Pressure smoothing method suitable for water-sand two-phase flow weak compressible smoothed particle fluid dynamic model

Abstract

The invention discloses a pressure intensity smoothing method suitable for a water-sand two-phase flow weak compressible smoothed particle fluid dynamic model, and the method comprises the steps: firstly assuming that the water phase density before correction is rho w, considering the water phase density after correction of a water-sand two-phase delta-SPH pressure intensity smoothing scheme as # imgabs0 #, and carrying out the two-phase strategy of a water-sand two-phase flow WCSPH model; according to the characteristic that the water phase and the sediment phase are carried by the same group of SPH particles, the correlation between the water phase and the sediment phase is deduced according to the mass conservation of the water phase and the sediment phase, and the correction form of the mass conservation equation of the water phase and the correction form of the mass conservation equation of the corresponding sediment phase are further deduced. According to the method, on the premise of strictly ensuring mass conservation of the water phase and the sand phase, numerical oscillation is effectively eliminated, and the calculation stability and the calculation precision of the water-sand two-phase flow WCSPH model are improved.

Description

Technical Field

[0001] The present invention belongs to the field of computational fluid dynamics, and in particular relates to a pressure smoothing method for a water-sand two-phase flow weakly compressible smooth particle fluid dynamics model. Background Art

[0002] In real-world water disasters, sediment movement often plays a key role. For example, in dam-break floods, high-speed water flows carry large amounts of sediment, and this water-sediment interaction exhibits a two-way impact: sediment transport not only changes the course of the flood, but the action of the water flow also affects sediment transport, significantly reshaping the riverbed morphology. Similarly, debris flows triggered by heavy rains also exhibit complex water-sediment interaction characteristics, and their enormous kinetic energy poses a serious threat to downstream areas. Furthermore, underwater landslides, as one of the triggers of tsunami disasters, have movement characteristics that are closely related to the generation and propagation characteristics of tsunami waves. Against the backdrop of global climate change and rising sea levels, the accurate prediction and effective prevention and control of such sediment-laden water disasters are particularly important.

[0003] The inherently complex interaction mechanisms of water-sediment two-phase flow make it a challenging problem in disaster prediction. In recent years, with advances in computational fluid dynamics (CFD) and computer technology, researchers have developed a variety of water-sediment two-phase flow dynamics models, including both grid-based and meshless methods. Among these, the smoothed particle hydrodynamics (SPH) model has become a research hotspot due to its inherent advantages in capturing the violent deformation of the free surface, sediment transport, and the morphological changes of granular flow. In particular, the weakly compressible smoothed particle hydrodynamics (WCSPH) model, with its explicit pressure solution and good parallelization potential, has achieved outstanding computational efficiency and has therefore been widely used.

[0004] However, the WCSPH model suffers from numerical oscillations during numerical calculations, which directly impacts the stability and accuracy of the results. To address this issue, researchers have explored various approaches, including artificial viscosity terms, the δ-SPH method, the Riemann solver, and density reinitialization algorithms. These methods have achieved significant results in single- or two-fluid WCSPH models, effectively suppressing numerical oscillations and improving computational accuracy. However, existing pressure smoothing methods are mostly designed for single-phase WCSPH models or two-fluid WCSPH models where each phase is solved independently. These models are characterized by requiring only independent numerical corrections to each fluid term to satisfy conservation conditions, without considering the coupling between the different fluids. Currently, there is limited research on pressure smoothing techniques for water-sediment two-phase WCSPH models, particularly those using a set of particles to represent both the water and sediment phases. A challenge with these approaches is that numerical corrections to the water phase must simultaneously account for adjustments to the sediment phase, even though the sediment phase is generally assumed to be incompressible to ensure mass conservation in the two-phase system. This coupling constraint makes it difficult to directly apply pressure smoothing methods from existing single-phase flow models to water-sediment two-phase systems. Therefore, developing pressure smoothing algorithms suitable for WCSPH models of water-sediment two-phase flow has become a key scientific challenge to improve the computational accuracy and stability of such models. Solving this problem will directly impact the reliability of WCSPH models of water-sediment two-phase flow and has important theoretical and applied value for the prediction and prevention of related sediment-laden water disasters. Summary of the Invention

