Two-dimensional Obstacle Research Articles

Fast Mapping of Obstacles Into Configuration Space

This paper presents an approach to fast mapping of obstacles from a workspace into a configuration space, based on defining some specific points in the workspace as fundamental obstacles. In order to achieve this objective, we propose efficient algorithms for mapping complex Cartesian obstacles by determining their critical points. The computational time required for mapping a two-dimensional obstacle is approximately 5 ms with a 33 Mhz 80486 CPU. Obstacle mapping in this paper is based on a CAD model of an environment that provide global information. The proposed mapping procedure and algorithms are suitable for industrial manipulators with a symmetric workspace occupied by polygonal obstacles.

Comparison of Multigrid Methods for Neutral and Stably Stratified Flows over Two-Dimensional Obstacles

Coupled and decoupled methods for solving the Navier–Stokes equations are compared as underlying smoothers for a multigrid algorithm. Numerical results for the benchmark problem of the lid-driven cavity confirm that residual reduction factors per multigrid cycle for the coupled method are superior to those of the decoupled method. The line-wise implementation of the coupled method is shown to be more efficient than the cell-wise while retaining good convergence rates and is the fastest method for this problem. Both approaches are applied to the more challenging problem of homogeneous and inhomogeneous viscous flow past obstacles (a vertical barrier and a cosine-shaped bump), where the flow is largely unidirectional and good convergence rates for the coupled method can now only be achieved by solving the coupled equations in vertical lines. The convergence rates of both methods are shown to deteriorate for these flows, compared with those for the lid-driven cavity, but the deterioration is generally less for the decoupled method, however, and the relative efficiency of the decoupled method means that execution times are significantly less than those required for the coupled method. In the case of stratified flows convergence difficulties are found for the coupled approach when a high order discretisation is used for the density transport equation. Strategies developed to overcome this, based on the use of double discretisation techniques, are described.

The Three-Dimensionalization of Stratified Flow over Two-Dimensional Topography

The authors present a series of new analyses of the problem of stratified flow over a localized two-dimensional obstacle, focusing upon the detailed dynamical characteristics of the flows that develop when the Froude number is such that the forced internal waves “break” above their topographic source. Results demonstrate that when the flow is restricted to evolve in two space dimensions, then the intensity of the Kelvin–Helmholtz-like (K–H) perturbations that form in the downstream shear layer that separates the accelerated low-level jet in the lee of the obstacle and the overlying region of decelerated flow increases dramatically with the governing parameter NU/g (U and N are, respectively, the velocity and buoyancy frequency characteristic of the upstream incident flow, while g is the gravitational acceleration). This nondimensional parameter represents the ratio of the acceleration that a fluid particle feels in the wave to the gravitational acceleration and measures the importance of non-Boussinesq effects. A marked change in the global characteristics of the flow is shown to occur with increasing NU/g, characteristics that include the speed of downstream propagation of the so-called chinook front, the drag exerted by the flow on the obstacle, and the intensity of the K–H instability induced pulsations of the surface velocity field. When the flow is allowed to access the third spatial dimension, the authors demonstrate that it develops intense three-dimensional motions in the regions where overturning of the isentropes in the otherwise stably stratified fluid takes place. An instability of convective type first appears in the form of streamwise-oriented vortices of alternating sign. This instability erodes the downstream propagating K–H billows, eventually leading to the complete arrest of their continued propagation as they “dissolve” into fully developed turbulent flow.

Open Channel Flow Over Submerged Obstructions: An Experimental and Numerical Study

This paper reports on experimental and computational investigations of water flow over two-dimensional obstacles in an open channel. Both triangular and semi-circular obstacles are considered in order to study the effects of obstacle type and size on the flow. The upstream flow is subcritical while the downstream flow is supercritical in all the cases discussed. The results of the experiments indicate that the downstream Reynolds number increases and appears to approach an asymptotic value as the obstacle height is increased. The upstream Reynolds number, on the other hand, decreases linearly as the obstacle height is increased. For the cases involving triangular obstacles, comparisons are presented between the results of the experimental measurements and the computational simulations as well as with available analytical solutions for inviscid flow. The comparisons point to the conclusion that the fluid rotational activity has a negligible effect on the overall flow in the open channel.

