From Direct Numerical Simulations to Data-Driven Models: Insights into Mean Velocity Profiles and Turbulent Stresses in Channel Flows

Abstract

In this paper, we compare three mathematical models for the mean velocity and Reynolds stress profiles for fully developed pressure-driven turbulent channel flow with the aim of assessing the level of accuracy of each model. Each model is valid over the whole boundary layer thickness (0 $\le y\le$ $\delta$), and it is formulated in terms of a law of the wall and a law of the wake. To calibrate the mathematical models, we use data obtained by direct numerical simulations (DNS) of pressure-driven turbulent channel flow in the range 182 $\le {Re}_{\tau }\le$10,049. The models selected for performance evaluation are two models (Musker’s and AL84) originally developed based on high Reynolds boundary layer experimental data and Luchini’s model, which was developed when some DNS data were also available for wall-bounded turbulent flows. Differences are quantified in terms of local relative or absolute errors. Luchini’s model outperforms the other two models in the “low” and “intermediate” Reynolds number cases (${Re}_{\tau }=$ 182 to 5186). However, for the “high” Reynolds number cases (${Re}_{\tau }=$ 8016 and ${Re}_{\tau }=$ 10,049). Luchini’s model exhibits larger errors than the other two models. Both Musker’s and AL84 models exhibit comparable accuracy levels when compared with the DNS datasets, and their performance improves as the Reynolds number increases.

1. Introduction

Wall turbulence is one of the most challenging unsolved problems in Fluid Dynamics. It is the subject of intense theoretical, experimental, and computational research efforts in the past hundred years. Some recent publications confirm the continuing interest in the subject [1,2,3,4,5,6,7,8]. Beyond its theoretical interest, the subject has important implications in understanding flows in nature as well as in the design and operation of artificial devices. For example, the accurate modeling and possible control of turbulence near solid surfaces of transportation vehicles can greatly reduce fuel consumption with obvious economic and environmental benefits, considering that about 30$\%$ of global energy consumption is “spent” on transport [8,9,10]. Furthermore, the interest in turbulent flow arises from its significant impact on various practical applications, especially in aerodynamics. If the flows were laminar instead of turbulent, fuel consumption, costs, and pollution would decrease. In large commercial aircraft, turbulence in the boundary layer—where airflow transitions near the wing and fuselage—significantly increases drag. Conspicuously, turbulence accounts for approximately 50% of fuel consumption in commercial aircraft [7,11]. Supplementary to what has been said, turbulence is a crucial factor in many mechanical and industrial applications, significantly affecting efficiency, performance and safety. However, excessive turbulence can lead to an increase in pressure losses in piping systems and material fatigue, thus making precise control strategies essential [12,13,14]. In addition, turbulence plays a catalytic role in the dispersion of droplets. Turbulence influences in different ways the efficiency of pesticide application, in the case of agricultural spraying [15], but also the efficiency of combustion in a fuel injection system in industrial spraying [16,17]. Finally, we cannot neglect its relevance to medical aerosol treatments [18,19]. Consequently, understanding and modeling turbulence remain crucial for improving designs, optimizing energy consumption, and diminishing adverse effects in these diverse applications.

Most of the research in turbulent flows is conducted in the framework of the statistical theory of turbulence, and more specifically, it is based on Reynolds averaging, where ensemble averages are replaced by time averages. Engineering computations are usually based on the Reynolds-averaged Navier–Stokes (RANS) equations, which require the adoption of a turbulence model [20,21]. In contradistinction, fundamental computational work is conducted in the framework of direct numerical simulations (DNS), where the unsteady Navier$–$Stokes equations are solved, and averaging is performed as a post-processing step. Carefully conducted DNS provide us with detailed information about the turbulent flow thanks to the highly resolved (in space and time) solutions achieved. DNS are very demanding in computational resources and so this methodology is primarily applied to flows in simple geometries and relatively moderate Reynolds numbers [22,23,24,25].

A small number of technologically important turbulent flows in simple geometries (which are frequently called canonical flows) is the focus of fundamental wall turbulence research. They include planar channel flows (Couette and Poiseuille type), pipe flow, open channel flow, and external flow over a flat plate at zero pressure gradient. For these types of flow, an extensive body of DNS datasets has accumulated over the years, starting from the seminal work of Kim, Moin, and Moser [26] up to the recent work of Hoyas et al. [27]. These DNS datasets provide a wealth of information regarding the flow statistics and the coherent structures observed in wall turbulence.

Although a lot of work has been performed on understanding and modeling canonical flows, there is still a gap in leveraging all the wealth of information provided by the relevant Direct Numerical Simulations. In this paper, we focus on the case of pressure-driven turbulent channel flow. Three well-documented composite models of the mean velocity profiles (MVPs) are compared for wall-bounded turbulent flow. The composite models approximate the MVP as a superposition of a law of the wall expression and a wake function. Two of them (Musker’s [28] and Liakopoulos’ [29]) preserve a form initially proposed for external turbulent boundary layers and adjust the strength of the wake function depending on the flow Reynolds number. On the other hand, Luchini’s model [30,31] proposes a wake function that is independent of the Reynolds number. After optimization of any free parameter in each MVP model, their accuracy is assessed for pressure-driven channel flow where high-quality DNS datasets are available for friction Reynolds numbers in the range 182$-$10,049. The planar pressure-driven channel flow is selected due to the fact that this type of flow presents fewer challenges for the DNS computation methodology compared e.g., to turbulent pipe flow [32,33,34,35].

A notable addition to the study of wall-bounded turbulent flows is Heinz’s model [36,37,38,39], which is based on the asymptotic analysis of canonical turbulent flows and provides a theoretically rigorous framework for describing mean velocity profiles (MVPs) at high Reynolds numbers. This recent contribution is particularly significant as it offers a probabilistic velocity model (PVM) that demonstrates excellent agreement with direct numerical simulation (DNS) and experimental data throughout various canonical flows. By incorporating a physically justified probabilistic interpretation, Heinz’s model ensures low predictive error and aligns with fundamental turbulence characteristics, making it a valuable addition to the relevant literature.

