1 Introduction
In a wall-bounded turbulent flow, the mean skin-friction drag has been identified to be much higher than that in a laminar case at the same Reynolds number. It contributes to the total drag up to 50 % for commercial aircraft, 90 % for submarines and almost 100 % for long pipe and channel flows (Gad-el Hak Reference Gad-el Hak1994). Understanding the mean skin-friction drag generation and its associated near-wall dynamical system is of fundamental and practical importance, particularly for the evaluation of aerodynamic/hydrodynamic performance and design of drag reduction approaches.
Although the mean skin-friction drag is a wall property, as can be directly calculated from the normal gradient of the mean streamwise velocity at the wall, it is connected to the mean and statistical turbulence quantities across the wall layer and can be further decomposed into various physics-informed components according to different mathematical derivations and physical interpretations. Fukagata, Iwamoto & Kasagi (Reference Fukagata, Iwamoto and Kasagi2002) derived a simple relationship (also referred to as the FIK identity) between the skin-friction coefficient and the Reynolds shear stress distribution for three canonical wall-bounded turbulent flows, with three successive integrations of the mean momentum balance equation. For instance, in a smooth incompressible turbulent channel flow the relation can be cast as
$$\begin{eqnarray}\displaystyle C_{f}=\underbrace{\frac{6}{Re_{b}}}_{C_{f1,FIK}}+\underbrace{\frac{6}{u_{b}^{2}}\int _{0}^{h}\left(1-\frac{y}{h}\right)(-\langle u^{\prime }v^{\prime }\rangle )\,\frac{\text{d}y}{h}}_{C_{f2,FIK}}, & & \displaystyle\end{eqnarray}$$
where
$Re_{b}=hu_{b}/\unicode[STIX]{x1D708}$
is the bulk Reynolds number (with
$h$
being the channel half-height,
$u_{b}$
the bulk velocity,
$\unicode[STIX]{x1D708}$
the kinematic viscosity),
$y$
is the distance from the wall surface and
$-\langle u^{\prime }v^{\prime }\rangle$
is the Reynolds shear stress. Equation (1.1) provides a decomposition of the skin-friction coefficient into a ‘laminar’ (
$C_{f1,FIK}$
) and a ‘turbulent’ (
$C_{f2,FIK}$
) contribution, allowing us to quantify the effect of the relative amount of skin-friction drag directly associated with the Reynolds stress. Several modifications to the FIK identity have been proposed in the literature. For instance, Peet & Sagaut (Reference Peet and Sagaut2009) and Bannier, Garnier & Sagaut (Reference Bannier, Garnier and Sagaut2015) extended the FIK identity to complex geometries, to investigate the skin-friction drag reduction with riblets. Modesti et al. (Reference Modesti, Pirozzoli, Orlandi and Grasso2018) generalized the FIK identity to arbitrarily complex geometries by interpreting the mean momentum balance equation as a Poisson equation for the mean velocity and using it to study the skin-friction drag generation in square duct flows. For flat-plate boundary layers, Mehdi & White (Reference Mehdi and White2011) modified the FIK identity by replacing the explicit streamwise gradients with the wall-normal gradient of the total stress on the basis of the Navier–Stokes equation, in order to avoid experimental measurements of streamwise gradients affected by large uncertainties. Moreover, to overcome the difficulty in measuring complete statistics across the whole boundary layer, Mehdi et al. (Reference Mehdi, Johansson, White and Naughton2014) proposed another modified FIK identity in which the upper integration bound may end at any arbitrary location within the boundary layer. Over the years, the FIK identity has been widely used in numerous studies: Iwamoto et al. (Reference Iwamoto, Fukagata, Kasagi and Suzuki2005), Deck et al. (Reference Deck, Renard, Laraufie and Weiss2014), Kametani et al. (Reference Kametani, Fukagata, Örlü and Schlatter2015), de Giovanetti, Hwang & Choi (Reference de Giovanetti, Hwang and Choi2016), to name a few.
Despite the additional insights provided by (1.1), the FIK identity can be difficult to physically interpret (Renard & Deck Reference Renard and Deck2016). One of the key controversial issues is that there is no simple interpretation for the three successive integrations and no physics-informed explanation for the linearly weighted Reynolds shear stress. An alternative and more objective skin-friction drag decomposition method was proposed by Renard & Deck (Reference Renard and Deck2016), referred to as the RD identity hereafter. The RD identity was derived from the mean streamwise kinetic-energy equation in an absolute reference frame in which the undisturbed fluid is not moving. It characterizes the power of skin-friction drag as an energy transfer from the wall to the fluid by means of dissipation of molecular viscosity and turbulent kinetic energy (TKE) production. The authors showed that this method overcomes some of the drawbacks of the FIK identity, allowing an improved physical interpretation of the skin-friction drag generation mechanism. In the case of turbulent channel flow the RD identity can be written as
$$\begin{eqnarray}\displaystyle C_{f}=\underbrace{\frac{2}{u_{b}^{3}}\int _{0}^{h}\unicode[STIX]{x1D708}\left(\frac{\unicode[STIX]{x2202}\langle u\rangle }{\unicode[STIX]{x2202}y}\right)^{2}\,\text{d}y}_{C_{f1,RD}}+\underbrace{\frac{2}{u_{b}^{3}}\int _{0}^{h}(-\langle u^{\prime }v^{\prime }\rangle )\frac{\unicode[STIX]{x2202}\langle u\rangle }{\unicode[STIX]{x2202}y}\,\text{d}y}_{C_{f2,RD}}, & & \displaystyle\end{eqnarray}$$
where
$\langle u\rangle$
is the mean streamwise velocity,
$C_{f1,RD}$
represents the contribution from direct molecular viscous dissipation and
$C_{f2,RD}$
characterizes the contribution associated with the production of TKE. Renard & Deck (Reference Renard and Deck2016) further analysed the Reynolds number dependence of the decomposed terms, and shed light on the importance of logarithmic-layer dynamics in the skin-friction drag generation.
Yoon et al. (Reference Yoon, Ahn, Hwang and Sung2016) related the mean skin-friction drag to the motions of vortical structures by integrating the mean vorticity transport equation, and identified contributions from advective vorticity transport, vortex stretching, viscosity and heterogeneity. Hwang & Sung (Reference Hwang and Sung2017) employed this method to examine the significant contributions associated with large-scale turbulent motions. Kim et al. (Reference Kim, Hwang, Yoon, Ahn and Sung2017) used this method to evaluate the influence of a large-eddy breakup device on the near-wall turbulence and skin-friction drag reduction.
These mean skin-friction drag decomposition methods have all been developed for incompressible flows, whereas only few studies can be found for compressible wall-bounded turbulence. Gomez, Flutet & Sagaut (Reference Gomez, Flutet and Sagaut2009) generalized the FIK identity to compressible flows and studied the compressibility effects on the mean skin-friction drag generation, but they actually have not fully clarified the effect of compressibility on the different contributions. In particular, in their study only the components associated with viscosity variations are ascribed to compressible effects, but compressibility also remarkably impacts the density and therefore these contributions from density variations should be marked as ‘compressible’ contributions as well. One objective of this study is to extend the physics-informed RD identity into a compressible form and use it to evaluate compressibility effects more precisely.
As for the compressibility effects, they can be classified into two kinds: (i) indirect effects due to the variation of the thermodynamic properties, such as density and viscosity; (ii) genuine effects caused by dilatational velocity fluctuations and thermodynamic fluctuations. Based on the Morkovin hypothesis (Morkovin Reference Morkovin1962), the latter could be neglected for non-hypersonic boundary layers (say, Mach number
$M<5$
), since the local root mean square (r.m.s.) fluctuating Mach number is negligible. The effect of density and viscosity variations in wall-bounded turbulence can be accounted for by using appropriate transformations of the mean streamwise velocity and Reynolds stresses, allowing us to collapse the compressible flow statistics onto the ‘universal’ incompressible distributions. A classical transformation of the mean streamwise velocity and Reynolds shear stress has been developed by van Driest (Reference van Driest1951) using inner-layer similarity arguments. This transformation is rather accurate in the case of adiabatic walls but fails on isothermal walls (Huang & Coleman Reference Huang and Coleman1994), especially for cases with strong heat transfer. Huang, Coleman & Bradshaw (Reference Huang, Coleman and Bradshaw1995) proposed a semi-local scaling for transforming the turbulent stresses which yields better collapse onto the corresponding incompressible distributions in a wide Mach number range,
$0.3$
–
$3.5$
(Foysi, Sarkar & Friedrich Reference Foysi, Sarkar and Friedrich2004). Patel et al. (Reference Patel, Peeters, Boersma and Pecnik2015), Trettel & Larsson (Reference Trettel and Larsson2016) showed that the semi-local Reynolds number is the appropriate equivalent Reynolds number to compare flows across Mach numbers. Moreover, Patel, Boersma & Pecnik (Reference Patel, Boersma and Pecnik2016) showed that near-wall streaks are less coherent in the cases where the semi-local Reynolds number increases away from the wall. Pecnik & Patel (Reference Pecnik and Patel2017) also applied the semi-local scaling for deriving transformed conservation equation of the turbulent kinetic energy, which shows that the ‘leading-order effect’ of variable density and viscosity on turbulence in wall-bounded flows can effectively be characterized by the semi-local Reynolds number. Recently, Trettel & Larsson (Reference Trettel and Larsson2016) proposed another transformation based on the logarithmic-layer scaling and near-wall momentum conservation which shows satisfactory agreements with incompressible flow data in a wide range of Reynolds and Mach numbers (Modesti & Pirozzoli Reference Modesti and Pirozzoli2016). The good accuracy of Trettel & Larsson (Reference Trettel and Larsson2016) transformation suggests that it can be used to scale the decomposed skin-friction constituents, to obtain ‘universal’ distributions that match the incompressible ones.
