Steady Flow Results Research Articles

Aquifer parameter estimation by extended Kalman filtering and boundary elements

The extended Kalman filtering (EKF) method is used as an automatic calibration procedure for the estimation of hydrogeological parameters based on groundwater head observation. The numerical tool employed is the dual reciprocity boundary element method (DRBEM). Numerical results for steady and unsteady flow in heterogeneous aquifers have demonstrated the potential of this combined procedure for groundwater applications.

Two-dimensional implicit flux split steady and unsteady Euler calculations using unstructured moving grids

AbstractTwo-dimensional Euler equations are solved on unstructured triangular meshes using commonly available minicomputers. The driving algorithm is an upwind biased implicit cell-centred finite volume scheme. The spatial discretisation involves a naturally dissipative flux-split approach that sharply captures Shockwaves. The grid generation method uses a type of advancing front technique which triangulates a given set of points. To demonstrate the application, the steady flow results are presented for single and two component aerofoils. The unsteady results are obtained using a dynamic mesh algorithm for an aerofoil pitching harmonically about the quarter chord. The paper presents a description of the grid generation and movement algorithm and of the Euler solver, along with results and comparison that assess their capabilities.

A Finite Element Model for the Unsteady Groundwater Flow over Sloping Beds

A numerical solution of the nonlinear two-dimensional unsteady groundwater flow over sloping beds, using the Galerkin finite element method, is presented. The applied differential equation is based on the assumption that the streamlines are parallel to the sloping bed while the conventional differential equation is based on the Dupuit–Forchheimer assumption of horizontal flow. Furthermore, the gradient of the piezometric head is evaluated as a function of the unknown slope of the groundwater free surface and not simply as the absolute slope of the water table. Water table profiles and seepage rates obtained from the model are compared with those obtained by analytical solutions and experimental results for steady one-dimensional flow and for different values of bed slope. Results of the present model compares reasonably well with experimental results and with the results of Childs' analytical solution. The unsteady-state solution for different values of bed slope are compared to the results of a model based on the assumption that the water table slope is equal to the bed slope. The differences in the predicted water table by these two models are insignificant for small slopes but increase with increasing slope.

Free-Surface Wave-Induced Separation

Free-surface wave-induced separation is studied for a surface-piercing NACA 0024 foil over a range of Froude numbers (0, .2, .37, .55) through computational fluid dynamics of the unsteady Reynolds-averaged Navier-Stokes and the continuity equations with the Baldwin-Lomax turbulence model, exact nonlinear kinematic and approximate dynamic free-surface boundary conditions, and a body/free-surface conforming grid. The flow conditions and uncertainty analysis are discussed. A topological rule for a surface-piercing body is derived and verified. Steady-flow results are presented and analyzed with regard to the wave and viscous flow and the nature of the separation.

Comparison of series expansions and finite-difference computations of internal laminar flow separation

We present computed results for steady viscous flow in a two-dimensional channel with one plane wall and the other wall parallel far upstream but not parallel elsewhere. The particular example studied most fully is that of gradual exponential divergence. The main aim is to see whether a series expansion approach is as good as a finite difference method in computing flow separation in a diverging channel at moderately large values of the Reynolds number Re. It is assumed that the wall slope parameter e is small but that eRe is not. The leading term of the series expansion is that given by lubrication theory, and the expansion proceeds in powers of eRe; two versions of the series are considered, one based on the boundary layer equations and one in which the leading effect of cross-stream pressure gradient is included. Convergence of the series is enhanced by the use of Pade approximants. Estimates are made of the position of the separation point, x s , at various values of eRe (= 1, 3, 5); the values of x s on the two walls are somewhat different if the cross-stream pressure gradient is included. The results are compared with those of finite difference solutions of ( a ) the boundary layer equations and ( b ) the full Navier-Stokes equations. The boundary-layer and series-expansion results (with no cross-stream pressure gradient) agree extremely well, but to use the series method for more general wall geometries would be forbiddingly cumbersome, especially with cross-stream pressure gradient included. The Navier Stokes computations fail to provide a converged symmetric solution in which separation occurs at approximately the same value x s of on each wall, but gives two converged solutions with the first separation point on one wall or the other. If too crude a convergence criterion had been used, a ‘symmetric’ solution would have been found. Such non-uniqueness is to be expected; the ‘symmetric’ flow (the only one achieveable by the series method) would presumably be unstable. None of the computed solutions predict that separation occurs as early as has been seen experimentally in a slightly different geometry.

