Unsteady Terms Research Articles

Blowdown experimental study on butene ejector

We present the results of a blowdown experiment to investigate the performance of a butene ejector. The blowdown configuration was selected for its flexibility and simplicity allowing for fast experimentation. The experiment featured an ejector where the primary, secondary and back pressure conditions are simulating those present in an ejector refrigeration cycle. The ejector was first tested with air and then 1butene refrigerant. Results showed that single- and double-choking conditions, as well as other well-known ejector features, could be achieved under various operating conditions. The validity of the acquired experimental results, based on the blowdown principle, was confirmed when the order of magnitude of the unsteady terms in the flow governing equations were found to be 105 to 107 times less than the steady terms. Displacing the primary nozzle positively towards the constant area section of the ejector increased both of the entrainment and compression ratios until an optimum position is reached, beyond which the entrainment ratio diminished. Based on the present experimental data, the estimated COP and mechanical COP values for a basic butene ejector cycle suitable for an indoor ice rink application ranged from 0.1 to 0.15, and from 9.2 to 13.7, respectively, when the evaporation temperature varied between −14.5 °C and −6.1 °C.

Simulation of unsteady flow in viscoelastic pipes

The paper focuses on modeling of unsteady flows in a hydraulic system built of a pressure tank, a plastic pipeline and a quick-closing valve. The influence of unsteady friction as well as the experimentally obtained creep function (necessary for modeling retarded strains) on simulation results was investigated. In addition, the effectiveness of the dimensionless parameter P known from the literature was analyzed, especially in the context of rejecting an unsteady term decision. Detailed investigations on the variability of terms describing unsteady friction have shown that one should still look for a dimensionless parameter with the help of which it will be possible to decide on the friction model before making the calculations. The quantitative analysis carried out showed that the use of unsteady hydraulic resistance in simulated runs brought simulated results closer to experimental results.

Modeling of the Instantaneous Transvalvular Pressure Gradient in Aortic Stenosis.

The simplified and modified Bernoulli equations break down in estimating the true pressure gradient across the stenotic aortic valve due to their over simplifying assumptions of steady and inviscid conditions as well as the fundamental nature in which aortic valves are different than idealized orifices. Nevertheless, despite having newer models based on time-dependent momentum balance across an orifice, the simplified and modified Bernoulli continue to be the clinical standard because to date, they remain the only models clinically implementable. The objective of this study is to (1) illustrate the fundamental considerations necessary to accurately model the time-dependent instantaneous pressure gradient across a fixed orifice and (2) propose empirical corrections when applying orifice based models to severely stenotic aortic valves. We introduce a general model to predict the time-dependent instantaneous pressure gradient across an orifice that explicitly model the Reynolds number dependence of both the steady and unsteady terms. The accuracy of this general model is assessed with respect to previous models through pulse duplicator experiments on a round orifice model as well as an explanted stenotic surgical aortic valve both with geometric areas of 0.6cm2 (less than 1cm2 which is the threshold for stenosis determination) over cardiac outputs of 3 and 5L/min and heart rates of 60, 90 and 120bpm. The model and the raw experimental data corresponding to the orifice showed good agreement over a wide range of cardiac outputs and heart rates (R2 exceeding 0.91). The derived average and peak transvalvular pressure gradients also demonstrated good agreement with no significant differences between the respective peaks (p > 0.09). The new model proposed holds promise with its physical and closed form representation of pressure drop, however accurate modeling of the time-variability of the valve area is necessary for the model to be applied on stenotic valves.

A novel Roe solver for incompressible two-phase flow problems

In this paper, a novel Roe-type Riemann solver is developed for solving incompressible two-phase flow problems. The two-phase flow is modeled by coupling the artificial compressibility based Navier-Stokes equations and an interface advection equation in a conservative level set framework. The proposed Roe solver does not assume a locally constant density field while evaluating the flux Jacobian. As a result, the solver is capable of capturing the exact jump conditions across the contact and shear waves, which in turn helps in enhancing the accuracy of the interface motion. Moreover, mass conservation characteristics are significantly improved by adopting conservative level set framework together with finite volume discretization of the system of equations. A dual time stepping method with explicit pseudo-time marching scheme is used for modeling the unsteady terms. The efficacy of the proposed solver is demonstrated using various standard incompressible two-phase flow problems, involving low amplitude sloshing, broken dam, Rayleigh-Taylor instability and droplet splash.

