On Aerodynamic Forces for Viscous Compressible Flow

The paper is aimed at reviewing and adding some new results to our recent work on a force theory for viscous compressible flows around a finite body. It has been proposed to analyze aerodynamic forces directly in terms of fluid elements of nonzero vorticity and density gradient. Let ρ denote the density, u the velocity, and ω the vorticity. It is demonstrated that for largely separated flows about bluff bodies, there are two major source elements: R _e(x) =−?u ²∇ρ·∇ϕ and V _e(x) =−u×ω·∇ϕ, where ϕ is an acyclic potential, generated by the solid body moving with unit velocity in the negative direction of the force considered. In particular, under mild conditions, the (unique) choice of ϕ enforces that the elements R _e(x) and V _e(x) decay rapidly away from the body. Four kinds of finite body are considered: a circular cylinder, a sphere, a hemi sphere-cylinder, and a delta wing of elliptic section—all in transonic-to-supersonic regimes. From an extensive numerical study carried out for these bodies, it is found that these two elements contribute to 95% or more of the total drag or lift for all the cases under consideration. Moreover, R _e(x) due to density gradient becomes progressively important relative to V _e(x) due to vorticity as the Mach number increases. The present method of force analysis enables effective analysis and assessment of relative importance of aerodynamics forces, contributed from individual flow structures. The analysis could therefore be very much useful in view of the rapid growth in numerical fluid dynamics; detailed (either local or global) flow information is often available. The paper is dedicated to Sir James Lighthill in honor of his great contributions to aeronautics on the occasion of the publication of his collected works. Received 3 January 1997 and accepted 11 April 1997     

Vorticities in a LES Model for 3D Periodic Turbulent Flows

This paper is devoted to the study of a LES model to simulate turbulent 3D periodic flow. We focus our attention on the vorticity equation derived from this LES model for small values of the numerical grid size δ. We obtain entropy inequalities for the sequence of corresponding vorticities and corresponding pressures independent of δ, provided the initial velocity u₀ is in L_x² while the initial vorticity ω₀ = ∇ × u₀ is in L_x¹. When δ tends to zero, we show convergence, in a distributional sense, of the corresponding equations for the vorticities to the classical 3D equation for the vorticity.     

A Note on the Existence of Solutions to the Oseen System in Lipschitz Domains

We study the boundary-value problem associated with the Oseen system in the exterior of m Lipschitz domains of an euclidean point space We show, among other things, that there are two positive constants and α depending on the Lipschitz character of Ω such that: (i) if the boundary datum a belongs to L^q(∂Ω), with q ∈ [2,+∞), then there exists a solution (u, p), with and u ∈ L^∞(Ω) if a ∈ L^∞(∂Ω), expressed by a simple layer potential plus a linear combination of regular explicit functions; as a consequence, u tends nontangentially to a almost everywhere on ∂Ω; (ii) if a ∈ W^1-1/q,q(∂Ω), with then ∇u, p ∈ L^q(Ω) and if a ∈ C^0,μ(∂Ω), with μ ∈ [0, α), then also, natural estimates holds.     

A Navier–Stokes Approximation of the 3D Euler Equation with the Zero Flux on the Boundary

Under assumptions on smoothness of the initial velocity and the external body force, we prove that there exists T ₀ > 0, ν ₀ > 0 and a unique continuous family of strong solutions u ^ν (0 ≤ ν < ν ₀) of the Euler or Navier–Stokes initial-boundary value problem on the time interval (0, T ₀). In addition to the condition of the zero flux, the solutions of the Navier–Stokes equation satisfy certain natural boundary conditions imposed on curl u ^ν and curl ² u ^ν .      

Symmetry of Mountain Pass Solutions of Some Vector Field Equations

We prove radial symmetry (or axial symmetry) of the mountain pass solution of variational elliptic systems − AΔu(x) + ∇ F(u(x)) = 0 (or − ∇.(A(r) ∇ u(x)) + ∇ F(r,u(x)) = 0,) u(x) = (u ₁(x),...,u _N (x)), where A (or A(r)) is a symmetric positive definite matrix. The solutions are defined in a domain Ω which can be , a ball, an annulus or the exterior of a ball. The boundary conditions are either Dirichlet or Neumann (or any one which is invariant under rotation). The mountain pass solutions studied here are given by constrained minimization on the Nehari manifold. We prove symmetry using the reflection method introduced in Lopes [(1996), J. Diff. Eq. 124, 378–388; (1996), Eletron. J. Diff. Eq. 3, 1–14].     

