User Manual for the SU2 EQUiPS Module: Enabling Quantification of Uncertainty in Physics Simulations
UUser Manual for the SU2 EQUiPS Module: EnablingQuantification of Uncertainty in Physics Simulations
Jayant Mukhopadhaya
Department of Aeronautics & Astronautics, Stanford University, Stanford, California-94305, USA a r X i v : . [ c s . C E ] S e p HAPTER 1
Introduction
This document serves as the manual for using the EQUiPS (Enabling Quantification of Uncertaintyin Physics Simulations) module in SU2. The EQUiPS module uses the Eigenspace Perturbationmethodology [ ] to provide interval bounds on Quantities of Interest (QoIs) that capture epistemicuncertainties arising from assumptions made in RANS turbulence models. This has been imple-mented and tested in SU2 [ ] for a variety of benchmark turbulence cases as well as flows ofaerodynamic interest.One of the key features of the EQUiPS module that we strove to achieve in its implementationis versatility : ensuring that anyone, regardless of their background in turbulence modeling, canutilize this module. The module can be used without explicit knowledge of the physics behind themethodology but some basic details are needed to understand the outputs of the simulations, andhow to create the interval bounds on quantities of interest.The methodology requires 5 perturbed simulations, in addition to a baseline unperturbed simulation,to characterize the epistemic uncertainties due to turbulence modeling. This baseline simulationrefers to the regular turbulent RANS simulation that would be performed in SU2. The perturbedsimulations are performed sequentially by a python script, the instructions for which are detailedin chapter 4. Each perturbed simulation results in a different realization of the flow field, and byextension, a different realization of the QoIs. The interval bounds are formed by the maximum andminimum values the QoIs resulting from these 6 simulations. It is important to note that the UQfunctionality is only available for the k − ω SST turbulence model, at present.This manual starts by explaining the theory underlying the Eigenspace Perturbation Framework thatis implemented in the EQUiPS module. Then, it walks the reader through the process of installingSU2 in chapter 3. This is followed by instructions on using the EQUiPS module by either runningall the perturbed simulations sequentially in chapter 4, or running them individually in chapter 5.These instructions are made concrete with examples included in chapter 6. Finally, we highlight theuse of this module in published literature in chapter 8. HAPTER 2
Introduction to the Eigenspace Perturbation Framework
In this chapter, we provide the mathematical and computational background for the Eigenspaceperturbation methodology. We introduce each sequentially, starting from the eigen-decompositionof the modeled Reynolds stresses, the introduction of the perturbations into the eigenvalues andeigenvectors to account for structural uncertainty, and deriving the uncertainty estimates from theperturbed simulations.
Representation of the uncertainty: Eigenspace expression of Reynoldsstresses
The Reynolds stress tensor, R ij = (cid:104) u i u j (cid:105) , is a primary Quantity of Interest for turbulence modeling.This can be decomposed into facets that determine the shape, the orientation and the amplitude ofthe Reynolds stress ellipsoid. To this end, the Reynolds stress tensor can be decomposed into theanisotropic and deviatoric components as(1) R ij = 2 k ( b ij + δ ij . Here, k (= R ii ) is the turbulent kinetic energy and b ij (= R ij k − δ ij ) is the Reynolds stress anisotropytensor. The Reynolds stress anisotropy tensor can be expressed as(2) b in v nl = v in Λ nl , where v nl is the matrix of orthonormal eigenvectors and Λ nl is the traceless diagonal matrix ofeigenvalues λ k . Multiplication by v jl yields b ij = v in Λ nl v jl . This is substituted into Equation (1) toyield(3) R ij = 2 k ( v in Λ nl v jl + δ ij . The tensors v and Λ are ordered such that λ ≥ λ ≥ λ . In this representation, the shape,the orientation and the amplitude of the Reynolds stress ellipsoid are directly represented by theturbulence anisotropy eigenvalues λ l , eigenvectors v ij and the turbulent kinetic energy k , respectively.The limitations of classical turbulence models can be re-expressed using this decomposition. For in-stance, one of the key ramifications of the eddy-viscosity hypothesis is that it obligates the modeledReynolds stress to share its eigen-directions with the mean rate of strain tensor. Consequently, theeigenvectors of the modeled Reynolds stresses are co-incident with those of the mean rate of strain.While this is true in simple shear flows, it is limited in complex engineering flows. Similarly, assump-tions made in the gradient diffusion hypothesis lead to imperfect representation of the amplitude of