Introducing a modified exact difference method for incorporating unsteady and non-uniform force terms in lattice kinetic models

The exact difference method (EDM) of Kupershtokh (2004), in this work, is modified to accurately incorporate unsteady non-uniform force terms into lattice kinetic models. We use the characteristic scheme for integrating the Boltzmann equation along with the trapezoidal rule to estimate the integration results. The method is based on discretizing the total derivative of the equilibrium distribution function (EDF) in the EDM, using a central second-order scheme, instead of a first order method. It is shown that the assumption of constant density in calculating the total derivative of the EDF is not necessary in the limit of incompressible fluid flows. Using the Chapman–Enskog analysis, it is shown that the present method can be applied to both the standard lattice Boltzmann method and the recently introduced constant speed kinetic model (CSKM) of Zadehgol and Ashrafizaadeh (2014). The following benchmark flows are simulated: (i) Womersley flow, (ii) Hartmann flow, (iii) unsteady Taylor–Green vortex flow, and (iv) the circular interface, using the CSKM, and the results are in excellent agreement with the analytical solutions.

Simulation of Marangoni convection effects on the hydrodynamics of liquid–liquid extraction drops

Concentration induced Marangoni convection (MC) effect has many applications in the chemical engineering processes. In this study, the MC effects have been investigated in the toluene/acetone/water (T/A/W) liquid–liquid extraction system, focusing on the mass transfer and hydrodynamics. For the first time, the VOF-CSS model has been used, and the mass transfer equation has been coupled with the hydrodynamic equations of the VOF model, using the surface tension force model of the CSS. A single mass transfer equation has been solved for the two phases, modifying the unsteady, convective, and diffusive terms based on the equilibrium distribution coefficient. The MC effect has been implemented in the CSS surface tension model with a concentration-dependent surface tension coefficient. Simulations have been carried out in 1, 1.5, and 2 mm drops in the wide range of concentration (0.9–30 g/L). Good agreements have been achieved for the concentration and reduced velocity values compared with the existing experimental data. Simulation results showed that the reduced velocity values remain almost constant for every drop diameter, in the initial concentration range from 3.7 to 30 g/L. The reacceleration time for the three drop sizes with the different initial concentrations has been obtained from the simulation results and compared. The MC effects on the axial velocity profiles and drag coefficients were also reported providing comprehensive information about the MC effects on the hydrodynamics.

A new form of equation for force determination based on Navier-Stokes equations

This work is focused on calculating the force effects of an incompressible homogeneous liquid on a surface of a rigid or a flexible tube. An unsteady flow induced by differential pressure at the beginning and at the end of the tube is assumed. The pressure difference for the unsteady flow is determined experimentally. The mathematical model is based on modified Navier-Stokes equations. The unsteady term is modified in order to be able to use the Gauss-Ostrogradsky theorem to calculate the force. This method of solution will allow the calculation of the force by integration of the Navier-Stokes equations, which will help to refine and simplify the calculations. In the article, both methods of force calculation will be presented and compared both through the ANSYS FEA and CFD ANSYS Fluent solvers and by the integration of the Navier-Stokes equation. The calculation will not only respect the compliance of the tube but also its movement status.

