Nonparallel Stability Research Articles

Aerodynamic drag modification induced by free-stream turbulence effects on a simplified road vehicle

We report an extensive experimental investigation into the effects of inflow turbulence on a simplified road vehicle, the so-called square back Ahmed body. Variations reaching up to +16% and −17% of the drag coefficient are observed for free-stream turbulence representative of open-road conditions [J. W. Saunders and R. B. Mansour, “On-road and wind tunnel turbulence and its measurement using a four-hole dynamic probe ahead of several cars,” SAE Trans. 109, 477 (2000)]. Regular turbulence grids are mounted upstream the Ahmed body. The turbulence intensity and the integral length scale of turbulence are varied using different mesh, bar sizes, and solidity. The boundary layer developing around the body together with the structure of the wake is strongly altered by free-stream turbulence where both the length of the recirculation and the shear layer characteristics are modified. A weakly non-parallel stability analysis of the shear layers together with a momentum budget, both bounding the recirculation region, shows that coherent structures, traced through the Reynolds stresses and streamwise turbulent fluctuations, are the key mechanisms that control drag. Subsequently, the analysis of the shear layer together with the stability analysis demonstrate that the mean vertical shear is the key component that controls the Reynolds stresses and thereby the drag experienced by the vehicle. These findings raise the question of the importance of free-stream turbulence when considering studies dedicated to car aerodynamics and subsequent control strategies, most of which neglect the influence of inflow conditions. This issue is also of major importance for guiding the design of the next generation of control strategies for drag reduction.

Secondary instability in thin film flows under an inclined plane: growth of lenses on spatially developing rivulets

The response of a thin film flowing under an inclined plane, modelled using the lubrication equation, is studied. The flow at the inlet is perturbed by the superimposition of a spanwise-periodic steady modulation and a decoupled temporally periodic but spatially homogeneous perturbation. As the consequence of the spanwise inlet forcing, the so-called rivulets grow downstream and eventually reach a streamwise-invariant state, modulated along the direction perpendicular to the flow. The linearized dynamics in the presence of a time-harmonic inlet forcing shows the emergence of a time-periodic flow characterized by drop-like structures (so-called lenses) that travel on the rivulet. The spatial evolution is rationalized by a weakly non-parallel stability analysis. The occurrence of the lenses, their spacing and thickness profile, is controlled by the inclination angle, flow rate, and the frequency and amplitude of the time-harmonic inlet forcing. The faithfulness of the linear analyses is verified by nonlinear simulations. The results of the linear simulations with inlet forcing are combined with the computations of nonlinear travelling lenses solutions in a double-periodic domain to obtain an estimate of the dripping length, for a large range of conditions.

Improvement for expansion of parabolized stability equation method in boundary layer stability analysis

An improved expansion of the parabolized stability equation (iEPSE) method is proposed for the accurate linear instability prediction in boundary layers. It is a local eigenvalue problem, and the streamwise wavenumber α and its streamwise gradient dα/dx are unknown variables. This eigenvalue problem is solved for the eigenvalue dα/dx with an initial α, and the correction of α is performed with the conservation relation used in the PSE. The iEPSE is validated in several compressible and incompressible boundary layers. The computational results show that the prediction accuracy of the iEPSE is significantly higher than that of the ESPE, and it is in excellent agreement with the PSE which is regarded as the baseline for comparison. In addition, the unphysical multiple eigenmode problem in the EPSE is solved by using the iEPSE. As a local non-parallel stability analysis tool, the iEPSE has great potential application in the eN transition prediction in general three-dimensional boundary layers.

