Stochastic turbulence modeling in RANS simulations via multilevel Monte Carlo

A multilevel Monte Carlo (MLMC) method for quantifying model-form uncertainties associated with the Reynolds-Averaged Navier-Stokes (RANS) simulations is presented. Two, high-dimensional, stochastic extensions of the RANS equations are considered to demonstrate the applicability of the MLMC method. The first approach is based on global perturbation of the baseline eddy viscosity field using a lognormal random field. A more general second extension is considered based on the work of [Xiao et al. (2017)], where the entire Reynolds Stress Tensor (RST) is perturbed while maintaining realizability. For two fundamental flows, we show that the MLMC method based on a hierarchy of meshes is asymptotically faster than plain Monte Carlo. Additionally, we demonstrate that for some flows an optimal multilevel estimator can be obtained for which the cost scales with the same order as a single CFD solve on the finest grid level

A multigrid multilevel Monte Carlo method using high-order finite-volume scheme for lognormal diffusion problems

The aim of this paper is to show that a high-order discretization can be used to improve the convergence of a multilevel Monte Carlo method for elliptic partial differential equations with lognormal random coefficients in combination with the multigrid solution method. To demonstrate this, we consider a fourth-order accurate finite-volume discretization. With the help of the Matérn family of covariance functions, we simulate the coefficient field with different degrees of smoothness. The idea behind using a fourth-order scheme is to capture the additional regularity in the solution introduced due to higher smoothness of the random field. Second-order schemes previously utilized for these types of problems are not able to fully exploit this additional regularity. We also propose a practical way of combining a full multigrid solver with the multilevel Monte Carlo estimator constructed on the same mesh hierarchy. Through this integration, one full multigrid solve at any level provides a valid sample for all the preceding Monte Carlo levels. The numerical results show that the fourth-order multilevel estimator consistently outperforms the second-order variant. In addition, we observe an asymptotic gain for the standard Monte Carlo estimator

Non-intrusive uncertainty quantification using reduced cubature rules

For the purpose of uncertainty quantification with collocation, a method is proposed for generating families of one-dimensional nested quadrature rules with positive weights and symmetric nodes. This is achieved through a reduction procedure: we start with a high-degree quadrature rule with positive weights and remove nodes while preserving symmetry and positivity. This is shown to be always possible, by a lemma depending primarily on Carathéodory's theorem. The resulting one-dimensional rules can be used within a Smolyak procedure to produce sparse multi-dimensional rules, but weight positivity is lost then. As a remedy, the reduction procedure is directly applied to multi-dimensional tensor-product cubature rules. This allows to produce a family of sparse cubature rules with positive weights, competitive with Smolyak rules. Finally the positivity constraint is relaxed to allow more flexibility in the removal of nodes. This gives a second family of sparse cubature rules, in which iteratively as many nodes as possible are removed. The new quadrature and cubature rules are applied to test problems from mathematics and fluid dynamics. Their performance is compared with that of the tensor-product and standard Clenshaw–Curtis Smolyak cubature rule

Influence of turbulence anisotropy on RANS predictions of wind-turbine wakes

Simulating wind-turbines in Reynolds-averaged Navier-Stokes (RANS) codes is highly challenging, at least partly due to the importance of turbulence anisotropy in the evolution of the wake. We present a preliminary investigation into the role of anisotropy in RANS simulations of vertical-axis turbines, by comparison with LES. Firstly an LES data-set serving as our ground-truth is generated, and verified against previously published works. This data-set provides raw turbulence anisotropy fields for several turbine configurations. This anisotropy is injected into RANS simulations of identical configurations to determine the extent to which it influences (i) the production of turbulence kinetic energy, (ii) the turbulence momentum forcing, and finally (iii) the mean-flow. In all these quantities we observe the anisotropy has a surprisingly limited effect, and is certainly not the leading-order error in Boussinesq RANS for these cases. Nevertheless we go on to show that it is feasible to predict anisotropy fields for unseen configurations based only on the mean-flow, by using a tensorized version of random-forest regression. </p

Data assimilation for Navier-Stokes using the least-squares finite-element method

We investigate theoretically and numerically the use of the least-squares finite-element method (LSFEM) to approach data-assimilation problems for the steady-state, incompressible Navier-Stokes equations. Our LSFEM discretization is based on a stress-velocity-pressure (S-V-P) first-order formulation, using discrete counterparts of the Sobolev spaces H(div) ×H1 ×L2 for the variables respectively. In general, S-V-P formulations are promising when the stresses are of special interest, e.g., for non-Newtonian, multiphase or turbulent flows. Resolution of the system is via minimization of a least-squares functional representing the magnitude of the residual of the equations. A simple and immediate approach to extend this solver to data assimilation is to add a data-discrepancy term to the functional. Whereas most data assimilation techniques require a large number of evaluations of the forward simulation and are therefore very expensive, the approach proposed in this work uniquely has the same cost as a single forward run. However, the question arises: what is the statistical model implied by this choice? We answer this within the Bayesian framework, establishing the latent background covariance model and the likelihood. Further we demonstrate that—in the linear case—the method is equivalent to application of the Kalman filter, and derive the posterior covariance. We practically demonstrate the capabilities of our method on a backward-facing step case. Our LSFEM formulation (without data) is shown to have good approximation quality, even on relatively coarse meshes—in particular with respect to mass conservation and reattachment location. Adding limited velocity measurements from experiment, we show that the method is able to correct for discretization error on very coarse meshes, as well as correct for the influence of unknown and uncertain boundary conditions.Aerodynamic

Influence of turbulence anisotropy on RANS predictions of wind-turbine wakes

Simulating wind-turbines in Reynolds-averaged Navier-Stokes (RANS) codes is highly challenging, at least partly due to the importance of turbulence anisotropy in the evolution of the wake. We present a preliminary investigation into the role of anisotropy in RANS simulations of vertical-axis turbines, by comparison with LES. Firstly an LES data-set serving as our ground-truth is generated, and verified against previously published works. This data-set provides raw turbulence anisotropy fields for several turbine configurations. This anisotropy is injected into RANS simulations of identical configurations to determine the extent to which it influences (i) the production of turbulence kinetic energy, (ii) the turbulence momentum forcing, and finally (iii) the mean-flow. In all these quantities we observe the anisotropy has a surprisingly limited effect, and is certainly not the leading-order error in Boussinesq RANS for these cases. Nevertheless we go on to show that it is feasible to predict anisotropy fields for unseen configurations based only on the mean-flow, by using a tensorized version of random-forest regression. Aerodynamic

Aerodynamic Shape Optimization Using the Discrete Adjoint of the Navier-Stokes Equations: Applications towards Complex 3D Configurations

Within the next few years, numerical shape optimization based on high fidelity methods is likely to play a strategic role in future aircraft design. In this context, suitable tools have to be developed for solving aerodynamic shape optimization problems, and the adjoint approach - which allows fast and accurate evaluations of the gradients with respect to the design parameters - is seen as a promising strategy. After describing the theory of the viscous discrete adjoint method and its implementation within the unstructured RANS solver TAU, this paper describes application for aerodynamic shape optimization. First wing and fuselage designs of the DLR-F6 wing-body aircraft are presented. A step forward in complexity is considered by applying the adjoint for flap and slat optimal settings of the DLR-F11 model, a wing-body aircraft in high-lift configuration. On all cases presented, optimization were successfully performed within a limited number of flows evaluations.Aerodynamics & Wind EnergyAerospace Engineerin