Arbitrary Pressure Gradient Research Articles

Viscous interlayer structure and transport properties in von Kármán swirling flows

This paper is a discussion of the thin viscous regions that separate the nearly inviscid cellular domains arising at large Reynolds number in von Kármán swirling flow. A review and extension are given of previous analyses of the inviscid structure and the associated critical-layer structure that occurs in the regime of small positive radial pressure gradients. It is first shown that the inviscid perturbation analysis can be cast into an analytical closed form, from which an explicit expression for the first-order viscous correction at arbitrary pressure gradients is obtained. By then allowing for negative pressure gradients, of the kind associated with weakly stagnating flows, the present analysis serves to describe the transition from the Prandtl-layer structure characteristic of stagnation-dominated flows to a diffuse viscous damping associated with rotation-dominated flows. Finally, it is shown that, in a discretely stratified flow, the hydrodynamic transition to the critical-layer structure involves a hydrodynamic ‘‘freezing’’ of the interface, which is accompanied by an interesting transition in the interfacial mass-transport characteristics, with the asymptotic Nusselt number changing from (Pe)1/2 abruptly to (Pe)1/3 dependency on the Péclet number Pe.

Eigenvalues of the time—dependent fluid flow problem I

The direct and inverse boundary value problems for the linear unsteady viscous fluid flow through a closed conduit of a circular annular cross‐section Ω with arbitrary time‐dependent pressure gradient under the third boundary conditions have been investigated.

Wall-wakes in moderate adverse pressure gradients

The flow past an obstacle attached perpendiculary to a surface (wall) is postulated to possess the characteristics of boundary layer flow in the vicinity of the wall and free-wake flow away from the wall. Analytical analyses of wall-wake flows with adverse pressure gradient (without causing separaton of the boundary layer) are made to obtain expressions for variations of free stream velocity and the chatacteristic length and velocity scales of the self preserving flow. Experiments are conducted in a wind tunnel for three arbitrary adverse pressure gradient flows. The experimental observations clearly indicate that wall-wakes in adverse pressure gradient can be adequately described by the two-layer model proposed.

Interaction of peristaltic flow with pulsatile flow in a circular cylindrical tube

The effect of pulsatile flow on peristaltic transport in a circular cylindrical tube is analysed. The flow of a Newtonian viscous incompressible fluid in a flexible circular cylindrical tube on which an axisymmetric travelling sinusoidal wave is imposed, is considered. The initial flow in the tube is induced by an arbitrary periodic pressure gradient. A perturbation solution with amplitude ratio (wave amplitude/tube radius) as a parameter is obtained when the frequency of the travelling wave and that of the imposed pressure gradient are equal. The interaction effects of periodic wall induced flow and periodic pressure imposed flow are visualized through the presence of substantially different components of steady and higher harmonic oscillating flow in the first order flow solution. Numerical results show a strong variation of steady state velocity profiles with boundary wave number and Reynolds number and a strong phase shift behaviour of the flow in the radial direction.

Structure of the transient wall-friction law in one-dimensional models of laminar pipe flows

The problem of describing an unsteady cylindrical pipe flow with one-dimensional equations is investigated, and an exact method for obtaining a closure relationship is proposed for the transient shear stress in a laminar flow submitted to an arbitrary transient pressure gradient. Extensive comparisons are given for a step or a harmonic pressure gradient between the approximate solution derived from this method, some results of the literature and exact solutions of the Navier–Stokes equations.

Model for Unsteadiness in Lateral Dynamics for Use in Parameter Estimation

Cohen, C.B. and Reshotko, E., The Compressible Laminar Boundary Layer with Heat Transfer and Arbitrary Pressure Gradient, NACA Rept. 1294, 1956. Schlichting, H., Boundary Layer Theory, Sixth Ed., McGraw Hill Book Co., New York, 1968. McGhee, R.J. and Beasley, W.D., Low-Speed Aerodynamic Characteristics of a 17-Percent-Thick Airfoil Section Designed for General Aviation Application, NASA TN D-7428, 1973.

