Unsteady Euler Equations Research Articles

Flows of real gas in nozzles with unsteady local energy supply

When gas flows at a high speed in a channel with a variable cross sectional area and high-intensity energy supply, it experiences complicated physical and chemical processes producing high-temperature gas effects. High-temperature gas effects are a key issue related to design and optimization of nozzles of plasmatron of alternating current. The finite volume method is applied to solve unsteady compressible Euler equations with high-temperature gas effects. Solutions of some benchmark test cases are reported, and comparison between computational results of chemically equilibrium and perfect air flowfields is performed. The results of numerical simulation of one-dimensional and two-dimensional under- and over-expanded nozzle flows with a moving region of energy supply are presented. Output nozzle parameters are calculated as functions of a number and time of burning of plasmatron arcs. The results obtained show a qualitative pattern of gas dynamics and thermal processes in the nozzle with unsteady energy supply demonstrating the displacement of the nozzle shock wave towards the nozzle outlet in the over-expanded nozzle flow in comparison to perfect gas flow.

Development of a Parallel Explicit Finite-Volume Euler Equation Solver using the Immersed Boundary Method with Hybrid MPI-CUDA Paradigm

ABSTRACTThis study proposed the application of a novel immersed boundary method (IBM) for the treatment of irregular geometries using Cartesian computational grids for high speed compressible gas flows modelled using the unsteady Euler equations. Furthermore, the method is accelerated through the use of multiple Graphics Processing Units – specifically using Nvidia’s CUDA together with MPI - due to the computationally intensive nature associated with the numerical solution to multi-dimensional continuity equations. Due to the high degree of locality required for efficient multiple GPU computation, the Split Harten-Lax-van-Leer (SHLL) scheme is employed for vector splitting of fluxes across cell interfaces. NVIDIA visual profiler shows that our proposed method having a computational speed of 98.6 GFLOPS and 61% efficiency based on the Roofline analysis that provides the theoretical computing speed of reaching 160 GLOPS with an average 2.225 operations/byte. To demonstrate the validity of the method, results from several benchmark problems covering both subsonic and supersonic flow regimes are presented. Performance testing using 96 GPU devices demonstrates a speed up of 89 times that of a single GPU (i.e. 92% efficiency) for a benchmark problem employing 48 million cells. Discussions regarding communication overhead and parallel efficiency for varying problem sizes are also presented.

Effects of rotational motion on dynamic aeroelasticity of flexible spinning missile with large slenderness ratio

Abstract The structural rigidity of a spinning missile with large slenderness ratio is usually small, and the structural deformation and rate should not be ignored. Furthermore, rotational motion makes the aeroelasticity more complicated. Therefore, unsteady Euler equations and generalized dynamic aeroelastic equations are coupled simultaneously to simulate the dynamic aeroelastic response of a spinning missile with large slenderness ratio using rigid-motion mesh and radial-basis-function (RBF) morphing mesh techniques. The unsteady Euler equations are solved by computational fluid dynamics (CFD) technique by the in-house code. The Coriolis term and centrifugal loading term due to rotational motion are both considered in the generalized dynamic aeroelastic equations. The rigid-motion mesh and RBF morphing mesh techniques are both based on unstructured mesh, and the rigid-motion mesh is adopted to treat the rigid motion due to rotational motion, while the RBF morphing mesh is employed for flexible structural deformation caused by aeroelasticity. Numerical results of aeroelastic case are well agreed with the experimental results, which validates the numerical method. A missile model with X-X configuration is constructed to investigate the effects of rotational motion on dynamic aeroelasticity. The dynamic aeroelastic responses of the missile with and without rotational motion are simulated, respectively. Comparison results show that the lateral modes and longitudinal modes are coupled together because of rotational motion. In addition, the structural natural frequencies are changed due to rotational motion. In the end, detailed numerical analysis of the generalized dynamic aeroelastic equations used in this paper indicates the mechanism by which the rotational motion leads to the coupling of lateral modes and longitudinal modes and changes the structural natural frequencies.

