Royal Society Publishing

On the ejection-induced instability in Navier–Stokes solutions of unsteady separation

Aleksandr V Obabko, Kevin W Cassel


Numerical solutions of the flow induced by a thick-core vortex have been obtained using the unsteady, two-dimensional Navier–Stokes equations. The presence of the vortex causes an adverse pressure gradient along the surface, which leads to unsteady separation. The calculations by Brinckman and Walker for a similar flow identify a possible instability, purported to be an inviscid Rayleigh instability, in the region where ejection of near-wall vorticity occurs during the unsteady separation process. In results for a range of Reynolds numbers in the present investigation, the oscillations are also found to occur. However, they can be eliminated with increased grid resolution. Despite this behaviour, the instability may be physical but requires a sufficient amplitude of disturbances to be realized.


1. Introduction

Unsteady separation is a high Reynolds number phenomenon involving the formation and detachment of a shear layer from a surface. As such, it would not be surprising to find that this is a highly unstable event. In this investigation, a possible instability is considered that may arise in solutions of the full two-dimensional Navier–Stokes equations for flows undergoing unsteady separation. Brinckman & Walker (2001) have computed the unsteady Navier–Stokes equations for the prescribed spanwise flow induced by longitudinal vortices that extend in the streamwise direction, as is the case within a turbulent boundary layer. Their results reveal the presence of a high-frequency instability in regions where the primary recirculation region is split by the ejection of near-wall vorticity. The instability is observed in the form of oscillations in the vorticity field that are found to be well resolved. An estimate of the wavelengths of the oscillations suggests that the instability may be of the Rayleigh type. In the present results for the flow induced by a thick-core vortex above a plane wall, oscillations appear in the same location and behave in a similar fashion to those observed by Brinckman and Walker. However, it is found that the oscillations may be removed by increasing the resolution by a sufficient amount.

The instability identified by Brinckman & Walker (2001), here referred to as the ejection-induced instability, is postulated to be a Rayleigh instability (e.g. Smith & Bodonyi 1985; Tutty & Cowley 1986). The Rayleigh instability is an inviscid instability that may occur when disturbances are present that have a streamwise length-scale, with a wavenumber, α, that is of the same size as the thickness of the viscous boundary layer (i.e. 1/α=O(Re−1/2)). The flow within the boundary layer is unstable to Rayleigh modes when a sufficiently strong inflection point is present within the streamwise velocity profiles. In a related study by Cassel & Obabko (submitted), it has been found that another Rayleigh instability is observed in Navier–Stokes solutions for the same flow owing to the thick-core vortex. This instability, however, is observed at higher Reynolds numbers than that required for the ejection-induced instability, but it begins earlier in time. In fact, at high enough Reynolds numbers, the instability appears well before the primary recirculation region even forms within the boundary layer.

The same thick-core vortex model problem treated in the present investigation has been used to consider other aspects of the unsteady separation process. For example, Cassel (2000) and Obabko & Cassel (2002) have considered how solutions of the Navier–Stokes equations compare with the theoretical description of unsteady separation as formulated asymptotically in the limit as Re goes to infinity. Based on these results, it has been hypothesized that viscous–inviscid interaction occurs over two distinct streamwise scales depending upon the Re of the flow. At very high Reynolds numbers, where the asymptotic theory applies, a sharp spike provokes a small-scale interaction. At moderate Reynolds numbers, a large-scale interaction occurs owing to the development of strong outflows from within the boundary layer prior to the formation of the spike. The growing spike then provokes the small-scale interaction. It is in this regime that the ejection-induced instability observed by Brinckman & Walker (2001) occurs. At low Reynolds numbers, there is only large-scale interaction and no spike formation and subsequent small-scale interaction. In this investigation, the behaviour of the ejection-induced instability in two-dimensional calculations of the Navier–Stokes equations for the flow owing to the thick-core vortex is illustrated and discussed.

