**3.1 Frequency analysis**

The Fast Fourier Transform (FFT) decomposes a signal in the time domain into sines and cosines, i.e. complex exponentials, in order to evaluate its frequency content. Specifically the FFT was developed by Cooley & Tukey (1965) to calculate the Fourier Transform of a K samples series with O(Klog2K) operations. Thus FFT is a powerful tool with low computational demand, but it can be performed only over data evenly distributed in time. In case of LDA recordings, these should be resampled and the original raw time series replaced with series uniform in time. As for the resampling techniques, simple methods like

have to be employed. These may be classified in: Unsteady Reynolds-averaged Navier-Stokes equations (URANS), Large Eddy Simulation (LES) and Direct Numerical Simulation

URANS employs the usual Reynolds decomposition, leading to the Reynolds-averaged Navier-Stokes equations, but with the transient (unsteady) term retained. Subsequently the dependent variables are not only a function of the space coordinates, but also a function of time. Moreover, part of the turbulence is modelled and part resolved. URANS have been applied to study stirred vessels by Torré et al. (2007) who found indications on the presence of flow instabilities from their computations; however their approach was not able to

LES consists of a filtering operation, so that the Navier-Stokes are averaged over the part of the energy spectrum which is not computed, that is over the smaller scales. Since the remaining large-scale turbulent fluctuations are directly resolved, LES is well suited for capturing flow instabilities in stirred vessels, although it is very computationally expensive. This has been shown for both single-phase (for example Roussinova et al., 2003, Hartmann

DNS consists on the full resolution of the turbulent flow field. The technique has been applied by Lavezzo et al. (2009) to an unbaffled stirred vessel with Re = 1686 providing

The above experimental or modelling investigations have to be analysed with suited tools in order to get information on flow instabilities. These consist mainly of signal processing techniques, which are applied to raw data, such as LDA recordings of the instantaneous

Among the techniques which have been employed in literature for the characterization of


Whereas the first two techniques have been largely used for the determination of the flow instability frequency, the latter has been used to evaluate the impact of flow instabilities on

The Fast Fourier Transform (FFT) decomposes a signal in the time domain into sines and cosines, i.e. complex exponentials, in order to evaluate its frequency content. Specifically the FFT was developed by Cooley & Tukey (1965) to calculate the Fourier Transform of a K samples series with O(Klog2K) operations. Thus FFT is a powerful tool with low computational demand, but it can be performed only over data evenly distributed in time. In case of LDA recordings, these should be resampled and the original raw time series replaced with series uniform in time. As for the resampling techniques, simple methods like

et al., 2004, Nurtono et al. 2009) and multi-phase (Hartmann et al., 2006) flows.

velocity, in order to gain information on the characteristics of flow instabilities.


Two kinds of information have been extracted so far:


flow instabilities in stirred vessels, there are (see Fig. 1b):



the motion through the analysis of the most energetic modes of the flow.

(DNS)

identify precessional flow instabilities.

evidence of flow instabilities.

**3. Analysis techniques** 

periodogram method);

**3.1 Frequency analysis** 

the "Nearest Neighbour" or the "Sample and Hold" should be preferred over complex methods (e.g. "Linear Interpolation", "Spline Interpolation"), because the latter bias the variance of the signal. It should be noticed that the resampled series contains complete information about the spectral components up to the Nyquist critical frequency fc=1/2Δ where Δ in the sampling interval. At frequencies larger than the Nyquist frequency the information on the spectral components is aliased.

The Lomb-Scargle Periodogram (LSP) method (Lomb, 1976, Scargle, 1982) performs directly on unevenly sampled data. It allows analysing frequency components larger than the Nyquist critical frequency: this is possible because in irregularly spaced series there are a few data spaced much closer than the average sampling rate, removing ambiguity from any aliasing. The method is much more computational expensive than FFTs, requiring O(102K2) operations.

It is worthwhile discussing the suitability of the analysis techniques described above for the investigation of flow instabilities and what are the main parameters to be considered. Flow instabilities are low frequency phenomena, therefore we are interested in the low frequency region of the frequency spectrum. The lowest frequency which can be resolved with both the FFT and Lomb-Scargle method is inversely related to the acquisition time; hence longer sampling times yield better frequency resolutions. This explains the long observations made for flow instabilities detection in stirred vessels. In our works on flow instabilities we have used typically LDA recordings at least 800 s long. In other words the sampling time should be long enough to cover a few flow instabilities cycles. As the time span covered by a series is proportional to the number of samples, the application of the LSP to long series requires strong computational effort.

