Series solution of the temperature distribution in the Falkner–Skan wedge flow by the homotopy analysis method

A new analytical method, namely the homotopy analysis method (HAM), has been applied to investigate the temperature field associated with the Falkner–Skan boundary-layer problem, and a series solution is provided in this paper. The results of the present work show agreement with those of numerical solutions in a large range of Prandtl numbers (0 < Pr ≤ 100), which demonstrates the validity of the present analysis.     

Correlations between porous medium flow data and turbulent tube flow results for aqueous hydroxypropylguar solutions (Part 2)

Turbulent tube flow and the flow through a porous medium of aqueous hydroxypropylguar (HPG) solutions in concentrations from 100 wppm to 5000 wppm is investigated. Taking the rheological flow curves into account reveals that the effectiveness in turbulent tube flow and the efficiency for the flow through a porous medium both start at the same onset wall shear stress of 1.3 Pa. The similarity of the curves = ( _w ) and = ( _w ), respectively, leads to a simple linear relation / =k, where the constantk or proportionality depends uponc. This offers the possibility to deduce (for turbulent tube flow) from (for flow through a porous medium). In conjunction with rheological data, will reveal whether, and if yes to what extent, drag reduction will take place (even at high concentrations).The relation of our treatment to the model-based Deborah number concept is shown and a scale-up formula for the onset in turbulent tube flow is deduced as well.     

Unsteady heat transfer in impulsive Falkner–Skan flows: Constant wall temperature case

A theoretical study of the velocity and thermal boundary-layer growth resulting from an impulsively started Falkner–Skan flow is presented in this paper. The forced convection, thermal boundary-layer is produced by the sudden increase of the surface temperature as it is set into motion. Analytical solutions for the simultaneous development of the thermal and momentum boundary layers are obtained for both small (initial, unsteady flow) and large (steady-state flow) times. These solutions are then matched numerically using a very efficient finite-difference scheme. Some considerable attention to the steady-state flow solution (large time) is also given in this paper. Results of the calculations are presented for a range of values of the Falkner–Skan exponent m and the Prandtl number Pr.     

Velocity and temperature distributions on a semi-infinite flat plate embedded in a saturated porous medium

In this paper the velocity and temperature distributions on a semi-infinite flat plate embedded in a saturated porous medium are obtained for the governing equations (Kaviany [7]) following the technique adopted by Chandrashekara [2] which are concerned with the interesting situations of the existence of transverse, velocity and thermal boundary layers. Here the pressure gradient is just balanced by the first and second order solid matrix resistances for small permeability and observed that by increasing of the flow resistance the asymptotic value for the heat transfer rate increases. Further we concluded that the transverse boundary layers are thicker than that of axial boundary layers. Hence we evaluated the expressions for the boundary layer thickness, the shear stress at the semi-infinite plate and _T (the ratio of the thicknesses of the thermal boundary layer and momentum boundary layer). The variations of these quantities for different values of the porous parameterB and the flow resistanceF have been discussed in detail with the help of tables. The curves for velocity and temperature distributions have been plotted for different values ofB andF.Lastly we have evaluated the heat fluxq(x) and found that it depends entirely upon the Reynolds numberRe, Prandtl numberPr,B andF.     

Investigation of Dynamic Stability Regimes for Axisymmetric Bodies in Supersonic Flow

The problem of supersonic flow past a slender blunt cone with allowance for the reverse boundary-layer effect on the outer flow is solved with the aim of studying the influence of the boundary layer on the damping coefficient of axisymmetric body oscillations. It is assumed that the body executes plane angular, both low-amplitude and low-velocity, oscillations about a center of rotation. A modified version of the method [1] is applied for calculating the time-dependent flow past a body with the viscosity effect taken into account. The high accuracy of the flow parameter determination provided by this technique is confirmed by wind- tunnel experiments on a large-scale cone model (L1 m) at Mach numbers M=4 and 6. The agreement between the calculated and measured data forms the basis for the numerical investigation of the blunt-cone damping coefficient over a wide range of freestream Mach (M=4–20) and Reynolds (Re _L =10⁶–10⁸) numbers. At moderate freestream Mach numbers (M=4 and 6) an appreciable Re _L effect on the damping coefficient was not detected. However, on the hypersonic range this effect manifests itself more strongly, especially when there is gas injection into the boundary layer from the vehicle surface.     

