calculation of naca 0012 airfoil through roe’s scheme...

Abstract— The present work performs the Roe’s scheme in

solving Euler equation, applied to the solution of flow over 2D airfoil

NACA 0012 and compares to experimental data and XFOIL

software. As one of the approximation solvers for Riemann scheme,

Roe’s scheme is extended to second order through MUSCL scheme.

MUSCL scheme has been applied in order to eliminate the spurious

oscillation due to shock wave presence. In this work, flow is treated

as compressible with Mach number around 0.3 to 0.8 and Reynolds

number is set at 3 million. Results show a good potential have been

made by present method especially at low angle of attack and low

Mach number. To improve the accuracy of solution, present study

proposed viscous effect should be included where viscosity plays a

major role in determination of aerodynamic characteristics

particularly for high speed aerodynamic.

Keywords— Inviscid flow, Euler solver, Roe’s scheme, and

MUSCL scheme.

I. INTRODUCTION

INCE early 20th

century, aerodynamics progressively

evolved in a wide range and become very interesting

subject in engineering and mathematic. Study on flow

behavior for single airfoil is neither new nor primitive subject

in aerodynamics. Since Eastman and Abbott [1], investigation

has been made on various types of NACA airfoil through

experiment within variable-density wind tunnel.

Consequently, Eastman made another attempt which more

systematic experimentation with relating a number of

N.A.C.A airfoil in a wide range of Reynolds number [2]. In

other occasions, experimental data on single airfoil also been

carried out by Wenzinger [3], Eastman and William [4],

Abbott, Von Doenhoeff, and Stivers [5], Moyers [6], Ferri [7],

and many others.

Numerical solution on 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 [8]. Subsequently, many works

was made by researchers to examine the capability of solution

methods. Lax, Roe, McCormack, Godunov, Ritchmeyer and

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]).

Rusanov who invented applicable schemes frequently gained

attention by succeeding researchers. Taylor explained some of

favorite schemes thoroughly in dealing with such serious

difficulties of aerodynamic problem [9]. 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 namely potential flow, Euler

solver and Navier stokes solution.

Behind these advances, experimental approach exhibited

similar improvement as computational one since the

technology of wind tunnel experienced successive

modification. Gregory and Wilby [10] on their study

aerodynamics characteristics of airfoil NPL9615 and NACA

0012 provided a complete data set for these airfoils at various

Mach number of subsonic flow. Consequently, Gregory made

another effort with O’Riley on NACA 0012 which including

effect of upper surface roughness [11]. Another excellent

accomplishment has been done by Harris [12] via experiment

on two-dimensional NACA 0012 within Langley 8-foot wind

tunnel where measurement is conducted at subsonic speed and

relatively high Reynolds number. Experiments on NACA

0012 were also made by Langley [13], Nash, Quincey,

Callinan [14]and Murdin[15]. All those experimental results

show uncertainty difference of each other and it caused

difficulty to validate CFD results.

Jameson carried on aerodynamics discovery with his work

on airfoils through numerical potential flow solution [16]. This

work offers the solution of flow at sonic Mach number and

also implements artificial viscosity as a shockwave treatment.

The great work by Jameson on Euler methods can be found in

[17] 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 dealing with Euler equation

is by Arias et. al[18]. In this research finite volume has been

simulated for a 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;

Calculation of NACA 0012 Airfoil through

Roe’s Scheme Method

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

S

International Conference Recent treads in Engineering & Technology (ICRET’2014) Feb 13-14, 2014 Batam (Indonesia)

http://dx.doi.org/10.15242/IIE.E0214521 51

mailto:[email protected]

(iii) and finally the Shu scheme, which uses a variation of the

Jameson time integration, in order to better capture of shock

waves.

Another effort that related to present work is explained by

Maciel [19] which demonstrated several high resolution of

TVD’s scheme to be dealt with two-dimension aerodynamic

problem. There are six schemes employed here namely Roe’s,

Van Leer vector splitting, Harten-Yee, Yee-Kutler and,

Hugson-Beran. In order to reach accuracy of second order,

Roe and Van Leer scheme apply MUSCL approach and other

schemes used Harten’s ideas of the construction of a modified

flux function to obtain second order accuracy and TVD

characteristics. This paper also offered solution in both of

formulation: implicit and explicit. Implicit solved through

ADI (“Alternating Direction Implicit”) approximate

factorization while the explicit one’s used a time splitting

method. Lastly, study concluded that Roe’s scheme exhibit the

best agreement to the experimental results both in the implicit

and explicit formulation due to the best estimative of the

shock angle.

