Mean Drag Force Research Articles

Double Averaging Analysis Applied to a Large Eddy Simulation of Coupled Turbulent Overlying and Porewater Flow

AbstractFreestream turbulence in rivers is a key contributor to the flux of dissolved nutrients, carbon, and other ecologically important solutes into porewater. To advance understanding of turbulent hyporheic exchange and porewater transport, we investigate flow over and through a rough bed of spheres using large eddy simulation (LES). We apply double averaging (combined space and time averaging) to the LES results to determine the mean velocity distribution, momentum balance, and drag forces. Our simulations show large‐scale freestream structures interacting strongly with vortices generated at the surfaces of individual spheres to control turbulent momentum fluxes into the bed. The transition between turbulent flow and Darcy flow occurs over the first row of spheres, where turbulence decays rapidly and turbulent kinetic energy, Reynolds stress, and drag forces peak. Below this region, turbulence is only present in the high‐velocity flow in open pore throats. Experimental observations suggest that minimum mean porewater velocity occurs in the first open pore space below the transition region, but our results show that the minimum occurs between the first and second pore spaces. The simulation mean porewater velocities are approximately half those captured in measurements because the model resolves the entire flow continuum while measurements can access high‐velocity fluid in open pores. The high‐resolution dual time‐space averaging of the LES resolves both turbulent and mean flow features that are important to interfacial solute and particle fluxes, providing a means to include turbulent hyporheic exchange in upscaled river models, which has not been achieved to date.

Transient simulation of an atmospheric boundary layer flow past a heliostat using the Scale-Adaptive Simulation turbulence model

Heliostat fields are exposed to changing climatic conditions as they are mostly erected in open environments where the wind naturally features a high unsteadiness at low altitude due to the ground effects. Much of the computational fluid dynamics (CFD) content in the open literature is focused on Reynolds–averaged-Navier–Stokes (RANS) simulations, which can only predict mean loads. This paper considers an isolated heliostat in worst-case orientation. The drag force is numerically modelled by means of a Scale-Resolving Simulation (SRS) in ANSYS v19. This paper firstly deals with two different methods that generate perturbations at the inlet boundary: the spectral synthesiser and the vortex method. In an empty domain, an atmospheric boundary layer (ABL) profile is modelled based on a wind tunnel experiment. Secondly, the wind tunnel test of a single heliostat model in upright orientation is replicated, aiming to model the mean and peak drag forces. Applicable for highly separated flows, the Scale-Adaptive Simulation (SAS) turbulence model is employed as it is computationally more affordable than a Detached Eddy Simulation (DES) approach. The latter would require a higher grid resolution and a reduced time step size. The SAS showed little but acceptable decay of the inlet profiles whilst achieving lateral homogeneity. The mean and root-mean-square error of the drag force signal showed a deviation with the experiment of 0.04% and 5.8%, respectively, whereas the error on the peak drag forces was around 18%, possibly mostly due to the under-prediction of the turbulent integral length scale at the model location.

Quantification of aerodynamic forces for truss bridge-girders based on wind tunnel test and kriging surrogate model

This study presents an investigation to quantify the aerodynamics of truss bridge-girders via wind tunnel test and kriging surrogate model. Currently, the conventional methods documented in design specifications only take into consideration the mean drag force at null attack angle. To gain an in-depth understanding on the aerodynamics of truss bridge-girders, experiments on simplified bridge-girder models with various geometric parameters were carried out in uniform flow. A total of 15 truss bridge-girder models with aspect ratio (the ratio of width to height) B/D = 1.0, 1.3, 1.6, 1.9, and 2.2, solidity ratio (the ratio of projected to envelope areas) Φ = 0.20, 0.25, 0.30, 0.35, and 0.40, and two typical truss topologies (Warren and Pratt trusses) were examined in the most concerned range of wind angle of attack α = [–6°, 6°]. These truss bridge-girder models cover most of the high-speed railway bridges widely used in China. Experimental results show that the truss topology has limited effects on the aerodynamics of truss bridge-girders, whereas the effects of α, B/D, and Φ are significant. Based on these wind tunnel results, the ordinary kriging surrogate model was utilized to approximate the aerodynamics of truss bridge-girders. In using this model, aerodynamic force values for test cases can be interpolated with zero variance and uncertainties in unsampled design zones where geometric parameters can be quantified with Gaussian variance.

