Sickness-keeping numerical calculation method for long-term evolution simulation of viscous fluid

Abstract

The invention discloses a pungency-keeping numerical calculation method for long-term evolution simulation of viscous fluid, and belongs to the technical field of computational fluid mechanics. The method comprises the following steps: firstly, constructing a dissipative system kinetic model containing a Navier-Stokes equation; based on a time average least square principle, through defining and minimizing a residual function, converting an original dissipation system into an equivalent high-order Hamiltonian system, and endowing the equivalent high-order Hamiltonian system with a sympathetic structure; and finally, based on the singing structure, designing an unconditionally stable Pocking integrator, and deducing a numerical value updating formula for solving. According to the method, the symplectic structure of the system can be strictly kept, numerical dissipation and dispersion are effectively inhibited, the precision and physical consistency of long-term simulation are remarkably improved, and the method can be widely applied to high-precision fluid simulation in the fields of aerospace, water conservancy projects and the like.

Description

Technical Field

[0001] This invention belongs to the field of computational fluid dynamics technology, specifically relating to a symplectic numerical calculation method for simulating the long-term evolution of viscous fluids. It can be applied to high-precision numerical simulation and engineering design of viscous fluid dynamics in fields such as aerospace, water conservancy engineering, and energy equipment. Background Technology

[0002] In computational fluid dynamics, high-precision, long-term numerical simulations of the dynamic behavior of viscous fluids are a critical requirement in fields such as aerospace and hydraulic engineering. These problems are typically described by the Navier-Stokes equations, which physically classify them as typical dissipative systems where total energy is not conserved.

[0003] Traditional numerical methods for solving such problems can be broadly categorized into two types. The first type consists of high-order explicit methods, such as the fourth-order Runge-Kutta method. While these methods offer high accuracy at small time steps, their stability is conditionally limited. In long-term simulations or with large time steps, errors easily accumulate and diverge, resulting in low computational efficiency. The second type comprises low-order implicit methods, such as the implicit Euler method. Although these methods possess unconditional stability, they only offer first-order accuracy, suffer from severe numerical dissipation, and introduce non-physical numerical damping during the simulation, leading to premature decay of kinetic energy and distorting the long-term evolution of the flow, making them unsuitable for high-precision simulations.

[0004] The symplectic integrator is a specialized numerical method designed for Hamiltonian conservative systems. Because it strictly preserves the symplectic structure of the system, it offers unparalleled advantages in long-term simulations, such as bounded energy error and high phase accuracy, and has been successfully applied in fields such as celestial mechanics. However, the theoretical basis of the symplectic integrator requires the system to possess a Hamiltonian structure, while dissipative systems described by the Navier-Stokes equations naturally do not possess this symplectic structure. This prevents the direct application of this advanced algorithm to the broad and important engineering field of viscous fluid simulation.

[0005] Existing research has attempted to construct temporary variational structures for some dissipative systems using pseudo-variable principles, but these methods lack universality and have not yet been successfully applied to the Navier-Stokes equations. Therefore, developing a general numerical algorithm that can introduce the advantages of Boctine integrators into dissipative systems, particularly applicable to the Navier-Stokes equations, and possessing both unconditional stability and high accuracy, has become a pressing technical challenge in this field. Summary of the Invention

[0006] To address the aforementioned technical problems, this invention provides a symplectic numerical calculation method for simulating the long-term evolution of viscous fluids. Based on a higher-order dynamic variational structure using the least squares principle, the dissipative system is transformed into an equivalent Hamiltonian system. An unconditionally stable and high-precision symplectic solution algorithm is developed and applied for the first time to viscous fluid problems described by the Navier-Stokes equations.