4. INTRODUCTION TO THE EIGENSPACE PERTURBATION FRAMEWORK 5 x x x x x x x x B x x x x x* (a) (b) (c) Figure 1.
Schematic outline of the eigenvalue perturbations on the barycentrictriangle, starting from an arbitrary state.the Reynolds stress ellipsoid and the form of the eddy-viscosity hypothesis leads to unsatisfactoryexpression for the Reynolds stress anisotropy eigenvalues.
Application of the perturbations: Eigenvalue & Eigenvector perturbations
To account for the errors due to closure assumptions, this eigenspace representation of the Reynoldsstress tensor is perturbed. These perturbations are injected directly into the modeled Reynoldsstress during the CFD solution iterations. This perturbed form is expressed as:(4) R ∗ ij = 2 k ∗ ( δ ij v ∗ in Λ ∗ nl v ∗ lj )where ∗ represents the perturbed quantities. The perturbations to the eigenvalues, Λ, correspondto varying the componentiality of the flow (or the shape of the Reynolds stress ellipsoid). Similarly,the perturbations to the eigenvectors and the turbulent kinetic energy vary the orientation andamplitude of the Reynolds stress ellipsoid. These perturbations are sequentially applied to themodeled Reynolds stress tensor. Eigenvalue perturbation : The eigenvalue perturbation can be represented on the barycentric map. Inthis representation, all realizable states of the Reynolds stress tensor lie on or inside the barycentrictriangle. The vertices of this triangle, labeled x , x and x in Fig. 1, represent the one, two andthree component limiting states of the turbulent flow field. A linear map between the co-ordinateson this triangle x and the Reynolds stress anisotropy eigenvalues λ i is defined by(5) x = x ( λ − λ ) + x (2 λ − λ ) + x (3 λ − . This linear transformation can be expressed as x = B λ . In physical terms, this invertible, one-to-onemapping expresses any realizable state of the Reynolds stress eigenvalues as a convex combinationof the three limiting states of turbulence.The projection of the eigenvalue perturbation in the barycentric map has both a direction and amagnitude, as is exhibited in Fig. 1. In this application, the perturbations are aligned towards thevertices of the barycentric triangle (or the limiting states of turbulence), as shown in in Fig. 1. Themagnitude of the eigenvalue perturbation in the barycentric triangle is represented by ∆ B ∈ [0 , x ∗ are given by
2. INTRODUCTION TO THE EIGENSPACE PERTURBATION FRAMEWORK
Arbitrary Reynolds Stress, Target Componentiality:
1C Reynolds stress, Target Alignment: v max Eigenvalue Perturbation Eigenvector Perturbation Eigenvalue Perturbation Eigenvector Perturbation t=1C, Δ B =1.0, v=v max (a1) (a2) (b1) (c1) (b2) (c2) Figure 2.
Schematic outline of Eigenspace perturbations from an arbitrary stateof the Reynolds stress. x ∗ = x + ∆ B ( x t − x ), where x t denotes the target vertex (representing one of the one-, two-, orthree-component limiting states) and x is the unperturbed model prediction. Thus, ∆ B = 0 wouldleave the state unperturbed and ∆ B = 1 would perturb any arbitrary state to the vertices of thebarycentric triangle. In Fig. 1, this eigenvalue perturbation methodology is illustrated, starting froman arbitrary Reynolds stress componentiality. For this illustration, the direction of the perturbation x t is chosen toward x and the magnitude of perturbation ∆ B is chosen as 0 .
