Uncertainty Quantification in Hypersonic Reentry

Flows Due to Aleatory and Epistemic Uncertainties

Benjamin R. Bettis∗ and Serhat Hosder†

Missouri University of Science and Technology, Rolla, MO, 65409

The objective of this study was to demonstrate an efficient methodology for the quan-tification of mixed (aleatory and epistemic) uncertainties in hypersonic flow computations.In particular, the approach was used to quantify the uncertainty in surface heat flux tothe spherical non-ablating heat-shield of a reentry vehicle at zero-angle of attack due toepistemic and aleatory uncertainties that may exist in various parameters used in the nu-merical solution of hypersonic, viscous, laminar blunt-body flows with thermo-chemicalnon-equilibrium. Three main uncertainty sources were treated in the computational fluiddynamics (CFD) simulations: (1) aleatory uncertainty in the freestream velocity, (2) epis-temic uncertainty in the recombination efficiency for a partially catalytic wall boundarycondition, and (3) epistemic uncertainty in the binary-collision integrals. Second-OrderProbability utilizing a stochastic response surface obtained with quadrature-based non-intrusive polynomial chaos was used for the propagation of mixed uncertainties. The un-certainty quantification approach was validated on a stochastic model problem with mixeduncertainties for the prediction of stagnation point heat transfer with Fay-Riddell relation,which included the comparison with direct Monte Carlo sampling results. In the stochasticCFD problem, the uncertainty in surface heat transfer was obtained in terms of intervalsat different probability levels at various locations including the stagnation point and theshoulder region. A non-linear global sensitivity analysis based on Sobol indices showed thatthe velocity and recombination efficiency were the major contributors to the uncertaintyin the stagnation point heat flux for the reentry case considered in this study.

Nomenclature

C = Mass Fraction Δhf = Heat of formation (J/kg)

CoV = Coefficient of Variation, D1/2

μα∗ μ = Coefficient of viscosity (kg/m·s)D = Statistical variance μα∗ = Mean

h = Enthalpy (J/kg) ξ = Standard random variable

Le = Lewis number �ξa = Standard aleatory uncertain variable

n = Number of random variables �ξe = Standard epistemic uncertain variable

p = Pressure (N/m2) ρ = Density (kg/m3)

Pr = Prandtl number Ψ = Random basis function

RN = Radius of curvature (m) Ω1,1 = Diffusion collision integral

S = Sobol indice Ω2,2 = Viscosity collision integral

ST = Total Sobol indice Subscripts

T = Temperature (K) e = Boundary layer edge

V = Velocity (m/s) o = Total or stagnation condition

α = Spectral modes sp = Stagnation point

α∗ = Stochastic output variable w = Wall

γ = Recombination efficiency ∞ = Freestream

∗Ph.D. Student, Aerospace Engineering, 400 West 13th Street, Rolla, MO 65409, Student AIAA Member.†Assistant Professor, Aerospace Engineering, 400 West 13th Street, Rolla, MO 65409, AIAA Senior Member.

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49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition4 - 7 January 2011, Orlando, Florida

AIAA 2011-252

Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

I. Introduction

Uncertainties are generally ubiquitous in the analysis and design of highly complex engineering systems.Uncertainties can arise from the lack of knowledge in physical modeling (epistemic uncertainty), inherentvariations in the systems (aleatory uncertainty), and numerical errors in the computational procedures usedfor analysis. It is important to account for all of these uncertainties in applications such as robust and reliabledesign of multi-disciplinary aerospace systems. One application is the design of a thermal protection system(TPS) for an atmospheric reentry vehicle. Orbital vehicles travel at very high velocities when reenteringthe Earth’s atmosphere and will experience a significant magnitude of aeroheating. In order to designand fabricate a reliable TPS for a reentry vehicle, engineers must have a tool set for accurate predictionof the surface heat flux during atmospheric reentry. Due to the high enthalpy and velocity requirementsfor most hypersonic flow simulations including reentry flows, there are few facilities where experimentscan be performed. These experiments also cover a limited reentry envelope with very small operating times.Therefore, computational fluid dynamics (CFD) methods play an important role in the prediction of the flowfield and the surface heat flux for atmospheric reentry, and for hypersonic applications in general. Accuratenumerical prediction of hypersonic flow fields are challenging due to the complex nature of the physics suchas strong shock waves, viscous shock layers, and non-equilibrium thermo-chemistry. Various uncertaintiesassociated with high fidelity hypersonic flow simulations can have significant effects on the accuracy of theresults including the surface heat flux. Therefore, it is important to include these uncertainties in thesimulations to assess the accuracy of the results and to obtain robust and/or reliable reentry vehicle designs.

The objective of this study is to demonstrate an efficient methodology for the quantification of mixed(aleatory and epistemic) uncertainties in hypersonic flow computations. In particular, the approach is usedto quantify the uncertainty in surface heat flux to the spherical non-ablating heat-shield of a reentry vehicleat zero-angle of attack due to epistemic and aleatory uncertainties that may exist in various parametersused in the numerical solution of hypersonic, viscous, laminar blunt-body flows with thermo-chemical non-equilibrium. In this study, the freestream velocity (V∞), binary collision integrals for the N2−O interaction,and the recombination efficiency (γ) of oxygen and nitrogen atoms used in the description of catalytic wallboundary condition1 were treated as uncertain variables. A recent work by MacLean et. al.,2 which includedboth experimental and numerical studies on the hypersonic aerodynamic heating of spherical capsule geome-tries, demonstrated a significant variation of the surface heat-flux with varying recombination efficiencies(e.g., catalytic wall conditions) and freestream velocity. Another study by Bose et al.3 demonstrated theimportance of uncertainty in the binary collision integrals for a Martian entry problem. The uncertaintyquantification in CFD simulations of the current study was performed for a particular test case and capsulegeometry selected from the work of MacLean et al.,2 where the freestream velocity for the experiment was4167 m/s at a stagnation enthalpy of 9.9 MJ/kg.

In the current work, the freestream velocity was modeled as an inherent uncertain variable describedwith a normal probability distribution. The recombination efficiency and binary collision integrals weremodeled as epistemic uncertain variables, since their uncertainty originates due to the lack of knowledge ina physical model,4 and they were represented as an interval with specified bounds. For the quantificationof mixed (aleatory-epistemic) uncertainty, Second-Order Probability Theory was used.5,6 The Quadrature-Based Non-Intrusive Polynomial Chaos (NIPC) Method (Hosder and Walters7) was utilized to propagatethe input uncertainties in the freestream velocity (inherent uncertainty), binary collision integrals (epistemicuncertainty), and the recombination efficiency (epistemic uncertainty) for the overall quantification of uncer-tainty in surface heat flux. In general, the NIPC methods, which are based on the spectral representation ofuncertainty, are computationally more efficient than traditional Monte Carlo methods for moderate numberof uncertain variables and can give highly accurate estimates of various uncertainty metrics. In addition,they treat the deterministic model (e.g, the CFD code) as a black box and the uncertainty information in theoutput is approximated with a polynomial expansion, which is constructed using a number of deterministicsolutions each corresponding to a sample point in random space. Therefore, the NIPC methods become aperfect candidate for the uncertainty quantification in the numerical solutions of viscous, non-equilibriumhypersonic flows, which are computationally expensive and complex. More information on the uncertaintyquantification in fluid dynamics with NIPC methods can be found in a recent review by Hosder and Walters.8