Formation of spanwise vorticity in oblique turbulent bands of transitional plane Couette flow, part 2: Modelling and stability analysis

This article presents a modelling of the formation of spanwise vorticity in the turbulent streaks of the oblique bands and spots of transitional plane Couette flow. A functional model is designed to mimic the coherent flow in the streaks. The control parameters of the model are extracted from Direct Numerical Simulations (DNS) statistical data. A Reynolds stress is proposed to study the effect on the instability of this additional force maintaining the baseflow. Local (quasi-parallel) temporal stability analysis is performed on that model to investigate the linear development of the spanwise vorticity. Results show that average profiles, even if they have an inflection, are stable: the shear layers inside the velocity streaks are responsible for the vorticity formation. Emphasis is put on the convective or absolute nature of the instability, depending on the location in the band. This shows that a transition from a convective to an absolute instability occurs in the zone in between fully turbulent and laminar flow. The group velocity of the most unstable modes compares well to the advection velocity of spanwise vorticity measured in DNS. This investigation is completed by the global (non-parallel) stability analysis of a typical band case. Eventually, the possible cycles of sustainment of localised low Reynolds number turbulence of shear flows are discussed in the light of these results.

Hydrodynamically locked morphogenesis in karst and ice flutings

Two of the most widespread and fascinating patterns observed on cave walls and icefalls – karst and ice flutings – are demonstrated to share the same morphogenesis, whose core is a water film-induced locking mechanism. Creeping flow-based parallel and non-parallel stability analyses are developed through a numerical and analytical approach. These instabilities are shown to develop at inverted overhung conditions. A sharp transition between fluting and ripple-like patterns is presented. The non-parallel problem is solved with the use of Papkovich–Neuber solutions in order to obtain a finite wavelength selection close to the critical conditions. The method and results can be extended to similar problems where the temporal evolution of the interface is linearly related to the film depth.

Stability, separation and close body interactions. Introduction.

Stability, separation and close body interactions. Introduction.

The Instability of Flow Through a Slowly Diverging Pipe With Viscous Heating

The nonparallel linear stability analysis of flow through a slowly diverging pipe undergoing viscous heating is considered. The pipe wall is maintained at constant temperatures and Nahme’s law is applied to model the temperature dependence of the fluid viscosity. A one-parameter family of velocity profiles for the basic state is obtained for small angles of divergence. The nonparallel stability equations for the disturbance velocity coupled to a linearized energy equation are derived and solved using a spectral collocation method. Our results indicate that increasing viscous heating, characterized by increasing Nahme number, is destabilizing. The Prandtl number has a negligible effect on the linear stability characteristics. The Grashof number stablizes the flow for Gr>106, below which it has a negligible effect.

The relative roles of divergence and velocity slip in the stability of plane channel flow

Wall slip and wall divergence are known to have large and opposing effects on the stability of flow in a two-dimensional channel. While divergence hugely destabilises, slip dramatically stabilizes the linear mode. In a non-parallel stability analysis, we study a combination of these two effects, since both will coexist in small-scale flows with wall roughness. Our main results are (i) that the stabilising effect of slip is reversed at higher angles of divergence, (ii) transient growth of disturbances is unaffected by either wall-divergence, or by slip at any divergence. Moreover, at the Reynolds numbers relevant here, transient growth is too low to be a significant player in transition to turbulence, which is more likely to be driven by linear instability.

Spatially convective global modes in a boundary layer

The linear stability of a weakly nonparallel flow, the case of a flat plate boundary layer, is revisited by a linear global stability approach where the two spatial directions are taken as inhomogeneous, leading to a fully nonparallel stability method. The resulting discrete eigenvalues obtained by the fully nonparallel approach seem to be in agreement with classical Tollmien–Schlichting waves. Then the different modes are compared with classical linear stability approach and weakly nonparallel method based on linear parabolized stability equations (PSEs). It is illustrated that the nonparallel correction provided by the linear global stability approach is well matched by linear PSE. Furthermore, physical interpretation of these spatio-temporal global modes is given where a real pulsation, which has more physical interest, is considered. In particular the use of a Gaster transformation and the pseudospectrum illustrate the local and global properties of these Tollmien–Schlichting modes. Finally, the contribution of different components of global modes (normal and streamwise) in the transient amplifying behavior associated with the convectively unstable boundary layer is analyzed and compared with a classical steepest descent method. Then, a discussion of an equivalent of the continuous branch is given.

