New momentum integral equation applicable to boundary layer flows under arbitrary pressure gradients

Abstract

By incorporating the traditionally overlooked advective term in the wall-normal momentum equation, a new momentum integral equation is developed for two-dimensional incompressible turbulent boundary layers under arbitrary pressure gradients. The classical Kármán's integral arises as a special instance of the new momentum integral equation when the pressure gradient is weak. The new momentum integral equation's validity is substantiated by direct numerical simulation data. Unlike the classical Kármán's integral, which is limited to predicting wall shear stress within mild pressure gradients, the new momentum integral equation accurately computes wall shear stress across a broad range of pressure gradients, even in the presence of strong adverse pressure gradients that lead to flow separation. Moreover, a new pressure parameter $\beta _\kappa$ is introduced through examining terms in the new momentum integral equation. This parameter naturally quantifies the pressure gradient's influence on turbulent boundary layers and offers guidance for applying the classical Kármán's integral. Additionally, to facilitate experimental determination of wall shear stress under strong pressure gradients, an approximate integral equation is proposed that relies solely on easily measurable variables. Validation against direct numerical simulation data demonstrates that this simplified equation provides reasonably accurate estimates of wall shear stress in turbulent boundary layers experiencing strong pressure gradients.

1. Introduction

Pohlhausen (1921), von Kármán (1921) and Gruschwitz (1931) pioneered integral analysis of the momentum equation in boundary layer flows, leading to the discovery of the momentum integral equation – a significant advancement in the field (see Schlichting 1979). The classic form of the momentum integral equation is expressed as follows (refer to (8.32) in Schlichting (1979)):

(1.1)\begin{equation} \frac{\tau_{wall}}{\rho} \approx U_e \frac{\mathrm{d} U_e}{\mathrm{d}\kern0.06em x} \delta_1 + \frac{\mathrm{d}(U^2_e \delta_2)}{\mathrm{d}\kern0.06em x}, \end{equation}

where $\tau _{wall}$ is the wall shear stress, $\rho$ is the fluid density, $U_e$ is the mean streamwise velocity at the boundary layer edge, $\delta _1$ is the mass displacement thickness and $\delta _2$ is the momentum thickness. The momentum integral equation in the form of (1.1) was first developed by Gruschwitz (1931), as noted by Schlichting (1979). However, in the literature, (1.1) is often referred to as Kármán's integral, for example in the books by Pope (2000) and Kundu, Cohen & Dowling (2012).

The classical momentum integral equation, commonly employed in previous studies, is derived under assumptions suitable for boundary layer flows with mild pressure gradients. However, these assumptions break down for flows under strong adverse pressure gradients, making Kármán's integral ineffective in predicting wall shear stress, as demonstrated in figure 1. The figure compares Kármán's integral with the directly calculated wall shear stress obtained from the direct numerical simulation (DNS) conducted by Coleman, Rumsey & Spalart (2018). The DNS covers a family of boundary layer flows that contain a small separation bubble. The pressure gradients were induced by a transpiration profile $V_{top}(x)$ acting through a virtual parallel plane offset a fixed distance $Y$ from the flat no-slip surface. Each simulation consists of a well-developed entry region with a negligible pressure gradient, followed by an adverse pressure gradient and then a favourable pressure gradient. For evaluation, we consider Case C of simulations, which is accessible in the NASA repository. Based on friction coefficient data, turbulent boundary layer separates at $x/Y \approx -1.4$ and subsequently reattaches at $x/Y \approx 0.4$. Using pressure gradient data, the favourable pressure gradient region begins approximately at $x/Y \approx 0.87$ (see Coleman et al. 2018).

Figure 1. (a) Comparison between Kármán's integral and the directly computed wall shear stress from the DNS data of Coleman et al. (2018). (b) Variation of the Rotta–Clauser pressure gradient parameter along the $x$ direction. Vertical dashed lines indicate $x$ positions with $|\beta _{RC}|=10$. The larger $\beta _{RC}$ values around the separation region are shown in figure 3. The streamwise location is normalized by the simulation domain height in the wall-normal direction, denoted as $Y$.

