**2. Investigation techniques**

Researchers have employed a large variety of investigation techniques for the detection of flow instabilities. As such techniques should allow identifying flow instabilities, they should be able to detect a change of the flow field (or other relevant variables) with time. Moreover a good time resolution is required to allow an accurate signal processing. Regarding this point, actually flow instabilities in stirred vessels are generally low frequencies phenomena as their frequency is much smaller than the impeller rotational frequency N; so, effectively, the needed temporal resolution is not so high. Anyway the acquisition frequency should at least fulfil the Nyquist criterion.

The graph of Fig. 1a summarises the main techniques, classified as experimental and numerical, employed so far for the investigation of flow instabilities. A brief description of the techniques will be given in the following text in order to highlight the peculiarities of their applications to stirred vessels.

Fig. 1. Overview of investigation (a) and analysis (b) techniques for flow instability characterization in stirred vessels.

### **2.1 Experimental techniques**

Laser Doppler anemometry (LDA) is one of the mostly used experimental technique for flow instability detection. LDA is an optical non-intrusive technique for the measurement of

which can influence the flow motion in different manners. Their knowledge and comprehension is still far from complete, however the mixing optimisation and safe

The present chapter aims at summarizing and discussing flow instabilities in mechanically agitated stirred vessels trying to highlight findings from our research as well as from other relevant works in literature. The topic is extremely wide as flow instabilities have been detected with different investigation techniques (both experimental and numerical) and

Thus investigation techniques and related analysis for the flow instability detection will be firstly overviewed. Then a possible classification of flow instabilities will be proposed and relevant studies in literature will be discussed. Finally, examples of findings on different

Researchers have employed a large variety of investigation techniques for the detection of flow instabilities. As such techniques should allow identifying flow instabilities, they should be able to detect a change of the flow field (or other relevant variables) with time. Moreover a good time resolution is required to allow an accurate signal processing. Regarding this point, actually flow instabilities in stirred vessels are generally low frequencies phenomena as their frequency is much smaller than the impeller rotational frequency N; so, effectively, the needed temporal resolution is not so high. Anyway the acquisition frequency should at

The graph of Fig. 1a summarises the main techniques, classified as experimental and numerical, employed so far for the investigation of flow instabilities. A brief description of the techniques will be given in the following text in order to highlight the peculiarities of

(a) (b)

Laser Doppler anemometry (LDA) is one of the mostly used experimental technique for flow instability detection. LDA is an optical non-intrusive technique for the measurement of

Fig. 1. Overview of investigation (a) and analysis (b) techniques for flow instability

operation of the stirred vessel should take into account such flow variations.

analysis tools, in different stirred vessel/impeller configurations.

**2. Investigation techniques** 

least fulfil the Nyquist criterion.

their applications to stirred vessels.

characterization in stirred vessels.

**2.1 Experimental techniques** 

flow instabilities and their effects on the mixing process will be shown.

the fluid velocity. It is based on the Doppler shift of the light scattered from a 'seeding' particle, which is chosen to be nearly neutrally buoyant and to efficiently scatter light. LDA does not need any calibration and resolves unambiguously the direction of the velocity. Moreover it provides high spatial and temporal resolutions. These are very important for flow instability detection. In addition, more than one laser Doppler anemometer can be combined to perform multi-component measurements. The application of LDA to cylindrical stirred vessels requires some arrangement in order to minimize refraction effects at the tank walls, so often the cylindrical vessel is placed inside a square trough.

Particle Image Velocimetry (PIV) is also an optical technique which allows the velocity of a fluid to be simultaneously measured throughout a region illuminated by a two-dimensional light sheet, thus enabling the instantaneous measurements of two velocity components. However recently the use of a stereoscopic approach allows all three velocity components to be recorded. So far the temporal resolution of PIV measurements has been limited because the update rate of velocity measurements, governed by the camera frame rate and the laser pulse rate, was too low. Thus PIV was not suited for the investigation of flow instabilities. However recently, high-frame rate PIV systems have been developed allowing flow measurements with very high update rates (more than 10 kHz); thus its use for the analysis of flow instabilities in stirred vessels has been explored by some investigators. Similarly to LDA, also PIV requires the fluid and vessel walls to be transparent as well as actions to minimize refraction effects at the tank curvature.

Different flow visualization techniques have also been used to help clarifying the mechanism of flow instabilities. Such flow visualization techniques may simply consist of tracing the fluid with particles and recording with a camera a region of the flow illuminated by a laser sheet. More sophisticated techniques are able of providing also concentration distribution: for example Laser Induced Fluorescence, LIF, uses a fluorescent marker and a camera (equipped with a filter corresponding to the wave of fluorescence) which detects the fluorescence levels in the liquid.

In addition to such optical instruments, different mechanical devices have been used in literature for the detection of flow instabilities. Such devices are based on the measurements of the effect of flow instabilities on some variables. Bruha et al. (1995) employed a "tornadometer", that is a device which allows measuring the temporal variation of the force acting on a small target placed into the flow where instabilities are thought to occur. Paglianti et al. (2006) proved that flow instabilities in stirred vessels could be detected by pressure transducers positioned at the tank walls. The pressure transducers provided time series of pressure with a temporal resolution suited for the flow instability detection. Such a technique is particularly interesting as it is well suited for industrial applications. Haam et al. (1992) identified flow instabilities from the measurement of heat flux and temperature at the walls through heat flux sensors and thermocouples. Hasal et al. (2004) measured the tangential force acting on the baffles as a function of time by means of mechanical devices. Also power number measurements (as for instance through strain gauge techniques) have been found to give an indication of flow instabilities related to change in the circulation loop (Distelhoff et al., 1995).

#### **2.2 Numerical techniques**

Numerical models have also been used for the investigation of flow instabilities in stirred vessels, especially because of the increasing role of Computational Fluid Dynamics (CFD). Logically, since the not steady nature of such instabilities, transient calculation techniques

Flow Instabilities in Mechanically Agitated Stirred Vessels 231

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

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)

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

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.

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

RT, D/T = 0.33, C/T = 0.5, Re = 27,000. Galletti (2005).

information on the spectral components is aliased.

operations.

strong computational effort.

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

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 identify precessional flow instabilities.

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

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 evidence of flow instabilities.
