Arbitrary Pressure Gradient Research Articles

INVERSE APPROACH SOLUTION FOR TURBULENT BOUNDARY LAYERS SUBJECTED TO ARBITRARY PRESSURE GRADIENTS

This paper deals with the study of the turbulent boundary layer development on two dimensional flows subjected to steady incompressible oncoming stream. In the turbulent flow, the boundary layer is divided into two layers, a wall layer and an outer layer. It is well known that the wall layer is universal but the outer layer depends on the pressure gradient. The outer layer is modeled as a nonlinear turbulent wake and consequently a constant eddy viscosity model for Clauser is adopted. The outer layer problem is shown to be governed by the classical Falkner – Skan equation for the laminar wake, the difference being in a turbulent condition which requires a finite stress at its inner boundary instead of zero stress in laminar flows. For the computations it was convenient to convert the above direct problem (for a given stress, find the slip velocity at the wall) to an inverse problem (given the slip velocity at the wall find the stress). Riley & Weidman (1989) dealt with the solution of the above inverse problem, but had not however covered the entire range of solutions. In the present work a much wider range solution to the inverse problem using a finite domain transformation is obtained by Runge-Kutta method. The results are displayed graphically and discussed critically. The solution of the inner and outer layers put together describes the turbulent boundary layer subjected to arbitrary pressure gradients.

Unsteady flow of viscoelastic fluid with fractional Maxwell model in a channel

The unsteady flow of viscoelastic fluid with the fractional derivative Maxwell model (FDMM) in a channel is studied in this note. The exact solutions are obtained for an arbitrary pressure gradient by means of the finite Fourier cosine transform and the Laplace transform. Two special cases of pressure gradient are discussed. Some results given by the classical models with integer-order are included in this note.

Thermal Characteristics of Mixed Electroosmotic and Pressure-Driven Microflows

Analytical solutions for the temperature distribution, heat transfer coefficient, and Nusselt number of steady electroosmotic flows with an arbitrary pressure gradient are obtained for two-dimensional straight microchannels. The thermal analysis considers interaction among advective, diffusive, and Joule heating terms in order to obtain the thermally developing behavior of mixed electroosmotic and pressure-driven flows with isothermal boundary conditions. Heat transfer characteristics are obtained for low Reynolds number microflows where the viscous and electric field terms are very dominant. The electroosmotic component of the flow velocity is modelled with Helmholtz-Smoluchowski slip velocity, and the mixed flow velocity is presented as linear superposition of pure electroosmotic velocity and plane Poiseuille velocity. In mixed flow cases, the governing equation for energy is not separable in general. Therefore, we introduced a method that considers the extended Graetz problem. Analytical results show that the heat transfer coefficient of mixed electroosmotic and pressure-driven flow is smaller than that of pure electroosmotic flow. For the parameter range studied here (Re < 0.7), the fully developed Nusselt number is independent of the thermal Peclet number for both pure electroosmotic and mixed electroosmotic-pressure driven microflows. In mixed electroosmotic and pressure-driven flows, the thermal entrance length increases with the imposed pressure gradient.

Heat Transfer Characteristics of Mixed Electroosmotic and Pressure Driven Micro-Flows

We analyze heat transfer characteristics of steady electroosmotic flows with an arbitrary pressure gradient in two-dimensional straight microchannels considering the effects of Joule heating in electroosmotic pumping. Both the temperature distribution and local Nusselt number are mathematically derived in this study. The thermal analysis takes into consideration of the interaction among advective, diffusive, and Joule heating terms to obtain the thermally developing behavior. Unlike macro-scale pipes, axial conduction in micro-scale cannot be negligible, and the governing energy equation is not separable. Thus, a method that considers an extended Graetz problem is introduced. Analytical results show that the Nusselt number of pure electrooosmotic flow is higher than that of plane Poiseulle flow. Moreover, when the electroosmotic flow and pressure driven flow coexist, it is found that adverse pressure gradient to the electroosmotic flow makes the thermal entrance length smaller and the heat transfer ability stronger than pure electroosmotic flow case.

