Unstructured mesh drag prediction based on drag decomposition

UNSTRUCTURED MESH DRAG PREDICTION BASED ON

DRAG DECOMPOSITION

Wataru YAMAZAKI*, Kisa MATSUSHIMA* and Kazuhiro NAKAHASHI* *Dept. of Aerospace Engineering, Tohoku University,

Aramaki-Aza-Aoba 6-6-01, Aoba-Ward, Sendai, 980-8579, Japan e-mail: yamazaki@ad.mech.tohoku.ac.jp

Web page: http://www.ad.mech.tohoku.ac.jp/yamazaki

Key words: Unstructured Mesh CFD, Drag Prediction and Decomposition, Spurious Drag,

Pure Drag, Mesh Resolution, U-MUSCL

Abstract. Unstructured mesh drag prediction is discussed using a drag decomposition

method. This method decomposes total drag into wave, profile, induced and spurious drag components, the latter resulting from numerical diffusion and errors. Computational results show that the method reliably predicts drag and is capable of meaningful drag decomposition. The accuracy of drag prediction is increased by eliminating the spurious drag component from the total drag. It is also confirmed that the physical drag components are almost independent of the mesh resolution or scheme modification.

NOMENCLATURE

a = sonic speed

Body = aircraft surface

CD = drag coefficient

CL = lift coefficient

Cp = pressure coefficient

D = drag force

(∆ ,s∆H)

Fr = entropy & enthalpy drag seed vector ind

Fr = induced drag seed vector

M = Mach number

(

nx,ny,nz

)

=

nr = outward unit normal vector to a surface

P = pressure

Q = variables

R = gas constant

ij

rr = vector between node i and j

∞

S = closed boundary surface of V

(

ux,uy,uz

)

=

ur = velocity vector

V = flow field around aircraft

∆ = perturbation term H

∆ = stagnation enthalpy variation

s

∆ = entropy variation

γ = specific heat ratio

κ = U-MUSCL parameter

l

µ = laminar viscosity coefficient t

µ = eddy viscosity coefficient

ρ = density

(

)

T z y x τ τ τ τrr = r ,r ,r = stress tensor Ψ = limiter Subscripts

▒_∞ = free stream value

▒_{x ,,yz} = orthogonal coordinate system with x axis points to the free stream flow direction

1 INTRODUCTION

The lift and drag of an aircraft in the cruising condition are known to be the most important parameters affecting its aerodynamic performance. Recently, owing to advances in numerical schemes and the rapid growth of computing power, Computational Fluid Dynamics (CFD) has achieved significant progress. However, the accurate drag prediction in CFD is still a major challenge today, as was pointed out at the meeting of AIAA Drag Prediction Workshop II1 in 20032. In fact, drag prediction accuracy within 1 count (1x10-4, about 0.4% in total drag of a typical transonic aircraft) has still not been achieved. In particular, the scatter band of drag prediction is bigger in unstructured mesh computation than in structured mesh computation because of the larger numerical diffusion. Therefore, various methods for more accurate drag prediction are being investigated.

Traditionally, surface integration of the pressure and stress tensor on the surface of the aircraft body, which is called ‘Surface Integration’ or ‘Near-Field Method’, is used for drag prediction in CFD computations. However, it has been pointed out that the total drag computed by the near-field method includes inaccuracies relating to numerical diffusion and errors, and that such inaccuracies can not be isolated from the total drag.

Other advantages of the mid-field method are that it enables the drag to be decomposed and visualized. By using the mid-field method, the total drag can be decomposed into three physical components of wave, profile and induced drag, and one spurious drag component. In Fig.1, these drag components and the classification are summarized. (Note that in this paper, profile drag is defined as a drag component based on the entropy production due to the effect of the boundary layer and wake.) Moreover, the drag strength and the generated positions can be visualized in the flow field because the integrand of the volume integral formula indicates the drag production rate per unit volume. Recently, aerodynamic shape design and optimizations using CFD are widely conducted, and these require detailed analysis of the drag reduction level, mechanisms and reliability. The ability to decompose and visualize drag would be most useful for such investigations.

