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Separating, transitional flow affected by various inflow oscillation regimes
Applied Mathematical Modelling 30 (2006) 1134–1142 www.elsevier.com/locate/apm
Separating, transitional flow affected by various inflow oscillation regimes J.G. Wissink
*
Institute for Hydromechanics, University of Karlsruhe, P.O. Box 6980, Kaiserstrasse 12, 76128 Karlsruhe, Germany Received 15 December 2004; received in revised form 23 February 2005; accepted 25 February 2005 Available online 25 October 2005
Abstract A series of direct numerical simulations of laminar separation bubble flow with oscillating oncoming flow has been performed. A study is made of the influence of the amplitude and the period of the inflow oscillation on the dynamics of the separation bubble. The oscillating inflow is found to cause the location of separation and the location of transition to alternately move upstream and downstream. For all simulations, in every period a new separation bubble is formed as the inflow decelerates. The growth-rate of the triggered Kelvin–Helmholtz (KH) instability is found to depend on both the amplitude and the period of the inflow oscillation. 2005 Elsevier Inc. All rights reserved. Keywords: CFD; Boundary layer separation; Kelvin–Helmholtz instability
1. Introduction Boundary layer flows influenced by a streamwise pressure gradient that turns from favourable to adverse are, for instance, found along the suction side of low-pressure turbine blades. Large-scale action of passing wakes, generated by an upstream row of blades, induces a periodic fluctuation in this pressure gradient. To perform a detailed study of the effect of such an oscillation on a separated boundary layer, it was decided to perform a series of direct numerical simulations (DNS) of a separating flat plate boundary layer with oscillating inflow. First results [1,2] showed evidence of a Kelvin–Helmholtz instability, resulting in the periodic roll-up of the separated shear layer. To be able to obtain more detailed information about the influence of the period and amplitude of the inflow oscillation on boundary layer separation, two more DNS were performed. The comparison of the results of these DNS with the results of earlier simulations is reported in this paper. In the presence of a strong enough adverse pressure gradient, a laminar boundary layer over a flat plate will separate. The separated boundary layer (shear layer) is very unstable and will usually undergo rapid transition
*
Tel.: +49 721 608 3536; fax: +49 721 661 686. E-mail address: [email protected]
0307-904X/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2005.02.016
J.G. Wissink / Applied Mathematical Modelling 30 (2006) 1134–1142
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to turbulence. Previous DNS of laminar separation bubble (LSB) flow with constant free stream have been performed by Alam and Sandham [3], Maucher et al. [4], Spalart and Strelets [5] and Wissink and Rodi [6]. To generate an adverse pressure gradient, required for the formation of a LSB, both Alam and Sandham as well as Spalart and Strelets used suction through the upper boundary, Maucher et al. applied a special boundary condition for the free-stream velocity and Wissink and Rodi employed the special shape of the upper wall (see also Fig. 1). By explicitly introducing disturbances into the boundary layer prior to separation, Alam and Sandham observed that the separated shear layer undergoes transition via oblique modes and K-vortex induced breakdown. They also showed that the LSB flow becomes absolutely unstable when the reverse flow exceeds 15–20% of the local free-stream velocity. As Alam and Sandham, Maucher et al. explicitly added disturbances to the attached laminar boundary layer as they focused on the early stages of transition. Both Spalart and Strelets and Wissink and Rodi relied on small numerical round-off error present in any numerical simulation to trigger natural modes. They detected a Kelvin–Helmholtz (KH) instability, characterized by a quasi-periodic shedding of vortices from the separated boundary layer. The addition of inflow oscillations induces an oscillating pressure gradient which causes both the location of separation and the location of transition to move back and forth. As in the simulations with steady inflow (see above) the streamwise pressure gradient along the flat plate—that turns from favourable to adverse—is generated by the special shape of the upper wall of the computational domain (see Fig. 1). A total of four DNS of such separating flow over a flat plate with oscillating inflow have been performed. An overview of the performed simulations is provided in Table 1. Both the amplitude a and the period T of the inflow oscillation were varied in order to be able to study their influence on the dynamics of the LSB. Results of Simulation A have been reported in Wissink and Rodi [1]. It was shown that every period a new separation bubble appeared as the inflow started to decelerate. Once the inflow was about to accelerate, a roll of re-circulating flow was shed which very rapidly turned turbulent. In this fast transition to turbulence two types of elliptic instabilities were found to play a role, the bending wave (BW) instability and the Core Dynamics (CD) instability. The BW instability is characterized by the presence of counter-rotating swirls perpendicular to the axis of vortex (see also Lundgren and Mansour [7]). The CD instability is described in Pradeep and Hussain [8]. In this instability (1) a thin annular vortex sheet is formed that surrounds a low-enstrophy bubble and eventually rolls-up into smaller ‘‘vortexlets’’ and (2) the folding and re-connection of core vortex filaments gives rise to additional fine-grained random vorticity within the tube of re-circulating flow. For Simulation B, the results of a study of passive heat transfer are reported in Wissink et al. [2]. It was shown that for a laminar separation bubble flow neither the turbulent viscosity hypothesis nor the temperature-gradient diffusion hypothesis holds (see also Pope [9]).
u=U0(1+a sin 2πt ) T v=0 w=0
free-slip
Convective outflow
free-slip
no-slip
-0.5
0
0.5
1
x/L Fig. 1. Spanwise slice through the computational domain.
