- Email: [email protected]
Calculations of massive separation around landing-gear-like geometries
9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China
883
2010, 22(5), supplement: 926-931 DOI: 10.1016/S1001-6058(10)60054-6
Calculations of massive separation around landing-gear-like geometries Zhi-xiang Xiao, Jian Liu, Song Fu * School of Aerospace Engineering, Tsinghua University, Beijing, China *Email: [email protected] ABSTRACT: The massive separating flows around landinggear-like configurations, such as Rudimentary landing gear (RLG) and tandem cylinders (TC), are calculated using unsteady Reynolds averaged Navier-Stokes (URANS) and delayed-detached-eddy simulation (DDES) based on k-ω-SST model. A number of numerical schemes and dissipation are applied in an effort to compare the averaged as well as the instantaneous flow-fields with the available measurements. It is shown that high-order and low-dissipation scheme is necessary to calculate the small scale structures. Furthermore, DDES delivered better results than URANS.
due to one region of the flow. Other reasons are the higher geometric complexity. The mechanism of the strong interactions among the different components is not clear.
KEY WORDS: massive separation, RLG, TC, DDES, numerical scheme and dissipation
However, it would be very difficult to distinguish the errors caused by the calculation itself and those caused by the acoustics components. It’s a great challenge to accurately calculate the massive separations around the LG and TCs.
1 INTRODUCTION In the flight phases of taking-off and approaching, landing gears (LG) are known to be one of the major generators of airframe noise. LG Noise reduction is one of the important “wars on noise”. Flows around LG are usually characterized by turbulent boundary layer separation, unsteady vortex shedding, and intensive unsteady inter-components interactions. The calculation of flow past LG is helpful to investigate and understand the turbulent flow-generated noise. The Boeing “Rudimentary Landing Gear” (RLG) configuration [1] has been selected as one of test cases in this article, because it is supported by an experimental project funded by Boeing and carried out at NAL (India). The RLG configuration has large axles with square cross-sections and tripping the wheel boundary layers, which ensures the flow are fully-turbulence. And the Reynolds number in experiment is more relevant to the realistic high Reynolds number. LG-noise CFD calculation remains very incomplete, and has few “success stories.” One reason is the absence of experimental data which isolate the noise
A canonical problem to develop modeling methods for multi-elements interactions corresponds to the tandem cylinder (TC) arrangement [2-6]. TCs with similar diameters can be found on a landing gear, such as the oleo and hoses.
If traditional URANS (unsteady Reynolds averaged Navier-Stokes) methods with turbulence models are applied to the massive separations, only large scales of structures can be calculated due to excessive dissipation, which prevents the formation of smallscale structures. LES (large eddy simulation) is a naturally accurate calculation method for separation, because the large scale of structures are resolved accurately, only very small-scale motions are modeled by sub-scale grid model. However, it’s time consuming to calculate the flow at high Reynolds number in industry applications using LES. Limited by computational resource, the combination of LES with RANS can achieve both reasonably well with high efficiency and numerical accuracy when calculateing the flow with massive separation. RANS/LES hybrid methods (originally proposed by Spalart, [7]), has widely used to calculate the unsteady and geometry-dependent separating flows. Such hybrid methods combine a high-efficiency turbulence model near the wall, where the flow is dominated by small scale motion with an LES-type treatment for the large scale motion in the flow region far away from
9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China
884
the wall. In fact, the convective scheme is also another very important factor, which will smooth the flow with few small scale structures if it is excessively dissipative. To capture higher frequency and small scale of motion, it’s necessary to decrease the numerical dissipation of scheme. In this article, RLG is preliminarily calculated using URANS and TCs is simulated using DDES[8-9]. The numerical simulations are compared with the available time-averaged and instantaneous flow-fields. 2 TURBULENCE MODEL AND DDES The fundamental turbulence model in this article is the two-equation k-ω-SST model, which combines the standard k-ε model with the k-ω model without the free-stream values dependence. The switching is realized by a flow dependent blending function. The SST model also limits the eddy viscosity by forcing the turbulent shear stress to be bounded by constant times the turbulent kinetic energy inside boundary layers (a realizability constraint). This modification improves the model’s performance on flows with strong adverse pressure gradients and separation. Due to its good numerical robustness and performance for most of flows, it’s widely used in industries. The detailed about SST model can be found in its original reference. It’s not listed here. To construct a DES-type hybrid method based on twoequation models, some transformation is adopted for the dissipation term in the turbulent kinetic energy transport equation. After introducing a length scale, the kinetic energy equation can be written as ∂ ( ρk ) ∂ρu j k ∂ ⎛ ∂k ⎞ * + = ⎜ ( μ + σ k μt ) ⎟ +τ ij Sij − β ρkωFDES (1) ∂t
∂x j
∂x j ⎜⎝
∂x j ⎟⎠
3 NUMERICAL SCHEMES 3.1 Spatial Scheme for Convection The computations are all based on a compressible solver using Roe flux-difference splitting scheme with 3rd-order monotone upstream centered scheme for conservation laws (MUSCL) and 5th order weighted essentially non-oscillatory (WENO) with R-S entropy fix in a cell-centered finite-volume formulation[10]. The flux of the convection is obtained in Roe as F
i+
1 2
=
1⎡ F ( q L ) + F ( q R ) − ε × A inv ( q R − q L ) ⎤ ⎦ 2⎣
(2)
the turbulence length scale Lt is defined as Lt = k1/2 (β*ω) ; CDES=0.78×F1+ 0.61×(1-F1); Δ is the grid scale defined as Δ=max(Δ ξ, Δη, Δζ); FSST can be taken as 0, F1 or F2, where F1 and F2 are the two blending functions in the SST model. If FDES=1, equation (1) is the same as the kinetic equation of SST model. If FSST = 0, the hybrid method reverts to a Strelets-type DES method. If FSST = F1 or F2, then, this hybrid approach is called the delayedDES method. Due to the numerical properties of F1 and F2, (1-FSST) approaches zero near the wall and the DDES will act in the RANS mode. At the same time, (1-FSST) becomes one out of the boundary layer and the DDES goes to the original Strelets’ DES model.
