# On the Composite Velocity Profile in Zero Pressure Gradient Turbulent Boundary Layer: Comparison with DNS Datasets

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## Abstract

**:**

_{θ}≤ 6500. The mathematical model is based on the superposition of an accurate description of the inner law and Coles’ wake function with appropriately chosen parameters. It is found that there is excellent agreement between the mathematical model and the DNS data in the inner layer when the Reynolds number based on momentum thickness, Re

_{θ}, is greater than 1000. Furthermore, there is very good agreement over the entire boundary layer thickness, when Re

_{θ}is greater than 2000. The diagnostic functions Ξ and Γ based on DNS data are examined and their characteristics are discussed in relation to the existence of a logarithmic layer or a power law behavior of the MVP. The diagnostic functions predicted by the mathematical model are also presented.

## 1. Introduction

## 2. Modelling the Mean Velocity Profile Methodology

^{+}= $\frac{\mathrm{y}{\mathrm{u}}_{\mathsf{\tau}}}{\nu}$ and u

^{+}= $\frac{\mathrm{u}}{{\mathrm{u}}_{\mathsf{\tau}}}$, where y is the distance normal to the solid surface, u is the velocity component in the main direction of flow, ν is the kinematic viscosity coefficient and u

_{τ}is the shear velocity at the wall [5,6]. It is worth pointing out that the logarithmic law u

^{+}= $\frac{1}{\kappa}$lny

^{+}+ Β is valid in the interval [${\mathrm{y}}_{\mathrm{l}\mathrm{o}\mathrm{w}}^{+}$, ${\mathrm{y}}_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}^{+}$]. The values of ${\mathrm{y}}_{\mathrm{l}\mathrm{o}\mathrm{w}}^{+}$ and ${\mathrm{y}}_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}^{+}$ depend on the value of the Reynolds number Re

_{θ}(defined by the boundary layer momentum thickness θ) and differ between researchers. A common approach when analyzing experimental data for turbulent boundary layer is to use the values κ = 0.40 or 0.41 for von Kármán’s constant and Β = 5.0.

^{+}≳ 65 and provides an accurate fit to the experimental velocity data for turbulent boundary layers.

^{+}) is an approximation of the inner law which is valid over the inner layer including the overlap layer. In order for Equations (1) and (2) to be compatible in the outer zone, the function f(y

^{+}) must tend asymptotically to the logarithmic law $\frac{1}{\kappa}$lny

^{+}+ Β, for high values of y

^{+}. Therefore, to approximate the composite velocity profile, we are required to determine two functions, f and g, whose sum must accurately approximate the velocity u

^{+}for any y

^{+}, in the interval [0, δ

^{+}].

#### 2.1. Inner Layer

^{+}) are proposed:

^{+}= 10 and takes the value 0.32. For values of y

^{+}≥ 50, both functions tend asymptotically to the logarithmic law with κ = 0.41 and B = 5.0, therefore any selection between Equations (3) and (4) has no significant effect on the shape of the mean velocity profile.

#### 2.2. Outer Layer

## 3. Mean Velocity Profiles

#### 3.1. Inner Region: Comparison of f(y^{+}) with ū(y^{+}) Calculated by DNS

_{θ}in the range 1000 to 4060. Equation (4) was used to plot the profiles in Figure 3.

^{+}≳ 270 (Re

_{θ}= 4060) simply demonstrates the need to add the g function (see Section 2.1). The difference [u

^{+}(y

^{+}) − f(y

^{+})] close to the wall is plotted in Figure 4. The corresponding error statistics are listed in Table 1.

#### 3.2. Comparison of the AL84 Composite Profile (f + g) with DNS Results over the Whole Boundary Layer (0 < y < δ)

_{θ}= 1000, 2000, and 4060. During the evaluation of the AL84 model, κ and Π were assigned values of 0.41 and 0.55, respectively.

_{θ}= 1000 and improves significantly as Re

_{θ}increases. The agreement for the moderate Reynolds number Re

_{θ}= 4060 is excellent. On the scale of Figure 5c, the differences between the two curves are not noticeable except in the interval 20 ≲ y

^{+}≲ 60.

^{+}) = u

^{+}− [f + g] where u

^{+}is the DNS mean velocity profile considered here as the “true” velocity profile. The graphical representation of the difference is shown in Figure 6 while the error quantification is summarized in Table 2.

