Velocity-pressure Gradient Correlation Research Articles

Entropy: An Inspiring Tool for Characterizing Turbulence-Combustion Interaction in Swirling Flames via Direct Numerical Simulations of Non-Premixed and Premixed Flames.

This article focuses on entropy generation in the combustion field, which serves as a useful indicator to quantify the interaction between turbulence and combustion. The study is performed on the direct numerical simulations (DNS) of high pressure non-premixed and premixed swirling flames. By analyzing the entropy generation in thermal transport, mass transport, and chemical reactions, it is found that the thermal transport, driven by the temperature gradient, plays a dominant role. The enstrophy transport analysis reveals that the responses of individual terms to combustion can be measured by the entropy: the vortex stretching and the dissipation terms increase monotonically with the increasing entropy. In high entropy regions, the turbulence behaves as the "cigar shaped" state in the non-premixed flame, while as the axisymmetric state in the premixed flame. A substantial increase in the normal Reynolds stress with the entropy is observed. This is due to the competition between two terms promoted by the entropy, i.e., the velocity-pressure gradient correlation term and the shear production term. As a result, the velocity-pressure gradient correlation tends to isotropize turbulence by transferring energy increasingly from the largest streamwise component to the other smaller normal components of Reynolds stress and is dominated by the fluctuating pressure gradient that increases along the entropy. The shear production term increases with the entropy due to the upgrading alignment of the eigenvectors of strain rate and Reynolds stress tensors.

Direct numerical simulation of turbulence modulation by premixed flames in a model annular swirling combustor

Direct numerical simulations (DNS) of two adjacent turbulent swirling kerosene/air premixed flames in a model annular combustor under aircraft engine-like conditions are carried out to investigate turbulence modulation by flames. One inert case and two reactive cases with different Reynolds numbers and Karlovitz numbers are studied. Most characteristic non-dimensional parameters of the reactive cases are designed to be close to those of the realistic aircraft engine-like conditions. The DNS results suggest that combustion increases the mean velocity of reactants, while the turbulence in the main swirling region and the shear layers is weakened. The recirculation in the center region is enhanced due to the stronger shearing. Analysis of the TKE balance equation demonstrates that the terms of convection, turbulent transport, and velocity-pressure gradient correlation play important roles in the current swirling flames with relatively high Karlovitz numbers, which is different from that of the planar jet flames. Compared to inert flows, the convection and turbulent transport terms in reactive flows are suppressed in the inner shear layer and enhanced in the main swirling region, while the velocity-pressure gradient correlation term changes little due to the weak pressure dilatation in the present configuration. The vorticity vector is preferentially aligning with intermediate principal strain rate in swirling flows, and this alignment is enhanced by flames in the low turbulent intensity case. When this preferential alignment works, dilatation term in reactive flows remarkably promotes the production of enstrophy.

Heat release effects on the Reynolds stress budgets in turbulent premixed jet flames at low and high Karlovitz numbers

In turbulent premixed flames at low Karlovitz number, combustion heat release can have a significant impact on turbulence. Thermal expansion in flame induces dilatation, and the corresponding pressure–dilatation correlation acts as a primary source of turbulent kinetic energy (TKE). As a consequence, the flame-normal component of the normal Reynolds stresses significantly increases. Additionally, for sheared flames, typical of jet flames, the shear component of the Reynolds stresses exhibits counter-Boussinesq behavior. For flames at low Karlovitz number, where these effects dominate, no models have successfully predicted all Reynolds stress components. To develop more complete turbulence models, heat release effects on the evolution of all Reynolds stress components need to be analyzed. In this work, Reynolds stress budgets are evaluated from Direct Numerical Simulation (DNS) databases of spatially-evolving turbulent premixed planar jet flames at low and high Karlovitz numbers. In the Reynolds stress budgets, the velocity–pressure gradient correlation term is important at both Karlovitz numbers but serves fundamentally different roles in each case. In the budgets for the normal components, the velocity–pressure gradient correlation term is decomposed into a redistribution term and an isotropic term, where the redistribution term acts to redistribute energy between the Reynolds stress components and the isotropic term is the pressure–dilatation source term of turbulent kinetic energy. At high Karlovitz number, the isotropic term is negligible, and the redistribution term acts to isotropize the turbulence as in non-reacting flows. Conversely, at low Karlovitz number, the isotropic term acts as a large source, and the redistribution term preferentially injects energy into the flame-normal component at the expense of other components which acts to make the turbulence less isotropic. In the budget for the shear component, the shear production term dominates the velocity–pressure gradient correlation term at high Karlovitz number, but the opposite is observed at low Karlovitz number. The dominance of the velocity–pressure gradient correlation term at low Karlovitz number is primarily induced by the flame-generated mean pressure gradient and ultimately leads to the counter-Boussinesq behavior of the shear component. The overall analysis indicates that any turbulence model that relies on small-scale isotropy and/or rapid isotropization will fail to capture the heat release effects on turbulence at low Karlovitz number.

