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Induced co-flow and laminar-to-turbulent transition with synthetic jets
Computers & Fluids 38 (2009) 1011–1017
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Computers & Fluids j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p fl u i d
Induced co-flow and laminar-to-turbulent transition with synthetic jets John Abraham *, AnnMarie Thomas University of St. Thomas, School of Engineering, 2115 Summit Avenue, St. Paul, MN 55105-1079, United States
a r t i c l e
i n f o
Article history: Received 18 August 2007 Received in revised form 14 December 2007 Accepted 20 January 2008 Available online 14 April 2008
a b s t r a c t A detailed numerical solution of the fluid flow patterns engendered by a synthetic jet has been carried out. The synthetic jet is caused by a reciprocating piston assembly which is attached to a large, stationary cavity. It was found that a significant momentum efflux is produced by the synthetic jet assembly. Also, fluid entrainment by the synthetic jet causes a coincident flow around the exterior of the cavity (selfinduced co-flow). Numerical solutions allow the investigation of the effect of reciprocation stroke length and piston speed on the resulting flow patterns and momentum flows. For all investigated cases, the contribution made by the co-flow to the momentum flowrate is found to be small. In order to account for the simultaneous existence of both laminar and turbulent regions, two numerical approaches were taken. One approach used the Shear Stress Transport (SST) turbulence model while the other used a newly devised transitional turbulence model. Of particular interest was a comparison between the predicted locations of the laminar-to-turbulent transition based on the two independent models. The excellent agreement between the two models reinforces the use of the SST model throughout the domain. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Synthetic jets can be used to provide a means of control for vehicles or for causing a directed flow within a larger fluid region. The jet is created by a vibrating wall or piston which is attached to a cavity. One wall of the cavity is an orifice plate through which fluid is allowed to flow during the reciprocating cycles. Inspiration for the use of synthetic jets for this purpose is taken from observation of living animals which use similar jets for their locomotion [1–5]. Alternating compression and expansion strokes by the wall or piston engender the fluid motion required for the formation of the pulsating jet. Throughout an entire reciprocating cycle, there is no net mass flow out of the cavity. Despite this fact, the jet creates a net flow of momentum in the direction of motion of the underwater body. Pioneering work on synthetic jets has provided experimental visualization of the flow patterns [6–9]. This body of work includes synthetic jets created by vibrating motion and by acoustic excitation. In [10], Schlieren and smoke tracing showed the vortex dynamics of the synthetic jet. More recently [11,12], simulations of synthetic jets have been compared to experiments. In [11] for instance, the solution domain did not extend into the cavity proper. Rather, the presence of the cavity was simulated by an imposed velocity which was applied to a replica of the orifice. In addition, the lateral extent of the cavity
* Corresponding author. Tel.: +1 651 962 5766; fax: +1 651 962 6419. E-mail address: [email protected] (J. Abraham). 0045-7930/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compfluid.2008.01.028
was infinite so that entrained fluid was drawn in only in the lateral direction. A number of works have been exclusively numerical. Included here are calculations on a shallow, wide cavity [13] with a planar geometry and an orifice plate of infinite size. In [14], calculations show the effect of cavity size on the jet. In addition, interactions between the synthetic jet and a crossflow were presented. The geometry of [14] was planar and again the orifice plate was infinite in extent. To the best knowledge of the authors, the numerical research reported in the literature has focused primarily on orifice plates whose lateral dimension is infinite. The infinite extent of the orifice prohibits flow from passing around the body of the cavity and thereby eliminates the possibility of the development of co-flow. An exception to this is the work done on the effects of a synthetic jet on a bump. Synthetic jets which are created by wall bumps are partially wall-bounded flows and exhibit entirely different flow characteristics than synthetic jets created by orifice cavities [15]. The present investigation makes use of information from an experimental investigation carried out with water as the working fluid [16]. There, a cavity was connected to a piston-cylinder device, as shown in Fig. 1. The experimental results provided conclusive evidence of the existence of a co-flow around the exterior of the device. The experiments were performed with a single operational protocol which is one case among many considered here. The numerical work to be presented will provide a detailed analysis of the fluid flow associated with a synthetic jet. The calculations will investigate the effect of operating parameters on the key features of the flow. Consideration of the transitional nature of the flow will be made and comparisons of results obtained from
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Nomenclature A dc do E F1 g hc k _ M p P r Reht S t T t
area channel width orifice width destruction term turbulence model blending function gravitational acceleration cavity height turbulent kinetic energy momentum flowrate pressure production term radial coordinate transition onset Reynolds number turbulence model constant time period velocity
a purely turbulent simulation model and a newly developed transitional model will be given. An assessment of the effect of the coflow on the propulsive ability of the synthetic jet will also be made. The calculations will span a fourfold variation in both the stroke length and the reciprocation period. The effect of these variations on the fluid flow and momentum transfer will be assessed. 2. Details of the numerical solution
wc xi xpiston z
cavity width tensor coordinate position of piston axial coordinate
Greek symbols a turbulence model constant b turbulence model constant c turbulence intermittency q Density r Prandtl number x specific turbulent dissipation rate Subscripts t turbulent c turbulence intermittency transition onset ht
symbolic representations of detached vortices which are created as flow passes into and out of the cavity. The annotations of Fig. 2 show some of the key boundaries of the fluid domain which are relevant to the computational work. As mentioned in the preceding section, the channel is bounded at the top by a moving plate which is allowed to reciprocate vertically. The other bounding walls of the cavity and channel are nonmoving and the no-slip condition is enforced there. At the upper right- and left-hand sides of the domain, fluid is allowed to enter into the zone while at the bot-
A generic description of the expected flow patterns is depicted in Fig. 2. Arrows show the patterns of co-flow around the cavity body and the jet flow emanating from the orifice. Also shown are
Fig. 1. Schematic diagram of the synthetic jet apparatus and relevant operating nomenclature.
