1. INTRODUCTION Computational fluid dynamics (CFD) will increasingly be used as a tool in the design engineering of advanced nuclear systems, such as the Very High Temperature Reactor (VHTR). Codes such as STAR-CD, FLUENT and several others can facilitate evaluation of thermohydraulic performance of various sub-systems and regions of the reactor system. One such CFD application is of flow in the lower-plenum of the VHTR. However, as design certification is called upon by nuclear industry and regulatory authority, any first-of-a-kind application of CFD to plant design needs to undergo a formal validation and verification (V&V) procedure. Benchmarking a CFD code against a generally accepted dataset from an experiment with similar flow phenomena is an important step in V&V. In the present study, we sought to simulate flow over a tube bank (with tubes of one diameter), for which an accepted velocimetric database exists (Simonin and Barcouda). Using both STAR-CD and FLUENT independently, under steady and transient conditions, and with various turbulence models, we investigated the flow over a single and multiple tube bundle. Some initial results from the simulations and analyses thereof will be described.
2. SCOPE OF WORK A tube bank is most commonly encountered in thermal engineering components in heat exchangers. Flow over tube banks (row of aligned or staggered rows of tubes) may consist of both internal and external flow; that is, inside and outside the tube. Heat transfer can be enhanced with judicious design of both the internal/external flow channels; for a given internal flow, performance can be improved on careful arrangement of tube array. In the present work, we sought to begin verifying and validating the FLUENT and STAR-CD codes against the benchmark experiment of flow through a staggered tube bundle, as reported by Simonin and Barcouda. Under steady-state conditions, and with four turbulence models (standard ε−k , Renormalized(-ation) Group (RNG)
ε−k , “realizable” ε−k , and SST ω−k ), we learned the following in summarized form. CONCLUSIONS TO DATE In the simple but important flow over a cylinder, we have learned the following:
o That the mean streamwise and spanwise velocities can be simulated using the standard and realizable ε−k models but perform marginally to poorly for flow regions with periodic and/or flow structures such as a wake.
o The SST ω−k model can simulate some of the same “physics” contained in the RNG ε−k model; thus generate similar trends but was not found to compare well to experimental data.
o The RNG ε−k model qualitatively gives results most similar to Simonin and Barcouda. However, it too cannot reproduce steep changes in the Reynolds shear stress, vu ′′ .
o For spatio-temporal phenomena, phased averaged velocimetric quantities should reveal additional contrasts and similarities amongst turbulence models.
The V&V exercise is as stated ongoing. We plan to report additional results at a future venue
Figure. Representative magnitude of velocity path lines of flow around a cylinder using the RNG ε−k Turbulent Model. The cylinders are duplicated for visual effect. REFERENCES Buyruk, E., Heat Transfer and Flow Structure Around Circular Cylinders in Cross-Flow. J. of Engineering and Environmental Science, 23, 299-315, 1999. FLUENT, Inc., As of 1/14/07 at URL: http:// www. fluent.com/software/fluent/index.htm Launder, B.E. and D. B. Spalding, Lectures in Mathematical Models of Turbulence, Academic Press, London, England, 1972 Menter, F.R., Two-equation eddy viscosity turbulence models for engineering applications, AIAA Journal, 32, 269-289, 1994. Rodi, W., Turbulence models and their applications to hydraulics, IAHR Monograph, Taylor Francis, ISBN: 9789054101505, 1993
ABSTRACT Computational fluid dynamics (CFD) will increasingly be used as a tool in the design engineering of advanced nuclear systems, such as the Very High Temperature Reactor (VHTR). Codes such as STAR-CD, FLUENT and several others can facilitate evaluation of thermohydraulic performance of various sub-systems and regions of the reactor system. One such CFD application is of flow in the lower-plenum of the VHTR. However, as design certification is called upon by nuclear industry and regulatory authority, any first-of-a-kind application of CFD to plant design needs to undergo a formal validation and verification (V&V) procedure. Benchmarking a CFD code against a generally accepted dataset from an experiment with similar flow phenomena is an important step in V&V. In the present study, we sought to simulate flow over a tube bank (with tubes of one diameter), for which an accepted velocimetric database exists (Simonin and Barcouda). Using both STAR-CD and FLUENT independently, under steady and transient conditions, and with various turbulence models, we investigated the flow over a single and multiple tube bundle. Some initial results from the simulations and analyses thereof will be described.
