized flow;

[0018] Among them, for the dynamic coupling gas model, a discrete Euler gas model is established:

[0019]

[0020]

[0021] Among them, is the air density; u is the air flow velocity; p is the air pressure head, p = ρα 2 ; α is the speed of sound waves in air; S d1 、S d2 and S fa are the source terms in the continuity equation, the momentum equation, and the loss between the air flow and the pipe wall, respectively.

[0022] As a preferred solution of the present invention, among them: States are assigned to the next time level based on a random sampling process:

[0023]

[0024] Among them, the superscript n represents the nth calculation time step; the subscript i represents the ith calculation control volume; β n is a random number. Due to the advantages of low discrepancy and uniform distribution, the van der Corput sequence of pseudorandom numbers is recommended as the best random sequence in the sampling process. The van der Corput sequence is expressed as:

[0025]

[0026] A i = k 2 a i (mod k 1 ) (9)

[0027]

[0028] Among them, k 1 and k 2 are two relatively prime numbers, k 1 > k 2 > 0; mod represents the remainder function; m, ai , A i are all intermediate generated parameters.

[0029] Among them, using the Godunov-type finite volume method, an accurate Riemann solver is adopted to handle the discontinuous problem:

[0030] f(A * ) ≡ f L (A * , A L ) + f R (A * , A R ) + u R - u L = 0

[0031]

[0032] Among them, u R and u L are the velocity variables on the left and right sides of the node respectively; f K (K = L or R) is the expression that establishes the relationship between the two sides and the intermediate state through the left or right non-linear wave; the superscript * is the solution value in the discontinuous region (i.e., the star region) of the Riemann problem; A K (K = L or R) is the area in the fluxes on the left and right sides; I 1 is the integral form of the pressure term for an interface of any shape; is a function of the area and can be calculated through existing literature.

[0033] Among them, the expression of the flow rate Q * in the star region of the Riemann problem:

[0034]

[0035] Among them, the solution A * of the flow area at the discontinuity is obtained by iteration using the Newton-Raphson method:

[0036]

[0037] Among them, f' represents the derivative of the flux.

[0038] As a preferred solution of the present invention, among them, considering the influence of gas release from multiple vertical shafts: the boundary conditions for vertical shaft exhaust apply the continuity equation and the orifice equation, and the continuity equation of this boundary condition is:

[0039]

[0040] Among them, the subscript air represents the corresponding variables of the gas; the subscript 1 represents the first computational control volume at the boundary; Mairout represents the mass of air discharged through the ventilation hole, which is calculated using Equation (15) and solved by using Riemann invariants to solve the isothermal, one-dimensional, and primitive-form Euler equations.

[0041]

[0042] Among them, the subscript 2 represents the second control volume adjacent to the first computational control volume.

[0043] Quantitatively considering the size effect of the system and designing according to the same Froude number Fr of the original model, we have:

[0044]

[0045]

[0046]

[0047]

[0048] Among them, Fr is the Froude number; U represents the water body flow velocity; g is the acceleration due to gravity; L is the system length; λ represents the similarity scale of each variable prototype compared to the model; Q is the flow rate; p is the pressure; H is the pressure head; ρ is the density; the subscript Y represents the prototype, the subscript M represents the model, the subscript u represents the velocity variable, the subscript l represents the length variable, the subscript t represents the time variable, the subscript r represents the characteristic physical quantity, the subscript Q represents the flow rate variable, and the subscript H represents the head variable.

[0049] Advantages of the present invention:

[0050] 1. Establish a high-precision one-dimensional multiphase model, combine the two-phase component water body simulation method and the discrete gas Euler simulation method to simulate the water-air two-phase transient flow, and can achieve accurate simulation of the mixed flow in the water conveyance pipeline.

[0051] 2. Importantly, the influence of the multi-shaft layout on the water-filling two-phase flow and the influence of the system size effect on the model applicability are carefully explored.