Numerical Simulation of a Cylinder in Uniform Flow: Application of a Virtual Boundary Method

In this study, a virtual boundary technique is applied to the numerical simulation of stationary and moving cylinders in uniform flow. This approach readily allows the imposition of a no-slip boundary within the flow field by a feedback forcing term added to the momentum equations. In the present work, this technique is used with a high-order finite difference method, effectively eliminating spurious oscillations caused by the feedback forcing when used with spectrally discretized flow solvers. Very good agreement is found between the present calculations and previous computational and experimental results for steady and time-dependent flow at low Reynolds numbers.

Nonlocal Hydrodynamic Theory of Flow in Polymer Layers

We present a nonlocal hydrodynamic theory for flow through an adsorbed layer of homopolymer. Our theory is based upon a multiple-scattering treatment in which the polymer is decomposed into loops, each attached to the surface at a point and obeying Zimm dynamics. The multiple scattering series is converted into an integral equation which we solve numerically. The dependence of our results on the truncation procedure in the sum over Zimm modes is discussed. Unlike previous models, which rely on a local porosity description, our theory explicitly incorporates polymer dynamics and is hence applicable to finite frequencies and (in principle) nonlinear responses. For an isolated layer, we present results for weak oscillatory and steady shear flows at ϑ conditions. We also give results for steady flow between a pair of parallel plates coated with homopolymer, which may be relevant to the dynamics of polymer-coated colloids undergoing hydrodynamic collisions.

Fluid mechanics of stenosed arteries

A finite difference scheme is used to investigate flow in a tube with an occlusion. The results are interpreted in the context of blood flow in stenosed arteries. Numerical results for steady and pulsatile flows confirm, in a quantitative sense, that a high shear stress is not likely to initiate atherosclerosis lesions. The study of unsteady flow reveals several interesting new features. It appears that there is a correlation between regions of recirculation, which are a prominent feature of the unsteady flow, and the location of lesions. Experimental measurements for steady flow complement the numerical study and show qualitative agreement.

Analysis of low Reynolds number airfoil flows

Compressible steady and unsteady flowfields over a NACA 0012 airfoil at transitional Reynolds numbers are investigated. Comparisons with recently obtained experimental data are used to evaluate the ability of a numerical solution based on the compressible thin layer Navier-Stokes approximation, augmented with a transition model, to simulate transitional flow features. The discretization is obtained with an upwind-biased, factorized, iterative scheme. Transition onset is estimated using an empirical criterion based on the computed mean flow boundary- layer quantities. The transition length is computed from an empirical formula. The incorporation of transition modeling enables the prediction of the experimentally observed leading-edge separation bubbles. Results for steady airfoil flows at fixed angles of attack and for oscillating airfoils are presented. HE prediction of steady, inviscid flows over aerodynamic configurations is performed routinely nowadays. The computation of flows with separation bubbles or of fully sep- arated flows, on the other hand, is still a very challenging problem. For many practical applications, the assumption of fully developed turbulent flow yields good predictions of the flowfield. In other circumstances, such as the leading-edge dynamic stall flow, this assumption is not valid and compu- tations of such flows need improved methods. A characteristic feature of the dynamic stall flow is the onset of compressibility effects at a very low freestream Mach num- ber of O.2. 1-2 In addition, at transitional Reynolds numbers, it has been shown3-4 that the dynamic stall events are closely governed by the formation of a laminar separation bubble and its subsequent bursting. In fact, the above-cited studies demonstrate that the failure of the separated shear layer to reattach initiates dynamic stall, leading to the formation of the dynamic stall vortex. Further complications arise when the locally supersonic flow forms shocks that interact with the local boundary layer. It is important to recognize that the scales of the flow are very small here and the flow physics is not very clear. It is obvious that an accurate computational study of the problem demands a proper modeling of the phys- ics of the local compressible flow. In an effort to reach this eventual goal, it is necessary to include the transition physics that plays a key role in the dynamic stall process. The study to be reported represents a step in this direction. It is well known that prediction of the transition point and the transition length in such strongly adverse pressure gradient driven flows with current methods is difficult and involves uncertainties. Although several methods are available, the engineering prediction of transition relies on empirical for- mulation for boundary-layer flows. Whereas these methods have been moderately successful in steady and subsonic flows,

