Analysis of turbulence anisotropy in a mixing tank

Analysis of turbulence anisotropy in a mixing tank

Chemical Engineering Science 61 (2006) 2771 – 2779 www.elsevier.com/locate/ces Analysis of turbulence anisotropy in a mixing tank Renaud Escudié, Ala...

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Chemical Engineering Science 61 (2006) 2771 – 2779 www.elsevier.com/locate/ces

Analysis of turbulence anisotropy in a mixing tank Renaud Escudié, Alain Liné∗ Laboratoire d’Ingénierie des Procédés de l’Environnement, I.N.S.A., 135 Avenue de Rangueil, 31077 Toulouse, France Received 16 June 2005; received in revised form 20 September 2005; accepted 23 September 2005 Available online 20 December 2005

Abstract The objective of this paper is to analyze the gap between the local turbulence and the 3D isotropic state in a mixing tank stirred with a Rushton turbine. Experiments were carried out using the PIV technique, and phase average treatment permits the calculation of the six components of the Reynolds stress tensor. In order to quantify the gap between the measured local turbulence and the 3D isotropic state, methods based on tensorial analysis are developed. In a turbulent flow, three representations are able to characterize the turbulence anisotropy: representation based on two eigenvalues (s.t) of the anisotropy tensor; the Lumley–Newman triangle based on two invariants (II – III) of the anisotropy tensor; a representation based on the axisymmetry invariant A and the two-dimensional invariant G. These three visual methods, that quantify the degree of anisotropy from two independent variables are developed and used in the impeller stream region, and the distribution of the turbulence state is thus described. A parameter is proposed to quantify the anisotropy level. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Stirred vessel; Hydrodynamics; Turbulence; Anisotropy; Mixing

1. Introduction In a stirred vessel, the flow field generated by a Rushton turbine is complex. The discharge flow of the impeller is characterized both by a high level of turbulence and by coherent vortical structures induced by the blade motion. Since the beginning of the 1970s, a lot of studies have been focused on the measurement of the components of the Reynolds stress tensor. In a tank stirred by a Rushton turbine, the velocity fluctuations due to the turbulence need to be separated from the organized motion generated by the impeller-blade (i.e., the trailing vortices). Several techniques were used: the velocity correlation function (Mujumbar et al., 1970; Wu and Patterson, 1989), the energy spectrum (Cutter, 1966; Van der Molen and Van Maanen, 1978; Costes and Couderc, 1988; Mahouast et al., 1989), synchronization of the data acquisition with the blade position (Van der Molen and Van Maanen, 1978; Yianneskis et al., 1987; Yianneskis and Whitelaw, 1993; Stoots and Calabrese, 1995; Schäfer et al., 1997; Sharp and Adrian, 2001; Escudié and Liné, 2003). The six components of the turbulent kinetic ∗ Corresponding author. Tel.: +33 5 61 55 97 86; fax : +33 5 61 55 97 60.

E-mail address: [email protected] (A. Liné). 0009-2509/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2005.09.022

energy tensor and especially the non-diagonal terms were only measured in few studies (Derksen et al., 1999; Escudié and Liné, 2003; Baldi and Yianneskis, 2004). The knowledge of the six components of the turbulent kinetic energy tensor constitutes a significant data bank in order to understand the physics and then to validate CFD simulations. (k, ) model is mainly used in CFD for sake of simplicity in the simulation of complex geometries. However, the (k, ) model is based on turbulent viscosity concept, which assumes basically that the model is isotropic. The goal of the present work is to revisit this assumption of isotropic turbulence. The complete experimental determination of the Reynolds tensor is the basis to estimate the level of anisotropy of the turbulence. In the flow field within the stirred tanks, the turbulence is clearly anisotropic, especially close to the region of the jet generated by the impeller, where the turbulence level is important. Bugay (1998) applied the method proposed by Lumley and Newman (1977), to characterize the turbulence in a stirred vessel equipped with an axial flow impeller (Lightnin A310). The anisotropy of the turbulence is important in the region just below the impeller (i.e., the axial jet generated by the impeller); it is small in the upper part of the tank (Escudié, 1998). In the case of a Rushton turbine, Derksen et al. (1999)

