A Newton method for the resolution of steady stochastic Navier-Stokes equations

We present a Newton method to compute the stochastic solution of the steady incompressible Navier-Stokes equations with random data (boundary conditions, forcing term, fluid properties). The method assumes a spectral discretization at the stochastic level involving a orthogonal basis of random functionals (such as Polynomial Chaos or stochastic multi-wavelets bases). The Newton method uses the unsteady equations to derive a linear equation for the stochastic Newton increments. This linear equation is subsequently solved following a matrix-free strategy, where the iterations consist in performing integrations of the linearized unsteady Navier-Stokes equations, with an appropriate time scheme to allow for a decoupled integration of the stochastic modes. Various examples are provided to demonstrate the efficiency of the method in determining stochastic steady solution, even for regimes where it is likely unstable.     

The coupling of boundary integral and finite element methods for the exterior steady Oseen problem

In this article, we present a new numerical method for solving the steady Oseen equations in an unbounded plane domain. The technique consists in coupling the boundary integral and the finite element methods. An artificial smooth boundary is introduced separating an interior inhomogeneous region from an exterior homogeneous one. The solution in exterior region is represented by an integral equation over the artificial boundary. This integral equation is incorporated into a velocity-pressure formulation for the interior region, and a finite element method is used to approximate the resulting variational problem. Finally, the optimal error estimates of the numerical solution are derived.Computer results will be discussed in a forthcoming paper.     

Steady Navier–Stokes equations in a domain with piecewise smooth boundary

We are concerned with the boundary value problem for the steady Navier–Stokes equations in a 2D bounded domain with piecewise smooth boundary. Existence and uniqueness of the solution to the above problem is proved in weighted Sobolev spaces by means of the Mellin transform and the regularizer method.     

Boundary conditions for subsonic compressible Navier-Stokes calculations

A systematic study of inflow and outflow boundary conditions for the numerical solution of the compressible Navier-Stokes equations is presented. Combinations of several representative inflow and outflow boundary conditions are applied in the solution of subsonic flow over a flat plate in a finite computational domain. These boundary conditions are evaluated in terms of their effect on the accuracy of the solution and the rate of convergence to a steady state. It is shown that errors in the data specified at the inflow boundary can produce significant errors in the computed flow field. It is also shown that a non-reflecting outflow boundary condition can significantly reduce the total computational time required.     

Resonant oscillations of a plate in an electrically conducting rotating Johnson-Segalman fluid

An analysis of hydromagnetic flow is examined in a semi-infinite expanse of electrically conducting rotating Johnson-Segalman fluid bounded by nonconducting plate in the presence of a transverse magnetic field and the governing equations are modeled first time. The structure of the velocity distribution and the associated hydromagnetic boundary layers are investigated including the case of resonant oscillations. It is shown that unlike the hydrodynamic situation for the case of resonance, the hydromagnetic steady solution satisfies the boundary condition at infinity. The inherent difficulty involved in the hydrodynamic resonance case has been resolved in the presence analysis.     

On the steady solution of non-linear advection equations in steady recirculating flows

Many physical problems like short fiber suspensions flows or viscoelastic flows are modeled by linear and non-linear advection equations. Many of the experimental and industrial flows show often steady recirculating areas which introduce some additional difficulties in the numerical simulation. Actually, the advection equation is supposed to have a steady solution in these steady recirculating flows but neither boundary conditions nor initial conditions are known in such flows. In this paper, we present accurate techniques to solve non-linear advection equations defined in steady recirculating flows. These techniques combine a standard treatment of the non-linearity with a more original treatment of the associated linear problems.     

Variational sensitivity analysis and design optimization

Variational method (VM) is employed to derive the co-state equations, boundary (transversality) conditions, and functional sensitivity derivatives. The converged solutions of the state equations together with the steady state solution of the co-state equations are integrated along the domain boundary to uniquely determine the functional sensitivity derivatives with respect to the design function. The application of the variational method to aerodynamic shape optimization problems is demonstrated on internal flow problems at supersonic Mach number range of 1.5. Optimization results for flows with and without shock phenomena are presented. The study shows that while maintaining the accuracy of aerodynamical objective function and constraint within the reasonable range for engineering prediction purposes, variational method provides a substantial gain in computational efficiency, i.e., computer time and memory, when compared with the finite difference computations.     

