analysis of rae 2822 airfoil with direct viscous-inviscid ... · simulated for an attached flow...

Abstract—The present work demonstrates a computational of

aerodynamics characteristics on RAE 2822 airfoil. This analysis will be through a combination between viscous and inviscid solution namely viscous-inviscid interaction method. In this analysis, the computation will be divided into two approaches which related to flow domain. For the viscous domain which is referred to close region the airfoil surface, it will be analyzed by using Keller Box Method which provided by Cebeci and Bradshaw. In this calculation, boundary layer characteristics will be prescribed based on local velocity that given from inviscid computation. For inviscid domain, a numerical approach is performed namely Roe’s scheme in solving Euler equation and MUSCL’s interpolation scheme is applied to diminishing spurious oscillation due to shock wave presence. The analysis is made at various angles of attack exclusively at high subsonic flow while Reynolds number is set at 5 to 6 million. Results from inviscid solution is compared to experimental data which taken from numerous sources.

Keywords—Viscous-Inviscid, Keller Box, Cebeci-Smith, Euler solver, Roe’s scheme, and MUSCL scheme.

I. INTRODUCTION TUDY on aerodynamics characteristics is frequently fulfilled through investigation on flow over airfoil geometry either experimentally or analytically. In decade

70’s, numerical solution of Navier Stokes equation gave major influence in aerodynamics analysis. As example, an investigation was made by Korn on shock-free transonic around airfoil by applying numerical method in solving linear partial differential equations [1]. Afterwards, many works was made by researchers to examine the capability of solution methods. Lax, Roe, McCormack, Godunov, Ritchmeyer and Rusanov who invented applicable schemes frequently gained attention by succeeding researchers. Taylor explained some favorite schemes thoroughly in dealing with such serious difficulties of aerodynamic problem [2]. This work engaged with a wide scope of aerodynamics properties such as subsonic, transonic and supersonic speed, viscous and inviscid, compressibility effect, high Reynolds numbers and various approach of solution including potential flow, Euler solver and Navier stokes solution.

Mohd Faizal bin Che Mohd Husin is a PHD student of University Tun Hussein Onn Malaysia, 86400 Johor, Malaysia (corresponding author’s phone: +6019-9789769; e-mail: [email protected] ).

Dr. Ir. Bambang Basuno is a senior lecturer at Department of Aeronautic, University Tun Hussein Onn Malaysia, 86400 Johor, Malaysia (e-mail: [email protected]).

Dr. Zamri bin Omar is a senior lecturer at Department of Aeronautic, University Tun Hussein Onn Malaysia, 86400 Johor, Malaysia (e-mail: [email protected] ).

Jameson carried on aerodynamics discovery with his work on airfoils through numerical potential flow solution [3]. This work offers the solution of flow at sonic Mach number and also implements artificial viscosity as a shock wave treatment. The great work by Jameson on Euler methods can be found in [4] by solving Euler equation with finite volume methods. Those methods were solved by Runge-Kutta time stepping schemes. Engaged with time stepping schemes, accuracy of solution was improved and the stability region can be extended.

The latest work which applied Euler solution is found from Arias et. al [5]. In this research finite volume has been simulated for an attached flow over airfoil NACA 0012 by using Jameson, MacCormack, Shue, and TVD schemes. This work presented two computer codes where both approach implement finite volume methods to solve Euler equations. First code namely ITA works on two-dimensional structured grid and it possess the capacity to work with three different schemes: (i) the Jameson scheme using a five stage Runge-Kutta time integration; (ii) the MacCormack scheme, based upon the predictor and corrector strategy to advance in time; (iii) and finally the Shu scheme, which uses a variation of the Jameson time integration, in order to better capture of shock waves.

In order to improve accuracy, viscosity effect should be included based on boundary layer theory as referred in [6], where boundary layer parameter plays significant role in determining aerodynamics characteristics especially for case high Reynolds Number. To that end, Keller Box method will be employed based on finite difference which has been successively applied by Cebeci on numerous works [7]. As attached flow in boundary layer considered change from laminar to turbulent, a turbulence modelling is used to treat such flow namely Cebeci-Smith model [8]. For the time being, present study will calculate boundary layer characteristics such as momentum thickness, displacement thickness, and friction coefficient.