Correcting the Semi-Implicit Numerical Scheme Incorporated in the KORSAR Code Two-Liquid Model

Two algorithms used to correct the semi-implicit numerical scheme for time integration of the conservation equations incorporated into the KORSAR/GP computer code two-liquid model are presented. The KORSAR/GP code has been developed jointly by specialists of the Federal State Unitary Enterprise Aleksandrov Research Institute of Technology and Gidropress Experimental Design Office; in 2009, the code was certified at the Federal Service for Environmental, Technological, and Nuclear Supervision (Rostekhnadzor) as applied to numerical safety assessment of VVER-based reactor plants. In the semi-implicit scheme, the convective terms appearing in the phase momentum conservation equations are written in an explicit form. The mass and energy flows of the phases are represented implicitly with respect to phase velocities, and the transferred donor quantities are calculated based on the parameters taken from the previous time layer. Owing to linearization of unsteady and source terms, the linear system of finite difference equations is solved in a noniteration manner. The first of the presented algorithms compensates for numerical imbalances of phase masses and energies resulting from linearization of unsteady terms appearing in the discrete equations and ensures conservativeness of the scheme. The second algorithm introduces correction for the nonphysical redistribution of coolant mass and energy over the computational cells if a change occurs in the phase motion direction within a time step when the scheme of approximating the convective terms becomes “antidonor” with respect to flow. The numerical imbalances and refinements taking into account the correction of donor quantities are computed at each time step according to the proposed correlations and are used for making compensation at the next time step as supplementary sources in the conservation equations. Results from testing the correction algorithms in the KORSAR/GP code are presented. The effectiveness of the imbalance compensation algorithm is confirmed by solving a problem involving the natural circulation loop heating process. The adequacy of the “antidonor” scheme correction algorithm is demonstrated on problems involving steplike initial distribution of the scalar parameters of a stagnant single-phase gaseous or water coolant in a horizontal tube when the flowrate at the tube inlet has different signs at different time steps.

Application of modified Navier-Stokes equations to determine the unsteady force effects of a heterogeneous liquid

The article is focused on calculating the force effects of a heterogeneous liquid on pipe walls. The solution is based on the concentration of solid particles. The base fluid is assumed to be incompressible. The solution will apply Euler-Lagrange's solution principle. Two tasks will be solved; with a rigid and a flexible tube wall. The solution will be carried out with non-stationary boundary conditions that were determined experimentally. Interaction of a heterogeneous fluid with a flexible wall assumes its deformation. The force effects will be solved by two methods; FSI simulation using ANSYS FEA solvers and CFD solvers ANSYS Fluent and using Navier-Stokes equations by direct integration through liquid volume. In this case, the unsteady term of the Navier-Stokes equations will be modified so that the Gauss Ostrogradsky theorem can be used to calculate the force. At the end, the force effects on the rigid and compliant wall will be compared with the unsteady turbulent flow of the heterogeneous liquid.

Assessment of unsteady Brinkman’s model for flow in karst aquifers

The Brinkman’s equation simplifies the numerical modeling of karst aquifers by allowing the use of a single transport equation to model the flow of fluids in both the free-flow and porous regions, in effect reducing the error arising from improper modeling of the interface between the two regions. Most equations available to model flow within karst aquifers deal with steady flow conditions. This may not be accurate in aquifers where unsteady conditions exist. We considered the effects of unsteady flow conditions in karst aquifers by assessing the addition of an unsteady flow term to the Brinkman’s equation. We solved the coupled mass conservation-transport equations that models unsteady fluid transport in karst aquifers and studied the effects of unsteady flow conditions on tracer transport in two different sample aquifers and compared to the results obtained from the steady flow Brinkman’s equation. The solution method adopted is sequential and it involves solving the unsteady Brinkman’s model first, followed by the advection-diffusion-adsorption equation using the cell-centered finite volume approach. The first example presented here is a simple aquifer model consisting of a single conduit surrounded by porous regions. The second example is a complicated structure consisting of complex geometrical caves embedded in a highly heterogeneous porous media. The results show that, inside the caves, the unsteady Brinkman’s model yielded lower tracer concentrations at early times when compared to the steady flow model. At longer times, both models produced almost similar results. In particular, the results obtained from the simplified example case (Example 1) indicate that the velocity profiles for unsteady flow within open conduits do not instantly yield a parabolic shape expected from the Brinkman’s equation, but gradually develops into one starting from a linear profile. Results obtained also show that the addition of unsteady flow term to the Brinkman’s model does not affect the flow of tracer within porous media in any significantly observable manner.

