Fully Developed Mixed Convection Flow Between Inclined Parallel Plates Filled with a Porous Medium

A fully developed mixed convection flow between inclined parallel flat plates filled with a porous medium is considered through which there is a constant flow rate and with heat being supplied to the fluid by the same uniform heat flux on each plate. The equations governing this flow are made non-dimensional and are seen to depend on two dimensionless parameters, a mixed convection parameter λ and the Péclet number Pe, as well as the inclination γ of the plates to the horizontal. The velocity and temperature profiles are obtained in terms of λ, Pe and γ when the channel is inclined in an upwards direction as well as for horizontal channels. The limiting cases of small and large λ and small Pe are considered with boundary-layer structures being seen to develop on the plates for large values of λ.     

Numerical and experimental investigation of transverse injection flows

The flow field resulting from a transverse injection through a slot into supersonic flow is numerically simulated by solving Favre-averaged Navier–Stokes equations with κ − ω SST turbulence model with corrections for compressibility and transition. Numerical results are compared to experimental data in terms of surface pressure profiles, boundary layer separation location, transition location, and flow structures at the upstream and downstream of the jet. Results show good agreement with experimental data for a wide range of pressure ratios and transition locations are captured with acceptable accuracy. κ − ω SST model provides quite accurate results for such a complex flow field. Moreover, few experiments involving a sonic round jet injected on a flat plate into high-speed crossflow at Mach 5 are carried out. These experiments are three-dimensional in nature. The effect of pressure ratio on three-dimensional jet interaction dynamics is sought. Jet penetration is found to be a non-linear function of jet to free stream momentum flux ratio.     

Pulsatile entrance flow in curved pipes: effect of various parameters

This paper presents the results of an experimental study on the developing pulsatile flow in curved pipes with a long, straight pipe upstream. In order to examine the dependence of flow-field development on the governing parameters, LDV measurements were conducted systematically for six cases of flow, where the Womersley number α was varied from 5.5 to 18, the mean Dean number D _m was 200 and 300, the flow rate ratio η was 0.5 and 1, and the curvature radius ratio Rc was 10 and 30. Peculiar flow phenomena, such as flow reversal for all values of α and a depression in the axial velocity profile for α = 10, were analyzed by decomposing the axial velocity into a time-mean and a varying component, as well as by obtaining the bias of their profiles. The velocity distributions abruptly change with the phase at turn angles Ω of 15–30°, corresponding to the nondimensional axial length z′ ≅ 1–2 from the bend entrance, and their development along the pipe axis is the most complicated for the flow at a moderate α of 10 and large η of 1. The entrance length in the case of pulsatile flow is shorter than that for steady flow with the same flow rate as the maximum pulsatile flow rate.     

Prediction of Flow and Heat Transfer through Stationary and Rotating Ribbed Ducts Using a Non-linear <Emphasis Type="Italic">k</Emphasis>−<Emphasis Type="Italic">ε</Emphasis> Model

The present paper deals with the prediction of three-dimensional fluid flow and heat transfer in rib-roughened ducts of square cross-section, which are either stationary, or rotate in orthogonal mode. The main objective is to assess how a recently developed variant of a cubic non-linear k−ε model (proposed by Craft et al. Flow Turbul Combust 63:59–80, 1999) can predict three-dimensional flow and heat transfer characteristics through stationary and rotating ribbed ducts. The present paper discusses turbulent air flow and heat transfer through two different configurations, namely: (I) a stationary square duct with “in-line” normal and (II) a square duct with normal ribs in a “staggered” arrangement under stationary and rotating conditions, with the axis of rotation normal to the flow direction and parallel to the ribs. In this paper the flow and thermal predictions of the linear k−ε model (EVM) are also included, as a set of baseline predictions. The mean flow predictions show that both linear and non-linear k−ε models can successfully reproduce most of the measured data for stream-wise and cross-stream velocity components. Moreover, the non-linear model is able to produce better results for the turbulent stresses. The heat transfer predictions show that both EVM and NLEVM2, the more recent variant of the non-linear k−ε, with the algebraic length-scale correction term, overestimate the measured Nusselt numbers for both geometries examined. While the EVM with the differential length-scale correction term underestimates heat transfer levels, the Nusselt number predictions with the NLEVM2 and the ‘NYP’ term are in close agreements with the measured data. Comparisons with our earlier work, Iacovides and Raisee (Int J Heat Fluid Flow, 20:320–328, 1999), show that the NLEVM2 thermal predictions are of similar quality to those of a second-moment closure.     

