Squire Transformation Research Articles

Magnetohydrodynamic stability of pressure-driven flow in an anisotropic porous channel: Accurate solution

The stability of fully developed pressure-driven flow of an electrically conducting fluid through a channel filled with a saturated anisotropic porous medium is studied under the influence of a uniform transverse magnetic field using a modified Brinkman equation. An analogue of Squire's transformation is used to show that two-dimensional motions are more unstable than three-dimensional ones. The modified Orr–Sommerfeld equation for the problem is solved numerically and a more accurate solution is obtained using the Chebyshev collocation method combined with Newton's and golden section search methods. The critical Reynolds number Rc and the corresponding critical wave number αc are computed for a wide range of porous parameter σp, the ratio of effective viscosity to the fluid viscosityΛ, the mechanical anisotropy parameter K1, the porosity ε and the Hartman number M. It is found that the system remains unconditionally stable to small-amplitude disturbances for the Darcy case and the energy stability analysis is also performed to corroborate this fact.

On the Squire's transformation for stratified two-phase flows in inclined channels

The applicability of the Squire's transformation for stability analysis of stratified two-phase flow in horizontal and inclined channels is examined. It is shown that for the considered flow such a transformation requires some additional constraints on the change of the inclination angle and flow rates of each of the phases. While the Squire's theorem (on the two-dimensionality of the critical disturbances) rigorously holds for the horizontal two-phase flow, for the inclined flow an exact mathematical theorem cannot be formulated. Nevertheless, it has been proven that 2D perturbations are the critical ones also for the case of inclined channel, since the transformation of a 3D stability problem to its 2D analog is associated with a stabilizing effect of reducing the system inclination, in addition to the reduction of the phases flow rates as in the case of horizontal flows.

Linear onset of convective instability for Rayleigh-Bénard-Couette flows of viscoelastic fluids

A linear stability analysis of a plane Couette flow in the presence of an unstable vertical temperature gradient is investigated numerically for viscoelastic fluids using the Oldroyd-B model. Three main configurations are considered. In the first configuration, named case (i), both walls move with the same velocity, resulting in a shear free flow. In the second configuration, named case (ii), the lower wall is kept fixed while the upper wall moves with a constant velocity parallel to the x axis, i.e there is a non-zero net flow. In the third configuration, named case (iii), the horizontal plates move in opposite directions but with the same velocity magnitude, i.e. there is a zero net flow. The effects of the Reynolds number Re, relaxation time or elasticity number λ1, ratio Γ between retardation and relaxation times and the ratio VR between lower and upper plate velocities on the critical Rayleigh numbers, critical frequencies and convection cell sizes are examined. When Newtonian fluids are considered, the critical Rayleigh number for the appearance of longitudinal rolls (LRs), which are rolls with axes parallel to the x axis, is independent of Re. The same cannot be said about viscoelastic fluids. There is a range of λ1 and Γ values, which defines a strongly elastic regime, where the system experiences a Hopf bifurcation giving rise to oscillatory LRs. On the other hand, in the parametric region outside of this range, which defines a weakly elastic regime, LRs are stationary. Since there is a Squire transformation for this problem, the linear characteristics of rolls with arbitrary orientation can be deduced from those determined for transverse rolls (TRs), which are rolls with axes perpendicular to the x axis. In the weakly elastic regime, TRs are stabilized by an increasing Re. This effect is stronger in case (ii) than in case (iii). Furthermore, TRs are stationary in case (iii) but oscillatory in case (ii), since the net flow breaks the reflexion symmetry x→(−x) and selects a travelling wave propagating in the main flow direction. Finally, elasticity has a stabilizing effect whereas concentrated polymeric solutions are less stable than diluted ones. On the other hand, in the strongly elastic regime, the constant shear flow promotes the simultaneous appearance of two travelling waves (TW). Because of the mirror symmetry in case (iii), these TWs propagate in opposite directions but with the same phase velocity magnitude. In case (ii), the nonzero net flow induces different values for the TW phase velocities. Finally, λ1 and Γ have opposite effects. Although increasing λ1 (Γ) destabilizes (stabilizes) TWs for small Re, it stabilizes (destabilizes) them for moderate and high values of Re.