[0052] 3. In addition, combining the random selection method with the Godunov-type finite volume method can achieve accurate and stable solution of the model, which is beneficial to ensuring the hydraulic safety prediction, operation scheduling, and prevention and control in the water conveyance pipeline. Description of the Drawings

[0053] To more clearly illustrate the technical solutions of the embodiments of the present invention, the following will briefly introduce the accompanying drawings required for the description of the embodiments. Obviously, the accompanying drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other accompanying drawings can be obtained based on these drawings. Among them:

[0054] Figure 1 It is a flowchart for implementation provided by the present invention.

[0055] Figure 2 It is a schematic diagram of the experimental system provided by the present invention.

[0056] Figure 3 It is a comparison diagram of the multi-shaft mixed flow model, the non-aerated model and the experimental results provided by the present invention.

[0057] Figure 4 It is an analysis diagram of the multi-shaft exhaust effect of the model conditions in the multi-shaft mixed flow model provided by the present invention.

[0058] Figure 5 It is an analysis diagram of the scale effect of the large-scale prototype conditions of the multi-shaft mixed flow model provided by the present invention. Specific Embodiments

[0059] To make the above objects, features and advantages of the present invention more obvious and understandable, the following will make a detailed description of the specific embodiments of the present invention in conjunction with the accompanying drawings of the specification. Obviously, the described embodiments are part of the embodiments of the present invention, rather than all embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the scope of protection of the present invention.

[0060] Consider an improved mixed flow simulation method for large multi-shaft pipelines, which is carried out according to the following steps:

[0061] S1. Construct a two-phase component model of the water body, calculate the mass and momentum fluxes of the main body of the pipeline. When calculating the mass and momentum fluxes of the main body of the pipeline, the traditional pressure head is subdivided into the head between the centroid of the cross-section and the free surface, the additional head above the cross-section, and the pressurized air head;

[0062] S2. Dynamically couple the discrete gas model, and separately solve the mass and momentum fluxes of the gas by combining the discrete gas Euler method, where the source terms in the continuity equation, momentum equation, and the loss between the gas flow and the pipe wall are considered simultaneously;

[0063] S3. Introduce the random selection method and the Godunov-type finite volume method to achieve accurate and stable solution of the model. In the random selection method, the states are mainly assigned to the next time level based on the van der Corput numbers through a random sampling process. The Godunov-type finite volume method uses an exact Riemann solver to handle discontinuous problems.

[0064] S4. Process the calculation results and verify the prediction results of the proposed model against the experimental data. The verification cases and results can be found in Figure 2 and Figure 3 ;

[0065] S5. Quantitatively consider the effects of multi-shaft gas emissions and system scale. For the gas emission effect, the continuity equation and the orifice equation are applied. For the system scale effect, the scale conversion relationships of various variables are quantitatively considered.

[0066] Examples:

[0067] To verify and analyze the simulation effect of the improved hybrid flow simulation method considering large multi-shaft pipelines of the present invention, in Example 1, an experimental system as shown in Figure 2 is selected. This experimental system is derived from an actual long-distance large-scale water transfer project, and is a typical local pipeline converted from the main part of the actual pipeline according to a standard ratio of 1:34. In the experimental system, the pipeline length is 2243.82 m, the pipe diameter is 0.3 m, the upstream water level is constant, and there is an initial submerged depth at the downstream end with a constant water level. The remaining part of the pipeline is an initially empty pipe with six vertical shafts having the same diameter as the main pipe. In addition, seven pressure sensors are installed at the intersections of each shaft and the main pipe and at the downstream outlet, all located at the top of the pipeline; four flow sensors are installed in the main pipe, located at Q1(x = 8 m), Q2(x = 790 m), Q3(x = 1598.20 m), and Q4(x = 2446.82 m) respectively. Based on this Figure 2 experimental system, in Example 1, the multi-shaft hybrid flow model of the present invention is used for simulation, and the experimental result data is compared with the traditional gas-free model and the experimental results. The data comparison chart can be seen in Figure 3 .

