Wei-wen Zhao, Yi-qian Wang, Song-tao Chen, Chun-hui Ma, De-cheng Wan

1. Computational Marine Hydrodynamics Lab (CMHL), School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

2. School of Mathematical Science, Soochow University, Suzhou 215006, China

Abstract: The motion of fluid consists of different scales of coherent vortical structures. These vortical structures determine the characteristics of fluid motion and are key to understand fluid dynamics. In this paper, we study the fine control method of vortical structures based on the Liutex force field model. This is achieved by constructing a source term using Liutex and directly add it to the Navier-Stokes equations. To investigate the influence of the constructed Liutex force model on vortical structures, a flow past a cylinder at Reynolds number of 100 is numerically studied with different source term magnitude and region. The drag and lift forces on the cylinder, as well as the flow field near and behind the cylinder are compared and analyzed. Results show that Liutex force model can effectively strengthen or weaken the vortical structures based on different purpose.

Key words: Liutex vector, vortex dynamics, vortex control

Introduction

Vortex plays an important role in fluid motions.Understanding the dynamics of vortices is of significance to reveal the characteristics of flow field.For practical flow problems, engineers often design and construct special components to strengthen or weaken the vortical structures in the wake region.These designs could change the overall aerodynamic or hydrodynamic performance of the structures. For instances, in aeronautical engineering field, a leading edge on wings can increase the maximum lift due to the presence of leading edge vortex. In ocean engineering field, the helical strakes installed on the Spar can break up the formation of coherent vortex structures along vertical direction to mitigate the large amplitude vortex-induced motions[1]. These methods change the shape of structures to alter the flow field and vortex structures which requires a great deal of industrial experience.

Recently, some researchers try to directly control the vortices numerically. To control vortices directly,an accurate definition of vortex should first be given.According to Liu et al.[2], the vortex identification (VI)methods can be divided into three generations. The first generation methods, based on vorticity, are insufficient to represent the vortical structures due to the weak correlation between vorticity and vortex,especially in the near wall region for wall-bounded flows. The second generation VI methods were proposed to overcome this issue. Examples are Q,2λ, Δ,ciλ and other methods[3-7]. While most of these methods calculate a scale field from the velocity gradient and identify vortical structures by the iso-surface of these scalars with an arbitrarily chosen threshold value. The threshold somewhat represents the swirling strength of vortices, but the physical meanings of these scalars remain unclear. Moreover,these methods more or less involve shear and stretch contamination problems, in which the shearing and stretching are counted as a part of vortex strength. To resolve the above issues, Liu et al.[8]proposed the conception of Liutex. Unlike the previous second generation VI methods, the Liutex method is an eigenvalue-based method which is local, accurate, and unique[9]. Liutex is defined as a vector. The magnitude of Liutex represents the rotational strength of the fluid rotation, and the direction of Liutex represents the axis of local fluid rotation. Based on this definition, several methods including the Liutex method, objective Liutex method[10], Liutex-Ω method[11-12], Liutex core line method[13], which are classified as the third generation VI methods, have been proposed in recent years.

Based on the Liutex method, Yu and Wang[14]performed a preliminary study of directly numerically manipulating vortices and thus obtain vortex dynamics and deduce control strategies. In their studies, two vortex control models, namely the centripetal force field model and the counter-rotation force field model, were proposed to be added as a source term into the Navier-Stokes equations.

In this paper, we first introduce the Liutex and Liutex-based force field models and implement the force models with the OpenFOAM toolbox. Then the force field models are applied to flow past a circular cylinder at Reynolds number of 100. Influence on the strength and the region size of the Liutex-based force field are discussed.

1. Liutex vector and the Liutex force field model

The Liutex vector was first proposed by Liu et al.[8]. Initially, the Liutex vector is determined by rotating the velocity gradient tensor to a special coordinate system in which the rotation axis is coincident with the local z- axis. The rotation axis is also the real eigenvector of the velocity gradient tensor given that the other two corresponding eigenvalues are complex conjugate[9]. However, this definition is sophisticated and difficult to implement numerically. To simplify the calculation of Liutex, an explicit expression is derived

where R is the Liutex vector, R is the magnitude of Liutex vector, r is the real eigenvector, ω is the vorticity vector andciλ is the imaginary part of the complex eigenvalue.

