Large Eddy Simulation of Turbulent Flow in a Confined Square

International Journal of Computational Fluid Dynamics, October 2003 Vol. 17(5), pp. 339–356

Large Eddy Simulation of Turbulent Flow in a Confined Square Coaxial Jet HONGYI XUa,*, MAHMOOD KHALIDa and ANDREW POLLARDb a

Institute for Aerospace Research, National Research Council of Canada, Ottawa, Ontario, Canada K1A OR6; bComputational and Experimental Fluid Dynamics Laboratory, Department of Mechanical Engineering, Queen’s University, Kingston, Ontario, Canada K7L 3N6 (Received 1 February 2001; Revised 27 October 2001; In final form 7 January 2002)

Large eddy simulation (LES) is performed to investigate the spatial evolution of the turbulent flow in a confined square coaxial jet. The turbulent inflow conditions in the jet are obtained from the LES in both square and annular ducts using the temporal approach. Such prescription of the inflow boundary condition faithfully represents the turbulent inlet conditions and makes it possible to realistically investigate two types of turbulent mixing mechanisms originated from the streamwise shear, caused by streamwise velocity difference, and the secondary shear, induced by the turbulence-driven secondary flows. The turbulent mixing properties in the confined square coaxial jet are studied by analyzing the spatial evolvement of the mean flow field and the second-order turbulence statistics. The simulation results present reasonable agreement with the experimental data from a square free jet and the measurements of a confined plane jet. The turbulent mixing phenomena are interrogated using the streamwise vorticity distributions on various section planes of the instantaneous flow field. The principle of vx-dynamics in [K.B.M.Q. Zaman, (1996). “Axis switching and spreading of an asymmetric jet: the role of coherent structure dynamics”, J. Fluid Mech., 316, pp. 1 – 17] is used to understand the effects of the turbulence-driven secondary flow and to explain the observation found in the simulation. Keywords: Large eddy simulation (LES); Mixing; Time-dependent inlet conditions; Vortex dynamics

INTRODUCTION In this paper, the turbulent flow in a confined square coaxial jet is considered. While coaxial confined jets are found in ejectors and other industrial equipment, the motivation for this work is to consider the interplay between two streams of fluid as a precursor to free, coaxial jet flows where ultimately the streams have different properties. Early experimental investigations on circular jets in circular ducts were made by Hembold et al. (1954), Mikhail (1960) and Becker et al. (1962). Craya and Curtet (1955) studied plane and axisymmetric confined jets using a theory based on the two-dimensional (2D) Reynoldsaveraged Navier –Stokes (RANS) equations. Barchilon and Curtet (1964) made use of the theory to experimentally investigate the mean flow structures in an axisymmetric confined jet. Some distinctive flow patterns were studied, particularly separated flow in a confined jet. Novick et al. (1979), Syed and Sturgess (1980) and Sturgess et al. (1983) modeled a confined jet flow using RANS. It was found that none of these calculations was capable of qualitatively predicting even the dominant *Corresponding author. ISSN 1061-8562 print/ISSN 1029-0257 online q 2003 Taylor & Francis Ltd DOI: 10.1080/1061856031000083477

features of the flow, the effect of inlet conditions being a possible cause (Nallasamy, 1986). Roquemore et al. (1991) considered a combusting flow using the isotropic k 2 1 closure model (RANS). The results were in qualitative agreement with experiments, but were unable to give accurate predictions of the fuel concentration in the recirculation zone and hence failed to provide a correct attached flame condition. Failure was attributed to the use of the steady RANS model for the highly anisotropic and transient turbulent flow phenomena. Akselvoll and Moin (1995) performed large eddy simulation (LES) of a confined co-annular jet flowing into a larger circular duct. The reattachment point for the sudden pipe expansion flow and the merging length of the shear layer between the fluid streams in the inner and outer pipes were consistent with the experimental results from Roquemore et al. (1991). On the other hand, the turbulent free jets have been extensively investigated for a variety of jet configurations. These include the investigations of a round jet by Ricou and Spalding (1961) and Liepmann and Gharib (1992), a square jet by Quinn (1988), a rectangular jet by Quinn et al. (1983a,b); (1985) and Schwab and Pollard (1988) and an

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FIGURE 1 Schema and geometric notation of square duct

