Reynolds Stress Anisotropy Research Articles

An explicit algebraic Reynolds stress and heat flux model for incompressible turbulence: Part I Non-isothermal flow

Tensor representation theory is used to derive an explicit algebraic model that consists of an explicit algebraic stress model (EASM) and an explicit algebraic heat flux model (EAHFM) for two-dimensional (2-D) incompressible non-isothermal turbulent flows. The representation methodology used for the heat flux vector is adapted from that used for the polynomial representation of the Reynolds stress anisotropy tensor. Since the methodology is based on the formation of invariants from either vector or tensor basis sets, it is possible to derive explicit polynomial vector expansions for the heat flux vector. The resulting EAHFM is necessarily coupled with the turbulent velocity field through an EASM for the Reynolds stress anisotropy. An EASM has previously been derived by Jongen and Gatski [10]. Therefore, it is used in conjunction with the derived EAHFM to form the explicit algebraic model for incompressible 2-D flows. This explicit algebraic model is analyzed and compared with previous formulations including its ability to approximate the commonly accepted value for the turbulent Prandtl number. The effect of pressure-scrambling vector model calibration on predictive performance is also assessed. Finally, the explicit algebraic model is validated against a 2-D homogeneous shear flow with a variety of thermal gradients.

Structure of turbulent channel flow with square bars on one wall

The organised motion in a turbulent channel flow with a succession of square bars on the bottom wall has been investigated using direct numerical simulations. Several values of the ratio w/ k, where k is the bar height and w is the longitudinal separation between consecutive bars have been examined in detail. Relative to a smooth surface, the streamwise extent of the near-wall structures is decreased while their spanwise extent is increased. As w/ k increases, the coherence decreases in the streamwise direction having a minimum for w/ k=7. This is due to the outward motion occurring, most of all, near the leading edge of the elements. The Reynolds stress anisotropy tensor and its invariants show a closer approach to isotropy over the rough wall than over a smooth wall.

Pressure-strain correlation in homogeneous anisotropic turbulence subject to rapid strain-dominated distortion

Rapid distortion calculations of initially anisotropic turbulence are performed to better understand the physics of the pressure-strain correlation in strain-dominated mean flows. Based on the results of simulations we infer important physical characteristics of the “rapid” pressure-strain correlation Φij(r) in such flows: (i) it vanishes when there is no production of anisotropy, (ii) in the proximity of two-componential state it tends to decrease Reynolds stress anisotropy, and (iii) its magnitude is generally smaller than that of production. The observed characteristics are proposed as criteria that pressure-strain correlation models may be required to satisfy. All of the current popular models violate the above criteria for a sizeable subset of anisotropic initial conditions. Reynolds stress transport model calculations show that unphysical and unrealizable model behavior can be directly attributed to these violations.

Large-eddy simulations of ducts with a free surface

This work studies the momentum and energy transport mechanisms in the corner between a free surface and a solid wall. We perform large-eddy simulations of the incompressible fully developed turbulent flow in a square duct bounded above by a free-slip wall, for Reynolds numbers based on the mean friction velocity and the duct width equal to 360, 600 and 1000. The flow in the corner is strongly affected by the advection due to two counter-rotating secondary-flow regions present immediately below the free surface. Because of the convection of the inner eddy, as the free surface is approached, the friction velocity on the sidewall first decreases, then increases again. A similar behaviour is observed for the surface-parallel Reynolds-stress components, which first decrease and then increase again very close to the surface. The budgets of the Reynolds stresses show a strong reduction of all terms of the dissipation tensor in both the inner and outer near-corner regions. They exhibit a reduction in both production and dissipation towards the free surface. Very close to the solid boundary, within 15–20 viscous lengths of the sidewall, the turbulent kinetic energy production and the surface-parallel fluctuations rebound in the thin layer adjacent to the free surface. The Reynolds-stress anisotropy appears to be the main factor in the generation of the mean secondary flow. The multi-layer structure of the boundary layer near the free surface is also discussed.

