Emmanuel Creusé andiraj Mortazavi

Transcription

1 AMRX Applied Mathematics Research express 2004, No. 4 Simulation of Low Reynolds Number Flow Control over a Backward-Facing Step Using Pulsed Inlet Velocities Emmanuel Creusé andiraj Mortazavi 1 Introduction Flow control is a rapidly developing area of fluid dynamics that offers the possibility of significant improvements in flow behavior applied to better engineering capabilities. It might be used to postpone the transition from laminar to turbulent regime, to reduce the skin friction drag and to delay the separation. In case studies where there are regions of separation, flow control can be used to encourage earlier reattachment or, more globally, to improve the shedding and transport phenomena in both internal and external flows (see, e.g.,[3]). One of the classical benchmark case studies of the internal separated flows with a rich comparing possibility to the literature is the flow over a backward-facing step. This flow has been analyzed either for laminar evolutions (see, e.g., [1, 4, 7]) or for higher Reynolds numbers (see, e.g., [6]) experimentally and numerically. An advantage of this case study is its quite simplified geometry that can be used to investigate several engineering problems, like flow in the combustion chambers, power plants, biomedical flows, and so forth. In this geometry, the behavior of the recirculation zone is a quantitative measure that should give strong data on the flow. Furthermore, it gives the opportunity to explore various possibilities to manipulate eddies shedding and transport. The flow studied in this work is laminar. The laminar flow is essentially steady and two-dimensional. This property has already been investigated by several authors. Armaly, Durst, Pereira, and Schonung [1] found a spatial variation of about 1% forthe Received 26 September Revision received 27 May Communicated by Jean-Michel Coron.

2 134 E. CreuséandI.Mortazavi downstream velocity in the spanwise direction. Otherwise, Denham and Patrick [4] observed less than 2% of spatial variation in the total flow rate in the streamwise direction along the central plane. Furthermore, the flow is reasonably stable and steady. This flow is characterized by a single, elliptically shaped recirculation zone (see, e.g.,[1, 4]). These characteristics make this flow very convenient for two-dimensional simulations, with a high accuracy and realistic physical features. For this flow, the data on the reattachment length characteristics as well as on the dynamical shedding and transport procedure give a large capability to begin with control-off simulations and then to move quickly towards a wide range of control-on investigations. This paper is organized as follows. In Section 2, the simulation of the flow is described. The compressible Navier-Stokes equations are recalled and the direct numerical simulation scheme used to solve them in a laminar (Re = 191) and slightly compressible (M = 0.1) evolution is presented. This Reynolds number is chosen because of the large experimental and numerical comparison possibilities with the existing literature. Then the computational domain corresponding to the above-mentioned geometry as well as the associated boundary conditions are specified. Section 3 is devoted to the validation of the uncontrolled simulation, and the good correspondence of obtained results with experimental and numerical available data. In the following, a control procedure basically generated by an oscillating inlet velocity with variable frequency and amplitude values is described to modify the dynamical shedding, the transport behavior, and the flow steadiness. Three coefficients used to characterize the control effect are introduced. Then, a control process cost is derived and the control target is carefully presented. In Section 5, the effects of the frequency and the amplitude on the breaking of the recirculation zone and the transport in the channel are investigated in detail. 2 Simulation of the flow 2.1 Governing equations The dynamics of the flow is governed by the two-dimensional compressible Navier-Stokes equations, given in their nondimensionalized formulation, in an orthonormal system, and with usual notations by U t + F x(u) + F y(u) = 1 ( Gx (U, U) + G ) y(u, U), x y Re x y (2.1) U(t = 0) = U 0