Wing Flutter Prediction by a Small-Disturbance Euler Method on Body-Fitted Curvilinear Grids

Time-accurate coupled fluid and structure solutions for the unsteady Euler equations are too expensive for routine aeroelastic computation. The small-disturbance Euler (SD-Euler) method is shown to be an efficient and accurate method for computation of the frequency-domain generalized aerodynamics forces, even in the transonic region. The SD-Euler method reduces the unsteady problem to a steady flow solution for the perturbation so that the generalized aerodynamics forces can be evaluated directly, enabling substantial reduction of computational time. A SD-Euler solver on body-fitted curvilinear grids has been developed and validated using two-dimensional airfoil and three-dimensional wing rigid-body pitching test cases. Frequency-domain flutter analyses of the Isogai wing model and the AGARD 445.6 weakened wing are performed using this new solver. Flutter boundary predictions by the SD-Euler method agree well with the results by a coupled fluid–structure method. The flutter mode shift phenomenon for the Isogai wing model and the AGARD 445.6 weakened wing is also demonstrated by using the frequency-domain method.

Study of new accurate supplementary energy modeling with multi-components based on newly developed high resolution scheme

In this work, we developed a code of finite volume method for 1D unsteady Euler equations to carry out the flow field problem due to moving contact discontinuity interface between two different species on both sides of the shock tube. The derived additional energy conservation equation would produce a non-conservative term. It is important to reduce the non-physical numerical oscillation problem and improve the capturing-shock ability and computing accuracy of high resolution problem. This study focuses on the following two parts. The first part, we expanded the energy equation of unsteady Euler equations into the form of i species. Further, we verified the feasibility and reliability of the new numerical energy model under the dual-species condition, and then we executed a series of systematic research to carry out the some experimental verification. Moreover, we adopted the most suitable numerical scheme for new Model B3 and validated with the old energy model. Second, we developed 9 new models of flux limiter and validated these limiters through the condition of dual species.For the validated results, the new energy Model B3 can effectively improve the numerical oscillations and capture phenomenon quite well, which just adopt the lower-order difference method. The validated results of new developed limiter models show that their capturing effects of some limiters are very well.

Acceleration of a 2D unsteady Euler solver with GPU on nested Cartesian grid

Graphics processing unit (GPU) parallel computation is used to accelerate a solver, which computes the 2D unsteady compressible Euler equation discretized with the finite-volume method on a nested Cartesian grid. In this solver, a second-order accurate upwind scheme is adopted, with explicit time-stepping by using the third-order total-variation-diminishing Runge–Kutta method. An improved parallel strategy is implemented, and through dealing with a test case on a three-level nested Cartesian grid, speedup ratios of 9.98–14.04 are achieved respectively at different grid sizes. Furtherly, through a numerical experiment, the relation between the kernel performance and the execution configuration at different grid sizes is examined by monitoring and analyzing the performance indicators. Moreover, approaches to improve kernel performances are explored.

A two-stage fourth-order discontinuous Galerkin method based on the GRP solver for the compressible euler equations

We develop a new fourth-order discontinuous Galerkin method using the generalized Riemann problem (GRP) solver based on the framework of the two-stage fourth-order accurate temporal discretization, with special application to compressible Euler equations. The appealing advantage of the two-stage fourth-order accurate temporal discretization is that it only takes a two-stage time-stepping to achieve the expected fourth-order accuracy. It turns out that the computational cost can be considerably reduced by more than 50% compared with the same order multi-stage SSP Runge–Kutta DG method. A number of test cases are presented to assess the accuracy and the efficiency of the two-stage fourth-order GRP-DG method, which demonstrates its appealing features to be an alternative approach for solving the unsteady compressible Euler equations.

