Twenty-Fourth Symposium on Naval Hydrodynamics (2003)

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24~ Symposium on Naval Hydrodynamics Fukuoka, JAPAN, 8-13 July 2002 Calculations of Flows over Underwater Appended Bodies with High Resolution ENO Schemes Zhen-yu Huang, Hong-rong Cheng, Lian-di Zhou (China Ship Scientific research Center, Wu Xi, 214082, China) ABSTRACT The numerical method based on flux difference splitting LU decomposition; implicit high- resolution third order Essentially Non-oscillatory (ENO) scheme is constructed for efficient computations of steady-state solution to three- dimensional, incompressible Navier-Stokes equations in general coordinates. The flowfields over underwater axisymmetric bodies, full-appended axisymmetric bodies and axisymetric bodies with a ring-wing duct are simulated. The method has been shown to be capable of predicting the circumferential-mean velocity distribution at model scale to accuracies of around 3% of measured values, and of predicting some details of flow feature, for example the wake harmonics. KEYWORDS Essentially Non-Oscillatory (ENO) Schemes, Flux Splitting, LU Decomposition, Computational Fluid Dynamics (CFD) INTRODUCTION The flow over an underwater appended vessel during level flight is characterized by the development of thick boundary layers, flow separation, flow into and through the propulsor, vortices formed at the root and tip of appendages and appendage turbulent wake. The spatial nonuniformity and temporal fluctuations of flow into vessel propulsor significantly affect propulsor noise. Whilst the fluid flow can be measured by using a scale model of the vessel in a towing tank or wind tunnel, there are errors associated with applying results obtained at model scale to full scale. Furthermore, the experiment requires many detailed measurements to be taken in the regions of interest. This is costly and time consuming. Recent advances in CFD have been developed, which have been shown to be capable of predicting flow over full-appended underwater bodiesEHuang, 20011. This paper presents a high-resolution numerical method, which have been used to calculate the flowfields over appended underwater bodies. The appendages include a fairwater, four-wing stern appendages and a ring- wing duct. The numerical scheme based on flux- difference splitting, LU decomposition and implicit high resolution third-order ENO achieved through flux reconstruction, are constructed for efficient simulation of steady-state solution to three- dimensional, incompressible conservative Navier- Stokes equations in general coordinates. The schemes have been shown to be capable of predicting the circumferential-mean velocity distributions at the propulsor plane of full appended submarine-like bodies to accuracies of around 3% of measured values in wind tunnel, and of predicting some details of flow feature. The computer code has been applied to evaluate and optimize the flow characteristics over single axisymmetric bodies and underwater full- appended bodies and to simulate axisymmetric bodies with a ring-wing duct. GOVERNING EQUATIONS . The governing equations, which describe the motion of viscous, incompressible fluid, are Navier- Stokes equations and the equation of continuity. Artificial compressibility method, which adds a time- derivative of the pressure to the continuity equation, can couple the equations of motion with the . . ,

continuity equation. Then the most efficient implicit time-dependent methods can be applied to the incompressible Navier-Stokes equations, the complete set of governing equations can be solved simultaneously~Peter, 19881. The Navier-Stokes equations in conservation law form for an incompressible, three-dimensional flow are written as Q'+(E -EV)X +(F -FV)Y +(G -GV)Z =0 With the inviscid flux vectors E = ( pu, u + pi us, uw) F = ( pV,Uv, V2 + petit (2b) G = (~,uw,~,w + p) (2c) And the shear flux vectors EV = Re~~0~2ux~uy +vx~uz + WX)T FV = Rem (0, uy + Ax, ,2vy, vz + wy )T GV = Rem (0, uz + wx, vz + wy,2wz AT Where Re is Reynolds number. Following the artificial compressibility method, the dependent vector Q in Eq. (1) is defined as Q=(p,u,v,w) (4) Considering a coordinate transformation of the form ~ =~(x,y,z), 77 =r1(x,y,z), and 5 =5(x,y,z), Equal) can be rewritten in strong conversation law form. (QIJ)t +(E-EV): +(F-FV)n +(G-GV)5 =0 (5) The flux vectors E, F. G are linear combination of E ,F ,G in Eq. (11. For example, E can be written as E = (4X I J)E + (;Y I J)F + (¢Z I J)G (6) Where J is the Jacobean of the coordinate transfor- mation. HIGH RESOLUTION SCHEMES FOR INVISCID FLUX Because of the complicity of flowfield structure around the underwater bodies with full appendages, the Essentially Non-oscillatory (ENO) schemes [Harten,1987], which were developed by Harten et al, are applied in discretization of inviscid flux of three-dimensional incompressible Navier- Stokes equations. The ENO schemes, which use adapted stencil, are uniformly high-order accuracy throughout even at critical points. Following Yang [Yang 1992,Huang 1999, 2000], third-order nonoscillatory schemes are given `2ay below. TakedE`:in direction ~ as an example, let A = bE/3Q, (A E ji+l/2 = (71, /2, 73, /4 ~ iS the eigenvalue diagonal matrix of A. R and L is right and left eigenvector matrices of AK. Then one can get A=RAEL. The spatial difference of E`: can be reached by using finite volume method (3a) (FVM)[Huang 1999, 20001. (3b) (3c) And E`.EN,°32 = 2 (Ei + Em + Ri+~,2O ~EN1O,32 ) (8) E: = EEN03 _ EEN03 (7) Let ~i+l/2 = L(Qi+~ - Qi), the components of ~~.E+l03 can be defined as ~+l/2 = ~~/i+l/2~i + fit) (/i+l/2 tori + ~i+ - ~~;i+l/2 + ii+l/2 + ii+l/2 ~ai+l/2'~ai-1/2 ~ < ~ai+l/ + A ;i+l/2 tori + ~i+1 ~ - ~~/i+l/2 + ii+l/2 + ti+l/2 ~ai+ll2,1ai ll2 | > |ai+ll2 | Where 6, 6, and ~ functions are given by: 6(z) = 2 (#r(z) - BZ2 ) (10) (Z) = 6 (2|z| - 3BIzl2 + B2lzl3 ) ~(Z) = ~ (B2 1Z13 - 1Zl) (12) (11) { ( Z + £ 2 ~ / £ k At' £

