Chinese Journal of Aeronautics

Chinese Journal of Aeronautics 21(2008) 19-27www.elsevier.com/locate/cja

A Structured Mesh Euler and Interactive Boundary Layer Method for Wing/Body Configurations

Li Jie*, Zhou Zhou School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China

Received 21 May 2007; accepted 03 December 2007

Abstract

To compute transonic flows over a complex 3D aircraft configuration, a viscous/inviscid interaction method is developed by cou-

pling an integral boundary-layer solver with an Eluer solver in a “semi-inverse” manner. For the turbulent boundary-layer, an integral

method using Green’s lag equation is coupled with the outer inviscid flow. A blowing velocity approach is used to simulate the dis-

placement effects of the boundary layer. To predict the aerodynamic drag, it is developed a numerical technique called far-field method

that is based on the momentum theorem, in which the total drag is divided into three component drags, i.e. viscous, induced and

wave-formed. Consequently, it can provide more physical insight into the drag sources than the often-used surface integral technique.

The drag decomposition can be achieved with help of the second law of thermodynamics, which implies that entropy increases and total

pressure decreases only across shock wave along a streamline of an inviscid non-isentropic flow. This method has been applied to the

DLR-F4 wing/body configuration showing results in good agreement with the wind tunnel data.

Keywords: viscous/inviscid interaction; far-field drag prediction; transonic flow; wing/body configuration

1 Introduction*

Assessing aerodynamic performance is essen-tial in the designing and developing of aircraft. One of the most important parameters in the aerody-namic design is the lift-to-drag ratio, L/D, under cruising conditions, which dominates the efficiency of an aircraft. Therefore, both the lift and the drag must be accurately predicted so as to maximize the L/D by improving the shape[1].

In general, since an accurate lift can be ob-tained by using computational fluid dynamics (CFD) methods, the accuracy of a drag prediction tech-nique plays a key role in the evaluation procedure. Especially, in an early design stage, the reliable drag prediction is of first magnitude when various con- *Corresponding author. Tel.: +86-29-88460448. E-mail address: leejay1981@163.com Foundation item: Aeronautics Science Foundation (2006ZA53009)

figurations are available for assessment. Viscous/inviscid interaction methods based on

the Euler and boundary layer equations have grown to maturity for many years and have been proved to be accurate for transonic wing/body and isolated nacelle geometries on structured grids. Under the cruising conditions, the results of their application are comparable to Navier-Stokes solutions at a cost of only a fraction of the latter. On the other hand, both Euler and Navier-Stokes solutions in the lift prediction can provide reasonably good results[1].

Traditionally, the drag prediction is both ex-perimentally and numerically one of the major chal-lenges in aerodynamics study. In experiments, it is difficult to simulate all of the features of a physical problem such as the Reynolds number. Moreover, other problems arise due to interference effects, in particular with the model support, and the difficulty

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in measurement of the figures that are much smaller than others in the test[2-4].

In the case of numerical technique, the CFD methods that have achieved tremendous progresses recently could not yet ensure sufficiently accurate drag predictions[5-6].

The most common technique, called near-field technique, to calculate the total drag is established on the base of the integration of pressure and skin friction on the aircraft surface. The two main rea-sons to cause inaccuracy in drag prediction are: first, the total value of the pressure drag coefficient is obtained by almost omitting a large component force in the thrust direction and a slightly larger component force in the drag direction[7]; second, the numerical viscosity inherent in the Euler or N-S methods affects the surface pressures especially in the stagnation region near the leading edge and in the recovery region near the trailing edge. It is stated that errors in the surface pressure in these two regions hardly affect the lift prediction, but signifi-cantly affect the drag prediction[1]. Therefore, the near-field technique could be helpful only when an accurate and detailed pressure distribution along the surface has been predicted[7].

On the other hand, there is a need in design as well as in analysis to identify the drag sources, but the near-field technique is unable to separate the drag as an integrated value into physical compo-nents.

