Helicopter blade flapping with and without small angle assumption in the presence of dynamic stall

The flapping equation for a rotating rigid helicopter blade is typically derived by considering (1)small flap angle, (2) small induced angle of attack and (3) linear aerodynamics. However, the use of nonlinear aerodynamics such as dynamic stall can make the assumptions of small angles suspect as shown in this paper. A general equation describing helicopter blade flap dynamics for large flap angle and large induced inflow angle of attack is derived. A semi-empirical dynamic stall aerodynamics model (ONERA model) is used. Numerical simulations are performed by solving the nonlinear flapping ordinary differential equation for steady state conditions and the validity of the small angle approximations are examined. It is shown that the small flapping assumption, and to a lesser extent, the small induced angle ofattack assumption, can lead to inaccurate predictions of the blade flap response in certain flight conditions for some rotors when nonlinear aerodynamics is considered. (C) 2010 Elsevier Inc. All rights reserved.

Unsteady MHD forced flow due to a point sink

An analysis is performed to study the unsteady laminar incompressible boundary-layer flow of an electrically conducting fluid in a cone due to a point sink with an applied magnetic field. The unsteadiness in the flow is considered for two types of motion, viz. the motion arising due to the free stream velocity varying continuously with time and the transient motion occurring due to an impulsive change either in the strength of the point sink or in the wall temperature. The partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme in combination with the quasilinearization technique. The magnetic field increases the skin friction but reduces heat transfer. The heat transfer and temperature field are strongly influenced by the viscous dissipation and Prandtl number. The velocity field is more affected at the early stage of the transient motion, caused by an impulsive change in the strength of the point sink, as compared to the temperature field. When the transient motion is caused by a sudden change in the wall temperature, both skin friction and heat transfer take more time to reach a new steady state. The transient nature of the flow and heat transfer is active for a short time in the case of suction and for a long time in the case of injection. The viscous dissipation prolongs the transient behavior of the flow.

Semi-similar solutions of the unsteady compressible second-order boundary layer flow at the stagnation point

Semi-similar solutions of the unsteady compressible laminar boundary layer flow over two-dimensional and axisymmetric bodies at the stagnation point with mass transfer are studied for all the second-order boundary layer effects when the free stream velocity varies arbitrarily with time. The set of partial differential equations governing the unsteady compressible second-order boundary layers representing all the effects are derived for the first time. These partial differential equations are solved numerically using an implicit finite-difference scheme. The results are obtained for two particular unsteady free stream velocity distributions: (a) an accelerating stream and (b) a fluctuating stream. It is observed that the total skin friction and heat transfer are strongly affected by the surface mass transfer and wall temperature. However, their variation with time is significant only for large times. The second-order boundary layer effects are found to be more pronounced in the case of no mass transfer or injection as compared to that for suction. Résumé Des solutions semi-similaires d'écoulement variable compressible de couche limite sur des corps bi-dimensionnels thermique, sont étudiées pour tous les effets de couche limite du second ordre, lorsque la vitesse de l'écoulement libre varie arbitrairement avec le temps. Le systéme d'équations aux dérivées partielles représentant tous les effets est écrit pour la premiére fois. On le résout numériquement á l'aide d'un schéma implicite aux différences finies. Les résultats sont obtenus pour deux cas de vitesse variable d'écoulement libre: (a) un écoulement accéléré et (b) un écoulement fluctuant. On observe que le frottement pariétal total et le transfert de chaleur sont fortement affectés par le transfert de masse et la température pariétaux. Néanmoins, leur variation avec le temps est sensible seulement pour des grandes durées. Les effets sont trouvés plus prononcés dans le cas de l'absence du transfert de masse ou de l'injection par rapport au cas de l'aspiration.

Combined free and forced convection along a rotating vertical cylinder

The steady incompressible laminar mixed convection boundary layer flow along a rotating slender vertical cylinder with an isothermal wall has been studied. The transformed coupled nonlinear partial differential equations have been solved numerically using the Keller box method. In general, the rotation of the cylinder, the buoyancy forces and the curvature parameter are found to significantly affect the skin friction, heat transfer, velocity and temperature profiles as well as the pressure distribution. The buoyancy forces cause an overshoot in the axial velocity profile but the rotation and curvature parameters reduce it.