Plane jets impinging on porous walls

The flow of a two-dimensional plane turbulent jet impinging on a porous screen has been studied experimentally. It is shown how the overall flow structure depends on the porosity of the surface. For low screen porosity (β < 0.41, say), transverse wall jets can be formed on both sides of the screen and in extreme cases the axial momentum flux some way downstream of the screen falls to zero, so that the screen has the same drag as would a solid wall. For high screen porosity (β > 0.57, say) the axial volume flux is largely preserved through the screen, but the dominant eddy structures present in the upstream jet are largely destroyed, so that entrainment rates downstream of the screen can be very low. The relatively small, intermediate range of porosities (0.41 < β < 0.57, where β is the screen open area ratio) is associated with dramatic changes in flow pattern and recirculating regions can exist on the upstream side of the screen. These flows, although all geometrically very simple, provide a serious challenge for computational modelling. Received: 25 May 2000 / Accepted: 22 February 2001     

Numerical study of self-sustained oscillatory flows and heat transfer in channels with a tandem of transverse vortex generators

A numerical investigation was conducted into channel flows with a tandem of transverse vortex generators in the form of rectangular cylinders. The oscillatory behavior of the flow is studied. Data for heat transfer and flow losses are presented for 100≤Re≤400 and cylinder separation distances 1≤S/H≤4. The results are obtained by numerical solution of the full Navier-Stokes equations and the energy equation. Self-sustained flow oscillations are found for Re>100. Alternate and dynamic shedding of large vortex structures from the cylinders is observed by visualization of the numerically determined flow field. A heat transfer enhancement up to a factor 1.78 compared to plane channel flow is observed. Received on 16 July 1997     

Flow of power-law fluids over a moving wedge surface with wall mass injection

The boundary layer problem of a power-law fluid flow with fluid injection on a wedge whose surface is moving with a constant velocity in the opposite direction to that of the uniform mainstream is analyzed. The free stream velocity, the injection velocity at the surface, moving velocity of the wedge surface, the wedge angle and the power law index of non-Newtonian fluid are assumed variables. The fourth order Runge–Kutta method modified by Gill is used to solve the non-dimensional boundary layer equations for non-Newtonian flow field. Without fluid injection, for every angle of wedge β, a limiting value for velocity ratio λ _cr (velocity of the wedge surface/velocity of the uniform flow) is found for each power-law index n. The value of λ _cr increases with the increasing wedge angle β. The value of wedge angle also restricts the physical characteristics of the fluid to be used. The effects of the different parameters on velocity profile and on skin friction are studied and the drag reduction is discussed. In case of C = 2.5 and velocity ratio λ = 0.2 for wedge angle β = 0.5 with the fluid with power law-index n = 0.5, 48.8% drag reduction is obtained.     

