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Showing papers on "Pressure gradient published in 1974"


Journal ArticleDOI
TL;DR: In this paper, a survey of mean flow and fluctuating quantities in a turbulent boundary layer developing on a smooth wall in a pressure domain P(x), where both dP/dx and d2p/dx2 are positive (increasingly adverse) is presented.
Abstract: This paper deals with a survey of mean flow and fluctuating quantities in a turbulent boundary layer developing on a smooth wall in a pressure domain P(x), where both dP/dx and d2P/dx2 are positive (increasingly adverse). The two-dimensional nature of the flow field was checked by momentum balance, as well as velocity traverses either side of the working section centre-line. Using the integrated form of the momentum integral equation, it was found that the skinfriction term and the summed momentum and pressure terms differed by at most 19%; but for the majority of measuring points they differed by less than 14%. The off-centre-line velocity profiles were indistinguishable from those taken on the centre-line. The flow field was also surveyed for fluctuating components , as well as for u1 spectra. Wherever possible, the results were compared with existing models of boundary-layer development. These comparisons indicated that the only all-embracing model for boundary-layer development is the law of the wall.

194 citations


Journal ArticleDOI
TL;DR: Computed and in vivo intravascular pressures show a marked disparity on the arterial side of the true capillaries and a fair agreement on the venous side, attributed to the effects of precapillary sphincter action and non-Newtonian rheological behavior.

138 citations


Journal ArticleDOI
TL;DR: In this article, an integral model of the steady-state dynamics of a shallow, small-scale oceanic front is developed, which shows that interfacial friction and/or upward mass entrainment is required to balance the net pressure gradient produced by the sloping sea surface and frontal interface in the light water pool.
Abstract: An integral model of the steady-state dynamics of a shallow, small-scale oceanic front is developed. Such fronts have been observed at the boundaries of river plumes discharging into coastal sea water. They share with larger scale oceanic fronts the features of persistence in time, despite sharp horizontal gradients in properties, and strong horizontal convergence at the surface front with consequent sinking. For a steady state to exist in a reference frame moving with the front, the model shows that interfacial friction and/or upward mass entrainment is required to balance the net pressure gradient produced by the sloping sea surface and frontal interface in the light water pool. Maintenance of this balance dictates that the Richardson number be of order unity; thus, friction and entrainment coefficients are kept low allowing sharp property gradients in the steady state. Strong surface convergence is also a prominent feature of the model dynamics. Comparisons are made with the observations of Ga...

117 citations


Journal ArticleDOI
TL;DR: In this paper, two time-dependent cross-section models were applied to wind-driven currents in Lake Ontario, one linear frictionless, two-level model and a numerical model which includes both friction and nonlinear terms.
Abstract: Two time-dependent “vertical cross section models” are analyzed and applied to wind-driven currents in Lake Ontario. The models are: 1) a linear frictionless, two-level model, and 2) a numerical model which includes both friction and nonlinear terms. They predict current and temperature under the assumption that all variables except pressure are independent of the longshore coordinate. The longshore pressure gradient is computed from the condition that the volume transport normal to the cross section is zero. First, the quasi-static response of the linear frictionless model is studied to isolate the effects of topography and stratification on the structure of the coastal currents. It predicts that the vertically averaged longshore current is independent of both rotation and stratification, being in the direction of the wind where the water is shallower than average and opposite the wind in the deep water. Under homogeneous conditions, the strongest currents are confined to a thin (∼3 km wide for ...

102 citations


Journal ArticleDOI
29 Jan 1974
TL;DR: In this article, the authors considered the effect of the flow induced within a porous sand bed during the passage of near-breaking or breaking waves and found that the horizontal flow rather than the vertical flow within the bed may affect its stability.
Abstract: The possible effect on the stability of a porous sand bed of the flow induced within the bed during the passage of near-breaking or breaking waves is considered. It is found that the horizontal flow rather than the vertical flow within the bed may affect its stability. An approximate analysis, used in geotechnical computations of slope stability, indicates that a momentary bed failure is likely to occur during the passage of the steep front slope of a near-breaking wave. Experimental results for the pressure gradient along the bottom under near-breaking waves are presented. These results indicate that the pressure gradient is indeed of sufficient magnitude to cause the momentary failure suggested by the theoretical analysis. The loss of stability of the bed material due to the flow induced within the bed itself may affect the amount of material set in motion during the passage of a near-breaking or breaking wave, in particular, in model tests employing light weight bed material. The failure mechanism considered here is also used as the basis for a hypothesis for the depth of disturbance of the bed in the surf zone. The flow induced in a porous bed is concluded to be an important mechanism which should be considered when dealing with the wave-sediment interaction in the surf zone.

