The Navier–Stokes equations are nonlinear, and their solutions are non-unique in nature. Therefore, simplifications are often made when describing or simulating turbulent flows, and here, we focus on literature on grid-generated turbulence (GGT). Over the decades, GGT has been investigated intensely, and different works looked into the empirical and analytical description of the evolution of the turbulence decay. According to , the decay of the turbulence intensity can be described by means of a power law. It has further been discussed that the inlet conditions play a role in the decay of the turbulence . The works of show experimentally that the ratio between TKE (k=12(u12‾+u22‾+u32‾) with the velocity fluctuations u1, u2, and u3 in three axis directions) and dissipation (ϵ=30νu12‾λ2, with the kinematic viscosity ν and the Taylor micro-scale λ) evolves linearly in the downstream direction. showed that grid turbulence is Saffman turbulence . They improved the decay exponent of TKE from 1.2, which Saffman gave for perfectly homogeneous, isotropic turbulence, to 1.1 for GGT. performed experiments with one grid for different grid-mesh-size-based Reynolds numbers (ReM=UMν, where M is the grid mesh size, and U the mean velocity) and found that the decay exponent of the TKE was equal to 1.18. They also showed that the decay exponent was independent of ReM. A literature review can be found in . We will use the existing framework that was very briefly summarized above as a starting point for our theoretical framework, and a detailed description of both is given in Sect. .

Simulating the inflow environment using computational fluid dynamics (CFD) has been the subject of many research works, and different turbulent formulations can be used that are summarized below from the most expensive computational effort to the least one, i.e. RANS models.

performed DNS to simulate the turbulent mixing layer for grid-generated turbulence, and they replicated typical GGT including the shear mixing layer. Continuing this work, showed that for a given mesh Reynolds number (ReM), turbulent mixing is enhanced for fractal grids compared to regular grids. simulated both a regular and different fractal grids using DNS, and they confirmed the characteristic regions of turbulence production and decay downstream of fractal grids. Continuing their work, studied inter-scale energy transfer of decaying turbulence for fractal grids. Efforts have also been made to use LES to simulate GGT. For example, developed a grid inlet technique to generate high-intensity turbulence for given length scales in LESs. With this, they successfully imitated the independence of the decay rate of the turbulence intensity (TI) from the mesh-size-based Reynolds number ReM. compared two LES models, namely the sigma model and the Smagorinsky model , to simulate grid-generated turbulence. They found that the sigma model is a good alternative to the static Smagorinsky model and comparable to the dynamic Smagorinsky model . Further, employed 3D LES to simulate GGT, and they found the same trend of absolute values of the mean TI. used the lattice Boltzmann method to simulate GGT, and they simulated the decay power law over a short distance from the grid but found it difficult to find the decay exponent.

Finally, have investigated the performance of various two-equation eddy viscosity models for predicting the decay of the turbulent kinetic energy downstream of a regular grid by means of RANS. They calculated the dissipation rate using an integral length scale equal to the grid mesh size. The simulation results were compared with experimental results, and a conclusion of their work was that eddy viscosity models were over-predicting the turbulent kinetic energy in comparison to the experiments. As an alternative to two-equation eddy viscosity models, Reynolds stress transport models (RSTMs) can also be used where transport equations for each of the Reynolds stress terms are solved. For example, performed RANS simulations using RSTMs. However, because of the closure problem, using RSTMs is quite complex as the nine non-closed components of the nine Reynolds stress transport equations have to be modelled.