In addition to the skin-friction drag, wall heat transfer is another subject of concern in this paper. Wall heat transfer can be directly linked to the skin-friction drag using temperature–velocity correlations, such as the one proposed by Walz (Reference Walz1959). Direct numerical simulation data have shown that Walz’s relation is rather accurate on adiabatic walls, whereas large deviations are observed for increasing wall heat transfer (Duan, Beekman & Martin Reference Duan, Beekman and Martin2010, Reference Duan, Beekman and Martin2011; Zhang et al. Reference Zhang, Bi, Hussain and She2014). Zhang et al. (Reference Zhang, Bi, Hussain and She2014) replaced the recovery factor in Walz’s equation and generalized it for non-adiabatic flows. Although these correlations give an analogy between the temperature and velocity, they are derived empirically under specific conditions or assumptions. Regarding incompressible boundary layers, Ebadi, Mehdi & White (Reference Ebadi, Mehdi and White2015) proposed an integral formula of wall heat flux, which ascribes the wall heat flux to contributions related to mean temperature profile, turbulent heat flux and gradient of total (molecular and turbulent) heat flux. In the present study, we will provide an exact correlation between the wall heat flux and skin-friction drag coefficient, from the energy conservation equation of compressible turbulent channel flows, giving the relationship between momentum transport and heat transfer at the wall.
This paper is organized in the following way. A compressible counterpart of the RD identity (1.2) for the skin-friction coefficient is derived in § 2, as well as the formulation for the wall heat flux. Direct numerical simulations of five supersonic turbulent channel cases are described in § 3. In § 4, contributions of the molecular viscous dissipation and the TKE production are quantified. We investigate the Reynolds number and Mach number dependence of the mean skin-friction drag generation via the assessment of the turbulence quantities across the channel. In order to obtain universal distributions such as those in incompressible channel flows and capture the ‘compressible’ contributions caused by the density and viscosity variations, compressibility transformations are employed to scale the decomposed constituents. Finally, concluding remarks are given in § 5.
2 Decomposition of the mean skin-friction drag and wall heat flux in compressible turbulent channel flows
Firstly, the RD identity is generalized into a compressible form for compressible turbulent channel flows, and the physical interpretations of each component are discussed. For flat-plate boundary layers, the compressible RD identity is given in appendix A. Secondly, we derive an exact correlation between the wall heat flux coefficient and the skin-friction drag coefficient.
2.1 Mean skin-friction drag decomposition
The mean skin-friction drag coefficient
$C_{f}$
can be cast as
$$\begin{eqnarray}\displaystyle C_{f}=\frac{2\unicode[STIX]{x1D70F}_{w}}{\unicode[STIX]{x1D70C}_{b}u_{b}^{2}}=\frac{2}{Re_{b}}\frac{\unicode[STIX]{x2202}(\langle u\rangle /u_{b})}{\unicode[STIX]{x2202}(y/h)}|_{wall}, & & \displaystyle\end{eqnarray}$$
where
$\langle \cdot \rangle$
is the Reynolds averaging operator,
$\unicode[STIX]{x1D70F}_{w}$
is the wall shear stress,
$u$
is the streamwise velocity,
$y$
is the wall distance and
$Re_{b}$
is the bulk Reynolds number. In compressible channel flows,
$Re_{b}$
(
$=\unicode[STIX]{x1D70C}_{b}u_{b}h/\unicode[STIX]{x1D707}_{w}$
) is based on the channel half-height
$h$
, bulk density
$\unicode[STIX]{x1D70C}_{b}=(1/h)\int _{0}^{h}\langle \unicode[STIX]{x1D70C}\rangle \,\text{d}y$
, bulk velocity
$u_{b}=(1/\unicode[STIX]{x1D70C}_{b}h)\int _{0}^{h}\langle \unicode[STIX]{x1D70C}u\rangle \,\text{d}y$
and dynamic viscosity at the wall
$\unicode[STIX]{x1D707}_{w}$
. Hereafter,
$x$
,
$y$
and
$z$
stand for the streamwise, wall-normal and spanwise directions, respectively.
Considering the channel flow, we assume (i) no-slip condition at the wall surfaces, (ii) statistical homogeneity in the spanwise and streamwise directions and (iii) symmetry with respect to the central plane of the channel. Under these hypotheses the compressible Reynolds-averaged momentum equation in the streamwise (
$x$
-) direction is
$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D70C}u\rangle }{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D70C}uv\rangle }{\unicode[STIX]{x2202}y}=\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D70F}_{yx}\rangle }{\unicode[STIX]{x2202}y}+\langle \unicode[STIX]{x1D70C}f\rangle , & & \displaystyle\end{eqnarray}$$
where
$u$
and
$v$
are respectively the streamwise and wall-normal components of the transient velocity,
$\unicode[STIX]{x1D70C}$
is density,
$t$
is time and
$\unicode[STIX]{x1D70F}_{yx}$
is the shear stress in the streamwise direction. A uniform body force
$f$
is added to drive the flow in the streamwise direction (Huang et al.
Reference Huang, Coleman and Bradshaw1995). Integrating (2.2) from the wall surface to the central plane gives
$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}Q}{\unicode[STIX]{x2202}t}=-\unicode[STIX]{x1D70F}_{w}+\unicode[STIX]{x1D70C}_{b}hf, & & \displaystyle\end{eqnarray}$$
where
$Q=\int _{0}^{h}\langle \unicode[STIX]{x1D70C}u\rangle \,\text{d}y$
is the mass flow rate across the traverse plane. To ensure the flow stability and continuity,
$\unicode[STIX]{x2202}Q/\unicode[STIX]{x2202}t=0$
is set up. Then we get
$$\begin{eqnarray}\displaystyle f=\frac{\unicode[STIX]{x1D70F}_{w}}{\unicode[STIX]{x1D70C}_{b}h}. & & \displaystyle\end{eqnarray}$$
For an arbitrary variable
$\unicode[STIX]{x1D719}$
, its Favre average
$\{\unicode[STIX]{x1D719}\}$
is defined as
$\langle \unicode[STIX]{x1D70C}\unicode[STIX]{x1D719}\rangle /\langle \unicode[STIX]{x1D70C}\rangle$
, and the double prime
$^{\prime \prime }$
denotes the turbulent fluctuations with respect to the Favre average, i.e.