The numerical solution of two-dimensional flow in a branching channel

A numerical method for treating the steady two-dimensional flow of a viscous incompressible fluid in a branching channel is given. The Navier-Stokes equations are written in terms of the stream function and vorticity, giving the usual two coupled partial differential equations. These equations are solved using the difference scheme of Dennis and Hudson [Proc. 1^st Cong. Num. Met. Laminar and Turbulent Flow, p. 69. Pentech Press, London (1978)] and a solution to the resulting large system of algebraic equations is obtained using a Gauss-Seidel iteration technique. The upstream and downstream boundary conditions are discussed and a logarithmic transformation is applied to the coordinate measuring distance downstream in order to extend the numerical solution far enough downstream. Two methods are presented for dealing with the singularity in the vorticity at the sharp corners where the channel bifurcates. The numerical solution is obtained for three separate grid sizes for two different widths of channel downstream of the channel branch. The effect on flow separation of both the variation of Reynolds number and the relative channel width upstream and downstream of the branch are discussed.     

Boundary Conditions and Estimates for the Steady Stokes Equations on Staggered Grids

We consider the steady state Stokes equations, describing low speed flow and derive estimates of the solution for various types of boundary conditions. We formulate the boundary conditions in a new way, such that the boundary value problem becomes non-singular. By using a difference approximation on a staggered grid we are able to derive a non-singular approximation in a direct way. Furthermore, we derive the same type of estimates as for the continuous case. Numerical experiments confirm the theoretical results.     

Existence and uniqueness theorems for the solution of boundary integral equations for linearized elastostatics

The paper consists of both theory and application. The existence and uniqueness theorems for the solution of boundary integral equations for linearized elastostatics are proposed, and computational models for direct formulation of boundary integral equations are presented. By using the existence and uniqueness theorems of this paper, the existence and uniqueness of the solution of the boundary integral equations are determined, showing that the method used here is successful. The computational models presented are suited to the solution of mixed boundary value problems and linear elasticity problems in geomechanics.     

High-speed simulation of discrete dynamic probabilistic systems

The theory and design of a special purpose stochastic computer for the high-speed simulation of Markov chains and random walks is described. Experimental results are presented for the transient and steady response to Markov systems and for fundamental studies of random walks. The paper conlcludes with a discussion of the extension of the system to the Monte-Carlo solution of partial differential equations with arbitrary boundary conditions.     

Numerical modelling of incompressible flows for Newtonian and non-Newtonian fluids

This paper deals with numerical solution of two-dimensional and three-dimensional steady and unsteady laminar incompressible flows for Newtonian and non-Newtonian shear thickening fluids flow through a branching channel. The mathematical model used in this work is the generalized system of Navier–Stokes equations. The right hand side of this system is defined by the power-law model. The finite volume method combined with artificial compressibility method is used for numerical simulations of generalized Newtonian fluids flow. Numerical solution is divided into two parts, steady state and unsteady. Steady state solution is achieved for t→∞$t\to \infty$ using steady boundary conditions and followed by steady residual behaviour. For unsteady solution high artificial compressibility coefficient β²${\beta }^2$ is considered. An artificial compressibility method with a pulsation of the pressure in the outlet boundary is used.     

Fourth order “mehrstellen”-integration for three-dimensional turbulent boundary layers

A new finite difference solution for Prandtl's boundary-layer equations is described in detail for steady, incompressible luminar and turbulent flows. Only boundary sheets will be considered and curvature effects in the direction normal to the wall will be neglected. The governing equations are presented in form of a vector equation. Their numerical stability is discussed for an elementary finite difference molecule. Improved finite-difference approximations with a truncation error of fourth order are then introduced to enable either increased accuracy or shortened calculation times, in particular, for three-dimensional problems. Detailed studies of the behaviour of the overall error of the solution and several applications to real flow situations supplement the general considerations.     

