uncertainty quantification for high enthalpy …uncertainty quantification for high enthalpy...

UNCERTAINTY QUANTIFICATION FOR HIGH ENTHALPY FACILITIES

By

MARGARET A. OWEN

A THESIS PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2019

To David Keister (now Owen-Keister) my husband, my UF and VKI friends/colleagueswho helped me and shaped me as a person, and Nicholas Arnold-Medabalimi whose

LaTeX skills saved this thesis

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ACKNOWLEDGMENTS

I would like to thank Thierry Magin for his support and mentorship. I would like to

thank Ana Isabel del val Benitez (Anabel) for being the best commander a co-pilot could

ask for, for sharing the eternal love of the Apollo era, and being a fellow SPACE nerd. I

would also like to thank the friends I have made here for all of their support. This place

has truly amazed me in every way possible.

4

TABLE OF CONTENTSpage

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Uncertainty: A Potential Mission Killer . . . . . . . . . . . . . . . . . . . . . . 10Background Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Cabaret Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Theory Behind Cabaret . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Assumptions regarding the modeling of plasma . . . . . . . . . . . 12Rankine-Hugoniot relations . . . . . . . . . . . . . . . . . . . . . . 13Statistical Thermodynamics . . . . . . . . . . . . . . . . . . . . . . 14Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Total quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Fay and Riddell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Code execution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Plasmatron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Longshot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Numerical Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

ICP computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Mutation ++ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2 TOOLS, FACILITIES, AND PROCEDURES . . . . . . . . . . . . . . . . . . . 27

CABARET For The Plasmatron . . . . . . . . . . . . . . . . . . . . . . . . . . 27Reproducing Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

CABARET For The Longshot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Differences Between The Facilities . . . . . . . . . . . . . . . . . . . . . . . 28Obtaining The Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Sensitivity Analysis For The Longshot . . . . . . . . . . . . . . . . . . . . . . . 29

3 RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Plasmatron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Longshot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

CABARET Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5

4 CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . . . . . . . . . . . 34

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

APPENDIX: SUPPLEMENTARY INFORMATION . . . . . . . . . . . . . . . . . . 35

Newton-Raphson Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Original Values Entered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Reproducing Plasmatron Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Reproducing Longshot Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Longshot Sensitivity Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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LIST OF TABLESTable page

3-1 The standard deviation of the Cabaret vs the ICP computation . . . . . . . . . 30

3-2 Standard Deviation of Cabaret vs Longshot test data . . . . . . . . . . . . . . . 30

3-3 Results of the 10% dispersion to Beta . . . . . . . . . . . . . . . . . . . . . . . 31

3-4 Results of Inverse problem with Heat Flux and Stagnation Presssure varyingfrom their nominal value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

A-1 Original values plugged into Cabaret . . . . . . . . . . . . . . . . . . . . . . . . 36

A-2 Plasmatron Reproduction Results . . . . . . . . . . . . . . . . . . . . . . . . . . 36

A-3 Longshot Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

A-4 Long Shot Sensitivity Analysis Results . . . . . . . . . . . . . . . . . . . . . . . 38

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LIST OF FIGURESFigure page

1-1 Orion Capsule re-entering the Earth’s atmosphere. Photo permitted by NASA . 10

1-2 Inputs and outputs for solving the problem . . . . . . . . . . . . . . . . . . . . . 12

1-3 Mixture Energy of Air 11 (example) . . . . . . . . . . . . . . . . . . . . . . . . 15

1-4 Mixture Enthalpy of N2 at 1atm (example) . . . . . . . . . . . . . . . . . . . . 16

1-5 Dynamic Viscosity of air 11 at 1 and 0.5 atm . . . . . . . . . . . . . . . . . . . 18

1-6 Thermal Conductivity of air 11 at 1 atm . . . . . . . . . . . . . . . . . . . . . . 19

1-7 The flow diagram of how CABARET works and executes . . . . . . . . . . . . . 21

1-8 The VKI Plasmatron. Photo permitted by the von Karman Institute . . . . . . 22

1-9 Plasmatron set up for experiments with probe . . . . . . . . . . . . . . . . . . . 22

1-10 Deviation of the free stream conditions compared to the ICP computation(Probe radius: 25 mm, Plasma power: 100 kW, mass flow: 5 g/s, reservoir pres-sure: 13170 Pa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1-11 The VKI Longshot. Photo permitted by the von Karman Institute . . . . . . . 24

1-12 Schematic of the VKI Longshot . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2-1 Temperature distribution in supersonic plasma flow in Plasmatron facility (Proberadius: 25mm, plasma power: 80kW, mass flow: 5g/s, reservoir pressure: 15550Pa,static pressure in the chamber: 1200Pa) . . . . . . . . . . . . . . . . . . . . . . 27

3-1 Sensitivity Analysis Graph 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3-2 Sensitivity Analysis Graph 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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Abstract of Thesis Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of the

Requirements for the Degree of Master of Science

UNCERTAINTY QUANTIFICATION FOR HIGH ENTHALPY FACILITIES

By

Margaret A. Owen

August 2019

Chair: David W. HahnMajor: Aerospace Engineering

At this point in time, there are still many problems associated with the re-entry of

space vehicles and Thermal Protection System (TPS) design. More accurate models,

better prediction methods, and better simulations are some of the issues that need to

be addressed. Ground testing is a key way to improve and solve these problems. These

facilities and software are used to simulate the harsh environment of re-entry and rebuild

the free stream flow parameters. With rebuilding, there is a certain level of uncertainty

that can propagate from the measurements to the results, causing skewed data. The

goal of this project is to start the uncertainty analysis for high enthalpy facilities by

reproducing results previously done, and to conduct a sensitivity analysis on the Longshot

using a tool created by VKI. The reproduced results and those newly obtained followed

trends that were expected and provided validation on some of the assumptions used and

previous work.

