Uncertainty Quanti�cation of Structural Uncertainties

in RANS Simulations of Complex Flows

Eric Dow�and Qiqi Wangy

Massachusetts Institute of Technology, Cambridge, MA, 02139

A method is presented for building a statistical model for the structural uncertaintiesin RANS turbulence models. This approach compares the results of RANS calculationsto direct numerical simulation for a collection of simple, randomly generated geometries.The adjoint method is used to e�ciently solve an inverse problem for each geometry todetermine the RANS turbulent viscosity �eld that most accurately reproduces the mean ow �eld computed by direct numerical simulation. The discrepancy between the inferredturbulent viscosity and the turbulent viscosity predicted by RANS is modeled as a Gaussianrandom �eld that is scaled according to the local ow �eld properties. The uncertainty inthe turbulent viscosity �eld is then propagated to the engineering quantities of interest.

I. Introduction

Computational methods based on solving the Reynolds-Averaged Navier-Stokes (RANS) equations arecurrently the most popular choice for simulating ow problems that involve turbulence. Solving the RANSequations determines the statistically-averaged ow �eld without regard to the �ne scale turbulent structures,allowing a relatively coarse mesh to be used, thus reducing the computational e�ort as compared to methodsthat attempt to resolve the �ne scale turbulent motions. This reduction in computational cost makes RANSideal for use in the engineering design process, where the ow around numerous design iterations must besimulated during an optimization procedure.

In RANS simulations, the e�ect of the unresolved components of turbulence on the statistically averaged,or mean, ow �eld is represented by the Reynolds stress tensor. One speci�c class of turbulence models,the Boussinesq turbulent viscosity models, approximate the e�ect of the Reynolds stresses by prescribinga turbulent viscosity acting on the mean ow �eld.1 Within the class of Boussinesq models, a varietyof methods have been proposed for estimating the turbulent viscosity �eld. For simple ows, these modelstypically produce good estimates of the e�ect of turbulence. However, for more complex ows, the mean ow�elds computed using turbulent viscosity models show signi�cant discrepancies with experimental results.Turbulent viscosity models are especially inaccurate for ows that experience or are close to separation, orwhere the streamline curvature is large.2�4 We show that a large part of these discrepancies can be viewedas a result of uncertainty in the estimation of the turbulent viscosity �eld, and re ect the inability of theReynolds stress tensor to be approximated accurately by solving for a small number of transport scalars, e.g.the turbulence kinetic energy and speci�c dissipation rate in the k � ! model. The uncertainties resultingfrom these inexact estimates are referred to as model uncertainties or structural uncertainties. Since thisform of uncertainty can theoretically be reduced (for example, by devising better models for estimating theturbulent viscosity), structural uncertainties represent an epistemic uncertainty. A number of approacheshave emerged to estimate the nature of structural uncertainties in RANS simulations.5�8 However, theseworks focus on simple ows, which do not exhibit the complex ow characteristics commonly observed inengineering systems.

We propose an approach for quantifying structural uncertainty in RANS simulations of complex ows.This approach consists of two steps: an inverse modeling step and a statistical modeling step. The inversemodeling step generates data which is in turn used to construct a statistical model of structural uncertainties.

�PhD student, Department of Aeronautics and Astronautics, Room 37-442, MIT, Cambridge, MA 02139, AIAA StudentMemberyProfessor, Department of Aeronautics and Astronautics, Room 37-408, MIT, Cambridge, MA 02139, AIAA Member

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20th AIAA Computational Fluid Dynamics Conference27 - 30 June 2011, Honolulu, Hawaii

AIAA 2011-3865

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

In a previous work,9 we presented our approach for quantifying structural uncertainty for ow through astraight channel. The aim of this work is to generalize the approach presented there to more complicated ows. The data required to build our statistical model is generated by considering ow through a collection ofchannels with randomly shaped walls. These geometries contain geometric features that are likely to appearin complex engineering geometries. For each geometry, a direct numerical simulation (DNS) is performed,and we determine the \true" RANS turbulent viscosity that most accurately reproduces the DNS ow �eld.To invert the turbulent viscosity �eld, we �rst reformulate the inverse problem as a constrained optimizationproblem, and solve the resulting optimization problem using gradient based optimization techniques. Theadjoint method is used to compute the sensitivity gradient. The true turbulent viscosity �elds are storedtogether with the mean ow �eld and turbulent properties. The inverse modeling step reduces the problem ofquantifying the sources of uncertainty to a statistical data analysis problem. In this statistical modeling step,we analyze the data generated in the inverse modeling step to construct a statistical model of the uncertaintyin the calculated RANS turbulent viscosity. The key assumption made in formulating our approach is thatthe uncertainty in RANS computations can be largely attributed to the inability of current RANS models toestimate the true turbulent viscosity. This assumption is validated by considering the results of the inversemodeling step, and motivates our approach of characterizing the discrepancy between the computed RANSturbulent viscosity and the true turbulent viscosity �elds.

