Vortical Disturbances Research Articles

Continuous mode transition and the effects of pressure gradient

Continuous mode transition is an instance of the bypass route to boundary-layer turbulence. The stages that precede breakdown are explained in terms of continuous Orr–Sommerfeld and Squire spectra. In that context, the role of pressure gradient is less evident than it is in natural transition. Its role is investigated using linear theory and numerical simulations. Both approaches demonstrate that adverse pressure gradients enhance the coupling of low-frequency vortical disturbances to the boundary-layer shear. The result is stronger boundary-layer perturbation jets – or Klebanoff distortions. The correlation between the intensity of the perturbation jets and transition location is tested by direct numerical simulations of pairwise continuous mode interactions; such interactions can reproduce the entire transition process. The results confirm that stronger perturbation jets are more unstable, and hence provoke early transition in adverse pressure gradient. This is also consistent with the experimental observation that transition becomes independent of pressure gradient at high turbulent intensities. Under such conditions, boundary-layer streaks are highly unstable and transition is achieved swiftly, independent of the mean gradient in pressure.

Evolution of a localized vortex in plane nonparallel viscous flows with constant velocity shear. I. Hyperbolic flow

The framework of the linear theory is employed to study the evolution of an initial compact vortical disturbance in unbounded plane nonparallel viscous incompressible flows with constant velocity gradients. Two types of such flows are known to be possible: hyperbolical and elliptical (as well as an intermediate case of the well-studied parallel Couette flow). The results presented here are obtained for a hyperbolical flow. (Results concerning the elliptical flow are to be issued in a separate publication.) This paper is a development of earlier work by R. R. Lagnado, N. Phan-Thien, and L. G. Leal [Phys. Fluids 27, 1094 (1984)] studying the stability of a hyperbolical flow relative to the simplest perturbations in the form of plane waves with a time-dependent wave vector. The dynamics of vortex intensity is investigated as well as the evolution of its geometrical form and orientation. The results are discussed in the context of the problem of hairpin vortex formation.

Distributed two-dimensional boundary-layer receptivity to non-stationary vortical disturbances in the presence of surface roughness

The distributed (over the longitudinal coordinate) excitation of two-dimensional (2D) Tollmien — Schlichting (TS) waves by weak non-stationary free-stream vortices propagating along the edge of the laminar boundary layer developing over a surface with small-amplitude 2D roughness is examined. The vorticity vector of the free-stream vortices was oriented over the span of the model, i. e., did not depend on the transverse coordinate. The theoretical analysis of the excitation mechanism reported in [1] was refined to develop, around it, a procedure making it possible to experimentally determine the coefficients of distributed vortical receptivity of the flow and the coefficients of “vortex-roughness” receptivity by fitting the experimental distributions with analytical solutions. Under conditions with controllable excitation of disturbances, a detailed hot-wire study of free-stream disturbances and boundary-layer disturbances at several vortex frequencies and at several surface-roughness periods was performed, and the shape of the controllable surface roughness was measured. The point-source method was employed to experimentally examine the characteristics of linear three-dimensional (3D) stability of the flow to TS waves, necessary for determination of the coefficients of distributed receptivity. It was found that the free-stream vortices with transverse orientation of the vorticity vector excited boundary-layer TS waves via two receptivity mechanisms: (a) on the smooth surface (due to natural non-uniformity of the flow) and (b) during interaction of the vortices with the surface roughness. The developed approach was used to experimentally estimate the amplitudes and phases of the coefficients of both types of distributed vortical receptivity as dependent on problem parameters. The absolute values of both types of receptivity coefficients were found to rapidly grow in value with increasing vortex frequency. It is shown that the most efficient excitation of TS waves is observed in the situation with satisfied resonance conditions for streamwise vortex, surface-roughness, and TS-wave streamwise wavenumbers, resulting in strong deviation of the increments of the TS waves from the linear-stability increments. Under no-resonance conditions, only amplitude beats of boundary-layer disturbances were observed.