3 Flow Control over a Backward-Facing Step 135 with where ρ ρu ρv U = ρu ρv, F x(u) = ρu 2 + p ρuv, F y(u) = ρuv ρv 2 + p, ρe β x (ρe + p)u 0 0 G x (U, U) = σ xx σ xy, G y(u, U) = σ xy σ yy, σ xx = 4 u 3 x 2 3 v y, σ yy = 4 v 3 y 2 u 3 x, σ xy = u y + v x, β x = uσ xx + vσ xy + γk T Pr x, β y = uσ xy + vσ yy + γk T Pr y ; β y (ρe + p)v (2.2) (2.3) γ, k, and Pr are, respectively, the perfect gas constant, the thermal conductivity coefficient, and the Prandtl number, that are constant in the simulations we are dealing with. In order to close the system, the following two relations have to be added: p = (γ 1)ρT ρe = ρ( u 2 + v 2) 2 (state equation) + p γ 1. (2.4) The Reynolds number of the flow, Re, is defined by Re = u cl c ν, (2.5) where u c, l c, and ν are, respectively, a characteristic velocity, a characteristic length, and the viscosity of the fluid.

4 136 E. CreuséandI.Mortazavi The evolution considered in this paper is a laminar (Re = 191) and slightly compressible one (Mach number M = 0.1). 2.2 Numerical scheme The Navier-Stokes equations are solved using a direct numerical simulation method based on unstructured triangular grids. The numerical scheme is a mixed finite volumefinite element method, with a second-order accuracy in both space and time, fully explicit in time. In order to obtain the solution U n+1 l at node A l and time t n+1, we need to evaluate the global space derivatives of the flux D(U n l ) at node A l and time t n : D ( U n l ( ) Fx = x + F ) n y + 1 ( Gx y l Re x + G ) n y. (2.6) y l The convective derivative ( F x / x + F y / y) n l is evaluated by a vertex-based finite volume method associated to a Roe solver. In order to obtain the second-order accuracy in space, a monotone upstream centred scheme for conservation laws procedure is added to increase the order of the interpolation. No flux limiter is needed, since the solutions considered in this paper do not have any discontinuity. The diffusive derivative ( G x / x + G y / y) n l is evaluated by a classical P 1 finite element method. This is a centered process, with the second-order accuracy in space. Finally, the temporal integration of the global space derivatives is performed with a modified Euler scheme (Heun method), with second-order accuracy in time. A mass lumping procedure is applied to make the matrix of the linear system diagonal, and a Courant-Friedrichs-Levy condition (imposing a maximal value of the time step as a function of the space discretization) has to be respected to ensure the numerical stability of the scheme. We note that this well-known and frequently used numerical scheme was chosen because of its efficiency for the simulation of such flows [5]. 2.3 Computational domain The computational domain D is a channel with a rearward-facing step (Figure 2.1). The numerical scheme is applied to compute the flow field in the channel at moderate Reynolds numbers, based on the height of the step and on the inlet velocity U 0 (Re = U 0 H s /ν). This flow can be considered as an entry flow in a channel with two parallel walls, followed by a confined recirculating flow behind a rearward-facing step.

5 Flow Control over a Backward-Facing Step 137 (1) H 1 (2) (3) H s (2) X min B X max C Figure 2.1 Computational domain. We note that the computational cost allows to run the code on a standard workstation: to give an idea, with the mesh used in this work (composed of nodes), the time needed to simulate the transport of a freestream flow from the inlet boundary (1) to the outlet boundary (3)(see Figure 2.1) at the velocity U 0 is roughly equal to half an hour on a Compaq Alpha Server ES Boundary conditions As the domain is bounded, it is necessary to specify some boundary conditions at the frontier. In this work, three boundary conditions are used (see Figure 2.1). (1) The boundary (1) is a subsonic inflow with velocity and temperature imposed: u = U 0, v = 0, T = T 0. (2.7) (2) The boundary (2) is an isothermal no-slip wall: u = 0, v = 0, T = T 0. (2.8) (3) The boundary (3) is a nonreflecting subsonic outflow. All these boundary conditions are precisely described in [11]. For the boundary conditions (1) and (2), the density ρ is computed from the continuity equation on the boundary itself, and the pressure is deduced from the state equation. For the boundary condition (3), the amplitude of the unique entering characteristic wave is cancelled, and a pressure recall coefficient is used to prevent the flow from any pressure derivation [2]. It should be noticed that for the boundary condition (1), only three conditions are used, while Strikwerda [12] claims that four conditions are needed for a two-dimensional subsonic inflow. This feature is pointed out by Poinsot and Lele [11], and comes from their method used to derive this subsonic inflow boundary condition. Thus, this technique is used here for its efficient numerical behavior in numerous benchmark configurations.