Table 1 Overview of the simulations performed Simulation
Spanwise size
Grid
a
T
A B C D
0.12L 0.08L 0.12L 0.12L
966 · 226 · 128 1286 · 310 · 128 966 · 226 · 128 966 · 226 · 128
0.20 0.10 0.05 0.05
0.61L/U0 0.30L/U0 0.30L/U0 0.15L/U0
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In all simulations, the primary instability-mechanism for the transition to turbulence was found to be a two-dimensional KH instability of the separated shear layer which was triggered by the inflow oscillation. The KH instability caused the shear layer to roll-up. Inside the rolled-up shear layer, entrained disturbances triggered elliptic instabilities which led to a rapid transition to fully three-dimensional (3D) turbulence [1,2]. As a result, a rotating tube of turbulent flow was obtained. When the period and the amplitude of the inflow oscillation was large enough, each period a new LSB was formed, which was convected downstream as soon as the inflow began to accelerate. 1.1. Computational details The computational domain is illustrated in Fig. 1. The adverse pressure gradient is induced by the special shape of the upper wall. In the spanwise direction periodic boundary conditions are employed, while at the outlet a convective outflow boundary condition is used. Along the upper boundary and along the lower boundary for x < 0 a free-slip boundary condition is prescribed. A no-slip boundary condition is employed along the lower boundary for x P 0. Finally, at the inlet an oscillating flow ðu; v; wÞt ¼ U 0 ð1 þ a sin 2pt ; 0; 0Þt , where a T is the amplitude and T is the period of the inflow, is employed. The Reynolds number of the flow problem, Re = 60,000, is based on the mean inflow velocity U0 and the length of the flat plate L, as chosen in companion experiments performed at the Technical University of Berlin [10]. The three-dimensional, incompressible Navier Stokes equations, defined by: 1. The continuity equation for the conservation of mass oui ¼ 0: oxi 2. The momentum equation oui o op 1 þ ðui uj Þ ¼ þ Dui ; oxi Re ot oxj were discretised using a finite-volume approach, employing a central, second-order accurate discretisation in space combined with a three-stage Runge–Kutta method for the time-integration. A collocated variable arrangement was chosen. To avoid a decoupling of the velocity and the pressure field, the momentum-interpolation procedure of Rhie and Chow [11] was employed. For a more detailed description of the flow solver used see Breuer and Rodi [12]. The computations were performed on the IBM SP-SMP supercomputer in Karlsruhe using up to 128 processors and 51 · 106 grid points. To optimize load balancing, the computational domain was subdivided into subdomains of equal size which were each assigned to their own unique processor. Communication between processors was performed through the standard message passing interface (MPI) protocol. 1 Phase-averaging was performed by subdividing each period into 256 equal phases / ¼ 0; 256 ; . . . ; 255 at 256 which statistics were gathered. To speed up the convergence of the statistics, phase-averaging was combined with averaging in the homogeneous spanwise direction. In each simulation, phase-averaging has been performed during at least six non-dimensional time-units L/U0. Notation: hfi is the phase-averaged signal of the quantity f. In Section 2, a detailed comparative study of the dynamics of LSB flow under different inflow oscillation regimes will be presented by using both instantaneous flow fields and phase-averaged statistics. A further discussion of the results and the conclusions can be found in Section 3.
2. Results In this section, key results of the simulations listed in Table 1 will be presented. The aim is to elucidate the influence of the amplitude and period of the oscillating inflow on the dynamics of a laminar separation bubble flow.
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Fig. 2 shows the phase-averaged location of separation during two periods as a function of phase /. Sep-ffi qffiffiffiffiffiffiffiffi aration is identified by grey contours, corresponding to negative values of the friction velocity hus i ¼ m ohui . oy Each plot, corresponding to one of the Simulations A–D (see Table 1), shows the formation of a new separation bubble as the inflow starts to—or is about to start to—decelerate (/ 0.25, 1.25). As the inflow decelerates further, the separation bubble grows in both the upstream and downstream direction. As the inflow is about to accelerate (/ 0.75 mod 1), a large roll of re-circulating flow—identified by label ‘‘I’’—is shed. The remainder of the separation bubble—identified by label ‘‘II’’—is only visible in the graphs corresponding to Simulations A and B. While in Simulation A parts of separation bubbles stemming from different periods
Fig. 2. Space–time plot showing two periods of the phase-averaged location of separation as a function of phase / for Simulations A–D.