To control vortices directly, the first thought is to modify the Navier-Stokes equations by adding an extra source term on the right hand side. This can be written by the following formula

The above equations describe the fluid motions for incompressible flow. Here, u is the velocity vector field, while p, ρ and ν represent the pressure, fluid density and kinematic viscosity,respectively, c is a coefficient to flexibly control the magintude or strength of the added force field and a is the Liutex-based force field source term to be constructed.

In previous work of Yu and Wang[14], two Liutex-based force field models are proposed. The first one is a centripetal force model, given by

where l is a vector that starting at any field point P and ending at the local mimimum pressure point P0which is regarded as the vortex core center point.R / 2 is half of the Liutex vector and represents the angular velocity of the rigid rotation part at P around vortex core center0P.

Another force field model that involves a time scale is given by

The time scale τ acts like a relaxation time during which the rigid rotation of the fluid is ceased gradually and the velocity is decreased to zero.

2. The implementation of force field model

In the present study, we provided two strategies to explore the possibilities of flow control by utilizing the Liutex-based force field. The definition of length scale l in force field model is simplified to l = u / ∇u for general consideration.

The first strategy is similar to the centripetal force model proposed by Yu and Wang[14]. The only difference between the present method and Yu and Wang’s model is the definition of length scale l. The main objective of this strategy is to study the vortex dynamics by observing the response of flow to see whether the Liutex-based force field model could strengthening or weakening vortices in certain regions.The incompressible Navier-Stokes solver is modified to include the source term in the momentum equations.The solving procedures are as follows: the Liutex vector and corresponding centripetal acceleration vector is first calculated at the current time step for each cell, and then the modified Navier-Stokes equations with additional source terms are solved to obtain the velocity and pressure field for the next time step.

The second strategy is different from the first one.It involves two individual computational meshes, one with objects and the other without objects. The main procedure is to extract the force field source term from the first mesh with objects and map the source term to the second mesh without objects. The domain sizes of the two grids are the same. At the current time step,the flow fields on the first mesh with objects are first obtained by solving the original unmodified Navier-Stokes equations, then the source term is calculated and mapped as a vector field to the grid without objects. The modified Navier-Stokes equations with force field source term are then solved on the mesh without objects to obtain the flow field driven by the Liutex force field.

3. Flow past a cylinder at Reynolds number of 100

In this section, Liutex force model applied to a two-dimensional flow past a cylinder at Reynolds number of 100 is going to be detailed. Figure 1 shows the computational domain with a circular cylinder in the center of domain. As illustrated in the figure, the domain size extends -10D to 20D in x-direction,-1 0D 10D in y-direction. The computational grid is shown in Fig. 2 and the total cell number is 58 520.

Fig. 1 (Color online) The computational domain of flow past a circular cylinder at Reynolds number of 100

Fig. 2 The computational grid of flow past a circular cylinder at Reynolds number of 100

3.1 Centripetal force model

For the centripetal force model, we setup three different control regions (CRs) to explore practical vortex control strategies. The locations and sizes of the CRs are illustrated in Fig. 3. CR1 is a small box that wraps the whole boundary of the circular cylinder.CR2 is located in the wake region behind the cylinder and extends to the downstream outlet boundary. And CR3 is a small and short box in the wake region.

Fig. 3 (Color online) Sketch of the control regions inside which the centripetal forces are added to the Navier-Stokes equations

Table 1 shows the statistical flow quantities of flow past the circular cylinder for different c of CR1 cases. These quantities include the time-averaged drag coefficient, the root-mean-square (rms) of the lift coefficient, the Strouhal number, the flow separation angle and the base suction pressure coefficient. When the centripetal force is added to the Navier-Stokes equations, e.g., c＞0, the separation angle is moving towards the stagnation point, and the drag and lift coefficients as well as the base suction pressure coefficient are increasing. If centripetal force is subtracted from the equations, e.g., c＜0, the drag and lift force coefficients are decreasing. When c=2,the lift force coefficient is 0.0562 which is a considerable huge reduction compared with the original solution. The reduction of lift oscillation can be up to 76.5%.

