Validation of an Eddy-Viscosity-Based Roughness Model Using High-Fidelity Simulations ^†

Abstract

In this study, the modeling of rough surfaces by eddy-viscosity-based roughness models is investigated, specifically focusing on surfaces representative of deterioration in aero-engines. In order to test these models, experimental measurements from a rough T106C blade section at a Reynolds number of 400 K are adopted. The modeling framework is based on the k-ω-SST with Dassler’s roughness transition model. The roughness model is recalibrated for the k-ω-SST model. As a complement to the available experimental data, a high-fidelity test rig designed for scale-resolving simulations is built. This allows us to examine the local flow phenomenon in detail, enabling the identification and rectification of shortcomings in the current RANS models. The scale-resolving simulations feature a high-order flux-reconstruction scheme, which enables the use of curved element faces to match the roughness geometry. The wake-loss predictions, as well as blade pressure profiles, show good agreement, especially between LES and the model-based RANS. The slight deviation from the experimental measurements can be attributed to the inherent uncertainties in the experiment, such as the end-wall effects. The outcomes of this study lend credibility to the roughness models proposed. In fact, these models have the potential to quantify the influence of roughness on the aerodynamics and the aero-acoustics of aero-engines, an area that remains an open question in the maintenance, repair, and overhaul (MRO) of aero-engines.

1. Introduction

During the operation of turbomachinery components, their surfaces are often worn due to erosion and particle deposition. The wear on the blade surfaces is usually accompanied by an increase in surface roughness, which mainly causes a decrease in the turbine and compressor efficiency [1,2]. However, at low Reynolds numbers, surface roughness can actually reduce aerodynamic losses by eliminating laminar separation [3]. Besides the effects on frictional losses, roughness influences the heat transfer at the surface of the blade. For instance, surface roughness can lead to an early transition, enhancing the overall heat transfer compared to a smooth surface [4].

The importance of considering the transition from laminar to turbulent flow in gas turbines has been highlighted by Mayle [5]. Several transition modes, including the natural and bypass transition, the detachment-induced transition, and the wake-induced transition, have been identified. Various approaches have been developed to model them [6,7,8,9,10]. However, it is worth noting that most of these models are based on the correlations introduced by AbuGhannam and Shaw [11]. Incorporating the transition into the simulations leads to a better prediction of the boundary layer parameters and the losses, even at high Reynolds numbers like 500,000 [12].

In order to improve the accuracy of deterioration-based performance loss predictions, extensions of the transition models by secondary effects, such as surface roughness, are needed [3,13]. Incorporating roughness effects into the $\gamma \text{-}{\mathrm{Re}}_{\theta }$ model [9] is proposed by Dassler [14], where an additional transport equation for roughness amplification is used. Apart from this, Stripf et al. [13] apply a rather direct approach by adding source terms to the momentum and energy conservation equations to model the influence of roughness in turbulent boundary layers and propose an alternative transition onset correlation.

Another challenge for the appropriate modeling of the roughness effect is the wide range of different types of roughness. For instance, Gilge et al. [15] optically measure surfaces of V2500 compressor blades after 20,000 cycles of operation. The structures vary a lot along the blades and across the stages. They also identify anisotropic roughness structures at the leading edge of a front stage. Using these measurements, Seehausen et al. [16] showed in a numerical study that real surface roughness causes locally increased flow losses, which are amplified by the multistage nature of the compressor and lead to a performance loss of up to 2.5% points. It was consequently found that taking into account the surface roughness in gas turbines is crucial for optimizing their performance and efficiency and ultimately reducing operational costs. Since fuel costs are still responsible for approx. 30% of the overall airline costs, performance optimization is important not only for ecological reasons but also for economic ones. This is true for the aerodynamics, aero-acoustics, and aero-elasticity of aero-engines and is therefore vital for the design of new engines as well as for MRO workflows. However, it requires roughness models with the capability of capturing the local effects accurately.

