comparison of different models for evaluating the velocity .... basic. appl...ramana prasada [18]...

J. Basic. Appl. Sci. Res., 3(12)273-288, 2013

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ISSN 2090-4304 Journal of Basic and Applied

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*Corresponding Author: Hossein Bonakdari, Department of Civil Engineering, Faculty of Engineering, Razi University, Kermanshah, Iran. Email: [email protected]

Comparison of Different Models for Evaluating the Velocity Profiles in Narrow Sewers

Hossein Bonakdari1,2*, Mohammad Saeed Ahadi2

1Department of Civil Engineering, Faculty of Engineering, Razi University, Kermanshah, Iran 2Water and Wastewater Research Center, Razi University, Kermanshah, Iran

Received: October 15 2013 Accepted: November 13 2013

ABSTRACT This study provides a comparison of various existing methods for computing the longitudinal velocity profile distribution in narrow open channels. Four models for computing velocity profile in narrow channels were selected such as Chiu’s, Sarma’s, Yang’s and Bonakdari’s model. A comparison was made between the experimental data obtained from an actual measurement site in narrow sewer. A wide range of longitudinal velocity profiles for narrow open channels were used to validate and compare these models. The comparison indicated that all models provide reliable results in the central region of the channel. However, the Yang’s model gives the best prediction of the velocity near the sidewalls. KEYWORDS: Navier– Stokes, flow, discharge, turbulence, open channel.

1. INTRODUCTION The velocity distribution is one of the most important issues in the study of flow in open channels and is

necessary for determining mean and maximum velocities, flow rate or discharge, shear stress on the bottom and sidewalls, and sediment transport [1-3]. As a result, the longitudinal velocity profile of flow has been studied in channel cross sections over a long period of time. Prandtl [4] carried out theoretical studies on velocity distribution. Von Karman [5] presented experimental studies, which showed the relations between velocity distribution and hydraulic resistance in turbulent flows on smooth surfaces in circular pipes. The experimental studies of Nikuradse [6] have led to rational formulas for velocity distribution and hydraulic resistance for turbulent flows over flat plates and in circular pipes. These formulas were first extended to open-channel flow by Keulegan [7]. Perston [8] presented an indirect method to determine the shear stress distribution on a wetted perimeter of a channel and its relation with velocity distribution; Thereafter, Bagnold [9] and Coles [10] improved logarithmic law, and some other researchers such as Chow [11] and Henderson [12] studied velocity distribution more comprehensively. These laws show a positive gradient of velocity in the flow depth and the maximum velocity occurs near the free surface.

During 1980s, the development and use of laser technology enabled more accurate measurements of the flow velocity components. The three-dimensional turbulent structure and turbulence-induced secondary currents in open channels have been studied experimentally by many investigators [13-17]. Nezu and Rodi [13] found that in narrow channels with an aspect ratio less than 5 (Ar < 5 = the ratio of channel width to flow depth < 5), the maximum velocity occurs below the free surface, which is called the dip phenomenon or negative gradient of velocity next to the free surface. Ramana Prasada [18] showed that the dip phenomenon position is sensitive to aspect ratio. Nezu et al., [19] showed that a dip phenomenon exists in a cross-section with with an Ar of 2.5 in any velocity field.

Various authors have obtained results based on their simplification of the relevant equations to describe the velocity profiles in narrow channels [20-23]. Chiu [24-27] made use of probability concepts and the entropy maximization law and presented a velocity distribution equation which can determine velocity distribution in vertical and lateral directions. Sarma et al., [28] suggested some other laws by dividing the cross-section into four regions and determining velocity distribution equations in each region in order to show the longitudinal velocity profile over the entire cross-section. Yang et al., [29] proposed a new law that is capable of describing the dip phenomenon and was applicable to the velocity profile in the region from the near-bed region to just below the free surface, and transversely, from the center line to the near-wall region of the narrow channel. Bonakdari [30, 31] presented a new formulation of the vertical velocity profile in the central portion of the steady, fully developed, turbulent open-channel flows, based on an analysis of the Navier-Stokes equations.

