1 Introduction

Vortex is a common structure of fluid flow. It is viewed as "sinews and muscles of fluid motions" by Küchemann ^[1]. Vortex scale ranges from millimeters in small vortex filament of turbulence ^[2] to kilometers of tornadoes ^[3] and vortex in ocean or even huge vortex of galaxy ^[4]. Vortex has many applications in engineering. The vortex in the lee-side of a delta wing can generate nonlinear lift. Vortex can be used to increase the mixing efficiency of fuel and air in a combustion chamber. However, vortex is destructive in some cases, e.g., tornado ^[5]. The efficiency of either the usage or control of vortex depends on the structure of the vortex, and the mathematical formulation is still poorly described in complex flows ^[6].

The structures of vortex are often visualized by plotting three-dimensional (3D) streamlines, sectional streamlines, and friction lines. Sectional streamlines are plane curves, in which the tangential directions are the same with the vector containing two components of the velocity in the cross section. Friction lines are limits of streamlines to the wall. In practice, sectional streamlines are often plotted and studied theoretically ^[7-9], numerically ^[10], and experimentally through the digital measurement for the velocity components with particle image velocity (PIV) ^[11]. It has been shown that there are limit cycles in sectional streamlines and friction lines of 3D vortical flows ^[7-11]. However, sectional streamlines are not physically observable lines except in very special situations. Whether a limit cycle appearing in these vector lines is physical or only a mathematical artifact due to the cut plane visualization is a valid question. In this paper, we report a new physical structure for vortical flow, i.e., tubular limiting stream surface (TLSS). The TLSS is a tornado-like structure, which has a close relationship with a limit cycle. The work here is devoted to a universal structure which is nevertheless relevant to the physical flows in real systems.

2 Tubular limiting stream surface

We study the structure of vortex from the streamlines defined by

(1)

where X is the position of the fluid particle, U=(u, v, w) is the vector of the velocity which has three components, i.e., u, v, and w, corresponding to the coordinates x, y, and z, respectively. The local velocity field around a point X can be expressed by

(2)

where D=∇U is the velocity gradient tensor, which can describe many fundamental and intrinsic properties of small-scale motions in turbulence and local flow topology structure ^[12-13]. Its characteristic equation is

(3)

where

They are the three invariants of the velocity gradient tensor. The discriminant of the characteristic equation (3) is

where

In the sequel, we choose a special section to define the sectional streamlines, which is spanned by the pair of complex eigenvalues, corresponding to the vortex defined by Chong et al. ^[14]. In this case, the discrimination Δ of the characteristic equation (3) is positive, and the velocity gradient tensor can be decomposed as follows:

(4)

where λ_r is the real eigenvalue with a corresponding eigenvector v_r, and λ_cr ±iλ_ci are the conjugate pair of the complex eigenvalues with the complex eigenvectors v_cr±iv_ci. In a local coordinate (y₁, y₂, y₃) system defined by the three vectors v_r, v_cr, and v_cr, the local streamlines can be expressed as follows ^[15]:

(5)

where c_r, c_c⁽¹⁾, and c_c⁽²⁾ are constants. Figure 1 contains the 3D streamlines and sectional streamlines in the cut planes (y₂, y₃) and (y₁, y₂), where

The local flow is either stretched or compressed along the axis v_r. While, there are essential differences in the flow patterns of the sectional streamlines in different cut planes. On the plane spanned by the vectors v_cr and v_cr, the flow is swirling. The sign of λ_cr determines the spiral direction of the vortex. If λ_cr >0, the vortex spins inward. If λ_cr < 0, the vortex spins outward.

Therefore, the particular sectional streamlines of interest lie in the cross section spanned by the vectors v_cr and v_ci. For convenience, we use (x, y, z) to represent (y₁, y₂, y₃) in the following content. Denote the vortex axis by the x-axis. Then, our sectional streamline can be given by

(6)

According to the critical point theory for ordinary differential equations ^[16], the sectional streamline pattern in the vicinity of the vortex axis depends mainly on the following two parameters:

There are three kinds of critical points, i.e., saddle, node, and focus. For a swirling flow, the sectional streamline pattern should be a focus in the near region of the vortex axis ^[14]. From the continuity equation, Zhang et al. ^[14] obtained that the spiral direction could be determined by

where the subscript o represents values on the vortex axis. If λ(x, t)>0, the sectional streamlines in the cross-section perpendicular to the vortex axis spiral inward. If λ(x, t) =0, the sectional streamlines in this cross-section form a nonlinear center. If λ(x, t) < 0, the sectional streamlines in this cross-section spiral outward. If λ changes its sign along the vortex axis, one or more limit cycles will appear in the cross-section.

