A family of high-order accuracy explicit difference schemes with branching stability for solving 3-D parabolic partial differential equation

ＩｎｔｒｏｄｕｃｔｉｏｎＷｅｏｆｔｅｎｍｅｅｔｔｈｅｐｒｏｂｌｅｍｏｆｓｏｌｖｉｎｇｅｑｕａｔｉｏｎｏｆｐａｒａｂｏｌｉｃｔｙｐｅｉｎｍａｎｙｆｉｅｌｄｓｓｕｃｈａｓｓｅｅｐａｇｅ ,ｄｉｆｆｕｓｉｏｎ ,ｈｅａｔｃｏｎｄｕｃｔｉｏｎａｎｄｓｏｏｎ .Ｉｎｔｈｅｃａｓｅｏｆ3＿ｄｉｍｅｎｓｉｏｎ ,ｔｈｅｍｏｄｅｌｉｓａｎｉｎｉｔｉａｌａｎｄｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍａｓｆｏｌｌｏｗｓ: ｕ ｔ = 2 ｕ ｘ2 2 ｕ ｙ2 2 ｕ ｚ2 　　　　 (0 <ｘ,ｙ,ｚ<1 ;ｔ>0 ) ,ｕ(ｘ ,ｙ,ｚ,0 ) =φ(ｘ ,ｙ ,ｚ)…     

A class of two-level explicit difference schemes for solving three dimensional heat conduction equation

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｉｓｐａｐｅｒｄｅａｌｓｗｉｔｈｔｈｅｉｎｉｔｉａｌ＿ｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｏｆｔｈｒｅｅ＿ｄｉｍｅｎｓｉｏｎａｌｈｅａｔｃｏｎｄｕｃｔｉｏｎｅｑｕａｔｉｏｎｉｎｔｈｅｒｅｇｉｏｎＤ :0≤ｘ,ｙ ,ｚ≤Ｌ ,0 ≤ｔ≤Ｔ ｕ ｔ= 2 ｕ ｘ2 2 ｕ ｙ2 2 ｕ ｚ2 ,ｕ｜ｘ=0 =ｆ1(ｙ,ｚ,ｔ) ,　ｕ｜ｘ=Ｌ =ｆ2 (ｙ ,ｚ,ｔ) ,ｕ｜ｙ=0 =ｇ1(ｚ,ｘ,ｔ) ,　ｕ｜ｙ=Ｌ =ｇ2 (ｚ,ｘ,ｔ) ,ｕ｜ｚ=0 =ｈ1(ｘ ,ｙ ,ｔ) ,　ｕ｜ｚ=Ｌ =ｈ2 (ｘ ,ｙ ,ｔ) ,ｕ｜ｔ=0 =φ(ｘ ,ｙ,ｚ) .(1 )(2 )…     

The high accuracy explicit difference scheme for solving parabolic equations 3-dimension

ＩｎｔｒｏｄｕｃｔｉｏｎＦｒｏｍｐｒａｃｔｉｃａｌｐｒｏｂｌｅｍ,ｗｅｃｏｕｌｄｃｏｎｃｌｕｄｅｍａｎｙｐｒｏｂｌｅｍｓａｂｏｕｔｓｏｌｖｉｎｇｐａｒａｂｏｌｉｃｐａｒｔｉａｌｄｉｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎ．Ｎｏｗｔｈｅｒｅａｒｅｍａｎｙｎｕ...     

A class of three-level explicit schemes with higher stability properties for a dispersive equation ut=auxxx

In this paper, a class of three level explicit schemes for a dispersive equation u_t=au_xxx with stability condition |r|=|α|Δt/(Δx)³≤2.382484, are considered. The stability condition for this class of schemes is much better than |r|≤0.3849 in [1], [2] and |r|≤0.701659 in [3], and |r|≤1.1851 in [4].     

