Weak Singularities of Planar Vector Fields Under Weak Perturbation

We characterize the elements of the set H _n of degree n homogeneous polynomial vector fields that are structurally stable with respect to perturbation in H _n , both on the plane and on the Poincaré sphere. We use this information to characterize elements of the set W _n of smooth vector fields on ² beginning with terms of order n at (0, 0) that are structurally stable in a neighborhood of (0, 0) under perturbation in W _n . We also determine the set of elements of H _n that are determining for topological equivalence at (0, 0), in the sense that the topological type of the singularity at (0, 0) is invariant under the addition of higher order terms.     

Parameter Dependence of Stable Manifolds for Delay Equations with Polynomial Dichotomies

We establish the existence of Lipschitz stable invariant manifolds for semiflows generated by a delay equation x′ = L(t)x _t + f (t, x _t , λ), assuming that the linear equation x′ = L(t)x _t admits a polynomial dichotomy and that f is a sufficiently small Lipschitz perturbation. Moreover, we show that the stable invariant manifolds are Lipschitz in the parameter λ. We also consider the general case of nonuniform polynomial dichotomies.     

Lift-induced drag of a cambered wing for Re ≤ 1 × 106

An investigation of the dependence of the lift-induced drag coefficient C _Di of a square-tipped, cambered wing model on Reynolds number for Re ≤ 1 × 10⁶ was conducted. Computed based on the vorticity distribution inferred from the near-field cross-flow velocity measurements of the tip vortex, different C _Di prediction schemes were used. The effect of measurement plane size and grid resolution on the C _Di calculations was also identified. The C _Di estimated by the integral method was found to increase with increasing Re and was below the C _Di = C _l²/πeAR prediction. Limits on the measurement plane size and grid resolution were determined to be at least 40% larger than the vortex outside diameter and no larger than 0.63% chord, respectively, in order to provide a good estimate of the induced drag.     

Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications

The basic problem of order reduction of nonlinear systems with time periodic coefficients is considered in state space and in direct second order (structural) form. In state space order reduction methods, the equations of motion are expressed as a set of first order equations and transformed using the Lyapunov–Floquet (L–F) transformation such that the linear parts of new set of equations are time invariant. At this stage, four order reduction methodologies, namely linear, nonlinear projection via singular perturbation, post-processing approach and invariant manifold technique, are suggested. The invariant manifold technique yields a unique ‘reducibility condition’ that provides the conditions under which an accurate nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An alternate approach of deriving reduced order models in direct second order form is also presented. Here the system is converted into an equivalent second order nonlinear system with time invariant linear system matrices and periodically modulated nonlinearities via the L–F and other canonical transformations. Then a master-slave separation of degrees of freedom is used and a nonlinear relation between the slave coordinates and the master coordinates is constructed. This method yields the same ‘reducibility conditions’ obtained by invariant manifold approach in state space. Some examples are given to show potential applications to real problems using above mentioned methodologies. Order reduction possibilities and results for various cases including ‘parametric’, ‘internal’, ‘true internal’ and ‘true combination resonances’ are discussed. A generalization of these ideas to periodic-quasiperiodic systems is included and demonstrated by means of an example.     

Number of Invariant Straight Lines for Homogeneous Polynomial Vector Fields of Arbitrary Degree and Dimension

We study the number of invariant straight lines through the origin of the homogeneous polynomial differential systems of degree m in \mathbb R^d{{\mathbb R}^d} or \mathbb C^d{{\mathbb C}^d} , when this number is finite. This notion extends in the natural way the classical notion of eigenvectors of homogeneous linear differential systems to homogeneous polynomial differential systems. This number provides un upper bound for the number of infinite singular points of the polynomial differential systems of degree m in \mathbb R^d{{\mathbb R}^d} . This upper bound is reached if all the invariant straight lines through the origin are real.     