93 citations


Journal ArticleDOI
TL;DR: In this article, the maintenance of an elevated inversion in steady flow above a cold, rotating surface is shown to be possible for a certain range of the buoyancy number, where b is the buoyant acceleration appropriate to the density deficiency of the fluid above the inversion, f is Coriolis parameter and V676gis geostrophic velocity.
Abstract: The maintenance of an elevated inversion in steady flow above a cold, rotating surface is shown to be possible for a certain range of the ‘buoyancy number’ bfV g, where b is the buoyant acceleration appropriate to the density deficiency of the fluid above the inversion, f is Coriolis parameter and V gis geostrophic velocity (so that fV gis also horizontal pressure gradient in kinematic units) The height of the inversion lid is determined by a balance of surface stress and buoyancy, in a way which may be deduced from laboratory experiments With the aid of such empirical evidence a theory is constructed for the layer below the inversion lid The cross-isobar angle of ground-level stress is found to increase with the buoyancy number, to a limiting value of 90‡, by which time the inversion descends to the ground Under typical conditions, a temperature difference of order 10‡C is necessary to eliminate the possibility of an equilibrium, elevated inversion lid and reduce ground level wind stress to a vanishingly small value

79 citations


Journal ArticleDOI
TL;DR: In this article, the results of an experimental investigation of the development of the turbulent boundary layer with surface mass injection were reported, and two injectants were employed, air and freon, with normalized injection rates up to 0.8 per cent of the free-stream mass flux.

48 citations


Journal ArticleDOI
TL;DR: In this article, the authors reported mean velocity and concentration measurements for the injection of carbon dioxide through a plate prepared with discrete holes aligned at 45° with the plate and past which an external stream is flowing.

47 citations


01 May 1974
TL;DR: In this paper, the relationship between the turbulent transport of heat and momentum in an adverse pressure gradient boundary layer was studied, and the superposition principle was used to accurately predict the Stanton number variation for two cases of arbitrarily varying wall temperature.
Abstract: The relationship between the turbulent transport of heat and momentum in an adverse pressure gradient boundary layer was studied. An experimental study was conducted of turbulent boundary layers subject to strong adverse pressure gradients with suction. Near-equilibrium flows were attained, evidenced by outer-region similarity in terms of defect temperature and defect velocity profiles. The relationship between Stanton number and enthalpy thickness was shown to be the same as for a flat plate flow both for constant wall temperature boundary conditions and for steps in wall temperature. The superposition principle used with the step-wall-temperature experimental result was shown to accurately predict the Stanton number variation for two cases of arbitrarily varying wall temperature. The Reynolds stress tensor components were measured for strong adverse pressure gradient conditions and different suction rates. Two peaks of turbulence intensity were found: one in the inner and one in the outer regions. The outer peak is shown to be displaced outward by an adverse pressure gradient and suppressed by suction.