The application of Moire interferometry in the measurement of displacement field and strain field at notch-tip and crack-tip

This paper presents the application of Moire interferometry in measuring the displacement and strain field at notch-tip and crack-tip before and after crack propagation. The experiment is carried out using a three point bending beam with a notch. TheN _x andN _y fringe patterns representing displacement field, and the ΔN _x/Δx and ΔN _γ/Δγ fringe patterns representing the strain field are obtained. The sensitivity of the measured displacement is 0.417 μm per fringe order. The displacement and strain distribution along the sectionx=0 have been worked out according toN _x andN _γ fringe patterns. The project supported by Chinese Academy of Sciences and National Natural Science Foundation of China     

Nonsimilarity solutions for non-Darcy mixed convection from horizontal surfaces in a porous medium

Non-Darcy mixed convection in a porous medium from horizontal surfaces with variable surface heat flux of the power-law distribution is analyzed. The entire mixed convection regime is divided into two regions. The first region covers the forced convection dominated regime where the dimensionless parameter ζ _f =Ra^* _x /Pe² _x is found to characterize the effect of buoyancy forces on the forced convection with K ^′ U _∞/ν characterizing the effect of inertia resistance. The second region covers the natural convection dominated regime where the dimensionless parameter ζ _n =Pe _x /Ra^*1/2 _x is found to characterize the effect of the forced flow on the natural convection, with (K ^′ U _∞/ν)Ra^*1/2 _x /Pe _x characterizing the effect of inertia resistance. To obtain the solution that covers the entire mixed convection regime the solution of the first regime is carried out for ζ _f =0, the pure forced convection limit, to ζ _f =1 and the solution of the second is carried out for ζ _n =0, the pure natural convection limit, to ζ _n =1. The two solutions meet and match at ζ _f =ζ _n =1, and R ^* _h =G ^* _h . Also a non-Darcy model was used to analyze mixed convection in a porous medium from horizontal surfaces with variable wall temperature of the power-law form. The entire mixed convection regime is divided into two regions. The first region covers the forced convection dominated regime where the dimensionless parameter ξ _f =Ra _x /Pe _x ^3/2 is found to measure the buoyancy effects on mixed convection with Da _x Pe _x /ɛ as the wall effects. The second region covers the natural convection dominated region where ξ _n =Pe _x /Ra _x ^2/3 is found to measure the force effects on mixed convection with Da _x Ra _x ^2/3/ɛ as the wall effects. Numerical results for different inertia, wall, variable surface heat flux and variable wall temperature exponents are presented. Received on 8 July 1996     

A Similarity Subgrid Model for Premixed Turbulent Combustion

The accuracy of large-eddy simulation (LES) of a turbulent premixed Bunsen flame is investigated in this paper. To distinguish between discretization and modeling errors, multiple LES, using different grid sizes h but the same filterwidth Δ, are compared with the direct numerical simulation (DNS). In addition, LES using various values of Δ but the same ratio Δ/h are compared. The chemistry in the LES and DNS is parametrized with the standard steady premixed flamelet for stochiometric methane-air combustion. The subgrid terms are closed with an eddy-viscosity or eddy-diffusivity approach, with an exception of the dominant subgrid term, which is the subgrid part of the chemical source term. The latter subgrid contribution is modeled by a similarity model based upon 2Δ, which is found to be superior to such a model based upon Δ. Using the 2Δ similarity model for the subgrid chemistry the LES produces good results, certainly in view of the fact that the LES is completely wrong if the subgrid chemistry model is omitted. The grid refinements of the LES show that the results for Δ = h do depend on the numerical scheme, much more than for h = Δ/2 and h = Δ/4. Nevertheless, modeling errors and discretization error may partially cancel each other; occasionally the Δ = h results were more accurate than the h ≤ Δ results. Finally, for this flame LES results obtained with the present similarity model are shown to be slightly better than those obtained with standard β-pdf closure for the subgrid chemistry.     

