Time-dependent Incompressible Navier-Stokes Equations Research Articles

Characteristic Local Discontinuous Galerkin Methods for Incompressible Navier-Stokes Equations

AbstractBy combining the characteristic method and the local discontinuous Galerkin method with carefully constructing numerical fluxes, variational formulations are established for time-dependent incompressible Navier-Stokes equations in ℝ2. The nonlinear stability is proved for the proposed symmetric variational formulation. Moreover, for general triangulations the priori estimates for the L2–norm of the errors in both velocity and pressure are derived. Some numerical experiments are performed to verify theoretical results.

Effects of temperature-dependent contact angle on the flow dynamics of an impinging droplet on a hot solid substrate

A temperature-dependent dynamic contact angle as a function of temperature-dependent surface tension and reference equilibrium contact angle is proposed for modeling of moving contact line flows, in particular, for computations of liquid droplet impingement on a hot solid substrate. The fluid flow in the liquid droplet is described by the time-dependent incompressible Navier–Stokes equations, whereas the heat transfer in the liquid droplet and in the solid substrate is described by the energy equation. The arbitrary Lagrangian–Eulerian (ALE) approach together with the finite element method is used to solve the governing equations in a time-dependent domain. Further, the Marangoni effects are incorporated into the model without evaluating the tangential derivatives of the temperature on the free surface. The effects of temperature-dependent contact angle on the flow dynamics of the droplet and on the heat transfer from the solid substrate into the liquid droplet are studied for different Reynolds numbers, Weber numbers, solid phase Peclet numbers, solid phase initial temperatures and reference equilibrium contact angles. Numerical studies show that the influence of the temperature-dependent contact angle is negligible in partially wetting droplets, whereas the effects on the wetting diameter and on the total heat transfer are 10.79% and 7.36% respectively in the considered highly wetting and non–wetting droplets.

Linear stability and energetics of rotating radial horizontal convection

The effect of rotation on horizontal convection in a cylindrical enclosure is investigated numerically. The thermal forcing is applied radially on the bottom boundary from the coincident axes of rotation and geometric symmetry of the enclosure. First, a spectral element method is used to obtain axisymmetric basic flow solutions to the time-dependent incompressible Navier–Stokes equations coupled via a Boussinesq approximation to a thermal transport equation for temperature. Solutions are obtained primarily at Rayleigh number $\mathit{Ra}=10^{9}$ and rotation parameters up to $Q=60$ (where $Q$ is a non-dimensional ratio between thermal boundary layer thickness and Ekman layer depth) at a fixed Prandtl number $\mathit{Pr}=6.14$ representative of water and enclosure height-to-radius ratio $H/R=0.4$. The axisymmetric solutions are consistently steady state at these parameters, and transition from a regime unaffected by rotation to an intermediate regime occurs at $Q\approx 1$ in which variation in thermal boundary layer thickness and Nusselt number are shown to be governed by a scaling proposed by Stern (1975, Ocean Circulation Physics. Academic). In this regime an increase in $Q$ sees the flow accumulate available potential energy and more strongly satisfy an inviscid change in potential energy criterion for baroclinic instability. At the strongest $Q$ the flow is dominated by rotation, accumulation of available potential energy ceases and horizontal convection is suppressed. A linear stability analysis reveals several instability mode branches, with dominant wavenumbers typically scaling with $Q$. Analysis of contributing terms of an azimuthally averaged perturbation kinetic energy equation applied to instability eigenmodes reveals that energy production by shear in the axisymmetric mean flow is negligible relative to that produced by conversion of available potential energy from the mean flow. An evolution equation for the quantity that facilitates this exchange, the vertical advective buoyancy flux, reveals that a baroclinic instability mechanism dominates over $5\lesssim Q\lesssim 30$, whereas stronger and weaker rotations are destabilised by vertical thermal gradients in the mean flow.

A Stabilized Characteristic-Nonconforming Finite Element Method for Time-Dependent Incompressible Navier–Stokes Equations

In this paper, we study a stabilized characteristic-nonconforming finite element method to solve the time-dependent incompressible Navier–Stokes equations. The characteristic scheme is used to deal with advection term and temporal differentiation, which avoid some difficulties caused by trilinear terms. The space discretization utilizes the nonconforming lowest equal-order pair of mixed finite elements (i.e. \(\textit{NCP}_1-{\mathbf {P}}_1\)). The stability analysis and optimal-order error estimates for velocity and pressure are presented. Numerical results are also provided to verify theory analysis.

