Investigation into a new simple DBD-plasma actuation model

Transcription

1 Faculty of Engineering Technology (CTW) Investigation into a new simple DBD-plasma actuation model Report Internship ISAS/JAXA s503 Thijs Bouwhuis

2 Contents General Introduction Introduction to the research Research method Body force distribution Parametric study Normalization of results Results Alternative body force distribution Conclusion and recommendations Bibliography

3 General Introduction In this report, one can read the results of the study conducted at the Institute of Space and Astronautical Sciences, ISAS, located in Japan. This study is part of my internship, a key element in the educational program Master Mechanical Engineering, in which skills and knowledge acquired in previous years have to be used in a real life situation at a graduate level. In this context, I have visited Japan from October first 205 till Februari fourth 206. In this period, I have worked as an intern at the Institute of Space and Astronautical Sciences, ISAS, located near Sagamihara, Kanagawa in Japan. ISAS is part of the Japan Aerospace Exploration Agency, JAXA, founded in 2003 as a fusion of the three major aerospace agencies of Japan: Institute of Space and Astronautical Science (ISAS), National Aerospace Laboratory (NAL) and National Space Development Agency of Japan (NASDA). At ISAS I worked at the department of Space Flight Systems. More specific, I was part of the Fujii-Oyama lab.this lab worked on three topics: Acoustics of rocket jets, a mars airplane and flow separation control. The last subject, flow separation control, was my topic of research for four months. I am very grateful to Fujii-sensei and Oyama-sensei for accepting me in their research group. I also like to thanks Professor Hoeijmakers, for introducing me at ISAS/JAXA and without whom I would not have been able to experience this great opportunity. Words of gratitude and thanks go to Mr. Nonomura-san, Ms. Yakeno-san and Mr. Abe-san for their great help during my research and many fruitful discussions. Thanks also goes to Ms. Tamura-san, for she helped me a lot with arranging all the practical conditions making this internship possible. Finally, words of thanks go to all members of the Fujii-Oyama lab since I had an educative and great time, Arigatou gozaimasu. Thijs Bouwhuis

4 2 Introduction to the research Active flow separation control using dielectric barrier discharge plasma actuators (hereafter: PA) has been studied intensively during the last decade, with the aim of improving performance and/or efficiency of a wide variety of fluid machinery. Flow separation occurs when a boundary layer flow is unable to overcome an adverse pressure gradient and the velocity of the fluid with respect to the boundary falls to zero. The flow becomes detached. This phenomenon can occur with a flow around an airfoil at a high angle of attack. The angle at which the flow starts to separate from the airfoil is called the stall angle. As a result of the detached flow, a lot of undesirable effects occur. Among others: a decrease in lift and an increase in drag. By controlling this flow separation, large improvements in efficiency and performance of airfoils can be achieved. Flow separation can be controlled by using a PA. The PA consists of two electrodes with a dielectric material in between. When a high voltage O(0 3 V ), high frequency AC is applied between the two electrodes a plasma is created. This plasma induces a wall jet which can be utilized in flow separation control. A schematic representation of the PA can be observed in figure. Figure : Schematic representation of the Dielectric Barier Discharge Plasma Actuator (PA) The PA has different mechanisms influencing the flow separation. First, the momentum in the boundary layer increases by the wall jet behaviour of the PA. Secondly relative large vortices introduce additional momentum to the flow and thirdly small scale vortices, turbulence, is introduced which can transition the laminar flow to turbulent flow. A conventional numerical method for the PA, which gives results in good agreement with experimental results [2], solves the flow equations and an additional two equations on an additional grid. It is known as the Suzen model []. The resulting body force field, shown is figure 2, of this model is coupled to the flow equations. Much experimental and numerical research is focused on the relation between the operational parameters of the PA (voltage, base- and burst frequencies etc.) and the performance of the PA. However, the dominant parameters of the body force which determine the induced flow have not been identified so far. The main focus of this study will be to identify the dominant parameters of the body force field, determining the induced flow. To this end the body force field of the Suzen model will be reduced in complexity and a Gaussian distributed body force will be studied. A future objective is to propose an new simple PA model, based on the dominant parameters, for which the induced flow is similar to that of existing models but without additional equations to be solved on an extra grid. In this report an outline of the research method is given: first a body force distribution is determined based on the Suzen model. This body force distribution is used in a numerical simulation for two 2