With increasing supply of resources, scale-resolving simulations have gradually become accessible enough to serve as high-fidelity test rigs. The effect of roughness on turbulent boundary layers in direct numerical simulations of channel flows has widely been investigated using immersed-boundary methods (IBM) [17,18,19]. Alternatively, use of high-order curved elements for representing the rough boundaries (e.g., on the T106C blade) has been proposed recently [20]. In a previous study [21], the potential of using high-order curved elements to represent roughness was scrutinized. It was shown that this could be an efficient way to resolve the roughness together with the flow around the blades. Compared to our previous approach using a finite volume method with IBM and using a CPU cluster; the new approach, using the flux-reconstruction method, curved elements, and GPU-cards, reduces the required amount of computational resources by 75%. Independent of the approach, high-fidelity simulations provide immense information about the flow, deeming them precious for developing and tuning roughness models in the Reynolds-averaged Navier-Stokes (RANS) framework.

In the present study (initially presented in [22]), a RANS test rig for roughness investigations, whereby the roughness effect is modeled, is developed. A significant shortcoming of transition and turbulence models, that is, the prediction of the isotropic roughness effect, is addressed. This shortcoming is to some extent overcome through recalibration of a transition roughness model. In order to improve the models, a high-fidelity test rig using Large-eddy Simulation (LES) is also developed. Finally, the results of the high-fidelity test rig are utilized for validating the recalibrated model.

2. Test Case

This study focuses on the T106C low-pressure turbine cascade, the measurements and simulations of which have become ubiquitous (see Figure 1). In fact, it has been investigated for a wide range of Mach and Reynolds numbers, subject to various turbulence levels. Montis et al. [23] extended the measurements with a variation of the surface roughness on the entire blade surface. They investigated three different roughness heights ($k_s$ = 4.83 µm, $k_s$ = 38.9 µm, and $k_s$ = 187 µm). The surface structure was described as “sand-blasted.” The measurements were later used for model development purposes, focusing on better prediction of the roughness effect using RANS models [14,24]. To isolate the roughness effects on the boundary layer, a separation-free case is essential. Thus, we adopt the Reynolds number $\mathrm{Re}=\mathrm{400,000}$ and $\mathrm{Tu}=6\%$ with the highest roughness $k_s$ = 187 µm.

3. Methodology

Preliminary simulations, both RANS and LES, showed lower pressure than the experiment on the pressure side of the blade section. Thus, a one degree higher incidence angle (i.e., ${38.7}^{\circ }$) is adopted to match the experiment. This mismatch is attributed to the uncertainties of the experiment, such as a possible contraction effect because of the end-wall boundary layers. In light of these, we keep the experimental measurements only for completeness. More details about the RANS & LES flow conditions are shown in

Table 1. Flow Conditions for the RANS and LES simulations.

Parameter Group	Values
Gas Constants	$\gamma =1.4$, $R_{gas}=287.05$ $\mathrm{J} {\mathrm{k}\mathrm{g}}^{-1} {\mathrm{K}}^{-1}$, $Pr=0.72$
Inlet Conditions	$T_{tot,1}=303.15$ K, $p_{tot,1}=\mathrm{36,106.62}$ $\mathrm{Pa}$
Outlet Tank Conditions	$p_{2\mathrm{t}\mathrm{h}}=\mathrm{28,307.73}$ $\mathrm{Pa}$, $T_{2\mathrm{t}\mathrm{h}}=282.79$ K
Incidence Angle	38.7° (1° larger than the experiment)
Sutherland’s Law Constants	$S=110.4$ $\mathrm{K}$, $T_{ref}=273.15$ $\mathrm{K}$, ${\mu }_{ref}=1.716$ × ${10}^{-5}$ $\mathrm{k}\mathrm{g} {\mathrm{m}}^{-1} {\mathrm{s}}^{-1}$

.

A three-step process is planned: (1) Verify the large-eddy simulations (LES) with the experiments to establish a realistic high-fidelity test rig; (2) Analyze the current behavior of the RANS roughness model by Dassler [14] that was built and calibrated with focus on integral data; (3) Devise a recalibrated transition model for rough surfaces by adding a local dimension to it.

3.1. LES

The scale-resolving simulations are conducted using the open-source flow solver called PyFR (version 2.0.0) [25], which is based on a high-order flux-reconstruction schemes. Through using a specific flux correction function, the scheme is similar to the discontinuous Galerkin methods.

