Euler Equations

As widely accepted, justified by the historical developments of physics, the background for standard formulation of postulates of physical theories leading to equations of motion, or even the form of equations of motion themselves, come from empirical experience. Equations of motion are then a starting point for obtaining specific conservation laws, as, for example, the well-known conservation laws of momenta and mechanical energy in mechanics. On the other hand, there are numerous examples of physical laws or equations of motion which can be obtained from a certain variational principle as Euler-Lagrange equations and their solutions, meaning that the \true trajectories" of the physical systems represent stationary points of the corresponding functionals. It turns out that equations of motion in most of the fundamental theories of physics (as e.g. classical mechanics, mechanics of continuous media or fluids, electrodynamics, quantum mechanics, string theory, etc.), are Euler-L...

he method of lines approach for solving hyperbolic conservation laws is based on the idea of splitting the discretization process in two stages. First, the spatial discretization is performed by leaving the system continuous in time. This approximation is usually developed in a non-oscillatory manner with a satisfactory spatial accuracy. The obtained semi-discrete system of ordinary differential equations (ODE) is then solved by using some standard time integration method. In the last few years, a series of papers appeared, dealing with the high order strong stability preserving (SSP) time integration methods that maintain the total variation diminishing (TVD) property of the first order forward Euler method. In this work the optimal SSP Runge--Kutta methods of different order are considered in combination with the finite volume weighted essentially non-oscillatory (WENO) discretization. Furthermore, a new semi--implicit WENO scheme is presented and its properties in combination wit...

On the Basis of Syllabus Provide by Tu_IOE
This practical session covers
1. IMPACT OF JET
2. STUDY OF FLOW THROUGH ORIFICE
3. STABILITY OF FLOATING BODY
4. VERIFICATION OF BERNOULLI’S THEROEM
5. STUDY OF FLOW THROUGH BROAD CRESTED WEIR
The practical exercises provide a practical bridge between theoretical concepts and real-world  applications

Over the last few years, the discontinuous Galerkin method (DGM) has demonstrated its excellence in accurate, higher-order numerical simulations for a wide range of applications in computational physics. However, the development of practical, computationally efficient flow solvers for industrial applications is still in the focus of active research. This paper deals with solving the Navier-Stokes equations describing the motion of three-dimensional, viscous compressible fluids. We present details of the PADGE code under development at the German Aerospace Center (DLR) that is aimed at largescale applications in aerospace engineering. The discussion covers several advanced aspects like the solution of the Reynolds-averaged Navier-Stokes and k-ω turbulence model equations, a curved boundary representation, anisotropic mesh adaptation for reducing output error and techniques for solving the nonlinear algebraic equations. The performance of the solver is assessed for a set of test cases.

In this paper, we consider the pressureless Euler equations with a congestion constraint. This system still raises many open questions and, aside from its one-dimensional version, very few is known concerning its solutions. The strategy that we propose relies on previous works on crowd motion models with congestion in the framework of the Wasserstein space, and on a microscopic granular model with nonelastic collisions. We illustrate the approach by preliminary numerical simulations in the two-dimensional setting.

Non-classical non-linear waves exist in dense gases at high pressure in the region close to a thermodynamical critical point. These waves behave precisely opposite to the classical non-linear waves (shocks and expansion fans) and do not violate entropy conditions. More complex EOS other than the ideal or perfect gas equation of state (EOS) is used in describing dense gases. Algorithm development with non-ideal/real gas EOS and application to dense gasses is gaining importance from a numerical perspective. Algorithms designed for perfect gas EOS can not be extended directly to arbitrary real gas EOS with known EOS formulation. Most of the algorithms designed with prefect gas EOS are modified significantly when applied to real gas EOS with the known formulation. These algorithms can become complicated and some times impossible based on the EOS under consideration. The objective of the present work is to develop central solvers with smart diffusion capabilities independent of the eigen...

