Problem Of Uniform Flow Research Articles

The theory and application of Navier-Stokeslets (NSlets)

Consider a closed body moving in an unbounded fluid that decays to rest in the far-field and governed by the incompressible Navier-Stokes equations. By considering a translating reference frame, this is equivalent to a uniform flow past the body. A velocity representation is given as an integral distribution of Green’s functions of the Navier-Stokes equations which we shall call NSlets. The strength of the NSlets is the same as the force distribution over the body boundary. An expansion for the NSlet is given with the leading-order term being the Oseenlet. To test the theory, the following three two-dimensional steady flow benchmark applications are considered. First, consider uniform flow past a circular cylinder for three cases: low Reynolds number, high Reynolds number, and also intermediate Reynolds numbers at values 26 and 36. These values are chosen because the flow is still steady and has not yet become unsteady. For the low Reynolds number, approximate the NSlet by the leading order Oseenlet term. For the high Reynolds number, approximate the NSlet by the Eulerlet which is the leading order Oseenlet in the high Reynolds number limit. For the intermediate Reynolds numbers, approximate the NSlet by an Eulerlet close to its origin and an Oseenlet further away. Second, consider uniform flow past a slender body with elliptical cross section with Reynolds number Re ∼ 106 and approximate the NSlet by the Eulerlet. Finally, consider the Blasius problem of uniform flow past a semi-infinite flat plate and consider the first three terms in the NSlet approximation.

A fast moving least squares approximation with adaptive Lagrangian mesh refinement for large scale immersed boundary simulations

In this paper we propose and test the validity of simple and easy-to-implement algorithms within the immersed boundary framework geared towards large scale simulations involving thousands of deformable bodies in highly turbulent flows. First, we introduce a fast moving least squares (fast-MLS) approximation technique with which we speed up the process of building transfer functions during the simulations which leads to considerable reductions in computational time. We compare the accuracy of the fast-MLS against the exact moving least squares (MLS) for the standard problem of uniform flow over a sphere. In order to overcome the restrictions set by the resolution coupling of the Lagrangian and Eulerian meshes in this particular immersed boundary method, we present an adaptive Lagrangian mesh refinement procedure that is capable of drastically reducing the number of required nodes of the basic Lagrangian mesh when the immersed boundaries can move and deform. Finally, a coarse-grained collision detection algorithm is presented which can detect collision events between several Lagrangian markers residing on separate complex geometries with minimal computational overhead.

Uniform flow past a periodic array of cylinders

The problem of uniform flow past a linear periodic array, or grating, of circular cylinders is solved by a new transform technique recently devised by the author. The solutions are expressed as an integral representation dependent on a set of coefficients which are the solutions of an explicit and well-conditioned linear system that is easily solved. We also give new explicit approximation formulas for the so-called blockage coefficients that are accurate well beyond the dilute limit relevant to cylinders that are small relative to their spacing. Finally, we describe an alternative approach to the flow problems through conformal mapping theory and produce a transform representation of a useful conformal mapping from the disc-in-channel geometry to a concentric annulus. The latter result is expected to be useful in many other applications.

Manning’s equation and two-dimensional flow analogs

summary Two-dimensional (2D) flow models based on the well-known governing 2D flow equations are applied to floodplain analysis purposes. These 2D models numerically solve the governing flow equations simultaneously or explicitly on a discretization of the floodplain using grid tiles or similar tile cell geometry, called ‘‘elements”. By use of automated information systems such as digital terrain modeling, digital elevation models, and GIS, large-scale topographic floodplain maps can be readily discretized into thousands of elements that densely cover the floodplain in an edge-to-edge form. However, the assumed principal flow directions of the flow model analog, as applied across an array of elements, typically do not align with the floodplain flow streamlines. This paper examines the mathematical underpinnings of a four-direction flow analog using an array of square elements with respect to floodplain flow streamlines that are not in alignment with the analog’s principal flow directions. It is determined that application of Manning’s equation to estimate the friction slope terms of the governing flow equations, in directions that are not coincident with the flow streamlines, may introduce a bias in modeling results, in the form of slight underestimation of flow depths. It is also determined that the maximum theoretical bias, occurs when a single square element is rotated by about 13, and not 45 as would be intuitively thought. The bias as a function of rotation angle for an array of square elements follows approximately the bias for a single square element. For both the theoretical single square element and an array of square elements, the bias as a function of alignment angle follows a relatively constant value from about 5 to about 85, centered at about 45. This bias was first noted about a decade prior to the present paper, and the magnitude of this bias was estimated then to be about 20% at about 10 misalignment. An adjustment of Manning’s n is investigated based on a considered steady state uniform flow problem, but the magnitude of the adjustment (about 20%) is on the order of the magnitude of the accepted ranges of friction factors. For usual cases where random streamline trajectory variability within the floodplain flow is greater than a few degrees from perfect alignment, the apparent bias appears to be implicitly included in the Manning’s n values. It can be concluded that the array of square elements may be applied over the digital terrain model without respect to topographic flow directions.

