LES Analysis on Shock-Vortex Ring Interaction Yong Yang Jie Tang Chaoqun Liu Technical Report 2015-08 http://www.uta.edu/math/preprint/
LES Analysis on Shock-Vortex Ring Interaction Yong Yang 1, Jie Tang 2, Chaoqun Liu 3 University of Texas at Arlington, Arlington, Texas 76019 The mechanism why MVG can reduce flow separation is widely accepted as that the MVG can generate streamwise vortex which strongly mix the boundary layer and make the velocity profile fuller at the bottom. Therefore, the boundary layer becomes more capable to resistant the strong adverse pressure gradient caused by shocks to keep attached. However, this is not the case. The mechanism of reduction of shock induced flow separation by MVG is that the shock wave breaks down and disappears when the ring-like vortices generated by MVG are passing through the shock and the vortex structures never break down which is influenced very little when they pass the shock wave. In this paper, the interaction between supersonic boundary layer and the shock wave in a ramp flow at M=2.5 and Re=1440 is investigated by using a high order large eddy simulation code (LESUTA) with the 5th order Bandwidth-optimized WENO scheme developed in UTA. A quantitative investigation on the interaction between ring-like vortices and shock wave by tracking one ring-like vortex in three time steps is carried out. Details are reported in this paper. Keywords: MVG, LES, Shock, Vortex Ring, Flow Separation Nomenclature MVG = micro ramp vortex generator M = Mach number Re θ = Reynolds number based on momentum thickness h = micro ramp height δ = incompressible boundary-layer nominal thickness x, y, z = spanwise, normal and streamwise coordinate axes u,v,w = spanwise, normal and streamwise velocity LES = large eddy simulation Subscript 0 = inlet w = wall = free stream I. Introduction t is known that shock boundary layer interaction (SBLI), for supersonic ramp flow, could significantly degrade Ithe quantity of flow field by inducing large scale separations. In order to control separations induced by SBLI, vortex generators (VG) have been used widely since 1940s 1. As a kind of low-profile passive device designed for the boundary layer control, micro vortex generator (MVG) is only with a height approximate 20-40% (more or less) of the boundary layer thickness, which usually get the efficiency similar to conventional VGs while with less compensation led into. The mechanism why MVG can reduce flow separation is widely accepted as that the MVG can generate streamwise vortex which strongly mix the boundary layer and make the velocity profile fuller at the 1 PhD student, University of Texas at Arlington. 2 PhD Student, University of Texas at Arlington. 3 Professor, University of Texas at Arlington, AIAA Associate Fellow.
bottom 1. Therefore, the boundary layer becomes more capable to resistant the strong adverse pressure gradient caused by shocks to keep attached. However, recent researches show that MVG has the ability to break shock waves which induce boundary layer separation 2. Li and Liu 3, 4 already showed the real mechanism of MVG is to generate a strong momentum deficit zone behind MVG (Fig.1). This momentum deficit zone can generate a strong shear layer and further a chain of vortex rings (Fig.2 and Fig.3) due to the K-H type instability. This new finding has been confirmed by experiments 5 (Fig.4). The generated vortex rings will immediately interact with shock wave induced by the ramp and shock wave will be distorted badly and even be destroyed 6 (Fig.5). (a) Figure 1. (a) Momentum deficit behind MVG; The distribution of the time-averaged streamwise velocity with and without MVG Figure 2. A chain of vortex rings generated by MVG
Figure 3. Sketch of vortex ring generation by shear layer with low speed streaks (a) LES Simulation Experiment Result by (Sun et al 2011) Figure 4. Distribution of Kelvin-Helmholtz vortices and streamwise vortices (a) Figure 5. Pressure gradient distribution (a) without ring-vortex shock interaction and with ring-vortex shock interaction
