Unsteady Equations Research Articles

Influence of elasticity of high-concentration paste on unsteady flow in pipeline transportation

In high-pressure pulsatile pipeline transport of high-concentration viscous paste, severe backflow is generated when there are sudden pressure drops due to the paste's elastic recovery, resulting in strong impact and damage to the pipeline system. The paste demonstrates high viscosity, yield stress, compressibility, and shear elasticity. Nevertheless, the effect of the paste's elastoviscoplasticity properties on unsteady flow during pipeline transport remains uncertain. Consequently, we derived the unsteady flow control equations of the paste that accounts for the elastoviscoplasticity behavior and linear compressibility. We conducted experimental research by building a circulating pipeline transportation system and performed numerical analysis using the finite-volume method (modified RheoFoam solver). The results demonstrate a substantial increase in pressure fluctuations within the pipeline during rapid pressure drops due to the elasticity of the paste. When accounting for elasticity, the maximum backflow velocity increases by 32.4% compared to cases where elasticity is not considered. The backflow velocity during piston pump reversal is influenced by the pressure level and the rate of pressure drop. Pressure fluctuations are more likely to propagate along the pipeline during high-pressure than during low-pressure periods. These findings offer guidance for analyzing unsteady flow characteristics in paste pipeline transport and designing pipeline systems.

Numerical Studies of the Influence of the Nature of the In-Chamber Flow on the Initial Period of Operation of the Gas Generator

The ignition of a fuel pellet in a gas generator chamber is a time-consuming complex physico-chemical process. During the ignition of the checker, the operating conditions of the device differ significantly from the conditions of its operation in the main area of operation. In particular, an important characteristic of the initial stage is the pattern of flame propagation over the surface of the fuel. A computational algorithm for modeling unsteady turbulent flows of compressible viscous gas based on numerical integration of unsteady equations obtained using physical conservation laws was used to study the vortex flow of gas in the gas generator path and the ignition process of the fuel block. The method used a system of equations written in a cylindrical coordinate system. The computational algorithm is based on a modified flow vector splitting scheme (Steger – Warming scheme). The algorithm has the 2nd order of accuracy in space. The obtained results of numerical simulation of the gas flow in the gas generator allow us to investigate the patterns of development of the ignition process of the fuel block in time, depending on the nature of the gas flow in the combustion chamber.

Analyzing the effects of suction and injection Reynolds number on the transport process in a hydromagnetic flow through a channel of reactive porous walls

The present paper analyzes the effects of suction and injection Reynolds number on the dispersion process of solute in an incompressible magneto-hydrodynamics flow through a channel. The walls of the channel are permeable and a first order heterogeneous reaction is applied on it. Also, an inclined magnetic field is employed at both walls of the channel. The unsteady convection-diffusion equation is solved semi-analytically to find out the coefficient of dispersion, skewness and kurtosis of the solute distribution for all time period. Using Hermite polynomial representation, the distribution of mean concentration is obtained. The effects of various parameters such as suction Reynolds number (R), injection Reynolds number (R′), boundary absorption parameter (β), inclination angle of the magnetic field (α), Hartmann number (M) and dispersion time (t) on dispersion process of the solute are examined. It is shown that with the increment of suction Reynolds number, the peak of the mean concentration of solute enhances prominently. But, when injection Reynolds number increases, the amplitude of the mean concentration decreases. The current research gives an insight to understand the complex behavior of transport process of the solute in a hydro-magnetic channel flow with reactive porous walls.

Unsteady drag force on an immersed sphere oscillating near a wall

The unsteady hydrodynamic drag exerted on an oscillating sphere near a planar wall is addressed experimentally, theoretically and numerically. The experiments are performed by using colloidal-probe atomic force microscopy in thermal noise mode. The resonance frequencies and quality factors are extracted from the measurement of the power spectrum density of the probe oscillation for a broad range of gap distances and Womersley numbers. The shift in the resonance frequency of the colloidal probe as the probe goes close to a solid wall infers the wall-induced variations of the effective mass of the probe. Interestingly, a crossover from a positive to a negative shift is observed as the Womersley number increases. In order to rationalize the results, the confined unsteady Stokes equation is solved numerically using a finite-element method, as well as asymptotic calculations. The in-phase and out-of-phase terms of the hydrodynamic drag acting on the sphere are obtained and agree well with the experimental results. All together, the experimental, theoretical and numerical results show that the hydrodynamic force felt by an immersed sphere oscillating near a wall is highly dependent on the Womersley number.