A standard alternative to the near-field tech-nique is the far-field integration technique, by which a total precise drag can be predicted, and the physi-cal drag sources could be identified. This capability of separating a drag is especially important in the design stage when the performance of an aircraft is being maximized by improving its shape. The effi-ciency of this design process can be greatly en-hanced if the physical mechanism, from which the drag has formed, can be identified [8-9].

In this paper, an integral boundary-layer solver is coupled with an Eluer solver in a “semi-inverse” manner. A blowing velocity approach is used to simulate the displacement effects of the boundary

layer. Far-field technique is applied as an improve-ment of the code for the purpose of predicting drag in the wing/body combined configuration. The main advantage of this technique lies in none of require-ment for detailed information on the surface geome-try of the configuration. It also enables the drag to be decomposed into physical components, which means that the drag could be expressed as a sum of the following three component-drags: wave, in-duced and viscous. This decomposition is useful for understanding the sources of drags.

2 Governing Equation

The fluid motion is governed by the time de-pendent Euler equations for an ideal gas which ex-press the conservation of mass, momentum, and energy for a compressible inviscid nonconductive adiabatic fluid in the absence of external forces. The equations given below are in an integral form for a bounded domain Ω with a boundary Ω∂

d ( ) d 0V St Ω Ω∂

∂+ ⋅ =

∂ ∫∫∫ ∫∫Q F Q n (1)

where

uvwe

ρρρρ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Q

0

( ) ( )

0

x

y

z

nunpv

w ne p

ρρρρ

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⋅ = ⋅ +⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥+⎣ ⎦ ⎣ ⎦

F Q n V n

ρ, u, v, w, p and e represent the density, velocity in x-direction, velocity in y-direction, velocity in z-direction, pressure and total energy respectively. nx, ny and nz are the components of the grid vector n in the x-, y-, z-direction. On the assumption of ideal gas, the pressure and total enthalpy can be ex-pressed as

2 2 21( 1) ( )2

p e u v wγ ρ⎡ ⎤= − − + +⎢ ⎥⎣ ⎦ (2)

Li Jie et al. / Chinese Journal of Aeronautics 21(2008) 19-27 · 21 ·

where γ is the ratio of specific heats and for air it is equal to 1.4.

3 Integral Boundary-layer Method

By taking into consideration the computational cost and uncertainties of turbulence modeling in-volved in a finite differential method, an integral boundary-layer method is used to compensate for the viscous effects. The classical boundary-layer calculation is to solve the boundary-layer thickness by using the boundary-layer edge pressure gradient obtained from the outer inviscid flow solver. How-ever, this direct method of boundary-layer calcula-tion fails to fit in with the flows involving strong inviscid/viscous interactions, especially when sepa-ration exists[10-12]. Thus the inverse boundary-layer calculation is coupled with the outer inviscid flow solution. In the inverse boundary-layer calculation, the edge pressure or velocity is determined by a given distribution of boundary-layer displacement thickness. For convenience, according to Cater [13], the perturbation mass flow parameter *

e em Uρ δ= is introduced. For a given distribution of m along the wall, the boundary-layer edge velocity, Ue, can be solved as follows:

Let * Hδ θ= , by expanding can be obtained

( )e eddd d

U Hms s

ρ θ= (3)

and

( )2 ee

e

d1 d 1 d 1 d 11d d d d

Um H Mam s H s s U s

θθ

= + + − (4)

where δ and θ denote the boundary-layer displace-ments and momentum thicknesses; ρe, Ue and Mae local air density, velocity and Mach number at the boundary-layer edge, respectively; s the streamwise coordinate along the wall or wake; H the bound-ary-layer shape factor.

Considering the correlation between the shape factor H and the kinematics shape factor H , i.e.,

( )1 1 1H R H= + − , can be obtained

( ) 3 e1

e

dd d 1d d d

R UH HR Hs s U s

= + + (5)

Thus, Eq.(4) becomes

( ) ( )1

2 e3 e

e

d d dd d d

d1 1

d

H m HH Rm s s s

UH R H Ma

U s

θ θ θ

θ

= + +

⎡ ⎤+ + −⎣ ⎦ (6)

where, R1, R2 and R3 are three parameters which are defined to be related to the ratio of specific heats γ, temperature recovery factor r, and the local boun- dary-layer edge Mach number Mae, respectively.

