Finite size analysis of a double crossover in transitional wall turbulence

Turbulent–laminar intermittency, typically in the form of bands and spots, is a ubiquitous feature of the route to turbulence in wall-bounded shear flows. Here we study the idealised shear between stress-free boundaries driven by a sinusoidal body force and demonstrate quantitative agreement between turbulence in this flow and that found in the interior of plane Couette flow – the region excluding the boundary layers. Exploiting the absence of boundary layers, we construct a model flow that uses only four Fourier modes in the shear direction and yet robustly captures the range of spatiotemporal phenomena observed in transition, from spot growth to turbulent bands and uniform turbulence. The model substantially reduces the cost of simulating intermittent turbulent structures while maintaining the essential physics and a direct connection to the Navier–Stokes equations. We demonstrate the generic nature of this process by introducing stress-free equivalent flows for plane Poiseuille and pipe flows that again capture the turbulent–laminar structures seen in transition.

In this paper the development and a partially verification of an anisotropic modification of linear and nonlineare models of turbulence is presented. This modification departs from the observation that there are some regions of fluid domains in which the Reynolds stresses do not depend on the magnitude of the mean flow gradients. The model is based on the hypothesis that actions of fluctuating motion on mean motion are almost exclusively due to large turbulent eddies, and that the latter contain the same kinetic energy of the small eddies of mean flow. The contribution of large turbulent eddies to the Reynolds stresses is then modelled by a linear combination of an isotropic term and an anisotropic term. The latter is obtained by multiplying a quote of the turbulent kinetic energy by a normalised redistribution tensor. The performances of the model are tested calculating the Reynolds stresses in a fully developed turbulent channel flow, and simulating the separated turbulent flow over a backward-facing step, and a fully developed turbulent flow in a squared duct. The results show that the modified models improve the prediction of the normal Reynolds stresses in a bidimensional channel and of the reattachment point in a backward-facing step. On the other hand, these models do not supply any improvement of the secondary flow in a squared duct. INTRODUCTION Algebraic two-equation models of turbulence for Reynolds averaged Navier-Stokes equations are still, today, among the most widely used in the field of engineering design. The two major alternative to such models are the second-order closure models and large-eddy simulations (LES). The weak points of the first approach are high computational cost, due to the five more equations to be solved, and wall-bounded turbulent flows. On the other hand, LES requires much more computational time than second-order closure models, and improved models for the smallscale and the wall region are needed. For these reasons these two approaches have not become yet a useful engineering tool. The aim of this work is to develop an anisotropic modification for linear and nonlinear turbulence model which improves the prediction of normal Reynolds stresses, without increasing the number of differential equations to be solved. More precisely, the objective is to simulate the contribution to the anisotropy of the turbulent stresses observable in areas characterised by low mean velocity gradients. This phenomenon, which can be noted, for example, in experimental data of the turbulent channel flow in figure 5, is not actually described by standard, explicit algebraic models. The starting point of this model consists in the substitution of turbulent kinetic energy by an appropriate vectorial magnitude which provides information about the anistropy of the turbulent stresses. The quantity is identified by the field of velocity of the small eddies of mean motion. The kinetic energy of these small eddies is obtained by assuming that these t Dr, Center of Studies and Activities for Space (CISAS). Dr, Department of Mechanical Engineering. ' Professor, Department of Mechanical Engineering. Copyright © 2001 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. 1 American Institute of Aeronautics and Astronautics (c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. have the same kinetic energy content of the large turbulent eddies: the two fields of motion are said to be similar. It has been demonstrated that the field of motion of large turbulent eddies, or coherent structures^ can be obtained directly from measurements of instantaneous velocity once the quota of turbulent kinetic energy has been defined. The models obtained provides improvements in the predictions of normal Reynolds stresses present in the turbulent flow of a bidimensional channel; as for the separated turbulent flow in a backward-facing step appreciable improved results of mean flow and of normal Reynolds stresses have been attained compared to standard algebraic models. On the other side, as it will be shown in the following paragraphs, the model of the turbulent kinetic energy redistribution tensor does not supply any improvement in the case of the fully developed turbulent flow in a squared duct. ANISOTROPIC MODIFICATION OF THE REYNOLDS STRESSES Let us consider a field of turbulent motion of a viscous, homogenous and incompressible fluid; the instantaneous velocity v and the instantaneous pressure p can be set out as follows in a mean and fluctuating parts: v = v -f v' p-p + p (1) The evolution of the velocities and mean pressures can be described completely by the well-known Reynolds equations:

In this paper, the effects of adiabatic and isothermal conditions on the statistics in compressible turbulent channel flow are investigated using direct numerical simulation (DNS). DNS of two compressible turbulent channel flows (Cases 1 and 2) are performed using a mixed Fourier Galerkin and B-spline collocation method. Case 1 is compressible turbulent channel flow between isothermal walls, which corresponds to DNS performed by Coleman et al. (1995). Case 2 is the flow between adiabatic and isothermal walls. The flow of Case 2 can be a very useful framework for the present objective, since it is the simplest turbulent channel flow with an adiabatic wall and provides ideal information for modelling the compressible turbulent flow near the adiabatic wall. Note that compressible turbulent channel flow between adiabatic walls is not stationary if there is no sink of heat. In Cases 1 and 2, the Mach number based on the bulk velocity and sound speed at the isothermal wall is 1.5, and the Reynolds number based on the bulk density, bulk velocity, channel half-width, and viscosity at the isothermal wall is 3000. To compare compressible and incompressible turbulent flows, DNS of two incompressible turbulent channel flows with passive scalar transport (Cases A and B) are performed using a mixed Fourier Galerkin and Chebyshev tau method. The wall boundary conditions of Cases A and B correspond to those of Cases 1 and 2, respectively. Case A corresponds to the DNS of Kim & Moin (1989). In Cases A and B, the Reynolds number based on the friction velocity, the channel half-width, and the kinematic viscosity is 150. The mean velocity and temperature near adiabatic and isothermal walls for compressible turbulent channel flow can be explained using the non-dimensional heat flux and the friction Mach number. It is found that Morkovin's hypothesis is not applicable to the near-wall asymptotic behaviour of the wall-normal turbulence intensity even if the variable property effect is taken into account. The mechanism of the energy transfers among the internal energy, mean and turbulent kinetic energiesis investigated, and the difference between the energy transfers near isothermal and adiabatic walls is revealed. Morkovin's hypothesis is not applicable to the correlation coeffcient between velocity and temperature fluctuations near the adiabatic wall.

Nowadays, the global increase of energy demand and the necessity to satisfy high safety standards have led engineers and scientists to focus their efforts in order to understand and describe fundamental phenomena that are crucial for a correct design of the new generation nuclear power plants. In this framework, the present thesis aims at providing a first insight of the mechanisms of deposition of aerosol particles inside a closed geometry where relatively strong currents are present due to turbulent natural convective flows. Direct Numerical Simulations were conducted coupling high-order pseudo-spectral code with a Lagrangian particle tracker. Laminar flows were computed in two and three dimensions in order to benchmark the code with published reference data. A parametric study was performed for three different aerosol micro-size particle diameters and two super-critical Rayleigh numbers in a square cavity. An extended analysis of the turbulent flows is provided in terms of first and second order statistics, time-averaged momentum and energy budgets, and moreover, important terms appearing in the transport equations of turbulent kinetic energy and temperature variance are also briefly discussed. Furthermore, the evolution in time of particle concentration for the three different diameters is considered. The text provides information about the deposition velocity, the deposition patterns on the cavity surfaces, the influence of lift and thermophoretic forces and the fractal dimension. The same size dependent parametric study for the three different sets of micro-size particles was carried out in a fully three-dimensional closed cubic cavity for one super-critical Rayleigh number. A detailed investigation of the turbulence was performed by means of statistical quantities, signal processing and conditional averaging, in order to get a general view of the complexity of the flow and its characteristics. Further on, the sedimentation process is studied in the same way as for the two dimensional case. Finally a simple theoretical deposition model is provided in order to interpret the numerical results for the aerosol phase.

The turbulence in plane Couette flow subjected to system rotation is investigated.The anti-cyclonic rotation rate is well above the range in which roll-cells occur and close to the upper bound, beyond which no stationary turbulent states of motion exist.The mean velocity profile exhibits a linear region over 80% of the cross-section, in which the mean absolute vorticity is driven to zero.Viscous effects still prevail in narrow regions next to the walls, whereas the quasi-homogeneous central core exhibits abnormal anisotropies of the Reynolds stress tensor, the vorticity tensor and the energy dissipation rate tensor.In spite of the distinctly higher turbulence level observed, a 13% drag reduction is found.This paradoxical finding is ascribed to configurational changes in the turbulence field brought about by the system rotation.

