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1 International Journal of Heat and Fluid Flow 3 () Contents lists available at ScienceDirect International Journal of Heat and Fluid Flow journal homepage: Effects of inflow pulsation on a turbulent coaxial jet Seong Jae Jang, Hyung Jin Sung * Department of Mechanical Engineering, KAIST, 335 Gwahangno, Yuseong-Gu, Daejeon 35-7, Republic of Korea article info abstract Article history: Received 4 August 9 Received in revised form January Accepted 6 January Available online 9 January Keywords: Coaxial jet Mixing enhancement LES Pulsation frequency Turbulent pulsating flow Kelvin Helmholtz vortices The effects of inflow pulsation on the flow characteristics and mixing properties of turbulent confined coaxial jet flows have been studied. Large eddy simulations were performed at Re = 9 and the mean velocity ratio of the central to annular jet, U i /U o, was.6. Pulsation was generated in the inflow jet by varying the flow rates. First, inflow pulsation was applied at frequencies in the range. < St <.9 while other parameters were fixed. The pulsation frequency responses were scrutinized by examining the phase- and time-averaged turbulence statistics. The pulsation frequencies St = and.37 were found to produce the largest enhancement in mixing and the largest reduction in the reattachment length, respectively. The effects of the phase difference between the two inflow jets at these two optimal frequencies were then investigated. The optimal phase difference conditions for mixing enhancement and the reduction in the reattachment length were obtained when the strength of the outer vortices was high. Further, we found that the strength of the inner vortices was reduced by varying the phase difference, and the reattachment length was minimized, and that if the strength of the inner vortices was increased, mixing was enhanced. Ó Elsevier Inc. All rights reserved.. Introduction Coaxial jets in which a central jet is surrounded by an annular jet are commonly utilized where mixing between the two jet streams is desired, and are often encountered in engineering applications such as gas turbine combustors, jet pumps, and chemical reactors. Two turbulent shear layers that undergo interaction and mixing are present in coaxial jet flows, and the near-field structures of such jets are very complex. The mixing between two streams is determined by the dynamics and interactions of the vortical structures in the shear layers that develop between the two jets and also between the annular jet and the ambient flow. The vortical structures can be altered by varying the inflow conditions, such as the velocity ratio of the two jets and the inflow pulsation frequency. Knowledge of the dynamics of coaxial jets and suitable pulsation conditions should be useful to efforts to improve mixing efficiencies in combustion devices and industrial chemical systems. Many studies of the reactions of jets to changes in the velocity ratio and to periodic forcing have been performed. Dahm et al. (99) investigated the near-field vortex structures and dynamics of a coaxial, naturally developing jet for various velocity ratios. They referred to the vortex patterns in the inner layer as wake-like if they involved vorticity of opposing signs and shear-layer-like if they involved vorticity of only one sign. It was also found that * Corresponding author. Tel.: ; fax: address: hjsung@kaist.ac.kr (H.J. Sung). the coaxial near-field vortical structures in the inner and outer layers do not develop independently and that there is strong coupling between the two layers. The near-field vortex structures and dynamics were found to be functions of the velocity ratio and the absolute velocities of the two streams. Wicker and Eaton (994) studied the responses of coaxial jets to axial acoustic excitation. They found that excitation of the inner jet produces periodic structures in the inner layer, but has no significant effects on the evolution of the outer layers. Both studies (Dahm et al., 99; Wicker and Eaton, 994) found that the vortical motion is dominated by the vortices emerging in the outer shear layer when the annular jet velocity is larger than the central jet velocity. Rehab et al. (997) studied the effects of the velocity ratio on the flow regimes in the near-field. In particular, these authors have shown how the velocity ratio determines the inner jet potential core length and the conditions for a recirculating, wake-like transition, which occurs when the velocity ratio is increased above a critical value. Villermaux and Rehab () focused on the role played by coherent vortices in near-field mixing in coaxial jets. They showed that the area of the interface between the two streams increases with the instability of the outer shear layer and thus that the vorticity thickness of the outer shear layer is an important parameter. Several studies have focused on the manipulation of flow vortices through active control. The control of mixing in synthetic jets was studied by Ritchie et al. (). They found that large scale structure modifications, especially in the outer mixing layer, have large effects on the mixing properties of coaxial jets. Angele et al. 4-77X/$ - see front matter Ó Elsevier Inc. All rights reserved. doi:.6/j.ijheatfluidflow...3

