Stochastic Flow Field Research Articles

Stochastic Characteristic Analysis of Lid-Driven Cavity Flow

The stochastic characteristic of lid-driven cavity flow under the condition of Re=100 is investigated in this paper by a non-intrusive polynomial chaos method. The lid-driven velocity and viscosity coefficient are assumed to be two independent stochastic input variables with Gaussian distributions. The velocities at every grid points are the response of interest and no longer deterministic but stochastic. Their stochastic characteristics are represented by Hermite polynomials with interaction effects between two input variables considered. The statistical results, including mean value, variance and the relative contribution of each input variable to the variance of outputs are analysed. It is found that the variance of velocity shows similar structures with its magnitude of mean value and the correlation coefficient is larger than 0.96. The nearly linear correlation coefficient shows that the lid-driven velocity is the main factor affecting the velocity distribution and the influence of viscosity coefficient on most regions is unimportant. The analysis of stochastic flow field helps understand the inherent mechanism in the lid-driven cavity flow.

One-dimensional solute transport in open channel flow from a stochastic systematic perspective

Solute transport by river and stream flows in natural environment has significant implication on water quality and the transport process is full of uncertainties. In this study, a stochastic one-dimensional solute transport model under uncertain open-channel flow conditions is developed. The proposed solute transport model is developed by upscaling the stochastic partial differential equations through their one-to-one correspondence to the nonlocal Lagrangian–Eulerian Fokker–Planck equations. The resulting Fokker–Planck equation is a linear and deterministic differential equation, and this equation can provide a comprehensive probabilistic description of the spatiotemporal evolutionary probability distribution of the underlying solute transport process by one single numerical realization, rather than requiring thousands of simulations in the Monte Carlo simulation. Consequently, the ensemble behavior of the solute transport process can also be obtained based on the probability distribution. To illustrate the capabilities of the proposed stochastic solute transport model, various steady and unsteady uncertain flow conditions are applied. The Monte Carlo simulation with stochastic Saint–Venant flow and solute transport model is used to provide the stochastic flow field for the solute transport process, and further to validate the numerical solute transport results provided by the derived Fokker–Planck equations. The comparison of the numerical results by the Monte Carlo simulation and the Fokker–Planck equation approach indicated that the proposed model can adequately characterize the ensemble behavior of the solute transport process under uncertain flow conditions via the evolutionary probability distribution in space and time of the transport process.

Lagrangian tracking in stochastic fields with application to an ensemble of velocity fields in the Red Sea

Lagrangian tracking of passive tracers in a stochastic velocity field within a sequential ensemble data assimilation framework is challenging due to the exponential growth in the number of particles. This growth arises from describing the behavior of velocity over time as a set of possible combinations of the different realizations, before and after each assimilation cycle. This paper addresses the problem of efficiently advecting particles in stochastic flow fields, whose statistics are prescribed by an underlying ensemble, in a parallel computational framework (openMP). To this end, an efficient algorithm for forward and backward tracking of passive particles in stochastic flow-fields is presented. The algorithm, which employs higher order particle advection schemes, presents a mechanism for controlling the growth in the number of particles. The mechanism uses an adaptive binning procedure, while conserving the zeroth, first and second moments of probability (total probability, mean position, and variance). The adaptive binning process offers a tradeoff between speed and accuracy by limiting the number of particles to a desired maximum. To validate our method, we conducted various forward and backward particles tracking experiments within a realistic high-resolution ensemble assimilation setting of the Red Sea, focusing on the effect of the maximum number of particles, the time step, the variance of the ensemble, the travel time, the source location, and history of transport.

Isothermal coherent structures and turbulent flow produced by a gas turbine combustor lean pre-mixed swirl fuel nozzle

The steady and unsteady isothermal fluid dynamics generated by an industrial low emission, lean premixed, fuel swirl nozzle designed by Solar Turbines Incorporated were investigated in this study. The experiments were carried out in a model optical can combustor operating at atmospheric pressures. Non-time resolved, planar Particle Image Velocimetry (PIV) measurements were taken at Reynolds numbers with respect to the nozzle throat diameter of ∼50000, ∼100000, and ∼180000. The time-averaged velocity fields were approximately self-similar, with the highest mass flow exhibiting a central recirculation zone (CRZ) with a slightly larger diameter. The results were analyzed using a methodology based on Proper Orthogonal Decomposition (POD) to extract the periodic structures in the flow and obtain the underlying stochastic turbulence field. This distinction between stochastic and coherent fluctuations is critical to properly model combustor flows. Coherent flow instabilities such as the precessing vortex core (PVC) and the propagation of axial/radial vortices were observed to significantly contribute to the mixing between the nozzle exit flow and the recirculated mass flow. Over 30% of the total fluctuation (difference between instantaneous and time-averaged velocity fields) kinetic energy was attributed to coherent structures throughout the inner shear layer between the swirling jet exiting the nozzle and the CRZ. Stochastic variability was prevalent close the liner wall and throughout the combustor domain after the swirling jet impinged on the wall, with <20% of the total fluctuation attributed to coherent structures. The normalized coherent and stochastic flow fields were also approximately self-similar with Reynolds number.