Introducing unsteady and nonuniform source terms in entropic lattice kinetic models using Fourier series

Introducing unsteady and nonuniform source terms in entropic lattice kinetic models using Fourier series

Development and application of a dynamic model for a solar assisted liquid desiccant air conditioning system

The investigation showed that there were few research works about the real dynamic performance of solar-assisted liquid desiccant air conditioning systems with an unsteady term in the governing equations. Regarding the unsteady properties of the components, a novel dynamic model was developed for a solar-assisted internally cooled liquid desiccant air conditioning system. In the model, the variable t (time) was involved in the governing equations of each component to take into consideration the influence of the previous condition on the following state of the system. The accuracy of the above mathematical model was verified partly with experimental results of some key components. Then the model was applied to a real solar-assisted internally cooled liquid desiccant air conditioning system for an office building with the capacity for eight people in Hong Kong. The transient performance of the system on a summer design day in Hong Kong was investigated. The results reflected the necessity of considering the influence of the former state on the following running of the system. In conclusion, the novel model could be a reliable tool to predicate the system operation and provide useful information for better control of the system.

On the impact of the ISS reboosting maneuvers during thermodiffusion experiments of ternary liquid systems: Pure diffusion

In the present study, the effects of reboosting maneuvers on the diffusion process during the thermodiffusion experiments known as Diffusion and thermodiffusion Coefficients Measurements in ternary liquid mIXtures (DCMIX) are analyzed. These experiments have systematically conducted in the International Space Station (ISS). Six different compositions of a 1,2,3,4-TetraHydroNaphtalene (THN), IsoButylBenzene (IBB) and n-Dodecane (nC12) mixture have been studied by means of numerical simulations. Real acceleration levels have been used to compute the unsteady buoyancy term in the momentum equation. Different gravity scenarios have been considered by changing both the intensity and the duration of the reboosting as well as its orientation, parallel or perpendicular to the concentration gradients. In these thermodiffusion experiments, the different components of the mixture separate in a parallelepipedic Soret cell, with a constant externally imposed temperature gradient between two opposite walls. When the temperature and concentration distributions reach the steady state, and consequently linear concentration profiles between hot and cold walls are achieved, the temperature difference is switched off and the components of the mixture diffuse until uniform concentration distributions are obtained. The present simulations only reproduce this last pure diffusive part of the thermodiffusion experiment. Results show that when a reboosting occurs, errors in the obtaining of the diffusion coefficients arise because of the convection generated. The lower values of the diffusion coefficient show a larger error since they are more sensitive to disturbances. The maximum error is obtained when the acceleration is acting perpendicular to the concentration gradient, which is the most harmful orientation.

Numerical Analysis of the Pressure Peak Position Shift With Deadrise Angle in Two-Dimensional Wedge Water Entry

This study deals with the pressure peak position shift with deadrise angle during the initial phase of a two-dimensional (2D) wedge water entry. The finite volume method with volume of fluid (VOF) and dynamic mesh technique is used to simulate the water entry process of the 2D wedges with the moderate deadrise angles within the range α = 20 deg–60 deg. The results show that with the increasing deadrise angle, the pressure peak position shifts from the spray root to the wedge apex. And, the critical deadrise angle of pressure peak position shift is identified in the range between 40.8 deg and 41 deg, which is more precise than previous studies. In the initial stage of water entry of a 2D wedge, the pressure on wedge side is determined by the dynamic pressure term and unsteady term simultaneously. For the spray root position, at small deadrise angles, the unsteady term is stronger than the dynamic pressure term; at large deadrise angles, the former is weaker than the later.