## 4. Results Diagnostic Functions

^{+}with respect to y

^{+}is calculated using the following formula (Equation (7)) for unequally spaced data:

^{+}(y

^{+}) acts as an error amplifier and thus, differences in MVPs are accentuated.

#### 4.1. The Diagnostic Function Ξ

^{+}) = y

^{+}$\frac{{\mathrm{d}\mathrm{u}}^{+}}{{\mathrm{d}\mathrm{y}}^{+}}$ serves as a tool for answering some fundamental questions concerning the mathematical form of the inner law. In the first place, as a diagnostic tool, we can easily prove that if a mean velocity profile (MVP) includes an interval [${\mathrm{y}}_{\mathrm{l}\mathrm{o}\mathrm{w}}^{+}$, ${\mathrm{y}}_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}^{+}$] where the classical logarithmic law (see Section 2) holds, then the function Ξ(y

^{+}) must attain a constant value, equal to 1/κ, in the interval [${\mathrm{y}}_{\mathrm{l}\mathrm{o}\mathrm{w}}^{+}$, ${\mathrm{y}}_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}^{+}$]. Examining the semi-log plot of function Ξ in Figure 7, we conclude that the MVP reported by Schlatter and Örlü [13] for the moderately large Re

_{θ}= 4060 does not exhibit an interval where Ξ(y

^{+}) is strictly constant. Thus, the existence of a logarithmic layer is justified only as an approximation in this set of DNS data. This issue is further discussed in Section 4.1.3.

#### 4.1.1. The Diagnostic Function Ξ as Predicted by the AL84 Model

_{θ}, is presupposed in the AL84 model is verified in Figure 8. The function Ξ

_{1}= y

^{+}$\frac{{\mathrm{d}\mathrm{f}}^{+}}{{\mathrm{d}\mathrm{y}}^{+}}$ of AL84 (Equation (4)) is shown for Re

_{θ}in the range [1000, 6500]. When y

^{+}≥ 150, the function Ξ

_{1}becomes equal to 2.45 for all values of Re

_{θ}.

_{θ}as can be seen in Figure 9 where Ξ

_{2}= y

^{+}$\frac{{\mathrm{d}\mathrm{g}}^{+}}{{\mathrm{d}\mathrm{y}}^{+}}$ is plotted.

#### 4.1.2. The Function Ξ Based Exclusively on the DNS Data

^{+}≤ 100 all curves collapse on a single curve according to the classical view of an inner law independent of Re

_{θ}. Some minor differences appear near the first local maximum of Ξ (“inner peak”) at approximately y

^{+}≈ 9.5. The magnitude of the inner peak as well as the position where it is located are listed in Table 4. The “inner peak” of Ξ is located within the buffer zone of the MVP and tends to “oscillate” slightly with respect to its mean location (see Table 4).

^{+}≳ 100 the graphs of Ξ separate in accordance with the view that in the outer layer, there are evident Re

_{θ}effects on MVPs when plotted in inner law variables. A second local maximum of Ξ (“outer peak”) is formed around y

^{+}≈ 1150. Table 5 summarizes the “outer peak” values of Ξ as well as the location of the peak for Re

_{θ}in the range 4060 to 6500.

_{θ}= 4060. In addition, the rough character of the computed Ξ in the neighborhood of the “outer peak” may also contribute to the listed values of Ξ

_{max}(Re

_{θ}). We stress here that no local smoothing of function Ξ was applied. As the momentum Reynolds number increases, the location of the “outer peak” is always shifted towards higher values of y

^{+}(see Figure 12).

#### 4.1.3. Search for a Logarithmic Layer

_{θ}[3030, 6500] we focused on the y

^{+}interval [60, 250]. In this interval, a local minimum of Ξ is attained and possibly an approximate plateau is formed. Relevant results are summarized in Table 6 and Figure 13. Based on the data under consideration, it is observed that increasing the Re

_{θ}number leads to a larger interval of nearly constant value of Ξ and thus, to a possible determination of the von Kármán’s constant.

