Numerical simulation method, numerical simulation apparatus, and control program

Abstract

To provide a numerical simulation method configured to reduce computational load and enable stable computation.SOLUTION: A numerical simulation method is a method for numerically simulating a flow of an incompressible viscoelastic fluid iteratively calculates a viscoelastic stress calculation step (S3), a provisional velocity calculation step (S4), a pressure calculation step (S5), and a velocity update step (S6) while advancing time steps. The viscoelastic stress calculation step (S3) includes calculating viscoelastic stress at the (n+1)th step by applying viscoelastic stress at the n-th step to a time-discretized constitutive model equation representing viscoelastic stress. The provisional velocity calculation step (S4) includes calculating a provisional velocity by applying a well-known velocity at the n-th step to an equation that constitutes a discretized form of a momentum conservation equation. The pressure calculation step (S5) includes calculating pressure at the (n+1)th step. The velocity update step (S6) includes calculating a correct velocity at the (n+1)th step.SELECTED DRAWING: Figure 4

Description

[Technical Field]

[0001] The present invention relates to a method for numerically simulating the flow of an incompressible viscoelastic fluid, an apparatus for numerically simulating the flow of an incompressible viscoelastic fluid, and a control program. [Background technology]

[0002] Conventionally, various methods have been proposed for numerical analysis of incompressible viscous fluids, such as the finite element method, the finite difference method, the finite volume method, and the boundary element method (see Non-Patent Document 1). For example, Patent Documents 1 and 2 disclose simulation methods for predicting the behavior of viscoelastic bodies such as unvulcanized rubber using the finite element method or the like. [Prior art documents] [Patent documents]

[0003] [Patent Document 1] Japanese Patent Application Publication No. 2017-189957 [Patent Document 2] Japanese Patent Application Laid-Open No. 2007-190827 [Non-patent literature]

[0004] [Non-Patent Document 1] Toshio Kobayashi et al., "Handbook of Computational Fluid Dynamics," Maruzen, March 2003, pp. 20-34 Summary of the Invention [Problem to be solved by the invention]

[0005] However, in the simulation methods of Patent Documents 1 and 2, it is necessary to divide the area to be calculated into multiple meshes and then assign a large number of variables to each mesh, which increases the calculation load. Therefore, there is a problem that further measures such as relaxation coefficients and re-correction, as well as calculation time, are required to perform the calculation more stably.

[0006] An object of one aspect of the present invention is to provide a numerical simulation method, a numerical simulation device, and a control program that can reduce the calculation load and perform calculations stably. [Means for solving the problem]

[0007] In order to solve the above-mentioned problems, a numerical simulation method according to one aspect of the present invention is a method for numerically simulating a flow of an incompressible viscoelastic fluid, comprising: a viscoelastic stress calculation step of calculating the viscoelastic stress at n+1 steps by applying the viscoelastic stress at n steps (n is the number of time steps) to an equation obtained by discretizing a constitutive model equation indicating the viscoelastic stress of the incompressible viscoelastic fluid with respect to time; a tentative velocity calculation step of calculating a tentative velocity by applying a known velocity at n steps to an equation constituting an equation obtained by discretizing a momentum conservation equation for the incompressible viscoelastic fluid; The method includes a pressure calculation step of calculating a pressure at the n+1 step using the viscoelastic stress calculated at the force calculation step, an equation obtained by discretizing the mass conservation equation, and the tentative velocity calculated at the tentative velocity calculation step, and a velocity update step of calculating a correct velocity at the n+1 step using the viscoelastic stress calculated at the viscoelastic stress calculation step, the tentative velocity calculated at the tentative velocity calculation step, and the pressure calculated at the pressure calculation step, and the viscoelastic stress calculation step, the tentative velocity calculation step, the pressure calculation step, and the velocity update step are iteratively calculated while advancing the time step.