Extended Squire’s transformation and its consequences for transient growth in a confined shear flow

AbstractThe classical Squire transformation is extended to the entire eigenfunction structure of both Orr–Sommerfeld and Squire modes. For arbitrary Reynolds numbers $\mathit{Re}$, this transformation allows the solution of the initial-value problem for an arbitrary three-dimensional (3D) disturbance via a two-dimensional (2D) initial-value problem at a smaller Reynolds number $\mathit{Re}_{2D}$. Its implications for the transient growth of arbitrary 3D disturbances is studied. Using the Squire transformation, the general solution of the initial-value problem is shown to predict large-Reynolds-number scaling for the optimal gain at all optimization times $t$ with ${t}/{\mathit{Re}}$ finite or large. This result is an extension of the well-known scaling laws first obtained by Gustavsson (J. Fluid Mech., vol. 224, 1991, pp. 241–260) and Reddy & Henningson (J. Fluid Mech., vol. 252, 1993, pp. 209–238) for arbitrary $\alpha \mathit{Re}$, where $\alpha $ is the streamwise wavenumber. The Squire transformation is also extended to the adjoint problem and, hence, the adjoint Orr–Sommerfeld and Squire modes. It is, thus, demonstrated that the long-time optimal growth of 3D perturbations as given by the exponential growth (or decay) of the leading eigenmode times an extra gain representing its receptivity, may be decomposed as a product of the gains arising from purely 2D mechanisms and an analytical contribution representing 3D growth mechanisms equal to $1+ \left (\beta \mathit{Re}/\mathit{Re}_{2D}\right )^2 \mathcal{G}$, where $\beta $ is the spanwise wavenumber and $\mathcal{G}$ is a known expression. For example, when the leading eigenmode is an Orr–Sommerfeld mode, it is given by the product of respective gains from the 2D Orr mechanism and an analytical expression representing the 3D lift-up mechanism. Whereas if the leading eigenmode is a Squire mode, the extra gain is shown to be solely due to the 3D lift-up mechanism. Direct numerical solutions of the optimal gain for plane Poiseuille and plane Couette flow confirm the novel predictions of the Squire transformation extended to the initial-value problem. These results are also extended to confined shear flows in the presence of a temperature gradient.

On the stability of plane Couette–Poiseuille flow with uniform crossflow

We present a detailed study of the linear stability of the plane Couette–Poiseuille flow in the presence of a crossflow. The base flow is characterized by the crossflow Reynolds number Rinj and the dimensionless wall velocity k. Squire's transformation may be applied to the linear stability equations and we therefore consider two-dimensional (spanwise-independent) perturbations. Corresponding to each dimensionless wall velocity, k ∈ [0, 1], two ranges of Rinj exist where unconditional stability is observed. In the lower range of Rinj, for modest k we have a stabilization of long wavelengths leading to a cutoff Rinj. This lower cutoff results from skewing of the velocity profile away from a Poiseuille profile, shifting of the critical layers and the gradual decrease of energy production. Crossflow stabilization and Couette stabilization appear to act via very similar mechanisms in this range, leading to the potential for a robust compensatory design of flow stabilization using either mechanism. As Rinj is increased, we see first destabilization and then stabilization at very large Rinj. The instability is again a long-wavelength mechanism. An analysis of the eigenspectrum suggests the cause of instability is due to resonant interactions of Tollmien–Schlichting waves. A linear energy analysis reveals that in this range the Reynolds stress becomes amplified, the critical layer is irrelevant and viscous dissipation is completely dominated by the energy production/negation, which approximately balances at criticality. The stabilization at very large Rinj appears to be due to decay in energy production, which diminishes like Rinj−1. Our study is limited to two-dimensional, spanwise-independent perturbations.

Applicability of the squire transformation in linearized problems on shear stability

In eigenvalue problems for the Orr-Sommerfeld equation with nonclassical boundary conditions (which differ from the adhesion conditions), the applicability of the Squire transformation is studied (this transformation enables one to reduce the three-dimensional picture of perturbations to the two-dimensional picture in the plane of the principal shear. Linearized equations are derived that generalize the Orr-Sommerfeld equations to tensor linear non-Newtonian media. It is shown that the Squire transformation algorithm is not applicable to stability problems for shear flows of these media.

