Some generalizations of the Schwarz-Christoffel mapping formula

Summary The Schwarz-Christoffel formula for the mapping of a polygon in thez-plane on an upper half-plane (thew-plane) is extended to deal with singlyconnected domains of quite general shape. The mapping problem in the general ease is shown to depend on the solution of an awkward integrodifferential equation and an iterative method of finding this solution is indicated. Two further generalizations are made to the formula; these are (i) the boundary of the singly-connected domain in thez-plane is mapped on to afinite interval of the real axis of thew-plane instead of the whole of it, and (ii) the formula is extended to deal with doubly-connected domains.Paper, read at the first annual general meeting of the Australian Mathematical Society at Sydney, August, 1957.     

Use of a Monte Carlo PDF method in a study of the influence of turbulent fluctuations on selectivity in a jet-stirred reactor

A model for single-phase turbulent reacting flow is presented and a solution algorithm is described. The model combines the standardk - model for the velocity field with a transport equation for the probability density function (PDF) of the thermochemical variables. In this equation terms describing spatial transport by velocity fluctuations and mixing on the smallest scales are modelled. The essential advantage of this approach is that the effect of nonlinear kinetics appears in closed form and that the influence of turbulent fluctuations on mean reaction rates is included. A stochastic algorithm for the solution of the PDF transport equation, essentially due to Pope, is described. Cylindrical symmetry is assumed. The PDF is represented by ensembles ofN representative values of the thermochemical variables in each cell of a nonuniform finite-difference grid and operations on these elements representing convection, diffusion, mixing and reaction are derived. A simplified model and solution algorithm which neglects the influence of turbulent fluctuations on mean reaction rates is also described. Both algorithms are applied to a selectivity problem in a real reactor studied earlier by Liu and Barkelew. Spatial profiles of mean species mole fractions and of relative selectivity to the target product are obtained. The profiles are clearly different in both models but at the end of the reactor the same selectivity is predicted.Presented at the Shell Conference on Computational Fluid Dynamics for Petrochemical Process Equipment, Hoenderloo, December 10–12, 1989.     

Laminar flow stability in a mass force field

A study is made of the stability of a certain class of velocity profiles in a two-dimensional channel on the basis of a new numerical algorithm for the solution of the Orr-Sommerfeld equation. A profile is found which has (in a certain sense) the property of maximum stability. In form this profile is very similar to the averaged velocity profile of turbulent flow in a two-dimensional pipe.     

Parabolic method of solving problems of the flow of a sonic gas stream around a thin symmetric profile

A parabolic method consisting of replacement of the stream acceleration ?_xx in the non-linear member of (1.1) by a specially chosen constant has been proposed [1] for the solution of the mixed-type transonic equation with boundary conditions on the profile, and the solution of the linear parabolic-type equation obtained can be considered as a certain approximation to the solution of the initial problem. An improvement of the parabolic method is the method of local linearization [2] (see [3] also), in which the acceleration ?_xx fixed from the beginning is replaced by a function of the coordinate x which satisfies some condition. An ordinary first-order differential equation is obtained for the velocity distribution along the profile in [2]. Another method of “defrosting” the acceleration ?_xx “frozen” from the beginning is proposed in this paper; a second-order ordinary differential equation is obtained for the velocity on the profile, which permits getting rid of some disadvantages of the local linearization method. Several solutions of (2.3) are presented in comparison to known exact solutions.     

Velocity-defect scaling for turbulent boundary layers with a range of relative roughness

Velocity profile measurements in zero pressure gradient, turbulent boundary layer flow were made on a smooth wall and on two types of rough walls with a wide range of roughness heights. The ratio of the boundary layer thickness (δ) to the roughness height (k) was 16≤δ/k≤110 in the present study, while the ratio of δ to the equivalent sand roughness height (k _s) ranged from 6≤δ/k _s≤91. The results show that the mean velocity profiles for all the test surfaces agree within experimental uncertainty in velocity-defect form in the overlap and outer layer when normalized by the friction velocity obtained using two different methods. The velocity-defect profiles also agree when normalized with the velocity scale proposed by Zagarola and Smits (J Fluid Mech 373:33–70, 1998). The results provide evidence that roughness effects on the mean flow are confined to the inner layer, and outer layer similarity of the mean velocity profile applies even for relatively large roughness.     

