Boundary Layer Method Research Articles

Vortical wakes over a prolate spheroid

The vortex-lattice method is employed to calculate the inviscid flow and the development of separated vortex sheets over a prolate spheroid. An approximate boundary-layer method based on the assumption of local similarity is used to calculate the line of open separation. A condition of vortex shedding along the separation line is proposed. The two methods of calculation, viscous and inviscid, interact through the line of separation that is allowed to displace as the wake grows. Results are compared with flow visualization data for laminar separation and pressure data for turbulent separation.

Shooting method in singular perturbation problem of ordinary differential equations with boundary layers

By using the adaptive steplength integration scheme with a shooting technique, a rather difficult singular perturbation problem of ordinary differential equations with boundary layers can be calculated effectively. Computing examples are given in this paper which show the convergence within one iteration of the method in the case of a linear problem, the efficiency of the method for many boundary layers and turning points, especially the convenience in calculating multiple solutions. A comparison with traditional difference method is given at the end of this paper.

Asymptotic integration of non-linear equations of cylindrical panel vibrations

Complete asymptotic expansions of the solution of the two-dimensional problem of the non-linear vibrations of a cylindrical panel with free curvilinear boundaries are constructed using the boundary layer method /1, 2/ in the case when the parameter δ, equal to the ratio between the lengths of the clamped and free sides, is fairly small. The principal term of the expansion for the deflection function is determined from the known non-linear integrodifferential equation of arch vibrations. The discrepancies in satisfying the boundary conditions on the clamped boundaries turn out to be higher-order infinitesimals in δ and are compensated by boundary layer functions that are determined from linear boundary value problems for a biharmonic operator in a half-strip. Calculations are performed by using finite differences for elastic, elastoplastic cylindrical panels, arches, and rectangular plates subjected to an instantaneously applied transverse step load, and the limits of applicability are established for a monomial expansion. Questions on passage to the limit from the three-dimensional equations of the theory of elasticity to two-dimensional equations in the case of thin domains were investigated in /3–5/ for non-linear problems.

Carrier mediated transport of amino acids, small peptides, and their drug analogs

The oral absorption of amino acids, small peptides and their analogs has been studied in rats using single pass intestinal perfusion. The experimental data was analyzed using a modified boundary layer method permitting the calculation of intrinsic membrane absorption parameters. Cefatrizine, l-Dopa, α-methyldopa, and l-PHE demonstrated concentration dependent absorption. Furthermore, the absorption of the amino acid analog, l-Dopa, was significantly inhibited by l-Leucine. Cefatrizine absorption was significantly inhibited by the peptide PHE-PHE (20 mM whereas the amino acid l-PHE (100 mM) did not inhibit its absorption. The results clearly demonstrate that l-Dopa and α-methyldopa oral absorption occurs via the amino acid pathway, while cefatrizine absorption occurs via the peptide pathway. These results indicate that nutrient pathways may function as a possible route for the oral delivery of selected drugs.

Recognition of spatial structures and ensemble averages in the benthic boundary layer

Simultaneous high‐frequency spatial and temporal measurements of benthic boundary layer (BBL) velocity and sediment concentration generate complex data sets. In order to investigate the various spatial and temporal structures, a pattern recognition method is, for the first time, adapted and used to investigate different and similar structures (patterns) of velocity and acoustically measured suspended sediment concentration profiles in the lowest 2 m of the BBL at Tiana Beach, Long Island. This layer is subject to interactions between differently scaled forcing fields which result in structures difficult to predict using existing time average or boundary layer methods. Under the assumption of no a priori knowledge about either the spatial or temporal structures in the BBL, a deterministic mathematical methodology is implemented to recognize spatial structures in the data and classify them in an optimum number of classes. The spatial pattern analysis in this paper indicates that ∼ 70% of the above mentioned data are represented by only 5–20% of its size. Spatial patterns different from those predicted from theory or experience are recognized.