Fluid hammer analysis with unsteady flow friction model in coiled tubing drilling

Hydraulic vibrator is a useful tool to reduce the mechanical friction and increase the lateral-reach in coiled tubing drilling (CTD). This paper proposes an unsteady friction model to study the interaction between the fluid hammer and the axial vibration of CT with multi-boundary conditions. The proposed model considers the unsteady friction term and comprises the continuity and momentum conserve equations, which are solved by the method of characteristics. Then a series of laboratory experiments with a hydraulic pulse cavitation vibration tool was conducted to validate the model, the simulated results are in good agreement with the experimental measurement. Finally, the characteristic of periodic flow generation and propagation, as well as the effect of crucial flow parameters on the axial vibration are analyzed with the proposed model. The results show that the actual pressure pulse at different positions is the sum of the forward-traveling wave and return wave. Although the magnitude of drag force induced by hydraulic vibrator is not affected by the frequency of flow area, it increases exponentially with flow rate while decreases exponentially with minimum flow area ratio.

Unsteady loads for coaxial rotors in forward flight computed using a vortex particle method

ABSTRACTRecent advances in coaxial rotor design have shown benefits of this configuration. Nevertheless, issues related to rotor-head drag, aerodynamic performance, wake interference, and vibration should also be considered. Simulating the unsteady aerodynamic loads for a coaxial rotor, including the aerodynamic interactions between rotors and rotor blades, is an essential part of analysing their vibration characteristics. In this article, an unsteady aerodynamic analysis based on a vortex particle method is presented. In this method, a reversed-flow model for the retreating side of the coaxial rotor is proposed based on an unsteady panel technique. To account for reversed flow, shedding a vortex from the leading edge is used rather than from the trailing edge. Moreover, vortex-blade aerodynamic interactions are accounted for. The model considers the unsteady pressure term induced on a blade by tip vortices of other blades, and thus accounts for the aerodynamic interaction between the rotors and its contribution to the unsteady airloads. Coupling the reversed-flow model and the vortex-blade aerodynamic interaction model with the viscous vortex-particle method is used to simulate the complex wake of the coaxial rotor. The unsteady aerodynamic loads on the X2 coaxial rotor are simulated in forward flight, and compared with the results of PRASADUM (Parallelized Rotorcraft Analysis for Simulation And Design, developed at the University of Maryland) and CFD/CSD computations with the OVERFLOW and the CREATE-AV Helios tools. The results of the present method agree with the results of the CFD/CSD method, and compare to it better than the PRASADUM solutions. Furthermore, the influence of the aerodynamic interaction between the coaxial rotors on the unsteady airloads, frequency, wake structure, induced flow, and force distributions are analysed. Additionally, the results are also compared against computations for a single-rotor case, simulated at similar conditions as the coaxial rotor. It is shown that the effect of tip vortex interaction plays a significant role in unsteady airloads of coaxial rotors at low speeds, while the rotor blade passing effect is obviously strengthened at high-speed.

On the Approximation of Two-Dimensional Transient Pipe Flow Using a Modified Wave Propagation Algorithm

In this study, a second-order accurate Godunov-type finite volume method is used for the solution of the two-dimensional (2D) water hammer problem. The numerical scheme applied here is well balanced and is able to treat the unsteady friction terms, together with the convective terms, within the differences between fluxes of neighboring computational cells. In order to consider the effect of unsteady friction terms during the water hammer process, k−ε and k−ω turbulence models are employed. The performance of the proposed method with the choice of different turbulence models is evaluated using experimental data obtained from one low and one high Reynolds-number turbulent test cases. In addition to velocity and pressure distributions, the turbulence characteristics of each variant of the model, including eddy viscosity, dissipation rate, and turbulent kinetic energy during the water hammer process are fully analyzed. It is found that the inclusion of the convective inertia terms leads to more accurate pressure profiles. The results also show that using a relatively high Courant–Friedrichs–Lewy (CFL) number close to unity, the introduced numerical solver with both choices of turbulence models provides reasonable and acceptable predictions for the studied flows.