In this paper, therefore, the mid-field drag decomposition method is applied to unstructured mesh CFD results and its capability is analyzed.

Fig.1: Drag Components and the Classification 2 DRAG DECOMPOSITION METHODS

This chapter outlines the concept and computational method of the three drag prediction methods.

2.1 Near-field method

In the near-field method, the drag force is computed as follows:

[

P P n

]

ds D Body x x

∫∫

− − + ⋅ = ( ∞) τr nr ₍₁₎

The integral area ‘ Body ’ indicates the surface of the aircraft. The first and second terms correspond to the pressure drag component and skin friction drag component, respectively.

2.2 Far-field method

(

)(

)

_∫∫

(

)

_∫∫

(

)

∫∫

− ⋅ − − + ⋅ − = ∞ ∞ S S S n n u ds P P n ds ds u u D ρ x r r x τrx r ₍₂₎

The integral area ‘S’ indicates an arbitrary closed surface around the aircraft. It is known that Eq.(2) can be transformed as follows by using the small perturbation approximation3:

(

)

( )

∫∫

∆ −

∫∫

∆ +

∫∫

⋅ + ∆ = _{∞ ∞} WA WA ind WA O ds Hds ds R s P D _{ρ F}r _nr 2

(

) (

)

(

)

[

]

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ _{+ − − ∆ − ∆ − ∆} = ∞ y z ∞ x ∞ y x ∞ z x ind u u M u ρ u u ρ u u ρ , , 1 2 2 2 2 2 Fr (3)

The integral area ‘WA’ indicates a wake plane normal to the free stream flow direction, as shown schematically in Fig.2. The first term of Eq.(3) corresponds to entropy drag which includes the wave, profile and spurious drag components. The second term including ∆ can H be neglected in cases where external work is not supplied in flow. The third term including

ind

Fr originates in the vorticity, which corresponds to induced drag.

Fig.2: Schematic Sketch for Far- & Mid-Field Method 2.3 Mid-field method

As mentioned in chapter 1, the mid-field method is derived from the far-field method by applying the divergence theorem, also known as Gauss’ theorem. First, the concept is explained using the entropy and enthalpy term. By using the divergence theorem, the entropy and enthalpy term of the far-field method can be transformed as follows:

(∆ ∆ ) =

∫∫

(∆ ∆ )⋅ ≅

∫∫

(∆ ∆ )⋅ =

∫∫∫

∇⋅ (∆ ∆ ) ∞ V S F n F n F ds ds dv D _{s H s H} WA H s H s, , , , r r r r r (4)

surface of V. In other words, S_∞consists of the wake plane of ‘WA’ and the far-field surfaces of the upstream/lateral regions. Fr_{(∆ ,s∆H)} is the entropy & enthalpy drag seed vector, which is defined from Ref.7 as follows:

( ) ( )( )

[

]

(

)

∞ ∞ ∆ − ∞ ∞ ∆ ∆ − − − − ∆ + = ∆ ∆ − = u M e u H u u u R s H s 2 1 2 , 1 1 2 2 1

γ

ρ

γ γ u Fr r (5) u

∆ of Eq.(5) can be expanded in Taylor’s series as follows:

(

)

(

)

(

)

(

)

(

)

(

2

) ( )

3 2 2 2 2 2 1 2 2 1 ∆ + ∆ ∆ + ∆ + ∆ + ∆ + ∆ = ∆ ∞ ∞ ∞ ∞ O u H R s f u H f u H f R s f R s f u u sH H H s s (6) Here, 2 1 1 ∞ − = M f_s γ ,

(

)

4 2 2 2 2 1 1 ∞ ∞ − + − = M M f_s γ γ , ……….. (7)

Now, calculate the first-order term of entropy variation as follows:

u Fr

ρ

r

γ

R s M u s ∆ = ∞ ∞ ∆1 2 (8)

Eq.(8) is the Oswatitsch formula itself. Moreover, using the small perturbation approximation, Eq.(8) can be rewritten as follows:

(

)(

)

_∫∫

∫∫

∆ +∆ +∆ ≅ ∆ = _{∞ ∞ ∞} ∞ ∞ ∞ ∞ WA Entropy ds R s P ds R s u P D S n u ur r r

ρ

ρ

ρ

(9)

Eq.(9) is the first term of Eq.(3) itself.