5. The initial x andperturbed x ∗ states are exhibited in the figure, along with the transition. Eigenvector perturbations : The eigenvector perturbations vary the alignment of the Reynolds stressellipsoid. These are guided by the turbulence production mechanism, P = − R ij ∂U i ∂x j . The eigenvec-tor perturbations seek to modulate turbulence production by varying the Frobenius inner product (cid:104) A, R (cid:105) = tr ( AR ), where A is the mean velocity gradient and R is the Reynolds stress tensor. For thepurposes of bounding all permissible dynamics, we seek the extremal values of this inner product. Inthe coordinate system defined by the eigenvectors of the rate of strain tensor, the critical alignmentsof the Reynolds stress eigenvectors are given by v max = and v min = . Therange of this inner product is [ λ γ + λ γ + λ γ , λ γ + λ γ + λ γ ], where γ ≥ γ ≥ γ are theeigenvalues of the symmetric component of A . . INTRODUCTION TO THE EIGENSPACE PERTURBATION FRAMEWORK 7 In physical terms, the eigenvalue perturbation changes the shape of the Reynolds stress ellipsoid andthe eigenvector perturbation changes its relative alignment with the principal axes of the mean rateof strain tensor. To illustrate this eigenspace perturbation framework, we outline a representativecase schematically in Fig. 2. In the upper row, ( a , b , c ), we represent the Reynolds stress tensor ata specific physical location in barycentric coordinates and in the lower row, ( a , b , c ), we visualizethe Reynolds stress ellipsoid in a coordinate system defined by the mean rate of strain eigenvectors.These are arranged so that λ ≥ λ ≥ λ , . Thus, the 1-axis is the stretching eigendirection and the3-axis is the compressive eigendirection of the mean velocity gradient.Initially, the Reynolds stress predicted by an arbitrary model is exhibited in the first column, Fig. 2, a a
2. The eigenvalue perturbation methodology seeks to sample from the extremal states of thepossible Reynolds stress componentiality. Thus, we may, for instance, translate the Reynolds stressfrom this state to the 1 C state, exhibited in the transition from Fig. 2, a b
1. This translationchanges the shape of the Reynolds stress ellipsoid from a a tri-axial ellipsoid to a prolate ellipsoid,exhibited in the transition from Fig. 2, a b b c C, C, C ) and 2 extremal alignments of the Reynolds stress eigenvectors,( v min , v max ). For the 3 C limiting state, the Reynolds stress ellipsoid is spherical. Due to rotationalsymmetry, all alignments of this spherical Reynolds stress ellipsoid are identical and eigenvectorperturbations are superfluous. Determining the uncertainty: Uncertainty estimates
In this subsection, we outline how the uncertainty estimates are engendered from the set of perturbedCFD simulations. This process is schematically exhibited in Fig. 4. The illustrative flow used is thecanonical case of separated turbulent flow in a planar diffuser. The conditions and the experimentaldata are from the experimental study of [ ].The central panel of Fig. 4 outlines the unperturbed, baseline CFD solution. Using the k − ω SSTmodel, this leads to a unique flow field realization in the flow domain. To illustrate the compositionof the uncertainty bounds, we choose a specific location in the domain, specifically at x/H = 24which is marked in the figures. This unique flow field realization from the SST model leads to asingleton profile for the mean velocity, u i /u , shown in panel C with the solid gray line.The upper and lower panels of the figure outline perturbed solutions. While there are 5 perturbedstates as discussed in the last subsection, we exhibit only 2 of these in the illustration. Each ofthese perturbed solutions leads to a different realization of the flow field, as is illustrated in panel
2. INTRODUCTION TO THE EIGENSPACE PERTURBATION FRAMEWORK (b) (a) (d) (e) (c)
Figure 3.