Equilibrium states of turbulent homogeneous buoyant flows

The equilibrium states of homogeneous turbulent buoyant flows are investigated through a fixed-point analysis of the evolution equations for the Reynolds stress anisotropy tensor and the scaled heat flux vector. The mean velocity and thermal fields are assumed to be two-dimensional. Scalar invariants formed from the Reynolds stress anisotropy tensor, the scaled heat flux vector, and the strain rate and rotation rate tensors are governed by a closed set of algebraic equations derived for the stress anisotropy and scaled heat flux under a (weak) equilibrium assumption. Six equilibrium state variables are identified for the buoyant case and contrasted with the corresponding two state variables obtained for the non-buoyant homogeneous turbulence case. These results, while dependent on the functional forms of the models for the pressure-strain rate correlation tensor and the pressure-scalar-gradient correlation and viscous dissipation vector, can be used as in the non-buoyant case to either calibrate new closure models or validate the performance of existing models

Modulation of Near-Wall Turbulence Structure with Wall Blowing and Suction

The effects of wall blowing and suction on a fully developed, equilibrium turbulent channel flow are studied numerically. Direct numerical simulation data are analyzed to investigate the. modulation of near-wall turbulence anisotropy after the sudden application of wall blowing and suction. The effects on. the near-wall turbulence structure are examined in terms of the asymptotic near-wall behavior of various turbulence quantities and the turbulence anisotropy. It is found that blowing makes the near-wall flow more isotropic and enhances the transverse (nu' and w') components of velocity fluctuations. A significant increase in the anisotropy of the near-wall region is found in the suction case. The anisotropy invariant map for the Reynolds stress anisotropy tensor indicates that the relaxation processes of the anisotropy of the near-wall turbulence are different for the blowing and suction cases, respectively. The response of the flow to the sudden application of wall blowing and suction occurs earlier. for blowing than for suction, although there is a delayed response in both cases.

Reynolds stress anisotropy of turbulent rough wall layers

The current classification of turbulent boundary layers over different wall surfaces is based on the effect the roughness has on the mean velocity, via the roughness function ΔU+. Previous hot-wire measurements have shown that turbulent boundary layers over different rough wall surfaces, but with identical ΔU+, contain significant differences in the Reynolds stresses throughout the layer. This suggests that a detailed documentation of the effect of the roughness on the anisotropic Reynolds stress tensor may pave the way for a more general classification. This paper examines experimental (boundary layer) and direct numerical simulation (channel flow) data for the invariants of the Reynolds stress tensor. Although only a limited number of rough surfaces have been examined, the results indicate that, relative to a smooth wall, the roughness reduces the level of anisotropy. This is more prominent for k-type roughnesses, the anisotropic invariant map (AIM) signature of the d-type roughness being closer to that for a smooth surface. In the vicinity of the roughness, the AIM signature varies dramatically, within one roughness wavelength, reflecting the significant changes in the characteristics of the turbulence field.

Accounting for Buoyancy Effects in the Explicit Algebraic Stress Model: Homogeneous Turbulent Shear Flows

Most explicit algebraic stress models are formulated for turbulent shear flows without accounting for external body force effects, such as the buoyant force. These models yield fairly good predictions of the turbulence field generated by mean shear. As for thermal turbulence generated by the buoyant force, the models fail to give satisfactory results. The reason is that the models do not explicitly account for buoyancy effects, which interact with the mean shear to enhance or suppress turbulent mixing. Since applicable, coupled differential equations have been developed describing these thermal turbulent fields, it is possible to develop corresponding explicit algebraic stress models using tensor representation theory. While the procedure to be followed has been employed previously, unique challenges arise in extending the procedure for developing the algebraic representations to turbulent buoyant flows. In this paper the development of an explicit algebraic stress model (EASM) is confined to the homogeneous buoyant shear flow case to illustrate the methodology needed to develop the proper polynomial representations. The derivation is based on the implicit formulation of the Reynolds stress anisotropy at buoyant equilibrium. A five-term representation is found to be necessary to account properly for the effect of the thermal field. Thus derived, external buoyancy effects are represented in the scalar coefficients of the basis tensors, and structural buoyancy effects are accounted for in additional terms in the stress anisotropy tensor. These terms will not vanish even in the absence of mean shear. The performance of the new EASM, together with a two-equation (2-Eq) model, the non-buoyant EASM of Gatski and Speziale (1993) and a full second-order model, is assessed against direct numerical simulations of homogeneous, buoyant shear flows at two different Richardson numbers representing weak and strong buoyancy effects. The calculations show that this five-term representation yields better results than the 2-Eq model and the EASM of Gatski and Speziale where buoyancy effects are not explicitly accounted for.