[0007] To achieve the above objectives, the present invention adopts the following technical solution:

[0008] A basicus numerical calculation method for simulating the long-term evolution of viscous fluids includes:

[0009] Step 1: Construct a dynamic model of the dissipative system, including the first-order dissipative differential equation and the residual form of the Navier-Stokes equation;

[0010] Step 2: Based on the time-averaged least squares principle, the dissipative system is transformed into an equivalent high-order Hamiltonian system functional. By defining the residual and minimizing its square integral over the time interval, the dissipative system is given a symplectic structure.

[0011] Step 3: Simplify the Navier-Stokes equations by combining the physical constraints of the fluid mechanics problem, and derive the equivalent Hamiltonian system of the simplified Navier-Stokes equations based on the equivalent high-order Hamiltonian system functional.

[0012] Step 4: Based on the Bausch integrator, develop and adopt an unconditionally stable numerical solution scheme. By setting discrete momentum conditions, obtain the update formula of the state variables of the equivalent Hamiltonian system of the simplified Navier-Stokes equations, and complete the numerical simulation.

[0013] Furthermore, in step 1, constructing the dynamic model of the dissipative system specifically involves:

[0014] The first-order dissipative differential equation is in the form that the first derivative of the state variable is equal to the dynamic function containing the dissipative term;

[0015] The Navier-Stokes equations employ an unsteady Poiseuille flow model to describe the motion of a viscous fluid driven by a constant pressure gradient between two infinitely parallel plates. The governing equations are dimensionless to obtain residual form, and it is assumed that the velocity profile is parabolic, with the centerline velocity being the variable to be solved.

[0016] Furthermore, in step 2, transforming the dissipative system into an equivalent higher-order Hamiltonian system functional specifically includes:

[0017] For the first-order dissipative differential equation, its residual is defined as the difference between the first derivative of the state variable and the dynamic function containing the dissipative term;

[0018] By minimizing the square integral of the residual over a given time interval using the variational method, a second-order dynamic equation without explicit first-order derivative terms is derived.

[0019] Based on the second-order dynamic equations, a corresponding Hamiltonian function is constructed, such that the system dynamics are described by the Hamiltonian canonical equations, thereby introducing a symplectic structure into the original dissipative system.

[0020] Furthermore, step 3 specifically includes:

[0021] Substitute the parabolic velocity profile and its second spatial derivative into the Navier-Stokes equations;

[0022] By integrating and averaging the flow field cross section, the spatial dimension is eliminated, resulting in a first-order dissipative ordinary differential equation that only concerns the centerline velocity.

[0023] Substitute the obtained first-order equation into the functional transformation framework established in step 2 to elevate it to an equivalent second-order Hamiltonian equation.

[0024] By defining a generalized momentum related to the centerline velocity, we construct the system’s exclusive Hamiltonian function and verify that it satisfies the Hamiltonian canonical equation, thus completing the transformation from the Navier-Stokes equations to the equivalent Hamiltonian system.

[0025] Furthermore, in step 4, the developed unconditionally stable numerical solution scheme includes two methods:

[0026] The first method involves setting a first discrete momentum condition, combining it with the Bausch integrator scheme and the Lagrangian of the equivalent Hamiltonian system, and deriving an explicit update formula for the centerline velocity.

[0027] The second method involves setting a second discrete momentum condition and deriving another form of the centerline velocity update formula based on it.

[0028] Furthermore, both update formulas include higher-order correction terms determined by the equivalent Hamiltonian system, thereby ensuring the unconditional stability of the numerical solution.

[0029] Furthermore, the centerline velocity update formula derived by the first method has the following characteristics:

[0030] The centerline velocity value at step n+1 is equal to the velocity value at step n multiplied by a coefficient consisting of the time step and the system viscosity parameter, plus a driving term consisting of the constant pressure gradient and the time step.

[0031] Furthermore, as the time step approaches infinity, the steady-state solution calculated by the update formula of the first method is exactly consistent with the known steady-state velocity determined by physical laws.

[0032] In a second aspect, the present invention provides an electronic device, comprising: one or more processors; and a memory for storing one or more programs; wherein, when the one or more programs are executed by the one or more processors, the one or more processors implement the aforementioned method for basic numerical calculation of long-term evolution simulation of viscous fluids.