The Generalized Riemann Problem (GRP) for nonlinear hyperbolic system of m balance laws (or alternatively "quasi-conservative" laws) in one space dimension is formulated as follows: Given initial-data which are analytic on two sides of a discontinuity, determine the time evolution of the solution at the discontinuity. In particular, the GRP numerical scheme (second-order high resolution) is based on an analytical evaluation of the first time derivative. The analytical solution is readily obtained for a single equation (m = 1) and, more generally, if the system is endowed with a complete (coordinate) set of Riemann invariants. In this case it can be "diagonalized" and reduced to the scalar case. However, most systems with m > 2 do not admit such a set of Riemann invariants. This paper introduces a generalization of this concept: "Weakly Coupled Systems" (WCS). Such systems have only "partial set" of Riemann invariants, but these sets are weakly coupled in a way which enables a "diagonalized" treatment of the GRP. An important example of a WCS is the Euler system of compressible, nonisentropic fluid flow (m = 3). The solution of the GRP discussed here is based on a careful analysis of rarefaction waves. A "propagation of singularities" argument is applied to appropriate Riemann invariants across the rarefaction fan. It serves to "rotate" initial spatial slopes into "time derivative". In particular, the case of a "sonic point" is incorporated easily into the general treatment. A GRP scheme based on this solution is derived, and several numerical examples are presented. Special attention is given to the "acoustic approximation" of the analytical solution. It can be viewed as a proper linearization (different from the approach of Roe) of the nonlinear system. The resulting numerical scheme is the simplest (second-order, high-resolution) generalization of the Godunov scheme.

An analytical linear solution of the fully compressible Euler equations is found, in the particular case of a stationary two dimensional flow that passes over an orographic feature with small height-width ratio. A method based on the covariant formulation of the Euler equations is used, and the analytical vertical velocity as well as the horizontal velocity, density and pressure, are obtained. The analytical solution is tested against a numerical model in three different regimes, hydrostatic, non-hydrostatic and potential flow. The model used is a non-hydrostatic spectral semi-implicit model, with a heightbased vertical coordinate. It is shown that there is a clear and consistent convergence of the numerical solution towards the analytical solution, when the resolution increases. The method described is intended to be used as an idealized test for numerical weather models.

Most of the dynamical cores of operational global models can be broadly classified according to the spatial discretisation into two categories: spectral models with mass-based vertical coordinate and grid point models with height-based vertical coordinate. This article describes a new non-hydrostatic dynamical core for a global model that uses the spectral transform method for the horizontal directions and a height-based vertical coordinate. Velocity is expressed in the contravariant basis (instead of the geographical orthonormal basis pointing to the East, North and Zenith directions) so that the expressions of the boundary conditions and the divergence of the velocity are simpler. Prognostic variables in our model are the contravariant components of the velocity, the logarithm of pressure and the logarithm of temperature. Covariant tensor analysis is used to derive the differential operators of the prognostic equations, such as the curl, gradient, divergence and covariant derivative of the contravariant velocity. A Lorenz type grid is used in the vertical direction, with the vertical contravariant velocity staggered with respect to the other prognostic variables. High-order vertical operators are constructed following the finite difference technique. Time stepping is semi-implicit because it allows for long time steps that compensates the cost of the spectral transformations. A set of experiments reported in the literature is implemented so as to confirm the accuracy and efficiency of the new dynamical core.

