Stokes Solver Research Articles

Numerical simulation of unsteady hydrodynamic performance of CRP

Strong hydrodynamic interference between forward and aft contra-rotating propellers (CRP) can result in load fluctuations, local blade vibrations, and noise. This study adopts a Reynolds-averaged Navier–Stokes solver (RANS) and a shear stress transport (SST) k−ω turbulence model to investigate the unsteady hydrodynamic performance of CRPs during uniform flow. The influences of time step and grid scale on the numerical simulation are discussed first. The numerical results of the open-water hydrodynamic performances of the CRPs agree well with experimental data. The fluctuation patterns of the hydrodynamic loads (i.e., force and torque) and propulsion efficiency of CRPs having different advance coefficients induced by the mutual hydrodynamic interference between the forward and aft propeller are investigated. A series of probes that rotate with the propellers are set upon the surface of the blades to analyze the unsteady local pressures under the hydrodynamic interference between the forward and aft propellers.

Parallel defect-correction methods for incompressible flows with friction boundary conditions

We study three parallel defect-correction methods based on finite element approximations for the incompressible Navier–Stokes problem with friction boundary conditions and high Reynolds numbers in this work, where a fully overlapping domain decomposition is considered for parallelization. In the proposed methods, with a global multiscale grid that builds a fine grid around its own subdomain and coarse elsewhere, we iteratively solve an artificial viscosity nonlinear variational inequality problem in a defect step, and then compute the residual by the linearized variational inequality problems in the r-step corrections. The studied methods are easy to implement on the basis of the existing Navier–Stokes solver and possess less communication complexity. We provide a rigorously theoretical derivation for the error estimates of the one-step correction solutions from the proposed methods under some stable conditions, and derive scalings of the algorithmic parameters. We demonstrate by a series of numerical experiments that the velocity and pressure errors computed by our parallel defect-correction methods are comparable to those of the standard defect-correction method, while our present methods reduce the computational cost.

Stationary Stokes solver for single-phase flow in porous media: A blastingly fast solution based on Algebraic Multigrid Method using GPU

The paper is focused on high efficiency Stokes solver that is applied to the incompressible flow in porous media. Computational domains are represented by binarized 3D computed tomography voxel models. The solution procedure is constructed around a fast algebraic multigrid solver that utilizes the power of graphics processing units (GPUs). In order to minimize memory footprint and accelerate the solver, a simple MAC-type staggered finite difference discretization is used and a coupled Stokes saddle-point type system is solved directly on a GPU. The MAC discretization is discussed, taking particular care of internal solid boundaries around pores. The method includes topological domain analysis on a GPU, which removes isolated volumes with no flow in the domain (based on the connected component labeling), depending on the boundary conditions. We consider various types of boundary conditions and efficient parallel strategies for the GPUs, including fast matrix assembly and residual regularization. Our method is extensively benchmarked both against analytical solutions and applied problems from digital rock physics in terms of computational wall time, precision, approximation order, and convergence. We demonstrate that it takes up to 5–23 s on a modern GPU card to obtain a solution with 1⋅10−6 error residuals for 3D geometries with 300–4503 voxels and a porosity range of 5–37%.

Numerical study of the interaction between the pure loss of stability, surf-riding, and broaching on ship capsizing

Ship capsizing in stern quartering wave is very complicate with a strong correlation to the stability failure modes proposed in second-generation intact stability criteria. It is of great importance to shed some light on this complex phenomenon. In this study, the capsizing behavior of a free-running ONR Tumblehome model in stern quartering seas is investigated by an unsteady Reynolds-averaged Navier–Stokes (URANS) solver coupled with dynamic overset grid approach. Validation study of the URANS solver is performed for the prediction of extreme roll motion of the ship. The comparisons show satisfactory agreement, with an average difference less than 3% between the simulation and experimental results. The correlation between the occurrence of stability failure modes (surf-riding, pure loss of stability, and broaching) and ship capsizing is then assessed under different ship speed and heading angle conditions. It is concluded that the capsizing occurs as the free-running ship sailing with a high speed, which is caused by the interaction between the broaching and severe stability loss. In addition, influenced by the wave-induced moment, both the pure loss of stability and surf-riding modes may cause the broaching, indicating the similar occurring condition and strong relation between these stability failure modes.