Fig. 2. Expected flow patterns and relevant components of the solution domain.
J. Abraham, A. Thomas / Computers & Fluids 38 (2009) 1011–1017 Table 1 Values of geometric parameters Parameter
Symbol
Value (cm)
Cavity width Cavity height Channel width Orifice width
wc hc dc do
7.0 9.9 2.5 1.9
Stroke (cm) Period (s)
Case number 1 2
3
4
5
5.1 0.83
5.1 3.32
2.5 0.83
1.3 0.83
5.1 1.66
tom of the diagram, the fluid exit is marked where the flow leaves the region. The right and left vertical edges are non-moving walls and the no-slip condition is enforced there. A listing of values of dimensional and operational parameters are set forth in Tables 1 and 2, respectively. It can be seen from Table 2 that the investigation spans a four-fold range for both the reciprocation period and the stroke length. The effect of varying these parameters on the flow patterns and momentum flowrates will be assessed in Section 3 of this report. 2.1. The governing equations The complicated flow patterns and regions of free shear virtually guarantee that portions of the fluid will be turbulent. This prediction is reinforced by the flow separation which will occur near the orifice. Furthermore, the pulsating motion of the jet created by the piston reciprocation requires the solution be carried out in an unsteady fashion. In order to accommodate these features, the unsteady form of the Shear Stress Transport (SST) model [17] was utilized. This model combines features of the popular k–e model of [18] with the k–x taken from [19]. The combination of these approaches is performed in such a manner that the k–x equations dominate in the near-wall region [20] while k–e holds away from the wall. In this way, the advantage of the near-wall calculations of k–x are realized yet its sensitivity to freestream values of the turbulence level is mitigated [21]. It has been shown that the SST approach provides superior results for near-wall and separated flow calculations [22–27]. The unsteady RANS equations of motion are written in tensor form as oui ¼ 0; oxi ouj op o ouj ¼ þ qg i þ ðl þ lt Þ q ui oxi oxi oxi oxi
ð1Þ j ¼ 1; 2; 3;
ð2Þ
where lt is the turbulent viscosity. Additional transport equations are presented for the turbulent kinetic energy, k, and the specific dissipation rate, x. Those transport equations, written in tensor form are oðqkÞ oðqti kÞ o ¼ Pk bqkx þ þ ot oxi oxi
In these expressions, q is the fluid density, r represents the Prandtl number associated with the transport of k or x. The terms a, S, and b are model constants, and Pk is a production term for turbulent kinetic energy. F1 is a blending function that takes on a value of zero away from solid surfaces and one near surfaces. The effect of blending is that the SST model behaves as k–e away from the wall and k– x within the boundary layer. 2.2. Transitional flow
Table 2 Values of the stroke and reciprocation period Parameter
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l ok lþ t rk oxi
ð3Þ
and oðqxÞ oðqti xÞ o l ox ¼ aqS2 bqx2 þ lþ t þ rx oxi ot oxi oxi 1 ok ox : þ 2ð1 F 1 Þqrx2 x oxi oxi
ð4Þ
Eddy-viscosity turbulence models such as the k–e, k–x, or their derivative (SST) have been developed and evaluated for situations in which the freestream flow was either fully laminar or turbulent. Their ability to model flow that is undergoing laminar-to-turbulent transition has been a source of doubt for some time. This inability has spawned a very extensive body of research, particularly in recent years which has resulted in a new approach to modeling flow that expresses laminar and turbulent regions within a single fluid domain. Pioneering work such as that carried out by [28–31] has demonstrated various modes of laminar–turbulent transition including natural transition, bypass transition, separation-induced transition, and wake-induced transition. Efforts to predict the location at which laminar boundary-layer flow begins to break down has been based on empirical correlations which relate the freestream turbulence levels to the transition Reynolds number based on the momentum thickness [32]. These correlations, coupled with the concept of flow intermittency first set forth in [33,34] have enabled a new approach to be developed in which a new transport equation is proposed for intermittency, c [35– 41]. Of the various models, the most viable for implementation in a modern CFD environment with an unstructured mesh and parallel processing, is that of [41]. That model requires the solution of additional transport equations for the intermittency, c, and for the transition onset Reynolds number, Reht. The new equations are oðqcÞ oðqti cÞ o l oc ð5Þ ¼ Pc Ec þ lþ t þ rc oxi ot oxi oxi and oðqReht Þ oðqti Reht Þ o oReht : ¼ P ht þ rht ðl þ lt Þ þ oxi ot oxi oxi
ð6Þ