A staggered control volume scheme for unstructured triangular grids

SUMMARY The purpose of this work is to introduce and validate a new staggered control volume method for the simulation of 2D=axisymmetric incompressible flows. The present study introduces a numerical procedure for solving the Navier‐Stokes equations using the primitive variable formulation. The proposed method is an extension of the staggered grid methodology to unstructured triangular meshes for a control volume approach which features ease of handling of irregularly shaped domains. Two alternative elements are studied: transported scalars are stored either at the sides of an element or at its vertices, while the pressure is always stored at the centre of an element. Two interpolation functions were investigated for the integration of the momentum equations: a skewed mass-weighted upwind function and a flow-oriented exponential shape function. The momentum equations are solved over the covolume of a side or of a vertex and the pressure‐velocity coupling makes use of a localized linear reconstruction of the discontinuous pressure field surrounding an element in order to obtain the pressure gradient terms. The pressure equation is obtained through a discretization of the continuity equation which uses the triangular element itself as the control volume. The method is applied to the simulation of the following test cases: backward-facing step flow, flow over a two-dimensional obstacle and flow in a pipe with sudden contraction of cross-sectional area. All numerical investigations are compared with experimental data from the literature. A grid convergence and error analysis study is also carried out for flow in a driven cavity. Results compared favourably with experimental data and so the new control volume scheme is deemed well suited for the prediction of incompressible flows in complex geometries. # 1997 John Wiley & Sons, Ltd.

Multigrid Computation of Stratified Flow over Two-Dimensional Obstacles

A robust multigrid method for the incompressible Navier–Stokes equations is presented and applied to the computation of viscous flow over obstacles in a bounded domain under conditions of neutral stability and stable density stratification. Two obstacle shapes have been used, namely a vertical barrier, for which the grid is Cartesian, and a smooth cosine-shaped obstacle, for which a boundary-conforming transformation is incorporated. Results are given for laminar flows at low Reynolds numbers and turbulent flows at a high Reynolds number, when a simple mixing length turbulence model is included. The multigrid algorithm is used to compute steady flows for each obstacle at low and high Reynolds numbers in conditions of weak static stability, defined byK=ND/πU≤ 1, whereU,N, andDare the upstream velocity, bouyancy frequency, and domain height respectively. Results are also presented for the vertical barrier at low and high Reynolds number in conditions of strong static stability,K> 1, when lee wave motions ensure that the flow is unsteady, and the multigrid algorithm is used to compute the flow at each timestep.

Gel electrophoresis and diffusion of ring-shaped DNA

A model for the motion of ring-shaped DNA in a gel is introduced and studied by numerical simulations and a mean-field approximation. The ring motion is mediated by finger-shaped loops that move in an amoebalike fashion around the gel obstructions. This constitutes an extension of previous reptation tube treatments. It is shown that tension is essential for describing the dynamics in the presence of loops. It is included in the model as long-range interactions over stretched DNA regions. The mobility of ring-shaped DNA is found to saturate much as in the well-studied case of linear DNA. Experiments in agarose gels, however, show that the mobility drops exponentially with the DNA ring size. This is commonly attributed to dangling ends in the gel that can impale the ring. The predictions of the present model are expected to apply to artificial two-dimensional obstacle arrays [W. D. Volkmuth and R. H. Austin, Nature 358, 600 (1992)] which have no dangling ends. In the zero-field case an exact solution of the model steady state is obtained, and quantities such as the average ring size are calculated. An approximate treatment of the ring dynamics is given, and the diffusion coefficient is derived. The model is also discussed in the context of spontaneous symmetry breaking in one dimension.

Topographic effects in stratified flows

1. Background 2. The flow of a homogeneous layer with a free surface 3. Two-layer flows 4. Waves in stratified fluids 5. Stratified flow over two-dimensional obstacles 6. Stratified flow past three-dimensional topography 7. Applications to practical modelling of flow over complex terrain.

