Two-dimensional Navier-Stokes Equations Research Articles

Numerical investigation of local scour around a vibration pipeline under free surface wave condition

A two-dimensional finite element numerical model was developed to predict local scour around a vibration submarine pipeline under free surface wave condition. The flow governing equations are the two-dimensional Navier-Stokes equations which were closed by using the Shear-Stress Transport (SST) k-ω turbulence model. Both suspended load and bed load sediment transportations are considered in the scour model. The Arbitrary Lagrangian-Eulerian (ALE) method was used to track the moving boundaries induced by the vibration pipeline, the free wave surface, and the evolution of the seabed due to local scour. Satisfactory results were obtained after validations carried out. The numerical model is then used to investigate local scour around a vibration pipeline under free surface wave conditions. Present numerical results show that vibration makes the scour depth below the pipeline deeper, scour scope around the pipeline wider.

Efficient semi-implicit coupling fluid-structure interaction analysis via model-order reduction of dynamic grids

In this paper, a fluid-structure interaction (FSI) using the model-order reduction (MOR) of dynamic grids is proposed. Within the interface problem between the fluid and the structure, the grid deformation is based on the spring analogy, and it is further extended in order for the efficient computation. In this procedure, the projection-based MOR technique is employed. The relevant MOR technique is realized by using the proper orthogonal decomposition and the discrete empirical interpolation method (POD/DEIM). The resulting FSI framework is based on a semi-implicit coupling approach. In order to achieve this, a characteristic-based split (CBS) scheme is used for an arbitrary Lagrangian–Eulerian (ALE) formulation of the two-dimensional Navier-Stokes equation. Also, a structural analysis is based on the co-rotational (CR) formulation in order to simulate the geometrically nonlinear behavior. Both the fluid and structural analyses are combined with the semi-implicit coupling approach, and the relevant algorithm is embedded in the ALE/CBS scheme. Then, the present FSI analysis is verified using three examples, and the computational efficiency is evaluated. The resulting numerical data demonstrated the validity of the present analysis, and it is found that a significant reduction in the computational cost is possibly achieved with the proposed approach.

Hydrodynamics and Heat Transfer Analysis of Airflow in a Sinusoidally Curved Channel

For heat transfer enhancement in heat exchangers, different types of channels are often tested. The performance of heat exchangers can be made better by considering geometry composed of sinusoidally curved walls. This research studies the modeling and simulation of airflow through a units long sinusoidally curved wavy channel. For the purpose, two-dimensional Navier Stokes equations along with heat equations are under consideration. To simulate the fluid flow problem, the finite element-based software COMSOL Multiphysics is used. The parametric study for Reynolds number from to and the period of vibration <i>P</i> from to are observed. The surface plots, streamline patterns, contours, and graphs are presented for the velocity field magnitude, temperature, and pressure against the Reynolds number as well as period of vibration. The results are compared with various literature. It is found that due to the creation of periodic contraction regions the velocity magnitude of the flow is continuously increasing with the increase of Reynolds number, on the contrary the pressure is decreasing from inlet to outlet of the channel. Also, a periodic variation in the pressure distribution along the vibrating boundaries has been found with an average increase of for the high Reynolds number. A novel work was done by expressing the rotation rate per second in terms of local Reynolds number for the recirculating regions found due to the periodic oscillation of the boundaries. The average temperature near the outlet where a fixed temperature is imposed initially is decreasing with an increase in Reynolds number. The convection process is weakened due to an increase of periodic vibration of boundaries.

Convective instability in multicomponent mixtures with Soret effect

We study the two-dimensional modes of gravity-dependent convective instability arising in a ternary mixture in a plane horizontal layer. The gravity force is oriented perpendicular to the layer heated from below. It is assumed that the system is in the state of mechanical equilibrium of a mixture considered as the base state of the system, which can become unstable under certain conditions. The mathematical model uses two-dimensional Navier-Stokes equations, transfer equations for describing mixture components with the Soret effect, and a heat transfer equation. Under the influence of a concentration gradient, the heat transfer symmetric to the Soret effect is neglected because it is usually small in water-based mixtures. The possible concentration-dependent diffusion and cross-diffusion of dissolved components are also neglected. We investigate a convective stability using linear approximation and under finite-amplitude convection regimes. The stability analysis of the base state after linearization of the governing equations with respect to the state of mechanical equilibrium predicts various convection modes. For each case, a neutral curve and the dependences of the disturbance growth decrements on both the Rayleigh number and the wave number are plotted. We show that in the case of the layer being heated from below, there are ranges of governing parameters where both long-wave and short-wave modes of oscillatory instability occur. The numerical analysis of a nonlinear coupling between these modes has revealed that such instabilities develop in the ternary mixtures whose components are arranged (due to the Soret effect) in different directions with respect to the temperature gradient. The effect of the Rayleigh number on the flow structure and heat mass transfer is demonstrated.

