Wake Evolution of High-Lift Configuration from Roll-Up to Vortex Decay

Transcription

1 Wake Evolution of High-Lift Configuration from Roll-Up to Vortex Decay Takashi Misaka, Frank Holzäpfel and Thomas Gerz Deutsches Zentrum für Luft- und Raumfahrt (DLR), Oberpfaffenhofen, Germany The development of aircraft s wake in high-lift configuration is studied from the rollup until vortex decay. An aircraft model and the surrounding flow field obtained from high-fidelity Reynolds-averaged Navier-Stokes simulation are swept through a ground-fixed computational domain to initialize the wake. After the wake initialization, the large-eddy simulation (LES) of the vortical wake is performed until vortex decay. In this paper we consider an aircraft model in high-lift configuration where flaps and slats are deployed. In a few wingspan lengths downstream, the complex vorticity structures of the aircraft wake focus on co-rotating vortex pairs originated from wing- and flap-tips. The co-rotating vortex pairs merge at t =0.25 and a common counter-rotating vortex pair is formed at around t =1.0 entraining the turbulent aircraft wake into the vortex system. The rolled-up vortex features axial velocity in the vortex core with a magnitude of a half of the maximum tangential velocity. Behavior of the formed vortex pair is similar to that of a vortex pair defined by a vortex model such as the Lamb-Oseen model. The initial descent speed of the vortices is slightly faster than that of the reference descent speed, i.e., w =1.1w 0. In addition it is confirmed that the ambient turbulence accelerates vortex linking and decay as in the case of a coherent vortex pair. The overall behavior of wake vortices is not affected by the different mesh resolutions considered here, except for vortex core radius. I. Introduction Wake vortices generated by a flying aircraft pose a potential risk for following aircraft due to the strong and coherent vortical flow structure. 1 In addition, it is pointed out that condensation trails (contrails) originating from the jet exhaust which interacts with wake vortices and the environmental atmosphere may trigger the formation of cirrus clouds (contrail cirrus) which have influence on the climate. 2, 3 Wake vortex is related to a broad scale of flows, that is, flows around aircraft s main wing, fuselage, slat, flap, jet engine and tail plane, and their interactions may affect the generation of wake vortex in particular in a highlift configuration. 4 On the other hand, contrails generated by cruising aircraft may persist several tens of kilometers. The evolution of aircraft s wake can be divided into several phases, for example, (1) roll-up phase, (2) vortex phase, and (3) dissipation phase. Although numerical simulation is one of the effective approaches to tackle this problem, the applicable flow scale of a numerical simulation is usually limited to only one of those regimes. High-fidelity Reynolds-averaged Navier-Stokes (RANS) simulation can handle flows around aircraft and subsequent roll-up process of wake vortex in the jet regime. 5 In addition, experimental measurements of near field wake evolutions have been conducted. 4, 6 On the other hand, the dynamics of rolled-up wake vortex in the vortex and dissipation regime has been studied mainly by large-eddy simulation (LES) or direct numerical simulation (DNS). In these researches, detailed time evolution of a vortex pair with an initially longitudinally constant velocity profile is investigated, where short-wave (elliptic) instability 7 9 and Crow instability 10, 11 may develop. In addition, various atmospheric conditions of turbulence, stability and wind shear are considered to assess the effect of these factors on wake vortex evolution and decay The LES of wake vortex in the late dissipation and diffusion regimes is performed along with microphysical Research Assistant, Institut für Physik der Atmosphäre, Münchner Straße 20, Member AIAA. Currently, Institute of Fluid Science, Tohoku University, Japan. Research Scientist, Institut für Physik der Atmosphäre, Münchner Straße 20, Member AIAA. 1of11