[0005] The present invention addresses the numerical oscillation problem existing in the weakly compressible smooth particle hydrodynamic model (WCSPH) of water-sand two-phase flow with only one set of SPH particle strategies, and proposes a pressure smoothing method suitable for the weakly compressible smooth particle hydrodynamic model of water-sand two-phase flow. According to the mass conservation of the water phase and the sediment phase, the correlation between the water phase and the sediment phase is derived, and the modified form of the mass conservation equation of the water phase and the corresponding modified form of the mass conservation equation of the sediment phase are further derived. This effectively eliminates the numerical oscillation while strictly ensuring the mass conservation of the water phase and the sediment phase, thereby improving the computational stability and accuracy of the water-sand two-phase flow WCSPH model.

[0006] The present invention provides a pressure smoothing method applicable to a water-sand two-phase flow WCSPH model, specifically a water-sand two-phase δ-SPH pressure smoothing method, comprising the following steps:

[0007] (1) The numerical dissipation term D is calculated according to the single-phase δ-SPH method (Molteni, D., and Colagrossi, A. (2009). A simple procedure to improve the pressure evaluation in hydrodynamic context using the SPH. Computer Physics Communications, 180(6), 861-872.) a .

[0008]

[0009] In the formula, subscript "a" is the target particle, subscript "b" is the neighboring particle of the target particle, and the neighboring particle "b" in the kernel function domain of the target particle "a" is retrieved (see Figure 1 ), the adjacent particles are defined as all particles within a radius of 2h of the target particle "a"; δ is a coefficient, and studies have shown that δ = 0.1; h is the smoothing length, which is taken as h = 1.3Δ; Δ is the initial spacing between SPH particles; c0 is the numerical sound velocity, which is generally taken as 10 times the maximum velocity of the water phase; is the gradient of the distance kernel function, see Figure 1 ; V b is the volume of particle “b”; ρ w is the density of the water phase; (ρ w ) a is the water phase density of particle “a”; (ρ w ) b is the water phase density of particle “b”; r ab is the distance vector, which refers to the relative position vector between the target particle "a" and the adjacent particle "b"; ||r ab || is the modulus of the distance vector; r ab is the relative distance between particles; ε is a minimum value to prevent singularities.

[0010] In this description, the quotation marks in particle "a" and particle "b" indicate that ab is a code, not a variable, and the quotation marks are used to distinguish it from the variable.

[0011] (2) The water phase mass conservation equation for “a” particles is calculated as follows:

[0012]

[0013] In the formula, all variables refer to the variables of particle "a", α w is the volume fraction of the water phase, defined as m w / ρ w V a ; V ais the volume of particle "a"; ρ w is the density of the water phase; m w is the mass of the water phase; the rightmost item is an additional term (i.e., a numerical dissipation term added to alleviate numerical oscillations), representing the water phase mass density correction term proposed to address the numerical oscillation problem of water phase density in the WCSPH model of water-sand two-phase flow; u w is the water phase velocity; D a It is the numerical correction term in the single-phase δ-SPH method proposed by Molteni and Colagrossi (Molteni, D., and Colagrossi, A. (2009). A simple procedure to improve the pressure evaluation in hydrodynamic context using the SPH. Computer Physics Communications, 180(6), 861-872.), see formula (1).

[0014] (3) The sediment mass conservation equation for particle “a” is calculated as follows:

[0015]

[0016] In the formula, all variables refer to the variables of particle "a", and the rightmost term is the sediment phase correction corresponding to the water phase correction to ensure the conservation of sediment phase mass; α s is the volume fraction of the sediment phase, defined as m s / ρ s V a According to the numerical strategy of the water-sand two-phase flow WCSPH model, α w +α s =1;ρ s is the density of sediment phase; m s is the mass of sediment phase; u s is the sediment phase velocity; D a It is the numerical correction term in the single-phase δ-SPH method proposed by Molteni and Colagrossi, see formula (1).

[0017] (4) The momentum conservation equations for the water and sediment phases of particle “a” are:

[0018]

[0019] In the formula, all variables refer to the variables of particle "a", p w is the water phase pressure; τ wis the water phase shear force; g is the acceleration of gravity; F is other body forces; τ s is the sediment phase shear force; p s It is the sediment phase pressure.

[0020] (5) The water phase state equation is as follows, which can be solved by closing the control equation group to obtain the basic variables of the two phases.

