Numerical study of the exact solution of the Navier–Stokes equations describing free-boundary fluid flow

This paper presents a numerical study of the partially invariant solution of the Navier–Stokes equations for the plane case which describes unsteady flow in a layer bounded by a straight solid wall and a free boundary parallel to it. It is found that for different initial flow velocities, a steady state can be established with a decrease or an increase in the initial layer thickness or the layer thickness can be increased infinitely due to fluid inflow from infinity.     

Global convergence strategies for a spectral-element space-time discontinuous-Galerkin discretization of the Navier Stokes–equations

ABSTRACT

This paper presents two global convergence strategies for a spectral-element, space-time discretisation of the Navier–Stokes equations. The first employs a hierarchical temporal mesh subdivision and polynomial order reduction to approximate the high-order solution. The second generalises Pseudo-Transient Continuation for steady problems to a space-time system.     

Viscous boundary layers for the Navier–Stokes equations with the Navier slip conditions

We tackle the issue of the inviscid limit of the incompressible Navier–Stokes equations when the Navier slip-with-friction conditions are prescribed on impermeable boundaries. We justify an asymptotic expansion which involves a weak amplitude boundary layer, with the same thickness as in Prandtl’s theory and a linear behavior. This analysis holds for general regular domains, in both dimensions two and three.     

Flooding in two-phase counter-current flows: Numerical investigation of the gas–liquid wavy interface using the Navier–Stokes equations

This paper is devoted to a theoretical analysis of counter-current gas–liquid wavy film flow between vertical plates. We consider two-dimensional nonlinear waves on the interface over a wide variation of parameters. We use the Navier–Stokes equations in their full statement to describe the liquid phase hydrodynamics. For the gas phase equations, we use the Benjamin-Miles approach where the liquid phase is a small disturbance for the turbulent gas flow. We find a region of the superficial velocity where we have two solutions at one set of the problem parameters and where the flooding takes place. We calculate the flooding dependences on the gas/liquid physical properties, on the liquid Reynolds number and on the distance between the plates. These computations allow us to present the correlation for the onset of flooding that based on the fundamental equations and principles.     

A combination of parabolized Navier–Stokes equations and level-set method for stratified two-phase internal flow

A novel methodology is proposed for the numerical computation of pressure-driven gravity-stratified flows along channels comprising two immiscible phases. The parabolized Navier–Stokes equations are combined with the level set approach, resulting into a downstream-marching problem in which the solution is computed at each cross-section based on upstream information only. A main difficulty in the implementation of the approach for internal flows is the conservation of the mass flow rates, which is addressed by extending to two-phase flows the method proposed by Patankar and Spalding (1972) and Raythby and Schneider (1979), and by adding an explicit forcing term in the equation for the advection of the level function. The combination of high-order finite differences and sparse storage and algebra used here allows a fully-coupled integration of the parabolized equations, as opposed to the more classical segregated approaches. This enables a very efficient calculation of the complete downstream-developing flow field.     

Couette–Poiseuille flow of Bingham fluids between two porous parallel plates with slip conditions

Analytical solutions of Couette–Poiseuille flow of Bingham fluids between two porous parallel plates are derived. This study extends the work of Tsangaris et al. [S. Tsangaris, C. Nikas, G. Tsangaris, P. Neofytou, Couette flow of a Bingham plastic in a channel with equally porous parallel walls, J. Non-Newtonian Fluid Mech. 144 (2007) 42–48] to a general situation where the slip effect at the porous walls is considered. It is found that the form of the flow inside the channel depends not only on the Bingham number Bn, the Couette number Co (related to the moving wall) and the transverse Reynolds number Re, but also on the slip parameter Cs at the porous walls. In both the Co–Re diagram and the Co–Bn diagram, the region where plug flow appears enlarges as the slip effect increases, especially in the case where Co is negative. In the case where plug flow and double shear flow coexist, the transverse position of the plug flow and the shear rate at the boundaries exhibit two opposite behaviors when Cs increases, depending on the value of the other three dimensionless numbers. In other cases, slippage always weakens the shearing deformation of the flow.     

