An inviscid model of unsteady aerofoil flow with fixed upper surface separation

Presented in this paper is a new method for the prediction of unsteady, incompressible separated flow over a two-dimensional aerofoil. The algorithm was developed from an existing unsteady potential flow model¹ and makes use of an inviscid formulation for the flow field. The aerofoil is represented by vortex panels of linearly varying strength which are piecewise continuous at the corners. Discrete vortices with finite cores are used to model the separating shear layers. Following a brief summary of unsteady separation modelling, the theoretical framework is presented and the subsequent numerical implementation is discussed in detail. Results are given for flows which tend asymptotically to the steady state and conclusions are drawn regarding the usefulness of the method.     

An experimental investigation of pulsatile flow through a smooth constriction

Measurements of flow disturbances in the downstream region of modeled stenoses in a rigid tube, with upstream pulsatile flow are reported. Experiments were conducted over physiologically relevant mean Reynolds numbers of 600; based on the tube diameter and the time-averaged value of upstream centerline velocity. Contoured constrictions with 25%, 50% and 75% area reductions were investigated and velocity data were obtained from ensemble averaging techniques (phase-locked waveform). Experimental data over extensive spatial regions of poststenotic fields were taken, employing a two-component laser Doppler velocimeter LDV. Constant time sampling techniques for performing data or frequency analyses were used to avoid velocity bias and to study the evolution of poststenotic flow disturbances. It is found that different types of flow disturbances exist downstream of the constriction. Data analysis methods with the aid of flow visualization allow accurate classification of the disturbances which are sensitive indicators of mild to moderate constrictions. Although the present study was motivated by a biological situation, sufficient data were reported in detail that they may also be used by investigators working in computational fluid dynamics.     

Boundary layer flow and heat transfer over an unsteady stretching vertical surface

The solution to the unsteady mixed convection boundary layer flow and heat transfer problem due to a stretching vertical surface is presented in this paper. The unsteadiness in the flow and temperature fields is caused by the time-dependent of the stretching velocity and the surface temperature. The governing partial differential equations with three independent variables are first transformed into ordinary differential equations, before they are solved numerically by a finite-difference scheme. The effects of the unsteadiness parameter, buoyancy parameter and Prandtl number on the flow and heat transfer characteristics are thoroughly examined. Both assisting and opposing buoyant flows are considered. It is observed that for assisting flow, the solutions exist for all values of buoyancy parameter, whereas for opposing flow, they exist only if the magnitude of the buoyancy parameter is small. Comparison with known results for steady-state flow is excellent.     

Exact solution for the unsteady flow of a semi-infinite micropolar fluid

The unsteady motion of an incompressible micropolar fluid filling a half-space bounded by a horizontal infinite plate that started to move suddenly is considered. Laplace transform techniques are used. The solution in the Laplace transform domain is obtained by using a direct approach. The inverse Laplace transforms are obtained in an exact manner using the complex inversion formula of the transform together with contour integration techniques. The solution in the case of classical viscous fluids is recovered as a special case of this work when the micropolarity coecient is assumed to be zero. Numerical computations are carried out and represented graphically.     

On the comparison of two different solutions in the form of series of the governing equation of an unsteady flow of a second grade fluid

In this paper, two different solutions in the form of series of the governing equation of unsteady flow of a second grade fluid are considered. These are series expansions with respect to inverse power of time and a perturbation expansion. Two illustrative examples are given. One of them is the unsteady flow of a second grade fluid over a plane wall suddenly set in motion and the other is the diffusion of a line vortex in a fluid of second grade. It is a remarkable fact that the expression of the series expansion with respect to inverse power of time is exactly in the same form as that of the perturbation expansion. Thus, it is possible to replace a series expansion with respect to inverse power of time with a perturbation expansion.     