The rest of this paper is organized as follows: section II describes our approach for inferring the structuraluncertainties. Section III describes our method of generating random geometries and the various ow andadjoint solvers used in this work. Sections IV presents the results of the inverse modeling of the RANS tur-bulent viscosity �eld. Section V presents the results of the statistical modeling and uncertainty propagationsteps. Section VI presents our conclusions and suggestions for future work.

II. Estimating the structural uncertainty in RANS models

A. Inverse problem formulation

For the class of inverse problems that we are interested in solving, the goal is to determine the set of modelparameters m that yields the closest agreement between the output of our system and the observables d.Abstractly, since the system of interest is governed by a collection of PDEs, i.e. the RANS mean owequations, the objective of our inverse problem is to determine the set of parameters such that

d = G(m);

where G(m) represents the evaluation of our PDEs with the model parameters m. In this work, the modelparameters are the values of the turbulent viscosity everywhere in the domain. The model output we havechosen to consider is the mean velocity �eld. To e�ciently solve this inverse problem, we �rst cast theinverse problem as an optimization problem. In this optimization problem, the objective function is chosento measure the di�erence between the observables and the output of our model evaluated for some choice ofmodel parameters ~m, i.e.

J( ~m) = jjd�G( ~m)jj22:

When the norm of the di�erence between the model output and the observables is zero, we conclude thatthe two must agree, and that the inverse problem has been solved. The solution to the inverse problem isthen de�ned as

m = argmin~m

J( ~m): (1)

A simple procedure to compute the optimal solution to (1) is to �rst compute a descent direction @J=@m,which represents the sensitivity gradient of the objective function with respect to the model parameters.Then, by choosing an appropriately small step size �, a simple approach for updating the estimate of themodel parameters is obtained by setting

mk+1 = mk � � @J@m

:

To �rst order

J + �J = J +@J

@m

T

�m = J� � @J@m

T @J

@m;

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and thus there exists some � such that the objective function value is reduced. The key di�culty in thisapproach is evaluating the sensitivity gradient when the number of model parameters is large. For complex ows, the turbulent viscosity is typically a complex spatially varying �eld, and the discretized representationof this �eld is high-dimensional. Thus, computing the sensitivity gradient using �nite di�erences, where a new ow solution must be computed to estimate the sensitivity of the model output to each model parameter, isprohibitively expensive. Instead, we address the problem of high-dimensionality by using the adjoint method.

B. Computing the sensitivity gradient

The adjoint method provides an e�cient approach for computing the sensitivity gradient that does not relyon repeatedly evaluating the objective function.10 We �rst present an abstract description of how the adjointmethod can be used to compute the sensitivity gradient. The objective function that is to be minimized isa function of both the model parameters and the solution to the governing PDEs evaluated with the modelparameters, referred to as u(m). In this case,

J = J(u(m);m);

and a change in the model parameters results in a change in the objective function value

�J =

�@J

@u

�T�u+

�@J

@m

�T�m;

where �J, �u, and �m are in�nitesimal. We assume that the governing equation for the PDE that controlsour system can be written as

R(u;m) = 0:

Linearizing the governing equation, the variation �R can be written as

�R =

�@R

@u

��u+

�@R

@m

��m = 0: (2)

This variation is zero, so we can multiply the linearized governing equation by a costate and introduce thelinearized equation as a \constraint" in the minimization problem. Equation (2) can thus be replaced by

�J =@J

@u

T

�u+@J

@m

T

�m� T��

@R

@u

��u+

�@R

@m

��m

�=

@J

@u

T

� T�@R

@u

�!�u+

@J

@m

T

� T�@R

@m

�!�m

To eliminate the direct dependence of the objective function on the solution u, we choose the costate tosatisfy the adjoint equation: �

@R

@u

�T =

@J

@u: (3)

If satis�es (3), which is a linear PDE, the variation in the objective function becomes

�J = G �m;

where the sensitivity gradient G is de�ned as

G =@J

@m

T

� T�@R

@m

�:

The sensitivity gradient can be computed by solving the governing PDE once, followed by one additionalsolve of the adjoint system. The computational cost of solving (3) is roughly the same as the cost of solvingthe original PDE. Thus, the sensitivity gradient with respect to all of the model parameters can be computedat roughly twice the cost of solving the governing PDE.