His a small positive constant. B is the value of At I A; . One can find that the flux OiE+N/ 3 iS self- adapted with the distribution of the flowfield physics. And pi = min mod[(xi+ll2 '(7i-l/2 ] (14) ~ =m[^ Hi lid, /~+ai lit] lai lit| <|ai+l,2| (15) pi=m[~-ai+ll2' ^+`xi+l/2] lail,2|>|ai+l,21 (16) Ill/ 2 = ~ (ai+l/ 2 )~(§i+1 ~ i) / ~i+l/2 ai+ll2 ~ O ~ O otherwise _ _ ~i+l/2 = ~ (ai+l/2 ) ('i+1 -§ i) /~i+112 ~i+l/2 ~ O I O otherwise (18) A ~ i+l/2 = ~ (ai+l/2 ) (§i+1 -I i) /ai+ll2 ai+l/2 ~ O otherwise (19) H = it + jH(~) + kH(5 H(~)=E-R Ev H(~) = F- R Fv H( ) =G- ~ G (22) If Eq. (21) is applied over a hexahedral cell (/~=/~=~\5=1), temporal derivative discretiza- tion uses the first-order-accurate differencing, inviscid terms use implicit differencing, viscous (17) terms use explicit central differencing, Eq. (4) will be discretized as J ~ (Qn+~ _ Qn ~ + Att [E.n+~/2—E'n+~, + fF,.n++,.,2—F,.n+~,2 ~ + EGkn++~,2—Gkn+i, = ~ ~ LEV .n+l,2—EVin I,2 ~ + EFvj+~/2—Fv~-~/2 + [GVk+~/2 GVk-~/2 ~ ~ (23) {Z eye> ~Z~ (20) The definition of (A~)i+~,2 and A = (dE I aQ)i+~,2 are same as the last section. Let The remaining fluxes of three-dimensional Navier-Stokes equations (Eq.(5)) can be defined with the similar way. LU DECOMPOSITION FOR TEMPORAL DERIVATIVE (AK )i+l/2 = [(AK )i+l/2 + |(AE )i+l/2 |] / 2 (24) If LE ~ RE are the left and right eigenvector matrices of Jocob matrix A, one can get By using Finite Volume Method (FVM), the AN = ReAi LE (25a) integrated form of Eq. (5) can be rewritten as[Huang and Zhou. 2000,2001] Similarly —[— -IQ V]+— H.dS =— D V Bi =RFA-FLF (25b) ~tV; Al Vl Where S is the surface around the cell, do is the normal vector of each surface, H is the tensor whose vector components in three directions are (21) C = RGAGLG (25c) Let Qn+1 _ On = i\Qn LU decomposition formula of Eq.(5) caI1 be expressed as

