Contribution of interphase force curl to rigid vorticity transport in water-sand two-phase flow with fine particles

Rigid vorticity transport equation is an effective tool for describing the intuitive vortex evolution characteristics. Compared to single-phase flows, the curl of the interphase force appears as a new source term of this equation under the condition of multiphase flows, which may cause additional contributions. However, the effects of the interaction force on rigid vorticity transport in water-sand two-phase flows with fine particles are still unclear. In this article, taking the Karman vortices induced by a hydrofoil as a typical case, the distributions of rigid vorticity in single-phase flows and two-phase flows were compared, and the dynamics mechanism of the dominant interaction force was analyzed. The following notable results are obtained. Firstly, the drag force can be regarded as the dominant interaction force. Secondly, the effect of the drag force on a vortex tube is mainly manifested as inducing normal strain and the contribution is relatively low. Thirdly, there are only slight differences in the waveform, amplitude, frequency of rigid vorticity and apparent vortical structures between the single-phase flows and the fine-particle two-phase flows. These new findings are helpful for understanding the vortex evolution in water-sand two-phase flows with fine particles.


Introduction
Vorticity is a fundamental concept in the study of fluid dynamics [1], offering valuable insights into the behavior and characteristics of flowing fluids.It is a measure of the local rotation of fluid, providing a quantitative understanding of the swirling motion present in a fluid.Rigid vorticity is the part of vorticity that characterizes rigid rotation [2].It captures the basic characteristics of vorticity and has been successfully used for vorticity identification and visualization under the Euler framework.Rigid vorticity transport equation is an effective tool to describe the intuitive evolution characteristics of vortices [3].Moreover, water-sand two-phase flows are widely existed in hydraulic engineering and chemical engineering.These flows involve the simultaneous flow of water and sand particles, characterized by complex interaction mechanisms [4].The presence of sand particles introduces additional complexities rather than single-phase flow, such as interphase forces [5][6] and turbulence modulation [7].These interactions may lead to the change of vortical structures and affect the characteristics of rigid vorticity.It can be known that there are additional deformation terms-interphase force curls in rigid vorticity transport equation under the condition of two-phase flow.However, the effect of interphase forces on rigid vorticity in water sand two-phase flow with fine particles is still unclear.Hence, conducting a comprehensive investigation into the vorticity characteristics in water-sand two-phase flow and analyzing the contribution of interphase forces to vorticity can yield profound insights into the underlying mechanisms governing vortex structure formation.This paper uses the numerical simulation of hydrofoil turbulence as a case to study the contribution of interphase force curls to rigid vorticity transport in water-sand two-phase flow with fine particles.Firstly, the dominant interphase force and interphase force curl are determined by the comparison of magnitude among the interphase force concerned in this case.Secondly, the deformation terms of rigid vorticity are analyzed to determine the role of the dominant interphase force curl.Finally, the contribution of dominant interphase force curl is determined by the comparison of vortical structure characteristics of single-phase flow and two-phase flow.