A New Solution of a Nonlinear Model of Upwelling

Abstract A two-dimensional, frictionless, nonlinear model of coastal upwelling is reexamined. The model has been solved previously at steady state and as an initial-value problem. The previous solution to the initial-value problem is inconsistent with the steady-state solution. A new solution to the spinup problem is presented that approaches the existing steady-state solution. In the new solution, a surface equatorward jet develops more rapidly than a poleward undercurrent, but the surface jet is of limited strength so that the undercurrent velocity eventually surpasses that of the surface flow. Consideration of dimensional scales implies that the magnitude of the wind stress determines how quickly steady state is approached but does not affect the steady-state fields. Exact solutions found with an arbitrary alongshore pressure gradient imply that there is no poleward flow without a poleward pressure gradient.

Effect of Initial Temperature Profile on Recovery Factor Computations

Effect of Initial Temperature Profile on Recovery Factor Computations

Pressure Gradients in the Regenerator and Overall Pulse-Tube Refrigerator Performance

Significant pressure drops in the regenerator are typical in pulse-tube cryocoolers, with significant impact on performance. Irreversibilities due to viscous friction obviously lower efficiency, but in the pulse tube, this is not necessarily the most crucial issue. Indeed, by virtue of having only one driven element (the compressor), the pulse tube is a rather inflexible device from a design standpoint. Pressure and velocity amplitudes and phases determine energy fluxes. Impedances ultimately determine how large these fluxes are, hence how good a refrigerator a given design will produce. Impedance values are determined by the volume distribution, the orifice resistance, and the effect of viscous friction in the regenerator. The focus is on friction, which is difficult to deal with, especially if the device includes a bypass. An asymptotically consistent analysis has been developed, in which the regenerator is represented as an arbitrary porous medium. In contrast with most models, the analysis initially assumes arbitrary large pressure gradients and, of course, arbitrarily large temporal pressure fluctuations. The model thus obtained shows that when pressure differences due to viscous friction are comparable with the amplitude of temporal variations, viscous irreversibilities are much larger than the thermal ones. The regenerator formulation is then incorporated within a small-amplitude, harmonic model of the overall device, including the bypass, if any. For simple assumptions with respect to the temperature profile along the regenerator, such as linear and exponential profiles, closed-form solutions are obtained. Finally, the results are analyzed and their relevance is discussed.

On the streamwise evolution of turbulent boundary layers in arbitrary pressure gradients

A new approach to the classic closure problem for turbulent boundary layers is presented. This involves, first, using the well-known mean-flow scaling laws such as the log law of the wall and the law of the wake of Coles (1956) together with the mean continuity and the mean momentum differential and integral equations. The important parameters governing the flow in the general non-equilibrium case are identified and are used for establishing a framework for closure. Initially closure is achieved here empirically and the potential for achieving closure in the future using the wall-wake attached eddy model of Perry &amp; Marusic (1995) is outlined. Comparisons are made with experiments covering adverse-pressure-gradient flows in relaxing and developing states and flows approaching equilibrium sink flow. Mean velocity profiles, total shear stress and Reynolds stress profiles can be computed for different streamwise stations, given an initial upstream mean velocity profile and the streamwise variation of free-stream velocity. The attached eddy model of Perry &amp; Marusic (1995) can then be utilized, with some refinement, to compute the remaining unknown quantities such as Reynolds normal stresses and associated spectra and cross-power spectra in the fully turbulent part of the flow.

압력구배가 변하는 표면 위의 Bypass 천이 유동의 예측

A modified k-ε model is proposed for calculation of transitional boundary-layer flows with changing pressure gradient. In order to develop the model for this problem, the flow is divided into three regions; pre-transition region, transition region and fully turbulent region. The effect of pressure gradient is taken into account in stream-wise intermittency factor, which bridges the eddy-viscosity models in the pre-transition region and the fully turbulent region. From intermittency data in various flows, Narashima's intermittency function, F(ｒ), has been found to be proportional to xⁿ according to the extent of pressure gradient. Three empirical correlations of intermittency factor being analyzed, the best one was chosen to calculate three benchmark cases of bypass transition flows with different<br/> free-stream turbulence intensity under arbitrary pressure gradient. It was found that the variations of skin friction and shape factor as well as the profiles of mean velocity in the transition <br/>

New evolution equations for turbulent boundary layers in arbitrary pressure gradients

A new approach to the classical closure problem for turbulent boundary layers is presented. This involves using the well-known mean-flow scaling laws such as Prandtl’s law of the wall and the law of the wake of Coles together with the mean continuity and the mean momentum differential and integral equations. The important parameters governing the flow in the general non-equilibrium case are identified and are used for establishing a framework for closure. Initially, closure is done here empirically from the data but the framework is most suitable for applying the attached eddy hypothesis in future work. How this might be done is indicated here.

