Unsteady Terms Research Articles

Multiple-relaxation-time lattice Boltzmann simulation for flow, mass transfer, and adsorption in porous media.

In this paper, to predict the dynamics behaviors of flow and mass transfer with adsorption phenomena in porous media at the representative elementary volume (REV) scale, a multiple-relaxation-time (MRT) lattice Boltzmann (LB) model for the convection-diffusion equation is developed to solve thetransfer problem with an unsteady source term in porous media. Utilizing the Chapman-Enskog analysis, the modified MRT-LB model can recover the macroscopic governing equations at the REV scale. The coupled MRT-LB model for momentum and mass transfer is validated by comparing with the finite-difference method and the analytical solution. Moreover, using the MRT-LB method coupled with the linear driving force model, the fluid transfer and adsorption behaviors of the carbon dioxide in a porous fixed bed are explored. The breakthrough curve of adsorption from MRT-LB simulation is compared with the experimental data and the finite-element solution, and the transient concentration distributions of the carbon dioxide along the porous fixed bed are elaborated upon in detail. In addition, the MRT-LB simulation results show that the appearance time of the breakthrough point in the breakthrough curve is advanced as the mass transfer resistance in the linear driving force model increases; however, the saturation point is prolonged inversely.

Two-Scale Methodology for URANS/Large Eddy Simulation Solutions of Unsteady Turbomachinery Flows

A general issue in turbomachinery flow computations is how to capture and resolve two kinds of unsteadiness efficiently and accurately: (a) deterministic disturbances with temporal and spatial periodicities linked to blade count and rotational speed and (b) nondeterministic disturbances including turbulence and self-excited coherent patterns (e.g., vortex shedding, shear layer instabilities, etc.) with temporal and spatial wave lengths unrelated to blade count and rotational speed. In particular, the high cost of large eddy simulations (LES) is further compounded by the need to capture the deterministic unsteadiness of bladerow interactions in computational domains with large number of blade passages. This work addresses this challenge by developing a multiscale solution approach. The framework is based on an ensemble-averaging to split deterministic and nondeterministic disturbances. The two types of disturbances can be solved in suitably selected computational domains and solvers, respectively. The local fine mesh is used for nondeterministic turbulence eddies and vortex shedding, while the global coarse mesh is for deterministic unsteadiness. A key enabler is that the unsteady stress terms (UST) of the nondeterministic disturbances are obtained only in a small set of blade passages and propagated to the whole domain with many more passages by a block spectral mapping. This distinctive multiscale treatment makes it possible to achieve a high-resolution unsteady Reynolds-averaged Navier–Stokes (URANS)/LES solution in a multipassage/whole annulus domain very efficiently. The method description will be followed by test cases demonstrating the validity and potential of the proposed methodology.

Implicit Virtual Boundary Method for Moving Boundary Problems on Non-Staggered Cartesian Patch Grids

AbstractA simple numerical method is proposed in the paper for fluid flow around a moving boundary of irregular shape. The unsteady term is discretized with the implicit scheme such that large time step is allowed. All of the computations are performed on non-staggered Cartesian grid system. Fine Cartesian patch grid system covering the moving object is employed to resolve the solution around the solid body. A closed curve defined by connecting the solid grid points adjacent to the solid-liquid interface is referred to as virtual boundary. The narrow irregular strip of solid between the virtual boundary and the actual solid-fluid interface is called pseudo-fluid. The general fluid region consisting of both fluid and pseudo-fluid is a regular domain that can be efficiently solved with conventional numerical method. In this connection, external force is imposed at each fluid grid point adjacent to the solid-fluid interface to compensate for the numerical error arising from the assumption of pseudo-fluid. The solution procedure is iterated until the required external force converges. Accuracy of the new numerical method is validated through three test problems. The numerical method then is used to investigate the flow induced by the flapping wings of a tethered dragonfly in literature. The corresponding CFL numbers of the four examples are infinity, 20, 100, and 3.29.

