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Numerical investigation of steady and periodically unsteady flow for various separation distances between a wall jet and an offset jet Tanmoy Mondal, Manab Kumar Das n, Abhijit Guha Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, West Bengal 721302, India

a r t i c l e in f o

abstract

Article history: Received 16 May 2013 Accepted 12 July 2014 Available online 20 August 2014

The effects of separation distance between two jets in the near field region of turbulent dual jet flow consisting of a wall jet flow and an offset jet flow have been investigated using two-dimensional unsteady RANS equations. The unsteady fluid mechanics of two interacting jets have been illustrated by means of time series analysis of the instantaneous flow field. For a comprehensive understanding of the fluid mechanics, computation has been carried out at a large number of values of the ratio of separation distance (d) and jet width (w) for finely resolving the relevant range of d/w. Results show that when d/w lies in the range 0:7 r d=w r 2:1, the near field of the flow domain is characterized by a periodic large scale von Kármán-like vortex shedding phenomenon similar to what would be expected in the wake of a bluff body. On the contrary, for d=w r 0:6 and d=w Z 2:2, no periodic vortex shedding is detected; rather, a pair of steady counter rotating stable vortices are formed in between the two jets close to the nozzle plate. Fast Fourier transform of the velocity signals provides vortex shedding frequency corresponding to the Strouhal number that decreases with the increase in separation distance between the two jets. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Wall jet Offset jet Unsteady RANS Periodic vortex shedding Strouhal number

1. Introduction Turbulent jets have attracted considerable research attention due to their intriguing physics as well as many engineering applications. The theory of turbulent jets has been thoroughly discussed by Abramovich (1963). Rajaratnam (1976) has provided comprehensive experimental and theoretical treatments of the characteristics of various types of turbulent jets. The present study involves a wall jet and an offset jet. A wall jet is generated when a fluid is discharged from a nozzle tangentially along a solid wall. Therefore, the wall jet is bounded by a solid wall on one side and the quiescent fluid on the other. Forthmann (1936) has presented the first authoritative analysis of turbulent wall jet. Since then many studies have been carried out on various aspects of turbulent wall jet flow and the subject continues to be an active area of research up to the present time; see, for example, the works of Glauert (1956), Myers et al. (1963), Chandrasekhara Swamy and Bandyopadhyay

n

Corresponding author. Tel.: þ91 3222 282924; fax: þ 91 3222 282728. E-mail address: [email protected] (M.K. Das).

http://dx.doi.org/10.1016/j.jfluidstructs.2014.07.009 0889-9746/& 2014 Elsevier Ltd. All rights reserved.

T. Mondal et al. / Journal of Fluids and Structures 50 (2014) 528–546

Nomenclature Roman symbols c1ϵ ; c2ϵ cμ d f G I k N P p Re St T t Ui ui uτ w Xi xi

constants in the model equation for ϵ turbulent viscosity constant of the k ϵ model separation distance between two jets (dimensional) frequency (dimensional) production of turbulent kinetic energy (non-dimensional) turbulence intensity turbulent kinetic energy (dimensional) number of grids non-dimensional pressure dimensional pressure Reynolds number ðu0 w=νÞ Strouhal number ðfw=u0 Þ time period (non-dimensional) time (dimensional) non-dimensional Cartesian mean velocity components (U, V) dimensional Cartesian mean velocity components (u, v) friction velocity (dimensional) nozzle width (dimensional) non-dimensional Cartesian coordinates (X, Y) dimensional Cartesian coordinates (x, y)

Δτ δ ϵ

ν ρ σk; σϵ τ ω

529

time step size (non-dimensional) dimensional wall boundary layer thickness rate of dissipation of turbulent kinetic energy (dimensional) laminar kinematic viscosity (dimensional) fluid density (dimensional) turbulent Prandtl numbers for kinetic energy and dissipation, respectively time (non-dimensional) vorticity (non-dimensional)

Subscripts 0 0.5 cp i; j m mp n t x y

inlet, ambient half of maximum combined point indices maximum quantity merging point non-dimensional quantity turbulent x-direction y-direction

Overbar ðÞ

time averaged quantity

Greek symbols

Δ

non-dimensional wall boundary layer thickness

(1975), Launder and Rodi (1981), Abrahamsson et al. (1994), Gogineni and Shih (1997), Eriksson et al. (1998), George et al. (2000), Dejoan and Leschziner (2005) and Agelin-Chaab and Tachie (2011b). An offset jet refers to a jet whose axis, at the inlet of the flow domain, is offset by a distance from the solid wall, similar to the flow over a backward facing step or that in a sudden expansion. A low pressure region where the pressure is lower than the ambient forms (near the equivalent of the backward facing step) underneath the offset jet; as a result, the jet deflects towards the solid wall and eventually attaches to the surface. In the literature (Nasr and Lai, 1997), this point is usually referred to as the reattachment point. The flow has a strong interaction with the solid wall in the reattachment region which occurs downstream of the reattachment point. Further downstream, in the wall jet region, the flow continues to develop and displays the characteristics of the classical wall jet flow described in the previous paragraph. The details of the mean flow and turbulence characteristics of an offset jet flow have been studied by several authors, e.g., Bourque and Newman (1960), Sawyer (1960, 1963), Rajaratnam and Subramanya (1968), Hoch and Jiji (1981), Pelfrey and Liburdy (1986b,1986a), Koo and Park (1992), Gu (1996), Gao and Ewing (2007) and Agelin-Chaab and Tachie (2011a). While several studies have been conducted on either a wall jet or an offset jet separately, only a modest number of literature are available on the interaction of two jets. The most common topic in the latter category is the study of two parallel jets. In this flow configuration, two turbulent jets are issued parallel to each other from two nozzles which are set on a common end wall. As a result of mutual entrainment, the two jets deflect to each other and merge together at the merging point (mp); the region between the nozzle exit and the merging point is known as the converging region. This is followed by the merging region which extends up to the combined point (cp). In this region, vigorous interaction of the two jets takes place. The two jets combine at the combined point, develop as a single jet in the combined region, and eventually display the characteristics of the free jet at a far downstream position. Miller and Comings (1960), Tanaka (1970, 1974), Lin and Sheu (1990, 1991), Nasr and Lai (1997), Anderson and Spall (2001), Anderson et al. (2003), Spall et al. (2004) and Bunderson and Smith (2005) have studied the flow and turbulence characteristics of two parallel jets. Only two studies (Anderson et al., 2003; Bunderson and Smith, 2005) out of the cited references report the unsteady behavior of two parallel jets, although the nature of the unsteadiness is entirely different in both the cases. Anderson et al. (2003) have found that the near field region is characterized by a periodic vortex shedding phenomenon similar to what would be

