Linear stability analysis of flow in a periodically grooved channel

We have conducted the linear stability analysis of flow in a channel with periodically grooved parts by using the spectral element method. The channel is composed of parallel plates with rectangular grooves on one side in a streamwise direction. The flow field is assumed to be two‐dimensional and fully developed. At a relatively small Reynolds number, the flow is in a steady‐state, whereas a self‐sustained oscillatory flow occurs at a critical Reynolds number as a result of Hopf bifurcation due to an oscillatory instability mode. In order to evaluate the critical Reynolds number, the linear stability theory is applied to the complex laminar flow in the periodically grooved channel by constituting the generalized eigenvalue problem of matrix form using a penalty‐function method. The critical Reynolds number can be determined by the sign of a linear growth rate of the eigenvalues. It is found that the bifurcation occurs due to the oscillatory instability mode which has a period two times as long as the channel period. Copyright © 2003 John Wiley & Sons, Ltd.     

The Absolute Instability of Thin Wakes in an Incompressible/Compressible Fluid

RID="ID=" Communicated by P. HallAbstract:The absolute/convective instability of two-dimensional wakes forming behind a flat plate and near the trailing-edge of a thin wedge-shaped aerofoil in an incompressible/compressible fluid is investigated. The mean velocity profiles are obtained by solving numerically the classical compressible boundary-layer equations with a negative pressure gradient for the flat plate case, and the incompressible triple-deck equations for a thin wedge-shaped trailing-edge. In addition for a Joukowski aerofoil the incompressible mean boundary-layer flow in the vicinity of the trailing-edge is also calculated by solving the interactive boundary-layer equations. A linear stability analysis of the boundary-layer profiles shows that a pocket of absolute instability occurs downstream of the trailing-edge with the extent of the instability region increasing with more adverse pressure gradients. The region of absolute instability persists along the near-wake axis, while the majority of the wake is convectively unstable. For a thin wedge-shaped trailing-edge in an incompressible fluid, a similar stability analysis of the velocity profiles obtained via a composite expansion, also shows the occurrence of absolute instability behind the trailing-edge for a wedge angle greater than a critical value. For increasing values of the wedge angle and for thicker aerofoils, separation takes place near the trailing-edge and the extent of absolute instability increases. Calculations also show that for insulated plates compressibility has a stabilizing effect but cooling the wall destabilizes the flow unlike wall heating.} Received 11 May 1998 and accepted 25 February 1999     

On the absolute instability in a boundary-layer flow with compliant coatings

The question of absolute instabilities occuring in a boundary-layer flow with compliant coatings is reassessed. Compliant coatings of the Kramer's type are considered. Performing a local, linear absolute/convective stability analysis, a family of spring-backed elastic plates with damping is shown to be absolutely unstable for sufficiently thin plates. The absolute instability arises from the coalescence between an upstream propagating evanescent mode and the Tollmien–Schlichting wave. To reinforce the local, linear stability results the global stability behaviour of the system is investigated, integrating numerically the full nonparallel and nonlinear two-dimensional Navier–Stokes system coupled to the dynamical model. Injecting Gaussian-type, spatially localized flow disturbances as initial conditions, the spatio-temporal evolution of wave packets is computed. The absolute stability behaviour is retrieved in the global system, for a compliant panel of finite length. It is demonstrated numerically that the global stability behaviour of the wall, triggered by finite-end-effects, may be independent of the disturbance propagation in the flow.     

Stability of Hartmann flows

The stability of Hartmann flows for arbitrary magnetic Reynolds numbers is investigated in the framework of linear theory. The initial three-dimensional problem reduces to the equivalent two-dimensional problem. Perturbation theory is used to find asymptotic expressions for the eigenvalues. Distinguishing two types of disturbances — magnetic and hydrodynamic — is shown to be advantageous in a number of cases. Simple features of the stability are considered for particular cases. The well-know Lundquist result is generalized. An energy approach is applied to the problem of stability. The results of simulations involving the solution of the linear stability problem are described. A distinctive picture of stability is developed. There are several types of instability and they can develop simultaneously. The hydrodynamic and magnetic phenomena interact with each other in a very complex fashion. The magnetic field can either enhance flow stability or reduce it.Novosibirsk. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 6, pp. 17–31, November–December, 1972.     

