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Computers & Fluids j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p fl u i d

Mean ﬂow characteristics of a turbulent dual jet consisting of a plane wall jet and a parallel offset jet Amitesh Kumar ⇑ Department of Mechanical Engineering, National Institute of Technology, Rourkela 769008, India

a r t i c l e

i n f o

Article history: Received 11 July 2014 Received in revised form 23 September 2014 Accepted 25 February 2015 Available online 5 March 2015 Keywords: Turbulent jet Wall jet Offset jet Twin parallel jet Dual jet Numerical simulation

a b s t r a c t An exhaustive study is carried out to highlight the ﬂow features arising due to a dual jet consisting of a plane wall turbulent jet and a parallel turbulent offset jet (hereafter this combination is termed as a dual jet). The standard high Reynolds number two equation k–e model is used to solve the two-dimensional, incompressible, turbulent ﬂow. A detailed analysis is carried out to compare the ﬂow characteristics of the offset jet and the dual jet for an offset ratio between 3 and 15 at an interval of 2. It is noticed that the presence of wall jet in addition to the parallel offset jet has a tremendous effect on the ﬂow characteristics. In the present study, the location of reattachment point and vortex centre of an offset jet and merge point, combined point, and the vortex centres of a dual jet are identiﬁed and expressed by correlations which depend on the offset ratio. It is interesting to observe that the cross-stream movement of vortex centre is linear with the change in offset ratio for both kind of jets either an offset jet or a dual jet. However, the streamwise movement of vortex centre follows a parabolic trend with respect to the offset ratio with an exception of combined point which is weakly dependent on the cubic power of offset ratio. The jet half width of a dual jet is compared for different offset ratios by a unique scaling method for which the correlation is also proposed; it is observed that the variation is parabolic with respect to the axial location. The total momentum ﬂux of a dual jet is found to decrease with the increase in the offset ratio as opposed to almost insigniﬁcant change in the value for an offset jet. And, the value of integral constant is on the lower side as compared to the corresponding case of either an offset jet or a parallel jet. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Studying the ﬂow characteristics of a turbulent jet is an important topic amongst researchers across the world because of its various applications in industries. Some of the common examples include: cooling of a combustion chamber wall in a gas turbine, the use of an air deﬂector as a circulation controller, an automobile demister, ﬁlm cooling, evaporation enhancement, entrainment and mixing process in gas turbine and boiler combustion chambers, heat exchangers, carburetor systems, environmental dischargers, burners, thrust-augmenting ejector for V/STOL aircrafts, heating, ventilation, and air-conditioning [2,6,12,14,16,24,30]. Turbulent jet can be broadly classiﬁed into following categories: free jet, wall jet, wall attaching offset jet, two parallel jets, and dual jet consisting of a wall jet and a parallel offset jet. These classiﬁcations are done based on the position and presence of the horizontal impingement wall in the domain. The discussion on various types of jet can be found in Rajaratnam [28]. ⇑ Tel.: +91 661 2462532; fax: +91 661 2462549. E-mail address: [email protected] http://dx.doi.org/10.1016/j.compﬂuid.2015.02.017 0045-7930/Ó 2015 Elsevier Ltd. All rights reserved.

Free jet is characterised by a ﬂow which is unbounded from all the sides but one. A wall jet is a jet ﬂow which is bounded by a wall from one side and the ﬂow is directed along this wall. As a result, a boundary layer develops over the wall where the variation of streamwise velocity takes place and above which there exists a free region. The wall jet ﬂow has been extensively studied by many researchers [7–10,13,17]. When the jet is placed at an offset of some distance from the wall, it is termed as an offset jet. Such a jet is shown in Fig. 1(b), where D is the jet offset and a is the jet width at the exit. The offset ratio (OR) is deﬁned as jet offset D to the jet width a, i.e. OR ¼ D=a. This deﬁnition of offset ratio is different than the deﬁnition which is usually found in the literature; the reason will be discussed in the next section. When the offset jet is issued from the nozzle, the jet at some downstream location attaches to the impingement wall because of the Coanda effect [36]. This effect takes place when there is an imbalance between the entrained masses from above and below of the issuing jet. The jet after impinging onto the wall changes its direction and further downstream behaves like a wall jet. There have been many studies on a planar offset jet [3,11,22,24–27,31,32,37]. Various experimental techniques have been utilised to study this

49

A. Kumar / Computers & Fluids 114 (2015) 48–65

y Converging Region

Merging Region

Combined Region

U0

Merge Point (mp) (y0.5) 1

a

Combined Point (cp)

(y0.5) 2

D U0

(y0.5) 3

a

x Recirculation Region

(a) a dual jet y Outer Shear Layer Jet Centreline Inner Shear Layer U0 a

D

Recirculation Zone

Reattachment Point Recirculation Region

Impingement Region

x Wall Region

(b) an offset jet Fig. 1. Schematic diagram of turbulent jets.

phenomenon which includes pressure tap [3,11], single hot wire [11], Prandtl type pitot static tube [29], one component Laser Doppler Anemometer (LDA) system [26,27], and two-component LDA system [24]. The effect of Reynolds number and offset ratio on the reattachment length and wall static pressure in the converging region is studied by Bourque and Newmann [3]. Pelfrey and Liburdy [27] studied the mean ﬂow characteristics of a turbulent offset jet for a Reynolds number of 15,000 and an offset ratio of 7. They deﬁned the offset ratio as the jet centre line height to the jet (nozzle) width. They noted that the jet decay and spread rates are similar to those of a plane jet if appropriate curved coordinates are used. They also mentioned that the magnitude of curvature strain rate, due to jet deﬂection, is signiﬁcant, whose effect has been studied in Pelfrey and Liburdy [26]. Nasr and Lai [25] studied the turbulent ﬂow characteristics of an offset jet for a small offset ratio of 2.125 using two-component LDA system. Their deﬁnition of offset ratio is similar to the one deﬁned by Pelfrey and Liburdy [26]. For this small offset ratio, they noted a high turbulence in a region close to the nozzle plate between the jet and the offset plate. The ﬂow ﬁeld was also studied numerically using three turbulence models: standard k–e, RNG, and Reynolds stress. The

numerical results were compared to the experimental results and the conclusion withdrawn is that the standard k–e turbulence model best agreed with the experimentally determined values. Chaab and Tachie [6,7] have studied the characteristics of three-dimensional wall jet and offset jet using Particle Image Velocimetry (PIV). They considered three Reynolds numbers equal to 5000, 10,000, and 20,000 and four offset heights of 0.5, 1.0, 2.0, and 4.0. They also noted that the location of reattachment point is independent of the Reynolds number. The decay and spread rates were found to be independent of Reynolds number at certain downstream distance for small offset ratios (<2). The comparison of two low Reynolds number turbulence models has recently been done by Rathore and Das [30] for studying the ﬂow characteristics of an offset jet with an offset ratio of 3, 5, 7, 9, and 11. They compared the different turbulence models for a Reynolds number of 15,000; their geometry is similar to the geometry of Pelfrey and Liburdy [27]. The Moffatt vortices (secondary recirculation region) were identiﬁed near the corner of the wall with low Reynolds number turbulence modelling. Their study reveals that apart from near wall region, standard k–e model can give reliable result with less computing time.