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expected in the wake of a bluff body. The Strouhal number (St) based on the vortex shedding frequency increases from 0.243 to 0.334 as the ratio d/w is increased from 0.5 to 1.67. They have attributed this unsteady phenomenon to the confining effect of the outer shear layers of two jets – as d/w increases, for a fixed value of d, the outer shear layers are brought more close to the vortex shedding region between the two jets resulting in an increase of the vortex shedding frequency. Bunderson and Smith (2005), on the other hand, have observed periodic oscillation (termed as “flapping” in the literature) of two jets columns for relatively larger separation distances between the two jets (d/w¼7–25). The Strouhal number corresponding to the flapping frequency decreases from 0.18 to 0.1 as the ratio d/w is increased from 7 to 25. While a modest pool of literature exists on the topic of two parallel jets as discussed above, only four literatures are traceable on the area of combined wall jet and offset jet flow – the subject matter of the present study. Wang and Tan (2007) have presented the first work in the area of combined wall jet and offset jet flow with remarkable results. Using particle image velocimetry (PIV), they have observed a periodic vortex shedding phenomenon in the near field region of this flow (termed as dual jet flow in the literature) similar to what would be expected in the wake of a bluff body. They have conducted the experiment for a fixed separation distance between two jets (d/w¼1). A few years later, Kumar and Das (2011) have numerically simulated the dual jet flow using Reynolds-averaged Navier–Stokes (RANS) equations for a relatively larger value of separation distance (d/w¼8). Their results focus on the mean flow and turbulence characteristics of dual jet. Later on, the work of Wang and Tan (2007) has been revisited by Li et al. (2011) using large eddy simulation (LES). They have described the unsteady interaction between the wall jet and the offset jet by means of statistical analysis. The results also include mixing characteristics of the two jets, but do not contain any information associated with the vortex shedding phenomenon. In another study, Li et al. (2012) have investigated the interaction of two jets for various turbulence models. Additionally, they have shown the effects of different combinations of the inlet velocity ratio and separation distance on the mean flow and mixing characteristics of a dual jet using a particular turbulence model (realizable k ϵ model). They have presented only the steady state results. A thorough literature review on the combined wall jet and offset jet flow indicates that except the work of Wang and Tan (2007), none of the previous studies report the periodic vortex shedding phenomenon. However, the study of Wang and Tan (2007) is limited to a fixed value of d/w (i.e., d/w ¼1). Although Li et al. (2012) have considered various values of d/w, they have shown the results of two interacting jets in the form of steady state solutions. The objective of the present work is to establish the range of d/w for which periodic unsteadiness arises in the flow field and to computationally study the details of the flow physics. To the best of the authors’ knowledge, this would be the first CFD solutions of the unsteady interaction of a wall jet and an offset jet, involving vortex shedding, for variable separation distances between the two jets. For this purpose, unsteady Reynolds-averaged Navier–Stokes (URANS) equations are used to simulate the dual jet flow. The computation is carried out at a large number of values of d/w to resolve the range of d/w on a finer scale. Total 14 cases of d/w (i.e., d/w¼0.5, 0.6, 0.7, 0.8, 1, 1.3, 1.5, 1.8, 2, 2.1, 2.2, 2.3, 2.4 and 2.5) are computed. Each simulation (i.e., for a particular value of d/w) takes approximately 168 h of CPU time on an Intel core i7 3.06 GHz machine on the Linux computing platform. 2. Problem description Schematic diagram of the dual jet flow is shown in Fig. 1. Two plane, incompressible and turbulent jets with identical inlet velocity (u0) and width (w) are issued into a quiescent ambient. Two two-dimensional (2D) nozzles are separated by a distance d. The origin of the coordinate system is located at the intersection of the jet exit plane (y-axis) and the bottom wall

y Converging region

Merging region

Potential core

w

u0

Jet centre−line

0.5 u m

um Recirculation zone

d

mp

w

u0

Combined region

cp

0.5 u m 0.5 u m

um

y3 y2

y1

0.5 0.5

0.5

Wall boundary layer

δ

x Fig. 1. Schematic diagram of a plane turbulent dual jet comprising a wall jet and an offset jet (adapted from Wang and Tan, 2007).

