Numerical Study of High-Lift Hydrofoil Near Free Surface at Moderate Froude Number *

Home , Lift (force)

Available online at https://link.springer.com/journal/42241 http://www.jhydrodynamics.com Journal of Hydrodynamics, 2020, 32(1): 44-53 https://doi.org/10.1007/s42241-019-0095-0

Numerical study of high-lift hydrofoil near free surface at moderate Froude number *

Tao Xing1, Konstantin I. Matveev2, Miles P. Wheeler2 1. Department of Mechanical Engineering, The University of Idaho, Moscow, Idaho, USA 2. School of Mechanical and Materials Engineering, Washington State University, Pullman, Washington, USA

(Received February 21, 2019, Revised September 20, 2019, Accepted September 21, 2019, Published online December 11, 2019) ©China Ship Scientific Research Center 2020

Abstract: Shallow hydrofoils are known to produce lower lift in normal operating conditions in comparison with deep hydrofoils. However, the maximum lift capability of shallow hydrofoils at moderate speeds, which is important for transitional regimes of hydrofoil boats, is studied insufficiently. In this work, two-dimensional flow around a high-lift hydrofoil at a moderate Froude number is numerically simulated in a broad range attack angles up to the stall occurrence in both single-phase fluid and in the vicinity of free surface. It is found that nearly the same maximum lift coefficient can be produced by the shallow foil in the modeled condition as by the deep foil, but much higher attack angle is required near the free surface, which also results in larger drag. Additionally, it is shown that higher Reynolds numbers lead to higher lift coefficients, especially at large attack angles.

Key words: Hydrofoil, numerical simulation, free surface effect, stall

Introduction  susceptible to air ventilation, when suction of the Hydrofoils are some of the most hydrodyna- atmospheric air to the lifting foil surface reduces the mically efficient bodies that can generate high lift at generated lift[2]. These hydrofoils are also beneficial low drag when moving through the water. Fast boats for delaying cavitation to higher speeds. At lower equipped hydrofoils can elevate their hulls so that the speed, however, thin low-camber profiles cannot hull’s water drag becomes small, thus allowing these produce enough lift, which leads to higher operational boats to achieve high speeds. Hydrofoil ships were speeds that require much greater propulsion power. developed and widely used in the second half of the Thicker, more cambered airfoil-type sections, last century for passenger transportation[1]. Their such as shown in Fig. 1(b), can potentially produce typical speeds and capacities are around 30 kn-40 kn higher lift and increase lift-drag ratio at lower speeds and 100-250 passengers, respectively. However, in and thus reduce operational speeds (or Froude case of heavy-payload requirements these boats are numbers) at which hydrofoils can still be beneficial. not so efficient, and other types of marine craft, such This can help foil-assisted boats be more economical as large multi-hulls, proved to be more suitable for at higher payloads and/or lower speeds. In principle, combined passenger/car transportation, especially the air ventilation problem can be mitigated with when the intended relative speeds (Froude numbers) application of fences or other techniques[3-4]. Although are lower. Hydrofoil profiles used in common such hydrofoils would be prone to cavitation at high surface-piercing foil systems are usually thin and speeds, in case of relatively low operational speeds the often have flat lower surfaces (Fig. 1(a)). Although cavitation is not an issue. Another hydrodynamic such shapes are not the most efficient for single-phase consideration important for hydrofoils operating near flows, their characteristics at high speeds in water are a free surface is the effect of the shallow submergence adequate, and these low-profile sections are less on the occurrence of stall and specifically on the maximum lift coefficient of the foil, which is a vital property for elevating the boat hull out of water and * Biography: Tao Xing (1973-), Male, Ph. D., P. E., overcoming the boat hump drag. Associate Professor It is well known that lift of hydrofoils generally Corresponding author: Tao Xing, E-mail: [email protected] recedes with approaching the free surface at moderate [5] attack angles and typical speeds . However, accor-

