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A variant of the inverse method of characteristics is presented, in the algorithm of which an additional fractional time step is introduced, which makes it possible to carry out calculations with a large time step without loss of accuracy and stability. Calculation formulas of the modified inverse method of characteristics are given for a single-velocity generalized equilibrium model of a gas–liquid mixture. When calculating multidimensional problems, the original system of equations is split into a number of one-dimensional subsystems, for the calculation of which a modified inverse method of characteristics is applied. Using the proposed method, the spreading of a liquid column under the influence of gravitational forces for flat and axisymmetric cases, as well as a drop of liquid falling into a hard smooth barrier were calculated. The calculation results are compared with the available experimental data.

To calculate flows of a gas-liquid mixture, a modified inverse method of characteristics is proposed. An additional fractional time step is introduced in its algorithm, which makes it possible to carry out calculations with a large time step without loss of accuracy and stability. A formulation of boundary conditions on curvilinear walls is discussed in relation to a multidimensional nodal method of characteristics which is based on splitting along the coordinate directions of the original system of equations into a number of one-dimensional subsystems. For the boundary points located on curvilinear impenetrable surfaces, a calculation method based on a method of fictitious nodes is proposed. When testing the modified method, a supersonic interaction of a homogeneous dispersed flow with a barrier is calculated for a flow regime with an attached shock wave. Problems of steady mixture flows near an external obtuse angle, as well as near a cone, which are analogues of Prandtl–Meyer and Busemann flows in gas dynamics, are solved. The calculation results are compared with available self-similar solutions, and a satisfactory agreement is reached.

The method of characteristics type is especially effective for convection-dominated diffusion problems. Due to the nature of characteristic temporal discretization, the method allows one to use a large time step in many practical computations, while all previous theoretical analyses always required certain restrictions on the time stepsize. Here, we present a new analysis to establish unconditionally optimal error estimates for a modified method of characteristics with a mixed finite element approximation to the miscible displacement problem in ℝd (d = 2, 3). For this purpose, we introduce a new characteristic time-discrete system. We prove that the L2 error bound the characteristic time-discrete systemof the fully discrete method of characteristics to the time-discrete system is τ-independent and the numerical solution is bounded in W1,∞-norm unconditionally. With the boundedness, optimal error estimates are established in a traditional manner. Numerical results confirm our theoretical analysis and clearly show the unconditional stability.

We consider a Friedmann-Lemaître-Robertson-Walker physical metric g , an auxiliary metric q with a relativistic charged and colliding plasma in presence of a massive scalar field in Eddington-inspired-Born-Infeld theory of gravity. We first derive a governing system of second order nonlinear partial differential equations. By a judicious change of variables, we manage to build a system of partial differential equations of the first order equivalent to the system of the second order previously found. Then, by the method of characteristics applied to the Boltzmann equation which is a first order hyperbolic equation having as unknown the distribution function f , we construct an iterated sequence and prove the existence and uniqueness of a local solution on a positive time interval [0,  T ). Then, by the continuation criterion and under certain assumptions of smallness on the initial data, the Eddington parameter k and the dimensionless parameter λ , we show that this local solution is global in time.

An accurate prediction of propagation speed and the magnitude induced pressure in water hammer is very critical for the analysis, design, and operation of pipeline transmissions. A new numerical scheme based on an implicit discretization of the method of characteristics solution (IMOC) was proposed in this study. The numerical results were compared with the experimental data and the explicitly discretized solution of the method of characteristics (MOC). Due to the transient nature of the problem, accurate estimation of the head loss can significantly alter the performance of numerical models. In this regards, the performance of both numerical models (i.e., MOC and IMOC) were tested. Four equations were employed based on the steady, quasi-steady, unsteady, and the simplified unsteady algorithms. An acceptable prediction of pressure head with a minimum relative error of −2% and a maximum error of 11% was achieved by using a quasi-steady algorithm for prediction of head loss term in the IMOC model whereas, the relative error of the MOC model was between −11% and +30%. The IMOC model in combination with the simplified unsteady algorithm for estimation of the head loss term had an error range between −20% and +2%, whereas the error range of the MOC model was between −35% and −3%. The numerical results indicated that the MOC model is less accurate than the IMOC in properly modeling the wave propagation speed which resulted in a time lag accumulation between numerical results and the measurements. It was found that the best combination of the numerical scheme and the head loss prediction equation was the IMOC model and the unsteady algorithm, which accurately simulated the time-history of pressure head within ±3% error.

