Partial validation and verification of the Neumann-Michell theory of ship waves

Proceedings of the 11 ' International Conference on Hydrodynamics (ICHD 2014) October 19-24, 2014 Singapore

PARTIAL VALIDATION AND V E R I F I C A T I O N OF T H E NEUMANN-MICHELL T H E O R Y

OF SHIP WAVES

CL. ZHANG, F. NOBLESSE, D.C. WAN

Stale Key Laboratoiy of Ocean Engineering, School of Naval Architecture, Ocean & Civil Engineering, Slianghai Jiao Tong University, Dongclman Road 800, Shanghai, China

F.X. HUANG, C. YANG

School of Pliysics, Astronomy & Computational Sciences, George Mason University Fairfax VA, USA

The results o f numerical studies directed toward the verification and the validation o f a practical method, called Neumann-Michell theory, for evaluating the flow around a ship hull that steadily advances in calm water o f effectively infinite depth and lateral extent are reported. Numerical predictions given by two completely independent numerical implementations of the theory, performed at the Shanghai Jiao Tong University and at the George Mason University, are compared with experimental measurements for four ship hull fornis (the Wigley hull, and the Series 60, DTMB and KCS models) over the relatively wide range of Froude numbers 0.1 < F < 0.45. The independent numerical results are found to be in good agreement on the whole, and to be consistent with experimental measurements.

1. Introduction

The drag of a ship hull that steadily advances at constant speed along a straight path in calm water of effectively infinite depth and lateral extent is of considerable practical importance because the drag is a critical element of ship design. Accordingly, prediction of the flow around a ship hull is a classical basic ship-hydrodynamics problem that has been widely considered in a huge body of literature, and a number of alternative theoretical and numerical methods have been developed to compute the flow around a ship hull. A brief review of these alternative methods may be found in e.g. [1].

Ship design (especially early design) and hull-form optimization (increasingly used for ship design) require computational methods that are realistic, i.e. that account for all dominant flow physics, yet are robust and practical; indeed, robustness and efflciency are of particular importance for optimization, which requires computations for a very large number of alternative ship hulls with shapes that may be widely different. Accuracy (on one hand) and robustness/efflciency/practicality (on the other hand) are essential, although competing and even contradictory, requirements for a computational tool to be useful for routine applications to design and optimization. In particular, a complete measure of success for an accurate CFD method — that fully accounts for flow unsteadiness and turbulence (needed to properly model bow waves and flows near ship stems) and can predict the flow around a smooth or rough ship hull at full scale, as required for accuracy — is not whether the method yields more accurate predictions than approximate methods based on simplifying assumptions (evidently to be expected), but whether the method is sufficiently robust and practical to be used for systematic parametric studies, essentially required for optimization and early design. Similarly, a highly efficient approximate theory is of limited practical usefulness i f it does not yield realistic predictions.

The total 'cahn-water drag' of a smooth ship hull can be realistically estimated from the sum of the fricdon drag given by the classical ITTC friction formula and the wave drag that can be obtained via potential flow theory with the linearized ICelvin-Michell boundaiy condition at the free surface, as shown in e.g. [2]. The modification, expounded in [1,3-5] and called Neumann-Michell (NM) theory, of the classical Neumann-Kelvin theory of ship waves is used here to evaluate linear potential flow around a ship hull and the corresponding wave drag. This practical theory provides a simple correction to the waves predicted by the classical Hogner approximation, which is defined explicitly in tenns of the ship speed, length, and hull form [1].

Verification and validation are essential elements of the numerical implementation of a theoiy, or the development of a numerical method. Strict verification is not possible because no exact analytical solution for three-dimensional flow around a ship hull exists. An imperfect but nevertheless useful step toward the verification of two completely independent numerical implementations of the N M theoiy is then considered here. Specifically, the numerical predictions recently obtained by Chenliang Zhang at the Shanghai Jiao Tong University (SJTU) are compared to the predictions previously obtained by Fuxin Huang of the George Mason University (GMU) and reported in [5].