[0068] In Example 2, the actual large-scale project corresponding to the above case is used to analyze the influence of system size on the model. In the actual project, the pipeline length is 194.3 km, the pipe diameter is 10.2 m, the highest operating water level at the inlet is 173.3 m, and the downstream design water level is 88.3 m. Based on this actual large-scale project, in Example 2, the multi-shaft hybrid flow model of the present invention is used for simulation, and data display and analysis are carried out for the exhaust effect Figure 4 and data display and analysis are carried out for the size effect respectively Figure 5 .

[0069] The specific implementation steps are as follows:

[0070] Step 1:

[0071] Construct a two-phase component simulation model for the water body part and calculate the mass and momentum fluxes of the main pipeline:

[0072]

[0073] In the formula

[0074]

[0075] where U is the vector of conserved variables; F is the vector of conserved variable fluxes; S is the vector of source terms; x is the distance along the pipe axis; t is the time; A is the flow cross-sectional area; Q is the flow rate; g is the acceleration due to gravity; h u is the distance between the centroid of the cross-section and the free surface; h p is the additional head; h air is the pressurized air head; θ = π - arccos[(y - D / 2)D / 2]; D is the pipe diameter; A pipe is the cross-sectional area; a is the wave speed in the pressurized flow.

[0076] Step 2:

[0077] Dynamically couple the gas model to establish a discrete Euler gas model:

[0078]

[0079]

[0080] where ρ is the air density; u is the air velocity; p is the air pressure head, p = ρα 2 ; α is the speed of sound waves in the air; S d1 , S d2 and S fa are the source terms in the continuity equation, momentum equation, and losses between the air flow and the pipe wall, respectively.

[0081] Step 3:

[0082] To achieve accurate and stable solution of the model, introduce the random selection method and the Godunov-type finite volume method: The random selection method is used to solve the discontinuity problem in the interaction of complex waves. Different from the integral averaging in the Godunov method, the random selection method mainly assigns states to the next time level based on the Van der Corput number through a random sampling process:

[0083]

[0084] where the superscript n represents the n-th computational time step; the subscript i represents the i-th computational control volume; β n is a random number. Due to the advantages of low discrepancy and uniform distribution, the van der Corput number pseudo-random sequence is recommended as the best random sequence in the sampling process. The van der Corput sequence is expressed as:

[0085]

[0086] A i = k 2 a i (mod k 1 )(9)

[0087]

[0088] where k 1 and k 2 are two relatively prime numbers, k 1 > k 2 > 0; mod represents the remainder function; m, a i , A i are all intermediate generated parameters.

[0089] Using the Godunov-type finite volume method, an exact Riemann solver is adopted to handle the discontinuous problem, and its formula is as follows:

[0090] f(A * ) ≡ f L (A * , A L ) + f R (A * , A R ) + u R - u L = 0

[0091]

[0092] where u R and u L are the velocity variables on the left and right sides of the node respectively; f K (K = L or R) is the expression that establishes the relationship between the left and right sides and the intermediate state through the non-linear wave on the left or right side; the superscript * is the solution value in the discontinuous region (i.e., the star region) of the Riemann problem; A K (K = L or R) is the area in the flux on the left and right sides; I 1 is the integral form of the pressure term for an interface of any shape; is a function of the area.

[0093] Subsequently, the expression for the flux Q* in the star region of the Riemann problem can be obtained:

[0094]

[0095] Further, the flow area solution A at the discontinuity is obtained by iteration using the Newton-Raphson method * :

[0096]

[0097] where f' represents the derivative of the flux.

[0098] Step 4:

[0099] Process the calculation results, and verify the prediction results of the proposed model against the experimental data. See the detailed calculation result diagrams in Example 1 and Figure 2 and Figure 3 .

