Complementary numerical–experimental benchmarking for shape optimization and validation of structures subjected to wave and current forces
Introduction
The utilization of numerical methods to solve fluid dynamics problems continues to be a topic of high interest in research and development. Despite rigorous and ongoing efforts in the field, physically accurate and efficient modeling of fluid flows remains to be a challenging task. Nevertheless, many of the developed methods are gaining in importance beyond pure research efforts, as they are applied industrially in various fields of engineering. As a result of developments in computer technology, improvements in the numerical methods underlying commonly used fluid software, and simpler handling of commercial and open source programs, numerical methods are increasingly utilized to solve complex design problems. However, the readiness to produce results does not always reflect the complexity of the problems and the knowledge required to apply the models appropriately. Therefore, as new methods and software packages emerge, verification and validation is ever more important.
For the purpose of testing new and established methods, a great variety of benchmark studies have been proposed and analyzed. Numerical and experimental results for a number of fundamental flow problems were documented by Freitas [14]. These include the two dimensional flow over a backward-facing step and the three dimensional flow in a shear-driven cubical cavity. Concerning flows around a structure, the cylinder is arguably the best documented reference problem, including studies at various flow conditions, orientations and aspect ratios (e.g. [48], [9], [40], [18], [41], [55], [56], [3]). For computational wind engineering applications, the Silsoe cube, a 6 m cube exposed to a physical wind field, has been studied vigorously both experimentally and numerically (e.g. [25], [46], [21]). For the purpose of code refinements and validation of aeroelastic problems, NASA carried out a program that established the NACA 0012 air foil benchmark, which has been the subject of various experimental investigations (e.g. [37], [50]) and numerical studies (e.g. [44], [10]). Fluid Structure Interaction (FSI) codes are frequently verified based on the so-called “Turek benchmark”, consisting of an elastic object in an incompressible flow [53], [54]. For free surface methods, a dam-break flow [43], breaking wave impact [19] and sloshing in tanks [29] serve as popular validation problems.
The aforementioned studies serve as an example of the multitude of different benchmark tests that provide excellent information for validation and verification of different fluid related solution methods and modeling aspects. However, only few studies have been carried out that specifically target optimization problems, where the primary concern is to capture the differences between the individual optimization runs and to attain a precise optimal solution. Even fewer studies focus specifically on shape optimization in ocean engineering related topics, which typically involve not only steady current flows but also dynamic wave loading. Those studies concerned with ocean engineering related applications have mainly targeted the shape optimization of ship hulls (e.g. [45], [5], [51]). As part of these studies, the results of the numerical optimization runs were validated by carrying out a few selected experimental reference tests. Although the approach was very successful, the complexity of the analyzed problems renders these studies not ideally suited as general benchmark problems for the validation of codes and methods. This pertains particularly to the complex three dimensional hull geometries that require a significant discretization effort when utilizing mesh based numerical methods. Furthermore, the solution of the ensuing three dimensional flow problem involves substantial computational costs. Complex optimization problems often also suffer from being highly none-convex, which may be problematic when testing certain optimization approaches.
This paper addresses the need for a simple optimization benchmark problem that can efficiently be applied as part of fluid related validation and verification studies. A successful benchmark should exhibit certain characteristics in terms of the degree of complexity and the required resources necessary to solve the benchmark problem. While an oversimplification of the problem to guarantee easy reproducibility may result in a lack of useful results, overly complex benchmark problems become superfluous if a solution cannot be attained with reasonable effort, or if the likelihood of producing invalid results is too high [13]. Therefore, the benchmark problem should be complex enough to reflect the essential characteristics of the application at hand, while simultaneously allowing for a solution of the problem within a reasonable amount of time and effort. In this particular case, the essential physical phenomenon chosen to be modeled and verified is the horizontal load-sensitivity to shape variations of bottom mounted structures subjected to inertia and drag forces during wave and current loading. These forces are of utmost importance in the design process of reliable offshore structures. The essentially very complex problem definition is simplified through two main limitations in the definition of the benchmark problem: First of all, only one parameter is chosen for variation of the structure geometry, reducing the modeled objective function to a 1D problem in terms of design variables. Secondly, the problem is configured such that the overall domain can be modeled in a 2D environment in terms of space coordinates, which significantly reduces the computational effort required to solve the problem. Under both of these problem definitions, the physical characteristics of wave and current induced inertia and drag forces can be retained, therefore preserving the initial characteristics of the problem.
The benchmark problem definition of this paper is distinctive in that it allows for testing of highly complex flow conditions in a two-dimensional framework. Typically, 2D benchmarks are presented for relatively simple flow problems, such as the aforementioned flow over a backward-facing step or the modeling of the von Kármán vortex street about a cylinder. On the other hand, more complex flow fields are usually given in the literature in conjunction with complex 3D models, such as the mentioned ship hull investigations. By presenting experimental reference results of a complex wave–current flow that can be modeled numerically in a 2D environment, a highly efficient problem setup is created that is ideally suited as an optimization benchmark problem solved by analyzing multiple geometry variations.
