89 research outputs found
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On the effects of suction and injection on the absolute instability of the rotating-disk boundary layer
In this paper we are concerned with the theoretical behavior of the laminar von Kármán boundary-layer flow, extending the work presented by Lingwood [J. Fluid Mech. 299, 17 (1995); 314, 373 (1996)] to the flow with mass transfer at the surface of the disk. It is known that, within specific regions of the parameter space, the flow is absolutely unstable in the radial direction, i.e. disturbances grow in time at every radial location within these regions. Uniform suction through the disk is shown to delay the onset of absolute instability, while uniform injection promotes the onset. By comparing suction and injection velocities of the same magnitude, it is shown that suction has a greater stabilizing effect on the absolute instability than the destabilizing effect of injection. Suction is also strongly stabilizing to both stationary and travelling inviscidly unstable branch-1 modes; injection is destabilizing. Stationary viscously unstable branch-2 modes are strongly stabilized and destabilized by suction and injection, respectively, but travelling branch-2 modes are shown to be much less sensitive to mass transfer through the disk
On the global nonlinear instability of the rotating-disk flow over a finite domain
Direct numerical simulations based on the incompressible nonlinear Navier–Stokes equations
of the flow over the surface of a rotating disk have been conducted. An impulsive
disturbance was introduced and its development as it travelled radially outwards
and ultimately transitioned to turbulence has been analysed. Of particular interest was
whether the nonlinear stability is related to the linear stability properties. Specifically
three disk-edge conditions were considered; (i) a sponge region forcing the flow back to
laminar flow, (ii) a disk edge, where the disk was assumed to be infinitely thin, and
(iii) a physically-realistic disk edge of finite thickness. This work expands on the linear
simulations presented by Appelquist et al. (J. Fluid. Mech., vol. 765, 2015, pp. 612-631),
where, for case (i), this configuration was shown to be globally linearly unstable when
the sponge region effectively models the influence of the turbulence on the flow field. In
contrast, case (ii) was mentioned there to be linearly globally stable, and here, where
nonlinearity is included, it is shown that both case (ii) and (iii) are nonlinearly globally
unstable. The simulations show that the flow can be globally linearly stable if the linear
wavepacket has a positive front velocity. However, in the same flow field, a nonlinear
global instability can emerge, which is shown to depend on the outer turbulent region
generating a linear inward-travelling mode that sustains a transition-front within the
domain. The results show that the front position does not approach the critical Reynolds
number for the local absolute instability, R = 507. Instead, the front approaches R = 583
and both the temporal frequency and spatial growth rate correspond to a global mode
originating at this position.Swedish Research Counci
On the stability of the Hartmann layer
In this paper we are concerned with the theoretical stability of the laminar Hartmann layer, which forms at the boundary of any electrically conducting fluid flow under a steady magnetic field at high Hartmann number. We perform both linear and energetic stability analyses to investigate the stability of the Hartmann layer to both infinitesimal and finite perturbations. We find that there is more than three orders of magnitude between the critical Reynolds numbers from these two analyses. Our interest is motivated by experimental results on the laminar–turbulent transition of ducted magnetohydrodynamics flows. Importantly, all existing experiments have considered the laminarization of a turbulent flow, rather than transition to turbulence. The fact that experiments have considered laminarization, rather than transition, implies that the threshold value of the Reynolds number for stability of the Hartmann layer to finite-amplitude, rather than infinitesimal, disturbances is in better agreement with the experimental threshold values. In fact, the critical Reynolds number for linear instability of the Hartmann layer is more than two orders of magnitude larger than experimentally observed threshold values. It seems that this large discrepancy has led to the belief that stability or instability of the Hartmann layer has no bearing on whether the flow is laminar or turbulent. In this paper, we give support to Lock’s hypothesis [Proc. R. Soc. London, Ser. A 233, 105 (1955)] that “transition” is due to the stability characteristics of the Hartmann layer with respect to large-amplitude disturbances
A model for the turbulent Hartmann layer
Here we study the Hartmann layer, which forms at the boundary of any electrically-conducting fluid
flow under a steady magnetic field at high Hartmann number provided the magnetic field is not
parallel to the wall. The Hartmann layer has a well-known form when laminar. In this paper we
develop a model for the turbulent Hartmann layer based on Prandtl’s mixing-length model without
adding arbitrary parameters, other than those already included in the log-law. We find an exact
expression for the displacement thickness of the turbulent Hartmann layer @also given by Tennekes,
Phys. Fluids 9, 1876 ~1966!#, which supports our assertion that a fully-developed turbulent
Hartmann layer of finite extent exists. Leading from this expression, we show that the interaction
parameter is small compared with unity and that therefore the Lorentz force is negligible compared
with inertia. Hence, we suggest that the turbulence present in the Hartmann layer is of classical type
and not affected by the imposed magnetic field, so justifying use of a Prandtl model. A major result
is a simple implicit relationship between the Reynolds number and the friction coefficient for the
turbulent Hartmann layer in the limit of large Reynolds number. By considering the distance over
which the stress decays, we find a condition for the two opposite Hartmann layers in duct flows to
be isolated (nonoverlapping)
A new way to describe the transition characteristics of a rotating-disk boundary-layer flow
A new method of graphically representing the transition stages of a rotating-disk
flow is presented. The probability density function contour map of the fluctuating
azimuthal disturbance velocity is used to show the characteristics of the boundarylayer
flow over the rotating disk as a function of Reynolds numbers. Compared with
the variation of the disturbance amplitude (rms) or spectral distribution, this map more
clearly shows the changing flow characteristics through the laminar, transitional, and
turbulent regions. This method may also be useful to characterize the different stages