[0008] In order to solve the above-mentioned problems, a numerical simulation device according to one aspect of the present invention is a numerical simulation device for a flow of an incompressible viscoelastic fluid, and includes a viscoelastic stress calculation unit that calculates the viscoelastic stress at n+1 steps by applying the viscoelastic stress at n steps (n is the number of time steps) to an equation obtained by discretizing a constitutive model equation indicating the viscoelastic stress of the incompressible viscoelastic fluid with respect to time, a tentative velocity calculation unit that calculates a tentative velocity by applying a known velocity at n steps to an equation constituting an equation obtained by discretizing a momentum conservation equation for the incompressible viscoelastic fluid, and a viscoelastic stress calculation unit that calculates a tentative velocity by applying a known velocity at n steps to an equation constituting an equation obtained by discretizing a momentum conservation equation for the incompressible viscoelastic fluid. The present invention includes a pressure calculation unit that calculates the pressure at the n+1 step using the viscoelastic stress calculated by the stress calculation unit, a discretized equation of the mass conservation equation, and the tentative velocity calculated by the tentative velocity calculation unit, and a velocity update unit that calculates the correct velocity at the n+1 step using the viscoelastic stress calculated by the viscoelastic stress calculation unit, the tentative velocity calculated by the tentative velocity calculation unit, and the pressure calculated by the pressure calculation unit, and the viscoelastic stress calculation unit, the tentative velocity calculation unit, the pressure calculation unit, and the velocity update unit perform iterative calculations while advancing the time steps. [Effects of the Invention]

[0009] According to one aspect of the present invention, the calculation load can be reduced. [Brief explanation of the drawings]

[0010] [Figure 1] 1 is a block diagram showing a configuration of a numerical simulation device according to an embodiment of the present invention. [Figure 2] FIG. 1 is a diagram illustrating elements of the Maxwell model. [Figure 3] FIG. 10 is a diagram illustrating a staggered grid. [Figure 4] 1 is a flowchart showing an example of the flow of a numerical simulation method according to the present embodiment. [Figure 5] FIG. 3 is a diagram showing a result of a numerical simulation performed by the numerical simulation device in the first embodiment. [Figure 6]FIG. 10 shows experimental results for the flow of an incompressible viscoelastic fluid in an abrupt contraction channel. [Figure 7] FIG. 10 is a diagram illustrating a calculation target of the numerical simulation device according to the second embodiment. [Figure 8] FIG. 10 is a diagram comparing the results of a numerical simulation in Example 2 with the results of a numerical simulation using a conventional finite element method and experimental results. DETAILED DESCRIPTION OF THE INVENTION

[0011] [Embodiment] A numerical simulation method using a numerical simulation device 1 according to an embodiment of the present invention will be described below with reference to FIGS.

[0012] [Calculation method] A description will be given of a calculation method used in the numerical simulation method executed by the numerical simulation device 1 shown in Fig. 1. In the numerical simulation method of this embodiment, the flow of an incompressible viscoelastic fluid is analyzed by the fractional step method.

[0013] Here, an incompressible viscoelastic fluid is a non-Newtonian fluid whose density changes little when pressure is applied and which simultaneously exhibits viscous and elastic properties. A non-Newtonian fluid is a fluid that does not obey Newton's law of viscosity, i.e., a fluid in which shear stress is not proportional to the velocity gradient of the flow.

[0014] The numerical simulation method of this embodiment can be applied to, for example, analysis of flows related to functional material processes including polymers, fillers, and slurries, and bioprocesses. Hereinafter, for convenience of explanation, an incompressible viscoelastic fluid may be simply referred to as a fluid.

[0015] (Derivation of the formula) First, the derivation of the equations used in the numerical simulation method for the flow of an incompressible viscoelastic fluid in this embodiment will be described. The basic equations for an incompressible viscoelastic fluid can be expressed as the following equations (1) and (2). Note that the following equations (1) to (8) use tensor notation.

[0016]

number

[0017]

number

[0018] In the numerical simulation method of this embodiment, τ in the above equation (2) ij By calculating the above and taking into account stresses including elasticity, it is possible to perform highly accurate analysis of incompressible viscoelastic fluids.

[0019] The above equations (1) and (2) are partial differential equations, so the fluid velocity u i In order to obtain the pressure p of the fluid, it is necessary to perform discretization. First, the above equation (1) is discretized using the Euler explicit method with respect to time.

[0020] Here, the time step size is Δt, and the variable u that you want to change after time Δt is i n+1 , known variable u at time nΔt i n Then, the following equation (3) is derived from Taylor expansion.

[0021]

number

[0022] By modifying the above formula (3), the following formula (4) is obtained.

number

[0023] Here, O(Δt) is the first-order error for Δt. Next, by applying equation (4) to the above equation (2), the following equation (5) is obtained. However, p and τ ij is evaluated in n+1 steps, and the other terms are evaluated in n steps.