J-l[I+AtJ(A+ +B+ +C+ )]x [I—~J(A- + B- + C- )]~`Qin; =—Ad{ [El+ll2—El-ll2) + (FJ+l/2—FJ-l/2 ) + (Gk+l,2—Gk l,2 )] } + R { [EVI+1I2—Evl-l/2 ] + [Fv)+l/2—Fv) l/2 ] + [Gvk+l/2—Gvk_l/2 ] } TURBULENCE MODEL The effects of turbulence fluctuations, which can be solved on the computational grid, are approximated with simple Smagorinsky model. A, = (CsL)2lSijI (27) Sip = 2 (Oxj + Oxi ) In which the model coefficient Cs = 0~05 ~ 0.1 and length scale is geometric mean of the grid spacing L = (~c/\yAz)l/3 . RESULTS AND DISCUSSIONS The flowfields around the appended underwater axisymmetric bodies are simulated as steady-state solutions to the incompressible Navier- Stokes equations. The appendages include a fairwater, four-wing stern appendages and a ring-wing duct. Three kinds of flowfields are included in the numerical simulation. They are the flowfield over the axisymmetric bodies, the axisymmetric bodies appended with a fairwater and four-wing stern appendages and the axisymmetric bodies with a ring wing duct. The main bodies of most modern underwater vessels are axisymmetric or cylinder-like shapes, such as submarine, torpedo or other deep-sea vehicles. For validation of numerical method, the flowfield around the axisymmetric bodies are simulated firstly. Fig.1 shows the comparison of the numerical velocity distributions at difference positions with the experimental data, which are measured in wind tunnel. The numerical results agree with the measured very well, which indicates that the third-order ENO schemes are high resolution, and can be used in the numerical simulation of three- dimensional incompressible Navier-Stokes equations. In general, the geometry of underwater vehicles are very complex. There are various appendages attached to the main body. Take submarines as an example, the appendages include a fairwater, rudder, several wing-like stern appendages, and even a ring-wing-like duct. The flowfields around underwater axisymm- etric bodies with a fairwater and four-wing stern appendages are simulated with the third-order high- resolution ENO schemes. The complete calculations (26) are carried out by two steps. The flowfields around the body and a fariwater are numerically simulated firstly, which can provide the inlet boundary condition for the following fine simulation of flowfield around stern part of body and four-wing stern appendages. The numerical dimensionless circumferential -mean velocity along the radius of the propeller are presented in Fig. 2, which are in good agreement with the experimental data. The details of numerical dimensionless circumferential-mean velocity and the experimental data can also been found in Table 1. 28 Except one points, the relative error between the numerical results and the experimental data is less than 3%, their average relative error is only 2.107%, the accurate numerical nominal wake at propeller can be used as input data of vehicle propulsor blade design. Also the numerical dimensionless circumferential velocity distributions at different radius station are showed in Fig. 3(a)-Fig. 3(e). There are difference between the numerical results and the experimental data, but their phases are similar, so the calculated circumferential velocity can be applied to the optimization and evaluation of hydrodynamics noise of vehicle propulsor. The same accuracy has been reached from the other numerical simulations over different models. The code developed in this paper has been used in the design and optimization of new underwater bodies with full appendages. Table 1. Comparison of Numerical circumfere- ntial velocity with experimental data r/R 0.217 0.300 0.400 0.518 0.678 0.840 1.000 Numerical results 0. 4866 0. 5177 0. 5471 0. 5935 0. 6681 0. 7387 0. 7933 Experiment data 0. 4833 0. 4911 0. 5319 0. 5993 T o. 6814 0. 7479 _ 0. 8064 Error (%) . 0.68 . 5.42 2.88 -0.97 -1.95 -1.23 -1.62