Influence of wavy surfaces on coherent structures in a turbulent flow

We describe how outer flow turbulence phenomena depend on the interaction with the wall. We investigate coherent structures in turbulent flows over different wavy surfaces and specify the influence of the different surface geometries on the coherent structures. The most important contribution to the turbulent momentum transport is attributed to these structures, therefore this flow configuration is of large engineering interest. In order to achieve a homogeneous and inhomogeneous reference flow situation two different types of surface geometries are considered: (1) three sinusoidal bottom wall profiles with different amplitude-to-wavelength ratios of α = 2a/Λ = 0.2 (Λ = 30 mm), α = 0.2 (Λ = 15 mm), and α = 0.1 (Λ = 30 mm); and (2) a profile consisting of two superimposed sinusoidal waves with α = 0.1 (Λ = 30 mm). Measurements are carried out in a wide water channel facility (aspect ratio 12:1). Digital particle image velocimetry (PIV) is performed to examine the spatial variation of the streamwise, spanwise and wall-normal velocity components in three measurement planes. Measurements are performed at a Reynolds number of 11,200, defined with the half channel height h and the bulk velocity U _B. We apply the method of snapshots and perform a proper orthogonal decomposition (POD) of the streamwise, spanwise, and wall-normal velocity components to extract the most dominant flow structures. The structure of the most dominant eigenmode is related to counter-rotating, streamwise-oriented vortices. A qualitative comparison of the eigenfunctions for different sinusoidal wall profiles shows similar structures and comparable characteristic spanwise scales Λ _z = 1.5 H in the spanwise direction for each mode. The scale is observed to be slightly smaller for α = 0.2 (Λ = 15 mm) and slightly larger for α = 0.2 (Λ = 30 mm). This scaling for the flow over the basic wave geometries indicates that the size of the largest structures is neither directly linked to the solid wave amplitude, nor to the wavelength. The characteristic spanwise scale of the dominant eigenmode for the developed flow over the surface consisting of two superimposed waves reduces to 0.85 H. However, a scale in the order of 1.3 H is identified for the second mode. The eigenvalue spectra for the superimposed waves is much broader, more modes contribute to the energy-containing range. The turbulent flow with increased complexity of the bottom surface is characterized by an increased number of dominant large-scale structures with different spanwise scales.     

An experimental investigation of moderate reynolds number flow in a T-Channel

An experimental investigation of water flow in a T-shaped channel with rectangular cross section (20 × 20 mm inlet ID and 20 × 40 mm outlet ID) has been conducted for a Reynolds number Re range of 56–422, based on inlet diameter. Dynamical conditions and the T-channel geometry of the current study are applicable to the microscale. 2-D planar particle imaging velocimetry (PIV) and laser-induced fluorescence (LIF) were used in multiple locations of the T-channel to investigate local dynamical behaviors. Steady symmetric and asymmetric flow regimes predicted in the literature, which is largely numerical, are experimentally verified. Unsteady flow regimes, which are numerically predicted to occur at higher Re but have not yet been experimentally characterized, are also examined, and real-time LIF results illuminate the evolution of unsteady structure. Experimental data of the present resolution and scope are not presently available for unsteady flow regimes. Time scales are presented for unsteady flow regimes, which are found to exhibit periodic behavior and to occur for Re  ≥ 195. An unsteady symmetrical regime is identified for Re ≥ 350 that is detrimental to mixing. Momentum fields and dynamical behaviors of all flow regimes are characterized in detail, such that published mixing trends may be better understood. Results of all experimental trials were used to construct a regime map. A symmetric topology is found to be dominant for Re from 56 to 116, when flow is steady, and 350 to 422, when flow is characterized by unsteady stagnation-point oscillation in the T-channel junction. Asymmetric flow, which is positively indicated for mixing, is dominant for Re between 142 and 298, and the fluid interface exhibits both steady (two standing vortices) and unsteady (shear-layer type roll-up) behaviors. This result is based on multiple experiments and suggests a practical operating range of 142  ≤ Re ≤ 298 where asymmetric flow is highly likely to experimentally occur. The identification of an upper limit on Re,  beyond which mixing appears negatively impacted by a more symmetrical momentum field, is practically important as pressure drops on the microscale are significant.     

Uniqueness of free-boundary flows under gravity

A class of steady potential flows of an ideal fluid is considered in which the fluid flows between fixed boundaries and then emerges as a jet with one free boundary. Gravity acts on the fluid perpendicularly to the direction of the jet at infinity downstream. An inverse Froude number α is defined in terms of the flux Q and the depth d of the fluid at the separation point. It is proved that for each α>0 there is at most one flow which reaches to a supercritical uniform stream depth at infinity downstream. Monotonicity properties are proved for various flow parameters, and the behaviour of the flow as α → 0 is described.     

