DNS of Transition to Turbulence in a Linear Compressor Cascade T.A. Zaki, P.A. Durbin, J. Wissink and W. Rodi
Abstract A series of direct numerical simulations were carried out of the flow through a compressor passage. The behavior of the flow in the presence and absence of free-stream turbulent fluctuations is contrasted. In the former case, both the pressure and suction surface undergo separation due to the adverse pressure gradient in the passage. In the presence of free-stream turbulent fluctuation, the pressure surface boundary layer transitions to turbulence upstream of the laminar separation point and, hence, remains attached. The suction surface, however, undergoes separation independent of the free-stream perturbation. The frequency of shedding, however, and the mechanics of the separation region, are dependent on the turbulence level in the free-stream. Further simulations at higher turbulence intensities are planned.
1 Introduction In zero pressure gradient, laminar boundary layers are inviscidly stable, but possess a weak viscous instability. The instability modes, known as Tollmien-Schlichting waves, are associated with the orderly route to transition, which takes place on a slow viscous time-scale. In the presence of free-stream turbulence, however, this process is bypassed and the interaction of the turbulence in the outer flow with the underlying laminar layer causes breakdown on an inertial time-scale; a process known as bypass transition. In practical applications, particularly in turbo-machinery, the boundary layer is generally subject to curvature and streamwise pressure gradients. Both effects introduce additional instabilities and can accelerate breakdown of the laminar layer, even T.A. Zaki Imperial College London, London, UK e-mail:
[email protected] P.A. Durbin Iowa State University, Ames, IA, USA e-mail:
[email protected] J. Wissink Brunel University, Uxbridge, UK e-mail:
[email protected] W. Rodi University of Karlsruhe, Karlsruhe, Germany e-mail:
[email protected] 431
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in the absence of free-stream turbulence. Significant flow deceleration, however, can also induce separation of the laminar boundary layer. The objective of our direct numerical simulations (DNS) is to study the interaction of free-stream turbulence with the adverse pressure gradient boundary layer in a compressor passage. The presence of free-stream turbulence can accelerate transition, thus preventing separation of the boundary layer. Alternatively, should separation take place first, the interaction with the turbulent free-stream can be significant in ensuring the reattachment of the flow. In addition to the pressure gradient, the presence of a leading edge and surface curvature can also contribute to the instability of the flow. Three simulations were carried out, and are reported in detail in [7]: The first is a laminar computation, with no free-stream disturbances. The results from the laminar computation help establish the required grid resolution and the base state of the flow. Two further simulations were computed on the LRZ system, both with free-stream turbulence, but at different grid resolutions in order to ensure grid independence.
1.1 Simulation Setup A schematic of the computational domain is shown in Fig. 1. The blade geometry is designated V103 as used in the experiments of Hilgenfeld and Pfitzner [1]. The simulation Reynolds number, based on the mean inflow velocity U0 and the axial chord L, is Re = 138,500. The simulation setup in Fig. 1 is equivalent to an infinite linear cascade, similar to the setting using by Kalitzin et al. [2] and Wissink et al. [4] for the simulation of a turbine passage: Periodic boundary conditions are applied in
Fig. 1 Cross section through the computational domain at midspan
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Fig. 2 Computational grid, showing every 8th line in x and y
the y-direction upstream and downstream of the blade surface. The periodic regions correspond to x/L < 0 and x/L > 1. The separation of the top and bottom computational boundaries is one blade pitch, P = 0.59L. The streamwise extent of the domain is 1.9L. At the inflow plane, x/L = −0.4, a mean velocity (Uo cos(α), Uo sin(α), 0) is prescribed, where the angle of attack α = 42o . Inflow perturbations, u , v , w , can be superimposed to the mean flow at the inlet plane. Convective boundary conditions are applied at the outflow, x/L = 1.5. Finally, periodic boundary conditions are enforced in the spanwise direction, the extent of which is 0.20L. Initially, a laminar simulation, without inflow perturbations, was carried out on a coarse mesh comprising 624×288×8 grid cells in the x, y, and spanwise directions, respectively. The simulation helps identify the base properties, such as separation locations and required grid resolution for the laminar separation bubbles. The same (x, y) grid as the laminar case, but with increased spanwise resolution of 64 grid cells, was used for a preliminary simulation with inflow turbulence. This grid is hereinafter referred to as the coarse mesh. Based on the results from the coarse mesh computation, the grid was refined in order to fully resolve both the boundary layer on both the pressure and suction sides. This exercise was particularly important on the suction surface due to the presence of a thin separation bubble. The fine mesh, composed of 1024 × 640 × 128 grid cells, also provides higher resolution outside the boundary layers in order to avoid any excessive decay of the free-stream turbulence. Figure 2 illustrates the final mesh, with only every 8th grid line plotted.