Figure 1(b) illustrates the variation of the Rotta–Clauser pressure gradient parameter $\beta _{RC}$ along the $x$ direction. Widely employed in the investigation of turbulent boundary layer under pressure gradient, $\beta _{RC}$ is defined as $(\delta _1/\tau _{wall})({\rm d}P/{{\rm d}\kern0.06em x})$ (Clauser 1954). In regions under mild pressure gradients, where $|\beta _{RC}| \ll 10$ (e.g. $-13 < x/Y < -8$ in the DNS data), classical Kármán's integral accurately predicts wall shear stress, as shown in figure 1. However, in the presence of strong adverse pressure gradients, the traditional Kármán's integral proves inadequate for predicting wall shear stress.

An effort to extend Kármán's integral equation was made by Coleman et al. (2018), who integrated the $x$-momentum equation from the inflow station to an arbitrary location $x$. The resultant mean integral–momentum balance consists of the conventional two terms involving $\delta _1$ and $\delta _2$ in (1.1), as well as three additional terms that account for turbulence effects and pressure variations within the boundary layer. While these additional terms are often negligible under boundary layer assumptions, they become significant for flows subject to strong adverse pressure gradients (see Appendix C in Coleman et al. (2018)).

In this work, we derive a general momentum integral equation that is applicable to boundary layer flows under arbitrary pressure gradients. By removing the limitations of the conventional momentum integral analysis, this new equation provides a more accurate representation of boundary layer flows across a broader range of conditions.

2. New momentum integral equation for boundary layer flows

The governing equations for a statistically two-dimensional incompressible turbulent boundary layer flow are (e.g. Townsend 1956; Schlichting 1979; Pope 2000)

(2.1)$$\begin{gather} 0 = \frac{\partial U}{\partial x} + \frac{\partial V}{\partial y}, \end{gather}$$

(2.2)$$\begin{gather}0 = {-U\frac{\partial U}{\partial x}-V\frac{\partial U}{\partial y}} + {\nu \frac{\partial^2 U}{\partial x^2} +\nu \frac{\partial^2 U}{\partial y^2}} + {\frac{\partial R_{uu}}{\partial x} + \frac{\partial R_{uv}}{\partial y}} {-\frac{\partial}{\partial x}\left(\frac{P}{\rho} \right)}, \end{gather}$$

(2.3)$$\begin{gather}0 = {-U\frac{\partial V}{\partial x}-V\frac{\partial V}{\partial y}} + {\nu \frac{\partial^2 V}{\partial x^2} + \nu \frac{\partial^2 V}{\partial y^2}} + {\frac{\partial R_{uv}}{\partial x} + \frac{\partial R_{vv}}{\partial y} } {-\frac{\partial }{\partial y}\left(\frac{P}{\rho} \right)}. \end{gather}$$

Here, $U$ and $V$ represent the mean velocity component in the streamwise $x$ direction and wall-normal $y$ direction, respectively, while $u$ and $v$ denote the corresponding velocity fluctuations. The fluid kinematic viscosity is denoted by $\nu$. The kinematic Reynolds shear stress is denoted as $R_{uv}=-\langle uv \rangle$, and the kinematic Reynolds normal stresses in the streamwise and wall-normal directions are denoted as $R_{uu}=-\langle uu \rangle$ and $R_{vv}=-\langle vv \rangle$, respectively, with the angle brackets denoting the Reynolds averaging operator.

