Lower Branch Modes of the Compressible Boundary Layer Flow Due to a Rotating-Disk

In this article, a theoretical study is pursued to investigate the structure of the lower branch neutral stability modes of three-dimensional small disturbances imposed on the compressible boundary layer flow due to a rotating-disk. Special attention is focused on to the short-wavelength stationary/nonstationary compressible crossflow vortex modes at sufficiently high Reynolds numbers with reasonably small scaled frequencies. Following closely the asymptotic framework introduced in [ 1 ] for the incompressible stationary modes, it is demonstrated here that the compressible modes having sufficiently long time scale can also be described by an asymptotic expansion procedure based on the triple-deck approach. Making use of this rational asymptotic technique, which rigorously takes into account the nonparallel effects, the asymptotic structure of the nonstationary modes is shown to be adjusted by a balance between viscous and Coriolis forces, and resulted from the fact of vanishing shear stress at the disk surface, as in the incompressible Von Karman's flow. As a consequence of matching successive regions in the asymptotic procedure, it is found that the wavenumber and the orientation of the compressible lower branch modes are governed by an eigenrelation, which is akin to the one obtained previously in [ 1 ] for the incompressible stationary mode and in [ 2 ] for the compressible stationary modes. The nonparallel influences are toward destabilizing all the modes, though the wall insulation and heating are relatively stabilizing for the modes in the vicinity of the stationary mode, unlike the wall cooling. The asymptotic compressible data obtained at high Reynolds number limit compares fairly well with the numerical results generated directly solving the linearized compressible system with usual parallel flow approximation.     

Effects of suction/blowing on the lower branch modes of the incompressible boundary layer flow due to a rotating-disk

In this study a theoretical approach is pursued to investigate the effects of suction and blowing on the structure of the lower branch neutral stability modes of three-dimensional small disturbances imposed on the incompressible von Karman’s boundary layer flow induced by a rotating-disk. Particular interest is placed upon the short-wavelength, non-linear and nonstationary crossflow vortex modes developing within the presence of suction/blowing at sufficiently high Reynolds numbers with reasonably small scaled frequencies. Following closely the asymptotic framework introduced in [1], the role of suction on the non-linear disturbances of the lower branch described first in [2] for the stationary modes only, is extended in order to obtain an understanding of the behavior of non-stationary perturbations. The analysis using the rational asymptotic technique based on the triple-deck theory enables us to derive initially an eigenrelation which describes the evolution of linear modes. The asymptotic linear modes calculated at high Reynolds number limit are found to be destabilizing as far as the non-parallelism accounted by the approach is concerned, and they compare fairly well with the numerical results generated directly by solving the linearized system with the usual parallel flow approximation. An amplitude equation is derived next to account for the effects of non-linearity. Even though the form of this equation is the same as that of found in [2] for no suction, it is under the strong influence of suction and blowing. This amplitude equation is shown to be adjusted by a balance between viscous and Coriolis forces, and it describes the evolution of not only the stationary but also the non-stationary modes for both suction and injection applied at the disk surface. A close investigation of the amplitude equation shows that the non-linearity is highly destabilizing for both positive and negative frequency waves, though finite amplitude growth of a disturbance having positive frequency close to the neutral location is more effective at destabilization of the flow under consideration. Finally, a smaller initial amplitude of a disturbance is found to be sufficient for the non-linear amplification of the modes in the case of suction, whereas a larger amplitude is required if injection is active on the surface of the disk.     

Influence of Finite Amplitude Disturbances on the Nonstationary Modes of a Compressible Boundary Layer Flow

A weakly nonlinear stability analysis is performed to search for the effects of compressibility on a mode of instability of the three-dimensional boundary layer flow due to a rotating disk. The motivation is to extend the stationary work of [ 1 ] (hereafter referred to as S90) to incorporate into the nonstationary mode so that it will be investigated whether the finite amplitude destabilization of the boundary layer is owing to this mode or the mode of S90. Therefore, the basic compressible flow obtained in the large Reynolds number limit is perturbed by disturbances that are nonlinear and also time dependent. In this connection, the effects of nonlinearity are explored allowing the finite amplitude growth of a disturbance close to the neutral location and thus, a finite amplitude equation governing the evolution of the nonlinear lower branch modes is obtained. The coefficients of this evolution equation clearly demonstrate that the nonlinearity is destabilizing for all the modes, the effect of which is higher for the nonstationary waves as compared to the stationary waves. Some modes particularly having positive frequency, regardless of the adiabatic or wall heating/cooling conditions, are always found to be unstable, which are apparently more important than those stationary modes determined in S90. The solution of the asymptotic amplitude equation reveals that compressibility as the local Mach number increases, has the influence of stabilization by requiring smaller initial amplitude of the disturbance for the laminar rotating disk boundary layer flow to become unstable. Apart from the already unstable positive frequency waves, perturbations with positive frequency are always seen to compete to lead the solution to unstable state before the negative frequency waves do. Also, cooling the surface of the disk will be apparently ineffective to suppress the instability mechanisms operating in this boundary layer flow.     

