Existence and Stability of Solutions for Steady Flows of Fibre Suspension Flows

We establish existence, uniqueness, convergence and stability of solutions to the equations of steady flows of fibre suspension flows. The existence of a unique steady solution is proven by using an iterative scheme. One of the restrictions imposed on the data confirms a well known fact proven in Galdi and Reddy (J Non-Newtonian Fluid Mech 83:205–230, 1999), Munganga and Reddy (Math Models Methods Appl Sci 12:1177–1203, 2002) and Munganga et al. (J Non-Newtonian fluid Mech 92:135–150, 2000) that the particle number N _p must be less than 35/2. Exact solutions are calculated for Couette and Poiseuille flows. Solutions of Poiseuille flows are shown to be more accurate than those of Couette flow when a time perturbation is considered.     

Existence and Stability of Supersonic Euler Flows Past Lipschitz Wedges

It is well known that, when the vertex angle of a straight wedge is less than the critical angle, there exists a shock-front emanating from the wedge vertex so that the constant states on both sides of the shock-front are supersonic. Since the shock-front at the vertex is usually strong, especially when the vertex angle of the wedge is large, then a global flow is physically required to be governed by the isentropic or adiabatic Euler equations. In this paper, we systematically study two-dimensional steady supersonic Euler (i.e. nonpotential) flows past Lipschitz wedges and establish the existence and stability of supersonic Euler flows when the total variation of the tangent angle functions along the wedge boundaries is suitably small. We develop a modified Glimm difference scheme and identify a Glimm-type functional, by naturally incorporating the Lipschitz wedge boundary and the strong shock-front and by tracing the interaction not only between the boundary and weak waves, but also between the strong shock-front and weak waves, to obtain the required BV estimates. These estimates are then employed to establish the convergence of both approximate solutions to a global entropy solution and corresponding approximate strong shock-fronts emanating from the vertex to the strong shock-front of the entropy solution. The regularity of strong shock-fronts emanating from the wedge vertex and the asymptotic stability of entropy solutions in the flow direction are also established.     

Existence and Stability of Multidimensional Shock Fronts in the Vanishing Viscosity Limit

In this paper we present a new approach to the study of linear and nonlinear stability of inviscid multidimensional shock waves under small viscosity perturbation, yielding optimal estimates and eventually an extension to the viscous case of the celebrated theorem of Majda on existence and stability of multidimensional shock waves. More precisely, given a curved Lax shock solution u⁰ to a hyperbolic system of conservation laws, we construct nearby viscous shock solutions u to a parabolic viscous perturbation of the hyperbolic system which converge to u⁰ as viscosity 0 and satisfy an appropriate (conormal) version of Majdas stability estimate.The main new feature of the paper is the derivation of maximal and optimal estimates for the linearization of the parabolic problem about a highly singular approximate solution. These estimates are more robust than the singular estimates obtained in our previous work, and permit us to remove an earlier assumption limiting how much the inviscid shock we start with can deviate from flatness.The key to the new approach is to work with the full linearization of the parabolic problem, that is, the linearization with respect to both u and the unknown viscous front, and to allow variation of the front at all stages – not only in the construction of the approximate solution as done in previous work, but also in the final error equation. After reformulating the problem as a transmission problem, we show that the linearized problem can be desingularized and optimal estimates obtained by adding an appropriate extra boundary condition involving the front. The extra condition determines a local evolution rule for the viscous front.Acknowledgement The work of O.G. was partially supported by European network HYKE, HPRN-CT-2002-00282. The work of G.M. was partially supported by European network HYKE, HPRN-CT-2002-00282. The work of M.W. was partially supported by NSF grant DMS-0070684. The work of K.Z. was partially supported by NSF grant DMS-0070765.     

