Boundary Layer Type Flow Research Articles

Drag Reduction in Aerated Chute Flow: Role of Bottom Air Concentration

Wall shear stress modification in self-aerated flows has been observed in boundary layer–type flows mainly in the form of drag reduction. Due to its complexity, drag reduction has predominantly been studied using experimental techniques and empirical analyses, whereas the development of conceptual approaches is limited. In this study, two-phase open-channel flow equations with variable density are revisited and bottom air concentration is identified as a key parameter for drag reduction in air-water flows. The application of this new theory to smooth and stepped chute flows is introduced, and implications for the determination of friction factors are discussed. A modified version of the Manning-Strickler formula is proposed to model friction factors across a wide range of roughness and bottom air concentration levels.

RETRACTED: Heat transfer and circular flow around a hydrodynamic turning point through a porous annular tube

Abstract In this paper, the circular flow and heat transfer in a uniformly porous annular tube are investigated for the purpose of examining the influence of the fluid distribution and the effects due to different values of the control numbers on the variation of temperature. The attention is focused on the hydrodynamic turning point highlighted inside the annular conduct and formed by three solution branches that are the first branch Fr as well as two secondary branches S1 and S2. So, the flow structures and heat transfer patterns are analyzed through each branch of the turning point diagram. It is found that the opposite behaviors between the first branch Fr and the secondary branch of type S2 raised in the fluid distribution are also revealed in the heat transfer. The secondary branch of type S1 where a boundary layer type flow takes place by increasing the Reynolds number is also the seat of the thermal boundary layer as the Peclet number is close to the value of 10. In all the cases, the influence of the Peclet number is great compared to that of the Reynolds number on the variation of temperature.

Non-Ideal Compressible Fluid Dynamics: A Challenge for Theory

The possibility that compression as well as rarefaction shocks may form in single phase vapours was envisaged first by Bethe (1942). However calculations based on the Van der Waals equation of state indicated that the latter type of shock is possible only if the specific heat at constant volume cv divided by the universal gas constant R is larger than about 17.5 which he considered too large to be satisfied by real fluids. This conclusion was contested by Thompson (1971) who showed that the type of shock capable of forming in arbitrary fluids is determined by the sign of the thermodynamic quantityto which he referred to as fundamental derivative of gas dynamics. Here v, p, s and c denote the specific volume, the pressure, the entropy and the speed of sound. Thompson and co-workers also showed that the required condition for the existence of rarefaction shocks, that Γ may take on negative values, is indeed satisfied for a number of hydrocarbon and fluorocarbon vapours. This finding spawned a burst of theoretical studies elaborating on the unusual and often counterintuitive behaviour of shocks with rarefaction shocks present. These produced both results of theoretical character but also results suggesting the practical importance of Non-Ideal Compressible Fluid Dynamics in general. The present paper addresses some of the challenges encountered in connection with the theoretical treatment of the associated flow behaviour. Weakly nonlinear acoustic waves of finite amplitude serve as a starting point. Here mixed rather than strictly positive nonlinearity generates a wealth of phenomena not possible in perfect gases. Examples of steady flows where these non-classical effects play a decisive role (and which may be useful also for future experimental work) are quasi one-dimensional nozzle flows and transonic two-dimensional flows past corners. The study of viscous effects concentrates on laminar flows of boundary layer type. Here non-classical phenomena are caused by the uncommon smallness of the Eckert number but also by the unconventional Mach number dependence on p in the external inviscid flow region.

Reexamination on methane/oxygen combustion in a rapidly mixed type tubular flame burner

