Numerical simulation of fluid-structure interaction in a...

Journal of Mechanical Science and Technology 25 (7) (2011) 1749~1760

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-011-0514-9

Numerical simulation of fluid-structure interaction in a stirred

vessel equipped with an anchor impeller† Sarhan Karray*, Zied Driss, Hedi Kchaou and Mohamed Salah Abid

Laboratory of Eletro-Mechanic Systems, National School of Engineers of Sfax, University of Sfax, B.P 1173, Km 3.5 Soukra, 3038 Sfax, Tunisia

(Manuscript Received December 21, 2010; Revised March 30, 2011; Accepted April 9, 2011)

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Abstract A coupling algorithm is used to compute the equilibrium of a flexible anchor impeller in a stirred vessel. This coupling algorithm is

based on a partitioned approach, which consists of three relatively independent modules: the computational fluid dynamics (CFD), the computational structure dynamics (CSD) and the interface. In the CFD module, the Euler formulation was used to account for the mov-ing boundary. In the CSD module, the updated Lagrangian formulation for solving the motion of non-linear structure was used and a static study was adopted. In the interface module, an exchange of the forces and displacements was allowed. The numerical results, such as the velocity field, the turbulent kinetic energy, its dissipation rate, the turbulent viscosity and the mechanical deformation, have been presented. Particularly, we are interested in the study of the static behavior of the anchor impeller and the evolution of the displacement field of the arms during various iterations of our coupling algorithm. Accordingly, if the anchor impeller undergoes a deformation due to the flexion of the arms of the anchor impeller, the numerical results changes slightly from iteration to another. At the end of certain itera-tion, the anchor impeller becomes deformed and the velocity field is preserved. These results confirm that the fluid has a significant effect on the deformation of the arms of the anchor impeller during mixing depending on the velocity of the anchor impeller and the fluid flow. The numerical results were validated by a comparison with literature data.

Keywords: Anchor impeller; Computational fluid dynamics (CFD); Computational structure dynamics (CSD); Coupling algorithm; Fluid-structure interac-

tion (FSI) ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction

Fluid-structure interaction (FSI) problems involve the cou-pling of CFD (computational fluid dynamics) and CSD (com-putational structure dynamics) codes. This plays a very impor-tant role in many industrial applications as diverse as aero-space, automotive, biomedical, civil engineering and nuclear engineering.

Though efficient solvers for both the fluid and the structural dynamics exist, the development of tools for modeling various FSI problems remains a challenge. As was already mentioned and in order to describe these problems, an exchange of data has to take place between the fluid and structural fields. Based on this data exchange, the methods for solving fluid-structure interaction problems can be divided into monolithic [1-3], and partitioned couplings [4-6]. The first consists in solving both parts of fluid and structure in the same system of equations. The choice of time step is only limited by the required preci-

sion. However, this type of algorithm requires the develop-ment of a computer code. Moreover, the numerical methods employed for the fluid and the structure domains are different and are difficult to place within the same code, hence the util-ity of the partitioned algorithm methodology. In the parti-tioned method, the fluid and structural parts are solved sepa-rately. The main advantage of this method is that it allows already developing efficient and well validated solvers for both the fluid and structure subtasks to be combined. Some theoretical and numerical studies of partitioned coupling algo-rithms for one and two dimensional problems can be found [7]. Particularly, commercial codes for combining existing solvers have been developed. This method has been introduced by Park and Felippa [8], and further investigated by Wood [9].

Various experimental and numerical simulations study the fluid behavior, when the movement of the structure is pre-scribed analytically [10-12]. Other researchers concentrated on the fluid part, while a simple structural model for a rigid body was used [13]. Further simplifications have been done by neglecting the dynamic effects and simulating static FSI. For example, Beckert et al. [14] used a multivariate interpolation scheme for coupling fluid and structural models in three-

† This paper was recommended for publication in revised form by Associate Editor Oh Chae Kwon

*Corresponding author. Tel.: +216 74 274 409, Fax.: +216 74 275 595 E-mail address: [email protected], [email protected]

© KSME & Springer 2011

1750 S. Karray et al. / Journal of Mechanical Science and Technology 25 (7) (2011) 1749~1760

dimensional spaces. They applied it to static aeroelastic prob-lems in order to predict the equilibrium of elastic wing models in transonic fluid flow. Seiber [15] developed an efficient cou-pling algorithm combining FASTEST-3D code and FEAP code for solving various fluid-structure interaction problems in three-dimensional domains for arbitrary elastic structures. Glück et al. [16] applied a partitioned coupling between the CFD code FASTEST-3D and the CSD code ASE to thin shells and membranous structures with large displacements. Bucchignani et al. [17] presented a numerical code to study the problem of an incompressible flow in a stirred vessel. It was based on a method of a partition treatment type, with the fluid and structural fields resolved by coupling two distinct models. Wang [18] provided an effective new idea to solve aeroelasticity problems, in which the tools Fluent and ABAQUS/ANSYS are employed.

