Unsteady Equations Of Motion Research Articles

Influence of different delays on mixed types of oscillations under limited excitation

This review summarizes the findings of some published studies that have explored the influence of various delays (elasticity, damping, and self-oscillatory mechanism of excitation) on the dynamics of classes (or types) of mixed oscillations (MO) without and with consideration of the interaction between the oscillating system and the energy source. A general holistic framework was provided for how such delays, both separately and in combination, affect the dynamics of MOs. A unified computational scheme (model) used in the works studied made it easy to understand and compare the results of this influence on different types of MOs. With the account of the interaction with the energy source, the known calculation scheme (or model) of a mechanical frictional self-oscillating system serves as a unified basis for considering all types of MOs. Nonlinear differential equations of motion valid for all types of MOs with their respective solutions were presented, from which the relations for any certain type of MO are derived as special cases. Equations of unsteady motion and relations to calculate the amplitude and phase of stationary oscillations, the velocity of the energy source and the load of the oscillating system on it, as well as the stability conditions of stationary oscillations were given. The results of the calculations carried out to gain insight into the influence of delays on the system dynamics were discussed. Overall, the calculations show that the interaction between the forces with delay and the forces in the energy source is at the core of a variety of phenomena. Different delays in the same system change the shape of the amplitude-frequency curves, shift them, and influence the stability of motion.

Three dimensional viscous flow due to a squeezing porous slider

Three dimensional viscous flow as a result of expanding or contracting porous slider is undertaken in the present analysis. The fluid injection to levitate the slider is not taken constant, but it changes with time in accordance with the location of the slider at any time. The unsteady equations of motion corresponding to the fluid flow between the slider and the ground are transformed into their similarity form with the help of two dimensionless parameters; Reynolds number and dilation parameter. In the limiting case of Reynolds number tending to zero, closed-form expressions are derived. Otherwise, the system of equations is treated numerically, and how the up and down movement of the porous slider contributes to the flow field as well as to the lift and drag properties are studied in detail. It is found that flat slider expansion leads to suppression of lift and drag, with an opposite impact of slider contraction. The former is of practical significance for encountering less frictional resistance while operating the slider.

Transition boundary between regular and Mach reflections for a moving shock interacting with a wedge in inviscid and polytropic air

The transition boundary separating the regions of regular and Mach reflections for a planar shock moving in argon and interacting with an inclined wedge in a shock tube is investigated using flow-field simulations produced by high-resolution computational fluid dynamics (CFD). The transition boundary is determined numerically using a modern and reliable CFD algorithm to solve Euler’s inviscid equations of unsteady motion in two spatial dimensions with argon treated as a polytropic gas. This numerically computed transition boundary for inviscid flow, without a combined thermal and viscous boundary layer on the wedge surface, is determined by post-processing many closely stationed flow-field simulations to accurately determine the transition-boundary point when the Mach stem of the Mach-reflection pattern just disappears, and this pattern then transcends into that of regular reflection. The new numerical transition boundary for argon is shown to agree well with von Neumann’s closely spaced sonic and extreme-angle boundaries for weak incident shock Mach numbers from 1.0 to 1.55, but it deviates upward and above the closely spaced sonic and extreme-angle boundaries by almost $$2^\circ $$ at larger shock Mach numbers from 1.55 to 4.0. This upward trend of the numerical transition boundary for this sequel case with monatomic gases like argon ( $$\gamma =5/3$$ ) and no boundary layer on the wedge surface (inviscid flow) is similar to the previous finding for the case of diatomic gases and air ( $$\gamma =7/5$$ ). An alternative method used to determine one point on the transition boundary between regular and Mach reflections, from a collection of Mach-reflection patterns with a constant-strength shock and different far-field wedge angles, by linear and higher-order polynomial extrapolations to zero for triple-point trajectories versus wedge angle, is compared to the present method of using near-field data that are close to and surround the new transition boundary. Such extrapolation methods are shown to yield a different transition-boundary estimate that corresponds to the mechanical-equilibrium boundary of von Neumann. Finally, the significance of the computed inviscid transition boundary between regular and Mach reflections for monatomic and diatomic gases is explained relative to the case of viscous flow with a combined thermal and viscous boundary layer on the wedge surface.

