1 Introduction

Due to its complexity in physics, the turbulent wake from an underwater vehicle’s sail and sailplanes remains an area with very little information published in the public domain. However, they are extremely important to the behavior of the vehicle due to their effect on the vehicle’s hydrodynamics, acoustics, and maneuvering characteristics. For example, they will affect the flow structure around the vehicle as well as the flow onto aft appendages and into the propeller (see Fig. 1). One of the main vortices found in the sail wake is the horseshoe vortex (HSV) formed at the sail’s base when the upstream turbulent boundary layer interacts with the sail and introduces a strong adverse pressure gradient. The vortex formed is then stretched and wrapped along the sail body, forming a necklace trace that affects the inflow into the vehicle’s aft appendages [1]. Therefore, it is important to understand the wake behavior of the sail and its components, and the effect of the sail wake interaction with the aft appendages and the propeller to characterize and enhance the hydrodynamic-related performances.

Fig. 1
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Appendages of an underwater vehicle and its wake structure [2]

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Experimental and numerical methods are the two common methodologies adopted to study the turbulent wake behavior of sail-like structures [1, 3,4,5]. Both methods have demonstrated their capabilities in describing the wake flow of a foil geometry, but with the experimental studies limited to model scale. With the ongoing development of high-performance computing capabilities and numerical codes to predict fluid flows and pressure fields, there has been an increasing number of studies in recent years on computational fluid dynamics (CFD) methods in predicting the wake features of underwater vehicles. The three CFD schemes: Reynolds averaging Navier–Strokes (RANS), hybrid RANS–LES, and large Eddy simulation (LES), have been compared and evaluated for their capabilities in predicting and quantifying a turbulent wake behavior for the two established generic underwater vehicle models, i.e. the SUBOFF by DARPA [6,7,8,9] and the BB2 by DSTG/MARIN [10,11,12]. Although these studies have demonstrated that CFD-based methods can predict the general wake formation of the vehicle, it is worth noting that studies describing the wake formed by the vehicle’s forward appendages and the influence of sail-sailplane coupling toward the wake flow remain limited and forms the long-term focus of this study. Based on the literature review, the investigation of a fully appended underwater vehicle is computationally expensive and complex. Hence, a build-up approach by first investigating a simplified geometry (i.e. the Rood wing) that is analogous to an underwater vehicle sail due to its similar aspect ratio forms the focus of this study. The Rood wing was also used in this study to analyze and validate the numerical models' ability to accurately predict the flow features of a fin-like appendage given the experimental data available.

The Rood wing model (see Fig. 2) developed by Devenport et al. [13] consists of a wing body mounted on a flat ground plate. Junction flows around the Rood wing have been heavily studied both experimentally [14,15,16,17,18,19,20,21] and numerically [22,23,24,25,26,27,28,29,30,31,32]. The numerical studies indicate steady-state turbulence models are unable to predict the turbulence properties of the horseshoe vortex (HSV), given its anisotropy and highly unsteady nature. However, these numerical studies were focused on the wing’s leading edge and near-field flow that surrounds the body. To date, there is limited literature available that investigates the ability of the RANS-based turbulence models in predicting the far-field downstream junction wake which is more relevant to the focus of this study. In addition, the finding from Fleming et al. [18] further suggested that the downstream junction wake consisted of a more stable wake flow compared to the leading edge and the near-field wake, highlighting the possibility to adopt a less sophisticated numerical model such as RANS- and hybrid RANS–LES-based turbulence models in analyzing the far-field junction wake.

