2005 FM 1
AR&DB Centre of Exellene for Aerospae CFD
Department of Aerospae Engineering
Indian Institute of Siene
Fluid Mechanics Report
Mathematical Analysis of Dissipation in m−kf vs Method
N Anil
AR&DB Centre of Excellence for Aerospace CFD
Department of Aerospace Engineering
Indian Institute of Science
Bangalore 560012
NKS Rajan
CGPL
Department of Aerospace Engg.
Indian Institute of Science
Bangalore 560012
Report 2005 FM 1
May 2005
SM Deshpande
Engineering Mechanics Unit
Jawaharlal Nehru Centre for
Advanced Scientific Research
Jakkur, Bangalore 560064
Fluid Mechanics Report
Mathematical Analysis of Dissipation in m − kf vs Method
N Anil
AR&DB CFD Centre,
Department of Aerospace Engg.,
Indian Institute of Science,
Bangalore 560012, India
Email: anil@aero.iisc.ernet.in
NKS Rajan
Combustion Gasification & Propulsion Laboratory,
Department of Aerospace Engg.,
Indian Institute of Science,
Bangalore 560012, India
Email: rajan@cgpl.iisc.ernet.in
SM Deshpande
Engineering Mechanics Unit,
Jawaharlal Nehru Centre for Advanced Scientific Research,
Bangalore 560064, India
Email: smd@jncasr.ac.in
1
Introduction
The Modified Kinetic Flux Vector Split Method, in short called as the m − kf vs method [1,
2] belongs to the family of low dissipative kinetic schemes [11, 8, 1] where a molecular velocity
dependent function has been used as a dissipation control parameter. In this report we have
presented some of the interesting mathematical properties of the m − kf vs method. In section 2
we have presented briefly the basic theory behind the m − kf vs method. The modified kinetic
split fluxes and some of its nice properties are studied in Section 3. The m − kf vs flux Jacobians
and the corresponding eigenvalues are computed numerically in Section 4. The coefficient of
numerical dissipation is presented in Section 5.
2
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Anil, Rajan and Deshpande
m-kfvs
The kinetic schemes [6, 10, 5] exploit the connection between the Boltzmann equation of the
kinetic theory of gases and the Euler or Navier-Stokes equations of the continuum mechanics.
In these schemes, upwinding is enforced at the Boltzmann level and taking moments with a
suitable distribution function, we arrive at an upwind scheme for the Euler or Navier-Stokes
equations. This is called the moment-method-strategy by Deshpande [5].
In this section we explain the basics of yet another kinetic scheme called m − kf vs method
with respect to 1D Euler equations. Consider the 1D Boltzmann equation in the Euler limit
∂F
∂F
+v
=0
∂t
∂x
(1)
where F is the Maxwellian velocity distribution function given by
ρ
F =
I0
r
β
exp −β(v − u)2 − I/I0
π
(2)
Here, ρ density, v molecular velocity, u fluid velocity, I internal energy variable, R gas constant
1
per unit mass, β = 2RT
, I0 internal energy due to nontranslational degrees of freedom, given by
I0 =
3−γ
4(γ − 1)β
The Euler equations can then be obtained as
∂U
∂G
∂F
∂F
+
= Ψ,
+v
= 0
∂t
∂x
∂t
∂x
(3)
where U is the vector of conserved variables and G is the flux vector. The inner product Ψ, f
is defined by
Z
Ψ, f =
Ψf (v)dvdI
(4)
R+ ×R
The moment function vector Ψ is defined by