43 citations


Journal ArticleDOI
TL;DR: In this article, a new and reliable method for measuring V is described, based on the simultaneous recording of traces from two capacitance-type volume fraction sensors, and a modified pressure-drop calculation for two-phase slug flow in an uphill pipe is presented.
Abstract: Experiments were performed with the co-current flow of two-phase \"mixtures of air and a light oil in a transparent 1-inch-diameter\" pipe at angles of inclination varying from 0 to 10 degrees above the horizontal. Visual observation and numerical data were obtained for in-situ. liquid 'Volume fractions, slug velocities, bubble rise velocity in stagnant oil and pressure drop. These data were used to generate expressions for V the slug translational velocity, and E , the -in-situ liquid volume fraction, and to develop a pressure gradient prediction scheme. A new and reliable method for measuring V is described, based on the simultaneous 'recording of traces from two capacitance-type volume fraction sensors. Introduction WITH CROSS-COUNTRY PIPELINES and oil and gas field gathering lines, purely horizontal orientations are rare. Most of the grades encountered are relatively shallow, however and the pipe is inclined only slightly to the horizontal. For example, in a pipeline test section studied by Flanigan , the steepest angle was 8 ? degrees from the horizontal. In downhill sections, a two-phase (gas and liquid) mixture will usually adopt the stratified flow configuration whereas in uphill sections the slug flow 1'egime predominates. The in-situ liquid volume fractions and pressure drop characteristics of these two flow patterns are quite different. The most important consequence of this, from an economic point of view, is that, unlike single-phase liquid flow, the hydrostatic component of the pressure drop in the uphill section of the pipeline is not recovered in the following downhill portion. Thus, considering only the net change in elevation between the inlet and outlet of the pipeline in calculating a hydrostatic head term for a pressure-drop prediction is not valid. In the prediction of over-all pressure losses for two-phase flow in cross-country pipelines, the hydrostatic head component in the pressure-drop equations is usually the major contributor to the total pressure loss. This term depe.nds directly on the mixture density, which in turn depends on the insitu liquid volume fraction or holdup, With the constant need for improvement in pipeline design techniques and prediction methods, there has been an increase in the development and use of mechanistic models. These models are generally more amenable to extrapolation and variation in fluid properties than the earlier and generally less reliable correlation methods. Testing these mechanistic models over a wide range of conditions is often hampered by a lack of reliable data regarding in-situ liquid volume fractions, slug velocities and pressure gradient for a variety of flow rate ranges. The presentation of some new data and a modified pressure-drop calculation for two-phase slug flow in an uphill pipe are the subjects of this paper. Previous Work Two-phase flow in horizontal and vertical pipes has been studied extensively, as evidenced in, for example. the recent book by Govier and Aziz . This is not the case for such flows in inclined pipes. Very few studies have been published; most are briefly summarized in the following paragraphs. Flanigan suggested that the total pressure drop consists of two principal contributions: b L b

40 citations


01 Mar 1974
TL;DR: A survey of turbulence measurements in compressible flows is presented in this paper, where the majority of the measurements at super-and hypersonic speeds were made for the zero pressure gradient, turbulent boundary layer.
Abstract: A survey of turbulence measurements in compressible flows is presented. The majority of turbulence measurements at super- and hypersonic speeds were made for the zero pressure gradient, turbulent boundary layer. It was found that the nondimensional turbulent stress terms for the zero pressure gradient flow appear to agree closely with equivalent incompressible measurements in the outer part of the boundary layer. The stress terms were nondimensionalized by the wall value of shear stress and plotted versus the distance from the wall, nondimensionalized by the boundary-layer thickness. Indirect evaluation of the total shear stress distribution from mean velocity measurements for both super- and hypersonic flows (zero pressure gradient, two-dimensional flows) indicate a near universal distribution. These total shear stress curves also agree very closely with measured incompressible shear stress distributions. Recent laser anemometer measurements of the turbulent Reynolds shear stress (puv), reported by Johnson and Rose for a Mach number 2.9 flow, are in reasonable agreement with the expected total shear stress curve over the outer 60% of the boundary layer.

25 Mar 1974
TL;DR: In this paper, an extended version of the wall law is derived for three-dimensional flows and the effects of both the pressure gradient and the inertial forces have been taken into account.
Abstract: An extended law of the wall is derived for three-dimensional flows. This law of the wall describes the variation of the magnitude and the direction of the velocity close to the wall. The effects of both the pressure gradient and the inertial forces have been taken into account. The derived wall law is valid only when the deviations from the simple law of the wall are not large. The most important feature of a three-dimenBional wall law is the prediction of the rotation of the velocity vector near the wall. Comparisons of the flow angle variations predicted by the present wall law with the few available experimental data show good agreement

Journal ArticleDOI
TL;DR: Using temperature fields derived from the Mariner 9 infrared spectroscopy experiment, the Martian atmospheric tidal pressure and wind fields are calculated by using a spherical harmonic expansion, retaining one symmetric and one asymmetric term for wenumber zero and wavenumber one.
Abstract: Using temperature fields derived from the Mariner 9 infrared spectroscopy experiment, the Martian atmospheric tidal pressure and wind fields are calculated. Temperature as a function of local time, latitude, and atmospheric pressure level is obtained by secular and longitudinal averaging of the data. The resulting temperature field is approximated by a spherical harmonic expansion, retaining one symmetric and one asymmetric term for wavenumber zero and wavenumber one. Vertical averaging of the linearized momentum and continuity equations results in an inhomogeneous tidal equation for surface pressure fluctuations with the driving function related to the temperature field through the geopotential function and the hydrostatic equation. Solutions of the tidal equation show a diurnal fractional pressure amplitude approximately equal to one half of the vertically averaged diurnal fractional temperature amplitude.