i+
1 2
(3)
where qL and qR are the original variable at the left and right side of the face ()i+1/2 using MUSCL or WENO interpolation. A inv is the matrix of Roe averaged. For the original Roe scheme, ε is taken as 1.
Furthermore, in order to isolate the errors of dispersion and dissipation, a higher-order symmetric total variation diminishing (STVD) scheme, which is the combination of central scheme with the numerical dissipation of the Roe, is used to solve the NavierStokes equations. In this article, S6-WENO5 is taken as the main scheme for the TCs case. Fi + 1 = 2
Fsymmtric, i+ 1 2 6th order symmetric scheme
where FDES is the hybrid function defined as ⎡ ⎤ L FDES = max ⎢(1 − FSST ) ⋅ t ; 1⎥ CDES Δ ⎦ ⎣
Therefore, the delayed-DES can ensure itself to act in the RANS mode near the wall without the effects on the locally clustered grid scales. In other words, this hybrid method can delay the switching from RANS to LES near the wall due to the grid scales, especially the locally refined grids in the streamwise and spanwise direction for the complex aircraft configurations. In this paper, FSST is taken as F2.
1 − ε × ⎡⎣ Ainv ( q R − q L )⎤⎦ 1 i+ 2 2
(4)
5th order WENO
In fact, when high-order scheme is applied in largereddy simulation, the convective scheme is over dissipative. Then, only low dissipation scheme can be taken as the appropriate when LES or DES is applied to calculate massive separations. The simplest method is adding a coefficient (ε)[11], which is less than 1 and greater than 0, to the dissipation terms in Equ.(4). 3.2 Temporal Scheme A modified fully implicit LU-SGS with Newton-like sub-iteration in pseudo time is taken as the time marching method when solving the mean flow equations and the turbulence model equations. Global non-dimensional time stepping is implemented to capture the unsteady properties of the separate flows. It indicates that the RANS method here is the unsteady-RANS. For the DDES, about 30 subiterations are taken to ensure the maximum residuals
9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China drop two orders. Implicit residual smoothing is employed to accelerate the convergence in the sub-iteration. The approach is parallelized using domain-decomposition and message-passing-interface strategies for the platform on PC clusters with 64-bit XEON-E5520.
885
nominal front-view area for RLG model. When the computations are running after 16,000 steps, the fluctuation amplitude of Cd is very small, and the URANS is stopped and DES is started. However, the results by DES are not presented this time because it is under-going.
4 RESULTS DISCUSSION 4.1 Rudimentary Landing Gear (RLG) Description: Unsteady flows past RLG, dominated by massive separation and interactioons among different components, are calculated using URANS based on SST model. In addition, the geometry of RLG is also very complex. Then, it’s very difficult to accurately calculate the unsteady flows and airframe noise using CFD and CAA methods. The computational grids contain 55 blocks and the overall cells are 10,772,708, shown in figure 1. The Reynolds number based on the wheel diameter (D, which is equal to 0.4064m) and reference velocity (U0 = 40 m/s) is equal to 106. The unsteady flows are calculated by solving URANS equations with SST turbulence model. Normalized time step Δt*=Δt×U/D =0.005, which is corresponding to 5.08×10-5 s for each step in physical time. Then, no instantaneous results, such as instantaneous pressure fluctuations, frequency on some samples, are presented using URANS.
Fig. 2 History of Cd
Figure 3 presents the comparisons on the timeaveraged surface pressure coefficients from two angles of view, where the contours have used the same legend as for the experimental measurement. The left figures are the measurements, while the right figures are our numerical results. The overall URANS results match the measurements very well, and local difference can be observed, especially on the surface of the streaimwise strut between the four wheels.