Symbolic computation of flow in a rotating pipe

A perturbation solution of the fully developed flow through a pipe of circular cross-section, which rotates uniformly around an axis oriented perpendicularly to its own, is considered. The perturbation parameter is given by R = 2Ωa²/ν in terms of the angular velocity Ω, the pipe radius a and the kinematic viscosity ν of the fluid. The two coupled non-linear equations for the axial velocity ω and the streamfunction ? of the transverse (secondary) flow lead to an infinite system of linear equations. This system allows first the computation of a given order ?_n, n ? 1, of the perturbation expansion ? = ∑ Rⁿ?_n in terms of ω_n-1, the (n-1)-th order of the expansion ω = ∑ Rⁿω_n, and of the lower orders ?₁,…,?_{n ? 1}. Then it permits the computation of ω_n from ω₀,…,ω_{n ? 1} and ?₁,…,?;_n. The computation starts from the Hagen–Poiseuille flow ω₀, i.e. the perturbation is around this flow. The computations are performed analytically by computer, with the REDUCE and MAPLE systems. The essential elements for this are the appropriate co-ordinates: in the complex co-ordinates chosen the two-dimensional harmonic (Laplace, Δ) and biharmonic (Δ²) operators are ideally suited for (symbolic) quadratures. Symmetry considerations as well as analysis of the equations for ω_n, ?_n and of the boundary conditions lead to general (polynomial) formulae for these functions, with coeffcients to be determined. Their determination, order by order, implies, in complex co-ordinates, only (symbolic) differentiation and quadratures. The coefficients themselves are polynomials in the Reynolds number c of the (unperturbed) Hagen–Poiseuille flow. They are tabulated in the paper for the orders n ? 6 of the perturbation expansion.     

Inviscid evolution of incompressible swirling flows in pipes: The dependence of the flow structure upon the inlet velocity field

This paper analyses the influence of the inlet swirl on the structure of incompressible inviscid flows in pipes. To that end, the inviscid evolution along a pipe of varying radius with a central body situated inside the pipe is studied for three different inlet swirling flows by solving the Bragg–Hawthorne equation both asymptotically and numerically. The downstream structure of the flow changes abruptly above certain threshold values of the swirl parameter (L). In particular, there exist a value L_r above which a near-wall region of flow reversal is formed downstream, and a critical value L_f above which the axial vortex flow breaks down. It is shown that the dependence upon the pipe geometry of these critical values of the swirl parameter varies strongly with the inlet azimuthal velocity profile considered. An excellent agreement between asymptotic and numerical results is found.     

Reynolds Number Dependence of Energy Spectra in the Overlap Region of Isotropic Turbulence

A Near-Asymptotics analysis of the turbulence energy spectrum is presented that accounts for the effects of finite Reynolds number recently reported by Mydlarski and Warhaft [21]. From dimensional and physical considerations (following Kolmogorov and von Karman), proper scalings are defined for both low and high wavenumbers, but with functions describing the entire range of the spectrum. The scaling for low wavenumbers uses the kinetic energy and the integral scale, L, based on the integral of the correlation function. The fact that the two scaled profiles describe the entire spectrum for finite values of Reynolds number, but reduce to different profiles in the limit, is used to determine their functional forms in the “overlap” region that both retain in the limit. The spectra in the overlap follow a power law, E(k) =Ck _{−5/3 + μ}, where μ and C are Reynolds number dependent. In the limit of infinite Reynolds number, μ → 0 and C → constant, so the Kolmogorov/Obukhov theory is recovered in the limit. Explicit expressions for μ and the other parameters are obtained, and these are compared to the Mydlarski/Warhaft data. To get a better estimate of the exponent from the experimental data, existing models for low and high wavenumbers are modified to account for the Reynolds number dependence. They are then used to build a spectral model covering all the range of wavenumbers at every Reynolds number. Experimental data from grid-generated turbulence are examined and found to be in good agreement with the theory and the model. Finally, from the theory and data, an explicit form for the Reynolds number dependence of φ = ɛL/u ₃ is obtained. This revised version was published online in July 2006 with corrections to the Cover Date.     

Invertibility and dichotomy of differential operators on a half-line

A linear ordinary differential operator with bounded coefficients satisfying certain homogeneous initial conditions is shown to be invertible onL _n ² (0, ) if and only if the underlying system of differential equations has a dichotomy. Moreover, in that case the operator is proved to be a direct sum of two infinitesimal generators ofC ₀-semigroups, one of which has support on the negative half-line and the other on the positive half-line. The effect of perturbations of the initial values on the dichotomy is also described.     

An efficient and accurate fully discrete finite element method for unsteady incompressible Oldroyd fluids with large time step

This paper proposes a second‐order accuracy in time fully discrete finite element method for the Oldroyd fluids of order one. This new approach is based on a finite element approximation for the space discretization, the Crank–Nicolson/Adams–Bashforth scheme for the time discretization and the trapezoid rule for the integral term discretization. It reduces the nonlinear equations to almost unconditionally stable and convergent systems of linear equations that can be solved efficiently and accurately. Here, the numerical simulations for L², H¹ error estimates of the velocity and L² error estimates of the pressure at different values of viscoelastic viscosities α, different values of relaxation time λ₁, different values of null viscosity coefficient μ₀ are shown. In addition, two benchmark problems of Oldroyd fluids with different solvent viscosity μ and different relaxation time λ₁ are simulated. All numerical results perfectly match with the theoretical analysis and show that the developed approach gives a high accuracy to simulate the Oldroyd fluids under a large time step. Furthermore, the difference and the connection between the Newton fluids and the viscoelastic Oldroyd fluids are displayed. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence,uniqueness and asymptotics of steady jets

