Deconvolution‐based nonlinear filtering for incompressible flows at moderately large Reynolds numbers

We consider a Leray model with a deconvolution‐based indicator function for the simulation of incompressible fluid flow at moderately large Reynolds number (in the range of a few thousands) with under‐resolved meshes. For the implementation of the model, we adopt a three‐step algorithm called evolve–filter–relax that requires (i) the solution of a Navier–Stokes problem, (ii) the solution of a Stokes‐like problem to filter the Navier–Stokes velocity field, and (iii) a final relaxation step. We take advantage of a reformulation of the evolve–filter–relax algorithm as an operator‐splitting method to analyze the impact of the filter on the final solution versus a direct simulation of the Navier–Stokes equations. In addition, we provide some direction for tuning the parameters involved in the model based on physical and numerical arguments. Our approach is validated against experimental data for fluid flow in an idealized medical device (consisting of a conical convergent, a narrow throat, and a sudden expansion, as recommended by the U.S. Food and Drug Administration). Numerical results are in good quantitative agreement with the measured axial components of the velocity and pressures for two different flow rates corresponding to turbulent regimes, even for meshes with a mesh size more than 40 times larger than the smallest turbulent scale. After several numerical experiments, we perform a preliminary sensitivity analysis of the computed solution to the parameters involved in the model. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical study of breaking waves by a two‐phase flow model

A two‐phase flow model, which solves the flow in the air and water simultaneously, is presented for modelling breaking waves in deep and shallow water, including wave pre‐breaking, overturning and post‐breaking processes. The model is based on the Reynolds‐averaged Navier–Stokes equations with the k ?ε turbulence model. The governing equations are solved by the finite volume method in a Cartesian staggered grid and the partial cell treatment is implemented to deal with complex geometries. The SIMPLE algorithm is utilised for the pressure‐velocity coupling and the air‐water interface is modelled by the interface capturing method via a high resolution volume of fluid scheme. The numerical model is validated by simulating overturning waves on a sloping beach and over a reef, and deep‐water breaking waves in a periodic domain, in which good agreement between numerical results and available experimental measurements for the water surface profiles during wave overturning is obtained. The overturning jet, air entrainment and splash‐up during wave breaking have been captured by the two‐phase flow model, which demonstrates the capability of the model to simulate free surface flow and wave breaking problems.Copyright © 2011 John Wiley & Sons, Ltd.     

On coupling the Reynolds‐averaged Navier–Stokes equations with two‐equation turbulence model equations

Two methods for coupling the Reynolds‐averaged Navier–Stokes equations with the q–ω turbulence model equations on structured grid systems have been studied; namely a loosely coupled method and a strongly coupled method. The loosely coupled method first solves the Navier–Stokes equations with the turbulent viscosity fixed. In a subsequent step, the turbulence model equations are solved with all flow quantities fixed. On the other hand, the strongly coupled method solves the Reynolds‐averaged Navier–Stokes equations and the turbulence model equations simultaneously. In this paper, numerical stabilities of both methods in conjunction with the approximated factorization‐alternative direction implicit method are analysed. The effect of the turbulent kinetic energy terms in the governing equations on the convergence characteristics is also studied. The performance of the two methods is compared for several two‐ and three‐dimensional problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Simulation of viscous water column collapse using adapting hierarchical grids

An adaptive hierarchical grid‐based method for predicting complex free surface flows is used to simulate collapse of a water column. Adapting quadtree grids are combined with a high‐resolution interface‐capturing approach and pressure‐based coupling of the Navier–Stokes equations. The Navier–Stokes flow solution scheme is verified for simulation of flow in a lid‐driven cavity at Re=1000. Two approaches to the coupling of the Navier–Stokes equations are investigated as are alternative face velocity and hanging node interpolations. Collapse of a water column as well as collapse of a water column and its subsequent interaction with an obstacle are simulated. The calculations are made on uniform and adapting quadtree grids, and the accuracy of the quadtree calculations is shown to be the same as those made on the equivalent uniform grids. Results are in excellent agreement with experimental and other numerical data. A sharp interface is maintained at the free surface. The new adapting quadtree‐based method achieves a considerable saving in the size of the computational grid and CPU time in comparison with calculations made on equivalent uniform grids. Copyright © 2005 John Wiley & Sons, Ltd.     

