Computation of the turbulent plane plume using the k±±t 02 ±c model

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1 Applied Mathematical Modelling 24 (2000) 815±826 Computation of the turbulent plane plume using the k±±t 02 ±c model Kalyan Kalita, Anupam Dewan *, Anoop K. Dass Department of Mechanical Engineering, Indian Institute of Technology, Guwahati, Guwahati , India Received 15 April 1999; received in revised form 21 February 2000; accepted 7 March 2000 Abstract The k±±t 02 ±c turbulence model is used to predict the self-similar plane plume in a quiescent environment. This model has been recently used to predict the turbulent axisymmetric plume. Modelled transport equations for the turbulent kinetic energy k, its dissipation, mean square temperature uctuations t 02 and intermittency factor c have been solved numerically along with the equations for the mean quantities. A small change in one of the model constants, incorporation of the dissipation term in the intermittency transport equation and withdrawal of the intermittency interaction invariant term from the dissipation equation yield predictions of mean and turbulent quantities including intermittency that are in good agreement with the experimental data. Ó 2000 Elsevier Science Inc. All rights reserved. Keywords: Buoyancy driven ow; Turbulence model; Intermittency; Entrainment 1. Introduction A continuous source of buoyancy generates a laminar plume which becomes turbulent at some distance from the point of discharge. Such ows are encountered in the eld of pollution control, cooling of electronic equipment, etc. The knowledge of plume behaviour is useful for modelling more complex buoyancy in uenced ows and to illustrate the e ect of buoyancy on turbulence. The two-equation standard k± model has some limitations in predicting di erent free shear ows. To overcome the limitations some modi cations to the standard k± model have been proposed in the literature [1±3]. Cho and Chung [3] proposed the k±±c model where they had incorporated intermittency interaction invariant term in dissipation equation, correlated the eddy viscosity with the intermittency factor and considered a transport equation for intermittency. The k±±c model was tested successfully for di erent turbulent nonbuoyant free shear ows by Ahn and Sung [4] and Kim and Chung [5]. To take care of the e ect of buoyancy the standard k± model has been extended by Lumley [6] and Launder [7]. Hossain and Rodi [8] reviewed the model proposals of Lumley [6] and Launder [7]. Gibson and Launder [9] extended Launder's model [7] to predict horizontal surface jet and mixing layer. Chen and Rodi [10] proposed the k±±t 02 model based on Launder's model [7] and * Corresponding author. Tel.: ; fax: address: adewan@iitg.ernet.in (A. Dewan) X/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved. PII: S X ( 0 0 )

2 816 K. Kalita et al. / Appl. Math. Modelling 24 (2000) 815±826 Nomenclature B buoyancy flux ˆ R 1 u Dt dy 1 q C E entrainment coefficient ˆ 1 u m c's model constants D nozzle width at exit G k production of k due to buoyancy g acceleration due to gravity k turbulent kinetic energy m local volume ux P k production of k by shear stress P t production of temperature uctuations Pr t turbulent Prandtl number T mean temperature DT ˆ T T 1 t 02 mean square temperature uctuations u streamwise mean velocity v cross-stream mean velocity u 0 v 0 Reynolds shear stress u 0 t 0 streamwise heat ux v 0 t 0 cross-stream heat ux u 02 streamwise velocity uctuations v 02 cross-stream velocity uctuations x streamwise coordinate y cross-stream coordinate b coe cient of volumetric expansion d u velocity halfwidth d t temperature halfwidth rate of dissipation of k t rate of dissipation of temperature uctuations c intermittency factor C intermittency interaction invariant m t eddy viscosity q density t speci c weight Dt excess speci c weight above ambient r u normalised centreline velocity ˆ u0 B r 1=3 t centreline temperature decay constant ˆ bg DT m x B 2=3 Subscripts 0 centreline value 1 ambient temperature m maximum value j discharge value dm dx used it to predict the far eld behaviour of vertical buoyant jets. Chen and Chen [11] used the model of Chen and Rodi [10] to predict centreline mean and turbulent quantities of plane and axisymmetric buoyant jets from the exit to the self-similar region for di erent values of Froude