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never merge, in Simulation B the newly formed separation bubble is found to merge with Region II from the bubble that was formed during the previous period. The maximum streamwise extent of the bubble is observed to depend on the period T, which determines the duration of the deceleration period, and the inflow-amplitude a, which determines the strength of the deceleration. Compared to Simulation C, the larger inflow oscillationamplitude that is employed in Simulation B induces a stronger deceleration for 0.25 < / < 0.75. As a consequence, in Simulation B one can observe a faster growth of the separation bubble during the initial phases of inflow-deceleration as well as a periodic re-circulation ‘‘revival’’ of Region II. Also, the increased triggering of the KH instability in Simulation B is found to lead to a faster roll-up of the shear layer—illustrated by the fact that the maximum reverse flow (labeled ‘‘M’’) is reached at an earlier phase—and to promote the subsequent destruction of the old separation bubble (Region I) as it is convected downstream (compare Fig. 2; Simulations B and C). The effect of decreasing the inflow oscillation period T is studied by comparing the results of Simulations C and D in Fig. 2. The decreased period in Simulation D is observed to result in a merger of Region I of the old bubble with the newly formed bubble. Also, the relatively small period does not allow enough time for the recirculating flow associated with the KH instability to become very strong. As a result, the accelerating inflow—which displaces the velocity field inside the channel—is strong enough to make the re-circulating region invisible for 1 < / < 1.2. As in Simulations B and C, the maximum reverse flow is reached in the period following the period in which the bubble was formed. In Fig. 3 series of snapshots of the instantaneous spanwise vorticity are shown, illustrating the evolution of the separation bubble in Simulations C (left) and D (right) during one inflow oscillation period. In Simulation C a small area of becalmed flow is found to be present in between the remains of separation bubbles formed at subsequent periods. For both simulations, the snapshots clearly illustrate the roll-up of the separated boundary layer and the subsequent shedding of this spanwise roll of re-circulating flow. As soon as the roll is shed, the re-circulating flow very quickly turns turbulent [5,1,2]. The large spanwise rolls of re-circulating flow correspond to the Regions I as displayed in Fig. 2. Owing to the decreased period in Simulation D, compared to Simulation C the size of the separation bubble is observed to decrease. In Fig. 4 the magnitude of the phase-averaged reverse flow—as a percentage of the local free-stream velocity—and its streamwise location have been plotted as a function of phase for all simulations. In Simulations B–D the maximum reverse flow reaches values above 20% of the free-stream velocity for all phases (see Fig. 4b–d). Consequently, the flow in Simulations B–D is absolutely unstable [3] at all times and self-sustained turbulence can exist for all phases. In Simulation A the maximum reverse flow is above 20%, and is thus absolutely unstable, for 0 6 / < 0.07 and 0.45 < / 6 1 (see Fig. 4a). The sudden jumps in the graphs of the location of maximum reverse flow are a consequence of the reverse flow of a newly formed separation bubble growing stronger than the reverse flow of the bubble formed during the last period. Making use of the information about the location of the separation bubbles provided in Fig. 2, it is possible to determine the time interval Dt between the appearance of a new separation bubble and the phase at which the re-circulating flow reaches its maximum. As already noted before, the location of this absolute maximum is identified by the label ‘‘M’’ in Fig. 2. The first appearance of the separation bubble is around / = 0.25. In Simulation A the maximum reverse flow is reached at / = 0.74, giving Dt = 0.49T 0.30L/U0. In Simulation B, the maximum is reached at / = 1.22, giving Dt = 0.29L/U0. The reverse flow in Simulation C is maximum at / 1.60 such that Dt = 0.40L/U0, while in Simulation D the maximum is reached at / 1.92 giving Dt = 0.25L/U0. This result is also illustrated in Fig. 5, showing the maximum reverse flow of all simulations versus Dt. It confirms that a smaller amplitude of the inflow oscillation results in a slower development of the KH instability which is responsible for the roll-up of the shear layer (compare Simulation B to Simulation C). Decreasing the inflow oscillation period—while keeping its amplitude constant—on the other hand is observed to result in a stronger triggering of the KH instability (compare Simulation D to Simulation C). This stronger triggering of the KH instability in Simulation D, as compared to Simulation C, leads to a reduction in both the maximum reverse flow (see Fig. 5) and the size of the separation bubble (see Fig. 3). This is explained partially by the fact that at the time the maximum reverse flow is reached, the inflow in Simulation C decelerates, while the inflow in Simulation D accelerates. The former significantly contributes to the magnitude of the maximum reverse flow in Simulation C, while the latter reduces the maximum reverse flow obtained in Simulation D. There is also a possible contribution from the fact that the smaller size of the separation bubble in
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Fig. 3. Series of snapshots showing the instantaneous spanwise vorticity, left: Simulation C, right: Simulation D.
Simulation D hinders the further development of the KH instability which is responsible for the generation of reverse flow. Why a reduction in the period T leads to a stronger triggering of the KH instability is explained in Fig. 6. In this figure, for both Simulations C and D, the frequency spectra of the v-velocities at four points P1, . . . , P4, located at midspan are shown. The points are identified in the lower graphs showing contours of the mean spanwise vorticity. In Simulation C, the first peak in the spectrum at f = 3.33U0/L corresponds to the inflow oscillation frequency. The most amplified mode of the KH instability is found to be the second harmonic of this fundamental frequency fmax = 6.67U0/L. In Simulation D, the inflow oscillation frequency f = 6.67U0/L is identical to the most amplified mode of the KH instability. This direct triggering of the most unstable KH
J.G. Wissink / Applied Mathematical Modelling 30 (2006) 1134–1142 180
0.9
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0.4
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(b)
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φ
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0.8
0.5 0
1
0
0.2
(d)
0.4
φ
0.6
0.8
1
Fig. 4. Phase-averaged maximum reverse flow as a percentage of the local free-stream velocity and location of maximum reverse flow both as a function of phase. (a) Simulation A, (b) Simulation B, (c) Simulation C, (d) Simulation D.