On the crest factor and its relevance in detecting turbulent behaviour in solutions of partial differential equations.

In this work we investigate the connection between two fundamental features of solutions of partial differential equations (PDEs), namely the crest factor and the length scale associated to each solution. We illustrate how the crest factor of solutions of some linear and non-linear PDEs, including the incompressible two-dimensional Navier-Stokes equations, has the capability for detecting turbulent and non-turbulent behaviour.

Finite-dimensional negatively invariant subsets of Banach spaces

We give a simple proof of a result due to Mañé (1981) [17] that a compact subset A of a Banach space that is negatively invariant for a map S is finite-dimensional if DS(x)=C(x)+L(x), where C is compact and L is a contraction (and both are linear). In particular, we show that if S is compact and differentiable then A is finite-dimensional. We also prove some results (following Málek et al. (1994) [15] and Zelik (2000) [23]) that give bounds on the (box-counting) dimension of such sets assuming a ‘smoothing property’: in its simplest form this requires S to be Lipschitz from X into another Banach space Z that is compactly embedded in X. The resulting bounds depend on the Kolmogorov ε-entropy of the embedding of Z into X. We give applications to an abstract semilinear parabolic equation and the two-dimensional Navier–Stokes equations on a periodic domain.

An Lp spaces-based mixed virtual element method for the two-dimensional Navier–Stokes equations

In this paper we extend the utilization of the Banach spaces-based formulations, usually employed for solving diverse nonlinear problems in continuum mechanics via primal and mixed finite element methods, to the virtual element method (VEM) framework and its respective applications. More precisely, we propose and analyze an [Formula: see text] spaces-based mixed virtual element method for a pseudostress-velocity formulation of the two-dimensional Navier–Stokes equations with Dirichlet boundary conditions. To this end, a dual-mixed approach determined by the introduction of a nonlinear tensor linking the usual pseudostress for the Stokes equations with the convective term is employed. As a consequence, this new tensor, say [Formula: see text], and the velocity [Formula: see text] of the fluid constitute the unknowns of the formulation, whereas the pressure is computed via a post-processing formula. The simplicity of the resulting VEM scheme is reflected by the absence of augmented terms, on the contrary to previous works on this and related models, and by the incorporation in it of only the projector onto the piecewise polynomial tensors and the usual stabilizer depending on the degrees of freedom of the virtual element subspace approximating [Formula: see text]. In turn, the non-virtual but explicit subspace given by the piecewise polynomial vectors of degree [Formula: see text] is employed to approximate [Formula: see text]. The corresponding solvability analysis is carried out by using appropriate fixed-point arguments, along with the discrete versions of the Babuška–Brezzi theory and the Banach–Nečas–Babuška theorem, both in subspaces of Banach spaces. A Strang-type lemma is applied to derive the a priori error estimates for the virtual element solution as well as for the fully computable approximation of [Formula: see text], the post-processed pressure, and a second post-processed approximation of [Formula: see text]. Finally, several numerical results illustrating the performance of the mixed-VEM scheme and confirming the rates of convergence predicted by the theory are reported.

A new direct discontinuous Galerkin method with interface correction for two-dimensional compressible Navier-Stokes equations

We propose a new formula for the nonlinear viscous numerical flux and extend the direct discontinuous Galerkin method with interface correction (DDGIC) of Liu and Yan [1] to compressible Navier-Stokes equations. The new DDGIC framework is based on the observation that the nonlinear diffusion can be represented as a sum of multiple individual diffusion processes corresponding to each conserved variable. A set of direction vectors corresponding to each individual diffusion process is defined as a function of the exact solution and approximated using the average value of the numerical solution at the cell interfaces. The new framework only requires the computation of conserved variables' gradient, which is linear and approximated by the original direct DG numerical flux formula. The proposed method greatly simplifies the implementation, and thus, can be easily extended to general equations and turbulence models. Numerical experiments with P1, P2, P3 and P4 polynomial approximations are performed to verify the optimal (k+1)th high-order accuracy of the method. The new DDGIC method is shown to be able to accurately calculate physical quantities such as lift, drag, and friction coefficients as well as separation angle and Strouhal number.