Journal ArticleDOI
TL;DR: An experimental procedure is described for determining the six independent conductance coefficients that control the fluxes of ions and solvent in a system of a single salt and water through an ionic membrane and it is indicated that the postulate of local linear behaviour is obeyed.
Abstract: An experimental procedure is described for determining the six independent conductance coefficients that control the fluxes of ions and solvent in a system of a single salt and water through an ionic membrane. Some of the data required are electrical and have been published already. Additionally, measurements are needed on flows under a pressure gradient. A cell is described for measuring the volume and salt fluxes through the membrane in the pressure range up to 1 MN m -2 . Data are given at various concentrations up to 1 mol dm -3 , for NaBr, CsBr and SrBr 2 in the cation permeable membrane Zeo-Karb 315 at 25 oC. The calculation of the non-equilibrium thermodynamic conductance coefficients is described and their values tabulated. It is indicated that the postulate of local linear behaviour is obeyed for electrical potential and osmotic gradients. For pressure gradients, the filtration coefficients L p decreased by about 1% per atmosphere across a membrane 1.4 mm thick. This effect correlated reasonably well with the observed effect of pressure on the membrane thickness if an idealized capillary model was used to describe flow. The differential conductance coefficients are strong functions of solution concentration. This dependence effectively limits use of integral linear flux equations to extremely small concentration intervals across the membrane.

Journal ArticleDOI
TL;DR: In this article, a model for cocurrent gas liquid flow through a packed bed is proposed for predicting pressure gradient and liquid holdup for both cocurrent upflow and downflow.

Journal ArticleDOI
X.B. Reed1, E. Riolo1, S. Hartland1
TL;DR: In this article, the effect of hydrodynamic coupling of adjacent phases on axisymmetric drainage of thin films is examined using a prototype model of coalescence for long times, pressure forces in the film dominate flow in all three regions, and finally all move effectively as one, whereas for short times, profiles are sharp and initial flow differences in the three regions can dominate pressure effects.

Journal ArticleDOI
TL;DR: In this article, a differential equation of the kinetic energy balance of turbulent flows is used to close the equations describing average motion in turbulent flows. But the authors do not consider the effect of the pressure gradient on the average flow and turbulent viscosity.
Abstract: A differential equation of the kinetic-energy balance of turbulence is used in a number of papers to close the equations describing average motion in turbulent flows. On the basis of this relation, a differential equation for turbulent viscosity is obtained herein. Numerical computations are carried out for incompressible non-self-similar turbulent and transition flows in awake, a jet, and a boundary layer; universal constants in the equation for the viscosity are refined. The flow in a wake and boundary layer with high longitudinal pressure gradients is investigated by analytical and numerical methods. Dimensionless criteria determining the nature of the effect of the pressure gradient on the average flow and turbulent viscosity are obtained.

Journal ArticleDOI
TL;DR: In this article, the early and intermediate development of a highly accelerated (or decelerated) turbulent boundary layer is analyzed, and the theoretical results for the evolution of the mean velocity in favorable and adverse pressure gradients agree well with experiment for the cases considered.
Abstract: The early and intermediate development of a highly accelerated (or decelerated) turbulent boundary layer is analyzed. For sufficiently large accelerations (or pressure gradients) and for total normal strains which are not excessive, the equation for the Reynolds shear stress simplifies to give a stress that remains approximately constant as it is convected along streamlines. The theoretical results for the evolution of the mean velocity in favourable and adverse pressure gradients agree well with experiment for the cases considered. A calculation which includes mass injection at the wall is also given.

Journal ArticleDOI
TL;DR: In this article, a general procedure to predict the dynamic response, including resonance of hyperbolic cooling towers to turbulent winds, was presented. But the authors did not consider the effect of the windward and wake regions.
Abstract: The paper outlines a general procedure to predict the dynamic response, including resonance of hyperbolic cooling towers to turbulent winds. Pressure spectra on the tower surface were measured in a boundary layer wind tunnel. Simplifying assumptions have led to the description of pressure spectra and response spectra by means of compact formulation. Application to full-scale towers, including Ferrybridge, is examined. It is concluded that the windward and wake regions are not correlated; the quasi-steady response resides mainly in the coupled harmonics m=1, 2, 3; and the resonant response resides mainly in the harmonics m=4, 5, 6. It is also concluded that while the quasi-steady response increases as wind velocity squared, the resonant response increases faster than wind velocity cubed.

