Nonstationary Flow Field Research Articles

Modulation effect on rotor-stator interaction subjected to fluctuating rotation speed in a centrifugal pump

In the paper, the modulation effect of rotor-stator interaction subjected to fluctuating rotation speed is investigated numerically. A quasi-bivariate variable mode decomposition method for analyzing non-stationary flow fields is proposed for the first time. Different from the previous studies, this paper emphasizes the local features of the transient flow field at fluctuating rotation speed. The results show that the impact of fluctuating rotation speed on the impeller force is limited, but it significantly impacts the pressure pulsation at fluctuating frequency. The correlation between amplitude and fluctuation ratio is positive. Meanwhile, the fluctuating rotation speed also induces sidebands centered on the blade passing frequency and its harmonics. The periodic changes in rotational speed lead to frequency and amplitude modulation of pressure pulsations. The magnitude of the instantaneous frequency and amplitude increases as the fluctuation ratio grows. The change in pressure field is well synchronized with the rotational speed, while the change in velocity field lags behind the rotational speed. The variation of pressure field with rotational speed is mainly attributed to two aspects. One is the flow structure at the fluctuating frequency, and the other is the instantaneous component of the flow structure at blade passing frequency.

Quantitation of the uncertainty in the prediction of flow fields induced by the spatial variation of the fracture aperture

The problem of fluid flow in a single fracture with variable aperture is analyzed from a stochastic point of view. The spatially distributed fracture aperture is treated as a random variable and the random aperture process can be characterized as a zero-order intrinsic random process (nonstationary random process with homogeneous increments), resulting in nonstationary flow fields. To characterize the stochastic behavior of the flow process, the general solutions for the mean values of and variances of the pressure head and the integrated flux in the mean flow direction are derived. These derivations are performed using the Fourier-Stieltjes representation approach. The mean is an unbiased estimator and the variance can be used to characterize the uncertainty of the mean model. The theory is applied to the case where the process of random aperture can be described as fractal Brownian motion. The analysis shows that the larger the parameter H, the greater the variance of pressure head and integrated flux. The transfer function helps smooth out much of the small-scale variation caused by the random aperture field. The variance (uncertainty) of the integrated flux in x-direction around the mean flux (or the classical deterministic model solution) increases with distance. Estimation of total flux through a natural fractured system is often evaluated for the purpose of rational management of groundwater resources. The proposed stochastic theory for flow prediction under uncertainty under the natural field conditions as input to the flow model for a fractured system should provide useful information for regional groundwater resource management. To our knowledge, the subject of fluid flow in nonstationary random flow fields has not yet been addressed in fracture. Our findings will be useful for the interpretation of real field data and as a stimulus for further research in this area.

Monopoles in a zonal flow with constant shear on a quasi-geostrophic f-plane: Effects of Galilean non-invariance

Building on the work of Kravtsov and Reznik [J. Fluid Mech. 909, A23 (2021); hereafter KR21], we studied the interactions of a localized monopole with a rectilinear, constant-shear flow in a 1½-layer, f-plane, quasi-geostrophic model. The non-invariance of this model with respect to Galilean transformations plays a crucial role in the dynamics of such interactions. Of particular importance here are two configurations in which the center of the vortex is located on the line of zero zonal current and remains motionless in the background of a nonstationary flow field generated via interactions of the vortex with the zonal flow. In configuration I (II), the vortex is prograde (retrograde), that is, its vorticity is of the same (opposite) sign with the vorticity of the background flow. Configuration I is stable, whereas configuration II eventually breaks down: a retrograde vortex drifts off of the zero-current line, rapidly accelerates and radiates intense Rossby waves, which results in a gradual weakening of the vortex. Naturally, the same scenario plays out if a retrograde vortex is initially off of the zero-current line. On the other hand, a prograde vortex initially located at some distance from the zero-current line drifts toward this line, albeit at a rate that decreases with time, so the solution always tends to configuration I. Therefore, the line of zero zonal current “attracts” prograde vortices and “repels” retrograde vortices. The present numerical experiments with singular vortices, using the scheme developed in KR21, confirm the above scenarios and clarify their dynamics.

