Pressure Diffusion Term Research Articles

Momentum transport process in the quasi self-similar region of free shear mixing layer

In this study, we performed a direct numerical simulation (DNS) of a spatially developing shear mixing layer covering both developing and developed regions. The aim of this study is to clarify the driving mechanism and the vortical structure of the partial counter-gradient momentum transport (CGMT) appearing in the quasi self-similar region. In the present DNS, the self-similarity is confirmed in x/L ≥ 0.67 (x/δU0 ≥ 137), where L and δU0 are the vertical length of the computational domain and the initial momentum thickness, respectively. However, the trend of CGMT is observed at around kδU = 0.075 and 0.15, where k is the wavenumber, δU is the normalized momentum thickness at x/L = 0.78 (x/δU0 = 160), and kδU = 0.075 corresponds to the distance between the vortical/stretching regions of the coherent structure. The budget analysis for the Reynolds shear stress reveals that it is caused by the pressure diffusion term at the off-central region and by −p(∂u/∂y)¯ in the pressure-strain correlation term at the central region. As the flow moves toward the downstream direction, the appearance of those terms becomes random and the unique trend of CGMT at the specific wavenumber bands disappears. Furthermore, we investigated the relationship between the CGMT and vorticity distribution in the vortex region of the mixing layer, in association with the spatial development. In the upstream location, the high-vorticity region appears in the boundary between the areas of gradient momentum transport and CGMT, although the high-vorticity region is not actively producing turbulence. The negative production area gradually spreads by flowing toward the downstream direction, and subsequently, the fluid mass with high-vorticity is transported from the forehead stretching region toward the counter-gradient direction. In this location, the velocity fluctuation in the high-vorticity region is large and turbulence is actively produced. In view of this, the trend of negative production appears in the flow where the turbulence production and non-turbulent regions mix. Then, the non-turbulent region and CGMT almost simultaneously disappear in the fully developed region.

Budgets of turbulent kinetic energy, Reynolds stresses, variance of temperature fluctuations and turbulent heat fluxes in a round jet

The self-preserving region of a free round turbulent air jet at high Reynolds number is investigated experimentally (at$x/D=30$,$\mathit{Re}_{D}=1.4\times 10^{5}$and$\mathit{Re}_{{\it\lambda}}=548$). Air is slightly heated ($20\,^{\circ }\text{C}$above ambient) in order to use temperature as a passive scalar. Laser doppler velocimetry and simultaneous laser doppler velocimetry–cold-wire thermometry measurements are used to evaluate turbulent kinetic energy and temperature variance budgets in identical flow conditions. Special attention is paid to the control of initial conditions and the statistical convergence of the data acquired. Measurements of the variance, third-order moments and mixed correlations of velocity and temperature are provided (including$\overline{vw^{2}}$,$\overline{u{\it\theta}^{2}}$,$\overline{v{\it\theta}^{2}}$,$\overline{u^{2}{\it\theta}}$,$\overline{v^{2}{\it\theta}}$and$\overline{uv{\it\theta}}$). The agreement of the present results with the analytical expressions given by the continuity, mean momentum and mean enthalpy equations supports their consistency. The turbulent kinetic energy transport budget is established using Lumley’s model for the pressure diffusion term. Dissipation is inferred as the closing balance. The transport budgets of the$\overline{u_{i}u_{j}}$components are also determined, which enables analysis of the turbulent kinetic energy redistribution mechanisms. The impact of the surrogacy$\overline{vw^{2}}=\overline{v^{3}}$is then analysed in detail. In addition, the present data offer an opportunity to evaluate every single term of the passive scalar transport budget, except for the dissipation, which is also inferred as the closing balance. Hence, estimates of the dissipation rates of turbulent kinetic energy and temperature fluctuations (${\it\epsilon}_{k}$and${\it\epsilon}_{{\it\theta}}$) are proposed here for use in future studies of the passive scalar in a turbulent round jet. Finally, the budgets of turbulent heat fluxes ($\overline{u_{i}{\it\theta}}$) are presented.

