Uncertainty Quantification in Computational Fluid Dynamics

Abstract

When modeling physical systems, several sources of uncertainty are present. For example, variability in boundary conditions like free stream velocity or ambient pressure are always present. Furthermore, uncertainties in geometry arise from production tolerances, wear or unknown deformations under loading. Uncertainties in computational fluid dynamics (CFD) simulations can have a significant impact on the computed aerodynamic performance. Since CFD simulations are computationally intensive, an efficient uncertainty quantification approach is required.The main objective of this research is to obtain an efficient approach for uncertainty quantification in CFD simulations. This was achieved by focusing on efficient uncertainty propagation and the practical applicability to a wide range of test cases. The Probabilistic Collocation method was developed as an efficient non-intrusive uncertainty propagation method. It is based on the polynomial chaos framework and shows spectral convergence with respect to the polynomial chaos order. Its effectiveness was demonstrated on several flow cases using a commercial CFD solver.For cases with a discontinuous response or involving long time integration, modifications of the Probabilistic Collocation method were used to efficiently propagate the uncertainties. A Multi-element formulation was successfully applied to capture the discontinuous response of a stall flutter problem. Furthermore, a time-independent parameterization was used to efficiently propagate uncertainties in case of vortex shedding behind a circular cylinder, which required long time integration.Geometric uncertainties were shown to have a significant influence on the aerodynamic performance. Since geometric uncertainties affect the shape, a new computational grid should be computed for every collocation point in the Probabilistic Collocation method. To efficiently treat geometric uncertainties in CFD, a grid deformation technique was used.Most CFD simulations in this thesis involved solving the Reynolds-averaged Navier-Stokes equations. This required a turbulence model to close the system of equations. Turbulence models often contain several parameters that are tuned to computed or measured simplified flow problems, which introduces uncertainty in the model. Uncertainty quantification was applied to the parameters of the k-ǫ turbulence model in combination with wall functions in the cases of flow over a flat plate and flow around a NACA0012 airfoil. The drag coefficient showed a coefficient of variation of 3-4% for both cases. The wall function parameters κ and C and the model parameter Cμ proved to affect the solution most. General conclusions require more test cases, like a shear layer and an expanding jet.Compressor rotors are components of a gasturbine that are highly sensitive to operational and geometrical uncertainties. Operational uncertainties like static outlet pressure and the total pressure profile at the inlet of the rotor were considered. The Probabilistic Collocation method was validated using a Monte Carlo simulation using 10,000 Latin Hypercube samples. It was shown that the mass flow was most sensitive to the uncertainty in the total pressure profile at the inlet.Multiple uncertainties were shown to be effectively handled using a two-step approach. The first step was a screening of the parameters. A sensitivity analysis was used to identify the most important parameters of the problem. Here it was assumed that all parameters are independent and have no combined effects. Secondly, the probability density functions of the most important parameters are propagated using the Probabilistic Collocation method. The Probabilistic Radial Basis Function approach was developed as an alternative efficient approach for multiple uncertain parameters. To obtain an accuracy of 0.01-0.001 for the mean and variance, the CFD test cases required 10-35 support points for 3 uncertain parameters. Close agreement between the Probabilistic Radial Basis Function approach and a Monte Carlo simulation using 10,000 Latin Hypercube samples was shown for flow around a RAE2822 airfoil with three uncertain parameters.It can be concluded that the Probabilistic Collocation method and adapted versions are capable of efficiently propagating uncertainties in CFD simulations. The development of the Probabilistic Radial Basis Function approach provided an efficient alternative for cases with multiple uncertain parameters. From the test cases it became clear that there is not a single method that is most efficient for all possible cases.Uncertainty quantification increases the reliability of CFD computations, since the effect of uncertain parameters on the output of interest is quantified. It was shown that small coefficients of variation of uncertain parameters can lead to a significant variability of the aerodynamic performance. Taking uncertainties into account in CFD simulation is therefore of great importance and with the current state of technology feasible for many real world applications.