AIAC-2009-033
4. ANKARA INTERNATIONAL AEROSPACE CONFERENCE
17-19 August, 2009 - METU, Ankara
NUMERICAL ANALYSIS OF UNSTEADY FLOW AROUND A OSCILLATOR AIRFOIL WITH
MOVING STRUCTURED ADAPTIVE GRID BY USING CENTRAL AND UPWIND SCHEMES
Mahmood Pasandideh Fard1 and Ali Heidary2
Majid Malekjafarian3
Ferdosi University
of Mashhad, Iran
University of Birjand
Birjand, Iran
ABSTRACT
This paper deals with an adaptive dynamic grid around oscillationg airfoils and compares Roe upwind
methods, with scalar and CUSP central difference schemes. Euler equations have been solved for the
external invisid compressible flow. Thus, at first, the equations were written into the integrated form, and
then in CUSP and Scalar schemes, by means of the central difference method, and in Roe scheme, by
means of the upwind difference method, they were made into the discritized form. In this paper, these
schemes will be introduced, and the results of these methods will be compared.
To adapt grids, when bodies move or rotate, the spring network analisys is used. In this paper edges are
replaced with linear springs. Some numerical solutions are run to balance forces, applies to springs that
connected to a node, by displacements. a logical definition is assumed for linear spring coefficient and
displacement of nodes. All we have done, are for structured meshes for airfoil. Then boundaries are moved
or rotated and meshes are refined by using spring network analogy.
Keywords: Adaptive mesh, Structured grid, spring analogy, Roe upwind method, scalar and cusp central
difference schemes
INTRODUCTION
In this work we consider the problem of deforming structured two-dimensional grids. Deforming meshes
are used in many computational applications including problems with moving boundaries and interfaces, and
for adaption. In all these cases, the motion of a portion of the domain boundary is known, and one wants to
deform the rest of the mesh in order to accommodate these imposed displacements.
The most commonly used technique for grid deformation is based on the spring analogy method, whose
basic idea is to create a network of springs connecting all nodes in the grid. In the first and simplest of this
class of methods each mesh edge is replaced by a spring, whose stiffness is inversely proportional to the
edge length. This way, longer edges will be softer, while shorter ones will be stiffer, somewhat preventing the
collision of neighboring vertices.[1]This method is accurate for small displacements that called linear spring
analogy. Unfortunately, in many practical cases the necessary grid displacements are not small, for example
when conducting implicit coupled fluid structure interaction simulations. Furthermore, even if the
displacements are small, the edge spring method can not prevent the creation of nearly flat elements and
also it will lead to collaps mesh networks. Therefore, there is a need for methods that can specifically deal
with large deformation problems. To address this issue, torsional springs were added to the linear edge
springs in References [2,3,4] in order to avoid the possible collapse mechanisms of mesh network. With the
same goal, in this work we propose a new simple method of controlling collapse mechanisms in structured
meshes. This method is based on the idea of complementing the linear edge springs with secondary linear
springs in two dimensions.
Furthermore, we will study the performance of central and upwind schemes in solving Euler and Navier-stoks
equations. Since, the central difference schemes do not imply enough dissipation, the artificial dissipation,
which has an important effect on the accuracy of the solutions, should be added to the equations. The nonoscillating difference schemes were established two decades ago. One of the foremost suggested schemes
in this field was represented by A Jameson, in which too much dissipation should be added, whereas it is
needed to choose the amount of the coefficients as little as possible that they can only damp the oscillations
[6]. Considering this, Jameson et al has recently introduced CUSP scheme that will be studied in this paper
[7].
1
2
3
Prof. Associate. Department of Mechanical Engineering, Email: fardmp@yahoo.com
Phd student, Email: heydary.a@gmail.com
Prof. Assistant. Department of Mechanical Engineering, Email:mmjafarian@excite.com
Generally, the central difference schemes are based on a symmetrical estimate of the information, which are
saved in the points adjacent to the surfaces of the cell. Therefore, the central difference methods do not
consider the direction, from which the information comes to the cell, and ignore the hyperbolical nature of the
problems.
The second group is the upwind methods. These methods are based on the distribution of the flow
information along the defined directions in the physical domain. So, these methods have a good coincidence
with the physics of the flow information all over the field of the fluid flow. Moreover, the distributional nature
of these methods has made them reliable. But on the other hand, the upwind schemes are much more
complex in programming, so that they need a larger memory to determine the velocity of the sonic waves
(eigen values), which define the distribution direction of the information [8].
GOVERNING EQUATIONS AND DIFFERENT SCHEMES
The following equation shows the general form used for a one-dimensional flow:
w