In this paper, a coupling algorithm is developed to predict the equilibrium of the elastic arms of the anchor impeller. Particularly, we have studied the turbulent flow in a stirred vessel as well as the mechanical deformation of the structure. So, it’s necessary to consider this phenomenon in the design of the industrial process. The content of this paper is organized as follows: Section 2 describes the geometry of the system and provides the governing equations of the fluid and the structure. Section 3 describes the computational simulation for both parts. Section 4 presents the numerical results. Section 5 con-tains concluding remarks.

2. Governing equations

It was already pointed out in section 1 that an FSI problem is actually a two-field problem. Therefore, its mathematical description involves the governing equations of the fluid and the structural parts, which will be given in sections 2.1 and 2.2, respectively. The fluid-structure interaction is modeled via special boundary conditions on the interface for each subprob-lem which will be presented in section 2.3. In the following investigations, the fluid used in the CFD simulation is the water defined by the density and the viscosity are equal, re-spectively to ρ = 1000 Kg/m3, μ = 1 mPa.s. The flow is fully turbulent, where the Reynolds number and the Froude number are equal, respectively, to Re = 104 and Fr = 0.02. The struc-ture is assumed to be isotropic linear and elastic material law is applied. Its mechanical characteristics are defined by a Young modulus equal to E = 210 MPa, a Poisson ratio equal to ν=0.28 and a yield stress equal to σ = 215 MPa. The system was made of a flat bottom cylindrical vessel of diameter D = 0.3 m, and the liquid height in the vessel was H = D. An an-chor impeller of thickness e = 1 mm was attached on the in-side wall of the cylindrical tank. It has two arms of diameter d = 0.9 D (Fig. 1). The anchor impeller arms were made of steel. The clearance between the bottom of the mixing vessel and the blade tip was h = 0.75 D. The anchor rotated in a clock-wise direction when viewed from above. The origin of the coordinate system used is located in the bottom of the vessel

(Fig. 1). The geometry of the system resembles the one al-ready experimentally studied by Nagata [19].

2.1 Fluid fields

The problem is governed by the continuity equation and the incompressible momentum equations; both in three dimen-sional forms:

div V 0= . (1) The momentum equations are a statement of conservation

of momentum in each of the three component directions. These equations are written in a rotating frame of reference. Therefore, the centrifugal and the Coriolis accelerations terms are added to the momentum equations. The dimensionless equations are written as follows:

2

e

e2 2 2 2

e e

U 2 d 1 pdiv VU υ gradUt π D Re r

U 1 V 1 U-2 rυr r rr r2 d 1 V r 2V

π D Re rV Wυ r υr r r z r

⎡ ⎤∂ ∂⎛ ⎞⎢ ⎥+ − = −⎜ ⎟∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦⎡ ⎤∂ ∂ ∂⎡ ⎤ ⎡ ⎤+ +⎢ ⎥⎢ ⎥ ⎢ ⎥∂θ ∂ ∂⎣ ⎦ ⎣ ⎦⎛ ⎞ ⎢ ⎥+ + + +⎜ ⎟ ⎢ ⎥⎡ ⎤∂ ∂ ∂ ∂⎛ ⎞ ⎡ ⎤⎝ ⎠ ⎢ ⎥+ +⎢ ⎥⎜ ⎟ ⎢ ⎥∂θ ∂ ∂ ∂⎢ ⎥⎝ ⎠ ⎣ ⎦⎣ ⎦⎣ ⎦