50

A. Kumar / Computers & Fluids 114 (2015) 48–65

There are plenty of literature on the two plane parallel jets [1,2,15,21,23,24,34,35]. The ﬂow characteristics in the merging and combined regions are studied experimentally by Miller and Comings [23]. They used disk type static pressure probe and hot wire for the data acquisition. They found a highly convergent ﬂow near the nozzles and in this region the ﬂow structures are similar to the ﬂow structures of a free jet provided due account is taken of the mean ﬂow accelerations attributable to a radically different static pressure ﬁeld. In a more exhaustive study, Tanaka [34,35] explored the streamwise velocity, turbulence intensity, and pressure distribution in the merging and converging regions for offset ratio between 8.5 to 26.5. He found that the locations of merge point and combined point follow a power law with an exponent of 0.27. Also, it is noticed that the deﬂection of the two jets follows a circular arc. The momentum ﬂux along the axial locations is also calculated by Tanaka [35], where it is observed that because of the sub-atmospheric region in the merging zone the total momentum ﬂux does not remain a constant. Till 90s most of the investigators utilised static pressure measurements to ﬁnd out the reattachment length. Pelfrey and Liburdy [26,27] ﬁrst time used single component LDA system to better resolve the recirculation region. The accuracy in measuring the ﬂow ﬁled characteristics inside the recirculation region is further improved by using two-component LDA system [24]. The ﬂow ﬁeld of a turbulent offset jet and a twin parallel jet is compared using two-component LDA system. It is noticed by Nasr and Lai [24] that in the wall jet region the offset jet behaves like a wall jet while a twin parallel jet behaves like a free jet. Similar observations have also been noted by Anderson and Spall [2], where they studied the case of a twin parallel jet with a jet separation ratio ranging between 9 and 18.25. They considered a Reynolds number of 6000 based on the single jet width which relates to an initial jet velocity of approximately 18 m/s. The turbulence intensity at the jet exit was considered as 3.6%. They used a TSI IFA-300 constant temperature anemometer for data acquiring and velocity conversions were done using TSI ThermalPro software. The numerical predictions were obtained by standard k–e model and differential Reynolds stress model. It was concluded that both the turbulence models predicted the characteristics of merge point and the combined point to a good accuracy. Lai and Nasr [18] have also concluded after comparing the three turbulence models, standard k–e, RNG k–e, and Reynolds stress model, that standard k–e model performed at par with the other two models in predicting the ﬂow characteristics of a twin parallel jet. Low aspect ratio parallel jets have been studied by Anderson et al. [1]. The aspect ratio considered was in the range of 0:6 6 w=d 2:0, where w is the nozzle width and d is the width of the spacer separating the two jets. They noted a vortex shedding in the ﬂow ﬁeld at such a small aspect ratio because of the spacer acting as a bluff body. Detailed literature survey reveals that there are plethora of works on wall jet, offset jet, and a twin parallel jet. Very little attention is paid to the dual jet which consists of a wall jet and a parallel offset jet. The ﬁrst reported work on this topic is that of Wang and Tan [39], who experimentally studied the ﬂow characteristics of a dual jet using PIV. They considered an offset ratio of 2 with a Reynolds number equals to 104 . Similar to the observation of Anderson et al. [1], they also noted a periodic large-scale Karman-like vortex shedding similar to what would be expected in the wake of a bluff body. The study of ﬂow characteristics of a dual jet for higher offset ratio is missing in the literature and the present investigation is aimed to bridge this gap. The emphasis is placed on identifying the locations of merge point, combined point, and vortex centres of a dual jet. For this, standard high Reynolds number k–e turbulence model is used for numerical simulation, as literature survey also revealed that this model gives reliable

result at par with RSM [2], RNG k–e [18], and low Reynolds number models [30]. The dual jet characteristics is entirely different than the wall jet, offset jet, and the parallel jet as shown in Fig. 1(a). This ﬁgure reveals the three jet half widths which are: the jet half width for the outer layer of the upper offset jet ðY 0:5 Þ1 , the jet half width for the inner layer of the upper offset jet ðY 0:5 Þ2 , and the jet half width for the outer layer of the lower wall jet ðY 0:5 Þ3 . To compare the jet half widths for different offset ratios, a unique scaling is done and correlations are proposed for the scaled jet half widths. These correlations are obtained as a function of offset ratio OR and the axial location X. The developed correlations for minimum pressure in the recirculation region, jet half widths, X and Y locations of merge point, combined point, and two vortex centres will certainly provide a quick estimate of these parameters for better understanding of the jet mixing process. 2. Mathematical model The physical model considered for the present study is that of a dual jet consisting of a wall jet and a parallel jet at a certain offset height D. Fig. 1 shows the schematic diagram of a dual jet and an offset jet. To compare the jet characteristics of an offset jet and a dual jet, offset ratio (OR) is deﬁned as the ratio of jet offset D to the jet width a, i.e. OR ¼ D=a. It should be mentioned here that this deﬁnition of offset ratio is different than the deﬁnition usually found in most of the literature. However, this is the ﬁrst time when the parametric study of comparison of an offset jet and a dual jet is reported for an offset ratio ranging between 3 and 15. As mentioned earlier, the dual jet characteristics differ from the offset jet characteristics in many ways. For example, in recirculation region, instead of one vortex observed in offset jet there are two counter-rotating vortices for the case of dual jet. These two counter-rotating vortices attract the upper offset jet (of a dual jet) more towards the lower wall. The offset jet attaches to the lower wall because of the Coanda effect [36] and thereafter, it behaves as a wall jet. However, for the case of a dual jet, the two jets attract, merge, and combine together in the converging region, merging region, and the combined region respectively. After combined region, the dual jet starts to behave as a wall jet. Therefore, it is expected that onset of wall jet characteristics will be delayed for the dual jet in comparison to the case of offset jet. 2.1. Governing equations The Reynolds number is calculated based on the jet inﬂow velocity U 0 (for the present case both the jets are assumed to have same inﬂow velocity) and the jet width a. The Reynolds number of issuing jet is considered in a manner to have turbulent ﬂow throughout the computational domain; the Reynolds number considered for the present study is 15000. Two-dimensional ﬂow of incompressible ﬂuid is considered. The jet inﬂow velocity U 0 and the jet width a are taken as the normalising parameters for velocity and length respectively. With these scales, the normalising parameters are:

u v x y ; V¼ ; X¼ ; Y¼ U0 a a U0 p p0 k e ; kn ¼ 2 ; en ¼ 3 ; P¼ 2 U0 qU 0 U 0 =a

U¼

mn ¼

mt m

ð1Þ

In the above equation, x; y, u; v ; p; k; e are the dimensional quantities representing co-ordinate along x-axis, co-ordinate along y-axis, velocity along x-axis, velocity along y-axis, static pressure, turbulent kinetic energy, and the dissipation respectively; m and mt are the dimensional kinematic viscosity and the turbulent viscosity

51

A. Kumar / Computers & Fluids 114 (2015) 48–65

1.25

12

Fu , 160×128 Fp , 160×128 Fu , 200×160 Fp , 200×160 Fu , 240×192 Fp , 240×192

1 0.75

10

8

Y

0.5 160×128 200×160 240×192

6

0.25 4 0 2

-0.25 -0.5

0

10

20

30

40

50

60

0

70

-0.2

0

0.2

0.4

0.6

0.8

1

X

U

(a) Momentum fluxes for different grid densities

(b) Spanwise variation of mean streamwise velocity at X = 7

Fig. 2. Grid independence test.

Turbulent kinetic energy (kn ) equation:

@ðUkn Þ @ðVkn Þ 1 @ mt;n @kn 1 þ ¼ 1þ þ @X @Y Re @X Re rk @X @ mt;n @kn þ Gn en 1þ @Y rk @Y

Table 1 Calculation of discretisation error. / = dimensionless axial location of merge point N1 , N2 ; N3

20,480, 32,000, 46,080

/21 ext

9.85

r21

1.25

e21 a

3.7%

r32 /1

1.2 10.01

e21 ext

1.6% 2.0%

/2

10.38

10.33

/3

10.46

/32 ext e32 a

p

5.378

e32 ext

0.4%

GCI32 fine

0.5%

GCI21 fine

Rate of dissipation (en ) equation is:

0.8%

rp s p ¼ ln ð1r21 Þ jln je32 =e21 j þ qðpÞj; qðpÞ ¼ ln rp21 s ; s ¼ 1 sgnðe32 =e21 Þ, 32 21 p p 21 /1 /2 21 / / 1:25e21 21 /21 ; eext ¼ ext/21 1 ; GCIfine ¼ rp 1a . ext ¼ r 21 /1 /2 = r 21 1 ; ea ¼ / 1

ext

ð5Þ

@ðU en Þ @ðV en Þ 1 @ mt;n @ en 1 þ þ ¼ 1þ @X @Y Re @X Re re @X @ mt;n @ en en e2 þ C 1e Gn C 2e n 1þ @Y re @Y kn kn

where Gn is the production rate of turbulent kinetic energy, which is given as follows: Production (Gn ):

21

Gn ¼

" 2 2 # 2 @U @V @U @V 2 þ2 þ þ @X @Y @Y @X Re

mt;n

ð7Þ

respectively. Corresponding uppercase letters without bar are the non-dimensional parameters. Based on the above normalisation, the nondimensional form of continuity and momentum equations is given as, Continuity equation:

and ﬁnally, the eddy viscosity is related to k and Kolmogorov–Prandtl relation, as follows: Eddy viscosity (mt;n ):

@U @V þ ¼0 @X @Y

mt;n ¼ C l Re

ð2Þ

X-momentum equation:

@ðUÞ2 @ðUVÞ @ 2 1 @ @U P þ kn þ þ ¼ ð1 þ mn Þ @Y @X 3 Re @X @X @X 1 @ @U ð1 þ mn Þ þ Re @Y @Y

ð3Þ

Y-momentum equation:

@ðUVÞ @ðVÞ @ 2 1 @ @V þ ¼ ð1 þ mn Þ P þ kn þ @X @Y 3 Re @X @X @Y 1 @ @V ð1 þ mn Þ þ Re @Y @Y