1. INTRODUCTION A tube bank is most commonly encountered in
thermal engineering components in heat exchangers. Flow over tube banks (row of aligned or staggered rows of tubes) may consist of both internal and external flow; that is, inside and outside the tube. Heat transfer can be enhanced with judicious design of both the internal/external flow channels; for a given internal flow, heat transfer performance can be improved on careful arrangement of tube array (Buyruk, 1999; Tang, 2006).
The fluid dynamics of rod bundle flow are characterized by curvature effects, pressure gradients, turbulent flow separation and reattachment, and wake flow
local (downstream) dynamics as noted by a number of researchers; Buyruk, Sumner (2000), Seetharamu (2001), and Li et al. (2003). Thus, one key to optimizing heat transfer performance is to understand the complex flow around not only one, but many cylinders making up the tube bundle. A particular challenge in turbulent flow around a tube bundle is the accurate spatio-temporal prediction of turbulent flow. As the accurate simulation is sensitive to the turbulence model used in the simulation, the exercise itself is one of evaluating the application of the appropriate turbulence model versus the computational load (time required per simulation).
In order to validate and verify commercial computational fluid dynamics (CFD) codes for nuclear reactor design certification, one of the objectives of the present work it to explore the predictive ability of four popular turbulent models, as follows: the standard ε−k , the RNG ε−k , the realizable ε−k model, and the Shear Stress Transport (SST) ω−k turbulent models. Each model is applied to turbulent flow around a staggered tube bundle using two codes, FLUENT and STAR-CD. 2. TURBULENCE MODELS In the present work, we began with the Navier-Stokes equations (NSEs) and then applied a traditional Reynolds averaging method. For brevity, we refer the reader to classic texts on turbulent theory and models such as Tennekes and Lumley (1976) and Rodi (1993). We consulted additional brief notes on turbulent flow models in manuals for FLUENT (2005) and STAR-CD (2005). We briefly present the governing equations and the turbulent flow models used to date. 2.1 Flow Governing Equations
The modified NSEs, often called the Reynolds-Averaged Navier-Stokes (RANS) equations, can be written in the tensor form as,
Proceedings of ICONE15:15th International Conference on Nuclear Engineering
Here, the unknown terms are called the Reynolds Stress terms and defined as,
jiij uu ′⋅′⋅−= ρτ 2.2 Boussinesq Hypothesis
We take the traditional approach in modeling Reynolds Stress and employ the Boussinesq hypothesis in order to relate the Reynolds Stress with the velocity gradients as shown below:
iji
it
i
j
j
itji x
ukxu
xuuu δμρμρ ⋅⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂⋅+⋅⋅−⎟
⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂∂
⋅=′⋅′⋅−32
(3) 2.3 Modeling turbulent eddy viscosity
In order to simulate turbulent flow, the turbulent eddy motion is here described b the turbulent viscosity term ( )tμ . As well known, in contrast to the molecular viscosity (�), the eddy viscosity depends on the flow dynamics. Therefore, selection of the appropriate turbulence model characterizing the eddy viscosity is directly linked to the accuracy of the simulation relative to a consensus benchmark experiment or simulation.