T. Mondal et al. / Journal of Fluids and Structures 50 (2014) 528–546

531

(x-axis). As a result of the mutual entrainment between the two jets, a lower pressure zone (where the pressure is below the atmospheric pressure) is created close to the nozzle plate. This lower pressure zone causes the individual jet to deflect towards each other and then merge together at the merging point (mp); the region between the jet exit to the merging point is known as the converging region. Downstream of the merging point, in the merging region, two jets continue to interact and combine together at combined point (cp). The streamwise distances from the nozzle plate to the merging point and combined point are referred to as the merging length (Xmp) and combined length (Xcp), respectively. Further downstream, in the combined region, the combined jet displays the characteristics of a single wall jet flow. Due to the flow complexity, several shear layers are formed in the dual jet flow configuration, such as Layer 1 (outer layer of offset jet), Layer 2 (inner layer of offset jet) and Layer 3 (inner layer of wall jet). Besides, unlike conventional unbounded parallel jets where the outer shear layers are freely developing, a wall boundary layer exists in the dual jet flow because of the presence of the bottom wall. To illustrate the jet development, three jet half-widths y10:5 , y20:5 and y30:5 , respectively, for Layer 1, Layer 2 and Layer 3 are denoted in Fig. 1. The wall boundary layer thickness δ is also shown in the same figure. The jet half width y0:5 is defined as the vertical distance from the bottom wall to the location where the streamwise velocity u is half of its local maximum um and δ is the location where u ¼0.99um. 3. Mathematical formulation The flow is assumed to be unsteady, two dimensional, turbulent and the fluid is incompressible. The body forces are neglected and the fluid properties are assumed to be constant. The unsteady Reynolds-averaged Navier–Stokes (URANS) equations are used for predicting the turbulent flow. The Boussinesq hypothesis is incorporated to relate the Reynolds stresses to the mean velocity gradients. The high Reynolds number two-equation k ϵ turbulence model is used for the Reynolds stresses. Following Biswas and Eswaran (2002), the governing equations are expressed in non-dimensional form. The dimensionless variables are defined as follows: Ui ¼

ui ; u0

Xi ¼

xi ; w

τ¼

t w=u0

P¼

p p0 ; ρu20

kn ¼

k ; u20

ϵn ¼

ϵ

; u30 =w

ν νt;n ¼ t : ν

The non-dimensionalized equations are expressed in terms of indicial notation: Continuity equation: ∂U i ¼ 0: ∂X i

ð1Þ

Momentum equations: ∂U i ∂U i ∂ðU j U i Þ ∂ 2 1 ∂ þ ¼ P þ kn þ 1 þ νt;n : ∂X i 3 Re ∂X j ∂τ ∂X j ∂X j Turbulent transport equations (ϕ ¼ kn, ϵn): ∂ϕ ∂ðU j ϕÞ 1 ∂ νt;n ∂ϕ þ ¼ þ S1 þ S2 ; Re ∂X j σ ∂X j ∂τ ∂X j

ð2Þ

ð3Þ

where σ, S1 and 8 respectively for kn equation > < σ k ; G and ϵn ; ϵn ϵ2n S2 ¼ > : σ ϵ ; C 1ϵ k G and C 2ϵ k ; respectively for ϵn equation: n n Production: νt;n ∂U i ∂U j ∂U i G¼ þ : Re ∂X j ∂X i ∂X j Eddy viscosity: k

2

νt;n ¼ C μ Re n ϵn According to the indicial notation used above, X, Y, U and V are synonymous with X1, X2, U1 and U2, respectively. The model constants are given as follows: σ k ¼ 1:0; σ ϵ ¼ 1:30; C 1ϵ ¼ 1:44; C 2ϵ ¼ 1:92 and C μ ¼0.09 (Launder and Spalding, 1974). 4. Numerical schemes, methods of solution and boundary conditions In order to solve the above governing equations an in-house computational program is developed (written in C language). Following Patankar (1980), the governing equations are discretized using the Finite Volume Method (FVM). The central difference scheme is used to discretize diffusive terms and the power-law upwind scheme is used to discretize the

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T. Mondal et al. / Journal of Fluids and Structures 50 (2014) 528–546

convective terms to ensure the stability of the solution. To avoid the fine mesh required to resolve the viscous sub-layer near the solid boundary, the wall function method of Launder and Spalding (1974) which is appropriate for the high Reynolds number flows is employed. The SIMPLE algorithm of Patankar (1980) is followed to couple the velocity and pressure equations. The temporal term is discretized using a first-order (backward) differencing fully implicit scheme given in Versteeg and Malalasekera (2010) with the optimized time step size Δτ ¼ 0:01. All the variables ðU i ; P; kn ; ϵn Þ are initialized (at τ ¼0) with zero (except at the boundaries where the appropriate boundary conditions are applied). The computed data are stored at every alternate time step for post-processing. An under-relaxation factor of 0.2 is used for the pressure correction equation. The detailed discretization of the equations for turbulent kinetic energy (kn) and the rate of dissipation (ϵn) is performed following Biswas and Eswaran (2002). The computational results are considered to be converged when the mass residual is smaller than 1 10 6 for continuity equation and 1 10 4 for other equations. The residual is defined as the square root of the summation of the squares of the difference between right and left side of the discretized equations for a single control volume (Doormaal and Raithby, 1984). The summation is performed over all the control volumes in the computational domain. All the computations are conducted on an Intel core i7 3.06 GHz machine on the Linux platform. A typical run with Δτ ¼ 0:01 takes about 12 h of CPU time per 104 time-steps (which, according to Fig. 17(a), corresponds to about 10 vortex shedding cycles). The size of the computational domain is taken as 60 30 (along X, Y directions, respectively). The Reynolds number based on the jet width w at inlet is taken as 10 000. At inlet, both jets have the same non-dimensional streamwise velocity U0 ¼1, whereas the transverse velocity is set to zero for each jet. For the turbulent kinetic energy equation, the boundary condition at the inlets is set as kn ¼ 1:5I 2 , where I is the turbulence intensity and equals 0.05 (Biswas and Eswaran, 2002). 3=2 For the rate of dissipation equation, the inlet boundary condition is set as ϵn ¼ ðkn C 3=4 μ Þ=l (Biswas and Eswaran, 2002), where l ¼0.07 is considered. The no-slip boundary condition is adopted for the solid walls and the Neumann boundary condition ð∂ϕ=∂Y ¼ 0Þ is provided for the top boundary (entrainment side). At the exit boundary, a developed condition of ∂ϕ=∂X ¼ 0 is considered, where ϕ ¼ U i , kn and ϵn. It is ensured that the first grid point near the solid wall falls in the logarithmic region, i.e., 30 oY þ o 100, where Y þ ¼ yuτ =ν and uτ is the dimensional friction velocity. 5. Grid independence test and time step sensitivity study For grid independence test, three sets of grid density ðNx Ny Þ of 171 131 (coarse grid), 221 181 (medium grid) and 271 231 (fine grid) are considered. Here, Nx and Ny are the number of grids used in X and Y directions, respectively. Therefore, the total number of nodes for coarse, medium and fine grid densities are 22 401, 40 001 and 62 601, respectively. The results of the grid independence test based on these three grid densities are shown in Fig. 2 for the flow configuration of d/w¼1. Fig. 2 shows the mean streamwise velocity ðU Þ profile at a fixed streamwise location of X ¼25 for each grid density. It is evident from Fig. 2 that the change in the solutions becomes minimum as the grid density is refined from medium to fine. It can therefore be said that a grid independent solution may be said to be achieved for the grid density of 221 181. A typical layout of the adopted grid density (221 181) is shown in Fig. 3. Along the Y direction, in the region of wall jet, offset jet and intermediate region between the two jets, an equal number of grids (Ny ¼20 nodes) are used. Above the offset jet, the remaining grids are non-uniformly distributed with the expansion ratio of 1.0212. Along the X direction, near to the vertical wall, relatively dense grids (Nx ¼ 40 nodes) with equal spacing are provided up to X ¼2 and X ¼2 onwards, nonuniform grids are located with the expansion factor of 1.0171. Geometric expansion scheme ∑ ¼ að1 r N Þ=ð1 rÞ is used for 12