Fig. 1 Computational Grids
Boundary Condition: Uniform inflow is applied to the upstream boundary; Outflow is taken downstream boundary with zero pressure gradients. On the side boundary, the flow is assumed as symmetry. On the top and bottom boundaries, invicid wall is taken as the boundary conditions. No-slip conditions are applied on the wheels and struts wall. The flow is calculated using URANS with initial uniform flows. The history of streamwise force (Cx or Cd) is presented in Fig.2. The reference area here is taken as 1.06675 D2, which is sum of the
Fig. 3 Comparisons on Cp (Left: Exp; Right: URANS)
Figure 4 presents the time-averaged surface friction patterns in comparison with the experimental oil visualization. It is shown that the URANS computations have produced very similar mean flow features over the outward wheel surfaces, where the flow is attached and being visualized in the experiment. URANS results also present a small scope of second separation over the upstream wheel.
9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China
886
shear layer instability, the interaction of unsteady wake of the front cylinder with the downstream one and unsteady massively separated flow between the cylinders and in the wake of the rear cylinder, etc. In this article, typically unsteady-RANS and advanced DDES with higher-order low-dissipation scheme are applied to calculate the unsteady pressure fields.
Fig. 4 Comparisons on surface streamlines (Left: Exp; Right: URANS)
The flow separation on the backsides of both the front and the rear wheels by computations differ from those of experimental visualization. Then it’s a great challenge to well calculate the massive separation behind the wheels and struts. Figure 5 illustrates the resolved vortices by different modeling approaches. One is SA-DDES by Spalart and another is our URANS based on SST. As expected, SST-DDES are able to resolve much richer vortex motions, whereas the URANS simulation has seemingly claimed rather “stiff” shear layers after the flow is detached from bluff-body surfaces. The main reason is the excessively viscous when URANS is applied to the massive separation flow. Then the future work is to calculate this flow using advanced RANS/LES hybrid methods with higher-order and low-dissipation scheme.
Figure 6 presents the sketch map of the tandem cylinder with the same diameter (D). The space (L) between the two cylinder centers is 3.7D. The velocity of freestream is 44m/s and the Reynolds number based on the cylinder diameter is 1.66×105, and the angle of attack is 0 degree. y
Ma
θ
O
x
L D
Fig.6 Model of tandem cylinder
The mesh provided by NTS is shown in Fig.7. The total size of the grid in the XY plane is 82,000 cells. The grid size between the cylinders is almost isotropic (about 0.01D) to calculate the small scale vortices. The spanwise size of the domain for Roe scheme are 1D and 4D with Δz=0.033D and for STVD is 3D with Δz=0.02D. Then the overall grids are 2.78 million for 1D, 10.9 million for 4D and 12.4 million for 3D, respectively. At the time of submitting this paper, the DDES computation with low-dissipation and highorder scheme for the last grids is under-going and only some instantaneous results are presented here.
Fig.7 2-D grids of tandem cylinder Fig. 5 Resolved instantaneous vortex motions (Left: reference by DDES; Right: URANS)
4.2 Tandem Cylinders (TCs) The tandem-cylinder is a prototype for interaction problems commonly encountered in airframe noise configurations (e.g., the oleo and hoses on a landing gear). The flow has been studied in a series of experiments performed in NASA Langley Research Center[2,3]. Simulation of TCs can help testing the capability of turbulence modeling approaches, spatial and temporal methods to reproduce properly complex flow phenomena, such as the transition on the two cylinders, separation of turbulent boundary layer, free
Fig.8 comparisons on pressure coefficients on surface
Figure 8 presents the comparisons on the timeaveraged pressure coefficients over the surface using DDES with 3rd MUSCL interpolation. The symbols are the experimental results and dashed line represents the computational results of 1D and solid line represents the 4D numerical results. From the comparisons, we can find that the spanwise length has a significant influence on the pressure distribution on
9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China the surface. Longer spanwise length can obtained more reasonable results. Of course, to achieve the well matched results with the measurements, longer spanwise length should be applied. As we known, the flow is extremely unsteady. Then the pressure fluctuations on the surface or in the wake should be investigated to verify the calculation methods. The root mean square (RMS) of the pressure on the rear cylinder is presented in figure 9a. From this figure, we can see that the smaller spanwise calculate the Cp,rms much higher results than that of 4D and experiments. At the same time, we can find that the Cp,rms shows two peak values where θ are 45° (due to vortices shedding from the front cylinder) and 120° (due to the separation), respectively. Figure 9b presents the comparisons on frequency and amplitude of the pressure fluctuation for the sample where θ is equal to 45° on the rear cylinder surface. From this figure, the results of frequency, especially the amplitude of the power-spectral-density (PSD) for 4D spanwise length can match the measurements much better than that of 1D spanwise length. And the primary frequency is about 180Hz, which is corresponding to the vortices shedding frequency from the front cylinder and also is the frequency of impingement on the rear cylinder surface by the former-mentioned vortices.