Although turbulent flows are common in the world around us, a solution to the fundamental equations that govern turbulence still eludes the scientific community. Turbulence has often been called one of the last unsolved problem in classical physics, yet it is clear that the need to accurately predict the effect of turbulent flows impacts virtually every field of science and engineering. As an example, a critical step in making modern computational tools useful in designing aircraft is to be able to accurately predict the lift, drag, and other aerodynamic characteristics in numerical simulations in a reasonable amount of time. Simulations that take months to years to complete are much less useful to the design cycle. Much work has been done toward this goal (Lee-Rausch et al. 2003, Jameson 2003) and as cost effective accurate tools for simulating turbulent flows evolve, we will all benefit from new scientific and engineering breakthroughs. The problem of simulating high Reynolds number (Re) turbulent flows of engineering and scientific interest would have been solved with the advent of Direct Numerical Simulation (DNS) techniques if unlimited computing power, memory, and time could be applied to each particular problem. Yet, given the current and near future computational resources that exist and a reasonable limit on the amount of time an engineer or scientist can wait for a result, the DNS technique will not be useful for more than 'unit' problems for the foreseeable future (Moin & Kim 1997, Jimenez & Moin 1991). The high computational cost for the DNS of three dimensional turbulent flows results from the fact that they have eddies of significant energy in a range of scales from the characteristic length scale of the flow all the way down to the Kolmogorov length scale. The actual cost of doing a three dimensional DNS scales as Re{sup 9/4} due to the large disparity in scales that need to be fully resolved. State-of-the-art DNS calculations of isotropic turbulence have recently been completed at the Japanese Earth Simulator (Yokokawa et al. 2002, Kaneda et al. 2003) using a resolution of 40963 (approximately 10{sup 11}) grid points with a Taylor-scale Reynolds number of 1217 (Re {approx} 10{sup 6}). Impressive as these calculations are, performed on one of the world's fastest super computers, more brute computational power would be needed to simulate the flow over the fuselage of a commercial aircraft at cruising speed. Such a calculation would require on the order of 10{sup 16} grid points and would have a Reynolds number in the range of 108. Such a calculation would take several thousand years to simulate one minute of flight time on today's fastest super computers (Moin & Kim 1997). Even using state-of-the-art zonal approaches, which allow DNS calculations that resolve the necessary range of scales within predefined 'zones' in the flow domain, this calculation would take far too long for the result to be of engineering interest when it is finally obtained. Since computing power, memory, and time are all scarce resources, the problem of simulating turbulent flows has become one of how to abstract or simplify the complexity of the physics represented in the full Navier-Stokes (NS) equations in such a way that the 'important' physics of the problem is captured at a lower cost. To do this, a portion of the modes of the turbulent flow field needs to be approximated by a low order model that is cheaper than the full NS calculation. This model can then be used along with a numerical simulation of the 'important' modes of the problem that cannot be well represented by the model. The decision of what part of the physics to model and what kind of model to use has to be based on what physical properties are considered 'important' for the problem. It should be noted that 'nothing is free', so any use of a low order model will by definition lose some information about the original flow.

Compressible large eddy simulations of wall-bounded turbulent flows using a semi-implicit numerical scheme for low Mach number aeroacoustics

We present results of performance optimization for direct numerical simulation (DNS) of wall bounded turbulent flow (channel flow). DNS is a technique in which the fluid flow equations are solved without subgrid modeling. Of particular interest are high Reynolds number (Re) turbulent flows over walls, because of their importance in technological applications. Simulating high Re turbulence is a challenging computational problem, due to the high spatial and temporal resolution requirements. An optimized code was developed using spectral methods, the method of choice for turbulent flows. Optimization was performed to address three major issues: efficiency of banded matrix linear algebra, cache reuse and memory access, and communication for the global data transposes. Results show that performance is highly dependent on characteristics of the communication network, rather than singlecore performance. In our tests, it exhibits approximately 80% strong scaling parallel efficiency at 786K cores relative to performance on 65K cores.

