Equilibrium Turbulent Flow Research Articles

Influence of riblet geometry and freestream turbulence on boundary layer transition

High-fidelity large eddy simulations are used to investigate the influence of distributed surface-mounted riblets on spatially developing laminar boundary layer under varying freestream turbulence (FST). The spanwise homogenous riblets are employed in the study, with varying cross-sectional shapes: Square and semi-circular represented as SQ and SC, respectively. The inlet Reynolds number based on momentum thickness and freestream velocity is 360. The flow features exhibit considerable differences between the two riblet geometries under the influence of FST. Instantaneous results reveal the development of Λ-vortices as a genesis of flow transition over SQ-riblets across all the FST levels. While the streamwise streaks are evident over SC-riblets, where spot-like perturbations are observed at an FST of 6.0%. Here, the streaks are susceptible to sinuous secondary instability with subsequent breakdown into small-scale eddies. Thus, the novelty of this study lies in the identification of the modes of transition and flow structures, altered substantially by the riblet geometry and flow environments. Moreover, the turbulent kinetic energy budget is discussed, where an imbalance between production and dissipation is apparent in the transitional region, while a balance between them is established in the equilibrium turbulent flow.

Current trends in modelling research for turbulent aerodynamic flows

The engineering tools of choice for the computation of practical engineering flows have begun to migrate from those based on the traditional Reynolds-averaged Navier-Stokes approach to methodologies capable, in theory if not in practice, of accurately predicting some instantaneous scales of motion in the flow. The migration has largely been driven by both the success of Reynolds-averaged methods over a wide variety of flows and the inherent limitations of the method itself. Practitioners, emboldened by their ability to predict a wide variety of statistically steady equilibrium turbulent flows, have now turned their attention to flow control and non-equilibrium flows, i.e. separation control. This review gives some current priorities in traditional Reynolds-averaged modelling research as well as some methodologies being applied to a new class of turbulent flow control problem.

Stochastic modelling of inertial particle dispersion by subgrid motion for LES of high Reynolds number pipe flow

Aiming at the better prediction of inertial particle dispersion in LES of turbulent shear flow, a stochastic diffusion process of the Langevin type is adopted to model the time increments of the fluid velocity seen by inertial particles. This modelling is particularly crucial for dispersion and deposition of inertial particles with small relaxation times compared to the smallest LES-resolved turbulence time scales. Simple closures for drift and diffusion terms are described. The effects of inertia and external forces on particle trajectories are modelled. To numerically validate the proposed model, LES calculations are performed to track solid particles (glass beads in air with different Stokes' numbers) in a high Reynolds number, equilibrium turbulent pipe flow (Re = 50 000 based on the maximum velocity and the pipe diameter). LES predictions are compared to RANS results and experimental observations. Simulation findings demonstrate the superiority of LES compared to RANS in predicting particle dispersion statistics. More importantly, the use of a stochastic approach to model the subgrid scale fluctuations has proven crucial for results concerning the small-Stokes-number particles. The influence of the model on particle deposition needs to be assessed and additional validation for non-equilibrium turbulent flows is required.

Modeling of Wall Pressure Fluctuations Based on Time Mean Flow Field

Time-mean flow fields and turbulent flow characteristics obtained from solving the Reynolds averaged Navier-Stokes equations with a k‐ε turbulence model are used to predict the frequency spectrum of wall pressure fluctuations. The vertical turbulent velocity is represented by the turbulent kinetic energy contained in the local flow. An anisotropic distribution of the turbulent kinetic energy is implemented based on an equilibrium turbulent shear flow, which assumes flow with a zero streamwise pressure gradient. The spectral correlation model for predicting the wall pressure fluctuation is obtained through a Green’s function formulation and modeling of the streamwise and spanwise wave number spectra. Predictions for equilibrium flow agree well with measurements and demonstrate that when outer-flow and inner-flow activity contribute significantly, an overlap region exists in which the pressure spectrum scales as the inverse of frequency. Predictions of the surface pressure spectrum for flow over a backward-facing step are used to validate the current approach for a nonequilibrium flow.