The terms Pc, Pht, and Ec, are production and destruction terms, respectively. Details for their evaluation can be obtained from [41]. For the calculations which are provided in the present report, a comparison between results obtained from the SST and the transitional models will be made. The comparison will demonstrate whether a traditional two-equation turbulence model such as the SST is suitable for capturing a flow that exhibits both laminar and turbulent characteristics. 2.3. Initial and boundary conditions In order to complete the description of the fluid flow problem, specification of the flow at all boundaries is required. At all walls, the no-slip condition was enforced so that the fluid was either stationary or moved with the velocity of the wall. The reciprocating motion of the piston was sinusoidal so that Stroke 2p sin t ; ð7Þ xpiston ¼ 2 T where T is the period of the motion. At the rightmost and leftmost boundaries, a wall was employed so that fluid there would be
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motionless. At the lower boundary, an exit condition was given with weak closure conditions enforced on the second derivatives in the flow direction. Initially, at t = 0, the entire fluid region was at rest. The simulation spanned ten full reciprocating cycles of the piston to ensure that timewise periodic flow was achieved. 2.4. The numerical method The circular geometry of the cavity, channel, and the surrounding water was initially constructed using more than 80,000 elements to resolve the spatial domain. These elements were preferentially deployed in regions where large variations in the transported variables were expected. In addition, thin prismatic elements were deployed along interfaces between the fluid and solid surfaces to ensure proper resolution of the boundary layers. The unsteady computations were performed with 100 timesteps for each reciprocating cycle. All timesteps contained 10 iter-
ations, each of which was performed using a two-step, multi-grid computational algorithm. The time stepping was carried out using an Euler backward scheme of second-order accuracy. Coupling of the velocity–pressure equations was achieved on a non-staggered, collocated grid using the techniques developed by Rhie and Chow [42] and Majumdar [43]. The inclusion of pressure-smoothing terms in the mass conservation equation suppresses oscillations which can occur when both the velocity and pressure are evaluated at coincident locations. The advection term in the momentum equations was evaluated by using the upwind values of the momentum flux, supplemented with an advection-correction term. The correction term reduces the occurrence of numerical diffusion and is of second-order accuracy. Further details of the advection treatment can be found in [44]. Mesh and time-step values were sufficiently small to ensure a solution that was independent of their values. The selected values resulted from a thorough independence study during which both the element sizes and time steps were halved and results were
Fig. 3. SST velocity magnitude contours obtained for as the piston is (a) moving downwards, (b) at bottom-dead center, (c) is moving upwards, and (d) is at top-dead center.
J. Abraham, A. Thomas / Computers & Fluids 38 (2009) 1011–1017
compared. When the sequential reductions failed to yield noticeable changes in the results, it was determined that the settings were sufficiently refined. 3. Results and discussion 3.1. Comparison of SST and transitional results Fig. 3a–d show a series of global velocity magnitude contour diagrams. The figures correspond to results obtained using the SST model for turbulent simulation. In Fig. 3a, the piston is passing downward through its center point, and a mass of fluid is seen to be ejected from the cavity at relatively high speed as indicated by the contour. In Fig. 3b, the fast-moving fluid is seen to be detached from the orifice and is moving downwards, away from the cavity. In Fig. 3c the piston is traveling upwards, drawing fluid into the cavity from the surrounding region. A remnant jet of fluid is seen to be continuing its downwards motion, away from the cav-
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ity. Finally, when the piston is at a top-dead-center position in Fig. 3d, a fast-moving mass of fluid is observed being pulled into the cavity. An overall view of the velocity patterns from Fig. 3 depict the oscillating inflow and outflow during an entire reciprocating cycle. The contour is scaled by the legend at the left-hand side of the figures. Fig. 4a–d show the counterpart results based on the transitional model. An examination of Figs. 3 and 4 reveals a close similarity between the results for the SST and transitional approaches. This similarity includes both the flow patterns and velocity magnitudes inside and outside of the cavity space. These results strongly support the congruence of the SST and transitional models. Similar agreement was seen with the other cases indicated in Table 2. As a consequence, the remaining discussion will solely make use of results obtained from the SST model. That discussion will focus on the mass and momentum flowrates engendered by the synthetic jet throughout an entire reciprocating cycle.
Fig. 4. Transition velocity magnitude contours obtained for as the piston is (a) moving downwards, (b) at bottom-dead center, (c) is moving upwards, and (d) is at top-dead center.