f ( w)  0
(1)
t
x
In CUSP and Scalar central schemes the numerical fluid flow between the cells i and i+1 is obtained as
following:
1
 f
 f  d
( 2)
f
2
di+1/2 is the dissipation, which is added to the equations to prevent the oscillations.
The basis of CUSP scheme is on separating the pressure terms of the momentum fluxes. The dissipation
term in this scheme is defined as the following form [7]:
i 1 / 2
d
i 1 / 2

1
2

i 1
*
i 1 / 2

CQ
i 1
i 1 / 2
i

Q 
i
1
2

i 1 / 2
f
i 1

f
i

(3)
In which, by separating the pressure term in the flux vector we have:
f
i 1

f

i


 u .(W i 1  Wi )  w.(ui 1  ui )  (

1
(Wi 1  Wi )
2
w
u
,
f
)
p i 1
 (
f
)
(4)
p i
1
( u i 1  u i )
2
(5)
The coefficients α*c and β in equation (3) are given in ref. [7]:
In order to have a more precise result, the artificial dissipation terms are needed by larger amounts adjacent
to the shocks, and by less amounts in the rest of the solution domain. For this purpose, a limiter function,
L (u , v ) , is used to determine the flow characteristics as below:

. L(u , v)  1 1  u  v

2
q
u v


(u  v )


(6)
The power q is optional and is between 2 and 3.
R
Now W
i+1
and W
should be replaced by the new amounts wE and
i
w
L
E
. These changes should be added
to all over the domain of the solution.
R
w
E
L
w
E
 wi 1 
 wi 
1
L( wi  3 / 2 ,  wi 1 / 2)
2
1
L( wi  3 / 2 ,  wi 1 / 2)
2
(7)
In Scalar scheme the dissipation flux is defined as following :
d
i 1 / 2
The coefficients



2
i 1 / 2
,
2
i 1 / 2

Wi 1 / 2 
4
i 1 / 2

4
i 1 / 2
( Wi 3 / 2  2Wi 1 / 2  Wi 1 / 2 )
(8)
are constants which are well known and are given in refs [1,4,5].
This scheme is designed such that in the vicinity of the shocks much more artificial dissipations are added to
the governing equations.
In Roe scheme the numerical flux, is defined as :
f
i 1 / 2

1
2
 f
R

f
L
  


A



Q
R
Q
L

(9)
Where A is the Roe matrix. , In first-order Roe scheme it is assumed that the initial variables inside the
control volume are constant, and their amounts are related to the both sides of each cell surface, by a zero
degree polynomial for the points ( j, k ) and ( j+1, k ). The second and third order schemes are obtained
using a linear averaging of Q R and Q L respectively.
PROPOSED CUSP DISSIPATION SCHEME
As it has been mentioned, in order to apply the advantages of CUSP artificial dissipations method with the
high Mach numbers, it is necessary to imply some corrections. These corrections have been done by means
of Roe methods with more than one- order accuracy.