Regarding RANS equations, studies focus mostly on improving the eddy viscous model, while not much effort was put on the choice of the length scale that is usually assumed to have an order of magnitude of the integral length scale. Two quantities are generally used at the inlet of RANS simulations: the turbulence intensity and the length scale. Indeed, when using the most popular two-equation eddy viscosity models, namely the k-ϵ and k-ω models , or a mixture of these two models, the k-ω shear stress transport (SST) Menter 1994 and 2003 models , the following quantities need to be provided at the inlet boundary: k, the turbulent kinetic energy, ϵ, the dissipation rate, and ω, the specific dissipation rate. k is generally given through a turbulence intensity quantity, and ϵ and ω are computed using k and a given characteristic turbulence length scale chosen individually for each flow configuration. For example, in the case of pipe flow, the turbulence length scale is empirically approximated as 0.07 times the diameter of the pipe .

We would like to emphasize that a correctly evaluated and properly defined turbulence length scale is a basic requirement to accurately capture the evolution of turbulence. Finally, in principle, if provided with an adequate closure, the RANS model should be able to capture statistical measures of turbulence and their evolution in the flow field. The present work will focus on improving the length scale choice to reproduce the correct evolution of k.

As stated above, the third challenge of the work presented here is the reproduction of aerodynamic loads on a 2D airfoil section by means of RANS simulations in particular for low Reynolds numbers. There are many experimental works that describe the evolution of the lift and drag coefficients with the variation of the angle of attack (AoA) of the airfoil, and a summary of the design and aerodynamics for wind turbine blades can for example be found in . For this work, the impact of low Reynolds numbers and the impact of turbulence on the performance of 2D blade sections are of interest. Different studies on the influence of the turbulence intensity in the inflow on the aerodynamics of a 2D blade section are summarized in . There, an NREL S826 profile is exposed to different turbulence levels up to 5.4 % for a chord-based Reynolds number of Rec=4⋅105 defined as Rec=c⋅U∞ν, where c denotes the chord length, U∞ denotes the inflow velocity, and ν the kinematic viscosity. They found that the slope of linear part of the lift curve increases with increasing turbulence intensity and that turbulence intensities of up to 1.6 % led to a reduction of the maximum lift, whereas turbulence intensities of 2.1 % and higher led to an increase in lift compared to the reference case. and studied the influence of turbulence intensities up to 16 % on the aerodynamics of a 2D NACA65(4)421 blade section. The chord-based Reynolds numbers were 1⋅105≤Rec≤7⋅105. showed that increasing the turbulence intensity shifts the stall angle towards higher angles of attack. This is attributed to a turbulent boundary layer flow that is known to be less prompt to flow separation, which also displaces the transition towards the trailing edge. found that the fluctuations of the surface pressure measurements, characterized by the standard deviation, increased in the separated flow region with increasing turbulence intensity. The average location of the separation line was not affected.

Simulating turbulent flow upstream of the airfoil often requires the use of LES, as demonstrated in the work of . In this paper, we will show that with appropriate boundary conditions at the inlet, RANS simulations can yield an accurate evolution of turbulence properties.

Finally, fewer investigations of the impact of turbulence have been performed at higher Reynolds numbers due to experimental complexity. The present digital twin model will pave the way for such studies.

As stated above, the aim of this work is the creation of a digital twin of a low-Reynolds-number wind tunnel where turbulence is generated with grids. For this, first, a theoretical framework is developed in Sect.  to demonstrate that the Taylor micro-scale (λ) is the correct length scale to be used as the turbulence length scale in RANS simulations for homogeneous, isotropic turbulent flows. Additionally, we show that the spatial and temporal decays of turbulent kinetic energy are directly dependent on the Taylor micro-scale, and we derive a relation between the Taylor micro-scale and the downstream position. In Sect. , the theoretical framework is then validated using RANS simulations and experiments behind a homogeneous grid; the experimental and numerical set-ups are detailed in Sects. and . A validation and the results are presented in Sect. . In the next step, the digital twin is used to perform aerodynamic simulations which are compared to experiments. The numerical and experimental set-ups are explained in Sects.  and , and the results are compared in Sect. . Finally, conclusions and perspectives are given in Sect. .

