Singularities in Three-Dimensional Compressible Euler Flows with Vorticity

It is known from earlier work that three-dimensional incompressible Euler flows with vorticity can develop a singularity in a finite time, at least if the initial conditions are of a certain class. Here we discuss corresponding possibilities for flows with compressibility. Naturally, it is known that the shock-wave phenomenon represents an important singular field in compressible fluid dynamics especially in the irrotational case. However, here we are concerned not with that phenomenon but rather with compressible flows where any singularity is associated with the presence of vorticity. In particular we expose the role played by the ratio of specific heats in an adiabatic flow field. Received 9 December 1996 and accepted 4 April 1997     

Global Existence and Large-Time Behavior of Solutions to the Three-Dimensional Equations of Compressible Magnetohydrodynamic Flows

The three-dimensional equations of compressible magnetohydrodynamic isentropic flows are considered. An initial-boundary value problem is studied in a bounded domain with large data. The existence and large-time behavior of global weak solutions are established through a three-level approximation, energy estimates, and weak convergence for the adiabatic exponent ${gamma > frac 32}${gamma > frac 32} and constant viscosity coefficients.     

Centralised Versus Decentralised Active Control of Boundary Layer Instabilities

We use linear control theory to construct an output feedback controller for the attenuation of small-amplitude three-dimensional Tollmien-Schlichting (TS) wavepackets in a flat-plate boundary layer. A three-dimensional viscous, incompressible flow developing on a zero-pressure gradient boundary layer in a low Reynolds number environment is analyzed using direct numerical simulations. In this configuration, we distribute evenly in the spanwise direction up to 72 localised objects near the wall (18 disturbances sources, 18 actuators, 18 estimation sensors and 18 objective sensors). In a fully three-dimensional configuration, the interconnection between inputs and outputs becomes quickly unfeasible when the number of actuators and sensors increases in the spanwise direction. The objective of this work is to understand how an efficient controller may be designed by connecting only a subset of the actuators to sensors, thereby reducing the complexity of the controller, without comprising the efficiency. If n and m are the number of sensor-actuator pairs for the whole system and for a single control unit, respectively, then in a decentralised strategy, the number of interconnections deceases mn compared to a centralized strategy, which has n ² interconnections. We find that using a semi-decentralized approach – where small control units consisting of 3 estimation sensors connected to 3 actuators are replicated 6 times along the spanwise direction – results only in a 11 % reduction of control performance. We explain how “wide” in the spanwise direction a control unit should be for a satisfactory control performance. Moreover, the control unit should be designed to account for the perturbations that are coming from the lateral sides (crosstalk) of the estimation sensors. We have also found that the influence of crosstalk is not as essential as the spreading effect.     

Immersed Boundaries in Large Eddy Simulation of Compressible Flows

Methods to immerse walls in a structured mesh are examined in the context of fully compressible solutions of the Navier–Stokes equations. The ghost cell approach is tested along with compressible conservative immersed boundaries in canonical flow configurations; the reflexion of pressure waves on walls arbitrarily inclined on a cartesian mesh is studied, and mass conservation issues examined in both a channel flow inclined at various angles and flow past a cylinder. Then, results from Large Eddy Simulation of a flow past a rectangular cylinder and a transonic cavity flow are compared against experiments, using either a multi-block mesh conforming to the wall or immersed boundaries. Different strategies to account for unresolved transport by velocity fluctuations in LES are also compared. It is found that immersed boundaries allow for reproducing most of the coupling between flow instabilities and pressure-signal properties observed in the transonic cavity flow. To conclude, the complex geometry of a trapped vortex combustor, including a cavity, is simulated and results compared against experiments.     

Three-Dimensional Instability of Planar Flows

We study the stability of two-dimensional solutions of the three-dimensional Navier–Stokes equations, in the limit of small viscosity. We are interested in steady flows with locally closed streamlines. We consider the so-called elliptic and centrifugal instabilities, which correspond to the continuous spectrum of the underlying linearized Euler operator. Through the justification of highly oscillating Wentzel–Kramers–Brillouin expansions, we prove the nonlinear instability of such flows. The main difficulty is the control of nonoscillating and nonlocal perturbations issued from quadratic interactions.     

