2.1. Momentum equation

The double-averaged streamwise momentum equation for rough-wall flows (Raupach & Shaw Reference Raupach and Shaw1982; Nikora et al. Reference Nikora, Ballio, Coleman and Pokrajac2013, Reference Nikora, Stoesser, Cameron, Stewart, Papadopoulos, Ouro, McSherry, Zampiron, Marusic and Falconer2019) reads

(2.1)\begin{align} & \frac{\partial \langle \bar{u}_1\rangle}{\partial t} + \frac{1}{\phi}\,\frac{\partial \phi \langle \bar{u}_1 \rangle \langle \bar{u}_j \rangle }{\partial x_j} + \frac{1}{\phi}\,\frac{\partial \phi \langle \widetilde{\bar{u}}_1 \widetilde{\bar{u}}_j \rangle }{\partial x_j} + \frac{1}{\phi}\,\frac{\partial \phi \langle \overline{u_1' u_j'} \rangle }{\partial x_j} \nonumber\\ &\quad ={-} \frac{1}{\rho}\,\frac{1}{\phi}\,\frac{\partial \phi \langle \bar{p}\rangle}{\partial x_1} + \frac{1}{\rho}\,\frac{1}{A_f} \oint_S \bar{p} n_1 \,{\rm d}S + \frac{1}{\phi}\,\frac{\partial }{\partial x_j} \left( \phi \left\langle \nu\,\frac{\partial \bar{u}_1}{\partial x_j} \right\rangle \right) - \frac{1}{A_f} \oint_S \nu\,\frac{\partial \bar{u}_1}{\partial x_j}\,n_j \,{\rm d}S + f_x, \end{align}

where $x_1$, $x_2$, $x_3$ are the streamwise, spanwise and wall-normal directions, $u_1$, $u_2$ and $u_3$ are the instantaneous fluid velocities in the $x_1$, $x_2$, $x_3$ directions, $p$ is the pressure, $\rho$ is the density, $f_x$ is a body force that contains the mean pressure gradient, $\phi =A_f/A_0$ is the roughness porosity (with $A_f$ the fluid occupied planar area that is a function of $x_3$), $A_0$ is the total planar area, $S$ marks the boundary of the fluid area, and $n_j$ is the unit vector normal to the solid boundary directed into the fluid. Here, $\bar {\cdot }$ denotes time average, $\langle \cdot \rangle$ denotes the intrinsic spatial average, defined as

(2.2)\begin{equation} \langle \theta \rangle = \frac{1}{A_f} \iint_{A_f} \theta \,{\rm d}\kern0.7pt x_1 \,{\rm d}\kern0.7pt x_2, \end{equation}

$\theta '=\theta - \bar {\theta }$ is fluctuation about the time average, and $\widetilde {\bar {\theta }}=\bar {\theta } - \langle \bar {\theta } \rangle$ is the deviation of the time average from the double average. Figure 1 is a sketch illustrating these concepts.

Figure 1.

(a) An illustration of the multi-connected area for the spatial integration in the roughness layer. The planar area $A_0$ is bounded by the dashed line. The fluid area $A_f$ is blue and surrounds the white, roughness-occupied area. Their ratio $A_f/A_0$ gives the roughness porosity $\phi$. The fluid boundary $S$ consists of the outer boundary (dashed) and the inner boundary (solid). (b) An illustration of the roughness planar area $A_p$, i.e. the red area under the roughness elements. The ratio $A_p/A_0$ gives the planar coverage density $\lambda _p$.

We may rewrite the double-averaged $x_1$ momentum equation as

(2.3)\begin{equation} \frac{\partial }{\partial x_3} \left( \phi \left\langle \nu\, \frac{\partial \bar{u}_1 }{\partial x_3} \right\rangle \right) - \frac{\partial \phi \langle \overline{u_1'u_3'} \rangle}{\partial x_3} - \frac{\partial \phi \langle \widetilde{\bar{u}}_1 \widetilde{\bar{u}}_3 \rangle}{\partial x_3} - I_x + f_D + \phi f_x = 0, \end{equation}

where the term $\langle \overline {u_1'u_3'} \rangle$ is the Reynolds shear stress, $\langle \widetilde {\bar {u}}_1\widetilde {\bar {u}}_3 \rangle$ is the dispersive stress, $I_x$ contains the horizontal convection and dispersion,

(2.4)\begin{align} I_x &= \phi\,\frac{\partial \langle \bar{u}_1 \rangle}{\partial t} + \frac{\partial \phi \langle \bar{u}_1 \rangle \langle \bar{u}_j \rangle}{\partial x_j} + \frac{\partial \phi \langle \widetilde{\bar{u}}_1 \widetilde{\bar{u}}_1 \rangle}{\partial x_1} + \frac{\partial \phi \langle \widetilde{\bar{u}}_1 \widetilde{\bar{u}}_2 \rangle}{\partial x_2}+ \frac{\partial \phi \langle \overline{ u_1' u_1'} \rangle}{\partial x_1} \nonumber\\ &\quad + \frac{\partial \phi \langle \overline{ u_1' u_2'} \rangle}{\partial x_2} - \frac{\partial }{\partial x_1} \left( \phi \left\langle \nu\, \frac{\partial \bar{u}_1}{\partial x_1} \right\rangle \right) - \frac{\partial }{\partial x_2} \left( \phi \left\langle \nu\, \frac{\partial \bar{u}_1}{\partial x_2} \right\rangle \right) + \frac{1}{\rho}\,\frac{\partial \phi \langle \bar{p} \rangle}{\partial x_1}, \end{align}

and $f_D$ contains the form drag and viscous drag on the roughness elements per unit volume,

(2.5)\begin{equation} f_D = \underbrace{\phi\,\frac{1}{A_f}\,\frac{1}{\rho} \oint_{S} \bar{p} n_1 \,{\rm d}S}_{{form\ drag}} - \underbrace{\phi\,\frac{1}{A_f} \oint_{S} \nu\,\frac{\partial \bar{u}_1}{\partial x_j}\, n_j \,{\rm d}S }_{viscous\ drag}. \end{equation}

