1. Effect of N2 Vibrational Energy in
Supersonic flows
Sathyan Padmanabhan
Dept. of Aerospace Engineering, Pusan National University
Thesis advisor - Prof. Bernard Parent
Dept. of Aerospace Engineering, Pusan National University

2. High Temperature Gas Dynamics effect
High temperature gas dynamics effect include non
equilibrium thermodynamics and chemical reactions.
Pic courtesy : NASA

3. When Vibrational Energy becomes active ?
A simple diatomic molecule has several modes of energy
such as translational, rotational, vibrational and electronic
energy
In general all the modes of energy is equally distributed
To excite vibrational energy, diatomic molecule’s should
experience large number of collisions (nearly 20,000
collisions required to excite O2)
Number of collisions depends on the type of diatomic
molecule
But when the kinetic energy of the colliding partners is
high enough, number of collisions to excite the vibrational
energy will get reduced.

4. Variation of Ratio of Vibrational energy to total
Internal energy of N2
500 1000 1500 2000 2500 3000
0.05
0.1
0.15
0.2
T; K
evib=e

5. When Vibrational Energy experiences
nonequilibrium ?
In many problems in high speed gas dynamics, fluid does
not have enough time to come to equilibrium
Consider a volume of gas passes through a shock, various
energy form (translational,rotational,vibrational and
electronic) will transit from upstream equilibrium to
downstream equilibrium.
These energy form comes to equilibrium at different rates
trans < rot < vib < elec
Hence in a certain region behind shock there will be a
region of vibrational nonequilibrium.

6. Vibrational nonequilibrium in ground-based
facilities
Impulse facilities such as shock tunnels are widely used in
testing of scramjet
Vibrational excitation happens on the nozzle, because the
expansion is so rapid. Translational and rotational
temperature drops rapidly, but vibrational temperature
departs from equilibrium and freezes.
Thus the test conditions depart from desired flight
conditions, where there is complete thermo-chemical
equilibrium
So it is generally said that ground based facilities cannot
replicate the actual flight conditions.

7. Solving Vibrational nonequilibrium in CFD
Separate vibrational energy terms should be included to
capture the energy generated by vibrations
For modeling thermal nonequilibrium, we should include
one temperature for every mode of transfer
For the case considered here translational and vibrational
energy plays a major role
Translational and rotational modes have same temperature
And electronic modes are not active
So we assume there are two temperatures, T be the
translational temperature and Tv be vibrational
temperature.

8. Landau-Teller model
In air nearly 70 % of air is comprised of N2, and most of
vibrational energy transfer happens through N2
@
@t
ev C
dX
j D1
@
@xj
vj ev
dX
j D1
@
@xj
Â
Äev
@Tv
@xj
Ã
D Sev
where ev is the vibrational energy, Äev
is the effective
vibrational conductivity. Using Landau-Teller model Sev
can
be expressed as
Sev
D
.ev e0
v/
vt
e0
v = Vibrational energy in equilibrium
vt = Vibrational relaxation rate.

9. Vibrational relaxation rate
V T Transfer W A.n/ C A.n/ $ A.n ˙ 1/ C A.n/ KE
Macheret V-T correlation (M-V-T)
Empirical correlation from experiments in weakly Ionized
plasmas
Vibrationally excited N2 relaxes in collision with oxygen
molecule and atom
1
vt
D N
Ä
7 10 16
exp
Â
141
T 1=3
Ã
C ˛0 5 10 18
exp
Â
128
T 1=2
Ã
; s 1
N = Number density (1/m3
)
˛0 = mass fraction of atomic oxygen
T = Translational temperature in Kelvin

10. Millikan and White correlation (M-W)
This correlation is a commonly used vibrational relaxation
rate
Derived from shock tunnel experiments, and the general
formulation is applicable for all diatomic molecules
p s D exp As.T 1=3
Bs/ 18:42 ; atm:sec
The above correlation is used to find the relaxation rate for
individual collisions
N2 relaxation rate for the mixture is given as
1
vt
D
Â
N2
N2 N2
C
O2
N2 O2
C
O
N2 O
Ã
s = Mole fraction of species s

11. Sebacher experimental correlations (S-E)
Sebacher found the relaxation rate from expanding flows
(S-E-E) in air in wind tunnel.
Correlation is fitted for the experimental data using
Levenberg - Marquardt least square fitting method
1
vt
D N 3:2384 10 14
exp
Â
160:1098
T 1=3
Ã
Another correlation is also fitted for the data for the
experiments in shock tunnel (S-E-C)
1
vt
D N 1:0109 10 16
exp
Â
135:581
T 1=3
Ã
N = Number density (1/m3
)
T = Translational temperature in Kelvin