elliptic jet by Quinn (1989). One of the interesting flow phenomena in the free jet is the so-called axis switching, which is reported by Sforza et al. (1966), Krothapalli et al. (1981), and Quinn (1992) for the rectangular, elliptic and square free jets, respectively. As presented in Zaman (1996), the phenomenon can be explained using azimuthal vu- and axial vx-vorticity and their dynamics. These vorticity fields are not independent since the flow field is replete with both azimuthal and streamwise vortical structures, which continually interact, as has been shown recently by McIlwain et al. (2000) for a round jet. So far the axis switching has not been found and reported in a confined asymmetric jet. In the present study, the turbulent flow in a confined square coaxial jet is simulated using LES. The simulation involves two flow streams issuing from an inner square duct and a square annular duct into a square chamber. The spatial evolution of the flow mixing between the two streams is predicted. The mixing mechanisms come from two sources, namely the streamwise shear layer interaction between the two streams and the cross-streamwise mixing driven by the anisotropy of turbulence stresses that preexists in the square and annular ducts prior to the mixing. One of the most challenging issues in Computational Fluid Dynamics (CFD) is the prescription of the turbulent inflow and outflow conditions in the non-homogeneous directions, (Rogallo and Moin, 1984; Ciofalo, 1994). The inflow condition appears to be more troublesome since the influence of the upstream conditions persists well downstream in most convection-dominated flow cases. A common practice to prescribe the turbulent inflow is to use simulation results from a relevant temporal flow simulation, as seen in Akselvoll and Moin (1995) and Breuer and Rodi (1994). The investigation of the flows inside both square and annular ducts (Xu, 1997; Xu and Pollard, 2001), provides an important insight that demonstrates the need for precise specification of the turbulent inflow conditions for the confined square coaxial jet. The effects of both streamwise shear, caused by streamwise velocity difference, and secondary shear, induced by the turbulence-driven secondary flows, are investigated, and could not be done if the time-dependent inflow conditions were not copied from the temporal simulations. The principle of vx-dynamics in Zaman (1996) is used as an aide to understand the effects of the turbulence-driven secondary flows. The results and

FIGURE 2 Schema and geometric notation of annular duct

conclusions from the current investigation, we believe, provide some critical and useful insights into the features and drawbacks of the use of isotropic-type turbulence models. Also, the current investigation is an initial step towards using LES to realistically simulate turbulent flow past complex configurations.

PROBLEM DEFINITION AND THE GOVERNING EQUATIONS The problems considered here are best described using a Cartesian coordinate system. The physical situations considered culminate in a confined square coaxial flow, as depicted in Fig. 3. The geometries used to produce inflow conditions for this are shown in Figs. 1 and 2. Turbulent flows can be investigated using: (1) Direct Numerical Simulation (DNS); (2) Large Eddy Simulation (LES); and (3) Reynolds-averaged Navier– Stokes method (RANS). In the present LES calculation, the spatiallyfiltered, three-dimensional, time-dependent N – S equations are used, which are closed by an appropriate subgrid scale (SGS) eddy viscosity model. These equations can be written in the following non-dimensional form:

›u i ¼0 ›x i

ð1Þ

›u i ›u j u i ›p þ ¼2 ›t ›x j ›x i 1 › ›u i ›u j þ ð1 þ nt Þ þ ð2Þ Re ›xj ›x j ›x i where the indices i; j ¼ 1; 2; 3 refer to the x, y and z directions, respectively; x is the streamwise direction and y and z are the transverse directions, see Madabhushi and Vanka (1991) for further details. Before proceeding further, a number of issues need to be discussed that involve inlet boundary conditions, problem scaling and SGS modeling. Turbulent Inlet Boundary Conditions A challenging problem in turbulence simulation is the prescription of the turbulent inflow boundary conditions in

LES OF CONFINED COAXIAL JET

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For the annular duct in Fig. 2, the force balance can be written as: ð p1 2 p2 Þðd2o 2 d 2i Þ ¼ 4ðtwo d o þ twi di Þl

where p1 and p2 are pressures on the inlet and outlet, l is the length of the annular duct, do and di are the hydraulic diameters and t¯wo and twi are the mean wall shear stresses of the inner and outer ducts, respectively. An overall mean wall shear stress in the annular duct can then be defined as:

FIGURE 3 Schema and geometric notation of square coaxial jet

the non-homogeneous direction, see Rogallo and Moin (1984) and Ciofalo (1994). The turbulent inflow problem is troublesome, since the influence of the upstream conditions persists for a large distance downstream of the inlet plane. This issue has long been recognized in some schools, experimentalists included, but ignored in others. The sensitivity to inlet conditions has long been recognized by the present authors, see Latornell and Pollard (1986). Two flow simulation types can be used: those evolving in time (temporal) and those developing in space (spatial). In a temporal simulation, the flow in at least one direction is assumed to be homogeneous so that the periodic boundary conditions can be applied in that direction. In a spatial simulation, since the flow is non-homogeneous in three dimensions, the turbulent inflow conditions need to be prescribed on the inlet boundary. Breuer and Rodi (1994) and Akselvoll and Moin (1995) both used results from a temporal simulation to specify inlet conditions for the spatially evolving simulations. In the present study, the instantaneous temporal simulation results on one of the cross-sections of the square and annular ducts in Figs. 1 and 2 are imposed onto the inlet plane of the confined square coaxial jet in Fig. 3. Such prescription of the inflow conditions faithfully represents the turbulent inlet conditions and makes it possible to realistically investigate two types of turbulent mixing mechanisms that originate from the streamwise shear and the turbulence-driven secondary shear as discussed in the following sections.

tw ¼ di ðtwo do þ twi d i Þ=ðd 2o 2 d2i Þ:

ð5Þ

By choosing d/2 and di/2, (note in current application d=2 ¼ di =2), as the length scales for the square and the annular duct, respectively, and the mean frictional velocity, u t ¼ ðtw =rÞ1=2 ; as the velocity scale, the fullydeveloped laminar flows in both square and annular ducts are governed by the following relation: 1 ›2 u ›2 u ›p p2 2 p1 ¼ ¼ 22: ð6Þ þ ¼ Ret ›y 2 ›z 2 ›x l The governing equations for the turbulent flow in both square and annular ducts can then be written as:

›u i ¼0 ›x i

ð7Þ

›u i ›u j u i ›p þ ¼2 þ 2d1i ›t ›x j ›x i 1 › ›u i ›u j þ ð1 þ nt Þ þ : ð8Þ Ret ›xj ›x j ›x i The initial conditions of both square and annular ducts are generated from the fully developed laminar velocity upon which is superimposed the disturbances that are solutions to the O–S equation (Orr, 1907; Sommerfeld, 1908):

2 d2 2 2a f dy 2 2 d d2 u 2 2 iaRe ðu 2 cÞ 2 a f 2 dy 2 dy 2

Problem Scaling For the square duct in Fig. 1, the streamwise force balance gives: ð p1 2 p2 Þd 2 ¼ 4tw dl

ð4Þ

¼0

ð3Þ

ð9Þ

where the streamfunction cðyÞ ¼ AfðyÞeiaðx2ctÞ with f(y ) being the eigenfunction, u is the basic laminar velocity, Re is the Reynolds number, a is the spatial wave

where p1 and p2 are pressures on the inlet and outlet, l is the length of the square duct, d is the hydraulic diameter and tw is the mean wall shear stress.

TABLE I Computational parameters for LES of square and annular ducts Run Square Annular

Grid points 150,280 416,000

Ttotal 40 40

Dt 24

5.0 £ 10 5.0 £ 1024

Ret

Rebulk

Domain size

250 150

4921 2534

16p £ 2 £ 2 16p £ 4 £ 4

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H. XU et al. TABLE II Computational parameters for LES of confined square coaxial jet

Grid

Ttotal

Time Step*

ut h=n

U bulk – jet

ðU sq =U an Þbulk

M sq =M an †

566,280

6.75

0.0001

150

47.542

3.2367

1.079

* Variable time step is used and 0.0001 is the maximum one. † M is mass flow rate and subscripts “sq” and “an” mean square and annular, respectively. U bulk – jet : The bulk inlet velocity of the jet; ðU sq =U an Þbulk : the ratio of inlet bulk velocity and M sq =M an : the ratio of mass flow rates between the square and annular duct, respectively.

number, ac ¼ acr þ iaci with acr being the temporal frequency and aci being the temporal amplification factor. The lengths of the ducts are chosen as 16p to guarantee that it is larger than the characteristic lengths at which the correlation between fluctuating quantities becomes negligible (see Gavrilakis (1992)). Table I provides the relevant computational parameters in the temporal LES of both square and annular ducts. The total integration time, T total ; and the time step, Dt; are normalized by the large eddy turnover time (LETOT) h=ut : The Reynolds numbers, Ret and Rebulk ; are defined as Ret ¼ ut h=n and Rebulk ¼ Uh=n; respectively. For the LES in the confined square coaxial jet, Fig. 3, the length and velocity scales in the annular duct are used as the characteristic quantities. Therefore, the Reynolds number for the confined square coaxial jet, based on the mean wall shear velocity, is 150 and the inlet velocity from square duct has to be re-scaled by a factor of RetðsqÞ =RetðanÞ ¼ 5=3: The major computational parameters are presented in Table II for the LES of the confined square coaxial jet. SGS Modeling In the current investigation, the governing equations are closed using the Smagorinsky, 1963 SGS model. The eddy viscosity nt is calculated by: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ›u i ›u j 2 2 nt ¼ Ret ðCs DDÞ þ ð10Þ 2 ›x j ›x i ¼ ðhx hy hz Þ1=3 with hx ; hy and hz where C s ¼ 0:1 and D being the control volume size in x, y and z, respectively. A wall damping function is used (Kajishima and Miyake, 1992): þ yþ z D ¼ 1 2 exp 2 1 2 exp 2 ð11Þ 25 25 where y þ and z þ are the non-dimensional coordinates scaled by the mean frictional velocity u t : Equation (11) accounts for the damping effects from the two side-walls in the near corner region and the damping can automatically be reduced to the one used in the planewall case near the region of the wall away from the corner. The use of the Smagorinsky model and wall damping noted above is justified (Xu and Pollard, 2001) at least for the temporal simulations. As there are no wall normal flow separations (reversed flow regions, in the axial direction),

we have no reason to believe the present results have not captured the essential mechanisms that characterize a co-axial jet mixing.

NUMERICAL METHOD In this section, the basic algorithm used to solve the governing equations is described. Code validation and verification are also addressed. Spatial and Temporal Discretization The governing equations are spatially discretized by the second-order finite volume formulation based on a staggered grid system, which proved to be effective and practical in the DNS computation of square duct flow by Gavrilakis (1992). As demonstrated by Kim and Moin (1985), a common practice for the temporal discretization of the LES governing equations is to apply the secondorder Adams – Bashforth scheme for the convection terms and the second-order Adams – Moulton scheme for the diffusion terms. The explicit treatment of the non-linear terms eliminates the need for linearization and the implicit treatment of the diffusion terms eases the numerical stability restriction. The fractional step method in Kim and Moin (1985) is used to de-couple the pressure and velocity and to obtain the time-dependent pressure and the divergence-free velocity. With this approach, the discretized governing equations can then be expressed as: u~ nþ1 2 uni 1 1 ~ nþ1 i ¼ ð3Cni 2 C n21 Þ þ ðD þ Dni Þ i 2 2 i Dt unþ1 2 u~ nþ1 ›f nþ1 i i ¼2 Dt ›x i where

and

›unþ1 i ¼0 ›xi

› 1 › ›uj ðui uj Þ þ nt Ci ¼ 2 ›x j Re ›xj ›x i 1 › ›ui and Di ¼ ð n þ nt Þ : Re ›xj ›x j

