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Chapter 1
Uncertainty Quantification Applied to
Aeroacoustics of Wall-Bounded Flows
Julien Christophe*, Marlène Sanjose†, Jeroen Witteveen‡ and Stéphane Moreau†
*Institut von Karman, 72 Chaussée de Waterloo
B-1640 Rhode-Saint-Genese, Belgium, [email protected] †Université de Sherbrooke, 2500 boulevard de l’université
Sherbrooke, QC, Canada,, J1K2R1
‡Center for Mathematics and Computer Science (CWI)
Amsterdam, The Netherlands
The uncertainty quantification (UQ) related to the self-noise prediction based on a Reynolds-Averaged Navier-Stokes (RANS) flow computation of a low-subsonic axial fan has been achieved. As the methodology used for fan noise prediction is based on airfoil theories, the uncertainty quantification of a low-speed Controlled-Diffusion (CD) airfoil has been first considered. For both applications, deterministic incompressible flow solvers are coupled with a non-intrusive stochastic collocation method, found to be two orders-of-magnitude more efficient than a classical Monte Carlo simulation for the same accuracy. In the case of airfoil UQ, the effective flow angle is used as a random variable. Two wall-pressure reconstruction models are used to obtain necessary inputs of Amiet’s trailing-edge noise model: Rozenberg’s model has larger uncertainties at high frequencies because of the uncertainty on the wall-shear stress parameter required in the method, and Panton & Linebarger’s model is less accurate at low frequencies because of the slow statistical convergence of the integration involved in the model. Similar behaviours are observed in the fan UQ involving the volume flow-rate and the rotational speed as random variables. The stochastic mean sound spectra are found to be dominated by the tip strip and compare well with experimental data. Larger uncertainties are seen in the hub and tip regions, where large flow detachment and recirculation appear. The known uncertainties on flow rate yield larger uncertainties on sound than those on rotational speed.
1.Introduction
In modern rotating machines, significant effort has been put to reduce annoying tonal noise, either by passive devices or by active noise control. The next challenge is then to reduce the broadband contribution to decrease the overall noise level and meet increasingly stringent environmental noise regulations. A key source of broadband noise is the trailing-edge noise or self-noise, caused by the scattering of boundary-layer pressure fluctuations into acoustic waves at the trailing edge of any lifting surface. In the absence of any interaction noise source, it represents the dominant source of noise generated by rotating machines such as low-speed fans, high-speed turboengines,1 wind turbines2 and other high-lift devices.3 However, an accurate prediction of the sound by a rotating system still remains a daunting task by a direct computation (a compressible Large Eddy Simulation (LES) for instance). A hybrid approach combining a near-field turbulent flow simulation and an acoustic analogy for the sound propagation in the far-field is therefore preferred. Such a method has been thoroughly validated for broadband noise prediction on multiple airfoils in various flow conditions including blowing.4–13 The computational cost associated with unsteady turbulent flow simulations still limits most numerical studies to simpler geometries such as airfoils,8 even with sophisticated non-boundary-conforming methods such as the Lattice Boltzmann method and Immersed Boundary method,4,14 or the use of unstructured grid topologies.5 To meet industrial design constraints of rotating machines, approaches that model the pressure and velocity fluctuations needed for an acoustic analogy from steady RANS are often used.15–18 These methods add further levels of modeling and the associated uncertainties grow, which may make the final acoustic prediction of fan broadband self-noise inaccurate and unreliable.
To illustrate this point, some of the aleatory uncertainties associated with the prediction of trailing-edge noise are considered, first for the wall-bounded canonical case of airfoil noise as measured in an open-jet wind tunnel,6,19 and then for the actual complex case of a low-speed automotive engine cooling fan as tested in a reverberant wind tunnel.20 For the former the uncertainty propagation from uncertain inlet velocity profiles caused by inaccurate knowledge of the jet deflection induced by the airfoil is studied. For the latter the uncertainty propagation from uncertain operating conditions (both volume flow-rate through the fan and rotational speed) mainly caused by industrial process issues is considered. In doing so, two representative wall-pressure models derived from steady RANS are dealt with, and compared with direct unsteady LES predictions of the trailing-edge noise for the airfoil case. The sensitivity of the RANS and LES solutions to inlet conditions and the uncertainty introduced by wall-pressure models coupled with RANS on the prediction of the noise sources and the far-field pressure are then assessed. The difference is made here between uncertainties in the physical inputs to the problem (aleatoric uncertainties) and the constants or variables of the model used to solve for the flow, for instance all constants used in the turbulence modeling, or for the far-field noise (epistemic uncertainties).21
The methodology for uncertainty quantification based on simulations of either a standard experimental setup for trailing-edge airfoil noise or a wall-mounted fan in a standard interface is presented in Sec. 2. The present stochastic approach is outlined in Sec. 3. The sound prediction methods are then presented in Sec. 4. The next two sections, Secs. 5 and 6, show the uncertainty quantification for the airfoil and the fan cases respectively. For both examples the deterministic flow simulations are briefly outlined, the random variables are described, and the stochastic aerodynamic and acoustic results are compared with the available experimental data. Conclusions are finally drawn in Sec. 7.
2.Uncertainty Quantification Methodology
The methods involved in the fan noise prediction rely on airfoil aeroacoustic models,22 and the uncertainty related to such models is first assessed. The approaches to compute airfoil or fan trailing-edge noise are illustrated in Fig. 1.1. The directions of the arrows outline the logical sequence of the method. Starting from the fan blade geometry (grey square box), two different methodologies based on similar computational methods are used to study trailing-edge noise in the case of airfoil or fan applications.
Fig. 1.1.Uncertainty quantification methodology: the solid-line rectangular boxes refer to the different computational steps in the hybrid methodology, the dashed-line rectangular boxes denote the main inputs and the dash-dotted-line boxes denote the outputs at each step and the arrows indicate the various possible workflows to yield the final acoustic pressure. The round box refers to the point where uncertainty quantification is introduced.
On the one hand, a mid-span cut of an automotive cooling fan is performed to obtain a two-dimensional profile, and used to study uncertainty for airfoil trailing-edge noise. As in previous studies4–13 the same validated numerical method for predicting trailing-edge noise is used. A computation of the complete experimental setup of the large anechoic wind tunnel in Ecole Centrale de Lyon (LWT), including the nozzle and part of the anechoic chamber is first done in order to capture the strong jet-airfoil interaction and its impact on airfoil loading.23 The input parameters for this computation, termed “wind tunnel” in Fig. 1.1, are the wind tunnel velocity Ut, air density ρ, and kinematic viscosity ν. The computational setup, defined by the nozzle and the airfoil geometries as well as the position of the airfoil in the wind tunnel and its geometrical angle of attack αw with respect to the nozzle axis,23 is directly taken from measurements on the experimental setup, and is therefore considered as a mino...