Ducted propulsors are known to offer significant advantages for particular marine applications, such as increases in efficiency for high propeller loading with flow-accelerating ducts, or alternatively smaller propeller sizes; reduction of inflow velocity and, consequently, of cavitation and noise with flow-decelerating ducts; better control over the inflow to the propeller; improvement of maneuverability and position-keeping abilities of vessels; protection from damage to the propeller, etc. The ducted propeller has been installed in underwater vehicle, such as submarine and torpedo, to increase the vehicle speed and reduce the propeller hydrodynamics noise. The inflow velocity, which is related to the geometry of ducts and the interaction between duct and main body, is very important to the design of propulsors' blade. For the first step, the flowfields around the axisymmetric bodies with a ring-wing duct has been simulated. The numerical results are presented in Fig.4, which is compared with the experimental data measured in wind tunnel. Because the geometry used in the calculation is little different from the real model, which attached more appendages. Meanwhile the experimental data is very limited. The numerical results do not agree with the experimental very well. But, by the innovation of numerical schemes and multi-block grid system, the internal and external flows of the combinations of full-appended unterwater bodies and ring-wing duct will be simulated more accurately. It is sure that the numerical nominal wake in the duct would be used in the design of duct-propeller. CONCLUSION Based on flux splitting, implicit high- resolution schemes have been constructed for efficient calculations of steady-state solutions to the three dimensional, incompressible Navier-Stokes equations in curvilinear coordinates. The third-order- accurate efficient ENO has been applied in the calculations, which can capture the details of the flowfield around underwater bodies with full appendages. The numerical results agree quite well with the experimental data. The schemes and code developed in this paper can be applied in the design of underwater vehicle propulsor and in the optimization and evaluation of its hydrodynamics noise. Also the code can be used in the optimization and design of shapes of vehicle body and its appendages. ACKNOWLEDGE The project was supported by Shanghai Natural Science Fund (No.OOZF14065) and National Key Laboratory on Hydrodynamics (No. H9958~. REFERENCE Peter M. Hartwich, Chung Hao Hsu, "High - resolution upwind schemes for the three-dimensional incompressible Navier-Stokes equations", AILS J. Vol. 26,No.11,1988,pp.1321-1326 Harten A, Osher S. "Uniformly High-order Accurate Nonoscillatory Schemes I", STAM J. on Numer. Analysis, Vol. 24, No. 2, 1997, pp.279-309 Huang Zhenyu, "Numerical simulation of jet flows and nozzle flows", Doctorial dissertation, Beijing Institute of Technology~in Chinese), 1999 Yang J. Y. "High-resolution, nonoscillatory schemes for unsteady compressible flows", AI4A J. Vol.30, No.6, 1992,pp.570-1575 Huan~ ZhenYu. "Flowfield calculation with high resolution ENO", The proceedings of eighth international space conference of pacific-basin societies (8~ ISCOPS), Xitan China, l 999,pp.691- 697 Huang Zhenyu, Xu Wencan, "Flowfield calculation with high resolution ENO", ACTA Aerodynarnica SINICA, Vol.18, No.1, 2000, pp.14-21 (in Chinese) Huang Zhenyu, Xu Wencan, Numerical simulation of turbulent jet, Journal of Reijing, Institute of Technology, Vol.19, No.6, 1999, pp.691-695 (in Chinese) Huang Zhenyu, Zhou Liandi and Zhao Feng "High- resolution schemes in simulation of flowfield around submarines", Technical Report 00867, CSSRC, 2000 (in Chinese) Huang Zhenyu, Zhou Liandi, "Numerical Simulation of Flows over Underwater Axisymmetric Bodies with Full Appendages, .Shipbuilding of China, Vol.42, No.4, 2001, pp.6- 11 (in Chinese) Zhenyu Huang, Liandi Zhou, "Numerical Simulation of Flows over Underwater Axisymmetric Bodies with Full Appendages", proceedings of Fighth International Symposium on Practical Design of Sllips and Other Floating, Structure, Sept. 16-21, Shanghai, China, pp. 429-436

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.2 0.1 o 0.8 _ 0.6 0.4 0.2 _ 1 1 1 ~ ~ I , , , , I 0.15 0.2 0.25 ut R ~ig. l (a) x=-0.4333m _~ ~ — ~ ' ;1 Exp. Cal. . ~ 1 1 1 ~ I I I I,,,, I,,,, I 0.1 0.15 0.2 0.25 R Figl.(c) x=-O. 130 Figl. The comparison of velocity distribution at difference x-axis stationsk the x=0 is at the tail of body) 0.8t 0.75 0.7 .65 /,.... r Cal. /' / /K' ~ 06 0.5 0.4 0.6 0.8 1 Fig. 2 Comparison of numerical circumferential- mean velocity at propeller plane with experimental data — Exp. ~ C~. ~ , " ~ , ~/, n ,,,, .,,,, ',,,, ',,,, I 45 90 1 35 180 deg Fig.3(a) r/R=0.30U 0.8 0.6~ 0.4L 0.~ nQ ~/ — Exp. — C~. /, / ,, 1 1,, ,, 1, ,, , 1 90 135 180 deg Fig.3(b) r/R=0.518 '''~ WN W~ 3.6~l'' 0.4 0.2 Exp. C~. 1 , , , 1 1 45 90 1 35 . ~ Fig.3(c) r/R=0.678

1 0.8 0.4 o 1 0.8 0.6 0.4 0.2 , . . . . . . . . . ~ ~ 45 90 1 35 deg Fig.3(d) r/~=0.840 Ol . I , - Exp. — — — Cal l 180 i_ i= · Exp. _ _ _ — — Cal. l 135 180 deg Fig.3(e) r/R=l.OOO Fig.3 Circumferential variation of axial velocity at difference radius station of propeller plane 0.6 0.4 x 0.2 o 0.6 0.4 0.2 ,ii _ ~! / /f' ' —~ / . ~ / ' _ /, At' , . . . . , . . . . , 0.06 0.08 0. R Fig4.(a) In duct(x=-0.165) 0.8 X- _ ,, 1 0 O.05 , it, , . . . . , R . . . . . , 0.15 0.2 Fig4.(b) Behind the duct(x=0.03) Fig. 4 Comparison the velocity distributions internal and external of the ring-wing duct around axisymmetric bodies.