(a) (b) Fig.9 Cp,rms and PSD at location of 45 degree on rear cylinder
As mentioned before, the dissipation of the numerical scheme has a significant influence of the information of small scale of structures. The high-frequency pressure fluctuations are easily cut off when large numerical dissipation schemes are used to calculate the unsteady flow. Sometime, the numerical dissipation can exceed the physical viscosity. Then, the advanced DDES combined with large dissipation scheme maybe can’t obtain acceptable results and their performance look like those of URANS. Figure 10 presents the comparisons on spanwise vorticity (with the same color legends) among numerical simulations using MUSCL, S6WENO5 with lowdissipation and measurements. Fig. 9(a) is the result of Roe scheme with MUSCL interpolation, 9(b) is the results of S6WENO5 with low-dissipation and 9(c) is the experimental results. From the comparisons, although both numerical simulations are DDES, low-order upwind MUSCL scheme smoothes almost all the small scale
887
structures and only extreme large structures are reserved. The excited results can be obtained using S6WENO5 with ε=0.12. The spanwise vorticity can match the measurements much better than that of low-order scheme. Plenty of small scales of turbulence structures are captured although the vortices breakdowns a little downstream in the gap region.
(a)
(b)
(c) Fig. 10 Comparison of spanwise vorticity (a) Roe scheme with MUSCL interpolation; (b) S6WENO5 with low-dissiapation (c) Exp.
Figure 11 demonstrates four snapshots of iso-surface of Q, which is defined as -(Sij2-Wij2) and its value is 60. From these figures, very small scale of 3-D spanwise structures are presented colored by streamwise velocities. It visibly displays the capability of the simulations to resolve fine-grained turbulence (consistent with the grid used), and exhibits the complex although in general, similar vortical structures calculated by the different simulations. CONCLUSIONS Preliminary results about the flows are calculated using URANS around the rudimentary landing gear. At he same time, flows around the tandem cylinders are deeply investigated using DDES with different scheme and acceptable results, especially the instantaneous structures are obtained with high-order scheme with low-dissipation. a) Our in-house code (UNITS) is successfully applied to calculate the complex unsteady flows around the RLG configuration based on arbitrary multi-blocks of structured mesh. URANS can present reasonable pressure distribution on the RLG and it provides the initial flow-fields for the advanced turbulence modeling methods, such as DDES or iDDES in the future. b) DDES could be used to simulate the unsteady flow around the tandem cylinders, and spanwise length has
888
9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China
a great effect on the mean pressure coefficients. c) The spanwise length is also have a significant influence on Cp,rms and PSD on the cylinders surface. Larger spanwise length is more reasonable because Lz is equal to 14D in experiment. d) The order of scheme should be increased and the dissipation of scheme should be decreased when the spanwise correlation parameters and information, such as small vortex structures, Cp,rms, and so on, are hoped to calculate accurately. e) Transition trip on the front cylinder surface is applied in experiment but fully turbulence is assumed in our computations. Therefore, more attention on the fix-transition should be paid and more reasonable results are hoped to achieve.
Fig.11: Q=-60 iso-surface around tandem cylinders
ACKNOWLEDGEMENTS This project is supported by EU-FP7-ATAAC project and National Science Foundation of China under Contract 10932005. The author also thanks Prof. M. Strelets for his useful suggestion about the numerical scheme.
REFERENCES [1] P Spalart, M Shur, M Strelets, et al. Initial RANS and DDES of a rudimentary landing gear. 3rd symposium on RANS/LES hybrid methods, Gdansk, Poland, June, 2009. [2] L N Jenkins, et al. Characterization of unsteady flow structures around tandem cylinders for component interaction studies in airframe noise. AIAA paper, 20052812, 2005. [3] L N Jenkins, et al. Measurements of unsteady wake interference between tandem cylinders. AIAA paper, 20063202, 2006. [4] D P Lockard, et al. Tandem cylinder noise calculations. AIAA paper, 2007-3450, 2007. [5] M R Khorrami, et al. Unsteady flowfield around tandem cylinders as prototype component interaction in airframe noise. AIAA J., 2007,45(8). [6] D H Neuhart, et al. Measurements of the flowfield interaction between tandem cylinders. AIAA paper 20093275, 2009. [7] P R Spalart, W H Jou, M Strelets, et al. Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach. Advanced in DNS/LES, edited by C. Liu and Z. Liu, Greyden Press, Columbus, OH, 1997: 137–147; [8] Z X Xiao, et al. Study of delayed-detached eddy simulation with weakly nonlinear turbulence model. J. of Aircraft. 2006, 43(5): 1377-1385. [9] S Fu, Z X Xiao, et al. Simulation of wing-body junction flows with hybrid RANS/LES methods. Int. J. of Heat and Fluid Flow, 2007,28: 1379-1390. [10] Z X Xiao. Study of RANS/LES hybrid methods for aircraft design. Post-Doctor report, Tsinghua university, 2005. [11] T T Bui, et al. A Parallel, finite-volume algorithm for large-eddy simulation of turbulent flows. NASA/TM-1999206570, 1999.
