Steady-state Incompressible Navier-Stokes Equations Research Articles

An alternative full multigrid SIMPLEC approach for the incompressible Navier–Stokes equations

An alternative approach to solve the steady-state incompressible Navier–Stokes equations using the multigrid (MG) method is presented. The mathematical model is discretized using the finite volume method with second-order approximation schemes in a uniform collocated (nonstaggered) grid. MG is employed through a full approximation scheme-full MG algorithm based on V-cycles. Pressure-velocity coupling is ensured by means of a developed modified SIMPLEC algorithm which uses independent V-cycles for relaxing the pressure-correction and momentum equations. The coarser grids are used only internally in these cycles. All other original SIMPLEC steps can be performed only on the finest grid of the current full MG level. The model problem of the lid-driven flow in the unitary square cavity is used for the tests of the numerical model. Computational performance is measured through error and residual decays and execution times. Good performances were obtained for a wide range of Reynolds numbers, with speedups of orders as high as Linear relationships between execution times and grid sizes were observed for low and high Re values ( and 7,500). For intermediate Re values ( and 1,000), the linear trend was observed from more refined grids (512 onwards).

Very high-order accurate finite volume scheme for the steady-state incompressible Navier–Stokes equations with polygonal meshes on arbitrary curved boundaries

The numerical solution of the incompressible Navier–Stokes equations raises challenging numerical issues on the development of accurate and robust discretisation techniques due to the div-grad duality. Significant advances have been achieved in the context of high-order of convergence methods, but many questions remain unsolved in the quest of practical, accurate, and efficient approaches. In particular, the discretisation of problems with arbitrary curved boundaries has been the subject of intensive research, where the conventional practice consists in employing curved meshes to avoid accuracy deterioration. The present work proposes a novel approach, the reconstruction for off-site data method, which employs polygonal meshes to approximate arbitrary smooth curved boundaries with linear piecewise elements. The Navier–Stokes equations are discretised with a staggered finite volume method, and the numerical fluxes are computed on the polygonal mesh elements. Boundary conditions are taken into account via polynomial reconstructions with specific linear constraints defined for a set of points on the physical boundary. The very-high order of convergence is preserved without relying on the full curve parametrisation, avoiding the limitations of curved mesh approaches, such as sophisticated mesh generation algorithms, cumbersome quadrature rules, and complex non-linear transformations. A comprehensive verification benchmark is provided, with numerical test cases for several fluid flow problems, to demonstrate the capability of the proposed approach to achieve very high-orders of convergence.