Turbulence Kinetic Energy Balance in the Wake of a Sharp-edged Highly Swept Delta Wing

Hot-wire measurements in the wake of a flat plate delta wing, at a fixed angle of attack, were carried-out in order to investigate its turbulence kinetic energy budget. The purpose of this study is to help acquire more information about the vortex structure formed in the wake region of the delta wing and to provide data for turbulence models’ validation. Point measurements were conducted in five planes perpendicular to the flow, on a high spatial resolution two–dimensional measurement grid that allowed precise derivative calculations of the flow quantities needed for the derivation of the kinetic energy budget. The turbulence kinetic energy convection, production, viscous diffusion and turbulence diffusion terms were directly obtained from the experimental data, whereas dissipation and pressure diffusion terms were calculated indirectly using techniques presented and validated in previous studies. All of the terms of the turbulence kinetic energy budget, along with the velocity components and the Reynolds stresses, are presented and discussed.

Energy Redistribution Between the Mean and Pulsating Flow Field in a Separated Flow Region

One of the main features of the backward-facing step (BFS) low frequency pulsatile flow is the unsteadiness due to the convection of vortical (coherent) structures, which characterize the flow dynamics in the shear layer. The physics of the flow field is analyzed by looking at energy redistribution between the mean and pulsating flow field obtained via a particle image velocimeter (PIV) using the concept of a triple decomposition. The total fluctuating kinetic budget is calculated and discussed for a mean Reynolds number of 100 and for 0.035 ≤ St ≤ 2.19. The effects that these coherent structures have on the fluctuating kinetic energy production, dissipation, and transport mechanism are examined. The results provide insight into the physics of the flow and suggest reasons for vortex growth and decay. Fluctuating kinetic energy is generally produced at the separated shear layers and transported towards the core flow and then to the upper and lower walls where viscosity dissipates the energy. The remaining energy is transported streamwise and decays as it is convected downstream (St = 0.4 and 1 cases). It was also found that the pressure-velocity correlation diffusion plays a significant role in the transport of kinetic energy and Reynolds stresses, especially in the separated shear layer. More energy was dissipated at the walls for the high Strouhal number case St = 2.19 due to the transverse pressure diffusion term being increasingly dominant. This could be the reason why the convected primary vortices were much smaller in size and weaker with no upper wall vortices formed at this pulsation Strouhal number. The shear production for St = 0.035 was very minimal; thus, the vortices died down quickly even before the shedding could happen. Finally, the pressure-strain correlation term was found to be significant in redistributing the kinetic energy from u-component to v-component.

Experimental verification of the turbulence models for the pressure diffusion process in a turbulent plane jet

We performed simultaneous measurement of the three velocity components and the pressure in a plane turbulent jet as well as the verification of turbulence models related to the pressure diffusion process, such as the gradient-diffusion model and the model for the rapid and slow terms of the pressure diffusion term. The results show that the gradient-diffusion model developed in the previous studies are valid in the region where the turbulent intensity and the turbulent/non-turbulent intermittency are high and the production of the turbulent energy is dominant to the other processes such as the convection and diffusion of the turbulent energy in the turbulent energy budget. In addition, the pressure diffusion of the turbulent energy cannot be modeled accurately by using only the slow term, and its accuracy increases by considering both rapid and slow terms in the model. These results indicate that the modeling of the pressure diffusion process using only the slow term has a certain risk for causing a misunderstanding of the turbulent energy transport process.

A direct numerical simulation study of higher order statistics in a turbulent round jet

Up until recently direct numerical simulation (DNS) studies involving round turbulent jets have focused on first and second order statistics and vortical behavior near the source of the jet. The third order statistics necessary to compute the turbulent kinetic energy and Reynolds stress transport equations have been examined using LES studies. However, further examination with DNS is important as, on the subgrid scale, LES uses models for Reynolds stress. In this study a DNS of a turbulent free jet with a Reynolds number equal to ReJ = 2000 is computed using a second order accurate, time splitting finite volume scheme. First, second, and third order statistics are compared with previous experimental and numerical studies. All terms of the turbulent kinetic energy balance are calculated directly. The results are compared to experimental studies such as those of Hussein et al. [“Velocity measurements in a high-Reynolds-number, momentum-conserving, axisymmetric, turbulent jet,” J. Fluid Mech. 258, 31–75 (1994)], Panchapakesan and Lumley [“Turbulence measurements in axisymmetric jets of air and helium. Part 1. Air jet,” J. Fluid Mech. 246, 197–233 (1993)], and others. The assumptions made by the various experimental studies in order to solve the dissipation and pressure diffusion terms are discussed and examined using the data from the current study. The Reynolds stress transport equations are also calculated and discussed. Vortical structures are visualized by the λci method and discussed along with entrainment of ambient fluid into the jet. The one dimensional energy spectra in the azimuthal direction are calculated directly and are also discussed.

