Channel Half-width Research Articles

LES approach coupled with stochastic forcing of subgrid acceleration in a high-Reynolds-number channel flow

In this paper, the high-Reynolds-number channel flow is simulated by numerical approach at coarse resolution, in which the instantaneous acceleration is decomposed into filtered and subgrid parts, and then both components are modeled. The filtered acceleration is modeled in the framework of the large-eddy simulation approach. The model for the subgrid acceleration is based on two stochastic processes. The first is for its norm and is based on statistical universalities in fragmentation under scaling symmetry, providing correlation of subgrid forcing across the channel. The second is for its orientation and is based on the Brownian motion on a unit sphere in order to represent a stochastic relaxation toward full isotropy away from the wall. Two main parameters of the stochastic process include the Reynolds number based on the friction velocity and the channel half-width. In order to assess the capability of the model proposed, the paper illustrates its application versus recent high-Reynolds-number direct ...

Stagnation point von Kármán coefficient

On the basis of various direct numerical simulations (DNS) of turbulent channel flows the following picture is proposed. (i) At a distance y from either wall, the Taylor microscale lambda is proportional to the average distance l(s) between stagnation points of the fluctuating velocity field, i.e., lambda(y)=B(1)l(s)(y) with B(1) constant, for delta(nu) << y < or approximately equal to delta, where the wall unit delta(nu) is defined as the ratio of kinematic viscosity nu to skin friction velocity u(tau) and delta is the channel's half-width. (ii) The number density n(s) of stagnation points varies with height according to n(s)=(C(s)/delta(nu)(3))y(+)(-1) where y(+) identical with y/delta(nu) and C(s) is constant in the range delta(nu) << y < or approximately equal to delta. (iii) In that same range, the kinetic energy dissipation rate per unit mass, equals 2/3(E(+)((u(tau)(3)/kappa(s)y) where E(+) is the total kinetic energy per unit mass normalized by u(tau)(2) and kappa(s)=B(1)(2)/C(s) is the stagnation point von Kármán coefficient. (iv) In the limit of exceedingly large Reynolds numbers Re(tau) identical with delta/delta(nu), large enough for the Reynolds stress -(uv) to equal u(tau)(2) in the range delta(nu) << y << delta, and assuming that production of turbulent kinetic energy balances dissipation locally in that range and limit, the mean velocity U(+), normalized by u(tau), obeys (d/dy)U(+) approximately equal to 2/3(E(+)/kappa(s)y) in that same range. (v) It follows that the von Kármán coefficient kappa is a meaningful and well-defined coefficient and the log law holds in turbulent channel/pipe flows only if E(+) is independent of y(+) and Re(tau) in that range, in which case kappa approximately kappa(s). (vi) In support of (d/dy)U(+) approximately equal to 2/3(E(+)/kappa(s)y), DNS data of turbulent channel flows which include the highest currently available values of Re(tau) are best fitted by E(+) approximately equal to 2/3(B(4)y(+)(-2/15)) and (d/dy(+))U(+) approximately equal to (B(4)/kappa(s))y(+)(-1-2/15) with B4 independent of y in delta(nu) << y << delta if the significant departure from -(uv) approximately equal to u(tau)(2) at these Re(tau) values is taken into account.

Kolmogorov scaling of turbulent flow in the vicinity of the wall

How to scale even the simplest of turbulent flows continues to be a cause for considerable controversy. In the present research, a data base compiling results from channel flow direct numerical simulations and turbulent boundary layer experiments is employed to investigate the properties of shear and normal Reynolds stresses very close to the wall. Two types of scaling based on Kolmogorov length and velocity scales are analyzed. It is shown that it is highly likely that large length scales of the order of the channel half-width or the boundary layer thickness play an important role even in the innermost regions of wall-bounded turbulent flows, which hints at the persistence of Reynolds number effects in even high Reynolds number flows.

