Pressure Diffusion Term Research Articles

Validation of a Central-Difference Finite Volume Method for Analysing Wall Turbulent Heat Transfer Using Transport Equations for Turbulent Heat Fluxes

Abstract The aim of this study is to validate the finite volume method approach using the transport equations for turbulent heat fluxes in turbulent heat transfer at the wall surface. The finite volume method technique is a numerical analysis based on conservation laws of physical quantities in the flow field, with particular emphasis on the conservation of velocity fluctuation intensity in the flow field. The behaviour of turbulent heat flows is analysed in detail using this method and its validity is tested. In the study, heat transfer in channel turbulence is analysed under a constant heat flux heating condition. The individual terms of the temperature variance equation are analysed and the results are shown to be in agreement with a previous study. The residual is small, confirming the high accuracy of the results. The present study analyses the transport equation for the dissipation of the temperature variance. The transport equations for turbulent heat fluxes in the flow direction and in the vertical wall direction are analysed and show good agreement with those of the previous study. The results show that in the vertical wall direction, non-zero terms such as the pressure diffusion term appear to contribute and reduce the accuracy of the transport equations.

Non-homogeneous anisotropic bulk viscosity for acoustic wave attenuation in weakly compressible methods

A major limitation of the weakly compressible approaches to simulate incompressible flows is the appearance of artificial acoustic waves that introduce mass conservation errors and lead to spurious oscillations in the force coefficients. In this work, we propose a non-homogeneous anisotropic bulk viscosity term to effectively damp the acoustic waves. By implementing this term in a computational framework based on the recently proposed general pressure equation, we demonstrate that the non-homogeneous and anisotropic nature of the term makes it significantly more effective than the isotropic homogeneous version widely used in the literature. Moreover, it is computationally more efficient than the pressure (or mass) diffusion term, which is an alternative mechanism used to suppress acoustic waves. We simulate a range of benchmark problems to comprehensively investigate the performance of the bulk viscosity on the effective suppression of acoustic waves, mass conservation error, order of convergence of the solver, and computational efficiency. The proposed form of the bulk viscosity enables fairly accurate modelling of the initial transients of unsteady simulations, which is highly challenging for weakly compressible approaches, and to the best of our knowledge, existing approaches can't provide an accurate prediction of such transients.

A Novel Connection Element Method for Multiscale Numerical Simulation of Two-Phase Flow in Fractured Reservoirs

Summary This paper presents a novel approach to the numerical simulation of fractured reservoirs, called the connection element method (CEM), which differs from traditional grid-based methods. The reservoir computational domain is discretized into a series of nodes, and a system of connection elements is constructed based on the given connection lengths and angles. The pressure diffusion term is approximated using generalized finite difference theory. Meanwhile, the transmissibility and volume of the connection elements are determined, and pressure equations are solved discretely to obtain pressure at nodes to approximate the upstream flux along connection elements. Then, we solve the transport equation to obtain oil saturation profiles with low numerical diffusion, utilizing the discontinuous Galerkin (DG) method. Moreover, the flow path tracking algorithm is introduced to quantify the flow allocation factors between wells. In all, the pressure equation can be solved at a global coarse-scale point cloud and the saturation equation is calculated at a local fine-scale connection element. In other words, CEM is of multiscale characteristics relatively. Finally, several numerical examples are implemented to demonstrate that CEM can achieve a relatively better balance between computational accuracy and efficiency compared with embedded discrete fracture modeling (EDFM). Furthermore, CEM adopts flexible meshless nodes instead of grids with strong topology, making it more practical to handle complex reservoir geometry such as fractured reservoirs.

Pressure stabilization method for the 2D/3D time‐dependent natural convection equations with salt concentration

AbstractIn this paper, the pressure stabilization method for the 2D/3D time‐dependent natural convection equations with salt concentration (NCSC) is considered. The artificial pressure diffusion term is introduced to reduce the constraint of the incompressible condition. The first‐order backward difference formula is adopted to approximate the temporal derivative terms, the diffusion terms in NCSC are treated implicitly, the convective terms in momentum equation are treated semi‐implicitly, the convective terms in energy equation and salt concentration equation are treated implicitly. The error estimates for the semi‐discrete pressure stabilization method are given. Some numerical experiments are presented to verify the theoretical results and show the reliability of the proposed method.

