Unsteady Euler Equations Research Articles

Pressure-Sensitive-Paint-Driven Hybrid Unsteady Compressible Flow Simulation

A hybrid unsteady compressible computational fluid dynamics simulation (CFD) driven by pressure-sensitive paint (PSP) data is proposed, and the concept is demonstrated in two dimensions. By imposing the wall pressure acquired with PSP as the Dirichlet boundary condition, an unsteady compressible Euler equation solver is marched together with integral boundary-layer equations in time. As a result, an entire flowfield that fits the PSP measurement is created in the CFD domain. This concept is tested by taking PSP data from a past article of a wind-tunnel test using the NASA CRM65 airfoil and the steady and unsteady capabilities of this data-driven hybrid simulation are demonstrated for transonic flows in two dimensions, and the convergence characteristics are also investigated. It is found that a steady shock position, which cannot be accurately predicted by the Euler equation solver with a boundary layer, can be rectified by the hybrid simulation. Moreover, unsteady shock motion due to buffeting can also be captured well by the hybrid simulation.

Zero-frequency added mass of submerged bodies with near-surface effects

The added mass of ships, submarines, and marine platforms is a primary quantity of interest to understand vessel dynamics and motion. For low-frequency maneuvers at low forward speed, the water surface acts as a wall and the added-mass depends on the proximity to the free-surface. In this paper a method for efficiently determining the full six-degree-of-freedom zero-speed added-mass tensor is presented for bodies that are on or near a wall boundary. The method is based on solving the unsteady Euler equations from rest for one time step to find the generalized added-mass force. The method is based on the work of el Moctar (2022) but in this paper it is simplified to use a static grid, and thereby extended for general six degree-of-freedom motion. The method is validated for a series of canonical problems with analytical solutions that include a circular cylinder that is both deeply submerged and near a wall, and a spheroid that extends from a wall. The method is further validated by computing the added mass for a very-large-crude carrier that is in finite-depth water. Finally the method is used to compute the full six degree-of-freedom added-mass tensor for the Joubert BB2 Submarine for deep submergence and near the water surface.

Numerical Study of the Effects of Injection Conditions on Rotating Detonation Engine Propulsive Performance

A three-dimensional upwind conservation element and solution element method (CESE) in cylindrical coordinates is first developed to effectively solve the unsteady reactive Euler equations governing a hydrogen–air rotating detonation engine (RDE) with coaxial structures. The effects of the annular width on the structure of the detonation front and the relationship between the thrust and mass flow rate are then investigated. Additionally, RDEs with various injection conditions are systematically analyzed regarding flow patterns and propulsion performance. The results reveal a positive correlation between the specific impulse and the area ratio of the injection slot to the head-end wall. Nevertheless, the specific impulse shows minimal dependence on the injector slot’s location when the area ratio is constant. Ultimately, it is concluded that the area ratio between the injector and the head-end wall is critical in determining the loss of specific impulse.

A Large Time Step Wave-Adding Simple Scheme for Unsteady Compressible Flows

In this paper, we construct a large time step wave-adding scheme (LTS-WA) for unsteady Euler equations in compressible flows with a much simpler strategy in the wave-adding procedure, named LTS-WAS. The new method does not need to calculate coefficients of contribution and also add the waves according to their types and families, which makes the scheme easy for coding and optimisation. The new scheme has a high speedup ratio, and its formulation is the same in scalar and system cases. The scheme is extended to 3D compressible flows using the dimension split method. Numerical experiments show that the new scheme maintains the advantages of the original LTS-WA scheme at large CFL numbers, and has a higher efficiency.

THE APPLICATION OF THE EXTENDED CELLS METHOD TO SIMULATE THE FLOW OF COMBUSTION GASES IN THE LPRE CHAMBER


 
 
 Abstract. The numerical modeling of the process of two-dimensional axisymmetric flow of combustion gases in the chamber of a liquid rocket engine is considered in this study. In general, when solving such problems, meshes are used which lines coincide with the boundaries of the computational domain. However, an alternative solution is proposed here, which is to apply the extended cells method. It allows using rectangular Cartesian grids, which lines do not coincide with the boundaries of the computational domain, without reducing the stability of the numerical solution due to the fractional finite volumes. This also simplifies the setting of boundary conditions in such volumes. The advantage of the proposed approach over the generally accepted one is the absence of the global geometric transformations during the entire modelling process, which leads to a reduction in its duration. To perform the numerical modelling, an inviscid ideal compressible gas of constant chemical composition was chosen as a basic model of a continuum. It is described by a system of the unsteady Euler equations in integral form, which was closed by the Mendeleev-Clapeyron equation of state. For the numerical solution of this system, the finite volume method was used with the reconstruction of the flow parameters by the WENO algorithm of the third order of accuracy. The solution of the Riemann problem was carried out using the Lax-Friedrichs relations. Time integration of the system of equations was performed using the explicit Runge-Kutta method of the third order of accuracy. All calculations were carried out on a uniform rectangular Cartesian mesh, which lines did not coincide with the boundaries of the computational domain. The results were compared with the solution of the same problem using the ANSYS Fluent on an unstructured mesh coinciding with the boundaries of the computational domain. The value of the relative error obtained as a result of comparing both solutions did not exceed 0.05.
 
