Pressure fluctuations and their similarity to temperature fluctuations . 

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Pressure fluctuations and their similarity to temperature fluctuations

The velocity involved in the flow being more than two order of magnitude smaller the speed of sound, the incompressible limit may not be questioned. However, the presence of negative peaks in the temperature signal remind us of the pressure fluctuations in turbulent flows, which also exhibit sharp negative peaks [13]. We measure the temporal evolution of pressure fluctuations obtained from the pressure probe flushed to the wall, a typical time signal of which is shown in fig.II.12a. What is immediately noticeable is that pressure fluctuations also display negative peaks similar to what was seen for the temperature fluctuations. A zoom in of one of these peaks is shown in the inset figure of the time series. For the largescale scaling of the RMS of pressure fluctuations, we again turn to dimensional analysis which tells us, where the function j depending on the dimensionless numbers is an unknown function.
Similar to what was done in the case of Trms and other dimensional scalings that we obtained, we have hypothesized . temperature fluctuations, the existence of negative peaks in the time signal of pressure fluctuations is evidenced by the exponential negative tails observed in their PDFs (fig.II.13). However, there are two noteworthy differences between the statistics of temperature and pressure fluctuations:
• The RMS of temperature fluctuations deviate from the large scale scaling whereas the RMS of pressure fluctuations, like velocity, follow the large scale scaling.
• The exponential tails observed in the PDFs of temperature fluctuations normalized by their RMS values do not coincide. The PDFs for pressure fluctuations, though, coincide on normalization with their RMS values. Thus the energy in the pressure fluctuations and the negative peaks are a result of physical structures which are of comparable magnitude and size, which is not the case for temperature fluctuations.
The negative peaks or depressions in pressure fluctuations in a turbulent flow and their statistics have been fairly well-studied and so is the physical phenomenon associated with it. These have been attributed to vortex filaments [55],[58],[14],[60], [61] and were shown to result in the low frequency power spectrum of pressure [13]. All except one from the references mentioned are experimental studies and were performed on von Kármán swirling flow similar to ours albeit with some geometrical changes.

Joint PDFs of pressure and temperature fluctuations

As was seen in the previous section, both the pressure and temperature signals are observed to have negative peaks. One immediate question that arises is whether they are correlated and thus a result of the same physical structures in the turbulent flow. To understand this, we evaluate the joint PDFs between pressure and temperature signals. To do so, we placed one temperature probe as close as possible to the pressure probe (d 3 mm) which was still flushed to the wall. The sensing filament of the temperature probe was oriented along the midplane to capture the vorticity filaments being advected by the flow.
The joint PDF between pressure and temperature is defined as, (p0, T0,t) = probability p(t) = p0, T(t + t) = T0 for any time lag t between the pressure and temperature signals. The introduction of a time lag t is required to account for the mean time required for any fluctuation to advect from one probe to another. The correlation between pressure and temperature shows a maximum for a non zero time lag tmax, corresponding to a mean advection speed of d tmax O(1) m/s u0 rms.

Structure of negative peaks in pressure and temperature fluctuations

As we concluded in the last section, vorticity filaments result in the negative peaks or bursts observed in the time signals of pressure and temperature fluctuations. In this section, we obtain the average internal structure of these filaments. In succeeding to do so, we can obtain the order of magnitudes of their width and the temperature (and pressure) drops between their cores and the surrounding turbulent flow.
To obtain the average structure of the vorticity filaments, we use the method of coherent averaging and its definition as used by Labbé et al. [63]. In their article, Labbé et al. demonstrated the technique and used it to extract a periodic structure from a signal with noise (velocity, temperature). In the above mentioned study, the structures were periodic but in our case the structures (negative peaks) are not. Since the method is not constrained to structures that are periodic, we can apply it to our case with some modifications. The details of the technique are presented in appendix II.B. Applying this method to the temperature and pressure signals in our case allows us to obtain the average temporal (and spatial) structure of the peaks.
Figures II.15a and II.15b show the result of coherent averaging when applied to peaks in temperature and pressure respectively for a rotation rate of = 2000 rpm. We observe that the actual amplitude of the averaged structure for both pressure and temperature is roughly three times smaller than the amplitudes of the peaks seen from the actual time signal. This would mean that the turbulent flow over which the vorticity filaments are superimposed affect the magnitude of the peaks but the overall structure of the peaks remains unaffected. If we measure the difference in temperature and pressure between the core of the structure and its edge (T and p), we observe that, p T prms Trms 1 Pa/mK.

