Steady-state Euler Equations Research Articles

Steady-state simulation of Euler equations by the discontinuous Galerkin method with the hybrid limiter

In the realm of steady-state solutions of Euler equations, the challenge of achieving convergence of residue close to machine zero has long plagued high-order shock-capturing schemes, particularly in the presence of strong shock waves. In response to this issue, this paper presents a hybrid limiter designed specifically for the discontinuous Galerkin (DG) method which incorporates a pseudo-time marching method. This hybrid limiter, initially introduced in DG methods for unsteady problems, preserves the local data structure while enhancing resolution through effective and precise shock capturing. Notably, for the steady problem, the hybridization of the DG solution with the cell average is employed, deviating from the low-order limited DG solution typically used for unsteady problems. This approach seamlessly integrates the previously distinct troubled cell indicator and limiter components resulting in a more cohesive and efficient limiter. It obviates the need for characteristic decomposition and intercell communication, leading to substantial reductions in computational costs and enhancement in parallel efficiency. Numerical experiments are presented to demonstrate the robust performance of the hybrid limiter for steady-state Euler equations both on the structured and unstructured meshes.

Adjoint and Direct Characteristic Equations for Two-Dimensional Compressible Euler Flows

The method of characteristics is a classical method for gaining understanding in the solution of a partial differential equation. It has recently been applied to the adjoint equations of the 2D steady-state Euler equations and the first goal of this paper is to present a linear algebra analysis that greatly simplifies the discussion of the number of independent characteristic equations satisfied along a family of characteristic curves. This method may be applied for both the direct and the adjoint problem. Our second goal is to directly derive in conservative variables the characteristic equations of 2D compressible inviscid flows. Finally, the theoretical results are assessed for a nozzle flow with a classical scheme and its dual consistent discrete adjoint.

High-Order Cut-Cell Discontinuous Galerkin Difference Discretization

We present a high-order cut-cell method based on the discontinuous Galerkin difference (DGD) discretization. We leverage the inherent properties of the DGD basis functions to construct a cut-cell discretization that does not require special treatment to mitigate the small-cell problem. The paper describes how the DGD discretization can be constructed from an existing discontinuous Galerkin (DG) discretization, and we highlight differences between the DG and DGD methods. By performing condition-number studies on one- and two-dimensional model problems, we demonstrate that cut-cell DGD discretization remains well conditioned even when the cut-cell volume is orders of magnitude smaller than neighboring cells. We verify the high-order accuracy of the discretization by solving the two-dimensional steady-state Euler equations.

A hybrid WENO scheme for steady-state simulations of Euler equations

For strong shock waves in solutions of steady-state Euler equations, the high-order shock capturing schemes usually suffer from the difficulty of convergence of residue close to machine zero. Several new weighted essentially non-oscillatory (WENO) type schemes have recently been designed to overcome this long-standing issue. In this paper, a new hybrid strategy is proposed for the fifth-order WENO scheme to simulate steady-state solutions of Euler equations. Compared with the existing WENO schemes, the hybrid WENO scheme performs better steady-state convergence property with less dissipative and dispersive errors. Meanwhile, the essentially oscillation-free feature is kept. In the hybrid strategy, the stencil is distinguished into smooth, non-smooth, or transition regions, which is realized by a simple smoothness detector based on the smoothness indicators in the original WENO method. The linear reconstruction and the specific WENO reconstruction are applied to the smooth and non-smooth regions, respectively. In the transition region, the mixture of the linear and WENO reconstructions is adopted by a smooth transitive interpolation, which plays a vital role in the steady-state convergence for the hybrid scheme. Numerical comparisons and spectral analysis are presented to demonstrate the robust performance of the new hybrid scheme for steady-state Euler equations.