 

Symmetry group analysis of several coupled fractional partial differential equations

Under investigations were several coupled time–space fractional integrable equations which are called the (1+1)-dimensional KdV-type equations, the (2+1)-dimensional Burgers equations and the (3+1)-dimensional unsteady Euler equations of gas dynamics. In this article, these above considered time–space fractional models through the symmetry analysis approach were explored. As the most basic results, symmetries of these researched equations, were obtained. With the help of the multiple-parameters Erdélyi-Kober fractional operators, they can be reduced into lower dimensional systems. We noticed that this reduction process was somewhat different from the integer case. Besides, we also obtained the explicit power series solution and analyze the convergence to it. By considering the new fractional conservation theorem, conserved quantity of time and space of these time–space fractional systems, were also obtained.

Towards an open-source framework for Fluid–Structure Interaction using SU2, MBDyn and preCICE

Fluid–Structure Interaction (FSI) problems are a primary concern for the aerospace industry. Predicting the deformation of flexible structures due to aerodynamic forces is essential in different scenarios, e.g., for determining the performance of rotorcrafts and wind turbines. In this context, we present a new high-fidelity, open-source, modular, and user-friendly FSI simulation framework by coupling the well-established fluid solver SU2 with the multi-body structural solver MBDyn. The SU2 suite solves steady and unsteady Euler and RANS equations using a finite volume method on unstructured grids. The aerodynamic loads on the surfaces are then passed to MBDyn. Through a multi-body analysis, rotations, displacements, and the structural response of rotorcraft systems are efficiently computed and exchanged back with the fluid solver. A partitioned approach is used between the two codes, each specific for a single physical domain. Both explicit and implicit coupling were considered. The coupling is obtained through the library preCICE, which treats the two specialized solvers as “black boxes” and provides data mapping between non-matching grids. Classical FSI test cases are presented and results are compared with data from literature to validate the framework.

Явное представление сокращенных в размерности уравнений Эйлера сжимаемой жидкости и полной системы уравнений гидродинамики в интегральной форме

Various ways of reducing the complete system of hydrodynamic equations in volume to a system of equations on the surface are of great scientific interest. In this article, the unsteady Euler equations of a compressible fluid and the complete system of hydrodynamic equations are reduced to “stationary” integral-differential equations where time derivatives are absent. Such a procedure (3D → 2D, 2D → 1D) makes it possible to reduce the dimension of the problem by one, which significantly reduces the necessary computing power when modeling them. If to set the correct problem, then one can determine the entire unsteady flow in the volume without solving the unsteady problem. It is enough to set time-varying data only on some surface of this stream, which must be determined independently. The peculiarity of this work is that all equations reduced in dimension are obtained in explicit form, unlike previous works of the authors, where up to 200–500 equations with reduced dimension were proposed, which are very difficult to study and model. In previous authors’ work, dimensionality was reduced by reducing overdetermined systems of differential equations. In this method, in the case of a successful choice of an additional constraint equation for overdetermined systems of differential equations, there is a reduction to PDE systems of dimension less than that of the original PDE systems. The method that is used to reduce the dimension of the Euler equations and the complete system of hydrodynamic equations in this paper is actually a special case of the above. The technique of dimensional reduction in overdetermined systems of PDEs is obviously generalized to the case of integral-differential equations. The article also obtained integral equations in new variables (Lagrangian and pseudo-Lagrangian), which determine the evolution of the flow. Also, a new method is proposed for overdetermination of any system of PDEs using the general integral space relation, which is consequence of the Helmholtz decomposition theorem.