This paper is structured as follows. In Section 2, we present some aspects of the statistical theory of turbulent boundary layer and channel flow required for the presentation and discussion of our results. In the same section, we present the three MVP models chosen for comparison as well as the optimization of the free parameters in each model. In Section 3, we present the comparisons of mathematical model predictions in terms of MVP, viscous stress, and Reynolds shear stress, both qualitatively and quantitively. Finally, the conclusions are summarized in Section 4.

2. Mathematical Preliminaries

The channel geometry and the coordinate system used in this study are shown in Figure 1.

The streamwise, wall-normal, and spanwise averaged-in-time (mean) velocity components at a point are denoted by ($\overline{u},\overline{v},\overline{w})$. Correspondingly, ($u^{\prime },v^{\prime },w^{\prime })$ denote velocity fluctuations, and $\overline{p}$ the time-averaged local pressure. The time-averaged flow is parallel, i.e., the mean velocity field is ($\overline{u }$, 0, 0) and the averaged-in-time streamwise velocity, $\overline{u}$, is a function of the distance from the lower wall, $y$, only.

The flow in the lower half of the channel is a turbulent boundary layer of a constant thickness ($\delta$ $=$ $h$) and has the usual structure of the three-layer model, i.e., an inner layer, an outer layer, and an overlapping layer.

For the mathematical description of the inner zone, suitable dimensionless variables are $y^{+}=y{u}_{*}/\nu$ and $u^{+}=\overline{u}/u_{*}$, where ν is the kinematic viscosity coefficient and $u_{*}$ is friction velocity at the wall [40]. Appropriate variables for the outer layer are $\left(U-\overline{u}\right)/u_{*}$ and $y/h=y/\delta$. Here, $\delta =h$ is the boundary layer thickness, and $U$ the mean velocity at the channel midplane. With these dimensionless variables, the mean velocity profile is described in the inner layer by the law of the wall ($u^{+}={\phi }_1\left(y^{+}\right)$) and in the outer layer by the velocity defect law $(U-u)/u_{*}={\phi }_2\left(y/\delta \right)$. The law of the wall $u^{+}={\phi }_1\left(y^{+}\right)$ is strictly independent of the Reynolds number when the Reynolds number is sufficiently high. However, a widely accepted approximation of the buffer zone (4 $\lesssim y^{+}\le y_{*}^{+}$) is not available. An overlap layer, where the two laws are both valid, is assumed to give rise to the logarithmic law [40]

$$u^{+}=(1/\kappa )ln{y}^{+}+B \mathrm{for} y_{\mathrm{*}}^{+}\le y^{+}\le y_{high}^{+}$$

(1)

The logarithmic law (with constant $\kappa$ and ${\rm B}$) is valid asymptotically as the Reynolds number $Re \to \infty$ [41]. In practical terms, for each type of canonical flow, there exists a sufficiently high Reynolds number where $\kappa$ and ${\rm B}$ reach asymptotically their constant values. For lower values of $Re$ it is assumed, on dimensional grounds, that $\kappa =\kappa \left(Re\right)$ and $B=B\left(Re\right)$. This approach is treated differently in the three examined MVP models and is discussed in Section 2.1.

Coles, after carefully analyzing the available experimental data for external turbulent boundary layers, reformulated the velocity-defect law and instead proposed the “law of the wake”. Specifically, Coles introduced a wake function, $w$, to describe the deviation of the actual velocity distribution from the logarithmic law. In this formulation

$$u^{+}=(\mathrm{law} \mathrm{of} \mathrm{the} \mathrm{wall})+(\mathrm{law} \mathrm{of} \mathrm{the} \mathrm{wake})$$

(2)

where the law of the wake introduces an additional parameter, Π, to quantify the strength of the wake [42].

2.1. Composite MVPs

A number of authors have proposed composite models that approximate, with a single formula, the MVP over the whole boundary layer thickness, including the viscous sublayer ($y^{+}\lesssim$ 4) and the buffer layer (4 $\le y^{+}\le y_{*}^{+}$). The value of $y_{*}^{+}$ varies but is conventionally considered to be about 30 [41,43].

In this paper, we compare three such mathematical models of the MVP. They are listed below for easy reference.

 1.

Musker’s model

The composite MVP expression is given in the form [28]

$$\begin{gathered} u^{+}=5.424 {\mathit{tan}}^{-1}\left({\displaystyle \frac{{2y}^{+}-8.15}{16.7}}\right)+{\mathit{log}}_{10}\left({\displaystyle \frac{{\left(y^{+}+10.6\right)}^{9.6}}{{\left(y^{{+}^2}-8.15y^{+}+86\right)}^2}}\right)-3.52 \\ +2.44\left\{\Pi \left[6({{\displaystyle \frac{y}{\delta }})}^2-4({{\displaystyle \frac{y}{\delta }})}^3\right]+\left[({{\displaystyle \frac{y}{\delta }})}^2\left(1-{\displaystyle \frac{y}{\delta }}\right)\right]\right\} \end{gathered}$$

(3)

 2.

