Unsteady boundary layer and its separation over a heated circular cylinder

A numerical method is developed to solve the coupled unsteady laminar momentum and thermal boundary layers over a circular cylinder impulsively started from rest. The present non-iterative finite difference method, which requires relatively fewer grid points in the reversed flow region than any other method, can easily handle the separating boundary layer flows. The results indicate that the present method has accuracy comparable with the earlier methods, while consuming computer time approximately one order of magnitude less. The present numerical method allowed investigation of the effect of buoyancy parameter on the starting boundary layer. The time-dependent behaviour of the boundary layer is studied in terms of the appearance of the singularity, the distribution of skin friction and wall heat flux, and the wall position of the inflection point of the velocity profile. The transient as well as buoyancy-dependent patterns of the streamlines and isotherms are also studied.     

The boundary layer flow of a Reiner-Rivlin fluid near a spinning cone

Summary The steady motion of an incompressible inelastic Reiner-Rivlin fluid near a spinning cone has been studied and a similarity solution has been presented. It has been shown that the flow patterns can be obtained from Srivastava-Jain's work. But the pressure distribution is not the same and we have given the numerical values of the pressure in two tables and have also shown their variation in two figures.Sponsored by the Mathematics Research Center, United States Army, Madison Wisconsin under Contract No.: DA-11-022-ORD-2059.     

Unsteady mixed convection boundary layer flow over a vertical cone with non-uniform slot suction (injection)

The non-similar solution of an unsteady mixed convection laminar boundary layer flow over a vertical cone in the presence of non-uniform surface mass transfer through slot has been obtained while the axis of cone is inline with the flow. The unsteadiness is caused by the time dependent free stream velocity. The governing boundary layer equations are transformed into a non-dimensional form by a group of non-similar transformations. The resulting coupled non-linear partial differential equations have been solved numerically by the combination of quasi-linearization technique and an implicit finite difference scheme. Numerical computations are performed for different values of the parameters to display the velocity and temperature profiles graphically. Both accelerating and decelerating free stream velocities are considered. Numerical results are reported to display the effects of non-uniform single and double slot suction (injection) on skin friction and heat transfer coefficients at the wall. Further, the effects of Prandtl number, buoyancy and mass transfer (suction or injection) parameters at different stream-wise locations for various times on velocity and temperature profiles, and on skin friction and heat transfer coefficients are also presented in this paper.     

Unsteady compressible boundary layer flow in the stagnation region of a sphere with a magnetic field

Summary An analysis is performed to study the unsteady compressible laminar boundary layer flow in the forward stagnation-point region of a sphere with a magnetic field applied normal to the surface. We have considered the case where there is an initial steady state that is perturbed by the step change in the total enthalpy at the wall. The nonlinear coupled parabolic partial differential equations governing the flow and heat transfer have been solved numerically using a finite-difference scheme. The numerical results are presented, which show the temporal development of the boundary layer. The magnetic field in the presence of variable electrical conductivity causes an overshoot in the velocity profile. Also, when the total enthalpy at the wall is suddenly increased, there is a change in the direction of transfer of heat in a small interval of time. Received 15 January 1996; accepted for publication 21 November 1996     

Unsteady boundary layer on a circular cylinder embedded to a wedge

The boundary layer growth on a circular cylinder embedded to a wedge, when the motion is started impulsively from rest, is discussed using the method of inner and outer expansions. The equation for the time of separation involving the Reynolds number and the wedge angle is obtained. A uniformly valid solution is also found for the entire flow field. It is found that separation first occurs at the points where the cylinder meets the wedge. Also, the time of earliest separation decreases with increase in the Reynolds number and with increase in the wedge angle.     

Mixed convection boundary layer flow of a viscoelastic fluid over a horizontal circular cylinder

The steady mixed convection boundary layer flow of a viscoelastic fluid over a horizontal circular cylinder in a stream flowing vertically upwards is numerically studied for both cases of heated and cooled cylinders. The governing partial differential equations are transformed into dimensionless forms using an appropriate transformation and then solved numerically using the Keller-box method. The comparison between the solutions obtained and those for a Newtonian fluid is found to be very good. Effects of the mixed convection and elasticity parameters on the skin friction and heat transfer coefficients for a fluid having the Prandtl number equal to one are also discussed. It is found that for some values of the viscoelastic parameter and some negative values of the mixed convection parameter (opposing flow) the boundary layer separates from the cylinder. Heating the cylinder delays separation and can, if the cylinder is warm enough, suppress the separation completely. Similar to the case of a Newtonian fluid, cooling the cylinder brings the separation point nearer to the lower stagnation point. However, for a sufficiently cold cylinder there will not be a boundary layer.     

