Toward Simple Boundary Condition Representations of Zero-Net Mass-Flux Actuators in Grazing Flow

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1 39th AIAA Fluid Dynamics Conference - 5 une 9, San Antonio, Texas AIAA 9-48 Toward Simple Boundary Condition Representations of Zero-Net Mass-Flux Actuators in Grazing Flow Ehsan Aram, Rajat Mittal Department of Mechanical and Aerospace Engineering, The George Washington niversity, Washington, DC 5 and Louis Cattafesta 3 Interdisciplinary Microsystems Group Department of Mechanical and Aerospace Engineering, niversity of Florida, Gainesville, FL - 36 The paper explores a set of simple boundary conditions that can represent the flow emanating from zero-net mass-flux (ZNMF) jets in grazing flow. Results from numerical simulations of ZNMF jets in grazing flows are used to determine the key characteristics of the jet profile, and these are used to construct a series of boundary condition models. These various boundary conditions are then tested for a jet exhausting in an attached boundary layer as well as a boundary layer with an induced separation bubble. C uv = u momentum flux in y-direction C uv = Time average of C uv C vv = v momentum flux in y-direction C vv = Time average of d = Slot width C vv F + = Dimensionless forcing frequency, f / f f = Forcing frequency of ZNMF actuator f = Separation bubble frequency Sep H h Q Nomenclature = Cavity height = Slot height = olume flow rate amplitude of ZNMF actuator Re δ = Boundary layer Reynolds number, δ / ν Re = et Reynolds number, d / ν St = Strouhal number, π fd / = u-velocity amplitude at the jet exit = Free stream velocity Sep u = Streamwise velocity (in x-direction) Graduate Student, Department of Mechanical & Aerospace Engineering, Student Member AIAA. Professor, Department of Mechanical & Aerospace Engineering, Associate Fellow AIAA, mittal@gwu.edu 3 Professor, Department of Mechanical & Aerospace Engineering, Associate Fellow AIAA, cattafes@ufl.edu. Copyright 9 by Authors. Published by the, Inc., with permission.

2 u mean = Time average of u-velocity [ u ] mean = Time average of u () i = Fourier expansion amplitudes of v-velocity at left half of jet exit, i=,,3, () i = Fourier expansion amplitudes of v-velocity at right half of jet exit, i=,,3, = Time-space average of jet exit v-velocity during the expulsion phase = Cross-stream velocity (in y-direction) v = Time average of v-velocity mean [ v ] mean = Time average of v W = Cavity width x = Horizontal direction y = ertical direction δ = Boundary layer thickness ξ = Spanwise vorticity z [ ξ z ] mean = Time average of spanwise vorticity Ω = Spanwise vorticity flux Ω = Time average of spanwise vorticity flux ω = Angular forcing frequency of the ZNMF actuator, π f I. Introduction ero-net mass-flux (ZNMF) or synthetic jet actuators are versatile devices with numerous applications including Z active control of flow separation, mixing enhancement, heat transfer, and thrust vectoring of jets. -4 A number of parametric studies indicate that several geometrical and flow parameters govern the performance characteristics of ZNMF actuators and their control effectiveness. 5-7 The scales of the actuator are in general - -4 times smaller in size compared to the dominant length scale of the problem (such as the wing chord). Thus, conducting numerical flow simulations which include full representations of the actuator cavity is an extremely daunting proposition. 8 A more practical approach is to devise a simpler representation of the actuator and use that instead of simulating the full actuator. A variety of such approaches have been used in the past. For instance, Kral et al. 9 simulated the jet flow associated with a two-dimensional, incompressible synthetic jet actuator in the quiescent external flow by prescribing modified boundary conditions for velocity and pressure at the jet exit. Comparison between the computational results and experimental data for both steady and unsteady conditions showed good agreement in the turbulent case, but the model was not able to capture the breakdown of the vortex train in the laminar case. Rizzetta et al. used the recorded jet exit velocity profile of an isolated actuator as the boundary condition for the D and 3D external flow field. Other more sophisticated approaches have also been used with varying degrees of success. Filz et al. have modeled D isolated synthetic jets using deterministic source terms trained by a neural network. The source terms were calculated from the time average of the unsteady solution. This technique was able to reduce the CP time significantly and predict the momentum thickness of the flow with reasonable accuracy. A low-dimensional model of D and 3D synthetic jet actuator by applying the unsteady compressible quasi-one-dimensional Euler equations was proposed by Yamaleev and Carpenter and was tested in grazing flow. Their model satisfied the conservation laws and was able to predict the resonance characteristics of the actuator. Although this model reduced the computational cost by eliminating the cavity, it did not consider the multidimensional effects associated with the vortex generation, boundary layer, and the boundary-layer separation inside the actuator cavity. With regards to simplified actuator models, Rathnasingham and Breuer 3 developed a system of first-order, nonlinear differential equations which presented fluid structure interaction characteristics of the device in the quiescent external flow. The membrane was modeled as a circular plate, and the Bernoulli equation was used for flow in the slot. Comparison with experimental data showed that the semi-empirical model can accurately predict the time and frequency characteristics. Gallas et al. 5 and Agashe et al. 4 have developed Lumped Element Models (LEM) of piezoelectric and electrodynamic actuators in the quiescent external flow, respectively. In this method, each component of the actuator is modeled as an element of an equivalent circuit. Comparisons with the experimental