[0100] Step 5:

[0101] Quantitatively consider the influence effects of multi-shaft gas emissions and system scale:

[0102] The continuity equation and orifice equation are applied to the shaft exhaust boundary conditions. The continuity equation for this boundary condition is:

[0103]

[0104] where the subscript air represents the corresponding variables of the gas; the subscript 1 represents the first computational control volume at the boundary; Mairout represents the mass of air discharged through the ventilation hole and is calculated using Equation (15). The Riemann invariant is used to solve the isothermal, one-dimensional, primitive-form Euler equations

[0105] u air1 = u air2 - α(log ρ 2 - log ρ 1 ) (15)

[0106] where the subscript 2 represents the second control volume adjacent to the first computational control volume.

[0107] Quantitatively consider the size effect of the system. If it is designed according to the same Froude number Fr of the original model, then its formula is

[0108]

[0109] Among them, Fr is the Froude number; U represents the water body flow velocity; g is the acceleration due to gravity; L is the system length; λ represents the similarity scale ratio of the prototype of each variable to the model; Q is the flow rate; p is the pressure; H is the pressure head; ρ is the density; the subscript Y represents the prototype, the subscript M represents the model, the subscript u represents the velocity variable, the subscript l represents the length variable, the subscript t represents the time variable, the subscript r represents the characteristic physical quantity, the subscript Q represents the flow rate variable, and the subscript H represents the head variable.

[0110] Through the above formula calculation, through the above method programming calculation, based on this Figure 2 experimental system, Example 1 uses the multi-shaft mixed flow model of the present invention for simulation, and compares the experimental result data with the traditional gas-free model and experimental results. The data comparison diagram is shown in Figure 3 . Based on the actual large-scale project corresponding to a certain scale Figure 2 , Example 2 uses the multi-shaft mixed flow model of the present invention for simulation, discusses the influence of quantitatively considering the gas emission of multiple shafts and the influence effect of system scale in the improved mixed flow model, and conducts Figure 4 data display and analysis for the exhaust effect, and separately conducts Figure 5 data display and analysis for the size effect. It can be seen from Figure 3 that the improved mixed flow simulation method considering large multi-shaft pipelines has satisfactory accuracy in the complex layout of actual projects. It can be seen from Figure 4 , Figure 5 that the layout of multiple shafts will cause a staged exhaust phenomenon in the pipeline mixed flow, which has a great impact on the mixed flow simulation. Among them, the gas volume, pressure, and density show dynamic staged changes; compared with large-size actual engineering cases, it can be seen that this model considers the scale effect of the water-gas two-phase, and is also applicable to large-size mixed flow cases. Therefore, the improved mixed flow simulation considering large multi-shaft pipelines has satisfactory accuracy in the mixed flow simulation of actual projects with complex layouts.

[0111] Technical effects:

[0112] 1. Establish a high-precision one-dimensional multiphase model, combine the two-phase component water body simulation method and the discrete gas Euler simulation method to simulate the water-gas two-phase transient flow, and can achieve accurate simulation of the mixed flow in the water conveyance pipeline.

[0113] 2. Importantly, the influence of the multi-shaft layout on the water-filled two-phase flow and the influence of the system size effect on the model applicability are carefully discussed.

[0114] 3. In addition, combining the random selection method with the Godunov-type finite volume method can achieve accurate and stable solution of the model, which is beneficial to ensuring the hydraulic safety prediction, operation scheduling and prevention and control in the water conveyance pipeline.

[0115] It should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention rather than to limit them. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that the technical solutions of the present invention can be modified or equivalently replaced without departing from the spirit and scope of the technical solutions of the present invention, and they should all be covered within the scope of the claims of the present invention.