6 138 E. CreuséandI.Mortazavi Table 3.1 H 1 /H s 2 X min /H s 12 X max/h s 15 Evaluated quantities - Reattachment point - Velocity profiles - Recirculating area Height Vertical Section Location Figure 3.1 Velocity profiles at Re = Validation of the uncontrolled simulation Before explaining our control objectives, it is important to check if the uncontrolled simulation leads to physically coherent results. As a consequence, we compare results to the existing experimental and numerical literature. The Reynolds number Re = 191 corresponds to a quasisteady laminar evolution. At this Reynolds number, Denham and Patrick [4] and Armaly, Durst, Pereira, and Schonung [1] experimentally showed that the recirculating flow remains essentially two-dimensional. Table 3.1 summarizes the choice of parameters as well as the computed quantities, compared identically to some results in literature [7]. Computed results presented in this paper should be considered as converged results, that is, obtained by successive refinement of the numerical parameters until no further changes occurred. The reattachment length L r is defined as the point on the bottom wall where the streamwise velocity gradient u/ y becomes positive. When the steady state is reached, L r is equal to 8.6H s, which is very close to the value obtained by the experiments in [1, 4] and by another numerical simulation (see [7, page 367, Figure 11]), for which L r 8.6H s. The horizontal velocity profiles are shown graphically in Figure 3.1.

7 Flow Control over a Backward-Facing Step 139 Figure 3.2 Recirculating area (u<0) at Re = 191. The x-axis origin x = 0 corresponds to the point B fitted in Figure 2.1. Wenotethat these profiles have the same shape and slope as in the literature, even if their exact values are slightly different from the results obtained by Ghoniem and Gagnon [7, page 368, Figure 12], that are related to the low compressibility of our flow (Himdi [9] has already compared the slightly compressible and incompressible configurations, validating their similar behavior in spite of a few quantitative differences that are completely predictable). Finally, for the recirculating zone, defined as the area with negative horizontal velocity u (Figure 3.2), our numerical results show an excellent agreement with the incompressible results (see [7, page 367, Figure 10]). 4 Control procedure 4.1 Control method and coefficients The purpose is to control this laminar flow, implementing a time-dependent boundary condition at the entry of the computational domain. To do this manipulation, we take X min /H s = 3. Instead of the previous inflow boundary condition, we impose on the boundary (1) the following oscillating flow: ( )) 2πU0 ft u(t) = U 0 (1 + A cos, v = 0, H s (4.1) T = T 0. The two parameters A and f allow us to vary, respectively, the amplitude and the frequency of the signal. The period of this signal corresponds to the time needed by the fluid to sweep the distance H s /f with the uniform velocity U 0. The theory of oscillating flow in a general framework is described in [10].

