Three-dimensional Disturbances Research Articles

ON TURBULENCE MODELLING AND THE TRANSITION FROM LAMINAR TO TURBULENT FLOW

AbstractFluid turbulence is often modelled using equations derived from the Navier–Stokes equations, perhaps with some semi-heuristic closure model for the turbulent viscosity. This paper considers a possible alternative hypothesis. It is argued that regarding turbulence as a manifestation of non-Newtonian behaviour may be a viewpoint of at least comparable validity. For a general description of nonlinear viscosity in a Stokes fluid, it is shown that the flow patterns are indistinguishable from those predicted by the Navier–Stokes equation in one- or two-dimensional geometry, but that fully three-dimensional flows differ markedly. The stability of linearized plane Poiseuille flow to three-dimensional disturbances is then considered, in a Tollmien–Schlichting formulation. It is demonstrated that the flow may become unstable at significantly lower Reynolds numbers than those expected from Navier–Stokes theory. Although similar results are known in sections of the rheological literature, the present work attempts to advance the philosophical viewpoint that turbulence might always be regarded as a non-Newtonian effect, to a degree that is dependent only on the particular fluid in question. Such an approach could give a more satisfactory account of the underlying physics.

Dynamics of nonlinear three-dimensionalwaves on the interface between two fluids in a channel with low-sloping bottom and top

A combined approach is proposed to describe the transformation of three-dimensional disturbances of the interface between two incompressible immiscible fluids of different densities contained in a channel with fixed rigid top and bottom. It is assumed that the wavelengths are moderately large, the amplitudes are small but finite, the top and the bottom can be gently sloping, and capillary effects are small. The system of equations derived is applicable for modeling disturbances simultaneously scattering in arbitrary horizontal directions. Some typical wave problems are numerically solved and the effect of governing parameters is shown.

CURRENT SINGULARITIES IN LINE-TIED THREE-DIMENSIONAL MAGNETIC FIELDS

This paper considers the current distributions that derive from finite amplitude perturbations of line-tied magnetic fields comprising hyperbolic field structures. The initial equilibrium on which we principally focus is a planar magnetic X-point threaded by a uniform axial field. This field is line-tied on all surfaces but subject to three-dimensional (3D) disturbances that alter the initial topology. Results of ideal relaxation simulations are presented which illustrate how intense current structures form that can be related, through the influence of line-tying, to the quasi-separatrix layers (QSLs) of the initial configuration. It is demonstrated that the location within the QSL that attracts the current, and its scaling properties, are strongly dependent on the relative dimensions of the QSL with respect to the line-tied boundaries. These results are contrasted with the behavior of a line-tied 3D field containing an isolated null point. In this case, it is found that the dominant current always forms at the null, but that the collapse is inhibited when the null is closer to a line-tied boundary.

Tracking stages of transition in Couette flow analytically

AbstractThe current study focuses on a transition scenario in which the linear transient growth mechanism is initiated by four decaying normal modes. It is shown that the four modes, the initial structure of which corresponds to counter-rotating vortex pairs, are sufficient to capture the transient growth mechanism. More importantly, it is demonstrated that the kinetic energy growth of the initial disturbance is not the key parameter in this transition mechanism. Rather, it is the ability of the transient growth process to generate an inflection point in the wall-normal direction and consequently to make the flow susceptible to a three-dimensional disturbance leading to transition to turbulence. Because of the minimal number of modes participating in the transition process, it is possible to follow its earlier key stages analytically and to compare them with the results of direct numerical simulation. This procedure reveals the role of various flow parameters during the transition, such as the difference between symmetric and antisymmetric transient growth scenarios. Moreover, it is shown that the resulting modified base flow of the linear process is not sufficient to produce a significant localized maximum of the base-flow vorticity (i.e. a ‘strong’ inflection point), and it is only due to nonlinear effects that the base flow becomes unstable with respect to an infinitesimal three-dimensional disturbance. Finally, the physical mechanism during key stages of transition is well captured by the analytical expressions. Furthermore, the vortex dynamics during these stages is very similar to the model proposed by Cohen, Karp & Mehta (J. Fluid Mech., vol. 747, 2014, pp. 30–43) according to which streamwise variation of the initial counter-rotating vortex pair is required to generate concentrated spanwise vorticity, which together with the lift-up by the induced velocity and shear of the base flow generates packets of hairpins.