A Two-Grid Finite Element Method for Time-Dependent Incompressible Navier-Stokes Equations with Non-Smooth Initial Data

AbstractWe analyze here, a two-grid finite element method for the two dimensional time-dependent incompressible Navier-Stokes equations with non-smooth initial data. It involves solving the non-linear Navier-Stokes problem on a coarse grid of size H and solving a Stokes problem on a fine grid of size h, h « H. This method gives optimal convergence for velocity in H1-norm and for pressure in L2-norm. The analysis mainly focuses on the loss of regularity of the solution at t = 0 of the Navier-Stokes equations.

On the immersed boundary method: Finite element versus finite volume approach

Two immersed boundary methods (IBM) for the simulation of fluid flow problems with complex geometries are introduced: a direct-forcing immersed finite element method (IFEM) and a direct-forcing immersed finite volume method (IFVM).In the IFEM a projection approach is presented for the coupled system of time-dependent incompressible Navier–Stokes equations (NSEs) in conjunction with the IBM for solving fluid flow problems in the presence of rigid objects not represented by the underlying mesh. The IBM allows solving the flow for geometries with complex objects without the need of generating a body-fitted mesh. Dirichlet boundary constraints are satisfied applying a boundary force at the immersed body surface. Using projection and interpolation operators from the fluid volume mesh to the solid surface/volume mesh (i.e., the “immersed” boundary) and vice versa, it is possible to impose the extra constraint to the NSEs as a Lagrange multiplier in a fashion very similar to the effect that pressure has on the momentum equations to satisfy the divergence-free constraint. The IFEM approach presented shows third order accuracy in space and second order accuracy in time when the simulation results for the Taylor–Green decaying vortex are compared to the analytical solution.For the IFVM presented, a ghost-cell approach with sharp interface scheme is used to enforce the boundary condition at the fluid/solid interface. The interpolation procedure at the immersed boundary preserves the overall second order accuracy of the base solver. The developed ghost-cell method is applied on a staggered configuration with the Semi-Implicit Method for Pressure-Linked Equations Revised algorithm. Second order accuracy in space and first order accuracy in time are obtained when the Taylor–Green decaying vortex test case computed with the IFVM is compared to the analytical solution.Computations were performed using the IFEM and IFVM approaches for the two-dimensional flow over a backward-facing step, two-dimensional flow past a stationary circular cylinder, three-dimensional flow past a sphere and two-dimensional oscillating circular cylinder in a stationary square domain. The numerical results obtained with the discussed IFEM and IFVM were compared against other IBMs available in literature and simulations performed with the commercial computational fluid dynamics code STAR-CCM+/V7.04.006. The benchmark test cases showed that the numerical results obtained with the implemented immersed boundary methods are in good agreement with the predictions from STAR-CCM+ and the numerical data from the other IBMs. The immersed boundary method based on finite element approach is numerically more accurate than the IBM based on finite volume discretization. In contrast, the latter is computationally more efficient than the former.

A finite element variational multiscale method for computations of turbulent flow over an aerofoil

Numerical simulation of turbulent flows over different aerofoil configurations are presented in this paper. The incompressible fluid flow is described by the time-dependent incompressible Navier–Stokes equations. Further, a finite element variational multiscale method is used to simulate the turbulent flows. Computation over a cylinder and different variants of aerofoils are presented. The obtained numerical results demonstrate the capabilities of variational multiscale methods.

Modeling of the non-isothermal liquid droplet impact on a heated solid substrate with heterogeneous wettability