Результати проектування надзвукового сопла з подвійним розширенням для рідинного ракетного двигуна першого ступеня методами обчислювального аналізу

The subject of this study is dual-bell supersonic nozzles of liquid rocket engines. The goal is to design a dual-bell supersonic nozzle that will provide maximum thrust for a liquid rocket engine in the first stage of a launch vehicle by solving an optimization problem using numerical simulation methods. The goal of the study is to choose the exit pressures of each part of the dual-bell nozzle based on the known flight trajectory of the launch vehicle, set and solve the problem of optimizing the dual-bell nozzle contour, and assess the impact of the use of a dual-bell nozzle on the efficiency of a liquid-propellant rocket engine. The methods used are: numerical methods for solving the hyperbolic system of unsteady equations of gas dynamics and multidimensional minimization problems. The following results were obtained. The formulation and solution of the optimization problem of the contour of a dual-bell supersonic nozzle, considering the design limitations in the form of a fixed length of the nozzle, is carried out. By analyzing the flight path of the first stage of the launch vehicle, a first approximation of the nozzle contour was obtained. For further calculations, both sections were approximated by parabolas, the coefficients of which, together with the lengths of the sections, formed a vector of optimized parameters. An expression for the axial component of thrust with the opposite sign was used as the objective function to solve the minimization problem. Because of solving the optimization problem, a dual-bell nozzle contour was designed to provide maximum thrust. An assessment of the effectiveness of his work was also conducted. The calculated value of the average along the trajectory of the specific impulse when using a dual-bell nozzle was higher by 1.6% than when using a standard nozzle. Conclusions. The scientific novelty of the obtained results is as follows: the design of the dual-bell nozzle contour was performed by solving the problem of multidimensional optimization, taking into account the design constraints, and using computational analysis methods

Generation of microbubbles via a tapered capillary

We propose a novel method for efficient production of microbubbles based on a tapered capillary with an interiorly attached filament. When gas–liquid displacement driven by an input pressure occurs in the capillary, the gas cone ruptures close to the orifice of the capillary. The generated microbubbles can be pushed out of the capillary and collected by a liquid tank when the pressure is appropriately selected. A liquid column is employed in the straight part of the capillary, which can sustain the liquid film near the capillary orifice and hence the bubble generation by transporting liquid along the filament. Within the working pressure range, increasing the input air pressure leads to a decrease in the microbubble diameter. The minimum diameter of the microbubbles is approximately equal to the orifice diameter of the tapered capillary. In our experiments, microbubbles with a minimum diameter of 1.56 μm can be realized. Theoretically, we derive a one-dimensional unsteady lubrication equation describing the evolution of the gas–liquid interface in a tapered tube. The bubble pinch-off is justified by the numerical solution of the lubrication equation. In particular, the predicted bubble diameters are in agreement with the experimental measurements.

Discretization Error Estimation Using the Unsteady Error Transport Equations

Abstract Computational Fluid Dynamics (CFD) has gained significant utility in the analysis of diverse fluid flow scenarios, thanks to advances in computational power. Accurate error estimation techniques are crucial to ensure the reliability of CFD simulations, as errors can lead to misleading conclusions. This study focuses on the estimation of discretization errors in time-dependent simulations, building upon prior work addressing steady-state problems Wang et al. (2020, “Error Transport Equation Implementation in the Sensei CFD Code,” AIAA Paper No. 2020-1047). In this research, we employ unsteady error transport equations (ETE) to generate localized discretization error estimates within the framework of the finite volume CFD code SENSEI. For steady-state problems, the ETE only need to be solved once after the solution has converged, whereas the unsteady ETE need to be co-advanced with the primal solve. To enhance efficiency, we adopt a one-sided temporal stencil and develop a modified iterative correction process tailored to the unsteady ETE. The time-marching schemes utilized encompass second-order accurate singly diagonally implicit Runge–Kutta (SDIRK) and second-order backward differentiation formula (BDF2), both being implicit. To rigorously assess the accuracy of our error estimates, all test cases feature known analytical solutions, facilitating order-of-accuracy evaluations. Two test cases are considered: the 2D convected vortex for inviscid flow and a cross-term sinusoidal (CTS) manufactured solution for viscous flow. Results indicate higher-order convergence rates for the 2D convected vortex test case even when iterative correction is not applied, with similar observations in the CTS case, albeit not at the finest grid levels. Although the current implementation of iterative correction exhibits lower stability compared to the primal solve, it generally enhances the discretization error estimate. Notably, after iterative correction, the discretization error estimate for the unsteady ETE achieves higher-order accuracy across all grid levels in the 2D CTS manufactured solution.

