Accuracy Of Flow Simulations Research Articles

A simplified two-fluid model coupled with EMMS drag for gas-solid flows

A simplified two-fluid model (STFM) combined with energy-minimization multi-scale (EMMS) drag was proposed for accurate and fast simulation of gas-solid flows. In the proposed approach, the solid phase viscosity is neglected, the solid phase pressure is calculated with an empirical formulation, and the interphase momentum transfer is modeled with EMMS drag, which takes the effects of meso-scale structures into consideration. Three typical fluidization cases, namely, a 2D circulating fluidized bed, a 3D lab-scale bubbling fluidized bed, and a 3D lab-scale full-loop circulating fluidized bed, were successfully simulated with this approach. The numerical results are compared with those of full two-fluid model (FTFM, i.e., the two-fluid model using the kinetic theory for granular flow to close solid phase stress term), as well as experimental data. Predictions of STFM coupled with EMMS drag are comparable with those of FTFM coupled with EMMS drag, and both agree well with experimental data. However, computational cost of STFM is significantly reduced compared with that of FTFM. It is suggested that drag model has a dominant effect on gas-solid simulation, and the effect of solid phase stress term seems to play a minor role, demonstrating the feasibility and practicality of STFM with EMMS drag for describing the hydrodynamics of heterogeneous gas-solid flows.

Wind energy evaluation for a highly complex terrain using Computational Fluid Dynamics (CFD)

Computational Fluid Dynamics (CFD) modeling is becoming an important tool in the wind industry to study wind flow patterns. Accurate CFD simulations of wind flow are essential for the selection of wind farm locations as well as the design of appropriate wind turbines. This article validates the average wind power estimated by the state of the art CFD tool WindSim using on-site measurements from nine meteorological stations scattered around a highly complex terrain at several heights. It is known that the numerical solver is very sensitive to the wide number of computational parameters that have to be taken into consideration by the user. This paper investigates those computational parameters in details including a grid dependency test, the order of the discretization schemes, the turbulence models (Standard k-ε, k-ε with Yap corrections, RNG k-ε and Modified k-ε) and the iterative convergence criteria. The best model is employed to investigate major hot spots identified where wind farming is feasible in Mauritius with due consideration to land use and topographical requirements. Wind maps are produced at four levels which are of typical hub heights of commercial wind turbines. These maps can be used to assist in the decision-making process when locating best placements for wind farming.

Size-induced segregation during pharmaceutical particle die filling assessed by response surface methodology using discrete element method

Discrete element method (DEM) combined with response surface methodology (RSM) was carried out for investigating the segregation behaviour of binary particles during die filling of the pharmaceutical tableting machine, which comprised a horizontally moving die equipped with a vertically moving bottom punch. To achieve accurate particle flow simulations, parameters of the pharmaceutical spherical granules were calibrated using experimental bounced height, rolling distance and angle of repose. These parameters were implemented in the DEM simulations and compared with experimental high-speed camera observations and measurements of the filling ratio. Simulations and experimental results showed reasonable qualitative and quantitative agreement, validating the simulation model. The validated model was employed to evaluate the segregation of binary particle mixtures and RSM was used for analysing the DEM results of the segregation. The RSM revealed that, unlike the die velocity, small particle size, horizontal position and vertical position affected the segregation index significantly. The segregation resulted from percolation of small particles between large particles in the feed shoe and bottom back corner of the die. DEM combined with RSM approach will become significantly efficient method to understand the process for implementing the quality by design approach.

An automated multi-grid Chimera method based on the implicit hole technique

An automated multi-grid overset grid algorithm is presented for accurate simulation of viscous flows around complex configurations. The algorithm is based on the implicit hole cutting technique optimized with an overset grid construction strategy, a grid cutting criterion and a multi-level overset grid cutting method. The enhanced method is more general than the original method, while preserving the high degree of automation of the implicit hole cutting algorithm. Moreover, a mesh sequencing and multi-grid Chimera strategy is proposed to achieve convergence acceleration of the flow calculations. The present Chimera algorithm is demonstrated through the solution of the flows over the NASA Common Research Model configurations. The results show that the present mesh sequencing and multi-grid strategy in the overset grid framework are very effective in increasing convergence of solution in comparison with the standard three-level multi-grid. Wing pressure comparisons indicate the pressure distribution varies with grid solution at the shock, and a higher grid resolution improves shock definition. Abrupt reductions in lift and drag coefficients are correlated with the side-of-body separation bubble as angle of attack is increased. And various modeling and grid resolution have a strong impact on predicting the side-of-body separation. The side-of-body separation decreases with grid refinement. And the bubble size from spalart-allmaras (SA) modeling is larger than that from shear stress transport model.

