The meshless local Petrov-Galerkin method for simulating unsteady incompressible fluid flow

This article presents a numerical algorithm using the Meshless Local Petrov-Galerkin (MLPG) method for the incompressible Navier–Stokes equations. To deal with time derivatives, the forward time differences are employed yielding the Poisson’s equation. The MLPG method with the moving least-square (MLS) approximation for trial function is chosen to solve the Poisson’s equation. In numerical examples, the local symmetric weak form (LSWF) and the local unsymmetric weak form (LUSWF) with a classical Gaussian weight and an improved Gaussian weight on both regular and irregular nodes are demonstrated. It is found that LSWF1 with a classical Gaussian weight order 2 gives the most accurate result.     

Fractional calculus modelling for unsteady unidirectional flow of incompressible fluids with time-dependent viscosity

In this note we analyze a model for a unidirectional unsteady flow of a viscous incompressible fluid with time dependent viscosity. A possible way to take into account such behaviour is to introduce a memory formalism, including thus the time dependent viscosity by using an integro-differential term and therefore generalizing the classical equation of a Newtonian viscous fluid. A possible useful choice, in this framework, is to use a rheology based on stress/strain relation generalized by fractional calculus modelling. This is a model that can be used in applied problems, taking into account a power law time variability of the viscosity coefficient. We find analytic solutions of initial value problems in an unbounded and bounded domain. Furthermore, we discuss the explicit solution in a meaningful particular case.     

On two‐dimensional unsteady incompressible fluid flow in an infinite strip

In this paper, we study two‐dimensional incompressible fluid flow in an infinite strip. The stream function form of Navier–Stokes equation is considered, which keeps the physical boundary condition and avoids some difficulties in numerical simulations. The existence and uniqueness of global solution are proved. Some results on the regularity of solution are obtained. Copyright © 2000 John Wiley & Sons, Ltd.     

On the unsteady flow and temperature distribution of a viscous incompressible fluid between two parallel flat plates

The unsteady flow and temperature distribution of a viscous incompressible fluid between two parallel flat plates has been investigated. The pressure gradient is varying linearly with time. The velocity and temperature profiles for various values oft have been shown graphically. Fort = 0, the velocity and temperature profiles correspond to the plane Poiseuille flow. The velocity and temperature both increase with time and they are maximum on the central plane of the channel.     

Finite elements for incompressible flow

We describe in this paper a finite element method for the solution of viscous incompressible flow problems which incorporates an approximate form of the incompressibility condition automatically into the finite element basis. Several examples of such finite elements are presented and applied to a simple test problem.     

Analysis of a splitting method for incompressible inviscid rotational flow problems

This paper is devoted to analyze a splitting method for solving incompressible inviscid rotational flows. The problem is first recast into the velocity–vorticity–pressure formulation by introducing the additional vorticity variable, and then split into three consecutive subsystems. For each subsystem, the L²$L^2$ least-squares finite element approach is applied to attain accurate numerical solutions. We show that for each time step this splitting least-squares approach exhibits an optimal rate of convergence in the H¹$H^1$ norm for velocity and pressure, and a suboptimal rate in the L²$L^2$ norm for vorticity. A numerical example in two dimensions is presented, which confirms the theoretical error estimates.     

Numerical solution of unsteady incompressible Navier-Stokes equations using high-order method of lines

A high-order method of lines is devised for solving the unsteady incompressible Navier-Stokes equations in the vorticity-stream function formulation. The vorticity transport equation is solved by the eight- or tenth-order method of lines and the Poisson equation for the stream function is solved by a high-order multigrid method. The numerical results of the two-dimensional (2D) homogeneous isotropic turbulence and the turbulent mixing layer are presented. In the homogeneous isotropic turbulence with tenth order of spatial accuracy, the power law of the inertial energy spectrum at the climax stage coincides with the predictions by Batchelor, Leith and Kraichnan. In the turbulent mixing layer with eight order of spatial accuracy, the vortex pairing are reproduced and the coherent structure of the Reynolds stress at the pairing is noticed.     

Analysis of some projection method based preconditioners for models of incompressible flow

In this paper, several projection method based preconditioners for various incompressible flow models are studied. In the derivations of these projection method based preconditioners, we use three different types of the approximations of the inverse of the Schur complement, i.e., the exact inverse, the Cahouet–Chabard type approximation and the BFBt type approximation. We illuminate the connections and the distinctions between these projection method based preconditioners and those related preconditioners. For the preconditioners using the Cahouet–Chabard type approximation, we show that the eigenvalues of the preconditioned systems have uniform bounds independent of the parameters and most of them are equal to 1. The analysis is based on a detailed discussion of the commutator difference operator. Moreover, these results demonstrate the stability of the staggered grid discretization and reveal the effects of the boundary treatment. To further illustrate the effectiveness of these projection method based preconditioners, numerical experiments are given to compare their performances with those of the related preconditioners. Generalizations of the projection method based preconditioners to other saddle point problems are also discussed.     