Fast dynamo action in a steady flow

The existence of fast dynamos caused by steady motion of an electrically conducting fluid is established by consideration of a two-dimensional spatially periodic flow: the velocity, which is independent of the vertical coordinate z, is finite and continuous everywhere but the vorticity is infinite at the X-type stagnation points. A mean-field model is developed using boundary-layer methods valid in the limit of large magnetic Reynolds number R. The magnetic field is confined to sheets, width of order R−½. The mean magnetic field lies and is uniform on horizontal planes: its direction is independent of time but rotates once about the vertical axis over a short distance 2πl, where l−1 = R½β and β is a vertical stretched wavenumber independent of R. Its alternating direction gives it a rope-like structure within the sheets. An α-effect is calculated for the model, whose strength for a given flow is a function of β and R. Two sources of α-effect are isolated whose relative importance depends critically on the size of β. When the vorticity is finite everywhere and β [Lt ] 1, the dynamo is ‘almost’ fast with growth rates of order (ln R)−1. The maximum growth rate ln (ln R)/ln R occurs when, correct to leading order, β is (ln R)−½. The asymptotic results valid for large R compare excellently with Roberts (1972) modal analysis for finite R.

A method for calculating axisymmetric afterbody flows

A numerical method for calculating the flow around axisymmetric afterbodies with single or coaxial jets is presented. The solution technique used is based on the modification of an Euler method to include a representation of viscous effects on the afterbody surface and in the wake region. The development of the afterbody boundary layer and its continuation into the wake is predicted by an integral boundary-layer method used in inverse mode. The boundary-layer equations can thus be integrated through a region of separated flow. Jet entrainment due to turbulent mixing is modeled empirically. Results from the method and their comparison with experimental data are presented. The agreement obtained demonstrates that the method can be used in a civil or military design environment.

Fast viscous correction method for full-potential transonic wing analysis

An analysis of the transonic flowfield around a three-dimensional wing is carried out using a strip method. Attention is given to the boundary layer growth in the streamwise direction. A viscous correction technique is defined for the TWING code for solving the full potential equations. A viscous ramp at the base of a shock is superimposed on the boundary layer displacement thickness generated by an integral boundary layer method. A relationship is then obtained between the effective displacement thickness and a vertical component of the surface velocity, a transpirational boundary condition. The viscous correction is found to be unnecessary in weak shock conditions but gives a better shock position and pressure distribution in a strong shock condition when compared with data from an ONERA M6 airfoil and the Hinson and Burdges (1980) Wing A.

Geometrical Theory of Diffraction(GTD) Part II: Extensions and future trends of the theory

An analysis of the extensions and improvements of GTD since its early time is given. Especially the theoretical foundations of two powerful methods, namely the Boundary Layer Method and the Uniform Asymptotic Expansion Method, are outlined. Aspects of these theories which are of growing interest at present, for solving electromagentic diffraction and radiation problems are pointed out. The resulting solutions are described in a general framework including perfectly conducting and imperfectly conducting objects condition. Their limitations are discussed and indications for further improvements are given. Some new non published results on higher-order edges and coated metallic bodies are also included. The author concludes with a brief description of future trends in the development of GTD.

The proof of a quasi-equilibrium approximation in the theory of sorption dynamics

The first terms of the asymptotic expansion of the solution of the system of diffusion-sorption equations are obtained by the method of boundary-layer functions in the case when the sorption processes are fast compared with the transport. The asymptotic form contains regular terms, ordinary boundary layer functions describing the rapid establishment of a “quasi-equilibrium” state, and corner boundary layer functions. Estimates are obtained for the remainder terms in the asymptotic expansions. There is an interesting drop in the power of the corner boundary layer functions with respect to the stretched variables, which prevents higher-order asymptotic forms being obtained by the boundary-layer method.

Flow analysis in two-dimensional impeller of centrifugal pump by the surface singularity method combined with a flow-model of three-dimensional boundary layer.

An iterative solution method has been developed for calculating flows in two-dimensional centrifugal pump impellers. The method was formed by combining an integral calculation method for a three-dimensional boundary layer on the whole channel wall with a surface singularity method for a two-dimensional inviscid core-flow. In the inviscid analysis, the blade shape and the area of flow passage were modified by calculated displacement thickness of boundary layer, and then a finite number of vortices were distributed on the modified blade surfaces with the Kutta condition at the outlet edge on the suction surface. This method was applied to flows in two impellers with different blade thicknesses. Comparing the results of calculated and measured flows, both were qualitatively in good agreement, including an effect of the outlet blade surface with constant radius, which showed that relative velocity near the pressure surface was faster than one near the suction surface.