MA
160
Sim. A Sim. B Sim. C Sim. D
Max. reverse flow
140 120 100 80
MB MD
60
MC
40 20 0
0
0.2
Δt
0.4
0.6
Fig. 5. Phase-averaged maximum reverse flow as a percentage of the local free-stream velocity as a function of the time passed since the onset of inflow deceleration. Time has been made non-dimensional using U and L.
mode results in a rapid growth of the KH instability and explains the stronger triggering of the KH instability in Simulation D as compared to Simulation C. In Fig. 7 the location of transition is defined as the most upstream location where the phase-averaged spanwise fluctuations hwwi are larger than 0.01. In Simulation A, the large amplitude of the inflow oscillation induces a large fluctuation in the location of transition. The distance, Dx, between the most downstream location and the most upstream location of the onset of transition exceeds 0.14L. Compared to Simulation A, in Simulations B, C, D this distance Dx is an order of magnitude smaller. Because of the limited number of 10–20 periods employed in the phase-averaging procedure of Simulations A, B, C, the corresponding curves in Fig. 7 contain spurious oscillations. This is not a major problem in Simulation A, where Dx is much larger than the level of these spurious oscillations. In Simulations B and C, however, where Dx is only marginally larger, it is very difficult to obtain an accurate picture of the variation in the phase-averaged location of transition. Compared to Simulation B, for most phases of Simulation C the onset of transition is located slightly farther downstream, while the variation in the location of transition, Dx, is slightly smaller. Hence, the larger inflow oscillation amplitude prescribed in Simulation B both induces a larger variation in the location of transition and causes it to move slightly upstream. In Simulation D more than 70 periods of phase-averaging were performed. As a result, a very smooth sine-wave like curve is obtained with Dx 0.007L. This Dx is of the same order of magnitude as the Dx obtained
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Fig. 6. Frequency spectra of the instantaneous v-velocity at P1, P2, P3, P4. P1, . . . , P4 are identified in the lower graphs, showing contours of the time-averaged spanwise vorticity. (a) Simulation C, (b) Simulation D.
0.6 Sim. A Sim. B Sim. C Sim. D
0.58 0.56
x/L
0.54 0.52 0.5 0.48 0.46 0.44 0.42
0
0.5
1
φ
1.5
2
Fig. 7. Streamwise location of transition as a function of phase / for eight phases / = 0, 1/8, . . . , 7/8.
in Simulation C. The result that for all phases the onset of transition in Simulation D is located upstream of the onset of transition in Simulation C, confirms that the most effective triggering of the KH instability is provided by the inflow oscillation-frequency used in Simulation D (see also Fig. 5). Above, it was shown that the maximum streamwise extent of the separation bubble depends mostly on the inflow oscillation period. Increasing the amplitude of the oscillation, results in a stronger triggering of the KH instability which promotes the decay of the separation bubble and leads to earlier transition. Finally, a direct triggering of the most unstable KH mode by an appropriate choice of the inflow oscillation period was also found to significantly promote the decay of the LSB and to shift the location of transition upstream. 3. Conclusions Several DNS of laminar separation bubble flow with oscillating oncoming flow were performed. By varying both the amplitude and the period of this oscillation, its influence on the dynamics of the laminar separation bubble was studied. The results presented above lead to the following conclusions:
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• In each of the simulations a new separation bubble was found to be formed around / = 0.25, corresponding to the onset of inflow deceleration. • If the oscillation period is sufficiently long, the newly formed separation bubble will never merge with the remainder of bubbles formed during previous periods. • For all simulations the amplitude of the oscillation in the streamwise location of transition is found to increase with increasing inflow oscillation amplitude. • Increasing the amplitude of the inflow oscillation while keeping its period constant, leads to a stronger triggering of the KH instability which manifests itself by a faster roll-up of the separated shear layer and an earlier transition to turbulence. • Choosing the period of the inflow oscillation to match the one of the most unstable KH mode also leads to a faster roll-up of the separated shear layer and an upstream shift in the location of transition. • In all simulations, the very fast transition to turbulence is found to take place inside the rolled-up shear layer. • Except for a brief period around / = 0.25 in Simulation A, the maximum reverse flow in all simulations reaches values in excess of 20% of the free-stream velocity. According to Alam and Sandham [3] such flows are absolutely unstable and self-sustained turbulence can exist.
Acknowledgments The author wishes to thank the German Research Foundation (DFG) for funding this research and the Computing Centre of the University of Karlsruhe for their support. References [1] J.G. Wissink, W. Rodi, DNS of a laminar separation bubble in the presence of oscillating flow, Flow Turbul. Combust. 71 (2003) 311–331. [2] J.G. Wissink, V. Michelassi, W. Rodi, Heat transfer in a laminar separation bubble affected by oscillating external flow, Int. J. Heat Fluid Flow 25 (2004) 729–740. [3] M. Alam, N.D. Sandham, Direct numerical simulation of ÔshortÕ laminar separation bubbles with turbulent reattachment, J. Fluid Mech. 410 (2000) 1–28. [4] U. Maucher, U. Rist, M. Kloker, S. Wagner, DNS of laminar-turbulent transition in separation bubbles, in: E. Krause, W. Ja¨ger (Eds.), High-Performance Computing in Science and Engineering, Springer, 2000. [5] P.R. Spalart, M.Kh. Strelets, Mechanisms of transition and heat transfer in a separation bubble, J. Fluid Mech. 403 (2000) 329–349. [6] J.G. Wissink, W. Rodi, DNS of transition in a laminar separation bubble, in: I.P. Castro, P.E. Hancock (Eds.), Advances in Turbulence IX, Proceedings of the Ninth European Turbulence Conference, CIMNE, 2002. [7] T.S. Lundgren, N.N. Mansour, Transition to turbulence in an elliptic vortex, J. Fluid Mech. 307 (1996) 43–62. [8] D.S. Pradeep, F. Hussain, Core dynamics of a strained vortex: instability and transition, J. Fluid Mech. 396 (2001) 247–285. [9] S.B. Pope, Turbulent Flows, Cambridge University Press, 2000. [10] W. Lou, J. Hourmouziadis, Separation bubbles under steady and periodic-unsteady main flow conditions, in: Proceedings of the 45th ASME International Gas Turbine and Aeroengine Technical Congress, Munich, Germany, May 8–11, 2000. [11] C.M. Rhie, W.L. Chow, Numerical study of the turbulent flow past an airfoil with trailing edge separation, AIAA J. 21 (1983) 1525– 1532. [12] M. Breuer, W. Rodi, Large eddy simulation for complex turbulent flow of practical interest, in: Flow Simulation with Highpreformace Computers II Notes on Numerical Fluids Mechanics, Vieweg Verlag, 1996.
