Application of Weighted Least Squares Kinetic Upwind Method
using Eigendirections(WLSKUM-ED) to 3-D Flows
Konark Arora1 arora@aero.iisc.ernet.in
N.K.S. Rajan2 nksr@cgpl.iisc.ernet.in
S. M. Deshpande3 smd@jncasr.ac.in
1
2
Department of Aerospace Engineering,
Indian Institute of Science, Bangalore 560012.
Combustion Gasification and Propulsion Laboratory, Department of Aerospace Engineering,
Indian Institute of Science, Bangalore 560012.
3
Engineering Mechanics Unit,
Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560064.
Abstract. The WLSKUM-ED(Weighted Least Squares Kinetic Upwind Method) using the
EigenDirections has shown to be quite robust owing to its ability to work on any point distribution and avoiding the bad connectivity induced code divergence. The weights in the
WLSKUM-ED are calculated to diagonalize the least squares matrix such that the multidimensional least squares formulae reduce to the corresponding one dimensional formulae along
the eigendirections. This method has been successfully extended to 3-D flow problems and
applied to a wide variety of point distributions on various complex bodies ranging from the
simplest body of a hemisphere to a multibody configuration of ASLV with strap-on boosters.
Keywords: Gridfree, Connectivity, Eigenvector basis, Defect correction, WLSKUM-ED
1
Introduction
The grid free methods based on kinetic schemes[3, 4, 8, 1, 7] are gaining popularity over the conventional
methods in solving the flow field past complex configurations. The main reason for the growth of this
popularity is their ability to work on any point distribution. These methods donot require the grid for the
flow simulation, which is an essential requirement for all other conventional CFD methods. The grid-free
methods do require point distribution or clouds of points. The point distribution around any configuration
can be generated by either a random number generator or a combination of several simple grid generators.
However, an important issue of good connectivity generation for the points in the computational domain
cropps up in place of the grid generation. The connectivity of the nodes affects not only the robustness
but also the accuracy and the convergence characteristcs of the grid free solvers[7]. A new method
to deal with the issue of the robustness of the grid free methods has been developed and successfully
applied to 2-D flows using the WLSKUM-ED (Weighted Least Squares Kinetic Upwind Method using
Eigendirections) [2, 5]. Here, the same method has been successfully extended to the 3-D flows. In this
paper, a brief introduction to the WLSKUM-ED method will be given, followed by the description of
the method of calculation of the positive weights in 3-D. Finally the results on the 3-D test cases will be
discussed in support of our claim.
0 Submitted
to 9th Annual Aesi CFD Symposium, 11th -12th August,2006, Bangalore
2
Weighted Least Squares Method in 3-D
Consider Fig.1 showing connectivity of Po . To get the derivative of the function F (x, y) at point Po ,
we expand Fi around point Po in terms of Taylor series
y
P0
x
z
Figure 1: Connectivity for node o in 3 dimension
Fi = Fo + ∆xi Fxo + ∆yi Fyo + ∆zi Fzo + O (∆xi )2 , (∆yi )2 , (∆zi )2 , ∆xi ∆yi , ∆xi ∆zi , ∆yi ∆zi
(1)
Define the differences
∆xi = xi − xo ,
∆y i = yi − yo ,
∆z i = zi − zo , ∆F i = Fi − Fo
The weighted sum of the squares of errors at point Po (after truncating Taylor Series in Eq.(1)) is given
by
X
X
2
E=
wi e2i =
wi (∆Fi − ∆xi Fxo − ∆yi Fyo − ∆zi Fzo )
(2)
Minimizing the error E in Eq.(2) with respect to Fx o ,Fy o and Fz o , we get the following system of linear
algebraic equations
T
A (w) (∇F )o = b (w)
(3)
where
P
P

 P
w ∆x 2
w ∆x ∆y
w ∆x ∆z
P i i
Pi i 2 i P i i i
A (w) =  wi ∆xi ∆yi
w ∆y ∆z  ,
w ∆y
P
P i i
Pi i 2i
wi ∆xi ∆zi
wi ∆yi ∆zi
wi ∆zi


