Enhancement of turbulent mixing by embedded longitudinal vorticity: a numerical study and experimental comparison

(1)

Proceedings of ASME FEDSM2006 2006 ASME Joint U.S. - European Fluids Enigneering Summer Meeting July 17-20, Miami, FL

DRAFT PAPER FEDSM2006-98367

ENHANCEMENT OF TURBULENT MIXING BY EMBEDDED LONGITUDINAL VORTICITY: A NUMERICAL STUDY AND EXPERIMENTAL COMPARISON

Hakim MOHAND KACI

Thermofluids Complex Flows & Energy Research Group, Laboratoire de Thermocinétique, CNRS-UMR 6607, rue

Christian Pauc, BP 50609, F-44306 Nantes, France [email protected]

Thierry LEMENAND

Dominique DELLA VALLE

Thermofluids Complex Flows & Energy Research Group, ENITIAA, rue de la Géraudière, BP 82225, F-44322 Nantes,

France

[email protected]

Hassan PEERHOSSAINI

ABSTRACT

This work concerns the characterization of turbulent flow underlying the mixing phenomenon in a static mixer-reactor HEV (high-efficiency vortex). An experimental test section made of a cylindrical tube equipped with seven rows of vortex generators was designed and constructed for this purpose. Each row has four vortex generators fixed symmetrically on the tube wall. This new type of mixer generates coherent structures in the form of longitudinal counter-rotative vortices. The resulting flow enhances radial mass transfer and thus facilitates the dispersion and mixing of the particles. The energy cost of this mixer is 1000 times less than that of other mixers for a given interface area [1, 2].

The aim of this work is to study numerically and experimentally the turbulence structure of the flow generated by the mixer, in particular the more energetic structures present in the base flow. Numerical simulations of the velocity distribution and turbulence levels inside the static mixer were conducted for various turbulence models by using the commercial mesh-generator code Gambit coupled with the CFD package Fluent. Attention was focused on the evolution and distribution of the rate of turbulent kinetic energy dissipation as the underlying mechanism for turbulent mixing.

Experiments were carried out on the test section in a flow loop by using LDA. Mean and turbulent quantities were measured and numerical results were compared with experimental results.

This study provides a basis for understanding the physical mechanisms in the mixing and homogenising of the flow and therefore the efficiency of the mixer.

INTRODUCTION

Blending of chemical reactants is a common operation in chemical process industries. A number of unit operations are carried out: mixing, reaction, heat transfer, separation. Most of these operations are carried out separately and in different containers. Two types of reactors are generally used in industries: batch reactors and continuous reactors. The batch reactors are closed and are used to carry out the processes of mixing and reaction, while the continuous reactors allow uninterrupted fluids processes.

The use of the stirred tank as a batch reactor presents several disadvantages: abnormal fluid stagnation, the large consumption of energy required to maintain the rotational movement of the moving part, and the large dimensions of the devices. The continuous static mixer, which combines most of the unit operations within the same apparatus (Ferrouillat et al.

[3-5]), is an alternative to the discontinuous mixer. Indeed, static mixers offer several advantages such as the reduction of the overall device dimensions, the absence of moving parts consuming energy, better process control, shorter residence time, improvement of selectivity, and reduction of by-products.

The mixing process in mixers is carried out by turbulent vortices. However, the two configurations, stirred tank and continuous mixer, differ in how these vortices are generated. In stirred-tank mixers the dynamic of vortex creation is maintained as long as the moving part remains in rotation; in continuous mixers, on the other hand, this dynamic is intrinsic to the flow and requires no external energy contribution.

Many multiphase processes are carried out in stirred-tank reactors. Poor flow patterns and low inhomogeneous mixing are characteristic of stirred tank reactors and typically yield energy

(2)

dissipation rates in the range . High selectivity requires high micromixing rates, which need turbulent energy dissipations above . Therefore, fast exothermic reactions, when carried out in stirred tanks, start before mixing is complete, causing low apparent reaction rates and formation of by-products. This study focuses on the high-efficiency vortex (HEV) heat exchanger-reactor mixer, of interest because of its ability to generate large-scale vortex motions and enhance turbulent energy dissipation in the flow. Its design is based on curved tabs fixed on the tube walls; these tubes generate longitudinal vortices that increase transport phenomena over the simple pipe and even over some static mixers known for high efficiency.

1-10 .W kg-1

100 .W kg-1

This paper aims to contribute to the understanding and control of the flow within this static mixer, and to investigate the extent to which computational fluid dynamic (CFD) models can be used in the design of industrial reactors. The commercially available program Fluent™ is used to calculate flow patterns and transport in a HEV mixer. In order to characterize this mixer, we use an experimental approach by making LDA measurements and a numerical approach by applying various CFD models. The flow is modelled under stationary turbulent conditions for a three-dimensional geometry that can be reduced to one-eighth its size thanks to symmetries in geometry and flow. Computations are carried out in various turbulence models: the standard k-ε [6], RNG k-ε [7], standard k-ω [8, 9], k-ω SST, and RSM model [10, 11, 12]. The flow field is governed by the continuity and momentum conservation equations and energy equation, assuming constancy of the fluid properties.

The numerical results for the different models used are compared among themselves and to LDA measurements obtained for the instantaneous fluctuating and average velocity components. Measurements behind each tab show a recirculation zone there that increases the quality and intensity of mixing in the HEV mixer.

