Cartesian Cut Cell Research Articles

A 3D parallel particle-in-cell solver for extreme wave interaction with floating bodies

Floating structures are widely used for vessels, offshore platforms, and recently considered for deep water floating offshore wind system and wave energy devices. However, modelling complex wave interactions with floating structures, particularly under extreme conditions, remains an important challenge. Following the three-dimensional (3D) parallel particle-in-cell (PIC) model developed for simulating wave interaction with fixed bodies, this paper further extends the methodology and develops a new 3D parallel PIC model for applications to floating bodies. The PIC model uses both Lagrangian particles and Eulerian grid to solve the incompressible Navier-Stokes equations, attempting to combine both the Lagrangian flexibility for handling large free-surface deformations and Eulerian efficiency in terms of CPU cost. The wave-structure interaction is resolved via inclusion of a Cartesian cut cell method based two-way strong fluid-solid coupling algorithm that is both stable and efficient. The numerical model is validated against 3D experiments of focused wave interaction with a floating moored buoy. Good agreement between the numerical and experimental results has been achieved for the motion of the buoy and the mooring force. Additionally, the PIC model achieves a CPU efficiency of the same magnitude as that of the state-of-the-art OpenFOAM® model for an extreme wave-structure interaction scenario.

ON THE PERFORMANCE OF DIFFERENT RANS BASED MODELS TO DESCRIBE THE TURBULENT FLOW IN AN AGITATED VESSEL USING NON-STRUCTURED GRIDS AND PIV VALIDATION

ABSTRACT The performances of the Standard κ-e, RNG, Realizable, κ-ω and RSM turbulence models to describe the behavior of the flow in agitated tank reactors were evaluated. Because most of these tanks have complex geometries, structured meshing schemes are not a practical alternative in engineering. Two particular unstructured meshing schemes composed of either Cartesian Cut-Cell (CC) or Tetrahedral elements were tested in domains larger than 3 million cells. In this work, the blade thickness, which is usually disregarded, was accounted for. The performance of the turbulence models was validated by PIV measurements. A strong dependence on the grid size and the power number was found, with CC grids having good agreement with grids > 3.6 Million. The pumping number was independent of the cell number and the values agree with experimental data. The Realizable model resolved with a tetrahedral grid presented the overall best performance.

Applying the free-slip boundary condition with an adaptive Cartesian cut-cell method for complex geometries

The slip condition at the interface of a multiphase flow can occur in situations including micro-and nano-fluidic flow, flow over hydrophobic surfaces, rising bubbles in quiescent liquid, and polymer extrusion processes. The aim of this work is to implement the free-slip boundary condition with an adaptive Cartesian grid method. The Navier-Stokes (NS) equations are solved by a cell-centered collocated finite volume method with adaptive mesh refinement. The arbitrarily-shaped solids imbedded in the computational domain are treated by the cut-cell method where the geometric properties of cut-cells near the boundary are computed through robust geometric operations. In discretized NS equations, the second-order-accurate center difference method is used to estimate the surface fluxes of the regular Cartesian cells in the bulk region, whereas the least-squares method is used to estimate the fluxes of the cut-cells near the boundary. A local coordinate system aligned with the normal and tangential directions of the solid boundary is defined for each cut-cell in order to properly implement the free-slip condition. The tangential velocities at the curved solid boundary are obtained using the free-slip condition and the principal curvatures of the solid surface. The proposed numerical method is implemented in the open-source code Gerris. Numerical tests have been carried out to validate our method. The tests confirm the excellent performances of the proposed method. Although our work focuses on the free-slip condition, the extension of the proposed method to more general slip conditions is straightforward.