[0033] Thirdly, the present invention provides a computer-readable storage medium having executable instructions stored thereon, which, when executed by a processor, enable the processor to implement the aforementioned method for basic numerical calculation of long-term evolution simulation of viscous fluids.

[0034] The beneficial effects of this invention are as follows:

[0035] Breakthrough in theoretical application: For the first time, the Bausch integrator was successfully applied to the Navier-Stokes equations. An equivalent Hamiltonian structure for the dissipative system was constructed using the least squares variational principle, realizing the unification of conservative and dissipative systems within the Bausch computational framework. This solved the fundamental problem that this advanced algorithm could not be used for viscous fluid simulation for a long time.

[0036] Balancing high accuracy and high stability: Both developed symplectic numerical schemes possess unconditional stability, allowing for time steps much larger than those of traditional explicit methods. At the same time, they maintain second-order computational accuracy through higher-order correction terms, fundamentally overcoming the contradiction in traditional methods where "high accuracy leads to conditional stability, while unconditional stability leads to low accuracy."

[0037] Significantly enhances long-term simulation capabilities: The algorithm can strictly maintain the symplectic structure of the equivalent system, effectively suppressing the accumulation of numerical dissipation and dispersion during long-term evolution, so that physical quantities such as kinetic energy can still maintain the correct evolution trend over long time scales, greatly improving the physical consistency and reliability of long-term simulations.

[0038] It has broad engineering application prospects: This method provides a high-precision and high-efficiency numerical tool for engineering design involving viscous fluid dynamics, such as airfoil flow around aerospace and pipeline transportation of energy equipment. It can be directly applied to the research and development and optimization of related products and has important practical engineering value. Attached Figure Description

[0039] Figure 1 This is a flowchart of a basic numerical calculation method for simulating the long-term evolution of viscous fluids according to the present invention;

[0040] Figure 2 This is a graph showing the change of error over time obtained based on the method of this invention;

[0041] Figure 3 This is a graph showing the relationship between the maximum error and the inverse step size obtained based on the method of this invention.

[0042] Figure 4 This is a comparison chart of numerical solutions and exact solutions obtained at different time steps based on the method of this invention. Detailed Implementation

[0043] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0044] like Figure 1 As shown, this invention proposes a basalt numerical calculation method for simulating the long-term evolution of viscous fluids, comprising the following steps:

[0045] Step 1: Construct the dynamic model of the dissipative system, including the residual form of the first-order dissipative differential equation and the Navier-Stokes equation; where:

[0046] The first-order dissipative differential equation is in the form of ,in State variables of a dissipative system (e.g., those that change with time) Changing fluid velocity ), For dynamic functions containing dissipation terms (such as linear dissipation) Nonlinear secondary resistance The superscript · indicates the first derivative, and the superscript ·· indicates the second derivative;

[0047] Unsteady Poiseuille flow model based on Navier-Stokes equations: two infinitely parallel flat plates ( Intermittent viscous fluid, caused by a constant pressure gradient The dimensionless residuals of the driving and governing equations are: Assuming the velocity profile is parabolic... , Centerline ( )speed.

[0048] Step 2: Based on the time-averaged least squares principle, the dissipative system is transformed into an equivalent high-order Hamiltonian system functional. By defining the residuals and minimizing their square integral over the time interval, the dissipative system is given a symplectic structure; specifically including:

[0049] The residuals defined above are expressed using a dynamic function containing dissipation terms: ,in The first derivative of the state variable;

[0050] Minimize residuals over time interval Square integrals on The second-order dynamic equations are obtained through variational calculations. ,in ;

[0051] Introducing momentum Constructing Hamiltonian functions based on second-order equations (See step 3 for specific construction methods), so that the system satisfies the Hamiltonian canonical equations. This gives the dissipative system an symplectic structure.

