Comparative assessment and analysis of Rortex vortex in swirling jets *

Nan Gui, Liang Ge, Peng-xin Cheng, Xing-tuan Yang, Ji-yuan Tu, , Sheng-yao Jiang

1. Institute of Nuclear and New Energy Technology, Collaborative Innovation Center of Advanced Nuclear Energy Technology, Key Laboratory of Advanced Reactor Engineering and Safety of Ministry of Education,Tsinghua University, Beijing 100084, China 2. School of Engineering, RMIT University, Melbourne, VIC 3083, Australia

Abstract: Recently, a new definition, called Rortex, was proposed to quantify the purely rotational motion of fluids. In this work,based on the DNS data, the Rortex is used to assess and visualize the rotational motion and structure of the vortex in swirling jets in comparison with other kinds of vortex criteria, including Q, 2λ , vorticity and Ω criteria. The Rortex vector, Ω , Q and 2λ criteria are found to be better than the vorticity criterion for the vortex core identification. The vector triangle formed by the Rortex R, the nonrotational shear S, and the vorticity VΩ is analyzed to give mechanical explanations, especially of the effect of the non-rotational shear on the rotation of fluids. In addition, the probability density distributions (PDF) of the Rortex R, the nonrotational shear S, and the vorticity VΩ are computed. The peak value of the PDF of the vorticity could be used to explain the pure rigid rotation effect and the combination effects of the rigid rotation and the non-rotational shear.

Key words: Swirling flow, vortex, Rortex, direct numerical simulation, rigid rotation, non-rotational shear

Introduction

As well-known, the identification of vortex plays an important role in studying the feature of the fluid flow and uncovering the mechanisms of turbulence.The most conventional definition of vortex by thecriterion is based on the curl of the fluid velocity V , i.e.

whereVΩ is called the vorticity. Hence, a connectedis regarded as a vortex.More recently, based on the work of Chong et al.[1],with ?V being the local velocity gradient tensor, its characteristic equation is fluid region with

(1) Q- vortex[2]: It is defined by the positive second invariant, where A, B are the symmetric and anti-symmetric parts of the velocity gradient tensor ?V . Q represents the balance between the shear strain rate and the vorticity magnitude.

(2) λ2- vortex[3]: To avoid complex eigenvalues,Jeong and Hussain[3]defined the λ2- vortex by the characteristic equation of. The connected are coefficients defined after the rotation Q transformation of the velocity gradient tensor. Notice that R could be positive or negative or 0, indicating the clockwise and anti-clockwise rotations. When Eq. (2)has one real root and two complex roots, the minimum rotational strength R in the plane perpendicular to the eigenvector corresponds to the real eigenvalue(root of Eq. (2)).region with the second largest eigenvalueless than zero, i.e.,is regarded as a vortex.vortex[4]: When the characteristic equation Eq. (2) has two complex eigenvalues, the imaginary part of the complex eigenvalue of the velocity gradient tensor is also used to quantify the local swirling strength of the vortex.

(4) Omega vortex[5]

whereT=trace( )a A A ,T=trace( )b B B and ε is a small positive number used to avoid division by zero.

(5) Rortex vortex[6-8]: Liu et al.[6]used a new decomposition = + +?V A C D , where C is regarded a rigid rotation tensor with an angular speed of/2 R .

to define a pure rigid rotation, with

where

Motivated by the clear physical meaning of the Rortex-vortex, we would revisit the vortex in the swirling flow. Notice that the swirling flow is characterized by the motion of the fluid swirl imparted onto a directional jet flow or without the directional jet flow[9-10], it might be one of the flow patterns mostly consistent with the definition of the Rortex vortex. Therefore, the direct numerical simulation data of the swirling jet flows in a rectangular container are utilized here, and the comparison of the Rortex-vortex,the Q- vortex, thevortex, thevortex is carried out to assess the performance and the capacity of these vortex criteria.

