Stability of transonic shocks in supersonic flow past a wedge

In this paper we study the stability of transonic shocks in steady supersonic flow past a wedge. We take the potential flow equation as the mathematical model to describe the compressible flow. It is known that in generic case such a problem admits two possible location of shock, connecting the flow ahead it and behind it. They can be distinguished as supersonic-supersonic shock and supersonic-subsonic shock (or transonic shock). Both these possible shocks satisfy the Rankine-Hugoniot conditions and entropy condition. In this paper we prove that the transonic shock is also stable under perturbation of the coming flow provided the pressure at infinity is well controlled.     

Examples of entropy generating sequence

We introduce the notion of entropy generating sequence for infinite words and define its dimension when it exists. We construct an entropy generating sequence for each symbolic example constructed by Cassaigne such that the dimension of the sequence is the same as its topological entropy dimension. Hence the complexity can be measured via the dimension of an entropy generating sequence. Moreover, we construct a weakly mixing example with subexponential growth rate.     

Stability of transonic shocks for the full Euler system in supersonic flow past a wedge

We study the stability of transonic shocks in steady supersonic flow past a wedge. It is known that in generic case such a problem admits two possible locations of the shock front, connecting the flow ahead of it and behind it. They can be distinguished as supersonic–supersonic shock and supersonic–subsonic shock (or transonic shock). Both these possible shocks satisfy the Rankine–Hugoniot conditions and the entropy condition. We prove that the transonic shock is conditionally stable under perturbation of the upstream flow or perturbation of wedge boundary. Copyright © 2005 John Wiley & Sons, Ltd.     

Detailed computational and experimental fluid dynamics of fluidized beds

To describe the hydrodynamic phenomena prevailing in large industrial scale fluidized beds continuum models are required. The flow in these systems depends strongly on particle–particle interaction and gas–particle interaction. For this reason, proper closure relations for these two interactions are vital for reliable predictions on the basis of continuum models. Gas–particle interaction can be studied with the use of the lattice Boltzmann model (LBM), while the particle–particle interaction can suitably be studied with a discrete particle model. In this work it is shown that the discrete particle model, utilizing a LBM based drag model, has the capability to generate insight and eventually closure relations in processes such as mixing, segregation and homogeneous fluidization.     

Stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow

We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem (initial value problem) in the flow direction, and then develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than $1$, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.     

Triple-deck regularization of weak laminar hydraulic jumps in a two layer shallow water flow

The inviscid, weakly nonlinear, shallow-water limit of the Navier–Stokes equations leads to a hyperbolic conservation law. In certain cases of a two-layer flow the flux-function is non-convex, thus leading to the possibility of shocks (in physical terms hydraulic jumps) violating the Oleinik-entropy criteria, so called non-classical shocks. To rule out the inadmissible shocks their internal structure is studied, based on an asymptotic approach consistent with the Navier–Stokes equations. This leads to a triple-deck problem with a novel, non-linear interaction equation in the form of a forced, extended KdV-equation. The limit of vanishing and weak influence of the displacement effect is studied analytically, and in addition representative numerical solutions of the full problem are presented. Of particular interest is a solution, which has a pronounced, almost vanishing minimum in he wall shear. Its local structure is studied in some detail. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A reliability system under different types of shock governed by a Markovian arrival process and maintenance policy K

A reliability system subject to shocks producing damage and failure is considered. The source of shocks producing failures is governed by a Markovian arrival process. All the shocks produce deterioration and some of them failures, which can be repairable or non-repairable. Repair times are governed by a phase-type distribution. The number of deteriorating shocks that the system can stand is fixed. After a fatal failure the system is replaced by another identical one. For this model the availability, the reliability, and the rate of occurrence of the different types of failures are calculated. It is shown that this model extends other previously published in the literature.     

On shock dynamics

This is in continuation of our paper On the propagation of a multi-dimensional shock of arbitrary strength’ published earlier in this journal (Srinivasan and Prasad [9]). We had shown in our paper that Whitham’s shock dynamics, based on intuitive arguments, cannot be relied on for flows other than those involving weak shocks and that too with uniform flow behind the shock. Whitham [12] refers to this as misinterpretation of his approximation and claims that his theory is not only correct but also provides a natural closure of the open system of the equations of Maslov [3]. The main aim of this note is to refute Whitham’s claim with the help of an example and a numerical integration of a problem in gasdynamics.     