Influence of Boundary Layer and Wake on Free Surface Flow around a Ship Model

The Rankine-source method has been combined with an integral method for boundary layer and wake so as to investigate the influence of boundary layer and wake on the free surface flow around a ship model. The head loss due to viscosity has been taken into account in the equation of the boundary condition on the free surface. The numerical solution of the free surface flow with the boundary layer and wake around 6 m long Wigley model has been obtained with aid of the technique of matrix calculation for source densities on the panel array on the hull surface along with center plane in wake and on the free surface. The computed results as for pressure distributions on the hull surface, wave profiles along the hull side and pressure, frictional and total resistances have been shown in comparison with measurements. They display good results by considering the existence of boundary layer and wake around the hull of the model.

The boundary layer method for the solution of singular perturbation problem for the parabolic partial differential equation

In this paper, we discuss the singular perturbation problem of the parabolic partial differential equation. As usual, we must reduce the mesh size in the neighbourhood of the boundary layer so that typical feature of the boundary layer will not be lost. Then we need very large operational quantity when mesh sizes are getting too small.

Application of an inverse cascade design method to an axial fan.

In order to demonstrate the applicability of an inverse cascade design method for blade design for an axial turbomachine, rotor and stator blades of a single stage axial fan are designed and tested. This design concept, proposed in the previous reports, is based on an inverse boundary layer method and an inverse cascade method. Overall performance and the rotor exit flow field are measured and are compared with a conventional fan designed using NACA 65 series cascade. The inversely designed fan obtains higher efficiency and a wider operating range compared with the conventional one. The midspan wake of the rotor blade in the trailing edge region reveals an extremely low velocity defect, a low wake shape factor, and low momentum thickness. boundary layer separation on the rotor blade suction surface seems to be avoided and the losses are low. The experimental results show the validity of the practical aplication of the inverse design concept, especially for rotating blades.

Nonlinear mode II crack-tip fields for some Hookean materials

Abstract This paper presents a modified nonlinear Mode II crack model which is shown to satisfy the nonpenetrating crack surface boundary condition for homogeneous isotropic Hookean materials taking into account finite deformations. A recent investigation of the problem by Knowles [1] reveals apparent interpenetration of the crack surfaces which is considered nonphysical and therefore invalid. This observation is confirmed when a general solution based on Knowles's perturbation boundary layer method to characterize the finite deformation effects on Mode II crack-tip fields for the materials is derived. By deducing complete 2nd order solutions of the problem, the Poynting nonlinear effect becomes self-evident at the crack tip and for Hookean materials with v 1 2 there always exists the penetrating phenomenon between upper and lower crack surfaces.

Evaluation of a Surface/Vegetation Parameterization Using Satellite Measurements of Surface Temperature

Ground measurements of surface-sensible heat flux and soil moisture for a wheat-growing area of Beauce in France were compared with the values derived by inverting two boundary layer models with a surface/vegetation formulation using surface temperature measurements made from NOAA-AVHRR. The results indicated that the trends in the surface heat fluxes and soil moisture observed during the 5 days of the field experiment were effectively captured by the inversion method using the remotely measured radiative temperatures and either of the two boundary layer methods, both of which contain nearly identical vegetation parameterizations described by Taconet et al. (1986). The sensitivity of the results to errors in the initial sounding values or measured surface temperature was tested by varying the initial sounding temperature, dewpoint, and wind speed and the measured surface temperature by amounts corresponding to typical measurement error. In general, the vegetation component was more sensitive to error than the bare soil model.