5 dimensional flow, in order to carry out a parametric study on the parameters of the body force. After that an outline of the normalization is given, which is used to present the results afterwards. Some additional, more advanced, work is presented in which the body force distribution is somewhat adjusted.finally some concluding remarks are given. Additionally one appendix is added: An abstract of the most important topics of the research, submitted to th International ERCOFTAC Symposium on Engineering Turbulence Modelling and Measurements. 3

6 3 Research method A parametric study is carried out using numerical simulations for two dimensional flow. The flow field is described by the Navier-Stokes equations, in which the PA forcing is implemented as a source term. The number of grid points in the computational grid is Near the body force field, the grid is uniformly distributed and grid spacing is small, as illustrated in figure 2. The computational domain is taken sufficiently large to ensure the far field boundaries do not affect the induced flow. The grid coarsens further away from the body force field. The computational Mach number is taken as M a = 0.2 (incompressible upper limit) and the computational Reynolds number is taken as Re = A computational Reynolds number Re = is used as a validation later on. Sixth-order compact difference schemes [3] were used to discretize the spatial derivatives and ADI-SGS methods [4] were used for time integration. This is an inhouse CFD code of ISAS/JAXA. No further commands are made on these schemes and computational methods, since it is not in the scope of this research. The flow field which is numerically simulated, is the induced flow of a PA in quiescent air. In the simulation the PA is represented as a body force distribution, on which the next subsection (3.) will elaborate. In order to find the dominant parameters of this body force distribution, a parametric study is set up. How this parametric study is designed will be explained in subsection (3.2). Because we re dealing with an induced flow in quiescent air, there is no free stream velocity which can be used as a reference to normalize the results. To deal with this problem a new normalization is derived in subsection (3.3). 3. Body force distribution A conventional way to represent a PA in a numerical flow simulation, is by using the Suzen-Huang model. This model incorporates the effects of the PA on the external flow into the Navier Stokes computations as a body force vector. Two additional equations are solved in order to compute this body force vector. One equation for the electric field due to the applied AC voltage at the electrode of the PA and one equation for the charge density representing the ionized air. This numerical model is calibrated against an experiment having a PA driven flow in quiescent air. The numerical model shows good agreement with the experimental results. The body force field distribution is shown is figure 2. The components of the force in x- and z-direction are shown in figures 2(a) and 2(b). The amplitude of the force is in each point as: (F = f 2 x + f 2 z ). Spatial integration of the body force field over the whole flow domain yields the total induced momentum per second, which is denoted as C µ. In this research we do not use this Suzen-Huang model, but a more simple spatial Gaussian distributed body force field. This is a temporal constant body force, (a) x-component of Suzen model body force (b) z-component of Suzen model body force Figure 2: The commonly used Suzen-Huang model (c) Amplitude of Suzen model body force 4