Numerical study on the temporal variations and physics of heat transfer coefficient on a flat plate with unsteady thermal boundary conditions

Numerical simulations with unsteady RANS method have been performed to study the effect of unsteady thermal boundary conditions on the temporal behavior of heat transfer coefficient in unsteady heat transfer processes. Steady flow velocity condition of U = 21 m/s was used in the computational models, including conjugated model and unconjugated models, to focus on the effects of unsteady flow temperature and thermal boundary conditions on the wall. Especially for the unsteady heat flux boundary condition, analysis has been performed on the effects of pulsation frequency, mean value and amplitude which range from 0 to 8 Hz, 1000–4000W/m2 and 500–1000W/m2 respectively. Temporal behavior of two kinds of heat transfer coefficient with different reference temperature has been analyzed. The results show that the conventionally defined heat transfer coefficient hTg(τ) with the mainstream temperature pulsates intensely for the conditions with unsteady flow temperature. When the local adiabatic wall temperature Taw(τ), which has phase shift from local mainstream temperature Tg(τ), is used to define the heat transfer coefficient hTaw(τ), it can exclude the influence of unsteady flow temperature and make the hTaw(τ) be a good invariant descriptor in heat transfer under unsteady flow temperatures. Unsteady heat flux boundary condition, which can cause unsteady hTaw(τ) under flow conditions with steady velocity and temperature, is an important reason for the temporal variations of heat transfer coefficient in unsteady heat transfer processes. However, the temporal behavior of heat transfer coefficient keeps the same under different unsteady heat flux boundary conditions which are temporally similar. The reason for the effects of adiabatic wall temperature and heat flux boundary condition on the temporal behavior of heat transfer coefficient has been revealed with an analysis method on the spatial distribution of the “Unsteady Term” ∂T(y)/∂τ.

Low Reynolds number hydrodynamics and mesoscale simulations

Hydrodynamics and hydrodynamic interactions are fundamental for the motility of microswimmers. This includes the propulsion mechanism itself, the synchronized motion of flagella in flagellar bundles and beating cilia of cilia arrays, and even extends to collective behaviors. The general importance of hydrodynamics has stimulated the development of mesoscale simulation approaches to efficiently study dynamical properties of objects embedded in a fluid. In this minireview, the properties of flows at low Reynolds numbers are discussed, thereby the unsteady acceleration term is typically taken into account (Landau-Lifshitz Navier-Stokes equations). Specifically, the synchronization of microrotors by time-dependent hydrodynamic interactions is discussed and the propulsion of a rotating helix. Moreover, the multiparticle collisions dynamics method (MPC), a mesoscale simulation approach for fluids, is outlined. Simulation results for the flow field of a model E. Coli bacterium and its swimming behavior next to a surface are presented.

Finite Reynolds number corrections of the 4/5 law for decaying turbulence

We examine finite Reynolds number contributions to the inertial range solution of the third order structure functions D3,0 and D1,2 stemming from the unsteady and viscous terms. Under the assumption that the second order correlations f and g are self-similar under a coordinate change, we are able to rewrite the exact second order equations as a function of a normalized scale r̃ only. We close the resulting system of equations using a power law and an eddy-viscosity ansatz. If we further assume K41 scaling, we find the same Reynolds number dependence as previously in the literature. We proceed to extrapolate towards higher Reynolds numbers to examine the unsteady and viscous terms in more detail. We find that the intersection between the two terms, where their contribution to the solution of the structure function equations is relatively small, scales with the Taylor scale λ.2 MoreReceived 29 February 2016DOI:https://doi.org/10.1103/PhysRevFluids.1.064403Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasTurbulenceFluid Dynamics

Linearized Euler Equations for the Prediction of Linear High-Frequency Stability in Gas Turbine Combustors