Numerical Simulation of Compressible Viscous Flows around an Arbitrary Shape Body Using the Cartesian Grid System.

A new Cartesian grid system approach is proposed for the numerical simulation of two-dimensional compressible viscous flows around an arbitrary shape body. The grid system is constructed of patched blocks which is composed of the nested-spacing regular grid. The numerical procedure is based on the method of lines. For the spatial discretization, the neighboring-point local collocation method is developed and introduced for the grid points near the body, and the 2nd-order central difference method is used for the points far from the body. The 2-stage Runge-Kutta method (RK 2) is used for the time integration scheme. For the unsteady solution the nesting time step technique is combined with RK 2. Numerical results are obtained for subsonic steady and unsteady flows around a circular cylinder and compared with the other results. The results for transonic and supersonic steady flows over NACA 0012 are compared with the GAMM workshop ones. The present approach is confirmed to be applicable for the arbitrary shape body, and the computational cost is about 50% that of the conventional boundary-fitted grid approach.

On constitutive equations for electrorheological materials

Constitutive equations for electrorheological (ER) fluids have been based on experimental results for steady shearing flows and constant electric fields. The fluids have been modeled as being rigid until a yield stress is reached. Additional stress is then proportional to the shear rate. Recent experimental results indicate that ER materials have a regime of solid-like response when deformed from a rest state. They behave in a viscoelastic-like manner under sinusoidal shearing and exhibit time-dependent response under sudden changes in shear rate or electric field. In this work, a constitutive theory for ER materials is presented which accounts for these recent experimental observations. The stress is given by a functional of the deformation gradient history and the electric field vector. Using the methods of continuum mechanics, a general three-dimensional constitutive equation is obtained. A sample constitutive equation is introduced which is then used to determine the response of an ER material for different shear histories. The calculated shear response is shown to be qualitatively similar to that observed experimentally.

Some exact and numerical results for plane steady sheared flow of an incompressible inviscid fluid

Analytical and numerical solutions are presented for the steady flow of an inviscid fluid about symmetric lifting profiles at an angle of attack in a plane sheared onset flow for which conformal mapping plays a critical role. For uniform shear (i.e. the onset flow speed varies linearly with position) in two dimensions, the disturbance field is potential and hence a solution based on the conformal transformation technique may be constructed. The Moriya transformation, which employs a leading-term transformation coefficient that stretches and rotates the field at great distances from the foil (as distinct from other classical transformations which leave the far field unchanged) is used and, with a limited number of terms selected for the transformation, a simple elegant solution is obtained that may be easily evaluated at arbitrary points on the foil contour. An additional investigation is reported for the field solution — involving a locally similar but globally non-uniform sheared onset flow — about one of the foils for which a simple O-type grid is analytically generated from the mapping function. These data indicate that the uniform-shear solution overpredicts the lift and surface speed on the suction side of the foil relative to the more realistic onset flow: the numerical solution predicts surface speeds that generally lie between those for the uniform flow and the uniformly sheared flow solutions.