A convenient way of interpreting steady shear rheo-optical data of semi-dilute polymer solutions

Rheo-mechanical and rheo-optical investigations were carried out with the aim of determining the influence of deformation and orientation or disentangling of polymer coils on the flow behavior in the non-Newtonian region of the flow curve, for a moderately concentrated network solution. To avoid the influence of polydispersity this was done on a series of narrowly distributed polystyrene standards (dissolved in toluene). By using steady state shear flow measurements it was possible to detect qualitatively a reduction in the entanglement density within the non-Newtonian flow region. Birefringence experiments were able to show that deformation of the polymer coils also occurs in the Newtonian flow region, which has no effect on the flow behavior in this range, whereas in the non-Newtonian flow region the increase in deformation is lower than in the Newtonian range. The flow birefringence and its orientation can be described over the whole range of the flow curve with a newly developed equation system (Eq. 8 and 14) derived from the stress states of a sheared solution using the stress-optical rule. Starting from these equations, it could be shown, that in the Newtonian flow region a mastercurve in form of a reduced birefringence Δn′/η₀=f(γ˙) and a reduced orientation φ= f(γ˙/γ˙ _crit) can be plotted, independent from concentration and molar mass. A comparison of the experimentally determined orientation angle and birefringence curve form with theoretical deformations and orientations of polymer coils in a solution state, without intermolecular interactions, was able to demonstrate that the flow behavior of a moderately concentrated network solution is determined decisively (approximately to 85%) by the disentanglement. Received: 8 May 2000 Accepted: 12 September 2000     

Steady mixed convection boundary-layer flow over a vertical flat surface in a porous medium filled with water at 4°C: variable surface heat flux

The steady mixed convection boundary-layer flow over a vertical impermeable surface in a porous medium saturated with water at 4°C (maximum density) when the surface heat flux varies as x ^m and the velocity outside the boundary layer varies as x ^(1+2m)/2, where x measures the distance from the leading edge, is discussed. Assisting and opposing flows are considered with numerical solutions of the governing equations being obtained for general values of the flow parameters. For opposing flows, there are dual solutions when the mixed convection parameter λ is greater than some critical value λ _c (dependent on the power-law index m). For assisting flows, solutions are possible for all values of λ. A lower bound on m is found, m > −1 being required for solutions. The nature of the critical point λ _c is considered as well as various limiting forms; the forced convection limit (λ = 0), the free convection limit (λ → ∞) and the limits as m → ∞ and as m → −1.     

Laminar mixed convection from a continuously moving vertical surface with suction or injection

The boundary layer flow over a uniformly moving vertical surface with suction or injection is studied when the buoyancy forces assist or oppose the flow. Similarity solutions are obtained for the boundary layer equations subject to power law temperature and velocity boundary conditions. The effect is of various governing parameters, such as Prandtl number Pr, temperature exponent n, injection parameter d, and the mixed convection parameter λ=Gr/Re², which determine the velocity and temperature distributions and the heat transfer coefficient, are studied. The heat transfer coefficient increases as λ assisting the flow for all d at Pr=0.72 however, for n=−1 it decreases sharply with λ. On the other hand, increasing λ has no effect on heat transfer coefficient for Pr=10 at n=0, and 1 for almost all values of d studied. However, for n=−1 it has similar effect as for Pr=0.72. It is also found that Nusselt number increases as n increases for fixed λ and d. Received on 26 March 1997     

Maximum density effects on convective instability of horizontal plane poiseuille flows in the thermal entrance region