10

171x131 221x181 271x231

Y

8

6

4

2

0

0

0.2

0.4

0.6

0.8

1

U Fig. 2. Mean streamwise velocity profile of dual jet flow with d/w¼ 1 at X¼ 25 for different grid densities.

T. Mondal et al. / Journal of Fluids and Structures 50 (2014) 528–546

533

30

25

Y

20

15

10

5

0

0

10

20

30

40

50

60

X Fig. 3. Grid layout of the computational domain for dual jet flow with d/w¼1.

1

0.8

0.6

Δτ = 0.005 Δτ = 0.01 Δτ = 0.02

U

0.4

0.2

0

-0.2

-0.4 0

10

20

30

40

50

X Fig. 4. Mean streamwise velocity along the mid-plane of two jets (Y ¼1.5) of dual jet flow with d/w¼ 1 for different time step sizes.

generating non-uniform grids, where a is the size of the first grid, r is the geometric progression ratio for grid expansion, ∑ is the domain size and N is the number of grid in the domain. The time step sensitivity study is conducted using three different time step sizes of Δτ ¼ 0:005, 0.01 and 0.02 with the successive increment of the time step sizes by a factor of 2. Fig. 4 shows the variation of the mean streamwise velocity U along the mid-plane of two jets (Y¼ 1.5) for these three time step sizes. The computational results obtained for the time step sizes of Δτ ¼ 0:005 and 0.01 produce nearly close solutions indicating that there is no more change in the solution possible as the time step size becomes smaller than 0.005. For further analysis, absolute errors of the Strouhal numbers are calculated as the time step size increases from 0.005 to 0.01 and then from 0.01 to 0.02. It is found that absolute error of the Strouhal number (St) is 1.7% when the time step size Δτ ¼ 0:01 is halved ðΔτ ¼ 0:005Þ and 3% when it is doubled ðΔτ ¼ 0:02Þ. Therefore, to optimize the CPU resources with an acceptable level of accuracy, the remainder of the computations is carried out with the medium grid density (221 181) and time step size Δτ ¼ 0:01.

6. Validation of the computational results To assess the accuracy of the numerical solutions, computational results of the present study are compared with the experimental results (PIV) of Wang and Tan (2007) and computational results (LES) of Li et al. (2011) for dual jet flow with

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T. Mondal et al. / Journal of Fluids and Structures 50 (2014) 528–546

d/w¼1. Additionally, a comparison is shown between the present computational results and the experimental results (PIV) of Wang and Tan (2007) and computational results (LES) of Dejoan and Leschziner (2005) for the wall jet flow. Fig. 5 shows the variation of maximum mean streamwise velocity U m with the downstream distance X for dual jet flow. As seen, in the immediate downstream of the jet exit, the rate of decay of U m is slightly greater for the present computational results as compared to the other results up to X¼3. In the merging region, 5 rX r15, the simulated data agree well with the experimental data, although deviation is observed from X ¼20 onwards. Fig. 6 illustrates the rate of jet growth over the range X¼ 0–25 in terms of the half widths Y 10:5 , Y 20:5 , Y 30:5 and the wall boundary layer thickness Δ, as denoted in Fig. 1 in terms of the dimensional form. According to the present computational results, Y 10:5 decreases from X¼0 up to X¼3 and then increases gradually with the downstream distance; Y 20:5 and Y 30:5 vary monotonically with the downstream distance until they converge together at X¼2.8. It appears evident from Fig. 6 that the predicted variations of the jet half widths are found to be close to the corresponding experiment in most respects. For further comparison, the computed profiles of U =U m are plotted in Fig. 7 along with the experimental results of Wang and Tan (2007) at various downstream locations of the combined point i.e., X¼20, 25 and 30. As seen in Fig. 7, the computed profiles of U =U m exhibit self-similarity and compare well with the experimental results in the range 0 o Y=Y 0:5 r ¼ 1:1. Near the outer edge, Y=Y 0:5 41:1, the match between the computed and experimental profiles deteriorates. In this region, both computation and experiment show that the velocity profiles are not fully self-similar. Wygnanski et al. (1992) have attributed this lack of selfsimilarity to the effect of entrained flow or counter flow.