Large-eddy simulation of turbulent channel flows with conservative IDO scheme

In the present study, an experimental investigation is conducted for the research of the turbulent open channel flow field at the three directions with the aid of one component Laser Doppler Anemometer and x Hot Film sensors. The instrumentation is used in an efficient manner for the extraction of three turbulent statistics with one measurement. The application of the present technique is conducted on fully developed uniform open channel turbulent flow and on flow with suction from the bed. With the aid of assumption for zero correlation between velocity and direction fluctuations the required number of measurements in the plane of flow direction for turbulent statistics is reduced. Using the Hot film sensor in the streamwise- spanwise orientation and with the LDA beam plane inclined in streamwise - vertical plane for the extraction a component of streamwise velocity direction, the three turbulent statistics are estimated. Also, with the assumption of randomness of velocity fluctuations of the open channel flow in the zero correlation assumption, the dependence of turbulent statistics with the angle is possible to be calculated. The influence of suction from the bed on the uniform open channel flow, alters the turbulent flow field characteristics and in the present paper the distribution of turbulent intensities are presented for various suction rates in the flow depth. In the form of covariance and autocovariance, angle dependence is examined for uniform flow and for flow with suction from the bed.

Purpose – This study is concerned with the direct numerical simulation (DNS) of a turbulent channel flow by an improved vortex in cell (VIC) method. The paper aims to discuss these issues. Design/methodology/approach – First, two improvements for VIC method are proposed to heighten the numerical accuracy and efficiency. A discretization method employing a staggered grid is presented to ensure the consistency among the discretized equations as well as to prevent the numerical oscillation of the solution. A correction method for vorticity is also proposed to compute the vorticity field satisfying the solenoidal condition. Second, the DNS for a turbulent channel flow is conducted by the improved VIC method. The Reynolds number based on the friction velocity and the channel half width is 180. Findings – It is highlighted that the simulated turbulence statistics, such as the mean velocity, the Reynolds shear stress and the budget of the mean enstrophy, agree well with the existing DNS results. It is also shown that the organized flow structures in the near-wall region, such as the streaks and the streamwise vortices, are favourably captured. These demonstrate the high applicability of the improved VIC method to the DNS for wall turbulent flows. Originality/value – This study enables the VIC method to perform the DNS for wall turbulent flows.

Turbulent bubbly flows have attracted a lot of attention because of their importance for many practical applications such as flows in chemical plants and nuclear power plants. Enhancement of heat transfer by bubble-induced turbulence also attracts a lot of attention from the view point of energy saving. Many studies have been conducted for the motion of bubbles and the characteristics of heat-transfer in turbulent bubbly flows. It is expected that the research on turbulent bubbly flows is accelerated by fully resolved simulations of bubble-turbulence interaction (Tryggvason et al., 2011). The characteristics of bubbly upflow strongly depend on the motions of bubbles and resulting void fraction distribution in the flow. Serizawa et al. (1975) found that the local void fraction is high near the walls and is lower in the core region of upflow in a pipe. Liu (1993) also found in the experiments of turbulent bubbly upflow in a vertical channel that the void fraction has peaks near the walls for the bubbles smaller than 5-6mm, while it has a peak in the core of the channel for the bubbles larger than 5-6mm. Lu & Tryggvason (2008) also showed in their direct numerical simulations of turbulent bubbly upflow in a vertical channel that nearly spherical bubbles tend to concentrate on the near-wall regions, while strongly deformable bubbles tend to be expelled from the near-wall regions. They also showed that the turbulence structures are changed by the motions of bubbles. The detailed mechanism of turbulence modulation due to the bubbles, however, has not been fully clarified yet. Some experimental studies have been conducted for heat-transfer enhancement by the injection of bubbles. Tamari & Nishikawa (1976) showed in their experiments of laminar natural convection heat transfer in water from a vertical plate that the heat transfer is enhanced by the injection of air bubbles. The enhancement of heat transfer by bubble injection was studied further in detail by Tokuhiro & Lykoudis (1994) and Kitagawa et al. (2008, 2010). However, the mechanism of the heat-transfer enhancement has not yet been fully clarified especially in turbulent flows. In the present study, direct numerical simulations have been conducted for turbulent bubbly upflow between two parallel heating walls in order to clarify its heat transfer characteristics. The mechanism of the heat-transfer enhancement is examined by performing simulations with different values of control parameters. The performance of the heat-transfer enhancement is discussed based on the numerical results.