II. PROCEDURE FOR COMPUTATION

Present airfoil analysis is employing with Euler equation to

deal with two-dimension inviscid flow over airfoil NACA

0012. 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 increase

accuracy of second order formulation. 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

problem of single airfoil NACA 0012 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 [20].

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

the study 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 [20] is utilized namely AIRFOIL_ROE_SCHEME in

solving the objective of study.

There are plenty of experimental data can be used as a

weighing scale for analysis, however it must be chosen

depends on fundamental of experiment. The work of Gregory

and Wilby [10] and, Gregory and O’Riley [11] were reliable

and matched to be an assessment data set since experiment

was conducted at subsonic flow and Reynolds number about

2.88 X 106. Moreover, it will facilitate comparison process

where the results of experiment represent pressure coefficient

distribution CP [11], lift coefficient CL, and moment

coefficient CM [10]. For the high Mach number, experimental

data have been taken from Harris in [21] as suggested in [22].

Another comparison has been made with aerodynamics

software namely XFOIL by Drela where represents a

combination of panel method and global Newton method.

Resolution that offered by XFOIL mainly applicable for low

Reynolds number case, while at high Reynolds number case,

software exhibit inconsistence results. A brief description of

Euler solution and computer code is shown in the next section.

III. THE GOVERNING EQUATION

A. 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 is [23]:

Where:

[

] [

], [

] }

With:

(

)

}

Above equation is known as Compressible Euler Equation

and represents a highly nonlinear partial differential equation

and there is no analytical solution. denotes as the ratio of

specific heat capacities of the gas. In a two dimensional, Euler

equation is wrote in hyperbolic equation form.

Where A and B is Jacobian matrix system

[

]

}

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:

(

) [

]

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

derived as follow [24]:



(

)

(∑|

|

)

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. is eigenvector matrix

correspondents to matrix eigenvalues respects to similarity

such that [25]:

[

] [

]

D and X-1

are matrix diagonal and inverse eigenvector

matrix respectively and speed of sound, √ . All quantities with the hat that appears in (9) are evaluated by

Roe average:

√ √

√ √ √

√ √

√ √ √ ( )

√ √

√ √

√ √

√ √

As an Riemann approximation solver, Roe’s scheme reads

[26].

(

)

(∑| |

)

(

)

(∑| |

)

According to [24] and [27] Roe’s vector terms in (10-11) is

formulated as:

∑| |

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

( (

) (

))

( (

) (

))

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

formulated as follow [28].

Where:

[( )

( )

]

[( )

( )

]

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

, is a central difference approximation, multiplied by

, 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

[20]:

}

With

and

, and following definition:

}

Thus:

[( ) (

) ( ) ]

[( ) (

) ( ) ]

It can be written as:

With:

[( ) (

) ( ) ]



[( ) (

) ( ) ]

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

with the symmetry property as:

(

)

Thus, (15) becomes:

[ ]

[ ] }

With limiter function is defined as:

[( ) ( )]

MUSCL scheme is divided into two category where it is

determined by value of . MUSCL0 represents and

MUSCL3 for

For the MUSCL3 for

, Van Albada

flux limiter, and limiter function, followed below

expression.

}

For simplification purpose, Eq (26) is written in this form:

Where:

The additional parameter in (29) prevents the activation

of the limiter in smooth flow regions due to small-scale

oscillations [20]. This is sometimes necessary in order to

achieve a fully converged steady-state solution. For this

purpose, is set at 0.00001 while other alphabets are defined

as follow.

}

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 [18] with

applying a five-stage Runge-Kutta to advance the solution in

time. Updating solution due to Runge-Kutta methods, it

follows steps below.

Generating grid for computational space can be undergone

in various techniques. Present study uses structured grid C-

type as obtained by Blazek [20] 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 lines start at

the farfield ( , follow the wake, pass the trailing edge

(node b), surround the body in clockwise, then reach farfield

again at ( . For the other grid lines exudes

in normal direction from the body and wake. The coordinate

cut that is represented by segment of a-b of grid lines at

physically map onto two segments in the

computational space namely and 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:

(

)

(

)

(

)

(

)

Where metrics coefficient in equation are:



(

)

(

)

(

)

(

)

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 is portrayed in Fig 4.1, where it is a

result from C_GRID_GENERATOR code. Pressure

coefficient of airfoil NACA 0012 is observed as shown in Fig

4.2, an agreement between experiment and computation is

achieved in excellent manner for low angle of attack and low

Mach number (M = 0.3) cases. At high angle of attack within

low Mach number, Roe’s solver still provides fairly prediction

as XFOIL did. At Mach number 0.8, a wide deviation is

occurred due to shock wave presence. As shown in Fig 4.2 a

good agreement is achieved ahead shock wave come off, and

Roe’s scheme exposes poor capability to capture such

phenomenon. Nonetheless, compare to software XFOIL,

Roe’s scheme remains exceptional since XFOIL software was

limited to flow at considerably low speed.

