Plane Channel Flow Research Articles

Variational multiscale turbulence modelling in a high order spectral element method

In the variational multiscale (VMS) approach to large eddy simulation (LES), the governing equations are projected onto an a priori scale partitioning of the solution space. This gives an alternative framework for designing and analyzing turbulence models. We describe the implementation of the VMS LES methodology in a high order spectral element method with a nodal basis, and discuss the properties of the proposed scale partitioning. The spectral element code is first validated by doing a direct numerical simulation of fully developed plane channel flow. The performance of the turbulence model is then assessed by several coarse grid simulations of channel flow at different Reynolds numbers.

Turbulent momentum transport and kinetic energy production in plane-channel flows

Based on a momentum balance analysis, this paper basically presents and discusses experimental results from fully developed turbulent plane-channel flows utilizing hot-wire anemometer. The paper stresses the fact that in experimental investigations of two-dimensional shear flows it is difficult to obtain, with reasonable accuracy, all turbulent quantities of interest to fluid mechanics researchers. This makes it necessary to measure some relevant turbulent flow properties as a basis to drive others by utilizing theoretically derived relationships obtained from the mean momentum equation. It is demonstrated that both the turbulent Reynolds shear stress - ρ u 1 ′ u 2 ′ ¯ and the kinetic energy production - ρ u 1 ′ u 2 ′ ¯ ( d U ¯ / dy ) for two-dimensional fully developed turbulent flows can be predicted in this way through detailed mean velocity and mean pressure gradient measurements. In addition, some flow similarities and differences among channel, pipe and boundary layer flows are discussed, revealing effects of flow geometry and Reynolds number on the mean flow scaling.

Large-Eddy Simulation of Turbulence-Radiation Interactions in a Turbulent Planar Channel Flow

Large-eddy simulation (LES) has been performed for planar turbulent channel flow between two infinite, parallel, stationary plates. The capabilities and limitations of the LES code in predicting correct turbulent velocity and passive temperature field statistics have been established through comparison to direct numerical simulation data from the literature for nonreacting cases. Mixing and chemical reaction (infinitely fast) between a fuel stream and an oxidizer stream have been simulated to generate large composition and temperature fluctuations in the flow; here the composition and temperature do not affect the hydrodynamics (one-way coupling). The radiative transfer equation is solved using a spherical harmonics (P1) method, and radiation properties correspond to a fictitious gray gas with a composition- and temperature-dependent Planck-mean absorption coefficient that mimics that of typical hydrocarbon-air combustion products. Simulations have been performed for different optical thicknesses. In the absence of chemical reactions, radiation significantly modifies the mean temperature profiles, but temperature fluctuations and turbulence-radiation interactions (TRI) are small, consistent with earlier findings. Chemical reaction enhances the composition and temperature fluctuations and, hence, the importance of TRI. Contributions to emission and absorption TRI have been isolated and quantified as a function of optical thickness.

Refined cf relation for turbulent channels and consequences for high-Re experiments

There have been rising concerns regarding the accuracy of measurements in turbulent channel flows, in particular, measurements of the skin-friction results. In the present study, two different methods, namely, mean streamwise pressure gradient (PG) and oil film interferometry (OFI), are used to estimate the wall skin-friction relation, cf = f(Rem), for fully developed turbulent plane-channel flow over a wide range of Reynolds numbers. The channel skin-friction data are then fitted to the well-known logarithmic friction law, providing outstanding agreement with values for the constants of the logarithmic law of the mean velocity profile. A revised logarithmic skin-friction relation is developed, providing good agreement with our skin-friction results and data from the literature, when constants of the logarithmic friction relation adopted from the recent work of Zanoun et al (2003 Phys. Fluids 15 3079–89, 2005 4th Int. Conf. on Heat Transfer, Fluid Mechanics and Thermodynamics, HEAT2005, 19–22 September, Cairo, Egypt) are utilized. A new experimental channel facility is proposed, allowing measurements at high Reynolds numbers well beyond those achieved previously in laboratories, i.e. over five times the highest Rem reached in the present study, while maintaining sufficiently high spatial resolution.

Hydraulic analysis of averaged parameters of turbulent flows in circular pipes and plane channels

Variation in the position of three characteristic (conditional) layers of a flow is investigated.

Instability of a horizontal plane-channel flow of a dilute suspension

The stability of a horizontal plane-channel flow of a dilute suspension is studied theoretically. It is shown that the mechanism of action of the sedimenting particles on the flow stability parameters is equivalent to the effect of a distributed flow stratification and is attributable to the vertical nonuniformity of the body force induced by the excess weight of the sedimenting particles. A strong dependence of the disturbance growth rate on the location of the interface between the suspension and the pure liquid is detected.