Separating, transitional flow affected by various inflow oscillation regimes J.G. Wissink
*
Institute for Hydromechanics, University of Karlsruhe, P.O. Box 6980, Kaiserstrasse 12, 76128 Karlsruhe, Germany Received 15 December 2004; received in revised form 23 February 2005; accepted 25 February 2005 Available online 25 October 2005
Abstract A series of direct numerical simulations of laminar separation bubble flow with oscillating oncoming flow has been performed. A study is made of the influence of the amplitude and the period of the inflow oscillation on the dynamics of the separation bubble. The oscillating inflow is found to cause the location of separation and the location of transition to alternately move upstream and downstream. For all simulations, in every period a new separation bubble is formed as the inflow decelerates. The growth-rate of the triggered Kelvin–Helmholtz (KH) instability is found to depend on both the amplitude and the period of the inflow oscillation. 2005 Elsevier Inc. All rights reserved. Keywords: CFD; Boundary layer separation; Kelvin–Helmholtz instability
1. Introduction Boundary layer flows influenced by a streamwise pressure gradient that turns from favourable to adverse are, for instance, found along the suction side of low-pressure turbine blades. Large-scale action of passing wakes, generated by an upstream row of blades, induces a periodic fluctuation in this pressure gradient. To perform a detailed study of the effect of such an oscillation on a separated boundary layer, it was decided to perform a series of direct numerical simulations (DNS) of a separating flat plate boundary layer with oscillating inflow. First results [1,2] showed evidence of a Kelvin–Helmholtz instability, resulting in the periodic roll-up of the separated shear layer. To be able to obtain more detailed information about the influence of the period and amplitude of the inflow oscillation on boundary layer separation, two more DNS were performed. The comparison of the results of these DNS with the results of earlier simulations is reported in this paper. In the presence of a strong enough adverse pressure gradient, a laminar boundary layer over a flat plate will separate. The separated boundary layer (shear layer) is very unstable and will usually undergo rapid transition
*
Tel.: +49 721 608 3536; fax: +49 721 661 686. E-mail address: [email protected]
0307-904X/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2005.02.016
J.G. Wissink / Applied Mathematical Modelling 30 (2006) 1134–1142
1135
to turbulence. Previous DNS of laminar separation bubble (LSB) flow with constant free stream have been performed by Alam and Sandham [3], Maucher et al. [4], Spalart and Strelets [5] and Wissink and Rodi [6]. To generate an adverse pressure gradient, required for the formation of a LSB, both Alam and Sandham as well as Spalart and Strelets used suction through the upper boundary, Maucher et al. applied a special boundary condition for the free-stream velocity and Wissink and Rodi employed the special shape of the upper wall (see also Fig. 1). By explicitly introducing disturbances into the boundary layer prior to separation, Alam and Sandham observed that the separated shear layer undergoes transition via oblique modes and K-vortex induced breakdown. They also showed that the LSB flow becomes absolutely unstable when the reverse flow exceeds 15–20% of the local free-stream velocity. As Alam and Sandham, Maucher et al. explicitly added disturbances to the attached laminar boundary layer as they focused on the early stages of transition. Both Spalart and Strelets and Wissink and Rodi relied on small numerical round-off error present in any numerical simulation to trigger natural modes. They detected a Kelvin–Helmholtz (KH) instability, characterized by a quasi-periodic shedding of vortices from the separated boundary layer. The addition of inflow oscillations induces an oscillating pressure gradient which causes both the location of separation and the location of transition to move back and forth. As in the simulations with steady inflow (see above) the streamwise pressure gradient along the flat plate—that turns from favourable to adverse—is generated by the special shape of the upper wall of the computational domain (see Fig. 1). A total of four DNS of such separating flow over a flat plate with oscillating inflow have been performed. An overview of the performed simulations is provided in Table 1. Both the amplitude a and the period T of the inflow oscillation were varied in order to be able to study their influence on the dynamics of the LSB. Results of Simulation A have been reported in Wissink and Rodi [1]. It was shown that every period a new separation bubble appeared as the inflow started to decelerate. Once the inflow was about to accelerate, a roll of re-circulating flow was shed which very rapidly turned turbulent. In this fast transition to turbulence two types of elliptic instabilities were found to play a role, the bending wave (BW) instability and the Core Dynamics (CD) instability. The BW instability is characterized by the presence of counter-rotating swirls perpendicular to the axis of vortex (see also Lundgren and Mansour [7]). The CD instability is described in Pradeep and Hussain [8]. In this instability (1) a thin annular vortex sheet is formed that surrounds a low-enstrophy bubble and eventually rolls-up into smaller ‘‘vortexlets’’ and (2) the folding and re-connection of core vortex filaments gives rise to additional fine-grained random vorticity within the tube of re-circulating flow. For Simulation B, the results of a study of passive heat transfer are reported in Wissink et al. [2]. It was shown that for a laminar separation bubble flow neither the turbulent viscosity hypothesis nor the temperature-gradient diffusion hypothesis holds (see also Pope [9]).
u=U0(1+a sin 2πt ) T v=0 w=0
free-slip
Convective outflow
free-slip
no-slip
-0.5
0
0.5
1
x/L Fig. 1. Spanwise slice through the computational domain.