2 processes of contrails where the properties of ice crystals and thier effects on radiation exchange are of 15, 16 primary interest. The authors have been studying the feasibility of a wake initialization approach where realistic aircraft wake is generated in a LES domain by sweeping a high-fidelity RANS flow field through the domain. 17 Using this approach the simulation was performed from the roll-up of the DLR-F6 wing-body model until the vortex decay. The roll-up processes of the vorticity sheet emanating from the wing and tight vortices from the wing-tip were simulated. A stable vortex pair appeared after this. The growth of vortex core radius was small especially after the roll-up, where the vortex core radius still depends on the mesh resolution. Following the work of temporal and spatial simulations of boundary layer flows, 18 we refer to a common way of simulating a vortex pair in a periodic computational domain as the temporal simulation of wake vortex, while a typical high-fidelity RANS simulation around aircraft which employs an aircraft fixed domain with inflow boundary conditions could be called a spatial simulation because the spatial development of aircraft wake can be simulated. The present approach stands somewhere between those approaches by considering the spatial development of aircraft wake in the temporal simulation. Following the previous work, 17 this paper focuses on the evolution of wake vortex generated from a highlift configuration aircraft model. Here we use a RANS flow field with adaptive mesh refinement in the wake to avoid the dependency of the results on the RANS mesh resolution, which was an issue in the previous DLR-F6 model simulations. We investigate the evolution of vortex parameters such as the radially averaged circulation, vortex core radius and vortex positions from the wake roll-up to vortex decay as in the previous work. A series of study aiming for bridging the gap between the roll-up and the vortex phases shall provide more realistic insights into aircraft wake vortex evolution expressed in terms of e.g., vortex circulation and vortex core radius. II. Governing Equations and Numerical Methods The LES is performed by using incompressible Navier-Stokes code MGLET. 14, 19 An equation for potential temperature is also solved to take into account buoyancy effects of the atmosphere. u i t + (u iu j ) = 1 p +(ν + ν t ) 2 u i x j ρ 0 x i x 2 + g θ δ i3, (1) j θ 0 θ t + (u jθ ) =(κ + κ t ) 2 θ dθ s x j x 2 + u 3, j dx 3 (2) u j =0, x j (3) where u i, p and θ represent velocity components in three spatial directions (i=1, 2 or 3), pressure and potential temperature, respectively. Summation convention is used for velocity components u i and δ ij denotes Kronecker s delta. The primes for pressure and potential temperature show that these are defined by the deviation from the reference states: p = p 0 + p, θ = θ 0 + θ. In the Boussinesq approximation, the potential temperature is coupled to momentum equations through the vertical velocity component. Kinematic viscosity in Eq. (1) is given by the sum of molecular viscosity and eddy viscosity defined by a subgrid-scale model. Corresponding diffusion coefficient κ in Eq. (2) is obtained by assuming constant molecular and turbulent Prandtl numbers of 0.7 and 0.9, respectively. The above equations are solved by a finite-volume approach with the fourth-order finite-volume compact 20, 21 scheme. A split-interface algorithm is used for the parallelization of the tri-diagonal system, which realize smaller overhead time and scalability in parallel environment compared to the existing parallel tridiagonal matrix solvers. 22 In addition, a divergence free interpolation is employed for obtaining advection velocity, which ensures conservation of velocity and passive tracer fields. A pressure field is obtained by the velocity-pressure iteration method 23 with a multi-grid convergence acceleration technique. 24 The third-order Runge-Kutta method is used for the time integration. 25 The Lagrangian dynamic model is employed for a turbulence closure. 26 The Lagrangian dynamic model does not require specific direction for the averaging process of subgrid model coefficients which is usually required in dynamic-type models for stable computation, therefore, the Lagrangian dynamic model is appropriate for wake vortex simulation where there is no relevant direction for the averaging. Computations are performed in parallel by a domain decomposition approach. 2of11

