Laminar Solutions Research Articles

Flow Development in a Parallel Plate Channel

The paper presents a study of the flow that occurs in a finite, parallel plate channel. The experimental work consists of velocity profile measurements upstream of and inside the channel of a belt-type apparatus. Theoretical prediction of velocity profile development is made via numerical methods in both laminar and turbulent situations. In the laminar flow case analytical solution of a linearized form of the momentum equation was also possible. Good agreement is shown between prediction and experimental results for all Reynolds numbers tested; in turbulent flow this occurs particularly when employing Reichardt’s eddy viscosity model. For laminar flow the entrance length is estimated to be 0.008–0.01 times the Reynolds number, while in turbulent flow no essential difference was found between an entrance and the corresponding asymptotic profile. Upstream from the entrance the similar laminar profiles of Sakiadis were observed experimentally.

Stagnation-point heat transfer correlation for ionized gases

On the basis of published laminar boundary layer solutions for argon, xenon, nitrogen, and air, the effect of external‐flow ionization upon stagnation point heat transfer is correlated by the 0.7 power of the Prandtl number ratio across the boundary layer.

Thermal Boundary Conditions and Some Solutions for Laminar Duct Flow Forced Convection

The importance of laminar flow forced convection theoretical solutions for the design of compact heat exchangers has been well realized. Many of these theoretical solutions published in the literature are difficult to interpret by a designer, because thermal boundary conditions used in the analysis are generally not defined clearly and consistently. The objective of this paper is to clarify this situation by a systemized presentation. Starting with a general classification, nine specific thermal boundary conditions are categorized for singly connected ducts. At the end, the most useful fully developed laminar flow solutions for some important duct geometries are summarized for the designer.

Influence of Turbulence on Performance Characteristics of the Tilting Pad Thrust Bearing

The influence of fluid film turbulence on the performance of centrally-pivoted tilting pad thrust bearings was analyzed. Major features of the analysis are: (1) today’s two predominant “engineering” turbulent flow theories are delineated and their quantitative predictions compared; (2) a spherical pad profile was assumed, and (3) an equal area technique was used in the finite difference equations. The results confirmed earlier predictions of increases in power loss and load capacity when compared to a laminar solution. Also, no significant differences were found between the results predicted by the two predominant turbulent flow theories. Power loss, load capacity, and hydrodynamic oil flow are given for 13, 15, and 17 in. sizes. Comparisons of laminar and turbulent numerical results are presented.

Analysis of modulation spectra of stratified media

A general formulation is obtained which relates the relative change of reflectance observed in a normal-incidence modulated-reflectance experiment to the dielectric properties of the system for laminar configurations consisting of a substrate, ambient, and possible surface dielectric overlayers. The two general types of modulation interactions, perturbation-induced changes of the substrate dielectric function or of the thickness of one of the dielectric layers, are discussed as limiting cases. The modulated-substrate expressions are evaluated numerically for the case of optically thick Ni and Ni–Al203 films on a Ge substrate, typical examples of Schottky-barrier and MOS electroreflectance configurations. It is shown in the latter case that the intrinsic sensitivity obtainable with shot-noise-limited measurement systems is neither enhanced nor impaired for modulation measurements performed in the vicinity of the reflectance minima characteristic of these field-effect configurations, although operation near these minima should be avoided for other reasons. The line-shape evolution observed in reflectance measurements of the exciton polariton is shown to follow naturally from the interference effect of the small dielectric discontinuity within the sample. The approximate integral expression describing the effect of inhomogeneous perturbations is also related to the more-general laminar solution. It is shown that, for small modulations, the integral formulation is equivalent to treating the substrate as being uniform with respect to propagation, but producing a net reflected component of the field, obtained by summing the values at the surface of the reflected components generated at the small dielectric discontinuities between the hypothetical laminations in the limit that the lamination thickness approaches zero.

On a finite-difference approach to turbulence problems

Using a new, energy conserving finite-difference scheme for the unaveraged Navier-Stokes equations, pressure driven flow in a square duct is calculated for meshes of 7, 9, 11 and 17 points in each of three space directions. Both fully developed laminar and turbulent solutions are obtained as a function of the pressure drop imposed. The solutions are independent of initial conditions. Friction factor-Reynolds number relations, mean velocity profiles, Reynolds stress profiles and mean turbulence energy profiles qualitatively resemble data from physical flows. Improvement is noted as the mesh is refined. However, quantitative results would require an excessively fine mesh with impractically large computing requirements.