A direct numerical simulation of axisymmetric liquid–gas two‐phase laminar flows in a pipe

An accurate finite‐volume‐based numerical method for the simulation of an isothermal two‐phase flow, consisting of a liquid slug translating in a non‐reacting gas in a circular pipe is presented. This method is built on a sharp interface concept and developed on an Eulerian Cartesian fixed‐grid with a cut‐cell scheme and marker points to track the moving interface. The unsteady, axisymmetric Navier–Stokes equations in both liquid and gas phases are solved separately. The mass continuity and momentum flux conditions are explicitly matched at the true surface phase boundary to determine the interface shape and movement. A quadratic curve fitting algorithm with marker points is used to yield smooth and accurate information of the interface curvatures. It is uniquely demonstrated for the first time with the current method that conservation of mass is strictly enforced for continuous infusion of flow into the domain of computation. The method has been used to compute the velocity and pressure fields and the deformation of the liquid core. It is also shown that the current method is capable of producing accurate results for a wide range of Reynolds number, Re, Weber number, We, and large property jumps at the interface. Copyright © 2010 John Wiley & Sons, Ltd.     

A fourth‐order compact finite difference scheme for the steady stream function–vorticity formulation of the Navier–Stokes/Boussinesq equations

A fourth‐order compact finite difference scheme on the nine‐point 2D stencil is formulated for solving the steady‐state Navier–Stokes/Boussinesq equations for two‐dimensional, incompressible fluid flow and heat transfer using the stream function–vorticity formulation. The main feature of the new fourth‐order compact scheme is that it allows point‐successive overrelaxation (SOR) or point‐successive underrelaxation iteration for all Rayleigh numbers Ra of physical interest and all Prandtl numbers Pr attempted. Numerical solutions are obtained for the model problem of natural convection in a square cavity with benchmark solutions and compared with some of the accurate results available in the literature. Copyright © 2003 John Wiley & Sons, Ltd.     

Fourth‐order compact formulation of Navier–Stokes equations and driven cavity flow at high Reynolds numbers

A new fourth‐order compact formulation for the steady 2‐D incompressible Navier–Stokes equations is presented. The formulation is in the same form of the Navier–Stokes equations such that any numerical method that solve the Navier–Stokes equations can easily be applied to this fourth‐order compact formulation. In particular, in this work the formulation is solved with an efficient numerical method that requires the solution of tridiagonal systems using a fine grid mesh of 601 × 601. Using this formulation, the steady 2‐D incompressible flow in a driven cavity is solved up to Reynolds number with Re = 20 000 fourth‐order spatial accuracy. Detailed solutions are presented. Copyright © 2005 John Wiley & Sons, Ltd.     

Relaxation of the Navier–Stokes–Korteweg equations for compressible two‐phase flow with phase transition

The Navier–Stokes–Korteweg (NSK) system is a classical diffuse‐interface model for compressible two‐phase flow. However, the direct numerical simulation based on the NSK system is quite expensive and in some cases even not possible. We propose a lower‐order relaxation of the NSK system with hyperbolic first‐order part. This allows applying numerical methods for hyperbolic conservation laws and removing some of the difficulties of the original NSK system. To illustrate the new ansatz, we first present a local discontinuous Galerkin method in one and two spatial dimensions. It is shown that we can compute initial boundary value problems with realistic density ratios and perform stable computations for small interfacial widths. Second, we show that it is possible to construct a semi‐discrete finite‐volume scheme that satisfies a discrete entropy inequality. Copyright © 2015 John Wiley & Sons, Ltd.     