2 35 S.J. Jang, H.J. Sung / International Journal of Heat and Fluid Flow 3 () (6) experimentally investigated the evolution of a threedimensional structure in a free coaxial jet controlled by micro flap actuators. This study showed that non-axisymmetric forcing enables the faster development of streamwise vortices and leads to enhanced mixing. In these studies (Ritchie et al., ; Angele et al., 6), the forcing was only applied to the outer shear layer because of its dominant role in jet dynamics. Both studies found that the mixing rate of a coaxial jet flow can be altered by applying a suitable excitation to the outer jet. Recently, several interesting studies using direct numerical and large eddy simulations (DNS and LES) have been carried out: the DNS of a free coaxial jet by da Silva et al. (3), and the LES of a confined coannular jet at practically high Re by Akselvoll and Moin (996). These studies revealed the detailed dynamics of vortical structures in uncontrolled jets. Jahnke et al. (5) studied the influence of various parameters (Re, Sc, and the density ratio) by using LES. Balarac et al. (7b) demonstrated that the turbulent mixing process is strongly affected by upstream jet conditions, which modify the transitional state. Inflow conditions that favor the appearance of streamwise vortices were found to strongly enhance jet mixing properties. Balarac et al. (7a) and Mitsuishi et al. (7) studied the effects of excitations that were applied only to the outer jet and that mimic microactuators by using DNS. This study investigated the effects of inflow pulsation on the flow characteristics and mixing properties of turbulent confined coaxial jet flows. Large eddy simulations were carried out at Re = 9, based on the bulk velocity and the outer radius of the annular jet. The mean velocity ratio of the central to annular jet was.6. The main control parameters were the pulsation frequency (. < St <.9) and the phase difference (/) between the annular and central jets. The pulsations in the annular and central jets were generated by varying their flow rates by 5% and %, respectively. The pulsation frequency responses were scrutinized by examining the phase- and time-averaged turbulence statistics. The pulsation frequencies St = and.37 were found to result in the largest enhancement in mixing and the largest reduction in the reattachment length, respectively. The effects of varying the phase difference between the two inflow jets at the two optimal frequencies were then investigated. The inner and outer Kelvin Helmholtz vortices were investigated in detail to observe the effects of these coherent vortices on the mixing enhancement and the reduction in the reattachment length.. Numerical method Many finite difference schemes have been constructed in physical space with cylindrical coordinates. However, the mapping of independent variables is a useful tool for constructing finite difference methods on a non-uniform mesh (Morinishi et al., 4). In the present study, (x, r, h) coordinates in physical space are mapped into (f x, f r, f h ) in computational space. The scaling factors and Jacobians are defined as, h x ¼ dx df x ; h r ¼ dr df r ; h h ¼ r dh df h ; J ¼ h xh r h h : ðþ In the LES approach, flow variables are decomposed into largescale and subgrid-scale components via a filtering operation. The filtered variables are defined by, Fig.. Schematic diagram of the computational domain; inflow pulsation.

3 f ðxþ ¼ Z D f ðx ÞGðx; x Þdx ; where G(x) is the filtering function and D is the computational domain. The filtered and transformed governing equations for the continuity, incompressible Navier-Stokes, and mixture fraction transport equations þ þ þ J ¼ J j j u x h J j j u r h J j j u h h J j j f h j J h j u j h x u hu h x þ h j ¼ h r þ u ru h r h h ¼ h j þ q r r ; þ s rx r þ h h þ h j ð þ s rr s j r þ s rh r ; ðþ ð3þ ð4þ Þ ; ð5þ where u j (j = x, r, h) is the velocity, p is the pressure, and f is the mixture fraction (the filtering operation is omitted for simplicity). Here, the repeated indices j imply summation over j = x, r, h. The viscous shear stresses s ij (i, j = x, r, h) and the scalar fluxes q j (j = x, r, h) are given by, s xx ¼ m t x Re h x ; s rr ¼ m t r ð8þ Re h r ; s hh ¼ m t þ h h þ u r h r ; S.J. Jang, H.J. Sung / International Journal of Heat and Fluid Flow 3 () ð6þ ð7þ s xr ¼ m t r Re h x x h r ; ð9þ s xh ¼ m t h Re h x x ; h h s rh ¼ m t h Re h r u h r r ; ðþ h h q j ¼ a t ; ðj is not repeatableþ: ðþ ReSc h j All variables are non-dimensionalized by the bulk velocity (U o ) and the outer radius (R o ) of the annular jet. The Reynolds number based on the bulk velocity and the outer radius of the annular jet is 9. The mean velocity ratio of the central to annular jet, U i /U o,is.6. The working fluid is air, so the Schmidt number (Sc) is. A dynamic subgrid-scale stress model was used to account for subgrid-scale stresses. By using the eddy-viscosity assumption, the turbulent eddy viscosity m t can be expressed as m t ¼ CD j Sj, with j qffiffiffiffiffiffiffiffiffiffiffiffi Sj¼ S ij Sij. In this study, the model coefficient C was determined by using the dynamic eddy viscosity model proposed by Germano et al. (99), as modified and extended by Lilly (99). In this model, C is not given a priori, but is computed from the flow variables during each simulation. The model constant is averaged over the h direction. A detailed description of the method for determining the model coefficient can be found in the papers of.9 Re = =.44 x r / x ro U /U x i - Present Akselvoll and Moin (996) =5.5 =5.5 = St = St Fig. 3. Effects of varying the pulsation frequency on the mean reattachment length. u x /U i (c) f =5.5 =.44 =8.6 =.44 = r /R o Fig.. Comparison of mean velocity, turbulent intensity and mean mixture fraction. e(x).6.4. St =.37 St = St =.8 St =.6 e(x) / e(x).3.. St = St = St = St 5 5 Fig. 4. Effects of varying the pulsation frequency on the mixing efficiency.