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demonstrated that the turbulence is anisotropic in several regions close to the impeller (the near wake of an impeller blade, a region at disk level close to the disk, and the trailing vortex). In order to characterize the anisotropy with a single parameter, Derksen et al. (1999) proposed the distance from the isotropic state in the Lumley Newman triangle. Using the method proposed by Lumley and Newman (1977), Galletti et al. (2004) showed that high degrees of anisotropy of turbulence occur not only near the impeller, but also near the vessel bottom and in the proximity of the centers of the two circulation loops.

In addition to the well-known Lumley–Newman triangle, two other possible representations of turbulence characteristics are developed. The anisotropy of the turbulence then can be characterized by the three following techniques. The first representation is based on eigenvalues of the anisotropy tensor, noted s.t. The second representation was proposed by Lumley–Newman and is based on two invariants of the anisotropy tensor, noted II – III: it is the most commonly used representation (Pope proposed a slightly different representation). The third representation is based on two other characteristics, the axisymmetry invariant A and the two-dimensional invariant G. We will demonstrate that the turbulence state is 3D isotropic when G is close 1/9, whereas the turbulence state is axisymmetic when the invariant A is close to ±1. These different representations of the turbulence make the choice of a distance in Lumley–Newman triangle as a measure of anisotropy open to discussion. In any point within a turbulent flow, turbulence can be described by the Reynolds tensor expressed in a cartesian frame of reference (R) and noted rij = ui uj . In the case of nonisotropic turbulence, it is interesting to estimate the degree of anisotropy in terms of gap between anisotropy and 3D isotropy. The deviatoric tensor (bij ) defined as 2 3

kij ,

aij =

bij = k

k



2 2 • 2D isotropic: u2 I = uII and uIII = 0, 2 2 • 1D: u2 I  = 0 and uII = uIII = 0,

2 2 • 2D limit: u2 I  = 0, uII  = 0 and uIII = 0, 2 • axisymmetric limit: u2 I = uII .

Basically, the trace of the anisotropy tensor is null. Thus, only two independent variables s and t constitute the tensor 

0 t 0

s aij | = 0 0

0 0 −(s + t)

 (4)

with the convention s t  − (s + t). Obviously, the three di2 2 agonal component of the Reynolds stress tensor (u2 I , uII , uIII ) are positive, thus s and t variables must verify the following constraints: s  − 23 ; ⇒

t  − 23 ;

− 23 s  43 ;

−(s + t)  − 23 , − 23 t  23

− s.

(5)

These two variables enable to plot the anisotropy of the turbulence in a triangle (Fig. 1a). The physical meaning of the edges and vertices of this triangle is given in Table 1. 2.2. Lumley–Newman triangle (II – III) For any given tensor (tij ), three invariants (noted I, II and III ) can be defined as follows:

(1)

where k is the turbulent kinetic energy. The deviatoric tensor (bij ) estimates this gap. The anisotropy tensor noted (aij |R ) in the frame of reference (R) is obtained by dividing each component of the deviatoric tensor (bij ) by the turbulent kinetic energy (k): ui uj

2 2 • 3D isotropic: u2 I = uII = uIII ,

2.1. s.t representation: eigenvalues of the anisotropy tensor

2. Anisotropy of the turbulence

bij = ui uj −

The typical turbulence characteristics can be defined

2 ij . 3

3

tii ,

i=1

II =

3 3

tij tj i ,

i=1 j =1

(2)

This tensor is the basis of turbulence anisotropy analysis. However, the analysis of this anisotropy tensor remains difficult in the initial frame of reference (R). One can calculate the eigenvectors of the Reynolds stress tensor to define a new frame, called principal frame () in which the Reynolds stress tensor is diagonal. ⎤ ⎡ 2 uI 0 0 ⎥ ⎢ rij | = ⎣ 0 u2 (3) 0 ⎦. II 2 0 0 uIII 2 2 The following convention will be used u2 I uII uIII .