1 STATE OF THE ART 1.1 GOVERNING EQUATIONS

Turbulent flows of incompressible Newtonian fluids are governed by the Reynolds-averaged Navier-Stokes equations.

The continuity and momentum equations for the mean flow may be written in Cartesian tensor notation as

i 0

i

U x

∂ =

∂ (1)

(

i

) (

j i

)

^ij i

j i j

U U U P g

t x x τx Fⁱ

ρ ∂

∂ + ∂ = ∂ + +

∂ ∂ ∂ ∂ ρ + (2)

Here is the mean velocity in direction , is the mean pressure and

U

i i P

ρ is the density. The Reynolds stresses

τ

_ij

= − ρ u

_i

u

_j , resulting from the averaging of the nonlinear convective terms in the momentum equations, represent the effects of the turbulence on the mean flow. In the absence of gravitational body force and any external body force, the two last terms on the right side of Eq.(2) are zero.

The presence of turbulent stresses leads to an open system of

equations. A turbulence model, i.e. additional equations for these terms, must therefore be incorporated.

1.2 TURBULENCE MODELS

For fundamental study of a particular physical phenomenon, one often has recourse to direct numerical simulation (DNS) in which no explicit closure model of turbulence is employed.

This type of simulation appears very useful in understanding the complex physical phenomena present in turbulent flows.

However, the complete simulation at all scales in turbulent flow requires very fine computational grids and is thus strongly limited in real applications (complex geometry, high Reynolds number) by computer power. The extent of this limitation depends on the flow field size, grid smoothness and also on the range of length scales present in turbulent flows. However, the relationship between the smallest and largest size scales, on which the density of the grid depends, increases with the Reynolds number. The number of grid points necessary for the spatial representation of developed turbulence field is given by the relation n n n_{x y z} >R_t⁹⁴. This constraint can be partially alleviated by macro simulation, or large-scale LES (large eddy simulation), in which only the largest structures (the energy swirls) are solved, while the action of the small scales is modelled.

However, the processing time required by techniques such as DNS and LES is considerable, especially for complex geometries. Although these methods are sometimes used, most computational simulations are based on RANS (Reynolds- averaged Navier-Stokes) methods, in which all the variables describing the movement are decomposed into a mean value and a fluctuating value. This statistical processing of turbulence leads to additional unknown factors in the governing equations that are called Reynolds stresses. This problem is commonly called the closure problem.

The RANS approach requires modelling the Reynolds stress.

The method commonly used entails the Boussinesq assumption, which links the Reynolds stress to the velocity gradients via the turbulent viscosityμ_t :

2 3

i j

ij i j t ij

j i

U U

u u k

x x

τ ρ μ ⎛∂ ∂ ⎞ δ ρ

= − = ⎜⎜⎝∂ + ∂ ⎟⎟⎠− (3) The Boussinesq assumption is employed in Spalart-Allmaras, k– ε and k – ω models. The Spalart-Allmaras model requires the solution of an additional transport equation relating to turbulent viscosity, while the k– ε and k – ω models need the resolution of two additional transport equations related to the turbulent kinetic energy, k and the turbulence dissipation rate ε or the specific dissipation rate ω.

1.2.1 Standard k – ε model

The standard k–ε model, proposed by Launder & Spalding [6], uses dimensional arguments to describe the turbulent viscosityμ_tby a velocity scale and a length scale characterising the turbulence. The two scales are defined by the turbulent kinetic energy k and its dissipation rate ε at any given point and time in the fluid flow field as follows:

Velocity scale: V =k¹²

(3)

Length scale:

32

l k

= ε

From this, the turbulent viscosity is given by

2 t

C_μ k

μ ρ

= ε

The turbulent transport coefficient or diffusivity is related to the turbulent viscosity

Γφ

μtby the turbulent Prandtl/Schmidt number

σ

_φ as _φ ^t

φ

μ Γ =σ .

The model transport equations for the turbulent kinetic energy k and its dissipation rate ε may be written as

( ) (

j

)

^t k

j j k j

k U k k P

t x x x

ρ ρ μ ρ ε

σ

⎛ ⎞

∂ ∂ ∂ ∂

+ = ⎜⎜ ⎟⎟+

∂ ∂ ∂ ⎝ ∂ ⎠ − (4)

( ) ( )

2

1 2

t i

i j j

k

t x U x x

C P C

k k

ε

μ ε

ρ ε ρ ε

σ

ε ρε

⎛ ⎞

∂ + ∂ = ∂ ⎜⎜ ∂ ⎟⎟

∂ ∂ ∂ ⎝ ∂ ⎠

+ −

(5)

Here the production of turbulent kinetic energy is expressed by

Pk

i 2

k ij t

j

P U S

τ ^∂x μ

= =

∂

where the mean strain rate S is given by S= 2S S_ij _ij

The modelling constantsC_μ,

C

₁,

C

₂,

C

_σ ,

σ

_kand

σ

_εare given in Table 1.

The k –ε model suffers from the necessity of modelling a number of quantities for which there is little trustworthy experimental data.