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1. Adjoint system for the mean ow equations

To cast the RANS inverse problem as an optimization problem, we must form an objective function thatmeasures the di�erence between the RANS ow velocity u(�T ) computed with a speci�ed turbulent viscosityand the DNS ow velocity uDNS . The objective function is chosen as

J(u(�T )) = jju(�T )� uDNS jj2L2 (4)

For physical reasons, the turbulent viscosity is required to be non-negative, so our minimization statementis given as

min jju(�T )� uDNS jj2L2

s.t. �T � 0(5)

Although we ultimately seek to compute the adjoint sensitivity gradient for steady ow, we derive theadjoint system for the unsteady problem. This is motivated by the fact that the solver used in this workcomputes the adjoint solution to the steady problem by computing the steady state solution of the unsteadyadjoint equations. The mean ow equations take the form

@u

@t+ u � ru+rp�r � (�e�ru)� f = 0;

r � u = 0

in the spatial domain in a time interval [0; T ]. The e�ective viscosity �e� is the sum of the laminar andturbulent viscosities. In this work, we only consider ows with solid wall boundary conditions on the entireboundary @, so that the velocity satis�es the no slip condition on @. The turbulent viscosity must alsobe zero at the solid boundaries. Linearizing the mean ow equations about the ow solution u(�T ), we have

@�u

@t+ L(u;�T ) +r�p = 0

r � �u = 0

where the linearized operator L(u;�T ) is de�ned as

L(u;�T ) = u � r�u+ �u � ru�r � ((� + ��T )ru)�r � (�e�r�u)

The linearized objective function is then given by

�J =

Z T

0

ZZZ

2(u� uDNS) � �u dx dt:

Note that we integrate in time since we are deriving the unsteady adjoint system. Introducing the adjointvariables u and p, and combining the linearized objective function and mean ow equations:

�J =

Z T

0

ZZZ

2(u� uDNS) � �u dx dt

+

Z T

0

ZZZ

@�u

@t� u+ L(u;�T ) � u+ p(r � �u) dx dt

Integrating by parts in space and time, we arrive at the following relation:

�J =

Z T

0

ZZZ

2(u� uDNS) � �u dx dt

=

Z T

0

ZZZ

�u ���@u@t� u � ru+ru � u�r � (�e�ru) +rp

�� �pr � u dx dt

+

ZZZ

u(T ) � �u(T )� u(0) � �u(0) dx

+

Z T

0

ZZ@

�p u � ~n� �e�((r�u) � u) � ~n ds dt

+

Z T

0

ZZZ

��Tru : ru dx dt

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To determine the adjoint sensitivity gradient with respect to the turbulent viscosity, we force all termsinvolving �u and �p to vanish by choosing the adjoint variables to satisfy the continuous adjoint equations:

�@u@t� u � ru+ru � u�r � (�e�ru) +rp = 2(u� uDNS) (6)

r � u = 0:

In practice, a terminal condition must also enforced on the adjoint velocity. Since we seek to compute thesteady state solution of the adjoint equation, the choice of terminal condition is unimportant. For simplicity,we choose to enforce u(T ) = 0. Once the adjoint system has been solved, the sensitivity gradient is computedas

@J

@�T= ru : ru: (7)

Since we specify a terminal condition, equation (6) must be solved backward in time. In practice, we cancompute the adjoint solution forward in time by substituting � = T � t into equation (6).

2. Regularization of the inverse problem

For the inverse problem of interest, the sensitivity gradient computed by equation (7) can lead to an ill-posedproblem. If the velocity gradient tensor is identically zero somewhere in the ow, the objective function valueis insensitive to changing the turbulent viscosity at this location. This implies that the inverse problem isill-posed, as the model parameters at these locations can be changed arbitrarily without changing the modeloutput. Since we seek to uniquely invert the turbulent viscosity �eld, we introduce additional informationto regularize the solution. Speci�cally, we penalize the total variation of the turbulent viscosity �eld byintroducing an additional term to the objective function. The new objective function is

J(u(�T ); �T ) = jju(�T )� uDNS jj2L2 +d

"jjr�T jj2L2

where d is the distance to the nearest wall and the regularization parameter " is chosen to be large relative tothe channel width. Since the regularization term in the objective function does not involve the ow solutionu(�T ), it does not need to be included in the derivation of the adjoint equations. Instead, the sensitivitygradient of this term with respect to the turbulent viscosity can simply be added to the adjoint sensitivitygradient computed using equation (7). The contribution to the sensitivity gradient due to the regularizationterm is computed independently of the adjoint sensitivity gradient, and the two are added together whenperforming the optimization.