Analogy between velocity and scalar fields in a turbulent channel flow

The relationship between the fluctuating velocity vector and the temperature fluctuation has been examined using direct numerical simulation databases of a turbulent channel flow with passive scalar transport using a constant time-averaged heat flux at each wall for h+ = 180, 395, 640 and 1020 (where h is the channel half-width with the superscript denoting normalization by wall variables) at Prandtl number Pr=0.71. The analogy between spectra corresponding to the kinetic energy and scalar variance is reasonable in both inner and outer regions irrespective of whether the spectra are plotted in terms of kx or kz, the wavenumbers in the streamwise and spanwise directions respectively. Whereas all three velocity fluctuations contribute to the energy spectrum when kx is used, the longitudinal velocity fluctuation is the major contributor when kz is used. The quality of the analogy in the spectral domain is confirmed by visualizations in physical space and reflects differences between spatial organizations in the velocity and scalar fields. The similarity between the spectra corresponding to the enstrophy and scalar dissipation rate is not as good as that between the kinetic energy and scalar variance, emphasizing the prominence of the scalar sheets as the centre of the channel is approached. The ratio R between the characteristic time scales of the velocity and scalar fluctuations is approximately constant over a major part of the channel and independent of h+, when the latter is sufficiently large. This constancy, which is not observed in quantities such as the turbulent Prandtl number, follows from the spectral similarities discussed in this paper and has implications for turbulent heat transport models.

Correlation between small-scale velocity and scalar fluctuations in a turbulent channel flow

Direct numerical simulations of a turbulent channel flow with passive scalar transport are used to examine the relationship between small-scale velocity and scalar fields. The Reynolds number based on the friction velocity and the channel half-width is equal to 180, 395 and 640, and the molecular Prandtl number is 0.71. The focus is on the interrelationship between the components of the vorticity vector and those of the scalar derivative vector. Near the wall, there is close similarity between different components of the two vectors due to the almost perfect correspondence between the momentum and thermal streaks. With increasing distance from the wall, the magnitudes of the correlations become smaller but remain non-negligible everywhere in the channel owing to the presence of internal shear and scalar layers in the inner region and the backs of the large-scale motions in the outer region. The topology of the scalar dissipation rate, which is important for small-scale scalar mixing, is shown to be associated with the organized structures. The most preferential orientation of the scalar dissipation rate is the direction of the mean strain rate near the wall and that of the fluctuating compressive strain rate in the outer region. The latter region has many characteristics in common with several turbulent flows; viz. the dominant structures are sheetlike in form and better correlated with the energy dissipation rate than the enstrophy.

DNS of turbulent heat transfer in a channel flow with a high spatial resolution

Direct numerical simulations of turbulent heat transfer in a channel flow are performed to investigate the effects of Reynolds and Prandtl numbers on higher-order turbulence statistics such as a turbulent Prandtl number and the budget for the dissipation rate of the temperature variance. The Reynolds numbers based on the friction velocity and the channel half width are 180 and 395, and the molecular Prandtl numbers Pr’s 0.71–10.0. Careful attention is paid to ensure accuracy of the higher-order statistics through the use of a high spatial resolution comparable to Batchelor length scale. The wall-asymptotic value of the turbulent Prandtl number is mostly independent of Reynolds number for the current range of Pr’s. The budget for the dissipation rate of the temperature variance has been computed, and the negligible effect of a Reynolds number on the sum of all source and sink terms in near-wall region in the current computational range is found. This result is quite similar to the one in the budget for the dissipation rate of turbulent energy. In addition, a priori test for existing models is also performed to assess the Pr dependence on the individual terms and their summations in the budget.

Generalized lattice Boltzmann equation with forcing term for computation of wall-bounded turbulent flows

In this paper, we present a framework based on the generalized lattice Boltzmann equation (GLBE) using multiple relaxation times with forcing term for eddy capturing simulation of wall-bounded turbulent flows. Due to its flexibility in using disparate relaxation times, the GLBE is well suited to maintaining numerical stability on coarser grids and in obtaining improved solution fidelity of near-wall turbulent fluctuations. The subgrid scale (SGS) turbulence effects are represented by the standard Smagorinsky eddy viscosity model, which is modified by using the van Driest wall-damping function to account for reduction of turbulent length scales near walls. In order to be able to simulate a wider class of problems, we introduce forcing terms, which can represent the effects of general nonuniform forms of forces, in the natural moment space of the GLBE. Expressions for the strain rate tensor used in the SGS model are derived in terms of the nonequilibrium moments of the GLBE to include such forcing terms, which comprise a generalization of those presented in a recent work [Yu, Comput. Fluids 35, 957 (2006)]. Variable resolutions are introduced into this extended GLBE framework through a conservative multiblock approach. The approach, whose optimized implementation is also discussed, is assessed for two canonical flow problems bounded by walls, viz., fully developed turbulent channel flow at a shear or friction Reynolds number (Re) of 183.6 based on the channel half-width and three-dimensional (3D) shear-driven flows in a cubical cavity at a Re of 12 000 based on the side length of the cavity. Comparisons of detailed computed near-wall turbulent flow structure, given in terms of various turbulence statistics, with available data, including those from direct numerical simulations (DNS) and experiments showed good agreement. The GLBE approach also exhibited markedly better stability characteristics and avoided spurious near-wall turbulent fluctuations on coarser grids when compared with the single-relaxation-time (SRT)-based approach. Moreover, its implementation showed excellent parallel scalability on a large parallel cluster with over a thousand processors.