Comment on "Separation Laminar Boundary Solutions for Layer"

Comment on "Separation Laminar Boundary Solutions for Layer"

A note on the stability of convection in a vertical slot

The study of natural convection in a rectangular cavity whose side walls are maintained at different fixed temperatures can be regarded as one of the classical problems of thermal convection (see Batchelor 1954; Elder 1965 and Gill 1966). One aspect of this problem is the question of stability of the laminar flow solution for given values of the three governing parameters, the Rayleigh numberA= γgΔTL3/κν, the aspect ratioh=H/Land the Prandtl number σ = ν/κ. HereHis the height andLthe width of the cavity,Tis the temperature difference between the two walls andgthe acceleration due to gravity. The fluid filling the container has coefficient of expansion γ, thermal diffusivity κ and kinematic viscosity ν. Instead of studying the stability of the exact laminar flow solution, which is not a parallel flow, the stability of the following solution of the governing eauations is examined instead:

Separation solutions for the similar laminar boundary layer

The complete spectrum of similar laminar boundary-layer solutions exhibiting separation, pressure gradient, and mass transfer (blowing or suction) have been obtained. The resulting map includes variation of pressure gradient parameter and heat-transfer parameter as functions of mass transfer rate and wall enthalpy ratio. The solutions range from the limiting Blasius blow-off to an identification of asymptotic suction solutions. These latter solutions are obtained analytically. Nomenclature a,b = curve fit constants c = constant of integration / = transformed stream function g = stagnation enthalpy ratio, hs/hse h = static enthalpy qw = wall heat-transfer rate r = radius s = transformed streamwise coordinate u = streamwise velocity v = normal velocity x = streamwise coordinate y = normal coordinate /? = pressure gradient parameter 5* = displacement thickness t = transformed normal coordinate 0 = momentum thickness BH = energy thickness K — characteristic of asymptotic solution

Empirical Treatment of Hydrodynamic Journal Bearing Performance in the Superlaminar Regime

The application of a perturbation in terms of simple correlations for friction in turbulent Couette and ‘screw’ flows, together with a further empirical assumption consonant with the experimental work of Smith and Fuller (1), leads to a pressure field equation identical in form with the Reynolds equation. The load capacity of journal bearings throughout most of the superlaminar range may be represented by a single curve, and existing laminar solutions may be applied with the parameters modified by Reynolds number. The theory is compared with published experimental results, and with the most successful theoretical treatment (4). The correlations obtained confirm the adequacy of the theory to predict performance in the superlaminar régime.

Nonsimilar nature of the laminar boundary layer

Nonsimilar laminar boundary layer solutions with negative pressure gradient compared to experimental boundary layer velocity profiles, momentum and displacement thicknesses

Sink flow turbulent boundary layers

The study of sink flow turbulent boundary layers is of particular relevance to the problem of laminarization. The reason lies in the fact that the acceleration parameter which principally determines when a turbulent boundary layer will begin to revert towards laminar is, in these flows, constant from station to station. The paper presents theoretical solutions to this class of boundary layer by making use of the Prandtl mixing-length formula to relate the turbulent shear stress to the mean velocity gradient. Near the wall the Van Driest recommendation for mixing length is adopted and the Van Driest function, A+, is chosen such that the skin friction coefficient does not exceed a certain maximum value.The predicted solutions, which are in good agreement with available experimental data, display a plausible shift from the turbulent towards the laminar solution as the acceleration parameter is increased.

Laminar boundary-layer solutions with strong blowing.

Laminar boundary-layer solutions with strong blowing.

The buoyant motion within a hot gas plume in a horizontal wind

A smoke plume that has become nearly horizontal at some distance from the source behaves much like a ‘line-thermal’, for which, using a perturbation method, a solution in laminar flow may be obtained, on the supposition that excess temperatures are small and buoyant movements slow, i.e. that the Rayleigh number of the problem is suitably low. In analogy with some other problems in turbulent flow and turbulent diffusion, the laminar solution is then assumed to approximate what is observed in the turbulent case, provided that the rate of growth of the diffusing cloud is assessed realistically.The so-calculated pattern of streamlines in a cross-section of the plume agrees qualitatively with the observed behaviour of hot plumes and puffs, consisting of two vortex-like structures of opposite sense of rotation, lying on either side of the plume centre. The bodily upward movement of the plume is found to depend critically on the rate of growth of the plume. Thus when the plume diameter grows faster than linearly with distance (such behaviour characterizes the ‘quasi-asymptotic’ stage of relative diffusion predicted by Batchelor, 1952, for which Richardson's (4/3)-power law of eddy diffusivity holds) the plume tends to reach an asymptotic height. A crude theoretical estimate of the asymptotic height attained shows fair agreement with observations reported elsewhere. Although the plume nearly reaches this asymptotic height in the quasi-asymptotic phase, it retains a small gradient in the final phase which may be of importance at large distances from the source. The small-Rayleigh-number criterion restricts the validity of the solution to ‘weakly buoyant’ plumes.