3 results of prototype actuators reveal good agreement, but this method is incapable of providing detailed profiles of the flow quantities at the slot/orifice exit. Motivated by the LEM, Ravi 5 used numerical simulations to explore the idea of replacing the ZNMF jet in grazing flow with appropriate selected boundary conditions applied on the surface. He compared the simpler model with full simulations that include the cavity and slot. Several candidate profiles were considered. The overall conclusion was that none of these simple models were able to produce the effect of the jet on the cross flow boundary layer. Tang et al. 6 recently have predicted the instantaneous spatial average velocity at the orifice exit by three following models: Dynamic Incompressible Flow model, Static Compressible Flow Model and LEM. Comparison with the experimental and numerical results suggested that the LEM model performs more reliably than the Static Compressible Flow Model in different conditions. A simple "slot only" model based on including only the jet slot (and not the cavity) of the actuator in grazing flow was subsequently proposed by Raju et al. 8 Comparison with the full cavity and a simple sinusoidal plug flow velocity model showed that this simplified actuator model was able to capture more of the flow physics associated with the full cavity simulations and can predict the reduction of the separation bubble size in the low forcing frequency for a canonical separated flow with improved accuracy. Although the "slot-only" model represents a significant reduction in the computational complexity required to represent the actuator in flow simulations, it still requires simulation of the unsteady viscous flow in the slot. Furthermore, if the slot location is varied, a new grid may be required. If, however, the entire actuator can be replaced by an appropriate velocity boundary condition at the jet exit, there would be no need to include a slot, and that would significantly simplify the task of ZNMF flow control simulations. In the current paper, we explore the possibility of developing a simple boundary condition representation for ZNMF jet actuators in grazing external flow. II. Modified Boundary Condition Actuator Models The basic idea of this study is to use the jet exit profile of the full cavity simulation to construct a simple loworder description of the ZNMF jet. An attempt is made to reproduce the flow physics associated with the full cavity simulations using our proposed models. Since in the current study the oscillation frequency of the diaphragm is considered well below the Helmholtz frequency of the actuator, the incompressible flow assumption is fairly accurate inside the actuator and flow rate at the slot exit is equal to that produced by the diaphragm oscillation. tturkar et al. 7 showed that in the case of incompressible flow, the cavity shape has a limited effect on the jet exit flow. Figure (a) & (b) show a schematic of the two configurations considered in the current study: the Full Cavity (FC) and the Modified Boundary Condition (MBC) models. The FC model is considered the most accurate representation of the actuator, and the objective of the current study is to assess a variety of MBC models by comparing with the FC model. v u u ( x, t ) v ( x, t ) y x h d d H W Figure : Schematics of (a) full cavity (FC) and (b) modified boundary condition (MBC) models. 3

4 FC MBC FC MBC (a) (b) (c) (d) (e) (f) (g) (h) Figure : elocity vectors and instantaneous spanwise vorticity contours for the full cavity and MBC models at (a) ϕ = o, (b) ϕ = 45, (c) ϕ = 9, (d) ϕ = 35 o, (e) ϕ = 8 o, (f) ϕ = 5, (g) ϕ = 7, (h) ϕ = 35 for Re = 5, St =., / = 4. and δ / d = 3.. By far the simplest MBC model is one that assumes a sinusoidally varying plug flow for the vertical velocity profile at the jet exit during the expulsion and ingestion phase. Numerical simulations also require the specification of the horizontal velocity component in this simplest MBC model; we assume that the horizontal velocity is zero 4