883
2010, 22(5), supplement: 926-931 DOI: 10.1016/S1001-6058(10)60054-6
Calculations of massive separation around landing-gear-like geometries Zhi-xiang Xiao, Jian Liu, Song Fu * School of Aerospace Engineering, Tsinghua University, Beijing, China *Email: [email protected] ABSTRACT: The massive separating flows around landinggear-like configurations, such as Rudimentary landing gear (RLG) and tandem cylinders (TC), are calculated using unsteady Reynolds averaged Navier-Stokes (URANS) and delayed-detached-eddy simulation (DDES) based on k-ω-SST model. A number of numerical schemes and dissipation are applied in an effort to compare the averaged as well as the instantaneous flow-fields with the available measurements. It is shown that high-order and low-dissipation scheme is necessary to calculate the small scale structures. Furthermore, DDES delivered better results than URANS.
due to one region of the flow. Other reasons are the higher geometric complexity. The mechanism of the strong interactions among the different components is not clear.
KEY WORDS: massive separation, RLG, TC, DDES, numerical scheme and dissipation
However, it would be very difficult to distinguish the errors caused by the calculation itself and those caused by the acoustics components. It’s a great challenge to accurately calculate the massive separations around the LG and TCs.
1 INTRODUCTION In the flight phases of taking-off and approaching, landing gears (LG) are known to be one of the major generators of airframe noise. LG Noise reduction is one of the important “wars on noise”. Flows around LG are usually characterized by turbulent boundary layer separation, unsteady vortex shedding, and intensive unsteady inter-components interactions. The calculation of flow past LG is helpful to investigate and understand the turbulent flow-generated noise. The Boeing “Rudimentary Landing Gear” (RLG) configuration [1] has been selected as one of test cases in this article, because it is supported by an experimental project funded by Boeing and carried out at NAL (India). The RLG configuration has large axles with square cross-sections and tripping the wheel boundary layers, which ensures the flow are fully-turbulence. And the Reynolds number in experiment is more relevant to the realistic high Reynolds number. LG-noise CFD calculation remains very incomplete, and has few “success stories.” One reason is the absence of experimental data which isolate the noise
A canonical problem to develop modeling methods for multi-elements interactions corresponds to the tandem cylinder (TC) arrangement [2-6]. TCs with similar diameters can be found on a landing gear, such as the oleo and hoses.
If traditional URANS (unsteady Reynolds averaged Navier-Stokes) methods with turbulence models are applied to the massive separations, only large scales of structures can be calculated due to excessive dissipation, which prevents the formation of smallscale structures. LES (large eddy simulation) is a naturally accurate calculation method for separation, because the large scale of structures are resolved accurately, only very small-scale motions are modeled by sub-scale grid model. However, it’s time consuming to calculate the flow at high Reynolds number in industry applications using LES. Limited by computational resource, the combination of LES with RANS can achieve both reasonably well with high efficiency and numerical accuracy when calculateing the flow with massive separation. RANS/LES hybrid methods (originally proposed by Spalart, [7]), has widely used to calculate the unsteady and geometry-dependent separating flows. Such hybrid methods combine a high-efficiency turbulence model near the wall, where the flow is dominated by small scale motion with an LES-type treatment for the large scale motion in the flow region far away from
9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China
884
the wall. In fact, the convective scheme is also another very important factor, which will smooth the flow with few small scale structures if it is excessively dissipative. To capture higher frequency and small scale of motion, it’s necessary to decrease the numerical dissipation of scheme. In this article, RLG is preliminarily calculated using URANS and TCs is simulated using DDES[8-9]. The numerical simulations are compared with the available time-averaged and instantaneous flow-fields. 2 TURBULENCE MODEL AND DDES The fundamental turbulence model in this article is the two-equation k-ω-SST model, which combines the standard k-ε model with the k-ω model without the free-stream values dependence. The switching is realized by a flow dependent blending function. The SST model also limits the eddy viscosity by forcing the turbulent shear stress to be bounded by constant times the turbulent kinetic energy inside boundary layers (a realizability constraint). This modification improves the model’s performance on flows with strong adverse pressure gradients and separation. Due to its good numerical robustness and performance for most of flows, it’s widely used in industries. The detailed about SST model can be found in its original reference. It’s not listed here. To construct a DES-type hybrid method based on twoequation models, some transformation is adopted for the dissipation term in the turbulent kinetic energy transport equation. After introducing a length scale, the kinetic energy equation can be written as ∂ ( ρk ) ∂ρu j k ∂ ⎛ ∂k ⎞ * + = ⎜ ( μ + σ k μt ) ⎟ +τ ij Sij − β ρkωFDES (1) ∂t
∂x j
∂x j ⎜⎝
∂x j ⎟⎠
3 NUMERICAL SCHEMES 3.1 Spatial Scheme for Convection The computations are all based on a compressible solver using Roe flux-difference splitting scheme with 3rd-order monotone upstream centered scheme for conservation laws (MUSCL) and 5th order weighted essentially non-oscillatory (WENO) with R-S entropy fix in a cell-centered finite-volume formulation[10]. The flux of the convection is obtained in Roe as F
i+
1 2
=
1⎡ F ( q L ) + F ( q R ) − ε × A inv ( q R − q L ) ⎤ ⎦ 2⎣
(2)
the turbulence length scale Lt is defined as Lt = k1/2 (β*ω) ; CDES=0.78×F1+ 0.61×(1-F1); Δ is the grid scale defined as Δ=max(Δ ξ, Δη, Δζ); FSST can be taken as 0, F1 or F2, where F1 and F2 are the two blending functions in the SST model. If FDES=1, equation (1) is the same as the kinetic equation of SST model. If FSST = 0, the hybrid method reverts to a Strelets-type DES method. If FSST = F1 or F2, then, this hybrid approach is called the delayedDES method. Due to the numerical properties of F1 and F2, (1-FSST) approaches zero near the wall and the DDES will act in the RANS mode. At the same time, (1-FSST) becomes one out of the boundary layer and the DDES goes to the original Strelets’ DES model.