Gravity-driven flow of a shear-thinning power-law fluid over a permeable plane

The flow of a thin layer of power-law fluid on a porous inclined plane is considered. The unsteady equations of motion are depth-integrated according to the von Karman momentum integral method. The variation of the velocity distribution with the depth is accounted for, and it is furthermore assumed that the flow through the porous medium is governed by the modified Darcy’s law. The stability condition is deduced considering the hierarchy of kinematic and gravity waves. The response of the linearized model to a Dirac-delta disturbance in unbounded domain is analytically deduced, in both stable and unstable conditions 1624 C. Di Cristo, M. Iervolino and A. Vacca of flow. The influence of the effect of power-law exponent and bottom permeability on the disturbance propagation is finally analyzed, suggesting indications about the choice of the bottom permeability in order to improve the performance of industrial processes.

Asymptotic theory of turbulent bluff-body separation: A novel shear layer scaling deduced from an investigation of the unsteady motion

A rational treatment of time-mean separation of a nominally steady turbulent boundary layer from a smooth surface in the limit Re→∞, where Re denotes the globally defined Reynolds number, is presented. As a starting point, it is outlined why the “classical” concept of a small streamwise velocity deficit in the main portion of the oncoming boundary layer does not provide an appropriate basis for constructing an asymptotic theory of separation. Amongst others, the suggestion that the separation points on a two-dimensional blunt body is shifted to the rear stagnation point of the impressed potential bulk flow as Re→∞—which is expressed in a previous related study—is found to be incompatible with a self-consistent flow description. In order to achieve such a description, a novel scaling of the flow is introduced, which satisfies the necessary requirements for formulating a self-consistent theory of the separation process that distinctly contrasts former investigations of this problem. As a rather fundamental finding, it is demonstrated how the underlying asymptotic splitting of the time-mean flow can be traced back to a minimum of physical assumptions and, to a remarkably large extent, be derived rigorously from the unsteady equations of motion. Furthermore, first analytical and numerical results displaying some essential properties of the local rotational/irrotational interaction process of the separating shear layer with the external inviscid bulk flow are presented.

Turbulent shear‐layer scaling in the limit of infinite Reynolds number derived from the unsteady equations of motion

AbstractBy considering the time‐mean motion, an asymptotic description of turbulent shear flows in the limit of high Reynolds numbers, conveniently carried out by adopting the method of matched asymptotic expansions, is inherently based on (a few) assumptions regarding the essential physical properties of the flow in the respective layer considered. Although this procedure has substantiated some classical results, as, for instance, the logarithmic law of the wall, it is, nevertheless, not completely satisfactory from a rational point of view: for this reason, we first demonstrate that a rigorous application of the matching principle alone is not capable of deducing the aforementioned wall law in the case of boundary layer flows by taking into account the widely accepted presumption of the so‐called ‘inviscid’ nature of the fully turbulent main portion of the shear layer. Therefore, to some extent, the usual approaches turn out to be incomplete, for (i) the motivation for the assumptions commonly employed seems not stringent and, unfortunately, (ii) are, strictly speaking, insufficient for developing a complete asymptotic theory. To be more precise, assumptions that are usually expressed by dimensional considerations can not adequately enter the matching conditions derived from the non‐dimensional form of the governing equations, which, however, is required for an asymptotic analysis. Consequently, the time‐mean scaling of the asymptotic structure of the flow is seen to result from an asymptotic analysis of the Navier–Stokes equations by, as the only fundamental assumption, considering an asymptotically slender turbulent shear layer. Amongst others, it is demonstrated briefly how a combination of multiple‐scales techniques and matched asymptotic expansions leads to the internal boundary layer splitting. Also, it is outlined how the latter also reflects the time‐mean characteristics of the flow, which have already been adopted successfully in former studies, e.g. by using rather conventional mixing length arguments. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Instability of power-law fluid flows down an incline subjected to wind stress