5 during the expulsion phase. Our past studies (Raju et al. 8 ) have shown that during the ingestion phase, assuming a zero Neumann boundary condition for the horizontal velocity u is preferable versus any Dirichlet boundary condition, and so that is the condition employed for the horizontal velocity component during the ingestion phase. Note that the amplitude of the vertical velocity is chosen so as to match the volume flux of the FC model. We denote this model as MBC. In Figure (), we compare the instantaneous spanwise vorticity contours produced by the FC model to that of the MBC model for a case where the jet is emanating into a grazing laminar boundary layer. The dimensionless flow parameters for this comparison are Re = 5, St =., / = 4. and δ / d = 3.. A comparison of the external flow and the jet exit velocity profile suggests the following. First, the simple MBC type model produces an external flow that is different from that of the FC model, especially during the expulsion phase. Second, the vertical velocity shows significant variations along the slot width. Third, the flow at the slot exit has a non-zero horizontal velocity component which is important for determining the trajectory of the jet. We use these observations as a basis for exploring a set of MBC models in the following section. Two key factors need to be considered while designing viable MBC models for ZNMF jets in grazing flows. First, the models should be as simple as possible and have a relatively small number of parameters. Second, the model should provide a reasonably good approximation to the effects of the actual ZNMF jet. Keeping this in mind, we introduce models which have at most two spatial degrees of freedom for the vertical velocity. We prescribe the vertical velocity for at most two interior points across the slot exit. This allows us to prescribe a non-constant velocity profile across the slot. Furthermore at any given time, we assume a linear velocity variation between these two points and impose the no-slip boundary condition at the slot edges. Figure 3 shows the type of vertical velocity profile that is employed in our MBC models where () and are the points where the velocity is specified and points () and (4) are the slot edge points. The points () and can in principle be located anywhere, but we choose a simple prescription whereby the points are located at 5% and 75% locations along the slot exit. With regards to the horizontal velocity component, in the interest of simplicity, we assume that there is no spatial variation of this component of velocity during the expulsion phase. Note that the no-slip boundary condition is imposed at the slot edges for horizontal velocity during the expulsion and ingestion phases. We denote the vertical () velocity at the points () and as and, respectively, and the horizontal velocity as. x x x 3 x 4 x Figure 3: Assumed vertical velocity profile for MBC models. () The next consideration is the time variation of this assumed profile for, and. Given that ZNMF jet flows are usually periodic, it is natural to assume that the above velocity components are periodic and can be expressed in terms of a Fourier series in time as follows: ( t) = + sin( ωt + ϕ ) + sin( ωt + ϕ ) +... () () () () () () ( t) = + sin( ωt + ϕ ) + sin( ωt + ϕ ) +... ( t) = + sin( ωt + ψ ) + sin( ωt + ψ ) +... Next, assuming that the volume flux through the slot is sinusoidal, i.e. We obtain the following relationships for the above coefficients: Q( t) = Q sin( ωt) () 5 = = () ()

6 We now look at the FC model to see if any guidance can be obtained regarding the amplitudes and phases of the terms in the above Fourier series. Figure 4(a) shows the velocity averaged over the right and left halves of the slot exit as a function of time in addition to the average over the entire slot. If we assume that and are represented by the spatial averages over the left and right halves of the slot, respectively, then we find that vary mostly at the ZNMF jet frequency implying that higher harmonics are negligible () () n () () and >> for n > (4a) >> for n > (4b) n Second, the two average velocities are also approximately in phase with the volume flux suggesting that φ, φ (5) () Next, Figure 4(b) shows the spatial average of the horizontal u velocity component at the exit of the full cavity over one cycle. The spatial integral is calculated over the whole length of the exit. Two observations can be made from this plot. First, the time-space averaged horizontal velocity is non-zero. Second, the horizontal velocity component possesses a large Fourier amplitude at the first super harmonic of the jet frequency, i.e. ω with ψ. (a) (b) Figure 4: The spatial average of the jet velocity component along the slot exit of the full cavity model, (a) vertical v component of velocity, (b) horizontal u component of velocity. Given all of the above observations, the following reduced representation of the velocity components can be obtained for the two interior points in the slot: ( t) = + sin( ωt) () () ( t) = + sin( ωt) ( t) = + sin( ωt) (6) () Note that the above prescription has five coefficients. However, since and are related to the amplitude of the volume flux Q, only four of these coefficients are unknown. In order to assess the viability of the above prescription (as well as further simplifications to it) we have evaluated the four remaining coefficients from the FC model. This is done by extracting the velocity components at the slot exit as a function of space and time from the FC simulations, evaluating the spatial average of velocity along the slot exit for horizontal velocity and along each half of slot separately for vertical velocity and then subjecting these to a Fourier analysis. For instance, is evaluated as the spatial average of the jet exit velocity on the left half of the slot, whereas is the corresponding velocity for the right half of the slot. We then create a number of variations of the above model by making further simplifying assumption(s) and computing these flows for these simpler models. The computed flows are then compared with the FC model to assess the performance of each model. () 6

