Introduction For a long time, so-called wall functions has been used for calculating wall-bounded turbulent ows. It means that boundar conditions can

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1 Helsini Universit of Technolog CFD-group/ Laborator of Applied Thermodnamics MEMO No CFD/TERMO-9-96 DATE: Ma 24,996 TITLE TEST CALCULATIONS OF VARIOUS TWO-EQUATION LOW REYNOLDS NUMBER TURBULENCE MODELS AUTHOR(S) Patri Rautaheimo ABSTRACT Purpose of this wor is compare dierent two-equation low Renolds number turbulence model and their suitabilit to the dierent ow situations. There is comparison of old ; model, cross-diusion model and Gatsi et al. ARSM. Flow simulations are applied for low Renolds number channel ow, ow over at plate, wall jet and curved duct. MAIN RESULT Implementation of the dierent turbulence models PAGES 7 KEY WORDS two-equation turbulence models, low Renolds number, turbulence modeling. APPROVED BY Timo Siionen Ma 24, 996

2 Introduction For a long time, so-called wall functions has been used for calculating wall-bounded turbulent ows. It means that boundar conditions can be applied to points in the uid awa from the boundaries and thereb avoid the problem of modeling the direct inuence of viscosit. This procedure can be applied onl for the situation in which the universal wall functions are valid. However, for a complex ow situations, the calculation of the ow variables must be extended next to the solid wall. Wall functions do not generall appl to the separation, stream line curvature, a sstem rotation or surfaces with mass or heat transfer. Thus, the low Renolds number turbulence models have been applied in this wor. To calculate near wall turbulence quantities means generall use damping functions in both Renoldsstress models (RSM) and two equation models. These damping functions are depended the distance from the wall, unit wall normals or/and local shear stress in the closest solid wall. These relation are generall \ad hoc" in nature because the contain no turbulence phsics and are calibrated based on the equilibrium turbulent boundar laer. The desired turbulence model would be model that calculates values close to the wall without use of an damping functions. One of the such modelis ;! model of Wilcox []. However ;! model has some undesired features. For example, it is ver sensitive for a free stream turbulence. Other such model has been developed b UMIST group and also b Gatsi et al. [2]. This paper is concerned with an evaluation some of existing two-equation, low Renolds number turbulence model. Two models are ordinar isotropic two-equation models and one is explicid algebraic Renolds-stress model (ARSM). 2 Methods Three models, namel those of [3], cross-diusion modication of Yoon et al. [4] and Gatsi et al. [2] with low Renolds number correction b Abid et al. [5]. The rst one is classical low Renolds number ; model (CH), the second is ; model with cross-diusion modication (CD), and the third one is explicid ARSM (GS). The ; model can be written as u g00 i u00 j = F (S ij W ij T i ~ ( + T i T = c f 2 ~ (2) = ~ + D (3) i ( + T = + P ; i +c 3 i ~ + c i P ; c ~ 2 2 i ~ + E (5) Table summarizes functions and constants for a dierent turbulence models. Constants c and c 3 can be determined from the other constants [4] c = c 2 ; p 2 (6) c s! c 3 = ; ; 24c (7) 2 2 2

3 2 Table. : Functions and constants. Model D ~ w -B.C. c c c 2 c 3 CH 2 n =0 0:09 :44 :92 0:0 :0 :3 CD 2 n =0 0:09 :73 :92 ;:26 :0 0:75 GS. n 2 0:088 :39 :83 0:0 :0 :3 Model f f f 2 E CH :0 ; e ;0:005+ :0 :0 ; 0:22e ;Re2 T =36 ;2(~=n)e 2 ;0:5+ CD :0 ; e ;0:005+ :0 :0 ; 0:22e ;Re2 T =36 ;2(~=n)e 2 ;0:5+ GS.0 :0 :0 ; e ;Re=2:5 0:0 Various Renods numbers and dimensionless distances are dened as Re T = 2 ~ (8) Re = p n + = nu and n is normal distance from the wall. The production of turbulent inetic energ P is exact are modeled in CH and CD model b using Boussinesq ap- and Renolds stresses u g00 i u00 j proximation where S ij is the mean strain rate tensor (9) (0) u = = p w = () P = u 00 i i u00 j j u g00 i u00 j = ;2 T S ij ij (3) S ij j i With explicit RSM model of Gatsi and [2] (GS) the Renolds stresses are given b u g 00 i u00 j = 2 3 6( + 2 ) ij ; [ S ij ; 3 S ij + 4 (S iw j + S j W i ) ; 5 S i S j ; 3 S ls l ij ] (5) where W ij j i is the mean vorticit tensor. In Eq. (5) and are strain rate invariants dened b = 3 (S ij S ij ) 2 2 = 2 (W ij W ij ) 2 2 where, 2, 3, 4 and 5 are the constants that assume the values of 0:07, 0:00250, 0:090, 0:38 and 0:00, respectivel [5]. (4) (6) (7)