A comprehensive numerical investigation on the impingement and spreading of a non-isothermal liquid droplet on a solid substrate with heterogeneous wettability is presented in this work. The time-dependent incompressible Navier–Stokes equations are used to describe the fluid flow in the liquid droplet, whereas the heat transfer in the moving droplet and in the solid substrate is described by the energy equation. The arbitrary Lagrangian–Eulerian (ALE) formulation with finite elements is used to solve the time-dependent incompressible Navier–Stokes equation and the energy equation in the time-dependent moving domain. Moreover, the Marangoni convection is included in the variational form of the Navier–Stokes equations without calculating the partial derivatives of the temperature on the free surface. The heterogeneous wettability is incorporated into the numerical model by defining a space-dependent contact angle. An array of simulations for droplet impingement on a heated solid substrate with circular patterned heterogeneous wettability are presented. The numerical study includes the influence of wettability contrast, pattern diameter, Reynolds number and Weber number on the confinement of the spreading droplet within the inner region, which is more wettable than the outer region. Also, the influence of these parameters on the total heat transfer from the solid substrate to the liquid droplet is examined. We observe that the equilibrium position depends on the wettability contrast and the diameter of the inner surface. Consequently, the heat transfer is more when the wettability contrast is small and/or the diameter of inner region is large. The influence of the Weber number on the total heat transfer is more compared to the Reynolds number, and the total heat transfer increases when the Weber number increases.

Investigation on the unsteady aerodynamics of cycloidal propeller in hovering flight

The aerodynamics of a cycloidal propeller operating in hovering flight is systematically studied using numerical method. By solving the time-dependent incompressible Navier–Stokes equations and using a dynamic mesh technique, the complex unsteady flow field around the cycloidal propeller and the time-varying aerodynamic forces are investigated. The wake flow deflection can be observed in the flow field of the cycloidal propeller, which results in a deflection of the thrust. Strong downwash around cycloidal propeller and unsteady interference between blades are induced by the rotation and pitch motion of the blade. The instantaneous vertical, horizontal and resultant thrusts fluctuate cyclically with the azimuths in a pattern similar to sinusoid, and although the direction of instantaneous resultant thrust varies in a comparatively large region during rotating, the direction of averaged resultant thrust produced in each revolution is almost constant at different rotation speeds. This is beneficial to the flight stability and maneuverability of a cyclogyro. When one blade moves upward, the aerodynamic forces of the blade are much higher than that produced when moving downward. With the increase of the blade number, the unsteady interference between blades and the wakes of other blades tend to be stronger, producing great effects on aerodynamic forces of the blades as well as the aerodynamic performance of cycloidal propeller.

GPGPU implementation of mixed spectral-finite difference computational code for the numerical integration of the three-dimensional time-dependent incompressible Navier–Stokes equations

In the recent past, High-Performance-Computing machines and in particular Computational-Fluid-Dynamics hardware, have been dominated by architectures that mainly included clusters of processors communicating on high-speed interconnects.Since a few years though, cluster performance has reached an upper limit for many technical codes. The scalar-processor ability has saturated, and the sustained Flops often result to be only a fraction of the declared peak values. In an attempt to overcome these difficulties, the electronic industry has unveiled new trends. Since about 2004, there has been a transition taking place in high-performance scientific computing, with the advent of multicore- and manycore chips, so that the different software models for parallel programming needed to be revisited. In the present work these issues are addressed with reference to the field of fluid-flow investigation through the numerical integration of the Navier–Stokes equations. A mixed spectral-finite difference computational code for the numerical solution of the three-dimensional time-dependent incompressible Navier–Stokes equations (with no forcing), is implemented on a high-performance hybrid CPU/GPGPU computing system. The computational algorithm is thought for the execution of accurate calculations at high values of the flow Reynolds number, so allowing the investigation of turbulence with the method of the Direct Numerical Simulation (no turbulence models in the system of the governing equations). The performance of different code implementations has been measured as related to the case of the flow of a viscous incompressible fluid in a plane channel, a problem that has a long tradition in the field of wall-bounded turbulence research, so becoming a reference case for the study of turbulence via Direct Numerical Simulation. Results are presented in terms of parallel performance of the computational code on different hardware partitions of the computing system, as related to corresponding different sizes of the computational grid, showing that remarkably-high computational performance is reached, mainly in virtue of the presence of the GPGPU boards in the computing-hardware architecture.

Parallel spectral-element direction splitting method for incompressible Navier–Stokes equations

An efficient parallel algorithm for the time dependent incompressible Navier–Stokes equations is developed in this paper. The time discretization is based on a direction splitting method which only requires solving a sequence of one-dimensional Poisson type equations at each time step. Then, a spectral-element method is used to approximate these one-dimensional problems. A Schur-complement approach is used to decouple the computation of interface nodes from that of interior nodes, allowing an efficient parallel implementation. The unconditional stability of the full discretized scheme is rigorously proved for the two-dimensional case. Numerical results are presented to show that this algorithm retains the same order of accuracy as a usual spectral-element projection type schemes but it is much more efficient, particularly on massively parallel computers.