Table 1 Overview of the simulations performed Simulation
Spanwise size
Grid
a
T
A B C D
0.12L 0.08L 0.12L 0.12L
966 · 226 · 128 1286 · 310 · 128 966 · 226 · 128 966 · 226 · 128
0.20 0.10 0.05 0.05
0.61L/U0 0.30L/U0 0.30L/U0 0.15L/U0
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In all simulations, the primary instability-mechanism for the transition to turbulence was found to be a two-dimensional KH instability of the separated shear layer which was triggered by the inflow oscillation. The KH instability caused the shear layer to roll-up. Inside the rolled-up shear layer, entrained disturbances triggered elliptic instabilities which led to a rapid transition to fully three-dimensional (3D) turbulence [1,2]. As a result, a rotating tube of turbulent flow was obtained. When the period and the amplitude of the inflow oscillation was large enough, each period a new LSB was formed, which was convected downstream as soon as the inflow began to accelerate. 1.1. Computational details The computational domain is illustrated in Fig. 1. The adverse pressure gradient is induced by the special shape of the upper wall. In the spanwise direction periodic boundary conditions are employed, while at the outlet a convective outflow boundary condition is used. Along the upper boundary and along the lower boundary for x < 0 a free-slip boundary condition is prescribed. A no-slip boundary condition is employed along the lower boundary for x P 0. Finally, at the inlet an oscillating flow ðu; v; wÞt ¼ U 0 ð1 þ a sin 2pt ; 0; 0Þt , where a T is the amplitude and T is the period of the inflow, is employed. The Reynolds number of the flow problem, Re = 60,000, is based on the mean inflow velocity U0 and the length of the flat plate L, as chosen in companion experiments performed at the Technical University of Berlin [10]. The three-dimensional, incompressible Navier Stokes equations, defined by: 1. The continuity equation for the conservation of mass oui ¼ 0: oxi 2. The momentum equation oui o op 1 þ ðui uj Þ ¼ þ Dui ; oxi Re ot oxj were discretised using a finite-volume approach, employing a central, second-order accurate discretisation in space combined with a three-stage Runge–Kutta method for the time-integration. A collocated variable arrangement was chosen. To avoid a decoupling of the velocity and the pressure field, the momentum-interpolation procedure of Rhie and Chow [11] was employed. For a more detailed description of the flow solver used see Breuer and Rodi [12]. The computations were performed on the IBM SP-SMP supercomputer in Karlsruhe using up to 128 processors and 51 · 106 grid points. To optimize load balancing, the computational domain was subdivided into subdomains of equal size which were each assigned to their own unique processor. Communication between processors was performed through the standard message passing interface (MPI) protocol. 1 Phase-averaging was performed by subdividing each period into 256 equal phases / ¼ 0; 256 ; . . . ; 255 at 256 which statistics were gathered. To speed up the convergence of the statistics, phase-averaging was combined with averaging in the homogeneous spanwise direction. In each simulation, phase-averaging has been performed during at least six non-dimensional time-units L/U0. Notation: hfi is the phase-averaged signal of the quantity f. In Section 2, a detailed comparative study of the dynamics of LSB flow under different inflow oscillation regimes will be presented by using both instantaneous flow fields and phase-averaged statistics. A further discussion of the results and the conclusions can be found in Section 3.
2. Results In this section, key results of the simulations listed in Table 1 will be presented. The aim is to elucidate the influence of the amplitude and period of the oscillating inflow on the dynamics of a laminar separation bubble flow.
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Fig. 2 shows the phase-averaged location of separation during two periods as a function of phase /. Sep-ffi qffiffiffiffiffiffiffiffi aration is identified by grey contours, corresponding to negative values of the friction velocity hus i ¼ m ohui . oy Each plot, corresponding to one of the Simulations A–D (see Table 1), shows the formation of a new separation bubble as the inflow starts to—or is about to start to—decelerate (/ 0.25, 1.25). As the inflow decelerates further, the separation bubble grows in both the upstream and downstream direction. As the inflow is about to accelerate (/ 0.75 mod 1), a large roll of re-circulating flow—identified by label ‘‘I’’—is shed. The remainder of the separation bubble—identified by label ‘‘II’’—is only visible in the graphs corresponding to Simulations A and B. While in Simulation A parts of separation bubbles stemming from different periods
Fig. 2. Space–time plot showing two periods of the phase-averaged location of separation as a function of phase / for Simulations A–D.