uuuurr

,

(2) 2

e

e e2 2

e e

V 2 d 1 pdiv VV υ gradVt π D Re r

1 U V 1 Uυ υ Vr r r rr2 d 1 UV 2U

π D Re rV 2U Wυ υr r r z

⎡ ⎤∂ ∂⎛ ⎞⎢ ⎥+ − = −⎜ ⎟∂ ∂θ⎢ ⎥⎝ ⎠⎣ ⎦⎡ ⎤⎡ ⎤ ⎡ ⎤∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞+ + −⎢ ⎥⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟∂θ ∂ ∂ ∂θ⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎣ ⎦ ⎣ ⎦⎢ ⎥+ + −⎜ ⎟ ⎢ ⎥⎡ ⎤⎝ ⎠ ∂ ∂ ∂ ∂⎛ ⎞ ⎡ ⎤⎢ ⎥+ + +⎢ ⎥⎜ ⎟ ⎢ ⎥⎢ ⎥∂θ ∂θ ∂ ∂θ⎝ ⎠ ⎣ ⎦⎣ ⎦⎣ ⎦

uuuurr

,

(3) 2

e

e e2

e

W 2 d 1 pdiv VW υ gradWt π D Re z

1 U Vrυ υr r z r θ z2 d 1 1 .

π D Re FrWυz z

⎡ ⎤∂ ∂⎛ ⎞⎢ ⎥+ − = −⎜ ⎟∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦⎡ ⎤∂ ∂ ∂ ∂⎡ ⎤ ⎡ ⎤+⎢ ⎥⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂⎛ ⎞ ⎣ ⎦ ⎣ ⎦⎢ ⎥+ +⎜ ⎟ ⎢ ⎥∂ ∂⎡ ⎤⎝ ⎠ +⎢ ⎥⎢ ⎥∂ ∂⎢ ⎥⎣ ⎦⎣ ⎦

uuuurr

(4)

Fig. 1. Stirred vessel equipped with an anchor impeller.

S. Karray et al. / Journal of Mechanical Science and Technology 25 (7) (2011) 1749~1760 1751

The k-ε model is used in process engineering, and particu-larly in the simulation of mixing problems. The standard k-ε turbulence equations are given in the following form:

:

2 2υk 2 d 1 2 d 1tdiv Vk 1 grad k Gt π D Re π D Rek

⎡ ⎤⎛ ⎞∂ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟+ − + = − ε⎜ ⎟ ⎜ ⎟⎜ ⎟⎢ ⎥∂ σ⎝ ⎠ ⎝ ⎠⎝ ⎠⎣ ⎦

uuuurr

(5) 2 2υ2 d 1 2 d 1tdiv V 1 grad C G Cc1 c2t π D Re k π D Re

⎡ ⎤ ⎡ ⎤⎛ ⎞∂ε ε⎛ ⎞ ⎛ ⎞⎢ ⎥ ⎢ ⎥⎜ ⎟+ ε − + ε = − ε⎜ ⎟ ⎜ ⎟⎜ ⎟⎢ ⎥ ⎢ ⎥∂ σ⎝ ⎠ ⎝ ⎠ε⎝ ⎠⎣ ⎦ ⎣ ⎦

uuuurr .

(6) The turbulent kinetic energy production takes the following

form:

2 2 2U V U W2r r r z

G υ t 2 2 2V V U W V U Wr r r r z z r

⎡ ⎤⎡ ⎤∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎢ ⎥+ + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎢ ⎥∂ ∂θ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦⎢ ⎥=⎢ ⎥

∂ ∂ ∂ ∂ ∂ ∂⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥+ − + + + + +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥∂ ∂θ ∂θ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ .

(7) The k-ε model is based on the concept of turbulent viscosity

where k and ε are calculated by solving the equation above Eqs. (5) and (6). The value of the turbulent viscosity is de-duced from the following equations:

2 2π D kυ C Ret 2 d

⎛ ⎞= ⎜ ⎟μ ε⎝ ⎠.