Investigations of time-growing instabilities in laminar separation bubbles

The occurrence of temporally growing unstable disturbances is investigated based on eigenvalues with zero group velocity from linear stability theory (LST) and compared with observations of upstream travelling disturbances obtained in a two-dimensional direct numerical simulation (DNS) of an unsteady laminar separation bubble. Numerical solutions of the Orr–Sommerfeld equation using analytically constructed base-flow velocity profiles modelled by a modified hyperbolic tangent function help to identify the role of parameters, such as maximum reverse flow, wall distance and intensity of the shear layer, as well as Reynolds number on the possibility that a true time-growing instability occurs. Then, the viscous or inviscid nature of the solutions found is classified on the basis of their eigenfunctions. For large wall distances two unstable modes are found. Apart from a low-frequency motion of the bubble the DNS exhibit high-frequency oscillations which periodically appear and disappear. Part of these disturbances travel upstream and amplify with respect to time. Their initial occurrence and their frequency are in excellent agreement with the results of the parameter study based on LST and a closer examination of the disturbances yields insight into their spatial structure.     

On two-dimensional linear waves in Blasius boundary layer over viscoelastic layers

This work concerns the direct numerical simulation of small-amplitude two-dimensional ribbon-excited waves in Blasius boundary layer over viscoelastic compliant layers of finite length. A vorticity-streamfunction formulation is used, which assures divergence-free solutions for the evolving flow fields. Waves in the compliant panels are governed by the viscoelastic Navier's equations. The study shows that Tollmien–Schlichting (TS) waves and compliance-induced flow instability (CIFI) waves that are predicted by linear stability theory frequently coexist on viscoelastic layers of finite length. In general, the behaviour of the waves is consistent with the predictions of linear stability theory. The edges of the compliant panels, where abrupt changes in wall property occur, are an important source of waves when they are subjected to periodic excitation by the flow. The numerical results indicate that the non-parallel effect of boundary-layer growth is destabilizing on the TS instability. It is further demonstrated that viscoelastic layers with suitable properties are able to reduce the amplification of the TS waves, and that high levels of material damping are effective in controlling the propagating CIFI.     

Global stability analysis of axisymmetric boundary layer over a circular cylinder

This paper presents a linear global stability analysis of the incompressible axisymmetric boundary layer on a circular cylinder. The base flow is parallel to the axis of the cylinder at inflow boundary. The pressure gradient is zero in the streamwise direction. The base flow velocity profile is fully non-parallel and non-similar in nature. The boundary layer grows continuously in the spatial directions. Linearized Navier–Stokes (LNS) equations are derived for the disturbance flow quantities in the cylindrical polar coordinates. The LNS equations along with homogeneous boundary conditions forms a generalized eigenvalues problem. Since the base flow is axisymmetric, the disturbances are periodic in azimuthal direction. Chebyshev spectral collocation method and Arnoldi’s iterative algorithm is used for the solution of the general eigenvalues problem. The global temporal modes are computed for the range of Reynolds numbers and different azimuthal wave numbers. The largest imaginary part of the computed eigenmodes is negative, and hence, the flow is temporally stable. The spatial structure of the eigenmodes shows that the disturbance amplitudes grow in size and magnitude while they are moving towards downstream. The global modes of axisymmetric boundary layer are more stable than that of 2D flat-plate boundary layer at low Reynolds number. However, at higher Reynolds number they approach 2D flat-plate boundary layer. Thus, the damping effect of transverse curvature is significant at low Reynolds number. The wave-like nature of the disturbance amplitudes is found in the streamwise direction for the least stable eigenmodes.     

