Plane Channel Flow Research Articles

Direct and large-eddy simulation of turbulent flows on composite multi-resolution grids by the lattice Boltzmann method

In order to properly address the simulation of complex (weakly compressible) turbulent flows, the lattice Boltzmann method, originally designed for uniform structured grids, needs to be extended to composite multi-domain grids displaying various levels of spatial resolution. Therefore, physical conditions must be specified to determine the mapping of statistical information (about the populations of moving particles) at the interface between two domains of different resolutions. quite simply in terms of the probability distributions of the underlying discrete-velocity Boltzmann equation. Namely, the continuity of the mass density and fluid momentum is fulfilled by imposing the continuity of the equilibrium part of these distributions, whereas the discontinuity of the rate-of-strain tensor is ensured by applying a “spatial transformation” to the collision term of the discrete-velocity Boltzmann equation. The latter condition allows us to explicitly account for the subgrid-scale modeling in the treatment of resolution changes.Test computations of a turbulent plane-channel flow have been carried out. The lattice Boltzmann scheme relies on the standard D3Q19 lattice in a cell-vertex representation, and uses the BGK approximation for the collision term. A shear-improved variant of the Smagorinsky viscosity is used to account for possible subgrid-scale dynamics. In a quasi-direct numerical simulation at Reτ=180 with two levels of grid resolution, the results are found in excellent agreement with reference data obtained by a high-resolution pseudo-spectral simulation. In a Large-Eddy Simulation (LES) at Reτ=395 with three levels of grid resolution, the results compare reasonably well with high-resolution reference data. The accuracy is improved in comparison with a standard finite-volume LES performed with the same subgrid-scale viscosity and comparable grid resolution.By combining the simplicity and physical relevance of the shear-improved Smagorinsky model and the computational efficiency of the lattice Boltzmann scheme, here extended to composite multi-resolution grids, we believe that our proposal offers a high potential for the simulation of complex turbulent flows.

A fully divergence-free method for generation of inhomogeneous and anisotropic turbulence with large spatial variation

A fully divergence-free method is proposed for generation of inhomogeneous/anisotropic turbulence with large spatial variation. The method is based on the method of Smirnov et al., which is known to violate the divergence-free constraint when spatial variation of turbulence is present. In the proposed method a vector potential field is introduced; by taking the vector curl of the potential field one can generate a strictly divergence-free flow field. A novel formulation for scaling the vector potential field, together with a coordinate transformation strategy, is proposed in this work. The result is a six-step procedure for the generation of inhomogeneous turbulence fields. The proposed formulation is proven to reproduce the prescribed velocity correlation with energy spectrum at large Reynolds numbers. Four test cases are considered to evaluate the new method. First, the statistical quantities introduced in the proposed method are verified numerically in a classical homogeneous turbulence case. Then, the performance of the new method is demonstrated in three different inhomogeneous turbulence cases: a confined turbulent flow in a “slip-wall” box, a planar channel flow and an annular flow. It is shown that the unphysical patterns in the fluctuation fields and divergence errors produced with Smirnovʼs method are absent in the results from the new method. The accuracy of the proposed methods is verified by comparing the velocity correlations with the prescribed scaling functions. The proposed method is efficient and simple to implement.

Laminarization mechanisms and extreme-amplitude states in rapidly rotating plane channel flow