Flow of a Conducting Fluid with Suspension of Particles in Cylinders with Arbitrary Time Varying Pressure Gradient

The problem of unsteady flow of a viscous incompressible conducting fluid containing a dilute suspension of small inert shperical particles in cylinders with circular and sectorial cross-sections, in the presence of a radial magnetic field, have been discussed. The analysis applies to flows with pressure gradients which are arbitrary functions on time. Explicit expressions for the exact velocities of the fluid and that of the particles are obtained by using the methods of operational calculus-the Laplace transform, the finite Hankel transforms etc. Numerical results have been obtained for the developing flow due to a constant pressure gradient, when the flow is taking place in a coaxial circular cylinder. The effects of mass concentration and relaxation time of particles and the magnetic field on the velocities have also been discussed.

Unsteady flow of an inelastic power-law fluid in a circular tube

Flow of a power-law fluid with a suddenly applied arbitrary time-dependent pressure gradient in a horizontal circular tube is presented. The momentum equations are solved numerically using a implicit finite-difference technique for the time-dependent velocity profiles for both start up and oscillating pressure gradients. Dimensionless curves are presented showing velocity profile development and phase angle variation for a variety of power-law index values.

On the unsteady flow of viscous fluid between two plates

A solution is found for Navier-Stokes equations for the unsteady laminar flow of a viscous incompressible fluid between two horizontal parallel plates, the upper one is moving with an arbitrary velocityU(t) and the other at rest, under the action of an arbitrary pressure gradient.

Boundary Layer Calculation for Analysis and Design

A simple, semiempirical method for calculating the laminar, transition, and turbulent boundary layer with arbitrary free stream pressure gradient is developed. Good correlation is obtained with data on general two dimensional turbulent flows, diffuser flows, and the cylinder in cross-flow. However only for the diffuser has the boundary layer flow been coupled with the potential core so that only the inlet conditions and geometry are required. In other cases the free stream velocity distribution must be known or calculable. Skin friction coefficient, momentum thickness Reynolds number, and free stream pressure gradient parameter correlation employs a simple lag theory. With the integral momentum equation the complete boundary layer parameters are obtained as functions of the distance along a surface.

Convective heat transfer through boundary layers with arbitrary pressure gradient and non-isothermal surfaces

The solution method introduced by Chao and Cheema in their study of forced convection wedge flow with non-uniform surface temperature has been generalized to treat general two-dimensional and axisymmetric boundary-layer flows with non-uniform surface temperature. By the introduction of appropriate transformation variables, the equations for the temperature profile and the local wall heat flux can be expressed explicitly in terms of the Prandtl number and the wedge parameter for a step discontinuity in wall temperature. Numerical examples for an isothermal surface and for a wall temperature step change for a circular cylinder are given and are compared with values obtained from other formula available in the literature.

Hydromagnetic laminar flow of a conducting fluid in an annulus in presence of a radial magnetic field

The object of the present paper is to study the MHD effects on the laminar flow of a viscous, incompressible and conducting fluid in an annulus with arbitrary time-varying pressure gradient and arbitrary initial velocity in presence of a radial magnetic field. Using finite Hankel transform, solutions for both the unsteady and steady flows under different prescribed pressure gradients have been found out.

Measurements in adverse-pressure-gradient turbulent boundary layers with a step change in surface roughness

The response of turbulent boundary layers to sudden changes in surface roughness under adverse-pressure-gradient conditions has been studied experimentally. The roughness used was in the ‘d’ type array of Perry, Schofield & Joubert (1969). Two cases of a rough-to-smooth change in surface roughness were considered in the same arbitrary adverse pressure gradient. The two cases differed in the distance of the surface discontinuity from the leading edge and gave two sets of flow conditions for the establishment and growth of the internal layer which develops downstream from a change in surface roughness. These conditions were in turn different from those in the zero-pressure-gradient experiments of Antonia & Luxton. The results suggest that the growth of the new internal layer depends solely on the new conditions at the wall and scales with the local roughness length of that wall. Mean velocity profiles in the region after the step change in roughness were accurately described by Coles’ law of the wall-law of the wake combination, which contrasts with the zero-pressure-gradient results of Antonia & Luxton. The skin-friction coefficient after the step change in roughness did not overshoot the equilibrium distribution but made a slow adjustment downstream of the step. Comparisons of mean profiles indicate that similar defect profile shapes are produced in layers with arbitrary adverse pressure gradients at positions where the values of Clauser's equilibrium parameter β (= δ*τ−10dp/dx) are similar, provided that the pressure-gradient history and local values of the pressure gradient are also similar.