A linear stability analysis is used to study the conditions marking the onset of secondary flow in the form of longitudinal vortices for plane Poiseuille flow of water in the thermal entrance region of a horizontal parallel-plate channel by a numerical method. The water temperature range under consideration is 0∼30°C and the maximum density effect at 4°C is of primary interest. The basic flow solution for temperature includes axial heat conduction effect and the entrance temperature is taken to be uniform at far upstream location jackie=−∞ to allow for the upstream heat penetration through thermal entrance jackie=0. Numerical results for critical Rayleigh number are obtained for Peclet numbers 1, 10, 50 and thermal condition parameters (λ ₁, λ ₂) in the range of −2.0≤λ ₁≤−0.5 and −1.0≤λ ₂≤1.4. The analysis is motivated by a desire to determine the free convection effect on freezing or thawing in channel flow of water.     

T-RANS Simulations of Subcritical Flow with Heat Transfer Past a Circular Cylinder Surrounded by a Thin Porous Layer

We study flow and heat transfer to a cylinder in cross flow at Re = 3,900–80,000 by means of three-dimensional transient RANS (T-RANS) simulations, employing an RNG k − ε turbulence model. Both the case of a bare solid cylinder and that of a solid cylinder surrounded at some fixed distance by a thin porous layer have been studied. The latter configuration is a standard test geometry for measuring the insulating and protective performance of garments. In this geometry, the flow in the space between the solid cylinder and the porous layer is laminar but periodic, whereas the outer flow is transitional and characterized by vortex shedding in the wake of the cylinder. The results from the T-RANS simulations are validated against data from Direct Numerical Simulations and experiments. It is found that T-RANS is very well suited for simulating this type of flow. The transient nature of the flow underneath the porous layer is well reproduced, as well as the influence of vortex shedding on the heat transfer in the downstream stagnation zone. T-RANS results are found to be in much better agreement with DNS and experimental data than results from steady-state RANS.     

Flow of a viscous incompressible fluid of temperature dependent viscosity past a permeable wedge with uniform surface heat flux

 The effect of uniform suction on the steady two-dimensional laminar forced flow of a viscous incompressible fluid of temperature dependent viscosity past a wedge with uniform surface heat flux is considered. The governing equations for the flow are obtained by using suitable transformations and are solved by using an implicit finite difference method. Perturbation solutions are also obtained near the leading edge and in the downstream regime. The results are obtained in terms of the local skin friction coefficient and the rate of heat transfer for various values of the pertinent parameters, such as the Prandtl number, Pr, the velocity gradient parameter, m, the local suction parameter, ξ, and the viscosity variation parameter, ɛ. Perturbation solutions are compared with the finite difference solutions and are found to be in excellent agreement. The effect of ξ, m and ɛ on the dimensionless velocity profiles and viscosity distribution are also presented graphically for Pr = 0.7 and 7.0, which are the appropriate values for gases and water respectively. Received on 22 July 1999     

Experimental investigation on turbulent flow in a square-sectioned 90-degree bend

The development of steady, turbulent flow in a 90° section of a curved square duct was studied at a Reynolds number of 4 × 10⁴ by hot-wire anemometer. The curved duct has a cross-section measuring 80 × 80 mm and a curvature radius ratio of 4 and is connected with a long, straight duct at its both ends. The longitudinal and lateral components of mean and fluctuating velocities, and the Reynolds stresses were measured by the method of rotating a probe with an inclined hot-wire. The velocity fields of the primary and secondary flows, and the Reynolds stress distributions in the cross-section were illustrated in the form of contour map. The development of the primary flow was found to be connected with a strong pressure gradient near the outer and inner wall and a secondary flow induced in the cross-section of the bend by a pressure difference between the outer and inner wall and a centrifugal force acting on the fluid; the fluid is accelerated near the inner wall and decelerated near the outer wall between the bend angle ϕ ≅ 0° and ϕ ≅ 30°, but an increase and decrease of the fluid velocity are reversed between ϕ ≅ 30° and ϕ ≅ 90°. The fluctuating velocity correlations, i.e. the Reynolds stresses follow a complicated progress according to the complex development of the primary flow. The results obtained can be available to verify various types of turbulence models and to develop new models. Received: 10 May 1999/Accepted: 15 March 2000     