Consider the rough walls with discrete roughness elements. The friction consists of two parts: the roughness element drag and the skin friction on the bottom wall. The roughness element drag and the corresponding drag coefficient are defined as

(2.6a,b)\begin{equation} \tau_{R} ={-} \int_0^h \rho f_D \,{\rm d}\kern0.7pt x_3,\quad C_R = \frac{1}{\lambda_p}\,\frac{\tau_R}{0.5 \rho U_r^2}, \end{equation}

where $h$ is the roughness height, $U_r$ is some reference velocity, $\lambda _p=A_p/A_0$ is the planar roughness packing density and is not a function of $x_3$, and $A_p$ is the roughness planar area as shown in figure 1(b) and equals $1-\phi$ in for cubic roughness. The bottom-wall skin friction and the corresponding drag coefficient are defined as

(2.7a,b)\begin{equation} \tau_{S} = \left[ \rho \phi \left\langle \nu\, \frac{\partial \bar{u}_1 }{\partial x_3} \right\rangle \right]_{x_3=0},\quad C_S = \frac{\tau_S}{0.5 \rho U_r^2}. \end{equation}

The common choices of the reference velocity $U_r$ are the freestream velocity $U_\infty$, the centreline velocity $U_0$, and the bulk velocity $U_b=\int _0^1 \phi \langle \bar {u}_1 \rangle \,{\rm d}(x_3/\delta )$. Note that the freestream velocity of boundary layers is an external variable that stays the same independently of conditions of a boundary layer. In contrast, the centreline and bulk velocities in internal flows are not independent of the conditions on the wall and thus are internal variables. The overall drag is given by $\tau _R+\tau _S$, and the overall friction coefficient $C_f$ is

(2.8)\begin{equation} C_f= \frac{\tau_S+\tau_R}{0.5 \rho U_r^2}= C_S + \lambda_p C_R, \end{equation}

such that $C_f= C_S + \lambda _p C_R$ can be used to evaluate $C_f$ when $C_S$ and $C_R$ are available.

2.2. Force balance

The overall force balance can be obtained by integrating the double-averaged momentum equation (2.3) from $x_3=0$ to $x_3=\delta$:

(2.9)\begin{equation} \int_0^{\delta} \left[\frac{\partial }{\partial x_3} \left(\phi \left\langle \nu\, \frac{\partial \bar{u}_1 }{\partial x_3} \right\rangle \right) - \frac{\partial \phi \langle \overline{u_1'u_3'} \rangle}{\partial x_3} - \frac{\partial \phi \langle \widetilde{\bar{u}}_1 \widetilde{\bar{u}}_3 \rangle}{\partial x_3} - I_x + f_D + \phi f_x \right]{\rm d}\kern0.7pt x_3 = 0. \end{equation}

The first term gives the viscous friction on the flat bottom wall:

(2.10)\begin{equation} \int_0^{\delta} \frac{\partial }{\partial x_3}\left( \phi \left\langle \nu\, \frac{\partial \bar{u}_1 }{\partial x_3} \right\rangle\right) {\rm d}\kern0.7pt x_3 = \phi \left.\left\langle \nu\,\frac{\partial \bar{u}_1 }{\partial x_3}\right\rangle\right|_0^{\delta} ={-} \tau_S/\rho, \end{equation}

where we have neglected the velocity gradient at $x_3=\delta$. The second and third terms, i.e. the Reynolds and dispersive stress terms, are internal terms and do not contribute to the overall force balance. The integration of $f_D$ gives the roughness drag $-\tau _R/\rho$ per (2.6a,b). Hence the overall force balance is

(2.11)\begin{equation} {-\tau_S}/\rho - \tau_R/\rho + \int_0^\delta \phi f_x \,{\rm d}\kern0.7pt x_3- \int_0^{\delta}{I_x \,{\rm d}\kern0.7pt x_3} = 0. \end{equation}

Figure 2 provides a visual illustration of the force balance. The momentum flux on the left, right and top boundaries, denoted as $I_x$ in the figure, and the forcing, denoted by $f_x$ in the figure, are balanced by the wall shear stress and the roughness drag force, denoted by $\tau _S$ and $\tau _R$ in the figure. The viscous force, Reynolds stress and dispersive stress are internal forces; they redistribute the momentum within the flow, but do not contribute to the overall force balance at the scale of the whole flow.

Figure 2.

Schematic of flow over a rough surface: $f_x$, $I_x$, $\tau _S$, $\tau _R$ are the external forces; $\tau =\phi \langle \nu \,\partial \bar {u}_1/\partial x_3 \rangle - \phi \langle \overline {u_1'u_3'} \rangle - \phi \langle \widetilde {\bar {u}}_1\widetilde {\bar {u}}_3 \rangle$ is the internal fluid stress.

3.1. Momentum-based integral

The momentum-based integrals are pioneered by Fukagata et al. (Reference Fukagata, Iwamoto and Kasagi2002). Integrating the double-averaged momentum equation (2.3), we arrive at the following decomposition of the wall shear stress:

(3.1)\begin{align} \tau_S/\rho &= \frac{2 U_b^2}{Re_b}+2 \int_0^{1}{ \left( 1- \frac{x_3}{\delta} \right) (- \phi\langle \overline{u_1'u_3'} \rangle - \phi\langle \widetilde{\bar{u}}_1 \widetilde{\bar{u}}_3 \rangle)\,{\rm d} \frac{x_3}{\delta} } \nonumber\\ &\quad +\int_0^1{ \left( 1- \frac{x_3}{\delta} \right) ^2 (\phi f_x +f_D - I_x) \delta \,{\rm d} \frac{x_3}{\delta} }. \end{align}