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3.2. Mass and momentum flowrates through the orifice
Table 5 Momentum flowrate imparted to the co-flow during one cycle
While conservation of mass guarantees that there can be no net mass efflux from the cavity, the synthetic jet is capable of generating a net momentum flow in the streamwise direction. The net momentum flow is a critical measure of the propulsive capability of the synthetic jet device. The total momentum imparted to the fluid during a cycle was calculated by timewise integration of the local momentum variations at the orifice. The integral, expressed in both its integral form and its numerical analog is " # Z T Z t¼T X X 2 2 _ qt dA dt ¼ qti DAi Dti ; ð8Þ M orifice ¼
Case
Momentum flowrate per cycle (N)
Average momentum flowrate per second (N/s)
1 2 3 4 5
1.31 104 6.14 105 4.94 105 1.28 105 1.34 106
1.58 104 3.70 105 1.49 105 1.54 105 1.61 106
0
Orifice Area
t¼0
Orifice Area
where q is the fluid density, t is the streamwise component of the fluid velocity, and DA is the horizontal projection of the area of the elements which span the orifice. A tabulation of resulting momentum flowrates is presented in Table 3. The results are keyed to the five cases set forth in Table 2. In all cases, it was found that there was a net momentum flow in the streamwise direction. This finding is verified from experimental measurements made in [16]. The table presents two sets of results. The first results list the total momentum flowrate passing through the orifice throughout an entire reciprocation of the piston. The second set shows the time-averaged rate of momentum transfer to the fluid. From the tabulated results, it is seen that the variation of momentum flowrate very nearly follows the behavior 2
_ orifice stroke length : M 2 Period
ð9Þ
respective masses are divided by the corresponding time periods to obtain the average mass flowrate for the individual cases. It can be seen from the results that the effect of an extended time period (Cases 2 and 3) is to reduce the average mass flowrate. Likewise, a shorter stroke length gives rise to lesser mass flowrates. It is also seen that the mass flowrate is substantially more sensitive to stroke length than to reciprocation period. Next, attention is turned to the contribution of momentum by the entrained fluid in the co-flow region. The calculation of the co-flow momentum was performed by the numerical integration presented in Eq. (10). There, the timewise variation of the momentum flow was integrated spatially and over time to give the results shown in Table 5. Also shown in the table are the average momentum flow rates. Z T Z _ co-flow ¼ qt2 dA dt M 0 Co-flow Area " # t¼T X X 2 qti DAi Dti : ð10Þ ¼ t¼0
The dependence of momentum flowrate to the operating parameters indicated in Eq. (9) agrees qualitatively with results of experimentally observed in [16]. 3.3. Mass and momentum flowrates through the co-flow region The numerical simulation clearly shows a directed streamwise flow in the co-flow region throughout the entire period of reciprocation. This finding is in contradistinction to the observations made at the orifice where the direction of flow reverses with the direction of the piston. As a consequence, it is relevant to calculate the mass flowrate through the co-flow region throughout the reciprocation. The resulting mass flowrates are shown in Table 4. The table displays two sets of results. First, the total mass passing through the co-flow region for one cycle is presented. Next, the
Table 3 Momentum flowrate of fluid passing through the orifice throughout one cycle Case
Momentum flowrate per cycle (N)
Momentum flowrate per second (N/s)
1 2 3 4 5
4.1 103 2.0 103 9.5 104 9.5 104 2.0 104
5.0 103 1.2 103 2.9 104 1.1 103 2.4 104
Table 4 Mass passing through the co-flow region Case
Mass flow per cycle (kg)
Mass flow per second (kg/s)
1 2 3 4 5
0.0932 0.0914 0.137 0.0185 0.000509
0.112 0.0550 0.0414 0.0222 0.000614
Co-flow Area
Inspection of the table reveals key results regarding the importance of the co-flow region in overall momentum transfer. First, a comparison of Tables 3 and 5 show that for all cases, the contribution to the momentum transfer by the co-flow is seen to be much smaller than the jet flow. This finding suggests that the momentum flowrate in the co-flow region can be neglected with respect to its propulsive contribution. However, it raises the possibility that modifications to the cavity shape may enable an enhancement of the contribution of the co-flow to the overall momentum transfer. In addition, the results show the effect of reciprocation period and stroke length on momentum transfer. It is seen that when the time period increases (Cases 2 and 3), the momentum flowrate decreases due to the lower piston velocity. Also, when the stroke length decreases (Cases 4 and 5), the momentum flow decreases. The sensitivity of the momentum flowrate to stroke length is substantially greater than the sensitivity due to period. In fact, the variation of co-flow momentum with respect to operating parameters differs from that of the orifice momentum. 4. Concluding remarks A comprehensive numerical simulation was performed to investigate the fluid flow patterns associated with a synthetic jet. A reciprocating piston-cylinder assembly attached to a cavity creates a pulsating synthetic jet which passes through an orifice plate. The calculations set forth here shows evidence of the existence of a co-flow in the region surrounding the cavity. The co-flow, which is created by entrainment of fluid into the emerging jet was investigated in terms of its overall mass flowrate and its contribution to the momentum imparted to the fluid by the jet. The simulations showed the agreement between fully turbulent calculations with calculations made using a new two-equation laminar–turbulent transition model. The assessment was based on global and local velocity patterns.