A novel methodology for linear stability analysis of high-frequency thermoacoustic oscillations in gas turbine combustors is presented. The methodology is based on the linearized Euler equations (LEEs), which yield a high-fidelity description of acoustic wave propagation and damping in complex, nonuniform, reactive mean flow environments, such as encountered in gas turbine combustion chambers. Specifically, this work introduces three novelties to the community: (1) linear stability analysis on the basis of linearized Euler equations. (2) Explicit consideration of three-dimensional, acoustic oscillations at screech level frequencies, particularly the first-transversal mode. (3) Handling of noncompact flame coupling with LEE, that is, the spatially varying coupling dynamics between perturbation and unsteady flame response due to small acoustic wavelengths. Two different configurations of an experimental model combustor in terms of thermal power and mass flow rates are subject of the analysis. Linear flame driving is modeled by prescribing the unsteady heat release source term of the linearized Euler equations by local flame transfer functions, which are retrieved from first principles. The required steady-state flow field is numerically obtained via computational fluid dynamics (CFD), which is based on an extended flamelet-generated manifold (FGM) combustion model, taking into account heat transfer to the environment. The model is therefore highly suitable for such types of combustors. The configurations are simulated, and thermoacoustically characterized in terms of eigenfrequencies and growth rates associated with the first-transversal mode. The findings are validated against experimentally observed thermoacoustic stability characteristics. On the basis of the results, new insights into the acoustic field are discussed.

Detailed numerical simulation of unsteady drag coefficient of deformable droplet

Abstract The detailed numerical simulation of the unsteady drag coefficient of deformable droplet is investigated with a mass conserving level set (LS) method. The simulation results indicate that the unsteady drag coefficient is always larger than the steady standard drag coefficient for the present decelerating relative flow. The effects of Weber number, density ratio and viscosity ratio on the unsteady drag coefficient of drop deformation are studied. It is found that the unsteady term (including density ratio) has the most effect on the unsteady drag coefficient and Weber number secondly. The viscosity ratio has little effect on the unsteady drag coefficient due to low Ohnesorge number. The numerical results confirm that the difference between drag coefficient and standard steady drag coefficient has an approximately linear curve fit with the unsteady parameter for low Weber number. These results will lay the foundation for the modeling of unsteady drag coefficient in our further work.

On the onset of horizontal convection

The convective flow driven by a differential heating along a horizontal boundary, commonly referred to as horizontal convection, is investigated. In particular, the inception of the convection cell following impulsive application of heating, i.e. the onset of horizontal convection, is studied experimentally in a rectangular box filled with water. Piecewise boundary conditions of uniform/constant heat flux and temperature are imposed along the box length at its bottom wall. The convective heat transfer coefficient on the heated half of the base is evaluated using the heated thin foil sensor modified to account for the unsteady term of the energy balance. The experiments are carried out over a range of Rayleigh numbers (based on heat flux input, box length and initial fluid properties) from 1.6 × 1011 to 1.3 × 1012 and for a Prandtl number (based on the initial water temperature) of approximately 6. Flow visualizations with the shadowgraph technique are also performed to complement heat transfer measurements. Thermocouple data are acquired inside the domain for validation purposes. A scaling for the characteristic time of the transient is proposed and verified. In the range of the investigated Rayleigh and Prandtl numbers, three subsequent phases in the onset process are identified: pure heat conduction through the fluid layer, Rayleigh-Bènard convection with transition of the boundary layer; onset and time evolution of longitudinal rolls. The presence of longitudinal rolls is justified via an analogy with Görtler vortices theory and results show a Nusselt number enhancement on the heated side of the order of 200% with respect to that on the cold one.

Квазиодномерная модель гиперболического типа гидроразрыва пласта

Представлена квазиодномерная модель гиперболического типа развития трещины гидроразрыва пласта в предположении, что коэффициент интенсивности напряжений намного превосходит коэффициент трещиностойкости породы. Рассматриваемая модель, учитывающая конвективные и нестационарные слагаемые в уравнении движения жидкости, является обобщением локальной модели Перкинса--Керна--Нордгрена. Доказано, что полученная система дифференциальных уравнений является квазилинейной строго гиперболической системой, для которой найдены характеристики и соотношения на характеристиках. В случае пренебрежения поправкой Кориолиса найдены инварианты Римана. При пренебрежении фильтрационной утечкой и вязкостью закачиваемой жидкости определены волны Римана, аналогичные простым плоским волнам в газовой динамике, и изучены их свойства. Исследована эволюционность границ трещины гидроразрыва пласта. Поставлена начально-краевая задача развития трещины гидроразрыва пласта. Показано, что при пренебрежении диссипативными слагаемыми в представленной модели можно построить теорию простых волн, аналогичную теории одномерной газовой динамики с изоэнтропическими плоскими волнами.

