# Advances in the Prediction of the Statistical Properties of Wall-Pressure Fluctuations under Turbulent Boundary Layers

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## Abstract

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## 1. Introduction

## 2. Theoretical Background

## 3. Study of the Controlled-Diffusion Airfoil Test Case

#### 3.1. Large Eddy Simulation of the Flow around a Controlled-Diffusion Airfoil

^{X}, which offers a low dissipation cell-centered finite-volume scheme. The numerical solution method is second-order accurate in space and third-order accurate in time, using a low-storage Runge–Kutta–Wray scheme for explicit time advancement. The sub-grid turbulent scales (SGS) are represented by Vreman’s model. The unstructured computational grid has an O-grid topology and is made up of 225.5 M cells. The computational domain extends $20c$, where c is the airfoil chord, in both the streamwise and normal-to-wall directions and $0.1c$ in the transverse direction. The dimensionless wall distance, ${y}^{+}$, on the CD airfoil surface is below $1.12$. The normalized time step is $2.68\times {10}^{-5}$. The mesh resolution quality has been verified by means of Pope’s criterion [37], which measures the ratio of resolved to total (i.e., sum of resolved and SGS) turbulent kinetic energy. This ratio is found to be at least $0.94$ for all the WRLES simulations performed in the SCONE project, meaning that the resolution level is close to that of a DNS mesh.

#### 3.2. Boundary Layer Profiles Close to the Trailing Edge of the Airfoil

#### 3.3. Turbulence Statistics

#### 3.3.1. Streamwise and Transverse Velocity Correlation Coefficient

#### 3.3.2. Integral Length Scale and Anisotropy Coefficient

## 4. Direct Computation and Empirical Modeling of Wall-Pressure Statistics

## 5. Prediction of Wall-Pressure Statistics with the Analytical Model Based on the Poisson Equation

#### 5.1. Single-Point Statistics

#### 5.2. Multi-Point Statistics

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Reference frame for the calculation of wall-pressure statistics based on turbulence and mean flow statistics. The classical boundary layer model for which the mean flow speed is aligned with the ${x}_{1}$ direction, and it varies only in the normal-to-wall, ${x}_{2}$ direction, is also depicted.

**Figure 2.**Positions of the mesh layers on which the correlation coefficients have been computed, courtesy of ISAE-Supaero. The black circle indicates the position of pressure sensor number 26 in the experimental setup.

**Figure 10.**Integral length scale of turbulent structures. Blue: integral of streamwise velocity correlation coefficient. Orange: integral of transverse velocity correlation coefficient. Dashed line: hyperbolic tangent model used as a working hypothesis.

**Figure 12.**Examples of calculation of the phase of the wall-pressure streamwise correlation coefficient and linear interpolation for the calculation of the wall-pressure fluctuations convective speed. (

**a**) $St=4.678$; (

**b**) $St=9.357$; (

**c**) $St=14.035$; (

**d**) $St=18.713$.

**Figure 15.**Fit of the coherence plot with a log-normal distribution. Same data as in Figure 14.

**Figure 16.**Prediction of the wall-pressure PSD at $2\%$ of the chord upstream of the trailing edge, compared with measurement and LES computation.

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**MDPI and ACS Style**

Grasso, G.; Roger, M.; Moreau, S.
Advances in the Prediction of the Statistical Properties of Wall-Pressure Fluctuations under Turbulent Boundary Layers. *Fluids* **2022**, *7*, 161.
https://doi.org/10.3390/fluids7050161

**AMA Style**

Grasso G, Roger M, Moreau S.
Advances in the Prediction of the Statistical Properties of Wall-Pressure Fluctuations under Turbulent Boundary Layers. *Fluids*. 2022; 7(5):161.
https://doi.org/10.3390/fluids7050161

**Chicago/Turabian Style**

Grasso, Gabriele, Michel Roger, and Stéphane Moreau.
2022. "Advances in the Prediction of the Statistical Properties of Wall-Pressure Fluctuations under Turbulent Boundary Layers" *Fluids* 7, no. 5: 161.
https://doi.org/10.3390/fluids7050161