To objectively assess the accuracy of the turbulent hydrodynamic simulations of flow over a staggered tube bank, we initially chose four different turbulence models with varying degrees of sophistication and accessible documentation. The models in this study are as follows: standard ε−k , Renormalized (-ation) Group (RNG)
ε−k , “realizable” ε−k , and SST ω−k . The eddy viscosity turbulent model for each model is briefly described below. 2.3.1 Standard ε−k Turbulent Model
For the standard ε−k Model (Launder and Spalding, 1972), the turbulent viscosity is computed by a combination of the turbulent kinetic energy (TKE), k , and its rate of dissipation, ε , as follows,
ερμ μ
2kCt ⋅⋅= (4)
The quantities, k and ε , are obtained from the
following equations,
ερσ
μμρ ⋅−Ρ+
⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂∂
⋅⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+⋅
∂∂
=⋅ji xk
k
txDt
Dk (5)
and
kCP
kC
xxDtD
i
t
j
2
21ερεε
εσμ
μερ εε ⋅⋅−⋅⋅+⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂∂
⋅⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+⋅
∂∂
=⋅
(6)
Here, P, represents the production of turbulent kinetic energy. The terms, kσ , εσ , ε1C
and ε2C are constant and
take on values described in the Appendix. 2.3.2 Realizable ε−k Turbulent Model
The difference between the traditional ε−k model and the “realizable” ε−k model is that μC is itself considered a variable here and modeled as follows,
ε
μ *
0
1kUAA
C
S+
= (7)
The k and ε were obtained from the following
equations
ρεσ
μμρ −Ρ+
⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂∂
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+
∂∂
=ji xk
k
txDt
Dk (8)
and
υεερε
εσμμερ
+−
⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂∂
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+
∂∂
=k
CxxDt
D
i
t
j
2
2 (9)
In the above equations, P again represents generation
of turbulent kinetic energy. As evident, its formulation is identical to that shown in the standard ε−k model. We noted that, sA , 0A , ε1C , 2C , kσ , and εσ are model
constants, while *U is a function of mean strain rate and rotation rate. 2.3.3 RNG ε−k Turbulent Model
The RNG-based ε−k turbulent model is derived from a formulation of the instantaneous NSEs, using an analytical approach called a renormalized group (RNG) model (Yakhot and Orszag, 1986). The application of RNG theory yields a differential equation for the turbulent viscosity as follows,
where one uses an effective viscosity, �eff . Although Eq. (7) is unique to the model, the equations for the k and ε have appear in form similar to the standard ε−k model as follows,
ερμαρ ⋅−Ρ+⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂∂
⋅⋅⋅∂∂
=⋅j
effki x
kxDt
Dk (11)
And
kCP
kC
xxDtD
ieff
j
2
21ερεεμαερ εεε ⋅⋅−⋅⋅+⎥
⎦
⎤⎢⎣
⎡∂∂
⋅⋅⋅∂∂
=⋅
(12)
In the above equations, P , again represents generation of turbulent kinetic energy and has the same formulation as the standard ε−k model. In the above equations, P , again represents generation of turbulent kinetic energy and has the same formulation as the standard ε−k model.. The ε1C and ε2C are model constants. kα and εα are computed from the formulation derived by RNG theory,
03929.2
3929.23929.1
3929.13679.0
0
6321.0
0=
−−
−−
αα
αα (13)
2.3.4 SST ω−k Turbulent Model
The Shear Stress Transport (SST) ω−k model uses an expression for eddy viscosity as follows,
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⋅⋅Ω
⋅⋅=
ωα
ωρμ
1
2* ,
1max
1
aF
kt
(14)
The Ω represents a mean rate of rotation and 2F is the blending function used to ensure a transition between the
ω−k model at the wall and the in the ε−k model in the bulk flow. The term, *α , is given by,
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+
+⋅=
kt
kt
R
RRe1
Re*0
**α
αα (15)
where ωμ
ρ⋅⋅
=k
tRe , 6=kR , 3
*0
tβα = and 072.0=tβ
Besides the above relationships, the model has
equations for turbulent kinetic energy and the specific dissipation rate, k and ϖ respectively. Additional details are given my Menter (1994). ( )