1


Ψ= v 
2
I + v2
(5)
Using MCIR splitting [11, 12] of the molecular velocity v, eq. (1) becomes
∂F
v + |v|φ ∂F
v − |v|φ ∂F
+
+
=0
∂t
2
∂x
2
∂x
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Department of Aerospace Engg.
Mathematical Analysis of Dissipation in m − kf vs Method
3
where φ is a molecular velocity dependent dissipation control function. Note that, for the case of
φ = 1, the above equation reduces to the usual CIR split [4] Boltzmann equation. The different
choices [1] for the function φ are given by
α
− |v|
and
φ=e
φ = e−α|v|
(7)
where α could be a mesh or flow dependent function and has the dimension of velocity. A
detailed analysis on these choices and the corresponding physical arguments are presented in [1,
2]. Taking the Ψ moments of the eq. (6), we get the m − kf vs Euler equations
∂U
∂Gm+ ∂Gm+
+
+
=0
∂t
∂x
∂x
(8)
where Gm± are the modified split kinetic (m − kf vs) fluxes. Forward and backward difference
appproximations to the respective spatial derivatives and using Taylor series expansion, eq. (8)
gives the modified partial differential equation corresponding to the Euler equations
∂U
∂G
∆x ∂ 2
+
=
Gm+ − Gm− + O (∆x)2
2
∂t
∂x
2 ∂x
Z
∆x ∂ 2
=
Ψ|v|φF dvdI + O (∆x)2
2 ∂x2
(9)
R+ ×R
From the above equation it follows that when φ = 1, we get Gm± = G± , the usual kf vs fluxes
and usual upwind scheme based on kf vs is obtained. When φ = 0, we get a central differencing
scheme. Thus in m − kf vs method by tuning φ, the coefficent of numerical dissipation can be
controlled and hence the formal order of accuracy. In the next section we present the m − kf vs
fluxes and some of its nice mathematical properties.
3
Modified split kinetic fluxes
The modified split kinetic (m − kf vs) fluxes [1] for the case of 1D Euler equations are given by
Z
v ± |v|φ
±
Gm =
Ψ
F dvdI
(10)
2
R+ ×R
For the function φ, we have chosen φ = e−α|v| , for which the closed form expressions for the
split fluxes are available. Using this choice the m − kf vs fluxes are given by
2
α
α2
G 1
α
α
−αu
+αu
±
+
−
Gm = ± e 4β
G u−
− e 4β
G u+
(11)
2
2
2β
2β
where G is the unsplit flux (total flux) and G± are the kfvs fluxes for the 1D Euler equations
and are presented in Appendix A. It can be observed that the modified split fluxes depend on
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FM Report 1:2005
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Anil, Rajan and Deshpande
the non-dimensional parameter α̃ =
G 1
Gm =
± e
2
2
±
√α .
β
√
α̃2
−α̃u β
4
In terms of α̃, eq. 12 reduces to
α̃ G u− √
−e
2 β
+
√
α̃2
+α̃u β
4
α̃ G u+ √
2 β
−
(12)
Before studying the nature of the m−kf vs fluxes it is worthwhile to recall some of the important
properties of the kf vs fluxes [10, 3]. They can be summarised as follows:
1. The positive split fluxes will always remain positive and the negative split fluxes will always
be negative.
2. As Mach number increases beyond unity, the positive split fluxes asymptotically approach
the corresponding unsplit fluxes. On the other hand, the negative split fluxes continue to
decrease asymptotically and become negligibly small. That is, in the limit
M → ∞ : G+ → G and G− → 0
3. The kf vs fluxes have smooth transition characteristics throught the Mach number domain.
They are continuous differentiable functions, which is an extremely important property
both from mathematical and numerical point of view.
Fig. 1 confirms all the above nice properties of the kf vs fluxes. The m − kf vs fluxes are plotted
with respect to Mach number with varying α̃ and these plots are shown in Figs. 6 to ??. The
important features of these fluxes are
1. Like the kf vs fluxes, the modified split kinetic fluxes are also smooth throughout the Mach
number range. They are continuous differentiable functions.
2. As the value of α̃ increases by keeping M fixed, the m−kf vs fluxes asymptotically approach
half the value of corresponding unsplit fluxes. Numerical experiments show that around
α̃ = 4.0, they almost approached the value G2 . This clearly shows that the upwinding is
completely lost in this limit. To explain this in more detail, consider the flux evaluation
across a cell face
−
Gf ace = Gm+
L + GmR
For large values of α̃, we get
GL + GR
2
which is like a central differencing. That is, as the value of α̃ increases, the fully upwind