A correction method for measuring turbulence kinetic energy dissipation rate by PIV

The accuracy of turbulent kinetic energy (TKE) dissipation rate measured by PIV is studied. The critical issue for PIV-based dissipation measurements is the strong dependency on the spatial resolution, Δx, as reported by Saarenrinne and Piirto (Exp Fluids Suppl:S300–S307, 2000). When the PIV spacing is larger than the Kolmogorov scale, η, the dissipation is underestimated because the small scale fluctuations are filtered. For the case of Δx smaller than the Kolmogorov scale, the error rapidly increases due to noise. We introduce a correction method to eliminate the dominant error for the small Δx case. The correction method is validated by using a novel PIV benchmark, random Oseen vortices synthetic image test (ROST), in which quasi-turbulence is generated by randomly superposing multiple Oseen vortices. The error of the measured dissipation can be more than 1,000% of the analytical dissipation for the small Δx case, while the dissipation rate is underestimated for the large Δx case. Though the correction method does not correct the underestimate due to the low resolution, the dissipation was accurately obtained within a few percent of the true value by using the correction method for the optimal resolution of η/10 < Δx < η/2.     

An Efficient Derivation of the Aronsson Equation

For 1<p<∞, the equation which characterizes minima of the functional u↦∫ _U |Du| ^p ,dx subject to fixed values of u on ∂U is −Δ _p u=0. Here −Δ _p is the well-known ``p-Laplacian'. When p=∞ the corresponding functional is u↦|| |Du|<sup>2</sup>|| <sub>L∞(U)</sub> . A new feature arises in that minima are no longer unique unless U is allowed to vary, leading to the idea of ``absolute minimizers'. Aronsson showed that then the appropriate equation is −Δ_∞ u=0, that is, u is ``infinity harmonic' as explained below. Jensen showed that infinity harmonic functions, understood in the viscosity sense, are precisely the absolute minimizers. Here we advance results of Barron, Jensen and Wang concerning more general functionals u↦||f(x,u,Du)|| _L∞(U) by giving a simplified derivation of the corresponding necessary condition under weaker hypotheses. (Accepted September 6, 2002) Published online April 14, 2003 Communicated by S. Muller     

Periodic solutions of dynamical systems

Summary In this paper we look for T-periodic solutions of dynamical systems. Particularly we consider the system whereU ɛC ¹(ℝ ⁿ x x ℝ, ℝ),U(x, t + T)=U(x,t) ∀ x ℝ ⁿ , ∀t ɛ ℝ T>0. We assume that the problem is asymptotically linear with a bounded nonlinearity. Under a resonance assumption, we find a multiplicity of T-periodic solutions for T large enough.

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Sommario In questo lavoro si cercano soluzioni periodiche di periodo T assegnato di sistemi dinamici. In particolare si considera un sistema di n equazioni differenziali del secondo ordine del tipo doveU ɛC ¹(ℝ ⁿ x x ℝ, ℝ),U(x, t + T)=U(x,t) ∀ x ℝ ⁿ , ∀t ɛ ℝ T>0. Nel caso in cui il problema sia asintoticamente lineare, con termine nonlineare limitato e in condizioni di risonanza, troviamo che esiste tale che per il sistema ha una molteplicità di soluzioni.

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Presented at the VII A.I.M.E.T.A. and supported by M.P.I. (40% and 60%).     

Solution of the equations for a regular beam emitted by a curvilinear surface in the nonstationary case

An analytical solution is given of the equations for a regular beam (§1) emitted by an arbitrary surface in the nonstationary case and the - and T-limited states for nonzero initial velocity (§ §2-4). It is assumed that the emitter is the coordinate surface x¹=0 in the orthogonal system x¹ (i=1, 2, 3), and the current density J, the electric field , and the magnetic field H are given functions J(t, x²,x³), (t, x², x³), and H (x¹, x², x³). The solution is given in the form of series in terms of (X¹) with coefficients that are functions of t, x², and x³. These coefficients are determined from recurrence relations ( =1/3, 1/2, 1, depending on the emission conditions). Plane, cylindrical, and spherical diodes are considered in § 5 in the case in which the high-frequency component of the current density J is not small in comparison with its constant components.     

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.     