The main objectives of this study are; (1) to provide a quantitative assessment of these four models,

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suitable for describing the velocity profile in the outer region and, transversely, from the center line to the wall region in narrow sewers, and (2) to analyze the experimental data based on different methods to demonstrate their reliability. Finally, a comparison is made between the experimental data and the predicted results, using a distinctive method for obtaining the velocity profile in a narrow open channel, calculating the relative error of each model. 2. Field data

The experimental data collected from a real site have been used in this study in order to compare different models for the evaluation of velocity profile in the narrow channel. 2.1 Experimental site

The chosen experimental site is located in a region called Cordon Bleu, a few kilometers upstream from the treatment plant on the main sewer line of the City of Nantes (northwestern France). All the wastewater generated by a population of around 600,000 from the northern part of the urban district is conveyed through this line. The cross section corresponds to a narrow section made of an egg-shaped channel with a bank. The maximum vertical dimension of the sewer is DC = 2.86 m (see Figure 1).

Figure 1: Scheme of cross section.

2.2 Velocity measuring equipment

An experimental device, Cerberes, has been developed by Larrarte [32] to permit operations from the ground level in order to measure the velocity field in the entire cross-section (see Figure 2). Cerberes allows up to three simultaneous measurements along a vertical profile with three acoustic Doppler velocimeters located every 20 cm on the vertical jack. These sensors are remote-controlled from the ground level. Once the velocities have been simultaneously measured at 3 points, the carriage is moved 10 cm to perform 3 additional measurements. When the velocities are measured on a vertical profile, the carriage is shifted to the next profile location and so on. Thus, it becomes possible to scan the entire wetted area from the bottom to a maximum level of 1.5 m (which encompasses roughly 90% of the situations encountered during a given year). With Cerberes, five to ten minutes are required to obtain a vertical velocity profile (depending on the water level and clogging, with greater frequency when levels are high). The duration of mapping increases linearly with the maximum water level.

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Figure 2: “Cerberes”: the 2D remote controlled device during measurement. 3. Review of models

Four models for the computation of the velocity distribution flow in narrow open channel are compared against the experimental data, obtained from a narrow compound channel with turbulent flow. These four models were selected because they provide an approach sufficiently general to compute the velocity distribution and dip phenomenon, and because they are simple enough for engineering applications. 3.1 Chiu's model

The velocity distribution that results from entropy maximization is more accurate and useful than the logarithmic velocity distribution and power laws and is defined as follows: by applying the probability concept coupled with the principle of maximum information entropy, Chiu [24-26], a velocity distribution equation for fluid flows is analytically derived, the general form of which is:

Dz1e1

Muu Mmax (1)

where u = flow velocity in longitudinal direction, umax = maximum velocity in a channel cross section, M = entropy parameter, z = vertical coordinate parameter and D = water depth. The validation of Eq. (1) as velocity distribution law and its relation with the logarithmic velocity distribution law is determined by investigating the experimental information [33]. This includes the measured velocities near the channel bed with different roughness indices.

To consider the dip phenomenon, this model uses a new independent coordinate system, . It should be mentioned that in narrow channels where the dip phenomenon exists, the relation of the velocity and vertical coordinate parameter (z) is not one to one (Eq. 1); therefore, defining this coordinate system for each , there is a velocity measure and, as a result, it is possible to use the related entropy principle, changing the calculations and mathematical concepts [34]. Figure 3 shows the coordinate axis and the parameters of the model.

Figure 3: Patterns of velocity distribution and independent coordinate system [24].

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Using this coordinate system, the velocity distribution equation, Eq. (1), in narrow channels becomes:

0dip

0M

max1e1

M1

uu (2)

where 0 is a place where u = 0 and occurs at the bottom of rectangular channels, also dip is a place where u = umax. This model not only shows a velocity distribution similar to the logarithmic distribution near the bottom, but also represents the dip phenomenon in narrow channels. However, this model needs the evaluation of the entropy parameter. This parameter can be evaluated from the pairs of umean and umax , measured at the channel section. The entropy parameter remains constant for each channel while the velocity distribution varies [35-39].