II. PROCEDURE FOR COMPUTATION Euler equation will be treated in explicit formulation. Roe’s TVD scheme is utilized to resolve this explicit Euler equation with MUSCL’s scheme is exploited to eliminate spurious oscillation of second order formulation [9]. In order to apply these methods to complex geometric configurations, the finite volume formulation has been used to develop the space discretization, and allows the implementation of an arbitrary grid. Structured numerical grid generation is used since the

Analysis of RAE 2822 Airfoil with Direct Viscous-Inviscid Interaction

Mohd Faizal bin Che Mohd Husin , Dr. Ir. Bambang Basuno, and Dr. Zamri bin Omar

S

International Journal of Mining, Metallurgy & Mechanical Engineering (IJMMME) Volume 2, Issue 2 (2014) ISSN 2320–4060 (Online)

62

mailto:[email protected]

problem of single airfoil RAE 2822 is considered as relatively straightforward configuration. To accomplish the goal above, computer codes for TVD scheme and grid generation were utilized, which had taken from Blazek [10].

The criterion must be satisfied by grid generation process were 1) they domain is completely covered by the grid, 2) there is no free space left between the grid cells, and 3) the grid cells do not overlap each other. The detail about its governing equation would be described in the next section. The results of inviscid solution are mainly focused on pressure distribution, lift coefficient and moment coefficient. Due to Euler solution for inviscid flow domain, aerodynamics characteristics mentioned is sufficient enough without taking account of drag coefficient since the viscosity effect that affected the airfoil surface characteristics are neglected. Computer code that introduced in [10] is utilized namely AIRFOIL_ROE_SCHEME in solving the objective of study. Present study is separated in 4 study cases base on Mach number, M, angle of attack, AOA, and Reynolds number, Re which is listed as follow;

a) M = 0.676, AOA = -2.18, Re = 5.7 million b) M = 0.676, AOA = 1.90, Re = 5.7 million c) M = 0.725, AOA = 2.10, Re = 6.5 million d) M = 0.750, AOA = 2.10, Re = 6.5 million

For each case study, solution is extended to boundary layer calculation by integrating AIRFOIL_ROE_SCHEME of Roe’s scheme with computer code of Keller Box method. The governing equation of compressible flow is transformed to transformed coordinate via Falkner-Skan method. At this stage, flow will be dealt as laminar and turbulence due to tripped transition point. This computer code provides the data of boundary layer characteristics and will be compared to experimental data by Cook, McDonald, and Firmin [11]. In regard to previous work, the result will also be compared altogether to Navier-Stoke codes (ARC2D) [12] and Semi-Discrete Galerkin (SDG VII) [13] data respectively.

III. THE GOVERNING EQUATION Description of Euler Solution

The governing equation of inviscid flow domain for the case of compressible, non-viscous and two dimensional unsteady flows in conservative form are [14]:

∂Q∂t

+∂F∂x

+∂g∂y

= 0 (1)

Where:

Q= �

ρρuρvE�, F = �

ρuρu2 + pρuv

u(E + p)�, g = �

ρvρuv

ρv2 + pv(E + p)

� � ( 2)

With:

E = �e +12

u2� ρ e =p

(γ − 1)ρ � (3)

Above equation is known as Compressible Euler Equation

and represents a highly nonlinear partial differential equation

and there is no analytical solution. γ denoted as the ratio of specific heat capacities of the gas. In a two dimensional, Euler equation is wrote in hyperbolic equation form.

∂U∂t

+ A(U)∂U∂x

+ B(U)∂U∂y

= 0 (4)

Where A and B is Jacobian matrix system

A =∂F∂U

=

⎣⎢⎢⎢⎡∂f1∂u1

⋯∂f1∂u

⋮ ⋱ ⋮∂f1∂u4

⋯∂f4∂u4⎦

⎥⎥⎥⎤

B =∂G∂U

=

∂g1∂u1

⋯∂g1∂u4

⋮ ⋱ ⋮∂f1∂u4

⋯∂g4∂u4⎭

⎪⎬

⎪⎫

(5)

For more convenience, it is wise if Euler equation is derived

in one dimensional then for future use, one can simply extend to multi-dimensional. One dimensional explicit time stepping formulation read as:

Uin+1 = Ui

n −12�∆t∆x� �F

i+12

n − Fi−12

n � (6)

Following the step of Roe’s scheme, each term in (6) are

derived as follow [15]:

Fi+

12/i−

12

n =12�Fi+1/2

R + Fi+1/2L � +

12

��𝛌�(𝐤)� 𝚫𝐕(𝐤)𝐗�(𝐤)3

k=1

�i+

12/𝑖−12

(7)

First terms of right hand side equation represents convective

flux while the second terms are dissipative flux. Convective flux is treated by upwind scheme, and dissipative flux will follow Roe high resolution scheme. 𝐗� is eigenvector matrix correspondents to matrix eigenvalues 𝛌� respects to similarity such that [16]:

𝐀 = 𝐗𝐃𝐗−𝟏 (8)

𝐗� = �1 1 1

(u� − a�) u� (u� + a�)H� − u�a� 1

2u�2 H� + u�a�

�, 𝛌� = �u� − a� 0 0

0 u� 00 0 u� + a�

� (9)

D and X-1 are matrix diagonal and inverse eigenvector

matrix respectively and speed of sound, a = �γ(p/ρ) . All quantities with the hat that appears in (9) are evaluated by Roe average:

H� = HL�ρL + HR�ρR

�ρL + �ρR ρ� = �ρLρR

u� = uL�ρL + uR�ρR

�ρL + �ρR c� = �(γ − 1)�H� − q�2/2�


63

v� = vL�ρL + vR�ρR

�ρL + �ρR V� = u�nx + v�ny + w�nz

w� = wL�ρL + wR�ρR

�ρL + �ρR 𝑞�2 = 𝑢�2 + 𝑣�2 + 𝑤�2

As a Riemann approximation solver, Roe’s scheme reads [15].

Fi+12

n =12�F

i+12

R + Fi+12

L � +12

��𝛌� (𝐤)�𝚫𝐔(𝐤)𝐗�(𝐤)3

k=1

�i+12

(10)

Fi−12

n =12�F

i−12

R + Fi−12

L � +12

��𝛌� (𝐤)�𝚫𝐔(𝐤)𝐗�(𝐤)3

k=1

�i−12

(11)

According to [15] and [17] Roe’s vector terms in (10-11) is formulated as:

��𝛌� (𝐤)�𝚫𝐕(𝐤)𝐗�(𝐤)3

k=1

= ∆𝐅 = 𝐀�𝚫𝐔 = 𝐀�(𝐔𝐑 − 𝐔𝐋)

= 𝐗𝐃𝐗−𝟏(𝐔𝐑 − 𝐔𝐋) (12)

Hence, (10-11) turns to the following forms. Fi+12

n

=12�F �U

i+12

R � + F �Ui+12

L ��

+12

𝐗𝐃𝐗−𝟏(UR − UL)i+12

(13)

Fi−12

n =12�F �U

i−12

R � + F �Ui−12

L ��

+12

𝐗𝐃𝐗−𝟏(UR − UL)i−12

(14)

With MUSCL’s interpolation, velocity terms in (13-14) are

formulated as follow [17].

Ui+12

L = Uin +

12∆i+12

; Ui+12

R = Ui+1n −

12∆i+32

(15)

Where:

∆i+12

(L) =12��1 + k��ϕr

i+12

− ∆+(Ui)

+ �1 − k��ϕri−12

+ ∆−(Ui)� (16)

∆i+32

(R) =12��1 + k��ϕr

i+12

+ ∆−(Ui+1)

+ �1 − k��ϕri+32

− ∆+(Ui+1)� (17)

k� is a free parameter lies in interval [-1,1], where for

k� = 0, ∆i is a central difference approximation, multiplied by ∆x, to the first spatial derivative of the numerical of the numerical solution at the time level n. MUSCL’s interpolation can be more accurate with quadratic reconstruction, that are [10]:

ri+12

+ =Ui+2 − Ui+1

Ui+1 − Ui r

i+12

− =Ui − Ui−1

Ui+1 − Ui

ri−12

+ =Ui+1 − Ui

Ui − Ui−1 r

i+32

− =Ui+1 − Ui

Ui+2 − Ui+1 ⎭⎬

⎫ (18)