1.2 Process Topology and Algorithmic Performance The discretization of the incompressible Navier-Stokes equations is carried out on a staggered grid with a local volume flux formulation in curvilinear coordinates
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Fig. 3 Domain decomposition in “pencils”, or “drawers”
[3, 5]. Explicit time advancement of the convective terms is implemented using Adams-Bashforth. The pressure and diffusion terms are treated using implicit Euler and Crank-Nicolson, respectively. The implementation of the numerical scheme on the parallel computing facilities at LRZ is via Message Passing Interface (MPI). Two domain decompositions of the numerical method are implemented: The first is a two-dimensional decomposition in “pencils”, or “drawers”, extending through the span of the geometry (Fig. 3). The second decomposition is one-dimensional in spanwise planes and is, hence, limited in number of processors to the number of spanwise grid points (Figs. 4a–b). For a general three-dimensional simulation, one can adopt either one of the two available domain decompositions (Figs. 3 and 4a–b), or alternate among the two topologies. For advancement of the non-linear terms, the “pencil” implementation is more efficient, since we solve the same equations for multiple right-hand-sides, corresponding to the different spanwise locations. Adjacency of the terms which form the right-hand-side vector in memory results in high levels of cache hits and significant speedup of this portion of the computation, which occupies approximately 50% of the computing time. For optimal performance, the computation of the non-linear terms in “pencil” topology is followed by a transpose operation in order to obtain a one-dimensional decomposition in spanwise planes (Fig. 4). The plane topology is used for the solution of the pressure equation. The solution of the pressure Helmholtz equation is based on a multi-grid algorithm, with possibility of point or line relaxation. In addition, red-black coloring is implemented for both the point and line-relaxation (Figs. 4a and 4b respectively), in order to accelerates convergence of the Poisson solve. For line-relaxation, either x- or y-lines can be chosen, or the solver can alternate between the two implementations. It should be noted that the laminar simulation is two-dimensional and is therefore entirely computed using the “pencil” implementation for speed-up (see Fig. 3). In this setting, red-black coloring of the Helmholtz solver is essential for parallel speed-up, in order to ensure all processes can compute, for example, red lines, exchange data at the boundary, then compute black lines and vice versa. The optimal relaxation algorithm, number of grid levels, and multi-grid cycles is evaluated at the beginning of the calculation, and maintained for the remainder of the simulation. For the blade geometry of interest, x-line relaxation with red-black coloring is most effective for accelerating the pressure solve, which occupies 50%
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Fig. 4 Domain decomposition in planes, with (a) point and (b) line-relaxation solve
of the computational time. The superior performance of x-line relaxation is due to the strongest pressure gradient alignment in that direction. For the fully turbulent calculations, the global grid, 1024 × 640 × 128 grid cells, is distributed among 128 processors. The “pencil” processor topology includes 16 processors in streamwise direction, and 8 processors in the cross-stream direction. Therefore, each data block includes 64 × 80 × 128 cells, or 614,400 elements. For the incompressible solver, the memory requirements are approximately 32 M-bytes per processor, in contrast to 32 G-bytes for the overall computation. The choice of 128 processors ensures optimal performance of the algorithm. A smaller number of processors reduces performance, perhaps due to increase in cache misses. Larger number of processors increase the boundary exchanges among processes relative to the volume of the computations per processor. Using 128 dual-core processes, the full three-dimensional Navier-Stokes equations are advanced one time-step every 19 wall-clock seconds. Therefore, approximately 80 hours of run-time are required in order to compute the fully turbulent flow over the length of the blade, and 3 weeks for converged turbulent statistics after the initial transient has elapsed. These resources are required for every flow
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condition tested. Throughout the simulation, filtered velocity fields are output, in order to study the time evolution of the flow. The filtered fields include the velocity at every other point of the computational grid. This and all other read/write operation, for example of restart files, take place simultaneously by all processors to/from a single file using MPI-IO for optimal parallel performance.
