Decay Of Homogeneous Turbulence Research Articles

Flow models of perforated manifolds and plates for the design of a large thermal storage tank for district heating with minimal maldistribution and thermocline growth

Large water tanks are used as thermal energy storage components in district heating systems to store sensible heat produced by intermittent energy sources and to decouple the production of thermal energy from its demand. Good thermal stratification is crucial for energy storage efficiency, thus flow maldistribution and mixing of water layers at different temperatures should be minimized. This paper proposes an innovative internal flow distribution configuration for a large-size thermal energy storage, and develops new simplified analytical models for the choice of its design parameters. In the novel configuration, water is injected into (and collected from) the cap volumes of the tank by flowing radially inward (outward) through several small orifices of a peripheral toroidal manifold. Two horizontal perforated plates cover the full cross sections downstream of the manifolds and rectify the vertical flow, thus reducing mixing. Uniform perforation pitch was analytically demonstrated to be the most reasonable solution both for the toroidal distributors and for the rectifying plates. A 1D model was developed to predict the time evolution of the vertical temperature distribution in the tank. The turbulence-related parameters that could not be inferred from the existing fluid-mechanics literature were initially estimated with CFD simulations. The results of CFD-calibrated model were then compared to experimental data obtained from a full-scale large water-tank facility recently built in Brescia according to the proposed design. After a re-calibration of the exponent defining the decay of homogeneous turbulence downstream of the perforated plates, good agreement was found between measured and predicted vertical temperatures. With the novel inlet design, a thermocline of about 0.5 m is established immediately downstream of the perforated plate, and remains practically constant along time. The model is important to minimize and control the thermocline thickness so as to maximize the recoverable thermal energy, not only at the tank design stage but also to identify optimal loading and unloading protocols.

Numerical-based theoretical analysis on the decay of homogeneous turbulence affected by small strain based on constant and linear strain variations

This study clarifies the effects of small strain on the decay of homogeneous turbulence by focusing on the temporal profile of the small strain. Small strain is defined in such a way as to not affect the anisotropy of the homogeneous turbulence. We apply the framework of the standard k–ε model to examine the effects of small strain. Constant and linearly varying small strains are studied. The effects of linearly varying small strain are found to be greater than those of constant strain. To discuss the results, we derive an analytical solution that describes the effects of the two types of small strain. Although the form of the analytical solutions is the same for constant and linearly varying strains, the coefficients used in the analytical solutions, for which an equation is also obtained, differ. The difference observed in the effects of the two types of small strain is thus caused by the difference in coefficient.

Time-parallel simulation of the decay of homogeneous turbulence using Parareal with spatial coarsening

Direct Numerical Simulation of turbulent flows is a computationally demanding problem that requires efficient parallel algorithms. We investigate the applicability of the time-parallel Parareal algorithm to an instructional case study related to the simulation of the decay of homogeneous isotropic turbulence in three dimensions. We combine a Parareal variant based on explicit time integrators and spatial coarsening with the space-parallel Hybrid Navier–Stokes solver. We analyse the performance of this space–time parallel solver with respect to speedup and quality of the solution. The results are compared with reference data obtained with a classical explicit integration, using an error analysis which relies on the energetic content of the solution. We show that a single Parareal iteration is able to reproduce with high fidelity the main statistical quantities characterizing the turbulent flow field.