kkkj
kj
SYGxk
xDtkD
+−+⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
⋅Γ∂∂
=⋅ρ (16)
and
( )
ωωωωωωωρ SDYGxxDt
D
jj++−+⎟
⎟⎠
⎞⎜⎜⎝
⎛
∂∂
⋅Γ∂∂
=⋅ (17)
The terms kG and ωG represent the production of k
and ω , respectively, while kY and ωY present the dissipation of k and ω . The terms, kΓ , and ωΓ , are the effective diffusivity for k and ω . 3. CFD SIMULATION OF FLOW ACROSS TUBE BUNDLE As noted, in order to be able to use commercial CFD codes in the design engineering process of advanced nuclear systems, such as the VHTR, (hopefully) all codes need to be validated and verified (V&V) against experimental data from a flow configuration similar to the anticipated design flow. In the present case, we are interested in flow in the lower plenum of the VHTR. Here, flow of a multiple number of thermally stratified (unmixed) jets of unequal velocity enter the lower plenum from above (vertically) and is redirected 90° as it impinges on the lower surface of the lower plenum. The jets then encounter a tube bundle consisting of cylinders (pillars) of several different sizes on its way to the outlet. Thus the two simplified flow configurations and phenomena are envisioned as follows: 1) thermal mixing of multiple, thermally stratified jets impinging on the lower surface of a confined space, and 2) flow of partially mixed thermally stratified flow through a tube bundle defined by pillars of different sizes and configurations. Here, we report the results to date on flow through a tube bundle defined by uniform cylindrical elements (see Figure 1). The first flow configuration will be investigated separately.
Figure 1. The geometries of the actual staggered tube bundle array
The flow over the tube bundle shown schematically in Figure 1, reproduces the configuration of flow through a staggered tube bundle by Simonin and Barcouda (1986, 1988). This investigation and the dataset which is accessible is widely accepted as a good benchmark for
(competing) CFD codes. The model consists of seven horizontal staggered rows of rods with diameter of 21.7 mm. The staggered tubes were uniformly spaced in both streamwise and spanwise directions with the distance of 45 mm. Additional Details of the flow are given at the European Research Community on Fluid, Turbulence and Combustion (ERCOFTAC) Databas (2006). Rollet-Miet, Laurance and Ferziger (1999) provide additional information on the flow modeling. Figure 2 shows the unit cell and dimensions of the experimental configuration of Simonin and Barcouda.
In the present study, we mainly used the commercial CFD code, FLUENT (Version 6.2) but a parallel effort using Adapco’s STAR-CD, is also ongoing. With respect to FLUENT (and unless otherwise noted for STAR-CD), both codes use a time-independent incompressible Navier-Stokes formulation and the various turbulent models that are discretized using the finite volume method. Further, a Quadratic Upstream Interpolation for Convective Kinematics (QUICK) and central differencing flow numerical schemes were applied for convective and diffusive terms, respectively. The discrete nonlinear equations were implemented implicitly. To evaluate the pressure field, the pressure-velocity coupling algorithm Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) was selected. The linearized equations were solved by multi-grid method. The numerically approximated equations were performed on the collocated viscous hybrid mesh, approximately totaling 25,000 cells for the cell shown in Figure 2(b). For viscous hybrid mesh topology, the structured quadrilateral cells are generated in the boundary layer region, of which +y of all tube surfaces was less than 5 to ensure the resolution of the turbulent boundary layer, and the rest of the computational domain were unstructured quadrilaterals.
(a) (b)
(c)
Figure 2. Schematic configuration of (a) the staggered tube bundle, (b) the computational domain and unit cell and (c) the sub-unit cell with dimensions per Simonin and Barcouda.
