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Procedia Environmental Sciences 10 (2011) 753 – 758 Procedia Environmental Sciences 00 (2011) 000 000
2011 3rd International Conference on Environmental www.elsevier.com/locate/procedia Science and Information Application Technology (ESIAT 2011)
Numerical Simulation of 3D Density Flow by an Improved EASM Model L. H. Xinga, G.B. Huanga, Min Yanb a* a
Changjiang River Scientific Research Institute, No. 23 Huangpu street ,Wuhan,430010, CHINA b Hohai University, No. 1 Xikang street ,Nanjing,210098, CHINA
Abstract Reynolds stress turbulence models are still an effective method to solve thermal density flow in reservoir. Presently, these turbulence models have extensive research and play an important role in the project. However, the pursuit of a more general, economic and accurate Reynolds stress model is still necessary for buoyant turbulent flow. In this paper, an improved explicit algebraic Reynolds stress model was established accounting for buoyancy. The tensor representation of the Reynolds stresses are divided into two parts, the former is composed of EASM derived by Wallin & Johansson (2000) and latter is the formulations of buoyant stress. In the derivation of explicit algebraic active scalar flux model, WWJ (Wikström, Wallin and Johansson, 2000) model only for explicit tensor representation of passive scalar flux is improved by affiliating external buoyancy effects terms. The current EASM is discretized on three-dimensional unstructured grids and compared with experimental data of 3D thermal density flow. The calculations show that the model yields better results than LAHM and LARM. In addition, EASM model also has good stability and relative economic calculation CPU time-consuming. The current model would be promising in hydraulic engineering and environmental engineering. Keywords: Explicit algebraic Reynolds stress model; Density flow; Unstructured grids; 3D; Buoyancy;Explicit algebraic active scalar flux model
1. Introduction Water temperature is one of the most important factors affecting the water physical and chemical properties, aquatic life and crops irrigation. Thus, accurate prediction of water temperature distribution is so significant in stratified reservoir, which can benefit the optimal operation of reservoir, improvement of water environment and development of industrial and agricultural production. So far, the time-averaged Reynolds stress turbulence models are still the most effective methods to solve such problems. As for a larger depth reservoir, it may present strong thermal stratification in a particular season, the diffusive action in vertical may much stronger than that in homogeneous water. So a more accurate turbulence model is directly related to determine the appropriate vertical diffusion coefficient.
* Corresponding author. Tel.: +0-086-13545109714; fax:+0-086-82633828. E-mail address:
[email protected].
1878-0296 © 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and/or peer-review under responsibility of Conference ESIAT2011 Organization Committee. doi:10.1016/j.proenv.2011.09.122
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Currently, the turbulence models for prediction of stratified water temperature include empirical formula[1-5], vertical mixing scheme [6], isopycnal diffusion scheme [7], Eddy viscosity model [8] and the Reynolds stress model [9-10] etc. Among them, the algebraic Reynolds stress model (ARSM) has many particularly advantages, including integrating Reynolds stress model (RSM) of versatility and eddy viscosity models (EVM) of economy to a certain extent, taking into account the volume force (such as buoyancy, streamline bending, rotation, etc.). In this point of view, it is one of the most practical models for simulation of more complex turbulence flow. However, in the traditional ARSM model, an implicit relationship is existed between Reynolds stress and other tensors, including deformation rate tensor, strain rate tensor and buoyancy tensor. In a general three-dimensional problem, the convergence of algebraic equations in iteration is poor. Especially when there is a big strain flow, singularity may occur and can cause the calculation failure. In order to overcome the difficulties, explicit algebraic stress model (EASM) flourished [11-18] in recent years, and became an important branch of the development of turbulence models. From the differential form of the turbulent model equation, the Reynolds stress was derived to a strictly explicit algebraic expression and added by explicit nonlinear Reynolds stress term in a mature two-equation turbulence model. It is a nonlinear algebraic stress model with the aid of eddy viscosity assumption. In a sense, EASM model has closer numerical efficiency while compared to two-equation turbulence model, moreover, it can overcome some of the shortcomings of linear eddy viscosity model, reflect the anisotropy effect of Reynolds stress, avoid the numerical singularity effectively, improve the model stability and convergence. In a word, EASM would be so promising in practical engineering. 2. Improved EASM Model Accounting for Buoyancy 2.1 Explicit algebraic Reynolds stress model Presently, the Reynolds stress will be broken into two parts under the assumption of the principle of linear superposition. The first part is strain caused by stress and the second is strain caused by the buoyancy impaction, which can be gained from the relationship of the implicit algebraic stress model by _____ stress is_____ Rodi[2]. The final form of Reynolds _____ ui u j ui u j s ui u j b (1) Where, the non-buoyancy part contains isotropic part and the remaining part of the anisotropy, its expression is referenced to Wallin_____ & Johansson model[20]. (2) ui u j s k 2 / 3G ij 2CPeff Sij aij( an )
Effective coefficient CPeff can be expressed as CPeff written as
E IIZ E 1
6
/ 2;a
( an ) ij
is anisotropy part and can be
a( an) = E 2 s 2 - II S / 3, E 3 Ȧ 2 - IIZ / 3, E 4 sȦ - Ȧs E 5 s 2 Ȧ Ȧs 2 E 6 Ȧ 2 s sȦ 2 - 2/ 3IV , IIZ s E 7 Ȧ 2 s 2 - s 2 Ȧ 2 - 2 / 3V , E 8 sȦs 2 - s 2 Ȧs E 9 ȦsȦ 2 - Ȧ 2 sȦ E 10 Ȧs 2 Ȧ 2 - Ȧ 2 s 2 Ȧ The coefficients in detail can be seen from Wallin & Johansson [20]. The buoyancy part is _____ k § 2 · ui u j b (1- c3 ) G ¨ Gij / G - įij ¸ c1 ( P G) / ȡİ -1 İ © 3 ¹ Where, c3
0.55 , c1
0.22 , Gij C ( gi u j R g j ui R ) , G
_____
_____
UE gi uiT , P ui u j sU i / sxJ .