In this paper we investigate the effect of a prescribed superficial shear stress on the generation and structure of roll waves developing from infinitesimal disturbances on the surface of a power-law fluid layer flowing down an incline. The unsteady equations of motion are depth integrated according to the von Kármán momentum integral method to obtain a non-homogeneous system of nonlinear hyperbolic conservation laws governing the average flow rate and the thickness of the fluid layer. By conducting a linear stability analysis we obtain an analytical formula for the critical conditions for the onset of instability of a uniform and steady flow in terms of the prescribed surface shear stress. A nonlinear analysis is performed by numerically calculating the nonlinear evolution of a perturbed flow. The calculation is carried out using a high-resolution finite volume scheme. The source term is handled by implementing the quasi-steady wave propagation algorithm. Conclusions are drawn regarding the effect of the applied surface shear stress parameter and flow conditions on the development and characteristics of the roll waves arising from the instability. For a Newtonian flow subjected to a prescribed superficial shear stress, using an analytical theory, we show that the nonlinear governing equations do not admit roll waves solutions under conditions when the uniform and steady flow is linearly stable. For the case of a general power-law fluid flow with zero shear stress applied at the surface, the analytical investigation leads to a procedure for calculating the characteristics of a roll waves flow. These results are compared with those yielded by the numerical procedure.

A two-layer model for a non-Newtonian gravity current subjected to wind shear

A theoretical model is developed for the gravity current resulting from the sudden release of a fixed volume of fluid of non-Newtonian power law rheology on top of a slightly denser Newtonian fluid layer in the presence of wind stress. The model incorporates the flow of both layers and accounts for the effects of inertial and viscous forces, and is suited for moderate Reynolds number flows. The governing equations are obtained by depth-averaging the unsteady equations of motion in accordance with the von Karman's momentum integral method, and constitute a hyperbolic system of four equations for the flow rates and thicknesses of the fluid layers. Results are obtained by a well established numerical scheme for systems of nonlinear hyperbolic equations. For a particular case analytical results are obtained by employing an asymptotic matching approach. Good agreement is obtained between the numerical and analytical results. The effects of the thickness of the ambient layer, wind stress, Reynolds numbers, and rheology on the gravity current are discussed.

Models of steam force and torque of a rotor subjected to the leakage of tip clearance

The topic of this paper is focused on the models of steam force and torque of a rotor subjected to the leakage of tip clearance. The unsteady three-dimensional Navier-Stokes solver is used to simulate the flow field for a rotating blade row with a tip clearance. The basic algorithm time marches the fully three-dimensional unsteady equations of motion expressed in a finite volume form. The Baldwin-Lomax model is used as turbulence model. By simulating the complex flow field including oscillating tip clearance with rotor vibration, the instantaneous asymmetry and oscillatory pressures acting on the surfaces of the blade row can be acquired, and the instantaneous steam force and torque acting on the rotor can be calculated. A new nonlinear model of steam force is created in this paper, and a model of steam torque excited by leakage is also reported.

A model for two-layer moderate Reynolds number flow down an incline

In this paper we investigate the gravity current resulting from the spread of a fluid of non-Newtonian power law rheology down an incline of constant slope under a less dense shallow Newtonian fluid layer. The theoretical model employed in this investigation accounts for the effect of both inertial and viscous forces, and is suited for moderately high Reynolds number flows down gentle slopes. The governing equations are obtained from the unsteady equations of motion by applying von Kármán's momentum integral method. Results are obtained by a well-established numerical scheme for systems of non-linear hyperbolic conservation laws. For a particular case analytical results are obtained by employing an asymptotic matching approach. A good agreement is obtained between the numerical and analytical results. The effects of the thickness of the ambient layer, Reynolds number, and rheology on the gravity are discussed.

Numerical study of internal wave–wave interactions in a stratified fluid

A finite volume method is used to study the generation, propagation and interaction of internal waves in a linearly stratified fluid. The internal waves were generated using single and multiple momentum sources. The full unsteady equations of motion were solved using a SIMPLE scheme on a non-staggered grid. An open boundary, based on the Sommerfield radiation condition, allowed waves to propagate through the computational boundaries with minimum reflection and distortion. For the case of a single momentum source, the effects of viscosity and nonlinearity on the generation and propagation of internal waves were investigated.Internal wave–wave interactions between two wave rays were studied using two momentum sources. The rays generated travelled out from the sources and intersected in interaction regions where nonlinear interactions caused the waves to break. When two rays had identical properties but opposite horizontal phase velocities (symmetric interaction), the interactions were not described by a triad interaction mechanism. Instead, energy was transferred to smaller wavelengths and, a few periods later, to standing evanescent modes in multiples of the primary frequency (greater than the ambient buoyancy frequencies) in the interaction region. The accumulation of the energy caused by these trapped modes within the interaction region resulted in the overturning of the density field. When the two rays had different properties (apart from the multiples of the forcing frequencies) the divisions of the forcing frequencies as well as the combination of the different frequencies were observed within the interaction region.The model was validated by comparing the results with those from experimental studies. Further, the energy balance was conserved and the dissipation of energy was shown to be related to the degree of nonlinear interaction.