Squeeze Film Dampers Executing Small Amplitude Circular-Centered Orbits in High-Speed Turbomachinery

This work represents a pressure distribution model for finite length squeeze film dampers (SFDs) executing small amplitude circular-centered orbits (CCOs) with application in high-speed turbomachinery design. The proposed pressure distribution model only accounts for unsteady (temporal) inertia terms, since based on order of magnitude analysis, for small amplitude motions of the journal center, the effect of convective inertia is negligible relative to unsteady (temporal) inertia. In this work, the continuity equation and the momentum transport equations for incompressible lubricants are reduced by assuming that the shapes of the fluid velocity profiles are not strongly influenced by the inertia forces, obtaining an extended form of Reynolds equation for the hydrodynamic pressure distribution that accounts for fluid inertia effects. Furthermore, a numerical procedure is represented to discretize the model equations by applying finite difference approximation (FDA) and to numerically determine the pressure distribution and fluid film reaction forces in SFDs with significant accuracy. Finally, the proposed model is incorporated into a simulation model and the results are compared against existing SFD models. Based on the simulation results, the pressure distribution and fluid film reaction forces are significantly influenced by fluid inertia effects even at small and moderate Reynolds numbers.

A hybrid finite-volume/finite-difference-based one-dimensional Boussinesq model for waves attenuated by vegetation

A hybrid finite-volume/finite-difference scheme is proposed to solve the one-dimensional Boussinesq equations for wave attenuation by vegetation. The effect of vegetation is included as a source term in a form of drag force. The convective part of the equations is discretized by the finite-volume method, while the finite-difference method is used to discretize the remaining terms. The variable values for the local Riemann problem at each cell face are calculated by a fourth-order MUSCL reconstruction method. The source terms and the dispersion terms are discretized using the centered finite-difference schemes up to fourth-order accuracy. The unsteady terms are discretized by the second-order MUSCL-Hancock scheme. The discretized continuity equation is solved explicitly, while the discretized momentum equation is solved using the Thomas algorithm. The developed Boussinesq model is tested with analytical solutions and reported experimental data. To further validate the model, the computed results are compared with the experimental data observed in two vegetated wave flumes. It is demonstrated that the developed model is suitable for predicting wave propagation in vegetated water bodies.

Near field characteristics of buoyant helium plumes

Puffing and entrainment characteristics of helium plumes emanating out into ambient air from a circular orifice are investigated in the present study. Velocity and density fields are measured across a diametric plane using Particle Image Velocimetry (PIV) and Planar Laser Induced Fluorescence (PLIF) respectively in phase resolved manner. Experiments are performed in Froude numbers range 0.2–0.4 and for Reynolds numbers 58–248. Puffing frequency measurements reveal that the plume puffing frequencies are insensitive to the plume exit conditions, since the instability is buoyancy driven. The frequencies obtained in the present case are in agreement with frequencies obtained by Cetegen & Kasper (1996) for plumes originating from circular nozzles of various L/D ratios. Velocity and density measurements reveal that toroidal vortex formed during a puffing cycle entrains ambient air as it traverses downstream and this periodic engulfment governs the entrainment mechanism in pulsating plumes. The obtained velocity and density fields are used to calculate mass entrainment rates. It is revealed that though the flow is unsteady, the contribution of unsteady term in mass conservation to entrainment is negligible, and it becomes zero over a puff cycle. Finally, an empirical relation for variation of mass entrainment with height has been proposed, in which the non-dimensional mass entrainment is found to follow a power law with the non-dimensional height.