Modified advection upstream splitting method: Revolutionizing accuracy and convergence speed in low-Mach flows

The vital role of the numerical scheme is becoming increasingly critical as the use of computational fluid dynamics grows. To address the unfavorable effects experienced in low-speed flows when using the AUSM+M scheme (Improved Advection Upstream Splitting Method), the present paper presents an improved approach known as Modified-AUSM+M (M-AUSM+M). This novel method offers enhanced reliability in simulating low-Mach number flows, effectively mitigating the challenges associated with low-speed symptoms encountered in the original AUSM+M scheme. The novel scheme is facilitated by the parameter-free form of the pressure diffusion term in the mass flux and the low-dissipative form of the velocity diffusion term in the pressure flux. The impacts of these critical ingredients are then thoroughly evaluated, and the different characteristics are explored in terms of robustness and accuracy using a wide range of low-Mach test cases. The proposed scheme maintains a consistent correlation between accuracy and convergence speed. In addition, the recently devised technique demonstrates superior accuracy compared to AUSM+M and AUSM+UP schemes when dealing with low-Mach flows. Furthermore, the findings indicate an incredible reduction in iteration numbers, ranging from 30% to 80%, by employing the enhanced scheme in low-Mach domains. In the investigation of high-Mach test cases, the newly developed method preserves the accuracy achieved by AUSM+M in high-Mach flows.

An Assessment of Second Moment Closure Modeling for Stratified Wakes Using Direct Numerical Simulations Ensembles

Abstract Buoyant wakes encountered in the ocean environment are characterized by high Reynolds (Re) and Froude (Fr) numbers, leading to significant space–time resolution requirements for turbulence resolving CFD models (i.e., direct numerical simulations (DNS), large eddy simulations (LES)). Therefore, Reynolds-averaged Navier–Stokes (RANS) based models are attractive for these configurations. The inherently complex dynamics of stratified systems render eddy-viscosity-based modeling inappropriate. RANS second-moment closure (SMC) based modeling is more suitable because it accounts for flow anisotropy by solving the transport equations of important second-moment terms. Accordingly, eleven transport equations are solved at the SMC level, and a range of submodels are implemented for diffusion, pressure strain and scrambling, and dissipation terms. This work studies nonstratified and stratified towed wakes using SMC and DNS. Submodels in the SMC are evaluated in terms of how well their exact Reynolds averaged form impacts the accuracy of the full RANS closure. An ensemble average of 40 and 80–100 DNS realizations are required and conducted for these temporally evolving nonstratified and stratified wakes, respectively, to obtain converged higher-order statistics. SMC over-predicts wake height by over a factor of 2, and under-predicts defect velocity, wake width, and turbulent kinetic and potential energies by factors ranging from 1.3 to 3.5. Also, SMC predicts a near isotropic decay of normal Reynolds stresses (a33→−0.25), in contrast to the anisotropic decay (a33→−0.64) returned by DNS. The DNS data also provide important insights related to the inaccuracy of the dissipation rate isotropy assumption and the non-negligible contribution of pressure diffusion terms. These results lead to several important recommendations for SMC modeling improvement.

A connection element method: Both a new computational method and a physical data-driven framework—-Take subsurface two-phase flow as an example

In traditional reservoir simulation methods, the computational domain is often discretized into a grid system (FDM, FVM, FEM, etc.) or a mesh reduction system (point source solution, BEM, meshless methods, etc.). This paper develops a connection-based method, which uses connection elements to discretize the computational domain from the flow perspective (named as connection element method, CEM). Based on a generalized finite difference approximation of the pressure diffusion term, each connection element can be characterized by two parameters, namely connection transmissibility and connection volume. The connection transmissibility measures the diffusion capacity of the connection element, and the connection volume measures the volume of the connection element as a geometric entity. Then, the flow equation can be solved directly on the connection elements, by either a fully implicit scheme or a sequential coupled scheme. The fully implicit scheme solves pressures and saturations simultaneously with high accuracy, but the computational cost is very high when the number of connections is large. The sequential coupled scheme calculates the pressure at the nodes of connection elements at the coarse-scale, then solves the fine-scale saturation along the connection elements. In this way, high computational efficiency can be achieved with accuracy. Furthermore, CEM can efficiently and intuitively obtain the possible flow paths between connection elements and reveal the interaction between the nodes by the path-tracking algorithm in graph theory. In addition, CEM with a sequential coupled scheme can be regarded as a physical network framework with small degrees of freedom, as well as a good data-driven framework. In the end, two numerical examples and one field case are presented to demonstrate the superiority of the developed CEM.