4 3 3 Results 3. Channel Flow Fig. : Mean velocit proles in wall coordinates. The models were checed b calculating a full developed ow in a plane channel. The results were compared with the DNS data of Kim et al. [6], and the Renolds-stress budgets were compared with Mansour et al. [7] data. The DNS data is at Re m = u m = 2800 where u m, and are the mean velocit, thechannel half-width and molecular viscosit. The mesh is rectangular The height of the rst row of cells is =0:005 or + 0:9. Onl half of the channel is modeled. The length of the computational mesh is 32. The calculations were performed using cclic boundar conditions. After having a converged result, the solution was taen from the downstream boundar and utilized as the upstream boundar condition of the next run. The velocit proles are compared in Fig. in terms p of u +, which isauniversal dimensionless velocit dened as u + = u=u, where u = w = is a friction velocit. The velocit proles in a viscous sublaer agree well with DNS and universal proles. The velocit pro- les are not completel satisfactor in outer laers. Onl the CD model gives excellent results in this simpl case. The turbulent stress can be seen in Fig.2. It can be seen that CH and GS under predict the sin-friction and that is also the reason wh velocit distributions are so badl estimated. The turbulent stress should be almost straight line between + =50! 80. None of the model has that feature. The inetic energ of turbulence is shown in Fig.3. It can been seen that CH and CD models estimate the pea of the inetic energ of turbulence well in a close wall region whereas GS model do not estimate it at all. It also should be noted that GS model gives no inetic energ of turbulence at the ver close to the wall ( + < 4). Non-isotroph of the Renolds stresses in GS model is shown in Fig.3. It can be seen that the Renolds-stresses are ver close to each other. 3.2 Flat Plate The next test case was the ow over a at plate with high free-stream turbulent intensit. The test case was taen from ERCOFTAC Fluid Dnamics Database WWW Services ( ept b P.Voe. Measurements were made b John Coupland (Rolls-Roce).

5 4 Fig. 2: Comparison of the calculated turbulent stress and the DNS-data in a plane channel. Fig. 3: Comparison of the calculated inetic energ of turbulence and comparison of Renolds stress components with Gatsi model. Inlet velocit was 9:4m/s and the pressure gradient was zero. Measurements were made down to x = :495m that correspond to Re x Upstream turbulence intensit (at theq beginning of the at plate) was Tu = 6:0% where turbulence level is dened as q Tu = ug 00 3 i u00 i =U = 2=U. Dissipation is set so that deca of free-stream turbulence 3 is in balance. Also dissipation values could have been changed and that would have had some eect in boundar laer. These eects were not studied. Resulting inetic energ of turbulence, dissipation and turbulence coecient ( T =) at the free-stream can be seen in Fig.4. It can be seen that GS model has dierent behaviour than others. This is due to fact that c 2 has dierent values. The models are similar in a free-stream where there are no gradients. For example the turbulent inetic energ and the dissipation equation reduced in following form if stream wise diusion is = C 2 (8) where x is along to the ow.