( )

21 e

22 e

2e 2

31

112

112

1

R rMa

R Ma

rMa RR

R

γ

γ

γ

⎫− ⎪= +⎪⎪− ⎪= + ⎬⎪⎪−

= ⎪⎪⎭

(7)

For a turbulent boundary-layer, Head[14] intro-duced the entrainment coefficient CE, which repre-sents the rate at which the fluid enters the boun- dary-layer from the outer inviscid flow through the boundary-layer edge. By defining

( )e e 1E

e e

d1dU H

CU s

ρ θρ

= (8)

where H1 is Head’s shape factor, and by expanding the derivative, is obtained

( )2 e 1E 1 1 e

e

d dd d1d d d d

U H HC H H Mas U s H sθ θ θ= + − + (9)

In addition, according to the integral momen-tum equation for the compressible boundary-layer:

( )2 efe

dd 22 d de

UCH Ma

s U sθ θ

= + + − (10)

can be obtained a linear system of Eqs.(6), (7), (10)

pertinent to the three unknown derivatives: ddsθ ,

eddUs

and ddHs

. By solving the system, can be ob-

tained a system of three first-order ordinary differ-ential equations pertinent to the three boundary- layer parameters: θ, Ue and H .

Besides, the Green’s lag equation[15] is em-ployed to consider the history effects in non-equi-

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non-equilibrium turbulent boundary-layer:

( ) ( )0.5 0.5EEQ0

1

2e e1e 2

e eeEQ

d 2.8d

d d1 0.075

d d1 0.1

CF C C

s H H

U URMa

U s U sMa

τ τθ λ

θ θ

⎧ ⎡ ⎤= − +⎨ ⎣ ⎦+⎩⎫⎡ ⎤⎛ ⎞ ⎪− +⎢ ⎥ ⎬⎜ ⎟

+⎢ ⎥⎝ ⎠ ⎪⎣ ⎦ ⎭

(11)

where Cτ is the shear stress coefficient, λ the pa-rameter to consider secondary effects. The subscript EQ denotes the quantities evaluated under equilib-rium conditions with the shape factor and the en-trainment coefficient invariant, while EQ0 the quan-tities under an equilibrium flow free of secondary effects.

In summing up，a single system inclusive of four first-order ordinary differential equations per-tinent to the four unknown boundary-layer parame-ters is obtained. Given a distribution of m along the wall together with the initial values at a starting point, for example a fixed transition point, the four ordinary differential equations can be integrated by using Runge-Kutta method and solved for the four unknown boundary-layer parameters: θ, Ue, H and CE.

4 Inviscid/Viscous Coupling Procedure

Given the boundary-layer edge properties ob-tained from the outer inviscid solver, Thwaites’ method[16] can be used to calculate the laminar part of the boundary-layer starting from the stagnation point. Transition is by Michel’s formula[17-18]:

0.4622 4001.174 1 ss

Re ReReθ

⎛ ⎞> +⎜ ⎟

⎝ ⎠ (12)

In the case of the turbulent part, the boundary- layer calculation needs to be coupled with the outer equivalent inviscid flow (EIF) calculation, for which should be employed Carter’s “semi-inverse” coupling scheme[11], which involves the use of ρe and Ue from the preliminary inviscid calculation. Firstly, given a distribution of the boundary-layer displacement thickness *δ , an assumed perturba-tion mass-flow parameter m = *

e eUρ δ could be ob-

tained. Then the viscous version of the boundary- layer edge velocity Uev could be acquired through an inverse boundary-layer calculation introduced in the last section. Also from m , the wall and wake boundary-conditions for the EIF calculation could be derived. Solving the Euler equations under these boundary-conditions for the outer EIF results in an inviscid version of boundary-layer edge velocity Uei. Thus Carter’s relaxation scheme could be used to acquire an updated boundary-layer thickness:

*new ev*

eiold

1 1UU

δω

δ⎛ ⎞

= + −⎜ ⎟⎝ ⎠

(13)

where ω is an under-relaxation factor. In most cases, an amount of two-order-magnitude of the residual dropping between the two boundary-layer edge ve-locities, Uev and Uei, is enough to judge the conver-gence.