Convective cooling of tandem heated squares in a channel

This article elaborates on the numerical simulation of forced convective incompressi-ble flow in a horizontal plane channel and heat transfer over two isothermal tandem square cylinders. Finite-volume method was applied to discretize the momentum and energy equations. This study details the effects of S/ H (gap between two squares), Reynolds number, and blockage ratio on the characteristics of flow field and heat transfer. The blockage ratios and Reynolds numbers were chosen between 12.5 to 50 per cent and 100≤ Re≤300, respectively. Results are presented in the form of time-averaged streamlines, instantaneous contours of temperature and vorticity, and some characteristics of fluid flow and heat transfer such as time-averaged of the local and total Nusselt number, drag coefficient, and Strouhal number. The present results show that the gap spacing between squares at high blockage ratios does not have an obvious effect on heat transfer. On the other hand, this parameter plays a main role on the drag coefficient at a high value ( S/ H=4), which is because of the formation of the vortex cores in gap spacing and downstream of the second square. The heat transfer rate of the second square increases with the increase of S/ H. Increasing the blockage ratio has a negative effect on convective heat transfer of the second square. With the vortex shedding in the gap spacing, the drag coefficient and the Nusselt number both increase in the second square in comparison to the first square.

Dispersion of a buoyant plume in a turbulent pressure-driven channel flow

Direct numerical simulations of the turbulent dispersion of a buoyant line of hot fluid released at the inlet of a plane channel flow are reported ( Re τ = 180, Gr = 10 7 and Pr = 0.7). Results of turbulent dispersion of a neutrally buoyant scalar and mixed convection flow are also included. The buoyancy force induces a vertical movement that, although small in mean, exhibits a significant fluctuation in the vertical velocity component and deflects the plume with the consequent loss of symmetry found in the neutrally buoyant results. The modification of the budgets for the time averaged momentum and heat transport equations reflects the rearranging of the different contributions induced by the buoyancy force.

Investigation of the stability of a plane-channel suspension flow with account for finite particle volume fraction

Within the framework of the two-fluid approach, a variant of a heterogeneous-medium model which takes into account a finite volume fraction of the inclusions and a small but finite phase velocity slip is proposed. The interphase momentum exchange is described by the Stokes force with the Brinkman correction for the finite particle volume fraction. The suspension viscosity depends on the particle volume fraction in accordance with the Einstein formula. Within the framework of the model constructed, a formulation of the problem of linear stability of plane-parallel two-phase flows is proposed. As an example, the stability of a channel suspension flow is considered. The system of equations for small disturbances with the boundary conditions is reduced to an eigenvalue problem for a fourth-order ordinary differential equation. Using the orthogonalization method, the dependence of the critical Reynolds number on the governing nondimensional parameters of the problem is studied numerically. It is shown that taking a finite volume fraction of the inclusions into account significantly affects the laminar-turbulent transition limit.

Direct numerical simulation of turbulence in injection-driven plane channel flows

Compressible turbulent flow in a periodic plane channel with mass injecting walls is studied as a simplified model for core flow in a solid-propellant rocket motor with homogeneous propellant and other injection-driven internal flows. In this model problem, the streamwise direction was asymptotically homogenized by assuming that at large distances from the closed end, both the mean and rms of turbulent fluctuations evolve slowly in the streamwise direction when compared to the turbulent fluctuations themselves. The Navier–Stokes equations were then modified to account for this slow growth. A direct numerical simulation of the homogenized compressible injection-driven turbulent flow was then conducted for conditions occurring at a streamwise location situated 40 channel half-widths from the closed off end and at an injection Reynolds number of approximately 190. The turbulence in this model flow was found to be only weakly compressible, although significant compressibility existed in the mean flow. As in nontranspired channels, turbulence resulted in increased near-wall shear for the mean streamwise velocity. When normalized by the average rate of turbulence production, the magnitudes of near-wall velocity fluctuations were similar to those in the log region of nontranspired wall-bounded turbulence. However, the sharp peak in streamwise velocity fluctuations observed in nontranspired channels was absent. While streaks and inclined vortices were observed in the near-wall region, their structure was very similar to those observed in the log region of nontranspired channels. These differences are attributed to the absence of a viscous sublayer in the transpired case which in turn is the result of the fact that the no-slip condition for the transpired case is an inviscid boundary condition. That is, unlike nontranspired walls, with transpiration, zero tangential velocity boundary conditions can be imposed at the wall for the Euler (inviscid) equations. The results of this study have important implications on the ability of turbulence models to predict this flow.

On linear stability of Rayleigh–Bénard Poiseuille flow of viscoplastic fluids

The present paper deals with the onset of the two-dimensional Rayleigh–Bénard convection for a plane channel flow of viscoplastic fluid. The influence of the yield stress on the instability and stability conditions characterized by the Rayleigh numbers denoted, respectively, RaL and RaE is investigated in the framework of linear analysis using modal and energetic approaches. The results show that the yield stress, represented by the Bingham number B, delays the onset of convection. For low values of the Reynolds number Re, the critical conditions RaL and RaE tend to be equal and the difference RaL−RaE increases with increasing Re, highlighting the non-normality of the linear operator. For Re<1 and large B (B≥O(10)), it is shown that the critical Rayleigh number increases as B2 and the critical wave number evolves according to B1/4.