A general unsteady Fourier solution for orthotropic heat transfer in 2D functionally graded cylinders

In the present research, an analytical solution is presented for unsteady heat transfer in a two‐dimensional finite cylinder made of functionally graded Materials (FGM). The governing equation, as an unsteady partial differential equation (PDE), is solved with general boundary conditions (BCs). The thermal conductivity coefficients are considered as a power function of the radius, both in the longitudinal and radial directions of the FGM cylinder. The presence of non‐constant coefficients and general BCs will confront us with the challenge of solving an unsteady problem with high complexity. A combination of Laplace transform, Fourier transform, and meromorphic functions are utilized to deal with this complex PDE. The Laplace transform was applied to transform to the frequency domain, the Sturm–Liouville equation was used to extract the orthogonal basis Fourier series, and the meromorphic function was deployed to transform back from the frequency domain to the time domain. Results were well verified against previous research works. Results showed that the proposed analytical solution could well predict the temperature distribution. The current analytical solution is useful to better understand unsteady heat transfer in FGMs.

Using Working Parameters to Control Formation and Shape of Lotus-Type Pores

The development and final shape of lotus-type pores can be manipulated in advance during unidirectional solidification in this study. Independent parameters considered are solidification rate, ambient pressure, and factor accounting for solute concentration at a selected reference state in liquid deviated from that at the top surface. Lotus-type porous materials have contemporarily been used in heat sinks, energy including shock, vibration, and sound absorption in aircraft engines, etc., which strongly depend on directions, distributions, and pore shapes in solid. This model accounts for transient gas pressure in the pore affected by solute transfer, gas, capillary and hydrostatic pressures, and Sieverts’ law or Henry’s law at the bubble cap and top surface. Solute transport across the cap self-consistently accounts for solute convection at the cap based on a reference concentration deviated from that at the top free surface, the amount of solute rejected by the solidification front into the pore, and the convection-affected concentration at the solidification front. The resulting simultaneous systems of unsteady first-order ordinary differential equations are solved by a MATLAB code. The length of lotus-type pores is also interpreted by the conservation of the solute content in the system. The predicted final shapes of lotus-type pores agree with algebraic results previously confirmed by available experimental data.

An interwoven composite tailored finite point method for two dimensional unsteady Burgers' equation

This paper centers on constructing a lucid and utilitarian approach to tackle linear and non-linear two-dimensional partial differential equations. To test the applicability of the proposed algorithm a variant of the classical two-dimensional unsteady Burgers' equation is set up as a testing ground. The method in a nutshell reduces to solving one-dimensional Burgers' equations resulting from the application of appropriate operator splitting techniques in the temporal direction. In solving these one-dimensional Burgers' equations a refined tailored finite point method in conjunction with an apposite linearization to the purpose is employed. The conditional stability, consistency, and convergence of the method are established theoretically and the method is found to be first-order convergent in time and second-order convergent in space. To illustrate the accuracy of the scheme, divers examples have been solved and the results obtained prove that this method is top-notch in terms of cost-cutting and time efficiency through the sufficiency of coarse meshes.

A combined BEM and Laplace transform for unsteady modified-Helmholtz equation of time–space variable coefficients for anisotropic media

PurposeTwo-dimensional (2D) problems are governed by unsteady anisotropic modified-Helmholtz equation of time–space dependent coefficients are considered. The problems are transformed into a boundary-only integral equation which can be solved numerically using a standard boundary element method (BEM). Some examples are solved to show the validity of the analysis and examine the accuracy of the numerical method.Design/methodology/approach The 2D problems which are governed by unsteady anisotropic modified-Helmholtz equation of time–space dependent coefficients are solved using a combined BEM and Laplace transform. The time–space dependent coefficient equation is reduced to a time-dependent coefficient equation using an analytical transformation. Then, the time-dependent coefficient equation is Laplace transformed to get a constant coefficient equation, which can be written as a boundary-only integral equation. By utilizing a BEM, this integral equation is solved to find numerical solutions to the problems in the frame of the Laplace transform. These solutions are then inversely transformed numerically to obtain solutions in the original time–space frame.FindingsThe main finding of this research is the derivation of a boundary-only integral equation for the solutions of initial-boundary value problems governed by a modified-Helmholtz equation of time–space dependent coefficients for anisotropic functionally graded materials with time-dependent properties.Originality/valueThe originality of the research lies on the time dependency of properties of the functionally graded material under consideration.