We present a rigorous approach and related techniques to construct global solutions of the two-dimensional (2-D) Riemann problem with four-shock interactions for the Euler equations for potential flow. With the introduction of three critical angles: the vacuum critical angle from the compatibility conditions, the detachment angle, and the sonic angle, we clarify all configurations of the Riemann solutions for the interactions of two-forward and two-backward shocks, including the subsonic-subsonic reflection configuration that has not emerged in previous results. To achieve this, we first identify the three critical angles that determine the configurations, whose existence and uniqueness follow from our rigorous proof of the strict monotonicity of the steady detachment and sonic angles for 2-D steady potential flow with respect to the Mach number of the upstream state. Then we reformulate the 2-D Riemann problem into the shock reflection-diffraction problem with respect to a symmetric line, along with two independent incident angles and two sonic boundaries varying with the choice of incident angles. With these, the problem can be further reformulated as a free boundary problem for a second-order quasilinear equation of mixed elliptic-hyperbolic type. The difficulties arise from the degenerate ellipticity of the nonlinear equation near the sonic boundaries, the nonlinearity of the free boundary condition, the singularity of the solution near the corners of the domain, and the geometric properties of the free boundary. To solve the problem, we need to analyze the solutions for a quasilinear degenerate elliptic equation by the maximum principle of the mixed-boundary value problem, the theory of the oblique derivative boundary value problem, the uniform a priori estimates, and the iteration method. To the best of our knowledge, this is the first rigorous result for the 2-D Riemann problem with four-shock interactions for the Euler equations. The approach and techniques developed for the Riemann problem for four-wave interactions should be useful for solving other 2-D Riemann problems for more general Euler equations and related nonlinear hyperbolic systems of conservation laws.

We establish the optimal convergence rate to the hypersonic similarity law, which is also called the Mach number independence principle, for steady compressible full Euler flows over two-dimensional slender Lipschitz wedges. Mathematically, it can be formulated as the comparison of the entropy solutions in BV ∩ L 1 between the two initial-boundary value problems for the compressible full Euler equations with parameter τ > 0 and the hypersonic small-disturbance equations (the scaled compressible full Euler equations with parameter τ = 0) with curved characteristic boundaries. We establish the L 1-convergence estimate of these two solutions with the optimal convergence rate, which justifies the Van Dyke's similarity theory rigorously for the compressible full Euler flows. This is the first mathematical result on the comparison of two solutions of the compressible Euler equations with characteristic boundary conditions. To achieve this, we first employ the special structures of the two systems and establish the global existence and the L 1-stability of the entropy solutions via the wave-front tracking scheme un-* The corresponding author.

We establish the vanishing viscosity limit of the Navier-Stokes equations to the isentropic Euler equations for one-dimensional compressible fluid flow. For the Navier-Stokes equations, there exist no natural invariant regions for the equations with the real physical viscosity term so that the uniform sup-norm of solutions with respect to the physical viscosity coefficient may not be directly controllable and, furthermore, convex entropy-entropy flux pairs may not produce signed entropy dissipation measures. To overcome these difficulties, we first develop uniform energy-type estimates with respect to the viscosity coefficient for the solutions of the Navier-Stokes equations and establish the existence of measure-valued solutions of the isentropic Euler equations generated by the Navier-Stokes equations. Based on the uniform energy-type estimates and the features of the isentropic Euler equations, we establish that the entropy dissipation measures of the solutions of the Navier-Stokes equations for weak entropy-entropy flux pairs, generated by compactly supported C 2 test functions, are confined in a compact set in H −1 , which lead to the existence of measure-valued solutions that are confined by the Tartar-Murat commutator relation. A careful characterization of the unbounded support of the measure-valued solution confined by the commutator relation yields the reduction of the measure-valued solution to a Delta mass, which leads to the convergence of solutions of the Navier-Stokes equations to a finite-energy entropy solution of the isentropic Euler equations.

Dick, E., Second-order formulation of a multigrid method for steady Euler equations through defect-correction, Journal of Computational and Applied Mathematics 35 (1991) 159-168. A multigrid method for steady Euler equations based on polynomial flux-difference splitting is discussed. The multigrid procedure is applied to the first-order accurate discrete system. Second-order corrections are introduced through defect-correction.

We study the point spectrum of the linearisation of Euler's equation for the ideal fluid on the torus about a shear flow. By separation of variables the problem is reduced to the spectral theory of a complex Hill's equation. Using Hill's determinant an Evans function of the original Euler equation is constructed. The Evans function allows us to completely characterise the point spectrum of the linearisation, and to count the isolated eigenvalues with non-zero real part. We prove that the number of discrete eigenvalues of he linearised operator for a specific shear flow is exactly twice the number of non-zero integer lattice points inside the so-called unstable disk.

