On the stationarity of linearly forced turbulence in finite domains

A simple scheme of forcing turbulence away from decay was introduced by Lundgren some time ago, the `linear forcing', which amounts to a force term linear in the velocity field with a constant coefficient. The evolution of linearly forced turbulence towards a stationary final state, as indicated by direct numerical simulations (DNS), is examined from a theoretical point of view based on symmetry arguments. In order to follow closely the DNS the flow is assumed to live in a cubic domain with periodic boundary conditions. The simplicity of the linear forcing scheme allows one to re-write the problem as one of decaying turbulence with a decreasing viscosity. Scaling symmetry considerations suggest that the system evolves to a stationary state, evolution that may be understood as the gradual breaking of a larger approximate symmetry to a smaller exact symmetry. The same arguments show that the finiteness of the domain is intimately related to the evolution of the system to a stationary state at late times, as well as the consistency of this state with a high degree of isotropy imposed by the symmetries of the domain itself. The fluctuations observed in the DNS for all quantities in the stationary state can be associated with deviations from isotropy. Indeed, self-preserving isotropic turbulence models are used to study evolution from a direct dynamical point of view, emphasizing the naturalness of the Taylor microscale as a self-similarity scale in this system. In this context the stationary state emerges as a stable fixed point. Self-preservation seems to be the reason behind a noted similarity of the third order structure function between the linearly forced and freely decaying turbulence, where again the finiteness of the domain plays an significant role.Comment: 15 pages, 7 figures, changes in the discussion at the end of section VI, formula (60) correcte

A Preliminary Experimental Study of the Turbulence Decay in the Wake of a Hemisphere in Free-Surface Flow

Flows observed in open channels are usually turbulent. The word turbulent describes a motion in which an irregular velocity fluctuation (mixing or eddying motion) is superimposed on the main stream motion. The essential characteristic of turbulent flow is that the turbulent fluctuations are random in nature. If Reynold\u27s rule of averaging is used, the Navier-Stokes equations for laminar flow may be transformed into the Reynold\u27s equations, which hold true for turbulent mean motion. The solution of Reynold\u27s equations will properly describe turbulent flow. Unfortunately, the number of unknowns exceeds the number of equations; therefore the use of mathematical method to solve turbulent flow problems is extremely difficult and not possible at present. The details of turbulent flow are so complicated that a statistical approach must be used. Extensive research has been done in this regard during the last few decades. G. I. Taylor (1935) presented a statistical theory of turbulence based on the assumption of homogeneous isotropic turbulence. He introduced various concepts such as turbulence intensity, correlation coefficient, scale of turbulence, and energy spectrum. Kolmogoroff (1941) introduced the theory of locally isotropic turbulence. He postulated that turbulent motion at large Reynold\u27s number is locally isotropic whether or not the large scale motions are isotropic. He also introduced the concept that the small scale motions are mainly governed by viscous forces and the amounts of energy which are handed down to them from the large eddies. Other investigations of isotropic turbulence have been made by Karman (1938), Heisenberg (1948), Lin (1948), etc. Even for the simplest type of turbulence (i.e. isotropic turbulence) , it is not possible to obtain the general solution to the equations because the details of turbulence are so complicated. Present knowledge about the statistical distribution of non-homogeneous turbulence is even less extensive. Therefore, before real use can be made of the statistical theory of turbulence much work must be done

A Study of the Structure of Shear Turbulence in Free Surface Flows

Turbulence is a familiar phenomenon which gives rise to complicated problems in many branches of engineering. Hinze has set forth the following definition for turbulence: Turbulent fluid motion is an irregular condition of flow in which the various quantities show a random variation in time pnd space coordinates, so that statistically distinct average values can be discerned. Osborne Reynolds (1894) was the first to introduce the notion of statistical mean values into the study of turbulence. He visualized turbulent flow as the sum of mean and eddying motion. By introducing this sum of mean velocity and fluctuating velocity into the Navier-Stokes equations and with the aid of the continuity equation, he derived equations giving relationships between the various components of the fluctuating velocity. It was soon realized that before any further results could be obtained from a theoretical analysis of Reynold;s equations of motion, a mechanism had to be postulated for the ihteraction of fluctuating v~locity components at different points in the turbulent field. Consequently, three decades after Reynold\u27s: work, phenomenological theories of turbulence, such as the momentum-transfer theory of Prandtl (1926), the vorticity transport theory of Taylor (1932) and the similarity theory of Karman (1930) were introduced. Not only are they based on unrealistic physical models, but they do not furnish any information beyond temporal-mean velocity distribution. A complete theory of turbulence should describe the mechanism of production of turbulence, its convection, diffusion, distribution, and eventual dissipation for any given boundary conditions

The structure of turbulence in an open channel with large spherical roughness elements