Liakopoulos’ model

In this model, the MVP is expressed as [29]

$$u^{+}=f\left(y^{+}\right)+g\left(\Pi ,{\displaystyle \frac{y}{\delta }}\right)$$

(4)

where

$$\begin{gathered} f\left(y^{+}\right)=ln\left[{\displaystyle \frac{{\left(y^{+}+11\right)}^{4.02}}{{{(y}^{+2}-7.37y^{+}+83.3)}^{0.79}}}\right] \\ +5.63 {\mathit{tan}}^{-1}(0.12y^{+}-0.441)-3.81 \end{gathered}$$

(5)

and

$$g\left(\Pi ,{\displaystyle \frac{y}{\delta }}\right)={\displaystyle \frac{1}{\kappa }}\left(1+6\Pi \right){\left({\displaystyle \frac{y}{\delta }}\right)}^2-{\displaystyle \frac{1}{\kappa }}\left(1+4\Pi \right){\left({\displaystyle \frac{y}{\delta }}\right)}^3$$

(6)

The function $g$ defined in Equation (6) is related to the wake function, $w$, by the relation $g\left(\Pi ,y/\delta \right)=(\Pi /\kappa )w\left(y/\delta \right)$.

It is noted that the original Coles’ proposal for the wake function of external boundary layers [$w\left(y/\delta \right)=2{sin}^2\left(\pi y/2\delta \right)$] is not appropriate for wall-bounded flows as the one studied in this work. We also note that the wake functions in Musker and Liakopoulos expressions are identical [44].

Function $g$ in Equation (6) attains its maximum value

$$g_{max}={\displaystyle \frac{1}{\kappa }}{\displaystyle \frac{4}{27}}{\displaystyle \frac{{\left(1+6\Pi \right)}^3}{{\left(1+4\Pi \right)}^2}} \mathrm{a}\mathrm{t} {\displaystyle \frac{y}{\delta }}={\displaystyle \frac{2}{3}}\left[{\displaystyle \frac{\left(1+6\Pi \right)}{\left(1+4\Pi \right)}}\right]$$

(7)

The Taylor series expansion of $g_{max}$around $\Pi =$0, using [45], gives the power series

$$\begin{gathered} g_{max}={\displaystyle \frac{1}{\kappa }}{\displaystyle \frac{4}{27}}\left(1+10\Pi +12{\Pi }^2-40{\Pi }^3+128{\Pi }^4-384{\Pi }^5+1024{\Pi }^6 \\ -2048{\Pi }^7+\mathrm{32,768}{\Pi }^9-\mathrm{262,144}{\Pi }^{10}+\mathcal{O}({\Pi }^{11}\right)) \end{gathered}$$

(8)

As we shall see in Section 2.2., in the case of pressure-driven channel flow, 0.1 $\le \Pi \le$ 0.2 and consequently $g_{max}$ is, with good approximation, a linear function of $\Pi$.

In the remainder of this paper, we will frequently refer to this model as AL84 following references [46,47,48].

 3.

Luchini’s model

Luchini’s models for canonical wall-bounded flows are of the form [30,31]

$$u^{+}=\phi \left(y^{+}\right)+G\left({\displaystyle \frac{y}{\delta }}\right)$$

(9)

and incorporate different forms of “wake” functions for planar Poiseuille channel flow, Couette flow, open channel, and pipe flows [30,31].

For pressure-driven channel flow, Luchini assumes $\kappa =$ 0.392 and ${\rm B}=$ 4.48. These values are embedded in the mathematical expression for $\phi \left(y^{+}\right)$,

$$\begin{gathered} \phi \left(y^{+}\right)={\displaystyle \frac{\mathit{ln}\left(y^{+}+3.109\right)}{0.392}}+4.48 \\ -{\displaystyle \frac{7.3736+\left(0.493-0.0245y^{+}\right)y^{+}}{1+(0.05736+0.01101y^{+})y^{+}}}{e}^{-0.03385y^{+}} \end{gathered}$$

(10)

The corresponding expression for the wake function is

$$G\left({\displaystyle \frac{y}{\delta }}\right)={\displaystyle \frac{y}{\delta }}-0.57{\left({\displaystyle \frac{y}{\delta }}\right)}^7$$

(11)

Note that Luchini’s MVP model does not involve the Coles’ parameter $\Pi$ (at least explicitly) and that the $G$ function for planar Poiseuille flow is everywhere positive for 0 $<y/\delta \le$ 1 and attains its maximum value, $G_{max}=$ 0.681, at $y/\delta =$ 0.794.

2.2. DNS Data for Model Calibration

In this work, we used direct numerical simulation data published by Lee and Moser [49], Bernardini, Pirozzoli, and Orlandi [50], Lozano-Durán and Jimenez [51], Yamamoto and Tsuji [52], and Hoyas, Oberlack et al. [27] in order to optimize the parameter $\Pi$ in the first two MVP models for each Reynolds number considered. The specific datasets employed in the present work are listed in

Table 1. Datasets considered in the present study. ${Re}_{\tau }=u_{*}h/\nu$.

Case	Datasets	${\mathit{R}\mathit{e}}_{\mathit{\tau }}$
LM180	Lee and Moser, 2015 [49]	182
LM550	Lee and Moser, 2015 [49]	544
LM1000	Lee and Moser, 2015 [49]	1000
LM2000	Lee and Moser, 2015 [49]	1995
BPO4079	Bernardini, Pirozzoli and Orlandi, 2014 [50]	4079
LDJ4179	Lozano-Durán and Jiménez, 2014 [51]	4179
LM5200	Lee and Moser, 2015 [49]	5186
YT8016	Yamamoto and Tsuji, 2018 [52]	8016
HO10,000	Hoyas, Oberlack et al., 2022 [27]	10,049

, together with the friction Reynolds number, ${Re}_{\tau }$, corresponding to each dataset.

In Musker’s and Liakopoulos’ composite models, the values of the logarithmic law parameters ($\kappa$ and $B$) have been “hardcoded” in the mathematical expressions with fixed values. Consequently, the only parameter to be optimized is Coles’ wake strength parameter, $\Pi$. In order to optimize the value of $\Pi$ for each Reynolds number, we introduce the objective function, ${\rm E}$, and consider the minimization problem.

$$minimize {\rm E}\left(\Pi \right)=\sum _{i=1}^n{\left[u_{i,DNS}^{+}-u_{i,Model}^{+}\right]}^2$$

(12)

where $u_{i,Model}^{+}$ stands for $u_{i,Musker}^{+}$or $u_{i,AL84}^{+}$ and the summation index $i$ runs over these DNS data.

