Comparison of Five Two-Equation Turbulence Models for Calculation of Flow in 90 Curved Rectangular Ducts

Transcription

1 Journal o Applied Fluid Mechanics, Vol. 9, No. 6, pp , 6. Available online at ISSN , EISSN Comparison o Five Two-Equation Turbulence Models or Calculation o Flow in 9 Curved Rectangular Ducts K. Ma and H. Lai School o Mechanical and Power Engineering, East China University o Science and Technology, Shanghai, 37, P. R. China Corresponding Author hlai@ecust.edu.cn (Received September 6, 5; accepted March 9, 6) ABSTRACT This paper presents a comparative study o ive most widely used two-equation turbulence models in predicting the developing lows in two 9 curved rectangular ducts. These include the standard k-ε model, the shear stress transport k-ω model, and three low-reynolds number k-ε models by Jones and Launder, Launder and Sharma, and Nagano and Hishida, respectively. The computational time or convergent solutions, streamwise and secondary velocities, pressure distributions as well as the Reynolds-averaged turbulence quantities resolved by these models are compared and validated against available experimental data. The purpose o this paper are to provide a detailed comparative veriication or applying the ive most widely used two-equation turbulence models to predicting curved rectangular duct lows, which are a kind o proto type lows in luids engineering, and to provide a reerence or the selection o turbulence models in predicting such lows in industrial applications. Keywords: Curved duct lows; RANS (Reynolds-averaged Navier-Stokes); Validation; Predicting capability.. INTRODUCTION Turbulence in a 9 curved duct is a proto type low in luid dynamics and luids engineering. This kind o lows occurs widely in engineering devices such as the cooling coils o heat exchangers and the low passages in turbo-machinery. The streamline curvature brings great inluences on the redistribution o the low-ield. A well known phenomenon is that the turbulence intensity is suppressed by the convex curvature but ampliied by the concave curvature (Bradshaw 973). As the results o the curvature and the pressure gradient, secondary motions, low energy losses and enhanced heat transer occur in curved ducts. These phenomena are very dierent rom the lows in straight ducts, and are oten decisive to the general perormance o the engineering devices. Because o the academic and practical importance o the curved lows, researchers have devoted huge amount o eorts to study the problem, and a great number o theoretical and experimental investigations have been carried out. Benchmark experimental data o curved duct lows can be ound in Humphrey et al. (98), Taylor et al. (98), Kim and Patel (994), Suzuki and Kasagi (), Sudo et al. (), to name a ew. Simple and concise reviews o these experimental studies have been presented by Suzuki and Kasagi () and Sudo et al. (). Seeing in general, valuable experiments on the three-dimensional velocity components, pressure, heat transer and turbulent intensity have been made available or the curved ducts o either circular or rectangular sections, with some typical radial ratio values, Rc / d (where Rc is mean curvature radius o the bend and d is the hydraulic diameter o the duct), and in a sensible Reynolds number range widely seen in engineering. These experimental investigations have not only contributed to the understanding o the development o turbulent low in curved ducts, but also provided detailed observations o secondary motions and turbulence luctuations in a orm suitable or validation o numerical solution techniques and evaluation o turbulence models. Accurate prediction o the development o the turbulence in the curved ducts is still a challenging task or nowadays computational luid dynamics (CFD) because o the numerical method. Restricted by the computational resource, early numerical calculations o turbulent low in curved ducts solved the parabolic or partially parabolic RANS equations (Iacovides et al. 987), which assumed that the low was not returning even locally in the duct and the diusion in the streamwise direction was negligible. Apparently, these assumptions may

