Unsteady Euler Equations Research Articles

Solution of unsteady Euler equations: Gust–cascade interaction tones

The problem of interaction of a vortical gust with a two-dimensional cascade is considered. Full nonlinear time dependent Euler equations governing the flow are solved employing a 6th-order accurate spatial differencing scheme and a 4th-order accurate time marching technique. The vortical gust is represented by a Fourier series which includes three harmonics. The acoustic response of the cascade for single and multi frequency (vortical) excitations are calculated. The solutions show the generation and propagation of modes that are expected from the theory. It is demonstrated that at low amplitudes of excitation, the time domain analysis produces characteristics of the propagating modes such as the complex mode amplitudes, phase variations, axial waveforms, and tangential waveforms that are in very good agreement with those expected from the linear theory. The exponential decay of the cutoff modes of the first harmonic is also clearly observed. The sound pressure levels of the propagating modes obtained from the present nonlinear time domain analysis are compared with the results of a linearized Navier–Stokes solution and a linearized Euler solution (frequency domain analyses) and good agreement between the results is observed for all the propagating modes.

Thermal gradient effects on the propagation of nonlinear pressure waves in a gas

A numerical study of wave propagation through gases with nonuniform temperature distributions will be presented. The main objective of this study is to determine the impact of temperature gradients on high-intensity, initially sinusoidal pressure waves. Particular emphasis is paid to wave reflection, transmission, and any influence a high-temperature region may have on nonlinear behavior. Ultimately, the performance of thermal barriers in attenuating nonlinear waves is evaluated. The concept of using regions of hot gas inside an ambient environment has potential in aeroacoustic applications, such as jet screech mitigation. This analysis considers one-dimensional compressible unsteady Euler equations with an ideal gas state equation, applied over a uniform domain. The domain is composed of two regions with uniform and equal gas properties (ambient conditions) separated by a third region with higher gas temperature and, accordingly, lower density. Pressure is uniform throughout the domain. We introduce a high-intensity sinusoidal wave into this medium. The shape and extent of the high-temperature zone is varied to study the effect of this region on wave propagation. Further, wave reflection and transmission are studied for a range of wave and thermal field parameters. Results for nonlinear pressure waves are compared to linear acoustic waves.

A superlinearly convergent Mach-uniform finite volume method for the Euler equations on staggered unstructured grids

A Mach-uniform finite volume scheme for solving the unsteady Euler equations on staggered unstructured triangular grids that uses linear reconstruction is described. The scheme is applied to three benchmark problems and is found to be considerably more accurate than a similar scheme based on piecewise constant reconstruction.

Euler flow in a supersonic mixed-compression inlet

Numerical simulation of a two-dimensional mixed compression supersonic inlet is carried out by solving unsteady compressible Euler equations via a stabilized finite element method. The geometry of the inlet is similar to the one used by Anderson and Wong for experimental investigation for Mach 3 flow. The computations are capable of simulating the start-up problems associated with the inlet. The critical back pressure for the successful operation of the inlet is computed. The effect of inlet back pressure on the total pressure recovery and the flow distortion level is analysed. Contrary to the popular belief, it is found that in addition to the throat to inlet capture area ratio, the ramp geometry close to the throat plays an important role in the start-up dynamics. It is demonstrated via simulations that, everything else being same, the geometries of ramp upstream of the throat that are associated with a curvature higher than a certain threshold, result in unstarting the intake.

Application of space–time CE/SE method to shallow water magnetohydrodynamic equations

In this article we apply a space–time conservation element and solution element (CE/SE) method for the approximate solution of shallow water magnetohydrodynamic (SMHD) equations in one and two space dimensions. These equations model the dynamics of nearly incompressible conducting fluids for which the evolution is nearly two-dimensional with magnetic equilibrium in the third direction. In this article we are using a variant of the CE/SE method developed by Zhang et al. [A space–time conservation element and solution element method for solving the two-dimensional unsteady Euler equations using quadrilateral and hexahedral meshes, J. Comput. Phys. 175 (2002) 168–199]. This method uses structured and unstructured quadrilateral and hexahedral meshes in two and three space dimensions, respectively. In this method, a single conservation element at each grid point is employed for solving conservation laws no matter in one, two, and three space dimensions. The present scheme use the conservation element to calculate flow variables only, while the gradients of flow variables are calculated by central differencing reconstruction procedure. We give both one- and two-dimensional test computations. A qualitative comparison reveals an excellent agreement with previous published results of wave propagation method and evolution Galerkin schemes. The one- and two-dimensional computations reported in this paper demonstrate the remarkable versatility of the present CE/SE scheme.