简单周期结构波传播主动控制研究

研究了在梁结构插入压电材料的周期结构波传播的一般规律。以梁的弯曲波传播方程的解析解为基础，进一步分析了简单周期结构的波传播的一些固有特性。并在此基础上，以压电材料作为作动器，采用主动控制方法研究了调整结构传播波的带通、带阻的频率范围可行性和有效性。     

Existence and Stability of Compressible Current-Vortex Sheets in Three-Dimensional Magnetohydrodynamics

Compressible vortex sheets are fundamental waves, along with shocks and rarefaction waves, in entropy solutions to multidimensional hyperbolic systems of conservation laws. Understanding the behavior of compressible vortex sheets is an important step towards our full understanding of fluid motions and the behavior of entropy solutions. For the Euler equations in two-dimensional gas dynamics, the classical linearized stability analysis on compressible vortex sheets predicts stability when the Mach number \(M > \sqrt{2}\) and instability when \(M < \sqrt{2}\) ; and Artola and Majda’s analysis reveals that the nonlinear instability may occur if planar vortex sheets are perturbed by highly oscillatory waves even when \(M > \sqrt{2}\) . For the Euler equations in three dimensions, every compressible vortex sheet is violently unstable and this instability is the analogue of the Kelvin–Helmholtz instability for incompressible fluids. The purpose of this paper is to understand whether compressible vortex sheets in three dimensions, which are unstable in the regime of pure gas dynamics, become stable under the magnetic effect in three-dimensional magnetohydrodynamics (MHD). One of the main features is that the stability problem is equivalent to a free-boundary problem whose free boundary is a characteristic surface, which is more delicate than noncharacteristic free-boundary problems. Another feature is that the linearized problem for current-vortex sheets in MHD does not meet the uniform Kreiss–Lopatinskii condition. These features cause additional analytical difficulties and especially prevent a direct use of the standard Picard iteration to the nonlinear problem. In this paper, we develop a nonlinear approach to deal with these difficulties in three-dimensional MHD. We first carefully formulate the linearized problem for the current-vortex sheets to show rigorously that the magnetic effect makes the problem weakly stable and establish energy estimates, especially high-order energy estimates, in terms of the nonhomogeneous terms and variable coefficients. Then we exploit these results to develop a suitable iteration scheme of the Nash–Moser–Hörmander type to deal with the loss of the order of derivative in the nonlinear level and establish its convergence, which leads to the existence and stability of compressible current-vortex sheets, locally in time, in three-dimensional MHD.     

Computation of Internal and External Compressible Flows Using EDICT

Results are presented for finite element computations involving high speed, viscous compressible internal and external flows. The stabilized finite-element formulations for the Navier-Stokes equations in the conservation law form are solved using the conservation variables. To improve the accuracy of the base method, especially in the regions of flow that are associated with shocks, boundary-layers and their interactions, the Enhanced-Discretization Interface-Capturing Technique (EDICT) is utilized. An error indicator is employed to identify the regions in the computational domain that need enhanced discretization for increased accuracy. The method is implemented on a shared-memory parallel computer and is used to study complex flows, that involve shock-wave/boundary-layer interactions, in supersonic diffusers and wind-tunnels. The start-up problem in supersonic wind-tunnels, caused by a narrow second throat in the diffuser section, is simulated. This computation brings out some of the very interesting features of the unsteady dynamics of the start-up shock.     

Spiral Waves in Three-Dimensional Helical Viscous Flows

A study is made of an unbounded viscous incompressible flow in the case in which the vortex lines of the absolute motion coincide with the streamlines of the relative motion; after Joukowsky, this flow is called helical flow. The vector potential of the flow is constructed and the notion of the stream function is generalized to include three-dimensional uniform helical flows in an arbitrary orthogonal coordinate system. It is shown that both axisymmetric and asymmetric waves may propagate in a rotating fluid; the amplitude of these waves decays exponentially with time, the decrement being proportional to the square of the rotational velocity of the fluid and the kinematic viscosity and inversely proportional to the square of the phase velocity.     

Absolute and Convective Instabilities in the Compressible Boundary Layer on a Rotating Disk

A numerical study has been undertaken to investigate the nature of inviscid instability of the three-dimensional compressible boundary layer flow due to a rotating disk. The compressible Rayleigh equation is integrated using a spectral Chebyshev-collocation method together with a fourth-order Runge–Kutta integrator. In the context of spatio-temporal stability analysis, the singularities of the resulting dispersion relation are determined and the ones that satisfy the Briggs–Bers pinching criterion have been selected. In certain finite parameter regions of eigenvalues (wave numbers and wave angles, for instance) it is found that by varying the Mach number, absolute instability occurs in the compressible boundary layer on a rotating disk. The range corresponding to the incompressible flow case given in Lingwood (1995) (ε between 14.615° and 38.114°) is verified. The results of Cole (1995) are also verified. The overall effect of compressibility is to reduce the extent of absolute instability at higher Mach numbers. The effect of heating the wall is to enhance the absolute instability properties, however, cooling the wall is found to decrease greatly the region of absolute instability regime for the range of Mach numbers studied. It is also shown in this study that for non-insulated walls a direct spatial resonance of the eigenmodes is possible and this raises the possibility of large local algebraic growth of perturbations being important in some instances. Received 15 October 1999 and accepted 10 December 1999     

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.     