Unsteady mixed convection with double diffusion over a horizontal cylinder and a sphere within a porous medium

The combined effects of the permeability of the medium, magnetic field, buoyancy forces and dissipation on the unsteady mixed convection flow over a horizontal cylinder and a sphere embedded in a porous medium have been studied. The nonlinear coupled partial differential equations with three independent variables have been solved numerically using an implicit finite-difference scheme in combination with the quasilinearization technique. The skin friction, heat transfer and mass transfer increase with the permeability of the medium, magnetic field and buoyancy parameter. The heat and mass transfer continuously decrease with the stream-wise distance, whereas the skin friction increases from zero, attains a maximum and then decreases to zero. The skin friction, heat transfer and mass transfer are significantly affected by the free stream velocity distribution. The effect of dissipation parameter is found to be more pronounced on the heat transfer than on the skin friction and mass transfer

Second-order boundary-layer effects for the unsteady laminar incompressible three-dimensional stagnation-point flow

All the second-order boundary-layer effects on the unsteady laminar incompressible flow at the stagnation-point of a three-dimensional body for both nodal and saddle point regions have been studied. It has been assumed that the free-stream velocity, wall temperature and mass transfer vary arbitrarily with time. The effect of the Prandtl number has been taken into account. The partial differential equations governing the flow have been derived for the first time and then solved numerically unsteady free-stream velocity distributions, the nature of the using an implicit finite-difference scheme. It is found that the stagnation point and the mass transfer strongly affect the skin friction and heat transfer whereas the effects of the Prandtl number and the variation of the wall temperature with time are only on the heat transfer. The skin friction due to the combined effects of first- and second-order boundary layers is less than the skin friction due to, the first-order boundary layers whereas the heat transfer has the opposite behaviour. Suction increases the skin friction and heat transfer but injection does the opposite

Natural Convection on a Horizontal Cone in a Porous Medium with Non-Uniform Wall Temperature/Concentration or Heat/Mass Flux and Suction/Injection

The steady natural convection flow on a horizontal cone embedded in a saturated porous medium with non-uniform wall temperature/concentration or heat/mass flux and suction/injection has been investigated. Non-similar solutions have been obtained. The nonlinear couple differential equations under boundary layer approximations governing the flow have been numerically solved. The Nusselt and Sherwood numbers are found to depend on the buoyancy forces, suction/injection rates, variation of wall temperature/concentration or heat/mass flux, Lewis number and the non-Darcy parameter.

Compressible Boundary-Layer Swirling Flow in a Nozzle and a Diffuser With Highly Cooled Wall

The steady laminar compressible boundary-layer swirling flow with variable gas properties and mass transfer through a conical nozzle, and a diffuser with a highly cooled wall has been studied. The partial differential equations governing the nonsimilar flow have been transformed to a system of coordinates using modified Lees transformation. The resulting equations are transformed into coordinates having finite ranges by means of a transformation which maps an infinite region into a finite region. The ensuing equations are then solved numerically using an implicit finite-difference scheme. The results indicate that the variation of the density-viscosity product across the boundary layer and mass transfer have strong effect on the skin friction and heat transfer. Separationless flow along the entire length of the diffuser can be obtained by applying suction. The results are found to be in good agreement with those of the local nonsimilarity method but they differ appreciably from those of the local similarity method.

Investigation of the effect of nonlocal scale on ultrasonic wave dispersion characteristics of a monolayer graphene

In this article, an ultrasonic wave propagation in graphene sheet is studied using nonlocal elasticity theory incorporating small scale effects. The graphene sheet is modeled as an isotropic plate of one-atom thick. For this model, the nonlocal governing differential equations of motion are derived from the minimization of the total potential energy of the entire system. An ultrasonic type of wave propagation model is also derived for the graphene sheet. The nonlocal scale parameter introduces certain band gap region in in-plane and flexural wave modes where no wave propagation occurs. This is manifested in the wavenumber plots as the region where the wavenumber tends to infinite or wave speed tends to zero. The frequency at which this phenomenon occurs is called the escape frequency. The explicit expressions for cutoff frequencies and escape frequencies are derived. The escape frequencies are mainly introduced because of the nonlocal elasticity. Obviously these frequencies are function of nonlocal scaling parameter. It has also been obtained that these frequencies are independent of y-directional wavenumber. It means that for any type of nanostructure, the escape frequencies are purely a function of nonlocal scaling parameter only. It is also independent of the geometry of the structure. It has been found that the cutoff frequencies are function of nonlocal scaling parameter (e(0)a) and the y-directional wavenumber (k(y)). For a given nanostructure, nonlocal small scale coefficient can be obtained by matching the results from molecular dynamics (MD) simulations and the nonlocal elasticity calculations. At that value of the nonlocal scale coefficient, the waves will propagate in the nanostructure at that cut-off frequency. In the present paper, different values of e(o)a are used. One can get the exact e(0)a for a given graphene sheet by matching the MD simulation results of graphene with the results presented in this paper. (C) 2010 Elsevier B.V. All rights reserved.