The previous studies involving uncertainty quantification in different hypersonic re-entry problems includeBose et. al.3,,9 Weaver et. al.,10 and Ghaffari et. al.,11 which mainly treated all input uncertainties asprobabilistic. A recent study by Bettis and Hosder12 investigated the propagation of mixed uncertaintiesthrough high-fidelity CFD simulations for quantifying uncertainty in the aerodynamic heating of a hypersonic

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reentry vehicle. In that study, the freestream velocity was modeled as an inherent uncertain variable describedwith a uniform probability distribution, and the recombination efficiency at the wall (γ) was treated as anepistemic uncertainty represented with an interval. The Point-Collocation Non-Intrusive Polynomial Chaosmethod was used to propagate the uncertainty through the CFD simulations. The current work aims to buildupon the previous work by Bettis and Hosder.12 In specific, this study includes the modeling of additionalepistemic uncertainties (i.e., binary collision integrals) and involves a test case with a higher freestreamtotal enthalpy, which may further amplify the uncertainty in aerodynamic heating due to the increase in thecomplexity of aerothermodynamic phenomena. Furthermore, the mixed uncertainties are propagated usingQuadrature-Based NIPC to demonstrate that different NIPC methods can be implemented in the mixeduncertainty quantification framework that utilizes Second-order probability. Another aspect of this paperincludes non-linear sensitivity analysis (SA) with Sobol indices.13

In the following section, the approach for aleatory and epistemic uncertainty quantification using Quadra-ture Based NIPC and Second-Order Probability will be described. The uncertainty quantification approachwill be first applied to a model problem involving Fay-Riddell relation for approximating stagnation pointheat transfer on a blunt body in hypersonic flow (Section III). Due to the low computational cost of eval-uating the Fay-Riddell relation, the results will also be compared to Monte Carlo (MC) simulations forthe validation of the proposed uncertainty quantification methodology. In Section IV, all relevant modelingaspects for the high-fidelity CFD simulations will be outlined along with the description of the stochasticnature of the problem at hand. Then the uncertainty results will be presented and sensitivity analysis willbe conducted to describe the relative importance of each uncertainty source. The conclusions will be givenin Section V.

II. Uncertainty Quantification Approach

A. Types of Uncertainties in Computational Simulations

As described in Oberkampf et. al.,4 there can be three different types of uncertainty and error in a com-putational simulation: (1) aleatory uncertainty, (2) epistemic uncertainty, and (3) numerical error. Theterm aleatory uncertainty describes the inherent variation of a physical system. Such variation is due tothe random nature of input data and can be mathematically represented by a probability density functionif substantial experimental data are available for estimating the distribution (uniform, normal, etc.). Thevariation of the free-stream velocity or manufacturing tolerances can be given as examples for aleatory un-certainty in a stochastic external aerodynamics problem. The aleatory uncertainty is sometimes referred asirreducible uncertainty due to its nature.

Epistemic uncertainty in a non-deterministic system originates due to ignorance, lack of knowledge, orincomplete information (such as the values of transport quantities or catalytic wall recombination efficienciesin high temperature hypersonic flow simulations). The key feature of this definition is that the fundamentalcause is incomplete information of some characteristics of the system. As a result, an increase in knowledge orinformation can lead to a decrease in the epistemic uncertainty. Therefore, epistemic uncertainty is referredto as reducible uncertainty. As shown by Oberkampf and Helton,14 modeling of epistemic uncertainties withprobabilistic approaches may lead to inaccurate predictions in the amount of uncertainty in the responsesdue to the lack of information on the characterization of uncertainty as probabilistic. One approach tocharacterize the epistemic uncertain variables is to use intervals. The upper and lower bounds on theuncertain variable can be prescribed using either limited experimental data or expert judgment.

Numerical error is defined as a recognizable deficiency in any phase or activity of modeling and simulationthat is not due to the lack of knowledge. If errors cannot be well-characterized, then they must be treatedas part of the epistemic uncertainties. The discretization error in spatial or temporal domain originatingfrom the numerical solution of partial differential equations that describes a physical model in a discretizedcomputational space (mesh) can be given as an example of numerical uncertainty.

B. Mixed Aleatory-Epistemic Uncertainty Propagation

1. Second-Order Probability

In this study, Second-Order Probability5,6 was utilized to propagate mixed (aleatory and epistemic) un-certainty through CFD simulations and the Fay-Riddell model problem. Second-Order Probability uses an

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Figure 1. Schematic of second-order probability.

inner loop and an outer sampling loop as described in Figure 1. In the outer loop, a specific value for theepistemic variable is prescribed and then passed down to the inner loop. Any traditional aleatory uncertaintymethod may then be used to perform aleatory uncertainty analysis in the inner loop for the specified valueof the epistemic uncertain variable. The Second-Order Probability will give interval bounds for the outputvariable of interest at different probability levels. Each iteration of the outer loop will produce a cumulativedistribution function (CDF) based on the aleatory uncertainty analysis in the inner loop. Thus, if there are100 samples in the outer loop, then 100 different CDF curves will be generated. One major advantage ofSecond-Order Probability is that it is easy to separate and identify the aleatory and epistemic uncertainties.On the other hand, the two sampling loops can make this method computationally expensive especially iftraditional sampling techniques, such as Monte Carlo, are used for the uncertainty propagation.

Since this study is mainly focused on efficient uncertainty propagation, a quadrature-based NIPC methodwill be utilized to fit a stochastic response surface to the output quantity of interest (e.g., surface heat flux )as a function of both aleatory and epistemic uncertain variables. The Second-Order Probability approach willthen be implemented by sampling the epistemic uncertain variables (in the outer loop) and then samplingthe aleatory uncertain variables in the inner loop (for a fixed value of epistemic uncertain variable) andevaluating these sample points using the stochastic response surface approximation.

2. Quadrature-Based Non-Intrusive Polynomial Chaos

The Quadrature-Based Non-Intrusive Polynomial Chaos is derived from polynomial chaos theory, which isbased on the spectral representation of the uncertainty. An important aspect of spectral representationof uncertainty is that one may decompose a random function (or variable) into separable deterministicand stochastic components. For example, for any random variable (i.e., α∗ ) such as velocity, pressure, ortemperature in a stochastic fluid dynamics problem, one can write,

α∗(�x, �ξ) ≈P∑

j=0

αj(�x)Ψj(�ξ) (1)

where αj(�x) is the deterministic component and Ψj(�ξ) is the random basis function corresponding to thejth mode. Here we assume α∗ to be a function of the deterministic independent variable vector �x and then-dimensional random variable vector �ξ = (ξ1, ..., ξn), which has a specific probability distribution. In theory,the polynomial chaos expansion given by Equation 1 should include infinite number of terms, however inpractice a discrete sum is taken over a number of output modes. For a total order expansion, the number ofoutput modes is given by,

Nt = P + 1 =(n+ p)!

n!p!(2)

which is a function of the order of polynomial chaos (p) and the number of random dimensions (n). The basisfunction ideally takes the form of multi-dimensional Hermite Polynomial to span the n-dimensional randomspace when the input uncertainty is Gaussian (unbounded), which was first used by Wiener15 in his original

