Viscoelastic boundary-layer flows past wedges and cones

An implicit finite-difference method is used to solve the boundary-layer equations corresponding to the plane flow of a viscoelastic liquid past a symmetric wedge or the axisymmetric flow of such a fluid past a right circular cone. A representative sample of the computer data is displayed graphically and used to illustrate the interesting physical features of the problem.     

Unsteady two-dimensional and axisymmetric MHD boundary-layer flows

Summary Nonsimilar solution of the unsteady laminar incompressible magneto-hydrodynamic boundary layer flow and heat transfer for an electrically conducting fluid over two-dimensional and axisymmetric bodies in the presence of an applied magnetic field has been obtained. The effects of surface mass transfer, Joule heating and viscous dissipation are included in the analysis. Numerical computation have been carried out for the flow over a circular cylinder and a sphere using an implicit finite difference scheme in combination with a quasi-linearization technique. It is observed that magnetic field and suction cause the location of vanishing skin friction to move downstream while, the effect of injection is just the opposite. The effect of magnetic field on the skin friction is more pronounced as compared to its effect on the heat transfer. On the other hand, the heat transfer is strongly affected by the viscous dissipation and the effect is more for larte times. However, heat transfer responds comparatively less to the fluctuations of the free stream than the skin friction.     

A note concerning free-convective boundary-layer flows

Summary We show that the free-convective boundary-layer flow in a porous medium in the vicinity of a parameter value at which similarity solutions cease to exist is of semi-jet form.     

Second-order effects in self-similar laminar compressible boundary-layer flows

Second-order effects associated with longitudinal curvature, transverse curvature, entropy gradients, stagnation enthalpy gradient, velocity slip, temperature jump and displacement speed in self-similar two-dimensional and axisymmetric flows of a compressible fluid are studied. The self-similar governing equations for second-order effects show that the four parameters, due to longitudinal curvature, transverse curvature, displacement and body shape are needed to describe the flow. Numerical solutions and several closed form solutions are obtained. The present study is found to contain all the previous studies in the self-similar domain as special cases.     

Second-order boundary-layer theory of laminar jet flows

Summary Laminar jet flows are investigated by asymptotic expansions in terms of large Reynolds numbers and large distances from an orifice in a wall. For two-dimensional flow, previously known results are modified and extended. For axisymmetric flow, the expansion breaks down as the distance from the orifice tends to infinity. An error analysis indicates that the range of applicability of classical boundary layer theory is severely limited.With 4 Figures     

Unsteady nonsimilar compressible laminar two-dimensional and axisymmetric boundary-layer flows

Summary Unsteady nonsimilar laminar compressibletwo-dimensional and axisymmetric boundarylayer flows have been studied when external velocity varies arbitrarily with time and the flow is nonhomentropic. The governing nonlinear partial differential equations with three independent variables have been solved using an implicit finite difference scheme with quasilinearization technique from the origin to the point of zero skin-friction. The results have been obtained for (i) an accelerating stream and (ii) a fluctuating stream. The skin friction responds to the fluctuations in the free stream more compared to the heat transfer. It is observed that Mach number and hot wall cause the point of zero skin friction to occur earlier whereas cold wall delays it.With 16 Figures     

Turbulent spots in channel flows

A number of investigations into the formation and development of turbulent spots in plane Poiseuille, plane Couette and watertable flows are reviewed. Three main observations are drawn from this work. Firstly, the initial development is associated with the transient growth due to the three-dimensional lift-up effect. Secondly, the spreading and propagation velocities of the different spots are quite similar. Thirdly, the velocity field inside the spots show essentially all the characteristics of fully developed turbulent flow, albeit at low Reynolds numbers. The mechanism behind the rapid spreading of the spots, however, is not yet fully understood, although observations of wave activity, modifications of the surrounding flow field and stability calculations point in the direction of growth by destabilization.     

A finite difference-Galerkin finite element solution of compressible laminar boundary-layer flows

A finite difference-Galerkinfinite element method is presented for the solution of the two-dimensional compressible laminar boundary-layer flow problem. The streamwise derivatives in the momentum and energy equations are approximated by finite differences. An iterative scheme, due to the non-linearity of the problem, in conjunction with the Galerkin finite element method is then proposed for the solution of the problem through the boundary-layer thickness. Numerical results are presented and these are compared with other numerical and analytical solutions in order to show the applicability and the effectiveness of the proposed formulation. In all the cases here examined, the results obtained attained the same accuracy of other numerical methods for a much smaller number of points in the boundary-layer.     