An Approximate Method for the Drags of Two-Dimensional Obstacles at Low Reynolds Numbers

We propose a new simple method of computing the drag coefficients of two-dimensional obstacles symmetrical to the main-flow axis at Reynolds numbers less than 100. The governing equations employed in this method are the modified Oseen’s linearized equation of motion and continuity equation, and the computation is based on a discrete singularity method. As examples, simple obstacles such as circular cylinders, rectangular prisms, and symmetrical Zhukovskii aerofoils are considered. And it was confirmed that the computed drags agree well with experimental values. Besides optimum shapes of these geometries, which minimize the drag coefficients, are also determined at each Reynolds number.

A numerical study of wave-breaking in stratified flow over obstacles

This paper describes results of numerical computations of stratified flow of finite depth D over a variety of obstacles using the time-dependent Navier Stokes equations with stability-dependent eddy viscosity turbulence models. Most computations were performed with the Froude number F h in the range 0.5 ⩽ F h ⩽ 2 and the parameter K = D /( πF h ) in the range 1 ⩽ K ⩽ 10. Here F h = U / Nh , where U is the free stream velocity, N is the buoyancy frequency and h is the height of the body. The domain and boundary conditions correspond to those of a towing tank experiment and the initial conditions were ‘impulsive start’. Critical Froude numbers for wave-breaking over two-dimensional obstacles are compared with theoretical predictions and experimental results. Preliminary results with mixing length and one-equation turbulence models suggest that differences in the representation of flows with breaking waves with such models are small. The occurrence of ‘merged flow’, in which the breaking region joins with the secondary separation zone, and which has been seen in experiments, is found in the computations in three dimensions, but not those in two.

Numerical Simulation on Turbulent Shear Flows over Surface-Mounted Obstacles

A modified k- turbulence model having a generality is proposed in the present study, in which the constant in the -equation is simply changed as a functional form of a new parameter both satisfying the tensor invariant condition and representing the extra straining effect on complex shear flows. With this model turbulent shear flows over two-dimensional obstacles placed in a channel are numerically studied for different blockage ratios and aspect ratios. Comparing with the available experimental data, the predicted results with the present model provide definite improvements over the standard model's results and work fairly well with the experimental data on the size of the recirculation zone, as well as mean velocity, wall static pressure, turbulent kinetic energy and Reynolds stresses.

Numerical Simulations of Upstream Bores and Solitons in a Two-Layer Flow past an Obstacle

Abstract Numerical simulations of upstream propagating disturbances generated through the interaction of an inviscid, nearly two-layer flow past a two-dimensional obstacle are discussed. The experimental configuration is an appropriate one for flow past ice keels–the inverted ridges of submerged ice found in frozen seas. In the first series of simulations, a steady inflow is specified and properties of upstream propagating bores are compared with the predictions of hydrostatic theory. While the comparison shows a generally good agreement with theory, a dependence of the response an the height-to-width aspect ratio of the obstacle is identified. For obstacles with relatively large aspect ratios, bore amplitudes and propagation speed are overestimated by the theory as the Froude number becomes large. Conversely, bore amplitudes are slightly underestimated for obstacles with small aspect ratios. In a second series of simulations, flow past the obstacle is modulated at the M2 frequency over one-half tidal cyc...

Cavitation in the rotational structures of a turbulent wake

Experiments show that cavitation, if moderately developed, makes three kinds of vortical coherent structures visible inside the turbulent wake of a two-dimensional obstacle: Bénard–Kármán vortices, streamwise three-dimensional vortices and finally the vortices which appear on the borders of the very near wake. The latter, which are called here near-wake vortices, result by successive pairing in the first ones and there is some indication that they are also the origin of streamwise vortices. Cavitation is not a passive agent of visualization, as can be established on the basis of fundamental arguments, and it reacts with the flow as soon as it appears; when it is developed, it breaks the connection between the elongation rate and the vorticity rate of the vortex filaments. Then the subsequent evolution of a cavitating vortex and its final implosion are rather complicated. Despite its active character, cavitation in rotational structures, if properly interpreted, can give information of interest on the basic non-cavitating turbulent flow. By adapting a simple model due to Kermeen & Parkin (1957) and Arndt (1976), and counting near-wake vortices, it is possible to accurately predict the conditions of cavitation inception: consideration of coherent rotational structures is probably the best approach to explain, in an almost deterministic way, the large difference between the absolute value of the mean pressure coefficient at the obstacle base and the incipient cavitation number.

