Modelling Natural Transition onHydrofoils for Application in

Underwater GlidersSébastien Lemaire†,∗, Artur K. Lidtke∗,

Guilherme Vaz†, and Stephen R. Turnock∗

†MARIN, 6708 PM Wageningen, NL∗Fluid-Structure Interactions Group, University of

Southampton, SO16 7QF, UK

1 Introduction

Underwater gliders are a class of autonomous un-derwater vehicles (AUV) used for long-endurancemissions. They employ a buoyancy engine in orderto induce vertical motion through the water columnand by adopting an appropriate trim angle they pro-pel themselves forward using hydrofoils. The ve-locities these vessels reach are typically less than0.5 ms−1. Efficiency of their hydrofoils is of coursea key factor determining the overall system perfor-mance. Therefore, better understanding the natureof transition occurring on these foils is of signifi-cant importance for the design of next generationunderwater gliders.

Standard turbulence models are known to pre-dict transition onset too early in terms of Reynoldsnumber, mainly because they were first developedwith applications to fully turbulent flows in mind(Rosetti et al., 2016). For this reason a range ofmodels designed to predict transition have been in-troduced in the literature (Van Ingen, 2008).

The aim of this paper is to assess the usefulnessof the Local Correlation Transition Model (LCTM,also known as γ−Reθ, Langtry and Menter, 2009),implemented in the finite volume solver Re-FRESCO (Rosetti et al., 2016), for application toengineering problems involving laminar and tran-sitional Reynolds number regimes. The study willbe focusing on modelling the flow around 3D in-finite foils and underwater glider swept hydrofoilsto analyse transition to turbulence as well as thenature of the separation bubble. Development of abetter understanding of these phenomena will helpto achieve a more efficient design in the future.

2 Methodology

2.1 ReFRESCOReFRESCO (www.refresco.org) is a com-munity based open-usage CFD code for the Mar-itime World. It solves multiphase incompressibleviscous flows using the Navier-Stokes equations,

complemented with turbulence models, cavitationmodels and volume-fraction transport equations fordifferent phases (Vaz et al., 2009).

ReFRESCO is currently being developed, ver-ified and its several applications validated atMARIN (in the Netherlands) in collaborationwith many Universities including University ofSouthampton (Hawkes et al., 2015).

2.2 Local Correlation Transition ModelThis transition model was developed byLangtry and Menter, 2009 and has been vali-dated multiple times since its full disclosure(Langel et al., 2016), also in ReFRESCO inRosetti et al., 2016.

It involves solving the transport equations forturbulent kinetic energy, k, and specific dissi-pation rate, ω, similar to those presented byMenter et al., 2003. The model also solves two ad-ditional equations for intermittency, γ (Eq 1), andReynolds momentum thickness, Reθ,

∂(ργ)∂t+

∂

∂x j(ρU jγ) =

Pγ − Eγ +∂

∂x j

[(µt

σ f+ µ)

∂γ

∂x j

],

(1)

Pγ = Flengthca1ρS [γFonset]0.5(1 − ce1γ). (2)

The transport equation for Reθ converts non lo-cal correlations into local variables. In equation 2,Fonset and Flength control respectively the transitionlocation and its length; these variables are com-puted from empirical correlations based on Reθ.The intermittency is then computed to yield thelocal turbulence state of the flow. The key featurebrought by the LCTM is its local aspect.

Because this model adds two transport equationsto the set convergence can be more challenging(Rosetti et al., 2016). Moreover, since the formula-tion is local the sensitiveness to external parame-ters is increased. Furthermore, because the empir-ical correlations were determined experimentally,its limits and suitability of application have to bestudied with care.

