## Introduction

Rough surfaces influence the aerodynamics of turbomachinery, e.g. aero engines. In general, roughness increases losses in the flow due to higher production of turbulence: higher turbulence due to a change in momentum transfer between fluid and surface leads to higher losses. The increase of losses also leads to a decrease of the fluid’s velocity close to the wall and thereby an increase of the boundary layer thickness. Also other properties of the blade like the location of separation bubbles or the transition point are affected by the change of the boundary layer flow (Roberts and Yaras, 2005).

The velocity defect induced by the roughness leads to a deflection of the wake, and a change of incidence angle on the stage downstream. A disturbance of the incidence, for example in a compressor, can reduce the performance by up to 2.5% (Seehausen et al., 2020). Examples for the diverse types of roughness, found on worn blades of an airplane engine, are shown in Figure 1. The roughness depends on the location of the blade in the engine and on the position on the blade (Gilge et al., 2019a). In low-fidelity simulations like RANS, the boundary condition is modified to take the roughness of the wall into account. In the

To develop new models, or to improve the existing turbulence models and the

This brief (and by far not full overview) of recent research shows the many influencing factors that need to be considered for modeling the influence of roughness on wall bounded flows. In this paper, further investigations into the influence of roughness skewness are conducted, with the aim to find a new or to improve an existing ks-correlation for rough surfaces found on aero engine blades. To investigate the influence of single roughness parameters independently, direct numerical simulations of systematically varied rough surfaces are conducted.

## Systematically varied real roughness

Two roughness, found on compressor blades of a medium size high-bypass aircraft engine, are used as a basis for this investigation (Gilge et al., 2019b). An overview is shown in Figure 2. A confocal microscope is used to measure the surfaces height on equidistant scanning points *s* are non-dimensionalised by

with

##### Table 1.

### Isolation of roughness skewness

The skewness

### Variation of roughness height

To change the roughness height without changing it’s density, all dimensions are multiplied by a scaling factor

### Variation of roughness density

By multiplying only the horizontal roughness dimensions (

## Numerical setup

In this study, an immersed boundary method (IBM) is adopted for direct numerical simulation (DNS) of the flow over the rough surface. This method has some advantages over the conventional body-fitted approach. Firstly, the mesh is regular with perfect quality, being independent of the immersed body. Secondly, the mesh can also be used for other surfaces, simply because the introduction of the surface involves nothing but immersion in the computational domain. Last but not the least; the generation of the mesh requires very little human effort. The IBM approach used in this paper is provided in *foam-extend-4.0*. It is based on the polynomial fitting of the solution on the IB cells concerning the boundary condition on the wall. A detailed explanation of the method can be found in Senturk et al. (2019).

The CFD tool used, *foam-extend-4.0*, is based on a cell-centered finite volume discretization. For the incompressible solver based on *icoFoam*, Gauss linear scheme and backward temporal discretization are chosen, resulting in second-order accuracy both in time and space. PISO algorithm is used for the solution of the incompressible flow equations. For all the simulations, a maximum CFL number smaller than 0.5 is ensured.

A channel setup is convenient to investigate the flow over a roughness patch because the friction velocity Reynolds number

can easily be set through the incorporation of a constant body force in the momentum equations. Hence, the drag on the surface is balanced by this flow-driving force as long as the flow is fully developed. The boundary layer thickness

The mesh consists of

The DNS-IBM approach is validated with the

## Results

During the simulation, the friction velocity Reynolds number (Equation 2) is kept constant by applying a constant body force on the fluid. An increase of losses by the roughness is therefore shown as a decrease of the mass flow. In the velocity profiles, shown in Figure 5, this decrease of mass flow is reflected by a decrease of velocity. In comparison with the results of a smooth channel DNS, the simulations with a roughness show a velocity deficit

Figure 6 shows the velocity deficit

uses the mean roughness height

The correlations by Bons (2005)

use the shape and density parameter

by Dirling (1973). The shape and density parameter evaluates the density of the roughness elements and the gradient (shape) of the roughness elements. For the roughness height *k* the maximum roughness height

Within the

The constant *B* depends on the roughness of the wall. For smooth surfaces it is given as

For rough surfaces it is given as

with

The velocity difference for a given equivalent surface roughness

The accuracy of a ks-correlation can be evaluated by the fit of its prediction of

## Modelling the influence of skewness

A correlation matrix is used to identify or improve correlations from the roughness geometry to the DNS results. Figure 7 shows an excerpt of a correlation matrix for the given DNS results. Roughness parameters, calculated from the roughness geometry, are listed on the abscissa. On the ordinate, values deduced from the DNS are listed. Solving Equation 11 for the equivalent sand grain roughness

The coefficient of determination

Most DNS results are within the 95% functional confidence interval shown as a gray shadow. Based on the found correlation, the ks-correlation based on the shape and density parameter can be improved.

The equivalent sand grain roughness

calculated from the roughness geometry with the correlation based on the shape, density and skewness parameter is shown in Figure 9. The factor

## Conclusions

The aim of this investigation is to improve the equivalent sand grain roughness (ks) correlation for surfaces from aero engines. To achieve this, direct numerical simulations (DNS) of channel flows with systematically varied rough surfaces are carried out at a friction Reynolds number of

## Nomenclature

effective slope in x-direction (

equivalent sand grain roughness (equ. 12, 3, 5)

normalized equivalent sand grain roughness (

number of cells (−)

shear stress Reynolds number (

coefficient of determination (−)

mean roughness height (

scaling factor (−)

kurtosis of height values (

sknewess of height values (

root mean square roughness height (

maximum roughness height (−)

friction velocity (

Velocities (−)

normalized velocities (

spatial coordinates (−)

normalized spatial coordinates (

channel half height (−)

correlation function (

roughness function (−)

shape and density parameter (−)

kinematic viscosity (−)

Density (−)

wall shear stress (−)