Studies of vortex breakdown and its stability in a confined cylindrical container 7

CHAPTER 7* HARMONICALLY FORCING ON A STEADY ENCLOSED SWIRLING FLOW 7.1 Introduction In Chapter 6, the response of an axisymmetric time-periodic swirling flow in a confined cylinder t

CHAPTER 7*

HARMONICALLY FORCING ON A STEADY ENCLOSED

SWIRLING FLOW

7.1 Introduction

In Chapter 6, the response of an axisymmetric time-periodic swirling flow in a

confined cylinder to harmonically modulated rotation of the endwall has been

investigated Two things emerged from that study One was quite expected, that for

very low amplitude forcing, the response is well described by resonant behavior

However, the second finding was not directly obvious, and it seems to be unrelated to

resonances of the type described by the Arnold circle map model (Arnold 1965) To

help clarify the spatio-temporal responses at the slightly smaller forcing amplitudes,

we explore in this paper the response to the same type of harmonic forcing, but at

mean Re below the critical value for the Hopf bifurcation, so that we are harmonically

forcing a stable axisymmetric steady state

On the other hand, there has been much interest in the swirling flow in an enclosed

cylinder driven by the rotation of an endwall for applications where a high degree of

mixing is desired, such as in micro-bioreactors, albeit at a low level of shear stress (see

Yu et al 2005a, 2005b, 2007; Dusting et al 2006; Thouas et al 2007) The interest

stems from the very good mixing properties when the flow is operated above the

threshold for self-sustained oscillations as these provide chaotic mixing (Lopez and

Perry 1992) The concern is, of course, that the chaotic mixing is only present when

the Reynolds number is above a critical level for the Hopf bifurcation, and so one

would like to achieve comparable oscillations at lower Reynolds numbers, thereby

subjecting the biological material to lower damaging stress levels This motivated us to

explore the flow behavior of the steady state vortex flow under harmonic modulation

The study includes experimental investigation and numerical simulations of the

axisymmetric Navier-Stokes equations

7.2 Experimental Method

The experimental apparatus and technique used in this chapter are the same as

those presented in Chapter 2 In the present investigation, the Reynolds number was set

at 2600 and below with the modulation amplitude A varying from 0.005 to 0.04, and

the aspect ratio Λ was maintained at a constant value of 2.5 throughout Note that all

flow visualization photos were inverted for ease of comparison with numerical results

7.3 Numerical Method

In this study, the governing equations are the axisymmetric Navier-Stokes

equations, and they are solved using the streamfunction / vorticity / circulation form

with a predictor-corrector finite difference method as introduced in Chapter 3 The

computations presented in this chapter were performed by the author

7.4 Results and Discussions

First, the steady vortex breakdown state at Re = 2600 and Λ = 2.5 is described,

which is the basic state to be investigated under forcing This state is about 4% below

the onset of self-sustained oscillations, which set in at Re = 2710 for Λ = 2.5 via a

supercritical Hopf bifurcation with Hopf frequency ω0 ≈ 0.17 Dye flow visualization

together with computed streamfunction ψ, and azimuthal vorticity η of this basic state

are shown in Fig 7.1 The flow manifests a large steady axisymmetric vortex

breakdown recirculation zone on the axis For harmonically forcing, a wide range of

forcing frequencies was considered, with the forcing amplitude kept small, typically A

≤ 0.02 Having examined experimentally dozens of frequencies in the range ωf ∈[0.04,

0.5] for various amplitudes A, it is found that in all cases the power spectral densities

(PSD) from the time-series of the hot-film outputs only have power (above the

background noise level) at the forcing frequencies and its harmonics

Fig 7.1 The steady axisymmetric basic state at Re = 2600 and Λ = 2.5 (a) flow

visualization using food dye (only the axial region is shown), (b) computed streamlines

ψ, and (c) computed azimuthal component of vorticity η There are 10 positive (red)

and negative (blue) contours quadradically spaced, i.e contour levels are [min/max] x