I=

III =

3 3 3

tij tj l tli .

(6)

i=1 j =1 l=1

This definition can be applied to the anisotropy tensor. The first invariant is null since it expresses the trace of the tensor. The two others invariants are: II = s 2 + t 2 + (s + t)2 = 2(s 2 + t 2 + st), III = s 3 + t 3 − (s + t)3 = −3st(s + t).

(7)

The anisotropy of the turbulence can thus be identified in terms of these two invariants (II, III ). It corresponds to the

R. Escudié, A. Liné / Chemical Engineering Science 61 (2006) 2771 – 2779

Thus, the axisymmetric invariant is defined as

Two-dimensional limit Axisymmetric limit 1D Isotropic state 2D Isotropic state 3D Isotropic state

1

0.5

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A=

III/6 (II/6)3/2

.

(9)

t

In the plane II – III, the two-dimensional limit is II = III + 89 .

0

(10)

Thus, the two-dimensional invariant is defined as

-0.5

G = 18 (III − II ) + 19 . -1 -1

-0.5

0

0.5

(a)

The anisotropy of the turbulence can thus be identified in terms of these two invariants (A, G). It corresponds to rectangular domain plotted on Fig. 1c. The physical meaning of the edges and vertices of this rectangular is given in Table 1.

1

s 2 1D

Two-dimensional limit Axisymmetric limit 1D Isotropic state 2D Isotropic state 3D Isotropic state

1.5

3. Materials and methods

III

1

(11)

3.1. PIV technique 0.5 3D isotropic 0 2D isotropic -0.5 0

0.5

1

1.5 II

(b)

2

2.5

3

0.12 0.1 0.08

G

0.06

Two- dimensional limit Axisymmetric limit 1D Isotropic state 2D Isotropic state 3D Isotropic state

The principle of this technique consists in acquiring instantaneous 2D velocity fields. The PIV system used is the commercial system acquired from Dantec Measurement Technology. The system includes a laser (Mini Yag, 15 Hz, 30 mJ), a double frame image recorder camera (Kodak Megaplus ES 1.0, 1008 ∗ 1018 pixels), a dedicated processor (PIV 2000) and the software. The processor performs all the calculations in real time. As the processor produces vector maps, these are displayed and optionally stored by the software. The software also automatically generates all the synchronization signals for system integration. The seeding material is spherical glass hollow silvered particles from Dantec (p =1.4, 10 m < dp < 30 m).

0.04

3.2. Experimental apparatus 0.02 0

-1

-0.75

-0.5

(c)

-0.02 -0.25 0 A

0.25

0.5

0.75

1

Fig. 1. Representation of turbulence anisotropy: (a) s .t representation, (b) invariant II – III, Lumley–Newman representation and (c) invariant A.G representation.

well-known Lumley–Newman triangle plotted on Fig. 1b. The physical meaning of the edges and vertices of this triangle is given in Table 1. 2.3. Axisymmetry and two-dimensionality invariants (A–G)

The apparatus used in this study consists of a standard cylindrical tank equipped with a Rushton turbine. The cylindrical tank is made of glass (6 mm thick) with a diameter T =450 mm and a liquid height H = T = 450 mm. The cylindrical vessel is placed in a cubic tank filled with tap water to minimize optical refraction. Four equally spaced baffles made of glass (width B = 45 mm = T /10) are fitted along the internal surface of the vessel. The tank, filled with tap water as working fluid, is open at the top. The Rushton turbine is of standard design with a diameter D = T /3 = 150 mm. The clearance C is equal to the impeller diameter and is measured between the bottom of the mixing vessel and the impeller disk plane. The blade height w is 0.2D. Both the blade thickness tb and the disk thickness td are equal to 2 mm. Experiments were carried out at a single value of impeller rotational speed N = 150 rpm. The Reynolds number (Re = ND2 /) is then equal to 56.250. Flow regime is turbulent.