Numerical simulation of high peak overpressure blast wave through shock tube and its interaction with a rectangular object

The generation of high peak overpressure (>10 bar) blast wave inside the shock tube and its interaction with a rectangular object is examined in this paper using the in-house developed multi-component Navier–Stokes solver. It is difficult to generate these peak overpressure blast waves in the laboratory due to aerodynamic noise and strong reflected blast waves. Driver and driven section lengths of the shock tube are 0.8 m and 6 m respectively. A blast wave with a non-dimensional peak overpressure of 11.65 is generated inside the shock tube using helium as the driver section gas with a diaphragm pressure ratio of 57. The peak overpressure and pressure histories obtained from simulations are validated with shock tube experiments and found to be matching closely. Next, the development of a strong trailing jet after diffraction of the blast wave at the shock tube exit is studied. Finally, the interaction of blast wave with a rectangular object is studied using numerical schlieren images, line plots of pressure and density.

Comparison of aerodynamic models for horizontal axis wind turbine blades accounting for curved tip shapes

AbstractCurved tip extensions are among the rotor innovation concepts that can contribute to the higher performance and lower cost of horizontal axis wind turbines. One of the key drivers to exploit their advantages is the use of accurate and efficient computational aerodynamic models during the design stage. The present work gives an overview of the performance of different state‐of‐the‐art models. The following tools were employed, in descending order of complexity: (i) a blade‐resolved Navier Stokes solver, (ii) a lifting line model, (iii) a vortex‐based method coupling a near‐wake model with a far‐wake model, and (iv) two implementations of the widely used blade element momentum method (BEM), with and without radial induction. The predictions of the codes were compared when simulating the baseline geometry of a reference wind turbine and different tip extension designs with relatively large sweep angle and/or dihedral angle. Four load cases were selected for this comparison, to cover several aspects of the aerodynamic modeling: steady power curve, pitch step, extreme operating gust impact, and standstill in deep stall. The present study highlighted the limitations of the BEM‐based formulations to capture the trends attributed to the introduction of curvature at the tip. This was true even when using the radial induction submodel. The rest of the computational methods showed relatively good agreement in most of the studied load cases. An exception to this was the standstill configuration, as the blade‐resolved Navier‐Stokes solver was the only code able to capture the highly unsteady effects of deep stall.

A Momentum-Conserving Weakly Compressible Navier–Stokes Solver for Simulation of Violent Two-Phase Flows with High Density Ratio

A consistent and conservative formulation for mass and momentum transport is proposed in the context of simulating incompressible two-phase flows by using weakly compressible method. Combined with the evolving pressure projection method to prevent oscillation of the solution induced by the acoustic wave, this solver aims at a robust and accurate computation of violent two-phase flows with a high density ratio, while taking advantage of fully explicit time integration of the weakly compressible Navier–Stokes equations. Coupled with the volume of fluid method for capturing interfaces, the mass and momentum fluxes are evaluated in a consistent manner using the finite volume method. In addition, a special implementation of the pressure projection is devised to avoid velocity-pressure decoupling on a collocated grid. The solver's accuracy and stability are demonstrated through various two-phase flow simulations, including dam break and liquid jet atomization scenarios, emphasizing its momentum-conserving properties.

