1 Introduction

Unsteady flow separation has been an active research area for decades. In the study of unsteady flow separation, there are two different frames which are named as Lagrangian and Eulerian descriptions. In the Lagrangian perspective, the flow is described in terms of particle trajectories in the phase space while the Eulerian description is based on the quantities given at fixed locations in the space. For steady flows, the two descriptions give the same result. However, for unsteady flows, the two descriptions are very different. For example, the streamlines and streaklines of separation bubble shown in Fig. 1 give different separation points. The instantaneous streamlines vary largely as the time changes while the origin of the material spike given by streaklines remains fixed. In general, the instantaneous Eulerian description fails to predict the unsteady flow separation point correctly^[1-2].

In recent years, the Lagrangian approach has achieved a notable success in predicting the unsteady flow separation point. Using a Lagrangian definition that the separation point is a fixed point with an unstable manifold for the Poincaré map associated with the periodic flow, Shariff et al.^[3] showed that the separation point is located at the boundary point where the time average of the skin-friction vanishes. Yuster and Hackborn^[4] re-derived the zero-mean-friction principle for near-steady time-periodic incompressible flows in a rigorous mathematical way. Viewing separation profiles as non-hyperbolic unstable manifolds in the Lagrangian frame, Haller^[1] proposed an exact theory of unsteady separation for two-dimensional flows, which asserts that fixed flow separation occurs where weighted averaging skin-friction vanishes and admits a negative gradient, and this theory can be applied to both incompressible and compressible flows.

The finite-time Lyapunov exponent (FTLE) is inherently a Lagrangian description of flows because it measures the degree of particles' separation after a given interval of time. In particular, regions of high values for the maximal FTLE seem to be likely candidates for regions containing hyperbolic trajectories and their stable and unstable manifolds^[5]. Hence, the FTLE has been adopted in massive literatures to detect Lagrangian coherent structures (LCSs) and analyze the transport and mixing problems of flows. Considering that the separation profile behaves like an unstable manifold, Shadden et al.^[6] suggested that separation profiles can be captured by the FTLE fields computed by integrating backward in the time. However, the location of separation point was not given clearly. And the separation point at the wall always showed non-hyperbolicity while the ridges of FTLE in the flow field always contained hyperbolic trajectories. More recently, Lei et al.^[7] studied the unsteady flow separation of flows over an airfoil. The separation profile was captured by LCSs extracted from the FTLE field, and the separation point was identified by Haller's criterion.

Based on Haller's theory on the two-dimensional unsteady flow separation^[1] and motivated by the work^[6], we analyze the FTLE field along the wall for two-dimensional periodic flows and find that unsteady flow separation occurs at the minimal point of the FTLE along the wall. And a new Lagrangian criterion for unsteady flow separation in periodic flows is derived. The criterion is an extension of Haller's zero weighted averaging skin-friction principle, and makes it clear where the separation point is in the FTLE field, and thus can be used as a new analytical tool for locating flow separation. This criterion is verified with an analytical solution of time-periodic separation bubble and a numerical simulation of a time-periodic lid-driven cavity flow.

This paper is organized as follows. In Section 2, we derive the Lagrangian criterion of unsteady flow separation for two-dimensional periodic flows based on the principle of weighted averaging zero skin-friction given by Haller^[1]. In Section 3, we verify the criterion with two examples. Section 4 contains our concluding remarks.

2 A Lagrangian criterion of unsteady flow separation for two-dimensional periodic flows

Consider a two-dimensional velocity field v(x, t) = (u(x, t), v(x, t)) with the fluid particle motion satisfying

(1)

where x=(x(t; x₀, y₀, t₀), y(t; x₀, y₀, t₀)) denotes the trajectory of Eq. (1) passing through the point (x₀, y₀) at the time t₀. The no-slip boundary is applied to the flow at y=0 with

(2)

2.1 Haller's criterion

For a nonlinear dynamical system as Eq. (1) above, we first review Haller's theory on unsteady separation. According to Haller's theory, a necessary condition for separation can be expressed in the form of

(3)

leading to the Lagrangian form

(4)

For time-periodic flows, the separation criterion becomes

(5)

and the Lagrangian form is

(6)

2.2 The FTLE criterion

Based on Haller's theory above, we derive a new necessary criterion for periodic flows by analyzing the distribution of FTLE along the wall, and this criterion is an extension of Haller's theory.