Hydrodynamic Permeability of the Membrane as a System of Rigid Particles Covered with Porous Layer (Cell Model)

Problem of uniform flow of viscous, incompressible liquid around a rigid particle covered with porous layer and located inside the spherical cell was solved at small Reynolds numbers. In order to describe the motion of liquid within porous layer, the Brinkman equations were used. Based on the Happel and Brenner cell method, the hydrodynamic permeability of the membrane consisting of the system of porous particles with rigid impermeable cores was calculated.

셀중심법에 의한 축대칭 극소 로봇의 압축팽창운동에 대한 수치적인 연구

Flow mechanism of contractive and dilative motion is numerically investigated to obtain a propulsive force in highly viscous fluid. An axisymmetric code is developed with unstructured grid system based on cell-centered scheme. It is validated by comparing with the results of Stokes approximation for the problem of uniform flow past a sphere in low Reynolds number(R<TEX>$_{n}$</TEX>=1). The validated code is applied to the simulation of contractive and dilative periodic motion of body whose results are quantitatively compared with the two dimensional case. In order to investigate the grid dependency, two different grids are applied to the present computations. The present study provides key information for the development of an axisymmetric Micro-hydro-robot.t.

A panel-free method for time-domain analysis of the radiation problem

A panel-free method (PFM), based on the desingularized Green’s formulae proposed by Landweber and Macagno, has been developed to solve the radiation problem of a floating body in the time domain. The velocity potential due to a non-impulsive velocity is obtained by solving the boundary integral equation in terms of source strength distribution. The singularity in the Rankine source term of the time-dependent Green function is removed. The geometry of a body surface is mathematically represented by NURBS surfaces. The integral equation can be globally discretized over the body surface by Gaussian quadratures. No assumption is needed for certain degree of approximation of distributed source strength on the body surface. The accuracy of PFM was demonstrated by its application to a classical problem of uniform flow past a sphere. The response function of a hemisphere at zero speed was then computed by PFM. The computed response function, added-mass and damping coefficients are compared with other published results.

On the causal behaviour of flow over an elastic wall

In this paper we consider the causal response of the inviscid shear-layer flow over an elastic surface to excitation by a time-harmonic line force. In the case of uniform flow, Brazier-Smith &amp; Scott (1984) and Crighton &amp; Oswell (1991) have analysed the long-time limit of the response. They find that the system is absolutely unstable for sufficiently high flow speeds, and that at lower speeds there exist certain anomalous neutral modes with group velocity directed towards the driver (in contradiction of the usual radiation condition of out-going disturbances). Our aim in this paper is to repeat their analysis for more realistic shear profiles, and in particular to determine whether or not the uniform-flow results can be regained in the limit in which the shear-layer thickness on a length scale based on the fluid loading, denoted ε, becomes small. For a simple broken-line linear shear profile we find that the results are qualitatively similar to those for uniform flow. However, for the more realistic Blasius profile very significant differences arise, essentially due to the presence of the critical layer. In particular, we find that as ε → 0 the minimum flow speed required for absolute instability is pushed to considerably higher values than was found for uniform flow, leading us to conclude that the uniform-flow problem is an unattainable singular limit of our more general problem. In contrast, we find that the uniform-flow anomalous modes (written as exp (ikx − iωt), say) do persist for non-zero shear over a wide range of ε, although now becoming non-neutral. Unlike the case of uniform flow, however, the k-loci of these modes can now change direction more than once as the imaginary part of ω is increased, and we describe the connection between this behaviour and local properties of the dispersion function. Finally, in order to investigate whether or not these anomalous modes might be realizable at a finite time after the driver is switched on, we evaluate the double Fourier inversion integrals for the unsteady flow numerically. We find that the anomalous mode is indeed present at finite time, once initial transients have propagated away, not only for impulsive start-up but also when the forcing amplitude is allowed to grow slowly from a small value at some initial instant. This behaviour has significant implications for the application of standard radiation conditions in wave problems with mean flow.

Two-dimensional problem of a uniform flow of a two-layer fluid of finite depth past a circular cylinder

The linear steady problem of an irrotational uniform flow past a horizontal circular cylinder located in the upper or in the lower layer of a two-layer fluid is solved by the multipole-expansion method. The flow is perpendicular to the axis of the cylinder. The fluid is assumed to be inviscid and incompressible, and the flow in each layer is assumed to be potential. The upper layer can be bounded by a free surface or a solid lid, and the lower layer by a rigid horizontal bottom.