Influence of different interface component distributions on Richtmyer-Meshkov instability

In this paper, the Richtmyer-Meshkov instability is studied numerically by using the high-resolution Roe scheme based on the two-dimensional unsteady Euler equation, which is caused by the interaction between shock wave and the helium circular light gas cylinder with different component distributions. The numerical results are used to further discuss the deformation process of the gas cylinder and the wave structure of the flow field, and also to quantitatively analyze the characteristic dimensions (length, height and central axial width) of the gas cylinder, the time-dependent volume compression ratio of the cylinder. In addition, the flow mechanism of shock-driven interface gas mixing is analyzed from multiple perspectives by combining the flow field pressure, velocity, circulation and gas mixing rate. Then the effects of different initial component distribution conditions on interface instability are investigated. The results show that when the diffusion interface transforms into the sharp interface, the reflection coefficient gradually increases on both sides of interface. When the incident shock wave interacts with the cylinder, the transmission of the shock wave will transform from conventional transmission into unconventional transmission. At the same time, the reflected shock wave is gradually strengthened and the transmitted shock wave is gradually weakened, which leads the Richtmyer-Meshkov instability to be strengthened. Moreover, the Atwood numbers on both sides of the interface also increase as the diffusion interface transforms into the sharp interface, which leads the Rayleigh-Taylor instability and the Kelvin-Helmholtz instability to be strengthened. Therefore, the increase of instability will cause the circulation to increase, resulting in the increase of the growth rate of gas mixing rate.

Busemann diffuser for supersonic ramjet on detonation combustion of kerosene vapor

The possibility of using a convergent-divergent nozzle with a Busemann diffuser in ramjet engines for aerospace engineering applications is studied. Kerosene vapors are considered as a fuel, the combustion of which is carried out in the detonation mode at a hypersonic speed of the oncoming air flow. It is shown that at the altitude of about 40 km with the flight Mach number of 9, the Busemann diffuser can provide stable detonation combustion of kerosene with the efficiency of over 50%. Modeling was carried out on the basis of two-dimensional unsteady Euler equations of motion for axisymmetric flows of a multicomponent reacting gas using the modification of the Godunov numerical scheme, which provides the second order of the solution approximation in terms of spatial variables. The previously published kinetic scheme of ignition and combustion of kerosene vapors is supplemented by the processes of low-temperature combustion of the initial fuel.

Busemann diffuser for supersonic ramjet engine with detonation combustion of hydrogen-air mixture

The expediency of using a Busemann diffuser in a scramjet engine with detonation combustion of hydrogen-air mixtures is studied. A method is developed for calculating an air intake that provides spontaneous initiation of oblique detonation in a hydrogen-air mixture under conditions of rarefied atmosphere at hypersonic freestream Mach numbers. The geometry of the Laval nozzle (convergent-expanding) with the Busemann diffuser for the free flow Mach number 9 is determined. It is shown that at an altitude of about 40 km, the detonation combustion of the hydrogen-air mixture in this nozzle can provide more than 0.4 tons of thrust and more than 35% efficiency. The mathematical model is based on unsteady two-dimensional Euler equations for axisymmetric multicomponent reacting gas flow. To simulate chemical transformations, the detailed kinetic scheme is used that takes into account 33 non-equilibrium reactions for nine components of the mixture. The heat capacity, enthalpy, and entropy of a mixture are calculated using the reduced Gibbs energy of the gas components. Numerical modeling is carried out according to the modified Godunov scheme of the second order of approximation in spatial variables.

An ALE residual distribution scheme for the unsteady Euler equations over triangular grids with local mesh adaptation

This work presents a novel interpolation-free mesh adaptation technique for the Euler equations within the arbitrary Lagrangian–Eulerian framework. For the spatial discretization, we consider a residual distribution scheme, which provides a pretty simple way to achieve high order accuracy on unstructured grids. Thanks to a special interpretation of the mesh connectivity changes as a series of fictitious continuous deformations, we can enforce by construction the so-called geometric conservation law, which helps to avoid spurious oscillations while solving the governing equations over dynamic domains. This strategy preserves the numerical properties of the underlying, fixed-connectivity scheme, such as conservativeness and stability, as it avoids an explicit interpolation of the solution between different grids. The proposed approach is validated through the two-dimensional simulations of steady and unsteady flow problems over unstructured grids.

Multidimensional transonic shock waves and free boundary problems

We are concerned with free boundary problems arising from the analysis of multidimensional transonic shock waves for the Euler equations in compressible fluid dynamics. In this expository paper, we survey some recent developments in the analysis of multidimensional transonic shock waves and corresponding free boundary problems for the compressible Euler equations and related nonlinear partial differential equations (PDEs) of mixed type. The nonlinear PDEs under our analysis include the steady Euler equations for potential flow, the steady full Euler equations, the unsteady Euler equations for potential flow, and related nonlinear PDEs of mixed elliptic–hyperbolic type. The transonic shock problems include the problem of steady transonic flow past solid wedges, the von Neumann problem for shock reflection–diffraction, and the Prandtl–Meyer problem for unsteady supersonic flow onto solid wedges. We first show how these longstanding multidimensional transonic shock problems can be formulated as free boundary problems for the compressible Euler equations and related nonlinear PDEs of mixed type. Then we present an effective nonlinear method and related ideas and techniques to solve these free boundary problems. The method, ideas, and techniques should be useful to analyze other longstanding and newly emerging free boundary problems for nonlinear PDEs.