2. The Neumann-Michell theory and its numerical implementation

A main feature of the Michell (NM) theoiy expounded in [1] is that, unlike the classical Neumann-Kelvin (NK.) theoiy, the N M theoiy is based on a consistent linear flow model, and does not involve a line integral around the ship wateriine. Furthermore, the flow potential at the ship hull surface is determined via an integro-differential equation that expresses the flow as a coiTection to the Hogner explicit flow approximation. Other notable features of the theory are that the local-flow potential given by a dipole distribution over the ship hull surface is ignored, and that the analytical approximation to the local-flow component in the Green function (associated with the radiation condition and the linearized free-surface boundary condition) given in [3] is used.

The numerical implementation of the N M theory considered in [5] is followed here. In particular, the N M integro-differential equation that defines the flow potential and flow velocity over the ship hull surface is solved iteratively, and a low-order panel method based on piecewise linear variations within flat triangular panels is used. A n essential element of the numerical implementation of the N M theory reported in [5] is that short waves are removed via a filter function based on parabolic extrapolation, in the manner explained in [4,5]. The height o f the extrapolation layer associated with this filter function accounts for the fact that the bow wave created by a ship differs significantly from the waves along a ship hull surface aft of the bow wave.

Three sets of numerical results, identified as GMU, SJTU and SJTU*, are considered here. The numerical results GMU and SJTU, independently obtained by Fuxin Huang at GMU and Chenliang Zhang at SJTU, are both based on the short-wave filter function given by Eq. (16) in [5]. Thus, the results SJTU and GMU should be identical in principle, and differences between these two sets of numerical results stem from the independent numerical implementations reported here and in [5].

1.6 1.4 1.2 0.8 0 . 6 0.4 0.2 0 0 2 4 6 8 10 q

Figure 1. Filter function E(q) defined by Equation (1)

The results SJTU* consider a slightly modified short-wave filter function. Specifically, the parameter and C f in Eq. (26) of [5] are taken as = 0.1 and C f = 0.8 in GMU and SJTU but are chosen as = 0.08 and C f = 1 in SJTU*. Furthermore, the integrands of the Fourier representations of ship waves given by Eqs (9a) and (10) in [5] are multiplied by the additional short-wave filter function function E(q) defined as

The function E(q) is depicted in Fig. 1 for ^ c o ~ 3,5,7 and 9. The wavenumber k and the wavelength A that

correspond to q are given by F^k = l + and A = 2n/k = 2nF^/(l + q^). Fig.l also depicts = ( 3 / 2 ) / ( l + q^) where A'^"^'' = AnF'^denotes the wavelength at the cusp ofthe I<Celvin wake.

3. Partial verification and validation for four ship hull forms

The numerical results GMU, SJTU and SJTU* are depicted in Figs 3-9 for four ship hull forms: the Wigley parabolic hull and the Series 60, DTMB and KCS models. The positive halves of these four hull surfaces are approximated using 8000 fiat triangular panels for the Wigley hull and the Series 60 model, 9928 panels for the DTMB-5415 model and 9452 panels for the ICCS model for the numerical results GMU reported in [5]. For the computafions SJTU and SJTU*, these four hull surfaces are approximated by 7876, 10000, 11900 and 11400 flat triangular panels. Fig.2 shows side views of the panelized hull surfaces used for the computafions SJTU and SJTU*. The numerical predictions GMU, SJTU and SJTU* are also compared to the experimental measurements of sinkage, trim, residuary drag and wave profiles reported in [6-10].

Figs 3, 4 and 5 consider the sinkage, the trim and the (nearfield) wave drag of the Wigley hull (top left corner), the Series 60 model (top right), the DTMB model (bottom left) and the KCS model (bottom right). The numerical predictions are depicted for 0.1<F<0.4 for the Wigley hull and the Series 60 model, and for 0.1<F<0.45 or 0.1<F<0.3 for the DTIVIB or I<:CS models. Figs 6 and 7 show wave profiles for the Wigley hull and the Series 60 model for six Froude numbers within the ranges 0.25<F<0.408 or 0.18<F<0.34, and Figs 8 and 9 show wave profiles for the DTMB model at F=0.28 and 0.41 and for the KCS model at F=0.26.

