Inviscid Compressible Fluid Research Articles

A Mathematical Model for a Vibrating Human Head

In this paper, a mathematical model has been formulated to study the vibration of the human head. In the mathematical analysis of the model, the skull is considered as an anisotropic spherical shell and brain matter is represented as an inviscid compressible fluid. Also, in the model, the translational acceleration is considered as a general function of time. The authors use the method of Laplace transformation to achieve the analytical solution of the problem, while the analytical expressions have been used to compute the stress distribution in the system by resorting to numerical techniques.

Destabilizing effect of compressibility on Rayleigh–Taylor instability for fluids with fixed density profile

Analytical solutions for the growth rate of the Rayleigh–Taylor instability (RTI) for two superimposed fluids of exponentially varying densities are obtained by variational approach, which has been introduced to solve the eigenvalue problems related to RTI for inviscid compressible fluid. In order to distinguish the effect of density profile from compressibility effects, two independent parameters have been introduced to describe the static fluid states, one (Pa) is for the variation of pressure near the interface and the other (Pb) for setting the density profile. Three factors related to the RTI growth for compressible fluids have been discussed in detail. For incompressible fluids, the only factor to affect the RTI growth is the density profile. While for compressible fluids, besides the density profile, the specific heat ratio and the interface pressure also contribute to the growth rate of RTI. Analytic dispersion relations and growth rates have been compared to numerical results. It is found that compressibility alone acts a destabilizing role for fluids with fixed density profile.

Some applications of generalized moments of the density in an inviscid compressible flow

A generalized moment of an inviscid compressible fluid contained in a domain Ω is a function of the form I = ∫ Ω ρ Φ d V , where ρ is the density and Φ is an arbitrary, time-independent function. The moment I satisfies an evolution equation which relates the energy of the flow, the distribution of matter and the pressure at the boundary. By considering convex domains and choosing appropriately Φ we show that any flow which does not blow-up must satisfy a balance between these magnitudes which preclude e.g. a vacuum at the boundary. Another possibility is to take Φ = r α , where r is the distance to the center of a ball or to an axis; the classical moment of inertia corresponds to α = 2 . For this case we find estimates on the evolution of the growth of I which show that in general this growth rate decreases with α .

Specially orthotropic panel flutter control using PID controller

Controlling the flutter speed of a specially orthotropic plate of rectangular cross-section through which an inviscid compressible fluid flows is considered. A state-space model of the coupled aeroelastic system helps us to determine flutter boundaries and provides a method for applying modern control theory to the problem. So in the present study, flow is modelled by piston theory and the coupled state-space model of panel motion and flow is solved. After that, the coupled system is controlled with Proportional-Integral-Derivative controller. This makes it possible to compute the flutter dynamic pressure and so the flutter velocity at which unstable plate oscillations occur as a function of the gain coefficient. It is found that by changing the gain coefficient the flutter of the plate can indeed be pushed to higher velocities.

Adaptive mesh fluid simulations on GPU

We describe an implementation of compressible inviscid fluid solvers with block-structured adaptive mesh refinement on Graphics Processing Units using NVIDIA's CUDA. We show that a class of high resolution shock capturing schemes can be mapped naturally on this architecture. Using the method of lines approach with the second order total variation diminishing Runge-Kutta time integration scheme, piecewise linear reconstruction, and a Harten-Lax-van Leer Riemann solver, we achieve an overall speedup of approximately 10 times faster execution on one graphics card as compared to a single core on the host computer. We attain this speedup in uniform grid runs as well as in problems with deep AMR hierarchies. Our framework can readily be applied to more general systems of conservation laws and extended to higher order shock capturing schemes. This is shown directly by an implementation of a magneto-hydrodynamic solver and comparing its performance to the pure hydrodynamic case. Finally, we also combined our CUDA parallel scheme with MPI to make the code run on GPU clusters. Close to ideal speedup is observed on up to four GPUs.