(i/10)2, and ψ ∈[-0.00702, 8.305 x 10-5], η ∈[−4.12, 21.68]

The effects of the modulation amplitude with a forcing frequency not in resonance

with the natural frequency ω0 was first examined (in this case, ωf = 0.20, so ωf /ω0 ≈

1.17) Figure 7.2 presents PSD from hot-film outputs at forcing amplitudes A = 0.005,

0.01 and 0.02 These illustrate that the resultant flow is synchronous with the imposed

modulation frequency, even at very low forcing amplitudes This is in contrast to the

situation where a limit cycle flow is harmonically forced as presented in Chapter 6

There, the resultant flow is periodic for low forcing amplitudes, and the

quasi-periodic flow collapses to a quasi-periodic flow synchronous with the forcing frequency as

the forcing amplitude is increased above a critical level

Fig 7.2 Power spectral density from time-series of hot-film output for flows with Λ =

2.5, Re = 2600, ωf = 0.2 and forcing amplitudes A as indicated

Figure 7.3 presents PSD from the hot-film outputs at various forcing frequencies

with fixed A = 0.02 In all the experimental runs, it was checked that the hot-film

outputs from the two channels are in phase (peaks matching in time), providing

experimental evidence of the axisymmetric nature of the forced limit cycles Again, the

response in all cases is a flow synchronous with the forcing, but with more power

when ωf ≈ ω0

Fig 7.3 Power spectral density from time series of hot-film output for flows with Λ =

2.5, Re = 2600, A = 0.02 and forcing frequency ωf as indicated

The flow visualizations shown in Fig 7.4 at Λ = 2.5, Re = 2600, A = 0.01, and ωf =

0.171, 0.20, and 0.50 illustrate the enhanced oscillations when ωf = 0.171 ≈ ω0 In fact,

at ωf = 0.171, the forced synchronous flow is very similar to the natural limit cycle

flow for Re > Rec ≈ 2710, exhibiting axial pulsations For ωf = 0.20 the flow does not

exhibit as strong oscillations, but there are still observable movements of the dye sheet,

whereas for ωf = 0.50 the dye sheet is quite steady and very much like that in the A = 0

basic state shown in Fig 7.1

(a) ωf = 0.171

(b) ωf = 0.20

(c) ωf = 0.50

Fig 7.4 Dye flow visualization of the central vortex breakdown region at Re = 2600, Λ

= 2.5, A = 0.01 and ωf as indicated

In order to obtain a more quantitative measure of the amplitude of the forced

synchronous oscillations in the experiment, Figure 7.5 presents the peak-to-peak

amplitude of the hot-film output at Re = 2600, Λ = 2.5 over a wide range of ωf for

forcing amplitudes A = 0.01 and A = 0.02, along with that at Re = 2000 and A = 0.01

What is most striking is that the hot-film output amplitude spikes for ωf ≈ ω0 There

are also a number of other smaller spikes, the main ones at ωf ≈ 0.12 and ωf ≈ 0.22

These appear to be related to the 2:3 and 4:3 resonances with ω0, but if these other

spikes were simply other resonances with ω0, one would expect the 1:2, 1:3, 2:1

resonances to be at least comparable, but they are not evident

Fig 7.5 Peak-to-peak amplitudes of hot-film output with varying forcing frequency ωf

at Λ = 2.5, Re = 2600, and A = 0.01 and 0.02 The three dotted vertical lines indicate

the Hopf frequencies of the three most dangerous modes of the basic state at Re =

2600, as determined by Lopez et al 2001, where ωH1 = 0.1692, ωH2 = 0.1135 and ωH3

= 0.2181

Hence, it is conjectured that these other spikes in Fig 7.5 are 1:1 resonances with

secondary Hopf modes The frequencies associated with these secondary Hopf modes

were first detected experimentally by Stevens et al (1999), computed nonlinearly by