3 K. Kalita et al. / Appl. Math. Modelling 24 (2000) 815± number. Malin and Spalding [12] proposed the k±w ±t 02 model to predict plane and axisymmetric jets and plumes, where W is the mean square vorticity uctuations. Malin and Younis [13] predicted plane and axisymmetric plume using Reynolds stress and the turbulent heat ux transport model. Intermittency factor is an important quantity for turbulent ows and is de ned as the fraction of time that the ow is turbulent. Libby [14] proposed a guessed relation for the intermittency transport equation. Dopazo [15] proposed an exact transport equation for intermittency by conditioning the instantaneous continuity equation using an intermittency indicator function. Cho and Chung [3] proposed another transport equation for intermittency based on Byggstoyl and Kollmann's model [16,17] and used it to predict di erent nonbuoyant free shear ows (jet, wake and mixing layer). This was later extended by Dewan et al. [18] who proposed k±±t 02 ±c model to predict the characteristics of the self-similar axisymmetric plume. In this model transport equation for intermittency was considered along with the transport equations for turbulent kinetic energy, its dissipation and temperature uctuations. They considered an improved model for the buoyancy production of turbulent kinetic energy, accounted for the e ect of buoyancy on intermittency and showed the k±±t 02 ±c model to be superior to the existing models. The pro le of the intermittency factor for the plane plume has not been predicted so far. In the present work we have predicted the intermittency pro le across the plane plume in a quiescent environment using the modi ed k±±t 02 ±c model. The predictions of mean and turbulent quantities are also presented. The predictions have been compared with experimental data of Ramaprian and Chandrasekhara [19±21]. The turbulence model and numerical method are discussed in Section 2, and in Section 3 we present predictions using the model. 2. Governing equations and turbulence model 2.1. Mean ow equations The boundary layer forms of the Reynolds-averaged governing equations for mean velocity and temperature distribution for the turbulent plane plume are given below. The ow is assumed to be steady in mean and incompressible and the Boussinesq approximation is considered. Continuity: ou ox ov oy ˆ 0: 1 Streamwise momentum: u ou ox v ou oy ˆ o oy u0 v 0 gb T T 1 : 2 Mean temperature distribution: u ot ox v ot oy ˆ o oy v0 t 0 ; 3 where u is the mean velocity along the streamwise x direction and v is the mean velocity along the cross-stream y direction. T and T 1 are the mean temperature and ambient temperature, respectively.

4 818 K. Kalita et al. / Appl. Math. Modelling 24 (2000) 815± Turbulence model Turbulent transport quantities The Reynolds stress ( u 0 v 0 ) in the streamwise momentum equation and cross-stream heat ux ( v 0 t 0 ) in the mean temperature equation are modelled using the algebraic stress±heat ux model given as follows: (i) Reynolds stress: u 0 v 0 ˆ 1 c 0 v 02 1 c 1 k where kgb ot =oy k 2 ou c h ou=oy oy ; 4 v 02 ˆ c 2 k: 5 (ii) Cross-stream heat ux: v 0 t 0 ˆ v02 k 2 ot c h k oy : 6 (iii) Streamwise heat ux: The streamwise heat ux (u 0 t 0 ) which appears in the expression for the buoyancy production of turbulent kinetic energy G k is modelled as p u 0 t 0 ˆ k h kt 02 : 7 This model was used by Dewan et al. [18] for the axisymmetric plume and involves the meansquare temperature uctuations which is obtained by solving the transport equation for the same. We have adopted the same values of the model constants as used by Dewan et al. [18] (c 0 ˆ 0:55, c 1 ˆ 2:2, c 2 ˆ 0:53 and c h ˆ 3:2) except for k h. They considered k h ˆ 0:56 while studying axisymmetric plume. Malin and Spalding [12] suggested k h values from 0.4 to 0.7 on the basis of the experimental results for axisymmetric plume. In the present work we have taken k h ˆ 0: Transport equations for turbulent quantities (i) Turbulent kinetic energy k : "! # u ok ox v ok oy ˆ o v c 02 k ok k P k G k : 8 oy oy Here c k ˆ 0:225. P k and G k are the shear production and buoyancy production of turbulent kinetic energy, respectively, and these are given as P k ˆ u 0 v 0 ou=oy, G k ˆ gbu 0 t 0 : (ii) Rate of turbulent kinetic energy dissipation : u o ox v o oy ˆ o oy "! # v c 02 k o oy c 1 k P k G k c 2 2 k ; where c ˆ 0:15, c 1 ˆ 1:43 and c 2 ˆ 1:92. Dewan et al. [18] considered the intermittency invariant term in addition to the terms considered in the present work. We have not incorporated this term as its inclusion was found to have an adverse e ect on the growth rates of the plane plume. 9