7 A. Proposed Models Based on the above arguments, the following models are considered: Table : Different modified boundary condition (MBC) models examined in current study. Model Expulsion Ingestion nknowns Remarks MBC (plug flow) ( t ) =. () () ( t) = sin( ωt) ( t) = sin( ωt) () / Q d MBC ( t ) =. () () ( t) = sin( ωt) + ( t) = + sin( ωt) () MBC3 ( t) = + sin( ωt) () () ( t) = sin( ωt) + ( t) = () + sin( ωt) u / y = ( t) = sin( ωt) () () ( t) = sin( ωt) () / Q d u / y = ( t) = sin( ωt) () () + ( t) = + sin( ωt) () u / y = ( t) = sin( ωt) () () + ( t) = + sin( ωt) () No unknowns () () -Zero u-vel during the expulsion phase -niform v-vel. during the expulsion and ingestion phases -Simplest model -u-vel same as MBC -Nonuniform v-vel amplitude with non zero mean value in left and right half of slot (phase difference) during the expulsion and ingestion phases -Sinusoidal u-vel. with non zero mean value in expulsion phase -v-vel. same as MBC -Most general model MBC4 t) = + sin(ω ) ( t ( t) = sin( ωt) () () + ( t) = + sin( ωt) () MBC5 ( t) = + sin( ωt) () () ( t) = sin( ωt) ( t) = sin( ωt) () MBC6 ( t) = () () ( t) = sin( ωt) ( t) = sin( ωt) () u / y = ( t) = sin( ωt) () () + ( t) = + sin( ωt) () / Q d u / y = ( t) = sin( ωt) () () ( t) = sin( ωt) () / Q d u / y = ( t) = sin( ωt) () () ( t) = sin( ωt) () / Q d () () () -u-vel. same as MBC3 -v-vel. same as MBC3 during the expulsion phase -niform v-vel. amplitude with non zero mean value in left and right half of slot during the ingestion phase -u-vel. same as MBC4 -v-vel. same as MBC4 with zero mean value in both halves of slot during the expulsion phase -v-vel. same as MBC during ingestion phase -Constant u-vel. in expulsion phase -v-vel. same as MBC5 MBC7 ( t ) =. () () ( t) = sin( ωt) ( t) = sin( ωt) () u / y = ( t) = sin( ωt) () () ( t) = sin( ωt) () / Q d () -u-vel. same as MBC -v-vel. same as MBC6 7