In the present study, a polynomial degree of $p=3$ is adopted, yielding fourth-order spatial accuracy. Thanks to the subgrid-model-like dissipative characteristics of the numerical scheme, implicit large-eddy simulation (iLES) is adopted herein. On the element interfaces, a Rusanov Riemann solver is applied. A local discontinuous Galerkin (LDG) scheme, with the upwind and penalty parameters of $\beta =0.5$ and $\tau =0.1$, is used for the viscous fluxes. The time marching is based on the explicit RK45[2R+] scheme with proportional-integral (PI) adaptive time-step controlling. The Gauss–Legendre flux and solution point sets are used. Anti-aliasing through approximate $L^2$ projection of flux in the volume and on the face using tenth-order quadrature is activated. This is indeed considered necessary because aliasing-driven instabilities can become severe with highly curved elements.

The turbulence generation is based on a synthetic-eddy method [26], which introduces eddies in a given box using the source-term formulation. The center plane of the box is located at around $0.83l_{ax}$ upstream of the leading edge. The mesh is refined starting from that plane, as can be seen in Figure 2. The plane is perpendicular to the flow direction as required. The evolution of turbulence along a streamline that passes first through the turbulence generation plane, then through the middle of the passage, is shown in Figure 3. The mesh density in the wake region is also kept high down to a plane located approximately at $2.34l_{ax}$ downstream of the trailing edge. The resolution on the inlet and outlet boundaries is chosen low enough to eliminate any spurious reflections from the boundaries.

On the smooth blade, the maximum p3 element sizes in wall units in streamwise, wall-normal, and spanwise directions are found to be $\Delta {\xi }^{+}\approx 50$, $\Delta {\eta }^{+}\approx 8$, and $\Delta {\zeta }^{+}\approx 56$, respectively. Note that each element has four inner solution nodes in each direction, over which polynomials are defined. Hence, a representative roughness element with $k_s$ = 187 µm, corresponding to $k_s^{+}<34$ according to the RANS results, is resolved approximately by one element. A visual comparison of the discretized surface with the original surface is shown in Figure 4, indicating a fairly good representation of the original surface. Note that the spanwise resolution is uniform for each case due to the simple extrusion of the $xy$-mesh in the spanwise (z) direction. The extrusion generates 80 levels, traversing approximately over $0.25l$, resulting in $\mathrm{3.8M}$ high-order elements in total.

After washing away the initial transients, statistics are collected for approximately 18 convective time units based on the outlet velocity and the chord length. The statistical errors are estimated by the method used by Bergmann et al. [27]. The probe data at several wake and blade surface locations demonstrate that the statistical errors are estimated to be low, being smaller than the figure symbol sizes herein.

3.2. RANS

The non-commercial flow solver TRACE 9.1 [28] from the Institute of Propulsion Technology at the German Aerospace Center (DLR) is used for the steady-state CFD simulations. In this solver, the three-dimensional Favre-averaged Navier–Stokes equations are solved based on a finite-volume method on structured multi-block meshes. The convective fluxes are discretized with the second-order upwind scheme of Roe, and the diffusive fluxes are discretized by a central-differencing scheme. The time integration is performed with an implicit predictor-corrector method. For all the simulations, the turbulence model $k\text{-}\omega \mathrm{SST}$ of Menter et al. [29] in combination with the transition model $\gamma \text{-}{\mathrm{Re}}_{\theta }$ of Langtry and Menter [9] was used. The advantages of this combination are described by Kožulović et al. [24]. The influence of rough surfaces on the transition of boundary layers is taken into account with an additional transport equation for the amplification roughness ${\tilde{A}}_r$ developed by Dassler [14] (see Equation (1)). The model equation required is published by Kožulović et al. [24] for the first time.

$${\displaystyle \frac{\partial \left(\rho {\tilde{A}}_r\right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho U_j{\tilde{A}}_r\right)}{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left[{\sigma }_{A_r}\left(\mu +{\mu }_{\mathrm{t}}\right){\displaystyle \frac{\partial {\tilde{A}}_r}{\partial x_j}}\right]$$

(1)

A quasi-three-dimensional setup of T106C with 11,160 cells is selected, having only one cell in the spanwise direction. The distance of the first cell centers to the walls ensures $y^{+}<1$. To assess the convergence of the grid, a study was performed using the method of Roache [30]. The results show that the grid convergence index (GCI) for the grid considered is $\mathrm{GCI}=1\%$ for the normalized mean total pressure loss coefficient $\zeta /{\overline{\zeta }}_{\mathrm{ref}}$.

