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Numerical simulation of flow over topographic features by revised k–ε models
Journal of Wind Engineering and Industrial Aerodynamics 91 (2003) 231–245
Numerical simulation of flow over topographic features by revised k2e models Yu Fat Luna,*, Akashi Mochidaa, Shuzo Murakamib, Hiroshi Yoshinoa, Taichi Shirasawaa b
a Tohoku University, Aoba 06, Sendai 980-8579, Japan Keio University, 2-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
Abstract The accurate prediction of the wind energy distribution over terrains is important for making an appropriate selection of a suitable site for installing wind power plant. In this study, two-dimensional numerical simulations of flow over two common types of topographic features, i.e. a cliff and a hill, are presented. Three types of turbulence models, namely standard k2e and its revised linear form (Durbin model) as well as a revised nonlinear form (Shih model) are employed in this work. The performance of these models in predicting flow over these features is investigated. The accuracy of the prediction by using a z0 type wall function to reproduce the effect of surface roughness on flowfield from these turbulence models is also examined. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Flow over terrains; Cliff; Hill; Revised k2e; Durbin model; Shih model
1. Introduction Large amount of energy consumption caused by human activities have significantly changed our present global environment. For this reason, environmental concerns on the causes of global warming have led to many countries to introduce renewable energy technologies like wind power. Japan has a long coast and it is widely exposed to the sea, Sea of Japan on the west and Pacific Ocean on the east, undulating lands occupy a vast majority of spaces *Corresponding author. E-mail address: [email protected] (Y.F. Lun). 0167-6105/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 0 2 ) 0 0 3 4 8 - 3
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Nomenclature xi ui /S uH ut u* /uþ t S z0 xþ 2
spatial component (i ¼ 1 streamwise, i ¼ 2 vertical) velocity component in xi direction time averaged quantity, u0i ¼ ui /ui S u1 value at inflow of computational domain at height H tangential component of velocity vector friction velocity /ut S=/u * S roughness parameter wall co-ordinate
over the territory. Wind flows over terrains can gain an important advantage for installing wind power plants if the location of the site is correctly chosen. An appropriate selection of a suitable land for wind power plants can provide significant output of energy. However, it is not an easy task to choose a site for a wind farm because many factors have to be taken into account. Important factors such as change of wind speed near complex geometry of hilly terrain and the surface conditions of the lands need to be considered carefully in order to have an accurate evaluation of wind energy distribution. It is well known that the standard k2e turbulence model suffers from a serious drawback in overestimating turbulence kinetic energy, k; in the flow with impinging. In order to overcome this drawback, several revised k2e models have been proposed [1–3]. Among them, the results of a revised model proposed by Durbin [3] are clearly shown well in comparison to other revised k2e models when applying to flow over bluff bodies [4]. Recently, Ishihara et al. [5] applied a nonlinear revised k2e model proposed by Shih et al. [6,7] to flow over rough hill and the results showed good agreement with the experiment. The present work employed two modified k2e models, namely Durbin and Shih models, to simulate flow over topographic features such as a cliff and a hill. Complex terrain contains small objects such as trees, rocks that make the surface of the ground being less rough or more rough. Wind flow over terrain is strongly affected by these ground level objects. Numerical prediction of the effect of surface roughness on flowfield is one of the important areas within the wind engineering studies. In order to have an accurate prediction of flow around topographic features under the influence of such ground level objects, the boundary conditions for ground surface must be correctly selected. The aims of the present work are to investigate the performance of revised k2e models by Durbin [3] and Shih et al. [7] in predicting flow over topographic features, and to investigate the accuracy of the prediction using a z0 type wall function to reproduce the effect of surface roughness on flowfield over terrain.
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2. Outline of computations 2.1. Flowfields analyzed in this study Two different configurations of obstacle, steep curvature and small curvature, are considered. A two-dimensional step-shaped cliff is chosen as a model of steep curvature obstacle. The model geometry is based on the experimental work of Kato et al. [8], which is shown in Fig. 1a. The cliff height, H; is 75 mm and its length is 20H; the front section of the cliff is situated at 7H from the inlet and the end section of the cliff is placed at 7H from outlet. The vertical dimension of the flow domain is 24H: The computational flow domain for the case of cliff is discretized into 95ðX1 Þ 61ðX2 Þ grids and the minimum grid width is approximately 0.07H from the solid surface. Mesh density is placed at the front upper corner of the cliff. Fig. 2a shows an enlarge view of the computational grid around the front section of the model cliff. The second model is a two-dimensional hump with small curvature, e.g. a low hill. The model geometry is based on the wind tunnel investigation of Ishihara and Hibi [9] (cf. Fig. 1b). The height, H; and length, L; of the model hill are 40 and 200 mm (5H) respectively. The flow domain has dimension of 60H in the horizontal direction and 22:5H in vertical. The computational flow domain in this case is 110ðX1 Þ 50ðX2 Þ nodes. Grid density is enhanced around the hill and along the bottom surface of the flow domain. The minimum grid width in this model is approximately 0:04H: Fig. 2b shows an enlarge view of the computational mesh around the model hill.
u1 = (X2)0.143 Wind Cliff
1.8m
X2 H=75mm
X1
(a)
L=200mm
x2 = H
cos2 (x1
/ 2L) x2 H=40mm
(b)
x1
Fig. 1. Configuration of the two models used: (a) model geometry of cliff [8], (b) model geometry of hill [9].
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Fig. 2. Computational mesh for the two models, (a) cliff case: enlarge view around the front corner, (b) hill case: enlarge view around the hill.