Mathematical modelling of water-sand two-phase flow with fine particles
The numerical simulation framework for multiphase flow is generally determined based on the characteristics of sediment flow.For instance, considering the water-sand two-phase flow in the Yellow River region [8], the primary constituent of sand is quartz, with a density of 2650 kg/m³, and the median particle diameter typically falls around 25 µm.The volumetric concentration of sand passing through the pump does not exceed 1%, resulting in an approximate count of 10 12 sand particles.Consequently, the number of tracked particles is exceedingly large.However, the Euler-Lagrange framework usually employs parcels to delineate a multitude of particles [12] for engineering computations, and the number of tracked parcels is exceptionally limited (10 4 -10 5 ), which can control the computational costs but may be difficult to capture the detailed characteristics of the sand phase such as local velocity and sand concentration.By contrast, the Euler-Euler framework is commonly utilized for engineering problems involving relatively high sediment concentration with fine particles [9], where the balance between the computational cost and the prediction accuracy is ensured.
The flow around the hydrofoil displays various intricate flow phenomena, such as transition and vortex shedding.A combination of the time-scale-driven (TSD) hybrid URANS/LES model and the γ-Reθt transition model is more appropriate for capturing these characteristics, and the reliability of this transition scheme has been demonstrated in the previous studies [17].The newly TSD turbulence model employs the Spalart-Shur-correction-enhanced SST k-ω model as its baseline.The switch between the URANS mode and the LES-like mode is supported by a novel time-scale-driven damping function Df, which can automatically and dynamically adjust the turbulent viscosity μtf.Compared with the SST k-ω model, TSD achieves a broader span of the resolved structures, thereby obtaining higher prediction accuracy.Compared with the conventional hybrid URANS/LES models, TSD avoids the issue of nonmonotonic grid convergence and enhances the algorithm robustness [9][10].The governing equations of the TSD turbulence model are given by ( ) (1 ) where Df is the time-scale-driven damping function, Rt is the adaptive time scale ratio, and Hn is the normalized helicity, uf is the velocity of the fluid phase, ρf is the density of the fluid phase αf is the volumetric concentration of the fluid phase, kmf is the modeled turbulent kinetic energy of the fluid phase, ωmf is the modeled turbulent eddy frequency of the fluid phase.
This combined approach effectively represents the vorticity dynamics of the water phase, ensuring a comprehensive simulation of the complex flow phenomena involved in hydrofoil flow while maintaining a balance between prediction accuracy and computational cost.
In terms of the solid phase, the turbulent characteristics are captured using the classical Hinze-Tchen algebraic model [11], commonly known as the Ap model.This model is widely adopted in engineering applications for accurately representing the turbulent behavior of the solid phase.
The most important interaction between the water phase and the sand phase is the interphase force, which reflects the characteristics of interfacial momentum transfer.The following interphase forces are typically taken into account based on previous research: drag force (FD), virtual/added mass force (FVM), lift force (FL), turbulent dispersion force (FTD), and Basset force (FB).However, in engineering calculations, the FB is often neglected due to the computational complexities involved, despite its potential significant impact on particle motions.In this study, the improved Wen-Yu drag force model is utilized, as it is widely recognized for dilute solid-liquid two-phase flows and considers the influence of fluid turbulence and the wall surface [9].The virtual mass force is incorporated using the conventional Odar model, with a virtual mass force coefficient of 0.5, commonly employed in engineering calculations [12].Furthermore, the lift force model employed is the classical one, and the lift coefficient utilized is the commonly used value of 0.5 in engineering calculations [13].The turbulence diffusion force utilizes the classical Favre turbulent diffusion force model, which takes into account the turbulence diffusion coefficient representing the interaction between particles and turbulence eddies [9].

Rigid vorticity transport equation in water-sand two-phase flow
According to the binary decomposition theory of vorticity [14], the total vorticity ω can be decomposed into rigid vorticity ωr and deformation vorticity ωs.The former characterizing the rigid rotation of fluids and the latter characterizing the shear motion of fluids.The mathematical expression for this theory is To analysis the contribution of interphase force curl to rigid vorticity transport, the classical vorticity transport equation is equivalently transformed into a new form that explicitly emphasizes the rigid vorticity transport process.The rigid vorticity transport equation is expressed is given by [3] ( ) The first term on the right-hand side of Eq. ( 3) is denoted as the rigid vorticity stretching term (RST), which represents the normal strain (tension or compression) experienced by the vortex core line.The second term is referred to as the rigid vorticity dilatation term (RDT), which represents the tangential strain (torsion or bending) experienced by the vortex tube during the convection process.The third term is known as the curl term of the pseudo Lamb vector (RCT), which represents the change in intensity of rigid vorticity in accordance with the Biot-Savart law.The fourth term is the rigid vorticity viscous term (RVT), which accounts for the diffusion and dissipation of rigid vorticity caused by viscous effects.Compared with the single-phase flow, the interphase interaction is the key bond between the water phase and the fine particle phase.The final term corresponds to the curl of the interphase forces (Curl-Finterphase), including drag force curl (Curl-FD), virtual mass force curl (Curl-FVM), lift force curl (Curl-FL) and turbulent dispersion force curl (Curl-FTD), which signifies the contribution of interphase forces curl to rigid vorticity transport and serves as the primary focus of this research.

Research object and numerical simulation schemes 2.3.1. Computational domain and its spatial discretization
As depicted in figure 1, the computational domain for this flow case is determined based on experiments conducted at EPFL [15].A NACA0009 hydrofoil is employed to induce wake vortex shedding.The water tank has a length of 900 mm and a square cross-section measuring 150 mm×150 mm.The attack angle is α=0°, and the chord length L=100 mm.The Reynolds number, calculated using the free-stream velocity V=20 m/s, is approximately 2×10 6 .
To discretize the flow domain, high-quality hexahedral grids are utilized to guarantee the y+≈ 1 and the convergence analysis is conducted using the Grid Convergence Index (GCI) criterion recommended by ASME-FE [16].The final meshing results are presented in figure 1.