Schematic visualization of the extremal states, as Reynolds stress ellip-soids, in the eigenspace perturbation methodology.B. These flow realizations differ in essential aspects. For instance, the perturbation to the state(1
C, v max ) maximizes the turbulence production mechanism and thus, suppresses flow separation.The perturbation to the state (3
C, v min ) minimizes the turbulence production mechanism and thus,strengthens flow separation. This is evidenced in the variation of the separation zones in panel B.Each of these perturbations leads to a different flow field and consequently, the velocity profiles fromthese flow fields are different as well. The velocity profiles at x/H = 24 from the (1
C, v max ) and(3
C, v min ) are shown in panel C with the dashed and dot-dashed lines respectively. (panel C alsoshows the profiles from the (1
C, v min ), (2
C, v max ) and (2
C, v min ) perturbations using the dottedline, dotted line with circles and dotted lines with squares) The uncertainty estimates on the profilesof a quantity of interest (QoI) at a location are engendered by the union of all the states lying in theprofiles from this set of perturbed RANS simulations. This is illustrated by the gray shaded zone inFig. 4 panel C.At this juncture, it may be useful to outline what the uncertainty estimates are, and more impor-tantly, what they are not. These uncertainty estimates do not represent confidence or predictionintervals at any significance level. To generate such confidence or prediction intervals, one mayrequire data from high-fidelity realizations of the flow (Direct Numerical Simulation (DNS), LargeEddy Simulation (LES) or experimental studies) along with assumptions regarding the distributionsof the explanatory features. In a stereotypical design investigation, one has access to little or nohigh-fidelity data. Additionally, any assumptions about the distribution would require knowledge offor instance the history of the turbulent flow field, which is not feasible for most engineering appli-cations. The uncertainty estimates outlined in this investigation are data-free and rely on a purely . INTRODUCTION TO THE EIGENSPACE PERTURBATION FRAMEWORK 9 U np e r t u r b e d C F D S o l u t i o n P e r t u r b e d C F D S o l u t i o n , ( C , v m a x ) P e r t u r b e d C F D S o l u t i o n , ( C , v m i n ) P a n e l A P a n e l B P a n e l C Figure 4.
Schematic outlining stages in computation of uncertainty estimates.
Panel A :RANS simulations with perturbations;
Panel B :perturbed realizations ofturbulent flow fields;
Panel C :Compositions of uncertainty estimates from union ofperturbed QoI profiles. physics-based framework. In this context, the uncertainty estimates represent estimated ranges forthe values of a Quantity of Interest contingent upon the uncertainties and discrepancies arising dueto the model-form of eddy-viscosity based RANS closures.HAPTER 3
Downloading and Installing the EQUiPS Module
The EQUiPS module is implemented in SU2, an open-source multi-physics simulation software. Thesoftware binaries can be downloaded from the website. For best performance, consider building SU2from the source code. The repository for SU2 is hosted on GitHub. The best way to download thesource code would be to clone the repository with the command: git clone https://github.com/su2code/SU2.git
If you have already cloned the SU2 repository in the past, it is a good idea to update the currentversion with the latest changes in the remote repository using git pull
Detailed instructions to build SU2 from the source code are available here. HAPTER 4
Running the Python Script
As mentioned in the Introduction, the python script sequentially performs the 5 perturbed simu-lations that are required to inform the interval bounds. For smooth operation, it is best to haveperformed a baseline unperturbed simulation with SU2 and have achieved sufficient convergence.This ensures that the mesh file, and the input configuration file are well posed, and, if run throughthe Python script, can provide converged perturbed solutions.The Python script takes an input configuration file that identical to the one used to run the baselineCFD simulation in SU2. The options for the python script, and their uses are: • -f : Specifies the name of the configuration file to be used • -n : Sets the number of processors being used to run the simulations. A parallel build isrequired to use this option • -u : Sets the under-relaxation factor used in performing perturbation. This option neednot be changed unless the perturbation simulations are unstable. u ∈ [0 ,
1] and it’s defaultvalue is 0 .
1. This should not be set to < .
05 as the perturbations may not be completedby convergence. • -b : Sets the magnitude of perturbation. This option should not be touched without havingread the references on the Eigenspace Perturbation methodology [
8, 13 ]. b ∈ [0 ,
1] and it’sdefault value is 1 .
