The extended Prandtl closure model applied to the two-dimensional turbulent classical far wake

2021 ◽  
pp. 375-385 ◽  
Author(s):  
Ashleigh J. Hutchinson
Keyword(s):  
Two Dimensional ◽  
Closure Model ◽  
Far Wake

A perturbation analysis of the laminar far wake behind a symmetrical two-dimensional body in a uniform shear flow

1973 ◽  
Vol 61 (2) ◽  
pp. 305-321 ◽  
Author(s):  
Masaru Kiya ◽  
Mikio Arie
Keyword(s):  
Shear Flow ◽  
Stream Function ◽  
Perturbation Analysis ◽  
Maximum Velocity ◽  
Two Dimensional ◽  
Low Velocity ◽  
Velocity Defect ◽  
Uniform Shear ◽  
Uniform Shear Flow ◽  
Far Wake

An aspect of the laminar far wake behind a symmetrical two-dimensional body placed in a uniform shear flow is described theoretically by means of the Oseen type of successive approximation, in which the shear is regarded as a small perturbation on a uniform stream. The expression for the stream function is determined up to the third approximation both in and outside the wake region, and the region in which the results of the perturbation analysis are valid is also determined. The stream function is found to contain four constants which cannot be determined from the boundary conditions for the far wake. The analysis also shows that the spreading of the wake is greater towards the side of smaller velocity than the side of larger velocity, the asymmetrical feature of the velocity defect becoming more evident as the distance from the obstacle is increased: the point which shows the maximum velocity defect shifts to the low-velocity side.

Three-dimensional secondary instability of a spatially developing von Kármán vortex street in a far wake

2010 ◽  
Vol 467 (2127) ◽  
pp. 675-694 ◽  
Author(s):  
J. T. C. Liu ◽  
X. Yu
Keyword(s):  
Nonlinear Interaction ◽  
Three Dimensional ◽  
Secondary Instability ◽  
Vortex Street ◽  
Wake Flow ◽  
Two Dimensional ◽  
Primary Flow ◽  
Von Karman ◽  
Von Karman Vortex Street ◽  
Far Wake

This paper presents studies of a three-dimensional secondary instability of a spatially developing von Kármán vortex street. It develops owing to the nonlinear interaction between a two-dimensional mean far-wake flow and its most unstable disturbances. This forms a nonlinear primary wake flow. Sections of this flow are selected to perform a temporal secondary stability study under the assumption of parallel flow. The eigenvalue characteristics of the secondary instability are compared with the results from the use of a linear primary flow comprising unmodified mean wake flow coexisting with a linear primary fundamental disturbance with an empirical amplitude as a parameter, resulting in a simpler Floquet analysis. The maximum amplification rates occur at about the same spanwise wavenumber for both the nonlinear and linear primary flows, in qualitative agreement. But the amplification rate versus the spanwise wavenumber spectrum are both qualitatively and quantitatively different, the nonlinear primary flow results in a lower magnitude of the amplification rates. Some interpretations of controlled experiments are made, and it is concluded that the two- and three-dimensional disturbances so obtained appeared to be from the primary instability, where the amplification mechanisms come from the unmodified mean flow. A general discussion of the nonlinear interaction between the primary two-dimensional flow and the three-dimensional secondary instability is given, which may well form the basis for further nonlinear studies.

Acoustic forcing of oblique wave resonance in the far wake

1993 ◽  
Vol 256 ◽  
pp. 315-341 ◽  
Author(s):  
C. H. K. Williamson ◽  
A. Prasad
Keyword(s):  
Free Stream ◽  
Higher Order ◽  
Two Dimensional ◽  
Near Wake ◽  
Peak Response ◽  
Oblique Waves ◽  
Oblique Wave ◽  
Wave Resonance ◽  
Acoustic Forcing ◽  
Far Wake