APPLICATION OF THE DUAL RECIPROCITY BOUNDARY ELEMENT METHOD TO SOLUTION OF NONLINEAR DIFFERENTIAL EQUATION

In this paper the dual reciprocity boundary element method is employed to solve nonlinear differential equation ∇² u+u+ɛu ³ =b. Results obtained in an example have a good agreement with those by FEM and show the applicability and simplicity of dual reciprocity method(DRM)in solving nonlinear differential equations.     

Existence of the minimal positive solution of some nonlinear elliptic systems when the nonlinearity is the sum of a sublinear and a superlinear term

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A NOTE ON BIFURCATIONS OF u″+μ(u-uk)=0(4≤k∈Z+)

Bifurcations of one kind of reaction-diffusion equations, u″+μ(u-u^k)=0(μ is a parameter,4≤k∈Z⁺), with boundary value condition u(0)=u(π)=0 are discussed. By means of singularity theory based on the method of Liapunov-Schmidt reduction, satisfactory results can be acquired.     

Direction of Vorticity and Regularity up to the Boundary: On the Lipschitz-Continuous Case

In their famous 1993 paper, Constantin and Fefferman consider the evolution Navier–Stokes equations in the whole space R ³ and prove, essentially, that if the direction of the vorticity is Lipschitz continuous in the space variables, during a given time-interval, then the corresponding solution is regular. Since Lipschitz-continuity is a very natural, basic, property, it looks interesting to go further in this particular direction. In this paper, we consider the initial-boundary value problem for the Navier–Stokes equations in a regular, bounded, domain under a slip boundary condition, and prove regularity of the solution, up to the boundary, under a weakened Lipschitz-continuity assumption on the direction of the vorticity. The interest of our result highly relies on the fact that the Lipschitz-continuity coefficient g(x, t) is sharp. This means, in a sense, that our finding possesses the same level of accuracy as that of the classical “Prodi-Serrin” type conditions; see the introductory section. It should be remarked that a similar result was already obtained in the 2009 paper by Beirão da Veiga and Berselli. In the latter, the proof of an analogous sharp result was shown under the assumption of ${\frac12}$ -H?lder continuity on the direction of vorticity. The authors also claimed, correctly, that by the same ideas the proof of such a result could be extended to H?lder exponents ${\beta \in\,]\,0,\,1\,]}$ . However the proofs would be extremely involved. On the contrary, the proof followed in this paper treat the Lipschitz case is definitely more elementary than any other proof, even if restricted to the whole space case.     

A continuum mechanical theory for turbulence: a generalized Navier–Stokes-<Emphasis Type="Italic">α</Emphasis> equation with boundary conditions

We develop a continuum-mechanical formulation and generalization of the Navier–Stokes-α equation based on a recently developed framework for fluid-dynamical theories involving higher-order gradient dependencies. Our flow equation involves two length scales α and β. The first of these enters the theory through the specific free-energy α ²|D|², where D is the symmetric part of the gradient of the filtered velocity, and contributes a dispersive term to the flow equation. The remaining scale is associated with a dissipative hyperstress which depends linearly on the gradient of the filtered vorticity and which contributes a viscous term, with coefficient proportional to β ², to the flow equation. In contrast to Lagrangian averaging, our formulation delivers boundary conditions and a complete structure based on thermodynamics applied to an isothermal system. For a fixed surface without slip, the standard no-slip condition is augmented by a wall-eddy condition involving another length scale ℓ characteristic of eddies shed at the boundary and referred to as the wall-eddy length. As an application, we consider the classical problem of turbulent flow in a plane, rectangular channel of gap 2h with fixed, impermeable, slip-free walls and make comparisons with results obtained from direct numerical simulations. We find that α/β ~ Re ^0.470 and ℓ/h ~ Re ^−0.772, where Re is the Reynolds number. The first result, which arises as a consequence of identifying the specific free-energy with the specific turbulent kinetic energy, indicates that the choice β = α required to reduce our flow equation to the Navier–Stokes-α equation is likely to be problematic. The second result evinces the classical scaling relation η/L ~ Re ^−3/4 for the ratio of the Kolmogorov microscale η to the integral length scale L.      