Higher-order and length-scale statistics from DNS of a decelerated planar wall-bounded turbulent flow

A DNS database is presented to document third- and fourth-order moments and their budgets for fully developed plane-channel flow and for strained plane-channel flow. The effect of straining has a similar effect on statistics, such as the skewness and flatness of velocities, as in an adverse-pressure-gradient (APG) boundary layer. In addition to higher-order statistics, some modeling implications are also described, including issues related to decomposition of the velocity–pressure gradient correlations, the assumption of dissipation isotropy and a fourth-order turbulence time scale. An analysis of two-point correlations along the inhomogeneous direction is made to include the effects of APG straining. This reveals the advantage of a one-point length-scale based on wall-normal velocity fluctuations as an alternative to the traditionally used length-scale based on turbulence kinetic energy. The present study should prove to be useful for turbulence modelers in need of data to develop and assess higher-than-second-order Reynolds-averaged Navier–Stokes closure models.

Stagnation-point flow under free-stream turbulence

In this paper, the effects of free-stream turbulence on stagnation-point flow and heat transfer are investigated through large eddy simulation (LES) of homogeneous isotropic turbulence impinging upon an isothermal elliptical leading edge. Turbulent mean flow and Reynolds stress profiles along the stagnation streamline, where the mean flow is strain dominant, and at different downstream locations, where the mean flow gradually becomes shear-dominated, are used to characterize evolution of the free-stream turbulence. The Reynolds stress budgets are also obtained, and the turbulence anisotropy is analysed through the balance between the mean flow strain and the velocity pressure gradient correlation. In the presence of free-stream turbulence, intense quasi-streamwise vortices develop near the leading edge with a typical diameter of the order of the local boundary-layer thickness. These strong vortices cause the thermal fluxes to peak at a location much closer to the wall than that of the Reynolds stresses, resulting a greater sensitivity to free-stream turbulence for the heat transfer than the momentum transfer. The heat transfer enhancement obtained by the present LES agrees quantitatively with available experimental measurements. The present LES results are also used to examine the eddy viscosity and pressure-strain correlations in Reynolds stress turbulence models.

Measurements of velocity–acceleration statistics in turbulent boundary layers

An advanced laser Doppler velocimetry system is developed to acquire measurements of fluctuating velocity–acceleration statistics in turbulent boundary layers. These correlations are enabled by customized burst signal processing that estimates both the Doppler frequency and the rate-of-change of Doppler frequency, which are related to the particle velocity and acceleration by the interference fringe spacing. The measurements give important insight into the near-wall turbulence structure since the statistical correlations of interest, u i a j ¯ appear directly in the Reynolds stress transport equations as a sum of the velocity–pressure gradient correlation, - 1 ρ u i ∂ p ∂ x j + u j ∂ p ∂ x i ¯ , the dissipation rate, 2 ν ∂ u i ∂ x k ∂ u j ∂ x k ¯ , and the viscous diffusion, ν ∇ 2 u i u j ¯ . The immediate power of such measurements is that combinations of terms in the Reynolds stress transport equation may be characterized by a single statistical measurement at one location in the flow—no gradients need be computed. In the present paper, data are presented for a constant-pressure 2D turbulent boundary layer at Re θ = 6800. Near-wall results for the dominant term in the velocity–acceleration tensor, the streamwise correlation ua x ¯ , compare favorably with DNS for the same quantity at Re θ = 1410 and Re τ = 640; furthermore, the quantity exhibits no Reynolds number effects within experimental uncertainties. The balance of the velocity–acceleration equation in the streamwise direction is presented, giving the first measurements for the profile of the velocity–pressure gradient correlation with this technique. This study exhibits the potential of the technique to be applied to more complex flows, particularly those 3D separating flows in which the motions contributing to the velocity–acceleration correlations become dominant.