approximation tends to central differencing.
Gf ace →
3. An interesting property of these split fluxes is that, for a given α̃, as Mach number increases
the modified split kinetic fluxes asymptotically approach half the value of unsplit fluxes.
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Department of Aerospace Engg.
Mathematical Analysis of Dissipation in m − kf vs Method
5
4. Thus, by tuning α̃ in the m − kf vs fluxes, we can control the dissipation in the numerical
scheme and hence the order of accuracy.
4
Modified split kinetic flux Jacobians and their eigenvalues
Mandal [10] has studied in great depth the kf vs split flux Jacobians and the corresponding
eigenvalues using symmetric hyperbolic form [7] for the inviscid Euler equations. Chou [3]
has presented the corresponding split kinetic flux Jacobians and its eigenvalues for the NavierStokes equations based on Chapman-Enskog distribution function. In this section we derive
the m − kf vs flux Jacobians and analyze the corresponding eigenvalues based on numerical
experiments. The modified split kientic (m − kf vs) flux Jacobians are defined by
∂Gm±
(13)
∂U
where U is the vector of conserved variables and Gm± are the modified split kinetic fluxes. The
expressions for the Jacobians Am± are given in Appendix B. Since Gm± are functions of α̃,
and for the case of α̃ = 0, we get the Jacobians corresponding to kf vs fluxes. The expressions
for the Jacobians Am± are presented in Appendix B. It can be easily shown that the Jacobians
given by eq. (13) satisfy the property
Am± =
∂G
(14)
∂U
It clearly shows that the split flux Jacobians given by eq. (22) are very complex as they involve
a lot of error functions and exponentials. Because of this complexity it is very difficult to find
the eigenvlaues manually. However, we can use MATHEMATICA [9] to find the eigenvalues
analytically but the expressions are very tedious and complex so that a close study of the
eigenvalues is almost impossible. Figs. (8) and (9) show the eigenvalues of the positive split flux
Jacobian based on m − kf vs for different values of α̃. As α̃ increases, the eigenvalues of Am+
approach half the eigenvalues of the full flux Jacobian A. This is evident from the fact that
Am+ + Am− = A =
Gm± →
5
G
2
for large values of α̃
(15)
Coefficient of numerical dissipation
We now analyze the coefficient of numerical dissipation obtained based on m − kf vs method
for the 1D Euler equations. Consider the MCIR split Boltzmann equation (6). Forward and
backward differencing of the spatial derivatives accordingly as v < 0 and v > 0 on a three point
stencil (j − 1, j, j + 1), we get
∂F
v + |v|φ Fj − Fj−1 v − |v|φ Fj+1 − Fj
+
+
=0
∂t j
2
∆x
2
∆x
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FM Report 1:2005
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Anil, Rajan and Deshpande
Using Taylor series expansion and taking the Ψ - moments of the above equation, we get the
modified partial differential equation corresponding to 1D Euler equations
∂U
∂G
∆x ∂ 2
+
=
Gm+ − Gm− + O (∆x)2
2
∂t
∂x
2 ∂x
This can be rewritten in the artificial viscosity form as
∂U
∂U
∂U
∂ ∆x ∂
+A
=
Gm+ − Gm−
+ O (∆x)2
∂t
∂x
∂x
2 ∂U
∂x
(17)
(18)
From the above equation, the numerical dissipation matrix D is given by
∆x ∂
Gm+ − Gm−
2 ∂U
∆x
Am+ − Am−
=
2
D (U ) =
(19)
Here, A is the full flux Jacobian and Am± are Jacobians of modified split kinetic fluxes and
are presented in the Appendix B. For the upwind scheme based on m − kf vs to be stable,
the dissipation matrix D should be positive definite. That is, all the eigenvalues must be real,
positive and non-zero. Alternatively, we can choose the coefficient of dissipation as
ν = M ax (|λ1 |, |λ2 |, |λ3 |)
(20)
where λ1 , λ2 and λ3 are the eigenvalues of the dissipation matrix D. For a stable scheme the
coefficient ν should be positive. In the present work we have computed the eigenvalues of the
matrix D numerically. Figs. (10) and (11) show the eigenvalues of the dissipation matrix for
different vlaues of α̃. These eigenvalues are compared to the corresponding eigen values of the
dissipation matrix based on kf vs. It can be clearly seen that as α̃ increases the eigenvalues are
asymptotically tending to zero.
Appendix A
kfvs split fluxes in one dimension:
The kfvs flux expressions for the 1D Euler equations are given by


