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A COMPUTATIONAL STUDY OF SUBSONIC FLOWS OVER A MEDIUM RANGE CARGO AIRCRAFT
Murat Uygun *
* Turkish Air Force Academy Istanbul, Turkey
Ismail H. Tuncer †
† Middle East Technical University Ankara, Turkey
Inviscid and viscous flows over CN-235 cargo aircraft are computed at cruise conditions and at angles of attack ranging from 0 deg to 5 deg using CFD-FASTRAN Navier-Stokes solver. Computations are performed on a Pentium III based PC with a 1Gb memory. Grid sensitivity and verification studies are performed on clean-wing and wing-nacelle configurations to determine the effects of the surface grid density, the size of the computational domain and available numerical schemes. Inviscid and viscous flows with available turbulence models over the full aircraft are computed at a range of incidence angles using multi-block structured grids. Computed aerodynamic forces and moments are also compared with semi-empirical and numerical data available in the literature. It is shown that viscous flow solutions provide valuable aerodynamic data, which may be used in determining and/or altering the aerodynamic characteristics of the aircraft.
Introduction Due to rapid decrease in the cost of computations,
Computational Fluid Dynamics (CFD) methods are becoming economically cheaper than measuring the aerodynamic characteristics of an aircraft in a wind tunnel. Flow solutions with CFD computations can give detailed flow field information that may be too expensive to obtain in a wind tunnel. Consequently, CFD methods are becoming more popular and they are being used for complimenting the experimental studies and decreasing the number of the wind tunnel measurements.
CN-235 aircraft (Fig. 1 and Table 1) is a twin turboprop tactical transport aircraft having a maximum cruising speed of 455 km/h.1 It was initially designed and built by the cooperation between CASA (Spain) and IPTN (Indonesia). Later CASA developed its own * Research Engineer, Department of Aeronautical Engineering Email: [email protected] † Assoc. Prof., Department of Aerospace Engineering, AIAA Member Email: [email protected]
Fig. 1 Medium range cargo aircraft (CN-235)
Table 1 Geometrical description of the aircraft studied.
Parameter Value Overall length 21.353 m Full span 25.81 m Root chord 3.0 m Wing mean chord 2.62 m Wing gross area 59.1 m Wing incidence angle 3 deg Wing airfoil NACA653-218 Horizontal and Vertical tail airfoils NACA641-012
versions. CN-235 has been built by TAI (Turkey) under production license by CASA for several years.
Earlier, Karaagac2 computed the aerodynamic characteristics of the CN-235 aircraft using Advanced Aircraft Analysis (AAA) software (Version 2.2).3 AAA is a widely used software for preliminary design, stability and control analysis of a new and existing airplane. Design methodology used in AAA is based on empirical methods for airplane design, airplane flight dynamics & automatic flight controls and airplane aerodynamics and performance. Bahar obtained inviscid flow solutions for CN-235 aircraft using CFD-FASTRAN flow solver4 (Version 2.2) with unstructured grids.5 Kurtulus6 recently computed viscous and inviscid subsonic flows over the same aircraft at cruise, landing and take off conditions using
21st Applied Aerodynamics Conference23-26 June 2003, Orlando, Florida
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the VSAERO software (Version 6.2).7 VSAERO solver is used for calculating the nonlinear aerodynamic characteristics of arbitrary configurations in subsonic flow. The effects of viscosity are treated in an iterative loop coupling potential flow and integral boundary layer calculations. Kurtulus compared the potential flow solutions with the Euler solutions of Bahar and concluded that inviscid flow solutions computed by VSAERO and CFD-FASTRAN are similar.
In this study, inviscid and viscous, compressible flows over the CN-235 aircraft were computed at cruise conditions using the CFD-FASTRAN (Version 2.2) Navier-Stokes solver. Grid sensitivity of the flow solver and the capabilities of the CFD-GEOM grid generation software8 (Version 5) in creating various types of grids were determined on clean-wing and wing-nacelle configurations (Fig. 2). Inviscid and viscous flows over the full aircraft configuration were computed for a range of incidence angles using multi-block structured grids. Computed results are also compared with the numerical data available in the literature.