2.3 Simulation set upSteady and unsteady RANS simulations arefirst performed on the SD7003 foil section,shown in Fig. 1, using the k − ω SST model(Menter et al., 2003) with and without the LCTM.This infinite straight foil is set at an angle of inci-dence of 4◦ and sees a chord-based Reynolds num-ber of 60,000. The resultant flow features are com-pared against PIV measurements by Ol et al., 2005

and Burgmann and Schröder, 2008. Moreover, datafrom ILES simulations by Visbal et al., 2009 andUranga et al., 2011 are also referred to. Inlet turbu-lence intensity of 0.1% and eddy viscosity ratio of50 are used to match the experimental conditions.

Then, steady flow past a swept wing based on theSD8020 foil (Fig. 1) similar to those used on un-derwater gliders is simulated at a number of anglesof attack up to 4 degrees and at a fixed Reynoldsnumber of 60,000. The undertaken analysis entailscomparing computed forces to 2D data and analysisof the effect of sweep and tip vortex on transition.

Fig. 1: Foil sections used for validation purposes(Selig, 1995)

2.4 Numerical set upIn all simulations discretisation is performed withthe QUICK scheme for the momentum equationsand a first order upwind scheme for turbulence andtransition equations, both diffusion and discretisa-tion in time uses a 2nd order scheme.

Convergence is guaranteed by infinity norms ofresiduals below 10−7 for steady simulations and be-low 5·10−4 for unsteady ones. For unsteady simula-tions the Courant number is kept around 1, time av-eraging is performed on at least 20 chord based di-mensionless times. Because it is a requirement fortransition models all generated grid satisfy y+ ≤ 1.

2.5 Grid set upFor the SD7003 study a large 3D C-type domainwas generated (“L”). In this structured grid (Fig-ure not shown), the inflow and the outflow are at 15chords from the foil, and the top and bottom bound-aries are separated by 32c. To simulate an infinitefoil symmetric boundary conditions are set on bothspan-normal planes.

A grid convergence study has been conductedfor this case using five 2D grids with cell countsfrom 32,000 to 500,000. Eventually, the grid with130,000 was considered to be the best choicefor balancing accuracy and simulation cost as itshowed to differ from the finest grid only by0.082% in terms of lift and 0.82% for drag.

A span size of 0.3 chord was chosen because 3Deffects for this test case have been reported to besmaller than 0.2c (Burgmann and Schröder, 2008).

Six span resolutions with z+ from 360 to 1800were tested but because no evident differences werefound between these simulations a span-wise res-olution of 15 cells (leading to a z+ of 1200) wasused to obtain the presented results. The resultant3D mesh has 1,826,000 cells in total.

A second domain with a blockage ra-tio of 6.5%, the same as in the tests byBurgmann and Schröder, 2008, was prepared(“S”). This is shown in Fig. 2. The span size of thefoil was chosen to be 0.3c, the inflow and outfloware respectively at 3c and 6c from the foil, the topand bottom boundaries are separated by 1.25c andsimulated as slip walls; the outflow is treated asa pressure boundary condition. The same numberof points along the foil was used as for the “L”domain, leading to 64,000 cells per a 2D plan, and15 cells were used in the span-wise direction (z+

of 1200). The total cell count in 3D was 895,000elements.

For an underwater glider application a symmet-ric foil is needed, thus the second study is basedon the SD8020 foil (Fig. 1) from Selig, 1995. Thegeometry has been modelled with a round trailingedge to be more representative of a real foil. A 2DO-grid with a radius of 14 chords and 32,000 cellsis used (mesh not shown).

Fig. 2: Small domain (S) generated for the airfoilSD7003 following Burgmann and Schröder, 2008experimental set up

Fig. 3: Computational domain for the swept wing

The swept wing is based on the SD8020 foil witha round trailing edge, sweep angle of 35◦, taper ra-tio of 0.9, and a span of 1.6c. The generated domain(Fig. 3), is 27c long, 7.7c high and 6c wide. A 5Mcells structured hex mesh is used, with 125 cells

placed across the span and 170 along the chord ofthe foil. The wing is attached to one side of the do-main simulated by a symmetry boundary condition,each of the remaining sides is modelled by a slipwall and the outflow is a pressure boundary condi-tion.