With r

i+32

− = rR and ri−12

− = rL, and following definition:

Δ+Ui = Ui+1 − Ui Δ−Ui = Ui − Ui−1

Δ+Ui+1 = Ui+2 − Ui+1 Δ−Ui+1 = Ui+1 − Ui

� (19)

It can be written as:

∆i+12

(L) =12ψL(Ui − Ui−1), ∆

i+32

(R) =12ψR(Ui+2 − Ui+1) (20)

With:

ψL =12��1 + k��rLΦ �

1rL� + �1 − k��ΦrL� (21)

ψR = 12��1 + k��rRΦ �

1rR� + �1 − k��ΦrR� (22)

The (15) can be simplified if we consider slope limiters

with the symmetry property as:

Φ(r) = Φ�1r� (23)

Thus, (15) becomes:

Ui+12

L = Uin +

12

[ψL(Ui − Ui−1)];

Ui+12

R = Ui+1n −

12

[ψR(Ui+2 − Ui+1)] ⎭⎬

⎫ (24)

With limiter function is defined as:

ψL/R =12��1 + k��rL/R + �1 − k��ΦL/R (25)

MUSCL scheme is divided into two category where it is

determined by value of k�. MUSCL0 represents k� = 0 and MUSCL3 for k� = 1

3. For the MUSCL3 for k� = 1

3, Van Albada

flux limiter, Φ and limiter function, ψ followed below expression.

Φ =3r

2r2 − r + 2 , ψ =

2r2 + r2r2 − r + 2

� (26)

For simplification purpose, (24) is written in this form:

Ui+12

L = Uin +

12δ𝐿 (27)

Ui+12

R = Ui+1n −

12δ𝑅 (28)

Where:


64

δ =(2a + ϵ)b + (b2 + 2ϵ)a(2a2 + 2b2 − ab + 3ϵ)

(29)

The additional parameter 𝜖 in (27) prevents the activation

of the limiter in smooth flow regions due to small-scale oscillations [10]. This is sometimes necessary in order to achieve a fully converged steady-state solution. For this purpose, 𝜖 is set at 0.00001 while other alphabets are defined as follow.

aR = Δ + ui+1 aL = Δ + uibR = Δ − ui+1 bL = Δ − ui

�

In order to increase the accuracy and to extend the stability

region [17], solution is enhanced by Runge-Kutta multistep method. It first has been developed by Jameson [5] with applying a five-stage Runge-Kutta to advance the solution in time. Updating solution due to Runge-Kutta methods, it follows steps below.

Q(0) = Qn

Q(1) = Q(0) − α1∆t

s∆xQ(0)

Q(2) = Q(1) − α2∆t

s∆xQ(1)

Q(3) = Q(2) − α3∆t

s∆xQ(2)

Q(4) = Q(3) − α4∆t

s∆xQ(3)

Q(5) = Q(4) − α5∆t

s∆xQ(4)

Qn+1 = Q(5)

α1 =14

α2 =16

α3 =38

α4 =12

α5 =15

Description of Compressible Boundary Layer Equation Keller box is a method to solve partial differential equation which can be categorized as finite difference method. However, it is derived from the governing equation of compressible flow including mass, momentum, and energy equation with the algebraic eddy viscosity (εm) and turbulent Prandtl number (Prt ) formulated by Cebeci and Smith [8], they can be expressed as follows: ∂∂x

(ρu) +∂∂y

(ρv)�� = 0 (30)

ρu∂u∂x

+ ρv��∂u∂y

= ρeueduedx

+∂∂y�(µ + ρεm)

∂u∂y� (31)

ρu∂H∂x

+ ρv��∂H∂y

=∂∂y��k + cpρ

εmPrt

�∂T∂y

+ u(µ + ρεm)∂u∂y

� (32)

Where T is temperature, H is the total enthalpy given by:

H = cpT + u2

2 and ρv�� = ρv + ρ′v′��

The boundary conditions for an adiabatic surface are:

y = 0, u = 0, v = 0, ∂H∂y

= 0

y → ∞, u → ue(x), H → He

Grid Generation Generating grid for computational space can be undergone in various techniques. Present study uses structured grid C-type as obtained by Blazek [10] namely C_GRID_GENERATOR. This method is dealt with elliptic partial differential equation or specifically Poisson equation. C type is one of the grid topology which is enclosed by one family of grid lines and also forms the wake region. The situation is shown is Fig 3.1 where tangential lines( η =const. ) start at the farfield of perpendicular direction (ξ = 0), follow the wake, pass the trailing edge (node b), surround the body in clockwise, then reach farfield again at (ξ = 1). For the other grid lines( ξ = const. ) exudes in normal direction from the body and wake. The coordinate cut that is represented by segment of a-b of grid lines at ( η = 0) physically map onto two segments in the computational space namely a ≤ ξ ≤ b and b′ ≤ ξ ≤ a′ for lower space and upper space respectively.

Fig. 3.1 C-grid topology in two-dimension

Elliptic equations for the two-dimension grid generation are:

𝛼11 �∂2x∂ξ2

+ P∂x∂ξ� − 2𝛼12

∂2x∂ξ ∂η

+𝛼22 �∂2x∂η2

+ 𝑄∂x∂η� = 0 (33)

𝛼11 �∂2y∂ξ2

+ P∂y∂ξ� − 2𝛼12

∂2y∂ξ ∂η

+𝛼22 �∂2y∂η2

+ 𝑄∂y∂η� = 0 (34)

Where metrics coefficient in equation are:


65

(

)

(

)

(

)

(

)

IV. RESULT AND ANALYSIS

Numerical high resolution of Euler’s solver scheme namely

Roe’s scheme is represented in this section. An illustration

about computational space of RAE 2822 is shown in Fig 4.1,

where it is a result from C_GRID_GENERATOR code.

Fig 4.1 Structured grid of 2D RAE 2822 airfoil.

Case A

This test case performs of a RAE 2822 at Mach number (M)

of 0.676, angle of attack (AOA) of -2.18 and Reynolds

number (Re) of 5.7 million. The distribution of computed

pressure around airfoil surface is shown in Fig. 4.2, well

compared with experimental and Semi-Discrete-Galerkin

(SDG VII) data. Displacement thickness (DELS) and friction

coefficient (CF) of upper surface is shown in Fig. 4.3 and Fig.

4.4 respectively. All results for case A show a good agreement

with experimental data and mutually with SDG VII.

Fig 4.2 Pressure coefficient distribution along RAE 2822 airfoil

surface for case A.

Fig 4.3 Displacement thickness distribution along RAE 2822 airfoil

surface for case A.

Fig 4.4 Friction coefficient distribution along RAE 2822 airfoil

surface for case A.

Case B

At AOA of 1.90, M of 0.676, and Re of 5.7 million,

prediction of pressure coefficient as depicted in Fig. 4.5

becomes more deviant especially at upper surface. It should be

noted that for the upper surface flow achieved supersonic

speed and leads this miscomputation. The small errors also

found at lower surface near to trailing edge. However in

prediction of DELS and CF, it still gains well achievement as

shown in Fig. 4.6 and Fig. 4.7 respectively.

Fig. 4.5 Pressure coefficient distribution along RAE 2822 surface for

case B.


66

Fig 4.6 Displacement thickness distribution along RAE 2822 airfoil

surface for case B.


surface for case B.

Case C

As M and Re is increased, the graph fashion entirely

changed due to the presence of shock wave. Nonetheless, the

computation gives an excellent result despite very small

overpredicted occurred along upper surface as illustrated in

Fig. 4.8. It also shows Roe’s scheme is better than SDG VII

method in working on inviscid flow regime especially dealing

with shock wave while, MUSCL interpolation scheme also

display well ability in reducing spurious oscillation of shock

wave. The boundary layer characteristics also been affected by

shock wave as shown in Figs. 4.9 and 4.10 for DELS and CF

respectively.

Fig 4.8 Pressure coefficient distribution along RAE 2822 surface

for case C.

Fig 4.9 Displacement thickness distribution along RAE 2822

airfoil surface for case C.


surface for case C.