Identifying finite-time coherent sets from limited quantities of Lagrangian data.

A data-driven procedure for identifying the dominant transport barriers in a time-varying flow from limited quantities of Lagrangian data is presented. Our approach partitions state space into coherent pairs, which are sets of initial conditions chosen to minimize the number of trajectories that "leak" from one set to the other under the influence of a stochastic flow field during a pre-specified interval in time. In practice, this partition is computed by solving an optimization problem to obtain a pair of functions whose signs determine set membership. From prior experience with synthetic, "data rich" test problems, and conceptually related methods based on approximations of the Perron-Frobenius operator, we observe that the functions of interest typically appear to be smooth. We exploit this property by using the basis sets associated with spectral or "mesh-free" methods, and as a result, our approach has the potential to more accurately approximate these functions given a fixed amount of data. In practice, this could enable better approximations of the coherent pairs in problems with relatively limited quantities of Lagrangian data, which is usually the case with experimental geophysical data. We apply this method to three examples of increasing complexity: The first is the double gyre, the second is the Bickley Jet, and the third is data from numerically simulated drifters in the Sulu Sea.

Fractional Ensemble Average Governing Equations of Transport by Time-Space Nonstationary Stochastic Fractional Advective Velocity and Fractional Dispersion. II: Numerical Investigation

AbstractIn this paper, the second in a series of two, the theory developed in the companion paper is applied to transport by stationary and nonstationary stochastic advective flow fields. A numerical solution method is presented for the resulting fractional ensemble average transport equation (fEATE), which describes the evolution of the ensemble average contaminant concentration (EACC). The derived fEATE is evaluated for three different forms: (1) purely advective form of fEATE, (2) moment form of the fractional ensemble average advection-dispersion equation (fEAADE) form of fEATE, and (3) cumulant form of the fractional ensemble average advection-dispersion equation (fEAADE) form of fEATE. The Monte Carlo analysis of the fractional governing equation is then performed in a stochastic flow field, generated by a fractional Brownian motion for the stationary and nonstationary stochastic advection, in order to provide a benchmark for the results obtained from the fEATEs. When compared to the Monte Carlo sim...

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...

Wind turbine wakes for wind energy

Wind turbine wakes for wind energy

Heuristic space–time design of monitoring wells for contaminant plume characterization in stochastic flow fields

An optimization methodology for designing groundwater quality monitoring networks applicable to stochastic flow fields is presented and evaluated. The approach sets itself apart from previous techniques by incorporating the time dimension directly into the objective function. This function is extremized using a directed partial enumeration strategy guided by physical considerations related to transport processes. The result is a set of monitoring well locations and a sampling schedule that minimizes plume characterization error while satisfying constraints on the maximum number of wells and allowable number of active wells. The method is evaluated using hypothetical plumes with varying degrees of heterogeneity. Results indicate that the proposed approach is successful in generating near-optimal sampling networks that satisfy all imposed constraints. Monitoring networks with as little as three active wells and a total of 12 wells are found to provide adequate plume characterization for low toxicity contaminants.

The conformation change of model polymers in stochastic flow fields: Flow through fixed beds

