Stokes Solver Research Articles

Dynamics and energetics of a sphere during water exit

By using a multi-phase Navier–Stokes solver combining the volume of fluid approach for water–air interface, the large-eddy simulations method for turbulence effect, and the overset mesh technique for moving boundary, we investigate the dynamics and energetics of a neutrally buoyant sphere during water exit process. The sphere is launched vertically with various initial velocities. In our analysis, the water exit process is divided into three distinct phases, fully submerged, partially submerged, and airborne. The focus is on the roles of the gravity, buoyancy, viscous, and wave-radiation forces in determining the motion and energy exchange/dissipation. During the fully submerged phase, the energy loss of the sphere is caused by the viscous force and the wave-radiation force, with the former playing the dominating role. In the partially submerged phase, the buoyancy force decreases as the submerged volume is reduced. However, under certain conditions, there could be an additional supporting force on the sphere caused by upward water flow beneath it. Once the sphere is fully airborne, its motion is primarily governed by gravity, and the maximum height it attains correlates strongly with the water-exit speed. These findings offer deeper insights into optimizing underwater launch parameters and understanding energy transfer mechanisms in water exit scenarios.

Hydrodynamic Characteristics of the Pusher, Tractor and Schottel Types of the Azipod Propulsion System

Abstract The main purpose of this paper is to investigate the hydrodynamics performance of the three types (pusher, tractor and Schottel) of the azimuthing podded drive (AZIPOD) electric propulsion system. To evaluate the propulsive performance of the podded drive system, the Reynolds-Averaged Navier Stokes (RANS) solver is employed. KP-505 propeller as the research object, hydrodynamic open-water characteristics of this propeller was first calculated, and agreed well with test results. Then, numerical simulation of the thrust, torque and efficiency of the three types of the AZIPOD systems (Pusher, Tractor and Schottel) with KP-505 propeller at various yaw angles (from -30° to +30° with 15° increments) and different advance coefficients were compared. For the Schottel propulsion system, the effects of the number of propeller blades and the blade pitch-diameter ratio on performance are presented and discussed. Finally, the effect of sturt, support element and pod for pusher type on the pressure coefficient, thrust and torque of one blade and whole blades is investigated during one cycle.

A novel rigid-fluid method for landslide tsunami modeling

This study presents a three-dimensional numerical model with the novel rigid fluid method (RFM) to simulate slide-type landslides. RFM, coupled with the Navier–Stokes solver Splash3D, is developed to calculate the movement and rotation of a moving solid. With the pressure and shear stress data in each grid collected by the discrete element method, the moving solid is granted to be involved in the simulation. The model is validated against a three-dimensional experiment of a sphere entering water, demonstrating good agreement in terms of displacement and air column pinch-off time. Subsequently, the model is applied to simulate the process of a semispherical landslide tsunami, using a semispherical laboratory experiment as a benchmark. The numerical results show excellent agreement with the experimental data in terms of sliding track, velocity, dynamic pressure, and wave height. To investigate the applicability of the model to real-world scale events, the above-mentioned case is scaled up proportionally, and the time, flow velocity, dynamic pressure, and resultant force are compared to verify the Froude number similarity. The results indicate that the model can effectively capture the essential features of landslide-induce tsunamis and has the potential to be used for further studies on the impacts of such events.

A consistent diffuse-interface model for two-phase flow problems with rapid evaporation