A qualitative study on the pulsatile flow phenomenon in a dense fly ash pneumatic conveyor

Understanding of the dynamic particulate flow structures within a dense gas-fly ash pneumatic conveyor must be improved in order to better aid its design guidance.The complex pulsatile movement of the gas-fly ash mixture dominates the flow performance within the pipeline,and historically,non-invasive measurement devices such as the electrical capacitance tomography（ECT） were often used to sufficiently capture the flow dynamics.However,inadequate studies have been conducted on the pulsatile flow phenomenon,which directly relate to the gas-fly ash two-phase flow performance.This paper aims to investigate the pulsatile flows using an ECT device.Initially,pulsatile flow patterns under various experimental conditions were obtained through ECT.Pulses within a flow were then characterised into pulse growth and decay segments,which represent the superficial fluidisation and deaeration processes during conveying.Subsequently,structural and statistical analyses were performed on the pulse growth and decay segments.Results suggested that the increasing air mass flow rate led to the decrease of the superficial fluidisation/deaeration magnitude,however,the increase of the superficial fluidisation/deaeration durations.Also,the air mass flow rate was indicated as the dominant factor in determining the pulsing statistical parameters.This research provides fundamental insights for further modelling the dense fly ash pneumatic flows.     

Improvement of a pressure gradient method and its application to an unsteady flow problem

The pressure gradient method using velocity components and components of a pressure gradient as dependent variables has been modified to solve incompressible Newtonian fluid flow problems numerically. Applying this modified method to unsteady-state development of flow in a circular cavity shows that, at least for the case of a low Reynolds number flow, relative errors produced by the proposed method are smaller for most time intervals than those produced by the primitive velocity-pressure variable method and by the standard pressure gradient method. Also it is found that the modified and standard pressure gradient methods can be applied to the unsteady circular cavity flow at a moderate Reynolds number of at least up to 200.     

Numerical computation of unsteady incompressible flow in complex geometry using a composite multigrid technique

This paper presents a composite multigrid method and its application to a geometrically complex flow. The treatment of the interior boundary conditions within a composite multigrid strategy is described in detail for a 1D model equation. For the Navier-Stokes equations a staggered grid technique is adopted for spatial discretization and a fractional step method is used for the time advance. Lid-driven cavity flows are used to demonstrate the effectiveness of the method.     

Wall pressure fluctuations of a turbulent separated and reattaching flow affected by an unsteady wake

The spatio-temporal characteristics of the wall-pressure fluctuations in separated and reattaching flows over a backward-facing step were investigated through pressure-velocity joint measurements carried out using multiple-arrayed microphones and split-film probes. A spoke-wheel-type wake generator was installed upstream of the backward-facing step. The flow structure at the effective forcing frequency (St _f=0.2) was found to be well organized in terms of wall pressure spectrum, cross-correlation, wavenumber-frequency spectrum, and wavelet auto-correlation. Introduction of the unsteady wake (St _f=0.2) reduced the reattachment length by 10%. In addition, the unsteady wake enhanced the turbulence intensity near the separation edge and, as a consequence, enhanced the quadrupole sound sources; however, the turbulence intensity near the reattachment region was weakened and the overall flow noise was attenuated. The greater organization of the flow structure induced by the unsteady wake led to a weakening of the dipole sound sources, which are the dominant sound sources in this system. The dipole sound sources generated by wall pressure fluctuations were calculated using Curles integral formula.Abbreviations AR Aspect ratio - SBF Spatial box filtering Roman symbols C _p Wall pressure fluctuation coefficient, p/0.5U ² - H Step height of backward-facing step (mm) - H _s Shape factor (H _s = ^*/) - R _s Distance from acoustic source point to observation point (m) - Re _H Reynolds number, U H/ - St The reduced frequency, fH/U - St _f Normalized forcing frequency by unsteady wake, f _p H/U - T Vortex shedding period (s) - U Free-stream velocity (m/s) - a Speed of sound (m/s) - f Frequency (Hz) - f _p Wake passing frequency (Hz) - k Turbulent kinetic energy (m²/s²) - k _x Streamwise wave number (1/m) - k _z Spanwise wave number (1/m) - l _j Cosine of angle - p Instantaneous wall pressure (Pa) - p _rms Root-mean-square of wall pressure (Pa) - p _SBF Spatial box filtered wall pressure (Pa) - p _d Dipole sound source (Pa) - p _w Conditionally-averaged wall pressure (Pa) - q Dynamic pressure, 0.5U ² (Pa) - r Distance from origin to observation point (mm) - u _c Convection velocity (m/s) - u_max Root-mean-square of streamwise velocity (m/s) - x _R Time-mean reattachment length (mm) Greek symbols _p Forward-flow time fraction - Auto-correlation of pressure at x ₀ - Two-dimensional cross-correlation of pressure with streamwise separation interval , spanwise separation interval , and time delay , at (x ₀, z ₀) - Boundary layer thickness (mm, _99%) - ^* Displacement thickness (mm, ) - _ij Kroneckers delta function - Phase angle (°) - Wavelength (mm) - Momentum thickness (mm, ) - Angle between vertical axis and observation point (°) - Density (kg/m³) - Time delay (s) - Streamwise separation interval (m) - Spanwise separation interval (m) - _p (f; x ₀) Autospectrum of pressure measured at x ₀ (Pa² s) - _pp (, ; x ₀) Streamwise cross spectrum of pressure at x ₀ (Pa² s) - _pp (, , ; x ₀, z ₀) Streamwise and spanwise cross spectrum of pressure at (x ₀, z ₀) (Pa² s) - _pp (kx, ; x ₀) Streamwise wavenumber-frequency spectrum of pressure at x ₀ (Pa² s) - _pp (kx, kz, ; x ₀, z ₀) Two-dimensional wavenumber-frequency spectrum of pressure at (x ₀, z ₀) (Pa² s)     