A rescaled elliptic relaxation approach: Neutralizing the effect on the log layer

An alternative scaling for the relaxation function describing the velocity pressure–gradient correlation used in the elliptic relaxation procedure for both eddy-viscosity and Reynolds stress models is presented. While other alternatives have been proposed to neutralize the adverse effect on log-layer dynamics, they have relied on altering the original differential formulation. A simpler alternative is presented here that involves a rescaling of the relaxation function with the isotropic dissipation rate as well as the turbulent kinetic energy. Various comparative tests are made and the new rescaled formulation is shown to provide improved and accurate predictions for both the eddy-viscosity and Reynolds stress models.

Budgets of Reynolds stress, kinetic energy and streamwise enstrophy in viscoelastic turbulent channel flow

The budgets of the Reynolds stress, turbulent kinetic energy and streamwise enstrophy are evaluated through direct numerical simulations for the turbulent channel flow of a viscoelastic polymer solution modeled with the Finitely Extensible Nonlinear Elastic with the Peterlin approximation (FENE-P) constitutive equation. The influence of viscoelasticity on the budgets is examined through a comparison of the Newtonian and the viscoelastic budgets obtained for the same constant pressure drop across the channel. It is observed that as the extensional viscosity of the polymer solution increases there is a consistent decrease in the production of Reynolds stress in all components, as well as in the other terms in the budgets. In particular, the effect of the flow elasticity, which is associated with the reduction in the intensity of the velocity-pressure gradient correlations, potentially leads to a redistribution of the turbulent kinetic energy among the streamwise, the wall-normal and the spanwise directions. In this work, we also show that in the presence of viscoelasticity there is a significant reduction in all components of the production of streamwise enstrophy. This is consistent with a proposed mechanism for polymer-induced drag reduction through the inhibition of vortex stretching by the high extensional viscosity of the polymer solution.

A numerical study of strained three-dimensional wall-bounded turbulence

Channel flow, initially fully developed and two-dimensional, is subjected to mean strains that emulate the effect of rapid changes of streamwise and spanwise pressure gradients in three-dimensional boundary layers, ducts, or diffusers. As in previous studies of homogeneous turbulence, this is done by deforming the domain of a direct numerical simulation (DNS); here however the domain is periodic in only two directions and contains parallel walls. The velocity difference between the inner and outer layers is controlled by accelerating the channel walls in their own plane, as in earlier studies of three-dimensional channel flows. By simultaneously moving the walls and straining the domain we duplicate both the inner and outer regions of the spatially developing case. The results are used to address basic physics and modelling issues. Flows subject to impulsive mean three-dimensionality with and without the mean deceleration of an adverse pressure gradient (APG) are considered: strains imitating swept-wing and pure skewing (sideways turning) three-dimensional boundary layers are imposed. The APG influences the structure of the turbulence, measured for example by the ratio of shear stress to kinetic energy, much more than does the pure skewing. For both deformations, the evolution of the Reynolds stress is profoundly affected by changes to the velocity–pressure-gradient correlation Πij. This term – which represents the finite time required for the mean strain to modify the shape and orientation of the turbulent motions – is primarily responsible for the difference (lag) in direction between the mean shear and the turbulent shear stresses, a well-known feature of perturbed three-dimensional boundary layers. Files containing the DNS database and model-testing software are available from the authors for distribution, as tools for future closure-model testing.