Generation and characterization of a non-stationary flow field in a small-scale wind tunnel using a multi-blade flow device

Physical experimentation for non-stationary wind phenomena has been the focal point of much research. Replication of transient, non-synoptic wind events such as thunderstorm downbursts and tornadoes require innovative means that break from traditional boundary-layer wind tunnel experiments. This paper examines the feasibility of a multi-blade flow device capable of reproducing the gust front of a non-stationary outflow. The device is inserted into the test chamber of the small-scale, closed-circuit wind tunnel at Northeastern University. Replication of spatial and temporal features of a non-stationary gust front (slowly-varying flow and turbulence fluctuations) is investigated through velocity measurements at various elevations. The wind velocity record from the Andrews Air Force Base downburst is employed for verification of the experimental procedure; its physical effects on a tall building model are recorded. The results suggest that this device may be used to simulate the main features of simplified non-stationary gust fronts in a conventional wind tunnel without significant modifications to the test chamber.

Medium fidelity modelling of loads in wind farms under non-neutral ABL stability conditions – a full-scale validation study

The aim of the present paper is to demonstrate the capability of medium fidelity modelling of wind turbine component fatigue loading, when the wind turbines are subjected to wake affected non-stationary flow fields under non-neutral atmospheric stability conditions. To accomplish this we combine the classical Dynamic Wake Meandering model with a fundamental conjecture stating: Atmospheric boundary layer stability affects primary wake meandering dynamics driven by large turbulent scales, whereas wake expansion in the meandering frame of reference is hardly affected. Inclusion of stability (i.e. buoyancy) in description of both large- and small scale atmospheric boundary layer turbulence is facilitated by a generalization of the classical Mann spectral tensor, which consistently includes buoyancy effects. With non-stationary wind turbine inflow fields modelled as described above, fatigue loads are obtained using the state-of-the art aeroelastic model HAWC2.The Lillgrund offshore wind farm (WF) constitute an interesting case study for wind farm model validation, because the WT interspacing is small, which in turn means that wake effects are significant. A huge data set, comprising 5 years of blade and tower load recordings, is available for model validation. For a multitude of wake situations this data set displays a considerable scatter, which to a large degree seems to be caused by atmospheric boundary layer stability effects. Notable is also that rotating wind turbine components predominantly experience high fatigue loading for stable stratification with significant shear, whereas high fatigue loading of non-rotating wind turbine components are associated with unstable atmospheric boundary layer stratification.

Fractional Ensemble Average Governing Equations of Transport by Time-Space Nonstationary Stochastic Fractional Advective Velocity and Fractional Dispersion. I: Theory

AbstractIn this study, starting from a time-space nonstationary general random walk formulation, the pure advection and advection-dispersion forms of the fractional ensemble average governing equations of solute transport by time-space nonstationary stochastic flow fields were developed. In the case of the purely advective fractional ensemble average equation of transport, the advection coefficient is a fractional ensemble average advective flow velocity in fractional time and space that is dependent on both space and time. As such, in this case, the time-space nonstationarity of the stochastic advective flow velocity is directly reflected in terms of its mean behavior in the fractional ensemble average transport equation. In fact, the derived purely advective form represents the Lagrangian derivation of the ensemble average mass conservation equation for solute transport in fractional time-space. In the case of the fractional ensemble average advection-dispersion transport equation, the moment and cumula...

Investigation of solute transport in nonstationary unsaturated flow fields

Abstract. Most theoretical work on analyzing plume spreading at the field scale in a partially saturated heterogeneous formation assumes weak stationarity of velocity field. While this assumption is not applicable to the case of a bounded flow domain, the nonstationarity in fluctuations of unsaturated velocity fields is induced through the presence of the boundary conditions in the flow domain. In this work, we attempt to quantify the large-time macrodispersion in nonstationary unsaturated velocity fields caused by the presence of a fixed head boundary condition. Application of the perturbation-based nonstationary spectral approach leads to an analytical expression for the macrodispersion describing the field-scale dispersive solute flux in terms of the statistical moments of two formation parameters: the Gardener's parameter (α) and the saturated hydraulic conductivity (Ks). The results predicted from the expression indicate that the enhanced unsaturated plume spreading can arise from a larger correlation scale of ln Ks or ln α fields. In addition, the α-parameter takes the role in reducing the field-scale plume spreading.

Investigation of a nonstationary flow field generated by a dielectric barrier discharge

The search for ways of controlling a flow over flight vehicles is today an extremely topical issue, since the limiting capabilities of standard control methods are the bulk of the progress of space technology. A promising technique in this respect is the application of activators using a low-temperature plasma. Specifically, these electrogasdynamic devices can directly convert electrical energy to a mechanical action on the flow. An activator based on a dielectric barrier discharge is considered. The nonstationary behavior of the activator is studied by measuring the electrodynamic parameters of the discharge and also by measuring induced velocity fields with particle image velocimetry. The instantaneous and integral parameters of the gas acceleration stage are found, and the efficiency of the activator is determined.