We consider the three-dimensional flow through an aperture in a plane either with a prescribed flux or pressure drop condition. We discuss the existence and uniqueness of solutions for small data in weighted spaces and derive their complete asymptotic behaviour at infinity. Moreover, we show that each solution with a bounded Dirichlet integral, which has a certain weak additional decay, behaves like O(r ⁻²) as r=|x|→∞ and admits a wide jet region. These investigations are based on the solvability properties of the linear Stokes system in a half space ℝ ₊ ³ . To investigate the Stokes problem in ℝ ₊ ³ , we apply the Mellin transform technique and reduce the Stokes problem to the determination of the spectrum of the corresponding invariant Stokes-Beltrami operator on the hemisphere.     

Diffusion of macromolecular solutions in the turbulent boundary layer of a cylindrical pipe — Evolution of wall concentration

Summary In this paper, we are presenting a model of the evolution of the wall concentration of a macromolecular solution (PEO) annularly injected in a cylindrical pipe in a turbulent flow. This model valid for all diffusion zones is based on mathematical and physical considerations and proves to be in good agreement with the experimental data.

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Zusammenfassung Es wird ein Modell der Entwicklung der Wandkonzentration einer makromolekularen Lösung (PÄO) vorgestellt, die in einem wandnahen Ringspalt in die turbulente Strömung durch ein zylindrisches Rohr injiziert worden ist. Dieses für alle Diffusionszonen gültige Modell basiert auf mathematischen und physikalischen Betrachtungen und erweist sich für die Beschreibung der experimentellen Daten als gut geeignet.

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C _w wall concentration - C _i initial concentration before injection - L ₀ distance from the slot at which the wall concentration drops toe ^-1 of its original value - L _IT ,L _IF ,L _F characteristic lengths - L _I length scale of the second region - x downstream distance from the source - n _I ,n _T ,n _F characteristic exponents - K ₀,K _I ,K _F characteristic constants - ln natural logarithm - q _i flow rate of injection - Q _T flow rate - C _j =C _{i · q i} /Q _T concentration in homogeneous medium - A, B, C, m constants - p andq annex variables - Re Reynolds number With 7 figures     

Direct numerical simulation of the amplification of a 2D temporal disturbance in plane Poiseuille flow

A direct numerical scheme is developed to study the temporal amplification of a 2D disturbance in plane Poiseuille flow. The transient non-linear Navier–Stokes equations are applied in a region of wavelength moving with the wave propagation speed. The complex amplitude involved in the perturbation functions is considered as the initial input of the non-linear stability equations. In this study a fully implicit finite difference scheme with five points in the flow direction and three points in the normal direction is developed so that numerical simulation of the amplification of a two-dimensional temporal disturbance in plane Poiseuille flow can be investigated. The growth and decay of the disturbance with time are presented and neutral stability curves which are in good agreement with existing solutions can be determined. The critical conditions as a function of the amplitude A₀ of the disturbance are presented. Fixing the wavelength, the Navier–Stokes equations are solved up to Re=10,000 a friction factor increasing with Reynolds number is observed. The 2D non-linear behaviour of the streamfunction, vorticity and velocity components at Re=10,000 are also exhibited. © 1998 John Wiley & Sons, Ltd.     

Mixed boundary elements for laminar flows

This paper presents a mixed boundary element formulation of the boundary domain integral method (BDIM) for solving diffusion–convective transport problems. The basic idea of mixed elements is the use of a continuous interpolation polynomial for conservative field function approximation and a discontinuous interpolation polynomial for its normal derivative along the boundary element. In this way, the advantages of continuous field function approximation are retained and its conservation is preserved while the normal flux values are approximated by interpolation nodal points with a uniquely defined normal direction. Due to the use of mixed boundary elements, the final discretized matrix system is overdetermined and a special solver based on the least squares method is applied. Driven cavity, natural and forced convection in a closed cavity are studied. Driven cavity results at Re=100, 400 and 1000 agree better with the benchmark solution than Finite Element Method or Finite Volume Method results for the same grid density with 21×21 degrees of freedom. The average Nusselt number values for natural convection 10³≤Ra≤10⁶ agree better than 0.1% with benchmark solutions for maximal calculated grid densities 61×61 degrees of freedom. Copyright © 1999 John Wiley & Sons, Ltd.     