in the transition process not only for the rotating-disk flow but also for other flows.Swedish Research Counci
Boundary-layer transition over a rotating broad cone
The Swedish Research Council (VR
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Investigation of the structures in the unstable rotating-cone boundary layer
This work reports on the unstable region and the transition process of the boundary-layer flow induced by a rotating cone with a half apex angle of 60 degrees using the probability density function (PDF) contour map of the azimuthal velocity fluctuation, which was first used by Imayamaetal. (Physics of Fluids, vol.24, 2012, 031701) for the similar boundary-layer flow induced by a rotating disk. The PDF shows that the transition behavior of the rotating-cone flow is similar to that on the rotating disk. The effects of roughness elements on the cone surface have been examined. For the cone with roughnesses, we reconstructed the most probable vortex structure within the boundary layer from the hot-wire anemometry time signals. The results show that the PDF clearly describes the overturning process of the high-momentum upwelling of the spiral vortices, which due to vortex meandering cannot be detected in the phase-averaged velocity field reconstructed from the point measurements. At a late stage of the overturning process, our hot-wire measurements captured high-frequency oscillations, which may be related to secondary instability.Swedish Research Council (VR); Japan Society for the Promotion of Science (JSPS)
Turbulence in the rotating-disk boundary layer investigated through direct numerical simulations
Direct numerical simulations (DNS) are reported for the turbulent rotating-disk boundary layer for the
first time. Two turbulent simulations are presented with overlapping small and large Reynolds numbers,
where the largest corresponds to a momentum-loss Reynolds number of almost 2000. Simulation data are
compared with experimental data from the same flow case reported by Imayama et al. (2014), and also a
comparison is made with a numerical simulation of a two-dimensional turbulent boundary layer (2DTBL)
over a flat plate reported by Schlatter and Ă–rlĂĽ (2010). The agreement of the turbulent statistics between
experiments and simulations is in general very good, as well as the findings of a missing wake region and
a lower shape factor compared to the 2DTBL. The simulations also show rms-levels in the inner region
similar to the 2DTBL. The simulations validate Imayama et al.’s results showing that the rotating-disk
turbulent boundary layer in the near-wall region contains shorter streamwise (azimuthal) wavelengths
than the 2DTBL, probably due to the outward inclination of the low-speed streaks. Moreover, all velocity
components are available from the simulations, and hence the local flow angle, Reynolds stresses and
all terms in the turbulent kinetic energy equation are also discussed. However there are in general no
large differences compared to the 2DTBL, hence the three-dimensional effects seem to have only a small
influence on the turbulence.Swedish Research Counci
Investigation of the Global Instability of the Rotating-disk Boundary Layer
The development of the flow over a rotating disk is investigated by direct numerical simulations using both the linearized and fully
nonlinear incompressible Navier–Stokes equations. These simulations allow investigation of the transition to turbulence of the
realistic spatially-developing boundary layer. The current research aims to elucidate further the global linear stability properties
of the flow, and relate these to local analysis and discussions in literature. An investigation of the nonlinear upstream (inward)
influence is conducted by simulating a small azimuthal section of the disk (1/68). The simulations are initially perturbed by
an impulse disturbance where, after the initial transient behaviour, both the linear and nonlinear simulations show a temporally
growing upstream mode. This upstream global mode originates in the linear case close to the end of the domain, excited by
an absolute instability at this downstream position. In the nonlinear case, it instead originates where the linear region ends and
nonlinear harmonics enter the flow field, also where an absolute instability can be found. This upstream global mode can be
shown to match a theoretical mode from local linear theory involved in the absolute instability at either the end of the domain
(linear case) or where nonlinear harmonics enter the field (nonlinear case). The linear simulation grows continuously in time
whereas the nonlinear simulation saturates and the transition to turbulence moves slowly upstream towards smaller radial positions
asymptotically approaching a global upstream mode with zero temporal growth rate, which is estimated at a nondimensional radius
of 582.Swedish Research Counci
Transition to turbulence in the rotating-disk boundary-layer flow with stationary vortices
This paper proposes a resolution to the conundrum of the roles of convective and absolute
instability in transition of the rotating-disk boundary layer. It also draws some comparison
with swept-wing
ows. Direct numerical simulations based on the incompressible
Navier{Stokes equations of the
ow over the surface of a rotating disk with modelled
roughness elements are presented. The rotating-disk
ow has been of particular interest
for stability and transition research since the work by Lingwood (J. Fluid. Mech., vol. 299,
1995, pp. 17-33) where an absolute instability was found. Here stationary disturbances
develop from roughness elements on the disk and are followed from the linear stage, growing
to saturation and nally transition to turbulence. Several simulations are presented
with varying disturbance amplitudes. The lowest amplitude corresponds approximately
to the experiment by Imayama et al. (J. Fluid. Mech., vol. 745, 2014, pp. 132-163). For
all cases, the primary instability was found to be convectively unstable, and secondary
modes were found to be triggered spontaneously while the
ow was developing. The
secondary modes further stayed within the domain, and an explanation for this is a
proposed globally unstable secondary instability. For the low-amplitude roughness cases,
the disturbances propagate beyond the threshold for secondary global instability before
becoming turbulent, and for the high-amplitude roughness cases the transition scenario
gives a turbulent
ow directly at the critical Reynolds number for the secondary global
instability. These results correspond to the theory of Pier (J. Eng. Math, vol. 57,
2007, pp. 237-251) predicting a secondary absolute instability. In our simulations, high
temporal frequencies were found to grow with a large ampli cation rate where the secondary
global instability occurred. For smaller radial positions, low-frequency secondary
instabilities were observed, tripped by the global instability.Swedish Research Counci
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