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CHAPTER 1INTRODUCTION

Uncertainty: A Potential Mission Killer

Space is the final frontier. We have sent astronauts into this unforgiving environment

for 50 years now, and most have come back unharmed. It’s one thing to send an astronaut

up, but another thing entirely to bring them back home safely. Thermal Protection

Systems (TPS) protect the crew upon re-entry and ensure their safe return back to Earth.

These systems are designed to take the brunt of the aerodynamic heating and forces

that occur upon re-entry into the Earth’s atmosphere. The shuttle used a system of

surface tiles with underlying insulation, while the Apollo command modules used ablative

honeycomb materials. Until now we have only tested materials able to withstand re-entry

from something as close as the moon. What happens when humans explore other planets

such as Mars? The next generation capsule Orion had a flight test on December 5th, 2014,

to test the heat shields from interplanetary velocity. Although it was successful, there is

still much to learn. Fig.1-1 shows the Orion capsule re-entering the atmosphere.

Figure 1-1. Orion Capsule re-entering the Earth’s atmosphere. Photo permitted by NASA

Different space vehicles require different systems, materials, and solutions. Step 1

is understanding the problem. Step 2 as an engineer is to design it from the material, to

the structure. Designs have to be tested, computed, and proven. Unfortunately proving

these systems on a launch or re-entry vehicle is very expensive, nearly impossible without

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flight heritage, and these designs are not always perfect. Some aerospace mission killers

include radiative heat flux prediction, laminar- turbulent transition, and gas surface

interaction. In order to make sure TPS will work, ground facilities are used to stress

and heat conceptual designs to their limits. Ground facilities reproduce on Earth what

theoretically occurs in the atmosphere. First CFD simulations and code are used to

compute the free stream conditions necessary for ground testing. These free stream

conditions are then used to compute operating conditions of the facilities. Unfortunately

computer simulations and physical data from tests don’t always match up and are not

always accurate. This is where the problem of Uncertainty comes into play. How do

engineers know what is actually computed is within a reasonable limit for physical testing?

How do engineers know that the numbers they are getting are even remotely correct?

Background Information

The following question is important for us to define in our problem statement: To

what accuracy must quantities be measured, and what tolerances/uncertainties must

be applied? The von Karman Institute (VKI) has two facilities used to test TPS, the

Plasmatron and the Longshot. The Plasmatron tests materials at high temperature with

chemical equilibrium effects. The Longshot tests radiation and shock pressure effects

in the flow field on a material/body. Together these two facilities provide much insight

into the thermal and physical characteristics of TPS. CABARET, a program developed

for ground facilities, calculates free stream parameters for the flow by rebuilding the

conditions and solving an inverse problem.

Cabaret Code

CABARET, developed by Ana Isabel del val Benitez at the VKI [1] for induced

plasma flow, computes free stream conditions for the flow. Although it was originally

developed for the Plasmatron, the goal was to use it for many different ground testing

facilities. As such it has been updated to be used for the Longshot as well as the arc

heater in Cologne. It has three modes of solving for these conditions. The first mode

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solves the inverse problem with inputs of reservoir pressure, wall heat flux, and stagnation

pressure and outputs of free stream temperature, pressure, and mach number. The second

mode solves the forward problem with inputs of free stream pressure, temperature, and

mach number, and outputs of reservoir pressure, wall heat flux, and stagnation pressure.

The third mode, is used specifically for the Longshot. The Plasmatron directly measures

mass flow rate which is an input parameter to the first and second modes. However, the

Longshot measures free stream pressure, instead of mass flow rate, and requires another

mode to bypass this measurement in the calculation. A model of these inputs and outputs

for each mode can be seen in (1, 1, 1).

Input Inverse outputPo, Qw, Pt2 T1, P1,M1

Input Forward outputT1, P1,M1 Po, Qw, Pt2

Input Longshot outputP1, Qa, Ta, Pt2a T1,M1

Figure 1-2. Inputs and outputs for solving the problem

Theory Behind Cabaret

Assumptions regarding the modeling of plasma

In order to reproduce the free-stream conditions for ground facilities, the right

assumptions must be made. In Hypersonic Flows, high temperatures occur and dissociation

effects become an issue in the mixture. The specific heat ratio γ is also not constant, and

changes as a function of temperature. For the Plasmatron, the gas is assumed to be

air 11 and composed of multiple species (N2, NO,O2, N,O, etc). As the plasma (gas

mixture) instantaneously adapts its composition to changes in the flow, it can be thought

of as a single gas with a determined composition (a gas in chemical equilibrium). For

thermal equilibrium, pressure and temperature are enough to define the flow as it does

12

not depend on history but the current state (not work dependent). If both chemical and

thermal equilibrium exist, then the gas can be assumed to be in Local Thermodynamic

Equilibrium. This is the main assumption for the Plasmatron.

For the Longshot the flow is considered to be in equilibrium as well. The Free stream

pressure is directly measured.

Rankine-Hugoniot relations

In hypersonic flows, the free stream mach number is supersonic. In order for the

supersonic flow to move and adapt around a still model, a shockwave forms allowing the

flow to correct itself. The shock wave decelerates the flow, compresses it in an almost

infinitesimal space, and finally heats it up. It is analyzed locally in the stagnation region

of the model as a normal shock wave. The Navier-Stokes model is simplified using the

Rankine-Hugoniot relations to compute flow parameters behind the shock. The Rankine

Hugoniot relations are inviscid Navier-Stokes equations applied within a finite volume

where the shock is contained. The ”jump” properties are calculated using the conservation

of mass, momentum, and energy:

ρ1v1 = ρ2v2 (1–1)

p1 + ρ1v21 = p2 + ρ2v

22 (1–2)

(ρ1E1 + p1)v1 = (ρ2E2 + p2)v2 (1–3)

The total energy is:

E = e+1

2v2 (1–4)

Since the inviscid condition is imposed the equilibrium or non-equilibrium condition is

conserved. The internal energy and density are computed using MUTATION ++ (1) and

the non-linear terms of these equations are solved using a Newton-Raphson method (4).

The velocity reference frame before and after the shock is also incredibly important.