Turbulence without Richardson–Kolmogorov cascade

We investigate experimentally wind tunnel turbulence generated by multiscale/fractal grids pertaining to the same class of low-blockage space-filling fractal square grids. These grids are not active but nevertheless produce very much higher turbulence intensities u′/U and Reynolds numbers Reλ than higher blockage regular grids. Our hot wire anemometry confirms the existence of a protracted production region where turbulence intensity grows followed by a decay region where it decreases, as first reported by Hurst and Vassilicos [“Scalings and decay of fractal-generated turbulence,” Phys. Fluids 19, 035103 (2007)]. We introduce the wake-interaction length scale x⋆ and show that the peak of turbulence intensity demarcating these two regions along the centerline is positioned at about 0.5x⋆. The streamwise evolutions on the centerline of the streamwise mean flow and of various statistics of the streamwise fluctuating velocity all scale with x⋆. Mean flow and turbulence intensity profiles are inhomogeneous at streamwise distances from the fractal grid smaller than 0.5x⋆, but appear quite homogeneous beyond 0.5x⋆. The velocity fluctuations are highly non-Gaussian in the production region but approximately Gaussian in the decay region. Our results confirm the finding of Seoud and Vassilicos [“Dissipation and decay of fractal-generated turbulence,” Phys. Fluids 19, 105108 (2007)] that the ratio of the integral length-scale Lu to the Taylor microscale λ remains constant even though the Reynolds number Reλ decreases during turbulence decay in the region beyond 0.5x⋆. As a result, the scaling Lu/λ∼Reλ, which follows from the u′3/Lu scaling of the dissipation rate in boundary-free shear flows and in usual grid-generated turbulence, does not hold here. This extraordinary decoupling is consistent with a noncascading and instead self-preserving single-length scale type of decaying homogeneous turbulence proposed by George and Wang [“The exponential decay of homogeneous turbulence,” Phys. Fluids 21, 025108 (2009)], but we also show that Lu/λ is nevertheless an increasing function of the inlet Reynolds number Re0. Finally, we offer a detailed comparison of the main assumption and consequences of the George and Wang theory against our fractal-generated turbulence data.

Direct numerical simulation of stationary particles in homogeneous turbulence decay: Application of the k–ε model

This study focuses on understanding how the presence of particles, in homogeneous turbulence decay, affects the dissipation of dissipation coefficient within the volume averaged dissipation transport equation. In developing this equation, the coefficient for dissipation of dissipation was assumed to be the sum of the single phase coefficient and an additional coefficient that is related to the effects of the dispersed phase. Direct numerical simulation was used to isolate the effect of stationary particles in homogeneous turbulent decay at low Reynolds numbers (Re L = 3.3 and 12.5). The particles were positioned at each grid point and modeled as point forces and a comparison was made between a 64 3 and 128 3 domain. The results show that the dissipation of dissipation coefficient correlates well with a dimensionless parameter called the momentum coupling factor.

Large eddy simulation of a forward–backward facing step for acoustic source identification

The feasibility of using a commercial CFD code for large eddy simulation (LES) is investigated. A first test on homogeneous turbulence decay allows a fine-tuning of the eddy viscosity with respect to the numerical features of the code. Then, a flow over forward–backward facing step at Reynolds number Re h =1.7×10 5 is computed. The results found show good agreement with the new LDA data of Leclercq et al. [Forward backward facing step pair: aerodynamic flow, wall pressure and acoustic characterization. AIAA-2001-2249]. The acoustic source term, recorded from the LES and to be fed into a following acoustic propagation simulation, is found to be largest in the separation from the forward step. The source terms structures are similar to the vortical structures generated at the front edge of the obstacle and advected downstream. Structures generated from the backward step rapidly break down into smaller scale structures due to the background turbulence.

G.K. BATCHELOR AND THEHOMOGENIZATION OFTURBULENCE

▪ Abstract This essay is based on the G.K. Batchelor Memorial Lecture that I delivered in May 2000 at the Institute for Theoretical Physics (ITP), Santa Barbara, where two parallel programs on Turbulence and Astrophysical Turbulence were in progress. It focuses on George Batchelor's major contributions to the theory of turbulence, particularly during the postwar years when the emphasis was on the statistical theory of homogeneous turbulence. In all, his contributions span the period 1946–1992 and are for the most part concerned with the Kolmogorov theory of the small scales of motion, the decay of homogeneous turbulence, turbulent diffusion of a passive scalar field, magnetohydrodynamic turbulence, rapid distortion theory, two-dimensional turbulence, and buoyancy-driven turbulence.