2 K. Ma and H. Lai / JAFM, Vol. 9, No. 6, pp , 6. be conlicted in strongly curved duct lows. On the other hand, low resolution discretization schemes such as the irst-order hybrid scheme (Rhie 985) or the two-point backward dierence (Govindan et al. 99) were widely used in those calculations o turbulent lows in curved ducts. As well known to nowaday s CFD study, these simpliications on the governing equations and the low order discretization schemes are harmul to the resolution o numerical results. The second diiculty in predicting the curved-duct turbulent low is the selection o turbulence model. In order to model the eects o curvature on the turbulence ield, two-dimensional low cases were studied by Wilcox and Chambers (977) and Leschziner and Rodi (98), and the coeicients o the turbulence model were modiied by including the radius o curvature o the streamlines. These modeling methods are basically not applicable to complex three-dimensional lows because o the diiculties in numerically reconstructing the streamlines and deining the curvature. For curved duct turbulent lows, which are naturally threedimensional, simple eddy viscosity models (EVM) were widely employed by researchers in the early studies o predicting the low behavior and to reproduce the eects o curvature. Kreskovsky et al. (98) calculated the 9 bend turbulent low o Taylor et al. (98) using a mean-ield closure (one- equation) model. Although a relatively coarse mesh (3 points in radius and points in halspan) was used, satisactory agreement with the experimental data was still achieved over the irst 3 o the bend. Calculation o the same low was perormed by Govindan et al. (99), using a mixing length (zero-equation) model and a iner mesh which has nodes in the cross section. Comparisons o streamwise velocity contours were shown, and impressively, very well agreements between the calculation and experimental data were obtained or all cross sections rom the upstream to the downstream o the ducts. But unortunately, no urther results were shown by Kreskovsky et al. (98) and Govindan et al. (99). Especially, the wall pressure, the turbulence intensities such as the luctuating o velocity components and their cross correlations, which had been made available in Taylor et al. s measurements, were not calculated and compared. In the amily o RANS models or turbulence closure, two-equation eddy-viscosity models are widely employed in numerical analysis o curved duct lows. The k-ε type models are the most popular in the two-equation models. Humphrey et al. (98) and Chang et al. (983) employed the standard k-ε model (SKE) combined with the logarithm wall unction. Their computational grids were relatively coarse. For example, in Humphrey et al. s calculation o the 9 bend turbulent low o Taylor et al., the mesh had only nodes in hal o the span and 4 points in the radial direction in the cross section o the duct. In order to improve the prediction, Iacovides et al. (987) used the SKE in combination with a mixing-length hypothesis wall treatment. Their calculation used a 5 47 mesh to cover the hal cross section o the duct between the symmetry plane and the end wall; such a mesh is much iner than that in Humphrey et al. s (98) study. Meanwhile, the non-diusive QUICK scheme which was strongly recommended by Chang et al. (983) was applied to the discretization. The calculated streamwise and secondary velocities were presented, and good agreements with experimental data were achieved. Based on these studies, Iacovides et al. (99) employed an algebraic second-moment (ASM) model to compute the U-bend low. The ASM model obtains the Reynolds stresses rom an algebraic statement o their transports, it is still belonging to the amily o two-equation models. The high Reynolds number version o k-ε model cannot be applied in the immediate vicinity o the wall, so it has to be implemented with the so-called logarithm wall unction which may loss the universality or complex lows. Low Reynolds number k-ε models (also mentioned as near-wall k-ε models by researchers), however, directly model the turbulent viscosity by introducing a local turbulence Reynolds number and wall-damping unctions. These models allow integration o the transport equations or both k and ε to the wall, at a cost o more intensive computational task. Application o Launder and Sharma s low-re-number k-ε model (974) to 9 curved rectangular ducts has been attempted by Raisee et al. (6). In order to urther improve the predicted results, a cubic nonlinear low-re-number k-ε model (Crat et al. 996) was included in their calculations. The results showed that both linear and non-linear low-re-number k-ε models produced satisactory predictions o the mean low ield, while the non-linear model returned even better predictions o the turbulence intensity quantities. Although no-surprisingly nearwall second moment closures (Suga 3) showed superiority in the results o Reynolds stress, the question o whether it worth to proceed with a complex turbulence model was not answered yet by researchers. Meanwhile, or non-linear models, numerical instability induced by high-order derivatives has to be suppressed so as to guarantee the convergence o calculation (Speziale and Ngo 988). The side eect o such suppression or iltering on the resolution o numerical results is not known yet. Indeed, despite the well-known deiciencies primarily related to the isotropic behavior, linear two-equation models are still the irst choice or industrial purpose o a quick and stable prediction o complex low ields. Another important and attractive two-equation eddy-viscosity model is the shear stress transport k- ω model (SST). The model can also be integrated to the wall and does not require a wall unction. The SST model is originally developed or prediction o aeronautics lows with strong adverse pressure gradients and separation (Menter 993, 994), but has since made its way into most industrial, commercial and research codes. The SST model is also criticized or being not capable o capturing the eects o streamline curvature and system rotation, so modiying the model so as to make it sensible to 98