1.3 The Free-Stream Turbulence In order to prescribe vortical perturbations at the inlet of the computational domain, a separate DNS of homogeneous isotropic turbulence was carried out in a periodic box whose vertical and spanwise extents match the inflow plane of the compressor. By invoking Taylor’s hypothesis, the streamwise extent of the box of turbulence can be regarded as time. Two-dimensional planes of u , v , w can thus be extracted from the turbulent box as a function of time, and superimposed onto the mean flow at x/L = −0.4 in the compressor DNS. It should also be noted that the viscosity of both simulations must match in order to ensure a short adjustment length for the inflow turbulence once it enters the compressor computational domain. The pseudo-spectral implementation of the NavierStokes equations requires a resolution of 128 × 128 × 384, and is interpolated to the inflow non-uniform grid at every time-step. The turbulence intensity at the inflow of compressor computational domain was Tu = 3.5% of the mean velocity. This value decays to 2.5% at x/L = 0, the position of the leading edge of the blade. The freestream turbulence intensity at mid-pitch is shown in Fig. 5. The figure shows that the inflow turbulence decay does not require a significant adjustment length. This decay rate is reduced within the passage, and recovers a higher rate downstream of the leading edge in the outflow region.
Fig. 5 Turbulent intensity in the free-stream
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2 Results The surface pressure distribution from the laminar, coarse turbulent, and fine turbulent simulations are shown in Fig. 6. The pressure surface of the blade (top curve) is subject to an adverse pressure gradient (APG) from the leading edge till approximately 80% chord, after which the boundary layer is subject to favorable pressure gradient (FPG). The suction surface (lower curve) initially undergoes strong acceleration up to 20% chord, followed by strong APG that causes flow separation. The flow reattaches farther downstream, with the separation region extending over nearly 20–25% of the axial chord. The differences between the quiescent and turbulent free-stream computations are addressed in the subsequent sections, where the pressure and suction surface are discussed in detail.
2.1 The Pressure Surface The Cp distribution on the pressure surface of the blade (Fig. 6) indicates that a mild separation region exists on the pressure surface in the absence of any freestream turbulent forcing. In the presence of free-stream turbulence, however, that mild separation is absent; the boundary layer remains attached throughout the extent of the pressure surface. The change in the boundary layer behavior takes place due to transition to turbulence upstream of the separation point, and the energetic turbulent boundary layer remains attached. The mechanism of the boundary layer breakdown is of the bypass type, which is characterized by the formation of elongated disturbances in the boundary layer upstream of breakdown. These disturbances, known as Klebanoff distortions, are dominated by the streamwise velocity, which can reach on the order of 10% of the mean flow. The elongated disturbances develop secondary instabilities which lead to a sporadic breakdown into turbulent spots. The Klebanoff distortions are best visualized using contours of streamwise velocity perturbations, shown in Fig. 7. The turbulent spots, on the other hand, are most clear in contours of the spanwise
Fig. 6 Pressure coefficient around the blade surface. laminar, coarse fine mesh turbulent, mesh turbulent
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Fig. 7 Contours of the velocity perturbations tangential to the mean flow. The plane shown is inside the pressure surface boundary layer
Fig. 8 Contours of the spanwise velocity perturbations. The plane shown is inside the pressure surface boundary layer
or vertical velocity perturbations. The spanwise perturbations are shown in Fig. 8, and the patch of turbulence indicating transitional flow is marked. On the figure, the mean transitional onset and completion locations are also marked.
