Simplified Turbulence Models for Confined Swirling Flows

Engineering Applications of Computational Fluid Mechanics Vol. 2, No. 4, pp. 404–410 (2008)

SIMPLIFIED TURBULENCE MODELS FOR CONFINED SWIRLING FLOWS V. A. Bui Thermal & Fluids Engineering, CSIRO Materials Science & Engineering, PO Box 56, Graham Road, Highett VIC 3189, Australia E-Mail: [email protected] ABSTRACT: Turbulent flow in a stirred vessel is distinguished by the domination of the swirl velocity component, which makes the flow turbulence highly anisotropic. As a result, conventional turbulence models based on the eddy viscosity hypothesis provide poor predictions for not only turbulence characteristics, but also velocity distributions. Many suggestions to modify the conventional turbulence models that take into account the effect of flow swirl have been made. Even though a reasonable distribution of the velocity field can be obtained with the modified k-ε turbulence model proposed by Launder, Priddin and Sharma (1977), the approach fails to provide a good prediction for the turbulence characteristics, especially at the core region. The Reynolds stress transport model based on a second-order closure scheme seems to be able to provide much better prediction results without any ad hoc modification. However, the computational complexity with this model is considerable. The ordinary algebraic stress model, which stems from the Reynolds stress transport model, was found not working for the swirling flows in a stirred vessel. In this work, a modification to the algebraic stress model is proposed, which takes into account additional terms arising from the production terms of the Reynolds stress transport equations during the transformation of these equations from Cartesian to cylindrical coordinate system. Comparisons of the prediction results are made with available experimental data and with the results obtained by means of the differential Reynolds stress turbulence model. Keywords:

turbulence models, confined swirling flow, algebraic stress model, differential stress model

Derivation of the turbulence (Reynolds stress) equations for incompressible, rotating, and curved flows is provided in the work by Lakshminarayana (1986). These equations show that rotation has no effect on the transport equation for kinetic energy and that its effect is mainly to redistribute the kinetic energy in three coordinate directions. At the same time, both the flow curvature and rotation can influence the rate of turbulence energy dissipation.

1. INTRODUCTION 1.1 Characteristics of turbulent confined swirling flows Flows in stirred vessels are characterized as “complex”, where the mean flow curvature and centrifugal forces may change the turbulence structure of the flows. For flows with streamline curvature, it is evident that the centrifugal force suppresses the turbulence on a convex surface and amplifies the turbulence production on a concave surface. Experimental data by Kitoh (1991) show that swirling flow in a pipe comprises three regions based on the tangential velocity distribution: wall, annular and core. In the wall region, only the centrifugal destabilizing effect appears. The annular region is characterized by flow skewness and highly anisotropic turbulence. As a result, the Reynolds stress tensor is not aligned with the main strain tensor, which invalidates the application of eddy-viscosity assumption. In the core region, the centrifugal stabilizing effect becomes important and this part of the flow is characterized by a solid-body rotation.

1.2 Summary of the turbulence models applicable to swirling flows One of the major obstacles in numerical modeling of complex turbulent swirling flows is the unavailability of accurate turbulence closure models. The k-ε model (KEM) is widely used in the computation of turbulent flows. Although KEM performs well for simple flow cases (strictly valid for two-dimensional “simple shear flows”), it is not adequate for complex recirculating or swirling flows involving streamline curvature and rotation, owing to the shortcoming of the isotropic “eddy viscosity” assumption. It is reported that the standard KEM provides wrong predictions for both swirl and

Received: 23 Nov. 2007; Revised: 17 Mar. 2008; Accepted: 18 Apr. 2008 404

Engineering Applications of Computational Fluid Mechanics Vol. 2, No. 4 (2008)

the flow turbulence without ad hoc modifications. The method is capable of correctly describing the highly anisotropic turbulence in strongly swirling flows. Examples of successful DSM simulations of swirling flows are provided in Jones and Pascau (1989), Ohtsuka (1995), and Sharif and Wong (1995). In the Reynolds stress transport equations, there are three terms that need to be modeled, namely: (i) diffusion; (ii) pressure strain correlation; and (iii) viscous dissipation. These terms have to be expressed with respect to mean velocities, Reynolds stresses, and their derivatives. The dissipation term, for instance, can be modeled by the correlation proposed by Hanjalic and Launder (1972), which is more accurate for the complex turbulent flows, where the dissipative motions may not be locally isotropic. The Algebraic Stress Model (ASM) is a simplified version of DSM, which is usually based on the Rodi’s assumption that

