Fig. 1. The coordinate system and the directions of the velocities.

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1 ICMAR 201 FLOW STABILITY NEAR A PLATE WITH A SURFACE MOVING AGAINST AN INCOMING STREAM A.M. Gaifullin, N.N. Kiselev, A.A. Kornyakov The Central Aerohydrodynamic Institute 10180, Zhukovsky, Moscow Reg., Russia 1. Statement of the problem. Consider a flow around a stationary flat plate mounted at zero angle of attack to an incoming stream of incompressible fluid. We assume that the free-stream velocity and the velocity of the surface of the plate are different from zero when t 0. Linear dimensions are divided by the length of the plate l, speed is divided by the free-stream velocity u. Velocity of movement of the surface of the plate is directed against the velocity of the incoming stream and is equal to. The coordinate system is chosen so that its origin coincides with the middle of the plate, the axis x is directed along the plate, the axis y is directed along the normal to it (Fig. 1). Fig. 1. The coordinate system and the directions of the velocities. The problem of fluid flow around a flat plate with surfaces moving against the stream is considered in [1]-[]. In these studies there were shown that if the velocity of movement of the surface of the plate divided by the free stream velocity is equal , the boundary layer equations with zero pressure gradient have stationary solutions of this problem. Attempts to obtain the solution of the boundary layer equations when 1 had no success. It turned out that when 1 the solution of the problem ceases to correspond to the boundary layer equations, and it is necessary either to solve numerically full the Navier-Stokes equations or to consider an asymptotic solution of these equations with transition to various equations in the asymptotic regions selected. This problem was solved in [5, 6] for a finite plate and in [7] for a semiinfinite plate. The solution of the problems for a finite plate with 1 in [5], [6] was searched with one major assumption symmetry of separated zones above and below the plate were postulated, and, therefore, only the flow over the plate was considered. In this case the solution depends only on two parameters: and Re u l /. It was found that for a function of these parameters the flow reaches a steady or self-similar regime. In this paper a numerical study of flow around a finite plate with the surface moving against the flow when 1 is conducted. The results obtained by postulating symmetry of separated zones and without this assumption are compared. For the laminar case calculation of unsteady the Navier Stokes equations is performed, and for the turbulent case calculation using the model SST is performed. A.M. Gaifullin, N.N. Kiselev, A.A. Kornyakov, 201 1

When 0 1 the disturbed flow is described by the first approximation stationary boundary layer equations at zero pressure gradient. When 1 2 ( 1.6 2 2.

2 Section No. 2: Stability, Turbulence and Separation 2. Results of calculations if symmetry of vouchers zones is postulated. In case the flow is postulated to be symmetry when Re 1 depending on three types of flows are realized. When 0 1 the disturbed flow is described by the first approximation stationary boundary layer equations at zero pressure gradient. When 1 2 ( ) the flow of liquid retains a stationary character and in the most part is described by inviscid equations corresponding to vortex-potential flow with velocity discontinuity along the boundary between the vortex and the potential areas [8]. The limiting solution for this class of flows is the Lavrentev Shabat's decision, for which there is no velocity discontinuity at the boundary area between the potential flow and vortex recirculation flow [9]. In Fig 2, the color shows the vorticity field, and black lines are the current lines when 1 and 1.6 and Re = 10. Since the flow is symmetry, in the figures only the upper half of the flow is represented. In numerical calculations the flow is stable. Fig. 2. The vorticity field and the streamlines from after each 0. 0 when 1. Fig.. The vorticity field and the streamlines from after each 0. 0 when When 1 and the number Re is large the flow in the main part of the recirculation region is described by the first approximation solution of the Euler equations. The value of ( y) is constant in the main recirculating flow area and varies in magnitude in the thin mixing layers lying in the vicinity of a line 0 (Fig. ). This result is consistent with Batchelor s theorem about constancy of vorticity in a steady recirculating flow [10]. 2

ICMAR 201 y 1 0,5 1 0,5 Cf Re 0-8 - 0 Fig.. Vorticity dependence on the transverse coordinate y when x 0., 1. 6, Re 10. 0 0 0,5 1 Fig. 5. The coefficient of resistance of the plate.