Finite difference streamline diffusion method using nonconforming space for incompressible time-dependent Navier-Stokes equations

This paper proposes a new nonconforming finite difference streamline diffusion method to solve incompressible time-dependent Navier-Stokes equations with a high Reynolds number. The backwards difference in time and the Crouzeix-Raviart (CR) element combined with the P0 element in space are used. The result shows that this scheme has good stabilities and error estimates independent of the viscosity coefficient.

Time-dependent coupling of Navier–Stokes and Darcy flows

A weak solution of the coupling of time-dependent incompressible Navier–Stokes equations with Darcy equations is defined. The interface conditions include the Beavers–Joseph–Saffman condition. Existence and uniqueness of the weak solution are obtained by a constructive approach. The analysis is valid for weak regularity interfaces.

Superconvergence in velocity and pressure for the 3D time-dependent Navier-Stokes Equations

This work is devoted to the superconvergence in space approximation of a fully discrete scheme for the incompressible time-dependent Navier-Stokes Equations in three-dimensional domains. We discrete by Inf-Sup-stable Finite Element in space and by a semi-implicit backward Euler (linear) scheme in time.

Computational study on the hovering mechanisms of a chordwise flexible wing

The flexion effect of a flexible wing during the hovering flight is systematically studied by using computer simulations when Reynolds number is kept at 140. The aerodynamic forces and flow vortex are investigated by solving the two-dimensional time-dependent incompressible Navier–Stokes equations using the finite volume method. The wing is modelled as a thin plate which has a rigid leading portion and a flexible aft part. The simulations are performed for a variation of the location of rotational centre and the amplitude of the flexion, and the corresponding changes of the aerodynamic forces and vortex shedding mechanism in the wake structure are investigated. Form the quantitative comparisons, the present simulations reveal that a moderate flexion during the hovering fly can provide a better performance for a flapping wing, while an excessive flexion will lead the aerodynamic performance become worse. Though the flexion does not change the force trace of the hovering wing, the magnitude of the instantaneous peak forces and mean periodic forces are definitely changed by the chordwise deformation. Compared with a rigid hovering wing, the dynamic flexion during hovering alters the intensity of the leading edge vorticity and the transfer direction of the wake. Here, the present results indicate that the aerodynamic forces can be altered by adding some deformation characteristics to the trailing edge of the wing. Furthermore, the appropriate flexion can be used to alter the aerodynamic performance of a wing and will be helpful for design and control of the flexible microflying vehicles.

A multigrid-accelerated three-dimensional transient-flow code and its application to a new test problem

A multigrid-assisted solver for the three-dimensional time-dependent incompressible Navier-Stokes equations on graded Cartesian meshes is developed. The spatial accuracy is third-order for the convective terms and fourth-order for the viscous terms, and a fractional-step strategy ensures second-order time accuracy. To achieve good time-wise efficiency a multigrid technique is used to solve the highly time-consuming pressure-Poisson equation that requires to be solved at every time step. The speed-up achieved by multigrid is shown in tabular form. The performance and accuracy of the code are first ascertained by computing the flow in a single-sided lid-driven cubic cavity with good grid-economy and comparing the results available in the literature. The code, thus validated, is then applied to a new test problem we propose and various transient and asymptotically obtained steady-state results are presented. Given the care taken to establish the credibility of the code and the good spatio-temporal accuracy of the discretization, these results are accurate and may be used for ascertaining the performance of any computational algorithm applied to this test problem.