ð6Þ

2

kn

ð8Þ

en

In the above equations, Re is Reynolds number based on the jet inﬂow velocity U 0 and the jet width a. The standard high Reynolds number k–e model is suggested by Launder and Spalding [20] with the following values of constants: rk ¼ 1:0; re ¼ 1:30; C 1e ¼ 1:44; C 2e ¼ 1:92, and C l =0.09. Non-dimensional boundary conditions are: Wall jet and offset jet : U ¼ 1; V ¼ 0; kn ¼ 1:5I2 ;

2

Solid walls : U ¼ 0; V ¼ 0; kn ¼ 0;

In the above momentum equations, kn and mn are the nondimensional turbulent kinetic energy and the nondimensional eddy viscosity respectively. In deriving these equations, the concept of eddy viscosity is assumed. For handling the turbulent nature of the ﬂow, standard high Reynolds number two equation k–e model is utilised, which are given below:

3=4 en ¼ k3=2 n C l =0:07

en ¼ 0

@U ¼ 0; @Y @U @V ¼ 0; ¼ 0; Exit boundary ðat X ¼ 75Þ : @X @X

Entrainment boundary ðat Y ¼ 60Þ :

ð4Þ

e via

@V ¼ 0; @Y @kn ¼ 0; @X

@kn @ en ¼ 0; ¼0 @Y @Y @ en ¼0 @X

In the above expressions, I is the turbulent intensity which is assumed to be equal to 0.05. To avoid the ﬁne grid near the solid wall, it has been ensured that the ﬁrst grid point near the wall should fall in the logarithmic region, i.e., 30 < Y þ < 100 where Y þ ¼ yus =m; us being the friction velocity.

52

A. Kumar / Computers & Fluids 114 (2015) 48–65

15

15 Present Numerical Result Pelfrey and Liburdy (Exp.)

Present Numerical Result Pelfrey and Liburdy (Exp.)

10

Y

Y

10

5

5

0 -0.6 -0.4 -0.2

0

0.2

0.4

0.6

0.8

1

0 -0.6 -0.4 -0.2

1.2

0

0.2

0.4

0.6

U

U

(a) X = 3

(b) X = 6

0.8

1

1.2

15

15 Present Numerical Result Pelfrey and Liburdy (Exp.)

Present Numerical Result Pelfrey and Liburdy (Exp.)

10

Y

Y

10

5

5

0 -0.6 -0.4 -0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 -0.6 -0.4 -0.2

0

0.2

0.4

0.6

U

U

(c) X = 9

(d) X = 12

0.8

1

1.2

Fig. 3. U velocity proﬁle at downstream distances.

The two-dimensional Reynolds averaged Navier–Stokes (RANS) equations are solved using ﬁnite volume approach. The collocated arrangement of primitive variables is utilised on a structured grid system. The grids are laid in such a manner so that grid density should be higher near the wall and gradually becomes coarser towards the outﬂow boundary. To impose the inﬂow and outﬂow conditions the downstream boundary is taken at 75 times the nozzle width a while cross-stream boundary is taken at a distance of 60 times the nozzle width a. These distances are also found sufﬁcient to obtain the self similarity solution for the respective cases. Altogether, ten grid points are employed across the jet width a to resolve the turbulent jet characteristics near the jet. Because of the complex nature of ﬂow, the convective ﬂux is discretised using the power law upwind scheme. However, it has been shown by others [25] that this scheme gives reliable result. On the other hand, the diffusive term is discretised using central difference scheme to ensure second order accuracy.

ratio of 9. It may be noted that all sets show almost same value of momentum ﬂuxes (as shown in Fig. 2(a)). But, a more converged solution for streamwise mean velocity at a location of X ¼ 7 is obtained for grid densities of 200 160 and 240 192 (as reﬂected in Fig. 2(b)). In addition to the above tests, another test is carried out to ﬁnd the grid convergence index (GCI) as suggested by Celik 10 9

Present Numerical Result Tanaka (Exp.)

8 7 6

H/2a

2.2. Solution approach

5 4 3 2

2.3. Grid independency test

1

The best practice guidelines for CFD code veriﬁcation and validation is well explained by Casey and Wintergerste [4]. In order to obtain a grid independent solution, three sets of grid densities have been tested and the results are presented in Fig. 2 for an offset

0

0

2

4

6

8

10

12

14

x/a Fig. 4. Comparison of the central streamline of jet with Tanaka [34].

53

A. Kumar / Computers & Fluids 114 (2015) 48–65

1 0.8 0.6

Fu

Fu

Fp

Fp Fk

0.4

Fk Fr

0.2

Ftotal

Fr Ftotal

0 -0.2 -0.4

0

10

20

30

40

X Fig. 5. Term-by-term evaluation of the momentum ﬂuxes and comparison with Spall et al. [33].

et al. [5]. For this case, the axial location of merge point of a dual jet for an offset ratio of 9 is selected and the results are presented in Table 1 where r is the grid reﬁnement factor, p is the apparent order of the method, /ext is the extrapolated value of /, ea is the approximate relative error, eext is the extrapolated relative error, and GCIfine is the ﬁne grid convergence index. Table 1 suggests that the numerical uncertainty in the ﬁne grid solution for the axial location of merge point is GCI21 fine ¼ 2%. However, when it is evaluated for the ﬁner sets (set 2 and set 3, represented by the total number of control volumes N 2 and N 3 respectively) it comes out to be less than 1%. Therefore, a grid density of 240 192 has been considered for the rest of the simulations. 2.4. Code validation To the author’s best knowledge, there is no experimental study on the dual jet for large offset ratios. However, there are only few experimental/numerical studies on dual jet for small offset ratio [39,38]. Therefore, the developed numerical code for a dual jet is validated extensively with the published experimental results for an offset jet or a twin parallel jet. The computation of ﬂow ﬁeld is very important in studying the turbulent jet characteristics, that is why the streamwise velocity (U) proﬁle at certain downstream distances is compared with the experimental results of Pelfrey

and Liburdy [27] for the case of an offset jet, with an offset ratio of 7 as per their deﬁnition. It should be noted here that offset ratio of 7 of Pelfrey and Liburdy is actually offset ratio (OR) of 6.5 as per the present deﬁnition. Fig. 3 compares the numerical predictions with the experimental results of Pelfrey and Liburdy for an offset ratio of 7. A good agreement between the current numerical predictions and the experimental results can be seen in this ﬁgure. To check the capability of developed code for capturing the bend of an offset jet, the problem of two parallel jets as studied by Tanaka [34] has been considered for validation. Since the two parallel jet problem is symmetric about the mid plane, only onehalf of the solution domain with symmetric boundary condition at the bottom is considered. The central streamline of the jet for a jet spacing of H=2a ¼ 6 is shown in Fig. 4; here H is the distance between the central plane of two jets. The Reynolds number considered for this problem is 6000 and the turbulent quantities at the inlet are speciﬁed for an inlet turbulence intensity of 5%. As observed, a good agreement has been obtained. To further validate the numerical work, a comparison of the numerical results for the integral of the individual terms of Eq. (3) with the results of Spall et al. [33] is presented. The momentum ﬂux is calculated by integrating the X-momentum equation over a control volume that deﬁnes the computational domain which results

R Y max in: Y min dY ¼ constant, evaluated at the UU þ P þ 23 kn Re2 t @U @X planes of a particular axial lengthX. The Reynolds stress term was computed using the Boussinesq approximations. They have considered a dual jet comprising of two parallel plane jets separated by a distance of 9. The turbulence intensity was 5% and the jet Reynolds number was 50,000. Spall et al. [33] originally validated their numerical calculation with their experimental work. The indiR R R vidual components of F u ¼ UUdY , F p ¼ PdY, F k ¼ ð2=3Þkn dY R and F r ¼ ð2=Ret Þð@[email protected]ÞdY are computed and compared with those of Spall et al. [33] and shown in Fig. 5. It is to be noted that to obtain the momentum ﬂux for a symmetric single jet, the value obtained for dual jet is divided by 2. The continuous line is the result obtained from the present computation and the markers are the results of Spall et al. [33]. As observed, a very good agreement has been obtained.

3. Results and discussion The two-dimensional Reynolds averaged Navier–Stokes (RANS) equations are solved for studying the jet characteristics of a dual jet comprising of a wall jet and a parallel offset jet. The offset jet 30

30 Present Numerical Result Nasr and Lai (Correlation) Holland and Liburdy (Exp.) Borque and Newmann (Exp.) Sawyer (Exp.) Kim et al. (Exp.) Lund (Exp.)

25

Correlated Value

Xrp

20

Xrp

20

Numerical Value

25

15

15 10 10

5

Xrp=3.169×(OR)

0.768

5

2

4

6

8

10

12

14

16

0

2

4

6

8

10

12

14

OR

OR

(a) comparison with the experimental results

(b) correlation for reattachment length

Fig. 6. Variation of reattachment length with respect to offset ratio (OR).