The evolution of a finite-amplitude localized disturbance in an irrotational unbounded plane stagnation flow

The evolution of a finite-amplitude three-dimensional localized disturbance, having an initial dipole Gaussian vorticity distribution, embedded in an external unbounded irrotational plane stagnation flow (U = (Ay,Ax,0)) is investigated. Using the fluid impulse integral as a characteristic of such a disturbance, the viscous vorticity equation is integrated analytically. Accordingly, the fluid impulse associated with such disturbances decays and grows exponentially along the principal axes x=y and x=-y, respectively. Numerical simulations, carried out for both linear and nonlinear disturbances at a Reynolds number of 40, confirm the above predictions. The simulations have also been compared with the solution of the linear viscous vorticity disturbance equation. While the solution predicts the vorticity distribution for the linear case, it fails to predict the essential characteristics of a nonlinear disturbance associated with its self-induced movement. Finally, it is shown that the fluid impulse and the disturbance kinetic energy follow the same trend, i.e. when the fluid impulse increases with time so does the kinetic energy and vice versa. The correspondence between them suggests the use of the fluid impulse to predict the stability of a localized disturbance. © 2006 Cambridge University Press.

Numerical simulation of unsteady blade row interactions induced by passing wakes

A numerical study of the unsteady phenomena resulting of periodic passing wakes is presented. An unsteady passing wake boundary condition is implemented in a three-dimensional Navier–Stokes code. Unsteady computations are performed to evaluate the capability of the code to simulate the rotor–stator interaction flow. The analysis of the flow structures shows the vortical disturbances and the migration of the incoming wakes through the blade passage. This physical analysis allows to separate the main origins of the losses.

The generation of streaks and hairpin vortices from a localized vortex disturbance embedded in unbounded uniform shear flow

The similarity of the coherent structures (streaks and hairpin vortices) naturally occurring in different fully developed bounded turbulent shear flows as well as in transitional flows suggests the existence of a basic mechanism responsible for the formation of these structures, under various base flow conditions. The common elements for all such flows are the shear of the base flow and the presence of a localized vortical disturbance. The objective of the present numerical study is to examine the capability of a simple model of interaction, between a localized vortical disturbance and laminar uniform unbounded shear flow, to reproduce the generation mechanism and characteristics of the coherent structures that naturally occur in turbulent bounded shear flows. The effects of the disturbance ‘localized character’ in the stream-wise and spanwise directions as well as its initial orientation relative to the base flow are investigated by using several geometries of the initial disturbance. The results demonstrate that a small-amplitude initial disturbance (linear case) eventually evolves into a streaky structure independent of its initial geometry and orientation, whereas, a large-amplitude disturbance (strongly nonlinear case) evolves into a hairpin vortex (or a packet of hairpin vortices) independent of its geometry over a wide range of the initial disturbance orientations. The main nonlinear effects are: (i) self-induced motion, which results in the movement of the vortical structure relative to the base flow and the destruction of its streamwise symmetry, and (ii) the alignment of the vortical structure with the vorticity lines. This is unlike the linear case, where there is a strong deviation of the vorticity vector from the direction of the vortical structure. Qualitatively, the disturbance evolution is sufficiently independent of its initial geometry, whereas the associated quantitative characteristics, i.e. inclination angle, centre and strength (which is governed by the transient growth mechanism), strongly depend on the disturbance geometry. The Reynolds number is found to have a negligible effect on the kinematics of the vortical structure, but does have a significant effect on its transient growth. Finally, the formation of the asymmetric hairpin vortex, due to minor spanwise asymmetries of the initial disturbance, is demonstrated.

Controlling the Development of Stationary Longitudinal Structures in the Boundary Layer on a Flat Plate using Riblets

This paper reports results of experiments on controlling longitudinal structures in the boundary layer on a at plate. The longitudinal structures were generated by a controlled vortical disturbance of the external flow by means of a distributed susceptibility mechanism. It is shown that riblets reduce the intensity of both stationary and traveling disturbances. The linear and weakly linear stages in the development of disturbances in the boundary layer are the most favorable for the use of riblets.