Animproved ghost cell method for flow simulation involving static and moving boundary

In this paper, an improved ghost cell method is implemented to solve flow problems with static and moving boundary. The two-dimensional unsteady compressible Euler equations are discretized on a fixed Cartesian grid by the finite volume method, and the monotonic upstream-centered scheme for conservation law scheme is adopted to obtain the second-order precision by reconstructing the numerical fluxes computed by the AUSM + (advection upstream splitting method +) scheme. In time discretization, the explicit third-order total-variation-diminishing Runge-Kutta method is considered. To simplify the assignment method for the ghost cells and deal with the slit problem, the cells in the flow field are selected as the mirror points, which avoids complex interpolations. For preventing non-physical solutions when the mirror point is very close to the boundary, the current mirror point will be replaced by another one, which is regarded as the second flow field cell located in the direction away from the boundary. Moreover, properties of a ghost cell along the <i>X</i> and <i>Y</i> direction are computed respectively, and then the final property is obtained by using a weighted average method, where the weight is determined by the distance between the ghost cell and corresponding boundary point. In this method, the selection of mirror points does not change abruptly for adjacent ghost cells, and therefore, no kink occurs. Furthermore, considering the gradient of the variable near the boundary, an extended scheme of the improved ghost cell method is achieved, which can deal with the flow problems with moving boundaries. When computing the property of a ghost cell by using the extended scheme, the result may be wrong if the location of a shock is just between two mirror points. In view of the problem above, a shock monitor is implemented to switch to the appropriate approach, that is, when the monitor detects a shock between mirror points, the extended scheme will be replaced by the original ghost cell method. Two typical test cases are investigated to validate the accuracy of the proposed method. The first test case is the Schardin’s problem, in which a shock impinges on a finite wedge and is reflected and diffracted. The results at different grid sizes are obtained, and good agreement with experiment results as well as the previous numerical results is achieved, which shows that the improved ghost cell method can offer the same precision as the body-fitted grid method. The second test case is the cylinder lift-off problem involving moving boundaries and the slit problem. Good agreement with the previous results of a high-order complex ghost cell method shows that the improved simple ghost cell method can meet the requirement for dealing with flow problems with moving boundaries.

Development and validation of the hybrid code for numerical simulation of detonations

The present paper describes a numerical code for simulation of detonation flows on hybrid computational clusters (CPU/GPU). The code can solve 1D, 2D and 3D unsteady Euler equations for multispecies chemically reacting flows using high-order shock-capturing TVD schemes and a finite-rate chemistry solver. The implementation is based on the OpenMP, MPI and CUDA technologies allowing for both flexibility and computational efficiency of the code. The program is verified on the analytical Zeldovich-von Neumann-Doering solution for the 1D detonation wave in H2/O2 mixture. Some examples of 2D computations are also given.

An Optimized Symmetric WENO Method-Based Numerical Simulation of Intense Sound Field Generated by Underwater Plasma Sound Source

The intensive pulse sound wave can be generated by the underwater plasma sound source (UPSS) based on the discharge of the underwater high voltage. The distribution of the sound field is prominently nonlinear. In this paper, the sound field of the intensive UPSS is described by the integral two-dimensional axisymmetric unsteady Euler equations firstly. In order to solve the Euler equations numerically, an optimized fifth-order symmetric WENO (weighted essentially nonoscillatory) method based on the three templates is proposed which is called WENO-SYM3. Without increasing the number of candidate templates, a new symmetric template structure can be obtained by expanding the second template and shifting the third one backwards for one space. The method is validated through numerical examples and experiments, and the results show that WENO-SYM3 has a high distinguished accuracy; meanwhile, its nonphysical oscillations are not obvious. The experimental results are basically the same as the calculation results, and the maximum error is around 3%.