The GMU, SJTU and SJTU* predictions of sinkage shown in Fig.3 do not differ appreciably, and are in fairly good agreement with experimental measurements. Differences between the GMU predictions and the SJTU or SJTU* results are also relatively small for the trim of the Wigley hull and the Series 60 and DTMB models shown in Fig.4, and for the wave drag of the Wigley hull and the Series 60 model in Fig.5. However, larger differences between the GMU predictions and the SJTU and SJTU* predictions can be observed for the wave drag of the DTMB model in Fig.5, as well as for the wave drag and, especially, the trim ofthe KCS model. The GMU, SJTU and SJTU* predictions are consistent with experimental measurements of the wave drag and the trim on the whole, but discrepancies can be obsei-ved, notably in Fig.4 for the trim of the Series 60 model.

Figs 6 and 7 show that the GMU, SJTU and SJTU* wave profiles do not differ appreciably for the Wigley hull and the Ser ies 60 model. However, differences between the SJTU or SJTU* wave profiles and the GMU profiles are larger for the DTMB and KCS models considered in Figs 8 and 9. The GMU, SJTU and SJTU* predictions are consistent with experimental wave profiles on the whole, although some discrepancies can be observed.

Figure 2. Side views of tlie panelized surfaces o f the Wigley hull (top row), the Series 60 model (second row), the DTMB model (third row) and the KCS model (bottom row).

0.005 0 . 0 0 4 5 0.004 O . 0 0 3 5 a, 0.003 Dl 0 . 0 0 2 5 0.002 O . 0 0 1 5 0.001 0 . 0 0 0 5 1 GHU -i 1 1 1 S J T U •• V l i g l e y S J T U * — • - E x p . I H H I • E x p . S R I - 1 • _{• /'} - E x p . S R I - 2 • E x p . S R I - 3 r / • - E x p . U T - 1

f^/

Figure 3. Experimental measurements of sinkage, and corresponding numerical predictions given by the N M theory, for the Wigley hull and the Series 60, DTMB and KCS models.

1 GHU -- 1 1 1 I S J T U -- S e r i e s 60 S J T U * — E x p . I H H I * ƒ ^ - E x p . H M R I - 1 • ƒ ~ E x p . I U • E x p . K U - 1 E x p . U H - 1 _• E x p . U H - 2 A - E x p . S R S - 1 _fi Ji // * 1 1 0.25 0.3 J I I I I I 1 1 L 0.25 0.3 0 . 3 5 0.4 0 . 4 5 0.1 0.12 0.14 0 . 1 6 0 . 1 8 0.2 0.22 0.24 0 . 2 6 0 . 2 8 0.3

Figure 4. Experimental measurements of trim, and corresponding numerical predictions given by the N M theory, for the Wigley hull and the Series 60, DTMB and KCS models.

•O 0 . 0 0 1 5 u 0 . 0 0 0 5 0 . 0 0 5 5 0 . 0 0 5 0 . 0 0 4 5 0.004 0 . 0 0 3 5 0.003 0 . 0 0 2 5 0.002 0 . 0 0 1 5 0 . 0 0 1 0 , 0 0 0 5 GMU " S J T U _ S J T U t E x p T 1 1 r DTMB-5415 J I I l _ 0.1 0.15 0.2 0.25 0.3 0 . 3 5 0.4 0 . 4 5 0.1 0.12 0.14 0.16 0.18 0.2 O . 22 O . 24 O . 26 O . 28 0.3

Figure 5. Experimental measurements o f wave drag, and corresponding numerical predictions given by the N M theory, for the Wigley hull and the Series 60, D T M B and KCS models.

4. Conclusion

Figs 3-9 show that the three sets of numerical predictions — identified as GMU, SJTU and SJTU* — considered here are mostly consistent, and are also mostly consistent with experimental measurements, for four ship hull forms — the Wigley parabolic hull, and the Series 60, DTMB and K.CS models — over a relatively broad range of Froude numbers. The numerical results GMU and SJTU correspond to independent numerical implementations (by Fuxin Huang at GMU and Chenliang Zhang at SJTU) of the Neumann-Michell (NM) theoiy, in accordance with [5]. The overall consistency between the GMU and SJTU results displayed in Figs 3-9 means that both the GMU numerical results reported in [5] and the SJTU resuhs reported here are likely to represent 'correct' numerical implementations of the N M theory, and furthermore demonstrates that the information given in [5] is adequate to implement the N M theory.