The boundary conditions used in the research were as follows. Due to the repetitive configuration along the streamwise (x) and spanwise (y) directions, the computation was undertaken for only one set of the staggered tubes with the periodic condition as shown in Figure 2(c). A streamwise periodic condition was applied at the inlet and outlet boundaries. This simulated constant mass flow rate is based on the average velocity of water flow. The streamwise periodic boundary condition is taken as fully developed flow, while the spanwise-periodic lateral boundaries is assumed to be repetitive in the y-direction. 4. RESULTS AND DISCUSSIONS
To evaluate the predictive capability of noted turbulence models, results from the simulation and experiments are compared in plots as follows: 1) streamwise mean velocities U , 2) spanwise mean velocity V , 3) turbulent kinetic energy, 4) streamwise Reynolds stress, uu ′′ , 5) spanwise Reynolds stress profiles, vv ′′ , and 6) Reynolds shear stress, vu ′′ . The data is presented streamwise at x=0, 11 and 16.5 mm and also at y=0 and 22.5 mm. The locations of point, (x=11, y=22.5), corresponds to the front stagnation point of the upper cylinder; (x=16.5, y=0) the wake region of the lower (left cylinder in Figure 2(c)) cylinder , while the position, (x=0, y=22.5) locates the oncoming flow for the upper cylinder.
In order Figure 3 shows the above velocimetric quantities at x=-0 [all mm], Figure 4 at x=11 and Figure 5 at x=16.5. Starting with Figure 6, we show the same velocimetric data at y=0 and in Figure 7, at y=22.5. In each of these figures, we show the experimental data of Simonin and Barcouda along with the four turbulence models implemented in our study. Figures 8 and 9 respectively show color iso-contours of the magnitude of velocity along with path lines, and color iso-contours of the static pressure distribution of two adjacent cylinders.
Based on Figures 3-7, it is evident that the streamwise and spanwise mean velocity computations for the (x,y) positions shown in good agreement with the experimental data. The fluctuations shown in the experiment in Figure 3(b), 6(b) and 7(b) is likely due to the periodic vortex shedding in the wake of the upstream cylinder and demarcation of the near- and far-wake regions. Thus one expects a poorer agreement in terms of the mean; a phase averaged mean will be derived instead. We also note that generally, the streamwise and spanwise mean velocity profiles, as computed by use of the standard and realizable
ε−k turbulent models agreed well with the experimental data. As a starting point, these model provide a quick check on the capability of any CFD with such standard models.
With respect to simulation of the turbulent kinetic energy at coordinates (x=0, y) shown in Figure 3 (c), the agreement with experiment is good at (x=0,y), marginal at (x=11, y) and good up to y ~ 5mm but much less so (bad) beyond y > 5mm. As these plots respectively correspond to the oncoming flow, the very near wake region and the wake region approaching the upper cylinder, the dynamics
other than that driven by periodic vortex shedding should be different. Thus one might expect a turbulence model that ‘tracks” the abrupt changes locally, like the RNG model (that uses a differential approach), may reproduce the experimental data points (spanwise) relatively better than other models. This is partially substantiated at y=0 and y=22.5 where all the models, except the realizable
ε−k model predicts the experiment reasonably well along the rear centerline (y=0) and outside the from the frontal separation point to the other (upper cylinder). Regardless, phase-averaged velocimetric quantities based on unsteady (transient, periodic) simulations may be appropriate.
Figure 3-5, (c)-(f) show the Reynolds stress profiles at the line of x=0, 11 and 16.5 for various y-distances. In this Generally, we note that agreement with experiment is poorer as x increases. This is especially true at x=16.5 where the flow past the cylinder “reattaches” at the rear, centerline. In fact, the quantities, uu ′′ and vv ′′ , fair better than the Reynolds shear stress, vu ′′ , as the turbulence model in essence has to simulate the product of both fluctuating components of the turbulent flow. Here again, as above, the abrupt local changes in the flow dynamics can only be marginally (to good) simulated with a “differential” model such as the RNG ε−k model. This partial conclusion is also supported by Figures 6-7, (c)-(f), where again uu ′′ and vv ′′ , fair better than the Reynolds shear stress, vu ′′ . Again, phase-averaged quantities need to be considered.