Mode interaction and the bypass route to transition

The manner by which external vortical disturbances penetrate the laminar boundary layer and induce transition is explored. Linear theory suggests that the well-known Klebanoff mode precursor to transition can be understood as a superposition of Squire continuous modes. Shear sheltering influences the ability of free-stream disturbances to generate a packet of Squire modes. A coupling coefficient between continuous spectrum spectrum Orr–Sommerfeld and Squire modes is used to characterize the interaction. Full numerical simulations with prescribed modes at the inlet substantiate this approach. With two weakly coupled modes at the inlet, the boundary layer is little perturbed; with two strongly coupled modes, Klebanoff modes are produced; with one strongly coupled and one weakly coupled high-frequency mode, the complete transition process is simulated.

Numerical study of local boundary layer receptivity to freestream vortical disturbances

Method of direct numerical simulation was used for the investigation of the local receptivity of a 2-D boundary layer of a flat plate to harmonic vortical disturbances in the freestream. Under the interaction of the vortical disturbances in the freestream and the roughness element on the wall, Tollmein-Schlichting (T-S) waves were generated and detected in the boundary layer, thus confirming that the wavelength conversion mechanism and the local receptivity exist. Numerical simulations were performed to obtain the relations between the amplitude of the generated T-S wave and the amplitude of freestream disturbance, the roughness height, and the width of rectangular roughness elements, which agree with those obtained from experiments. Then the range of validity of the linear receptivity formula, which was determined by these relations, also agrees with that determined by experimental results.

Evolution of Localized Vortex Disturbance in Uniform Shear Flow: Numerical Investigation

The possibility that a simple model of interaction between a localized vortical disturbance and laminar uniform unbounded shear flow is capable of reproducing the generation mechanism and characteristics of coherent structures naturally occurring in wall bounded turbulent flows is examined numerically. Gaussian and toroidal vortices are used as the initial disturbances. The concentrated vorticity field of the Gaussian vortex is defined by a single length scale 6, which is assumed to be much smaller than the characteristic scale of the base flow. The toroidal disturbance is defined by two length scales r 0 and δ, corresponding to the radius and thickness of the torus, respectively

A visual study of vortex-induced subcritical instability on a flat plate laminar boundary layer

This paper reports results of our experimental investigation on flow instability on a flat plate laminar boundary layer caused by a “captive” vortex migrating far outside the boundary layer. Results show that the sign of the circulation associated with the vortex is the main determinant for the severity of the boundary layer instability. A captive vortex with an opposite sign to that of the unperturbed shear layer vorticity causes a breakdown ahead of it, while the one with the same sign as the unperturbed shear layer vorticity gives rise to weaker excitation trailing it. Additional parameters that influence the flow instability are the strength and distance of the vortical disturbance from the boundary layer, as well as the translational speed of the vortex. These experimental results compliment the corresponding theoretical analysis of Sengupta et al. (J Fluid Mech 493:277–286, 2003).

Hydrodynamic instability of fiber suspensions in channel flows

A linear stability analysis of channel flow in the presence of fiber suspensions is performed. Based on slender-body theory, a modified stability equation is derived by using the natural closure approximation to determine the fiber orientation. It is found that the flow instability of fiber suspensions is governed by two parameters: H, a ratio between the axial stretching resistance of fibers and the inertial force of the fluid, and the orientational diffusivity coefficient C I that accounts for inter-fiber hydrodynamic interactions. The increment of either parameter causes an increase of critical Reynolds number, a reduction of the maximum disturbance growth rate and a shift of the peak value of velocity disturbances away from the walls. Hence, the fiber additives effectively attenuate the instability of the flow. The orientation distribution state of fibers in the flow is also examined, and the influence of parameters is determined. An investigation of different terms in the vorticity disturbance equation and an enstrophy-based energy analysis reveal the sources of disturbance energy and the mechanisms behind the instability of fiber suspensions. Finally, an experiment is performed using the visualization technique. The comparative measurements of stable lengths in every type of flow show that fiber additives have a stabilizing effect on the flow, and this effect is clearer for higher values of Re. The conclusions of experiment are in agreement with those of linear instability analysis.