Instabilities in free-surface Hartmann flow at low magnetic Prandtl numbers

We study the linear stability of the flow of a viscous electrically conducting capillary fluid on a planar fixed plate in the presence of gravity and a uniform magnetic field, assuming that the plate is either a perfect electrical insulator or a perfect conductor. We first confirm that the Squire transformation for magnetohydrodynamics is compatible with the stress and insulating boundary conditions at the free surface but argue that unless the flow is driven at fixed Galilei and capillary numbers, respectively parameterizing gravity and surface tension, the critical mode is not necessarily two-dimensional. We then investigate numerically how a flow-normal magnetic field and the associated Hartmann steady state affect the soft and hard instability modes of free-surface flow, working in the low-magnetic-Prandtl-number regime of conducting laboratory fluids (Pm ≤ 10−4). Because it is a critical-layer instability (moderately modified by the presence of the free surface), the hard mode exhibits similar behaviour as the even unstable mode in channel Hartmann flow, in terms of both the weak influence of Pm on its neutral-stability curve and the dependence of its critical Reynolds number Rec on the Hartmann number Ha. In contrast, the structure of the soft mode's growth-rate contours in the (Re, α) plane, where α is the wavenumber, differs markedly between problems with small, but non-zero, Pm and their counterparts in the inductionless limit, Pm ↘ 0. As derived from large-wavelength approximations and confirmed numerically, the soft mode's critical Reynolds number grows exponentially with Ha in inductionless problems. However, when Pm is non-zero the Lorentz force originating from the steady-state current leads to a modification of Rec(Ha) to either a sub-linearly increasing or a decreasing function of Ha, respectively for problems with insulating or perfectly conducting walls. In insulating-wall problems we also observe pairs of counter-propagating Alfvén waves, the upstream-propagating wave undergoing an instability driven by energy transferred from the steady-state shear to both of the velocity and magnetic degrees of freedom. Movies are available with the online version of the paper.

A note on the linear stability of a two-dimensional Stokes layer

The fluid motion generated adjacent to an infinite flat plate undergoing orbital motion in its own plane is a generalization of the classical Stokes-layer profile. We show that the stabilities of these flows can be related to each other via two transformations directly analogous to the well-known Squire transformation. The main result obtained is that, in general, the two-dimensional Stokes layer is more stable than the corresponding unidirectional Stokes layer. A further by-product of the analysis is the construction of a shear flow having identical neutral stability conditions when subject to either two- or three-dimensional disturbances.

Effet d'un champ magnétique transversal sur la stabilité de l'écoulement de Hartmann : les modes tridimensionnels

A numerical study is conducted in order to determine the influence of a transverse magnetic field on the three-dimensional instabilities of the Hartmann flow. We prove that the Squire transformation currently used to characterize such instabilities give satisfactory results only in the case where Ha is weak and 3D modes are slightly deviated from the 2D modes. Such a study is justified by the fact that the two-dimensional instabilities could not always explain successfully the experimental observations relating to the transition phenomenon. To cite this article: M. Jédidi et al., C. R. Mecanique 333 (2005).

Gravity Wave Generation by a Three-Dimensional Thermal Forcing

Abstract Thermal forcing is one of the mechanisms of wave generation in convection. Although it does not account for all the wave generation mechanisms, thermal forcing is a good proxy for estimating the gravity wave spectrum forced by convection. This work presents a linear theory of gravity wave generation by a three-dimensional thermal forcing assuming constant mean wind conditions in the forcing region. Relationships are derived linking the amplitude and spectral characteristics of the generated gravity waves to the properties of the thermal forcing. It is shown that the three-dimensional problem can be viewed as a multiple two-dimensional problem by means of a squire transformation and that the gravity wave properties in a given azimuthal direction depend on the projection of the heating and environmental mean wind in that direction. Shown in detail is the sensitivity of the gravity wave characteristics in three dimensions to varying heating parameters, and it is concluded that the knowledge of both ...

Linear Stability of Parallel Inviscid Flows of Shallow Water and Bubbly Fluid

The extensions of Howard's semicircle theorem, Squire's transformation, and Rayleigh's stability criterion are obtained for the model of dispersive shallow water waves and the model of bubbly fluid.