We present accurate and mathematically consistent formulations of a diffuse-interface model for two-phase flow problems involving rapid evaporation. The model addresses challenges including discontinuities in the density field by several orders of magnitude, leading to high velocity and pressure jumps across the liquid–vapor interface, along with dynamically changing interface topologies. To this end, we integrate an incompressible Navier–Stokes solver combined with a conservative level-set formulation and a regularized, i.e., diffuse, representation of discontinuities into a matrix-free adaptive finite element framework. The achievements are three-fold: First, we propose mathematically consistent definitions for the level-set transport velocity in the diffuse interface region by extrapolating the velocity from the liquid or gas phase. They exhibit superior prediction accuracy for the evaporated mass and the resulting interface dynamics compared to a local velocity evaluation, especially for strongly curved interfaces.Second, we show that accurate prediction of the evaporation-induced pressure jump requires a consistent, namely a reciprocal, density interpolation across the interface, which satisfies local mass conservation. Third, the combination of diffuse interface models for evaporation with standard Stokes-type constitutive relations for viscous flows leads to significant pressure artifacts in the diffuse interface region. To mitigate these, we propose to introduce a correction term for such constitutive model types. Through selected analytical and numerical examples, the aforementioned properties are validated. The presented model promises new insights in simulation-based prediction of melt–vapor interactions in thermal multiphase flows such as in laser-based powder bed fusion of metals.

Resolving subgrid-scale structures for multiphase flows using a filament moment-of-fluid method

Multiphase flows are present in many industrial and engineering applications as well as in some physical phenomena. Capturing the interface between the phases for complex flows is challenging and requires an accurate method, especially to resolve fine-scale structures. The moment-of-fluid (MOF) method improves drastically the accuracy of interface reconstruction compared to previous geometrical methods. Instead of refining the mesh to capture increased levels of detail, the MOF method, which uses zeroth and first moments as well as a conglomeration algorithm, enables subgrid structures such as filaments to be captured at a small extra cost. Coupled to a finite volume Navier–Stokes solver, the MOF method has been tested on a fixed grid and validated using well-known benchmark problems such as dam break flows, the Rayleigh–Taylor and Kelvin–Helmholtz instability problems, and a rising bubble. The ability of the novel filament MOF method to capture the filamentary structures that eventually form for the Rayleigh–Taylor instability and rising bubble problems is assessed. Good agreement has been found with other numerical results and experimental measurements available in the literature.

A Study on the interaction of shock tube-generated blast waves with a circular object at different pressure ratios

The interaction of high peak overpressure blast waves with a circular object placed at two different axial locations from the shock tube exit is studied through numerical simulation using an in-house developed multi-component Navier–Stokes solver. The driver and driven sections of the shock tube were 0.8 m and 6 m, respectively. Helium is used in the driver section, while atmospheric air is used in the driven section and outside the shock tube. The evolution of blast waves inside an open-ended shock tube and its interaction with a rectangular object is reported in Murugan et al.. (2022). Here, the blast wave interacting with a circular object is examined for diaphragm pressure ratios of 13 and 57 by placing the objects at 250 mm and 500 mm from the shock tube exit. The flow field is evaluated through numerical Schlieren, vorticity, density, pressure plots, and the enstrophy plot, which shows the vortical structures that originated in the flow field. The blast load acting on the circular object is calculated for two diaphragm pressure ratios and axial locations. This study helps understand the reflection and diffraction of blast waves and associated flow fields around circular objects used in blast wave attenuation.

Influence of Rarefaction Degree and Aft-Body Geometry on Supersonic Flows

During atmospheric entry, super-/hypersonic vehicles cross distinct atmospheric layers characterized by large density variations and thus experience different flow regimes ranging from free molecular, transition, slip, to continuous regimes. Due to the distinct modeling strategy between these regimes and complex physical phenomena appearing near the vehicles (boundary-layer/shock interaction, base-flow recirculation, etc.), assessing their aerodynamic properties may be difficult. The present work focuses on supersonic flows around sharp-base geometries in both continuous and slip-flow regimes and aims at highlighting the influence of both rarefaction degree and base geometry on the vehicles’ aerodynamic features. For this purpose, three axisymmetric cone-cylinder geometries with right-angled, rounded, or flared rear parts are considered. Flow visualization, pressure, and drag measurements are carried out at Mach number Ma=4 and Knudsen numbers ranging from Kn=4×10−4 to 1×10−2 in the supersonic rarefied MARHy wind tunnel. The experimental data are compared with numerical results of simulations performed with a continuous-flow Navier–Stokes (NS) solver and two rarefied flows codes: a discrete-ordinate Bhatnagar–Gross–Krook (K) solver and a direct simulation Monte Carlo (SPARTA) solver. While the NS solver overestimates frictional drag as Kn rises, the rarefied K and SPARTA results show satisfactory agreement with experimental data. The latter numerical results highlight the main effects of rarefaction: as Kn increases, shocks become more diffuse, skin friction strengthens (leading to a significant increase in drag coefficients), and the extent of the base-recirculation decreases. Regarding the aft-body geometry, its influence on the base recirculation vanishes with increasing Kn.