Separation of variables in the hydrodynamic stability equations

The problem of variable separation in the linear stability equations, which govern the disturbance behaviour in viscous incompressible fluid flows, is discussed. The so-called direct approach, in which a form of the 'ansatz' for a solution with separated variables as well as a form of reduced ODEs are postulated from the beginning, is applied. The results of application of the method are the new coordinate systems and the most general forms of basic flows, which permit the postulated form of separation of variables. Then the basic flows are specified by the requirement that they themselves satisfy the Navier–Stokes equations. Calculations are made for the (1+3)-dimensional disturbance equations written in Cartesian and cylindrical coordinates. The fluid dynamics interpretation and stability properties of some classes of the exact solutions of the Navier–Stokes equations, defined as flows for which the stability analysis can be reduced via separation of variables to the eigenvalue problems of ordinary differential equations, are discussed. The eigenvalue problems are solved numerically with the help of the spectral collocation method based on Chebyshev polynomials. For some classes of perturbations, the eigenvalue problems can be solved analytically. Those unique examples of exact (explicit) solution of the nonparallel unsteady flow stability problems provide a very useful test for numerical methods of solution of eigenvalue problems, and for methods used in the hydrodynamic stability theory, in general.

A synthetic perturbative hypothesis for multiscale analysis of convective wake instability

The paper presents a nonparallel stability analysis of the intermediate region of the two-dimensional wake behind a bluff body. In particular, it analyzes the convective instabilities using a Wentzel-Kramers-Brillouin-Jeffreys method on a basic flow previously derived from intermediate asymptotics [D. Tordella and M. Belan, Phys. Fluids 15, 1897 (2003)]. The multiscaling is carried out to explicitly account for the effects associated to the lateral momentum dynamics at a given Reynolds number. These effects are an important feature of the base flow and are included in the perturbative equation as well as in the associated modulation equation. At the first order in the multiscaling, the disturbance is locally tuned to the property of the instability, as can be seen in the zero-order theory (near-parallel parametric Orr-Sommerfeld treatment). This leads to a synthetic analysis of the nonparallel correction of the instability characteristics. The system is, in fact, considered to be locally perturbed by waves with a wave number that varies along the intermediate wake and which is equal to the wave number of the dominant saddle point of the zero order dispersion relation, taken at different Reynolds numbers. In this study, the Reynolds number is thus the only parameter. It is shown that the corrections to the frequency, and to the temporal and spatial growth rates are remarkable in the first part of the intermediate wake and lead to absolute instability in regions that extend to about ten body scales. The correction increases with the Reynolds number and agrees with data from laboratory and numerical experiments in literature. An eigenfunction and eigenvalue asymptotic analysis for the far wake is included, which is in excellent agreement with the complete problem.

ON NONPARALLEL STABILITY OF THREE-DIMENSIONAL COMPRESSIBLE BOUNDARY LAYERS

Nonparallel stability of the compressible boundary layers for three-dimensional configurations having large curvature variation on the surface is investigated by using the parabolic stability equations, which are derived from the Navier-Stokes equations in the curvilinear coordinate system. The difference schemes with fourth-order accuracy can be used in the entire computational regions. The global method is combined with the local method using a new iterative formula, thus more precise eigenvalues are obtained, and fast convergences are achieved. Computed curves of the amplification factor and shape functions of disturbances show clearly variable process of the flow stability, and agree well with other available results.