For ground testing facilities, the shock wave is attached and shows no movement in

relation to the absolute axis to the ground. Is it therefore assumed that us = 0. For flight,

13

the shock wave is moving so us = 0. For flight:

v1 = us − u1, (1–5)

v2 = us − u2 (1–6)

For ground:

v1 = u1, (1–7)

v2 = u2 (1–8)

Now applying the Newton-Raphson method:

U =

ρ2

ρ2v2

ρ2E2

(1–9)

R =

ρ2v2 − ρ1v1

ρ2v22 + p2 − ρ1v

21 + p1

(ρ2E2 + p2) v2 − (ρ1E1 + p1) v1

(1–10)

The U and R matrices are plugged into the Newton Raphson method equation

described in the Appendix 4. An initial guess for U is needed, with the Jacobian matrix

and the residual being computed after this guess.

Statistical Thermodynamics

The internal energy as shown in equation 1–11 is calculated using the sum of five

components: translation, rotation, vibration, electronic, and formation energies. Each

energy state has to be taken into account as the temperatures are high enough that each

degree of freedom could potentially be excited.

ei(T ) = eTi (T ) + eTi (T ) + eEi (T ) + eRi (T ) + eVi (T ) + eFi (1–11)

14

where each state is:

eTi (T ) =32RT, i = electrons

eTi (T ) =32RT, i = atoms,molecules

eRi (T ) = RT(1− θR

θR+3T

), i = molecules

eVi (T ) = R

(θ

exp( θT )−1

), i = molecules

eEi (T ) = R

N∑i=1

geθE,eexp(

θE,eT

)N∑i=1

geexp(

θE,eT

) , i = atoms, molecules

(1–12)

Figure 1-3. Mixture Energy of Air 11 (example)

In Fig. 1-3 the energy for each degree of freedom in an air 11 mixture is plotted

versus temperature. The translation state is the lowest and can be easily excited.

Formation energy is the highest individual degree of freedom as it takes more energy

to form bonds. The formation energy is constant for each species.

The mixture energy overall is the summation of each state and is the highest curve

above formation. Finally the internal energy of the mixture overall is:

e =N∑i=1

yiei (1–13)

The yi in equation 1–13 is the mass fraction of species i

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Below the same principals with different equations apply for enthalpy:

hi(T ) = hTi (T ) + hE

i (Tv) + hRi (T ) + hV

i (Tv) + hFi (1–14)

hTi (T ) =

52RT, i = atoms,molecules

hRi (T ) =

32RT, i = molecules

hVi (T ) = R

(θ

e( θT )−1

), i = molecules

hEi (T ) = R

N∑i=1

geθE,ee(

θE,eT

)N∑i=1

gee(

θE,eT


hFi = constant, i = electrons, atoms, molecules

(1–15)

h =NS∑i=1

yi(p, T, xj)hi(T ) (1–16)

Figure 1-4. Mixture Enthalpy of N2 at 1atm (example)

The sum should be the highest enthalpy as it is the addition of each type. Translational

should be the second highest curve as it is easily excited. Rotational should be the third

highest with Vibrational being fourth. This is shown in Fig.1-4. With a molecule there are

two degrees of freedom about which to rotate. Vibrational and Rotational are exactly the

same curve seperated by a constant multiplier. Electronic is the lowest as it takes more

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energy to excite this degree of freedom. The characteristic temperature is shown at about

13000 K.

For entropy:

si = sTi + sRi + sVi + sEi (1–17)

sTi , i = electrons

sTi (T, P ) = 52R +Rln

[(2πmh2

) 32 k

52b

]+ 5

2RlnT −Rlnp, i= atoms, molecules

sRi (T, P ) = R(1 + ln T

σθR

)+Rln

(1 + θR

3TR

)− RθR

θR+3T, i = molecules

sVi (T, p) = R θT

1

exp( θT )−1

−Rln[1− exp

(θT

)], i = molecules

sEi (T, p) = Rln

(N∑i=1

geexp(θE ,eT

)

)+ R

T

(N∑i=1

geθE,eexp(θE,e

T)

)(

N∑i=1

geexp(θE,e

T)


(1–18)

ρs =NS∑i=1

ρisi + kB

NS∑i=1

niln(1

xi

) (1–19)

The entropy equation has an extra term accounting for the entropy of mixing. In the

following equations yi is the mass fraction of species i, xi is the mole fraction, kB is the

Botlzmann’s constant, and the entropy and enthalpy with subscript i are of species i.

Transport

The transport properties are computed through a multiscale Chapman-Enskog

perturbative solution of the Boltzmann equation. Dynamic viscosity and thermal

conductivity are needed. The dynamic viscosity is taken from the first and second

Laguerre-Sonine polynomial approximations of the Chapman-Enskog expansion.

NS∑j=1

Gµijα

µj = xi (1–20)

Taking the expression above in equation 1–20 and finding the solution of linear transport

systems, results in the dynamic viscosity.

µ =NS∑i=1

αµi xi (1–21)

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The µ and Gµ are transport matrices depending on species mole fractions and binary

collision integrals. MUTATION ++ computes dynamic viscosity.

Figure 1-5. Dynamic Viscosity of air 11 at 1 and 0.5 atm

Fig. 1-5 shows that decreasing the pressure decreases the maximum viscosity around

the same temperature of 1000 K.

The thermal conductivity is needed to compute the heat flux from the ICP data.

q = −λ∇T +NS∑i=1

ρiVihi (1–22)

The heat flux equation 1–22 accounts for the convective and diffusive terms as seen first

and second in the equation. The Radiation Heat Flux is not important here, hence why it

was not taken into account. Using Fick’s Law below:

ρiVi = −∑j

Dij∇xj (1–23)

where:

∇xj =∂xj

∂T∇T (1–24)

and combining these equations results in:

q = −(λ+ λR)∇T (1–25)

18

where:

λR =∑ij

hi(T )Dij∂xj

∂T(1–26)

The heat transfer equation 1–26 above is equivalent to the convective term of the

aforementioned heat flux equation but including the diffusivity. MUTATION ++

computes thermal conductivity using the equations below:

NS∑i=1

Gλijα

λj = xi (1–27)

λ =NS∑j=1

αλj xj (1–28)

Figure 1-6. Thermal Conductivity of air 11 at 1 atm

Each peak in Fig. 1-6 shows dissociation of species occurring.

