# Analytical Formula for the Mean Velocity Profile in a Pipe Derived on the Basis of a Spatial Polynomial Vorticity Distribution

## Abstract

**:**

_{(av)}, the pipe radius R, and Re to be able to predict the turbulent mean velocity profile in a pipe.

## 1. Introduction

## 2. Backgrounds

_{i}, which is defined as the curl of the velocity vector.

_{ijk}is a Levi-Civita tensor of 3

^{−d}order. For a 2D axis-symmetrical flow in a pipe, it is possible to write:

_{m}is the outward unit normal vector to the boundary surface of the vortex tube/filament element, S is the area of the vortex tube/filament cross-section. The intensity µ of the vortex tube/filament is equal to the circulation Γ over the boundary curve of the vortex tube/filament cross-section area. This is a result of the well-known Kelvin–Stokes theorem [6].

_{j}induces an infinitesimal velocity dv

_{i}in Figure 1. This velocity can be expressed by the Biot–Savart law applied to the vortex flow:

_{k}, x

_{k}is the vortex filament element location, x’

_{k}is the induced velocity location, ℓ

_{k}= x’

_{k}− x

_{k}, τ

_{j}is a unit vector tangential to the vortex filament. The vortex tube/filament has the same orientation as the vorticity vector Ω

_{m}.

_{j}is really infinitesimal, then it is possible to assume that the vectors Ω

_{m}, n

_{m}, and τ

_{j}are parallel. It is possible to write:

_{j}does not depend on the x

_{1}coordinate. It is the function of the angle φ. The location of the induced velocity is given by the coordinates x’

_{k}. These coordinates are constants from the view of the integration. The coordinates x

_{k}determine the location of the vorticity element on the vortex sheet. These coordinates are variables from the point of view of the integration. Variable ℓ is the distance between x

_{k}and x’

_{k}in accordance with the Expression (5). The integral in Expression (13) has a simple analytical solution (14). This is also mentioned in [7]. The solution is:

_{i}is a unit vector in the streamwise direction of the pipe axis. It means that such a circular vortex sheet induces a plug flow inside of the circular vortex sheet and no flow outside the circular vortex sheet. Another result is that this solution does not depend on the cross-section shape of the vortex sheet. The only condition is that the cross-section shape must be constant along the whole vortex sheet.

_{2}> is v = γ

_{1}+ γ

_{2}. Velocity for the radius interval <R

_{2}, R

_{1}> is v = γ

_{1}.

## 3. Basic Idea of Velocity Profile Derivation

_{w}).

_{i}is the streamwise oriented unit vector. This vector is constant over the whole domain; therefore, it is possible to neglect it for this case. The velocity magnitude will be solved only. Equation (15) will then be modified as follows:

_{*}is result of the effects of the vortex sheets with radius within the interval <r

_{*}, R>. This can be expressed by the integral:

_{(i)}are the free coefficients, i is an integer in the interval <0, ∞>. The number M can be an infinite value. It means that there are an infinite number of free coefficients A

_{(i)}in the polynomial. The number of free coefficients must be restricted. The Ω function is shown in Figure 4. It is possible to assume that it is an odd function in accordance with Figure 4. It means that only the odd values of i are taken into consideration. The free coefficients A

_{(i)}represent the values of i-th derivatives of the polynomial function in the pipe axis (r = 0). The idea about using the odd polynomial function is supported by the fact that Ω = 0 for r = 0. It means that A

_{(0)}= 0. This is also the condition of smoothness of the first velocity profile derivative in the pipe axis.

_{(1)}≠ 0. The third derivate is also chosen not to be zero, what means that A

_{(3)}≠ 0. The same situation is for the K-th and N-th derivatives, A

_{(K)}≠ 0 and A

_{(N)}≠ 0. All other derivatives in the pipe axis (free coefficients) are chosen to be zero.