Blackburn and Lopez (2002), but most significantly, positively correlated with

secondary axisymmetric Hopf bifurcations from the basic state via linear stability

analysis by Lopez et al (2001) The three vertical dotted lines in Fig 7.5 correspond to

ωf = ω0 = 0.1692, ωf = ω1 = 0.1135, and ωf = ω2 = 0.2182, where ω0, ω1, ω2 are the Hopf

frequencies of the first three Hopf modes bifurcating from the basic state The values

quoted are their values determined by linear stability analysis (Lopez et al 2001) at Re

= 2600 The first Hopf bifurcation is at Re = 2710, and the second and third occur at

Re = 3044 and 3122 Of course, the Hopf frequencies vary with parameters (Re and Λ,

as well as A and ωf), but these variations are quite small The good correspondence

between these Hopf frequencies and the spikes in the hot-film response to ωf lends

strong experimental evidence to the spikes being 1:1 resonances with the most

dangerous axisymmetric Hopf modes

In the experiment, there are only quantitative measurements of the oscillation

amplitudes at the location of the hot-film probes, which could give a skewed picture of

the response To get a global measure, we turn to the numerical simulations, where we

are able to measure the total kinetic energy (Ek) of the flow in the entire cylinder As a

measure of the oscillation amplitude, we use the peak-to-peak amplitude of the kinetic

energy, ΔE, normalized by the kinetic energy of the steady flow without modulation at

the mean Re, E0, and scaled with ωf0.5 Figure 7.6 shows how ωf0.5ΔE/E0 varies with ωf

for various mean Re, all with A = 0.01 The response for Re = 2600 shows the same

spikes response as that observed in the hot-film data, with minor spikes at the same

frequencies At Re = 2000, smaller spikes are evident at the same frequencies, and the

reduction in the spikes is comparable to that observed in the experiments (see Fig 7.5)

This all lends confidence that the local hot-film measurements are representative of the

global dynamics

0.0 0.1 0.2 0.3 0.4 0.5 0.000

0.004 0.008

0.012

Re = 2600 Re = 2000 Re = 800

ω

f

ω f

Fig 7.6 Computed variation with ωf of the peak-to-peak amplitude of the kinetic

energy relative to the kinetic energy of the basic state, ΔE/E0, and scaled by ωf0.5, of

the synchronous state for A = 0.01, Λ = 2.5 and various Re as indicated The three

dotted vertical lines indicate the Hopf frequencies of the three most dangerous modes

of the basic state at Re = 2600, as determined by Lopez et al 2001, where ωH1 =

0.1692, ωH2 = 0.1135 and ωH3 = 0.2181

A few sample solutions at Re = 2600, Λ = 2.5, A = 0.01 at various ωf are shown in

Fig 7.7 As was observed in the experiments, for very low ωf = 0.01, the flow

undergoes a quasi-static adjustment as shown in the instantaneous streamlines (Fig

7.7a) For high-ωf (ωf > 0.3) the results show that the axial region that includes the

vortex breakdown recirculation is essentially steady, with the streamlines virtually

identical to those of the A = 0 steady state shown in Fig 7.1, and all the oscillations are

concentrated in the bottom and sidewall boundary layers The ωf = 0.2 state shows a

pulsating vortex breakdown recirculation on the axis, and for ωf = 0.171 ≈ ω0, these

pulsations are significantly more pronounced, as was observed in the experiment

While the instantaneous streamlines and the experimental dye sheets are

convenient to visualize the vortex breakdown on the axis, they are not particularly

enlightening in identifying the boundary layer responses to the modulations It is found

that the relative azimuthal vorticity, i.e the difference between the instantaneous

azimuthal vorticity η(t) and the azimuthal vorticity of the steady state at A = 0, η0, is

much more informative Figure 7.8 shows snap-shots of η(t)-η0 for ωf = 0.01, 0.171,