In the plane II – III, the axisymmetric limit may be written

3/2 II III = ±6 . 6

3.3. Accuracy of data acquisition and data processing (8)

The accuracy of velocity measurements depends on different parameters such as the seeding concentration, the size of PIV

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Table 1 Characteristics of turbulence anisotropy

Diagonal Reynolds tensor

3D isotropic

2D isotropic

1D isotropic

2D limit

Axisymmetric limit

2 2 u2 I = uII = uIII

2 u2 I = uII

u2 I = 0

u2 I =0

2 u2 I = uII

u2 = 0 III

u2 = u2 = 0 II

III

u2 II  = 0 u2  = 0 III

s .t representation

s = t = −(s + t) = 0

s = t = 13 s + t = 23

s = 43 −t = s + t = 23

s = − 23

s=t

Lumley–Newman triangle

II = 0

II = 23

II = 83

III = 89 + II

III = −6

III = 6

III = 0

III = − 29

III = 16 9

G = 19 ∀A

G = 0, A = −1

G = 0, A = +1

G=0 −1 < A < + 1

A = −1 0 < G < 19

A = +1 0 < G < 19

A.G representation

measurement area, the time interval between two laser pulses and the spatial resolution. The reader can refer to previous papers to get more information on this topic (Escudié and Liné, 2003; Escudié et al., 2004). The calculation of statistical average was performed on a series of 1000 instantaneous velocity fields. Statistical convergence was checked on mean velocity and Reynolds stress components. Some vector measurements were considered to be spurious by the processor. They correspond to displacement vectors in the correlation plane with peak heights that are rejected. In the statistical averaging procedure, these spurious vectors are ignored and the statistical average is determined on a reduced number of data, which takes into account the missing data. In our experiments, raw data were used as instantaneous velocity fields, without filtering and without move-averaging validation. 3.4. Measurement planes The measurement plane is located in the region of the jet induced by the Rushton turbine, in a vertical plane of symmetry of the tank. The vertical plane is a bisector plane relative to two baffles. In this study, the three components of the velocity vector and the 6 components of the stress tensor were measured. As a consequence, three kinds of PIV measurement planes were performed: • a vertical “radial” plane in order to calculate the components and their gradient in X1 .X3 plane, • Nine vertical “tangential” planes in order to calculate the components and their gradient in X1 .X2 plane. Two successive planes are 5 mm apart, • Nine horizontal planes in order to calculate the components and their gradient in X2 .X3 plane. Two successive planes are also 5 mm apart. Consequently, experimental data are available in a vertical plane with the radial location ranging between r/R of 1.06 and 1.6,

t = −(s + t)