Based on the order-of-magnitude analysis and supported by the DNS data from Coleman et al. (2018) (see Appendix A), the viscous terms and $\partial R_{uv}/\partial x$ term are higher-order ones in the $y$-momentum equation (2.3). Consequently, the $y$-momentum equation (2.3) can be approximated as follows:

(2.4)\begin{equation} 0 \approx \left( - \frac{\partial (U V)}{\partial x} - \frac{\partial V^2}{\partial y} \right) + \frac{\partial R_{vv}}{\partial y} - \frac{\partial }{\partial y} \left( \frac{P}{\rho} \right)\!. \end{equation}

Integrating (2.4) along the $y$ direction yields the pressure distribution within the boundary layer as

(2.5)\begin{equation} \frac{P}{\rho} = \frac{P_{wall}}{\rho} + \left( R_{vv} - V^2 \right) - \int_0^y \frac{\partial (UV)}{\partial x} \mathrm{d}y. \end{equation}

This equation is equivalent to (5.5.17) in the book of Tennekes & Lumley (1972). Taking the derivative of (2.5) with respect to the $x$ direction yields

(2.6)\begin{equation} \frac{\partial }{\partial x} \left( \frac{P}{\rho} \right) = \frac{\mathrm{d}}{\mathrm{d}\kern0.06em x} \left(\frac{P_{wall}}{\rho} \right) + \frac{\partial (R_{vv} - V^2)}{\partial x} - \frac{\partial }{\partial x} \left( \int_0^y \frac{\partial (UV)}{\partial x} \mathrm{d}y \right)\!. \end{equation}

Substituting this result into the $x$-momentum equation (2.2) gives

(2.7)\begin{align} 0 &\approx \left( - \frac{\partial U^2}{\partial x} -\frac{\partial (UV)}{\partial y} \right) + \nu \frac{\partial^2 U }{\partial y^2} + \frac{\partial R_{uv} }{\partial y} - \frac{\mathrm{d}}{\mathrm{d}\kern0.06em x} \left(\frac{P_{wall}}{\rho} \right) + \frac{\partial (R_{uu}- R_{vv} + V^2)}{\partial x} \nonumber\\ &\quad + \frac{\partial }{\partial x} \left( \int_0^y \frac{\partial (UV)}{\partial x} \mathrm{d}y \right)\!. \end{align}

Note that the term $\nu \partial ^2 U/\partial x^2$ is neglected in the $x$-momentum equation (2.2) based on Prandtl's boundary layer theory applicable to thin boundary layers (see Schlichting 1979). This work reaffirms the validity of this omission even in boundary layers with separation, which are typically not thin. Integrating (2.7) from $y=0$ to $y=\delta _e$ yields a general momentum integral equation as

(2.8)\begin{align} \frac{\tau_{wall}}{\rho} &\approx- \int_0^{\delta_e} \frac{\partial U^2}{\partial x} \mathrm{d}y - U_e V_e - \frac{\mathrm{d}}{\mathrm{d}\kern0.06em x} \left(\frac{P_{wall}}{\rho}\right)\delta_e + \int_0^{\delta_e} \frac{\partial (R_{uu})}{\partial x} \mathrm{d}y \nonumber\\ &\quad + \int_0^{\delta_e} \frac{\partial ( V^2 - R_{vv})}{\partial x} \mathrm{d}y + \int_0^{\delta_e} \frac{\partial }{\partial x} \left( \int_0^y \frac{\partial (UV)}{\partial x} \mathrm{d}y \right) \mathrm{d}y . \end{align}

The new momentum integral equation (2.8) can also be expressed as (see Appendix B for details)