Physical properties of the flow in the separation region for three-dimensional interaction of a boundary layer with a shock wave

In a supersonic stream we consider the three-dimensional flow in the plane of symmetry in the region of interaction of a boundary layer with a shock wave which arises ahead of an obstacle mounted on a plate. The principal characteristic of this flow is the penetration of a filament of the ideal fluid within the separation zone and the formation on the surface of the plate and obstacle of narrow segments with high pressures, high velocity gradients, and large heat transfer coefficients.Pressure distribution measurements were made, shadow and schlieren photos were taken, and photographs of the flow pattern on the surface were made using dye coatings and low-melting models. The basic physical characteristics of the separation flow are established. The independence of the separation zone length of the boundary layer thickness is shown. Local supersonic flows are detected in the separation region, flow regimes are identified as a function of the angle of encounter of the separating flow with the obstacles, characteristic flow zones in the interaction region are identified.Notation s coordinate of separation point on the plate - l length of separation zone - H obstacle height - d obstacle transverse dimension - u freestream velocity - velocity gradient on stagnation line of obstacle - b jet width - compression shock standoff from the body - p static pressure - p_* pressure at stagnation point on obstacle - density - viscosity coefficient - boundary-layer thickness - compression shock angle - effective angle of separation zone - setting angle of obstacle on plate - M Mach number - R Reynolds number - P Prandtl number     

Local Nonsimilarity Solution for the Impact of a Chemical Reaction in an MHD Mixed Convection Heat and Mass Transfer Flow over a Porous Wedge in the Presence Of Suction/Injection

Combined heat and mass transfer in free, forced, and mixed convection flows along a porous wedge with a magnetic effect in the presence of a chemical reaction is investigated. The flow field characteristics are analyzed with the Runge—Kutta—Gill method in conjunction with the shooting method, and local nonsimilarity method. The governing boundary-layer equations are written in a dimensionless form with the use of the Falkner—Skan transformations. Owing to the effect of the buoyancy force, the power law of temperature and concentration, and suction/injection on the wall of the wedge, the flow field is locally nonsimilar. Numerical calculations up to the third-order level of truncation are carried out for different values of dimensionless parameters as a special case. Effects of the magnetic field strength in the presence of a chemical reaction with a variable wall temperature and concentration on the dimensionless velocity, temperature, and concentration profiles are shown graphically. Comparisons with previously published works are performed, and excellent agreement between the results is obtained.     

Laminar boundary layer on blunt bodies with account for vorticity of the external flow

This paper presents the technique for and results from numerical calculations of the hypersonic laminar boundary layer on blunted cones with account for the vorticity of the external flow caused by the curved bow shock wave. It is assumed that the air in the boundary layer is in the equilibrium dissociated state, but the Prandtl number is assumed constant, =0.72. The calculations were made in the range of velocities 3–8 km/sec, cone half-angles _k=0°–20°. With account for the vortical interaction of the boundary layer with the external flow, the distribution of the thermal flux and friction will depend on the freestream Reynolds number (other conditions being the same). In the calculations the Reynolds number R, calculated from the freestream parameters and the radius of the spherical blunting, varies from 2.5·10³ to 5.10⁴. For the smaller Reynolds numbers the boundary layer thickness on the blunting becomes comparable with the shock standoff, and for R<2.5·10³ it is apparent that we must reconsider the calculation scheme. With R>5·10⁴ for cones which are not very long the vortical interaction becomes relatively unimportant. The results of the calculations are processed in accordance with the similarity criteria for hypersonic viscous gas flow past slender blunted cones [1, 2].     

