Uncertainty quantification for RANS simulation of flow over a wavy wall

Uncertainty quantification for RANS simulation of flow over a wavy wall Catherine Gorlé 1,2,3, Riccardo Rossi 1,4, and Gianluca Iaccarino 1 1 Center for Turbulence Research, Stanford University, Stanford, CA, 9435, USA. gorle.catherine@gmail.com 2 Von Karman Institute for Fluid Dynamics, Sint-Genesius-Rode, Belgium. 3 EMAT, Department of Physics, University of Antwerp, Antwerp, Belgium. 4 Dipartimento di Ingegneria Energetica Nucleare e del Controllo Ambientale, Università di Bologna, Italy. Abstract The inability of Reynolds-averaged Navier-Stokes (RANS) simulations with two-equation linear eddy viscosity models to correctly predict flow separation and reattachment limits the reliability of such simulations in many wind engineering problems. A methodology to quantify the turbulence model form uncertainty in the RANS simulations would provide valuable additional information. In this study we consider this problem for the flow over a wavy wall. The methodology is based on perturbing the Reynolds stress tensor in the momentum equations, and was shown previously to correctly predict the uncertainty in the location of the reattachment point along the wavy wall for simulations with the SST k ω model Gorlé et al. (212). We first present further details on the comparison of the turbulence kinetic energy obtained from two RANS models, the SST k ω and the realizable k ɛ model, and from direct numerical simulation for the flow over the wavy wall. Subsequently the results of the uncertainty quantification are further evaluated by investigating the variation in the velocity field predicted by the different simulations. 1 Introduction The predictive capability of Computational Fluid Dynamics (CFD) for wind engineering problems is limited by uncertainties in the simulations. The incoming boundary layer characteristics are an obvious example of uncertain parameters for which the influence on the solution should be quantified if one is to base engineering decisions on the simulation outcome. An uncertainty quantification (UQ) study inevitably requires a large set of simulations to correctly capture the influence of the uncertainties and in order to keep the overall computational cost of such a study within reasonable limits, the cost of each individual simulation should be minimized. This can enforce the use of reduced-order models for physical processes that are expensive to resolve, such as turbulence and turbulent mixing. Reynoldsaveraged Navier-Stokes (RANS) models remain the most affordable technique for simulating complex turbulent flows and they are therefore well suited for performing a large set of simulations. However, the use of turbulence models introduces additional uncertainties in the simulation outcome and these epistemic, or model-form, uncertainties, need to be quantified as well. In Emory et al. (211) the idea to quantify the uncertainty related to the turbulence model through the introduction of perturbations in the Reynolds stress tensor was first introduced. The method defines the perturbations in terms of the magnitude of the tensor, i.e. the turbulence kinetic energy, and the eigenvalues and eigenvectors of the normalized anisotropy tensor, thereby being completely 1

