Flow around a Kriso Container Ship

The final validation case refers to the fully turbulent flow around an unappended Kriso container ship hull (KCS). Experimental resistance data and wave fields are published by Kim et al. [2001] for a 1:31.6 scale model and a large amount of comparative numerical data exists, e.g. Larsson et al. [2003], Kr¨oger et al. [2018], Kr¨oger [2016], Manzke et al.

[2012], Banks et al. [2010]. The distance between aft and front perpendiculars of the hull model serves as a reference length L^∗ = 7.2786 m (= L^∗pp). Other reference values refer to the gravity acceleration G^∗ = |g^∗₃|, the inflow velocity magnitude V = |v₁^∗| and the kinematic viscosity of the waterν^∗b. The model scale investigations refer to Reynolds- and Froude numbers of Re =V^∗L^∗/ν^∗b = 1.4×10⁷ and Fn = V^∗/√

G^∗L^∗ = 0.26. The hull is fixed at the full scale static draught with zero trim and the motion and propulsion of the ship are suppressed during the simulation and the experiments.

The numerical grid consists of approximately 14.6 million unstructured hexahedral cells.

The domains extends over 5L^∗, 1.75L^∗, 2.5L^∗ in longitudinal (x^∗₁), lateral (x^∗₂) and ver-tical (x^∗₃) direction. Due to symmetry, only half of the flow field is simulated. The inlet is located upstream at x^∗₁/L^∗pp = 3 and the free surface is initialized at x^∗₃/L^∗pp = 1.75 over the lower boundary of the domain. The surface of the hull is discretized with ap-proximately 300,000 surface elements. The wall normal resolution of the hull refers to a dimensionless wall distances of 30 ≤ y⁺ = x⁺₂ ≤ 100 and justifies the use of wall func-tions. The vertical resolution of the free surface region is constant throughout the domain and attempts to resolve the expected wave amplitude of 5·10⁻⁴L^∗pp by hundred cells in the immediate vicinity of the hull. The tangential resolution of the free surface is refined within a Kelvin-Wedge to capture the resulting wave pattern. Based on the current Froude number, a dimensionless wavelength of λ^∗/L^∗pp = 2πFn² = 0.4247 is expected, which is approximated with roughly 100 cells. Figure 2.24 indicates the different refinement levels for the near and the far field.

Figure 2.24: Kriso container ship case (Re = 1.4·10⁷ and Fn = 0.26): Illustration of the employed computational mesh along the still water plane in the vicinity of the vessel hull.

At the inlet, outlet, outer and lower boundaries, Dirichlet values for velocity and con-centration are specified, while the pressure is extrapolated. A reverse situation is given at the top face which corresponds to a pressure boundary. Symmetry and wall boundary conditions are declared along the mid-ship plane as well as the hull. Turbulence is mod-eled by a high-Rek^∗−ϵ^∗ model. Convective momentum transport is realized by a QUICK scheme. Similar to the 2D case, data obtained from CH-VoF simulation is compared with VoF results. CH-VoF calculations refer again to steady simulations using ˜M = 0.1 and the nonlinear EoS. VoF calculations employ time stepping based on ∆t^∗ = ∆x^∗_3,FS/V^∗ Co, where the Courant number is assigned to Co = 0.3 and ∆x^∗_3,FS denotes the vertical reso-lution of the free surface. All simulations are performed until the integrated forces on the hull converge.

Figure 2.25 depicts the evolution of total resistance over the wall-clock time. The pre-dicted total drag force coefficient is normalized with the static wetted surface of 9.5121 m² and converges toc_d = 3.68×10⁻³ andc_d= 3.66×10⁻³ for the CH-VoF and the VoF simu-lation. Both values differ by only 0.5% and compare favorable with the experimental value cd = 3.56×10⁻³ – subject to the influence of other aspects, e.g. turbulence modelling.

However, the CH-VoF approach clearly outperforms the VoF simulation with respect to computational time, while introducing only minor additional wave damping, cf. Fig. 2.26.

The wave elevation (x^∗_FS,3/L^∗pp) measured at three different lateral planes through the free surface, i.e. x^∗₂/L^∗pp = 0.0741, x^∗₂/L^∗pp = 0.1509 and x^∗₂/L^∗pp = 00.4224, is compared with experimental data in Fig. 2.27. The predictive discrepancy is generally small and the nonlinear CH-VoF tends to provide slightly larger amplitudes. Mind that the nonlinear EoS 2.9 leads to a significant sharpening of the density field, as illustrated by Fig. 2.28.

10⁻² 10⁰ 10² 10⁻³

10⁻² 10⁻¹

t^∗sim. [h]

2|F∗ 1| ρ∗bV∗2A∗w[-]

VoF (Lin.) CH-NS (N.-Lin.)

Figure 2.25: Kriso container ship case (Re = 1.4·10⁷ and Fn = 0.26): Evolution of the predicted drag force over the wall-clock simulation time for the VoF and CH-VoF simulation.

-0.01 x^∗_FS,3/L^∗pp 0.01 x^∗₁

x^∗₂

CH-VoF

VoF Figure 2.26: Kriso container ship case (Re = 1.4 ·10⁷ and Fn = 0.26): Comparison of

predicted wave field obtained by the VoF (bottom) and CH-VoF (top) ap-proach. White horizontal lines indicate evaluation planes used for the wave cuts displayed in Fig. 2.27.