Fig. 2
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Wake structures of the Rood wing

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In this paper, the ability of different numerical turbulence models in capturing the wing-body junction flow wake is evaluated. The numerical turbulence models include a selection of RANS and hybrid RANS–LES turbulence models and are listed in Table 1. The selected RANS-based numerical models are categorized into three groups: Boussinesq based, two Equation k-epsilon, and Reynolds stress-based model. Gand et al. [28, 31] indicated the possibility of the Reynolds stress model to predict the highly anisotropic flow features of a junction downstream wake, while RNG KE-EN is introduced in this paper due to the nature of the models, whereby an additional equation is included in the RNG model to predict the swirling effect within the flow [33]. This provided an opportunity to investigate the high swirling flow of the horseshoe vortex (HSV) properties within the wake. SAS was chosen among all the hybrid RANS–LES models as Menter and Egorov [34] reported that it behaves identically to detached Eddy simulation models but is without an explicit influence of the grid spacing on the RANS mode of the model. The comparison between the turbulence models is to identify the best modeling practices for flows featuring separation, stagnation, and a strong three-dimensional vortex interaction with the boundary layer. The turbulence model with the closest agreement to the experimental data in predicting the anisotropic properties of the wake flow features will then be the starting point for future work on more sophisticated simulation models of a full-scale appended underwater vehicle and the sail wake influences on the aft appendages.

Table 1 List of turbulence models used in the investigation of the wing-body wake
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2 Numerical domain configuration

The Rood wing geometry consists of a 3:2 elliptical wing nose and a NACA 0020 tail. The maximum thickness of the wing, T is 71.7 mm and the chord length, C, is 4.254 T. The numerical model replicated the experimental configuration, which included matching the Reynolds number (5.19 × 105) and wind tunnel domain boundary conditions. An overview of the setup is provided in Fig. 3 for boundary conditions, Fig. 4 for Rood wing details, Fig. 5 for mesh details, and Fig. 6 for inlet features. The inlet and outlet of the domain were placed at X/T = − 23.8 and X/T = 14.3, respectively, while the sidewalls were at Z/T = − 3.5 and 3.5. A special feature was adopted in the bottom plate whereby a free slip boundary was imposed at the beginning of the model (see Fig. 6). This setup is to ensure that a boundary layer would develop on the bottom plate that was identical to the experimental conditions.

Fig. 3
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Numerical domain and boundary conditions adapted throughout the investigation

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Fig. 4
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Dimensions and coordinates direction of the Rood wing

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Fig. 5
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Mesh details of the Rood wing

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Fig. 6
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Inlet location features of the computational domain

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A comparison was made on the momentum thickness of the bottom plate to ensure that the wall boundary layer represented the experimental tripped condition. The momentum thickness of the numerical and experimental data at X/T = − 3.2 were 35.9 and 36 mm, respectively. A velocity inlet and a pressure outlet were adopted for the domain, while symmetry was used for the domain’s top and spanwise boundary conditions. The wing-body surface and the bottom plate were both specified as non-slip walls with the non-dimensional first height node, y+, of 1 for the mesh.

3 Numerical setup

3.1 Solver settings and turbulence models

Simulations of the turbulence models outlined in Table 1 were carried out using the commercial CFD software, ANSYS Fluent 19.3. This section provides detailed information on the numerical method and the RANS and hybrid RANS–LES turbulence models used.

The RANS-based turbulence models involved in this study are based on two different Reynolds stress modeling approaches, the Boussinesq approach and the Reynolds stress transport model. The turbulence model that adapted the Boussinesq approach were the re-normalization group k-epsilon model with enhanced wall treatment (RNG KE-EN), the shear stress transport k-omega model (SST), and the shear stress transport k-omega model with curvature correction (SSTCC). The baseline Reynolds stress model (BSLRSM) on the other hand adapted the Reynolds stress transport model in solving their Reynolds stresses.

The simulations using the Boussinesq approach-based RANS turbulence model used the following simulation settings: steady state with the coupled pressure velocity scheme, Green-Gauss node cell gradient spatial discretisation, PRESTO! For pressure, third-order MUSCL for momentum, and second-order upwind for turbulence kinetic energy and specific dissipation rate. The selected schemes were employed to provide a more robust and efficient steady flow investigation as per the theoretical approaches suggested by ANSYS [33]. A similar scheme was adapted to the Reynolds stress model with the exception of the turbulence kinetic energy discretisation which was replaced by the Reynolds stresses discretisation due to the nature of the turbulence model.