Modeling nonequilibrium and history effects of homogeneous turbulence in a stably stratified medium

Abstract Homogeneous turbulence decay in a stably stratified flow has two distinct characteristics. One is countergradient fluxes that are developed to keep the energy budget in equilibrium. Another is the formation of gravity waves that contribute to the velocity variance with little vertical mixing. The prediction of these characteristics are investigated using a hierarchy of turbulence models. They include second-order models as well as models that solve the transport equations for the turbulent kinetic energy, its dissipation rate, the temperature variance and its dissipation rate. In the latter class of models, the vertical heat flux is calculated either from an algebraic equation or from a transport equation. The algebraic equation is derived by invoking equilibrium and nonequilibrium turbulence assumption. Thus, modeling level and the relative importance of nonequilibrium and history effects in the predictions of countergradient fluxes and gravity waves could be assessed. The investigation reveals that countergradient heat flux can be predicted even when the equilibrium assumption is invoked. However, the formation of gravity waves can be predicted only when the history effects of the vertical heat flux are accounted for properly. The decay rate of the total energy is very much affected by two model constants in the temperature variance dissipation rate equation. On the other hand, the calculated frequency and amplitude of the gravity waves are influenced by the model constants in the heat flux equation.

The decay of axisymmetric homogeneous turbulence

The decay of a homogeneous turbulence generated by an axisymmetric distribution of random impulsive forces acting at the initial instant is studied by means of large-eddy simulations. The impulsive forces may be either parallel or perpendicular to the symmetry axis. For impulsive forces, which result in a k4 low wave number energy spectrum of the turbulence, it is determined that the flow approaches isotropy on all scales of motion at long times, provided the Reynolds number is large. However, for the type of impulsive forces originally proposed by Saffman [J. Fluid Mech. 27, 581 (1967)], in which a k2 low wave number energy spectrum is produced, the turbulence approaches isotropy only at the smallest scales, and remains significantly anisotropic at the largest and energy-containing scales. Nevertheless, a similarity state of the flow field establishes itself asymptotically, in which the kinetic energy per unit mass of the turbulence decays as t−6/5.

Power-law decay of homogeneous turbulence at low Reynolds numbers

The decay of nominally isotropic, homogeneous incompressible turbulence is studied by direct numerical simulations for Reλ in the range (5–50) with 2563 spectral coefficients. A power-law decay of the turbulent energy is observed with exponents approximately equal to 1.5 and 1.25, apparently dependent on Reλ. A new complete similarity form for the double and triple velocity correlation functions, f(r,t) and k(r,t), is proposed for low to intermediate Reλ that is consistent with the Kármán–Howarth equation and the results of the numerical experiments. The results are also consistent with Saffman’s proposed asymptotic behavior of f(r,t) for large separation r for runs with a decay exponent of 1.5. The so-called final period of decay is not observed.

Renormalization-group analysis of turbulence.

Using renormalization-group methods and the postulated equivalence between the inertial-range structures of turbulent flows satisfying initial and boundary conditions and of flows driven by a random force, we evaluate the Kolmogorov constant (1.617) and Batchelor constant (1.161), skewness factor (0.4878), power-law exponent (1.3307) for the decay of homogeneous turbulence, turbulent Prandtl number (0.7179), and von K\'arm\'an constant (0.372). This renormalization-group technique has also been used to derive turbulent transport models.

Decay of homogeneous turbulence in three-range models

New models, with energy spectra which consist of three power-law ranges, are suggested to describe the free evolution of homogeneous and isotropic high-Reynolds-number turbulence. By selecting a single number, a characteristic of the energy spectrum in the model, very good agreement with experiments has been achieved for two independently measured parameters. Also, verifiable dependences of turbulence decay features on characteristics of the turbulence generation process are suggested.

Exact theory for the self‐similarity and decay of homogeneous turbulence

It is shown that space‐time dilatation invariance (x→ξ−1x, t→ξ−2t, with concomitant transformations for dependent variables) and linearity of the Φ‐equation engender an exact, time‐explicit generic form for the solution applicable to freely‐decaying homogeneous incompressible fluid turbulence. This solution features a summation over mutually independent dynamical modes labeled by the dilatation scaling‐index n(>1). Without the assumption of isotropy nor introduction of a closure approximation procedure, the theory provides an explanation for the experimentally observed self‐similarity of the correlation tensors and the decay laws 〈‖u(x, t)‖2〉∝t−n for the different types and decay stages of homogeneous turbulence.