8 Note that in the above profiles, if = then we have the so-called plug flow profile wherein the vertical () velocity is uniform across the slot exit at all time instants. Conversely, for models with (), we assume a non-uniform exit velocity profile. For models with = we are assuming that both halves of the jet slot separately satisfy the zero-net mass-flux constraint. The simulations to study the performance of the different models in the vicinity of the actuator have been carried out on the domain size of 8d 9.3d with a single grid of dimensions 57 6 for the MBCs, and a domain size of 8d 3d with a grid for the full cavity model. In all models, 5 grid points are used along the jet exit. The geometrical parameters for the full cavity are h / d =., W / d = 3. and H / d =.5 (defined in Figure ). The values of the dimensionless flow parameters for the preliminary results are δ / d = 3., / = 4., Re = 5 ( Reδ = 3 ) and St =.. A second order fractional step method is used to solve the incompressible Navier- Stokes equations in time. The solver employs a second-order Adams-Bashforth scheme for the convective terms and an implicit Crank-Nicolson scheme for diffusion terms on Cartesian grids. The solver has been tested extensively by 8, 9 comparisons with the established numerical and experimental data including ZNMF jets. B. Canonical Separated Flow Configuration One way to study the performance of potential models is to examine the effect of some of these models on a separated flow and understand to what extend the details of the actuator model are necessary to model the effects of the ZNMF jet on the separation bubble. In order to address this issue we have examined a case where a separation bubble is created downstream of the jet by prescribing a steady blowing-suction boundary condition on the top boundary of the domain. A zero-vorticity condition is applied on this boundary as follows: u dg v( x, Ly ) = G( x), = y dx ( x, Ly ) (7) where G( x ) is the prescribed blowing and suction velocity profile defined in the form of: π ( x xc ) G( x) = top sin e L β ( x x c ) α L where L is the length and x c is the center of the velocity profile. The parameters top, α and β are set to, and, respectively, which form a closed separation bubble on the flat plate. A schematic of the flow configuration is shown in Figure 5. The inflow is a cubic approximation to the Blasius boundary layer (8) u 3 y y = δ δ 3 (9) Slip condition Suction/Blowing Profile Blasius inflow Outflow ZNMF Actuator Separation Bubble No-slip condition Figure 5: Flow configuration used for simulating a canonical separated flow. 8

9 Separated flow simulations have been performed for the following dimensionless flow parameters: δ / d = 3., / = 4., Re = 87.5, 5 ( Reδ = 5,3 ), and three dimensionless forcing frequencies F + =.5,.,.5, where F + is the ratio of the jet frequency ( f ) to the shear layer frequency ( f Sep ). We have used a grid on the domain size of 96d 9d for MBC (plug flow), MBC6 and MBC7 models, and a domain size of 96d 3d with a 64 3 grid for the FC model. In all models, grid points are used along the jet exit. Note that all of the unknowns in MBC6 and MBC7 models are obtained from FC model in external flow with no separation bubble (part A). A. Model Performance in icinity of Actuator III. Results (FC) (MBC) (MBC) (MBC3) (MBC4) (MBC5) (MBC6) (MBC7) Figure 6: Instantaneous spanwise vorticity during peak expulsion. For the given set of conditions of St =., based on the criterion suggested by Holman et al 7 a strong jet is formed and it is able to influence the boundary layer at a large distance in the streamwise direction and impart a significant amount of the spanwise vorticity flux to the external flow. Figure 6 shows a comparison of the spanwise vorticity at the peak expulsion for all of the models at the following condition: Re = 5, St =., / = 4. and δ / d = 3.. The strength of the vortex at both edges of the slot exit in MBC (plug flow) is much less than the full cavity model, but the rest of the models provides a reasonable approximation to the full cavity model in this time 9

10 instance. Comparing MBC and MBC7 with the full cavity model shows that due to zero u-velocity for these two models in the expulsion phase, the vortex penetrates further in the y-direction, which leads to the shorter extension of the vortex in the streamwise direction. This is not the case for MBC4, MBC5 and MBC6. The performance of these models can also be examined based on the integral quantities such as spanwise vorticity magnitude flux and u and v momentum flux at the jet exit. The normalized instantaneous variations of these quantities are compared for the various proposed models in Figure 7. The definitions for these quantities are as follows: where ξ ( x, t) is the spanwise vorticity at the jet exit. z d C ( t) = u( x, t) v( x, t) dx () uv vv d C ( t) = v( x, t) v( x, t) dx () d Ω ( t) = ξ ( x, t) v( x, t) dx () z (a) (b) (c) Figure 7: Comparison of the normalized instantaneous (a) u momentum flux, (b) v momentum flux, and (c) spanwise vorticity flux It is seen that the MBC (plug flow) model significantly underpredicts all three quantities during the expulsion phase, but there is a good agreement between this model and the full cavity model during the ingestion phase. MBC and MBC3 overpredict the v momentum flux in the ingestion phase, which means that the two-degree of freedom profile is not a good candidate during the ingestion phase. It is also expected that the u momentum for MBC and MBC7 should be zero due to zero u component of velocity in the expulsion phase. MBC5 has the same performance as MBC4, even though the mean v-velocity has been neglected in this model. Comparing the MBC6 case and MBC5 model shows that neglecting the unsteady part of the u velocity does not have a significant effect on the stated quantities and it only slightly underpredicts the u momentum flux.