By the transformation to the volume integral form, further drag decomposition of the entropy drag term is possible by domain decomposition of the flow field V. Physically, entropy variation in the flow field should originate in the shock (Vshock) or wake/boundary layer region (Vprofile), so the entropy variation in the remaining region (Vspurious) is considered to be an unphysical (spurious) phenomenon. The domain decomposition of the flow field is shown schematically in Fig.2. Then Eq.(4) can be transformed as follows:

( ) ( ) ( ) ( ) spurious profile wave H s H s H s H s D D D dv dv dv D spurious profile shock + + = ⋅ ∇ + ⋅ ∇ + ⋅ ∇ =

_∫∫∫

_{∆ ∆}

_∫∫∫

_{∆ ∆}

_∫∫∫

_{∆ ∆} ∆ ∆ V V V F F F , , , , r r r (10)

where Dwave, Dprofile and Dspurious correspond to wave, profile and spurious drag component

the divergence theorem again as follows: (only wave drag term was described) ( )

∫∫

( )

∫∫∫

∇⋅ = ⋅ = _{∆ ∆ ∆ ∆} shock shock ds dv D_{wave s H s H} S V n F Fr _, r _, r ₍₁₁₎

where Sshock indicates the boundary surface of Vshock. As the reader can guess, spurious entropy drag may be generated in the shock and profile region, and the effect can not be isolated in this approach. However, it is known that the majority of the spurious drag is generated in a region around the leading edge which is outside of the profile (boundary layer) region. So the spurious drag generated in the shock and profile region is insignificant. The advantage of the mid-field method is that it can divide the entropy drag into the wave, profile and spurious drag components, and can visualize the generated positions and the strength of the drag in the flow field because the integrand of the volume integral form indicates the drag production rate per unit volume.

The domain decomposition of flow field is based on the following shock/profile detective functions. For the detection of the shock region, the following function is used:

(

P

)

(

a P

)

f_shock = ur⋅∇ ∇ (12)

For the detection of the wake and boundary layer region, the following function is used:

(

l t

) ( )

l

profile

f

=

µ

+

µ

µ

(13)

The regions which satisfy f_shock ≥1 and f_profile ≥C⋅

(

f_profile

)

_∞ are recognized as the upstream region of shock waves and the profile region, respectively. C is a cutoff value for selecting the profile region, and C=1.1 was used in this research.

Likewise, the induced drag can be evaluated as follows:

∫∫

∫∫

∞ ⋅ ≅ ⋅ = S n F n F ds ds D _ind WA ind induced r r r r (14) 3 FLOW SOLVER

For flow computations, three-dimensional flows were analyzed using the TAS (Tohoku University Aerodynamic Simulation) code. Compressible NS equations were solved by a finite-volume cell-vertex scheme on an unstructured hybrid mesh. The numerical flux normal to the control volume boundary was computed using an approximate Riemann solver of Harten-Lax-van-Leer-Einfelds-Wada (HLLEW)12. The LU-SGS implicit method for unstructured meshes13 was used for the time integration. The original Spalart-Allmaras model14 was adopted to treat turbulent boundary layers, and fully turbulent flow was assumed in the computation.

⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ∇ Ψ + = 2 ij i i i L ij Q Q Q rr (15) where L ij

Q is an extrapolated variable from the left side to the face between node i and j as shown in Fig.3. Recently, a more accurate method for the variable extrapolation, Unstructured MUSCL (U-MUSCL)16 has been suggested. In this method, using the variable of the neighboring node Q , more accurate extrapolation is realized as follows: _j

(

)

(

)

_⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ∇ − + − Ψ + = 2 1 2 ij i i j i i L ij Q Q Q Q Q κ κ rr (16)

where κ is a U-MUSCL parameter. If κ is set to 0, the conventional extrapolation formula, Eq.(15), is obtained. If κ is set to 1, the extrapolated variable can be obtained as the arithmetic mean of Q and _i Q_j (assuming no limiter). In this case, the scheme has the characteristics of the central difference scheme and becomes unstable. Third-order extrapolation (note: not third-order accuracy) can be achieved by setting κ to 0.5. The details are given in Ref.16. In section 4.3, the effectiveness of the U-MUSCL scheme is discussed using the mid-field drag decomposition method. The conventional reconstruction method of Eq.(15) is used in section 4.1 and 4.2.

Fig.3: Schematic Sketch for U-MUSCL Extrapolation

4 RESULTS & DISCUSSION

4.1 Validation study in ONERA M6 transonic computation

In order to predict the induced drag component, the diffusion of the wingtip vortex has to be considered. The wingtip vortex diffuses and transforms into entropy generation equivalently in the wake region. So the entropy drag term obtained by integrating from the wingtip to the downstream surface WA has to be considered as an additional term for the induced drag component. In Fig.6, the variation of drag components as the integral region extends to the far downstream is shown. With the extension, the ‘original’ induced drag term (MF_Induced*) was reduced and the ‘additional’ induced drag term (Wake’s Entropy Drag) was increased. Moreover, the sum of the original and additional terms was almost constant at arbitrary wake positions, meaning that the ‘total’ induced drag was almost conserved in the wake region.

The result of drag polar is shown in Fig.7. In this figure, prefixes NF_ and MF_ represent the near-field and mid-field drag predictions, respectively. It was confirmed that the total drag computed by the mid-field method (MF_Total) showed good agreement with that computed by the near-field method (NF_Total). In detail, the wave drag and induced drag were found to increase with higher angle of attack. The profile drag increased at high-lift conditions, which was considered to be the effect of massive separation. The spurious drag was almost constant in all cases. MF_Pure plot is defined as ‘pure’ drag, and indicates the sum of physical drag components, in other words, the remainder after subtracting spurious drag from the total drag. In Fig.7, a structured mesh near-field result11 is also plotted (NF_Structured). The plot of the pure drag showed better agreement with the structured mesh result compared with the total drag. This meant that more accurate unstructured mesh drag prediction was achieved by using the drag decomposition method.

The visualization of mesh, pressure, entropy variation and entropy drag strength ∇⋅F(∆s,∆H)

r

of Eq.(4) at the angle of attack of 3.06 degrees, 65% semi-span section is shown in Fig.8. The production of entropy drag at the leading/trailing edge, boundary layer, shock positions and wake region could be confirmed. The spurious drag was mainly generated around the leading edge, and was caused by the numerical diffusion relating to the mesh coarseness against the rapid change of the flow variables. The entropy drag production around the trailing edge was considered to be the effect of the wake’s massive diffusion. Thus, the visualization implies that for more accurate computation, it is necessary to increase the mesh resolution around the leading edge and wake region.

-0.06 0.04 0.14 0.24 0.34 0.44 0.54 -0.1 0.3 0.7 1.1 (x/c) (z/ c) -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 Cp Airfoil TAS Experiment

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0 1 2 3 4 5 6 7 8 Position of WA CD MF_Induced* Wake's Entropy Drag MF_Induced*+Wake's Wing 0.0 0.1 0.2 0.3 0.4 0.5 0.00 0.01 0.02 0.03 0.04 0.05 CD CL NF_Total MF_Total MF_Pure MF_Wave MF_Profile MF_Induced MF_Spurious NF_Structured

Fig.6: Induced Drag Prediction in Wake Region Fig.7: Drag Polar of ONERA M6 Wing @ Mach 0.84