Fig. 2 Wing-nacelle and clean-wing configurations
Numerical Methods
In this study, CFD-FASTRAN Navier-Stokes solver (Version 2.2) and the CFD-GEOM grid generation software (Version 5) are used to obtain inviscid and viscous flow solutions on unstructured, structured, hybrid and overset grid systems.
Flow Solver
CFD-FASTRAN is a Navier-Stokes solver based on a cell-centered finite volume discretization. Flow fields may be discretized with structured, unstructured or hybrid grids. Inviscid fluxes may be computed using Roe's flux difference splitting scheme9 and Van Leer's flux vector splitting scheme.10 Both schemes are spatially first order accurate unless higher order flux limiters are used. For Min-mod and Van leer limiter the accuracy is 2nd order. For Osher-Chakravarthy limiter the accuracy is up to third order. CFD-FASTRAN has an algebraic (Baldwin-Lomax11) and two equation (k −ε model12 and k −ω model13,14) turbulence models.
Grid Generator
Computational grids are generated using the CFD-GEOM grid generation software. CFD-GEOM can generate the multi-block structured and multi-domain unstructured grids with point match interface, overset (chimera) grids and hybrid grids (structured-
unstructured grid combinations). For structured grids, grid orthogonality at the wall boundaries may be enforced, and grid quality may be improved with elliptic smoothers.
In generating unstructured grids, CFD-GEOM provides an automatic and controllable surface triangulation and tetrahedral volume grid generation algorithms using the advancing front technique. However, it does not provide a user control over the grid spacing in the direction normal to the wall boundaries. The grid resolution in the boundary layers may therefore be poor, and unstructured grids may not be employed in viscous flow computations. Instead multi-block structured grids with the point match interfaces at the block boundaries may be used for viscous flow solutions. Overset (overlapping) grids also provide simplicity in grid generation around complex geometries.15,16 However, similar grid densities have to be provided at the overlapping intergrid boundaries. In addition, the Chimera hole-cutting process requires a large computer memory.
In this study, the origin of the coordinate system is located at the nose of the aircraft, on the symmetry axis. The x coordinate extends in the chordwise direction, and the z coordinate extends along the spanwise direction. Wing-nacelle configuration is obtained by extracting the wing and the nacelle from the full aircraft model. The clean-wing configuration is then obtained by excluding the nacelle.
Boundary Conditions
The computational grids were generated for the half configuration due to symmetric geometry and symmetry boundary conditions were implemented to the symmetry planes. Wall boundary conditions were implemented to the solid boundaries. Inflow/outflow boundary conditions that are based on the Riemann invariants corresponding to the incoming and outgoing waves were implemented to the rest of the boundaries.
Results and Discussion
In this study, inviscid and viscous flows were computed at the cruise condition of the aircraft (at Mach number 0.39 and at an altitude of 15000 feet). The flow cases studied are summarized in Table 2.
Inviscid flows were computed using structured, unstructured and hybrid grids. For inviscid flows, only lift forces were evaluated. Viscous flows were computed only on structured grids, and flows were assumed to be fully turbulent. For viscous flows, lift and drag forces were evaluated. Inviscid fluxes were computed using Roe flux differencing scheme and Van-Leer flux vector splitting scheme with fully implicit time integration scheme and using both first and third order spatial accuracy (with osher-chakravarty limiter). First order accurate solutions were used as an initial
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guess to high order accurate computations. CFL number was varied between 50-100, except for the viscous computations around the aircraft configuration it was set to 5. Computations were performed in double precision on a PC with a Pentium III 450 MHz processor and 1 GB memory running Windows NT.
The sensitivity of inviscid flow solutions to the surface grid density and the size of the computational domain was established by computing the inviscid flows over the clean-wing model using structured and unstructured grids.