Computational Model for Electrochemical Gas Separation and Inerting System

In July 1996, TWA flight 800 crashed into the Atlantic Ocean only 12 minutes after takeoff killing all 230 people on board. After a 4-year investigation into this catastrophe, the National Transportation Safety Board concluded that the accident was probably caused by the explosion of flammable fuel vapor in the Boeing 747’s center fuel tank. Following this landmark incident new regulations for aircraft fuel tank inerting were developed to prevent such explosions. Fuel tank inerting is the process of reducing the flammability of the fuel vapor by restricting the oxygen concentration in the ullage (air volume above the fuel) to a safe value – 12% for commercial aircraft and 9% for military aircraft. Generally, inerting is carried out by driving oxygen out the ullage using an inert gas like nitrogen.Existing inerting technologies include systems like liquid nitrogen, explosion suppressant foam, halon extinguishment, and on-board inert gas generation systems (OBIGGS). Compared to other techniques, OBIGGS is seen as viable option since nitrogen enriched air (NEA) can be produced during flight by supplying pressurized air into separation modules that preferentially retain nitrogen while allowing oxygen to escape. However, such systems are expensive and consume substantial amounts of energy, while also undergoing premature failure. Additionally, the NEA generation rate is fixed in such systems, whereas the actual NEA requirement depends on the aircraft’s flight stage. Thus, incumbent technologies are designed based on the highest demand making them overweight and expensive.Here, we propose a novel on-board electrochemical gas separation and inerting system (EGSIS) to overcome such problems. EGSIS is an electrically powered system that can be used to produce and supply NEA to the fuel tank. It combines a polymer electrolyte membrane (PEM) electrolyzer anode wherein water is dissociated to generate oxygen, and a PEM fuel cell cathode where oxygen from the inlet air inlet is reduced to produce humidified NEA as the output gas. After proper dehumidification, the NEA can be supplied directly to the ullage to reduce its oxygen concentration. Unlike current technologies, the NEA production rate in EGSIS can be easily varied with demand just by controlling the voltage applied to the system. Experimental investigations of EGSIS have been previously conducted by our group. This paper represents the first attempt to model and simulate EGSIS.We present a two-dimensional mathematical model of a single cell EGSIS that provides a foundation for the simulation and analysis of a full-scale EGSIS module. A single phase, two-dimensional, steady state, isothermal model is implemented in an EGSIS cell using COMSOL. Various EGSIS performance parameters such as current densities, reactant concentration distributions, and polarization curves are investigated as a function of operating voltage and temperature. The model involves the material properties of the various components including the polymer electrolyte membrane, the catalyst later and gas diffusion media; the transport of water vapor and air in the anode and cathode flow field channels, respectively; the transport of protons and water in the catalyst layer and the membrane; and the transport of electrical charge in the solid phase. The governing equations include continuity and steady-state incompressible Navier-Stokes equations. Additionally, species transport is modeled using the generalized Stefan-Maxwell equations whereas the Butler-Volmer equation is employed for charge transport. All of these coupled model equations are solved by the finite element method using the COMSOL Multi physics software package. The results from the computational model are then compared and validated with our experimental results for various operating conditions. Different operating strategies are explored with the goal of improving system performance. The simulated results reveal the influence of temperature, reaction flowrate, and optimized material properties on EGSIS performance. This model also provides a base for future research into the effect of alternate catalysts and gas diffusion media. It also serves as a foundation for the analysis of a practical EGSIS stack.

An unfitted HDG method for Oseen equations

We propose and analyze a high order unfitted hybridizable discontinuous Galerkin method to numerically solve Oseen equations in a domain Ω having a curved boundary. The domain is approximated by a polyhedral computational domain not necessarily fitting Ω. The boundary condition is transferred to the computational domain through line integrals over the approximation of the gradient of the velocity and a suitable decomposition of the pressure in the computational domain is employed to obtain an approximation of the pressure having zero-mean in the domain Ω. Under assumptions related to the distance between the computational boundary and Ω, we provide stability estimates of the solution that will lead us to the well-posedness of the scheme and also to the error estimates. In particular, we prove that the approximations of the pressure, velocity and its gradient are of order hk+1, where h is the meshsize and k the polynomial degree of the local discrete spaces. We provide numerical experiments validating the theory and also showing the performance of the method when applied to the steady-state incompressible Navier–Stokes equations.

Adaptive HDG Methods for the Steady-State Incompressible Navier–Stokes Equations

We consider a hybridizable discontinuous Galerkin method for the steady-state incompressible Navier–Stokes equations. We use polynomials of degree $$k+1$$ , k, k and k for approximations of the velocity, the velocity gradient, the pressure and the boundary traces. Some stability results for approximate solutions and some relationships between norms are provided. Moreover an a posteriori error estimator is introduced. By $$L^2$$ -projection and inf-sup condition, we prove that the error estimator is robust for the global $$L^2$$ errors in the velocity, the velocity gradient and the pressure. Finally, a Picard iteration method and an adaptive HDG algorithm are presented. Furthermore, several numerical examples are shown to validate the theoretical analysis.

Effects of boundary layer suction control on flow through an axisymmetric diverging channel

ABSTRACT The conical diffuser is one of the oldest and most versatile hydraulic devices, however its performance is often less than optimal. This is because boundary layer separation and instabilities in the flow lead to energy losses through internal friction. Boundary layer suction offers a method for controlling and improving this flow, and some experimental studies have shown this to be effective – though they leave questions as to its net efficiency. This study takes a more comprehensive approach to suction control of the conical diffuser through the use of numerical studies coupled with optimisation. The flow through a conical diffuser with semi-divergence angle was simulated by solving the full steady-state incompressible Navier-Stokes equations in the fully laminar Reynolds Number range . Uniform suction control was then applied in the divergent region, and optimisation methods used to determine the most effective suction strength for two objectives: (a) maximising the diffuser efficiency, and (b) minimising the separation length. It was found that suction control could increase the performance of the diffuser and was efficient even when considering the power to run the control. The results suggest that fluidic control of the diffuser should be reconsidered as it shows significant promise, particularly for applications where the constraints on the geometry of the diffuser are tight, for example in aircraft turbine engines.