High-Field Limit for the Vlasov-Poisson-Fokker-Planck System

This paper is concerned with the analysis of the stability of the Vlasov-Poisson-Fokker-Planck system with respect to the physical constants. If the scaled thermal mean free path converges to zero and the scaled thermal velocity remains constant, then a hyperbolic limit or equivalently a high-field limit equation is obtained for the mass density. The passage to the limit as well as the existence and uniqueness of solutions of the limit equation in L ¹, global or local in time, are analyzed according to the electrostatic or gravitational character of the field and to the space dimension. In the one-dimensional case a new concept of global solution is introduced. For the gravitational field this concept is shown to be equivalent to the concept of entropy solutions of hyperbolic systems of conservation laws. Accepted December 1, 2000?Published online April 23, 2001     

Use of tensor polynomials for constructing an equation for turbulence scale in semiempirical models

Thek-kl-model of turbulence based on the equation of second two-point moments (pointsA andB) of the fluctuating velocity field is presented. The second and third moments entering into this equation are expressed using polynomials whose terms are products of the tensor components characterizing a given turbulent motion and scalar functions of the distanceAB. ForAB=0 the equation obtained gives thek-turbulence energy balance equation and, on being integrated overAB from 0 to ∞, the transport equation for thekl-quantity (l is the integral turbulence scale). The model is used for calculating mixing layers, plane and circular jets, the wake behind a cylinder, tube and channel flows, and the boundary layer on a plate. The results of all the calculations agree well with the experimental data for a single set of empirical coefficients. St. Petersburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 51–64, July–August, 1994.     

Testing of Model Equations for the Mean Dissipation using Kolmogorov Flows

Direct Numerical Simulations (DNS) of Kolmogorov flows are performed at three different Reynolds numbers Re _λ between 110 and 190 by imposing a mean velocity profile in y-direction of the form U(y) = F sin(y) in a periodic box of volume (2π)³. After a few integral times the turbulent flow turns out to be statistically steady. Profiles of mean quantities are then obtained by averaging over planes at constant y. Based on these profiles two different model equations for the mean dissipation ε in the context of two-equation RANS (Reynolds Averaged Navier–Stokes) modelling of turbulence are compared to each other. The high Reynolds number version of the k-ε-model (Jones and Launder, Int J Heat Mass Transfer 15:301–314, 1972), to be called the standard model and a new model by Menter et al. (2006), to be called the Menter–Egorov model, are tested against the DNS results. Both models are solved numerically and it is found that the standard model does not provide a steady solution for the present case, while the Menter–Egorov model does. In addition a fairly good quantitative agreement of the model solution and the DNS data is found for the averaged profiles of the kinetic energy k and the dissipation ε. Furthermore, an analysis based on flow-inherent geometries, called dissipation elements (Wang and Peters, J Fluid Mech 608:113–138, 2008), is used to examine the Menter–Egorov ε model equation. An expression for the evolution of ε is derived by taking appropriate moments of the equation for the evolution of the probability density function (pdf) of the length of dissipation elements. A term-by-term comparison with the model equation allows a prediction of the constants, which with increasing Reynolds number approach the empirical values.     

An approximate solution for the Couette–Poiseuille flow of the Giesekus model between parallel plates

An approximate analytical solution is derived for the Couette–Poiseuille flow of a nonlinear viscoelastic fluid obeying the Giesekus constitutive equation between parallel plates for the case where the upper plate moves at constant velocity, and the lower one is at rest. Validity of this approximation is examined by comparison to the exact solution during a parametric study. The influence of Deborah number (De) and Giesekus model parameter (α) on the velocity profile, normal stress, and friction factor are investigated. Results show strong effects of viscoelastic parameters on velocity profile and normal stress. In addition, five velocity profile types were obtained for different values of α, De, and the dimensionless pressure gradient (G).     

