Two-dimensional Gas Dynamic Equations Research Articles

Conservation laws of the two-dimensional gas dynamics equations

Two-dimensional gas dynamics equations in mass Lagrangian coordinates are studied in this paper. The equations describing these flows are reduced to two Euler–Lagrange equations. Using group classification and Noether’s theorem, conservation laws are obtained. Their counterparts in Eulerian coordinates are given. Among these counterparts there are new conservation laws.

Involvement of Dyrk1A, a down syndrome candidate gene, in the thalamocortical connectivity

We present a new Lagrangian discontinuous Galerkin (DG) hydrodynamic method for solving the two-dimensional gas dynamic equations on unstructured hybrid meshes. The physical conservation laws for the momentum and total energy are discretized using a DG method based on linear Taylor expansions. Three different approaches are investigated for calculating the density variation over the element. The first approach evolves a Taylor expansion of the specific volume field. The second approach follows certain finite element methods and uses the strong mass conservation to calculate the density field at a location inside the element or on the element surface. The third approach evolves a Taylor expansion of the density field. The nodal velocity, and the corresponding forces, are explicitly calculated by solving a multidirectional approximate Riemann problem. An effective limiting strategy is presented that ensures monotonicity of the primitive variables. This new Lagrangian DG hydrodynamic method conserves mass, momentum, and total energy. Results from a suite of test problems are presented to demonstrate the robustness and expected second-order accuracy of this new method.

A Lagrangian discontinuous Galerkin hydrodynamic method

We present a new Lagrangian discontinuous Galerkin (DG) hydrodynamic method for solving the two-dimensional gas dynamic equations on unstructured hybrid meshes. The physical conservation laws for the momentum and total energy are discretized using a DG method based on linear Taylor expansions. Three different approaches are investigated for calculating the density variation over the element. The first approach evolves a Taylor expansion of the specific volume field. The second approach follows certain finite element methods and uses the strong mass conservation to calculate the density field at a location inside the element or on the element surface. The third approach evolves a Taylor expansion of the density field. The nodal velocity, and the corresponding forces, are explicitly calculated by solving a multidirectional approximate Riemann problem. An effective limiting strategy is presented that ensures monotonicity of the primitive variables. This new Lagrangian DG hydrodynamic method conserves mass, momentum, and total energy. Results from a suite of test problems are presented to demonstrate the robustness and expected second-order accuracy of this new method.

A discontinuous Galerkin discretization for solving the two-dimensional gas dynamics equations written under total Lagrangian formulation on general unstructured grids

Based on the total Lagrangian kinematical description, a discontinuous Galerkin (DG) discretization of the gas dynamics equations is developed for two-dimensional fluid flows on general unstructured grids. Contrary to the updated Lagrangian formulation, which refers to the current moving configuration of the flow, the total Lagrangian formulation refers to the fixed reference configuration, which is usually the initial one. In this framework, the Lagrangian and Eulerian descriptions of the kinematical and the physical variables are related by means of the Piola transformation. Here, we describe a cell-centered high-order DG discretization of the physical conservation laws. The geometrical conservation law, which governs the time evolution of the deformation gradient, is solved by means of a finite element discretization. This approach allows to satisfy exactly the Piola compatibility condition. Regarding the DG approach, it relies on the use of a polynomial space approximation which is spanned by a Taylor basis. The main advantage in using this type of basis relies on its adaptability regardless the shape of the cell. The numerical fluxes at the cell interfaces are computed employing a node-based solver which can be viewed as an approximate Riemann solver. We present numerical results to illustrate the robustness and the accuracy up to third-order of our DG method. First, we show its ability to accurately capture geometrical features of a flow region employing curvilinear grids. Second, we demonstrate the dramatic improvement in symmetry preservation for radial flows.

Cell-centered discontinuous Galerkin discretization for two-dimensional Lagrangian hydrodynamics

We present a cell-centered discontinuous Galerkin discretization for the two-dimensional gas dynamics equations written using the Lagrangian coordinates related to the initial configuration of the flow, on general unstructured grids. A finite element discretization of the deformation gradient tensor is performed ensuring the satisfaction of the Piola compatibility condition at the discrete level. A specific treatment of the geometry is done, using finite element functions to discretize the deformation gradient tensor. The Piola compatibility condition and the Geometric Conservation law are satisfied by construction of the scheme. The DG scheme is constructed by means of a cellwise polynomial basis of Taylor type. Numerical fluxes at cell interface are designed to enforce a local entropy inequality.

Runge-Kutta discontinuous Galerkin finite element method for two-dimensional gas dynamic equations in unified coordinate

In this paper we develop the Runge-Kutta discontinuous Galerkin finite element method for two-dimensional compressible gas dynamic equations in unified coordinate. The equations for fluid dynamics and geometry conservation laws are solved simultaneously. All the calculations can be performed on the fixed meshes. The information about the grid velocities is not needed in calculation. Some numerical examples are given to evaluate the efficiency and the reliability of the scheme. The numerical results show that the algorithm works well.