16

54

A. Kumar / Computers & Fluids 114 (2015) 48–65

6

8

0 5

Upper vortex centre 6

0

4

U-contour 00

U-contour

3

Y

Y

Upper vortex centre

V-contour

4

V-contour

0 Lower vortex centre

0 Merge point

2

0

0 2

0

0

1

0 Lower vortex centre 0

0

2

Merge point

4

6

0

2

4

6

X

X

(a) OR = 3

(b) OR = 7

12

8

12

Upper vortex centre 0

0 0

10

10

Upper vortex centre

U-contour

8

V-contour

8

0

6

Y

Y

U-contour V-contour

6

0 Lower vortex centre

Lower vortex centre

4

4

0

0

2

2

2

4

6

8

10

12

14

Merge point

0

Merge point 0

0 0

0

16

0

18

2

4

6

8

10

12

X

X

(c) OR = 11

(d) OR = 15

14

16

18

Fig. 7. Merge-point and vortex centres for the case of dual jet.

Table 2 Location of merge point, combined point, and vortex centres. OR

X mp

Y mp

X cp

Y cp

Xv c

Yvc

X lv c

Y lv c

X uv c

Y uv c

3.0 5.0 7.0 9.0 11.0 13.0 15.0

3.00 6.08 8.41 10.46 12.51 14.54 16.31

1.70 1.96 2.20 2.42 2.63 2.82 3.07

16.25 15.80 16.85 18.40 20.30 22.35 24.10

1.70 2.10 2.45 2.80 3.15 3.40 3.80

3.96 6.33 8.05 9.20 10.16 10.99 11.90

1.24 2.00 2.76 3.59 4.47 5.38 6.28

1.16 2.77 4.05 5.18 6.37 7.63 8.65

1.38 1.66 1.93 2.18 2.41 2.62 2.85

1.31 3.37 5.05 6.49 7.96 9.43 10.62

2.39 3.57 4.62 5.66 6.66 7.61 8.61

is kept at a distance of 3, 5, 7, 9, 11, 13, and 15 times the jet width a from the bottom wall making the offset ratio (OR) equals to 3, 5, 7, 9, 11, 13, and 15 respectively. The Reynolds number considered for this study is 15,000. An exhaustive study is carried out to study the effect of offset ratio on the jet characteristics in the presence and absence of a wall jet. The results are presented for mean ﬂow characteristics, pressure distribution, turbulent quantities, similarity solutions, decay/growth rate of streamwise maximum velocity etc. In addition, emphasis is also placed on proposing correlations for reattachment length (X rp ), location of vortex centre (X v c ; Y v c ), and minimum pressure (P min ) for the case of an offset jet; subscript rp stands for the reattachment point and subscript min stands for the minimum value. These correlations are obtained as a function of the offset ratio (OR). Similarly, correlations are also proposed for

the case of a dual jet. However, instead of reattachment length, correlations are obtained for minimum pressure (P min ), locations of merge point (X mp ; Y mp ), combined point (X cp , Y cp ), and vortex centres (X lv c ; Y lv c ) and (X uv c , Y uv c ) appearing in the recirculation region of the ﬂow ﬁeld of a dual jet. The subscripts mp and cp denote the merge point and the combined point respectively while subscripts lvc and uvc refer to the lower vortex centre and the upper vortex centre respectively for the case of a dual jet.

3.1. Reattachment, merge, and combined points and vortex centres For the case of an offset jet, the jet reattaches to the wall at a certain distance (X rp ) from the issuing point because of the

55

0

A. Kumar / Computers & Fluids 114 (2015) 48–65

8

0

8

-0.03

-0.03 0

12

0

-0.03

14

0.1

5

-0.15 -0.09-0.12 -0.06

-0.06 -0.12 -0.09 -0.09

16

18

20

6

-0.09 -0.03

4 0

16

Y

0 6

-0.03

0

-0

0. 03

.0 3

-0. 09 -0Combined .06

0.06

0

0.03 0 -0 -0 6 .03 -0.12 .0-0 -0.15 .09

20

-0.0

-0 12

22

12

-0.03

-0.06

-0.09

18

point

0.09

2

0 15

14

0.03 0 -0.06-0.03 -0.09

-0.06

-0.12

-0. 0

06

-0.03

-0.09

3

.06

2

0.06 -0.03 0 0.03 -0.09 -0.0 6 -0.12 -0.15

12

22

-0.09

0.1

-0.06 0 0.0 3

-0

06 0.

-0.12

-0.12

.09

8

Combined point

9

0 -0 0. .03 03

10

-0.09

-0. 0

-0 . 09

-0

0. 06

09 0.

-0. 0

6

0.12

2

12 -0.03

-0.06

09 0.

9

0.0

20

5 .1

0

-0.12

3

-0.09

18

(b) OR = 7

-0.0 9

-0. 0

16

2

(a) OR = 3

-0 . 12

3

-0.1

14

2

0.0

-0.15

-0.06

X

.06

0. 15

-0.0

-0.03 -0.06 -0.09

-0

0

12

0.1

0. 12

22

12 0.

-0

0 -0.12

-0.12 -0.09 -0.06 -0.03

0

X

-0.15

0. 09

2

-0.15

-0.15

-0.09 0.03

-0.06

-0. 0 06 0.0 3 -0. 12

-0 . 09

40.0

.12 -0-0 .0-0 6 .03 0.1 2

-0.15 -0.12 -0.09 -0.06 -0.03

0

-0.15

6

4

0

-0.06 -0.09

8

Combined point

0.06

0.03

-0.15

-0.15

Combined point

2

-0.12

-0.15

Y

Y

-0.12

-0.15

-0.09

-0.09 -0.12

-0.09

-0.09 -0.12

-0.09

4

6

-0.06

-0.03 -0.06

-0.06

-0.06

Y

6

0 -0.03

-0.06

14

16

18

20

X

X

(c) OR = 11

(d) OR = 15

22

0 09

24

26

Fig. 8. Estimation of combined point through the contours of dU=dY, dashed lines represent negative values.

-0.04 -0.06 -0.08

Pmin

-0.1 -0.12 Offset Jet Dual Jet

-0.14

Bourque and Newman Hoch and Jiji Lund

-0.16

Rajaratnam and Subramanya

-0.18

2

4

6

8

10

12

14

16

OR Fig. 9. Variation of minimum pressure with offset ratio.

Coanda effect [36]. The study of impingement characteristics is important for understanding the downstream development and the recirculating ﬂow. The reattachment point is characterised by zero wall shear stress after which the inner shear layer starts

behaving as the wall jet. There are many investigations so far on this issue [3,11,12,14,22,24,30,32]. Most of the results are presented in Fig. 6(a) along with the present results. This ﬁgure also contains the proposed correlation with a goodness of ﬁt 99.97%. The experimental results are shown with symbols while the present numerical result is shown with a solid line. The dash line represents the value obtained by Nasr and Lai’s [24] correlation. It can be seen that prediction of reattachment length by two correlations (Nasr and Lai and the present correlations) matches quite well for an offset ratio less than 8. However, for an offset ratio higher than 8, the present correlation predicts the reattachment length in very close agreement with the experimental results as compared to the prediction obtained by Nasr and Lai [24]. When the entrainment from the lower side of offset jet is increased by introducing a wall jet in conjugation with the offset jet, the ﬂow ﬁeld changes drastically. Because of the wall jet, the offset jet now no longer reattaches to the wall; instead the two jets attract and merge together at a certain downstream distance, known as merge point. Merge point is characterised by the point in the domain where velocity is zero. The very same idea is used to identify this point. Identiﬁcation of merge point and vortex centres is illustrated in Fig. 7 for an offset ratio of 3, 7, 11, and 15. Unlike the two parallel plane jets, where the merge point lies on the symmetry plane (having only axial coordinate of merge point), merge point of a dual jet also possess a y-coordinate. Table 2 presents the location of merge point, combined point, and the vortex

56

A. Kumar / Computers & Fluids 114 (2015) 48–65

35 Present Result

20

Present Result

Present Correlation

Present Correlation

30

Anderson and Spall (Exp)

Anderson and Spall (Exp) Anderson and Spall (RSM)

25

Anderson and Spall (k-ε) Miller and Comings

Nasr and Lai

20

Nasr and Lai

Xcp

Xmp

15

Anderson and Spall (RSM)

Anderson and Spall (k-ε) Miller and Comings Tanaka

10

15 10

5 Xmp=1.166+1.519(OR)-0.024(OR)2

0

2

4

6

8

10

12

Xcp=19.56-1.82(OR)+0.26(OR)2-7.64×10-3(OR)3

5 14

16

2

4

6

8

10

12

OR

OR

(a) merge point

(b) combined point

14

16

Fig. 10. Comparison of merge point and combined point.