In solving the Euler equations for the outer EIF, four boundary-conditions are needed to match the EIF with the viscous flow for a 2D problem. How-ever, as was proved by Sockol and Johnston[19], if the surface normal blowing velocity derived from the continuity equation is used as a boundary condi-tion, other matching requirements such as the nor-mal flux of streamwise momentum and total en-thalpy will automatically be satisfied. Taking ad-vantage of the first-order boundary-layer approxi-mation, the calculation of surface values of density, streamwise velocity and total enthalpy could be simplified by way of linear extrapolation from the adjacent grid to the wall. Therefore, the only change in solving the EIF is the need for adding a blowing velocity, which could be obtained from mass con-servation:

( )*e e

e e

1 d 1 d( )d dnV u ms s

ρ δρ ρ

= = (14)

As is known, the Kutta condition is automati-cally satisfied in Euler calculations, so, unlike the boundary-layer coupling with a potential code, us-ing a jump condition is no longer needed in the wake, which could be treated simply as two boun- dary-layers developed on both sides of its dividing

Li Jie et al. / Chinese Journal of Aeronautics 21(2008) 19-27 · 23 ·

streamline. In this paper, for convenience, this di-viding streamline is assumed to be the extension of the airfoil mean chord.

5 Far-field Drag Analysis Method

In principle, the far-field drag computation is based on the momentum theorem. However, direct application of the theorem to the Euler or N-S method only makes the total drag and the compo-nent drags unable to be extracted.

The drag decomposition can be achieved with help of the second law of thermodynamics, which implies that entropy increases and total pressure decreases only across shock wave along a stream-line of an inviscid non-isentropic flow. It is then possible to show that the drag can be expressed, not exactly but to a high degree of approximation, as the sum of the following three components[7]

w i vD D D DC C C C= + + (15)

Next, a brief description of these components will be given. For further details, see Ref.[7].

5.1 Wave drag

The wave drag is related to the entropy in-crease across the surface of a shock wave through the following Qswatitsch drag integral:

( )wavewave

dT

D sV

ρ σ∞

∞

= Δ ⋅∫ V n (16)

where s s s∞Δ = − represents the specific entropy produced by shocks; T∞ is the free-stream tempera-ture; V∞ the free-steam velocity, and dσ an ele-mental area on the shock.

This expression of the wave drag is obtained under the assumption that the static pressure is un-modified far downstream.

In Ref.[7], Lock gave a simple formula for wave drag, which involves only conditions just up-stream of the shock. According to Lock, the wave drag coefficient can be expressed in the form of

( ) ( )w 1 11 , dD nC F Ma G Ma MaS

σ= ⋅∫ (17)

where S is a wing area, and

321 0.22 MaF

Maγ∞

∞

⎛ ⎞+= ⎜ ⎟⎜ ⎟

⎝ ⎠

( )1

21

2 21 1

21

12 1 0.2

5 7 17 ln 5ln

66

n

n n

n

MaG

Ma

Ma MaMa

=+

⎡ ⎤⎛ ⎞ ⎛ ⎞+ −+⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

i

in which suffix 1 refers to conditions just upstream of the shock and Ma1n the component of upstream Mach number normal to the shock.

5.2 Induced drag

The induced drag can likewise be derived from the momentum theorem under the assumption that entropy does not vary along streamlines.