Wind turbine wakes for wind energy

Wind turbine wakes for wind energy

Nonstationary flow field generated by dielectric barrier discharge

The search for new methods of the flow control is of great importance because the capability of traditional control devices is one of the limits of the aircraft efficiency. A promising method of flow control involves application of electrogasdynamic devices based on the direct transformation of the electric energy to the force acting on the flow. An activator employing the principles of dielectric barrier discharge (DBD) was studied. Nonstationary characteristic of the activator have been studied by measuring the electrodynamic parameters of DBD and the induced velocity fields (particle image velocimetry). Instantaneous and integral parameters of the gas acceleration are determined and the activator efficiency is evaluated.

Reduced-order estimation of nonstationary flows with electrical impedance tomography

In this paper, we consider the simultaneous reconstruction of a nonstationary concentration distribution and the underlying nonstationary flow field. As the observation modality, we employ electrical impedance tomography. Earlier studies have shown that such an estimation scheme is in principle possible since the evolution of an inhomogeneous concentration carries information also on the velocity field. These results have, however, been restricted to either stationary velocity fields or simplified non-physical models. In the general case, the estimation of the velocity field up to the fine details of the flow with diffuse tomography is impossible. In this paper we show, however, that it is possible to estimate a reduced-order representation of a physical fluid dynamics model, here the Navier–Stokes model, simultaneously with the concentration. This is accomplished by considering a proper orthogonal decomposition representation for the velocity field, and careful modelling of the uncertainties of the models, in particular, the subspace of the velocity field that is not estimated. We assess the approach with two numerical examples in which a projection of a vortex flow is reconstructed.

An Eulerian time filtering technique to study large-scale transient flow phenomena

Unsteady fluctuating velocity fields can contain large-scale periodic motions with frequencies well separated from those of turbulence. Examples are the wake behind a cylinder or the processing vortex core in a swirling jet. These turbulent flow fields contain large-scale, low-frequency oscillations, which are obscured by turbulence, making it impossible to identify them. In this paper, we present an Eulerian time filtering (ETF) technique to extract the large-scale motions from unsteady statistical non-stationary velocity fields or flow fields with multiple phenomena that have sufficiently separated spectral content. The ETF method is based on non-causal time filtering of the velocity records in each point of the flow field. It is shown that the ETF technique gives good results, similar to the ones obtained by the phase-averaging method. In this paper, not only the influence of the temporal filter is checked, but also parameters such as the cut-off frequency and sampling frequency of the data are investigated. The technique is validated on a selected set of time-resolved stereoscopic particle image velocimetry measurements such as the initial region of an annular jet and the transition between flow patterns in an annular jet. The major advantage of the ETF method in the extraction of large scales is that it is computationally less expensive and it requires less measurement time compared to other extraction methods. Therefore, the technique is suitable in the startup phase of an experiment or in a measurement campaign where several experiments are needed such as parametric studies.

Numerical method of moments for solute transport in a nonstationary flow field conditioned on hydraulic conductivity and head measurements

The unconditional stochastic studies on groundwater flow and solute transport in a nonstationary conductivity field show that the standard deviations of the hydraulic head and solute flux are very large in comparison with their mean values (Zhang et al. in Water Resour Res 36:2107–2120, 2000; Wu et al. in J Hydrol 275:208–228, 2003; Hu et al. in Adv Water Resour 26:513–531, 2003). In this study, we develop a numerical method of moments conditioning on measurements of hydraulic conductivity and head to reduce the variances of the head and the solute flux. A Lagrangian perturbation method is applied to develop the framework for solute transport in a nonstationary flow field. Since analytically derived moments equations are too complicated to solve analytically, a numerical finite difference method is implemented to obtain the solutions. Instead of using an unconditional conductivity field as an input to calculate groundwater velocity, we combine a geostatistical method and a method of moment for flow to conditionally simulate the distributions of head and velocity based on the measurements of hydraulic conductivity and head at some points. The developed theory is applied in several case studies to investigate the influences of the measurements of hydraulic conductivity and/or the hydraulic head on the variances of the predictive head and the solute flux in nonstationary flow fields. The study results show that the conditional calculation will significantly reduce the head variance. Since the hydraulic head measurement points are treated as the interior boundary (Dirichlet boundary) conditions, conditioning on both the hydraulic conductivity and the head measurements is much better than conditioning only on conductivity measurements for reduction of head variance. However, for solute flux, variance reduction by the conditional study is not so significant.