## 4. Available Conditions for Free Coefficients Determining

- Zero velocity at the wall.
- Vorticity at the wall Ω
_{w}. This condition corresponds to the condition of pressure drop in the pipe. - Zero value of the vorticity Ω in the pipe axis. This corresponds to the smoothness condition of the first velocity profile derivative in the pipe axis.
- The first vorticity derivative in the pipe axis. This condition corresponds to the curvature radius.
- Maximal velocity in the pipe axis.
- The knowledge of flow rate in pipe.
- The knowledge of radius where the velocity is the same as the average velocity.
- All conditions are explained in detail in the next chapters.

#### 4.1. Zero Velocity at the Wall

#### 4.2. Vorticity Value at the Wall Ω_{w}

_{w}. The pressure force F

_{p}must be in balance with the friction force F

_{τ}as it is depicted in Figure 5.

_{w}by the force balance applying:

#### 4.3. Zero Value of the Vorticity Ω in the Pipe Axis

#### 4.4. The First Derivative of the Vorticity Function Ω in the Pipe Axis

_{(av)}and r’ is a radius normalized by the pipe radius R, r’ = r/R. Then, the first normalized velocity derivative has the following form:

#### 4.5. Maximal Velocity in the Pipe Axis

_{(max)}and v

_{(av)}can be normalized by the average velocity. This way, we obtain a value which is signed as n.

#### 4.6. The Knowledge of Flow Rate in Pipe

^{3}.s

^{−1}] is the flow rate, S is the cross-section area. If the cross-section has a circular shape, then the infinitesimal area dS can be expressed as:

#### 4.7. The Radius of the Average Velocity

## 5. Mean Velocity Profile Formula Derivation

_{(1)}, A

_{(3)}, A

_{(K)}, and A

_{(N)}. The condition of the first vorticity derivative, or the negative value of the second velocity derivative in the pipe axis can be expressed this way:

_{(1)}is the value of the first vorticity derivative. This value depends on Reynolds number. This dependence will be discussed later. Next condition is the vorticity value on the wall. It means the value of vorticity for r = R. The expression of this condition has the following form:

_{w}is derived from the pressure drop. The third condition is the condition of the maximal velocity in the pipe axis. It is derived from the Expression (43) for the case that r = 0.

_{(av)}

_{(1)}(D′

_{(1)}), Expression (62) becomes:

## 6. Exponents Determination

_{(av)}, v

_{(max)}, Ω

_{(w)}, D

_{(1)}) are listed in Table 1. They are listed for all PSP velocity profiles. The variables, which enter into the minimization process, are exponents N and K in this case. They are highlighted in grey in Table 1.

_{(av)}, v

_{(max)}are taken directly from the experimental data. The vorticity at the wall Ω

_{(w)}is calculated from the pressure drop through Expression (24) or (26). The normalized vorticity at the wall is expressed by (60). First, the vorticity derivative at the pipe axis is taken from the vorticity distribution calculated from the experimental data. The vorticity distribution near the pipe axis was approximated by a linear function from which the first vorticity derivative was calculated. The comparisons of the analytical velocity profiles and the experimental data are in Figure 11 and Figure 12. Figure 11 shows the results for the low Re. Results for the high Re are shown in Figure 12.

_{(exp)}is the experimental velocity at a given radius r, v

_{(an)}is the analytical velocity at a given radius r, v

_{(av)}is the average velocity, M is the number of measured velocities, v’

_{(exp)}is the normalized experimental velocity at a given radius r, and v

_{(an)}is the normalized analytical velocity at a given radius r. The value of s for the case of low Re (31,577) is s = 2.183 and for the case of high Re (13,598,000) is s = 0.364.

_{(w)}plays a crucial role in the accuracy of the analytical velocity profile, therefore the precise pressure drop measurement is essential. The problems with the mean velocity measurement are apparent from the vorticity diagram. The vorticity values calculated from the experimental data are very scattered. It is clear from Figure 11b and Figure 12b.