Numerical simulation of the unsteady three-dimensional flow in confined domains crossed by moving bodies

A numerical method devoted to the prediction of unsteady flows in complex domains with moving boundaries is presented. Based on the unsteady Euler equations with source terms to take diffusive effects into account as well as additional mass, momentum or enthalpy sources, it has been specially developed to model the thermal and dynamic behavior of the ambient air inside underground stations in the presence of moving trains. The numerical solution method is a unstructured finite-volume cell-centered scheme using the SIMPLE algorithm coupled with a second-order intermediate time stepping scheme. The spatial discretization is realized with an automatic Cartesian grid generator, complemented by a technique of sliding grids to handle straight moving bodies inside the domain.

Numerical Prediction of Wave Impact Loads on Multiple Rectangular Beams

Wave impact on one and two structural beams with rectangular cross section is simulated with a two-dimensional finite volume method, solving the unsteady Euler equations and employing a VOF-type method for the description of the free surface. Four different test series are carried out, each corresponding to a wave impact scenario in the experimental database of Sterndorff [2002]. For the case of wave impact on a single structural element the numerical results show good agreement, with measured force time histories. In the computations featuring two beams, the prediction of the shadowing effect of the first beam on the second is in reasonable agreement with the experimental data. However, the force peak on the second beam is somewhat over-predicted. The calculations successfully predict a second peak in the force time series of the second beam, which is caused by airborne water shipped over the first beam. Throughout the work, spurious spikes of very short duration appear in the computed load time series, originating from the changing of the flow separation location along the lower edge of the beams.

Calculation of Airfoil Flutterby an Euler Method with Approximate Boundary Conditions

A numerical method is demonstrated for solving the steady and unsteady Euler equations on stationary Cartersian grids for the purpose of time-domain simulation of aeroelastic problems. Wall boundary conditions are implemented on nonmoving mean chord positions by assuming the airfoil being thin and undergoing small deformation, whereas the full nonlinear Euler equations are used in the flowfield for accurate resolution of shock waves and vorticity. The method does not require the generation of moving body-fitted grids and thus can be easily deployed in any fluid-structure interaction problem involving relatively small deformation of a thin body. The first-order wall boundary conditions are used in solving the full Euler equations, and the results are compared with the Euler solutions using the exact boundary conditions and known experimental data. It is shown that the first-order boundary conditions are adequate to represent airfoils of typical thicknesses with small deformation for both steady and unsteady calculations. Flutter boundaries are accurately predicted by this method for the Isogai wing model test case.

The optical distortion mechanism in a nearly incompressible free shear layer

The aero-optical distortions caused by compressible flows have been used by researchers for flow diagnostics and accepted by designers of airborne optical systems as a performance penalty. In order to estimate these distortions, an understanding of the optical distortion mechanism is required. This article examines the mechanisms which produce a variable-density field (and accompanying index-of-refraction field) in a nearly incompressible shear-layer flow. The two-dimensional-shear-layer velocity field was approximated using a discrete vortex model. From this ‘known’ velocity field, the pressure and density fields were determined by iteratively solving the unsteady Euler equations. The resulting index-of-refraction field produced simulated schlieren images which closely resemble experimental schlierens. Optical wavefronts computed from the simulation reasonably match the behaviour of large-scale aberrations measured in a transonic wind tunnel. Small-scale distortions in the experimental data may have been caused by boundary layers on the splitter plate and tunnel walls or by three-dimensional effects that were not simulated.

A Numerical Method for Turbomachinery Aeroelasticity

This work provides an accurate and efficient numerical method for turbomachinery flutter. The unsteady Euler or Reynolds-averaged Navier-Stokes equations are solved in integral form, the blade passages being discretised using a background fixed C-grid and a body-fitted C-grid moving with the blade. In the overlapping region data are exchanged between the two grids at every time step, using bilinear interpolation. The method employs Roe’s second-order-accurate flux difference splitting scheme for the inviscid fluxes, a standard second-order discretisation of the viscous terms, and a three-level backward difference formula for the time derivatives. The dual-time-stepping technique is used to evaluate the nonlinear residual at each time step. The state-of-the-art second-order accuracy of unsteady transonic flow solvers is thus carried over to flutter computations. The code is proven to be accurate and efficient by computing the 4th Aeroelastic Standard Configuration, namely, the subsonic flow through a turbine cascade with flutter instability in the first bending mode, where viscous effect are found practically negligible. Then, for the very severe 11th Aeroelastic Standard Configuration, namely, transonic flow through a turbine cascade at off-design conditions, benchmark solutions are provided for various values of the inter-blade phase angle.