Analysis of the flow field induced by the sessile peritrichous ciliate Opercularia asymmetrica

The feeding mechanism of the sessile protozoon Opercularia asymmetrica (Oligohymenophorea, Peritrichia) relies on the cilia beat generating a flow field that convectively transports suspended particles and dissolved substances to the oral cavity of the organism. By use of optical micro-flow measurement and theoretical methods the flow environment of two neighbouring peritrichous ciliate cells is studied. Both, yeast cells ( Saccharomyces cerevisiae) and artificial flow tracers are used for the visualisation of the flow field. Artificial tracers are rejected by the protozoa and deviate from the fluid path lines, while yeast cells follow the flow almost perfectly. This is shown through a dimensional analysis of the involved hydrodynamic forces on the tracers. The measured flow field exhibits maximum velocities of 25 μm/s at around 20 μm distance ahead of an individual ciliate. The flow field extends 200 μm from the location of the ciliate. A nicking motion of the protozoon is observed and found not to obey any periodic law. Multiples of protozoa exhibit most commonly an alternating cilia beat regime generating a non-stationary flow field. It can be shown through theoretical methods that fluid exchange is enhanced in this alternating regime compared to a flow field generated by a single ciliate. Fluid exchange depends on the distance of the ciliates from each other and on the alteration frequency of the cilia beat. The comparison of an analytical Stokes’ flow solution with the observed fluid flow serves to determine the force required to maintain the flow field against viscous dissipation. The force magnitude is in the order of magnitude of 10–100 pN.

A Numerical Method of Moments for Solute Transport in Nonstationary Flow Fields

A Lagrangian perturbation approach has been applied to develop the method of moments for predicting mean and variance of solute flux through a three-dimensional nonstationary flow field. The flow nonstationarity may stem from medium nonstationarity, finite domain boundaries, and/or fluid pumping and injecting. The solute flux is described as a space–time process where time refers to the solute flux breakthrough and space refers to the transverse displacement distribution at the control plane. The analytically derived moment equations for solute transport in a nonstationary flow field are too complicated to solve analytically, a numerical finite difference method is implemented to obtain the solutions. This approach combines the stochastic model with the flexibility of the numerical method to boundary and initial conditions. This method is also compared with the numerical Monte Carlo method. The calculation results indicate the two methods match very well when the variance of log-conductivity is small, but the method of moment is more efficient in computation.

Reactive Solute Transport in a Nonstationary, Fractured Porous Medium: A Dual-Porosity Approach

A numerical method of moment is developed for solute flux through a nonstationary, fractured porous medium. Solute flux is described as a space-time process where time refers to the solute flux breakthrough and space refers to the transverse displacement distribution at a control plane. A first-order mass diffusion model is applied to describe interregional mass diffusion between fracture (advection) and matrix (nonadvection) regions. The chemical is under linear equilibrium sorption in both fracture and matrix regions. Hydraulic conductivity in the fracture region is assumed to be a spatial random variable. In this study, the general framework of Zhang et al.(2000) is adopted for solute flux in a nonstationary flow field. A time retention function related to physical and chemical sorption in the dual-porosity medium is developed and coupled with solute advection along random trajectories. The mean and variance of total solute flux are expressed in terms of the probability density function of the parcel travel time and transverse displacement. The influences of various factors on solute transport are investigated. These factors include the interregional mass diffusion rate between fracture and matrix regions, chemical sorption coefficients in both regions, water contents in both regions, and location of the solute source. In comparison with solute transport in a one-region medium, breakthrough curves of the mean and variance of the total solute flux in a two-region medium have lower peaks and longer tails. As compared with the classical stochastic studies on solute transport in fractured media, the numerical method of moment provides an approach for applying the stochastic method to study solute transport in more complicated fractured media.

Three-Dimensional Numerical Method of Moments for Linear Equilibrium-Adsorbing Solute Transport in Physically and Chemically Nonstationary Formations

A Lagrangian perturbation method is applied to develop a method of moments for reactive solute flux through a three-dimensional, nonstationary flow field. The flow nonstationarity may stem from medium nonstationarity, finite domain boundaries, and/or fluid pumping and injecting. The reactive solute flux is described as a space–time process where time refers to the solute flux breakthrough in a control plane at some distance downstream of the solute source and space refers to the transverse displacement distribution at the control plane. The analytically derived moments equations for solute transport in a nonstationary flow field are too complicated to solve analytically; therefore, a numerical finite difference method is implemented to obtain the solutions. This approach combines the stochastic model with the flexibility of the numerical method to boundary and initial conditions. The approach provides a tool to apply stochastic theory to reactive solute transport in complex subsurface environments. Several case studies have been conducted to investigate the influence of the physical and chemical heterogeneity of a medium on the reactive solute flux prediction in nonstationary flow field. It is found that both physical and chemical heterogeneity significantly affect solute transport behavior in a nonstationary flow field. The developed method is also applied to an environmental project for predicting solute flux in the saturated zone below the Yucca Mountain Project area, demonstrating the applicability of the method in practical environmental projects.