(2.9)\begin{align} \frac{\tau_{wall}}{\rho} & \approx \underbrace{\left( U_e \frac{\mathrm{d}U_e}{\mathrm{d}\kern0.06em x}\delta_1 + \frac{\mathrm{d}(U^2_e \delta_2)}{\mathrm{d}\kern0.06em x} \right )}_{\mathrm{I}} \underbrace{-\left( \frac{1}{\rho} \frac{\mathrm{d}P_{wall} }{\mathrm{d}\kern0.06em x} + U_e\frac{\mathrm{d}U_e}{\mathrm{d}\kern0.06em x} \right)\delta_e}_{\mathrm{II}} \nonumber\\ & \quad+ \underbrace{\int_0^{\delta_e} \frac{\partial (R_{uu})}{\partial x} \mathrm{d} y}_{\mathrm{III}} + \underbrace{\int_0^{\delta_e} \frac{\partial ( V^2 - R_{vv})}{\partial x} \mathrm{d}y}_{\mathrm{IV}} + \underbrace{\int_0^{\delta_e} \frac{\partial }{\partial x} \left( \int_0^y \frac{\partial (UV)}{\partial x} \mathrm{d}y \right) \mathrm{d}y}_{\mathrm{V}}. \end{align}

Notably, the first term $\mathrm {I}$ on the right-hand side of (2.9) corresponds to the classical Kármán's integral equation (1.1). Term $\mathrm {II}$ results from the difference between the wall pressure gradient and $U_e \,{\rm d}U_e/{{\rm d}\kern0.06em x}$ (or $-{\rm d}(P_e/\rho )/{{\rm d}\kern0.06em x}$), term $\mathrm {III}$ arises from the Reynolds normal stress term in the $x$-momentum equation, term $\mathrm {IV}$ includes the advective and Reynolds normal stress terms in the $y$-momentum equation and term $V$ is from the advective term in the $y$-momentum equation.

Figure 2 displays the terms in the new momentum integral equation (2.9) alongside the directly computed wall shear stress using the DNS data from Coleman et al. (2018). The figure highlights the new equation's ability to accurately predict wall shear stress, even in regions subjected to strong pressure gradients. Discrepancies and variations between the predicted and observed wall shear stress values might stem from uncertainties linked to the finite difference method used in computing derivatives of less than perfectly smooth data. This issue becomes particularly significant when calculating the term involving ${\partial (UV)}/{\partial x}$, as it requires computing $x$ derivatives twice.

Figure 2. Comparison of terms in the new momentum integral equation (2.9) and their summation with the directly computed wall shear stress. The DNS data are from Coleman et al. (2018), displaying every 50th grid point in the $x$ direction for clarity.

The new momentum integral equation (2.9) facilitates the definition of dimensionless parameters that characterize the pressure gradient's impact. For instance, the commonly used Rotta–Clauser pressure parameter $\beta _{RC}$ (Rotta 1950; Clauser 1954) can be interpreted within the context of the momentum integral equation as the ratio between the wall shear stress and the first component of term $\mathrm {I}$ on the right-hand side of (2.9) (or the first term on the right-hand side of (1.1)):

(2.10)\begin{equation} \beta_{RC} = \frac{\dfrac{1}{\rho} \dfrac{\mathrm{d} P_\infty}{\mathrm{d}\kern0.06em x} \delta_1}{\dfrac{\tau_{wall}}{\rho}} \approx-\frac{U_e \dfrac{\mathrm{d}U_e}{\mathrm{d}\kern0.06em x} \delta_1}{\dfrac{\tau_{wall}}{\rho}}. \end{equation}

Here, we introduce a new dimensionless pressure parameter by considering the ratio of term $\mathrm {II}$ to term $\mathrm {I}$ on the right-hand side of (2.9):

(2.11)\begin{equation} \beta_{\kappa} = \frac{ -\left( \dfrac{1}{\rho}\dfrac{\mathrm{d} P_{wall}}{\mathrm{d}\kern0.06em x}+U_e \dfrac{\mathrm{d}U_e}{\mathrm{d}\kern0.06em x}\right) \delta_e}{U_e \dfrac{\mathrm{d}U_e}{\mathrm{d}\kern0.06em x}\delta_1 + \dfrac{\mathrm{d}(U^2_e \delta_2)}{\mathrm{d}\kern0.06em x} }. \end{equation}