[0052] Step 3: Simplify the Navier-Stokes equations by incorporating the physical constraints of the fluid mechanics problem, and derive the equivalent Hamiltonian system of the simplified Navier-Stokes equations based on the functional of the equivalent higher-order Hamiltonian system; specifically including:

[0053] Expansion of the original Navier-Stokes equations: The dimensionless form of the Navier-Stokes equations for unsteady Poiseuille flow is as follows: ,in This is the time derivative term (including dissipative properties). For a constant pressure gradient, This is the viscous diffusion term;

[0054] Substituting physical constraints: combining plate boundary conditions (The flat plate is fixed and there is no slippage), assuming the velocity profile is parabolic. (Saving the boundary conditions and conforming to the laminar flow velocity distribution law), taking its second spatial derivative yields... ;

[0055] Transform into a first-order dissipative ODE: Substituting into the Navier-Stokes equation, we get the residual: ;

[0056] The Lagrange of the second-order formula can be obtained using the residuals in the following form:

[0057] ;

[0058] Furthermore, by eliminating the spatial variable y, we obtain information about the centerline velocity. First-order dissipative ODE: By domain integral averaging Simplified , recorded as (in ), independent of time t, );

[0059] Derivation of the equivalent second-order Hamiltonian equation: Based on the least-squares variational framework of the second step, Substitution ,have to ;

[0060] The Hamiltonian can be constructed using the second-order formula of the Lagrangian: defining momentum. Combining the equivalent second-order Hamiltonian equation Through Legendre transformation Construct Hamiltonian functions specific to Navier-Stokes equations Verify that it satisfies the Hamiltonian canonical equation: Consistent with the second-order equations, this completes the transformation of the Navier-Stokes equations into the equivalent Hamiltonian system.

[0061] Step 4: Based on the symplectic integrator, develop and adopt an unconditionally stable numerical solution scheme. By setting discrete momentum conditions, derive the update formula for the state variables of the equivalent Hamiltonian system of the simplified Navier-Stokes equations, and complete the numerical simulation. Specifically, the development of two symplectic numerical schemes includes:

[0062] Method 1: Set Discrete Momentum Conditions (Corresponding to a zero residual), combined with the discretization scheme of the symplectic integrator, that is:

[0063] ;

[0064] Substitute the Lagrange of the second-order formula derived in step 3:

[0065] ;

[0066] Derivation of the centerline velocity update formula: ,in For the first Step speed, For time step, A constant pressure gradient;

[0067] The test shows that this scheme is unconditionally stable because, in the limit of h→∞, we have:

[0068] ;

[0069] With known steady-state velocity Consistent.

[0070] Method 2: Setting Discrete Momentum Conditions Derivation of the speed update formula:

[0071] ;

[0072] Both schemes contain higher-order correction terms and satisfy the unconditional stability condition.

[0073] Example:

[0074] Unsteady Poiseuille flow, as a laminar flow mode of viscous fluid within a rigid circular pipe with a pressure gradient varying over time, has wide-ranging engineering applications in biomedicine (hemodynamic diagnosis, precision transport in microfluidic chips), petrochemicals (water hammer prevention in oil pipelines, optimization of multiphase flow transport), mechanical engineering (dynamic response design of hydraulic systems, precision control of machine tools), energy engineering (transient safety of nuclear reactor cooling, heat transfer fluid transport in solar thermal power generation), and food processing (intermittent transport and precision control of viscous fluids in filling). Its core value lies in providing theoretical support for safety risk prevention (such as water hammer in pipelines, reactor thermal accidents), operational precision optimization (such as blood flow diagnosis, food filling), and improved transport efficiency in various scenarios through quantitative analysis of transient velocity, pressure, and flow rate changes. Its applicable boundaries always revolve around the core characteristics of "circular pipe, laminar flow, viscosity dominance," and "time dependence of pressure gradient." To more fully understand the characteristics of this invention and its practical applicability to specific problems, this invention presents a numerical simulation example based on the Navier-Stokes equations for unsteady Poiseuille flow.