(3)
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2.2 Explicit algebraic active scalar flux model A more accurate way to model the passive scalar flux is use a model that does not assume the scalar flux to be aligned with the mean scalar gradient k w4 uiT 1 cT 4 Aij1 u j uk wxk H (4) The Eq. (4) is one such model, a so called explicit algebraic scalar flux model(EASFM) by Wikström, Wallin and Johansson[21]. It utilizes an algebraic relation for the passive scalar flux consisting of mean flow quantities by an equilibrium condition in the transport equations for the normalized passive scalar flux. In order to model the active scalar flux accurately, the buoyancy effects might be included in the algebraic relation. Applying the analogous approach as Wikström did, the explicit algebraic active scalar flux can be derived as k w4 k uiT 1 cT4 Aij1 u j uk Aij1 (cT6 1)E g j TT wxk H H (5) ____
Something should be noted that the scalar pulse-related item TT appeared above still needs to be modeled for fully explicit. ____One of the simple way is to use the assumption of local balance for the scalar transport equation, thus TT can be expressed as TT
Where, cT
k _____ w4 u jT cT H wx j
(6)
0.13 ( Launder 1975ǃ1976). Then, u i T is rewritten as uiT
k w4 cT6 1 1 k 2 w4 1 cT4 Aij1 u j uk Aij 2 E g j uk T wxk wxk H cT H
(7)
[20]
Some of the parameters can be found in EASFM model by Wikström et al
.
3. Numerical method In the present study, a method incorporating 3D unstructured grids along with a modified QUICK scheme and a pressure-correction equation is developed to solve incompressible flows. This approach can be fitted to arbitrary geometric shape easily. Generalized Minimum Residual (GMRES) method with the Incomplete LU (ILUT) precondition is used to accelerate the convergence speed for linear equations. 4. Numerical Simulation of Density Flow The validation of the modified EASM model is carried out on density flow by Johnson[21]. The experimental flume is 24.38m long with a 0.91mh0.91m cross section at the downstream end. The cross section at the upstream end is 0.30mh0.30m and linearly grows in width over the first 6.10m to a cross section 0.30m deep and 0.91m wide. The bottom of 0.61m linearly is horizontal for the first 6.10m and then drops a total of 0.61 linearly over the final 18.29m of the flume. The water in the flume was at rest and homogeneous at the initiation of the test, with the temperature being 21.44ć. Cold water was input at a temperature of 16.67ć. A baffle restricted the cold water to enter over about the bottom 0.15m of the cross section. The inflow rate was 0.00063m3/s with the outflow rate at the downstream end being the same. The outflow was removed from a port located 0.15m above the bottom of the flume. The computational region can be divided in arbitrary hexahedral mesh to fit the complexity of the border. Total number of nodes is 24,106 and total number of elements is 29,408. The under-relaxation
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Distance above bottom/m 0.50
Experiments LAHM LARM EASM
0.25
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
Velocity/(m/s)
Outflow Temperature/ď§Đ
factors for each discrete equation, namely the momentum equation, the turbulent kinetic energy equation, the turbulent dissipation rate equation, the scalar transport equation and the pressure correction equation, are 0.6, 0.4,0.4, 1.0 and 0.1. The flow velocities are zero at initial time, k and¦are set as 2.94h10-6m2/s2 and 1.18h10-7 m2/s3 preliminarily in inlet areas, all variables in output section are assumed as zero gradients. For numerical stability, the time step is fixed as 2s at first and gradually increased to 5s in the left time, total computation time is within 50min. In order to capture the accurate flow pattern, the maximum inner iteration is fixed as 20 at each time step in framework of SIMPLE algorithm. Illustration of velocity distribution along midperpendicular is plotted in Fig. 1 in cross section 11.43m at 11min. Experimental results show that the cold-water dives quickly at the bottom layer, correspondingly reverse vortex appears in the upper layer. The phenomenon is better predicted by EASM model than those of LARM and LAHM, wherever in the diving thickness or vertical velocity variation. 21.5 21.0
Experiments LAHM LARM EASM
20.5 20.0 19.5 19.0
Fig. 1. Illustration of velocity distribution at midperpendicular in cross section 11.43m at 11min
0
10
20
30
Time after initiation/min
40
50
Fig. 2. Illustration of temperature variation in outlet section
0
0
-0.2
-0.2
-0.4
-0.4
Z/m
Z/m
Illustration of temperature variation in outlet section is plotted in Fig. 2. Experimental data show that the outflow temperature degrades at 14.6min and becomes 19.4ć at 25.6min. The first cooling time is postponed to 16.6 min by EASM, 18.6 min by LAHM and 20.5 min by LARM. As compared to other two models, the EASM has much better precision at first stage. However, the outflow temperature calculated by EASM deviates from the experimental data after 25.6 min, which may be caused by sparse grids not enough to capture the density flow front.