1. Numerical method

1.1 Governing equations

In swirling jets[11-14], the Navier-Stokes equations for the time-dependent, incompressible viscous fluids, based on the conservation laws of mass and momentum, are expressed in the full three-dimensional dimensionless form as follows:

where u, p, Re are the fluid velocity vector, the pressure and the Reynolds number, respectively.=? e is the gradient operator. “?” is the inner product. Equatins (6) are solved directly on the structured grids with its resolution as required by a direct numerical simulation. The upwind compact schemes[15]are used to discretize the convection term.The fourth-order compact difference schemes[16]are applied for the space derivatives and the pressuregradient terms. The third-order explicit schemes are used to deal with the boundary points, maintaining the global fourth-order spatial accuracy. The fourth-order Runge-Kutta schemes[17]are used for time integration.The pressure-Poisson equation is solved to obtain the pressure by using the fourth-order finite difference method[18]. The present simulation tool has been validated[11].

1.2 Numerical setup and validation

In this work, a swirling air jet of diameter =1cm is issued from the surface center of a rectangular box of dimension 0.10 m×0.05 m×0.05 m at the inlet velocity. The flow domain is discretized by 512×256×256 Cartesian mesh grids. The Reynolds number is/ =3 000 ν , where the kinematic viscosity. The parameters used in the present simulation are listed in Table 1.

In the swirling flow, it is assumed that the rotational motion in the azimuthal direction is a translational streamwise motion. The inlet velocity profiles of the streamwise u , the azimuthal v velocities are shown in Figs. 1(a), 1(b), where the swirl number Snis defined as the ratio of the flowrate of the rotational momentum to the translational momentum as

Table 1 Real parameters and corresponding dimensionless values used in simulation

Fig. 1 (Color online) Velocity profiles at the inlet

In addition, the program for computing R vortex,vortex is provided by Professor Chaoqun Liu from the Department of Mathematics, University of Texas at Arlington, USA.

Fig. 2 (Color online) Snapshots of 3-D vorticity V Ω for= 0.36n S , and a central slice extracted from (c)

2. Numerical results

In this work, the features of the vorticity vortex( )VΩ , the Q- vortex, the λ2- vortex, the Omega vortex ( )

LΩ and the Rortex vortex R are compared.

2.1 Vorticity presentation

The vorticity presentation (Eq. (1)) in the case ofis shown in Fig. 2. The vortex is expanded in the lateral and spanwise directions after being initially issued into the flow domain at t=4(Fig. 2(a)), and expanded continuously to make a group of rather complicated twisted vortices ( t= 12,Fig. 2(b)). The fully developed swirling vortices( t=20) are shown in Fig. (2(c)) with a clear bubble vortex breakdown (VB for short[20-21], Fig. (2(d)) and the turbulent motions downstream the VB region.These features of the VB and the complicated vortices are generally consistent with our early results on the swirling flows[11-14,22].

Fig. 3 (Color online) Vortex structures of S n =0.36 at t=20

2.2 Comparison amongst Q, LΩ , VΩ , 2λ and R

Then, we take t=4 (Fig. 3), t=20 (Fig. 4)for the case study, to show the vortex presentations by Q (a),LΩ (b),VΩ (c),2λ (d), and R (e),respectively. It is clearly seen that all presentations have similar structures. They all indicate that the major ring-like vortex structure is formed perpendicular to the streamwise direction, where the interaction between such major streamwise vortex ring and the central axial direct vortex tube is through a circumferential array of secondary vortices. The secondary vortex rings follow a spiral distribution around the major vortex ring. The structure can all be identified in all kinds of vortex presentations. Thus, in such a free swirling flow, the Q,LΩ ,VΩ ,2λ and R presentations are all valid for the vortex study. It is also noticed that the diameter of the swirling jet visualized byVΩ near the immediate inlet of the flow (Fig. 3(c)) is larger than those obtained by other kinds of vortex presentations, especially larger than that presented byLΩ (Fig. 3(b)),2λ (Fig. 3(d)),and R (Fig. 3(e)). In fact, the diameter of the jet presented by the latter (e.g.2λ ) is realistic. It means that the vortex presented byVΩ has some shortcomings, for the possibility of making a fake recognition of the vortex cores.