A note on optimal transient growth in turbulent channel flows

We compute the optimal transient growth of perturbations sustained by a turbulent channel flow following the same approach recently used by del Álamo and Jiménez [J. Fluid Mech. 559, 205 (2006)]. Contrary to this previous analysis, we use generalized Orr–Sommerfeld and Squire operators consistent with previous investigations of mean flows with variable viscosity. The optimal perturbations are streamwise vortices evolving into streamwise streaks. In accordance with del Álamo and Jiménez, it is found that for very elongated structures and for sufficiently large Reynolds numbers, the optimal energy growth presents a primary peak in the spanwise wavelength, scaling in outer units, and a secondary peak scaling in inner units and corresponding to λz+≈100. Contrary to the previous results, however, it is found that the maximum energy growth associated with the primary peak increases with the Reynolds number. This growth, in a first approximation, scales linearly with an effective Reynolds number based on the centerline velocity, the channel half width and the maximum eddy viscosity associated. The optimal streaks associated with the primary peak have an optimal spacing of λz=4h and scale in outer units in the outer region and in wall units in the near wall region, where they still have up to 50% of their maximum amplitude near y+=10.

Direct numerical simulation of sensitivity coefficients in low Reynolds number turbulent channel flow

The sensitivity equation method (SEM) has been implemented in direct numerical simulations (DNS) of smooth-wall turbulent channel flow to quantify Reynolds number effects on the mean flow field. The present approach towards sensitivity analysis represents a departure from conventional parametric studies wherein multiple numerical simulations are performed with discrete increments in the value of the parameter of interest, in this case the Reynolds number. In the present study, sensitivity derivatives (or sensitivity coefficients) represent the rate of change of the primitive variables with respect to the Reynolds number, and are determined directly by numerically solving the discretized continuous sensitivity equations concurrently with the discretized Navier–Stokes equations. Simulations have been performed at Reynolds numbers of 100 and 180, based on the friction velocity and channel half-width, using a finite volume computational scheme. Wall-normal profiles of the mean streamwise velocity and Reynolds stresses compare very well to others in the literature at the same Reynolds numbers [19,20]. The SEM results correctly predict the expected change in both the mean streamwise velocity and Reynolds shear stress profiles with increasing Reynolds number. Furthermore, the mean SEM results correctly predict the local slope of the skin friction coefficient versus Reynolds number curve, which may have potential use in applications such as boundary layer control by surface modification. The present study is novel insofar as it constitutes the first attempt to implement SEM in an unsteady, fully turbulent, three-dimensional flow field. The additional computational expenses incurred in order to run the SEM simulations in this context are also discussed.

Direct numerical simulation of turbulent channel flows using a stabilized finite element method

Direct numerical simulations (DNS) of incompressible turbulent channel flows at Re τ = 180 and 395 (i.e., Reynolds number, based on the friction velocity and channel half-width) were performed using a stabilized finite element method (FEM). These simulations have been motivated by the fact that the use of stabilized finite element methods for DNS and LES is fairly recent and thus the question of how accurately these methods capture the wide range of scales in a turbulent flow remains open. To help address this question, we present converged results of turbulent channel flows under statistical equilibrium in terms of mean velocity, mean shear stresses, root mean square velocity fluctuations, autocorrelation coefficients, one-dimensional energy spectra and balances of the transport equation for turbulent kinetic energy. These results are consistent with previously published DNS results based on a pseudo-spectral method, thereby demonstrating the accuracy of the stabilized FEM for turbulence simulations.