Researchers in the physics and mathematics community have studied the case of homogeneous isotropic turbulence in great detail. proposed to call the Reynolds number based on the Taylor micro-scale (λ) the “Reynolds number of turbulence” and also suggested that λ is representative of the eddies of large wavenumber, i.e. small eddies, before viscosity becomes relevant. The mathematical definition and the experimental methodology for calculating λ are detailed in Sect. . In the theory of homogeneous isotropic turbulence, it is necessary to assume the similarity of turbulence at all stages of decay; i.e. changes in the structure of turbulence can be described by two parameters, namely a characteristic length and a characteristic velocity . Later on, proved that the characteristic length scale of the entire energy spectrum is the Taylor micro-scale. This was confirmed by , who also proposed that the turbulent kinetic energy and the Taylor micro-scale are the appropriate scaling parameters for all scales of motion. They proposed a set of differential equations to calculate the temporal decay of turbulent kinetic energy, k(t), and dissipation rate, ϵ(t) : 1dk(t)dt=ϵ(t),2dϵ(t)dt=-αϵ(t)2k(t).

Here, α is a constant. Equations () and () formulate an initial value problem which can easily be solved to give a temporal decay law for initial conditions at time t=0, k(t)=k(0), and ϵ(t)=ϵ(0); cf. , which gives 3k(t)=k(0)1+1αϵ(0)k(0)t-α.

With the equation from , 4k(0)ϵ(0)=(λ(0))220ν, we can write Eq. () as 5k(t)=k(0)1+1α20ν(λ(0))2t-α.

Looking at Eq. (), we can clearly say that the temporal decay of the turbulent kinetic energy has a direct dependence on the initial Taylor micro-scale λ(0). This is comforted by , who empirically found a relationship between the energy decay exponent and the Taylor Reynolds number Rλ.

The dependence of the downstream evolution of k on λ is demonstrated here within the framework of homogeneous, isotropic turbulence. For the steady-state case, the transport equation for k can be written as (here, we follow Einstein's summation convention) 6Uj∂k∂xj=∂∂xj1ρujp‾+12uiuiuj‾-2νuisij‾-uiuj‾Sij-ϵ.

Uj and uj are the mean velocity and the velocity fluctuation, respectively, with i,j=1,2,3. The pressure fluctuations are denoted by p, and ν is the kinematic viscosity of the fluid. sij is the strain rate tensor for velocity fluctuations, and Sij is the strain rate tensor for the mean flow, defined as 7Sij=12∂Ui∂xj+∂Uj∂xi,8sij=12∂ui∂xj+∂uj∂xi.

The turbulent kinetic energy is defined as 9k=12u12‾+u22‾+u32‾.

It should be noted that for grid-generated turbulence 10u12‾=1.2u22‾=1.2u32‾=u2‾. (cf. ).

After substituting () in (), we get 11k=43u2‾.

If the dominant flow direction is the x1 direction, then Eq. () can be written as 12U1∂k∂x1=∂∂x11ρu1p‾+12uiuiu1‾-2νuisi1‾︸transport-uiu1‾Si1︸production-ϵ.

If we assume the turbulence to be homogeneous and decaying, then the term representing the transport of k in an inhomogeneous field due to pressure fluctuation, the turbulence itself, and viscous stresses, and the term representing the production of k can be set to zero .

Therefore, Eq. () becomes 13∂k∂x=-ϵ/U.

Here, we have dropped the indices for the sake of simplification. For homogeneous, isotropic turbulence, the dissipation rate ϵ can be related to both the longitudinal Taylor micro-scale (λ1) and the transverse Taylor micro-scale (λ2) as 14ϵ=15νu2‾λ22=30νu2‾λ12.

By using Eq. (), we can write Eq. () as 15ϵ=11.25νkλ22=22.5νkλ12.

From here onwards, we will only use the relation corresponding to λ1, and for simplicity, we will drop the subscript “1”. Hence, the relation for the dissipation rate ϵ can be written as 16ϵ=22.5νkλ2.