2. Formulation

A thick-core vortex above a plane surface is considered in this investigation. The inviscid flow resulting from the thick-core vortex consists of a semicircular vortex, in which the vorticity is proportional to the stream function, surrounded by the irrotational flow from a circular cylinder in a uniform flow with speed U0 (Batchelor 1967). The thick-core vortex is shown schematically in figure 1 in a frame of reference moving with the vortex. Note that dimensional variables are denoted by an asterisk. The inviscid vortex has a velocity Vc relative to the wall owing to its self-induced flow and the oncoming uniform flow U0. To characterize the speed of the vortex, the following parameters are defined (see Doligalski & Walker 1984; Degani et al. 1998):Embedded Image(2.1)The fractional convection rate of the vortex relative to the uniform flow speed is denoted by α, and β is the non-dimensional wall speed in a frame of reference moving with the vortex. The uniform flow speed in this reference frame is U0(1−α). The case considered in this investigation is that with β=0 (α=0), which corresponds to a vortex with sufficient self-induced velocity to exactly balance the uniform flow and remain stationary relative to the wall (i.e. Vc=0) in the inviscid case. The characteristic length and velocity scales are taken to be the normal distance from the wall to the centre of the vortex, given by r0, and the velocity at infinity, which is U0(1−α), respectively. Therefore, Re is defined by Embedded Image, where ν is the kinematic viscosity.

Figure 1

Schematic of the thick-core vortex in a frame of reference moving with the vortex. In the present study, α=β=0.

The initial condition for the Navier–Stokes calculations is the inviscid solution for the thick-core vortex above an infinite plane wall. As shown in Cassel (2000), the non-dimensional stream function in cylindrical coordinates (r, θ) in a frame of reference moving with the vortex isEmbedded Image(2.2)Here, the radius of the vortex is taken as e (the base of the natural logarithm), and k and r0 are constants chosen such that ke and kr0 are the first zeros of the first-order Bessel function of the first kind and its derivative, respectively. The non-dimensional location of the centre of the thick-core vortex in a frame of reference moving with the vortex is (x,y)=(0,1), and the inviscid stagnation points are located at x=±2.0811. The local flow direction beneath the thick-core vortex is from right to left, and the pressure gradient along the surface is adverse between the origin and the left stagnation point.

The two-dimensional, incompressible Navier–Stokes calculations are initiated impulsively from the inviscid solution at t=0 and are carried out using the vorticity-stream function formulation. The vorticity transport equation isEmbedded Image(2.3)and the Poisson equation for the stream function is given byEmbedded Image(2.4)with boundary conditions being no-slip at the wall and an expanding Rayleigh layer for |x|→∞. A grid transformation is used to focus grid points near the wall and in the streamwise region where unsteady separation occurs. The vorticity transport equation is solved using a factored ADI algorithm with Jensen's method applied for the vorticity boundary condition at the surface, and the Poisson equation for stream function is solved using a multigrid algorithm. The latter algorithm is second-order accurate, while the factored ADI algorithm is O(ΔxΔt, ΔyΔt) accurate owing to the manner in which upwind–downwind differencing is implemented in the context of the Crank–Nicolson time-marching procedure for equation (2.3). (See Cassel (2000), Obabko (2001) and Obabko & Cassel (2002) for additional details about the formulation of the model problem, initial and boundary conditions, grid transformations and the numerical algorithms used.)

3. Results

Solutions to the Navier–Stokes equations (2.3) and (2.4) have been obtained numerically for a range of Reynolds numbers 103Re≤105. Grids with up to two million points having minimum Δx=1.53×10−4, Δy=2.75×10−4 and time-steps as small as 2×10−6 have been used to obtain the results shown, and all solutions have been tested for grid independence.

The boundary-layer flow induced by the presence of a vortex is characterized by an adverse streamwise pressure gradient that leads to the formation of a recirculation region at approximately t=0.4 for the thick-core vortex with β=0. This recirculation region is shown at t=1.3 for Re=104 in figure 2a. Above a Re of approximately 5×103, corresponding to the lower extent of the moderate Re regime (Obabko & Cassel 2002), a hump forms on the upstream side of the recirculation region as shown in figure 2a centred near x=0.8. As the flow evolves, secondary vorticity develops beneath the recirculation region owing to the locally adverse pressure gradient that is produced. This secondary vorticity, which is of opposite sign as the primary vorticity, is eventually ejected away from the surface and splits the recirculation region into two corotating eddies. This is shown for Re=104 in figure 2b near x=0.72 just prior to splitting of the primary recirculation region and figure 2c immediately after penetration of the ejection of secondary vorticity through the primary recirculation region. Note that the early stages of a second ejection and splitting of the upstream portion of the primary recirculation region are apparent near x=−0.55 in figure 2c at t=1.9. For comparison, the singularity time for the corresponding non-interacting boundary-layer solution is approximately t=1.4.

Figure 2

Instantaneous streamlines and vorticity contours for Re=104, β=0. (a) t=1.3, (b) t=1.65, (c) t=1.9.