P

Fx (1)
wi ∆xi ∆Fi
o
P


T
, b (w) =  wi ∆yi ∆Fi 
(∇ F )o = Fy (1)
o 
P
(1)
wi ∆zi ∆Fi
Fz o
(4)
The above system of linear algebraic equations can now be solved by any method to get the 3-D least
squares (LS) derivatives.
To derive the WLSKUM method for the Euler equations, the spatial derivatives in the 3-D CIR split
Boltzmann equation in the Euler limit are discretized by the weighted least squares method using the
approproate split stencils shown in the Fig.2, Fig.3 and Fig.4.
y
y
y
N(P3)
x
P0
x
P0
N(P6 )
P0
x
N(P1)
N(P2 )
N(P5 )
z
z
z
N(P4)
Figure 2: Split stencil in X - direction in 3-D
dF
dt
Figure 3: Split stencil in Y - direction in 3-D
v1 + |v1 | ∂F
+
2
∂x N (P1 )
v2 − |v2 | ∂F
+
2
∂y N (P4 )
+
v1 − |v1 | ∂F
+
2
∂x N (P2 )
v3 + |v3 | ∂F
+
2
∂z N (P5 )
Figure 4: Split stencil in Z - direction in 3-D
v2 + |v2 | ∂F
+
2
∂y N (P3 )
v3 − |v3 | ∂F
=
2
∂z N (P6 )
0 (5)
Now taking the moments of the discretized Boltzmann equation Eq.(5), we get the Weighted Least Squares
Kinetic Upwind Method (WLSKUM) for the Euler equation as given in Eq.(6)
dU
dt
+
+
∂
∂
∂
+
−
+
+
+
+
(GX)
(GX)
(GY )
∂x
∂x
∂y
N (P2 )
N (P1 )
N (P4 )
∂
∂
∂
−
+
−
+
+
=
(GY )
(GZ)
(GZ)
∂y
∂z
∂z
N (P3 )
N (P6 )
N (P5 )
+
+
0 (6)
where (GX) , (GX) , (GY ) , (GY ) , (GZ) and (GZ) are the split fluxes in the corresponding
directions.
The crux of the WLSKUM-ED lies in reducing multidimensional LS formulae for derivatives to
1-D formulae. For this purpose, it is necessary to diagonalize the LS matrix A (w) given in Eq.(4). As in
case of 2-D formulae [2, 5], this is achieved by choosing the weights such that the x, y and z directions
become the eigendirections of the LS matrix A (w). However, it must be observed that as against 2D LS matrices, the weights have to be obtained to simultaneously satisfy the following conditions for
diagonalization the 3-D LS matrix ie.
X
X
X
wi ∆xi ∆yi = 0
wi ∆xi ∆zi = 0
wi ∆yi ∆zi = 0
(7)
−
−
−
The method of determining the weights for satisfying the above condition along with the positivity
constraint has been explained in the next section. Now, if we put the weights so obtained in the Eqs.(3)
and (4), we get