On the computation of multi-material flows using ALE formulation

Computation of compressible multi-fluid flows with a general equation of state using interface tracking and moving grid approach is discussed in this paper. The AUSM+, HLLC, and Godunov methods are presented and implemented in the context of arbitrary Lagrangian–Eulerian formulation for solving the unsteady compressible Euler equations. The developed methods are fully conservative, and used to compute a variety of multi-component flow problems, where the equations of state can be drastically different and stiff. Numerical results indicate that both ALE HLLC and Godunov schemes demonstrate their simplicity and robustness for solving such multi-phase flow problems, and yet ALE AUSM+ scheme exhibits strong oscillations around material interfaces even using a first order monotone scheme and therefore is not suitable for this class of problems.

A study of the impulse wave discharged from the exits of two parallel tubes

The twin impulse wave leads to very complicated flow fields, such as Mach stem, spherical waves, and vortex ring. The twin impulse wave discharged from the exits of the two tubes placed in parallel is investigated to understand the detailed flow physics associated with the twin impulse wave, compared with those in a single impulse wave. In the current study, the merging phenomena and propagation characteristics of the impulse waves are investigated using a shock tube experiment and by numerical computations. The Harten-Yee’s total variation diminishing (TVD) scheme is used to solve the unsteady two-dimensional compressible Euler equations. The Mach number M s of incident shock wave is changed below 1.5 and the distance between two-parallel tubes, L/d, is changed from 1.2 to 4.0. In the shock tube experiment, the twin impulse waves are visualized by a Schlieren optical system for the purpose of validation of computational work. The results obtained show that on the symmetric axis between two-parallel tubes, the peak pressure produced by the twin impulse waves and its location strongly depend upon the distance between two-parallel tubes, L/d and the incident shock Mach number, M s. The predicted Schlieren images represent the measured twin-impulse wave with a good accuracy.

Numerical solution of the unsteady Euler equations for airfoils using approximate boundary conditions

This paper presents an efficient numerical method for solving the unsteady Euler equations on stationary rectilinear grids. Boundary conditions on the surface of an airfoil are implemented by using their first-order expansions on the mean chord line. The method is not restricted to flows with small disturbances since there are no restrictions on the mean angle of attack of the airfoil. The mathematical formulation and the numerical implementation of the wall boundary conditions in a fully implicit time-accurate finite-volume Euler scheme are described. Unsteady transonic flows about an oscillating NACA 0012 airfoil are calculated. Computational results compare well with Euler solutions by the full boundary conditions on a body-fitted curvilinear grid and published experimental data. This study establishes the feasibility for computing unsteady fluid-structure interaction problems, where the use of a stationary rectilinear grid offers substantial advantages in saving computer time and program design since it does not require the generation and implementation of time-dependent body-fitted grids.

Characteristic Euler shock‐fitting formulation for multi‐dimensional flows

AbstractA three‐dimensional explicit time marching algorithm has been developed for the numerical solution of inviscid internal flows. The formulation uses the natural streamline co‐ordinate system. The unsteady Euler equations in non‐conservative form are expressed in terms of the extended Riemann variables and the flow angles. Along the characteristic trajectories in the space–time domain, these equations reduce to a system of ordinary differential equations. Boundary conditions are also implemented in characteristic form. Shock waves are calculated after performing a one‐point shock correction that maintains conservation across the discontinuity. The algorithm has been applied to subsonic, transonic and supersonic test cases. Despite the wide range in the Mach number and the diversity of the tested flow geometries, close agreement have been obtained with available analytical and numerical results. Copyright © 2003 John Wiley & Sons, Ltd.

Optimal Transpiration Boundary Control for Aeroacoustics

We consider the optimal boundary control of aeroacoustic noise, governed by the two-dimensional unsteady compressible Euler equations. The control is the time- and space-varying wall normal velocity on a subset of an otherwise solid wall. The objective functional to be minimized is a measure of acoustic amplitude. Optimal transpiration boundary control of aeroacoustic noise introduces challenges beyond those encountered in direct aeroacoustic simulations or in many other optimization problems governed by compressible Euler equations. One nontrivial issue that arises in the optimal control problem is the formulation and implementation of transpiration boundary conditions. Because suction and blowing on the boundary are allowed, portions of the boundary may change from inflow to outflow, or vice versa, and different numbers of boundary conditions must be imposed at inflow vs outflow boundaries. Another important issue is the derivation of adjoint equations, which are required to compute the gradient of the objective function with respect to the control. Among other things, this is influenced by the choice of boundary conditions for the compressible Euler equations. The approaches to meet these challenges are described and results presented for three model problems. These problems are designed to validate the transpiration boundary conditions and their implementation, study the accuracy of gradient computations, and assess the performance of the computed controls.