An Assessment of Shock-Disturbances Interaction Considering Real Gas Effects

In this study, a theoretical analysis is performed to assess the interaction of freestream disturbances with a plane normal shock considering real gas effects. Such effects are important in a field with high velocities and high temperatures. To perform the theoretical analysis, the downstream disturbances field is expressed as a mathematical function of the upstream one by incorporating real gas effects in the formulation. Here, the linearized one-dimensional perturbed unsteady Euler equations are used for the classification of the downstream/upstream disturbances field and the linearized one-dimensional perturbed Rankine–Hugoniot equations are applied to provide a relationship between the disturbances field of two sides of the shock. To incorporate real gas effects in the formulation, real gas relations and equilibrium air curve-fits are used in the resulting system of equations. The general formulation presented here may be simplified to derive Morkovin's formulation by the perfect gas assumption. The magnitudes of downstream disturbances field resulting from different types of upstream disturbances field (entropy wave and fast/slow acoustic waves) with the shock are expressed by appropriate analytical relations. Results for different disturbance variables are presented for a wide range of upstream Mach number considering real gas effects and compared with those of the perfect gas and some conclusions are made. The effects of the presence of body are also studied theoretically and the analytical relations for the magnitude of the pressure disturbance at the body for different types of upstream disturbances field considering real gas effects are provided and their results are presented and discussed.

A numerical study on the instability of oblique detonation waves with a two-step induction–reaction kinetic model

In this study, the surface instability of oblique detonation waves (ODW) formed by two-dimensional, semi-infinite wedges is investigated numerically by solving the unsteady Euler equations with a two-step induction–reaction kinetic model. The chemical kinetic model introduces two length scales, namely, induction and reaction lengths, which can be varied independently to change the sensitivity of the chemical reaction and also the shape of the reaction zone structure. The present numerical results elucidate that both smooth and cellular ODW surfaces may appear after the initiation, and the surface becomes unstable when the reaction zone length decreases while keeping the induction zone the same as observed in normal detonation wave propagation. To investigate the degree of instability quantitatively, the oscillations of post-shock pressure inside the reaction zone are examined, and analyzed using Fast Fourier Transformation (FFT) to get the power spectral density (PSD). Results suggest that there are two types of unstable surfaces, one is dominated by random disturbances, without distinct large amplitude unstable modes, on the ODW surface due to the upstream perturbations interacting with the incoming flow and continuous generation within the structure, while the other formed from the inherent disturbances convected from upstream in the initiation region and later developed into dominant unstable modes via an apparent bifurcation pattern. Equivalent to normal detonations, the stability parameter χ as defined by the ratio of induction length over the reaction length multiplied by the global reduced activation energy can also be used to describe qualitatively the trends of the ODW surface instability observed in this study.

Transition Between Different Initiation Structures of Wedge-Induced Oblique Detonations

Oblique detonation waves (ODWs) have been widely studied due to their application potential for airbreathing hypersonic propulsion. Moreover, various formation structures of wedge-induced oblique detonation waves have been revealed in recent numerical investigations. Given the inflow conditions, the wave configuration is dependent on the wedge angle. Hence, any wedge-angle change will induce a transient ODW evolution to transition from one configuration to another. In this study, the transient development created by instantaneously changing the wedge angle is investigated numerically, based on the unsteady two-dimensional Euler equations and one-step irreversible Arrhenius chemical kinetics. The evolution caused by the abrupt wedge-angle change from one smooth initiation structure to another, both with a curved oblique shock/detonation surface at high-Mach-number regime, is investigated. Two processes are analyzed; the first consists of the downstream transition of the ODW initiation region the by decreasing the angle, and the second is the upstream transition by increasing the angle. In the downstream transition, the overall structure moves globally and readjusts continuously, generating an intermediate kinklike initiation structure. In the upstream transition, a localized reaction region forms and induces a more complex process, mainly derived from the different responding speeds of the oblique shock and detonation waves. To avoid the generation of the new localized explosion region, which causes an abrupt change in the initiation position and potentially affects the ODWE’s stability and performance, it is suggested to vary the wedge angle in incremental steps within a certain time interval.

Hysteresis phenomenon of the oblique detonation wave

Hysteresis phenomenon of the oblique detonation wave (ODW) is numerically studied. Two-dimensional unsteady reactive Euler equations are numerically solved as governing equations with a two-step reduced reaction mechanism. Wedge angle variation is realized by modifying inflow direction. It is found that hysteresis phenomenon does exist in ODW problem, i.e., the final state of the ODW is closely relevant to initial condition. Two types of hysteresis are discovered in this study: the hysteresis of upstream-downstream triple point and the hysteresis of smooth-abrupt transition pattern. Detonation/Shock polar analysis on primary triple point structure of abrupt ODW demonstrates that the precursor shock of the ODW near primary triple point is actually a strong shock solution and therefore characterized by local detachment behavior which is responsible for primary triple point's upstream moving. Similar to hysteresis of shock reflection, irreversibility is the mechanism for ODW's hysteresis. It is found that hysteresis will disappear when the wedge angle is smaller than a certain value, which means that it may be impossible to obtain a standing Chapman–Jouguet (CJ) ODW without ignition delay or with short ignition delay at a CJ wedge angle via hysteresis.