3. Optimization strategy

To solve the optimization problem e�ciently, the constrained optimization problem is transformed into anunconstrained problem by optimizing the log of the turbulent viscosity �eld. The transformed sensitivitygradient is computed from the original sensitivity gradient using the relation

@J

@ log(�T )= �T

@J

@�T:

Optimizing log(�T ) ensures that the turbulent viscosity will remain nonnegative assuming the initial RANSturbulent viscosity model produces a nonnegative turbulent viscosity.

Since the adjoint method provides only gradient information at a particular turbulent viscosity �eld,we have decided to use a quasi-Newton method to perform the optimization. Quasi-Newton methods con-struct an approximation to the Hessian matrix using only the sensitivity gradient. Using the additionalinformation provided by the Hessian matrix greatly accelerates convergence, especially once the gradient hasbeen su�ciently reduced. To reduce the memory requirements, we have used the low-memory extension ofthe Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm. This method computes an approximation tothe Hessian matrix using the gradient and position information at a small number of previous iterations,continuously replacing the information obtained at the oldest iteration with information from the currentiteration.11

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C. Statistical modeling and uncertainty propagation

The uncertainty in the RANS turbulent viscosity is modeled by comparing the RANS and optimized turbulentviscosity �elds computed by solving the inverse problem. We choose to model the discrepancy in the turbulentviscosity as a random �eld. The smoothness of this random �eld can be controlled by the functional formof the covariance function and correlation length, and the magnitude of the uncertainty can be controlledby the variance. The exact nature of this random �eld is determined by examining the data obtained in theinverse modeling step, so further discussion is presented in section V.

Ultimately, we would like to estimate the uncertainty in quantities related to the velocity �eld, since theseare the quantities of interest in engineering applications. This requires the uncertainty in the RANS turbulentviscosity to be propagated to the ow �eld. We employ the Monte Carlo method to estimate the mean andvariance of the RANS velocity �eld. Each turbulent viscosity �eld sample �sT is generated by multiplyingthe turbulent viscosity �eld computed using the k� ! model by the exponential of the log-discrepancy �eldds�T :

�sT = exp(ds�T ):

The corresponding ow solution sample is computed by prescribing the the sample turbulent viscosity �eld�sT and solving the mean ow equations. The collection of Monte Carlo results allows us to estimate theuncertainty in the RANS velocity �eld, as well as any quantities of interest that can be computed from the ow solution.

III. Numerical procedure

A. Random geometry generator

The geometries used to construct the database of ows are chosen to satisfy two important conditions:

1. They should be su�ciently simple. The direct numerical simulation requires a very �ne mesh toresolve the relevant scales of turbulent motion, and using simple geometries reduces the time requiredto construct the resulting meshes.

2. They should produce ow phenomena observed in complex engineering applications. Since the owsstored in this database are used to construct a statistical model for structural uncertainties arisingin complex ows, they should exhibit similar ow characteristics, including regions of separation,recirculation, and reattachment.

To satisfy these requirements, a random channel geometry generator has been developed. The channel wallsare generated by simulating a Gaussian process with the correlation function

C(d) = exp

��d2

c20 + c1jdj

�where d is the distance between two points on the boundary and the parameters c0 and c1 control thecorrelation length of the simulated �eld. This correlation function was chosen as it produces smoothlyvarying wall geometries. The Gaussian process is conditioned to have zero slope at the inlet and outletsections of the channel, and is simulated using the matrix factorization method.12

Unstructured meshes are used to compute the RANS and DNS solutions. Near the solid boundaries, themesh is re�ned to resolve the boundary layer. The interior of the domain is discretized with triangles. Twoexample meshes used for computing the DNS solution are shown in �gure 1. The solid boundaries are theupper and lower curved surfaces. The ow is computed on a periodic domain by connecting the left andright vertical faces of the channel geometry. The upper surface and lower surfaces are symmetric about thecenterline of the channel, as shown in the �gure. Since turbulence is inherently three-dimensional in nature,the meshes used to perform the direct numerical simulations must be three-dimensional. The two-dimensionalmeshes are translated 100 times in the z-direction with �z = 0:033 to create a three-dimensional mesh. Thetwo-dimensional meshes contained approximately 20,000 nodes, and the corresponding three-dimensionalmeshes contained roughly two million nodes. The mesh is made periodic in the z-direction by connectingthe front and back faces.