(8)

2.2 Structure fields

In this work, elastic isotropic structures were considered. The continuity equation and the balance of the momentum equation are given in the following form:

( )D s div V 0s sDt

ρ+ ρ =

r, (9)

DVs div fs Dt⎛ ⎞ρ + σ =⎜ ⎟⎝ ⎠

rruuur

. (10)

The hypoelastic law is presented in the following form:

2 tr Is⎛ ⎞σ = μ ε + λ ε⎜ ⎟⎝ ⎠

. (11)

The strain tensor is defined as follows:

( )T1 gradu gradu2⎛ ⎞

ε = +⎜ ⎟⎝ ⎠

uuuur uuuurr r (12)

with:

uVs t∂

=∂

rr. (13)

2.3 Fluid-structure interface

The fluid-structure interface represents the contact mechan-ics problem between an elastic structure and the fluid flow. This requires the characterization of boundary conditions ex-changed and describes the interaction between the fluid and the structure. We use two conditions on the level of the inter-face. These conditions are given in the following form:

Vn V ns=r rr r

, (14)

n nσ = τr r

(15)

with:

n n nf s= = −r r r

. (16)

3. Computational simulation

3.1 Coupling algorithm

To make a computational simulation for the fluid-structure interaction phenomena, we used a coupling algorithm to deal with problems of strongly non-linear FSI. This algorithm is based on a partitioned method that can also be seen as a con-ventional serial staggered procedure (CSS) used by Ref. [13]. This method determines the structural and flow solution vec-tors independently from each other, updating afterwards the relevant boundary conditions (Fig. 2). This algorithm requires a CSD code, a CFD code and a coupling interface.

3.1.1 CSD code

The CSD code allows the static calculation of the structure and solves the partial differential Eqs. (9) and (10) of the model chosen for the structure. More precisely, this code must be able to receive information representing the varying physi-cal forces in time exerted by the fluid on the fluid-structure interface. On the other hand, this code is used to determine the displacement of the nodes.

3.1.2 CFD code

The CFD code allows the hydrodynamic calculation of the fluid, and solves the momentum Eqs. (2), (3) and (4) for the model chosen for the fluid. The finite volumes method is adopted for the treatment of these equations. Initially, the fluid

Fig. 2. Conventional serial staggered procedure.


field is being divided into elementary volumes representing the geometry of the problem. Secondly, the differential equa-tions are discretized using the control volume approach [20]. The appropriate equations in the range of the turbulent incom-pressible Newtonian fluid flow are expressed in the general conservation from which can be written as:

dv divJ dv S dvD D Dt∂

Φ = − +∫∫∫ ∫∫∫ ∫∫∫Φ Φ∂

r

(17)

with:

J V gradΦ Φ= Φ − Γ Φ . (18) Several quantities can be found. These are the velocity field,

the pressure, the turbulent kinetic energy and its dissipation rate. In a fluid flow problem, we need to know the pressure value at the level of the fluid structure interface. Also, this code must be able to receive information representing dis-placement of the structure part to be used to solve the fluid flow problem.

3.1.3 Coupling interface

The coupling interface ensures the transfer of the forces and displacements mentioned above, when the mesh in each fluid and structure codes is different. It organizes the transfer of information in time and makes an effective coupling between the fluid and structure codes. For every iteration the force exerted by the fluid is calculated in the CFD code. The ge-ometry is updated at the interface to calculate the structural displacement by the CSD code. Therefore, the role of the cou-pling algorithm is to exchange nearly any kind of data be-tween the coupled codes. The iteration steps are stopped when the convergence is reached. The coupling algorithm was used to study an anchor impeller in a vessel tank (Fig. 3).

3.2 Update mesh of fluid domain In this work, the design software was used at first to con-

struct the anchor shape. In the second step, a list of nodes was defined that belong to the interface separating the structure domain from the fluid domain. Using this list, the mesh in the fluid domain is automatically generated for the three-dimensional simulations. Therefore, the region to be modeled is subdivided into a number of control volumes defined in a cylindrical coordinates system (r,θ,z). A staggered mesh is used in such a way that four different control volumes are defined for a given nodal point, one for each of the vector components and one for the scalar variable.

4. Simulation results

4.1 Hydrodynamic studies

Following the implementation of the CFD code developed in our laboratory LASEM, we present the hydrodynamic re-sults such as the velocity field and the turbulence characteris-tics.

4.1.1 Flow patterns in r-θ plane

Figs. 4-6 show a velocity vector plot (U,V) in the three r-θ planes for six iterations of the coupling algorithm. These planes are located, respectively, at the bottom, the top of the anchor impeller and at the mid-height of the tank. They are defined by the axial positions z=0.16, z=1.5 and z=1. These figures show that the velocity field is very high in magnitude beside the walls. It weakens as it approaches the axis of the anchor impeller.

Away from this field, the flow characteristics change con-siderably. Indeed, at the top of the anchor impeller, the flow beside the axis becomes centripetal (Fig. 5). Under these con-ditions, the values of the velocity remain weak. Near the bot-tom of the anchor impeller, we observe an important increase in the magnitude of the velocity field (Fig. 4).