AbstractFully developed plane channel flow rotating in the spanwise direction has been studied analytically and numerically. Linear stability analysis reveals that all cross-flow modes are stable for supercritical rotation numbers, $Ro\gt R{o}_{c} $, where $R{o}_{c} $ will approach 3 from below for increasing Reynolds number. Plane Tollmien–Schlichting (TS) waves are independent of rotation and always linearly unstable for supercritical Reynolds numbers. Direct numerical simulation (DNS) of the laminarization process reveals that the turbulence is damped when $Ro$ approaches $R{o}_{c} $. Hence, the laminarization is dominated by linear mechanisms. The flow becomes periodic for supercritical Reynolds numbers and rotation rates, i.e. when $Ro\gt R{o}_{c} $ and $Re\gt R{e}_{c} $. At such rotation rates, all oblique (cross-flow) modes are damped and when the disturbance amplitude becomes small enough, the TS modes start to grow exponentially. Secondary instabilities are initially blocked by the rotation since all cross-flow modes are linearly stable and the breakdown to turbulence will be strongly delayed. Hence, the TS waves will reach extremely high amplitudes, much higher than for typical turbulent fluctuations. Eventually, the extreme-amplitude state with TS-like waves will break down to turbulence and the flow will laminarize due to the influence of the rapid rotation, thus completing the cycle that will then be repeated. This flow is strongly dominated by linear mechanisms, which is remarkable considering the extremely high amplitudes involved in the processes of laminarization of the turbulence at $Ro\geq R{o}_{c} $ and the growth of the unstable TS waves.

Large-eddy simulation of turbulent flows in a heated streamwise rotating channel

In this paper, large-eddy simulation has been performed to investigate the impacts of the centrifugal and Coriolis forces on a heated plane channel flow subjected to streamwise system rotations. In order to study the rotation effects, a variety of rotation numbers Roτ ranging from 0 to 15 have been tested in conjunction with a fixed low Reynolds number Reτ=300. The subgrid-scale effects on the budget balance of turbulent stresses and heat fluxes have been investigated, and the physical mechanisms underlying the transport of resolved turbulent stresses under the influence of streamwise system rotations have been thoroughly analyzed. Numerical simulations were performed using two dynamic subgrid-scale stress models and two dynamic subgrid-scale heat flux models; namely, the conventional dynamic model (DM) and an advanced dynamic nonlinear model (DNM) for closure of the filtered momentum equation, and the conventional dynamic eddy thermal diffusivity model (DEDM) and an advanced dynamic full linear tensor thermal diffusivity model (DFLTDM) for closure of the filtered thermal energy equation. The predictive performances of the DM and DNM have been carefully compared in conjunction with the same subgrid-scale heat flux model DFLTDM. In order to validate the numerical approach, turbulent statistics obtained from the simulations have been compared with the available direct numerical simulation data.

Divergence-free turbulence inflow conditions for large-eddy simulations with incompressible flow solvers

Synthetic turbulence for inflow conditions formulated on a 2-D plane generally produces unphysically large pressure fluctuations in direct numerical and large-eddy simulations. To reduce such artificial fluctuations a divergence-free method is developed with incompressible flow solvers. The procedure of the velocity–pressure solvers is slightly modified on a vertical plane near (rather than at) the inlet by inserting the synthetic turbulence on that plane during the procedure. Simple analytic and numerical error estimations are used to show that the impact of the modified solvers on solution accuracy is small. The final synthetic turbulence satisfies the divergence-free condition. No additional CPU time is required to achieve this condition. The method was tested via simulations of a plane channel flow with Reτ=395. Reynolds stresses, wall skin friction and power spectra of velocity fluctuations are compared with those obtained from using periodic inlet–outlet boundary conditions. In particular, the variances and power spectra of pressure fluctuations are shown to be accurately predicted only when the divergence-free inlet condition is used.

Development of an explicit algebraic turbulence model for buoyant flows – Part 1: DNS analysis

An analysis of DNS databases of vertical plane channel flow for forced, mixed and natural convection is proposed. This analysis aims to assess the main features needed to develop an algebraic model for buoyant flows. First, the weak equilibrium assumption, at the root of algebraic models, is investigated. This hypothesis is shown to fail near the velocity maximum and close to the walls but remains valid otherwise, whatever the convection regime. The models for the redistribution term and the pressure scrambling term are then analyzed on the same configurations. A linear form of the Speziale et al. (1991) model is retained for the redistribution term. No model for the pressure scrambling term is fully satisfactory; nevertheless some models are recommended. The buoyant contributions to the pressure term are investigated. Finally, the generalized gradient diffusion hypothesis, which could be used to model the turbulent heat fluxes in order to avoid the coupling with the Reynolds stresses, is shown to be inaccurate.