In defining $\beta _\kappa$, Kármán's integral is used in the denominator rather than the wall shear stress. This choice is driven by the challenge of accurately determining wall shear stress in experimental studies. The variation of $\beta _\kappa$ in the DNS data by Coleman et al. (2018) is depicted in figure 3. For comparison, the conventional Rotta–Clauser parameter is also included in the figure. Near the inlet, where the mean pressure gradient is small, both $\beta _{RC}$ and $\beta _\kappa$ approach zero. As observed in figure 2, as the wall shear stress approaches zero (separation point), the left-hand side of (2.9) tends to zero, and terms $\mathrm {I}$ and $\mathrm {II}$ have similar magnitudes but opposite signs. Therefore, in the region near separation, $\beta _{\kappa }$ is approximately $-1$, as illustrated in figure 3.

Figure 3. Newly defined pressure parameter $\beta _\kappa$ and the conventional $\beta _{RC}$. The DNS data are from Coleman et al. (2018).

Figure 1 shows a substantial region around the separation point where the wall shear stress is approximately zero. Consequently, the Rotta–Clauser pressure parameter exhibits significantly high magnitudes in these regions. In contrast, the range of the new pressure parameter $\beta _{\kappa }$ is more confined. The two singular points in $\beta _{\kappa }$ correspond to the zero-crossing of term $\mathrm {I}$ in (2.9) (refer to figure 2). Unlike the wall shear stress, the region where term $\mathrm {I}$ approaches zero is much narrower.

3. Approximations of terms in the new momentum integral equation

Flow variables associated with terms $\mathrm {I}$ and $\mathrm {II}$ are relatively more accessible compared with those in other terms. However, obtaining precise measurements to evaluate terms $\mathrm {IV}$ and $\mathrm {V}$ accurately in experimental settings is extremely challenging. Therefore, deriving an approximation for these terms becomes highly desirable for practical applications.

Figure 2 demonstrates that under mild pressure gradients, terms $\mathrm {IV}$ and $\mathrm {V}$ are negligible (refer to Appendix C for an order-of-magnitude estimation). However, their magnitudes become notably larger than the wall shear stress when the boundary layer encounters a strong adverse pressure gradient. Thus, integrating both the advective and turbulence terms of the $y$-momentum equation into the integral momentum analysis becomes imperative for predicting wall shear stress accurately in turbulent boundary layer subjected to strong pressure gradients.

Examining the DNS data from Coleman et al. (2018) reveals an empirical observation: the profile shape of the sum of terms $\mathrm {IV}$ and $\mathrm {V}$ in (2.9) closely resembles that of term $\mathrm {II}$. Figure 4 demonstrates that this relationship can be approximated as

(3.1)\begin{align} \int_0^{\delta_e} \frac{\partial ( V^2 - R_{vv})}{\partial x} \mathrm{d} y + \int_0^{\delta_e} \frac{\partial }{\partial x} \left( \int_0^y \frac{\partial (UV)}{\partial x} \mathrm{d}y \right) \mathrm{d}y \approx 0.23 \left( \frac{\mathrm{d}}{\mathrm{d}\kern0.06em x} \left(\frac{P_{wall}}{\rho}\right) + U_e\frac{\mathrm{d}U_e}{\mathrm{d}\kern0.06em x} \right)\delta_e. \end{align}

Figure 4. Approximation of the sum of terms $\mathrm {IV}$ and $\mathrm {V}$ in (2.9). The DNS data are from Coleman et al. (2018).

To a large extent, as demonstrated in figure 4, the sum of terms $\mathrm {IV}$ and $\mathrm {V}$ in (2.9) can be fairly approximated by (3.1), in spite of some noticeable discrepancies observed near the rear and in the region of $-2 < x/Y < 0$.