Table 1 Test cases under consideration Case
Obstacle configuration
Turbulence model
Surface condition
1-1 1-2 1-3
Cliff
Standard k2e Durbin Shih
Smooth Smooth Smooth
2-1-1 2-1-2 2-1-3
Hill
Standard k2e Durbin Shih
Smooth Smooth Smooth
2-2-1 2-2-2 2-2-3
Hill
Standard k2e Durbin Shih
Rough Rough Rough
2.2. Cases of computations Table 1 lists all the cases conducted within this study. Two different surface conditions, smooth and rough, were applied to the hill case and only a smooth surface was considered in the cliff case. Standard k2e model (hereafter denoted by standard k2e), revised forms by Durbin and Shih (hereafter denoted by Durbin model and Shih model) were adopted for both cases of flow over a cliff and a hill. Durbin model employs the eddy viscosity concept based on the linear relations between the Reynolds stress and the strain rate, and its form of nt is modified to
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Table 2 Boundary conditions for both cases Location
Conditions
Inlet
(1) Cliff case [8] /u1 Spxn2 ; n¼ 0:143; k ¼ 0:016/uH S2 ; e ¼ Pk ; Cases 1-1B1-3 /uH S : /u1 S is the value at a height of H at inflow boundary H: cliff height (2) Hill case [9] /ut S 1 x2 ¼ ln ; /u * S k z0
/u2 ðx2 ÞS ¼ 0;
k¼
/u * S2 1=2 cm
;
e¼
/u * S3 kz
Smooth surface: /u * S ¼ 0:21 m=s; z0 ¼ 0:01 mm; Cases 2-1-122-1-3 Rough surface: /u * S ¼ 0:32 m=s; z0 ¼ 0:3 mm; Cases 2-2-122-2-3 Outlet
@=qx1 f/u1 S; /u2 S; k; eg ¼ 0
Top domain
/u2 S ¼ 0;
Solid surface
(1) Cliff case:
@=qx2 f/u1 S; k; eg ¼ 0
! /ðut Þp SðCm1=2 kÞ1=2 1 E x2 ðCm1=2 kÞ1=2 ¼ ln ; k tw =r n
E ¼ 9;
1 x2 ln ; k z0
qk ¼0 qx2
(2) Hill case: /uþ t S¼
e¼
3=2 c3=4 m k ; kx2
e¼
cm3=4 k3=2 ; kx2
qk ¼0 qx2
Smooth surface: z0 ¼ 0:01 mm Rough surface: z0 ¼ 0:3 mm k ¼ 0:4; Cm ¼ 0:09
impose the ‘realizability’ constraint as described in Section 2.3. On the other hand, Shih model uses a quadratic nonlinear eddy viscosity model. The boundary conditions used in this work are shown in Table 2. For the cliff case, a power law velocity profile, which corresponds to the inflow condition of the wind tunnel experiment [8], was used at the inlet and a generalized log law profile for the smooth surface was applied at the solid surface. For the case of hill, a log law velocity profile was used at the inlet and a z0 type wall function was employed at the solid surface in order to incorporate the effect of surface roughness. The values of surface roughness parameters were chosen from Ishihara and Hibi [9]. The QUICK scheme was employed for the convection terms in all the equations. 2.3. Governing equations As mentioned above, in the standard k e model and Durbin model, the unknown Reynolds stresses are obtained from the conventional Boussinesq linear
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stress–strain relationship viz: @/ui S @/uj S 2 0 0 þ kdij : /ui uj S ¼ nt qxj 3 qxi
ð1Þ
The eddy viscosity, nt is related to the turbulence velocity scale, /v02 S and its time scale, t as nt ¼ Cm /v02 St:
ð2Þ
In k2e models, /v S assumes to be k and Eq. (2) becomes 02
nt ¼ Cm kt:
ð3Þ
In the standard k2e model, t ¼ k=e: The traditional k2e model has been found to be inadequate in dealing with impinging flows, and is not appropriate for involving object such as bluff body. A modification, which addresses this stagnation flow anomaly, is due to Durbin who suggested imposing the ‘realizability’ constraint 2kX/u0a u0a SX0 via a bound on the time scale, t; where summation is not taken in the /u0a u0a S [3]. The proposed bound on the time scale is t ¼ min½TS ; TD ;
ð4Þ
where k TS ¼ ; e
ð5aÞ
pffiffiffi 2 pffiffiffiffiffiffiffiffi; TD ¼ 3Cm jS j2 2
ð5bÞ
where jSj2 ¼ Sij Sji and Sij is the rate of strain tensor given by @/ui S @/uj S þ : Sij ¼ qxj qxi Another turbulence model used in this study is a nonlinear model proposed by Shih and co-workers. Shih model adopts a nonlinear relationship between Reynolds stresses and the rate of strain as follows: /u0i u0j S 2 nt @/uk S nt nt 1 ¼ dij Sij þ C1 Sik Skj dij Skl Skl 3 k qxk 3 k k e
nt nt 1 þ C2 Oik Skj þ Ojk Ski þ C3 Oik Ojk dij Okl Okl ; ð6Þ 3 e e where Sij ¼
@/ui S @/uj S þ qxj qxi
and
Oij ¼
@/ui S @/uj S qxj qxi
are the elements of the mean strain tensor and vorticity tensor respectively. The details of the model and expressions of the coefficients are given by Shih et al. [7].
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3. Results and discussion 3.1. Computation of flow around cliff Fig. 3 shows the ratio of two time scales ðTD =TS Þ: The definitions of TS and TD are given in Eqs. (5a) and (5b) respectively. The shaded area with the value smaller than 1 in this figure indicates that the Durbin’s time scale, TD is adopted (cf. Eq. (4)). This figure means that the Durbin’s time scale is used only around the frontal corner of the cliff and the time scale in the standard k2e model, TS is applied in other regions. Distributions of k and mean velocity vector plots over the cliff, predicted by standard k2e; Durbin and Shih models, are shown in Figs. 4(a) and (b). It is now widely known that one of the shortcomings of the standard k2e model is an overestimation of the turbulent kinetic energy around regions of impingements. It is evidently shown at the windward corner in Case 1-1. An improvement of k at the same area is obtained both by Durbin and Shih models in Case 1-2 and Case 1-3 respectively. In comparison between the standard k2e and Durbin models, k is predicted to be smaller by Durbin model comparing to the standard k2e in the area where TD =TS p1:0; shown in Fig. 3. On the top of the cliff flow separation and reattachment are all found in these three models, as shown in Fig. 4b. Figs. 5a and b show vertical profiles of the normalized mean velocity and normalized k in the separation region, respectively, by these three models comparing with experimental data. Prediction of mean velocity obtained by Durbin model showed generally very good agreement with the experiment at the cliff top surface. 2.5
2 2
1
TD /TS < 1.0
x2 /H
1.5
(TD is applied)
3
4
23 4 0
1
2
3
54
5 5
1
2 0.5 1 0 -1.5
31 -1
-0.5
0
0 0.5
1
x1/H Fig. 3. Ratio of two time scales TD =TS (cf. Eqs. (5a) and (5b)).