Nonparallel stability of the flow in an axially rotating pipe

The linear stability of the developing flow in an axially rotating pipe is analyzed using parabolized stability equations (PSE). The results are compared with those obtained from a near-parallel stability approximation that only takes into account the axial variation of the basic flow. Though the PSE results obviously coincide with the near-parallel ones far downstream, when the flow has reached a Hagen-Poiseuille axial velocity profile with superimposed solid-body rotation, they differ significantly in the developing region. Therefore, the onset of instability strongly depends on the axial evolution of the perturbations. The PSE results are also compared with experimental data from Imao et al. [Exp. Fluids 12 (1992) 277], showing a good agreement in the frequencies and wavelengths of the unstable disturbances, that take the form of spiral waves. Finally, a simple method for detecting one of the conditions to characterize the onset of absolute instability using PSE is given.

Higher order stability effects in a natural convection boundary layer over a vertical heated wall

The stability of a laminar boundary layer flow under natural convection on a vertical isothermally heated wall is studied analytically. The analysis is performed by using two different two-dimensional linear models: (1) The non-parallel flow model in which the steady mean flow as well as the disturbance amplitude functions can change in the streamwise direction; (2) The parallel flow model in which the effects of the mean flow and disturbance changes in the streamwise direction are neglected. The linear non-parallel stability analysis is based on the so-called parabolised stability equations (PSEs) which have been successfully applied to the stability analysis of forced convection boundary layers. In this study the PSE equations are applied to natural convection boundary layers in order to show the difference between parallel and non-parallel stability analysis. A second part of this study deals with the effects of variable properties, which are always present in natural convection flows. They are analysed by an extended version of the Orr–Sommerfeld equation (EOSE).

On the nonparallel stability of the injection induced two-dimensional Taylor flow

In this article, the nonparallel stability of the so-called Taylor flow induced by wall injection is investigated. This flow is unstable due to nonparallel effects which have not been rigorously treated yet. Only inconsistent methods have been used. They give good results in comparison to the experiments but they suffer from a lack of justification and from the dependency of the results on the formulation. The necessity of using inconsistent methods instead of the Orr–Sommerfeld approach is justified in this article and a much more rigorous asymptotic analysis is performed in the limit of the large streamwise distances.

Non-parallel stability analysis of three-dimensional boundary layers along an infinite attachment line

Instability of a non-parallel similar-boundary-layer flow to small and wavy disturbances is governed by partial differential equations with respect to the non-dimensional vertical coordinate ζ and the local Reynolds number R1 based on chordwise velocity of external stream and a boundary-layer thickness. In the particular case of swept Hiemenz flow, the equations admit a series solution expanded in inverse powers of R12 and then are decomposed into an infinite sequence of ordinary differential systems with the leading one posing an eigenvalue problem to determine the first approximation to the complex dispersion relation. Numerical estimation of the series solution indicates a much lower critical Reynolds number of the so-called oblique-wave instability than the classical value Rc = 583 of the spanwise-traveling Tollmien–Schlichting instability. Extension of the formulation to general Falkner–Skan–Cooke boundary layers is proposed in the form of a double power series with respect to 1/R12 and a small parameter ε denoting the difference of the Falkner–Skan parameter m from the attachment-line value m = 1.

Nonparallel and nonlinear stability of supersonic jet flow

Linear and nonlinear evolution of disturbances in an axisymmetric, supersonic, low Reynolds number jet is studied using parabolized stability equations approach. Both axisymmetric and helical modes are considered and nonparallel effect is found to increase the disturbance growth rate, although there is very little effect on the wavenumber. Nonlinear interaction of the helical modes, which are the dominant instability modes of the jet, results in disturbance saturation, spectrum filling and large mean flow distortions. Similar to that for the supersonic boundary layer flow (Chang, C.-L., Malik, M.R., Oblique-mode breakdown and secondary instability in supersonic boundary layers. J. Fluid Mech. 273 (1994) 323–359), interaction of the helical modes induces streamwise vortices which cause significant mean flow distortion and growth of other harmonics. The computed evolution of disturbances is in reasonably good agreement with the experimental data of Morrison and McLaughlin (Morrison, G.L., McLaughlin, D.K., Instability process in low Reynolds number supersonic jets. AIAA J. 18(7) (1980) 793–800). The present marching scheme is able to compute evolution of supersonic disturbances without any numerical reflections and the induced far-field pressure disturbance is found to be almost in phase with the acoustic wave at ambient conditions.