Standing Waves for Nonlinear Schrödinger Equations with a General Nonlinearity

For elliptic equations ε²Δu − V(x) u + f(u) = 0, x ∈ R ^N , N ≧ 3, we develop a new variational approach to construct localized positive solutions which concentrate at an isolated component of positive local minimum points of V, as ε → 0, under conditions on f which we believe to be almost optimal. An erratum to this article can be found at     

An Experimental Investigation of Two-Point Correlations in Two- and Three-Dimensional Turbulent Boundary Layers

Two-point correlation measurements of the wall normal fluctuating velocities were made in two-dimensional (2-D) and pressure-driven three-dimensional (3-D) turbulent boundary layers. These data are needed for characterization and modeling of active-motion length scales, especially for 3-D flows. The fine-probe-volume data were measured using two custom-designed laser-Doppler-velocimeter fiber-optic probes. The data are relatively free of noise, signal broadening, and bias effects. Favorable comparisons with direct-numerical-simulation (DNS) results in the near-wall region of the 2-D flow validate the experimental techniques used here. For a given fixed probe location, non-dimensional correlation values scale best on the probe separation. For both the 2-D and 3-D cases, peak correlations lie along a line inclined away from the wall at 11° and 8°, respectively, which suggests the existence of an outgoing characteristic line affected by only the upstream flow. The decay of the correlation coefficient occurs nearer the wall than away from the wall relative to the fixed probe location. The variations for the 3-D flow correlations are similar to the 2-D variations, but with longer Δ x ⁺and Δ y ⁺ decay distances, probably because of the 3-D flowacceleration. While the spanwise variation of the correlationcoefficients is symmetric about the fixed point for the 2-D case asdictated by reciprocity, the 3-D case shows a large asymmetry for spanwise variations Δ z ⁺ < 68. The profiles at higher Δ z ⁺ are more symmetric. In general, at a given y the maximum correlation is skewed toa non-zero Δ z. It appears that the skewing of the correlation coefficient in the z direction tracks the sign of . This revised version was published online in July 2006 with corrections to the Cover Date.     

Existence of Singular Self-Similar Solutions of the Three-Dimensional Euler Equations in a Bounded Domain

A self-similar solution of the three-dimensional (3d) incompressible Euler equations is defined byu(x,t) =U(y)/t*-t) ^{α, y = x/(t* ~ t)β,α,β>} 0, whereU(y) satisfiesζU + βy. ΔU + U. VU + VP = 0,divU = 0. For α = β = 1/2, which is the limiting case of Leray’s self-similar Navier—Stokes equations, we prove the existence of(U,P) ε H³(Ω,R³ X R) in a smooth bounded domain Ω, with the inflow boundary data of non-zero vorticity. This implies the possibility that solutions of the Euler equations blow up at a timet = t*, t* < +∞.     

Average value of the safety coefficient in a framed structure corresponding to random distributions of yield stresses

Summary The safety coefficient relevant to the plastic collapse is considered for a framed structure, when the yield stresses of its beams are random quantities, however distributed. A value for the safety coefficient may be obtained through a deterministic calculation, assuming for each beam the yield stress equal to its average value. It is demonstrated that such a value of the safety coefficient is greater, or at least equal, to the average value of the safety coefficient of all the feasible structures corresponding to the various yield stresses of the beams.

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Sommario Si considera una struttura reticolare le cui aste presentino valori del carico di snervamento aleatori e comunque distribuiti attorno ai loro valori medi. Si dimostra che il coefficiente di sicurezza al collasso plastico calcolato deterministicamente, assumendo per ciascuna asta i carichi di snervamento pari ai loro valori medi, è maggiore od uguale al valor medio dei coefficienti di sicurezza di tutte le strutture che la aleatorietà dei carichi di snervamento rende possibili.

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The equationN _sj =x±/2 means thatN _sj assume one value in the intervalx–/2,x+/2.     

On the motion of a material point on a moving surface

Summary Sufficient conditions are given for the stability and instability of the equilibrium position x=y=z=0 in the mechanical system consisting of a material point constrained to move on the moving surface z=−λ(t)(x²+y²) (λ(t)>0) in a constant field of gravity (the axis 0z is directed vertically upward) under the action of viscous friction of total dissipation.