8 140 E. CreuséandI.Mortazavi Now, we note r dw (t), r recbot (t), and r rec (t), the three coefficients given by r dw (t) = 1 ( BC BC r recbot (t) = r rec (t) = u ) + (x, y, t) dx, y 1 ( ) +dx u(x, y, t) dy, area(d) D/y<H s 1 ( ) +dx u(x, y, t) dy, area(d) D (4.2) where for α R, α + = (α + α )/2. This notation permits to detect the recirculation zone that corresponds to negative u/ y and u values. Here, r dw (t) corresponds to the proportion at time t of the bottom detached vortical structure lengths versus the channel length beyond the step. This first coefficient is very important in studying the backward-facing step channel flow because it determines how the control procedure (the inlet pulsing condition) modifies the flow behavior at the recirculation zone. First, as we will show in the next section, an increase of the r dw (t) coefficient corresponds to a breakdown of the initial steady recirculation zone, generating fast-traveling separated vortical structures in the neighborhood of the bottom wall. On the other hand, a decrease of this coefficient is also meaningful and corresponds to a transformation of the initial steady recirculation zone to a wavy recirculation area containing small connected rotational structures which oscillate at positions approximately overlapping with the initial surface of the noncontrolled recirculation zone. The second coefficient r recbot (t) corresponds to the proportion at time t of the bottom recirculation areas (u <0) located in the part of the domain defined by y< H s to the channel area. This coefficient is directly related to r dw (t) and the evolution of the main recirculation area. As will be seen, this coefficient is very significant to focus on the evolution and the separation of the steady-state recirculation area, especially in what is related to the influence of the control on the number, size, and proportion of the newly generated vortices only due to the breakdown of the recirculation zone behind the step versus the global flow pattern. The data analyzed by this coefficient is complemented using the r rec (t) that corresponds to the proportion of the total recirculation areas (u <0) to the channel area. To be noted is that in the noncontrolled case, because the vortical structures are uniquely concentrated in the bottom recirculation area, r recbot (t) and r rec (t) are equal. However, pulsing the flow, new eddies appear on the top side of the channel that mainly characterize the effect of the control on the vortex shedding. In fact, the eddy generation is governed

9 Flow Control over a Backward-Facing Step 141 by the frequency and amplitude of the oscillations not only creating detached vortices, but also inducing some regular rotational surfaces in the upper wall. This is why the study of r rec (t) is necessary to measure the relationship between the property of these pulses and evolution of these secondary top-wall structures. Matching these three coefficients to dynamical flow visualizations gives us necessary tools to understand the influence of the inlet velocity oscillation control method on the flow behavior. This control will be performed in function of two frequency and amplitude parameters. Furthermore, it is interesting in our case to explore if we can relate these parameters to a cost function, and if yes, what it means numerically. 4.2 Control process cost We now evaluate the control process cost. At the entry of the computational domain, the velocity of the fluid is ( )) 2πU0 ft U(t) = U 0 (1 + A cos. (4.3) H S The instantaneous acceleration of the fluid at a given time t is then equal to U t (t) = 2πAfU2 0 sin H S ( 2πU0 ft H S ). (4.4) Consequently, the global force needed at time t to ensure this acceleration in the whole inflow section S is F(t) = S ρ U (t)ds, (4.5) t where ρ is the density of the fluid. Since the Mach number is very low, ρ can be considered as a constant. Moreover, ( U/ t)(t) does not depend on the space variables. Thus, we can write F(t) = ρs U (t). (4.6) t The first control process cost on a given period [0, T F ] will be defined by J c(f, A) = F(t) 2 L 2 [0,T F ], (4.7)

10 142 E. CreuséandI.Mortazavi that is to say J c(f, A) = 2ρ2 S 2 π 2 A 2 f 2 U 4 [ 0 H 2 T F H S S 4πU 0 f sin ( 4πU0 ft F H S )]. (4.8) In this work, tests will be performed with several values of the parameter f in the set {f 1,f 2,...,f n } N n. Setting T F = H S /U 0, we get J ( c fj,a ) = 2ρ2 S 2 π 2 A 2 f 2 j U4 0 T F H 2 S which is proportional to the final control process cost function (1 j n), (4.9) J c ( fj,a ) = A 2 f 2 j. (4.10) We remark that the value of J c (f j,a) is given explicitly by the knowledge of f j and of A. What is important to remember from this analysis is that for two couples of variables (f 1,A 1 ) and (f 2,A 2 ) leading to the same effect on the fluid, it is more ingenious to choose the one for which the product f 2 i A2 i is the smaller one (1 i 2). 4.3 Control target The control purpose in this paper is mainly focused on generating the vortex breakdown mechanism in the recirculation zone. This procedure can create fast-traveling rotational structures in the channel and help the trapped particles in the recirculation zone to leave this region and to be transported in the channel flow reaching the exit boundaries. This procedure is a typical benchmark to study the flow control in the combustion problems (see, e.g., [8]). Theefficiency of this control strategy depends on the pulsing frequency and amplitude values imposed at the inlet section. The flow behavior related to this target can be analyzed using the three parameters r dw, r recbot, and r rec defined, respectively, as the average temporal values of r dw (t), r recbot (t), and r rec (t) on the period [0, T F ], which directly exhibit the modification of the rotational areas and their relationship to the flow map. Moreover, a correlation between these parameters and the instantaneous visualization of the flow behavior modifications is very important to better investigate the flow control procedure efficiency. The instantaneous flow field is a good tool to understand the dynamics of the vortical structures evolution, which is directly related to the purpose of our manipulations.