Although the well-know logarithmic law of Prandtl-Von Karman

0

*

zzlnuu (z0 = distance at which the

velocity is hypothetically equal to zero, [4, 5]) might be similar to Eq. (2), it does not estimate the actual amount of velocity in the vicinity of the boundaries near the bed and the free water surface (where the velocity gradient is negative). 3.2 Sarma's model

Sarma et al., [28] fitted the velocity distribution using a combination of the log law of the inner region with the binary law of the outer region. In this model, the flow is divided into four regions, and according to the logarithmic or power laws, there is a velocity distribution for each region, which are modified. Figure 4 shows these regions in any flow section [40].

Region 1: The velocity distribution vertically in the wall region of the bed is known to follow the logarithmic law or equally well the one-seventh power law. The velocity distribution horizontally in this region is assumed to follow the parabolic law.

2wy,zz,C Y1CUU (3)

in which z*z,Cz,C u/uU , z,Cu = velocity at a distance of z in a central vertical, z*u = shear velocity on the

sidewall at a distance of z from the bed, z*y,z u/uU , b/y2Y and wC is a constant.

Figure 4: Four regions defined in the Sarma's model.

Region 2: The flow in this corner region is influenced by both the bed and the sidewalls. The proposed model assumes that the log-law and one-seventh power law are valid independently in both vertical and horizontal directions. Region 3: The velocity distribution in the vertical direction in this region requires modification for taking into account the phenomenon of dip. The proposed modified form is:

2ybz,ym,y ZD1CUU (4)

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in which y*m,ym,y u/uU , m,yu = the maximum velocity in the vertical direction at a distance of y from the

sidewall, d/zZ , h/DDy the dimensionless depth below the free surface, at which the maximum velocity occurs and where Cb is a constant. The velocity distribution in the horizontal direction is given by either the logarithmic law or the one-seventh power law. Region 4: This region is outside the inner regions of both the bed and sidewalls. The vertical and horizontal velocity profiles are described by Eq.(4) and Eq.(3), respectively [28]. However, the analysis would require a priori knowledge of the velocity distribution. Thus, good agreement between the measured velocity profiles and Sarma's binary law is not unexpected, because the measured velocity profiles were used to determine the coefficients in the binary law. 3.3 Yang's model

Yang et al. [29] suggested a modified logarithmic law of velocity distribution for the dip phenomenon in open channels. This new model is able to define the dip phenomenon and the velocity profile from the bottom to the water surface and from the central axis to the sidewalls of the channel in a lateral direction. It is found that the velocity deviation from the log law is linearly proportional to the logarithmic distance, D/z1ln , from the free surface. The dip-modified log law consists of two logarithmic distances, one from the bed, /zuln z* , and the other from the free surface, D/z1ln , respectively, and a dip-correction factor, α. Yang et al. [29] suggested the following equation:

Dz1ln

zzln1

uu

0* (5)

in which = Karman constant; 5.0* gRSu ; z*0 cu/z distance at which the velocity is hypothetically

equal to zero; and c = constant. In this model the dip correction factor ( ) value is very important. Based on the measured data, these authors proposed a relationship between and the aspect ratio (Ar). If 0 , Eq. (5) reverts to the classical log law. The second term on the right-hand side of Eq. (5) plays an important role in the outer region, but it is negligible in the inner region, since 0D/z1ln . 3.4 Bonakdari's model

Bonakdari [30, 31], based on an analysis of the Navier-Stokes equations, developed a new formulation to compute the vertical velocity profile in the central region of steady, fully developed, turbulent flows in open channels. This formulation is able to predict time-averaged primary velocity in the outer region of the turbulent boundary layer for both narrow and wide-open channels in the central region of an open channel. The proposed law is a modification of the well-known logarithmic law, which involves an additional parameter, depending on the position of the maximum velocity and the roughness height [31]. Using the non-slip boundary condition, (u = 0) at 0 , where 10 , the distribution of the mean velocity in the central channel, and out of the inner region, ( > 0.2), can be determined by:

0

22

*lnAC25.0

AC5.01

uu (6)

where U is the longitudinal velocity, *u shear velocity at the bottom, 2*u

ghS ,

Dz

and C(A) is a

parameter. This parameter is defined by an analytical equation between the dip phenomenon location, (dip), and C(A) for the different values of the roughness height, (Ks), by a derivative from Eq. (6). The evolution of the C(A) parameter with the dip phenomenon location for different roughness heights is given in Figure 5.