EFFECT OF NONEQUILIBRIUM PHONONS ON THE ELECTRON RELAXATION AND TRANSPORT

We review the recent theoretical study of the effect of nonequilibrium phonons on hot-carrier relaxation and transport. In a quantum well, the proper treatment of the electron-phonon coupling between electrons confined to two dimensions (2-D) by phonons traveling freely in three dimensions (3D) requires special care because phonon heating produces a bottleneck in the rate of transfer of energy from the carriers to the phonons. Because the carriers interact with phonons primarily when the latter are close to the quantum well, the latter should be described, not by plane waves, but by packets adapted to the shape of the carrier confinement. A quasi-equilibrium technique that retains off-diagonal elements in the phonon wave-vector permits an unrestricted treatment of the density operator equation. That in turn leads to a choice of wave packet that comes from solving the integrodifferential equations rather than by imposition. Moreover, if the carrier distribution is assumed in quasi-equilibrium with a given drift and temperature, the coupled partial differential equations are reduced to coupled ordinary differential equations that can be solved with modest computer power. Comparison with experimental results for steady flow of energy from carriers to phonons, and for time-dependent relaxation yields quantitative agreement.

Multilevel adaptive methods for incompressible flow in grooved channels

Incompressible moderate-Reynolds-number flow in a periodically grooved channel is investigated by direct numerical simulation using the finite-volume method on a staggered grid. A second-order, fully-implicit time-marching scheme is used together with a multigrid full approximation scheme (FAS) to accelerate the convergence process. Convergence factors of about 0.15 for each V(2,2) cycle are observed. The computational results for steady flow show good agreement with that of Ghaddar (1986). A local fine grid is placed about the cavity to achieve better accuracy, without the need for a global fine grid. Both FAC (see McCormick (1989)) and MLAT (see Brandt (1984)) adaptive techniques show optimal efficiency as solvers for the resulting composite grid problem.

Accuracy of an unstructured-grid upwind-Euler algorithm for the ONERA M6 wing

Improved algorithms for the solution of the three-dimensional, time-dependent Euler equations are presented for aerodynamic analysis involving unstructured dynamic meshes. The improvements have been developed recently to the spatial and temporal discretizations used by unstructured-grid flow solvers. The spatial discretization involves a flux-split approach that is naturally dissipative and captures shock waves sharply with at most one grid point within the shock structure. The temporal discretization involves either an explicit time-integration scheme using a multistage Runge-Kutta procedure or an implicit time-integration scheme using a Gauss-Seidel relaxation procedure, which is computationally efficient for either steady or unsteady flow problems. With the implicit Gauss-Seidel procedure, very large time steps may be used for rapid convergence to steady state, and the step size for unsteady cases may be selected for temporal accuracy rather than for numerical stability. Steady flow results are presented for both the NACA 0012 airfoil and the Office National d'Etudes et de Recherches Aerospatiales M6 wing to demonstrate applications of the new Euler solvers. The paper presents a description of the Euler solvers along with results and comparisons that assess the capability.

Simple theories of dynamic stall that are helpful in interpreting computational results

This paper briefly reviews the phenomena of dynamic stall and comments on some of the theoretical work performed to help understand why the phenomena occurs for two-dimensional flow past an airfoil undergoing a pitching motion. A first look at how these understandings might be extended to three-dimensional flow fields past finite-span wings is also presented. In particular, a simple theory is developed that provides a rationale for why the experimental results from studies of finite-span wings undergoing dynamic stall closely resemble the dynamic-stall results for two-dimensional airfoils, while steady-flow results from finite-span wing studies are so dissimilar from those for two-dimensional airfoils.