II. Numerical methods, grid generation and turbulent inflow In this study, numerical simulations are made on supersonic ramp flow at 2.5 and 1440. In order to make simulations, a kind of large eddy simulation method is used by solving the unfiltered form of the Navier- Stokes equations with the 5th order bandwidth-optimized WENO scheme, which is generally referred to the socalled implicitly implemented LES. Fig.5 (a) shows the geometry of MVG with back edge declining angle 70. The other geometric parameters in the figure are the same as that given by Babinsky 7, i.e., 7.2,24 and 7.5. The geometries for case is shown in Fig.5, where the height of MVG is assumed to be /2. The grid number for the whole system is: n n! n # 1371921600. (a) Figure 5. The geometry of (a) MVG and case The details about the geometric objects, grid generation, computational domain, etc, which are introduced in our previous paper 3, 4, 8, will not be repeated here. The adiabatic, zero-gradient of pressure and non-slipping conditions are adopted at the wall. To avoid possible wave reflection, the non-reflecting boundary conditions are used on the upper boundary. The boundary conditions at the front and back boundary surfaces in the spanwise direction are treated as the periodic condition, which is under the consideration that the problem is about the flow around MVG arrays and only one MVG is simulated. The outflow boundary conditions are specified as a kind of characteristic-based condition, which can handle the outgoing flow without reflection. To generate the true turbulent inlet, twenty thousand turbulent profiles are obtained from previous DNS simulation and used as the time dependent inflow 9. In our previous paper 10-13, the flow properties were also checked. The inflow boundary layer velocity profile agrees with the analytical profile as well. To check the flow properties before the MVG, we analyzed the relevant flow parameters on a spanwise cross section which is 11.97h ahead the apex of MVG. As a result, the displacement thickness δ * = 0.371h, the momentum thickness θ = 0.275h, nominal boundary layer thickness δ = 2.36h. Thus, we can obtain a shape factor H as about 1.35, which shows the flow before the MVG is fully developed turbulence flow. Fig.6 shows the inflow boundary layer velocity profile in log-coordinates on the same cross section. There is a well-defined log region and the agreement with the analytical profile is well established. These results are typical for a naturally grown turbulent boundary layer in equilibrium.
Figure 6. Inflow boundary-layer profile comparison with Guarini et al s 14 III. Detailed studies on interaction between ring-like vortices and shock A. Quantitative investigation by tracking one ring-like vortex. The generated vortex rings appear mainly near the central plane along the spanwise direction. The central plane (I=69) is chosen to show several quantities of the head of ring-like vortex (ring head). Figure 7. The position of central plane (blue plane) and iso-surface of ' ( One ring-like vortex is tracked in three time steps, which are in front of shock, interacting with shock and behind shock. 1. Time step 460000 Fig.8 gives the distribution of vorticity magnitude (VM) and spanwise vorticity around the ring head and shock. The set of black lines are contour lines of pressure gradient magnitude, which indicates the position of shock wave. At this time step, the ring head is in front of shock and shock keeps its typology. The point P whose coordinator is (20.86, 3.02) has maximal VM and spanwise vorticity around ring head, which are 9.52 and 8.77 respectively. It shows spanwise vorticity is the main component of vorticity.
(a) Figure 8. Distribution of (a) Vorticity Magnitude (VM) and X-Vorticity (i.e. spanwise vorticity) Fig.9 shows distribution of pressure around ring head and shock. It is easy to see that ring head is a low pressure center. At point P, pressure is 0.94, which is significantly lower than the pressure behind shock wave. This kind of low pressure center will play an important role in breaking shock wave down. Figure 9. Pressure distribution around ring head and shock Fig.10 gives streamwise velocity distribution around ring head and shock. Red rectangular area is a low speed zone (streamwise velocity is around 1.1), which is at the center of the ring-like vortex. This zone also play an important role in breaking shock down. Actually, from Fig.10, we could see shock wave has already been broken down in the low speed zone.