Pulsatile electroosmotic flow of a Maxwell fluid in a parallel flat plate microchannel with asymmetric zeta potentials

The pulsatile electroosmotic flow (PEOF) of a Maxwell fluid in a parallel flat plate microchannel with asymmetric wall zeta potentials is theoretically analyzed. By combining the linear Maxwell viscoelastic model, the Cauchy equation, and the electric field solution obtained from the linearized Poisson-Boltzmann equation, a hyperbolic partial differential equation is obtained to derive the flow field. The PEOF is controlled by the angular Reynolds number, the ratio of the zeta potentials of the microchannel walls, the electrokinetic parameter, and the elasticity number. The main results obtained from this analysis show strong oscillations in the velocity profiles when the values of the elasticity number and the angular Reynolds number increase due to the competition among the elastic, viscous, inertial, and electric forces in the flow.     

On a nonlinear hyperbolic variational equation: II. The zero-viscosity and dispersion limits

We consider viscosity and dispersion regularizations of the nonlinear hyperbolic partial differential equation (u _t+uu_x)_x=1/2u _x ² with the simplest initial data such that u _x blows up in finite time. We prove that the zero-viscosity limit selects a unique global weak solution of the partial differential equation without viscosity. We also present numerical experiments which indicate that the zero-dispersion limit selects a different global weak solution of the same initial-value problem.     

On the numerical stability of mixed finite-element methods for viscoelastic flows governed by differential constitutive equations

Mixed finite-element methods for computation of viscoelastic flows governed by differential constitutive equations vary by the polynomial approximations used for the velocity, pressure, and stress fields, and by the weighted residual methods used to discretize the momentum, continuity, and constitutive equations. This paper focuses on computation of the linear stability of the planar Couette flow as a test of the numerical stability for solution of the upper-convected Maxwell model. Previous theoretical results prove this inertialess flow to be always stable, but that accurate calculation is difficult at high De because eigenvalues with fine spatial structure and high temporal frequency approach neutral stability. Computations with the much used biquadratic finite-element approximations for velocity and deviatoric stress and bilinear interpolation for pressure demonstrate numerical instability beyond a critical value of De for either the explicitly elliptic momentum equation (EEME) or elastic-viscous split-stress (EVSS) formulations, applying Galerkin's method for solution of the momentum and continuity equations, and using streamline upwind Petrov-Galerkin (SUPG) method for solution of the hyperbolic constitutive equation. The disturbance that causes the instability is concentrated near the stationary streamline of the base flow. The removal of this instability in a slightly modified form of the EEME formulation suggests that the instability results from coupling the approximations to the variables. A new mixed finite-element method, EVSS-G, is presented that includes smooth interpolation of the velocity gradients in the constitutive equation that is compatible with bilinear interpolation of the stress field. This formulation is tested with SUPG, streamline upwinding (SU), and Galerkin least squares (GLS) discretization of the constitutive equation. The EVSS-G/SUPG and EVSS-G/SU do not have the numerical instability described above; linear stability calculations for planar Couette flow are stable to values of De in excess of 50 and converge with mesh and time step. Calculations for the steady-state flow and its linear stability for a sphere falling in a tube demonstrate the appearance of linear instability to a time-periodic instability simultaneously with the apparent loss of existence of the steady-state solution. The instability appears as finely structured secondary cells that move from the front to the back of the sphere.Financial support for this research was given by the National Science Foundation, the Office of Naval Research, and the Defense Research Projects Agency. Computational resources were supplied by a grant from the Pittsburgh National Supercomputer Center and by the MIT Supercomputer Facility.     

Solution of the heat-transfer equation for equilibrium turbulent boundary layers when the temperature distribution of the streamlined surface is arbitrary

We give an approximate solution of the heat-transfer equation for equilibrium turbulent boundary layers for which the velocity distribution and the coefficient of turbulent viscosity can be described by functions of two parameters. In [1–4] equilibrium turbulent boundary layers characterized by a constant dimensionless pressure gradient were investigated. The $$\beta = \frac{{\delta ^{* \circ } }}{{\tau _w ^ \circ }}\left( {\frac{{dP}}{{dx^ \circ }}} \right)$$ profile of the velocity defect was calculated in [4] for such layers throughout the whole range ?0.5≤β≤∞, while a method was indicated in [5] for combining the defect velocity profiles with the universal profiles of the wall law, and a composite function defining the coefficient of turbulent viscosity was proposed. In this paper we construct the solution of the heat-transfer equation for equilibrium boundary layers under the assumption that the velocity distribution in the layer and the coefficient of turbulent viscosity are described by functions, obtained in [4, 5], of the dimensionless coordinateη=y/Δ, depending on two parametersβ and Re_*, while the turbulent Prandtl number Pr_t is either constant or is also a known function of η and the parametersβ and Re_*. The temperature of the surface T_w(x) is assumed to be an arbitrary function of the longitudinal coordinate and the solution is constructed in the form of series in the form parameters containing the derivatives of T_w(x). These form parameters are similar to those used in [6–9] to construct exact solutions of the equations of the laminar boundary layer.     