Rectification,mixing, self-induced oscillations,measurement and control in asymmetrically grooved channels

Two separate constructions used in advanced microfluidics are combined to achieve controlled mixing and mass transport at maximum efficiency over minimal distance. One is the use of grooves to enhance mixing – an intensively investigated technique employed in electronic components cooling. So far, only grooves of ectangular cross–sections were used. The other construction builds on the well known effect of partial rectification in axially asymmetric channels and has been employed for valvless pumping. It is now shown that a cascade of axially asymmetric grooves retains and even improves the rectification efficiency of a single nozzle while offering the potential of simultaneous mixing enhancement by a factor of more than 2. The latter is achieved in a certain range of moderate Reynolds numbers characterized by self–induced oscillations at much higher frequency than that of flow actuation. Tuning the pressure drop provides precise control of the effective flow rate, up to suppression or reversion. The duration and intensity of mixing and shearing can thus be adjusted within a broad range and effected in very short channels without additional actuators. In the regime of self–induced oscillations, a few identical sensors with sufficient temporal resolution for temperature or concentration allow reliable determination of the flow rate as well as of the admixture composition of the transported fluid. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Entropy Production and Admissibility of Shocks

In shock wave theory there are two considerations in selecting the physically relevant shock waves.There is the admissibility criterion for the well-posedness of hyperbolic conservation laws.Another consideraztion concerns the entropy production across the shochs.The latter is natural from the physical point of view,but is not sufficient in its straightforward formulation,if the system is not genuinely nonlinear.In this paper we propose the principles of increasing entropy production and that of the superposition of shocks.These principles arc shown to be equivalent to the admissibility criterion.     

Aus- und Eintrittsstösse an Schaufelgittern

Summary Though the flow in the cylindrical surface of turbines or compressors may be treated as flow in a plane by development of the cylinder into a plane, the double connection once around the cylinder introduces difficulties in the law of forbidden signals for supersonic flow with a subsonic axial component. In cases of large gaps between the stages and blades of zero thickness and straight exit and entry zones, the methods of steady one-dimensional flow can shed light on the adaptability of the flow to the transient conditions by means of special kinds of shock polars for the exit and entry flow at given blade anglesβ (Figures 7 and 8). The shocks may sometimes be a combination ofHugoniot's discontinuities andCarnot's transient zones, but are in all cases governed by mass, impulse and energy balances with an increase in entropy.      

First-order L1-convergence for relaxation approximations to conservation laws

We derive a first-order rate of L¹-convergence for stiff relaxation approximations to its equilibrium solutions, i.e., piecewise smooth entropy solutions with finitely many discontinuities for scalar, convex conservation laws. The piecewise smooth solutions include initial central rarefaction waves, initial shocks, possible spontaneous formation of shocks in a future time, and interactions of all these patterns. A rigorous analysis shows that the relaxation approximations to approach the piecewise smooth entropy solutions have L¹-error bound of O(ε|log ε| + ε), where ε is the stiff relaxation coefficient. The first-order L¹-convergence rate is an improvement on the error bound. If neither central rarefaction waves nor spontaneous shocks occur, the error bound is improved to O(ε). © 1998 John Wiley & Sons, Inc.     

Convergence rates to discrete shocks for nonconvex conservation laws

Summary. This paper is concerned with polynomial decay rates of perturbations to stationary discrete shocks for the Lax-Friedrichs scheme approximating non-convex scalar conservation laws. We assume that the discrete initial data tend to constant states as , respectively, and that the Riemann problem for the corresponding hyperbolic equation admits a stationary shock wave. If the summation of the initial perturbation over is small and decays with an algebraic rate as , then the perturbations to discrete shocks are shown to decay with the corresponding rate as . The proof is given by applying weighted energy estimates. A discrete weight function, which depends on the space-time variables for the decay rate and the state of the discrete shocks in order to treat the non-convexity, plays a crucial role. Received November 25, 1998 / Published online November 8, 2000     

On the local and global behavior of weak shocks and induced discontinuities in a class of materials with internal state variables

We examine in this paper coupled governing differential equations of weak shocks and induced discontinuities in a class of materials with internal state variables, and under physically reasonable conditions deduce definite specific results concerning their local and global behavior. It is shown that all admissible choices of material parameters can be divided into four classes depending on their magnitudes, and the local and global behavior of shocks and induced discontinuities can have disparate features among the classes.     

Viscous limits for piecewise smooth solutions of the p-system

We study the zero-dissipation problem for a one-dimensional model system for the isentropic flow of a compressible viscous gas, the so-called p-system with viscosity. When the solution of the inviscid problem is piecewise smooth and having finitely many noninteracting shocks satisfying the entropy condition, there exists unique solution to the viscous problem which converges to the given inviscid solution away from shock discontinuities at a rate of order ε as the viscosity coefficient ε goes to zero. The proof is given by a matched asymptotic analysis and an elementary energy method. And we do not need the smallness condition on the shock strength.     