Direct Spatial Resonance in the Compressible Boundary Layer on a Rotating-disk

This paper focuses on a resonance mechanism that can lead to significant disturbance amplification at conditions which are sub-critical to nonlinear instabilities. Particularly, direct spatial resonance instability is investigated, which is present in the basic three-dimensional viscous compressible boundary-layer flow due to a rotating-disk. Within this purpose, the linearized system of stability equations is treated numerically making use of a spectral Chebyshev collocation method. The analysis provides critical resonant Reynolds numbers above which growth occurs. Amplitudes of the response of the degeneracies decaying rapidly due to their high damping rates are shown to exist for small enough Reynolds numbers while the flow is still in the laminar state. If the flow is restricted to the incompressible case, the results of Turkyilmazoglu and Gajjar (in Sadhana Acad P Engs 25:601–617, 2000) are completely reproduced. The influences of compressibility are then explored by means of varying the Mach and Prandtl numbers in the cases of heating/cooling the wall as well as the isothermal wall. In general, compressibility effects are found strongly in favor of stabilizing as the Mach number increases, while a strong destabilization is observed by lowering the critical values of Reynolds numbers in the cases of wall heating and insulation. The modal interaction and coalescence of the eigenmodes calculated here create local algebraic growth by rapid development of relatively large amplitudes which might then provide the onset of nonlinear effects followed by transition.     

The Laminar Boundary Layer over a Permeable Wall

An analysis is given of the laminar boundary layer over a permeable/porous wall. The porous wall is passive in the sense that no suction or blowing velocity is imposed. To describe the flow inside and above the porous wall a continuum approach is employed based on the Volume-Averaging Method (S. Whitaker The method of volume averaging). With help of an order-of-magnitude analysis the boundary-layer equations are derived. The analysis is constrained by: (a) a low wall permeability; (b) a low Reynolds number for the flow inside the porous wall; (c) a sufficiently high Reynolds number for the freestream flow above the porous wall. Two boundary layers lying on top of each other can be distinguished: the Prandtl boundary layer above the porous wall, and the Brinkman boundary layer inside the porous wall. Based on the analytical solution for the Brinkman boundary layer in combination with the momentum transfer model of Ochoa-Tapia and Whitaker (Int. J. Heat Mass Transfer 38 (1995) 2635). for the interface region, a closed set of equations is derived for the Prandtl boundary layer. For the stream function a power series expansion in the perturbation parameter is adopted, where is proportional to ratio of the Brinkman to the Prandtl boundary-layer thickness. A generalization of the Falkner–Skan equation for boundary-layer flow past a wedge is derived, in which wall permeability is incorporated. Numerical solutions of the Falkner–Skan equation for various wedge angles are presented. Up to the first order in wall permeability causes a positive streamwise velocity at the interface and inside the porous wall, but a wall-normal interface velocity is a second-order effect. Furthermore, wall permeability causes a decrease in the wall shear stress when the freestream flow accelerates, but an increase in the wall shear stress when the freestream flow decelerates. From the latter it follows that separation, as indicated by zero wall shear stress, is delayed to a larger positive pressure gradient.     

A pseudospectral approach applicable for time integration of linearized N-S operator that removes pole singularity and physically spurious eigenmodes

This paper addresses a modified singularity removal technique for the eigenvalue or optimal mode problems in pipe flow using a pseudospectral method. The current approach results in the linear stability operator to be devoid of any unstable physically spurious modes, and thus, it provides higher numerical stability during time-based integration. The correctness of the numerical operator is established by calculating the known eigenvalues of pipe Poiseuille flow. Subsequently, the optimal modes are determined with Farrell's approach and compared with the existing literature. The usefulness of this approach is further demonstrated in the time-based numerical integration of the linearized Navier-Stokes operator for the adjoint method–based optimal mode determination. The numerical scheme is implemented with the radial velocity-radial vorticity formulation. Even number of Chebyshev-Lobatto grid points are distributed over the domain r∈[−1,1] omitting the centerline, which also efficiently provides higher resolution near the wall boundary. The boundary conditions are imposed with homogeneous wall boundary conditions, whereas the analytic nature of a proper set of base functions enforces correct centerline conditions. The resulting redundancy introduced in the process is eliminated with the proper usage of parity.     