3 III. Description of Present Approach The present approach which is schematically shown in Fig. 1 may be considered as a numerical realization of the catapult facility. 27 The numerical approach has several advantages for investigating aircraft wake. Decay of a vortex pair strongly depends on environmental conditions such as ambient turbulence, temperature stratification and wind shear. Therefore the control of these conditions is crucially important to assess the influence of the ambient conditions on vortex decay. Unlike the consideration of realistic inflow conditions in an aircraft fixed LES domain, the generation of controlled turbulence fields in the ground fixed LES domain is straightforward. The other reason is that the present approach does not need a long computational domain in the flight direction for obtaining large vortex ages compared to an aircraft fixed LES domain. Ambient turbulence Wake initialization (~seconds) Time integration until decay (~ minutes) Figure 1. Schematic of the present approach, (a) wake initialization in the order of seconds, (b) wake evolution in the order of several minutes until vortex decay. III.A. Wake initialization using RANS flow field An aircraft model and the surrounding flow field obtained from high-fidelity RANS simulation are swept through a ground fixed LES domain to initialize the aircraft s wake. 17 The RANS flow field is provided as a forcing term of Navier-Stokes equations in the LES. Similar approach might be referred to as the fortified solution algorithm (FSA), 28 or a nudging technique used in data assimilation. 29 The resulting velocity field is represented by the weighting sum of LES velocity field V LES and RANS velocity field V RANS, V = f(y, α, β)v LES +[1 f(y, α, β)] V RANS. (4) In this study, V RANS is provided as a constant forcing term of Navier-Stokes equations solved in the LES as a one-way coupling. Since the aircraft model is swept through a computational domain, the forcing term acts as a moving boundary condition for the LES. The weighting function f(y, α, β) could be a smooth function of the wall-distance y, or of other physical quantities such as velocity magnitude. Here, we employ the following function of wall-distance to realize smooth transition between the RANS and LES flow fields, [ f(y, α, β) = 1 [ ( y tanh α 2 β β )] ] +1.0, (5) y where the constants α and β represent the slope of the transition and the wall-distance where solutions of RANS and LES are equally weighted, respectively. These constants can be determined by trial and error, as well as by optimization techniques. 17 The mapping of the RANS flow field onto the Cartesian LES mesh is performed by a linear interpolation only once before the wake initialization. An additional computer memory is prepared to store the mapped RANS flow field, however, the additional computational cost for the forcing term is minimal. The forward movement of an aircraft is represented by simply shifting the mapped flow field for a certain mesh spacing, which is also possible for a decomposed LES domain if the increments of the advancement is smaller than the overlap region of the domain decomposition for parallel computation. 3of11

4 y f(y,α,β) LES RANS Figure 2. Schematic of a weighting function for a combination of RANS and LES flow fields. III.B. Reproduction of eddy viscosity Since we only use a RANS velocity field to initialize the wake, the eddy viscosity in the LES domain appears to be low compared to that in the original RANS flow field. Therefore it is required to reproduce velocity fluctuations modeled in the RANS flow field. It is pointed out that the correct representation of eddy viscosity in the wake is important to simulate the wake evolution. 30 Most crude but still useful representation of such velocity fluctuations may be a white noise. Here we add a white noise to the RANS flow field in the region of RANS-LES transition so that the time averaged LES turbulent kinetic energy matches the RANS turbulent kinetic energy in the wake. The magnitude of the fluctuations is modified by the proportional-integral (PI) controller during the advancement of the model through the LES domain. V RANS+WN = V RANS + KV WN, (6) K = a 1 ( kt,les k t,rans ) + a2 ( kt,les k t,rans ) dt, (7) where V WN is a white noise field and K is a gain to control the magnitude of the velocity fluctuations. The gain is defined by the difference between the time-averaged LES turbulent kinetic energy k t,les and RANS turbulent kinetic energy k t,rans. These turbulent kinetic energies for calculating the gain are integrated in the wake region with the weighting of the RANS turbulent kinetic energy. The magnitude of the added white noise is also weighted locally using the RANS turbulent kinetic energy. The constants a 1 and a 2 are set according to the convergence of the gain and numerical stability but the results are not very sensitive to these values. IV. Computational Setting We use a RANS flow field of AWIATOR long range aircraft model obtained by the DLR TAU-code with the adaptive mesh refinement for wing- and flap-tip vortices as well as for the fuselage wake. 31 Flow conditions of the RANS simulation were the same as those of the ONERA s catapult facility experiment, i.e., Reynolds number Re = , freestream velocity U =25m/s, and a lift coefficient C L =1.4. The 1/27 scaled model has a wingspan of m. For the normalization of quantities we use the following reference values assuming an elliptic load distribution, 1, 4 Γ 0 = 2C LU b πλ, b 0 = π 4 b, w 0 = Γ 0, t 0 = b 0, (8) 2πb 0 w 0 where the wing aspect ratio is Λ = 9.3. The resulting reference values for the normalization are Γ 0 = 5.36 m 2 /s for circulation, b 0 =1.756 m for length, w 0 =0.49 m/s for velocity, and t 0 =3.617 s for time. Two different mesh resolutions are considered in LES. One mesh has the resolution of dy = dz =0.009 in a plane perpendicular to the flight path and of dx = in the flight direction, which is called Δ =0.009 mesh in this paper. Another mesh has the spacing of dy = dz =0.005 on the plane, while the mesh spacing in the flight direction is the same as the former mesh (Δ =0.005 mesh). The computational domains are bounded by the lengths of 8.8b 0,5.8b 0,and4.7b 0 in flight, span and vertical directions for the Δ =0.009 mesh, and of 8.8b 0,5.8b 0,and4.4b 0 for the Δ =0.005 mesh. Uniform mesh spacing is employed for all three spatial directions in the Δ =0.009 mesh, while stretched mesh is used only in span direction in the Δ =0.005 mesh. 4of11