0.2, and 0.5 respectively These are the variations in the azimuthal vorticity

distribution (see Fig 7.1c for the mean η distribution) due to the modulations

(a) ωf = 0.01 (T ≈ 628.32)

(b) ωf = 0.171 (T ≈ 36.74)

(c) ωf = 0.2 (T ≈ 31.42)

(d) ωf = 0.50 (T ≈ 12.57)

t = 0 T t ≈ 0.17 T t ≈ 0.43 T t ≈ 0.68T t ≈ 0.86 T

Fig 7.7 Time sequences of contours of ψ at Re = 2600, Λ = 2.5, ε = 0.01 and ωf as

indicated; there are 16 positive (red) and negative (blue) contours quadradically

spaced, i.e contour levels are [min/max]x (i/16)2, and ψ ∈ [-0.00757, 0.0002972]

ωf = 0.01 ωf = 0.171 ωf = 0.20 ωf = 0.50

Fig 7.8 Snap-shots of the azimuthal vorticity modulation, η(t)-η0 (where η0 is the

steady η for A = 0), at various ωf as indicated, all at Re = 2600, Λ = 2.5, A = 0.01 and

at the same phase in the forced modulation There are 15 positive (blue) and 15

negative (red) contour levels with η∈[−0.2, 0.2]; some clipping particularly for the ωf

= 0.171 case is clearly evident

A number of salient features become immediately obvious One of them is the

alteration in the structure of the disk and sidewall boundary layers, particularly near

the corner where the disk meets the sidewall These alterations can be interpreted as

the formation of junction vortices (Allen and Lopez 2007) between the stationary

sidewall and the modulated rotating disk Another salient feature which is evident from

Fig 7.8 is the way that the sequence of junction vortices propagate up the sidewall and

collide at the axis near the top and combine to enhance the vortex breakdown

recirculation and amplify its pulsations This is particularly dramatic at the 1:1

resonance with ωf = 0.171 ≈ ω0 To illustrate this 1:1 resonance, a comparison was

made for the value of η(t)-η0 for Re = 2600, Λ = 2.5 (which without modulation

corresponds to the steady vortex breakdown solution in Fig 7.1) at A = 0.01 and ωf =

0.171, with the natural limit cycle solution at Re = 2800, Λ = 2.5, A = 0 Snap-shots of

these two solutions at five phases over one period are shown in Fig 7.9 Note that for

the natural limit cycle at Re = 2800, this Re is only about 3.3% above critical for the

Hopf bifurcation, and so η(t)-η0 is a very good approximation to the η-Hopf

eigenfunction (Lopz et al 2001) What is evident, particularly from the detailed time

sequence images, is that the length scale of the junction vortex scales with ωf-1 (Fig

7.8), and that for ωf ≈ ω0 the structure of the junction vortices are very similar to the

vortex structure of the Hopf eigenfunction (Fig 7.9) Furthermore, Lopez et al (2001)

have previously found that the length scales of the secondary Hopf vorticity structures

scale inversely with their Hopf frequencies, and hence the very good correspondence

between the imposed ωf and the length scales of the modulation-induced junction

vortices leading to the other 1:1 resonance spikes in the experimental (Fig 7.5) and

numerical (Fig 7.6) response diagrams

t = 0 T t ≈ 0.17 T t ≈ 0.43 T t ≈ 0.68T t ≈ 0.86 T

Fig 7.9 Snap-shots of the azimuthal vorticity modulation, η(t)-η0 (where η0 is the

steady η for A = 0) for (top row) the natural limit cycle at Re = 2800 and Λ = 2.5, and

(bottom row) the synchronous state at Re = 2600, Λ = 2.5, A = 0.01 and ωf = 0.171

There are 15 positive (blue) and 15 negative (red) contour levels with η ∈[−0.2, 0.2]

These actions of the modulation-induced junction vortices at Re = 2600 are

complicated by the resonant interaction with the nearby Hopf modes For lower Re, the

small amplitude modulations (A = 0.01) do not resonate with the Hopf modes (their