Figure 2 is an example of an unstable limit cycle in the sectional streamlines on a typical cross section x=-5.0 perpendicular to the vortex axis in the interaction of a shock wave and an isentropic vortex ^[10]. If the flow is steady, the first term of λ(x, t) vanishes, and the function λ(x, t) can be rewritten as follows:

If the flow is isentropic,

where a is the sound speed of the fluid. If the viscous effect is neglected,

and λ(x, t) can be represented by

which was firstly obtained by Zhang ^[17]. Here, M_x² is the Mach number along the vortex axis. From this equation, we find that, there is an essential difference between the structures of a subsonic vortex and a supersonic vortex. For a subsonic swirling flow, the sectional streamlines in the vicinity of the vortex axis spin inward in a locally favorable pressure gradient region and outward in a locally adverse pressure gradient region. However, for a supersonic swirling flow, the sectional streamlines in the same vortex region go in the opposite directions, e.g., spin outward in the locally favorable pressure gradient region and inward in the locally adverse pressure gradient region.

If the velocity components (v, w) in Eq. (6) are replaced by the skin friction forces (τ_y, τ_z) on a wall, the vector lines represent the skin friction lines, which are often used to visualize the pattern of the flow separation in experiments ^[18]. In certain conditions, there are also one or more limit cycles in the skin-friction lines ^[20].

From the viewpoint of nonlinear dynamical systems of ordinary differential equations, it is easy to understand that there are one or more limit cycles ^[10] in the sectional streamlines and skin-friction lines given by Eq. (6). The question is that whether these limit cycles are physical or only a mathematical artifact due to the cut plane visualization. In fact, because the velocity component perpendicular to the cross section may be non-zero and most flows are not axis-symmetric, the limit cycle appearing in the cross section perpendicular to the vortex axis may not be physical. Besides, the limit cycle of the skin friction on a wall is physical, which can be visualized in experiments. However, no matter whether the limit cycle in a cross section or on a wall is physical or not, it definitely has physical meanings, i.e., the appearance of a limit cycle in the sectional streamlines represents the fact that the radial velocity is in the opposite directions on the two sides of the limit cycle. Therefore, there is a TLSS, which can separate the vortex into two regions, i.e., the inner region near the vortex axis and the outer region away from the vortex axis. The flow particles in these two regions can approach to (or leave) the TLSS, but never could reach it. The structure of the TLSS is schematically shown in Fig. 3(a). It is similar to a "tornado" (see Fig. 3(b)). A TLSS will capture and swallow the passively floating debris nearby. If a picture is taken toward the vortex axis, a dark region, which is viewed as a "black hole", in the case of stable limit cycles (lighting region in the case of unstable limit cycles) will be observed in the near region of the vortex core.

3 Theoretical and numerical examples

The following contents are five examples containing TLSSes. The first example is the Sullivan vortex ^[19]. The velocity field is given by

(7)

where

and u(r), v(r), and w(r, x) are the velocity components in the streamwise, radial, and circumferential directions of the cylinder coordinates (x, r, θ), respectively. It is an exact solution of the Navier-Stokes equations.

Figure 4 contains a TLSS (grey), the sectional streamlines (red) in two typical cross sections, and the 3D streamlines (blue) on the two sides of the TLSS. The Sullivan solution is axis-symmetric. The limit cycle matches exactly with the TLSS.

In the second example, a TLSS starts at a limit cycle of the skin friction lines on a wall. The velocity field is given by

(8)

It is a theoretical solution of the velocity near the wall given by Surana et al. ^[20]. The skin friction field is

(9)

If μ>0, an attracting limit cycle appears at y²+z²=μ in the skin-friction lines of the skin vectors of Eq. (9). There is a limit cycle in the sectional streamlines of the two velocity components (v, w). The skin-friction line is the limit of the sectional streamlines toward the wall. In Fig. 5(a), we plot the skin-friction lines. A stable limit cycle is clearly visible. In Fig. 5(b), we plot the TLSS (grey), the 3D streamlines (yellow), the limit cycle (blue) in the skin friction, and a limit cycle (red) in a sectional streamlines of a typical cross section off the wall. We can observe that the TLSS starts exactly from the limit cycle of the skin-friction on the wall. The 3D streamlines starting near the wall on the two sides of the limit cycle approach to the TLSS. However, it never could reach it. The limit cycle in the sectional streamlines of the cross section off the wall does not match the TLSS. This is a typical flow separation starting from a limit cycle on the wall that may be one typical formation of hairpin vortex in the turbulent boundary layer ^[21-22].

There are three numerical examples in the following contents, which are obtained by solving 3D unsteady Navier-Stokes equations, in which the nonlinear terms are discretized by a second-order no-parameter non-oscillatory dissipation (NND) scheme ^[23] (the first example) and a fifth-order finite difference weighted essentially non-oscillatory (WENO) scheme ^[24] (the second and third examples). The viscous terms are discretized by a fourth-order central difference scheme, and the time derivative is discretized by the third-order TVD Runge-Kutta method in the direct numerical simulation (DNS) for the second and third numerical examples. For the first numerical example, a dual-time step with the LU-SGS method is used in the time integration.