12. Comparison between relaxation models
500 1000 1500 2000 2500
10
-6
10
-5
10-4
10-3
10
-2
10
-1
100
101
T; K
P;atm:sec
M V T
M W
S E E
S E C

13. Governing equations
@bQ
@t
C
dX
jD1
2
4 @bF j
@xj
dX
iD1
@
@xj
0
@bKij
@bG
@xi
1
A
3
5 D bSt
bQ D
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
c1
:
:
:
cns
v1
:
:
:
vd
E
k
!
ev
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
; bF j D
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
vj c1
:
:
:
vj cns
vj v1 C ıK
1j P ?
:
:
:
vj vd C ıK
dj P ?
vj E C vj P ?
vj k
vj !
vj ev
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
; bG D
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
c1
:
:
:
cns
v1
:
:
:
vd
T
k
!
Tv
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
; bSt D
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
W1
:
:
:
Wns
0
:
:
:
0
0
Sk
S!
Sev
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
;

14. Reaction Mechanism

15. Computational Methodology
WARP solves N2 vibrational energy equation along with
multi species continuity, momentum, energy, turbulent
kinetic energy and specific dissipation rate. And the
equations are favre averaged
Vibrational relaxation rate is modeled with Macheret V-T
relaxation model
All terms are discretized using second order accurate
central difference stencil, except the convective derivative,
which is discretized using Yee-Roe scheme
Solution is brought to steady state by using approximate
factorization. And the solution is accelerated using domain
decomposition methods such as multizone cycle

16. Validation of Macheret V-T relaxation model
q1 D 2000 m=s
outflow
outflow
outflow
L
H D L=2
Tt
Tv

17. Relaxation history of vibrational nonequilibrium
and definition of relaxation distance
0 1E-06 2E-06
2000
2500
3000
time; sec
T;K
Tt
Tv
0 0.002 0.004
0
0.2
0.4
0.6
0.8
1
x; m
TTv
TiTv;i
Lr

18. Temperature Ti,K Kg=m3
vt.Computational/; sec vt.Macheret/; sec
1000 0.01 1.108E-03 1.781E-03
1000 0.1 2.304E-04 1.781E-04
1000 1.0 3.716E-05 1.781E-05
1000 5 7.097E-06 3.563E-06
1500 0.01 7.735E-04 6.254E-04
1500 0.1 8.283E-05 6.254E-05
1500 1.0 1.137E-05 6.254E-06
1500 5 2.670E-06 1.250E-06
2000 0.01 4.239E-04 2.873E-04
2000 0.1 4.281E-05 2.873E-05
2000 1.0 4.885E-06 2.873E-06
2000 5.0 1.210E-06 5.746E-07
2500 0.01 2.391E-04 1.547E-04
2500 0.1 2.539E-05 1.547E-05
2500 1.0 2.690E-06 1.547E-06
2500 5.0 6.280E-07 3.094E-07
3000 0.01 1.634E-04 9.315E-05
3000 0.1 1.639E-05 9.315E-06
3000 1.0 1.687E-06 9.315E-07
3000 5.0 3.700E-07 1.863E-07
3500 0.01 1.116E-04 7.984E-05
3500 0.1 1.122E-05 7.984E-06
3500 1.0 1.137E-06 7.984E-07
3500 5.0 2.410E-07 1.596E-07

19. Validation of k ! (1988) turbulence model
M1 D 2:84
T1 D 98 K
2:2
0:126 outflow
adiabatic wall
20ı
outflow

20. Comparison with Settles Experimental data
-2 0 2 4 6
0
0.0005
0.001
0.0015
s
Cf
1760 880
Settles
-2 0 2 4 6
1
1.5
2
2.5
3
3.5
s
1760 880
Settles
effectivepressure

21. Vibrational energy Transport Studies on Blunt
body
air
M1 D 12:0
P1 D 377:1 pa
T1 D 244:8 K
adiabatic wall
outflow
20
30
13.5
40
symmetry
X,m
Y,m
0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Streamline - 1
Streamline - 2

22. Temperature profile for streamline - 1
0.045 0.05 0.055 0.06 0.065 0.07
500
1000
1500
2000
2500
3000
3500
4000
4500
X; m
T;K
With vib
No vib
Tv

23. Temperature profile for streamline - 2
0.05 0.055 0.06 0.065 0.07
500
1000
1500
2000
2500
3000
X; m
T;K
With vib
No vib
Tv

24. Pressure contour on the blunt body

25. Concluding remarks
Computed relaxation rate shows a better agreement with
Macheret V-T correlation
In vibrational transport studies on blunt body, when grid
convergence is done. Even for the finer mesh grid induced
error is higher
Computing next grid level might yield a grid converged
result
Thermal state for both the models are completely different
Exclusion of vibrational models will provide aphysical
results
Inclusion of vibrational-vibrational transfer between
diatomic molecules will affect relaxation rate