A geometry independent near-wall Reynolds-stress closure

Models proposed for high-Reynolds-number Reynolds–stress closures are, in general, geometry independent. Therefore, they are most suitable for flows with complex geometries. With the advent of supercomputers, the near-wall flow can be calculated with greater efficiency to give more accurate results on surface properties. However, with modifications made to the various models to account for near-wall behavior, the closures are no longer geometry independent. The cause is the presence of wall distance and wall unit normals in the proposed corrections for either the velocity–pressure-gradient correlation tensor, the dissipation rate tensor, the dissipation rate equation, or all of them. In this paper, a proposal is put forward where the modifications made to these tensors and equation are free of wall distance and wall unit normals. The closure is validated against a wide variety of simple flows and good agreement with data is obtained. Further, the closure is applied to calculate backstep flows and developing square duct flows. The results are compared with direct numerical simulation and experimental data. In addition, they are also compared with the calculations of another near-wall Reynolds–stress closure with wall unit normal dependence and is known to give good correlations with backstep flow and square duct flow data. The present closure is found to yield results that are in good agreement with the data and the calculations of the model with wall unit normals. As a result, a viable geometry independent near-wall Reynolds–stress closure is available.

Structure of Reynolds shear stress in the central region of plane Couette flow

The structures and mechanisms for maintaining a high Reynolds shear stress throughout the core region of plane turbulent Couette flow were examined by means of databases originating from a direct numerical simulation. At the relatively low Reynolds number considered, the mean shear rate and some one-point statistics exhibited a non-negligible variation in the core region, which conflicted with the postulated homogeneity. The slope of the mean velocity profile was below a theoretically established lower bound for the limit of infinite Re. Analysis of the time-averaged and structural information obtained by conditional sampling showed that mean shear generation and velocity-pressure gradient correlations played a crucial role in the generation and annihilation of the Reynolds shear stress. It was moreover observed that strong, very localized velocity fluctuations in the wall-normal direction were essential in both processes. A simple conceptual model was proposed to explain the physical significance of the pressure field associated with quadrant 2 events.

Direct numerical simulation of turbulent flow in a square duct

A direct numerical simulation of a fully developed, low-Reynolds-number turbulent flow in a square duct is presented. The numerical scheme employs a time-splitting method to integrate the three-dimensional, incompressible Navier-Stokes equations using spectral/high-order finite-difference discretization on a staggered mesh; the nonlinear terms are represented by fifth-order upwind-biased finite differences. The unsteady flow field was simulated at a Reynolds number of 600 based on the mean friction velocity and the duct width, using 96 × 101 × 101 grid points. Turbulence statistics from the fully developed turbulent field are compared with existing experimental and numerical square duct data, providing good qualitative agreement. Results from the present study furnish the details of the corner effects and near-wall effects in this complex turbulent flow field; also included is a detailed description of the terms in the Reynolds-averaged streamwise momentum and vorticity equations. Mechanisms responsible for the generation of the stress-driven secondary flow are studied by quadrant analysis and by analysing the instantaneous turbulence structures. It is demonstrated that the mean secondary flow pattern, the distorted isotachs and the anisotropic Reynolds stress distribution can be explained by the preferred location of an ejection structure near the corner and the interaction between bursts from the two intersecting walls. Corner effects are also manifested in the behaviour of the pressure-strain and velocity-pressure gradient correlations.

Further Velocity Measurements in a Turbulent Diffusion Flame with Moderate Swirl

This paper is a sequel to previously published work (Stårner and Bilger 1988) on a swirling turbulent hydrogen diffusion flame in a co-flowing stream. Improved nozzle design and altered boundary conditions have resulted in much stronger influence of the swirl, whilst maintaining the simple chemistry and the parabolic character of the flow. A new set of measurements is presented, including all components of the stress tensor. Results show a shortened, broaded flame with inverted excess velocity profile, marked turbulence suppression and a reduction in the u-v stress correlation coefficient in downstream profiles. Estimates of source terms in the transport equations for turbulence kinetic energy and u-v stress indicate significant suppression of both quantities by the velocity pressure-gradient correlations. This set of data is intended as a detailed base for direct testing of turbulence submodels in swirling flows.