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−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

−0.5 0 0.5 1 1.5 2 2.5 3 3.5

0

0.5

1

1.5

2

2.5

3

Figure 1. Sample DNS meshes

B. Flow solvers

To compute the RANS mean ow �eld and turbulent viscosity �eld, the \Joe" ow solver from Stanford’sCenter for Turbulence Research was used. This code solves the compressible RANS equations on unstructuredmeshes using a second order accurate �nite volume scheme, and includes a number of RANS turbulencemodels. All RANS solutions were computed using the Wilcox k � ! two-equation model, one of the mostpopular RANS turbulence models used in industry.13 A unit body force in the positive x direction is appliedto drive the ow, and the laminar viscosity was set to �‘ = 2:0� 10�3. It is important to note that althoughwe have chosen to only consider the k�! model, the approach described in this work is entirely generalizableto any eddy viscosity model, both linear and nonlinear. Our approach only requires the turbulent viscosity�eld computed by the turbulence model. How the turbulent viscosity �eld is computed is irrelevant.

The CDP code, also developed at the CTR, was used to perform the direct numerical simulations. Thiscode uses a second order accurate node based �nite volume method, and handles unstructured meshes. The ow solution is advanced in time using the Crank-Nicholson scheme, and the code is fully parallel. Unlikethe Joe code, CDP is an incompressible code. We therefore consider ows with very low Mach number (onthe order of M = 0:1) to ensure that the e�ects of compressibility are minimal, allowing the comparisonof the velocity �elds computed by Joe and CDP. To compare the results of DNS to the mean velocity �eldcomputed using RANS, the unsteady DNS velocity �eld must be averaged. The averaging procedure isperformed both in space and in time. As described above, fully three-dimensional DNS meshes are createdby extruding the two-dimensional mesh in the z-direction. A spatial averaging is performed by averagingthe ow �eld over all of the two-dimensional \slices" created by translating the mesh. Since the geometriesare symmetric about the centerline, we also average the values above and below the centerline. The spatiallyaveraged solution is averaged in time over 50,000 iterations with a �xed timestep of �t = 7:5 � 10�4 toproduce the mean velocity �eld uDNS. The same laminar viscosity and forcing used to compute the RANSsolution was used for the DNS simulations.

The adjoint solution is computed using a code derived from CDP. This code uses the same spatialdiscretization and timestepping methods as CDP to solve the adjoint system given by (6). The adjointsensitivity gradient was compared against a �nite di�erence estimate, and the two results showed very goodagreement. Once the adjoint system has been solved, the sensitivity gradient is computed and the turbulentviscosity �eld is updated using the L-BFGS update procedure. The NLopt library, which includes an e�cientimplementation of the L-BFGS algorithm, was used to perform the optimization in this work.14 Since onlythe mean velocity and pressure need to be recomputed and the turbulent viscosity is prescribed, there is noneed to solve the scalar transport equations. Therefore, once the turbulent viscosity �eld is updated, thevelocity �eld is recomputed using CDP. The turbulent viscosity �eld is updated until the mean velocity �eldcomputed using the optimized turbulent viscosity agrees su�ciently well with the DNS mean velocity �eld.This optimized viscosity �eld represents the true turbulent viscosity �eld for the corresponding ow, and isused in constructing the statistical model.

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IV. Inverse modeling results

A. Comparison of RANS and DNS results

For each of the geometries considered, the RANS equations were solved, and the turbulent viscosity pro�lecomputed using the k � ! model was stored for use in the optimization step. The mean DNS ow �eldwas also computed and stored. In all simulations, a unit body force is applied in the positive x directionto drive the ow. Figure 2 shows a comparison between the mean velocity �elds computed using directnumerical simulation and by solving the RANS equations for one of the geometries considered. The velocity�elds computed using the two methods exhibit the same basic ow features, including the complex region ofrecirculation that forms as a result of the \bump" in the center of the geometry. However, while the y-velocity�elds computed using the two methods agree fairly well, the x-velocity component shows a large discrepancy,especially near the center of the channel. The largest discrepancy in the x-velocity is approximately 30% ofthe RANS velocity. Since the RANS velocity is larger than the mean DNS velocity in most of the channel,it is apparent that the k � ! model is underestimating the level of turbulent dissipation in this ow.

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Figure 2. Comparison between the mean DNS (left) and RANS (right) x-velocity �elds (upper) and y-velocity�elds (lower).