An Optimization Method in Inverse Acoustic Scattering by an Elastic Obstacle

We consider the interaction between an elastic body and a compressible inviscid fluid, which occupies the unbounded exterior domain. The inverse problem of determining the shape of such an elastic scatterer from the measured far field pattern of the scattered fluid pressure field is of central importance in detecting and identifying submerged objects. Following a method proposed by Kirsch and Kress, we approximate the acoustic and elastodynamic wave by potentials over auxiliary surfaces, and we reformulate the inverse problem as an optimization problem. The objective function to be minimized is the sum of three terms. The first is the deviation of the approximate far field pattern from the measured one, the second is a regularization term, and the last a control term for the transmission condition. We prove that the optimization problem has a solution and that, for the regularization parameter tending to zero, the minimizers tend to a solution of the inverse problem. In contrast to a numerical method from a previous paper, the presented method requires neither a direct solution method nor an additional treatment of possible Jones modes.

Numerical solution of transonic and subsonic flow of compressible inviscid and viscous fluid

AbstractThe work presents two numerical solutions of compressible flows problems with high and very low Mach numbers. Both problems are numerically solved by finite volume method and the explicit MacCormack scheme using a grid of quadrilateral cells. Moved grid of quadrilateral cells is considered in the form of conservation laws using Arbitrary Lagrangian–Eulerian method. In the first case, inviscid transonic flow through cascade DCA 8% is presented and the numerical results are compared to experimental data. The second case, numerical solution of unsteady viscous flow in the channel for upstream Mach number M∞=0.012 and frequency of the wall motions 100 Hz is presented. The unsteady case can represent a simplified model of airflow coming from the trachea, through the glottal region with periodically vibrating vocal folds to the human vocal tract. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A generalization of the Runge–Kutta iteration

Iterative solvers in combination with multi-grid have been used extensively to solve large algebraic systems. One of the best known is the Runge–Kutta iteration. We show that a generally used formulation [A. Jameson, Numerical solution of the Euler equations for compressible inviscid fluids, in: F. Angrand, A. Dervieux, J.A. Désidéri, R. Glowinski (Eds.), Numerical Methods for the Euler Equations of Fluid Dynamics, SIAM, Philadelphia, 1985, pp. 199–245] does not allow to form all possible polynomial transmittance functions and we propose a new formulation to remedy this, without using an excessive number of coefficients. After having converted the optimal parameters found in previous studies (e.g. [B. Van Leer, C.H. Tai, K.G. Powell, Design of optimally smoothing multi-stage schemes for the Euler equations, AIAA Paper 89–1923, 1989]) we compare them with those that we obtain when we optimize for an integrated 2-grid V -cycle and show that this results in superior performance using a low number of stages. We also propose a variant of our new formulation that roughly follows the idea of the Martinelli–Jameson scheme [A. Jameson, Analysis and design of numerical schemes for gas dynamics 1, artificial diffusion, upwind biasing, limiter and their effect on multigrid convergence, Int. J. Comput. Fluid Dyn. 4 (1995) 171–218; J.V. Lassaline, Optimal multistage relaxation coefficients for multigrid flow solvers. http://www.ryerson.ca/~jvl/papers/cfd2005.pdf] used on the advection–diffusion equation which that can be extended to other types. Gains in the order of 30%–50% have been shown with respect to classical iterative schemes on the advection equation. Better results were also obtained on the advection–diffusion equation than with the Martinelli–Jameson coefficients, but with less than half the number of matrix-vector multiplications.

Formal solution of ill-posedness initial-boundary value problem for Euler equations of compressible adiabatic inviscid fluid flow

It is proved that the ill-posed initial value problem of the Euler equations for compressible adiabatic inviscid fluid flow can be only formally solved. The necessary and sufficient conditions for existence of formal solution of some representative ill-posedness initial-boundary value problem are presented. Finally, an example is also given.

Variational Approach to the Inviscid Compressible Fluid

It is well-known that not every partial differential equation admits a variational formula. A rigorous proof of the existence of a variational principle is very difficult. In this paper, the semi-inverse method proposed by Ji-Huan He is used to construct a variational principle for a one-dimensional inviscid compressible fluid.