On Lagrangian time scales and particle dispersion modeling in equilibrium turbulent shear flows

As intermediate quantities available from various existing numerical computations, the fluid Lagrangian time scales are of primary importance in the development of probability density function models for turbulent flows. Similarly, the time scales of the fluid seen by discrete particles in two-phase flows are essential for the development of dispersion models based on stochastic differential equations. Such time scales are obviously depending not only on the particle properties but also on the fluid Lagrangian and Eulerian time scales. A model is proposed here to estimate the directional dependence of the fluid Lagrangian time scales and drift coefficients in one-directional equilibrium turbulent shear flows, based on a local homogeneity assumption in the frame of the generalized Langevin model. Through comparison with available direct and large eddy simulation predictions in channel flows and in a homogeneous shear flow, the model is shown to lead to significant improvements in the streamwise and spanwise directions, where the existing empirical laws for the Lagrangian time scales are far from being satisfactory. We examine the way this model can be used to build a suitable stochastic process for the fluid seen by inertial particles in such basic turbulent shear flows.

Modeling of wall-pressure fluctuation based on time-mean flow field

Time-mean flow field and turbulent flow characteristics from solving the Navier–Stokes equations are used to predict the flow direction wave number spectrum of wall-pressure fluctuations. The vertical turbulent velocity is represented by the kinetic energy contained in the local turbulent flow. A redistribution of the turbulent kinetic energy to account for the anisotropic turbulence is implemented based on the equilibrium turbulent shear flow. The spectral correlation model in predicting the wall-pressure fluctuation is obtained through the Green’s function formulation, and the streamwise and spanwise wave number spectrum modelings. Predictions for the equilibrium flow agree well with measurements, and reveal an overlap region of an inverse of the frequency. Predictions for a 2-D reattached flow field after flow separation and a 3-D flow field on an airfoil with focus on the trailing edge and airfoil tip are discussed and compared to measurements. [Work supported by ONR.]

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.

Scaling properties of turbulent pipe flow at low Reynolds number

Direct numerical simulation of an equilibrium turbulent pipe flow at a Reynolds number of 2500, based on bulk velocity and pipe diameter, has been carried out with a grid of about 7.4×10 5 points and a fully resolved wall layer. The simulation fills the existing gap in the availability of accurate turbulence data in the low Reynolds number range, so that the sensitivity of the main statistical quantities to the Reynolds number effects could be completed. The systematic dependence of the turbulent stress tensor components upon Re in the near wall region, when the normalization is based upon the wall shear stress and kinematic viscosity, is confirmed. Normalization based on the Kolmogorov velocity and length scales as suggested by Antonia and Kim [1] in the context of plane channel flow, proves effective also for pipe flow at lower Re. The high values attained by the fourth-order moments of the radial and azimuthal velocities are shown to be of truly physical nature and to be related with rare and energetic events characterizing the whole wall layer. Vorticity dynamics analysis suggests that the origin of the high flatness factors is associated with the interaction of counter-rotating streamwise vortices with the wall. Analysis of the transport equations of the turbulent kinetic energy shows that the production, turbulent dissipation and viscous diffusion rates, the leading terms of the budget, increase considerably with Re for y +<20.

Simulation of Second Moments of Fluctuations of the Velocity and Temperature of Particles in Equilibrium Turbulent Flows

An analysis is performed of equilibrium solutions of equations for the second moments of fluctuations of the velocity and temperature of particles in a homogeneous shear flow, in a homogeneous flow subjected to tensile or compressive strain, and in a wall layer. The stability of equilibrium solutions to small perturbations is investigated. Algebraic models are given for turbulent stresses in the dispersed phase.

Relaxation of Turbulent Channel Flow through a Contracted Region. Turbulent Structure.

Effects of impulsive change in pressure gradient in a two-dimensional turbulent channel flow have been experimentally investigated. The rapid change of pressure gradient was obtained by passage through a contracted region. Turbulent intensities of velocity fluctuations and Reynolds shear stress following the removal of rapid pressure change were measured by means of a hot-wire anemometer. The statistical properties, i.e., energy spectra, length scales of turbulence, skewness / flatness factors and turbulent diffusions, were investigated. Turbulence intensities and Reynolds shear stress increase due to pressure gradient, however, the correlation coefficient of Reynolds shear stress depends slightly on pressure gradient and is constant across the chennel. The length scales are considerably smaller than those in equilibrium turbulent flows. The turbulent diffusions((U2V)^^- - (UV2)^^-) are negative in the near-wall region. The results reveal that the sweep event of bursting is strengthened by the rapid change in pressure gradient.

Near-wall integration of Reynolds stress turbulence closures with no wall damping

The explicit algebraic stress model of Gatski and Speziale is considered. This constitutes a two-equation model with an anisotropic eddy viscosity that is systematically derived from the SSG second-order model via the algebraic stress approximation for equilibrium turbulent flows.