Moreover, the strong vortex breakdown in the downstream of the bubble can be clearly seen from Fig. 4. The bubble shape can be more clearly recognized by Q (Fig. 4(a)),2λ (Fig. 4(d)), and R(Fig. 4(e)). The strongly kinked small scale vortices in the downstream of the bubble are rather complex.Their structures are very similar, especially those presented by Q (Fig. 4(a)),2λ (Fig. 4(d)) and R(Fig. 4(e)). In Fig. 4(b) forLΩ , the small scale vortices are somewhat different from those presented by Q (Fig. 4(a)),2λ (Fig. 4(d)) and R (Fig. 4(e)),since more small scale vortices can be seen immediately after the bubble VB, masked as the VB.Moreover, some additional small vortices can be observed in the central and further downstream regions (Fig. 4(b)). Notice that all the vortex criterion presentations are computed from the same data. They are at the same time point under the same flow condition, including the same level of the swirl number Sn. It means that theLΩ definition can recognize the additional small scale vortices, which other criteria cannot, except the R criterion.

In Fig. 4(e), the additional small scale vortices can also be recognized by the R criterion, and very completely since the R level (=2) may be a bit larger. In other words, the R vortex can be used to clearly visualize the large scale bubble VB, and the strongly kinked small scale vortices and the additional small scale vortices in the further downstream region.It would be one of the best candidate for the vortex presentation. ComparingLΩ with R, it is shown that the latter is somewhat better, since some additional small scale vortex can still be observed in the immediate inlet of the flow domain, which are not recognized by other criteria except theVΩ in Fig.4(c).

Fig. 4 (Color online) Vortex structures of =0.36 at t=20

TheVΩ (Fig. 4(c)) may be not a good candidate for the vortex presentation although it has been used the most widely with a longest history. It is seen from Fig. 4(c) that the diameter of the jet near the inlet is larger than those obtained by other vortex presentations. It is in fact a fake since the jet inlet is not so large, and it is caused by the shearing (larger radial velocity gradient) around the periphery of the main jet.Moreover, the 3-D bubble VB is not so clear as presented in other cases.

Fig. 5 (Color online) Visualization of ΩV (red-vectors), R(green-vectors) at the locations of R ≠0

2.3 Difference between VΩ , R: The S vector

According to the former analysis, more attention is paid to the difference between the “worst” and “best”candidates of the vortex criterion, i.e.,VΩ , R. At first, taking the case of =0.36at t=16 for example, theVΩ ,R vectors on the location of R ≠ 0 are shown in Fig. 5. In Fig. 5(a), the red and green vectors are theVΩ , R vectors, respectively, at the same location at the same time. Particular attention is paid to the local region-A (Fig. 5(b)) and region-B(Fig. 5(c)) for a detailed inspection, and it is clear that VΩ , R results are always different in both magnitudes and directions, although they sometimes look consistent with each other.

Fig. 6 (Color online) The relation between

Notice that Liu et al.[5,23-24]and Liu et al.[6]proposed an equation

where S is the difference betweenVΩ , R, which means the non-rotational shear vector in the vorticity.Based on the vector Eq. (8),VΩ , R and S form a vector triangle, which is completely defined by the length of the vector magnitudes. Therefore, we just compute the ratios of the vector norms, e.g.,,, as shown in Fig. 6.

It is interesting to see from Fig. 6 that,have very similar distributions amongst the flows of various Snand time. We have just fitted the enveloping curves (see C1, C2 and C3 curves in Fig.6(f)) in the last case of =0.36at t=40. It is seen that other cases, including the cases of =0(Fig. 6(a)),=0.10(Fig. 6(b)), =0.20(Fig.6(c)), and at earlier time t=10 of =0.36(Fig.6(e)) are all within the three enveloping fitted curves:

Regarding the geometric relation between the sides of a triangle, Curve C1 indicates the limiting relation

which means that the non-rotational pure shear vector S is in the opposite direction of the pure rigid rotational motion vector R (Fig. 7(a)). In physics,for example, if a local fluid point is rotational clock-wise, the induced anti-symmetrical deformation is anti-clockwise, which partly counteracts the rigid rotational effect of the fluids. In other words, the shear motion plays the role of the flow resistance to the rotational motion of the fluids.

Curve C2 indicates the limiting relation

It means that the pure rigid rotational vector R is in the same direction of the local non-rotational shear vector S (Fig. 7(b)). It means that the rotational effect is caused by the shear deformation of the fluids.In other words, the shear deformation plays the role of accelerating the rotational flow to the fluid points.