The hybrid RANS–LES turbulence model utilized for this study is the scaled adaptive simulation (SAS) combination with BSLRSM turbulence model as its RANS coupled. The simulations used for the hybrid model adapted the following simulation settings: transient with a time step of 0.01 s and 20 inner iteration loops, coupled pressure–velocity scheme, least-square cell gradient spatial discretisation, PRESTO! for pressure, MUSCL for momentum, turbulence kinetic energy and specific dissipation rate. The time was discretized using the bounded second-order implicit scheme.

3.2 Mesh independent study

The mesh independent study was carried out with the BSLSRM and focused on two key areas, first, the near-field (including surface mesh) of the Rood wing, and then the wake region downstream.

The ideal mesh for the near-field of the Rood wing was ascertained via a drag coefficient study defined by Eq. 1, which involved four fully structured block mesh models with different densities, i.e. 2, 4, 5.8, 8.3, 15, and 18 million cells. The drag coefficient from each mesh model is calculated using Eq. 1 and is displayed in Fig. 7. Mesh-independent results were shown to be achieved with the 8.3 million cells mesh model which has a relative error of 2.5% when compared to the finest mesh.

$${C}_{D}=\frac{{\text{F}}_{\text{D}} }{0.5\rho {V}^{2}S},$$
(1)

where \({C}_{D}\) represents the drag coefficient, \({ F}_{D}\) is the drag force, \(\rho\) the water density, V the inlet velocity and S the surface area of a Rood wing.

Fig. 7
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Rood wing’s drag coefficient grid independent study

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The mesh model with 8.3 million cells was then used for the downstream wake refinement study which involved increasing the mesh resolution at the downstream wake region and observing the location and strength of the turbulence intensities captured. The mesh model was refined to mesh densities of 15 million, 30 million, and 37 million. Figure 8 shows the downstream wake turbulence profile captured by the three different mesh models at the vertical plane located at the furthest investigation plane, i.e. Plane 13 (see Fig. 12). It is observed that turbulence properties such as the peak RMS Reynolds normal stress, u’/Uref strength of the 30 million cells mesh model is identical to the 37 million cells mesh model while providing a similar peak vortex core location.

Fig. 8
figure 8

Comparison of RMS Reynolds normal stress u’/Uref contour at Plane 13 for different mesh densities: a 15 million mesh model, b 30 million mesh model and c 37 million mesh model

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A convergence analysis of the local peak core location and u’/Uref properties with respect to mesh density is shown in Fig. 9, highlighting the turbulence properties predicted by different mesh densities in terms of the peak u’/Uref strength and locations. It is observed that at 30 million cells, the mesh exhibits a difference of 0.5% for the Z/T location and 1.5% for the Y/T location compared to the 37 million cell mesh. This, hence, suggested 30 million cells mesh model is capable to predict the Rood wing’s junction wake profile.

Fig. 9
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Mesh convergence studies for the location of peak vortex core turbulence properties and the strength with a Z/T location, b Y/T location and c Peak u’/Uref strength

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4 Evaluation of the turbulence models

To evaluate the accuracy of the schemes used, three main parameters were investigated and compared against experimental data at the three downstream planes. These were the pressure distribution along the wing’s body, the mean streamwise Velocity U/Uref and its RMS Reynolds normal stress, u’/Uref.

4.1 Mean pressure distribution on the wing body

Preliminary checks were required to verify the validity of the wake comparisons between the experimental, RANS, and hybrid RANS–LES simulations. The pressure coefficient on the wing’s body was chosen for the preliminary validation to ensure the accuracy of the physics involved. Figure 10 shows pressure measurement points along Y/T = 0.1328, 0.3984, 1.4607, and 1.7263, while Fig. 11 compares the predicted pressure coefficient (Cp) calculated using Eq. 2 against measurements by Devenport and Simpson [15].

$${C}_{p}=\frac{p }{{p}_{\infty }},$$
(2)

where \(p\) is the measured pressure at the location and \({p}_{\infty }\)= 0.5ρU2 is the freestream pressure calculated using the free stream velocity, U and water density, ρ.