1.5
x2/H
x2/H
(a)
x2/H
-1
-1
0.02
-1
0.02
0.16
0.03 0.04
-0.5 0 Case1-3 (Shih)
0.01
0.02
-0.5 0 Case1-2 (Durbin)
0
0
0.14
0.04
-0.5 0 Case1-1 (k-ε)
0.04
0.06 0.1
0.5
0.5
0.08
0.5
0
1
0.02
0.1
0.06 0.04
1
1 x1/H
0.04
x1/H
0.08 0.06
0
0.02
0.12
x1/H
0.08 0.1 0.12
0.06
0.08 0.1 0.06 0.04
0.12
0.08
1.5
1.5
1.5
(b)
0 -1.5
0.5
1
1.5
2
2.5
0 -1.5
0.5
1
1.5
2
2.5
0 -1.5
0.5
1
1.5
2
2.5
-1
-1
-1
-0.5 0 Case1-3 (Shih)
-0.5 0 Case1-2 (Durbin)
-0.5 0 Case1-1 (k-ε)
0.5
0.5
0.5
1
x1/H
x1/H
x1/H
1
1
Fig. 4. Predictions by the three models for the cliff case: (a) distribution of k=huH i2 ; (b) mean velocity vector.
0 -1.5
0.5
1
1.5
2
2.5
0 -1.5
0.5
1
1.5
2
2.5
0 -1.5
0.5
1
1.5
2
2.5
x2/H x2/H x2/H
1.5
1.5
1.5
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Plotting position X1 H o Exp.
Durbin
----- k-ε + Shih
2
2
1.75
1.75
Vertical height X2/h
Vertical height X2/h
X1=0.5H
1.5
1.25
(a)
1 -0.5
0
0.5
1
1.5
(b)
1.5
1.25
1
0
0.05
0.1
0.15
0.2
Fig. 5. Comparison of vertical profiles of hu1 i and k just after the front corner at 0:5H from the front corner: (a) normalized mean velocity hu1 i=huH i ; (b) normalized k distribution k=huH i2 :
The distributions of k predicted by Durbin model also showed well with the experiment, below the height of 1:25H; close to the surface of the cliff but the model of Shih did not perform well at this region. However, Shih model is in better agreement with the experiment regarding the height where the maximum peak of k exists. The performance by standard k2e started showing poorly, to upper region of the flow domain, from the height where the peak of experiment located. It is pointed out that the standard k2e greatly overestimates k in the upwind corner of the bluff body. On the other hand, the values of k in the modified k2e evaluated to be smaller than that of experiment and this underestimation of k caused the re-circulation region on the roof larger than the standard k2e: 3.2. Computation of flow over hills 3.2.1. Distributions of k and mean velocity vectors Fig. 6 shows the computation results of normalized k distributions and mean velocity vector for the three models of the smooth hill case. It can be seen that the predictions of k at the hillcrest area by Durbin and Shih models are found smaller than the standard k2e; and these were also observed in the cliff case showing previously. Separation point and re-circulation zone are both obtained by the three models, along the hill slope and behind the hill respectively. Position of the separation points for the turbulence models comparing with the experiment is shown
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Fig. 6. Predictions by the three models for the smooth hill case: (a) distribution of k=huH i2 ; (b) mean velocity vector.
Table 3 Normalized separation point and re-circulation length of the hill cases in comparison with experiment Case
Experiment [9] Stadark k2e Durbin k2e Shih k2e
Xs =H (Separation)
Xr =H (Re-circulation)
Smooth
Rough
Rough
Smooth
1.1 1.3 1.0 1.2
0.6 0.8 0.5 0.5
6.4 5.4 13.7 6.6
4.6 3.6 8.5 4.5
H
Xs 0 Xr
in Table 3. In this figure, the re-circulation generated by standard k2e is smaller in size than the models of Durbin and Shih. When comparing the models of Durbin and Shih, Shih model predicts the reattachment length more accurately. The re-circulation predicted by Durbin model has a larger area than Shih model behind the hill but Shih model corresponds very well to the experiment with the size of 6:6H:
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Fig. 7. Predictions by the three models for the rough hill case: (a) distribution of k=huH i2 ; (b) mean velocity vector.
Fig. 7 shows the computation results of normalized k distributions and mean velocity vector for the case of rough hill by the three models. Similar to the smooth case, k is overestimated around the hillcrest by the standard k2e; and this fault is corrected by the two revised k2e models. As will be discussed later in detail in Section 3.2.3, it is clearly observed that change in position of the separation point leads to the change in reattachment length (smooth and rough cases). 3.2.2. Comparison with the experiment Fig. 8 shows the vertical profiles of normalized k at various positions over the rough surface hill. It can be seen that the predictions of k by Durbin model showed very good agreement with the experiment in the front section of the hill, at X1 =H ¼ 1:25 and X1 =H ¼ 0; whilst standard k2e and Shih models overestimated k at the same area. Along the downstream behind the hill, Durbin model underestimated the k but the results of Shih model agreed well with the experiment in this region. Predicted profiles of streamwise and vertical velocity by standard k2e; Durbin and Shih models, at different positions across the rough hill are shown in Fig. 9. Shih model showed to be generally close with the experiment, but the result of Durbin model deviated from experiment in the region behind the hill. A large re-circulation
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Vertical Height, X2/H
4
3
2
1
H 0
-3.75
-2.5
0
-1.25
0
1.25
2.5
3.75
5
6.25
o Exp.