axial velocities for strongly swirling flows (Abujelala and Lilley, 1984; Jones and Pascau, 1989). Preliminary CFD results obtained by the author for a simple swirling flow (see below) indicated the tendency of the KEM to produce an excessive solid-body component type of rotation (Fig. 1). Various modifications to the standard KEM have been proposed, such as Richardson number corrections for streamline curvature, to account for the effect of turbulent anisotropy, which is inherent in swirling flows (see Nallasamy, 1987; Lakshminarayana, 1986; Abujelala and Lilley, 1984; Pourahmadi and Humphrey, 1983; Kim and Chung, 1987). However, those modifications are rather ad-hoc and case-based. In these modifications, the constant C μ , used to calculate turbulent viscosity ( vt = C μ k 2 / ε ), or constants C1 / C 2 , appearing in the transport equation for turbulent energy dissipation rate, are redefined as functions of a Richardson number, Ri. Note that the definition of Richardson number can be different in these modifications.

Fig. 1

S (ui u j ) =

ui u j k

S (k ) ,

(1)

where S (ui u j ) and S (k ) are the source terms of the Reynolds stress and kinetic energy transport equations. However, successful application of ASM to confined swirling flows in an axisymmetric geometry seems to be very limited. As pointed out by Fu et al. (1988), the limited predictive capability of the ASM for swirling flows is, possibly, caused by mis-representation of the diffusive transport of the shear stress in axisymmetric flows. The work by Weber, Visser and Boysan (1990), however, reported good predictions of confined swirling flows with recirculation with the ASM model, provided that fine numerical grids were used. It is noteworthy that in that work the extra convective terms, appearing in the Reynolds stress equations in the cylindrical coordinate system, were taken into account during derivation of the algebraic stress correlations. In order to improve the predictive capability of ASM model for swirling flows, some modifications to the stress-strain relations are proposed in Wall and Taulbee (1996) and Zhang et al. (1997). Suggestion to modify the algebraic correlations for Reynolds stresses, based on Richardson numbers, can also be found in Parchen and Steenbergen (1998). In this work, the predictive capability of some modified k-ε models for confined swirling flows will be examined. The computational results were validated using the experimental data obtained at the Thermal & Fluids Engineering Laboratory, CSIRO (Shepherd et al., 2003). A simple

Comparison of DSM and standard KEM predictions for the swirl velocity of a confined swirling flow. The DSM result is in good agreement with the experimental data (Shepherd et al., 2003).

Reynolds Stress Transport or Differential Stress model (DSM) is an implementation based on the solutions of the modeled partial differential equations for the components of the stress. Even though it is complex and computationally expensive, it seems to be the most capable tool for simulation of confined swirling flows since it allows modeling of the anisotropic structure of 405


boundary layer flow types, the proposed C2 formulation is

cylindrical vessel with a strong swirling flow in it was considered. The flow was generated by an impeller, which was located near the top of the vessel as seen in Fig. 2. Water was used as the working fluid and the flow field was measured by the PIV technique. A derived Algebraic Stress Model will also be presented and tested against available experimental data and the DSM prediction results. All calculations are performed using a commercial CFD code (CFX-4.2). In the following derivations, U, V, W represent the axial, radial, and azimuthal velocities respectively, while r is the radius. The CFD simulations were conducted on an axisymmetric cylindrical domain as schematically shown in Fig. 2. The domain had a radius of 0.195 m and a height of 0.93 m, which was represented by a 70×132 computational mesh. In the simulations, free-slip boundary condition was applied at the top surface, whereas the nonslip condition was used at the walls. The flow characteristics at the impeller outlet were set in accordance with the experimental data.