_{(w)}and the first vorticity derivative in the pipe axis D

_{(1)}, as a new variable, are added to the exponents K and N. Table 2 shows the parameters obtained, for this case, by the velocity difference absolute value minimization. The grey columns emphasize the parameters which are taken as a variable in the minimization process.

^{+}and y

^{+}. The definition of u

^{+}is in (64) and the definition of y

^{+}is in (65).

_{τ}is the friction velocity, τ

_{w}is the wall shear stress, ρ is the density, and ν is the kinematic viscosity. All velocity profiles in coordinates u

^{+}, y

^{+}, for the case of two variables N and K are drawn in Figure 15a. Each of the profiles is shifted about u

^{+}= 6 from the previous one. The solid lines represent the analytical velocity profiles, and the circles represent experimental data.

^{+}, y

^{+}for the case of four variables Ω

_{(w)}, D

_{(1)}, N, and K are drawn in Figure 15b. It is apparent that the analytical velocity profiles for low Re fit much better with the experimental data than in the previous case.

_{(w)}. It is possible to compare the wall vorticity for the cases of two and four variables which are coming into the optimization process. The wall vorticity in the case of two variables is taken from the pressure drop obtained by the experiment. In the case of four variables, the wall vorticity is one of the variables which are obtained by the velocity difference minimization. It is better to compare the normalized wall vorticity which is expressed by Expression (60). The comparison is in Figure 16.

_{(1)}, N, and K on Re to be able to predict the velocity profile without actual experimental data.

## 7. Normalized Wall Vorticity Ω′_{(w)} as a Function of Re

_{(w)}. Even though the normalized wall vorticity can be expressed with the help of the friction factor, it will be useful to find an expression for the dependence of Ω′

_{(w)}on the Re. The interval of Re will be divided into two parts. The functions will be defined separately for each interval. The general form of the function is:

_{(Ωw)}and B

_{(Ωw)}depend on the interval of Re. For the interval Re < 31,500, 550,000>, the constants are A

_{(Ωw)}= 0.810279 and B

_{(Ωw)}= 100.1872, and for the interval Re <550,000, 35,259,000>, the constants are A

_{(Ωw)}= 0.868399 and B

_{(Ωw)}= 227.1003. These constants are derived on the basis of the comparison of these functions with the experimental data of (PSP) [3]. The comparison of the empirical expression and data from the experiment is in Figure 17. The agreement seems to be good.

## 8. The Dependence of the First Vorticity Derivative at the Pipe Axis on Re

_{(1)}) depends on the average velocity and the density. When the density is constant, then the dependence on the average velocity is near to be linear. This is apparent from Figure 7. It is better to work with the normalized vorticity to express the derivative (D′

_{(1)}) as a Re function. The relationship between D

_{(1)}and D′

_{(1)}is expressed by (63). The dependence of D′

_{(1)}on the Re is in Figure 18. The dashed line with the stars represents the data obtained from the experiments. The solid line is the analytical expression of this dependence.

_{(D1)}= 2.832 and B

_{(D1)}= −1.053. It seems that the analytical expression agrees rather well with the experimental data.

## 9. The Dependence of Exponents K and N on the Re

_{(eK)}= 0.757628 and B

_{(eK)}= 620.062.

_{(eN}

_{1)}= 0.9038382, B

_{(eN}

_{1)}= 205.532, A

_{(eN}

_{2)}= 0.823883 and B

_{(eN}

_{2)}= 75.3454.

## 10. Mean Velocity Profile Prediction

_{(av)}, the pipe radius R, and viscosity are known. All other parameters are obtained with the help of empirical expressions. This solution for low and high Re is shown in Figure 22 and Figure 23. The analytical mean velocity precession is evaluated through the average absolute difference percentage s (64). The precession comparison is in Table 3.