An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier–Stokes equations

We present an implicit high-order hybridizable discontinuous Galerkin method for the steady-state and time-dependent incompressible Navier–Stokes equations. The method is devised by using the discontinuous Galerkin discretization for a velocity gradient-pressure–velocity formulation of the incompressible Navier–Stokes equations with a special choice of the numerical traces. The method possesses several unique features which distinguish itself from other discontinuous Galerkin methods. First, it reduces the globally coupled unknowns to the approximate trace of the velocity and the mean of the pressure on element boundaries, thereby leading to a significant reduction in the degrees of freedom. Moreover, if the augmented Lagrangian method is used to solve the linearized system, the globally coupled unknowns become the approximate trace of the velocity only. Second, it provides, for smooth viscous-dominated problems, approximations of the velocity, pressure, and velocity gradient which converge with the optimal order of k+1 in the L2-norm, when polynomials of degree k⩾0 are used for all components of the approximate solution. And third, it displays superconvergence properties that allow us to use the above-mentioned optimal convergence properties to define an element-by-element postprocessing scheme to compute a new and better approximate velocity. Indeed, this new approximation is exactly divergence-free, H(div)-conforming, and converges with order k+2 for k⩾1 and with order 1 for k=0 in the L2-norm. Moreover, a novel and systematic way is proposed for imposing boundary conditions for the stress, viscous stress, vorticity and pressure which are not naturally associated with the weak formulation of the method. This can be done on different parts of the boundary and does not result in the degradation of the optimal order of convergence properties of the method. Extensive numerical results are presented to demonstrate the convergence and accuracy properties of the method for a wide range of Reynolds numbers and for various polynomial degrees.

Stabilized finite element method for incompressible flows with high Reynolds number

In the following paper, we discuss the exhaustive use and implementation of stabilization finite element methods for the resolution of the 3D time-dependent incompressible Navier–Stokes equations. The proposed method starts by the use of a finite element variational multiscale (VMS) method, which consists in here of a decomposition for both the velocity and the pressure fields into coarse/resolved scales and fine/unresolved scales. This choice of decomposition is shown to be favorable for simulating flows at high Reynolds number. We explore the behaviour and accuracy of the proposed approximation on three test cases. First, the lid-driven square cavity at Reynolds number up to 50,000 is compared with the highly resolved numerical simulations and second, the lid-driven cubic cavity up to Re = 12,000 is compared with the experimental data. Finally, we study the flow over a 2D backward-facing step at Re = 42,000. Results show that the present implementation is able to exhibit good stability and accuracy properties for high Reynolds number flows with unstructured meshes.

New error estimates for a viscosity-splitting scheme in time for the three-dimensional Navier-Stokes equations

This work is devoted to obtaining optimal error estimates (for the velocity and the pressure) for a first-order time-discrete splitting scheme (using decomposition of the viscosity) for solving the incompressible time-dependent Navier-Stokes equations in three-dimensional domains. This scheme has been previously studied by other authors (Blasco et al. 1997 Int. J. Numer. Methods Fluids, 28, 1391-1419; Blasco & Codina, 2004, Appl. Numer. Math., 51, 1-17), but the main novelty of this paper is to establish optimal error estimates for the pressure. This behaviour has been numerically observed, but never hitherto proved. Moreover, owing to the introduction of a weight for the initial steps, these optimal error estimates are obtained without imposing either constraints on the time step or global compatibility conditions for the pressure at the initial time (related to further regularity hypotheses on the exact solution). © The author 2010. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A Multigrid-Accelerated Code on Graded Cartesian Meshes for 2D Time-Dependent Incompressible Viscous Flows

The present work deals with the development of a multigrid-assisted solver for the 2D time-dependent incompressible Navier-Stokes equations on graded Cartesian meshes. As finite difference method is used to discretize the governing equations on nonuniform staggered grids, a transformation of the governing equations from the physical space to the computational space is performed. To obtain second-order time accuracy a fractional-step method is employed. The convective terms are discretized using a third-order accurate upwind scheme and the viscous terms are centrally differenced to fourth-order accuracy. To improve the time-wise efficiency of the code a multigrid technique is employed to solve the pressure-Poisson equation that is required to be solved at every time-step. To establish the accuracy and performance of the code the standard 2D lid-driven cavity flow is computed for unsteady, periodic and asymptotically obtained steady solution for a wide range of Reynolds numbers (Re). The code is then used to compute the transient and asymptotically approached steady flows in a hitherto unexamined problem of two-sided lid-driven square cavity, which involves gradual development of a free shear layer and accompanying off-corner vortices. The computations also show that for this configuration, at Re=2000, there exists a steady solution, about which there was some doubt earlier. The reliability of all known and unknown results in the paper is carefully established and efficiency of the method in respect of grid economy is demonstrated.