For Curve C3, we may linearize the fitted curve of Eq. (9) as follows

where γ is a coefficient. It means that both the non-rotational shear vector and the pure rotational vector has a partial contribution to the vorticity,depending on the constant γ . For a geometrical explanation, as shown in Fig. 7(c), the non-rotational shear vector S has combinational effects: one is through accelerating (or decelerating) the rotation of the fluids (lengthening or shortening R caused byThe other one is through changing the direction of rotation of the fluids (changing direction of R caused by S⊥).

Fig. 7 Sketches of the geometrical configuration amongst ΩV ,R and S

2.4 Number distribution of criteria for the points of R ≠ 0

Then the number distribution (ND) of the vortex criteria at the fluid points of R ≠0 is quantified,which is defined as.=number of points having the same values of x:

where the variable x can be R, S andVΩ , here.

Figure 8 for NR(a), NS(b) shows the lg-lg plots offor different Sn. Just like the division of the energy-spectrum, the number distributions of R and S can also be divided similarly into three parts. Particular interest is paid to the central part with an almost linear relation betweenas shown in the inset of Fig. 8(a) forand the inset of Fig. 8(b) for (1≤, respectively. In this part, the number distribution nearly follows a power-law as

where the exponent K is in the range of -3.3 ＜for NSas obtained by a linear fitting.for,

Fig. 8(a) (Color online) The lg-lg plot of number distribution of N R for different S n (main frame) on R≠0 points, and local linear part of lg()-lg( (leftdown inset)

Fig. 8(b) (Color online) The lg-lg Plot of number distribu-tion of N S for different S n (main frame) on R≠0 points, and local linear part of lg-lg (upright inset)

Besides, the comparison of Nxamongst,andVΩ is shown in Fig. 9. It is seen that only lg(-lg( )R , lg(-lg( )S have a local linear lg-lg distribution, whereas no such relations are found for NVΩ. lg(always has a peak value almost at the largest S, which means that there are a large proportion of NVΩlocated within the region with pure R ≠ 0 and without S, namely, the pure rigid rotational region without non-rotational deformation(as denoted in Fig. 9(a)). Otherwise, both the rigid rotational and non-rotational deformation may coexist.Additionally, when Snbecomes large, the peak ofbecomes wider but the largestVlocates almost at the same value ofimmediately beyond the largest S. In other words, the most probability ofis always around the region of nearly zero S.

Fig. 9(a) (Color online) The lg-lg plot of number distributions of and on R ≠ 0 points for=0 at t=40

Fig. 9(b) (Color online) The lg-lg plot of number distributions of and V on R ≠ 0 points for=0.36 at t=40

3. Conclusions

In this work, various vortex criteria are applied to identify the vortex structures in swirling flows. Based on the observations and analyses, the key features and conclusions of vortex criteria can be summarized as follows:

(1) All vortex criteria can be generally used to characterize the basic feature of the vortex structures,such as the bubble vortex breakdown, the axial major vortex ring, and the spiral distribution of the secondary vortex ring, in the swirling flows. A phenomenological comparison shows that the vorticity criterion VΩ performs worse than others since it cannot correctly recognize the jet inlet diameter, and it cannot distinguish the shear from the pure rotational motion.Instead, theLΩ ,2λ , Q and R criteria can identify the vortex core and the jet diameter correctly.

(2) TheVΩ , R criteria have the ability to identify more additional secondary small scale vortices immediately after the VB and in the far downstream, which cannot be clearly seen by the2λ ,Q criteria at the same levels of strongly kinked small scale vortices.

(3) TheVΩ , R and S vectors form an interesting geometrical triangle. They have two limiting cases, i.e., S is in the opposite direction of R ,which indicates the role of the non-rotational shear in either resisting or promoting the rotational motion, i.e.,decelerating or accelerating the rotation of the fluids,respectively. Otherwise, S may have combinational effects not only on accelerating or decelerating the rotation strength of the fluids R (caused by the parallel component of S||) but also on changing the direction of rotation (caused by the perpendicular component S⊥).

(4) The number distribution function has parts of linear relation between,in the lg-lg presentation. This indicates the power-law distribution characteristics of NR, NS, whereas there is no such relation forVΩ . However, NVΩhas the most high probability on the locations with a pure rigid rotation (R ≠0) without non-rotational shear( S=0).