Effect of vertical porous baffle on sloshing mitigation of two-layered liquid in a swaying tank

The sloshing behavior of two-layered liquid in a rectangular sloshing tank with a vertical porous baffle under sway excitation has been investigated using both analytical and numerical methods. The analytical model is developed using the matched eigenfunction expansion method in conjunction with linear potential flow theory. The existence of vertical porous baffle results in energy dissipation by applying the boundary condition that pressure drop across the baffle is proportional only to the quadratic drag term. For comparison, numerical simulations are conducted using computational fluid dynamics (CFD) with unsteady, implicit, and incompressible fluid equations. The study explores the effects of the liquid density ratio, the baffle's porosity, and its submergence depth, on various parameters including the free surface and interfacial elevations, hydrodynamic force on the tank surface, and horizontal force on the baffle. The trends observed in the analytical model align well with the CFD results and experiments. It is found that a fully submerged vertical porous baffle with a porosity close to P = 0.1 leads to stable and less dynamically active sloshing behavior in a two-layered liquid tank.

Stability and boundedness of regular solutions for a Sisko flow in an infinite annular porous space

The present article is intended to study the boundedness of solutions for an unsteady non-Newtonian flow, whose strain–stress relationship is provided by the Sisko fluid model. Such kind of flow can appear in different scenarios, but we make it particular to the field of magnetohydrodynamics, as it is a remarkable area of research in its own right. The ideas exposed in this work can be extended, in a similar manner, to a wide range of applications. To make our fluid further general, we use the Darcy’s law to characterize the porous space in which the fluid is flowing. We develop the boundedness criteria provided that the velocity w and the function g=−∂w/∂r satisfy w,g,∂g/∂r,∂2g/∂r2∈L2(0,T;BMO), where BMO means bounded mean oscillation space. For this purpose, we will consider energy estimates in Sobolev spaces and develop the boundedness criteria for the resulting unsteady parabolic nonlinear equation.

An analytical model of the dynamics of reattaching jets

A jet that emerges from a nozzle and attaches to an inclined, setback wall is studied. The reattaching jet is the canonical element of most fluidic devices, and the aim of this paper is to improve the understanding of its dynamics. An analytical model is developed that describes the dynamic behavior of the jet position in response to acoustic excitation. A novel, unsteady jet curvature equation is derived from first principles, which forms the basis of the model. The Görtler velocity profile for a plane, turbulent jet is assumed, and an alternative approach to determining its virtual origin is used. A novel approach to modeling the momentum balance at the jet reattachment point is demonstrated to predict the observed behavior in response to acoustic excitation. The resulting model is linearized at a series of operating points informed by data from the literature at a range of flow rates, wall setback distances, and gas types. A Monte Carlo analysis is conducted to quantify the model sensitivity to parameter uncertainty. The bandwidth of the jet response to acoustic excitation is demonstrated to depend linearly on jet velocity.

BOUNDARY VALUE PROBLEMS WITH AN INTEGRO-DIFFERENTIAL NON-LOCAL CONDITION FOR COMPOSITE TYPE DIFFERENTIAL EQUATIONS OF THE FOURTH ORDER

The paper studies new nonlocal boundary value problems with an integro-differential boundary condition for unsteady differential equations of the Sobolev type of the fourth order. The peculiarity of the studied problems is that they contain derivatives both in spatial variables and derivatives in time variables in the boundary condition. For the problems under study, the existence and uniqueness theorems of regular solutions are proved – solutions having all derivatives generalized by S.L. Sobolev included in the corresponding equations.

Evolution of liquid–vapour interface during film boiling on sphere

The dynamics of the liquid–vapour interface during stable liquid film boiling on a spherical heater is described by formulating a complete system of equations. The dynamics of the interface surface is analysed using the Rayleigh equation considering the non-equilibrium effects near the vapour–liquid interface in the boundary conditions. Heat transfer in the vapour phase is determined by the thermal conductivity and radiation and that in liquid is described by solving the unsteady thermal conductivity equation. The closed mathematical description is reduced to a system of ordinary differential equations with corresponding initial and boundary conditions under certain assumptions. For high efficiency liquid heat transfer cases, the system of equations can be simplified owing to zero mass flux across the interface. Finally, the system of equations is solved for sub-cooled water and superfluid helium; the corresponding temporal dependences of the vapour films radiuses are presented.The aim of this work is to propose a method for calculating the evolution of the vapour film during film boiling of highly thermally conductive and subcooled liquids, taking into account the molecular-kinetic features of heat and mass transfer processes on the liquid-vapour interfaces.