[0075] In the embodiment of the quasi-static problem, the core parameter (unsteady Poiseuille flow) is: constant pressure gradient. Simulated final time Time step (Covering both large and small step size scenarios); Accurate solution for the number of terms in Fourier series (ensure Time error ).

[0076] Figure 2 The error versus time comparison graphs ((a)-(f) correspond to multiple time steps respectively): the absolute errors of Method I (blue solid line) and Method II (red dashed line) mentioned in step 4 are significantly lower than those of the implicit Euler method (green dotted line) and the RK4 method (black dotted line), demonstrating the high accuracy and stability of the algorithm of this invention under large time steps, and solving the problem of divergence of traditional methods (such as RK4) under large time steps. Small time steps ( All methods reduced the error, but Method I still maintained the lowest error, verifying the effect of its higher-order correction terms on improving accuracy. With small step size, the upper limit of accuracy is better than that of traditional methods.

[0077] Figure 3 The relationship between the maximum absolute error and the reciprocal of the time step (convergence analysis). Comparison of convergence order and upper limit of accuracy: Error of Method I as... The growth slope is close to second-order, significantly outperforming the first-order convergence of the implicit Euler method; when When the error approaches a non-zero plateau (due to the physical assumptions of the approximate shape function), method I has the lowest plateau value, proving that its upper limit of accuracy is higher and it can still maintain optimal accuracy under the engineering approximation model. Method comparison: RK4 with large step size ( The error spikes dramatically in small steps, while the method of the present invention (method I / II) has controllable error across the entire step range, demonstrating the core advantage of unconditional stability.

[0078] Figure 4 A comparison of the responses of numerical and exact solutions (physical consistency verification) is shown in the dynamic response matching graph: At different time steps, the dynamic trends of the numerical solutions and exact solutions (black solid lines) of Method I and Method II are highly consistent, especially at large time steps. When the flow reaches a certain threshold (h), traditional methods (such as implicit Euler and RK4) show significant deviations from the exact solution. However, the algorithm of this invention can accurately capture the transient development and steady-state convergence process of the flow at high h, verifying the physical rationality of the equivalent Hamiltonian transformation. Steady-state convergence: All methods eventually converge to the same steady-state value (h). This demonstrates that the steady-state accuracy of the algorithm in this invention is consistent with physical properties, and meets the accuracy requirements of engineering fluid mechanics for steady-state solutions.

[0079] In a second aspect, the present invention provides an electronic device, comprising: one or more processors; and a memory for storing one or more programs; wherein, when the one or more programs are executed by the one or more processors, the one or more processors implement the aforementioned method for basic numerical calculation of long-term evolution simulation of viscous fluids.

[0080] Thirdly, the present invention provides a computer-readable storage medium having executable instructions stored thereon, which, when executed by a processor, enable the processor to implement the aforementioned method for basic numerical calculation of long-term evolution simulation of viscous fluids.

[0081] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above descriptions are merely specific embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A basicus numerical calculation method for simulating the long-term evolution of viscous fluids, characterized in that, include: Step 1: Construct a dynamic model of the dissipative system, including the first-order dissipative differential equation and the residual form of the Navier-Stokes equation; Step 2: Based on the time-averaged least squares principle, the dissipative system is transformed into an equivalent high-order Hamiltonian system functional. By defining the residual and minimizing its square integral over the time interval, the dissipative system is given a symplectic structure. Step 3: Simplify the Navier-Stokes equations by combining the physical constraints of the fluid mechanics problem, and derive the equivalent Hamiltonian system of the simplified Navier-Stokes equations based on the equivalent high-order Hamiltonian system functional. Step 4: Based on the Bausch integrator, develop and adopt an unconditionally stable numerical solution scheme. By setting discrete momentum conditions, obtain the update formula of the state variables of the equivalent Hamiltonian system of the simplified Navier-Stokes equations, and complete the numerical simulation.