4 354 S.J. Jang, H.J. Sung / International Journal of Heat and Fluid Flow 3 () Germano et al. (99) and Lilly (99). The grid filter width D is set equal to the grid spacing. The test filter is a box filter in real space, applied by using three-point averaging with quadratic interpolation. The box filter is applied in the streamwise and azimuthal directions. No explicit test filtering is applied in the radial direction. The total viscosity /Re + m t is constrained to be non-negative to ensure the numerical stability of time integration, i.e., / Re + m t = when /Re + m t <. The total viscosity is forced to be zero at any point where the model returns a negative value. Following the dynamic subgrid-scale model for scalar transport, the subgrid-scale mixture fraction transport was modeled by assuming it is aligned with the scalar gradient vector. The turbulent eddy diffusivity a t is expressed as a t ¼ C f D j Sj, where j qffiffiffiffiffiffiffiffiffiffiffiffi Sj¼ S ij Sij and C f is a model constant for mixture fraction transport. The test filter and the clipping procedure for mixture fraction transport are the same as those of momentum transport. Details can be found in a paper of Moin et al. (99). The governing equations were integrated in time by using the fractional step method with the implicit velocity decoupling procedure proposed by Kim et al. (). In this approach, the terms are first discretized in time using the Crank Nicholson method, and then the coupled velocity components in the convection terms are decoupled by using the implicit velocity decoupling procedure. St = x r =9.97 r / Ro St =.8 x r =8.79 r / Ro (c) St = x r = r / Ro (d) St =.37 x r =6.473 r / Ro (e) St =.6 x r = 8.44 r / Ro x / R o Fig. 5. Time-averaged streamlines.

5 S.J. Jang, H.J. Sung / International Journal of Heat and Fluid Flow 3 () The decoupled velocity components are then solved without iteration. Since the implicit decoupling procedure relieves the Courant Friedrichs Lewy restriction, the computation time is reduced significantly. The overall accuracy in time is second-order. All the terms in Eq. (3) (6) and the diffusion terms in Eq. (7) are resolved using a second-order central difference scheme in space with a staggered mesh. The bounded QUICK scheme proposed by Herrmann and Blanquart (6) is used to resolve the convection terms in Eq. (7). Note that the central difference scheme applied to the scalar equation with inflow-outflow boundary conditions might lead to numerical instability (Akselvoll and Moin, 995). Further details of the numerical algorithm can be found in a paper of Kim et al. (). By using the homogeneity in the azimuthal direction, the three-dimensional Poisson equation was reduced to a set of decoupled two-dimensional Helmholtz equations through Fourier decomposition. Each Fourier mode was solved with a multigrid algorithm to accelerate the convergence of the iterative procedure. A schematic diagram of the computational domain and the scheme for inflow pulsation are shown in Fig.. The numbers of grid points in the x, r, and h directions were , respectively. 8 of the 35 axial grid points were used to cover the inlet section (upstream of the expansion). The axial grid points were compressed around the expansion point and stretched on either side. The radial grid points were clustered along all solid walls. The grid resolutions were Dx þ min ¼ :; Dxþ max ¼ 8:; Dr þ min ¼ :5; Drþ max ¼ 5:35; and R odh þ ¼ :7 based on u s for the chamber. The time step was.58r o /U o and the total averaging time was 6R o /U o. At each time step, stored two-dimensional instantaneous velocity data obtained from a simulation of periodic turbulent pipe and annular flow with pulsation were provided at the inlet of the main simulation. Pulsation was generated in the inflow jets by varying the flow rates according to the equations: Q annular = Q o { + a o sin(pf p t)}, Q central = Q i { + a i sin(pf p t + /)}, where / is the phase difference between the annular and central jets and f p is the pulsation frequency. The pulsation amplitudes of the annular and central jets were 5% (a o ) and % (a i ), respectively. The pulsation frequency was varied in the range. < St <.9 where the definition of St is St = f p R o /U o, while other parameters were fixed. The convective boundary condition ou/ot + U c ou/ox = was used at the exit, where U c is the mean exit velocity. Periodic conditions were applied in the azimuthal direction for the velocity components, and no-slip boundary conditions were imposed at the solid walls. For the mixture fraction, periodic condition was used in the azimuthal direction. The Neumann condition was applied along all solid walls. The inflow boundary conditions were determined by using Dirichlet conditions: f = at the central jet, and f = at the annular jet. The convective boundary condition was the same as for the momentum equations. To ascertain the reliability and accuracy of the present numerical simulation, we simulated the confined coaxial jet flow with the same conditions of Akselvoll and Moin (996), i.e., Re = 48, and U o /U i = 3.3. As shown in Fig., the mean velocity, turbulent intensity and mean mixture fraction are in good agreement with those of Akselvoll and Moin (996). 3. Effects of varying the pulsation frequency First, we examined the effects of varying the inflow pulsation frequency on the mean reattachment length without phase difference. The variation of the mean reattachment length with pulsation frequency is shown in Fig. 3. The pulsation frequency was varied in the range. < St <.9 while other parameters were fixed. The reattachment length was normalized by that found in M i St increases St = St St =.9 St increases St St =.37 St =.37 St = St =.37. M o.5 St increases St = St increases St =.9 St = Fig. 6. Momentum flux: inner shear layer; outer shear layer.