ρuA± ± √ρπβ B


2 )A± ± √ρu B

(p
+
ρu
G± = 


πβ
p
(p + ρe)uA± ± √ρπβ ( 2ρ
+ e)B
where
1 ± erf (s)
e−s
A =
, B=
2
2
±
FM Report 1:2005
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2
p
and s = u β.
Department of Aerospace Engg.
Mathematical Analysis of Dissipation in m − kf vs Method
7
Appendix B
The modified split kinetic flux Jacobians Am± are given by
1
1
P+ − P− Q
Am± = A ±
2
2
(22)
where A is the full flux Jacobian given by

0

1
2
A=
2 (γ − 3) u
1
a2 u
3
2 (γ − 2) u − γ−1
1
(3 − u) u
1
2
2 (3 − 2γ) u +
a2
γ−1
the matrices P ± are given by

I1±
2βr1
s1
ρ


I2±
±
P =
2βr2
s
ρ
h
2
 ±
I
±
1
3
2βI0 r1 + βr3 I0 s1 + β1 − 2I0 I1± −
ρ I0 I1 + 2
and the Jacobian Q is given by

Q=

∂V
=

∂U
− 2p12
1
− uρ
(γ − 1) 12 ρu2 −

0

γ − 1
γu

I3±
2
i
+ 12 s3 +
0
p
γ−1
0
0
1
ρ
1
2p2
(23)
(γ − 1) ρu
− 2p12



± 
(24)
I3
β




(γ − 1) ρ
(25)
where V is the primitive vector defined by
h
iT
V = ρ u β
(26)
Expressions for In± :
The functions In± , n = 1, .., 5 are defined by
In+
=
Z∞
n −αv
v e
F̃ dv and
In−
=
Z0
v n eαv F̃ dv
(27)
−∞
0
where F̃ is defined by
ρ
F̃ =
I0
r
β
exp −β(v − u)2
π
(28)
We can evaluate these integrals using MATHEMATICA [9] or can derive closed form expressions
manually.
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FM Report 1:2005
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Anil, Rajan and Deshpande
References
[1] Anil N. and Deshpande S.M. Low Dissipative Modified KFVS (M-KFVS) Method. Report
2004 FM 22, Dept. of Aerospace Engg., IISc, Bangalore, India.
[2] Anil N. and Deshpande S.M. Modified KFVS (M-KFVS) Method. Proceedings of the Workshop on Modeling and Simulation in Life Sciences, Materials and Technology, in honour of
Prof. Helmut Neunzert, IIT Madras, Dec 2004.
[3] Harrison S.Y. Chou. On the Mathematical Properties of Split Kinetic Fluxes. AIAA paper
2000-0921.
[4] Courant R., Isaacson E. and Rees M. On the solution of Nonlinear Hyperbolic Differential
Equations by Finite Differences. Comm. Pure Appl. Math., 5, 243-255, 1952.
[5] Deshpande S.M. Kinetic flux splitting schemes. Computational Fluid Dynamics Review
1995, (eds.) M.M. Hafez and K. Oshima, John Wiley & Sons Ltd., 1995.
[6] Deshpande S.M. A second order accurate Kinetic Theory based method for inviscid compressible flows. NASA Technical Paper 2613, NASA Langely Research Centre, Hampton,
Virginia, 1986.
[7] Deshpande S.M. On the Maxwellian Distribution, Symmetric Form, and Entropy Conservation for the Euler Equations. NASA Technical Paper 2583, NASA Langely Research Centre,
Hampton, Virginia, 1986.
[8] Jaisankar S. and Raghurama Rao S.V. A Low Dissipative Peculiar Velocity Based Upwind
Method. 7th AeSI CFD symposium, August 2004, Bangalore, India.
[9] Wolfram S. Mathematica. 2nd Ed., Addison-Wesley Publishing Company, Inc., 1991.
[10] Mandal J.C. Kinetic Upwind Method for Inviscid Compressible Flows. Ph.D. Thesis, Dept.
of Aerospace Engg., IISc, Bangalore, India, 1989.
[11] Ramesh V. and Deshpande S.M. Least squares kinetic upwind method with modified CIR
splitting. 7th AeSI CFD symposium, August 2004, Bangalore, India.