Applying the approximate equation (3.1), the wall shear stress can be estimated as

(3.2)\begin{equation} \frac{\tau_{wall}}{\rho} \approx \left\{ U_e \frac{\mathrm{d}U_e}{\mathrm{d}\kern0.06em x}\delta_1 + \frac{\mathrm{d}(U^2_e \delta_2)}{\mathrm{d}\kern0.06em x} \right\} - 0.77 \left\{\frac{1}{\rho}\frac{\mathrm{d} P_{wall}}{\mathrm{d}\kern0.06em x} + U_e \frac{\mathrm{d}U_e}{\mathrm{d}\kern0.06em x} \right\}\delta_e + \int_0^{\delta_e} \frac{\partial R_{uu}}{\partial x} \mathrm{d}y. \end{equation}

In experimental studies, accurately measuring $P_{wall}$ is feasible. Experimental investigations utilizing particle image velocimetry can provide high-resolution measurements of $U$ and $R_{uu}$ across a range of $x$ locations. While obtaining accurate measurements in the near-wall region is often challenging in experiments, determining $\delta _1$ and $\delta _2$ does not necessitate high-resolution measurements of $U$ in this region, because the thin near-wall region contributes minimally to the integral quantities. Therefore, obtaining necessary data for approximate equation (3.2) is readily achievable in experimental studies.

Figure 5 presents the directly calculated wall shear stress alongside predictions from the new momentum integral equation (2.9), the traditional Kármán's integral and the approximate equation (3.2). While discrepancies between the directly calculated and approximated wall shear stress are evident within the range $-2 < x/Y < 0$ and near the rear at $x/Y >2.5$, overall, the approximate equation (3.2) demonstrates fairly accurate predictions, particularly in the first half of the domain. The relatively large discrepancies between the directly calculated wall shear stress and that predicted by the approximate integral equation primarily stem from the use of empirical equation (3.1) to model the summation of terms IV and V. Empirical equation (3.1) demonstrates poor estimation in regions proximal to flow separation or during rapid changes in pressure gradients ($-2 < x/Y < 0$) and towards the simulation domain's end ($x/Y > 2.5$), where the influence of outer flow boundary conditions could be significant. To establish the broader applicability of the approximate equation, additional evaluation using more experimental and simulation data of turbulent boundary layer flows under pressure gradients is necessary.

Figure 5. Comparison of directly calculated wall shear stress with predictions from the Kármán integral equation (1.1), the new momentum integral equation (2.9) and the approximate equation (3.2). The DNS data are from Coleman et al. (2018).

Figure 6 shows a remarkable similarity between the shapes of terms $\mathrm {II}$ and $\mathrm {III}$ on the right-hand side of the new momentum integral equation (2.9). Intriguingly, a better correlation in their shapes is observed when the $x$ axis is shifted. The origin of this shift is presently unclear. Nevertheless, figure 6 suggests that the magnitude of term $\mathrm {III}$ can be estimated as a fraction of term $\mathrm {II}$:

(3.3)\begin{equation} \int_0^{\delta_e} \frac{\partial R_{uu}}{\partial x} \mathrm{d}y \sim O \left(0.15 \left( \frac{1}{\rho}\frac{\mathrm{d} P_{wall}}{\mathrm{d}\kern0.06em x}+ U_e \frac{\mathrm{d}U_e}{\mathrm{d}\kern0.06em x}\right) \delta_e \right)\!. \end{equation}

Figure 6. Relation between term $\mathrm {III}$ and term $\mathrm {II}$ in the new momentum integral equation (2.9). The DNS data are from Coleman et al. (2018).

Thus, the wall shear stress can be estimated as

(3.4)\begin{equation} \frac{\tau_{wall}}{\rho} \sim \left(U_e \frac{\mathrm{d}U_e}{\mathrm{d}\kern0.06em x}\delta_1 + \frac{\mathrm{d}(U^2_e \delta_2)}{\mathrm{d}\kern0.06em x} \right) (1 + \beta_\kappa - c \beta_\kappa), \end{equation}

where $c$ is a factor of about $0.38$ for the DNS data of Coleman et al. (2018). For small $|\beta _\kappa | \lesssim 0.1$, the classical Kármán's integral provides accurate prediction of the wall shear stress, but its validity diminishes at larger values of $\beta _\kappa$.