[0024]

number

[0025] The above equation (5) is expressed as u i n+1 By solving for, we obtain the following equation (6).

number

[0026] Here, in order to simplify equation (6), the third and fourth terms on the right side of equation (6) are replaced as shown in the following equation (7), which can be calculated from known values.

number

[0027] Next, when, for example, the Maxwell model is used as a constitutive model equation expressing the properties of an incompressible viscoelastic fluid, the following equation (8) is obtained.

number

[0028] Figure 2 is a diagram showing the elements of the Maxwell model. As shown in Figure 2, the Maxwell model is represented by an element in which a dashpod exhibiting viscosity and a spring exhibiting elasticity are connected in series. In Figure 2, μ represents the viscosity of the fluid, γ1 represents the strain of the dashpod, G represents the elastic modulus of the fluid, and γ2 represents the strain of the spring. The relaxation time λ is the time it takes for the stress acting on the dashpod and spring to become 1 / e of the initial stress when strain γ is applied to the element shown in Figure 2.

[0029] As a constitutive model formula for an incompressible viscoelastic fluid, the Oldroyd model, the Giesekus model, the Larson model, and the Phan-Thien-Tanner model may also be used.

[0030] The constitutive model equation is preferably set according to the value of the Weissenberg number Wi. The Oldroyd model takes into account the delay time λ' in addition to the relaxation time λ. The delay time λ' is the time elapsed until the strain reaches a specific magnitude when a constant external force is applied to an incompressible viscoelastic fluid. The Oldroyd model is suitable for fluids that are prone to change over time.

[0031] The Giesekus model is suitable for representing the first and second normal stress differences and the growth and relaxation of shear viscosity that occur specifically in incompressible viscoelastic fluids such as polymer solutions. The Larson model is suitable for fluids that well represent the properties of linear polymers. The Phan-Thien-Tanner model is suitable for fluids that represent the nonlinear properties of viscoelastic fluids.

[0032] Next, when the above-mentioned equation (8) is discretized with respect to time, the following equation (9) is obtained. Here, n represents the number of time steps. Note that the following equations (9) to (16) use tensor notation.

number

[0033] Furthermore, the above equation (9) is ij n+1Solving for this, the following equation (10) is derived.

number

[0034] Next, to calculate the velocity of the fluid, the first and second terms on the right side of the above equation (6) are removed, and the following equation (11) is used to calculate the virtual velocity u i * Let's say.

[0035]

number

[0036] The above equation (11) can also be obtained by discretizing the time derivative of the following equation (12) and introducing a virtual value for pressure. Note that equation (12) is derived from the above equation (2).

number

[0037] Next, by subtracting equation (11) from equation (6) above, the following equation (13) is obtained.

number

[0038] By spatially differentiating both sides of the above equation (13), the following equation (14) is derived.

number

[0039] Here, when the above formula (1) is evaluated with respect to time, the following formula (15) is obtained.

number

[0040] Substituting equation (15) into equation (14) above yields equation (16) below.

number

[0041] p in the above formula (16) n+1 can be found by direct or iterative methods. Here, by iterative methods, p n+1 We will explain how to find this.

[0042] The left side of equation (16) is expressed as the Cartesian coordinate x i =(x,y,z),u i * When (u, v, w) are discretized, the following equation (17) is derived.

number

[0043] Furthermore, when the right side of equation (16) is discretized in the same manner as the left side, the following equation (18) is derived.

number

[0044] By spatially discretizing Equation (17) and Equation (18) and using a variable p arranged in a mesh, the following can be expressed:

number

[0045] The above equation (19) can be solved, for example, by using the BiCGSTAB (Bi-Conjugate Gradient Stabilized Method), which is one of the iterative methods. The BiCGSTAB method is known to have high convergence and is suitable for analyzing incompressible viscoelastic fluids. By solving equation (19), p n+1Alternatively, by solving equation (16) directly, p n+1 You can also ask for:

[0046] Other iterative methods may also be used, such as the SOR (Successive Over-relaxation) method. Since the pressure calculation unit 33 requires iterative calculations, a parallel calculation method using multiple calculation units and RAMs can be used to speed up the calculations, thereby achieving higher calculation performance. The same is true for the other calculation units 31, 32, and 34.

[0047] Next, by subtracting equation (11) from equation (6), the following equation (20) is obtained. n+1 can be obtained.

number

[0048] (Fractional Step Method) The fractional step method is a calculation method that is a modification of the MAC (Marker and Cell) method. The calculation procedure of the fractional step method is as follows: In step 1, the viscoelastic stress τ of the fluid at the n+1 step is calculated using the above equation (10). n+1 Next, in step 2, the virtual velocity u of the fluid is calculated using the above equation (11). * Calculate.

[0049] Next, in step 3, the pressure p of the fluid at step n+1 is calculated using the above equation (16). n+1 Then, in step 4, the fluid velocity u at the n+1 step is calculated using the above equation (20). n+1 After calculating, the time step number n is advanced and the process returns to step 1.