Closure of the Reynolds stress and scalar flux equations

A second-order, single-point closure model for calculating the transport of momentum and passive scalar quantities in turbulent flows is described. Of the unknown terms that appear in the Reynolds stress and scalar flux balance equations, it is those which involve the fluctuating pressure that exert a dominant influence in the majority of turbulent flows. A closure approximation (linear in the Reynolds stress) has been formulated for the velocity-pressure gradient correlation appearing in the Reynolds stress equation. When this is used in conjunction with previous proposals for the other unknown terms in the stress equation, the proposed model closely simulates most of the data on high Reynolds number homogeneous turbulent flows. For the fluctuating scalar-pressure gradient correlation appearing in the scalar flux equation, an approximation has been devised that satisfies the linear transformation properties of the exact equation. Additional characteristics of the fluctuating scalar field are obtained from the solution of modeled balance equations for the scalar variance and its ‘‘dissipation’’ rate. The resulting complete scalar field model is capable of reproducing measured data in decaying scalar grid turbulence and strongly sheared, nearly homogeneous flow in the presence of a mean scalar gradient. In addition, applications to the thermal mixing layer developing downstream from a partially heated grid and to a slightly heated plane jet issuing into stagnant surrounds result in calculated profiles in close agreement with those measured.

Joint measurements of velocity and scalars in a turbulent diffusion flame with moderate swirl

Simultaneous measurements by two-component LDA and Mie scattering methods have been made in a moderately swirling, turbulent hydrogen diffusion flame in a horizontal co-flowing stream. Mean temperature is obtained separately by thermocouple. The three velocity components are measured in pairs so that all components of the stress tensor are obtained. The Mie signal is used to obtain scalar time traces. The flame expands into the freestream, which does not act as a confinement, and is shortened one third and widened around one quarter by the swirl. This contrasts with results in the literature for confined swirling flames which lengthen with increasing swirl. The difference is thought to be caused mainly by the action of the velocity pressure-gradient correlations: in a confined flame the radial pressure gradient is maintained high downstream, so these terms appear to suppress the u-v stress. In the present work, the upstream decay of streamwise velocity is increased, as is the turbulence intensity, whilst far downstream the normalized data look similar to those of unswirled flames. There is less penetration of external fluid towards the flame axis than in non-swirling flames. The v-w and u-w stress components have similar magnitudes, and it appears that the v-w stress may be dealt with by gradient transport modelling where the radial pressure gradient is not too high.

Some Budgets of Turbulent Stresses in Round Jets and Diffusion Flames

Abstract Measurements are presented of budgets of shear stress, normal stresses and turbulent kinetic energy in a round air jet and in two H2 diffusion flames, all in a co-flowing air stream. The results are based on simultaneous two-component LDA and Mie-scattering measurements, and are presented in Favre averaged form. The flames show basically similar features to the cold flow, but overall magnitudes of the normalized terms are lower, and the advection contribution to turbulent kinetic energy is much reduced. The turbulence dissipation rate e is obtained from the budgets of turbulent kinetic energy by difference, and is used later to obtain values of the velocity pressure-gradient correlations u ∂p/∂x and ν ∂p/∂r, again by difference. The results are consistent with other published estimates of these terms; whilst they seem small in the turbulent kinetic energy balance, they dominate in the budgets for the stresses, and are largely balanced by production. A comparison with a well-publicized numerical m...

Estimation of Some Correlations in a Premixed Reactive Turbulent Flow

Abstract Analysis is made of the rotational and potential fields in an open premixed turbulent flame to deduce the relative strengths of these fluctuating fields. The potential field, loosely described as the acoustic field, is found to be as strong as the rotational field for sufficiently high heat release, in accordance with prior estimates for ducted flames. The potential field decouples from the vortical field enabling separate calculations of correlations involving the potential field which are of importance in turbulence stress equations. For the case of a plane flame orthogonal to the oncoming stream and in the limit of a weak potential field these results are used to form a model for the velocity-pressure gradient correlation. The model differs from and is believed superior to others reported in the literature.

Duality, dilatation, diffusion and dissipation in reacting turbulent flows

Several areas of current interest in the analytical description of reacting turbulent flowsare reviewed, with varying detail. Included are a) an assessment of the competing roles of the vortical and potential fields and their interactions, b) gradient modelling of turbulent transport, c) new roles for dissipation descriptions and d) velocity-pressure gradient correlations as they affect second order closure methods and pdf evolution techniques. It is concluded that in all these areas there are great uncertainties in modelling and they offer rich research opportunities.