3 ICMAR 201 y 1 0,5 1 0,5 Cf Re Fig.. Vorticity dependence on the transverse coordinate y when x 0., 1. 6, Re ,5 1 Fig. 5. The coefficient of resistance of the plate. Resistance of the plate is determined by momentum which potential flow loses in the mixing layer on the dividing streamline 0. Loss of momentum is proportional to the velocity discontinuity crossing the line 0. Consequently, when 2 the resistance value of the plate must be zero in order O (Re 1/ 2 ). Monotonic decrease of the drag coefficient of the plate with increasing parameter, obtained in the numerical calculations ( Re 10 ), is an indirect confirmation of the validity of this conclusion (Fig. 5). Thus, for a given number Re with increasing parameter within 2 the recirculation zone size increases and the velocity discontinuity at the boundary of the vortex and potential flow decreases. When 2 this gap vanishes. There is a question, what happens if is made more than 2. It turned out that the flow along the plate is unstable. At large times the transverse and longitudinal dimensions of the recirculation flow increase proportional to t, and the recirculation region moves away from the plate upstream with velocity proportional to 1 / t. The color at Fig. 6 shows the vorticity field, and the black lines shows current lines when 5, Re 10, t 25. Figure 6. The vorticity field and the streamlines from after each 1 when 5 Re 10, t 25.. Results of calculation of the complete problem in case of the laminar flow. Solution of the problem without the symmetry condition differs significantly from the above solution. It was carried out in two ways: in case of the laminar flow by a program the authors created specifically for this problem and in case of turbulent flow by a calculation using a commercial code. In both cases in a sufficiently wide range of the parameter instability of the symmetric solutions and the flow reaches the unsteady one, periodically changing, was observed. Although the plate flows about zero angle of attack, symmetry absence makes the lifting force different from zero.

Section No. 2: Stability, Turbulence and Separation In case of the laminar flow the solution for 0. 5, 1, 1.6, 5 is presented in Figs. 7-10. When 0.5 the flow is stable, corresponding to 1 and 1.

As be- fore the color shows change of the vorticity field, and the lines shows the lines current. Comparing Figs. 2, and 6 with Figs.

4 Section No. 2: Stability, Turbulence and Separation In case of the laminar flow the solution for 0. 5, 1, 1.6, 5 is presented in Figs When 0.5 the flow is stable, corresponding to 1 and 1. 6 the period of unsteady flow variations is equal to T 2. 2 and T Calculations were made with the number Re 10. As be- fore the color shows change of the vorticity field, and the lines shows the lines current. Comparing Figs. 2, and 6 with Figs shows that some features of the symmetric flow are remained in the unstable flow. Thus, evolution of vortex structures when 1and 1. 6 takes place around the plate, and when 5 the vortex flow eventualy departs from the plate and moves erratically upstream. Fig. 7. The vorticity field and the streamlines from 0. 2 to 0. 2 after each when 0.5 Re 10, t 100. Fig. 8. Change of flow characteristics (left to right) when 1 in a time interval t 1. 1(half-cycle). The vorticity field and the streamlines from 0. to 0. after each 0. 05

ICMAR 201 Fig. 9. Change of flow characteristics (left to right) when 1. 6 in a time interval t 1.25 (half-cycle). The vorticity field and the streamlines from 0. 6 to 0. 6 after each 0. 05.