6 th European and African Wind Engineering Conference 2 independent of the initial turbulence model form. In Gorlé & Iaccarino (213) this framework was extended to include UQ of turbulent scalar flux models for RANS simulations of the mixing of a jet in a supersonic crossflow. It was shown that the uncertainty in a quantity of interest related to the downstream mixing could be captured when defining the range of discrepancies considered based on a LES/RANS comparison of the decomposed Reynolds stresses. The results indicated a number of remaining questions for future research. In particular, the definition of the perturbation functions in a more affordable and general way, relying less on the availability of high-fidelity data, was identified as the main challenge. In Gorlé et al. (212), this problem was further addressed for the simulations of the flow over a wavy wall using the SST k ω model, and the uncertainty in the location of the reattachment point for the flow over the wavy wall was successfully quantified. The present paper presents further details and additional results of this uncertainty quantification study for the flow over a wavy wall. Two turbulence models are considered: the realizable k ɛ and the SST k ω models. The comparison between the turbulence kinetic energy, k, obtained by both RANS models and that computed by direct numerical simulations (DNS) for the flow over a wavy wall (Rossi, 211) is presented. Three different sets of simulations were performed. First, both the RANS flow and turbulence equations were solved. Second, the effect of the coupling to the mean flow was eliminated by freezing the flow to the time-averaged DNS flow field, solving the transport equations for the turbulence quantities only. Third, the turbulence kinetic energy production term was also frozen to the production term calculated from the DNS. Based on this analysis the definition of a perturbation function for k that will be used in the model form UQ study for the flow over the wavy wall was established. Finally, the variation in the velocity field obtained from this model form UQ study for both turbulence models is presented. The details of the configuration and the RANS simulations are presented in Section 2. The methodology used for the model form UQ is summarized in the third section. An overview of the results of the comparative analysis for the turbulence kinetic energy is presented in the Section 4. Section 5 of this paper discusses how the full UQ study was formulated and presents the resulting variation in the velocity field. 2 Flow setup, DNS data and RANS simulations The flow setup considered in the present study is identical to the one used in the DNS simulations by Rossi (211) and is shown in Figure 1.The geometry is defined as a two-dimensional channel, with a flat upper wall and a wavy surface at the lower wall defined by: ( y = A cos 2πx λ ), (1) with A the amplitude and λ the wavelength. The wave steepness is defined by 2A/λ and is equal to.1. The DNS database includes results for the flow and for scalar dispersion at Re = 685, based on the bulk velocity U b at the wave crest and the average channel height H. The comparison presented in Section 4 focuses on the magnitude of the Reynolds stress tensor, which is given by the turbulence kinetic energy, k. In Section 5 the time-averaged velocity is compared to the results obtained with the UQ study. The Reynolds stresses and the velocity obtained from the time-averaged DNS solution were averaged in the spanwise direction and over the 4 wave crests, since the flow is assumed to be fully developed and hence periodic in both directions. The RANS simulations were performed on a two-dimensional mesh of 256 x 192 cells, and the near wall resolution was sufficient (y + 1) to avoid the use of wall functions. In the streamwise and spanwise direction periodic conditions are applied. Two different two-equation turbulence models based on the turbulent viscosity hypothesis have been tested: the SST k-ω and the realizable k-ɛ model.

6 th European and African Wind Engineering Conference 3 Figure 1: Flow configuration (Rossi, 211). The uncertainty introduced by these models will be investigated using the methodology from Emory et al. (211), summarized in the following section. 3 Epistemic UQ methodology In the past, the influence of turbulence models has mainly been investigated by comparing different models, often belonging to the same class of two-equation linear eddy viscosity models. Such traditional sensitivity studies can not capture the full model form uncertainty, because this uncertainty is largely determined by the fundamental model assumptions, i.e. those made in the turbulent viscosity hypothesis: u i u j = 2 3 kδ ij 2ν t S ij, (2) where u i u j are the Reynolds stresses, ν t is the turbulent viscosity and S ij is the strain rate tensor. To overcome this limitation, a framework for the epistemic UQ of the Reynolds stresses independent of the initial model form was established in Emory et al. (211). The method consists in directly introducing perturbations in the Reynolds stress tensor computed by the model and used in the momentum equations. The definition of the perturbations is based on the eigenvalue decomposition of the normalized anisotropy tensor a ij : a ij = R ij 2k 1 2 δ ij = v ik Λ kl v jl, (3) where v ij is the matrix of orthonormal eigenvectors and Λ kl is the diagonal matrix of eigenvalues λ l. This decomposition allows reformulating the Reynolds stress tensor as ( ) 1 R ij = 2k 3 δ ij + v ik Λ kl v jl. (4) The above formulation does not involve any modeling assumptions, thereby presenting a general way of introducing epistemic uncertainty in the Reynolds stress tensor by writing the perturbed Reynolds stresses as: ( ) 1 Rij = 2k 3 δ ij + vikλ klvjl. (5) The perturbations are thus specified in terms of a discrepancy in the turbulence kinetic energy k = k + k, the diagonal matrix Λ kl of perturbed eigenvalues λ l, and perturbed eigenvectors v ij = q ikv kj, where q ik is an orthonormal rotation matrix. As stated in the Section 1, the main challenge in this approach is the definition of the perturbation functions in an affordable and general way. For the eigenvalues, the perturbations will be defined in