Channelized melt flow in downwelling mantle: Implications for 226Ra‐210Pb disequilibria in arc magmas

We present the results of an analytical model of porous flow of viscous melt into a steadily dilating “channel” (defined as a cluster of smaller veins) in downwelling subarc mantle. The model predicts the pressure drop in the mantle wedge matrix surrounding the channel needed to drive melt flow as a function of position and time. Melt is sucked toward the dilatant region at a near‐constant velocity (10−5 s−1) until veins comprising the channel stop opening (t = τ). Fluid elements that complete their journey within the time span t &lt; τ arrive at a channel. Our results make it possible to calculate the region of influence sampled by melt that surrounds the channel. This region is large compared to the model size of the channelized region driving flow. For a baseline dilation time of 1 year and channel half width of 2 m, melt can be sampled over an 80‐m radius and has the opportunity to sample matrix material with potentially contrasting chemistry on geologically short timescales. Our mechanical results are consistent with a downgoing arc mantle wedge source region where melting and melt extraction by porous flow to a channel network are sufficiently rapid to preserve source‐derived 238U‐230Th‐226Ra, and potentially also 226 Ra‐210Pb, disequilibria, prior to magma ascent to the surface. Since this is the rate‐determining step in the overall process, it allows the possibility that such short‐lived disequilibria measured in arc rocks at the surface are derived from deep in the mantle wedge. Stresses due to partial melting do not appear capable of producing the desired sucking effect, while the order of magnitude rate of shear required to drive dilation of ∼10−7 s−1 is much larger than values resulting from steady state subduction. We conclude that local deformation rates in excess of background plate tectonic rates are needed to “switch on” the dilatant channel network and to initiate the sucking effect.

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.

Turbulent structures and statistics in turbulent channel flow with two-dimensional slits

Direct numerical simulation (DNS, hereafter) of turbulent channel flow with periodic two-dimensional slits has been performed in order to investigate the turbulent statistics and the turbulent structures behind the slits. The Reynolds numbers based on the friction velocity and the channel half width are 10–1500. In the wake region, the mean flow becomes asymmetric with respect to the centerline of the geometry through the Coanda effect. Large-scale vortices are generated at the height of the slit edges. These vortices become deformed in various scenarios and break up into disordered small-scale structures in the shear layers behind the slit. The small-scale vortices are convected toward the channel center. The budgets of the Reynolds stresses have been computed. The significant differences are found between the budgets in this study and those in a backward-facing step turbulence. The positive Reynolds shear stress u ′ v ′ ¯ is observed owing to the flow contraction just behind the slit. The wake region was classified into several categories based upon the budgets of the Reynolds stresses and turbulent structures.

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.

Comparison between experiments and direct numerical simulations in a channel flow with roughness on one wall

The turbulent flow in a two-dimensional channel with roughness on one wall is investigated using experiments and direct numerical simulations (DNS). The elements have a square cross-section with height k=0.1H (H is the channel half-width) and a streamwise spacing of 4k. The Reynolds number Reτr, based on the friction velocity at the rough wall and H, is in the range 300–1100. Particular attention is given to the rough-wall side. Measured turbulence intensities, length scales, leading terms in the turbulent kinetic energy budget, and velocity spectra are compared with those obtained from the DNS. Close agreement is found, yielding support for the simplifying assumptions in the experiment (notably local isotropy and Taylor's hypothesis) and the adequacy of the spatial resolution in the simulation. Overall, the profiles of the Reynolds normal stresses on the roughness side are almost independent of Reτr, when normalized by outer variables. Energy spectra at different locations above the rough wall collapse well at high wavenumbers, when normalized by Kolmogorov scales. In contrast to previous studies, a region of negative energy production near the location of the maximum streamwise velocity is not observed. Comparison with a smooth-wall channel, at similar values of the friction-velocity Reynolds number, highlights differences only in the streamwise velocity component near the wall.

Evolutionary optimization of an anisotropic compliant surface for turbulent friction drag reduction

Direct numerical simulation (DNS) of the channel flow with an anisotropic compliant surface is performed in order to investigate its drag reduction effect in a fully developed turbulent flow. The computational domain is set to be 3δ×2δ×3δ, where δ is the channel half-width. The surface is passively driven by the pressure and wall-shear stress fluctuations, and the surface velocity provides a boundary condition for the fluid velocity field. An evolutionary optimization method (CMA-ES) is used to optimize the parameters of the anisotropic compliant surface. The optimization identifies several sets of parameters that result in a reduction of the friction drag with a maximum reduction rate of 8%. The primary mechanism for drag reduction is attributed to the decrease of the Reynolds shear stress (RSS) near the wall induced by the kinematics of the surface. The resultant wall motion is a uniform wave traveling downstream. The compliant wall, with the parameters found in the optimization study, is also tested in a computational domain that is doubled in the streamwise direction. The drag, however, is found to increase in the larger computational domain due to excessively large wall-normal velocity fluctuations.