The present status of knowledge of turbulent flow is inadequate, especially in the case of rough open channels, for the formulation of a general theory. It is believed that more experimental data and the subsequent interpretation of these data are necessary before a workable theory can be formulated. Hence, a description of the turbulence present in a rough open channel can be valuable. For this study an artificially roughened bed 48 feet in length was placed in a channel 8 feet wide and 6 feet deep. Measurements were made of the following properties of turbulence at three different slopes: 1. Intensity of turbulence. 2. Autocorrelation and cross-correlation coefficients. 3. Microscale and macroscale of turbulence. 4. One-dimensional energy and dissipation spectra. All the measurements were taken with a piezoelectric total head tube in combination with necessary electronics equipment. The results have been presented in the form of dimensionless curves as far as possible. These curves are compared with published data

Aerodynamics of a wing in turbulent bluff body wakes

The aerodynamics of a stationary wing in a turbulent wake are investigated. Force and velocity measurements are used to describe the unsteady flow. Various wakes are studied with different dominant frequencies and length scales. In contrast to the pre-stall angles of attack, the time-averaged lift increases substantially at post-stall angles of attack as the wing interacts with the von Kármán vortex street and experiences temporal variations of the effective angle of attack. At an optimal offset distance from the wake centreline, the time-averaged lift becomes maximum despite of small amplitude oscillations in the effective angle of attack. The stall angle of attack can reach 20° and the maximum lift coefficient can reach 64 % higher than that in the freestream. Whereas large velocity fluctuations at the wake centreline cause excursions into the fully attached and separated flows during the cycle, small-amplitude oscillations at the optimal location result in periodic shedding of leading edge vortices. These vortices may produce large separation bubbles with reattachment near the trailing-edge. Vorticity roll-up, strength and size of the vortices increase with increasing wavelength and period of the von Kármán vortex street, which also coincides with an increase in the spanwise length scale of the incident wake, and all contribute to the remarkable increase in lift.</p

Spectral imbalance and the normalized dissipation rate of turbulence

The normalized turbulent dissipation rate $C_\epsilon$ is studied in decaying and forced turbulence by direct numerical simulations, large-eddy simulations, and closure calculations. A large difference in the values of $C_\epsilon$ is observed for the two types of turbulence. This difference is found at moderate Reynolds number, and it is shown that it persists at high Reynolds number, where the value of $C_\epsilon$ becomes independent of the Reynolds number, but is still not unique. This difference can be explained by the influence of the nonlinear cascade time that introduces a spectral disequilibrium for statistically nonstationary turbulence. Phenomenological analysis yields simple analytical models that satisfactorily reproduce the numerical results. These simple spectral models also reproduce and explain the increase of $C_\epsilon$ at low Reynolds number that is observed in the simulations

The role of dissipation in flexural wave turbulence: from experimental spectrum to Kolmogorov-Zakharov spectrum

The Weak Turbulence Theory has been applied to waves in thin elastic plates obeying the F\"oppl-Von K\'arm\'an dynamical equations. Subsequent experiments have shown a strong discrepancy between the theoretical predictions and the measurements. Both the dynamical equations and the Weak Turbulence Theory treatment require some restrictive hypotheses. Here a direct numerical simulation of the F\"oppl-Von K\'arm\'an equations is performed and reproduces qualitatively and quantitatively the experimental results when the experimentally measured damping rate of waves $\gamma_\mathbf{k}= a + bk^2$ is used. This confirms that the F\"oppl-Von K\'arm\'an equations are a valid theoretical framework to describe our experiments. When we progressively tune the dissipation so that to localize it at the smallest scales, we observe a gradual transition between the experimental spectrum and the Kolmogorov-Zakharov prediction. Thus it is shown dissipation has a major influence on the scaling properties stationary solutions of weakly non linear wave turbulence.Comment: 10 pages, 11 figure

Reduced-order modeling and dynamics of nonlinear acoustic waves in a combustion chamber

For understanding the fundamental properties of unsteady motions in combustion chambers, and for applications of active feedback control, reduced-order models occupy a uniquely important position. A framework exists for transforming the representation of general behavior by a set of infinite-dimensional partial differential equations to a finite set of nonlinear second-order ordinary differential equations in time. The procedure rests on an expansion of the pressure and velocity fields in modal or basis functions, followed by spatial averaging to give the set of second-order equations in time. Nonlinear gasdynamics is accounted for explicitly, but all other contributing processes require modeling. Reduced-order models of the global behavior of the chamber dynamics, most importantly of the pressure, are obtained simply by truncating the modal expansion to the desired number of terms. Central to the procedures is a criterion for deciding how many modes must be retained to give accurate results. Addressing that problem is the principal purpose of this paper. Our analysis shows that, in case of longitudinal modes, a first mode instability problem requires a minimum of four modes in the modal truncation whereas, for a second mode instability, one needs to retain at least the first eight modes. A second important problem concerns the conditions under which a linearly stable system becomes unstable to sufficiently large disturbances. Previous work has given a partial answer, suggesting that nonlinear gasdynamics alone cannot produce pulsed or 'triggered' true nonlinear instabilities; that suggestion is now theoretically established. Also, a certain form of the nonlinear energy addition by combustion processes is known to lead to stable limit cycles in a linearly stable system. A second form of nonlinear combustion dynamics with a new velocity coupling function that naturally displays a threshold character is shown here also to produce triggered limit cycle behavior