6 th European and African Wind Engineering Conference 4 Figure 2: Eigenvalue perturbation defined as a displacement in the Barycentric map (Banerjee et al., 27). terms of the location of the Reynolds stress tensor in the Barycentric map (Banerjee et al., 27), by specifying a displacement in the map as shown in Figure 2. This imposes clear bounds on the possible perturbations, since all realizable states of turbulence are located inside the triangular map. For the eigenvectors, the perturbations consist of rotations, which are bound between and 36. Since the flow field is two-dimensional, only one eigenvector rotation angle has to be considered. For the turbulence kinetic energy, the bounds are less easily defined, which motivated the investigation presented in the following section, focusing on the evaluation of the discrepancies in k between the RANS and DNS simulations. 4 Analysis of discrepancies in turbulence kinetic energy Figure 3 presents the comparison for the turbulence kinetic energy from the DNS and from the SST k ω and the realizable k ɛ model for the solutions obtained when freezing the flow field to the time-averaged DNS solution (case A) and when combining this with the exact production term P k = u i u j U i x j computed from the DNS (case B). The difference between both sets of simulations is that when only freezing the DNS flow field, P k is still determined from the inexact Reynolds stresses as computed using the turbulent viscosity hypothesis, while in the second case the exact Reynolds stresses are used. The result shows an overprediction of up to 1% in k when using the correct flow field (case A) with the realizable k ɛ model and an underprediction of the same order of magnitude with the SST k ω model. However, when also using the correct production term for k (case B) both models predict k with much higher accuracy. The comparison of the turbulence kinetic energy field presented in Figure 3 shows that when the exact production term is used in the transport equations for the turbulence quantities, a much more accurate solution for the turbulence kinetic energy can be obtained. Based on this result, the UQ study was performed with perturbations introduced in the eigenvalues and eigenvectors of the Reynolds stress tensor, and subsequently the turbulence kinetic energy was perturbed indirectly using the updated Reynolds stress tensor to compute the production term in the transport equations for the turbulent quantities.

6 th European and African Wind Engineering Conference 5.4 k DNS k.6 k.5.2.4.5 1.2.5 k RKE case A k RKE case B k RKE case A k RKE case B.4.4.4.4.2.2.2.2.5 1.5 1.5 1.5 1 k SSTKO case A k SSTKO case B k SSTKO case A k SSTKO case B.4.4.4.4.2.2.2.2.5 1.5 1.5 1.5 1 Figure 3: Comparison of k from DNS, realizable k ɛ (RKE) and SST k ω (SSTKO) models for solutions to the turbulence quantity transport equations based on the DNS flow field (case A) and based on the DNS flow field and DNS turbulence kinetic energy production term (case B). 5 UQ study 5.1 Introducing the perturbations First, the introduction of uncertainties in regions of the flow where the turbulence model performs sufficiently accurate was avoided by defining a marker. This marker identifies regions that deviate from parallel shear flow as regions where perturbations should be introduced, since the models are not expected to provide accurate predictions in those regions. The exact definition and validation of the marker can be found in Gorlé et al. (212). Second, instead of exploring the entire possible space of eigenvalues and eigenvectors, only two sets of perturbations were selected. Those corresponded to the perturbations that produce the maximum and minimum turbulence kinetic energy production term integrated over the marked region, since these can be expected to provide the smallest and largest length of the separated region. In Gorlé et al. (212), it was shown that the maximum and minimum integrated production terms are found when moving the eigenvalues to the one- and three-component corners of the map, respectively, without a rotation of the eigenvectors. 5.2 Uncertainty predicted in the flow field In Gorlé et al. (212), the two perturbed simulations as defined above were shown to capture the uncertainty in the size of the separated region and give a very realistic representation of the uncertainty in the streamwise wall shear stress for the SST k ω model. The same can be demonstrated for the realizable k ɛ model. In Figures 4 and 5 we further explore this result by considering the contour plots for the horizontal and vertical velocities obtained from the DNS, from the original realizable k ɛ (RKE) and SST k ω (SSTKO) models, and from the perturbed cases for both models. As mentioned in Section 5.1