A spectral difference lattice Boltzmann method for solution of inviscid compressible flows on structured grids

In this work, a spectral difference lattice Boltzmann method (SDLBM) is developed and applied for an accurate simulation of two-dimensional inviscid compressible flows on structured grids. The compressible form of the discrete Boltzmann–BGK equation is used in which multiple particle speeds have to be employed to correctly model the compressibility in a thermal fluid. Here, the 2D compressible Lattice Boltzmann (LB) model proposed by Watari is applied. The spectral difference (SD) method is implemented for the solution of the LB equation in which the particle distribution function is stored at the solution points while the fluxes are computed at the flux points for calculating the particle distribution function. For time accurate solutions, the fourth-order Runge–Kutta scheme is used to discretize the temporal term in the LB equation. Note that the procedure of implementing the SD method to solve the LB equation is nearly the same as that procedure developed in the literature for the solution of the Euler equations. The accuracy and robustness of the present solution methodology are demonstrated by simulating different benchmark compressible flow problems. A sensitivity study is also conducted to evaluate the effects of the numerical parameters and the grid size/distribution on the accuracy and performance of the solution. At first, three inviscid compressible flow problems, namely, the stationary isentropic vortex, the shock tube and the shock–vortex interaction are solved by using the SDLBM to demonstrate the accuracy and robustness of the present solution methodology. Results computed for these test cases are in good agreement with the analytical and the available numerical solutions. To more assess the accuracy and robustness of the SDLBM, a third-order finite volume LBM (FVLBM) is also developed and the solutions obtained by these two methods for the test cases simulated are thoroughly compared with each other. It is demonstrated that the SDLBM is more accurate and robust compressible inviscid flow solver that can be used for evaluating the other compressible LB-based flow solvers. As the final problem, the supersonic flow past a bump is simulated to demonstrate the accuracy and robustness of the SDLBM in reaching to a steady-state solution.

A Multi-Dimensional Virtual Characteristic Scheme for Laminar and Turbulent Incompressible Flows

A multi-dimensional virtual characteristic-based scheme (MVCB) by the aid of artificial compressibility is developed for convective fluxes in laminar and turbulent incompressible flows. The proper combinations of compatibility equations are determined to obtain primitive variables on cell interfaces. The Reynolds averaged Navier-Stokes equations joined with Spalart-Allmaras turbulent model are solved by finite volumes. This scheme is applied to the flows between parallel plates, over backward-facing step, and in square lid-driven cavity at a wide range of Reynolds numbers. A FORTRAN 90 code has been written and all the results have come out from this code. Several comparisons confirm the scheme ability in accurate simulation of flows without need to any artificial viscosity in laminar and turbulent regimes.

The Depth-Averaged Numerical Simulation of Laminar Thin-Film Flows With Capillary Waves

Thin-film flows encountered in engineering systems such as aero-engine bearing chambers often exhibit capillary waves and occur within a moderate to high Weber number range. Although the depth-averaged simulation of these thin-film flows is computationally efficient relative to traditional volume-of-fluid (VOF) methods, numerical challenges remain particularly for solutions involving capillary waves and in the higher Weber number, low surface tension range. A depth-averaged approximation of the Navier–Stokes equations has been used to explore the effect of surface tension, grid resolution, and inertia on thin-film rimming solution accuracy and numerical stability. In shock and pooling solutions where capillary ripples are present, solution stability, and accuracy are shown to be highly sensitive to surface tension. The common practice in analytical studies of enforcing unphysical low Weber number stability constraints is shown to stabilize the solution by artificially damping capillary oscillations. This approach, however, although providing stable solutions is shown to adversely affect solution accuracy. An alternative grid resolution-based stability criterion is demonstrated and used to obtain numerically stable shock and pooling solutions without recourse to unphysical surface tension values. This allows for the accurate simulation of thin-film flows with capillary waves within the constrained parameter space corresponding to physical material and flow properties. Results obtained using the proposed formulation and solution strategy show good agreement with available experimental data from literature for low Re coating flows and moderate to high Re falling wavy film flows.

