Donald Coles

The Law of the Wake in the Turbulent Boundary Layer

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.

TRANSITION IN CIRCULAR COUETTE FLOW

Two distinct kinds of transition have been identified in Couette flow between concentric rotating cylinders. The first, which will be called transition by spectral evolution, is characteristic of the motion when the inner cylinder has a larger angular velocity than the outer one. As the speed increases, a succession of secondary modes is excited; the first is the Taylor motion (periodic in the axial direction), and the second is a pattern of travelling waves (periodic in the circumferential direction). Higher modes correspond to harmonics of the two fundamental frequencies of the doubly-periodic flow. This kind of transition may be viewed as a cascade process in which energy is transferred by non-linear interactions through a discrete spectrum to progressively higher frequencies in a two-dimensional wave-number space. At sufficiently large Reynolds numbers the discrete spectrum changes gradually and reversibly to a continuous one by broadening of the initially sharp spectral lines.

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

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.

An experimental study of a turbulent vortex ring

A turbulent vortex ring having a relatively thin core is formed in water by a momentary jet discharge from an orifice in a submerged plate. The necessary impulse is provided by a pressurized reservoir and is controlled by a fast programmable solenoid valve. The main aim of the research is to verify the similarity properties of the mean flow, as defined by ensemble averaging, and to find the distribution of mean vorticity, turbulent energy, and other quantities in the appropriate non-steady similarity coordinates. The velocity field of the vortex is measured for numerous realizations with the aid of a two-channel tracking laser-Doppler velocimeter. The problem of dispersion in the trajectories of the individual rings is overcome by development of a signature-recognition technique in two variables. It is found that the turbulence intensity is largest near the vortex core and that at least the radial component is not negligible in the near wake. The slow growth of the ring structure is controlled by a slight excess of entrainment over de-entrainment. An important inference is that the growth process and the process of turbulence production probably involve secondary vortices wrapped around the core in azimuthal planes.

Measurements of Turbulent Friction on a Smooth Flat Plate in Supersonic Flow

Direct measurements of supersonic local skin friction, using the floating-element technique, are presented for Mach Numbers from 2.0 to 4.5 and Reynolds Numbers from 3 X 10^5 to 9 X 10^6. Turbulent flow and transition are emphasized, although some measurements in the laminar regime are included. The observed effect of compressibility is to reduce the magnitude of turbulent skin friction by a factor of two at a Mach Number of 4.5 and a Reynolds number of about 10^7. The boundary-layer momentum-integral equation for constant pressure is verified within a few per cent by two experimental methods. Typical static pressure measurements are presented to show that transition can be detected by observing disturbances in pressure associated with changes in displacement thickness of the boundary layer. It is found that the turbulent boundary layer cannot be defined experimentally for values of u_1 θ/v_1 less than about 2,000, where θ is the momentum thickness. For larger values of u_1 θ/v_1, there is a unique relationship between local friction coefficient and momentum-thickness Reynolds Number at a fixed Mach Number. The Appendix compares the present measurements at M = 2.5 with experimental data from other sources.

The law of the wall in turbulent shear flow

The boundary-layer equations of continuity and momentum are integrated for a general turbulent shear flow whose mean-velocity profile is given by the law of the wall, u/u τ — f (yu τ /v), with u = v = 0 at y = 0. It is found that u/u τ and yu τ /v are constant on streamlines of the mean flow.