To fundamentally elucidate the requirement for an inherently safe technique of rapidly mixed type tubular flame combustion, experiments have been made to investigate (1) the mixing process of fuel and oxidizer, and (2) the appearances of methane flames under various oxygen mole fractions. Three optically accessible quartz burners of different slit widths were made for measuring the mixing layer thickness with a PIV system. Under various rates of flow of the oxidizer to the fuel, a boundary layer type flow is recognized to dominate the mixing of fuel and oxidizer around the exit of the injection slit, namely the mixing layer thickness is inversely proportional to the square root of mean injection velocity. Using two stainless steel burners, combustion tests were conducted with the oxidizers of oxygen/air mixtures. To quantitatively investigate the requirement for tubular flame establishment, the Damköhler number, which is the ratio of characteristic mixing time to characteristic chemical reaction time, has been discussed in detail. The mixing time was calculated according to estimated mixing layer thickness, while the chemical reaction time was computed with the Chemkin code. The Damköhler number has proved to be a useful measure for success/failure of tubular flame combustion. When the Damköhler number is larger than unity, chemical reaction starts before complete fuel/air mixing and the tubular flame fails to be established; when the Damköhler number is much smaller than unity, the fuel and the oxidizer are completely mixed before the onset of reaction, resulting in successful tubular flame combustion. The results confirm our hypothesis in a previous study. Furthermore, based on the concept of Damköhler number, the minimum flow rate to achieve the tubular flame combustion could be estimated.

Methane/oxygen combustion in a rapidly mixed type tubular flame burner

An inherently safe technique of rapidly mixed type tubular flame combustion has been applied to pure oxygen combustion. The flame characteristics under various oxygen mole fractions are experimentally investigated with use of burners of different slit widths, and in addition, with optically accessible quartz burners. To fundamentally understand the requirements for rapidly mixed pure oxygen tubular flame combustion, the characteristic reaction times were calculated with Chemkin code, while the characteristic mixing times were estimated based on the width of mixing layer determined. Results show that the rapidly mixed type tubular flame combustion has been obtained under oxygen-enriched and even pure oxygen conditions. At high oxygen mole fraction, diffusion flames are anchored at the exits of the fuel slits, which restrains mixing between fuel and oxidizer, resulting in a failure of tubular flame combustion. By reducing the slit width, however, the diffusion flame is inhibited and a uniform tubular flame can be obtained from 0.11 to 0.18 in overall equivalence ratio. To quantify the mixing process, the flow field was visualized, and it has been found that there exist two types of flows, a boundary layer type flow near the exits of the slits close to the wall and an axisymmetric potential flow around the axis of rotation. Based on the mixing layer width of the boundary layer type flow, the Damkohler numbers are obtained and it is clearly shown that when the Damkohler number is less than unity, the mixing is completed before the onset of reactions, resulting in establishment of tubular flame combustion.

Numerical simulation of dynamical aspects of natural convection flow in a double-skin façade

A dynamical study of the flow in a double-skin façade was carried out numerically for modified Rayleigh numbers in a range corresponding to the boundary layer-type flow in the laminar regime. The façade is modeled by an asymmetrically heated vertical plane channel. 2D numerical simulations have been carried out with the finite volume approach. Velocity profiles show the asymmetric character of the flow. The presence of a recirculation zone at the channel outlet near the unheated wall has been highlighted. Numerical results are compared with experiments carried out in water. This simulation allows finding the patterns of the recirculation zone observed in the experiments. Numerically predicted recirculation lengths are overestimated by about 6% compared with the experimental results.

Turbulent free convection in a vertical converging channel

This paper presents the results of experimental and numerical investigations of the problem of turbulent natural convection in a converging-plate vertical channel. The channel has two isothermally heated inclined walls and two adiabatic vertical side walls. The parameters involved in this study are the channel geometry represented by the channel width at exit, the inclination of the heated walls and the temperature difference between the heated walls and the ambient. The investigation covered modified Rayleigh numbers up to 108 in the computational study and up to 9.3 × 106 in the experimental work. The experimental measurements focused on the velocity field and were carried out using a PIV system and included measurements of the mean velocity profiles as well as the root-mean-square velocity and shear stress profiles. The experiments were conducted for an inclination angle of 30°, a gap width of 10 mm and two temperature differences (∆T=25.4°C and 49.8°C). The velocity profiles in the lower part of the channel indicated the presence of two distinct layers. The first layer is adjacent to the heated plate and driven by buoyancy forces while the second layer extends from the point of maximum velocity to the channel center plane and driven mainly by shear forces. The velocity profile at the upper portion of the channel has shown the merging of the two boundary layers growing over the two heated walls. The measured values of the Reynolds shear stress and root mean square of the horizontal and vertical velocity fluctuation components have reached their maximum near the wall while having smaller values in the core region. The computational results have shown that the average Nusselt number increases approximately linearly with the increase of the modified Rayleigh number when plotted on log–log scale. The variation of the local Nusselt number indicated infinite values at the channel inlet (leading edge effect) and high values at the channel exit (trailing edge effect). For a fixed value of the top channel opening, the increase of the inclination angle tended to reduce flow velocity at the inlet section while changing the flow structure near the heated plates in such a way to create boundary-layer type flow. The maximum value of the average Nusselt number occurs when θ = 0 and decreases with the increase of the inclination angle. On the other hand, the increase of the channel width at exit for the same inclination angle caused a monotonic increase in the flow velocity at the channel inlet.