X1/H
0.05
Durbin
7.5
8.75
----- k-ε × Shih
Fig. 8. Vertical profiles of turbulent kinetic energy k=huH i2 at various positions for rough surface hill.
Vertical Height, X2/H
4
3
2
1
H 0
-2.5
-3.75
(a)
0
-1.25
0
1.25
2.5
3.75
X1/H
0.1
5
o Exp.
6.25
Durbin
7.5
8.75
----- k-ε × Shih
Vertical Height, X2/H
4
3
2
1
H 0
(b)
-3.75
-2.5
0
-1.25
0.5
0
1.25
2.5
X1/H
3.75
5
o Exp.
6.25
Durbin
7.5
8.75
----- k-ε × Shih
Fig. 9. Profiles of streamwise and vertical velocity at various positions for rough surface hill: (a) streamwise velocity hu1 i=uH ; (b) vertical velocity hu2 i=uH :
occurred by the Durbin prediction along downstream of the hill. This discrepancy is closely related to the underestimation of k by Durbin model in this region. This is of the reason that the small values of k obtained by Durbin model lead to small turbulence diffusion causing large re-circulation generated behind the hill. This
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indicated a limitation of Durbin model, which is based on the linear eddy viscosity model.1 3.2.3. Effects of surface roughness on flow pattern Table 3 shows a summary of the computation predictions so far, in terms of normalized separation point and re-circulation length for the hill cases in comparing with the experiment. Separation point and re-circulation length were both measured from the center of the hill. The comparison is based on the mean velocity vector at about H=8 height in corresponding to the lowest measuring point in the experiment. Ishihara et al. [5] also conducted calculations for the case with rough surface using standard k2e and Shih [6] models. The Shih model adopted in their work was slightly different from the one used in this study. However, Ishihara’s results and our results are very close. The re-circulation lengths predicted by standard k2e model and Shih model for the rough hill in their study were 5:3H and 6:5H; respectively, while the re-circulation lengths obtained in this current study were 5:4H and 6:6H: As already shown in Section 3.2.1, the surface roughness has a great influence on the flow separation point and re-circulation. It can be seen that separation point tends to move to the upstream direction, by increasing the roughness parameter and the position of the separation point occurred earlier with rougher surface leading to a larger re-circulation area produced at the back of the hill. All the turbulence models under investigation in this study reproduced this tendency. Re-circulation region simulated by Shih model shows closer agreement with experiment, whilst standard k2e and Durbin models, under-predicted and over-predicted the size, respectively. 4. Conclusions (1) The performance of three turbulence models when applying to flow over topographic features, cliff and hill, with different surface conditions is highlighted. The surface roughness has a great influence on the flow separation point and re-circulation, by increasing the roughness parameters. The position of the separation point occurred earlier with rougher surface leading to a larger re-circulation area produced at the back of the hill. (2) Performance of these models, standard k2e and its revised form proposed by Durbin and Shih, for step geometry (cliff case) is shown. The shortcoming of an 1 The performance of Durbin model in predicting flow over small curvature (i.e. a hill) is shown in Section 3.2, and it is found out that this model cannot perform well at the downstream region. Additional work has been tried on applying the Durbin model to object with sharper edges (i.e. a step) in order to investigate its performance at the rear region. The model geometry was based on the experimental work of Moss and Baker [10]. The boundary conditions at the inlet are specified in correspondence with the experiment. Fig. 10 shows the change of the mean velocity of flow over a step model, by the case of Durbin model at various converge criterion. It can be seen that the flow separated on the windward corner and reattached at the roof. As the calculation proceeded further with smaller specified residual, reattachment started moving downstream of the roof causing numerical instability consequently leaded to diverged solution. Thus it is deduced, in this case, that Durbin model has a difficulty in convergence behind twodimensional object with steep edges.
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Fig. 10. Change of mean velocity vector over a step at various converged criterion of Durbin case: (a) residual=0.007, (b) residual=0.006, (c) residual=0.005.
overestimation on the turbulent kinetic energy around regions of impingement with standard k2e is seen. This fault is corrected by the revised k2e models. For the front step model, performance by the Durbin model shows to be in good agreement with the experiment and is better than the Shih model.
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(3) In the case of geometry with small curvature, the predictions obtained by the Durbin model perform well near the hillcrest than standard k2e and Shih models, but show difference in the downstream.1 This indicated a limitation of Durbin model, which is based on the linear eddy viscosity model. In the region behind the hill, predictions by Shih model show very close agreement with the experiment. (4) The current study covers the initial stage of wind assessment and identifies key elements associated with various turbulence models, surface boundary conditions, providing a basis for future work.
Acknowledgements The authors would like to express their gratitude to Prof. Shinsuke Kato of IIS, University of Tokyo and Prof. Takeshi Ishihara of University of Tokyo for their valuable information on the experiments of cliff and hill models, respectively. This research is supported by the New Energy and Industrial Technology Development Organization (NEDO) of Japan and the work is carried out as part of the research project ‘‘Development of the Local Area Wind Energy Prediction Model’’.