Fig. 10
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Experimental measurement points on the Rood wing for pressure [15]

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Fig. 11
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Rood wing surface pressure coefficient comparison between the turbulence models and experimental data at a Y/T = 0.1328, b Y/T = 0.3984, c Y/T = 1.4607, d Y/T = 1.7263

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The comparison shows a good agreement between the numerical and experimental results for the pressure coefficient except for the measured point near the experimental artificial turbulence tripping device (X/T = 0.6). This discrepancy is within reason as the tripping device is absent in the numerical model as turbulence is modeled in the numerical simulation. The strong correlation between the numerical and experimental results provides further confidence in the accuracy of the numerical simulations. Therefore, further analysis into the capabilities of the numerical methodology investigating the wake flow field generated by the wing-body junction flow can be conducted with reasonable confidence.

4.2 Velocity U/U ref flow field

Figure 12 displays the three different measured planes, Planes 10, 11, and 13, located along the downstream wake, with distances of X/C of 1.05, 1.5, and 3, respectively. Planes 10 and 11 are both considered to be within the near-field wake, while Plane 13 is considered a far-field wake. The three planes with different distances are examined in terms of their mean Velocity U/Uref and the RMS Reynolds normal stress, u’/Uref. Velocity U/Uref is assessed to understand the accuracy of each turbulence model when predicting the inflow into the appendage, and the RMS Reynolds normal stress, u/Uref’ is to clarify the accuracy of each turbulence model predicting the vortex structure, mainly on the horseshoe vortex (HSV) properties.

Fig. 12
figure 12

Location of wake measurement planes

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4.2.1 Near-field wake

Figure 13 illustrates the experimental Velocity U/Uref flow field measured by Fleming et al. [16] at Plane 10, indicating that the flow field consists of three main flow features which influence the wake. The three flow features are the corner flow, which is located near the Rood wing’s boundary layer, followed by a plume of low momentum fluid between Z/T of 0.25 and 0.65 and the Horseshoe vortex (HSV) between Z/T of 0.65 and 1.2.

Fig. 13
figure 13

Experimental measured Velocity U/Uref contour plot [16] for the Rood wing at Plane 10, with the incoming velocity as the reference velocity

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Figure 14 illustrates the comparison between the Plane 10 numerical prediction and the experimentally measured Velocity U/Uref contour. The discrepancies between the various turbulence models and experimental measurements are categorized within the corner flow and the HSV areas. Although all turbulence models capture the trend, the location and size of the corner flow differ from the experimental data. The turbulence models predicted the presence of corner flow located between Z/T of 0 and 0.15 and Y/T of 0 and 0.2, while the experimental measured data observed it between Y/T of 0.1 and 0.2 and Z/T of 0 and 0.18. The difference in the location of corner flow introduces a difference in the low momentum fluid plume size and subsequently influences the location of the HSV hump. The peak of the HSV, located between Z/T of 0.65 and 1.2 predicted by the turbulence models differs from the experimentally measured data. A comparison between the predicted and the measured result shown in Fig. 14 indicates that the turbulence models tend to overpredict the HSV height and underpredict the width of the HSV. It is observed that BSLRSM and SAS models predicted the location of the HSV with the highest accuracy, followed by the RNG KE-EN model. An offset in the HSV peak location is observed when the RSM and Boussinesq based turbulence models are applied.

Fig. 14
figure 14

Comparison of Velocity U ‘s contour between different turbulence models (black line) and experiment [16], (Purple Line) at Plane 10, non-dimensionalized with reference incoming velocity, Uref

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Figure 15 illustrates the difference in the Velocity U/Uref contour between the numerical models and the experimental data [18] at Plane 11. Similar discrepancies to the previously measured plane were observed. The turbulence models predicted the presence of corner flow located at Z/T between 0 and 0.15 and Y/T between 0 and 0.2, while the experimental measured data observed the corner flow located between Y/T of 0.1 and 0.2 and Z/T of 0 and 0.18. A comparison of the HSV height located between Z/T of 0.65 and 1.2 indicates all turbulence models tend to overpredict the HSV height and underpredict the width of the HSV. Despite the qualitative discrepancy, it is observed that the RNG KE-EN model and SAS model do provide close prediction on the HSV hump size and location, suggesting that the models do capture the influence of the HSV transverse motion on the boundary layer fluid. The accuracy reduces with the BSLRSM model and the Boussinesq-based models, SST and SSTCC.