C 2 = C 2o (1 − C ′Ri )

(2)

where Ri is a Richardson number defined by Ri =

k 2 W ∂ ( rW ) . ε 2 r 2 ∂r

(3)

A streamline curvature correction was proposed by Leschziner and Rodi (1984), derived from the Reynolds stress equations, which relates lengthscale changes to the extra production of radial fluctuations due to the centripetal body-force term. The correction is presented as a modification of the turbulence parameter C1 given by C1 = C1o (1 + 0.9 Ri )

(4)

where the Richardson number is defined as ⎡ ∂ ⎛ W ⎞⎤ Ri = ⎢2νtW ⎜ ⎟ /P ∂r ⎝ r ⎠⎥⎦ ⎣

(5)

and P is the turbulence production. Abujelala and Lilley (1984) proposed a modified k-ε model where the turbulent constant C2 is modified by a Richardson number defined as 2 Ri =

Fig. 2

⎛ ∂U ⎜⎜ ⎝ ∂r

2

W ∂ ( rW ) r2 ∂ r

⎡ ∂ ⎛ W ⎞⎤ ⎞ ⎟⎟ + ⎢ r ⎜ ⎟⎥ ⎠ ⎣ ∂ r ⎝ r ⎠⎦

2

.

(6)

Our simulations employing the modifications suggested by Leschziner and Rodi (1984) and Abujelala and Lilley (1984) are shown not being able to correctly predict swirl and axial velocities for confined swirling flows. However, the modification proposed by Launder, Priddin and Sharma (1977) seems to work well and is able to provide the “solid-body rotation” at the core of the flows. The results of computations using the modification by Launder, Priddin and Sharma are presented in Fig. 3 in comparison with a set of experimental data. The constant C ′ was chosen equal to 0.14 in these calculations. The figure shows that the modified k-ε model can provide reasonably good predictions for both swirl and vertical velocities. It seems to be markedly better than the DSM without wall corrections in predicting axial velocity. It is also worth noting that calculation using the k-ε turbulence model has good convergence rate and does not require as much refinement of computational grid as the calculation using the DSM does.

Computational domain.

2. VERIFICATION OF THE MODIFIED k − ε TURBULENCE MODELS The standard k-ε model can be modified to account for the effect of curvature and rotation. Several modifications to the equation of turbulence dissipation are suggested by Launder, Priddin and Sharma (1977), Leschziner and Rodi (1984), and Abujelala and Lilley (1984). Launder, Priddin and Sharma (1977) incorporates streamline curvature effects on the turbulence structure in the k-ε model. The direct effect of curvature on the model is limited to a single empirical coefficient, whose magnitude is directly proportional to a Richardson number based on a time scale of the energy-containing eddies. For 406

Engineering Applications of Computational Fluid Mechanics Vol. 2, No. 4 (2008) Table 1 Extra convection terms in the stress transport equations. Extra convection term (Eij)

uu

–

vv

- 2W (vw/r )

ww

2W (vw/r )

uv

- W (uw/r )

vw

W (vv − ww)/r

uw

W (uv/r )

The extra convection terms, however, can easily be taken into account during derivation of the algebraic stress correlations. The algebraic stress correlation (1) then becomes:

r [m]

Fig. 3

Equation

Predictions of the swirl (solid line) and axial (dash line) velocities by the modified k-ε model in comparison with the experimental data (Shepherd et al., 2003).

ui u j k

In order to improve the prediction capability of the k-ε turbulence model for the flows with low Reynolds number and in near-wall regions, a wall treatment using the law-of-wall or a low Reynolds number modification to the k-ε turbulence model is needed. In this work, the low Reynolds-number k-ε turbulence model proposed by Lam and Bremhorst (1981) is used. Even though the above-mentioned modified k-ε turbulence model works well for some simple geometries, its application for confined swirling flows in more complex vessel geometries seems to have limited success. The modified k-ε turbulence model is shown not being able to predict the structure of swirling flows in irregular recirculation zones.