Symmetry of turbulent boundary-layer flows: Investigation of different eddy viscosity models

Summary The symmetrical properties of the turbulent boundary-layer flows and other turbulent flows are studied utilizing the Lie group theory technique. The self-similar forms of the indepedent variables and the solution function for the turbulent boundary layer flows with three different models of the turbulent (eddy) viscosity are obtained. Proceeding from this analysis, a simple numerical method for computation of turbulent flows is developed.     

Stability of two-layered fluid flows in an inclined channel

A linear stability analysis of two-layer fluid flows in an inclined channel geometry has been carried out. The onset of flow transitions and the spatio-temporal characteristics of secondary flows produced by the flow instabilities have been examined. The effects of density and viscosity stratifications and surface tension on flow structures also have been investigated at various values of Froude numbers (channel inclinations). Multi-domain Chebyshev–Tau spectral methods along with MATLAB QZ eigenvalue solver are used to determine the whole spectrum of the eigenvalues and associated eigenfunctions. The neutral stability diagrams and stability boundaries are constructed for various values of flow parameters. The onset of flow transitions and flow structures predicted by linear stability analysis are compared against experimental results and they agree reasonably well. The results presented in the present paper imply that the shear mode of flow transitions is the one likely to be identified in experiments.     

Free convection in hydromagnetic flows in a vertical wavy channel

A study is made of the free convection in hydromagnetic flows in a vertical wavy channel in the presence of heat source or sink. The governing equations for the hydromagnetic fluid flow and the heat transfer are solved subject to the relevant boundary conditions with the assumption that the solution consists of a mean part and a perturbed part. The zeroth-order, the first order and the total solution of the problem are numerically evaluated for various values of the magnetic parameter, heat source/sink parameter, wall-waviness parameter, and free convection parameter. The velocity and the temperature profiles are graphically represented for these parameters. The qualitative features of the hydromagnetic solution are discussed. A comparison is made between the hydromagnetic and the hydrodynamic solutions. The numerical values of the skin friction and the Nusselt number are tabulated for various parameters involved in the analysis. Special attention is given on the characteristic features of the flow, heat transfer, skin friction and the Nusselt numbers at the walls.     

A perturbation solution for compressible viscous channel flows

The two-dimensional compressible Navier-Stokes equations are solved by a perturbation expansion in the parameter (Mach number)²/Reynolds number. A fortieth-order solution is generated by a computer algorithm. These series are then summed as convergent series of diagonal Padé approximants. Effectively-exact solutions have been found for Reynolds numbers between zero and 1000 and a range of subsonic Mach numbers in the case of fully-developed isothermal flow between parallel side walls. Choking of the flow is shown to occur for a moderate value of channel Reynolds number. The two-dimensional velocity and pressure fields are obtained. The engineering assumption that friction factor is sensibly independent of Mach number may lead to significant underprediction of head loss in the laminar flow regime.     

Calculation of unsteady flows due to small motions of cylinders in a viscous fluid

Summary The problem of small oscillations of a cylinder of general cross-section in a viscous fluid is formulated in terms of integral equations. Numerical solutions of the integral equation are presented for the special case of a ribbon of zero thickness.This work has been carried out under the support of the Office of Naval Research, Contract N0014-67-A-0094-0011.     

Applications of conformal mapping to diverging open channel flows

A conformal mapping technique is used for predicting the width of the separation zone at a diverging open channel flow. The Schwarz-Christoffel transformation is used to transform the physical boundaries of the flow into a complex plane and the flow field is solved using modified boundary conditions utilizing a complex velocity potential in the resulting hodograph plane. The final solution gives the width of the separation zone in nondimensionalized form and provides an inviscid solution for comparative study.     

Granular configurations, motions, and correlations in slow uniform flows driven by an inclined conveyor belt

The present experimental study examines the behaviour of slow granular flows, focusing on the details of particle patterns and motions over the depth of a sheared layer. A conveyor belt circuit enclosed in an inclined flume is used to generate steady uniform open-channel flows of dry granules. Particle positions near the transparent sidewall are extracted from video sequences. The Voronoï diagram is then used to characterise the configurations formed by neighbouring grains and to assist particle tracking over successive frames. This allows a qualitative visualisation of the internal structure of the flowing layer, as well as quantitative measurements of lattice defect density and granular velocities at different depths. The response of the depth profiles to different conveyor belt speeds is examined. In addition to the mean and fluctuating velocities, we probe the time and space correlations of the fluctuations.     