By substituting ϵ from Eq. () into Eq. (), we arrive at 17∂k∂x=-22.5νUkλ2.

Since the evolution of k is only a function of x, for the given case, the partial differential Eq. () becomes an ordinary differential equation, 18dkdx=-22.5νUkλ2.

Solving the differential Eq. () , 19k(x)=C1exp-22.5νU∫1λ2dx.

Equation () shows that the evolution of k in the downstream direction has a direct dependence on the Taylor micro-scale λ. Here, C1 is a constant.

To find the final solution, we need to understand the dependence of λ on x. This is performed in the following section.

From Eq. () we know that λ2∝k/ϵ. Experiments have shown that for homogeneous isotropic turbulence, k/ϵ evolves linearly in the downstream direction (e.g. ). Thus, we can write 20kϵ=kinϵin+mUx, where m/U is the slope. Now, using Eqs. () and (), we can write 21λ2=22.5νkϵ=22.5νkinϵin+mUx.

kin and ϵin are the values of the turbulent kinetic energy and the dissipation rate at the starting point x=0 from where the TKE's decay is calculated. Substituting Eq. () into Eq. (), performing integration, and subjecting it to the boundary condition at the starting point, k(x=0)=kin, gives 22k(x)=kin1+mϵinkinxU-1/m.

Using relation (Eq. ), we can rewrite Eq. () as 23k(x)=kin1+m22.5νλin2xU-1/m, where λ(x=0)=λin. It should be noted that for Eq. (), the starting point, i.e. x=0, and, thus, the origin of the coordinate system, lies at the point of measurement which is used to give the value of kin and λin. Looking at Eq. (), we can say that the evolution of k in the downstream direction can be defined for a homogeneous, isotropic flow, provided we have the values of kin and λin at one point.

From the framework of the k-ω series of models for RANS simulations, one can also derive the following evolution equation for k for decaying, homogeneous, isotropic turbulence , 24k(x)=kin1+ωinβxU-β∗/β, where ω is the so-called turbulence frequency that is related to ϵ and k by 25ω=ϵβ∗k, and β and β∗ are constants with values of 0.0828 and 0.09, respectively . The value of ω at the inlet boundary is given by ωin. By substituting Eq. () into Eq. (), we have 26k(x)=kin1+ββ∗ϵinkinxU-β∗/β.

Using the relation in Eq. (), we can rewrite Eq. () as follows: 27k(x)=kin1+ββ∗22.5νλin2xU-β∗/β.

Equations () and () have a similar form. Even if Eq. () comes from the k-ω model, believing in its proven applicability, we can assume that 28m=ββ∗.

Therefore, the value of m can be taken as 0.92. It is important to notice that further investigations are required to find the correct value of the parameter m theoretically. The authors wish to emphasize to the readers that, unlike the equations commonly encountered in prior literature, such as those referenced in , , , or , Eq. () does not have any fitting parameter, and it is neither an empirical equation nor does it have any virtual origin. The only assumption taken while deriving Eq. () is statistical stationarity of fully developed GGT, which is believed to start from x/M≈20 downstream of a grid . Upstream of x/M≈20, one may expect some changes in the form of Eqs. (), (), and (), which are used to derive Eq. (). However, between x/M≈10 and x/M≈20, the turbulent flow can be considered approximately developed . Therefore, for all practical purposes, Eq. () is valid downstream of x/M≈10. For the cases where the assumption of Taylor's hypothesis is valid, i.e. time t equals x/U, the spatial evolution equation of k, Eq. (), becomes equivalent to the temporal decay equation of k (cf. Eq. ). Hence, measurements of k and λ at a given position in a regular-grid-generated turbulent flow will enable us to obtain the evolution of k when solving the RANS equations. This is essential to reproduce realistic regular-grid-generated turbulent inflow conditions at the inlet boundary of RANS simulations. The experimental and numerical set-ups used to validate the spatial evolution in Eq. () are presented in the following section.