Solver setup and monitoring scheme
A transient numerical simulation is performed utilizing the ANSYS CFX code.The time step size, denoted as δt, is set to 5×10 -6 s in order to satisfy the CFL condition, resulting in an average Courant number of approximately 0.25.The liquid phase is water, while the solid phase is sand with a density of 2650kg•m -3 , a particle diameter of 25μm and a particle volumetric concentration of 0.38%.Regarding the discretization scheme, the convection term employs a second-order upwind scheme, the transient term adopts a second-order Euler post difference scheme, and the diffusion term utilizes a central difference scheme.As for the boundary conditions, the inlet of the computational domain applies a velocity boundary with a medium turbulence intensity of 5%, the outlet of the domain employs a static pressure boundary, and the wall surface adopts a non-slip boundary condition for water phase while a free-slip boundary conditions for sand phase.The calculation method employs a fully implicit coupling solution technique, and the convergence residual threshold is set at 1.0E-05.
The grid-independent verification and the results of numerical prediction accuracy in single phase flow can be found in references [3,17] and this study is based on these references.
The monitoring points (A, B and C) are depicted in figure 2. They have been chosen to facilitate a more accurate quantitative analysis of the contribution of interphase forces to rigid vorticity.These points are positioned near the trailing edge of the hydrofoil and are distributed along the y-axis (span direction) from top to bottom.It has been observed that a Karman vortex street is formed at the trailing edge through previous studies of hydrofoil flows.The upper vortex street, originating from the upper hydrofoil, passes through monitoring point A, while the lower vortex street, arising from the lower hydrofoil, traverses monitoring point C. Monitoring point B, located between A and C, experiences the influence of both the upper and lower vortex street.The vorticity characteristics in the vicinity of these three monitoring points exhibit distinct patterns, and the analysis of various physical quantities measured at these points can provide insights into underlying laws or trends.

Magnitude analysis of the interphase force and its curl: To highlight the dominant interphase interaction
The previous studies have been shown that the importance of each interphase force is different in different simulated cases.There is no consensus as to which interphase force dominates in numerous cases with different flow conditions.Some minor interphase forces are not taken into account in order to focus on the key issues and save the computational cost in numerical simulations.Hence, it is crucial to identify the dominant force among the four interphase forces considered in this case to simplify the analysis.A suitable approach is to compare the magnitudes of the interphase forces and the interphase force curls in this study, which is direct and simple to determine the significant interphase force that has the greatest impact on the rigid vorticity.On the whole, it is determined that the drag force curl plays a significant role among interphase force.Therefore, the other interphase force curls can be disregarded in subsequent analyses.

Transport characteristics analysis of rigid vorticity: To determine the role of the dominant interphase force curl
The determination of the contribution of the most significant interphase force, which is drag force, to the transport of rigid vorticity can be assessed by considering both the direction and magnitude of the drag force curl.Figure 6 displays the contour of the cosine value depicting the angle between the drag force curl and the rigid vorticity, along with the vector diagram illustrating the drag force curl in the context of watersand two-phase flow.The images predominantly exhibit shades of blue and red, indicating that the direction of the drag force curl aligns predominantly with the direction of the rigid vorticity.This observation is further supported by the vector diagram.Consequently, it can be concluded that the impact of the drag force curl on the vortex tube primarily manifests as induced positive strain, resembling a tension effect, thus further confirming its contribution.
Figure 7 (a)-(c) are the time dependent curves of deformation terms of rigid vorticity in monitoring points.It is obvious that deformation terms of rigid vorticity are larger in monitoring point B than these in other two monitoring points because of the effect of the upper and lower vortex street.The RCT and RDT play important roles in rigid vorticity transport from the average value, which means the generation of vortex and the torsion and bending deformation of vortex tube are more prominent in rigid vorticity transport.The RST and RVT are relative lower than RCT and RDT, which means tensile deformation and viscous effect are relatively weak in rigid vorticity transport.The curl-FD is consistent with RST in magnitude, but it is lowest among deformation terms, which means it plays a low role in rigid vorticity transport and has little effect on tension and compression of rigid vorticity.
On the whole, it can be concluded that RCT (O(10 6 )) > RDT (O(10 6 )) > RST (O(10 4 )) >RVT (O(10 4 ) >Curl-FD (O(10 4 ) from the magnitudes of deformation terms in this comparison, which means the contribution of the drag force curl to the transport of rigid vorticity is relatively low compared to the other deformation terms and RCT is still largest deformation term in rigid vorticity transport.Considering the direction of the curl of drag force, its impact on the vortex tube primarily manifests as induced positive strain, resembling a tension effect rather than a compression effect.Figure 11 illustrates the iso-surfaces (the threshold is ||ωR||2 = 600s −1 ) of rigid vorticity under both single-phase and two-phase flow conditions.A comparative analysis of the apparent vortex structure reveals a notable similarity in terms of the number and shape of the vortex tubes, which also reflected drag force curl has limited effect on rigid vorticity.On the whole, there are slight decrements of average and fluctuation of rigid vorticity in monitoring point A and C under water-sand two-phase flow compared with single-phase flow while a slight increment of amplitudes in dominant frequency.The presence of sand particles changes the dominant frequency in monitoring point B, but does not change the dominant frequency in monitoring point A and C. The overall macroscopic features of both conditions are roughly same from the rigid vorticity structure.