The effects of the surface grid density were studied using six different surface grids having 17774, 25388 and 32876 triangles for unstructured grids and 5200, 8330 and 12150 quadrilaterals for structured grids (Fig. 3). They are denoted as coarse, fine and finer grids depending on the number of the surface grid elements. The effects of the size of the computational domain were studied using 6 different computational boundaries located at 6, 12 and 15 chord lengths away from the clean-wing for unstructured grids and 4, 6 and 10 chord lengths away from the clean-wing for structured grids in both x and y directions (Fig. 4). They are denoted as small, medium and large boundaries depending on the distance between the computational boundaries and clean-wing. In spanwise direction, for both structured and unstructured grids, computational boundary is located at 2.5 wing span away from the wing root.
Using triangular and quadrilateral surface grids with medium outer boundary, six volume grids having 359916, 526090, 798257 tetrahedrals and 399620, 529830 and 702844 hexahedrals were obtained to determine the surface grid dependency. Computed results are given in Table 3. The lift forces computed using the fine and the finer triangular surface grids agree well with each other. Similarly, the fine and the finer quadrilateral surface grids yielded similar lift forces. It was concluded that the finer triangular and quadrilateral surface grids were the most appropriate surface representation for the optimum memory usage, CPU time and solution quality. Four additional volume
grids having, 418679 and 614258 tetrahedrals and 292030 and 882180 hexahedrals were obtained using finer surface grids with small and large boundaries to determine the size of the computational domain. Since the results computed using medium and large outer boundaries agree well, it was concluded that the medium outer boundary location provides a solution independent of outer boundary location.
It was observed that in the unstructured grid solutions the lift is reduced by about 7% compared to structured grid ones. Also the pressure distribution at the upper surface of the wing is different from structured grid computations (Fig. 5). This difference is attributed to the difference in the leading edge resolution of structured and unstructured grids. Computed results indicated that unstructured grids might be loose in the resolution of surfaces with high curvature like leading edges depending on the number of triangular surface grids clustered around such regions.
Fig. 6 and 7 show the convergence histories of the inviscid flows computed using structured and unstructured grids. The L2-norm residual decreased 3 orders of magnitude for structured grids whereas it decreased only 2 orders of magnitudes for unstructured grids. Inviscid flow studies indicated that the structured grids are about 60% more efficient in CPU time and 43% more efficient in memory usage in comparison to unstructured grids.
The sensitivity of the solutions to the turbulence models was established by computing viscous flows over the clean-wing configuration using the Baldwin-Lomax, k −ω and k −ε models. The optimum grid distributions on the wing surface and the location of the outer boundary were established in the earlier studies for grid independent solutions.17,5 Viscous grid spacing at the wall (dn) was computed using
=
+
2Re
1
fCydn
(1)
where Cf is the skin friction coefficient, which is empirically approximated by
( )Re006.0ln455.0
2=fC (2)
11 =+y was used for Baldwin-Lomax and k −ω models, and 401 =+y for k −ε model, which implements the law of the wall explicitly. 147x167x30 size grid was used, and a typical solution required 380 Mbytes of memory and took about 50 hours of CPU time.
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Table 3 The sensitivity study of inviscid flows to the surface grid density and the size of the computational domain
Pressure distributions computed with each of the turbulence models are shown in Fig. 8. As shown, all the models predict about the same pressure variation as the inviscid flows, except k −ω model slightly differs from all the others near the upper wing tip regions and at the stagnation point. The computed lift forces are given in Table 4. k −ω model yields a lift force about 2.2% smaller than the others.
The spanwise and chordwise skin friction distributions as a function of +
1y values are given in Fig. 9. Although all the models predict about the same trend, the magnitudes differ as much as 25%. The solution with the k −ε model for 2001 =+y indicates the presence of a recirculating region at about 20% chord location. The total drag force and viscous drag component for the different +
1y locations are given in Table 5. The drag force based on the Baldwin-Lomax model is about 16% greater that of the k −ε model, and 17% smaller than that of the k −ω model. It is also observed that the drag forces based on each turbulence model decrease as much as 15% as the +
1y values are increased beyond the proper values.