A One-Dimensional Inviscid and Compressible Fluid in a Harmonic Potential Well

A Hamiltonian model is analyzed for a one-dimensional inviscid compressible fluid. The space–time evolution of the fluid is governed by the following system of the Hamilton–Jacobi and the continuity equations: $$\begin{array}{rl}S_{t}+\frac{1}{2}(S_{x}^{2}+\omega^{2}x^{2})&=0,\qquad S(x,0)=S_{0}(x);\\\rho_{t}+(\rho S_{x})_{x}&=0,\qquad \rho(x,0)=\rho_{0}(x).\end{array}$$ Here S and ρ designate the velocity potential and the mass density, respectively. Unless S 0 is convex, shocks form and the velocity S x becomes discontinuous in {0<ω t<π/2}. It is demonstrated that there nevertheless exists a unique viscosity–measure solution (S,ρ) when S 0 is globally Lipschitz continuous and locally semi-concave while ρ 0 is a finite Borel measure. The structure of the velocity and the density is exhibited. For initial data correlated in a certain sense, a class of classical solutions (S,ρ) is given. Negative time is also considered, and illustrating examples are given.

On the well-posedness for the Euler-Korteweg model in several space dimensions

The Euler-Korreweg model results from a modification of the standard Euler equations governing the motion of compressible inviscid fluids through the adjunction of the Korteweg stress tensor, which takes into account capillarity effects in regions where the density experiences large variations, typically across interfaces for fluids exhibiting phase changes. One of the main difficulties in the analysis of the Cauchy problem for this model, a third order system of conservation laws, is the absence of dissipative regularization, since viscosity is neglected. The Cauchy problem for isothermal fluids in one space dimension has been addressed by the authors in an earlier paper, using Lagrangian coordinates. Here the Cauchy problem is investigated in arbitrary space dimension N, still for isothermal fluids, and a variable capillarity coefficient. A local well-posedness result is obtained in Sobolev spaces as though the density gradient and the velocity field were solutions of a symmetrizable hyperbolic system. More precisely, well-posedness is shown for Hs+1 x H-s (s > N/2 + 1) perturbations of smooth global solutions, either constant states or traveling profiles. In addition, almost-global existence is proved for small enough perturbations, and a blow-up criterion is shown. Proofs rely on a suitable extended formulation of the system, which turns out to amount to a nonlinear degenerate Schrodinger equation coupled with a transport equation, and on a priori estimates without loss of derivatives for the extended system, which necessitate various 'gauge' functions to cancel out bad commutators.

2D Slightly Compressible Ideal Flow in an Exterior Domain

We consider the Euler equations of barotropic inviscid compressible fluids in the exterior domain. It is well known that, as the Mach number goes to zero, the compressible flows approximate the solution of the equations of motion of inviscid, incompressible fluids. In dimension 2 such limit solution exists on any arbitrary time interval, with no restriction on the size of the initial data. It is then natural to expect the same for the compressible solution, if the Mach number is sufficiently small.

Hydroelastic vibration of two annular plates coupled with a bounded compressible fluid

Abstract A theoretical study on a linear hydroelastic vibration of two annular plates coupled with a bounded fluid is presented. The proposed method, based on the Rayleigh–Ritz method and the finite Hankel transform, is verified through a finite element analysis by using a commercial computer code, with an excellent accuracy. It is assumed that plates with an unequal thickness and with an unequal inner radius are clamped along their edges and an inviscid compressible fluid fills the space between the annular plates and the outer rigid vessel. When the two annular plates are identical, distinct in-phase and out-of-phase modes are observed. By increasing the difference in the plate thickness, the symmetric in-phase and out-of-phase modes with respect to the middle plane of the system are gradually shifted to pseudo in-phase and out-of-phase modes, and eventually they are changed to mixed modes. It is found that the natural frequencies decrease with an increase of the fluid compressibility, and additional modes due to a fluid concentration are observed when the plates are coupled with a compressible fluid. The fluid compressibility effect on the natural frequency is dominant in the out-of-phase modes and the higher modes. Also, the effects of the fluid thickness or the distance between the plates and the inner radius of the plates on the natural frequencies of the wet modes are investigated.