2. The method for basicus numerical calculation of viscous fluid long-term evolution simulation according to claim 1, characterized in that, In step 1, constructing the dynamic model of the dissipative system specifically involves: The first-order dissipative differential equation is in the form that the first derivative of the state variable is equal to the dynamic function containing the dissipative term; The Navier-Stokes equations employ an unsteady Poiseuille flow model to describe the motion of a viscous fluid driven by a constant pressure gradient between two infinitely parallel plates. The governing equations are dimensionless to obtain residual form, and it is assumed that the velocity profile is parabolic, with the centerline velocity being the variable to be solved.

3. The method for basicus numerical calculation for simulating the long-term evolution of viscous fluids according to claim 1, characterized in that, Step 2, which transforms the dissipative system into an equivalent high-order Hamiltonian system functional, specifically includes: For the first-order dissipative differential equation, its residual is defined as the difference between the first derivative of the state variable and the dynamic function containing the dissipative term; By minimizing the square integral of the residual over a given time interval using the variational method, a second-order dynamic equation without explicit first-order derivative terms is derived. Based on the second-order dynamic equations, a corresponding Hamiltonian function is constructed, such that the system dynamics are described by the Hamiltonian canonical equations, thereby introducing a symplectic structure into the original dissipative system.

4. The method for basicus numerical calculation for simulating the long-term evolution of viscous fluids according to claim 1, characterized in that, Step 3 specifically includes: Substitute the parabolic velocity profile and its second spatial derivative into the Navier-Stokes equations; By integrating and averaging the flow field cross section, the spatial dimension is eliminated, resulting in a first-order dissipative ordinary differential equation that only concerns the centerline velocity. Substitute the obtained first-order equation into the functional transformation framework established in step 2 to elevate it to an equivalent second-order Hamiltonian equation. By defining a generalized momentum related to the centerline velocity, we construct the system’s exclusive Hamiltonian function and verify that it satisfies the Hamiltonian canonical equation, thus completing the transformation from the Navier-Stokes equations to the equivalent Hamiltonian system.

5. The method for calculating the basalt numerical evolution of viscous fluids according to claim 1, characterized in that, In step 4, the developed unconditionally stable numerical solution scheme includes two methods: The first method involves setting a first discrete momentum condition, combining it with the Bausch integrator scheme and the Lagrangian of the equivalent Hamiltonian system, and deriving an explicit update formula for the centerline velocity. The second method involves setting a second discrete momentum condition and deriving another form of the centerline velocity update formula based on it.

6. The method for calculating the basalt numerical evolution of viscous fluids according to claim 5, characterized in that, Both update formulas include higher-order correction terms determined by the equivalent Hamiltonian system, thereby ensuring the unconditional stability of the numerical solution.

7. The method for calculating the basalt numerical evolution of viscous fluids according to claim 5, characterized in that, The centerline velocity update formula derived by the first method has the following characteristics: The centerline velocity value at step n+1 is equal to the velocity value at step n multiplied by a coefficient consisting of the time step and the system viscosity parameter, plus a driving term consisting of the constant pressure gradient and the time step.

8. The method for calculating the basalt numerical evolution of viscous fluids according to claim 7, characterized in that, As the time step approaches infinity, the steady-state solution calculated by the update formula of the first method is exactly consistent with the known steady-state velocity determined by physical laws.

9. An electronic device, characterized in that, include: One or more processors; Memory, used to store one or more programs; When one or more programs are executed by the one or more processors, the one or more processors implement the Bausch numerical calculation method for long-term evolution simulation of viscous fluids as described in any one of claims 1-8.

10. A computer-readable storage medium, characterized in that, It stores executable instructions that, when executed by a processor, enable the processor to implement the Bausch numerical calculation method for simulating the long-term evolution of viscous fluids as described in any one of claims 1-8.

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