A benchmark between the two methods is provided in Galletti (2005) and shown in Fig. 2.

Fig. 2. Frequency of the main and the secondary peak in the low frequency region of the spectrum calculated with the Lomb-Scargle method as a function of the number of samples. RT, D/T = 0.33, C/T = 0.5, Re = 27,000. Galletti (2005).

The solid squares show the frequency f of the main peak identified in the spectrum calculated with the LSP as function of the number of samples used for the analysis. It can be observed that f is scattered for low numbers of samples, and it approaches asymptotically the value of f = 0.073 Hz (the same of the FFT analysis over the whole acquisition time of 800 s with 644,000

Flow Instabilities in Mechanically Agitated Stirred Vessels 233

A first kind of flow instability (see left-hand side of the diagram of Fig. 3) manifests as a real change in the circulation pattern inside the tank. Two main sources of such a change have been identified: a variation of the Reynolds number (Re) or a variation of the

In relation to the former source, Nouri & Whitelaw (1990) reported a transition due to Re variations in the flow pattern induced by a 60° PBT with D = T/3 set at C = T/3 in a vessel of T = 0.144 m. For non-Newtonian fluids a flow pattern transition from a radial to an axial flow was observed as the Re was increased up to Re = 4,800. For Newtonian fluids the authors observed that the flow pattern transition occurred at about Re = 650. This value was also confirmed by the power number measurements through strain-gauges carried out by Distelhoff et al. (1995). Similar investigations on such transition may be found in the works

Schäfer et al. (1998) observed by means of flow visualisation the flow discharged by a 45° PBT to be directed axially at higher Re and radially at lower Re. The flow stream direction was unstable, varying from radial to axial, for Re = 490-510. A similar flow transition was also indicated by Bakker et al. (1997) who predicted with CFD techniques the flow pattern generated from a 4-bladed 45° PBT of diameter D = T/3 and set at C = T/3 inside a tank of T = 0.3 m. The regime was laminar, the Reynolds number being varied between 40 and 1,200. The impeller discharge stream was directed radially for low Re numbers, however for Re larger than 400 the flow became more axial, impinging on the vessel base rather than on the

A second source of instabilities, manifesting as a flow pattern change, is associated with variations of the impeller/vessel geometrical configuration, which means either variations of the distance of the impeller from the vessel bottom (C/T) or variation of the impeller

This kind of instabilities were firstly noticed by Nienow (1968) who observed a dependency on the clearance of the impeller rotational speed required to suspend the particles (Njs) in a solid-liquid vessel equipped with a D = 0.35T RT. He observed that for C < T/6 the pattern was different (the discharge stream was directed downwards towards the vessel corners) from the typical radial flow pattern, providing low Njs values. Baldi et al. (1978) also observed a decrease of the Njs with the impeller off-bottom clearance for a 8-bladed turbine. Conti et al. (1981) found a sudden decrease of the power consumption associated with the change in the circulation pattern when lowering the impeller clearance of a 8-bladed turbine. The aforementioned authors concluded that the equation given by Zwietering (1958) for the calculation of the Njs should be corrected in order to take into account the

The dependency of the power number on the impeller off-bottom clearance was also observed by Tiljander & Theliander (1993), who measured the power consumption of two PBT of different sizes, i.e. D = T/3 and D = T/2, and a high flow impeller of D = T/2. The visual observation of the flow pattern revealed that at the transition point between the axial

Ibrahim & Nienow (1996) investigated the efficiency of different impellers, i.e. a RT, a PBT pumping either upwards or downwards, a Chemineer HE3 and a Lightnin A310 hydrofoil pumping downwards and a Ekato Intermig agitator, for solids suspension. For the RT, the aforementioned authors observed a sudden decrease of both the impeller speed and the mean dissipation rate required to just suspend the particles as the clearance was decreased

and the radial flow patterns, the circulation inside the vessel appears chaotic.

**4.1 Change in circulation pattern** 

impeller/vessel geometrical configuration.

of Hockey (1990) and Hockey & Nouri (1996).

diameter (D/T) or a combination of both variations.

walls.

dependency on C/T.

samples) as the number of samples increases. The empty triangles indicate the presence of further low frequency peaks. The main fact to be aware of is that low time intervals conceal the flow instabilities by covering only a portion of the fluctuations.