Stochastic delineation of well capture zones

In this work, we describe a stochastic method for delineating well capture zones in randomly heterogeneous porous media. We use a moment equation (ME) approach to derive the time-dependent mean capture zones and their associated uncertainties. The mean capture zones are determined by reversely tracking the non-reactive particles released at a small circle around each pumping well. The uncertainty associated with the mean capture zones is calculated based on the particle displacement covariances for nonstationary flow fields. The flow statistics are obtained either by directly solving the flow moment equations derived with a first-order ME approach or from Monte Carlo simulations (MCS) of flow. The former constitutes a full ME approach, and the latter is a hybrid ME-MCS approach. This hybrid approach is invoked to examine the validity of the transport component of the stochastic method by ensuring that the ME and MC transport approaches have the same underlying flow statistics. We compared both the full ME and the hybrid ME-MCS results with those obtained with a full MCS approach. It has been found that the three approaches are in excellent agreement when the variability of hydrologic conductivity is small (σ Y 2=0.16). At a moderate variability (σ Y 2=0.5), the hybrid ME-MCS and the full MCS results are in excellent agreement whereas the results from the full ME approach deviate slightly from the full MCS results. This indicates that the (first-order) ME transport approach renders a good approximation at this level of variability and that the first-order ME flow approximation may not be sufficiently accurate at this variability in the case of divergent/convergent flow. The first-order ME flow approach may need to be corrected with higher-order terms even for moderate σ Y 2 although the literature results reveal that the first-order ME flow approach is robust for uniform mean flow (i.e., giving accurate results even with σ Y 2 as large as four).

A numerical method of moments for solute transport in physically and chemically nonstationary formations: linear equilibrium sorption with random K d

A Lagrangian perturbation method is applied to develop a method of moments for solute flux through a three-dimensional nonstationary flow field. The flow nonstationarity stems from medium nonstationarity and internal and external boundaries of the study domain. The solute flux is described as a space-time process where time refers to the solute flux breakthrough through a control plane (CP) at some distance downstream of the solute source and space refers to the transverse displacement distribution at the CP. The analytically derived moment equations for solute transport in a nonstationarity flow field are too complicated to solve analytically, a numerical finite difference method is implemented to obtain the solutions. This approach combines the stochastic model with the flexibility of the numerical method to boundary and initial conditions. The developed method is applied to study the effects of heterogeneity and nonstationarity of the hydraulic conductivity and chemical sorption coefficient on solute transport. The study results indicate all these factors will significantly influence the mean and variance of solute flux.

Applications of nonstationary stochastic theory to solute transport in multi-scale geological media

In this study, we make use of a nonstationary stochastic theory in studying solute flux through spatially nonstationary flows in porous media. The nonstationarity of flow stems from various sources, such as multi-scale, nonstationary medium features and complex hydraulic boundary conditions. These flow nonstationarities are beyond the applicable range of the ‘classical’ stochastic theory for stationary flow fields, but widely exist in natural media. In this study, the stochastic frames for flow and transport are developed through an analytical analysis while the solutions are obtained with a numerical method. This approach combines the stochastic concept with the flexibility of the numerical method in handling medium nonstationarity and boundary/initial conditions. It provides a practical way for applying stochastic theory to solute transport in complex groundwater environments. This approach is demonstrated through some synthetic cases of solute transport in multi-scale media as well as some hypothetical scenarios of solute transport in the groundwater below the Yucca Mountain project area. It is shown that the spatial variations of mean log-conductivity and correlation function significantly affect the mean and variance of solute flux. Even for a stationary medium, complex hydraulic boundary conditions may result in a nonstationary flow field. Flow nonstationarity and/or nonuniform distribution of initial plume (geometry and/or density) may lead to nonGaussian behaviors (with multiple peaks) for mean and variance of the solute flux. The calculated standard deviation of solute flux is generally larger than its mean value, which implies that real solute fluxes may significantly deviate from the mean predictions.