3/2 II 6

3/2 II 6

and with normalized axial direction ranging between 2z/w of −1.3 and 1.3. The impeller blade tip is situated at the radius r/R = 1, for axial position 2z/w ranging between −1 and 1. 3.5. Phase average treatment Angle resolved measurements are made in each plane in order to determine the three components of the velocity vector and the six components of the Reynolds stress tensor. The region corresponding to the jet flow induced by the impeller is analyzed in detail. This choice is related to the analysis of different types of fluctuations induced by the impeller. In the experiments, data acquisitions must be synchronized with the position of impeller blades in order to perform the triple decomposition. The Rushton turbine is a six blade impeller, the blades being equally spaced. As a consequence, given 1◦ angle-resolved measurements, it is necessary to measure in 60 planes to reconstruct the flow between two successive blades. An encoder, mounted on the impeller shaft, enables to synchronize the velocity field measurement with one of the six blades of the impeller. PIV provides instantaneous velocity fields in one plane. The reader can refer to previous papers to get more information on this topic (Escudié and Liné, 2003; Escudié et al., 2004). 4. Results on isotropy of the turbulence The anisotropy of the turbulence is based on the values of the Reynolds stress averaged over all the positions of the blade of the impeller. The six components of the Reynolds stress tensor have been determined in a vertical plane, in the domain: 1.07 r/R 1.6, −1.3 2z/w 1.3. The data acquired in this region are plotted in terms of invariants s.t (Fig. 2a), II – III (Fig. 2b) and A.G (Fig. 2c). The distribution of the data plotted on Fig. 2a is limited to one 2 2 of the six triangles because we have imposed u2 I uII uIII . In this case, s t  − (s + t). The distribution of the data in the Lumley–Newman triangle (invariant II – III ) presented in Fig. 2b seems to be close to 3D

R. Escudié, A. Liné / Chemical Engineering Science 61 (2006) 2771 – 2779

2775 G

Zone 1 Zone 2 Zone 3

1

0.11

1.3

0.1 1

0.09

0.5 t

0.08 0

2z/w

0.07

-0.5

0

0.05

-1 -1

(a)

-0.5

0

0.5

1

0.03

s 2 1D

Zone 1 Zone 2 Zone 3

1.5

-1 0 -1.3 1.07

1 III

1.4

1.2

(a)

1.6

r/R

0.5

A

1.3

3D isotropic 0

0.8

1 2D isotropic

-0.5 0

0.5

1

0.6

1

1.5 II

(b)

2

2.5

3

0.4 0.2

G

Zone 1 Zone 2 Zone 3

2z/w

0.12

-0.2

0.08

-0.4 -0.6

0.06

-1

(c)

-0.5

-0.02 -0.25 0 A

-1

-1.3 1.07

(b)

0

-0.75

-0.8

0.04 0.02

-1

0

0

0.1

0.25

0.5

0.75

1

Fig. 2. Representation of turbulence anisotropy in a mixing tank: (a) s .t representation, (b) invariant II – III, Lumley–Newman representation and (c) invariant A.G representation.

isotropy. In fact, the plots in planes A.G and s.t enable to better understand the anisotropy of the flow. Indeed, a large amount of data is located along the axisymmetric limit, as mentioned by Derksen et al. (1999) between 3D isotropy and 1D state. Some data have a distinct behavior and will be discussed below. In order to define different zones in the region of the jet generated by the Rushton turbine, the invariants G and A are plotted on Figs. 3a and b, in the vertical plane. The invariant G plotted on Fig. 3a enable to localize the zones close to 3D isotropic turbulence (red) and far from 3D isotropy (blue). In the plane of measurement, G is mainly larger than 0.08: thus, the turbulence is relatively close to 3D isotropy (G = 1/9 =

1.4

1.2

1.6

r/R

Fig. 3. 2D invariant G and axisymmetry invariant A in the vertical plane of measurement: (a) 2D invariant G and (b) axisymmetry invariant A.

0.11). Close to the impeller tip, the values of G are significantly smaller: this zone will be referred zone 1, close to the impeller tip. Fig. 3b shows that the turbulence is close to axisymmetric limit (A = 1) in a large zone, referred as zone 3, external to the liquid jet. In the region of the liquid jet axis, noted zone 2, the axisymmetric invariant is close to zero. The three zones defined in terms of different levels of anisotropy of turbulence are located on Fig. 4. Each zone will be analyzed. 5. Discussion of the results 5.1. Physical analysis of invariant The first part of the discussion will be related to the physical meaning of invariants II and III. Let consider first the invariants of the Reynolds stress tensor written in the principal frame

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II∗ (r) = −II(r)/2; it represents the sum of the three external surfaces of the box. Consider a modified third invariant defined as III∗ (r) = III(r)/3: it represents the volume of the box. Given these remarks, we can return to the invariants of the dimensionless anisotropy tensor a. II(a) invariant can be rewritten