## 11. Conclusions

_{(w)}, average velocity v

_{(av)}, maximal velocity v

_{(max)}, and the first vorticity derivative in the pipe axis D

_{(1)}. The condition of the average velocity radius has not been used yet. The analytical expression of the mean velocity profile is defined by the Expressions (48)–(57). There are two unknown exponents K and N in the formula. The values of these exponents can be obtained by the minimization process of the velocity difference between the analytical velocity and the experimental data. The important fact is that during the minimization process the mean velocity profile characteristics are constant. The exponents K and N can be expressed as Re functions by the empirical formulas (69) and (70). If only the average velocity, pipe radius, and Re are known, then it is possible to use Expressions (36), (67), and (68) for the values of v

_{’(max)}, Ω′

_{(w)}, and D′

_{(1)}. Two examples of mean velocity profile prediction are demonstrated at the end of the paper. It was found that the prediction precision, in comparison with experimental data PSP, is very good.

## 12. Future Work

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 6.**Normalized mean velocity profile acquired by experiment (PSP) and the corresponding normalized vorticity distribution over cross-section for Re = 230,460. (

**a**) Normalized mean velocity profile; (

**b**) Normalized vorticity distribution.

**Figure 7.**The dependence of the first vorticity derivative in the axis on the average velocity for different densities.

**Figure 9.**All mean velocity profiles measured by Zagarola [3] on PSP. There are 26 profiles for the Reynold’s number range <31,577, 35,259,000>.

**Figure 10.**The average velocity radius for all 26 velocity profiles measured by Zagarola [3] on PSP.

**Figure 11.**The analytical formula comparison with the experimental data (PSP) for Re = 31,577, using K and N as the variables. Dashed line with circles represents the experimental data, the solid line is the analytical profile. (

**a**) Normalized mean velocity profile, (

**b**) Normalized vorticity distribution.

**Figure 12.**The analytical formula comparison with the experimental data (PSP) for Re = 13,598,000, using K and N as the variables. Dashed line with circles represents the experimental data, the solid line is the analytical profile. (

**a**) Normalized mean velocity profile, (

**b**) Normalized vorticity distribution.

**Figure 13.**The analytical formula comparison with the experimental data (PSP) for Re = 31,577, using variables Ω

_{(w)}, D

_{(1)}, K, and N. Dashed line with circles represents the experimental data, the solid line is the analytical profile. (

**a**) Normalized mean velocity profile, (

**b**) Normalized vorticity distribution.

**Figure 14.**The analytical formula comparison with the experimental data (PSP) for Re = 13,598,000, using variables Ω

_{(w)}, D

_{(1)}, K, and N. Dashed line with circles represents the experimental data, the solid line is the analytical profile. (

**a**) Normalized mean velocity profile, (

**b**) Normalized vorticity distribution.

**Figure 15.**The velocity profiles in the coordinates u

^{+}, y

^{+}. (

**a**) The case of two variables N and K in the optimization process. (

**b**) The case of four variables Ω

_{(w)}, D

_{(1)}, N, and K in the optimization process.

**Figure 16.**The comparison of the normalized wall vorticity obtained by four variables optimization with experimental data.

**Figure 20.**The experimental data (PSP) for Re = 31,577 comparison with predicted mean velocity profile. The empirical formulas for exponents K and N are used only. Dashed line with circles represents the experimental data, the solid line is the analytical profile. (

**a**) Normalized mean velocity profile, (

**b**) Normalized vorticity distribution.

**Figure 21.**The experimental data (PSP) for Re = 13,598,000 comparison with predicted mean velocity profile. The empirical formulas for exponents K and N are used only. Dashed line with circles represents the experimental data, the solid line is the analytical profile. (

**a**) Normalized mean velocity profile, (

**b**) Normalized vorticity distribution analytical.

**Figure 22.**The experimental data (PSP) for Re = 31,577 comparison with predicted mean velocity profile. The average velocity and Re are known in this case. All other parameters are evaluated from the empirical expressions. Dashed line with circles represents the experimental data, the solid line is the analytical profile. (