Squire theorem with variable physical properties

The Squire theorem or Squire transformation is studied under the influence of variable physical properties. It is demonstrated that the Squire theorem can still hold if the parallel base flow and the base temperature are weakly dependent on the flow direction.

Stability of inviscid flow in a flexible tube

The equivalents of the classical theorems of hydrodynamic stability are derived for inviscid flow through a flexible tube. An important difference between flows in plane and cylindrical geometries is that the Squire transformation, which states that two-dimensional perturbations in plane parallel flows are always more unstable than three-dimensional perturbations, is not valid for tube flows. Therefore, it is necessary to analyse both axisymmetric and non-axisymmetric perturbations in flows in cylindrical geometries. Perturbations of the form exp[ik(χ –ct) + inϕ] are imposed on a steady axisymmetric mean flow V(r), and the stability of the mean velocity profiles and bounds for the phase velocity of the unstable modes are determined. Here r, ϕ and x are the radial, polar and axial directions, and k and c are the wavenumber and phase velocity. The flexible wall is represented by a standard constitutive equation which contains inertial, elastic and dissipative terms. Results for general velocity profiles are derived in two limiting cases: axisymmetric flows (n = 0) and highly non-axisymmetric flows (n [Gt ] k). The results indicate that axisymmetric perturbations are always stable for (V″ – r−1V′) V < 0 and could be unstable for (V″ – r−1V′) V < 0, while highly non-axisymmetric perturbations are always stable for (V″ + r−1V′) V [ges ] 0 and could be unstable for (V″ + r−1V′) V < 0. In addition, bounds on the real part (cr) and imaginary part (ci) of the phase velocity are also derived. For the practically important case of Hagen–Poiseuille flow, the present analysis indicates that axisymmetric perturbations are always stable, while highly non-axisymmetric perturbations could be unstable. This is in contrast to plane parallel flows where two-dimensional disturbances are always more unstable than three-dimensional ones.

Stability of mixed-convection flow in a tall vertical channel under non-boussinesq conditions

We have examined the linear stability of the fully developed mixed-convection flow in a differentially heated tall vertical channel under non-Boussinesq conditions. The Three-dimensional analysis of the stability problem was reduced to an equivalent two-dimensional one by the use of Squire's transformation. The resulting eigenvalue problem was solved using an integral Chebyshev pseudo-spectral method. Although Squire's theorem cannot be proved analytically, two-dimensional disturbances are found to be the most unstable in all cases. The influence of the non-Boussinesq effects on the stability was studied. We have investigated the dependence of the critical Grashof and Reynolds numbers on the temperature difference. The results show that four different modes of instability are possible, two of which are new and due entirely to non-Boussinesq effects.

Stability of unstably stratified shear flow in a channel under non-Boussinesq conditions

We investigate the linear stability of unstably stratified Poiseuille flow between two horizontal parallel plates under non-Boussinesq conditions. It is shown that Squire's transformation can be used to reduce the three-dimensional stability problem to an equivalent two-dimensional one. The eigenvalue problem, consisting of the generalized Orr-Sommerfeld equations, is solved numerically using an integral Chebyshev pseudo-spectral method. The influence of the non-Boussinesq effects on stability is studied. The dependence of the critical Rayleigh number on the Reynolds number and temperature difference parameter is obtained. As in the Boussinesq case, results show that the most unstable mode is that of longitudinal rolls. However, in contrast to the Boussinesq case, the rolls are highly distorted for large temperature differences. In addition, the critical Rayleigh number increases with the increase of the temperature difference and is independent of the Reynolds number.