Figure 5 shows the profile pressure distribution in terms of $C_p(={\displaystyle \frac{p-p_2}{p_{tot,1}-p_2}})$ for the smooth surface. No separation or reattachment is observed, which was found to be the ideal condition for the roughness model development. As it was shown experimentally by Montis et al. [23], surface roughness has a negligible effect on the pressure distribution. This was confirmed by the present iLES and is therefore omitted here for clarity. The agreement between RANS and LES is quite remarkable. The deviation from the experiment on the suction side might be stemming from the different incidence angle or the end-wall effects. Based on the accuracy of the measurement system specified by Montis et al. [23], an uncertainty was estimated for the experiment. This uncertainty is within the symbol size.

3.3. Proposed Recalibration of the Roughness Model

Dassler [14] originally calibrated the roughness model within the $\gamma \text{-}{\mathrm{Re}}_{\theta }$ transition model of Langtry and Menter [9] in combination with the $k\text{-}\omega$ turbulence model by Wilcox [31]. The surface roughness in terms of the equivalent sand-grain roughness $k_s$ serves as a wall boundary condition for the transport equation of the amplification roughness ${\tilde{A}}_r$. In the turbulent region of the boundary layer, the equivalent sand-grain roughness $k_s$ is also set as a Dirichlet wall boundary condition and corrects the boundary condition $\omega$ for the transport equation of specific dissipation rate [31]. On one hand, Alldieck et al. [32] show that the effect of $k\text{-}\omega \mathrm{SST}$ (turbulence model) on the transition is negligible when using a rough wall boundary condition, which is a major advantage over $k\text{-}\omega$. On the other hand, Hellsten and Laine [33] describe that the eddy-viscosity limiter ensuring the Bradshaw assumption of the $k\text{-}\omega \mathrm{SST}$ model can disable the rough-wall condition of the $k\text{-}\omega$ formulation. Herein, the surface roughness taken into account is below the fully rough regime and therefore does not belong to this category.

However, recalibrating the roughness transition model within the $\gamma \text{-}{\mathrm{Re}}_{\theta }$ transition model is recommended because of the different interactions in combination with the $k\text{-}\omega \mathrm{SST}$ and $k\text{-}\omega$ model. The recalibration is based on a local flow field measurement over rough surfaces recorded with particle image velocimetry (planned to be published) and integral data correlating the transition location with surface roughness [34]. Use of local flow fields adds a dimension that serves to increase the reliability of the transition model for isotropic surface roughness. Figure 6 provides a schematic overview of the general cause-and-effect chain of the roughness model, which was developed by Dassler [14]. The proposed modification for the amplification roughness (see Equation (1)) at the wall $A_{r,W}$ follows:

$$A_{r,W}=\left\{\begin{array}{cc}8·k_s^{+}\hfill & \mathrm{if}{k}_s^{+}<L_{A_r}\hfill \\ C_{A_r}·k_s^{+}+8·L_{A_r}\hfill & \mathrm{if}{k}_s^{+}\ge L_{A_r}\hfill \end{array}\right.$$

(2)

with $L_{A_r}=12$ and $C_{A_r}=1.3$. The scalar value of the transported quantity ${\tilde{A}}_r$ correlates with the term $Ar{g}_r$ in the source term $P_{\theta t}$ of the transport equation for ${\widetilde{\mathrm{Re}}}_{\theta t}$ (see Equation (3)). Since the term $F_{\theta t}$ basically deactivates the production of ${\widetilde{\mathrm{Re}}}_{\theta t}$ within the boundary layer, only the influence of roughness can be taken into account without conflicting with the empirical correlations of the outer flow. The term $Ar{g}_r$ is amplified by a factor of $C_{Ar{g}_r}=4.1$.

$$P_{\theta t}=c_{\theta t}{\displaystyle \frac{\rho }{t}}\left[\left(R{e}_{\theta t}-{\widetilde{Re}}_{\theta t}\right)\left(1-F_{\theta t}\right)-C_{Ar{g}_r}·Ar{g}_r\right].$$

(3)

Additionally, the diffusion constant ${\sigma }_{A_r}$ in Equation (1) is set to ${\sigma }_{A_r}=20$. The calibration constants and the diffusion constant were determined using a Latin hypercube sampling. The combination that proved to have the smallest deviation from the reference measurements (based on the transition location and local flow) in the entirety of the test cases was chosen.