The pressure, p, is linked with the pressure potential, f, by the following formulation: p¼f2

1 Dt › ›f ð n þ nt Þ : 2 Re ›xj ›xj


Regarding the accuracy of the fractional step method, it is commonly accepted that the accuracy of pressure is first order in time and the first-order accuracy of pressure does not influence the second-order temporal accuracy of velocity. The explicit scheme applied to the non-linear convective terms imposes a restriction on the largest time step. In the current application, the time step, Dt; is restricted to a CFL value that satisfies:

CFLmax

juj jvj jwj þ þ ¼ Dt max # 0:6 Dx Dy Dz

which is comparable to the one used in Madabhushi and Vanka (1991). Code Verification and Validation The governing equations were solved using an in-house developed computer code. On each time level, the three momentum equations are satisfied to computer round-off error. A robust multigrid acceleration technique is applied to accelerate the convergence of the pressure Poisson equation, which guarantees the residue of the continuity equation being driven down five to six orders of magnitude on each time level. The code was first verified against the analytical solutions of Steinman et al. (1994), see Xu (1997), with excellent agreement. It was then critically compared against the existing data on the square duct flow. These data include the DNS results from Gavrilakis (1992) and Huser and Biringen (1993), LES results from Madabhushi and Vanka (1991) and the experimental data from Niederschulte (1989), etc. Because of the lack of experimental data and computational results for flow in a square annular duct, a grid refinement study was performed by doubling the grid size from 130 £ 66 £ 66 to 130 £ 130 £ 130 in the temporal simulation. Again, the results in the square annular duct were carefully compared with the existing relevant data and were critically judged by the appropriate physics, see Xu and Pollard (2001). The grid-dependence of the flow field on grid density reveals that the distributions of mean streamwise velocity varied only by about 10% even though the mesh is refined by a factor of four. In the present case of spatially developing confined co-axial jet flow, an accurate description of spatial and temporal inlet conditions is required, which can only be obtained from the temporal simulations in both square and annular ducts. Thus, the current spatial simulation, in the cross-streamwise directions, is dependent on the fidelity of the resolution of the temporal simulations. Given that the temporal simulations have been validated against the available experimental data, see Xu and Pollard (2001), and that the code has been verified against analytical solutions to the N – S equations, see Xu (1997), the spatial resolution in the cross-streamwise directions can be considered reliable. The streamwise grid resolution in the current spatial simulation will be

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discussed in the “Spatial LES in confined square coaxial jet” section.

PRESENTATION AND DISCUSSION OF RESULTS Temporal LES in Square and Annular Ducts Following the code verification and validation work, the overviews of the general flow features are demonstrated on the inlet surface of the confined square coaxial jet. These distinctive flow features, including the vortical turbulence-driven secondary flow and the strong anisotropic distributions of Reynolds stresses, justify the necessity of using temporal LES to generate the realistic turbulent inflow conditions, which can otherwise be a major difficulty in a RANS calculation. Figure 4 presents the isovels of the mean streamwise velocity on the inlet surface, which consists of the flows from both square and annular ducts with a strong streamwise velocity shear near the inner square duct. Figure 5 gives the mean secondary flow pattern on the inlet surface of the confined square coaxial jet. The mean secondary flow in both square and annular ducts produces a chain of counter-rotating vortex pairs. It is noteworthy that the eight vortices inside the inner square duct (concave corners), when combined with the corresponding eight vortices from the outside wall surface of the inner square duct (convex corners), form eight shear layers distal to the exit of the two jets. These shear layers are referred to as the secondary shear and their strengths are much weaker than the streamwise shear. However, the influence of the secondary shear extends well into the spatially evolving flow in the confined square coaxial jet, as seen in the following flow field analysis. For future reference, Figs. 6 – 8 provide the typical turbulence statistics of u0 u0 ; u0 v0 and v0 v0 distributions on the inlet, which demonstrate the strong anisotropic and flowconfiguration dependent structures.

Spatial LES in Confined Square Coaxial Jet The initial condition inside the confined square coaxial jet is set as a uniform flow, which satisfies the continuity requirement imposed by the turbulent inlet boundary conditions. The grid density is increased around the inner square duct, in anticipation of resolving the regions with high turbulence energy production. The DNS and LES calculations of the square duct flows by Gavrilakis (1992) and Madabhushi and Vanka (1991) indicate that a minimum grid spacing of about y þ # 2 is needed in the cross-streamwise direction to resolve the flow near a concave 908 corner. The LES results in a square annular duct (see Xu and Pollard (2001)) demonstrate that the grid points need to be distributed well into the region of y þ # 1 to adequately resolve the flow near a convex 908 corner and a grid resolution of 1 # y þ # 2 used in