Moment Equations for Polyatomic Gases

The aim of this paper is to analyze the moment equations for polyatomic gases whose internal degrees of freedom are modeled by a continuous internal energy function. The closure problem is resolved using the maximum entropy principle. The macroscopic equations are divided in two hierarchies—“momentum” and “energy” one. As an example, the system of 14 moments equations is studied. The main new result is determination of the production terms which contain two parameters. They can be adapted to fit the expected values of Prandtl number and/or temperature dependence of the viscosity. The ratios of relaxation times are also discussed.     

Saturated-unsaturated flow in a fractured unconfined aquifer

Saturated-unsaturated flow is investigated in an unconfined aquifer containing a distribution of fractures such that on some macroscopic scale there can be defined a smooth averaged hydraulic conductivity. After constructing the conditions which relate the hydraulic properties between the unsaturated and saturated zones, it is shown that the flow in the unsaturated zone is essentially vertical. Below the water table the macropores are conduits for groundwater and the flow has a large horizontal component. Some important implications on previous models is discussed.     

Systems of conservation laws of mixed type

A local theory of weak solutions of first-order nonlinear systems of conservation laws is presented. In the systems considered, two of the characteristic speeds become complex for some achieved values of the dependent variable. The transonic “small disturbance” equation is an example of this class of systems. Some familiar concepts from the purely hyperbolic case are extended to such systems of mixed type, including genuine nonlinearity, classification of shocks into distinct fields and entropy inequalities. However, the associated entropy functions are not everywhere locally convex, shock and characteristic speeds are not bounded in the usual sense, and closed loops and disjoint segments are possible in the set of states which can be connected to a given state by a shock. With various assumptions, we show (1) that the state on one side of a shock plus the shock speed determine the state on the other side uniquely, as in the hyperbolic case; (2) that the “small disturbance” equation is a local model for a class of such systems; and (3) that entropy inequalities and/or the existence of viscous profiles can still be used to select the “physically relevant” weak solution of such a system.     

The fluid-dynamic limit of the Broadwell model of the nonlinear Boltzmann equation in the presence of shocks

This paper studies the asymptotic equivalence of the Broadwell model of the nonlinear Boltzmann equation to its corresponding Euler equation of compressible gas dynamics in the limit of small mean free path ε. It is shown that the fluid dynamical approximation is valid even if there are shocks in the fluid flow, although there are thin shock layers in which the convergence does not hold. More precisely, by assuming that the fluid solution is piecewise smooth with a finite number of noninteracting shocks and suitably small oscillations, we can show that there exist solutions to the Broadwell equations such that the Broadwell solutions converge to the fluid dynamical solutions away from the shocks at a rate of order (ε) as the mean free path ε goes to zero. For the proof, we first construct a formal solution for the Broadwell equation by matching the truncated Hilbert expansion and shock layer expansion. Then the existence of Broadwell solutions and its convergence to the fluid dynamic solution is reduced to the stability analysis for the approximate solution. We use an energy method which makes full use of the inner structure of time dependent shock profiles for the Broadwell equations.     

On the development and generalizations of Cahn–Hilliard equations within a thermodynamic framework

We provide a thermodynamic basis for the development of models that are usually referred to as ??phase-field models?? for compressible, incompressible, and quasi-incompressible fluids. Using the theory of mixtures as a starting point, we develop a framework within which we can derive ??phase-field models?? both for mixtures of two constituents and for mixtures of arbitrarily many fluids. In order to obtain the constitutive equations, we appeal to the requirement that among all admissible constitutive relations that which is appropriate maximizes the rate of entropy production (see Rajagopal and Srinivasa in Proc R Soc Lond A 460:631?C651, 2004). The procedure has the advantage that the theory is based on prescribing the constitutive equations for only two scalars: the entropy and the entropy production. Unlike the assumption made in the case of the Navier?CStokes?CFourier fluids, we suppose that the entropy is not only a function of the internal energy and the density but also of gradients of the partial densities or the concentration gradients. The form for the rate of entropy production is the same as that for the Navier?CStokes?CFourier fluid. As observed earlier in Heida and Málek (Int J Eng Sci 48(11):1313?C1324, 2010), it turns out that the dependence of the rate of entropy production on the thermodynamical fluxes is crucial. The resulting equations are of the Cahn?CHilliard?CNavier?CStokes type and can be expressed both in terms of density gradients or concentration gradients. As particular cases, we will obtain the Cahn?CHilliard?CNavier?CStokes system as well as the Korteweg equation. Compared to earlier approaches, our methodology has the advantage that it directly takes into account the rate of entropy production and can take into consideration any constitutive assumption for the internal energy (or entropy).