The problem of the turbulent boundary layer

SummaryExisting measurements of low-speed turbulent surface friction on a flat plate, in the absence of pressure gradient and roughness, are shown to be consistent with a simple analysis based on functional similarity in the velocity profile. In particular, the fully developed turbulent boundary layer is found to be unique within the accuracy of the experimental data, with uniqueness defined as the existence of a definite correspondence between local friction coefficient and momentum thickness Reynolds number. The relationships known as the law of the wall and the velocity defect law are found to describe the turbulent velocity profiles accurately for a considerable range of Reynolds numbers, and an effort is made to clarify the physical significance of these formulae. Finally, the proper definition of a length Reynolds number is discussed in terms of the asymptotic local properties of the ideal boundary layer, and numerical values for ideal mean and local friction coefficients are tabulated against Reynolds numbers based on momentum thickness and on distance from the leading edge.ZusammenfassungEs wird gezeigt, dass vorhandene Messungen der turbulenten Wandschubspannung an der glatten ebenen Platte in inkompressibler Strömung ohne Druckgradient durch eine einfache Berechnung in Übereinstimmung gebracht werden können. Die Rechnung beruht auf einer funktionellen Ähnlichkeit der Geschwindigkeitsverteilung. Es wird im besonderen gefunden, dass die vollentwickelte turbulente Grenzschicht innerhalb der Messgenauigkeit einem eindeutigen Zusammenhang zwischen dem örtlichen Reibungskoeffizienten und der Reynoldsschen Zahl, bezogen auf die Impulsdicke, folgt. Die Beziehungen, die als Wandgesetz und Mittengesetz bekannt sind, beschreiben die Geschwindigkeitsverteilung genau innerhalb eines erheblichen Bereiches Reynoldsscher Zahlen, und es wird versucht, den physikalischen Inhalt dieser Gesetzmässigkeiten zu vertiefen. Abschliessend wird eine zweckmässige Definition der auf Plattenlänge bezogenen Reynoldsschen Zahl diskutiert, die auf dem asymptotischen örtlichen Zustand der idealen Grenzschicht beruht. Rechenwerte der idealen, mittleren und örtlichen Reibungskoeffizienten, bezogen auf beide obigen Definitionen der Reynoldsschen Zahl, werden tabelliert.

Prospects for useful research on coherent structure in turbulent shear flow

Six different flows involving coherent structures are discussed with varying amounts of detail. These are the puff in a pipe, the turbulent spot, the spiral turbulence, the vortex ring, the vortex street, and the mixing layer. One central theme is that non-steady similarity arguments and topology are of the essence of coherent structure. Another is that the Reynolds equations, which are sterile when applied to a structureless mean flow, may be quite productive when applied to a single structure. A third theme is the prospect for at least partial control of technically important flows by exploiting the concept of coherent structure.

A 17‐Inch Diameter Shock Tube for Studies in Rarefied Gasdynamics

A shock tube for studying problems in rarefied gasdynamics is described. The motivation for operating at low density (to increase the length and time scales of certain interesting flows) and the effect of low density on the performance and design of the shock tube are discussed. In order to guarantee uniform and reproducible shock waves of moderate strength, the configuration of the tube is conventional. However, innovations are introduced (for example in the suspension, the pumping system, and the diaphragm loading and rupturing mechanism) to simplify the operation of the large facility. Care in the design of the tube as a vacuum system has resulted in a leak rate of less than 0.01 μ Hg per hour. A series of shakedown runs at relatively high pressures has shown, for example, that the reproducibility of a given shock Mach number is ±0.6%.

Remarks on the Equilibrium Turbulent Boundary Layer

Two similarity laws are known for the mean-velocity profile in a turbulent boundary layer with constant pressure. These are Prandtls law of the wall and Karmans momentum-defect law. The first law has recently been generalized empirically to flows with arbitrary pressure gradient by Ludwieg and Tillmann, and the second law to a certain class of equilibrium flows by F. Clauser. In the present paper it is shown that the pressure distribution corresponding to a given equilibrium flow cam be computed by assuming that a certain parameter D = (τ_w/q)dq/dτ_w is constant, where q and τ_w are the dynamic pressure in the free stream and the shearing stress at the wall, respectively. The hypothesis D = constant is suggested by a study of the integrated continuity equation and is supported by a rigorous analogy between the class of equilibrium flows defined by Clauser and the class of laminar flows studied by Falkner and Skan. The hypothesis D = constant is also verified using experimental data for several equilibrium turbulent flows and is interpreted physically from a kinematic point of view. Two hypothetical limiting cases of equilibrium flow are described. At one extreme is the boundary Layer in a sink flow, with a completely logarithmic mean-velocity profile outside the sublayer. At the other extreme is a continuously separating boundary layer in a dimensionless pressure gradient (x/q)dq/dx approximately twice that for the corresponding laminar flow. Typical shearing-stress profiles are computed for several equilibrium turbulent flows, including the two limiting cases.