4.5

3.5 Numerical Value Correlated Value

Numerical Value Correlated Value

4

3 3.5 2.5

Ycp

Ymp

3 2.5

Ymp=1.394+0.112(OR)

2

Ycp=1.205+0.173(OR)

2 1.5 1.5 1

2

4

6

8

10

12

14

16

1

2

4

6

8

10

12

OR

OR

(a) Merge point

(b) Combined point

14

16

Fig. 11. Y location of merge point and combined point.

centres. It is noted that axial location of merge point increases with the increase in offset ratio, which is also noted by others for the case of two parallel plane jets [2,21,35]. Also, the cross-stream distance of merge point increases with increase in the offset ratio. On the other hand, combined point shows different patterns. Combined point is deﬁned as the point where the deﬂection in the mid part of axial velocity proﬁle (after merge point) disappears, i.e. the maximum axial extent of dU=dY ¼ 0 line. This logic is used to identify the combined point in the domain. Fig. 8 shows the contours of dU=dY for an offset ratio of 3, 7, 11, and 15. This ﬁgure clearly indicates the dU=dY ¼ 0 line which separates the positive and negative values. As per the deﬁnition of combined point, extremity of dU=dY ¼ 0 gives the location of combined point. As observed for merge point, the cross-stream location is found to increase monotonically with increase in the offset ratio. However, the axial location of combined point X cp ﬁrst decreases, reaches to a minimum near offset ratio of 5, and again increases thereafter (see Table 2). The location of combined point depends on the ﬂow ﬁeld in the converging region. There are two factors which affect the ﬂow ﬁeld: the inertia of the jets and the depression of pressure in the recirculation region. Fig. 9 shows the variation of minimum pressure with respect to the offset ratio for both the cases, an offset jet and a dual jet. The minimum pressure

decreases almost linearly with the decrease in offset ratio till OR ¼ 5; then, there is a sudden drop in pressure. However, this sudden drop is not noticed for an offset jet. The sudden drop in pressure causes both the jets to attract more towards each other; the jets merge together with high intensity, which ultimately delays the disappearance of the deﬂection in mid part of axial velocity proﬁle. With increase in the offset ratio to 5, the minimum pressure increases to 0.124; thereafter, minimum pressure increases at a steady rate. And, the effect of inertia dominates the effect of pressure on the location of combined point for an offset ratio greater than 5. But, cross-stream distance of combined point increases monotonically with increase in the offset ratio. For comparison, normalised minimum static gauge pressure in the recirculation zone obtained experimentally by Bourque and Newman [3], Hoch and Jiji [11], Lund [22], and Rajaratanam and Subramanya [29] is also presented in Fig. 9 for an offset jet. The present numerical result compares reasonably well with the result of Lund, Hoch and Jiji, and Rajaratanam and Subramanya. However, the difference is large when compared with the result of Bourque and Newman. The discrepancy is attributed to the low Reynolds number of 2760–7500 considered by Bourque and Newman. It is to be noted here that turbulent ﬂow characteristics become independent of Reynolds number beyond Re ¼ 10000 [27].

57

A. Kumar / Computers & Fluids 114 (2015) 48–65

10 14

Xvc (Numerical)

Yvc (Numerical)

Xlvc (Numerical)

Ylvc (Numerical)

Xuvc (Numerical)

12 10

Xuvc (Correlation)

Yvc (Correlation) Ylvc (Correlation)

Y

Yuvc (Correlation)

6

Y

8

X

Yuvc (Numerical)

8

Xvc (Correlation) Xlvc (Correlation)

Yv

63

+

R) 4 (O

c uv

.4 =0 c

R) 1(O

4

6

) 22(OR 53+0.1 Y lvc=1.0

4

Xvc=0.644+1.28(OR)-3.61×10-2(OR)2

2

Xuvc=-1.62+1.053(OR)-0.016(OR)2

2

-3

Xlvc=-1.0+0.774(OR)-8.752×10 (OR)

0

.9 =0

1 0.5

2

4

6

8

10

12

14

2

0

16

2

4

6

8

10

12

OR

OR

(a) X-coordinate

(b) Y-coordinate

14

16

Fig. 12. X- and Y-coordinates of vortex centres for an offset jet and a dual jet.

2

2 X=25 X=30 X=35 X=40 X=45

1

0

0.2

0.4

0.6

0.8

0

1

0

0.2

0.4

0.6

U/Umax

U/Umax

(a) OR = 3

(b) OR = 7

0.8

1

2

2 X=35 X=40 X=45 X=50 X=55

1

X=40 X=50 X=55 X=60 X=70

1.5

Y/Y0.5

1.5

Y/Y0.5

1

0.5

0.5

0

1.5

Y/Y0.5

Y/Y0.5

1.5

X=30 X=35 X=40 X=45 X=50

1

0.5

0.5

0

0 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

U/Umax

U/Umax

(c) OR = 11

(d) OR = 15

0.8

1

Fig. 13. Similarity solution for OR ¼ 3, 7, 11, and 15.

The axial location of merge point and combined point is compared in Fig. 10. The experimental results of Tanaka [35], Miller and Comings [23], Nasr and Lai [24], and Anderson and Spall [2] are considered for comparison. They have considered the case of

two parallel plane jets. The present numerical result for merge point matches quite well with the RSM model of Anderson and Spall. They have found experimentally the merge point at an axial location of 11.71 and 15.07 for the offset ratio of 9 and 13

58

A. Kumar / Computers & Fluids 114 (2015) 48–65

2

Y/Y0.5

1.5

w w w w w w x w ww ww w w ww ww w wx w ww

1

0.5

0x 0

wwww w

0.2

Offset Jet, OR=3 Offset Jet, OR=7 Offset Jet, OR=11 Offset Jet, OR=15 Dual Jet, OR=3 Dual Jet, OR=7 Dual Jet, OR=11

ww ww Dual Jet, OR=15 x w ww Wygnanski et al. w w x w w ww ww w w wxw ww ww ww ww w ww w x w ww w ww w ww w w w w xw w wwwww

0.4

0.6

0.8

1

U/Umax Fig. 14. Comparison of self-similarity solution of the offset jets and the dual jets.

respectively. The percentage errors between the experimental result of Anderson and Spall and the present numerical result are 10.67% and 7.39% respectively for an offset ratio of 9 and 13. However, these errors are 3.23% and 2.54%, and 11% and 14.48% when compared with RSM and k–e models of Anderson and Spall. The present result over-predicts the results obtained by Tanaka for any offset ratio. But, the numerical result compares very well with the experimental result of Miller and Comings for OR ¼ 6, the difference being 4.17%. The value for comparison (OR ¼ 6) is obtained using the given correlation in Fig. 10(a). Also, the present numerical result slightly over-predicts the observation by Nasr and Lai for OR ¼ 4:25. The difference noticed between the experimental results may be attributed to the different measuring techniques and the operating conditions. But, it is interesting to observe the similar trend shown by the dual jet and the two parallel plane jets as far as variation of merge point is concerned. The developed correlation for X mp is also shown in Fig. 10(a); this correlation is valid for 3 6 OR 15. Wang and Tan [39] observed a value of 0.75 as X mp for an offset ratio of 2. They have used PIV for studying the ﬂow characteristics of a dual jet. Using the developed correlation, X mp for OR ¼ 2 is 1.77 which is on the higher side. The possible reason for discrepancy is the vortex shedding and unsteadiness in the ﬂow ﬁeld at small offset ratio, as noted by Anderson et al. [1]. The comparison of axial location of combined point X cp with the experimental results are presented in Fig. 10(b). For this case also, the numerical result matches quite well with the results of Anderson and Spall for an offset ratio of 9 and 13. In fact, the errors are 4.66% and 0.53% with the experimental results respectively for an offset ratio equal to 9 and 13. These ﬁgures are: 1% and 0.77% for k–e model and 0.98% and 0.36% for RSM model of Anderson and Spall. However, discrepancy is more for lower offset ratios. The trend starts deviating from the trend of two parallel plane jets when OR falls below 7. The reason may be attributed to the relative strength of sub-atmospheric pressure in the recirculation region and the inertia of jets. The developed correlation for combined point is weakly dependent on the cubic power of offset ratio. But, the parabolic dependence is prominent as compared to the dependence of merge point on offset ratio. It should be noted that these correlations are obtained with a minimum of 99.9% goodness of ﬁt. Fig. 11 shows the cross-stream location of merge point and the combined point for different offset ratios. It is noticed that the variation is linear with respect to the offset ratio and it increases with the increase in the offset ratio. The cross-stream location of combined point moves faster than that of the merge point as reﬂected