Table 1 : Numerical values for constants used in the k –ε model

Cμ

C

₁

C

₂

σ

_k

σ

_ε

0.09 1.44 1.92 1.0 1.3 1.2.2 RNG k – ε model

A popular alternative to the standard k–ε -model is the so- called RNG k–ε model, which is derived from the instantaneous Navier-Stokes equations by using a mathematical technique called the “renormalization group” (RNG) method [7]. The RNG procedure systematically removes the small scales of motion from the governing equations by expressing their effects in terms of larger-scale motion and a modified viscosity (Versteeg & Malalasekera, 1996). The analytical derivation yields a model with constants different from those in the standard k-ε model, and additional terms and functions in the transport equations for k and ε appear.

The RNG k-ε model is similar in form to the standard k- ε model. The transport equation for the dissipation of turbulent kinetic energy may then be written as

( ) ( )

0

*

1 1 3

2 2

1

1 1

i t

i j j

k

t x U x x

C C P

k

C k

ε

μ ε

ρ ε ρ ε

σ η η

η ε

βη

ρε

⎛ ⎞

∂ + ∂ = ∂ ⎜⎜ ∂ ⎟⎟

∂ ∂ ∂ ⎝ ∂ ⎠

⎛ ⎛ ⎞⎞

⎜ ⎜ − ⎟⎟

⎜ ⎝ ⎠⎟

+ ⎜⎜ − + ⎟⎟

⎜ ⎟

⎝ ⎠

−

(6)

where η S ^k

= ε is a timescale ratio between the turbulence and the mean flow. The modelling constant takes the values

*

C1

* 1

1

C 1

=C . The numerical values for constants used in the RNG- k- ε model are shown in Table 2.

Table 2 : Numerical values for constants used in the RNG-k –ε model

Cμ

C

₁

C

₂

σ

_k

σ

_ε

η

₀ ^β

0.0845 1.42 1.68 0.72 0.72 4.377 0.012 1.2.3 Standard k – ω model

The k–ω model, proposed by Wilcox [8, 9], is gaining in popularity. In this model the standard k equation is solved but ω (often called specific dissipation rate from its definition

k

ω ≡ ε ) is used as a characteristic fluctuation frequency. The model’s k and ω equation are

( ) ( )

* t i

i j k

k

k U k k

t x x

P k

ω

ρ ρ μ μ

σ β ω

⎡⎛ ⎞ ⎤

xj

∂ ∂ ∂

+ = ⎢⎜ + ⎟ ⎥

∂ ∂ ∂ ⎢⎣⎝ ⎠∂ ⎥⎦

+ −

∂

(7)

( ) ( )

(

1 2

)

t i

i j

w k

t x U x x

C P C k

k

ω ω

μ ω

ρ ω ρ ω μ

σ ω

j

ρ ω

⎡⎛ ⎞ ⎤

∂ + ∂ = ∂ ⎢⎜ + ⎟∂ ⎥

∂ ∂ ∂ ⎢⎣⎝ ⎠∂ ⎥⎦

+ −

(8)

where

ρ ω

μ

_t = ^k and

ε = β

^*

ω k

The numerical values for the constants in the standard k – ω model are given in Table 3 below.

Table 3 : Numerical values for constants used in the k –ω model

1

Cw C_w₂ σ_ω σ_k^ω β^*

5/9 3/40 2.0 2.0 0.09 1.2.4 RSM Model

The disadvantage of the Boussinesq hypothesis is the assumption that the turbulent viscosity is an isotropic scalar quantity, since this is not completely and always true. The RSM model [10, 11, 12] presents an alternative approach, while proposing to solve transport equations for each term of the Reynolds stress tensor. This means that four additional

(4)

transport equations must be solved in the case of two- dimensional flow and seven for three-dimensional flow.

In most cases the models based on the Boussinesq hypothesis produce satisfactory results, and recourse to RSM model is not necessary. However, the RSM model is generally preferred for situations in which the anisotropy of turbulence dominates the average flow.

1.3 WALL TREATMENT

Turbulent flows are significantly affected by the presence of walls. There are two ways to take wall effects into account: a wall-function model and a low-Reynold-number turbulence model.

a) Wall-function model

This method assumes that the flow near the wall behaves like a fully developed turbulent boundary layer and prescribes boundary conditions employing wall functions. Considering the effects of the wall for the k–ε model, the mean velocity distribution is given by

* 1ln( . )*

U E

=κ y

where is the dimensionless distance to the wall and is given by :

y*

1 1

4 2

* C k yp p

y ρ ^μ

= μ

von Kàrmàn constant ( 0.42) empirical constant ( 9.81)

kinetic energy of turbulence at position P wall distance of position P

p p

E k y

κ = =

= =

=

This logarithmic law for the mean velocity is known to be valid for about 30 to 60. The log law is employed when . When the mesh is such that at the wall-adjacent cells, we can apply the laminar stress-strain relationship U y .

y*

* 11.225

y > y^*<11.225

*= *

b) Two-layer zonal model

When the low-Reynolds-number effects are important in the flow domain, the hypothesis underlying the wall functions ceases to be valid and models like the two-layer model can be used. However, the two-layer model has the restriction that the near-wall mesh must be sufficiently fine everywhere to resolve the laminar sublayer (typically ), which might impose excessive computational requirements.