Table 4 Computed Forces
Turbulence Model Lift, N B-L Model ( +
1y =1) 92359 k −ε Model ( +
1y =40) 92380 k −ω Model ( +
1y =1) 90387 Inviscid Flow 93381
The variation of lift, drag and pitching moment coefficients with respect to angle of attack, and the drag polar are given in Fig. 10. The lift and the pitching moment coefficients based on all the turbulence models agree reasonable well among themselves and with the inviscid flow prediction. Yet, as the angle of attack increases, the k −ω model predicts slightly lower values. The pitching moment values at the mean quarter chord increase slightly with the angle of attack, which indicates that the aerodynamic center is slightly off of the quarter chord location. Similarly, large differences in drag predictions are observed at α =5 deg as well.
Fig. 11 shows the velocity vectors around the trailing edge at 95% wingspan for the inviscid and viscous flows computed with all the turbulence models. The flow based on the k −ε model has a shallow flow separation while the other models predict fully attached flow.
Table 5 +1y sensitivity study of turbulence models
Inviscid Flow Solutions over Wing-Nacelle Configuration
In the wing-nacelle configuration, a multi-block structured grid could not be generated due to geometric constraints. Instead, an overset (Chimera) grid was considered (Fig. 12). However, CFD-FASTRAN does not allow wall-to-wall overlap in the overset grid methodology. Therefore viscous flow analysis around wing-nacelle model could not be performed.
For the inviscid flow solutions over the wing-nacelle configuration, a hybrid grid system with an unstructured subgrid around the nacelle and a structured subgrid for the rest of the domain is considered (Fig. 12). The final volume grid contains one unstructured domain with 143497 tetrahedrals and 20 structured blocks with 843904 hexahedral. Computations required 342 Mbytes of memory and 62 hours of CPU time.
The computed pressure distribution on the wing-nacelle configuration is given in Fig. 13. The chordwise pressure distributions at 35% wingspan, which corresponds to the symmetry axis of the nacelle, are compared in Fig. 14. Two pressure spikes observed in the pressure distributions correspond to the points where the nacelle geometry is attached to the upper and lower wing surfaces. The total lift on the wing-nacelle configuration is computed to be 86315 N, which is
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about 8% less than that of the clean wing configuration (93381 N).
Flow Solutions over Medium Range Cargo Aircraft
Since a viscous grid could not be created around the wing-nacelle configuration, the nacelle was removed from the aircraft configuration in the analysis of the full aircraft. A multi-block structured grid was then created around the aircraft (Fig. 15). It consisted of 122 blocks. Degenerate blocks where one of the block edges collapses to a point were avoided to improve the convergence of the solution. The grid generation process took about 4 man weeks. The same multi-block topology was used to generate both inviscid and viscous grids by adjusting the grid spacing normal to the wall. The inviscid and viscous grids had about 1.8 and 2.3 million hexahedrals, respectively.
First, inviscid flows are computed at 0, 3 and 5 deg angles of attack. The solution required 686 Mbytes of RAM and about 200 hours of CPU time. Fig. 16 shows the convergence history of the inviscid solution and the lift history at α = 0 deg. The computed pressure distribution is given in Fig. 17. The lift computed on the full aircraft is found to be about 35% less than that on the clean-wing at α = 0 deg and 6% less at α = 5 deg, as shown in Fig. 18. The reduced lift is attributed to the 10% reduction on the wingspan of the aircraft compared to the wingspan of the clean-wing model, and the aircraft fuselage interference. The negative incidence angle of the horizontal tail also adds to the loss of lift. As shown in Fig. 18, Bahar’s inviscid computations on unstructured grids yields smaller lift values than the present computations on structured grids, which may be attributed to the finer resolution of the flow field with structured grids.
For the viscous flow solutions with the k −ε turbulence model, the grid blocks were refined at the walls to provide a sufficient grid resolution in boundary layers. In the computations, the double precision version of the flow solver was used, and the CFL number was set in the 1-5 range. The computations took about 300 hours of CPU time.