3 Results and Discussion

3.1 Infinite straight foil

(a) Friction coefficient

(b) Pressure coefficient

Fig. 4: Surface coefficients for simulations of theSD7003

According to Table 1, the present model showssimilar trends to the measurements in terms of pre-dicting the location of separation, which may alsobe seen in Fig. 5, although the experimental datashows a large amount of scatter and differencesbetween various facilities. Onset of transition es-timates from the present simulations, made usingnormalised Reynolds Stress of 0.001, analogouslyto how the analysis was made in the experimen-tal studies, fall within the range described by theexperimental studies. Reattachment is predicted atx/c ≈ 0.7, which is slightly further downstream ofthe measured location.

Furthermore, the pure k − ω SST simulationstrigger transition earlier than the LCTM, aroundxt/c ≈ 0.3. Consequently, they do not predict thepresence of a separation bubble (Fig. 5a). It is alsoworth noting that the high blockage ratio of thesmall domain (“S”) has a significant impact on thepredicted force coefficients, leading to higher liftand drag (Fig. 4b). It does not, however, appear

(a) pure k − ω SST

(b) LCTM

(c) Experiment TU-BS, Ol et al., 2005

Fig. 5: Velocity field around the foil SD7003,transition is shown by the red line (normalisedReynolds stress of 0.001)

to affect the nature of the separation bubble anddownstream vortices.

Fig. 6: Instantaneous surface of Q criterion aroundthe SD7003 foil (in red), µt/µ ≥ 1 displayed in theforeground to visualise the onset of transition forthe LCTM simulation

Moreover, steady and unsteady simulationsyielded similar results because turbulence is notsolved directly and only modelled, and thereforethe flow does not experience much unsteadiness.This is one of the fundamental and well understoodlimitations of the RANS approach. Experimentaldata by Burgmann and Schröder, 2008 show a lotof small flow features and unsteadiness, whereasthe LCTM simulations predict only large driftingvortices with no 3D effects (Fig. 6). Moreover,these large vortices drift downstream faster and arecreated with a higher frequency compared to whatwas reported in the experiments, as seen Table 1.

Table 1: Integral quantities of SD7003 compared with experiments from Burgmann and Schröder, 2008and Ol et al., 2005, XFOIL and ILES simulations (Visbal et al., 2009 and Uranga et al., 2011)

Type Grid CL CD Sep Trans Reattach vortex freq vortex drift vel(xs/c) (xt/c) (xr/c) ( fv · c/uin f ) (uv/uin f )

steady pure k − ω SST L 0.561 0.0194 no 0.304 nosteady pure k − ω SST S 0.723 0.0214 no 0.290 nosteady LCTM L 0.5807 0.0239 0.2085 0.529 0.7299unsteady LCTM L 0.5851 0.0239 0.2075 0.523 0.7307 5.2 0.675steady LCTM S 0.7357 0.0272 0.2121 0.503 0.6971unsteady LCTM S 0.7401 0.0267 0.2132 0.496 0.6967 5.5 0.698Visbal (ILES) 0.23 0.55 0.65Uranga (ILES) 0.6122 0.0241 0.21 0.53 0.67XFOIL 0.625 0.019 0.21 0.57 0.59Burgmann 0.390 0.470 0.515 7.8 ≈0.55IAR 0.33 0.57 0.63TU-BS 0.30 0.53 0.62AFRL 0.18 0.47 0.58

The drift velocity has been computed using crosscorrelation of the velocity field in two mesh pointson the vortex path.

Finally, ILES results by Visbal et al., 2009 andUranga et al., 2011 (Table 1) agree quite well withthe present LCTM simulations in terms of separa-tion bubble characteristics and force coefficients.

(a) pure k − ω SST

(b) LCTM

Fig. 7: Wall shear stress on the swept wing, streamwise velocity and µt/µ = 1 in red (turbulence onset)at an angle of attack of 4◦.