Scattering of incident disturbances by an annular cascade in a swirling flow

Analytical and numerical analyses are developed for the interaction and scattering of incident acoustic and vortical disturbances by an unloaded annular cascade in a swirling flow. The mathematical formulation uses the Euler equations linearized about an axial and swirling mean flow. The incident disturbances are decomposed into nearly sonic and nearly convected disturbances using the results of a normal-mode analysis, namely the unsteady pressure is predominantly associated with the former. Exact non-reflecting inflow/outflow conditions are derived in terms of the normal modes using the group velocity to segregate the modes propagating downstream and upstream. An inflow condition is also derived for the nearly convected disturbances. An explicit primitive-variable scheme is implemented and validated by comparison with the uniform flow and narrow annulus limits. Acoustic and aerodynamic results are presented to examine how swirl modifies the scattering from that of the uniform flow and narrow annulus limits and to determine the conditions leading to strong scattering. The results indicate that the swirl changes the physics of the scattering in three major ways: (i) it modifies the number of acoustic modes in the duct, (ii) it changes their duct radial profile, and (iii) it causes significant amplitude and radial phase variations of the incident disturbance. The results also show that when the radial phase of the incident disturbance is different from that of the duct modes, weak scattering into the duct acoustic modes occurs. These results suggest that analysis of the radial variation of the incident disturbance and duct modes can provide an indication of the efficiency of the scattering process.

Combustion instability due to the nonlinear interaction between sound and flame

It has been observed in experiments that significant levels of sound may be produced when a curved flame propagates downwards along a tube in a gravity field. In this paper, we present a mathematical description of this acoustic amplification process, which represents a simple form of combustion instability. First, based on the large-activation-energy and small-Mach-number assumptions, a general asymptotic formulation is derived, in which the nature of flame–sound coupling is brought out explicitly. This framework is then employed to study the weakly nonlinear coupling between a Darrieus–Landau (D-L) instability mode of the flame and an acoustic mode of the tube, which is the main mechanism for sound generation in the experiments. In order to provide a somewhat unified description, the linear coupling via the direct pressure effect has also been included in our analysis. A set of coupled equations which govern the evolution of the acoustic and D-L modes was derived. The solutions show that the nonlinear coupling leads to very rapid amplification of sound. After reaching an appreciable level, the sound inhibits the flame, causing the latter to flatten. The sound then saturates at an almost constant level, or continues to grow at a smaller rate owing to the pressure effect. The above theoretical predictions are in good qualitative agreement with experiments. The present study also considered the influence of weak vortical disturbances in the oncoming flow. It is shown that certain components in these perturbations may form resonant triads with the acoustic and D-L modes, thereby providing an additional coupling mechanism.

On the linear stability of swept attachment-line boundary layer flow. Part 2. Non-modal effects and receptivity

Following the study of the spectral properties of linearized swept Hiemenz flow (see Part 1, Obrist & Schmid 2003) we investigate the potential of swept Hiemenz flow to support transiently growing perturbations owing to the non-normal nature of the underlying linear stability operator. Transient amplification of perturbation energy is found for polynomial orders higher than zero, and a catalytic role of the continuous modes in increasing transient growth is demonstrated. The adjoint stability equations are derived and used in a numerical receptivity experiment to illustrate the scattering of vortical free-stream disturbances into the least stable boundary layer mode.

Numerical Unsteady Flow Analysis of a Turbine Stage With Extremely Large Blade Loads

This paper presents the detailed numerical analysis including parametric studies on the aerodynamic excitation mechanisms in a turbine stage due to the unsteady stator-rotor interaction. The work is part of the predesign study of a high-pressure subsonic turbine for a rocket engine turbopump. The pressure level in such turbines can be remarkably high (in this case 54 MPa inlet total pressure). Hence, large unsteady rotor blade loads can be expected, which impose difficult design requirements. The parameter studies are performed at midspan with the numerical flow solver UNSFLO, a 2-D/Q3-D unsteady hybrid Euler/Navier-Stokes solver. Comparisons to 2-D and steady 3-D results obtained with a fully viscous solver, VOLSOL, are made. The investigated design parameters are the axial gap (∼8–29 percent of rotor axial chord length) and the stator vane size and count (stator-rotor pitch ratio ∼1–2.75). For the nominal case the numerical solution is analyzed regarding the contributions of potential and vortical flow disturbances at the rotor inlet using rotor gust computations. It was found that gust calculations were not capable to capture the complexity of the detected excitation mechanisms, but the possibility to reduce excitations by enforcing cancellation of the vortical and potential effects has been elaborated. The potential excitation mechanism in the present turbine stage is found dominant compared to relatively small and local wake excitation effects. The parameter studies indicate design recommendations for the axial gap and the stator size regarding the unsteady rotor load.