Decay of homogeneous turbulence from a given state at higher Reynolds number

The hierarchy of turbulence equations is closed by several methods, but good results for higher Reynolds numbers are obtained only for a closure by specification of initial conditions. Four terms of an exponential series, which is an iterative solution of the Navier–Stokes equation, are used in the closure. A complete solution, in the sense that the evolution of all of the initially specified spectra can be calculated, is obtained. This cannot be done by using a Taylor series, regardless of how many terms are retained. The results indicate that purely viscous decay becomes negligible compared with inertial decay as Reynolds number becomes large.

Similar Solutions of the Mean Velocity, Turbulent Energy and Length Scale Equation

A semiempineal approximation of the equations for second order correlations and moments was quantitatively investigated. The resulting partial differential equations for mean velocity, turbulent kinetic energy and length scale are supposed to be valid for plane and axisymmetric free as well as bounded turbulent flows. They were converted into ordinary differential equations by means of similarity transformations for six different flow cases (plane free jet, plane asymptotic wake, free mixing layer, plane channel flow, axisymmetric free jet, axisymmetric pipe flow) and then solved numerically. The values of the coefficients introduced for the closure of the equations were determined from investigations of the dependence of the solutions on them and by comparison with experimental results. A single set of coefficients was obtained, which reproduces the measurements of mean velocity for all six treated cases and the decay of homogeneous turbulence with satisfactory agreement.

Remarks on the decay of homogeneous turbulence from a given state

A previous theory which did not require a usual closure assumption required three or more initial spectra. By allowing a simple physical assumption (a modification of Kovasznay's hypothesis), the required number of spectra is reduced to two. Agreement with experiment is good.

Multi-point, multi-time concentration correlations and decay of homogeneous turbulence

Following Deissler's approach[1], four-point, four-time correlation equations for the concentration of a dilute contaminant undergoing a first order chemical reaction, have been obtained by neglecting the quintuple correlations. The solution obtained gives the state of concentration fluctuation in a homogeneous turbulence between its initial and final periods.

Use of similarity hypotheses for solving the problem of homogeneous turbulence decay: PMM vol. 37, n≗5, 1973, pp. 864–881

The feasibility of defining a homogeneous, and in particular, an isotropic turbulence, using only the hypotheses of similarity of spectral, correlation, or structural functions is considered. Hypotheses of similarity (“self-preservation”) of correlation functions were used in [1] and in many subsequent works in conjunction with other assumptions. Sedov [2], using only these hypotheses for analyzing the problem of decay of homogeneous and isotropic turbulent motions of a viscous incompressible fluid, had shown possible alternatives: either the energy of turbulent velocity pulsations decrease in inverse proportion to time and the linear scale increases in proportion to the square root of time, when the determination of dimensionless correlation functions requires additional assumptions; or the complete solution can be derived (and was derived by Sedov) without having to resort to supplementary assumptions. It is shown here that this solution is in good agreement with experimental data on the decay of turbulent motions of air and water downstream of a grid. With the use of the hypothesis of self-preservation of spectral tensors Sedov's solution is extended to homogeneous nonisotropic turbulence, and with the use of hypotheses of self-preservation of structural functions isotropic turbulence. In both cases alternatives similar to those of Sedov were obtained.

The relaxation of stress in a ≠-fluid with reference to the decay of homogeneous turbulence

It was suggested by Proudman (1970) that many of the phenomena of turbulence at high Reynolds number could be modelled by a suitably chosen member of a class of non-Newtonian fluids, ν-fluids, all of whose properties depend only on a single dimensional constant with the dimensions of viscosity. This paper investigates the relaxation of homogeneous stress in a doubly degenerate third-order ν-fluid (which is the simplest member of the class that can possibly be used to model turbulence) in the limit ν → 0.The equation which governs the stress tensor S is of the form \[ AS\ddot{S} + B\dot{S}^2 = 0, \] where A and B are isotropic tensor constants of the fluid; its differential structure can be of four distinct types. A list of properties required of the solutions of the equation is set out, and it is shown that only one of the four types has all the properties demanded of it. The behaviour of solutions of this equation is found to be consistent with the theoretical and experimental results on the decay of homogeneous turbulence.

Further Comparison of Theory and Experiment for Decay of Homogeneous Turbulence

An analysis for the decay of homogeneous turbulence from a given initial state is compared with the water-channel data of Ling and Huang. Good agreement is indicated.