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FIGURE 4 Mean streamwise velocity contour at inlet colour-mapped by its magnitude U=U cl

the current calculation can only resolve the flow qualitatively. The grid in the streamwise direction is refined near the inlet boundary to resolve the near-field flow and its development. This grid refinement is done in reference to a similar LES calculation of a confined circular coaxial jet,

performed by Akselvoll and Moin (1995). The minimum and maximum streamwise grid spacings are Dxþ min ¼ 13:8 and Dxþ max ¼ 53:1; respectively, in the current simulation, þ whereas Dxþ min ¼ 10:4 and Dxmax ¼ 152:1 in Akselvoll and Moin (1995). The following analyses indicate that the current streamwise grid resolution is sufficient to resolve

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi FIGURE 5 Mean secondary flow vector-plot at inlet colour-mapped by secondary velocity magnitude v 2 þ w 2 =U cl


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pffiffiffiffiffiffiffiffiffi FIGURE 6 Streamwise turbulence intensity u 0 u 0 =U cl at inlet

the flow in the near field and to accurately predict the streamwise velocity decay. Figures 9 and 10 present the grid size variations in both streamwise and cross-streamwise directions, respectively. The whole simulation domain contains 130 £ 66 £ 66 control volumes. The initial time step is kept small (at about 2:5 £ 1025 ). With the inlet flow evolving from

a perturbed laminar flow to a fully developed turbulence, the velocity magnitude is drastically reduced, as indicated in Fig. 11, and the time step can then be increased (to about 1:0 £ 1024 ) during the course of the calculations. The flow information is collected over 80,000 time steps to get the mean flow field and various turbulence statistics (this is equivalent to 6.75 LETOTs). As is well

FIGURE 7 Reynolds shear stress

pffiffiffiffiffiffiffiffiffi u 0 v 0 =U cl at inlet

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FIGURE 8 Transverse turbulence intensity

pffiffiffiffiffiffiffiffiffi v 0 v 0 =U cl at inlet

the mean flow field and the turbulence statistics are quadrant-averaged.

known, the larger the sample size, the more reliable the statistics. To increase the sampling size for calculating the turbulence statistics, the flow field is averaged in the homogeneous direction. See, for example, Gavrilakis (1992) and Breuer and Rodi (1994). The quadrantaveraging procedure in Gavrilakis (1992) and Huser and Biringen (1993) makes use of the geometric symmetry properties of the flow to increase the sampling size. Here,

Figures 12 –14 present the turbulence signals of u and v at three locations in the near field, respectively. These signals represent the output from a “virtual” velocity

FIGURE 9 Grid size distribution in cross-streamwise direction

FIGURE 10 Grid size distribution in streamwise direction

Instantaneous Flow Field


FIGURE 11 History of mean streamwise velocity on wall bisector

anemometer. These locations are typically at ðx; y; zÞ ¼ ð0:33; 0:094; 0:053Þ; which is close to the inner corner of the outer wall, and ðx; y; zÞ ¼ ð0:33; 0:95; 0:95Þ; which is close to the outer corner of the inner square duct and

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ðx; y; zÞ ¼ ð0:33; 1:71; 1:41Þ; which is close to the centerline of the confined square coaxial jet. From Fig. 14, it can be seen that the turbulence becomes statistically stationary after approximately 30,000 time-steps. Therefore, the mean flow field and the turbulence statistics are taken from the flow information between 30,000 and 80,000 time steps. A comparison of Figs. 12 and 13 with Fig. 14, suggests that the wavelength of the signals tends to be larger in the near-wall or in the near-corner regions, due to the wall or corner damping effects. In addition, the qualitative comparison of the signal wavelength between Figs. 12 and 13 indicates that the concave 908 corner of the outer duct more vigorously damps the turbulence than the convex 908 corner of the inner duct. This observation is consistent with the quantitative analysis in Xu and Pollard (2001), where a “universal law”, which is similar to the “law of the wall”, is proposed for the streamwise velocity near both concave and convex 908 corners. The distributions of streamwise vorticity on two longitudinal planes are presented in Figs. 15 and 16. Figure 15 is on the wall bisector (at y ¼ 2h; see Fig. 3) and Fig. 16 is on a plane parallel to one side of the inner square duct (at y ¼ 1h; see Fig. 3). It can be seen, in Fig. 15, that the distributions of streamwise vorticity on the plane of y ¼ 2h are characterized by streaky

FIGURE 12 Turbulence signals of u and v close to the corner of outer wall in near field

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FIGURE 13 Turbulence signals of u and v close to the corner of inner square duct in near field

FIGURE 14 Turbulence signals of u and v close to center of the square coaxial jet in near field


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FIGURE 15 Instantaneous distribution of streamwise vorticity on plane of y = 2

FIGURE 16 Instantaneous distribution of streamwise vorticity on plane of y = 1

LES, confined square jet, present Linear decay region I Linear decay region II Exp, free square jet, Quinn(1988) Linear-regression Exp (1), rectangular confined jet, Chua & Lua Linear-regression Exp. (2), rectangular confined jet, Chua & Lua Measurements (1), rectangular confined jet, Chua & Lua [1998] Measurements (2), rectangular confined jet, Chua & Lua [1998]