A numerical study of solidification and viscous dissipation effects on polymer melt flow in plane channels

Abstract The combined effects of solidification and viscous dissipation on the hydrodynamic and thermal behavior of polymer melt flow during the injection process in a straight plane channel of constant cross section are numerically studied by considering the shear-rate and temperature-dependent viscosity and transient-phase change behavior. A numerical finite volume method, in conjunction with a modified form of the Cross constitutive equation to account for shear rate, temperature-dependent viscosity changes and a slightly modified form of the method proposed by Voller and Prakash to account for solidification of the liquid phase, is used and a validation with an analytical solution is presented for viscous heating effects. The hydrodynamic and solidified layers growth under the influence of a transient phase-change process and viscous dissipation, are analyzed for a commercial polymer melt flow, polypropylene (PP) for different parametric conditions namely, inflow velocity, polymer injection (inflow) temperature, the channel wall temperature, and the channel height. The results demonstrate that the proposed numerical formulations, including conjugate effects of viscous heating and transient-solidification on the present thermal transport process, can provide an accurate and realistic representation of polymer melt flow behavior during the injection molding process in plane channels with less simplifying assumptions.

A Direct Numerical Simulation Method for Flow of Brownian Fiber Suspensions in Complex Geometries

A two-way coupled, direct simulation technique is proposed for the numerical solution of Brownian fiber suspension flows in complex geometries. The isothermal, incompressible, non-Newtonian Navier-Stokes equations are solved in an Eulerian framework using the finite volume method for the spatial discretization and a third-order Runge-Kutta scheme for the time integration. A conservative immersed boundary method is employed for the treatment of complex geometries. The fibers are treated in a Lagrangian manner. Therefore, complex geometries are retrieved naturally. The conformation of fibers is obtained by solving Jeffery's equation for an ensemble of rigid fibers. Brownian motion is simulated by a three-dimensional Wiener process. The proposed method does not require a moment closure model. The simulator is validated in a plane channel flow and a cylinder flow at the limit of extremely strong Brownian motion. Then, we use it to solve four problems, that is, a circular cylinder in a cross flow, the flow in a channel with periodic constrictions, the flow in a 4:1 contraction channel and the flow in a rectangular pipe with cylindrical constrictions.

Wall effects on pressure fluctuations in turbulent channel flow

AbstractThe purpose of the present paper is to study the influence of wall echo on pressure fluctuations${p}^{\prime } $, and on statistical correlations containing${p}^{\prime } $, namely, redistribution${\phi }_{ij} $, pressure diffusion${ d}_{ij}^{(p)} $and velocity pressure-gradient${\Pi }_{ij} $. We extend the usual analysis of turbulent correlations containing pressure fluctuations in wall-bounded direct numerical simulations (Kim,J. Fluid Mech., vol. 205, 1989, pp. 421–451), separating${p}^{\prime } $not only into rapid${ p}_{(r)}^{\prime } $and slow${ p}_{(s)}^{\prime } $parts (Chou,Q. Appl. Maths, vol. 3, 1945, pp. 38–54), but also further into volume (${ p}_{(r; \mathfrak{V})}^{\prime } $and${ p}_{(s; \mathfrak{V})}^{\prime } $) and surface (wall echo,${ p}_{(r; w)}^{\prime } $and${ p}_{(s; w)}^{\prime } $) terms. An algorithm, based on a Green’s function approach, is developed to compute the above splittings for various correlations containing pressure fluctuations (redistribution, pressure diffusion, velocity pressure-gradient), in fully developed turbulent plane channel flow. This exact analysis confirms previous results based on a method-of-images approximation (Manceau, Wang & Laurence,J. Fluid Mech., vol. 438, 2001, pp. 307–338) showing that, at the wall,${ p}_{(\mathfrak{V})}^{\prime } $and${ p}_{(w)}^{\prime } $are usually of the same sign and approximately equal. The above results are then used to study the contribution of each mechanism to the pressure correlations in low-Reynolds-number plane channel flow, and to discuss standard second-moment-closure modelling practices.