Third-order and fifth-order upwind compact finite difference schemes based on flux-difference splitting are proposed for solving the incompressible Navier-Stokes equations in conjunction with the artificial compressibility (AC) method. Since the governing equations in the AC method are hyperbolic, fluxdifference splitting (FDS) originally developed for the compressible Euler equations can be used. In the present upwind compact schemes, the split derivatives for the convective terms at grid points are linked to the differences of split fluxes between neighboring grid points, and these differences are computed by using FDS. The viscous terms are approximated with a sixth-order central compact scheme. Comparisons with 2D benchmark solutions demonstrate that the present compact schemes are simple, efficient, and high-order accurate.

We study the solutions of the equation φ(Cm)/φ(Cn) = r, where r is a fixed rational number, C k is the kth Catalan number and φ is the Euler function. We note that the number r = 4 is special for this problem and for it we construct solutions (m, n) to the above equation which are related to primes p such that 2p − 1 or 4p − 3 is also prime.

We relate the concept of measure valued solutions to conservation laws, introduced by DiPerna, to the concept of generalized function solutions arising in a differential algebra containing the distributions and having the algebra of smooth functions as a subalgebra. As an example, following results of DiPerna and Majda on measure valued solutions, we construct generalized solutions to the Euler equations.

Mitigating the impact of waves leaving a numerical domain has been a persistent challenge in numerical modeling. Reducing wave reflection at the domain boundary is crucial for accurate simulations. Absorbing layers, while common, often incur significant computational costs. This paper introduces an efficient application of a Legendre-Laguerre basis for absorbing layers for two-dimensional non-linear compressible Euler equations. The method couples a spectral-element bounded domain with a semi-infinite region, employing a tensor product of Lagrange and scaled Laguerre basis functions. Semi-infinite elements are used in the absorbing layer with Rayleigh damping. In comparison to existing methods with similar absorbing layer extensions, this approach, a pioneering application to the Euler equations of compressible and stratified flows, demonstrates substantial computational savings. The study marks the first application of semi-infinite elements to mitigate wave reflection in the solution of the Euler equations, particularly in nonhydrostatic atmospheric modeling. A comprehensive set of tests demonstrates the method's versatility for general systems of conservation laws, with a focus on its effectiveness in damping vertically propagating mountain gravity waves, a benchmark for atmospheric models. Across all tests, the model presented in this paper consistently exhibits notable performance improvements compared to a traditional Rayleigh damping approach.

A substantial literature has been generated on the estimation of allocative and technical inefficiency using static production, cost, profit, and distance functions. We develop a dynamic shadow distance system that integrates dynamic adjustment costs into a long-run shadow cost-minimization problem, which allows us to distinguish static allocative distortions from short-run inefficiencies that arise due to period-to-period adjustment costs. The set of estimating equations is comprised of the first-order conditions from the short-run shadow cost-minimization problem for the variable shadow input quantities, a set of Euler equations derived from subsequent shadow cost minimization with respect to the quasifixed inputs, and the input distance function, expressed in terms of shadow quantities. This system nests within it the static model with zero adjustment costs. Using panel data on U.S. electric utilities, we contrast the results of static and dynamic shadow distance systems. First, the zero-adjustment-cost restriction is strongly rejected. Second, we find that adjustment costs represent about .42 percent of total cost and about 1.26 percent of capital costs. Third, while both models reveal that labor is not utilized efficiently, the dynamic model indicates a longer period of over-use and less variance over time in the degree of inefficiency. With the dynamic model, productivity growth is larger but more stable.

