Two-dimensional Navier Research Articles

A Nonlinear Computational Model of Tethered Underwater Kites for Power Generation

The dynamic motion of tethered undersea kites (TUSK) is studied using numerical simulations. TUSK systems consist of a rigid winged-shaped kite moving in an ocean current. The kite is connected by tethers to a platform on the ocean surface or anchored to the seabed. Hydrodynamic forces generated by the kite are transmitted through the tethers to a generator on the platform to produce electricity. TUSK systems are being considered as an alternative to marine turbines since the kite can move at a high-speed, thereby increasing power production compared to conventional marine turbines. The two-dimensional Navier–Stokes equations are solved on a regular structured grid to resolve the ocean current flow, and a fictitious domain-immersed boundary method is used for the rigid kite. A projection method along with open multiprocessing (OpenMP) is employed to solve the flow equations. The reel-out and reel-in velocities of the two tethers are adjusted to control the kite angle of attack and the resultant hydrodynamic forces. A baseline simulation, where a high net power output was achieved during successive kite power and retraction phases, is examined in detail. The effects of different key design parameters in TUSK systems, such as the ratio of tether to current velocity, kite weight, current velocity, and the tether to kite chord length ratio, are then further studied. System power output, vorticity flow fields, tether tensions, and hydrodynamic coefficients for the kite are determined. The power output results are shown to be in good agreement with the established theoretical results for a kite moving in two dimensions.

Global strong solutions to radial symmetric compressible Navier–Stokes equations with free boundary

In this paper, we consider the two-dimensional barotropic compressible Navier–Stokes equations with stress free boundary condition imposed on the free surface. As the viscosity coefficients satisfies μ(ρ)=2μ, λ(ρ)=ρβ, β>1, we establish the existence of global strong solution for arbitrarily large spherical symmetric initial data even if the density vanishes across the free boundary. In particular, we show that the density is strictly positive and bounded from the above and below in any finite time if the initial density is strictly positive, and the free boundary propagates along the particle path and expand outwards at an algebraic rate.

Weak solutions to the two-dimensional steady compressible Navier–Stokes equations

We study motions of the steady compressible viscous isothermal fluids in a bounded two dimensional domain governed by the Navier–Stokes equations. We obtain that there exists at least one weak solution to such a problem satisfying some integrability. The proof of this result is based on delicate estimates for the momentum equations and a bootstrapping argument.

Effect of variation of length-to-depth ratio and Mach number on the performance of a typical double cavity scramjet combustor

The two equation standard k–ɛ turbulence model and the two-dimensional compressible Reynolds-Averaged Navier–Stokes (RANS) equations have been used to computationally simulate the double cavity scramjet combustor. Here all the simulations are performed by using ANSYS 14-FLUENT code. At the same time, the validation of the present numerical simulation for double cavity has been performed by comparing its result with the available experimental data which is in accordance with the literature. The results are in good agreement with the schlieren image and the pressure distribution curve obtained experimentally. However, the pressure distribution curve obtained numerically is under-predicted in 5 locations by numerical calculation. Further, investigations on the variations of the effects of the length-to-depth ratio of cavity and Mach number on the combustion characteristics has been carried out. The present results show that there is an optimal length-to-depth ratio for the cavity for which the performance of combustor significantly improves and also efficient combustion takes place within the combustor region. Also, the shifting of the location of incident oblique shock took place in the downstream of the H2 inlet when the Mach number value increases. But after achieving a critical Mach number range of 2–2.5, the further increase in Mach number results in lower combustion efficiency which may deteriorate the performance of combustor.

On the Numerical Controllability of the Two-Dimensional Heat, Stokes and Navier–Stokes Equations

The aim of this work is to present some strategies to solve numerically controllability problems for the two-dimensional heat equation, the Stokes equations and the Navier–Stokes equations with Dirichlet boundary conditions. The main idea is to adapt the Fursikov–Imanuvilov formulation, see Fursikov and Imanuvilov (Controllability of Evolutions Equations, Lectures Notes Series, vol 34, Seoul National University, 1996); this approach has been followed recently for the one-dimensional heat equation by the first two authors. More precisely, we minimize over the class of admissible null controls a functional that involves weighted integrals of the state and the control, with weights that blow up near the final time. The associated optimality conditions can be viewed as a differential system in the three variables $$x_1$$ , $$x_2$$ and t that is second-order in time and fourth-order in space, completed with appropriate boundary conditions. We present several mixed formulations of the problems and, then, associated mixed finite element Lagrangian approximations that are relatively easy to handle. Finally, we exhibit some numerical experiments.