On the passive control of aeroacoustics noise behind a square cylinder

This study investigated the application of a splitter plate as a passive control method for reducing the aerodynamic sound of square cylinders. To control the noise and vortex shedding, the splitter plates attached to the downstream of the square cylinder have been used. The flow is assumed to be unsteady and it is simulated using the URANS equations model k-ω-SST. Aerodynamic noise calculations are carried out by the use of F-fowcs Williams-Hawkins analogy. To verify numerical results, the sound pressure level (SPL) obtained in different situations (Microphone position) is compared with the experimental results, and a good agreement is observed. The simulation is carried out with different lengths of the splitter, and the results can be divided into three groups besides the baseline configuration as well. In the first group, when the length of the splitter varies from 0.2 to 2D, noise is noticeably reduced by about 23 dB. In the second group, the length of the splitter increases from 3 to 5D, and the SPL is increased by about 42 dB. In the third group, the length of the splitter varies from 6 to 8D, which in turn the noise is decreased by about 4 dB (Compared to the second group). Results show that adding the splitter reduces the mean drag force generally. But the changes in the lift fluctuations are similar to the SPL variations. The results reveal that the optimal splitter length through which noise, RMS lift, and drag coefficients are reduced, occurs for a Splitter in the first category (i.e., L = 0.2–2D).

Fluid–particle drag and particle–particle drag in low-Reynolds-number bidisperse gas–solid suspensions

Particle-resolved direct numerical simulations (PR-DNSs) of dynamic bidisperse gas–solid suspensions are performed at low particle Reynolds numbers. Unlike the fixed-bed suspensions, the mobility of particles allows particles of different size types to develop different slip velocities relative to the fluid phase. The scaled slip velocity, defined as the ratio of the slip velocity of one particle type to the mean slip velocity of the mixture, varies profoundly depending on the specific properties of the bidisperse mixture. For large particles, the drag force, scaled by the mean drag force of the mixture, is reasonably predicted by the models obtained from fixed-bed suspensions, while for small particles, these models tend to underestimate the scaled drag force as the scaled slip velocity decreases. By introducing the scaled slip velocity, a new model for the fluid–particle drag on each particle type is proposed and agrees well with the PR-DNS data. For the situation where the monodisperse drag models are employed to predict the mixture mean drag force, a new mean diameter that is variant with the total solid volume fraction is suggested. This diameter increases as the total solid volume fraction decreases and approaches the Sauter mean diameter in the close-packed volume fraction. In dilute suspensions, due to the strong influence of surrounding fluids on the particle phase, the simulated particle–particle drag is significantly smaller than the predictions of models based on kinetic theory of granular flow. Based on the PR-DNS results, new relations for particle–particle drag are also proposed.

Shear-induced lift force on spheres in a viscous linear shear flow at finite volume fractions

Several studies have shown a significant increase in drag on a distribution of solid spherical particles within a fluid with increasing particle volume fraction. As a result, many empirical drag laws accounting for the dependence on the Reynolds number and volume fraction can be found in the literature. This study investigates the possibility of a similar effect of the particle volume fraction on the mean hydrodynamic lift force on randomly distributed spherical particles in a linear shear flow. Particle-resolved direct numerical simulations are performed to evaluate the mean lift force, and the results are compared with the case of an isolated particle in a linear shear flow for the same Reynolds number and shear rate. The mean lift force acting on the particles appears to remain nearly the same as that on an isolated particle. However, due to the influence of neighboring particles, there is a substantial force variation in transverse directions on each individual particle, whose magnitude is comparable to the mean drag force. The distribution of drag force in a linear shear flow is shown to be nearly the same as in a uniform flow at the same volume fraction and Reynolds number. A simple stochastic model based on a Gaussian distribution is presented for the lift force variation, and its performance is compared to the prediction of the deterministic pairwise interaction extended point-particle model.