An adaptive Cartesian cut-cell method for conjugate heat transfer on arbitrarily moving fluid-solid interfaces

This study developed and validated a numerical method to handle two-dimensional mixed-convection conjugate heat transfer (CHT) over arbitrarily moving/deforming fluid-solid interfaces. We will base the development on the finite-volume adaptive Cartesian grid method with the cut cell approach developed and matured previously. Few studies in literature tackled the issue of CHT across moving/deforming interfaces, especially in the community of Cartesian grid methods. The present work benefited from the merit of the cut cell approach, i.e., adapting to various types of boundary condition in a unified manner, to implement the boundary conditions required of the CHT. A temperal discretization scheme near moving interfaces is proposed aiming to achieve nominally second-order accuracy in time. Some cases were simulated to demonstrate the validity and spatial/temperal order of accuracy of the numerical method. Results show that the method has a globally second-order accuracy in space if the time step size varies in some fashion with the mesh size and, in a local sense (i.e., near the interface), both the spatial and temperal accuracies are superior or equal to first order for all the considered flow and thermal variables except that the temperal accuracy of pressure is mildly worse than first order for complicated moving-interface problems.

A dimensionally split Cartesian cut cell method for the compressible Navier–Stokes equations

We present a dimensionally split method for computing solutions to the compressible Navier–Stokes equations on Cartesian cut cell meshes. The method is globally second order accurate in the L1 norm, fully conservative, and allows the use of time steps determined by the regular grid spacing. We provide a description of the three-dimensional implementation of the method and evaluate its numerical performance by computing solutions to a number of test problems ranging from the nearly incompressible to the highly compressible flow regimes. All the computed results show good agreement with reference results from theory, experiment and previous numerical studies. To the best of our knowledge, this is the first presentation of a dimensionally split cut cell method for the compressible Navier–Stokes equations in the literature.

Modeling a dry running twin-screw expander using a coupled thermal-fluid solver with automatic mesh generation

Understanding the details of the internal flow processes in screw compressors and expanders is very important for their efficient and robust design. Computational fluid dynamics (CFD) provides full access to a modeled three dimensional flow field and its variation in time. However, the application of CFD to screw compressors and expanders can be difficult because of the complicated geometries involved and the need to supply the computational grid on which the modeled equations are solved over a large number of time steps. While the majority of previous research on CFD applications to screw machines features inventive techniques for generating meshes that adequately resolve the flows in the small clearances, an alternate approach is demonstrated in this work. The screw expander SE 51.2 from TU Dortmund University is analyzed here through a CFD model which generates the grid automatically based on a modified Cartesian cut-cell approach. The grid is then adaptively refined based on local gradients of velocity and temperature. At each time step, the grid is regenerated based on the geometry motion. As opposed to resolving the flow in the clearances, a model is applied so that the cells in the clearance can remain relatively large. The detailed measurements of the screw expander are used to validate the model. The operating conditions investigated include the expansion of dry air at a four-to-one pressure ratio for four different rotational speeds. The measured internal chamber pressures are compared to the results from the model, as are the average mass flow rate, indicated power, and outlet temperature. A coupled thermal-fluid approach is used to model the rotor temperatures and corresponding thermal deformation of the rotors and housing. In this approach, the fluid and solid temperatures are solved together; to deal with the problem of the disparate time scales between the fluid and solid heat transfer, the solids are periodically solved to steady state using heat transfer coefficients and near-wall temperatures computed from an energy conserving averaging over several cycles. The effects of the various leakage flow paths in the model, including the rotor-to-rotor, rotor-tip-to-housing, and bearing leakage are demonstrated and quantified. Finally, simulation and experimental results are compared in terms of different rotor-tip-to-housing clearance heights. The model considering the appropriate thermal deformation of the rotors is shown to yield the best agreement with the measurements, however there is work remaining to reduce the model calculation time, especially at low rotational speeds.

Large eddy simulation of tip leakage cavitating flow focusing on cavitation-vortex interaction with Cartesian cut-cell mesh method

In the present paper, flow configurations of cavitating flow around a straight NACA0009 foil with a gap between the foil tip and sidewall are investigated numerically by large eddy simulation (LES) coupled with Zwart-Gerber-Belamri (ZGB) cavitation model. A Cartesian cut-cell method is used for mesh generation, which is of good orthogonality and high quality. A good agreement is obtained between simulation and experiment. Two influencing factors on vorticity distributions, the interaction between different vortices and the occurrence of cavitation, are discussed in detail based on the numerical results. A series of Ω- shaped loops are observed during the development of the induced vortex, which is a result of the instability of vortex pair. This finding may provide a new viewpoint to control the evolution of tip-leakage vortex (TLV) cavitation. Moreover, it is found that the dilatation term plays a much more important role in the evolution of TLV cavitation compared with that in sheet cavitation.