J. Abraham, A. Thomas / Computers & Fluids 38 (2009) 1011–1017
The complete calculations were carried out using a range of operating parameters. The effect of operating parameters was classified by reciprocation period and piston stroke length. It was seen that both at the orifice and in the co-flow region, the momentum transfer depended strongly on these parameters. In both regions, the momentum flow decreased when the period increased or when the stroke length of the piston decreased. In general, the momentum flow was more sensitive to changes in the stroke length than to changes in the time period. Acknowledgement Support of H. Birali Runesha and the Supercomputing Institute for Digital Simulation & Advanced Computation at the University of Minnesota is gratefully acknowledged. References [1] Dabiri J, Colin S, Costello J, Gharib M. Flow patterns generated by oblate medusan jellyfish: field measurements and laboratory analyses. J Exp Biol 2005;208:1257–65. [2] Seikmann J. On a pulsating jet from the end of a tube, with applications to the propulsion of certain aquatic animals. J Fluid Mech 1963;15:399–418. [3] Weihs D. Periodic jet propulsion of aquatic creatures. Fortsch Der Zoologie 1977;24:171–5. [4] Mohseni K. Pulsatile vortex generators for low-speed maneuvering of small underwater vehicles. Ocean Eng 2006;33:2209–23. [5] Thomas A, Milano M, Grazier G’Sell M, Fischer K, Burdick J. Synthetic jet propulsion for small underwater vehicles. In: Proceedings of IEEE ICRA; 2005. [6] Davidson D, Riley N. Jets induced by oscillatory motion. J Fluid Mech 1972;53:287–303. [7] Kovasznay L, Fujita H, Lee L. Unsteady turbulent puffs. Adv Geophys 1973;18B:253–63. [8] Mednikov E, Novitski B. Experimental study of intense acoustic streaming. Sov Phys Acoust 1975;21:152–4. [9] James R, Jacobs J, Glezer A. A round turbulent jet produced by an oscillating diaphragm. Phys Fluids 1996;8:2484–95. [10] Smith B, Glezer A. The formation and evolution of synthetic jets. Phys Fluids 1998;10:2281–97. [11] Mallinson S, Reizes J, Hong G. An experimental and numerical study of synthetic jet flow. AIAA Journal Paper No. 2533; 2001. pp. 41–9. [12] Lee C, Goldstein D. Two-dimensional synthetic jet simulation. AIAA J 2002;40:510–6. [13] Rizzetta D, Visbal M, Stanek M. Numerical investigation of synthetic-jet flowfields. AIAA J 1999;37:919–27. [14] Utturkar Y, Mittal R, Rampunggoon P, Cattafest L. Sensitivity of synthetic jets to the design of the jet cavity. AIAA Paper 2002-0124, Reno, NV; 2002. [15] Suzuki T. Effects of a synthetic jet acting on a separated flow over a hump. J Fluid Mech 2006;546:331–59. [16] Thomas A. Exploration into the feasibility of underwater synthetic jet propulsion. PhD thesis, California Institute of Technology; 2006. [17] Menter F. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J 1994;32:1585–605. [18] Launder B, Spalding B. Numerical computation of turbulent flows. Comput Meth Appl Mech Eng 1974;3:269–89.
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Computers & Fluids j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p fl u i d
Induced co-flow and laminar-to-turbulent transition with synthetic jets John Abraham *, AnnMarie Thomas University of St. Thomas, School of Engineering, 2115 Summit Avenue, St. Paul, MN 55105-1079, United States
a r t i c l e
i n f o
Article history: Received 18 August 2007 Received in revised form 14 December 2007 Accepted 20 January 2008 Available online 14 April 2008
a b s t r a c t A detailed numerical solution of the fluid flow patterns engendered by a synthetic jet has been carried out. The synthetic jet is caused by a reciprocating piston assembly which is attached to a large, stationary cavity. It was found that a significant momentum efflux is produced by the synthetic jet assembly. Also, fluid entrainment by the synthetic jet causes a coincident flow around the exterior of the cavity (selfinduced co-flow). Numerical solutions allow the investigation of the effect of reciprocation stroke length and piston speed on the resulting flow patterns and momentum flows. For all investigated cases, the contribution made by the co-flow to the momentum flowrate is found to be small. In order to account for the simultaneous existence of both laminar and turbulent regions, two numerical approaches were taken. One approach used the Shear Stress Transport (SST) turbulence model while the other used a newly devised transitional turbulence model. Of particular interest was a comparison between the predicted locations of the laminar-to-turbulent transition based on the two independent models. The excellent agreement between the two models reinforces the use of the SST model throughout the domain. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Synthetic jets can be used to provide a means of control for vehicles or for causing a directed flow within a larger fluid region. The jet is created by a vibrating wall or piston which is attached to a cavity. One wall of the cavity is an orifice plate through which fluid is allowed to flow during the reciprocating cycles. Inspiration for the use of synthetic jets for this purpose is taken from observation of living animals which use similar jets for their locomotion [1–5]. Alternating compression and expansion strokes by the wall or piston engender the fluid motion required for the formation of the pulsating jet. Throughout an entire reciprocating cycle, there is no net mass flow out of the cavity. Despite this fact, the jet creates a net flow of momentum in the direction of motion of the underwater body. Pioneering work on synthetic jets has provided experimental visualization of the flow patterns [6–9]. This body of work includes synthetic jets created by vibrating motion and by acoustic excitation. In [10], Schlieren and smoke tracing showed the vortex dynamics of the synthetic jet. More recently [11,12], simulations of synthetic jets have been compared to experiments. In [11] for instance, the solution domain did not extend into the cavity proper. Rather, the presence of the cavity was simulated by an imposed velocity which was applied to a replica of the orifice. In addition, the lateral extent of the cavity
* Corresponding author. Tel.: +1 651 962 5766; fax: +1 651 962 6419. E-mail address: [email protected] (J. Abraham). 0045-7930/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compfluid.2008.01.028