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never merge, in Simulation B the newly formed separation bubble is found to merge with Region II from the bubble that was formed during the previous period. The maximum streamwise extent of the bubble is observed to depend on the period T, which determines the duration of the deceleration period, and the inflow-amplitude a, which determines the strength of the deceleration. Compared to Simulation C, the larger inflow oscillationamplitude that is employed in Simulation B induces a stronger deceleration for 0.25 < / < 0.75. As a consequence, in Simulation B one can observe a faster growth of the separation bubble during the initial phases of inflow-deceleration as well as a periodic re-circulation ‘‘revival’’ of Region II. Also, the increased triggering of the KH instability in Simulation B is found to lead to a faster roll-up of the shear layer—illustrated by the fact that the maximum reverse flow (labeled ‘‘M’’) is reached at an earlier phase—and to promote the subsequent destruction of the old separation bubble (Region I) as it is convected downstream (compare Fig. 2; Simulations B and C). The effect of decreasing the inflow oscillation period T is studied by comparing the results of Simulations C and D in Fig. 2. The decreased period in Simulation D is observed to result in a merger of Region I of the old bubble with the newly formed bubble. Also, the relatively small period does not allow enough time for the recirculating flow associated with the KH instability to become very strong. As a result, the accelerating inflow—which displaces the velocity field inside the channel—is strong enough to make the re-circulating region invisible for 1 < / < 1.2. As in Simulations B and C, the maximum reverse flow is reached in the period following the period in which the bubble was formed. In Fig. 3 series of snapshots of the instantaneous spanwise vorticity are shown, illustrating the evolution of the separation bubble in Simulations C (left) and D (right) during one inflow oscillation period. In Simulation C a small area of becalmed flow is found to be present in between the remains of separation bubbles formed at subsequent periods. For both simulations, the snapshots clearly illustrate the roll-up of the separated boundary layer and the subsequent shedding of this spanwise roll of re-circulating flow. As soon as the roll is shed, the re-circulating flow very quickly turns turbulent [5,1,2]. The large spanwise rolls of re-circulating flow correspond to the Regions I as displayed in Fig. 2. Owing to the decreased period in Simulation D, compared to Simulation C the size of the separation bubble is observed to decrease. In Fig. 4 the magnitude of the phase-averaged reverse flow—as a percentage of the local free-stream velocity—and its streamwise location have been plotted as a function of phase for all simulations. In Simulations B–D the maximum reverse flow reaches values above 20% of the free-stream velocity for all phases (see Fig. 4b–d). Consequently, the flow in Simulations B–D is absolutely unstable [3] at all times and self-sustained turbulence can exist for all phases. In Simulation A the maximum reverse flow is above 20%, and is thus absolutely unstable, for 0 6 / < 0.07 and 0.45 < / 6 1 (see Fig. 4a). The sudden jumps in the graphs of the location of maximum reverse flow are a consequence of the reverse flow of a newly formed separation bubble growing stronger than the reverse flow of the bubble formed during the last period. Making use of the information about the location of the separation bubbles provided in Fig. 2, it is possible to determine the time interval Dt between the appearance of a new separation bubble and the phase at which the re-circulating flow reaches its maximum. As already noted before, the location of this absolute maximum is identified by the label ‘‘M’’ in Fig. 2. The first appearance of the separation bubble is around / = 0.25. In Simulation A the maximum reverse flow is reached at / = 0.74, giving Dt = 0.49T 0.30L/U0. In Simulation B, the maximum is reached at / = 1.22, giving Dt = 0.29L/U0. The reverse flow in Simulation C is maximum at / 1.60 such that Dt = 0.40L/U0, while in Simulation D the maximum is reached at / 1.92 giving Dt = 0.25L/U0. This result is also illustrated in Fig. 5, showing the maximum reverse flow of all simulations versus Dt. It confirms that a smaller amplitude of the inflow oscillation results in a slower development of the KH instability which is responsible for the roll-up of the shear layer (compare Simulation B to Simulation C). Decreasing the inflow oscillation period—while keeping its amplitude constant—on the other hand is observed to result in a stronger triggering of the KH instability (compare Simulation D to Simulation C). This stronger triggering of the KH instability in Simulation D, as compared to Simulation C, leads to a reduction in both the maximum reverse flow (see Fig. 5) and the size of the separation bubble (see Fig. 3). This is explained partially by the fact that at the time the maximum reverse flow is reached, the inflow in Simulation C decelerates, while the inflow in Simulation D accelerates. The former significantly contributes to the magnitude of the maximum reverse flow in Simulation C, while the latter reduces the maximum reverse flow obtained in Simulation D. There is also a possible contribution from the fact that the smaller size of the separation bubble in
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Fig. 3. Series of snapshots showing the instantaneous spanwise vorticity, left: Simulation C, right: Simulation D.
Simulation D hinders the further development of the KH instability which is responsible for the generation of reverse flow. Why a reduction in the period T leads to a stronger triggering of the KH instability is explained in Fig. 6. In this figure, for both Simulations C and D, the frequency spectra of the v-velocities at four points P1, . . . , P4, located at midspan are shown. The points are identified in the lower graphs showing contours of the mean spanwise vorticity. In Simulation C, the first peak in the spectrum at f = 3.33U0/L corresponds to the inflow oscillation frequency. The most amplified mode of the KH instability is found to be the second harmonic of this fundamental frequency fmax = 6.67U0/L. In Simulation D, the inflow oscillation frequency f = 6.67U0/L is identical to the most amplified mode of the KH instability. This direct triggering of the most unstable KH
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mode results in a rapid growth of the KH instability and explains the stronger triggering of the KH instability in Simulation D as compared to Simulation C. In Fig. 7 the location of transition is defined as the most upstream location where the phase-averaged spanwise fluctuations hwwi are larger than 0.01. In Simulation A, the large amplitude of the inflow oscillation induces a large fluctuation in the location of transition. The distance, Dx, between the most downstream location and the most upstream location of the onset of transition exceeds 0.14L. Compared to Simulation A, in Simulations B, C, D this distance Dx is an order of magnitude smaller. Because of the limited number of 10–20 periods employed in the phase-averaging procedure of Simulations A, B, C, the corresponding curves in Fig. 7 contain spurious oscillations. This is not a major problem in Simulation A, where Dx is much larger than the level of these spurious oscillations. In Simulations B and C, however, where Dx is only marginally larger, it is very difficult to obtain an accurate picture of the variation in the phase-averaged location of transition. Compared to Simulation B, for most phases of Simulation C the onset of transition is located slightly farther downstream, while the variation in the location of transition, Dx, is slightly smaller. Hence, the larger inflow oscillation amplitude prescribed in Simulation B both induces a larger variation in the location of transition and causes it to move slightly upstream. In Simulation D more than 70 periods of phase-averaging were performed. As a result, a very smooth sine-wave like curve is obtained with Dx 0.007L. This Dx is of the same order of magnitude as the Dx obtained
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Fig. 6. Frequency spectra of the instantaneous v-velocity at P1, P2, P3, P4. P1, . . . , P4 are identified in the lower graphs, showing contours of the time-averaged spanwise vorticity. (a) Simulation C, (b) Simulation D.