3.4. Surface Roughness

For the LES, the grit-blasted surface “s8”, published by Thakkar et al. [17], is applied to the blade. The surface is found to be comparable to the one used by Montis et al. [23] because it is a sand-blasted surface with similar roughness parameters. In order to achieve the targeted roughness height $S_a$, the surface is scaled down and tiled $50\times 10$ onto the blade section (see Figure 4).

The surface roughness in eddy-viscosity-based RANS models is incorporated via an equivalent sand-grain roughness $k_s$. The equivalent sand-grain roughness $k_s$ was first introduced by Schlichting [35] to derive a correlation between arbitrary roughness and sand-grain roughness, which was also used in the experiments of Nikuradse [36]. Since then, there has been an abundance of approaches to obtain the equivalent sand-grain roughness from measurements of other roughness parameters, e.g., $Ra$ or $Rz$ for surface-specific correlations [37]. A broader application for stochastic surfaces is the evaluation of three-dimensional roughness elements with optical measurements of the surface. For instance, Montis et al. [23] used the Shape and Density parameter ${\mathsf{\Lambda }}_s$ of Sigal and Danberg [38] to determine $k_s$. Bons [39] adjusts the correlation based on the shape and density parameter for real turbine blade roughness

$$log\left({\displaystyle \frac{{\mathrm{ks}}_{\mathrm{SD}}}{k}}\right)=-0.43·log\left({\mathsf{\Lambda }}_S\right)+0.82$$

(4)

and suggests to use $Rz$ ($S{z}_5\mathrm{x}5$) for the roughness height k. Kurth et al. [18] identified in a systematic variation of surfaces in a DNS channel with fully turbulent flow that the skewness of the surface $Ssk$ is also required for the accurate determination of $k_s$ of real surfaces (see Equation (5)).

$${\displaystyle \frac{{\mathrm{ks}}_{\mathrm{SDS}}}{k}}={ϵ}_{\mathrm{ks},\mathrm{SD}}·{10}^{-0.43·{log}_{10}\left({\mathsf{\Lambda }}_S\right)+0.82}$$

(5)

Similar findings regarding the skewness of a surface were already shown by Flack et al. [40]. However, they did not build on real surface structures. For completeness, we refer to further approaches [13], choosing a deterministic approach to evaluate a surface and transform it into a $k_s$. They also propose an alternative simulation approach of the discrete element method for predicting the roughness effect.

The focus of the present paper is rather on the prediction of the roughness effect by the eddy-viscosity-based RANS models. In the paper by Montis et al. [23], the surface roughness on the blade was applied through a sandblasting process. Thakkar et al. [17] also investigated such a structure in a DNS study with fully turbulent flow to determine the influence of different types of surface roughness on the flow. Based on the given surface structure, it is possible to estimate a $k_s$ for the three-dimensional surface structure. In this study, three different approaches to calculating $k_s$ are taken into account.

Table 2. Roughness parameters of the investigated roughness based on different correlations. The $k_s$ value used in the present study is shown in bold font.

	Montis et al. [23]	Thakkar et al. [41]	Equation (5)
$k_s$	187 µm	127 µm	101 µm
${\mathsf{\Lambda }}_s$	11.2	99	99
k	96.6 µm	147 µm	147 µm

summarizes $k_s$ obtained using the correlation of Montis et al. [23], the skewness-based $k_s$ correlation by Kurth et al. [18], and $k_s=0.87S{z}_{5\times 5}$ based on Thakkar et al. [41].

It is worth mentioning that the approach of Thakkar et al. [41] is specified for roughness of the Nikuradse type, whereas the skewness-based approach of Kurth et al. [18] is made for a wider range of roughness structures. Surprisingly, the skewness scaling for the surface under consideration results in ${ϵ}_{\mathrm{ks},\mathrm{SD}}=0.73$, which is not far off the scaling proposed by Thakkar et al. [41]. Considering the approach of Kurth et al. [18], the roughness value in terms of wall units is $k_s^{+}\approx 20\dots 25$ in the laminar boundary layer on the suction side.