The radial component shows a flow moving towards the walls. Also, it is noted that the magnitude of velocity field is

Fig. 3. Coupling algorithm.

Fig. 4. Flow patterns induced in the r-θ plane defined by z=0.16.


high near the bottom of the anchor impeller. This result is already expected because of the effect of the lower anchor impeller arm. In the middle of the tank, a progressive decel-eration of the flow against the axis of the anchor impeller was observed (Fig. 6). Moreover, if the anchor impeller undergoes a deformation due to the flexion of the arms of the anchor impeller, the velocity field changes slightly from iteration to another. In fact, at the end of a certain iteration (ni>3), the anchor impeller becomes deformed, and the velocity field is preserved.

4.1.2 Flow patterns in the r-z plane

Figs. 7-9 show the secondary flow (U,W) in the different r-z planes defined by the angular coordinates equal to θ=45°, θ=35° and θ=55°, respectively. The plane defined by the angu-lar position θ=45° corresponds to the plane which coincides with the anchor impeller plane in its initial state (Fig. 7(a)). This figure shows correctly the non-sliding condition, which assumes that the velocity field is null. In the following itera-tions, this zone disappears due to the deformation of the an-chor impeller arm. Also, we observed a strong penetration of

the fluid towards the tank interior (Fig. 7). Moreover, the phe-nomenon of brewing is clearly observed with the flexion of the arm. This phenomenon was very clear during the second iteration. This result is already expected, since the anchor im-peller arms undergo a large deformation. In the upstream of the anchor impeller (Fig. 8), the hydrodynamic flow was char-acterized by the presence of a circulation zone near the bottom of the tank. Near the walls, it created an ascending vertical jet that moves towards the second circulation zone. This jet was also transformed into a descending jet near to the axis of the anchor impeller. If the anchor geometry undergoes a deforma-tion due to the flexion of the arm, the circulation zones would change their position from one iteration to another. Moreover,

Fig. 7. Flow patterns induced in the r-z plane defined by θ=45°.


Fig. 5. Flow patterns induced in the r-θ plane defined by z=1.5.

Fig. 6. Flow patterns induced in the r-θ plane defined by z=1.


the centers of the lower circulation zones move towards the walls of the tank due to the flexion effect of the arm. At the same time, the centers of the higher circulation zones come near to the anchor impeller axis. The positions of these centers were stabilized in the second iteration (Fig. 8(c)).

In the downstream of the anchor impeller (Fig. 9), the ve-locity field converges towards the axis of the anchor impeller because of the depression which is located behind the anchor impeller arms. The brewing phenomenon was detected along the arm. This phenomenon was developed into two opposite directions. The first moves towards the walls of the tank while the second moves towards the anchor axis. In the third itera-tion, the flow has only a lower circulation zone (Figs. 9(d), 9(e) and 9(f)).

4.1.3 Axial profiles of the radial velocity component

Fig. 10 illustrates the predicted axial profiles of the dimen-sionless radial velocity component U(z). The dimensionless radial coordinates used are equal to r=0.5, r=0.8 and r=0.9. The superimposed profiles correspond to the second and the fourth iteration of the coupling algorithm. These profiles show a parabolic shape function which develops with the level of the horizontal arm of the anchor impeller. The extremum of these functions is, respectively, defined by the axial positions equal to z=0.33, z=0.37 and z=0.35. During these various iterations, the intensity of the radial jet decreases with the increase of the horizontal arm retreated. This result is already confirmed within a retreated blade paddle [21].

4.1.4 Distribution of the turbulent kinetic energy in the r-θ

plane Figs. 11 and 12 presented, respectively, the evolution of the

distribution of the turbulent kinetic energy k in the horizontal planes located in the middle height of the tank and in the bot-tom of the anchor impeller. These planes are defined by the

axial positions z=1 and z=0.16. These results show that the area defined by the maximum

values is located in the wake which develops at the arm end of the anchor. These maximum values are reached during the first iteration and are about k=0.35 and k=0.1, respectively. In addition, the wake zone is extended at the initial state in the mid-height of the tank (Fig. 11(a)). In these conditions, the angular sector is equal to αin=90°. This sector decreases from one iteration to another. Indeed, for the first iteration, the value of the angular sector is equal to α1it=45° (Fig. 11(b)). This value decreases further during the various iterations, and stabilizes at the third iteration. In these conditions, the angular sector is equal to α3it=23° (Figs. 11(d), (e), (f)).