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work of polynomial chaos. To extend the application of the polynomial chaos theory to the propagationof continuous non-normal input uncertainty distributions, Xiu and Karniadakis16 used a set of polynomialsknown as the Askey scheme to obtain the ”Wiener-Askey Generalized Polynomial Chaos”. The Legendreand Laguerre polynomials, which are among the polynomials included in the Askey scheme are optimal basisfunctions for bounded (uniform) and semi-bounded (exponential) input uncertainty distributions respectivelyin terms of the convergence of the statistics. The multivariate basis functions can be obtained from theproduct of univariate orthogonal polynomials (see Eldred et al.17). If the probability distribution of eachrandom variable is different, then the optimal multivariate basis functions can be again obtained by theproduct of univariate orthogonal polynomials employing the optimal univariate polynomial at each randomdimension. This approach requires that the input uncertainties are independent standard random variables,which also allows the calculation of the multivariate weight functions by the product of univariate weightfunctions associated with the probability distribution at each random dimension. The detailed informationon polynomial chaos expansions can be found in Walters and Huyse,18 Najm,19 and Hosder and Walters.8

To model the uncertainty propagation in computational simulations via polynomial chaos with the intru-sive approach, all dependent variables and random parameters in the governing equations are replaced withtheir polynomial chaos expansions. Taking the inner product of the equations, (or projecting each equationonto jth basis) yields P + 1 times the number of deterministic equations which can be solved by the samenumerical methods applied to the original deterministic system. Although straightforward in theory, anintrusive formulation for complex problems can be relatively difficult, expensive, and time consuming to im-plement. To overcome such inconveniences associated with the intrusive approach, non-intrusive polynomialchaos formulations have been considered for uncertainty propagation.

The quadrature-based NIPC method uses spectral projection to find the polynomial coefficients αk =αk (�x) in Equation (1). This equation is projected onto the kth basis:

⟨α∗(�x, �ξ),Ψk(�ξ)

⟩=

⟨P∑

j=0

αj(�x)Ψj(�ξ)Ψk(�ξ)

⟩(3)

Then, by the virtue of orthogonality,⟨α∗(�x, �ξ),Ψk(�ξ)

⟩= αk(�x)

⟨Ψ2

k(�ξ)⟩

(4)

which leads to

αk(�x, �ξ) =

⟨α∗(�x, �ξ),Ψk(�ξ)

⟩⟨Ψ2

k(�ξ)⟩ =

1⟨Ψ2

k(�ξ)⟩ ∫

R

α∗(�x, �ξ)Ψk(�ξ)p(�ξ)d�ξ (5)

The objective of the spectral projection methods is to predict the polynomial coefficients by evaluating

the numerator (⟨α∗(�x, �ξ),Ψk(�ξ)

⟩) in Equation (5), since the term in the denominator (

⟨Ψ2

k(�ξ)⟩) can be

computed analytically for multivariate orthogonal polynomials.With the quadrature-based non-intrusive approach, the multi-dimensional integral in the numerator

term of Equation (5) is evaluated with numerical quadrature17 in the support region (R) defined by thebounds of input uncertain variables. For the integration of one-dimensional problems, the straight-forwardapproach will be to use Gaussian quadrature points, which are the zeros of orthogonal polynomials thatare optimal for the given input uncertainty distribution (i.e., Gauss-Hermite, Gauss-Legendre, and Gauss-Laguerre points for normal, uniform, and exponential distributions, respectively). The extension of thisapproach to multidimensional problems can be achieved via tensor product of one-dimensional quadratureformulas. For a one-dimensional problem, if the polynomial degree for the chaos expansion is chosen asnp, then the minimum number of quadrature points required for the exact evaluation of the integral willbe np + 1, since a Gauss quadrature formula of np points will evaluate a polynomial of 2np − 1 degreeor less exactly and the polynomial degree of the product of the function approximation and the basis inthe integrand of the numerator term in Equation (5) will be 2np for the evaluation of the highest degreecoefficient. For stochastic problems with small number of input uncertain variables (i.e, n ≤ 4) this approachwill be computationally efficient compared to the expense of a typical Monte Carlo simulation. However, formultidimensional problems with large number of uncertain variables, the computational cost may becomesignificant due to its exponential growth with the number of random dimensions, since the required numberof deterministic function evaluations will be (np + 1)n for a stochastic problem with n random variables

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Innerloop

Select Collocation Points

Evaluate Deterministic Code at Selected Collocation Points

Formulate Stochastic Response Surface Using NIPC

Sample epistemic variables

Sample aleatory variables

Evaluate stochastic response surface

Second Order Probability Sampling Loops

Outerloop

Figure 2. Flowchart describing the procedure for propagating mixed aleatory-epistemic uncertainties withSecond-Order Probability and NIPC response surface.

having the same degree of polynomial expansion (np) in each dimension. It is important to emphasize thatcomputational expense of propagating mixed input uncertainty will be high even if the number of epistemicand aleatory uncertain variables is not large as explained in the previous section. Therefore utilizing astochastic (polynomial chaos) response surface will significantly reduce the required number of deterministicCFD evaluations for the propagation of mixed uncertainties.

3. Second-Order Probability with Stochastic Response Surface

The current study utilizes an efficient approach for the propagation of mixed uncertainties using the frame-work based on Second-Order Probability. With this approach, the stochastic response (e.g., the surfaceheat transfer in the current study) is represented with a polynomial chaos expansion both on aleatory andepistemic variables. In this study, quadrature-based NIPC was used to construct the stochastic responsesurface, however the other NIPC methods (i.e., point-collocation or sampling based) can also be used. Theoptimal basis functions are used for the aleatory variables whereas Legendre polynomials are used for theepistemic uncertain variables. It should be noted that the use of Legendre polynomials should not imply auniform probability assignment to the epistemic variables. This choice is made due to the bounded natureof epistemic uncertain variables. Once the stochastic response surface is formed, at fixed values of epistemicuncertain variables, the stochastic response values can be evaluated for a large number of samples randomlyproduced based on the probability distributions of the aleatory input uncertainties (inner loop of Second-Order Probability). This procedure will produce a single cumulative distribution function. By repeating theinner loop procedure for a large number of epistemic uncertain variables sampled from their correspondingintervals (outer loop of Second-Order Probability), a population of cumulative distribution functions canbe obtained which can be used to calculate the bounds of the stochastic response at different probabilitylevels. A flowchart of the entire process of propagating mixed aleatory-epistemic uncertainties is shown inFigure 2. Due to the analytical nature (polynomial) of the stochastic response, the described procedurewill be computationally efficient, especially compared to the approaches based on direct MC sampling whichrequire a large number of deterministic CFD simulations.