High-order ALE method for the Navier–Stokes equations on a moving hybrid unstructured mesh using flux reconstruction method

The flux reconstruction (FR) formulation can unify several popular discontinuous basis high-order methods for fluid dynamics, including the discontinuous Galerkin method, in a simple, efficient form. An arbitrary Lagrangian–Eulerian (ALE) extension to the high-order FR scheme is developed here for moving mesh fluid flow problems. The ALE Navier–Stokes equations are derived by introducing a grid velocity. The conservation law are spatially discretised on hybrid unstructured meshes using Huynh’s scheme (Huynh 2007) on anisotropic elements (quadrilaterals) and using Correction Procedure via Reconstruction scheme on isotropic elements (triangles). The temporal discretisation uses both explicit and implicit treatments. The mesh movement is described by node positions given as a time series, instead of an analytical formula. The geometric conservation law is tested using free stream preservation problem. An isentropic vortex propagation test case is performed to show the high-order accuracy of the developed method on both moving and fixed hybrid meshes. Flow around an oscillating cylinder shows the capability of the method to solve moving boundary viscous flow problems, with the numeric method further verified by comparison of the result on a smoothly deforming mesh and a rigid moving mesh.     

Higher order spectral/hp finite element models of the Navier–Stokes equations

In this paper, we present higher order least-squares finite element formulations for viscous, incompressible, isothermal Navier–Stokes equations using spectral/hp basis functions. The second-order Navier–Stokes equations are recast as first-order system of equations using stresses as auxiliary variables. Both steady-state and transient problems are considered. For a better coupling of pressure and velocity, especially in transient flows, an iterative penalisation strategy is employed. The outflow-type boundary conditions are applied in a weak sense through the least-squares functional. The formulation is verified by solving various benchmark problems like the lid-driven cavity, backward-facing step and flow over cylinder problems using direct serial solver UMFPACK.     

Shape optimisation problem for stability of Navier–Stokes flow field

ABSTRACT

This paper presents the numerical results of a shape optimisation problem with regard to delaying the transition of a Navier–Stokes flow field from laminar to turbulent by using the theory developed by Nakazawa and Azegami. The theory was reviewed within the framework of functional analysis and updated with another expression of the shape derivative with respect to the objective function. A computer program was developed with the FreeFEM++. Numerical analyses were performed for two types of problems: a two-dimensional Poiseuille flow field with a sudden expansion and a two-dimensional uniform flow field around an isolated body. From the first example, two local minimum points of symmetric and asymmetric flow fields were determined, and the asymmetric flow field was found to be more stable. With regard to the second example, we reached the local minimum point of an elliptical shape, and infrequently determined a solution converging to an elliptical shape with the bluff in the leeward direction. By comparison, the superiority of the elliptical shape was obvious.     

Simulations of granular flow along an inclined plane using the Savage–Hutter model

In this work a velocity-dependent friction is introduced into a depth-averaged Savage–Hutter dynamical model for shallow granular flows. The process of granular material flowing along an inclined plane and then depositing on a horizontal plane is simulated. The surface profiles and evolution of various types of energy are investigated and compared when using the standard Coulomb-type friction versus velocity-dependent friction. Interestingly, there is a small difference between the two different types of friction.     

Dynamic flight stability of hovering model insects: theory versus simulation using equations of motion coupled with Navier–Stokes equations

In the present paper, the longitudinal dynamic flight stability properties of two model insects are predicted by an approximate theory and computed by numerical sim- ulation. The theory is based on the averaged model （which assumes that the frequency of wingbeat is sufficiently higher than that of the body motion, so that the flapping wings＇ degrees of freedom relative to the body can be dropped and the wings can be replaced by wingbeat-cycle-average forces and moments）; the simulation solves the complete equations of motion coupled with the Navier-Stokes equations. Comparison between the theory and the simulation provides a test to the validity of the assumptions in the theory. One of the insects is a model dronefly which has relatively high wingbeat frequency （164 Hz） and the other is a model hawkmoth which has relatively low wingbeat frequency （26 Hz）. The results show that the averaged model is valid for the hawkmoth as well as for the dronefly. Since the wingbeat frequency of the hawkmoth is relatively low （the characteristic times of the natural modes of motion of the body divided by wingbeat period are relatively large） compared with many other insects, that the theory based on the averaged model is valid for the hawkmoth means that it could be valid for many insects.     