Journal ArticleDOI
TL;DR: In this paper, the Orr-Sommerfeld equation is solved numerically for the boundary layer profiles which are the solutions to Stewartson's branch of the Falkner-Skan equation.
Abstract: The Orr‐Sommerfeld equation is solved numerically for the boundary layer profiles which are the solutions to Stewartson's branch of the Falkner‐Skan equation. These profiles are of significance in describing post‐separation flows with negative skin friction. The cases considered are for the Hartree pressure gradient parameter β = −0.18, −0.15, −0.10, and −0.05. The profiles become more stable as β decreases from ‐0.05 to ‐0.18 in contrast to the results of the Falkner‐Skan similarity profiles with positive skin friction.

Journal ArticleDOI
TL;DR: In this article, a new and simple version of the Prandtl mixing length hypothesis for flow past smooth surfaces was proposed, which has been used in obtaining extensive predictions of boundary layers on porous surfaces including flows with steep density and pressure gradients measured in the present study.

Journal ArticleDOI
TL;DR: In this paper, the mean and fluctuating flow properties of a compressible turbulent boundary layer in a shock-wave-induced adverse pressure gradient were investigated, and it was concluded that the shockwave/boundary-layer interaction significantly alters the shear-stress characteristics of the boundary layer.
Abstract: The results of an experimental investigation of the mean- and fluctuating-flow properties of a compressible turbulent boundary layer in a shock-wave-induced adverse pressure gradient are presented. The turbulent boundary layer developed on the wall of an axially symmetric nozzle and test section whose nominal free-stream Mach number and boundary-layer-thickness Reynolds number were 4 and 105, respectively. The adverse pressure gradient was induced by an externally generated, conical shock wave.Mean and time-averaged fluctuating-flow data, including the experimental Reynolds shear stresses and experimental turbulent heat-transfer rates, are presented for the boundary layer and external flow, upstream, within and downstream of the pressure gradient. The turbulent mixing properties of the flow were determined experimentally with a hot-wire anemometer. The calibration of the wires and the interpretation of the data are discussed.From the results of the investigation, it is concluded that the shock-wave/boundary-layer interaction significantly alters the shear-stress characteristics of the boundary layer.

Journal ArticleDOI
TL;DR: In this paper, a circular flow tank facility has been developed which features extremely low acoustic and vibration ambient conditions, and operating procedures associated with its use are presented, and limitations in the measurement procedures are mentioned.
Abstract: A circular flow tank facility has been developed which features extremely low acoustic and vibration ambient conditions. This facility is described, operating procedures associated with its use are presented, and limitations in the measurement procedures are mentioned. The flow facility has been used to measure the noise output of pressure and pressure gradient hydrophones during actual flow conditions and results of these tests are presented and discussed.

01 Nov 1974
TL;DR: An integral method for predicting boundary layer development in turbulent flow regions on two-dimensional or axisymmetric bodies was developed in this paper, which has the capability of approximating nonequilibrium velocity profiles as well as the local surface friction in the presence of a pressure gradient.
Abstract: An integral method for predicting boundary layer development in turbulent flow regions on two-dimensional or axisymmetric bodies was developed. The method has the capability of approximating nonequilibrium velocity profiles as well as the local surface friction in the presence of a pressure gradient. An approach was developed for the problem of predicting the heat transfer in a turbulent boundary layer in the presence of a high pressure gradient. The solution was derived with particular emphasis on its applicability to supersonic combustion; thus, the effects of real gas flows were included. The resulting integrodifferential boundary layer method permits the estimation of cooling reguirements for scramjet engines. Theoretical heat transfer results are compared with experimental combustor and noncombustor heat transfer data. The heat transfer method was used in the development of engine design concepts which will produce an engine with reduced cooling requirements. The Langley scramjet engine module was designed by utilizing these design concepts and this engine design is discussed along with its corresponding cooling requirements. The heat transfer method was also used to develop a combustor cooling correlation for a combustor whose local properties are computed one dimensionally by assuming a linear area variation and a given heat release schedule.

