Two new two-equation eddy-viscosity turbulence models will be presented. They combine different elements of existing models that are considered superior to their alternatives. The first model, referred to as the baseline (BSL) model, utilizes the original k-ω model of Wilcox in the inner region of the boundary layer and switches to the standard k-e model in the outer region and in free shear flows. It has a performance similar to the Wilcox model, but avoids that model's strong freestream sensitivity

/pdf/two-equation-eddy-viscosity-turbulence-models-for-54j7shl2v6.pdf

Two-equation eddy-viscosity turbulence models for engineering applications

Considerable confusion surrounds the longstanding question of what constitutes a vortex, especially in a turbulent flow. This question, frequently misunderstood as academic, has recently acquired particular significance since coherent structures (CS) in turbulent flows are now commonly regarded as vortices. An objective definition of a vortex should permit the use of vortex dynamics concepts to educe CS, to explain formation and evolutionary dynamics of CS, to explore the role of CS in turbulence phenomena, and to develop viable turbulence models and control strategies for turbulence phenomena. We propose a definition of a vortex in an incompressible flow in terms of the eigenvalues of the symmetric tensor ${\bm {\cal S}}^2 + {\bm \Omega}^2$ are respectively the symmetric and antisymmetric parts of the velocity gradient tensor ${\bm \Delta}{\bm u}$. This definition captures the pressure minimum in a plane perpendicular to the vortex axis at high Reynolds numbers, and also accurately defines vortex cores at low Reynolds numbers, unlike a pressure-minimum criterion. We compare our definition with prior schemes/definitions using exact and numerical solutions of the Euler and Navier–Stokes equations for a variety of laminar and turbulent flows. In contrast to definitions based on the positive second invariant of ${\bm \Delta}{\bm u}$ or the complex eigenvalues of ${\bm \Delta}{\bm u}$, our definition accurately identifies the vortex core in flows where the vortex geometry is intuitively clear.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112095000462

On the identification of a vortex

Presentation d'une methode numerique par volume fini pour la resolution des equations de Navier-Stokes bidimensionnelles, incompressibles, et stationnaires, en coordonnees generales curvilignes Application de la methode aux ecoulements turbulents sur des profils avec et sans separation au bord de sortie posterieur Comparaison des calculs avec des donnees experimentales

Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Separation

The prediction of laminarization with a two-equation model of turbulence

Since the review of periodic flow phenomena by Berger & Wille (1972) in this journal, over twenty years ago, there has been a surge of activity regarding bluff body wakes. Many of the questions regarding wake vortex dynamics from the earlier review have now been answered in the literature, and perhaps an essential key to our new understandings (and indeed to new questions) has been the recent focus, over the past eight years, on the three-dimensional aspects of nominally two-dimensional wake flows. New techniques in experiment, using laser-induced fluorescence and PIV (Particle-Image-Velocimetry), are vigorously being applied to wakes, but interestingly, several of the new discoveries have come from careful use of classical methods. There is no question that strides forward in understanding of the wake problem are being made possible by ongoing three- dimensional direct numerical simulations, as well as by the surprisingly successful use of analytical modeling in these flows, and by secondary stability analyses. These new developments, and the discoveries of several new phenomena in wakes, are presented in this review.

Vortex Dynamics in the Cylinder Wake

After an extensive survey of mean-velocity profile measurements
in various two-dimensional incompressible turbulent boundary-layer
flows, it is proposed to represent the profile by a linear
combination of two universal functions. One is the well-known
law of the wall. The other, called the law of the wake, is
characterized by the profile at a point of separation or reattachment.
These functions are considered to be established empirically, by
a study of the mean-velocity profile, without reference to any
hypothetical mechanism of turbulence. Using the resulting
complete analytic representation for the mean-velocity field,
the shearing-stress field for several flows is computed from the
boundary-layer equations and compared with experimental data.
The development of a turbulent boundary layer is ultimately
interpreted in terms of an equivalent wake profile, which supposedly
represents the large-eddy structure and is a consequence of the
constraint provided by inertia. This equivalent wake profile is
modified by the presence of a wall, at which a further constraint is
provided by viscosity. The wall constraint, although it penetrates
the entire boundary layer, is manifested chiefly in the sublayer flow
and in the logarithmic profile near the wall.
Finally, it is suggested that yawed or three-dimensional flows
may be usefully represented by the same two universal functions,
considered as vector rather than scalar quantities. If the wall
component is defined to be in the direction of the surface shearing
stress, then the wake component, at least in the few cases studied,
is found to be very nearly parallel to the gradient of the pressure.