1 y⁺ ≤

2 EXPERIMENTAL SETUP AND METHODS 2.1 GEOMETRY

The problem to be analysed is the turbulent flow in a HEV test section consisting of a duct of circular cross section in which seven tab arrays are fixed. Each array is composed of trapezoidal tabs that are rotated 90° with respect to each other.

Each tab is tilted at 30° relative to the wall. The test section is 180 mm long and the distance between two successive rows of tabs is 20 mm. Figure 1 shows a schematic view of the physical problem. The z coordinate is the main flow direction and the x and y coordinates are the transverse coordinates.

(a)

(b)

Test section 1^st tab array

(c) 7 mm

7 mm

5 mm

Figure 1: Geometry of the test section: (a) side view of the mixer, (b) front view of tab array, (c) dimensions of one tab

The test section is connected to a preconditioner and postconditioner, 30 cm straight pipes of circular inner cross section 20 mm that are used to provide a fully developed flow at the test section inlet. Thanks to symmetries of the geometry and the flow field, the computational domain was reduced to one-eighth of the cross section in this simulation.

The Reynolds number associated with the geometry is defined as, Re ρU D

= ν where D is the inner diameter of the cross section, U the axial mean velocity, ν the kinematic viscosity and ρ the density of the fluid. Water is the working fluid in this study, with density 998.2 kg and viscosity 10 . The experiments and simulations were applied to a stationary turbulent flow with Reynolds numbers varying from 7500 to 15000.

/m3 ⁻³kg m s/ .

2.2 EXPERIMENTAL METHOD

Mean and turbulent flow quantities such as mean velocity, root mean square fluctuation and turbulence characteristics were measured by LDA over a grid of points across the flow field.

Figure 2 shows a schematic diagram of the test section and LDA system.

Velocity measurements were made by an LDA system equipped with a 10W argon-ion laser source. The laser beam is divided into two beams in the laser transmitter, one with direct frequency and the other with shifted frequency. Each beam is then separated into two colours, green (λ = 514.5 nm) and blue (λ = 488 nm), and each colour is used for measuring one velocity component

(5)

Figure 2: Schematic diagram of the test section and LDA system.

The accuracy of results depends strongly on the measurement volume, which varies proportionally to the focal length. The measurement volume must be sufficiently small relative to the flow scale in order for the measurement to be considered local.

At the same time, it should be large enough to increase the acquisition frequency in order to detect the faster velocity fluctuations. Tacking into account this restriction, we use a lens of 160 mm focal distance. Table 4 describes the other characteristics of the laser Doppler anemometry instrument used for the measurements.

Table 4: Measurement system specifications LDA

Laser type argon-ion

Laser max power 10 W

Laser wavelength 514.5 nm (green); 488 nm (blue)

Beam splitting Dantec 60X24

Velocity sign detection Bragg cell (40 MHz)

Positioning Manual (3D) +

automatic

Data acquisition and processing Dantec BSA Flow Software

Beam diameter 38 mm

Beam half angle 6.772 degree Dimensions of measurement volume

Length 404.0 µm

Width 48.0 µm

Height 48.0 µm

Number of fringes in measurement volume 21

Fringe spacing 2.182 µm

Because the static mixer is cylindrical, the optical refraction problem needs particular treatment. The laser beams traverse air, mixer wall, and water, media of different refraction indexes, and are bent depending on these refraction indexes.

This difficulty has been investigated and a solution proposed by Gardavský et al. [13]. To remove this constraint, we immersed the test section into a container filled with the working fluid (water in this study), thus rectifying the optical refraction effects.

2.3 NUMERICAL PROCEDURE

The solver used for the flow computation is the CFD code FLUENT, which uses a Eulerian approach to solve the equations on a computational mesh, based on a cell-centred finite volume discretization. The solver is a double-precision, segregated, implicit, second-order upwind [14], finite volume solver. Using the segregated approach, the governing equations are solved sequentially. To obtain second-order accuracy, quantities at the cell faces are computed using a multidimensional linear reconstruction approach [14, 15].

Pressure-velocity coupling is achieved by the SIMPLE algorithm [16].

2.3.1 Initial and boundary conditions

In the entrance region, a fully developed turbulent flow velocity profile in a pipe was prescribed. The theoretical fully developed turbulent profile is expressed by the power law as

( )¹7 max

1 y

mean R

u =u × −⎛⎜⎝ ⎞⎟⎠ (m/s). Non-slip and impermability conditions were applied on all solid boundaries. The turbulence kinetic energy and its dissipation rate are used to specify the turbulence boundary conditions at the flow inlet.

Kinetic energy and dissipation rate are prescribed, respectively, as

( )

²

3 2 ^mean k= u I

where I is the turbulence intensity. The turbulence dissipation rate

ε

is computed as

3 3

4 2

C k

l ε = μ

l

, the characteristic length of turbulence, is a physical quantity related to the size of the large scales that contain the main energy in the turbulent flow. In fully developed channel flows, is limited by the size of the channel (since the largest scales of turbulence cannot be larger than the channel size). An approximate ratio between the physical size of the channel and the characteristic length

l

is:

l

0.07 l= D

For outflow in which the details of flow velocity and pressure are not known a priori, no conditions are defined: Fluent software extrapolates the required information from the interior. In order to determine convergence, a criterion of 10⁻6was used for each residual quantity. To perform a convergence test, we ran a simple test case using a criterion of 10⁻9; no significant changes were observed in either velocity field or the turbulence quantities. Moreover, the standard wall- function approach and the low-Reynolds-number correction method are applied at the near-wall region.