Let (Δu, Δv, Δw) be the non-dimensional per-turbation velocity, i.e.,

u uu

u∞

∞

−Δ = , v v

vu

∞

∞

−Δ = , w w

wu

∞

∞

−Δ =

Then the induced drag can be expressed as fol-lows[7-8]

( )

( )

( )

2 2induced

2 2

d2

1 d

2 2 d

VD v w

Ma u

u v u v

σ

σ

σ

ρσ

σ

σ

∞ ∞

∞

⎧⎪ ⎡ ⎤= Δ + Δ −⎨ ⎣ ⎦⎪⎩⎡ ⎤− Δ −⎣ ⎦

⎫⎪Δ Δ + Δ Δ ⎬⎪⎭

∫

∫

∫

x n

x n

y z n (18)

in which, σ is a transverse surface downstream away from where no streamwise pressure gradient exists.

Thus, the induced drag coefficient can be ex-pressed as

i induced22

DC DV Sρ∞ ∞

= (19)

5.3 Viscous drag

Because the viscous effects on the wing sur-faces are simulated by means of the viscous/inviscid interaction technique, it can be obtained from the boundary layer parameters in accordance with the derivation given by Lock in Ref.[7,9].

According to Lock, the viscous drag due to the wing is predicted by calculating the components of

· 24 · Li Jie et al. / Chinese Journal of Aeronautics 21(2008) 19-27

momentum thickness of the boundary layers on both upper and lower surfaces at the wing trailing edge. The local values of viscous drag coefficient, CDv(η), are then obtained by using the Squire and Young’s method, which has been modified to allow for the effects of compressibility and trailing edge sweep[7,9] .

Now, the total viscous drag coefficient can be acquired by integrating the viscous drag contribu-tions along the span direction η.

v vspan

( ) ( )dD DcC C

cη η η= ∫ (20)

where c(η) is the local chord and c the mean chord.

6 Results and Discussions

The flow around DLR-F4 under the transonic condition was calculated to evaluate the efficiency of the flow solver. The mesh was generated by us-ing the method that fits in with the wing/body con-figuration. In solving the elliptic grid generation with an algebraic method, there was firstly gener-ated a series of 2D structured grid along the meri- dian of the body by using an elliptic method with Higenstock source correct technical, and then ad-justed to be a 3D structured grid smoothly by using an algebraic method, the grid is shown in Fig.1. The method is easy to use and can keep the grid as rigid as possible in the near-wall regions. The calculation was performed under the following condition: Ma = 0.75, Re = 3×106 and Cl = 0.6.

Fig.1 Structured grid over DLR F4 wing-body configura-

tion.

The computational results are compared to the experimental results obtained in three different wind tunnels[20]: the high speed wind tunnel (HST) of the NLR, the ONERA-S2MA wind tunnel of the ON-ERA and the 8-foot wind tunnel of the DRA.

Fig.2 shows Cp(x/c) at three spanwise locations on the wing at Cl = 0.6. Fig.2(a) demonstrates a good agreement between the computational and the experimental results at the 18.5% span location with a little lower suction peak. When the observed sec-tion of the wing is moving outwards, the two kinds of results become to discord. For example, as is shown in Fig.2(b), at the 51.2% span, the shock wave lags behind the experimental one, but at the 84.4% span, as is shown in Fig.2(c), they are be-coming to accord with each other again.

The comparison of Cl-Cd with the wind tunnel data is shown in Fig.3, from which a good agree-ment exists between the calculated and experimental data.

Li Jie et al. / Chinese Journal of Aeronautics 21(2008) 19-27 · 25 ·

Fig.2 Comparison of pressure distribution.

Fig.3 Comparison of Cl-Cd with wind tunnel data.

The lift coefficients and the drag coefficients of the three components are presented in Table 1, where Cdw, Cdi and Cdv represent wave drag, induced drag and viscous drag respectively. CFb is the body surface friction drag calculated with an engineering estimation method and Cdexp the experimental drag data from the ONERA wind tunnel. The results demonstrate the capability of the far-field drag method to evaluate component drags. Moreover, the comparison of drags there proves to be a quite good agreement between the calculated and the experi-mental data.