Pressure gain combustion is a promising alternative to conventional gas turbine technologies and within this class the Rotating Detonation Engine has the greatest potential. The Fickett–Jacobs cycle can theoretically increase the efficiency by 15% for medium pressure ratios, but the combustion chamber delivers a strongly non-uniform flow; in these conditions, conventionally designed turbines are inadequate with an efficiency below 30%. In this paper, an original mean-line code was developed to perform an advanced preliminary design of a supersonic turbine; self-starting capability of the supersonic channel has been verified through Kantrowitz and Donaldson theory; the design of the supersonic profile was carried out employing the Method of Characteristics; an accurate evaluation of the aerodynamic losses has been achieved by considering shock waves, profile, and mixing losses. Afterwards, an automated Computational Fluid Dynamics (CFD) based optimization process was developed to find the optimal loading condition that minimizes losses while delivering a sufficiently uniform flow at outlet. Finally, a novel parametric analysis was performed considering the effect of inlet angle, Mach number, reaction degree, peripheral velocity, and blade height ratio on the turbine stage performance. This analysis has revealed for the first time, in authors knowledge, that this type of machines can achieve efficiencies over 70%.

A conservative semi-Lagrangian finite volume method is presented for the numerical solution of convection–diffusion problems on unstructured grids. The new method consists of combining the modified method of characteristics with a cell-centered finite volume discretization in a fractional-step manner where the convection part and the diffusion part are treated separately. The implementation of the proposed semi-Lagrangian finite volume method differs from its Eulerian counterpart in the fact that the present method is applied at each time step along the characteristic curves rather than in the time direction. To ensure conservation of mass at each time step, we adopt the adjusted advection techniques for unstructured triangular grids. The focus is on constructing efficient solvers with large stability regions and fully conservative to solve convection-dominated flow problems. We verify the performance of our semi-Lagrangian finite volume method for a class of advection–diffusion equations with known analytical solutions. We also present numerical results for a transport problem in the Mediterranean sea.

For a stationary potential 2D planar open high-velocity water flow of the ideal liquid, we propose a closed system of nonlinear equations considering the resistance forces to the flow from the channel bottom. Tangential stresses on jet interfaces are ignored. The resistance force components are expressed in terms of velocity components. In this case, the flow equations can be solved through the method of characteristics, and the surface forces are reduced to equivalent volumetric forces. The system of non-linear equations is solved in the velocity hodograph plane; further, the transition to the physical plane takes place. Since the value of the hydrodynamic pressure decreases downstream of the flow, the friction forces to the flow in the first approximation can be considered by using the integral laws of resistance. At that, the form of the equations of motion in the plane of the velocity hodograph does not change. This fact is proved in the article. An example of calculating the water flow is provided. The kinecity, ordinates, and velocities of the flow along its extreme line are calculated without considering resistance forces. Validation of the model in the real flow is performed. Acceptable accuracy relative to experimental data is obtained.

Pigging is a routine operation in the oil and gas industry. In this paper, the governing equation of pig speed was combined with the gas flow equations. The transient equations of gas flow are solved by the method of characteristics (MOC). An experiment was carried out to test the proposed pigging model. The measured speed of the pig coincides with the calculated speed well. The process of a pig carrying a brake unit to pass over a hilly gas pipeline is simulated. The results indicate that the brake unit would lead to a sharp increase of the pressure on the tail of the pig, because the pig is dragged by the brake unit and thus prevented to accelerate together with the gas column in a downhill gas pipeline. This way, the pig speed in a downhill gas pipeline is much lower by using a brake unit, but the speed of pig still can hardly be controlled in the desired range. Furthermore, response surface methodology (RSM) is used to study the maximum speed of pig with/without a brake unit in downhill gas pipeline. Based on the results of the RSM simulations, two equations are present to predict the maximum speed of a pig in a downhill gas pipeline.

This investigation examines the influence of Laval nozzle geometry on thrust production using the method of characteristics. It explores various divergent section configurations, analyzing their impact on nozzle design and performance. Comparative analysis of software tools, study of supersonic flow phenomena, and application of the method of characteristics form the core of this research. Findings showcase strong agreement between computational tools, emphasizing the Mach number's role in divergent section shape variations and thrust force. Additionally, the study scrutinizes over-expansion and under-expansion phenomena, validated through computational simulations. Future research should be aimed at enhancing computational methodologies and investigating additional parameters affecting nozzle performance, promising advancements in rocket propulsion technology.