11 Flow Control over a Backward-Facing Step 143 Remark 4.1. We could also introduce a global cost function defined by J ( f j,a ) = α r + (1 α)j c ( fj,a ), (4.11) with α a real parameter to be chosen between 0 and 1, and r which would be either r dw, r recbot, or r rec. The idea would be to minimize this functional. Nevertheless, this is not the aim of this work: our objective is not to minimize a given quantity, but to try to understand the effect of the control on the vortex shedding and the transport phenomena. 5 Numerical results Numerical results are basically obtained to fulfill a parametric study on two variables that characterize such a pulsed flow: frequency and amplitude. The variation of these two parameters is systematically related to the flow oscillations that are the cornerstone of our control study. We separately vary these parameters and observe their effect on the flow map and the three coefficients r dw, r recbot, and r rec. 5.1 Variation of the frequency We plot on Figure 5.1 the variation of r dw, r recbot, and r rec as a function of the frequency f for three values of the amplitude parameter (A = , A = 0.15, and A = 0.5). We will see below that the frequency parameter is the main tuning source of our manipulation strategy. For a better understanding of the dynamics of the flow, the isovalues of u at a given time of the simulation as well as the recirculating area are displayed for f = 3, f = 6, f = 9, f = 12, f = 16, and f = 32, with A = 0.5 (Figures 5.2, 5.3, 5.4, 5.5, 5.6, and 5.7). These visualizations help us to better translate the above plots. All numerical observations confirm that for f 20, the oscillations have almost no influence on the recirculation zone status and the flow remains entirely steady without any vortical transport in the channel (Figure 5.7). For f 20, we recognize two general behaviors. For approximately f 6, r dw is rapidly increasing as a function of the frequency despite the decrease of r rec (Figures 5.1(a) and 5.1(c)). This result occurs because of the breakdown of the recirculation zone to a number of separated fast-traveling recirculating structures. These large convective structures are transported inside the channel and have smaller interaction with each other (Figure 5.2). In other words, the recirculation zone is broken in front of the step and large structures are shedding regularly inside the flow. Increasing the frequency value

12 144 E. CreuséandI.Mortazavi A = A = 0.15 A = rdw rrecbot rrec (a) f (b) f A = A = 0.15 A = 0.5 A = A = 0.15 A = f (c) Figure 5.1 (a) r dw,(b) r recbot, and (c) r rec for A = , A = 0.15, and A = 0.5 versus the frequency.

13 Flow Control over a Backward-Facing Step 145 Figure 5.2 Isovalues of u(top) and recirculation areas (bottom) for A = 0.5 and f = 3. Figure 5.3 Isovalues of u(top) and recirculation areas (bottom) for A = 0.5 and f = 6. Figure 5.4 for A = 0.5 and f = 9. Isovalues of u(top) and recirculation areas (bottom)

14 146 E. CreuséandI.Mortazavi Figure 5.5 Isovalues of u(top) and recirculation areas (bottom) for A = 0.5 and f = 12. Figure 5.6 Isovalues of u(top) and recirculation areas (bottom) for A = 0.5 and f = 16. Figure 5.7 for A = 0.5 and f = 32. Isovalues of u(top) and recirculation areas (bottom)