Fig. 15
figure 15

Comparison of Velocity U ‘s contour between different turbulence models (Black Line) and experiment [16], (Purple Line) at Plane 11, non-dimensionalized with reference incoming velocity, Uref

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Comparison plots from Fig. 15 do suggest the presence of a large discrepancy between the numerical and experiment when the measurement is taken between the region of Z/T of 0–0.4 and Y/T of 0–0.2. The large discrepancy will be mentioned in the aft section, Sect. 4.3.2.

4.2.2 Far-field wake

Figure 16 shows the experimental measured Velocity U/Uref at the junction wake of Plane 13 from [18]. It is observed that the HSV shifted further from the centerline (Z/T = 0), suggesting that the HSV movement has diverged from the centerline further downstream. In addition, it is also observed that there is a significant thickening on the HSV hump size compared to the previous near-field planes. This finding indicates that the transverse HSV motion further lifts the fluid upwards to the free stream.

Fig. 16
figure 16

Experimentally measured mean Velocity U/ Uref contour plot for Rood wing at Plane 13 [16]

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Figure 17 compares the predicted and measured streamwise velocity contour plots. The most obvious difference between the different turbulence model predictions and the experimental measurement is the size and location of the hump. Among the turbulence models, the SAS model shows the most accurate prediction to the experimental plot. This is followed by the BSLRSM model. The SST and SSTCC models underpredict the HSV size and strength, while RNG KE-EN predicted an offset in the hump location. There are a few reasons that contributed to the difference between the numerical and experimental and are discussed in Sect. 4.3.2.

Fig. 17
figure 17

Comparison of Velocity U ‘s contour between different turbulence models (Black Line) and experiment, [16] (Purple Line) at Plane 13, non-dimensionalized with reference incoming velocity, Uref

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4.3 RMS Reynolds normal stress, u’/U ref at different downstream planes

4.3.1 Near-field wake

Figure 18 outlines the experimental results that characterize the RMS Reynolds normal stress, u’/Uref contour for the near field (Plane 10). Four key parameters of interest that will be used for the discussion within this section are outlined within the figure: the HSV peak u’/Uref, the wing wake, the bottom wake height, and the corner flow.

Fig. 18
figure 18

Experimental measured RMS Reynolds normal stress, u’/Uref contour plot for the Rood wing at Plane 10 [16]

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Figures 19 and 20 compare the measured and predicted u’/Uref contour for the two near-field planes, i.e. Planes 10 and 11. The results show the experiment measured a thicker wing wake than the numerical prediction (see Z/T = 0–0.2). There is also a difference between the measured and predicted hump height with the prediction showing a larger hump height than the experiment. With regard to the HSV peak u’/Uref, the numerical prediction is a lower value and is located with an offset in comparison to the measured. A more detailed quantification of this will be discussed in Sect. 4.3.3.

Fig. 19
figure 19

Comparison of the RMS Reynolds normal stress, u’ contour between different turbulence models (black line) and experiment, [16] (purple line) at Plane 10, non-dimensionalized with reference incoming velocity, Uref

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Fig. 20
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Comparison of the RMS Reynolds normal stress, u’ contour between different turbulence models (black line) and experiment, [16] (purple line) at Plane 11, non-dimensionalized with reference incoming velocity, Uref

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For both Plane 10 (Fig. 19) and Plane 11 (Fig. 20), the RNG KE-EN model and the SAS model show good accuracy in terms of the HSV peak u’/Uref and the wake features, followed by the BSLRSM, SST and SSTCC models. This implies that the turbulence models predicted the flow field characteristics, although with an offset against the experiment, with detailed quantification of the errors in Sect. 4.3.3. Similar behavior is observed when Gand et al. [31] compared their RANS predicted results with their experiment conducted with a lower Reynolds number, citing an error of 10–25% observed when RANS turbulence models predict the flow.