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Figure 5: Evolution of the C(A) parameter versus dip phenomenon location.

The main advantage of this expression is that it is general and can be calculated in all cases from the

physical features of the channel (roughness, slope, etc.) and on the coefficient C(A). However, this model needs the value of the dip phenomenon position, ( dip ), which depends on the aspect ratio [41]. There is no physical

relation between Ar and dip. Wang et al., [42] have fitted experimental results with an empirical relation and given the location of the dip phenomenon for the central vertical profile in a narrow cross section by :

Ar6.2

2sin05.0Ar106.044.0dip with Ar < 5.2 (7)

For this relation, the dip phenomenon is located at =0.68 when Ar is equal to 2, whereas the results of Nezu and Rodi [13] give a maximum velocity at =0.5. Moreover, this relation underestimates the experimental results obtained from the Cordon Bleu experimental site. Yang et al. [29] have used full of measurements in a rectangular smooth open channel and proposed, for the central vertical profile in a narrow cross section, the following equation:

2Arexp2.11

1dip

(8)

In order to fit the experimental results obtained in the literature [13, 17, 32, 43, 44], a sigmoid model as follows can be proposed, (see Fig. 6):

3

3

C4

C21

dipArC

ArCC

(9)

where the four constant parameters are : C1 = 42.4, C2 = 1.0, C3 = 4.2, C4 = 94.7. Figure 6 shows the dip phenomenon as a function of the aspect ratio.

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Figure 6: Dip phenomenon position as a function of the aspect ratio, Ar.

Moreover, it is important to note that the proposed equation (8) gives an asymptotical trend (for Ar > 6), which is in agreement with the physical principles of flow in open channels. Effectively, when the channel becomes wide, the dip phenomenon takes place in a position in the vicinity of the free surface and thus dip tends towards 1.

Table 1: Summary of selected models for computing the velocity distribution.

Model Cross- sectional shape

Boundary roughness distribution

Dip Phenomenon

Method

Chiu's model General Smooth and rough Yes Entropy principal Yang's model Rectangular Smooth and rough Yes Modified Log law Sarma's model General Smooth and rough Yes Modified Log law

and Power law Bonakdari's model General Smooth and rough Yes Simplify Navier-

Stokes Eqs. In this section, a review of various existing models for computing the velocity profiles in a narrow open channel with turbulent flow was presented. Table 1 summarizes these four models.

4. RESULTS AND DISCUSSION

During most of the dry weather conditions, the water level remained under the bank and the section was almost trapezoid, a situation which will be referred to as low water situation (d < 0.9 m), hereafter. At the peak of the daily flow level in dry weather, or during wet weather conditions, the water level rose over the bank and the section became compound, a situation here referred to as high water situation (d > 0.9 m). The velocity profiles, according to given models and experimental data obtained from the Cordon Bleu site with Cerberes, were plotted and analyzed. For each case, six different distances, from y = 0.5 m to y = 1.5 m stepwise 0.2 m apart, were studied. 4.1 Low water situation

Figure 7(a-f) shows the local primary velocity profiles along a lateral section (y) normalized by the mean velocity, umean, against the span-wise direction, z, for a low water level, (D = 0.65 m). The profiles present the dip phenomenon with a maximum velocity located at a mid-water level. The black points on the profile demonstrate the experimental results obtained by Larrarte [32]. As y increases, the discrepancy of the models to fit the experimental data escalates, which might be attributed to the greater impact of the secondary current in the different depths, in the one compared to the deeper one studied [45]. It was found that the Bonakdari model shows a good fitness and represents the dip phenomenon acceptably at the central portion. It was also shown to

enjoy a good fitness with the data at Dz

less than 0.6, whereas this model overestimates the velocity profile

when y increases. In comparison to this, Chiu's model responds in a wider range and next to the walls; it only shows high errors in y = 1.50 m, which results from the difficulty in estimating parameters in the model, and to