11 Table : dimensionless quantities of time average of u momentum, v momentum and spanwise vorticity flux along the jet exit over one period. ρcuv ρc Ω vv ρ d ρ d FC MBC(plug flow) MBC MBC MBC MBC MBC MBC Table shows the dimensionless time average of spanwise vorticity magnitude flux, u and v momentum flux at the slot exit over one period. As expected MBC3 (the most complicated model) gives the closest value of mean u momentum flux to FC model, whereas MBC (plug flow), MBC, MBC7 significantly underpredict this due to zero u-velocity for these models during the expulsion phase. The mean v momentum flux for MBC4-7 has good agreement with FC model, while MBC and MBC3 overpredict this quantity. All the models except MBC (plug flow) are able to estimate the spatial-time average of spanwise vorticity flux along the jet exit with reasonable accuracy. Based on the these limited observations, the MBC6 model with only two unknown parameters ( and () / ) may be a good candidate for our simulations of the actuator in a separated flow. We also consider MBC7 () with only one unknown ( / ) in our study as the simplest proposed model among the nonuniform vertical velocity models and compare its performance with the FC model. B. Model Performance for the canonical separated flow Figure 8 shows the formation of a separation bubble by implementing the steady suction and blowing on the top boundary downstream of actuator for / = 4., δ / d = 3., Reδ = 5 and Reδ = 3. The adverse pressure gradient due to the suction on the top boundary reduces the momentum of the flow and eventually leads to boundary layer separation. The length of the separation bubble based on the mean flow shown by the streamlines in the Figure are 4.8d and 4.9d for Reδ = 5 and Reδ = 3, respectively. The contours of instantaneous spanwise vorticity also indicate that the separation bubble in these flow configurations is unsteady. (a)

12 (b) Figure 8: Instantaneous spanwise vorticity and mean streamlines for the unforced separated flow condition for / = 4., δ / d = 3., (a) Reδ = 5 and (b) Reδ = 3. The performance of selected proposed models MBC, MBC6 and MBC7, in this canonical separated flow has been compared with the FC model in Figure 9. The dimensionless flow parameters are / = 4., δ / d = 3., Re = 87.5 with forcing frequency F + =. and Re = 5 with forcing frequencies F + =.5, F + =.. This figure shows the mean streamlines in the vicinity of the separation bubble for (a) Re = 87.5, F + =., (b) Re = 5, F + =.5, and (c) Re = 5, F + =.. It is clear that MBC model is not able to capture the secondary separation bubble near the flow detachment point for all presented cases, while the MBC6 and MBC7 models are able to reproduce this feature. It is also seen that for both forcing frequencies of the Re = 5 case, the MBC model cannot predict the detachment and reattachment of the flow accurately, whereas the location of these points by using MBC6 and MBC7 models are well matched with the FC model. A comparison between the maximum separation bubble height for the MBC s models and FC model shows that MBC significantly underpredicts this quantity, whereas the two other models provide a reasonable prediction of this quantity. One reason might be that the v momentum flux for MBC (plug flow) at the jet exit is less than the FC model during the expulsion phase. (FC) (MBC) (MBC6) (MBC7) (a)

13 (FC) (MBC) (MBC6) (MBC7) (b) (FC) (MBC) (MBC6) (MBC7) (c) Figure 9: Mean streamlines of separated flow for / = 4., δ / d = 3., (a) Re = 87.5, F + =. (b) Re = 5, F + =.5, and (c) Re = 5, F + =.. Figure shows the time average values of u-velocity, u, v-velocity, v and spanwise vorticity (umean, [u]mean, vmean, [v]mean, and [ξ z ]mean ) along the y-direction at x/d=. located just upstream of the separation bubble and downstream of actuator over one period (the first figure on the top left of each part shows the exact location of the data extraction). The dimensionless flow parameters for this flow are / = 4., δ / d = 3., Re = 87.5 and Re = 5 with a forcing frequency F + =.. The general observation that can be made from these plots is that MBC (plug flow) results in the worst comparisons with the FC case whereas the comparisons with MBC6 and MBC7 are markedly better. Also, as expected, MBC6 is somewhat better than MBC7. 3

14 (a) (b) Figure : Location of extracting data, time average of u, u momentum, v, v momentum and spanwise vorticity along the y-direction at x/d=. in (a) Re = 87.5, F + =., (b) Re = 5, F + =.. 4