3 K. Ma and H. Lai / JAFM, Vol. 9, No. 6, pp , 6. rotation and curvature has been attempted by researchers in these years (Shur et al. ; Dhakal and Walters ). By so ar, application o the original SST model to calculate lows with streamline curvature is mainly or two-dimensional (D) cases such as the low in U-turn (Dhakal and Walters ). Shur et al. () compared the calculations o Kim and Patel s 9 rectangular bend using the SST and three other models, the results o skin-riction distribution were presented and compared with experimental data. The study supported that a scalar eddy-viscosity model was able to treat the rotation and curvature properly i the sensitization method by Spalart and Shur (997) was applied. However, no urther calculated result about the curved duct was presented by Shur et al. (). Thereore, the validation and comparison by Shur et al. were too incomplete to provide a reerence or selection o turbulence models in calculating curved duct lows. Based on the status quo o the research on curved duct lows mentioned in above, the present paper presents a comparative study o ive most widely used two-equation turbulence models in predicting the developing lows in two 9 curved rectangular ducts. The models are the standard k-ε model (SKE), the shear stress transport k-ω model (SST), and three low-reynolds number k-ε models by Jones and Launder (97; JL), Launder and Sharma (974; LS) and Nagano and Hishida (987; NH). Two 9 curved ducts which have benchmark experimental data measured by Taylor et al. (98) and Kim and Patel (994), respectively, are selected as testing cases or consideration the eects o geometrical aspect ratio on the low ields. In a curved duct o large aspect ratio, the low is nominally two-dimensional, while the low in a square duct is ully three-dimensional. The major dierence between these two cases is the scale and the inluenced region o the secondary motions. The main objective o this paper is to compare the perormances o these ive simple two-equation eddy-viscosity models in predicting the curved duct lows. The computational time or convergent solutions, the streamwise and secondary velocities, the pressure distributions as well as the Reynoldsaveraged turbulence quantities resolved by these models will be compared and discussed.. GOVERNING EQUATIONS AND NUMERICAL METHOD Consider the Reynolds-averaged Navier-Stokes equations or three-dimensional, incompressible and steady lows, the conservation laws o mass and momentum are written as, u j () x j uu j i p ui ( - uu i j) x x x x j i j j () where p is the pressure, is the kinematic viscosity, u and i u denote the mean and i luctuating velocities, respectively. uu i j is the Reynolds stress which may be calculated using the Boussinesq eddy-viscosity-approximation: u j ui uu i j ijk t( ) 3 x x i j (3) where the turbulence kinetic energy k u v w, is the eddy viscosity. t For the closure o the above equations, a turbulence model must be introduced. In the present work, ive turbulence models are used; they are the SKE, SST, JL, LS and NH. These models are all among the most widely used and are based on the eddyviscosity assumption. The details o the SST model can be ound in Menter s study (994). In the our k-ε type models (ε is the dissipation rate o k), the transport equations or k and ε, respectively, are as ollows: ( uk j ) t k u u i j ui [( ) ] t ( ) x j xj k xj xj xi xj ( D) ( u ) u [( ) ] C uu x x x k x C E k j t ' ' i t i j j j j j where C, C, C, and k (4) (5) are the model constants, the eddy viscosity t is then calculated by: k t C (6), and are damping unctions while D and E are additional terms or consideration o the wall eects. The ormula and constants or models can be ound in relevant literature. Eqs. (), (), (4) and (5) can be written in a general orm as ollows: u Γ S (7) where u is the velocity vector, Γ is the diusion coeicient o, and S is the source term. represents the dependent variable (which is, u, k i or ε) according to transport equations. For the momentum equation, the pressure gradient is included in the S. Integrating Eq. (7) over the control volume surrounding the node P and shown in Fig., we have: C D S V (8) P where the summation is applied to all the acets o 99

