Assessment of the finite volume method applied to the v2 − f model

The objective of this paper is to assess the accuracy of low‐order finite volume (FV) methods applied to the v² ? f turbulence model of Durbin (Theoret. Comput. Fluid Dyn. 1991; 3 :1–13) in the near vicinity of solid walls. We are not (like many others) concerned with the stability of solvers ‐ the topic at hand is simply whether the mathematical properties of the v² ? f model can be captured by the given, widespread, numerical method. The v² ? f model is integrated all the way up to solid walls, where steep gradients in turbulence parameters are observed. The full resolution of wall gradients imposes quite high demands on the numerical schemes and it is not evident that common (second order) FV codes can fully cope with such demands. The v² ? f model is studied in a statistically one‐dimensional, fully developed channel flow where we compare FV schemes with a highly accurate spectral element reference implementation. For the FV method a higher‐order face interpolation scheme, using Lagrange interpolation polynomials up to arbitrary order, is described. It is concluded that a regular second‐order FV scheme cannot give an accurate representation of all model parameters, independent of mesh density. To match the spectral element solution an extended source treatment (we use three‐point Gauss–Lobatto quadrature), as well as a higher‐order discretization of diffusion is required. Furthermore, it is found that the location of the first internal node need to be well within y⁺=1. Copyright © 2009 John Wiley & Sons, Ltd.     

Uniqueness of positive radial solutions of △u+g(r)u+h(r)u p =0 in Rn

Positive radial solutions of a semilinear elliptic equation u+g(r)u+h(r)u ^p =0, where r=|x|, xR ⁿ , and p>1, are studied in balls with zero Dirichlet boundary condition. By means of a generalized Pohoaev identity which includes a real parameter, the uniqueness of the solution is established under quite general assumptions on g(r) and h(r). This result applies to Matukuma's equation and the scalar field equation and is shown to be sharp for these equations.     

An optimized Schwarz method with two‐sided Robin transmission conditions for the Helmholtz equation

Optimized Schwarz methods are working like classical Schwarz methods, but they are exchanging physically more valuable information between subdomains and hence have better convergence behaviour. The new transmission conditions include also derivative information, not just function values, and optimized Schwarz methods can be used without overlap. In this paper, we present a new optimized Schwarz method without overlap in the 2d case, which uses a different Robin condition for neighbouring subdomains at their common interface, and which we call two‐sided Robin condition. We optimize the parameters in the Robin conditions and show that for a fixed frequency an asymptotic convergence factor of 1 – O(h^1/4) in the mesh parameter h can be achieved. If the frequency is related to the mesh parameter h, h = O(1/ω^γ) for γ?1, then the optimized asymptotic convergence factor is 1 – O(ω^(1–2γ)/8). We illustrate our analysis with 2d numerical experiments. Copyright © 2007 John Wiley & Sons, Ltd.     

Superconvergence analysis of bi-<Emphasis Type="Italic">k</Emphasis>-degree rectangular elements for two-dimensional time-dependent Schrödinger equation

Superconvergence has been studied for long, and many different numerical methods have been analyzed. This paper is concerned with the problem of superconvergence for a two-dimensional time-dependent linear Schrödinger equation with the finite element method. The error estimate and superconvergence property with order O(h^k+1) in the H¹ norm are given by using the elliptic projection operator in the semi-discrete scheme. The global superconvergence is derived by the interpolation post-processing technique. The superconvergence result with order O(h^k+1 + τ²) in the H¹ norm can be obtained in the Crank-Nicolson fully discrete scheme.     