_{θ}= 1000, there is a substantial difference in the DNS data of Schlatter and Örlü [13] and Abe [15]. The minimum values, Ξ

_{min}, are 2.461 and 2.490 corresponding to estimates of the maximum values of von Kármán’s constant equal to 0.406 and 0.402 respectively. Furthermore, the minimum value of Ξ appears at y

^{+}= 53.3 and y

^{+}= 54.8 respectively (see Table 6). For the DNS data in the range Re

_{θ}= 4000 to Re

_{θ}= 6500, the minimum value of Ξ appears at y

^{+}≈ 74. The value of Ξ

_{min}shows a remarkable consistency (Ξ

_{min}≈ 2.27) corresponding to a maximum possible value κ ≈ 0.44 for the von Kármán’s constant. This is remarkable since the DNS results of Schlatter and Örlü [13] were obtained by a different numerical method than those of Borrell et al. [14]. However, even for these relatively large values of Re

_{θ}, there is no clear evidence of a logarithmic layer (Ξ ≈ const.).

_{θ}increases, the slope of function Ξ gradually decreases. This behavior may reflect the initial stages of a process in which the graph of function Ξ reaches gradually a plateau and Ξ converges slowly to a value 1/κ = constant in the limit Re

_{θ}→ ∞. However, this is a purely speculative remark in view of the limited number of DNS-calculated MVPs published at present and the well—known limitations of DNS in computing flows at high Re

_{θ}. The graph of the derivative dΞ/dy

^{+}versus Re

_{θ}(Figure 14) shows quantitatively the diminishing slope but reveals nothing with respect to the asymptotic behavior of Ξ as Re

_{θ}→ ∞.

_{θ}≤ 6500, is found to be equal to κ ≈ 0.393. The dispersion of the data is shown graphically in Figure 15.

#### 4.2. The Diagnostic Function Γ

^{+}= Ay

^{+λ}(A and λ constants for a particular value of Re

_{θ}) approximates the function u

^{+}(y

^{+}) in an interval [${\mathrm{y}}_{\mathrm{l}\mathrm{o}\mathrm{w}}^{+}$, ${\mathrm{y}}_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}^{+}$], then Γ attains a constant value equal to λ in that interval.

_{θ}numbers) are λ = 0.155 for the interval 100 ≤ y

^{+}≤ 190 and λ ≈ 0.2 = 1/5 for 900 ≤ y

^{+}≤ 1200 (see Figure 17).

#### 4.3. A Note on the Accuracy of the DNS Profiles

_{θ}= 1000. It is evident that there are some noticeable differences. Comparisons of Ξ(y

^{+}) and Γ(y

^{+}) present stringent accuracy tests because the numerical differentiation of u

^{+}(y

^{+}) acts as an error amplifier, and thus, differences between computed MVPs are accentuated.

## 5. Conclusions

_{θ}number on the turbulent velocity profile is analyzed in the range [1000, 6500]. It was deemed useful to divide the presentation into two parts. One concerns the interval where the inner law applies, while in a second stage, the comparison is made over the entire thickness of the boundary layer.

^{+}) of Equation (4) was used. It was found that there is a very good agreement between the results of the numerical simulation of Schlatter and Örlü [13] and the AL84 model as long as the Reynolds number Re

_{θ}is greater than 1000. For y

^{+}≳ 150, the function f(y

^{+}) tends asymptotically to the logarithmic law.

^{+}). For this purpose, in the second part of the comparison, the use of Equations (2), (4), and (6) is proposed. It turns out that there is agreement between the composite model AL84 and the DNS results, as long as the Reynolds number based on the momentum thickness of the boundary layer is relatively high. When Re

_{θ}≥ 2000, we have very good agreement between the model and the results of the direct numerical simulation of the turbulent boundary layer.

_{θ}studied. The possibility of the formation of a logarithmic layer as Re

_{θ}→ ∞ is discussed and approximate values for the Kármán constant are estimated for each Re

_{θ}analyzed. An overall average value of the Karman constant in the range 1000 ≤ Re

_{θ}≤ 6500 is estimated ≈ 0.39. The power law assumption is better supported by the analysis of the diagnostic function Γ. A clear plateau is formed in the interval 70 ≲ y

^{+}≲ 250 for 4060 ≤ Re

_{θ}≤ 6500 corresponding to a power law exponent λ = 0.145 = 1/6.89 ≈ 1/7. In comparison for large Re

_{θ}the composite AL84 model exhibits a logarithmic behavior with κ ≈ 0.382 in the interval 80 ≲ y

^{+}≲ 150 and a power law behavior in the interval 100 ≤ y

^{+}≤ 190 with λ = 0.155.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Graphical comparison of Equations (3) and (4). (

**b**) Plot of the difference between Equations (3) and (4).

**Figure 3.**Comparison of f(y

^{+}) given by Equation (4) with the DNS MVPs of Schlatter and Örlü [13]. Re

_{θ}= 1000 (

**a**), 2000 (

**b**), 4060 (

**c**). The deviation from the logarithmic law (for ${\mathrm{y}}^{+}$ ≥ ${\mathrm{y}}_{\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}}^{+}$) points to the need for the inclusion of a wake function.