6 5 Fig. 4: Deca of turbulence quantities at free-stream along x-axis. Fig. 5: Grid of the at plate calculations. The calculation was started 6cm before the at plate. The length of the at plate was :6m. The height was 30cm and the height of the rst row ofcells was 2:5 0 ;5 m that is equal to + 0:7 atmost of the domain (at the leading edge + =2:). Grid is heavil clustered to the wall expect the rst three rows are ept constant. The ratio between neighboring cells is n+ = n =:25. The grid size is The grid can be seen in Fig. 5. Inlet conditions were uniform velocit distribution and pressure is extrapolated from the computational domain. Smmetr conditions were applied before the at plate. At the at plate, the velocities and inetic energ of turbulence were set zero. Dissipation was threated as written in Table. Zero gradients were assumed at the outlet. The pressure was given. Boundar parallel to the wall was assumed to be zero gradient, except u-velocit component and pressure was given. The second-order upwind scheme was used with Roe's ux splitting [8]. Universal boundar laer parameters are shown in Fig.6. Boundar laer parameters are in gure from left to right: friction coecient c f, boundar laer thicness at point where velocit is 99% of the free stream velocit 99, displacement thicness 99, momentum thicness and shape function H = 99=. These variables are dened as c f = w = 2 U e 2 (9) = = Z 99 0 Z 99 0 ( ; u() U e )d (20) u() ( ; u() )d (2) U e U e where U e velocit at the edge of the boundar laer. Overall performance of the friction coecient is the best for CD model, but a interesting feature is that GS model gets the transition in a right place. Even it is not strong enough this is promising because no transition model were used. Mabe CH model also get the transition but it begins too earl and it is too wea. At the begin of the at plate all the models get too low friction coecients but the recover at the station x 0:8m. The boundar laer thicness is best estimated b GSmodel. Displacement and momentum thicness are estimated ver well b CD model except at the beginning of the at plate, where GS performs ver well. This is

7 6 Fig. 6: Universal boundar laer parameters. From left to right: friction coecient, boundar laer thicness, displacement thicness, momentum thicness and shape function. due to fact that GS model predict the laminar part at the beginning of the boundar laer. Velocit proles are shown in Fig.7. In the gure -axis is the nondimensional velocit (u + = u=u ) and x-axis is nondimensional distance from solid wall ( + ). Smbols are same with Fig.6 with models and dashed-doted line is universal velocit distributions in two regions, viscous shear laer and overlap laer. Those are dened as u + = + when + < 5 (22) u + = 0:4 + +5:5 when 35 < + < 350 (23) (24) It can be seen in Fig.7 same thing as was seen in Fig.6. The GS model predicts the laminar prole beginning of the at plate as CD is turbulent allwa down. However after transition, sa station x =0:95m, the CD model performs superiorl agains others. At the last station velocit proles are similar with dierent models. q 2 3 =U e) proles are shown in Fig.8. None of the models Turbulence intensities (Tu = estimate the inetic energ levels well. Worst is the GS model, that is surprising, because it is the newest model and performed well in velocit proles. Also it is well seen in proles that GS model predict zero degree of turbulence up to + 3fromthewall. 3.3 Wall Jet Next test case was a two-dimensional plane wall jet enters along the bottom of a large pool, through a thin slot. Results were compared agains LDV measurements [9]. Flow was entering in cm slot. The dimensions of the pool was 7:45m :35m. The inlet velocit was m/s that correspond the inlet Renolds number Re = U 0 b= =0000. Kinematic viscosit ofwater at 8 o Cis =:05 0 ;6 m 2 =s. This case was also one of test cases of 5th ERCOFTAC-IAHR Worshop on Rened Modelling of Turbulent Flows (996). At the inlet, onl u velocit and u uctuation component were measured. At the inow slot, the measured velocit prole is specied. Since w 0 (rms) and actuall also v 0 (rms) are unnown at the inlet, their mutual contribution to the turbulence inetic energ is approximated to be equal to the contribution of u 0 (rms) alone. Thus, is approximated as 0:65u 0 u 0 (v 0 = 0 and w 0 0:3u 0 ). Dissipation is calculated so that the dissipation equation is

8 7 : Velicit proles for di erent down-stream stations. Solid line is CH, dash line is CD, doted line is GS, dashed-doted line is universal velicit and the dots are the experiments. Fig. 7 : Turbulent intensit proles for di erent down-stream stations. Fig. 8