1.1

1

PIV (Wang and Tan, 2007) LES (Li et al., 2011) k-ε (present)

0.9

Um

0.8

0.7

0.6

0.5

0.4 0

5

10

15

20

25

X Fig. 5. Variation of U m with downstream distance X for dual jet flow with d/w¼ 1.

4 3.5

1 Y0.5

3

Y0.5

2.5 PIV (Wang and Tan, 2007) LES (Li et al., 2011) k-ε (present)

2

Y 0.5

2 1.5

3

Y0.5

1 0.5 0

0

5

10

15

20

25

X Fig. 6. Development of Y 10:5 , Y 20:5 , Y 30:5 and Δ with downstream distance X for dual jet flow with d/w¼1.

T. Mondal et al. / Journal of Fluids and Structures 50 (2014) 528–546

2

1.6

X=20 X=25 X=30 X=20 X=25 X=30

} }

0.6

0.7

535

k-ε (present) PIV (Wang and Tan, 2007)

Y/Y0.5

1.2

0.8

0.4

0 -0.1

0

0.1

0.2

0.3

0.4

0.5

0.8

0.9

1

1.1

U/Um Fig. 7. Profiles of U =U m at various downstream locations of the combined point for dual jet flow with d/w¼ 1.

1.1

1

0.9

Um

0.8

0.7

PIV (Wang and Tan, 2007) LES (Dejoan and Leschziner, 2005) k-ε (present)

0.6

0.5

0.4 0

5

10

15

20

25

X Fig. 8. Variation of U m with downstream distance X for a single wall jet flow.

In case of the wall jet flow, the variation of maximum mean streamwise velocity U m with the downstream distance X is shown in Fig. 8. The computational results are found in a good agreement with the corresponding experimental results showing almost equal length of the potential core (a downstream distance from the jet exit to the location where jet centreline velocity remains to be constant), but variation is noticed downstream of X¼13. In the range 10 r X r 25, the present computation shows that U m decays according to the power law U m p X 0:52 ; Rajaratnam (1976) has reported this exponent value as 0.5 for the plane turbulent wall jet flow. It should be noted that Dejoan and Leschziner (2005) have conducted the LES with Re¼ 9600 which is close to Re¼10 000 used both in the work of Wang and Tan (2007) and the present computation for the wall jet flow.

7. Results and discussion 7.1. Mean velocity vectors Mean velocity vectors demonstrate the spatial development of the mean flow. Fig. 9 shows the mean velocity vectors of the dual jet flow for d/w¼1. The converging, merging and combined regions are evident from Fig. 9. The mean streamwise velocity is negative close to the nozzle plate indicating the flow reversal and the presence of a recirculation zone. The locations of the merging point (mp) and combined point (cp) are identified in the flow field. The merging point is a point

536

T. Mondal et al. / Journal of Fluids and Structures 50 (2014) 528–546

8

Xmp=1.54, Xcp=16.72

Y

6

cp

mp

4

2

0

0

5

10

15

20

25

30

X Fig. 9. Mean velocity vectors of dual jet flow with d/w¼1.

where the mean streamwise velocity ðU Þ is zero. The combined point corresponds to a point downstream of the merging point where two peaks of U velocity disappear to a single peak and U velocity attains its maximum. The mean velocity vectors are also obtained for d/w¼0.5, 1.5, 2 and 2.5, but they are not shown in the present communication due to the brevity. The merging length is found to be Xmp ¼0.68, 1.54, 2.29, 3.19 and 4.03 for d/w ¼0.5, 1, 1.5, 2 and 2.5, respectively. The computed value of Xmp is reasonably comparable with Xmp ¼0.75 in Wang and Tan (2007) for d/w ¼1, Xmp ¼1.75 in Ko and Lau (1989) for d/w¼1.5 and Xmp ¼2.97 in Spall et al. (2004) for d/w¼2. For visual inspection, these results are plotted in Fig. 10. As seen, for relatively smaller values of d/w, the lesser amount of entrainment results in a much shorter merging length. A best fit curve of Xmp versus d/w in the range of d/w¼0.5 to 2.5 gives a linear relationship: Xmp ¼ 0.16 þ1.67(d/w). The combined lengths for d/w¼0.5, 1, 1.5, 2 and 2.5 are found to be Xcp ¼16.15, 16.72, 17.31, 17.92 and 18.54, respectively. In comparison, these values of Xcp are considerably larger than Xcp ¼6.4 in Wang and Tan (2007) and Xcp ¼10.5 in Ko and Lau (1989). No physical reason could be ascribed to this fact. However, it may be mentioned to this connection that Lai and Nasr (1998) have found a similar fact in a study of two parallel jets. Compared to the measured combined length of Xcp ¼8, their computational results using standard k ϵ, RNG and RS models give Xcp ¼ 19, 18.17 and 17.33, respectively, for d/w¼ 3.25. No further explanation has been given in the literature of Lai and Nasr (1998). In another study of two parallel jets, Anderson and Spall (2001) have found a little variation between the computational (using standard k ϵ and RSM models) and experimental values of Xcp for a fixed d/w. These values of Xcp, however, vary significantly when they are compared with the reported values (experimental) of the other investigators for the same value of d/w. They have attributed this discrepancy to the facility dependent phenomenon, for example, the plenum conditions upstream of the nozzle. According to their literature, the possible location of the combined point depends on the initial spreading rate of each jet which is sensitive to the facility dependent influence. Note that Ko and Lau (1989) have considered the inlet turbulence intensity as 5%, which is same as the present study. However, the value of inlet turbulence intensity is nowhere mentioned in the literature of Wang and Tan (2007).