6 356 S.J. Jang, H.J. Sung / International Journal of Heat and Fluid Flow 3 () the case of no pulsation, x r /R o = 9.97 at St =. We found that the reattachment length is optimally reduced at St =.37 by 34.7% with respect to that of the case of no pulsation. When St is less (more) than.37, the reattachment length increases as St decreases (increases). The reduction in the reattachment length is more than 3% in the range < St <.36; this range is the same as the quarter- to half-harmonic St range predicted by the linear stability theory (Ho and Huerre, 984), for which St o =f p h o / U =.3 (St =.7). The variation with pulsation frequency in the mixing efficiency along the streamwise direction is shown in Fig. 4. The mixing efficiency is defined as, R jf ðx; rþ f jda eðxþ ¼ R jf ðx; rþ f jda ; ðþ.5.5 r / R o.5 r / R o St =.8 St = St =.37.5 St =.6 = x /R. o = u x / Uo u r / U o (c).5 r / R o.5 = u / U θ o Fig. 7. Resolved turbulence intensities..3 (i) (ii) St =.8 St = St =.37 St =.6 u x max / U o...3 (i) (ii) u x max / U o Fig. 8. Maximum turbulence intensities: (i) random fluctuation; (ii) the oscillating component of the phase-averaged velocity. Streamwise direction; radial direction.

7 S.J. Jang, H.J. Sung / International Journal of Heat and Fluid Flow 3 () ; 6 r=ro 6 and f is the mixture fraction ; otherwise associated with a completely mixed state, f =.667. The mixing efficiency monotonically increases from the expansion point along the streamwise direction and gradually approaches, the completely mixed state, at the end of the simulation domain in all cases. Near the expansion point (x/ro < 3), the mixing efficiency increases as St decreases. For lower St (St =.8, ), the mixing efficiency undergoes rapid growth in the range 3 < x/ro < 6. To investigate the overall effects on mixing enhancement of varying the pulsation frequency, we defined the global mixing efficiency as the integral of the mixing efficiency along the streamwise direction from the where f ðx; rþ ¼ 357 expansion point to the end of domain. The global mixing efficiency was normalized with respect to that of the case of no pulsation. The global mixing efficiency of all inflow pulsation cases was found to be higher than that of the case of no pulsation. In the inset in Fig. 4, it can be seen that the mixing is most enhanced at St =. When St is greater than, the global mixing efficiency decreases as St increases. The optimal pulsation frequency, St =, for the mixing enhancement is the same as the frequency of the general wake instability. Time-averaged streamlines for various St values are shown in Fig. 5. As shown in Fig. 3, the recirculation zone is smallest for St =.37. The average mass-flow in the recirculation zone is Fig. 9. Streamwise distributions of ur. The white and black lines represent the positive and negative distributions of radial velocity.

8 358 S.J. Jang, H.J. Sung / International Journal of Heat and Fluid Flow 3 () λ i.5..5 St = λ i Σλ i St =.37 λi Σλ i 3 4 i.6.4. Σλ i a (t) St = St = a (t) Fig.. Relationship of eigenmodes for St =,.37: level of eigenvalues; phase diagram of time-varying coefficients a (t) and a (t). approximately 3% of the total mass-flow for low St (.8, ) and % for high St (.37,.6). The entrainment of fluid into the recirculation zone occurs from the center of the recirculation zone (indicated by + ) to the reattachment point (indicated by the vertical dashed line). As St decreases, the center of the recirculation zone shifts downstream and the distance between the center of the recirculation zone and the reattachment point decreases. Therefore, a large positive radial velocity occurs in this region for low St. The variation of the momentum flux along the streamwise direction is shown in Fig. 6. We define the quantities M i for the inner shear layer and M o for the outer shear layer as follows: M i ðxþ ¼ Z Z 3 4 R o ðu x A þ u r þ u h Þrdrdh; ð3þ Z Z Ro M o ðxþ ¼ A ðu x þ u r þ u h 3 4 Ro Þrdrdh; ð4þ where A ¼ pðr o Þ is the cross-sectional area of the chamber. The momentum flux in the inner shear layer is smallest at St =.37, when the reattachment length is smallest, due to the better homogeneity of the axial velocity component. The momentum flux in the outer shear layer is largest at St =, when the mixing is most enhanced. The x-location for the maximum M o shifts downstream as St decreases for St <.45. The global trends in M i are similar to the variations in the reattachment length shown in Fig. 3, and the variation in the maximum values of M o is similar to that of the mixing enhancement. Thus, when the pulsation frequency is the only control parameter, M i and M o seem to track the variations of the reattachment length and the mixing enhancement, respectively. Fig. 7 shows the time-averaged streamwise, radial and azimuthal turbulence intensities. The global features of u x, u r, and u h are similar. The turbulence intensities have a dominant peak in each shear layer near the expansion point. As the outer shear layer grows, two peaks merge to one peak in the outer shear layer. For high St (.37,.6), the turbulence intensities inside the recirculation zone increase significantly near the expansion point (x/r o < ). For St =.37, which is the optimal St for the minimum reattachment length, all turbulence intensities inside the recirculation zone for the range x/r o < 3 are maximum. The local maximum values of the streamwise and radial turbulence intensities are shown in Fig. 8(i). The local maximum turbulence intensities are largest at St =.37, and smallest beyond x/ R o = 6. The x-locations for the largest values shift downstream with decreasing St. Fig. 8(ii) shows the local maximum velocity profiles of the r.m.s. of the oscillating component, which is defined as the local maximum of: qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~u x ¼ ð~u x Þ ¼ ðhu x i u x Þ ; ð5þ qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~u r ¼ ð~u r Þ ¼ ðhu r i u r Þ : ð6þ This quantity has a clear dependence on St. The rates of increase of ~u x;max and ~u r;max increase with increasing St. However, the rates of decrease are similar downstream. Thus, the size of the effects on ~u x;max and ~u r;max of the Kelvin Helmholtz vortex increases as St decreases. The x-locations for the largest values shift downstream with decreasing St,asinFig. 8(i). The locations of ~u r;max shift further downstream (.5R o ) than those of ~u x;max. To investigate the characteristics of vortical structures in the inner and outer shear layers, the streamwise distributions of u r as a function of time at the fixed radial location (r/r o =.5 and. for the inner and outer shear layers) are shown in Fig. 9. The largescale azimuthal vortical structures, such as the Kelvin Helmholtz vortices due to the inflow pulsation, convect with broad positive and negative u r at the front and back of the vortical structures in the outer shear layer and at the back and front of the vortical structures in the inner shear layer, respectively. In the inner shear layer (r/r o =.5), the inner vortices move faster than the outer vortices close to the expansion point with the same velocity (A I in Fig 9) for three cases (St =, and.37). However, as flow goes downstream x/r o > 6, 4 and for St =, and.37, respectively, the inner vortices slow down with the same velocity as that of the outer vortices. The outer vortices move with the same velocity A o in Fig. 9a and c for St = and.37, respectively. However, for St =, the velocity of the outer vortices (B o in Fig. 9b) is slow in comparison with the velocity for St = and.37, i.e., the slope B o is larger than the slope A o and the velocity is a half of the velocity of the inner vortices near the expansion point (A I ). Thus, the vortical structures for St = and.37 are closely related to the characteristics of the inner and outer shear layers of St =, respectively. The snapshot-based POD (Proper Orthogonal Decomposition) analysis (Lee et al., 8) is carried out by using 48 instantaneous snapshots of u r fluctuations to elucidate quantitative characteristics of large-scale vortical structures. The contribution of each mode is displayed in Fig. a, which is represented by the level of eigenvalues with circle symbols normalized by the total sum of all eigenvalues. The accumulation of eigenvalues at each mode is represented by the lines, which are also normalized by the total