Asymptotic calculation of inviscidly absolutely unstable modes of the compressible boundary layer on a rotating disk

In this work a long-wavelength asymptotic approach is used to analyze the region of absolute instability in the compressible rotating disk boundary layer flow. Theoretically determined values of branch points for the occurrence of absolute instability in the compressible flow are shown to match onto the ones which are obtained via a numerical solution of the linear inviscid compressible Rayleigh equations.     

Compressible Modes of the Rotating-Disk Boundary-Layer Flow Leading to Absolute Instability

This work is devoted to the clarification of the viscous compressible modes particularly leading to absolute instability of the three-dimensional generalized Von Karman's boundary-layer flow due to a rotating disk. The infinitesimally small perturbations are superimposed onto the basic Von Karman's flow to achieve linearized viscous compressible stability equations. A numerical treatment of these equations is then undertaken to search for the modes causing absolute instability within the principle of Briggs–Bers pinching. Having verified the earlier incompressible and inviscid compressible results of [ 1–3 ], and also confirming the correct match of the viscous modes onto the inviscid ones in the large Reynolds number limit, the influences of the compressibility on the subject matter are investigated taking into consideration both the wall insulation and heat transfer. Results clearly demonstrate that compressibility, as the Mach number increases, acts in favor of stabilizing the boundary-layer flow, especially in the inviscid limit, as far as the absolute instability is concerned, although wall heating and insulation greatly enhances the viscous absolutely unstable modes (even more dramatic in the case of wall insulation) by lowering down the critical Reynolds number for the onset of instability, unlike the wall cooling.     

Upper branch nonstationary modes of the boundary layer due to a rotating disk

A treatment of asymptotic calculation of upper branch nonstationary instability modes is undertaken in the boundary layer flow due to a rotating disk. A numerical spectral solution of the eigenvalue problem shows good agreement with the results of a rational asymptotic approach, based on the extension of the multideck theory of [1].     

Self-Similar "Stagnation Point" Boundary Layer Flows with Suction or Injection

Multiple solutions are reported for the two-dimensional boundary layer flow of a viscous fluid near a permeable wall through which fluid is uniformly withdrawn. In the limit of large wall suction, three flows of similarity form are found: the first is the well-known monotonic solution of Terrill; the second exhibits flow reversal, with the streamlines being separated into three distinct cells; the third also exhibits flow reversal, but has multiple cells only when the fluid withdrawal speed is less than some threshold. The wall injection problem is also briefly studied, only Terrill's branch of solutions being found. Numerical and asymptotic solutions are presented and compared; the large-suction asymptotics of the third solution branch are found to be rather subtle.     

Nonlinear Stability of Nonstationary Cross-Flow Vortices in Compressible Boundary Layers

The nonlinear evolution of long-wavelength non stationary cross-flow vortices in a compressible boundary layer is investigated; the work extends that of Gajjar [1] to flows involving multiple critical layers. The basic flow profile considered in this paper is that appropriate for a fully three-dimensional boundary layer with O(1) Mach number and with wall heating or cooling. The governing equations for the evolution of the cross-flow vortex are obtained, and some special cases are discussed. One special case includes linear theory, where exact analytic expressions for the growth rate of the vortices are obtained. Another special case is a generalization of the Bassom and Gajjar [2] results for neutral waves to compressible flows. The viscous correction to the growth rate is derived, and it is shown how the unsteady nonlinear critical layer structure merges with that for a Haberman type of viscous critical layer.     

On the steady flow between a rotating and a stationary disk with a uniform suction at the stationary disk

Summary The steady axisymmetric flow of a viscous incompressible fluid between two coaxial disks, one rotating and the other stationary, with uniform suction at the stationary disk is discussed. Similarity solutions of Navier-Stokes equations are obtained by the method of regular perturbation for small suction Reynolds number and numerically for an arbitrary suction Reynolds number by a series expansion method and also by integrating directly by a Newton-Raphson technique. It is found that for a fixed rotational Reynolds number and varying suction Reynolds number there is an overall increase in the velocity components as the suction increases.