This FIK decomposition proves to be a valuable diagnostic tool, and many have applied it for flow analysis. Deck et al. (Reference Deck, Renard, Laraufie and Weiss2014) examined the integrals for flat-plate boundaries for a wide range of Reynolds numbers ($3060 \le Re_{\theta } \le 13\,650$). They found that about 80 % of the skin friction in high-Reynolds-number boundary layers is due to turbulent motions. Moreover, the large-scale motions with streamwise wavelengths $\lambda _x > \delta$ and $\lambda _x > 2 \delta$ account for approximately 60 % and 45 % of the skin friction, respectively. de Giovanetti, Hwang & Choi (Reference de Giovanetti, Hwang and Choi2016) studied the skin friction generated by self-similar energy-containing motions up to a friction Reynolds number of approximately $4000$. Their findings indicated that the removal of very-large-scale and large-scale motions resulted in only a minimal 5 %–8 % reduction in skin friction. The utility of the FIK method is not limited to canonical boundary layer flows; it also serves as a reliable tool for assessing the effectiveness of drag-reduction techniques. For example, an analysis by Iwamoto et al. (Reference Iwamoto, Fukagata, Kasagi and Suzuki2005) suggested that eliminating near-wall turbulence within $x_3^+<10$ could lead to a substantial 35 % drag reduction at $Re_{\tau }=10^5$. The FIK integral method has also been used for rough-wall boundary layers (Bannier, Garnier & Sagaut Reference Bannier, Garnier and Sagaut2015; Nikora et al. Reference Nikora, Stoesser, Cameron, Stewart, Papadopoulos, Ouro, McSherry, Zampiron, Marusic and Falconer2019; Zhang et al. Reference Zhang, Wan, Xia, Wang, Lu and Chen2021) and compressible flows (Gomez, Flutet & Sagaut Reference Gomez, Flutet and Sagaut2009), where additional terms emerge due to the complexity of these flow regimes. In § 3.2, we will delve further into the integral method proposed by Nikora et al. (Reference Nikora, Stoesser, Cameron, Stewart, Papadopoulos, Ouro, McSherry, Zampiron, Marusic and Falconer2019) for rough walls.

The FIK integral has received criticism, and alternative integrals have been explored. In the following, we review these criticisms and the alternatives integrals. The first criticism concerns the triple integration. Renard & Deck (Reference Renard and Deck2016) found the triple integration hard to interpret. They argued that the product of a force and a length in the second integration has the dimension of energy, and developed integrals grounded in kinetic energy. Elnahhas & Johnson (Reference Elnahhas and Johnson2022) pointed out that the triple integration gives the second moment of momentum. They argued that instead of the second moment, the first moment of momentum is more straightforward to interpret. The second criticism concerns the integration limit in the case of boundary layer flow. Ricco & Skote (Reference Ricco and Skote2022) showed that the value of the terms in the FIK integral depends on the upper integration limit, which is a source of uncertainty. In addition to the usual $\delta _{99}$, Wenzel, Gibis & Kloker (Reference Wenzel, Gibis and Kloker2022) compared different choices of the upper integration limit, and found that their results do not depend qualitatively on the choice of the integration limit. Xu, Wang & Chen (Reference Xu, Wang and Chen2022) also noticed that the exact value of the upper integration limit is not critical for drawing general conclusions, at least in their study of hypersonic boundary layers. Nonetheless, to circumvent the issue of the integration limit, Renard & Deck (Reference Renard and Deck2016) turned to the moving reference frame, and Elnahhas & Johnson (Reference Elnahhas and Johnson2022) resorted to the deficit equation. In both scenarios, the contributions of the far field to the terms in the integrals approach 0 as the upper integration limits are approaching infinity. The third criticism concerns the weighting, i.e. $(1- x_3/\delta )$ and $(1- x_3/\delta )^2$ in the integral (Fukagata et al. Reference Fukagata, Iwamoto and Hasegawa2024). This weighting is a result of the triple integration. It emphasizes the wall layer, and de-emphasizes the logarithmic and outer layers. Whether such weighting is physical is debated. The existing literature shows that as the Reynolds number increases, the turbulent motions in the logarithmic and outer layers become increasingly more important (Smits, McKeon & Marusic Reference Smits, McKeon and Marusic2011; Marusic & Monty Reference Marusic and Monty2019). These large-scale and vary-large-scale motions influence the inner dynamics not only through superposition, but also through amplitude modulation (Marusic, Mathis & Hutchins Reference Marusic, Mathis and Hutchins2010; Yang & Howland Reference Yang and Howland2018). Besides, the weighting in the FIK integral poses challenges to laboratory data, which usually incur large uncertainties near the wall (Mehdi et al. Reference Mehdi, Johansson, White and Naughton2014; Volino & Schultz Reference Volino and Schultz2018; Xia, Zhang & Yang Reference Xia, Zhang and Yang2021). To overcome this issue, Volino & Schultz (Reference Volino and Schultz2018) and Xia et al. (Reference Xia, Zhang and Yang2021) proposed methods that allow one to determine the wall shear stress without access to the data in the near-wall region. The last criticism concerns the use of the overall force balance in the derivation, which was already highlighted in § 1.

3.2. The NSC method

The NSC method was introduced in Nikora et al. (Reference Nikora, Stoesser, Cameron, Stewart, Papadopoulos, Ouro, McSherry, Zampiron, Marusic and Falconer2019). The method extends the FIK integral to rough-wall boundary layers. The integral reads