The new geometric criterion of polynomial stability

In this paper, we apply the coningacy and boundedness of the zeros for a polynomial f_n(z) with real coefficienta (i=0,1,2,…,n). A new simple geometric criterion for stability of f_n(z) is given which is very convenient for application.     

VIBRATION OF CAVITATING ELASTIC WING IN A PERIODICALLY PERTURBED FLOW: EXCITATION OF SUBHARMONICS

The vibration of an elastic wing with an attached cavity in periodically perturbed flows is analyzed. Because the cavity thickness and length L also are perturbed, an excitation with a fixed frequency ω leads to a parametric vibration of the wing, and the amplitudes and spectra of its vibration have nonlinear dependencies on the amplitude of the perturbation. Numerical analysis was carried out for a two-dimensional flow of ideal fluid. Wing vibration was described by means of the beam equation. As a result, two frequency bands of a significant vibration increase were found. A high-frequency band is associated mainly with an elastic resonance of the wing, and a cavity can add a certain damping. A low-frequency band is associated with cavity-volume oscillations. The governing parameter for the low-frequency vibration is the cavity length-based Strouhal number St_C=ωL/U, where U is the free-stream speed. The most significant vibration in the low-frequency band corresponds to approximately constant values of Sh_Cand has the most extensive subharmonics.     

Bifurcation of Limit Cycles from a Polynomial Non-global Center

Consider the planar ordinary differential equation , where the set consists of k non-zero points. In this paper we perturb this vector field with a general polynomial perturbation of degree n and study how many limit cycles bifurcate from the period annulus of the origin in terms of k and n. One of the key points of our approach is that the Abelian integral that controls the bifurcation can be explicitly obtained as an application of the integral representation formula of harmonic functions through the Poisson kernel. Dedicated to Professor Zhifen Zhang on the occasion of her 80th birthday     

Creep behavior of crack near bi-material interface characterized by integral parameters

Creep behavior of crack in dissimilar materials is studied using steady-state C^* path independent integral and ABAQUS finite element code. The specific geometry involves an edge crack parallel to the interface of a bi-material tensile specimen at high temperature. Under extensive creep, the C^* value for the bi-material specimen can be significantly higher than that for the homogeneous specimen. For small-scale creep material mismatch has little influence on the transient integral designated by C_t. The integral parameters C^* or C_t are shown to depend on the inhomogeneity of the system and cannot characterize the creep behavior of cracks.The approach is extended to creep crack growth in a welded compact tension specimen. Modification factors are introduced for different crack and weld interface geometries.     

The Hopf bifurcation with symmetry for the Navier-Stokes equations in (Lp(Ω)) n,with application to plane poiseuille flow

The existence of periodic solutions of the Navier-Stokes equations in function spaces based upon (L _p())ⁿis proved. The paper has three parts, (a) A proof of the existence of strong solutions of the evolution equation with initial data in a solenoidal subspace of (L _p())ⁿ. (b) The evolution equation is restricted to a space of time periodic functions and a Fredholm integral equation on this space is formed. The Lyapunov-Schmidt method is applied to prove the existence of bifurcating time periodic solutions in the presence of symmetry. (c) The theory is applied to the bifurcation of periodic solutions from planar Poiseuille flow in the presence of symmetry (SO(2) x O(2) x S ¹) yielding new results for this classic problem. The O(2) invariance is in the spanwise direction. With the periodicity in time and in the streamwise direction we find that generically there is a bifurcation to both oblique travelling waves and to travelling waves that are stationary in the spanwise direction. There are however points of degeneracy on the neutral surface. A numerical method is used to identify these points and an analysis in the neighborhood of the degenerate points yields more complex periodic solutions as well as branches of quasi-periodic solutions.     

Normalizations with Exponentially Small Remainders for Nonautonomous Analytic Periodic Vector Fields

In this paper we deal with analytic nonautonomous vector fields with a periodic time-dependency, that we study near an equilibrium point. In a first part, we assume that the linearized system is split in two invariant subspaces E ₀ and E ₁. Under light diophantine conditions on the eigenvalues of the linear part, we prove that there is a polynomial change of coordinates in E ₁ allowing to eliminate up to a finite polynomial order all terms depending only on the coordinate u₀ ? E₀{u_0 \in E_0} in the E ₁ component of the vector field. We moreover show that, optimizing the choice of the degree of the polynomial change of coordinates, we get an exponentially small remainder. In the second part, we prove a normal form theorem with exponentially small remainder. Similar theorems have been proved before in the autonomous case: this paper generalizes those results to the nonautonomous periodic case.