Table 1 The statistical flow quantities of flow past a circular cylinder for CR1

Figure 4 shows the pressure coefficients and wall shear stress on the cylinder surface for CR1 cases. The wall shear stress in Fig. 4(b) is nondimensionalized by the density and velocity with the following formula

Fig. 4 (Color online) Time-averaged quantities on the cylinder surface for CR1, (a) The surface pressure coefficients and (b) The wall shear stress

The surface pressure is obviously altered by the force field. A larger negative surface pressure is obtained with c＞0. While for negative c, the surface pressure magnitude is reduced to a smaller value. As for the wall shear stress, similar trends can be observed for different c values.

It is shown in Fig. 5 that the vortices are weakened with a negativec, while they can be strengthened with a positivec. In addition, phase shift have been observed, i.e., the distance between a pair of vortices is slightly different for three cases. This indicates that the force field model will slightly alter the shedding frequency of the cylinder. Taking (a) and(b) as examples. The distance between two neighbor vortex core center in (b) is smaller than that in (a),which implies that a negativecvalue should decrease the shedding period and increase the shedding frequency. The above tendency are consistent with the results of Yu and Wang[14].

Fig. 5 (Color online) Instantaneous spanwise Liutex contour(z-direction) for CR1 with different c values

Fig. 6 (Color online) Time-averaged quantities on the cylinder surface for CR2, (a) The surface pressure coefficients and (b) The wall shear stress

For CR2 cases the results are surprisingly in contradiction with CR1 cases. The statistical flow quantities for CR2 cases are listed in Table 2.Recalling that CR2 is located behind the cylinder and does not intersect with cylinder wall surface. For cases withc＞0, the drag and lift forces are decreasing,while for cases withc＜0, the drag and lift turns to be smaller compared with the uncontrolled cases. For the specific case ofc=2, the oscillating of lift force coefficient decreased from 0.239 to 0.1327 which takes up to 44.5% oscillation reduction. It can be also observed that unlike CR1 cases, when the CR is setup in the wake region behind the cylinder, the separation angle is insensitive to the coefficientc.

Table 2 The statistical flow quantities of flow past a circular cylinder for CR2

Figure 6 shows the pressure coefficient and wall shear stress on the cylinder surface for CR2 cases.Unlike CR1 cases, the surface pressures in CR2 cases are insensitive to the sign of coefficientc, e.g., the surface pressure distribution ofc=1 (red line) andc= -1 (blue line) are almost identical. However, it is sensitive to the magnitude ofc. With largercmagnitude, the surface pressure magnitude is becoming smaller, which accounts for the reduction of lift oscillation.

From the flow field visualized by the spanwise Liutex contour for CR2 as shown in Fig. 7, the opposite effect of the samecvalue for different CRs can be seen more clearly and vividly. Figures 5(b) and 7 both illustrates the result ofc=-2. However, for CR1 the negativecvalue will weaken the vortices, while for CR2 the effects are opposite. In addition, thec=-2 case not only changes the shedding frequency but also broadening the wake. On the other hand, adding centripetal force (c=2) ameliorate the situation by weakening the vortices and narrowing the wake.

The size of CR3 is 1.5D(length)×2.4D(height). For this specific CR3 we’ve setup four cases with different distances from CR to the cylinder center.The horizontal distance between the most left corner of the CR and the center of circular cylinder is 1D,2D, 3Dand 4D, and noted as cases A, B, C and D,respectively. The flow quantities are listed in Table 3.The separation angle is almost the same for four cases.As the distance increasing, the changes on lift and drag are becoming smaller.