In this paper, we investigate to what extent the far-wake ‘signature’ of the near-wake vortex dynamics of a nominally two-dimensional bluff body is affected by the character of the free-stream noise. We confirm the existence of an oblique wave resonance (at frequency, fK–fT), which is caused by nonlinear ‘quadratic’ interactions between primary oblique shedding waves (fK) and secondary two-dimensional waves (fT), which are amplified from free-stream disturbances. In this work, oblique wave resonance is induced by acoustic forcing of two-dimensional waves. The use of acoustic forcing reveals a set of higher-order oblique wave resonances corresponding to frequencies (fK–nfT), where n is an integer. We find from visualization that, even when the secondary two-dimensional waves have the same frequency as the oblique waves, it is the oblique waves that are preferentially amplified. Oblique wave angles up to 74° have been observed. The response of the wake to a large range of forcing frequencies shows a broad region of peak response, centred around F = (fT/fK) = 0.55, and is in reasonable agreement with predictions from linear stability analysis. A similar broad response is found for each of the higher-order oblique wave modes. Simple equations for the oblique waves yield approximate conditions for maximum wake response, with a frequency for peak response given by Fmax = 1/2n = 1/2, 1/4, 1/6,…, and an oblique wave angle given by θmax = 2θK, where θK is the angle of oblique vortex shedding. An increase in forcing amplitude has the effect of bringing the nonlinear wave interactions, leading to oblique wave resonance, further upstream. Paradoxically, the effect of an increase in amplitude (A) of the two-dimensional wave forcing is to further amplify the oblique waves in preference to the two-dimensional waves and, under some conditions, to inhibit the appearance of prominent two-dimensional waves where they would otherwise appear. With a variation in forcing amplitude, the amplitude of oblique wave response is found to be closely proportional to A½. In summary, this investigation confirms the surprising result that it is only through the existence of noise in the free stream that the far wake is ‘connected’ to the near wake.

Vorticity and passive-scalar dynamics in two-dimensional turbulence

1987 ◽  
Vol 183 ◽  
pp. 379-397 ◽  
Author(s):  
Armando Babiano ◽  
Claude Basdevant ◽  
Bernard Legras ◽  
Robert Sadourny
Keyword(s):  
Similarity Theory ◽  
Physical Space ◽  
Passive Scalar ◽  
Inertial Range ◽  
Flow Dynamics ◽  
Two Dimensional ◽  
Closure Model ◽  
Coherent Vortices ◽  
Vorticity Transport ◽  
Kolmogorov Theory

The dynamics of vorticity in two-dimensional turbulence is studied by means of semi-direct numerical simulations, in parallel with passive-scalar dynamics. It is shown that a passive scalar forced and dissipated in the same conditions as vorticity, has a quite different behaviour. The passive scalar obeys the similarity theory à la Kolmogorov, while the enstrophy spectrum is much steeper, owing to a hierarchy of strong coherent vortices. The condensation of vorticity into such vortices depends critically both on the existence of an energy invariant (intimately related to the feedback of vorticity transport on velocity, absent in passive-scalar dynamics, and neglected in the Kolmogorov theory of the enstrophy inertial range); and on the localness of flow dynamics in physical space (again not considered by the Kolmogorov theory, and not accessible to closure model simulations). When space localness is artificially destroyed, the enstrophy spectrum again obeys a k−1 law like a passive scalar. In the wavenumber range accessible to our experiments, two-dimensional turbulence can be described as a hierarchy of strong coherent vortices superimposed on a weak vorticity continuum which behaves like a passive scalar.