Direct numerical simulations of laminar separation bubbles: investigation of absolute instability and active flow control of transition to turbulence

AbstractLaminar separation bubbles generated on a flat plate by an adverse pressure gradient are investigated using direct numerical simulations (DNSs). Two-dimensional periodic forcing is applied at a blowing/suction slot upstream of separation. Control of separation through forcing with various frequencies and amplitudes is examined. For the investigation of absolute instability mechanisms, baseflows provided by two-dimensional Navier–Stokes calculations are analysed by introducing pulse disturbances and computing the three-dimensional flow response using DNS. The primary instability of the time-averaged flow is investigated with a local linear stability analysis. Employing a steady flow solution as baseflow, the nonlinear and non-parallel effects on the self-sustained disturbance development are illustrated, and a feedback mechanism facilitated by the upstream flow deformation is identified. Secondary instability is investigated locally using spatially periodic baseflows. The flow response to pulsed forcing indicates the existence of an absolute secondary instability mechanism, and the results indicate that this mechanism is dependent on the periodic forcing. Results from three-dimensional DNS provide insight into the global instability mechanisms of separation bubbles and complement the local analysis. A forcing strategy was devised that suppresses the temporal growth of three-dimensional disturbances, and as a consequence, breakdown to turbulence does not occur. Even for a separation bubble that has transitioned to turbulence, the flow relaminarizes when applying two-dimensional periodic forcing with proper frequencies and amplitudes.

Mixed convection of a low Prandtl fluid with spatially periodic lower wall heating in the presence of a wall-normal magnetic field

Numerical simulations of the combined natural and forced convection flow in a horizontal channel have been carried out for a fluid of low Prandtl number in the presence of a uniform wall-normal magnetic field. The upper wall is maintained at a constant temperature, while a spatially periodic temperature is imposed at the lower wall. A stability diagram is created based on the results from two- and three-dimensional numerical simulations in order to examine whether the present two-dimensional flows are unstable to three-dimensional disturbances. At the range of parameters studied (5000⩽Ra⩽150,000-Rayleigh number, 5⩽Re⩽500-Reynolds number, and 0⩽Ha⩽20-Hartmann number), no stable two-dimensional unsteady flow is observed. All mixed convection flows are suppressed by the action of sufficiently strong electromagnetic forces and, in contrast to the hydrodynamic cases, the direct transition to unsteadiness is favored with an apparent lack of steady three-dimensional magnetohydrodynamic flows. The velocity and temperature features of the various flow regimes are discussed, addressing also the differences in the predictions produced by the two- and three-dimensional numerical simulations.

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.

Waves on a Stuart vortex

We consider the linear stability of Stuart vortices, an exact periodic solution of the two-dimensional inviscid Euler equations of hydrodynamics consisting of an infinite row of co-rotating vortices, to wavelike disturbances when the vortex amplitude is small but still much larger than the disturbances. An unsteady critical layer analysis is used, with outer and inner expansions away from and near the vortex cores which we match to obtain disturbance growth rates. Disturbance-vortex interactions occur inside the critical layer and are neglected elsewhere. We consider α+-1∼O(1) and α+≈1 separately, where α+ is ratio of the disturbance wavenumber to that of the periodic vortex row. We consider primarily three-dimensional (oblique) disturbances but touch upon two-dimensional disturbances in the vortex plane. There are resonance mechanisms for oblique disturbances when k+⩽1, with k+ the ratio of the streamwise disturbance wavenumber to that of the periodic vortex row. When k+=1, this is fundamental mode or Benney–Lin instability, thought to be connected to streamwise streaks seen in experiments, and when k+=1/2, this is subharmonic instability or helical pairing, thought to be connected to vortex pairing.

Experimental investigation of freely falling thin disks. Part 2. Transition of three-dimensional motion from zigzag to spiral

AbstractThe free-fall motion of a thin disk with small dimensionless moments of inertia (${I}^{\ast } \lt 1{0}^{- 3} $) was investigated experimentally. The transition from two-dimensional zigzag motion to three-dimensional spiral motion occurs due to the growth of three-dimensional disturbances. Oscillations in the direction normal to the zigzag plane increase with the development of this instability. At the same time, the oscillation of the nutation angle decreases to zero and the angle remains constant. The effects of initial conditions (release angle) were investigated. Two kinds of transition modes, zigzag–spiral transition and zigzag–spiral–zigzag intermittence transition, were observed to be separated by a critical Reynolds number. In addition, the solution of the generalized Kirchhoff equations shows that the small ${I}^{\ast } $ is responsible for the growth of disturbances in the third dimension (perpendicular to the planar motion).

Optimal disturbances above and upstream of a flat plate with an elliptic-type leading edge

Adjoint-based iterative methods are employed to compute linear optimal disturbances in a spatially growing boundary layer around an elliptic leading edge. The Lagrangian approach is used where an objective function is chosen and constraints are assigned. The optimisation problem is solved using power iterations combined with a matrix-free formulation, where the state is marched forward in time with a standard direct numerical simulation solver and backward with the adjoint solver until a chosen convergence criterion is fulfilled. We consider the global and, more relevant to receptivity studies, the upstream localised optimal initial condition leading to the largest possible energy amplification at time T. We find that the two-dimensional initial condition with the largest potential for growth is a Tollmien–Schlichting-like wave packet that includes the Orr mechanism and is located inside the boundary layer downstream of the leading edge. Three-dimensional optimal disturbances induce streaks by the lift-up mechanism. Requiring the optimal initial condition to be localised upstream of the plate enables us to better study the effects of the leading edge on the boundary layer receptivity mechanisms. Two-dimensional upstream disturbances are inefficient at triggering unstable eigenmodes, whereas three-dimensional disturbances induce streamwise streaks with significant growth.

