Simulation-based determination of systematic errors of flow meters due to uncertain inflow conditions

Computational fluid dynamics (CFD) provides well-established tools for the prediction of the velocity profiles in turbulent pipe flows. As far as industrial pipe and district heating systems are concerned, combinations of elbows are the most common pipe assemblies. Among the different pipe combinations, double elbows out-of-plane are of special interest, since they introduce strong disturbances into the flow profile and have a strong influence on many common types of flow meters. In front of a double elbow there is often another flow-disturbing installation. As a result the upstream conditions are unknown and an investigation of the resulting systematic bias on the measurement of the flow rate and the associated contribution to its measurement uncertainty is necessary. We demonstrate here that this can be achieved by a variation of the inlet profile in terms of swirls and asymmetry components. In particular, an ultrasonic and an electromagnetic flow meter are modeled in order to quantify the systematic errors stemming from uncertain inflow conditions. For this purpose, a generalized non-intrusive polynomial chaos method has been used in conjunction with a commercial CFD code. As the most influential parameters on the measured volume flow, the distance between the double elbow and the flow meter as well as the orientation of the flow meter are considered as random variables in the polynomial chaos approach. This approach allowed us to obtain accurate prediction of the systematic error for the ultrasonic and electromagnetic meter as functions of the distance to the double elbow. The resulting bias in the flow rate has been found to be in the range of 1.5–4.5% (0.1–0.5%) with a systematic uncertainty contribution of 2–2.4% (0.6–0.7%) for the ultrasonic (electromagnetic) flow meter if the distance to the double elbow is smaller than 40 pipe diameters. Moreover, it is demonstrated that placing the flow meters in a Venturi constriction leads to substantial decrease of the bias and the contribution to the measurement uncertainty stemming from the uncertain inflow condition

Uncertainty quantification integrated to computational fluid dynamic modeling of synthetic jet actuators

The Point Collocation Non-Intrusive Polynomial Chaos (NIPC) method was applied to a stochastic synthetic jet actuator problem to demonstrate the integration of computationally efficient uncertainty quantification to the high-fidelity CFD modeling of Synthetic Jet Actuators. The uncertainty quantification approach was first implemented in two stochastic model problem cases for the prediction of peak exit plane velocity using a Fluid Dynamic Based analytical model of the Synthetic Jet Actuator, which is computationally less expensive than CFD simulations. The NIPC results were compared with direct Monte Carlo sampling results. To demonstrate the efficient uncertainty quantification in CFD modeling of synthetic jet actuators, a test case, Case 1 (synthetic jet issuing into quiescent air), was selected from the CFDVal2004 workshop. In the stochastic CFD problem, the NIPC method was used to quantify the uncertainty in the long-time averaged u and v-velocities at several locations in the flow field, due to the uncertainty in the amplitude and frequency of the oscillation of the piezo-electric membrane. Fifth order NIPC expansions were used to obtain the uncertainty information which showed that the variation in the v-velocity is high in the region directly above the jet slot and the variation in the u-velocity is maximum in the region immediately adjacent to the slot. Even with a ten percent variation in the amplitude and frequency, the long-time averaged u and v-velocity profiles could not match the experimental measurements at y = 0.1mm above the slot, indicating that the discrepancy may be due to other uncertainty sources in CFD or measurement errors. A global sensitivity analysis using linear regression approach indicated that the frequency had a stronger contribution to the overall uncertainty in the long-time averaged flow field velocity for the range of input uncertainties considered in this study. Overall, the results obtained in this study showed the potential of Non-Intrusive Polynomial Chaos as an effective uncertainty quantification method for computationally expensive high-fidelity CFD simulations applied to the stochastic modeling of synthetic jet flow fields --Abstract, page iii

Sensitivity-enhanced generalized polynomial chaos for efficient uncertainty quantification

We present an enriched formulation of the Least Squares (LSQ) regression method for Uncertainty Quantification (UQ) using generalised polynomial chaos (gPC). More specifically, we enrich the linear system with additional equations for the gradient (or sensitivity) of the Quantity of Interest with respect to the stochastic variables. This sensitivity is computed very efficiently for all variables by solving an adjoint system of equations at each sampling point of the stochastic space. The associated computational cost is similar to one solution of the direct problem. For the selection of the sampling points, we apply a greedy algorithm which is based on the pivoted QR decomposition of the measurement matrix. We call the new approach sensitivity-enhanced generalised polynomial chaos, or se-gPC. We apply the method to several test cases to test accuracy and convergence with increasing chaos order, including an aerodynamic case with $40$ stochastic parameters. The method is found to produce accurate estimations of the statistical moments using the minimum number of sampling points. The computational cost scales as $\sim m^{p-1}$, instead of $\sim m^p$ of the standard LSQ formulation, where $m$ is the number of stochastic variables and $p$ the chaos order. The solution of the adjoint system of equations is implemented in many computational mechanics packages, thus the infrastructure exists for the application of the method to a wide variety of engineering problems

Compressive sensing adaptation for polynomial chaos expansions

Basis adaptation in Homogeneous Chaos spaces rely on a suitable rotation of the underlying Gaussian germ. Several rotations have been proposed in the literature resulting in adaptations with different convergence properties. In this paper we present a new adaptation mechanism that builds on compressive sensing algorithms, resulting in a reduced polynomial chaos approximation with optimal sparsity. The developed adaptation algorithm consists of a two-step optimization procedure that computes the optimal coefficients and the input projection matrix of a low dimensional chaos expansion with respect to an optimally rotated basis. We demonstrate the attractive features of our algorithm through several numerical examples including the application on Large-Eddy Simulation (LES) calculations of turbulent combustion in a HIFiRE scramjet engine.Comment: Submitted to Journal of Computational Physic

An efficient technique based on polynomial chaos to model the uncertainty in the resonance frequency of textile antennas due to bending

The generalized polynomial chaos theory is combined with a dedicated cavity model for curved textile antennas to statistically quantify variations in the antenna's resonance frequency under randomly varying bending conditions. The nonintrusive stochastic method solves the dispersion relation for the resonance frequencies of a set of radius of curvature realizations corresponding to the Gauss quadrature points belonging to the orthogonal polynomials having the probability density function of the random variable as a weighting function. The formalism is applied to different distributions for the radius of curvature, either using a priori known or on-the-fly constructed sets of orthogonal polynomials. Numerical and experimental validation shows that the new approach is at least as accurate as Monte Carlo simulations while being at least 100 times faster. This makes the method especially suited as a design tool to account for performance variability when textile antennas are deployed on persons with varying body morphology

Inverse problems and uncertainty quantification

In a Bayesian setting, inverse problems and uncertainty quantification (UQ) - the propagation of uncertainty through a computational (forward) model - are strongly connected. In the form of conditional expectation the Bayesian update becomes computationally attractive. This is especially the case as together with a functional or spectral approach for the forward UQ there is no need for time-consuming and slowly convergent Monte Carlo sampling. The developed sampling-free non-linear Bayesian update is derived from the variational problem associated with conditional expectation. This formulation in general calls for further discretisation to make the computation possible, and we choose a polynomial approximation. After giving details on the actual computation in the framework of functional or spectral approximations, we demonstrate the workings of the algorithm on a number of examples of increasing complexity. At last, we compare the linear and quadratic Bayesian update on the small but taxing example of the chaotic Lorenz 84 model, where we experiment with the influence of different observation or measurement operators on the update.Comment: 25 pages, 17 figures. arXiv admin note: text overlap with arXiv:1201.404