Simultaneous Measurement of the Velocity and Pressure in a Plane Jet (Estimation of Conditional Averaged Value and Diffusion Term in the Intermittent Region)

The simultaneous measurement of the velocity and pressure in a plane jet has been performed by using a combined probe that consists of an X-type hot-wire probe and a static pressure probe. Firstly, the conditional ensemble averaged statics are investigated on the basis of the intermittency function obtained from the velocity signal by the hot-wire sensor set in the intermittent region of a plane jet. Secondly, the validity of the models for a turbulent and pressure diffusion term is investigated. The measurement results show that the cross-streamwise profile of the diffusion term in the turbulent energy transport equation differs significantly, according to the state of the flow (turbulent or non-turbulent flow) in the intermittent region. By decomposing the diffusion term into two terms, i.e., the turbulent diffusion term and the pressure diffusion term and estimating each term separately, it is found that the cross-streamwise profile of the pressure diffusion term is virtually the same, irrespective of the state of the flow in the intermittent region. This indicates that the difference in the cross-steramwise profile of the diffusion term is cause by the difference in the turbulent diffusion term. In addition, the evaluation of the two kinds of gradient diffusion models of the Reynolds stress transport equation shows that the model constants become constant and the validity of these models can be confirmed except for the region near the jet centerline and intermittent region. Finally, it is shown that both slow-part and rapid-part of the velocity-pressure correlation term must be modeled appropriately to predict the velocity-pressure correlation correctly.

Multi-scale MHD analysis incorporating pressure transport equation for beta-increasing LHD plasma

A multi-scale MHD numerical scheme is developed for analysis of nonlinear evolution of a beta-increasing plasma. The scheme is based on iterative calculations of nonlinear dynamics based on the reduced MHD (RMHD) equations and three-dimensional static equilibrium. The equation for average pressure in the RMHD equations plays the role of a transport equation that involves a heat source term and background pressure diffusion terms. The heat source term is controlled so that the beta value should be increased at a constant rate. The scheme is applied to a Large Helical Device (LHD) plasma up to average beta of 1.05%, which is unstable against linear ideal interchange modes while beta values much higher than the stability limit are obtained in the experiments. The result with the multi-scale scheme indicates that many local flat regions are generated in the background pressure profile in the nonlinear evolution of the interchange modes. This structure of the pressure profile suppresses disruptive phenomena because it reduces the driving force of the modes at higher beta value. Such self-organization in the pressure profile is considered to be the stabilizing mechanism in the plasma.

Transient response of Reynolds stress transport to opposition control in turbulent channel flow

The mechanism for turbulence suppression by opposition control developed by Choi is studied via the analysis of Reynolds stress transport in the initial period of control. It is found that the pressure-strain and pressure-diffusion terms first break up the balance in vertical normal Reynolds stress transport, and result in the final global suppression of turbulence intensities. The transient growth in spanwise normal Reynolds stress is also caused by the pressure-strain term.

Numerical modelling of pore pressure variations due to time varying loads using a hybrid technique: the case of the Itoiz reservoir (Northern Spain)

SUMMARY In this work, we present a hybrid technique to estimate the pore pressure variations at the neighbourhood of dams due to time varying water loads in their reservoirs. When considering flow boundary conditions, the solution of the equations of the problem can be obtained as the superposition of (1) the part computed from a homogeneous diffusion equation with the Dirichlet boundary condition (this is due to the pore pressure diffusion) and (2) the solution of an initial value problem in an inhomogeneous diffusion equation in which the inhomogeneous term is related with the stress variations due to the water loads (the solution due to the compression in the medium). Here, two different techniques are joined to calculate each one of these partial solutions: the pore pressure diffusion term is obtained by using the Green's function of the problem, whereas the second contribution due to stress time changes is computed with a finite difference method. This hybrid technique has been used to compute the pore pressure variations produced by the initial impounding of the Itoiz reservoir, northern Spain. The possible relation between the reservoir and a close seismic series occurred on 2004 September, 8 months after the beginning of its impounding, is investigated. We pay special attention to the pore pressure changes at the hypocentre location of the main shock (with magnitude Mw= 4.5), and also evaluate the change of the Coulomb Failure Stress (ΔCFS) produced by the water loads in the reservoir over the fault responsible of this main shock, obtaining a maximum change of 0.5 kPa in the best of the cases. Accordingly, it seems that the role of the impounding of the reservoir to the main shock was marginal, and that the main load on the origin of the triggered seismicity could well be related to the regional state of stresses of the Pyrenees range and adjacent zones.