Flow and heat transfer in steady laminar compressible boundary layer swirling flow in a nozzle

The axisymmetric steady laminar compressible boundary layer swirling flow of a gas with variable properties in a nozzle has been investigated. The partial differential equations governing the non-similar flow have been transformed into new co-ordinates having finite ranges by means of a transformation which maps an infinite range into a finite one. The resulting equations have been solved numerically using an implicit finite-difference scheme. The computations have been carried out for compressible swirling flow through a convergent conical nozzle. The results indicate that the swirl exerts a strong influence on the longitudinal skin friction, but its effect on the tangential skin friction and heat transfer is comparatively small. The effect of the variation of the density-viscosity product across the boundary layer is appreciable only at low-wall temperature. The results are in good agreement with those of the local-similarity method for small values of the longitudinal distance.

Free vibration of circular plates of arbitrary thickness

Free vibration of circular plates of arbitrary thickness is investigated using the method of initial functions. State-space approach is used to derive the governing equations of the above method. The formulation is such that theories of any desired order can be obtained by deleting higher terms in the infinite-order differential equations. Numerical results are obtained for flexural and extensional vibration of circular plates. Results are also computed using Mindlin's theory and they are in agreement with the present analysis.

Unsteady three-dimensional boundary layer flow over a flat plate with attached cylinder

The unsteady laminar incompressible three-dimensional boundary layer flow and heat transfer on a flat plate with an attached cylinder have been studied when the free stream velocity components and wall temperature vary inversely as linear and quadratic functions of time, respectively. The governing semisimilar partial differential equations with three independent variables have been solved numerically using a quasilinear finite-difference scheme. The results indicate that the skin friction increases with parameter ? which characterizes the unsteadiness in the free stream velocity and the streamwise distance Image , but the heat transfer decreases. However, the skin friction and heat transfer are found to change little along Image . The effect of the Prandtl number on the heat transfer is found to be more pronounced when ? is small, whereas the effect of the dissipation parameter is more pronounced when ? is comparatively large.

Unsteady mixed convection over two-dimensional and axisymmetric bodies

The unsteady laminar incompressible mixed convection flow over a two-dimensional body (cylinder) and an axisymmetric body (sphere) has been studied when the buboyancy forces arise from both thermal and mass diffusion and the unsteadiness in the flow field is introduced by the time dependent free stream velocity. The nonlinear partial differential equations with three independent variables governing the flow have been solved numerically using an implicit finite-difference scheme in combination with the quasilinearization technique. The results indicate that for the thermally assisting flow the local skin friction, heat transfer and mass diffusion are enhanced when the buoyancy force from mass diffusion assists the thermal buoyancy force. But this trend is opposite for the thermally opposing flow. The point of zero skin friction moves upstream due to unsteadiness. No singularity is observed at the point of zero skin friction for unsteady flow unlike steady flow. The flow reversal is observed after a certain instant of time. The velocity overshoot occurs for assisting flows.

Unsteady boundary layer flow on a cylinder in a channel

The unsteady laminar incompressible nonsimilar boundary layer flow over a circular cylinder placed symmetrically inside a channel has been studied when the unsteadiness and nonsimilarity are due to the free stream velocity. The nonlinear partial differential equations with three independent variables have been solved numerically using an implicit finite-difference in combination with the quasilinearization technique. It is found that the channel blockage parameter controls the transfer of heat from the cylinder and delays separation. The skin friction and heat transfer are significantly affected by the free stream velocity distributions.

Unsteady laminar compressible boundary-layer flow at a three-dimensional stagnation point

Unsteady laminar compressible boundary-layer flow with variable properties at a three-dimensional stagnation point for both cold and hot walls has been studied for the case when the velocity of the incident stream varies arbitrarily with time. The partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme. Computations have been carried out for two particular unsteady free-stream velocity distributions: (i) an accelerating stream and (ii) a fluctuating stream. The results indicate that the variation of the density-viscosity product across the boundary layer, the wall temperature and the nature of stagnation point significantly affect the skin friction and heat transfer.