Progress on numerical simulation of yield stress fluid flows (Part I): Correlating thermosolutal coefficients of Bingham plastics within a porous annulus of a circular shape

The main purpose of our work is to show the impact of some pertinent parameters; such Lewis and solutal Rayleigh numbers as well as the buoyancy and the aspect ratios; on a viscoplastic materials’ double-diffusive convection which occurs within an SiC annulus; contained between a cold “and less concentric” outer circular cylinder and a hot “and concentric” inner one, to come out with innovative correlations what predict the mean transfer rates into such annulus. To do so, the physical model for the momentum conservation equation is made using the Brinkman extension of the classical Darcy equation. The set of coupled equations is solved using the finite volume method and the SIMPLER algorithm. To handle the cylinder shape in Cartesian coordinates; the Cartesian Cut-Cell approach was adopted. Subsequently, inclusive correlations of overall transfers within such viscoplastic porous annulus are set forth as a function of the governing parameters; which foresee with ±2 to ±3% the numerical predictions. Noted that the validity of the computing code was ascertained by comparing its results with experimental data and numerical ones; already available in the literature.

A moving boundary flux stabilization method for Cartesian cut-cell grids using directional operator splitting

An explicit moving boundary method for the numerical solution of time-dependent hyperbolic conservation laws on grids produced by the intersection of complex geometries with a regular Cartesian grid is presented. As it employs directional operator splitting, implementation of the scheme is rather straightforward. Extending the method for static walls from Klein et al. (2009) [37], the scheme calculates fluxes needed for a conservative update of the near-wall cut-cells as linear combinations of “standard fluxes” from a one-dimensional extended stencil. Here the standard fluxes are those obtained without regard to the small sub-cell problem, and the linear combination weights involve detailed information regarding the cut-cell geometry. This linear combination of standard fluxes stabilizes the updates such that the time-step yielding marginal stability for arbitrarily small cut-cells is of the same order as that for regular cells. Moreover, it renders the approach compatible with a wide range of existing numerical flux-approximation methods. The scheme is extended here to time dependent rigid boundaries by reformulating the linear combination weights of the stabilizing flux stencil to account for the time dependence of cut-cell volume and interface area fractions. The two-dimensional tests discussed include advection in a channel oriented at an oblique angle to the Cartesian computational mesh, cylinders with circular and triangular cross-section passing through a stationary shock wave, a piston moving through an open-ended shock tube, and the flow around an oscillating NACA 0012 aerofoil profile.

A dimensionally split Cartesian cut cell method for hyperbolic conservation laws

We present a dimensionally split method for solving hyperbolic conservation laws on Cartesian cut cell meshes. The approach combines local geometric and wave speed information to determine a novel stabilised cut cell flux, and we provide a full description of its three-dimensional implementation in the dimensionally split framework of Klein et al. [1]. The convergence and stability of the method are proved for the one-dimensional linear advection equation, while its multi-dimensional numerical performance is investigated through the computation of solutions to a number of test problems for the linear advection and Euler equations. When compared to the cut cell flux of Klein et al., it was found that the new flux alleviates the problem of oscillatory boundary solutions produced by the former at higher Courant numbers, and also enables the computation of more accurate solutions near stagnation points. Being dimensionally split, the method is simple to implement and extends readily to multiple dimensions.

Simulation of moving boundaries interacting with compressible reacting flows using a second-order adaptive Cartesian cut-cell method