[0021]

[0022] Where p w0 is the density of water phase at standard atmospheric pressure, p w0 =1000kg / m 3 ; ξ is the model parameter, generally taken as ξ=7.

[0023] (6) Solve the modified mass conservation equations (4) to (5), the original momentum conservation equations (13) to (14) and the water phase state equation (15) to obtain the basic variables of the two phases carried by each SPH particle, that is, the calculation results.

[0024] Furthermore, in the above method, the formula derivation process of steps (2) and (3) is as follows:

[0025] First, assume that the density of the water phase before correction is ρ w , considering the water-sand two-phase δ-SPH pressure smoothing scheme, the water phase density is corrected as follows: According to the two-phase strategy of the water-sediment two-phase flow WCSPH model, that is, the water phase and sediment phase are carried by the same set of SPH particles, the following relationship can be obtained according to the conservation of mass of the water phase and sediment phase:

[0026]

[0027] Wherein, the superscript 'δ' represents the variable after correction using the two-phase δ-SPH pressure smoothing scheme proposed in the present invention. The unsuperscripted variable represents the variable before correction.

[0028] According to the above formula, the changes in mass density of the water phase and the sediment phase caused by the correction of the two-phase δ-SPH pressure smoothing scheme can be obtained:

[0029]

[0030] According to the definition of the numerical correction term in the single-phase δ-SPH method, the following relationship exists:

[0031]

[0032] Subsequently, the mass conservation equation of the water-sand two-phase flow WCSPH model can be written as follows:

[0033]

[0034] Due to the weak compressibility of water, it is generally believed that the density of water fluctuates within 1%, so Δρ w is a very small value; and because α s <1, so α s Δρ w It is also a very small value. Here we assume that α in equations (11) to (12) s Δρ w ≈0, and substituting into formula (10) we get formulas (4) to (5):

[0035]

[0036] Furthermore, in the above method, in steps (5) and (6), the control equations (4) to (5) and equations (13) to (14) and the state equation (15) are solved according to the conventional SPH strategy, and the time iterative calculation is performed according to the time integration strategy of the prediction and correction method.

[0037] Compared with the prior art, the present invention has the following beneficial effects:

[0038] 1. The method of the present invention proposes a pressure smoothing method suitable for the water-sand two-phase flow WCSPH model, which can effectively eliminate the numerical oscillation of the water-sand two-phase flow WCSPH model and improve the calculation stability and accuracy of the model.

[0039] 2. The method of the present invention reduces numerical dissipation while ensuring the accuracy of pressure calculation, which is beneficial to calculation convergence and stability.

[0040] 3. The method of the present invention can greatly improve the accuracy of capturing the instantaneous impact pressure of the water-sand two-phase flow WCSPH model. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 Schematic diagram of the kernel function calculation domain;

[0042] Figure 2 Calculation of pressure for Example 1 - Hydrostatic Experiment; Figure (a) shows the evolution of bottom pressure, and Figure (b) shows the evolution of surface pressure.

[0043] Figure 3 Calculations for pressure distribution in Example 2, a dam-break water impact experiment. Figures (a) and (b) show the results without the pressure smoothing method. Figures (c) and (d) show the results using the two-phase δ-SPH pressure smoothing method.

[0044] Figure 4 This is a comparison chart of the evolution of the pressure at the measuring point in Example 2 - dam-break water flow impact experiment and the pressure calculated using the two-phase δ-SPH pressure smoothing method. DETAILED DESCRIPTION

[0045] The pressure smoothing method for the weakly compressible smooth particle hydrodynamic model of water-sand two-phase flow according to the present invention will be further described below with reference to specific embodiments.

[0046] Example 1

[0047] A water tank with a length of 0.4m, a width of 0.2m and a height of 0.4m is filled with water with a depth of h. w = 0.3m of tap water. When the water is completely still, the pressure distribution at this time conforms to the theoretical hydrostatic pressure, that is, p w =ρ w Ugh.