In the anchor impeller bottom, it is shown that through the field swept by the vertical arms of the anchor, the turbulent kinetic energy remains rather high. Outside this area, the tur-bulent kinetic energy becomes gradually very weak (Fig. 12).

4.1.5 Distribution of the turbulent kinetic energy in the r-z

plane Fig. 13 presents the distribution of the turbulent kinetic en-

ergy k in the r-z plane located at the level of the anchor impel-ler and defined by the angular position equal to θ=45°. This


Fig. 10. Axial profiles of the radial velocity.

Fig. 11. Turbulent kinetic energy k in the r-θ plane defined by z=1.


figure shows that the turbulent kinetic energy is high between the walls and the anchor impeller. Outside this area, k de-creases. Therefore, we can easily note that these results roughly reproduce the geometrical shape of the impeller. Dur-ing these various iterations, where the arms of the anchor have undergone a deformation due to the flexion, the maximum value changes. Indeed, it is noted that the zone, where the turbulent kinetic energy of the fluid is null, changed after the first iteration (Fig. 13). These results are obviously due to the flexion of the arm.

4.1.6 Distribution of dissipation rate of the turbulent kinetic

energy in the r-θ plane Figs. 14 and 15 show the distribution of the dissipation rate

of the turbulent kinetic energy ε in the horizontal planes lo-cated in the mid-height of the tank and in the bottom of the anchor impeller during the iterations of the coupling algorithm. These planes correspond to various axial positions defined respectively by z=1 and z=0.16. According to these results, we observe a distribution similar to that of the turbulent kinetic energy. Indeed, it is noted that the area of maximum values is

located in the wake which develops near the end of the arm. However, the dissipation rate of the turbulent kinetic energy becomes very weak outside of the field swept by the arm. The greatest rate is reached in the fourth and the fifth iteration and it is equal to ε=0.6 (Fig. 14). In the plane located just in the bottom of the anchor (z=0.16), a weakening of the dissipation rate of the turbulent kinetic energy takes place. In the same way, the maximum value of ε decreases from iteration to an-other (Fig. 15).

4.1.7 Distribution of dissipation rate of the turbulent kinetic

energy in the r-z plane Fig. 16 shows the distribution of the dissipation rate of the

turbulent kinetic energy in the r-z plane for an angular position equal to θ=45°. This plane is located at the level of the anchor impeller. This figure shows that the dissipation rate of the turbulent kinetic energy is high between the walls and the anchor impeller. It undergoes a very fast drop outside this field. During these iterations, the maximum value changes due to the flexion of the anchor impeller arm. Indeed, the maximum value of the dissipation rate of the turbulent kinetic energy is reached before the third iteration and is equal to 0.3 (Fig. 16(d)). It is noted that the end of the deformed arms reaches

Fig. 12. Turbulent kinetic energy k in the r-θ plane defined by z=0.16.

Fig. 13. Turbulent kinetic energy k in the r-z plane defined by θ=45°.

Fig. 14. Dissipation rate of the turbulent kinetic energy ε in the r-θ plane defined by z=1.

Fig. 15. Dissipation rate of the turbulent kinetic energy ε in the r-θ plane defined by z=0.16.


the representation plane. This result is visualized by the pres-ence of a small zone where the dissipation rate of the turbulent kinetic energy of the fluid is null. This zone has changed in the first iteration, due obviously to the flexion of the arms (Fig. 16(b)).

4.1.8 Distribution of the turbulent viscosity in the r-θ plane

Figs. 17 and 18 show the evolution of the distribution of the turbulent viscosity in the horizontal plane located in the bot-tom of the anchor impeller and in the mid-height of the tank during the iterations of the coupling algorithm. These presen-tations planes correspond to the axial positions equal, respec-tively, to z=0.16 and z=1. In the field swept by the arms of the anchor, the turbulent viscosity remains rather high. Outside this area, it is noted that the turbulent viscosity undergoes a very fast drop (Fig. 18), due to the deceleration of the flow. In the bottom of the anchor, the turbulent viscosity undergoes a progressive reduction outside this area swept by the arms of the anchor impeller (Fig. 17). By comparing these results from iteration to another, it’s noted that the maximum value is reached during the second iteration and it is equal to υt = 1100 (Fig. 18(b)).