Total quantities

A thermally perfect gas is in thermodynamic equilibrium, not chemically reacting,

and does not have a constant gamma, or specific heat ratio γ. Therefore the general

conservation equations are used to calculate the total quantities along the stagnation line.

This model is used to calculate the total quantities needed to solve the reservoir and the

heat flux.

h(Tt, pt) = h(T, p) +v2

2(1–29)

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s(Tt, pt) = s(T, p) (1–30)

Conservation of total enthalpy and entropy can be applied along the stagnation line,

through nozzle expansion, to convert free stream to reservoir conditions. Another Newton

Raphson iterative method is applied at this point.

h(T0, p0, ) +v202

= h(T1, p1) +v212

(1–31)

s(T0, p0) = s(T1, p1) (1–32)

The velocity in the reservoir is not necessarily zero. To fulfill this condition, the mass flow

is needed.

m = ρ0(T0, p0)v0S0 (1–33)

Fay and Riddell

The stagnation point heat flux is one of the most important parts of the entire

design problem of TPS. The heat flux is a direct result of decelerating the space vehicle

from orbital/interplanetary velocities (from 8 km/s to 11 km/s) to zero. This is the

transfer from potential energy to kinetic energy. To facilitate this heat transfer and energy

change most spacecraft are usually designed as blunt bodies. The higher the temperature

of the flow, the lower the temperature of the space vehicle. The heat flux presents a

multi-component problem consisting of space craft shape, enthalpy of the flow with

thermo-chemical non-equilibrium aspects, and material choice.

The equation for local heat transfer to the body is the sum of two contributions, the

convective and diffusive components.

qw =

[k

(∂T

∂y

)]y=0

+

{∑i

ρ(hi − h0

i

) [Di

(∂xi

∂y

)+

(DT

i xi

T

)(∂T

∂y

)]}y=0

(1–34)

At the stagnation point the Reynolds and Nusselt numbers are:

Nu =qxCpw

kw(hs − hw)(1–35)

20

Re =uex

vw(1–36)

Fay and Riddell implemented a similar equation to obtain a certain range of solutions

and accuracy for a generic blunt body. The formula is approximated for the heat trasnfer

at the wall in equilibrium boundary layer conditions. Equation 1–37 below was used in

computations.

qw = 0.763Pr−0.6(ρwµw)0.1(ρeµe)

0.4(He −Hw)√

βe

[1 + (Le0.52 − 1)

hd

H2

](1–37)

The e is the boundary layer edge, the w is the model wall, and the βe is shown below:

βe =

(∂u

∂x

)e

=1

rcurve

√2(pt2 − p1)

ρt2(1–38)

The βe is the velocity gradient at the edge of the boundary layer. This term provides the

most uncertainty. The density, viscosity, and enthalpy are computed using MUTATION++.

The difference between Equation 1–34 versus Equation 1–37 is the lack of ablation/catalysis,

and thermo-chemical non equilibrium effects in the boundary layer (the diffusion term).

The Lewis number is assumed to be close to one.

Code execution

Figure 1-7. The flow diagram of how CABARET works and executes[1]

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Using the theory described in this section, and a combination of MUTATION ++

(1), the flow diagram (Fig.1-7) shows how the CABARET code works. There are three

Newton-Raphson iterations, one solving the total quantities, one solving the conservation

equations for the nozzle, and one for the Rankine-Hugoniot equations. An explanation on

the Newton-Raphson iterative method can be found in the Appendix. Depending on the

mode, an initial estimated guess must be made to start the iterations, and the output is

the free stream conditions needed.

Plasmatron

Figure 1-8. The VKI Plasmatron. Photo permitted by the von Karman Institute[1]

Figure 1-9. Plasmatron set up for experiments with probe[2]

22

The Plasmatron is an Inductively-Coupled Plasma (ICP) torch, that generates plasma

flow through an electric arc capable of dissociating air at high power, and providing high

temperatures in the flow. The flow generated by ICP torches is a high purity and high

density plasma flow that allows for equilibrium conditions. The ICP torch (with a 160mm

diameter) works using a wound coil, connected to a high-frequency,high power, and high

voltage (400kHz, 1.2MW , 2kV ) generator surrounding a quartz tube. The different

testing gases (ranging from argon, air, and CO2) are then injected into the test section.

Once initial ionization occurs, the coil induces eddy currents in the conducting gas,

therefore transferring energy and maintaining the gas in a plasma state. When a probe is

present, the measurements taken are heat flux, and pressure at the wall of the probe. The

setup with a probe can be seen in Fig. 1-9. The pressure at the wall is measured using a

calorimeter and a Pitot probe, together with the pressure in the reservoir. The operational

parameters are the mass flow axially injected into the torch (m), the electrical power to

the coil (W ), and the static pressure of the test chamber (Ps).

Estimate of uncertainties

Using physical Pitot probes, for a large and small Pitot probe without Homann’s

correction, the relative error on velocity is 12% and 21% respectively. Relative error on

the velocity using Homann’s correction and the large probe: 1% (cooled) to 3% (heated).

[3] The deviation of free stream conditions compared to ICP computations can be seen in

Fig 1-10. [4]

Longshot

The Longshot is a short duration free piston tunnel used to reproduce high Reynolds

number hypersonic flows. The goal is to reproduce the flow-field occurring in the re-entry

of space craft through the atmosphere. As can be seen in (Fig. 1-11) there are five sections

to the Longshot: a driver tube, a driven tube, a nozzle, a piston, and a test section. The

driver tube is filled with Nitrogen(N2) at high pressure (350x105 Pa), and the driven

tube contains the gas for the test at low pressure and close to ambient temperature. The

23

Figure 1-10. Deviation of the free stream conditions compared to the ICP computation(Probe radius: 25 mm, Plasma power: 100 kW, mass flow: 5 g/s, reservoirpressure: 13170 Pa)

[4]

Figure 1-11. The VKI Longshot. Photo permitted by the von Karman Institute[5]

piston, initially between both tubes, is shot to the end of the driven tube where as a result

the second diaphragm reaches a high pressure forming what is known as the reservoir.