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in Simulation C. The result that for all phases the onset of transition in Simulation D is located upstream of the onset of transition in Simulation C, confirms that the most effective triggering of the KH instability is provided by the inflow oscillation-frequency used in Simulation D (see also Fig. 5). Above, it was shown that the maximum streamwise extent of the separation bubble depends mostly on the inflow oscillation period. Increasing the amplitude of the oscillation, results in a stronger triggering of the KH instability which promotes the decay of the separation bubble and leads to earlier transition. Finally, a direct triggering of the most unstable KH mode by an appropriate choice of the inflow oscillation period was also found to significantly promote the decay of the LSB and to shift the location of transition upstream. 3. Conclusions Several DNS of laminar separation bubble flow with oscillating oncoming flow were performed. By varying both the amplitude and the period of this oscillation, its influence on the dynamics of the laminar separation bubble was studied. The results presented above lead to the following conclusions:
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• In each of the simulations a new separation bubble was found to be formed around / = 0.25, corresponding to the onset of inflow deceleration. • If the oscillation period is sufficiently long, the newly formed separation bubble will never merge with the remainder of bubbles formed during previous periods. • For all simulations the amplitude of the oscillation in the streamwise location of transition is found to increase with increasing inflow oscillation amplitude. • Increasing the amplitude of the inflow oscillation while keeping its period constant, leads to a stronger triggering of the KH instability which manifests itself by a faster roll-up of the separated shear layer and an earlier transition to turbulence. • Choosing the period of the inflow oscillation to match the one of the most unstable KH mode also leads to a faster roll-up of the separated shear layer and an upstream shift in the location of transition. • In all simulations, the very fast transition to turbulence is found to take place inside the rolled-up shear layer. • Except for a brief period around / = 0.25 in Simulation A, the maximum reverse flow in all simulations reaches values in excess of 20% of the free-stream velocity. According to Alam and Sandham [3] such flows are absolutely unstable and self-sustained turbulence can exist.
Acknowledgments The author wishes to thank the German Research Foundation (DFG) for funding this research and the Computing Centre of the University of Karlsruhe for their support. References [1] J.G. Wissink, W. Rodi, DNS of a laminar separation bubble in the presence of oscillating flow, Flow Turbul. Combust. 71 (2003) 311–331. [2] J.G. Wissink, V. Michelassi, W. Rodi, Heat transfer in a laminar separation bubble affected by oscillating external flow, Int. J. Heat Fluid Flow 25 (2004) 729–740. [3] M. Alam, N.D. Sandham, Direct numerical simulation of ÔshortÕ laminar separation bubbles with turbulent reattachment, J. Fluid Mech. 410 (2000) 1–28. [4] U. Maucher, U. Rist, M. Kloker, S. Wagner, DNS of laminar-turbulent transition in separation bubbles, in: E. Krause, W. Ja¨ger (Eds.), High-Performance Computing in Science and Engineering, Springer, 2000. [5] P.R. Spalart, M.Kh. Strelets, Mechanisms of transition and heat transfer in a separation bubble, J. Fluid Mech. 403 (2000) 329–349. [6] J.G. Wissink, W. Rodi, DNS of transition in a laminar separation bubble, in: I.P. Castro, P.E. Hancock (Eds.), Advances in Turbulence IX, Proceedings of the Ninth European Turbulence Conference, CIMNE, 2002. [7] T.S. Lundgren, N.N. Mansour, Transition to turbulence in an elliptic vortex, J. Fluid Mech. 307 (1996) 43–62. [8] D.S. Pradeep, F. Hussain, Core dynamics of a strained vortex: instability and transition, J. Fluid Mech. 396 (2001) 247–285. [9] S.B. Pope, Turbulent Flows, Cambridge University Press, 2000. [10] W. Lou, J. Hourmouziadis, Separation bubbles under steady and periodic-unsteady main flow conditions, in: Proceedings of the 45th ASME International Gas Turbine and Aeroengine Technical Congress, Munich, Germany, May 8–11, 2000. [11] C.M. Rhie, W.L. Chow, Numerical study of the turbulent flow past an airfoil with trailing edge separation, AIAA J. 21 (1983) 1525– 1532. [12] M. Breuer, W. Rodi, Large eddy simulation for complex turbulent flow of practical interest, in: Flow Simulation with Highpreformace Computers II Notes on Numerical Fluids Mechanics, Vieweg Verlag, 1996.