Modeling drag force in ellipsoidal particle suspensions with preferential orientation

High aspect ratio ellipsoidal particle suspensions exhibit preferential orientation under the action of external forces and in the presence of walls. The preferential orientation of the particles has a large effect on the drag force experienced by the suspension. In this paper particle resolved simulations are conducted on ellipsoidal particles of aspect ratio 10 and 2.5 in the Reynolds number range 10 ≤ Re ≤ 200 and solid fractions 0.10 ≤ φ ≤ 0.30. An orientational order parameter −2.0 ≤ S ≤ 1 is used to define the preferential orientation of the suspension. It is shown that in spite of up to 400% variation in mean drag force with S, a simple linear relationship between S = −2 and 1 can describe the variation of drag force. This is a significant finding which establishes that only two data points at S = −2 and 1 are needed to define mean drag at any preferential orientation exhibited by the suspension. It is also shown that the inclination dependent mean drag is quite independent of S at low Reynolds number and solid fraction but shows increasing dependence at higher values.

Development of drag correlation for suspensions of ellipsoidal particles

To model drag the current state-of-the-art is to use isolated non-spherical particle drag correlations modified by a solid fraction correlation that is based on experimental or simulation results of spherical particle suspensions. It is shown that this practice can lead to substantial inaccuracies when the particle geometry deviates significantly from a spherical geometry. In this paper particle resolved simulations (PRS) are conducted for ellipsoids of aspect ratio 5 (AR5) and 10 (AR10) in random suspensions with no preferential orientation. Simulations are performed for a Reynolds number Re = 10 to 200, and solid fraction φ = 0.1 to 0.3 and 0.1 to 0.2 for AR5 and AR10 suspensions, respectively. Combined with PRS data from past studies for spherical particle suspensions and ellipsoids with AR2.5, a drag correlation is developed for the mean drag force in suspension as a function of Re, φ , aspect ratio, and inclination angle θ.

Lattice Boltzmann investigation on fluid flows through packed beds: Interaction between fluid rheology and bed properties

Non-Newtonian fluid flows through packed beds are common in many industries. Our understanding of this flow system is very limited, and the correlations for describing the fluid-particle interaction are not fully established. To overcome these problems, this paper presents a comprehensive study of this system on a sub-particle scale, with a special reference to the interaction between fluid rheology and bed properties. This is done by conducting about five hundred Lattice Boltzmann simulations under different conditions. The fluid rheology is represented by the power-law model to consider the shear-thinning, shear thickening and Newtonian behaviors of fluids. The simulation condition covers a wide range of bed porosity, particle size distribution and Reynolds number (Re). The results show that the effect of fluid rheology on the fluid behavior is strong. This effect varies significantly with bed porosity which is a function of particle size distribution. The interplay between fluid rheology and bed properties is however not strong in determining the distributions of particle-fluid interaction force. Based on the simulation data, a new drag correlation is established and validated against the experimental data in the literature. This correlation is more accurate and consistent than those reported in the past. It can estimate the mean drag forces on individual particles of different sizes, and is recommended to be used generally in the modeling of particle-fluid flows either for Newtonian fluids or for non-Newtonian fluids obeying the power-law model.

Application of Helical Strakes for Suppressing the Flow-Induced Vibration of Two Side-by-Side Long Flexible Cylinders

Helical strakes have been widely applied for suppressing the vibration of flexible cylinders undergoing vortex-shedding in offshore engineering. However, most research works have concerned on the application of helical strakes for the isolated flexible cylinder subjected to vortex-induced vibration (VIV). The effectiveness of helical strakes attached to side-by-side flexible cylinders in vibration reduction is still unclear. In this paper, the response characteristics of two side-by-side flexible cylinders with and without helical strakes were experimentally investigated in a towing tank. The configuration of the helical strakes used in the experiment had a pitch of 17.5D and a height of 0.25D (where D is the cylinder diameter), which is usually considered the most effective for VIV suppression of isolated marine risers and tendons. The center-to-center distance of the two cylinders was 3.0D. The uniform flow with a velocity ranging from 0.05 m/s to 1.0 m/s was generated by towing the cylinder models along the tank. Experimental results, including the displacement amplitude, the dominant frequency, the dominant mode, and the mean drag force coefficient, were summarized and discussed. For the case where only one cylinder in the two-cylinder system had helical strakes, the experimental results indicated that helical strakes can remarkably reduce the flow-induced vibration (FIV) of the staked cylinder. For the case of two straked cylinders in a side-by-side arrangement, it was found that the performance of helical strakes in suppressing the FIV is as good as that for the isolated cylinder.