  (1)  P
P
Fx o
wi ∆xi ∆Fi
wi ∆xi 2
0.0
0.0
P
P
 


(8)
=
w ∆y ∆Fi 
0.0
wi ∆yi 2
0.0
Fy (1)
o 
P i i
P
2
(1)
wi ∆zi ∆Fi
0.0
0.0
wi ∆zi
Fz o
Sr. No.
1.
2.
3.
4.
Quadrant
I
II
III
IV
∆x∆y
+
+
-
∆y∆z
+
+
-
∆z∆x
+
+
Table 1: Signs of Cross-Products in the Quadrants of Split Stencil
The solution of this system of equations is obviously ,
P
P
wi ∆xi ∆Fi
wi ∆yi ∆Fi
(1)
(1)
Fxo = P
Fyo = P
,
2 ,
wi ∆xi
wi ∆yi 2
Fzo
(1)
P
wi ∆zi ∆Fi
= P
wi ∆zi 2
(9)
It is thus observed that all the multidimensional least squares formulae have reduced to the corresponding
one dimensional formulae.
3
Calculation of weights for three dimensional least squares formulae
(I) One method of calculation of the weights for the three dimensional formulae is similar to that
of the two dimensional formulae, but the weights have to be calculated such that the products
P
P
P
wi (∆xi ∆yi ), wi (∆y∆z i ) and wi (∆z i ∆xi ) simultaneously vanish. The least squares matrix
A (w) operates on a 3D linear vector space of <3 . It has been proved that the eigenvectors of A (w)
provide an orthogonal basis for <3 . Thus mathematically, the above method of calculation of weights
amounts to decomposing <3 as <3 = e1 ⊕e2 ⊕e3 , where e1 , e2 and e3 are the eigenvectors of A (w).
Consider Fig.5 which shows the quadrants of split stencil necessary for enforcing upwinding for the
Z Axis : Normal to the paper
Y
II
∆ x < 0 , ∆ x ∗∆ y < 0
I
∆ x > 0 , ∆ x ∗ ∆ y> 0
∆ y> 0 , ∆ y∗∆ z > 0
∆ y> 0 , ∆ y∗∆ z > 0
∆z > 0 , ∆x ∗ ∆z< 0
∆z > 0 , ∆x ∗ ∆z> 0
∆ x < 0 , ∆ x ∗ ∆ y> 0
∆ x > 0 , ∆ x ∗∆ y < 0
∆ y< 0 , ∆ y ∗ ∆ z < 0
∆ y< 0 , ∆ y ∗ ∆ z < 0
∆z > 0 , ∆x ∗ ∆z< 0
∆z > 0 , ∆x ∗ ∆z> 0
X
III
IV
Figure 5: Quadrants of the split stencil in 3-Dimension
case ∆zi > 0.This split stencil is used for calculating the derivative of negative flux GZ − in the
z direction making use of the points in all the four quadrants shown as I, II,III and IV in the Fig.5.
Here we observe that each of the products ∆xi ∆yi , ∆yi ∆zi and ∆zi ∆xi is always positive in two
of the quadrants while negative in the remaining two. This fact is used to calculate the weights for
the corresponding points in the quadrants. Let wI , wII , wIII and wIV be the weights assigned to
the points in each of the quadrants I, II, III and IV respectively. The signs of cross products in the
quadrants are shown in Table 1. Now the conditions to be satisfied are :
P
P
wi (∆xi ∆yi ) = 0,
wi (∆y i ∆z i ) = 0,
P
wi (∆z i ∆xi ) = 0.
We obtain the following equations
I
II
III
IV
wI Cxy
+ wII Cxy
+ wIII Cxy
+ wIV Cxy
=0
(10)
I
II
III
IV
wI Cyz
+ wII Cyz
+ wIII Cyz
+ wIV Cyz
=0
(11)
I
wI Czx
(12)
+
II
wII Czx
+
III
wIII Czx
+
IV
wIV Czx
=0
where Cpj is the notation for the cross product in the j th quadrant in the pth plane. For example,
II
taking j = II and p = yz, we get Cyz
. Writing the system of equations in matrix form

I
Cxy
I
Cyz
I
Czx
II
Cxy
II
Cyz
II
Czx
III
Cxy
III
Cyz
III
Czx

  
wI
0
IV 
Cxy
 wII  0
IV  
  
Cyz
wIII  = 0
IV
Czx
wIV
0
(13)
Looking at Eq.(13), we see that we have to solve a system of three equations for the four unknowns
viz. wI , wII , wIII and wIV Using the signs given in Table 1, the above system of equations (13)
can be written as
  

 I
0
wI
II
III
IV 
|Cxy | −|Cxy | |Cxy | −|Cxy | 



w
I
II
III
IV   II  = 0
|Cyz
(14)
| |Cyz
| −|Cyz
| −|Cyz
| 


0
wIII
I
II
III
IV
|Czx
| −|Czx
| −|Czx
| |Czx
|
0
wIV
Now, the system of equations in (14) can be written in the following possible ways