Accuracy of out-of-plane vorticity measurements derived from in-plane velocity field data

 A study of the errors in out-of-plane vorticity (ω _z ) calculated using a local χ² fitting of the measured velocity field and analytic differentiation has been carried out. The primary factors of spatial velocity sampling separation and random velocity measurement error have been investigated. In principle the ω _z error can be decomposed into a bias error contribution and a random error contribution. Theoretical expressions for the transmission of the random velocity error into the random vorticity error have been derived. The velocity and vorticity field of the Oseen vortex has been used as a typical vortex structure in this study. Data of different quality, ranging from exact velocity vectors of analytically defined flow fields (Oseen vortex flow) sampled at discrete locations to computer generated digital image frames analysed using cross-correlation DPIV, have been investigated in this study. This data has been used to provide support for the theoretical random error results, to isolate the different sources of error and to determine their effect on ω _z measurements. A method for estimating in-situ the velocity random error is presented. This estimate coupled with the theoretically derived random error transmission results for the χ² vorticity calculation method can be used a priori to estimate the magnitude of the random error in ω _z . This random error is independent of a particular flow field. The velocity sampling separation is found to have a profound effect on the precise determination of ω _z by introducing a bias error. This bias error results in an underestimation of the peak vorticity. Simple equations, which are based on a local model of the Oseen vortex around the peak vorticity region, allowing the prediction of the ω _z bias error for the χ² vorticity calculation method, are presented. An important conclusion of this study is that the random error transmission factor and the bias error cannot be minimised simultaneously. Both depend on the velocity sampling separation, but with opposing effects. The application of the random and bias vorticity error predictions are illustrated by application to experimental velocity data determined using cross-correlation DPIV (CCDPIV) analysis of digital images of a laminar vortex ring. Received: 31 October 1997/Accepted: 6 February 1998     

A Sharp-Interface Limit for a Two-Well Problem in Geometrically Linear Elasticity

In the theory of solid-solid phase transitions the deformation of an elastic body is determined via a functional containing a nonconvex energy density and a singular perturbation. We study Frame indifference, within a linearized setting, requires that W depends only on the symmetric part of ∇u. The potential W is nonnegative and vanishes on two wells, i.e., for d = 2, on two lines in the space of matrices. We determine, for d = 2, the Gamma limit I₀ = Γ− lim _ɛ→0 I_ɛ. The limit I₀[u] is finite only for deformations u that fulfill W(∇u)=0 almost everywhere and have sharp interfaces where ∇u has jumps. For these u, I₀[u] equals the line integral over the interfaces of a surface energy density.     

Streamwise evolution of an inclined cylinder wake

The streamwise evolution of an inclined circular cylinder wake was investigated by measuring all three velocity and vorticity components using an eight-hotwire vorticity probe in a wind tunnel at a Reynolds number Re_d of 7,200 based on free stream velocity (U _∞) and cylinder diameter (d). The measurements were conducted at four different inclination angles (α), namely 0°, 15°, 30°, and 45° and at three downstream locations, i.e., x/d = 10, 20, and 40 from the cylinder. At x/d = 10, the effects of α on the three coherent vorticity components are negligibly small for α ≤ 15°. When α increases further to 45°, the maximum of coherent spanwise vorticity reduces by about 50%, while that of the streamwise vorticity increases by about 70%. Similar results are found at x/d = 20, indicating the impaired spanwise vortices and the enhancement of the three-dimensionality of the wake with increasing α. The streamwise decay rate of the coherent spanwise vorticity is smaller for a larger α. This is because the streamwise spacing between the spanwise vortices is bigger for a larger α, resulting in a weak interaction between the vortices and hence slower decaying rate in the streamwise direction. For all tested α, the coherent contribution to [`(v<sup>2</sup>)] \overline{{v^{2}}} is remarkable at x/d = 10 and 20 and significantly larger than that to [`(u²)] \overline{{u^{2}}} and [`(w<sup>2</sup>)]. \overline{{w^{2}}}. This contribution to all three Reynolds normal stresses becomes negligibly small at x/d = 40. The coherent contribution to [`(u²)] \overline{{u^{2}}} and [`(v²)] \overline{{v^{2}}} decays slower as moving downstream for a larger α, consistent with the slow decay of the coherent spanwise vorticity for a larger α.     

Existence of Weak Solutions to the Equations of Non-Stationary Motion of Non-Newtonian Fluids with Shear Rate Dependent Viscosity

In the present paper we prove the existence of weak solutions to the equations of non-stationary motion of an incompressible fluid with shear rate dependent viscosity in a cylinder Q = Ω × (0,T), where denotes an open set. For the power-low model with we are able to construct a weak solution with ∇ · u = 0.     