5 K. Kalita et al. / Appl. Math. Modelling 24 (2000) 815± (iii) Temperature uctuations t 02 : The mean-square temperature uctuations (t 02 ) are computed from the following modelled transport equation: u ot02 ox v ot02 oy ˆ o oy " # k 2 ot 02 c T oy P t t : 10 Here P t and t are the production and dissipation rate of temperature uctuations and given as P t ˆ 2v 0 t 0 ot, oy t ˆ c T 1 t 02 =k where c T 1 ˆ 1:79 and c T ˆ 0:13. (iv) Transport equation for intermittency factor c : In the present work we have computed the intermittency factor using the following intermittency transport equation: u oc ox v oc oy ˆ D g S g : 11 The term D g represents the transport of c due to the mean velocity gradient across the plume. We have used the same modelled relation for D g as proposed by Dewan et al. [18] D g ˆ o oy 1 c m t oc ; 12 r g oy where the model constant r g ˆ 1:0. S g represents the rate of conversion of nonturbulent uid into turbulent uid through the entrainment process and S g is considered as follows: S g ˆ c g1 c 1 c P k k c k 2 oc oc g2 oy oy c g3c 1 c k C c gb1c 1 c G k k c g4c 1 c k : 13 The rst three terms on the RHS were proposed by Cho and Chung [3]. C represents intermittency interaction invariant term and is modelled as C ˆ k5=2 2 ou oy oc oy : Cho and Chung [3] omitted the last term after assuming that the dissipation of turbulent kinetic energy is almost equal to its production. They accounted for the e ect of dissipation by reducing the value of the model constant c g1. Dewan et al. [18] used Cho and Chung's model [3] and incorporated fourth term on the RHS to consider the e ect of buoyancy on intermittency. In the present work we have retained dissipation term in the intermittency equation. The values of the model constants are c g1 ˆ 1:6, c g2 ˆ 0:15, c g3 ˆ 0:16, c gb1 ˆ 0:25. The new constant c g4 considered in this work is given a value of Boundary conditions The ow is assumed to be symmetric about the mid-plane of the plane plume and thus the computations were performed only over one half of the plume and zero ux boundary conditions were speci ed along the plane of symmetry. At the plume edge all the quantities, viz. u, k,, t 02 and c, were set to zero except the mean temperature T which was set equal to the ambient temperature T 1. At the discharge of the plume, top hat pro les for all the variables were speci ed. The computations were terminated when the ow eld became self-similar. The self-similar results are independent of the discharge conditions.

6 820 K. Kalita et al. / Appl. Math. Modelling 24 (2000) 815± Numerical method and code validation The governing equations and turbulence model presented in Sections 2.1 and 2.2 form a closed system which includes seven coupled partial di erential equations and these equations were solved using the nite volume method [22]. For this purpose a computer code has been developed in FORTRAN 77 which is a part of an ongoing project undertaken by the authors to develop a software for investigating free shear and wall bounded turbulent ows in various geometries. Each governing equation was discretized using upwind scheme in the streamwise x direction and power law scheme [22] in the cross-stream y direction. The numerical solution for the steady ow was obtained by marching in the streamwise x direction starting from the jet exit where values of all the dependent variables were prescribed. The solution procedure involves solution of sets of simultaneous algebraic equations at each streamwise location by tridiagonal matrix (TDMA). We have used 100 nonuniform grid-points along the normal y direction and a step size of 6% of the local velocity halfwidth (d u ) along streamwise x direction. Two cases were selected to validate the code, viz., the plane jet using the k±±c model of Cho and Chung [3] and the plane plume using the k±±t 02 model as used by Malin [23]. Computations were made with the discharge conditions such that the height nondimensionalised by the Mortan length scale [18] was much less than unity for the plane jet and more than 15 for the plane plume. The comparison of the present computations with these computations enables us the validation of all the modi cations to the k± model employed in the present work. The mean and turbulent quantities for the two cases are found to be within 2% of those predicted by Cho and Chung [3] and Malin [23]. The grid independence of the code was tested by varying the number of grid-points along the normal direction from 100 to 150 while keeping a xed step size of 6% of local velocity halfwidth d u in the streamwise direction. The independence of the streamwise step size was tested by varying the step size from 2% to 8% in an increment of 2% with xed 100 nonuniform grids in the normal direction. In these tests the variation of the mean and turbulent quantities were found to be within 2%. 3. Comparison of predictions with measurements Measured data for the self-similar turbulent plumes up to 1980 has been reviewed by Chen and Rodi [24] and by List [25] up to Subsequently Ramaprian and Chandrasekhara [19±21] carried out experiments on the turbulent plane plume. In the present work the data given by Ramaprian and Chandrasekhara [19±21] has been chosen for the comparison as this data is known to be more reliable than those of earlier experiments. Mean and turbulent quantities predicted in the present work are tabulated (Table 1) along with the predictions using k±w ±t 02 model [12], Reynolds stress and heat ux transport model of Malin and Younis [13] and the measured data [20]. In Figs. 1±9 predicted pro les of mean and turbulent quantities by the present model have been plotted together with predictions of k±w ±t 02 model [12] and the measured data [19±21] Mean ow quantities The calculated values of the normalised centreline velocity r u and centreline temperature decay constant r t (both de ned in the Nomenclature) are 2:01 and 0:33, respectively, in good agreement with the measured values [20] of 2:13 and 0:39. The present value of the velocity growth rate