Bifurcation Study of Three-Dimensional Solutions of the Curved Square-Duct Flow

Steady, periodic and traveling-wave solutions of the flow through a curved square duct are obtained numerically, and their linear stability to two- and three-dimensional disturbances is investigated. It is found that a steady traveling-wave (STW) solution bifurcates from the symmetric steady solution branch and is linearly stable in the intermediate range between low and high pressure gradient regions. Two stable periodic traveling-wave (PTW) solutions are also found, which have oscillating components with different symmetry. One keeps the same symmetry as the STW solution and the other breaks it. They seem to bifurcate from the STW solution branch via Hopf bifurcation although their solution branches are not obtained exactly. It is strongly suggested that a two-dimensional time-periodic (2DP) solution is created via heteroclinic bifurcation, not from local bifurcation. 2DP solution is always linearly unstable against three-dimensional disturbances though it has linearly stable region to two-dimensional disturbances.

WAVE-VORTEX MODE COUPLING IN ASTROPHYSICAL ACCRETION DISKS UNDER COMBINED RADIAL AND VERTICAL STRATIFICATION

We examine accretion disk flow under combined radial and vertical stratification utilizing a local Cartesian (or shearing box) approximation. We investigate both axisymmetric and nonaxisymmetric disturbances with the Boussinesq approximation. Under axisymmetric disturbances, a new dispersion relation is derived. It reduces to the Solberg-Hoiland criterion in the case without vertical stratification. It shows that, asymptotically, stable radial and vertical stratification cannot induce any linear instability; Keplerian flow is accordingly stable. Previous investigations strongly suggest that the so-called bypass concept of turbulence (i.e., that fine-tuned disturbances of any inviscid smooth shear flow can reach arbitrarily large transient growth) can also be applied to Keplerian disks. We present an analysis of this process for three-dimensional plane-wave disturbances comoving with the shear flow of a general rotating shear flow under combined stable radial and vertical rotation. We demonstrate that large transient growth occurs for K 2/k 1 1 and k 3 = 0 or k 1 ~ k 3, where k 1, K 2, and k 3 are the azimuthal, radial, and vertical components of the initial wave vector, respectively. By using a generalized wave-vortex decomposition of the disturbance, we show that the large transient energy growth in a Keplerian disk is mainly generated by the transient dynamics of the vortex mode. The analysis of the power spectrum of total (kinetic+potential) energy in the azimuthal or vertical directions shows that the contribution coming from the vortex mode is dominant at large scales, while the contribution coming from the wave mode is important at small scales. These findings may be confirmed by appropriate numerical simulations in the high Reynolds number regime.

Time-averaged and time-resolved volumetric velocimetry measurements of a laminar separation bubble on an airfoil

Measurements of a laminar separation bubble on an NACA4412 airfoil were performed using volumetric three-component velocimetry (V3V) at a Reynolds number of 50,000 and different angles of attack. The time-averaged V3V results were analyzed to estimate the locations of transition onset and reattachment by interpreting the relationships of the different Reynolds normal and shear stresses to these locations. More specifically, the locations of separation and mean reattachment are defined as the beginning and end of the reverse-flow region, respectively. The onset of transition is located by finding the point where the growth rates of the Reynolds shear stresses changes significantly. The results show that the spanwise Reynolds normal stresses and the Reynolds shear stresses grow in the transition and reattachment regions, respectively, suggesting that three-dimensional disturbances grow before the complete breakdown to turbulence. Counter-rotating streamwise and wall-normal vorticity exist in the reattachment region. Organized relations are found between the streamwise vorticity, vortex cores, and ejection and sweep events.

Linear stability of flow in rectangular ducts in the vicinity of the critical aspect ratio

We have performed three-dimensional BiGlobal linear stability analyses of flow in rectangular ducts by using a spectral element method. A linear stability theory is applied to a steady laminar solution and an eigenvalue problem is generated in the matrix form. The critical Reynolds numbers are evaluated for some aspect ratios to describe the neutral curves. It is found that the neutral curves consist of two bifurcation branches. One is the first bifurcation branch where the flow loses its stability with respect to three-dimensional disturbances, while the other is the second bifurcation branch where it recovers its stability. The first and second branches connect at the critical point of As=3.19 and Re=2.47×105, where the saddle–node bifurcation occurs. We have revealed that the discontinuous divergence of the critical Reynolds number to infinity in the vicinity of the critical aspect ratio As is caused by the saddle–node bifurcation.