4. Discussion

Turbulent boundary layer flows under pressure gradients, especially adverse ones, find extensive applications in aircraft, ships, wind turbines and various fluid systems. Understanding their behaviour is crucial, significantly influencing system performance and energy efficiency. Extensive research, theoretical, experimental and numerical (Rotta 1950; Clauser 1954; Townsend 1956; Stratford 1959; Mellor 1966; Mellor & Gibson 1966; Skote, Henningson & Henkes 1998; Castillo & George 2001; Bobke et al. 2017; Kitsios et al. 2017; Coleman et al. 2018; Maciel et al. 2018; Devenport & Lowe 2022; Subrahmanyam, Cantwell & Alonso 2022), has focused on unravelling turbulent boundary layers under pressure gradients. Accurate determination of wall shear stress remains pivotal in this study.

Wall shear stress is a fundamental parameter in the analysis of wall-bounded flows. It serves to quantify the drag experienced by a surface and plays an essential role in the scaling and comprehension of flow dynamics. Several methods, as reviewed by Winter (1979), Haritonidis (1989), Naughton & Sheplak (2002) and Tavoularis (2005), have been employed to directly measure or infer wall shear stress. However, obtaining an accurate determination of wall shear stress remains a challenge in practical applications.

The integral of the mean momentum equation offers a theoretical approach to indirectly determine wall shear stress (or friction coefficient $C_f = 2 \tau _{wall}/U^2_e$) given all pertinent terms are accurately measured. For example, Ligrani & Moffat (1986) employed the momentum integral equation to determine wall shear stress on sand grain roughness by measuring mean streamwise velocity and Reynolds shear stress profiles at various $x$ stations. Brzek et al. (2007) utilized closely spaced streamwise measurements of $\delta _2$ to determine wall shear stress in zero pressure gradient turbulent boundary layers. However, directly applying the momentum integral equation often involves streamwise gradient terms that are challenging to acquire with the necessary precision from experimental data.

Mehdi et al. (2014) developed an integral method based on the triple integration of the mean momentum equation. By replacing streamwise gradient terms with total stress gradient terms in the wall-normal direction, they determined $C_f$ using experimental data of mean velocity and Reynolds shear stress acquired at only one streamwise location. Given the inherent difficulty in measuring Reynolds shear stress very near the wall, they employed a fitting technique based on the expected shape of the total shear stress profile to smooth the experimental data and obtain the gradient.

More recently, Volino & Schultz (2018) introduced an integral equation to determine wall shear stress without making assumptions about the shape of the mean velocity profile or relying on fitting experimental data to expected functions. Their approach involves transforming the integral equation into wall coordinates, separating terms dependent on streamwise gradients from those that are not. Although this method requires mean velocity and Reynolds shear stress profiles from at least two streamwise stations, it significantly diminishes reliance on streamwise gradients. Moreover, their methodology does not require data from very near the wall to determine wall shear stress. Their evaluation across diverse experimental and numerical datasets showcased the close agreement between the wall shear stress determined through their method and that obtained in the original studies.

In previous applications of the momentum integral for predicting wall shear stress, the advective terms in the $y$-momentum equation (2.4) were typically neglected, resulting in an approximation of the pressure as (see e.g. Rotta 1962)

(4.1)\begin{equation} \frac{P}{\rho} \approx \frac{P_{wall}}{\rho} + R_{vv}, \quad \mathrm{traditional\ analysis}. \end{equation}

At the boundary layer edge, where $R_{vv} \approx 0$, the pressure is approximated as $P_e \approx P_{wall}$. Moreover, in the conventional analysis, it is typically assumed that $-\partial (P/\rho )/\partial x|_e \approx U_e \,\mathrm {d}U_e/\mathrm{d}\kern0.06em x$. In essence, the second term $\mathrm {II}$ on the right-hand side of the new momentum integral equation (2.9) is assumed negligible in traditional analyses (Townsend 1956; Rotta 1962).