**a**) Normalized mean velocity profile, (

**b**) Normalized vorticity distribution.

**Figure 23.**The experimental data (PSP) for Re = 13,598,000 comparison with predicted mean velocity profile. The average velocity and Re are known in this case. All other parameters are evaluated from the empirical expressions. Dashed line with circles represents the experimental data, the solid line is the analytical profile. (

**a**) Normalized mean velocity profile, (

**b**) Normalized vorticity distribution.

**Table 1.**Parameters of the analytical velocity profile. Exponents K and N are obtained from comparison with the experimental data.

p.n. | Re [1] | v_{(av)}[m.s ^{−1}] | v_{(max)}[m.s ^{−1}] | Ω_{(w}_{)}[s ^{−1}] | D_{(1)}[m ^{−1}.s^{−1}] | N [-] | K [-] | Ω′_{(w}_{)}[1] | D‘_{(1)}[1] | v‘_{(max)}[1] |
---|---|---|---|---|---|---|---|---|---|---|

01 | 31,577 | 3.876 | 4.821 | 2748 | 775 | 55.3 | 4.5 | 45.9 | 0.8363 | 1.2439 |

02 | 41,727 | 5.132 | 6.351 | 4523 | 974 | 71.5 | 5.0 | 57.0 | 0.7939 | 1.2374 |

03 | 56,677 | 6.845 | 8.410 | 7639 | 1299 | 95.7 | 6.0 | 72.2 | 0.7941 | 1.2287 |

04 | 74,293 | 8.952 | 10.926 | 12,288 | 1618 | 120.9 | 9.0 | 88.8 | 0.7562 | 1.2204 |

05 | 98,811 | 11.896 | 14.404 | 20,447 | 2032 | 159.9 | 12.7 | 111.3 | 0.7151 | 1.2118 |

06 | 145,790 | 17.532 | 21.011 | 41,029 | 2853 | 223.5 | 13.1 | 151.4 | 0.6808 | 1.1984 |

07 | 185,430 | 22.623 | 26.976 | 64,189 | 3567 | 280.5 | 17.1 | 183.5 | 0.6596 | 1.1924 |

08 | 230,460 | 9.751 | 11.58 | 32,974 | 1565 | 340.1 | 18.6 | 218.7 | 0.6715 | 1.1875 |

09 | 309,500 | 13.110 | 15.489 | 56,885 | 1971 | 449.8 | 24.4 | 280.7 | 0.6289 | 1.1815 |

10 | 409,290 | 17.357 | 20.405 | 94,436 | 2537 | 575.6 | 28.8 | 351.9 | 0.6116 | 1.1756 |

11 | 539,090 | 22.840 | 26.768 | 156,752 | 3225 | 737.3 | 34.3 | 443.9 | 0.5908 | 1.1720 |

12 | 751820 | 6.271 | 7.327 | 56,744 | 863 | 955.1 | 35.8 | 585.2 | 0.5757 | 1.1683 |

13 | 1,023,800 | 8.440 | 9.820 | 98,752 | 1161 | 1208.0 | 32.4 | 756.9 | 0.5754 | 1.1636 |

14 | 1,340,400 | 11.071 | 12.851 | 162,559 | 1461 | 1504.3 | 35.9 | 949.7 | 0.5519 | 1.1608 |

15 | 1,787,500 | 14.769 | 17.110 | 276,184 | 1940 | 1915.6 | 36.8 | 1209.5 | 0.5495 | 1.1585 |

16 | 2,345,000 | 19.478 | 22.493 | 456,420 | 2506 | 2354.0 | 37.3 | 1515.6 | 0.5382 | 1.1548 |

17 | 3,098,100 | 13.268 | 15.280 | 395,546 | 1652 | 2974.4 | 39.0 | 1928.2 | 0.5208 | 1.1516 |

18 | 4,420,300 | 5.081 | 5.830 | 205,830 | 597 | 4064.6 | 43.7 | 2620.3 | 0.4920 | 1.1474 |