i+
1 2
(3)
where qL and qR are the original variable at the left and right side of the face ()i+1/2 using MUSCL or WENO interpolation. A inv is the matrix of Roe averaged. For the original Roe scheme, ε is taken as 1.
Furthermore, in order to isolate the errors of dispersion and dissipation, a higher-order symmetric total variation diminishing (STVD) scheme, which is the combination of central scheme with the numerical dissipation of the Roe, is used to solve the NavierStokes equations. In this article, S6-WENO5 is taken as the main scheme for the TCs case. Fi + 1 = 2
Fsymmtric, i+ 1 2 6th order symmetric scheme
where FDES is the hybrid function defined as ⎡ ⎤ L FDES = max ⎢(1 − FSST ) ⋅ t ; 1⎥ CDES Δ ⎦ ⎣
Therefore, the delayed-DES can ensure itself to act in the RANS mode near the wall without the effects on the locally clustered grid scales. In other words, this hybrid method can delay the switching from RANS to LES near the wall due to the grid scales, especially the locally refined grids in the streamwise and spanwise direction for the complex aircraft configurations. In this paper, FSST is taken as F2.
1 − ε × ⎡⎣ Ainv ( q R − q L )⎤⎦ 1 i+ 2 2
(4)
5th order WENO
In fact, when high-order scheme is applied in largereddy simulation, the convective scheme is over dissipative. Then, only low dissipation scheme can be taken as the appropriate when LES or DES is applied to calculate massive separations. The simplest method is adding a coefficient (ε)[11], which is less than 1 and greater than 0, to the dissipation terms in Equ.(4). 3.2 Temporal Scheme A modified fully implicit LU-SGS with Newton-like sub-iteration in pseudo time is taken as the time marching method when solving the mean flow equations and the turbulence model equations. Global non-dimensional time stepping is implemented to capture the unsteady properties of the separate flows. It indicates that the RANS method here is the unsteady-RANS. For the DDES, about 30 subiterations are taken to ensure the maximum residuals
9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China drop two orders. Implicit residual smoothing is employed to accelerate the convergence in the sub-iteration. The approach is parallelized using domain-decomposition and message-passing-interface strategies for the platform on PC clusters with 64-bit XEON-E5520.
885
nominal front-view area for RLG model. When the computations are running after 16,000 steps, the fluctuation amplitude of Cd is very small, and the URANS is stopped and DES is started. However, the results by DES are not presented this time because it is under-going.
4 RESULTS DISCUSSION 4.1 Rudimentary Landing Gear (RLG) Description: Unsteady flows past RLG, dominated by massive separation and interactioons among different components, are calculated using URANS based on SST model. In addition, the geometry of RLG is also very complex. Then, it’s very difficult to accurately calculate the unsteady flows and airframe noise using CFD and CAA methods. The computational grids contain 55 blocks and the overall cells are 10,772,708, shown in figure 1. The Reynolds number based on the wheel diameter (D, which is equal to 0.4064m) and reference velocity (U0 = 40 m/s) is equal to 106. The unsteady flows are calculated by solving URANS equations with SST turbulence model. Normalized time step Δt*=Δt×U/D =0.005, which is corresponding to 5.08×10-5 s for each step in physical time. Then, no instantaneous results, such as instantaneous pressure fluctuations, frequency on some samples, are presented using URANS.
Fig. 2 History of Cd
Figure 3 presents the comparisons on the timeaveraged surface pressure coefficients from two angles of view, where the contours have used the same legend as for the experimental measurement. The left figures are the measurements, while the right figures are our numerical results. The overall URANS results match the measurements very well, and local difference can be observed, especially on the surface of the streaimwise strut between the four wheels.