7 in general computed by: ( ( (x x0 ) 2 f(x, z) x = a exp 2σx 2 + (z z 0) 2 )) 2σz 2 This model can be implemented in the Navier Stokes computations as a body force vector. As described in the introduction a gaussian body force model will be implemented in the governing equations. Equation (3.) is first simplified by setting the aspect ratio to unity (σ x = σ z = σ) and setting x 0 = 0 and z 0 = 0, thus obtaining the following body force distribution: ( ( x 2 + z 2 )) f(x, z) x = a exp 2σ 2 (3.2) In here the parameters a and σ have to be determined and will be based on the amplitude of the body force of the suzen model (figure 2). In the amplitude plot (figure 2(c)) one can observe a very high peak in the amplitude on the left (upstream) side, which will be ommited in the modeling of the Gaussian profile. Instead we focus on the larger region right of this high peak. Here we determine the maximum amplitude, which is directly implemented in the Gaussian distribution as the amplitude (a). The standard deviation (σ) is determined from the C µ value of the Suzen-Huang model, to ensure the Gaussian model has the same C µ value as the Suzen-Huang model. Note: When determining the Half Width at Half Maximum (HWHM) as a characteristic length of the Suzen-Huang model and ensuring the Gaussian model has got the same characteristic length, by HWHM 2 log 2 setting the standard deviation as σ =, we end up with a Gaussian model which does not significantly differ from the one determined above. Approximate values, used in the numerical model for a and σ are and 4.27E 4 respectivily. This gaussian distribution will be referred to as the basic gaussian. A spatial body force distribution of this basic gaussian can be observed in figure 3. (3.) Figure 3: Basic gaussian, described in equation Parametric study A parametric study is performed by deviating from the basic gaussian, by variating the parameters a and σ. The values for σ are chosen as follows (normalised with σ from basic gaussian): 0.50, 0.75,.00,.50, 2, 00 For a the following (normalised) values are selected: = 0.25, 0.44, =.00,.78, = By combining these parameters we obtain 25 cases. Since C µ aσ 2 the normalised C µ values can be easily obtained. The parameters are determined in such a way, that we obtain same C µ cases. The C µ values for all cases are summerized in the table below: 5

8 σ a E-2.4E- 2.50E- 5.62E E- 2.50E- 4.44E E- 5.62E E E Table : C µ of all 25 computational cases with the corresponding a and σ parameter. All values are normalised using the basic gaussian. 3.3 Normalization of results Since we are dealing with a numerical simulation of an PA induced flow in quiescent air, there is no free stream velocity with which the results can be normalized. In this subsection a new normalization is derived by nondimensionalizing the Navier-Stokes equations and additionaly, from a steady state form of the Navier-Stokes equations an approximation for the velocity of the induced flow is derived. Nondimensionalization We start with the following dimensional form of the Navier-Stokes equations (3.3): ( ) u i t + u u i j = p + µ 2 u i x j ρ x i ρ x 2 + f i (3.3) j We now define the reference parameters for bodyforce(3.4), density(3.5), lengthscale(3.6) and viscocity(3.7) with the following dimensions: [ ] ML f ref = T 2 L 3 = [ ML 2 T 2] (3.4) [ ] M ρ ref = L 3 = [ ML 3] (3.5) σ ref = [L] (3.6) [ ] M µ ref = = [ ML T ] (3.7) LT The reference bodyforce and lengthscale are based on the amplitude and body force distribution of the plasma actuator. Using the reference parameters we define a reference velocity(3.8), lengthscale(3.9) and timescale(3.0). U ref = Using this U ref, L ref and T ref we define dimensionless variables as: f ref σ ref ρ ref (3.8) L ref = σ ref (3.9) T ref = L ref U ref (3.0) 6

9 u i t = ũ i U ref t T ref u i ũ i U ref u j = ũ j x j x j T ref p = p U ref ρ x i x i T ref ( ) ( ) µ 2 u i 2 ũ i µ ref U ref ρ x 2 = j x 2 j ρ ref L 2 ref f i = f i U ref T ref Here, the variables with a tilde on top are dimensionless. Substitution of these variables in the Navier- Stokes equations (3.3) and multiplying with T ref U ref yields the following: ( ) ũ i t + ũ ũ i j = p + µ ref U ref T ref 2 ũ i x j x i ρ ref L 2 ref U ref x 2 + f i (3.) j Here we find an expression for the Reynoldsnumber in front of the second derivative term as: µ ref U ref T ref ρ ref L 2 ref U ref = µ ref T ref ρ ref L 2 ref = µ ref ρ ref L ref U ref = µ ref = fref σ ρ ref σ ref ref ρ ref Inverting yields the Reynoldsnumber we will use from now on: µ ref f ref ρ ref σ 3 ref = Re f Re f = µ ref f ref ρ ref σ 3 ref (3.2) Using this Reynoldsnumber, we nondimensionalized the Navier-Stokes equations (3.3) and obtained the following equation (note: from this point on we ommit the tilde signs on the variables, but keep in mind that they are dimensionless variables): u i t + u u i j = p + 2 u i x j x i Re f x 2 j + f i (3.3) Steady state Starting from equation (3.3) we define the following: u i = ū i + u i, p i = p i + p i (3.4) Here ū & p are time mean values and u i & p i are fluctuations. Substitution in equation (3.3) and taken the time average yields, after ommiting (near)zero terms, the steady state equation for momentum: ( ) ūi ū j = p + 2 ū i x j x i Re f x 2 + f i (3.5) j 7