4. Global Sensitivity Analysis with Sobol Indices

One of the main goals of global sensitivity analysis is to demonstrate the relative importance of each inputuncertainty to the overall uncertainty in the output variable of interest from a simulation code. In applicationswhere multiple uncertain parameters are present, it is often useful to rank the contribution of each uncertainty

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to the overall uncertainty in the output. In the current study, Sobol13 indices will be used to accomplish thistask. The Sobol Decomposition is a variance-based global sensitivity method and once the polynomial chaos(PC) expansions are formed (using Equation (5)) for output uncertain variables, it is fairly straightforwardto derive the associated Sobol indices. First, the total variance (D) can be written in terms of the PCexpansion:

D =

P∑j=1

α2j (�x)

⟨Ψ2

j (�ξ)⟩

(6)

Then, as shown by Sudret,20 Crestaux et. al.,21 and Ghaffari et. al.,11 the total variance can be decomposedas:

D =

i=n∑i=1

Di +

i=n−1∑1≤i<j≤n

Di,j +

i=n−2∑1≤i<j<k≤n

Di,j,k + · · ·+D1,2,...,n (7)

where the partial variances (Di1,...,is) are given by:

Di1,...,is =∑

β∈{i1,...,is}α2β

⟨Ψ2

β(�ξ)⟩, 1 ≤ i1 < . . . < is ≤ n (8)

Then the Sobol indices (Si1···is) are defined with,

Si1···is =Di1,...,is

D(9)

which satisfiesi=n∑i=1

Si +

i=n−1∑1≤i<j≤n

Si,j +

i=n−2∑1≤i<j<k≤n

Si,j,k + · · ·+ S1,2,...,n = 1.0 (10)

The Sobol indices are a measure of the relative contribution to the total variance due to input variables{ξi1 · · · ξis}. Theses indices provide a sensitivity measure due to individual contribution from each inputuncertainty (Si), as well as the mixed contributions ({Si,j}, {Si,j,k}, · · · ). As shown by Sudret20 and Ghaffariet. al.,11 the total (combined) effect (STi

) of an input parameter i is defined as the summation of the partialSobol indices that include that particular parameter:

STi =∑Li

Di1,...,is

D; Li = {(i1, . . . , is) : ∃ k, 1 ≤ k ≤ s, ik = i} (11)

For example with n = 3, the total contribution to the overall variance from the first uncertain variable(i = 1) can be written as:

ST1= S1 + S1,2 + S1,3 + S1,2,3 (12)

From these formulations, it can be seen that the Sobol indices can be used to provide a relative ranking ofeach input uncertainty to the overall variation in the output with the consideration of non-linear correlationbetween input variables and output quantities of interest. One of the goals of the current work is to calculateSobol indices with the PC formulation and then use them to rank the relative importance of each inputuncertainty.

III. Stochastic Model Problem for Stagnation Point Heat Transfer

A. Description of Deterministic Fay-Riddell Correlation

Before the high-fidelity hypersonic CFD problem, the mixed uncertainty quantification approach (the NIPCmethod and Second-Order Probability) was applied to a model problem which included the prediction ofstagnation point heat flux on a blunt body. For this model problem, it was assumed that the boundary layerwas laminar, flow was in equilibrium, and the vehicle’s wall was fully catalytic. With these assumptions, ananalytical correlation for the stagnation point heat flux was derived by Fay and Riddell:22

˙qsp = 0.76 (Prw)−0.6

(ρwμw)0.1

(ρeμe)0.4

√(dUe

dx

)sp

(hoe − hw)

[1 +

(Le0.52 − 1

)( hD

hoe

)](13)

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Table 1. Table outlining the free stream conditions for the CFD simulations and the model problem.

H0 (MJ/kg) V∞ (m/s) T (K) ρN2(kg/m3) ρO2

(kg/m3) ρNO (kg/m3) ρO (kg/m3)

9.9 4167 522 0.001168 0.0002719 0.0001041 0.00004596

where (dUe

dx

)sp

=1

RN

√2pe − p∞

ρe(14)

hD =∑i

Cie (Δhf )◦i (15)

In Equation (13), Pr symbolizes the Prandtl Number, which was assumed to be 0.714, and Le symbolizesthe Lewis number. The subscripts e and w represent the property at the edge of the boundary layer andat the wall of the vehicle respectively. In Equation (14), RN represents the radius of curvature of thespherical capsule geometry used in the experiments by MacLean et. al.2 In Equation (15), Ci represents themass fraction of the chemical species behind the normal shock wave which was calculated using statisticalthermodynamics.23 The heats of formation at absolute zero, (Δhf )

◦i , were taken as zero for the molecules.

The properties behind the normal shock were found with equilibrium air assumption using thermodynamiccurve fits by Srinivasan et. al.24 The freestream conditions correspond to one of the experimental testcases conducted by MacLean et. al.2 (Table 1) and the wall temperature was held constant at 300K (cold-wall boundary condition), consistent with the experiment. It should be noted that the conditions of theexperimental test case were used just as reference values for the model problem and not for comparison,since the actual flow in the tests are in thermo-chemical non-equilibrium.

B. Description of the Stochastic Problem

For this case, the freestream velocity, radius of curvature (RN ), Lewis number (Le), and the dynamicviscosity at the boundary layer edge (μe) were treated as random variables within the Fay-Riddell relation.The freestream velocity and the radius of curvature were assumed to be inherent uncertain variables. Thecoefficient of viscosity and Lewis number (physical model parameters) were assumed to be epistemic uncertainvariables. The dynamic viscosity was modeled using Sutherland’s Law. It is known that the accuracy ofSutherland’s Law degrades at high temperatures (beyond 3000K as shown in Anderson25) due to dissociationand ionization effects. One can use high-order models or curve-fits to increase the prediction accuracy ofviscosity at high temperatures. However, by retaining Sutherland’s law in this study, an epistemic uncertaintyis intentionally introduced to the model problem. In specific, the coefficient of viscosity was modeled as anepistemic variable through the introduction of a factor (k) which is multiplied with the value obtained withSutherland’s Law (e.g., μe = k × μeref ). This factor is treated as an epistemic uncertain variable with aspecified interval which had the upper and lower bounds approximated using the following procedure: First,a stagnation temperature of 4,388K behind a normal shock wave was obtained with the equilibrium aircalculations using the mean freestream velocity. Then, a chart in Anderson25 was utilized to approximatethe range of variation for the coefficient of viscosity at the calculated temperature relative to the valuecalculated by the Sutherland’s Law. This gave a lower and upper bound of 1.0 and 1.15 for the multiplier,which has been used in the calculations. For the model problem, the Lewis number is also modeled as anepistemic uncertain variable with the lower and upper bounds of 1.3 and 1.5, respectively. The freestreamvelocity was assumed to have a normal distribution with a mean of 4167 m/s (Table 1), which was the nominalvelocity in the test section of the wind tunnel for the experiments from MacLean et. al.2 A coefficient ofvariance (CoV ) of 1% was assigned to the freestream velocity, which corresponds to a standard deviation of41.67 m/s. The radius of curvature (RN ) was assumed to have a normal distribution with a mean of 0.17526m and a CoV of 1%. An overview of each of the uncertain variables is given in Table 2.

C. Mixed Aleatory-Epistemic Uncertainty Quantification

The approach described in Section II was followed to propagate the mixed (aleatory and epistemic) uncer-tainty through the Fay-Riddell relation. Convergence studies were carried out and it was found that a 3rd

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Table 2. Uncertainty ranges for the parameters used in the Fay-Riddell model problem

Uncertain Parameter Uncertainty Type Uncertainty Range

V∞ Aleatory (normal) μα∗ = 4167 m/s, CoV = 1%

RN Aleatory (normal) μα∗ = 0.17526 m, CoV = 1%

k (μ/μref ) Epistemic [1.0,1.15]

Le Epistemic [1.3,1.5]

order polynomial chaos was sufficient for the convergence of the NIPC response surface which required atotal of 256 deterministic function evaluations. A Latin Hypercube Sample (LHS) of size 10,000 was usedfor sampling the epistemic uncertain variables in the outer loop of Second-Order Probability. The NIPCresponse surface was utilized for the inner-loop of Second-Order Probability (aleatory UQ), where a totalof 5,000 samples were taken for the aleatory uncertain variables. Again, it is stressed that each of these5,000 function evaluations were evaluated using the stochastic response surface, not the original Fay-Riddellrelation. Each iteration of the outer-loop in Second-Order Probability produced one cumulative distribu-tion function (CDF) curve. Therefore, the overall Second-Order Probability analysis produced 10,000 CDFcurves.