The second diaphragm houses a system of 48 valves used to trap the gas in the reservoir.

Following the reservoir is a contoured nozzle which expands the flow to reach hyper sonic

velocities, into the test section where the model is placed. The operating conditions of

the longshot are as follows: mach number of 14, unit Reynolds numbers above 10 million,

N2 and CO2 gases. Three quantities are measured in the test section: the stagation point

pressure pt2, the stagnation point heat flux qw, and the free stream static pressure p0e. A

hemispherical stagnation probe with a radius of 12.7 mm was used to find the stagnation

24

point pressure and the heat flux, while a Nagamatsu probe was used to find the free

stream static pressure.

Figure 1-12. Schematic of the VKI Longshot[6]

Estimate of uncertainties

The heat flux can be measured with σ = ±5% accuracy. [6] The stagnation pressure

can be measured with a σ = 0.2% accuracy. The free stream static pressure can be

measured within σ = 5% . [7]

Numerical Tools

ICP computation

The ICP computation (by Van der Haegen at VKI [8]) is a simulation of plasma

flow at the supersonic regime in the Plasmatron. The simulation has been proven to be

consistent with previous studies and analytical results. Therefore instead of re-running

tests on the Plasmatron this flow field is used to reproduce the results obtained by del Val

Benitez.

Mutation ++

Mutation is the ”Multicomponent Thermodynamic And Transport properties

for Ionized gases in C++” library, developed by Scoggins at VKI [9]. It is used as a

database or library to provide algorithms for the accurate computation of transport and

thermodynamic properties of ionized gases in CABARET. This includes equilibrium

compositions and species production rates due to finite-rate elementary reactions. The gas

25

model currently used is assumed to be in thermodynamic equilibrium but not in chemical

equilibrium. Mutation has three separate functions for thermodynamic properties:

”mppequil, bprime, and checkmix.” The transport properties are derived from kinetic

theory providing relationships for macroscopic transport based on microscopic properties.

Objectives

The objective of this paper and work is to start the process of Uncertainty Quantification

for High-Enthalpy Facilities at VKI with applications to many other facilities around the

world. The end goal is to have one generic numerical program that can calculate free

stream parameters for most if not all ground facilities, specifically the Plasmatron and

Longshot at VKI. In order to make sure the results of this program are accurate, an

Uncertainty Analysis needs to be done. The work of this paper was started and done in

the following way:

1. Application of the CABARET code to the Plasmatron free stream conditionscharacterization.

2. Compare the results obtained with the previous work of del Val Benitez.

3. Application of the CABARET code to the Longshot free stream conditionscharacterization. This step has never been done before.

4. Review the results obtained.

5. Sensitivity Analysis for the stochastic variables qw, Pt2, and β.

26

CHAPTER 2TOOLS, FACILITIES, AND PROCEDURES

CABARET For The Plasmatron

Reproducing Results

The ICP computation was used as the data fed into CABARET to test the sensitivity

of the results and the program. The flow field was loaded into Tecplot, a piece of software

used to show the flow-field as seen in the graphic.

Figure 2-1. Temperature distribution in supersonic plasma flow in Plasmatron facility(Probe radius: 25mm, plasma power: 80kW, mass flow: 5g/s, reservoirpressure: 15550Pa, static pressure in the chamber: 1200Pa)

A one dimensional zone was then created, with line interpolation of that zone shortly

following. The interpolation data was then exported to MATLAB where a code extracted

the Pressure, Temperature, Density, and mach number at the wall and slightly before the

wall. The code also extracted the free stream conditions based off of the flow field to later

compare to the CABARET results. The conditions found at the wall are then plugged

into MUTATION++ using the mppequil function to find λw. Plugging in the conditions

at the wall, slightly before the wall, and the lambda, the heat flux is then found using the

following equation:

27

qw = λw ∗ δT

δxo

= λw ∗ T2 − T1

x2 − x1

(2–1)

After calculating the heat flux, the conditions at the wall (including the heat flux) are

plugged into inverse mode in CABARET. The results from CABARET are then compared

to the results extracted from the free stream of the flow field, just before the shockwave,

found earlier. The error is calculated and the results are analyzed. In total this process

was repeated for 7 test cases found from the Plasmatron flow field. The Prandtl number

was set to 0.713 [1].

CABARET For The Longshot

Differences Between The Facilities

Unlike the Plasmatron, the Longshot has a reservoir that is losing mass. This means

the mass flow rate m is not constant. One test is done at a time, and the timing is about

3 ms. A Nagamatsu probe is used to measure the free stream pressure. In order to have

the same amount of data points as the Plasmatron, the total time was divided into 8

equal increments and each point in time was put through CABARET at different testing

conditions. The mass flow rate is not constant as the deposit was emptied, which meant

the CABARET code needed to be changed to work around this parameter.

Obtaining The Results

A Longshot test provided by Grossir was fed into a post processing code developed

by Grossir and Diaz. This code outputs many variables and measurements that represent

the flow field. The measurements from the probes on the Longshot were fed into the

CABARET code, and compared to the actual post processed data obtained by Grossir. A

percent error was then calculated between CABARET and the actual Free stream. The

equation used for percent error is below.

% error =

∣∣∣∣old− new

old∗ 100

∣∣∣∣ (2–2)

28

Sensitivity Analysis For The Longshot

Using the same data provided (by Grossir) of this parameter on the heat flux

computation, using the Fay and Riddell Equation to determine the influence, a sensitivity

analysis was started on the velocity gradient (β) term of the heat flux equation.