Implementation of a parallel high-order WENO-type Euler equation solver using a CUDA PTX paradigm

ABSTRACT This study proposes the optimization of a low-level assembly code to reconstruct the flux for a splitting flux Harten–Lax–van Leer (SHLL) scheme on high-end graphic processing units. The proposed solver is implemented using the weighted essentially non-oscillatory reconstruction method to simulate compressible gas flows that are derived using an unsteady Euler equation. Instructions in the low-level assembly code, i.e. parallel thread execution and instruction set architecture in compute unified device architecture (CUDA), are used to optimize the CUDA kernel for the flux reconstruction method. The flux reconstruction method is a fifth-order one that is used to process the high-resolution intercell flux for achieving a highly localized scheme, such as the high-order implementation of SHLL scheme. Many benchmarking test cases including shock-tube and four-shock problems are demonstrated and compared. The results show that the reconstruction method is computationally very intensive and can achieve excellent performance up to 5183 GFLOP/s, ∼66% of peak performance of NVIDIA V100, using the low-level CUDA assembly code. The computational efficiency is twice the value as compared with the previous studies. The CUDA assembly code reduces 26.7% calculation and increases 37.5% bandwidth. The results show that the optimal kernel reaches up to 990 GB/s for the bandwidth. The overall efficiency of bandwidth and computation performance achieves 127% of the predicted performance based on the HBM2-memory roofline model estimated by Empirical Roofline Tool.

Numerical study of shock-disturbances interaction in hypersonic inviscid flows with real gas effects using high-order WENO scheme

In the present study, the shock-disturbances interaction in hypersonic inviscid flows with real gas effects is studied by applying a high-order accurate numerical method with the shock capturing technique. To consider real gas effects, the equilibrium air model is utilized here. The strong conservative form of the unsteady compressible Euler equations in the 2D generalized curvilinear coordinates is formulated and the resulting system of equations for the equilibrium air model is discretized by using the fifth-order finite-difference WENO scheme in space and the explicit third-order TVD Runge–Kutta scheme in time to provide a highly accurate and robust equilibrium airflow solver. The solution methodology adopted based on the high-order WENO scheme with the shock capturing technique can reasonably simulate the strong discontinuities in the solution domain without needing any numerical artifact and this feature makes the solution method suitable for the shock-disturbances interaction in hypersonic flows in which real gas effects are to be considered. To study the physical phenomenon of the shock-disturbances interaction problem in high-speed inviscid equilibrium airflows, two test cases are simulated by applying the solution methodology adopted. In the first test case, the interaction between the incoming fast acoustic wave with the bow shock formed around a cylinder with the nose radius of RN=0.0381 m at M∞=8.03 and T∞=182.333 K is simulated. In the second test case, the cylinder radius is taken RN=2.54×10−2 m and the upstream velocity and temperature are u∞=5590m/s and T∞=1833 K, respectively, and all kinds of free stream disturbances, including the entropy wave and the fast and slow acoustic waves, for different wave numbers are considered. The computations are carried out for both the perfect gas and equilibrium air models and the effects of real gas on both the mean and disturbances fields are also studied. The present study introduces the first-known numerical investigation of the shock-disturbances interaction in hypersonic flow over blunt noses with real gas effects in which the numerical results obtained are thoroughly validated by the analytical/theoretical ones and good agreement is found.

Simulation of propagation and diffraction of a shock wave in a planar curvilinear channel