CFD investigation on scour beneath different configurations of piggyback pipelines under steady current flow

Scour beneath piggyback pipelines laid on a sandy seabed is numerically investigated by solving two-dimensional (2D) Unsteady Reynolds-averaged Navier-Stokes (URANS) equations. The fluid flow and the sediment transport are resolved using a two-phase flow Eulerian-Eulerian numerical solver SedFoam based on the open-source code OpenFOAM. A parametric study is performed for different Shields parameters defined as θ=Uf2/(s−1)gd50 of 0.068, 0.18 and 0.33 where Uf denotes the shear friction velocity on the seabed, the specific density of the sediment grains is s=ρs/ρf where ρs and ρf are the density of the solid and fluid phases, respectively, g is the gravitational acceleration and d50 is the median grain diameter. Scour processes beneath piggyback pipelines under steady current flow condition are studied for different gap ratios of G/D=0,0.15,0.25 and 0.35 between the main cylinder and the additional ones with a diameter ratio of d/D=0.3 between them. Mesh convergence studies based on the scour depth and the sediment profile are performed to obtain an appropriate grid resolution. A validation study is performed by comparing the scour depth of the single cylinder cases with the published experimental results obtained by Mao (1986). The effects of different θ and geometry configurations on the scour depth and the sediment profile are obtained and discussed.

Wetting and non-wetting liquid flow in nanopores: Wettability, non-Poiseuille profile and intermolecular fundamentals

Wetting and non-wetting liquid flow in nanoscale pores or channels is of great interest in many science and technology fields, such as biotechnology, energy storage/conversion and micro-devices. It is also of considerable significance in geosciences and geoengineering. For instance, for the geological disposal of contaminated waste understanding flow and transport in nanoporous rocks is critical for accurately evaluating potential environmental hazards. Similarly, for oil recovery from tight reservoirs, understanding flow mechanisms in nanopores may improve production forecasts and enable the development of strategies to enhance production.This paper presents a comprehensive study on wetting and non-wetting liquid flow in nanopores, including the development of a new nanoflow model and its demonstration using published findings from molecular dynamics simulations and laboratory tests. The new nanoflow model is formulated based on the general two-dimensional Navier-Stokes equations. The novel aspects include: 1) the model incorporates the parameters relating to the intermolecular interactions between the fluid and the solid wall in a nanopore, which are lacking in traditional continuum models; 2) the model contains the effects of wettability and the nanochannel’s dimension on flowrate, which are unique to nanoflows and not considered in traditional continuum models; and 3) the model is based on a two-dimensional non-Poiseuille flow model described by a set of coupled ordinary differential equations. Therefore, the numerical solutions are much easier to obtain than the partial differential equations of non-Poiseuille flow models. The classic Poiseuille flow is its limiting case.Three examples are presented to demonstrate the model’s capability. The first is the determination of the contact angle of water in hydrophobic carbon nanochannels or nanotubes using published results from MD simulations and laboratory tests. The second example uses the new nanoflow model to simulate non-wetting water flow in hydrophobic nanochannels or nanotubes, including flow in a nanoscale graphene slit representative of those present in graphene batteries for energy storage. The results obtained are compared with those obtained from MD simulations. The last example discusses the wetting or partially wetting liquid flow profiles in nanoscale pores. The three examples provide new insights into wetting and non-wetting liquid flow behaviour in nanopores.

Numerical investigation on implication of innovative hydrogen strut injector on performance and combustion characteristics in a scramjet combustor

Abstract A detailed numerical analysis on a scramjet combustor is carried out by introducing an innovative shaped strut in place of the conventional strut. The design of newly added strut aids in generating intense vorticity which helps in efficient mixing of fuel and oxidizer. The air from the isolator enters the combustor at Mach 2.0, whereas fuel enters from the trailing edge of the strut sonically. In this study the flow dynamics with finite volume approach on commercial software Ansys-Fluent 20.0 to solve the two-dimensional Reynolds average Navier Stokes equation (RANS) with compressible fluid flow by considering the density-based solver with SST k-ε turbulent model. The species transport model with volumetric reaction and finite rate/eddy dissipation turbulence chemistry interaction is adopted to study the combustion phenomena and validated with the experimental results, and it is found that the interaction of the shear shock layer enhances the mixing rate by intensifying turbulence. The modified strut injector’s mixing efficiency is compared to the base strut and observed that with a 40% reduction in length, the modified strut injection technique exhibited a mixing efficiency of &gt;95%. The combustion efficiency is then estimated streamwise, and the plot follows the same pattern as the mixing efficiency with fuel burns down completely when x = 150 mm for the modified strut whereas x = 200 mm for the base strut. This can compact the combustion chamber and increases the thrust-to-weight ratio. So, the innovative strut adopted can improvise the combustion efficiency.