Three dimensional phase-field investigation of droplet formation in microfluidic flow focusing devices with experimental validation

In this paper, the droplet formation process at a low capillary number in a flow focusing micro-channel is studied by performing a three-dimensional phase field benchmark based on the Cahn–Hilliard Navier–Stokes equations and the finite element method. Dynamic moving contact line and wetting condition are considered, and generalized Navier boundary condition (GNBC) is utilized to demonstrate the dynamic motion of the interface on wall surface. It is found that the mobility parameter plays a very critical role in the squeezing and breakup process to control the shape and size of droplets. We define the characteristic mobility M_c to represent the correct relaxation time of the interface. We also demonstrate that the characteristic mobility is associated with the physical process and should be kept as a constant as the product of the mobility tuning parameter χ and the square of interfacial thickness ε². This criterion is applied for different interfacial thicknesses to correctly capture the physical process of droplet formation. Moreover, the size of the droplet, the velocity of the droplet along the downstream, and the period of droplet formation are compared between the numerical and experimental results which agree with each other both qualitatively and quantitatively. The presented model and criterion can be used to predict the dynamic behavior and movement of multiphase flows.     

A discontinuous Galerkin‐based sharp‐interface method to simulate three‐dimensional compressible two‐phase flow

A numerical method for the simulation of compressible two‐phase flows is presented in this paper. The sharp‐interface approach consists of several components: a discontinuous Galerkin solver for compressible fluid flow, a level‐set tracking algorithm to follow the movement of the interface and a coupling of both by a ghost‐fluid approach with use of a local Riemann solver at the interface. There are several novel techniques used: the discontinuous Galerkin scheme allows locally a subcell resolution to enhance the interface resolution and an interior finite volume Total Variation Diminishing (TVD) approximation at the interface. The level‐set equation is solved by the same discontinuous Galerkin scheme. To obtain a very good approximation of the interface curvature, the accuracy of the level‐set field is improved and smoothed by an additional P_NP_M‐reconstruction. The capabilities of the method for the simulation of compressible two‐phase flow are demonstrated for a droplet at equilibrium, an oscillating ellipsoidal droplet, and a shock‐droplet interaction problem at Mach 3. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical simulation of droplet ejection of thermal inkjet printheads

In this study, we present a method to predict the droplet ejection in thermal inkjet printheads including the growth and collapse of a vapor bubble and refill of the firing chamber. The three‐dimensional Navier–Stokes equations are solved using a finite‐volume approach with a fixed Cartesian mesh. The piecewise‐linear interface calculation‐based volume‐of‐fluid method is employed to track and reconstruct the ink–air interface. A geometrical computation based on Lagrangian advection is used to compute the mass flux and advance the interface. A simple and efficient model for the bubble dynamics is employed to model the effect of ink vapor on the adjacent ink liquid. To solve the surface tension‐dominated flow accurately, a hierarchical curvature‐estimation method is proposed to adapt to the local grid resolution. The numerical methods mentioned earlier have been implemented in an internal simulation code, CFD3. The numerical examples presented in the study show good performance of CFD3 in prediction of surface tension‐dominated free‐surface flows, for example, droplet ejection in thermal inkjet printing. Currently, CFD3 is used extensively for printhead development within Hewlett‐Packard. Copyright © 2015 John Wiley & Sons, Ltd.     

Weak imposition of boundary conditions for the Navier–Stokes equations by a penalty method

We prove convergence of the finite element method for the Navier–Stokes equations in which the no‐slip condition and no‐penetration condition on the flow boundary are imposed via a penalty method. This approach has been previously studied for the Stokes problem by Liakos (Weak imposition of boundary conditions in the Stokes problem. Ph.D. Thesis, University of Pittsburgh, 1999). Since, in most realistic applications, inertial effects dominate, it is crucial to extend the validity of the method to the nonlinear Navier–Stokes case. This report includes the analysis of this extension, as well as numerical results validating their analytical counterparts. Specifically, we show that optimal order of convergence can be achieved if the computational boundary follows the real flow boundary exactly. Copyright © 2008 John Wiley & Sons, Ltd.     

Finite element modified method of characteristics for the Navier–Stokes equations

An algorithm based on the finite element modified method of characteristics (FEMMC) is presented to solve convection–diffusion, Burgers and unsteady incompressible Navier–Stokes equations for laminar flow. Solutions for these progressively more involved problems are presented so as to give numerical evidence for the robustness, good error characteristics and accuracy of our method. To solve the Navier–Stokes equations, an approach that can be conceived as a fractional step method is used. The innovative first stage of our method is a backward search and interpolation at the foot of the characteristics, which we identify as the convective step. In this particular work, this step is followed by a conjugate gradient solution of the remaining Stokes problem. Numerical results are presented for:

• a Convection–diffusion equation. Gaussian hill in a uniform rotating field.
• b Burgers equations with viscosity.
• c Navier–Stokes solution of lid‐driven cavity flow at relatively high Reynolds numbers.
• d Navier–Stokes solution of flow around a circular cylinder at Re=100.