The optimization algorithm has been developed exclusively in a Python environment (Version 3.9.13). In order to achieve that, the L-BFGS-B (Limited-memory-Broyden–Fletcher–Goldfarb–Shanno) Python routine has been utilized. This routine is included in the more general SciPy Python library and is used to minimize a scalar function of one or more variables using the L-BFGS-B algorithm. The L-BFGS-B algorithm was originally written in Fortran by [53] with the help of [54]. The algorithm follows an iterative procedure in order to achieve a bound constrained minimization of the given function and produces as output the optimized variables. Finally, the iterative procedure terminates when

$${\displaystyle \frac{\left(f_{\kappa }-f_{\kappa +1}\right)}{max\left(\left|f_{\kappa +1}\right|,\left|f_{\kappa }\right|,1\right)}}\le \mathrm{f}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{r}\ast \mathrm{e}\mathrm{p}\mathrm{s}$$

(13)

Condition (13) is satisfied, where eps is the machine precision. For $\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{r}$ the value of 1e7 has been used in order to achieve moderate accuracy.

The optimal values of $\Pi$ for each friction Reynolds number are listed in

Table 2. Parameters of the Musker’s and AL84 model (fixed $\kappa$, ${\rm B}$ and optimized Coles’ parameter $\Pi$).

Case	${\mathit{R}\mathit{e}}_{\mathit{\tau }}$	$\mathit{\kappa }$	$\mathit{B}$	$\mathit{\Pi }$ (Musker)	$\mathit{\Pi }$ (AL84)
LM180	182	0.41	5.0	0.16	0.15
LM550	544	0.41	5.0	0.14	0.12
LM1000	1000	0.41	5.0	0.17	0.14
LM2000	1995	0.41	5.0	0.20	0.18
BPO4079	4079	0.41	5.0	0.17	0.14
LDJ4179	4179	0.41	5.0	0.16	0.13
LM5200	5186	0.41	5.0	0.18	0.15
YT8016	8016	0.41	5.0	0.13	0.10
HO10,000	10,049	0.41	5.0	0.14	0.11

.

2.3. RANS Equations for Pressure-Driven Channel Flow

For the channel flow studied, after an order of magnitude analysis and some mathematical manipulations, the $x$-component RANS equation takes the form

$$\nu {\displaystyle \frac{d\overline{u}}{dy}}+\overline{u^{\prime }{v}^{\prime }}=u_{\mathrm{*}}^2\left(1-{\displaystyle \frac{y}{h}}\right)$$

(14)

where $u_{*}=\sqrt{{\tau }_w/\rho }$ is the friction velocity, ${\tau }_w$is the mean (averaged-in-time) shear stress at the wall, $\nu$ the kinematic viscosity and $\rho$ the fluid density. The wall pressure gradient in the streamwise direction, $d{p}_w/dx$, is related to the mean shear stress at the wall, ${\tau }_w$, by the relation [55]

$${\tau }_w=-h{\displaystyle \frac{{dp}_w}{dx}}$$

(15)

Equation (14), expressed in terms of inner variables $y^{+}=y{u}_{*}/\nu$, $u^{+}=\overline{u}/u_{*}$, takes the dimensionless form

$${\displaystyle \frac{d{u}^{+}}{d{y}^{+}}}+({-\overline{u^{\prime }{v}^{\prime }})}^{+}=1-{\displaystyle \frac{y^{+}}{{Re}_{\tau }}}$$

(16)

where ${Re}_{\tau }=u_{*}h/\nu$ is the friction Reynolds number formed using the channel half height (as characteristic length) and the friction velocity (as characteristic velocity). The term $(-{\overline{u^{\prime }{v}^{\prime }})}^{+}=-\overline{u^{\prime }{v}^{\prime }}/u_{*}^{2 }$ is the normalized covariance of the fluctuations $u^{\prime }$ and $v^{\prime }$, and corresponds to the normalized Reynolds shear stress $\left(-\rho \overline{u^{\prime }{v}^{\prime }}\right)/u_{*}^2$.

3. Results

3.1. MVPs

The clustering method used here is the K-Means algorithm, a widely applied unsupervised machine learning technique. The algorithm groups data into a predefined number of clusters based on the similarity of a chosen feature—in this case, the Reynolds number. The method successfully classifies these data into three distinct groups, effectively separating the flow regimes by ${Re}_{\tau }$ and providing insights into trends within each cluster (see Figure 2). This approach is valuable for understanding and analyzing turbulent flow behavior in varying Reynolds number ranges.

A cluster analysis of the datasets listed in Table 1 leads to a classification of these data into three subgroups: “low” [182–1995], “intermediate” [4079–5186], and “high” Reynolds number [8016–10,049] cases (see Figure 3). It is noted that the ${Re}_{\tau }=$ 182 datasets can be considered as a separate case of very low ${Re}_{\tau }$ turbulent flow. We evaluate below the accuracy of each model’s predictions for each subgroup of direct numerical simulations.

For the “low” Reynolds number cluster, Luchini’s model’s accuracy is very good, even for the ${Re}_{\tau }=$ 182. The accuracy of Musker’s and AL84 model increases gradually as the Reynolds number increases. In the interval 1 $\le$ $y^{+}\le$ 15 and for $y/h\ge$0.2 all three models agree with these DNS data.

For the “intermediate” Reynolds number cases, the agreement between Luchini’s model and direct numerical simulation data is still excellent. Musker’s and AL84 model estimating the time-averaged velocity profile are in full agreement with DNS data over the whole thickness of the boundary layer except for the region 25 $\le$ $y^{+}\le$ 50. It should be noted that all three mathematical models approximate very well these DNS data over the whole boundary layer thickness in the case of BPO4079.