A high-order adaptive Cartesian cut-cell method, developed in the past by the authors [1] for simulation of compressible viscous flow over static embedded boundaries, is now extended for reacting flow simulations over moving interfaces. The main difficulty related to simulation of moving boundary problems using immersed boundary techniques is the loss of conservation of mass, momentum and energy during the transition of numerical grid cells from solid to fluid and vice versa. Gas phase reactions near solid boundaries can produce huge source terms to the governing equations, which if not properly treated for moving boundaries, can result in inaccuracies in numerical predictions. The small cell clustering algorithm proposed in our previous work is now extended to handle moving boundaries enforcing strict conservation. In addition, the cell clustering algorithm also preserves the smoothness of solution near moving surfaces. A second order Runge–Kutta scheme where the boundaries are allowed to change during the sub-time steps is employed. This scheme improves the time accuracy of the calculations when the body motion is driven by hydrodynamic forces. Simple one dimensional reacting and non-reacting studies of moving piston are first performed in order to demonstrate the accuracy of the proposed method. Results are then reported for flow past moving cylinders at subsonic and supersonic velocities in a viscous compressible flow and are compared with theoretical and previously available experimental data. The ability of the scheme to handle deforming boundaries and interaction of hydrodynamic forces with rigid body motion is demonstrated using different test cases. Finally, the method is applied to investigate the detonation initiation and stabilization mechanisms on a cylinder and a sphere, when they are launched into a detonable mixture. The effect of the filling pressure on the detonation stabilization mechanisms over a hyper-velocity sphere launched into a hydrogen–oxygen–argon mixture is studied and a qualitative comparison of the results with the experimental data are made. Results indicate that the current method is able to correctly reproduce the different regimes of combustion observed in the experiments. Through the various examples it is demonstrated that our method is robust and accurate for simulation of compressible viscous reacting flow problems with moving/deforming boundaries.

A 3-D numerical study of solitary wave interaction with vertical cylinders using a parallelised particle-in-cell solver

This paper aims to provide a better understanding of the interaction between solitary waves and vertical circular cylinders. This is achieved via process based numerical modelling using the parallel particle-in-cell based incompressible flow solver PICIN. The numerical model solves the Navier-Stokes equations for free-surface flows and incorporates a Cartesian cut cell method for fluid-structure interaction. Solitary waves are generated using a piston-type wave paddle. The PICIN model is first validated using a test case that involves solitary wave scattering by a single vertical cylinder. Comparisons between the present results and experimental data show good agreement for the free surface elevations around the cylinder and the horizontal wave force on the cylinder. The model is then employed to investigate solitary wave interaction with a group of eleven vertical cylinders. The wave run-up and wave forces on the cylinders are discussed.

An ODE-Based Wall Model for Turbulent Flow Simulations

This paper presents a new wall model to compute turbulent boundary layers using the Reynolds-averaged Navier–Stokes equations in high Reynolds number flows. The model solves a two-point boundary value problem for a coupled set of equations for the streamwise velocity and the turbulent viscosity. Since it includes both the pressure gradient and the momentum balance of the full Navier–Stokes system, the ordinary differential equation is valid farther from the wall. We implement the model within a Cartesian cut-cell method and use one-dimensional linelets in each cut cell to avoid the excessive mesh refinement that would otherwise be needed. The linelets are coupled to the outer Cartesian grid in a fully conservative manner, with two-way interaction between the linelets and the background grid. Detailed comparisons of velocity and eddy viscosity with three well-studied examples from the Turbulence Modeling Resource website are presented to demonstrate the model’s performance in two space dimensions, including an example with smooth-body separation. The results show the new model gives excellent results even when the value of the first point is in the wake layer.

Direct particle–fluid simulation of Kolmogorov-length-scale size particles in decaying isotropic turbulence

The modulation of decaying isotropic turbulence by 45 000 spherical particles of Kolmogorov-length-scale size is studied using direct particle–fluid simulations, i.e. the flow field over each particle is fully resolved by direct numerical simulations of the conservation equations. A Cartesian cut-cell method is used by which the exchange of momentum and energy at the fluid–particle interfaces is strictly conserved. It is shown that the particles absorb energy from the large scales of the carrier flow while the small-scale turbulent motion is determined by the inertial particle dynamics. Whereas the viscous dissipation rate of the bulk flow is attenuated, the particles locally increase the level of dissipation due to the intense strain rate generated near the particle surfaces due to the crossing-trajectory effect. Analogously, the rotational motion of the particles decouples from the local fluid vorticity and strain-rate field at increasing particle inertia. The high level of dissipation is partially compensated by the transfer of momentum to the fluid via forces acting at the particle surfaces. The spectral analysis of the kinetic energy budget is supported by the average flow pattern about the particles showing a nearly universal strain-rate distribution. An analytical expression for the instantaneous rate of viscous dissipation induced by each particle is derived and subsequently verified numerically. Using this equation, the local balance of fluid kinetic energy around a particle of arbitrary shape can be precisely determined. It follows that two-way coupled point-particle models implicitly account for the particle-induced dissipation rate via the momentum-coupling terms; however, they disregard the actual length scales of the interaction. Finally, an analysis of the small-scale flow topology shows that the strength of vortex stretching in the bulk flow is mitigated due to the presence of the particles. This effect is associated with the energy conversion at small wavenumbers and the reduced level of dissipation at intermediate wavenumbers. Consequently, it damps the spectral flux of energy to the small scales.