References [1] B.E. Launder, M. Kato, Modelling flow-induced oscillations in turbulent flow around a square cylinder, ASME Fluid Engineering Conference, Washington DC, June 1993. [2] M. Tsuchiya, S. Murakami, A. Mochida, K. Kondo, Y. Ishida, Development of a new k–e model for flow and pressure fields around bluff body, J. Wind Eng. Ind. Aerodyn. 67&68 (1997) 169–182. [3] P.A. Durbin, Technical note: on the k–e stagnation point anomaly, Int. J. Heat Fluid Flow 17 (1996) 89–90. [4] A. Mochida, Y. Tominaga, S. Murakami, R. Yoshie, T. Ishihara, R. Ooka, Comparison of various k–e models and DSM applied to flow around a high-rise building—report on AIJ cooperative project for CFD prediction of wind environment, Wind Struct. 5 (2–4) (2002) 227–244. [5] T. Ishihara, K. Hibi, Numerical study of turbulent wake flow behind a three-dimensional steep hill, Wind Struct. 5 (2–4) (2002) 317–328. [6] T.H. Shih, J. Zhu, J.L. Lumley, Anew Reynolds stress algebraic equation model, Comput. Methods Appl. Mech. Eng. 125 (1995) 287–302. [7] T.H. Shih, J. Zhu, J.L. Lumley, A realizable Reynolds stress algebraic equation model, NASA TM105993, 1993. [8] M.F. Yassin, S. Kato, S. Murakami, R. Ooka, T. Takahashi, T. Ohtsu, Wind tunnel study on prediction method of wind characteristics over local topography for suitable site of wind power station (Part 3): turbulence characteristics of flow over two-dimensional step model, Summaries of Technical Papers of Annual Meeting, AIJ, Environmental Engineering II, September 2001, pp. 757–758. [9] Y. Meng, K. Hibi, An experimental study of turbulent boundary layer over steep hills, Proceedings of the 15th National Symposium on Wind Engineering, April 1998, pp. 61–66 (in Japanese). [10] W.D. Moss, S. Baker, Re-circulating flows associated with two-dimensional steps, Aeronaut. Q. 31 (1980) 151–172.
Numerical simulation of flow over topographic features by revised k2e models Yu Fat Luna,*, Akashi Mochidaa, Shuzo Murakamib, Hiroshi Yoshinoa, Taichi Shirasawaa b
a Tohoku University, Aoba 06, Sendai 980-8579, Japan Keio University, 2-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
Abstract The accurate prediction of the wind energy distribution over terrains is important for making an appropriate selection of a suitable site for installing wind power plant. In this study, two-dimensional numerical simulations of flow over two common types of topographic features, i.e. a cliff and a hill, are presented. Three types of turbulence models, namely standard k2e and its revised linear form (Durbin model) as well as a revised nonlinear form (Shih model) are employed in this work. The performance of these models in predicting flow over these features is investigated. The accuracy of the prediction by using a z0 type wall function to reproduce the effect of surface roughness on flowfield from these turbulence models is also examined. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Flow over terrains; Cliff; Hill; Revised k2e; Durbin model; Shih model
1. Introduction Large amount of energy consumption caused by human activities have significantly changed our present global environment. For this reason, environmental concerns on the causes of global warming have led to many countries to introduce renewable energy technologies like wind power. Japan has a long coast and it is widely exposed to the sea, Sea of Japan on the west and Pacific Ocean on the east, undulating lands occupy a vast majority of spaces *Corresponding author. E-mail address: [email protected] (Y.F. Lun). 0167-6105/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 0 2 ) 0 0 3 4 8 - 3
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Nomenclature xi ui /S uH ut u* /uþ t S z0 xþ 2
spatial component (i ¼ 1 streamwise, i ¼ 2 vertical) velocity component in xi direction time averaged quantity, u0i ¼ ui /ui S u1 value at inflow of computational domain at height H tangential component of velocity vector friction velocity /ut S=/u * S roughness parameter wall co-ordinate
over the territory. Wind flows over terrains can gain an important advantage for installing wind power plants if the location of the site is correctly chosen. An appropriate selection of a suitable land for wind power plants can provide significant output of energy. However, it is not an easy task to choose a site for a wind farm because many factors have to be taken into account. Important factors such as change of wind speed near complex geometry of hilly terrain and the surface conditions of the lands need to be considered carefully in order to have an accurate evaluation of wind energy distribution. It is well known that the standard k2e turbulence model suffers from a serious drawback in overestimating turbulence kinetic energy, k; in the flow with impinging. In order to overcome this drawback, several revised k2e models have been proposed [1–3]. Among them, the results of a revised model proposed by Durbin [3] are clearly shown well in comparison to other revised k2e models when applying to flow over bluff bodies [4]. Recently, Ishihara et al. [5] applied a nonlinear revised k2e model proposed by Shih et al. [6,7] to flow over rough hill and the results showed good agreement with the experiment. The present work employed two modified k2e models, namely Durbin and Shih models, to simulate flow over topographic features such as a cliff and a hill. Complex terrain contains small objects such as trees, rocks that make the surface of the ground being less rough or more rough. Wind flow over terrain is strongly affected by these ground level objects. Numerical prediction of the effect of surface roughness on flowfield is one of the important areas within the wind engineering studies. In order to have an accurate prediction of flow around topographic features under the influence of such ground level objects, the boundary conditions for ground surface must be correctly selected. The aims of the present work are to investigate the performance of revised k2e models by Durbin [3] and Shih et al. [7] in predicting flow over topographic features, and to investigate the accuracy of the prediction using a z0 type wall function to reproduce the effect of surface roughness on flowfield over terrain.
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2. Outline of computations 2.1. Flowfields analyzed in this study Two different configurations of obstacle, steep curvature and small curvature, are considered. A two-dimensional step-shaped cliff is chosen as a model of steep curvature obstacle. The model geometry is based on the experimental work of Kato et al. [8], which is shown in Fig. 1a. The cliff height, H; is 75 mm and its length is 20H; the front section of the cliff is situated at 7H from the inlet and the end section of the cliff is placed at 7H from outlet. The vertical dimension of the flow domain is 24H: The computational flow domain for the case of cliff is discretized into 95ðX1 Þ 61ðX2 Þ grids and the minimum grid width is approximately 0.07H from the solid surface. Mesh density is placed at the front upper corner of the cliff. Fig. 2a shows an enlarge view of the computational grid around the front section of the model cliff. The second model is a two-dimensional hump with small curvature, e.g. a low hill. The model geometry is based on the wind tunnel investigation of Ishihara and Hibi [9] (cf. Fig. 1b). The height, H; and length, L; of the model hill are 40 and 200 mm (5H) respectively. The flow domain has dimension of 60H in the horizontal direction and 22:5H in vertical. The computational flow domain in this case is 110ðX1 Þ 50ðX2 Þ nodes. Grid density is enhanced around the hill and along the bottom surface of the flow domain. The minimum grid width in this model is approximately 0:04H: Fig. 2b shows an enlarge view of the computational mesh around the model hill.
u1 = (X2)0.143 Wind Cliff
1.8m
X2 H=75mm
X1
(a)
L=200mm
x2 = H
cos2 (x1
/ 2L) x2 H=40mm
(b)
x1
Fig. 1. Configuration of the two models used: (a) model geometry of cliff [8], (b) model geometry of hill [9].