5 ICMAR 201 Fig. 9. Change of flow characteristics (left to right) when 1. 6 in a time interval t 1.25 (half-cycle). The vorticity field and the streamlines from 0. 6 to 0. 6 after each Results of the calculation of the full problem in case of the turbulent flow. As in the laminar case the vortex structure around the plate with the surface moving against the flow is unstable with respect to asymmetric perturbations. However, the flow pattern is significantly different from the laminar flow at the same relative velocity of the surface of the plate. Calculation of unsteady flow was produced using the model SST. Evolution over time of the vorticity field when 2 and 5 is shown in Figs. 11, 12. Dependence of the lift and drag coefficients of the plate on time when 2 and 5, Re is shown in Figs. 1, 1. On the same figures resistance of the surface of the fixed plate in case of the turbulent boundary layer around is presented. 5

6 Section No. 2: Stability, Turbulence and Separation Fig. 10. Change of flow characteristics when 5 in the moments t 0. 05, t 0. 10, t 0. 15, t The vorticity field and the streamlines from to after each

Evolution of the vorticity field in time. 5, Re 6.5 10. Fig.

7 ICMAR 201 Fig. 11. Evolution of the vorticity field in time. 2, Re Fig. 12. Evolution of the vorticity field in time. 5, Re Fig. 1. Coefficient of resistance (left) and the coefficient of lift (right) in time, 2, Re

8 Section No. 2: Stability, Turbulence and Separation Fig. 1. Coefficient of resistance (left) and the coefficient of lift (right) in time, 5, Re Conclusion. With the assumption of symmetry of the flow and without this assumption the solution of the problem of flow around a plate mounted at zero angle of attack, with a moving against the flow surface in wide range of parameter was obtained. There were shown that the rejection of the assumption of symmetry of the flow significantly alters the flow characteristics that are periodic in time. This periodic change in the structure of the flow around the plate was not known previously. Since recirculation vortex structure sufficiently extended in space, this frequency has a finite period. On the upper half-cycle vortex is more intense than the lower, and in the next half cycle, conversely lower vortex is more intense. This study has shown how important it is to produce a study of the complete problem by using unsteady equations of motion, without additional assumptions about the condition of symmetry. This work was supported by the RFBR (project and ). REFERENCES 1. Klemp J.B., Acrivos A. A method for integrating the boundary-layer equations through a region of reverse flow // J. Fluid Mech Vol. 5. Iss. 1. P Cherny G.G. Boundary layer on a traveling surface // Selected Problems of Applied Mechanics: Collected Articles Devoted to the Sixtieth Birthday of Acad. V.N. Chelomei. Moscow: VINITI, 197. P [in Russian].. Cherny G.G. Boundary layer on a traveling surface // Aeromechanics: to the Sixtieth Birthday of Acad. V.V. Struminsky [in Russian]. Moscow: Nauka, P Klemp J.B., Acrivos A.A. A moving-wall boundary-layer with reverse flow // J. Fluid Mech Vol. 76, Iss. 2. P Gaifullin A.M. Flow past a plate with an upstream-moving surface // Fluid Dynamics Vol. 1, Iss. P [in Russian] 6. Gaifullin A.M., Zubtsov A.V. Flow past a plate with a moving surface // Fluid Dynamics Vol., Iss.. P [in Russian] 7. Gaifullin A.M., Zubtsov A.V. Asymptotic structure of unsteady flow over a semi-infinite plate with a moving surface // Fluid Dynamics Vol. 8, Iss. 1. P [in Russian] 8. Sadowsky V.S. Flat vortex-potential inviscid flows and their applications // Trudy TsAGI Vol p. [in Russian] 9. Lavrentiev M.A., Shabat B.V. Problems of hydrodynamics and their mathematical models. Moscow: Nauka, p. [in Russian] 10. Batchelor G.K. On Steady laminar flow with closed streamlines at large Reynolds number // J. Fluid Mech Vol. 1, Iss. 2. P

EXCITATION OF GÖRTLER-INSTABILITY MODES IN CONCAVE-WALL BOUNDARY LAYER BY LONGITUDINAL FREESTREAM VORTICES

ICMAR 2014 EXCITATION OF GÖRTLER-INSTABILITY MODES IN CONCAVE-WALL BOUNDARY LAYER BY LONGITUDINAL FREESTREAM VORTICES Introduction A.V. Ivanov, Y.S. Kachanov, D.A. Mischenko Khristianovich Institute of