19 | 6,072,700 | 6.928 | 7.931 | 368,398 | 839 | 5126.3 | 35.2 | 3439.3 | 0.5068 | 1.1447 |

20 | 7,714,700 | 6.580 | 7.523 | 427,544 | 777 | 6262.6 | 38.8 | 4202.4 | 0.4941 | 1.1432 |

21 | 10,249,000 | 8.703 | 9.917 | 719,643 | 1001 | 7815.3 | 35.5 | 5348.3 | 0.4810 | 1.1395 |

22 | 13,598,000 | 11.504 | 13.083 | 1,220,866 | 1302 | 10,002.3 | 37.5 | 6864.2 | 0.4733 | 1.1373 |

23 | 18,196,000 | 15.345 | 17.358 | 2,093,107 | 1705 | 12,541.5 | 32.8 | 8822.6 | 0.4647 | 1.1312 |

24 | 23,977,000 | 20.149 | 22.758 | 3,499,514 | 2115 | 15,818.9 | 35.7 | 11,233.7 | 0.4391 | 1.1295 |

25 | 29,927,000 | 25.043 | 28.265 | 5,295,477 | 2623 | 19,157.9 | 36.3 | 13,676.9 | 0.4382 | 1.1287 |

26 | 35,259,000 | 29.306 | 33.087 | 7,180,224 | 3010 | 22,306.5 | 40.6 | 15,847.2 | 0.4297 | 1.1290 |

**Table 2.**Parameters of the analytical velocity profile. Quantities Ω

_{(w)}, D

_{(1)}, N, and K are obtained from comparison with the experimental data.

p.n. | Re [1] | v_{(av)}[m.s ^{−1}] | v_{(max)}[m.s ^{−1}] | Ω_{(w)}[s ^{−1}] | D_{(1)}[m ^{−1}.s^{−1}] | N [-] | K [-] | Ω′_{(w)}[1] | D′_{(1)}[1] | v′_{(max)}[1] |
---|---|---|---|---|---|---|---|---|---|---|

01 | 31,577 | 3.876 | 4.821 | 3606 | 1156 | 82.2 | 4.0 | 60.2 | 1.2475 | 1.2439 |

02 | 41,727 | 5.132 | 6.351 | 6183 | 1420 | 109.9 | 5.1 | 77.9 | 1.1579 | 1.2374 |

03 | 56,677 | 6.845 | 8.410 | 10,480 | 1560 | 146.8 | 10.0 | 99.0 | 0.9532 | 1.2287 |

04 | 74,293 | 8.953 | 10.926 | 16,677 | 1881 | 180.1 | 11.0 | 120.5 | 0.8789 | 1.2204 |

05 | 98,811 | 11.886 | 14.404 | 27,207 | 1969 | 229.3 | 19.7 | 148.1 | 0.6929 | 1.2118 |

06 | 145,790 | 17.532 | 21.011 | 52,502 | 2654 | 307.2 | 22.7 | 193.7 | 0.6332 | 1.1984 |

07 | 185,430 | 22.623 | 26.976 | 73,314 | 3313 | 329.6 | 22.2 | 209.6 | 0.6126 | 1.1924 |

08 | 230,460 | 9.752 | 11.580 | 36,037 | 1353 | 382.3 | 26.2 | 239.0 | 0.5805 | 1.1875 |

09 | 309,500 | 13.110 | 15.489 | 51,500 | 1825 | 400.6 | 25.2 | 254.1 | 0.5824 | 1.1815 |

10 | 409,290 | 17.357 | 20.405 | 68,893 | 2310 | 401.3 | 27.1 | 256.7 | 0.5567 | 1.1756 |

11 | 539,090 | 22.840 | 26.768 | 94,562 | 3076 | 422.1 | 30.3 | 267.8 | 0.5634 | 1.1720 |