2.3.2 Wall treatment

The criterion that lets us fix the mesh size in the near-wall zone is the dimensionless distance y⁺, written as

Beams interference

Laser backscattered light Flow direction

Bragg Cell Detector

(6)

periodicity of the geometry under study, we carried out a series of simulations for only one HEV element. The mesh with lower node density was used to generate and simulate the computations for the seven HEV elements.

. 78 -0,2

2

+ t f

t f

yu C

y = u = u C = 0 0 R

ν ^e

Thus the size of the first mesh can be expressed as

0.078Re 0,2

Re 2

y D+

y= − (9)

We used both approaches to assign a mesh to the near-wall zone. Considering the conditions required for the two approaches and in accordance with Eq.(9), the size of the grid adjacent to the wall was fixed at about 0.2 mm and 0.0177 mm for the wall-functions and the two-layer model approaches respectively.

2.3.3 Grid dependence test

To determine the appropriate mesh density, the code was run for several mesh cases. Studying the sensitivity of the grid involves setting a starting grid that is increased gradually until no effect on the result quality is seen. Taking into account the

Figure 3: Mesh distribution of a 3D computational domain for one HEV element.

0,000 0,002 0,004 0,006 0,008 0,010

-0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4

1,6 M1

M2 M3 M4 M5 M6

axial velocity (m/s)

Axial position (m)

0,000 0,005 0,010 0,015 0,020

0 2 4 6 8 10 12

Turbulence Dissipation rate (m2/s3)

Axiale position (m)

M1 M2 M3 M4 M5 M6

(a) (b)

0,000 0,005 0,010 0,015 0,020

0 2 4 6 8 10 12 14 16 18 20

Wall y+

Axial position (m)

M1 M2 M3 M4 M5 M6

0,000 0,005 0,010 0,015 0,020

0,00 0,02 0,04 0,06 0,08 0,10 0,12 0,14

Friction Factor

Axial position (m)

M1 M2 M3 M4 M5 M6

(c) (d)

Figure 4: Grid dependence test: (a) variation of velocity (b) variation of turbulence dissipation rate (c)

y

⁺variation along the wall (d) axial profile of the friction factor

(7)

An unstructured mesh with hexahedral volumes was constructed and refined at all solid boundaries, with volume growing gradually with distance from the wall, as shown in Figure 3. The refinement is necessary to resolve correctly the strong velocity and turbulence gradients in this region. The mesh size is controlled by adjusting the number of the nodes in the radial directionN_r, on the periphery of elementN_p, and on the axial length per elementN_z. Simulations were carried out for 3D grids containing 16454, 32909, 61871, 102140, 183872, 283440 cells, labelled M1, M2, M3, M4, M5, and M6 respectively.

The criterion of the grid sensitivity test should be chosen with some caution. A comparison is more effective and more significant when the quantities compared have high gradients and significant variations. Therefore, we chose to compare the velocity profiles, turbulence dissipation rate, wally⁺variation and the friction factor at 1.5 mm downstream from the tab, where the flow undergoes strong shear due to the tab. Because the near-wall model is sensitive to the grid resolution, comparisons were made for the mesh grids considering the parameter, which lets us assess the near-wall model sensitivity on the wall.

y⁺

Figure 4 shows how the axial velocity, turbulence dissipation rate, skin friction factor and y⁺unit evolve with

N

_r N_pandN_z. As we can see, the rough grid (i.e., M1) gives a different development of the velocity profile near the wall, but the converged value is obtained as the grid density increases. The dimensionless distance of the computational cell centre from the solid wall was monitored in order to measure the mesh resolution in the near-wall region. It was observed that

y⁺

y⁺is equal or less than 5 for all the mesh grids used in this study, except for the first one which gives a very large value.

Considering the trade off between accuracy and computational cost, mesh grid M3 was chosen for flow simulation in the HEV mixer. The resulting grid allocation is maintained throughout this study.

The equiangle skew parameter quantifies the mesh quality (0 is best; 1 is worst). The equiangle skew QEAS is a normalized measure of skewness defined as follows:

max min

max ,

180

eq eq

EAS

eq eq

Q θ θ θ θ

θ θ

⎧ − − ⎫

⎪ ⎪

= ⎨⎪⎩ − ⎬⎪⎭

where θmaxand θminare the maximum and minimum angles (in degrees) between the edges of the element and θ_eqis the characteristic angle corresponding to an equilateral cell of similar form. For triangular and tetrahedral elements, θ_eq =60, and for quadrilateral and hexahedral elements, θ_eq =90. By definition , where describes an equilateral element and describes a completely degenerate element.

0≤QEAS ≤1 QEAS =0

EAS 1 Q =

For the mesh adopted in this study, the equiangle skew parameter was always smaller than 0.75 for all cells. The value distribution of this parameter is:

- 2.5% of the meshes of quality ranging between 0.5 and 0.75, which corresponds to average quality

- 23.64% of the meshes of quality ranging between 0.25 and 0.5, which corresponds to good quality

- 73.86% of meshes of quality ranging between 0.0 and 0.25, which corresponds to high quality.