Design of a ducted wind turbine for offshore floating platforms

Within the Marine Energy Laboratory project, funded by the Italian Ministry for Education, University and Research, one of the first offshore wind plants on floating hulls, hosting ducted wind turbines, has been considered. The confinement of horizontal-axis wind turbines inside divergent ducts is reconsidered, in light of material innovation and direct drive coupling. Ducted wind turbines can take advantage of the flow rate increase due to the effect of the divergent shrouds. The conventional blade element momentum theory has been reformulated in order to deal with ducted turbines. Furthermore, computational fluid dynamics simulations have been carried out based on the solution of the steady two-dimensional Reynolds-averaged Navier–Stokes equations for axisymmetric swirling flows. In order to avoid any expensive mesh refinement near the actual rotor blades, the turbine effect on the flow field is taken into account by means of source terms for the momentum equations solved inside the domain swept by the rotor. This technique allowed us to optimize the geometry of the ducted wind turbine in an extremely effective way.

A global Stokes method of approximated particular solutions for unsteady two-dimensional Navier–Stokes system of equations

ABSTRACTThe unsteady two-dimensional Navier–Stokes system of equations, for viscous incompressible fluids are solved using a global method of approximated particular solutions (MAPS) in terms of a Stokes formulation, where the velocity and pressure fields are approximated from a linear superposition of particular solutions of a non-homogeneous Stokes system of equations, with a multiquadric (MQ) radial basis function (RBF) as non-homogeneous term. Steady-state solution of the flow problems considered in this work can be unstable at high Reynolds numbers (Re), corresponding to bifurcation of solutions that result in the appearance of new stable steady-state or periodic solutions. The main objective of this work is to present a global meshless numerical scheme able to predict these bifurcation points and concurrent new stable or periodic solutions. This is well known to be a very difficult task for any numerical scheme. An implicit first-order time-stepping scheme is used to approximate the transient term and the obtained nonlinear system of algebraic equations is solved by a Newton–Raphson method with variable step. Two steady-state and two transient problems are considered to validate the numerical scheme: the lid-driven cavity and backward-facing step (BFS) flows (steady-state problems) and the decaying Taylor–Green vortex and two-sided lid-driven cavity flows (transient problems). The first two problems are solved up to Re=10,000 and 2300, respectively. Results obtained are compared with corresponding benchmark numerical solutions, showing excellent agreement. Obtained numerical solutions for the decaying vortices at Re=100 shown excellent agreement with the corresponding analytical results. The transient problem of a rectangular two-sided lid-driven cavity flow is solved at Re=700. The influence of the cavity length, l, in determining the different structures of the flow pattern is studied for values of , showing that the scheme is able to reproduce the previously reported change in the flow pattern when l=2. Finally, the global Stokes MAPS are used to carry out nonlinear stability analyses of three steady-state problems: the sudden expansion, lid-driven cavity and BFS flows. Stable and unstable steady-state solutions at Re values greater than critical are predicted with the proposed numerical scheme. Our numerical results are consistent with previously stability analysis reported in the literature.

Adaptive simulations of viscous detonations initiated by a hot jet using a high-order hybrid WENO–CD scheme

In the present work a sixth-order hybrid WENO–Centered Difference (CD) scheme with an adaptive mesh refinement method is employed to investigate gaseous detonation by injecting a hot jet into a hydrogen–oxygen combustible mixture flowing at supersonic speed. Two-dimensional reactive Navier–Stokes (NS) equations with one-step two-species chemistry model are solved numerically. The comparison between viscous and inviscid detonation structures shows that due to the absence of both the physical viscosity in Euler equations and minimization of numerical dissipation in the hybrid WENO–CD scheme, very small-scale vortices can be observed behind the detonation front. The diffusion effect in the NS equations suppresses the small-scale vortices, but it has negligible influence on the large-scale vortices generated by Richtmyer–Meshkov (RM) instability and those along the highly unstable shear layers induced by Kelvin–Helmholtz (KH) instability. When studying the same setup in an expanding channel and beyond the point of detonation initiation, it is found that because of the diffusion effect of detached shear layers, any unburned jet flow is consumed quickly and then additional energy is released periodically. Because of the formation of multiple secondary triple points and subsequent shear layers after the shutdown of the hot jet, a highly turbulent flow is produced behind the detonation front. Rather than the commonly known RM instability, the large-scale vortices involved in the highly unstable shear layers dominate the formation of the turbulent flow and the rapid turbulent mixing between the unburned jet flow and burned product. It is found that the size of unburned jets and vortices due to KH instability is growing for larger expansion angles. The further generated turbulent flow resulting from larger sized vortices, significantly enhances the mixing rate behind the Mach stem, leading to rapid consumption of the unburned reactants. Therefore, detonations propagate faster in channels with larger expansion angle and higher expansion ratio.