Numerical study of periodic long wave run-up on a rigid vegetation sloping beach

Coastal vegetation can reduce long wave run-up on beaches and inland propagation distances and thus mitigate these hazards. This paper investigates periodic long wave run-up on coastal rigid vegetation sloping beaches via a numerical study. Rigid vegetation is approximated as rigid sticks, and the numerical model is based on an implementation of Morison's formulation [21] for rigid structures induced inertia and drag stresses in the nonlinear shallow water equations. The numerical model is solved via a finite volume method on a Cartesian cut cell mesh. The accuracy of the numerical model is validated by comparison with experimental results. The model is then applied to simulate various hypothetical cases of long periodic wave run-up on a sloping vegetated beach with different plant diameters and densities, and incident long waves with different periods. The sensitivity of long wave run-up to plant diameter, stem density and wave period is investigated by comparison of the numerical results for different vegetation characteristics and different wave periods. The numerical results show that rigid vegetation can effectively reduce long wave run-up and that wave run-up is decreased with increase of plant diameter and stem density. Moreover, the attenuation of long periodic wave run-up due to vegetation is sensitive to the variation of the incident wave period, and the attenuation of wave run-up is not increased or decreased monotonically with incident wave period.

An Explicit Implicit Scheme for Cut Cells in Embedded Boundary Meshes

We present a new mixed explicit implicit time stepping scheme for solving the linear advection equation on a Cartesian cut cell mesh. We use a standard second-order explicit scheme on Cartesian cells away from the embedded boundary. On cut cells, we use an implicit scheme for stability. This approach overcomes the small cell problem--that standard schemes are not stable on the arbitrarily small cut cells--while keeping the cost fairly low. We examine several approaches for coupling the schemes in one dimension. For one of them, which we refer to as flux bounding, we can show a TVD result for using a first-order implicit scheme. We also describe a mixed scheme using a second-order implicit scheme and combine both mixed schemes by an FCT approach to retain monotonicity. In the second part of this paper, extensions of the second-order mixed scheme to two and three dimensions are discussed and the corresponding numerical results are presented. These indicate that this mixed scheme is second-order accurate in $$L^1$$L1 and between first- and second-order accurate along the embedded boundary in two and three dimensions.

A high-order adaptive Cartesian cut-cell method for simulation of compressible viscous flow over immersed bodies

A new adaptive finite volume conservative cut-cell method that is third-order accurate for simulation of compressible viscous flows is presented. A high-order reconstruction approach using cell centered piecewise polynomial approximation of flow quantities, developed in the past for body-fitted grids, is now extended to the Cartesian based cut-cell method. It is shown that the presence of cut-cells of very low volume results in numerical oscillations in the flow solution near the embedded boundaries when standard small cell treatment techniques are employed. A novel cell clustering approach for polynomial reconstruction in the vicinity of the small cells is proposed and is shown to achieve smooth representation of flow field quantities and their derivatives on immersed interfaces. It is further shown through numerical examples that the proposed clustering method achieves the design order of accuracy and is fairly insensitive to the cluster size. Results are presented for canonical flow past a single cylinder and a sphere at different flow Reynolds numbers to verify the accuracy of the scheme. Investigations are then performed for flow over two staggered cylinders and the results are compared with prior data for the same configuration. All the simulations are carried out with both quadratic and cubic reconstruction, and the results indicate a clear improvement with the cubic reconstruction. The new cut-cell approach with cell clustering is able to predict accurate results even at relatively low resolutions. The ability of the high-order cut-cell method in handling sharp geometrical corners and narrow gaps is also demonstrated using various examples. Finally, three-dimensional flow interactions between a pair of spheres in cross flow is investigated using the proposed cut-cell scheme. The results are shown to be in excellent agreement with past studies, which employed body-fitted grids for studying this complex case.