Prediction of Transpired Turbulent Boundary Layers With Arbitrary Pressure Gradients

An integral method, using Coles combined inner and outer law as the velocity profile, is developed for calculation of turbulent boundary layers with blowing or suction and pressure gradients. The resulting ordinary differential equations are solved numerically for the distribution of skin friction coefficient and integral thickness along the surface. Comparisons of predicted skin friction coefficients with experimental data are made for a wide range of blowing and suction rates and for various pressure gradients, including adverse, zero and a strong favorable gradient. In addition to good agreement with experimental data for constant blowing fractions F, the method is also successfully tested on cases where the blowing fraction is variable with position. Predictions, in general, exhibit satisfactory agreement with the data. The integral method predictions are comparable to, or better than, a number of finite difference procedures in a limited number of cases where comparisons were made.

Transport in mixed ionic electronic conductor solid oxide membranes with porous electrodes: Large pressure gradient

We extend the model of a mixed ionic electronic conductor (MIEC) oxygen perm-selective solid oxide membrane with porous electrodes, previously treated in the limit of small pressure gradient, to the case of arbitrary pressure gradient. The model is based on the assumption that the oxygen vacancy concentration varies only weakly with gas pressure and that the surface exchange reaction rate varies as √ p where p is the oxygen gas pressure. While the model applies specifically to SrCo 0.8Fe 0.2O 3 − δ it is expected to be applicable also to other mixed ionic electronic conductor materials with high acceptor dopant concentrations. Numerical solution of the diffusion-reaction equations, in the case of a large pressure gradient, show that the enhancement in oxygen permeability due to the porous electrodes scales with the simple analytical expression for the flux enhancement obtained in the limit of a small pressure gradient. Enhancement of over two orders of magnitude in oxygen flux are predicted for membranes with large surface area porous electrodes.

On finite-length cracks and inclusions in poroelasticity

A procedure is devised for the asymptotic solution of finite-length crack and inclusion problems under arbitrary time-dependent stress, pore pressure or pressure gradient loadings in porous elastic materials. The stress-intensity factors are identified explicitly as an expansion in real time for model problems involving uniformly loaded cracks. In the most general case this involves matched asymptotic expansions and rescalings that take advantage of the singular perturbation nature of the governing equations. The problems are written in terms of coupled Cauchy singular integral equations containing a small parameter ; these can be combined to form a single subsidiary integral equation. This approach is used to split the problems into pieces driven by elastic, or pore-pressure, dominated solutions. Outer solutions that are driven by elastic solutions and eigensolutions are identified ; these are matched to inner problems. The inner problems require the solution of a system of coupled integral equations ; these are formulated in terms of functional equations. Fortunately the system uncouples and allows the inner solutions to be found explicitly. The pore-pressure driven pieces are deduced, to leading order, by a direct rescaling.

An experimental study on the transitional boundary layer developing on NACA0012 airfoil

A study on the transitional boundary layer with arbitrary pressure gradient under various upstream conditions is very important for engineering applications like the performance predictions of the turbomachineries under various and strong disturbances. Experimental data on the transitional boundary layer for real cascades of the turbomachinery are rare because of difficulties in boundary layer experiments. Flow on NACA0012 airfoil is more similar to the real case than that on the flat plate with which many researches are done. The data of the transitional flow on the airfoil could be used to verify or to develop a turbulence model for numerical simulations. The experiment was performed with two cases of Reynolds number at a=0 and one case of Reynolds number at a=5 . The measured data are the transition length and the wall shear stresses. These two characteristic values are measured within 25%~90% of the airfoil chord by Computation Preston tube Method(CPM) proposed by Nitsche et al.(1983). At a=0 , transition occured at 70% and 55% of chord length when R =6*10 and 8* 10 , respectively. It started when R {\theta}=500 regardless of R , and ended when R {\theta}=750, and 850 respectively. The transition length was 15~20% of the chord length. At a=5 (R =6*10 ), boundary layer on the pressure side does not undergo transition, but on the suction side transition occured at .chi./c=0.16 and ended at .chi./c=0.22.c//c=0.22./c=0.22.c//c=0.22.