After developing the grid for one tab element, we generalized the mesh to the whole static mixer. We recall that simulations were carried out for Reynolds numbers of 7500, 10000, 12500 and 15000, and that all parameters represented below are taken 1.5 mm behind each tab. Also, we note that the dimensionless coordinate y is defined as 1 y

y= −R, where is the radius of the static mixer and is the distance from the tube centre.

R y

3 RESULTS AND ANALYSIS 3.1 NUMERICAL ANALYSIS

In order to compare the accuracy of various RANS models (Standard K-ε, RNG K-ε, Standard K-ω, SST K-ω) and RSM model, velocity and experimental profiles are compared. As remarked above concerning the wall treatment, the choice is governed by the nondimensional wall distance y⁺ of the mesh.

The two-layer zonal model is used with computed wall distance of mesh size less then 5 and the standard wall function is used with computed wall distance of mesh size about 30.

Analysis of the various velocity profiles plotted in Figure 5 for Re=15000 shows that the results are more sensitive to the turbulence model than to the wall treatment. The main difference lies in the prediction of the recirculation zone, which is minimized by K-ω variant models behind the first tab.

Downstream from the last tab, the two K-ω models overestimate the velocity gradient between the tube centre and the recirculation zone compared to the experimental results.

The velocity profiles predicted by all models decay faster than the experimental data except for the K-ε model, which is in good agreement with the experiments. In addition, Figure 5a shows that all the model predictions are in very good agreement with experimental results for the main flow downstream the first tab. However, behind the last tab, a slight velocity decrease in the central flow region is obtained in all models, as shown in Figure 5c. The fluctuating z-velocity component predicted by the RSM model is in equally good agreement with experimental results in the main flow zone downstream from the first and the last tab, as Figure 5b and Figure 5d show. The numerical and experimental results are in good agreement behind the first tab.

Behind the last tab, the numerical profile underestimates the intensity of the fluctuating velocity. In the near-wall region, the predicted and measured profiles resemble each other qualitatively. Quantitatively, we notice a slight difference between the results provided by LDA and RSM models that can be explained by the difficulty of making accurate LDA measurements in the near-wall zone where the signal-to-noise ratio is large.

This result suggests that the sublayer velocity distribution deviates from the logarithmic assumption for this flow field, which includes strong secondary flow extending into the sublayer zone. Therefore it can be argued that a more rigorous determination of the sublayer is required than that provided by simple wall functions. The two-layer model was found to predict the velocity profiles and recirculation zones more accurately. Taking into account the results discussed above, one

(8)

0,0 0,2 0,4 0,6 0,8 1,0 -0,2

0,0 0,2 0,4 0,6 0,8 1,0 1,2

LDA result K-Epsilon-Standard (TL) K-Epsilon-Standard (WF) K-Epsilon-RNG (TL) K-Epsilon-RNG (WF) K-Omega-Standard (TL) K-Omega-Standard (WF) K-Omega-SST (TL) K-Omega-SST (WF) RSM

Axial Velocity (m/s)

y 0,0 0,2 0,4 0,6 0,8 1,0

0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35

0,40 Experimental result

RSM model result

fluctuanting axial velocity (m/s)

y

(a) (b)

0,0 0,2 0,4 0,6 0,8 1,0

-0,2 0,0 0,2 0,4 0,6 0,8 1,0 1,2

y

Axial Velocity (m/s)

LDA result K-Epsilon-Standard (TL) K-Epsilon-Standard (WF) K-Epsilon-RNG (TL) K-Epsilon-RNG (WF) K-Omega-Standard (TL) K-Omega-Standard (WF) K-Omega-SST (TL) K-Omega-SST (WF) RSM

y

0,0 0,2 0,4 0,6 0,8 1,0

0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40

fluctuanting axial velocity (m/s)

Experimental result RSM model result

y

(c) (d)

Figure 5: Axial velocity profile downstream from the first tab (a) and the last tab (c).

Fluctuating z-velocity components downstream from the first tab (b) and the last tab (d), Re=15000.

can conclude that the k-ε model with two-layer wall-treatment model is in good agreement with experimental data and that the k-ω model is less useful for our case.

The short development length upstream of the mixer in this study may be the source of the anomalous and deformed velocity profiles produced by the k-ω model in the flow centre. Another explanation of this poor performance may be the model’s high sensitivity to the values specified for ω at the inlet. The RSM may not always yield results that are so clearly superior to the simpler models in all classes of flows as to warrant the additional computational expense. However, use of the RSM is recommended when the flow features of interest are the result of anisotropy in the Reynolds stresses. In the circumstances of our study, the RSM model brings no improvement for the quality of the results. These conclusions are similar in all respects for all Reynolds numbers used in this study.

It seems very difficult to make a good choice of turbulence models or wall treatment by merely analyzing velocity and nondimensional wall distance profiles. But according to the flow measurements, one can conclude that the K-ε model with

two-layer zonal model is the more appropriate approach in the present study.

3.2 PHYSICAL ANALYSIS

Providing a clear illustration of a three-dimensional velocity vector field is not a trivial task. Figure 6b and Figure 6c show one-dimensional profiles of axial components of mean and fluctuating velocities along the lines situated behind the tabs.