Extension of Ovsyannikov's Analytical Solutions to Transonic Flows

The solution of the equation of the velocity potential of a steady axisymmetric ideal-gas flow in the neighborhood of a given point at the axis of symmetry in the form of a double series in powers of the distance to the axis of symmetry and its logarithm is considered. Recurrent chains of equations with arbitrariness in two analytical functions of the streamwise variable are obtained for coefficients of the series. Convergence of the constructed series is proved by the method of special majorants. The theorem of existence and uniqueness of the solution of the initial-boundary problem for this nonlinear differential equation in partial derivatives with a singularity at the axis of symmetry is obtained as an analog of Kovalevskaya's and Ovsyannikov's theorems. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 14–25, November–December, 2005.     

Structure of Time-Dependent Transonic Flows in Plane Channels

The evolution and interaction of supersonic zones and the shock waves closing them are considered on the basis of the Lin-Reissner-Tsien equation for flow in plane channels with a local bottleneck.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, 2005, pp. 168–179.Original Russian Text Copyright © 2005 by Bibik, Duesperov, and Popov.     

以压力为基本求解变量数值模拟粘性超、跨音速流动

应用以压力为基本求解变量的SIMPLE方法 ,对一双喉喷管中的层流超音速流动和一扩压器中的紊流跨音速流动进行了数值计算。计算结果显示 ,本文的计算结果与文献数据及实验结果相符很好。表明本文方法对可压缩流动有很高的模拟精度。进而表明经过可压缩推广的SIMPLE方法适用于任何马赫数的流动计算     

Slow Viscous Flow through Arbitrary Triangular Tubes and Its Application in Modelling Porous Media Flows

In this article we extend the analytical solution for viscous flow in an equilateral triangular tube to irregular triangular tubes. The validity of the solution is examined and proved by comparison with the numerical simulation results. With the new extension of the equations, the average velocity of viscous flow through an arbitrary triangular tube can be readily calculated as a function of inscribed radius of the triangular cross-section of the tube, and the volumetric flow rate is computed as a function of inscribed radius and the cross- sectional area. To illustrate the advantages in using an arbitrary triangular tube for modelling a porous medium, we present examples of tube bundle models, which give a wide range of variation in porosity and permeability with a fixed pore size distribution, by using various combinations of three types of triangular tubes.     

Subsonic Euler Flows with Large Vorticity Through an Infinitely Long Axisymmetric Nozzle

This paper is a sequel to the earlier work Du and Duan (J Diff Equ 250:813–847, 2011) on well-posedness of steady subsonic Euler flows through infinitely long three-dimensional axisymmetric nozzles. In Du and Duan (J Diff Equ 250:813–847, 2011), the authors showed the existence and uniqueness of the global subsonic Euler flows through an infinitely long axisymmetric nozzle, when the variation of Bernoulli’s function in the upstream is sufficiently small and the mass flux of the incoming flow is less than some critical value. The smallness of the variation of Bernoulli’s function in the upstream prevents the attendance of the possible singularity in the nozzles, however, at the same time it also leads that the vorticity of the ideal flow is sufficiently small in the whole nozzle and the flows are indeed adjacent to axisymmetric potential flows. The purpose of this paper is to investigate the effects of the vorticity for the smooth subsonic ideal flows in infinitely long axisymmetric nozzles. We modify the formulation of the problem in the previous work Du and Duan (J Diff Equ 250:813–847, 2011) and the existence and uniqueness results on the smooth subsonic ideal polytropic flows in infinitely long axisymmetric nozzles without the restriction on the smallness of the vorticity are shown in this paper.     

跨音速压气机级的三维周期性非定常流动计算

对跨音速压气机级动静叶排相干形成的三维非定常流场进行了数值研究，利用时间推进ＬＵ－ＳＧＳ稳式迭代法求解三维非定常欧拉方程，对流项采用高分辨率ＮＮＤ格式离散。对某压气机第一级动静叶排相干非定常流场的计算结果表明，本文方法不仅在收敛速度上明显地优于一般显式方法，而且保持了流场中激波的高分辨率，适于推广到计算量巨大的的多级轴流压气机三维非定常流场的数值分析问题     

Global Uniqueness of Transonic Shocks in Two-Dimensional Steady Compressible Euler Flows