A wall-wake model for the turbulence structure of boundary layers. Part 1. Extension of the attached eddy hypothesis

The attached eddy hypothesis developed for zero pressure gradient boundary layers and for pipe flow is extended here to boundary layers with arbitrary streamwise pressure gradients, both favourable and adverse. It is found that in order to obtain the correct quantitative results for all components of the Reynolds stresses, two basic types of eddy structure geometries are required. The first type, called type-A, is interpreted to give a ‘wall structure’ and the second, referred to as type-B, gives a ‘wake structure’. This is in analogy with the conventional mean velocity formulation of Coles where the velocity is decomposed into a law of the wall and a law of the wake.If the above mean velocity formulation is accepted, then in principle, once the eddy geometries are fixed for the two eddy types, all Reynolds stresses and associated spectra contributed from the attached eddies can be computed without any further empirical constants. This is done by using the momentum equation and certain convolution integrals developed here based on the attached eddy hypothesis. The theory is developed using data from equilibrium and quasi-equilibrium flows. In Part 2 the authors’ non-equilibrium data are used.

Wall turbulence closure based on classical similarity laws and the attached eddy hypothesis

A new look at the closure problem of turbulent boundary layers is taken here using recently derived analytical expressions for the shear stress distributions. These expressions are based on the logarithmic law of the wall and law of the wake formulation of Coles [J. Fluid Mech. 1, 191 (1956)] with the mean continuity and the mean momentum differential and integral equations. The concept of equilibrium layers of Clauser [Adv. Mech. 4, 1 (1956)] is extended and using similar ideas as Rotta [Prog. Aeronaut. Sci. 2, 1 (1962)] for self-similarity, a closure scheme is proposed for layers developing in arbitrary pressure gradients for the case where the streamwise derivative of the Coles wake factor is not too large. For a given flow case, this Coles wake condition can be tested with internal consistency checks. The mathematical framework is most suitable for incorporating Townsend’s attached eddy hypothesis as recently developed by Perry, Li, and Marušić [Philos. Trans. R. Soc. London Ser. A 336, 67 (1991)] for closure. This gives an opportunity to incorporate coherent structure concepts into closure schemes. Possible ways of handling the difficult case where the streamwise derivative of the Coles wake factor is significant are discussed.

Eigen-Functions of Linearized Unsteady Boundary Layer Equations

The Lam and Rott theory of linearized unsteady boundary layers is revisited, and some new results are obtained. The exact outer eigen-solution for a flat plate found in the original paper is shown to be a special case of the Prandtl-Glauert transposition theorem. The streamwise coordinate-dependent factor of the inner eigen-solutions, first found by M. E. Goldstein for the flat plate, is generalized for arbitrary pressure gradients.

The Role of Laminar-Turbulent Transition in Gas Turbine Engines: A Discussion

An extended discussion of Mayle’s (1991) critical study of transition phenomena in gas turbine engines is presented. Attention is focused on transition in decelerating flow regions, which are the major sources of loss production for axial turbomachine blades. The following points are examined in detail: (a) the physics of transition and its implications for the correlation of various transition phenomena; (b) the relative importance of pressure gradient and free-stream turbulence in controlling transition; (c) the influence of pressure gradient on periodic-unsteady transition; (d) the correlation of transition length under conditions of arbitrary pressure gradient and free-stream turbulence level; and (e) transition behavior in laminar separation bubbles. The discussion examines various differences in philosophy concerning the above phenomena and makes further suggestions for transition research, which may assist in resolving the issues raised.

Unsteady conducting dusty gas flow through a circular pipe in the presence of an applied and induced magnetic field

The unsteady motion of an electrically conducting dusty gas flowing through a circular pipe in the presence of a non-uniform imposed magnetic field is studied. The induced magnetic field is assumed non-negligible. Analytical expressions for the gas and dust velocities and magnetic field are obtained in the case where the dusty gas is set into motion by an arbitrary pressure gradient. The wall of the pipe is assumed non-conducting. The particular case when the pressure decreases exponentially in time is examined.

Unsteady flow of an elastico-viscous fluid in tubes of uniform cross-section

In this paper, the uniqueness of solution for internal bounded unsteady flows of a shortmemory fluid is first established. Closed-form solutions are then obtained for the equations characterizing flows of such fluids in circular and rectangular tubes of uniform cross-section under an arbitrary pressure gradient. Special cases including the oscillatory flow between two parallel plates are discussed.