The lines are normal to the mixer centreline and the position of the lines in the mixer cross-section is illustrated in Figure 6a. We observe that the flow in the static mixer is composed of two distinct zones: a first zone ranging between the axis of the mixer (y= 1) and the position y> 0.5, where the mean velocity is high (about 1.12 m/s) and almost constant and the fluctuating velocity rather weak (between 0.00103 and 0.01052 m/s). In the second zone, delimited by y< 0.4 and the mixer wall, the mean axial velocity is weak and negative and the fluctuating velocity is twice as significant as in the first zone.

The first zone is the zone in which most of the fluid flows; the second zone is a zone of flow recirculation, induced by the

(9)

presence of the tab, that traps the fluid and subjects it to intense fluctuations. These two zones are separated by a thin layer (0.4< y <0.5) in which the gradients of the mean axial velocity and fluctuating velocity are very significant. We note that the fluctuating velocity reaches its maximum at

0.4

y≅ and the axial velocity undergoes a significant acceleration arising from the reduction of the passage cross section due to the presence of the tabs at this position.

(a)

-0,25 0,00 0,25 0,50 0,75 1,00 1,25

Z-Velocity (m/s)

Inlet Tab 1 Tab 2 Tab 3 Tab 4 Tab 5 Tab 6 Tab 7

Radial Position

(b)

(c)

Figure 6: Velocity profiles along a normal line at planes located as shown in (a). (b) mean axial velocity. (c) fluctuating z-velocity

component of the axial velocity

Figure 7 shows vector plots of the x and y mean velocities in the planes normal to the flow. The colours of the velocity contours and vectors indicate the magnitude of the velocity (red denotes fast flow, dark blue denotes the slow flow region, and other colours indicate intermediate velocities). The first normal plane is situated at the flow inlet and the others are taken at 1.5 mm downstream from each tab. Downstream from the first tab a pair of vortices is generated that promotes radial mixing; the velocity in the centre flow being very high compared to the velocity at the top wall. This causes a pressure increase in the upper zone that generates movement of the fluid from the wall zone towards the central zone of the flow.

Tab 1

Tab 2

Tab 3

Tab 4

(For caption see next page)

(10)

Tab 5

Tab 6

Tab 7

Figure 7: Velocity vectors in the mixer cross section downstream from different tabs for Re=15000.

Figure 8 represents the contours of the mean axial velocity in the planes positioned as described previously. A recirculation zone characterized by negative signs clearly appears behind each tab. This zone is a dead zone in which the fluid particles are engulfed and trapped. The contours of the axial velocity field behind the tabs reflect an acceleration of the flow in the central zone and a widening of the recirculation zone until the passage of the fluid on the fourth tab. The buffer zone, which is the transition between the central zone and the recirculation zone, is very thin behind the first tab and the axial velocity in this plane decreases very quickly. This thin zone gradually widens and the axial velocity gradient also gradually decreases until the flow reaches the fourth tab. After this tab, no

significant changes are observed and the flow field appears stable.

Figure 8: Cross-sectional contours of axial velocity for Re=15000.

In light of this result, we conclude that beyond the fourth tab, the tabs merely maintain the flow created by the preceding ones.

Figure 9 shows the longitudinal evolution of the turbulent kinetic energy from upstream of the static mixer to the seventh tab array in the axial plane flow. We notice a strong modification in the distribution of turbulent kinetic energy after crossing the first tab array. The effect of radial flow generated by the streamwise vortices is visible here: fluid particles close to the wall are pushed towards the inner region of the mixer, increasing the turbulence intensity in the centre of the mixer to above that at the preconditioner tube. At the same time, the turbulence kinetic energy increases with the turbulence intensity. Note that most of the turbulence is produced in the buffer zone where the velocity gradients are the highest.

Figure 9: Axial evolution of the turbulence kinetic energy plotted in a longitudinal plane.

After passing the second, third and forth tabs, the turbulence kinetic energy gains in intensity and seems to be diffused in a significant way over a broad area to reach the central zone of the flow. Beyond the fourth tab, the kinetic energy of turbulence no longer increases and diffuses. Just as for velocity field, we note that the four arrays of tabs generate and intensify turbulence within the flow, while the other tab arrays maintain it.

Figure 10: Contour of turbulence dissipation rate in the flow direction in a plane situated at x=0, Re=15000.

Figure 10 shows eight profiles of the turbulence dissipation rate in the plane located as in Figure 6a for a Reynolds number of 15000. This representation lets us appreciate the way in which

(11)

the hydrodynamic structures created by the turbulence promoters affect the dissipation rate in the mixer. At the mixer inlet the profile of the dissipation rate is almost stable and homogeneous. After the first row of tabs the dissipation rate undergoes significant modification and becomes very inhomogeneous. The dissipation rate presents two zones of strong activity: the first zone coincides at the level of the tab and the second zone coincides with the recirculation zone. As described previously, in these two zones the flow presents very intense hydrodynamics structures, principal sources of turbulence energy that explain the intensification of the dissipation rate (which reaches its maximum at the top of the tab). In the flow centre, the dissipation rate does not undergo intensification and remains almost constant.