Inertia-driven enhancement of mixing efficiency in microfluidic cross-junctions: a combined Eulerian/Lagrangian approach

Mixing of a diffusing species entrained in a three-dimensional microfluidic flow-focusing cross-junction is numerically investigated at low Reynolds numbers, $$1 \le Re \le 150$$ , for a value of the Schmidt number representative of a small solute molecule in water, $$Sc = 10^3$$ . Accurate three-dimensional simulations of the steady-state incompressible Navier–Stokes equations confirm recent results reported in the literature highlighting the occurrence of different qualitative structures of the flow geometry, whose range of existence depends on Re and on the ratio, R, between the volumetric flowrates of the impinging currents. At low values of R and increasing Re, the flux tube enclosing the solute-rich stream undergoes a topological transition, from the classical flow-focused structure to a multi-branched shape. We here show that this transition causes a nonmonotonic behavior of mixing efficiency with Re at constant flow ratio. The increase in efficiency is the consequence of a progressive compression of the cross-sectional diffusional lengthscale, which provides the mechanism sustaining the transversal Fickian flux even when the Peclet number, $${Pe=Re \, Sc}$$ , characterizing mass transport, becomes higher due to the increase in Re. The quantitative assessment of mixing efficiency at the considerably high values of the Peclet number considered ( $$10^3 \le Pe \le 1.5 \times 10^5$$ ) is here made possible by a novel method of reconstruction of steady-state cross-sectional concentration maps from velocity-weighted ensemble statistics of noisy trajectories, which does away with the severe numerical diffusion shortcomings associated with classical Eulerian approaches to mass transport in complex 3d flows.

Analysis of a hybridizable discontinuous Galerkin method for the steady-state incompressible Navier-Stokes equations

Analysis of a hybridizable discontinuous Galerkin method for the steady-state incompressible Navier-Stokes equations

Large scale three-dimensional topology optimisation of heat sinks cooled by natural convection

This work presents the application of density-based topology optimisation to the design of three-dimensional heat sinks cooled by natural convection. The governing equations are the steady-state incompressible Navier–Stokes equations coupled to the thermal convection–diffusion equation through the Bousinessq approximation. The fully coupled non-linear multiphysics system is solved using stabilised trilinear equal-order finite elements in a parallel framework allowing for the optimisation of large scale problems with order of 20–330 million state degrees of freedom. The flow is assumed to be laminar and several optimised designs are presented for Grashof numbers between 103 and 106. Interestingly, it is observed that the number of branches in the optimised design increases with increasing Grashof numbers, which is opposite to two-dimensional topology optimised designs. Furthermore, the obtained topologies verify prior conclusions regarding fin length/thickness ratios and Biot numbers, but also indicate that carefully tailored and complex geometries may improve cooling behaviour considerably compared to simple heat fin geometries.

A superconvergent HDG method for the incompressible Navier–Stokes equations on general polyhedral meshes

We present a superconvergent hybridizable discontinuous Galerkin (HDG) method for the steady-state incompressible Navier-Stokes equations on general polyhedral meshes. For arbitrary conforming polyhedral mesh, we use polynomials of degree k+1, k, k to approximate the velocity, velocity gradient and pressure, respectively. In contrast, we only use polynomials of degree k to approximate the numerical trace of the velocity on the interfaces. Since the numerical trace of the velocity field is the only globally coupled unknown, this scheme allows a very efficient implementation of the method. For the stationary case, and under the usual smallness condition for the source term, we prove that the method is well defined and that the global L2-norm of the error in each of the above-mentioned variables and the discrete H1-norm of the error in the velocity converge with the order of k+1 for k>=0. We also show that for k>=1, the global L2-norm of the error in velocity converges with the order of k+2. From the point of view of degrees of freedom of the globally coupled unknown: numerical trace, this method achieves optimal convergence for all the above-mentioned variables in L2-norm for k>=0, superconvergence for the velocity in the discrete H1-norm without postprocessing for k>=0, and superconvergence for the velocity in L2-norm without postprocessing for k>=1.

