Frederick Stern

Comprehensive Approach to Verification and Validation of CFD Simulations—Part 1: Methodology and Procedures

We present a comprehensive approach to verification and validation methodology and procedures for CFD simulations from an already developed CFD code applied without requiring availability of the source code for specified objectives, geometry, conditions, and available benchmark information. Concepts, definitions, and equations derived for simulation errors and uncertainties provide the overall mathematical framework. Verification is defined as a process for assessing simulation numerical uncertainty and, when conditions permit, estimating the sign and magnitude of the numerical error itself and the uncertainty in that error estimate. The approach for estimating errors and uncertainties includes (1) the option of treating the numerical error as deterministic or stochastic, (2) the use of generalized Richardson extrapolation for J input parameters, and (3) the concept of correction factors based on analytical benchmarks, which provides a quantitative metric to determine proximity of the solutions to the asymptotic range, accounts for the effects of higher-order terms, and are used for defining and estimating errors and uncertainties

Uncertainties and CFD Code Validation

A new approach to computational fluid dynamics code validation is developed that gives proper consideration to experimental and simulation uncertainties. The comparison error is defined as the difference between the data and simulation values and represents the combination of all errors. The validation uncertainly is defined as the combination of the uncertainties in the experimental data and the portion of the uncertainties in the CFD prediction that can be estimated. This validation uncertainty sets the level at which validation can be achieved. The criterion for validation is that the magnitude of the comparison error must be less than the validation uncertainty. If validation is not accomplished, the magnitude and sign of the comparison error can be used to improve the mathematical modeling. Consideration is given to validation procedures for a single code, multiple codes and/or models, and predictions of trends. Example results of verification/validation are presented for a single computational fluid dynamics code and for a comparison of multiple turbulence models. The results demonstrate the usefulness of the proposed validation strategy. This new approach for validation should be useful in guiding future developments in computational fluid dynamics through validation studies and in the transition of computational fluid dynamics codes to design.

Comprehensive Approach to Verification and Validation of CFD Simulations—Part 2: Application for Rans Simulation of a Cargo/Container Ship

Part 2 of this two-part paper provides an example case study following the recently developed comprehensive verification and validation approach presented in Part 1. The case study is for a RANS simulation of an established benchmark for ship hydrodynamics using a ship hydrodynamics CFD code. Verification of the resistance (integral variable) and wave profile (point variable) indicates iterative uncertainties much less than grid uncertainties and simulation numerical uncertainties of about 2%S 1 (S1 is the simulation value for the finest grid). Validation of the resistance and wave profile shows modeling errors of about 8%D (D is the measured resistance) and 6% z max (z max is the maximum wave elevation), which should be addressed for possible validation at the 3%D and 4%z max levels. Reducing the level of validation primarily requires reduction in experimental uncertainties. The reduction of both modeling errors and experimental uncertainties will produce verified and validated solutions at low levels for this application using the present CFD code. Although there are many issues for practical applications, the methodology and procedures are shown to be successful for assessing levels of verification and validation and identifying modeling errors in some cases. For practical applications, solutions are far from the asymptotic range; therefore, analysis and interpretation of the results are shown to be important in assessing variability for order of accuracy, levels of verification, and strategies for reducing numerical and modeling errors and uncertainties. @DOI: 10.1115/1.1412236#