Effect of weak fluid inertia upon Jeffery orbits.

We consider the rotation of small neutrally buoyant axisymmetric particles in a viscous steady shear flow. When inertial effects are negligible the problem exhibits infinitely many periodic solutions, the "Jeffery orbits." We compute how inertial effects lift their degeneracy by perturbatively solving the coupled particle-flow equations. We obtain an equation of motion valid at small shear Reynolds numbers, for spheroidal particles with arbitrary aspect ratios. We analyze how the linear stability of the "log-rolling" orbit depends on particle shape and find it to be unstable for prolate spheroids. This resolves a puzzle in the interpretation of direct numerical simulations of the problem. In general, both unsteady and nonlinear terms in the Navier-Stokes equations are important.

Aerodynamic external pressure loads on a semi-circular bluff body under wind gusts

This brief communication concerns the unsteady aerodynamic external pressure loads acting on a semi-circular bluff body lying on a floor under wind gusts and describes the theoretical model, experimental setup, and experimental results obtained. The experimental setup is based on an open circuit, closed test section, low speed wind tunnel, which includes a sinusoidal gust generating mechanism, designed and built at the Instituto de Microgravedad “Ignacio Da Riva” of the Universidad Politécnica de Madrid (IDR/UPM). Based on the potential flow theory, a theoretical model has been proposed to analyse the problem, and experimental tests have been performed to study the unsteady aerodynamic loads on a semi-circular bluff body. By fitting the theoretical model predictions with the experimental results, influencing parameters of the unsteady aerodynamic loads are ascertained. The values of these parameters can help in clarifying the phenomenon of the external pressure loads on semi-circular bluff body under various gust frequencies. The theoretical model proposed allows the pressure variation to be split into two contributions, a quasi-steady term and an unsteady term with a simple physical meaning.

重み関数モデルを用いた特性曲線法の気体管路内微小振幅波の計算誤差

The method of characteristics is suitable to calculate small amplitude wave in gas pipeline. The method is superior in a case that the boundary conditions have many frequency components, while requires a computation of unsteady friction terms. The unsteady friction terms include convolution integral of which calculation is time-consuming. Reduction of the computation load with maintenance of its accuracy, which is essential for a practical computation, requires a weighting function-based (WFB) approximation which approximates the weighting functions by sums of exponential terms, and of convolution integrals. However, even the WFB model contains approximation errors which sometimes cause computation error. The approximation errors in liquid pipeline have been evaluated in Fourier domain. However, the errors in gas pipeline have not been evaluated. In this paper, we present a discussion and quantification of the numerical errors that occur when using the method for the simulation of small amplitude wave in gas pipeline. Comparisons of numerical error caused by approximations are made in the Fourier domain where exact solutions can be determined.

G1800202 スラブ軌道トンネル内バラスト敷設による微気圧波対策の数値解析

The magnitude of an impulsive pressure wave (micro-pressure wave) emitted from a tunnel portal is approximately proportional to the pressure gradient of a compression wave at the tunnel exit portal. It is confirmed that ballast layers installed in a slab-track tunnel have effect on the reduction of the micro-pressure wave by a field measurement. In this study, method of a numerical simulation is investigated for estimating the effect of the ballast layers on the reduction of the micro-pressure wave. The results indicates that the effect of the ballast layers can be simulate by calibrating a parameter of a unsteady friction term.

Numerical Solution of Solar Energy Absorbed in Porous Medium with a New Approach for Vapor Pressure Calculation and Consideration of Solute Crystallization

The goal of the study is to enhance the productivity of solar stills using an unsaturated porous medium initially saturated by salty water and using concentrating reflector. This paper concentrates only on the mathematical model for the porous medium and its solution using a finite-volume approach. The previous studies dealt with wick medium with high water content and liquid saturation in the wick medium was not determined. A physical model for the initially saturated porous medium was developed. The model takes into consideration the salt concentration in the solution, surface and internal water diffusions to humid air with vapor pressure determined from vapor mass balance. The system of transient one-dimensional differential equations was developed together with the boundaries and initial conditions. A finite-volume method was used for discretisation of the differential equations. A fully-implicit scheme was used for unsteady term discretisation while the convective terms (liquid solution, vapor and dry air) in the energy equation are handled by an upwind scheme method. The nonlinear equations are solved simultaneously by updating the coefficients matrix at one time step until the five variables converge to prescribed tolerance. Matlab was used as a programming tool. Solution of the model is obtained and discussed.