Wall-Driven Versus Pressure Gradient-Driven Flows in Porous Media

The heat transfer characteristics of two boundary layer flows past an isothermal plane surface adjacent to a saturated Darcy–Brinkman porous medium is compared to each other in this paper. The flows are driven either by a stretching of the adjacent plane boundary, or by an external pressure gradient. It is found that below a threshold value $\tilde{P}r_{*} $ of the modified Prandtl number $\tilde{P}r$ , the Nussselt number in case of the pressure gradient-driven flow is larger than in case of the wall- driven flow, while for $\tilde{P}r>\tilde{P}r_{*} $ the flow driven by the moving wall provides a more efficient heat transfer mechanism. The dependence of $\tilde{P}r_{*} $ on the Darcy number is also discussed in detail.     

Flow of a viscous fluid past a heterogeneous porous sphere at low Reynolds numbers

A flow past a heterogeneous porous sphere is investigated by using the perturbation theory. The flow through the sphere is divided into two zones, which are fully saturated with the viscous fluid, and the flow in these zones is governed by the Brinkman equation. The space outside the sphere, where a clear fluid flows, is also divided into two zones: the Navier–Stokes zone and the Oseen flow zone. The solutions on the interface inside the sphere are matched with the condition proposed by Merrikh and Mohammad. The stream function in the Navier–Stokes zone is matched with that on the sphere surface by the condition proposed by Ochoa-Tapia and Whitaker. It is found that the drag on the spherical shell decreases as the permeability toward the sphere boundary increases.     

A Robust High-Resolution Split-Type Compact FD Scheme for Spatial Direct Numerical Simulation of Boundary-Layer Transition

In this paper an efficient split-type Finite-Difference (FD) scheme with high modal resolution – most important for the streamwise convection terms that cause wave transport and interaction – is derived for a mixed Fourier-spectral/FD method that is designed for the spatial direct numerical simulation (DNS) of boundary-layer transition and turbulence. Using a relatively simple but thorough and instructive modal analysis we discuss some principal trouble sources of the related FD discretization. The new scheme is based on a 6th-order compact FD discretization in streamwise and wall-normal direction and the classical 4th-order Runge–Kutta time-integration scheme with symmetrical final corrector step. Exemplary results of a fundamental-(K-) type breakdown simulation of a strongly decelerated Falkner–Skan boundary layer (Hartree parameter _H = – 0.18) using 70 mega grid points in space are presented up to the early turbulent regime (Re_,turb 820). The adverse pressure gradient gives rise to local separation zones during the breakdown stage and intensifies final breakdown by strong amplification of (background) disturbances thus enabling rapid transition at moderate Reynolds number. The appearance and dynamics of small-scale vortical structures in early turbulence basically similar to the large-scale structures at transition can be observed corroborating Kachanov's hypothesis on the importance of the K-regime of breakdown for coherent structures in turbulence.     

Uniqueness of the self-similar solutions of three-dimensional laminar boundary-layer equations

The equations of the three-dimensional laminar boundary layer on lines of flow outflow and inflow are studied for conical outer flow under the assumption that the Prandtl number and the productρμ are constant. It is shown that in the case of a positive velocity gradient of the secondary flow (α₁>0) the additional conditions which result from the physical flow pattern determine a unique solution of the system of boundary-layer equations. For a negative velocity gradient of the secondary flow (α₁≤0) these conditions are satisfied by two solutions. An approximate solution is obtained for the boundary layer equations which is in rather good agreement with the numerical integration results. Compressible gas flow in a three-dimensional laminar boundary layer is described by a system of nonlinear differential equations whose solution is not unique for given boundary conditions. Therefore additional conditions resulting from the physical pattern of the gas flow are imposed on the resulting solution. In the solution of problems with a negative pressure gradient these additional conditions are sufficient for a unique selection of the solution of the boundary-layer equations. However, in the case of a positive pressure gradient the solution of the boundary-layer equations satisfying the boundary and additional conditions may not be unique. In particular, in [1] in a study of a three-dimensional laminar boundary layer in the vicinity of the stagnation point it was shown that for $$c = {{\frac{{\partial v_e }}{{\partial y}}} \mathord{\left/ {\vphantom {{\frac{{\partial v_e }}{{\partial y}}} {\frac{{\partial u_e }}{{\partial x}}}}} \right. \kern-\nulldelimiterspace} {\frac{{\partial u_e }}{{\partial x}}}} > 0$$ the solution is unique, while for c<0 there are two solutions. In the present paper we study the question of the uniqueness of the self-similar solution of the three-dimensional laminar boundary-layer equations on lines of flow outflow and inflow for a conical outer flow.     