was infinite so that entrained fluid was drawn in only in the lateral direction. A number of works have been exclusively numerical. Included here are calculations on a shallow, wide cavity [13] with a planar geometry and an orifice plate of infinite size. In [14], calculations show the effect of cavity size on the jet. In addition, interactions between the synthetic jet and a crossflow were presented. The geometry of [14] was planar and again the orifice plate was infinite in extent. To the best knowledge of the authors, the numerical research reported in the literature has focused primarily on orifice plates whose lateral dimension is infinite. The infinite extent of the orifice prohibits flow from passing around the body of the cavity and thereby eliminates the possibility of the development of co-flow. An exception to this is the work done on the effects of a synthetic jet on a bump. Synthetic jets which are created by wall bumps are partially wall-bounded flows and exhibit entirely different flow characteristics than synthetic jets created by orifice cavities [15]. The present investigation makes use of information from an experimental investigation carried out with water as the working fluid [16]. There, a cavity was connected to a piston-cylinder device, as shown in Fig. 1. The experimental results provided conclusive evidence of the existence of a co-flow around the exterior of the device. The experiments were performed with a single operational protocol which is one case among many considered here. The numerical work to be presented will provide a detailed analysis of the fluid flow associated with a synthetic jet. The calculations will investigate the effect of operating parameters on the key features of the flow. Consideration of the transitional nature of the flow will be made and comparisons of results obtained from
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Nomenclature A dc do E F1 g hc k _ M p P r Reht S t T t
area channel width orifice width destruction term turbulence model blending function gravitational acceleration cavity height turbulent kinetic energy momentum flowrate pressure production term radial coordinate transition onset Reynolds number turbulence model constant time period velocity
a purely turbulent simulation model and a newly developed transitional model will be given. An assessment of the effect of the coflow on the propulsive ability of the synthetic jet will also be made. The calculations will span a fourfold variation in both the stroke length and the reciprocation period. The effect of these variations on the fluid flow and momentum transfer will be assessed. 2. Details of the numerical solution
wc xi xpiston z
cavity width tensor coordinate position of piston axial coordinate
Greek symbols a turbulence model constant b turbulence model constant c turbulence intermittency q Density r Prandtl number x specific turbulent dissipation rate Subscripts t turbulent c turbulence intermittency transition onset ht
symbolic representations of detached vortices which are created as flow passes into and out of the cavity. The annotations of Fig. 2 show some of the key boundaries of the fluid domain which are relevant to the computational work. As mentioned in the preceding section, the channel is bounded at the top by a moving plate which is allowed to reciprocate vertically. The other bounding walls of the cavity and channel are nonmoving and the no-slip condition is enforced there. At the upper right- and left-hand sides of the domain, fluid is allowed to enter into the zone while at the bot-
A generic description of the expected flow patterns is depicted in Fig. 2. Arrows show the patterns of co-flow around the cavity body and the jet flow emanating from the orifice. Also shown are
Fig. 1. Schematic diagram of the synthetic jet apparatus and relevant operating nomenclature.
Fig. 2. Expected flow patterns and relevant components of the solution domain.
J. Abraham, A. Thomas / Computers & Fluids 38 (2009) 1011–1017 Table 1 Values of geometric parameters Parameter
Symbol
Value (cm)
Cavity width Cavity height Channel width Orifice width
wc hc dc do
7.0 9.9 2.5 1.9
Stroke (cm) Period (s)
Case number 1 2
3
4
5
5.1 0.83
5.1 3.32
2.5 0.83
1.3 0.83
5.1 1.66
tom of the diagram, the fluid exit is marked where the flow leaves the region. The right and left vertical edges are non-moving walls and the no-slip condition is enforced there. A listing of values of dimensional and operational parameters are set forth in Tables 1 and 2, respectively. It can be seen from Table 2 that the investigation spans a four-fold range for both the reciprocation period and the stroke length. The effect of varying these parameters on the flow patterns and momentum flowrates will be assessed in Section 3 of this report. 2.1. The governing equations The complicated flow patterns and regions of free shear virtually guarantee that portions of the fluid will be turbulent. This prediction is reinforced by the flow separation which will occur near the orifice. Furthermore, the pulsating motion of the jet created by the piston reciprocation requires the solution be carried out in an unsteady fashion. In order to accommodate these features, the unsteady form of the Shear Stress Transport (SST) model [17] was utilized. This model combines features of the popular k–e model of [18] with the k–x taken from [19]. The combination of these approaches is performed in such a manner that the k–x equations dominate in the near-wall region [20] while k–e holds away from the wall. In this way, the advantage of the near-wall calculations of k–x are realized yet its sensitivity to freestream values of the turbulence level is mitigated [21]. It has been shown that the SST approach provides superior results for near-wall and separated flow calculations [22–27]. The unsteady RANS equations of motion are written in tensor form as oui ¼ 0; oxi ouj op o ouj ¼ þ qg i þ ðl þ lt Þ q ui oxi oxi oxi oxi
ð1Þ j ¼ 1; 2; 3;
ð2Þ
where lt is the turbulent viscosity. Additional transport equations are presented for the turbulent kinetic energy, k, and the specific dissipation rate, x. Those transport equations, written in tensor form are oðqkÞ oðqti kÞ o ¼ Pk bqkx þ þ ot oxi oxi
In these expressions, q is the fluid density, r represents the Prandtl number associated with the transport of k or x. The terms a, S, and b are model constants, and Pk is a production term for turbulent kinetic energy. F1 is a blending function that takes on a value of zero away from solid surfaces and one near surfaces. The effect of blending is that the SST model behaves as k–e away from the wall and k– x within the boundary layer. 2.2. Transitional flow
Table 2 Values of the stroke and reciprocation period Parameter
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l ok lþ t rk oxi
ð3Þ
and oðqxÞ oðqti xÞ o l ox ¼ aqS2 bqx2 þ lþ t þ rx oxi ot oxi oxi 1 ok ox : þ 2ð1 F 1 Þqrx2 x oxi oxi
ð4Þ