Figure 3 shows the mixed uncertainty results. The left plot shows the results obtained with Second-OrderProbability approach with the NIPC response surface formulation and right plot gives the results obtainedwith a direct Monte Carlo (MC) approach that utilized 10,000 samples for the outer-loop and 5,000 samplesfor the inner loop (a total number of 5 × 107 Fay-Riddell evaluations). By comparing the results of NIPCand MC, it can clearly be seen that the NIPC results compare well with MC. These results indicate thatthe Second-Order Probability with stochastic response surface approach performs well for mixed uncertaintypropagation and provide confidence for using the same method in CFD simulations (to be discussed in thenext section).

Stagnation heat flux information at particular probability levels is shown in Table 3. In this table, theheat flux uncertainty results obtained from Second-Order Probability are reported using intervals at eachprobability level. The second column in the table is for the results obtained with the NIPC response surfaceformulation for uncertainty propagation and the third column shows results obtained with the MC. Onceagain, the NIPC results are consistent with the MC which demonstrates the effectiveness of the NIPC method.The fourth column lists the results from a pure aleatory uncertainty analysis that modeled the coefficientof viscosity and the Lewis number as uniform random variables. The same 3rd order NIPC response surfacewas used to propagate the aleatory uncertainty. Although it may not be appropriate to treat the coefficientof viscosity and Lewis number as a probabilistic uncertainties due to their nature, the results are shownhere for the purpose of comparison to mixed uncertainty results. It can be seen that only a single value isavailable (not an interval) at each probability level for the aleatory NIPC results. Overall, the results of the

Figure 3. Horse-tail plots representing mixed aleatory-epistemic uncertainty results from the Fay-Riddellmodel problem for the NIPC method (left) and the MCS (right).

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Table 3. Stagnation point heat transfer (W/cm2) at different probability levels for the model problem.

Probability Level Second-Order Probability (NIPC) Second-Order Probability (MC) Aleatory (NIPC)

P = 0.0 [103.65, 119.45] [107.47, 117.963] 109.36

P = 0.2 [118.60, 129.96] [118.63, 128.99] 123.75

P = 0.4 [120.80, 132.25] [120.88, 131.34] 126.37

P = 0.6 [122.74, 134.22] [122.82, 133.48] 128.74

P = 0.8 [124.96, 136.44] [125.01, 135.99] 131.46

P = 1.0 [135.32, 150.52] [136.25, 152.73] 148.64

model problem indicate that Second-Order Probability with the NIPC stochastic response surface performswell when compared with the MCS results. This helps validate the methodology so that it can be appliedto more complex problems, such as high-fidelity CFD simulations.

1. Global Sensitivity Analysis with Linear Regression

The purpose of global sensitivity analysis (SA) is to quantify and rank the importance of individual uncertainrandom variables on the overall uncertainty in an output variable of interest from a simulation code. For themodel problem, a global SA approach was used to provide the relative importance of each of the uncertainvariables on the stagnation point heat transfer uncertainty. Helton et. al.26 describes a sampling-based SAprocedure using linear regression for calculating correlation coefficients and interpreting the results based onthese coefficients. Bose et. al.3,9 considered a similar SA approach in their uncertainty quantification studiesof hypersonic entry into Martian and Titan atmospheres. In this study, the same linear global SA methodwas used by creating a total number of 20,000 samples from the 3rd order stochastic response obtained forthe uncertainty analysis described in the previous section.

The SA results are shown in the form of scatter plots in Figure 4. Qualitatively, one can see the relativeimportance simply by observing the thickness of the band in the scatter plots. It is obvious that thefreestream velocity has a more drastic impact on stagnation point heat transfer for the Fay-Riddell modelproblem. Furthermore, the correlation coefficient was calculated using linear regression26 and is imposed onthe plots in Figure 4. The correlation coefficient (CC) gives an indication of the linear relationship betweenthe stochastic inputs and the output variable of interest. Using the CC in Figure 4, one can rank therelative importance of the input parameters as velocity, dynamics viscosity (k), Lewis Number, and radiusof curvature (RN ) for this model problem.

2. Global Sensitivity Analysis with Sobol Indices

Sobol indices were also computed to include the non-linear model contributions in the global sensitivityanalysis and to have a more complete description of the ranking of importance of each input uncertainty.The Sobol indices for each uncertain parameter are shown in Table 4. Only the individual Sobol indicesare shown, rather than the total Sobol indices (ST ), because all mixed contributions were less than 0.04%of the entire contribution. For this model problem, the total variance in the heat flux could be explainedalmost entirely due to individual contributions from each input variable including the non-linear effects. Theresults in Table 4 were put in a more convenient form, Figure 5, to show the relative ranking of each inputuncertainty to the overall uncertainty in stagnation point heat transfer. The results from the Sobol indicescalculations are consistent with the relative ranking given by the linear regression analysis. Once again,it can be seen that the freestream velocity has the largest contribution to the overall uncertainty for thisparticular problem with the given input uncertainty ranges.

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Figure 4. Scatter plots demonstrating the influence of each uncertain parameter on the overall uncertainty inthe stagnation heat flux for the model problem.

Table 4. Sobol indices for each uncertain parameter in the Fay-Riddell model problem

Index Uncertain Parameter Sobol Indices

S1 Velocity 0.68335

S2 Dynamic Viscosity 0.20047

S3 Lewis Number 0.09657

S4 RN 0.01929

Figure 5. Relative contributions of each input uncertainty to the overall uncertainty in the stagnation pointheat transfer for the model problem. Contributions were calculated using the Sobol indices.

IV. Uncertainty Quantification in High-Fidelity CFD Simulations

A. Computational Modeling

1. Numerical Scheme and Physical Models

The high-fidelity CFD simulations were performed with the Data-Parallel Line-Relaxation (DPLR)27 code ofNASA Ames Research Center, which is a three-dimensional, structured, finite-volume code capable of solvingthe Navier-Stokes equations for reacting high-temperature flows with non-equilibrium thermo-chemistry. Thecode uses the Steger-Warming28 approach for flux splitting for the convective terms, and central differencingfor the diffusion terms. In accordance with the experimental test case (Table 1) selected from MacLean et. al.2

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MacLean et. al. MacLean et. al.

x

r

Figure 6. Geometry and coordinate nomenclature for the reentry vehicle model used in the experiments byMacLean et. al.2

which is used in the current stochastic CFD study, laminar flow was assumed for modeling the viscous terms.A five species (O2, O, N2, N , NO) model by Park29 was selected to model the high temperature air withfinite rate chemistry. Furthermore, the Landau-Teller30 finite rate vibrational relaxation model was selected.Vibrational energy relaxation rates are modeled using the formulation given by Millikan and White.31 Forhigh temperature non-equilibrium flows, DPLR models the transport quantities using binary collision-integralbased mixing rules from Gupta et al.32 The self-consistent effective binary diffusion method is used to modelthe diffusion coefficients. At the particular velocity range considered in this study, the main mechanismsthat contribute to the total surface heat flux will be heat conduction to the surface via translational andvibrational modes and the diffusion of chemical energy flux to the surface which will depend on the surfacecatalysis. The radiation heat transfer is not modeled in the present work.