0. The default value corresponds to a full perturbation and is requiredto correctly characterize the epistemic uncertaintiesThe most common use of this script would be: compute uncertainty.py -f turb naca0012.cfg -n 8
This will run the 5 perturbed simulations for the case defined in the turb naca0012.cfg configura-tion file on 8 processors. It creates a new directory for each new simulation, and outputs the resultsin the respective directories. The directories are named: and p1c2 . Each flowsolution is an instantiation of the flow field that must be post-processed to extract the necessarymodel form uncertainty information.It is important to note that this UQ functionality is only available with the SST turbulence model. HAPTER 5
Running Individual Perturbations
The python script is an easy interface to run the perturbed simulations sequentially. In case thereis a need to perform the simulations separately (for example to run them in parallel, or on differentmachines), individual perturbations can be performed by setting options within the configurationfile. The list of the different options available is given below. • USING UQ : Boolean that ensures EQUiPS module is used • UQ COMPONENT : Number that specifies the eigenvalue perturbation to be performed • UQ PERMUTE : Boolean that indicates whether eigenvector permutation needs to be per-formed • UQ URLX : Sets the under-relaxation factor used in performing perturbation. This optionneed not be changed unless the perturbation simulations are unstable. u ∈ [0 ,
1] and it’sdefault value is 0 .
1. This should not be set to < .
05 as the perturbations may not becompleted by convergence. • UQ DELTA B : Sets the magnitude of perturbation. This option should not be touchedwithout having read the references on the Eigenspace Perturbation methodology [
8, 13 ].∆ b ∈ [0 ,
1] and it’s default value is 1 .
0. The default value corresponds to a full perturbationand is required to correctly characterize the epistemic uncertaintiesAn example of how the configuration options would look, is shown below: % ------------------ UNCERTAINTY QUANTIFICATION DEFINITION ------------------%%% Using uncertainty quantification module (YES, NO). Only available with SSTUSING_UQ= YES%% Eigenvalue perturbation definition (1, 2, or 3)UQ_COMPONENT= 1%% Permuting eigenvectors (YES, NO)UQ_PERMUTE= NO%% Under-relaxation factor (float [0,1], default = 0.1)UQ_URLX= 0.1%% Perturbation magnitude (float [0,1], default= 1.0)UQ_DELTA_B= 1.0
134 5. RUNNING INDIVIDUAL PERTURBATIONS
Table 1.
Combination of options required to perform perturbed simulationsPerturbation
UQ COMPONENT UQ PERMUTE1c NO2c NO3c NOp1c1 YESp1c2 YES
Even though each perturbed simulation can be performed individually, all 5 perturbed simulations,in addition to the baseline unperturbed simulation are required to characterize the interval bounds.A combination of
UQ COMPONENT and
UQ PERMUTE options are required to perform the 5 differentperturbations. These combinations are enumerated in Table 1. These simulations can be runindependently from each other which allows for parallelization of the simulations. This also allowsfor the tuning of the convergence parameters for the different simulations.HAPTER 6
NACA0012 Example
To illustrate the capabilities of the EQUiPS module, some simple test cases are explored. The firsttest case concerns flow over a NACA0012 airfoil at a range of angles attack from 0 ◦ to 20 ◦ . Thisis a simple 2D geometry that stalls, and exhibits separated flow, at high angles of attack. It is aubiquitous geometry that has significant amounts of experimental data available that allows for thecomparison of lower fidelity RANS CFD simulations, to the higher fidelity wind tunnel tests thathave been conducted.This tutorial is also available through the SU2 website and the relevant configuration and mesh filesare available from the GitHub repository. Problem setup
This problem will solve the flow past the airfoil with the conditions shown in Table 1.Although this particular case simulates flow at 15 ◦ , the same simulation can be run at varying anglesof attack. The results section also presents analyses from performing the simulations at a range ofangles of attack which allows the exploration of the various flow regimes that occur. At low anglesof attack, the flow stays attached and RANS simulations are quite accurate in predicting the flow.At higher angles of attack, the onset of stall causes flow separation which leads to inaccuracies inflow predictions. Mesh Description
The mesh is a structured C-grid. The farfield boundary extends 500c away from the airfoil surface.A magnified view of the mesh near the wall can be seen in Fig. 1.
Table 1.
Simulation conditions for the NASA CRM.Mach Number 0 . × Reference chord length 1 . . α − ◦ ≤ α ≤ ◦
156 6. NACA0012 EXAMPLE
Figure 1.