Numerical study on effects of leading-edge manufacturing defects on cavitation performance of a full-scale propeller. I. Simulation for the model- and full-scale propellers without defect

Paper I of this two-part paper focuses on predicting the open-water cavitation performance of the model- and full-scale propellers based on the geometry of David Taylor model basin 5168 propeller without leading-edge (LE) defect. Simulations were conducted using the three-dimensional (3D) steady Reynolds-Averaged Navier–Stokes solver in the commercial software package Star-CCM+. Effects of simulation parameters, including domain size, grid size, stretch ratio, first-grid spacing, y+, and turbulence model on the solutions were carefully examined and the best-practice settings for the model propeller and the full-scale propeller with no LE defect were developed. Validation studies were carried out for the propeller model against the experimental data. Results for the full-scale propeller were compared with those of the model-scale propeller. The differences in model and full-scale predictions are generally insignificant. To further confirm this finding, the scale effect was investigated with four additional scale ratios, λ = 2, 3, 4, and 5. The results showed that the scale effect is in general minimal and, therefore, the developed best-practice settings can be applied to the open-water simulation of the full-scale propeller. Paper II presents the simulation for the full-scale propellers with LE defects.

Numerical study on effects of leading-edge manufacturing defects on cavitation performance of a full-scale propeller. II. Simulation for the full-scale propeller with defects

Paper II of this two-part paper investigated the effects of leading-edge (LE) manufacturing defects on the open-water cavitation performance of a full-scale propeller based on the geometry of David Taylor Model Basin propeller by using the three-dimensional (3D) steady Reynolds-Averaged Navier–Stokes solver. Various simulation parameters, including domain size, grid size, stretch ratio, first-grid spacing, y+, and turbulence model, were carefully examined for their effects on the solutions, leading to the development of the best modeling practices for the full-scale propeller with LE defects. Employing these recommended best-practice settings, simulations were conducted on the full-scale propellers with 0.10, 0.25, and 0.50 mm LE defects. Compared to the predictions from Paper I [Jin et al., “Numerical study on effects of leading-edge manufacturing defects on cavitation performance of a full-scale propeller—Paper I: Simulation for the model- and full-scale propellers without defect,” Phys. Fluids 36, 105179 (2024).], which did not account for LE defects, the results showed that the LE defects within International Standards Organization (ISO) 484 Class S tolerances narrow the cavitation buckets. As a consequence, such LE defects can result in more than 40% reduction in cavitation inception speed, which is similar to the conclusions drawn from earlier two-dimensional (2D) studies [Jin et al., “2D CFD studies on effects of leading-edge propeller manufacturing defects on cavitation performance,” in SNAME Maritime Convention (The Society of Naval Architects and Marine Engineers, 2020).]. Note that Paper I [Jin et al., “Numerical study on effects of leading-edge manufacturing defects on cavitation performance of a full-scale propeller—Paper I: Simulation for the model- and full-scale propellers without defect,” Phys. Fluids 36, 105179 (2024).] presents the simulations for the model- and full-scale propellers without LE defects.