4 CONCLUSIONS

Numerical simulations of a HEV static mixer with advanced turbulence models and refined mesh grids give significant qualitative and quantitative results. RANS predictions based on two-equation transport models are used, in addition to predictions from a Reynolds stress model. Computations were evaluated via a comparison of the results of the different models and a comparison against experimental measurements of the velocity field. The flow field predictions were generally similar, especially in the leading flow region. In the wall- bounded zone some differences were noted in the velocity profiles predicted by the models. The two-layer model approach appears to be the best way to model the wall-bounded zone for moderate Reynolds number flow.

The k-ε models and the RSM model produced better results for the flow dynamics than the other models. The Reynolds stress transport model employed in this investigation did not offer strong advantages in the shearing region, since simpler models accurately describe flow in this region. One can thus conclude that the k-ε standard model is the model that best predicts the flow. In addition, it is the model that requires less memory and computing time. However, it is difficult to draw conclusions from the comparison between k-ε and RSM models because the more general RSM model presents additional information concerning the fluctuating velocities that the k-ε does not yield.

The three-dimensional nature of the turbulent flow inside the static mixer is underlined. The mixing mechanism is the longitudinal vortices generated behind each row of tabs. When extra shear is generated in the flow, vortices increase the turbulent energy dissipation. Insertion of the tabs on the wall of the static mixer creates a secondary flow in the plane perpendicular to the axial direction. Fluid particles in motion in the wall vicinity are directed to the central region by the tabs, creating a pressure gradient between ascending and downward flows. This pressure difference causes radial flow around the opposite tabs’ sides, which produces counter-rotating vortices whose axis of rotation is in the axial direction. These vortices induce a transverse flow that leads to acceleration and homogenization in the mixing process.

The present study improves our understanding of turbulent flow development and the mixing ability of the HEV static mixer, and shows the utility of advanced numerical methods in studying the flow inside complex geometries. This work has enabled us to choose the turbulence model that best corresponds to the flow in the HEV mixer. This model will be used in the framework of a more complete study to integrate the

mass and heat transfer characterization in HEV mixer, with the ultimate objective of defining the capacities of this mixer as a multifounctional heat exchanger in industrial applications.

REFERENCES

[1] T. Lemenand, D. Della Valle, Y. Zellouf, and H.

Peerhossaini, “Droplets formation in turbulent mixing of two immiscible fluids”, Int. J. Multiphase Flow, 29, pp. 813-840, 2003.

[2] T. Lemenand, P. Dupont, D. Della Valle, and H.

Peerhossaini, “Turbulent mixing of two immiscible fluids”, ASME Journal of Fluids Engineering, 127, pp. 1132-1139, 2005.

[3] S. Ferrouillat, P. Tochan, C. Garnier and H. Peerhossaini,

« Intensification of heat transfer and mixing in multifunctional heat exchangers by artificially generated stream-wise vorticity”, Applied Thermal Engineering (in press), 2006

[4] S. Ferrouillat, P. Tochan, and H. Peerhossaini, « Micro- mixing enhancement by turbulence: application to multifunctional heat exchangers”, Chemical Engineering and Processing (in press), 2006

[5] S. Ferrouillat, P. Tochan, D. Della Valle, and H.

Peerhossaini, « Open-loop thermal control of exothermal chemical reactions in multifunctional heat exchangers”, Int. J.

Heat Mass Transfer (in press), 2006

[6] B.E. Launder and D.B. Spalding, “Lectures in Mathematical Models of Turbulence”, Academic Press, London, England, 1972.

[7] V. Yakhot and S.A. Orzag, “Renormalization group analysis of turbulence: I. Basic theory”, J. Sci. Computing 1, pp. 1-51, 1986

[8] D. Wilcox, “Turbulence Modeling for CFD”, DCW Industries, Inc., 5354 Palm Drive, La Canada, California 91011, 1993.

[9] D. Wilcox, “Simulation of transition with a two-equation turbulence model”, AIAA Journal 32, pp. 247.255, 1994.

[10] M.M. Gibson and B.E. Launder, “Ground Effects on Pressure Fluctuations in the atmospheric Boundary Layer”, J.

Fluid Mech., 86:491-511, 1978.

[11] B. E. Launder, “Second-Moment Closure: Present... and Future?” Inter. J. Heat Fluid Flow, 10(4):282-300, 1989.

[12] B. E. Launder, G. J. Reece, and W. Rodi, “Progress in the Development of a Reynolds-Stress Turbulence Closure”, J.

Fluid Mech., 68(3):537-566, April 1975.

[13] J. Gardavský, J. Hrvek, Z. Chára, and M. Severa,

“Refraction correction for LDA measurements in circular tubes within rectangular optical boxes”, Dantec Information N°9, 1990.

[14] R.F. Warming, and R.M. Beam, “Upwind Second-order Difference Schemes and Applications in Unsteady Aerodynamic Flows”, Proc. AIAA 2^nd Computational Fluid Dynamics Conference, Hartford, Connecticut, pp. 17-28, 1975.

[15] J.T. Barth and D. Jespersen, “The design and application of upwind schemes on unstructured meshes”, Technical Report AIAA-89-0366, AIAA 27th Aerospace Sciences Meeting, Reno, Nevada, 1989.

[16] J.P. Vandoormaal, and G.D. Raithby, “Enhancements of the SIMPLE Method for Predicting Incompressible Fluid Flows”, Numerical Heat Transfer, 7, pp.147-163. 1984.