Supersonic jet and nozzle flows in uniform-flow and free-vortex aerodynamic windows of gas lasers

The mathematical models and computational tools for design, analysis and predictions of supersonic jet and nozzle flows in the aerodynamic windows of high-power gas lasers are considered. The steady-state Euler equations describing strong shock waves, contact discontinuities, rarefaction waves and their interactions are solved with the finite-volume solver and space-marching method. The results of numerical simulation of steady-state supersonic flows of inviscid compressible gas in nozzles and under- and over-expanded jets are obtained and analyzed for different pressure ratios in the laser cavity and ambient atmosphere. The flowfields corresponding to the uniform velocity profile and free-vortex velocity profile in the outlet nozzle boundary are compared. Nozzle profiling tools are developed on the basis of numerical solution of a sequence of direct problems. The aerodynamic performance of the window is evaluated in terms of the simulated laser cavity pressure and plenum pressure of the free-vortex supply nozzle. The pressure support characteristic for the aerodynamic window is established by determining the ambient to cavity pressure ratio over a range of aerodynamic window supply pressures.

An Adjoint-Based h-Adaptive Reconstructed Discontinuous Galerkin Method for the Steady-State Compressible Euler Equations

An Adjoint-Based h-Adaptive Reconstructed Discontinuous Galerkin Method for the Steady-State Compressible Euler Equations

On continuation of inviscid vortex patches

Abstract This investigation concerns solutions of the steady-state Euler equations in two dimensions featuring finite area regions with constant vorticity embedded in a potential flow. Using elementary methods of the functional analysis we derive precise conditions under which such solutions can be uniquely continued with respect to their parameters, valid also in the presence of the Kutta condition concerning a fixed separation point. Our approach is based on the Implicit Function Theorem and perturbation equations derived using shape-differentiation methods. These theoretical results are illustrated with careful numerical computations carried out using the Steklov–Poincare method which show the existence of a global manifold of solutions connecting the point vortex and the Prandtl–Batchelor solution, each of which satisfies the Kutta condition.

Analytical Euler Solution for Two Dimensional Compressible Ramp Flow with Experimental Comparison

A newly developed procedure for obtaining analytical asymptotic solutions of the two-dimensional steadystate Euler equations is applied to compressible flow past a ramp to further demonstrate its general utility. The procedure has been shown to be applicable into the low transonic range (shock free). The equations are written in natural streamline coordinates with mass flux and flow angle as dependent variables. Higher-order effects, for example, compressibility, appear as nonhomogeneous forcing terms. This new solution procedure does not require aG reen’s function for the forcing terms and has general applicability to many other disciplines, for example, heat transfer, besides fluid dynamics. Application of the new approach to flow problems having geometric corners, for example, ramp, reveals the typical singularity compounding at higher order. The analytical nature of the solution guides implementation of a nonconformal mapping and a new type of coordinate straining strategy to control the phenomenon and ensure uniform validity. Understanding of the nature of the inviscid flow near a geometric corner can be used to devise improved computational fluid dynamics surface boundary conditions at such singular points, for example, airfoil trailing edge. The analytical validity of the ramp solution is corroborated for this purpose by

Convergence acceleration for the steady-state Euler equations

We consider the iterative solution of systems of equations arising from discretizations of the non-linear Euler equations governing compressible flow. The differential equation is discretized on a structured grid, and the steady-state solution is computed by a time-marching method. A convergence acceleration technique based on semicirculant approximations of the difference operator or the Jacobian is used. Implementation issues and variants of the scheme allowing for a reduction of the arithmetic complexity and memory requirement are discussed. The technique can be combined with a variety of iterative solvers, but we focus on non-linear explicit Runge–Kutta time-integration schemes. The results show that the single-stage forward Euler method can be used, and that the time step is not limited by a CFL-criterion. This results in that the arithmetic work required for computing the solution is equivalent to the work required for a fixed number of residual evaluations.

The fluid mechanics of fire whirls: An inviscid model

Whirling fire plumes are known to increase the danger of naturally occurring or post-disaster fires. In order for a fire whirl to exist, there must be an organized source of angular momentum to produce the large swirl velocities as air is entrained into the fire plume. These vorticity-driven fires occur over a large range of length and velocity scales, and significantly alter the entrainment and combustion dynamics. A new model is derived for a buoyant plume that incorporates angular rotation and neglects dissipation; the result is a form of the steady state Euler equations. Included is a general solution for large density and temperature variations. Results are presented that identify the mechanisms and their effects toward creating a fire whirl.