Figure 7 shows the second term $\mathrm {II}$ of (2.9) alongside its two components and the directly computed wall shear stress from the DNS data of Coleman et al. (2018). The figure indicates that when the pressure gradient is small, term $\mathrm {II}$ in (2.9) is negligible. However, in the presence of strong pressure gradients, term $\mathrm {II}$ becomes significant, greatly exceeding the magnitude of the wall shear stress. This highlights the crucial role of term $\mathrm {II}$ on the right-hand side of the new momentum integral equation (2.9) in predicting wall shear stress under strong adverse pressure gradients. Consequently, in experimental studies of turbulent boundary layers under such conditions, accurate measurement of wall pressure distribution is imperative and should not be approximated from $U_e(x)$ measurements.

Figure 7. Components of the second term $\mathrm {II}$ in (2.9) alongside the directly computed wall shear stress. The DNS data are from Coleman et al. (2018).

The new momentum integral equation (2.9) exhibits a close connection to the mean integral–momentum balance equation formulated by Coleman et al. (2018). However, our approach differs in two main aspects. Firstly, while Coleman et al. (2018) integrated the mean $x$-momentum equation along the $x$ direction, our derivation is conducted at a fixed $x$ location, akin to the conventional momentum integral analysis. Secondly, we leveraged the $y$-momentum equation to represent $P$ as a function of $P_{wall}$, $R_{vv}$, $V^2$ and $UV$ (refer to (2.5)), which further clarify the pressure variation's influence.

5. Summary

This study reveals the primary causes of the classical Kármán integral's failure in predicting wall shear stress accurately within turbulent boundary layers subjected to strong pressure gradients. Specifically, in the presence of strong adverse pressure gradients, the advective terms in the wall-normal momentum equation can significantly alter pressure distribution across the boundary layer. Consequently, the conventional approximation $-\partial (P/\rho )/\partial x \approx U_e \,\mathrm {d}U_e/\mathrm{d}\kern0.06em x$, typically employed in traditional analysis, becomes erroneous.

By incorporating the $y$-momentum equation's advective terms and retaining the often neglected Reynolds normal stress term in the $x$-momentum equation, we derive a more general momentum integral equation for wall shear stress calculation. Importantly, the classical Kármán's integral emerges as a special instance under minimal or zero pressure gradient conditions. The new momentum integral equation's predictive precision for wall shear stress within turbulent boundary layers is substantiated by its excellent agreement with DNS data. This agreement spans a wide range of pressure gradients, including even severe adverse pressure gradients that lead to flow separation. Additionally, the new momentum integral equation can be used to assess the quality of numerical simulations of turbulent boundary layers under strong pressure gradients.

Although the new momentum integral equation exhibits robust accuracy in predicting wall shear stress within turbulent boundary layers under arbitrary pressure gradients, its direct application in experimental practices faces challenges due to the difficulty and impracticality of accurately measuring $V$ and $R_{vv}$ in experiments. To address this, we introduce an empirical approximate equation for wall shear stress estimation, relying solely on easily measurable variables like $P_{wall}$, $U$ and $R_{uu}$. Despite the use of fewer data measurements, the resulting wall shear stress estimation demonstrates reasonable agreement with DNS data in turbulent boundary layers experiencing strong pressure gradients.

Furthermore, a novel dimensionless parameter, $\beta _\kappa$, is defined based on the new momentum integral equation to quantify the influence of pressure gradients on turbulent boundary layers. Our findings reveal that $\beta _\kappa$ remains close to zero in regions of weak pressure gradients but surpasses $O(1)$ in regions under strong pressure gradients. Consequently, the predictions of the classical Kármán's integral lose reliability in regions with a substantial magnitude of $\beta _\kappa$ – such as the $|\beta _\kappa | \gtrsim 0.1$ observed in this study.