Modeling and parallel computation of the non-linear interaction of rigid bodies with incompressible multi-phase flow

A computational tool is developed to capture the interaction of solid object with two-phase flow. The full two-dimensional Navier–Stokes equations are solved on a regular structured grid to resolve the flow field. The level set and the immersed boundary methods are used to capture the free surface of a fluid and a solid object, respectively. A two-step projection method along with Multi-Processing (OpenMP) is employed to solve the flow equations. The computational tool is verified based on numerical and experimental data with three scenarios: a cylinder falling into a rectangular domain due to gravity, transient vertical oscillation of a cylinder by releasing above its equilibrium position, and a dam breaking in the presence of a fixed obstacle. In the first two validation simulations, the accuracy of the immersed boundary method is verified. However the accuracy of the level set method while the computational tool can model the high density ratio is confirmed in the dam breaking simulation. The results obtained from the current method are in good agreement with experimental data and other numerical studies. The applicability of the current computational tool for the interaction of a buoy in a water wave tank with two types of waves; symmetrical and asymmetrical waves; has also been studied.

A 2 D parallel high-order sliding and deforming spectral difference method

Abstract In this paper, we present a novel parallel high-order sliding and deforming spectral difference (SD 2 ) method for solving the two-dimensional compressible Navier–Stokes equations on unstructured quadrilateral grids. In the SD 2 method, an arbitrary Lagrangian–Eulerian (ALE) approach is combined with a novel sliding interface approach to separately handle rotational and translational motions on a dynamic grid. The SD 2 method is shown to be capable of completely removing grid skewness caused by rotating boundaries to improve solution qualities. The SD 2 method is verified as high-order accurate and is scalable for parallel computing on a distributed-memory computer. The SD 2 method is also proven to be more robust and more accurate than deforming-only ALE approach for problems involving large-amplitude rotating motions.

Turbulent 2.5-dimensional dynamos

We study the linear stage of the dynamo instability of a turbulent two-dimensional flow with three components $(u(x,y,t),v(x,y,t),w(x,y,t))$ that is sometimes referred to as a 2.5-dimensional (2.5-D) flow. The flow evolves based on the two-dimensional Navier–Stokes equations in the presence of a large-scale drag force that leads to the steady state of a turbulent inverse cascade. These flows provide an approximation to very fast rotating flows often observed in nature. The low dimensionality of the system allows for the realization of a large number of numerical simulations and thus the investigation of a wide range of fluid Reynolds numbers $Re$, magnetic Reynolds numbers $Rm$ and forcing length scales. This allows for the examination of dynamo properties at different limits that cannot be achieved with three-dimensional simulations. We examine dynamos for both large and small magnetic Prandtl-number turbulent flows $Pm=Rm/Re$, close to and away from the dynamo onset, as well as dynamos in the presence of scale separation. In particular, we determine the properties of the dynamo onset as a function of $Re$ and the asymptotic behaviour in the large $Rm$ limit. We are thus able to give a complete description of the dynamo properties of these turbulent 2.5-D flows.

Wedge-stabilized oblique detonation in an inhomogeneous hydrogen–air mixture

We performed numerical simulation against oblique detonation formed in an inhomogeneous hydrogen–air mixture. Planar two-dimensional Navier–Stokes equations were solved with the chemical source terms calculated with a detailed chemical kinetics including 9 species and 27 elementary reactions. Fuel concentration gradients described by the Gaussian function were introduced into two different conditions: (A) a Mach 8 flow onto a 28.20° wedge with a static pressure of 8.50kPa, and (B) a Mach 8 flow onto a 23.8° wedge with a static pressure of 34.0kPa. Fuel-rich conditions on the centerline generated V-shaped deflagration front under the former condition, and V + Y Mach stem under the latter due to the strong concentration gradients near the surface. These shapes caused a couple of triple points to appear on the incident shock front. Transition to detonation occurred twice along the front in some cases. The apex of V-shape was always located in a similar position due to almost constant thermochemical properties of the incoming streamline. The threshold for emergence of these V-shapes can be reasonably attained by the magnitude of induction length gradient close to the wedge.