$\unicode[STIX]{x1D719}^{\prime \prime }=\unicode[STIX]{x1D719}-\{\unicode[STIX]{x1D719}\}$
. If we rewrite the left-hand side of (2.2) in the form of Favre average, we have
$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D70C}u\rangle }{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D70C}uv\rangle }{\unicode[STIX]{x2202}y}=\{u\}\left(\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D70C}\rangle }{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D70C}v\rangle }{\unicode[STIX]{x2202}y}\right)+\langle \unicode[STIX]{x1D70C}\rangle \left(\frac{\unicode[STIX]{x2202}\{u\}}{\unicode[STIX]{x2202}t}+\{v\}\frac{\unicode[STIX]{x2202}\{u\}}{\unicode[STIX]{x2202}y}\right)+\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D70C}\rangle \{u^{\prime \prime }v^{\prime \prime }\}}{\unicode[STIX]{x2202}y}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
Using the continuity equation and the total derivative
$D\{\cdot \}/Dt=\unicode[STIX]{x2202}\{\cdot \}/\unicode[STIX]{x2202}t+\{v\}(\unicode[STIX]{x2202}\{\cdot \}/\unicode[STIX]{x2202}y)$
and substituting (2.4) and (2.5) into (2.2), we have
$$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D70C}\rangle \frac{D\{u\}}{Dt}=-\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D70C}\rangle \{u^{\prime \prime }v^{\prime \prime }\}}{\unicode[STIX]{x2202}y}+\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D70F}_{yx}\rangle }{\unicode[STIX]{x2202}y}+\frac{\langle \unicode[STIX]{x1D70C}\rangle }{\unicode[STIX]{x1D70C}_{b}h}\unicode[STIX]{x1D70F}_{w}. & & \displaystyle\end{eqnarray}$$
Following the derivation of the original RD identity (Renard & Deck Reference Renard and Deck2016), we transform the initial reference frame (attached to the wall) into an absolute reference frame, where the wall is moving at the speed of
$-u_{b}$
. Let the subscript
$a$
represent the variables in the absolute frame. Then the time
$t_{a}$
, density
$\unicode[STIX]{x1D70C}_{a}$
, coordinates
$x_{a}$
and
$y_{a}$
and velocity
$u_{a}$
and
$v_{a}$
satisfy
$$\begin{eqnarray}t_{a}=t,\quad \unicode[STIX]{x1D70C}_{a}=\unicode[STIX]{x1D70C},\quad x_{a}=x-u_{b}t,\quad y_{a}=y,\quad u_{a}=u-u_{b},\quad v_{a}=v.\end{eqnarray}$$
Substituting (2.7) into (2.6) yields
$$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D70C}_{a}\rangle \frac{D\{u_{a}\}}{Dt_{a}}=-\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D70C}_{a}\rangle \{u_{a}^{\prime \prime }v_{a}^{\prime \prime }\}}{\unicode[STIX]{x2202}y_{a}}+\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D70F}_{yx}\rangle }{\unicode[STIX]{x2202}y_{a}}+\frac{\langle \unicode[STIX]{x1D70C}_{a}\rangle }{\unicode[STIX]{x1D70C}_{b}h}\unicode[STIX]{x1D70F}_{w}. & & \displaystyle\end{eqnarray}$$
In the absolute reference frame, the averaged streamwise kinetic energy of unit mass
$\{K_{a}\}$
is defined as
${\{u_{a}\}}^{2}/2$
. Thus multiplying both sides of (2.8) by
$\{u_{a}\}$
, we have the energy budget equation,
$$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D70C}_{a}\rangle \frac{D\{K_{a}\}}{Dt_{a}}=-\{u_{a}\}\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D70C}_{a}\rangle \{u_{a}^{\prime \prime }v_{a}^{\prime \prime }\}}{\unicode[STIX]{x2202}y_{a}}+\{u_{a}\}\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D70F}_{yx}\rangle }{\unicode[STIX]{x2202}y_{a}}+\{u_{a}\}\frac{\langle \unicode[STIX]{x1D70C}_{a}\rangle }{\unicode[STIX]{x1D70C}_{b}h}\unicode[STIX]{x1D70F}_{w}. & & \displaystyle\end{eqnarray}$$
Moreover, in statistically temporally and spatially homogeneous channel flows, the rate of mean streamwise kinetic energy in the absolute frame,
$D\{K_{a}\}/Dt_{a}$
, is definitely zero for that
$\{v\}=0$
.
A single integration over the half-channel is then performed on (2.9), using no-slip boundary conditions at the wall and symmetry boundary conditions at the centreline leads to the formula of the skin-friction coefficient in the absolute reference frame,
$$\begin{eqnarray}\displaystyle C_{f}=\underbrace{\frac{2}{\unicode[STIX]{x1D70C}_{b}u_{b}^{3}}\int _{0}^{h}\langle \unicode[STIX]{x1D70F}_{yx}\rangle \frac{\unicode[STIX]{x2202}\{u_{a}\}}{\unicode[STIX]{x2202}y_{a}}\,\text{d}y_{a}}_{C_{f1}}+\underbrace{\frac{2}{\unicode[STIX]{x1D70C}_{b}u_{b}^{3}}\int _{0}^{h}\langle \unicode[STIX]{x1D70C}_{a}\rangle \{-u_{a}^{\prime \prime }v_{a}^{\prime \prime }\}\frac{\unicode[STIX]{x2202}\{u_{a}\}}{\unicode[STIX]{x2202}y_{a}}\,\text{d}y_{a}}_{C_{f2}}. & & \displaystyle\end{eqnarray}$$
The skin-friction coefficient has been decomposed into two components,
$C_{f1}$
represents the direct molecular viscous dissipation, transforming the power of the skin-friction drag into heat, and
$C_{f2}$
represents the power converted into turbulent kinetic energy production induced by turbulent fluctuations, before being dissipated.
For practical reasons, (2.10) is reformulated back into the initial reference frame, and the direct viscous dissipation explicitly associated with the thermodynamic fluctuations is isolated. Thus the skin-friction drag is expressed as
$$\begin{eqnarray}\displaystyle C_{f}=\underbrace{\frac{2}{\unicode[STIX]{x1D70C}_{b}u_{b}^{3}}\int _{0}^{h}\left\langle \unicode[STIX]{x1D707}\left(\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}y}+\frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}x}\right)\right\rangle \frac{\unicode[STIX]{x2202}\{u\}}{\unicode[STIX]{x2202}y}\,\text{d}y}_{C_{f1}}+\underbrace{\frac{2}{\unicode[STIX]{x1D70C}_{b}u_{b}^{3}}\int _{0}^{h}\langle \unicode[STIX]{x1D70C}\rangle \{-u^{\prime \prime }v^{\prime \prime }\}\frac{\unicode[STIX]{x2202}\{u\}}{\unicode[STIX]{x2202}y}\,\text{d}y}_{C_{f2}}, & & \displaystyle\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle C_{f1}=\underbrace{\frac{2}{\unicode[STIX]{x1D70C}_{b}u_{b}^{3}}\int _{0}^{h}\langle \unicode[STIX]{x1D707}\rangle \frac{\unicode[STIX]{x2202}\langle u\rangle }{\unicode[STIX]{x2202}y}\frac{\unicode[STIX]{x2202}\{u\}}{\unicode[STIX]{x2202}y}\,\text{d}y}_{C_{f1,m}}+\underbrace{\frac{2}{\unicode[STIX]{x1D70C}_{b}u_{b}^{3}}\int _{0}^{h}\left\langle \unicode[STIX]{x1D707}^{\prime }\frac{\unicode[STIX]{x2202}u^{\prime }}{\unicode[STIX]{x2202}y}+\unicode[STIX]{x1D707}^{\prime }\frac{\unicode[STIX]{x2202}v^{\prime }}{\unicode[STIX]{x2202}x}\right\rangle \frac{\unicode[STIX]{x2202}\{u\}}{\unicode[STIX]{x2202}y}\,\text{d}y}_{C_{f1,f}}. & & \displaystyle\end{eqnarray}$$
Recalling the incompressible RD identity (1.2) (Renard & Deck Reference Renard and Deck2016), similarities and differences are observed with the compressible formulation (2.11). In particular, as in the incompressible case, the skin-friction drag is decomposed into two branches, associated with molecular viscosity dissipation and TKE production. On the other hand differences can be found in the additional term
$C_{f1,f}$
in (2.12) which embeds the viscosity variations. In compressible isothermal channel flows, direct viscous dissipation consists of
$C_{f1,m}$
and
$C_{f1,f}$
, which are respectively dependent on the mean flow and thermodynamic fluctuations. In § 4.3, we apply classical compressibility transformation to take into account the mean density and viscosity variations and compare the contributions in the compressible cases to those in the incompressible ones.