CABARET was put in Forward mode, Beta was multiplied by +10%, and the results

were recorded. Beta was then multiplied by −10%, and the results were recorded. The

percent error and standard deviation can be seen in the results section. The next step

was to see how fluctuating the input variables by a certain percentage would affect the

output in Inverse mode. The two variables fluctuated were heat flux and stagnation

pressure. First the heat flux recorded from the previous step was fed into CABARET

in inverse mode. The results for temperature and mach number were then recorded.

Second, the stagnation pressure was multiplied by ±5%, ran through CABARET, and

the temperature and mach number were recorded. Collecting the results together, the

percent error and standard deviation were taken for the temperature and mach number.

The equation used for standard deviation is below.

σ =

√∑(x− x)2

n− 1(2–3)

The x is the sample mean average and n is the sample size.

29

CHAPTER 3RESULTS

Plasmatron

As seen from the table of standard deviations 3-1 and the percent error for each

case in the appendix A-2 the results match up with what del Val Benitez obtained. The

Temperatures, and mach numbers have significantly smaller percent error and standard

deviation compared to the freestream pressure. This is as a result of the assumptions

initially made while running the program. [1] [4] The real results with percent errors can

be seen in the AppendixA-2.

Table 3-1. The standard deviation of the Cabaret vs the ICP computationMeasurement Cabaret ICPT1 [K] 256.036 415.517P1 [Pa] 634.222 822.004M1 0.055 0.013V1 [m/s] 213.534 246.117

Longshot

CABARET Code

Table 3-2. Standard Deviation of Cabaret vs Longshot test dataMeasurement Cabaret TestT1 [K] 5.358 2.530P1 [Pa] 35.872 35.872M1 0.228 0.100

Comparing the results of the Longshot with the Plasmatron, the temperature and

mach number have the smallest standard deviation, with the pressure being significantly

higher. When looking at the test case results from the Longshot (as seen in the appendix

A-3), the percent error is much higher for temperature than mach number and pressure.

Seeing as the free stream pressure is directly measured, the large uncertainties in pressure

seen in the Plasmatron do not occur here. Directly measuring the free stream pressure,

by-passes the assumptions made for the Plasmatron which no longer affect the results.

30

Sensitivity Analysis

As seen in Table 3.3, when changing the beta by ±10%, the heat flux pops out a

percent error between 4.9 and 5.1%. So fluctuating Beta results in a direct increase by

half.

Table 3-3. Results of the 10% dispersion to Beta

Time FORWARD Beta + 10% Beta – 10% Error + 10% Error - 10%Q (W/m2) P0 PT2 Q (W/m2) Q (W/m2) Error Error

0.82 2058075.8 20536797.1 127379.5 2158295.3 1952251.5 4.9 5.12.94 1436219.9 15670048.9 92097.7 1506129.7 1362345.6 4.9 5.12.96 1435981.8 15652332.3 91980.2 1505880.2 1362119.9 4.9 5.12.98 1435830.3 15636278.5 91868.0 1505721.4 1361976.2 4.9 5.13.00 1435819.4 15616471.2 91743.3 1505710.0 1361966.0 4.9 5.13.02 1435951.0 15595606.2 91614.8 1505848.2 1362091.0 4.9 5.13.04 1436216.0 15571281.8 91474.7 1506126.2 1362342.4 4.9 5.13.06 1436589.2 15541705.1 91317.3 1506517.7 1362696.5 4.9 5.1

Average 1587528.6 1435973.6 4.9 5.1Standard Deviation 230624.7 208607.9 0.0 0.0

Relative Standard Deviation 14.5 14.5 0.0 0.0

When varying the heat flux, the largest percent error is between 18 to 19% percent

for the temperature. When varying the stagnation pressure the largest percent error is

between 26 to 45%. When varying both parameters (3-4), the largest standard deviation

and error can be seen in the temperature. This could be due to the assumptions made for

the temperature in the longshot calculations.

Table 3-4. Results of Inverse problem with Heat Flux and Stagnation Presssure varyingfrom their nominal value

Time (ms)Q + 5% Q - 5% Pt2 + 5% Pt2 - 5%% Error % Error % Error % Error

T1 M1 T1 M1 T1 M1 T1 M10.82 7.295 1.279 0.969 1.244 100.000 100.000 100.000 100.0002.94 12.416 6.254 18.393 6.243 45.635 8.218 45.138 8.1022.96 23.683 8.210 18.780 6.429 26.575 3.353 45.125 8.1022.98 12.575 6.528 19.057 6.565 26.485 3.354 45.038 8.102

3 12.610 6.544 19.090 6.581 26.407 3.353 44.949 8.1023.02 12.383 6.442 18.881 6.479 26.345 3.353 44.882 8.1023.04 12.255 6.386 18.764 6.423 26.303 3.353 44.845 8.1023.06 24.213 8.269 18.665 6.375 100.000 100.000 100.000 100.000

Average 14.679 6.239 16.575 5.792 47.219 28.123 58.747 31.077Standard Deviation 8.541 0.263 6.511 0.223 4.938 0.235 0.026 0.001

Relative Standard Deviation 58.188 4.220 39.285 3.845 10.457 0.837 0.045 0.002

31

The graphs in (Fig.3-1) show a visual representation of how the heat flux changes

with time as a result of ±10% dispersions to the beta term. Looking at the graphs for

heat flux, one can see the linear correlation between the positive and negative ten percent.

The figures shown in Fig.3-2 are a visual representation of dispersions of ±5% to the heat

flux and stagnation pressure. They are meant to show focus on the effects to temperature

and mach number as a result of this change.