( P − ε ) = Pij + φij −

2 δij ε − Eij , 3

(7)

where: Pij = − u i u k

∂U j ∂ xk

− u j uk

∂ Ui , ∂ xk

ε 2 2 φ ij = − c1 ( u i u j − δij k ) − c 2 ( Pij − δij P ), k 3 3

(8)

indicate the production and pressure strain tensors, respectively. The production rate of k is given by P = − u k u l (∂ U k /∂ xl ) and the extra convection terms Eij are given in Table 1. Turbulence constants c1 and c2 are equal to 1.8 and 0.6, respectively. Eq. (7) can be simplified and expressed as ui u j

3. A MODIFIED ALGEBRAIC STRESS TURBULENCE MODEL

k

= φ1 Pij + φ 2 δij − φ3 Eij ,

(9)

where 1 , φ1 = (1 − c 2 ) ⋅ φ3 , P + c1ε − ε 2 and φ 2 = ( c 2 P + c1ε − ε ) ⋅ φ3 . 3

The standard ASM model, as supplied in the CFX-4.2 code, was found not working well for axisymmetric swirling flows. In general, the model provided predictions for the swirl and axial velocities that were very similar to those obtained with the standard k-ε model. During transformation of the Reynolds stress equations from the Cartesian to cylindrical coordinate systems, there appear additional terms in the convection, diffusion, and production terms of the equations (see Fu et al. (1988) for more information). While the extra production terms can be taken into account in the (Rodi’s) ASM derivation, other extra terms are commonly neglected.

φ3 =

After some manipulations, algebraic expressions for Reynolds stress components are obtained as follows: ⎛ ∂ U φ 2 ⎞⎟ uu = − C u ⎜ 2uv − ⎜ ∂ r φ 1 ⎟⎠ ⎝ ⎡ ⎛ φ3 ⎞ φ2 ⎤ ∂V W ⎟− ⎥ vv = − C v ⎢ 2 uv − 2 vw⎜ 1 + ⎜ ⎟ φ1 ⎥ r ∂z φ ⎢⎣ 1 ⎝ ⎠ ⎦ ⎡ ⎤ ⎛ ⎞ φ ∂W ∂W 3 W ⎟ φ2 ⎥ ww = − C w ⎢ 2uw + 2 vw ⎜ + − ⎜ ⎟ ∂z ⎢⎣ ⎝ ∂ r φ 1 r ⎠ φ 1 ⎥⎦

407


φ 1k uv =

W r

⎡ ⎤⎛ φ3 ∂U vw − uw ⎥ ⎜ 1 + ⎢ 2C v ∂r φ1 ⎣ ⎦ ⎜⎝

⎞ ⎟ + φ 2 k ⎛⎜ C u ∂ V + C v ∂ U ⎞⎟ ⎜ ⎟ ∂z ∂ r ⎟⎠ ⎝ ⎠ ⎛ ∂U ∂V ⎞ ∂U ∂V ⎟⎟ − 2φ 1 k 1 + φ 1 k ⎜⎜ (C + C v ) + ∂ ∂ ∂r ∂z u z r ⎝ ⎠

⎧⎪

⎡∂W

⎪⎩

⎢⎣ ∂ z

⎡

⎛ ∂W φ3 W ∂W ∂U ∂ U ∂ W ⎞⎟ ⎤ ⎥ + φ 2 kC u + − 2C u vw + uv ⎜ ⎜ ⎟ ∂r ∂ r ∂ z ⎠⎥ ∂z ⎝ ∂ r φ1 r ⎦ ⎛ ∂U V ⎞ 1 + φ 1 k ⎜⎜ + ⎟⎟ ⎝∂z r ⎠