From Figs 4.2-4.5 it can be observed that two parameters

impede computational are high angle of attack and large Mach

number. Higher angle of attack affected calculation with over-

prediction occurred in tracking maximum pressure coefficient,

CP. In similar manner as present method, software XFOIL

also poses alike fashion even in determining the point of

maximum CP, XFOIL remains greater than prediction of

present method. From point of view, it simply can be realized

that the present method with no viscosity effect exhibit a good

quality in emulating experimental data. Another parameter

mentioned is Mach number. As depicted in Fig 4.4 and 4.5 for

Mach number 0.7 and 0.799 respectively, error deviates

proportionally to Mach number, where at Mach number 0.799,

with existence of shock wave, error radically exceeded 20%

chord. It implies for relatively larger Mach number as

transonic, present method remains unrealistic to be applied.

On the other hand, an excellent work done can be seen in the

Fig 4.5 where assessment of MUSCL’s interpolation scheme

plays role in diminishing spurious oscillation of shock wave.

Fig 4.1: View of structured grid about NACA 0012.

Fig. 4.2: Pressure coefficient distribution along NACA 0012

surface at angle of attack 6 and Mach number 0.3.

Fig 4.3: Pressure coefficient distribution along NACA 0012

airfoil surface at angle of attack 16.5 and Mach number 0.3.

Fig 4.4: Pressure coefficient distribution along NACA 0012




Fig 4.5: Distribution of pressure coefficient along NACA 0012


For more general, it is convenient if we look at flow

behavior by viewing lift coefficient characteristic at various

angles of attack. Fig 4.6 illustrates lift data in range of angle of

attack between 0 to12 for three difference approach. Roe’s

scheme shows relatively good estimation to XFOIL for Mach

number 0.3. As angle of attack arose, difference between

present method and experiment becoming more evident and

gains its peak at maximum lift point. Nevertheless, this fine

performance of present method descending as Mach number is

increased towards to transonic flow. Apparently, it can be seen

in the Fig 4.7 how the Roe’s scheme performed a better

prediction than XFOIL software in presuming lift coefficient,

CL.

Fig 4.6: Distribution of lift coefficient along NACA 0012 airfoil

surface at various angles of attack and Mach number 0.3.


surface at various angles of attack and various Mach number.



The same tendency of CL can be found for the moment

coefficient CM tracing, where inadequate of prediction as

shown in Fig 4.8 happened at relatively high angle of attack

and at large Mach number. Typically for this study, neglecting

of viscous effect is identified as a major factor for this lacking

since viscosity plays a big part for compressible flow

especially at high Mach number. Fig 4.8 also shows XFOIL

made a better imitation than present method while at Mach

number equal to 8, as depicted in Fig 4.9 XFOIL remains

unreliable method to be used in computing such flow

behavior.





V. CONCLUSION

Present study has proposed Roe’s scheme as a

computational method to deal with flow around 2D airfoil

NACA 0012 at Reynolds number 3 million and Mach number

from 0.3 to 0.8. As discussed in previous section, present

method reveals a good ability in emulating experimental

results as provided by Gregory and Wilby [10], and Harris

[12]. Viscosity effect is detected as causal factor for inaccurate

prediction particularly for the high Mach number occasion.

Generally, computational results were outstanding instead of

XFOIL software. In linearizing the second order of Roe’s

scheme, MUSCL with Van Albada limiter exposed an

excellent performance due to diminishing spurious oscillation.

Artificial viscosity is suggested to be included in governing

equation to pursue the accuracy or another technique can be

used is interaction boundary layer approach as alternative.

ACKNOWLEDGMENT

This research is sponsored by University Tun Hussein Onn

Malaysia under postgraduate faculty. Present work is provided

according to requirement of International Institute of

Engineering (IIENG) and to be presented at Batam, Indonesia.

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calculation of naca 0012 airfoil through roe’s scheme...

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