Unsteady Hartmann flow in a rotating channel with arbitrary conducting walls

Unsteady hydromagnetic flow of a viscous incompressible electrically conducting fluid in a rotating channel with finitely conducting walls, induced due to an oscillating pressure gradient, in the presence of a uniform transverse magnetic field is studied. An exact solution of the governing equations is obtained in closed form. The expression for the shear stress at the walls due to primary and secondary flows is also derived. The solutions valid for small and vanishing magnetic Prandtl number P m are derived from the general solution and the asymptotic solutions valid for a large value of frequency parameter ω are also obtained in both the cases to study the nature of boundary layer type flow arising due to the large frequency of oscillations.

Numerical Simulation of Buoyancy-Induced Turbulent Flow Between Two Concentric Isothermal Spheres

Buoyancy-induced turbulent flow and natural convection heat transfer between two differentially heated concentric isothermal spheres is studied numerically. The low-Reynolds-number k–ω model is used for turbulence modeling. The two-dimensional governing equations are discretized using control volume method and solved by employing the alternating direction implicit scheme. Results are presented in the form of streamline and temperature patterns, and local and average Nusselt numbers, over the heated and cooled boundaries for a wide range of Rayleigh numbers (102–1010), extending the previous studies to the turbulent flow regime and for the radius ratio of 2. The results of the flow pattern and average Nusselt numbers were compared with the previously published experimental and numerical investigations and very good agreements were observed. For low values of Rayleigh numbers, regions with conduction-dominated flow pattern accompanied with low values of Nusselt numbers were observed, while for higher Rayleigh numbers, the flow pattern was changed to the convection dominated boundary layer type flow, resulting in an increase in the rate of heat transfer and flow velocities adjacent to both inner and outer boundaries. The average Nusselt numbers were correlated against Rayleigh number and a 1/4 power dependence of Ra in both laminar and turbulent regimes is obtained.

Flow between two stretchable disks—An exact solution of the Navier–Stokes equations

In this work, an exact solution for the steady state Navier–Stokes equations in cylindrical coordinates is presented by similarity transformation technique. The solution involves the flow between two stretchable infinite disks with accelerated stretching velocity. The similarity equation was solved numerically and the effects of disk stretching parameter and stretching Reynolds number were studied. With the increase of the stretching Reynolds numbers, the fluid begins with a creeping type flow at R = 0 to a typical boundary layer type flow for large Reynolds numbers. The pressure parameter β changes from a positive number to a negative value with the increase of non-zero stretching parameter. The upper wall stretching parameter also greatly affects the velocity distribution between the two disks with a downward net flow for γ ≠ 1. The results are also useful as a benchmark problem for the validation of three-dimensional numerical computation code.

The thermagoustic irreversibility for a single-plate thermoacoustic system

The problem of thermagoustic ( thermo- magneto-ac oustic) irreversibility near a single-plate, representative of the stack in a thermoacoustic engine, is modeled and analyzed in this paper. Assumptions (long wave, short stack, small amplitude oscillation, boundary layer type flow, etc.) are made to enable simplification of the governing continuity, momentum, and energy equations to achieve analytical solutions for velocity, temperature, and pressure. A general equation for entropy generation is derived from first principles that accounts for the transverse magnetic force present in a magnetohydrodynamic system. This entropy generation equation is then simplified in order to model the specific thermoacoustic situation considered in this paper. Finally, a time-averaged entropy generation rate followed by a global entropy generation rate are calculated and graphically presented for further analysis.