Columnar vortices in isotropic turbulence

The structure of the intense vorticity regions is studied in numerically simulated homogeneous, isotropic, equilibrium turbulent flow fields at four different Reynolds numbers, in the rangeRe λ=35–170, and is found to be organized in coherent, cylindrical or ribbon-like, vortices (‘worms’). At the Reynolds numbers studied, they are responsible for much of the extreme intermittent tails observed in the statistics of the velocity gradients, but their importance seems to decrease at higherRe λ. Their radii scale with the Kolmogorov microscale and their lengths with the integral scale of the flow, while their circulation increases monotonically withRe λ. An explanation is offered for this latter scaling, based in the assumed presence of axial inertial waves along their cores, excited by a random background strain of the order of the root mean square vorticity. This explanation is consistent with the presence of comparable amounts of stretching and compression along the vortex cores.

Lag model for turbulent boundary layers over rough bleed surfaces

Boundary-layer mass removal (bleed) through spanwise bands of holes on a surface is used to prevent or control separation and to stabilize the normal shock in supersonic inlets. The addition of a transport equation lag relationship for eddy viscosity to the rough wall algebraic turbulence model of Cebeci and Chang was found to improve agreement between predicted and measured mean velocity distributions downstream of a bleed band. The model was demonstrated for a range of bleed configurations, bleed rates, and local freestream Mach numbers. In addition, the model was applied to the boundary-layer development over acoustic lining materials for the inlets and nozzles of commercial aircraft. The model was found to yield accurate results for integral boundary-layer properties unless there was a strong adverse pressure gradient. Nomenclature A = constant (26.0) in van Driest's damping function, Eq. (3) ks = equivalent sand-grain roughness height k+ = roughness Reynolds number, ksuT/v k^ = constant (0.02) in outer layer eddy viscosity model, Eq. (4) k3 = constant (0.5) in the lag model, Eq. (7) / = length scale rabl = bleed mass flow rate p = static pressure R = augmented mixing length due to surface roughness effect Re = Reynolds number M, v = velocity components in the streamwise and the normal to surface directions UT = frictional velocity *, y = coordinates in the streamwise and the normal to boundary-layer surface directions T = intermittency factor in Eq. (4), 0 < F < 1 8* = displacement thickness 6 = momentum thickness K = von Karman constant, 0.40, Eq. (3) fji — viscosity p = density v = kinematic viscosity of the fluid Subscripts e = outer region of boundary layer eq = equilibrium turbulent flow t = turbulent flow Superscripts ' = fluctuating turbulent quantity + = quantity normalized by vluT

Statistical Characteristics of Turbulence in a Periodically Diverging-Converging Channel Flow.

The statistical properties of a periodically diverging-converging turbulent channel flow are examined experimentally. Air flow is alternately decelerated through a linearly diverging section, and accelerated through a converging section. Measurements on the energy spectrum, auto-correlation, probability density, skewness/flatness factors and space-time correlation are made in two channels with different diverging-converging angle. The inertial subrange can be recognized for the diverging and converging sections. Large differences are observed in the skewness and flatness factors compared with the equilibrium turbulent flows with no pressure gradient. The eddy scales in the converging section are considerably smaller than those in the diverging section.

The structure of intense vorticity in isotropic turbulence

The structure of the intense-vorticity regions is studied in numerically simulated homogeneous, isotropic, equilibrium turbulent flow fields at four different Reynolds numbers, in the range Re, = 35-170. In accordance with previous investigators this vorticity is found to be organized in coherent, cylindrical or ribbon-like, vortices (‘worms’). A statistical study suggests that they are simply especially intense features of the background, O(o’), vorticity. Their radii scale with the Kolmogorov microscale and their lengths with the integral scale of the flow. An interesting observation is that the Reynolds number y/v, based on the circulation of the intense vortices, increases monotonically with ReA, raising the question of the stability of the structures in the limit of Re, --z co. Conversely, the average rate of stretching of these vortices increases only slowly with their peak vorticity, suggesting that self-stretching is not important in their evolution. One- and two-dimensional statistics of vorticity and strain are presented; they are non-Gaussian and the behaviour of their tails depends strongly on the Reynolds number. There is no evidence of convergence to a limiting distribution in this range of Re,, even though the energy spectra and the energy dissipation rate show good asymptotic properties in the higher-Reynolds-number cases. Evidence is presented to show that worms are natural features of the flow and that they do not depend on the particular forcing scheme.