The computed pressure distribution at α =0 deg is given in Fig. 19. The integrated aerodynamic loads are given in Fig. 20. It is observed that in the viscous flow solutions the lift is reduced by about 3.6% at α =3 deg and 7% at α =5 deg, which may be attributed to the boundary layer growth. The viscous drag force is about 18% higher than that of the inviscid flow solution. The aerodynamic loads computed by AAA and VSAERO underpredict those of the CFD-FASTRAN solutions, but they all show the same trend. The predictions of the pitching moment at the mean quarter chord agree well.
The particle traces at the tip region do not show a significant variation than that of the inviscid flow
(Figure 21). It again suggests that the tip vortex is pressure driven, and can be captured in inviscid flows.
Conclusions
In this study, low speed, viscous flows over the CN-235 cargo aircraft are computed using CFD-FASTRAN Navier-Stokes flow solver. Preliminary studies are performed on the clean-wing and the wing-nacelle configurations. Inviscid and viscous flows are then computed on the aircraft configuration at the cruise condition. Although the CFD-FASTRAN has proven to be a viable software in the computation of flows over complex aircraft configurations, the unstructured grid methodology in CFD-GEOM lacks the effective user level controls to provide sufficient grid resolution in the boundary layers. Viscous flows are only studied using the multi-block structured grids, which are highly time consuming to produce over complex aircraft configurations. The computed lift and moment coefficients based on viscous flows agree well with the inviscid flow predictions as well as the panel code and the semi-empirical predictions. Drag values based on the available turbulence models yielded slightly different values, which need to be compared to the experimental data for further assessment. It is shown that viscous flow analysis for a full aircraft configuration is now feasible on today's powerful PCs.
Acknowledgement
The authors gratefully acknowledge the support provided by ASELSAN Inc. Ankara, Turkey.
References 1Jane's All the World's Aircraft, Jane's Information
Group, Sentinel House, 163 Brighton Road, Coulsdon, Surrey CR5 2NH, UK, ISBN for 1991-1992 issue: 07106 0965 5.
2Karaagac C., “The Aerodynamics, Flight Mechanics and Performance Predictions for a Medium Range Cargo Aircraft”, M.S. Thesis, Middle East Technical University, Turkey, 1998.
3Advanced Aircraft Analysis Software (AAA) by Design, Analysis and Research Corporation (DAR Corporation).
4CFD-FASTRAN User Manual, Version 2.2, CFD Research Corporation, 1998.
5Bahar, C., “Euler Solutions for a Medium Range Cargo Aircraft”, M.S. Thesis, Middle East Technical University, Turkey, 2001.
6Kurtulus D.F. “Aerodynamic Analysis of a Medium Range Cargo Aircraft Using a Panel Method”, M.S. Thesis, Middle East Technical University, Turkey, 2002.
7Nathman, J.K., VSAERO, “A computer Program for calculating the Nonlinear Characteristics of arbitrary Configurations, User’s Manual, 1999.
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8CFD-GEOM User Manual, Version 5, CFD Research Corporation, 1998.
9P.L. Roe, "Approximate Reimann Solvers, Parameter Vectors and Difference Schemes", Journal of Computational Physics, Vol. 43, 1981, pp. 357- 372.
10B. Van Leer, "Flux-Vector Splitting for the Euler Equations.", Lecture Notes in Physics, Vol. 170, 1982, pp.507-512.
11Baldwin, B.S. and H. Lomax, "Thin Layer Approximation and Algebraic Model for Separated Flows", AIAA-78-257, 1978.
12Speziale, C., Abi, R., and Anderson E., “A Critical Evaluation of Two-Equation Models for Near-Wall Turbulence”, AIAA paper 90-1481, 1990.
13Wilcox, D. C., "A Half Century Historical Review of the k − ω Model", AIAA- 91-0615, 1991.
14Wilcox, D.C., "Turbulence Modeling for CFD", DCW Industries, 1998.