7.2. Instantaneous streamline patterns After issuing from the nozzles, two jets merge together in the converging region forming a recirculation zone between them, as indicated in Fig. 1. The present study reveals that the flow features of the recirculation zone depend on the separation distance between two jets (d/w). With the help of instantaneous streamline patterns, the flow features of the recirculation zone are described for various values of d/w. Fig. 11(a) shows a closed recirculation zone comprising a pair of stable vortices formed one above the other for d/w ¼0.6. When the value of d/w is slightly increased to 0.7, on the contrary, this closed recirculation zone opens up and becomes absolutely unstable (Fig. 11(b)) via Hopf bifurcation – a phenomenon of transition from a stationary state to a time dependable state. A flow is termed as absolutely unstable if an impulsively generated, small amplitude transient grows exponentially at the location of its generation (Williamson, 1996). As a result, an unsteady time-periodic flow phenomenon occurs close to the nozzle plate similar to the periodic vortex shedding phenomenon found in the near wake region of a bluff body. Likewise, if the value of d/w is progressively increased, according to the present computational results, this periodic phenomenon is discernible up to d/w¼2.1, as shown in Fig. 12(a). With further increase of d/w to 2.2 (Fig. 12(b)), the vortex shedding phenomenon gradually diminishes with time and eventually the recirculation zone becomes stable similar to the case of d/w¼ 0.6.

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6 Experiment on parallel jets (Ko and Lau, 1989) Computation on parallel jets (Spall et al., 2004) Experiment on dual jet (Wang and Tan, 2007) Present computation Best fit curve (linear)

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Fig. 13(a)–(d) demonstrates a sequence of instantaneous streamline patterns for d/w ¼1.5 within a time period T (nondimensional) at an interval of T/4. A clockwise rotating vortex is shed at T/2 from the topside of the nozzle plate followed by an anti-clockwise rotating vortex is shed at T from the bottom side of the nozzle plate. While the clockwise rotating vortex shedding from the top side (Fig. 13(a) and (b)), the fluid is drawn up into the recirculation zone from below the nozzle plate. The anti-clockwise vortex that forms on the bottom side of the nozzle plate continues to grow in size until the fluid from the opposite side is incorporated into the vortex to initiate the shedding process (Fig. 13(c) and (d)). 7.3. Instantaneous vorticity contours To explain the underlying flow physics of periodic vortex shedding phenomenon, the instantaneous vorticity contours are constructed. For the present 2D flow, the vorticity (ω) is computed using the expression ω ¼ ð∂V=∂X ∂U=∂YÞ. Fig. 14(a)–(d) illustrates the instantaneous vorticity contours during a time period T at an interval of T/4 for d/w¼1.5. As depicted, the large scale von Kármán-like discrete vortices (both positive and negative) are formed in the interacting region of two jets. These discrete vortices are recognized by the regions of isolated and concentrated vorticity in the instantaneous flow field. Within time period T, an upper row of negative vortices (labeled as B) and a lower row of positive vortices (labeled as A) are alternately grow, shed and convected downstream forming a well-organized vortex street. According to Fig. 14(a) and (b),

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vortex B in the upper row is seen to grow at T/4 but is detached at T/2; similarly, according to Fig. 14(c) and (d), vortex A in the lower row is seen to grow at 3T/4 but is detached at T. This phenomenon repeats itself in the successive time periods. It is noted that although the discrete vortices exist in the far downstream region of the flow field, the strength of the vortices decays with the downstream distance due to the effect of viscosity, which can be appreciated from the vorticity contour values. The von Kármán-like vortices are generated because of the mutual interaction of two inner shear layers i.e., Layer 2 and Layer 3. When the offset jet separates at the top edge of the nozzle plate, Layer 2 is formed. Similarly, the wall jet separates at the bottom edge of the nozzle plate resulting in the formation of Layer 3. Each of the shear layers rolls up into vortex due to the continuous feeding of vorticity generated at the separation point. Vortices B and A are generated due to the roll up of Layer 2 and Layer 3, respectively. These two vortices eventually interact with each other such that one vortex grows faster than the other. The larger vortex B ðω o0Þ grows in size and strength until it draws the opposing vortex A ðω o0Þ across the mid plane of two jets. As a result, the opposing vortex A cuts off the further supply of the vorticity to vortex B. This is the instant when vortex B is shed. Being a free vortex, vortex B is then convected away towards the downstream direction. Following the shedding of vortex B, a new vortex C forms at the same side of the nozzle plate. Vortex A plays the same roll as vortex B: it grows in size and strength before it becomes strong enough to draw vortex C. The above sequence of fluid flow phenomena repeats itself generating the alternate shedding of vortices from the two sides of the nozzle plate. 7.4.