[12] Ramesh V. and Deshpande S.M. Low dissipation grid free upwind kinetic scheme with
modified CIR splitting. Report 2004 FM 20, Dept. of Aerospace Engg., IISc, Bangalore,
India.
FM Report 1:2005
Department of Aerospace Engg.
Mathematical Analysis of Dissipation in m − kf vs Method
9
KFVS positive split flux
KFVS negative split flux
5
5
4
3
3
Non−dimensional mass flux
4
2
1
0
2
1
0
−1
−1
−2
−2
−3
−4
Non−dimensional momentum flux
G1−
G1
−3
−2
−1
0
1
2
3
−3
−4
4
−2
−1
0
1
Mach number
KFVS positive split flux
KFVS negative split flux
5
5
4
4
3
G2+
G2
2
1
0
−1
−2
−3
−4
−3
Mach number
Non−dimensional momentum flux
Non−dimensional mass flux
G1+
G1
3
2
3
4
2
3
4
G2−
G2
2
1
0
−1
−2
−3
−2
−1
0
1
2
3
−3
−4
4
−3
−2
−1
Mach number
0
1
Mach number
KFVS negative split flux
KFVS positive split flux
5
5
G3−
G3
4
4
3
3
Non−dimensional energy flux
Non−dimensional energy flux
G3+
G3
2
1
0
2
1
0
−1
−1
−2
−2
−3
−4
−3
−2
−1
0
Mach number
1
2
3
4
−3
−4
−3
−2
−1
0
1
2
3
4
Mach number
Figure 1: The kf vs split fluxes are plotted with respect to Mach number.
Indian Institute of Science
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Anil, Rajan and Deshpande
m−KFVS positive split flux at alpha−tilde = 0.2
m−KFVS positive split flux at alpha−tilde = 1.0
5
4
5
Gm1+
G1+
0.5*G1
4
3
Non−dimensional mass flux
Non−dimensional mass flux
3
2
1
0
2
1
0
−1
−1
−2
−2
−3
−4
−3
−2
−1
0
1
2
3
−3
−4
4
Gm1+
G1+
0.5*G1
−3
−2
−1
Mach number
m−KFVS positive split flux at alpha−tilde = 2.0
4
3
4
3
4
Gm1+
G1+
0.5*G1
3
Non−dimensional mass flux
Non−dimensional mass flux
2
5
Gm1+
G1+
0.5*G1
3
2
1
0
2
1
0
−1
−1
−2
−2
−3
−4
1
m−KFVS positive split flux at alpha−tilde = 4.0
5
4
0
Mach number
−3
−2
−1
0
Mach number
1
2
3
4
−3
−4
−3
−2
−1
0
1
2
Mach number
Figure 2: The m − kf vs positive mass split fluxes are plotted with respect to Mach number.
FM Report 1:2005
Department of Aerospace Engg.
Mathematical Analysis of Dissipation in m − kf vs Method
m−KFVS negative split flux at alpha−tilde = 0.2
5
4
4
2
1
0
2
1
0
−1
−1
−2
−2
−3
−4
−3
−2
−1
0
1
Gm1−
G1−
0.5*G1
3
Gm1−
G1−
0.5*G1
Non−dimensional mass flux
Non−dimensional mass flux
m−KFVS negative split flux at alpha = 1.0
5
3
11
2
3
−3
−4
4
−3
−2
−1
Mach number
m−KFVS negative split flux at alpha−tilde = 2.0
2
3
4
5
Gm1−
G1−
0.5*G1
4
Gm1−
G1−
0.5*G1
4
3
3
Non−dimensional mass flux
Non−dimensional mass flux
1
m−KFVS negative split flux at alpha−tilde = 4.0
5
2
1
0
2
1
0
−1
−1
−2
−2
−3
−4
0
Mach number
−3
−2
−1
0
Mach number
1
2
3
4
−3
−4
−3
−2
−1
0
1
2
3
4
Mach number
Figure 3: The m − kf vs negative mass split fluxes are plotted with respect to Mach number.
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FM Report 1:2005
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Anil, Rajan and Deshpande
m−KFVS positive split flux at alpha−tilde = 1.0
5
4
4
Non−dimensional momentum flux
Non−dimensional momentum flux
m−KFVS positive split flux at alpha−tilde = 0.2
5
3
2
1
0
−1
−2
−3
−4
−2
−1
0
1
2
3
2
1
0
−1
−2
Gm2+
G2+
0.5*G2
−3
3
−3
−4
4
Gm2+
G2+
0.5*G2
−3
−2
−1
Mach number
4
4
3
2
1
0
−1
−2
−2
−1
0