An energy-dissipative remedy against carbuncle: Application to hypersonic flows around blunt bodies

The carbuncle phenomenon is a critical numerical instability preventing the accurate simulation of hypersonic flows around blunted configurations. After reviewing the known remedies to handle numerically this instability (designed, all of them, for Finite Volume methods), we present and analyze in depth the only effective cure reported in literature for Residual Distribution schemes.The shock fix, which is formulated as a locally active artificial diffusive term, depends on a single controlling parameter ϵs. Extensive testing of the fix in combination with the Residual Distribution technique allows for the determination of an optimal value for ϵs, which can be related to a local Péclet-like number.After testing the performance of the fix in combination with the Residual Distribution technique it was originally designed for, the flexibility of the fix is demonstrated by coupling it with a cell-centered Finite Volume discretization: carbuncle instabilities are equally avoided in this case.Finally, a number of limitations identified in the first testing phase lead to the derivation of a more physically consistent, energy dissipative family of carbuncle fixes, whose improved capabilities are demonstrated.

Wind flow simulations in idealized and real built environments with models of various level of complexity

The suitability of Computational Fluid Dynamics (CFD) simulations on the built environment for the purpose of estimating average roughness characteristics and for studying wind flow patterns within the environment is assessed. Urban models of various levels of complexity are considered including an empty domain, array of obstacles arranged in regular and staggered manners, in-homogeneous roughness with multiple patches, a semi-idealized built environment, and finally a real built environment. For each of the test cases, we conducted CFD simulations using RANS turbulence closure and validated the results against appropriate methods: existing empirical formulas for the homogeneous roughness case, empirical wind speed models for the in-homogeneous roughness case, and wind tunnel tests for the semi-idealized built environment case. In general, results obtained from the CFD simulations show good agreement with the corresponding validation methods, thereby, giving further evidence to the suitability of CFD simulations for built environment studies consisting of wide-ranging roughness. This work also provides a comprehensive overview of roughness modeling in CFD-from the simplest approach of modeling roughness implicitly through wall functions to the most elaborate approach of modeling roughness explicitly for the sake of accurate wind flow simulations within the built environment.

Shadowgraph visualization of condensing R134a flow through ejectors

Accurate simulation of ejector flows has been challenging due to the difficulties in modeling two-phase supersonic flows of fluids other than air or steam at high temperature and pressure conditions. Such modeling is necessary for ejector optimization, but the models currently available have yet to be validated. A transparent ejector test section is designed, fabricated, and operated in an ejector-based chiller loop using R134a as the working fluid. Undistorted visual access allows for detailed shadowgraph visualization of the motive jet in the mixing section at various degrees of condensation. High-speed imaging is used with measured temperatures and pressures at the ejector inlets to validate available analytical and CFD models. It is found that the best modeling techniques assume that the motive flow exits the motive nozzle at full thermodynamic equilibrium, and that the k-ε RNG turbulence model in ANSYS FLUENT captures these equilibrium conditions sufficiently.

A p-adaptation method for compressible flow problems using a goal-based error indicator

An accurate calculation of aerodynamic force coefficients for a given geometry is of fundamental importance for aircraft design. High-order spectral/hp element methods, which use a discontinuous Galerkin discretisation of the compressible Navier–Stokes equations, are now increasingly being used to improve the accuracy of flow simulations and thus the force coefficients. To reduce error in the calculated force coefficients whilst keeping computational cost minimal, we propose a p-adaptation method where the degree of the approximating polynomial is locally increased in the regions of the flow where low resolution is identified using a goal-based error estimator as follows.Given an objective functional such as the aerodynamic force coefficients, we use control theory to derive an adjoint problem which provides the sensitivity of the functional with respect to changes in the flow variables, and assume that these changes are represented by the local truncation error. In its final form, the goal-based error indicator represents the effect of truncation error on the objective functional, suitably weighted by the adjoint solution. Both flow governing and adjoint equations are solved by the same high-order method, where we allow the degree of the polynomial within an element to vary across the mesh.We initially calculate a steady-state solution to the governing equations using a low polynomial order and use the goal-based error indicator to identify parts of the computational domain that require improved solution accuracy which is achieved by increasing the approximation order. We demonstrate the cost-effectiveness of our method across a range of polynomial orders by considering a number of examples in two- and three-dimensions and in subsonic and transonic flow regimes. Reductions in both the number of degrees of freedom required to resolve the force coefficients to a given error, as well as the computational cost, are both observed in using the p-adaptive technique.