Mixed convection boundary layer flow over a horizontal circular cylinder with Newtonian heating

The steady mixed convection boundary layer flow over a horizontal circular cylinder, generated by Newtonian heating in which the heat transfer from the surface is proportional to the local surface temperature, is considered in this study. The governing boundary layer equations are first transformed into a system of non-dimensional equations via the non-dimensional variables, and then into non-similar equations before they are solved numerically using a numerical scheme known as the Keller-box method. Numerical solutions are obtained for the skin friction coefficient Re ^1/2 C _f and the local wall temperature θ _w (x) as well as the velocity and temperature profiles with two parameters, namely the mixed convection parameter λ and the Prandtl number Pr.     

Experimental study of compressible boundary layer on a cone at angles of attack

Experimental study was conducted for boundarylayers on a sharp 5° half-angle cone of 400mm length at angles of attack. The model was tested in the T-326 hypersonic wind tunnel （ITAM） at freestream Mach number M = 5.95. Mean and fluctuation wall characteristics of the boundary layer are measured at 0°, 2°, 3° and 4° angles of attack for different stagnation pressures. Pulsation measurements are carried out by means of ALTP sensor. Pressure and temperature distributions along the model are obtained, and transition beginning and end locations have been found. Boundary layer stabilization with the increase of angle of attack and the decrease of stagnation pressure is observed. High frequency pulsations inherent to hypersonic boundary layer （second mode） have been detected.     

Effects of blowing or suction on the laminar boundary layer flow over a rotating cone

The effects of suction or blowing at the surface of a rotating cone in a quiet fluid on the skin friction and heat transfer are described. The equations which govern the fluid motion and thermal energy transfer are transformed by the boundary layer approximations and the resulting equations are solved under the condition that the suction or blowing velocity varies as x ⁿ (x: distance measured from the apex of the cone, n: arbitrary constant). The solutions are obtained as a perturbation from the basic laminar flow of an incompressible viscous fluid over the impermeable rotating cone. Detailed numerical calculations are performed for the case of an isothermal rotating cone with uniform blowing or suction, i.e. n=0, the Prandtl number being 0.72. Results are given for the shear stress, heat transfer and velocity and temperature fields. It is shown from the analysis that suction sharply increases the circumferential shear stress and the heat transfer at the surface.Nomenclature c proportional constant - C _fx dimensionless skin friction factor, _x /(V ²) - C _fx0 dimensionless skin friction factor for an impermeable cone - C _fy dimensionless circumferential skin friction factor, _y /(V ²) - C _fy0 dimensionless circumferential skin friction factor for an impermeable cone - c _p specific heat at constant pressure - f _k function of - g _k function of - h heat transfer coefficient, q/(T _w–T ) - k thermal conductivity of fluid - n arbitrary constant - Nu _x local Nusselt number, hx/k - Nu _x0 local Nusselt number for an impermeable cone - Pr Prandtl number - q heat transfer rate - r radius of a circular cross section of the cone, x sin - R _x Reynolds number, Vx/ - T temperature - T _w surface temperature of the cone - T temperature of the surrounding fluid - u fluid velocity in x-direction - v fluid velocity in y-direction - V circumferential velocity at the cone surface, r - w fluid velocity in z-direction - x coordinate along meridional section - y coordinate along a circular cross section - z coordinate perpendicular to both x and y - perturbation parameter, cx ⁿ /(x sin )^1/2 - dimensionless z-coordinate, z( sin /)^1/2 - _k function of - kinematic viscosity - density of fluid - _x skin friction in x-direction - _y circumferential skin friction - stream function - angular speed of the cone     

Stability of a compressible boundary layer

The problem of stability in a compressible boundary layer, as opposed to an incompressible layer, involves many parameters and requires consideration of three-dimensional perturbations. The transverse component of the velocity, the thermal regime at the wall, etc., take on great significance. Investigation of all aspects of this problem requires systematic calculations performed by electronic computers. There do exist a few calculations of stability of a compressible boundary layer with respect to three-dimensional disturbances for particular cases. It follows from those studies (see, for example, [1]) that consideration of three-dimensional perturbations and of the transverse component of the basic flow velocity is important. Many aspects of this problem remain uninvestigated. Aside from the sheer cumbersomeness of the problem, there exist purely mathematical difficulties connected with the presence of a small parameter with higher derivatives in the differential equations for the perturbations, which causes losses in accuracy of calculation. In this present study an algorithm will be developed for solution of the problem of stability of a compressible boundary layer relative to three-dimensional disturbances with consideration of the transverse component of the basic velocity. Calculations are performed for a boundary layer on a plane thermally insulating plate, and the effects of the transverse velocity component and the three-dimensionality of the perturbations on stability at various Mach numbers are demonstrated.     