Mach number
2
3
4
1
2
3
3
4
3
2
1
0
−1
−2
Gm2+
G2+
0.5*G2
−3
1
m−KFVS positive split flux at alpha−tilde = 4.0
5
Non−dimensional momentum flux
Non−dimensional momentum flux
m−KFVS positive split flux at alpha−tilde = 2.0
5
−3
−4
0
Mach number
4
−3
−4
Gm2+
G2+
0.5*G2
−3
−2
−1
0
1
2
Mach number
Figure 4: The m − kf vs positive mass split fluxes are plotted with respect to Mach number.
FM Report 1:2005
Department of Aerospace Engg.
Mathematical Analysis of Dissipation in m − kf vs Method
m−KFVS negative split flux at alpha−tilde = 1.0
5
5
4
4
Non−dimensional momentum flux
Non−dimensional momentum flux
m−KFVS negative split flux at alpha−tilde = 0.2
3
2
1
0
−1
−2
−3
−4
−2
−1
0
1
2
3
3
2
1
0
−1
−2
Gm2−
G2−
0.5*G2
−3
13
−3
−4
4
Gm2−
G2−
0.5*G2
−3
−2
−1
Mach number
4
4
3
2
1
0
−1
−2
−2
−1
0
Mach number
2
3
4
1
2
3
3
2
1
0
−1
−2
Gm2−
G2−
0.5*G2
−3
1
m−KFVS negative split flux at alpha−tilde = 4.0
5
Non−dimensional momentum flux
Non−dimensional momentum flux
m−KFVS negative split flux at alpha−tilde = 2.0
5
−3
−4
0
Mach number
4
−3
−4
Gm2−
G2−
0.5*G2
−3
−2
−1
0
1
2
3
4
Mach number
Figure 5: The m − kf vs negative mass split fluxes are plotted with respect to Mach number.
Indian Institute of Science
FM Report 1:2005
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Anil, Rajan and Deshpande
m−KFVS positive split flux at alpha−tilde = 1.0
5
4
4
3
3
Non−dimensional energy flux
Non−dimensional energy flux
m−KFVS positive split flux at alpha−tilde = 0.2
5
2
1
0
−1
1
0
−1
−2
−3
−4
2
−2
Gm3+
G3+
0.5*G3
−3
−2
−1
0
1
2
3
−3
−4
4
Gm3+
G3+
0.5*G3
−3
−2
−1
Mach number
1
2
3
4
m−KFVS positive split flux at alpha−tilde = 4.0
5
4
4
3
3
Non−dimensional energy flux
Non−dimensional energy flux
m−KFVS positive split flux at alpha−tilde = 2.0
5
2
1
0
−1
2
1
0
−1
−2
−3
−4
0
Mach number
−2
Gm3+
G3+
0.5*G3
−3
−2
−1
0
Mach number
1
2
3
4
−3
−4
Gm3+
G3+
0.58G3
−3
−2
−1
0
1
2
3
4
Mach number
Figure 6: The m − kf vs positive mass split fluxes are plotted with respect to Mach number.
FM Report 1:2005
Department of Aerospace Engg.
Mathematical Analysis of Dissipation in m − kf vs Method
m−KFVS negative split flux at alpha−tilde = 0.2
m−KFVS negative split flux at alpha−tilde = 1.0
5
5
Gm3−
G3−
0.5*G3
4
3
Non−dimensional energy flux
Non−dimensional energy flux
4
2
1
0
2
1
0
−1
−2
−2
−3
−2
−1
0
1
2
3
−3
−4
4
Gm3−
G3−
0.5*G3
3
−1
−3
−4
15
−3
−2
−1
Mach number
m−KFVS negative split flux at alpha−tilde = 2.0
4
Non−dimensional energy flux
Non−dimensional energy flux
2
3
4
3
4
5
Gm3−
G3−
0.5*G3
3
2
1
0
2
1
0
−1
−2
−2
−3
−2
−1
0
Mach number
1
2
3
4
Gm3−
G3−
0.5*G3
3
−1
−3
−4
1
m−KFVS negative split flux at alpha−tilde = 4.0
5
4
0
Mach number
−3
−4
−3
−2
−1
0
1
2
Mach number
Figure 7: The m − kf vs negative mass split fluxes are plotted with respect to Mach number.
Indian Institute of Science
FM Report 1:2005
16
Anil, Rajan and Deshpande
Eigen values of 0.5*A
Eigen values of the A+ matrix when alpha−tilde = 0.0
6
6
e1
e2
e3
e1
e2
e3
5
5
4
Eigen values
Eigenvalues
4
3
3
2
2
1
1
0
0
0
0.5
1
1.5
2
2.5
3
3.5
4
0
0.5
1
Mach number
Eigen values of Am+ matrix when alpha−tilde = 0.5