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A steady solution for single weakly compressible adiabatic vortex

One striking feature of turbulence is the existence of coherent elongated structures in the flow along which vorticity is concentrated called vorticity filaments. Since their discovery [64], they have been observed both numerically and experimentally [65, 62, 55] and are also proposed to be inherent to the physical picture of Kolmogorov’s cascade of energy to small scales [66]. As a consequence of its definition, vorticity in a turbulent flow is a solenoidal (or divergence-free) field and thus vorticity filaments would form closed loops in a turbulent flow far from the boundaries that bound the turbulent flow. In the close vicinity of these vorticity filaments, we can thus consider them as cylindrical, tubular structures with a certain radius within which vorticity is concentrated. This physical picture forms the basis of our current model of a single steady vortex.
As a result of the physical picture given above, we proceed by writing the governing equations for compressible, viscous flow in cylindrical co-ordinates with the radial, azimuthal and axial components of the velocity denoted by ur, u and uz respectively,

Table of contents :

I General Introduction 
II Temperature fluctuations in turbulent flows 
II.1 Theoretical background
II.1.1 The governing equations
II.1.2 Limit I: Incompressible flow
II.1.3 Limit II: Acoustic wave
II.1.4 Limit III: Perfect fluid
II.1.5 Temperature fluctuations in incompressible, homogeneous and isotropic turbulent flows: Theory
II.1.6 Temperature fluctuations in incompressible, homogeneous and isotropic turbulent flows: Energy spectrum and RMS
II.2 Temperature fluctuations due to viscous dissipation
II.2.1 The von Kármán swirling flow
II.2.2 Experimental setup
II.2.3 Characterization of the flow
II.3 Experimental Results
II.3.1 Statistics of temperature fluctuations
II.3.2 Pressure fluctuations and their similarity to temperature fluctuations .
II.3.3 Joint PDFs of pressure and temperature fluctuations
II.3.4 Structure of negative peaks in pressure and temperature fluctuations
II.3.5 A steady solution for single weakly compressible adiabatic vortex
II.3.6 The energy spectra of temperature fluctuations
II.4 Conclusion
II.A Estimation of RMS of velocity fluctuations from 1D hot-wire probe
II.B Method of coherent averaging
II.B.1 Demonstration
IIICoherence of velocity fluctuations in turbulent flows 
III.1 Theoretical background
III.1.1 The sweeping effect and the energy spectrum
III.1.2 The sweeping effect and coherence
III.2 Experimental setup I
III.2.1 Characterization of the flow
III.3 Experimental results of setup I
III.4 Experimental setup II
III.4.1 Forcing mechanism: Helices and motors
III.4.2 Working fluid and its properties
III.4.3 The setup of the PIV technique
III.4.4 Characterization of the flow
III.5 Experimental results of setup II
III.6 Conclusion
IVAcoustic scattering by turbulent flows 
IV.1 Theoretical background
IV.1.1 Vortex sound
IV.1.2 Acoustic scattering
IV.1.3 The geometrical acoustic limit
IV.2 Acoustic scattering in Line Of Sight (LOS) propagation
IV.2.1 Parameter fluctuations of incident wave in LOS propagation
IV.2.2 Statistical properties of phase fluctuations of the incident wave .
IV.2.3 Statistical properties of log-amplitude fluctuations of the incident wave
IV.2.4 Coherent wave propagation and change in speed of sound
IV.3 Experimental setup
IV.4 Experimental results
IV.4.1 Parameter fluctuations of incident wave in LOS propagation
IV.4.2 Change in speed of sound due to turbulent flow
IV.5 Conclusion
IV.A The governing equation of vortex sound
IV.B The governing equation of acoustic scattering
IV.C The governing equation for the geometrical acoustic limit
IV.D The governing equations for log-amplitude and phase fluctuations
V General conclusion and perspectives 
V.1 Perspectives
A Calibration of hot-wire probes 
A.1 King’s law
A.2 Setup
A.3 Results
Bibliography

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