Inflow/Outflow Conditions for Time-Harmonic Internal Flows

Inflow/Outflow conditions are formulated for time-harmonic waves in a duct governed by the Euler equations. These conditions are used to compute the propagation of acoustic and vortical disturbances and the scattering of vortical waves into acoustic waves by an annular cascade. The outflow condition is expressed in terms of the pressure, thus avoiding the velocity discontinuity across any vortex sheets. The numerical solutions are compared with the analytical solutions for acoustic and vortical wave propagation with and without the presence of vortex sheets. Grid resolution studies are also carried out to discern the truncation error of the numerical scheme from the error associated with numerical reflections at the boundary. It is observed that even with the use of exponentially accurate boundary conditions, the dispersive characteristics of the numerical scheme may result in small reflections from the boundary that slow convergence. Finally, the three-dimensional interaction of a wake with a flat plate cascade is computed and the aerodynamic and aeroacoustic results are compared with those of lifting surface methods.

INFLOW/OUTFLOW CONDITIONS FOR TIME-HARMONIC INTERNAL FLOWS

Inflow/Outflow conditions are formulated for time-harmonic waves in a duct governed by the Euler equations. These conditions are used to compute the propagation of acoustic and vortical disturbances and the scattering of vortical waves into acoustic waves by an annular cascade. The outflow condition is expressed in terms of the pressure, thus avoiding the velocity discontinuity across any vortex sheets. The numerical solutions are compared with the analytical solutions for acoustic and vortical wave propagation with and without the presence of vortex sheets. Grid resolution studies are also carried out to discern the truncation error of the numerical scheme from the error associated with numerical reflections at the boundary. It is observed that even with the use of exponentially accurate boundary conditions, the dispersive characteristics of the numerical scheme may result in small reflections from the boundary that slow convergence. Finally, the three-dimensional interaction of a wake with a flat plate cascade is computed and the aerodynamic and aeroacoustic results are compared with those of lifting surface methods.

Conservative Hybrid Compact-WENO Schemes for Shock-Turbulence Interaction

In the present paper an efficient hybrid compact-WENO scheme is proposed to obtain high resolution in shock-turbulence interaction problems. The algorithm is based on a fifth-order compact upwind algorithm in conservation form to solve for the smooth part of the flow field, which is coupled with a high-resolution weighted essentially nonoscillatory (WENO) scheme to capture the discontinuities. The derivation of the compact scheme is discussed in detail, and a stability study of the full discretization is included. The performance of the numerical algorithm has been assessed by performing preliminary simulations on benchmark problems, such as the interaction of a shock wave with entropy and vortical disturbances. The algorithm here developed is proven to have better resolution properties than standard WENO schemes and hybrid compact-ENO schemes as well, at a lower computational cost. In addition, the application to more realistic shock-turbulence interaction problems is discussed.

Stability of miscible displacements of shear thinning fluids in a Hele-Shaw cell

A linear stability analysis of the viscous fingering of miscible non-Newtonian flow displacements in a rectilinear Hele-Shaw cell is presented. The shear-thinning character of the non-Newtonian fluid is described using the Carreau model which involves two rheological parameters De and n. Flows where either the displacing or displaced phase has a shear-thinning behavior are examined and compared with those of Newtonian flows. It is found that the shear-thinning character of the non-Newtonian fluid has an important effect on the flow instability. In particular, a flow where the driving fluid is shear-thinning is always more unstable than its Newtonian counterpart. For this flow, the maximum growth rate and the spectrum of unstable wave numbers are larger than in the Newtonian case which suggests that more ramified structures will develop as the finger instability grows. On the other hand, when the displaced fluid is non-Newtonian, a stronger shear-thinning rheological behavior leads in general to a less unstable flow. The mechanisms responsible for the changes in the flow instability are explained in terms of the different sources contributing to the generation of the vorticity disturbance.