Turbulence structure in boundary layers over different types of surface roughness

The classical treatment of rough wall turbulent boundary layers consists in determining the effect the roughness has on the mean velocity profile. This effect is usually described in terms of the roughness function ΔU+. The general implication is that different roughness geometries with the same ΔU+ will have similar turbulence characteristics, at least at a sufficient distance from the roughness elements. Measurements over two different surface geometries (a mesh roughness and spanwise circular rods regularly spaced in the streamwise direction) with nominally the same ΔU+ indicate significant differences in the Reynolds stresses, especially those involving the wall-normal velocity fluctuation, over the outer region. The differences are such that the Reynolds stress anisotropy is smaller over the mesh roughness than the rod roughness. The Reynolds stress anisotropy is largest for a smooth wall. The small-scale anisotropy and intermittency exhibit much smaller differences when the Taylor microscale Reynolds number and the Kolmogorov-normalized mean shear are nominally the same. There is nonetheless evidence that the small-scale structure over the three-dimensional mesh roughness conforms more closely with isotropy than that over the rod-roughened and smooth walls.

Improved Prediction of Turbomachinery Flows Using Near-Wall Reynolds-Stress Model

In this paper an assessment of the improvement in the prediction of complex turbomachinery flows using a new near-wall Reynolds-stress model is attempted. The turbulence closure used is a near-wall low-turbulence-Reynolds-number Reynolds-stress model, that is independent of the distance-from-the-wall and of the normal-to-the-wall direction. The model takes into account the Coriolis redistribution effect on the Reynolds-stresses. The five mean flow equations and the seven turbulence model equations are solved using an implicit coupled OΔx3 upwind-biased solver. Results are compared with experimental data for three turbomachinery configurations: the NTUA high subsonic annular cascade, the NASA_37 rotor, and the RWTH 1 1/2 stage turbine. A detailed analysis of the flowfield is given. It is seen that the new model that takes into account the Reynolds-stress anisotropy substantially improves the agreement with experimental data, particularily for flows with large separation, while being only 30 percent more expensive than the k−ε model (thanks to an efficient implicit implementation). It is believed that further work on advanced turbulence models will substantially enhance the predictive capability of complex turbulent flows in turbomachinery.

Turbulent supersonic channel flow

The effects of compressibility are studied in low Reynolds number turbulent supersonic channel flow via a direct numerical simulation. A pressure-velocity-entropy formulation of the compressible Navier-Stokes equations which is cast in a characteristic, non-conservative form and allows one to specify exact wall boundary conditions, consistent with the field equations, is integrated using a fifth-order compact upwind scheme for the Euler part, a fourth-order Padé scheme for the viscous terms and a third-order low-storage Runge-Kutta time integration method. Coleman et al fully developed supersonic channel flow at M?=?1.5 and Re?=?3000 is used to test the method. The nature of fluctuating variables is investigated in detail for the wall layer and the core region based on scatter plots. Fluctuations conditioned on sweeps and ejections in the wall layer are especially instructive, showing that positive temperature, entropy and total temperature fluctuations are mainly due to sweep events in this specific situation of wall cooling. The effect of compressibility on the turbulence structure is in many respects similar to that found in homogeneous shear turbulence and in mixing layers. The normal components of the Reynolds stress anisotropy tensor are increased due to compressibility, while the shear stress component is slightly reduced. Characteristic of the Reynolds stress transport is a suppression of the production of the longitudinal and the shear stress component, a suppression of all velocity-pressure-gradient correlations and most of the dissipation rates. Comparison with incompressible channel flow data reveals that compressibility effects manifest themselves in the wall layer only.

Effects of System Rotation on Anisotropy of Homogeneous Decaying Turbulence.

The linear effect of system rotation and the implicit effect of nonlinear interactions modified by the rotation on anisotropic Reynolds stress tensor bij for homogeneous decaying turbulence are studied in this paper. A linear solution of the anisotropic tensor in rotating system is presented. After proposed a method to introduce anisotropy into an isotropic turbulence, direct numerical simulation (DNS) of homogeneous decaying turbulence with or without initially anisotropy is carried out. Theoretical analysis indicates that the anisotropic tensor bij can be split into two parts : bijz and bije. DNS results show that bijz decays rapidly by system rotation at the initial period of evolution and its evolution agress well with the linear solution. On the other hand, bije reflects the implicit effect of the nonlinear interactions modified by the rotation.