φ 1 k ⎨ uv ⎢ vw =

φ 1 k ⎢C v ⎢⎣

uw =

− 2C v

∂V ∂z

⎡ ⎡ ⎛ φ ⎞ ⎛ ⎛ φ ⎞⎤ ⎫ ⎛ ∂ W φ 3 W ⎞⎤ φ ⎟ ⎥ + uw ⎢ ∂ V + 2 C w W ∂ W ⎜ 1 + 3 ⎟ ⎥ ⎪⎬ + ...φ 2 k ⎢ C v ⎜ ∂ W + 3 W ⎟ − C w W ⎜ 1 + 3 ⎜ + ⎜ ⎟ ⎟ ⎜ ⎜ ⎜ ∂ r φ 1 r ⎟⎥ z r z r r r ∂ ∂ φ ∂ φ φ1 ⎢ ⎢ ⎥ 1 1 ⎠ ⎦ ⎪ ⎝ ⎠ ⎝ ⎝ ⎠⎦ ⎝ ⎣ ⎣ ⎭ ⎛ φ3 ⎞W ⎛ ∂W φ3 W ⎞ ⎛ ∂V V ⎞ ⎟ ⎜ ⎟ (C + C w ) + ⎟⎟ + 2φ 1 k ⎜ 1 + + 1 + φ 1 k ⎜⎜ ⎜ φ 1 ⎟⎠ r ⎜⎝ ∂ r φ 1 r ⎟⎠ v ⎝∂r r ⎠ ⎝

⎞⎤ ⎟⎥ ⎟⎥ ⎠⎦

(10)

where Cu =

φ 1k ∂U 1 + 2φ 1 k ∂z

, Cv =

φ 1k ∂V 1 + 2φ 1 k ∂r

, and C w =

φ 1k 1 + 2φ 1 k

V r

.

Computational results for the swirl and axial velocities of a confined swirling flow in a simple vessel geometry, shown in Fig. 4, are in reasonably good agreement with available experimental data and DSM results. Remarkably, the ASM has been shown to work well for more complex vessel geometries.

If the Reynolds stresses calculated from above expressions are used directly in the momentum equations, numerical instabilities become apparent. In order to alleviate this problem, only the deviations of the stresses from their isotropic values (obtained from the eddy-viscosity model) are introduced to the momentum equations, which still have their diffusion terms based on the isotropic turbulence viscosity. The calculated Reynolds stresses are also used to redefine production term in turbulence energy and dissipation equations.

4. LOW REYNOLDS-NUMBER MODIFICATION TO ALGEBRAIC STRESS TURBULENCE MODEL In derivation of the above algebraic stress model, the wall-reflection redistribution terms were purposedly omitted. These terms are formulated using the wall-normal vector and the distance from the wall, and are included in the stress transport equations in order to correct the pressure/strain interaction term, φ ij , in the nearwall region. Due to their complexity and geometric dependency, the wall-reflection terms are often ignored in ASM formulations, and their use for complex wall geometries is also limited. The effect of the wall can however also be accounted for in the Reynolds stress transport equations using the modification proposed by Shima (1998). In this modification, the turbulence constants c1 and c 2 are formulated as functions of A, A2 and ReT , where A and A2 are the invariants of the stress anisotropy tensor a ij = u i u j /k − 2 δ ij /3 defined by

r [m]

Fig. 4

Predictions for a typical confined swirling flow by the new ASM model.

408


A A2 A3

= 1 − 9 A2 /8 + 9 A3 /8, = a ij a ji , = a ij a jk a ki

(11) 3.

and ReT = k 2 /νε is the turbulence Reynolds number. The turbulence coefficients c1 and c 2 are then determined by

4.

c1 = 1 + 2.45 A21/4 A 3/4 {1 − exp[ − (7 A) 2 ]} {1 − exp[ − ( ReT /60) 2 ]}, c 2 = 0.7 A.

5.

(12)

5. CONCLUDING REMARKS

6.

Due to the complex nature of turbulence in confined swirling flows, turbulence models based on the isotropic eddy-viscosity assumptions are not generally in the position to provide reasonable prediction of flow characteristics. The Reynolds stress transport model, which seems to have all facilities needed for simulation of highlyanisotropic turbulent flows, is almost the only choice for such complicated flows, since its simplified counterpart—the standard algebraic stress model—still does not work well. However, the Reynolds stress transport model, available in CFX-4.2 code, is often too unstable when used for simulation of confined swirling flows. In addition, the Reynolds stress transport model is computationally expensive, which limits its application in many practical problems. Among all suggested modifications to the standard k-ε turbulence model, only modification suggested by Launder, Priddin and Sharma (1977) seems to work for the swirling flow under investigation. However, its deficiency becomes apparent when being applied to complex vessel geometries. The algebraic stress model proposed in this work seems to be able to provide reasonable predictions for highly swirling flows in stirred vessels at a relatively low computational cost.

7. 8. 9.

10.

11. 12.

13.

14.

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