Climatic Change of Summer Wind Direction and Its Impact on Hydrodynamic Circulation in the Pearl River Estuary

Assessing the trend of sea surface wind is important for understanding the response of the marine environment to climate change. Analysis of wind data reveals that the summer wind direction in the Pearl River Estuary (PRE) shifts anticlockwise at a rate of −0.36°yr−1 over the past 42 years (1979–2020). The mean wind direction in July shifts from 183.6° (in 1979) to 169.3° (in 2020) and is predicted as 142.1° by 2100. How this long-term wind direction change affects the PRE hydrodynamic circulation structure has not been examined yet. A fully calibrated high resolution 3D hydrodynamic model is used to evaluate the response of local hydrodynamics to wind direction shifting in this study. The model results indicate that both the cross-channel wind-driven transport and along-channel seaward flow are weakened as wind direction shifts. Consequently, the lateral circulation is slowed down significantly while the longitudinal exchange flow is weakened slightly. A remarkable increase in stratification occurs in the coastal sea adjacent to the Modaomen where hypoxia has been frequently reported. The residence time of Lingding Bay increases slightly. The Momentum budget indicates the wind direction shifting can cause major changes in the barotropic pressure term, which is mainly balanced by the baroclinic pressure term and diffusion term.

Dynamics of a single bubble rising in a quiescent medium

In the present work, an experimental analysis was performed to characterise the flow field around a single bubble of different diameters ∼ 2.77–3.53 mm) rising in a quiescent medium aiming to determine the effect of bubble size on kinetic energy distribution. The velocity field was measured using a non-intrusive particle image velocimetry (PIV) technique and kinetic energy spectrum was determined in both transverse and longitudinal directions applying a Fast Fourier Transformation (FFT). Both small- and large-scale motions of the flow field were identified and separated using a discreate wavelet transformation (DWT) method. It was found that the energy spectrum of the large-scale motions depended on the bubble size while the small-scale energy spectrum was nearly independent of it. The slopes of the energy spectrum were found to be close to -5/3 and -3 for the large- and small-scale regimes, respectively and the transition of slope was observed to occur at the wavenumber corresponding to the bubble diameter. Using the measured velocity field data, a turbulence kinetic energy (TKE) budget analysis was performed involving five components namely kinetic energy production, turbulent transport, pressure diffusion, viscous diffusion, and energy dissipation. Overall, it was observed that in the vicinity of bubble surface, turbulence production term was not entirely balanced by the dissipation term; and turbulent transport and pressure diffusion term also had significant contributions.

Large-eddy-simulation-based analysis of Reynolds-stress budgets for a round impinging jet

A large-eddy simulation is used to assess the Reynolds-stress budgets of a round impinging jet in the context of a second-moment closure of turbulence. The present work focuses on the stagnation region where no data are available in the literature except in the wall vicinity. Inside the stagnation region, it is shown that the pressure terms are dominant in the budgets. They balance the Reynolds-stress production and the convective fluxes. A visualization of this equilibrium through a specific indicator reveals that impingement effects extend to less than one nozzle diameter in the wall-normal direction and to about one diameter radially. This study underlines the role of the pressure diffusion term that conveys energy to the wall, balancing the high production rates (both positive or negative). Finally, the failure of turbulence models is explained by the absence of appropriate modeling of this pressure diffusion term leading to excessive Reynolds-stress values inside the impingement region.