Reynolds-averaged equations for free-surface flows with application to high-Froude-number jet spreading

The goals of this study were to develop a set of Reynolds-averaged governing equations for turbulent free-surface flow, and to use the resulting equations to determine the origin of the surface current in high-Froude-number jet flows. To develop the Reynolds-averaged equations, free-surface turbulent flow is treated as a two-fluid flow separated by an interface. It is shown that the general Navier–Stokes equations written for variable property flow embody the field equations applicable to each fluid, as well as the boundary conditions for the interface and, therefore, can be applied across the entire fluid domain, including the interface. With this as a starting point, a formulation of the Reynolds-averaged governing equations for turbulent free-surface flows can be developed rigorously. The resulting Reynolds-averaged equations are written in terms of density-weighted averages, their derivatives, and the probability density function for the free-surface position. These equations are similar to the conventional Reynolds-averaged equations, but include additional terms which represent the average effect of the forces acting instantaneously on the free surface, forces normally associated with the boundary conditions. These averaged equations are applied to the interaction of a turbulent jet with the free surface in order to establish, for arbitrary-Froude-number flows, the origin of the surface current, the large outward velocity which occurs in a thin layer adjacent to the surface. It is shown via an order-of-magnitude analysis that the outward acceleration associated with the surface current results from a combination of the Reynolds-stress anisotropy and the free-surface fluctuations. For low Froude number, the surface current is mainly driven by the Reynolds stress anisotropy, consistent with the results of Walker (1997); when the Froude number is large, the Reynolds-stress anisotropy is smaller and the free-surface fluctuations make a significant contribution.

Turbulence Model Predictions of Strongly Curved Flow in a U-Duct

The ability of three types of turbulence models to accurately predict the effects of curvature on the flow in a U-duct is studied. An explicit algebraic stress model performs slightly better than one- or two-equation linear eddy viscosity models, although it is necessary to fully account for the variation of the production-to-dissipation-rate ratio in the algebraic stress model formulation. In their original formulations, none of these turbulence models fully captures the suppressed turbulence near the convex wall, whereas a full Reynolds stress model does. Some of the underlying assumptions used in the development of algebraic stress models are investigated and compared with the computed flowfield from the full Reynolds stress model. Through this analysis, the assumption of Reynolds stress anisotropy equilibrium used in the algebraic stress model formulation is found to be incorrect in regions of strong curvature. By the accounting for the local variation of the principal axes of the strain rate tensor, the explicit algebraic stress model correctly predicts the suppressed turbulence in the outer part of the boundary layer near the convex wall.

LDV Measurement of Confined Parallel Jet Mixing

Laser Doppler Velocimetry (LDV) measurements were taken in a confinement, bounded by two parallel walls, into which issues a row of parallel jets. Two-component measurements were taken of two mean velocity components and three Reynolds stress components. As observed in isolated three-dimensional wall bounded jets, the transverse diffusion of the jets is quite large. The data indicate that this rapid mixing process is due to strong secondary flows, transport of large inlet intensities, and Reynolds stress anisotropy effects.

Derivation and investigation of a new explicit algebraic model for the passive scalar flux

An algebraic relation for the scalar flux, in terms of mean flow quantities, is formed by applying an equilibrium condition in the transport equations for the normalized scalar flux. This modeling approach is analogous to explicit algebraic Reynolds stress modeling (EARSM) for the Reynolds stress anisotropies. The assumption of negligible advection and diffusion of the normalized passive scalar flux gives, in general, an implicit, nonlinear set of algebraic equations. A method to solve this implicit relation in a fully explicit form is proposed, where the nonlinearity in the scalar-production-to-dissipation ratio is considered and solved. The nonlinearity, in the algebraic equations for the normalized scalar fluxes, may be eliminated directly by using a nonlinear term in the model of the pressure scalar-gradient correlation and the destruction and thus results in a much simpler model for both two-and three-dimensional mean flows. The performance of the present model is investigated in three different flow situations. These are homogeneous shear flow with an imposed mean scalar gradient, turbulent channel flow, and the flow field downstream a heated cylinder. The direct numerical simulation (DNS) data are used to analyze the passive scalar flux in the homogeneous shear and channel flow cases and experimental data are used in the case of the heated cylinder wake. Sets of parameter values giving very good predictions in all three cases are found.