Experimental study on flow and local heat/mass transfer characteristics inside corrugated duct

The present study investigates the flow and heat/mass transfer characteristics of wavy duct for the primary surface heat exchanger application. Local heat/mass transfer coefficients on the corrugated duct side walls are measured using a naphthalene sublimation technique. The flow visualization technique and a numerical analysis using a commercial code, FLUENT, are used to understand the overall flow structures inside the duct. The corrugation angle of the wavy duct is 145° and the duct aspect ratio is 7.3. The Reynolds numbers, based on the duct hydraulic diameter, vary from 100 to 5000. The results show that complex secondary flows and transfer processes exist inside the wavy duct resulting in non-uniform distributions of the heat/mass transfer coefficients on the duct side walls. At low Reynolds numbers ( Re ⩽ 1000), relatively high heat/mass transfer regions like cell shapes appear on both pressure- and suction-side walls due to the secondary vortex flows called Taylor–Görtler vortices perpendicular to the main flow direction. However, at high Reynolds numbers ( Re > 1000), these secondary flow cells disappear and boundary layer type flow develop on pressure-side wall. On the suction-side wall, high heat/mass transfer region appears by the flow reattachment. The average heat/mass transfer coefficients are higher than those of the smooth circular duct and pressure drop increases due to the secondary flows inside wavy duct.

Blow‐up phenomena in marginally separated boundary layer type flows

AbstractIt is well‐known that unsteady effects may lead to finite time break‐down of the solutions to the fundamental equation following from the interaction theory of marginal separation which describes e.g. the leading edge flow of a slender airfoil at a small angle of attack. Recent studies dealing with the analysis of a bifurcation problem associated with this integral equation which arises if the angle of attack reaches a critical value showed that this equation can be simplified significantly for small deviations from this limiting value. The resulting evolution equation is a nonlinear partial differential equation whose local blow‐up behaviour is analyzed in detail. The existence of singular travelling waves strongly suggests the possible continuation of the solution beyond the blow‐up time.

On a steady, viscous flow in two-dimensional collapsible channels

Two characteristic cases of steady, viscous flow in 2-D collapsible channels are considered in the paper: low Reynolds number, lubrication type flow, and moderately high Reynolds number, boundary-layer type flow. In the latter case the boundary layer occupies the whole cross-section of the channel, and, in order to treat the flow analytically, we apply an approximate method based strongly on the Karman-Pohlhausen method, known from early developments of the classical boundary-layer theory. In modeling the elastic behavior of channel walls, we show how the careful scaling employed in studying the fluid flow imposes the application of geometrically nonlinear Karman shell theory. For the lubrication type flow we derive a single integro-differential equation describing the wall configuration, while in the boundary-layer type flow a similar integro-differential equation is supplemented by two first order, nonlinear ordinary differential equations. In both cases the governing equations are solved numerically by finite differences, for a wide range of control parameters. At that, steady flow was found to exist in all cases considered (we did not perform the stability analysis of the obtained solutions). Also, in the boundary-layer type flow the boundary layer was shown to be very resistant to separation, so that the presented theory was valid over the whole length of the channel. The effect of control parameters, in particular of the volume flow rate and the entrance transmural pressure, upon the wall configuration was discussed.

Transient flow rate behaviour in an external natural convection boundary layer

A theoretical approach is proposed to investigate the transient dynamic behaviour of a free convection boundary layer-type flow. The set of continuity, momentum and energy equations are solved with the classical Boussinesq approximation using the Karman–Pohlhausen integral method. Applying a step variation of the uniform heat flux on a vertical wall, the boundary layer thickness and velocity profiles within the viscous layer, streamline patterns and volumetric flow rate are evaluated as a function of time. In addition, corresponding fully analytical asymptotic solutions are derived to be readily used in engineering applications.