1.3

Zone 3

0

II(a) = tr(a 2 ) − [tr(a)]2 = tr(a 2 ) ⎞2 ⎛ ⎞2

2 ⎛ 2 2 u u u2 2 2 2 I + ⎝ II − ⎠ + ⎝ III − ⎠ − = 3 3 3 k k k

Zone 2

Zone 1

2z/w

1

=

Zone 3

tr(r 2 ) 4 II(r) + [tr(r)]2 4 − = − k2 3 k2 3

II(r) 8 2II∗ (r) 8 + =− + 2 k 3 k2 3

2 2 4k = 2 − II∗ (r) . k 3 =

-1

-1.3 1.07

1.2

1.4

1.6

r/R

Fig. 4. Zoning of turbulence in the plane of measurement.

(Eq. (3)).The three invariants of this tensor are 2 2 I (r) = tr(r) = u2 I + uII + uIII = 2k, 2 2 2 2 2 II(r) = tr(r 2 ) − [tr(r)]2 = −2(u2 I uII + uI uIII + uII uIII ), 2 2 III(r) = 3 det(r) = 3u2 I uII uIII .

(12)

In order to propose a physical meaning to these invariants, consider in the fluid a point M and a facet with a normal n. The normal n can describe all the directions around M. When the orientation of the facet varies, the extremity of the turbulent stress vector s(M, n) = r(M) n describes a surface which is an ellipsoid, the equivalent of a Lamé ellipsoid in elasticity. When the turbulence in M is 3D isotropic, this ellipsoid reduces to a 2 2 sphere (with a radius u2 I = uII = uIII ). When the turbulence in M is 2D isotropic, the ellipsoid reduces to a disk (with a 2 2 radius u2 I = uII and no thickness uIII = 0). When the turbulence becomes 1D, the ellipsoid reduces to a segment, with a length u2 I  = 0 and no dimensions in the two other directions 2 (u2 II = uIII = 0). In the Lumley–Newman triangle, two edges of the curvilinear triangle are related to axisymmetric limit: when 2 2 u2 I =uII > uIII , it corresponds to the edge between 3D isotropic

2 2 and 2D isotropic; the ellipsoid is oblate. When u2 I > uII = uIII , it corresponds to the edge between 3D isotropic and 1D; the ellipsoid is prolate. The 2D limit corresponds to ellipses between a disk (2D) and an segment (1D). In the general case, the ellipsoid has three semi-axis values: 2 2 u2 I  = uII  = uIII . Let us imagine a box containing on eighth

2 2 of the ellipsoid: it is a box whose edges are u2 I , uII and uIII . The first invariant of r is thus related to the sum of the edges of the box. Consider a modified second invariant defined as

(13)

When the turbulence is isotropic, the box is cubic and II∗ (r) is equal to 3(2k/3)2 = 4k 2 /3. Thus II(a) is a measure of the difference of surface of the box in a given state of turbulence II∗ (r) compared to the surface of the cubic box in 3D isotropic state 4k 2 /3, for a given turbulent kinetic energy k. III(a) invariant can be expressed by ⎞⎛ ⎞

⎛ 2 2 u u u2 2 2 2 I ⎝ II − ⎠ ⎝ III − ⎠ III(a) = 3 det(a) = 3 − k 3 3 3 k k III(r) II(r) 4 I (r) 8 + 2 + − k3 k 3 k 9    