Finally, for the group designated as “high” Reynolds number cases, all three models work very well close to the wall in the interval 1 $\le$ $y^{+}\le$ 15. Luchini’s model overestimates the MVP for $y^{+}\ge$ 15. AL84 shows an excellent agreement throughout the thickness of the boundary layer for the case of YT8016. In addition, AL84 provides more accurate results than Musker’s model for the “high” Reynolds number cases (YT8016 and HO10,000).

These differences are quantified in Section “Profiles of Local Relative Error”, where the relative errors for each model are discussed.

Profiles of Local Relative Error

In order to quantify the degree of approximation achieved in each case, we introduce the local relative error defined as $e_1\left(y^{+}\right)=\left|u_{DNS}^{+}\left(y^{+}\right)-u_{Model}^{+}\left(y^{+}\right)\right|/u_{DNS}^{+}\left(y^{+}\right)$.

For “low” Reynolds number cases, the relative error that Musker’s and AL84 models exhibit is about 4% in the inner law region (with the sole exception of the case of LM180, see Figure 4a, where the Reynolds number is quite low), while in the outer region, the relative error observed decreases significantly to less than 2%. Analogous behavior is observed in all profiles for 544 $\le {Re}_{\tau }\le$ 1995. Figure 4, specifically cases (b), (c) and (d), illustrates that Luchini’s mathematical model exhibits a distinct behavior in contrast to the other two models, with an error rate of less than 1% throughout the entire boundary layer thickness.

For the “intermediate” Reynolds number cases—Figure 4e–g—the relative error remains low for all three mathematical models ($\lesssim$2%). Luchini’s model shows a smaller error in the inner layer compared with the other two, but it slightly underperforms far from the wall.

Finally, when we focus on “high” Reynolds number cases, the performance trends of the mathematical models change. In Musker’s and Liakopoulos’ models, the errors become smaller as the Reynolds number increases (see Figure 4h,i). In contradistinction, Luchini’s model does not follow this trend displayed by Musker’s and AL84 models. Specifically, the case of YT8016 shows a constant value of 1.5% in the interval 0.1 $\le y/h\le$ 1, while for the case of HO10,000, the relative error is approximately 1% in the interval 0.3 $\le y/h\le$ 1.

To quantify the performance of each model, we compare two key statistical indices: the Root Mean Square Error ($RMSE$), which captures the overall deviation from these DNS data, and the Nash–Sutcliffe Coefficient ($NSC$), which evaluates the predictive ability relative to the DNS baseline. These metrics provide a comprehensive evaluation of the accuracy and reliability of the models, as shown in

Table 3. Evaluation indices for model validation: Root Mean Square Error ($RMSE$) and Nash–Sutcliffe Coefficient ($NSC$) for each mathematical model.

Case	${\mathit{R}\mathit{e}}_{\mathit{\tau }}$	Model	$\mathit{R}\mathit{M}\mathit{S}\mathit{E}$	$\mathit{N}\mathit{S}\mathit{C}$
LM180	182	Musker	0.3535	0.9965
LM550	544	Musker	0.1765	0.9991
LM1000	1000	Musker	0.1603	0.9992
LM2000	1995	Musker	0.1691	0.9991
BPO4079	4079	Musker	0.1441	0.9993
LDJ4179	4179	Musker	0.2121	0.9976
LM5200	5186	Musker	0.2127	0.9980
YT8016	8016	Musker	0.1162	0.9992
HO10,000	10,049	Musker	0.2047	0.9974
LM180	182	AL84	0.3627	0.9963
LM550	544	AL84	0.1766	0.9991
LM1000	1000	AL84	0.1518	0.9993
LM2000	1995	AL84	0.1441	0.9994
BPO4079	4079	AL84	0.1066	0.9996
LDJ4179	4179	AL84	0.1711	0.9984
LM5200	5186	AL84	0.1699	0.9987
YT8016	8016	AL84	0.0703	0.9997
HO10,000	10,049	AL84	0.1557	0.9985
LM180	182	Luchini	0.1663	0.9992
LM550	544	Luchini	0.0266	1.0000
LM1000	1000	Luchini	0.0309	1.0000
LM2000	1995	Luchini	0.0552	0.9999
BPO4079	4079	Luchini	0.1023	0.9996
LDJ4179	4179	Luchini	0.1287	0.9991
LM5200	5186	Luchini	0.0813	0.9997
YT8016	8016	Luchini	0.2799	0.9954
HO10,000	10,049	Luchini	0.2681	0.9956

.

$$RMSE=\sqrt{\sum _{i=1}^n{\displaystyle \frac{{\left(u_{Model}-u_{DNS}\right)}^2}{n}}}$$

(17)

$$NSC=1-{\displaystyle \frac{\sum _{i=1}^n{\left(u_{DNS}-u_{Model}\right)}^2}{\sum _{i=1}^n{\left(u_{DNS}-{\overline{u}}_{DNS}\right)}^2}}$$

(18)

3.2. Viscous and Reynolds Shear Stress Profiles

As mentioned in Section 2.3., simplifying the Reynolds-averaged Navier$–$Stokes (RANS) equations for fully developed, incompressible, pressure-driven turbulent channel flow, one obtains Equation (16) [55]. The first term on the left-hand side of Equation (16) represents the dimensionless viscous stress, while the second term is the normalized Reynolds shear stress ($\rho =$ 1).