9 8 Fig. 9: Maximum velocit and sperthing rate of the jet. in balance with the distributions of velocit and inetic energ of turbulence. B repeating the calculations with several inow -distributions, the ow eld was found to be insensitive to the inlet -conditions. If given turbulent inetic energ values were used the ow had a oscillation at the rst measurement stations. Ten times larger uctuating values did not have an eect after about 0 slot height, but larger uctuating values have a signicant eect in stabilit of the problem. With higher uctuation values the ow was stable also close to the jet entrance. In the nal calculation the larger values were used. At the outow slot, the upper right corner of the pool, the static pressure is xed. Mirror condition were used at the surface of the pool. An orthogonal grid is used to model the whole pool. The grid consists of 64 cells in the direction of the jet (i) and 80 cells in the jet normal direction (j). The grid is heavil clustered in j-direction to the pool bottom wall and 32 cells were placed inside the inlet slot height. There is also clustering in i-direction so that the majorit of the cells in the i-direction lie inside the area of interest. Near-wall cells in j-direction extend to a distance of about 3 0 ;5 mandini-direction extend to a distance of about : 0 ;3 m The spreading of the jet was totall wrong with the CD model. After some test modications results were not obtained with that model. Onl CH and GS model will be presented. Modifing CD model to estimate the jet correctl is leaved to be the future wor. Maximum velocit and the spreading rate of the jet can be seen in Fig.9. GS model does little bit better than CH model. Various proles for dierent stations can be seen in Appendix A. These proles indicates that both models wor prett well. GS model does little bit better again in these gures. For rst stations, both model wor quite badl. This is because of the transition that taes place close to entrance of the jet. After station 03 both models simulate ow ver well. 3.4 Curved rectangular duct Last test case was ow in a 90 o curved duct of rectangular cross-section with aspect ratio 6. Experiments were done b Kim et al. [0]. How-wire velocit measurements have been carried out using a miniature X-wire probe for the turbulence quantities. Mean velocit measurements have been carried out using a ve-hole pressure probe of a diameter of 3mm. This case was also one of test cases of 5th ERCOFTAC-IAHR Worshop on Rened Modelling of Turbulent Flows (996). Duct was with two straight and one curved sections. In Fig.0 can be seen geometr of the duct. Also measurement stations (U, U2, 5, 45, 75, D and D2) are specied in the gure. Inlet conditions were given in station U. Onl dissipation must be approximated. It was approximated so that the dissipation equation is in balance with the distributions of velocit and inetic energ of turbulence. The inlet velocit was 6m/s that correspond the inlet

10 H H 4.5H H Outlet D2 D R i =3H U2 6H H=2 cm U z x Inlet Fig. 0: Geometr of the curved duct H Renolds number Re = U 0 b= = Kinematic viscosit of air is =:4550 ;5 m 2 =s. The duct is interesting test case because there was turbulence driving vortices because of nonisotrop of Renolds stresses and also pressure driving vortices in the curved section. An orthogonal grid is used to model the whole duct. Mesh consists of cells with a total dimension of From U to the bending there is 48 cells, at the bending there is 48 cells and after bending to the outlet there is 32 cells. The computational domain is divided into 4 blocs in order to utilize parallel computation. The calculations begin in station U and ends in x =28:5H. Near-wall cells in and z direction extend to a distance of about 4 0 ;5 m. Pressure distribution along the duct smmetr plane is shown in Fig.. Both models show good agreement with experiments. Friction coecient from inner wall to outer wall is presented in Fig.2. Friction coecients are not so well estimated as pressure coecients. Expeciall close to the corner models have problems to estimate right friction coecients. Fig. : Pressure distribution along the duct smmetr plane.

11 0 Fig. 2: Friction coecient from inner wall to outer wall. The bottom wall is badl estimated all most in ever stations. Velocit fringes for GS model can be seen in Fig.3. It should be noted that colors are not exactl same for experiments and calculations all though the number of colors are same. Fringes of inetic energ of turbulence for GS model can be seen in Fig.4. Velocit andinetic energ of turbulence distributions were ver close to each other with GS and CH models, and thus no gures of CH model is printed. Kinetic energ of turbulence is not so well estimated at the corners. Secondar motion and turbulent viscosit T = is presented in Fig.5. Onl small dierences can be seen between models. Although experiments are not shown here the GS model mae bit better estimate for secondar motion. 4 Discussion There is no big dierencies between models in these calculations. GS model estimates ow eld best of these three models. Because its nonisotroph, it is promising model for rotating ows. CD model has problems with spreading of the jet but it estimates close wall behaviour ver well. This might beinterest of the future research. For simple cases, channel ow, boundar laer and wall jet, all models performed well, but for more complicated case, in the curved duct, models have big diculties. All thought none of the model were not made to simulate transition, the CH and GS models predict transition close to the right place in the boundar laer problem. Transition were not strong enough. References [] Wilcox, D., Turbulence Modeling for CFD. La Canada: DCW Industries, Inc., 993. ISBN 0{963605{0{0. [2] Gatsi, T. and, C., \On explicit algebraic stress models for complex turbulent ows," Journal of Fluid Mechanics, Vol. 254, 993, pp. 59{78. [3], K.-Y., \Predictions of Channel and Boundar-Laer Flows with a Low-Renods- Number Turbulence Model," AIAA Journal, Vol. 20, Jan 982, pp. 33{38.