A substantial literature has been generated on the estimation of allocative and technical inefficiency using static production, cost, profit, and distance functions. We develop a dynamic shadow distance system that integrates dynamic adjustment costs into a long-run shadow cost-minimization problem, which allows us to distinguish static allocative distortions from short-run inefficiencies that arise due to period-to-period adjustment costs. The set of estimating equations is comprised of the first-order conditions from the short-run shadow cost-minimization problem for the variable shadow input quantities, a set of Euler equations derived from subsequent shadow cost minimization with respect to the quasifixed inputs, and the input distance function, expressed in terms of shadow quantities. This system nests within it the static model with zero adjustment costs. Using panel data on U.S. electric utilities, we contrast the results of static and dynamic shadow distance systems. First, the zero-adjustment-cost restriction is strongly rejected. Second, we find that adjustment costs represent about .42 percent of total cost and about 1.26 percent of capital costs. Third, while both models reveal that labor is not utilized efficiently, the dynamic model indicates a longer period of over-use and less variance over time in the degree of inefficiency. With the dynamic model, productivity growth is larger but more stable.

SummaryThis paper proposes a new kinetic‐theory‐based high‐resolution scheme for the Euler equations of gas dynamics. The scheme uses the well‐known connection that the Euler equations are suitable moments of the collisionless Boltzmann equation of kinetic theory. The collisionless Boltzmann equation is discretized using Sweby's flux‐limited method and the moment of this Boltzmann level formulation gives a Euler level scheme. It is demonstrated how conventional limiters and an extremum‐preserving limiter can be adapted for use in the scheme to achieve a desired effect. A simple total variation diminishing criteria relaxing parameter results in improving the resolution of the discontinuities in a significant way. A 1D scheme is formulated first and an extension to 2D on Cartesian meshes is carried out next. Accuracy analysis suggests that the scheme achieves between first‐ and second‐order accuracy as is expected for any second‐order flux‐limited method. The simplicity and the ex...

It has been 40 years since Whitney's seminal paper [W1] on the structure of real algebraic varieties. This paper supplied proofs of a number of results in the folklore, but stopped short of any treatment of local properties. Some years later, Whitney [W2], [W3] returned to study local properties, distinguishing carefully between a number of different tangent cones, and relating them to stratification properties of varieties. In so doing, however, he confined himself largely to the complex analytic category. In the early $1980' \mathrm{s}$ , L\^e, Teissier, and others [HL], [L], [LT] worked out the structure of spaces of limiting tangent spaces at a singularity, again in the complex analytic setting. However, there has, to our knowledge, been no attempt to extend these results to real varieties. This is striking because the tangent cone and the space of limiting tangent spaces

In this paper we give optimal lower bounds for the blow-up rate of theḢ s T 3-norm, 1 2 < s < 5 2 , of a putative singular solution of the Navier-Stokes equations, and we also present an elementary proof for a lower bound on blow-up rate of the Sobolev norms of possible singular solutions to the Euler equations when s > 5 2 .

We consider the Cauchy problem with smooth data for compressible Euler equations in many dimensions and concentrate on two cases: solutions with finite mass and energy and solutions corresponding to a compact perturbation of a nontrivial stationary state. We prove the blowup results using the characteristics of the propagation of the solution in space and find upper and lower bounds for the density of a smooth solution in a given region of space in terms of the initial data. To solve the problems, we introduce a special family of integral functionals and study their temporal dynamics.

It is proved that the radially symmetric solutions of the repulsive Euler-Poisson equations with a non-zero background, corresponding to cold plasma oscillations blow up in many spatial dimensions except for d = 4 for almost all initial data. The initial data, for which the solution may not blow up, correspond to simple waves. Moreover, if a solution is globally smooth in time, then it is either affine or tends to affine as t → ∞.

Exact expressions for the parameters of Stevens Hamiltonian are derived within the framework of a specific model that assumes uniform character of charge density distribution in a certain direction over crystalline lattice. The new model is developed to strengthen the possibilities for analysis of experimental data obtained by inelastic neutron scattering. It is particularly valuable for studying the charge distribution in the layered perovskite-like compounds doped with the rare-earth ions being used as a local probe.

This article deals with the vortex patch problem of the two-dimensional Euler–Boussinesq system. This system couples the incompressible Euler equation for the velocity and a transport diffusion for the temperature with rough initial data of the Yudovich type.