Computational Investigation of Blade–Vortex Interaction of Coaxial Rotors for eVTOL Vehicles

In the design of electric vertical takeoff and landing (eVTOL) vehicles, coaxial rotors have garnered significant attention due to their superior space usage and aerodynamic efficiency compared to standard rotors. However, it is challenging to study the flow field near the rotors due to the blade–vortex interface (BVI) and vortex–vortex contact between two rotors. Using sliding mesh technology and Reynolds-averaged Navier–Stokes (RANS) solvers, a numerical method was established to simulate the flow field of a coaxial rotor in hover, which was verified by experiments. Using this method, this paper analyzes the relationship between position and intensity of the tip vortex of the upper rotor, the axial velocity of induced flow and the load distribution on the blades at the azimuth when the BVI phenomenon occurs with a difference in rotational speed and rotor spacing. The results indicate that, when the BVI phenomenon appears, the blade-tip vortex of the top rotor rapidly dissipates, and the load distribution of the lower blade changes due to the induced flow of the vortex. When the rotational speed increases, the spanwise thrust coefficient of each rotor changes slightly. The vortex–vortex interaction becomes stronger, which leads to vortex pairing. When the distance between the rotors decreases, the BVI phenomenon occurs at an earlier azimuth and the location of the BVI moves towards the tip of the lower blade. The vortex–vortex interaction is also enhanced, which leads to vortex pairing and vortex merging.

NekRS, a GPU-accelerated spectral element Navier–Stokes solver

The development of NekRS, a GPU-oriented thermal-fluids simulation code based on the spectral element method (SEM) is described. For performance portability, the code is based on the open concurrent compute abstraction and leverages scalable developments in the SEM code Nek5000 and in libParanumal, which is a library of high-performance kernels for high-order discretizations and PDE-based miniapps. Critical performance sections of the Navier–Stokes time advancement are addressed. Performance results on several platforms are presented, including scaling to 27,648 V100s on OLCF Summit, for calculations of up to 60B grid points (240B degrees-of-freedom).

Prediction of Unstable Hydrodynamic Forces on Submerged Structures under the Water Surface Using a Data-Driven Modeling Approach

Catastrophic failures of partially or fully submerged structures, e.g., offshore platforms, hydrokinetic turbine blades, bridge decks, etc., due to the dynamic impact of free surface flows such as waves or floods have revealed the need to evaluate their reliability. In this respect, an accurate estimation of hydrodynamic forces and their relationship to instability in structures is required. The computational fluid dynamics (CFD) solver is known as a powerful tool to identify dynamic characteristics of flow; however, it commonly consumes a huge computational cost, especially in cases of re-simulations needed. In this paper, an efficient surrogate model based on the Gaussian process is developed to rapidly predict the nonlinear hydrodynamic pressure coefficients on submerged bodies near the water surface. For this purpose, a CFD model is first developed, which is based on a two-dimensional incompressible Navier–Stokes solver incorporating free surface treatment and turbulent flow models. Then, an experimental design is adopted to generate initial training samples considering the effect of the submerged body shape ratio and flow Re number. Surrogate models of hydrodynamic pressure coefficients and their instability based on Gaussian process modeling are established using the outcome from the CFD simulations, where optimal trend and correlation functions are also investigated. Once surrogate models are obtained, the mean and oscillation amplitudes of hydrodynamic pressure coefficients on a submerged rectangular body, which represents the shape of most civil structures, can be rapidly predicted without the attempt at re-simulation. The findings can be practically applied in rapidly assessing hydrodynamic forces and their instability of existing submerged civil structures or in designing new structures, where a suitable shape ratio should be adopted to avoid flow-induced instability of hydrodynamic forces.

Hydrodynamic interaction study during surface ship overtaking submarine

Submarine is easily disturbed by the passing surface ships when submarine sailing near the sea surface, the disturbance can greatly affect the hydrodynamic performance and sailing status of the submarine during navigation. In this study, A model-scale tanker (KVLCC2,LPP = 32m) and a model-scale submarine (SUBOFF,Loa = 6.534m) were used to study the hydrodynamic interaction to the submarine when the surface ship overtakes the submarine sailing below it. An in-house unsteady Reynolds-averaged Navier–Stokes solver was applied to the simulation. First, uncertainty analysis of the computational fluid dynamics (CFD) solver was conducted, and validation study based on available experimental data showed good agreement. The hydrodynamic interaction to the submarine during the overtaking process was then investigated numerically with different submarine forward speed, tanker forward speed and different tanker-submarine vertical distance. The results showed that the vertical distance between tanker and submarine has great influence on submarine, the peak values of vertical force and pitching moment of submarine were obviously affected by the forward speed of tanker and submarine, meanwhile, The interaction force on submarine resistance was less effected by submarine forward speed, but the forward speed of tanker was more effected.