Direct Numerical Simulations of Horizontally Oblique Flows Past Three-Dimensional Circular Cylinder Near a Plane Boundary

Abstract Understanding hydrodynamics of a free-spanning pipeline subjected to omni-directional flows is important to engineering design. In this study, horizontally oblique flows past a three-dimensional circular cylinder in the vicinity of a plane boundary are numerically investigated using direct numerical simulations. Parametric studies are carried out at the normal Reynolds number of 500, a fixed gap-to-diameter ratio of 0.8 and five flow inclination angles (α) ranging from 0 deg to 60 deg with an increment of 15 deg. Two distinct vortex-shedding modes are observed: parallel (α ≤ 15 deg) and oblique (α ≥ 30 deg) vortex shedding. The wake evolution is further divided into two or three stages depending on α. The occurrence of the oblique vortex shedding is accompanied by the base pressure gradient along the cylinder span and the resultant axial flows near the cylinder base. The total hydrodynamic drag and lift force coefficients decrease from being the parallel mode to the oblique mode, owing to the intensified three-dimensionality of wake flows and the phase differences in the spanwise vortex shedding. The independence principle (IP) is found to be valid in predicting hydrodynamic forces and wake patterns when α ≤ 15 deg. This IP might produce unacceptable errors when α > 15 deg. In comparison with the mean drag force, the fluctuating lift force is more sensitive to the inclination angle. The IP validity range is substantially smaller than that in the case of flow past a wall-free cylinder. Such finding would be practically useful for vortex-induced vibration prediction.

Numerical Simulation of Drag Reduction on a Square Rod Detached with Two Control Rods at Various Gap Spacing via Lattice Boltzmann Method

Numerical simulations are performed to examine the effect of size of control rods (d1) and spacing ratio (g) on flow around a square rod with upstream and downstream control rods aligned in-line using the lattice Boltzmann method (LBM). The Reynolds number (Re) is fixed at Re = 160, while the spacing between the main rod and control rods is taken in the range 1 ≤ g ≤ 5 and the size of the control rod is varied between 4 and 20. Seven different types of flow mods are observed in this study at different values of g and d1. Variation in force statistics, like mean drag coefficient (Cdmean), Strouhal number (St), root mean square values of drag (Cdrms) and lift coefficients (Clrms), and percentage reduction in mean drag coefficient is discussed in detail. It was examined that vortex shedding completely suppressed at (g, d1) = (1, 12), (2, 12), and (2, 16) where steady flow mode exists. Moreover, it was found that at large gap spacing, where g = 5, the effect of control rods on the main rod vanishes. Due to this strong vortex shedding produced and as a result, maximum value of Cdmean is found at (g, d1) = (5, 8). The negative values of mean drag force are also observed at some gap spacing and size of control rods are due to the effect of thrust. Furthermore, the maximum percentage reduction in Cdmean is 121%, found at (g, d1) = (2, 20).

Crosswind aerodynamic characteristics of a stationary interior railway carriage through a long-span truss-girder bridge

This paper investigates the aerodynamics of a stationary model of an interior railway carriage inside a long-span truss-girder bridge in a wind tunnel. Effects of principle parameters of the truss bridge-girder configuration on the aerodynamics were examined, including bridge wind angle of attack (α), truss bridge-girder solidity ratio (Φ, projected area to envelope area ratio), and aspect ratio (B/D, truss bridge-girder width to height ratio). By varying α, the tested pressure distribution of the carriage model was analyzed in terms of the underlying flow physics of the carriage-bridge interaction. Consequently, the schematic flow patterns were inferred and subsequently utilized to expound the variations in the mean and fluctuating aerodynamic force coefficients. The results indicate that qualitatively the aerodynamic interaction of the truss bridge-girder on the carriage model is characterized by the shielding effect caused by upstream truss members, the suppression of underbody vortex-shedding due to the girder lower deck, and the quasi-Reynolds number effect on the upstream carriage shoulder. Compared to the results of the carriage model without the truss bridge-girder, the shielding effect reduces the mean drag force, whereas the suppression of underbody vortex shedding decreases the mean rolling moment and all the fluctuating forces. The quasi-Reynolds number effect surfaces as α varies given that the inflow Reynolds number is held constant, thus altering the aerodynamic forces abruptly. For the variation of the other two truss bridge-girder configuration parameters (Φ and B/D), qualitatively similar carriage-bridge interaction on the carriage aerodynamics was observed, although with some quantitative differences. The imperfect spanwise correlation of sectional aerodynamics was also observed as mostly owing to the shielding effect formed by the diagonal truss members. In addition, the influence of carriage-bridge interaction on the bridge aerodynamics was also examined and briefly discussed.