Magnified view of the NACA0012 mesh near the wall
Running the Module
The module is built to be versatile, such that it can be used by experts and non-experts alike. Asimple Python script abstracts away the details of the perturbations (componentality, eigenvectorpermutations) and sequentially performs the perturbed simulations. The script requires a mesh andconfiguration file that are identical to ones that are needed to run a baseline RANS CFD simulation.For smooth operation, it is best to have performed the baseline simulation with SU2 and haveachieved sufficient convergence. This ensures that the configuration file and mesh are well posed,and, if run through the Python script, can provide converged, perturbed simulations. Details in thenext section on Configuration File Options are not required to run the Python script. Unless thereis a need to perform the perturbations individually, you can move to the Running SU2 section.
Configuration File Options
If there is a need to perform the perturbations individually (for example to run them in parallel, oron different machines), configuration options need to be set to specify the kind of perturbation toperform. . NACA0012 EXAMPLE 17 % ------------------- UNCERTAINTY QUANTIFICATION DEFINITION -------------------%% Using uncertainty quantification module (YES, NO). Only available with SSTUSING_UQ= YES%% Eigenvalue perturbation definition (1, 2, or 3)UQ_COMPONENT= 1%% Permuting eigenvectors (YES, NO)UQ_PERMUTE= NO%% Under-relaxation factor (float [0,1], default = 0.1)UQ_URLX= 0.1%% Perturbation magnitude (float [0,1], default= 1.0)UQ_DELTA_B= 1.0
Results
In order to obtain the interval bounds of a QOI, all 6 instantiations of the flow solution (1 baselineand 5 perturbed) must be analyzed. To illustrate how the bounds are formed, we use the exampleof the C P distribution along the upper surface of the airfoil. In Fig. 2(a) the C P distributions ofeach perturbed simulation is plotted along with the baseline simulation, experimental data, and theuncertainty bounds. In Fig. 2(b), only the individual perturbation data is hidden. The uncertaintybounds are formed by a union of all the states the QOI predicted by the module. It is interestingto see the bounds are larger in areas with correspondingly large discrepancy between the baselinesimulation, and the experimental data.As we can see in Fig. 2(b), the predictions of the RANS model are not in perfect agreement withthe experimental data. This lack of agreement is even more severe near x/c = 0. However, theuncertainty estimates from the EQUiPS module account for this discrepancy and the experimentaldata lies in the uncertainty estimates. Analysing further from Fig. 2(a), we observe that theexperimental data are in agreement with the 1 C perturbations. These represent limiting states ofthe Reynolds stress anisotropy and eddy-viscosity based models are not able to predict such extremestates of anisotropy as their predictions are restricted to the plane strain line of the barycentrictriangle.At an angle of attack of 10deg, the baseline RANS model is able to accurately predict the C P distribution. If the UQ module is run at this angle, it is seen that the uncertainty bounds are muchsmaller. This case can be run simply using the steps as above, only changing the AOA option forthe files. This is illustrated in Fig. 3.Similarly, if the module is run for a number of angles of attack, the predicted lift curve can beplotted. This showcases the robustness of the model in different flow situations. Fig. 4 illustratesthe results from a angle of attack sweep from 0 to 20 degrees. At low angles of attack, there is almostno discernible difference between the RANS predictions and the experimental data. Accordingly, x/c - C p -2024681012 Uncertainty boundsBaselineExperimental1c2c3cp1c1p1c2 x/c - C p -2024681012 Uncertainty boundsBaselineExperimental
Figure 2. C P distribution along upper surface for the NACA0012 airfoil at 15degAOA (a) with individual perturbations included, (b) with only the resulting intervalbounds. . NACA0012 EXAMPLE 19 x/c - C p -10123456 Uncertainty boundsBaselineExperimental
Figure 3. C P distribution along upper surface for the NACA0012 airfoil at 10degAOA with predicted interval boundshere the uncertainty bounds from the EQUiPS module are negligible. At higher angles of attackcloser to stall, there is substantial discrepancy between the RANS predictions and the high fidelitydata. For these values of the angle of attack, the uncertainty bounds are substantial as well. Atall values of the angle of attack, the uncertainty bounds from the EQUiPS module envelope theexperimental dataTo highlight the robustness of the EQUiPS module in handling different configurations, we includea similar figure for the lift curve of the NACA4412 airfoil in Fig. 5. , C L Uncertainty boundsBaselineExperimental Data
Figure 4.