Edge-Based Viscous Method for Node-Centered Finite-Volume Formulation on Tetrahedra

This paper presents a novel, efficient, edge-based viscous (EBV) discretization for finite-volume, node-centered formulations on tetrahedral grids. This new method is implemented in a practical, unstructured-grid Reynolds-averaged Navier–Stokes solver and applied to viscous-kernel computations that include evaluations of mean flow viscous fluxes, turbulence-model diffusion terms, and the corresponding Jacobian contributions. The EBV method uses an efficient loop over edges and features a compact discretization stencil based on nearest neighbors. This study verifies the EBV method by comparing its accuracy and iterative convergence with those of a well-established method based on a cell-based approach to discretization of viscous fluxes. Multifold efficiency gains for all viscous-kernel computations on tetrahedral grids are reported, leading to a significant reduction of the overall time to solution.

Ensemble Kalman method for learning turbulence models from indirect observation data

In this work, we propose using an ensemble Kalman method to learn a nonlinear eddy viscosity model, represented as a tensor basis neural network, from velocity data. Data-driven turbulence models have emerged as a promising alternative to traditional models for providing closure mapping from the mean velocities to Reynolds stresses. Most data-driven models in this category need full-field Reynolds stress data for training, which not only places stringent demand on the data generation but also makes the trained model ill-conditioned and lacks robustness. This difficulty can be alleviated by incorporating the Reynolds-averaged Navier–Stokes (RANS) solver in the training process. However, this would necessitate developing adjoint solvers of the RANS model, which requires extra effort in code development and maintenance. Given this difficulty, we present an ensemble Kalman method with an adaptive step size to train a neural-network-based turbulence model by using indirect observation data. To our knowledge, this is the first such attempt in turbulence modelling. The ensemble method is first verified on the flow in a square duct, where it correctly learns the underlying turbulence models from velocity data. Then the generalizability of the learned model is evaluated on a family of separated flows over periodic hills. It is demonstrated that the turbulence model learned in one flow can predict flows in similar configurations with varying slopes.

Wave impacts on a solid deck in transient wave groups

Wave impacts on an elevated solid deck due to transient focused wave groups are studied numerically. Previously reported experiments with and without an I-beam grillage beneath the deck by Santo et al. (2020) are reproduced successfully in a three-dimensional numerical wave tank based on a two-phase Navier–Stokes solver. The impact loads on the solid deck are relatively simple. The three-dimensional horizontal force, characterised by a single peak in time, is close to two-dimensional, whereas the vertical force consists of upward force and downward suction force. The downward suction force is related to triangularisation of the wetted area underneath the deck and dominated by the added mass effect, which is a three-dimensional effect and therefore any two-dimensional simulation will overpredict the strongly three-dimensional process and the vertical impact loads. The wave impact loads on the solid deck with a grillage are more complicated, with successive force spikes observed for both the horizontal and vertical loads. The significance of entrapped air pockets in the grillage to global wave impact loads is ascertained through interrogating flow field of numerical experiments. It is found that large upward vertical impulsive forces are caused by high local pressures when entrapped down-wave air-pockets are formed, while large downward suction forces are resulted from both high-frequency up-wave air-pocket effect and the low-frequency added mass effect. Large horizontal impulsive forces are due to the combined effects of the down-wave air-pocket and the upward jetting motion of the wave crests. The entrapped air-pocket effects are found to be more important for vertical than horizontal forces.