High Angle-of-Attack Aerodynamics on a Slender Body with a Jet Plume

Turbulent Boundary Layer on a Smooth Flat Plate With and Without Heat Transfer, Journal of Fluid Mechanics, Vol. 18, Pt. 1, Jan. 1964, pp. 117-143. * Reshotko, E. and Tucker, M., Approximate of the Compressible Turbulent Boundary Layer With Heat Transfer and Arbitrary Pressure Gradient, TN 4154, 1957, NACA. 12 Dhawan, S. and Narasimha, R., Some Properties of Boundary Layer Flow During the Transition From Laminar to Turbulent Motion, Journal of Fluid Mechanics, Vol. 3, No. 4, April 1958, pp. 418-436. 13 Maslen, S. H., Inviscid Hypersonic Flow Past Smooth Symmetric Bodies, AIAA Journal, Vol. 2, No. 6, June, 1964, pp. 1055-1061. 14 Maslen, S. H., Asymmetric Hypersonic Flow, CR-2123, Sept. 1972, NASA. 15 DeJarnette, F. R., Calculation of Heat Transfer on ShuttleType Configurations Including the of Variable Entropy at Boundary Layer Edge, CR-112180, Oct. 1972, NASA. 16 Hopkins, E. J. and Inouye, M., An Evaluation of Theories for Predicting Turbulent Skin Friction and Heat Transfer on Flat Plates at Supersonic and Hypersonic Mach Numbers, AIAA Journal, Vol. 9, No. 6, June, 1971, pp. 993-1003. 17 Cleary, J. W., Effects of Angle of Attack and Bluntness on Laminar Heating-Rate Distributions of a 15° Cone at a Mach Number of 10.6, TN D-5450, 1969, NASA. 18 Hillsamer, M. E., and Rhudy, J. P., Heat-Transfer and Shadowgraph Tests of Several Elliptical Lifting Bodies at Mach 10, U.S. Air Force, AEDC-TDR-64-19,1964, Arnold Engineering Development Center, Tullahoma, Tenn.

A Reference Temperature Method for Compressible Laminar Boundary Layers

A calculation procedure for the solution of two-dimensional and axi-symmetric laminar boundary layers in compressible flow has been developed. The method is an extension of the integral approach of Tani to include compressibility effects by means of a reference temperature. Arbitrary pressure gradients and wall temperature can be specified. Comparisons with experiments obtained for supersonic flows over a flat plate indicate that the method yields adequate results. The method is then applied to the solution of the boundary layer on a Basemann inlet.