Space-Marching Method for Calculating Steady Supersonic Flows on a Grid Adapted to the Solution

A noniterative implicit space-marching method based on an adaptive grid approach is developed for solving the 2D steady state Euler equations describing supersonic gas flows without streamwise separation. A grid adapted to the solution is generated by using a variational method optimizing such important properties of a grid as smoothness, orthogonality, and grid cell volume variation simultaneously. The weight function in the integral of adaptation is constructed, so that the grid lines are accumulated near strong gradients and curvatures of the solution curve. A special treatment of the boundary points is proposed to determine the location of nodes near slope discontinuities of the boundary mesh line. The noniterative implicit space-marching method of Y. C. Vigneronet al. is applied to solve the Euler equations written in an arbitrary curvilinear system of coordinates. The streamwise and transverse derivatives in the governing equations are approximated by the second-order upwind Richardson and second-order symmetric TVD schemes, respectively. Implicit boundary condition procedure based on the theory of characteristics for hyperbolic systems of equations is employed. Numerical calculations show that the resolution of strong gradient flow fields can significantly be improved by using the grid adapted to the solution, while the number of grid nodes is the same.

Global Far-Field Computational Boundary Conditions for C- and 0-Grid Topologies

Global far-field computational boundary conditions for inviscid external flow problems have been developed for C- and O-grid topologies. This analysis represents a unified approach for two-dimensional external flow problems. These boundary conditions are derived from analytic solutions of an asymptotic form of the steady-state Euler equations and have improved accuracy compared to characteristic boundary conditions commonly used in practice. The Euler equations are asymptotically linearized about a constant pressure, rectilinear flow condition, which is the true boundary condition at infinity. Previous work had developed higher-order boundary conditions for C-grid topologies by assuming small perturbations in both pressure and flow direction at and beyond the computational boundary, by decoupling the inflow and outflow analyses, and by linearizing the thermodynamic relations. This work lifts these restrictions, although some higher-order compressibility effects are neglected. It is based on a global mapping of the boundary and solution of the resulting Dirichlet-Neumann problem. Because the Euler equations are used to develop the boundary conditions, the flow crossing the boundary can be rotational. The boundary conditions can be used with any numerical Euler solution method and allow computational boundaries to be located very close to the nonlinear region of interest. This leads to a significant reduction in the number of grid points required for numerical solution. Numerical results are presented that show that the boundary conditions and far-field analytic solutions provide a smooth transition across a computational boundary to the true far-field conditions at infinity. They also demonstrate the synergism that can be realized from coupling analytic and computational methods.

On a steady state euler-poisson model for semiconduc

The Euler-Poisson model, a simplified version of the hydrodynamic model for semiconductor devices, consists of a first order nonlinear system coupled to Poisson`s equation. The nonlinear system is just the steady state Euler equations of gas dynamics for a gas of charged particles in a electric field that includes a velocity relaxation term. In the one-dimensional steady state case, the differential equations are elliptic in the subsonic regions, and hyperbolic/elliptic in the transonic regions solution in the one-dimensional case, we refer researchers. For a transonic solution in the one-dimensional case are discussed. They prescribed boundary conditions for electron density and are discussed. They prescribed boundary conditions for electron density and electrostatic potential, and showed that if the current density is small enough, a unique subsonic solution exists. In it was shown that the existence of a subsonic solution in the two-dimensional case, though uniqueness of solution is unknown. Furthermore, in existence and uniqueness of a subsonic solution were proved for potential flow in the three-dimensional case. Without the potential flow assumption, we study in this paper the existence of a subsonic solution in the three-dimensional case. By prescribing a vorticity on inflow boundary, a small normal component of velocity onmore » the whole boundary, and assuming a small variation of the velocity relaxation time in the entire semiconductor domain, we prove the existence of a smooth subsonic solution by the Schauder fixed point theorem. Then, a way of prescribing admissible vorticities on the inflow boundary so that subsonic solutions exist is discussed.« less