279


corresponding wall roughness. Sarma's model determines the negative longitudinal velocity gradient and the dip phenomenon properly; however, it is not valid near the bed and walls and underestimates the velocities, using the logarithmic law in these regions. Yang suggested a model which represents the negative longitudinal velocity gradient and accordingly, shows more correspondence to the experimental results all over the channel compared with the other models mentioned above. There are some difficulties using this model, as it is important to determine the parameters; however, considering the reforms which Yang suggested in the logarithmic law, his model describes best the dip phenomenon in narrow channels and next to the walls.

a)

b)

c)

d)

280


e)

f)

Figure 7: Comparison between non dimensional velocity profiles obtained with the different models (water level 0.65 m) against relative depth

4.2 High water situation

Figure 8(a-f) shows a normalized velocity distribution at different distances of y, from y = 0.5 to 1.5 m for a water level of 0.91 m. At this water level, the water comes up to the passing bank level but the section is not compound, yet. From the passing bank, one can realize secondary and vortex flows near the walls, and their effects on the positioning of maximum velocity is obvious in the (a-f) of this Figure. Accordingly, in comparison with three other models, Yang's model shows more correspondence to the measured results. All the models present proper descriptions of the velocity profile in the central region.

a)

281 273


b)

c)

d)

e)

282 273


f)

Figure 8: Comparison between non dimensional velocity profiles obtained with the different models (water level 0.91 m) against relative depth.

Figure 9(a-f) presents the longitudinal velocity profiles at the maximum water level and the compound

cross sections for D = 1.21 m. As the water level heightens, the dip phenomenon and the maximum velocity occur under the water surface again, obviously. As the secondary flows seem stronger here, the error rate increases in the models and is more significant at the maximum water level. In these profiles, the positioning of the maximum velocity is affected by the secondary flows on the passing walkway or bank, and there is a slight difference between the measured and calculated results. At this level, as the cross section changes to compound, the results of the models are more relevant in the central portion in comparison with low water level, because of the smaller ratio of width to depth (Ar). Accordingly, in comparison to the three other models, Yang's model shows better correspondence with the measured results.

a)

b)

283


c)

d)

e)

f)

Figure 9: Comparison between non dimensional velocity profiles obtained with the different models (water level 1.19 m) against relative depth

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4.3 DISCUSSION

By comparing the figures, as the depth in the central portion (0.7 m < y < 1.1 m) increases, the effect of non-isotropy of turbulence diminishes because of the smaller aspect ratio. While in all depths the discrepancy between the models and the data appear to be high at y equal to 1.5m, it might be due to the difficulty of measuring at that location during the experiment. Also, as a result of changes in the water level during the measurement, some other errors might result from vibrations produced in the measurement system by the flow. To quantitatively compare the performance of each model in the entire cross section as compared to the measured data, two indicators, viz., relative error (Rerr) and root mean squared error (RMSE) were used as follows:

exp

expmo

uuu

Rerr

(10)

n

1i

2

exp

expmo

uuu

n1RMSE (11)

where n is the total point measurement in the cross section, umo = estimated velocity according to models for the different points and uexp = velocity resulted from measurement in the cross section. A perfect agreement between the calculated case and the actual measured data correspond to Rerr=0 and RMSE=0. Table 2 shows that errors of each model correspond to the measured data in the channel.

Table 2: Computed error of each model

Bonakdari's model Chiu`s model Yang`s model Sarma`s model y (m) D D D D

1.19 m

0.91 m

0.65 m

1.19 m

0.91 m

0.65 m

1.19 m

0.91 m

0.65 m

1.19 m

0.91 m

0.65 m

0.4 2.6 1.7 0.2 0.2 0.1 0.3 0.5 0.1 0.4 1.8 2.5 0.5 0.1 1.3 0.3 0.1 0.4 0.2 0.5 0.3 0.2 0.6 1.0 2.9 0.7 0.1 0.1 0.2 0.2 0.2 0.1 0.8 0.6 0.2 0.8 0.6 0.3 0.9 0.9 0.2 0.2 0.2 0.2 0.2 0.4 0.8 0.5 0.5 0.4 1.2 1.1 2.2 0.6 0.7 0.1 0.5 0.5 0.3 0.9 1.5 0.2 1.0 1.3 1.3 4.1 1.1 4.2 0.9 0.5 3.0 0.6 0.6 4.7 2.1 1.7 3.1 1.5