IV 
I
II
III  
−|Cxy
|
wI
|Cxy
| −|Cxy
| |Cxy
|
IV 
I
II
III  
|Cyz
|
| |Cyz
| −|Cyz
|
wII  = −wIV −|Cyz
IV
I
II
III
|Czx
|
|Czx
| −|Czx
| −|Czx
| wIII


 III 
I
II
IV  
|Cxy |
wI
|Cxy
| −|Cxy
| −|Cxy
|
III 
I
II
IV  
|Cyz
|
| |Cyz
| −|Cyz
|
wII  = −wIII −|Cyz
III
I
II
IV
−|Czx |
wIV
|Czx | −|Czx | |Czx |
 I


III
IV  
II 
wI
|Cxy | |Cxy
| −|Cxy
|
−|Cxy
|
I
III
IV  
II
|Cyz


| −|Cyz | −|Cyz |
wIII = −wII |Cyz | 
I
III
IV
II
|Czx | −|Czx | |Czx |
−|Czx
|
wIV





I 
II
III
IV
|Cxy
|
wII
−|Cxy
| |Cxy
| −|Cxy
|
I 
II
III
IV  
 |Cyz
wIII  = −wI |Cyz
|
| −|Cyz
| −|Cyz
|
I
II
III
IV
wIV
|Czx
|
−|Czx
| −|Czx
| |Czx
|

(15)
(16)
(17)
(18)
We can use any of the above linear set of Eqs.(15), (16), (17) or (18) to obtain weights by assuming
the free parameter as unity. For example, we can use the system (15) and assume w IV = 1.0 and
then obtain wI ,wII and wIII as the solution of (15). It has been shown [2] sometimes weights
become negative, if say we solve (15). In such a case, we can consider then another system (16)
and keep shifting the system to be solved till we get positive weights. This method has been shown
to work successfully in many cases [2].
(II) In the second method, we are not interested in finding the weights such that all the (x, y, z) directions
become the eigen directions as done in the first method described above. Suppose, we want to
calculate the x derivative using the given connectivity of the node. In order to reduce the three
dimensional formula for the x derivative to the one dimensional formula for the x derivative at
the node, we require that only the x direction be made the eigen-direction of the least squares
matrix A (w). This condition is less restrictive and gives more flexibility in calculating the weights
ensuring the positivity constraint. Mathematically, this method of weights calculation amounts to
decomposing <3 as <3 = e1 ⊕ 2-D vector space <2 . Thus if we minimize the weighted sum of
the squares of the error in Eq.(2) with respect to Fx o , we get
X
dE 2
= −2
wi ∆Fi − ∆xi Fxo − ∆yi Fy o − ∆zi Fz o (∆xi )
dFxo
X
= −2
wi ∆Fi ∆xi − wi ∆xi 2 Fxo − wi ∆yi ∆xi Fy o − wi ∆zi ∆xi Fz o
(19)
P
P
From Eq.(19), we see that if we satisfy the conditions
(wi ∆yi ∆xi ) = 0 and
(wi ∆zi ∆xi ) = 0,
we will be able to use only a 1-D formula for the x derivative instead of a 3-D formula. Now, using
the notations described above, we have to solve the following two equations for the four unknown
weights:
I
II
III
IV
wI Cxy
+ wII Cxy
+ wIII Cxy
+ wIV Cxy
=0
(20)
I
II
III
IV
wI Czx
+ wII Czx
+ wIII Czx
+ wIV Czx
=0
(21)
The system of equations (21) can be written in matrix form as
I
Cxy
I
Czx
II
Cxy
II
Czx
III
Cxy
III
Czx
IV
Cxy
IV
Czx

  
wI
0
 wII  0

  
wIII  = 0
0
wIV
(22)
Using the signs given in Table 1, the above system of equations (22) reduces to