(3.2)\begin{align} \frac{8 \tau_R}{\rho U_b^2} &= \frac{1}{N}\,\frac{48 }{Re_b} + \frac{1}{N}\,\frac{48}{\delta^2 U_b^2}\int_0^{\delta} (\delta - x_3) ( - \phi\langle \overline{u_1'u_3'} \rangle - \phi\langle \widetilde{\bar{u}}_1 \widetilde{\bar{u}}_3 \rangle)\,{\rm d}\kern0.7pt x_3 \nonumber\\ &\quad + \frac{1}{N}\,\frac{24}{\delta^2 U_b^2}\int_0^{\delta} (\delta - x_3)^2 \left( \phi f_x - I_x - \frac{1}{\delta} \int_0^{\delta} (\phi f_x - I_x )\,{\rm d}\kern0.7pt x_3 \right) {\rm d}\kern0.7pt x_3, \end{align}

where

(3.3)\begin{equation} N = \left({ 2 \tau_S - \tau_R - 3 \int_0^{\delta} \rho f_D (1 - x_3/\delta)^2\,{\rm d}\kern0.7pt x_3 }\right)/{\tau_R } \end{equation}

is interpreted as the flow–rough-wall interaction (Nikora et al. Reference Nikora, Stoesser, Cameron, Stewart, Papadopoulos, Ouro, McSherry, Zampiron, Marusic and Falconer2019). Nikora et al. (Reference Nikora, Stoesser, Cameron, Stewart, Papadopoulos, Ouro, McSherry, Zampiron, Marusic and Falconer2019) applied the decomposition in (3.2) and analysed the river-bed friction. They found that both the roughness-induced and large-scale secondary-currents-induced dispersive stress may play significant roles in generating bed friction at sufficiently high Reynolds numbers.

Equation (3.2) has the same weaknesses as FIK. Additionally, the decomposition in (3.2) does not distinguish between the skin friction on the bottom wall and the drag force on the roughness elements.

3.3. Angular momentum integral

The angular momentum integral (AMI) is introduced in Elnahhas & Johnson (Reference Elnahhas and Johnson2022). To derive the AMI, we subtract the freestream momentum equation

(3.4)\begin{equation} \frac{\partial U_\infty}{\partial t} + U_\infty\,\frac{\partial U_\infty}{\partial x_1} ={-}\frac{1}{\rho}\, \frac{\partial P_\infty}{\partial x_1} \end{equation}

from the double-averaged streamwise momentum equation (2.3). This gives

(3.5)\begin{align} & \frac{\partial }{\partial x_3} \left( \phi \left\langle \nu\, \frac{\partial \bar{u}_1 }{\partial x_3} \right\rangle \right) - \frac{\partial \phi \langle \overline{u_1'u_3'} \rangle}{\partial x_3} - \frac{\partial \phi \langle \widetilde{\bar{u}}_1 \widetilde{\bar{u}}_3 \rangle}{\partial x_3} \nonumber\\ &\quad +\left(\frac{\partial (U_\infty - \langle \bar{u}_1 \rangle)\phi \langle \bar{u}_j \rangle } {\partial x_j}+ (U_\infty - \phi \langle \bar{u}_1 \rangle)\,\frac{\partial U_\infty}{\partial x_1}\right) - I_{x,d}+ f_D + \phi f_x = 0, \end{align}

where $I_{x,d}$ contains both the streamwise inhomogeneity terms and the freestream terms:

(3.6)\begin{align} I_{x,d} &={-}\left(\frac{\partial (U_\infty - \phi \langle \bar{u}_1 \rangle)}{\partial t} + \frac{1}{\rho}\,\frac{\partial( P_\infty - \phi \langle \bar{p} \rangle)}{\partial x_1}\right) \nonumber\\ &\quad + \frac{\partial \phi \langle \widetilde{\bar{u}}_1 \widetilde{\bar{u}}_1 \rangle}{\partial x_1} + \frac{\partial \phi \langle \widetilde{\bar{u}}_1 \widetilde{\bar{u}}_2 \rangle}{\partial x_2} + \frac{\partial \phi \langle \overline{ u_1' u_1'} \rangle}{\partial x_1} + \frac{\partial \phi \langle \overline{ u_1' u_2'} \rangle}{\partial x_2} \nonumber\\ &\quad - \frac{\partial }{\partial x_1} \left( \phi \left\langle \nu\, \frac{\partial \bar{u}_1}{\partial x_1} \right\rangle \right) - \frac{\partial }{\partial x_2} \left( \phi \left\langle \nu\, \frac{\partial \bar{u}_1}{\partial x_2} \right\rangle \right). \end{align}

Integrating the deficit momentum equation (3.5) premultiplied by $(x_3-l)$ directly gives the AMI:

(3.7)\begin{align} \frac{\tau_S}{\rho U_\infty^2} &= \frac{1}{Re_l} +\int_0^\infty\frac{( - \phi \langle \overline{u_1'u_3'} \rangle - \phi \langle \widetilde{\bar{u}}_1 \widetilde{\bar{u}}_3 \rangle )}{U_\infty^2 l}\,{\rm d}\kern0.7pt x_3 \nonumber\\ &\quad +\left(\frac{\partial \theta_l}{\partial x_1} + \frac{\theta_l-\theta}{l}\,\frac{\partial l}{\partial x_1} + \frac{2\theta_l}{U_\infty}\,\frac{\partial U_\infty}{\partial x_1}\right) +\frac{\partial \theta_{l2}}{\partial x_2}+\frac{\theta_3}{l} + \frac{\delta^*_l}{U_\infty}\,\frac{\partial U_\infty}{\partial x_1} \nonumber\\ &\quad + \frac{1}{U_\infty^2} \int_0^\infty \left(1-\frac{x_3}{l}\right) (- I_{x,d}+ f_D + \phi f_x)\,{\rm d}\kern0.7pt x_3. \end{align}

Here, $Re_l$ is the Reynolds number based on the freestream velocity $U_\infty$ and the origin location $l$, $\theta$ and $\theta _3$ are the momentum thicknesses,

(3.8a,b)\begin{equation} \theta=\int_0^\infty\left(1 - \frac{\langle \bar{u}_1 \rangle}{U_\infty}\right) \dfrac{\phi \langle \bar{u}_1 \rangle}{U_\infty}\,{\rm d}\kern0.7pt x_3,\quad \theta_3 = \int_0^\infty \left(1 - \frac{\langle \bar{u}_1 \rangle}{U_\infty} \right) \frac{\phi \langle \bar{u}_3 \rangle }{U_\infty}\,{\rm d}\kern0.7pt x_3, \end{equation}

and $\delta _l^*$, $\theta _l$, $\theta _{l2}$ are the modified displacement and momentum thicknesses:

