Equivalence of u–p and ζ–ψ formulations of the time-dependent Navier–Stokes equations

This paper deals with the non-stationary incompressible Navier--Stokes equations for two-dimensional flows expressed in terms of the velocity and pressure and of the vorticity and streamfunction. The equivalence of the two formulations is demonstrated, both formally and rigorously, by virtue of a condition of compatibility between the boundary and initial values of the normal component of velocity. This condition is shown to be the only compatibility condition necessary to allow for solutions of a minimal regularity, namely H¹ for the velocity, as in most current numerical schemes relying on spatial discretizations of local type.     

Analysis of plane and axisymmetric flows of incompressible fluids with the stream tube method: Numerical simulation by trust-region optimization algorithm

New concepts for the study of incompressible plane or axisymmetric flows are analysed by the stream tube method. Flows without eddies and pure vortex flows are considered in a transformed domain where the mapped streamlines are rectilinear or circular. The transformation between the physical domain and the computational domain is an unknown of the problem. In order to solve the non-linear set of relevant equations, we present a new algorithm based on a trust region technique which is effective for non-convex optimization problems. Experimental results show that the new algorithm is more robust compared to the Newton-Raphson method.     

A theorem on integer flows on cartesian products of graphs

It is shown that the Cartesian product of two nontrivial connected graphs admits a nowhere‐zero 4‐flow. If both factors are bipartite, then the product admits a nowhere‐zero 3‐flow. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 93–98, 2003     

Calculation of impulsively started incompressible viscous flows

The paper's focus is the calculation of unsteady incompressible 2D flows past airfoils. In the framework of the primitive variable Navier–Stokes equations, the initial and boundary conditions must be assigned so as to be compatible, to assure the correct prediction of the flow evolution. This requirement, typical of all incompressible flows, viscous or inviscid, is often violated when modelling the flow past immersed bodies impulsively started from rest. Its fulfillment can however be restored by means of a procedure enforcing compatibility, consisting in a pre‐processing of the initial velocity field, here described in detail. Numerical solutions for an impulsively started multiple airfoil have been obtained using a finite element incremental projection method. The spatial discretization chosen for the velocity and pressure are of different order to satisfy the inf–sup condition and obtain a smooth pressure field. Results are provided to illustrate the effect of employing or not the compatibility procedure, and are found in good agreement with those obtained with a non‐primitive variable solver. In addition, we introduce a post‐processing procedure to evaluate an alternative pressure field which is found to be more accurate than the one resulting from the projection method. This is achieved by considering an appropriate ‘unsplit’ version of the momentum equation, where the velocity solution of the projection method is substituted. Copyright © 2004 John Wiley & Sons, Ltd.     

Passive control of the flow around a square cylinder using porous media

The passive control of bluff body flows using porous media is investigated by means of the penalization method. This method is used to create intermediate porous media between solid obstacles and the fluid in order to modify the boundary layer behaviour. The study covers a wide range of two‐dimensional flows from low transitional flow to fully established turbulence by direct numerical simulation of incompressible Navier–Stokes equations. A parametric study is performed to illustrate the effect of the porous layer permeability and thickness on the passive control. The numerical results reveal the ability of porous media to both regularize the flow and to reduce the drag forces up to 30%. Copyright © 2004 John Wiley & Sons, Ltd.     

Monotone Mappings and Flows of Viscous Media

We establish solvability of the boundary-value problems describing stationary and periodic flows of viscoplastic media. In the case of stationary flows we study the question of convergence of the Galerkin method. For the problem of periodic flows we prove a version of the second Bogolyubov theorem.     

Temperature factor effect on the structure of the separated flow within a supersonic gas stream

The effect of the temperature factor, that is, the ratio of the body temperature to the freestream stagnation temperature, on the structure of the separated flow formed in the presence of a concave corner in a supersonic stream is studied. The strong influence of the temperature factor on the separation zone length and the flow-generated aerodynamic characteristics is established. It is shown that for fairly large deflection angles this flow cannot be described by free interaction, or triple deck, theory.     

Rigorous results in selection of steady needle crystals

This paper concerns the existence of solutions to a steady needle crystal growth problem in a one-sided model. We rigorously prove that for small nonzero anisotropy γ, analytic symmetric needle crystal solutions exist in the limit of surface tension ε² if only if the stokes constant S for a relatively simple nonlinear differential equation is zero. This Stokes constant S depends on the parameter β=2^9/7γε^−8/7 and earlier numerical calculations by a number of investigators have shown this to be zero for a discrete set of values of β. It is also proven that for γ=0, there can be no symmetric needle crystal solution in the considered space.The methodology consists of two steps. First, the original problem is reduced to a weak half-strip problem for any γ in a compact set of [0,1) by relaxation of the symmetry condition. The weak problem is shown to have a unique solution in the function space considered for any γ∈[0,γ_m] for some γ_m>0. When a symmetry is invoked, the weak problem is shown equivalent to the original needle crystal problem. Next, by considering the behavior of the solution in neighborhood of an appropriate complex turning point for γ∈(0,γ_m], we extract an exponentially small term in ε as ε→0⁺ that generally violates the symmetric condition. We prove that the symmetry condition is satisfied for small ε when the parameter β is constrained appropriately.     

Rheological behaviour and shear thickening exhibited by aqueous CTAB micellar solutions

In this experimental work we carefully investigate the rheological behaviour and in particular the shear thickening exhibited by aqueous micellar solutions of CTAB with NaSal as added counterion. We are particularly interested in the evolution of the critical shear rate (at which shear thickening occurs) versus C _D , the surfactant concentration. We show that , at fixed salt concentration C _S , increases with C _D following a power law evolution with a positive exponent of + 5.8. On the other hand we show that if the ratio C _D / C _S is fixed, decreases with C _D with a negative exponent of -2.0. Nevertheless investigations of the zero shear viscosity indicate that in all situations (implying variation of the surfactant concentration C _D , or the salt concentration C _S or the temperature) is a decreasing function of the length of the micelles. All these evolutions are compatible with a gelation mechanism which could possibly be associated with entanglement effects of large interacting flowing structures. Received: 3 March 1998 / Revised: 16 June 1998 / Accepted: 3 July 1998