**Figure 4.**Behavior of the difference [u

^{+}(y

^{+}) − f(y

^{+})] in the inner layer. Re

_{θ}= 1000 (

**a**), 3030 (

**b**), 4060 (

**c**).

**Figure 5.**Mean velocity profile over the whole boundary layer (0 ≤ y ≤ δ). Comparison of the composite profile, calculated by AL84 model [f + g], and the DNS results reported by Schlatter and Örlü [13] for Re

_{θ}= 1000 (

**a**), 2000 (

**b**), 4060 (

**c**). κ = 0.41, Π = 0.55.

**Figure 6.**Graphical representation of the error defined as e(y

^{+}) = u

^{+}(y

^{+}) − [f(y

^{+}) + g($\frac{\mathrm{y}}{\mathsf{\delta}}$, Π)] over the entire boundary layer thickness 0 ≤ y ≤ δ. κ = 0.41, Π = 0.55. Re

_{θ}= 1000 (

**a**), 3030 (

**b**), 4060 (

**c**).

**Figure 7.**Schlatter and Örlü [13] data do not exhibit a logarithmic layer for the moderately large Re

_{θ}= 4060.

**Figure 8.**The function Ξ

_{1}= y

^{+}$\frac{{\mathrm{d}\mathrm{f}}^{+}}{{\mathrm{d}\mathrm{y}}^{+}}$ as predicted by AL84. For y

^{+}≥ 150 Ξ

_{1}= constant = 2.45.

**Figure 9.**The function Ξ

_{2}= y

^{+}$\frac{\mathrm{d}\mathrm{g}}{{\mathrm{d}\mathrm{y}}^{+}}$ as predicted by AL84.

**Figure 14.**Blue dots: estimates of the gradient of the diagnostic function for various Re

_{θ}. Dotted black straight line: least square fit.

**Figure 15.**Upper and lower bounds of the von Kármán’s constant. The black line represents the mean value, κ ≈ 0.393.

Statistics | Re_{θ} = 1000 | Re_{θ} = 3030 | Re_{θ} = 4060 |
---|---|---|---|

Mean | 0.1619 | 0.1518 | 0.1727 |

Standard Error | 0.0162 | 0.0127 | 0.0127 |

Root Mean Square Error | 0.2011 | 0.1928 | 0.2145 |

Mean Square Deviation | 0.1205 | 0.1195 | 0.1279 |

Variance | 0.0145 | 0.0143 | 0.0164 |

Range | 0.3413 | 0.3849 | 0.4244 |

Min | −0.015 | −0.0177 | −0.0182 |

Max | 0.3264 | 0.3673 | 0.4062 |

Number of data points | 55 | 89 | 102 |

Statistics | Re_{θ} = 1000 | Re_{θ} = 3030 | Re_{θ} = 4060 |
---|---|---|---|

Mean | −0.3399 | −0.0736 | 0.0255 |

Standard Error | 0.0409 | 0.0143 | 0.0122 |

Root Mean Square Error | 0.5273 | 0.204 | 0.179 |

Mean Square Deviation | 0.4053 | 0.1908 | 0.1776 |

Variance | 0.1643 | 0.0364 | 0.0315 |

Range | 1.1498 | 0.6367 | 0.6561 |

Min | −0.8778 | −0.2781 | −0.1496 |

Max | 0.272 | 0.3586 | 0.5065 |

Number of data points | 98 | 179 | 212 |

Reference | AL84 Parameters | DNS Parameters | ||||
---|---|---|---|---|---|---|

κ | Π | Re_{θ} | Re_{τ} | u_{τ} | θ | |

Borrell, Sillero and Jimenez, 2013 [14] | 0.41 | 0.55 | 4500 | 1437.066 | 0.0384 | 1.2836 |

0.41 | 0.55 | 5500 | 1709.493 | 0.0374 | 1.5691 | |

0.41 | 0.55 | 6000 | 1847.654 | 0.0371 | 1.712 | |

0.41 | 0.55 | 6500 | 1989.472 | 0.0368 | 1.855 | |

Reference | AL84 Parameters | DNS Parameters | ||||

κ | Π | Re_{θ} | Re_{δ*} | Re_{τ} | C_{f} | |

Schlatter and Örlü, 2010 [13] | 0.41 | 0.55 | 1000 | 1459.397 | 359.38 | 0.0043 |