Anisotropic goal‐oriented error analysis for a third‐order accurate CENO Euler discretization

SummaryIn this paper, a central essentially non‐oscillatory approximation based on a quadratic polynomial reconstruction is considered for solving the unsteady 2D Euler equations. The scheme is third‐order accurate on irregular unstructured meshes. The paper concentrates on a method for a metric‐based goal‐oriented mesh adaptation. For this purpose, an a priori error analysis for this central essentially non‐oscillatory scheme is proposed. It allows us to get an estimate depending on the polynomial reconstruction error. As a third‐order error is not naturally expressed in terms of a metric, we propose a least‐square method to approach a third‐order error by a quadratic term. Then an optimization problem for the best mesh metric is obtained and analytically solved. The resulting mesh optimality system is discretized and solved using a global unsteady fixed‐point algorithm. The method is applied to an acoustic propagation benchmark.

Low-Order Modeling of Combustion Noise in an Aero-Engine: The Effect of Entropy Dispersion

The present work studies the effect of entropy dispersion on the level of combustion noise at the turbine outlet of the Rolls-Royce ANTLE aero-engine. A new model for the decay of entropy waves, based on modeling dispersion effects, is developed and utilized in a low-order network model of the combustor (i.e., LOTAN code that solves the unsteady Euler equations). The proposed model for the dispersion of entropy waves only requires the mean velocity field as an input, obtained by Reynolds-averaged Navier–Stokes (RANS) computations of the demonstrator combustor. LOTAN is then coupled with a low-order model code (LINEARB) based on the semi-actuator disk model that studies propagation of combustion noise through turbine blades. Thus, by combining LOTAN and LINERAB, the combustion noise and its counterparts, direct and indirect noise, generated at the turbine exit are predicted. In comparison with experimental data, it is found that without the inclusion of entropy dispersion, the level of combustion noise at the turbine exit is overpredicted by almost 2 orders of magnitude. The introduction of entropy dispersion in LOTAN results in a much better agreement with the experimental data, highlighting the importance of entropy wave dispersion for the prediction of combustion noise in real engines. In more detail, the agreement with the experiment for high and low frequencies was very good. At intermediate frequencies, the experimental measurements are still overpredicted; however, the predicted noise is much smaller compared to the case without entropy dispersion. This discrepancy is attributed to (i) the role of turbulent mixing in the overall decay of the entropy fluctuations inside the combustor, not considered in the model developed for the decay of entropy waves, and (ii) the absence of a proper model in LINEARB for the decay of entropy waves as they pass through the turbine blade rows. These are areas that still need further development to improve the prediction of low-order network codes.

An interpolation-free ALE scheme for unsteady inviscid flows computations with large boundary displacements over three-dimensional adaptive grids

A novel strategy to solve the finite volume discretization of the unsteady Euler equations within the Arbitrary Lagrangian–Eulerian framework over tetrahedral adaptive grids is proposed. The volume changes due to local mesh adaptation are treated as continuous deformations of the finite volumes and they are taken into account by adding fictitious numerical fluxes to the governing equation. This peculiar interpretation enables to avoid any explicit interpolation of the solution between different grids and to compute grid velocities so that the Geometric Conservation Law is automatically fulfilled also for connectivity changes. The solution on the new grid is obtained through standard ALE techniques, thus preserving the underlying scheme properties, such as conservativeness, stability and monotonicity. The adaptation procedure includes node insertion, node deletion, edge swapping and points relocation and it is exploited both to enhance grid quality after the boundary movement and to modify the grid spacing to increase solution accuracy. The presented approach is assessed by three-dimensional simulations of steady and unsteady flow fields. The capability of dealing with large boundary displacements is demonstrated by computing the flow around the translating infinite- and finite-span NACA 0012 wing moving through the domain at the flight speed. The proposed adaptive scheme is applied also to the simulation of a pitching infinite-span wing, where the bi-dimensional character of the flow is well reproduced despite the three-dimensional unstructured grid. Finally, the scheme is exploited in a piston-induced shock-tube problem to take into account simultaneously the large deformation of the domain and the shock wave. In all tests, mesh adaptation plays a crucial role.