Effects of the side walls on the unsteady flow of a second-grade fluid in a duct of uniform cross-section

In this paper, the effects of the side walls on the unsteady flow of a second-grade fluid in a duct of rectangular cross-section are considered. Two types of unsteady flows are investigated. One of them is the unsteady flow in a duct of rectangular cross-section moving parallel to its length and the other is the unsteady flow due to an applied pressure gradient in a duct of rectangular cross-section whose sides are at rest. It is shown that a Newtonian fluid reaches steady-state earlier than a second-grade fluid and the effect of the side walls on a second-grade fluid is more effective than that on a Newtonian fluid.     

Effects of the side walls on unsteady flow of a second grade fluid over a plane wall

The effects of the side walls on unsteady flow of a second grade fluid over a plan wall are considered. The solution of the governing equation for velocity is obtained by the sine transform method. This gives a correct result for the shear stress at the bottom wall. The shear stress at the bottom wall is minimum at the middle of the plate and it increases near the side walls. It is shown that the mean thickness of the layer of the liquid over the plate increases with time and the ratio of the mean thickness to the distance between the side walls becomes ultimately 0.2714.     

A two-fluid model for pulsatile flow in catheterized blood vessels

The pulsatile flow of blood through a catheterized artery is analyzed, assuming the blood as a two-fluid model with the suspension of all the erythrocytes in the core region as a Casson fluid and the peripheral region of plasma as a Newtonian fluid. The resulting non-linear implicit system of partial differential equations is solved using perturbation method. The expressions for shear stress, velocity, flow rate, wall shear stress and longitudinal impedance are obtained. The variations of these flow quantities with yield stress, catheter radius ratio, amplitude, pulsatile Reynolds number ratio and peripheral layer thickness are discussed. It is observed that the velocity distribution and flow rate decrease, while, the wall shear, width of the plug flow region and longitudinal impedance increase when the yield stress increases. It is also found that the velocity increases, but, the longitudinal impedance decreases when the thickness of the peripheral layer increases. The wall shear stress decreases non-linearly, while, the longitudinal impedance increases non-linearly when the catheter radius ratio increases. The estimates of the increase in the longitudinal impedance are considerably lower for the present two-fluid model than those of the single-fluid model.     

Stability of fluid flow in the presence of a compliant surface

This paper is concerned with the dynamic behaviour of a thin elastic plate in the presence of flow. The plate is excited by a time varying force applied on a line which is at right angles to the direction of flow. Causality is invoked to make the solution unique. It is found that the system responds differently for flow speeds above and below a critical value. Above this value, a temporal instability occurs and the amplitude of the motion grows with time at each point on the plate. Below this value there are travelling waves upstream of the driver, whilst downstream the behaviour is dominated by a disturbance which does not grow with time at any one position, but nevertheless grows as it is convected downstream. This type of instability is less severe than the temperal one and is of the type generally referred to as convective.     

Unsteady boundary layer flow due to a stretching surface in a rotating fluid

The induced unsteady flow due to a stretching surface in a rotating fluid, where the unsteadiness is caused by the suddenly stretched surface is studied in this paper. After a similarity transformation, the unsteady Navier–Stokes equations have been solved numerically using the Keller-box method. Also, the perturbation solution for small times as well as the asymptotic solution for large times, when the flow becomes steady, has been obtained. It is found that there is a smooth transition from the small time solution to the large time or steady state solution.     