BOUNDARY LAYER TRANSITION- FREESTREAM TURBULENCE AND PRESSURE GRADIENT EFFECTS

Abstract : A theory of boundary layer transition is presented which includes the effects of free-stream turbulence and pressure gradient. The basic notion underlying the development concerns the remoteness of the boundary layer vorticity from the surface. The theory agrees well with experimental data. The laminar boundary layer solutions of Pohlhausen and Falkner-Skan are used to calculate the effect of pressure gradient on transition. Engineering formulas are presented and one example is calculated showing the application of the theory to a body of revolution. The theory can also indicate the effects of Mach number and heat transfer. (Author)

The Unsteady Laminar Boundary Layer on a Flat Plate

Abstract The problem of a semi-infinite flat plate started initially from rest in an incompressible fluid with an arbitrary velocity without flow reversal is considered in this paper. In reference (1) the unsteady laminar boundary layer on a semi-infinite flat plate started to move with a velocity proportional to tn, where n is positive, is analyzed in detail. It is shown that a solution, constructed from the leading edge and properly continued into the far downstream region of the leading edge, possesses the proper limiting behavior that either at small t or in the far downstream region the flow field is substantially independent of x and is given by the Rayleigh-type solution. This solution also possesses the proper behavior at very large times. In view of the fact that serious difficulties have been encountered in trying to perturb the Rayleigh-type solution either to obtain solutions in the upstream region or to obtain solutions valid at large times, it is of interest to see whether the method of constructing the unsteady laminar boundary-layer solution as shown in reference (1) can be applied to arbitrary plate velocity and still lead to solutions with proper limiting behavior. This is found to be so if the plate was started initially from rest. In particular, the limiting form of the results for large t produces a solution for the motion of an oscillating plate in a uniform stream. This is of interest at least in so far as the theoretical results can be checked by experiment, not mentioning its connection with a number of practical problems. The effects of compressibility have been considered briefly with Howarth’s transformation in reference (2). Unfortunately, the simple energy integrals of Busemann and of Crocco when Pr = 1 cannot be extended to unsteady flow with physically sensible thermal condition on the plate. It seems that the temperature field would be better determined numerically for each specific example in question and is omitted from the present paper.

Mass transfer between co-current fluid streams and boundary layer solutions

Approximate laminar boundary layer solutions for mass transfer across the plane interface between two co-current parallel fluid streams are derived in the form ▪ where p and q are not constants but are functions of U 2/ U 1 and μ 2ϱ 2/μ 1ϱ 1. It is suggested that the boundary layer theory provides a more realistic physical picture than either the L ewis and W hitman stagnant film theory or the H igbie Penetration Theory.

The Boundary Layer Inside a Conical Surface Due to Swirl

Abstract An analysis of the axially symmetric boundary layer on the inside of a conical surface due to swirl flow about the conical axis is presented. The momentum integrals of the boundary layer are derived, and both the laminar and turbulent boundary-layer solutions are obtained numerically. In the free stream the flow is assumed to be a potential vortex. A comparison between theory and experiment is made by using the particle paths on the conical surface of a cyclone dust separator.

Heat transfer from a rotating disk

The flow due to a disk rotating in its own plane has been investigated theoretically by von Kármán, Goldstein, and others, but little has been published on the heat transfer. For laminar conditions theoretical solutions have been given by Millsaps & Pohlhausen and by Wagner, but for conditions when the flow is turbulent over the outer part of the disk there is no previous information. The present paper describes an experimental investigation of the heat transfer for a range of conditions from entirely laminar flow to conditions when the outer 80% of the disk area is under turbulence. For laminar flow the heat transfer agrees with Wagner’s results, but Millsap’s theory is found to give too low values and an explanation is given. For the turbulent case, which occurs in most practical applications, values are given for the heat transfer which is found to approach the expression N = 0∙015 R 0∙8 for all-turbulent flow. An attempt is made to deduce the turbulent flow heat transfer theoretically by assuming a 1/7 power law of temperature distribution, but this gives too low a value. Some measurements of the velocity and temperature profiles both for laminar and for turbulent conditions are given. For laminar flow these show fair agreement with the theoretical values. For turbulent flow the temperature ratios are higher than those of velocity, which explains the low heat transfer values calculated assuming a 1/7 power temperature distribution. The relation between heat transfer and friction torque is also discussed.

Engineering Relations for Heat Transfer and Friction in High-Velocity Laminar and Turbulent Boundary-Layer Flow Over Surfaces With Constant Pressure and Temperature

Abstract Relations are presented which permit a rapid calculation of friction and heat transfer from two-dimensional high-velocity flow to surfaces with locally constant pressure and temperature and with laminar and turbulent boundary layers. The calculation procedure is one which has been well established in the field of heat transfer, namely, to use equations for friction and heat-transfer parameters which have been developed for constant-property fluids and to adapt them to conditions where properties vary in such a way that those properties are introduced into the equations at a properly determined reference temperature. Relations are developed for this reference temperature which make the results of the outlined method agree best with published laminar boundary-layer solutions. The same relations turn out to give also good representation of measured results on turbulent boundary layers. The following advantages are connected with this particular method. It gives answers by simple calculations and it is quite generally applicable. The friction and heat-transfer equations are valid for any fluid gas or liquid. The relation for the reference temperature (or enthalpy), gives very good agreement with boundary-layer solutions regardless of the specific variation of the property values on which the calculations had been based. Therefore it may be expected that they are not only valid for air at any pressure and temperature level (as long as slip flow and dissociation are avoided) but approximate real conditions well also for other gases.