A low-diffusion robust flux splitting scheme towards wide-ranging Mach number flows

This paper develops a low-diffusion robust flux splitting scheme termed TVAP to achieve the simulation of wide-ranging Mach number flows. Based on Toro-Vázquez splitting approach, the new scheme splits inviscid flux into convective system and pressure system. This method introduces Mach number splitting function and numerical sound speed to evaluate advection system. Meanwhile, pressure diffusion term, pressure momentum flux, interface pressure and interface velocity are constructed to measure pressure system. Then, typical test problems are utilized to systematically assess the robustness and accuracy of the resulting scheme. Matrix stability analysis and a series of numerical cases, such as double shear-layer problem and hypersonic viscous flow over blunt cone, demonstrate that TVAP scheme achieves excellent low diffusion, shock stability, contact discontinuity and low-speed resolution, and is potentially a good candidate for wide-ranging Mach number flows.

Quantification of turbulent structures in and around the boundary region of a turbulent round jet released into counter-flow

This experimental study reports the turbulent flow structure of the interaction of turbulent round jet issued into a counter turbulent flow for improved understanding of the turbulent mixing characteristics including pollutant dispersal at the estuarine region for tidal river flows. A high-frequency data acquisition system (Acoustic Doppler velocimeter) was employed to quantify the turbulent flow structures within the mixing region of the turbulent jet. It was found that the cross-sectional mean axial jet velocity decayed faster in counter turbulence compared to a quiescent background. A reduced occurrence of engulfment and nibbling processes were evident which signified that the entrainment rate of the moving fluid reduced owing to the jet released into the counter flow. Further, the intermittency and diffusion depth of eddies decreased for jets issued against the current. The kinetic energy budget revealed that the pressure diffusion term of the turbulent kinetic energy budget equation supplied energy to the jet from the mean background flow at the self-similar region of the jet.

Analysis of Compressibility Corrections for Turbulence Models in Hypersonic Boundary-Layer Applications

To accurately predict the hypersonic turbulent boundary layer, the compressibility corrections for a two-equation turbulence model are investigated substantially. The two-equation shear-stress-transport model based on the assumption of an eddy-viscous coefficient is adopted. The closure approximations for pressure work, pressure dilatation, and dilatation dissipation are examined by a priori and a posteriori analyses. The direct numerical simulation (DNS) of a hypersonic boundary layer shows that the pressure work term has little influence on the turbulent energy transport, whereas the effect of pressure expansion and pressure diffusion terms can be counteracted. The individual closure assumptions are evaluated by using the DNS data. Furthermore, Reynolds-averaged Navier–Stokes simulations with compressibility corrections are conducted for the hypersonic boundary layers of a flat plate and a cone. Comparisons with the experimental data and engineering correlations show that the original turbulence model overestimates the heat flux, and the pressure dilation and dilation dissipation corrections lead to a decrease in heating transfer rates. The combination of a priori and a posteriori analyses comes to the conclusion that the Zeman correction of the dilation dissipation term is applicable in hypersonic boundary layers.

An improved AUSM-family scheme with robustness and accuracy for all Mach number flows

With the increasing popularity of Computational Fluid Dynamics (CFD), the reliability of numerical scheme becomes prominent. The work presents a newly improved scheme more reliable in all Mach number regimes to circumvent some typical symptoms of the previous AUSM-family schemes observed in hypersonic and very low speeds. This scheme is facilitated by reconstructing pressure diffusion term in mass flux, velocity diffusion term in pressure flux and numerical sound speed. Then, a variety of benchmark test cases are selected to systematically assess the effects of these key ingredients and investigate the additional features in terms of robustness and accuracy. The proposed scheme attains stronger shock robustness against carbuncle instability, better low-speed accuracy and higher resolution of oblique shocks compared with many existing upwind schemes. Moreover, it can exactly resolve contact discontinuity, preserve positivity, damp numerical overshoots and avert the global cut-off strategy. Numerical results for a wide spectrum of Mach numbers indicate its potential and reliable application to all Mach number flows.