[0050] In this way, in the fractional step method, steps 1 to 4 are repeated over time starting from the initial conditions, thereby sequentially calculating the velocity and pressure of the fluid at each time.

[0051] When the numerical simulation device 1 calculates the velocity, pressure, etc. of a fluid in a calculation domain using the above-mentioned equations, the calculation domain is divided into a plurality of calculation grids (cells) and calculations are performed. In the fractional step method, for example, a staggered grid is used.

[0052] Figure 3 is a diagram illustrating a staggered grid. In Figure 3, u represents the fluid velocity in the x direction, v represents the fluid velocity in the y direction, and p represents the fluid pressure. i represents the cell order in the x direction, and j represents the cell order in the y direction. By using this staggered grid, the mass conservation equation can be naturally expressed in one cell, and the property of the above fundamental equation, in which the pressure gradient in each direction determines the velocity in that direction, can also be naturally expressed.

[0053] For example, v i,j The value of u at the defining point is calculated by averaging four points around the defining point, as shown in equation (21) below.

number

[0054] Also, u i,j The value of v at the defining point is calculated by averaging four points around the defining point, as shown in the following equation (22).

number

[0055]

number

[0056]

number

[0057] By using such a calculation method using a staggered grid and calculating the above-mentioned equations with the numerical simulation device 1, it is possible to calculate the velocity, pressure, etc. of the fluid in each cell of the calculation domain.

[0058] [Configuration of the numerical simulation device] Next, the configuration of a numerical simulation device 1 according to this embodiment will be described with reference to Fig. 1. Fig. 1 is a block diagram showing the configuration of the numerical simulation device 1. As shown in Fig. 1, the numerical simulation device 1 includes a control unit 10 and a storage unit 20. An input device 2 and a display device 3 are connected to the numerical simulation device 1.

[0059] The input device 2 is a device for operating the numerical simulation device 1. The input device 2 may be any device that can accept input operations for operating the numerical simulation device 1 and transmit operation data indicating the content of the accepted input operation to the numerical simulation device 1, and may be, for example, a mouse, a keyboard, or the like.

[0060] The display device 3 is a device that displays an image using display data output by the numerical simulation device 1. The display device 3 may be any device that can display an image, and may be a liquid crystal display device, an organic EL (Electro Luminescence) display device, or the like. The display device 3 displays an operation screen for operating the numerical simulation device 1, numerical simulation results, and the like.

[0061] The control unit 10 comprehensively controls the operation of the numerical simulation device 1. The control unit 10 performs the above control by reading out a program stored in the storage unit 20 into, for example, a RAM (Random Access Memory) and executing it.

[0062] The storage unit 20 stores data, programs, etc. necessary for the operation of the numerical simulation device 1. The storage unit 20 also stores intermediate results calculated during the numerical simulation process, final results of the simulation, etc.

[0063] The control unit 10 has a simulation calculation unit 30, a simulation control unit 40, and a display control unit 50. The simulation calculation unit 30 has a viscoelastic stress calculation unit 31, a tentative velocity calculation unit 32, a pressure calculation unit 33, and a velocity update unit 34.

[0064] The viscoelastic stress calculation unit 31 calculates the viscoelastic stress τ in n steps by dividing the equation (8), which is a constitutive model equation showing the viscoelastic stress of the incompressible viscoelastic fluid, into the equation (10), which is an equation discretized with respect to time. n Applying the viscoelastic stress τ at the n+1 step n+1 Calculate.

[0065] The virtual velocity calculation unit 32 calculates the momentum conservation equation for an incompressible viscoelastic fluid by dividing the momentum conservation equation (2) into equation (11) which is a component of equation (6) obtained from the discretized equation, and then adding the known velocity u n Applying this, the virtual velocity u * Calculate.

[0066] The pressure calculation unit 33 calculates the viscoelastic stress τ n+1 , Equation (16) obtained by discretizing the mass conservation equation, and the tentative velocity u calculated by the tentative velocity calculation unit 32 * Using this, the pressure p n+1 Calculate.

[0067] The velocity update unit 34 updates the viscoelastic stress τ calculated by the viscoelastic stress calculation unit 31. n+1 , the tentative velocity u calculated by the tentative velocity calculation unit 32 * , and the pressure p calculated by the pressure calculation unit 33 n+1 Using this, the correct velocity u at n+1 steps n+1 Calculate.