Note on ship oscillations and wake solitons

1991 ◽  
Vol 432 (1886) ◽  
pp. 535-546 ◽  
Keyword(s):  
Schrödinger Equation ◽  
Simple Form ◽  
Schrodinger Equation ◽  
Field Experiments ◽  
Nonlinear Schrodinger Equation ◽  
Source Mechanism ◽  
Two Dimensional ◽  
Sea Waves ◽  
Long Distance ◽  
Far Wake

Recent field experiments have shown that inside the Kelvin wake of a moving ship there can be two soliton-like envelopes, which extend to a long distance behind the ship. The source mechanism of these solitons has so far been attributed to the steady advance of a ship. In this note we suggest that ship oscillations, induced by ambient sea waves, provide a more plausible explanation for the initiation. Reducing the two-dimensional nonlinear Schrödinger equation to a simple form, we use existing theories to predict the wave envelopes in the far wake of a travelling ship in heave or pitch oscillations. While available data are insufficient to give a conclusive check of the theory, qualitative relevance is discussed.

scholarly journals Large structure in the far wakes of two-dimensional bluff bodies

1988 ◽  
Vol 190 ◽  
pp. 265-298 ◽  
Author(s):  
John M. Cimbala ◽  
Hassan M. Nagib ◽  
Anatol Roshko
Keyword(s):  
Circular Cylinder ◽  
Hydrodynamic Instability ◽  
Broad Band ◽  
Parametric Instability ◽  
Driving Mechanism ◽  
Bluff Bodies ◽  
Two Dimensional ◽  
Large Structure ◽  
Flat Plates ◽  
Far Wake

Smoke-wire flow visualization and hot-wire anemometry have been used to study near and far wakes of two-dimensional bluff bodies. For the case of a circular cylinder at 70 < Re < 2000, a very rapid (exponential) decay of velocity fluctuations at the Kármán-vortex-street frequency is observed. Beyond this region of decay, larger-scale (lower wavenumber) structure can be seen. In the far wake (beyond one hundred diameters) a broad band of frequencies is selectively amplified and then damped, the centre of the band shifting to lower frequencies as downstream distance is increased.The far-wake structure does not depend directly on the scale or frequency of Kármán vortices shed from the cylinder; i.e. it does not result from amalgamation of shed vortices. The growth of this structure is due to hydrodynamic instability of the developing mean wake profile. Under certain conditions amalgamation can take place, but is purely incidental, and is not the driving mechanism responsible for the growth of larger-scale structure. Similar large structure is observed downstream of porous flat plates (Re ≈ 6000), which do not initially shed Kármán-type vortices into the wake.Measured prominent frequencies in the far cylinder wake are in good agreement with those estimated by two-dimensional locally parallel inviscid linear stability theory, when streamwise growth of wake width is taken into account. Finally, three-dimensionality in the far wake of a circular cylinder is briefly discussed and a mechanism for its development is suggested based on a secondary parametric instability of the subharmonic type.

scholarly journals Prandtl’s extended mixing length model applied to the two-dimensional turbulent classical far wake

2021 ◽  
Vol 477 (2246) ◽  
pp. 20200875
Author(s):  
Ashleigh J. Hutchinson ◽  
Nicholas Hale ◽  
Kendall Born ◽  
David P. Mason
Keyword(s):  
Differential Equation ◽  
Kinematic Viscosity ◽  
Turbulent Viscosity ◽  
Nonlinear Ordinary Differential Equation ◽  
Original Model ◽  
Extended Model ◽  
Mixing Length ◽  
Two Dimensional ◽  
Mean Velocity ◽  
Far Wake

Despite its limitations, Prandtl’s mixing length model is widely applied in modelling turbulent free shear flows. Prandtl’s extended model addresses many of the shortfalls of the original model, but is not so widely used, in part due to additional mathematical complexities that arise in its derivation and implementation. Furthermore, in both models, Prandtl neglects the kinematic viscosity on the basis that it is much smaller in magnitude than the turbulent viscosity. Recent work has shown that including the kinematic viscosity in the original model has both mathematical and physical advantages. In the present work, a novel derivation of the extended model is provided, and it is demonstrated that similar advantages are again obtained when the kinematic viscosity is included. Additionally, through the use of scaling techniques, similarity mean velocity profiles of the extended model are derived, resulting in a single nonlinear ordinary differential equation that is solved numerically with a Hermite spectral method. The computed profiles for the normalized similarity mean velocity and shear stress are compared with experimental observations and shown to be in excellent agreement.

Sign in / Sign up

Export Citation Format

Share Document