Our first numerical example is a 3D flow over a prolate spheroid. The ratio of the major to minor axes is 6. The computational domain is ten times the major axis in the upstream of the prolate spheroid, thirty times in the downstream, and ten times in the radial direction. The grid density is 400 × 120 × 200 in the streamwise, circumferential, and normal directions, respectively. The code validation can be found in our previous works ^[25-26]. The Mach number of the free stream is 0.3. The attack angle is 40°, and the Reynolds number is 10⁶. Figure 6 contains the flow structure. Figure 6(a) contains the 3D streamlines (blue), the iso-surface of density (grey) with ρ=0.968 7, which represents two separated vortices in the lee-side of the prolate spheroid and a tornado-like structure (red). Figure 6(b) contains the skin-friction lines on the surface of the prolate spheroid. A stable limit cycle is observed in the boxed region (see Fig. 6(c)). Figure 6(d) is the enlarged figure for the skin-friction lines (blue) in the boxed region of Fig. 6(b) as well as 3D streamlines (yellow). A tornado-like structure is clearly observed, which is similar to that given in Fig. 5. As far as we know, this is the first time to find this kind of flow separation.

The second numerical example is the interaction of a plane shock wave and a 3D isentropic vortex ^[10]. Following the simulation in Ref. [27], a stationary shock is initially located at the x=0 plane. In the upstream of the shock (x < 0),

(10)

and the downstream mean solution is (x>0)

(11)

An isentropic vortex is superimposed on the mean flow upstream of the shock. The axis of the vortex is along the x-axis (y=z=0). The analytical forms of such vortices with arbitrary radial profiles are steady state solutions of the Euler equations. The perturbations of the azimuthal velocity u'_θ and the temperature associated with the vortex are given by

(12)

where is the radius to the vortex axis, r₀ is the vortex core radius, and ϵ is a non-dimensional circulation at r=1, which is related to the dimensional circulation Γ by

(13)

The axial and radial velocity components u'_x and u'_r are zero, the entropy,

is constant inside the vortex, and u'_θ is maximum at r=1.

The computational domain is set to be

The validation of our code and the grid convergence has been given in Ref. [10]. In this case, the Mach number of the normal shock wave is M₁=2, and the strength of the vortex is ϵ=7. The grid density is 300 × 120 × 120, corresponding to the x-, y-, and z-directions, respectively.

Figure 7 contains the TLSS and 3D streamlines around the TLSS at four typical instants t=2.0, 3.0, 5.0, and 11.0, respectively. In Fig. 7(d), we plot the sectional streamlines at a cross section of x=-6.2 perpendicular to the vortex axis. We can observe that there is a limit cycle in the sectional streamlines. But the limit cycle does not match with the TLSS.

Figures 4-7 show the instantaneous flow structures described by the 3D streamlines of Eq. (1), which may not be a material line for unsteady flow. The material line or material surface can be shown by the trajectory of the fluid particle described by

(14)

whose solution is denoted by X(t; t₀, X₀), where X₀ refers to the initial position of the particle at the time t₀. A streamline (Eulerian frame) is obtained by integrating Eq. (1) over space by the icing time. Moreover, a trajectory line (Lagrangian frame) is obtained by integrating Eq. (14) over the space-time domain. Only in steady case, a streamline is the same with the trajectory starting from the same initial point. The evolution of fluid elements in the space-time domain is described by the flow map

(15)

which takes any initial position X₀ to a later position at the time t. The Lagrangian strain in the flow is often characterized by the right Cauchy-Green strain tensor field

In Fig. 8, we present the time evolution of particles, which are injected from a small circle near the vortex core in the plane x=-7.5. The particles move with the flow. When they arrive at the critical point on the vortex axis, they move along the TLSS. The material surface takes the structure of the TLSS. Therefore, the TLSS can be observed in both the Eulerian frame and the Lagrangian frame. In the near region of the vortex axis, we do not find any particle, which means that there is a "black hole".

Our last example is a direct numerical simulation of 3D decaying compressible isotropic turbulence. The computational domain is a cube with 2 π periodic boundary conditions. The initial turbulent Mach number is M_t=0.3, and the initial Reynolds number based on the Taylor micro scale is Re_λ=72. The density of the numerical grid is chosen as 512 × 512 × 512 after a grid convergence study. The code is run to , where τ is the initial large-eddy-turn over time. We refer to Ref. [28] for more details. Figure 9(a) presents the 3D streamlines. We can observe that there are many TLSS structures, e.g., the marked regions A, B, C, D, E, and F. Figure 9(b) is a zoomed figure of the marked region A, which takes the pattern of a TLSS. It should be the smallest tornado in the world. The TLSS is abundant in turbulent flow, and it might be the origin of the vortex structure ^[2]. It plays a very important role in the turbulence evolution, the statistical properties, and the turbulent mixing noise.

4 Conclusions

The revelation of the tubular limit stream surface has the potential in providing a new concept to control the vortical flow and in better understanding many flow problems. For example, the appearance of a TLSS is a disaster for combustion because it prevents the mixing of materials on the two sides of the TLSS. Therefore, it should be avoided. TLSS is the origin of the appearance of "black hole", which usually appears in experiments of fluid dynamics ^[29]. Moreover, the revelation of TLSS offers us better understanding for vortex dynamics and turbulent structure, such that the formation and evolution of hairpin vortices ^[21-22]. The contribution of a TLSS to the statistic property and mixing noise of turbulence should be further studied.