1.7 1.6

y=0.185x-0.0277

U max/U cl

1.5

y=0.0463x+0.9623 y=0.0512x+0.8879

1.4 1.3 1.2

linear decay region II: y=0.045x+0.835

1.1 linear decay region I: y=0.008x+1.0

1

0

2

4

6

8

10

12

x/D e FIGURE 17 Mean streamwise velocity decay on the jet centerline

FIGURE 18 Isogram contours of u 0 w 0 on the wall bisector of the jet

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structures both in the regions of near-wall and near-shear layer. These streaky structures have a tendency to tilt with the leading front in the higher velocity inner flow and the trailing front in the lower velocity flow that emanates from the annulus or the walls. The distribution of streamwise vorticity on the plane of y ¼ 1h in Fig. 16 is not that stretched and elongated in the near field because of the upstream sidewall effects of the inner square duct. In the near-wall region, the turbulent upwash and down-wash phenomena, which appear burstlike (see Robinson (1991)) can be clearly seen through the animation of the instantaneous flow field. Mean Flow Field The decay of the mean streamwise velocity along the centerline of the jet is presented in Fig. 17. Here, U max is the maximum value of the mean streamwise velocity on the jet centerline at the inlet. Two linear decay regions can be found, namely the linear decay region in 0:0 , x=De , 3:6 and the linear decay region II in 5:8 , x=De : Here, De is the equivalent diameter defined as the diameter of a round slot with the same exit area as the square slot. By linear regression, see, for example, Quinn (1989) and Chua and Lua (1998), the two linear decay regions can be expressed as U max =U cl ¼ K u ðx=De þ C u Þ; where K u is the mean streamwise velocity decay rate on the jet centerline and Cu is the kinematic virtual origin of the jet. Linear regression gives K u ¼ 0:008; Cu ¼ 125 for linear decay region I and K u ¼ 0:045; Cu ¼ 18:55 for linear decay region II. The decay rate in region I is much lower than that in region II, indicating that the major turbulent mixing takes place in the region 5:8 , x=De : Figure 17 clearly indicates that confining the jets causes the mean centerline velocity to decay much slower than a free jet. The current prediction gives a centerline velocity decay rate of 0.045 for the confined square jet, which is in good agreement with the experimental decay rate at 0.046 from Chua and Lua (1998) for a confined plane jet. The parallel shift of the two curves (the difference in the regression constants) is quite probably attributable to the different flow configurations used in the computation and the experiment. That is, Chua and Lua (1998) used a 6:1 aspect ratio jet issuing into a 20:1 aspect ratio receiving chamber, whereas here we consider a square configuration. The turbulence statistics, u0 v0 and u0 w0 ; are a measure of the turbulence mixing in directions perpendicular to the predominant flow direction. Figure 18 presents the contours of u0 w0 on the wall-bisector of the x 2 z plane. It can be seen that the potential core region vanishes at about x=De ¼ 3:6 with the elimination of the mixing layer on either side of the jet. After some transition distance ð3:6 , x=De , 5:8Þ; the mixing is complete over the entire cross-section of the jet and, therefore, the streamwise velocity reaches the linear decay region II. The confinement effect of the outer square duct produces

a streamwise velocity decay rate K u ¼ 0:045; which is significantly lower when compared with free jet results, such as K u ¼ 0:152 from Ho and Gutmark (1987) and K u ¼ 0:185 from Quinn (1988). This finding is further confirmed in Fig. 17 by comparing the streamwise velocity decay between the current LES results and the experimental data from Quinn (1988). On the other hand, the length of the potential core, which is of order x=De ¼ 3:6 is close to that of an elliptic jet ðx=De ¼ 3:5Þ and the sharp-edged round jet ðx=De ¼ 4:4Þ; Quinn (1989). Also, the experimental data from Chua and Lua (1998) suggests that the transition from decay region I to decay region II roughly occurs in the region 4:0 , x=De , 6:0: The mean streamwise velocity profiles in the central x 2 z plane are compared with the experimental results from Quinn (1988) in Fig. 19. Pollard and Iwaniw (1985) and Quinn et al. (1985), and others, noted the off-centre peaks (saddle-backed velocity profile) in the mean streamwise velocity in the very near field; these are evident in the data from Quinn (1988), as seen in Fig. 19(a). The saddle-backed profile in the mean velocity is due to induction from the circumferential vortex rings, which are generated in the initial region of jet exit. The effects of the circumferential vortex rings do not exist in the current simulation, since the co-flow stream removes the large velocity gradient between the jet and its surroundings. Figures 19(a)– (d) present the axial evolution of the mean streamwise velocity, which are compared to the experiments of Quinn (1988). The velocity profiles, particularly near the centre of the square duct, bear some resemblance to those in the square free jet, at least in the near field, x=De , 2:658; where De is the diameter of an equivalent round jet. The velocity profiles in the confined square jet depart from the square free jet data beyond x=De , 2:658; as expected. The streamwise development of the jet is highlighted using four cross-stream contour diagrams, Figs. 20(a) – (d), which includes the cross-stream velocity vectors. In the very near field ðx=De ¼ 0:37Þ; the square shape of the jet is evident. The gradients across the shear layer between the square and annular duct flow, and particularly around the four corner regions, suggest strong mixing. The mean secondary velocity vectors indicate that, within a small distance from the inlet, the fluid from the annular duct starts to be entrained by the high-speed stream from the inner square duct. The vortices at the convex corner side of the annular duct vanish quickly due to the entrainment. However, the vortex pair at the inner square duct side, although distorted by the entrainment, still persists since the pressure gradient at the square duct side favors the vortex flow in the diagonal direction. Downstream, at x=De ¼ 1:41; the gradients in the transverse directions are relaxed somewhat and the entrainment along the four sideedges of the square shape is reduced when compared with that at x=De ¼ 0:37: However, the entrainment near the four corner regions remains strong, which distorts the velocity contours around the corner regions by “pushing”