The experiments were performed in the aerodynamic closed-loop low-Reynolds-number wind tunnel facility at the LHEEA (Laboratoire de recherche en Hydrodynamique, Énergétique et Environnment Atmosphérique) laboratory of CNRS and Centrale Nantes (see Fig. ). This wind tunnel has a cross section of 0.5 m × 0.5 m and a test section length of 2.3 m with a maximum inflow velocity of 40 ms-1 and a turbulence intensity of less than 0.3 %.

To induce a turbulent inflow, a regular wooden grid with square bars was used. The cross section of the bars used for the frame is 11 mm × 10.5 mm and of the ones used for creating the mesh d = 6 mm × 6 mm. The blockage (b) of the grid is 16 %, and the grid mesh size M is 70 mm × 70 mm.

To measure the downstream evolution of turbulence properties of the inflow, a 1D hot-wire probe of type 55P11 with a wire length of 1.25 mm from Dantec Dynamics was used. It was operated using a DISA55M01 unit. The hot-wire was calibrated in the velocity range 0.5 ms-1 ≤U≤ 40 ms-1 applying the temperature correction suggested by . During the measurements, the mean velocity was approximately 25 ms-1. The data were recorded at a sampling frequency of 25 kHz for 10 s for calibration and 20 s for the measurements. A hardware low-pass filter with a cut-off frequency of 10 kHz was used. The downstream evolution of k was measured at the centre line of the wind tunnel at nine downstream positions that are indicated in Fig. .

The simulations are performed using the finite-volume-based ISIS-CFD incompressible unsteady Reynolds-averaged Navier–Stokes (URANS) solver. This solver, developed by CNRS and Centrale Nantes, also available as a part of the FINE™/Marine computing suite worldwide distributed by Cadence Design Systems, uses an incompressible URANS method. The solver is based on a finite volume method to build the spatial discretization of the transport equations. The unstructured discretization is face-based, which means that cells with an arbitrary number of arbitrarily shaped faces are accepted. A second-order backward difference scheme is used to discretize time. All flow variables are stored at the geometric centre of arbitrarily shaped cells. Volume and surface integrals are evaluated with second-order accurate approximations. As the method is face-based, numerical fluxes are reconstructed on the mesh faces by linear extrapolation of the integrand from the neighbouring cell centres. A centred scheme is used for the diffusion terms, whereas for the convective fluxes, a blended scheme with 80 % central and 20 % upwind is used. In the case of turbulent flows, additional transport equations for the variables in the turbulence model are added. In the following, the transport equations for URANS simulations are presented:

The momentum conservation equation is 29ρ∂U∂t+ρ(U⋅∇U)=ρg-∇p‾+μ△U-ρ∇⋅R‾‾.

The continuity equation for the mean component is 30∇⋅U=0.

The continuity equation for fluctuations is 31∇⋅u=0.

The Reynolds stress tensor R‾‾ present in Eq. () is defined as 32R‾‾=μtu⊗u‾, where U and u are the mean and fluctuating component of the velocity, and μt is the turbulent viscosity. There are many ways to calculate the value of μt, and in this paper, a two-equation eddy viscosity method, namely k-ω SST Menter (2003), has been used to do so. In the following, a description of this model is given.

The transport equation for the turbulent kinetic energy k is 33∂(ρk)∂t+∂(ρUik)∂xi=Pk̃-β∗ρkω+∂∂xi(μ+σkμt)∂k∂xi.

The transport equation for the turbulence frequency ω is 34∂(ρω)∂t+∂(ρu‾iω)∂xi=αρS2-βρω2+∂∂xi(μ+σωμt)∂ω∂xi+2(1-F1)ρσw21ω∂k∂xi∂ω∂xi.