Claims

1. An improved mixed flow simulation method considering large multi-shaft pipelines, characterized in that: It includes dynamically coupling gas and fluid transient parameters, quantitatively analyzing multi-shaft gas emissions and system size effects, and improving the model's reproduction accuracy and stability. The specific steps are as follows: S1: Construct a two-phase component model of the water body and calculate the mass and momentum flux of the pipeline body; S2: Dynamically coupled discrete gas model combined with discrete gas Euler method; S3: Introduce random selection method and Godnov type finite volume method to achieve accurate and stable solution of the model; S4: Process the calculation results and verify the prediction results of the proposed model with experimental data; S5: Quantitatively consider the effects of multi-shaft gas emissions and system scale; Among them, a two-phase component simulation model is established for the water part: Where: Where U is the vector of conserved variables; F is the vector of conserved variable fluxes; S is the vector of source terms; x is the distance along the tube axis; t is time; A is the flow cross-sectional area; Q is the flow rate; g is the gravitational acceleration; h u is the distance between the centroid of the cross section and the free surface; h p is the additional pressure head; h air is the pressurized air head; θ = π-arccos[(yD / 2)D / 2]; D is the pipe diameter; A pipe is the cross-sectional area; a is the wave velocity in the pressurized flow; Among them, the gas model is dynamically coupled to establish a discrete Euler gas model: in, is the air density; u is the air velocity; p is the air pressure head, α is the speed of sound waves in air; S d1 , S d2 and S fa They are the source terms in the continuity equation, momentum equation and the loss between the airflow and the pipe wall.

2. The improved mixed flow simulation method considering large multi-shaft pipelines according to claim 1 is characterized in that: A random selection method is adopted to resolve discontinuities in complex wave interactions, assigning states to the next time level based on a random sampling process: Where, the superscript n represents the nth computation time step; the subscript i represents the i-th computation control volume; β n For random numbers, the van der Korput number pseudo-random sequence is used as the best random sequence in the sampling process due to its advantages of low variance and uniform distribution; The van der Korput sequence is expressed as: A i =k2a i (modk1) (9) Where k1 and k2 are two relatively prime numbers, k1>k2>0; mod represents the remainder function; m, ai, A i All are intermediate generated parameters; Among them, the discontinuity problem is handled by using the Godnov type finite volume method with an exact Riemann solver: f(A * )=f L (A * ,A L )+f R (A * ,A R )+u R -u L =0 Among them, u R and u L are the velocity variables on the left and right sides of the node respectively; f K (K=L or R) is the expression that passes through the left or right nonlinear wave and establishes the relationship between the two sides and the middle state; the superscript * is the solution value of the discontinuity area (i.e., the asterisk area) in the Riemann problem; A K (K=L or R) is the area in the flux on the left and right sides; I1 is the integral form of the pressure term of the interface of any shape; is a function of area; Among them, the expression of the flow Q* in the asterisk area in the Riemann problem is: Among them, the flow area solution A at the discontinuity is obtained by iteration of the Newton-Raphson method * : Where f' represents the flux derivative.

3. The improved mixed flow simulation method considering large multi-shaft pipelines according to claim 2 is characterized in that: Considering the impact of gas release from multiple shafts: The shaft exhaust boundary condition applies the continuity equation and the orifice equation. The continuity equation of this boundary condition is: Wherein, the subscript air represents the corresponding variable of the gas; the subscript 1 represents the first computational control volume of the boundary; Mairout represents the mass of air discharged through the vent, which is calculated using equation (15); the isothermal, one-dimensional, original format Euler equation is solved using the Riemann invariant you air1 =u air2 -α(logρ2-logρ1) (15) Among them, the subscript 2 represents the second control body adjacent to the first calculation control body; Considering the size effect of the system quantitatively, the design is based on the same Froeder number Fr of the original model, then: Among them, Fr is the Froude number; U represents the water flow velocity; g is the gravitational acceleration; L is the system length; λ represents the similarity ratio of the prototype to the model of each variable; Q is the flow rate; p is the pressure; H is the pressure head; ρ is the density; subscript Y represents the prototype, subscript M represents the model, subscript u represents the velocity variable, subscript l represents the length variable, subscript t represents the time variable, subscript r represents the characteristic physical quantity, subscript Q represents the flow rate variable, and subscript H represents the pressure head variable.

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