12 | 751,820 | 6.271 | 7.327 | 39,572 | 815 | 660.0 | 37.2 | 408.1 | 0.5440 | 1.1683 |

13 | 1,023,800 | 8.439 | 9.820 | 80,881 | 1133 | 994.2 | 34.9 | 619.9 | 0.5615 | 1.1636 |

14 | 1,340,400 | 11.071 | 12.851 | 135,828 | 1456 | 1254.0 | 35.8 | 793.5 | 0.5502 | 1.1608 |

15 | 1,787,500 | 14.769 | 17.110 | 226,335 | 1952 | 1563.6 | 35.8 | 991.2 | 0.5528 | 1.1585 |

16 | 2,345,000 | 19.478 | 22.493 | 374,371 | 2527 | 1926.3 | 36.7 | 1,243.2 | 0.5427 | 1.1548 |

17 | 3,098,100 | 13.268 | 15.280 | 353,477 | 1658 | 2652.4 | 38.6 | 1,723.2 | 0.5227 | 1.1516 |

18 | 4,420,300 | 5.081 | 5.830 | 181,425 | 617 | 3554.7 | 40.9 | 2,309.6 | 0.5084 | 1.1474 |

19 | 6,072,700 | 6.928 | 7.931 | 318,384 | 835 | 4427.3 | 35.3 | 2,972.4 | 0.5044 | 1.1447 |

20 | 7,714,700 | 6.580 | 7.523 | 375,852 | 781 | 5501.9 | 38.5 | 3,694.3 | 0.4963 | 1.1432 |

21 | 10,249,000 | 8.703 | 9.917 | 597,357 | 1010 | 6495.2 | 35.6 | 4,439.5 | 0.4853 | 1.1395 |

22 | 13,598,000 | 11.504 | 13.083 | 1,104,237 | 1332 | 8983.2 | 34.6 | 6,208.5 | 0.4844 | 1.1373 |

23 | 18,196,000 | 15.345 | 17.358 | 1,812,336 | 1669 | 10,839.6 | 32.7 | 7,639.1 | 0.4550 | 1.1312 |

24 | 23,977,000 | 20.149 | 22.758 | 2,476,268 | 2041 | 11,258.8 | 40.3 | 7,949.0 | 0.4238 | 1.1295 |

25 | 29,927,000 | 25.043 | 28.265 | 4,757,948 | 2661 | 17,202.4 | 35.5 | 12,288.6 | 0.4446 | 1.1287 |

26 | 35,259,000 | 29.306 | 33.087 | 6,549,529 | 3293 | 20,042.4 | 31.9 | 14,455.2 | 0.4701 | 1.1290 |

**Table 3.**The mean velocity profile precession comparison for different ways of prediction. The comparison is done through the value δ.

Analytical Velocity Profile | δ [% of v_{(av)}] | |
---|---|---|

Re = 31,577 | Re = 13,598,000 | |

Minimization variables K, and N | 2.183 | 0.364 |

Minimization variables Ω_{(w)}, D_{(1)}, K, and N | 0.494 | 0.362 |

The first prediction way | 2.210 | 0.377 |

The second prediction way | 2.783 | 0.339 |

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Štigler, J. Analytical Formula for the Mean Velocity Profile in a Pipe Derived on the Basis of a Spatial Polynomial Vorticity Distribution. *Water* **2021**, *13*, 1372.
https://doi.org/10.3390/w13101372

**AMA Style**

Štigler J. Analytical Formula for the Mean Velocity Profile in a Pipe Derived on the Basis of a Spatial Polynomial Vorticity Distribution. *Water*. 2021; 13(10):1372.
https://doi.org/10.3390/w13101372

**Chicago/Turabian Style**

Štigler, Jaroslav. 2021. "Analytical Formula for the Mean Velocity Profile in a Pipe Derived on the Basis of a Spatial Polynomial Vorticity Distribution" *Water* 13, no. 10: 1372.
https://doi.org/10.3390/w13101372