15Benek, J.A., Buning, P.G., and Steger, J.L., “A 3D Chimera Grid Embedding Technique”, AIAA Paper 85-1523, July 1985.
16Tuncer I.H., “A 2-D Navier-Stokes Solution Method with Overset Grids”, American Society of Mechanical Engineers, Paper 96-GT-400, June 1996.
17Uygun, M., “A Computational Study of Subsonic Flows over A Medium Range Cargo Aircraft”, M.S. Thesis, Middle East Technical University, Turkey, 2002.
Fig. 3 Triangular and quadrilateral surface grids on the clean-wing configuration
Fig. 4 Computational boundaries around the clean-wing configuration for unstructured and structured grids
x/c
Cp
0 0.25 0.5 0.75 1
-1
-0.5
0
0.5
1
Structured gridUnstructured grid
Fig. 5 Pressure distributions at 75% span on the clean-wing configuration, P=57207 Pa, M=0.39, α=3°
N iter
Lo
g(
L2
Nor
mof
∆ρ)
0 500 100010-3
10-2
10-1
100
101
High order soln.
Fig. 6 Convergence history of inviscid flow solution around clean-wing configuration with structured grid
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N iter
Log
(L2
No
rmof
∆ρ)
0 500 1000 1500
10-1
100
101
High order soln.
Fig. 7 Convergence history of inviscid flow solution around clean-wing configuration with unstructured grid
x/c
Cp
0 0.25 0.5 0.75 1
-1
-0.5
0
0.5
1
1.5
Baldwin-Lomaxk-εk-ωInviscid
z/b
Cp
0 0.2 0.4 0.6 0.8 1
-1
-0.5
0
0.5
Baldwin-Lomax
k-εk-ωInviscid
Fig. 8 Pressure distribution along the chord at the 75% span location and along the span at the midchord, P=57207 Pa, M=0.39, α=3°
z/b
Cf
0 0.25 0.5 0.75 10
0.002
0.004
0.006
0.008
0.01
y1+ = 1
y1+ = 5
Baldwin-Lomax Model
z/b
Cf
0 0.25 0.5 0.75 10
0.002
0.004
0.006
0.008
0.01
y1+ = 40
y1+ = 100
y1+ = 200
k-ε Model
z/b
Cf
0 0.25 0.5 0.75 10
0.002
0.004
0.006
0.008
0.01
y1+ = 1
y1+ = 5
k-ω Model
x/c
Cf
0 0.25 0.5 0.75 10
0.003
0.006
0.009
0.012
y1+ = 1
y1+ = 5
Baldwin-Lomax Model
x/c
Cf
0 0.25 0.5 0.75 10
0.003
0.006
0.009
0.012
y1+ = 40
y1+ = 100
y1+ = 200
k-ε Model
x/c
Cf
0 0.25 0.5 0.75 10
0.003
0.006
0.009
0.012
y1+ = 1
y1+ = 5
k-ω Model
Fig. 9 Skin friction distribution along the span at the midchord and along the chord at the 75% span location, P=57207 Pa, M=0.39, α=3°
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Aoa (degree)
CL
2 3 4 5 6 7 8 90.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Baldwin-Lomax modelk-ε modelk-ω modelInviscid
Aoa (degree)
CD
2 3 4 5 6 7 8 90
0.01
0.02
0.03
0.04
0.05
Baldwin-Lomax modelk-ε modelk-ω modelInviscid
Aoa (degree)
CM
c/4
2 3 4 5 6 7 8 90
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Baldwin-Lomax modelk-e modelk-w modelInviscid
CL
CD
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0.055
Baldwin-Lomax modelk-ε modelk-ω modelInviscid
Fig. 10 Variation of the aerodynamic forces and moments for the clean-wing configuration. Inviscid flow Viscous flow (Baldwin-Lomax model)
Fig. 17 Inviscid pressure distribution over the aircraft configuration, Fig. 18 Lift forces for inviscid flows P=57207 Pa, M=0.39, α=0° over the aircraft and the clean-wing configurations at α=0°, 3°and 5°
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