S055011 矩形断面曲がり管内流の分岐解析([S055-01]内部流れ(1))

Steady, periodic and traveling-wave solutions of the flow through a curved square duct are obtained numerically, and their linear stability to two- and three-dimensional disturbances is investigated. It is found that a steady traveling-wave (STW) solution bifurcates from the symmetric steady solution branch and is linearly stable in the intermediate range between low and high pressure gradient regions. Two stable periodic traveling-wave (PTW) solutions are also found, which have oscillating components with different symmetry. One keeps the same symmetry as the STW solution and the other breaks it. It is strongly suggested that a two-dimensional time-periodic (2DP) solution is created via heteroclinic bifurcation, not from local bifurcation. 2DP solution is always linearly unstable against three-dimensional disturbances though it has linearly stable region to two-dimensional disturbances.

Linear Instability Analysis of Confined Compressible Boundary/Mixing Layers Flow

A compressible supersonic confluent flow composed of boundary layers and mixing layers are studied by linear stability theory. The flow is confined in a two-dimensional adiabatic channel. A slower flow lying in the center mixes with faster boundary layer flows on both sides and two mixing layers are evolved near the centerline. Different unstable modes were discovered and the first mode was found to be most unstable. Three-dimensional disturbances were investigated and comparison of instability features was made with unconfined boundary layer flows. The investigation of different slow flow widths was also made and a smaller spacing between the boundary layer and mixing layer was found to suppress the growth of disturbance.

An asymptotic approach to transition

A unified theory of large-sized three-dimensional disturbances set up is aimed at developing a dynamical system of transition embracing incompressible fluid as well as subsonic and transonic flows. In appropriately scaled and normalised variables, incompressible and subsonic pulsations obey the same evolution equation for the instantaneous displacement thickness alone. The dynamical system for three-dimensional transonic non-linear waves involves the self-induced pressure as an additional desired function. The notion of the inviscid/inviscid interaction between the upper potential region and an adjustment sublayer at the bottom underlies the theory. In accord with available observational data, the coexistense of organised motion and irregular disturbances is the main conclusion to be drawn from the new asymptotic approach.

Three-dimensional disturbances of a plane-parallel two-layer flow of a viscous, heat-conducting fluid

The stability of the Marangoni-Poiseuille flow in an inclined channel against three-dimensional disturbances is studied. It is shown that the nature of instability is determined by wall heating conditions and the system orientation with respect to its horizontal position. With variation in the angle of inclination of the system spiral disturbances leading to the system crisis can develop.

Three Dimensional Film Stability and Draw Resonance

In order to understand the effects of inertia and gravity on draw resonance and on the physical mechanism of draw resonance in three-dimensional Newtonian film casting, a linear stability analysis has been conducted. An eigenvalue problem resulting from the linear stability analysis is formulated and solved as a nonlinear two-point boundary value problem to determine the critical draw ratios. Neutral stability curves are plotted to separate the stable/unstable domain in different appropriate parameter spaces. Both inertia and gravity stabilize the process and the process is more unstable to two- than to three-dimensional disturbances. The effects of inertia and gravity on the physical mechanism of draw resonance have been investigated using the eigenfunctions from the eigenvalue problem. A new approach is introduced in order to evaluate the traveling times of kinematic waves from the perturbed thickness at the take-up, which satisfies the same stability criterion illustrating the general stability of the system.

Discrete linear local eigenmodes in a separating laminar boundary layer

AbstractThe evolution of two- and three-dimensional small-amplitude disturbances in the laminar part of a laminar separation bubble is investigated in detail. We apply a combination of local linear stability theory, results from different experimental measurement campaigns and direct numerical simulations to identify two different discrete eigenmodes in the laminar part of the bubble. A stable eigenmode, the outer mode, governs unsteady oscillations in the upstream part of the bubble. However, this perturbation is quickly overtaken by an unstable eigenmode, the inner mode, which eventually leads to transition of the detached shear layer. Such a behaviour is observed due to an acceleration region with a favourable pressure gradient preceding the adverse-pressure-gradient region. The flow is stable in the acceleration region, in which the outer mode is only moderately damped, while the inner mode is strongly damped. At the onset of instability for the unstable eigenmode upstream of separation, both viscous Tollmien–Schlichting and inviscid Kelvin–Helmholtz instability mechanisms contribute to amplification, while deeper inside the bubble only the inviscid mechanism is active. If the explicit forcing is moved to a region downstream of the favourable pressure gradient, only the unstable eigenmode appears. The same behaviour is found for two-dimensional and weakly oblique waves.