Proyecto 4 (Mountain Lodge)

As previously stated, one intriguing feature of the turbulence generated by square space-filling fractal grids highlighted by the pioneering experiments of Hurst & Vassilicos (2007) is the apparent violation of the commonly accepted turbulence dissipation scaling _{. An immediate implication of this}

scaling, when combined the relation (coming from the isotropy assumption in (7.4) and ) is that:

(7.20)

The (7.20) shows that, according to the so-called dissipation anomaly, the ratio should increase as increases. The assertion (7.20) is tested in Fig. 7.15, where the ratio is plotted as a function of . The function is

underestimated due to the effects of limited spatial resolution discussed in Sec. 7.3.3; on the other hand, is proportional to the Taylor microscale as well, and for this reason it is overestimated in the same measure, thus retaining the proportionality scaling (7.20). Dashed lines obtained from (7.20) with different values of the constant are included in the figure for reference. The plot indicates

Chapter 7 – The decay of fractal-generated turbulence

that the dissipation anomaly does not hold, since the ratio is approximately constant along the streamwise direction while decreases rapidly due to the fast decay of the turbulent kinetic energy.

A different way of testing the validity of the dissipation scaling is to determine from measurement of the kinetic energy, dissipation and integral length scale. The turbulent dissipation is notoriously difficult to be measured accurately in anisotropic and inhomogeneous flows, and for this reason three methods are implemented. In the first method, the dissipation is found directly from the measurements of the mean squared velocity gradient:

(7.21)

The turbulent rate-of-strain tensor is defined as follows:

(7.22)

The great advantage of 3D3C measurements is the availability of all the components of the velocity gradient tensor in the same location. The scenario is rather different with respect to the case of the hot-wire measurement performed by Seoud & Vassilicos (2007), in which the small-scale isotropy assumption is invoked, or by Valente & Vassilicos (2011), who used a x-wire with separation of 10 times the Kolmogorov scale, thus leaving some questions open about the validity of the dissipation measurement. On the other hand, the limited spatial resolution is an obstacle to the successful measurement of the spatial derivatives of the velocity fluctuations at the smaller turbulent scales. On top of this, in order to reduce the degrading effect of noise in the derivative computation, the velocity fluctuations fields are low-pass filtered by a local 2nd _{order polynomial fitting function on a 5 x 5}

kernel.

The second method extracts the turbulent dissipation from the rate of decay of the turbulent kinetic energy, as is commonly done for homogeneous grid turbulence. In the present case, the inhomogeneous nature of the flow recommends to use the full turbulent kinetic energy balance. The kinetic energy balance (7.12) can be expressed on the centreline as in the following (supposing :

1 +1 +1 + 2 2+ 2 2+ 2 2

(7.23)

The first term in the right hand side dominates the balance, so the others can be treated as small corrections in the evaluation of the dissipation. The term in curly brackets (i.e. the sum of the triple correlation transport and the pressure transport) is evaluated by using the second moment closure model suggested by Daly & Harlow (1970).

Fig. 7.16 Dissipation obtained with direct measurement , energy balance and Large Eddy PIV . a) Streamise profile of the dissipation along the centerline; b) as a function of the local .

The third method is often referred as Large Eddy PIV (Sheng et al 2000), as it assumes the equilibrium of the transfer of energy from the large lengthscales (production) to the small lengthscales (dissipation). The energy flux is evaluated in terms of the resolved rate of strain tensor and a subgrid stress (SGS) tensor :

(7.24)

The SGS must be modelled by a small-scale turbulence model, as in Large Eddy Simulations. In this case, a Smagorinsky eddy viscosity model (Smagorinsky 1963) is used:

(7.25)

where is the Smagorinsky constant.

The dissipation obtained with the three methods is plotted as a function of the streamwise location in Fig. 7.16. As expected, the direct measurement with (7.21) achieves the smallest values due to the lack of resolution at the smaller scales where the dissipation takes place. The Large Eddy PIV technique results in a significant improvement of the dissipation measurement. The gap between the direct measurement and the dissipation obtained with the energy balance decreases along the decay. Indeed, while the spatial resolution is the same, the Kolmogorov scale (and so the dissipative scales) increases along the decay, as it depends on _{( , Kolmogorov 1941). The Kolmogorov scale}

obtained using the dissipation measured with the (7.23) ranges between and 96 (i.e. about 7.6 and 5.9 times the effective , thus justifying the significant underestimation of the dissipation with the direct measurement).

The scaling _{is applied to evaluate (Fig. 7.16b). The results}

from the direct measurement show a significant variation of along the decay, most likely to be addressed to the increasing Kolmogorov scale along the

Chapter 7 – The decay of fractal-generated turbulence

longitudinal direction (consider that smaller correspond to larger , as the turbulent kinetic energy strongly decays while the Taylor microscale shows only a weak rate of growth. The variation of observed with the data obtained with the Large Eddy PIV method and the energy balance is less remarkable, but still relevant (about 20% over the investigated range). Such variation is substantially at odds with RANS modelling in which the semi-empirical quantity is assumed to be constant.