Exploiting Axisymmetry to Optimize CFD Simulations—Heave Motion and Wave Radiation of a Spherical Buoy

Simulating the free decay motion and wave radiation from a heaving semi-submerged sphere poses significant computational challenges due to its three-dimensional complexity. By leveraging axisymmetry, we reduce the problem to a two-dimensional simulation, significantly decreasing computational demands while maintaining accuracy. In this paper, we exploit axisymmetry to perform a large ensemble of Computational Fluid Dynamics (CFDs) simulations, aiming to evaluate and maximize both accuracy and efficiency, using the Reynolds Averaged Navier–Stokes (RANS) solver interFOAM, in the opensource finite volume CFD software OpenFOAM. Validated against highly accurate experimental data, extensive parametric studies are conducted, previously limited by computational constraints, which facilitate the refinement of simulation setups. More than 50 iterations of the same heaving sphere simulation are performed, informing efficient trade-offs between computational cost and accuracy across various simulation parameters and mesh configurations. Ultimately, by employing axisymmetry, this research contributes to the development of more accurate and efficient numerical modeling in ocean engineering.

Blended-Wing-Body Regional Aircraft Optimization with High-Fidelity Aerodynamics and Critical Design Requirements

Conventional tube-and-wing and proposed blended-wing-body airliners must satisfy several design requirements, but the latter configuration is tightly integrated and sensitive to these requirements. In this work, blended-wing-body regional aircraft are investigated using a gradient-based mixed-fidelity multidisciplinary optimization framework centered on a Reynolds-averaged Navier–Stokes solver. In addition to sizing and cruise trim, design requirements considered include one-engine-inoperative directional trim, takeoff rotation ability, takeoff field length, initial climb performance, low-speed trim and static margin, and top-of-climb rate of climb. Results show how optimal design features vary and performance is overpredicted if critical design requirements are excluded and how key elements of geometric freedom help realize the potential of the configuration. A 4.8% block fuel burn benefit is enabled by pivot-piston variable-length landing gear even with low-mounted engines and high design freedom. The off-design constraints penalize block fuel burn by 3.2% if variable-length landing gear is considered, but this value reaches 7.6% if lower geometric freedom that inhibits tight cabin contouring and the formation of a novel forebody ridge is removed. Leading-edge carving is found to be optimal. The high design freedom and high-fidelity aerodynamics model help efficiently satisfy the design requirements, resulting in a cruise lift-to-drag ratio of 21.7 at 36,000 ft and Mach 0.78.

Coupling a time-domain code with a RANS solver for slamming analysis

ABSTRACT A time domain hydroelastic code (TD) based on strip theory is two-way coupled with a Reynolds Averaged Navier−Stokes (RANS) solver to capture the bottom slamming effects on ships. The study is conducted on an Ultra Large Container Ship (ULCS). The methodology considers the hull’s flexibility and analyses longitudinal bending in head seas. Ship motions are calculated using a body non-linear time-domain method based on strip theory. The equations of motion are solved, and time integration is performed using the fourth-order Runge–Kutta method. When a slamming event is identified, the TD code activates a RANS solver to determine the bottom slamming force. The simulations are conducted under extreme sea conditions, and the hull experiences springing and whipping responses. The numerical results of vertical bending moments are compared with experimental data. The bottom and bow flare slamming are identified from the simulations.

Direct simulations of external flow and noise radiation using the generalized interpolation-supplemented cascaded lattice Boltzmann method

Direct simulations of external flow and the associated noise radiation are studied by an improved lattice Boltzmann method, i.e., the generalized interpolation-supplemented cascaded lattice Boltzmann method (GICLBM). In this method, the cascaded collision scheme is used to improve the numerical stability of the conventional collision schemes, and the generalized interpolation approach is used in the particle streaming process so as to allow a non-uniform and body-fitted mesh partition. With that, both near- and far-field flow dynamics and noise radiation are resolved simultaneously. In order to capture sound waves, the perfectly matched layer is also implemented so as to avoid waves reflecting to and polluting the inner acoustic field. Moreover, a novel index technique is developed for the GICLBM to enable implicit streaming, which brings an efficient memory reduction. Three cases are then performed to showcase the feasibility, accuracy, extensibility, and efficiency of the present framework, including flow past a square cylinder, flow past an elliptic cylinder, and flow past a NACA 0012 airfoil, each implemented with a type of body-fitted mesh. Both the fluid dynamic and noise radiation are found to be in good agreement with results using the Navier–Stokes solvers. This study demonstrates the potential of the GICLBM for accurately and efficiently simulating external problems as well as sound generation and propagation.