3 3 2 2k 2 2k ∗ ∗ = 3 III (r) − + 2 3 − II (r) . k 3 k 3 =

(14) When the turbulence is isotropic, the volume of the cubic box can be expressed by III∗ (r) equal to (2k/3)3 . Thus, the first term of III(a) is a measure of the difference of volume of the box in a given state of turbulence III∗ (r) compared to the volume of the cubic box in 3D isotropic state (2k/3)3 , for a given turbulent kinetic energy k. In 3D isotropic turbulence, the invariants of the anisotropy tensor, II(a) and III(a) are null. In 2D isotropic turbulence, 2 2 u2 I = uII = k and uIII = 0, the ellipsoid vanishes to a disk of ∗ radius k. Thus, II (r) is equal to k 2 (surface of a square of edge k), and

2 4k 2 2 ∗ II(a) = 2 − II (r) = . k 3 3 Since the ellipsoid is reduced to a disk, its volume III∗ (r) is null. Thus, the second invariant 

3  2k 2 3 + II(a) = − . III(a) = 3 0 − k 3 9

R. Escudié, A. Liné / Chemical Engineering Science 61 (2006) 2771 – 2779

In 1D isotropic turbulence, both the surface II∗ (r) and the volume III∗ (r) are null. Thus, II(a) =

2 k2



4k 2 8 −0 = 3 3

and 

3  2k 16 3 + II(a) = . III(a) = 3 0 − k 3 9 Another aspect of the interpretation of the turbulence is based on the plot of the turbulent stress vector s(M, n) in the plane (, ) where  is the normal component of s(M, n) and  is its tangential component. This representation is analogous to Mohr circles in continuum mechanics. In order to quantify the different cases of turbulence, corresponding to 3D isotropy, axisymmetric limit and 2D limit, we propose to use the dimensionless maximum turbulent shear stress is defined as 2 u2 max I − uIII = k 2k

(15)

its maximum value being 100% and the anisotropy coefficient of Taylor as K=

2 u2 I − uIII 2 u2 I + uIII

.

(16)

These parameters will be used in the following discussion. 5.2. Analysis of anisotropy of the turbulence in the mixing tank 5.2.1. Zone 3 Zone 3 corresponds to the majority of the data outside the liquid jet. Figs. 3a–c represent the characteristics of the turbulence for data located in two regions: −0.72z/w 0 and rR 1.7, 0.7 2z/w 1.3 and r/R 1.7. The plot of anisotropy in zone 3 shows a typical axisymmetric behavior. The turbulence is clearly three dimensional. Two diagonal components of the stress tensor are close to each other 2 2 and the third diagonal component is larger: u2 I > uII =uIII > 0. The ellipsoid is prolate. The experimental data are distributed along the axisymmetric limit between two points having the following co-ordinates in the (s, t) plane: • Point A, relatively close to 3D isotropy: s =0.25, t =−0.125 and −(s +t)=−0.125. We obtain a maximum dimensionless turbulent shear stress max /k =0.18 and an anisotropy coefficient K =0.27. In this case, II=0.1 and III  is negligible. The criteria of Derksen et al. (1999) lead to |I |= II2 + III2 0.1 which is quite far from K. • Point B : s =0.5, t =−0.25 and −(s +t)=−0.25. We obtain a maximum dimensionless turbulent shear stress max /k = 0.375 and an anisotropy coefficient K = 0.5. In this case,

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II=0.375 andIII=0.05. The criteria of Derksen et al. (1999) lead to |I | = II2 + III2 0.38 which is smaller than K. 5.2.2. Zone 1 The zone 1 is located close to the impeller tip. Given a radial position r/R = 1.07, the data acquired on a vertical line (−1 2z/w 1) have been plotted on Figs. 3a–c in the three kinds of representation. The distribution of the data acquired on the vertical line shows symmetry with respect to the jet axis located at 2z/w = 1.083 (Fig. 3c). In the center of the jet, near the impeller, the turbulence is far from being isotropic. The value of the invariant G is low: 0.02 and the axisymmetry invariant A is null. Far from the axis of the jet, the turbulence tends to 3D isotropy or to axisymmetric limit. In zone 1, two distinct behaviors are observed: • far from the axis of symmetry of the impeller, the behavior follows the axisymmetric limit as observed in zone 3, outside the jet, • close to the axis of symmetry, the structure is more complex, • on s.t representation, the point corresponding to the measurement in the plane of symmetry closest to the impeller tip has the following co-ordinates: s = 0.5 and t = 0, thus −(s + t) = −0.5. It means that the normal stresses have the 7 2 1 2 2 following values: u2 I = 6 k, uII = 3 k and uIII = 6 k. In this case, the maximum shear stress is thus max /k = 21 and the Taylor coefficient of anisotropy is K=