A study on ship performance in waves using a RANS solver, part 2: Comparison of added resistance performance in various regular and irregular waves

Commercial ships frequently operate under irregular sea conditions; hence, their performance must be estimated through simulations in a more realistic environment. To date, a number of experiments and numerical studies have been conducted using regular waves to estimate added resistance. In this study, the performance of the KVLCC2 hull considering the added resistance in various regular and irregular waves was evaluated using an unsteady Reynolds-averaged Navier–Stokes solver. Resistance and motion responses were observed for regular waves with various wavelengths, regular waves with various wave steepness, and bichromatic waves superposed with regular waves. The change in resistance performance and damping with respect to motion response based on wave height was analyzed. An irregular wave corresponding to the joint North Sea wave project spectrum under sea state 6 was derived; based on this derived wave, the added resistance in irregular waves was estimated. The added resistance was compared with the estimated value using the spectral method, which was also employed to derive the added resistance under the target sea state using regular wave results. The estimated values differ from the direct calculation results owing to the nonlinearity of resistance in the irregular wave. However, the motion responses showed a similar tendency to those of the regular wave (except for those in the proximity of natural frequency that were relatively large and could be influenced by the wave height), as observed when various wave steepness values were used.

On coherent vortical structures in wave breaking

The flow generated by the breaking of free-surface waves in a periodic domain is simulated numerically with a gas–liquid Navier–Stokes solver. The solver relies on the volume-of-fluid method to account for different phases, and the interface tracking is carried out by using novel schemes based on a tailored total-variation-diminishing limiter. The numerical solver is proved to be characterized by a low numerical dissipation, thanks to the use of a scheme that guarantees energy conservation in the discrete form. Both two- and three-dimensional simulations have been performed, and the analysis is presented in terms of energy dissipation, air entrainment, bubble fragmentation, statistics and distribution. Particular attention is paid to the analysis of the mechanisms of viscous dissipation. To this purpose, coherent vortical structures, such as vortex tubes and vortex sheets, are identified, and the different behaviours of the vortex sheets and tubes at various Reynolds numbers are highlighted. The correlation between vortical structures and energy dissipation demonstrates clearly their close link both in the mixing zone and in the pure water domain, where the coherent structures propagate as a consequence of the downward transport. Notably, it is found that the dissipation is identified primarily by the vortex sheets, whereas the vortex tubes govern mainly the intermittency.

Wake Effects on Wave-Induced Loads Acting on Cylinders in a Tripod Configuration

Comparative hydrodynamic loads caused by a focused wave acting on differently sized slender vertical cylinders placed in a wave canal were predicted at model scale using an unsteady Reynolds-averaged Navier–Stokes (URANS) solver and the Morison equation. This paper focused, first, on wake effects that need to be considered for the structural design of tripod configurations and, second, on the influence of these wake effects on wave-induced loads. For the Morison approach, flow velocities were gathered at incremental representative locations along the length of the cylinders, whereby the use of a central differencing approach obtained the associated flow accelerations. A first scenario dealt with a small- and a large-diameter cylinder, implemented individually at two immersion depths. A second scenario considered medium- and large-diameter cylinders arranged in a tripod configuration, where the upstream and downstream cylinders had the same diameter. For the single cylinder, impact loads obtained by both methods compared favorably to experimental measurements. For the tripod configuration, however, impact loads obtained by both methods differed due to the influence of wake effects. Thus, it was necessary to determine the influence of wake effects to assess the structural integrity of tripod configurations.