(3.9)$$\begin{gather} \delta^*_l =\int_0^\infty \left(1-\frac{x_3}{l}\right) \left(1 - \frac{\phi \langle \bar{u}_1 \rangle}{U_\infty}\right){\rm d}\kern0.7pt x_3, \end{gather}$$

(3.10a,b)$$\begin{gather}\theta_l= \int_0^\infty \left(1-\frac{x_3}{l}\right) \left(1 - \frac{\langle \bar{u}_1 \rangle}{U_\infty}\right) \frac{\phi \langle \bar{u}_1 \rangle}{U_\infty}\,{\rm d}\kern0.7pt x_3,\nonumber\\ \theta_{l2}=\int_0^\infty \left(1 - \frac{x_3}{l}\right) \left(1 - \frac{\langle \bar{u}_1 \rangle}{U_\infty}\right) \frac{\phi \langle \bar{u}_2 \rangle}{U_\infty}\,{\rm d}x_3. \end{gather}$$

Equation (3.7) holds for arbitrary non-zero $l$. Elnahhas & Johnson (Reference Elnahhas and Johnson2022) propose to pick $l$ such that

(3.11)\begin{equation} \frac{1}{Re_l} = \frac{0.332}{\sqrt{Re_x}} = \frac{0.221}{Re_\theta} =\frac{0.571}{Re_{\delta^*}} = \frac{1.63}{Re_\delta}, \end{equation}

which is the friction coefficient of a laminar flat-plate boundary layer. Here, $Re_x$, $Re_\theta$, $Re_{\delta ^*}$, $Re_\delta$ are the Reynolds numbers based on the streamwise distance, the momentum thickness, the displacement thickness and the boundary layer thickness, respectively. This choice allows one to compare the laminar and turbulent boundary layers.

3.4. Kinetic energy integral

The triple integration in Fukagata et al. (Reference Fukagata, Iwamoto and Kasagi2002) gives rise to weightings that de-emphasize the logarithmic layer, which, as Renard & Deck (Reference Renard and Deck2016) argued, is undesirable. Attempts have been made so that fewer integrations are needed. A noteworthy work along this direction is the kinetic energy integral in Renard & Deck (Reference Renard and Deck2016), which requires a single integration. The ‘trick’ is to integrate the premultiplied mean momentum equation in a reference frame moving with the freestream. The resulting integral can be interpreted as the mean kinetic energy. The integral has been applied to study the effects of compressibility (Li et al. Reference Li, Fan, Modesti and Cheng2019), mean pressure gradient (Fan et al. Reference Fan, Li, Atzori, Pozuelo, Schlatter and Vinuesa2020), chemical reactions (Passiatore et al. Reference Passiatore, Sciacovelli, Cinnella and Pascazio2021) and the transition (Marxen & Zaki Reference Marxen and Zaki2019).

Again, we re-derive the integral based on the double-averaged Navier–Stokes equation. Following Renard & Deck (Reference Renard and Deck2016), we premultiply the double-averaged Navier–Stokes equation by $(\phi \langle \bar {u}_{1} \rangle - U_\infty )$, and integrate the equation. This leads to

(3.12)\begin{align} U_\infty \tau_S/\rho &= \int_0^\infty \nu \left(\frac{\partial \phi\langle \bar{u}_{1} \rangle}{\partial x_{3}} \right)^2 {\rm d}\kern0.7pt x_{3}+ \int_0^\infty(- \phi \langle \overline{u_{1}'u_{3}'} \rangle - \phi \langle \widetilde{\bar{u}}_{1} \widetilde{\bar{u}}_{3} \rangle)\, \frac{\partial \phi \langle \bar{u}_{1} \rangle }{\partial x_{3}}\, {\rm d}\kern0.7pt x_{3} \nonumber\\ &\quad - \int_0^\infty (\phi \langle \bar{u}_1 \rangle- U_\infty) (- I_x + f_D +\phi f_x )\,{\rm d}\kern0.7pt x_3. \end{align}

The first term on the right-hand side is the modified viscous dissipation, the second term is the modified production of the turbulent and dispersive kinetic energy, and the third term is the additional energy loss due to $(- I_x + f_D +\phi f_x )$. For channel flows, the integration of (2.3) should be performed by premultiplying $(\phi \langle \bar {u}_{1} \rangle - U_b)$ (Renard & Deck Reference Renard and Deck2016), which gives

(3.13)\begin{align} U_b \tau_S/\rho &= \int_0^\delta \nu \left(\frac{\partial \phi \langle \bar{u}_{1} \rangle}{\partial x_{3}} \right)^2 {\rm d}\kern0.7pt x_{3}+ \int_0^\delta(- \phi \langle \overline{u_{1}'u_{3}'} \rangle - \phi \langle \widetilde{\bar{u}}_{1} \widetilde{\bar{u}}_{3} \rangle)\, \frac{\partial \phi \langle \bar{u}_{1} \rangle }{\partial x_{3}}\,{\rm d}\kern0.7pt x_{3} \nonumber\\ &\quad -\int_0^\delta (\phi \langle \bar{u}_1 \rangle- U_b) (- I_x + f_D +\phi f_x )\,{\rm d}\kern0.7pt x_3. \end{align}

The integral holds for any $U_r$ that one puts in the premultiplier $(\phi \langle \bar {u}_1\rangle -U_r)$. Like the prefactor $\alpha$ in the FIK integral, and the length scale $l$ in the AMI, $U_r$ here is an adjustable parameter.