Figure 10. Streamwise velocity distribution around ring head and shock At point P, streamwise velocity is ) * 1.76 and normal velocity is + * 0.09. Fig.11 shows stream traces around point P after streamwise velocity subtracting ) * and normal velocity subtracting + *. We could find that the stream traces near point P is isolated, which are independent on stream traces away from P. 2. Time step 461000 (a) Figure 11. Stream traces (a) near P and away from P. Fig.12 gives the distribution of vorticity magnitude (VM) and spanwise vorticity around the ring head and shock. The set of black lines are contour lines of pressure gradient magnitude, which indicates the position of shock wave. At this time step, the ring head is penetrating shock and shock still keeps its typology. The point P whose coordinator is (21.08, 3.11) has maximal VM and spanwise vorticity around ring head, which are 10.65 and 9.84 respectively. It shows spanwise vorticity is still the main component of vorticity.
(a) Figure 12. Distribution of (a) Vorticity Magnitude (VM) and X-Vorticity (i.e. spanwise vorticity) Fig.13 shows distribution of pressure around ring head and shock. It is easy to see that ring head is not a low pressure center any more. At point P, pressure is 1.35, which increases about 50% than last time step. Figure 13. Pressure distribution around ring head and shock Fig.14 gives streamwise velocity distribution around ring head and shock. Red rectangular area is also a low speed zone (streamwise velocity is around 1.0), which is at the center of the ring-like vortex.
Figure 14. Streamwise velocity distribution around ring head and shock At point P, streamwise velocity is ) * 1.72 and normal velocity is + * 0.23. Fig.15 shows stream traces around point P after streamwise velocity subtracting ) * and normal velocity subtracting + *. We could find that the stream traces near point P is not isolated, which are dependent on stream traces away from P. The reason is still not clear. Figure 15. Stream traces around P. 3. Time step 461500 Fig.16 gives the distribution of vorticity magnitude (VM) and spanwise vorticity around the ring head and shock. The set of black lines are contour lines of pressure gradient magnitude, which indicates the position of shock wave. At this time step, the ring head is behind shock and shock is broken down. The point P whose coordinator is (21.17, 3.14) has maximal VM and spanwise vorticity around ring head, which are 12.37 and 11.32 respectively. It shows spanwise vorticity is still the main component of vorticity.
(a) Figure 16. Distribution of (a) Vorticity Magnitude (VM) and X-Vorticity (i.e. spanwise vorticity) Fig.17 shows distribution of pressure around ring head and shock. At point P, pressure is 1.51, which increases a lot than previous time steps. But we still see P is a low pressure center around P. Even though the pressure at P increases, it changes the distribution around shock by decreasing the pressure behind the shock wave. The difference of pressure front and behind shock almost vanishes, so that the discontinuity of pressure disappear. As a result, shock wave is distorted badly even broken down. Figure 17. Pressure distribution around ring head and shock Fig.18 gives streamwise velocity distribution around ring head and shock. Red rectangular area is a low speed zone (streamwise velocity is around 1.0), which is at the center of the ring-like vortex.
Figure 18. Streamwise velocity distribution around ring head and shock At point P, streamwise velocity is ) * 1.61 and normal velocity is + * 0.39. Fig.19 shows stream traces around point P after streamwise velocity subtracting ) * and normal velocity subtracting + *. We could find that the stream traces near point P is isolated, which are independent with stream traces away from P. (a) Figure 11. Stream traces (a) near P and away from P. B. Future study A detailed study about the role of low speed zone in breaking shock wave down is in progress. The reasons of why it has a different type stream traces when the ring head penetrates in the shock and why the stream traces is isolated near ring head when ring head is in front of and behind shock wave need to be investigated in details. Acknowledgments The work was supported by Department of Mathematics at University of Texas at Arlington. The authors are grateful to Texas Advanced Computing Center (TACC) for the computation hours provided. This work is accomplished by using Code LESUTA developed by Dr. Qin Li and Dr. Chaoqun Liu at University of Texas at Arlington in 2009.
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