[0048] The two-phase δ-SPH pressure smoothing method proposed in this paper is coupled to a water-sand two-phase flow WCSPH model (refer to Shi, H., Si, P., Dong, P. Yu, X. (2019). A two-phase SPH model for massive sediment motion in free surface flows. Advanced Water Resources, 129, 80-98.) to simulate the pressure distribution in the tank. In this case, α w =1.0 and α s = 0.0, the initial spacing of SPH particles is Δ = 5 mm. Considering the convergence condition, the calculation step size is Δt = 10 -5 s. According to the WCSPH model strategy (Shi, H., Yu, X., & Dalrymple, RA (2017). Development of a two-phase SPH model for sediment ladenflows. Computer Physics Communications, 221, 259-272.), the modified mass conservation equations (4) to (5), momentum conservation equations (13) to (14), and state equation (15) are solved simultaneously. According to the prediction and correction method (Monaghan, JJ (1989). On the problem of penetration in particle methods. Journal of Computational physics, 82 (1), 1-15.), the time-dependent iterative calculation is performed to obtain the calculation results.

[0049] Figure 2Comparison of calculated pressures from a hydrostatic experiment. Clearly, without the pressure smoothing scheme, the calculated bottom and surface pressures fluctuate significantly, due to mathematical limitations of the WCSPH model. After employing the two-phase δ-SPH pressure smoothing method, the calculated pressures stabilize significantly and agree with the theoretical values. This demonstrates that the proposed two-phase δ-SPH pressure smoothing method successfully eliminates numerical fluctuations and significantly improves the computational stability and accuracy of the weakly compressible smooth particle model for water-sand two-phase flow.

[0050] Example 2

[0051] A 1.2m long and 0.6m high water body was confined to the left side of a flat-bottomed flume by a gate. By momentarily raising the gate, the evolution and impact of a dam-break flow were simulated. During the experiment, a pressure sensor was placed on the right wall to measure the evolution of the pressure exerted by the dam-break flow on the side wall.

[0052] The two-phase δ-SPH pressure smoothing method proposed in this paper is coupled to a water-sand two-phase flow WCSPH model (refer to Shi, H., Si, P., Dong, P. Yu, X. (2019). A two-phase SPH model for massive sediment motion in free surface flows. Advanced Water Resources, 129, 80-98.) to simulate the evolution of dam break water flow and impact pressure. Based on the sensitivity analysis and considering the convergence conditions, the initial spacing of the SPH particles in this case is Δ = 10 mm, and the calculation step size is Δt = 10 -5 s. According to the WCSPH model strategy (Shi, H., Yu, X., & Dalrymple, RA (2017). Development of a two-phase SPH model for sedimentladen flows. Computer Physics Communications, 221, 259-272.), the modified mass conservation equations (4) to (5), momentum conservation equations (13) to (14), and state equation (15) are solved simultaneously. According to the prediction and correction method (Monaghan, JJ (1989). On the problem of penetration in particle methods. Journal of Computational physics, 82 (1), 1-15.), the time-dependent iterative calculation is performed to obtain the calculated results. At the same time, the results are compared with the calculation results without considering pressure smoothing.

[0053] Figure 3 The pressure distribution is calculated for the dam break water impact experiment, and the calculation results of the coupled two-phase δ-SPH pressure smoothing method are compared with the calculation results without considering pressure smoothing. Figure 3 As can be seen from (a) and (b), without considering pressure smoothing, the calculated pressure distribution is chaotic, seriously affecting the stability and accuracy of the calculation. However, with the two-phase δ-SPH pressure smoothing method proposed in this paper, the calculated pressure distribution is smooth and accurately reflects the pressure increase at the main impact point. Figure 4 The evolution of the pressure measured at the pressure sensor was further compared. As can be seen from the figure, the dam-breaking water flow reaches the side wall at about t=0.68s, and jets upward along the wall. At this time, the pressure at the measuring point instantly increases to the first pressure peak. As the water flow attached to the wall falls, surges and water jumps gradually form, and then the second peak of pressure is reached at t=1.50s. Although the calculated second peak pressure is slightly delayed, the two-phase δ-SPH pressure smoothing method proposed in the present invention can generally capture the changes in the pressure of the measuring point more accurately, accurately estimate the arrival time of the flood and the first peak time, and the error is within an acceptable range. This shows that the two-phase δ-SPH pressure smoothing method proposed in the present invention can effectively eliminate numerical oscillations and improve calculation stability and accuracy.