The unsteady separation process described above occurs on a cascade of scales at these Reynolds numbers. This is illustrated in figure 2b for Re=104 at t=1.65. It is the thick-core vortex (not shown on this scale), which has negative circulation, which produces the adverse pressure gradient, leading to formation of the primary recirculation region as shown in figure 2a. By t=1.65, the primary recirculation region splits into three corotating eddies centred near x=−1.20, −0.85 and −0.65 owing to the influence of large-scale interaction (Obabko & Cassel 2002). Each of these eddies has positive circulation and induces its own locally adverse pressure gradient along the surface beneath it. The central eddy, having the strongest adverse pressure gradient, is the first to induce its own secondary eddy near x=−0.72. This secondary eddy then induces its own locally adverse pressure gradient along the surface that leads to formation of the tertiary eddy near x=−0.75. Note that for each generation of eddy, the adverse pressure gradient and the subsequent splitting is on the downstream side of the eddy with respect to the local flow direction adjacent to the wall. In the moderate Re regime, this process is observed repeatedly on the upstream side of the primary recirculation region, for example at x=−0.55 in figure 2c, and is discussed in more detail in Obabko & Cassel (2002). Similar behaviour has also been observed in numerical calculations of a vortex-induced flow by Brinckman & Walker (2001) and in the unsteady separation behind a circular cylinder computed by Koumoutsakos & Leonard (1995), see figs. 21 and 26).

The Navier–Stokes calculations of Brinckman & Walker (2001) contain the ejection-induced instability in vorticity in the region where the ejection occurs (e.g. figure 2c). This same behaviour has been observed in the present calculations. An example of this oscillatory behaviour is shown in figure 3a for the case with Re=104. Note that the vorticity contours in this figure are shown at a time slightly before that shown in figure 2c for the same case, but on a coarser grid. Brinckman & Walker (2001) graphically estimated the wavelength of the oscillations and found that the wavenumber scales close to O(Re1/2), which is consistent with a Rayleigh instability. Note, however, that the Re range used to obtain their scaling spans less than one order of magnitude. In addition, they claim that the oscillatory behaviour is grid independent.

Figure 3

Instantaneous vorticity contours for Re=104, β=0 at t=1.8. The zero vorticity contour is bold. (a) 513×513 grid, (b) 1025×513 grid, (c) 2049×513 grid.

In the present investigation, however, the oscillations could be removed by increasing the grid resolution sufficiently. For example, results for the same case as shown in figure 3a, but with two times and four times the streamwise resolution, are shown in figure 3b,c, respectively. Doubling the streamwise resolution significantly reduces the oscillations, and they are eliminated completely in the solution with four times the streamwise resolution. The degree of grid independence achieved for the overall flow field with these grid resolutions is illustrated in Appendix A of Obabko (2001). Figure 4 shows a similar set of plots of vorticity for Re=3×104 in which two regions of instability are observed in figure 4a owing to the occurrence of two simultaneous ejections. Figure 4b shows the results for the solution with double the streamwise resolution in which the oscillations in the region of the right ejection are eliminated, and figure 4c shows the results for the solution with four times the streamwise resolution in which all of the oscillations are eliminated. No attempt has been made to estimate the dominant wavenumbers of the oscillations in the present results because they are clearly different for each of the two ejections in the Re=3×104 case (as shown in figure 4a), making determination of Re scaling impossible.

Figure 4

Instantaneous vorticity contours for Re=3×104, β=0 at t=1.7. The zero vorticity contour is bold. (a) 1025×513 grid, (b) 2049×513 grid, (c) 4097×513 grid.

4. Discussion

The ejection-induced instability described in this investigation for the flow induced by a thick-core vortex was first observed by Brinckman & Walker (2001) in numerical calculations of a similar flow. The instability is integrally connected with the unsteady ejection event described in §3 and in more detail in Obabko & Cassel (2002). The adverse pressure gradient produced by the thick-core vortex leads to formation of a recirculation region. In what Obabko & Cassel refer to as the moderate Re regime, marked by the presence of both small-and large-scale interactions, this recirculation region is eventually split into multiple eddies owing to ejections of secondary vorticity. Brinckman & Walker (2001) refer to the region where the ejection occurs as an alley-way, and it is in this region where oscillations in vorticity develop. The authors' estimates of the wavenumber of this oscillation suggest that the instability may be a Rayleigh instability, for which the wavenumber is O(Re1/2).