Ill-posedness for the Cauchy problem of the two-dimensional compressible Navier-Stokes equations for an ideal gas

Ill-posedness for the Cauchy problem of the two-dimensional compressible Navier-Stokes equations for an ideal gas

Linearly implicit variable step-size BDF schemes with Fourier pseudospectral approximation for incompressible Navier-Stokes equations

In this paper, linearly implicit backward differentiation formulas with variable step-sizes are proposed to solve numerically the two-dimensional incompressible Navier-Stokes equations (formulated in vorticity-stream function). With Fourier pseudospectral methods for spatial discretization, the diffusion term is discretized implicitly and the nonlinear convection term is treated by a combination of implicit and explicit discretizations. As a result, only linear solvers are needed at each time step to achieve the desired temporal accuracy. With the help of a priori assumption and aliasing error control techniques, the error estimates for one-step and two-step backward differentiation formulas are established in several norms under appropriate step-size constraints. Compared with the numerical results of implicit-explicit (the nonlinear convection term is treated explicitly) BDF2 method and fully implicit Crank-Nicolson method, it demonstrates that the proposed linearly implicit variable step-size BDF2 method is effective and robust.

Computer Simulation of Water Flow Animation Based on Two-Dimensional Navier-Stokes Equations

Simulation of water flow animation is a significant and challenging subject in computer graphics. With the continuous development of computational fluid dynamics and computer graphics, many more effective simulation methods have been developed, and fluid animation simulation has developed rapidly. In order to obtain realistic flow animation, one of the key aspects is to simulate flow motion. Based on the two-dimensional Navier-Stokes equations, a mathematical model is established to solve the boundary conditions required by the physical flow field of water. The coordinate transformation formula is introduced to transform the irregular physical area into a regular square calculation area, and then, the specific expressions of the liberalized Navier-Stokes equation, continuity equation, pressure Poisson equation, and dimensionless boundary conditions are given. Using animation software to sequence graphics and images of all kinds of control and direct operation of the relevant instructions, through the computer technology to simulate the flow of animation, based on the stability of fluid simulation method and simulation efficiency, so as to make realistic flow animation. The results show that FluidsNet has considerable performance in accelerating large scene animation simulation on the basis of ensuring the rationality of prediction, and the motion of water wave is realistic. The application of computer successfully simulates water flow.

Mean Current Profile over Rippled-Beds in the Presence of Non-Breaking Waves and Analysis of Its Influencing Factors

Classical eddy viscosity model deviates from the actual mean current profiles, when calculating the mean current profiles over rippled-beds in the presence of non-breaking waves, owing to the neglect of the enhancement of the wave boundary layer thickness by ripples and the wave-induced shear stress (the radiation stress and the wave Reynolds stress). Considering these shortcomings, a semi-empirical one-dimensional vertical (1DV) model is presented in this study. The present model was obtained using the two-dimensional Navier–Stokes equations and eddy viscosity assumptions, which differ from those of previous researchers, while a top-to-bottom sequence was adopted to calculate the mean current profiles. Empirical formulae were derived from the laboratory measurements and used in the present model to accurately predict the wave boundary layer thickness and bed roughness. The present model is in satisfactory agreement with the data from laboratory experiments. The factors influencing the mean current profiles were analyzed also. The wave-induced second-order shear stresses were found to be the principal reason for the deviations of the mean current profiles in the near-surface layer; as the influencing factors of wave-induced shear stress, the intensity of the wave relative to the current, the angle between the wave and current, and the size of ripples can also have a non-negligible effect on the mean current profiles.

Well-posedness and large deviations for 2D stochastic constrained Navier-Stokes equations driven by Lévy noise in the Marcus canonical form

We consider stochastic two-dimensional constrained Navier-Stokes equations driven by Lévy noise in the Marcus canonical form. The aim of this work is two-fold. At first, we prove the existence of a martingale solution based on the construction relying on classical Faedo-Galerkin approximations, compactness method and the Jakubowski's version of Skorokhod representation theorem for non-metric spaces. We further prove that the martingale solution is pathwise unique and deduces the existence of a strong solution. In the second part of the paper, we establish a Wentzell-Freidlin type large deviations principle for the small noise asymptotic of solutions using weak convergence method.