Adaptive strategy of transonic flows over vibrating blades with interblade phase angles

AbstractAn error indicator and a locally implicit scheme with anisotropic dissipation model on dynamic quadri‐ lateral–triangular mesh are developed to study transonic flows over vibrating blades with interblade phase angles. In the Cartesian co‐ordinate system, the unsteady Euler equations with moving domain effects are solved. The error indicator, in which unified magnitudes of dynamic grid speed, substantial derivative of pressure, and substantial derivative of vorticity magnitude are incorporated to capture the unsteady wave behaviours and vortex‐shedding phenomena due to unsteadiness. To assess the accuracy of the locally implicit scheme with anisotropic dissipation model on quadrilateral–triangular mesh, two flow calculations are performed. Based on the comparison with the related numerical and experimental data, the accuracy of the present approach is confirmed. According to the high‐resolutional result on the adaptive mesh, the unsteady pressure wave, shock and vortex‐shedding behaviours are clearly demonstrated. Copyright © 2003 John Wiley & Sons, Ltd.

Prediction of Damping Coefficients Using the Unsteady Euler Equations

A prediction method for dynamic damping coefficients using the unsteady Euler equations is presented. Direct unsteady simulation can be used to compute the pitch-damping moment without any geometric approximations when compared to the steady methods using the coning motions. A forced harmonic pitching motion is employed to generate the pitch-damping moments. To compute the pitch- and the roll-damping moments for the basic finner, a dual-time stepping algorithm combined with an implicit multigrid method is applied. The computed coefficients show good agreement with the experimental data. Grid refinement and parametric studies are performed to assess the accuracy of the numerical method. The linearity of the angular rates and the variation with Mach numbers are examined for both pitch- and roll-damping moment coefficients. Through analysis of the pressure distributions at various Mach numbers, the large variations of roll-damping moment coefficient in the transonic region are explained in detail.

Parallelized Three-Dimensional Unstructured Euler Solver for Unsteady Aerodynamics

A parallel algorithm for the solution of unsteady Euler equations on unstructured and moving meshes is developed. A cell-centered finite volume scheme is used. The temporal discretization involves an implicit time-integration scheme based on backward-Euler time differencing. The movement of the computational mesh is accomplished by means of a dynamically deforming mesh algorithm. The parallelization is based on decomposition of the domain into a series of subdomains with overlapped interfaces. The scheme is computationally efficient, time accurate, and stable for large time increments. Detailed descriptions of the solution algorithm are given, and computations for airflow around a NACA0012 airfoil and a missile configuration are presented to demonstrate the applications

Application of the method of lines to unsteady compressible Euler equations

AbstractIn the sense of method of lines, numerical solution of the unsteady compressible Euler equations in 1D, 2D and 3D is split into three steps: First, space discretization is performed by the first‐order finite volume method using several approximate Riemann solvers. Second, smoothness and Lipschitz continuity of RHS of the arising system of ordinary dimensional equations (ODEs) is analysed and its solvability is discussed. Finally, the system of ODEs is integrated in time by means of implicit and explicit higher‐order adaptive schemes offered by ODE packages ODEPACK and DDASPK, by a backward Euler scheme based on the linearization of the RHS and by higher‐order explicit Runge–Kutta methods. Time integrators are compared from several points of view, their applicability to various types of problems is discussed, and 1D, 2D and 3D numerical examples are presented. Copyright © 2003 John Wiley & Sons, Ltd.

Computational simulation of wing rock in three degrees-of-freedom for a generic fighter with chine-shaped forebody

AbstractModern fighter aircraft have been associated with lateral self-excited limit cycle oscillation known as ‘wing rock’. Simulations of wing rock have been encouraged to develop a complete understanding of the fluid mechanism that triggers and drives the oscillation, as well as for prediction purposes. Previous simulations of wing rock in wind/water tunnels were almost exclusively limited to a single degree-of-freedom in roll, due to the difficulty encountered in mounting the model to freely oscillate in more than one degree-of-freedom. Numerical simulations, utilising computational fluid dynamics, were also limited to roll-only degree-of-freedom. The loss of simulation accuracy due to the reduction of the actual wing rock degrees-of-freedom to roll-only has not as yet been fully investigated. In this study wing rock is numerically simulated in three degrees-of-freedom: roll, sideslip, and vertical motion for a generic fighter model. The unsteady Euler equations are coupled with the rigid-body dynamic equations through an innovative sub-iteration algorithm to simultaneously solve the coupled equations. The effect of including the sideslip and vertical degrees-of-freedom was found to delay the onset angle-of-attack of wing rock by 5° and reduce the limit cycle amplitude by about 50% with the frequency remained almost unchanged.