An investigation of transition to turbulence in bounded oscillatory Stokes flows Part 1. Experiments

Experimental results on flow-field statistics are presented for turbulent oscillatory flow in a circular pipe for the range of Reynolds numbers Reδ = U0δ/ν (U0 = amplitude of cross-sectional mean velocity, δ = (2ν/ω)½) = Stokes layer thickness) from 550 to 2000 and Stokes parameters Λ = R/δ (R = radius of the pipe) from 5 to 10. Axial and radial velocity components were measured simultaneously using a two-colour laser-Doppler anemometer, providing information on ensemble-averaged velocity profiles as well as various turbulence statistics for different phases during the cycle. In all flows studied, turbulence appeared explosively towards the end of the acceleration phase of the cycle and was sustained throughout the deceleration phase. During the turbulent portion of the cycle, production of turbulence was restricted to the wall region of the pipe and was the result of turbulent bursts. The statistics of the resulting turbulent flow showed a great deal of similarity to results for steady turbulent pipe flows; in particular the three-layer description of the flow consisting of a viscous sublayer, a logarithmic layer (with von Kármán constant = 0.4) and an outer wake could be identified at each phase if the corresponding ensemble-averaged wall-friction velocities were used for normalization. Consideration of similarity laws for these flows reveals that the existence of a logarithmic layer is a dimensional necessity whenever at least two of the scales R, u*/ω and ν/u* are widely separated; with the exact structure of the flow being dependent upon the parameters u*/Rω and u2*/ων. During the initial part of the acceleration phase, production of turbulence as well as turbulent Reynolds stresses were reduced to very low levels and the velocity profiles were in agreement with laminar theory. Nevertheless, the fluctuations retained a small but finite energy. In Part 2 of this paper, the major features observed in these experiments are used as a guideline, in conjunction with direct numerical simulations of the ‘perturbed’ Navier–Stokes equations for oscillatory flow in a channel, to identify the nature of the instability that is most likely to be responsible for transition in this class of flows.

COUPLING BETWEEN 2‐D ELECTRONS IN QUANTUM WELLS AND 3‐D PHONONS

The proper treatment of the electron‐phonon coupling between electrons confined to two dimensions (2‐D) by phonons traveling freely in three dimensions (3‐D) requires special care because phonon heating produces a bottleneck in the rate of transfer of energy from the carriers to the phonons. Because the carriers interact with phonons primarily when the latter are close to the quantum well, the latter should be described, not by plane waves, but by packets adapted to the shape of the carrier confinement. A quasi‐equilibrium technique that retains off‐diagonal elements in the phonon wave‐vector permits an unrestricted treatment of the density operator equation. That in turn leads to a choice of wave packet that comes from solving the integrodifferential equations rather than by imposition. Moreover, if the carrier distribution is assumed in quasi‐equilibrium with a given drift and temperature, the coupled partial differential equations are reduced to coupled ordinary differential equations that can be solved with modest computer power. Comparison with experimental results for steady flow of energy from carriers to phonons, and for time‐dependent relaxation yields quantitative agreement.

A mathematical model of flow through a collapsible tube—I. Model and steady flow results

A one-dimensional model is presented to describe the flow through a collapsible tube whose ends are tethered to rigid tubes and which is enclosed in a pressurized chamber. Results are presented for the special case of steady flow. Predicted pressure drop versus flow rate (Δ P- Q) characteristics agree qualitatively with available experimental data. The significance of the model and of various physical parameters, in regard to the shape of these characteristics, is discussed.

Computation of Viscous Flow Around Propeller-Shaft Configurations

A viscous-flow method for predicting propeller-hull interaction is validated by detailed comparisons with available experimental data for the relatively simple case of propeller-shaft configurations. The steady-flow results are in excellent agreement with the data and show that the present procedures are able to accurately predict many details of the flow field. The dependence of the flow field on propeller loading, including the formation of the hub vortex, is accurately simulated. Also, the robustness of the solution procedure is demonstrated by performing calculations for off-design (large-loading) conditions. The unsteady-flow calculations, which simulate the fanning action of a rotating finite-bladed propeller and which show reasonable agreement with certain general aspects of the experimental data, point out the difficulties of accurately resolving the complex blade-to-blade flow.