New numerical method for partial differential equations. 1: Application to the diffusion equation

In this paper a new, highly accurate method called PH is presented for the numerical integration of partial differential equations. The method is applied for the solution of the one-dimensional diffusion equation. Upon integrating the equation within a subdomain of space and time using the prismoidal approximation, a three-point implicit scheme is obtained with a truncation error of order O(k⁴, h⁶), where k and h represent the time and space steps respectively. The method is stable under the condition s = αk/h² ? S(δ), where the function S(δ) increases as the parameter δ decreases from 1/12 to negative values. In practice the method behaves as unconditionally stable upon choosing an appropriate value for δ. A new formula is also adopted for the implementation of a Neumann boundary condition, introducing a truncation error of order O(h⁴). Numerical solutions are obtained incorporating Dirichlet and Neumann boundary conditions. The results prove that our method is far more accurate than any other-implicit or explicit method.     

A note on the stability and accuracy of Cn finite elements in steady diffusion-convection problems

The paper is concerned with stability and accuracy of an nth order Lagrangian family of finite element steady-state solutions of the diffusion-convection equation, and furthermore is concerned with the stability and the accuracy of on mth kind Hermitian family of finite element solutions. We discuss the stability of the numerical solution based on the fact that the characteristic finite element solution can be expressed approximately as a rational function of cell Peclet number Pe_c ( = uh/k). Moreover, it is shown that by eliminating derivatives and by using the interpolation method over elements a stable solution is obtained over the domain independent of Pe_c for P^1,3, and for P^2,5 the stable solution is obtained for Pe_c less than 44.4.     

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     

Particle statistics in x-ray diffractometry

Summary Expressions are derived for the relative r.m.s. error of diffractometer intensity measurements. The result for stationary specimens:=4R[sin/m w h N _eff]^1/2, withh=1/2(h _F+h_S) andN _eff=cAv/v², is identical with the result of Alexander c.s.¹, except for a slight difference in the numerical constant and in the definition ofw. The value of this parameter is found to lie betweenR+(w_F, w_S)_min (the last term indicating the smallest of the widthsw _F andw _S) andR+1/2(w _F+w_S); it reaches the latter limit in the case of integrated intensities being measured by totalizing counts while scanning through a line. For rotating specimens the particle statistics error turns out to be almost independent ofw. The following approximative formula is established:=6.5R sin/h(mN _eff)^1/2, showing that the factor of improvement resulting from specimen rotation is of the order of (h/w)^1/2.Part. II: Experiments, by P. M. de Wolff, Jeanne M. Taylor and W. Parrish, is in the course of preparation.Work done when on leave of absence (Nov. 1954–May 1955) from Technisch Physische Dienst T.N.O. and T.H., Delft, Netherlands.     

An hp‐adaptive discontinuous Galerkin method for shallow water flows

An adaptive spectral/hp discontinuous Galerkin method for the two‐dimensional shallow water equations is presented. The model uses an orthogonal modal basis of arbitrary polynomial order p defined on unstructured, possibly non‐conforming, triangular elements for the spatial discretization. Based on a simple error indicator constructed by the solutions of approximation order p and p?1, we allow both for the mesh size, h, and polynomial approximation order to dynamically change during the simulation. For the h‐type refinement, the parent element is subdivided into four similar sibling elements. The time‐stepping is performed using a third‐order Runge–Kutta scheme. The performance of the hp‐adaptivity is illustrated for several test cases. It is found that for the case of smooth flows, p‐adaptivity is more efficient than h‐adaptivity with respect to degrees of freedom and computational time. Copyright © 2010 John Wiley & Sons, Ltd.     

The effect of 1,3:2,4-<Emphasis Type="Italic">bis</Emphasis>-<Emphasis Type="Italic">O</Emphasis>-(<Emphasis Type="Italic">p</Emphasis>-methylbenzylidene)-<Emphasis Type="SmallCaps">d</Emphasis>-sorbitol (PDTS) on uniaxial elongational viscosity of polypropylene