ding to two-dimensional linearized potential-flow concepts of planing hulls and SWATH equipped with predictions[6], the hydrofoil lift may actually increase foils were studied in Refs. [11-13], whereas the auto- at shallow submergence if Froude number is matic grid refinement for surface-piercing foils was sufficiently low. The finite submergence effects on the investigated in Ref. [10]. Harwood et al.[14] reported a foil hydrodynamics at high attack angles beyond the simulation of a ventilated hydrofoil vertically crossing range of the constant lift slope are not well established. the water surface. Duncan’s wave measurements[15] In the past experiments at relatively low Reynolds behind a shallow hydrofoil are also used for CFD numbers[7], it was noticed that the stall occurrence was validation[16]. delayed to higher attack angles and the maximum lift coefficient could be even higher in the presence of the free surface and at low Froude numbers. Other studies 1. Computational methods focused on flow structures and/or hydrodynamics of The commercial CFD software, Ansys Fluent cavitating flows around hydrofoil without the version 19.1[17] is used for all simulations in this study. presence of a free surface[8-9]. Fluent is a finite volume solver which provides a suite of numerical schemes and turbulence modeling options. In this work, single-phase simulations employed the pressure-based solver option, which is the typical predictor-corrector method with solution of pressure via a pressure Poisson equation to enforce mass conservation. Pressure-velocity coupling is performed using the SIMPLE scheme. The convective terms in the momentum equation are discretized with the third-order MUSL scheme. While the present simulations involved a transient period, the focus of this work is only on the steady-state regimes, when hydrodynamic characteristics or their time-average values do not evolve. Unsteady terms are discretized using a second order implicit scheme. A sufficient number of iterations per time step ensured that the minimum residuals were lower than 105 for the continuity and three momentum equations for all cases. The simulations have been conducted using ANSYS Academic Research CFD package with high Fig. 1 (a) typical thin hydrofoil profile often used on boats with performance computing on a Dell Precision T7810 surface-piercing foil systems. (b) high-lift airfoil section that has 40 cores and 64 GB RAM. E603. Coordinates are normalized by the foil chord c

1.1 Governing equations The focus of the present work is to numerically This study first evaluated two turbulence models, explore variations of hydrodynamic characteristics of a thick, cambered foil profile at a relatively low the realizable k - model that is widely used for free-surface flows and the Menter’s SST k - Froude number in a broad range of attack angles up to [18] the stall occurrence. Also, two characteristic Reynolds turbulence model . Since the Menter’s SST k - numbers are investigated here to determine the scale turbulence model showed better agreement with effect that can be important when extrapolating available experimental data, as discussed below, it experimental data from model tests to large-scale was used for all subsequent simulations in this study. systems or designing boats of very different scales. The continuity and momentum equations, and The foil profile E603 selected for the present study transport equations of the turbulent kinetic energy k (Fig. 1(b)) has high lift characteristics and soft-stall and the specific dissipation rate  are presented properties in a single-phase fluid[10]. The two-dimen- below for two-phase flow but they are also applicable sional computational fluid dynamics (CFD) simula- to single-phase flows with minor adaptions. tions conducted in this work employ ANSYS The continuum equation for mixture is FLUENT software and use RANS-based approach. With general increase of available computational  ()+( v )=0 (1) power, CFD is now routinely applied in marine mmm t hydrodynamics. Simulations of hydrofoils near the free surface are sometimes used as validation cases where v is the mass-averaged velocity and for assessing specific designs. For example, novel m

and ,2 =1.168 . The blending function F1 is n kkkv given by v =  k =1 (2) m m 4 F11=tanh( ) (10) and m is the mixture density  kk500 4   =min max , ,  (11) n 1 0.09yy2+2 Dy  ,2  mkk=  (3) k =1  11k   D+10=max 2 ,10 (12) where k is the volume fraction of phase k , n is     ,2 xxjj  the number of phases.   The equation for the conservation of momentum + can be written as where y is the distance to the next surface, D is the positive portion of the cross-diffusion term  v T mm+(vv )=+[( p   v +  v )]+ t mmm m m m k 1  = (13) t   1 SF2  max * ,  m gF+ (4)  1 

where p is the gage pressure, F is a body force, where S is the strain rate magnitude,  * damps g is the gravitational acceleration and m is the the turbulent viscosity causing a low-Reynolds viscosity of the mixture number correction:

n uu  =  (5) uuiij j mkk S  ++ (14) k =1 x xxx ji ji

Governing equations for the Menter’s SST k - * turbulence model: **0 +/Retk R =   (15)

1+Retk / R    t ()+(mmmmkkv )=   +  k + t  k * k  Re = , R =6,  =0.024 (16) t  k 0

GYSkk + k (6) 2 F22=tanh( ) (17)

   ()+( v )=   +t   + mmmm  k 500  t     2 =max 2 ,  (18) 0.09 yy2  

GYDS  ++ (7) In these equations, the term Gk and G repre- where Sk and S are user-defined source terms. sent the production of turbulence kinetic energy k and generation of  , respectively. 1  k = (8) 2 GSkt=  (19) FF1,1/+(1)/ kk 1 ,2

 1  2 GG ==k   S (20)  = (9)  t FF1,11/+(1)/ ,2

* Ykk =  (21) Model constants are  k,1 =1.176, ,1 =2,  k,2 =1

Table 1 Foil geometries and flow conditions Case Geometry Re Fr AOA/ Submerged depth