[0068] The simulation control unit 40 controls the simulation calculation unit 30 to perform calculations for each time point in each cell of the calculation domain as described above, thereby executing a numerical simulation of the fluid to be calculated, and transmits the execution results to the display control unit 50. The simulation control unit 40 may store the execution results of the numerical simulation in the storage unit 20.

[0069] More specifically, when the simulation calculation unit 30 finishes the calculation for one cell, the simulation control unit 40 updates the cell to be calculated and performs the calculation for the updated cell. In this way, the simulation control unit 40 updates the number of time steps n and performs the calculation for all cells up to the maximum number of time steps n. max The process is repeated while updating the time step number n until

[0070] The display control unit 50 displays the results of the numerical simulation executed by the simulation control unit 40 on the display device 3. The display control unit 50 also reads display data to be displayed from the storage unit 20 and displays it on the display device 3. That is, the display control unit 50 displays the results of the numerical simulation performed by the numerical simulation device 1 on the display device 3, and reads operation screen data for operating the numerical simulation device 1 from the storage unit 20 and displays it on the display device 3.

[0071] [Flow of numerical simulation method using numerical simulation equipment] Next, the flow of a numerical simulation method performed by the numerical simulation device 1 will be described with reference to Fig. 4. In this embodiment, the numerical simulation device 1 performs a numerical simulation regarding the flow of an incompressible viscoelastic fluid.

[0072] 4 is a flowchart showing an example of the flow of the numerical simulation method according to this embodiment. In the flowchart shown in FIG. 4, the numerical simulation device 1 reads calculation data from the storage unit 20 (S1). A user is assumed to input calculation data to be calculated in the numerical simulation into the storage unit 20 of the numerical simulation device 1 in advance via the input device 2. The calculation data includes data on the size of the flow path through which the fluid flows, the number of cells, the time step width Δt, the number of time steps n, the Reynolds number Re, initial conditions, boundary conditions, the order of calculation commands, and physical properties of the fluid such as the density ρ and viscosity μ.

[0073] In S1, the user selects a program corresponding to the Weissenberg number Wi of the incompressible viscoelastic fluid. Note that the storage unit 20 is assumed to have pre-stored therein programs employing each constitutive model formula corresponding to the Weissenberg number Wi.

[0074] Here, the Weissenberg number Wi is a dimensionless number that represents the ratio of elastic force to viscous force. The larger the value of the Weissenberg number Wi, the greater the influence of elastic force. If the value of the Weissenberg number Wi is in the range of 1 to 500, calculations can be stabilized.

[0075] In S1, a program corresponding to the Weissenberg number Wi of the incompressible viscoelastic fluid may be automatically selected.

[0076] After S1, the simulation calculation unit 30 acquires initial conditions from the storage unit 20 (S2). The initial conditions include an initial velocity u0 of the fluid, a viscoelastic stress τ0, etc. The boundary conditions include the velocity of the fluid on the wall surfaces that form the flow path, etc. In S1, the user may input the initial conditions to the simulation calculation unit 30 via the input device 2.

[0077] After S2, the viscoelastic stress calculation unit 31 of the simulation calculation unit 30 calculates the relaxation time λ, the time step width Δt, the fluid velocity u at n steps in the above equation (13). n, viscoelastic stress τ at n steps n By substituting, the viscoelastic stress τ n+1 (S3: viscoelastic stress calculation step). In the first step, the initial velocity u0 and the viscoelastic stress τ0 are substituted into equation (13).

[0078] As described above, Equation (10) is obtained by applying the Maxwell model as a constitutive model equation for an incompressible viscoelastic fluid. In this embodiment, the application of a simple Maxwell model simplifies the calculation and shortens the calculation time.

[0079] After S3, the virtual velocity calculation unit 32 of the simulation calculation unit 30 calculates the known velocity u at n steps in the above equation (11). n Substituting the above, the virtual velocity of the fluid in n steps, u * (S4: tentative speed calculation step). In S4, the tentative speed calculation unit 32 calculates the tentative speed u using the formula (11). * By using equation (7) when calculating, it is possible to perform highly accurate discretization.

[0080] After S4, the pressure calculation unit 33 of the simulation calculation unit 30 calculates the viscoelastic stress τ at the n+1 step calculated in the viscoelastic stress calculation step S3 in the above-mentioned equation (16). n+1 , and the tentative velocity u calculated in the tentative velocity calculation step S4 * Using this, the pressure p of the fluid at the n+1 step n+1 is calculated (S5: pressure calculation step).

[0081] In S5, the pressure calculation unit 33 solves the above equation (19) using, for example, the BiCGSTAB method, to calculate the pressure p of the fluid at the n+1 step. n+1 Calculate.