A perturbative model for predicting the high‐Reynolds‐number behaviour of the streamwise travelling waves technique in turbulent drag reduction

AbstractThe background of this work is the problem of reducing the aerodynamic turbulent friction drag, which is an important source of energy waste in innumerable technological fields (transportation being probably the most important). We develop a theoretical framework aimed at predicting the behaviour of existing drag reduction techniques when used at the large values of the Reynolds numbers Re which are typical of applications. We focus on one recently proposed and very promising technique, which consists in creating at the wall streamwise‐travelling waves of spanwise velocity (M. Quadrio, P. Ricco, and C. Viotti, J. Fluid Mech. 627, 161–178, 2009). A perturbation analysis of the Navier‐Stokes equations that govern the fluid motion is carried out, for the simplest wall‐bounded flow geometry, i.e. the plane channel flow. The streamwise base flow is perturbed by the spanwise time‐varying base flow induced by the travelling waves. An asymptotic expansion is then carried out with respect to the velocity amplitude of the travelling wave. The analysis, although based on several assumptions, leads to predictions of drag reduction that agree well with the measurements available in literature and mostly computed through Direct Numerical Simulations (DNS) of the full Navier–Stokes equations. New DNS data are produced on purpose in this work to validate our method further. The method is then applied to predict the drag‐reducing performance of the streamwise‐travelling waves at increasing Re, where comparison data are not available. The current belief, based on a Re‐range of about one decade only above the transitional value, that drag reduction obtained at low Re is deemed to decrease as Re is increased is fully confirmed by our results. From a quantitative standpoint, however, our outlook based on several decades of increase in Re is much less pessimistic than other existing estimates, and motivates further, more accurate studies on the present subject.

Normal Mode Analysis of the High Speed Channel Flow in a Bidisperse Porous Medium

The linear Darcy–Brinkman model of the high speed flow in a bidisperse porous medium proposed by Nield and Kuznetsov (Transport Phenomena in Porous Media, 2005) is revisited in this paper. For the steady unidirectional flow in a parallel plane channel the exact analytical solutions for the fluid velocities are worked out by the normal-mode reduction of the governing equations. The limiting cases of the weak and strong momentum transfer between the flows in the fracture and porous phases are discussed in detail. A comparison to the nonlinear Forchheimer extension of the model proposed recently by Nield and Kuznetsov (Transport Porous Media, 2013) shows that, in the considered parameter range, the nonlinear effect of the Forchheimer drag is negligibly small. Even the simplest zero-momentum transfer solution yields an acceptable approximation.

Computational Modeling of the Effects of Viscous Dissipation on Polymer Melt Flow Behavior During Injection Molding Process in Plane Channels

The present finite volume method based fluid flow solutions investigate the boundary-layer flow and heat transfer characteristics of polymer melt flow in a rectangular plane channel in the presence of the effect of viscous dissipation and heat transfer by considering the viscosity and density variations in the flow. For different inflow velocity boundary conditions and the injection polymer melt temperatures, the viscous dissipation effects on the velocity and temperature distributions are studied extensively to analyze the degree of interactions of thermal flow field dominated by the viscous heating and momentum diffusion mechanism with varying boundary conditions. The modified forms of Cross constitutive equation and Tait equation of state are adopted for the representation of viscosity variations and density change, respectively, in the polymer melt flow. These models together with the viscous dissipation terms are successfully incorporated into the finite volume method based fluid flow solutions to realistically represent the heat effects in the plane channel. The numerical results presented for two different commercial polymer melt flows, namely, polymer Polyacetal POM-M90-44 and polypropylene (PP), demonstrate that proposed mathematical formulations for viscosity and density variations including viscous heating terms into the energy equations, which are fully coupled with momentum equations, lead to more accurate representation of the fluid flow and heat transfer phenomena for the polymer melt flows in plane channels.