Semi-analytic solution of the two-dimensional turbulent energy equation in round tubes using an analytic velocity profile and its experimental validation

 A semi-analytic solution of the temperature development of single-phase, turbulent viscous flows inside smooth round tubes is performed. The special feature of the theoretical analysis revolves around two single universal functions of analytic form for the accurate characterization of the turbulent diffusivity of momentum and the turbulent velocity profile in the entire cross-section of a round tube. Using this valuable information that emanates from the analytic solution of the one-dimensional momentum balance equation, the two-dimensional energy balance equation was reformulated into an adjoint system of ordinary differential equations of first–order with constant coefficients. Each equation in the system of differential equations governs the axial variation of the average temperature of a finite volume of fluid of annular shape. Exploiting the linearity of the system of differential equations, an analytic solution of it was obtained via the matrix eigenvalue method with LAPACK, a library of Fortran 77 subroutines for numerical linear algebra. Reliable series have been determined for the axial variation of the two thermal quantities of importance: (a) the time-mean bulk temperature and (b) the local Nusselt number. The semi-analytic nature of the local Nusselt number distribution is advantageous because it may be viewed as an analytic-based correlation equation. Prediction of the local Nusselt numbers for turbulent air flows compare satisfactorily with the comprehensive correlation equations and the abundant experimental data that are accessible from the literature. The air flows are regulated by a wide spectrum of turbulent Reynolds numbers. Received on 4 June 2001 RID="★" ID="★" Current address Mechanical Engineering Dept. The University of Vermont Burlington, VT 05405, USA     

On a nonlinear hyperbolic variational equation: I. Global existence of weak solutions

We study the nonlinear hyperbolic partial differential equation, (u _t+uu_x)_x=1/2u _x ² . This partial differential equation is the canonical asymptotic equation for weakly nonlinear solutions of a class of hyperbolic equations derived from variational principles. In particular, it describes waves in a massive director field of a nematic liquid crystal.Global smooth solutions of the partial differential equation do not exist, since their derivatives blow up in finite time, while weak solutions are not unique. We therefore define two distinct classes of admissible weak solutions, which we call dissipative and conservative solutions. We prove the global existence of each type of admissible weak solution, provided that the derivative of the initial data has bounded variation and compact support. These solutions remain continuous, despite the fact that their derivatives blow up.There are no a priori estimates on the second derivatives in any L ^p space, so the existence of weak solutions cannot be deduced by using Sobolev-type arguments. Instead, we prove existence by establishing detailed estimates on the blowup singularity for explicit approximate solutions of the partial differential equation.We also describe the qualitative properties of the partial differential equation, including a comparison with the Burgers equation for inviscid fluids and a number of illustrative examples of explicit solutions. We show that conservative weak solutions are obtained as a limit of solutions obtained by the regularized method of characteristics, and we prove that the large-time asymptotic behavior of dissipative solutions is a special piecewise linear solution which we call a kink-wave.     

Phenomenological Modeling of Electrorheological Fluids with an Extended Casson-Model

In this paper we propose a phenomenological theory for electrorheological fluids. In general these are suspensions which undergo dramatic changes in their material properties if they are exposed to an electric field. In the context of continuum mechanics these fluids can be modeled as non-Newtonian fluids. Recalling the governing equations of rational thermodynamics and electrodynamics of moving media (Maxwell-Minkowski-equations), we derive suitable governing equations of electrorheology using essentially two assumptions concerning magnetic quantities. Furthermore we introduce a 3-dimensional nonlinear constitutive equation for the Cauchy stress tensor which is an extension of the model proposed by Ružička (see [14]). Assuming a viscometric flow, we compare the shear stress of our model with other well known models and fit the parameters by using measurements that were obtained in a rotational viscometer. Excellent agreement between model and measurements is achieved. On the basis of these results we propose a 3-dimensional model, the so-called extended Casson -model. This model is investigated further for a channel flow configuration with a homogeneous electric field. We determine analytical solutions for the electric field, the velocity and the volumetric flow rate and illustrate the velocity profiles and the predicted pressure drop. The velocity profiles are flattened compared to parabolic profiles and become more flat if the electric field increases. Received March 21, 2000     