Aspects of linear and nonlinear instabilities leading to transition in pipe and channel flows

The failure of normal-mode linear stability analysis to predict a transition Reynolds number (Retr) in pipe flow and subcritical transition in plane Poiseuille flow (PPF) has led to the search of other scenarios to explain transition to turbulence in both flows. In this work, various results associated with linear and nonlinear mechanisms of both flows are presented. The results that combine analytical and experimental approaches indicate the strong link between the mechanisms governing the transition of both flows. It is demonstrated that the linear transient growth mechanism is based on the existence of a pair of least stable nearly parallel modes (having opposite phases and almost identical amplitude distributions). The analysis that has been applied previously to pipe flow is extended here to a fully developed channel flow predicting the shape of the optimized initial disturbance (a pair of counter-rotating vortices, CVP), time for maximum energy amplification and the dependence of the latter on Re. The results agree with previous predictions based on many modes. Furthermore, the shape of the optimized initial disturbance is similar in both flows and has been visualized experimentally. The analysis reveals that in pipe flow, the transient growth is a consequence of two opposite running modes decaying with an equal decay rate whereas in PPF it is due to two stationary modes decaying with different decay rates. In the first nonlinear scenario, the breakdown of the CVPs (produced by the linear transient growth mechanism) into hairpin vortices is followed experimentally. The associated scaling laws, relating the minimal disturbance amplitude required for the initiation of hairpins and the Re, are found experimentally for both PPF and pipe flow. The scaling law associated with PPF agrees well with the previous predictions of Chapman, whereas the scaling of the pipe flow is the same as the one previously obtained by Hof et al. indicating transition to a turbulent state. In the second nonlinear scenario, the base flow of pipe when it is mildly deviated from the Poiseuille profile by an axisymmetric distortion is examined. The nonlinear features reveal a Retr of approximately 2000 associated with the bifurcation between two deviation solutions.     

Meshfree particle simulation of micro channel flows with surface tension

This paper presents a study of micro channel flows using a meshfree particle approach. The approach is based on smoothed particle hydrodynamics (SPH) and its variant, adaptive smoothed particle hydrodynamics (ASPH). The incompressible flow in the micro channels is modeled as an artificially compressible flow. The surface tension is incorporated into the equations of motion. The classic Poiseuille flow and a practical micro channel flow problem of flip-chip underfill encapsulation process are investigated. It is found that the adaptive kernel can well match the computational geometry with long channels and can greatly save computational time. The simulation results are in close agreement with the analytical solutions.     

Direct numerical simulation of two-dimensional channel flows of electro-rheological fluids

Direct numerical simulation (DNS) of electro-rheological (ER) fluid flows in two-dimensional (2D) electrode channel has been performed by adopting a combined finite element method (FEM). Hydrodynamic interactions between the particles and the fluid are described by the Navier-Stokes equations for the fluid in combination with the equations of motion for the particles, while the multi-body electrostatic interaction is represented by the point-dipole model.ER effects on the plane channel flow for a given pressure gradient have been studied by varying the Mason number and volume fraction of the particles, and interrogating the motion of the particles in views of the formation of ER chain structures, the fluid velocity profile in the channel, and the shear stress versus the shear rate. As the Mason number decreases and volume fraction increases, the tendency that particles align to form chain structures becomes stronger. The yield stress of the ER fluid increases with the electric field intensity and the particle concentration. The quadratic correlation between the yield stress and the electric field intensity has been extracted from the present direct numerical simulation. Lastly, it has been shown that the yield stress linearly increases with the volume fraction in the intermediate range.     

Direct numerical simulation of stably and unstably stratified turbulent open channel flows

Summary Direct numerical simulation of stably and unstably stratified turbulent open channel flow is performed. The three-dimensional Navier-Stokes and energy equations under the Boussinesq approximation are numerically solved using a fractional-step method based on high-order accurate spatial schemes. The objective of this study is to reveal the effects of thermally stable and unstable stratification on the characteristics of turbulent flow and heat transfer and on turbulence structures near the free surface of open channel flow. Here, fully developed weakly stratified turbulent open channel flows are calculated for the Richardson number ranging from 20 (stably stratified flow) to 0 (unstratified flow) and to −10 (unstably stratified flow), the Reynolds number 180 based on the wall friction velocity and the channel depth, and the Prandtl number 1. To elucidate the turbulent flow and heat transfer behaviors, typical quantities including the mean velocity, temperature and their fluctuations, turbulent heat fluxes, and the structures of velocity and temperature fluctuations are analyzed.     

Turbulent fluid flows in a circular pipe and plane channel and models of mesoscale turbulence

Based on the model of anisotropic wall turbulence in the near-wall layer and the momentum model in the flow core, velocity profiles in the entire region of the turbulent flow of an incompressible fluid in a circular pipe and plane channel have been obtained. The differences in the profiles among the layers are due to the change in the structure of turbulent vortices.