Session H21: Turbulence Simulation: Sensitivity/Uncertainty

Stabilized sensitivity analysis of scalar mixing in laminar and turbulent jet in crossflow

Solutions to both linearized Navier-Stokes equation and its adjoint equation grow exponentially in high Reynolds number turbulent flows, making sensitivity analysis of long time averaged, statistical quantities difficult. This talk presents a stabilization scheme by adding numerical viscosity to stabilize these equations. This adjoint stabilization scheme is applied to scalar mixing in laminar and turbulent jets in cross flow. We vary two parameters, the Reynolds number Re and the jet-to-crossflow velocity ratio R. At low Re and low R, steady flow field and adjoint solution is obtained. At medium Re and low R, the flow is unsteady and the adjoint solution grows; we analyze the effect of additional stabilizing viscosity in the linearized and adjoint equations to the accuracy of computed sensitivity. The effect of numerical viscosity at high Re and large R, when the flow field becomes turbulent and chaotic, is also analyzed. By comparing these cases, we summarize the performance of the stabilization scheme.   

Impact of numerical errors on the turbulent mixing of high Schmidt number passive scalars

Numerical errors associated with scalar transport schemes can affect significantly the mixing of high Schmidt number passive scalars. In this work, we present an analysis of the impact of these errors on the scalar transport characteristics in homogeneous isotropic turbulence and turbulent mixing layers. These two configurations are selected as representatives of different regions of a reacting turbulent jet. We evaluate scalar energy and dissipation spectra, as well as the probability density functions of the scalar and its dissipation rate. This analysis is performed at various grid resolutions, using several different Eulerian and Semi-Lagrangian transport schemes. The results are used to establish the accuracy of these schemes in capturing and preserving the small-scale turbulent structures. It is shown that Eulerian schemes require comparatively higher grid resolution to produce results independent of the mesh size. Conversely, semi-Lagrangian schemes are capable of achieving comparable accuracy at lower grid resolutions, resulting in significant reductions in computational cost. We use the results to propose grid resolution criteria to ensure scheme independent results for high Schmidt number scalar transport in homogeneous isotropic turbulence and turbulent mixing layers.   

Adjoint Sensitivity Computation for Unsteady, Periodic Fluid Flows

Adjoint sensitivity analysis is an important computational method to assist in engineering design and optimization problems, and can be used to efficiently compute the sensitivity of an objective function with respect to many parameters simultaneously. While these methods are popular for steady problems, there are issues when extending them to periodic and chaotic systems. Some techniques, such as windowing, have made progress at computing time average sensitivities for periodic systems; however, they do not provide a time accurate representation of the desired sensitivity. We propose a new method of adjoint computation for periodic systems that produces a time accurate sensitivity by computing and correcting two adjoint systems simultaneously. By decomposing input perturbations into components that produce pure phase shifts and no phase shifts, we derive the governing equations for the time accurate adjoint. We then propose an algorithm for computing this adjoint solution with the added overhead of storing and computing one additional adjoint variable. This algorithm is tested on the Van der Pol oscillator and a CFD simulation of vortex shedding behind a cylinder. Possible extensions to chaotic systems, such as turbulent fluid flows, will also be examined.   

Estimating Uncertainties in Statistics Computed from DNS

Direct numerical simulation (DNS) of turbulence is a critical tool for investigating the physics of turbulent flows and for informing and developing engineering turbulence models. For instance, flow statistics obtained from DNS are commonly used as ``truth data'' for the calibration and evaluation of turbulence models. Thus, like experimental data, uncertainty estimates are a necessary component of the reported output. In DNS, uncertainties in the computed statistics arise from two sources: finite sampling and the discretization of the Navier-Stokes equations. Here, we apply estimators for both sources of error. Finite sampling errors are estimated using the ``effective sample size,'' which accounts for the fact that the instantaneous data are correlated. Discretization errors are estimated using data from simulations with varying time step and mesh spacing. The performance of these estimators is tested for several statistics using DNS of turbulent channel flow at low Reynolds number ($Re_{\tau} \approx 180$).   

Representing Turbulence Model Uncertainty with Stochastic PDEs

Validation of and uncertainty quantification for extrapolative predictions of RANS turbulence models are necessary to ensure that the models are not used outside of their domain of applicability and to properly inform decisions based on such predictions. In previous work, we have developed and calibrated statistical models for these purposes, but it has been found that incorporating all the knowledge of a domain expert---e.g., realizability, spatial smoothness, and known scalings---in such models is difficult. Here, we explore the use of stochastic PDEs for this purpose. The goal of this formulation is to pose the uncertainty model in a setting where it is easier for physical modelers to express what is known. To explore the approach, multiple stochastic models describing the error in the Reynolds stress are coupled with multiple deterministic turbulence models to make uncertain predictions of channel flow. These predictions are compared with DNS data to assess their credibility. This work is supported by the Department of Energy [National Nuclear Security Administration] under Award Number [DE-FC52-08NA28615].   