We prove that for the two-dimensional steady complete compressible Euler system, with given uniform upcoming supersonic flows, the following three fundamental flow patterns (special solutions) in gas dynamics involving transonic shocks are all unique in the class of piecewise C ¹ smooth functions, under appropriate conditions on the downstream subsonic flows: (i) the normal transonic shocks in a straight duct with finite or infinite length, after fixing a point the shock-front passing through; (ii) the oblique transonic shocks attached to an infinite wedge; (iii) a flat Mach configuration containing one supersonic shock, two transonic shocks, and a contact discontinuity, after fixing a point where the four discontinuities intersect. These special solutions are constructed traditionally under the assumption that they are piecewise constant, and they have played important roles in the studies of mathematical gas dynamics. Our results show that the assumption of a piecewise constant can be replaced by some weaker assumptions on the downstream subsonic flows, which are sufficient to uniquely determine these special solutions. Mathematically, these are uniqueness results on solutions of free boundary problems of a quasi-linear system of elliptic-hyperbolic composite-mixed type in bounded or unbounded planar domains, without any assumptions on smallness. The proof relies on an elliptic system of pressure p and the tangent of the flow angle w = v/u obtained by decomposition of the Euler system in Lagrangian coordinates, and a newly developed method for the L ^∞ estimate that is independent of the free boundaries, by combining the maximum principles of elliptic equations, and careful analysis of the shock polar applied on the (maybe curved) shock-fronts.     

Profiling the Supersonic Part of a Plug Nozzle with a Nonuniform Transonic Flow

The effect of transonic flow nonuniformity on the profiling of optimal plug nozzles is studied in the inviscid gas approximation. Sonic and supersonic regions providing maximum thrust for given nozzle dimensions and a given outer pressure are designed for given subsonic contours and calculated nonuniform transonic flows. As in the case of uniform flow on a cylindrical sonic surface, the initial regions of the designed contours satisfy the condition that in these regions the flow Mach number is unity or near-unity. In all the examples calculated, the optimal plug nozzles produce a greater thrust than the optimal axisymmetric and annular nozzles with a near-axial flow for the same lengths and the same gas flow rates through the nozzle. It is established that contouring without regard for transonic flow nonuniformity can result in considerable thrust losses. However, these losses are due only to a decrease in the flow rate, while the specific thrust may even increase slightly.     

Local Existence and Finite-time Blow-up in Multidimensional Radiation Hydrodynamics

We first prove the local existence of smooth solutions to the Cauchy problem for the equations of multidimensional radiation hydrodynamics which are a hyperbolic-Boltzmann coupled system. Then, we show that a smooth solution will blow up in finite time if the initial data are large. Moreover, the property of finite propagation speed is obtained simultaneously. Supported by the NSF of Jiangxi Province, the Special Funds for Major State Basic Research Projects, the NSFC (Grant No. 10225105) and the CAEP (Grant No. 2003-R-02).     

翼型跨音速流动计算和外形设计的变域变分有限元方法

在气体动力学问题研究中经常会碰到诸如激波、翼型设计等未知界面问题。未知界面的存在为该类问题的理论分析和数值求解带来了很大困难。刘高联针对未知界面问题发展了一种变域变分有限元方法,该方法将未知界面看作是一个变化区域的边界,采用变域变分将未知界面结合在变分泛函中,使其与求解流场的控制方程结合起来,从而将未知界面的求解和流场的求解完全耦合进行,因而是一种处理未知界面的独特工具,极适合于气动外形的设计求解。本文运用变域变分有限元方法对翼型跨音速流动正、反命题进行了数值研究。由于在跨音速翼型绕流中存在激波,所以为了得到压缩激波解,采用了“人工密度”办法。几个算例均得到了满意的计算结果和设计结果,证明了本文方法的有效性和优越性。     

Assessment of Uncertainties in Modeling of Laminar to Turbulent Transition for Transonic Flows