A numerical study of a viscous flow past a right circular cylinder on a β-plane

Abstract The viscous flow past a right circular cylinder on a β-plane is numerically studied. The basic flow, unbounded laterally, is assumed to be uniform. The explored parameter space is Ro<O(1), Ek≪O(1), ∼O(1) and δ=2, where Ro is the Rossby number, Ek the Ekman number, the beta parameter, and δ the aspect ratio. Under these conditions, the quasigeostrophic potential vorticity equation is integrated numerically. It is found that an almost exact correspondence exists for steady flows between the numerical solutions and the flow-patterns observed in the laboratory experiments of Boyer and Davies (1982). For a prograde flow, in addition to the upstream influence, the accelerated jet extends to the rear of the_ cylinder because of the existence of Rossby lee-waves. This tendency increases as increases. The separation points shift toward the rear of the cylinder with increasing . As a result the size of the standing eddy is reduced. In addition, the total drag coefficient Cd increases as increases. For a retrograde flow, the location of the separation point is not so sensitive to in contrast to the prograde case. This result is consistent with Merkine's (1980) theoretical prediction based on a boundary layer method. The retrograde flow pattern is marked by the formation of an elongated blocking area behind the cylinder. This is mainly because the eddy behind the cylinder propagates as a long Rossby wave. The total drag coefficient Cd increases rapidly as increases due to the excessive vorticity generation. It is also found that the nonlinear Ekman friction can generate a noticeable asymmetry in the wake. The β-effect suppresses the unsteadiness for a prograde flow, but enhances it for a retrograde flow. As a result, the Strouhal number increases as increases for a retrograde flow. When eddy shedding occurs the anticyclonic eddy is stronger than the cyclonic one due to the asymmetric property of the nonlinear Ekman friction. However, this final result is not consistent with that observed by Boyer and Davies (1982) and needs further investigation.

Aerodynamic tradeoff study of conventional, canard, and trisurface aircraft systems

Conventional, canard, and three-surface aircraft configurations are investigated analytically to determine each configuration's induced and viscous drag under trimmed conditions. A three-surface vortex lattice method is used to trim the aircraft, as well as to predict the induced drag of each configuration. A vortex panel method in conjunction with the momentum integral boundary-layer method is used to predict inviscid and viscous characteristics. Parameters varied including wing to stabilator surface area ratio, static margin, canard to tail loading ratio, and CL^ . For all of the parameters considered, the conventional configuration had the highest ^tnm^' ^ e stabilator aspect ratios, the CL . /CD of the conventional aircraft was the highest, whereas for the highest stabilator aspect ratio considered the canard configuration had the highest CL /Cp. The trisurface was superior to the canard at the lower aspect ratio with the canard becoming superior at the higher values.

Airfoil trailing-edge flow measurements

Trailing-edge data for boundary-layer/near-wake thickness parameters are given for airfoils and flat plates. Reynolds number effects are examined as a function of model size, velocity and boundary-layer tripping. Reference comparisons are made with boundary-layer calculations using potential-flow modeling and a finitedifference method for laminar and turbulent boundary layers. Open wind-tunnel corrections to angle of attack and camber are incorporated into the potential-flow modeling. It was found that the open tunnel flow turbulence affects early boundary-layer transition for the higher velocities. The theory still brackets the data allowing some assessment of boundary-laye r transition behavior.

Steady Infiltration From Buried Discs and Other Sources

Scattering analog techniques are used to analyze steady quasilinear infiltration from buried discs and to develop related results. The exact solution involves spheroidal wave functions; but, more usefully, we express the far‐field wetting function and the discharge function as expansions appropriate for small and large values of s, the dimensionless disc radius. Other simple techniques for estimating the moisture field include boundary layer and Green's function methods. Further aspects of the study include: useful comparisons with Wooding's classical results for the surface disc; and small s expansions, not hitherto available, for steady infiltration from cylindrical and spherical cavities. Buried discs (with aspect ratio h = 0), oblate spheroids (0 &lt; h &lt; 1), spheres (h = 1), and prolate spheroids (h &gt; 1) form an ordered sequence. Both capillary effects and the difficulty of matching small and large s expansions increase markedly with h. The surface disc is an exceptional “least capillarity” case. The foregoing considerations lead to design principles for permeameters to measure the saturated hydraulic conductivity of unsaturated field soils. The customary borehole permeameter has h large and consequently strong capillary effects; but saturated conductivity has greatest influence on permeameter discharge when capillary effects are least. Accordingly, the optimal design is the surface disc permeameter, with h = 0 and minimal capillary effects.