Subject of Research. Numerical simulation of a shock wave propagation in a plane curved channel is considered on the basis of numerical simulation data. Method. Calculations of an inviscid compressible gas were carried out on the basis of unsteady two-dimensional Euler equations. Discretization of the basic equations was carried out using the finite volume method. Calculations were carried out for different channels with different radius of curvature and Mach numbers of the initial wave. To find the angular position of the front at the current time, the absolute value of the derivative of the density with respect to the angular coordinate was used. The calculation results were compared with the data of a physical experiment. Main Results. The features of the emerging shock-wave flow pattern and its development in time are discussed. The shock-wave configuration observed in channels with different radii of curvature is compared. Some differences in the curvature change of the front of shock waves formed in channels with different radius of curvature are shown. The size of the Mach leg and its change with time depending on the intensity of the initial wave and the size of the annular gap is the angular coordinate function corresponding to the position of the shock wave at the current time. While the maximum Mach number on the outer wall is relatively weakly dependent on the initial wave velocity, the Mach number on the bottom wall decreases with increasing Mach number at the channel entrance. The performed numerical studies show that in all variants there are no non-physical oscillations of the solution. Practical Relevance. The study of shock-wave and detonation processes is of interest for using their potential in pulsed installations and power systems for aircraft and rockets. The calculation results are important for the search of the new flow patterns that guarantee the formation of self-sustained detonation combustion in the combustion chambers of promising propulsion systems. Adjusting the size of the annular gap gives the possibility to select a geometric configuration that will provide the formation of an optimal triple shock wave structure, as well as the required intensity and size of the Mach wave.

Numerical Simulation of Detonation Combustion of Kerosene Vapors in an Expanding Nozzle

The initiation and stabilization of detonation combustion of kerosene vapor in a supersonic air flow entering an expanding axisymmetric nozzle with a coaxial central body is numerically studied. Calculations are based on a reduced kinetic model of combustion, which includes 68 reactions for 44 components. Enthalpy and entropy of components are determined using approximating polynomials from the NASA database. A hydrodynamic model is based on two-dimensional unsteady Euler equations for an axisymmetric flow of a multicomponent reacting gas. Calculations are performed using the Godunov finite-difference scheme and its $$\beta$$ -modification of improved accuracy on smooth solutions. The flow parameters that ensure the steady detonation combustion of kerosene vapor in the nozzle under consideration are determined. The detonation combustion of kerosene has higher thrust than hydrogen combustion, but is noticeably inferior with regard to specific thrust. The calculations are performed on the “Lomonosov" supercomputer at the Lomonosov Moscow State University.

Comparison of the Noise Directivity Pattern of the Main Rotor of a Helicopter for Flight and Hover Modes

The noise of the main rotor of a helicopter in hover and horizontal flight modes is numerically simulated. To calculate the acoustic characteristics caused by the nonstationary load on the blades, a software package is used, validated with experimental data in hover mode. A hybrid calculation method is used, according to which the near field is calculated first, followed by the acoustic characteristics in the far field. In the near field, the values of the field parameters near the rotating blade are found by direct solution of the unsteady Euler equations in a noninertial coordinate system. To calculate the far field, the integral Ffowcs Williams–Hawkings method is applied using permeable control surfaces enclosing the blade.

Simulating aeroelastic response of a self-sustained oscillating rigid airfoil using a boundary oscillation method

The present study presents a new nonlinear unsteady aerodynamic model to investigate the aeroelastic behavior of a self-sustained oscillating rigid airfoil. Here the unsteady Euler equations are considered simulating inviscid compressible transonic flow over an oscillating airfoil. In regard to providing boundedness criteria, a high-order technique based on a normalized variable diagram scheme has been presented. Since using dynamic mesh for simulating flow over a dynamic airfoil is too complex and requires many computational efforts, the current paper proposes a nonorthogonal and static mesh with oscillation of flow boundary. The results are compared with both well-validated numerical methods and experimental data. A time-marching method is employed to determine system responses. The predicted flutter boundary for NACA0012 airfoil at different free-stream Mach numbers is in fair agreement with direct flutter tools of the Hopf bifurcation points. Finally, the influences of the center of mass and elastic axis position on the system aeroelastic behavior are examined.

WENO schemes on unstructured meshes using a relaxed a posteriori MOOD limiting approach

In this paper a relaxed formulation of the a posteriori Multi-dimensional Optimal Order Detection (MOOD) limiting approach is introduced for weighted essentially non-oscillatory (WENO) finite volume schemes on unstructured meshes. The main goal is to minimise the computational footprint of the MOOD limiting approach by employing WENO schemes—by virtue of requiring a smaller number of cells to reduce their order of accuracy compared to an unlimited scheme. The key characteristic of the present relaxed MOOD formulation is that the Numerical Admissible Detector (NAD) is not uniquely defined for all orders of spatial accuracy, and it is relaxed when reaching a 2nd-order of accuracy. The augmented numerical schemes are applied to the 2D unsteady Euler equations for a multitude of test problems including the 2D vortex evolution, cylindrical explosion, double-Mach reflection, and an implosion. It is observed that in many events, the implemented MOOD paradigm manages to preserve symmetry of the forming structures in simulations, an improvement comparing to the non-MOOD limited counterparts which cannot be easily obtained due to the multi-dimensional reconstruction nature of the schemes.