0.5 1 1.5 2 2.5 3 3.5

Time (ms)

1.2

1.4

1.6

1.8

2

2.2

2.4

Heat F

lux (

W/m

2)

×106 Heat flux Beta+10%

OriginalNew

0.5 1 1.5 2 2.5 3 3.5

Time (ms)

1

1.2

1.4

1.6

1.8

2

2.2

Heat F

lux (

W/m

2)

×106 Heat flux Beta-10%

OriginalNew

Figure 3-1. Sensitivity Analysis Graph 1

32

0 1 2 3 4

Time (ms)

20

40

60

80

100T

em

pe

ratu

re (

K)

Heat flux +5%

OriginalNew

0 1 2 3 4

Time (ms)

11.5

12

12.5

13

13.5

Ma

ch

#

Heat flux +5% Mach

0 1 2 3 4

Time (ms)

20

40

60

80

100

Te

mp

era

ture

(K

)

Heat flux -5%

0 1 2 3 4

Time (ms)

11.5

12

12.5

13

13.5

Ma

ch

#

Heat flux -5%

2.9 2.95 3 3.05

Time (ms)

60

70

80

90

100

Te

mp

era

ture

(K

)

Stagnation Pressure +5%

2.9 2.95 3 3.05

Time (ms)

10.5

11

11.5

12

Ma

ch

#

Stagnation Pressure +5%

2.9 2.95 3 3.05

Time (ms)

60

70

80

90

100

Te

mp

era

ture

(K

)

Stagnation Pressure -5%

2.9 2.95 3 3.05

Time (ms)

10.5

11

11.5

12

Ma

ch

#

Stagnation Pressure -5%

Figure 3-2. Sensitivity Analysis Graph 233

CHAPTER 4CONCLUSION

Conclusion

The results of the Plasmatron align with what del Val Benitez predicted and tested

previously. This was purely used as a baseline to verify and validate the previous results.

Fig. 3-1 validates that adding a ±10% results in a ±5% change in the heat flux.

Taking the ±5% heat flux and feeding it back into Cabaret gives different results within

the range expected shown in 3-2. The stagnation pressure shows curves that are different

compared to the original results. The stagnation pressure exponentially grows and decays

until it reaches a stable solution.

Future Work

The following is future work that needs to be completed, with no specific order.

First, a Monte-Carlo simulation for UQ needs to be run on each of these parameters

and compared with the flow field. Second, more analysis using the Plasmatron with real

time data (instead of the ICP code), needs to be done. Third, the actual Uncertainty

Quantification needs to be finished.

34

APPENDIXSUPPLEMENTARY INFORMATION

Newton-Raphson Method

To solve the rebuilding problem, the Rankine-Hugoniot relations, and the total

quantities, the Newton-Raphson method for non-linear equations was chosen as the

iterative numerical method.

Xn+1 = Xn + δX (A–1)

JδX = −R(Xn) (A–2)

J =

δR1

δX1

δR1

δX2

δR1

δX3

δR2

δX1

δR2

δX2

δR2

δX3

δR3

δX1

δR3

δX2

δR3

δX3

(A–3)

Equation A–1 represents an iteration of the Newton-Raphson method. The variable X

in equation A–1 is the vector containing the variables to be computed after the iterative

method. The value of X has a determined decrement added to it in the term δX which is

calculated through the Jacobian of the matrix in equations A.2 and A.3 . The residual R

is the ”distance between” the measured parameters and the parameters obtained in each

iteration, having different sets of variables while converging to the final solution/variables.

The residual should gradually get smaller and smaller while the iteration process is taking

place. Taking the residual and plugging it into A–1 yields the following:

R(Un+1) = R(Un) + JδU (A–4)

Original Values Entered

The following values were used in both the Plasmatron and the Longshot analysis.

35

Table A-1. Original values plugged into CabaretOriginal Values Used

Time (ms) Pt2a (Pa) Ta (K) Qa (W/m2) Pinf (Pa) Tinf (K) Mach (inf) Hflux0.8 124203.0 305.3 2472070.0 668.7 70.7 12.1 2492778.02.9 103840.0 309.7 1829320.0 569.7 63.4 11.8 1856670.03.0 104069.0 309.8 1828740.0 568.9 63.4 11.8 1856142.03.0 104206.0 309.8 1827120.0 568.1 63.5 11.8 1854560.03.0 104096.0 309.8 1825660.0 567.3 63.5 11.8 1853135.03.0 103753.0 309.8 1824700.0 566.5 63.5 11.8 1852213.03.0 103485.0 309.9 1824360.0 565.7 63.6 11.8 1851924.03.1 103214.0 309.9 1823240.0 564.9 63.7 11.8 1850846.0