9 S.J. Jang, H.J. Sung / International Journal of Heat and Fluid Flow 3 () sum. As shown in Fig. a, the contribution of the first two modes is similar, which is approximately 7% of total u r fluctuations for St = and % for St =.37. Close values of k and k support that the two modes are originated from identical motions. The relationship of the two modes is clearly shown in the phase diagram of Fig. b, where the horizontal and vertical axes represent the time-varying coefficients a (t) and a (t), respectively. The timevarying coefficient a m (t) of the mth mode is calculated from the following formulation: Z a m ðtþ ¼ u r ðx; tþw m ðxþdx ð7þ X where w m (x) is the mth eigenmode. It is interesting to note that a torus is generated by the phase diagram of a (t) and a (t). Distortion of the circular shape represents the variation of phase difference between the two eigenmodes, whereas the dispersion of circulation is caused by the amplitude change of oscillations. The present torus shape means that small fluctuations of oscillating amplitudes are expected in the first two modes while the phase difference between the two eigenmodes is nearly constant. The first and second eigenmodes of u r fluctuations are shown in Fig.. As mentioned above, the first and second modes are originated from identical motions and the phase difference between the two modes is estimated to be a quarter of one period. These two eigenmodes clearly show the most energetic motions of vortical structures due to the inflow pulsation. For St =, the energetic motions appear in the range of < x/r o < 7 and < x/r o < 5 for St =.37 and ~u r;max has a large value in this range as shown in Fig. 8. In the calculation of the kernel matrix of POD using the instantaneous snapshots, we have tested three sub-domains. One is to use the entire radial domain with.5 < x/r o < 6. Others are to use the restricted radial domains in the inner and outer shear layers,.48 < r/r o <.5 and.97 < r/r o <.3, in the range.5 < x/r o < 6. The first two modes for three sub-domains are almost the same as shown in Fig. b d. Thus, the most energetic motions of the vortical structures in the inner and outer shear layers are almost identical. The figures on the left and right of Fig. show the time- and phase-averaged mixture fractions, respectively. The snapshots of the phase-averaged mixture fractions were obtained at t = /4T, Fig.. Eigenmodes of u r fluctuations. The gray and black contours represent the positive and negative distributions of eigenmodes.

10 36 S.J. Jang, H.J. Sung / International Journal of Heat and Fluid Flow 3 () where T is the pulsation period. It can be seen that the stream issuing from the central jet moves further towards the corner of the expansion point as St decreases. Thus, the mixing efficiency becomes higher near the expansion point as St decreases, as shown in Fig. 4. The vortex identification method of Zhou et al. (999) was adopted to visualize the vortical structures by using a contour of swirling strength k ci, where k ci is the imaginary part of the complex eigenvalue of the velocity gradient tensor. The swirling strength is a quantity that can be used to detect vortex cores and to distinguish vortical structures from shear regions. In the figures on the right of Fig., the two-dimensional swirling strength k ci (indicated by solid lines) clearly shows the azimuthal vortical structures due to the inflow pulsation; these vortices convect downstream over time. The rotation direction of the azimuthal vortices is indicated in Fig. c. The inner vortex rotates in the opposite direction to the outer vortex. The inner vortex is trapped in the free space between two consecutive outer vortices in Fig. b d; this is the locking phenomenon reported by many previous studies. At St =.6, the inner vortex is divided into two parts due to the strong outer vortex: one is trapped between outer vortices as in the other cases, and the other vortex is located beneath the outer vortex, as shown in Fig. e. The first outer azimuthal vortex forms faster and the distance between the two outer vortices decreases as St increases. However, the radius of the outer vortical structure increases as St decreases. The dashed line is the mixture fraction of the completely mixed state, f =.667. It can be seen that the outer vortices pinch the inner shear layer, as indicated by the dashed line. This pinching effect Fig.. Averaged mixture fractions and spanwise vortical structures (k ci ). Left: time-averaged; right: phase-averaged. Dashed line: f =.667; Solid line: k ci.