4. Results

The LES results provide the data that allow a detailed comparison of the flow field between RANS and LES. This is especially useful for assessing the validity of the RANS model components. If the integral quantities from some measurements were taken as the only basis for assessment, it would be difficult to determine which components of a model are functioning properly and which are under- or overestimating the flow features. In this section, the uncertainty estimates of the LES represent the spatial error across the blade span. Probe data from several locations on the wake and blade surface demonstrate that estimated statistical errors are lower than the spatial errors.

Figure 7 shows contours of the local total pressure loss of the RANS and LES for the smooth blade. There is a fundamental agreement between RANS and LES. Downstream of the trailing edge, the LES predicts a thicker wake compared to RANS, probably due to a different development of the turbulent mixing in the wake. This is indeed typical of RANS models, which mostly lack anisotropy and accurate modeling of turbulent mixing. Both simulation methods show a loss intensification on the suction side of the wake, which is also the case for the rough surface. It is clear that the roughness increases the loss in the wake.

In order to investigate the flow in the front region of the profile in detail, flow variables and model variables are scrutinized along wall normal lines at point A ($0.1l_{\mathrm{ax}}$), B ($0.15l_{\mathrm{ax}}$), and C ($0.2l_{\mathrm{ax}}$) along the blade’s suction surface (see Figure 7). The profiles of wall-tangent velocity, normalized with the boundary-layer-edge velocity, $u_{\Vert }/Ue$, at different positions provide information about the accuracy of the flow modeling within the RANS simulation. The velocity profiles at positions A, B, and C, shown in Figure 8, illustrate a good agreement between LES and RANS. Even close to the leading edge (wall normal A), the LES velocity profile shape over the rough surface has more turbulent character than that over the smooth surface (see also Figure 9). This shows that the presence of roughness increases the production of turbulence right from the leading edge. The velocity profile by the RANS simulation over a rough surface still agrees with the LES at positions A and B. At position C, however, a slightly more turbulent-like velocity profile is seen over the rough surface in the RANS simulation, compared to the LES. A possible explanation is that the roughness is physically present in the LES and is therefore subject to local fluctuations in comparison to the rough RANS simulations. No significant difference between the profiles by the roughness model variants is recognizable, both of which indicate an over-prediction of the roughness effect.

For a better visualization of the boundary layer development, the shape factor and the skin-friction coefficient are analyzed along the blade’s suction side. On the right side of Figure 10 the coefficient of friction $C_{\mathrm{f}}$ is shown for the different surface types and RANS models. A good agreement between RANS and LES can be observed for both the smooth surface and partly for the rough surface. The transition (abrupt increase of $C_{\mathrm{f}}$) in the RANS simulation for a smooth surface can be located at a position of $x/l_{\mathrm{ax}}=0.75$. In contrast, the transition of the boundary layer with a rough surface is already initiated at a position of $x/l_{\mathrm{ax}}=0.3$ (rough, recalibrated model). The roughness model by Dassler [14] predicts the location of the transition to be at $x/l_{\mathrm{ax}}=0.45$ (see $C_{\mathrm{f}}$).

The skin friction coefficient reveals a clear distinction between the roughness models. In the initial 20% of the blade’s suction side, the recalibrated roughness model aligns closely with the LES results. Downstream of this region, the friction coefficient decreases, reaching a minimum at approximately $x/l_{\mathrm{ax}}=0.35$ for the LES over a rough surface. In contrast to the smooth surface, the friction coefficient appears to increase slightly up to $x/l_{\mathrm{ax}}=0.5$, indicating a transition to a turbulent boundary layer. This is followed by a slight stabilization of the boundary-layer flow up to $x/l_{\mathrm{ax}}=0.6$, driven by the acceleration of the flow. The boundary layer becomes fully turbulent at $x/l_{\mathrm{ax}}=0.7$. Both RANS transition models have already transitioned by this point and therefore fail to capture the stabilization observed between $0.5<x/l_{\mathrm{ax}}<0.7$ in the LES.

The behavior of the shape factor on the left side of Figure 10 reinforces the previous observations. Although there is no clear indication of a transition for the LES curves, the reduced shape factor already indicates a higher momentum loss for the rough surface. This is the result of a change in the development of the displacement and momentum thickness during the transition process. Near the trailing edge of the blade suction side ($x/l_{\mathrm{ax}}>0.8$), the boundary layer flow is turbulent, with a significantly smaller shape factor for the LES than for the RANS. Moreover, the stabilization of the boundary layer at $x/l_{\mathrm{ax}}=0.6$ is visible due to the increase of the shape factor for the rough surface in the LES.