2. Geometry and Computational Grid

The computational grid used for the CFD simulations was provided by MacLean et. al.2 The original griddimensions were 257 grid points in the streamwise direction and 229 in the normal direction and the geometryof the vehicle was based upon the experiments performed by MacLean et. al.2 which utilized the capsulegeometry shown in Figure 6. Grid convergence studies were conducted to find the optimum grid mesh sizein terms of minimizing the discretization error and computational expense by dividing the original grid intocoarser grid levels by skipping every other grid point in both the normal and stream-wise directions. Thefinal grid used for all the CFD simulations is shown in Figure 7. Since the CFD runs were conducted for testcases involving the capsule geometry at zero degrees angle of attack, the numerical solutions were obtainedwith an axis-symmetric flow assumption. The left side of Figure 7 is the entire domain of the grid. On theright side of Figure 7 is a zoomed in view of the stagnation line in order to help visualize the grid spacing.A contour plot of the pressure is shown in Figure 8 to indicate a well converged CFD solution had beenobtained.

Figure 7. Computational grid for the 2-D axis-symmetric spherical capsule.

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3. Boundary Conditions

Figure 8. Pressure contour plot indi-cating a converged CFD solution.

A no-slip non-ablating boundary condition was specified at the cap-sule wall, and the wall temperature was held at 300K to enforce acold-wall boundary condition, which is consistent with the experi-ment. Simulations were conducted at zero degrees angle of attackwhich allowed an axis-symmetric flow assumption. The freestreamwas fixed at the values shown in Table 1, and a 1st order extrapola-tion was specified for the outflow.

An important aspect of DPLR for this study is the capability ofmodeling wall recombination efficiencies (γ) for partially catalyticwalls. The DPLR code utilizes the method described by Milos et.al.1 to model the non-ablating finite rate catalytic wall boundarycondition that requires the specification of recombination efficiency(γ) for Nitrogen and Oxygen atoms.33–35 For the current study, thecatalytic wall represents recombination of dissociated oxygen andnitrogen species on the wall with a certain percentage. The limitingcase of a fully-catalytic wall represents complete recombination atthe wall (100% efficiency), and the non-catalytic wall represents zerorecombination (0% efficiency). In terms of the heat transfer to thevehicle, a fully-catalytic wall provides the highest heat transfer dueto the exothermic nature of the recombination process and a non-catalytic wall provides the lowest amount of heat transfer. Thus,fully-catalytic and non-catalytic walls represent the theoretical up-per and lower bounds of the heat flux to the vehicle for a given set of flight conditions. The catalytic wallmodel does not include the surface reactions that include the recombination of the nitric oxide (NO) species.

B. Description of the Stochastic Problem

In general, it is difficult to obtain the exact values of the recombination efficiencies for different wall materials,temperatures, and gas species, therefore γ is considered as one of the uncertainty sources in this study.Recombination efficiency is mainly a physical modeling parameter so it is appropriate to treat it as anepistemic uncertain variable. In this study, the same recombination efficiency for oxygen and nitrogen wereused, therefore γ should be considered as a single epistemic uncertain variable. The binary collision integralshave also been shown to be an important modeling parameter in high temperature flows. A study by Wrightet. al.36 indicates that there can be approximately 25% uncertainty associated with the binary collisionintegral data. There have been other studies by Palmer,37 Bose et. al.,9 and Bose and Wright3 which treatthe binary collision integrals as uncertain. For this study, the uncertainty in binary collision integrals weremodeled as an epistemic uncertainty since it originates due to a lack of knowledge in a physical model. Theuncertainty in the binary collision integrals were implemented through the use of a single parameter, A as:

Ω1,1 = Af1(T ) (16)

Ω2,2 = Af2(T ) (17)

where this parameter (multiplier) was treated as the uncertain variable with lower and upper bounds of 0.75and 1.25, respectively. This corresponds to a ±25% uncertainty in the collision integrals. For this study, thecollision integrals for the N2 − O interaction were taken to be uncertain. In Equation 16, Ω1,1 is used tocalculate the diffusion coefficients. Similarly in Equation 17, Ω2,2 is the collision integral equation used tocalculate the viscosity coefficients.

Heat transfer to the surface of the vehicle is also strongly dependent on the total enthalpy of the flow,hence the free stream velocity. The variation in free stream velocity can be described through probabilisticmeasures due to its inherent nature. For the current study, the freestream velocity input to the CFDsimulation is treated as an aleatory uncertain variable with a normal distribution with a mean of 4167 m/s(freestream condition given in Table 1). Three different coefficient of variances (CoV ) were considered inthis study: 1%, 2%, and 3%. The purpose of using multiple CoV was to study the relative contribution ofeach uncertainty source to the overall uncertainty in the surface heat flux at three uncertainty levels for the

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Table 5. Uncertainty ranges for the parameters used in CFD simulations

Uncertain Parameter Uncertainty Type Uncertainty Range

V∞ Aleatory (normal) μα∗ = 4167 m/s, CoV = 1%, 2%, and 3%

A, (Ω1,1,Ω2,2 = Af1,2(T )) Epistemic [0.75, 1.25]

log10(γ) Epistemic [-5,-0.301]

free stream velocity. Sensitivity analysis will be conducted on all three CoV values. However, the mixeduncertainty quantification results will be demonstrated only for the CoV = 2% case.

Bose et. al.3 showed that the largest variation in heat flux to a Mars entry vehicle due to wall catalyticparameters occurred in moderately catalytic wall regime where γcat in their reactions ranged between 10−3

and 10−1. Preliminary results of the current study have also demonstrated the same type of trend for air.For the velocity range and the wall temperature considered in this study, the change in surface heat transferwas found to be negligible for γ values above 0.5. Based upon these results, the interval bounds for γ wastaken to be 0.00001 and 0.5. The heat transfer to the surface has an exponential growth due to changesin γ in the moderately catalytic wall regime. Therefore, the parameter log10(γ) was taken as the epistemicuncertain parameter rather than purely γ. This approach improved the quality and the convergence ofstochastic response surface obtained with the NIPC approach. An overview of the uncertainties consideredwithin this CFD study are given in Table 5.

A total number of 216 DPLR runs were necessary to obtain a 5th order stochastic response surface heattransfer with the Quadrature-Based NIPC method. Each DPLR CFD simulation was performed on a Linuxcomputing cluster utilizing 12 processors. One simulation required a wall clock time of approximately 20minutes.

C. Uncertainty Quantification in Aerodynamic Heating

1. Results with Purely Aleatoric Uncertainty Assumption

Before the mixed uncertainty analysis, uncertainty quantification was conducted with the assumption ofpurely aleatoric input uncertainty. The results presented in this section are later compared to the mixeduncertainty quantification results to show the difference between two uncertainty quantification approaches.For purely aleatoric uncertainty modeling, besides freestream velocity modeled with a normal distribution(CoV = 2%), log10(γ) and A (multiplier in the collision integral expressions) were assumed to have uniformdistributions with the bounds given in the previous section. The Quadrature-Based NIPC method wasutilized to propagate the input uncertainties through the CFD simulations. It is important to ensure thatthe polynomial order is sufficient to capture the non-linear effects of the uncertainty in the output variable

Figure 9. Convergence of NIPC responsesurface.