7 K. Kalita et al. / Appl. Math. Modelling 24 (2000) 815± Table 1 Predicted a mean and turbulent quantities for the plane plume compared with measurements Quantities Present model k±w ±t 02 Model [12] Model of Malin and Younis [13] Measured data [20] r u (2.03) ± 2.13 r t (0.40) ± 0.39 od u =ox (0.12) od t =ox (0.116) u 0 v 0 max=u (0.031) v 0 t 0 max= u 0 DT (0.55) u 0 t 0 max= u 0 DT 0 q (0.078) =DT (0.46) t 02 max a Malin and Spalding have reported the predictions for a variable and xed Prt. The values within parentheses are for Pr t ˆ 0:5. and the others are for variable Pr t. Fig. 1. Predicted mean velocity pro les for the turbulent plane plume compared with the measurements. (0:111) is in good agreement with the measured value [20] of 0:11. The temperature growth rate 0:11 from the present calculation is underpredicted by 17%. However, considering the large scatter (0.119±0.147) in the experimental data [20] the agreement is satisfactory. The predicted value of the entrainment coe cient (C E ) from the present model (0:231) is in good agreement with the measured value [20] (0:225). The measured and predicted pro les of streamwise mean velocity and temperature (in nondimensional form) are compared in Figs. 1 and 2, respectively. The present predictions agree well with the measured pro le [20] Turbulent quantities The pro les of the Reynolds shear stress, cross- ow turbulent heat ux and streamwise turbulent heat ux (in nondimensional form) are shown in Figs. 3±5, respectively. The Reynolds stress pro le agrees well with measured data [20]. Fig. 4 shows that the present prediction of cross- ow turbulent heat ux is in good agreement with measurement. The predicted streamwise turbulent

8 822 K. Kalita et al. / Appl. Math. Modelling 24 (2000) 815±826 Fig. 2. Predicted mean temperature pro les for the turbulent plane plume compared with the measurements. Fig. 3. Predicted Reynolds shear stress pro les for the turbulent plane plume compared with the measurements. heat ux pro le is in good agreement with measured data [20] (Fig. 5) and is also superior to the prediction using k±w ±t 02 model [12]. The present predictions of the turbulent transport quantities for the plane plume are signi cantly higher compared to the corresponding values for the nonbuoyant jets, supporting the earlier conclusion [20] that buoyancy causes a signi cant increase in the turbulent intensities and turbulent uxes. The temperature uctuation pro les are compared in Fig. 6. The present prediction agrees well with the measurement [19] and is superior to the prediction of the k±w ±t 02 model [12]. No measured data is available for the turbulent kinetic energy pro le. It is, however,

9 K. Kalita et al. / Appl. Math. Modelling 24 (2000) 815± Fig. 4. Predicted cross-stream turbulent heat ux pro les for the turbulent plane plume compared with the measurements. Fig. 5. Predicted streamwise turbulent heat ux pro les for the turbulent plane plume compared with the measurements. clear that the predictions of k by the present model and Malin and Spalding's model [12] are almost coincident (Fig. 7). The predicted pro les of the terms of turbulent kinetic energy equation are compared in Fig. 8 with the measured data [20]. No measured data for the dissipation term has been reported in the literature. The comparison shows that the present predictions are in good agreement with the measured data [20]. No such term by term comparison has been attempted earlier. The predicted pro le of turbulent kinetic energy due to buoyancy (G k ) agrees well with the experimental observation [20]. Thus the value of k h adopted in the present work is appropriate.

10 824 K. Kalita et al. / Appl. Math. Modelling 24 (2000) 815±826 Fig. 6. Predicted temperature uctuations pro les for the turbulent plane plume compared with the measurements Intermittency factor Fig. 7. Predicted kinetic energy pro les for the turbulent plane plume. The pro le of intermittency factor predicted in the present work is of bell-shape, unlike top-hat shape pro le of nonbuoyant jet [21]. Fig. 9 shows that the predicted values of the intermittency factor agree well with the measured data [21]. Thus the present model supports the view [21] that there is a signi cant amount of direct transport of ambient uids by large eddies even in the region of maximum Reynolds stress, unlike nonbuoyant jet where eddies are unable to penetrate the region of maximum Reynolds stress. Hence the modi cation made to the intermittency factor transport equation is justi ed.

11 K. Kalita et al. / Appl. Math. Modelling 24 (2000) 815± Fig. 8. Predicted pro les of the terms of the kinetic energy equation for the turbulent plane plume compared with the measurements. Fig. 9. Predicted pro le of the intermittency for the turbulent plane plume compared with the measurements. 4. Conclusions The self-similar vertical turbulent plane plume has been studied using the modi ed k±±t 02 ±c model. The modi cations made to the model are appropriate for the plane plume. The predicted pro le of intermittency factor agrees well with the measured data and the predictions of mean and other turbulent quantities are also in good agreement with experimental observations and superior to the predictions using previous models.

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