For a deeper insight into the behavior of the roughness transition model, the model quantity used in transition model by Langtry and Menter [9], which detects the transition onset, is analyzed. Based on an idea of Van Driest [43], the $\gamma \text{-}{\mathrm{Re}}_{\theta }$ transition model correlates the vorticity Reynolds number ${\mathrm{Re}}_V$ with the transition-onset location using flow variables in the free stream. This is possible through empirical correlations, which are included in the transport variables ${\widetilde{\mathrm{Re}}}_{\theta t}$, and introduce the influence of pressure gradients as well as of the turbulence intensity. This approach can be described as a physics-based one, even though transition models are taken only as triggers for turbulence models. The increase of the intermittency $\gamma$ due to acting quantities on the boundary layer is affected by the ratio of the vorticity Reynolds number ${\mathrm{Re}}_V$ to a critical Reynolds number ${\mathrm{Re}}_{{\theta }_c}$ determined from the transport variable ${\widetilde{\mathrm{Re}}}_{\theta t}$. This ratio is expressed as $F_{\mathrm{onset},1}$

$$F_{\mathrm{onset}},1={\displaystyle \frac{{\mathrm{Re}}_V}{2.193{\mathrm{Re}}_{\theta c}}}$$

(6)

with

$${\mathrm{Re}}_V={\displaystyle \frac{\rho y^{2}S}{\mu }},{\mathrm{Re}}_{\theta c}=f\left({\tilde{R}e}_{\theta t}\right).$$

(7)

Figure 11 illustrates schematically the development of $F_{\mathrm{onset},1}$ within the boundary layer on a flat plate. The maximum of $F_{\mathrm{onset},1}$ is at approximately 50% boundary-layer thickness. With growing disturbance (i.e., smaller ${\mathrm{Re}}_{{\theta }_c}$) of the laminar boundary-layer flow, $F_{\mathrm{onset},1}$ increases. In the roughness model extension of Dassler [14], the transport variable ${\widetilde{\mathrm{Re}}}_{\theta t}$ is decreased within the boundary layer depending on the rough-wall boundary condition. This causes an increase in $F_{\mathrm{onset},1}$, which is proportional to an increase of the intermittency. Assuming that the surface roughness has no influence on the free stream, the value of $F_{\mathrm{onset},1}$ can be estimated from the LES results using the critical Reynolds number ${\mathrm{Re}}_{{\theta }_c}$ at the boundary-layer edge of the RANS results:

$$F_{\mathrm{onset}},1,\mathrm{LES}={\displaystyle \frac{{\mathrm{Re}}_V}{2.193{\mathrm{Re}}_{\theta c,\delta ,\mathrm{RANS}}}}.$$

(8)

The development of the maximum of $F_{\mathrm{onset},1}$ within the boundary layer is visualized along the blade suction side in Figure 12. While $F_{\mathrm{onset},1,\mathrm{LES}}$ remains almost constant across the three positions for the smooth surface, it increases drastically for the rough surfaces. The recalibrated roughness model can reproduce this local behavior very well. In contrast, only a small roughness effect on the parameter $F_{\mathrm{onset},1}$ can be detected by the Dassler’s roughness model [14]. An increase of $F_{\mathrm{onset},1}$ for Dassler’s roughness model [14] is observed at a further downstream location. This correlates with the development of the skin-friction coefficient and the shape factor along the blade’s suction side. In Figure 10, the skin-friction coefficient for the RANS model of Dassler [14] is comparable to that of the smooth surface up to $x/l_{\mathrm{ax}}=0.3$. Downstream of this location, the curves for the RANS model and the smooth surface begin to diverge.

The current investigation has revealed that the RANS simulation for a smooth surface agrees well with the LES simulation. This can also be observed in the integral total pressure loss values in

Table 3. Normalized integral total pressure loss $\overline{\zeta }/{\overline{\zeta }}_{\mathrm{ref}}$ . Relative deviation of RANS from LES in brackets. Uncertainty corresponds to a 95% confidence interval.