1 E603 1106 N/A (no gravity) 0, 4, 8, 12, 16, 20 N/A (single-phase fluid)

2 E603 5106 N/A (no gravity) 0, 4, 8, 12, 16, 20 N/A (single-phase fluid) 3 NACA12 1.66105 0.567 5 1.286 chord length 6 4 E603 110 1.596 0, 4, 8, 12, 16, 20, 24, 28 1 chord length 5 E603 5106 1.596 0, 4, 8, 12, 16, 20, 24, 28 1 chord length

The variable input parameters include the foil submer- 2 gence and Reynolds and Froude numbers: Y = i (22)

Uc ii=+(1)FF1,1 1 i ,2 (23) Re = (29) 

1 k  DF=2(1 1,2 ) (24) U  x j x j Fr = (30) gc where  * =0.09,  = 0.075 ,  = 0.0828 . i,1 i,2 The tracking of the interface between the phases is accomplished by solving the continuity equation for the volume fraction of one of the phases. For qth phase, this equation is

1  n ()+(qq  qqqv )=(mm pq  qp )(25)  t  q p=1

where m qp is the mass transfer from phase q to

 phase p , mpq is the mass transfer from phase p to phase q . The volume fraction equation will not be solved for the primary phase, the primary-phase volume fraction is computed based on the following

n q =1 (26) q=1

The two most important hydrodynamic characteristics of hydrofoil performance, the lift and drag coefficients, are used for presenting results in the next sections. These metrics are defined as follows: Fig. 2 Mesh design for cases 1, 2 L C = (27) L 0.5Uc2 1.2 Geometry, grid, flow parameter, and simulation design

The two-dimensional foil profiles and simulation D parameters are summarized in Table 1. Cases 1, 3 are CD = 2 (28) 0.5Uc used for validation of the CFD models, including single-phase flows over a hydrofoil E603 at six angles where L and D are the lift and drag forces, U is of attack (AOA) and a two-phase flow for a the incident flow velocity and c is the foil chord. submerged NACA 12 at AOA 5[15]. Cases 2, 4 and 5

employed the validated CFD models to investigate the The factor of safety method[19-20] is used to estimate effect of AOA, submergence and Re on the lift and the grid uncertainties and results are summarized in drag coefficients of the high-lift hydrofoil E603. Table 2. For cases 1, 2, a two-dimensional O-type grid is created to model the flow (Fig. 2). Compared with a Table 2 Solution verification for lift coefficient for case 1 at C-type grid, the O-type grid has advantages of AOA = 16 and case 5 at AOA = 20 Mesh number specifying boundary conditions at any AOAs. The Case R P US(% ) fine grid has a total of 770 788 cells. For solution 1 2 3 G 1 verification, additional two coarser grids are created 1 1.3095 1.3096 1.3237 0.0071 7.14 0.01 systematically using a constant grid refinement ratio 5 1.4500 1.4380 1.4140 0.5000 1.00 1.33

2 in all three spatial directions. For case 3, mixed The fine grid (mesh 1) has 770 788, 106 122 structured and unstructured grid is used. Additional points for cases 1, 5, respectively. Meshes 2, 3 grid points near the foil and free surface are added to represent the medium and coarse grids, respectively. resolve the high velocity and pressure gradients. Cases The solutions on the fine, medium, and coarse grids 4, 5 have similar grid topology as case 3 except that are S , S and S , respectively. Solution changes an O-type mesh similar to that in cases 1, 2 is used to 1 2 3 resolve the E603 geometry, which preserves high  for medium-fine and coarse-medium solutions and quality resolution of the boundary layer around sub- the convergence ratio R are defined as follows: merged hydrofoil for all the AOA simulated (Fig. 3).   = SS ,  = SS , R = 21 (31) 21 2 1 32 3 2  32

When 0

the estimated order of accuracy pRE , error  RE , and grid uncertainty US(% ) G 1

ln(32 / 21 ) pRE = (32) ln(r )

 21  RE = (33) r pRE 1

When solutions are in the asymptotic range, then pRE is equal to the theoretical order of accuracy of the numerical scheme p used. In this study, p =2 as th th a 2nd order accuracy scheme is used; however, in many Fig. 3 Mesh design for cases 4, 5 with AOA = 8 . Y =0 cor- circumstances, especially for industrial applications, responds to the undisturbed free surface solutions are far from the asymptotic range such that p is greater or smaller than p [21]. The ratio of RE th p to p is used as the distance metric 2. Results and discussion RE th

