A Kriging-based elliptic extended anisotropic model for the turbulent boundary layer wall pressure spectrum

Abstract

The present study addresses the computation of the wall pressure spectrum for a turbulent boundary layer flow without pressure gradient, at high Reynolds numbers, using a new model, the Kriging-based elliptic extended anisotropic model (KEEAM). A space–time solution to the Poisson equation for the wall pressure fluctuations is used. Both the turbulence–turbulence and turbulence–mean shear interactions are taken into account. It involves the mean velocity field and space–time velocity correlations which are modelled using Reynolds stresses and velocity correlation coefficients. We propose a new model, referred to as the extended anisotropic model, to evaluate the latter in all regions of the boundary layer. This model is an extension of the simplified anisotropic model of Gavin (PhD thesis, 2002, The Pennsylvania State University, University Park, PA) which was developed for the outer part of the boundary layer. It relies on a new expression for the spatial velocity correlation function and new parameters calibrated using the direct numerical simulation results of Sillero et al. (Phys. Fluids, vol. 26, 2014, 105109). Spatial correlation coefficients are related to space–time coefficients with the elliptic model of He & Zhang (Phys. Rev. E, vol. 73, 2006, 055303). The turbulent quantities necessary for the pressure computation are obtained by Reynolds-averaged Navier–Stokes solutions with a Reynolds stress turbulence model. Then, the pressure correlations are evaluated with a self-adaptive sampling strategy based on Kriging in order to reduce the computation time. The frequency and wavenumber–frequency wall pressure spectra obtained with the KEEAM agree well with empirical models developed for turbulent boundary layer flows without pressure gradient.

Analytical models of the wall-pressure spectrum under a turbulent boundary layer with adverse pressure gradient

“…The turbulence-turbulence (TT) curves presented in figure 17 exhibit a plateau for , a trend that can also be observed in figure 17 of Slama et al. (2018). However, this conclusion must be taken with caution since it is obtained by making such a critical hypothesis as that of a normal probability distribution and isotropic turbulence.…”

“…The solution proposed by Peltier & Hambric (2007) was Gauss–Legendre integration, while Slama et al. (2018) used a Kriging-based algorithm. In order to provide a further alternative, this paper investigates the application of one of the Monte Carlo integration methods, for which the number of samples of the integrand function, , and the rate of convergence of the estimation of the integral are independent of the number of dimensions.…”

“…Peltier & Hambric (2007) and Slama et al. (2018) selected equation (2.7). However, Hu, Reiche & Ewert (2017) argued that (2.6) provides the best numerical realisation of the turbulence–turbulence interaction contribution.…”

“…However, this aspect has been taken into account in the models of Peltier & Hambric (2007) and Slama et al. (2018).…”

“…This also represents an improvement, since most currently used models are calibrated on zero-pressure-gradient boundary layers (see Slama et al. 2018). Turbulence and wall-pressure statistics are extracted from a state-of-the-art DNS in order to validate the different steps in the construction of the model.…”