Influence of inertial and centrifugal forces on rate and flow patterns in natural fracture networks

Fluid production from fractured rock masses readily induces fracture flow velocities of meters per second. Yet, most discrete fracture flow models treat flow as laminar creeping flow or account for inertia effects only by single-fracture constitutive relationships.This numeric simulation study investigates water flow patterns and spatial velocity variations in a natural fracture network with mm-wide open fractures, studying the transition from laminar creeping to turbulent flow. After verification with a fracture intersection model, a Reynolds-time-averaged Navier Stokes solver serves to analyse flow regimes and velocity distribution. Our results show that for fracture flow velocities greater than ∼1-cm/s, fluid inertia begins to markedly alter flow patterns and the overall velocity distribution in the network. The pressure-gradient-flow relationship therefore becomes non-linear long before the flow in straight fractures enters the weak inertia regime. This prominence of inertia effects highlights the need to improve fracture network flow models.

Usage of high-fidelity large eddy simulation to improve the turbulence modeling of Reynolds averaged navier stokes simulation in film cooling applications via a neural network

In this study, high-Fidelity Large eddy simulation (LES) data was used to create a surrogate turbulence model to enhance the estimation of turbulent heat flux in a Reynolds averaged Navier Stokes (RANS) solver for film cooling applications. The LES simulation (A flat plate film cooling application with a Reynolds number of 17,382, a blowing ratio of 1 and a density ratio of 2) was validated by comparison with experimental data. A correlation analysis was performed to identify the most influential parameters, and temperature gradient, velocity gradient, turbulent dissipation rate, shear strain rate, and turbulent viscosity ratio were selected as the most effective parameters. A second model was developed to create a Galilean-invariant model that remains unaffected by changes in the coordinate system. Neural network was applied to the LES training data to propose a modified surrogate dynamic turbulent Prandtl number to be applied to the k − ε realizable RANS simulation. The first surrogate model was implemented to two different geometries, one geometry was a flat plate film cooling similar to the training data but with different flow conditions, and another one was a more complex geometry with pressure gradient. The second model was only implemented on the second geometry and produced similar results. The surrogate models predicted a more accurate thermal field near the cooling wall (up-to 58 % error reduction) with a computational time complexity similar to the base RANS solver.

Wave–structure interaction by a two–way coupling between a fully nonlinear potential flow model and a Navier–Stokes solver

A two-way domain decomposition coupling procedure between a fully nonlinear potential flow model and a Navier–Stokes solver capturing the free surface with a Volume of Fluid method is used to study wave–structure interaction applied to offshore wind turbines. Away from the structure, the large-scale inviscid wave field is modeled by the potential code. Wave generation and absorption in this 3D hybrid model take place in the outer potential domain. The codes exchange data in the region around their common boundaries. Through the two-way coupling, waves propagate in and out of the viscous subdomain, making the hybrid algorithm suitable to study wave diffraction on marine structures, while keeping the viscous subdomain small. Each code uses its own mesh and time step. Subdomains are overlapping, therefore continuity conditions on velocity and free surface have to be verified on two distinct coupling surfaces at any time. Parallel implementation with communications between the models relying on the Message Passing Interface library allows calculations on large spatial and temporal scales. The coupling algorithm is first tested for regular nonlinear waves and then applied to simulate wave loads exerted on a vertical monopile in 3D. Attention is paid to the high-order components of the horizontal force.

Performance improvement of a Vestas V52 850kW wind turbine by retrofitting passive flow control devices

This study presents the results of a collaborative effort between academia and industry aimed at further enhancing the benefits provided by Vortex Generators and Gurney Flaps. To achieve this objective, an integrated approach was employed, involving wind tunnel experiments, on-site measurements, and computational simulations to design devices tailored for an onshore (Vestas V52, 850 kW) turbine and assess their influence on turbine performance. Device selection was based on wind tunnel measurements, while their positioning on the blade was based on infrared thermography images from the field. A Reynolds Averaged Navier Stokes solver was used to predict the performance of the devices on both airfoil and blade level. The final assessment of the upgrade pack was based on SCADA data and Lidar measurements. The results show that an Annual Energy Production uplift of 5.77% is measured for this turbine.