Transient buoyant convection of a power-law non-Newtonian fluid in an enclosure

Transient buoyant convection in a square enclosure of a non-Newtonian fluid is studied. A simple power-law fluid model, with the power-law index n and the consistency coefficient K, is adopted. Flow is initiated from the motionless isothermal initial state by abruptly raising the temperature at one vertical sidewall and lowering the temperature at the opposite vertical sidewall. The appropriately defined overall Rayleigh number Ra is large to render a boundary layer-type flow. A scale analysis is performed to gain a rudimentary understanding of the evolution process. Principal force balances are considered in each of the transient stages. Comprehensive numerical solutions are acquired to the governing equations. The transient flow and thermal characteristics, both in the boundary layers and in the interior, are portrayed. The effects of Ra and of n on the behavior of the Nusselt number are delineated. The system-wide heat transport at the steady state is calculated. Based on the numerical results, the Nusselt number correlations are proposed. The numerical solutions are in broad qualitative agreement with the descriptions obtainable from the scale analysis.

On the spectral problem for boundary-layer flows

In this article, we derive an asymptotic approximation to the eigenvalue relation for boundary-layer flows with a relative error of order (αRe)−1/2 using only first-order outer (WKB) approximations which are valid in the “complete” sense. This approximation involves only elementary transcendental functions and it appears to be capable of providing an accurate global description of both the temporal and spatial spectra for flows of boundary-layer type.

Natural convection in a liquid metal heated from above and influenced by a magnetic field

Natural convection in a liquid metal heated locally at its upper surface and affected by a vertical magnetic field is investigated both experimentally and numerically. The experiments are conducted in a cylindrical test cell of large aspect ratio which is typical for application. The cell is filled with the liquid alloy GaInSn in eutectic composition. Temperature and velocity are measured using thermocouples and an electric potential probe, respectively. In the absence of the magnetic field the experimental results indicate a dependence of the Nusselt number on the Rayleigh number according to the law Nu∝ Ra 0.191. The particular value of the scaling exponent is in excellent agreement with the prediction of a scaling analysis for laminar, boundary layer-type flow in a low-Prandtl number fluid. Furthermore the experiments demonstrate that the Nusselt number and therefore the convective heat losses can be decreased by about 20% when a magnetic field of moderate strength ( B=0.1 T) is present. The numerical simulations solve the Boussinesq equations in an axisymmetric geometry using a finite element method. The results of the simulations are both quantitatively and qualitatively in good agreement with the experimental observations. Deviations are attributed to the three-dimensional characteristics of the flow.

Near-Wall Modeling of Plane Turbulent Wall Jets

In most two-dimensional simple turbulent flows, the location of zero shear usually coincides with that of vanishing mean velocity gradient. However, such is not the case for plane turbulent wall jets. This could be due to the fact that the driving potential is the jet exit momentum, which gives rise to an outer region that resembles a free jet and an inner layer that is similar to a boundary layer. The interaction of a free-jet like flow with a boundary-layer type flow distinguishes the plane wall jet from other simple flows. Consequently, in the past, two-equation turbulence models are seldom able to predict the jet spread correctly. The present study investigates the appropriateness of two-equation modeling; particularly the importance of near-wall modeling and the validity of the equilibrium turbulence assumption. An improved near-wall model and three others are analyzed and their predictions are compared with recent measurements of plane wall jets. The jet spread is calculated correctly by the improved model, which is able to replicate the mixing behavior between the outer jet-like and inner wall layer and is asymptotically consistent. Good agreement with other measured quantities is also obtained. However, other near-wall models tested are also capable of reproducing the Reynolds-number effects of plane wall jets, but their predictions of the jet spread are incorrect.

Effects of spatial order of accuracy on the computation of vortical flowfields

The objective of the present investigation is to assess the effect of the spatial order of accuracy used for the evaluation of the inviscid fluxes on the resolution of higher order quantitites, such as velocity gradients. The viscous terms are computed as second-order accurate with central difference formulas, even though for the explicit part of the algorithm higher order approximations may be used. A viscous/inviscid method is used, and the outer part of the flowfield is computed with the inviscid flow equations. The viscous boundary-layer type flow region close to the body surface is computed with an algebraic eddy viscosity model. Results obtained with the conservative and nonconservative formulations and the viscous/inviscid approach are compared with available experimental data. The effect of grid refinement on the accuracy of the solution is also presented.