λ2 contours

The vorticity contours alone may not reveal all the flow features. For example, the study of vorticity contours cannot distinguish the swirling regions from the shearing regions. Therefore, the construction of λ2 contours is required to capture swirling regions in the flow field. A swirling region appears for non-real eigenvalues of the gradient matrix of velocity (g). In the X–Y plane, this gradient matrix can be expressed in terms of a second-order tensor ! ∂U=∂X ∂U=∂Y g¼ : ∂V=∂X ∂V=∂Y The discriminant λ2 of non-real eigenvalues of g separates a swirling region from a shearing region. It is obtained from λ2 ¼ ðtrace gÞ2 4 det g ¼ ð∂U=∂X þ ∂V=∂YÞ2 4ð∂U=∂X ∂V=∂Y ∂U=∂Y ∂V=∂XÞ (Vollmers, 2001). The swirling region exists in the core of the vortices. In the swirling region, λ2 is negative; on the contrary, in the shearing region, λ2 is positive. Fig. 15 shows the contours of λ2 at a time instant 3T/4 during a time period T of the periodic vortex shedding. The same vortices A and B, shown in the vorticity contour plot of Fig. 14(c) at the same instant for which Fig. 15 is drawn, are identified in the swirling region ðλ2 o 0Þ of the λ2 contours plot of Fig. 15. Fig. 15, however, also identifies the shearing regions (λ2 4 0) in addition to the swirling regions. According to the vorticity contours plots shown in Fig. 14, the outer shear layer of the offset jet (Layer 1) consists of a row of positive vortices along Y 3. These vortices are identified in the form of swirling regions in Fig. 15. Fig. 15 also shows the presence of a shearing region between two adjacent swirling regions along Y 3. These facts indicate the Kelvin–Helmholtz roll up of the outer shear layer of the offset jet. Fig. 14 shows negative vorticity in the wall boundary layer region ðY o0:5Þ. In this region, Fig. 15 shows three discrete swirling regions attached to the bottom wall at XE1, 2.5 and 5.5, respectively. Although the shearing regions are visible in the wall boundary layer region, they are considerably smaller than the swirling regions.

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7.5. Velocity signals To perform the time series analysis, the time series data of streamwise (U) and transverse (V) velocity components are collected over a successive time interval of Δτ ¼ 0:02 at a stationary point (X¼Xmp, Y¼1 þd/w) in the near flow field region for a given d/w. This location corresponds to a point located at a streamwise distance of the merging point straight downstream of the inner lip of the offset jet nozzle. The location of the stationary point is chosen based on the following facts. For a given d/w, the present study reveals that (figure not shown) the transverse component of the normal Reynolds stress reaches its maximum value at the merging point (Xmp, Ymp), a result consistent with the findings of Nasr and Lai (1997) for two parallel jets. Along Y¼Ymp, the streamwise velocity signal has two dominant frequency peaks: one is the vortex shedding frequency and another is twice the vortex shedding frequency. However, along Y¼1 þd/w, the streamwise velocity signal has a single dominant frequency peak which is the vortex shedding frequency. Fig. 16(a)–(d) displays the time series of U and V velocity signals for d/w¼0.6, 0.7, 2.1 and 2.2, respectively. Except the cases of d/w¼0.6 and 2.2, for the other two cases (i.e., d/w¼ 0.7 and 2.1), both the velocity signals exhibit a trend of sinusoidal oscillation. It is recalled that the instantaneous streamline plots display periodic vortex shedding for d/w¼0.7 and 2.1. Therefore, it can be concluded that the periodic vortex shedding phenomenon is responsible for the sinusoidal oscillation of the near flow field. On the other hand, for d/w ¼0.6, two velocity signals reach to a steady state condition after a small initial transience, as shown in Fig. 16(a). In the case of d/w¼2.2, the velocity signals undergo a sustainable underdamped oscillation before reaching to a steady state behavior, as depicted in Fig. 16(d). A closer look on Fig. 16(b) reveals that

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U velocity signal is formed by the superposition of two sinusoidal signals, whereas V velocity signal is purely sinusoidal. For d/w¼2.1 (Fig. 16(c)), both the velocity signals are purely sinusoidal, but the peak to peak amplitude of each signal is smaller than that for d/w¼0.7. Fig. 17(a) and (b), respectively, shows the velocity signals at a point in the near field (X¼2.29, Y¼ 2.5) and far field (X¼9, Y¼ 2.5) regions of the dual jet flow for d/w¼1.5. For any given Y, X¼9 is close to the mid-point between X ¼2.29 (streamwise co-ordinate of the merging point) and X¼17.31 (streamwise co-ordinate of the combined point). In the far field region (Fig. 17(b)), U and V velocity signals display sinusoidal oscillation similar to that in the near field region (Fig. 17(a)), but the peak to peak amplitude of both the signals is significantly reduced. The presence of unsteadiness in the far field region signifies the existence of large scale von Kármán-like vortices, which can be appreciated from the vorticity contours plots (Fig. 14). Fig. 17(b) also shows that the peak to peak amplitude of U velocity signal is greater than that of V velocity signal. This is due to the fact that the flow is mainly dominated by the U component of velocity in the far field region. 7.6. Fast Fourier transform Fast Fourier transform (FFT) is a faster version of the discrete Fourier transform (DFT) by which a time domain signal can be converted into a frequency domain signal. Fig. 18(a) and (b), respectively, displays the FFT of U and V velocity signals for d/w¼0.7, 1, 1.5 and 2.1. Because U and V velocities are non-dimensional quantities, the FFT of each velocity signal gives the