B. Results for the RANS inverse problem

The initial turbulent viscosity �eld computed by solving the RANS equations was optimized to determinethe turbulent viscosity �eld �?T that produces a mean velocity �eld that agrees more closely with the meanDNS velocity �eld. Figure 3 depicts the mean velocity �eld produced by prescribing the optimized turbulent

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viscosity �eld for the same geometry depicted in �gure 2. Comparing �gures 2 and 3, it is clear that theoptimized mean velocity �eld computed using �?T shows much better agreement with the mean DNS velocity�eld than the original velocity �eld computed using the k�! model. We note that the regions where the owis reversed (blue regions in the x-velocity contours) still show some disagreement after the turbulent viscosityhas been optimized. Typically, the optimization of the turbulent viscosity required roughly 30 iterations toachieve a high level of agreement between the RANS and DNS ow �elds, which involves tuning thousands ofnodal values for the turbulent viscosity. This demonstrates the e�ciency of the adjoint approach for solvinglarge-scale inverse problems.

We can quantify the level of agreement for the two velocity �elds by comparing the value of the objectivefunction J, which measures the di�erence between the RANS mean ow �eld and the DNS mean ow �eld.For the geometry shown in �gures 2 and 3, the initial objective function value was J(�k�!T ) = 7:35. For theoptimized turbulent viscosity �eld, the objective function value was J(�?T ) = 0:0840. This level of reductionwas typical of the geometries considered, as indicated by table 1. The last column of table 1 indicates the

percentage change in the norm of the velocity discrepancy, i.e. 1 �q

J(�?T )=J(�k�!T ), which represents the

percentage of the velocity discrepancy that can be attributed to uncertainty in the turbulent viscosity �eld.There are a number of possible sources for disagreement between the RANS and mean DNS ow solutions.

These sources include the statistical noise introduced by the averaging of the DNS solution; the e�ect ofcompressibility not captured by the incompressible DNS simulation; the di�erences between the numericalschemes used to compute the RANS and DNS solutions; the assumption of mean rate-of-strain/Reynoldsstress anisotropy alignment made by the Boussinesq hypothesis; and the uncertainty in the turbulent viscosity�eld. The substantial reductions in the velocity discrepancy presented in table 1, which were obtained byonly varying the turbulent viscosity �eld, suggest that the uncertainty in the ow solution can be largelyattributed to the inability of the k � ! model to estimate the true turbulent viscosity. This supports theassumption made earlier that the discrepancy between the RANS and DNS results is primarily due to theuncertainty in the turbulent viscosity. These results also quantify the level of uncertainty introduced bythe sources of uncertainty not related to uncertainty in the turbulent viscosity. Since the RANS velocity�eld cannot be made to match the DNS mean velocity �eld exactly by changing the turbulent viscosity, theother sources of uncertainty are not negligible. This level of uncertainty can be quanti�ed by consideringthe discrepancy in the RANS and DNS velocity �elds after the turbulent viscosity has been optimized.

J(�k�!T ) J(�?T ) % discrepancy due to �T

Geometry 1 23.5 0.148 92.1%

Geometry 2 0.515 0.0449 70.5%

Geometry 3 16.2 0.254 87.5%

Geometry 4 6.19 0.0.117 86.3%

Geometry 5 7.15 0.0646 90.5%

Geometry 6 0.165 0.0127 72.3%

Geometry 7 15.9 0.308 86.1%

Geometry 8 7.35 0.840 89.3%

Table 1. Comparison between the velocity discrepancies for the velocities computed using the k�! model andthe optimized turbulent viscosity.

For reference, we also plot the turbulent viscosity �eld computed using the k�! model and the optimizedturbulent viscosity pro�le in �gure 4. To highlight the di�erences between the two �elds, the log-discrepancybetween the two �elds, de�ned as log(�?T =�

k�!T ), is plotted in �gure 5. The log-discrepancy �eld depicted in

�gure 5 is typical of the geometries considered. We note that the largest changes in the turbulent viscosity�eld, corresponding to the areas where the log-discrepancy magnitude is largest, are made around the \bump"in the geometry, where the ow separates from the wall. It is clear that the presence of separation in the owintroduces a great deal of uncertainty in the estimate of the turbulent viscosity �eld. We also note that this�eld is highly anisotropic and non-stationary. Near the wall, the correlation length between the values oflog-discrepancy in the streamwise direction is much larger than in the direction normal to the solid boundary.This non-stationarity is consistent with the results present in our previous study of ow through a straight

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U-Y

1.41.210.80.60.40.20

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Figure 3. Comparison between the mean DNS (left) and optimized (right) x-velocity �elds (upper) and y-velocity �elds (lower).