4 K. Ma and H. Lai / JAFM, Vol. 9, No. 6, pp , 6. the control volume V, i.e., = n, s, e, w, b, and t (see Fig. ). C and D are the transport luxes through the ace due to convection and diusion, respectively: C u A F (9) D Γ A () where A is the area o. Fig.. Control volume surrounding node P. The diusive terms D is calculated by the central dierence scheme. For considerations o both high order accuracy o numerical results and stability o calculation, the convection term is discretised by using the third-order SMART scheme (Gaskell and Lau 988) and is implemented with a so-called deerred correction method. Denote the cell-ace values o calculated by the irst-order upwind (UPW) scheme and a high-order scheme with UPW and H, respectively, the convection term becomes: C F F F () H UPW H UPW With these treatments, Eq. (7) is discretised into the inal orm as ollows: a PP a b N i N () i where the subscript Ni denotes the neighbor nodes N, S, E, W, B and T, shown in Fig.. The coeicients a, P a and N i b in discretization Eq. () and details about solution procedure are given by Lai et al. (). The solution o the governing equations is based on the SIMPLE algorithm (Patankar 98). The calculations in this paper are carried out using an in-house code which uses a inite-volume grid arrangement in non-orthogonal system. The code is written using the FORTRAN computer language, and is parallelized according to the Open MP protocol or running on a small computer workstation which has processing elements. For the simplicity o comparing the computational time in the present study, all the algebraic equations in the orm o Eq. () are solved using the tridiagonal matrix method (TDMA) and applying an alternating direction implicit (ADI) strategy in the three coordinate directions. During a solution step, the coeicients in Eq. () are kept ixed or three to ive iteration steps using the TDMA and ADI method, then the temperate results o P are employed to update the coeicients o the discrete Eq. () so as to carry out a new solution step. Meanwhile, under relaxation is applied. The relaxation actors or pressure, velocity and turbulence quantities are set to be the same or calculations o using all the ive turbulence models, namely, p.7, u.3, and turb..3, respectively. Further, or checking the convergence o calculations, denote the low rate o a curved duct by Q, the calculated low rate at a cross-section as Q, convergence o calculation is judged by satisying the criteria: max Q Q E Q mp 4 E Q maxmp 6 E3 Q, 3 (3) where mp is the unbalanced mass source o the pressure correction equation. The values o mp or all grid nodes should be zero when the continuity equation is satisied (Patankar 98). As pointed out by Shur et al. (), diiculties or imposing the inlet boundary conditions are encountered or computing these lows, due to the lack o experimental data on the mean and turbulent low quantities in the near-wall region o the reerence section o the duct. In this paper, the inlet boundary conditions are imposed by set the inlow velocity U equal to the bulk velocity Uc, the velocity components in the cross-section and normal to the bulk low. Secondary velocities V and W at the inlet boundary are set to be zero. The inlow conditions or k and ε are imposed according to ully developed pipe low, i.e., kin= (.5~.5)% Uc /, and εin = Cμkin 3/ /(.3L), where L is the characteristic length o the low. At the downstream boundary, zero gradient conditions are imposed or all variables except pressure. The noslip condition is assigned or the velocity on solid walls, while wall conditions applied or k and ε are not summarized here or limited space. 3. TESTING CASES AND RESULTS For convenience o description, depict the two 9 curved ducts experimentally studied by Taylor et al. (98) and Kim and Patel (994), respectively, as Case A and Case B. Fig. (a) shows the geometry o Case A. The bound o square cross-section is d = 4 mm, and the mean bend radius is 9 mm, which result in a radius ratio o.3. The upstream and downstream tangent lengths are.3 m and.8 m, respectively. The bulk velocity Uc = m/s, 9

5 K. Ma and H. Lai / JAFM, Vol. 9, No. 6, pp , 6. corresponding to the Reynolds number, Re, o 4. Fig. (b) is the coniguration o Case B. The curved duct is o rectangular cross-section with an aspect ratio o 6 and the duct width H =.3 m. The lengths o the duct beore and ater the curved section are 8H and 6H respectively. The bulk velocity is Uc = 6 m/s, and then Re = 4. XH is the coordinate in upstream (XH takes negative value) and downstream (XH is positive) tangents o the bend, normalized by the hydraulic diameter d. In Case A, r * is the normalized radial coordinate, r * =(r-ro)/(ri-ro); z * is the normalized spanwise coordinate, z * =z/(.5d), so z * = is the symmetry plane o curved duct while z * = is the side wall. In Case B, Y is normalized radial coordinate, so Y= and are the outer and inner walls, respectively. And z * is the normalized by the duct width H in this case, z * =z/h. number o nodes is summarized in Table. All the tested grids are stretched in cross-sections so as to resolve the low ields near the walls. Figure. 3 is the example o checking the grids or Case A. The streamwise velocity proiles calculated using the LS at θ=3 are shown. At this plane, the core luid still remains at the inner wall. It can be seen that grid-reining rom Mesh A to A has produced a noticeable dierence in the velocity proiles. However, when the grids are urther reined, rom Mesh A to A3, the plotting dierences between the two numerical results are negligible. Thereore, Mesh A is believed to be ine enough to produce a grid-independent numerical solution. Similarly, Mesh B is selected or Case B. Table Mesh or grid-dependency check Number o grid points Case Radial Spanwise Streamwise Mesh A 3 3 Mesh A Mesh A Mesh B Mesh B Mesh B (a) Case A Mesh A Mesh A Mesh A Z* (a) θ=3, r * =.3.4. (b) Case B Fig.. Geometry o two curved ducts. 3. Grid-Dependency Check Grid-dependency check is perormed beore comparing the results o turbulence models. Three grids are tested or each case, respectively, and the Mesh A Mesh A Mesh A3..4 Z*.6.8 (b) θ=3, r * =.7 Fig. 3. Streamwise velocity proiles calculated by three meshes in Case A. 9