10 Assuming a 2D flow, so there are no fluctuations in the y-direction (i = 2). With this we obtain two equations, for x- and z- direction respectively: ū ū ū + w x z = p x + ( 2ū Re f ū w x + w w z Analytical derivation of induced flow = p z + Re f x ū z 2 ( 2 w x w z 2 ) + f x (3.6) ) + f z (3.7) From the steady state momentum equation in streamwise direction (equation (3.6), we attempt to find so expression for the induced flow. For low Reynoldsnumber, the momentum equation in streamwise direction (3.6) reduces to: Re f 2 ū z 2 = f x (3.8) If we now assume, as a most simplyfied model, a constant body force in the wall normal direction we can solve equation 3.8 by twice integrating with respect to z: Applying the boundary conditions ū(z) = Re f f x 2 z2 + c z + c 2 (3.9) z = 0, ū = 0 (3.20) z =, ū = U ref (3.2) yields the following relationship for ū: ū(z) = Re f 2 f x( z)z + U ref z (3.22) In absence of a free stream, U ref = 0, one can state: ū(z) Re f (3.23) 8

11 4 Results The results of the flow computation are quasi steady, as can be seen in figure 4. The results for 0 < T ft < 2 are not utilized. For further analyses instantaneous flow fields (for T ft > 2) are used unless stated otherwise. The maximum induced velocity is presented in figure 5. Here we see for an increase in induced velocity for both increasing amplitude a and increasing standard deviation σ of the body force distribution. Both of these results are as expected, since the momentum added to the flow increases in both cases. Rearranging the results yields figure 6. We observe for low C µ that U induced,max is constant. On the other hand, for high C µ, U induced,max decreases with increasing σ. The velocity profiles of five C µ =, are shown in figure 7. The shape of the five profiles show similarities, only the magnitude of the velocity varies with Re f. Normalization of figure 5 using U ref and arranging the results by Re f yields figure 8. It can be observed that all results collapse into one curve, depending on Re f only. On this curve two regions can be distinguished. Below Re f 00, all results are on the line Umax U ref Re f denoting a Stokes flow regime. This was also derived in equation Above Re f 00 the results start deviating from this straight line and tend to U max U ref. From figure 8 it is derived for low Re f : U max Cµ µ ref. For high Re f, assuming asymptotic behaviour, the following relation is derived: U max σ ref. To check if the assumption of asymptotic behaviour is legitimate, additional computations at a high Re f number are done. These additional results are plotted in figure 9 In order to validate the results shown in figure 0 the simulation was recomputed on a different computational Reynolds number. All previous results were computation at a computational Reynolds number of Re = 63000, the validation is done at Re = It can be seen, that the results for the new computational Reynolds number collapse with the origional data of the parametric study. U max,induced [m/s] T ft [-] C µ = /6 C µ = /4 C µ = C µ = 4 C µ = 6 Figure 4: The maximum induced velocity as a function of flow trough time T ft. 9