11 S.J. Jang, H.J. Sung / International Journal of Heat and Fluid Flow 3 () St =.8 St = St =.37 St =.6 3 St increases St increases Fig. 3. Swirling strength of the vortex center: outer shear layer; inner shear layer. r / R o.5 St =.8 St =.5 St =.37 St =.6 = Fig. 4. Fluctuations in the r.m.s. of the mixture fraction. f means that the central jet expands in the radial direction at the front of the outer vortex and the annular jet invades at the back of the outer vortex. Thus, the mushroom-shaped structures are formed by the dashed line (f =.667). When the mushroomshaped structures are generated, the mixture fraction in the central region of the structures diffuses quickly in the radial direction. The area of the interface between the two inflow jets increases because the stream issuing from the central jet invades the annular jet region and the annular jet penetrates deeply in the central jet area. This structure enhances the large scale mixing that arises due to the Kelvin Helmholtz vortices. The variation of the swirling strength of the vortex center due to inflow pulsation is shown in Fig. 3. The x-location of the maximum swirling strength of the outer vortex shifts downstream with decreasing St. This maximum location is the same as that of ~u x shown in Fig. 8a. The maximum value and the increase rate of the swirling strength of the outer vortex increase with increasing St. The swirling strength of the outer vortex is twice as large as that of the inner vortex. Fig. 4 shows the fluctuations in the r.m.s. mixture fraction. The peak in f in the inner shear layer diffuses to the center line and chamber wall as the streamwise distance increases. Downstream the peak moves from the inner shear layer to the center line of the chamber. For the r.m.s. velocity fluctuations shown in Fig. 7, the r-locations of the maximum value are in the outer shear layer due to the large shear and the azimuthal vortical structures in the outer shear layer. Even though the turbulence intensities increase inside the recirculation zone due to the inflow pulsation, f is very small due to the lack of mixture fraction in the outer shear layer near the expansion point. The local maximum in the fluctuations of the r.m.s. mixture fraction is shown in Fig. 5a. fmax describes small scale mixing and seems to be less sensitive to the pulsation St than ~u r;max near the expansion point, as shown in Fig. 8(ii). Fig. 5b shows the local maximum mixture fraction profiles of the r.m.s. of the oscillating component. ~ f max describes large scale mixing and has a clear dependence on St, in contrast to fmax. The locations of largest ~ f max are the same as the maximum swirling strength locations of the outer vortex, as shown in Fig. 3a. The largest value of ~ f max increases with increasing St..4.3 St =.8 St = St =.37 St =.6 f max Fig. 5. Maximum r.m.s. of the mixture fraction: random fluctuation; oscillating component of the phase-averaged mixture fraction.

12 36 S.J. Jang, H.J. Sung / International Journal of Heat and Fluid Flow 3 () x r / x ro x r / x ro φ ( ) St = φ ( ) Fig. 6. Effects of varying the phase difference on the mean reattachment length for St =.37. e(x).6.4. St =&φ =33 St =&φ =9 e(x) / e(x).3.. St = St = St =&φ = φ ( ) 5 5 Fig. 7. Effects of varying the phase difference on the mixing efficiency for St =. 4. Effects of the varying the phase difference between the two inflow jets To investigate the effects of varying the phase difference between the two inflow jets, we carried out large eddy simulations with the same pulsation inflow and 3 increments of the phase difference for the two optimal pulsation frequencies: St =.37 and. The variation of the reattachment length with the phase difference for St =.37 is shown in Fig. 6. We found that the reattachment length is most reduced at / =3 by 34.9%. A phase difference of / = 3 reduces the reattachment length by an amount that is.6r o (.4%) more than the reduction at / =. It can be seen that the reattachment length increases significantly in the range 5 < / < 4. For / =, the reattachment length is almost the same as that of the case with no pulsation. The variation seems to be symmetric about / =3 (or ). The variation of the mixing efficiency with the phase difference for St = is shown in Fig. 7. The most effective phase difference for mixing enhancement is / = 33. / = 33 increases the mixing efficiency by an amount that is.6% more than that at / = in the range < x/r o < 8. For / =, the trend in the mixing efficiency is similar to that in the case with no pulsation. However, the mixing efficiency is higher than that found in the no pulsation case by 5%. The global mixing efficiency gradually decreases as / increases from 33 ( 3 ) to, and abruptly increases as / increases from to 33. The vortical structures and the iso-surfaces of the mean mixture fraction (f =.667) are shown in Fig. 8. To visualize the vortical structures, we used an iso-surface of swirling strength k ci = 3.. Four snapshots taken at intervals of T/4 are shown, where T is the pulsation period. For St =.37 and / =3 in Fig. 8b, both of the streamwise and radial vortical structures are more activated near the expansion point and rapidly weaken beyond x/r o = 5. The large scale spanwise vortical structures consist of small scale vortices crowded in the outer shear layer between the annular jet and the expansion chamber, and are observed in the range < x/r o <4. Due to the streamwise and spanwise vortical structures near the expansion point, the center of the recirculation region is biased towards the expansion point and the maximum reduction in reat- Fig. 8. Vortical structures and iso-surfaces of the mean mixture fraction (k ci = 3.).