Claims

1. A pressure smoothing method for a weakly compressible smooth particle hydrodynamic model of water-sand two-phase flow, characterized by: (1) Calculate the numerical dissipation term D according to the single-phase δ-SPH method a Where subscript "a" is the target particle, subscript "b" is the neighboring particle of the target particle, and the neighboring particle "b" within the kernel function domain of the target particle "a" is retrieved. The neighboring particles are defined as all particles within a radius of 2h of the target particle "a"; δ is the coefficient, δ = 0.1; h is the smoothing length, which is set to h = 1.3Δ; Δ is the initial spacing between SPH particles; c0 is the numerical sound velocity, which is set to 10 times the maximum velocity of the water phase; ▽ a W ab is the gradient of the distance kernel function; V b is the volume of particle "b"; ρ w is the density of the water phase; (ρ w ) a is the water phase density of particle "a"; (ρ w ) b is the water phase density of particle "b"; r ab is the distance vector, which refers to the relative position vector between the target particle "a" and the adjacent particle "b"; ||r ab || is the modulus of the distance vector; r ab is the relative distance between particles; ε is a minimum value; (2) The water phase mass conservation equation for "a" particles is calculated as follows: In the formula, all variables refer to the variables of particle "a", α w is the volume fraction of the water phase, defined as m w ρ w V a ; V a is the volume of particle "a"; ρ w is the density of the water phase; m w is the mass of the water phase; the rightmost item To characterize the water phase mass density correction term proposed for the numerical oscillation problem of water phase density in the WCSPH model of water-sand two-phase flow; u w is the water phase velocity; D a is the numerical correction term in the single-phase δ-SPH method; (3) The sediment mass conservation equation for particle "a" is calculated as follows: In the formula, all variables refer to the variables of particle "a", and the rightmost term is the sediment phase correction corresponding to the water phase correction to ensure the conservation of sediment phase mass; α s is the volume fraction of the sediment phase, defined as m s ρ s V a According to the numerical strategy of the water-sand two-phase flow WCSPH model, α w +α s =1;ρ s is the density of sediment phase; m s is the mass of sediment phase; u s is the sediment phase velocity; D a is the numerical correction term in the single-phase δ-SPH method; (4) The momentum conservation equations for the water and sediment phases of particle "a" are: In the formula, all variables refer to the variables of particle "a", p w is the water phase pressure; τ w is the water phase shear force; g is the acceleration of gravity; F is other body forces; τ s is the sediment phase shear force; p s is the sediment phase pressure; (5) The water phase state equation is as follows. By closing the control equation, the basic variables of the two phases can be solved: Where p w0 is the density of water phase at standard atmospheric pressure, p w0 =1000kgm 3 ξ is a model parameter, usually ξ=7; (6) Solve the modified mass conservation equations (4) to (5), the original momentum conservation equations (13) to (14) and the water phase state equation (15) to obtain the basic variables of the two phases carried by each SPH particle, that is, the calculation results.

2. The method according to claim 1, characterized in that The formula derivation process of steps (2) and (3) is as follows: First, assume that the density of the water phase before correction is ρ w , considering the water-sand two-phase δ-SPH pressure smoothing scheme, the water phase density is corrected as follows: According to the two-phase strategy of the water-sediment two-phase flow WCSPH model, that is, the water phase and sediment phase are carried by the same set of SPH particles, the following relationship can be obtained according to the conservation of mass of the water phase and sediment phase: Wherein, the superscript 'δ' represents the modified variable using the two-phase δ-SPH pressure smoothing scheme proposed in the present invention; Unsubstantiated variables represent variables before correction; According to the above formula, the changes in mass density of water phase and sediment phase caused by the correction of the two-phase δ-SPH pressure smoothing scheme can be obtained: According to the definition of the numerical correction term in the single-phase δ-SPH method, the following relationship exists: Subsequently, the mass conservation equation of the water-sand two-phase flow WCSPH model can be written as follows: Due to the weak compressibility of water, the density of water is considered to fluctuate within 1%, so Δρ w is a very small value; and because α s <1, so α s Δρ w is also a very small value; assuming that α in equations (11) to (12) s Δρ w ≈0, and substituting into formula (10) we get formulas (4) to (5):

3. The method according to claim 1, characterized in that In the above method, in steps (5) and (6), the control equations (4) to (5) and equations (13) to (14) and the state equation (15) are solved according to the conventional SPH strategy, and the time iterative calculation is performed according to the prediction correction method.

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