Direct Numerical Simulation of a Spatially Advancing Turbulent Flow in a Two-Dimensional Curved Channel

Direct numerical simulation was performed for a spatially advancing turbulent flow in a two-dimensional curved channel. Fully developed turbulence generated by straight channel simulation was used as the inlet value of curved part simulation. The radius ratio of the curved part, α, was set 0.92, the same as Kobayashi et al.'s experiment. However, the frictional Reynolds number, Reτ0, was assigned 150, which was roughly quarter the experiment and allowed direct simulation of all the essential scale of turbulence. Computation was made for three cases changing domain size, and mainly discussed was the results from the computational volume extending 150° with spanwise length of 7.2 times channel half width. In this case, total number of 512×61×128 grid points was allocated. Numerically solved mean velocity showed trends consistent with the experiment in spite of the Reynolds number difference. Mean velocity normalized by the local frictional velocity indicated upward and downward offset of the logarithmic region, respectively, in the inner and outer side of the channel, but simulation showed stronger laminarization near the inner wall due to the low Reynolds number effect. Numerical data of mean velocity field illustrated that characteristic behavior of ejection from the outer wall and streamwise vortices appeared with advancement of large scale vortices. Power spectrum analysis implied that coherent structures near the outer wall were related to the birth and formation of large scale vortices.

Probability density function modeling of scalar mixing from concentrated sources in turbulent channel flow

Dispersion of a passive scalar from concentrated sources in fully developed turbulent channel flow is studied with the probability density function (PDF) method. The joint PDF of velocity, turbulent frequency and scalar concentration is represented by a large number of Lagrangian particles. A stochastic near-wall PDF model combines the generalized Langevin model of Haworth and Pope [Phys. Fluids 29, 387 (1986)] with Durbin's [J. Fluid Mech. 249, 465 (1993)] method of elliptic relaxation to provide a mathematically exact treatment of convective and viscous transport with a nonlocal representation of the near-wall Reynolds stress anisotropy. The presence of walls is incorporated through the imposition of no-slip and impermeability conditions on particles without the use of damping or wall-functions. Information on the turbulent time scale is supplied by the gamma-distribution model of van Slooten et al. [Phys. Fluids 10, 246 (1998)]. Two different micromixing models are compared that incorporate the effect of small scale mixing on the transported scalar: the widely used interaction by exchange with the mean and the interaction by exchange with the conditional mean model. Single-point velocity and concentration statistics are compared to direct numerical simulation and experimental data at Reτ=1080 based on the friction velocity and the channel half width. The joint model accurately reproduces a wide variety of conditional and unconditional statistics in both physical and composition space.

Stability of a binary mixture of a chemically reacting incompressible fluid in a plane couette flow

The stability of a binary mixture of a chemically reacting incompressible fluid in plane couette flow is considered with respect to a two-dimensional infinitesimal disturbance. The eighenvalues are found with the aid of microcomputer. Neutral stability curves for various Schmidt numbers at Reynolds numbers up to 150 are obtained by Gallagher method. It is found that the principle of the exchange of stabilities holds for this problem. Thus, using the basic orthogonal functions, the differential equations are reduced to a set of paired matrix equations with the conclusion that all fluids are unstable at different values of stability parameter C=1.40 x 10 to 2.0 x 10 for wave number (a) = 0.2 to 2 (based on half-channel width and half the relative velocity of the plates). Keywords: Incompressible fluid, binary mixture, wave number, couette flow and stability parameter. Nigerian Journal of Physics Vol. 18 (2) 2006: pp. 155-160

DNS of heat transfer in turbulent and transitional channel flow obstructed by rectangular prisms

Direct numerical simulation (DNS) of heat transfer in a channel flow obstructed by rectangular prisms has been performed for Re τ = 80–20, where Re τ is based on the friction velocity, the channel half width and the kinematic viscosity. The molecular Prandtl number is set to be 0.71. The flow remains unsteady down to Re τ = 40 owing to the disturbance induced by the prism. For Re τ = 30 and 20, the flow results in a steady laminar flow. In the vicinity of the prism, the three-dimensional complex vortices are generated and heat transfer is enhanced. The Reynolds number effect on the time-averaged vortex structure and the local Nusselt number are investigated. The mechanism of the heat transfer enhancement is discussed. In addition, the mean flow parameters such as the friction factor and the Nusselt number are examined in comparison with existing DNS and experimental data.