Vortical Solutions in Supersonic Corner Flows

Vortical solutions are investigated for inviscid supersonic steady flows, described by the steady-state Euler equations, that occur in the corner region generated by two orthogonal ramps. Complex shock interactions appear, with the formation of a vorticity field in the corner region and the vorticity itself converging into spiral singularities. Symmetric or asymmetric flow configurations are generated within symmetric corners. The investigation is carried out by a numerical technique based on a space-marching procedure with a finite volume approximation. The integration of the conservation laws allows for the correct numerical capturing of shocks and contact surfaces. A flux-differenc e-splitting procedure is used for the calculation of the fluxes on the side walls of the volumes, based on the hyperbolicity of the Euler equations for steady supersonic regimes. A high-order accuracy scheme is introduced founded on the essentially nonoscillatory (ENO) scheme. Numerical results are presented and discussed with reference to similar problems investigated by other authors. HIS paper analyzes inviscid, supersonic conical corner flows generated by two intersecting wedges. Such config- urations are typical of supersonic inlet flows and are similar to one of the four corners of the box-type inlets. The corner flows are conical if the two intersecting walls are plane. Then all flow variables (pressure, velocity, and entropy) will be independent of the spherical radial coordinate centered at the origin of the corner flow, and these flows will depend only on two space dimensions. Moreover, the topology of a conical flow is characterized by its conical or crossflow streamlines (the projections of the three-dimensional streamlines onto a spherical surface centered at the origin of the flow). Two possible shock configurations could be generated by the intersection of two compressive wedges. One is character- ized by a regular reflection and the other by the presence of a Mach disk (irregular reflection) at the intersection of the two- dimensional shocks produced by each wedge; Such possibili- ties have been investigated for a symmetrical geometry (two wedges of equal deflection), and the regular reflection shock configuration has been predicted using linearized theory.1 The Mach disk configuration has been detected experimentally2 and in the numerical solutions of the Euler equations using a shock-capturing approach3 and a shock-fitting approach.4 Figure 1 from Ref. 4 shows in the range of wall deflections (6) and upstream conditions (Mo,) where the regular or the Mach disk configuration is expected to occur. Note that, for deflec- tions above 5 deg, the only possible solution is- the Mach disk shock configuration. Contact surfaces are generated at points where shocks interact because of the different entropies pro- duced by the wave systems on either side of the interaction. However, in the present case of symmetrical geometry and for regular interaction, the strength of the contact surface van- ishes. Conversely, in the cases of irregular reflection, two symmetric contact surfaces are generated at the two triple points that bound the Mach disk. For moderate values of upstream Mach number and deflection, the two contact sur- faces converge on the corner. According to the results of the present study, when the Mach number or the deflection in- creases, the configuration may drastically change.

Far-field computational boundary conditions for two-dimensional external flow problems

Far-field computational boundary conditions for external flow problems have been developed from analytic solutions of the linearized steady-state Euler equations. These boundary conditions have improved accuracy compared with commonly used characteristic boundary conditions. The Euler equations are linearized about a constant pressure, rectilinear flow condition that truly represents conditions at infinity. Because the Euler equations are used to develop the boundary conditions, the flow crossing the boundary can be rotational

Numerical computations of steady transonic and supersonic flow fields

A prediction procedure for computing the shock and expansion waves associated with transonic and supersonic flow is presented. In order to assess the capability of the model, calculations are carried out for a number of benchmark cases, for which analytical solutions or published data are available. The cases considered are: supersonic flow over jet vanes; transonic shocked flow in a nozzle; over- and under-expanded free jets with Mach disc formation; and supersonic flow in a cascade of wedges with shock reflection and Prandtl-Meyer expansion. The model, which includes the solution of the steady-state Euler equations as well as the full Navier-Stokes equations, utilizes some modifications made to an existing finite-volume formulation in order to ensure momentum and energy conservation through the shock waves. In all the cases, the calculations compare favourably with the analytical and experimental results.

A TWO-DIMENSIONAL FLOW PROBLEM FOR STEADY-STATE EULER EQUATIONS

A boundary value problem for plane and spatial axially symmetric flows of an ideal (inviscid and incompressible) fluid in a bounded domain is considered. Bibliography: 35 titles.