6 K. Ma and H. Lai / JAFM, Vol. 9, No. 6, pp , 6. The ive models have been careully tested or the grid-dependency, and the results show that Mesh A and Mesh B, or Case A and Case B respectively, are ine enough or all the ive models. So or the simplicity and clarity o comparison between turbulence models, the numerical results and analyses in the ollowed sections are based on using these selected meshes, i.e., Mesh A and Mesh B or Case A and Case B, respectively. 3. Computational Time The needed iteration steps or convergence, tolerance and computational time or all the tested models are summarized and presented in Table. In Case A, the needed iteration steps or solution convergence using SST, JL, LS and NH models are less than SKE model, however, the computational time or SKE is less than other models, because o the lighter computational task or the SKE in a solution step. In Case B, low-reynolds-number k-ε models and SST model have similar iteration steps or convergence. On the other hand, the number o needed solution steps or SKE is slightly more than other models. Identical to the situation in Case A, the computational eiciency o the SKE is the highest among the ive turbulence models. Table Computational eiciency o ive models Models Iter. steps E3 Time SKE.E-8.5h SST 5.E-8.6h JL 5.E-8.6h LS 5.E-8.6h NH 5.E-8.6h (a) Case A Models Iter. steps E3 Time SKE 48.E-8 8h SST 4.E-8 8.h JL 4.E-8 8.h LS 4.E-8 8.h NH 4.E-8 8.h (b) Case B 3.3 Basic Features o the Two Cases Beore comparing the results predicted by dierent models, the basic eatures o the low ields calculated by using the LS in the two cases are compared and presented in Fig. 4 to 6. Figure. 4 compares the streamwise velocity proiles in the symmetric planes o the ducts, at θ=45 and 9. At θ=45, the maximum velocity o two ducts occurs both near the inner wall o the bends (where r=ri or Y=, see Fig. ). The radial gradients o the velocity near the outer side o the bends (where r=ro or Y=, also please see Fig. ) are less steep than near the inner wall. At the outlet o the bends, θ=9, the velocity proile or Case B remains similar to that at θ=45, but its value becomes more uniorm along the radial direction. For Case A, the peak o streamwise velocity at θ=9 has moved to the duct center, the velocity gradient at the outer wall becomes steeper than at the inner wall. Seeing rom velocity proiles in Fig. 4, the low in Case A is more complex and ully three-dimensional. On the other hand, the low in Case B has the characteristics o nominally two-dimensional low. The pressure distributions along the centerlines o the outer and inner walls are shown in Fig. 5, where the pressure coeicient C, is deined as:.5 p re c p C p p U (4) It can be seen that the radial pressure gradient (the dierence o pressure coeicient between the outer and inner walls at every streamwise location) in Case A is steeper than that in Case B. On the inner wall, Case A s minimum pressure coeicient occurs at θ=3, indicating that the avorable pressure gradient turns into the reversed at the location. Such trough o the C p also means the low is changed rom accelerating to decelerating. But in case B, the pressure coeicient decreases mildly along the walls, except in the regions near the inlet and outlet o the bend, means the low velocity in Case B is remained nearly uniorm along the streamwise o bend Case A Case B r*() (a) θ= Case A Case B..4.6 r*().8 (b) θ=9 Fig. 4. Streamwise velocity proiles along symmetrical centerline calculated by LS model. 9