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FIGURE 19 Spatial evolvement of streamwise velocity in the central x - z plane, compared with experimental results from Quinn [1988] for square free jet

them inward towards the centerline of the jet. Downstream at x=De ¼ 3:91; the square shape of the jet vanishes and the turbulent mixing spreads across the entire transverse section of the jet. At x=De ¼ 7:42; the flow begins to look similar to that in a square duct, which is characterized by four counter-rotating vortex pairs near the corners of the outer square duct. One of the interesting flow phenomena in a turbulent jet is the axis switching, in which the cross-section of an asymmetric jet evolves in such a manner that, after a certain distance from the nozzle, the major and minor axes are interchanged. The axis-switching phenomenon, so far, has been investigated mostly in the cases of free jets. See Quinn (1992) for a square free jet and Quinn et al. (1983a,b); (1985), Tsuchiya and Horikoshi (1986), Zaman (1996) and others for rectangular free jets. This phenomenon is not found in the current numerical

simulation, the explanation of which might be attributed to: (1) the inlet flow conditions used in the simulation and (2) the confinement effects of the outer square duct. Quinn (1992) noted that the vorticity distribution at the jet inlet of his experiments did not exhibit the typical fully developed secondary flow pattern in a square duct. This is not surprising since he used a flat plate on the end of a square settling chamber into which was machined a sharpedged orifice. However, the streamwise vorticity distributions in the near field, calculated from mean velocity gradients, was found to be dominated by four sets of counter-rotating vortex pairs, as sketched in Fig. 21. Quinn (1992) argues that these streamwise vortices are generated from distributed vorticity shed from the four corners of the slot by skewing of the shear layers as a result of a vena-contracta. It can be seen clearly from comparing Fig. 5 with Fig. 21 that the streamwise vortices

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FIGURE 20 Mean streamwise velocity contours overlapped by cross-stream velocity vector

rotate in different directions. Zaman (1996) considers the streamwise vorticity, vx and noted that each counterrotating vortex pair found in the Quinn’s experiment at first moves towards the jet centerline. The two vortices on one side form an outflow pair. The four resulting pairs then move away from the jet centerline, which causes the 458 axis switching downstream. The direction of rotation of the turbulence-driven secondary flows in both square and annular ducts tend, in the first place, to move away from the centerline rather than to move towards the jet centerline (Quinn (1992)). Thus the secondary flows at the jet inlet in the present simulations resist the formation of the axis switching, as explained by Zaman (1996). The confinement effects of the outer square duct on the axis switching are still not quite clear since free jets are

the normal models used for this type of investigation. Some comparative studies are needed between confined and free jets to investigate this effect. Turbulence Statistics The streamwise and spanwise turbulence stresses, urms and vrms ; in the central x 2 z plane, are presented in Figs. 22 and 23, respectively. These stresses exhibit similar behavior to the experimental data of Quinn (1988). The minima are found on the centerline and the peaks are predicted in the shear layer region, where the production of turbulence is high due to the high local gradients in the mean streamwise velocity and the high momentum exchange in the shear layer. As mentioned earlier, the flow


FIGURE 21 Schematic drawing of vortex pairs promoting axisswitching

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in the far field approaches that of a square duct. Therefore, as seen in Figs. 22(c),(d) and Figs. 23(c),(d), the peaks in the stress profiles, particularly urms ; are found to move toward the near-wall region of the outer square duct. The normalized turbulence shear stress 2u0 w0 =U 2cl in the central x 2 z plane is presented in Fig. 24. On the jet centerline, where there is no local shear in the mean streamwise velocity, the shear is zero. The peaks are in the high shear layer region where the local gradient of the mean streamwise velocity reaches its maximum value. At the jet exit, the shear stress exhibits a linear distribution typical of fully developed turbulence around the centerline of the inner square duct. The peak of the shear stress, as presented in Fig. 24(a), rises steeply from 0.002 at the jet exit to 0.008 at x=De ¼ 0:28 without much change in the linear distribution around the centerline of the jet. This indicates that the shear in the streamwise velocity drastically intensifies the local turbulent mixing. As the flow evolves downstream from x=De ¼ 0:28 to x=De ¼ 4:484; see Figs. 24(b) – (d), the shear layer

FIGURE 22 Streamwise turbulence intensity u 0 u 0 profiles in central x—z plane, compared with experimental results from Quinn [1988] for square free jet

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FIGURE 23 Spanwise turbulence intensity v 0 v 0 profiles in central x—z plane compared with experimental results from Quinn [1988] for square free jet

spreads. The peak in the shear stress decreases to a value of 0.005, which is significantly lower than 0.015 and 0.016 reported by Quinn (1988) and Wygnanski and Fiedler (1969). The shear stress around the centerline of the jet becomes linear with a slope that is clearly different to a free jet and is greater than that at x=De ¼ 0:

CONCLUSIONS LES is performed for a confined square coaxial jet. The fully developed turbulent conditions are imposed on the inlet of the jet. The following conclusions can be drawn from the current numerical prediction: (1) The center-velocity decay rate for the confined square jet is in good agreement with the measurement from Chua and Lua (1998) for a confined plane jet. The decay rate in the confined square coaxial jet is

found significantly lower than the one in a free jet. The current prediction gives a potential core length close to those in both free and confined jets. (2) The influence of the turbulence-driven secondary flows in the both square and annular ducts persists for quite long distances downstream inside the confined square coaxial jet. These pre-existing secondary flows around the concave and convex corners in both square and annular ducts tends to resist the formation of axis-switching, which is deemed to be the major reason that no axis switch is observed in the current simulation. (3) The turbulence statistics are consistent with the behavior of the mean streamwise velocity, namely, high values of these statistics are found in the region of high local shear in the mean streamwise velocity, where the high production of turbulence is present. The turbulence statistics in the confined square coaxial jets bear some resemblance in the near field


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FIGURE 24 Turbulent shear stress profiles in the central x—z plane cmpared with experimental results from Quinn [1988] for square free jet

to those in a square free jet, while some significant deviations are found in the far field. (4) There exist strong vortical turbulence-driven secondary flow and highly anisotropic distributions of Reynolds stresses at the inlet of the confined square coaxial jet. These conditions can be generated using a temporal LES, which suggests that future LES studies should carefully consider this option rather than employing either presumed distributions or characteristics for a flow. Indeed, a RANS calculation may be improved by using inlet conditions for the mean flow field and the turbulence derived from the current work. Acknowledgements The current work was partially supported by the Natural Science and Engineering Research Council of Canada. Thanks are also due to Dr W.R. Quinn for providing his experimental data.

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Quinn, W.R., Pollard, A. and Marsters, G.F. (1983b) “On saddle-backed velocity distributions in a three dimensional turbulent free jet”, AIAA83-1677. Quinn, W.R., Pollard, A. and Marsters, G.F. (1985) “Mean velocity and static pressure distributions in a three-dimensional turbulent free jet”, AIAA J. 23(6), 971–973. Ricou, F.P. and Spalding, D.B. (1961) “Measurements of entrainment by axisymmetrical turbulent jets”, J. Fluid Mech. 11, 21–32. Robinson, S.K. (1991) The kinematics of turbulent boundary layer structure, NASA Technical Memoranda, TM 103859. Rogallo, R.S. and Moin, P. (1984) “Numerical simulation of turbulent flows”, Ann. Rev. Fluid Mech. 16, 99–137. Roquemore, W.M., Reddy, V.K., Hedman, P.O., Post, M.E., Chen, T.H., Goss, L.P., Thump, D., Vilimpoc, V., Sturgess, G.J. (1991) Experimental and theoretical studies in a gas-fuel research combustor, AIAA paper 91-0639, 29th Aerospace Sciences Meeting Reno, Nevada. Schwab, R.R., Pollard, A. (1988) “The near field behaviour of rectangular free jets: An experimental and numerical study”, Proceedings 1st World Conference on Expt’l heat Transfer, Fluid Mechanics and Thermodynamics, Dubrovnik, Yugoslavia, pp. 1510–1517. Sforza, M.P., Steiger, H.M. and Trentacoste, N. (1966) “Studies on threedimensional viscous jets”, AIAA J. 4, 800 –806. Smagorinsky, J. (1963) “General circulation experiments with primitive equations”, Mon. Weather Rev., 93. Sommerfeld, A. (1908) Atti 4th Congr. Int. Math., Rome, pp. 116 –124. Steinman, D.A., Ethier, C.R., Zhang, X. and Karpik, S.R. (1994) “Code testing with exact solution to 3D Navier –Stokes equations”, In: Gottlieb, J. and Ethier, C.R., eds, 2nd Annual Conference of the CFD Society of Canada, Toronto, Canada, pp. 115– 122. Sturgess, G.J., Syed, S.A. and McManus, K.R. (1983) “Importance of inlet boundary condition for numerical simulation of combustor flows”, AIAA paper, AIAA-83-1263. Syed, S.A. and Sturgess, G.J. (1980) “Validation studies of turbulence and combustion models for aircraft gas turbine combustor”, Momentum and heat Transfer Process in Recirculating Flows (ASME, HTD, New York) Vol. 13, pp. 71– 89. Tsuchiya, Y. and Horikoshi, C. (1986) “On the spread of rectangular jets”, Exp. Fluids 4, 197–204. Wygnanski, I. and Fiedler, H. (1969) “Some measurements in the selfpreserving jet”, J. Fluid Mech. 38, 577. Xu, H. (1997) Large Eddy Simulation of Turbulent Flows in Square and Annular Ducts and Confined Square Coaxial Jet PhD thesis, Dept. of Mech. Eng., Queen’s University). Xu, H. and Pollard, A. (2001) “Large eddy simulation of turbulent flows in a square annular duct”, Phys. Fluids 13, 1. Zaman, K.B.M.Q. (1996) “Axis switching and spreading of an asymmetric jet: the role of coherent structure dynamics”, J. Fluid Mech. 316, 1–17.