Simulation of 2D fluid–structure interaction in inviscid compressible flows using a cell-vertex central difference finite volume method

In the present study, the applicability and accuracy of a cell-vertex finite volume method developed are assessed in simulating 2D fluid–structure interaction in inviscid compressible flows where the nonlinear phenomena exist in both the unsteady transonic fluid flows and the large nonlinear deformation of solid structures. The unsteady Euler equations are considered as the governing equations of the fluid flow in the arbitrary Lagrangian–Eulerian form and the large nonlinear deformation of the solid structure is considered to be governed by the Cauchy equations in the total Lagrangian form. Both the domains are discretized by a second-order central-difference cell-vertex finite volume method in space on arbitrary quadrilateral grids and a second-order implicit method in time. For each time step, the resulting set of coupled implicit nonlinear equations in either domains is solved iteratively in the pseudo-time. The fluidic and structural parts are coupled in a partitioned manner for the simulation of fluid–structure interaction problems where at the interface the grids of the two domains are coincident. The formulation for the structural dynamics applied here can accurately simulate different types of boundary conditions. It can also properly model the panel thickness and therefore the stress distribution within the structure can be reasonably obtained. Some benchmark test cases are studied and the results obtained are compared with the available experimental/numerical results. Simplicity of the implementation, similarity of the discretization and accuracy of the solution demonstrate the capability and robustness of the solution methodology proposed in simulating fluid–structure interaction problems in compressible flows.

Quasi-steady State Simulation of Rotating Detonation Engine

We performed a numerical simulation based on the two-dimensional (2-D) unsteady Euler's equation with a single-step Arrhenius reaction model in order to investigate the detonation wave front propagation of an Argon (Ar) diluted oxy-hydrogen mixture (<TEX>$2H_2+O_2+12Ar$</TEX>). This simulation operates in the detonation frame of reference. We examine the effect of grid size and the performance impact of integrated quantities such as mass flow. For a given set of baseline conditions, the minimal and maximum grid resolutions required to simulate the respective detonation waves and the detonation cell structures are determined. Tertiary shock wave behavior for various grids and pre-exponential factors are analyzed. We found that particle fluctuation can be weakened by controlling the mass flow going through the oblique shock waves.

A high-order time formulation of the RBC schemes for unsteady compressible Euler equations

Residual-Based Compact (RBC) schemes can approximate the compressible Euler equations with a high space-accuracy on a very compact stencil. For instance on a 2-D Cartesian mesh, the 5th- and 7th-order accuracy can be reached on a 5×5-point stencil. The time integration of the RBC schemes uses a fully implicit method of 2nd-order accuracy (Gear method) usually solved by a dual-time approach. This method is efficient for computing compressible flows in slow unsteady regimes, but for quick unsteady flows, it may be costly and not accurate enough. A new time-formulation is proposed in the present paper. Unusually, in a RBC scheme the time derivative occurs, through linear discrete operators due to compactness, not only in the main residual but also in the other two residuals (in 2-D) involved in the numerical dissipation. To extract the time derivative, a space-factorization method which preserves the high accuracy in space is developed for reducing the algebra to the direct solution of simple linear systems on the mesh lines. Then a time-integration of high accuracy is selected for the RBC schemes by comparing the efficiency of four classes of explicit methods. The new time-formulation is validated for the diagonal advection of a Gaussian shape, the rotation of a hump, the advection of a vortex for a long time and the interaction of a vortex with a shock.