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Fig. 2. Computational mesh for the two models, (a) cliff case: enlarge view around the front corner, (b) hill case: enlarge view around the hill.
Table 1 Test cases under consideration Case
Obstacle configuration
Turbulence model
Surface condition
1-1 1-2 1-3
Cliff
Standard k2e Durbin Shih
Smooth Smooth Smooth
2-1-1 2-1-2 2-1-3
Hill
Standard k2e Durbin Shih
Smooth Smooth Smooth
2-2-1 2-2-2 2-2-3
Hill
Standard k2e Durbin Shih
Rough Rough Rough
2.2. Cases of computations Table 1 lists all the cases conducted within this study. Two different surface conditions, smooth and rough, were applied to the hill case and only a smooth surface was considered in the cliff case. Standard k2e model (hereafter denoted by standard k2e), revised forms by Durbin and Shih (hereafter denoted by Durbin model and Shih model) were adopted for both cases of flow over a cliff and a hill. Durbin model employs the eddy viscosity concept based on the linear relations between the Reynolds stress and the strain rate, and its form of nt is modified to
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Table 2 Boundary conditions for both cases Location
Conditions
Inlet
(1) Cliff case [8] /u1 Spxn2 ; n¼ 0:143; k ¼ 0:016/uH S2 ; e ¼ Pk ; Cases 1-1B1-3 /uH S : /u1 S is the value at a height of H at inflow boundary H: cliff height (2) Hill case [9] /ut S 1 x2 ¼ ln ; /u * S k z0
/u2 ðx2 ÞS ¼ 0;
k¼
/u * S2 1=2 cm
;
e¼
/u * S3 kz
Smooth surface: /u * S ¼ 0:21 m=s; z0 ¼ 0:01 mm; Cases 2-1-122-1-3 Rough surface: /u * S ¼ 0:32 m=s; z0 ¼ 0:3 mm; Cases 2-2-122-2-3 Outlet
@=qx1 f/u1 S; /u2 S; k; eg ¼ 0
Top domain
/u2 S ¼ 0;
Solid surface
(1) Cliff case:
@=qx2 f/u1 S; k; eg ¼ 0
! /ðut Þp SðCm1=2 kÞ1=2 1 E x2 ðCm1=2 kÞ1=2 ¼ ln ; k tw =r n
E ¼ 9;
1 x2 ln ; k z0
qk ¼0 qx2
(2) Hill case: /uþ t S¼
e¼
3=2 c3=4 m k ; kx2
e¼
cm3=4 k3=2 ; kx2
qk ¼0 qx2
Smooth surface: z0 ¼ 0:01 mm Rough surface: z0 ¼ 0:3 mm k ¼ 0:4; Cm ¼ 0:09
impose the ‘realizability’ constraint as described in Section 2.3. On the other hand, Shih model uses a quadratic nonlinear eddy viscosity model. The boundary conditions used in this work are shown in Table 2. For the cliff case, a power law velocity profile, which corresponds to the inflow condition of the wind tunnel experiment [8], was used at the inlet and a generalized log law profile for the smooth surface was applied at the solid surface. For the case of hill, a log law velocity profile was used at the inlet and a z0 type wall function was employed at the solid surface in order to incorporate the effect of surface roughness. The values of surface roughness parameters were chosen from Ishihara and Hibi [9]. The QUICK scheme was employed for the convection terms in all the equations. 2.3. Governing equations As mentioned above, in the standard k e model and Durbin model, the unknown Reynolds stresses are obtained from the conventional Boussinesq linear
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stress–strain relationship viz: @/ui S @/uj S 2 0 0 þ kdij : /ui uj S ¼ nt qxj 3 qxi
ð1Þ
The eddy viscosity, nt is related to the turbulence velocity scale, /v02 S and its time scale, t as nt ¼ Cm /v02 St:
ð2Þ
In k2e models, /v S assumes to be k and Eq. (2) becomes 02
nt ¼ Cm kt:
ð3Þ
In the standard k2e model, t ¼ k=e: The traditional k2e model has been found to be inadequate in dealing with impinging flows, and is not appropriate for involving object such as bluff body. A modification, which addresses this stagnation flow anomaly, is due to Durbin who suggested imposing the ‘realizability’ constraint 2kX/u0a u0a SX0 via a bound on the time scale, t; where summation is not taken in the /u0a u0a S [3]. The proposed bound on the time scale is t ¼ min½TS ; TD ;
ð4Þ
where k TS ¼ ; e
ð5aÞ
pffiffiffi 2 pffiffiffiffiffiffiffiffi; TD ¼ 3Cm jS j2 2
ð5bÞ
where jSj2 ¼ Sij Sji and Sij is the rate of strain tensor given by @/ui S @/uj S þ : Sij ¼ qxj qxi Another turbulence model used in this study is a nonlinear model proposed by Shih and co-workers. Shih model adopts a nonlinear relationship between Reynolds stresses and the rate of strain as follows: /u0i u0j S 2 nt @/uk S nt nt 1 ¼ dij Sij þ C1 Sik Skj dij Skl Skl 3 k qxk 3 k k e
nt nt 1 þ C2 Oik Skj þ Ojk Ski þ C3 Oik Ojk dij Okl Okl ; ð6Þ 3 e e where Sij ¼
@/ui S @/uj S þ qxj qxi
and
Oij ¼
@/ui S @/uj S qxj qxi
are the elements of the mean strain tensor and vorticity tensor respectively. The details of the model and expressions of the coefficients are given by Shih et al. [7].