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non-dimensional frequency of that signal. The Strouhal number, measure of the non-dimensional frequency (St ¼ fw=u0 , where f is the dimensional frequency), represents the frequency axis of the FFT plots. According to Fig. 18(a), the FFT of U velocity signals for d/w¼0.7 and 1 contains two dominant frequency peaks: one is the fundamental frequency and other is its first harmonic (twice the fundamental frequency). It is important to note that U velocity signals for d/w ¼0.7 (Fig. 16(b)) and 1 (figure not shown) are formed because of the superposition of two sinusoids. The appearance of an additional frequency peak (first harmonic) for d/w¼0.7 and 1 is due to the influence of the opposite shear layer (inner shear layer of the wall jet or Layer 3) on the stationary point at which the time series data of velocities are collected. In comparison to d/w¼1, 1.5 and 2.1, the location of the stationary point is nearest to Layer 3 for d/w¼ 0.7. On the contrary, the FFT of U velocity signals for d/w¼1.5 and 2.1 shows a single dominant peak. In Fig. 18(b), FFT of V velocity signals for all the cases of d/w shows a single dominant frequency peak corresponding to the non-dimensional vortex shedding frequency (or Strouhal number). The single frequency peak of V velocity signal suggests the periodic nature of the shedding in the near field region. For a given d/w, the fundamental frequency peak of U and V velocity signals is found to occur at the same value of St. From FFT plots, the values of St for d/w¼0.7, 1, 1.5 and 2.1 are found to be 0.29, 0.22, 0.16 and 0.13, respectively. Note that the Strouhal number for d/w ¼1 i.e., St¼0.22 is close to St¼0.25 reported by Wang and Tan (2007) for similar flow configuration (d/w¼1). Therefore, the present CFD solution captures the fluid dynamics well. 7.7. Phase diagrams Phase diagrams usually demonstrate the relationship of two sinusoids. In the present study, the phase diagrams are constructed by plotting U (X-axis) and V (Y-axis) velocity data within a complete time period. Fig. 19 shows the phase diagram of U and V velocity signals shown in Fig. 17(a) for d/w¼1.5. Phase diagrams for d/w¼0.7, 1 and 2.1 are also constructed, but figures are not shown in this communication because they reveal same information like the case of d/w¼1.5. As seen, Fig. 19 consists of a simple closed curve indicating a well-behaved periodic flow in the near field region. The shape of the phase diagram can be interpreted by comparing it with the “Lissajous pattern” (Maor, 1998) which are most frequently useful for the problems in nonlinear dynamics. According to Fig. 19, an elliptic shape of phase diagram signifies that U and V velocity signals have same frequency but different amplitudes. FFT plots also convey the same message about the frequency values of the two velocity signals. The fact of difference in peak to peak amplitude of two velocity signals is confirmed by Fig. 17(a). 7.8. Probability density function The probability density function (PDF) describes the relative likelihood of a random variable to lie within a particular range of values. The PDF is non-negative everywhere and its integral over the entire region is equal to one. If z is a continuous random Rb variable, then the PDF of z is a function f(z) such that for any two numbers a and b with a rb; PDF ¼ a f ðzÞ dz in the interval R1 [a, b]. For all z, f ðzÞ Z0 and 1 f ðzÞ dz ¼ 1. Fig. 20 shows the PDF of V velocity signals for d/w¼0.7, 1, 1.5 and 2.1. For all values of d/w, the shape of PDF exhibits a bimodal distribution (two peaks) instead of a Gaussian distribution which has always a single peak. The deviation of PDF distribution from the Gaussian behavior indicates that the balance between the energy input to the turbulent vortices and the energy dissipation by viscosity is lost. The bimodal distribution of PDF shows that the probability is

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high near the minimum and maximum values of the V velocity signals as most of the possible values occur towards these two extreme values. A similar trend of PDF distribution is found for any sinusoidal function. The peak to peak amplitude of the V velocity signal (difference between two peaks of PDF) is maximum for d/w¼0.7 (i.e., 0.23) and minimum for d/w¼2.1 (i.e., 0.14).

7.9. Relationship between St and d/w The values of Strouhal number (St) for various values of separation distance between two jets (d/w) are presented in Table 1. For the purpose of qualitative analysis, a plot of St versus d/w is shown in Fig. 21. It is noticed that the Strouhal number gradually decreases with the progressive increase in d/w from 0.7 to 2.1. This could be explained from the flow physics of the vortex shedding mechanism. When d/w is increased progressively, the intermediate distance between two inner shear layers (Layer 2 and Layer 3) also increases accordingly. As a result, a vortex that forms due to roll up of each inner shear layer takes longer time to grow in size and strength to draw the opposing vortex formed in the other inner shear layer before it (former vortex) sheds away. This phenomenon delays the shedding process, and therefore the vortex shedding frequency (or Strouhal number) decreases with the increase in d/w. In other words, if d/w is decreased, the inner shear layers are brought closer together, their interaction is facilitated and thus the periodic time is shortened. This fact is consistent with Roshko (1955)'s observation for flow over the cylinder with different bluffnesses that if the bluffness of the cylinder increases, the vortex shedding frequency decreases and correspondingly Strouhal numbers decreases.

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Table 1 Values of St for various values of d/w. d/w

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8. Conclusions In the present study, two-dimensional unsteady RANS equations are solved to explore the effects of separation distance (d/w) between two jets on the turbulent dual jet flow consisting of a wall jet flow and an offset jet flow. Different flow phenomena are identified for various values of d/w. When the values of d/w lie in the range 0.7 r d=w r 2.1, the periodic

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vortex shedding phenomenon is observed near the nozzle plate between two jets. Large scale von Kármán-like vortices alternately form, grow and shed away from two sides of the nozzle plate similar to what would be expected in the flow behind a bluff body. The periodic flow does not occur if d/w is either 0.6 or 2.2. Computational results are shown for d/w¼0.6 and d/w¼2.2, where instead of any periodic flow, a pair of stable counter rotating vortices are formed close to the nozzle plate. Time series of U and V signals, within the range for which vortex shedding is observed, exhibit sinusoidal oscillation. FFT analysis of U velocity signals shows two dominant frequency peaks for d/w¼0.7 and 1; other cases of d/w (i.e., d/w¼1.5 and 2.1) show a single dominant frequency peak. For all cases of d/w, FFT analysis of V velocity signals shows a single dominant frequency peak corresponding to the Strouhal number of the vortex shedding phenomenon. The periodic nature of U and V signals is confirmed by the close-looped phase diagram. The shape of the phase diagram indicates that both the signals have identical frequency but different amplitudes. The probability density function (PDF) of V velocity signals in the near flow field region displays bimodal distribution. A progressive increase in separation distance between two jets (d/w) results in a gradual decrease in Strouhal numbers (St).

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