4.1.9 Distribution of the turbulent viscosity in the r-z plane

Fig. 19 presents the distribution of the turbulent viscosity in the angular position equal to θ=45°. It corresponds to a plane located at the level of the anchor impeller. According to these results, the vertical arm deformation reaches the representation plane. This is visualized by the presence of a zone where the turbulent viscosity of the fluid is null. This zone corresponds to the arm of the anchor in the initial state (Fig. 19(a)). Then, it decreases from iteration to another. The maximum values remain high inside the tank. They are reached in the second iteration and is equal to υt = 1100 (Fig. 19(c)).

4.1.10 Comparison with anterior results The power number calculated within an anchor impeller is

equal to Np=0.52. This value, obtained by a volume integra-tion of the field of the dissipation energy, is compared to the experimental results found in the literature. In fact, the values

Fig. 16. Dissipation rate of the turbulent kinetic energy ε in the r-z plane defined by θ=45°.

Fig. 17. Turbulent viscosity in the r-θ plane defined by z=0.16.

Fig. 18. Turbulent viscosity in the r-θ plane defined by z=1.

Fig. 19. Turbulent viscosity in the r-z plane defined by θ=45°.


found by Pollard and Kantyka [22], and Nagata [19] range between 0.5 and 0.6. The comparison of the power number with the experimental results shows a good agreement. Al-though in Fig. 20, the radial profiles of the dimensionless an-gular velocity component V(r) are presented in the laminar flow. In these conditions, the angular coordinate are equal to θ= 90°. The results founded by Murthy et al. [23] are super-posed over an average error of 8%. The good agreement con-firms the validity of the numerical method.

4.2 Static studies

In the static studies, we are interested in the structural part of the mixing system. More particularly, the static behavior of the anchor impeller and the deformation of the anchor impel-ler arms of under the flow effect have been studied. The dis-cretization is ensured by finite element method. In these con-ditions, 5163 nodes and 6580 elements are obtained. 2784 elements are quadratic type on the level of the anchor arms and 3796 elements are tetrahedral type on the anchor axis (Fig. 21).

Fig. 22 presents the results from the CSD code. Particularly, we are interested in the study of the static behavior of the an-chor impeller and the evolution of the displacement field of the arms during various iterations of our coupling algorithm. Particularly, we are interested in the nodal displacement of the anchor arm to allow us to mesh the fluid domain. This opera-

tion is repeated several times until equilibrium is obtained. Under these conditions, it is supposed that the deformation of the anchor axis due to the torsion is negligible. To study the displacement variations, six points placed exactly in the an-chor impeller are especially monitored. Their positions can be seen in Fig. 23. We can follow the radial and angular dis-placement of these points. The time history of these points is shown in Figs. 24 and 25. In the beginning of the simulation, all displacements reach their absolute maxima. Through the different iterations, they start oscillating around some values. Obviously, the biggest radial variation displacement Δr is at the point 1. It is fluctuating around Δr=0.03 m. On the other hand, the smallest displacements are at the point 5 and are equal to Δr=0.0005 m. The angular variation displacement is biggest at the point 2 and fluctuates around Δθ=14°. Also, the value of point 5 and 6 are rather small and is equal to Δθ=3.5°.

5. Conclusion

Coupling codes are used for solving fluid-structure interac-tion (FSI) problems in the stirred tank equipped by an anchor impeller. A computational fluid dynamics (CFD) code and a

Fig. 20. Radial profiles of the angular velocity.

Fig. 21. Meshing and boundary conditions.

Fig. 22. Displacement field of the anchor impeller.

Fig. 23. Points positions.


computational structural dynamics (CSD) code have been coupled by using an efficient coupling interface. Particularly, the fluid field in stirred tank was solved by the CFD code in turbulent regime. The deformation of the anchor impeller was solved by the CSD code. Specific techniques of rearrangement of grid and treatment of the boundary conditions are used to follow the behavior of the system. This method takes advan-tage of the parallel process involved within each analysis code. This allows both parts of the fluid-structure interaction prob-lem to be solved in the best possible way: a finite volume method for the fluid dynamics and a finite element method for the structure. The CFD results obtained allow a visualizing of the velocity field, the turbulent kinetic energy, the dissipation rate of the turbulent kinetic energy, the turbulent viscosity and the mechanical deformation. These results prove that the fluid flow can have a significant effect on the deformation of the anchor impeller arms. So, we can conclude that it’s fundamen-tal to consider this phenomenon in the design of the industrial process. The comparison of the power number and the veloc-ity profile with the results found in the literature shows a good comparison. This proves the validity of the numerical method.