Heat transfer effects on the stability of low speed plane Couette-Poiseuille flow

The stability problem of low-speed plane Couette-Poiseuille flow of air under heat transfer effects is solved numerically using the linear stability theory. Stability equations obtained from two-dimensional equations of motion and their boundary conditions result in an eigenvalue problem that is solved using an efficient shoot-search technique. Variable fluid properties are accounted for both in the basic flow and the perturbation (stability) equations. A parametric study is performed in order to assess the roles of moving wall velocity and heat transfer. It is found that the moving wall velocity and the location of the critical layers play decisive roles in the instability mechanism. The flow becomes unconditionally stable whenever the moving wall velocity exceeds half of the maximum velocity in the channel. With wall heating and Mach number effects included, the flow is stabilized.     

Absolute and Convective Instabilities in the Compressible Boundary Layer on a Rotating Disk

A numerical study has been undertaken to investigate the nature of inviscid instability of the three-dimensional compressible boundary layer flow due to a rotating disk. The compressible Rayleigh equation is integrated using a spectral Chebyshev-collocation method together with a fourth-order Runge–Kutta integrator. In the context of spatio-temporal stability analysis, the singularities of the resulting dispersion relation are determined and the ones that satisfy the Briggs–Bers pinching criterion have been selected. In certain finite parameter regions of eigenvalues (wave numbers and wave angles, for instance) it is found that by varying the Mach number, absolute instability occurs in the compressible boundary layer on a rotating disk. The range corresponding to the incompressible flow case given in Lingwood (1995) (ε between 14.615° and 38.114°) is verified. The results of Cole (1995) are also verified. The overall effect of compressibility is to reduce the extent of absolute instability at higher Mach numbers. The effect of heating the wall is to enhance the absolute instability properties, however, cooling the wall is found to decrease greatly the region of absolute instability regime for the range of Mach numbers studied. It is also shown in this study that for non-insulated walls a direct spatial resonance of the eigenmodes is possible and this raises the possibility of large local algebraic growth of perturbations being important in some instances. Received 15 October 1999 and accepted 10 December 1999     

On the stability of plane parallel viscoelastic shear flows in the limit of infinite Weissenberg and Reynolds numbers

Elastic effects on the hydrodynamic instability of inviscid parallel shear flows are investigated through a linear stability analysis. We focus on the upper convected Maxwell model in the limit of infinite Weissenberg and Reynolds numbers. We study the effects of elasticity on the instability of a few classes of simple parallel flows, specifically plane Poiseuille and Couette flows, the hyperbolic-tangent shear layer and the Bickley jet.The equation for stability is derived and solved numerically using the spectral Chebyshev collocation method. This algorithm is computationally efficient and accurate in reproducing the eigenvalues. We consider flows bounded by walls as well as flows bounded by free surfaces. In the inviscid, nonelastic case all the flows we study are unstable for free surfaces. In the case of wall bounded flow, there are instabilities in the shear layer and Bickley jet flows. In all cases, the effect of elasticity is to reduce and ultimately suppress the inviscid instability.     

Effect of Inertial Terms on Fluid–Porous Medium Flow Coupling

The study considers an effect of the nonlinear inertial terms in the Brinkman filtration equation on the characteristics of coupled flows in a pure fluid and porous medium in the frameworks of two independent problems. The first problem is the forced boundary-layer flow overlying the Darcy–Brinkman porous medium. The Prandtl theory is used, and the self-similar equations are built to describe it. It is shown that the inertial terms have a valuable effect on the boundary-layer structure because of the large velocity gradient in the transition zone. The boundary-layer thickness in a porous medium rapidly grows at large Reynolds numbers. The velocity magnitude and gradient at the interface also change. The second independent problem is an analysis of the inertial terms effect on the flow stability. The neutral curves of the full and linearized flow models are built using the shooting method. They have different short-wave asymptotic, but there are no significant changes in the critical Reynolds numbers and corresponding wave numbers.     