Stability of two-dimensional solutions to the Navier–Stokes equations in cylindrical domains under Navier boundary conditions

The Navier–Stokes motions in a cylindrical domain with Navier boundary conditions are considered. First the existence of global regular two-dimensional solutions is proved. The solutions are such that norms bounded with respect to time are controlled by the same constant for all t∈R+. Assuming that the initial velocity and the external force are sufficiently close to the initial velocity and the external force of a two-dimensional solution, we prove existence of global three-dimensional solutions which remain close to the two-dimensional solution for all time. In this sense we have stability of two-dimensional solutions. Thanks to the Navier boundary conditions the nonlinear term in the two-dimensional Navier–Stokes equations does not influence the energy estimate. This implies that the global two-dimensional solution is proved without any structural restrictions on the external force, initial data or viscosity.

Numerical modeling of local scour and forces for submarine pipeline under surface waves

A two-dimensional numerical model is developed to predict local scour around submarine pipelines induced by the orbital fluid motion under surface water waves. Instead of being simplified to oscillatory flow, the wave motion is modeled using a fully nonlinear wave model. The numerical model is based on the two-dimensional Reynolds-Averaged Navier–Stokes (RANS) equations with a Shear-Stress Transport (SST) k-ω turbulence closure. Both suspended load and bed load sediment transportations are considered. The moving boundaries of free surface and the evolution of seabed due to local scour are tracked using the Arbitrary Lagrangian–Eulerian (ALE) method. The Streamline Upwind Petrov–Galerkin Finite Element Method (SUPG-FEM) is used to discretize the governing equations. The numerical model is validated against the benchmarks of linear and nonlinear wave propagations and their interactions with submerged structures as well as local scour around submarine pipeline in steady current. Comparisons between the numerical results and available theoretical, numerical and experimental data show satisfactory agreements. The proposed numerical model is then used to investigate the nonlinear wave-induced local scour around pipelines placed flat and sloping seabed. The effects of wave height and wave period on local scour and wave forces on the pipelines are examined. The numerical investigations suggest the necessity of utilizing the free surface wave model rather than the simplified oscillatory flow model for the problem of local scour around submarine pipelines in case of large amplitude nonlinear waves and pipelines over uneven seabed.

A non-conforming finite volume element method for the two-dimensional Navier–Stokes/Darcy system

We consider the discretization of the stationary Navier–Stokes/Darcy system in a two-dimensional domain by the non-conforming finite volume element method. We use the standard formulation of the Navier–Stokes/Darcy system in the primitive variables and take as approximation space the non-conforming \(P_{1}\) elements for velocity and piezometric head and piecewise constant elements for the hydrostatic pressure. We prove that the unique solution of the non-conforming finite volume element method converges to the true solution with optimal order for velocity and piezometric head in discrete \(H^{1}\) norm and for pressure in discrete \(L^{2}\) norm, respectively. Finally, some numerical experiments are presented to validate our theoretical results.

Unsteady analysis on the instantaneous forces and moment arms acting on a novel Savonius-style wind turbine

This paper aims to present a transient analysis on the forces acting on a novel two-bladed Savonius-style wind turbine. Two-dimensional unsteady Reynolds Averaged Navier Stokes equations are solved using shear stress transport k–ω turbulence model at a Reynolds number of 1.23×105. The instantaneous longitudinal drag and lateral lift forces acting on each of the blades and their acting points are calculated. The corresponding moment arms responsible for the torque generation are obtained. Further, the effect of tip speed ratio on the force coefficients, moment arms and overall turbine performances are observed. Throughout the paper, the obtained results for the new design are discussed with reference to conventional semi-circular design of Savonius turbines. A significant performance improvement is achieved with the new design due to its increased lift and moment arm contribution as compared to the conventional design. More interestingly, the present study sets a platform for future aerodynamic research and improvements for Savonius-style wind turbines.