2.2 Correlation between wall heat flux and skin-friction drag coefficient
Heat transfer in the compressible isothermal channel flow is taken into consideration in this subsection. The Reynolds-averaged energy equation incorporating the channel-flow hypotheses gives
$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D70C}v(e+V^{2}/2)\rangle }{\unicode[STIX]{x2202}y} & = & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\left(K\frac{\unicode[STIX]{x2202}\langle T\rangle }{\unicode[STIX]{x2202}y}\right)-\frac{\unicode[STIX]{x2202}\langle vp\rangle }{\unicode[STIX]{x2202}y}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\unicode[STIX]{x2202}\langle u\unicode[STIX]{x1D70F}_{yx}\rangle }{\unicode[STIX]{x2202}y}+\frac{\unicode[STIX]{x2202}\langle v\unicode[STIX]{x1D70F}_{yy}\rangle }{\unicode[STIX]{x2202}y}+\frac{\unicode[STIX]{x2202}\langle w\unicode[STIX]{x1D70F}_{yz}\rangle }{\unicode[STIX]{x2202}y}+\langle \unicode[STIX]{x1D70C}fu\rangle ,\end{eqnarray}$$
where
$e$
and
$V^{2}/2$
are respectively the internal energy and kinetic energy per unit mass,
$K$
is thermal conductivity,
$T$
is temperature,
$p$
is static pressure,
$w$
is spanwise velocity and
$\unicode[STIX]{x1D70F}_{yx}$
,
$\unicode[STIX]{x1D70F}_{yy}$
,
$\unicode[STIX]{x1D70F}_{yz}$
are viscous stresses respectively along the
$x$
-,
$y$
-,
$z$
-direction. Wall heat flux
$q_{w}$
is defined as the local power per unit area transferred between the wall and fluid, and calculated by
$-K(\unicode[STIX]{x2202}\langle T\rangle /\unicode[STIX]{x2202}y)|_{wall}$
. To obtain the wall heat flux
$q_{w}$
, we integrate (2.13) from the wall surface to the central plane, leading to
$$\begin{eqnarray}\displaystyle q_{w}=-\unicode[STIX]{x1D70C}_{b}u_{b}hf. & & \displaystyle\end{eqnarray}$$
Equation (2.14) shows that the heat transfer to the wall comes from the power done by the external body force, which is balanced by the skin-friction drag. Substituting (2.4) into (2.14) gives
$$\begin{eqnarray}\displaystyle q_{w}=-u_{b}\unicode[STIX]{x1D70F}_{w}. & & \displaystyle\end{eqnarray}$$
As for the wall heat flux coefficient
$B_{q}$
, it is defined to be
$q_{w}/(\unicode[STIX]{x1D70C}_{w}C_{p}u_{\unicode[STIX]{x1D70F}}T_{w})$
, where subscript
$w$
denotes variables at the wall surface,
$C_{p}$
is the specific heat at constant pressure and
$u_{\unicode[STIX]{x1D70F}}$
is friction velocity defined as
$\sqrt{(\unicode[STIX]{x1D70F}_{w}/\unicode[STIX]{x1D70C}_{w})}$
(
$\unicode[STIX]{x1D70F}_{w}$
denotes the wall shear stress). Then we have
$$\begin{eqnarray}\displaystyle B_{q}=-\frac{0.5{M_{b}}^{2}(\unicode[STIX]{x1D6FE}-1)}{(\unicode[STIX]{x1D70C}_{w}u_{\unicode[STIX]{x1D70F}})/(\unicode[STIX]{x1D70C}_{b}u_{b})}C_{f}, & & \displaystyle\end{eqnarray}$$
where
$M_{b}$
is the bulk Mach number based on the bulk velocity and the sound speed at wall temperature, and
$\unicode[STIX]{x1D6FE}$
is the specific heat ratio. Equation (2.16) constitutes a direct relationship between
$B_{q}$
and
$C_{f}$
, allowing us to predict the wall heat flux coefficient and decompose it into different contributions, such as the skin-friction drag coefficient. Thus considering (2.11),
$B_{q}$
is also related to the statistical quantities across the whole wall layer, and no longer strongly dependent on the temperature gradient at the wall surface, which is difficult to accurately obtain.
3 Direct numerical simulation of compressible turbulent channel flows
Direct numerical simulations (DNSs) of compressible turbulent channel flows have been performed with a finite difference code, by solving the three-dimensional unsteady compressible Navier–Stokes equations. The convective terms are discretized with a modified seventh-order weighted compact nonlinear scheme (WCNS) (Nonomura & Fujii Reference Nonomura and Fujii2009; Nonomura, Iizuka & Fujii Reference Nonomura, Iizuka and Fujii2010; Nonomura et al.
Reference Nonomura, Li, Goto and Fujii2011). The viscous terms are evaluated with an eighth-order central difference scheme. A third-order total-variation-diminishing Runge–Kutta scheme (Jiang & Shu Reference Jiang and Shu1996) is adopted for the time integration, and the time step is set to ensure the Courant–Friedrichs–Lewy number is less than unity. A constant molecular Prandtl number
$Pr$
of 0.72 and specific heat ratio
$\unicode[STIX]{x1D6FE}$
of 1.4 are used. To drive the channel flow, a body force is imposed in the streamwise direction, maintaining a constant mass flow rate (Huang et al.
Reference Huang, Coleman and Bradshaw1995).
Table 1. Details of flow conditions, computational domains and grid resolutions.
We carry out five DNSs to investigate the effects of Reynolds and Mach number variation in supersonic isothermal channel flows. Details of the flow conditions, computational domains and grid resolutions are listed in table 1. In particular we carry out three simulations at a bulk Mach number
$M_{b}=u_{b}/c_{w}=1.5$
(where
$c_{w}$
is the speed of sound at wall temperature), and bulk Reynolds numbers
$Re_{b}=$
3000, 9400 and 20 000, respectively. As discussed in Modesti & Pirozzoli (Reference Modesti and Pirozzoli2016), compressible wall-bounded flows can be compared to incompressible flow data at the matching equivalent Reynolds number. To this end we introduce the following mapping for the wall-normal coordinate (Huang et al.
Reference Huang, Coleman and Bradshaw1995; Trettel & Larsson Reference Trettel and Larsson2016),
$$\begin{eqnarray}\displaystyle y_{T}(y)=\frac{(\langle \unicode[STIX]{x1D70C}\rangle /\unicode[STIX]{x1D70C}_{w})^{1/2}}{\langle \unicode[STIX]{x1D707}\rangle /\unicode[STIX]{x1D707}_{w}}y, & & \displaystyle\end{eqnarray}$$
and the transformed friction Reynolds number is defined accordingly (Modesti & Pirozzoli Reference Modesti and Pirozzoli2019),
$$\begin{eqnarray}\displaystyle Re_{\unicode[STIX]{x1D70F}}^{\ast }=y_{T}(h)/\unicode[STIX]{x1D6FF}_{v}. & & \displaystyle\end{eqnarray}$$
The modified wall distance in viscous units is indicated with a plus superscript,
$y_{T}^{+}=y_{T}/\unicode[STIX]{x1D6FF}_{v}$
.
Based on the local and wall properties, we also define the semi-local friction velocity and viscous length scale as
$u_{\unicode[STIX]{x1D70F}}^{\ast }=\sqrt{(\unicode[STIX]{x1D70C}_{w}/\langle \unicode[STIX]{x1D70C}\rangle )}u_{\unicode[STIX]{x1D70F}}$
and
$\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}^{\ast }=(\langle \unicode[STIX]{x1D707}\rangle /\unicode[STIX]{x1D707}_{w}/\sqrt{\langle \unicode[STIX]{x1D70C}\rangle /\unicode[STIX]{x1D70C}_{w}})\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$
, respectively (Huang et al.
Reference Huang, Coleman and Bradshaw1995). In the first three cases, the transformed Reynolds numbers are
$Re_{\unicode[STIX]{x1D70F}}^{\ast }=$
140, 400 and 800, respectively. Moreover we carry out two flow cases at
$M_{b}=3.0$
, and
$Re_{b}=4880$
and
$14\,000$
(
$Re_{\unicode[STIX]{x1D70F}}^{\ast }=150$
and
$400$
), respectively.
The computational domain for the cases at
$M_{b}=1.5$
,
$Re_{b}=3000$
and
$M_{b}=3.0$
,
$Re_{b}=4880$
is
$L_{x}\times L_{y}\times L_{z}=4\unicode[STIX]{x03C0}h\times 2h\times 3\unicode[STIX]{x03C0}h/2$
. For the other cases simulated, the computational domain is set as
$L_{x}\times L_{y}\times L_{z}=2\unicode[STIX]{x03C0}h\times 2h\times \unicode[STIX]{x03C0}h$
. According to del Álamo et al. (Reference del Álamo, Jiménez, Zandonade and Moser2004) and Lozano-Durán & Jiménez (Reference Lozano-Durán and Jiménez2014), this computational domain is sufficiently large to reproduce accurate turbulent statistics in the current range of
$Re_{\unicode[STIX]{x1D70F}}$
. To prove the adequacy of the domain size, we check the two-point correlations both in the streamwise and spanwise directions, as discussed in appendix B. The computational domain is discretized using
$N_{x}\times N_{y}\times N_{z}$
grid points in the
$x$
-,
$y$
- and
$z$
-directions, respectively. A uniform mesh spacing is employed in the spanwise and streamwise directions, whereas in the wall-normal direction the mesh is hyperbolically clustered towards the walls. The minimum and maximum wall-normal grid spacing in classical viscous units are
$\unicode[STIX]{x0394}y_{min}^{+}\approx 0.5$
and
$\unicode[STIX]{x0394}y_{max}^{+}\approx 10$
for all flow cases;
$\unicode[STIX]{x0394}x^{+}$
and
$\unicode[STIX]{x0394}z^{+}$
are the uniform grid spacings in the streamwise and spanwise directions, respectively. Table 1 also gives the grid spacing normalized by the transformed Reynolds number (as in (3.2)), i.e.