On the robust, flexible and consistent implementation of time domain impedance boundary conditions for compressible flow simulations

The accurate simulation of compressible flows requires the appropriate modeling of the reflection of acoustic waves at the boundaries. In the present study we discuss time domain impedance boundary conditions (TDIBC). The formulation proposed allows to impose a desired reflection coefficient at the inflow and outflow boundaries. Our formulation is an extension of the well known Navier-Stokes characteristic boundary conditions. The frequency dependent reflections at the boundaries are implemented with a state-space model in the time domain. We provide a comprehensive discussion on how such state-space models can be constructed and interpreted. This discussion shows that the state-space description allows a robust and flexible implementation. It allows to consider complex reflection coefficients and account for non-constant CFD time steps in a straight forward manner. Furthermore, we prove analytically and demonstrate numerically that the formulation proposed is consistent, i.e. the formulation ensures that the flow simulation exhibits the reflection coefficient imposed accurately, as long as the waves impinging on the boundary are plane, and it prohibits drift of the mean flow variables. Finally, the boundary conditions are tested successfully for laminar and turbulent flows.

Effect of Valve Opening/Closing Setup on Computational Fluid Dynamics Prediction of Engine Flows

In computational fluid dynamics (CFD) simulations of internal combustion engines, one of the critical modeling parameters is the valve setup. A standard workaround is to keep the valve opens at a certain clearance (minimum valve lift), while imposing a solid boundary to mimic valve closure. This method would yield a step change in valve lift during opening and closing event, and different valve event timing than hardware. Two parametric studies were performed to examine (a) the effect of the minimum valve lift and (b) the effect of grid resolution at the minimum valve lift on predicted in-cylinder flow fields in Reynolds-averaged Navier–Stokes (RANS) simulations. The simulation results were compared with the state-of-the-art particle image velocimetry (PIV) measurement from a two-valve transparent combustion chamber (TCC-3) engine. The comparisons revealed that the accuracy of flow simulation is sensitive to the choice of minimum valve lift and grid resolution in the valve seat region. In particular, the predicted in-cylinder flow field during the intake process was found to be very sensitive to the valve setup. A best practice CFD valve setup strategy is proposed as a result of these parametric studies. The proposed CFD valve setup was applied to large eddy simulation (LES) of TCC-3 engine and preliminary results showed noticeable improvement already. Further evaluation of the valve setup strategy for LES simulations is ongoing and will be reported in a separate report.

Influence of rough and smooth walls on macroscale flows in tumblers.

Walls in discrete element method simulations of granular flows are sometimes modeled as a closely packed monolayer of fixed particles, resulting in a rough wall rather than a geometrically smooth wall. An implicit assumption is that the resulting rough wall differs from a smooth wall only locally at the particle scale. Here we test this assumption by considering the impact of the wall roughness at the periphery of the flowing layer on the flow of monodisperse particles in a rotating spherical tumbler. We find that varying the wall roughness significantly alters average particle trajectories even far from the wall. Rough walls induce greater poleward axial drift of particles near the flowing layer surface but decrease the curvature of the trajectories. Increasing the volume fill level in the tumbler has little effect on the axial drift for rough walls but increases the drift while reducing curvature of the particle trajectories for smooth walls. The mechanism for these effects is related to the degree of local slip at the bounding wall, which alters the flowing layer thickness near the walls, affecting the particle trajectories even far from the walls near the equator of the tumbler. Thus, the proper choice of wall conditions is important in the accurate simulation of granular flows, even far from the bounding wall.

A Numerical Model for Storm Surges that Involve the Inundation of Complex Landscapes

A numerical model for the prediction of storm surges over complex landscapes is assembled using state-of-the-art techniques. The model is based on the full nonlinear depth- averaged long wave equations solved via an explicit finite volume scheme. The interfacial fluxes are computed from slope limited reconstructions of the state variables using Osher's approximate Riemann solver. The geometric source terms are treated in a simple manner that is well balanced and consistent with the numerical flux. The numerical solution technique has been chosen to enable the accurate simulation of flows over complex topography. Importantly, the scheme conserves both mass and momentum even when the flow involves wetting—drying fronts. This approach makes the model extremely robust in cases that involve bathymetries which include large spatial gradients and even discontinuities. Another important feature of the model is the use of a simple underlying Cartesian mesh with quad-tree-based static and dynamic refinement. This permits the simulation of highly unsteady flows over complex landscapes (including localized features such as canals) by locally increasing (or relaxing) grid resolution in a dynamic fashion. If it is deemed desirable, the numerical and physical boundaries can be aligned. This alignment is achieved by a cut-cell-type approach in combination with quad-tree-based mesh refinement. In this paper, we describe the model and then present a number of tests that have been carefully chosen to verify various aspects of the solution algorithm.