The paper provides not only an experimental reference solution, but also analyzes the potential of Computational Fluid Dynamics (CFD) as a powerful tool as part of shape optimization studies of structures. Both the experimental and numerical setup are described in detail in order to allow for and encourage future comparative validation and verification studies. A detailed description of the measured and simulated boundary conditions is given, including turbulence data for the considered flow conditions. Following, results for peak horizontal loads on the geometries as well as selected load time series are presented and analyzed. Objective function approximations are compared, showing the capabilities of numerical methods to predict changes in loading as a result of shape variations. Furthermore, data is provided to validate wave–current interaction studies. Finally, the study is extended to a consideration of 3D flow scenarios, as a reference for computationally demanding 3D validation studies. Overall, the presented work gives a comprehensive description of a reproducible benchmark problem for shape optimization, embracing a brought spectrum of different flow conditions, motivated by structural design requirements in ocean engineering.
Section snippets
Benchmark geometry specification
The body to be analyzed in the benchmark study is a bottom mounted, fully submerged structure that is varied in one design variable only. A clear focus is put on simplicity and reproducibility, in order to minimize preparation required to carry out comparison studies. The benchmark geometry consists of a polygonal shape that is parametrized using the design variable , the leading and trailing inclination angle of the geometries, as shown in Fig. 1a. The point coordinates to are
Flume
The experimental investigations for the introduced benchmark study were carried out in the Aalborg University (Denmark) wave–current flume. The flume consists of a piston wave maker as well as a recirculating water pump that drives a current from the outlet to the inlet of the flume through pipes located below the flume floor. A layout of the flume is given in Fig. 2. The flume bottom is horizontal in the test section and slopes towards the wave paddle at an inclination angle of approximately 3
Fundamental equations
Computational Fluid Dynamics (CFD) is utilized in order to carry out the numerical benchmark simulations of waves and currents. A solution of the problem is attained by solving the URANS equations in combination with a Volume of Fluid approach. The governing equations are derived by retaining the rate of change in the incompressible RANS equations, leading to the following averaged continuity equation and averaged system of momentum equations:
Flow kinematics
The flow conditions considered in the optimization benchmark include both current and wave flows, as well as the combination of both scenarios. Overall, three current velocities and three wave heights are modeled both physically and experimentally, as defined in Table 4a and b, respectively. Here, the velocity corresponds to the current surface velocity and denotes to the depth averaged current velocity. The water depth d for all cases is equal to 0.35 m. This depth was mainly dictated by
Conclusions
The presented optimization benchmark is a result of an elaborate experimental and numerical investigation, including data from over 450 experimental test runs and more than 200 numerical simulations. Based on this data set, a great variety of different test scenarios was presented that allows for a widespread validation and verification related to the following topics:
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Open channel flow kinematics of waves, currents, and combined wave–current conditions.
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Maximum horizontal forces on bottom
Acknowledgements
The authors wish to express their sincere gratitude to the International Graduate School of Science and Engineering (IGSSE project team 7.05) at TU München as well as the Marine Renewables Infrastructure Network (MARINET funding code TA1-ShapeOptGravityBase) for supporting the research documented in this paper.
References (60)
- et al.
Shape optimization in ship hydrodynamics using computational fluid dynamics
Comput Methods Appl Mech Eng
(2006) The numerical solution of steady water wave problems
Comput Geosci
(1988)- et al.
Turbulence levels in a flume compared to the field: implications for larval settlement studies
J Sea Res
(2006) - et al.
Volume of fluid (VOF) for the dynamics of free boundaries
J Comput Phys
(1981) - et al.
Towards quality assurance for wind tunnel tests: a comparative testing program of the Windtechnologische Gesellschaft
J Wind Eng Ind Aerodyn
(1998) - et al.
Hydrodynamic coefficients induced by waves and currents for submerged circular cylinder
Proc Eng
(2010) - et al.
A numerical study of three-dimensional liquid sloshing in tanks
J Comput Phys
(2008) - et al.
A virtual free surface (VFS) model for efficient wave–current CFD simulation of fully submerged structures
Coast Eng
(2014) - et al.
Damping of unwanted turbulence in wave–current experiments
Coast Eng
(2015) - et al.
A numerical investigation of combined wave–current loads on tidal stream generators
Ocean Eng
(2013)
Flow around rectangular cylinders: pressure forces and wake frequencies
J Wind Eng Ind Aerodyn
Hydrodynamic optimization of ship hull forms
Appl Ocean Res
Wind pressures on a 6m cube
J Wind Eng Ind Aerodyn
Neural-network-based d-step-ahead predictors for nonlinear systems with time delay
Eng Appl Artif Intell
An experimental investigation of hydrodynamic coefficients for a vertical truncated rectangular cylinder due to regular and random waves
Ocean Eng
An introduction to fluid mechanics
J Appl Mech
Benchmark computations of 3D laminar flow around a cylinder with CFX, OpenFOAM and FeatFlow
Int J Comput Sci Eng
Direkte Numerische Simulation und Large-Eddy Simulation turbulenter Strömungen auf Hochleistungsrechnern
On the interaction between a turbulent open channel flow and an axial-flow turbine
J Fluid Mech
Stream function representation of nonlinear ocean waves
J Geophys Res
Numerical solutions for steady flow past a circular cylinder at reynolds numbers up to 100
J Fluid Mech
Evaluation of the turbulence models for the simulation of the flow over a national advisory committee for aeronautics (NACA) 0012 airfoil
J Mech Eng Res
Neural networks in civil engineering. i: Principles and understanding
J Comput Civ Eng
Perspective: selected benchmarks from commercial CFD codes
J Fluids Eng
Turbulence measurements with acoustic doppler velocimeters
J Hydraul Eng
Despiking acoustic doppler velocimeter data
J Hydraul Eng
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