Detached eddy simulation of turbulent flow around square and circular cylinders on Cartesian cut cells

Square and circular cylinders in three-dimensional turbulent flows are studied numerically using the LES and DES turbulence models. One aim of the present study is to implement the LES and DES turbulence models in a cell-centered finite volume method (FVM) developed for solving the Navier–Stokes equations on Cartesian cut cells. The Cartesian cut cell approach is known to be robust for problems in geometrically complex domains with fixed or moving boundaries. For the purpose of validating the present numerical model, the current flow past fixed square and circular cylinders at moderate Reynolds numbers is tested first. Comparison of the computed results with experimental data reveals that the DES models are superior to the conventional LES and RANS models. The second aim of the present study is to assess the performance of different RANS based DES turbulence models. By means of the comparison of results obtained with the 0-equation mixing-length, 1-equation S–A and 2-equation k–ω based DES models for the flow over the same circular cylinder, some recommendations are proposed. According to the present study, in terms of accuracy the 1-equation S–A based DES model is very promising. Beside this, if the computational cost is the main concern, the 0-equation mixing-length based DES model might be an ideal option, achieving a good balance between accuracy and efficiency.

A Cartesian cut cell based two-way strong fluid–solid coupling algorithm for 2D floating bodies

In a recent paper Kelly et al. (2015) [PICIN: A Particle-In-Cell solver for incompressible free surface flows with two-way fluid–solid coupling. SIAM Journal on Scientific Computing 37 (3), B403–24.] detailed the PICIN full particle Particle-In-Cell (PIC) solver for incompressible free-surface flows. The model described in that paper employed a tailored version of the Distributed Lagrange Multiplier (DLM) method for the strong coupling of fluid–solid interaction. In this paper we propose an alternative strong fluid–solid coupling algorithm based on a modification to the cut cell methodology that is informed by the variational approach. The solid velocity flux/integral on the boundary is expressed purely in terms of pressure leading to a revised pressure Poisson equation that is discretised in a finite volume sense. This approach allows the PICIN model to simulate the motion of floating bodies of arbitrary configuration. 2D test cases involving floating bodies with one or more degrees of freedom (DoF) are used to validate the modified PICIN model. The results presented show that the modified PICIN model is able to both efficiently and robustly predict the motions of surface-piercing floating structures under either regular or extreme wave action.

An efficient conservative cut-cell method for rigid bodies interacting with viscous compressible flows

A Cartesian cut-cell method for viscous flows interacting with freely moving boundaries is presented. The method enables a sharp resolution of the embedded boundaries and strictly conserves mass, momentum, and energy. A new explicit Runge-Kutta scheme (PC-RK) is introduced by which the overall computational time is reduced by a factor of up to 2.5. The new scheme is a predictor-corrector type reformulation of a popular class of Runge-Kutta methods which substantially reduces the computational effort for tracking the moving boundaries and subsequently reinitializing the solver impairing neither stability nor accuracy. The structural motion is computed by an implicit scheme with good stability properties due to a strong-coupling strategy and the conservative discretization of the flow solver at the material interfaces. A new formulation for the treatment of small cut cells is proposed with high accuracy and robustness for arbitrary geometries based on a weighted Taylor-series approach solved via singular-value decomposition. The efficiency and the accuracy of the new method are demonstrated for several three-dimensional cases of laminar and turbulent particulate flow. It is shown that the new method remains fully conservative even for large displacements of the boundaries leading to a fast convergence of the fluid-solid coupling while spurious force oscillations inherent to this class of methods are effectively suppressed. The results substantiate the good stability and accuracy properties of the scheme even on relatively coarse meshes. An efficient Cartesian cut-cell method for freely moving solid boundaries is presented.The new predictor-corrector Runge-Kutta scheme (PC-RK) enables simulation speedups of up to 2.5.Mass, momentum, and energy are strictly conserved at the sharply resolved solid-fluid interfaces.The method exhibits very good stability and accuracy properties even for large displacements of the boundaries.Spurious force oscillations inherent to this class of methods are suppressed.