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channel. For the straight channel, the log-discrepancy �eld was highly non-stationary. Values near the wallwere more highly correlated than values far from the wall, as observed here. For the geometries consideredin this work, the magnitude of the variations is also much larger near the wall than it is far away from thewall. Conversely, near the centerline of the channel, the shear strain-rate is very small relative to the shearstrain rate near the solid boundaries, and the corresponding log-discrepancy magnitude is small. It is clearthat the ow is most sensitive to changes in the turbulent viscosity in regions where the shear strain rate islargest. It is these characteristics that we seek to capture with the statistical model of the log-discrepancy.

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0.190.180.170.160.150.140.130.120.110.10.090.080.070.060.050.040.030.020.01

Figure 4. Comparison between k � ! (left) and optimized (right) turbulent viscosity �elds.

Figure 5. log-discrepancy between the optimized and k � ! turbulent viscosities.

V. Statistical modeling and uncertainty propagation results

A. Statistical modeling

The apparent correlation between the magnitude of the log-discrepancy in the turbulent viscosity and theshear-strain rate was captured in our statistical model by performing a linear regression on the data obtainedfor the random geometries. Speci�cally, we considered the relation between the magnitude of the log-discrepancy and the corrected velocity strain-rate norm, de�ned as jjSjj2(1 � exp(�d=d0)), where S is thevelocity strain-rate tensor, d is the distance from the solid wall, and d0 is the distance from the wall ofthe point where jjSjj2 is largest. The norm of the velocity strain rate measures the shear stress at a givenlocation. A plot of the log-discrepancy versus the corrected velocity strain-rate norm is shown in �gure 6.

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We note that the magnitude of the log-discrepancy does show a positive correlation with the corrected shearstrain-rate norm. The correction factor (1 � exp(�d=d0)) weights the points furthest from the wall, wherejjSjj2 is smallest, more heavily than points near the wall, where d is small. For points very near the wall, themagnitude log-discrepancy is very small, since the log-discrepancy must go to zero at the wall. The correctionfactor thus limits the impact of points with large jjSjj2 and small discrepancy on the linear regression. Sincethe log-discrepancy must go to zero at the wall, we perform a linear regression that is constrained to passthrough the origin. The resulting regression is plotted in red in �gure 6.

0 10 20 30 40 50 60 70 80

||S||(1−exp(−d/d0 ))0

1

2

3

4

5

|log(ν T

/νk−ω

T)|

Figure 6. log-discrepancy plotted against the corrected velocity strain-rate norm.

The results presented in the previous section indicate that the log-discrepancy �eld is highly anisotropicand non-stationary, and it is insu�cient to simply model this random �eld as stationary and isotropic.Instead, the log-discrepancy is then model by scaling an isotropic, stationary Gaussian random �eld basedon the local corrected velocity strain-rate norm. To generate sample log-discrepancy �elds, a zero meanGaussian random �eld with covariance function

C(x1;x2) = exp

��jjx1 � x2jj22

2�2

�was simulated on a uniform grid with a correlation length of � = 0:2. The �eld was simulated using theKarhunen-Lo�eve expansion of the covariance matrix.15 Each random �eld realization was then interpolatedto the mesh points of the channel grid. The value of the random �eld was scaled according to the correctedRANS velocity strain-rate norm using the linear regression estimate depicted in 6. Since the geometriesconsidered were symmetric, we expect the realizations of turbulent viscosity �eld to be symmetric about thecenter of the channel. This symmetry was explicitly enforced for all random turbulent viscosity realizationsby setting the values of the log-discrepancy to be equal above and below the center of the channel. Acollection of random turbulent viscosity log-discrepancies is shown in �gure 7. For the samples shown, thelocations where the magnitude of the log-discrepancy is largest correspond to locations of large RANS velocitystrain-rate, i.e. around the \bump" in the geometry at x = 1:5, and the log-discrepancy goes to zero alongthe centerline of the channel at y = 1:5. Also, the correlation length in the streamwise direction is muchsmaller than in the wall normal direction. All of these features match the observations of the log-discrepancyrealizations made when comparing the k � ! and optimized turbulent viscosity �elds.

B. Uncertainty propagation

To propagate the uncertainty in the turbulent viscosity to the RANS velocity �eld, 500 Monte Carlo simula-tions were performed. For each Monte Carlo sample, a random log-discrepancy �eld was simulated using the

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Figure 7. Sample realizations of the turbulent viscosity log-discrepancy �eld.