Reynolds-Stress Budgets in an Impinging Shock-Wave/Boundary-Layer Interaction

Wall-resolved large-eddy simulations of an impinging shock-wave/boundary-layer interaction (SBLI) are performed to investigate the evolution of Reynolds stresses. After an assessment of accuracy by comparison with experimental data, mechanisms in the Reynolds-stress transport equation, including production, diffusion, strain, and dissipation are examined. In the upstream boundary layer, these quantities are validated with high confidence against published results from the literature. Concerning the SBLI region, all Reynolds stresses exhibit significant growth, consistent with the enhancement of turbulence in adverse pressure gradients. Of particular note is the pressure strain term, which plays a key role in energy redistribution from to and . The pressure diffusion term, commonly assumed to be negligible or combined with turbulent diffusion in Reynolds-stress models, becomes larger than the production and pressure strain terms in the region where the impinging shock crosses the shear layer above the separation bubble. Power spectra of in this region show links to shear layer flapping and Kelvin–Helmholtz-type instability. In the vicinity of the growth in pressure diffusion, turbulent mass flux also exhibits amplification. Critically, in the budget, pressure diffusion and turbulent mass flux behavior for suggests that there may be a benefit in considering the terms individually in modeling efforts.

Local equilibrium in the inertial layer of wall bounded turbulence

ABSTRACTRecently Brouwers [Dissipation equals production in the log layer of wall-induced turbulence. Phys Fluids. 2007;19:101702] carried out an asymptotic analysis using the RANS based turbulence energy transport equation and showed that the energy dissipation equals its production in the inertial layer of wall-induced turbulence. Assuming log-law profile to the mean velocity, pressure, viscous and energy diffusion terms were estimated and shown to be negligibly small compared to the production and dissipation terms thereby proving local equilibrium. However, based on scale relations Tennekes and Lumley [A first course in turbulence. Cambridge (MA): MIT Press; 1994] have already established that the pressure and energy diffusion terms appearing in the energy transport equation are of the same order of magnitude especially in the inertial layer thus leading to a contradiction. Hence we have attempted here to re-estimate the turbulence energy budgets in a different way by invoking the Kolomogrov’s similarity hypotheses and (4/5)th law. Magnitudes of pressure and energy diffusion terms are determined explicitly and found to match well with the DNS data. The striking point of the present analysis is that no prior assumption is enforced on the mean velocity profile. Further, two main advantages of the present study are, (i) reasonable estimates for both the diffusion terms are obtained explicitly that were unavailable before and (ii) these estimates help us to tweak the production/dissipation terms to reflect the influence of turbulent diffusion mechanisms without the necessity to model them as in the case of elliptic relaxation and Reynold stress RANS models.

Turbulence energetics in an axisymmetric impinging jet flow

Turbulence energetics is investigated in the impingement and wall jet regions of an axisymmetric impinging jet flow, by using the two-dimensional particle-image velocimetry measurement technique. The inlet-based Reynolds number is equal to 5200, and the height to the diameter ratio is equal to 5.95. This study shows that turbulence kinetic energy balances in outer, middle, and inner layers of the impingement region differ greatly from each other. The balances in near-field and far-field wall jet regions differ, as well. This study also provides various useful insights into the directional characteristics of the energy transports. For example, it has revealed that the turbulence diffusion process transports energy in the upstream direction in the jet core, but obliquely in the mixing layers. The balance terms are decomposed into physically meaningful spatial components, which, among various other insights, show that the negative production rate that is earlier reported in the stagnation region is caused by the negative work contributions of the radial and the tangential Reynolds stresses. This study, overall, lets us have a thorough understanding of the turbulence production, the convection, the turbulence diffusion, and the viscous diffusion processes, while the turbulence dissipation and the pressure diffusion terms are clubbed together and obtained as the residual of the energy balance equation. Information that is extracted from the residual has demonstrated the failure of Lumley’s model to estimate the pressure diffusion rate near the impingement surface.

Calculation of cavitation evolution and associated turbulent kinetic energy transport around a NACA66 hydrofoil