Approximate solution of the boundary problem for the equations of a stationary electrostatic charged-particle beam

The solution is given of the equations of a three-dimensional stationary electrostatic beam of charged particles of like sign filling the region between two nearby curvilinear surfaces. We assume that the flow is nonrotational and nonrelativistic and that the velocity vector is a single-valued function. The solution is constructed in the form of an asymptotic series in powers of the small parameter , which is the ratio of the characteristic transverse (a) and longitudinal (l) dimensions of the problem. The first dimension is taken to be the distance between the electrodes, andl defines the scale at which the geometric and physical parameters (emitter curvature, electric field E on the emitter, and the emission current density J) change noticeably. The emission regimes limited by the space charge (-regime), temperature (T-regime), and the case of nonzero initial velocity (U-regime) are studied. The asymptotic behavior is given by the formulas for the corresponding one-dimensional flow between parallel surface.The solution of the boundary problem for emission in the-regime reduces to determination of the emission current density J for fixed electrode geometry and given accelerating voltage. The corresponding formulas are presented, retaining terms of order ³.Two approximations with respect to are performed for the T- and U-regimes. Here the unknown quantity for given properties of the emitting surface (J) will be the electric field E.The results provided by the constructed expansions are compared with the exact solution for flow from a planar emitter along circular trajectories [1]. As an example we examine the two-dimensional problem of flow between two nearby circular cylindrical electrodes with disruption of the coaxiality.The conventional tensor notations are used.     

Propagating phase boundaries: Formulation of the problem and existence via the Glimm method

This paper treats the hyperbolic-elliptic system of two conservation laws which describes the dynamics of an elastic material having a non-monotone strain-stress function. FollowingAbeyaratne &Knowles, we propose a notion of admissible weak solution for this system in the class of functions of bounded variation. The formulation includes an entropy inequality, a kinetic relation (imposed along any subsonic phase boundary) and an initiation criterion (for the appearance of new phase boundaries). We prove theL ¹-continuous dependence of the solution to the Riemann problem. Our main result yields the existence and the stability of propagating phase boundaries. The proofs are based onGlimm's scheme and in particular on the techniques ofGlimm andLax. In order to deal with the kinetic relation, we prove a result of pointwise convergence of the phase boundary.     

Turbulent friction in the flow of liquid in tubes and channels

A calculating relationship is presented for turbulent flow; it takes a unique form over the whole cross section of the flow. A relationship is also derived between turbulent friction and the mean velocity profile on the basis of the equation for the maximum turbulent friction, which follows directly from the equation of motion. The proportionality factor in this relationship is obtained with due allowance for twelve boundary conditions relating to the turbulent flow, the mean velocity, and their derivatives. The resultant turbulent-friction profiles agree with the experimental data of Laufer. The profile parameters may be related to the Reynolds number.Leningrad. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 140–145, March–April, 1972.     

Hyperbolic Principal Subsystems: Entropy Convexity and Subcharacteristic Conditions

We consider a system of N balance laws compatible with an entropy principle and convex entropy density. Using the special symmetric form induced by the main field, we define the concept of principal subsystem associated with the system. We prove that the 2 ^N −2 principal subsystems are also symmetric hyperbolic and satisfy a subentropy law. Moreover we can verify that for each principal subsystem the maximum (minimum) characteristic velocity is not larger (smaller) than the maximum (minimum) characteristic velocity of the full system. These are the subcharacteristic conditions. We present some simple examples in the case of the Euler fluid. Then in the case of dissipative hyperbolic systems we consider an equilibrium principal subsystem and we discuss the consequences in the setting of extended thermodynamics. Finally in the moments approach to the Boltzmann equation we prove, as a consequence of the previous result, that the maximum characteristic velocity evaluated at the equilibrium state does not decrease when the number of moments increases. (Accepted October 6, 1995)