Proceedings ArticleDOI
30 Mar 1974
TL;DR: In this paper, it was shown that Archimedes type forces may either increase or decrease the cooling efficiency of a cooling film, according to the value of the mass flow rate ratio, the temperature ratio of the flows and the sign of the curvature of the wall.
Abstract: Analysis of the equations governing the equilibrium conditions of a cool gas film injected along a curved wall in a hot gas stream shows that Archimedes type forces may either increase or decrease the cooling efficiency of the film, according to the value of the mass flow rate ratio, the temperature ratio of the flows and the sign of the curvature of the wall. Experiments have been performed to verify these predictions: in the case of a concave wall and relatively high mass flow rate ratios, a gain in performance has been obtained, if compared to the constant pressure flat plate; in the case of a convex wall, the cooling efficiency of the film is less than that of a film on a plate with the same pressure gradient, although it is still higher, for the same mass flow ratio than the efficiency of a film on a constant pressure plate.Copyright © 1974 by ASME

Journal ArticleDOI
TL;DR: In this paper, a method has been outlined for the calculation of the pressure gradient that can exist within the reacted shell when a spherical pellet reacts with a gas and undergoes a transport controlled topochemical reaction.
Abstract: A method has been outlined for the calculation of the pressure gradient that can exist within the reacted shell when a spherical pellet reacts with a gas and undergoes a transport controlled topochemical reaction. It is known that pressure gradients can arise because of Knudsen flow existing in the reacted shell with small pores and the reactant gas having a different diffusivity than that of the product gas. The phenomena can be represented by a boundary value problem involving a set of partial differential equations with a moving boundary, incorporating time and positional dependence of diffusivities of the reactant and product gases. In the present work, the resulting equations have been solved numerically. A study has been made of the influence of the relevant parameters like total and Knudsen diffusivity ratios of the reactant and product gases, the porosity to tortuosity ratio of the reacted shell, the Biot modulus, the equilibrium constant of the reaction and the viscous flow parameter on the pressure build up inside the reacted shell.

Journal ArticleDOI
TL;DR: In this paper, a self-consistent theoretical model of the dynamical processes occurring in the Earth's upper atmosphere has been developed based on magnetohydrodynamic concepts, and the May 1967 geomagnetic storm data recorded at College, Alaska, have been used in the resulting set of non-linear, partial differential magnetodynamic equations, to calculate variations due to the storm in the amount of Joule heating and the winds at an altitude of approximately 140 km as a function of time.


Journal ArticleDOI
TL;DR: In this paper, the entrance problem of convective magnetohydrodynamic channel flow between two parallel plates subjected simultaneously to an axial temperature gradient and a pressure gradient is investigated numerically, and the solutions match to the fully developed solutions after a certain entrance length.

Journal ArticleDOI
TL;DR: In this paper, the stability of an incompressible flat plate boundary layer with pressure gradient (Hartree pressure gradient parameter β = 1.4) was computed from the linearized complete small disturbance equations, and the analysis is nevertheless a quasiparallel treatment because although boundary layer growth is accounted for, the disturbance wave function is correct only in a strictly parallel flow.
Abstract: The stability of an incompressible flat plate boundary layer with pressure gradient (Hartree pressure gradient parameter β = 1.0, 0.6, 0.4, 0.2, 0, −0.1, −0.14, and −0.1988) is computed from the linearized complete small disturbance equations. The analysis is nevertheless a quasiparallel treatment because although boundary layer growth is accounted for, the disturbance wave function is correct only in a strictly parallel flow. It is found that the nonparallel flow effect has a negligible influence on the critical Reynolds number Rδ*−c for 0.4 ≤ β ≤ 1.0, but leads to a decrease in Rδ*−c in the range −0.1988 ≤ β ≤ 0.4. The V terms lead to an increase in Rδ*−c in the range 0.4 ≤ β ≤ 1.0, and to a decrease in the range −0.1988 ≤ β ≤ 0.4. The stream tube stretching term led to a decrease in Rδ*−c in the range 0.4 ≤ β ≤ 1.0, and to an increase in the range 0.4 ≥ β ≥ −0.1988. The effect of the V terms and the stream tube stretching term ∂2U/∂x∂y appear to dominate all other boundary layer growth terms; near Rδ*−...