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Sommario Si danno condizioni sufficienti per la stabilità e la instabilità della posizione di equilibrio x=y=z=0 nel sistema meccanico che consiste di un punto materiale vincolato a muoversi sulla superficie mobile z=−λ(t)(x²+y²) (λ(t)>0) in un campo di gravità costante (l'asse 0z è diretto verticalmente e orientato verso l'alto) sotto l'azione di attriti viscosi con dissipazione completa.

     

Similarities Between Rotation and Stratification Effects on Homogeneous Shear Flow

Linear theory is applied to examine rotation and buoyancy effects on homogeneous turbulent shear flows with given vertical velocity shear, S=d/dx ₃. In the rotating shear case (where the rotation vector is perpendicular to the plane of the mean flow, Ω _i =Ωδ _{i 2}), general solutions for the Fourier components of the fluctuating velocity are proposed. These solutions are compared with those proposed in the literature for the Fourier components of the fluctuating velocity and density in the case of a homogeneous stratified shear flow with vertical density gradient, S _ρ=d/dx ₃. It is shown that from the normal mode stability stand point the Bradshaw parameter B=2Ω/S(1+2Ω/S) (in the rotating shear case) and the Richardson number R _i (in the statified shear case) play similar roles in identifying the stability for all the wave components except in the case where Ω·k=0, for which rotation has no effects on the flow. Analysis of the long-time behavior of the non-dimensional spectral density of energy, e ^g , is carried out. In the stable case, e ^g has decaying oscillations or undergoes a power law decay in time. Analytical solutions for the streamwise two-dimensional energy ℰ _ii ¹/2 (i.e. the limit at k ₁=0 of the one-dimensional energy spectra) are proposed. At large time, ℰ _ii ¹(t)/ℰ _ii ¹(0) oscillates around the value (3R _i +1)/(4R _i ) except at R _i =1 it stays constant in time. Similar behavior for ℰ _ii ¹(t)/ℰ _ii ¹(0) is also observed in the rotating shear case (ℰ _ii ¹(t)/ℰ _ii ¹(0) oscillates around the value (1+4B)/(4B)). Due to the behavior of the dimensionless spectral density of energy in both flow cases, the turbulent kinetic energy, /2, the production rate, ?, and the rate due to the buoyancy forces, ℬ, are split into two parts, , ?=?₁+?₂, ℬ=ℬ₁+ℬ₂ (in the stratified shear case, both ?₁ and ℬ₁ vanish when R _i >?, while in the rotating shear case one has ℬ=0). It is shown that when rotation is “cyclonic” (i.e. Ω/S>0), part reaches maximum magnitudes at St ≈2, independent of the B value, and the first time to a zero crossing of ?₂ occurs at this particular value. When rotation is “anticyclonic” (i.e. Ω/S<0) one finds St ≈1.6 instead of St ≈2. In the stratified shear case, both ?₂ and ℬ₂ cross zero at Nt=St ≈2, and part reaches maximum magnitudes at this particular value. These results and in particular those for the turbulent kinetic energy are compared with previous direct numerical simulation (DNS) results in homogeneous stratified shear flows. Received 30 July 2001 and accepted 19 February 2002     

A FINITE DIFFERENCE METHOD FOR SIMULATING TRANSVERSE VIBRATIONS OF AN AXIALLY MOVING VISCOELASTIC STRING

A finite difference method is presented to simulate transverse vibrations of an axially moving string.By discretizing the governing equation and the equation of stress- strain relation at different frictional knots,two linear sparse finite difference equation systems are obtained.The two explicit difference schemes can be calculated alternatively, which make the computation much more efficient.The numerical method makes the nonlinear model easier to deal with and of truncation errors,O(Δt~2 Δx~2).It also shows quite good stability for small initial values.Numerical examples are presented to demonstrate the efficiency and the stability of the algorithm,and dynamic analysis of a viscoelastic string is given by using the numerical results.