12 U U D D : Velocit fringes for GS model. Upper is GS model and lower experiments. Fig. 3 U U D D Fig. 4: Fringes of inetic energ of turbulence for GS model. Upper is GS model and lower experiments.

13 2 U U D D2 Fig. 5: Secondar motion in CH (upper) and GS (lower) model. Colors are turbulent viscosit T =, blue color correspond zero value and red color 350. [4] Yoon, B. and M.K., C., \Computation of Compression Ramp Flow with a Cross- Diusion Modied ; Model," AIAA Journal, Vol. 33, No. 8, 995, pp. 58{52. [5] Abid, R., Rumse, C., and Gatsi, T., \Prediction of Nonequilibrium Turbulent Flows with Explicit Algebraic Stress Models," AIAA, Vol. 33, No., 995, pp. 2026{203. [6] Kim, J., Moin, P., and Moser, R., \Turbulence Statistics in Full Developed Channel Flow at Low Renolds Number," Journal of Fluid Mechanics, Vol. 77, 987, pp. 33{ 66. [7] Mansour, N., Kim, J., and Moin, P., \Renolds-stress and Dissipation-rate Budgets in a Turbulent Channel Flow," Journal of Fluid Mechanics, Vol. 94, 988, pp. 5{44. [8] Roe, P., \Approximate Riemann Solvers, Parameter Vectors, and Dierence Schemes," Journal of Computational Phsics, Vol. 43, 98, pp. 357{372. [9] Karlsson, R., Erisson, J., and Persson, J., \LDV Measurements in a Plane Wall Jet in a Large Enclosure," in 6th International Smposium on Applications of Laser Techniques to Fluid Mechanics, (Lisbon, Portugal), pp..5.{.5.6, Jul 992. [0] Kim, W. and Patel, V., \Origin and Deca of Longitudinal Vortices in Developing Flow in a Curved Rectangular Duct," Journal of Fluid Engineerin, Vol. 6, March 994, pp. 45{52.

14 3 A Results for wall jet Profile of mean components of velocit and fluctuating values at Station 0 (x/b= 0) Profile of mean components of velocit and fluctuating values at Station 02 (x/b= 5) and in mm and in mm. Profile of mean components of velocit and fluctuating values at Station 03 (x/b= 0) Profile of mean components of velocit and fluctuating values at Station 04 (x/b= 20) and in mm. 0 and in mm

15 4 Profile of mean components of velocit and fluctuating values at Station 05 (x/b= 40) Profile of mean components of velocit and fluctuating values at Station 06 (x/b= 70) and in mm. 00 and in mm Profile of mean components of velocit and fluctuating values at Station 07 (x/b= 00) Profile of mean components of velocit and fluctuating values at Station 08 (x/b= 50) and in mm. 00 and in mm

16 5 0 Profile of mean components of velocit and fluctuating values at Station 09 (x/b= 200) Profile of u components of velocit and turbulence quantitives at Station 0 (x/b= 0) Turbulent viscosit Epsilon w rms and in mm. and in mm Profile of u components of velocit and turbulence quantitives at Station 02 (x/b= 5) Turbulent viscosit Profile of u components of velocit and turbulence quantitives at Station 03 (x/b= 0) Turbulent viscosit Epsilon Epsilon w rms w rms and in mm. and in mm

17 6 Profile of mean components of velocit and fluctuating values at Station 04 (x/b= 20) Profile of u components of velocit and turbulence quantitives at Station 05 (x/b= 40) Turbulent viscosit Epsilon w rms and in mm. and in mm Profile of u components of velocit and turbulence quantitives at Station 06 (x/b= 70) Turbulent viscosit Profile of u components of velocit and turbulence quantitives at Station 07 (x/b= 00) Turbulent viscosit Epsilon Epsilon w rms w rms and in mm. 2 and in mm

18 7 0 Profile of u components of velocit and turbulence quantitives at Station 08 (x/b= 50) Turbulent viscosit Profile of u components of velocit and turbulence quantitives at Station 09 (x/b= 200) Turbulent viscosit Epsilon Epsilon w rms w rms and in mm. and in mm