In this paper, we revisit the eddy viscosity formulation to highlight a number of important issues that have direct implications for the prediction of near-wall turbulence. For steady wall-bounded turbulent flows, we make the equilibrium assumption between rates of production ($P$) and dissipation ($\epsilon $) of turbulent kinetic energy ($k$) in the near-wall region to propose that the eddy viscosity should be given by${\nu }_{t} \approx \epsilon / {S}^{2} $, where$S$is the mean shear rate. We then argue that the appropriate velocity scale is given by$\mathop{(S{T}_{L} )}\nolimits ^{- 1/ 2} {k}^{1/ 2} $where${T}_{L} = k/ \epsilon $is the turbulence (decay) time scale. The difference between this velocity scale and the commonly assumed velocity scale of${k}^{1/ 2} $is subtle but the consequences are significant for near-wall effects. We then extend our discussion to show that the fundamental length and time scales that capture the near-wall behaviour in wall-bounded shear flows are the shear mixing length scale${L}_{S} = \mathop{(\epsilon / {S}^{3} )}\nolimits ^{1/ 2} $and the mean shear time scale$1/ S$, respectively. With these appropriate length and time scales (or equivalently velocity and time scales), the eddy viscosity can be rewritten in the familiar form of the$k$–$\epsilon $model as${\nu }_{t} = \mathop{(1/ S{T}_{L} )}\nolimits ^{2} {k}^{2} / \epsilon $. We use the direct numerical simulation (DNS) data of turbulent channel flow of Hoyas & Jiménez (Phys. Fluids, vol. 18, 2006, 011702) and the turbulent boundary layer flow of Jiménezet al. (J. Fluid Mech.vol. 657, 2010, pp. 335–360) to perform‘a priori’tests to check the validity of the revised eddy viscosity formulation. The comparisons with the exact computations from the DNS data are remarkable and highlight how well the equilibrium assumption holds in the near-wall region. These findings could prove to be useful in near-wall modelling of turbulent flows.

This article is concerned with the time evolution of the oblique laminar-turbulent bands of transitional plane Couette flow under the influence of turbulent noise. Our study is focused on the amplitude of modulation of turbulence (the bands). In order to guide the numerical study of the flow, we first perform an analytical and numerical analysis of a Stochastic Ginzburg-Landau (GL) equation for a complex order parameter. The modulus of this order parameter models the amplitude of modulation of turbulence. Firstly, we compute the autocorrelation function of said modulus once the band is established. Secondly, we perform a calculation of average and fluctuations around the exponential growth of the order parameter. This type of analysis is similar to the Stochastic Structural Stability Theory (S3T). We then perform numerical simulations of the Navier-Stokes equations in order to confront these predictions with the actual behaviour of the bands. Computation of the autocorrelation function of the modulation of turbulence shows quantitative agreement with the model: in the established band regime, the amplitude of modulation follows an Ornstein-Uhlenbeck process. In order to test the S3T predictions, we perform quench experiments, sudden decreases of the Reynolds number from uniform turbulence, in which modulation appears. We compute the average evolution of the amplitude of modulation and the fluctuations around it. We find good agreement between numerics and modeling. The average trajectory grows exponentially, at a rate clearly smaller than that of the formation of laminar holes. Meanwhile, the actual time evolution remains in a flaring envelope, centered on the average, and expanding at the same rate. These results provide further validation of the stochastic modeling for the time evolution of the bands for further studies. Besides, they stress on the difference between the oblique band formation and the formation of laminar holes.

The effects of the variable-property on the profiles of the turbulence statistics and the near-wall asymptotic behavior for the compressible turbulent channel flows are investigated using direct numerical simulation (DNS). Two cases of DNS are carried out. The first case is the compressible turbulent flow between isothermal walls, and the second case is one between adiabatic and isothermal walls. In the DNS, the Mach number based on the bulk velocity and sound speed at the isothermal wall is 1.5 and the Reynolds number based on the bulk density, bulk velocity, channel half-width, and viscosity at the isothermal wall is 3000. The main results are as follows : (1) When the velocity fluctuations, kinetic energy, and turbulent dissipation profiles are scaled by the wall variables, these profiles are larger near the isothermal wall and smaller near the adiabatic wall, comparing with those of the incompressible turbulent channel flow. On the other hand, these profiles agree well with the data of the incompressible turbulent channel flow, when these profiles are scaled by the local mean variables. It means Morkovin's hypothesis is applicable to these profiles. (2) The rms temperature fluctuation profile scaled by the mean temperature is almost the same as the rms density fluctuation profile scaled by the mean density, except that the rms temperature fluctuation on the isothermal wall is zero. (3) Differences between Favre and Reynolds averages on the Reynolds stresses are negligible in the wall bounded flow. (4) Morkovin's hypothesis is not applicable to the wall asymptotic behavior. (5) The rms temperature fluctuation profile scaled by the wall temperature varies linearly with the wall unit y+ near the isothermal wall and approaches a constant non-zero value with the decrease of y+ near the adiabatic wall.