Eddy-viscosity turbulence models such as the k–e, k–x, or their derivative (SST) have been developed and evaluated for situations in which the freestream flow was either fully laminar or turbulent. Their ability to model flow that is undergoing laminar-to-turbulent transition has been a source of doubt for some time. This inability has spawned a very extensive body of research, particularly in recent years which has resulted in a new approach to modeling flow that expresses laminar and turbulent regions within a single fluid domain. Pioneering work such as that carried out by [28–31] has demonstrated various modes of laminar–turbulent transition including natural transition, bypass transition, separation-induced transition, and wake-induced transition. Efforts to predict the location at which laminar boundary-layer flow begins to break down has been based on empirical correlations which relate the freestream turbulence levels to the transition Reynolds number based on the momentum thickness [32]. These correlations, coupled with the concept of flow intermittency first set forth in [33,34] have enabled a new approach to be developed in which a new transport equation is proposed for intermittency, c [35– 41]. Of the various models, the most viable for implementation in a modern CFD environment with an unstructured mesh and parallel processing, is that of [41]. That model requires the solution of additional transport equations for the intermittency, c, and for the transition onset Reynolds number, Reht. The new equations are oðqcÞ oðqti cÞ o l oc ð5Þ ¼ Pc Ec þ lþ t þ rc oxi ot oxi oxi and oðqReht Þ oðqti Reht Þ o oReht : ¼ P ht þ rht ðl þ lt Þ þ oxi ot oxi oxi
ð6Þ
The terms Pc, Pht, and Ec, are production and destruction terms, respectively. Details for their evaluation can be obtained from [41]. For the calculations which are provided in the present report, a comparison between results obtained from the SST and the transitional models will be made. The comparison will demonstrate whether a traditional two-equation turbulence model such as the SST is suitable for capturing a flow that exhibits both laminar and turbulent characteristics. 2.3. Initial and boundary conditions In order to complete the description of the fluid flow problem, specification of the flow at all boundaries is required. At all walls, the no-slip condition was enforced so that the fluid was either stationary or moved with the velocity of the wall. The reciprocating motion of the piston was sinusoidal so that Stroke 2p sin t ; ð7Þ xpiston ¼ 2 T where T is the period of the motion. At the rightmost and leftmost boundaries, a wall was employed so that fluid there would be
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motionless. At the lower boundary, an exit condition was given with weak closure conditions enforced on the second derivatives in the flow direction. Initially, at t = 0, the entire fluid region was at rest. The simulation spanned ten full reciprocating cycles of the piston to ensure that timewise periodic flow was achieved. 2.4. The numerical method The circular geometry of the cavity, channel, and the surrounding water was initially constructed using more than 80,000 elements to resolve the spatial domain. These elements were preferentially deployed in regions where large variations in the transported variables were expected. In addition, thin prismatic elements were deployed along interfaces between the fluid and solid surfaces to ensure proper resolution of the boundary layers. The unsteady computations were performed with 100 timesteps for each reciprocating cycle. All timesteps contained 10 iter-
ations, each of which was performed using a two-step, multi-grid computational algorithm. The time stepping was carried out using an Euler backward scheme of second-order accuracy. Coupling of the velocity–pressure equations was achieved on a non-staggered, collocated grid using the techniques developed by Rhie and Chow [42] and Majumdar [43]. The inclusion of pressure-smoothing terms in the mass conservation equation suppresses oscillations which can occur when both the velocity and pressure are evaluated at coincident locations. The advection term in the momentum equations was evaluated by using the upwind values of the momentum flux, supplemented with an advection-correction term. The correction term reduces the occurrence of numerical diffusion and is of second-order accuracy. Further details of the advection treatment can be found in [44]. Mesh and time-step values were sufficiently small to ensure a solution that was independent of their values. The selected values resulted from a thorough independence study during which both the element sizes and time steps were halved and results were
Fig. 3. SST velocity magnitude contours obtained for as the piston is (a) moving downwards, (b) at bottom-dead center, (c) is moving upwards, and (d) is at top-dead center.
J. Abraham, A. Thomas / Computers & Fluids 38 (2009) 1011–1017
compared. When the sequential reductions failed to yield noticeable changes in the results, it was determined that the settings were sufficiently refined. 3. Results and discussion 3.1. Comparison of SST and transitional results Fig. 3a–d show a series of global velocity magnitude contour diagrams. The figures correspond to results obtained using the SST model for turbulent simulation. In Fig. 3a, the piston is passing downward through its center point, and a mass of fluid is seen to be ejected from the cavity at relatively high speed as indicated by the contour. In Fig. 3b, the fast-moving fluid is seen to be detached from the orifice and is moving downwards, away from the cavity. In Fig. 3c the piston is traveling upwards, drawing fluid into the cavity from the surrounding region. A remnant jet of fluid is seen to be continuing its downwards motion, away from the cav-
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ity. Finally, when the piston is at a top-dead-center position in Fig. 3d, a fast-moving mass of fluid is observed being pulled into the cavity. An overall view of the velocity patterns from Fig. 3 depict the oscillating inflow and outflow during an entire reciprocating cycle. The contour is scaled by the legend at the left-hand side of the figures. Fig. 4a–d show the counterpart results based on the transitional model. An examination of Figs. 3 and 4 reveals a close similarity between the results for the SST and transitional approaches. This similarity includes both the flow patterns and velocity magnitudes inside and outside of the cavity space. These results strongly support the congruence of the SST and transitional models. Similar agreement was seen with the other cases indicated in Table 2. As a consequence, the remaining discussion will solely make use of results obtained from the SST model. That discussion will focus on the mass and momentum flowrates engendered by the synthetic jet throughout an entire reciprocating cycle.