Penalty-free discontinuous Galerkin methods for incompressible Navier–Stokes equations

A first-order discontinuous Galerkin method is proposed for solving the steady-state incompressible Navier–Stokes equations. The stability of this penalty-free method is obtained by locally enriching the discrete space with a quadratic polynomial. A priori error estimates are derived. Numerical examples confirm the theoretical convergence.

On some new exact solutions of incompressible steady state Navier-Stokes equations

In the present work, Lie symmetries are constructed for the steady state incompressible Navier-Stokes equations in two dimensional space. The Navier-Stokes equations are transformed into a fourth order quasilinear partial differential equation through well known approach of stream function. Seven similarity solutions are obtained using one-parameter Lie group of transformations with commuting infinitesimal operators. Among the seven solutions so obtained, three are new and analysed physically. Also, the remaining four solutions are reported in Polyanin and Zaitsev (Handbook of nonlinear partial differential equations, Chapman and Hall/CRC Press, Boca Raton, pp. 607–608, 2004) and interpreted hydrostatically.

Numerical Simulation on Effect of Obstructing Airflow by Different Outlet Parameters of Mine Air Curtain

According to characteristics of air curtain obstructing airflow in mine tunnel, utilized turbulent flow dynamic viscosity method, and three-dimensional steady-state incompressible Navier-Stokes equations of computational fluid mechanics, the simulation of flow field in mine tunnel was made by changing outlet parameters of air curtain. The results show that the obstructing airflow capability increase with the increase of the outlet airflow velocity, the obstructing airflow capability decrease with the increase of the outlet width when the amount of wind is a certain value, the obstructing airflow capability increase at first, then decrease with the increase of the outlet angle. When the parameters are the same, the obstructing airflow capability of acyclic air curtain is better than that of circular air curtain.The result of simulation analysis of mine air curtain can be used as the basis for optimized design.

On approximated ILU and UGS preconditioning methods for linearized discretized steady incompressible Navier-Stokes equations

When the artificial compressibility method in conjunction with high-order upwind compact finite difference schemes is employed to discretize the steady-state incompressible Navier-Stokes equations, in each pseudo-time step we need to solve a structured system of linear equations approximately by, for example, a Krylov subspace method such as the preconditioned GMRES. In this paper, based on the special structure and concrete property of the linear system we construct a structured preconditioner for its coefficient matrix and estimate eigenvalue bounds of the correspondingly preconditioned matrix. Numerical examples are given to illustrate the effectiveness of the proposed preconditioning methods.

A Sobolev gradient method for treating the steady-state incompressible Navier-Stokes equations

Abstract The velocity-vorticity-pressure formulation of the steady-state incompressible Navier-Stokes equations in two dimensions is cast as a nonlinear least squares problem in which the functional is a weighted sum of squared residuals. A finite element discretization of the functional is minimized by a trust-region method in which the trustregion radius is defined by a Sobolev norm and the trust-region subproblems are solved by a dogleg method. Numerical test results show the method to be effective.

Steady‐state solution of incompressible Navier–Stokes equations using discrete least‐squares meshless method

AbstractA fractional step method for the solution of the steady state incompressible Navier–Stokes equations is proposed in this paper in conjunction with a meshless method, named discrete least‐squares meshless (DLSM). The proposed fractional step method is a first‐order accurate scheme, named semi‐incremental fractional step method, which is a general form of the previous first‐order fractional step methods, i.e. non‐incremental and incremental schemes. One of the most important advantages of the proposed scheme is its capability to use large time step sizes for the solution of incompressible Navier–Stokes equations. DLSM method uses moving least‐squares shape functions for function approximation and discrete least‐squares technique for discretization of the governing differential equations and their boundary conditions. As there is no need for a background mesh, the DLSM method can be called a truly meshless method and enjoys symmetric and positive‐definite properties. Several numerical examples are used to demonstrate the ability and the efficiency of the proposed scheme and the discrete least‐squares meshless method. The results are shown to compare favorably with those of the previously published works. Copyright © 2010 John Wiley & Sons, Ltd.