On the correlation between enstrophy and energy dissipation rate in a turbulent wake

All three components of the vorticity fluctuation have been measured simultaneously in a turbulent wake using a new eight-sensor vorticity probe. The vorticity fluctuation spectra agree reasonably well with those from a direct numerical simulation of a turbulent channel flow at high wavenumbers. The similarity between the instantaneous energy dissipation rate ε and the instantaneous enstrophy ω² is examined using spectra and probability density functions. The correlation between ω² and ε is evaluated in some detail. The homogeneous value of ε is strongly correlated with ω². The full value of ε and, more especially its isotropic value, are less well correlated with the enstrophy. Conditional averaging indicates that high enstrophy regions are associated with high energy dissipation rate regions.     

The Dirichlet Problem for H-Systems with Small Boundary Data: BlowUp Phenomena and Nonexistence Results

Given H:ℝ³→ℝ of class C¹ and bounded, we consider a sequence (uⁿ) of solutions of the H-system in the unit open disc satisfying the boundary condition uⁿ=γⁿ on ∂. In the first part of this paper, assuming that (uⁿ) is bounded in H¹(,ℝ³) we study the behavior of (uⁿ) when the boundary data γⁿ shrink to zero. We show that either uⁿ→0 strongly in H¹(,ℝ³) or uⁿ blows up at least one H-bubble ω, namely a nonconstant, conformal solution of the H-system on ℝ². Under additional assumptions on H, we can obtain more precise information on the blow up. In the second part of this paper we investigate the multiplicity of solutions for the Dirichlet problem on the disc with small boundary datum. We detect a family of nonconstant functions H (even close to a nonzero constant in any reasonable topology) for which the Dirichlet problem cannot admit a ``large' solution at a mountain pass level when the boundary datum is small.     

Well-posedness and Regularity for the Elasticity Equation with Mixed Boundary Conditions on Polyhedral Domains and Domains with Cracks

We prove a regularity result for the anisotropic linear elasticity equation ${P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f}We prove a regularity result for the anisotropic linear elasticity equationP u : = div ( C ·?u) = f{P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f} , with mixed (displacement and traction) boundary conditions on a curved polyhedral domain W ì \mathbbR³{\Omega \subset \mathbb{R}^3} in weighted Sobolev spaces K^m+1_a+1(W){\mathcal {K}^{m+1}_{a+1}(\Omega)} , for which the weight is given by the distance to the set of edges. In particular, we show that there is no loss of K^m_a{\mathcal {K}^{m}_{a}} -regularity. Our curved polyhedral domains are allowed to have cracks. We establish a well-posedness result when there are no neighboring traction boundary conditions and |a| < η, for some small η > 0 that depends on P, on the boundary conditions, and on the domain Ω. Our results extend to other strongly elliptic systems and higher dimensions.     

The uniqueness and existence of solution of the characteristic problem on the generalized KdV equation

The generalized KdV equationu ₁+auu_a+μu_a3+eu_a5=0^[1] is a typical integrable equation. It is derived studying the dissemination of magnet sound wave in cold plasma^[2], the isolated wave in transmission line^[3], and the isolated wave in the boundary surface of the divided layer fluid^[4]. For the characteristic problem of the generalized KdV equation, this paper, based on the Riemann function, designs a suitable structure, then changes the characteristic problem to an equivalent integral and differential equation whose corresponding fixed point, the above integral differential equation has a unique regular solution, so the characteristic problem of the generalized KdV equation has a unique solution. The iteration solution derived from the integral differential equation sequence is uniformly convegent in .     

The hölder continuity of generalized solutions of a class quasilinear parabolic equations

Let A cud B satisfy the Structural conditions (2), the local Hölder continuityinterior to Q=G×(0, T) is proved for the generalized solutions of quasilinearparabolic equations as follows: u₂ - divA(x, t,u,∇u) + B(x, t, u, ∇u)=0     

A New Class of Weak Solutions of the Navier–Stokes Equations with Nonhomogeneous Data

We investigate a class of weak solutions, the so-called very weak solutions, to stationary and nonstationary Navier–Stokes equations in a bounded domain . This notion was introduced by Amann [3], [4] for the nonstationary case with nonhomogeneous boundary data leading to a very large solution class of low regularity. Here we are mainly interested in the investigation of the “largest possible” class of solutions u for the more general problem with arbitrary divergence k = div u, boundary data g = u|_∂Ω and an external force f, as weak as possible, but maintaining uniqueness. In principle, we will follow Amann’s approach.