We investigated the dynamic viscoelasticity and elongational viscosity of polypropylene (PP) containing 0.5 wt% of 1,3:2,4-bis-O-(p-methylbenzylidene)-d-sorbitol (PDTS). The PP/PDTS system exhibited a sol–gel transition (T _gel) at 193 °C. The critical exponent n was nearly equal to 2/3, in agreement with the value predicted by a percolation theory. This critical gel is due to a three-dimensional network structure of PDTS crystals. The elongational viscosity behavior of neat PP followed the linear viscosity growth function 3η ⁺ (t), where η ⁺ (t) is the shear stress growth function in the linear viscoelastic region. The elongational viscosity of the PP/PDTS system also followed the 3η ⁺ (t) above T _gel but did not follow the 3η ⁺ (t) and exhibited strong strain-softening behavior below T _gel. This strain softening can be attributed to breakage of the network structure of PDTS with a critical stress (σ _c) of about 10⁴ Pa.     

Mechanical quadrature methods and extrapolation for solving nonlinear boundary Helmholtz integral equations

This paper presents mechanical quadrature methods (MQMs) for solving nonlinear boundary Helmholtz integral equations. The methods have high accuracy of order O(h ³) and low computation complexity. Moreover, the mechanical quadrature methods are simple without computing any singular integration. A nonlinear system is constructed by discretizing the nonlinear boundary integral equations. The stability and convergence of the system are proved based on an asymptotical compact theory and the Stepleman theorem. Using the h ³-Richardson extrapolation algorithms (EAs), the accuracy to the order of O(h ⁵) is improved. To slove the nonlinear system, the Newton iteration is discussed extensively by using the Ostrowski fixed point theorem. The efficiency of the algorithms is illustrated by numerical examples.     

A numerical analysis of a class of generalized Boussinesq‐type equations using continuous/discontinuous FEM

An improved class of Boussinesq systems of an arbitrary order using a wave surface elevation and velocity potential formulation is derived. Dissipative effects and wave generation due to a time‐dependent varying seabed are included. Thus, high‐order source functions are considered. For the reduction of the system order and maintenance of some dispersive characteristics of the higher‐order models, an extra O(μ²ⁿ⁺²) term (n ∈ ?) is included in the velocity potential expansion. We introduce a nonlocal continuous/discontinuous Galerkin FEM with inner penalty terms to calculate the numerical solutions of the improved fourth‐order models. The discretization of the spatial variables is made using continuous P₂ Lagrange elements. A predictor‐corrector scheme with an initialization given by an explicit Runge–Kutta method is also used for the time‐variable integration. Moreover, a CFL‐type condition is deduced for the linear problem with a constant bathymetry. To demonstrate the applicability of the model, we considered several test cases. Improved stability is achieved. Copyright © 2011 John Wiley & Sons, Ltd.     

Non-Darcy mixed convection from a vertical plate in saturated porous media-variable surface heat flux

Nonsimilarity solutions for non-Darcy mixed convection from a vertical impermeable surface embedded in a saturated porous medium are presented for variable surface heat flux (VHF) of the power-law form. The entire mixed convection region is divided into two regimes. One region covers the forced convection dominated regime and the other one covers the natural convection dominated regime. The governing equations are first transformed into a dimensionless form by the nonsimilar transformation and then solved by a finite-difference scheme. Computations are based on Keller Box method and a tolerance of iteration of 10⁻⁵ as a criterion for convergence. Three physical aspects are introduced. One measures the strength of mixed convection where the dimensionless parameter Ra^* _x /Pe^3/2 _x characterizes the effect of buoyancy forces on the forced convection; while the parameter Pe _x /Ra^*2/3 _x characterizes the effect of forced flow on the natural convection. The second aspect represents the effect of the inertial resistance where the parameter K′U _∞/ν is found to characterize the effect of inertial force in the forced convection dominated regime, while the parameter (K′U _∞/ν)(Ra^*2/3 _x /Pe _x ) characterizes the effect of inertial force in the natural convection dominated regime. The third aspect is the effect of the heating condition at the wall on the mixed convection, which is presented by m, the power index of the power-law form heating condition. Numerical results for both heating conditions are carried out. Distributions of dimensionless temperature and velocity profiles for both Darcy and non-Darcy models are presented. Received on 26 May 1997     