Fig. 8: Iso-surface of the Q criterion at the tip of theswept wing (in red) and µt/µ ≥ 1 slices (showingturbulent flow location) from the LCTM simulationat an angle of attack of 4◦.

3.2 Underwater glider swept wingFig. 7a shows wall shear stress on the wing as com-puted using the pure k − ω SST model at an an-gle of attack of 4◦. Similarly to the SD7003 pre-dictions, this simulation did not produce a separa-tion bubble and the flow transitioned to turbulent atxt/c ≈ 0.35, as estimated using the eddy viscosity.

Fig. 7b presents the same data but computed us-ing the LCTM. One may note that the root of thefoil has been predicted to be subject to fully lam-inar flow and no separation bubble has been com-puted. At mid-span, however, the flow may be seento separate at approximately 30% of chord, reat-taches at x/c ≈ 0.75 and becomes turbulent atx/c ≈ 0.45, similarly to the 2D sections investi-gated before. This span-wise variation of the flowregime may also be seen in skin friction and pres-sure distributions sampled on the foil surface andshown in Fig. 9.

Finally, the tip of the wing has been predicted

(a) Skin friction coefficient

(b) Pressure coefficient

Fig. 9: Surface coefficients at various span loca-tions from the LCTM simulation of the swept wingat an angle of attack of 4◦.

to generate a substantial tip vortex (Fig. 8). Thisalso has been observed to interact with the laminarseparation, leading to a complex flow pattern on thesuction side close to the tip, as seen in Fig. 7.

It is important to understand the exact outcomeof three-dimensional effects on the performance ofthe wing. For this reason, the force coefficients ofthe swept foil are compared against 2D data ob-tained for the SD8020 using the both the LCTMand pure k−ω SST models in Fig. 10. It is noted that2D LCTM and XFOIL simulations show a delayedincrease of the lift compared to the experimentaldata. On the other hand, the pure k − ω SST sim-ulations produce almost a linear curve for the lift,indicating that no laminar separation is predicted.For angles of attack above 3◦ the LCTM simula-tions tend to predict a linear relation between liftand the angle of attack which has also been re-ported in the experiments. All of the predictionsconsistently over estimate the lift coefficient but theLCTM appears to stand in a better agreement withthe measured performance data than XFOIL. No-ticeably, the lift coefficient of the swept wing ap-pears to follow a linear trend with an increasingangle of attack, unlike the 2D predictions for thesame foil section.

Fig. 10: Force coefficients at various angles of at-tack for LCTM and pure k − ω SST simulationsof the swept wing and two-dimensional SD8020foil. Experimental measurements and XFOIL pre-dictions for the 2D scenario are also shown

4 Concluding remarks

Overall, it has been noted that the use of the LCTMsignificantly improves the accuracy of predictionsmade for hydrofoils at Reynolds numbers below100,000 in comparison to more standard RANSmodels. The key flow features, such as the lami-nar separation, transition taking place on top of theseparation bubble, and reattachment have all beenpredicted with reasonable accuracy.

The considered SD7003 foil has been reportedto experience substantial amounts of unsteady flow.Reportedly, it is possible to capture these featuresrelatively well using advanced turbulence mod-elling techniques, such as ILES. The LCTM doesnot, however, solve the turbulent motions and thuseven when a three-dimensional simulation is per-formed the unsteadiness is not captured well.

The LCTM model appears to account for thethree-dimensional nature and the flow which hasbeen predicted to significantly affect transitionon the swept glider wing. However, because theRANS approach is used and due to the empiri-cal relationships assumed, the influence of the tipvortex and span-wise instabilities may not be verywell computed in detail. Overcoming this directlywould require the turbulence to be resolved and notmodelled, for instance through the use of meth-ods such as LES or DNS, which is not yet feasi-ble at practical Reynolds numbers. Further studiesinvolving complex geometries are therefore nec-essary to truly understand the limitations of the

present model in this context and will likely needto be supported by dedicated experimental obser-vations. Nonetheless, the acquired data has beenseen to agree with experiments well enough to al-low confident application to engineering problems.