The far-field drag prediction technique is de-rived by applying the conservation law of momen-tum to the control volume enclosing the entire con-figuration: the wave drag is obtained from the inte-gration of the entropy jump over the shocks; the induced drag is obtained from the integration of the kinetic energy of the cross-flow induced by the trailing vortex sheet in a transverse plane far down-stream where there is no streamwise pressure gra-dient and the viscous drag due to the wing is ob-tained by using the method of Squire and Young by calculating the components of momentum thickness

Table 1 Lift coefficients and the drag coefficients

Cl Cdw Cdi Cdv CFb Cd Cdexp

0.048 9 0.000 13 0.001 10 0.007 70 0.009 5 0.018 43 0.018

0.108 1 0.000 05 0.001 48 0.007 68 0.009 5 0.018 71 0.019

0.167 1 0.000 05 0.002 09 0.007 69 0.009 5 0.019 33 0.019

0.196 5 0.000 05 0.002 47 0.007 71 0.009 5 0.019 73 0.020

0.226 0 0.000 05 0.002 91 0.007 74 0.009 5 0.020 20 0.020

0.285 3 0.000 14 0.004 01 0.007 80 0.009 5 0.021 45 0.021

0.344 5 0.000 22 0.005 25 0.007 87 0.009 5 0.022 84 0.023

0.401 6 0.000 27 0.006 71 0.007 98 0.009 5 0.024 46 0.025

0.455 6 0.000 31 0.008 27 0.008 12 0.009 5 0.026 20 0.026

0.506 9 0.000 38 0.009 92 0.008 26 0.009 5 0.028 06 0.028

0.552 8 0.000 51 0.011 61 0.008 59 0.009 5 0.030 21 0.030

0.566 4 0.000 54 0.012 16 0.008 71 0.009 5 0.030 91 0.031

0.598 5 0.001 31 0.013 52 0.008 88 0.009 5 0.033 21 0.033

0.621 8 0.001 67 0.014 56 0.008 92 0.009 5 0.034 65 0.035

0.637 7 0.002 24 0.015 31 0.008 98 0.009 5 0.036 03 0.036

· 26 · Li Jie et al. / Chinese Journal of Aeronautics 21(2008) 19-27

of the boundary layers on both upper and lower surfaces at the wing trailing edge. The main advan-tage of this technique lies in avoidance of the needs of detailed information on the surface geometry of the configuration. Therefore, even though the shock wave is proved to lag behind the experimental data at 51.2% span, it stands to reason that the drag agrees well with the wind tunnel data as shown in Table 1.

7 Conclusions

The object of this contribution is to introduce an efficient interactive boundary-layer method. An integral boundary-layer method using Green’s lag equation is coupled with the outer inviscid flow in a “semi-inverse” manner. The classical boundary- layer calculation is to solve the boundary-layer thickness by using the boundary-layer edge pressure gradient obtained from the outer inviscid flow solver. However, this so-called direct method of boundary-layer calculation is unable to be applied to the flows involving strong inviscid-viscous interac-tions, especially when separation exists. Therefore, the inverse boundary-layer calculation is coupled with the outer inviscid flow solution. This method is more robust than the direct one because the edge pressure or velocity is solved from a given distribu-tion of boundary-layer displacement thickness. Another object of this paper is to present a far-field drag prediction technique, which identifies various physical drag sources and ensures higher accuracy than the often-used near-field approach. Further-more, this technique is less dependent on spatial discretization and numerical schemes in the Euler solver, and it has the capability of decomposing the drag into component drags: wave, induced and vis-cous thereby enabling aerodynamic designers to grasp detailed information about the drag sources. As a result, the efficiency of the process to design civil aircraft can be greatly enhanced.

By taking a DLR-F4 wing body configuration as an example, the method is proved and the results agree well with the wind tunnel data.

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Biography: Li Jie Born in 1981, he received B.S. and M.S. from the

Aircraft Department of Northwestern Polytechnical Univer-

sity in 2003 and 2006 respectively, and is now pursuing a

doctorate in the same institution.

E-mail: leejay1981@163.com