A high‐order discontinuous Galerkin method for all‐speed flows

SUMMARYIn this article, we present a discontinuous Galerkin (DG) method designed to improve the accuracy and efficiency of steady solutions of the compressible fully coupled Reynolds‐averaged Navier–Stokes and k − ω turbulence model equations for solving all‐speed flows. The system of equations is iterated to steady state by means of an implicit scheme. The DG solution is extended to the incompressible limit by implementing a low Mach number preconditioning technique. A full preconditioning approach is adopted, which modifies both the unsteady terms of the governing equations and the dissipative term of the numerical flux function by means of a new preconditioner, on the basis of a modified version of Turkel's preconditioning matrix. At sonic speed the preconditioner reduces to the identity matrix thus recovering the non‐preconditioned DG discretization. An artificial viscosity term is added to the DG discretized equations to stabilize the solution in the presence of shocks when piecewise approximations of order of accuracy higher than 1 are used. Moreover, several rescaling techniques are implemented in order to overcome ill‐conditioning problems that, in addition to the low Mach number stiffness, can limit the performance of the flow solver. These approaches, through a proper manipulation of the governing equations, reduce unbalances between residuals as a result of the dependence on the size of elements in the computational mesh and because of the inherent differences between turbulent and mean‐flow variables, influencing both the evolution of the Courant Friedrichs Lewy (CFL) number and the inexact solution of the linear systems. The performance of the method is demonstrated by solving three turbulent aerodynamic test cases: the flat plate, the L1T2 high‐lift configuration and the RAE2822 airfoil (Case 9). The computations are performed at different Mach numbers using various degrees of polynomial approximations to analyze the influence of the proposed numerical strategies on the accuracy, efficiency and robustness of a high‐order DG solver at different flow regimes. Copyright © 2014 John Wiley & Sons, Ltd.

Spinning detonation, cross-currents, and the Chapman–Jouguet velocity

AbstractInterestingly, the Chapman–Jouguet detonation velocity ($\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}D_{CJ}$) based on a one-dimensional and steady model compares well with the measured data. For the spinning detonation, in particular, this agreement is particularly notable, since the flow is highly three-dimensional and unsteady; perpendicular to the leading shock front, a transverse detonation wave (TDW) spins periodically. In the wake of this TDW, a secondary flow, called here the cross-current, appears which is orthogonal to the leading shock. Despite the presence of these cross-currents, the $D_{CJ}$ agreement remains remarkably satisfactory, and we investigate the reason for this, for spinning detonation in a tube. First, we focus on the origin of the cross-current. The cross-current, driven by the shock pressure, arises initially across a warped shock frontal surface, and both its sign and magnitude depend on the local slopes of the shock surface. The cross-current undergoes further pressure-driven transitions, with its magnitude eventually diminishing downstream and reducing the flow to a quasi-one-dimensional one. Second, regarding the unsteadiness, under the assumptions that the TDW spins at constant angular wave speed and the flow is steady in the frame rotating with it, the unsteady energy equation becomes integrable, resulting in the invariance of the so-called rothalpy. Also, in the integral forms of the mass and momentum balance, the unsteady terms drop out. Taken together, in the far field the governing equations are reduced to being one-dimensional and steady. From these the $D_{CJ}$ follows immediately, which appears to be the reason for the enduring usefulness of the $D_{CJ}$. The results of the analysis are confirmed with computational fluid dynamics (CFD). Additionally, the area-averaged flow profiles are found to display more than a passing resemblance to the Zeldovitch–Von Neumann–Doering (ZND) model.