[0082] After S5, the velocity update unit 34 of the simulation calculation unit 30 applies the tentative velocity u at n steps calculated in the tentative velocity calculation step S4 to the above-mentioned equation (20). *, time step width Δt, density ρ, pressure p at n+1 step calculated in pressure calculation step S5 n+1 , the viscoelastic stress τ at the n+1 step calculated in the viscoelastic stress calculation step S3 n+1 By substituting, the correct velocity of the fluid at step n+1, u n+1 is calculated (S6: speed update step).

[0083] After S6, the simulation calculation unit 30 advances the number of time steps n by one step (S7), and the number of time steps n reaches the maximum number of time steps n max It is determined whether it is smaller than (S8).

[0084] The number of time steps n is the maximum number of time steps n max If n is smaller than the maximum number of time steps n (S8: YES), the simulation calculation unit 30 repeats S3 to S7. max The calculations of S3 to S6 are repeated until

[0085] The simulation control unit 40 then calculates the time step number n when the time step number n reaches the maximum time step number n max If the time reaches (S8: NO), that is, if the calculation is completed up to the end time of the numerical simulation, the numerical simulation by the simulation calculation unit 30 is ended, and the calculation result is output (S9).

[0086] In S9, the simulation control unit 40 transmits the calculation results of the numerical simulation of the fluid performed by the simulation calculation unit 30 to the display control unit 50. The display control unit 50 causes the display device 3 to display the output results of the numerical simulation performed by the simulation control unit 40. The display device 3 visualizes and displays, for example, changes in the speed, pressure, viscosity, etc. of the fluid flowing through the flow path.

[0087] The numerical simulation method using the numerical simulation device 1 described above can reduce the calculation load and shorten the calculation time compared to conventional numerical simulations using the finite element method. That is, in conventional numerical simulation methods, the region to be calculated needs to be divided into multiple meshes, and each mesh needs to have a large number of variables, which increases the calculation load. Therefore, there is a problem that further measures such as relaxation coefficients and re-correction are required, and calculation time is required, in order to execute the calculation more stably. The present numerical simulation method can solve the above problem.

[0088] Furthermore, by calculating the stress τ including elasticity using the above equation (10) and applying it to the above equations (16) and (20), it is possible to obtain a stable solution for the flow of an incompressible viscoelastic fluid with elastic properties even when using the fractional step method instead of the finite element method.

[0089] Furthermore, in discretizing the above equation (2), a higher-order discretization is performed than in the conventional finite element method, which allows for highly accurate analysis of incompressible viscoelastic fluids.Furthermore, since a method is used that directly discretizes and solves the fundamental equations for incompressible viscoelastic fluids, conservation of mass, momentum, and energy can be highly guaranteed.

[0090] This allows for highly accurate analysis of the flow of incompressible viscoelastic fluids such as molten resins and metals inside an extrusion molding machine, which can be useful for investigating the flow behavior of new materials such as resins and metals, and for clarifying problems that occur during extrusion molding.

[0091] Example 1 Next, the results of the numerical simulation performed by the numerical simulation device 1 in the first embodiment will be described with reference to FIGS.

[0092] [Calculation conditions] Fig. 5 is a diagram showing the results of a numerical simulation performed by the numerical simulation device 1 in Example 1. Fig. 6 is a diagram showing the results of an experiment on the flow of an incompressible viscoelastic fluid in an abruptly contracting channel.

[0093] The flow path to be calculated in Figure 5 is one in which the width of the flow path becomes one-quarter midway. The calculation was performed by dividing the calculation area into 8,000 0.5 mm square calculation grids (mesh). The flow path was assumed to be symmetrical from top to bottom, and only the flow in the upper part was calculated. Molten aluminum steel was used as the incompressible viscoelastic fluid.

[0094] In this calculation, the relaxation time λ was set to 1000 [s]. The time step width Δt was set to 1.0 × 10 ー4 [s]. Reynolds number Re = 1.0 × 10 -5 The maximum number of time steps is n max is 1 x 10 5 As an initial condition, the initial velocity of the fluid, u0, was set to 0 [m / s]. As a boundary condition, the velocity of the fluid at the wall of the channel was set to 0 [m / s].

[0095] [Calculation result] As shown in Figure 6, the experimental results showed that the fluid flowed backward at the corners of the flow channel, generating vortices, and the fluid velocity increased near the abrupt contraction where the width of the flow channel suddenly narrowed.