Experimental and numerical investigation on flame propagation and transition to detonation in curved channel

The Deflagration to Detonation Transition (DDT) process in a curved channel was numerically and experimentally investigated for stoichiometric ethylene/oxygen and acetylene/oxygen/argon mixtures. The numerical method was based on the two-dimensional Navier–Stokes equations with detailed chemical mechanisms. Flame propagation through the curved channel was observed using a high-speed camera. Four distinctive flame patterns were identified, appearing in the following order: spherical flame, finger flame, tongue-like flame, and detonation. The development of the tongue-like flame occupied most of time of the DDT process and played an important role in the onset of detonation. The formation of the tongue-like flame was analyzed in detail and its interactions with boundary layers enhanced the flame surface as well as the flame velocity. The modes of detonation initiation in irregular and regular cellular detonation systems were compared. Although the local explosion always occurred at the outer wall regardless of the mixture type, the detonation was caused by the amplification of instabilities for irregular system at the outer region of the tongue-like flame, whereas shock focusing-ignition occurred at the outer wall for regular system. Furthermore, the DDT run-up distance LDDT was shorter for the irregular system than for the regular system with the same detonation sensitivity.

Influence of turbulence intensity on vortex pattern for a rigid cylinder with turbulence stimulation in flow induced oscillations

The effect of ambient turbulence intensity on the flow induced oscillations of rigid circular cylinders with symmetric and asymmetric local turbulence stimulation is studied. Cylinders oscillate in one degree of freedom transversely to a steady uniform flow. Two-dimensional unsteady Reynolds-Average Navier-Stokes equations with the Spalart-Allmaras turbulence model are used to solve the problem numerically. Three freestream turbulence intensity values (0.2%, 1%, 5%) are used to study the influence of turbulence intensity on vortex pattern. Simulation results are compared with experimental data measured in the Marine Renewable Energy Laboratory of the University of Michigan in the range of 30,000 ≤ Re ≤ 110,000. The amplitude ratio, lift coefficient, vortex modes, and the interactions between vortices and cylinders are observed and discussed. Major conclusions are: (a) Vortex patterns strongly depend on the freestream ambient turbulence intensity. (b) Low turbulence intensity can generate multi-vortex patterns at high Re. (c) High turbulence intensity results in fewer separations of a shed vortex into multiple vortices inducing emergence of elongated vortex patterns.

Numerical investigation of scaling effect in two-dimensional oscillating water column wave energy devices for harvesting wave energy

Oscillating water column (OWC) devices utilise wave-induced oscillation of water column in partially submerged chambers to extract energy of waves in ocean. The principal aim of this paper is to investigate scaling effects of two-dimensional OWC through numerical simulations. The fluid motion under waves is simulated by solving the two-dimensional incompressible Reynolds-Averaged Navier-Stokes equations using finite element method under Arbitrary Lagrangian-Eulerian scheme. To investigate scaling effects, the waves and the OWC are scaled up or down according to Froude number similarity. The turbine is scaled up or down using three different scaling methods: (1) Turbine coefficient remains unchanged, i.e. not scaling the turbine at all; (2) Equivalent pneumatic damping coefficient of the turbine is constant and (3) the two-dimensional turbine coefficient is constant. The numerical results show that method (2) has the smallest effects on the best hydraulic efficiency and method (3) has the smallest effects on the best performance wave number, i.e. the wave number where the OWC receives the largest percentage of the wave energy. The worst scaling method is not scaling the turbine. Because the best performance wave number varies with the model scale in method (1) and (2), it is very likely the scaling effects of the model scale on the hydrodynamic performance at different wave numbers are different from each other. In addition, the effects of the wave height and the chamber volume are studied. Increasing chamber volume causes a reduction of the best performance, however it may improve general performance of OWC because it increases the frequency band of waves with high hydraulic efficiency. Increasing wave height causes a reduction of the hydraulic efficiency.

Initial boundary value problems for two-dimensional isentropic compressible Navier-Stokes equations with rotating effect terms

We consider the initial boundary value problem for two-dimensional isentropic compressible Navier-Stokes equations with rotating effect terms. This problem is related to the motion of the compressible viscous flow surrounding a rotating obstacle. We show the global existence and uniqueness of strong solutions with initial data close to some approximate solution, provided that the friction coefficient α attached to the Navier-slip boundary condition is small. The large time behavior of the solutions is also studied.