Dynamic behaviour simulation of a liquid desiccant dehumidification system

Among heat driven air conditioning systems, liquid desiccant air conditioning system can be assisted by low water temperature solar collectors. This can reduce the electricity demand of the air conditioning systems. Furthermore, these systems can provide possibility of energy storage in the form of chemical energy in the liquid desiccant solution or thermal energy in the solar collector hot water. This article simulates a liquid desiccant dehumidification cycle by two dynamic models: transient and discrete-steady state model. The transient model takes into consider the unsteady term in the governing equations for both the dehumidifier/regenerator model and also liquid-liquid heat exchanger model. While, in the other dynamic model, the steady state terms are used only. Correspondingly, a liquid desiccant dehumidification system was constructed, troubleshoot and field tested at summer conditions. According to the simulation results, the dynamic models are compatible with the variances in operating conditions and supply heating water. Also, the deviation between the humidity ratio results from simulation and field test effects on the simulated solution concentration with less than 4% deviation. The maximum deviation between the moisture removal rate results from simulation and test are −0.21 g/s in the dehumidifier and −0.5 g/s in the regenerator. Thus, the models of liquid desiccant system have reliability to predict the performance with solution storage mode.The comparison between simulation results shows that the disagreement between the two employed dynamic models is insignificant. However, the transient model reduces the computational time by 21% less than the discrete-steady state model. This can speed up the processing time.

Simulation of turbulent boundary layer wall pressure fluctuations via Poisson equation and synthetic turbulence

Flat plate turbulent boundary layers under zero pressure gradient are simulated using synthetic turbulence generated by the fast random particle–mesh method. The stochastic realisation is based on time-averaged turbulence statistics derived from Reynolds-averaged Navier–Stokes simulation of flat plate turbulent boundary layers at Reynolds numbers $\mathit{Re}_{\unicode[STIX]{x1D70F}}=2513$ and $\mathit{Re}_{\unicode[STIX]{x1D70F}}=4357$. To determine fluctuating pressure, a Poisson equation is solved with an unsteady right-hand side source term derived from the synthetic turbulence realisation. The Poisson equation is solved via fast Fourier transform using Hockney’s method. Due to its efficiency, the applied procedure enables us to study, for high Reynolds number flow, the effect of variations of the modelled turbulence characteristics on the resulting wall pressure spectrum. The contributions to wall pressure fluctuations from the mean-shear turbulence interaction term and the turbulence–turbulence interaction term are studied separately. The results show that both contributions have the same order of magnitude. Simulated one-point spectra and two-point cross-correlations of wall pressure fluctuations are analysed in detail. Convective features of the fluctuating pressure field are well determined. Good agreement for the characteristics of the wall pressure fluctuations is found between the present results and databases from other investigators.

Mathematical modeling of macrosegregation during solidification of binary alloy by Control Volume Finite Element Method

• A mathematical model for solidification with CVFEM is implemented in a computer code. • Coupled heat and mass transfer in solidification of lead-tin alloy is described. • Thermo-solutal convection and freckle formation in phase-change are reported. • Effects of cavity aspect ratio and cooling direction in solidification are studied. • Macrosegregation maps for Pb-48%Sn alloy in rectangular cavities is investigated. Unsteady coupled natural convective heat and mass transfer with liquid to solid phase change is described to analyze the macrosegregation of a Pb–Sn alloy in a rectangular two-dimensional cavity. The effect of cooling direction is investigated, by comparing the numerical simulation of solidification in rectangular molds with one vertical side cooled and with bottom wall cooling, both with the cold wall temperature changing with time. Fluid mechanics, heat and mass transfer coupled equations with unsteady, convective, diffusion and sources terms are solved with the Control Volume Finite Element Method (CVFEM) in cavities with aspect ratios A = H / B = 1 / 3 , 0.6 and 3. Results for the evolution of the distribution of streamlines, solidification front, isotherms and iso-concentrations are obtained with a code developed to characterize coupled fluid, heat and mass transport phenomena. Complex physical processes such as thermo-solutal convection during solidification, freckle formation and the final pattern of macrosegregation are described in terms of the cavity aspect ratio by the mathematical model. A good description of freckle formation and its relation to direction of cooling and aspect ratio has been obtained by the transient-convective–diffusive mathematical model solved by CVFEM.