Predicting the Collapse of Turbulence in Stably Stratified Boundary Layers

The collapse of turbulence in a plane channel flow is studied, as a simple analogy of stably stratified atmospheric flow. Turbulence is parameterized by first-order closure and the surface heat flux is prescribed, together with the wind speed and temperature at the model top. To study the collapse phenomenon both numerical simulations and linear stability analysis are used. The stability analysis is nonclassical in a sense that the stability of a parameterized set of equations of a turbulent flow is analyzed instead of a particular laminar flow solution. The analytical theory predicts a collapse of turbulence when a certain critical value of the stability parameter δ/L (typically O(0.5–1)) is exceeded, with δ the depth of the channel and L the Obukhov length. The exact critical value depends on channel roughness to depth ratio z ₀/δ. The analytical predictions are validated by the numerical simulations, and good agreement is found. As such, for the flow configuration considered, the present framework provides both a tool and a physical explanation for the collapse phenomenon.     

Combined heat and mass transfer along a vertical moving cylinder with a free stream

The mixed convection flow over a continuous moving vertical slender cylinder under the combined buoyancy effect of thermal and mass diffusion has been studied. Both uniform wall temperature (concentration) and uniform heat (mass) flux cases are included in the analysis. The problem is formulated in such a manner that when the ratio λ(= u _w/(u _w + u _∞), where u _w and u _∞ are the wall and free stream velocities, is zero, the problem reduces to the flow over a stationary cylinder, and when λ = 1 it reduces to the flow over a moving cylinder in an ambient fluid. The partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme. We have also obtained the solution using a perturbation technique with Shanks transformation. This transformation has been used to increase the range of the validity of the solution. For some particular cases closed form solutions are obtained. The surface skin friction, heat transfer and mass transfer increase with the buoyancy forces. The buoyancy forces cause considerable overshoot in the velocity profiles. The Prandtl number and the Schmidt number strongly affect the surface heat transfer and the mass transfer, respectively. The surface skin friction decreases as the relative velocity between the surface and free stream decreases. Received on 17 May 1999     

Secondary flow patterns and mixing in laminar pulsating flow through a curved pipe

Mixing by secondary flow is studied by particle image velocimetry (PIV) in a developing laminar pulsating flow through a circular curved pipe. The pipe curvature ratio is η = r ₀/r _c  = 0.09, and the curvature angle is 90°. Different secondary flow patterns are formed during an oscillation period due to competition among the centrifugal, inertial, and viscous forces. These different secondary-flow structures lead to different transverse-mixing schemes in the flow. Here, transverse mixing enhancement is investigated by imposing different pulsating conditions (Dean number, velocity ratio, and frequency parameter); favorable pulsating conditions for mixing are introduced. To obviate light-refraction effects during PIV measurements, a T-shaped structure is installed downstream of the curved pipe. Experiments are carried out for the Reynolds numbers range 420 ≤ Re_st ≤ 1,000 (Dean numbers 126.6 ≤ Dn ≤ 301.5) corresponding to non-oscillating flow, velocity component ratios 1 ≤ (β = U _max,osc/U _m,st) ≤ 4 (the ratio of velocity amplitude of oscillations to the mean velocity without oscillations), and frequency parameters 8.37 < (α = r ₀(ω/ν)^0.5) < 24.5, where α² is the ratio of viscous diffusion time over the pipe radius to the characteristic oscillation time. The variations in cross-sectional average values of absolute axial vorticity (|ζ|) and transverse strain rate (|ε|) are analyzed in order to quantify mixing. The effects of each parameter (Re_st, β, and α) on transverse mixing are discussed by comparing the dimensionless vorticities (|ζ _P |/|ζ _S |) and dimensionless transverse strain rates (|ε _P |/|ε _S |) during a complete oscillation period.