Fig.8: Flow Field Visualization @ 65% semi-span, Upper/Left: computational mesh, Upper/Right: pressure, Lower/Left: entropy variation, Lower/Right: entropy drag strength 4.2 Mesh resolution analysis in DPW-2 problem

In this section, the mesh resolution effect is analyzed using DLR-F6 transonic computation. The geometry, flow conditions and so on were the same as used in the second AIAA Drag Prediction Workshop1 held in 2003. In this analysis, the two computational meshes provided as the official unstructured meshes for DPW-2 were used for the computation. The numbers of mesh points were 1 and 3 million, respectively. The details of the mesh generation methods are given in Ref.21. In this section, the 1 and 3 million meshes are called ‘coarse’ mesh and ‘medium’ mesh, respectively. These meshes are visualized in Fig.9. The polar curve was computed at Mach number of 0.75.

comparison of Cp distribution at the angle of attack of 0.49 degrees, 33% semi-span section are shown with experimental data1. In Fig.10, the structured mesh result of ONERA provided at DPW-28 is also included. The total drag showed a difference of about 20cts between the coarse and medium meshes, while the skin friction drag showed good agreement. This implied that the spurious drag in the pressure drag of the coarse mesh was greater than that of the medium mesh.

In Fig.12, the flow field visualizations at the angle of attack of 0.49 degrees, 33% semi-span section are shown. It was confirmed that the spurious entropy (and spurious entropy drag) production around the leading edge and the wake’s diffusion were reduced with the increase of the mesh resolution. In Fig.13, entropy drag production maps on the x-y plane are shown. To create this figure, first a uniform Cartesian mesh was made on the x-y plane, then the entropy drag production of the unstructured mesh result was integrated to each cell of the Cartesian mesh referring to the x and y coordinates. The red/blue points at the left side indicate the diffusion of wingtip vortex and wake. From the figure, the reduction of (spurious) entropy drag around the leading/trailing edge was confirmed with the increase of the mesh resolution.

In Fig.14, the drag decomposition results are shown with the result of ONERA8. The physical drag components - wave, profile and induced drag - showed good agreement between the results of the coarse mesh and medium mesh. This meant that the physical drag components could be predicted almost independent of the mesh resolution by using the drag decomposition method. On the other hand, the spurious drag component was reduced from about 30cts to 10cts with the increase of the mesh resolution. In Fig.15, the pure drag is plotted with the near-field results and the experimental data. The pure drag of coarse and medium meshes showed much better agreement with the experimental data than the near-field drag prediction.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.01 0.02 0.03 0.04 CD CL Experiment ONERA Coarse Medium Coarse_friction Medium_friction -1.5 -1.0 -0.5 0.0 0.5 1.0 -0.1 0.3 0.7 1.1 (x/c) Cp Coarse Mesh Medium Mesh Experiment

Fig.10: Near-Field Drag Prediction Fig.11: Cp Distribution @ 33% semi-span Coarse Mesh’s

Medium Mesh’s

Fig.12: Flow Field Visualization @ 33% semi-span

From Left to Right: pressure, domain decomposition, entropy variation, entropy drag strength

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.000 0.005 0.010 0.015 0.020 0.025 CD CL Coarse_Wave Coarse_Profile Coarse_Induced Coarse_Spurious Medium_Wave Medium_Profile Medium_Induced Medium_Spurious ONERA_Wave ONERA_Profile ONERA_Induced _0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.015 0.020 0.025 0.030 0.035 0.040 CD CL Experiment NF_Coarse NF_Medium MF_Pure_Coarse MF_Pure_Medium

Fig.14: Drag Decomposition Results Fig.15: Pure Drag Prediction 4.3 U-MUSCL scheme analysis

In this section, the effectiveness of the U-MUSCL scheme is analyzed using DLR-F6 transonic computation. The flow conditions were the Mach number of 0.75 and angle of attack of 0.49 degrees. The coarse computational mesh introduced in the previous section was used for the analysis. The number of mesh points was about 1 million. The transonic computations were then conduced, while changing the U-MUSCL parameter κ from 0 to 0.9 (unstable at 0.95).