4.1. Kinetic-energy-based integral

We multiply the mean momentum equation (2.3) with $(\phi \langle \bar {u}_1 \rangle -U_w)$ and then integrate:

(4.1)\begin{align} & U_w \tau_S/\rho + \int_0^{\delta}(\phi \langle \bar{u}_1 \rangle - U_w ) f_D \,{\rm d}\kern0.7pt x_3 + \int_0^{\delta}(\phi \langle \bar{u}_1 \rangle - U_w ) \phi f_x \,{\rm d}\kern0.7pt x_3 - \int_0^{\delta}(\phi \langle \bar{u}_1 \rangle - U_w ) I_x \,{\rm d}\kern0.7pt x_3 \nonumber\\ &\quad =\int_0^{\delta}\left[ \nu \left( \frac{\partial \phi \langle \bar{u}_1 \rangle }{\partial x_3} \right)^2 - \phi \langle \overline{u_1'u_3'} \rangle\,\frac{\partial \phi \langle \bar{u}_1\rangle }{\partial x_3} - \phi \langle \widetilde{\bar{u}}_1 \widetilde{\bar{u}}_3 \rangle\, \frac{\partial \phi \langle \bar{u}_1 \rangle }{\partial x_3} \right]{\rm d}\kern0.7pt x_3. \end{align}

Here, the flow type (i.e. whether the flow is a boundary layer or a channel) is left unspecified. We will see soon that our decompositions do not depend on the flow type. In the integrals above, $\delta$ is the boundary layer height and should be sufficiently high such that the Reynolds shear stress and the dispersive stress are approximately 0 there. A convenient definition could be $\delta =1.5\delta _{99}$ (Wenzel et al. Reference Wenzel, Gibis and Kloker2022; Xu et al. Reference Xu, Wang and Chen2022) to enable qualitative results. Considering that the far-field contribution to these two stresses is 0, the integration will not depend sensitively on the value of $\delta$ as long as it is sufficiently large. For now, $U_w$ is left undetermined. Equation (4.1) can be interpreted as the kinetic energy equation in the reference frame that moves at speed $U_w$ in the $+x_1$ direction. The terms on the left- and right-hand sides are the external energy source terms and internal energy loss terms, respectively. The external energy source terms are dependent on the choice of the reference frame. The internal energy loss terms are due to viscosity, turbulence and mean flow inhomogeneity, and therefore do not depend on the reference frame. The first two terms on the left-hand side are the terms due to the bottom viscous friction and the roughness drag force, and they contain the skin friction coefficient and the roughness drag coefficient information. The third and fourth terms are energy inputs due to the forcing term and the streamwise evolution of the flow. The terms on the left-hand side are positive or negative depending on $U_w$. When a term is positive, it represents a positive source; and when a term is negative, it represents a negative source. The terms on the right-hand side are almost always positive, representing energy loss due to mean flow dissipation, turbulent production, and the production of dispersive kinetic energy.

The next step is to remove terms that represent external effects and isolate the surface skin friction term and the roughness drag coefficient term. We do that via $U_w$. Specifically, the following $U_w$ eliminates the terms representing external effects and isolates the skin friction term:

(4.2) \begin{align} U_w\equiv U_{w,S} &= \frac{\displaystyle\int_0^{\delta} \phi \langle \bar{u}_1 \rangle (f_D+\phi f_x - I_x)\,{\rm d}\kern0.7pt x_3} {\displaystyle\int_0^{\delta} (\,f_D+\phi f_x - I_x )\,{\rm d}\kern0.7pt x_3} \nonumber\\ &= \frac{1}{\tau_S } \int_0^{\delta} \rho \phi \langle\bar{u}_1 \rangle (f_D +\phi f_x -I_x)\,{\rm d}\kern0.7pt x_3, \end{align}

where we have invoked the force balance in (2.11). The values of $U_{w,S}$ for some specific flow scenarios are listed in table 1. For a fully developed plane channel flow, $f_D=0$, $I_x=0$, and (4.2) gives $U_{w,S}=U_b$. For a zero pressure gradient flat-plate boundary layer, $f_D=0$, $f_x=0$, and (4.2) gives $U_{w,S} = \int _0^{\delta } ({\rho \phi \langle \bar {u}_1 \rangle (-I_x)}/{\tau _S})\,{\rm d}\kern0.7pt x_3$. For a fully developed channel flow in the fully rough regime with $h\ll \delta$, $I_x=0$, the integration of $\phi \langle \bar {u}_1\rangle f_D$ is small, and (4.2) gives $U_{w,S}\approx (\tau _R/\tau _S+1)U_b$. Equations (4.1) and (4.2) together give the following decomposition of the bottom viscous friction coefficient $C_S$:

(4.3)\begin{equation} (0.5 U_{w,S}U_r^2) C_S= \int_0^{\delta}\left[ \nu \left( \frac{\partial \phi \langle \bar{u}_1 \rangle }{\partial x_3} \right)^2 - \phi \langle \overline{u_1'u_3'} \rangle\, \frac{\partial \phi \langle \bar{u}_1 \rangle }{\partial x_3} - \phi \langle \widetilde{\bar{u}}_1 \widetilde{\bar{u}}_3 \rangle\, \frac{\partial \phi \langle \bar{u}_1 \rangle }{\partial x_3} \right]{\rm d}\kern0.7pt x_3, \end{equation}

where $U_r$ is some reference velocity.

Table 1.

Values of $U_{w,S}$ and $U_{w,R}$ for some special flow scenarios. We assume that the flows are fully developed. ZPGBL is short for zero pressure gradient boundary layer.