Higher-order effects in rarefied channel flows

The regularized 13 moment (R13) equations and their boundary conditions are considered for plane channel flows. Chapman-Enskog scaling based on the Knudsen number is used to reduce the equations. The reduced equations yield second-order slip conditions, and allow us to describe the characteristic dip in the temperature profile observed in force driven flow. Due to the scaling, the R13 equations' ability to describe Knudsen layers is lost. Solutions with Knudsen layers are discussed as well, and it is shown that these give a better match to direct solutions of the Boltzmann equations than the reduced equations without Knudsen layers. For a radiatively heated gas the R13 equations predict a dependence of the average gas temperature on the Knudsen number with a distinct minimum around Kn = 0.2 , similar to the well-known Knudsen minimum for Poiseuille flow.

Inflow conditions for inhomogeneous turbulent flows

AbstractA cost‐effective method to generate inflow conditions for direct numerical simulations of wall‐bounded flows is presented. The method recycles a finite‐length time series of instantaneous velocity planes extracted from a precursor simulation and has earlier proved efficient for free shear layers. Now a spatially developing plane channel flow is considered. Different durations ts of the time series are tested and compared. Excellent agreement with fully developed channel flow statistics is observed when ts equals or exceeds the large‐eddy turnover time scale. The present results are more realistic than those obtained with synthetic turbulence generation and at the same time substantially cheaper than running an auxiliary simulation in parallel. Copyright © 2008 John Wiley & Sons, Ltd.

The relative roles of divergence and velocity slip in the stability of plane channel flow

Wall slip and wall divergence are known to have large and opposing effects on the stability of flow in a two-dimensional channel. While divergence hugely destabilises, slip dramatically stabilizes the linear mode. In a non-parallel stability analysis, we study a combination of these two effects, since both will coexist in small-scale flows with wall roughness. Our main results are (i) that the stabilising effect of slip is reversed at higher angles of divergence, (ii) transient growth of disturbances is unaffected by either wall-divergence, or by slip at any divergence. Moreover, at the Reynolds numbers relevant here, transient growth is too low to be a significant player in transition to turbulence, which is more likely to be driven by linear instability.

Study of a staggered fourth‐order compact scheme for unsteady incompressible viscous flows

AbstractThe objective of this paper is the development and assessment of a fourth‐order compact scheme for unsteady incompressible viscous flows. A brief review of the main developments of compact and high‐order schemes for incompressible flows is given. A numerical method is then presented for the simulation of unsteady incompressible flows based on fourth‐order compact discretization with physical boundary conditions implemented directly into the scheme. The equations are discretized on a staggered Cartesian non‐uniform grid and preserve a form of kinetic energy in the inviscid limit when a skew‐symmetric form of the convective terms is used.The accuracy and efficiency of the method are demonstrated in several inviscid and viscous flow problems. Results obtained with different combinations of second‐ and fourth‐order spatial discretizations and together with either the skew‐symmetric or divergence form of the convective term are compared. The performance of these schemes is further demonstrated by two challenging flow problems, linear instability in plane channel flow and a two‐dimensional dipole–wall interaction. Results show that the compact scheme is efficient and that the divergence and skew‐symmetric forms of the convective terms produce very similar results. In some but not all cases, a gain in accuracy and computational time is obtained with a high‐order discretization of only the convective and diffusive terms. Finally, the benefits of compact schemes with respect to second‐order schemes is discussed in the case of the fully developed turbulent channel flow. Copyright © 2008 John Wiley & Sons, Ltd.

Application of an Explicit Algebraic Reynolds Stress Model within a Hybrid LES–RANS Method

Large-eddy simulations (LES) still suffer from extremely large resources required for the resolution of the near-wall region, especially for high-Re flows. That is the main motivation for setting up hybrid LES–RANS methods. Meanwhile a variety of different hybrid concepts were proposed mostly relying on linear eddy-viscosity models. In the present study a hybrid approach based on an explicit algebraic Reynolds stress model (EARSM) is suggested. The model is applied in the RANS mode with the aim of accounting for the Reynolds stress anisotropy emerging especially in the near-wall region. For the implementation into a CFD code this anisotropy-resolving closure can be formally expressed in terms of a non-linear eddy-viscosity model (NLEVM). Its extra computational effort is small, still requiring solely the solution of one additional transport equation for the turbulent kinetic energy. In addition to this EARSM approach, a linear eddy-viscosity model (LEVM) is used in order to verify and emphasize the advantages of the non-linear model. In the present formulation the predefinition of RANS and LES regions is avoided and a gradual transition between both methods is assured. A dynamic interface criterion is suggested which relies on the modeled turbulent kinetic energy and the wall distance and thus automatically accounts for the characteristic properties of the flow. Furthermore, an enhanced version guaranteeing a sharp interface is proposed. The interface behavior is thoroughly investigated and it is shown how the method reacts on dynamic variations of the flow field. Both model variants, i.e. LEVM and EARSM, have been tested on the basis of the standard plane channel flow and even more detailed on the flow over a periodic arrangement of hills using fine and coarse grids.