Due to its sweep, the underwater glider wing hasbeen predicted to experience a significant amountof span-wise flow running from the root towardsthe tip. The tip vortex has also been computed toact to induce circulation around the tip and thusinduces local span-wise velocities, particularly onthe suction side. The combination of the two phe-nomena disturbs the flow and affects the transitiononset. Thus, modifying the tip shape to reduce theadverse impact of the tip vortex and controlling thespan-wise sweep and chord distributions to encour-age laminar flow are likely the two most importantdesign decisions.

Although not considered here, the design of thejoint between the wing and the glider hull may beexpected to substantially affect the amount of lam-inar flow near the root of the foil. This may in turninduce turbulent flow over a much larger area of theappendage than what was predicted here for a sim-ply supported foil. This highlights the importanceof carefully analysing local design features whendealing with natural laminar flow.

Acknowledgements

The authors would like to acknowledge the useof ReFRESCO 2.3.0 and the Iridis 4 super-computer of the University of Southampton forall of the presented simulations. The presentedwork has been funded as a part of the BRIDGESProject (http://www.bridges-h2020.eu).This Project has received funding from the Euro-pean Unions Horizon 2020 research and innovationprogramme under grant agreement No 635359.

References

Burgmann, S. and Schröder, W. (2008). Investiga-tion of the vortex induced unsteadiness of a sepa-ration bubble via time-resolved and scanning PIVmeasurements. Experiments in fluids, 45(4):675–691.

Hawkes, J. N., Turnock, S. R., Vaz, G., Cox, S. J.,and Philips, A. B. (2015). Chaotic linear equation-system solvers for unsteady CFD. VI Interna-tional Conference on Computational Methods inMarine Engineering MARINE 2015.

Langel, C. M., Chow, R., and Van Dam, C. C. P.(2016). A comparison of transition predictionmethodologies applied to high Reynolds numberexternal flows. In 54th AIAA Aerospace SciencesMeeting, page 0551.

Langtry, R. B. and Menter, F. R. (2009).Correlation-based transition modeling for un-structured parallelized computational fluid dy-namics codes. AIAA journal, 47(12):2894–2906.

Menter, F., Kuntz, M., and Langtry, R. (2003). Tenyears of industrial experience with the SST turbu-lence model. Turbulence, heat and mass transfer,4(1).

Ol, M. V., McAuliffe, B. R., Hanff, E. S., Scholz,U., and Kähler, C. (2005). Comparison of lam-inar separation bubble measurements on a lowReynolds number airfoil in three facilities. AIAApaper, 5149(1):2005.

Rosetti, G. F., Vaz, G., Hawkes, J., and Fujarra, A.(2016). Modeling transition using turbulence andtransition models for a flat plate flow: Uncertainty,verification and sensitivity. Journal of Fluids En-gineering.

Selig, M. S. (1995). Summary of low speed airfoildata, volume 1. SoarTech.

Uranga, A., Persson, P.-O., Drela, M., and Peraire,J. (2011). Implicit large eddy simulation of transi-tion to turbulence at low Reynolds numbers usinga discontinuous Galerkin method. InternationalJournal for Numerical Methods in Engineering,87(1-5):232–261.

Van Ingen, J. (2008). The eN method for transitionprediction. historical review of work at TU Delft.In 38th Fluid Dynamics Conference and Exhibit,pages 23–26.

Vaz, G., Jaouen, F., and Hoekstra, M. (2009).Free-surface viscous flow computations: Valida-tion of URANS Code FreSCo. In ASME 200928th International Conference on Ocean, Off-shore and Arctic Engineering, pages 425–437.American Society of Mechanical Engineers.

Visbal, M. R., Gordnier, R. E., and Galbraith,M. C. (2009). High-fidelity simulations of mov-ing and flexible airfoils at low Reynolds numbers.Experiments in Fluids, 46(5):903–922.