Similarly, we can isolate the roughness drag and eliminate the terms that represent external effects by choosing the following $U_w$:

(4.4)\begin{equation} U_w\equiv U_{w,R}= \frac{1}{\tau_R} \int_0^{\delta} \rho \phi \langle \bar{u}_1 \rangle (f_D+\phi f_x -I_x)\,{\rm d}\kern0.7pt x_3. \end{equation}

Again, we have also invoked the force balance in (2.11). This velocity obviously exists only for a rough-wall flow, for which $\tau _R\neq 0$. Its values for a few specific flow scenarios are listed in table 1. For a fully-developed channel flow in the transitionally rough regime with $h\ll \delta$, the integration of $\rho \phi \langle \bar {u}_1\rangle f_D$ is small, $I_x=0$, and we have $U_{w,R}\approx (1+\tau _S/\tau _R) U_b$. For fully developed channels in the fully rough regime with $h\ll \delta$, $I_x=0$, the integration of $\rho \phi \langle \bar {u}_1\rangle f_D$ is small, $\tau _R \gg \tau _S$, and we have $U_{w,R}\approx U_b$. Invoking (4.1) and (4.4), we have the following decomposition of the roughness element drag coefficient:

(4.5)\begin{align} (0.5U_{w,R}U_r^2) \lambda_p C_R = \int_0^{\delta} \left[\nu \left(\frac{\partial \phi \langle \bar{u}_1 \rangle }{\partial x_3}\right)^2 - \phi \langle \overline{u_1'u_3'} \rangle\, \frac{\partial \phi \langle \bar{u}_1 \rangle }{\partial x_3} - \phi \langle \widetilde{\bar{u}}_1 \widetilde{\bar{u}}_3 \rangle\, \frac{\partial \phi \langle \bar{u}_1 \rangle }{\partial x_3} \right]{\rm d}\kern0.7pt x_3. \end{align}

We now assess whether the decompositions presented in (4.3) and (4.5) meet our requirements. First and foremost, (4.3) and (4.5) stand as distinct decompositions for the skin friction coefficient $C_S$ and the roughness drag coefficient $C_R$, respectively. Second, ambiguities in the derivation process are removed, and we have unique choices for $U_w$. Third, we have eliminated terms dependent on the reference frame, retaining only those effects that are internal to the flow. Finally, the first term in both equations is the sole surviving term when the two decompositions are evaluated for laminar flows above a flat plate, regardless of whether the flow occurs in a channel or a boundary layer.

Before we proceed to the next subsection, we comment on the choice of keeping the terms that represent internal effects and eliminating the terms that represent external effects in the decompositions in (4.3) and (4.5). First and foremost, we should note that the external effects are only formally eliminated, as they are still contained in our specific choices of $U_w$ (or $l$ and $\alpha$ in the following sections). We can also explicitly keep the terms that represent external effects and eliminate the terms that represent internal effects. To do that, we divide both sides of (4.1) by $U_w$, and take $U_w$ to infinity. The resulting expression, however, reduces to the overall force balance, or Kármán's equation, which is not very instructive.

4.2. Angular-momentum-based integral

We premultiply the mean momentum equation with $(x_3- {l})$ and integrate:

(4.6)\begin{align} & {l} \tau_S/\rho+ \int_0^{\delta}{(x_3 - {l}) f_D \,{\rm d}\kern0.7pt x_3} + \int_0^{\delta}{(x_3 - {l}) \phi f_x\,{\rm d}\kern0.7pt x_3} + \int_0^{\delta}{(x_3 - {l}) ({-}I_x)\,{\rm d}\kern0.7pt x_3} \nonumber\\ &\quad = \int_0^{\delta} \left(\nu\,\frac{\partial \phi \langle \bar{u}_1 \rangle}{\partial x_3} - \phi \langle \overline{u_1'u_3'} \rangle - \phi \langle \widetilde{\bar{u}}_1 \widetilde{\bar{u}}_3 \rangle\right) {\rm d}\kern0.7pt x_3. \end{align}

Here, $l$ is the location of the rotation axis. The terms on the left-hand side encompass the external sources of angular momentum, which include contributions from the surface skin friction, the roughness drag force, the evolution of the flow in the streamwise direction, and external forcing. These terms depend on the location of the axis $l$. In contrast, the terms on the right-hand side represent internal losses of angular momentum due to mean flow gradient, Reynolds shear stress and dispersive stress, and they do not depend on the axis location.

As before, the next step is to eliminate the terms representing external effects and isolate the skin friction and drag force terms via the free parameter in the integral $l$. The following $l$ isolates $\tau _S$:

(4.7)\begin{equation} l\equiv {l}_{S} = \frac{1}{\tau_S} \int_0^{\delta}{ \rho x_3 (f_D +\phi f_x- I_x)\,{\rm d}\kern0.7pt x_3}. \end{equation}

The values of $l_S$ for a few specific flow scenarios are listed in table 2. Equations (4.6) and (4.7) together give the angular-momentum-based decomposition of the skin friction coefficient

(4.8)\begin{equation} C_S =\frac{2}{Re_{{l},S}}\,\frac{U_\delta}{U_r}+ \frac{2}{U_r^2{l}_{S}}\int_0^{\delta} (- \phi \langle \overline{u_1'u_3'} \rangle - \phi \langle \widetilde{\bar{u}}_1 \widetilde{\bar{u}}_3 \rangle)\,{\rm d}\kern0.7pt x_3, \end{equation}

where $U_\delta$ is the velocity at $x_3=\delta$, and $Re_{{l},S}={l}_{S}U_r/\nu$. The first term on the right-hand side of (4.8) represents the laminar contribution and is the only term surviving when the equation is evaluated for laminar-flow and flat-plate scenarios.

Table 2.

Examples of $l_{S}$ and $l_{R}$ in extreme cases; ZPGBL is short for zero pressure gradient boundary layer.

Similarly, we can isolate the roughness drag coefficient and eliminate the terms representing external effects by taking the following ${l}$:

(4.9)\begin{equation} l\equiv {l}_{R}= \frac{1}{\tau_R/\rho} \left(\int_0^{\delta}{ x_3 (f_D + \phi f_x - I_x) \,{\rm d}\kern0.7pt x_3}\right). \end{equation}

The values of $l_R$ for a few specific flow scenarios are listed in table 2. Equations (4.6) and (4.9) together give

(4.10)\begin{equation} \lambda_p C_R= \frac{2}{Re_{{l},R}}\,\frac{U_\delta}{U_r}+\frac{2}{ U_r^2 {l}_{R}}\int_0^{\delta} \left(- \phi \langle \overline{u_1'u_3'} \rangle - \phi \langle \widetilde{\bar{u}}_1 \widetilde{\bar{u}}_3 \rangle\right) {\rm d}\kern0.7pt x_3, \end{equation}

where $Re_{{l},R}=U_r {l}_{R}/\nu$. Again, the first term represents the laminar contribution and is the only surviving term when the equation is evaluated for laminar flow above a flat plate.