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Zusammenfassung Es wird die stationäre axisymmetrische Strömung zwischen zwei koaxialen Platten untersucht, von denen die eine rotiert und die andere stationär ist, wobei an der stationären Platte eine gleichförmige Absaugung stattfindet. Ähnlichkeitslösungen der Navier-Stokesschen Gleichungen werden erhalten, durch eine reguläre Störungsrechnung bei kleinen Absaug-Reynoldszahlen, während bei beliebigen Werten dieses Parameters numerische Resultate sowohl durch eine Reihenentwicklung wie auch durch direkte Integration mit Hilfe der Newton-Raphson-Technik gefunden werden. Bei fester Rotations-Reynoldszahl und variabler Absaugung findet man eine allgemeine Erhöhung der Geschwindigkeits-komponenten mit der Absaugung.

     

Wall effects on the slow steady motion of a particle in a viscous incompressible fluid

In this paper, asymptotic expansions with respect to small Reynolds numbers are proved for the slow steady motion of an arbitrary particle in a viscous, incompressible fluid past a single plane wall. The flow problem is modelled by a certain boundary value problem for the stationary, nonlinear Navier-Stokes equations. The coefficients of these expansions are the solutions of various, linear Stokes problems which can be constructed by single layer potentials and corresponding boundary integral equations on the boundary surface of the particle. Furthermore, some asymptotic estimates at small Reynolds numbers are presented for the slow steady motion of an arbitrary particle in a viscous, incompressible fluid between two parallel, plane walls and in an infinitely long, rectilinear cylinder of arbitrary cross section. In the case of the flow problem with a single plane wall, the paper is based on a thesis, which the author has written under the guidance of Professor Dr. Wolfgang L. Wendland.     

On Two-Phase Flow in a Rotating Boundary Layer

The two-phase flow induced by a rotating disk in a stationary unbounded mixture is considered. The generalized similarity assumption of von Karman reduces the averaged equations of motion with a linear drag between the phases to a system of ordinary differential equations. These are investigated by asymptotic and numerical techniques. The equations display a nontrivial behavior in a sublayer near the boundary, whose thickness is of the order of the particle size. The volume fraction of the dispersed phase is singular unless a small suction is applied on the disk or a small diffusion term is added to the continuity equations. Outside this sublayer, the velocity field is quite similar to a rescaled classical von Karman flow. Good agreement between asymptotic and numerical solution is obtained, although there is considerable stiffness in the equations. The motion of a solid particle in a von Karman flow is also discussed, but the present investigation is restricted to small radii because the shear-lift force is neglected.     

Asymptotic study of turbulent Poiseuille flow with wall transpiration

An incompressible, pressure–driven, fully developed turbulent flow between two parallel walls, with an extra constant transverse velocity component, is considered. A closure condition is formulated, which relates the shear stress to the first and second derivatives of the longitudinal mean velocity. The closure condition is derived without invoking any special hypotheses on the nature of turbulent motion, only taking advantage of the fact that the flow depends on a finite number of governing parameters. By virtue of the closure condition, the momentum equation is reduced to the boundary–value problem for a second–order differential equation, which is solved by the method of matched asymptotic expansions at high values of the logarithm of the Reynolds number based on the friction velocity. A limiting transpiration velocity is obtained, such that the shear stress at the injection wall vanishes, while the maximum point on the velocity profile approaches the suction wall. In this case, a sublayer near the suction wall appears where the mean velocity is proportional to the square root of the distance from the wall. A friction law for Poiseuille flow with transpiration is found, which makes it possible to describe the relation between the wall shear stress, the Reynolds number, and the transpiration velocity by a function of one variable. A velocity defect law, which generalizes the classical law for the core region in a channel with impermeable walls to the case of transpiration, is also established. In similarity variables, the mean velocity profiles across the whole channel width outside viscous sublayers can be described by a one–parameter family of curves. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Triple-deck theory in transonic flows and boundary layer stability

An analysis of the lower branch of the neutral curve for the Blasius boundary layer leads to a perturbed velocity field with a triple-deck structure, which is a rather unexpected result. It is the asymptotic treatment of the stability problem that has a rational basis, since it is in the limit of high Reynolds numbers that the basic flow has the form of a boundary layer. The principles for constructing a boundary layer stability theory based on the triple-deck theory are proposed. Although most attention is focused on transonic outer flows, a comparative analysis with the asymptotic theory of boundary layer stability in subsonic flows is given. The parameters of internal waves near the lower branch of the neutral curve are associated with a certain perturbation field pattern. These parameters satisfy dispersion relations derived by solving eigenvalue problems. The dispersion relations are investigated in complex planes.     

Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line

We study a large time behavior of a solution to the initial boundary value problem for an isentropic and compressible viscous fluid in a one-dimensional half space. The unique existence and the asymptotic stability of a stationary solution are proved by S. Kawashima, S. Nishibata and P. Zhu for an outflow problem where the fluid blows out through the boundary. The main concern of the present paper is to investigate a convergence rate of a solution toward the stationary solution. For the supersonic flow at spatial infinity, we obtain an algebraic or an exponential decay rate. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in the spatial asymptotic point, the solution converges to the corresponding stationary solution with the same rate in time as time tends to infinity. An algebraic convergence rate is also obtained for the transonic flow. These results are proved by the weighted energy method.     

Two-dimensional rotating waves in a functional-differential diffusion equation with rotation of spatial arguments and time delay

We consider the Neumann boundary value problem for a parabolic functional-differential equation in a disk. We describe spatially inhomogeneous solutions in the form of rotating waves branching from the homogeneous stationary solution in the case of an Andronov-Hopf bifurcation. By passing to a moving coordinate system and by reducing the original problem to a stationary boundary value problem for a partial differential equation with a deviating argument, we prove the existence of rotating waves appearing in the disk under the Andronov-Hopf bifurcation.     

Effect of Hall current on the unsteady MHD flow due to a rotating disk with uniform suction or injection

The unsteady flow of a viscous conducting fluid due to the rotation of an infinite, non-conducting, porous disk in the presence of an axial uniform steady magnetic field is studied without neglecting the Hall effect. The fluid is acted upon by a uniform injection or suction through the disk. The relevant equations are solved numerically with a special technique to resolve the conflict between the initial and boundary conditions. The solution shows that the inclusion of the injection or suction through the surface of the disk in addition to the Hall current gives some interesting results.     

Laws of the wall for velocity and temperature in supersonic turbulent boundary layers

Scaling laws for velocity and temperature profiles in the near-wall region of sub- and supersonic turbulent boundary layers have been developed, which allow us to represent velocity and temperature profiles in compressible gas stream in terms of those in an incompressible boundary layer. They are obtained as asymptotic expansions of the solutions to the Reynolds equations in a small parameter — the Mach number based on the friction velocity and gas enthalpy on the wall. The leading term of the expansion for velocity corresponds to known Van Driest's formula. However, the obtained solution contains additional terms of order unity, which explains the contradiction between Van Driest's formula and experimental data. The law of the wall for temperature, which has been formulated for the first time, has an analogous structure. Besides the von Kármán constant and the turbulent Prandtl number in the logarithmic region, known for incompressible flow, the obtained relations contain three new universal constants, which do not depend on gas molecular properties and the specific heat ratio. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence and a priori bounds for steady stagnation flow toward a stretching cylinder

We investigate the nonlinear boundary value problem (BVP) that is derived from a similarity transformation of the Navier-Stokes equations governing fluid flow toward a stretching permeable cylinder. Existence of a solution is proven for all values of the Reynolds number and for both suction and injection, and uniqueness results are obtained in the case of a monotonic solution. A priori bounds on the skin friction coefficient are also obtained. These bounds achieve any desired order of accuracy as the injection parameter tends to negative infinity.     

An Analytic Approach for Calculating Absolutely Unstable Inviscid Modes of the Boundary Layer on a Rotating Disk

An analytical treatment of inviscidly absolutely unstable modes is pursued using the long-wavelength asymptotic approach. It is shown using the inviscid Rayleigh scalings in conjunction with the linear critical layer theory that the rotating-disk boundary layer flow undergoes a region of absolute instability for some small azimuthal wave numbers. The analytically calculated branch points for the absolute instability are found to be in good agreement with those obtained via a numerical solution of the inviscid Rayleigh equation.     

Nonlinear Spatial Evolution of Small Disturbances from Boundary Conditions in Flat Inclined-Channel Flow

The asymptotic behavior of small disturbances as they evolve spatially from boundary conditions in a flat inclined channel is determined. These small disturbances develop into traveling waves called roll waves, first discussed by Dressler in 1949. Roll waves exist if the Froude number F exceeds 2, which consist of a periodic pattern of bores, or discontinuities. After confirming the instability condition for   F > 2  for the linearized equations in the boundary value case, the nonlinear boundary value problem for the weakly unstable region of F slightly larger than 2 is studied. Multiple scales and the Fredholm alternative theorem are applied to determine the evolution of the solution in space. It is found that the solution is dominated by the evolution of the disturbance along one characteristic. The shock conditions governing the asymptotic solution are determined and these conditions are used to determine the approximate shape of the resulting traveling wave from the solution. Both asymptotic and numerical results for periodic disturbances are presented.