Figures 7, 8 and 9 compare the different models using the experimental results. These models represent the

velocity profile properly in the external region of the channel center and show a smaller estimation error in comparison to that in the side regions. In the central portion, Bonakdari's model shows a relative error < 10% for a high water level situation (Table 3). Sarma’s model shows the same relative error for all cases, although the RMSE value is more than 10 percent. That means that there is a greater variance of individual errors in this model. Yang's model presents acceptable results throughout the channel, and has the advantage of showing the dip phenomenon in its entirety. Sarma's and Chiu's models may imply more acceptable results next to walls if some modifications are considered in the side regions. Sarma's model describes best the location of the longitudinal over maximum velocity, lending it an efficient one in narrow channels. There are fewer parameters in Bonakdari's model, also easily determined; but it shows poor results in the corner section portion. The advantage of Bonakdari’s model, in comparison with the three other models is that it shows the dip phenomenon well in the central portion and also its equation includes fewer parameters which can be easily calculated. However, this model is not efficient in the side sections portions of channels.

Table 3: Comparison of relative error and root mean squared error for different models. h = 0.65 m h = 0.91 m h = 1.21 m CM1 YM2 SM3 BM4 CM YM SM BM CM YM SM BM Rerr 5.3 7.0 9.5 12.6 6.6 6.00 9.1 7.8 5.7 8.3 8.6 8.6 RMSE 6.0 9.4 12.3 13.3 7.8 8.4 10.7 9.4 7.1 10.1 13.3 9.6

1: Chiu’s model, 2: Yang’s model, 3: Sarma’s model, 4: Bonakdari’s model

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5. CONCLUSIONS

This study provides a comparison between various existing methods for computing the longitudinal velocity profile distribution in narrow open channels. Four models (Chiu's, Sarma's, Yang's and Bonakdari's) were selected and tested. They were selected because they provide an approach sufficiently general to compute the velocity distribution and the dip phenomenon and are simple enough for engineering applications. The models are described and results compared with the experimental data obtained from the actual site of the measurements. All models present proper descriptions of the velocity profiles in the central sections and they are close to the experimental data. Yang's model provides overall good predictions of the velocity distribution and presents acceptable results throughout the channel, having the advantage of showing the dip phenomenon entirely. Chiu’s model provides the best predictions of the velocity distribution and presents acceptable results throughout the channel with minimum relative error and root mean squared error in the entire cross section. It was found that the Chiu’s and Bonakdari's models showed a good fitness and represent, with acceptable accuracy, the dip phenomenon at the central portion

Acknowledgment The authors declare that they have no conflicts of interest in this research.

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Basic and Applied Scientific Research, 3(8): 294-299. Nomenclature b free surface width (m) D water depth (m) DC maximum vertical dimension of experimental channel (m)

287


g gravitational acceleration (m/s2) h parameter in (m) Ks roughness height (m) R hydraulic radius (m) u primary flow velocity at (m/s) umean mean cross sectional velocity (m/s) umax maximum velocity (m/s)

z,Cu velocity at a distance of z in a central vertical (m/s)

m,yu maximum velocity in a vertical at a distance of y (m/s) u* shear velocity (m/s)

z*u shear velocity on the sidewall at a distance of z from the bed (m/s) expnu measured velocity (m/s) pronu estimated velocity (m/s)

y transverse spanwise distance (horizontal coordinate) (m) z vertical spanwise distance (vertical coordinate) (m) z0 distance at which the velocity is hypothetically equal to zero (m)

dynamic viscosity (m2/s) yi , parameter of isovels ( ) equation (m)

M dimensionless entropy parameter , coordinates based on isovels

0 minimum value of dip dip phenomenon location or maximum value of Karman constant S channel slop

h/DDy

z*z,Cz,C u/uU

y*m,ym,y u/uU

z*y,z u/uU b/y2Y

D/zZ

288

comparison of different models for evaluating the velocity .... basic. appl...ramana prasada [18]...

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