Rayleigh Conductivity and Self-Sustained Oscillations

The theory of self-sustaining oscillations of low Mach number, high Reynolds number shear layers, and jets impinging on edges and corners is discussed. Such oscillations generate narrow band sound, and are usually attributed to the formation of discrete vortices whose interactions with the edge or corner produce impulsive pressures that trigger the cyclic formation of new vorticity. A linearized analysis of these interactions is described in which free shear layers are treated as vortex sheets. Details are given for shear flow over wall apertures and shallow cavities, and for jet–edge interactions. The operating stages of the oscillations correspond to complex eigenvalues of the linear theory: for wall apertures and edge tones they are poles in the upper half of the complex frequency plane of the Rayleigh conductivity of the “window” spanned by the shear flow; for shallow wall cavities they are poles of a frequency-dependent drag coefficient. It is argued that the frequencies defined by the real parts of the complex frequencies at these poles determine the operating stage Strouhal numbers observed experimentally. Strouhal number predictions for a shallow wall cavity are in good agreement with data extrapolated to zero Mach number from measurements in air; edge tone predictions are in excellent accord with data from various sources in the literature. Received 3 January 1997 and accepted 13 February 1997     

Stability of plane poiseuille flow of a highly elastic liquid

The stability of fully developed pressure driven plane laminar flow of a Maxwell fluid has been studied using linear hydrodynamic stability theory. Elasticity is destabilizing in the inertial regime, but the flow is found to be stable to infinitesimal disturbances at low Reynolds numbers. This result contradicts previous calculations, which predicted a low Reynolds number flow instability at a critical recoverable shear of order unity. The previous calculations were carried out using less accurate numerical methods; the eigenvalue problem which must be solved is a delicate one, requiring sophisticated umerical techniques in order to avoid the calculation of spurious unstable modes.This work has direct bearing on the question of the mechanism of a low Reynolds number extrusion instability known as “melt fracture”. It is observed that the intensity of melt fracture increases with increasing die length for high density polyethylene, and it is therfore believed by some experimentalists that fully-developed die flow is unstable for this polymer above a critical recoverable shear. The analysis appears to be at variance with this interpretation of the experimental results.     

A novel implicit numerical treatment for non‐linear turbulence models using high and low Reynolds number formulations

Non‐linear turbulence models can be seen as an improvement of the classical eddy‐viscosity concept due to their better capacity to simulate characteristics of important flows. However, application of non‐linear models demand robustness of the numerical method applied, requiring a stable discretization scheme for convergence of all variables involved. Usually, non‐linear terms are handled in an explicit manner leading to possible numerical instabilities. Thus, the present work shows the steps taken to adapt a general non‐linear constitutive equation using a new semi‐implicit numerical treatment for the non‐linear diffusion terms. The objective is to increase the degree of implicitness of the solution algorithm to enhance convergence characteristics. Flow over a backward‐facing step was computed using the control volume method applied to a boundary‐fitted coordinate system. The SIMPLE algorithm was used to relax the algebraic equations. Classical wall function and a low Reynolds number model were employed to describe the flow near the wall. The results showed that for certain combination of relaxation parameters, the semi‐implicit treatment proposed here was the sole successful treatment in order to achieve solution convergence. Also, application of the implicit method described here shows that the stability of the solution either increases (high Reynolds with non‐orthogonal mesh) or preserves the same (low Reynolds number applications). Additional advantages of the procedure proposed here lie in the possibility of testing different non‐linear expressions if one considers the enhanced robustness and stability obtained for the entire numerical algorithm. Copyright © 2010 John Wiley & Sons, Ltd.