Factors of Safety for Richardson Extrapolation

A factor of safety method for quantitative estimates of grid-spacing and time-step uncertainties fbr solution verification is developed. It removes the two deficiencies of the grid convergence index and correction factor methods, namely, unreasonably small uncertainty when the estimated order of accuracy using the Richardson extrapolation method is greater than the theoretical order of accuracy and lack of statistical evidence that the interval of uncertainty at the 95% confidence level bounds the comparison error. Different error estimates are evaluated using the effectivity index. The uncertainty estimate builds on the correction factor method, but with significant improvements. The ratio of the estimated order of accuracy and theoretical order of accuracy P instead of the correction factor is used as the distance metric to the asymptotic range. The best error estimate is used to construct the uncertainty estimate. The assumption that the factor of safety is symmetric with respect to the asymptotic range was removed through the use of three instead of two factor of safety coefficients. The factor of safety method is validated using statistical analysis of 25 samples with different sizes based on 17 studies covering fluids, thermal, and structure disciplines. Only the factor of safety method, compared with the grid convergence index and correction factor methods, provides a reliability larger than 95% and a lower confidence limit greater than or equal to 1.2 at the 95% confidence level for the true mean of the parent population of the actual factor of safety. This conclusion is true for different studies, variables, ranges of P values, and single P values where multiple actual factors of safety are available. The number of samples is large and the range of P values is wide such that the factor of safety method is also valid for other applications including results not in the asymptotic range, which is typical in industrial and fluid engineering applications. An example for ship hydrodynamics is provided.

Sharp interface immersed-boundary/level-set method for wave-body interactions

A sharp interface Cartesian grid method for the large-eddy simulation of two-phase turbulent flows interacting with moving bodies is presented. The overall approach uses a sharp interface immersed boundary formulation and a level-set/ghost-fluid method for solid-fluid and fluid-fluid interface treatments, respectively. A four-step fractional-step method is used for velocity-pressure coupling, and a Lagrangian dynamic Smagorinsky subgrid-scale model is adopted for large-eddy simulations. A simple contact angle boundary condition treatment that conforms to the immersed boundary formulation is developed. A variety of test cases of different scales ranging from bubble dynamics, water entry and exit, landslide-generated waves, to ship hydrodynamics are performed for validation. Extensions for high Reynolds number ship flows using wall-layer models are also considered.

A simple and efficient direct forcing immersed boundary framework for fluid-structure interactions

A direct forcing immersed boundary framework is presented for the simple and efficient simulation of strongly coupled fluid-structure interactions. The immersed boundary method developed by Yang and Balaras [J. Yang, E. Balaras, An embedded-boundary formulation for large-eddy simulation of turbulent flows interacting with moving boundaries, J. Comput. Phys. 215 (1) (2006) 12-40] is greatly simplified by eliminating several complicated geometric procedures without sacrificing the overall accuracy. The fluid-structure coupling scheme of Yang et al. [J. Yang, S. Preidikman, E. Balaras, A strongly-coupled, embedded-boundary method for fluid-structure interactions of elastically mounted rigid bodies, J. Fluids Struct. 24 (2008) 167-182] is also significantly expedited by moving the fluid solver out of the predictor-corrector iterative loop without altering the strong coupling property. Central to these improvements are the reformulation of the field extension strategy and the evaluation of fluid force and moment exerted on the immersed bodies, by taking advantage of the direct forcing idea in a fractional-step method. Several cases with prescribed motions are examined first to validate the simplified field extension approach. Then, a variety of strongly coupled fluid-structure interaction problems, including vortex-induced vibrations of a circular cylinder, transverse and rotational galloping of rectangular bodies, and fluttering and tumbling of rectangular plates, are computed. The excellent agreement between the present results and the reference data from experiments and other simulations demonstrates the accuracy, simplicity, and efficiency of the new method and its applicability in a wide range of complicated fluid-structure interaction problems.

Numerical Ship Hydrodynamics - An Assessment of the Gothenburg 2010 Workshop

The Gothenburg 2010 Workshop on CFD in Ship Hydrodynamics was the sixth in a series starting in 1980. The purpose of the Workshops is to assess the state of the art in CFD for hydrodynamic applications. Active researchers in the field worldwide are invited to provide computed results for a number of well specified test cases, and the organizers collect and present the results such that comparisons between different methods can be made easily. Detailed information about each method is also reported via a questionnaire provided by the organizers. All results are discussed at a meeting, and a final assessment of the workshop is made by the organizers. The present workshop attracted 33 groups from all over the world, and different types of computations were carried out for three hulls. It was by far the largest of the workshops in the series so far. All computed results were compiled in a volume, called Proceedings II, and distributed at the meeting, which was held in Gothenburg 8-12 December 2010. Unlike previous workshops, there was no presentation of submitted papers. Instead, the three main organizers gave reviews of the submitted results at the meeting, and most of the time was spent on discussions of these reviews. In this book, updated versions of the reviews are presented, together with a verification and validation study of the submitted resistance predictions, as well as new measurement data obtained after the workshop and a comprehensive set of additional computations carried out by the organizers to investigate topics of particular interest found at the meeting. The book has been written by the three main organizers and their co-workers. Together with supplementary information on the web site SpringerExtras (for address, se book cover) the book constitutes the final documentation of the Gothenburg 2010 Workshop and gives a state-of-the-art assessment of the CFD capabilities within the area of Ship Hydrodynamics.