Simulated and experimental two-layer flows past isolated two-dimensional obstacles

Laboratory flows of a two-layer fluid past an isolated two-dimensional obstacle are compared with simulations from a numerical model based on a streamfunction-vorticity formulation of the governing equations. A detailed model-data comparison of the interface position and drag force at varying Froude numbers is discussed. Inviscid runs with the numerical model accurately simulate the upstream propagating disturbances and reproduce the interface displacement in the vicinity of the obstacle. Model drag forces from a Bernoulli calculation agree with the experimental forces over part of the Froude number range considered. However, at higher Froude numbers the obstacle drag is underestimated by the model due to neglect of viscous effects.

Numerical simulations of stratified inviscid flow over a smooth obstacle

Results of numerical simulations of the flow of a non-rotating, inviscid, Boussinesq fluid over smooth two-dimensional obstacles are described. The fluid has finite depth and a rigid lid. Far upstream of the obstacle the horizontal velocity and buoyancy frequency are uniform. Comparisons with linear theory for small-amplitude obstacles are made and the long-time behaviour is compared with steady-state Long's model solutions. Comparisons with the time-dependent results of Baines (1979) are done. For Froude numbers between ½ and 1 the obstacle amplitude is varied in order to determine the amplitudes needed to initiate wave breaking. These results are compared with the predictions of Long's model and with the experimental results of Baines (1977) showing good agreement with the latter. It is found that wave breaking occurs for amplitudes significantly lower than Long's model predicts for a large range of Froude numbers. This is shown to be the result of the generation of large-amplitude lee waves with wavelengths longer than that of stationary lee waves, but not long enough to propagate upstream. The behaviour of these waves is coupled to the generation of both longer mode-one waves which do propagate upstream from the obstacle and to mode-two waves which propagate against the flow as they are advected downstream. It is also coupled to oscillations in the wave drag. The periods of the wave drag oscillations are compared to experimental results showing good agreement with cases for which oscillations have been observed. The behaviour of these large transient lee waves is compared with the theoretical results contained in Grimshaw & Yi (1991), showing some similarities. As the Froude number approaches 0.5 the breaking behaviour is no longer due to these large waves, with the result that wave breaking occurs much later.

The hydraulics of steady two-layer flow over a fixed obstacle

This paper reports the results of a theoretical and experimental study of steady two-layer flow over a fixed two-dimensional obstacle. A classification scheme to predict the regime of flow given the maximum height of the obstacle, the total depth of flow, and the density and flow rate of each layer, is presented with experimental confirmation. There are differences between this classification scheme and that derived for flow over a towed obstacle by Baines (1984, 1987). These differences are due to the motion of upstream disturbances in towed obstacle flows. Approach-controlled flows, i.e. flows with an internal hydraulic control in the flow just upstream of the obstacle are studied in detail for the first time. This study reveals that non-hydrostatic forces, rather than a shock solution (called an internal hydraulic drop by previous investigators), need to be considered to explain the behaviour of Approach-controlled flows.