Fig. 4. Transition velocity magnitude contours obtained for as the piston is (a) moving downwards, (b) at bottom-dead center, (c) is moving upwards, and (d) is at top-dead center.
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3.2. Mass and momentum flowrates through the orifice
Table 5 Momentum flowrate imparted to the co-flow during one cycle
While conservation of mass guarantees that there can be no net mass efflux from the cavity, the synthetic jet is capable of generating a net momentum flow in the streamwise direction. The net momentum flow is a critical measure of the propulsive capability of the synthetic jet device. The total momentum imparted to the fluid during a cycle was calculated by timewise integration of the local momentum variations at the orifice. The integral, expressed in both its integral form and its numerical analog is " # Z T Z t¼T X X 2 2 _ qt dA dt ¼ qti DAi Dti ; ð8Þ M orifice ¼
Case
Momentum flowrate per cycle (N)
Average momentum flowrate per second (N/s)
1 2 3 4 5
1.31 104 6.14 105 4.94 105 1.28 105 1.34 106
1.58 104 3.70 105 1.49 105 1.54 105 1.61 106
0
Orifice Area
t¼0
Orifice Area
where q is the fluid density, t is the streamwise component of the fluid velocity, and DA is the horizontal projection of the area of the elements which span the orifice. A tabulation of resulting momentum flowrates is presented in Table 3. The results are keyed to the five cases set forth in Table 2. In all cases, it was found that there was a net momentum flow in the streamwise direction. This finding is verified from experimental measurements made in [16]. The table presents two sets of results. The first results list the total momentum flowrate passing through the orifice throughout an entire reciprocation of the piston. The second set shows the time-averaged rate of momentum transfer to the fluid. From the tabulated results, it is seen that the variation of momentum flowrate very nearly follows the behavior 2
_ orifice stroke length : M 2 Period
ð9Þ
respective masses are divided by the corresponding time periods to obtain the average mass flowrate for the individual cases. It can be seen from the results that the effect of an extended time period (Cases 2 and 3) is to reduce the average mass flowrate. Likewise, a shorter stroke length gives rise to lesser mass flowrates. It is also seen that the mass flowrate is substantially more sensitive to stroke length than to reciprocation period. Next, attention is turned to the contribution of momentum by the entrained fluid in the co-flow region. The calculation of the co-flow momentum was performed by the numerical integration presented in Eq. (10). There, the timewise variation of the momentum flow was integrated spatially and over time to give the results shown in Table 5. Also shown in the table are the average momentum flow rates. Z T Z _ co-flow ¼ qt2 dA dt M 0 Co-flow Area " # t¼T X X 2 qti DAi Dti : ð10Þ ¼ t¼0
The dependence of momentum flowrate to the operating parameters indicated in Eq. (9) agrees qualitatively with results of experimentally observed in [16]. 3.3. Mass and momentum flowrates through the co-flow region The numerical simulation clearly shows a directed streamwise flow in the co-flow region throughout the entire period of reciprocation. This finding is in contradistinction to the observations made at the orifice where the direction of flow reverses with the direction of the piston. As a consequence, it is relevant to calculate the mass flowrate through the co-flow region throughout the reciprocation. The resulting mass flowrates are shown in Table 4. The table displays two sets of results. First, the total mass passing through the co-flow region for one cycle is presented. Next, the
Table 3 Momentum flowrate of fluid passing through the orifice throughout one cycle Case
Momentum flowrate per cycle (N)
Momentum flowrate per second (N/s)
1 2 3 4 5
4.1 103 2.0 103 9.5 104 9.5 104 2.0 104
5.0 103 1.2 103 2.9 104 1.1 103 2.4 104
Table 4 Mass passing through the co-flow region Case
Mass flow per cycle (kg)
Mass flow per second (kg/s)
1 2 3 4 5
0.0932 0.0914 0.137 0.0185 0.000509
0.112 0.0550 0.0414 0.0222 0.000614
Co-flow Area
Inspection of the table reveals key results regarding the importance of the co-flow region in overall momentum transfer. First, a comparison of Tables 3 and 5 show that for all cases, the contribution to the momentum transfer by the co-flow is seen to be much smaller than the jet flow. This finding suggests that the momentum flowrate in the co-flow region can be neglected with respect to its propulsive contribution. However, it raises the possibility that modifications to the cavity shape may enable an enhancement of the contribution of the co-flow to the overall momentum transfer. In addition, the results show the effect of reciprocation period and stroke length on momentum transfer. It is seen that when the time period increases (Cases 2 and 3), the momentum flowrate decreases due to the lower piston velocity. Also, when the stroke length decreases (Cases 4 and 5), the momentum flow decreases. The sensitivity of the momentum flowrate to stroke length is substantially greater than the sensitivity due to period. In fact, the variation of co-flow momentum with respect to operating parameters differs from that of the orifice momentum. 4. Concluding remarks A comprehensive numerical simulation was performed to investigate the fluid flow patterns associated with a synthetic jet. A reciprocating piston-cylinder assembly attached to a cavity creates a pulsating synthetic jet which passes through an orifice plate. The calculations set forth here shows evidence of the existence of a co-flow in the region surrounding the cavity. The co-flow, which is created by entrainment of fluid into the emerging jet was investigated in terms of its overall mass flowrate and its contribution to the momentum imparted to the fluid by the jet. The simulations showed the agreement between fully turbulent calculations with calculations made using a new two-equation laminar–turbulent transition model. The assessment was based on global and local velocity patterns.
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