Equations (4.8) and (4.10) provide angular-momentum-based decompositions for the skin friction coefficient $C_S$ and roughness drag coefficient $C_R$. Just like the kinetic-energy-based decompositions in (4.3) and (4.5), the decompositions in (4.8) and (4.10) fulfil the requirements that we set forth. As before, we have kept the terms that represent internal effects and eliminated the terms that represent external effects. To keep the terms that represent external effects and eliminate the terms that represent internal effects, we need only divide both sides of (4.6) by $l$, and take $l$ to infinity. The equation again degenerates to the overall force balance.

4.3. Momentum-based integral

We multiply the force balance equation with $\alpha /\delta$ and subtract it from the mean momentum equation, where $\alpha$ is left undetermined for now. Integrating the resulting equation leads to

(4.11)\begin{align} & \left( 1- \frac{\alpha}{3} \right)\tau_S/\rho + \left( - \frac{\alpha}{3} \right)\tau_R/\rho \nonumber\\ &\quad -\int_0^{\delta } \left( 1- \frac{x_3}{\delta} \right)^2 (f_D + \phi f_x - I_x)\,{\rm d}\kern0.7pt x_3+ \left( \frac{\alpha}{3} \right) \left(\int_0^{\delta} (\phi f_x - I_x ) \,{\rm d}\kern0.7pt x_3 \right) \nonumber\\ &\qquad = \frac{2 \nu U_b}{\delta}+ \frac{2}{\delta} \int_0^{\delta} \left( 1- \frac{x_3}{\delta} \right) (- \phi \langle \overline{u_1'u_3'} \rangle - \phi \langle \widetilde{\bar{u}}_1 \widetilde{\bar{u}}_3 \rangle) \,{\rm d}\kern0.7pt x_3. \end{align}

The next step is to eliminate terms representing external effects, and isolate the skin friction and drag terms. The following $\alpha$ isolates $\tau _S$:

(4.12) \begin{align} \alpha_S \equiv \alpha &= \frac{\displaystyle\int_0^{\delta } (1- {x_3}/{\delta})^2 (f_D + \phi f_x - I_x) \,{\rm d}\kern0.7pt x_3} { (1/3) \displaystyle\int_0^{\delta } (f_D + \phi f_x - I_x) \,{\rm d}\kern0.7pt x_3} \nonumber\\ &= \frac{3}{ \tau_S/\rho} \int_0^{\delta } (1- {x_3}/{\delta})^2 (f_D + \phi f_x - I_x) \,{\rm d}\kern0.7pt x_3. \end{align}

The resulting decomposition of the skin friction coefficient $C_S$ is

(4.13)\begin{align} (0.5 U_r^2) C_S = \frac{1}{1 - \alpha_S/3} \left( 2 \nu\,\frac{ U_b}{\delta} + \frac{2}{\delta} \int_0^{\delta} \left( 1- \frac{x_3}{\delta} \right) (- \phi \langle \overline{u_1'u_3'} \rangle - \phi \langle \widetilde{\bar{u}}_1 \widetilde{\bar{u}}_3 \rangle)\,{\rm d}\kern0.7pt x_3 \right), \end{align}

where the first term on the right-hand side is the only term surviving when the decomposition is evaluated for laminar flow above flat plates.

Similarly, we can isolate $\tau _R$ by choosing the following $\alpha$:

(4.14) \begin{align} \alpha_R \equiv \alpha &= \frac{\displaystyle\int_0^{\delta } ( 1- {x_3}/{\delta})^2 (f_D + \phi f_x - I_x) \,{\rm d}\kern0.7pt x_3 - \tau_S/\rho }{({1}/{3}) \left(\displaystyle\int_0^{\delta} ( \phi f_x - I_x ) \,{\rm d}\kern0.7pt x_3 - \tau_S/\rho \right)} \nonumber\\ &= \frac{3}{\tau_R/\rho}\left( \int_0^{\delta } (1- {x_3}/{\delta})^2 (f_D + \phi f_x - I_x) \,{\rm d}\kern0.7pt x_3 - \tau_S/\rho \right). \end{align}

The resulting decomposition of the roughness drag coefficient $C_R$ is

(4.15)\begin{align} (0.5 U_r^2) \lambda_p C_R = \left(- \frac{3}{\alpha_R} \right) \left( 2 \nu\,\frac{ U_b}{\delta} + \frac{2}{\delta} \int_0^{\delta} \left( 1- \frac{x_3}{\delta} \right) (- \phi \langle \overline{u_1'u_3'} \rangle - \phi \langle \widetilde{\bar{u}}_1 \widetilde{\bar{u}}_3 \rangle)\,{\rm d}\kern0.7pt x_3 \right), \end{align}

where $U_r$ is the reference velocity.

The values of $\alpha _S$ and $\alpha _R$ for some specific flow scenarios are tabulated in table 3 and are not elaborated here for brevity. Equations (4.13) and (4.15) give momentum-based decompositions of $C_S$ and $C_R$, and they satisfy the requirements that we set forth. Again, if we were to keep the terms that represent external effects and eliminate terms that represent internal effects, then we would end up with the overall force balance.

Table 3.

Examples of $\alpha _{S}$ and $\alpha _{R}$ in extreme cases; ZPGBL is short for zero pressure gradient boundary layer.

As a final remark for this section, we note that the integrals obtained here should be viewed as alternatives to the existing ones in Fukagata et al. (Reference Fukagata, Iwamoto and Kasagi2002), Nikora et al. (Reference Nikora, Stoesser, Cameron, Stewart, Papadopoulos, Ouro, McSherry, Zampiron, Marusic and Falconer2019), Renard & Deck (Reference Renard and Deck2016) and Elnahhas & Johnson (Reference Elnahhas and Johnson2022), and provide useful extensions of the integral methods to the rough-wall flows.