13 S.J. Jang, H.J. Sung / International Journal of Heat and Fluid Flow 3 () tachment length occurs. In Fig. 8c, the large scale spanwise vortical structures are observed in the range < x/r o < 6. Near x/r o =6, the streamwise vortical structures are observed on the center line. The iso-surfaces of the mixture fraction contain a mushroomshaped structure in the range 3 < x/r o < 6. The mushroom-shaped structure is formed in the free space between two consecutive outer spanwise vortical structures and indicates the ejection of mixture fraction seeded in the central jet towards the lateral wall of the expansion chamber. This event corresponds to large scale mixing due to the Kelvin Helmholtz coherent vortices. There are small-scale streamwise and radial vortices crowded into the center and shoulder regions of the mushroom-shaped structure, respectively. These small scale vortices promote turbulent scale mixing. The same POD analysis is examined by using 48 instantaneous snapshots of u r fluctuations at the optimal phase difference for mixing enhancement at St = and.37, respectively. In Fig. 9, the contribution of each mode and the phase diagram between the first two eigenmodes are similar to those of no phase difference at the same St as shown in Fig.. The first and second modes are originated from identical motions and the phase difference between the two modes is estimated to be a quarter of one period. In Fig., the first two eigenmodes at the optimal phase difference are expanded to the center line in comparison with those of no phase difference as shown in Fig. This means that the large-scale vortical structures near the center line, i.e., inner vortices in the inner shear layer, are strengthened by the phase difference. Moreover, in the first two eigenmodes for St = with / = 33, the vortical structures move through the center line near x/r o = 6. The mixing is enhanced by these motions which cut the central jet with the annular jet. Fig. a and b show the time- and phase-averaged mixture fractions at the optimal phase differences for mixing enhancement at. 4.5 λ i..5 St =&φ = 33 λ i Σλ i St =.37&φ = 7 λ i Σλ i 3 4 i.6.4. Σλ i a (t) - -4 St = & φ =33 St =.37 & φ = a (t) Fig. 9. Relationship of eigenmodes for St = and / = 33, St =.37 and / = 7: level of eigenvalues; phase diagram of time-varying coefficients a (t) and a (t). Fig.. Eigenmodes of u r fluctuations. The gray and black contours represent the positive and negative distributions of eigenmodes.

14 364 S.J. Jang, H.J. Sung / International Journal of Heat and Fluid Flow 3 () St = and.37, respectively. The global mixing efficiencies are.5 for St = with / = 33 and. for St =.37 with / = 7. For St = with / = 33, which are the conditions resulting in the most enhanced mixing, the mushroom-shaped structure is detached from the main central jet stream near x/ R o = 5. When this structure is detached, the outer jet penetrates to the center line. For St =.37 with / = 7, the stream issuing from the central jet is closer to the corner of the expansion point than in Fig. d. A mushroom-shaped structure is also observed. The variation of the swirling strength of the vortex center with phase difference is shown in Fig.. When the reattachment length is minimized (indicated by the dashed-dotted line; St = and / =9, St =.37 and / =3 ), it can be seen that the outer swirling strength is large and the inner swirling strength Fig.. Averaged mixture fractions and spanwise vortical structures (k ci ). Left: time-averaged; right: phase-averaged. Dashed line: f =.667; Solid line: k ci..5 (i) (ii) St = φ =9 φ = φ = (i) (ii) St =.37 φ =3 φ = φ = Fig.. Swirling strength of the vortex center: St = ; St =.37. (i) Outer shear layer; (ii) inner shear layer.

15 S.J. Jang, H.J. Sung / International Journal of Heat and Fluid Flow 3 () is small. When the mixing is enhanced (indicated by the solid line; St = and / = 33, St =.37 and / = 7 ), both the outer and inner swirling strengths are somewhat large. In this case, the large positive and negative u r, which is located at the front and back of the outer vortex, is expanded near the center line by the strong inner vortex. Thus, the central jet is expanded towards the lateral wall of the chamber and the annular jet penetrates to the center line. As a result, mushroom-shaped structures form. At / = for both St = and.37, the outer swirling strength is the lowest and the effects of the inflow pulsation are the weakest. However, the inner swirling strength for St =.37 is large. Thus, the effects of the phase difference are greater for the outer vortex. The phase-averaged axial velocities are shown in Fig. 3; the two axial velocities are those of the annular and central jets. U x,- max and U x,centerline denote the maximum axial velocity and the axial velocity of the center line along the streamwise direction, respectively. The dashed lines are the time-averaged results. In Fig. 3a, the effects of inflow pulsation vanish beyond x/r o =7. The spatial peaks of the two velocities are not matched up to approximately x/r o = 5. In the region < x/r o < 5, the number of local peaks of U x,centerline is reduced to one, which is different to that of U x,max. This result suggests that the strong outer vortex does not strongly influence the central jet, because the local maximum positions of U x,max are linked to the center positions of the outer vortex. In Fig. 3b, the effect of inflow pulsation disappears beyond x/r o = and the spatial peaks of the two velocities are well matched along the streamwise direction. This phenomenon is closely related to the mushroom-shaped structures. In the neck and head regions of the mushroom-shaped structures, the two velocities become the local maximum and minimum, respectively. This behavior is also observed at St =.37 and / = 7..4 St =.37& φ= 3 St = & φ= 33. U x,max t = /4T U x,max t = /4T.6.4 U x,centerline U x,centerline.4. t = /4T t = /4T t = /4T t = /4T t = 3/4T t = 3/4T Fig. 3. Phase-averaged streamwise velocity profiles.