Fig. 1 Computational Grids
Boundary Condition: Uniform inflow is applied to the upstream boundary; Outflow is taken downstream boundary with zero pressure gradients. On the side boundary, the flow is assumed as symmetry. On the top and bottom boundaries, invicid wall is taken as the boundary conditions. No-slip conditions are applied on the wheels and struts wall. The flow is calculated using URANS with initial uniform flows. The history of streamwise force (Cx or Cd) is presented in Fig.2. The reference area here is taken as 1.06675 D2, which is sum of the
Fig. 3 Comparisons on Cp (Left: Exp; Right: URANS)
Figure 4 presents the time-averaged surface friction patterns in comparison with the experimental oil visualization. It is shown that the URANS computations have produced very similar mean flow features over the outward wheel surfaces, where the flow is attached and being visualized in the experiment. URANS results also present a small scope of second separation over the upstream wheel.
9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China
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shear layer instability, the interaction of unsteady wake of the front cylinder with the downstream one and unsteady massively separated flow between the cylinders and in the wake of the rear cylinder, etc. In this article, typically unsteady-RANS and advanced DDES with higher-order low-dissipation scheme are applied to calculate the unsteady pressure fields.
Fig. 4 Comparisons on surface streamlines (Left: Exp; Right: URANS)
The flow separation on the backsides of both the front and the rear wheels by computations differ from those of experimental visualization. Then it’s a great challenge to well calculate the massive separation behind the wheels and struts. Figure 5 illustrates the resolved vortices by different modeling approaches. One is SA-DDES by Spalart and another is our URANS based on SST. As expected, SST-DDES are able to resolve much richer vortex motions, whereas the URANS simulation has seemingly claimed rather “stiff” shear layers after the flow is detached from bluff-body surfaces. The main reason is the excessively viscous when URANS is applied to the massive separation flow. Then the future work is to calculate this flow using advanced RANS/LES hybrid methods with higher-order and low-dissipation scheme.
Figure 6 presents the sketch map of the tandem cylinder with the same diameter (D). The space (L) between the two cylinder centers is 3.7D. The velocity of freestream is 44m/s and the Reynolds number based on the cylinder diameter is 1.66×105, and the angle of attack is 0 degree. y
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Fig.6 Model of tandem cylinder
The mesh provided by NTS is shown in Fig.7. The total size of the grid in the XY plane is 82,000 cells. The grid size between the cylinders is almost isotropic (about 0.01D) to calculate the small scale vortices. The spanwise size of the domain for Roe scheme are 1D and 4D with Δz=0.033D and for STVD is 3D with Δz=0.02D. Then the overall grids are 2.78 million for 1D, 10.9 million for 4D and 12.4 million for 3D, respectively. At the time of submitting this paper, the DDES computation with low-dissipation and highorder scheme for the last grids is under-going and only some instantaneous results are presented here.
Fig.7 2-D grids of tandem cylinder Fig. 5 Resolved instantaneous vortex motions (Left: reference by DDES; Right: URANS)
4.2 Tandem Cylinders (TCs) The tandem-cylinder is a prototype for interaction problems commonly encountered in airframe noise configurations (e.g., the oleo and hoses on a landing gear). The flow has been studied in a series of experiments performed in NASA Langley Research Center[2,3]. Simulation of TCs can help testing the capability of turbulence modeling approaches, spatial and temporal methods to reproduce properly complex flow phenomena, such as the transition on the two cylinders, separation of turbulent boundary layer, free
Fig.8 comparisons on pressure coefficients on surface
Figure 8 presents the comparisons on the timeaveraged pressure coefficients over the surface using DDES with 3rd MUSCL interpolation. The symbols are the experimental results and dashed line represents the computational results of 1D and solid line represents the 4D numerical results. From the comparisons, we can find that the spanwise length has a significant influence on the pressure distribution on
9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China the surface. Longer spanwise length can obtained more reasonable results. Of course, to achieve the well matched results with the measurements, longer spanwise length should be applied. As we known, the flow is extremely unsteady. Then the pressure fluctuations on the surface or in the wake should be investigated to verify the calculation methods. The root mean square (RMS) of the pressure on the rear cylinder is presented in figure 9a. From this figure, we can see that the smaller spanwise calculate the Cp,rms much higher results than that of 4D and experiments. At the same time, we can find that the Cp,rms shows two peak values where θ are 45° (due to vortices shedding from the front cylinder) and 120° (due to the separation), respectively. Figure 9b presents the comparisons on frequency and amplitude of the pressure fluctuation for the sample where θ is equal to 45° on the rear cylinder surface. From this figure, the results of frequency, especially the amplitude of the power-spectral-density (PSD) for 4D spanwise length can match the measurements much better than that of 1D spanwise length. And the primary frequency is about 180Hz, which is corresponding to the vortices shedding frequency from the front cylinder and also is the frequency of impingement on the rear cylinder surface by the former-mentioned vortices.