Diffusion regulation for Euler solvers

A diffusion regulation parameter, which operates based on the jump in the Mach number, is presented for implementation in Euler solvers. This diffusion regulation parameter adjusts itself automatically in different regimes of the flow and leads to the exact capturing of steady contact discontinuities which are aligned with the grid-lines. This diffusion regulator parameter reduces numerical dissipation, is very simple and can be easily incorporated in any Euler solver. By coupling such a parameter with a simple numerical method like the Local Lax-Friedrichs (Rusanov) method, an accurate and yet simple numerical method is developed for the numerical simulation of inviscid compressible fluid flows. To demonstrate the applicability of this approach to any Euler solver, the diffusion regulation parameter is also applied in the framework of a Kinetic Scheme which is very diffusive and the improvements in the accuracy for both the methods are demonstrated through several bench-mark test problems.

The field due to a pair of line vortices in a compressible fluid

The problem considered is that of a self-propagating pair of line vortices, of equal and opposite strengths, in a compressible inviscid fluid. An asymptotic solution is sought in the limit of small propagation speed compared with the sound speed in the medium. In most of the fluid region, the elementary incompressible flow solution limit is enhanced by a Rayleigh–Janzen approximation. This approximation fails at points that are either far from, or very close to, a vortex. For distant points, rescaled outer variables lead to an approximation that corresponds to the field induced by a moving dipole in compressible fluid. The main approximation also fails near each vortex line, where the analysis of Barsony-Nagy, Er-El &amp; Yungster (J. Fluid Mech. vol. 178, 1987, p. 367) is used to express the local flow field in terms of a hypergeometric function. A particular feature of the problem is the propagation parameter .

Lagrangian averaging for the 1D compressible Euler equations

We consider a $1$-dimensional Lagrangian averaged model for an inviscid compressible fluid. As previously introduced in the literature, such equations are designed to model the effect of fluctuations upon the mean flow in compressible fluids. This paper presents a traveling wave analysis and a numerical study for such a model. The discussion is centered around two issues. One relates to the intriguing wave motions supported by this model. The other is the appropriateness of using Lagrangian-averaged models for compressible flow to approximate shock wave solutions of the compressible Euler equations.

Compressible Fluid Model for the Seismic Waves Generated by A Sunquake

The solar convection zone is modeled as a horizontally stratified atmosphere with a constant gravitational field and an adiabatic temperature gradient (a neutrally stratified polytrope). At equilibrium, the gas pressure and density decreases to zero at the solar surface so that the solar surface is treated as a free surface which is bounded by vacuum. The evolution of small amplitude perturbations about the equilibrium state is described by the linearized Euler equations for an inviscid compressible fluid with an adiabatic equation of state. A sunquake is initiated at time zero by means of an initial perturbation with a Gaussian velocity profile and the exact solution of the initial value problem is obtained in terms of a Fourier integral. Comparisons between theory and observations indicate that this highly simplified model is able to predict the propagation of sunquake waves across the solar surface with an error of approximately 10% or 20%.

Aeroelastic flutter of a disk rotating in an unbounded acoustic medium

The linear aeroelastic stability of an unbaffled flexible disk rotating in an unbounded fluid is investigated by modeling the disk–fluid system as a rotating Kirchhoff plate coupled to the irrotational motions of a compressible inviscid fluid. A perturbed eigenvalue formulation is used to compute systematically the coupled system eigenvalues. Both a semi-analytical and a numerical method are employed to solve the fluid boundary value problem. The semi-analytical approach involves a perturbation series solution of the dual integral equations arising from the fluid boundary value problem. The numerical approach is a boundary element method based on the Hadamard finite part. Unlike previous works, it is found that a disk with zero material damping destabilizes immediately beyond its lowest critical speed. Upon the inclusion of small disk material damping, the flutter speeds become supercritical and increase with decreasing fluid density. The competing effects of radiation damping into the surrounding fluid and disk material damping control the onset of flutter at supercritical speed. The results are expected to be relevant for the design of rotating disk systems in data storage, turbomachinery and manufacturing applications.

Well-posedness of initial value problem for Euler equations of inviscid compressible adiabatic fluid

The well-posedness of the initial value problem of the Euler equations was mainly discussed based on the stratification theory, and the necessary and sufficient conditions of well-posedness are presented for some representative initial or boundary value problem, thus the structure of solution space for local (exact) solution of the Euler equations is determined. Moreover the computation formulas of the analytical solution of the well-posed problem are also given.