A path-conservative finite volume scheme for compressible multi-phase flows with surface tension

The accurate simulation of compressible multi-phase flows with surface tension effects is currently still one of the most challenging problems in computational fluid dynamics (CFD). The basic difficulties are the capturing of the correct interface dynamics between the two fluids as well as the computation of the interface curvature. In this paper, we present a novel path-conservative finite volume discretization of the continuum surface force method (CSF) of Brackbill et al. to account for the surface tension effect due to curvature of the phase interface. This is achieved in the context of a diffuse interface approach, based on the seven equation Baer–Nunziato model of compressible multi-phase flows. Such diffuse interface methods for compressible multi-phase flows including capillary effects have first been proposed by Perigaud and Saurel. In the CSF method, the surface tension effect is replaced by a volume force, which is usually integrated as a classical volume source term. However, since this source term contains the gradient of a color function that is convected with the flow velocity, we propose to integrate the CSF source term as a non-conservative product and not simply as a source term, following the ideas on path-conservative finite volume schemes put forward by Castro and Parés.For that purpose, we use the new generalized Osher-type Riemann solver (DOT), recently proposed by Dumbser and Toro and compare it with a path-conservative Roe and Rusanov scheme. Via numerical evidence we can show that if the curvature computation is exact, our new scheme is well-balanced for a steady circular bubble in equilibrium according to the Young–Laplace law. This means that the pressure jump term across the interface in the momentum equation is exactly balanced with the surface tension force.We apply our scheme to several one- and two-dimensional test problems, including 1D Riemann problems, an oscillating droplet, as well as a deforming droplet initially attached to a wall and which is subject to gravity and surface tension forces. Finally, we also check the rising of a gas bubble due to buoyancy forces. We compare our numerical results with those published previously in the literature.

CFD simulation of wind flow over natural complex terrain: Case study with validation by field measurements for Ria de Ferrol, Galicia, Spain

Accurate and reliable Computational Fluid Dynamics (CFD) simulations of wind flow over natural complex terrain are important for a wide range of applications including dispersion of pollutants, wind energy resource assessment and ship manoeuvring in channels and near harbours. In the past 50 years, a very large number of CFD studies of wind flow over hills have been performed. However, a detailed review of the literature shows a lack of CFD studies including validation by field measurements for natural complex terrain beyond the case of isolated hills. Therefore, this paper presents a CFD study with field measurement validation for natural complex terrain that consists of an irregular succession of hills and valleys surrounding a narrow entrance channel. The aim of the study is twofold: (1) to evaluate the accuracy of 3D steady Reynolds-averaged Navier–Stokes (RANS) simulations with a revised k–ε model for calculating mean wind-velocity patterns over this type of natural complex terrain; and (2) to provide mean velocity data that can be used as input for real-time ship manoeuvring simulations to evaluate accessing the LNG terminal with larger LNG carriers. The irregular hilly terrain is expected to yield complex wind environmental conditions in the channel and complex forces on the LNG carriers. The study focuses on high wind speed conditions, for which the atmospheric boundary layer exhibits neutral stratification. The simulations are performed with 3D steady RANS and the realisable k–ε model for 12 wind directions. Special attention is given to surface roughness parameterisation and specification. The simulation results of mean wind speed and wind direction are generally within 10–20% of the corresponding measurement values. The results show that for wind directions 60° and 90°, the funnelling effect leads to an increase of wind speed in the channel compared to the wind speed over open sea. For other wind directions, the topography leads to a reduction of the wind speed in the channel, but also to strong wind speed gradients along the channel axis, which are important for ship manoeuvring. The study shows that for the present application, the 3D steady RANS approach with the realisable k–ε model can provide an accurate assessment of the complex mean wind-flow patterns and the funnelling effect by the natural complex topography on the wind.