Damping of surface pressure fluctuations in hypersonic turbulent flow past expansion corners

Surface pressure fluctuations of Mach 8 turbulent flow past a 2.5- and a 4.25-deg expansion corner maintained a Gaussian distribution but were severely attenuated by the expansion process. The pressure fluctuations did not recover to those of an equilibrium turbulent flow even though the mean pressures reached downstream inviscid values in four to six boundary-layer thicknesses. The fluctuations were convected with a velocity comparable to that on a flat plate, and they maintained their identities longer for the stronger expansion. The damping of pressure fluctuations at hypersonic Mach numbers, even by small corner angles, may be exploited in fatigue design.

Statistical Characteristics in a Periodically Diverging-Converging Turbulent Channel Flow.

The statistical properties of a periodically diverging-converging turbulent channel flow were examined experimentally. Air flow was decelerated through a linearly diverging section, and then accelerated through a converging section. The measurement of the energy spectrum, autocorrelation, probability density, skewness/flatness factors and space-time correlation were made on the two different channels. The inertial subrange can be recognized in the diverging and converging sections. Large differences are observed in the skewness and flatness factors compared with the equilibrium turbulent flows. The eddy scales in the converging section are considerably smaller than those in the diverging section.

An asymptotic theory of turbulent separation

The problem addressed in this paper is the turbulent counterpart of the classical Goldstein analysis of laminar separation points. In the present paper we present a new asymptotic theory of turbulent boundary layers and we use it to determine the singular behavior of solutions of the boundary layer equations at the point of zero skin friction in a prescribed pressure distribution. In the first part of the paper we describe a new asymptotic theory of turbulent boundary layers which is based on formal expansions in terms of two parameters: Reynolds number Re → ∞ and α → 0 , where α is a turbulence model constant appearing in the zero equation turbulence model employed in the present study. In order to establish the accuracy of the theory, solutions obtained with the new theory for equilibrium turbulent flow are compared with “exact” solutions of Mellor and Gibson. In the second part of the paper the local behavior of solutions of the turbulent boundary layer equations with a zero equation turbulent model is established using the new asymptotic theory. It is demonstrated that the solution of the turbulent boundary layer equations are singular at separation points if the pressure distribution is prescribed just as in the laminar case. However, the nature of the singularity is different in laminar and turbulent flow. We show that the approach to zero skin friction is linear in turbulent flow as compared to the square root behavior of the laminar “Goldstein” singularity.

Maintenance of oscillations in a turbulent Ekman layer

The turbulent Ekman boundary layer is modelled by prescribing a particular behavior for the eddy viscosity and thereby making it possible to calculate velocity profiles. This flow is then examined with respect to stability. Comparisons for this system are made with the more substantial documented features of a conventional flat-plate turbulent boundary layer without rotation and the accompanying stability investigations that have been made. Unlike the results known for the nonrotating layer, it appears that viscous oscillations are possible when rotation is present. General arguments are made in terms of the physics to substantiate this finding and one conjecture is that the modelled system does not represent a fully developed or equilibrium turbulent flow.

The Turbulent Boundary Layer With Mass Transfer and Pressure Gradient

An analysis of the incompressible, turbulent boundary layer, including the combined effects of mass transfer and pressure gradient is presented in this paper. An integral method employing the integral mechanical energy equation forms the basis of the analysis. Stevenson’s velocity profiles are used to obtain the functional dependence of the integral properties and also obtain a skin-friction law. A definition of an equilibrium flow with mass transfer and pressure gradient is given in order to evaluate the dissipation integral (CD) which appears in the integral mechanical energy equation. This definition requires a pressure gradient parameter similar to Clauser’s βT with a modification to include the effect of mass transfer to be held constant. An expression for CD in the case of equilibrium turbulent flows is then obtained which depends directly on this new pressure gradient parameter (βT*). In order to treat the general case of nonequilibrium flows, this expression for CD is uncoupled from βT*, through the use of a single empirical curve fit of the existing no mass transfer equilibrium flow data relating βT to the Clauser shape parameter. In addition to unhooking CD from the pressure gradient parameter, several specified variations in mass transfer rate are assumed in order to obtain an expression for CD which is not a function of the mass transfer rate derivative. The numerical results are found to be weakly dependent on which of these variations is used. Comparisons of the numerical results with a wide variety of experimental data, including cases where the blowing rate and pressure are varying simultaneously, show good agreement. In addition, several problems with discontinuities in blowing or suction are solved and seem to be in good agreement with the data.