Figure 10. PDF curve for 5th order NIPC.

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Figure 11. Mean and 95% C.I. for surface heat flux distribution (purely aleatoric uncertainty assumption).

of interest. Therefore, a convergence study was conducted in which the polynomial order was increasedup to 5 and the stagnation point heat transfer was analyzed at each order. Figure 9 shows the CDF ofstagnation point heat transfer for each polynomial order. There is no noticeable difference in CDF’s beyonda polynomial order of 3. Therefore, it is clear that NIPC response surface was converged at the 5th order.

Figure 10 displays the probability density function (PDF) for the stagnation point heat transfer using 5th

order NIPC. This distribution is fairly non-linear and skewed which demonstrates the non-linear relationshipbetween the uncertain variables and the stagnation point heat transfer, mostly due to γ. Other statisticalinformation can be calculated using NIPC, such as the mean and the standard deviation, since this analysisis made with the assumption of purely aleatory uncertainty. The mean stagnation point heat transfer wasfound to be 147.14 W/cm2 and the standard deviation was calculated to be 19.28 W/cm2 (e.g., a coefficientof variation of 13.1%). The 95% confidence interval (CI) for stagnation heat transfer was calculated as[121.16, 179.99] W/cm2. Figure 11 displays the mean heat transfer along the surface of the vehicle alongwith the 95% confidence intervals (C.I.) at selected points. The fairly large standard deviation and CI valuesare indicative of the large amount of uncertainty in the heat flux to the surface of the vehicle.

2. Results with Mixed (Aleatory-Epistemic) Uncertainty Assumption

For the mixed uncertainty propagation, the same stochastic response surface used for purely aleatory uncer-tainty quantification (5th degree polynomial chaos expansion for heat transfer) was utilized in Second-OrderProbability approach. The outer loop utilized 10,000 values for the epistemic uncertain variable sampledfrom their specified interval (Table 5). In the inner loop, for each value of the epistemic uncertain variable,the stochastic response surface was evaluated with a total number of 5,000 randomly produced samples basedon the standard probability distribution of the aleatoric input uncertainty (standard normal distribution inthis problem due to the normal distribution assumption made for the velocity). This procedure was used toproduce 10,000 CDF’s, which were then evaluated to find the upper and the lower bounds of the interval forheat transfer at each probability level.

The mixed uncertainty results for the heat transfer at the stagnation point and shoulder regions areplotted in Figure 12. Note that at a particular probability level, the variation in the the heat transfer isdue to the epistemic uncertainties, which is represented by the interval bounded by the maximum and theminimum heat transfer values obtained from the CDF samples at the same probability level. The width ofthe interval at each probability level is fairly constant for the stagnation and shoulder regions. However, theinterval bounds are slightly larger for the shoulder region at all probability levels. Quantitative results can beseen in Table 6. The pure aleatory results from the CFD simulations are also listed in Table 6 for comparisonpurposes. As expected, the stagnation heat flux from the purely aleatory uncertainty quantification lieswithin the bounds of the Second-Order Probability results.

The interval bounds for the heat transfer were plotted at selected points across the surface of the reentryvehicle at the 2.5%, 50%, and 97.5% probability levels. The resulting plots are shown in Figure 13. Further-more, the pure aleatory NIPC results are also shown in the figures at the corresponding probability levels.Notice that the pure aleatory result is once again just a single value at each probability value at each pointalong the surface. In contrast, the mixed uncertainty results are given as intervals along the surface of the

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(a) Stagnation region (b) Shoulder region

Figure 12. Second-order probability results for surface heat transfer at two locations.

Table 6. Surface heat transfer (W/cm2) for the stagnation point and the shoulder region at different probabilitylevels for the CFD problem.

Probability Stagnation Point Stagnation Point Shoulder Shoulder

Level (Mixed Uncertainty) (Aleatory Uncertainty) (Mixed Uncertainty) (Aleatory Uncertainty)

P = 0.0 [101.38, 140.81] 103.43 [88.53, 150.18] 99.41

P = 0.2 [124.67, 163.79] 132.73 [114.51, 177.78] 122.79

P = 0.4 [129.65, 169.81] 140.07 [119.32, 184.40] 130.54

P = 0.6 [133.88, 174.72] 148.96 [123.60, 190.24] 142.14

P = 0.8 [139.04, 180.82] 161.55 [128.58, 197.19] 168.21

P = 1.0 [162.97, 210.29] 204.97 [148.86, 248.93] 226.26

reentry vehicle (shaded region in the plots). At probability level 2.5%, the pure aleatory values stay almostin the bottom portion of the mixed uncertainty for all points along the surface of the vehicle. At probabilitylevel 50%, the pure aleatory values skew towards the center of the mixed uncertainty interval. The size ofthe interval is larger for 50% when compared to the 2.5% probability level. At probability level 97.5%, thepure aleatory values lie almost at the upper limits of the aleatory-epistemic interval. There is a significantincrease in the size of the interval near the shoulder region for all three probability levels, which is consistentwith the observation made from Figure 12.

The results of the mixed (aleatory-epistemic) uncertainty quantification can be used in the assessmentof the robustness or the reliability of a given hypersonic vehicle system such as TPS. For example, in arobust design study where aleatory and epistemic uncertainties are present, one possible approach would beto minimize the variation (interval) at the mean probability level (p=50%). By shrinking this interval, thedesign sensitivity due to the epistemic uncertainties would be reduced. One method for reducing the intervalis by gaining a better fundamental understanding of the physics associated with the epistemic uncertainty,and developing more accurate physical models. Alternatively, the designs that are robust to the uncertaintyin physical models can be developed. In a reliability-based assessment, a large interval (high epistemicuncertainty) at a specified probability level may indicate a larger design failure region for a given vehicleconfiguration and flight condition, which has to be addressed again with a stochastic design framework.

D. Sensitivity Analysis

1. Global Sensitivity Analysis with Linear Regression

The global linear SA was also used for the CFD problem to provide the relative importance of each of theuncertain variables on the overall uncertainty in the stagnation point heat transfer. To be consistent withthe uncertainty analysis described above, a 5th order NIPC response surface was used for the Monte Carlo(MC) simulation with 20,000 samples to obtain SA results. As described earlier, three different uncertainty

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(a) Probability level 2.5% (b) Probability level 50% (c) Probability level 97.5%

Figure 13. Comparison of pure aleatory and mixed aleatory-epistemic uncertainty results for surface heattransfer.

ranges for the velocity were considered (CoV = 1%, 2%, and 3%). Scatter plots are shown in Figure 14 alongwith the correlation coefficients (CC) for each of these cases. When there is a low uncertainty associatedwith the freestream velocity (CoV = 1%), the main contribution to output (heat-flux) uncertainty comesfrom log10(γ). However, as the uncertainty in the freestream velocity becomes larger, the contribution fromthe freestream velocity increases and becomes the dominant source at CoV = 3% case. The uncertaintyin stagnation point heat transfer due to log10(γ) is larger in the current study compared to the results ofa similar study by Bettis and Hosder.12 This is mainly due to the higher freestream enthalpy consideredin the current study, which causes more dissociation in the flowfield. More dissociation consequently makesthe effects of log10(γ) more drastic. The collision integrals have the least contribution to the uncertainty instagnation point heat transfer for this particular test case and the range of uncertainties considered.