Numerical coupling strategy using HOS-OpenFOAM-MoorDyn for OC3 Hywind SPAR type platform

The coupled (Potential theory and Navier–Stokes) solver is applied to simulate the interaction of sea waves with substructure of floating offshore wind turbines (FOWT) platform, notably similar to the OC3 Hywind SPAR structure. The intention is to develop a numerical tool that allows the study of the survivability of floating structures in extreme sea states. In this study, the moorings are modeled in two ways. One is by considering the mooring lines as a linear spring with defined spring stiffness, and another is by coupling the solver (foamStar) with a lumped-mass mooring dynamics model (MoorDyn). MoorDyn represents mooring line behavior subject to axial elasticity, hydrodynamic forces, and vertical contact forces with the seabed. The coupled model has been validated against the experiments carried out as part of the SOFTWIND project. The numerical model results of free surface elevation, floating body motions and mooring tensions are compared with the experiments. For wave cases with mild and moderate amplitudes, mooring in the form of a stiffness matrix is sufficient. However, dynamic mooring simulation (MoorDyn) is required for the extreme sea state conditions.

Chordwise flexible aft-tail suppresses jet-switching by reinstating wake periodicity in a flapping foil

The effect of a chordwise flexible aft-tail of a rigid heaving aerofoil on the dynamical transitions of the trailing-wake is studied here. The two-way coupled fluid–solid dynamics is simulated using an in-house fluid–structure interaction (FSI) platform, comprising a discrete forcing immersed boundary method based incompressible Navier–Stokes solver, weakly coupled with a finite difference method based structural solver. The FSI dynamics is studied in comparison to the corresponding rigid tail configuration. For the latter, mild jet-switching due to quasi-periodic movement of the wake vortices gives way to vigorous jet-switching as the dynamics transitions to a state of intermittency, where the quasi-periodic behaviour gets interspersed with chaotic windows. Introduction of a moderately flexible tail regularises this intermittent dynamics, eliminating jet-switching. The wake exhibits a deflected reverse Kármán pattern with fluctuating angles, governed by quasi-periodicity. With a highly flexible tail (very low rigidity), the wake shows almost a symmetric reverse Kármán street as periodicity is restored. Flexibility of the aft-tail is next controlled by changing its length, and flow is regularised and periodicity retained for moderate rigidity for increased length. Different dynamical states are established through robust nonlinear dynamical tools. The underlying flow-field behaviour, instrumental in suppressing the jet-switching phenomenon, is identified through a detailed investigation of the near-field vortex interactions dictated by the dynamics. A suite of measures has also been derived from the unsteady flow field to quantify the interactions of the key near-field vortices with a view to understanding the mechanism of switching and its subsequent suppression through flexibility.

Trailing-edge jets for UCAV's flight control over a wide speed range

With fluidic control effectors, the flying-wing aircraft have been expected to achieve flapless fight which can reduce the radar cross section (RCS). However, most researches are focused on their control capability at low speed (Ma<0.3). This paper numerically investigates using trailing-edge jets (TE jets) to control a flying-wing UCAV over a wide speed range (Ma=0.145∼0.7) to evaluate 3-axis flight control capability, bleeding air impacts on engine performance, and stealth improvement. The three-dimensional flow is simulated by a Reynolds–averaged Navier–Stokes (RANS) solver with k−ω SST turbulence model. The power consumption of bleeding air is converted into additional drag, and the control efficiency factor is proposed to compare the efficiency of fluidic and solid control effectors under a unified framework. The RCS is evaluated by the Large Element Physical Optics (LEPO) solver. The results show that, for pitch and roll control, TE jets relying on circulation control (CC) can produce the same moment as control surfaces deflected 30°, 20°, and 10° at Ma=0.145,0.4, and 0.7, respectively. Meanwhile, TE jets have less effect on the engine mass flow and thrust, and are more efficient at producing control moments than control surfaces. For yaw control, relying on the recoil effect, TE jets' control capability is equal to that of split flaps at low speed with relatively higher bleeding air impacts on engine performance, but will not be affected by the flow separation at the high angle of attack. For stealth characteristics, the RCS of jet-controlled UCAV in frontal, lateral, and some specific azimuth ranges reduces remarkably. This study indicates that TE jets are highly promising pitch and roll control effectors for a flying-wing UCAV to enhance survivability from low speed to high subsonic speed.