Comparative study for modeling reactor internal geometry in CFD simulation of PWR and PHWR internal flow – Nuclear regulatory perspective

In this study, in order to examine the effect of reactor internal geometry modeling method on the prediction accuracy for PWR (Pressurized Water Reactor) and PHWR (Pressurized Heavy Water Reactor) internal flow distribution, simulations with real geometry modeling were conducted with the commercial CFD (Computational Fluid Dynamics) software, ANSYS CFX R.14 and FLUENT R.14. The predicted results were compared with those of the porous medium assumption and the measured data. Test cases were 1/5 scaled-down APR+ (Advanced Power Reactor Plus) model for PWR and moderator test vessel (Stern Laboratories Inc.) for PHWR. It was concluded that core inlet flow distribution could be predicted more accurately by considering the real geometry of the scale-down APR + internal structures, located in the upstream of core inlet. Therefore, if the sufficient computation resource is available, an exact representation of these internal structures, for examples lower support structure bottom plate and ICI (In-Core Instrumentation) nozzle support plate, is essential for the accurate simulation of PWR internal flow. For PHWR, an approach to consider the real geometry of tubes predicted the local velocity and temperature distribution better than the Porous Medium Model. However, the benefits (e.g. improvements of prediction accuracy) against the computation cost were small; compared with PWR case. The reason may be that relatively simple PHWR internal structure can make it possible to establish a generally applicable correlation for the pressure loss coefficient.

Heterogeneous computing on mixed unstructured grids with PyFR

PyFR is an open-source high-order accurate computational fluid dynamics solver for unstructured grids. In this paper we detail how PyFR has been extended to run on mixed element meshes, and a range of hardware platforms, including heterogeneous multi-node systems. Performance of our implementation is benchmarked using pure hexahedral and mixed prismatic-tetrahedral meshes of the void space around a circular cylinder. Specifically, for each mesh performance is assessed at various orders of accuracy on three different hardware platforms; an NVIDIA Tesla K40c GPU, an Intel Xeon E5-2697 v2 CPU, and an AMD FirePro W9100 GPU. Performance is then assessed on a heterogeneous multi-node system constructed from a mix of the aforementioned hardware. Results demonstrate that PyFR achieves performance portability across various hardware platforms. In particular, the ability of PyFR to target individual platforms with their ‘native’ language leads to significantly enhanced performance cf. targeting each platform with OpenCL alone. PyFR is also found to be performant on the heterogeneous multi-node system, achieving a significant fraction of the available FLOP/s. Finally, each mesh is used to undertake nominally fifth-order accurate long-time simulations of unsteady flow over a circular cylinder at a Reynolds number of 3900 using a cluster of NVIDIA K20c GPUs. Long-time dynamics of the wake are studied in detail, and results are found to be in excellent agreement with previous experimental/numerical data. All results were obtained with PyFR v0.2.2, which is freely available under a 3-Clause New Style BSD license (see www.pyfr.org).

Validation of a closing procedure for fourth-order RANS turbulence models with DNS data in an incompressible zero-pressure-gradient turbulent boundary layer

Among factors affecting the accuracy of flow simulations with Reynolds-Averaged Navier–Stokes turbulence models is modeling turbulent diffusion processes. With the use of the Gram–Charlier series expansions, the turbulent diffusion in fourth-order one-point statistical closures of the Reynolds-Averaged Navier–Stokes equations can be modeled without introducing unknown model coefficients and without assuming turbulence being Gaussian. Terms representing turbulent diffusion processes in transport equations for second- and third-order velocity correlations do not require any modeling in such closures. In this regard, fourth-order closures are a more accurate alternative to lower-order closures where turbulent diffusion is modeled on semi-empirical or Gaussian turbulence assumptions. In the current paper, the accuracy of the closing procedure based on the Gram–Charlier series expansions is evaluated using data of direct numerical simulations in an incompressible zero-pressure-gradient turbulent boundary layer over a flat plate. One-point third-, fourth-, and fifth-order velocity moments were extracted for this purpose from the dataset collected by the Fluid Dynamics Group at the Universidad Politécnica de Madrid at two streamwise locations Reθ=4101 and 5200 that correspond to channels and pipes at δ+=1331 and 1626. Results demonstrate that the truncated Gram–Charlier series expansions are an accurate approximation of the fifth-order velocity moments in the considered flow.