Nomenclature------------------------------------------------------------------------

d : Turbine diameter, m D : Internal diameter of the vessel tank, m

Fr : Froude number, dimensionless, ( )22 N dFr

gπ

=

G : Turbulent kinetic energy production, dimensionless g : Gravity acceleration, m2.s-1 h : Turbine position, m H : Vessel tank height, m k : Turbulent kinetic energy, dimensionless N : Velocity of anchor impeller, rad.s-1 Np : Power number, dimensionless, PNp 3 5N d

=ρ

P : Power, W p : Pressure, dimensionless Re : Reynolds number, dimensionless r : Radial coordinate, dimensionless s : Shaft diameter, m SФ : Sink term, dimensionless t : Time, s U : Radial velocity components, dimensionless V : Angular velocity components, dimensionless W : Axial velocity components, dimensionless z : Axial coordinate, dimensionless ur

: Displacement vector Vr

: Velocity vector of the fluid sVr

: Velocity vector of the structure JΦr

: Flux term vector I : Identity tensor fr

: Force vector nr

: Normal vector Greek symbols

μs : Structure viscosity, Pa.s μ : Fluid viscosity, Pa.s ρs : Structure density, kg.m3 ρ : Fluid density, kg.m3 ε : Dissipation rate of the turbulent kinetic energy, dimen-

sionless θ : Angular coordinate, rad υt : Turbulent viscosity, dimensionless σk : Constant in the standard k-ε model ГΦ : Diffusion coefficient, dimensionless Ф : General transport parameter, dimensionless σ : Stress tensor of the structure ε : Strain tensor of the structure τ : Stress tensor of the fluid

Indices

f : Fluid s : Structure

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Fig. 24. Variation of the radial displacement Δr.

Fig. 25. Variation of the angular displacement Δθ.


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Sarhan Karray received from National School of Engineers of Sfax Tunisia his Electromechanical Engineering diploma in 2004 and his Master’s in Mechanical Engineering in 2006. He is interested in numerical analysis of mechanical problems and in computational solid dynamics (CSD). Also, his research has

been focused on the interaction between CFD and computational solid dynamics (CSD). Currently, Mr. Karray is member in the Laboratory of Electromechanical Systems at Sfax.

Zied Driss received his Electrome-chanical Engineering diploma in 2001, his Master’s degree in Mechanical En-gineering in 2003 and a PhD in Applied Mechanics in 2008 from National School of Engineers of Sfax, Tunisia. He is interested in analysis of fluid me-

chanical problems and energy applications. Also, his research has been focused on the interaction between computational fluid dynamics (CFD) and computational structure dynamics (CSD) codes. Currently, Dr. Driss is Assistant Professor of Mechanical Engineering at National School of Engineers of Sfax. He is a Chief of project at the Laboratory of Electrome-chanical Systems at University of Sfax.


Hedi Kchaou received his Bachelor’s degree in Mechanical engineering from Superior Normal school of Technical teaching of Tunisia in 1983, his Master’s degree in Mechanical Engineering from National School of Engineers of Tunis Tunisia in 1988 and a PhD in Applied Mechanics from

National School of Engineers of Sfax Tunisia in 1996. Current and previous research interests are in mixing, renewable energy, fluid flow and numerical computation. Currently, Dr. Kchaou is Associate Professor of Mechanical engineering at Preparatory School Institute of Engineering Studies at Sfax, Tunisia. He is a member in the Laboratory of Electromechanical Systems at Sfax.

Mohamed Salah Abid received his Bachelor’s degree in Chemical Engineering in 1981, his Doctor-engineer in 1984 and his “Doctorat d’état” in Chemical Engineering in 1988 from Institute Polytechnique of Toulouse INPT France. He is interested in mixing, computational fluid dynamics

(CFD) and energy. Currently, Dr. Abid is Professor of Mechanical Engineering at National School of Engineers of Sfax. He is a head of the Laboratory of Electromechanical Systems University of Sfax.

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