15 Flow Control over a Backward-Facing Step 147 up tof 6, the number of these traveling structures increases while their size decreases (Figure 5.3). Therefore, for this range of frequencies, one can control the size and number of advected vortices as well as the length and surface of the recirculation zone. Once the frequency reached a critical value around f = 6, a new behavior appears. The vortices join again together and generate a wavy oscillating recirculation area corresponding approximately to the boundaries of the initial noncontrolled steady recirculation zone (Figures 5.4, 5.5, and 5.6). This behavior coincides with a sharp decrease of r dw and a decrease of r recbot (Figures 5.1(a) and 5.1(b)). Therefore, we can consider that this range of frequencies still permits to avoid the generation of the rotational transport phenomena breaking however the steady recirculating zone. On the other hand, the range of frequency f 12 corresponds to a higher r rec value compared to the noncontrolled case (Figure 5.1(c)), justified by the vortex generation at the vicinity of the upper wall (Figures 5.2, 5.3, 5.4, and 5.5). 5.2 Variation of the amplitude We plot on Figure 5.8 the variation of the average values of r dw, r recbot, and r rec as a function of the amplitude A for three values of the frequency parameter (f = 4, f = 12, and f = 32, respectively). The first observation that confirms the previous results is that f = 32 generates no sensible effect on the flow events (Figure 5.8). In other words, over reasonable values of the oscillating frequency, the flow behavior returns to the initial steady case. So, our discussion will be related to f = 4 and f = 12 which are in the sensible range of the flow oscillations and permit us to focus on the amplitude effects. For f = 4, the r dw and r rec values are higher compared to the uncontrolled simulation. Moreover, these values are regularly increasing as a function of A (Figures 5.8(a) and 5.8(c)). This behavior is explained by the fact that once A is larger enough to ensure a control effect (approximately A 0.2), the size of the convected vortices increases as a function of A even if their number remains the same (Figures 5.9, 5.10, and 5.11). The evolution of r recbot (Figure 5.8(b)) indicates that the size of the bottom vortices significantly increases for A 0.5. So, the choice of the amplitude allows to control the size of the traveling eddies in an increasing order. For f = 12, the r dw, r recbot, and r rec values are lower compared to the uncontrolled simulation. Moreover, these values are regularly decreasing as a function of A up to A 0.5, and then achieve a plateau (Figure 5.8). This shows that, for A 0.2, increasing the amplitude strengthens the expected waviness in this range of frequencies (Figures 5.12, 5.13, and 5.14).

16 148 E. CreuséandI.Mortazavi rdw f = 4 f = 12 f = 32 rrecbot rrec A (a) (b) A f = 4 f = 12 f = 32 f = 4 f = 12 f = (c) A Figure 5.8 (a) r dw,(b) r recbot, and (c) r rec for f = 4, f = 12, and f = 32 versus the amplitude.

17 Flow Control over a Backward-Facing Step 149 Figure 5.9 Isovalues of u(top) and recirculation areas (bottom) for A = 0.1 and f = 4. Figure 5.10 Isovalues of u (top) and recirculation areas (bottom) for A = 0.4 and f = 4. Figure 5.11 for A = 0.9 and f = 4. Isovalues of u (top) and recirculation areas (bottom)

18 150 E. CreuséandI.Mortazavi Figure 5.12 Isovalues of u (top) and recirculation areas (bottom) for A = 0.1 and f = 12. Figure 5.13 Isovalues of u (top) and recirculation areas (bottom) for A = 0.4 and f = 12. Figure 5.14 for A = 0.9 and f = 12. Isovalues of u (top) and recirculation areas (bottom)