$\unicode[STIX]{x0394}y_{T,min}^{+}\lessapprox 0.5$
,
$\unicode[STIX]{x0394}y_{T,max}^{+}\lessapprox 10$
,
$\unicode[STIX]{x0394}x_{T}^{+}\lessapprox 10$
and
$\unicode[STIX]{x0394}z_{T}^{+}\lessapprox 5$
. Periodic boundary conditions are employed in the spanwise and streamwise directions, and isothermal no-slip conditions are imposed at the walls. The flow is initialized with a parabolic velocity profile with random perturbations superposed, and with uniform values of density and temperature (Modesti & Pirozzoli Reference Modesti and Pirozzoli2016).
DNS data at
$M_{b}=1.5$
,
$Re_{b}=3000$
and
$M_{b}=3.0$
,
$Re_{b}=4880$
are compared to flow statistics of Huang et al. (Reference Huang, Coleman and Bradshaw1995), Morinishi, Tamano & Nakabayashi (Reference Morinishi, Tamano and Nakabayashi2004) and Modesti & Pirozzoli (Reference Modesti and Pirozzoli2016). Figure 1 shows the profiles of mean streamwise velocity, temperature and Reynolds shear stress, which agree well with the previous studies. Flow cases at
$M_{b}=1.5$
,
$Re_{b}=9400$
and
$M_{b}=1.5$
,
$Re_{b}=20\,000$
are compared to approximately matching reference data of Modesti & Pirozzoli (Reference Modesti and Pirozzoli2016) at
$M_{b}=1.5$
,
$Re_{b}=7667$
and
$M_{b}=1.5$
,
$Re_{b}=17\,000$
, respectively, as shown in figure 2. Minor differences between the distributions are observed due to the difference in bulk Reynolds numbers, thus confirming the accuracy of the present dataset.
Figure 1. Profiles of (a) mean streamwise velocity, (b) temperature, (c) Reynolds shear stress for the two cases at
$M_{b}=1.5$
,
$Re_{b}=3000$
and
$M_{b}=3.0$
,
$Re_{b}=4880$
.
Figure 2. Profiles of (a) mean streamwise velocity, (b) temperature, (c) Reynolds shear stress for the two cases at
$M_{b}=1.5$
,
$Re_{b}=9400$
and
$M_{b}=1.5$
,
$Re_{b}=20\,000$
.
Figure 3. Mean skin-friction drag coefficients for compressible turbulent channel flows with regard to (a)
$Re_{\unicode[STIX]{x1D70F}}$
and (b)
$Re_{\unicode[STIX]{x1D70F}}^{\ast }$
compared with the incompressible empirical results.
4 Results and discussion
The relations between the mean skin-friction drag coefficients and Reynolds numbers are plotted in figure 3, in which the mean skin-friction drag coefficients are directly calculated from the normal gradients of mean streamwise velocity at the wall. They are compared to the empirical correlation of incompressible turbulent channel flows given by Abe & Antonia (Reference Abe and Antonia2016),
$C_{f}=2/(2.54\ln (Re_{\unicode[STIX]{x1D70F}})+2.41)^{2}$
. Both the friction Reynolds number
$Re_{\unicode[STIX]{x1D70F}}$
and the transformed Reynolds number
$Re_{\unicode[STIX]{x1D70F}}^{\ast }$
are used to check the
$C_{f}-Re$
correlation. Figure 3(a,b) shows that the compressible friction coefficient increases for increasing Mach number, at given
$Re_{\unicode[STIX]{x1D70F}}$
, and using the equivalent Reynolds number
$Re_{\unicode[STIX]{x1D70F}}^{\ast }$
is not sufficient to account for the thermodynamic property variation effects.
According to the theory developed in § 2.1, the mean skin-friction drag coefficients are decomposed into three contributing constituents,
$C_{f1,m}$
,
$C_{f1,f}$
and
$C_{f2}$
. Their proportions in terms of the total skin-friction drag coefficient, as well as
$C_{f1}/C_{f}$
(
$C_{f1}=C_{f1,m}+C_{f1,f}$
), are listed in table 2. The relative errors,
$[(C_{f1}+C_{f2})-C_{f}]/C_{f}$
, are confined within
$\pm 2.82\,\%$
, indicating that the decomposition method is fairly reliable in estimating the mean skin-friction drag coefficients. The decomposed results suggest that at low Reynolds numbers the direct viscous dissipation
$C_{f1}$
is the predominant drag component, reaching approximately 63 % at
$Re_{\unicode[STIX]{x1D70F}}^{\ast }\approx 150$
. As the Reynolds number increases, the predominance of
$C_{f1}$
is gradually overtaken by the component of TKE production
$C_{f2}$
. This phenomenon is consistent with the study of Renard & Deck (Reference Renard and Deck2016) in incompressible flows. At the same
$Re_{\unicode[STIX]{x1D70F}}^{\ast }$
, varying the bulk Mach number
$M_{b}$
from 1.5 to 3.0 does not have any significant influence on the decomposed results.
To further quantify the effects of Reynolds and Mach number on the mean skin-friction drag generation, in the following two subsections we will investigate the profiles of
$C_{f1}/C_{f}$
and
$C_{f2}/C_{f}$
across the channel.
Table 2. Contributions of the decomposed skin-friction drag components.
4.1 Effects of Reynolds number on the mean skin-friction drag generation
The formulas for
$C_{f1}/C_{f}$
and
$C_{f2}/C_{f}$
are rewritten in the intrinsic scales, viz.
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{C_{f1}}{C_{f}}=\int _{0}^{Re_{\unicode[STIX]{x1D70F}}}\frac{u_{\unicode[STIX]{x1D70F}}}{u_{b}}\left\langle \frac{\unicode[STIX]{x1D707}}{\unicode[STIX]{x1D707}_{w}}\left(\frac{\unicode[STIX]{x2202}u^{+}}{\unicode[STIX]{x2202}y^{+}}+\frac{\unicode[STIX]{x2202}v^{+}}{\unicode[STIX]{x2202}x^{+}}\right)\right\rangle \frac{\unicode[STIX]{x2202}\{u\}^{+}}{\unicode[STIX]{x2202}y^{+}}\,\text{d}y^{+}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{C_{f2}}{C_{f}}=\int _{0}^{Re_{\unicode[STIX]{x1D70F}}}\frac{u_{\unicode[STIX]{x1D70F}}}{u_{b}}\frac{\langle \unicode[STIX]{x1D70C}\rangle }{\unicode[STIX]{x1D70C}_{w}}\{-u^{\prime \prime }{v^{\prime \prime }\}}^{+}\frac{\unicode[STIX]{x2202}\{u\}^{+}}{\unicode[STIX]{x2202}y^{+}}\,\text{d}y^{+}, & \displaystyle\end{eqnarray}$$
where the superscript
$+$
denotes normalization with viscous variables.
To assess the effects of Reynolds number on the mean skin-friction drag generation, profiles of the pre-multiplied integrands in (4.1) and (4.2) are plotted in figures 4 and 5, respectively, as a function of viscous wall distance
$y^{+}$
(
$=y/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}},\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}=\unicode[STIX]{x1D708}_{w}/u_{\unicode[STIX]{x1D70F}}$
). Integrating the pre-multiplied integrands over the wall layer gives the corresponding proportions of
$C_{f1}$
and
$C_{f2}$
in the total skin-friction drag.
As shown in figure 4, the pre-multiplied integrands of
$C_{f1}/C_{f}$
share similarity in curve shape and peak at almost the same
$y^{+}$
location for the flows at the same bulk Mach number. Their peaks are reduced when we increase the Reynolds number, indicating a negative correlation between the contribution of viscous dissipation and the Reynolds number. The contribution of viscous dissipation is mainly located in the near-wall region, for instance
$y^{+}\lessapprox 30$
, as seen in figure 4.
Figure 4. Pre-multiplied integrands of
$C_{f1}/C_{f}$
as a function of
$y^{+}$
. (a)
$M_{b}=1.5$
and (b)
$M_{b}=3.0$
.
Figure 5. Pre-multiplied integrands of
$C_{f2}/C_{f}$
as a function of
$y^{+}$
. (a)
$M_{b}=1.5$
and (b)
$M_{b}=3.0$
.