Copyright © 2000 John Wiley & Sons, Ltd.     

Implicit application of non‐reflective boundary conditions for Navier–Stokes equations in generalized coordinates

The non‐reflective boundary conditions (NRBC) for Navier–Stokes equations originally suggested by Poinsot and Lele (J. Comput. Phys. 1992; 101 :104–129) in Cartesian coordinates are extended to generalized coordinates. The characteristic form Navier–Stokes equations in conservative variables are given. In this characteristic‐based method, the NRBC is implicitly coupled with the Navier–Stokes flow solver and are solved simultaneously with the flow solver. The calculations are conducted for a subsonic vortex propagating flow and the steady and unsteady transonic inlet‐diffuser flows. The results indicate that the present method is accurate and robust, and the NRBC are essential for unsteady flow calculations. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical simulation of 3D free surface flows,with multiple incompressible immiscible phases. Applications to impulse waves

A numerical method for the solution to the density‐dependent incompressible Navier–Stokes equations modeling the flow of N immiscible incompressible liquid phases with a free surface is proposed. It allows to model the flow of an arbitrary number of liquid phases together with an additional vacuum phase separated with a free surface. It is based on a volume‐of‐fluid approach involving N indicator functions (one per phase, identified by its density) that guarantees mass conservation within each phase. An additional indicator function for the whole liquid domain allows to treat boundary conditions at the interface between the liquid domain and a vacuum. The system of partial differential equations is solved by implicit operator splitting at each time step: first, transport equations are solved by a forward characteristics method on a fine Cartesian grid to predict the new location of each liquid phase; second, a generalized Stokes problem with a density‐dependent viscosity is solved with a FEM on a coarser mesh of the liquid domain. A novel algorithm ensuring the maximum principle and limiting the numerical diffusion for the transport of the N phases is validated on benchmark flows. Then, we focus on a novel application and compare the numerical and physical simulations of impulse waves, that is, waves generated at the free surface of a water basin initially at rest after the impact of a denser phase. A particularly useful application in hydraulic engineering is to predict the effects of a landslide‐generated impulse wave in a reservoir. Copyright © 2014 John Wiley & Sons, Ltd.     

A simple sliding‐mesh interface procedure and its application to the CFD simulation of a tidal‐stream turbine

An effective way of using computational fluid dynamics (CFD) to simulate flow about a rotating device—for example, a wind or marine turbine—is to embed a rotating region of cells inside a larger, stationary domain, with a sliding interface between. This paper describes a simple but effective method for implementing this as an internal Dirichlet boundary condition, with interfacial values obtained by interpolation from halo nodes. The method is tested in two finite‐volume codes: one using block‐structured meshes and the other unstructured meshes. Validation is performed for flow around simple, isolated, rotating shapes (cylinder, sphere and cube), comparing, where possible, with experiment and the alternative CFD approach of fixed grid with moving walls. Flow variables are shown to vary smoothly across the sliding interface. Simulations of a tidal‐stream turbine, including both rotor and support, are then performed and compared with towing‐tank experiments. Comparison between CFD and experiment is made for thrust and power coefficients as a function of tip‐speed ratio (TSR) using Reynolds‐averaged Navier–Stokes turbulence models and large‐eddy simulation (LES). Performance of most models is good near the optimal TSR, but simulations underestimate mean thrust and power coefficients in off‐design conditions, with the standard k– ? turbulence model performing noticeably worse than shear stress transport k– ω and Reynolds‐stress‐transport closures. LES gave good predictions of mean load coefficients and vital information about wake structures but at substantial computational cost. Grid‐sensitivity studies suggest that Reynolds‐averaged Navier–Stokes models give acceptable predictions of mean power and thrust coefficients on a single device using a mesh of about 4 million cells. Copyright © 2013 John Wiley & Sons, Ltd.     