Table 1 summarizes these results using a simple qualitative assessment of good (g), marginal (m) and poor(p). As even this g-m-p scale is subjective, we did not attempt to further instill subtle judgment. It does however, provide a quick overview.
While plots of turbulent quantities show useful information, color iso-contours of velocimetric quantities reveal information in another manner. In Figures 8 and 9 we respectively show color distributions of the magnitude of velocity, using flow path lines, and the static pressure distribution. Both are shown with each of the four turbulence models. For enhanced visual effect, we show a copy of the unit cell as if viewing two adjacent cylinders. While all of the turbulence models provide the generally similar flow field structures, the there are noticeable differences in magnitude and distribution, including the size of the wake region, amongst the models. In fact, the RNG ε−k model exhibits the largest range of small to large magnitudes and distributions, followed by SST ϖ−k , standard ε−k , and realizable ε−k models. From Figures 3-7, we generally know that the RNG model most accurately simulates the “physics” shown by the experimental data
3(a) 3(b)
3(c) 3(d)
3(e) 3(f) Figure 3. The comparison of (a) Streamwise Mean Velocity, (b) Spanwise Mean Velocity, (c) Turbulent Kinetic Energy, (d) Streamwise Normal Reynolds Stress, (e) Spanwise Normal Reynold Stress, and (f) Shear Reynolds Stress at x=0 mm
4(a) 4(b)
4(c) 4(d)
4(e) 4(f) Figure 4. The comparison of (a) Streamwise Mean Velocity, (b) Spanwise Mean Velocity, (c) Turbulent Kinetic Energy, (d) Streamwise Normal Reynolds Stress, (e) Spanwise Normal Reynold Stress, and (f) Shear Reynolds Stress at x=11mm
5(e) 5(f) Figure 5. The comparison of (a) Streamwise Mean Velocity, (b) Spanwise Mean Velocity, (c) Turbulent Kinetic Energy (k), (d) Streamwise Normal Reynolds Stress, (e) Spanwise Normal Reynold Stress, and (f) Shear Reynolds Stress at x=16.5mm
6(a) 6(b)
6(c) 6(d)
6(e) 6(f) Figure 6. The comparison of (a) Streamwise Mean Velocity, (b) Spanwise Mean Velocity, (c) Turbulent Kinetic Energy, (d) Streamwise Normal Reynolds Stress, (e) Spanwise Normal Reynold Stress, and (f) Shear Reynolds Stress at y=0mm
7(a) 7(b)
7(c) 7(d)
7(e) 7(f) Figure 7. The comparison of (a) Streamwise Mean Velocity, (b) Spanwise Mean Velocity, (c) Turbulent Kinetic Energy (d) Streamwise Normal Reynolds Stress, (e) Spanwise Normal Reynold Stress, and (f) Shear Reynolds Stress at y=22.5mm
Figure 8. The comparison of flow patterns by velocity magnitude path line plot (a) Standard ε−k Turbulent Model, (b) Realizable ε−k Turbulent Model, (c) RNG
ε−k Turbulent Model and (d) SST ϖ−k Turbulent Model
(a)
(b)
(c)
(d)
Figure 9. The comparison of flow patterns by static pressure contour plot (a) Standard ε−k Turbulent Model, (b) Realizable ε−k Turbulent Model, (c) RNG ε−k Turbulent Model and (d) SST ϖ−k Turbulent Model 5. CONCLUSIONS TO DATE
In order to use CFD codes with confidence as a design tool for future advanced nuclear systems, it is of paramount importance to validate and verify the codes against accepted benchmark experiments similar to the anticipated flow configuration. Of special interest in this exercise is learning the capability of various turbulence models to accurately simulate the anticipated complex spatio-temporal flows.