As for the pre-multiplied integrands of
$C_{f2}/C_{f}$
, a similar feature of the peak location and peak value is observed for flows at the same bulk Mach number, as displayed in figure 5. The majority of TKE-production contributions are generated in the region where
$y^{+}\gtrapprox 10$
. The profiles in the near-wall region (
$y^{+}\lessapprox 30$
) collapse well onto a single curve, indicating that the Reynold number has little impact on the TKE production in this region. A secondary peak emerges in the outer layer as the Reynolds number increases, which is ascribed to the generation of the large-scale turbulent motions. The existence of large-scale turbulent motions in the outer layers at large Reynolds numbers has been confirmed in many studies, e.g. Kim & Adrian (Reference Kim and Adrian1999), Balakumar & Adrian (Reference Balakumar and Adrian2007), Hutchins & Marusic (Reference Hutchins and Marusic2007a
), Monty et al. (Reference Monty, Hutchins, Ng, Marusic and Chong2009) and Lee & Sung (Reference Lee and Sung2011). The large-scale turbulent motions lead to the generation of large amounts of turbulent kinetic energy in the outer layer, and correspondingly contribute to the generation of the skin-friction drag. Meanwhile, large-scale turbulent motions have actions that modulate the small-scale turbulent motions in the near-wall region (Hutchins & Marusic Reference Hutchins and Marusic2007b
), which leads to the redistribution of turbulent kinetic energy between the near-wall region and the outer region. This possibly explains the decrease of the (first) peak values in figure 5 as the Reynolds number increases.
4.2 Effects of Mach number on the mean skin-friction drag generation
Keeping the transformed Reynolds number the same, we plot profiles of the pre-multiplied integrands of
$C_{f1}/C_{f}$
and
$C_{f2}/C_{f}$
in figures 6 and 7, respectively, aiming to discuss the effects of Mach number on the mean skin-friction drag generation. Two incompressible turbulent channel flows at
$Re_{\unicode[STIX]{x1D70F}}=$
180 and 395 (Moser, Kim & Mansour Reference Moser, Kim and Mansour1999) are also included for discussion.
Figure 6. Pre-multiplied integrands of
$C_{f1}/C_{f}$
as a function of
$y^{+}$
. (a)
$Re_{\unicode[STIX]{x1D70F}}^{\ast }\approx 150$
and (b)
$Re_{\unicode[STIX]{x1D70F}}^{\ast }\approx 400$
.
Figure 7. Pre-multiplied integrands of
$C_{f2}/C_{f}$
as a function of
$y^{+}$
. (a)
$Re_{\unicode[STIX]{x1D70F}}^{\ast }\approx 150$
and (b)
$Re_{\unicode[STIX]{x1D70F}}^{\ast }\approx 400$
.
The area below the integrands describes the contributions of molecular viscous dissipation or TKE production to the total skin-friction drag. If the transformed Reynolds number is the same, varying the bulk Mach number from 1.5 to 3.0 has little influence on the total area below the integrands, which corresponds to the results in table 2. However, as the Mach number increases, the peaks of the pre-multiplied integrands apparently shift away from the wall as shown in figures 6 and 7. It indicates that the thermodynamic property variations have non-negligible impacts on the distributions of both decomposed constituents respectively related to the molecular viscous dissipation and the TKE production. Figure 8 quantifies the variation of dynamic viscosity with respect to the Mach number. It is a vital parameter in the viscous dissipation distributions and is positively relevant to the temperature through Sutherland’s law. In turbulent isothermal channel flows, the viscosity coefficient increases dramatically in a wider region of
$y^{+}$
, as we increase the Mach number. For instance, the dramatic rise of
$\langle \unicode[STIX]{x1D707}\rangle /\unicode[STIX]{x1D707}_{w}$
is limited in the region of
$y^{+}\lessapprox 30$
at
$M_{b}=1.5$
; if we change the bulk Mach number to 3.0, this region is extended to
$y^{+}\lessapprox 100$
. Thus the maximum of the direct viscous dissipation contribution shows up at a larger
$y^{+}$
. Correspondingly, the dissipation is no longer limited in the inner region (
$y^{+}\lessapprox 30$
), as the bulk Mach number reaches high values, which means that Mach number increase leads to the reinforcement of dissipative effect across the channel and thus the intensification of heat transfer.
Figure 8. Profiles of dynamic viscosity.
4.3 ‘Compressible’ contributions to the skin-friction drag generation
As mentioned in § 2.1,
$C_{f}$
is split into
$C_{f1,m}$
,
$C_{f1,f}$
and
$C_{f2}$
, where
$C_{f1,f}$
is generated due to the fluid compressibility as it contains viscosity fluctuations, and it cannot be transformed into an ‘incompressible’ contribution. Whereas, the effect of density and viscosity variations on
$C_{f1,m}$
and
$C_{f2}$
can be accounted for by applying the compressibility transformations to the streamwise velocity and Reynolds shear stress, viz.
$$\begin{eqnarray}\displaystyle u_{T}^{+}=\int _{0}^{u^{+}}\frac{\langle \unicode[STIX]{x1D707}\rangle }{\unicode[STIX]{x1D707}_{w}}\frac{\text{d}y_{T}^{+}}{\text{d}y^{+}}\,\text{d}u^{+}\quad \text{and}\quad \unicode[STIX]{x1D70F}_{T}^{+}=\frac{-\langle \unicode[STIX]{x1D70C}u^{\prime \prime }v^{\prime \prime }\rangle }{\unicode[STIX]{x1D70C}_{w}u_{\unicode[STIX]{x1D70F}}^{2}}. & & \displaystyle\end{eqnarray}$$
These transformations should allow us to obtain the Mach-number-invariant contributions of
$C_{f1,m}$
and
$C_{f2}$
to the total skin-friction drag coefficient.
Substituting these transformations in (2.12) and (2.11),
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{C_{f1,m}}{C_{f}}=\int _{y=0}^{y=h}\frac{\unicode[STIX]{x2202}\langle u\rangle _{T}^{+}}{\unicode[STIX]{x2202}y_{T}^{+}}\frac{\unicode[STIX]{x2202}\{u\}_{T}^{+}}{\unicode[STIX]{x2202}y_{T}^{+}}\frac{u_{\unicode[STIX]{x1D70F}}^{\ast }}{u_{b}}\,\text{d}y_{T}^{+}+\int _{y=0}^{y=h}\frac{\unicode[STIX]{x2202}\langle u\rangle _{T}^{+}}{\unicode[STIX]{x2202}y_{T}^{+}}\frac{\unicode[STIX]{x2202}\{u\}_{T}^{+}}{\unicode[STIX]{x2202}y_{T}^{+}}\frac{u_{\unicode[STIX]{x1D70F}}^{\ast }}{u_{b}}\frac{y_{T}^{+}}{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}^{\ast }}\,\text{d}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}^{\ast }, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{C_{f2}}{C_{f}}=\int _{y=0}^{y=h}\unicode[STIX]{x1D70F}_{T}^{+}\frac{\unicode[STIX]{x2202}\{u\}_{T}^{+}}{\unicode[STIX]{x2202}y_{T}^{+}}\frac{u_{\unicode[STIX]{x1D70F}}^{\ast }}{u_{b}}\,\text{d}y_{T}^{+}+\int _{y=0}^{y=h}\unicode[STIX]{x1D70F}_{T}^{+}\frac{\unicode[STIX]{x2202}\{u\}_{T}^{+}}{\unicode[STIX]{x2202}y_{T}^{+}}\frac{u_{\unicode[STIX]{x1D70F}}^{\ast }}{u_{b}}\frac{y_{T}^{+}}{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}^{\ast }}\,\text{d}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}^{\ast }. & \displaystyle\end{eqnarray}$$
The integrands in (4.4) and (4.5) depend both on the Mach and Reynolds numbers through the ratio
