ТЕОРЕТИЧЕСКИЕ ОСНОВЫ ХИМИЧЕСКОЙ ТЕХНОЛОГИИ, 2015, том 49, № 2, с. 151-162
ESTIMATING THE LOCAL TURBULENT ENERGY DISSIPATION RATE USING 2-D PIV MEASUREMENTS AND A 1-D ENERGY
SPECTRUM FUNCTION © 2015 г. R. Sulc, V. Pesava, P. Ditl
Czech Technical University in Prague, Faculty of Mechanical Engineering, Department of Process Engineering, Technicka 4, Prague, Czech Republic e-mail: firstname.lastname@example.org Received 07.10.2014
A simple method for estimating the local turbulent energy dissipation rate for a non-isotropic state is proposed. The proposed method is based on an estimation of the isotropic dissipation rate under the assumption of local isotropy, which is subsequently corrected for the actual conditions. The level of anisotropy is characterized by the fluctuation velocity components. The isotropic dissipation rates are obtained by energy spectrum function fitting. A data block averaging technique is used to smooth the spectrum. The effect of the data number within block on the calculated turbulent energy dissipation rate is taken into account. The proposed method has been tested on data obtained by a 2-D time-resolved PIV method. The effect of the spatial resolution of PIV on the estimation of the dissipation rate is also taken into account, using the correction proposed by Delafosse et al. (2011). The estimate of the local turbulent energy dissipation was found to be the same irrespective of the fluctuation velocity component that was taken for calculating the energy spectrum. This accords with the scalar character of the dissipation rate. The method also enables an estimate to be made of the integral length scale components.
Keywords: mixing, Rushton turbine, particle image velocimetry, local turbulent energy dissipation rate, energy spectrum function, fluctuation velocity.
The theory of mixing operations such as liquid — liquid, gas — liquid, powder dispersion and also floc-culation and the scale of segregation in chemical reactions taking place in a turbulent regime is based on the Kolmogorov theory of turbulence, the basic parameter of which is the local dissipation rate of turbulent energy s. Unfortunately, it is not possible to determine the turbulent energy dissipation rate directly. The turbulent energy dissipation rate s is defined for a Newtonian fluid from the turbulent velocity gradients, using Eq. (1) :
duj + dUj dxj dxj
where v is kinematic viscosity, and du/dXj and Ouj/dxi are the fluctuation velocity gradients. One of indirect methods for estimating the dissipation rate is therefore based on fluctuation velocity gradient measurements. For three-dimensional flow, the estimate of the dissipation rate requires nine squared velocity gradients and the three cross-product velocity gradients to be determined. For example, the use of 2D-PIV enables only four velocity gradients to be measured, and so only 5 of the 12 gradient terms can be estimated . The
remaining velocity gradients have to be estimated using some of the relationships proposed in the literature, which are based on an assumption of isotropy. Homogeneous and fully isotropic turbulence is usually assumed, and then the dissipation rate is estimated from measurements of the velocity gradient in one direction only, using the following equation:
e = 15v| —
Sharp and Adrian  investigated the local isotropy condition in a vessel 150 mm in diameter agitated by a Rushton turbine in a region surrounding the blade tips, using the 2-D PIV method. They concluded that the type of assumption of isotropy used for estimating unknown velocity gradients has a significant effect on the estimate of the dissipation rate. The results exhibit deviation of approx. 30—50% between the individual assumptions used by the authors for estimating the velocity gradient. The gap between the local turbulence and the 3D isotropic state in the impeller stream region in a mixing tank stirred by a Rushton turbine was also investigated in . The authors developed a method based on tensorial analysis for quantifying the level of anisotropy. Delafosse et al.  tested assumptions of local isotropy in the impeller discharge in a vessel
stirred by an axial Mixel TTP impeller. As expected, they found that the turbulence in the impeller discharge is far from isotropy.
The local turbulent kinetic energy dissipation rate s can also be estimated by a spectral fitting method from the energy spectrum function in the inertial sub-range. This evaluation method enables an estimate to be made of the local energy dissipation rates without knowing the local velocity gradients.