Experimental study of an impinging jet with different swirl rates

A stereo PIV technique using advanced pre- and post-processing algorithms is implemented for the experimental study of the local structure of turbulent swirling impinging jets. The main emphasis of the present work is the analysis of the influence of swirl rate on the flow structure. During measurements, the Reynolds number was 8900, the nozzle-to-plate distance was equal to three nozzle diameters and the swirl rate was varied from 0 to 1.0. For the studied flows, spatial distributions of the mean velocity and statistical moments (including triple moments) of turbulent pulsations were measured. The influence of the PIV finite spatial resolution on the measured dissipation rate and velocity moments was analyzed and compared with theoretical predictions. For this purpose, a special series of 2D PIV measurements was carried out with vector spacing up to several Kolmogorov lengthscales. All terms of the axial mean momentum and the turbulent kinetic energy budget equations were obtained for the cross-section located one nozzle diameter from the impinging plate. For the TKE budget, the dissipation term was directly calculated from the instantaneous velocity fields, thereby allowing the pressure diffusion term to be found as a residual one. It was found that the magnitude of pressure diffusion decreased with the growth of the swirl rate. In general, the studied swirling impinging jets had a greater spread rate and a more rapid decay in absolute velocity when compared to the non-swirling jet.

Reynolds-Stress Modeling of Three-Dimensional Secondary Flows With Emphasis on Turbulent Diffusion Closure

Abstract The purpose of this paper is to assess the importance of the explicit dependence of turbulent diffusion on the gradients of mean-velocity modeling in second moment closures on three-dimensional (3D) detached and secondary flows prediction. Following recent theoretical work of Younis, Gatski, and Speziale, 2000, [Proc. Royal Society Lon. A, 456, pp. 909–920], we propose a triple-velocity correlation model, including the effects of the spatial gradients of mean velocity. A model for both the slow and rapid parts of the pressure-diffusion term was also developed and added to a wall-normal-free Reynolds-stress model. The present model is validated against 3D detached and secondary flows. Further developments, especially on the echo terms (which should appear in the formulation of pressure-velocity correlation), are discussed.

Guidelines for Modeling a 2D Rough Wall Channel Flow

The effects of 2-D roughness elements on the Reynolds stress anisotropy tensor, \(b_{ij}\) and of the energy dissipation rate anisotropy tensor \(d_{ij}\) of a turbulent channel flow are investigated using data obtained from direct numerical simulations (DNSs). The roughness elements consist of transverse square rods of size \(k\), placed on one wall of the channel only. While \(k\) is kept constant (\(k/h = 0.1\), \(h\) is the half-width of the channel), the spacing \(w\) between the rods is varied from \(w/k = 1\) to \(w/k = 59\). The results show that the variation in \(w/k\) can dramatically change the structure of the wall region flow. The modeling of the near-wall region needs to reflect the structural changes caused by the variation in \(w/k\). On the basis of the Reynolds stress budgets, attention needs to be given to the turbulent energy and pressure diffusion terms while local isotropy may be a reasonable approximation for the energy dissipation rate, especially over a range of \(w/k\) for which the drag is near its maximum.

Measurement of the turbulent kinetic energy budget of a planar wake flow in pressure gradients

Turbulent kinetic energy (TKE) budget measurements were conducted for a symmetric turbulent planar wake flow subjected to constant zero, favorable, and adverse pressure gradients. The purpose of this study is to clarify the flow physics issues underlying the demonstrated influence of pressure gradient on wake development, and provide experimental support for turbulence modeling. To ensure the reliability of these notoriously difficult measurements, the experimental procedure was carefully designed on the basis of an uncertainty analysis. Three different approaches were applied for the estimate of the dissipation term. An approach for the determination of the pressure diffusion term together with correction of the bias error associated with the dissipation estimate is proposed and validated with the DNS results of Moser et al (J Fluid Mech (1998) 367:255–289). This paper presents the results of the turbulent kinetic energy budget measurement and discusses their implications for the development of strained turbulent wakes.

乱流予混合燃焼DNSに基づくκとε輸送方程式の検討(オーガナイズドセッション6 乱流燃焼研究の最前線)

We have modeled the combustion-related terms in the transport equation of the turbulent kinetic energy for turbulent premixed combustion. In this paper, the modeling of diffusion terms was discussed to close the κ-equation. The turbulent diffusion, pressure diffusion and viscous diffusion terms were modeled together as a gradient diffusion type model. And the diffusion model was compared with diffusion terms estimated from DNS data of turbulent premixed flames, where the model constants were applied general experimental values. Then this result predicts that the experimental model constants are acceptable for κ-equation model of the turbulent premixed combustion.