  
wI
0
I
II
III
IV
 0
|Cxy
| −|Cxy
| |Cxy
| −|Cxy
| 
w
II

= 
I
II
III
IV
|Czx
| −|Czx
| −|Czx
| |Czx
| wIII  0
wIV
0
(23)
Now there are again various combinations in which the system of equations in (23) can be written.
Different possible ways are
I
III
II
IV
|Cxy | −|Cxy
| wI
|Cxy | −|Cxy
| wIII
=
−
(24)
I
II
III
IV
|Czx
| −|Czx
| wII
−|Czx
| |Czx
|
wIV
I
III
II
IV
wI
|Cxy
| |Cxy
|
−|Cxy
| −|Cxy
| wII
=
−
I
III
II
IV
|Czx
| −|Czx
| wIII
−|Czx
| |Czx
|
wIV
I
II
III
IV
−|Cxy
| |Cxy
|
wII
|Cxy | −|Cxy
|
wI
=−
II
III
I
IV
−|Czx
| −|Czx
| wIII
wIV
|Czx
| |Czx
|
(25)
(26)
We observe that as against method (I), here we have two free variables for each of the system of
linear algebraic equations. We can form other combinations of the system of equations to solve for
Mach Contours : LSKUM
1
WLSKUM−ED : Hemisphere
LSKUM : Hemisphere
2
0.1
1
0.01
RESIDUE
0
0.001
-1
1e−04
-2
1e−05
-2.5-2-1.5-1-0.5 0
1e−06
Mach Contours : WLSKUM-ED
1.95
1.88
1.8
1.72
1.65
1.57
1.5
1.43
1.35
1.27
1.2
1.12
1.05
0.975
0.9
0.825
0.75
0.675
0.6
0.525
0.45
0.375
0.3
0.225
0.15
0.075
2
1
0
-1
-2
-2.5-2-1.5-1-0.5 0
1.95
1.88
1.8
1.72
1.65
1.57
1.5
1.43
1.35
1.27
1.2
1.12
1.05
0.975
0.9
0.825
0.75
0.675
0.6
0.525
0.45
0.375
0.3
0.225
0.15
0.075
1e−07
0
500
1000
1500
ITERATIONS
2000
2500
3000
Figure 7: Mach Contours :
WLSKUM-ED : M = 2.0
Figure 6: Residue drop comparison
LSKUM and
P
the weights, each such combination giving us one possible set of weights satisfying
wi ∆xi ∆y i =
P
wi ∆xi ∆z i = 0. Thus this method gives us more flexibility in calculating the weights enforcing
the positivity constraint.
Second order accuracy in case of least squares formulae is obatined by the defect correction method
[3]. The least squares formulae for the derivatives using the defect correction applied in 3-D become
Fxo
where
(1)
P
gi
wi ∆xi ∆F
,
= P
wi ∆xi 2
Fyo
(1)
P
gi
wi ∆yi ∆F
= P
,
wi ∆yi 2
Fzo
(1)
P
gi
wi ∆zi ∆F
= P
wi ∆zi 2
(27)
gi = ∆F i − ∆xi ∆Fxi − ∆y i ∆Fy − ∆z i ∆Fz i
(28)
∆F
i
2
2
2
We can also employ q-variables for achieving second order accuracy. We then get 3D q-WLSKUM.
4
Results and Discussion
Results of the 3-D test cases of the supersonic flow past a hemisphere, the supersonic flow past the
multibody ASLV (Augmented Satellite Launch Vehicle) configuration with strap-on boosters and the
transonic flow past M165 configuration have been presented. There are 26133 nodes in the computational
domain of the hemisphere. The flow conditions for this test case are free stream Mach No. of 2.0 and
angle of attack of 0.0o . The residue drop obtained using the LSKUM and WLSKUM-ED solvers for the
hemisphere test case is compared in the Fig.6. The residue drop comparison from both the solvers shows
the good convergence characteristics of the WLSKUM-ED solver as compared to that of the LSKUM
solver. The Mach contours over the hemisphere are shown in the Fig.7.
The next test case used for testing the 3-D WLSKUM-ED code is that of the supersonic flow past
the multibody configuration of the ASLV with two strap-on-boosters - one on each side of the core body
in the yaw plane. There are approximately 2.5 lakh nodes in the computational domain. The test case
has been run for the supersonic flow free stream Mach number of 2.09 at side-slip angle (β) of 0.0 o and
angle of attack (α) of 0.0o . The Mach contours on the body of ASLV at an angle of attack of 0.0o are