Plane problem of uniform flow of a two-layer fluid past a circular cylinder

An explicit solution is found for the problem of uniform horizontal flow of a two-layer fluid of infinite depth past a circular cylinder. The cylinder axis is perpendicular to the flow. The problem is solved within a linear formulation. The solution of the problem is expressed in the form of rapidly converging series with coefficients determined from a recurrence relation. The first seven terms of the series yield the values of the hydrodynamic loads with a relative accuracy of 10−6. The results are in good agreement with the known values for similar problems in a homogeneous fluid. Tables of the lift and wave drag are given for homogeneous and two-layer fluids.

Uniform flow past one (or two in tandem) finite length circular cylinder(s)

The problem of uniform flow past one (or two in tandem formation) finite length circular cylinder(s) is investigated experimentally in this paper. For the case of flow past a single finite length cylinder, it is found that the flow that separates from the free end interacts strongly with those that separate from the sides and results in a three-dimensional flow. This separated flow from the free end delays the interaction between the separated flows from the sides, resulting in a less negative wake pressure and a lower drag when compared with flow past an infinitely long cylinder. Spanwise effects were found to be stronger over the longest (h/d = 8) cylinder is investigated, with the spanwise drag coefficient increasing towards the free end. Vortex shedding with some regularity can be detected only at y/h = 0.95 and y/h < 0.5 (with weak spectral peak) of the h/d = 8 cylinder, with corresponding Strouhal numbers equal to 0.08 and 0.191, respectively. For the flow past two tandem finite length circular cylinders, although the transition from reattached flow to co- shedding flow is still observed, due to the highly three-dimensional nature of the flow, the change no longer takes place over the entire span simultaneously. When the dimensionless spacing between the cylinders increases from one (cylinders touching), the co-shedding flow structure was first detected near the free end and its extend later expands towards the mounted end with further increase in cylinders spacing, at the expense of the reattached flow region. At very small spacing between cylinders, a third flow structure which is characterised by having lower surface pressure over the windward part of the downstream cylinder was also observed. Spanwise variation of the drag that acts on both cylinders was observed. The magnitude of the drag coefficient was found to be strongly dependent on the flow structure involved and normally varies as a direct consequence of the variation in the wake pressure. The overall trend is that of higher local drag towards the free end.

Direct particle simulation on the Connection Machine CM-2

Particle simulation is a useful technique for analyzing low density flows. The Connection Machine CM-2 is a useful test bed for studying the fine-grain data objects decomposition and the coarse-grain domain decomposition single instruction multiple datastream (SIMD) approaches to particle simulation. Both approaches are investigated for the model problem of uniform flow through a channel and the algorithms required for the SIMD domain decomposition approach are presented. For this model problem, the domain decomposition approach is superior to the data objects decomposition approach with an average particle update time of 2.9 μs compared with 5.1 μs on an 8K processor CM-2. An unresolved issue with the domain decomposition approach is the effect of a poor partitioning on flows with real geometries. Initial results with the channel flow problem indicate that a poor partitioning has only a small (15%) detrimental effect on the overall performance.

Uniform Asymptotic Approximations for Duct Acoustic Modes in a Thin Boundary-Layer Flow

Analytical approximations for the acoustic modes in a duct carrying a uniform core flow with a thin shear layer at the walls are developed using the method of matched asymptotic expansions. Both two-dimensional and cylindrical duct propagation are considered. Numerical results for eigenvalues calculated using the theory are presented for the two-dimensional problem and compared with results from earlier analyses. It is found that the new approximations yield a significant increase in accuracy. HE problem of sound transmission through a duct carrying a uniform core flow with a thin boundary layer at the walls has been of interest for some time. Such a flow has practical applications in modeling a fully developed real flow profile. Perhaps the strongest motivation for studying the problem, however, is the possibility of obtaining analytical results using asymptotic analyses which can yield physical understanding of shear flow propagation phenomena in more realistic flows which must be treated numerically. Analytical studies based on the assumption of small boundary-layer thickness have been carried out by several researchers during the past decade. Eversman and Beckemeyer1 developed an inner expansion for acoustic pressure in the layer from which they were able to derive an approximate equivalent boundary condition applicable at the outer edge of the layer. Using this condition they showed that the formulation of the shear layer problem reduces to that of the associated uniform flow problem with the well-known particle displacement continuity boundary condition as the layer thickness vanishes. In a later paper, Eversman2 employed the equivalent boundary condition to formulate the eigenvalue problem for acoustic modes in an annular duct flow. Using the known uniform flow solutions in the core flow, an approximate eigenvalue equation is developed. Numerical solution of this single equation yields the acoustic wave numbers and circumvents the task of numerically integrating the differential equation and iterating for the shear flow eigenfunctions. The procedure is equally applicable to rectangular duct geometries, for which it leads to an eigenvalue equation equivalent to one later derived less formally by Tester. 3 A slightly different analysis by Swinbanks,4 also for rectangular ducts, yields similar approximate results. Other studies less directly related to the present work are discussed in Sec. IV.B ofRef. 5. The work reported in the present paper arose in the course of some research on shear flow propagation in which it was desirable to obtain analytical representations of the acoustic