On the dependence on the formulation of some nonparallel stability approaches applied to the Taylor flow

In a study of the stability of the nonparallel Taylor flow in a channel with wall injection, see Taylor [Proc. R. Soc. London, Ser. A 234, 456 (1956)], Lee and Beddini [The 35th AIAA/ASME/ASEE Joint Propulsion Conference and Exhibit, AIAA-99-2508, 20–24 June 1999, Los Angeles, California, 1999] recently reported some discrepancies between their results and those of Casalis et al. [Phys. Fluids 10, 2558 (1998)], whereas they seem to agree perfectly with those of Varapaev and Yagodkin [Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza 4, 91 (1969)]. In Casalis et al. [Phys. Fluids 10, 2558 (1998)], the difference with Varapaev and Yagodkin [Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza 4, 91 (1969)] has not been clarified, so that it seems important to explain that the discrepancies do not come from a mistake but from a different choice of formulation. In addition, it will be shown that these discrepancies are related to the inconsistent nature of the stability approach which is used in the three papers.

Solution of the eigenvalue problems resulting from global non-parallel flow stability analysis

The eigensolution method, more flexible and efficient than previously used for global non-parallel flow stability analysis, is presented. The differential eigenvalue problem, resulting from linear stability equations of non-parallel flow is discretized with the penalty FEM. The large-dimensional algebraic eigenvalue problem is solved with the use of Subspace Iteration Method. Separation of the eigenvalues interesting for the flow stability analysis is performed with the inverse Cayley transformation. It is shown that certain barriers, resulting from very large dimensions of the eigenvalue problem and limiting the non-parallel flow stability method can be overcome with this approach. To demonstrate the algorithm, the stability of the flow around circular cylinder is analyzed.

Characterization of droplet/vapor/vortex interactions in a two-dimensional shear layer

Vol. 31, 1980, pp. 523-530. Leibovich, S., and Stewartson, K., A Sufficient Condition for the Instability of Columnar Vortices, Journal of Fluid Mechanics, Vol. 126, 1983, pp. 335-356. Stewartson, K., and Capell, K., On the Stability of Ring Modes in a Trailing Line Vortex: the Upper Neutral Points, Journal of Fluid Mechanics, Vol. 156, 1985, pp. 369-386. Duck, P., The Inviscid Stability of Swirling Flows: Large Wavenumber Disturbances, Journal of Applied Mathematics and Physics, Vol. 37, May 1986, pp. 340-360. Saric, W. S., and Nayfey, A. H., Nonparallel Stability of Boundary-Layer Flows, Physics of Fluids, Vol. 8, Aug. 1975, pp. 945-950. Bertolotti, F. P., Boundary Layer Stability Analyzed With the PSE Equations, AIAA 22nd Fluid Dynamics, Plasma Dynamics and Lasers Conference, AIAA Paper 91-1637, Honolulu, HI, June 1991. Hall, M. G., Breakdown, Annual Review of Fluid Mechanics, Vol. 4, 1972, pp. 195-218. Hall, M. G., A Numerical Method for Solving the Equations for a Vortex Core, Royal Aircraft Establishment, Tech. Rept. No. 65106, Farnborough, England, UK, May 1965. Batchelor, G. K., and Gill, A. E., Analysis of the Stability of Axisymmetric Jets, Journal of Fluid Mechanics, Vol. 14, 1962, pp. 529-551. Malik, M. R., and Orszag, S. A., Efficient Computation of the Stability of Three-Dimensional Compressible Boundary Layers, AIAA 14th Fluid and Plasma Dynamics Conference, AIAA Paper 81-1277, Palo Alto, CA, June 1981.