Lift Curve of the NACA0012 with interval bounds predicted by theEQUiPS module. . NACA0012 EXAMPLE 21 , C L Uncertainty boundsBaselineExperimental Data
Figure 5.
Lift Curve of the NACA4412 with interval bounds predicted by theEQUiPS module.HAPTER 7
Advanced Applications
The EQUiPS module can be used for more complex flow configurations as well. To illustrate this,the module is applied to the NASA Common Research Model (CRM) which is an aircraft geometrythat was developed for applied CFD validation studies [
17, 16 ]. The design is based on a Boeing777 and is representative of a transonic commercial aircraft. It has a wealth of openly accessiblecomputational and experimental data.
Problem Setup
The simulation conditions are described in Table 1. Note that the simulations are run at a rangeof angles of attack, at a free stream Mach number of 0.85. These conditions lead to complex flowfeatures, such as shock induced separation, that can greatly increase the uncertainties in the RANSpredictions. Separated flow exists for α > ◦ at this Mach number. Mesh Description
Figure 1 shows details of the unstructured mesh that was used for the CFD simulations. Thecomputational domain is made of 11 . × mixed elements (4 . × nodes) which correspondsto a coarse mesh based on the grid convergence studies performed for multiple solvers and gridtopologies [ ]. Results
The EQUiPS module is applied to the pitch sweep and compared to wind tunnel data from the NASAAmes 11ft Wind Tunnel experiment [ ]. In Fig 2, the solid black line represents the predictionsmade by the baseline SST turbulence model, the grey area represents the interval bounds predicted Table 1.
Simulation conditions for the NASA CRM.Mach Number 0 . × Reference chord length 7 . .
928 K α − ◦ ≤ α ≤ ◦
22. ADVANCED APPLICATIONS 23 (a) Surface mesh of the NASA CRM. (b) Close up of the nose cone showing boundary layercells on the symmetry plane.(c) Details of the wing surface mesh.
Figure 1.
Images of the NASA CRM mesh that was used for the CFD simulations.by the EQUiPS module, and the black crosses represent the wind tunnel data. These wind tunneldata points have error bars associated with them but these are barely discernible on the scale of theplot.We compare the integrated quantities, specifically the coefficients of lift ( C L ), drag ( C D ), and longi-tudinal pitching moment ( C m ), predicted by CFD and the EQUiPS module, to those experimentallydetermined. Focusing on the C L vs. α plot in Figure 2(a). At low angles of attack, the flow re-mains well attached to the aircraft body and there aren’t any complex flow features that would bedifficult for the turbulence model to predict. The turbulence model does not introduce significantuncertainty in its predictions and, accordingly, the interval bounds predicted by the UQ moduleare relatively small. At higher angles of attack when there is flow separation over portions of the , C L Uncertainty boundsBaseline CFDExperimental Data (a) C L vs. α . , C D Uncertainty boundsBaseline CFDExperimental Data (b) C D vs. α . , C m -0.2-0.100.10.20.3 Uncertainty boundsBaseline CFDExperimental Data (c) C m vs. α . C D C L Uncertainty boundsBaseline CFDExperimental Data (d) C L vs. C D . Figure 2.
Uncertainty in force and moment coefficients as calculated by the RANSUQ methodology on the NASA CRM.aircraft, simplifying assumptions made in the turbulence models make it difficult to make accurateflow predictions. This is reflected in the growing uncertainty bounds predicted by the module. Thisoverall trend is seen in all of the plots in Figure 2.Ideally, we should see better agreement between the computational and experimental data at thelower angles of attack. At these angles, the model-form uncertainty introduced by the turbulencemodel, as predicted by the module, is small. This discrepancy is explained by geometrical differencesin the model that was experimented on, and the one that was used for simulations. The wind tunnel . ADVANCED APPLICATIONS 25 (a) α = 1 ◦ . (b) α = 2 . ◦ .(c) α = 3 ◦ . (d) α = 4 ◦ . Figure 3.