Another discrepancy observed from mean Velocity U/Uref and its u’/Uref contour plot is at the corner flow region. Corner flow is observed in both experimental and numerical modeling; however, there is an offset of 10–15% at Plane 10 and 10–21% at Plane 11. A large discrepancy is observed when the wing’s boundary layer intersected the inner layer of the bottom plate’s boundary layer. Gand et al. [31] mentioned that the intersected region consists of high anisotropic turbulence components which reached the limitations of the current RANS-based turbulence model’s capabilities. SAS model with its hybrid RANS–LES features also faces similar limitations when predicting the corner flow physics. The discrepancy decreases when the distance increases from the bottom plate as the turbulence components return to an isotropic state and re-allowing the numerical simulation to capture the physics of the flow features.

4.3.2 Far-field wake

Figure 21 describes the characteristics of the experiment-measured u’/Uref contour plot at a far-field wake region. Two main observations from the figure are the absence of the wing wake and the increase of bottom plate wake height due to the upward shifting of the peak u’/Uref location when compared to the near-field wake plot (Fig. 18). The two observations suggest that the HSV dominates the downstream far-field wake.

Fig. 21
figure 21

Experimentally measured RMS Reynolds normal stress, u’/Uref contour plot for the Rood wing at Plane 13 [16]

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Figure 22 compares the u’/Uref contours of the experimental and numerical results at Plane 13. When comparing the hump height between the numerical and experiment approach, all turbulence models tend to overpredict the HSV height, with the SAS showing the closest prediction against the experiment, followed by the BSLRSM and RNG KE-EN models. As for the HSV peak properties, all turbulence models predicted a similar peak value, each located at an offset from the experimental result. Detailed quantification of the HSV peak u’/Uref properties is discussed in Sect. 4.3.3.

Fig. 22
figure 22

Comparison of the RMS Reynolds normal stress, u’ contour between different turbulence models (black line) and experiment [16] (purple line) at Plane 13, non-dimensionalized with reference incoming velocity, Uref

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Another distinctive discrepancy observed between predicted and measured results is the flow within the region of Z/T of 1.5–2.5 and Y/T of 0.05 and 0.3. The main reason contributing to the error between the measured and predicted results is the resolution of the experiment dataset. Plane 13's experiment data set [16] shows a decrease in u’/Uref magnitude as the distance increases from the centerline (Z/T = 0). The reduction in experiment measurement intensities contributes to the lower resolution at the region and warrants further investigation to understand the underlying discrepancy that exists when the numerical approach is compared with the experiment-measured result.

Fig. 23
figure 23

Experiment measured [16] and numerical predicted Y/T coordinate of peak u’/Uref at different downstream planes with error bars indicating experimental uncertainty. Experimental uncertainty is ± 0.001

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4.3.3 Quantification of the turbulence models discrepancy

Two parameters, the location, and the strength of local peak u’/Uref are compared and presented to quantify the difference between the predicted and measured behavior. Figures 23 and 24 show the predicted local peak u’/Uref’ normal stress core location between the numerical and experimental results in the Y and Z directions, respectively. Among the turbulence models investigated, RNGKE-EN performed the best in the near field, i.e. closer to the Rood wing’s body with the lowest errors being 12% at Plane 10 and 8% at Plane 11, respectively. In the far field, Plane 13, SAS has the lowest error, 17% when compared with the experiment. Despite the prediction errors, all turbulence models show a similar path movement of the vortex as with the experiment, which diverges from the center (Z/T = 0) and lifts upward from the base plate (Y/T = 0) as it moves downstream.

Fig. 24
figure 24

Experiment measured [16] and numerical predicted Z/T coordinate of peak u’/Uref normal stress at different downstream planes with error bars indicating experimental uncertainty. Experimental uncertainty is ± 0.005

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Fig. 25
figure 25

Experiment measured [16] and numerical predicted Peak u’/Uref at the different downstream planes with error bar indicating experimental uncertainty. Experimental uncertainty is ± 0.001

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Figure 25 compares the predicted peak u’/Uref normal stress strength with experimental data. The comparison indicates that all turbulence models had the greatest percentage error for the local turbulence peak at Plane 10, with the largest error of 18% predicted by the SAS model. This indicates that the current mesh resolution and setup are capable to predict the local peak turbulence intensities. When the measurement planes move further downstream, the error between the numerical predictions and the experiment reduces from a maximum error of 18 to 6.7%. This suggests that the accuracy of the investigated turbulence models in capturing the trend of HSV properties increases when the measurement plane moves further downstream.