A142 角柱背後の瞬時速度-圧力勾配相関

The present study considers a possibility of modeling the correlation of velocity and pressure-gradient vectors, in contrast to the conventional approach where pressure-strain correlation and pressure-diffusion terms are treated individually. A hybrid approach combining the velocity measurement by PIV and the numerical procedure to evaluate instantaneous pressure field in the wake of a rectangular cylinder has indicated that the correlation between the velocity vector and pressure-gradient vector become significant particularly when they are perpendicular to each other. The series of evaluations in necessary to understand the wider variation of complex flow problems. Well designed DNS should provide useful data.

Measured Reynolds stress distributions and energy budgets of a fully pulsed round air jet

A detailed experimental examination of the turbulent kinetic energy budgets in a fully pulsed air jet exhausting into still air is presented. Through the pulse hot-wire anemometer measurements were made to obtain estimates of all measurable terms in the energy budget of the intrinsic or phase averaged turbulent kinetic energy. The region of measurements ranged from 50≤x/d≤70 which was previously identified to be the transition from pulse domination to steady jet behaviour. Pulsing was at very low Strouhal numbers to be in the quasi-steady jet regime. Measurements identified significant regions of negative production of turbulent kinetic energy as well as a domination of the energy budget by the time dependent term which is balanced by the advection term during the pulse. The pressure diffusion term is seen to play a significant role, changing from a positive contribution to a negative one as downstream distance increases. Axial diffusion of turbulent energy varies strongly with x/d but radial diffusion is almost self-similar for the region investigated. Results obtained have significant implications for application of turbulence models which have not yielded satisfactory predictions to date. This may be partly due to the finding that the ratio of production of turbulent kinetic energy to dissipation rate is much higher during the pulse dominated part of a cycle than is the case for steady jets thus invalidating a quasi-steady jet assumption which is contained in many model equations.

Revisiting the Reynolds-averaged energy equation in near-wall turbulence models

It is sometimes reported in the heat transfer studies of gaseous media using the near-wall turbulence models that their fully-developed Nu predictions deviate from the experimental correlations. This work approaches the rigorous formulation of the Reynolds-averaged equation in terms of temperature from a fundamental basis, in order to highlight the possible errors due to the usual overapproximation made for the Reynolds-averaged energy equation. It is suggested that Eq. (18) which is one of the rigorous forms of the Reynolds-averaged equation and free of the unknown correlation of the pressure diffusion term is suitable for the analysis of turbulent convective heat transfer in gaseous media.

Improved Low-Reynolds-Number k-e Model Based on Direct Numerical Simulation Data

An improved low-Reynolds-number κ-e turbulence model is proposed. The model is designed not only to conform with the near-wall characteristics obtained with direct numerical simulation data but also to possess the correct asymptotic behavior in the vicinity of the wall. The key features of this model are the adoption of the Taylor microscale in the damping function and the inclusion of the pressure diffusion terms in the κ and e equations. Further applications of the present model to a two-dimensional backward-facing step flow indicate that the present model reproduces correctly the skin-friction and near-wall heat transfer coefficients

Structure Analysis of Spatially Developing Turbulent Mixing Layer with Mixing Transition.

Direct numerical simulation of a three-dimensional spatially developing turbulent mixing layer with mixing transition was conducted to investigate the structures of the mixing layer and the budgets of the transport equations of kinetic turbulent energy and Reynolds stresses. In the transport equation of (2, 2) component of Reynolds stress, the pressure diffusion term has a large value compared with the other terms, whereas it is very small in the budgets of (1, 1) and (3, 3) components. These results suggest that the model for the pressure diffusion term is very important for predicting turbulent flows with high accuracy. The influence of structures in the turbulent mixing layer on the budgets was also discussed. The production of kinetic turbulent energy and Reynolds stresses is governed by large-scale structures and streamwise vortices even after the transition. However, the dissipation of kinetic turbulent energy after the transition is dominated by small-scale vortex tubes in large-scale structures and streamwise vortices in the braid regions. The dominant structure for the transport mechanism and the redistribution mechanism of turbulent energy change from large-scale structures to small-scale vortices with mixing transition.