A new volume-of-fluid method with a constructed distance function on general structured grids

A second-order volume-of-fluid method (VOF) is presented for interface tracking and sharp interface treatment on general structured grids. Central to the new method is a second-order distance function construction scheme on a general structured grid based on the reconstructed interface. A novel technique is developed for evaluating the interface normal vector using the distance function. With the normal vector, the interface is reconstructed from the volume fraction function via a piecewise linear interface calculation (PLIC) scheme on the computational domain. Several numerical tests are conducted to demonstrate the accuracy and efficiency of the present method. In general, the new VOF method is more efficient than both the high-order level set and the coupled level set and volume-of-fluid (CLSVOF) methods. The results from the new method are better than those from the benchmark VOF method, particularly in the under-resolved regions, and are comparable to those from the CLSVOF method. Breaking waves over a submerged bump and around a wedge-shaped bow are simulated to demonstrate the application of the new method and sharp interface treatment in a two-phase flow solver on curvilinear grids. The computational results are in good agreement with the available experimental measurements.

Computational Towing Tank Procedures for Single Run Curves of Resistance and Propulsion

A procedure is proposed to perform ship hydrodynamics computations for a wide range of velocities in a single run, herein called the computational towing tank. The method is based on solving the fluid flow equations using an inertial earth-fixed reference frame, and ramping up the ship speed slowly such that the time derivatives become negligible and the local solution corresponds to a quasi steady-state. The procedure is used for the computation of resistance and propulsion curves, in both cases allowing for dynamic calculation of the sinkage and trim. Computational tests are performed for the Athena R/V model DTMB 5365, in both bare hull with skeg and fully appended configurations, including two speed ramps and extensive comparison with experimental data. Comparison is also performed against steady-state points, demonstrating that the quasisteady solutions obtained match well the single-velocity computations. A verification study using seven systematically refined grids was performed for one Froude number, and grid convergence for resistance coefficient, sinkage, and trim were analyzed. The verification study concluded that finer grids are needed to reach the asymptotic range, though validation was achieved for resistance coefficient and sinkage but not for trim. Overall results prove that for medium and high Froude numbers the computational towing tank is an efficient and accurate tool to predict curves of resistance and propulsion for ship flows using a single run. The procedure is not possible or highly difficult using a physical towing tank suggesting a potential of using the computational towing tank to aid the design process.

Computational ship hydrodynamics: Nowadays and way forward

Computational fluid dynamics for ship hydrodynamics has made monumental progress over the last ten years, which is reaching the milestone of providing first-generation simulation-based design tools with vast capabilities for model- and full-scale simulations and optimization. This is due to the enabling technologies such as free surface tracking/capturing, turbulence modeling, 6 degree of freedom (DoF) motion prediction, dynamic overset grids, local/adaptive grid refinement, high performance computing, environmental modeling and optimization methods. Herein, various modeling, numerical methods, and high performance computing approaches for computational ship hydrodynamics are evaluated thereby providing a vision for the development of the next-generation high-fidelity simulation tools. Verification and validation procedures and their applications, including resistance and propulsion, seakeeping, maneuvering, and stability and capsizing, are reviewed. Issues, opportunities, and challenges for advancements in higher-fidelity two-phase flow are addressed. Fundamental studies for two-phase flows are also discussed. Conclusions and future directions are also provided.