Towards development of unsteady near-wall interface boundary conditions for turbulence modeling

In near-wall turbulence modeling it is required to resolve a thin layer nearby the solid boundary, which is characterized by high gradients of the solution. An accurate enough resolution of such a layer can take most computational time. The situation even becomes worse for unsteady problems. To avoid time-consuming computations, a new approach is developed, which is based on a non-overlapping domain decomposition. The boundary condition of Robin type at the interface boundary is achieved via transfer of the boundary condition from the wall. For the first time interface boundary conditions of Robin type are derived for a model nonstationary equation which simulates the key terms of the unsteady boundary layer equations. In the case of stationary solutions the approach is automatically reduced to the technique earlier developed for the steady problems. The considered test cases demonstrate that unsteady effects can be significant for near-wall domain decomposition. In particular, they can be important in the case of the wall-function-based approach.     

Numerical simulation of Newtonian and non-Newtonian flows in bypass

This paper deals with the numerical solution of Newtonian and non-Newtonian flows with biomedical applications. The flows are supposed to be laminar, viscous, incompressible and steady or unsteady with prescribed pressure variation at the outlet. The model used for non-Newtonian fluids is a variant of power law. Governing equations in this model are incompressible Navier–Stokes equations. For numerical solution we use artificial compressibility method with three stage Runge–Kutta method and finite volume method in cell centered formulation for discretization of space derivatives. The following cases of flows are solved: steady Newtonian and non-Newtonian flow through a bypass connected to main channel in 2D, steady Newtonian flow in angular bypass in 3D and unsteady non-Newtonian flow through bypass in 2D. Some 2D and 3D results that could have application in the area of biomedicine are presented.     

A new coupled high-order compact method for the three-dimensional nonlinear biharmonic equations

This paper presents a new family of fourth- compact finite difference schemes for the numerical solution of three-dimensional nonlinear biharmonic equations using coupled approach. The numerical solutions of unknown variable and its first- derivatives as well as v(=Δ u) are obtained not only in the interior but also at the boundary. A prominent contribution of this work is that the boundary conditions for the variable v are approximated more accurately, which plays an important role for the efficiency of calculation. Finally, numerical experiments are conducted to verify the feasibility of this new method and the high accuracy of these schemes, including the steady Navier–Stokes equation in terms of vorticity-stream function formulation.     

Modified integral equation solution of viscous flows near sharp corners

Solutions of the biharmonic equation governing steady two dimensional viscous flow of an incompressible newtonian fluid are obtained by employing a direct biharmonic boundary integral equation (BBIE) method in which Green's Theorem is used to reformulate the differential equation as a pair of coupled integral equations. The classical BBIE gives poor convergence in the presence of singularities arising in the solution domain. The rate of convergence is improved dramatically by including the analytic behaviour of the flow in the neighbourhood of the singularities. The modified BBIE (MBBIE) effectively ‘subtracts out’ this analytic behaviour in terms of a series representation whose coefficients are initially unknown. In this way the modified flow variables are regular throughout the entire solution domain. Also presented is a method for including the asymptotic nature of the flow when the solution domain is unbounded.     

New fast super-dashpot time-dependent techniques for the numerical simulation of steady flows—I: Numerical formulation

This paper describes new fast artificial time-dependent methods leading asymptotically, after a sufficiently long time, to the solution of any steady system of first-order equations. They can, namely, be very useful and efficient for computing steady inviscid transonic mixed flows, as well as for solving the steady hybrid equations of subsonic rotational flows. The time-dependent equations used by the new methods are constructed by adding a purely artificial unsteady operator to the steady physical equations. That operator introduces a strong internal damping of the perturbation waves similar to that due to dashpots on the surface of a vibrating membrane. As a result, a very large rate of convergence of the same order of magnitude as that of the over-relaxation techniques is obtained. The new methods can be applied to any conservative finite differences, finite volumes or finite element discretization of the steady equations. Their level of generality is comparable to that of the classical time-dependent techniques using the unsteady Euler equations, but they are much faster.