An improved anisotropy-resolving subgrid-scale model with the aid of a scale-similarity modeling concept

An improved subgrid-scale (SGS) model was proposed by combining an isotropic linear eddy-viscosity term with an extra anisotropic term. In the present study, primary attention was given to maintaining the computational stability while improving the predictive performance particularly for coarse grid resolution in the near-wall region. For the extra anisotropic term used for this purpose, the present study introduced a residual term after subtracting an eddy-viscosity form from the Bardina SGS-Reynolds-stress model [Bardina, J., Ferziger, J.H., Reynolds, W.C., 1980. Improved subgrid scale models for large eddy simulation. AIAA Paper 80-1357]. The resultant extra term yields no undesirable extra energy transfer between the grid-scale and SGS components that could cause numerical instability under coarse grid conditions. Therefore, this extra term is not expected to have any serious negative effects on the computational stability. In order to assess the performance, the proposed model was applied to the numerical simulation of fully-developed plane channel flows with various grid resolutions and at various Reynolds numbers. The computational results were considerably improved by the present SGS model and detailed investigations of the obtained results indicated the usefulness of the present model for engineering applications.

EDDY VISCOSITY PROFILES FOR WAVE BOUNDARY LAYERS: VALIDATION AND CALIBRATION BY A k-ω MODEL

Eddy viscosity in wave boundary layers is a key parameter in coastal engineering. Two analytical eddy viscosity profiles present a particular interest for practical applications: the parabolic-uniform profile (Myrhaug 1982, van Rijn 1993, Liu and Sato 2006) and the exponential-linear profile (Gelfenbaum and Smith 1986, Beach and Sternberg 1988, Hsu and Jan 1998, Absi 2010). The aim of our study is to assess and validate these two profiles by: (1) investigation of eddy viscosity in steady fully developed plane channel flow; (2) comparisons with numerical results of the two equation baseline (BSL) k-ω model (Menter 1994, Suntoyo and Tanaka 2009). Our study shows that these two profiles are able to describe the eddy viscosity distribution in the wave bottom boundary layer but for different wave conditions given by the parameter am/ks, where am is the wave orbital amplitude and ks the equivalent roughness. The exponential-linear profile is adequate for am/ks

Experimental investigation of relaminarizing and transitional channel flows

A hot-wire measurement was conducted in a planar channel flow that originated from a strongly disturbed flow in an entrance channel followed by an expansion channel used to reduce the Reynolds number (Re). From ceasing decrease of the streamwise velocity fluctuation energy and the linear extrapolation of the intermittency factor, the lower marginal Re, which is defined as the minimum Re for partial existence of sustainable turbulence, is estimated around 1400 based on the channel width and the bulk velocity. The upper marginal Re at which the intermittency factor reaches one is about 2600. The flow fields passing a turbulent patch were reconstructed with conditional sampling of the streamwise velocity data based on the time of laminar-turbulence interfaces and the reconstructed flow fields indicate a large-scale flow structure across laminar and turbulent parts. This large structure makes it possible for some regions to be at higher Re than the average, so that turbulence can partly survive. The moderate-scale disturbances larger than the turbulent one appear in the non-turbulent parts of the transitional flow, and in these cases the non-turbulent velocity profile is almost identical to the turbulent one. The large-scale fluctuation is observed even over Re = 2600. This leads to the conclusion that a turbulent channel flow close to the upper marginal Re becomes inhomogeneous.