−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -0.005

0.0 0.005

x∗ 3,FS/L∗pp [-] ^{VoF (Lin.)} CH-NS (N.-Lin.) Kim et. al (2001)

−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -0.004

0.0 0.004

x∗ 3,FS/L∗pp [-]

−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -0.002

0.0 0.002

x^∗₁/L^∗pp [-]

x∗ 3,FS/L∗pp [-]

Figure 2.27: Kriso container ship case (Re = 1.4·10⁷ and Fn = 0.26): Comparison of measured and predicted wave elevation in 3 lateral planes, i.e. close to the hull at (a)x^∗₂/L^∗pp = 0.0741, (b) atx^∗₂/L^∗pp= 0.1509 and at a remote position (c) x^∗₂/L^∗pp = 0.4224.

VoF

CH-VoF

c ρ^∗

Figure 2.28: Kriso container ship case (Re = 1.4·10⁷ and Fn = 0.26): Concentration (left) and density field (right) obtained from a VoF with linear EoS (top) and a CH-VoF with nonlinear EoS (bottom).

State and Geometrical Constraints

In the previous chapter, the governing equations for describing maritime turbulent engi-neering flows were derived and discussed. Besides, a discrete FV-approximation of the equations was presented, verified and validated. Therefore, it is assumed that the discrete representation of the continuous balance equations provides an acceptable way to evaluate marine engineering devices’ performance.

The third chapter deals with possibilities to improve the performance using shape mod-ifications. Two optimization strategies have been established in the engineering context, typically referred to as global (gradient-free, stochastic) and local (gradient-based, deter-ministic) methods. Prominent global methods are evolutionary algorithms, which rely on population-based, evolutionary optimization strategies to explore the entire design space, cf. Back [1996], Giannakoglou [2002], Th´evenin and Janiga [2008], Deb [2011]. They aim at global optima and can consider multi-criteria optimizations to determine, e.g. Pareto fronts of non-dominating solutions. The evaluation software (PDE solver) can conve-niently be invoked as a black box. A disadvantage of global optimization methods is the considerable computational effort due to the relatively large number of required per-formance evaluations. The number of evaluations scales nonlinearly with the number of design variables, which theoretically tends to infinity in parameter-free shape optimization.

Therefore, parameter-free shape optimizations based on gradient-free approaches become prohibitively expensive in complex engineering flows. On the contrary, local optimization methods start at a defined position in the design space and rely on local gradient informa-tion to advance the design. Thus, they operate close to an existing initial/previous design, cf. Luenberger et al. [1984], L¨ohner [2008], Th´evenin and Janiga [2008], Mohammadi and Pironneau [2010] and seek for a local optimum. The required descent information poses a challenge since the employed PDE solver usually does not provide derivative information.

Therefore, a black box approach can be used in rare cases only and –dependent on the design space size– specially tailored tools for the derivative approximation are needed.

This chapter first describes prominent methods for obtaining derivative or sensitivity information for shape optimization problems. Thereupon, the cost functionals investigated in this thesis are presented and discussed w.r.t. their formulation. After a presentation of the material derivative concept, the determination of shape derivatives is addressed in the context of a general shape calculus. Subsequently, techniques are presented to account for geometrical constraints and how to extract the shape gradient from the shape derivative.

The chapter concludes by presenting the overall employed optimization procedure in terms of a gradient-based steepest descent approach.

3.1 Generic Shape Optimization Problem

A shape-design performance is measured via a scalar cost functional J^∗ in terms of a fluid mechanically motivated objective that depends on the flow state vector φ^∗ from Cha. 2.

The objective shall be optimized by modifying a control vector u^∗, represented by either all or only parts of the walled flow boundary. A valid physical state requirement provides the following generic constraint shape optimization problem

min. J(φ^∗(u^∗), u^∗) s.t. R^∗φ(φ^∗(u^∗), u^∗) = 0, (3.1) where the cost functional can be either surface-based (e.g., total drag) or volume-based (e.g., prescribed wave field). The state is described by a –usually nonlinear– PDE R^∗φ that is valid in the entire domain and closed by boundary conditions B^∗φ. A typical process chain to classify the actual design performance and determine its position in the design space proceeds in the following three steps, viz.

1. Define/modify the control u^∗.

2. Estimate the state φ^∗(u^∗) via the approximation of R^∗φ(φ^∗(u^∗), u^∗) = 0.

3. Evaluate the objective J(φ^∗(u^∗), u^∗).

In this work, local optimization strategies are pursued, and the concept of sensitivity analysis is discussed in the following. Existing approaches to determine the sensitivity of J^∗ w.r.t. the control u –that describes the design boundary Γ^∗D– can be divided into direct (Jacobian) or indirect (adjoint, Lagrangian) methods. Direct approaches apply the sequence above in forward mode by perturbing the control δu^∗ (sender) to finally estimate the variation of the objective δJ^∗ (receiver) w.r.t. to the control, i.e. δJ^∗/δu^∗. Indirect, adjoint procedures reverse the procedural flow of direct approaches by swapping the sender-receiver relation.