7 K. Ma and H. Lai / JAFM, Vol. 9, No. 6, pp , Cp Outer surace Inner surace Case A.4 -. Cp Outer surace Inner surace Case B Fig. 5. Pressure coeicients along the centerline o the outer and inner walls, calculated by LS. The most noticeable dierence between two cases is the area inluenced by secondary motion shown in Fig. 6 (cross-section at θ=9, right side o the cross section is the inner surace o the bend; In Case B, only one third o the cross-section is shown or the secondary velocity vectors). Compared with that in the Case A, the secondary motion in the Case B is weaker and is conined to the corner regions near the sidewalls. Because o the large aspect ratio, the area inluenced by the secondary motion is very limited. Thereore, the low ield in Case B shows some nominally two-dimensional characters. In Case A, a pair o counter rotating vortices occupy the area o almost the whole cross section, and the magnitude o the secondary velocity is also larger. * z * z * r (a) Case A (b) Case B Fig. 6. Secondary low vectors. Y 3.4 Validation and Comparison Case A The streamwise velocity in cross-sections and calculated by using the ive turbulence models are compared with the experimental data and are shown in Fig. 7. The results o JL, LS and NH models almost duplicate with each other. It can be seen that the low is still developing in the upstream o the bend (Fig. 7(a), XH= -.5), and all the ive models give a similar proile in the core region. Within the curved section, at θ=3, the ive models perorm similarly except in close to the outer wall, i.e., at r * =. in Fig. 7(b), where none o the ive models agree well with the experimental data. At θ=6 (shown in Fig. 7(c)), the dierences between the models are mainly ound at in close to the inner wall, at r * =.9. Due to the secondary motion, the low-momentum luid particles are accumulated in this region. The NH model has obtained the worst prediction, as compared with the other our models and the experimental data. Such transport and accumulation o low-momentum luid particles by secondary motion (see Fig. 6) are carried on until the downstream o the curved section, at r * =.7 and.9 in the cross-section XH=.5 (Fig. 7 (d)). The streamwise velocity becomes very low in this region and low separation is possibly to occur locally, although it is not clearly observed in the present calculation. The SST predicted this trough o streamwise velocity quite well, and so did the low-re k-ε models except the NH. The SKE also predicted the streawise velocity correctly at r * =.7, but at r * =.9, the predicted velocity by the SKE is quite lower than the measured value. Figure.8 shows the calculated radial component o the secondary velocity, V. As compared with the experimental data, it can be seen that the our k-ε models return almost the same results in these cross-sections again. In all these cross-sections, the SST perorms the best in predicting the secondary motion. In Fig. 9, the predicted contours o the turbulent kinetic energy at XH=.5 are compared with the corresponding measured data. It can be seen that the results o the low-re k-ε models JL and LS are very similar to each other, and their predicted peak-value contour lines, scale o.4, are in good agreement with the experiment, either in terms o the location and the scale value. Such peak-value contour line is also predicted well by the SST. However, the low turbulent kinetic energy zones in near the middle o outer wall and in near the inner wall are over predicted by the SST; its predicted values are lower than the measured. On the other hand, the predictions or these low value zones using JL and LS are airly good and acceptable. The NH model under predicted the peak value o the kinetic energy though the location o the peak is also approximately correct. The SKE over predicted the peak o the kinetic energy; the area enclosed by the peak-value contour line predicted by SKE is ar larger than the experimental result o Taylor et al. (98). 93

8 K. Ma and H. Lai / JAFM, Vol. 9, No. 6, pp , 6. r*.9 r* z* (a) XH= -.5 (b) θ=3 r*.9 r* (c) θ=6 (d) XH=.5 Fig. 7. Predicted streamwise velocity o Case A. ( JL; LS; NH; SST; SKE; Taylor et al. 98) In the literature, the turbulence quantities or the present Case A are seldom presented in evaluating the turbulence models. The predicted contours o the turbulence cross-correlation and luctuating velocities, namely, uv and u in the cross-section XH=.5 are shown in Fig. and, respectively, and are compared with experimental data. For the cross-correlation uv shown in Fig., it seems all the results o the ive models have basically agreed with the measured distribution, the result o the LS seems the best in amongst o the ive models. For the luctuating intensities u shown in Fig., the situation is quite similar to the turbulent kinetic energy, shown in Fig. 9. The low-re k-ε models JL and LS return quite similar contours, which are in good agreement with the experiment, either in terms o the location and the scale values. The SST also has predicted the peak values o the intensities well. However, the low value zones in near the middle o outer wall and in near the inner wall predicted by the SST are lower than the measured. The predicted luctuating intensities by the NH model are airly acceptable as compared with the experimental data. The value levels o the contour lines predicted by the SKE are generally higher than the experimental results. 94

9 K. Ma and H. Lai / JAFM, Vol. 9, No. 6, pp , 6. Z* (a) θ= r* Z* (b) XH=.5 Fig. 8. Predicted radial velocity o Case A ( JL; LS; NH; SST; SKE; Taylor et al. 98). r* Case B Again, the predictions o both the mean low velocity and turbulent luctuating quantities are employed to evaluate the perormance o turbulence models in modeling low characteristics through the curved duct. Ure stands or the ree-stream velocity at the reerence station XH=-4.5. It is used to normalize the velocity and the Reynolds-averaged turbulence quantities. The predicted streamwise velocity proiles at some sections in Case B are shown in Fig.. To validate the present calculation, the numerical results by Raisee et al. (6) are included; their calculation also used the LS model. Fig. shows that present results o using the LS model have better agreements with the experimental data (Kim and Patel, 994) than those by Raisee et al. (6). The presently calculated proiles are more close to the measured results, possibly due to the high resolution numerical methods adopted here. There is very little notable dierence between turbulence models in the predicted streamwise velocities, except at downstream o the bend and close to the side wall, as shown in Fig. (d). At such location, none o the selected models makes a good agreement with the measured data. The early analysis shows that the secondary motion is strongest at about XH=.5 and z * =.5. The secondary vortex introduces a point o inlection in the velocity proile. According to the measured data, the maximum o the streamwise velocity appears at about =.7. The result o the SST model seems agreed slightly better than the other our models with the experimental data. In Fig. 3, the predicted and measured turbulent kinetic energy proiles are compared. At the curved section θ=45 (Fig. 3(a) and (b)), it is seen that all selected models under-predict the peak turbulence kinetic energy near the outer wall (Y=, i.e., the outer wall o the bend). In downstream o the curved section, at XH=.5 and z * =.5 (Fig. 3(c) and (d)), the discrepancies between predictions and experimental data are quite clear in near the outer wall o the bend. Only the tendency o variation is 95