The flow map associated with Eq. (1) is defined by

(7)

and the Jacobian of this map, usually called the deformation gradient, is given by

(8)

Differentiate Eq. (1) with respect to the initial positions (x₀, y₀), and we can obtain the matrix differential equation,

(9)

with the initial condition

(10)

We have u_x(x, 0, t)=v_x(x, 0, t)=0 along the wall because of the no-slip boundary condition (2). Then, along the wall, Eq. (9) is simplified to

(11)

Consider the continuity equation

(12)

At the wall, the continuity equation is simplified to

(13)

leading to the density relation,

(14)

We solve Eq. (11) with the direct integration. Then, we use Eq. (14) and the initial condition (10) to obtain

(15)

To simplify our notation, we let

(16)

and b(x₀, t₀, t) > 0 holds obviously. Along the wall, the Cauchy-Green strain tensor is denoted as

(17)

The tensor C_t0^t(x₀, 0) is symmetric and positive definite, and hence admits two real positive eigenvalues

(18)

The maximum eigenvalue of C_t0^t(x₀, 0) satisfies λ_max≥1, and λ_max = 1 holds only when a=0 and 0 < b≤1.

The FTLE of a flow from the time t₀ to the time t is denoted as

(19)

First, we consider the steady case. For a steady flow, according to Prandtl's separation criterion, at the separation point (γ, 0), we have

(20)

where τ_w is the skin friction, ν is the kinematic viscosity, and ρ is the density of fluid. Therefore, for any time interval, at the separation point, we have a(γ, t₀, t) = 0 and b(γ, t₀, t) = 1. The maximum eigenvalue of C_t0^t(x₀, 0) equals 1 only at the separation point. Therefore, the separation point admits a minimal FTLE along the wall for the steady case.

Then, we consider a periodic flow with a time period of T. At the separation point γ, according to Eqs. (15) and (6), we have a(γ, t₀, t₀+T) = 0 and b(γ, t₀, t₀+T) = 1, and hence the maximum eigenvalue of C_t0^t₀+T(x₀, 0) at the separation point equals 1. In the neighborhood of the separation point along the wall, the maximum eigenvalue of C_t0^t₀+T(x₀, 0) satisfies λ_max > 1 except for the separation point. The FTLE along the wall of a periodic flow is denoted as

(21)

Then, we can conclude that the separation point of a periodic flow admits a minimal FTLE along the wall, and thus we can detect the separation point by investigating the FTLE filed. Furthermore, according to Eq. (16), a(γ, t₀, t₀+T) and b(γ, t₀, t₀+T) are independent of t₀ for a periodic flow. Therefore, the separation criterion for the periodic flow can be written as

(22)

Notice that Eq. (5) is just a necessary condition for separation, and thus Eq. (22) is also a necessary condition for separation. For the reattachment point on the no-slip wall, Eqs. (5) and (6) also hold, and thus the reattachment point also gives σ₀^T(γ, 0) = 0. To further identify whether a critical point is the separation point or the reattachment point, we can visualize the flow field around the point by streaklines. At the separation point, a material spike will form, and particles leave the boundary to go into the interior flow.

The present zero FTLE criterion for the separation of periodic flows can be stated as the fact that flow separation occurs at the location of the minimal FTLE, which in general equals zero.

3 Verification for the Lagrangian criterion 3.1 Time-periodic separation bubble

We first test our criterion by a time-periodic separation bubble, which can be referred to Ref. [1] for detail. The original velocity field was derived by Ghosh et al.^[8] for the study of passive scalar transport near an unsteady separation bubble. The velocity field takes the form

(23)

with the wall located at y=0. We set ω=2π and β=3. Thus, the period of this separation bubble flow is T=1.

For nonlinear dynamical systems, stable manifolds produce ridges in the FTLE field computed using a forward integration in the time, and unstable manifolds are revealed with backward integration. Therefore, we compute the FTLE fields at the time t by backward integration in the time. A group of particles initially set around the grids are advected from the time t to the time t-T. A second-order finite-difference approximation of Eq. (8) can be expressed by these particles' trajectories as^[9-10]

(24)

where

(25)

Moreover, choosing δ_x and δ_y small enough will substantially increase the numerical accuracy of ▽F_t0^t(x₀, y₀)^[10] and that of the FTLE.

The distribution of FTLE along the wall computed at the typical starting time (t=0, 0.2T, 0.4T, 0.6T, 0.8T, and T) in a period is shown in Fig. 2. The left figure contains the global view of the distribution of the FTLE, and the right figure contains the zoomed view in the near region of x=-1. We can observe that, the distribution of the FTLE along the wall at the typical starting time is almost the same, which means that it is independent of the starting time for an incompressible periodic flow. A minimal FTLE is located at x=-1, which implies a separation point at x=-1. The location of separation predicted by the minimal FTLE principle is the same as that predicted by the weighted averaging zero skin-friction principle^[1] and that given by material lines shown in Fig. 1.