The effect of physical variability and uncertainty in model correlations on laminar-turbulent transition in transonic flows is computed using two different Stochastic Collocation methods. Physical variability in the boundary conditions is first investigated for a flow over a flat plate with and without pressure gradient to quantify the uncertainties on the skin friction distribution along the plate surface. Since the laboratory conditions for the flat plate test cases are well defined and the applied transition model has been tuned for these cases, good agreement with experiments is achieved and the variability in the output is low. The second investigated cases exhibit boundary layer transition on the surface of a highly loaded turbine guide vane under transonic flow conditions. Comparisons between the predicted and measured wall heat transfer are used to quantify uncertainties in the free stream turbulence and the model correlations that accounts for compressibility effects on the onset and extension of the bypass transition. The computational results show that the uncertainties have a significant impact on the transition location for the turbine guide vane simulations and, consequently, on the reliability of the predictions for compressible flows. The output uncertainty accounts to a large extent for the difference between the deterministic simulation and the experiments. The results from the Simplex Stochastic Collocation method are computationally more efficient than those of the Stochastic Collocation based on Clenshaw–Curtis quadrature.     

Viscous Standing Asymptotic States of Isentropic Compressible Flows Through a Nozzle

In this work we consider a viscous regularization of a well-known one-dimensional model for isentropic viscous compressible flows through a nozzle. For the existence and multiplicity of standing asymptotic states for a certain type of ducts, a complete analysis in a framework of dynamical systems is provided. As an application of the geometric singular perturbation theory, we show that all standing asymptotic states admit viscous profiles.     

Efficient Generation of Initial- and Inflow-Conditions for Transient Turbulent Flows in Arbitrary Geometries

A method is presented to artificially generate initial conditions and transient inflow-conditions for DNS and LES. It creates velocity fields that satisfy a given Reynolds-stress-tensor and length-scale. Compared to existing approaches, the new method features greater flexibility, efficiency and applicability. It is well suited for the complex geometries and for the arbitrary grids that occur in technical applications. This is demonstrated in connection with the generation of initial data for an internal combustion engine. To assess the accuracy and efficiency of the new approach, it is applied to the test-case of a non-premixed jet-flame, which is known to be sensitive to transient inflow-data.     

Existence of Stationary Supersonic Flows Past a Pointed Body

In this paper we study the mathematical aspects of the stationary supersonic flow past a non-axisymmetric curved pointed body. The flow is described by a steady potential flow equation, which is a quasilinear hyperbolic equation of second order. We prove the local existence of the solution to this problem with a pointed shock attached at the tip of the pointed body, provided the pointed body is a perturbation of a circular cone, and the vertex angle of the approximate cone of the pointed body is less than a critical value. The solution is smooth in between the shock and the surface of the body. Consequently, such a structure of flow near the tip of the pointed body and its stability is verified mathematically. Accepted October 13, 2000?Published online January 22, 2001     

Existence of Finite-Time Hyperbolic Trajectories for Planar Hamiltonian Flows

We present a shadowing theorem on the existence of hyperbolic trajectories on finite time intervals based on the EPH partition by George Haller and apply it to an example which is inspired by the problem of two-dimensional symmetric vortex merger.     

Stability of Semi-Discrete Shock Profiles by Means of an Evans Function in Infinite Dimensions

Semi-discrete shock profiles are traveling wave solutions of hyperbolic systems of conservation laws under discretization in space. The existence of semi-discrete shocks has been investigated in earlier papers. Here the spectral stability of those nonlinear waves is addressed, and formulated in terms of a variational delay differential operator. Constructing a generalized Evans function, in infinite dimensions, it is shown how to derive stability criteria. Some examples are given when the criterion is fully explicit, e.g., for extreme Lax shocks. Additionally, connection is made with the alternative approach proposed by Chow, Mallet-Paret, and Shen (Journal of Differential Equations 1998), regarding the stability of traveling waves in general Lattice Dynamical Systems.     

Stability of Creeping Flows of Maxwell Fluids

We derive rigorous criteria for the linear stability of viscoelastic flows under periodic boundary conditions. The results are based on recent work of R. Shvydkoy (Commun Math Phys 265:507–545, 2006).