10 K. Ma and H. Lai / JAFM, Vol. 9, No. 6, pp , 6. * z (a) Experimental data (Taylor et al. 98) * r (b) JL (c) LS (d) NH (e) SST () SKE Fig. 9. Predicted turbulent kinetic energy k/ U ( ) contours at XH=.5 or Case A. C (a) Experimental data (Taylor et al. 98) (b) JL (c) LS (d) NH (e) SST () SKE Fig.. Predicted contours o uv ''/ U ( ) at XH=.5. C 96

11 K. Ma and H. Lai / JAFM, Vol. 9, No. 6, pp , 6. (a) Experimental data (Taylor et al. 98) (b) JL (c) LS (d) NH (e) SST () SKE Fig.. Predicted contours o u/ U ( ) at XH=.5. C approximately agree with the experiment. In the ive models, the LS model produces a prediction relatively better than other models. Again, Raisee et al s (6) calculated results using the LS model are also included or comparison. Although the turbulence model or the two calculations is the same, the present calculation is more close to the measurement, possibly because o our high resolution code. The comparisons or turbulent stresses are shown in Fig. 4 and 5. Similar to turbulent kinetic energy, the results obtained by ive models are basically alike in the core region in the upstream o the curved section. The LS model perorms better than the other our models. Dierences between the ive models become noticeable near the outer and inner walls, especially in the curved section, at θ=45 or example. In downstream o the bend, the agreement between numerical and experimental results is generally poor, though the LS model results a slightly better prediction. 4. CONCLUSION Even in nowadays, two-equation RANS models are still important choices or industrial purpose o predicting complex low ields. In this paper, ive widely used two-equation turbulence models are compared in predicting the developing lows in two 9 curved ducts. The computational eiciency, the predicted streamwise and secondary velocities, the pressure distributions as well as the Reynoldsaveraged turbulence quantities resolved by these models are compared and validated against available experimental data. The main conclusions are summarized as ollows: a. The numerical results indicate that the computation eiciency o the SKE is the highest among the ive turbulence models or both the ully three-dimensional and the nominally twodimensional 9 -curved-duct lows. However, the predicting capability o the turbulence quantities by the SKE is generally poor. b. The velocity proiles predicted by the SST model show better agreements with experimental data than the other our models in the ully three-dimensional case. The model also obtains a slightly better prediction o the velocity proile in the secondary vortex center. But unortunately, the predicting capability o the turbulence quantities by the SST is also relatively poor. c. In the two tested cases, the predicted results by the JL and LS models are very similar. As compared with available data, the two low-re number models obtained better agreements than the SKE and the SST models. The LS model returns better predictions or the Reynolds stresses and the turbulent kinetic energy. Validations by experimental data show that the capability o predicting turbulence quantities is indeed a 97

12 K. Ma and H. Lai / JAFM, Vol. 9, No. 6, pp , 6..5 U/Ure Experimental data LS (Raisee et al. (6) ) LS JL NH SST SKE.5 U/Ure (a) θ=45, z * = (b) θ=45, z * = U/Ure U/Ure (c) XH=.5, z * = (d) XH=.5, z * =.5 Fig.. Predicted streamwise velocity proiles. k/ure^ Experimental data LS(Raisee et al. 6) LS JL NH SST SKE k/ure^ (a) θ=45, z * = (b) θ=45, z * = k/ure^.6.4 k/ure^ (c) XH=.5, z * = (d) XH=.5, z * =.5 Fig. 3. Predicted turbulent kinetic energy k/ U ( ). re 98

13 K. Ma and H. Lai / JAFM, Vol. 9, No. 6, pp , 6. Reynolds shear stress Experimental data LS(Raisee et al. 6) LS JL NH SST SKE Reynolds shearstress (a) θ=45, z * = (b) θ=45, z * = Reynolds shear stress - - Reynolds shear stress (c) XH=.5, z * = (d) XH=.5, z * =.5 Fig. 4. Predicted cross-correlation uv ''/ U ( ). re Reynolds stress Experimental data JL LS NH SST SKE Reynoldsstress (a) θ=45, z*= (b) θ=45, z*=.5.8 Reynolds stress.6.4. Reynoldsstress (c) XH=.5, z*= (d) XH=.5, z*=.5 Fig. 5. Distributions o Reynolds normal stress u'u' / U re ( ). 99