5 There is a difference of vortex age between both sides of the computational domain in the flight direction after wake initialization. To conduct LES stably for a long time of periods, an appropriate boundary condition is needed. Here the flow field is simply inverted slice-wise to close the domain periodically. As a result periodic boundary conditions can be employed for all boundaries as in the typical temporal LES for wake vortex. 14 The impact of ambient turbulence on wake vortex decay is taken into account by preparing homogeneous turbulence fields as an initial flow field for wake initialization. The turbulence field is generated by the LES of decaying turbulence characterized by eddy dissipation rate and Brunt-Väisälä frequency, though the effect of temperature stratification is not considered in this paper. V. Results V.A. Near- to mid-field wake evolution until t =1 Figure 3 shows near-field vorticity distributions obtained from the RANS simulation. The contours in blue and red respectively represent axial vorticity in clockwise and counter-clockwise directions viewed from the tail. The vorticity distribution is complex just behind the main wing in high-lift configuration. Nevertheless, only a few vortices remain at the position of tail wings, i.e., wing- and flap-tip vortices as well as vortices from the wing-fuselage junction. Only the vortex from the wing-fuselage junction has opposite rotational direction among vortices at the position of the tail wings. Vorticity from the tail wings appears much smaller than that of the main wing in this flow conditions. Figure 3. Near-field vorticity distribution around AWIATOR long range aircraft model obtained from RANS simulation. Figure 4 shows the temporal evolution of vorticity distribution in a plane perpendicular to flight direction. Strong vortices in the plane are correlated with the result of the RANS simulation in Fig. 3. Two vortices generated from outer edges of the flaps are about to merge at t =0.021 (x =1.0). The resulting co-rotating vortices from wing- and flap-tips rotate as in Figs. 4(c) to (f). It is also observed that an isolated vortex rotates around primary co-rotating vortices, which originates from the inner engine pylon. Vortices from wing-fuselage junction decay quickly due to the turbulent wake of the fuselage. Vortex merging of primary co-rotating vortices occurs at around t =0.253 (x =12.9). Finally the turbulent wake is entrained into the primary vortex realizing a fully rolled-up vortex pair at t =1.049 (x =53.5). Note that the switching from the RANS simulation to the LES is activated at the wall-distance of β =0.07, therefore, the RANS velocity fields with strong vorticity is transferred to the LES part. Mesh spacing of the LES for Δ = mesh case is approximately 4 times larger than that of the RANS simulation in vortex core areas where adaptive mesh refinement was applied. However, the compact finite volume scheme along with equidistant Cartesian mesh in the LES enables to simulate vortex evolution accurately. 5of11