12 U induced,max [m/s] a=0.25 a=0.44 a=.00 a=.77 a= e+00 2e-05 4e-05 6e-05 8e-05 e-04 σ [m] Figure 5: Maximum induced velocity as a function of σ for a number of amplitudes. The amplitude a is normalized with the maximum amplitude of the Suzen model. U induced,max [m/s] C µ =0.25 C µ =0.44 C µ =0.56 C µ =.00 C µ =.78 C µ =2.25 C µ = e+00 2e-05 4e-05 6e-05 8e-05 e-04 σ [m] Figure 6: Maximum induced velocity results, arranged by C µ (normalized with Suzen model C µ ) of the body force field 0

13 z / σ ref [-] Re f = 4.2 Re f = 22.3 Re f = 99.8 Re f = 86.5 Re f = U induced /U ref [-] Figure 7: Normalized velocity profiles, for C µ = at x = x umax U max / U ref [-] 0 0. a=0.25 a=0.44 a=.00 a=.78 a= Re f [-] Figure 8: Normalized maximum induced velocity arranged by Re f. All data of figure 5 collapse in one curve

14 Figure 9: Maximum induced velocity for high Re f computations, plotted together with the origional results. It can be observed that the results tending towards a horizontal asymptote. 0 Re stdin =63000 Re stdin =23000 U max / U ref [-] Re f [-] Figure 0: Normalized maximum induced velocity arranged by Re f for two computational Reynolds number. Re stdin represents the computational Reynolds number. 2

15 A momentum balance is made, in order to clarify the deviation from the line Umax U ref Re f for high Reynolds number in figure 8. To that end, a control volume is required. This is schematically depicted in figure. In this momentum balance analysis, time averaged results are used (time average taken for 2 < T ft < 7). For the momentum balance we have an input of momentum into the control volume from the body force and output of momentum as dissipation at the wall and induced momentum (p out p in ). Note: The pressure term in the momentum balance is neglected, since the flow velocity is very low. The results of the momentum balance analysis can be observed in figure 2. It can be observed that for Re f < 00 the wall dissipation equals C µ, so we have a dissipation dominated flow, a stokes flow. This corresponds to the stokes flow regime observed in figure 8. For higher Re f, we observe a relative decrease in wall dissipation and an increasing induced momentum. Figure : The control volume for the momentum balance. Their is momentum convection into the c.v. (p in ) and a source of momentum, the body force field. The quantity of momentum added to the c.v. by the body force is denoted with C µ. Momentum is convecting out of the c.v. and momentum is dissipated at the wall. Quiescent air is assumed at the top of the control volume. 3

16 .2 wall dsp. ind. mom. p s - /C µ [-] Re f [-] Figure 2: Momentum per second normalized with the body forces C µ as a function of Re f. 4

17 5 Alternative body force distribution Up to this point, all body force distributions had the maximum amplitude at the wall. This section briefly discusses an alternative body force distribution discribed in equation 5. forz 0 0 and depicted in figure 3. With this new body force distribution additional simulations are computed with the same settings as the origional parametric study. The results will be compared. Simulations are done for z 0 σ = 0,, 2, 3, 4 and 5. Parameters a and σ are similar to the values of the basic gaussian. Figure 3: Body force distribution for z 0 0 For the case for which z 0 0, the normalization is adjusted. U ref is adapted to take an varying C µ value into account. L ref is adapted to account for two lengthscales: The height from the wall, computed as a gravity point (eq. 5.4) of the distribution and a characteristic length of the body force distribution itself. Half the harmonic mean of the two separate lengthscales becomes the new L ref. With these new reference parameters the Navier-Stokes equations are non-dimensionalised and a new Reynoldsnumber is obtained, equation 5.5. The original results (figure 8) of the parametric study are also re-normalized with the new normalization and the results are shown in figure 4. It is shown that the results for both z 0 = 0 and z 0 0 collapse into one, Re f,new -dependent curve. ( ( x 2 + (z z 0 ) 2 )) f(x, z) = f ref exp 2σ 2 C µ U ref,new = ρ σ (5.) (5.2) L ref,new = σ + (5.3) z 0,g z f dxdz z 0,g = (5.4) f dxdz Re f,new = C µ ρ L 2 ref,new µ σ (5.5) 5