$u_{\unicode[STIX]{x1D70F}}^{\ast }/u_{b}$
. In order to isolate the effect of mean density and viscosity variations and obtain universal distributions across Mach and Reynolds numbers, we follow the
$C_{f}^{1.5}$
scaling as in Fan, Cheng & Li (Reference Fan, Cheng and Li2019) for incompressible channel flows. To this end, we divide the first terms on the right-hand sides of (4.4) and (4.5) by
$\sqrt{\unicode[STIX]{x1D70C}_{w}/\unicode[STIX]{x1D70C}_{b}(u_{\unicode[STIX]{x1D70F}}/u_{b})^{2}}$
(recall
$C_{f}=2\unicode[STIX]{x1D70C}_{w}/\unicode[STIX]{x1D70C}_{b}(u_{\unicode[STIX]{x1D70F}}/u_{b})^{2}$
), and recast (4.4) and (4.5) as
$$\begin{eqnarray}\displaystyle \frac{C_{f1,m}}{C_{f}^{1.5}} & = & \displaystyle C\int _{0}^{Re_{\unicode[STIX]{x1D70F}}^{\ast }}\frac{\unicode[STIX]{x2202}\langle u\rangle _{T}^{+}}{\unicode[STIX]{x2202}y_{T}^{+}}\frac{\unicode[STIX]{x2202}\{u\}_{T}^{+}}{\unicode[STIX]{x2202}y_{T}^{+}}\,\text{d}y_{T}^{+}\nonumber\\ \displaystyle & & \displaystyle +\,C\int _{0}^{Re_{\unicode[STIX]{x1D70F}}^{\ast }}\frac{\unicode[STIX]{x2202}\langle u\rangle _{T}^{+}}{\unicode[STIX]{x2202}y_{T}^{+}}\frac{\unicode[STIX]{x2202}\{u\}_{T}^{+}}{\unicode[STIX]{x2202}y_{T}^{+}}\left(\sqrt{\frac{\unicode[STIX]{x1D70C}_{b}}{\langle \unicode[STIX]{x1D70C}\rangle }}-1+\frac{y_{T}^{+}}{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}^{\ast }}\sqrt{\frac{\unicode[STIX]{x1D70C}_{b}}{\langle \unicode[STIX]{x1D70C}\rangle }}\frac{\text{d}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}^{\ast }}{\text{d}y_{T}^{+}}\right)\text{d}y_{T}^{+},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \frac{C_{f2}}{C_{f}^{1.5}} & = & \displaystyle C\int _{0}^{Re_{\unicode[STIX]{x1D70F}}^{\ast }}\unicode[STIX]{x1D70F}_{T}^{+}\frac{\unicode[STIX]{x2202}\{u\}_{T}^{+}}{\unicode[STIX]{x2202}y_{T}^{+}}\,\text{d}y_{T}^{+}\nonumber\\ \displaystyle & & \displaystyle +\,C\int _{0}^{Re_{\unicode[STIX]{x1D70F}}^{\ast }}\unicode[STIX]{x1D70F}_{T}^{+}\frac{\unicode[STIX]{x2202}\{u\}_{T}^{+}}{\unicode[STIX]{x2202}y_{T}^{+}}\left(\sqrt{\frac{\unicode[STIX]{x1D70C}_{b}}{\langle \unicode[STIX]{x1D70C}\rangle }}-1+\frac{y_{T}^{+}}{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}^{\ast }}\sqrt{\frac{\unicode[STIX]{x1D70C}_{b}}{\langle \unicode[STIX]{x1D70C}\rangle }}\frac{\text{d}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}^{\ast }}{\text{d}y_{T}^{+}}\right)\text{d}y_{T}^{+},\end{eqnarray}$$
where
$C=1/(\sqrt{2})$
.
Figure 9. Pre-multiplied integrands of (a)
$(C_{f1,m}/C_{f}^{1.5})_{incomp}$
and (b)
$(C_{f2}/C_{f}^{1.5})_{incomp}$
as a function of
$y_{T}^{+}$
.
Given the generally good accuracy of the compressibility transformation used (Modesti & Pirozzoli Reference Modesti and Pirozzoli2016), the first terms in (4.6) and (4.7) are expected to be Mach-number invariant and they are referred to as the incompressible components, and labelled as
$(C_{f1,m}/C_{f}^{1.5})_{incomp}$
and
$(C_{f2}/C_{f}^{1.5})_{incomp}$
. Distributions of these pre-multiplied integrands are given in figure 9, in which good collapse is observed for cases at the same transformed Reynolds number
$Re_{\unicode[STIX]{x1D70F}}^{\ast }$
. Areas beneath the distributions denote the specific values of
$(C_{f1,m}/C_{f}^{1.5})_{incomp}$
and
$(C_{f2}/C_{f}^{1.5})_{incomp}$
, respectively. The maximum mismatch of both
$(C_{f1,m}/C_{f}^{1.5})_{incomp}$
and
$(C_{f2}/C_{f}^{1.5})_{incomp}$
for cases at the same
$Re_{\unicode[STIX]{x1D70F}}^{\ast }$
but different
$M_{b}$
is within
$\pm 1.8\,\%$
, which indicates that the influence of the thermodynamic property variations on each contribution has been removed, as expected. In addition, coincidences of the peak locations at
$y_{T}^{+}\approx 6.5$
and
$17.0$
are observed in figures 9(a) and 9(b), respectively, regardless of the variation of Reynolds number and Mach number. This phenomenon is interestingly consistent with the results for incompressible boundary layers (figure 5 in Renard & Deck’s (Reference Renard and Deck2016) paper) and incompressible channel flows (figures 4b and 5b in Fan et al.’s (Reference Fan, Cheng and Li2019) paper).
Furthermore, we note that additional terms appear which depend on the mean density and viscosity, suggesting that compressibility transformations can only approximately account for mean thermodynamic property variation effects on each contributing term. The second terms in equations (4.6) and (4.7) are therefore denoted as
$(C_{f1,m}/C_{f}^{1.5})_{comp}$
and
$(C_{f2}/C_{f}^{1.5})_{comp}$
, as they cannot be exactly written in terms of transformed variables and they should be regarded as deviations from the standard compressibility transformation. Profiles of the pre-multiplied integrands of
$(C_{f1,m}/C_{f}^{1.5})_{comp}$
and
$(C_{f2}/C_{f}^{1.5})_{comp}$
are plotted in figure 10. Areas beneath the curves explicitly indicate the total values and the figure shows that the relative contributions of
$(C_{f1,m}/C_{f}^{1.5})_{comp}$
and
$(C_{f2}/C_{f}^{1.5})_{comp}$
are small compared to
$(C_{f1,m}/C_{f}^{1.5})_{incomp}$
and
$(C_{f2}/C_{f}^{1.5})_{incomp}$
. We note that the peak values of
$(C_{f1,m}/C_{f}^{1.5})_{comp}$
and
$(C_{f2}/C_{f}^{1.5})_{comp}$
occur in the near-wall region (
$y_{T}^{+}<30$
for instance) where density and viscosity variations are higher, and figure 10 also suggests that these contributions becomes negligible for increasing
$Re_{\unicode[STIX]{x1D70F}}^{\ast }$
. In order to quantify the total ‘compressible’ contribution to the skin-friction drag coefficient we introduce
$$\begin{eqnarray}\displaystyle \left(\frac{C_{f}}{C_{f}^{1.5}}\right)_{comp}=\left(\frac{C_{f1,m}}{C_{f}^{1.5}}\right)_{comp}+\left(\frac{C_{f2}}{C_{f}^{1.5}}\right)_{comp}+\frac{C_{f1,f}}{C_{f}^{1.5}}, & & \displaystyle\end{eqnarray}$$
and we introduce the normalized deviation with respect to the total friction coefficient
$$\begin{eqnarray}\displaystyle R_{comp}=\frac{(C_{f}/C_{f}^{1.5})_{comp}}{(C_{f}/C_{f}^{1.5})_{incomp}+(C_{f}/C_{f}^{1.5})_{comp}}. & & \displaystyle\end{eqnarray}$$
Figure 10. Pre-multiplied integrands of (a)
$(C_{f1,m}/C_{f}^{1.5})_{comp}$
and (b)
$(C_{f2}/C_{f}^{1.5})_{comp}$
as a function of
$y_{T}^{+}$
.
Table 3 shows the different contributions to the skin-friction drag coefficient and highlights that the terms
$(C_{f1,m}/C_{f}^{1.5})_{comp}$
,
$(C_{f2}/C_{f}^{1.5})_{comp}$
, and
$C_{f1,f}/C_{f}^{1.5}$
are negligible compared the other components. In particular the term
$(C_{f1,m}/C_{f}^{1.5})_{comp}$
, associated with deviations from compressibility transformations for the velocity, is found to be close to zero for all flow cases, whereas the term
$(C_{f2}/C_{f}^{1.5})_{comp}$
, associated with deviations from compressibility transformations for the Reynolds stress is found to be larger but decreases with
$Re_{\unicode[STIX]{x1D70F}}^{\ast }$
increasing and it is barely
$3\,\%$
at
$M_{b}=3$
,
$Re_{\unicode[STIX]{x1D70F}}^{\ast }=400$
.