On the Computation of Compressible Turbulent Flows on Unstructured Grids

An accurate, fast, matrix-free implicit method has been developed to solve compressible turbulent How problems using the Spalart and Allmaras one equation turbulence model on unstructured meshes. The mean-flow and turbulence-model equations are decoupled in the time integration in order to facilitate the incorporation of different turbulence models and reduce memory requirements. Both mean flow and turbulent equations are integrated in time using a linearized implicit scheme. A recently developed, fast, matrix-free implicit method, GMRES+LU-SGS, is then applied to solve the resultant system of linear equations. The spatial discretization is carried out using a hybrid finite volume and finite element method, where the finite volume approximation based on a containment dual control volume rather than the more popular median-dual control volume is used to discretize the inviscid fluxes, and the finite element approximation is used to evaluate the viscous flux terms. The developed method is used to compute a variety of turbulent flow problems in both 2D and 3D. The results obtained are in good agreement with theoretical and experimental data and indicate that the present method provides an accurate, fast, and robust algorithm for computing compressible turbulent flows on unstructured meshes.     

The Pressure Singularity for Compressible Stokes Flows in a Concave Polygon

A compressible Stokes system is studied in a polygon with one concave vertex. A corner singularity expansion is obtained up to second order. The expansion contains the usual corner singularity functions for the velocity plus an “associated” velocity singular function, and a pressure singular function. In particular the singularity of pressure is not local but occurs along the streamline emanating from the incoming concave vertex. It is observed that certain first derivatives of the pressure become infinite along the streamline of the ambient flow emanating from the concave vertex. Higher order regularity is shown for the remainder. This work was supported by the Com2MaC-SRC/ERC program of MOST/KOSEF (grant R11-1999-054), and by the U.S. National Science Foundation.     

Homoclinic Orbits and Entropy for Three-Dimensional Flows

We prove that every \(C^1\) three-dimensional flow with positive topological entropy can be \(C^1\) approximated by flows with homoclinic orbits.     

Sensitivity Analysis Using Adjoint Parabolized Stability Equations for Compressible Flows

An input/output framework is used to analyze the sensitivity of two- and three-dimensional disturbances in a compressible boundary layer for changes in wall and momentum forcing. The sensitivity is defined as the gradient of the kinetic disturbance energy at a given downstream position with respect to the forcing. The gradients are derived using the parabolized stability equations (PSE) and their adjoint (APSE). The adjoint equations are derived in a consistent way for a quasi-two-dimensional compressible flow in an orthogonal curvilinear coordinate system. The input/output framework provides a basis for optimal control studies. Analysis of two-dimensional boundary layers for Mach numbers between 0 and 1.2 show that wall and momentum forcing close to branch I of the neutral stability curve give the maximum magnitude of the gradient. Forcing at the wall gives the largest magnitude using the wall normal velocity component. In case of incompressible flow, the two-dimensional disturbances are the most sensitive ones to wall inhomogeneity. For compressible flow, the three-dimensional disturbances are the most sensitive ones. Further, it is shown that momentum forcing is most effectively done in the vicinity of the critical layer. This revised version was published online in July 2006 with corrections to the Cover Date.     

Using LES to Study Reacting Flows and Instabilities in Annular Combustion Chambers

Great prominence is put on the design of aeronautical gas turbines due to increasingly stringent regulations and the need to tackle rising fuel prices. This drive towards innovation has resulted sometimes in new concepts being prone to combustion instabilities. In the particular field of annular combustion chambers, these instabilities often take the form of azimuthal modes. To predict these modes, one must compute the full combustion chamber, which remained out of reach until very recently and the development of massively parallel computers. Since one of the most limiting factors in performing Large Eddy Simulation (LES) of real combustors is estimating the adequate grid, the effects of mesh resolution are investigated by computing full annular LES of a realistic helicopter combustion chamber on three grids, respectively made of 38, 93 and 336 million elements. Results are compared in terms of mean and fluctuating fields. LES captures self-established azimuthal modes. The presence and structure of the modes is discussed. This study therefore highlights the potential of LES for studying combustion instabilities in annular gas turbine combustors.     

Modeling of Three-Dimensional Turbulent Jet and Boundary-Layer Flows

New anisotropic algebraic constitutive relations for the Reynolds stress tensor are formulated. These relations make it possible to model correctly three-dimensional turbulent flows which cannot be described on the basis of traditional modern semi-empirical models of turbulence. Along with the well-known nonlinear Saffman term, these relations contain new nonlinear terms which take the wall effect into account. Several two- and three-dimensional turbulent flows are calculated nu\-merically using the averaged Navier-Stokes equations. The calculation results are compared with known experimental data.     

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.     

Blow Up Criterion for Compressible Nematic Liquid Crystal Flows in Dimension Three

In this paper, we consider the short time strong solution to a simplified hydrodynamic flow modeling compressible, nematic liquid crystal materials in dimension three. We establish a criterion for possible breakdown of such solutions at a finite time in terms of the temporal integral of both the maximum norm of the deformation tensor of the velocity gradient and the square of the maximum norm of the gradient of a liquid crystal director field.