3.2.1. Viscous Stress Profiles

For each of the three composite MVP models the viscous stress has been calculated analytically by differentiating the MVP expression using symbolic algebra [45]. For the DNS datasets, the viscous stress was calculated by numerical differentiation using Equation (19) for unequally spaced data.

$$\begin{gathered} {\displaystyle \frac{d{u}^{+}}{{dy}^{+}}}=u^{+}\left(y_{i-1}^{+}\right){\displaystyle \frac{{2y}^{+}-y_i^{+}-y_{i+1}^{+}}{\left(y_{i-1}^{+}-y_i^{+}\right)\left(y_{i-1}^{+}-y_{i+1}^{+}\right)}}+u^{+}\left(y_i^{+}\right){\displaystyle \frac{{2y}^{+}-y_{i-1}^{+}-y_{i+1}^{+}}{\left(y_i^{+}-y_{i-1}^{+}\right)\left(y_i^{+}-y_{i+1}^{+}\right)}} \\ +u^{+}\left(y_{i+1}^{+}\right){\displaystyle \frac{{2y}^{+}-y_{i-1}^{+}-y_i^{+}}{\left(y_{i+1}^{+}-y_{i-1}^{+}\right)\left(y_{i+1}^{+}-y_i^{+}\right)}} \end{gathered}$$

(19)

For the cases where the Reynolds number is “low” (see Figure 5a–d), the largest discrepancies between the mathematical models and direct numerical simulation data are observed in the interval 8 $\le y^{+}\le$ 20 (buffer layer). The largest deviations between the mathematical models and these DNS data are observed in the LM180 profile, where Luchini’s model performs better than the other two. Turning our attention to the cases LM550, LM1000, and LM2000, Luchini’s model outperforms the other two in the interval 10 $\le y^{+}\le$ 30. It is also mentioned that in these cases, it underestimates the viscous shear stress in the interval 1 $\le y^{+}\le$ 2.

It can be inferred that with increasing Reynolds number, there is very good agreement between the mathematical models and these DNS data in the group of “intermediate” Reynolds number cases (see Figure 5e,f). However, Luchini’s model still underestimates the viscous shear stress in the interval 1 $\le y^{+}\le$ 2.

At “high” Reynolds number cases (see Figure 5g,h), all three mathematical models overestimate the viscous shear stress in the interval 3 $\le y^{+}\le$ 12. Luchini’s model continues to underestimate the viscous shear stress in the interval 1 $\le y^{+}\le$ 3.

In general, for all Reynolds numbers (except LM180), we observe that in the scale of Figure 5, the three models seen to predict correctly the asymptotic behavior of viscous stress for $y^{+}\ge$ 100. This statement is reconsidered in Section 3.2.2.

3.2.2. Relative Error in Profiles of Viscous Stress

Similarly to Section “Profiles of Local Relative Error”, the local relative error is defined as $e_2\left(y^{+}\right)=\left|{\left({du}^{+}/{dy}^{+}\right)}_{DNS}-{\left({du}^{+}/{dy}^{+}\right)}_{Model}\right|/{\left({du}^{+}/{dy}^{+}\right)}_{DNS}$.

In the case of a “low” Reynolds number, we observe that Musker’s and AL84 models have similar behavior with no noticeable differences throughout the boundary layer thickness. In contrast to these two, Luchini’s model exhibits different behavior. For most of the boundary layer, it shows the smallest local relative error. However, in the interval 0.7 $\lesssim y/h\lesssim$ 0.9, the error becomes larger and continues its upward trend until $y/h=$ 1 (see Figure 6a–d).

If we focus on the “intermediate” and “high” Reynolds numbers (see Figure 6e–h), we observe that Luchini’s model exhibits the smallest local relative error for $y/h\lesssim$ 0.6. However, its error becomes sharply larger in the interval 0.6 $\lesssim y/h\lesssim$ 0.9 and maintains its upward trend up to $y/h=$ 1. As expected, the error for AL84 decreases with increasing Reynolds number. In particular, in the case of the LM5200 dataset, the difference compared with Musker’s model is noticeable. We also bring to the attention of the reader that the relative error of both Musker’s and AL84 models increases considerably in the interval 0.9 $\lesssim y/h\lesssim$ 1 in the case of YT8016.

A number of remarks should be made with respect to error behavior in viscous stress profiles. First of all, we remark that close to the channel midplane, $d{u}^{+}/d{y}^{+}$ is close to zero, so the relative errors are typically high. It is also important to assess the extent to which the mean velocity boundary condition at the channel midplane is being met. Writing the general form of the three mathematical models as $u^{+}=f\left(y^{+}\right)+g\left(\eta \right)$ where $\eta =y/\delta$, we find that

$${\displaystyle \frac{d{u}^{+}}{d{y}^{+}}}={\displaystyle \frac{df}{{dy}^{+}}}+{\displaystyle \frac{1}{{\delta }^{+}}}{\displaystyle \frac{dg}{d\eta }}$$

(20)

where ${\delta }^{+}=\delta u_{\mathrm{*}}/\nu ={Re}_{\tau }$. Furthermore,

$${\displaystyle \frac{d{u}^{+}}{d{y}^{+}}}={\displaystyle \frac{\nu }{u_{\mathrm{*}}^2}}{\displaystyle \frac{du}{dy}}$$

(21)

Due to the symmetry of the MVP profile, $du/dy=$ 0 at $y=h$. Consequently, the “correct” value of $d{u}^{+}/d{y}^{+}$ at $y^{+}={Re}_{\tau }$ should be zero. Instead, each model gives the values of $d{u}^{+}/d{y}^{+}$ at the midplane, as listed in

Table 4. Satisfying the symmetry boundary condition at the channel midplane. Comparison among the three models.