The boundary layers of an unsteady incompressible stagnation-point flow with mass transfer

In the current work, the boundary layers of an unsteady incompressible stagnation-point flow with mass transfer were further investigated. Similarity transformation technique was used and the similarity equation group was solved using numerical methods. Interesting observation is that there are multiple solutions seen for negative unsteadiness parameters, β. The influences of mass transfer, unsteadiness parameter, and Prandtl numbers on velocity and temperature profiles, wall drag, and wall heat fluxes were investigated and analyzed. The asymptotic behaviors for the similarity equations in limiting situations were theoretically analyzed. It is found that solutions exist for all mass transfer parameters for β≥−1. For a certain mass transfer parameter, there are two solutions when β_c<β<0; there is one solution for (β=β_c)∪(β≥0); there is no solution for β<β_c, where β_c is a critical unsteadiness parameter dependent on mass transfer parameter.     

Influence of unsteady wake on a turbulent separation bubble

 An experimental study was made of turbulent separated and reattaching flow over a blunt body, where unsteady wake was generated by a spoked-wheel type of wake generator with cylindrical rods. The influence of unsteady wake was scrutinized by altering the rotation direction (clockwise and counter-clockwise) and the normalized passing frequency (0 ≤ St _H  ≤ 0.20). The Reynolds number based on the cylindrical rod was Re _d =375. A phase-averaging technique was employed to characterize the unsteady wake. The effect of different rotation directions, which gave a significant reduction of x _R , was examined in detail. The wall pressure fluctuations on the blunt body were analyzed in terms of the spectrum and the coherence. Received: 15 January 2001 / Accepted: 17 July 2001     

Three-dimensional unsteady flow simulations: Alternative strategies for a volume-averaged calculation

This work builds on a SIMPLE-type code to produce two numerical codes of greatly improved speed and accuracy for solution of the Navier–Stokes equations. Both implicit and explicit codes employ an improved QUICK (quadratic upstream interpolation for convective kinematics) scheme to finite difference convective terms for non-uniform grids. The PRIME (update pressure implicit, momentum explicit) algorithm is used as the computational procedure for the implicit code. Use of both the ICCG (incomplete Cholesky decomposition, conjugate gradient) method and the MG (multigrid) technique to enhance solution execution speed is illustrated. While the implicit code is first-order in time, the explicit is second-order accurate. Two- and three-dimensional forced convection and sidewall-heated natural convection flows in a cavity are chosen as test cases. Predictions with the new schemes show substantial computational savings and very good agreement when compared to previous simulations and experimental data.     

The effect of the slip condition on unsteady flow due to non-coaxial rotations of disk and a fluid at infinity

An exact solution of the Navier-Stokes equation is constructed for the magnetohydrodynamic (MHD) flow. The flow is due to non-coaxially rotations of a porous disk with slip condition and a fluid at infinity. The solutions for steady and unsteady cases are obtained by Laplace transform method. The effects of magnetic field and slip parameters are shown and discussed.     

An unsteady analysis of non-Newtonian blood flow through tapered arteries with a stenosis

The problem of non-Newtonian and nonlinear blood flow through a stenosed artery is solved numerically where the non-Newtonian rheology of the flowing blood is characterised by the generalised Power-law model. An improved shape of the time-variant stenosis present in the tapered arterial lumen is given mathematically in order to update resemblance to the in vivo situation. The vascular wall deformability is taken to be elastic (moving wall), however a comparison has been made with nonlinear visco-elastic wall motion. Finite difference scheme has been used to solve the unsteady nonlinear Navier-Stokes equations in cylindrical coordinates system governing flow assuming axial symmetry under laminar flow condition so that the problem effectively becomes two-dimensional. The present analytical treatment bears the potential to calculate the rate of flow, the resistive impedance and the wall shear stress with minor significance of computational complexity by exploiting the appropriate physically realistic prescribed conditions. The model is also employed to study the effects of the taper angle, wall deformation, severity of the stenosis within its fixed length, steeper stenosis of the same severity, nonlinearity and non-Newtonian rheology of the flowing blood on the flow field. An extensive quantitative analysis is performed through numerical computations of the desired quantities having physiological relevance through their graphical representations so as to validate the applicability of the present model.