18 z 0 = 0 z 0 > 0 U max / U ref [-] Re f [-] Figure 4: The results of the origional parametric study are plotted together with the results for z 0 0. For the normalization equations 5.2 and 5.5 are used. 6

19 6 Conclusion and recommendations The flow field resulting from a Gaussian body force field with z 0 = 0, can be normalized using the proposed U ref (eq. 3.8) and Re f (eq. 3.2). This normalization yields a relation of U induced,max as a function of Re f. For low Re f, C µ is the dominant parameter and for high Re f, C µ and σ ref are the dominant parameters. Introducing an extra parameter (z 0 0) requires an adapted normalization, which yields a relation between U induced,max and Re f,new (eq. 5.5). This relationship shows that the effect of height above the wall decreases when the height from the wall becomes larger (prescribed by L ref,new, eq. 5.3). Some recommendations can be made on both this work and some future work. In this work, the momentum balance analysis could have been done more thoroughly. There is no balance between input (C µ ) and output (wall dissipation, induced momentum). This might improve by implementing the pressure term, although the flow velocities are quite low, and by not assuming quiescent air at the top of the control volume. For the body force field with z 0 0 a relation is found for U induced,max and Re f,new, but to achieve this a reference length is set (eq. 5.3) which holds no particular physical meaning. A more thorough theoretical background is desirable. For future work, a more simple plasma actuation model can be proposed based on the dominant parameters for the induced flow. With this it might be possible to have an plasma actuator model, which can be directly inplemented even in a course grid flow simulation. 7

20 7 Bibliography [] Suzen, Y.B., Huang, P.G.,Jacob, J.D. and Ashpis, D.E., Numerical simulations of plasma based flow control application, AIAA , (2005). [2] Aono, H., Sekimoto, S., Sato, M., Yakeno, A., Nonomura, T., Fujii, K., Computational and experimental analysis of flow structures induced by a plasma actuator with burst modulations in quiescent air, Mechanical Engineering Journal, 2.4 (205), [3] Lele, S.K., Compact finite difference scheme with spectral-like resolution, Journal of Computational Physics, Vol. 03, (992), pp [4] Hishida, H. and Nonomura, T., ADI-SGS scheme on ideal magnetohydrodynamics, Journal of Computational Physics,Vol. 228, (2009), pp