A short note on Squire's theorem for interfacial instabilities in a stratified flow of two superposed fluids

In a recent paper Schaflinger (1994) investigated interfacial instabilities in a stratified flow of two superposed fluids. The paper focused on a viscous resuspension flow (Schaflinger et al., 1990) in which a sediment of non-buoyant, solid particles is resuspended and driven by a clear fluid flow. At the suggestion of one reviewer, Schaflinger (1994) mentions that "[a]ccording to Squire (1933) three-dimensional disturbances in the classical, inviscid problem can be transformed into an equivalent two-dimensional problem, i.e. to each unstable three-dimensional disturbance there corresponds a more unstable two-dimensional one (Drazin and Reid, 1981, p. 129). Even though it seems to be imaginative to carry over Squire's theorem to the current study, the foundation supporting Squire's statement may not hold in the present problem. No studies dealing with this specific issue have been found in the literature. Therefore, this matter is subject to further, thorough investigations." Here we show that Squire's transformation indeed reduces the equations and the boundary conditions for a stratified flow to an equivalent two-dimensional problem.

Asymptotic evaluation of three-dimensional wave packets in parallel flows

This paper examines the three-dimensional wave packets which are generated by an initially localized pulse disturbance in an incompressible parallel flow and described by a double Fourier integral in the wavenumber space. It aims to clear up some confusion arising from the asymptotic evaluation of this integral by the method of steepest descent. In this asymptotic analysis, the calculation of the eigenvalues can be facilitated by making use of the Squire transformation. It is demonstrated that the use of the Squire transformation introduces branch points in the saddle-point equation that links the physical coordinates to the saddle-point value, regardless of whether the flow is viscous or inviscid. It is shown that the correct branch should be chosen according to the principle of analytic continuation. The saddle-point values for the three-dimensional problem should be considered to be the analytic continuation of those for the two-dimensional case where the saddle-point values can be uniquely determined. The three-dimensional wave packets in an inviscid wake flow are examined; their behaviour at large time is calculated asymptotically by the method of steepest descent in terms of the two-dimensional eigenvalue relation.

Three-dimensional linear instability of parallel shear flows

The three-dimensional linear instability of parallel shear flows is investigated through simple geometrical constructions based on the classical Squire transformation. It is shown that if a profile is unstable for a finite Reynolds number and some value of (total) horizontal wavenumber, this profile is also unstable for all larger Reynolds numbers and the same wavenumber. In particular, the direction of the unstable mode tends toward the perpendicular as the Reynolds number increases, thus providing the viscous/shear balance required for resistive instability. The example of plane Poiseuille flow is discussed in detail, and the results compared with classical viscous and inviscid stability criteria.

On the interaction between Tollmien-Schlichting and Rayleigh-Bénard instabilities in the presence of a longitudinal magnetic field

AbstractThe method used by Gage and Reid(10) to investigate hydrodynamic stability of thermally stratified fluid is extended to hydromagnetic stability to study the effect of aligned magnetic field on the stability of unstable thermal stratification under the assumption of small magnetic Reynolds number. The interaction between the Tollmien–Schlichting–Stuart mechanism of instability due to shear and magnetic field and Rayleigh–Bénard–Thompson mechanism of instability due to thermally unstable stratification and magnetic field is brought out in detail. It is shown that, although Squire's transformation can be used to reduce the three-dimensional problem to an equivalent two-dimensional one, Squire's theorem is not valid. This conclusion follows from the fact that in our analysis the Richardson number Ri ( < 0) will not be greater than the value −0·92 × 10−6. In particular, it is shown that for the values of stratification parameter n ≤ 0·6 the effect of magnetic field for small values of Stuart number N is to augment instability and impose the restriction on the validity of our numerical procedure. However, for η = 0·8 a sharp transition from unstable to stable flow takes place at N = 0·3. A physical explanation for this based on eddies is given.

Secondary convective motions in a plane inclined layer

A linear theory of stability of a plane-parallel convective flow between infinite isothermal planes heated to different temperature was developed in [1–6]. At moderate Pr values the instability is monotonic and leads to the development of steady secondary motions. These motions for the case of a vertical layer have been investigated by the net [7, 8] and small-parameter [9] methods. In this paper steady secondary motions in an inclined layer are investigated. The small-parameter and net methods are used. The hard nature of excitation of secondary motions in a defined range of tilt angles is established. There are two types of secondary motions, whose regions of existence overlap — vortices at the boundary of countercurrent streams and convection rolls; the hard instability is due to the development of convection rolls. The analog of the Squire transformation obtained in [4] for infinitely small disturbances of a plane-parallel convective flow is extended to secondary motions of finite amplitude.