F1 is the blending function. F1 goes to zero for the flow away from the surface. Hence the k-ϵ model is applied there, and it goes to one near the wall where the k-ω model is applied. The constant α is computed by α=α1F1+α2(1-F1). For this model, the values of constant are β∗=0.09, α1=5/9, β1=3/40, σk1=0.85, σω1=0.5, α2=0.44, β=0.0828, σk2=1, and σω2=0.856. Using these constants, both transport equations are solved and the turbulent viscosity is found by Eq. (). 35μt=ρa1kmax(a1ω,SF2), where S is the measure of the strain rate and F2 is the second blending function defined by 36F2=tanh⁡max2kβ∗ωy,500νy2ω2.

Using the above-presented set of equations, 2D simulations are performed for the domain, mimicking the wind tunnel shown in Fig. , with a domain length of 3.12 m, a width of 0.5 m (same as the width of the wind tunnel test section), and 8000 cells. Simulations were performed with a higher number of cells (10000 and 12000) as well, but no changes were observed. The top and the bottom walls in the simulations were put to the no-slip condition.

It should be noted that the values put as the boundary condition at the inlet boundary are those obtained at the first point of the measurement, 758 mm downstream of the wind tunnel inlet, which refers to x/M=11 in Figs.  and .

For homogeneous, isotropic turbulence, both the integral length (L) and λ can be easily estimated using the one-dimensional energy spectrum . L can be obtained by taking the limit of the energy spectrum E(f) in the frequency domain for f→0, 37L=lim⁡f→0E(f)⋅U4σ2.

Here, U is the mean streamwise velocity, u denotes the fluctuations, and σ is the standard deviation of the velocity time series U(t). It should be noticed here that only the frequency range where E(f)≈const just outside of the inertial sub-range is used to determine L.

λ is defined as follows: 38λ=σ2∂u∂x21/2, where 〈(∂u∂x)2〉 can be determined from the spectrum in the wavenumber (κ) domain, derived through hot-wire measurements using Taylor's hypothesis, 39〈∂u∂x2〉=∫κminκmaxκ2E(κ)dκ, where κmin and κmax are the wavenumber boundaries of the whole energy spectrum. At the downstream position x/M=11, which represents the first point of measurement in the wind tunnel, we calculated L/M, λ/M, and the Taylor Reynolds number (Reλ) using hot-wire measurements and Eqs. () and (). An exemplary spectrum is shown in Fig. . The calculated values were L/M=0.39, λ/M=0.034, and Reλ=180, respectively.

Figure  shows the downstream evolution of k obtained theoretically with Eq. (), experimentally using hot-wire measurements (Sect. ) and from simulation using the k-ω SST Menter 2003 model (see Sect. ) with both L and λ as boundary conditions at the inlet. First, we see that the theoretical equation is validated against experiments. The log–log equivalent of the curve is provided in the inset. Next, simulation results obtained using both the Taylor micro-scale and the integral length as the boundary condition at the inlet are compared with Eq. (). Here, we can clearly see that the simulation result obtained for the case where the Taylor micro-scale is used as the boundary condition matches very well with the theory. The TKE decay exponent was determined to be 1.087 in Eq. () and simulation, while in the experimental data, it was observed to be 1.09. These values are in close proximity to the Saffman decay exponent of 1.1 for grid-generated turbulence, as reported by . In contrast, when the integral length is used as a boundary condition, the simulation results do not match the theory or the experimental results. The justification for this observation comes directly from Eq. () where the derived 1D spatial evolution equation shows a direct dependence of the turbulent kinetic energy on the Taylor micro-scale.