[0096] The numerical simulation method using the numerical simulation device 1 of this embodiment was able to satisfactorily reproduce the fluid flow near the sudden contraction section observed in the experimental results, as shown in Fig. 5. This fluid flow near the sudden contraction section is caused by the elastic effect of the incompressible viscoelastic fluid.

[0097] Example 2 Next, the results of the numerical simulation performed by the numerical simulation device 1 in the second embodiment will be described with reference to FIGS.

[0098] [Calculation conditions] Fig. 7 is a diagram showing a calculation target of the numerical simulation device 1 in Example 2. As shown in Fig. 7, the calculation target in Example 2 is a flow path with a square cross section with one side measuring 5 mm. A portion where the width of the flow path is narrowed (hereinafter referred to as a contraction portion) is provided midway through this flow path. The contraction portion is provided in an area 40 mm to 50 mm from the inlet of the flow path, and the width of the flow path is set to 0.538 mm.

[0099] The incompressible viscoelastic fluid used was an aqueous solution of polyethylene glycol (PEG) and polyethylene oxide (PEO), with the mass ratios of PEG and PEO set to 34.0% and 0.0105%, respectively.

[0100] For the experiment, reference was made to the literature (Anazawa et al., "Pressure Loss in the Contraction Flow of Viscoelastic Fluid from the Distribution Chamber to the Slot in an Extrusion Die," Journal of the Japanese Society of Rheology, Vol. 40, No. 2, pp. 91-99 (2012)). Fluid analysis software (Ansys Polyflow) was used for calculations using the conventional finite element method.

[0101] In the calculations using this numerical simulation method, the calculation area was divided into 34,410 calculation grids (meshes). The relaxation time λ was set to 0.0272 s. The time step width Δt was set to 1.0 × 10 -8 The maximum number of time steps is n max is 1 x 10 9 As an initial condition, the initial velocity of the fluid, u0, was set to 0 [m / s]. As a boundary condition, the velocity of the fluid at the wall of the channel was set to 0 [m / s].

[0102] [Calculation result] Fig. 8 is a diagram comparing the results of a numerical simulation performed by the numerical simulation device 1 in Example 2 with the results of a numerical simulation performed using a conventional finite element method and experimental results. The vertical axis of Fig. 8 represents the pressure change ΔP between the pressure at the inlet and the pressure at the outlet of the flow path. The horizontal axis of Fig. 8 represents the Weissenberg number Wi.

[0103] The Weissenberg number Wi is Wi=2λ Zimm V / H, where λ Zimm is the relaxation time in the Zimm theory, V is the average velocity of the fluid in the contraction section, and H is the width of the contraction section. In the experiment, the Weissenberg number Wi was changed by changing the average velocity V, and the pressure change ΔP was measured. The pressure change ΔP was normalized by dividing the total pressure loss by the pressure loss in the contraction section.

[0104] As shown in Figure 8, the experimental results show that ΔP becomes large in the region where the Weissenberg number Wi is greater than 15. This is because the vortex transition occurs in the contraction section due to the viscoelasticity of the incompressible viscoelastic fluid.

[0105] According to the numerical simulation method using the numerical simulation device 1 of this embodiment, it was possible to obtain calculation results that are closer to experimental results than numerical simulation results using conventional software using the finite element method.

[0106] In particular, the present numerical simulation method was able to improve the quantitative accuracy compared to conventional numerical simulation methods when the Weissenberg number Wi was in the range of 1 to 25.

[0107] In the above-described Examples 1 and 2, the calculation object is a two-dimensional region, but this is not limited thereto, and the numerical simulation method using the numerical simulation device 1 of this embodiment can also be applied when the calculation object is a three-dimensional region.

[0108] [Software implementation example] The functions of the numerical simulation device 1 (hereinafter referred to as the "device") can be realized by a control program for causing a computer to function as the device, and by a control program for causing a computer to function as each control block of the device (in particular, the viscoelastic stress calculation unit 31, the provisional velocity calculation unit 32, the pressure calculation unit 33, and the velocity update unit 34).

[0109] In this case, the device includes a computer having at least one control device (e.g., a processor) and at least one storage device (e.g., a memory) as hardware for executing the control program. The control device and storage device execute the control program, thereby realizing each function described in the above embodiment.

[0110] The control program may be non-transitory and may be recorded on one or more computer-readable recording media. The recording media may or may not be included in the device. In the latter case, the control program may be supplied to the device via any wired or wireless transmission medium.

[0111] Furthermore, some or all of the functions of the control blocks can be realized by logic circuits. For example, an integrated circuit in which a logic circuit that functions as each of the control blocks is formed is also included in the scope of the present invention. In addition, the functions of the control blocks can also be realized by, for example, a quantum computer.