In the present work, we sought to begin verifying and validating the FLUENT and STAR-CD codes against the benchmark experiment of flow through a staggered tube bundle, as reported by Simonin and Barcouda. Under steady state conditions, and with four turbulence models (standard ε−k , Renormalized (-ation) Group (RNG)
ε−k , “realizable” ε−k , and SST ω−k ), we learned the following in summarized form: o That the mean streamwise and spanwise velocities can
be simulated using the standard and realizable ε−k models but expectedly perform marginally to poorly for flow regions with periodic(dynamic) and/or inherent flow structures such as a wake.
o The SST ω−k model can simulate some of the same “physics” contained in the RNG ε−k model and thus gives some of similar trends but was not found to compare well to experimental data.
o The RNG ε−k model qualitatively gives results most similar to that of Simonin and Barcouda. However, it too cannot reproduce spatially steep changes in the Reynolds shear stress, vu ′′ .
o For spatio-temporal phenomena in flow through a tube bundle (dynamic wake region coupled to periodic vortex shedding), phased averaged velocimetric quantities may reveal additional contrasts and similarities amongst turbulence models.
The V&V exercise is as stated ongoing. We plan to report additional results at a future venue.
REFERENCES Buyruk, E., Heat Transfer and Flow Structure Around Circular Cylinders in Cross-Flow. J. of Engineering and Environmental Science, 23, 299-315, 1999. FLUENT, Inc., As of 1/14/07 at URL: http:// www. fluent.com/software/fluent/index.htm Li, T. N.G. Deen and J.A.M. Kuipers. Numerical Study of Hydrodynamics and Mass Transfer of In-Line Fiber Arrays in Laminar Cross-Flow. Third International Conference on CFD in the Minerals and Process Industries CSIRO, Melbourne, Australia. 10-12th December 2003.
Launder, B.E. and D. B. Spalding, Lectures in Mathematical Models of Turbulence, Academic Press, London, England, 1972 Menter, F.R., Two-equation eddy viscosity turbulence models for engineering applications, AIAA Journal, 32, 269-289, 1994. Rodi, W., Turbulence models and their applications to hydraulics, IAHR Monograph, Taylor Francis, ISBN: 9789054101505, 1993 Seetharamu. K. N. Finite Element Simulation of Internal Flows with Heat Transfer using a Velocity Correction Approach. Sadhana, Vol. 26, Part 3, 251-283, June 2001. Simonin, O. and Barcouda, M., Flow through a staggered tube bundle, ERCOFTAC, As of 010807 at URL:http://cfd.me.umist.ac.uk/cgi-bin/cfddb/ ezdb.cgi?ercdb+search+retrieve&default
Table 1 Qualitative Assessment of the Suitability of Turbulence Models versus Flow Regions (Key: g-good, m-marginal, p-poor)
x,y Flow Region Turbulence Model U V k u׳u׳ v׳v׳ u׳v׳
Std k-ε g g g m g m
Real k-ε g g m m g M
RNG k-ε m g g g m m
Fig. 3 (0,y)
Separation pt. & above
SST k-ω m m m m m m Flow Region Turbulence Model U V k u׳u׳ v׳v׳ u׳v׳
Std k-ε g g p p p p
Real k-ε g g p p p p
RNG k-ε g g m p m m
Fig. 4 (11,y)
Wake region & above
SST k-ω g g p p p p Flow Region Turbulence Model U V k u׳u׳ v׳v׳ u׳v׳
Std k-ε g g p m p m
Real k-ε g g m m p m
RNG k-ε g g m m p p
Fig. 5 (16.5,y)
Downstream wake region &
above
SST k-ω g g m m p p Flow Region Turbulence Model U V k u׳u׳ vv׳ u׳v׳
Std k-ε g p m p m p
Real k-ε g p p m g p RNG k-ε m p m m p p
Fig. 6 (x,0)
Wake region & downstream
SST k-ω m p m p p p Flow Region Turbulence Model U V K u׳u׳ v׳v׳ u׳v׳
Std k-ε g m p p p p Real k-ε g m m p m p RNG k-ε m m g g p p
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