A low-cost, penalty parameter-free, and pressure-robust enriched Galerkin method for the Stokes equations

This paper proposes a low-cost, penalty parameter-free, and pressure-robust Stokes solver based on the enriched Galerkin (EG) method with a discontinuous velocity enrichment function. The EG method employs the interior penalty discontinuous Galerkin (IPDG) formulation to weakly impose the continuity of the velocity function. However, despite its advantage of symmetry, the symmetric IPDG formulation requires a lot of computational effort to choose an optimal penalty parameter and compute different trace terms. To reduce such effort, we replace the derivatives of the velocity function with its weak derivatives computed by the geometric data of elements. Therefore, our modified EG (mEG) method is a penalty parameter-free numerical scheme that has reduced computational complexity and conserves the optimal convergence orders. Moreover, we achieve pressure robustness for the mEG method by employing a velocity reconstruction operator on the load vector on the right-hand side of the discrete system. The theoretical results are confirmed through numerical experiments with two- and three-dimensional examples.

An Improved Body Force Model to Simulate Blade Loading Variation Under Various Operating Conditions in Transonic Axial Flow Compressor

Abstract In order to analyze the unsteady aerodynamic response of compressors with limited computational resources, the development of the body force (BF) model has been pursued for decades. This model can simulate the effects of the compressor blade on the flow by applying a three-dimensional (3D) force field. Indeed, the accuracy of the BF model simulation highly depends on the construction strategy of BF field. However, for a transonic compressor, the influence of shock waves on the spatial distribution of blade loading is significant, which makes the construction of an appropriate BF field challenging. To solve this issue, an accurate BF model is proposed to describe the effects of the blade on the flow with high spatial fidelity in transonic compressors. For verification, a typical transonic fan rotor, NASA-Rotor 67, is selected. The numerical results calculated by the new BF model are compared with those obtained by the Reynolds-averaged Navier–Stokes (RANS) solver. According to the results, the new model can accurately capture the distribution of blade loading in the transonic blade channel at various operating conditions, eventually leading to a good prediction of compressor performance and the radial distribution of flow parameters downstream of the rotor. Moreover, this study discusses the causes of modeling errors and presents the application of the model in inlet distortion simulation at the end.

An edge-based interface tracking (EBIT) method for multiphase-flow simulation with surface tension

We present a novel Front-Tracking method, the Edge-Based Interface Tracking (EBIT) method for multiphase flow simulations. In the EBIT method, the markers are located on the grid edges and the interface can be reconstructed without storing the connectivity of the markers. This feature makes the process of marker addition or removal easier than in the traditional Front-Tracking method. The EBIT method also allows almost automatic parallelization due to the lack of explicit connectivity.In a previous journal article we have presented the kinematic part of the EBIT method, that includes the algorithms for piecewise linear reconstruction and advection of the interface. Here, we complete the presentation of the EBIT method and combine the kinematic algorithm with a Navier–Stokes solver. A circle fit is now implemented to improve the accuracy of mass conservation in the reconstruction phase. Furthermore, to identify the reference phase and to distinguish ambiguous topological configurations, we introduce a new feature: the Color Vertex. For the coupling with the Navier–Stokes equations, we first calculate volume fractions from the position of the markers and the Color Vertex, then viscosity and density fields from the computed volume fractions and finally surface tension stresses with the Height-Function method. In addition, an automatic topology change algorithm is implemented into the EBIT method, making it possible the simulation of more complex flows. The two-dimensional version of the EBIT method has been implemented in the free Basilisk platform, and validated with seven standard test cases: stagnation flow, translation with uniform velocity, single vortex, Zalesak's disk, capillary wave, Rayleigh-Taylor instability and rising bubble. The results are compared with those obtained with the Volume-of-Fluid (VOF) method already implemented in Basilisk.