Equations (4) and (5) show an ordinary averaging has been done to calculate

u and w .
To increase the accuracy of the results, we can involve more points. So to obtain a higher orders we propose
that the following schemes should be applied.
1- Second-order accuracy
2- Third-order accuracy


These corrections are applied so that the amounts of
u
w
and
can be obtained by means of the parabolic
or linear symmetrical extrapolations around the surface of the cell as the following form:

1
w  2 (w
L
E

 wE )
1
u  2 (u
R
R
E
 uE)
L
R
If the parabolic extrapolation is used, the amounts of wE ,
(10)
L
R
w , uE
and
E
u
L
E
are calculated as the following
equations:
1
(  w j  2, k  5 w j 1, k  2 w j ,k )
6
1
R
wE  6 ( w j  2,k  5 w j 1,k  2 w j ,k )
1
L
u E  6 ( u j 2  5 u j 1  2 u j )
1
R
u E  6 ( u j  2  5 u j 1  2 u j )

L
w
E
(11)
And if a linear extrapolation is used, to get the second-order accuracy, the equations (11) are written as
below:
1
(3

)
2 w j w j 1
1
R
wE  2 (3 w j 1  w j  2)
1
L
u E  2 (3u j  u j 1)
1
R
u E  2 (3u j 1  u j  2)
L
w
E

(12)
LINEAR SPRING ANALOGY
In this section we briefly review the classical linear spring method. Assume grid points as following: defining i
in direct of body surface and j in direct of normal to body surface. Length of edge is calculated with eq.13
[13]
Dij 
x
i, j
 xi 1, j

2
  y i , j  y i 1, j

(13)
2
and spring coefficient definition as following:
ij
C linear

1
Dijp
(14)
Thus short edges are stiffer than longer ones, which provides a beneficial effect in the control of the local
element deformation. The position of each vertex is found by writing its equilibrium under the effect of all its
neighbour edge-connected springs.
ne
f
j 1
edge
ij
(15)
0
After writing equilibrium equation and simplifying it we have:




Ci , j S i, j  S i , j   S i1, j  S i 1, j   Ci 1, j S i1, j  S i 1, j   S i, j  S i , j 
(16)
Then by solving eq.16 using an implicit algorithm we can obtain the new position of each node.
Secondary Linear Spring
To avoid collapsing of grids and prevent passing a node across an edge we defined a secondary set of linear
springs that are in direct of normal to body surface(motion). We first deal with the equilibrium in springs that
situate on edges along the motion direct. Afterward the equilibrium on normal to direction of motion must be
determined.
FOLLOWING PROBLEMS
Assume a NACA0012 airfoil that fluctuate with equation :
 t    0   m Sint 
(17)
Coefficient of this equation for two problem is defined as following.
case
airfoil
M
0
m

xm
c
CT2
NACA0012
0.6
3.16
4.59
0.162
0.273
CT5
NACA0012
0.755
0.016
2.51
0.162
0.25
In these problems ω is the rotating velocity, Xm/c is a point that airfoil rotates around it,
angle of rotation and airfoil rotates around
m
is the maximum
 0 . All exprimental data was taken from [14]
RESULTS AND CONCLUSION
 For the impedented adaptive grid method 22 second time is enough to oscillate airfoil in 1000 iterate but if
we regenerate grids in each step, 970 seconds is required. To consider one period of oscillation with 50000
iterates, it will last 2 days for regenerating all grids.
 By investigation figs.1 that plot lift coeff. vs angle of rotation for 6 oscillation in CT5 problem and comparing
with fig 2 for CT2 problem we can understand by approaching to supersonic flow, ROE and CUSP become
more sensitive for oscillation. But scalar is not sensitive for this.CUSP scheme is closer to exprimental
results than other scheme.
 Figs. 5 to 9 show pressure coeff. vs X in different angles of attack. At trailing edge CUSP scheme is closer
to exprimental results than other schemes.
Fig 3:lift coeff. vs angle of rotation(CT5) scalar
scheme
Fig 7: zooming of fig 7
in the middle of airfoil
Fig 4: lift coeff. vs angle of rotation(CT2)
Fig 8: pressure coeff. vs x in angle of
2.38 downward(CT5)
Fig 9: zooming of fig 7
in the end of airfoil
Fig 10: pressure coeff. vs x in angle of -2.41
downward(CT5)
Fig 11: pressure coeff. vs x in angle of
-0.51upward(CT5)
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