A numerical solution for unsteady anisotropic diffusion reaction equation of time–space variable coefficients

Two‐dimensional transient problems for anisotropic functionally graded materials governed by a diffusion reaction equation of time–space dependent coefficients are considered. Using a transformation, the space–time dependent coefficients equation is reduced to a time‐dependent coefficients equation. Then taking a Laplace transform will result in a constant coefficients equation, which can be written in forms of boundary‐only integral equations and solved using a standard boundary element method for the solutions. These boundary element method solutions are then numerically inverse transformed to get the solutions in the time variable. Some problems of trigonometrically, exponentially and quadratically graded materials are solved. The numerical solutions obtained are quite accurate.

Parameterization optimal control of an unsteady partial differential equations with convection term by an improved three-term spectrum conjugate gradient algorithm

The unsteady partial differential equations (UPDE) with convection term gives a clear descriptions for the solidification process of a slab in dynamic production of continuous casting. To give a suitable setting value of secondary cooling water flow rate for the dynamic control system, this study investigates an optimal control problem (OCP) of UPDE with convection term. Firstly, control vector discretization of OCP and the solution of UPDE are given. Secondly, due to the rapidity for gradient, this paper analyzes the expression of the gradient calculation method based on Hamiltonian function costate system by approximate treatment, matrix calculation and composite trapezoidal integral method. Thirdly, an improved three-term spectrum conjugate gradient algorithm (ITSCGA) is proposed to solve the OCP of UPDE, and the global convergence of the ITSCGA is demonstrated. Lastly, the performance of ITSCGA is demonstrated by experimental simulations. The results demonstrate that the ITSCGA provides a smaller temperature fluctuations, and improves the quality of a slab.

The classical unsteady boundary layer: A numerical study

AbstractThe transition process from a laminar to turbulent flow near a solid wall (e.g., at the suction side of a slender airfoil) is still not fully understood. We focus on the early stage of the flow transition based on boundary layer (BL) theory, that is, high Reynolds number asymptotic expansions, which reduce the Navier–Stokes equations into simplified forms. While steady solutions of the resulting classical BL equations and corresponding regularisations based on viscous–inviscid interaction theory are studied quite extensively, there is only few information about the fully time‐dependent scenario. Hence, we numerically study the classical unsteady BL and its breakdown structure, the Van Dommelen–Shen singularity, on the simplest example we can imagine, the incompressible planar fluid flow through a rectangular channel with suction. The depicted BL builds at the lower channel wall and follows initially a generic Blasius behaviour, which gets temporally modulated through suction by means of a slot located at the upper wall. As usual, the main part of the flow is inviscid and governed by Laplace's equation. Remarkably, the wall‐slip velocity, or alternatively the pressure gradient imposed on the BL, can be calculated in closed form and we have full control of its temporal dependency through the variation of the applied suction strength. This specific flow configuration allows us to split off the initial Blasius solution and promises high‐accuracy solutions of the remaining unsteady BL equation in the scaled stream function formulation. The Chebyshev collocation is used in the wall‐normal direction, where the infinite domain is mapped onto [ − 1, 1]. On the contrary, we discretise the mainstream direction with a finite difference scheme of second‐order accuracy, which proved to be more precise near the build‐up of singularities. In order to trigger a dynamic breakdown of the unsteady BL equations, we increase the suction rate continuously from zero to a level slightly above the critical value (obtained from the steady case). As a consequence, the system cannot relax to a steady solution and instead blows up in finite time at some mainstream position. We try to resolve this process with high resolution and the final goal is to verify the inner structure (the two‐dimensional blow‐up profile) of the Van Dommelen–Shen singularity and its role in flow separation and transition.

A dynamic-mode-decomposition-based acceleration method for unsteady adjoint equations at low Reynolds numbers

The computational cost of unsteady adjoint equations remains high in adjoint-based unsteady aerodynamic optimization. In this letter, the solution of unsteady adjoint equations is accelerated by dynamic mode decomposition (DMD). The pseudo-time marching of every real-time step is approximated as an infinite-dimensional linear dynamical system. Thereafter, DMD is utilized to analyze the adjoint vectors sampled from these pseudo-time marching. First-order zero frequency mode is selected to accelerate the pseudo-time marching of unsteady adjoint equations in every real-time step. Through flow past a stationary circular cylinder and an unsteady aerodynamic shape optimization example, the efficiency of solving unsteady adjoint equations is significantly improved. Results show that one hundred adjoint vectors contains enough information about the pseudo-time dynamics, and the adjoint dominant mode can be precisely predicted only by five snapshots produced from the adjoint vectors, which indicates DMD analysis for pseudo-time marching of unsteady adjoint equations is efficient.