Figure 8: Mach Contours on
ASLV Body : M = 2.0o , α =
0.0o
Figure 9: Mach Contours in the
Pitch Plane of the ASLV Body
: M = 2.0o , α = 0.0o
Figure 10: Mach Contours in
the Yaw Plane of the ASLV
Body : M = 2.0o , α = 0.0o
shown in the Fig.8. The Mach contours in the pitch plane and the yaw plane are shown in the Figs.9 and
10 respectively for angle of attack of 0.0o . The detached bow shock is observed at the nose of the core
body of the ASLV. The oblique shock wave is seen at the nose of the boosters in the yaw plane, which
is impinging the core body of the ASLV. The cp plots over the core body and the booster of the ASLV
configuration are shown in the Figs. 11 and 12 respectively.
The next test case is that of the transonic flow past M165 configuration at a free stream Mach number
of 0.789, side-slip angle (β) of 0.00o and angle of attack (α) of 4.36o . This test case has been chosen to
demonstrate the ability of the WLSKUM-ED solver to work effectively on a point distribution obtained
from FAME (Feature Associated Mesh Embedding) Mesh [7].The scatter plot of Mach number distribution over the M165 configuration is shown in the Fig.13. The comparison of the Cp distribution obtained
from the WLSKUM-ED and the q-LSKUM solvers over the symmetry plane of the fuselage is shown in
the Fig.14. The Cp distribution obtained from WLSKUM-ED solver, q-LSKUM solver and FAME solver
of UK at the root and mid section of the M165 wing is compared in the Figs. 15 and 16 respectively.
A good comparison is obtained in both the cases at the chordwise location except at the suction surface
at the leading edge of the mid span of canard, where the FAME and q-LSKUM solvers are highly overpredicting as compared to the WLSKUM-ED solver. The Cp distribution at various spanwise locations
on the M165 wing has been shown in the Figs.17 to 22. It has been compared with the q-LSKUM results,
FAME results and the experimental results at all the six spanwise locations. It is observed that the
WLSKUM-ED results show a better comparison with the experimental results as compared to both the
FAME results and the q-LSKUM results which are highly overpredicting the Cp valus at the leading edge
at 19% and 30% span location. It is observed that the WLSKUM-ED solver gives better comparison
with the experimental results as compared to the q-LSKUM solver and the FAME solver.
5
Conclusion
The 3-D WLSKUM-ED method has been successfully applied to various 3-D test cases varying from
the simplest test case of a hemisphere to the complex multibody configurations. The results have been
compared with the experimental results and a good comparison with the experimental results has been
obtained. This method has been checked for its robustness by running on various types of point distributions ranging from the one obtained from structured grid generator to the one obtained from multiple
2
1.6
Cp
Cp
1.4
1.5
1.2
1
0.8
Cp
Cp
1
0.5
0.6
0.4
0.2
0
0
-0.2
-0.5
0
200
400
600
800
1000
-0.4
600
1200
700
800
900
X/C
1000
1100
1200
Figure 13: Scatter plot of Mach
no. distribution over M165 configuration
X/C
Figure 11: Cp plot over the core
body of ASLV Configuration
Figure 12: Cp plot over the
booster of ASLV Configuration
0.4
0.5
0.3
WLSKUM
q-LSKUM
WLSKUM
q-LSKUM
FAME
0.3
WLSKUM
q-LSKUM
FAME
0.4
0.2
0.2
0.3
0.1
0.1
0.2
0
0.1
Cp
Cp
Cp
0
-0.1
0
-0.1
-0.2
-0.1
-0.3
-0.2