On the evaluation of linear and non-linear models using DNS data of turbulent channel flows

In this paper, a priori and a posteriori analyses of algebraic linear and non-linear models are carried out in order to compare their ability to predict near wall turbulent flows. Tests were done using data from a direct numerical simulation (DNS) of a plane channel flow for three Reynolds numbers, based on the friction velocity, Reτ = 180, Reτ = 395 and Reτ = 590 . These models include the linear standard k - e model, the linear v2- f (Manceau et al., 2002) and the non-linear model of Shih (Shih et al., 1995). The results obtained are then compared with the DNS data of Moser et al. (1999). The comparisons are shown for the mean velocity profile, components of the Reynolds stress tensor, the turbulent kinetic energy (k), and the dissipation rate (e). The results suggest that the v2 - f is an efficient model to capture the turbulent shear stress component of the Reynolds stress near wall flows. However, it is unable to predict correctly the level of anisotropy between normal components of the Reynolds stress tensor. Furthermore, it is shown that the presence of non-linear terms in a turbulent model improves the ability to predict the anisotropy

Development Length in Planar Channel Flows of Newtonian Fluids Under the Influence of Wall Slip

This technical brief presents a numerical study regarding the required development length (L=Lfd/H) to reach fully developed flow conditions at the entrance of a planar channel for Newtonian fluids under the influence of slip boundary conditions. The linear Navier slip law is used with the dimensionless slip coefficient k¯l=kl(μ/H), varying in the range 0<k¯l≤1. The simulations were carried out for low Reynolds number flows in the range 0<Re≤100, making use of a rigorous mesh refinement with an accuracy error below 1%. The development length is found to be a nonmonotonic function of the slip velocity coefficient, increasing up to k¯l≈0.1-0.4 (depending on Re) and decreasing for higher k¯l. We present a new nonlinear relationship between L, Re, and k¯l that can accurately predict the development length for Newtonian fluid flows with slip velocity at the wall for Re of up to 100 and k¯l up to 1.

Comparative study of turbulent mass transfer in the viscous sublayer using electrochemical method and direct numerical simulations

In this study we present, analyze and compare the power spectral density of the wall shear stress in a turbulent plane channel flow obtained with different techniques. Experimentally the instantaneous wall shear stress was measured with the electrochemical technique using different probes, which give approximately the same results after applying the transfer function for correction of the probe’s inertia. Numerically, the time evolution of the wall shear stress has been determined using direct numerical simulations (DNS) and large eddy simulations (LES). The results of DNS are in a good agreement with the electrochemical flow measurements. However the power spectra of the wall shear stress obtained with LES shows deviations with respect to DNS at high frequencies because of the spatial filtering inherent to the LES technique.

Performance analysis of parallel Schwarz preconditioners in the LES of turbulent channel flows

We present a comparative study of parallel Schwarz preconditioners in the solution of linear systems arising in a Large Eddy Simulation (LES) procedure for turbulent plane channel flows. This procedure applies a time-splitting technique to suitably filtered Navier–Stokes equations, in order to decouple the continuity and momentum equations, and uses a semi-implicit scheme for time integration and finite volumes for space discretisation. This approach requires the solution of four sparse linear systems at each time step, accounting for a large part of the overall simulation; hence the linear system solvers are a crucial component in the whole procedure. Several preconditioners are applied in the simulation of a reference test case for the LES community, using discretisation grids of different sizes, with the aim of analysing the effects of different algorithmic choices defining the preconditioners, and identifying the most effective ones for the selected problem. The preconditioners, coupled with the GMRES method, are run within SParC-LES, a recently developed LES code based on the PSBLAS and MLD2P4 libraries for parallel sparse matrix computations and preconditioning.

A nonlinear weighted least-squares finite element method for the Oldroyd-B viscoelastic flow

This paper studies a nonlinear weighted least-squares finite element method for the steady Oldroyd-B viscoelastic flow. Our least-squares functional involves the L2-norm of the residuals of each equation multiplied by a proper weight. The weights include mass conservation constant, mesh dependent weight, and a nonlinear weighting function. We prove that the least-squares approximation converges to the solutions of the linearized versions of the viscoelastic fluid model at the best possible rate and then present the planar channel flow problem illustrating our theoretical results. Results of the least-squares approach indicate that with carefully chosen nonlinear weighting functions and linear basis functions, the numerical solution exhibits a second order convergence rate for velocity and viscous stress and superlinear convergence rate in polymeric stress and pressure. The method is applied to the 4-to-1 planar contraction problem. The effects of Weissenberg numbers are also presented in the work.