Simulations of the conformation change of model polymers in various steady, anisotropic Gaussian random flow fields are presented. These flow fields have been chosen because they are models for the flow through porous media and have been predicted to be “stochastic strong flows” according to the criteria developed by Shaqfeh and Koch [J. Fluid Mech. 244, 17 (1992)]. To be specific, beyond a certain Deborah number (based on the sampling time of a velocity fluctuation), large average conformation change in the polymer is predicted. In our simulations, the polymers are modeled as dumbbells, but beyond this restriction, the assumptions of the theory by Shaqfeh and Koch are removed. Many realizations of the Gaussian fields are synthesized spectrally following a modified version of the method developed by Kraichnan [Phys. Fluids 13, 22 (1970)]. Moreover, the ratio of the mean “plug” flow to the amplitude of the fluctuations is varied from mean-dominant to fluctuation-dominant flows. The simulated conformation change shows that, in fact, these flows are “strong” in the sense that the average second moment of the end-to-end distance becomes large (relative to equilibrium) beyond a critical value of the fluctuation Deborah number. Although qualitatively capturing these trends, the theory by Shaqfeh and Koch underestimates the strength of the flows and thus overestimates the critical Deborah number. We present a new theory which includes spring relaxation and Brownian motion in the sampling of a velocity fluctuation (two factors which were neglected in the existing theory), thereby breaking the fore–aft symmetry of the sampling, thus increasing the average polymer stretch. The new theory quantitatively predicts the simulation results. To the authors knowledge, this is the first evidence via direct simulation that these random flows can produce large conformation change in model polymer molecules, even when the mean flow would produce no such change.

Polymer stretch in dilute fixed beds of fibres or spheres

A theory is developed to describe the conformation change of polymers in flow through dilute, random fixed beds of spheres or fibres. The method of averaged equations is used to analyse the effect of the stochastic velocity fluctuations on polymer conformation via an approach similar to that used in our previous analysis of particle orientation in flow through these beds (Shaqfeh & Koch 1988 a, b ). The polymers are treated as passive tracers, i.e. the polymeric stress in the fluid is neglected in calculating the stochastic flow field. Simple dumbbell models (either linear or FENE) are used to model the polymer conformation change. In all cases we find that the long-range interactions provide the largest contribution (in the limit of vanishingly small bed volume fraction) to an evolution equation for the probability density of conformation. These interactions create a conformation-dependent diffusivity in such an equation. Solutions for the second moment of the distribution demonstrate that there is a critical pore-size Deborah number beyond which the radius of gyration of a linear dumbbell will grow indefinitely and that of the FENE dumbbell will grow to a large fraction of its maximum extensibility. This behaviour is shown to be related to the development of ‘algebraic tails’ in the distribution function. The physical reasons for this critical condition are examined and its dependence on bed structure is analysed. These results are shown to be equivalent to those which we derive by the consideration of a polymer in a class of anisotropic Gaussian flow fields. Thus, our results are explicitly related to recent work regarding polymer stretch in model turbulent flows. Finally, the effect of close interactions and their modification of our previous results is discussed.

A numerical procedure for obtaining the mean solution and variance of the two-dimensional stochastic convection equation

Under certain conditions the concentration of a substance moving in a stochastic flow field is described by the stochastic convection equation. A numerical method yielding the mean solution and variance of the two-dimensional problem is described here. First, the differential operator is replaced by a discrete linear operator based on finite differences. The resulting system of stochastic equations is then replaced by a system of equations whose solution is the mean concentration. The variance of the concentration can then be calculated. In addition, and example is given for which an approximate analytical solution and its variance is known. The numerical method is applied to the example and results compared to the approximate analytical solution and variance.

Turbulence induced change in the conformation of polymer molecules

An investigation into the conformation of a polymer molecule exposed to a stochastic flow field is made utilizing a cumulant expansion. The study begins by considering the full Kirkwood–Riseman n-bead chain with an arbitrary connector potential as a model for the polymer molecule, digressing to the two-bead dumbbell molecule for detailed calculations. It is found that the effect of the stochastic velocity field on the molecule can be interpreted as a ‘‘renormalization’’ of the connector potential. The introduction of this renormalized potential greatly simplifies the mathematics and lends an intuitive understanding to the effect of the turbulence on the polymer. It is shown that this analysis will recover the anomalous tendency of the radius of gyration to become infinite for the special case of a Hookean potential which has been observed by previous investigators. Further, this study involves the Eulerian velocity–velocity correlation function rather than the Lagrangian correlation used in the previous studies.

Analysis of a numerical solution to the one-dimensional stochastic convection equation

Under certain conditions the concentration and flux of a substance moving in a stochastic flow field are described by the stochastic convection equation. A numerical method for solving the one-dimensional problem is studied here. The differential operator is replaced by a discrete linear operator based on finite differences. The resulting system of stochastic equations is then replaced by a system of equations whose solution is the mean concentration or mean flux. This final system is analysed and conditions for a stable numerical solution are obtained. Finally, numerical examples are given and are compared to an approximate analytical solution to the stochastic convection equation.