Serge Toxopeus

In the present paper we focus on the simulation of viscous flows at high Reynolds numbers using the Reynolds-averaged Navier-Stokes (RANS) equations. In carefully performed steady simulations, round-off and iterative errors are reduced to negligible levels when compared to the discretization error and so the numerical error is dominated by the contribution of the discretization error. In grid refinement studies the discretization error of a quantity of interest is described as a function of the typical cell size by power series expansions. The estimation of the exact solution requires numerical solutions in more than one grid and so a family of (nearly) geometrical similar grids needs to be generated. The requirement of grid similarity is a consequence of the definition of the typical cell size. In the numerical solution of the RANS equations, the determination of the shear-stress at the wall τ_w can be performed in two alternative ways: directly from its definition, or using wall functions. The grid refinement strategy required by each case is significantly different. In the first option, the near-wall cell must be systematically refined as all the remaining grid cells. When wall functions are used, the size of the near-wall cell size should remain fixed. In this paper, we present the consequences of using the wrong refinement strategy for the flow over a flat plate at Reynolds numbers of 10⁷ and 10⁹. The results show that using the wrong grid refinement strategy can lead to misleading results.

Traditionally, submarine hydrodynamic design has focussed mainly on requirements regarding operational range, powering performance and manoeuvring ability for deeply submerged conditions. To improve the effectiveness of the boat, attention is also paid to operating near or at the surface and fortunately, computational tools and experimental methods are available to analyse the performance of submarines at these conditions. To advance submarine hydrodynamics knowledge and tools, DMO and MARIN have conducted a wide variety of bi-lateral or collaborative studies using potential and viscous flow methods and experiments on several submarine hull forms. In this article, several examples are presented of the development and use of hydrodynamic tools available during the design and assessment process of future submarines. These examples range from experimental and numerical studies into at-surface and periscope-depth resistance and powering, periscope-depth manoeuvring, high-fidelity flow around the boat during straight flight and manoeuvring motions, wakes of surface-piercing masts, to depth keeping under waves. It is demonstrated how state-of-the-art studies help in advancing the knowledge on submarine hydrodynamics and improving the overall design of modern submarines.

Hydrodynamic loads acting on a ship can nowadays be reliably obtained from Computational Fluid Dynamics (CFD) techniques. In particular for the determination of the hydrodynamic coefficients of a mathematical manoeuvring model, the forces and moments on a ship sailing at a drift angle or with a yaw rate can be computed efficiently with CFD. While computations with a drift angle are relatively straightforward, computations involving a yaw rate present a challenge. This challenge consists in how to deal with the grid, the setup and the ship encountering its own wake when rotating. A solution based on a single grid setup with consistent boundary conditions and utilising a body force wake damping zone to remedy this challenge is proposed in this paper, leading to an effective, fast, and accurate method to compute hydrodynamic loads of a ship in steady yaw manoeuvres.

The presence of complex fluids in nature and industrial applications combined with the rapid growth of computer power over the past decades has led to an increasing number of numerical studies of non-Newtonian flows. In most cases, non-Newtonian models can be implemented in existing Newtonian solvers by relatively simple modifications of the viscosity. However, due to the scarcity of analytical solutions for non-Newtonian fluid flows and the widespread use of regularization methods, performing rigorous code verification is a challenging task. The method of manufactured solutions (MMS) is a powerful tool to generate analytical solutions for code verification. In this article, we present and discuss the results of three verification exercises based on MMS: (i) steady single-phase flow; (ii) unsteady two-phase flow with a smooth interface; (iii) unsteady two-phase flow with a free surface. The first and second exercises showed that rigorous verification of non-Newtonian fluid solvers is possible both on single- and two-phase flows. The third exercise revealed that “spurious velocities” typical of free-surface calculations with the Volume-of-Fluid model lead to “spurious viscosities” in the non-Newtonian fluid. The procedure is illustrated herein on a second-order finite volume flow solver, using the regularized Herschel-Bulkley fluid model as an example. The same methodology is however applicable to any flow solver and to all the rheological models falling under the class of generalized Newtonian fluid models.

The seakeeping ability of ships is one of the aspects that needs to be assessed during the design phase of ships. Traditionally, potential flow calculations and model tests are employed to investigate whether the ship performs according to specified

Collaboration is described on assessment of computational fluid dynamics (CFD) predictions for surface combatant model 5415 at static drift β = 0 deg and 20 deg using recent tomographic particle image velocimetry (TPIV) experiments. Assessment includes N-version verification and validation to determine the confidence intervals for CFD solutions/codes, and vortex onset, progression, instability, and turbulent kinetic energy (TKE) budget analysis. The increase in β shows the following trends. Forces and moment increase quadratically/cubically, and become unsteady due to shear layer, Karman and flapping instabilities on the bow. Wave elevation becomes asymmetric; its amplitude increases, but the total wave elevation angle remains same. The vortex strength and TKE increase by about two orders of magnitude, and for large β, the primary vortices exhibit helical mode instability similar to those for delta wings. Forces and moment for both β and wave elevation for β = 0 deg are compared within 4% of the data, and are validated at 7% interval. Wave elevation for β = 20 deg, and vortex core location and velocities for both β are compared within 9% of the data, and are validated at 12% interval. The vortex strength and TKE predictions show large 70% errors and equally large scatter and are not validated. Thus, both errors and scatter need reduction. TKE budgets show transport of turbulence into the separation bubble similar to canonical cases, but pressure transport is dominant for ship flows. Improved CFD predictions require better grids and/or turbulence models. Investigations of solution-adaptive mesh refinement for better grid design and hybrid Reynolds-averaged Navier-Stokes/large eddy simulation models for improved turbulent flow predictions are highest priority.