For the highest Reynolds numbers considered by Brinckman & Walker (2001), the time at which the instability sets in is before the singularity time in the corresponding non-interacting boundary-layer solution. As postulated by Cowley (2001), this suggests that the boundary-layer solution may not be the high Re limit solution of the Navier–Stokes equations for flows involving unsteady separation. In other words, an instability may set in before the onset of the non-interacting boundary-layer singularity. In addition, this behaviour makes it less probable that the ejection-induced instability is a finite Re manifestation of the instability identified by Cassel et al. (1996) in the first interactive stage (Elliott et al. 1983), which is an asymptotic structure governing over a short time-scale surrounding the non-interactive singularity time.

The present investigation, which uses a somewhat different numerical algorithm to solve a different flow to Brinckman & Walker (2001), also exhibits the same ejection-induced instability. This provides additional evidence that the ejection-induced instability may be a physical instability. Although Brinckman & Walker's (2001) results have been reported to be grid independent, the present results show that the instability is removable with sufficient grid resolution. Apparently, refining the grid reduces inaccuracies evident near the saddle points in the vorticity field that occur in the region where the ejection takes place. It is possible that it is these inaccuracies, which provide the initial perturbations, then grow according to the instability observed by Brinckman & Walker (2001). This suggests that as the resolution of the numerical calculation is increased and the corresponding accuracy is improved, the disturbances that naturally arise in numerical calculations are reduced until the perturbations are insufficient to excite the instability over the appropriate time-scales. In other words, the conditions necessary for the onset of the instability may exist for only a short period of time in this highly unsteady event, and the amplitude of the perturbations must be sufficient to excite the instability within this short time frame. This could be confirmed by performing a high-resolution calculation (e.g. corresponding to figure 3c) and then introducing random perturbations in the solution at the appropriate time. The amplitude of the perturbations would be commensurate with the truncation error of the solution on a coarser grid (e.g. corresponding to figure 3a), and one would expect to observe oscillations comparable with those observed in a calculation performed solely on the coarse grid. An additional difference between the present study and Brinckman & Walker's (2001) results is that we found that the instability occurred well after the singularity time for the corresponding boundary-layer solution.

The different behaviour reported in the two studies with regard to the influence of grid refinement may be a result of two factors. First, the model problem considered by Brinckman & Walker (2001) is such that the external inviscid flow consisting of the periodic longitudinal vortices is uncoupled from the developing viscous near-wall flow that it induces. In fact, special conditions are required to provide an appropriate upper boundary condition for computation of the near-wall flow. In this investigation, however, the entire flow field (including the thick-core vortex) is computed. That is, the present model problem accounts for interaction between the boundary-layer flow and the thick-core vortex that is driving the flow, whereas the model problem considered by Brinckman & Walker does not. The second factor contributing to the inconsistent behaviour in the two sets of results may be differences in spatial and temporal resolution. For example, in the region where the ejection-induced instability occurs, the streamwise resolution used in the present computations for Re=104 is approximately a factor of two times that used in the simulations carried out by Brinckman & Walker (2001). Therefore, it is probable that the maximum resolutions used by Brinckman & Walker are not sufficient to remove the instability for the Reynolds numbers considered.

Finally, the relationship between the ejection-induced instability addressed in this investigation with a Rayleigh instability identified in the same model problem by Cassel & Obabko (submitted) is discussed. It has been found that for Re≥105 a Rayleigh instability is present within the Navier–Stokes solutions for the thick-core vortex. This instability arises much earlier than the ejection-induced instability. In fact, for the highest Reynolds numbers considered (i.e. Re=107, 108), the Rayleigh instability is manifest in the calculations even before initiation of the recirculation region that forms as a result of the adverse pressure gradient along the surface owing to the presence of the thick-core vortex. Therefore, the ejection-induced instability is pre-empted by the earlier Rayleigh instability for large Reynolds numbers and, if it is physical, the ejection-induced instability only appears to be the dominant instability for a fairly narrow Re range of 5×103<Re<105 for the flow induced by the thick-core vortex.


This work has been partly supported by the US Army Research Office under contract DAAG55-98-1-0384, Dr Thomas L. Doligalski, technical monitor. Travel funds have been provided by the Engineering and Physical Sciences Research Council. The authors thank R. I. Bowles, S. J. Cowley and F. T. Smith for valuable discussions regarding this work. This paper is dedicated to the memory of J. D. A. Walker.


  • Present address: Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637, USA.

  • One contribution of 19 to a Theme ‘New developments and applications in rapid fluid flows’.


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