Investigation of Nonlinear Eddy-Viscosity Turbulence Models in Shock/Boundary-Layer Interaction

Validation of nonlinear two- and three-equation, eddy-viscosity turbulence models (NLEVM)in transonic e ows featuring shock/boundary-layer interaction and separation is presented. The accuracy of the models is assessed againstexperimentalresultsfortwotransonice owsoverbumpgeometries.Moreover,theaccuracyandefe ciencyof NLEVMsisalsoassessedincontrasttonumericalpredictionsobtainedbyavarietyofothermodelsemployedinthis study. These include two linear eddy-viscosity k‐≤ models, the k‐! shear-stress transport model, and a nonlinear version of the k‐! model. Discretization of the mean e ow and turbulence transport equations is obtained by a characteristics-based scheme (Riemann solver ) in conjunction with an implicit unfactored method. The study shows that NLEVMs improve the numerical predictions in shock/boundary-layer interaction, compared to the linear models, but they require longer computing times. Nomenclature A = Lumley’ s e atness parameter, ´ 1 i 9 (A2 i A3) A2 = second invariant of Reynolds-stress anisotropy, ´ a ija ij A3 = third invariant of Reynolds-stress anisotropy, ´ a ika kja ji c

Computation of heat transfer in rotating two-pass square channels by a second-moment closure model

A multiblock numerical method has been employed for the calculation of three-dimensional flow and heat transfer in rotating two-pass square channels with smooth walls. The finite-analytic method solves Reynolds-averaged Navier–Stokes equations in conjunction with a near-wall second-order Reynolds stress (second-moment) closure model and a two-layer k–ε isotropic eddy viscosity model. Comparison of second-moment and two-layer calculations with experimental data clearly demonstrate that the secondary flows in rotating two-pass channels have been strongly influenced by the Reynolds stress anisotropy resulting from the Coriolis and centrifugal buoyancy forces as well as the 180° wall curvatures. The near-wall second-moment closure model provides accurate heat transfer predictions which agree well with measured data.

Effect of different surface roughnesses on a turbulent boundary layer

The classical treatment of rough wall turbulent boundary layers consists in determining the effect the roughness has on the mean velocity profile. This effect is usually described in terms of the roughness function D U+. The general implication is that different roughness geometries with the same D U+ will have similar turbulence characteristics, at least at a sufficient distance from the roughness elements. Measurements over two different surface geometries (a mesh roughness and spanwise circular rods regularly spaced in the streamwise direction) with nominally the same D U+ indicate significant differences in the Reynolds stresses, especially those involving the wall-normal velocity fluctuation, over the outer region. The differences are such that the Reynolds stress anisotropy is smaller over the mesh roughness than the rod roughness. The Reynolds stress anisotropy is largest for a smooth wall. The small-scale anisotropy and interniittency exhibit much smaller differences when the Taylor microscale Reynolds number and the Kolmogorov-normalized mean shear are nominally the same. There is nonetheless evidence that the small-scale structure over the three-dimensional mesh roughness conforms more closely with isotropy than that over the rod-roughened and smooth walls.

Turbulence model predictions of strongly curved flow in a U-duct

The ability of three types of turbulence models to accurately predict the effects of curvature on the flow in a U-duct is studied. An explicit algebraic stress model performs slightly better than one- or two-equation linear eddy viscosity models, although it is necessary to fully account for the variation of the production-to-dissipation-rate ratio in the algebraic stress model formulation. In their original formulations, none of these turbulence models fully captures the suppressed turbulence near the convex wall, whereas a full Reynolds stress model does. Some of the underlying assumptions used in the development of algebraic stress models are investigated and compared with the computed flowfield from the full Reynolds stress model. Through this analysis, the assumption of Reynolds stress anisotropy equilibrium used in the algebraic stress model formulation is found to be incorrect in regions of strong curvature. By the accounting for the local variation of the principal axes of the strain rate tensor, the explicit algebraic stress model correctly predicts the suppressed turbulence in the outer part of the boundary layer near the convex wall.