Effect of Inertial Terms on Fluid–Porous Medium Flow Coupling

The study considers an effect of the nonlinear inertial terms in the Brinkman filtration equation on the characteristics of coupled flows in a pure fluid and porous medium in the frameworks of two independent problems. The first problem is the forced boundary-layer flow overlying the Darcy–Brinkman porous medium. The Prandtl theory is used, and the self-similar equations are built to describe it. It is shown that the inertial terms have a valuable effect on the boundary-layer structure because of the large velocity gradient in the transition zone. The boundary-layer thickness in a porous medium rapidly grows at large Reynolds numbers. The velocity magnitude and gradient at the interface also change. The second independent problem is an analysis of the inertial terms effect on the flow stability. The neutral curves of the full and linearized flow models are built using the shooting method. They have different short-wave asymptotic, but there are no significant changes in the critical Reynolds numbers and corresponding wave numbers.     

A Two-Layer Mathematical Model of Blood Flow in Porous Constricted Blood Vessels

In this paper, we discussed a mathematical model for two-layered non-Newtonian blood flow through porous constricted blood vessels. The core region of blood flow contains the suspension of erythrocytes as non-Newtonian Casson fluid and the peripheral region contains the plasma flow as Newtonian fluid. The wall of porous constricted blood vessel configured as thin transition Brinkman layer over layered by Darcy region. The boundary of fluid layer is defined as stress jump condition of Ocha-Tapiya and Beavers–Joseph. In this paper, we obtained an analytic expression for velocity, flow rate, wall shear stress. The effect of permeability, plasma layer thickness, yield stress and shape of the constriction on velocity in core & peripheral region, wall shear stress and flow rate is discussed graphically. This is found throughout the discussion that permeability and plasma layer thickness have accountable effect on various flow parameters which gives an important observation for diseased blood vessels.     

Non-Darcy Free Convection Along a Horizontal Heated Surface

In this paper we analyse how the presence of inertia (Forchheimerform-drag) affects the steady free convective boundary layer flow over anupward-facing horizontal surface embedded in a porous medium. The surfacetemperature is assumed to display a power-law variation,x ⁿ with distance from the leading edge, x. It is shown thatthere are three distinct cases to consider: n<0.5, n=0.5 and0.5相似文献   

Generalised Couette flow of two immiscible viscous fluids with heat transfer using Brinkman model

The present study deals with generalised Couette flow of two viscous, incompressible, immiscible fluids with heat transfer in presence of heat source through two straight parallel horizontal walls. The lower wall is bounded below, by a naturally permeable material of high porosity and the flow inside the porous medium is assumed to be of moderate permeability, modelled by Brinkman equation. The flow domain is divided into three zones to obtain analytical solutions of the momentum and energy equations. To link various flow regions, appropriate matching conditions have been used. The effects of permeability parameter, Reynolds number and viscous parameter on velocity field and the effects of Reynolds number, viscous parameter, permeability parameter, constant heat source and Brinkman number on temperature distribution in different zones are discussed graphically. The mass flow rate, skin-friction factor and rates of heat transfer at the upper boundary and porous interface are discussed with the help of tables.     