A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes

We present a high-order cell-centered Lagrangian scheme for solving the two-dimensional gas dynamics equations on unstructured meshes. A node-based discretization of the numerical fluxes for the physical conservation laws allows to derive a scheme that is compatible with the geometric conservation law (GCL). Fluxes are computed using a nodal solver which can be viewed as a two-dimensional extension of an approximate Riemann solver. The first-order scheme is conservative for momentum and total energy, and satisfies a local entropy inequality in its semi-discrete form. The two-dimensional high-order extension is constructed employing the generalized Riemann problem (GRP) in the acoustic approximation. Many numerical tests are presented in order to assess this new scheme. The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of our new scheme.

Nonstationary flow in a channel with variable cross section under conditions of periodic pulsed energy supply

The effect of a periodic pulsed supply of energy, equal to that of hydrogen combustion in air, on the structure of a supersonic flow in a channel of variable cross section modeling a ramjet tract has been studied. The flow has been numerically simulated based on two-dimensional gasdynamic equations. Various flow regimes are obtained depending on the configuration of energy supply zones and the excess air ratio in the channel.

Modeling the evolution of small distortions of the sphericity of a collapsing bubble

Results of calculating the evolution of small axisymmetric distortions of the spherical shape of a bubble during its collapse are presented. The full model based on two-dimensional gas dynamics equations (gas and liquid are considered inviscid non-heat-conducting) and a number of simplified models are used. The latter are obtained from the full model by splitting the gas and liquid motion into a spherical component and its small non-spherical perturbation. The differences of the simplified models are defined by the assumptions used in the realization of the splitting.

Heat fluxes in plasmatron channel

Numerical simulation of a longitudinally blown electric arc by two-dimensional gas-dynamic equations made it possible to determine temperature and velocity fields, axial and radial components of convective, conductive and radiative heat fluxes, heat losses through a channel wall. The model takes into account radiation transfer in the actual spectrum of an electric-arc plasma. Using the results of calculations for argon and air, approximation expressions are obtained for the temperature dependences of the divergences of radiation fluxes.

An Implicit Lagrangian Method for Solving One- and Two-Dimensional Gasdynamic Equations

The gasdynamic equations are solved by using an implicit Lagrangian algorithm in one and two dimensions. The evolution equation of energy is replaced by the algebraic isentropic condition for each Lagrangian computational cell. The algorithm is essentially developed for isentropic flows but is also applicable to problems involving weak shocks where the entropy increase across the shock is fairly small. The algorithm can be used to predict shock tube problems provided that the entropy change of the shocked fluid is taken into account by incorporating the Rankine-Hugoniot condition. The present method does not require an added artificial viscosity since it contains a built-in mechanism to damp high-frequency disturbances behind shocks. The solution performance of the algorithm is assessed against the exact solution for two shock tube problems. Contact discontinuities are computed with infinite resolution (the number of cells over which the variation occurs is zero). Finally, the algorithm is applied to several gasdynamic problems.

A flux-coordinate-splitting technique for flows with shocks and contact discontinuities

The two-dimensional gasdynamic equations are solved everywhere in the flow field except in regions surrounding the contact discontinuites. A flux-vector-splitting (FVS) technique is applied to the Euler equations so that the directions of propagation of the signals and hence the shocks in the flow can be correctly captured. The split flux equations are solved using conventional second-order-accurate finite difference methods. In the regions surrounding the contact discontinuities, the gasdynamic equations are split into a set of one-dimensional equations. These are transformed in such a way that the density does not appear explicitly in the spatial derivatives of the resultant equations, which are of the Langrangian form. The equations are then solved using second-order-accurate finite difference schemes and numerical smearing of the contact discontinuities is avoided because the dependent variables are continuous across the discontinuities. Consequently, both shocks and contact discontinuities in a two-dimensional gasdynamic flow are accurately resolved. This flux-coordinate-splitting technique is used to calculate the gasdynamic flow in a shock tube, a converging cylindrical shock and the mixing of two supersonic streams. The results are compared with exact solutions and with those deduced from proven numerical techniques. Good correlations are obtained, especially in the sharp definition of contact discontinuities. Therefore, the proposed coordinate-splitting technique improves the resolution of contact discontinuities without affecting the overall calculations of the flow field. In view of this, the coordinate-splitting technique can also be used with other shock capturing techniques besides FVS to achieve the same results.

Computation of the critical gas layer thickness by means of the ratio to detonation wave propagation

The numerical modeling method of detonation wave propagation proposed in earlier work for gas volumes of finite thickness was used in this paper with a more adequate description of the chemical kinetics to compute the critical gas layer thickness of the mixtures 2H/sub 2/ + O/sub 2/, 29% H/sub 2/ + 71% air, 23% H/sub 2/ + 77% air, and 4H/sub 2/ + O/sub 2/. To model the detonation wave propagation, two-dimensional gas dynamic equations in the Lagrange description were taken, supplemented by the chemical kinetics equation in Arrhenius form. A formula was derived for computing specific heat liberation. Results are presented of computed detonation wave velocities for different values of the effective specific heat of combustion as well as for values of the Chapman-Jouguet velocity for the mixtures.