19 Flow Control over a Backward-Facing Step 151 These results show that even if the amplitude variations modify the flow trends in each range of behavior, the main flow characteristics essentially depend on the frequency choice. 6 Conclusion In this work, a direct numerical simulation technique is validated for a laminar slightly compressible two-dimensional flow in a backward-facing step channel. The low Reynolds number Re = 191 not only offered us large benchmarking possibilities, but also permitted us to be sure about the two-dimensional and stable behavior of the flow. Then, a control procedure basically governed by a pulsed inlet velocity with variable frequency and amplitude values is applied to generate the vortex breakdown mechanism in the recirculation zone, creating fast-traveling rotational structures in the channel and helping the trapped particles in the recirculation zone to leave this region and to be transported in the channel flow reaching the exit boundaries. This control strategy was then studied by a parametric study (frequency and amplitude), and three coefficients were defined to characterize the flow behavior. The control process cost, which is a quadratic function of the amplitude and the frequency, was also defined. During the parametric study, we observed that to break down the recirculation area beyond the step and generate more separate traveling structures, the frequency should be quite small (around f 6). The main characteristics of the vortex shedding, the recirculation zone breakdown, and the transport properties (like number of eddies) were almost depending on the frequency choice. Furthermore, raising the amplitude parameter values only permitted to get larger structures. Then, increasing the frequency to higher values, the detached vortices progressively merged again to each other, generating a wavy unstable recirculation zone. The rate of waviness can be tuned by the amplitude. For higher frequencies again, the uncontrolled flow was recovered. It should be outlined that the main modifications in the flow behavior are generated by the frequency changes. Even if the amplitude has an effect on the evolution trends of the structures size or waviness, it has no influence on the recirculation zone breakdown and separation. To conclude, with an appropriate definition of an oscillating inlet flow, we can control not only the length and the surface of the recirculation zone, but also the dynamical shedding and transport of the vortical structures. The next step of this work is the extension of the control procedure to transitional flows with unsteady and unstable mechanisms.

20 152 E. CreuséandI.Mortazavi Acknowledgments The authors are grateful to Professor Ahmed Ghoniem for his very interesting suggestions. They also wish to thank Professor Remi Abgrall and the referees for their comments to improve this work. References [1] B. F. Armaly, F. Durst, J. C. F. Pereira, and B. Schonung, Experimental and theoretical investigation of backward-facing step flow, J. Fluid Mech. 127 (1983), [2] C. H. Bruneau and E. Creusé, Towards a transparent boundary condition for compressible Navier-Stokes equations, Internat. J. Numer. Methods Fluids 36 (2001), no. 7, [3] E. Creusé, Comparison of active control techniques over a dihedral plane, ESAIM Control Optim. Calc. Var. 6 (2001), [4] M. K. Denham and M. A. Patrick, Laminar flow over a downstream-facing step in a twodimensional flow channel, Trans. Inst. Chem. Eng. 52 (1974), [5] L. Fezoui, S. Lanteri, B. Larrouturou, and C. Olivier, Résolution numérique des équations de Navier-Stokes pour un fluide compressible en maillage triangulaire, Tech. Report 1033, The French National Institute for Research in Computer Science and Control, France, [6] Y. Gagnon, A. Giovannini, and P. Hébrard, Numerical simulation and physical analysis of high Reynolds number recirculating flows behind sudden expansions, Phys. Fluids 5 (1993),no. 10, [7] A. F. Ghoniem and Y. Gagnon, Vortex simulation of laminar recirculating flow, J. Comput. Phys. 68 (1987), [8] A. F. Ghoniem, A. J. Chorin, and A. K. Oppenheim, Numerical modelling of turbulent flow in a combustion tunnel, Philos. Trans. Roy. Soc. London Ser. A 304 (1982), [9] M. Himdi, Contribution à la simulation numérique des écoulements de fluides compressibles et peu compressibles par le code de calcul KIVA-II, Ph.D. thesis, Laboratoire de Mécanique de Lille, France, [10] R. L. Panton, Incompressible Flow, Wiley-Interscience, John Wiley & Sons, New York, [11] T. J. Poinsot and S. K. Lele, Boundary conditions for direct simulations of compressible viscous flows, J. Comput. Phys. 101 (1992), no. 1, [12] J. C. Strikwerda, Initial boundary value problems for incompletely parabolic systems, Comm. Pure Appl. Math. 30 (1977), no. 6, Emmanuel Creusé: Mathématiques AppliquéesauCalcul ScientifiqueEA 3337, Universitéde Valenciennes, Le Mont Houy, Valenciennes Cedex 9, France address: ecreuse@univ-valenciennes.fr Iraj Mortazavi: Mathématiques Appliquées de Bordeaux, UMR-CNRS 5466, 351 Cours de La Libération, Talence, France address: mortaz@math.u-bordeaux.fr