In Fig.16, the Cp distributions for κ =0.0, 0.5 and 0.9 at 41% semi-span section are compared with experimental data. With the increase of κ , the shock wave was captured sharply. However, some oscillations around the shock wave and trailing edge were observed from the result for κ=0.9, showing the limitation of stable computation. In Fig.17, the near-field drag prediction is plotted with the experimental data. As κ was increased from 0 to 0.9, a total drag reduction of about 35cts and good agreement with experimental data were confirmed. The total drag reduction was mainly due to the reduction of the pressure drag term. This result implied that the accuracy could be improved by using the U-MUSCL scheme and that the spurious drag could be reduced by increasing κ.

In Fig.18, the drag decomposition results are shown. The physical drag components were almost constant and independent of the parameter κ . Although the values fluctuated somewhat around κ = 0.8 ~ 0.9, it was considered to be the effect of the computational instability. On the other hand, the spurious drag was reduced monotonically, and the value was negative at κ > 0.8. The phenomenon of the negative spurious drag was reported in Ref.10, and was caused by negative entropy production at the border of the boundary layer. According to Ref.10, this phenomenon appeared when central difference schemes were used for computations, the finding matched our results. In Fig.19, the mid-field pure drag prediction is plotted with the experimental data. The difference of pure drag between

7 . 0

0≤κ ≤ was only about 3cts, and showed good agreement with the experimental data. This result showed that the physical drag components could be predicted almost independent of the U-MUSCL parameter κ by using the drag decomposition method.

κ= 0.9, the reduction of spurious entropy drag at the leading edge and the reduction of the wake’s diffusion can be observed with the increase of κ.

-1.5 -1.0 -0.5 0.0 0.5 1.0 -0.1 0.3 0.7 1.1 (x/c) Cp Experiment κ=0.0 k=0.5 k=0.9 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.015 0.020 0.025 0.030 0.035 0.040 CD CL Experiment NF_Total κ=0.0 κ=0.9

Fig.16: Cp Distribution @ 41% semi-span Fig.17: Near-Field Drag Prediction

-0.005 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.0 0.2 0.4 0.6 0.8 1.0 κ CD MF_Total MF_Pure MF_Wave MF_Profile MF_Induced MF_Spurious 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.015 0.020 0.025 0.030 0.035 0.040 CD CL Experiment MF_Pure κ=0.0 κ=0.9

Fig.18: Drag Decomposition Results Fig.19: Pure Drag Prediction

5 CONCLUSIONS

In this paper, the mid-field drag decomposition method was applied to unstructured mesh CFD results of the TAS code. This method is able to decompose total drag into physical drag components, to exclude the spurious drag component due to numerical diffusion and to visualize the strength and generated positions of entropy drag in the flow field.

These results could not have been obtained without using the mid-field drag decomposition method. For more accurate drag prediction, detailed analysis of drag reduction mechanisms and detailed data-mining of CFD results, the drag decomposition method will be an essential tool for aircraft designers and CFD researchers.

0 . 0 =

κ , κ=0.5, κ=0.9

Fig.20: Flow Field Visualization,

ACKNOWLEDGMENTS

The authors would like to thank Dr. Kazuhiro Kusunose for his helpful advice. The authors also would like to thank Prof. Dimitri Mavriplis for providing the unstructured hybrid meshes of DLR-F6 geometry. This research was supported by a grant from the Japan Society for the Promotion of Science. The present computation was executed by using NEC SX-7 in Super-Computing System Information Synergy Center of Tohoku University. We sincerely thank all the staff.

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[20] Data available at http://www.grc.nasa.gov/WWW/wind/ valid/m6wing/m6wing.html [21] Lee-Rausch, E. M., Frink, N. T., Mavriplis, D. J., Rausch, R. D., and Milholen, W. E.,