To verify the generality of Eq. (), apart from the validation performed against the TKE decay data obtained in the LHEEA wind tunnel, we also compared it with the results from hot-wire experiments conducted independently in the wind tunnel at the University of Oldenburg and data given in . The Oldenburg experiments were performed using a passive regular grid with M=115 mm for 33 downstream positions spanning from x/M∼8 to x/M∼170 for two inflow speeds: 5 ms-1 and 10 ms-1. Figure  shows the log–log plot of the comparison of the TKE decay obtained experimentally with that from Eq. (). It can clearly be seen that the evolution of TKE given by Eq. () matches very well with the experimental data. Note that the deviations are over-accentuated when visualized in the log–log plot. Readers interested in knowing the details of the experiments performed at the University of Oldenburg may refer to Appendix .

It is also interesting to see how different k-ω models calculate the downstream evolution of k by using λ as the boundary condition. This is shown in Fig. . Here, we can see that the results from the k-ω SST Menter 2003, k-ω SST Menter 1994, and k-ω baseline (BSL) Menter overlap with only a little deviation for the oldest k-ω series of the turbulence models, the k-ω Wilcox model.

This emphasizes the very important role of the choice of the turbulent length scale at the simulation domain inlet as compared to the choice of the turbulent model. The following section showcases an instance of the digital twin's functionality, where an airfoil is subjected to testing in both the physical wind tunnel and the digital twin, with a subsequent comparison of the resulting loads.

Figure a, b, and d present the 3D view, front view, and side view of the numerical domain (including mesh), respectively, for an angle of attack of the 2D blade section of α = 0∘. The mesh consists of 3 million cells, and Fig. c displays a close-up of the mesh around the airfoil. In this study, we only simulated the transverse half of the wind tunnel, as shown in Fig. a. The shaded area in the figure was not simulated; instead, we applied a mirror boundary condition. The chord length of the airfoil is 0.125 m, which is the same as that used in the experiments. The airfoil is positioned at a downstream distance of 1.035 m from the domain's inlet, as the simulation domain starts at the position of the first measurement point, where the inlet conditions are obtained and from where the turbulence decays. Therefore, the downstream position of the airfoil in the simulation is 0.758 m less than that in the experiments, resulting in a downstream position of 1.035 m.

We have applied a Dirichlet boundary condition at the inlet, and the values are given in Table . These values correspond to the values obtained at x/M=11 from the hot-wire measurement (see Sect. ). For pressure, we applied the Neumann boundary condition at the inlet, dpdn=0, where n is the normal vector to the inlet. These same values have been used as the initial conditions as well. In addition, we also use the integral length scale (L=25 mm) to investigate the impact of using the “wrong” length scale at the simulation domain inlet. At the outlet, the velocity is found using Rhie–Chow interpolation. We applied the Dirichlet boundary condition at the outlet for pressure p=po, where po=0 by default. For TKE and turbulence frequency, we applied the Neumann boundary condition as dkdn=0 and dωdn=0, respectively. We have applied a no-slip boundary condition on the airfoil and imposed wall functions 42∂U∂y=τsκρcμ1/4kwyw, on the top wall (TW), bottom wall (BW), and side wall (SW) to avoid explicitly simulating the boundary layer. Here, U is the velocity, kw is the TKE at the cell centre of the first cell from the wall, yw is the perpendicular distance of the cell centre of the first cell from the wall, τs is the wall shear stress, κ=0.41, and cμ=0.09. The y+ values for airfoil, BW, TW, and SW are given in the Table .

We conducted unsteady 3D RANS simulations using the k-ω SST Menter 2003 model following the same procedure and using the same equations that we presented in Sect. . We performed these simulations using a standard AVLSMART numerical scheme. To improve the computational efficiency and to accurately capture the flow downstream of the airfoil, we utilized an in-house adaptive grid refinement (AGR) methodology . The total number of cells at the end of the simulation was approximately 16 million (see Fig. e).

To obtain well-converged results for each angle of attack (AoA), we conducted 3D simulations using 400 cores on the IDRIS Jeay Supercomputer. This process took almost 100 computing hours for each AoA.

In the following sections, we present a comparison between the force coefficients and pressure coefficients obtained from the experiments and simulations.