Case D

In this case, airfoil is set at M of 0.75, AOA of 2.72 and Re

of 6.5 million. From the Fig. 4.11, it shows overpredicted of

shock wave happened because of high Mach number then

produce very strong pressure gradient in the flow attached

upper surface. For the DELS and CF as depicted in Figs. 4.12

and 4.12 respectively, strong pressure gradient led flow

separation at 70% chord length and computation remains

failed to provide any boundary layer characteristics. ARC2D

demonstrates better method in dealing with such flow.

Fig 4.11 Pressure coefficient distribution along RAE 2822 surface

for case D.


67

Fig 4.12 Displacement thickness distribution along RAE 2822

airfoil surface for case D.


surface for case D.

V. CONCLUSION

Present study has proposed a combination of Roe’s scheme

and Keller Box method as a computational method to deal

with a flow around 2D airfoil RAE 2822. As discussed in

previous section, present method reveals a good ability in

emulating experimental results as provided by Cook,

McDonald, and Firmin [11]. Meanwhile, MUSCL

interpolation scheme exposed an excellent performance due to

diminishing spurious oscillation for all case. Furthermore the

results of present study can be improved through interaction of

viscous-inviscid by updating inviscid velocity with Veldman

interaction law [8].

ACKNOWLEDGMENT

This research is sponsored by University Tun Hussein Onn

Malaysia under postgraduate faculty. Present work is provided

according to requirement of International Scientific Academy

of engineering & Technology (ISAET).

REFERENCES

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Design”, AEC Research and Development Report, New York, Courant

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and Supersonic Flow”, France, Tech. Rep. AGARD-AG-187, 1974.

[3] A. Jameson, “Iterative Solution of Transonic Flows over Airfoils and Wings, Including Flows at Mach 1”, Courant Inst. of Mathematics and

Science, New York, 1974.

[4] A. Jameson, , W. Schimidt, , and E. Turkel, “Numerical solution of the

Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes”, AIAA Journal, pp.81-1259, 1981

[5] A. Arias, O. Falcinelli, N.J. Fico, and S. Elaskar, “Finite Volume

Simulation of a Flow Over a Naca 0012 Using Jameson, Maccormack, Shu and Tvd Esquemes”, Mechanical Computational, vol. 16, Córdoba,

pp. 3097-3116, 2007.

[6] H. Sclichting, and K, Gertsen, Bounday layer theory. 8th Revised and Enlarged Edition, Germany: Springer, pp.392-400 2000.

[7] T. Cebeci, P. Bradshaw, and J. Whitelaw, Engineering Calculation

Methods For Turbulence Flow, London: Academic Press, 1981. [8] T. Cebeci, and A.M.O. Smith, Analysis of Turbulence Boundary

Layers, London: Academic Press, 1974.

[9] P.L. Roe, “Approximate Riemann Solvers, Parameter Vectors, and Differences Schemes”, Journal of Computational Physics, United

Kingdom, no. 357-372, 1981.

[10] J. Blazek, Computational Fluid Dynamics: Principles and Applications, United Kingdom: Elsevier Science Ltd, Oxford , pp.93-115, 2001.

[11] P. H. Cook, M. A. McDonald and M. C. P. Firmin, “Airfoil RAE 2822 -

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Advisory Report No. 138, London, pp. A6(1)-A6(77), 1979.

[12] C.M. Maksymiuk, and T.H. Pulliam, “Viscous Transonic Airfoil Workshop Results Using Arc2d”, AIAA Journal, pp. 87-0415, 1987

[13] B.A Day, and A.J. Maede, “Semi-Discrete Galerkin Solution of

Compressible Boundary Layer Equations with Viscous-Inviscid Interaction,” in 11th AIAA Applied Aerodynamics Conf. California,

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Dynamics, Berlin: Springer-Verlag , pp. 345-372, 1999.

[15] H. Nishikawa, “A Comparison of Numerical Flux Formulas for the Euler Equation”, Final Assigment, 1998.

[16] K.A. Hoffmann, & S.T Chiang, Computational Fluid Dynamics, vol. 2,

Kansas: www.eesbooks.com, 2000. [17] C. Kroll, M. Aftosmis, and D. Gaitonde, “An Examination of Several

High Resolution Schemes Applied to Complex Problems in High Speed

Flows”, Final Report Wright Laboratory, Ohio, AD-A250 814, 1992.


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