Figure 14. Correlation plots demonstrating the influence of each uncertain parameter on the overall uncertaintyin the stagnation heat flux (first row CoV = 1%, second row CoV = 2%, third row CoV = 3%).

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2. Global Sensitivity Analysis with Sobol Indices

Sobol indices were computed to account for the non-linear dependencies between the input uncertaintiesand the uncertainty in stagnation point heat transfer. The indices are shown in Table 7. The Sobol indicesare shown for all three cases CoV = 1%, 2%, and 3%. As in the model problem, only the individual Sobolindices (Si) are shown, rather than the total Sobol indices (ST ) because the mixed contributions were lessthan 0.5% of the overall contribution. A more convenient representation of the Sobol indices is given inFigure 15. The Sobol indices provide a relative ranking of the importance of each input uncertainty on theoverall uncertainty in the stagnation point heat transfer. Thus, Figure 15 demonstrates a similar observationmade in the scatter plots of Figure 14. When there is a relatively low amount of uncertainty associated withthe freestream velocity, the uncertainty in log10(γ) will be the main contributor to the uncertainty in thestagnation point heat transfer. However, the freestream velocity will be the main contributor when there ismore uncertainty associated with the freestream velocity (CoV = 3%). Sobol indices also indicate that thecollision integrals do not give a significant contribution for this particular test case with the input uncertaintyranges considered.

These results demonstrate that it is important to have accurate characterization of the freestream velocityin numerical simulations and experiments for hypersonic reentry flows, since the variations in this quantitycan have a drastic impact on the predicted heating to the surface of the vehicle and can make the detectionand analysis of other uncertainty sources difficult. The uncertainty in the recombination efficiency log10(γ)can also have significant impact on the heating to the surface of the vehicle. Therefore, it will be important togain a better physical understanding of the recombination efficiency in order to reduce this epistemic (model-form) uncertainty. Although the uncertainty in collision integrals did not have a significant contribution tothe overall uncertainty in stagnation point heating for this particular study due to much larger uncertaintiesassociated with the freestream velocity and recombination efficiency, they may become important in flowswith higher temperatures (i.e., higher total enthalpy cases).

Table 7. Sobol indices for each uncertain parameter in the CFD simulations.

Velocity CoV Index Uncertain Parameter Sobol Indices

S1 Velocity 0.10537

1% S2 log10(γ) 0.88002

S3 A 0.01077

S1 Velocity 0.33095

2% S2 log10(γ) 0.65379

S3 A 0.01277

S1 Velocity 0.51232

3% S2 log10(γ) 0.47707

S3 A 0.00641

(a) CoV = 1% (b) CoV = 2% (c) CoV = 3%

Figure 15. Relative contributions of each input uncertainty to the overall uncertainty in the stagnation pointheat transfer. Contributions were calculated using the Sobol indices.

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V. Conclusions

The objective of this study was to demonstrate an efficient methodology for the quantification of mixed(aleatory and epistemic) uncertainties in hypersonic flow computations. In particular, the approach was usedto quantify the uncertainty in surface heat flux to the spherical non-ablating heat-shield of a reentry vehicleat zero-angle of attack due to epistemic and aleatory uncertainties that may exist in various parametersused in the numerical solution of hypersonic, viscous, laminar blunt-body flows with thermo-chemical non-equilibrium. In this study, the freestream velocity (V∞), binary collision integrals for the N2−O interaction,and the recombination efficiency (γ) of oxygen and nitrogen atoms used in the description of catalytic wallboundary condition were treated as uncertain variables. The freestream velocity was modeled as an inherentuncertain variable described with a normal probability distribution, whereas the recombination efficiencyand collision integrals were modeled as epistemic uncertain variables represented with an interval. For thequantification of mixed (aleatory-epistemic) uncertainty, Second-Order Probability Theory that utilized astochastic response surface obtained with quadrature-based Non-Intrusive Polynomial Chaos (NIPC) Methodwas used. In addition to linear sensitivity analysis, a non-linear global sensitivity analysis (SA) with SobolIndices was also performed to rank the contribution of each uncertainty source to the uncertainty in thesurface heat flux.

Before the implementation of the uncertainty quantification method to the stochastic high-fidelity CFDproblem, the approach was applied to a stochastic model problem for the prediction of stagnation point heattransfer with Fay-Riddell relation, which considered velocity and radius of curvature as inherent uncertainvariables and the boundary layer edge dynamic viscosity and Lewis Number as epistemic uncertain variables.For the model problem, Second-Order Probability was implemented with two different approaches for thepropagation of mixed uncertainty: (1) direct Monte Carlo sampling and (2) a 3rd order stochastic responsesurface obtained with the Quadrature-Based NIPC. The uncertainty results for the stagnation point heattransfer obtained with two approaches matched well indicating the computational efficiency and the accuracyof the NIPC approach for mixed uncertainty propagation.

The uncertainty quantification in CFD simulations of the current study was performed for a particulartest case and capsule geometry selected from the work of MacLean et. al.,2 where the freestream velocityfor the experiment was 4167 m/s at a stagnation enthalpy of 9.9 MJ/kg. For the stochastic CFD problem,the mixed uncertainty quantification approach was utilized with a 5th degree stochastic response surfaceobtained with the Quadrature-Based NIPC, which required 216 deterministic simulations. The uncertaintyin surface heat transfer was obtained in terms of intervals at different probability levels at various locationsincluding the stagnation point and the shoulder region. The mixed uncertainty results were compared to theresults obtained with a purely aleatory uncertainty analysis to show the difference between two uncertaintyquantification approaches. A linear and non-linear global sensitivity analysis indicated that the velocityand recombination efficiency had the highest contributions to the overall uncertainty in the stagnation pointheat transfer for the hypersonic reentry case considered in this study. When the freestream velocity had arelatively low amount of uncertainty (CoV = 1%), the recombination efficiency was the dominant contributorto the overall uncertainty. When the CoV was increased to 3%, the velocity became the most significantcontributor. These results demonstrated the importance of accurate characterization of the freestreamvelocity in numerical simulations and experiments for hypersonic reentry flows, since the variations in thisquantity can have a drastic impact on the predicted heating to the surface of the vehicle and can make thedetection and analysis of other uncertainty sources difficult.

Overall, the results obtained in this study show the potential of the uncertainty quantification approachthat utilizes Second-Order Probability and the Non-Intrusive Polynomial Chaos for efficient and effectivepropagation of mixed (aleatory and epistemic) uncertainties in high-fidelity hypersonic flow simulationsincluding re-entry problems and the prediction of uncertainty in aerodynamic heating, which can be usedfor the design of reliable and optimized thermal protection systems.

VI. Acknowledgments

The authors would like to thank Dr. Matthew MacLean (CUBRC, Buffalo, New York) for helpful discus-sions and providing the baseline grid used in the CFD simulations. This work was performed under a NASASTTR Grant NNX10CF64P in collaboration with M4 Engineering, Inc. The first author would like to alsoacknowledge NASA Aeronautics Scholarship Program for the partial support of this work.

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