	Experiment	LES	RANS Recalibrated Model	RANS Model Dassler [14]
smooth	$0.79\pm 0.04$	$0.91\pm 0.05$	0.9 (1.1%)
rough	$1.53\pm 0.04$	$1.4\pm 0.04$	1.44 (2.8%)	1.34 (−4.3%)

, which deviates by 1.1% for the smooth blade. Moreover, a good approximation for the profile pressure distribution of an experimental test case could be achieved. Nevertheless, the LES, with respect to the experiment, seems to overestimate the pressure loss by 15% for the smooth blade, whereas the pressure losses for the rough blade are underestimated by 8.5%. As discussed before, this can be due to the uncertainties in the experiment’s conditions. The recalibration of the roughness model within the transition model leads to a more accurate prediction of the integral pressure loss than the model of Dassler [14] when compared to the LES results. The deviation is within the 95% confidence interval of the LES results. To conclude, the correct representation of the local boundary-layer state can improve the integral loss prediction accuracy compared to the original model by Dassler. Nevertheless, the investigation also demonstrates that there is still potential for further refinement in transition modeling, particularly in optimizing the stabilization processes of the boundary layer prior to its fully turbulent state.

Several probe measurements in the LES flow field are also investigated (Figure 13). It should be noted that the pressure fluctuations inside a turbulent flow are a combination of the hydrodynamic, entropy, and acoustic waves, having various wave speeds and amplitudes. Since acoustic waves often have small amplitudes, discerning them is challenging. In fact, a proper visualization of the acoustic waves would require an advanced approach such as the Doak’s momentum potential theorem [44], use of which is left to another study. Herein, only the power spectral density of the pressure fluctuations is investigated.

Figure 14 demonstrates the spectral analysis of the LES pressure fluctuations in the three measurement locations shown in Figure 13. The spectral analysis consists of windows multiplied by the Hann function, whereby the bulk of the audible range is covered. Peaks with different frequencies can be observed between the three probes, probably because of the stretching of the eddies through the channel. Probe #3 shows a smaller range of fluctuation frequencies, possibly due to the less mature turbulence in this upstream position. The deviation in the spectrum of the rough case from the smooth case is the most evident on Probe #2, located at the exit of the passage, where higher peaks in the rough case can be observed. The higher peak at around 6500 Hz is particularly interesting, which could be of an acoustic origin because the trailing-edge-bluntness-vortex-shedding (TEB-VS) noise pertains to higher frequencies. For example, according to the semi-empirical formula by Brooks et al. [45], which was devised for single airfoil sections under laminar free-stream inflows, the peak frequency based on the trailing-edge-thickness of the present turbine blade would be estimated to be approximately $10$ KHz. In fact, TEB-VS noise must be the relevant source in the present setup because the trailing-edge bluntness is larger than the boundary-layer displacement thickness.

5. Conclusions

The present study addresses the fidelity of RANS transition models in predicting the effects of surface roughness on flow losses. RANS models generally fall short in predicting flow losses caused by surface roughness. In fact, the effect of roughness on transition to turbulence must also be taken into account before improving the roughness correlation for the turbulent flow. The LES setup offers further insights into the RANS deficits and serves as a reference for improving RANS models with respect to surface roughness.

To accurately predict the performance losses associated with surface deterioration, we propose and validate a recalibration to the roughness model. A comparison with the high-fidelity results of the LES shows that the proposed recalibrated roughness transition model reproduces the local roughness effect. In fact, the overall transition behavior observed within the LES results could not be significantly improved by the prediction with the recalibrated model compared to the RANS roughness model of Dassler [14]. In the LES with the isotropic rough surface, the flow acceleration causes the boundary layer to slightly stabilize before a fully turbulent boundary layer is reached. Both roughness models are not able to reproduce this behavior. Further investigation of this observed phenomenon is therefore recommended. Moreover, the proposed recalibration has to be validated with further test cases to demonstrate the generality of the improvement.

An initial examination of the pressure fluctuations in the passage suggests that the roughness may lead to increased trailing-edge-bluntness noise. However, this needs to be confirmed using more insightful methods.

A follow-up study will examine the influence of anisotropic roughness structures on the boundary-layer transition. Last but not least, the immense data extracted from the high-fidelity simulation will be carefully utilized to quantify the impact of surface roughness on aero-acoustics. This may lead to the development of quieter engine designs and enable a reassessment of MRO workflows concerning roughness-related noise.