[0112] The present invention is not limited to the above-described embodiments, and various modifications are possible within the scope of the claims. Embodiments obtained by appropriately combining the technical means disclosed in different embodiments are also included in the technical scope of the present invention. [Explanation of symbols]

[0113] 1. Numerical simulation equipment 2 Input devices 3 Display device 10 Control Unit 20 Memory section 30 Simulation calculation section 31 Viscoelastic stress calculation section 32 Provisional speed calculation unit 33 Pressure calculation unit 34 Speed ​​update section

Claims

1. A method for numerically simulating the flow of an incompressible viscoelastic fluid, comprising: a viscoelastic stress calculation step of calculating the viscoelastic stress at n+1 steps by applying the viscoelastic stress at n steps (n is the number of time steps) to an equation obtained by discretizing the constitutive model equation representing the viscoelastic stress of the incompressible viscoelastic fluid with respect to time; a tentative velocity calculation step of calculating a tentative velocity by applying known velocities at n steps to an equation constituting a discretized equation of the momentum conservation equation for the incompressible viscoelastic fluid; a pressure calculation step of calculating a pressure at n+1 steps using the viscoelastic stress calculated in the viscoelastic stress calculation step, a discretized equation of the mass conservation equation, and the tentative velocity calculated in the tentative velocity calculation step; a velocity update step of calculating a correct velocity at n+1 steps using the viscoelastic stress calculated in the viscoelastic stress calculation step, the tentative velocity calculated in the tentative velocity calculation step, and the pressure calculated in the pressure calculation step; Including, A numerical simulation method in which the viscoelastic stress calculation step, the tentative velocity calculation step, the pressure calculation step, and the velocity update step are iteratively calculated while advancing the time step.

2. In the tentative velocity calculation step, the tentative velocity at n+1 steps is calculated based on discretization of the time derivative from the following equation (1) and introduction of a tentative value into the pressure: [Equation 1] In the pressure calculation step, the pressure at the n+1 step is calculated based on the following formula (2): [Equation 2] In the speed update step, the correct speed at the n+1 step is calculated based on the following equation (3): [Equation 3] The numerical simulation method according to claim 1 .

3. In the tentative velocity calculation step, u i * is the virtual velocity, u i is the velocity vector, x i are the spatial coordinates, ρ is the density, g i is the gravitational acceleration, the virtual velocity in n steps is calculated based on the following equation (4): [Equation 4] The numerical simulation method according to claim 2 .

4. 3. The numerical simulation method according to claim 2, wherein in the viscoelastic stress calculation step, one of a Maxwell model, an Oldroyd model, a Giesekus model, a Larson model, and a Phan-Thien-Tanner model is selected as the constitutive model formula depending on the properties of the incompressible viscoelastic fluid to calculate the viscoelastic stress.

5. The numerical simulation method according to claim 1 , wherein the constitutive model formula is set according to a value of the Weissenberg number.

6. 6. The numerical simulation method according to claim 5, wherein the Weissenberg number is in the range of 1 to 500, and preferably in the range of 1 to 25.

7. A numerical simulation device for the flow of an incompressible viscoelastic fluid, comprising: a viscoelastic stress calculation unit that calculates the viscoelastic stress at n+1 steps by applying the viscoelastic stress at n steps (n is the number of time steps) to an equation obtained by discretizing the constitutive model equation indicating the viscoelastic stress of the incompressible viscoelastic fluid with respect to time; a tentative velocity calculation unit that calculates a tentative velocity by applying a known velocity at n steps to an equation that constitutes a discretized equation of the momentum conservation equation for the incompressible viscoelastic fluid; a pressure calculation unit that calculates a pressure at n+1 steps using the viscoelastic stress calculated by the viscoelastic stress calculation unit, a discretized equation of the mass conservation equation, and the provisional velocity calculated by the provisional velocity calculation unit; and a velocity update unit that calculates a correct velocity at n+1 steps using the viscoelastic stress calculated by the viscoelastic stress calculation unit, the tentative velocity calculated by the tentative velocity calculation unit, and the pressure calculated by the pressure calculation unit; and Equipped with A numerical simulation device that performs iterative calculations while advancing time steps using the viscoelastic stress calculation unit, the tentative velocity calculation unit, the pressure calculation unit, and the velocity update unit.

8. A control program for causing a computer to function as the numerical simulation apparatus according to claim 7, A control program for causing a computer to function as the viscoelastic stress calculation unit, the provisional velocity calculation unit, the pressure calculation unit, and the velocity update unit.

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