Unsteady wave flows of a gas suspension with account for phase changes

A system of equations of unsteady motion of a gas suspension with account for heat and mass transfer is presented. As a result of the computational experiment, the effect of the gas vibrations at the channel inlet on the motion of the gas suspension and the rate of heat and mass transfer is studied. Specifically, it is shown that, in the considered parametric region, the gas vibrations enhance the interphase heat and mass transfer.

Oscillations of a viscous fluid in a thin layer between two coaxial disks rotating together

A study is made of the problem of the motion of an incompressible viscous fluid in the space between two coaxial disks rotating together with constant angular velocity under the assumption that the pressure changes in time in accordance with a harmonic law. The problem is solved using the equations of unsteady motion of an incompressible viscous fluid in a thin layer. It is shown that the velocity field in this case is a superposition on a steady field of damped oscillations with cyclic frequency equal to twice the angular velocity of the disks and forced oscillations with cyclic frequency equal to the cyclic frequency of the oscillations of the pressure field. It is shown that the amplitude of the forced oscillations of the velocity field depends strongly on the ratio of the cyclic frequency of the oscillations of the pressure field to the angular velocity of the disks. It is shown that there is a certain value of the ratio at which the amplitude of the forced oscillations has a maximal value (resonance). It is shown that even for very small amplitudes of the pressure oscillations the amplitude of the oscillations of the relative velocity at resonance may reach values comparable with the mean velocity of the main flow.

Numerical Solution of Flow Near the Rotor of a Wind Turbine

A numerical procedure based on the full Navier-Stokes equations as applied to the flow near a wind turbine rotor is developed. The flow is assumed axisymmetric, and the unsteady equations of motion are cast in terms of a stream function, one vorticity component, and the peripheral velocity. The vorticity equation and the peripheral momentum equation are solved by an alternating difference implicit technique, and the Poisson equation for the stream function is solved by direct matrix reduction. The rotor is modeled as an actuator disk, and the direct simulation of a given actual wind turbine rotor is considered in detail. Turbulent transport is modeled by an integrated turbulent kinetic energy equation with a simple extension to represent the effects of swirl. Comparison with field measurements on a horizontal-axis, three-bladed Elektro 10-kW machine at a station right behind the rotor shows that a good prediction of the radial distribution of the axial velocity was obtained.

Numerical solution of a body-propeller combination flow including swirl and comparisons with data

A numerical procedure based on the full Navier-Stokes equations as applied to the flow near a body/propeller system is developed. The flow is assumed axisymmetric, and the unsteady equations of motion are cast in terms of a stream function, one vorticity component, and the peripheral velocity. The voricity equation and the peripheral momentum equation are solved by an Alternating Difference Implicit technique, and the Poisson equation for the stream function is solved by direct matrix reduction. The propeller is modeled as a an actuator disk, and the direct simulation of a given actual propeller is considered in detail. Turbulent transport is modeled by an integrated Turbulent Kinetic Energy equation with a simple extension to represent the effects of swirl. Detailed comparisons with wind tunnel measurements at x/D = 2 show a good prediction of the axial velocity but an underprediction of the swirl level.

Effect of a Plane Wall on the Impulsive Motion of a Sphere in a Viscous Fluid

The impulsive motion of a sphere along a plane wall in a viscous fluid is treated on the basis of the Stokes approximation. A solution to the equations for unsteady motion is obtained by applying the Laplace transformation with respect to time to the basic equations. Results obtained to a first approximation for the drag and the torque acting on the sphere show that: \renewcommand\labelenumi\arabicenumi.

Impulsive Motion of a Circular Cylinder in a Viscous Fluid at Small Reynolds Numbers

The drag of a circular cylinder set in uniform motion impulsively from rest in a viscous fluid is calculated. On the assumption of small Reynolds numbers, Oseen's equations for unsteady motion are employed. The two limiting cases corresponding to the initial and final periods, respectively, are investigated.