A postprocessing mixed finite element method for the Navier–Stokes equations

A fully discrete postprocessing mixed finite element scheme is considered for solving the time-dependent Navier–Stokes equations. In the PP method, we only consider a non-linear equation in the coarse-level subspace and a linear problem in the fine-level subspace. The analysis shows that the PP scheme can reach the same accuracy as the standard Galerkin method with a very fine mesh size h by an appropriate choice of H. Numerical examples are provided that confirm both the theoretical analysis and the corresponding improvement in computational efficiency.     

A spectral solver for the Navier–Stokes equations in the velocity–vorticity formulation

A numerical algorithm intended for the study of flows in a cylindrical container under laminar flow conditions is proposed. High resolution of the flow field, governed by the Navier–Stokes equations in velocity–vorticity formulation relative to a cylindrical frame of reference, is achieved through spatial discretisation by means of the spectral method. This method is based on a Fourier expansion in the azimuthal direction and an expansion in Chebyshev polynomials in the (nonperiodic) radial and axial directions. Several regularity constraints are used to take care of the coordinate singularity. These constraints are implemented, together with the boundary conditions at the top, bottom and mantle of the cylinder, via the tau method. The a priori unknown boundary values of the vorticity are evaluated by means of the influence-matrix technique. The compatibility between the mathematical and numerical formulation of the Navier–Stokes equations is established through a tau-correction procedure. The resolved flow field exhibits high-precision satisfaction of the incompressibility constraints for velocity and vorticity and the definition of the vorticity. The performance of the solver is illustrated by resolution of several configurations representative of generic three-dimensional laminar flows.     

Block-structured compressible Navier–Stokes solution using the OPS high-level abstraction

abstract

In this paper, we report the development and validation of a compressible solver with shock capturing, using a domain-specific high-level abstraction framework, OPS, that is being developed at the University of Oxford. OPS uses an active library approach for block-structured meshes, capable of generating codes for a variety of parallel implementations with different parallelisation strategies. Performance results on various architectures are reported for the 1D Shu–Osher test case.     

On the Time Decay of the Solutions of the Navier–Stokes System

We investigate the relationship between the time decay of the solutions u of the Navier–Stokes system on a bounded open subset of and the time decay of the right-hand sides f. In suitable function spaces, we prove that u always inherits at least part of the decay of f, up to exponential, and that the decay properties of u depend only upon the amount and type (e.g., exponential, or power-like) of decay of f. This is done by first making clear what is meant by “type” and “amount” of decay and by next elaborating upon recent abstract results pointing to the fact that, in linear and nonlinear PDEs, the decay of the solutions is often intimately related to the Fredholmness of the differential operator. This work was done while the second author was visiting the Bernoulli Center, EPFL, Switzerland, whose support is gratefully acknowledged.     

On the Local Smoothness of Solutions of the Navier–Stokes Equations

We consider the Cauchy problem for incompressible Navier–Stokes equations with initial data in , and study in some detail the smoothing effect of the equation. We prove that for T < ∞ and for any positive integers n and m we have , as long as stays finite.     

An approximate solution for the Couette–Poiseuille flow of the Giesekus model between parallel plates

An approximate analytical solution is derived for the Couette–Poiseuille flow of a nonlinear viscoelastic fluid obeying the Giesekus constitutive equation between parallel plates for the case where the upper plate moves at constant velocity, and the lower one is at rest. Validity of this approximation is examined by comparison to the exact solution during a parametric study. The influence of Deborah number (De) and Giesekus model parameter (α) on the velocity profile, normal stress, and friction factor are investigated. Results show strong effects of viscoelastic parameters on velocity profile and normal stress. In addition, five velocity profile types were obtained for different values of α, De, and the dimensionless pressure gradient (G).     

Approximation of a Solution to the Euler Equation by Solutions of the Navier–Stokes Equation

We show that a smooth solution u ⁰ of the Euler boundary value problem on a time interval (0, T ₀) can be approximated by a family of solutions of the Navier–Stokes problem in a topology of weak or strong solutions on the same time interval (0, T ₀). The solutions of the Navier–Stokes problem satisfy Navier’s boundary condition, which must be “naturally inhomogeneous” if we deal with the strong solutions. We provide information on the rate of convergence of the solutions of the Navier–Stokes problem to the solution of the Euler problem for ν → 0. We also discuss possibilities when Navier’s boundary condition becomes homogeneous.