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method described above, and the corresponding sample turbulent viscosity �eld was computed by scalingthe k� ! turbulent viscosity �eld. The RANS ow �eld was computed using the sample turbulent viscosityand stored. The mean of the Monte Carlo sample velocity �elds was found to match the k�! velocity veryclosely. The standard deviation of the Monte Carlo sample ow �elds are plotted in �gure 8. The region oflargest variation is observed just behind the \bump" in the mesh. The large variability in the ow �eld isdue to the relatively large uncertainty in the location of the separation point. The large variability in theturbulent viscosity discrepancy around the separation point results in uncertainty in the ow �eld in thisregion. This is consistent with the fact that RANS models typically fail to accurately estimate the separationpoint.

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U_STD-Y

0.280.260.240.220.20.180.160.140.120.10.080.060.040.02

Figure 8. Standard deviation of the x-velocity (left) and y-velocity �elds (right).

To more clearly display the Monte Carlo results, �gure 9 shows the RANS, DNS and Monte Carlovelocity pro�les at four di�erent x locations, representing a vertical slice through the domain. The MonteCarlo pro�le shows the mean of the Monte Carlo sample velocity pro�les, as well as the two standarddeviation error intervals around the mean velocity pro�le, representing the 95% con�dence intervals. Themean DNS x-velocity pro�les typically fall outside the 2� intervals, especially near the center of the channel.This means the estimated level of uncertainty in the turbulent viscosity is too low. However, the meanDNS y-velocity pro�les are mostly contained inside the 2� intervals. The 2� intervals are typically largestwhere the k � ! and mean DNS pro�les show the largest disagreement, showing that the general trend inthe uncertainty is being captured. We note that the maximum magnitude of the random log-discrepancysamples shown in �gure 7 are smaller than the observed log-discrepancy �eld shown in �gure 5. Clearly, ourmodel of the discrepancy fails to produce realizations with the proper discrepancy magnitude. Consideringthe linear �t shown in �gure 6, we see that best linear �t for the relation between the discrepancy and thecorrected strain-rate norm is quite at, implying that realizations with a large discrepancy magnitude arerelatively unlikely. This explains why the maximum magnitude of the log-discrepancy realizations is too low,thereby underestimating the level of uncertainty in the ow �eld.

VI. Conclusion and Future Work

We have presented a new approach for quantifying the structural uncertainties in RANS turbulencemodels. The approach is general enough to be applied to any RANS turbulent viscosity model, since onlythe model output is considered in estimating the uncertainty. The results of the RANS inverse modelingdemonstrate that a signi�cant portion of the uncertainty in RANS solutions can be attributed to uncertaintyin the turbulent viscosity �eld for the geometries considered in this work. The variability in the turbulentviscosity is correlated with the ow properties of the RANS velocity �eld, and these correlations can be usedto model of the uncertainty in the turbulent viscosity �eld. Future work will focus on developing more re�nedstatistical models of the turbulent viscosity discrepancy that reproduces the larger discrepancy magnitudesobserved in the numerical data. We also plan to explore more accurate and e�cient methods of propagatinguncertainty to the quantities of interest.

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0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5y5

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Figure 9. RANS, DNS, and Monte Carlo velocity pro�les plotted at x = 0:1, x = 1:1, x = 2:1, and x = 2:9 (fromtop to bottom).

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Acknowledgments

The authors would like to thank Professor Steven Johnson at MIT for his helpful suggestions regardingthe NLopt code. The authors are also grateful to Dr. Frank Ham for this assistance with the CDP code,and Professor Rene Pecnik for his help with the Joe code. This work was funded by Pratt and Whitney anda subcontract of the DOE Predictive Science Academic Alliance Program (PSAAP) from Stanford to MIT.

References

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12Davis, M. W., \Production of conditional simulations via the LU triangular decomposition of the covariance matrix,"Mathematical Geology, Vol. 19, No. 2, 1987, pp. 9198.

13Wilcox, D. C., Turbulence Modeling for CFD, Gri�n Printing, Glendale, CA, 1993.14Johnson, S. G., \The NLopt nonlinear-optimization package," http://ab-initio.mit.edu/nlopt.15Chen, H., Wang Q., Hu, R., and Constantine, P., \Conditional Sampling and Experiment Design For Quantifying

Manufacturing Error of Transonic Airfoil," 49th AIAA Aerospace Sciences Meeting, 2011, AIAA Paper 2011-658.

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