Aerodynamic forces and three-dimensional flow structures in the mean wake of a surface-mounted finite-height square prism

Different flow models have been proposed for the flow around surface-mounted finite-height square prisms, but there is still a lack of consensus about the origin and connection of the streamwise tip vortices with the other elements of the wake. This numerical study was performed to address this gap, in addition to clarifying the relationship of the near-wake structures with the far wake and the near-wall flow, which is associated with the fluid forces. A large-eddy simulation approach was adopted to solve the flow around a surface-mounted finite-height square prism with an aspect ratio of AR = 3 and a Reynolds number Re = 500. The mean drag and normal forces and the bending moment for the prism were quantitatively compared in terms of skin-friction and pressure contributions, and related to the near-wall flow. Both three-dimensional visualizations and planar projections of the time-averaged flow field were used to identify, qualitatively, the main structures of the wake, including the horseshoe vortex, corner vortices and regions of high streamwise vorticity in the upper part of the wake. These features showed the same qualitative behavior as reported in high Reynolds number studies. It was found that some regions of high streamwise vorticity magnitude, like the tip vortices, are associated with the three-dimensional bending of the flow, and the tip vortices did not continuously extend to the free end of the prism. The three-dimensional flow analysis, which integrated different observations of the flow field around surface-mounted finite-height square prisms, also revealed that the mean near-wake structure is composed of two sections of different origin and location of dominance.

Flow-induced vibrations of two side-by-side circular cylinders at low Reynolds numbers

Flow-induced vibrations of two side-by-side circular cylinders are numerically studied using the immersed boundary method for Reynolds numbers of 60–200. The two cylinders are constrained to vibrate only in the transverse direction. The center-to-center spacing ratio (s/D) is in the range 2.0–5.0, and the reduced velocity (Ur) is in the range 2.0–10.0. The smallest increments for s/D and Ur are 0.02 and 0.01, respectively. It was found that the vibration amplitudes, mean position shifts, lift and drag forces, spectral frequencies, and phase lags between the lift and displacement are significantly influenced by s/D and Ur, supported by (a) a wider lock-in region compared to the case of a single cylinder, (b) emergence of asymmetric vibration where the amplitudes of two cylinders are not identical, (c) the phase jump between the lift and displacement resulting from multiple harmonic frequencies, and (d) a significant drop in mean position shift accompanied by the changes in the frequency or wake pattern. Furthermore, the influences of the spacing ratio, Reynolds number, blockage ratio, and mass ratio, on asymmetric vibration with single-sided hysteresis (AV-I), are examined in detail. The AV-I with double-sided hysteresis was confirmed for the first time when the cylinder mass ratio is larger than 5.0.

Nonlinear dynamics of viscoelastic pipes conveying fluid placed within a uniform external cross flow

This paper investigates the nonlinear dynamic response of a viscoelastic pipe conveying fluid subjected to a uniform external cross flow based on the Euler-Bernoulli theory. The main objective of this work is to find the proper viscoelastic coefficients to mitigate the dynamic response of a marine riser. A nonlinear oscillator is utilized to simulate the mean drag force and the vortex-induced lift force. Also, the pipe material is assumed to be viscoelastic and consisted of the Kelvin-Voigt type. The extended Hamilton's principle along with the Galerkin discretization are employed to construct the nonlinear model of the coupled fluid-structure system. Moreover, the assumed mode method along with the Runge-Kutta integration algorithm is used as a solution method. The results from the developed numerical model are verified with those available in the literature. Then, the effects of internal fluid velocity, external fluid reduced velocity and viscoelastic damping on the vibration amplitude of the pipe are examined. It is noted that, when lock-in phenomena occur, the internal fluid velocity and viscoelastic damping have vital effects on the vibration amplitude and the lock-in region range of the pipe. It is however demonstrated that employing a proper viscoelastic material may lead to eliminating jump phenomenon.