Transport coefficients of an evaporating liquid drop in creeping flow

A theoretical analysis is made of the heat, mass and momentum transfer from an evaporative liquid sphere which is suddenly introduced into a parallel stream of fluid at a higher temperature. The velocity field around the liquid sphere is assumed to be steady and of the Hadamard-Rybczynski type. Numerical solutions of energy and the vapour mass continuity equations have been carried out using the alternate direction implicit scheme of finite difference method. Temporal histories of the average Nusselt and Sherwood numbers (Nu, Sh) alongwith the drag coefficient (C _D ) during the life time of an evaporating drop have been predicted in terms of the pertinent input parameters, namely, initial and instantaneous Peclet number (Pe _i ,Pe), Lewis number (Le), and the ratio of free stream to initial droplet temperature (T _a /T _i ). Variations of local Nusselt and Sherwood numbers withPe, in the region of steady state evaporation, have also been presented. Values ofNu for steady state droplet evaporation are found to be in fair agreement with the corresponding values evaluated from the empirical equation of Eisenklam [5].Es wurde eine theoretische Untersuchung der Wärme-, Massen- und Impulsübertragung eines verdampfenden kugelförmigen Fluidtropfens, welcher plötzlich in eine gleichgerichtete Fluidströmung höherer Temperatur eingeleitet wird, untersucht. Das Geschwindigkeitsprofil um den Fluidtropfen herum wurde als konstant und als ein Hadamard-Rybczynski-Profil angenommen. Unter Benutzung eines ADI-Schemas der Finiten-Differenzen-Methode wurden numerische Lösungen der Erhaltungsgleichungen für Energie und Dampfmasse gewonnen. Zeitliche Gesetzmäßigkeiten der durchschnittlichen Nusselt und Sherwood-Zahlen (Nu, Sh) und des Widerstandsbeiwertes (C _D ) bis zur vollständigen Verdampfung des Tropfens wurden in Abhängigkeit von den zugehörigen Eingabeparametern nämlich der Anfangs-und momentanen Peclet-Zahl (Pe _i ,Pe) der Lewis-Zahl und dem Verhältnis von freier Strömungstemperatur zur Eintrittstemperatur des Tropfens (T _a /T _i ) berechnet. Ebenso werden die lokalen Nusselt und Sherwood-Zahlen in Abhängigkeit von der Peclet-Zahl im Bereich der stationären Verdampfung dargestellt. Es wurde festgestellt, daß Werte der Nusselt-Zahl im Bereich der stationären Verdampfung von Tropfen in guter Übereinstimmung mit den entsprechenden berechneten Größen aus der empirischen Gleichung von Eisenklam liegen.     

Calculation of convective heat flux in the vicinity of the stagnation point for partially ionized gas flow past a body

From numerical solutions of the boundary layer equations for a four-component gas mixture (E, N⁺, N₂, and N) with gas injection, approximate formulas for the heat flux as a function of the variation of λρ/c_p and h^* across the boundary layer and the magnitude of the objection are obtained (λ is the thermal conductivity of the mixture,ρ is density, c_p is the specific heat, and h^* is the enthalpy of the ideal gas state of the mixture). An effective ambipolar diffusion coefficient D^(a)(i) is introduced, making possible finite formulas for the convective heat fluxes in the “frozen” boundary layer. We study the behavior of these coefficients within the boundary layer. A formula is obtained for convective heat flux to the wall from partially ionized air for a nine-component mixture (E, O⁺, N⁺, NO⁺, O, N, NO, O₂ N₂). Even for simpler four-component gas model three effective ambipolar diffusion coefficients are necessary: $$\begin{gathered} D^{(a)} (A) = D (A, M) D^{(a)} (I) = 2D (A, M), \hfill \\ D^{(a)} (M) = [ 1 + c_e (I)] D(A, M). \hfill \\ \end{gathered} $$ Here D(A, M) is the binary diffusion coefficient of the atoms into molecules, and c_e(I) is the ion concentration at the outer edge of the boundary layer. The assumption of an infinitely large charge-exchange cross section and the other simplifying assumptions used in [1] lead to overestimation of the magnitude of the dimensionless heat flux by 7–15% for the “frozen” boundary layer case.     