Meshfree Point Collocation Schemes for 2D Steady State Incompressible Navier-Stokes Equations in Velocity-Vorticity Formulation for High Values of Reynolds Number

A meshfree point collocation method has been developed for the velocity- vorticity formulation of two-dimensional, steady state incompressible Navier-Stokes equations. Particular emphasis was placed on the application of the velocity-correc- tion method, ensuring the continuity equation. The Moving Least Squares (MLS) approximation is employed for the construction of the shape functions, in conjunc- tion with the general framework of the point collocation method. Computations are obtained for regular and irregular nodal distributions, stressing the positivity con- ditions that make the matrix of the system stable and convergent. The accuracy and the stability of the proposed scheme are demonstrated through two representative, well-known, and established benchmark problems. The numerical scheme was also applied to a case with irregular geometry for marginally high Reynolds numbers.

Two-dimensional turbulent heat and fluid flow model to study effect of working gas and other process parameters on heat transfer in plasma furnace

A two-dimensional heat and fluid flow model was used to simulate the plasma arc furnace, where the flow is governed by the steady state incompressible Navier–Stokes equations. The flow has been taken as turbulent and the standard k-epsilon model was used to simulate the turbulence in the flow. The coupled non-linear differential equations were solved with suitable boundary conditions and temperature dependent plasma properties at atmospheric pressure by employing an efficient finite volume method. The calculations and heat transfer to various parts of the furnace were calculated for argon, nitrogen and hydrogen plasmas. The voltage–current characteristic for the different types of plasma and the effect of other process parameters on heat transfer are discussed.

Analysis of fluid flow and reaction kinetics in a flow injection analysis biosensor

This paper presents the development of a convection–diffusion-reaction model to simulate the behavior of a flow injection analysis (FIA) biosensor with respect to its design and operational parameters, such as flow rate, fluidic channel dimensions, flow cell geometry and volume of substrate and enzyme injected. The problem is described by the steady-state incompressible Navier–Stokes equations for fluid flow, and transient convection–diffusion-reaction equations for the different dissolved species. The enzyme–substrate interaction is described by Michaelis–Menten kinetics. The model was successfully validated on a glucose FIA-biosensor by comparing measured and simulated H 2O 2 concentration versus time profiles. Both peaks matched well, however, a slightly longer tailing was observed in the measurements due to simplifications in the geometrical model of the flow cell. The simulated and measured calibration curves for glucose closely matched in the range between 0 and 2.5 mM. The model was also applied to simulate the substrate, enzyme and product concentration profiles in the fluidic channels and the flow cell as a function of time. A parabolic velocity profile obtained from a pressure driven flow resulted in more peak broadening in the flow cell than a uniform velocity profile that occurs in packed column flow. Given the specified set of parameters, the concentration profiles of substrate, enzyme and product illustrated that not all substrate was converted into product when the flow reached the inlet of the flow cell, indicating a suboptimal sensitivity of the designed biosensor. This modeling approach has a high potential in the design and development of high accuracy FIA-biosensors, regardless of the chosen enzyme substrate system.

Efficient preconditioning of the discrete adjoint equations for the incompressible Navier–Stokes equations

AbstractPreconditioning of the discrete adjoint equations is closely related to preconditioning the linear systems arising in the Newton linearization of the discretized flow equations. We investigate the use of an optimal preconditioner for both problems on the example of a finite element discretization of the steady state incompressible Navier–Stokes equations. It is demonstrated that complications arising from the use of a zero mean pressure condition in the problem formulation can be overcome by modifying the preconditioner suitably. Copyright © 2005 John Wiley & Sons, Ltd.