Verification testing in computational fluid dynamics: an example using Reynolds‐averaged Navier–Stokes methods for two‐dimensional flow in the near wake of a circular cylinder

Verification testing was performed for various Reynolds‐averaged Navier–Stokes methods for uniform flow past a circular cylinder at Re= 5232. The standard and renormalized group (RNG) versions of the k–ε method were examined, along with the Boussinesq, Speziale and Launder constitutive relationships. Wind tunnel experiments for flow past a circular cylinder were also performed to obtain a comparative data set. Preliminary studies demonstrate poor convergence for the Speziale relationship. Verification testing with the standard and RNG k–ε models suggests that the simulations exhibit global monotonic convergence for the Boussinesq models. However, the global order of accuracy of the methods was much lower than the expected order of accuracy of 2. For this reason, pointwise convergence ratios and orders of accuracy were computed to show that not all sampling locations had converged (standard k–ε model: 19% failed to converge; RNG k–ε model: 14% failed to converge). When the non‐convergent points were removed from consideration, the average orders of accuracy are closer to the expected value (standard k–ε model: 1.41; RNG k–ε model: 1.27). Poor iterative and global grid convergence was found for the RNG k–ε/Launder model. The standard and RNG k–ε models with the Boussinesq relationship were compared with experimental data and yielded results significantly different from the experiments. Copyright © 2003 John Wiley & Sons, Ltd.     

A finite element thermally coupled flow formulation for phase‐change problems

A finite element, thermally coupled incompressible flow formulation considering phase‐change effects is presented. This formulation accounts for natural convection, temperature‐dependent material properties and isothermal and non‐isothermal phase‐change models. In this context, the full Navier–Stokes equations are solved using a generalized streamline operator (GSO) technique. The highly non‐linear phase‐change effects are treated with a temperature‐based algorithm, which provides stability and convergence of the numerical solution. The Boussinesq approximation is used in order to consider the temperature‐dependent density variation. Furthermore, the numerical solution of the coupled problem is approached with a staggered incremental‐iterative solution scheme, such that the convergence criteria are written in terms of the residual vectors. Finally, this formulation is used for the solutions of solidification and melting problems validating some numerical results with other existing solutions obtained with different methodologies. Copyright © 2000 John Wiley & Sons, Ltd.     

GPU‐accelerated direct numerical simulations of decaying compressible turbulence employing a GKM‐based solver

Gas Kinetic Method‐based flow solvers have become popular in recent years owing to their robustness in simulating high Mach number compressible flows. We evaluate the performance of the newly developed analytical gas kinetic method (AGKM) by Xuan et al. in performing direct numerical simulation of canonical compressible turbulent flow on graphical processing unit (GPU)s. We find that for a range of turbulent Mach numbers, AGKM results shows excellent agreement with high order accurate results obtained with traditional Navier–Stokes solvers in terms of key turbulence statistics. Further, AGKM is found to be more efficient as compared with the traditional gas kinetic method for GPU implementation. We present a brief overview of the optimizations performed on NVIDIA K20 GPU and show that GPU optimizations boost the speedup up‐to 40x as compared with single core CPU computations. Hence, AGKM can be used as an efficient method for performing fast and accurate direct numerical simulations of compressible turbulent flows on simple GPU‐based workstations. Copyright © 2016 John Wiley & Sons, Ltd.     

Calculation of three‐dimensional turbulent flow with a finite volume multigrid method

A numerical method for the efficient calculation of three‐dimensional incompressible turbulent flow in curvilinear co‐ordinates is presented. The mathematical model consists of the Reynolds averaged Navier–Stokes equations and the k–ε turbulence model. The numerical method is based on the SIMPLE pressure‐correction algorithm with finite volume discretization in curvilinear co‐ordinates. To accelerate the convergence of the solution method a full approximation scheme‐full multigrid (FAS‐FMG) method is utilized. The solution of the k–ε transport equations is embedded in the multigrid iteration. The improved convergence characteristic of the multigrid method is demonstrated by means of several calculations of three‐dimensional flow cases. Copyright © 1999 John Wiley & Sons, Ltd.