6 (a) =0.0 (b) =0.021 (c) =0.064 (d) =0.113 (e) =0.165 (f) =0.209 (g) =0.253 (h) =0.430 (i) =1.049 Figure 4. Time evolution of near- to mid-field vorticity distribution until t =1. Figure 5 shows the evolution of vortex parameters in the near-field until t =0.31 (x =16.0). Solid lines represent parameters evaluated for wing-tip vortex, while broken lines are obtained from flap-tip vortex. In addition, results from two different meshes as well as those of RANS simulation are plotted. Quantities in the plots are non-dimensionalized by using reference values mentioned above. The averaged circulation in Fig. 5(a) shows that the circulation of flap-tip vortex is stronger than that of wing-tip vortex. Two vortices from the flap-tip seen in Fig. 4(b) are quickly merged, therefore, they are evaluated as one vortex in the plots in Fig. 5. The convergence of the circulation values of flap-tip and wing-tip vortices indicates the completion of the merger of these vortices at around t =0.25 (x =12.9), which was also confirmed in Fig. 4(g). The agreement of circulations from RANS simulation and LES is also confirmed. Figure 5(b) represents the evolution of vortex core radius. The influence of LES mesh spacing appears in the vortex core radius, i.e., the core radius of Δ =0.005 mesh is smaller than that of Δ =0.009 mesh. On the other hand, their growth rate until x =9.0 is almost the same. The core radius of wing-tip vortex rapidly increase after x =12.0 because each vortex is not clearly identified during the vortex merging of wing- and flap-tip vortices. Figures 5(c) and (d) show the evolution of vortex positions in span and vertical directions, respectively, where trajectories of one pair of wing- and flap-tip vortices are shown. The wing- and flap-tip vortices rotate quite similarly for both mesh cases. Figure 6 shows the distribution of axial velocity which is colored by the magnitude of axial vorticity at the respective positions. At t =0.113 a co-rotating vortex pair in red is distinguishable from the disturbed 6of11

7 (a) (b) 0.07 Solid line: wingtip vortex 0.06 Broken line: flaptip vortex Γ Solid line: wingtip vortex Broken line: flaptip vortex r c * RANS RANS y* (c) Solid line: wingtip vortex Broken line: flaptip vortex RANS z* (d) Solid line: wingtip vortex Broken line: flaptip vortex -0.5 RANS Figure 5. Near-field vortex parameters, (a) averaged circulation, (b) vortex core radius, (c) vortex position in span direction, as well as (d) that in vertical direction. (a) =0.11 (b) =0.34 (c) =1.05 Figure 6. Distribution of axial velocity during wake roll-up where the surface is colored by the axial vorticity. 7of11

8 (a) =-0.1 (=-5.5) (b) =0.0 (=0.0) (c) =0.1 (=4.0) (d) =0.2 (=10.7) (e) =1.1 (=53.5) (f) =3.7 (=186.5) (g) =4.4 (=222.8) (h) =5.6 (=286.0) Figure 7. Perspective view of iso-vorticity surfaces from roll-up until vortex decay. 8 of 11

9 wake. The merger of co-rotating vortices occurs on one side of the vortices as shown in Fig. 6(b), which corresponds to Fig. 4(g). The merge of co-rotating vortices is already finished at t =1.049 posing a vortex pair with the axial velocity of approximately Vax = 4 in the vortex core. At this time the maximum tangential velocity is approximately Vr = 8 for both mesh cases. In addition, fluctuations in the remaining area are decreased after forming a vortex pair. V.B. Far-field wake evolution until vortex decay Figure 7 shows the time evolution of wake vortex from roll-up until vortex decay. The flow field is visualized by two levels of iso-vorticity surfaces (red: ω = 100, blue transparent: ω = 22). Here the ambient turbulence is characterized by eddy dissipation rate of ε =(εb 0 )61/3/w 0 =0.23, while temperature stratification is not considered. The origin of the non-dimensional time and the distance in Fig. 7 is set to the center of the computational domain in the flight direction. Since ambient turbulence is relatively strong, which mimics the turbulence in the atmospheric boundary layer near the ground, weak vorticity structures are visible in Fig. 7(a). The co-rotating vortices originated from wing- and flap-tips appear during the wake initialization in Fig. 7(c) as a helical co-rotating vortex pair. And the merger of the co-rotating vortices proceeds from the older part of vortices toward the newer part in Fig. 7(d). A counter-rotating vortex pair is formed after one vortex reference time t =1.1 in Fig. 7(e). Due to relatively strong ambient turbulence, the vortex pair is deformed as in Fig. 7(f). Finally the vortex pair quickly decays after vortex linking in Fig. 7(g). The evolution of the counter-rotating vortex pair until vortex decay is similar to that obtained from temporal LES of wake vortex Γ (a) 1.0,ε*= r c * (b) ,ε*= z* (c) b* (d) ,ε*= ,ε*= Figure 8. Evolution of vortex parameters, (a) averaged circulation, (b) vortex core radius, (c) vertical vortex position, and (d) vortex separation. 9of11