-0.6
-0.6
-0.8
t=3min 20
15
X/m
10
5
t=9min 25
0
0
0
-0.2
-0.2
-0.4
-0.4
Z/m
Z/m
25
-0.8
20
15
10
X/m
5
0
-0.6
-0.6
-0.8
25
20
15
X/m
10
t=21min
-0.8
t=15min 5
0
25
20
15
Fig. 3. Illustration of full flow fields in longitudinal profile
X/m
10
5
0
L.H. Xing et al. / Procedia Environmental Sciences 10 (2011) 753 – 758
Illustration of flow fields at t=3 min, 9 min, 15 min and 21 min in longitudinal profile is plotted in Fig. 3. It can be seen that the numerical results can well predict density flow development and its front changes. At first, the turbulent mixing occurs between the bottom cooling water and upper hot water, and then it forces the density flow front upwelling, at last large counterclockwise circulation flows are formed. Illustration of iso-surface temperature distribution of density flow is plotted in Fig. 4. Something should be noted that the density flow front is depicted by iso-surface temperature 21.42ć. The numerical results show that no obvious vertical diffusion and mixing after the cold water into the reservoir, the density flow front dives along the reservoir bottom and the turbulent mixing and exchange along vertical are apparently restricted. In the whole process, the reservoir temperature and flow field are strongly coupled, which further improves the efficiency of EASM model. The present simulation was carried out on CPU Q6600 with frequency 2.40GHz, total CPU times consumed are 3.3 hours. As for comparison, 5.56 hours are needed by RSM to calculate the whole process on equivalent number of grids [9]. Thus, EASM model can significantly save computer time-consuming, strengthen the numerical stability and convergence.
t=10min
t=20min
t=50min
Fig. 4. Illustration of iso-surface temperature distribution of density flow(18ć,21ćand21.42ć)
5. Conclusions A modified EASM model is proposed including explicit algebraic Reynolds stress model accounting buoyancy and explicit algebraic active scalar flux model inherited from Wikström, Wallin and Johansson model, and then it was successfully applied on 3D unstructured grids and can fit arbitrary geometric shape. In numerical simulation of thermal density flow, the calculation results by EASM can fit experimental data better than those of LAHM and LARM. The present model can figure out the cooling water diving along the bottom reservoir and depict counterclockwise circulation flow formation, also accurately reflect the features of temperature distribution of density flow. In addition, EASM model can significantly save computer time-consuming as compared to RSM. Furthermore, it can strengthen the numerical stability and can avoid the numerical singularity. It is so promising to apply the turbulent model to practical engineer usage. Acknowledgements This research was sponsored by National Public Research Institutes for Basic R & D Operating Expenses Special Project (No.YWF0901, No.YWF0905, No.CKSF2011013/SL and CKSF2010011); National Basic Research Program of China (No. 007CB714106, Subproject of No.2007CB714100). Ministry of Water Resources Special Funds for Scientific Research on Public Causes ˄ Grant No. 201001033˅ References [1]Pacanowski R.C, Philander G..Parameterization of vertical models of the tropical ocean, J. Phys. Oceanogr.,1981,11:14421451 [2]Rodi W. Turbulence models and their application in hydraulics. 3rd edition. Rotterdam: IAHR, A. A. Balkema,1993 [3]Engelund F. Effect of lateral wind on uniform channel flow. Progress. Report 45 Inst. Of Hydrodynamic and Hydraulic Engr Tech. Univ. of Denmark.1978. [4]French R H. Open-channel Hydraulics.New York: McGraw-Hillˈ1985
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