Large injection rates on unsteady compressible boundary layer flow over cones at angles of attack

Summary Three-dimensional unsteady laminar boundary layer near the planes of symmetry of sharp cones at angles of attack subject to large rates of injection is obtained numerically by using an implicit finite difference scheme in combination with the quasi-linearization technique. Several model gases are considered with Mach numbers, wall-to-total-enthalpy ratios, and cross-flow parameters spanning the ranges of main engineering interest. A detailed study has been made of the solutions in the symmetry plane, in order to increase the understanding of the problem. Various cases are considered, when the free-stream velocity and the surface mass transfer (injection) vary arbitrarily with time. The effects of viscous dissipation and the cross-flow parameter have also been discussed.This research has been partially supported by the Research and Development Centre for Iron and Steel, Steel Authority of India Ltd. The constructive comments of Professor G. Nath and Professor A. K. Lahiri are sincerely appreciated.     

Unsteady three-dimensional boundary layer flow due to a permeable shrinking sheet

The unsteady viscous flow over a continuously permeable shrinking surface is studied. Similarity equations are obtained through the application of similar transformation techniques. Numerical techniques are used to solve the similarity equations for different values of the unsteadiness parameter, the mass suction parameter, the shrinking parameter and the Prandtl number on the velocity and temperature profiles as well as the skin friction coefficient and the Nusselt number. It is found that, different from an unsteady stretching sheet, dual solutions exist in a certain range of mass suction and unsteadiness parameters.     

Non-uniform slot suction/injection into mixed convection boundary layer flow over vertical cone

The aim of this work is to study the effect of non-uniform single and double slot suction/injection into a steady mixed convection boundary layer flow over a vertical cone, while the axis of the cone is inline with the flow. The governing boundary layer equations are transformed into a non-dimensional form by a group of non-similar trans- formations. The resulting coupled non-linear partial differential equations are solved nu- merically by employing the quasi-linearization technique and an implicit finite-difference scheme. Numerical computations are performed for different values of the dimensionless parameters to display the velocity and temperature profiles graphically. Also, numerical results are presented for the skin friction and heat transfer coefficients. Results indicate that the skin friction and heat transfer coefficients increase with non-uniform slot suction, but the effect of non-uniform slot injection is just opposite.     

Global stability analysis of axisymmetric boundary layer over a circular cylinder

This paper presents a linear global stability analysis of the incompressible axisymmetric boundary layer on a circular cylinder. The base flow is parallel to the axis of the cylinder at inflow boundary. The pressure gradient is zero in the streamwise direction. The base flow velocity profile is fully non-parallel and non-similar in nature. The boundary layer grows continuously in the spatial directions. Linearized Navier–Stokes (LNS) equations are derived for the disturbance flow quantities in the cylindrical polar coordinates. The LNS equations along with homogeneous boundary conditions forms a generalized eigenvalues problem. Since the base flow is axisymmetric, the disturbances are periodic in azimuthal direction. Chebyshev spectral collocation method and Arnoldi’s iterative algorithm is used for the solution of the general eigenvalues problem. The global temporal modes are computed for the range of Reynolds numbers and different azimuthal wave numbers. The largest imaginary part of the computed eigenmodes is negative, and hence, the flow is temporally stable. The spatial structure of the eigenmodes shows that the disturbance amplitudes grow in size and magnitude while they are moving towards downstream. The global modes of axisymmetric boundary layer are more stable than that of 2D flat-plate boundary layer at low Reynolds number. However, at higher Reynolds number they approach 2D flat-plate boundary layer. Thus, the damping effect of transverse curvature is significant at low Reynolds number. The wave-like nature of the disturbance amplitudes is found in the streamwise direction for the least stable eigenmodes.     

Periodic boundary layer near an axisymmetric stagnation point on a circular cylinder

An analysis is presented to investigate the time-mean characteristics of the laminar boundary layer near an axisymmetric stagnation point when the velocity of the oncoming flow relative to the body oscillates. Different solutions are obtained for the small and high values of the reduced frequency parameter. The range of Reynolds number considered was from 0.01 to 100. Numerical solutions for the velocity functions are presented, and the wall values of the velocity gradients are tabulated.     

Flow characteristics around a circular cylinder subjected to vortex-induced vibration near a plane boundary

Although vortex-induced vibration (VIV) has been extensively studied, much of existing literature deals with uniform flow in the absence of a boundary. The VIV flow field of a structure close to a boundary generally remains unexplored, but it can have important engineering implications, such as pipeline scour if the boundary is an erodible seabed. In this paper, laboratory experiments are performed to investigate the flow characteristics of an elastically mounted circular cylinder undergoing VIV, and a rigid plane boundary is considered to simplify the problem. The initial gap-to-diameter ratio is fixed at 0.8, and six different reduced velocities are considered. The velocity field is measured using a high resolution particle image velocimetry (PIV) system, which has several advantages over traditional PIV systems, including high sampling rate and the ability to mitigate scatter of laser light near the boundary, allowing accurate measurements at the viscous sublayer. This paper presents the vibration amplitude and oscillation frequency for different V_r; in addition, the mean velocity field, turbulence characteristics, vortex behavior, gap flow velocity, and normal/shear stresses on the boundary were measured/calculated, leading to new insights on the flow field behavior.