21 Investigation into a new simple DBD-plasma actuation model T. Bouwhuis,2, Y. Abe 3, A. Yakeno 4, T. Nonomura 4, H.W.M. Hoeijmakers and K. Fujii 5 University of Twente, Faculty of Engineering Technology, Drienerlolaan 5, 7522 NB Enschede, The Netherlands 2 t.bouwhuis@student.utwente.nl 3 University of Tokyo, Japan 4 ISAS/JAXA, Sagamihara, Kanagawa, Japan 5 Tokyo University of Science, Japan Abstract The dominant factors of a body force field, representing a plasma actuator, are identified by means of a parametric numerical study. Two dimensional flow simulations have been performed for a plasma actuator operating in quiescent air. Because of the absence of a free stream the induced velocity is normalized with a proposed reference velocity, based on parameters of the body force field. The normalized maximum induced velocity depends on the Reynolds number.. BACKGROUND Active flow control using dielectric barrier discharge plasma actuators (hereafter: PA) has been studied intensively, with the aim of improving performance and/or efficiency of a wide variety of fluid machinery. The PA consists of two electrodes with a dielectric material in between. When a high voltage O(0 3 V ), high frequency AC is applied between the two electrodes a plasma is created. This plasma induces a wall jet which can be utilized in flow separation control. A conventional numerical method for the PA, which gives results in good agreement with experimental results [2], solves the flow equations and an additional two equations on an additional grid. It is known as the Suzen model []. The resulting body force field, shown is figure (a), of this model is coupled to the flow equations. Much experimental and numerical research is focused on the relation between the operational parameters of the PA (voltage, base- and burst frequencies etc.) and the performance of the PA. However, the dominant parameters of the body force which determine the induced flow have not been identified so far. The main focus of this study will be to identify the dominant parameters of the body force field, determining the induced flow. To this end the body force field of the Suzen model will be reduced in complexity and a Gaussian distributed body force will be studied. A future objective is to propose an new simple PA model, based on the dominant parameters, for which the induced flow is similar to that of existing models but without additional equations to be solved on an extra grid. 2. METHOD A parametric study is carried out using numerical simulations for two dimensional flow. The flow field is described by the Navier-Stokes equations, in which the PA forcing is implemented as a source term. The number of grid points in the computational grid is Near the body force field, the grid is uniformly distributed and grid spacing is small, as illustrated in figure (a). The computational domain is taken sufficiently large to ensure the far field boundaries do not affect the induced flow. The grid coarsens further away from the body force field. Sixth-order compact difference schemes [3] were used to discretize the spatial derivatives and ADI-SGS methods [4] were used for time integration. The Gaussian body force model will have the force pointing in x-direction only, in contrast to the Suzen model which features a multidirectional force field (x- and z-direction) with a complex distribution of the body force field. The amplitude (F = f 2 x + f 2 z ) of the force field produced by the Suzen model is plotted in figure (a). The Gaussian body force field used in the present study is prescribed by equation and shown in figure (b). In here the parameters f ref and σ are based on the local maximum and characteristic length of the Suzen model (fig: (a)). Note that the body force strength is independent of time. Spatial integration of the body force field over the whole flow domain yields the total induced momentum per second, which is denoted as C µ. The C µ of the Gaussian body force model is in good agreement with the C µ of the Suzen model (fig. (a)). (a) Amplitude of Suzen model body force (b) Body force distribution for z 0 = 0 (c) Body force distribution for z 0 0 Figure : Body force fields, all presented on the same scale. A length indicator is shown in figure (b)