16 366 S.J. Jang, H.J. Sung / International Journal of Heat and Fluid Flow 3 () The time-averaged mixture fractions at r/r o =,.75, and.5 are shown in Fig. 4. The mixture fractions of the cases with enhanced mixing (St = and / = 33, St =.37 and / = 7 ) diffuse most rapidly and approach f. The region with f = and for St =.37 is shorter than that for St = at the center of both the central and annular jets (r/r o = and.75). This St = φ =9 φ = φ = 33 St =.37 φ =3 φ = φ =7 f.6 r /R o = r /R o =.4... r /R o =.75 r /R o =.75.. r /R o =.5 r /R o = x / R x / R Fig. 4. Time-averaged mixture fractions: St = ; St = (i) (ii) St = φ =9 φ = φ = 33.3 f max.. f max (i) (ii) St =.37 φ =3 φ = φ = Fig. 5. R.m.s. of the mixture fraction. (i) Random fluctuation; (ii) oscillating component of the phase-averaged mixture fraction.

17 S.J. Jang, H.J. Sung / International Journal of Heat and Fluid Flow 3 () result arises because of the strong inner and outer vortices, which are formed earlier than those for St =. Beyond the core region of f, the high rate of variation of f for St = lasts longer than that for St =.37. However, f for St = approaches f faster than that for St =.37 beyond x/r o =6.Atr/R o =.5 up to approximately x/r o = 4 (inside the recirculation zone), f in the cases of enhanced mixing is higher than in other cases by 3% due to the penetration of f issued from the central jet because of the influence of the mushroom-shaped structure. Thus, the mixing efficiency in the case of enhanced mixing at the expansion point is higher than in other cases, even though f does not change at the center of either of the central or annular jets. The local maximum of the r.m.s. of the fluctuation of the mixture fraction is shown in Fig. 5. fmax for the cases of enhanced mixing is smaller than that for the cases of minimum reattachment length (St = and / =9, St =.37 and / =3 ) because of a reduction in the turbulent scale mixing due to the better homogenization of f. ~ f max for the cases of enhanced mixing is larger than that for the cases of minimum reattachment length due to their strong inner vortices and mushroom-shaped structures. The effects of the outer vortices on ~ f max are less than those of the inner vortices due to a lack of f in the outer shear layer. Thus, the effects of the turbulent scale mixing on the mixing of f are more dominant than on the large scale mixing in the cases of minimum reattachment length, and both the turbulent and the large scale mixing are dominant in the cases of enhanced mixing. 5. Summary and conclusions Large eddy simulations of turbulent flows through a coaxial jet were performed at Re = 9 to investigate the effects of inflow pulsation. The mean velocity ratio of the central to annular jet was.6. Pulsation was generated in the inflow jets by varying their flow rates. The pulsation amplitudes of the annular and central jets were 5% and %, respectively. To investigate the effects of varying the pulsation frequency, the pulsation frequency was varied in the range. < St <.9 while other parameters were fixed. The reduction in the reattachment length was more than 3% in the range between the quarter- (St = ) and half-harmonic (St =.36) frequencies described by linear stability theory. The maximum reduction in the reattachment length, 34.7%, was obtained at St =.37. At this frequency, the momentum flux in the inner shear layer was smallest and all the turbulence intensities inside the recirculation zone were maximized in the range x/r o < 3. The mixing was most enhanced at St =, which is the same as the frequency of the general wake instability. At this frequency, the momentum flux in the outer shear layer was largest and the pinching effect, which results in mushroom-shaped structures and enhances large scale mixing, was intensified. In order to determine the effects of varying the phase difference between the two inflow jets, we performed large eddy simulations by using the same pulsation inflow with 3 increments of the phase difference for the two pulsation frequencies: St =.37, which optimizes the minimum reattachment length, and St =, which optimizes the mixing enhancement. We found that the reattachment length was most reduced for St =.37 with / =3 by 34.9%. When the reattachment length is minimized, the outer swirling strength is large but the inner swirling strength is small. The most effective phase difference for mixing enhancement was obtained at St = with / = 33. In this case, the large positive and negative u r, which is located at the front and back of the outer vortex, was expanded near the center line by the strong inner and outer vortices, and mushroom-shaped structures formed. From the r.m.s of the mixture fraction, both turbulent and large scale mixing are dominant in the cases of enhanced mixing. The optimal phase difference conditions for mixing enhancement and the reduction in the reattachment length were obtained when the outer vortices were strong. Then, if the inner vortices are weakened by the phase difference, the reattachment length is minimized, and if the inner vortices are strong, the mixing is enhanced. Acknowledgements This work was supported by the Creative Research Initiatives (Center for Opto Fluid Flexible Body Interaction) of MEST/NRF and partially supported by KISTI under the Grand Challenge Supercomputing Program (KSC-8-G-). References Akselvoll, K., Moin, P., 995. 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