Modulation of aerodynamic characteristics of a finite wall-mounted square cylinder through steady jet injection

This paper presents the results of an experimental study that investigates the flow behavior around a finite wall-mounted square cylinder (with the aspect ratio of AR = 7) subjected to a steady jet issued from the cylinder’s front face. Two wavy and continuous jets were considered to examine the jet velocity distribution effect. The wind tunnel measurements consist of smoke flow visualization, mean drag force, mean surface pressure, and velocity data obtained from hotwire probe, mostly at Reynolds number (Red, based on the cylinder width and inlet velocity) of 1.2 × 104. The ratio of the jet flow to the freestream, Г, was changed as 0 ≤ Г < 5. Both the wavy and continuous jets reduce the downwash flow contribution with the flow from the lateral faces. An increase of Г continuously amplifies this effect, and improves the flow control performance in drag reduction. The maximum pressure drag reduction of 29% and 16% were obtained for the wavy and continuous jets, respectively. Two characteristic continuous jet flow modes, namely deflected jet (Г < 1) and deflected oscillating jet (1 < Г < 2), were identified. The both continuous jet deflection mode have a leading role in the pressure reduction on the front face, and subsequently in drag reduction. Oscillation of the deflected continuous jet at high-Г also significantly increases the vortex shedding frequency in the upper half of the cylinder, and makes the mean wake flow asymmetrical. On the other hand, the base pressure recovery is the major reason for the drag reduction when the wavy jet is applied. The wavy jet shows a slight deflection at high-Г; however, the mean wake flow remains symmetrical. Finally, the influence of jet flow control on the flow topology was discussed.

Aerodynamics of a scale model of a high-speed train on a streamlined deck in cross winds

The cross wind aerodynamics of a high-speed train (HST) on a streamlined deck was investigated using sectional models in a wind tunnel. Aerodynamics of the HST-only model was also measured as a benchmark under similar conditions. The experimental results indicate that when the HST is on the downstream track of the deck it experiences significant changes in its aerodynamics depending on the angle of attack (α) of the oncoming flow. With an increase in α beyond zero, the mean drag and mean lift forces on HST begin to decrease or increase rapidly. This observation is attributed to the variation in the flow regime around HST from subcritical to critical although the approaching flow Reynolds number (based on the free-stream oncoming flow velocity U∞ and the height of train d) remains unchanged. The separated shear flow from the leading edge of the bridge impinges on the upstream shoulder of the HST on the downstream track at a critical orientation of the approach flow angle. The resulting accelerated flow in the shear layer, characterized by higher turbulence intensity, prompts the flow regime transition near the upstream shoulder of the HST without a change in the inflow Reynolds number. This change in the pressure field over the upstream shoulder of HST when it is on the downstream track on the bridge bears strong similarity to the changes experienced by circular cross-sections or cross-sections with rounded corners with a change in the inflow Reynolds number.

Vortex induced vibration of a circular cylinder with a filament by using penalty immersed boundary method

In this paper, vortex induced vibration of a circular cylinder with an attached filament is numerically studied by a Penalty Immersed boundary method (PIBM). In the current work, we study the effects of a single filament on the flow wake of a fixed cylinder, as well as its dynamic response when it is free to move transversely. Two key parameters, the bending coefficient (Kb) and the length (L) of the filament are considered. We report that for a static cylinder, by increasing the length or the rigidity of the filament, both the mean drag force and the fluctuation amplitude of the lift force on the cylinder can be reduced. When the cylinder is free to move, it is found that the existence of a filament actually enhances the vibration of the cylinder by comparing to that of a bare cylinder, differentiating from the prediction of lift fluctuation on a fixed cylinder. In the meanwhile, it leads to a broader resonance region on the vibration response of the cylinder with respect to the reduced velocity. Moreover, various flow patterns are identified in the wake of cylinder for different filament lengths.

Numerical simulation of solitary wave attenuation by vegetation with non-hydrostatic model

In this paper solitary wave attenuation by vegetation is investigated by using numerical simulation. To take into account effects of dispersion, we use a dispersive and nonlinear wave mode; the non-hydrostatic with 1 vertical layer. The damping by vegetation is modelled by using mean drag force using Morison’s formula. The dissipative term is added in the momentum equation. The wave model is implemented numerically using finite volume with momentum conservative staggered grid scheme. To test the numerical scheme implementation, we simulate a physical experiment of solitary wave attenuation by vegetation of Huang et al. 2011. Comparison with experimental data shows that the results of simulation agree quite well.