	$\mathit{d}{\mathit{u}}^{+}/\mathit{d}{\mathit{y}}^{+}$$\mathbf{Value} \mathbf{at} {\mathit{y}}^{+}={\mathit{R}\mathit{e}}_{\mathit{\tau }}$
${\mathit{R}\mathit{e}}_{\mathit{\tau }}$	Musker	AL84	Luchini
182	$-$$6.88\times {10}^{-5}$	$5.86\times {10}^{-5}$	$-$$2.78\times {10}^{-3}$
544	$-$$2.73\times {10}^{-5}$	$-$$3.12\times {10}^{-6}$	$-$$8.34\times {10}^{-4}$
1000	$-$$1.20\times {10}^{-5}$	$-$$1.24\times {10}^{-6}$	$-$$4.47\times {10}^{-4}$
1995	$-$$4.88\times {10}^{-6}$	$-$$1.81\times {10}^{-7}$	$-$$2.22\times {10}^{-4}$
4079	$-$$2.06\times {10}^{-6}$	$6.49\times {10}^{-8}$	$-$$1.08\times {10}^{-4}$
5186	$-$$1.56\times {10}^{-6}$	$7.85\times {10}^{-8}$	$-$$8.49\times {10}^{-5}$
8016	$-$$9.61\times {10}^{-7}$	$7.48\times {10}^{-8}$	$-$$5.49\times {10}^{-5}$
10,049	$-$$7.52\times {10}^{-7}$	$6.70\times {10}^{-8}$	$-$$4.38\times {10}^{-5}$

.

A value of $d{u}^{+}/d{y}^{+}$ at $y^{+}={Re}_{\tau }$ based on the DNS results is not readily available because the point $y_{i, DNS,max}^{+}$ of the DNS simulations is near but not exactly at $y^{+}={\delta }^{+}={Re}_{\tau }$. Consequently, the comparison of each model’s prediction to the DNS results should be made at $y_{i,DNS,max}^{+}$ for each DNS dataset. These comparisons are listed in

Table 5. Viscous stress. Comparison of model predictions with DNS data close to channel midplane.

${\mathit{R}\mathit{e}}_{\mathit{\tau }}$	${\mathit{y}}_{\mathit{i},\mathit{D}\mathit{N}\mathit{S},\mathit{m}\mathit{a}\mathit{x}}^{+}$	Musker	AL84	Luchini	$\mathit{d}{\mathit{u}}^{+}/\mathit{d}{\mathit{y}}^{+}$ by DNS
182	180.6	7.01 $\times {10}^{-4}$	8.22 $\times {10}^{-4}$	$-$1.59 $\times {10}^{-3}$	6.40 $\times {10}^{-4}$
544	541.2	9.53 $\times {10}^{-5}$	1.17 $\times {10}^{-4}$	$-$6.33 $\times {10}^{-4}$	1.31 $\times {10}^{-4}$
1000	997.4	4.03 $\times {10}^{-5}$	4.93 $\times {10}^{-5}$	$-$3.65 $\times {10}^{-4}$	5.38 $\times {10}^{-5}$
1995	1990.6	1.33 $\times {10}^{-5}$	1.79 $\times {10}^{-5}$	$-$1.95 $\times {10}^{-4}$	1.68 $\times {10}^{-5}$
5186	5180.7	1.61 $\times {10}^{-6}$	3.27 $\times {10}^{-6}$	$-$7.99 $\times {10}^{-5}$	2.91 $\times {10}^{-6}$
8016	7996.0	3.89 $\times {10}^{-6}$	4.77 $\times {10}^{-6}$	$-$4.67 $\times {10}^{-5}$	7.76 $\times {10}^{-7}$
10,049	10049.3	$-$8.98 $\times {10}^{-7}$	5.98 $\times {10}^{-8}$	$-$4.38 $\times {10}^{-5}$	0

.

Near the channel wall, although the curves of $d{u}^{+}/d{y}^{+}$ predictions follow closely the DNS curves, the relative errors are quite high. This is explained by the fact the pointwise error (for each $y$ given in the DNS dataset) is strictly correct but, in a sense, misleading. Something similar is observed when we consider, for example, two sinusoidal curves of the same period and amplitude but with a small phase error. The pointwise error appears very large, although the closeness of the two curves is apparent.

3.2.3. Reynolds Shear Stress Profiles

Having expressed the viscous stress analytically for each mathematical model, one can use Equation (16) to express analytically, in closed form, the corresponding Reynolds shear stress. These analytic expressions can be evaluated at the $y_{i,DNS}^{+}$ points of the DNS simulations and be compared with the DNS Reynolds stress profiles reported in [27,49,52]. At first glance, this comparison is superfluous since Equation (16) is exact. However, the DNS computation of Reynolds stresses involves decisions about temporal/spatial averaging, the length of the time record analyzed to obtain statistically stable results, etc. This postprocessing step in DNS may introduce additional uncertainty or errors. Consequently, we have decided to compare each model’s prediction to the DNS results.

Examining Figure 7, the “low” Reynolds number cases, and more specifically, if we focus on the LM550, LM1000, and LM2000 cases, we find that Musker’s and AL84 mathematical models exhibit the same behavior with small deviations from these DNS data in the interval 8 $\lesssim y^{+}\lesssim$ 25 (buffer layer), while Luchini’s mathematical model gives better accuracy over the whole thickness of the boundary layer. The LM180 case requires “special treatment”; we report it although the Reynolds number is quite low, where the largest deviations of all three mathematical models occur in the interval 1 $\lesssim y^{+}\lesssim$ 30. In Section 3.2.4, we also present the error of each model with these DNS data for completeness.

For the “intermediate” Reynolds number cases, and more specifically in the case of BPO4079, we observe that in two intervals, 1 $\lesssim y^{+}\lesssim$ 7 and 150 $\lesssim y^{+}\lesssim$ 450, the three models give the same predictions, which differ from direct numerical simulation data. For the case of LM5200, Musker’s and AL84 models deviate from these DNS data in the interval 15 $\lesssim y^{+}\lesssim$ 30 while Luchini’s model has better behavior throughout the boundary layer thickness.