0.41 | 0.55 | 3030 | 4237.594 | 974.18 | 0.0032 | |

0.41 | 0.55 | 4060 | 5633.318 | 1271.54 | 0.003 |

Datasets | Re_{θ} | Position y^{+} Where Ξ_{max} Appears | Ξ_{max} |
---|---|---|---|

Schlatter and Örlü, 2010 [13] | 4060 | 9.427 | 5.599 |

Borrell, Sillero and Jimenez, 2013 [14] | 4500 | 9.263 | 5.604 |

Borrell, Sillero and Jimenez, 2013 [14] | 5500 | 9.041 | 5.595 |

Borrell, Sillero and Jimenez, 2013 [14] | 6000 | 10.225 | 5.598 |

Borrell, Sillero and Jimenez, 2013 [14] | 6500 | 10.149 | 5.606 |

Datasets | Re_{θ} | Position y^{+} Where Ξ_{max} Appears | Ξ_{max} |
---|---|---|---|

Schlatter and Örlü, 2010 [13] | 4060 | 916.807 | 5.397 |

Borrell, Sillero and Jimenez, 2013 [14] | 4500 | 932.620 | 5.198 |

Borrell, Sillero and Jimenez, 2013 [14] | 5500 | 1140.034 | 5.433 |

Borrell, Sillero and Jimenez, 2013 [14] | 6000 | 1316.612 | 5.466 |

Borrell, Sillero and Jimenez, 2013 [14] | 6500 | 1364.088 | 5.435 |

**Table 6.**Local minima of function Ξ in the inner layer. Ξ

_{min}values and y

^{+}values where Ξ

_{min}appears.

Datasets | Re_{θ} | Position y^{+} Where Ξ_{min} Appears | Ξ_{min} | “κ_{max}” |
---|---|---|---|---|

Schlatter and Örlü, 2010 [13] | 1000 | 53.321 | 2.461 | 0.406 |

H. Abe, 2020 [15] | 1000 | 54.819 | 2.490 | 0.402 |

Schlatter and Örlü, 2010 [13] | 4060 | 71.624 | 2.274 | 0.440 |

Borrell, Sillero and Jimenez, 2013 [14] | 4500 | 69.929 | 2.279 | 0.439 |

Borrell, Sillero and Jimenez, 2013 [14] | 5500 | 71.351 | 2.287 | 0.437 |

Borrell, Sillero and Jimenez, 2013 [14] | 6000 | 73.865 | 2.279 | 0.439 |

Borrell, Sillero and Jimenez, 2013 [14] | 6500 | 73.312 | 2.283 | 0.438 |

Datasets | Re_{θ} | κ |
---|---|---|

Schlatter and Örlü, 2010 [13] | 1000 | 0.402 |

H. Abe, 2020 [15] | 1000 | 0.398 |

Schlatter and Örlü, 2010 [13] | 3030 | 0.403 |

Schlatter and Örlü, 2010 [13] | 4060 | 0.392 |

Borrell, Sillero and Jimenez, 2013 [14] | 4500 | 0.392 |

Borrell, Sillero and Jimenez, 2013 [14] | 5500 | 0.379 |

Borrell, Sillero and Jimenez, 2013 [14] | 6000 | 0.386 |

Borrell, Sillero and Jimenez, 2013 [14] | 6500 | 0.391 |

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**MDPI and ACS Style**

Liakopoulos, A.; Palasis, A.
On the Composite Velocity Profile in Zero Pressure Gradient Turbulent Boundary Layer: Comparison with DNS Datasets. *Fluids* **2023**, *8*, 260.
https://doi.org/10.3390/fluids8100260

**AMA Style**

Liakopoulos A, Palasis A.
On the Composite Velocity Profile in Zero Pressure Gradient Turbulent Boundary Layer: Comparison with DNS Datasets. *Fluids*. 2023; 8(10):260.
https://doi.org/10.3390/fluids8100260

**Chicago/Turabian Style**

Liakopoulos, Antonios, and Apostolos Palasis.
2023. "On the Composite Velocity Profile in Zero Pressure Gradient Turbulent Boundary Layer: Comparison with DNS Datasets" *Fluids* 8, no. 10: 260.
https://doi.org/10.3390/fluids8100260