Reproducing Plasmatron Tables

Table A-2. Plasmatron Reproduction ResultsCASE2 Cabaret ICP % Error

T1 [K] 6026.83099 6375.3 5.465923329

P1 [Pa] 2666.7376 4154.5 35.81086533

M1 1.86656 1.52 -22.8

a [m/s] 1845.14 1955.03 5.620885613

V1 [m/s] 3450.41 2971.65 -16.11091481

H [J/kg] 2.92E+07 3.27E+07 10.73451165

CASE3 Cabaret ICP % Error

T1 [K] 5785.533 6041.5 4.23681205

P1 [Pa] 2665.2899 3601.7 25.99911431

M1 1.718657 1.52 -13.06953947

a [m/s] 1738.07 1826.97 4.865980284

V1 [m/s] 2989.5 2776.994 -7.652375194

H [J/kg] 2.47E+07 2.80E+07 11.74902971


T1 [K] 5488.10239 5624.2 2.419857224

P1 [Pa] 2160.94967 2948.09 26.70001018

M1 1.7316971 1.4996 -15.47726727

a [m/s] 1624.41 1660.52 2.174619999

V1 [m/s] 2813.48 2490.12 -12.98571956

H [J/kg] 2.01E+07 2.12E+07 5.199454081


T1 [K] 6051.7276 6710.2 9.813007064

P1 [Pa] 3967.12 5346.6 25.80106984

M1 1.71296 1.495 -14.57926421

a [m/s] 1823.46 2062.54 11.59153277

V1 [m/s] 3118.1 3083.5 -1.122101508

H [J/kg] 2.77E+07 3.58E+07 22.68234373


T1 [K] 6042.5394 6341.65 4.7166053

P1 [Pa] 3571.648 4786.4 25.37924118

M1 1.71174 1.52 -12.61447368

a [m/s] 1828.11 1930.55 5.306259874

V1 [m/s] 3126.1 2934.4 -6.53285169

H [J/kg] 2.80E+07 3.16E+07 11.33370045


T1 [K] 5796.643 6000.51 3.397494546

P1 [Pa] 3040.51 4128.86 26.35957625

M1 1.726 1.52 -13.55263158

a [m/s] 1732.66 1797.79 3.622781304

V1 [m/s] 2990.6 2732.6 -9.441557491

H [J/kg] 2.42E+07 2.65E+07 8.608457207


T1 [K] 5450.102 5561.5 2.003020768

P1 [Pa] 2501.465 3418.7 26.82993536

M1 1.7393 1.493 -16.49698593

a [m/s] 1600.26 1625.46 1.550330368

V1 [m/s] 2784.5 2426.8 -14.73957475

H [J/kg] 1.88E+07 1.95E+07 3.245696449

36

Reproducing Longshot Tables

Table A-3. Longshot Results

CASE1 Cabaret Test Error

T1 [K] 65.201 70.671 7.740

P1 [Pa] 668.657 668.657 0.000

M1 11.967 12.114 1.214


T1 [K] 50.319 63.433 20.674

P1 [Pa] 569.666 569.666 0.000

M1 12.586 11.829 -6.398


T1 [K] 50.074 63.449 21.080

P1 [Pa] 568.871 568.871 0.000

M1 12.608 11.830 -6.584


T1 [K] 49.878 63.463 21.406

P1 [Pa] 568.079 568.079 0.000

M1 12.626 11.831 -6.720


T1 [K] 49.851 63.494 21.488

P1 [Pa] 567.288 567.288 0.000

M1 12.628 11.831 -6.737


T1 [K] 49.987 63.534 21.323

P1 [Pa] 566.500 566.500 0.000

M1 12.616 11.831 -6.635


T1 [K] 50.087 63.590 21.233

P1 [Pa] 565.714 565.714 0.000

M1 12.608 11.830 -6.579


T1 [K] 50.175 63.666 21.190

P1 [Pa] 564.930 564.930 0.000

M1 12.600 11.828 -6.530

37

Longshot Sensitivity Tables

Table A-4. Long Shot Sensitivity Analysis ResultsInverse

Time (ms) Q + 5% T1 M1 V1 Q - 5% T1 M1 V1 Pt2 + 5% T1 M1 V1 Pt2 - 5% T1 M1 V1

0.82 2158295.3 75.8 12.0 2122.7 1952251.5 70.0 12.0 2040.1 133748.4 121010.5

2.94 1506129.7 55.6 12.6 1909.7 1362345.6 51.8 12.6 1843.2 96702.6 92.4 10.9 2127.1 87492.8 92.1 10.9 2126.1

2.96 1505880.2 48.4 12.8 1815.7 1362119.9 51.5 12.6 1842.3 96579.2 80.3 11.4 2088.5 87381.2 92.1 10.9 2126.4

2.98 1505721.4 55.5 12.6 1913.6 1361976.2 51.4 12.6 1841.9 96461.4 80.3 11.4 2088.2 87274.6 92.0 10.9 2126.2

3.00 1505710.0 55.5 12.6 1914.0 1361966.0 51.4 12.6 1842.3 96330.4 80.3 11.4 2088.1 87156.1 92.0 10.9 2126.1

3.02 1505848.2 55.7 12.6 1915.2 1362091.0 51.5 12.6 1843.5 96195.5 80.3 11.4 2088.2 87034.0 92.0 10.9 2126.3

3.04 1506126.2 55.8 12.6 1916.3 1362342.4 51.7 12.6 1844.5 96048.4 80.3 11.4 2088.6 86901.0 92.1 10.9 2126.8

3.06 1506517.7 48.3 12.8 1813.2 1362696.5 51.8 12.6 1845.6 95883.1 86751.4

38

REFERENCES

[1] A.-I. del Val Benitez, “Characterization of ground testing conditions in high enthalpyand plasma wind tunnels for aerospace missions,” Master’s thesis, UniversidadPolitecnica de Madrid, 2015, von Karman Institute for Fluid Dynamics.

[2] A. I. del val Benitez and T. Magin, “Uncertainty quantification on cabaret,” vonKarman Institute for Fluid Dynamics.

[3] T. Magin, “Cooled pitot probe in inductive air plasma jet,” Aerothermodynamicsand Fluid Mechanics Seminar, February 2012, presented at the University of Texas inAustin.

[4] del val Benitez, Magin, Diaz, and Chazot, “Characterization of ground testingconditions in high enthalpy and plasma wind tunnels for aerospace missions,” 2015,von Karman Institute for Fluid Dynamics.

[5] B. Dias, “Rebuilding of hypersonic free stream conditions based on heat flux andpressure measurements,” VKI SR, von Karman Institute for Fluid Dynamics, 2012.

[6] G. Grossir, “Longshot Hypersonic Wind Tunnel Flow Characterization and BoundaryLayer Stability Investigations,” Ph.D. dissertation, University Libre do Bruxelles, 2015.

[7] J. Muylaert, “Design and calibration of a flush air data system (fads) for prediction ofthe atmospheric properties during re-entry,” von Karman Institute for Fluid Dynamics,Chaussée de Waterloo 72, 1640 Rhode-St-Genèse, Belgium, Tech. Rep., December2012.

[8] V. Van Der Haegen, “Development of an all-speed approximate riemann solver appliedto supersonic plasmas,” 2013, vKI PR, von Karman Institute for Fluid Dynamics.

[9] J. Scoggins, “Development of mutation++: Multicomponent thermodynamicsand transport properties for ionized gases library in c++,” AIAA/ASME JointThermophysics and Heat Transfer Conference, June 2014, atlanta, GA.

39

BIOGRAPHICAL SKETCH

Margaret Owen graduated from UF with a Master of Science in aerospace engineering.

The research done for this thesis was performed at the von Karman Institute for Fluid

Dynamics in Belgium.

40

uncertainty quantification for high enthalpy …uncertainty quantification for high enthalpy...

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