15 I. Conclusions Different modified boundary condition models are proposed to reproduce the flow physics associated with the ZNMF jet in grazing flow using the jet exit profile of the full cavity simulation as the basis for comparison. With the objectives of simplicity as well as reasonable accuracy, the introduced models have zero or uniform horizontal velocity on the slot flow boundary and at most two spatial degrees of freedom for the vertical velocity, which can produce a nonuniform vertical velocity across the slot exit. A comparison of key integral quantities of u momentum, v momentum, and spanwise vorticity flux for these models along the jet exit shows that the models with nonuniform vertical velocity during the expulsion phase and uniform vertical velocity during the ingestion phase can predict these quantities with good accuracy, whereas MBC (plug flow) model with zero horizontal velocity and uniform vertical velocity underpredicts these three quantities during the expulsion phase. The performance of MBC (plug flow), MBC6 (constant u-velocity during expulsion phase, non-uniform v- velocity during explusion phase, uniform v-velocity during ingestion phase), and MBC7 (zero u-velocity during the expulsion phase, same v-velocity as MBC6) was also examined for a canonical separated flow at different forcing frequencies and was compared with the FC model. It was found that MBC6 and MBC7 were able to predict the detachment and attachment points of flow reasonably well, whereas this was not the case for the MBC model in most flow conditions. The MBC model also underpredicted the height of the separation bubble for all cases. Comparing the time average of u-velocity, v-velocity, u, v and spanwise vorticity along the y-direction downstream of the jet for selected models showed that MBC6 with nonzero horizontal velocity provided the best approximation of FC model near the bottom boundary. () Based on these results, the MBC6 model with only two unknown parameters ( and / ) may be a good candidate which is able to capture more of the flow physics associated with full cavity simulations. Future work in this direction will focus on determining the appropriate scaling laws for the two unknown parameters as a function of the dimensionless flow parameters. Acknowledgements The work presented here is supported by NASA under cooperative agreement NNX7AD94A, monitored by Dr. Brian Allan. References. Seifert, A., T. Bachar, D. Koss, M. Shepshelovich, and I. Wygnanski, Oscillatory Blowing: A Tool to Delay Boundary-Layer Separation. AIAA ournal, ol. 3, No., 993, pp Wang, H., and S. Menon, Fuel-Air Mixing Enhancement by Synthetic Microjets. AIAA ournal, ol. 39, No.,, pp Mahalingam, R., and Glezer, A., Design and Thermal Characteristics of a Synthetic et Ejector Heat Sink, ournal of Electronic Packaging, ol. 7(), 5, pp Smith, B. L., and A. Glezer, et ectoring using Synthetic ets, ournal of Fluid Mechanics, ol. 458,, pp Gallas, Q., Holman, R., Nishida, T., Carroll, B., Sheplak, M., and Cattafesta, L., Lumped Element Modeling of Piezoelectric-Driven Synthetic et Actuators, AIAA ournal, ol. 4, No., 3, pp Glezer, A., and Amitay, M., Synthetic ets, Annual Review of Fluid Mechanics,, ol. 34, pp Holman, R., tturkar, Y., Mittal, R., Smith, B.L., and Cattafesta, L., Formation Criterion for Synthetic ets, AIAA ournal, ol. 43, No., 5, pp Raju, R., E. Aram, R. Mittal, and L. N. Cattafesta, Reduced-order Models of Zero-Net Mass-Flux ets for Large-Scale Flow Control Simulations, AIAA Paper 8-644, Kral, L. D.,. F. Donovan, A. B. Cain, A. B., and A. W. Cary, Numerical Simulation of Synthetic et Actuators, AIAA Paper 97-84, Rizzetta, D. P., M. R. isbal, and M.. Stanek, Numerical Investigation of Synthetic-et Flow Fields, AIAA ournal, ol. 37, No. 8, 999, pp Filz, C., D. Lee, P. D. Orkwis, and M. G. Turner, Modeling of Two Dimensional directed Synthetic ets using Neural Network-based Detereministic Source Terms, AIAA Paper , 3.. Yamaleev, N. K. and M. H. Carpenter, A Reduced-Order Model for Efficient Simulation of Synthetic et Actuators, NASA Technical Reports, NASA/TM-3-664, Rathnasingham, R. and K. S. Breuer, Coupled Fluid-Structural Characteristics of Actuators for Flow Control, AIAA ournal, ol. 35, No.5, 997, pp

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