Plane 10 possesses the highest error among all planes investigated as it is located at a region where the HSV is interacting with a corner flow, suggesting a high level of anisotropy is involved. Gand and Deck [29] in their analysis of the turbulence models performances do observe a similar behavior of the Reynolds stresses, with the RANS-based model overpredicting and the LES-based model underestimating the stresses. The finding is further supported by Ghate and Housman [44], where their LES model overestimates the stresses at the corner flow region.

The ability of the turbulence model to capture the peak u’/Uref is collectively assessed via the RMS error percentage of the three properties of the HSV core, i.e. Y/T, Z/T and u’/Uref. For Plane 10, the RNG KE-EN model demonstrates the closest agreement with the experimental data, with an RMS error of 19.7% followed by the SAS model and BSLRSM model with RMS errors of 25.5% and 26.6%, respectively. For Plane 11, the RNGKE-EN model exhibits the smallest error, with a percentage of 12.4%, followed by the SAS model and SST model with RMS errors of 16% and 17.9%, respectively. A small difference in percentage error is observed when the measurement is taken at Plane 13. The SAS and RNGKE-EN performed with 19.4% and 19.6% errors while BSLRSM followed with a 20.8% error. The small RMS error between the models indicates the local peak u’/Uref predicted is similar for the three models in the far field. Overall, the combination assessment between the RMS Reynold normal stress, u’/Uref contours with its local peak u’/Uref suggests that the RNG KE-EN provides the best prediction when near-field wake is examined, while the SAS model performed with better accuracy in far-field wake.

5 Conclusion

The fidelity of different turbulence models in predicting the downstream wake properties of a zero-angled wing’s body have been presented in this paper. The turbulence models involved in the numerical investigation were RANS-based RNG KE-EN, SST, SSTCC, and BSLRSM turbulence models and a hybrid RANS–LES, SAS.

Three main features of the flow (pressure distribution along the wing’s body, the mean streamwise velocity U and its root mean square fluctuation u’) were compared between the numerically predicted results and the experimental data provided by Fleming et al. [16]. The pressure coefficient along different Y/T predicted by the turbulence models along the wing’s body agreed with the experimental results except for the tripping region, which was expected as the trip mechanism was absent in the numerical model.

When the turbulence models computed the velocity U/Uref flow field, it was observed that the RNGKE-EN model, which includes a swirling equation, can predict the near-field wake with the highest accuracy. This was followed by the SAS and the BSLRSM turbulence model. As for the far-field wake, it was observed that the SAS model showed the highest accuracy in terms of the flow interpretation due to its nature of lower numerical dissipation in comparison to the RANS-based turbulence models. The BSLRSM provided good accuracy of streamwise flow field interpretation but tended to overpredict the flow and size of the HSV.

Through the comparison of the u’/Uref flow and properties, it was observed that the turbulence models RNGKE-EN, BSLRSM, and SAS provided a better prediction of the HSV in the downstream wake over the other models. Although a notable discrepancy against the experimental data was observed in the near field, the predictions show similar behavior to other published studies suggesting further investigation is required to understand the behavior of the highly anisotropic flow at the trailing edge corner of the wing. When the measurement moves beyond the high anisotropy region (Plane 13), it is observed that good agreement is reached between the predicted and measured values, with the SAS model showing the highest accuracy against the experiment among the investigated models followed by the BSLRSM model.

The current study has validated the SAS, RNGKE-EN, and BSLRSM models to provide a reasonable prediction of the sail’s downstream wake, with the near field by the SAS and RNGKE-EN, and the SAS and BSLRSM prediction for the far-field wake. Further work is ongoing to extend the current study to more complicated wake phenomena associated with sail configurations such as the sail and sailplanes wake interaction and the wake effect due to the deflection of the appendages. A detailed understanding of the forward appendages' wake features will aid the understanding of its effects on the downstream components of the underwater vehicle such as the aft control surfaces and propeller.