Table 3 The statistical flow quantities of flow past a circular cylinder for CR3

3.2 Force field driven flow

Besides the centripetal force model, another Liutex-based force field model is proposed. In this model, two computational meshes are utilized, one with and the other without cylinder. The boundary conditions of two meshes are the same. The flow fields on two meshes are solved in sequences within each time step. Liutex-based force field is computed from the first mesh with cylinder and mapped onto the second mesh without cylinder. This is inspired by the immersed boundary method which does not require the explicit definition of geometry boundary. The two meshes are illustrated in Fig. 8. For the mesh without cylinder, the near cylinder region is refined in order to reach the same cell size level compared with the corresponding location in the mesh with cylinder. The total grid number for the mesh without cylinder is 30 892.

Fig. 7 (Color online) Instantaneous spanwise Liutex contour(z-direction) for CR2 with different c values

Fig. 8 Local view of the two computational mesh used in the simulation

The primary object of this section is to check when the cylinder is removed, whether the Liutex force field can be utilized to drive flow and form asimilar flow pattern like the flow field with presence of cylinder. Therefore, the CR is removed which means the force field is added to the whole computational domain. According to the preliminary analysis results in the previous section, if the cylinder boundary is inside the CR, the negative c should strengthen the vortical structures. Therefore, c=-1 is used for the following studies.

An example of solution mapping for Liutex source is illustrated in Fig. 9. The upper figure shows the instantaneous Liutex source contour which is obtained from a uncontrolled normal case of flow past circular cylinder. The lower figure is mapped from the upper solution. This is realized by the function of C++class meshToMesh in OpenFOAM which can perform field interpola- tion between different meshes. From the figure, we can see a very smooth mapped field for Liutex magnitude which is sufficient to apply to the mesh without cylinder.

Figure 10 is the streamwise velocity contour at four different time steps in one vortex shedding period.The velocity is nondimensionalized by dividing the free stream velocity at the inlet boundary. Similar flow patterns with alternative vortex shedding can be observed in both the uncontrolled flow past a circular cylinder and the Liutex-based force field driven flow without cylinder. Discrepancies present at several regions in the vicinity of the cylinder, especially at the locations behind the cylinder. The typical recirculation zone caused by the negative pressure zone behind the cylinder exists in the uncontrolled flow on the left figures, but cannot be observed in the force field driven flow on the right figures. This discrepancy can be also discovered in the nondimensionalized normal velocity contours, as shown in Fig. 11.

Fig. 9 (Color online) Instantaneous Liutex source magnitude contour in two grids. (a) is computed from flow field of a uncontrolled case, (b) is mapped from the solution of(a)

Fig. 10 (Color online)Instantaneous nondimensionalized streamwise velocity (x-direction) contour for flow field with and without cylinder at four different time steps in one vortex shedding period T

Fig. 11 (Color online) Instantaneous nondimensionalized normal velocity (y-direction) contour for flow field with and without cylinder at four different time steps in one vortex shedding period

4. Conclusions

Based on the Liutex method, two strategies of directly vortex control are proposed. The first strategy adds a centripetal force source term to the Navier-Stokes equations. And the second strategy solves the flow field on two individual computational meshes.An example of a two-dimensional flow past a circular cylinder in selected to demonstrate the effects of the two vortex control strategies. The main conclusion can be summarized as follows:

(1) For flow past a circular cylinder, whether the Liutex-based force field will strengthen or weaken the vortices is determined by the control region of the source term.

(2) The Liutex-based force field can be used to drive empty flow field without objects. The developed flow field has similar flow patterns as the flow field with objects.

Future work involves the application of Liutexbased vortex control methodologies to the threedimensional incompressible flows with more complex geometries, such as propeller, ship hull, and deepdraft column-stabilized floating structures. An effective strategy of vortex control is the key to improve the hydrodynamic performance of these marine structures. These should also be further investigated in future work.

Acknowledgment

Communications with Profs. Lian-di Zhou and Zheng Ma are highly appreciated.