Large-Eddy Simulation of Turbulent Obstacle Flow Using a Dynamic Subgrid-Scale Model

We apply the dynamic subgrid-scale model to a large-eddy simulation of turbulent channel flow with a square rib mounted on one wall. The Reynolds number Re is 3.21 x 10 3 based on the mean velocity above the obstacle and the obstacle's height. Near-wall structures are resolved with the no-slip boundary condition. The results show better agreement with direct numerical simulation than large-eddy simulation with a fixed model constant, verifying the value of the dynamic subgrid-scale model for simulating complex turbulent flows. ARGE-EDDY simulation (LES) is an accurate method of simulating complex turbulent flows in which the large flow structures are computed while small scales are modeled. The rationale behind this method is based on two observations: most of the turbulent energy is in the large structures, and the small scales are more isotropic and universal. Therefore, LES may be more general and less geometry dependent than Reynolds-averaged modeling, although it comes at higher cost. Even though LES has been used by many investigators, most research has been limited to flows with simple geometry. In engineering applications, however, one encounters more complicated geometries. Here we shall consider a rectangular parallelepiped mounted on a flat surface. Related flows are those over surfaces protruding from submarines (conning towers or control fins), wind flows around buildings, and airflows over computer chips, among others. The most distinctive features associated with these flows are three dimensionality, flow separation due to protruding surfaces, and large-scale unsteadiness. As a model flow, we consider a plane channel flow in which a two-dimensional obstacle is mounted on one surface (see Fig. 1). This relatively simple geometry contains flow separation and reattachment. Flow in this geometry has been studied by Tropea and Gackstatter1 for low Re and Werner and Wengle2 and Dimaczek et al.3 for high Re, among others. Recently, Germano et al. 4 suggested a dynamic subgridscale model (DSGSM) in which the model coefficient is dynamically computed as computation progresses rather than specified a priori. This approach is based on an algebraic identity between the subgrid-scale stresses at two different filter levels and the resolved turbulent stresses. They applied the model to transitional and fully turbulent channel flows and showed that the model contributes nothing in laminar flow and exhibits the correct asymptotic behavior in the nearwall region of turbulent flows without an ad hoc damping function. This is a significant improvement over conventional subgrid-scale modeling.

Upstream stagnation points in stratified flow past obstacles

Abstract An experimental study has been made of stagnation points and flow splitting on the upstream side of obstacles in uniformly stratified flow. A range from small to large values of Nh/U (where N is the buoyancy frequency, hm is the maximum obstacle height and U is the undisturbed fluid velocity) has been covered, for three obstacle shapes which are, respectively, axisymmetric, and elongated in the across-stream and in the downstream directions. Upstream stagnation for the first two of these models does not occur until Nhm/U > 1.05, where it occurs at z ≈ hm/2. On the central line below this point the flow descends and diverges, and we term this ‘flow splitting’. For the third model (elongated in the downstream direction), stagnation upstream first occurs at Nhm/U ≈ 1.43, at z ≈ 0. Results for this obstacle are not consistent with the ‘Sheppard criterion’, and this upstream flow stagnation is not apparently related to lee wave overturning, in contrast to flow over two-dimensional obstacles.

Analysis of wind tunnel dispersion data over two-dimensional obstacles

An analysis of wind tunnel data of dispersion from elevated point sources over a flat floor and gently sloping, 2-D hills is performed. (The data were obtained by Khurshudyan et al., 1981.) Formulas for the mixing lengths and the Lagrangian time scales are tested, suitable for use in various dispersion models. Some expressions for the vertical first moments of the plume concentration distribution suitable for shear flow (Hunt, 1985) are also tested. Then, a normalization is suggested, based on the source mean flow and turbulence parameters, for the ground-level concentration maximum value and position along the plume centerline. Using this normalization, the maximum position is almost constant, regardless of source height variation and of whether the hill is present or not, at least for source positions upstream or over the top. The maximum values allow the determination of normalized terrain amplification factors TAFN, which are shown to be in most cases closer to one than the corresponding TAF obtained without normalization.

Comparison between theoretical and experimental scattering of an s-polarized electromagnetic wave by a two-dimensional obstacle on a surface

The scattering of s-polarized electromagnetic waves by a two-dimensional scatterer upon a surface is considered. An integral approach previously reported is generalized in order to account for multilayer systems. In order to compare the theory with scattered-light measurements, we have fabricated two ribs upon a germanium substrate by means of a dry-etching technique. The height and width of the ribs are approximately 5 μm. The ribs are separated by 15 μm, whereas the incident wavelength is 10.6 μm. A good agreement is found between theory and experiment.