Summary and conclusions
In this study, the contribution of interphase force curl to rigid vorticity transport in water-sand two-phase flow with fine particles is studied by using a classic case of flow around a hydrofoil.The main conclusions are drawn as follows: (1) The dominant interphase force and its curl are highlighted.Among the four interphase forces considered in the water-sand two-phase flow with fine particles, the drag force emerges as the dominant interphase force.Similarly, when examining the interphase force curls, it becomes evident that the drag force curl takes precedence among the four interphase force curls.
(2) The role of the dominant interphase force curl is determined.The influence of the drag force curl on rigid vorticity is primarily observed in the form of normal strain, which is close to tension or compression motions.However, when considering the magnitudes of the rigid vorticity deformation terms, the contribution of the drag force curl is relatively low compared to other terms.
(3) The intuitive contribution of the dominant interphase force curl is presented.The time and frequency characteristics of vorticity display slight variations between single-phase flow and water-sand two-phase flow conditions.Furthermore, when examining the iso-surfaces of rigid vorticity, the overall macroscopic features of both conditions are roughly same.
Overall, although it proves that the effect of fine particles on the rigid vorticity transport of the water phase is limited, this study is still valuable for the application and development of the newly developed rigid vorticity theory under the multiphase flow conditions.As a prospect, more studies on coarser particles and relatively higher sand concentration can be carried out in the future because of the poorer following behaviors, stronger inertia and significant interactions of eddy-particle [18], thereby enriching the contribution to fluids engineering.

Figure 2 .
Figure 2. Monitoring points in the wake flow fields.

Figure 3 (
a)-5(a) are the time dependent curves of interphase forces.The curves of FD, FVM, and FL exhibit relatively small fluctuation amplitudes due to the low slip velocity.However, the FTD is influenced by the sand concentration gradient, resulting in a significantly larger fluctuation amplitude.Monitoring point B (figure 4(a)) exhibits a beat frequency signal attributed to the influence of vortex street.It can be concluded that FD (O(10 1 )) > FVM (O(10 0 )) > FL (O(10 0 )) >FTD (O(10 -2 ) from the magnitudes of interphase forces in this comparison.

Figure 3 ( 3 . 4 . 5 .
b)-5(b) illustrate the time dependent curves of interphase force curls.A similar pattern is found in the curl of the interphase force.It can also be concluded that Curl-FD (O(10 4 )) > Curl-FVM (O(10 3 )) > Curl-FL (O(10 3 )) > Curl-FTD (O(10 1 ) in this case.(a) Interphase force (b) Interphase force crul Figure Distributions of the interphase force and its curl at monitoring point A. (a) Interphase force (b) Interphase force curl Figure Distributions of the interphase force and its curl at monitoring point B. (a) Interphase force (b) Interphase force curl Figure Distributions of the interphase force and its curl at monitoring point C.

Figure 6 .
Figure 6.Vector angle cosine between the drag force curl and the rigid vorticity.

Figure 7 .
(a) Monitoring point A (b) Monitoring point B (c) Monitoring point C Deformation term distributions of rigid vorticity at different monitoring points.

3. 3 .
Comparison of the Karman vortices: To display the intuitive contribution of the dominant interphase force curlFigure8(a)-10(a) depicts the time dependent curves of rigid vorticity under single-phase and two-phase flow conditions.

8 . 9 .
(a) Time domain (b) Frequency domain Figure Distributions of the rigid vorticity under single-phase flow and two-phase flow conditions at monitoring point A. (a) Time domain (b) Frequency domain Figure Distributions of the rigid vorticity under single-phase flow and two-phase flow conditions at monitoring point B. (a) Time domain (b) Frequency domain Figure 10.Distributions of the rigid vorticity under single-phase flow and two-phase flow conditions at monitoring point C. The relative difference of the average rigid vorticity and fluctuation under two flow conditions in three monitoring points does not exceed 10% and the average level is about 5%.The changes vary at different monitoring points.Figure 8(b)-10(b) depicts the frequency dependent curves of rigid vorticity under single-phase and two-phase flow conditions.The dominant frequencies are same under two flow conditions in monitoring point A and monitoring point C. The dominant frequencies are different in monitoring point B with 10.41% difference, which means the presence of sand particles change the dominant frequency in monitoring point B. The amplitudes in dominant frequency are slightly increased under the condition of two-phase flow compared to the single-phase flow in monitoring point C while the amplitude in dominant frequency obviously increased in monitoring point A and monitoring point B under the condition of two-phase flow.