2.1 Solution verification p P = RE (34) Solution verification is a process for assessing pth simulation numerical errors and associated uncertainties. In this study, the discretization error due to and the grid uncertainty is estimated by limited number of grid points is the main source of numerical errors. The solution verification is perfor- UP= (2.45 0.85 )  , 0

1 (35b)

As shown in Table 2, monotonic convergence is the k -  SST model were evaluated, but they achieved for both cases with a low grid uncertainty of showed nearly identical results for this case. Therefore,

0.01%S1 and 1.33%S1 for Cases 1, 5, respectively. only the result from the k -  SST model is This suggests that the current fine grid resolution is presented in Fig. 5. Again, CFD agrees well with adequate, and this fine grid is used for all simulations. experiment, but slightly underpredicts the wave amplitudes. 2.2 Validation for E603 foil in single-phase fluid (case 1) 2.4 Effects of Free surface and Reynolds number Figure 4 shows the comparison of the lift Four curves in Fig. 6 represent numerical results coefficients obtained with the realizable k -  model obtained for the lift and drag coefficients of E603 and the k -  SST model against experimental hydrofoil under two Reynolds numbers for no-gravity, data[22]. Overall, both turbulence models agree single-phase situation (cases 1, 2) and for finite- reasonably with test data at low attack angles. As one gravity, finite-submergence condition (cases 4, 5). In can see in Fig. 4, the k -  SST model outperforms order to make proper comparison, a correction due to buoyancy has been added to the no-gravity, single- k -  model, since it is closer to the test data and also phase lift coefficients. better predicts the stalling regime that was failed to be properly captured using the k -  model. Thus, the k -  SST model was adopted for other simulations in this study.

Fig. 4 Lift coefficient of E603 foil at various AOA (case 1)

Fig. 5 (Color online) Comparison of wave profile between CFD Fig. 6 Lift and drag coefficient of E603 hydrofoil and experiment for case 3 For the single-phase situations, increase of Re 2.3 Validation of CFD model for submerged NACA 12 lead to higher magnitudes of CL , especially at large (case 2) AOAs near the stall regime (Fig. 6(a)). For the foil The CFD model is also validated against under a free surface, the effect of Re on CL is experimental data for wave profiles generated by a similar but slightly less pronounced. This observation submerged NACA0012 hydrofoil profile reported in suggests that model-scale boat hulls having hydrofoils Ref. [15]. In this study, the leading edge of foil was at high AOA (likely in the transition from hull-borne located at x =0, whereas the chord length of the foil to foil-borne state) would demonstrate inferior lift was taken as 0.203 m. The depth of submergence is performance in comparison to full-scale boats. Hence, 1.286c , which is defined as the vertical distance if a small-scale hull is able to overcome transitional between the free surface and the midpoint of the chord hump drag under given propulsion level, then line of the foil. Both the realizable k -  model and additional margin will be built up in the design and

will likely ensure successful transition to the foil- borne mode of the full-size hydrofoil boat. The effect of finite foil submergence at small AOA is a reduction of the lift slope (Fig. 6(a)), which is well-known phenomenon. However, the maximum

CL is not decreased, it just shifts to higher AOA, thus delaying stall to higher attack angles. These findings are in agreement with early experimental measurements conducted at lower Reynolds numbers[7]. The reduction of the lift slope when approaching the maximum lift condition is more gradual for the foil near a free surface. Hence, if one needs to produce high lift on a hydrofoil (e.g., while the boat transits from hull-borne to foil-borne state), much higher AOA than usual can be employed, since stall is delayed. The effect of Reynolds number on the drag coefficient of a two-dimensional foil is found to be relatively small in the present simulations (Fig. 6(b)). For single-phase situations, higher Re leads to generally lower C . In cases with the free surface, D drag at different Reynolds numbers is nearly the same at low AOA, while higher Re results in somewhat bigger CD near the stall region. It should be kept in mind that real finite-span hydrofoils would also have substantial lift-induced drag due to trailing vortices. With regard to the submergence effect on drag, hydrofoils near the free surface naturally have higher