On missing boundary conditions with unsteady incompressible Navier–Stokes flows

We investigate an unsteady viscous flow problem where ‘good’ boundary conditions are available on part of the boundary only. This problem appears when the flow phenomena one is interested in are concentrated on part of the flow region and, for reasons of computational economy, are numerically computed in this subregion only. Assuming that outside of the subregion the flow is not subjected to any acceleration forces, we develop an (abstract) combined finite-element/boundary element scheme to compute the flow approximately. This scheme leads to a proof of the existence of a weak solution of the corresponding Navier–Stokes problem as well.     

Analysis of fluid flow and heat transfer over an unsteady stretching surface

This article considers two situations involving unsteady laminar boundary layer flow due to a stretching surface in a quiescent viscous incompressible fluid. In one configuration, the surface is impermeable with prescribed heat flux, in the other, the surface is permeable with prescribed temperature. The boundary value problems governing a similarity reduction for each of these situations are investigated and the existence of a solution is proved for all relevant values of physical parameters. The uniqueness of the solution is also proved for some (but not all) values of the parameters. Finally, a priori bounds are obtained for the skin friction coefficient and local Nusselt number.     

On predicting unsteady non-Newtonian blood flow

The influence of the non-Newtonian stress–strain relation of blood on the oscillatory shear index (OSI) and mean wall shear stress (WSS) are described. A mathematical non-dimensional model based on the momentum equation for a modified Casson’s fluid is formulated in terms of the dimensionless yield shear stress . An original direct numerical procedure is presented to predict the flow patterns. Results obtained by using a finite difference approach show a difference in OSI when blood is assumed to be a Newtonian fluid instead of a modified Casson’s fluid. The calculation of the OSI in human normal conditions under the Newtonian approach differs in 5% from the result obtained from using the Casson model.     

Compressible subsonic jet flows issuing from a nozzle of arbitrary cross-section

We are concerned with the well-posedness theory of two-dimensional compressible subsonic jet flow issuing from a semi-infinitely long nozzle of arbitrary cross-section. Given any atmospheric pressure $p_0$, we show that there exists a critical mass flux $m_{cr}$ depending on $p_0$ and Ω, such that if the incoming mass flux $m_0$ is less than the critical value, then there exists a unique smooth subsonic jet flow, issuing from the given nozzle. The jet boundary is a free streamline, which initiates from the end point of the nozzle smoothly and extends to the infinity. One of the key observations in this paper is that the restriction of the incoming mass flux guarantees completely the subsonicity of the compressible jet in the whole flow field, which coincides with the observation on the compressible subsonic flows in an infinitely long nozzle without free boundary in [8].     

Convergence of gauge method for incompressible flow

A new formulation, a gauge formulation of the incompressible Navier-Stokes equations in terms of an auxiliary field and a gauge variable , , was proposed recently by E and Liu. This paper provides a theoretical analysis of their formulation and verifies the computational advantages. We discuss the implicit gauge method, which uses backward Euler or Crank-Nicolson in time discretization. However, the boundary conditions for the auxiliary field are implemented explicitly (vertical extrapolation). The resulting momentum equation is decoupled from the kinematic equation, and the computational cost is reduced to solving a standard heat and Poisson equation. Moreover, such explicit boundary conditions for the auxiliary field will be shown to be unconditionally stable for Stokes equations. For the full nonlinear Navier-Stokes equations the time stepping constraint is reduced to the standard CFL constraint . We also prove first order convergence of the gauge method when we use MAC grids as our spatial discretization. The optimal error estimate for the velocity field is also obtained.

     

Simulation of unsteady flow in alluvial streams

The paper is concerned with the modelling of variations of longitudinal river bed profile during long periods of unsteady flow. Alluvial rivers with the flow varying with discharge are the object of simulation.The emphasis of the paper is on the numerical difficulties of simulating such phenomena while the theories usually used by engineers are accepted without criticism to reproduce the sediment transport due to given flow conditions. Case studies are described.     

Viscous incompressible flow past a finite plate

We consider the problem of viscous incompressible flow past a finite plate. The apparatus of hydrodynamic potential reduces the problem to Volterra— Fredholm boundary-value equations. Numerical results are reported.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 64, pp. 103–107, 1988.