The physical mechanism of flow unsteadiness is one of the key problems in cavitating flow. Significant efforts have been exerted to explain the cavitation-vortex interaction mechanism. As well, the process of kinetic energy transport during the evolution of unsteady cavitating flow must be elucidated. In this work, 2D calculations of cavitating flow around a NACA66 hydrofoil were performed based on the open source software OpenFOAM. The modified shear stress transport k-ω turbulence model, which considers curvature and turbulent eddy viscosity corrections, was employed to close the governing equations. The Schnerr-Sauer cavitation model was adopted to capture the cavitation phase change process. Numerical results showed reasonable consistency with the results of the experiments conducted by Leroux et al. (2004). The results showed that cavitation promotes turbulence intensity and flow unsteadiness around the hydrofoil. During the attached sheet cavity growth stage, high-value regions of turbulent kinetic energy are located substantially at the interface of the cavity, particularly at the rear portion of the cavity region. During the cloud cavity shed-off stage, the cavity begins to break off and the maximum value of turbulent kinetic energy is observed inside the shed cavity. Finally, the influence of cavitation on the turbulence intensity is illustrated using the turbulent kinetic energy transport equation, which shows that the pressure diffusion and turbulent transport terms dominate as cavitation occurs. In addition, cavitation promotes turbulence production and increases dissipation with fluid viscosity and flow unsteadiness. The viscous transport term only acts in the cavitation shedding stage under large-scale vortex shedding. Overall, these findings are of considerable interest in engineering applications.

Energy transport due to pressure diffusion enhanced by helicity and system rotation in inhomogeneous turbulence

It is known that turbulent energy is rapidly transferred in the direction of the rotation axis in a rotating system, in comparison with the non-rotating case. In this study, this phenomenon is investigated as a problem of energy diffusion expressed by the Reynolds averaged Navier-Stokes (RANS) model. The conventional gradient-diffusion approximation for the turbulent energy flux cannot account for the enhanced energy transport observed in rotating inhomogeneous turbulence. In order to adequately describe the phenomenon, we propose a new model for the energy flux due to the pressure associated with the rotational motion of a fluid. The model of the energy flux is expressed to be proportional to the turbulent helicity. This property is closely related to the group velocity of inertial waves in a rapidly rotating fluid. The validity of the model is assessed using a direct numerical simulation (DNS) of inhomogeneous turbulence under rotation. It is shown that most of the turbulent energy transport enhanced by the system rotation is attributed to the pressure diffusion term. The spatial distribution of the energy flux due to the pressure related to the system rotation is similar to that of the turbulent helicity with negative coefficient. Hence, the new model which is proportional to the turbulent helicity is able to qualitatively account for the enhanced energy flux due to the system rotation. Finally, the helical Rossby number is proposed in order to estimate the relative importance of the energy flux enhanced by the turbulent helicity and the rotation, in comparison to the conventional gradient-diffusion approximation.

Reappraisal of the velocity derivative flatness factor in various turbulent flows

We first analytically show, starting with the Navier–Stokes equations, that the value of the derivative flatness is controlled by pressure diffusion of energy, viscous destructive effects and large-scale effects (decay and/or production). The latter two terms tend to zero when the Taylor-microscale Reynolds number$Re_{\unicode[STIX]{x1D706}}$is sufficiently large. We argue that the pressure-diffusion term should also tend to a constant at large$Re_{\unicode[STIX]{x1D706}}$. Available data for the velocity derivative flatness,$F$, in different turbulent flows are re-examined and interpreted in the light of the finite-Reynolds-number effect. It is found that$F$can differ from flow to flow at moderate$Re_{\unicode[STIX]{x1D706}}$; for a given flow,$F$may also depend on the initial conditions. The data for$F$in various flows, e.g. along the axis in the far field of plane and circular jets, and grid turbulence, show that it approaches a constant, with a value slightly larger than 10, when$Re_{\unicode[STIX]{x1D706}}$is sufficiently large. This behaviour for$F$is supported, at least qualitatively, by our analytical considerations. The constancy of$F$at large$Re_{\unicode[STIX]{x1D706}}$violates the refined similarity hypothesis introduced by Kolmogorov (J. Fluid Mech., vol. 13, 1962, pp. 82–85) to account for the intermittency of the energy dissipation rate. It is not, however, inconsistent with Kolmogorov’s original similarity hypothesis (Dokl. Akad. Nauk SSSR, vol. 30, 1941, pp. 299–303), although we contend that the power-law relation$F\sim Re_{\unicode[STIX]{x1D706}}^{\unicode[STIX]{x1D6FC}_{4}}$(Kolmogorov 1962), which is widely accepted in the literature, has in reality been almost invariably used to ‘model’ the finite-Reynolds-number effect for the laboratory data and has been strongly influenced by the weighting given to the atmospheric surface layer data. The inclusion of the latter data has misled previous investigations of how$F$varies with $Re_{\unicode[STIX]{x1D706}}$.