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3. Results and discussion 3.1. Computation of flow around cliff Fig. 3 shows the ratio of two time scales ðTD =TS Þ: The definitions of TS and TD are given in Eqs. (5a) and (5b) respectively. The shaded area with the value smaller than 1 in this figure indicates that the Durbin’s time scale, TD is adopted (cf. Eq. (4)). This figure means that the Durbin’s time scale is used only around the frontal corner of the cliff and the time scale in the standard k2e model, TS is applied in other regions. Distributions of k and mean velocity vector plots over the cliff, predicted by standard k2e; Durbin and Shih models, are shown in Figs. 4(a) and (b). It is now widely known that one of the shortcomings of the standard k2e model is an overestimation of the turbulent kinetic energy around regions of impingements. It is evidently shown at the windward corner in Case 1-1. An improvement of k at the same area is obtained both by Durbin and Shih models in Case 1-2 and Case 1-3 respectively. In comparison between the standard k2e and Durbin models, k is predicted to be smaller by Durbin model comparing to the standard k2e in the area where TD =TS p1:0; shown in Fig. 3. On the top of the cliff flow separation and reattachment are all found in these three models, as shown in Fig. 4b. Figs. 5a and b show vertical profiles of the normalized mean velocity and normalized k in the separation region, respectively, by these three models comparing with experimental data. Prediction of mean velocity obtained by Durbin model showed generally very good agreement with the experiment at the cliff top surface. 2.5
2 2
1
TD /TS < 1.0
x2 /H
1.5
(TD is applied)
3
4
23 4 0
1
2
3
54
5 5
1
2 0.5 1 0 -1.5
31 -1
-0.5
0
0 0.5
1
x1/H Fig. 3. Ratio of two time scales TD =TS (cf. Eqs. (5a) and (5b)).
1.5
x2/H
x2/H
(a)
x2/H
-1
-1
0.02
-1
0.02
0.16
0.03 0.04
-0.5 0 Case1-3 (Shih)
0.01
0.02
-0.5 0 Case1-2 (Durbin)
0
0
0.14
0.04
-0.5 0 Case1-1 (k-ε)
0.04
0.06 0.1
0.5
0.5
0.08
0.5
0
1
0.02
0.1
0.06 0.04
1
1 x1/H
0.04
x1/H
0.08 0.06
0
0.02
0.12
x1/H
0.08 0.1 0.12
0.06
0.08 0.1 0.06 0.04
0.12
0.08
1.5
1.5
1.5
(b)
0 -1.5
0.5
1
1.5
2
2.5
0 -1.5
0.5
1
1.5
2
2.5
0 -1.5
0.5
1
1.5
2
2.5
-1
-1
-1
-0.5 0 Case1-3 (Shih)
-0.5 0 Case1-2 (Durbin)
-0.5 0 Case1-1 (k-ε)
0.5
0.5
0.5
1
x1/H
x1/H
x1/H
1
1
Fig. 4. Predictions by the three models for the cliff case: (a) distribution of k=huH i2 ; (b) mean velocity vector.
0 -1.5
0.5
1
1.5
2
2.5
0 -1.5
0.5
1
1.5
2
2.5
0 -1.5
0.5
1
1.5
2
2.5
x2/H x2/H x2/H
1.5
1.5
1.5
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Plotting position X1 H o Exp.
Durbin
----- k-ε + Shih
2
2
1.75
1.75
Vertical height X2/h
Vertical height X2/h
X1=0.5H
1.5
1.25
(a)
1 -0.5
0
0.5
1
1.5
(b)
1.5
1.25
1
0
0.05
0.1
0.15
0.2
Fig. 5. Comparison of vertical profiles of hu1 i and k just after the front corner at 0:5H from the front corner: (a) normalized mean velocity hu1 i=huH i ; (b) normalized k distribution k=huH i2 :
The distributions of k predicted by Durbin model also showed well with the experiment, below the height of 1:25H; close to the surface of the cliff but the model of Shih did not perform well at this region. However, Shih model is in better agreement with the experiment regarding the height where the maximum peak of k exists. The performance by standard k2e started showing poorly, to upper region of the flow domain, from the height where the peak of experiment located. It is pointed out that the standard k2e greatly overestimates k in the upwind corner of the bluff body. On the other hand, the values of k in the modified k2e evaluated to be smaller than that of experiment and this underestimation of k caused the re-circulation region on the roof larger than the standard k2e: 3.2. Computation of flow over hills 3.2.1. Distributions of k and mean velocity vectors Fig. 6 shows the computation results of normalized k distributions and mean velocity vector for the three models of the smooth hill case. It can be seen that the predictions of k at the hillcrest area by Durbin and Shih models are found smaller than the standard k2e; and these were also observed in the cliff case showing previously. Separation point and re-circulation zone are both obtained by the three models, along the hill slope and behind the hill respectively. Position of the separation points for the turbulence models comparing with the experiment is shown
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Fig. 6. Predictions by the three models for the smooth hill case: (a) distribution of k=huH i2 ; (b) mean velocity vector.
Table 3 Normalized separation point and re-circulation length of the hill cases in comparison with experiment Case
Experiment [9] Stadark k2e Durbin k2e Shih k2e
Xs =H (Separation)
Xr =H (Re-circulation)
Smooth
Rough
Rough
Smooth
1.1 1.3 1.0 1.2
0.6 0.8 0.5 0.5
6.4 5.4 13.7 6.6
4.6 3.6 8.5 4.5
H
Xs 0 Xr
in Table 3. In this figure, the re-circulation generated by standard k2e is smaller in size than the models of Durbin and Shih. When comparing the models of Durbin and Shih, Shih model predicts the reattachment length more accurately. The re-circulation predicted by Durbin model has a larger area than Shih model behind the hill but Shih model corresponds very well to the experiment with the size of 6:6H:
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Fig. 7. Predictions by the three models for the rough hill case: (a) distribution of k=huH i2 ; (b) mean velocity vector.