3.2 Time-periodic lid-driven cavity flow 3.2.1 Numerical methods

The lid-driven cavity flow is a classical benchmark problem with complex separation and reattachment topologies^[11]. For two-dimensional flows, the nondimensionalized computational domain consists of a square with the length L=1, which is shown in Fig. 3. The top wall moves in the x-direction at a velocity of steady or unsteady, which will be specified later. We perform the numerical simulation by solving two-dimensional time-dependent Navier-Stokes equations. A fifth-order weighted essentially non-oscillatory (WENO) scheme^[12], a fourth-order central finite difference scheme, and a third-order total variation diminishing (TVD) Runge-Kutta method are used to discretize the convection term, the viscous term, and the time term of the Navier-Stokes equation, respectively. Several steady cases with Reynolds numbers of 100, 400, 1 000, and 3 200 have been calculated first to validate our code. In all these cases, the Mach number based on the wall temperature and the inflow bulk velocity is Ma=0.1, the nondimensionalized velocity at the top lid is set to be u=1 and v=0, and the Prandtl number is Pr=0.72. Figure 4 contains the numerical results and their comparison with those of Ghia et al.^[13]. Excellent agreement can be observed, which validates our code and guarantees a well-resolved cavity flow.

For the periodic unsteady flow, the Mach number is set to be Ma=0.3, rendering the flow practically compressible. The nondimensionalized velocity at the top lid is set to be . The Reynolds number is 1 200. Since the trajectories of particles are obtained based on the integration of time series discrete velocity data according to Eq. (1), the accuracy of interpolation of velocity and integration should be guaranteed. Thus, a fourth-order Lagrange's interpolation of velocity and a fourth-order Runge-Kutta method are applied to Eq. (1).

3.2.2 Numerical results and discussion

Velocity components of u and v at the center of computational domain are used as indicators of the convergence of the numerical solution. Figure 5 shows the time evolution of the velocity components at the center of the cavity. In this case, the period of the flow is T=3π.

Instantaneous streamlines at the several different time in a period are shown in Fig. 6, and an unsteady characteristic is obvious with the upper secondary vortex periodically appearing and disappearing and the two bottom vortices moving around periodically.

In Fig. 7, we plot the time evolution of two saddle points on the bottom of the cavity and the distribution of the integral of weighted averaging skin-friction calculated by Eq. (5). At the saddle points, the skin friction equals zero. It is the separation point or the reattachment point in the Eulerian frame. The location of separation point (the left saddle point) on the bottom wall ranges from x=0.194 to x=0.267, and the location of reattachment (the right saddle point) ranges from x=0.662 to x=0.719. However, according to Haller's criterion (5)^[1], the separation point remains fixed at x=0.242, and the reattachment point remains fixed at x=0.693.

Figure 8 contains the distribution of FTLE along the wall and their zoomed view, and the plots of FTLE at the typical starting time are almost the same. We can observe that there is a minimum of FTLE at x=0.242, which implies that the separation occurs at x=0.242 according to our criterion (22). The location of separation identified by the minimal FTLE is the same as that predicted by Haller's criterion (5).

To analyze the characteristics of unsteady flow separation, we focus on the left part of the bottom wall. Prandtl stated that, at the separation location, a fluid-sheet projects itself into the free flow. In other words, at the location of separation, material spikes will form and transport particles from the vicinity of the boundary to other flow regions. For fixed flow separations, material spikes can be observed at separation points in experimental flow visualizations^[14]. We thus simulate the flow motion of particles near the separation point to visualize the material spike and validate the fixed separation. In Fig. 9, we compare the material spike and the leading-order profile which is in the form of x = γ + yf₀(t₀), where γ is the location of separation point, and f₀(t₀) is calculated as

(26)

according to Haller's theory. And notice that all the quantities in Eq. (26) are of wall quantities. As shown in Fig. 9, the leading-order separation profile agrees well with the material spike near the wall. Although the angle of separation profile does vary as the time changes, the flow motions of particles exhibit fixed flow separation.

4 Conclusions

A Lagrangian necessary criterion based on the wall FTLE for time-periodic flows is proposed. It is the separation that occurs at the point of the zero FTLE on the wall. This criterion is an extension of Haller's criterion^[1] and can be applied to both incompressible and compressible flows. The criterion can be used as a new approach for locating the separation point and enrich the information revealed by the FTLE field. An analytical solution of separation bubble and a numerical solution of lid-driven cavity flows are used to verify our new criterion. However, this criterion is restricted to time-periodic flows. The criterion for general time-dependent unsteady flows is left for future work.