2 u2 I − uIII

u2 I

+ u2

III

3 = . 4

In the Lumley–Newman triangle, the co-ordinates of this point are II = 0.5 and III  = 0. The criteria of Derksen et al. (1999) lead to |I | = II2 + III2 0.5 which is quite far from K.

5.2.3. Zone 2 Zone 2 corresponds to the liquid jet generated by the Rushton turbine. Given a vertical position, 2z/w = 0.083 the radial position increases from the impeller tip to the lateral wall. The radial evolution of invariants is plotted on Figs. 3a–c. The plot of anisotropy on the plane of symmetry of the impeller reveals a state close to 3D isotropy. This is relevant to classical observations in jet flows, where the turbulence is quasi isotropic on the axis of the jet. The departure from 3D isotropy is mainly due to the fact that the jet induced by the Rushton turbine is not located in the horizontal plane of the impeller but is slightly curved and directed above this plane, the distance to the plane increasing with increasing radial position. 5.3. Anisotropy level parameter The last point in the discussion is devoted to a general representation of the degree of anisotropy in one of the three plots.

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R. Escudié, A. Liné / Chemical Engineering Science 61 (2006) 2771 – 2779

1

20%

40%

60%

80%

0.5

obtained in 1D isotropic limit, where II = 83 and III = 16 9 ; hence, its maximum value is 3.2 which seems difficult to analyze in terms of level of anisotropy.

t

6. Conclusion

0

-0.5

-1 -1

-0.5

0

0.5

1

s

1

20%

40%

60%

80%

t

0.5 0

An analysis of the anisotropy level based on the work presented by Lumley and Newman (1977), and two other representations of the turbulence were proposed. The characteristics of anisotropy and two-dimensionality are contrasted in the impeller tip region, in the region of the liquid jet axis and in the region external to the liquid jet axis. The turbulence is close to axisymmetric limit (A = 1) in a large zone far from the liquid jet generated by the Rushton turbine. In the region of the liquid jet axis, the turbulence is close to 3D isotropy. In the region close to the impeller tip, in the center of the jet, the turbulence is far from being isotropic. The physical meaning of the different invariants were exposed. A general analysis on the quantification of anisotropy shows that the maximum dimensionless turbulent shear stress is a good criterion.

-0.5

References -1 -1

-0.5

0

0.5

1

s

Fig. 5. Anisotropy level in s .t triangle: (a) maximum dimensionless turbulent shear stress and (b) Taylor coefficient.

For sake of simplicity, the maximum dimensionless turbulent shear stress 2 u2 max I − uIII = k 2k

and the Taylor coefficient K=

2 u2 I − uIII 2 u2 I + uIII

have been plotted for constant values between 0% and 100% (Figs. 5a and b, respectively). It can be easily shown that max t =s+ k 2

(17)

and K=

2s + t 4 3

−t

.

(18)

These two parameters are equal to zero in 3D isotropic turbulence. The maximum dimensionless turbulent shear stress is 2 2 equal to 100% in ID turbulance, since u2 I = 2k, uII = uIII = 0. The K parameter is equal to 100% over all the 2D limit, since 2 2 in this case, u2 I  = 0, uII  = 0 and uIII = 0. Therefore, the maximum dimensionless turbulent shear stress seems to be a good criterion to quantify the level of anisotropy. Concerning the criterion proposed by Derksen et al. (1999), its maximum value is

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