-0.2
-0.4
-0.3
-0.5
-0.6
-1
-0.5
0
0.5
1
1.5
2
-0.3
-0.4
-0.2
2.5
X
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
-0.4
-0.05
0.2
0
0.05
Figure 14: Cp distribution at
symmetry plane of fuselage
Figure 15: Cp distribution at
root section of canard
0.3
0.2
0.5
WLSKUM
FAME
q-LSKUM
EXP
0.3
0.2
0.15
Figure 16: Cp distribution at
mid span section of canard
0.4
WLSKUM
FAME
q-LSKUM
EXP
0.1
X
X
WLSKUM
FAME
q-LSKUM
EXP
0.4
0.3
0.2
0.1
0.2
Cp
Cp
Cp
0.1
0
0.1
0
0
-0.1
-0.1
-0.1
-0.2
-0.2
-0.3
-0.2
-0.3
0
0.2
0.4
0.6
0.8
1
-0.3
0
0.2
0.4
X
0.6
0.8
1
0
0.2
0.4
X
Figure 17: Cp distribution at
19 % wing span
Figure 18: Cp distribution at
30 % wing span
0.5
0.8
1
Figure 19: Cp distribution at
49 % wing span
0.6
WLSKUM
FAME
q-LSKUM
EXP
0.4
0.6
X
0.6
WLSKUM
FAME
q-LSKUM
EXP
0.5
WLSKUM
FAME
q-LSKUM
EXP
0.5
0.4
0.4
0.3
0.3
0.2
0.2
Cp
Cp
0.2
0.1
0
Cp
0.3
0.1
0.1
0
0
-0.1
-0.1
-0.2
-0.2
-0.1
-0.2
-0.3
-0.3
-0.3
-0.4
0
0.2
0.4
0.6
0.8
1
X
Figure 20: Cp distribution at
61 % wing span
-0.4
0
0.2
0.4
0.6
0.8
1
X
Figure 21: Cp distribution at
74 % wing span
0
0.2
0.4
0.6
0.8
1
X
Figure 22: Cp distribution at
86 % wing span
overlapping chimera grids like the FAME. It is observed that all the essential flow features have been
accurately captured by this method, thereby showing great promise of the method.
Acknowledgements
The authors are grateful to Dr. A.K. Ghosh, Group Director (MCA), ADA, and Dr. C. Praveen,
CTFD, NAL for providing the FAME mesh and its LSKUM data structure for the M165 configuration
and giving their kind consent for its use for the purpose of the validation of the 3-D code. The authors
also acknowledge the help received from Dr. V. Ramesh, CTFD, NAL and Dr. Anandhanarayan, CFD
Division, DRDL, Hyderabad in obtaining the point distributions for various 3-D test cases like that of
ASLV configuration along with its LSKUM data structure for the purpose of the 3-D code validation.
References
[1] Anandhanarayanan, K. (2003) Development and Applications of a Gridfree Kinetic Upwind Solver
to Multibody Configurations, PhD. Thesis, Department of Aerospace Engineering, Indian Institute
of Science, Bangalore, India.
[2] Arora, Konark and Deshpande, S.M. (2004) Weighted Least Squares Kinetic Upwind Method using
Eigenvector Basis, FM Report No. 2004 FM 17, Department of Aerospace Engineering, Indian
Institute of Science, Bangalore, India.
[3] Ghosh, A.K. (1996) Robust Least Squares Kinetic Upwind Method for Inviscid Compressible Flows,
PhD. Thesis, Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India.
[4] Ghosh, A.K. and Deshpande, S.M. (1995) Least Squares Kinetic Upwind Method for Inviscid Compressible Flows, AIAA Paper No. 95-1735.
[5] Konark Arora, N.K.S. Rajan and S.M. Deshpande (2005), Weighted Least Squares Kinetic Upwind
Method (WLSKUM) using Eigenvector Basis, presented at 8th Annual Aesi CFD Symposium, 11th 13th August,2005, Bangalore.
[6] Mandal, J.C. and Deshpande, S.M. (1994) Kinetic Flux Vector Splitting for Euler Equations, Computers and Fluids, Vol 23. No. 2, pp. 447-478.
[7] Praveen, C. (2004) Development and Applications of Kinetic Meshless Methods for Euler Equations,
PhD. Thesis, Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India.
[8] Ramesh, V. (2001) Least Squares Grid-Free Kinetic Upwind Method, PhD. Thesis, Department of
Aerospace Engineering, Indian Institute of Science, Bangalore, India.