2.2 The Suction Surface The Cp distribution on the suction surface of the blade (Fig. 6) indicates that separation takes place independent of the free-stream condition, be it laminar or turbulent. In the presence of free-stream turbulence, the location of separation onset, x/L ∼ 0.45, is nearly unchanged. Differences only emerge downstream of separation; In the presence of free-stream turbulence, the extent of the first separation bubble is reduced by approximately 15%. Reattachment takes place at x/L = 0.7. Also, the reattached boundary layer is turbulent. The persistence of laminar separation on the suction surface despite the freestream turbulent forcing raises the following question: Why does bypass transition not take place upstream of separation and, in a similar manner to the pressure surface, maintain an attached boundary layer? In order to address the above question, we consider the instantaneous perturbation field inside the suction surface boundary layer. Figure 9 is a plane view showing contours of the tangential velocity perturba-
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Fig. 9 Contours of tangential velocity perturbations in a plane inside the boundary/shear layer
Fig. 10 Modulation of the three-dimensional separation surface by the boundary layer perturbations. Light and dark contours correspond to positive and negative tangential disturbances, respectively. The three-dimensional separation surface (white region) is superimposed on the perturbation field
tions. The figure suggests that, on the suction surface, the boundary layer does not develop strong Klebanoff distortions in the initial 20% of chord, which corresponds to the FPG region (see Fig. 6). This result is consistent with the fundamental simulations of bypass transition in constant pressure gradient [6], where flow acceleration is observed to reduce the amplification of Klebanoff distortions, and thus stabilizes a laminar boundary layer with respect to bypass transition. However, in the current setting, the “stabilized” laminar flow separates in the downstream APG section. Had bypass transition taken place, a turbulent boundary layer might have prevented flow separation. The amplitude of the Klebanoff distortions is approximately 8% of the freestream mean speed at the separation point, and continues to intensify downstream. These elongated disturbances inside the separated shear layer are therefore substantial and can become a seat for an interaction with the free-stream disturbances. Figure 10 shows the effect of the Klebanoff distortions on the instantaneous separation surface. The instantaneous separation bubble is shown in white, superimposed on the perturbation field. It is clear from the figure that the separation surface is modulated by the perturbation jets: separation is shifted upstream in regions of negative jets and downstream due to the positive ones.
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3 Conclusion Transitional flow through a compressor passage was computed using DNS. In the absence of any inflow perturbations, the flow remains laminar and the boundary layer separates on both blade surfaces. In response to free-stream forcing, the pressure surface boundary layer transitions to turbulence according to the bypass mechanism. Breakdown in the current simulation is compared to fundamental DNS of bypass transition in flat plate boundary layers, and the similarity is highlighted: Elongated perturbation jets, or Klebanoff distortions, are observed inside the boundary layer. The jets amplify with downstream distance and their interaction with the high-frequency eddies in the free stream leads to the formation of turbulent spots. Despite a free-stream turbulent forcing identical to pressure surface, the suctionside boundary layer does not undergo bypass transition. Instead, it remains laminar up to the separation point. The results from the DNS indicate that the initial FPG region of the suction surface suppresses the amplification of Klebanoff distortions. This observation is in agreement with the fundamental simulations of flat plate boundary layers subject to constant pressure gradient. In the subsequent APG portion of the suction surface, the Klebanoff distortions amplify, but do not reach a sufficiently high amplitude to become seat for secondary instability and bypass transition. Instead, the laminar boundary layer separates. While the mean separation location seems unaffected by the presence of free-stream turbulence, the instantaneous velocity fields demonstrate a modulation of the separation surface by Klebanoff distortions: The separation surface moves downstream in response to forward perturbation jets, and upstream in the response to negative disturbances. Acknowledgements The authors wish to thank the German Research Foundation (DFG) for funding this project and the steering committee of the supercomputing facilities in Bavaria for granting computing time on the SGI-Altix system in Munich.
References 1. L. Hilgenfeld, M. Pfitzner, Unsteady boundary layer development due to wake passing effects on a highly loaded linear compressor cascade. ASME-GT2004-53186 (2004) 2. G. Kalitzin, X.H. Wu, P.A. Durbin, DNS of fully turbulent flow in a LPT passage. Int. J. Heat Fluid Flow 24, 636–644 (2003) 3. M. Rosenfeld, D. Kwak, M. Vinokur, A fractional step solution method for the unsteady incompressible Navier–Stokes equations in generalized coordinate systems. J. Comput. Phys. 94, 102–137 (1991) 4. J.G. Wissink, W. Rodi, H.P. Hodson, The influence of disturbances carried by periodically incoming wakes on the separating flow around a turbine blade. Int. J. Heat Fluid Flow (2005 submitted) 5. X. Wu, P.A. Durbin, Evidence of longitudinal vortices evolved from distorted wakes in a turbine passage. J. Fluid Mech. 446, 199–228 (2001) 6. T.A. Zaki, P.A. Durbin, Continuous mode transition and the effects of pressure gradient. J. Fluid Mech. 563, 357–388 (2006) 7. T.A. Zaki, P.A. Durbin, J. Wissink, W. Rodi, Direct numerical simulation of bypass and separation induced transition in a linear compressor cascade, in ASME Turbo Expo 2006: Power for Land, Sea, and Air, GT2006-90885 (2006)