Fig. 7 shows the computation results of normalized k distributions and mean velocity vector for the case of rough hill by the three models. Similar to the smooth case, k is overestimated around the hillcrest by the standard k2e; and this fault is corrected by the two revised k2e models. As will be discussed later in detail in Section 3.2.3, it is clearly observed that change in position of the separation point leads to the change in reattachment length (smooth and rough cases). 3.2.2. Comparison with the experiment Fig. 8 shows the vertical profiles of normalized k at various positions over the rough surface hill. It can be seen that the predictions of k by Durbin model showed very good agreement with the experiment in the front section of the hill, at X1 =H ¼ 1:25 and X1 =H ¼ 0; whilst standard k2e and Shih models overestimated k at the same area. Along the downstream behind the hill, Durbin model underestimated the k but the results of Shih model agreed well with the experiment in this region. Predicted profiles of streamwise and vertical velocity by standard k2e; Durbin and Shih models, at different positions across the rough hill are shown in Fig. 9. Shih model showed to be generally close with the experiment, but the result of Durbin model deviated from experiment in the region behind the hill. A large re-circulation
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Vertical Height, X2/H
4
3
2
1
H 0
-3.75
-2.5
0
-1.25
0
1.25
2.5
3.75
5
6.25
o Exp.
X1/H
0.05
Durbin
7.5
8.75
----- k-ε × Shih
Fig. 8. Vertical profiles of turbulent kinetic energy k=huH i2 at various positions for rough surface hill.
Vertical Height, X2/H
4
3
2
1
H 0
-2.5
-3.75
(a)
0
-1.25
0
1.25
2.5
3.75
X1/H
0.1
5
o Exp.
6.25
Durbin
7.5
8.75
----- k-ε × Shih
Vertical Height, X2/H
4
3
2
1
H 0
(b)
-3.75
-2.5
0
-1.25
0.5
0
1.25
2.5
X1/H
3.75
5
o Exp.
6.25
Durbin
7.5
8.75
----- k-ε × Shih
Fig. 9. Profiles of streamwise and vertical velocity at various positions for rough surface hill: (a) streamwise velocity hu1 i=uH ; (b) vertical velocity hu2 i=uH :
occurred by the Durbin prediction along downstream of the hill. This discrepancy is closely related to the underestimation of k by Durbin model in this region. This is of the reason that the small values of k obtained by Durbin model lead to small turbulence diffusion causing large re-circulation generated behind the hill. This
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indicated a limitation of Durbin model, which is based on the linear eddy viscosity model.1 3.2.3. Effects of surface roughness on flow pattern Table 3 shows a summary of the computation predictions so far, in terms of normalized separation point and re-circulation length for the hill cases in comparing with the experiment. Separation point and re-circulation length were both measured from the center of the hill. The comparison is based on the mean velocity vector at about H=8 height in corresponding to the lowest measuring point in the experiment. Ishihara et al. [5] also conducted calculations for the case with rough surface using standard k2e and Shih [6] models. The Shih model adopted in their work was slightly different from the one used in this study. However, Ishihara’s results and our results are very close. The re-circulation lengths predicted by standard k2e model and Shih model for the rough hill in their study were 5:3H and 6:5H; respectively, while the re-circulation lengths obtained in this current study were 5:4H and 6:6H: As already shown in Section 3.2.1, the surface roughness has a great influence on the flow separation point and re-circulation. It can be seen that separation point tends to move to the upstream direction, by increasing the roughness parameter and the position of the separation point occurred earlier with rougher surface leading to a larger re-circulation area produced at the back of the hill. All the turbulence models under investigation in this study reproduced this tendency. Re-circulation region simulated by Shih model shows closer agreement with experiment, whilst standard k2e and Durbin models, under-predicted and over-predicted the size, respectively. 4. Conclusions (1) The performance of three turbulence models when applying to flow over topographic features, cliff and hill, with different surface conditions is highlighted. The surface roughness has a great influence on the flow separation point and re-circulation, by increasing the roughness parameters. The position of the separation point occurred earlier with rougher surface leading to a larger re-circulation area produced at the back of the hill. (2) Performance of these models, standard k2e and its revised form proposed by Durbin and Shih, for step geometry (cliff case) is shown. The shortcoming of an 1 The performance of Durbin model in predicting flow over small curvature (i.e. a hill) is shown in Section 3.2, and it is found out that this model cannot perform well at the downstream region. Additional work has been tried on applying the Durbin model to object with sharper edges (i.e. a step) in order to investigate its performance at the rear region. The model geometry was based on the experimental work of Moss and Baker [10]. The boundary conditions at the inlet are specified in correspondence with the experiment. Fig. 10 shows the change of the mean velocity of flow over a step model, by the case of Durbin model at various converge criterion. It can be seen that the flow separated on the windward corner and reattached at the roof. As the calculation proceeded further with smaller specified residual, reattachment started moving downstream of the roof causing numerical instability consequently leaded to diverged solution. Thus it is deduced, in this case, that Durbin model has a difficulty in convergence behind twodimensional object with steep edges.
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Fig. 10. Change of mean velocity vector over a step at various converged criterion of Durbin case: (a) residual=0.007, (b) residual=0.006, (c) residual=0.005.
overestimation on the turbulent kinetic energy around regions of impingement with standard k2e is seen. This fault is corrected by the revised k2e models. For the front step model, performance by the Durbin model shows to be in good agreement with the experiment and is better than the Shih model.
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(3) In the case of geometry with small curvature, the predictions obtained by the Durbin model perform well near the hillcrest than standard k2e and Shih models, but show difference in the downstream.1 This indicated a limitation of Durbin model, which is based on the linear eddy viscosity model. In the region behind the hill, predictions by Shih model show very close agreement with the experiment. (4) The current study covers the initial stage of wind assessment and identifies key elements associated with various turbulence models, surface boundary conditions, providing a basis for future work.
Acknowledgements The authors would like to express their gratitude to Prof. Shinsuke Kato of IIS, University of Tokyo and Prof. Takeshi Ishihara of University of Tokyo for their valuable information on the experiments of cliff and hill models, respectively. This research is supported by the New Energy and Industrial Technology Development Organization (NEDO) of Japan and the work is carried out as part of the research project ‘‘Development of the Local Area Wind Energy Prediction Model’’.
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