Isosurfaces representing areas where local Mach variability M v = 0 . ].The EQUiPS module runs multiple RANS simulations that result in multiple realizations of theflow field. In addition to quantifying uncertainty estimates on integrated quantities, this data canprovide insight into flow features/areas that contribute to the uncertainty estimates. Fig. 3 showsiso-surfaces of areas where the local Mach number varies by greater than 0 . M v ) is defined at every point in the computational domain as M v = max ( M i ) − min ( M i ) where i refers to each realization of the flow field (5 perturbed + 1baseline flow fields) and M i represents the Mach number at each point in that flow field.At low angles of attack, Fig. 3(a), the Mach variability is low and limited to the junction regionsin the flow field. The eigenspace perturbations do not cause major changes in the flow, resulting insmaller uncertainty bounds. As the angle of attack increases, as shown in Fig. 3(b) and 3(c), largerareas of variability appear where the shock would be expected, at the upper surface of the wing andaway from the leading edge. This denotes an uncertainty in the shock location. This area growsrapidly until it reaches the leading edge in Fig. 3(d), signalling large uncertainty bounds and reducedconfidence in the CFD predictions. Such visualizations allow us to analyse the relationship betweenthe dominant flow features and the uncertainty that they introduce in the turbulence models.HAPTER 8 Usage of EQUiPS in Published Literature
Since its release, the EQUiPS module has been used extensively by researchers, and, has been ac-knowledged in literature. The developers of the library have used it to estimate uncertainties inRANS model predictions for benchmark flows [ ] and complex flows of engineering interest [ ].Research groups from the University of Colorado, University of Michigan, Stanford University andSandia National Laboratories used the EQUiPS module to study uncertainties in the simulation ofhigh speed aircraft nozzles[ ]. Additionally, researchers from the University of Greenwich have usedthe EQUiPS module to quantify mixed uncertainty in complex turbulent jets [
6, 7 ]. Researchersfrom the French Institute for Research in Computer Science and Automation (INRIA) have usedthe EQUiPS module to carry optimization under uncertainty of Organic Rankine Cycle (ORC) su-personic nozzles [
15, 5 ]. Researchers from Stanford University along with the Boeing Company,have used the EQUiPS module to generate probabilistic aerodynamic databases [ ]. Researchersfrom the University of Cambridge and the Massachusetts Institute of Technology used the EQUiPSmodule for design optimization under model form uncertainty for aerospace applications [
4, 3 ]. Re-searchers at the Universidad de M´alaga, Spain have used the EQUiPS module for studying heattransfer characteristics in turbulent jets [ ]. Similarly, investigators at the German Aerospace Cen-ter (DLR) are utilizing the EQUiPS module to investigate RANS predictions for turbomachineryapplications [ ]. Researchers at the University of Southampton used the EQUiPS module to studythe sensitivity of aerodynamic shape optimization [ ].In addition to these published studies, the EQUiPS module is being actively used and developedfurther by research groups. For instance, research groups at the Los Alamos National Laboratoryare extending the EQUiPS module for the uncertainty estimation for RANS model predictions forvariable-density flows. In a different vein, researchers at the Delft University of Technology areintegrating data driven models to automatically tune the parameters of the perturbations in themodule. We shall continue to update future versions of this document with additional researchworks that utilize the EQUiPS module . If you utilize the EQUiPS module in your research, please cite as:@article { mishra2019uncertainty,title= { Uncertainty estimation module for turbulence model predictions in SU2 } ,author= { Mishra, Aashwin Ananda and Mukhopadhaya, Jayant and Iaccarino, Gianluca and Alonso, Juan } ,journal= { AIAA Journal } ,volume= { } ,number= { } ,pages= { } ,year= { } ,publisher= { American Institute of Aeronautics and Astronautics }} ibliography
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