C due to generation of surface waves. D For single-phase flow cases 1, 2, the gravity force is excluded. Therefore, pressure coefficient is used for analysis

pp C = ref (36) p 0.5U 2

where p is the gage pressure on the foil surface,

pref is the reference gage pressure at inlet/far- field. For two-phase flow cases 4, 5, the gravity force must be included in the simulations due to the presence of a free surface. In that case, piezometric pressure coefficient is used

pz+  C p = 2 (37) 0.5U where  is the specific weight of water, z is the vertical depth of water that is zero at the free surface and points vertically upward. Some details of the velocity field, pressure Fig. 7 (Color online) Normalized velocity magnitude with stream- distribution and free surface elevations are shown for lines at maximum CL selected conditions in Figs. 7-10. The velocity magnitudes normalized by the incident flow speed and states, and the corresponding piezometric pressure coefficients are presented in Fig. 8. For single-phase streamlines are given in Fig. 7 at the maximum CL

pronounced recirculation zones are already formed above the downstream portions of the foil upper surfaces (Figs. 7(a), 7(b) and 8(a), 8(b)). However, the flow is still attached in the front portion of the foil. A low-pressure zone just exists just behind the leading edge on the top side and a high-pressure zone is present near the leading edge on the bottom side (Fig. 8). For hydrofoils with shallow submergence (Figs.

7(c), 7(d) and 8(c), 8(d)), the maximum CL takes place at the attack angle of 24. Although significant recirculation zones are also visible, they are smaller in size that for deeply submerged hydrofoils in the pre-stall states. The free surface allows more freedom for the water flow to deform, thus relaxing pressure gradients and delaying massive flow separation. The main effect of different Reynolds numbers is in the reduction of recirculation zones at higher Reynolds numbers (cases 2, 5), which is caused by more turbulent and more stable boundary layers. The wave patterns at the air-water interfaces for

the maximum C conditions are shown in Fig. 9. L Large water surface deformations are noticeable, such as increased elevation above the front part of the foil, water depression behind the foil, and formation of a breaking wave at the end of the wave hollow. Knowing characteristics of the flow behind hydrofoils is important for design of hydrofoils in tandem arrangements or for planing hulls assisted with fore hydrofoils; and CFD tools can help optimize such configurations.

Fig. 9 (Color online) Wave profiles at maximum CL . Blue color corresponds to the water flow and red to the air domain

Additional information about piezometric pressure coefficient distributions on hydrofoil surfaces is

given in Fig. 10. The lower curves with pre-domi- Fig. 8 (Color online) Piezometric pressure coefficient with nantly negative CP correspond to the top (suction) streamlines at maximum C L surfaces and upper curves with mainly positive C P to the bottom (pressure) sides. At relatively low attack (or deeply submerged) hydrofoils, the maximum CL angles in a single-phase flow (Figs. 10(a), 10(b)), the occurs at the attack angle of 16. One can notice that front half of the top portion of E603 profile has an

becomes much less uniform with the greatest suction occurring now near the leading edge, whereas positive

CP on the pressure side increases mainly in the front portion. Higher Reynolds number leads to slightly larger magnitudes of the pressure coefficient. When the hydrofoil operates near the free water

surface, substantial reductions of C magnitudes P take place at low attack angle of 4 (Figs. 10(c), 10(d)), leading to about 50% drop of the lift coefficient (Fig. 6(a)). At the pre-stall condition with

AOA of 24, magnitudes of CP on both pressure and suction side significantly increase, but mainly in the front portion. The flatter C zone on the rear part of P the foil top surface is visible at high attack angles (Fig.

10). It can be attributed to the formation of recir-

culation bubbles in pre-stall conditions (Figs. 7, 8).

3. Conclusions

Two-dimensional, RANS-based computational

fluid dynamics simulations have been conducted for a

high-lift hydrofoil in a single-phase fluid and in the

shallow submergence condition at a moderate Froude

number. This modeling approach produced results in

good agreement with test data for an airfoil up to the

stall regime and for water waves formed behind a

hydrofoil. Parametric simulations carried out in a

broad range of attack angles showed that the

maximum lift coefficient attained at the shallow

submergence was about the same as that of the deeply

submerged foil. However, the corresponding stall

attack angle was about 50% higher in the shallow case

than that at the deep submergence for the considered

foil profile and flow conditions. This implies that

shallow foils can be capable of achieving high lift,

albeit at substantially higher (two-dimensional) drag

coefficient. Also, the lift produced by hydrofoils at

higher Reynolds numbers was found be larger than at

smaller Re , especially near the maximum C L conditions. Hence, model-scale tests with hydrofoil boats likely generate conservative results with regard to the lift capability. Possible future extensions of this research that can be important for practical design of hydrofoil boats include three-dimensional hydrofoils, multi-foil setups, broad range of Froude numbers, and presence of sea waves. Simulations of post-stall regimes and dynamic stall phenomena will likely require more computationally demanding approaches, such as DES Fig. 10 (Color online) Piezometric pressure coefficient or LES-based methods. extended region with nearly uniform pressure coefficient below -1, producing a dominant contribution to Acknowledgement This material is based upon research supported the lift force. At a high attack angle, CP distribution by the U. S. Office of Naval Research under Award

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