22 2 A parametric study has been carried out varying the parameters f ref and σ, in the Gaussian model (eq. ). Because the C µ from a Gaussian function can be determined exactly, the parameters f ref and σ are determined such that the set of studies contains body force models with similar C µ, but different f ref and σ. The total number of flow computations is 25. Because of the absence of a free stream velocity, there is not a proper reference velocity that can be used to normalize the results. Therefore a set of 4 reference parameters is chosen: f ref (maximum value in the body force field), σ (standard deviation of body force), µ (dynamic viscosity determined from flow computation as µ ref = Ma CFD /Re CFD ) and ρ (constant). Using these reference parameters, a reference velocity is defined in equation 2. By using a reference length (eq. 3) and time T ref = Lref U ref, the Navier Stokes equation is non dimensionalized. This nondimensionalization yields a Reynolds number defined in equation 4. These equations (2, 3 & 4) are used to normalize the parametric study. Besides the parametric study, variations of the height z 0 of the center of the body force from the wall is studied. Using the formulation for the body force field given in equation 5. An example of such a body force field is depicted in figure (c). By introducing one extra parameter, an additional length-scale is obtained. Also, both the shape and C µ change. Therefore the reference velocity ( ( x 2 + z 2 )) f(x, z) = f ref exp 2σ 2 () f ref σ U ref = (2) ρ L ref = σ (3) Re f = f ref ρ σ µ 3 (4) ( ( x 2 + (z z 0 ) 2 )) f(x, z) = f ref exp 2σ 2 (5) C µ U ref,new = ρ σ (6) L ref,new = σ + (7) z 0,g z f dxdz z 0,g = (8) f dxdz Re f,new = C µ ρ L 2 ref,new (9) µ σ and reference length need to be redefined as defined in equations 6 and 7. In eq. 7, z 0,g is the gravity point, the weighted height from the wall, of the body force field. It is defined in equation 8. Normalization of the Navier-Stokes equations with the new reference dimensions yields a new Reynoldsnumber defined in equation RESULTS The results of the flow computation are quasi steady so for further analyses, instantaneous flow fields are used unless stated otherwise. The maximum induced velocity is presented in figure 2(a). Rearranging the results yields figure 2(b). We observe for low C µ that U induced,max is constant. On the other hand, for high C µ, U induced,max decreases with increasing σ. The velocity profiles of five C µ =, are shown in figure 2(c). The shape of the five profiles show similarities, only the magnitude of the velocity varies with Re f. Normalization of figure 2(a) using U ref and arranging the results by Re f yields figure 2(d). It can be observed that all results collapse into one curve, depending on Re f only. On this curve two regions can be distinguished. Below Re f 00, all results are on the line Umax U ref Re f denoting a Stokes flow regime. Above Re f 00 the results start deviating from this straight line and tend to U max U ref. From figure 2(d) it is derived for low Re f : U max Cµ µ ref. For high Re f, assuming asymptotic behaviour, the following relation is derived: U max σ ref For the case for which z 0 0, the normalization is adapted as given by equations 6, 7 and 9. The original results (figure 2(d)) of the parametric study are also re-normalized with the new normalization and the results are shown in figure 3. It is shown that the results for both z 0 = 0 and z 0 0 collapse into one, Re f,new -dependent curve. 4. CONCLUSION The flow field resulting from a Gaussian body force field with z 0 = 0, can be normalized using the proposed U ref (eq. 2) and Re f (eq. 4). This normalization yields a relation of U induced,max as a function of Re f. For low Re f, C µ is the dominant parameter and for high Re f, C µ and σ ref are the dominant parameters. Introducing an extra parameter (z 0 0) requires an adapted normalization, which yields a relation between U induced,max and Re f,new. This relationship shows that the effect of height above the wall decreases when the height from the wall becomes larger (prescribed by L ref,new, eq. 7). REFERENCES [] Suzen, Y.B., Huang, P.G.,Jacob, J.D. and Ashpis, D.E., Numerical simulations of plasma based flow control application, AIAA , (2005). [2] Aono, H., Sekimoto, S., Sato, M., Yakeno, A., Nonomura, T., Fujii, K., Computational and experimental analysis of flow structures induced by a plasma actuator with burst modulations in quiescent air, Mechanical Engineering Journal, 2.4 (205), [3] Lele, S.K., Compact finite difference scheme with spectral-like resolution, Journal of Computational Physics, Vol. 03, (992), pp [4] Hishida, H. and Nonomura, T., ADI-SGS scheme on ideal magnetohydrodynamics, Journal of Computational Physics,Vol. 228, (2009), pp

23 3 U induced,max [m/s] a=0.25 a=0.44 a=.00 a=.77 a= e+00 2e-05 4e-05 6e-05 8e-05 e-04 σ [m] U induced,max [m/s] 0 0e+00 2e-05 4e-05 6e-05 8e-05 e-04 σ [m] (a) Maximum induced velocity as a function of σ for a number (b) Maximum induced velocity results, arranged by C µ (normalized of amplitudes. The amplitude a is normalized with the maximum with Suzen model C µ) of the body force field amplitude of the Suzen model C µ =0.25 C µ =0.44 C µ =0.56 C µ =.00 C µ =.78 C µ =2.25 C µ =4.00 z / σ ref [-] Re f = 4.2 Re f = 22.3 Re f = 99.8 Re f = 86.5 Re f = 70.6 U max / U ref [-] 0 0. a=0.25 a=0.44 a=.00 a=.78 a= U induced /U ref [-] Re f [-] (c) Normalized velocity profiles, for C µ = at x = x umax (d) Normalized maximum induced velocity arranged by Re f Figure 2: Results of the parametric study. Figures (a) and (b) present the maximum induced velocity as a function of the characteristic length σ of the body force field. Figures (c) and (d) show the normalization using the proposed reference velocity. z 0 = 0 z 0 > 0 U max / U ref [-] Re f [-] Figure 3: The results of the origional parametric study are plotted together with the results for z 0 0. For the normalization equations 6 and 9 are used.