Finally, for the “high” Reynolds number subgroup, we distinguish between two cases: (i) all three mathematical models show discrepancies in the interval 1 $\lesssim y^{+}\lesssim$ 7 (viscous sublayer and lower part of the buffer layer) when compared with the YT8016 dataset and (ii) for the HO10,000 case, considerable differences are observed in the interval 10 $\lesssim y^{+}\lesssim$ 25 (well within the buffer layer) for Musker’s and AL84 models only.

3.2.4. Errors in Reynolds Shear Stress Profiles

In this section, we define a local absolute error $e_3\left(y^{+}\right)=\left|{{\left(-\overline{u^{\prime }{v}^{\prime }}\right)}^{+}}_{DNS}-{{\left(-\overline{u^{\prime }{v}^{\prime }}\right)}^{+}}_{Model}\right|$. In Figure 8, all three mathematical models show consistent agreement in predicting the Reynolds shear stress with minimal error, particularly for Reynolds numbers ranging from 1000 to 10,049. For Musker’s and AL84 models, the mean value of the maximum absolute error over this ${Re}_{\tau }$ range is 3.2 $\times {10}^{-2}$. In comparison, Luchini’s model achieves better accuracy with a corresponding value of 1.8 $\times {10}^{-2}$. It is also mentioned here that Musker’s and AL84 models have larger errors in the inner boundary layers in the cases LM180 and LM550.

4. Discussion and Conclusions

We have examined the performance of three mathematical models describing the mean velocity, viscous stress and Reynolds shear stress distributions in turbulent pressure-driven channel flow. Their performance was examined in the range of available DNS data, i.e., 182 $\le {Re}_{\tau }\le$ 10,049. Each model is valid over the whole boundary layer thickness (0 $\le y\le$ $h$), and it is formulated in terms of different forms of the law of the wall and the law of the wake. Derivatives of the mean velocity have been computed analytically using symbolic algebra. When applicable, Coles’ wake strength parameter has been optimized based on DNS data. In Musker’s and AL84’s models, the parameters of the logarithmic law are considered fixed ($\kappa =$ 0.41 and $B=$ 5.0). Luchini’s model, on the other hand, incorporates the values $\kappa =$ 0.392 and $B=$ 4.48 and introduces a parameter-free wake function. In the range of friction, Reynolds numbers ${Re}_{\tau }$ considered [182, 10,049], the parameters $\kappa$ and $B$ of the logarithmic law are not constant but deviate “slightly” from the usually quoted values ($\kappa =$ 0.41, $B=$ 5.0), which are widely accepted for high Reynolds number boundary layers. Since these deviations are relatively small, the models by Musker and Liakopoulos (AL84) perform very well as long as Coles’ parameter $\Pi$ is optimally adjusted for each ${Re}_{\tau }$.

The model predictions have been compared with high-quality direct numerical simulation data. Considering these DNS data as “the truth” (for the range of ${Re}_{\tau }$ examined) we can summarize the main findings as follows: Luchini’s model for the MVP outperforms the other two models in the “low” and “intermediate” Reynolds number cases (${Re}_{\tau }$ 182 to 5186). However, for the high Reynolds number of cases (${Re}_{\tau }=$ 8016 and ${Re}_{\tau }=$ 10,049). Luchini’s model exhibits notably greater deviations compared with the other two models. This may be explained by the fact that Luchini’s model calibration incorporated DNS data available before the year 2017 [56]. AL84 and Musker’s models fit very accurately the higher Reynolds number DNS data, especially for ${Re}_{\tau }=$ 8016. The viscous stress (essentially the accuracy of the derivative of the MVP) Luchini’s model is very accurate in the inner layer but deviates considerably from the DNS results in the region $y/\delta \gtrsim$ 0.85 for all Reynolds number cases. AL84 is slightly better than Musker’s model in this set of comparisons. Turning now our attention to the Reynolds shear stress profiles and focusing on the ${Re}_{\tau }$ range [1000–10,049], we can summarize our findings as follows: Musker’s and AL84 models achieve similar levels of accuracy with a maximum absolute error of approximately 3.2$\times {10}^{-2}$ concentrated in the buffer layer. In the same range of ${Re}_{\tau }$ values, Luchini’s model achieves better accuracy, yielding a maximum absolute error of $\approx$1.8 $\times {10}^{-2}$.

Both Musker’s and Liakopoulos’ (AL84) models share the same structure, and the accuracy levels of both models are similar when compared with these DNS data. The errors associated with these two models become smaller in the “high” Reynolds number cases. This is overall true for the MVP, viscous stress, as well Reynolds shear stress profiles. A notable exemption is the viscous stress calculation in the case of ${Re}_{\tau }=$ 5186, where AL84 outperforms Musker’s model. This is worth noting since the LM5200 DNS dataset is currently considered by most researchers, as very accurate and self-consistent [32]. It should be mentioned that according to Ref. [38] Heinz model is equally simple as the models accessed in this paper. The reader is also advised to consult Refs. [36,37] for a discussion on modeling turbulent canonical flows.

Having established the accuracy of the three models, we conclude that (with the limitations mentioned above) all three MVP models are useful in testing turbulence models very close to solid walls [57]. The findings of this work also support the thesis that the three models examined are all good choices in RANS-, LES-, and SPH-based [57,58] computations very close to the wall in formulations via wall functions. Finally, due to their explicit form, the three MVP models are ideal as initial conditions in marching numerical integrations schemes for parabolic type partial differential equations in Fluid Dynamics when a turbulent velocity, viscous stress, or Reynolds shear stress profile is required, as well as in training ANNs in the context of flow control [59].

The three models considered in this paper are good approximations of turbulent flows where boundary layers develop under weak favorable pressure gradient or weak adverse pressure gradient without flow separation. They can also be used judicially in approximate modeling and engineering calculations of flows over complex geometries in the case where a predominantly streamwise flow direction can be identified.