6 th European and African Wind Engineering Conference 6 DNS RKE RKE C1 RKE C3.4.2.5 1 1.5.4.2.5 1 SSTKO.4.2.5 1.4.2.5 1 SSTKO C1.4.2.5 1.4.2.5 1 SSTKO C3.4.2.5 1 Figure 4: Comparison of horizontal velocity from the DNS, the original realizable k ɛ (RKE) and SST k ω (SSTKO) models, and the perturbed cases (C1 and C3). DNS RKE RKE C1 RKE C3.4.2.5 1.1.1.4.2.5 1 SSTKO.4.2.5 1.4.2.5 1 SSTKO C1.4.2.5 1.4.2.5 1 SSTKO C3.4.2.5 1 Figure 5: Comparison of horizontal velocity from the DNS, the original realizable k ɛ (RKE) and SST k ω models (SSTKO), and the perturbed cases (C1 and C3). eigenvalue perturbations towards the one-component (C1) and three-component (C3) corners of the Barycentric map were considered. The plots show that quantitatively the uncertainty in the velocity field is well represented, especially in the region below the crest of the wave. In general, the variation obtained with the SST k ω model is larger than that obtained with the realizable k ɛ model. A more quantitative comparison of the horizontal velocity component is presented in Figure 6. The plots show the DNS profiles for the horizontal velocities, and the lines that present the maximum and minimum values obtained from the perturbed runs. The plots confirm the observations from Figure 4, with the uncertainty in the velocity field being well represented by the two perturbed runs at most locations. For the realizable k ɛ model the uncertainty in the velocity field is not fully captured at x/λ =.5, while for the SST k ω model it is only on top of the wave crest that the uncertainty in the velocity field is less well represented. 6 Conclusions The goal of the present work is to quantify turbulence model form uncertainty for RANS simulations of the flow over a wavy wall. The method presented in Emory et al. (211) is used, which is based on introducing perturbations in the Reynolds stress tensor in terms of the magnitude of the tensor, i.e. the turbulence kinetic energy, and the eigenvalues and eigenvectors of the normalized anisotropy tensor. In Gorlé et al. (212), it was shown that this method enables quantifying the uncertainty in the

6 th European and African Wind Engineering Conference 7 =. =.25 =.5 =.75.4.4.4.4.3.2.3.2.3.2.3.2.1.1.1.1.5 1 =..5 1 =.25.5 1 =.5.5 1 =.75.4.4.4.4.3.2.3.2.3.2.3.2.1.1.1.1.5 1.5 1.5 1.5 1 Figure 6: Comparison of horizontal velocity for the realizable k ɛ (top) SST k ω (bottom) models. Solid lines: DNS, black dotted lines: unperturbed models, blue dotted lines: minimum and maximum velocities from the perturbed runs. location of the reattachment point for the flow over the wavy wall for simulations with the SST k ω model. The present paper presents further details and additional results of this uncertainty quantification study considering two turbulence models: the realizable k ɛ and the SST k ω models. For both models a comparison between the modeled turbulence kinetic energy and that computed by direct numerical simulations shows that when the exact production term is used in the transport equations for the turbulence quantities, a much more accurate solution for the turbulence kinetic energy can be obtained. This supports the procedure of indirectly perturbing the turbulence kinetic energy, by using the Reynolds stress tensor obtained after perturbing the eigenvalues and eigenvectors to compute the production term in the transport equations for the turbulent quantities. It is shown that a good representation of the uncertainty in the velocity field can be obtained when performing simulations for the perturbations that result in the minimum and maximum integrated production terms. Acknowledgements This work was supported by the Department of Energy [NNSA] under Award Number NA28614. The first author is supported by a Pegasus-Marie Curie fellowship from the Research Foundation Flanders (FWO), co-funded by the Marie Curie action, since October 212. References Banerjee, S., Krahl, R., Durst, F., & Zenger, C. 27. Presentation of anisotropy properties of turbulence, invariants versus eigenvalue approaches. Journal of Turbulence, 8, 1 27. Emory, M., Pecnik, R., & Iaccarino, G. 211. Modeling Structural Uncertainties in Reynolds Averaged Computations of Shock/Boundary Layer Interactions. AIAA paper, 211-479. Gorlé, C., & Iaccarino, G. 213. A framework for epistemic uncertainty quantification of turbulent scalar flux models for Reynolds-averaged Navier-Stokes simulations. Physics of Fluids.

6 th European and African Wind Engineering Conference 8 Gorlé, C., Emory, M., Larsson, J., & Iaccarino, G. 212. Epistemic uncertainty quantification of RANS modeling of the flow over a wavy wall. Annual Research Briefs 212, Center for Turbulence Research, Stanford, CA, 81 91. Rossi, R. 211. A numerical study of algebraic flux models for heat and mass transport simulation in complex flows. International Journal of Heat and Mass Transfer, 53, 4511 4524.