However, discrepancies between the GMU and SJTU numerical predictions can be observed (notably for the wave drag of the DTMB model, as well as for the wave drag and, especially, the trim of the KCS model). The differences that can be observed, notably in Figs 4, 5, 8 and 9, illustrate that differences between independent numerical implementations of a theory or numerical method are to be expected, in much the same way that differences among independent experimental measurements are to be expected. Indeed, differences between the GMU and SJTU predictions for the Wigley hull and the Series 60 model are no larger than the differences that can be observed among the experimental measurements displayed in Figs 3-7 for these two hulls.

Figs 3-9 show that differences between the SJTU and SJTU* predictions, based on slightly different short-wave filters, are relatively small. This finding confims the conclusion, reached in [4,5], that while filtering of short waves is a necessary and essential element of the numerical implementation of the N M theory, numerical predictions are not overiy sensitive to the filtering of short waves (as long as a reasonable physics-based filter is used).

er O.OS O -0.05 -0.1 _1 : I . I . 1 --0.6 -0.4 -0.2 O X 0.2 0.4 0.6 — ' — 1 — • — 1 ' 1 ' 1 F = 0 . 4 0 8 1 . ' • • - fa •

-

--Tv

/

Figure 6. Experimental measurements of wave profiles, and corresponding numerical predictions given by the N M theory, along the Wigley hull at Froude numbers F=0.25, 0.267, 0.289, 0.316, 0.354 and 0.408.

Figs 3-9 also show that the numerical predictions GMU, SJTU and SJTU* are mostly consistent with experimental measurements, although some discrepancies can be observed (notably in Fig.4 for the trim of the Series 60 model). Thus, the N M theory is an approximate theory that accounts for the dominant flow physics, and accordingly yields realistic estimates of major features of the flow around a ship hull that steadily advances in calm water, as already shown in [5,2]. The numerical implementation of the theory given in [5] yields a simple and efficient computational tool (the flow around a ship hull surface approximated by 10,000 triangular panels can be evaluated in about 20sec per Froude number on a PC) that is well suited for hull-form design and optimization [11-17].

Acknowledgments

This work is supported by National Natural Science Foundation of China (Grant Nos.11072154, 51379125, 11272120), Foundation of State Key Laboratory of Ocean Engineering of China (GKZD010061), The National Key Basic Research Development Plan (973 Plan) Project of China (Grant No. 2013CB036103), High Technology of Marine Research Project of The Ministry of Industry and Information Technology of China, the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher

Learning(Grant No. 2013022), and Center for HPC of Shanghai Jiao Tong University, to which the authors are most grateful. -0.6 -0.4 -0.2 0.2 0.4 0.6 0.1 0.05 0 -0.05 1 ' 1 I ' l ' F = 0 . 2 5 1 ' -' \ -° Ï V • * j «

Figure 7. Experimental measurements of wave profiles, and corresponding numerical predictions given by the N M theory, 60 model at Froude numbers F=0.18, 0.20, 0.22,0.25, 0.30 and 0.34.

0.4 0.6

along the Series

0. 05 0 -0.05 1 1 i 1 1 1 1 GMU S J T U F = 0 .28 'SJTU* E x p • 1 ' r -0.6 -0.4 -0.2 0.2 0.4 0.6 -0.6 -0.4 -0.2 0.2 0.4 0.6

Figure 8. Experimental measurements o f wave profiles, and corresponding numerical predictions given by the N M theory, along the DTMB model at Froude numbers F=0.28 and F=0.41.

O . 1 ! ^ 0 . 0 5 O - 0 . 0 5 - 0 . 1 I - O 1 1 G M U S J T U S J T U * E x p • 1 • 1 F = 0 . 2 6 1 / \ " ff \\ -l \ ï.\

-\\

Figure 9. Experimental measurements of wave profiles, and corresponding numerical predictions given by the N M theory, along the KCS model at Froude number F=0.26

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