by the higher slope of the line. The developed correlation is also put in the ﬁgure. It is noticed that irrespective of the offset ratio, the minimum pressure for a dual jet is lower than the minimum pressure of corresponding case of an offset jet. This indicates the merge point to be at a shorter axial location than the axial location of the reattachment point (see Fig. 6 and Table 2). The wall jet forces offset jet upwards making the merge point move away from the lower wall. As mentioned earlier, the dual jet will have two counter-rotating vortices, that is why there are two locations of vortex centres: lower vortex centre (X lv c ; Y lv c ) and upper vortex centre (X uv c ; Y uv c ). Fig. 12 presents the location of vortex centres for an offset jet and a dual jet for the studied cases. It can be noticed that axial location of vortex centre of an offset jet (X v c ) is greater than the axial locations of any of the two vortex centres (X lv c and X uv c ) appearing in the case of a dual jet. One of the possible reasons for this may be the higher depression of pressure in the recirculation region in two vortices compared to the single vortex of an offset jet. Also, X uv c and X lv c are almost same for lower values of offset ratios which suggests that both the vortices have similar characteristics at small offset ratio. With the increasing offset ratio, X uv c increases more rapidly than X lv c . This increase comes at the cost of suppression of lower vortex by the upper vortex. This is happening because of two reasons: (1) the horizontal wall offers higher resistance to the lower vortex (compared to the resistance offered by surrounding air to the upper vortex) which reduces the strength with increasing axial location and (2) the deﬂection of offset jet increases with increase in the offset ratio. In the ﬁgure, proposed correlations are also provided for X v c ; X lv c , and X uv c with a goodness of ﬁt of 99.5%, 99.9%, and 99.9% respectively. The variation is parabolic in nature; the parabolic nature is more pronounced for X v c . On the other hand, the variation in the cross-stream direction is linear for all the three vortex centres (see Fig. 12(b)) with a higher gradient for Y uv c . Generally, Y v c falls between Y uv c and Y lv c for an offset ratio greater than 4. However, for an offset ratio of 3, Y lv c is higher than Y v c . This is attributed to the fact that at lower offset ratios, the size of the two vortices decreases considerably; therefore, resistance due to lower wall on the lower vortex is low which allows it to grow longer in size and, in turn, the lower vortex manages to push upper vortex away from the wall. 3.2. Similarity solution It has been shown earlier by other investigators that the ﬂow characteristics of two parallel plane jets assume the ﬂow characteristics of a free jet and that for a conﬁned jet it reduces to that of a wall jet at far downstream locations [24]. For the very same purpose, traditional outer scaling as proposed by Wygnanski et al. [40] is usually utilised where the self-similarity is checked by plotting curves between Y=Y 0:5 and U=U max at certain downstream locations. In the above nomenclatures, jet half-width Y 0:5 is the non-dimensional distance in the cross-stream direction where the axial ﬂow velocity is U ¼ U max =2. For the present case of a dual jet, similar study has been performed and the results are presented in Fig. 13. The results are shown for an offset ratio of 3, 7, 11, and 15. The scaled non-dimensional axial velocity proﬁles are plotted at X ¼ 25, 30, 35, 40, and 45 for OR = 3; X ¼ 30, 35, 40, 45, and 50 for OR = 7; X ¼ 35, 40, 45, 50, and 55 for OR = 11; X ¼ 40, 50, 55, 60, and 70 for OR = 15. It can be seen that all the velocity proﬁles merge together after an axial location of X = 35, 40, 50, and 55 respectively for an offset ratio of 3, 7, 11, and 15 which indicate that before this location the ﬂow is developing and it has not attained the self-similar solution. Similar ﬁndings have also been reported by Vishnuvardhanarao and Das [37] for an offset jet. For their case, velocity proﬁle showed self-similarity at X = 30, 40, and 50 for an offset ratio of 3, 7, and 11 respectively.

59

A. Kumar / Computers & Fluids 114 (2015) 48–65

1.1 1

1.1 3

3 5

OR=3 OR=5 OR=7 OR=9 OR=11 OR=13 OR=15

5

0.9

0.7

0.8

3 5

0.6

5

0.5

OR=3 OR=5 OR=7 OR=9 OR=11 OR=13 OR=15

5

0.9

Umax

Umax

0.8

3

1 53

3

3 5

0.7

3 5

0.6 0.5

0.4

5

0.4

3

0.3

0.3

0.2 0

20

40

0.2

60

0

20

40

X

X

(a) Single offset jet

(b) Dual jet

60

1.05 1 0.9

1

0.8 0.95

Umax

Umax

0.7 0.6

0.9

LWJ, OR=3 UOJ, OR=3 LWJ, OR=7 UOJ, OR=7 LWJ, OR=11 UOJ, OR=11 LWJ, OR=15 UOJ, OR=15

0.5 0.4 0.3

LWJ, OR=3 UOJ, OR=3 LWJ, OR=7 UOJ, OR=7 LWJ, OR=11 UOJ, OR=11 LWJ, OR=15 UOJ, OR=15

0.85

0.2 0

5

10

15

0

1

2

3

4

5

X

X

(c) Decay of lower wall jet (LWJ) and upper offset jet (UOJ) of a dual jet

(d) Zoomed view of figure 15(c)

1.2

1o

Present Offset Jet, OR=1.625 Present Offset Jet, OR=6.5 Offset Jet, Nasr and Lai Pelfrey and Liburdy

1

o

0.9

o

o

o

o

0.8

o

0.8

Umax

Umax

0.7 0.6

0.6 0.5

0.4

o

0.4

Present Dual Jet, OR=2 UOJ, OR=2 LWJ, OR=2 Dual Jet, Wang and Tan

0.2 0.3 0

0

5

10

15

20

X

(e) Comparison with the experimental result for the offset jet

0.2

0

5

10

15

20

25

30

X

(f) Comparison with the experimental result for the dual jet

Fig. 15. Decay of normalised maximum velocity for different offset ratios.

Also, it has been shown by Kumar and Das [16], for a dual jet with offset ratio 9, that self-similarity is achieved between X = 45 and 55. The self-similarity solution for all the cases including offset jet as well as dual jet is compared in Fig. 14; all the proﬁles are

obtained at X = 60. The result is also compared with the experimental result of Wygnanski et al. [40]. They studied the wall jet ﬂow at a Reynolds number of 19000. It is interesting to observe that all the proﬁles collapse in a single line in the near wall region. However, there is a little deviation in outer region when

60

A. Kumar / Computers & Fluids 114 (2015) 48–65

10

10

Y

15

Y

15

100

100

0

5

10

15

0

0.02

0.03

5

0.02

0.01

10

15

(a) Turbulent viscosity, OR=5

(b) Kinetic energy, OR=5 15

2 00

0.01 0.04 0.03

2 00

4 00

0.0 0.01 2

0.0

13

120

0

12

00

0 10

00

Y

40 00

0

20

60

0

0

2

02 0.

80

20

0.0 0.01 00..003 3 2 0.02 0.02

4 00

600

8 00 10 00

200

0

20

0.06 0.04 0.01

X

20 0

400 600

0.04

100200

0.01

0.02 3 0 0.0.02 4

X

15

10

300

400

0

0.0

0.05

400

300

0.01

1 0.0

200 400

100200

0.02 0.01

0.03

300

0.04

0. 01

300

500

5

0.02

400

0.03 30 0

100 200

10 0 200

10 02 0.

Y

5

40 0

1300

120

0

0

0.0

800

10

600 40

15

20

0.03 0.010.02

0

0

0.058 0.05 0.010.02

0.03

5

0.0 0.0 2 3

0.0 0. 04

1000

3

10

0.02

15

X

X

(c) Turbulent viscosity, OR=15

(d) Kinetic energy, OR=15

4

05 0.

600 400

5

5

0

1000

200

0

100

0.02

600 400

800 200

0

80

1200

0.01

1000 800 200

5

0 01

20

Fig. 16. Distribution of the turbulent viscosity and the kinetic energy.