2.5
3
3.5
4
3
3.5
4
Eigen values of 0.5*A
6
e1
e2
e3
e1
e2
e3
5
5
4
4
Eigen values
Eigen values
2
Mach number
6
3
2
3
2
1
1
0
0
0
1.5
0.5
1
1.5
2
2.5
Mach number
3
3.5
4
0
0.5
1
1.5
2
2.5
Mach number
Figure 8: Eigen values of the positive split flux Jacobian matrix based on m-kfvs are plotted
with Mach number for different values of α̃. These eigen values are compared with the eigen
values of half the full flux Jacobian matrix.
FM Report 1:2005
Department of Aerospace Engg.
Mathematical Analysis of Dissipation in m − kf vs Method
17
Eigen values of Am+ matrix when alpha−tilde = 1.0
Eigen values of 0.5*A
6
6
e1
e2
e3
5
5
4
4
Eigen values
Eigen values
e1
e2
e3
3
2
3
2
1
1
0
0
0
0.5
1
1.5
2
2.5
3
3.5
4
0
0.5
1
Mach number
Eigen values of Am+ matrix when alpha−tilde = 4.0
2.5
3
3.5
4
3
3.5
4
Eigen values of 0.5*A
6
e1
e2
e3
e1
e2
e3
5
5
4
4
Eigen values
Eigen values
2
Mach number
6
3
2
3
2
1
1
0
0
0
1.5
0.5
1
1.5
2
2.5
Mach number
3
3.5
4
0
0.5
1
1.5
2
2.5
Mach number
Figure 9: Eigen values of the positive split flux Jacobian matrix based on m-kfvs are plotted
with Mach number for different values of α̃. These eigen values are compared with the eigen
values of half the full flux Jacobian matrix.
Indian Institute of Science
FM Report 1:2005
18
Anil, Rajan and Deshpande
Eigen values of the D − matrix when alpha−tilde = 0.1
Eigen values of the D − matrix based on kfvs
3.5
5.5
e1
e2
e3
5
e1
e2
e3
3
4.5
4
Eigen values
Eigen values
2.5
2
3.5
3
2.5
1.5
2
1.5
1
1
0.5
0
0.5
1
1.5
2
2.5
3
3.5
0.5
0
4
0.5
1
Mach number
2
2.5
3
3.5
4
Mach number
Eigen values of the D − matrix when alpha−tilde = 0.4
Eigen values of the D − matrix based on kfvs
1.2
1.1
1.5
5.5
e1
e2
e3
5
e1
e2
e3
4.5
1
Eigen values
Eigen values
4
0.9
0.8
0.7
3.5
3
2.5
2
0.6
1.5
0.5
0.4
0
1
0.5
1
1.5
2
2.5
Mach number
3
3.5
4
0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
Mach number
Figure 10: Eigen values of the dissipation matrix based on m-kfvs are plotted with Mach number
for different values of α̃. These eigen values are compared with the eigen values of D-matrix
based on kfvs.
FM Report 1:2005
Department of Aerospace Engg.
Mathematical Analysis of Dissipation in m − kf vs Method
19
Eigen values of the D − matrix based on kfvs
Eigen values of the D − matrix when alpha−tilde = 1.0
5.5
0.45
5
0.4
e1
e2
e3
4.5
0.35
Eigen values
Eigen values
4
0.3
0.25
0.2
3.5
3
2.5
2
0.15
1.5
0.1
0.05
0
e1
e2
e3
0.5
1
1
1.5
2
2.5
3
3.5
0.5
0
4
0.5
1
1.5
2
2.5
3
3.5
4
Mach number
Mach number
Eigen values of the D − matrix based on kfvs
Eigen values of the D − matrix when alpha−tilde = 4.0
5.5
0.08
e1
e2
e3
0.07
5
e1
e2
e3
4.5
0.06
Eigen values
Eigen values
4
0.05
0.04
0.03
3.5
3
2.5
2
0.02
1.5
0.01
0
0
1
0.5
1
1.5
2
2.5
Mach number
3
3.5
4
0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
Mach number
Figure 11: Eigen values of the dissipation matrix based on m-kfvs are plotted with Mach number
for different values of α̃. These eigen values are compared with the eigen values of D-matrix
based on kfvs.
Indian Institute of Science
FM Report 1:2005