Numerical studies of the fingering phenomena for the thin film equation

We present a new interpretation of the fingering phenomena of the thin liquid film layer through numerical investigations. The governing partial differential equation is h_t + (h²?h³)_x = ??·(h³?Δh), which arises in the context of thin liquid films driven by a thermal gradient with a counteracting gravitational force, where h = h(x, y, t) is the liquid film height. A robust and accurate finite difference method is developed for the thin liquid film equation. For the advection part (h²?h³)_x, we use an implicit essentially non‐oscillatory (ENO)‐type scheme and get a good stability property. For the diffusion part ??·(h³?Δh), we use an implicit Euler's method. The resulting nonlinear discrete system is solved by an efficient nonlinear multigrid method. Numerical experiments indicate that higher the film thickness, the faster the film front evolves. The concave front has higher film thickness than the convex front. Therefore, the concave front has higher speed than the convex front and this leads to the fingering phenomena. Copyright © 2010 John Wiley & Sons, Ltd.     

A note on the steady-state advection-diffusion equation

In this note we show that the numerical solution of the advection-diffusion equation can be improved by considering the asymptotic behaviour of its analytical solution. This is accomplished by including a correction term based on the numerical differentiation of the asymptotic (Pe » 1, Pe being the Peclet number) solution. This correction forces the usual oscillations associated with centred schemes to disappear.     

Stability, Instability, and Error of the Force-based Quasicontinuum Approximation

Due to their algorithmic simplicity and high accuracy, force-based model coupling techniques are popular tools in computational physics. For example, the force-based quasicontinuum (QCF) approximation is the only known pointwise consistent quasicontinuum approximation for coupling a general atomistic model with a finite element continuum model. In this paper, we present a detailed stability and error analysis of this method. Our optimal order error estimates provide a theoretical justification for the high accuracy of the QCF approximation: they clearly demonstrate that the computational efficiency of continuum modeling can be utilized without a significant loss of accuracy if defects are captured in the atomistic region. The main challenge we need to overcome is the fact that the linearized QCF operator is typically not positive definite. Moreover, we prove that no uniform inf-sup stability condition holds for discrete versions of the W ^1,p -W ^1,q “duality pairing” with 1/p + 1/q = 1, if 1 ≤ p < ∞. However, we were able to establish an inf-sup stability condition for a discrete version of the W ^1,∞-W ^1,1 “duality pairing” which leads to optimal order error estimates in a discrete W ^1,∞-norm.     

High Péclet number heat exchange between cocurrent streams

Summary  This paper concentrates on the analysis of the heat transfer between two cocurrent laminar flows in parallel channels. For high values of the Péclet number Pe, a boundary layer arises near the wall separating the streams. Matched asymptotic expansions (MAE) are used to obtain approximate solutions. We consider arbitrary inlet temperatures and derive higher-order corrections of the boundary problem. The separating wall is supposed to be sufficiently thin to neglect the heat conduction in it. Analyticity and adiabatic conditions at the outer walls impose restrictions on the inlet temperatures. It turns out, however, that only the inlet temperatures at the wall separating the two fluids enter the leading-order problem. The Nusselt numbers thus calculated are in the leading order proportional to (Pe/x)^1/3, where x is the stream-wise coordinate. An estimate of the thickness of the separating wall to validate the MAE approach is obtained. It is also demonstrated that the MAE analysis is unable to describe the heat exchange of counterflowing fluids. Received 9 June 1999; accepted for publication 17 November 1999     

Third-order systems: Periodicity conditions

Recently a third-order existence theorem has been proven to establish the sufficient conditions of periodicity for the most general third-order ordinary differential equation

x+f(t,x,x^′,x^″)=0