A predictive wall model for large-eddy simulation based on optimal control techniques

Wall models for large-eddy simulation (LES) based on optimal control theory have so far been nonpredictive due to the need to prescribe a known mean velocity profile to the controller. In this study, LES is coupled with a near-wall Reynolds-averaged Navier–Stokes (RANS) model that provides a target velocity for the cost function. For the wall model to be accurate and robust, the LES and RANS must not only be tied together via the controller but directly coupled to each other through boundary conditions. The method proves to be accurate and robust over a wide range of Reynolds numbers in a plane channel flow. It is shown that the control reacts only locally in all spatial directions, justifying the current control formulation and suggesting directions for future model development. Further, instantaneous velocity fields of the coarse LES indicate that the dynamics of the near-wall flow are very dependent on the computational grid, demonstrating that a control strategy is required in addition to physical reasoning for wall modeling.

Interface-resolved direct numerical simulation of vertical particulate channel flow in the turbulent regime

We have conducted a direct numerical simulation study of dilute turbulent particulate flow in a vertical plane channel by considering thousands of finite-size rigid particles with resolved phase interfaces. The particle diameter corresponds to approximately 11 wall units and their terminal Reynolds number is set to 136. The fluid flow with a bulk Reynolds number of 2700 is directed upward, which maintains the particles suspended on average. Two density ratios were simulated, differing by a factor of 4.5. The corresponding Stokes numbers of the two flow cases were O(10) in the near-wall region and O(1) in the outer flow. We have observed the formation of large-scale elongated streaklike structures with streamwise dimensions of the order of eight channel half-widths and cross-stream dimensions of the order of one half-width. At the same time, we have found no evidence of significant formation of particle clusters, which suggests that the large structures are due to an intrinsic instability of the flow, which is triggered by the presence of the particles. It was found that the mean fluid velocity profile tends toward a concave shape, and the turbulence intensity and the normal stress anisotropy are strongly increased. The effect of varying the Stokes number while maintaining the buoyancy, particle size, and volume fraction constant was relatively weak.

Frequency response of an electrochemical probe to the wall shear stress fluctuations of a turbulent channel flow

We analyzed numerically the frequency response of the mass transfer rates produced at the surface of an electrochemical probe to the fluctuations of the wall shear stress obtained from a direct numerical simulation of a turbulent plane channel flow at Re τ = 150. At low and high frequencies the response to the turbulent wall shear stress is in accordance with the existing relations between the amplitudes of the wall shear stress and the mass transfer rates obtained for harmonic perturbations. At intermediate frequencies the turbulent wall shear stress fluctuations produce a larger damping in comparison with the harmonic perturbations. The phase angle of the response of mass transfer rates with respect to the wall shear stress shows important deviations in comparison with the phase angle obtained with harmonic perturbations, specially at high frequencies.

Efficient Generation of Inflow Conditions for Large Eddy Simulation of Street-Scale Flows

Using a numerical weather forecasting code to provide the dynamic large-scale inlet boundary conditions for the computation of small-scale urban canopy flows requires a continuous specification of appropriate inlet turbulence. For such computations to be practical, a very efficient method of generating such turbulence is needed. Correlation functions of typical turbulent shear flows have forms not too dissimilar to decaying exponentials. A digital-filter-based generation of turbulent inflow conditions exploiting this fact is presented as a suitable technique for large eddy simulations computation of spatially developing flows. The artificially generated turbulent inflows satisfy the prescribed integral length scales and Reynolds-stress-tensor. The method is much more efficient than, for example, Klein’s (J Comp Phys 186:652–665, 2003) or Kempf et al.’s (Flow Turbulence Combust, 74:67–84, 2005) methods because at every time step only one set of two-dimensional (rather than three-dimensional) random data is filtered to generate a set of two-dimensional data with the appropriate spatial correlations. These data are correlated with the data from the previous time step by using an exponential function based on two weight factors. The method is validated by simulating plane channel flows with smooth walls and flows over arrays of staggered cubes (a generic urban-type flow). Mean velocities, the Reynolds-stress-tensor and spectra are all shown to be comparable with those obtained using classical inlet-outlet periodic boundary conditions. Confidence has been gained in using this method to couple weather scale flows and street scale computations.