Transition of a Steady to a Periodically Unsteady Flow for Various Jet Widths of a Combined Wall Jet and Offset Jet

In the present paper, a dual jet consisting of a wall jet and an offset jet has been numerically simulated using two-dimensional unsteady Reynolds-Averaged Navier–Stokes (RANS) equations to examine the effects of jet width (w) variation on the near flow field region. The Reynolds number based on the separation distance between the two jets (d) has been considered to be Re = 10,000. According to the computational results, three distinct flow regimes have been identified as a function of w/d. For w/d ≤ 0.5, the flow field remains to be always steady with two counter-rotating stable vortices in between the two jets. On the contrary, within the range of 0.6 ≤ w/d < 1.6, the flow field reveals a periodic vortex shedding phenomenon similar to what would be observed in the wake of a two-dimensional bluff body. In this flow regime, the Strouhal number of vortex shedding frequency decreases monotonically with the progressive increase in the jet width. For w/d ≥ 1.6, the periodic vortex shedding is still evident, but the Strouhal number becomes insensitive to the variation of jet width.

The meshless local Petrov-Galerkin method based on moving Kriging interpolation for solving the time fractional Navier-Stokes equations.

In this paper, we present a numerical scheme used to solve the nonlinear time fractional Navier–Stokes equations in two dimensions. We first employ the meshless local Petrov–Galerkin (MLPG) method based on a local weak formulation to form the system of discretized equations and then we will approximate the time fractional derivative interpreted in the sense of Caputo by a simple quadrature formula. The moving Kriging interpolation which possesses the Kronecker delta property is applied to construct shape functions. This research aims to extend and develop further the applicability of the truly MLPG method to the generalized incompressible Navier–Stokes equations. Two numerical examples are provided to illustrate the accuracy and efficiency of the proposed algorithm. Very good agreement between the numerically and analytically computed solutions can be observed in the verification. The present MLPG method has proved its efficiency and reliability for solving the two-dimensional time fractional Navier–Stokes equations arising in fluid dynamics as well as several other problems in science and engineering.

Global well-posedness of strong solutions to the two-dimensional barotropic compressible Navier–Stokes equations with vacuum

The aim of this paper is to establish the global well-posedness and large-time asymptotic behavior of the strong solution to the Cauchy problem of the two-dimensional compressible Navier–Stokes equations with vacuum. It is proved that if the shear viscosity \({\mu}\) is a positive constant and the bulk viscosity \({\lambda}\) is the power function of the density, that is, \({\lambda=\rho^{\beta}}\) with \({\beta \in [0,1],}\) then the Cauchy problem of the two-dimensional compressible Navier–Stokes equations admits a unique global strong solution provided that the initial data are of small total energy. This result can be regarded as the extension of the well-posedness theory of classical compressible Navier–Stokes equations [such as Huang et al. (Commun Pure Appl Math 65:549–585, 2012) and Li and Xin (http://arxiv.org/abs/1310.1673) respectively]. Furthermore, the large-time behavior of the strong solution to the Cauchy problem of the two-dimensional barotropic compressible Navier–Stokes equations had been also obtained.

Numerical simulation and global linear stability analysis of low-Re flow past a heated circular cylinder

We perform two-dimensional unsteady Navier–Stokes simulation and global linear stability analysis of flow past a heated circular cylinder to investigate the effect of aided buoyancy on the stabilization of the flow. The Reynolds number of the incoming flow is fixed at 100, and the Richardson number characterizing the buoyancy is varied from 0.00 (buoyancy-free case) to 0.10 at which the flow is still unsteady. We investigate the effect of aided buoyancy in stabilizing the wake flow, identify the temporal and spatial characteristics of the growth of the perturbation, and quantify the contributions from various terms comprising the perturbed kinetic energy budget. Numerical results reveal that the increasing Ri decreases the fluctuation magnitude of the characteristic quantities monotonically, and the momentum deficit in the wake flow decays rapidly so that the flow velocity recovers to that of the free-stream; the strain on the wake flow is reduced in the region where the perturbation is the most greatly amplified. Global stability analysis shows that the temporal growth rate of the perturbation decreases monotonically with Ri, reflecting the stabilization of the flow due to aided buoyancy. The perturbation grows most significantly in the free shear layer separated from the cylinder. As Ri increases, the location of maximum perturbation growth moves closer to the cylinder and the perturbation decays more rapidly in the far wake. The introduction of the aided buoyancy alters the base flow, and destabilizes the near wake shear layer mainly through the strain-induced transfer term and the pressure term of the perturbed kinetic energy, whereas the flow is stabilized in the far wake as the strain is alleviated.