The distribution of turbulent kinetic energy among eddies of different size can be described by the energy spectrum function (more correctly, by the energy spectrum density function) E(k). Hence the following relation can be written:
j E (k)dk,
where q is total turbulent kinetic energy, E(k) is a three-dimensional energy spectrum density function, and k is wave number. Kolmogorov  derived in the inertial sub-range the following form of the spectrum function, known as the Kolmogorov —5/3 power law:
E(k) = Air_3D6 2/3k-5/3, (4)
where E(k) is the three-dimensional energy spectrum, Ar-3d is a constant, and k is a wave number. According to Kolmogorov, the constant ^IR-3D takes a value of1.5.
However, the problem is how to obtain the three-dimensional energy spectrum function. In the literature, we can find that local isotropy is assumed, and so a three-dimensional energy spectrum E(k) can be replaced by a one-dimensional spectrum :
EjD(ki) = AIR-IDE 2/3k-5/3. (5)
In the longitudinal direction, aligned with the mean flow, using the assumption of isotropy (cf.  Eqs. (3-72) and (3-73)), the constant^4ir_1D = =(18/55) x ^4ir_3d — 0.491, using the Kolmogorov inertial range constant 1.5, Grant et al.  evaluated the constant Air-1d as 0.47 ± 0.02, and Sreenivasan  presented a value of 0.53 as a universal empirical constant.
The one-dimensional energy spectrum function E1D(f1) is obtained from the time course of fluctuation velocity u1(t), using Fourier transformation. Then the one-dimensional energy spectrum function E1Df1) is transformed from the frequency domain f) to the wave-number domain (k) by the following Eqs. (6 and 7):
E1D(k) = f Em = ^ E1D(/i), k1 2n
where Uconv is the so-called convective velocity. The convective velocity method is a way to transform turbulent time scales into length scales  and enables a three—dimensional or highly turbulent flow field to be taken into account .
Since it is difficult to estimate the local velocity gradients using both LDA and PIV (e.g. [2, 10—12]), we estimate the local dissipation rate from the energy spectrum function.
In our previous work , we investigated the hydrodynamics and the flow field in a bulk zone under the impeller in a mechanically agitated vessel T = 300 mm in inner diameter stirred by a Rushton turbine using the 2-D time-resolved PIV (2-D TR PIV). The selected investigated point was relatively far from the impeller and outside the impeller discharge flow, so the authors expected the local isotropic state defined on the length-scale level corresponding to the integral length scale to be an equality of the fluctuation velocity components. However, this expectation was not confirmed. We found that the axial rms fluctuation velocity was approx. one half of the radial component. Further, the energy dissipation rate was estimated using a one-dimensional approach. The energy dissipation rate estimated using the one-dimensional approach was found not to be the same in each direction.
The aim of this paper is to propose a simple method for estimating the local turbulent energy dissipation rate from the isotropic dissipation rate estimated using energy spectrum function fitting. We assume a local state of isotropy defined on the length-scale level corresponding to the integral length scale to be an equality of the fluctuation velocity components. The proposed approach is based on the following idea: first, estimate the local dissipation rate under the assumption of local isotropy; second, correct this value onto the actual state of anisotropy defined by the fluctuation velocity components. Since the turbulent energy dissipation rate is a scalar property, the estimated local dissipation rate should be independent of the direction of measurement. The method proposed here was tested on data obtained by 2-D time-resolved PIV.
On the basis of dimensional analysis, under the assumption of equilibrium between the production and dissipation of turbulent kinetic energy, Batchelor  derived the following widely-used expression for the local turbulent energy dissipation rate per unit mass s in a stirred tank:
6 - Cuj j >
where u and L are the characteristic velocity and length scales of energy-containing eddies, and Cu/L is an empirical proportionality constant. The turbulent fluctuation velocity is used as the characteristic velocity, and the integral length scale A is used as the length scale. Various papers and authors differ in their definition of characteristic fluctuation velocity u and integral length scale A, and in their constant of proportionality.
Taking into account the anisotropy of the flow, all three components of velocity are needed, and the following equation is often used to estimate turbulent kinetic energy q3D for anisotropic flow:
e = Akl л _D • Л
e = C,
q3D = (1/2) (u2 + u2 + u2) = (1/2) u2D,
where q3D is the turbulent kinetic energy for anisotropic flow, ta is an integral time scale, Uj are fluctuation velocities in the ith direction of the Cartesian coordinate system, and u3D is the resultant fluctuation velocity. Cq/T is a proportionality constant assumed to be equal to 1.
The resultant fluctuation velocity u3D can be
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