10 Figure 8 shows the time evolution of vortex parameters until vortex decay. Here three cases of spatial LES are conducted, i.e., Δ =0.005 and Δ =0.009 mesh simulations as well as the Δ =0.009 mesh case with the ambient turbulence. Figure 8(a) shows the averaged circulation where the radially averaged circulation is further averaged along vortex centerlines in the domain. Initial value of the averaged circulation is 0.9, which is slightly smaller than the circulation estimated assuming the elliptic load distribution in Eq. (8). As in the temporal LES of a counter-rotating vortex pair, 14 the rapid decay phase of circulation comes after the initial gradual decay phase. In these conditions, the rapid decay of the averaged circulation corresponds to the vortex linking and the resulting vortex decay shown in Fig. 7(h). In addition, the ambient turbulence effectively decreases the onset time of the rapid decay. On the other hand, the initial decay rates of the circulation for Δ =0.005 and Δ =0.009 appear similarly. Figure 8(b) shows the time history of vortex core radius. The growth rate in the initial decay phase is similar for the three cases, while the core radius of fine mesh case is smaller than that of the other cases. Figure 8(c) shows the vertical position of vortices in the domain. In the initial decay phase vortices descend similarly in cases considered here. The descent speed of the vortex pair is 1.1w 0, which is slightly faster than the theoretical descent speed of a vortex pair. Figure 8(d) shows the evolution of the averaged vortex spacing, which is initially smaller than b 0, i.e., b =0.92b 0 between t = 1 to 2, and slightly increases during descent. Using the actual value of vortex separation 0.92b 0 for the definition of reference velocity, the descent speed of the vortex pair becomes w =1.012w 0. VI. Conclusions LES of wake vortex evolution from its generation until vortex decay was conducted by combining RANS and LES flow fields. The RANS flow field was employed in the LES as a forcing term sweeping through the ground-fixed LES domain. In addition, the eddy viscosity initialization to match the time-averaged LES and RANS turbulent kinetic energies were employed. In this paper we considered an aircraft model in high-lift configuration where flaps and slats are deployed. After a few wingspan lengths downstream, the complex vorticity structures of the aircraft wake focused on co-rotating vortex pairs originated from wing- and flap-tips. Each co-rotating vortex pair merged at t =0.25 and a common counter-rotating vortex pair was formed at around t =1.0 entraining turbulent wake into the vortex system. The rolled-up vortex had axial velocity in the vortex core whose magnitude was half of the maximum tangential velocity. Behavior of the formed vortex pair was similar to those of a typical vortex pair defined by a vortex model such as the Lamb-Oseen model. The descent speed of the vortices was slightly faster than that of reference vortex speed, i.e., w =1.1w 0. In addition it was confirmed that the ambient turbulence accelerates vortex linking and decay as in the temporal LES. The overall behavior of wake vortices was not affected by the difference of mesh resolution considered here, except for vortex core radius. Acknowledgments We would like to thank Stefan Melber (Institut für Aerodynamik und Strömungstechnik, DLR-Braunschweig) for providing the RANS data of the AWIATOR long range aircraft model. We also thank Airbus for the allowance to use it. We thank Prof. Michael Manhart and Dr. Florian Schwertfirm for the provision of the original version of the LES code MGLET. Computer time provided by Leibniz-Rechenzentrum (LRZ) is greatly acknowledged. The current work was conducted within the DLR project Wetter&Fliegen. References 1 Gerz, T., Holzäpfel, F., and Darracq, D., Commercial Aircraft Wake Vortices, Progress in Aerospace Science, Vol. 38, No. 3, 2002, pp Minnis, P., Young, D. F., Ngyuen, L., Garber, D. P., Jr., W. L. S., and Palikonda, R., Transformation of Contrails into Cirrus during SUCCESS, Geophysical Research Letters, Vol. 25, No. 8, 1998, pp Schumann, U., Graf, K., and Mannstein, H., Potential to Reduce the Climate Impact of Aviation by Flight Level Change, AIAA Paper , Breitsamter, C., Wake Vortex Characteristics of Transport Aircraft, Progress in Aerospace Science, Vol. 47, No. 2, 2011, pp Stumpf, E., Study of Four-Vortex Aircraft Wakes and Layout of Corresponding Aircraft Configurations, Journal of 10 of 11

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