Y=Y 0:5 > 1:0. Similar observations have also been noted by Kumar and Das [16] and Rathore and Das [30]. Also, the matching with the experimental result is quite good till Y=Y 0:5 < 1:3. Beyond this limit, the discrepancy is more between the numerical results and the experimental results. However, the error is attributed to the possible entrained ﬂow or counter ﬂow in the fully developed region as pointed out by Wygnanski et al. [40]. Similar behaviour is also observed by Wang and Tan [39]. Although, the offset ratio is small but they also noted the scattered experimental data beyond Y=Y 0:5 ¼ 1:3. 3.3. Decay of normalised streamwise maximum velocity It is expected that after issuing from the nozzle, jet maintains its potential core which remains unaffected by the viscous effect of the surrounding ﬂuid till certain downstream location. After this location, the maximum velocity decays with the downstream locations. Fig. 15 presents the decay of maximum streamwise velocity with respect to the axial distance for the studied cases. This ﬁgure also compares the present numerical result with the existing experimental results for an offset jet and the dual jet. It is observed that U max remains nearly equal to 1, which is the potential core velocity, to some distance after the jet exit; this distance is found to increase with the increase in the offset ratio. This observation is also consistent with the observation of others [30,37]. As noted by others, after the potential core is consumed, three regions can be identiﬁed in the plot of offset jet (Fig. 15(a)), viz. a decreasing region where the U max decreases very rapidly in the recirculation region, an increasing region where U max increases almost with a constant rate irrespective of the offset ratio, and then a decreasing

region which starts in the wall jet region. It should be noted that the decay rate in the wall jet region decreases with the increase in the offset ratio. But, in the similarity region, where the ﬂow characteristics resemble the ﬂow characteristics of a wall jet, all the proﬁles collapse into a single proﬁle giving a U max value of almost 0.4. Contrary to this, the trend for the dual jet is completely different, as depicted in Fig. 15(b). Although, for an offset ratio of 3, there are three regions; but, the minimum value of U max (near the merge point) is higher than the minimum value of U max which is observed for the corresponding case of the offset jet. The reason is quite obvious: the presence of another jet (a wall jet), in addition to the offset jet, makes the converging region more sub-atmospheric (see Fig. 9) which causes both the jets to collapse with high intensity in merging region. However, the three regions observed in other plots are absent when the offset ratio is increased beyond 3 (Fig. 15(b)). Now, the jet maximum velocity after issuing from the nozzle increases to some extent and then it starts decreasing with increase in the axial location. But, in the wall jet region the decay rate is almost same for all the cases of a dual jet. Also, it is noticed that U max at the outlet decreases with the increase in the offset ratio. The reason for this can be explained with the help of Fig. 16. This ﬁgure presents the distribution of turbulent viscosity and the kinetic energy for an offset ratio of 5 and 15. It can be seen that the turbulent viscosity is quite low in the outer layer of the wall jet and the inner layer of the offset jet with an offset ratio of 5 as compared to the offset ratio of 15. As a result, the dual jet with lower offset ratio experiences less resistance which results in higher kinetic energy as seen in the ﬁgure. Also, it is interesting to observe that the location of maximum kinetic energy is very close to the merge point.

A. Kumar / Computers & Fluids 114 (2015) 48–65

point. It should be noted here that similar results have also been obtained by Rathore and Das [30] and their deviations were approximately 34.2% for standard high Reynolds number k–e model and 48.7%, and 46% respectively for low Reynolds number k–e models proposed by Launder and Sharma [19] and Yang and Shih [41]. The difference in predictions is attributed to the isotropic modelling of turbulent viscosity which is inaccurate in the recirculation region, as recirculation region experiences high shear strain rate [27]. On the other hand, the numerical prediction matches quite well with the experimental results of Nasr and Lai [24] till the reattachment point. After that, the numerical results deviate from the experimental results till X ¼ 15 and again there is a good match between the two predictions. Nasr and Lai [24] have studied the case of an offset jet with small offset ratio and the chances of error introduced by the curvature effect is little. Hence, error may be attributed to the inaccurate measuring technique in the recirculation region. Fig. 15(f) presents the comparison of numerical prediction of decay of maximum streamwise velocity with the experimental result of Wang and Tan [39]. They have studied the case of a dual jet with an offset ratio of OR ¼ 2 using Particle Image Velocimetry (PIV). The ﬂow ﬁeld is obtained through ensemble averaging of 360 instantaneous ﬁelds. The uncertainty in mean velocities was estimated to be 5% while that of the turbulent quantities were 10%. Based on the exit velocity of the jet, the Reynolds number was tak-

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It should be noted here that unlike the offset jet, ﬂow characteristics of a dual jet are decided by the relative strength of both the jets, viz. the upper offset jet (UOJ) and the lower wall jet (LWJ). To explore this possibility, U max for both the jets is plotted in Fig. 15(c) for an offset ratio of 3, 7, 11, and 15. It is clear that the maximum streamwise velocity is decided by the lower wall jet (LWJ) when the offset ratio is greater than 3. But, for the offset ratio 3, there is a competition between the lower wall jet (LWJ) and the upper offset jet (UOJ) as reﬂected in Fig. 15(d). It is clear that till the upper vortex centre, strength of LWJ is more; thereafter, U max is decided by the UOJ till X ¼ 4, which is little downstream from the merge point, and once again the strength of LWJ supersedes the strength of UOJ which continues till the end. The comparison with the experimental results for offset jet is presented in Fig. 15(e). For this purpose, results of Nasr and Lai [24] and Pelfrey and Liburdy [27] are used. The present numerical prediction over-predicts the results obtained by Pelfrey and Liburdy [27] beyond X ¼ 5. The maximum deviation is 25.64% which occurs near the reattachment

en equal to 104 ; the working substance was water. For the comparison purpose, similar numerical study is carried out with the same input parameters using standard high Reynolds number k–e model. In the ﬁgure, the streamwise maximum velocity for both the jets (UOJ and LWJ) is also presented. As noted earlier, the three regions in the velocity proﬁle can be seen with a higher value of U max for UOJ compared to LWJ till the merging point. After that, U max is decided by the LWJ for the entire region. The numerical prediction does not match with the experimental result and the maximum error for the given range is approximately 32%. As mentioned earlier, the reason for this discrepancy is the unsteadiness and the vortex shedding at such a small offset ratio [1]. 3.4. Mean static pressure, turbulent viscosity, jet half widths, and growth of jet width For an external ﬂow, the ﬂow strength is decided by the pressure gradient and the viscous resistance inside the domain. Fig. 17 shows the contours of pressure and turbulent viscosity for OR ¼ 13 for both the cases: an offset jet and the dual jet. In the pressure contour plots, the sub-atmospheric pressures are denoted by dashed lines. The deﬂection of the offset jet depends on the strength of the sub-atmospheric pressure gradient in the recirculation region. It is reﬂected from the ﬁgure that recirculation region is large for offset jet compared to the corresponding case of the dual jet. Hence, it is expected that the minimum sub-atmospheric pressure is more for the dual jet which is already shown earlier in Fig. 9. A correlation is proposed for the minimum sub-atmospheric pressure in the recirculation region which is given by Pmin ¼ 0:175ðORÞ0:376 and Pmin ¼ 0:257ðORÞ0:431 respectively for the offset jet and the dual jet for 3 6 OR 15. These correlations are obtained with a goodness of ﬁt of 99.74% and 99.27% respectively. For the offset jet, the location of maximum pressure coincides with the location of reattachment point, as noted by others also. But, the location of maximum pressure shifts a little downstream to axial location of merge point for the dual jet. Also, it is interesting to observe that maximum turbulent viscosity appears very near the location of the minimum pressure in the recirculation region for the offset jet. And, this location is also the location of vortex centre. But, this observation is not noticed

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16

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14

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OR=15 OR=11 OR=7 OR=3

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(c) Correlation for (Y0.5 )1scaled and (Y0.5 )2scaled

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Fig. 18. Jet half widths, ðY 0:5 Þ1 ; ðY 0:5 Þ2 , and ðY 0:5 Þ3 .

for the dual jet case where maximum value of turbulent viscosity is seen at two locations in the recirculation region; one maximum is observed near the merge point and the other near the location of upper vortex centre for the dual jet case. However, it should be stressed here that these two locations disappear as the offset ratio is decreased below 11 and the maximum value of turbulent viscosity appears at only one location, near the merge point for the corresponding offset ratio. The co-ordinate of reattachment point for offset jet is equal to (22.72, 0.0) and merge point for the corresponding dual jet ðOR ¼ 13Þ is (14.54, 2.82) (see Fig. 6(b)) and Table 2). It is clear from Fig. 17(a) that downstream of the reattachment point the ﬂuid experiences a favourable pressure gradient which accelerates the ﬂow. Upstream of this location, there is an adverse pressure gradient that is why the maximum streamwise velocity attains a minimum value at that location. It can be seen in Fig. 15(a) that there is an increase in the maximum streamwise velocity after the reattachment point till nearly X ¼ 32; after that, U max starts to decrease. Fig. 17(a) clearly indicates that favourable pressure gradient exists till nearly X ¼ 32. Also, it is noted that beyond this location axial pressure gradient is almost zero (not shown here) and there exists a lateral favourable pressure gradient. But, at the same time, it can be seen that the turbulent viscosity increases downstream of this location which is nearly X ¼ 32. This increase lowers down the value of maximum streamwise velocity because

of the increased viscous stress with the axial location, which is reﬂected in Fig. 15(a). But, on the other hand, we noted earlier that for the dual jet case instead of the three regions we observed only two regions in the U max plot. In Fig. 17(c), we can see that the maximum pressure occurs somewhere near X ¼ 16:5 and as noted earlier the axial location of merge point is 14.54 for this case. Therefore, the ﬂuid after the merge location experiences an adverse pressure gradient till X ¼ 16:5. Also, turbulent viscosity value is quite high near the merge point as compared to the value near the reattachment point for the corresponding offset jet (see Fig. 17(b) and (d)). These two effects do not allow U max to increase in the merging region and, hence, U max decreases with increase in the axial location after that. Jet half width is a measure of the spreading of a jet. Since the studied cases are that of the dual jet, consequently three jet half widths are identiﬁed as shown in Fig. 1(a); they are: the jet half width for the outer layer of the upper offset jet ðY 0:5 Þ1 , the jet half width for the inner layer of the upper offset jet ðY 0:5 Þ2 , and the jet half width for the outer layer of the lower wall jet ðY 0:5 Þ3 . Fig. 18 presents the development of jet half widths with respect to the axial location for OR = 3, 7, 11, and 15. For comparison purpose, the jet half widths for offset jets have also been put in the Fig. 18(a) and (b). It is clear that ðY 0:5 Þ1 and ðY 0:5 Þ2 of offset jets are always greater than their dual jets’ counterpart throughout the development process. Also, the deﬂection of inner shear layer