14 K. Ma and H. Lai / JAFM, Vol. 9, No. 6, pp. x-x, 6. weakness or the ive two-equation models. However, the LS model has still provided relatively more avorable predictions than the other our models in this paper. ACKNOWLEDGEMENTS This study is unded by the National Natural Science Foundation o China (NSFC) under grants and REFERENCES Bradshaw, P. (973). Eects o streamline curvature on turbulent low. Advisory group or aerospace research and development, Paris, France. Chang, S. M., J. A. C. Humphrey and A. Modavi (983). Turbulent-low in a strongly curved U- bend and downstream tangent o square crosssections. PhysicoChemical Hydrodynamics 4(3), Crat, T. J., B. E. Launder and K. Suga (996). Development and application o a cubic eddyviscosity model o turbulence. International Journal o Heat and Fluid Flow 7(), 8-5. Dhakal, T. P. and D. K. Walters (). A threeequation variant o the SST k-ω model sensitized to rotation and curvature eects. Journal o Fluids Engineering 33(),. Gaskell, P. H. and A. K. C. Lau (988). Curvature compensated convective transport: SMART, A new boundednesspreserving transport algorithm. International Journal or Numerical Methods in Fluids 8(6), Govindan, T. R., W. R. Briley and H. McDonald (99). General three-dimensional viscous primary/secondary low analysis. AIAA journal 9(3), Humphrey, J. A. C., J. H. Whitelaw and G. Yee (98). Turbulent low in a square duct with strong curvature. Journal o Fluid Mechanics 3, Iacovides, H., B. E. Launder and P. A. Loizou (987). Numerical computation o turbulent low through a square-sectioned 9 bend. International Journal o Heat and Fluid Flow 8(4), Iacovides, H., B. E. Launder, P. A. Loizou and H. H. Zhao (99). Turbulentboundary-layer development around a square-sectioned U- bend: measurements and computation. Journal o Fluids Engineering (4), Jones, W. P. and B. E. Launder (97). The prediction o laminarization with a twoequation model o turbulence. International Journal o Heat and Mass Transer 5(), Kim, W. J. and V. C. Patel (994). Origin and decay o longitudinal vortices in developing low in a curved rectangular duct (Data bank contribution). Journal o Fluids Engineering 6(), Kreskovsky, J. P., W. R. Briley and H. McDonald (98). Prediction o laminar and turbulent primary and secondary lows in strongly curved ducts. NASA Report, No. CR Lai, H. X., G. L. Xing, S. D. Tu and L. Zhao (). A pressure-correction procedure with high-order schemes implemented on unstructured meshes. International Journal o Numerical Methods or Heat & Fluid Flow (3), Launder, B. E. and D. B. Spalding (974). The numerical computation o turbulent lows. Computer Methods in Applied Mechanics and Engineering 3(), Launder, B. E. and B. I. Sharma (974). Application o the energy-dissipation model o turbulence to the calculation o low near a spinning disc. Letters in Heat and Mass Transer (), Leschziner, M. A. and W. Rodi (98). Calculation o annular and twin parallel jets using various discretization schemes and turbulence-model variations. Journal o Fluids Engineering 3(), Menter, F. R. (993). Zonal two equation k ω models or aerodynamic lows. AIAA Paper Menter, F. R. (994). Two-equation Eddy-viscosity Turbulence Models or Engineering Applications. AIAA journal 3(8), Nagano Y. and M. Hishida (987). Improved Form o the k-ε Model or Wall Turbulent Shear Flows. Journal o Fluids Engineering 9(), Patankar, S. V. (98). Numerical Heat Transer and Fluid Flow, Taylor and Francis, Philadelphia, London. Raisee, M., H. Alemi and H. Iacovides (6). Prediction o developing turbulent low in 9 curved ducts using linear and nonlinear lowre k ε models. International Journal or Numeethods in Fluids 5(), Rhie, C. M. (985). A three-dimensional passage low analysis method aimed at centriugal impellers. Computers and Fluids 3(4), Shur, M. L., M. K. Strelets and A. K. Travin (). Turbulence modeling in rotating and curved channels: Assessing the Spalart-Shur correction. AIAA journal 38(5), Spalart, P. R. and M. Shur (997). On the sensitization o turbulence models to rotation and curvature. Aerospace Science and Technology (5),

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