Numerical realization of an iterative method for boundary-layer equations

TWO modifications of the analytic method of successive approximation for solving the boundary-layer system of equations, which enable the numerical iterative process to be continued without limit on a computer, are explained. In this paper we describe a numerical realization of the method of successive approximations for solving the boundary-layer equations, explained previously in [1, 2]. Various iterative methods have been applied repeatedly to solve the boundary-layer equations and to prove existence theorems of the solution of such equations (see [3, 4]). The method of successive approximations considered is intended for solving the systems of equations of the laminar multicomponent boundary layer with an arbitrary pressure gradient on plane or axisymmetric bodies (possibly, taking into account chemical reactions). It defines a uniform, infinitely continuable iterative process of the form U j+1 = T( U j ), which in principle enables any approximation U 1, U 2, … to be obtained in analytic form. The method of [1, 2] contains some ideas formally similar to Targi and Shvets' approximate methods of calculating the boundary layer [5, 6]. These method, using the idea of a layer of finite thickness, construct only the first approximation, and the question of finding successive approximations is not discussed in these papers. Theoretically, successive approximations cannot be calculated, since they lead to differential equations of high order (for the thickness of the boundary layer) and require non-obvious supplementary initial conditions. For sufficiently complex problems, it is possible by the method described in [1, 2] to obtain analytically two or three approximations of high accuracy. No theoretical difficulties arise in the calculation of subsequent approximations, but the volume of technical work increases strongly. It would be considerably easier to obtain distant approximations by computer. Moreover, the computer realization and analysis of a large number of approximations may clarify the question of the convergence of the method. In section 1 the idea of the method is demonstrated by the example of a non-selfsimilar equation. Then, also by simple examples, in sections 2 and 3 two modifications of the analytic method of [1, 2], useful for the numerical realization of this method for problems of the type indicated above, are explained. Section 4, the last, contains some recommendations on the individual stages of the computer realization of the method, and also a brief description of the results of the solution of some model problems.

Study of turbulent boundary layers on smooth and rough surfaces with arbitrary pressure gradients

The results of an experimental investigation into the steady-state plane turbulent boundary layer in an incompressible liquid at an impermeable wall are presented. Cases of flow at smooth and rough surfaces in the presence of a longitudinal pressure gradient are considered. The results of measurements of the turbulent structure of the flow at various distances from the channel inlet are presented. A detailed analysis of the kinematic and dynamic characteristics of the flow is given. Special attention is paid to the boundary region of the flow close to the wall. A universal law is proposed for the variation in the local resistance coefficient along the boundary layer.

A Simple Theory for the Two-Dimensional Compressible Turbulent Boundary Layer

A new approach is proposed for analyzing the compressible turbulent boundary layer with arbitrary pressure gradient. Utilizing a compressible law-of-the-wall and a Crocco energy approximation, the new theory integrates the momentum equation across the boundary layer in terms of inner variables only. The result is a single first-order ordinary differential equation for skin friction, devoid of integral thicknesses and shape factors. When analyzed for flat plate flow, this new equation has an exact solution apparently superior in accuracy to any other flat plate theory (Table 1). The new equation also agrees well with supersonic skin friction data in both favorable and adverse pressure gradients. The new theory contains an explicit separation criterion and is the simplest and possibly most accurate existing analysis for compressible turbulent flow.

Eine näherungslösung für die kondensation von laminar strömendem dampf mit beliebigen druckgradienten bei kleiner mach-zahl und konstanten stoffwerten

Vapor and condensate flows on an arbitrary body curved in flow direction. In the condensate layer we considered only viscous forces and heat flow by conduction. In the vapor layer inertia and friction forces as well as pressure forces were considered. The temperature variation in the vapor layer was neglected. The problem was solved by an integral method. The integral equations are valid for flows with arbitrary pressure gradients with heat transfer. The velocity profile in the vapor layer was assumed to be a function of two shape parameters. Both the quantities of the boundary layer and the heat transfer coefficient were calculated from the stagnation point to the separation point. The average heat transfer coefficient was also calculated, and it includes the properties of the vapor and a heat transfer parameter which were not considered in the work of Shekriladze and Gomelauri.

Laminar Flow of an Incompressible Fluid in a Conduit With Arbitrary Cross Section, Arbitrary Time-Varying Pressure Gradient, and Arbitrary Initial Velocity

A computer-oriented solution is given for the flow described in the title of the paper. The boundary shape is represented by specification of the coordinates of N points on the boundary; the initial velocity is represented by specification of L values of the velocity in the cross section at time zero; the arbitrary time-varying pressure gradient is implemented by use of Duhamel’s Theorem. In the solution method presented, boundary and initial conditions are satisfied in the least squares sense. The Gram determinant is used to determine eigenvalues and the Gram-Schmidt orthonormalizing procedure is used to construct a set of functions appropriate for a finite series solution. Computer programs are referenced which have been used to investigate the correctness of the solution and the accuracy obtainable with reasonable digital computational time.