Breakdown of Sensitivity Analysis in Chaotic, Turbulent Fluid Flows

Sensitivity analysis is a class of algorithms for calculating derivatives of output quantities with respect to input parameters in computational fluid dynamics simulations. It is an essential ingredient for data assimilation, aerodynamic design, uncertainty quantification and flow control. Sensitivity analysis in high fidelity simulations of turbulent flows (DNS or LES) is challenging. These simulations are true to the chaotic nature of turbulence: instantaneous flow fields are very sensitive to perturbations in parameters and geometry; while long time averaged, statistical quantities are often well behaved functions of parameters and geometry. Consequently, sensitivity of statistics cannot be computed by taking the statistics of sensitivity. For example, averaging the sensitivity of instantaneous drag over a long time do not give the sensitivity of the mean drag. This talk first discuss the mathematical reason of the breakdown of sensitivity analysis of statistical quantities in chaotic systems. We then demonstrate the breakdown on a number of examples, including airfoil at high angle of attack, a cylinder in crossflow, and a turbulent jet in a cross flow. We show that positive Lyapunov exponents in these systems lead to divergence of the conventional sensitivity analysis.   

New Methods for Sensitivity Analysis in Chaotic, Turbulent Fluid Flows

Computational methods for sensitivity analysis are invaluable tools for fluid mechanics research and engineering design. These methods are used in many applications, including aerodynamic shape optimization and adaptive grid refinement. However, traditional sensitivity analysis methods break down when applied to long-time averaged quantities in chaotic fluid flowfields, such as those obtained using high-fidelity turbulence simulations. Also, a number of dynamical properties of chaotic fluid flows, most notably the ``Butterfly Effect,'' make the formulation of new sensitivity analysis methods difficult. This talk will outline two chaotic sensitivity analysis methods. The first method, the Fokker-Planck adjoint method, forms a probability density function on the strange attractor associated with the system and uses its adjoint to find gradients. The second method, the Least Squares Sensitivity method, finds some ``shadow trajectory'' in phase space for which perturbations do not grow exponentially. This method is formulated as a quadratic programing problem with linear constraints. This talk is concluded with demonstrations of these new methods on some example problems, including the Lorenz attractor and flow around an airfoil at a high angle of attack.   

Quantification of epistemic uncertainties in RANS turbulence models

Thanks to its limited computational requirements, the RANS approach has extensively been used and is still used to predict the low-order statistics of high Reynolds number turbulent flows. The main drawback is that an universal setup of the closure turbulence models has proved to be elusive. The free parameters present in turbulence models are usually derived from estimated deterministic values of some properties of benchmark turbulent flows, as e.g. the energy power law exponent for decaying homogeneous isotropic turbulence or the value of the von Karman constant. The free parameters present in different well-known turbulence models are obtained herein by considering the underlying properties as random variables over a bounded range. This range has been recovered from the results reported in literature for the relevant properties, so that the considered epistemic uncertainty is realistic. The sensitivity to this uncertainty of the results of turbulent channel flow RANS simulations is then investigated for different Reynolds numbers. The solution over the continuous multi-dimensional uncertainty space of the considered random variables is reconstructed through the application of a surrogate model (response surface) obtained by means of generalized Polynomial Chaos.   

Evaluation and Quantification of Uncertainty of RANS turbulence and turbulent mixing models for a separated flow

The inability of the k-$\varepsilon $ and k-$\omega $ RANS turbulence models to correctly predict flow separation and reattachment limits the reliability of simulations of complex flows. When also predicting the turbulent diffusion of a scalar, algebraic models for the scalar fluxes, which rely on the turbulent viscosity or Reynolds stresses predicted by the turbulence model, introduce additional errors in the solution. In the present work these errors are evaluated by comparing the Reynolds stresses obtained from the RANS models to DNS results for the flow over a wavy wall. The effect of the coupling to the mean flow is eliminated by freezing the flow to the time-averaged DNS flow field and only solving the transport equations for the turbulence quantities. The goal of this analysis is to establish a statistical model for the errors in the modeled Reynolds stresses. The errors are investigated in terms of the turbulence kinetic energy and eigenvalues and eigenvectors of the anistropy tensor. The statistical models for these quantities are used to perturb the Reynolds stresses and quantify the uncertainty in the location of the reattachment point. By also introducing the perturbations in an algebraic model formulation for the scalar fluxes the capability of quantifying the uncertainty in scalar mixing is investigated.