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A. Kumar / Computers & Fluids 114 (2015) 48–65

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Table 3 Value of total momentum ﬂux. OR

Present dual jet

Present offset jet

Tanaka [35]

Spall et al. [33]

3 5 7 9 11 13 15

0.745 0.712 0.683 0.661 0.642 0.631 0.614

0.753 0.749 0.744 0.737 0.730 0.726 0.725

0.956 0.927 0.897 0.864 0.839 0.810 0.781

0.828 0.776 0.747 0.721 0.701

is more than the deﬂection of the outer shear layer. This observation is consistent with the ﬁndings of Sawyer [32] and Nasr and Lai [25] who have observed a higher spreading rate on the convex side of a curved jet compared to that on the concave side. The deﬂection of the jet increases with the increase in offset ratio. This is quite obvious because with the increase in offset ratio the sub-atmospheric pressure in the recirculation region decreases. Also, the difference in deﬂection between the offset jet and the dual jet increases with the increase in offset ratio. Fig. 18(c) shows the variation of scaled jet half widths ðY 0:5 Þscaled and ðY 0:5 Þscaled with 1 2 the axial location for OR = 3, 5, 7, 9, 11, 13, and 15 for the case of dual jet. The jet half width is plotted till the axial location which is little downstream from the merge point for the corresponding case. The scaled jet half widths are deﬁned as below:

ðY 0:5 Þscaled 2

¼ ðY 0:5 Þ2 =OR

ð9Þ ð10Þ

The scaling is done to study the relative deﬂection of outer shear layer and inner shear layer of the upper offset jet of a dual jet. Because of the scaling, ðY 0:5 Þ1 starts with a value of 2 and ðY 0:5 Þ2 with a value of 1. As mentioned earlier, the deﬂection of inner shear layer is more than that of the outer shear layer and also the deﬂection increases with the increase in offset ratio. This ﬁgure also presents the correlated value which is obtained by the following correlations:

ðY 0:5 Þscaled ¼ 2 þ A1 X þ B1 X 2 1

ð11Þ

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ð12Þ

¼ 1 þ A2 X þ B2 X

2

where, A1 ¼ 2:012 102 þ 4:194 103 ðORÞ 1:776 104 ðORÞ2 , B1 ¼ 6:339 103 þ 5:039 104 ðORÞ 1:117 105 ðORÞ2 , A2 ¼ 7:63 102 þ 1:194 102 ðORÞ 4:632 104 ðORÞ2 ,

and

B2 ¼ 7:36 103 þ 3:903 104 ðORÞ 2:661 106 ðORÞ2 . It is noted that the variation is parabolic in nature. The effect of the upper offset jet on the growth of jet half width of the lower wall jet, ðY 0:5 Þ3 , is shown in Fig. 18(d). This ﬁgure also compares the present numerical result with the existing experimental result of Forthmann [8], Hsiao and Sheu [13], and

A. Kumar / Computers & Fluids 114 (2015) 48–65

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Lai and Lu [17]. The offset ratio considered for this ﬁgure is OR = 3, 7, 11, and 15. The result for a single wall jet is also presented for comparison. It can be seen that with increase in the offset ratio the variation approaches the variation of the single wall jet. The present numerical result compares reasonably well with the experimental result of Forthmann [8] for a single wall jet case. 3.5. Momentum ﬂuxes Fig. 19 shows the contribution of different terms in the momentum ﬂux calculation as deﬁned in the subsection of code validation. This ﬁgure corresponds to the dual jet case with an offset ratio of 3, 7, 11, and 15. For these offset ratios, the axial locations of merge point and combined point are 3.00, 8.41, 12.51, and 16.31 and 16.25, 16.85, 20.30, and 24.10 respectively (Table 2). The general trend of F u is: it follows the trend of U max till the potential core is consumed completely, then it decreases, assumes a minimum value near the location of combined point and again the value increases till it reaches a plateau. The variation of F p is entirely opposite to the variation of F u [16,30,33,35]. This is quite obvious as the decrease in momentum is compensated by an increase in the pressure. F p assumes a maximum value at the same instant when the F u assumes a minimum value and this is the location of maximum pressure in the domain. Most of the variation takes place prior to the combined region and this variation is found to increase with the increase in the offset ratio. But, the variation in ﬂux due to Reynolds stress is very little for the entire studied cases. It is interesting to observe that the value of total ﬂux F total is only decided by the budget from the velocity, F u , once the ﬂow proceeds downstream from the combined point. The integral constant for the entire cases is presented in Table 3. For comparison purpose, the results of Tanaka [35] and Spall et al. [33] are also included in the table. They have studied the two parallel plane turbulent jets. It is seen that the value of integral constant is lower for dual jet as compared to the value obtained for offset jet and the two parallel jets. This observation is consistent with the ﬁndings of others [33,35]. Also, it is observed that the value of integral constant decreases with the increase in offset ratio for the case of dual jet and the two parallel plane jets. This is due to the fact that with increase in the offset ratio the size of recirculation region increases, thereby reducing the turbulence which, in turn, results in the decrease of momentum ﬂux. However, there is almost insigniﬁcant decrease in the momentum ﬂux for the case of offset jet within the studied range of offset ratios. The possible reason is presented in Fig. 20, which shows the variation of ﬂuxes due to velocity and

pressure, as these two components decide the value of integral constant. It is interesting to observe that although increase in U max is more for the dual jet as compared to the offset jet (see Fig. 15), till the potential core is consumed, momentum ﬂux shows opposite trend. The increase in the momentum ﬂux F u , after issuing from the nozzle, is more for the offset jet than that of the dual jet. It should be noted here that for comparison purpose the values of momentum ﬂuxes of a dual jet are divided by 2. Fig. 20 tells that F u and F p remain almost unaffected with the change in OR for offset jets in the wall jet region while there is a signiﬁcant decrease in F u with the increase in the offset ratio for the case of a dual jet. The momentum ﬂux due to pressure is almost independent for the dual jet case, that is why F total of the offset jet remains almost constant while it decreases for the dual jet with the increase in the offset ratio. 4. Conclusion Two dimensional turbulence model is developed to study the ﬂow characteristics of a dual jet consisting of a wall jet and a parallel offset jet. For turbulence closure standard k–e model is utilised with a Reynolds number of 15,000. The offset ratio is varied between 3 and 15 with an interval of 2. A detailed analysis is carried out to compare the ﬂow characteristics of a single offset jet and the dual jet. The reattachment, merge, and combined points and the vortex centres are identiﬁed and correlations have been proposed for the axial and cross-streamwise locations for these points with a goodness of ﬁt not less than 99.5%. An exhaustive comparison with the existing experimental and numerical results in the literature is carried out for reattachment point, merge point, combined point, minimum sub-atmospheric pressure, decay of maximum streamwise velocity, jet half widths, and the momentum ﬂuxes. It has been noted that the presence of wall jet in addition to the parallel offset jet makes the offset jet deﬂect more towards the impingement wall and as a result the offset jet collides with the wall jet with high intensity. This intensity is found to decrease with increase in the offset ratio. Like offset jet shows wall jet characteristics in the wall jet region, dual jet also showed similar characteristics. The distance, after which the self-similar solution is obtained, is found to increase with the increase in offset ratio. The decay of maximum streamwise velocity for dual jet showed different characteristics than the offset jet. The jet deﬂection of a dual jet showed similar characteristics as that of a single offset jet with more deﬂection in the inner shear layer than the deﬂection in outer shear layer. The jet deﬂection pattern of the

A. Kumar / Computers & Fluids 114 (2015) 48–65

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