Construction of upper and lower solutions for flow past a non-uniformly heated plate

The Oseen linearization and the modified Oseen linearization are often used in studying fluid mechanical problems, but whether the linearized solution is accurate is usually difficult to assess. For the sample problem of uniform flow past a plate, we use a comparison theorem to show that the Oseen linearization, used in two ways, gives both an upper and a lower solution. Further, we make use of the comparison theorem and the modified Oseen linearization to construct a sharper upper solution valid in the boundary layer. We then go on to consider the case when the plate temperature increases along the plate according to a power law. Upper and lower solutions for the temperature equation are constructed, and bounds on the temperature gradient at the plate are obtained. With a minor modification, similar results for the case when the logarithmic derivative of the plate temperature lies between two power law curves are obtained.

A moving-wall boundary layer with reverse flow

The problem of uniform flow past a flat plate whose surface has a constant velocity λU opposite in direction to that of the mainstream is considered for large values of the Reynolds number R. In a previous communication (Klemp &amp; Acrivos 1972) it was shown that, if the region of reverse flow which is established next to the plate as a consequence of its motion is O(R−1/2) in thickness, the appropriate laminar boundary-layer equations have a solution provided λ ≤ 0·3541. Here the analysis is extended to the range λ &gt; 0·3541, which cannot be treated using a conventional boundary-layer approach. Specifically, it is found that for λ &gt; 0·3541 the flow consists of three overlapping domains: (a) the external uniform flow; (b) a conventional boundary layer with reverse flow for xs &lt; x &lt; 1, where xs, refers to the point of detachment of the ψ = 0 streamline and x = 1 is the trailing edge of the plate; and (c) an inviscid collision region in the neighbourhood of xs, having dimensions O(R−1/2) in both the streamwise and the normal direction, within which the reverse moving stream collides with the uniform flow, turns around and then proceeds downstream. It is established furthermore that xs = 0 for 0 ≤ λ ≤ 1 and that xs &lt; 0 for λ &gt; 1.Also, detailed streamline patterns were obtained numerically for various λ's in the range of 0 ≤ λ ≤ 2 using a novel computational scheme which was found to be more efficient than that previously reported. Interestingly enough, the drag first decreased with λ, reached a minimum at λ = 0.3541, and then increased monotonically until, at λ = 2, it was found to have attained essentially the value predicted from the asymptotic λ → ∞ similarity solution available in the literature. Thus it is felt that the present numerical results plus the two similarity solutions for λ = 0 and for λ → ∞ fully describe the high-R steady flow for all non-negative values of λ.

Similarity properties with the flow of supersonic uniform and nonuniform flows of gas around a sphere

The article proposes an approximation formula for the velocity gradient at the critical point for ellipsoids of revolution, suitable for both an equilibrium and a perfect gas. It is shown that an approximate solution to the problem of the flow of a stream arriving from a supersonic source around a sphere can be obtained, having the solution of the problem of uniform flow around a sphere.

A method for integrating the boundary-layer equations through a region of reverse flow

Development of a procedure for numerically integrating the boundary-layer equations through a region of reverse flow which takes downstream influence into account. This method is applied to the problem of uniform flow past a parallel flat plate of finite length whose surface has a constant velocity directed opposite to that of the main stream. Although singularities occur at both the point of detachment (x sub s) and reattachment (x sub r) of the psi = 0 streamline, this integration technique provides a solution which ceases to apply only in the close proximity of these singular points. From this solution it is evident that, throughout a large portion of the separated region, the flow is strongly affected by conditions near x sub r, thereby demonstrating the importance of allowing information to be transmitted upstream in a region of backflow. Near x sub s, however, it is found that, in spite of the presence of reverse flow, the solution has a self-similar form in this particular example.

Heat and Mass Transfer from Small Spheres and Cylinders Freely Suspended in Shear Flow

The problem of heat and mass transfer from small spheres and cylinders freely suspended in a shear flow is considered in the limit of Reynolds number Re → 0. Asymptotic formulas are derived which relate the Nusselt number Nu to the Péclet number Pe in the limit Pe → 0, and for the case of the cylinder, Pe → ∞. At high Pe, the Nusselt number is found to approach a constant value, whereas, at low Pe it is shown to increase with Pe12 for the sphere and with —(log Pe)−1 for the cylinder. These results indicate the existence of a fundamental difference at high Pe between the shear flow problem studied here and the corresponding classical problem of uniform flow at infinity.