On the localness of the spectral energy transfer in turbulence

By utilizing available experimental data for net energy transfer spectra for homogeneous turbulence, contributions P(, ) to the energy transfer at a wavenumber from various other wavenumbers are calculated. This is done by fitting a truncated power-exponential series in and to the experimental data for the net energy transfer T(), and using known properties of P(, ). Although the contributions P(, ) obtained by using this procedure are not unique, the results obtained by using various assumptions do not differ significantly. It seems clear from the results that for a region where the energy entering a wavenumber band dominates that leaving, much of the energy entering the band comes from wavenumbers which are about an order of magnitude smaller. That is, the energy transfer is rather nonlocal. This result is not significantly dependent on Reynolds number (for turbulence Reynolds numbers based on microscale from 3 to 800). For lower wavenumbers, where more energy leaves than enters a wavenumber band, the energy transfer into the band is more local, but much of the energy then leaves at distant wavenumbers.     

Effects of wall permeability on turbulence

In order to understand the effects of the wall permeability on turbulence near a porous wall, flow field measurements are carried out for turbulent flows in a channel with a porous bottom wall by a two-component particle image velocimetry (PIV) system. The porous media used are three kinds of foamed ceramics which have almost the same porosity (0.8) but different permeability. It is confirmed that the flow becomes more turbulent over the porous wall and tends to be turbulent even at the bulk Reynolds number of Re_b=1300 in the most permeable wall case tested. Corresponding to laminar to turbulent transition, the magnitude of the slip velocity on the porous wall is found to increase drastically in a narrow range of the Reynolds number. To discuss the effects of the wall roughness and the wall permeability, detailed discussions are made of zero-plane displacement and equivalent wall roughness for porous media. The results clearly indicate that the turbulence is induced by not only the wall roughness but the wall permeability. The measurements have also revealed that as Re_b or the wall permeability increases, the wall normal fluctuating velocity near the porous wall is enhanced due to the effects of the wall permeability. This leads to the increase of the turbulent shear stress resulting in higher friction factors of turbulence over porous walls.     

A Comment on the Flow and Heat Transfer Past a Permeable Stretching/Shrinking Surface in a Porous Medium: Brinkman Model

An analytical solution is presented for the boundary-layer flow and heat transfer over a permeable stretching/shrinking surface embedded in a porous medium using the Brinkman model. The problem is seen to be characterized by the Prandtl number $Pr$ , a mass flux parameter $s$ , with $s>0$ for suction, $s=0$ for an impermeable surface, and $s<0$ for blowing, a viscosity ratio parameter $M$ , the porous medium parameter $\Lambda $ and a wall velocity parameter $\lambda $ . The analytical solution identifies critical values which agree with those previously determined numerically (Bachok et al. Proceedings of the fifth International Conference on Applications of Porous Media, 2013) and shows that these critical values, and the consequent dual solutions, can arise only when there is suction through the wall, $s>0$ .     

On the effects of a periodic, spanwise variation of suction upon asymptotic suction flow

Summary The effects of superposing streamwise vorticity, periodic in the lateral direction, upon two-dimensional asymptotic suction flow are analyzed. Such vorticity, generated by prescribing a spanwise variation in the suction velocity, is known to play an important role in unstable and turbulent boundary layers. The flow induced by the variation has been obtained for a freestream velocity which (i) is steady, (ii) oscillates periodically in time, (iii) changes impulsively from rest. For the oscillatory case it is shown that a frequency can exist which maximizes the induced, unsteady wall shear stress for a given spanwise period. For steady flow the heat transfer to, or from a wall at constant temperature has also been computed.Nomenclature (x, y, z) spatial coordinates - (u, v, w) corresponding components of velocity - (, , ) corresponding components of vorticity - t time - stream function for v and w - v _w mean wall suction velocity - nondimensional amplitude of variation in wall suction velocity - characteristic wavenumber for variation in direction of z - T temperature - P pressure - density - coefficient of kinematic viscosity - coefficient of thermal diffusivity - (/v _w)² - frequency of oscillation of freestream velocity - nondimensional amplitude of freestream oscillation - /v _w ² - z z - y –v _w y/ - v _w ² t/4 - /v _w - U ₀ characteristic freestream velocity - u/U ₀ - coefficient of viscosity - _w wall shear stress - Prandtl number (/) - q heat transfer to wall - T _w wall temperature - T (T _w–T)/(T _w–)