(a) (b) Fig.9 Cp,rms and PSD at location of 45 degree on rear cylinder
As mentioned before, the dissipation of the numerical scheme has a significant influence of the information of small scale of structures. The high-frequency pressure fluctuations are easily cut off when large numerical dissipation schemes are used to calculate the unsteady flow. Sometime, the numerical dissipation can exceed the physical viscosity. Then, the advanced DDES combined with large dissipation scheme maybe can’t obtain acceptable results and their performance look like those of URANS. Figure 10 presents the comparisons on spanwise vorticity (with the same color legends) among numerical simulations using MUSCL, S6WENO5 with lowdissipation and measurements. Fig. 9(a) is the result of Roe scheme with MUSCL interpolation, 9(b) is the results of S6WENO5 with low-dissipation and 9(c) is the experimental results. From the comparisons, although both numerical simulations are DDES, low-order upwind MUSCL scheme smoothes almost all the small scale
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structures and only extreme large structures are reserved. The excited results can be obtained using S6WENO5 with ε=0.12. The spanwise vorticity can match the measurements much better than that of low-order scheme. Plenty of small scales of turbulence structures are captured although the vortices breakdowns a little downstream in the gap region.
(a)
(b)
(c) Fig. 10 Comparison of spanwise vorticity (a) Roe scheme with MUSCL interpolation; (b) S6WENO5 with low-dissiapation (c) Exp.
Figure 11 demonstrates four snapshots of iso-surface of Q, which is defined as -(Sij2-Wij2) and its value is 60. From these figures, very small scale of 3-D spanwise structures are presented colored by streamwise velocities. It visibly displays the capability of the simulations to resolve fine-grained turbulence (consistent with the grid used), and exhibits the complex although in general, similar vortical structures calculated by the different simulations. CONCLUSIONS Preliminary results about the flows are calculated using URANS around the rudimentary landing gear. At he same time, flows around the tandem cylinders are deeply investigated using DDES with different scheme and acceptable results, especially the instantaneous structures are obtained with high-order scheme with low-dissipation. a) Our in-house code (UNITS) is successfully applied to calculate the complex unsteady flows around the RLG configuration based on arbitrary multi-blocks of structured mesh. URANS can present reasonable pressure distribution on the RLG and it provides the initial flow-fields for the advanced turbulence modeling methods, such as DDES or iDDES in the future. b) DDES could be used to simulate the unsteady flow around the tandem cylinders, and spanwise length has
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9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China
a great effect on the mean pressure coefficients. c) The spanwise length is also have a significant influence on Cp,rms and PSD on the cylinders surface. Larger spanwise length is more reasonable because Lz is equal to 14D in experiment. d) The order of scheme should be increased and the dissipation of scheme should be decreased when the spanwise correlation parameters and information, such as small vortex structures, Cp,rms, and so on, are hoped to calculate accurately. e) Transition trip on the front cylinder surface is applied in experiment but fully turbulence is assumed in our computations. Therefore, more attention on the fix-transition should be paid and more reasonable results are hoped to achieve.
Fig.11: Q=-60 iso-surface around tandem cylinders
ACKNOWLEDGEMENTS This project is supported by EU-FP7-ATAAC project and National Science Foundation of China under Contract 10932005. The author also thanks Prof. M. Strelets for his useful suggestion about the numerical scheme.
REFERENCES [1] P Spalart, M Shur, M Strelets, et al. Initial RANS and DDES of a rudimentary landing gear. 3rd symposium on RANS/LES hybrid methods, Gdansk, Poland, June, 2009. [2] L N Jenkins, et al. Characterization of unsteady flow structures around tandem cylinders for component interaction studies in airframe noise. AIAA paper, 20052812, 2005. [3] L N Jenkins, et al. Measurements of unsteady wake interference between tandem cylinders. AIAA paper, 20063202, 2006. [4] D P Lockard, et al. Tandem cylinder noise calculations. AIAA paper, 2007-3450, 2007. [5] M R Khorrami, et al. Unsteady flowfield around tandem cylinders as prototype component interaction in airframe noise. AIAA J., 2007,45(8). [6] D H Neuhart, et al. Measurements of the flowfield interaction between tandem cylinders. AIAA paper 20093275, 2009. [7] P R Spalart, W H Jou, M Strelets, et al. Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach. Advanced in DNS/LES, edited by C. Liu and Z. Liu, Greyden Press, Columbus, OH, 1997: 137–147; [8] Z X Xiao, et al. Study of delayed-detached eddy simulation with weakly nonlinear turbulence model. J. of Aircraft. 2006, 43(5): 1377-1385. [9] S Fu, Z X Xiao, et al. Simulation of wing-body junction flows with hybrid RANS/LES methods. Int. J. of Heat and Fluid Flow, 2007,28: 1379-1390. [10] Z X Xiao. Study of RANS/LES hybrid methods for aircraft design. Post-Doctor report, Tsinghua university, 2005. [11] T T Bui, et al. A Parallel, finite-volume algorithm for large-eddy simulation of turbulent flows. NASA/TM-1999206570, 1999.