Global relaxation procedure for compressible solutions of the steady-state euler equations

Transonic flow solutions of the steady-state Euler equations are obtained based on the global relaxation procedure developed by Rubin and Lin for the Reduced Navier-Stokes (RNS). The same code written for the viscous RNS equations is used to predict inviscid flows by neglecting the viscous term in these equations and deleting the no-slip boundary condition. Results for compressible flows have been computed without any stability limitation or need for explicit artificial viscosity or compressibility. A variety of solutions is presented in this paper. This method provides an alternative approach to solving the Euler equations for certain cases and for capturing of simple imbedded shock waves; although, the primary purpose of the present study is to demonstrate the applicability of this procedure for accurately representing inviscid behavior when the full RNS system is solved. The global relaxation procedure considered here has been developed primarily for solutions of the RNS equations describing flows in which strong viscous-inviscid interaction occurs. Although for the Euler equations viscous-inviscid interaction is not present, the same methodology is still applicable since the boundary value nature in these two sets of equations is predominantly associated with the unknown axial pressure gradient term (Px). Note that the RNS equations contain an exact (Euler) representation of the inviscid flowfield and the presence of the viscous term does not alter this fact. The boundary value character associated with the Px term is easily determined for the steady-state Euler equations (characteristic analysis), and is a result of wave (elliptic) propagation when this term is treated as unknown. Rubin and Lin [1, 21] presented a global relaxation technique for this elliptic problem by using a simple difference form for the numerical approximation of Px. Using boundary-layer type differencing for the convective terms, the method requires a downstream pressure boundary condition which is a characteristic of the boundary value problem. The resulting procedure consists of several forward marching sweeps for pressure relaxation. This procedure has been applied to many complex subsonic and transonic flows. With this simple treatment of the px term, the method is capable of capturing smooth imbedded shock waves, or smooth detached shock waves with imbedded subsonic regions. These results have been presented in Refs [2-5]. Application of the global relaxation procedure to inviscid flows has been demonstrated for incompressible flow over a boattail afterbody in Ref. [3]. This paper will focus on inviscid subsonic, transonic and supersonic solutions of the steady-state Euler equations, with a modified treatment of the Px term to include the compressibility effect at high Math number. The numerical method has been previously developed for viscous compressible flows. The viscous code developed in Ref. [5] is used to compute the inviscid solutions by ignoring the viscous term in the axial momentum equation and the associated no-slip boundary condition. The compressibility correction employed for the px term is based on the results obtained from the eigenvalue analysis by Vigneron et al. [6] and the stability

Stability of Shock Waves Attached to Wedges and Cones

A numerical study of a reduced set of the Euler equations shows that both weak and strong attached shock waves are stable configurations depending on the boundary conditions imposed. The results obtained are in good agreement with experimental observations and with the minimum entropy principle. A heuristic ex- planation of the behavior observed is given in terms of quasistatic considerations. OR flow over a wedge or cone, whose deflection angle is less than the angle associated with shock detachment for the incoming supersonic flow, it is a well-known fact that the steady-state Euler equations admit two different solutions. Each solution is distinguished by the strength of the attached oblique shock. Thus a solution is labeled strong or weak if the shock-wave inclination is, respectively, greater than or less than the shock-wave inclination at detachment. For the strong shock solution, the flow is always subsonic downstream of the shock; for the weak shock solution, the flow is supersonic downstream of the shock, except for a small range of deflection angles in the neighborhood of the detachment angle where the weak shock is followed by subsonic flow. This small region will be ignored in the present work. The existence of multiple solutions has attracted considerable theoretical scrutiny, which has mainly succeeded in obscuring the problem. Indeed, our understanding of this problem has changed little since 1948, when Courant and Friedrichs1 wrote .. .The question arises which of the two actually occurs. It has frequently been stated that the strong one is unstable and that, therefore, only the weak one could occur. A convincing proof of this instability has apparently never been given. Quite aside from the question of stability, the problem of determining which of the possible shocks occurs cannot be formulated and an- swered without taking the boundary conditions at infinity into account. The confusion has come about because the question posed by Courant and Friedrichs and every succeeding investigator has not been the proper question to ask. The question should not be which solution actually occurs, since we have ex- perimental evidence2 that both the weak and the strong solutions do occur, but under what conditions do one or the other solutions occur. The answer to this latter question obviously depends on the boundary conditions, as already indicated by Courant and Friedrichs. Moreover, since the nature of steady-state solution is elliptic for the strong shock and hyperbolic for the weak shock, we should suspect that each problem is governed by a different set of boundary conditions. The purpose of this paper is, therefore, to study the stability of the weak and strong solutions with boundary conditions which are appropriate to each case under in- vestigation.