/pdf/the-law-of-the-wake-in-the-turbulent-boundary-layer-2fb2enxzfl.pdf

The law of the wake in the turbulent boundary layer

This paper describes an experimental investigation of transport processes in the near wake of a circular cylinder at a Reynolds number of 140000. The flow in the first eight diameters of the wake was measured using X-array hot-wire probes mounted on a pair of whirling arms. This flying-hot-wire technique increases the relative velocity component along the probe axis and thus decreases the relative flow angle to usable values in regions where fluctuations in flow velocity and direction are large. One valuable fringe benefit of the technique is that rotation of the arms in a uniform flow applies a wide range of relative flow angles to the X-arrays, making them inherently self-calibrating in pitch. An analog circuit was used to generate an intermittency signal, and a fast surface-pressure sensor was used to generate a phase signal synchronized with the vortex-shedding process. The phase signal allowed sorting of the velocity data into 16 populations, each having essentially constant phase. An ensemble average for each population yielded a sequence of pictures of the instantaneous mean flow field, with the vortices frozen as they would be in a photograph. In addition to globally averaged data for velocity and stress, the measurements yield non-steady mean data (in the sense of an average a t constant phase) for velocity, intermittency, vorticity, stress and turbulent-energy production as a function of phase for the first eight diameters of the near wake. The stresses were resolved into a contribution from the periodic motion and a contribution from the random motion at constant phase. The two contributions are found to have comparable amplitudes but quite different geometries, and the time average of their sum (the conventional global Reynolds stress) therefore has a quite-complex structure. The non-steady mean-vorticity field is obtained with good resolution as the curl of the non-steady mean-velocity field. Less than half of the shed circulation appears in the vortices, and there is a slow decay of this circulation for each shed vortex as it moves downstream. In the discussion, considerable emphasis is put on the topology of the non-steady mean flow, which emerges as a pattern of centres and saddles in a frame of reference moving with the eddies. The kinematics of the vortex-formation process are described in terms of the formation and evolution of saddle points between vortices in the first few diameters of the near wake. One important conclusion is that a substantial part of the turbulence production is concentrated near the saddles and that the mechanism of turbulence production is probably vortex stretching at intermediate scales. Entrainment is also found to be closely associated with saddles and to be concentrated near the upstream-facing interface of each vortex.

/pdf/an-experimental-study-of-entrainment-and-transport-in-the-3lyfkxrywu.pdf

An experimental study of entrainment and transport in the turbulent near wake of a circular cylinder

A transformation is derived from first principles to reduce the boundary-layer equations for a general compressible two-dimensional flow to incompressible form. For the case of boundary-layer flow of a Newtonian fluid past a smooth wall, but with no other restrictions, it is shown that the combination (rho[infinity]µ[infinity]/rhowµw) CfRetheta is an invariant of the transformation. This result is called the law of corresponding stations. In order to apply the transformation to the problem of the turbulent boundary layer on a smooth wall, it is assumed that the sublayer Reynolds number is unaffected by compressibility or heat transfer provided the density and viscosity are evaluated at a mean sublayer temperature defined by the transformation. Explicit formulas are obtained for the effect of Mach number and heat transfer on surface friction when the fluid is a perfect gas, the pressure is constant, and the stagnation temperature is constant or linear in the velocity. An appendix contains a brief critical discussion of the mean-temperature hypothesis, the laminar-film hypothesis, and other analytical ideas related to the idea of a transformation.

/pdf/the-turbulent-boundary-layer-in-a-compressible-fluid-idbkw12pmy.pdf

The Turbulent Boundary Layer in a Compressible Fluid

Laser-Doppler velocity measurements in water are reported for the flow in the plane of symmetry of a turbulent spot. The unsteady mean flow, defined as an ensemble average, is fitted to a conical growth law by using data at three streamwise stations to determine the virtual origin in x and t. The two-dimensional unsteady stream function is expressed as ψ=U^2_∞tg(ξ,η) in conical similarity co-ordinates ζ = x/U_∞t and η = y/U_∞t. In these co-ordinates, the equations for the unsteady particle displacements reduce to an autonomous system. This system is integrated graphically to obtain particle trajectories in invariant form. Strong entrainment is found to occur along the outer part of the rear interface and also in front of the spot near the wall. The outer part of the forward interface is passive. In terms of particle trajectories in conical co-ordinates, the main vortex in the spot appears as a stable focus with celerity 0·77U_∞. A second stable focus with celerity 0·64U_∞ also appears near the wall at the rear of the spot.
Some results obtained by flow visualization with a dense, nearly opaque suspension of aluminium flakes are also reported. Photographs of the sublayer flow viewed through a glass wall show the expected longitudinal streaks. These are tentatively interpreted as longitudinal vortices caused by an instability of Taylor-Gortler type in the sublayer.

/pdf/structure-and-entrainment-in-the-plane-of-symmetry-of-a-5ggklrn62g.pdf

Structure and entrainment in the plane of symmetry of a turbulent spot

Hot-wire measurements have been made in the boundary layer, the separated region, and the near wake for
flow past an NACA 4412 airfoil at mad mum lift. The Reynolds number based on chord was about 1,500,000.
Special care was taken to achieve a two-dimensional mean flow. The main instrumentation was a flying hot wire;
that is, a hot-wire probe mounted on the end of a rotating arm. The probe velocity was sufficiently high to avoid
the usual rectification problem by keeping the relative flow direction always within a range of ± 30 deg from the
probe ads. A digital computer was used to control synchronized sampling of hot-wire data at closely spaced
points along the probe arc. Ensembles of data were obtained at several thousand locations in the flowfield. The
data include Intermittency, two components of mean velocity, and twelve mean values for double, triple, and
quadruple products of two velocity fluctuations. No Information was obtained about the third (spanwise)
velocity component. An unexpected effect of rotor interference was identified and brought under reasonable
control. The data are available on punched cards in raw form and also after use of smoothing and interpolation
routines to obtain values on a fine rectangular grid aligned with the airfoil chord. The data are displayed In the
paper as contour plots.

/pdf/flying-hot-wire-study-of-flow-past-an-naca-4412-airfoil-at-4ypktycoj8.pdf

Donald Coles

Papers

The law of the wake in the turbulent boundary layer

An experimental study of entrainment and transport in the turbulent near wake of a circular cylinder

The Turbulent Boundary Layer in a Compressible Fluid

Structure and entrainment in the plane of symmetry of a turbulent spot

Flying-Hot-wire Study of Flow Past an NACA 4412 Airfoil at Maximum Lift