As for the exactly transformed components, they are compared to those of incompressible channel flows in figure 11. Statistical data of the incompressible channel flows at
$Re_{\unicode[STIX]{x1D70F}}^{\ast }=180,550,1000,2000,5200$
are used (Lee & Moser Reference Lee and Moser2015). For the incompressible cases, in contrast with the constancy of
$C_{f1,RD}/C_{f}^{1.5}({\approx}6.5)$
,
$C_{f2,RD}/C_{f}^{1.5}$
has a logarithmic relationship with
$Re_{\unicode[STIX]{x1D70F}}^{\ast }$
, well fitted by
$C_{f2,RD}/C_{f}^{1.5}\approx 1.87\ln (Re_{\unicode[STIX]{x1D70F}}^{\ast })-5.29$
. This conclusion is consistent with the findings for incompressible flows of Laadhari (Reference Laadhari2007), Abe & Antonia (Reference Abe and Antonia2016), Renard & Deck (Reference Renard and Deck2016) and Fan et al. (Reference Fan, Cheng and Li2019). As for the five cases simulated in the present study, the results of
$(C_{f1,m}/C_{f}^{1.5})_{incomp}$
and
$(C_{f2}/C_{f}^{1.5})_{incomp}$
agree fairly well with the incompressible empirical rules, which confirms that the Trettel & Larsson (Reference Trettel and Larsson2016) transformation of velocity and Reynolds stress can accurately take into account the effect of density and viscosity variations.
Figure 11. Distributions of
$(C_{f1,m}/C_{f}^{1.5})_{incomp}$
and
$(C_{f2}/C_{f}^{1.5})_{incomp}$
with regard to
$Re_{\unicode[STIX]{x1D70F}}^{\ast }$
in contrast with the results of incompressible channel flows.
Table 3. ‘Incompressible’ and ‘compressible’ contributions to the total skin-friction drag coefficient.
4.4 Wall heat flux coefficient
Wall heat flux coefficient is an important wall property, which can be directly calculated by
$$\begin{eqnarray}B_{q,local}=-\frac{1}{Re_{b}Pr\unicode[STIX]{x1D70C}_{w}u_{\unicode[STIX]{x1D70F}}/(\unicode[STIX]{x1D70C}_{b}u_{b})}\left.\frac{\unicode[STIX]{x2202}T/T_{w}}{\unicode[STIX]{x2202}y/h}\right|_{y=0}.\end{eqnarray}$$
In § 2.2, we derive an exact relation between the wall heat flux coefficient and the skin-friction drag coefficient. The relation in (2.16) offers a means to predict
$B_{q}$
from
$C_{f}$
, termed
$B_{q,integral}$
. Results of
$B_{q,integral}$
,
$B_{q,local}$
and their relative errors,
$(B_{q,integral}-B_{q,local})/B_{q,local}$
, are listed in table 4. Fairly good agreements are observed in table 4, validating the accuracy of the (2.16).
The exact relation between the wall heat flux coefficient and the skin-friction drag coefficient in (2.16) indicates a similarity between the heat transfer and the momentum transport near the wall. This allows us to decompose the wall heat flux from the perspective of energy budget. Therefore, similar to the decomposition of the skin-friction drag, the wall heat transfer may be related to the molecular viscous dissipation and the turbulent kinetic energy production. Correspondingly, Reynolds and Mach number effects on the wall heat flux can be evaluated, as well. One of the advantages of (2.16) is that it allows for the physical interpretation and prediction of the wall heat flux coefficient based on turbulence quantities across the wall layer in turbulent channel flows.
Table 4. Wall heat flux coefficients at five operation conditions.
5 Conclusions
We focus on the decomposition of mean skin-friction drag in compressible turbulent channel flows. Firstly, we generalize the original RD identity (Renard & Deck Reference Renard and Deck2016) to a compressible form, obtaining two contributing components
$C_{f1}$
and
$C_{f2}$
, respectively associated with: (i) power of the friction transformed into heat via direct molecular viscous dissipation, and (ii) power converted into turbulent kinetic-energy production. Unlike the original RD identity, we further split
$C_{f1}$
into two parts,
$C_{f1,m}$
and
$C_{f1,f}$
, representing the contributions of the mean flow and the thermodynamic fluctuations, respectively.
Secondly, we investigate the Reynolds number dependence of the skin-friction contributions due to molecular viscosity and the TKE production. At low Reynolds number
$C_{f1}$
is dominant, whereas
$C_{f2}$
becomes dominant for increasing equivalent Reynolds number
$Re_{\unicode[STIX]{x1D70F}}^{\ast }$
, representing more than
$50\,\%$
of the total skin-friction coefficient.
Thirdly, we quantify the effect of density and viscosity variations on the skin-friction coefficient by using Trettel & Larsson’s (Reference Trettel and Larsson2016) transformation to separate the ‘incompressible’ and ‘compressible’ contributions. The ‘incompressible’ contributions,
$(C_{f1,m}/C_{f}^{1.5})_{incomp}$
and
$(C_{f2}/C_{f}^{1.5})_{incomp}$
, are found to be comparable to the equivalent terms in incompressible turbulent channel flows (Fan et al.
Reference Fan, Cheng and Li2019), i.e.
$(C_{f1,m}/C_{f}^{1.5})_{incomp}\approx 6.5$
and
$(C_{f2}/C_{f}^{1.5})_{incomp}\approx 1.87\ln (Re_{\unicode[STIX]{x1D70F}}^{\ast })-5.29$
. Upon application of the compressibility transformation, additional terms appear, which cannot be cast as equivalent incompressible contributions and constitute the total ‘compressible’ contribution together with
$C_{f1,f}$
, representing the thermodynamic fluctuations. The total ‘compressible’ contribution to
$C_{f}$
, mostly associated with an excess of TKE production, is quite significant at low Reynolds numbers (up to
$8\,\%$
at
$M_{b}=3.0,Re_{\unicode[STIX]{x1D70F}}^{\ast }=150$
), whereas it becomes negligible at sufficiently large
$Re_{\unicode[STIX]{x1D70F}}^{\ast }$
and low
$M_{b}$
(barely
$0.4\,\%$
at
$M_{b}=1.5,Re_{\unicode[STIX]{x1D70F}}^{\ast }=800$
), suggesting that Morkovin’s hypothesis holds and the effect of thermodynamic property variations can be accounted for simply by using compressibility transformations.
In addition, we derive and validate an exact relationship between the wall heat flux coefficient and the skin-friction drag coefficient, which allows us to relate the heat flux coefficient to the turbulence quantities across the wall layer in turbulent channel flows.
Acknowledgements
The funding support of National Natural Science Foundation of China (under the grant no. 11772194) is acknowledged. We also thank Tianhe-2 and the Center for High Performance Computing at Shanghai Jiao Tong University for supporting the large-scale computations.
Appendix A. Skin-friction decomposition in compressible flat-plate boundary layers
For the compressible turbulent boundary layers, Reynolds number is characterized by free-stream velocity
$u_{\infty }$
, free-stream density
$\unicode[STIX]{x1D70C}_{\infty }$
, boundary-layer thickness
$\unicode[STIX]{x1D6FF}$
and dynamic viscosity at free-stream temperature
$\unicode[STIX]{x1D707}_{\infty }$
. Three assumptions are made: (i) no-slip conditions at the wall surfaces; (ii) statistical homogeneity in the spanwise (
$z$
-) direction; (iii) no additional body force. Then the Reynolds-averaged momentum equation in the streamwise (
$x$
-) direction is formulated as
$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D70C}u\rangle }{\unicode[STIX]{x2202}t}+\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D70C}uu\rangle }{\unicode[STIX]{x2202}x}+\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D70C}uv\rangle }{\unicode[STIX]{x2202}y}=-\frac{\unicode[STIX]{x2202}\langle p\rangle }{\unicode[STIX]{x2202}x}+\left(\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D70F}_{xx}\rangle }{\unicode[STIX]{x2202}x}+\frac{\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D70F}_{yx}\rangle }{\unicode[STIX]{x2202}y}\right), & & \displaystyle\end{eqnarray}$$
where
$\unicode[STIX]{x1D70F}_{xx}$
is the normal stress in the
$x$
-direction.
Following the derivations in § 2.1, the mean skin-friction drag coefficient can be expressed in the absolute reference frame
$$\begin{eqnarray}\displaystyle C_{f} & = & \displaystyle \underbrace{\frac{2}{\unicode[STIX]{x1D70C}_{\infty }u_{\infty }^{3}}\int _{0}^{\unicode[STIX]{x1D6FF}}\langle \unicode[STIX]{x1D70F}_{yx}\rangle \frac{\unicode[STIX]{x2202}\{u_{a}\}}{\unicode[STIX]{x2202}y_{a}}\,\text{d}y_{a}}_{C_{f1}}+\underbrace{\frac{2}{\unicode[STIX]{x1D70C}_{\infty }u_{\infty }^{3}}\int _{0}^{\unicode[STIX]{x1D6FF}}\langle \unicode[STIX]{x1D70C}_{a}\rangle \{-u_{a}^{\prime \prime }v_{a}^{\prime \prime }\}\frac{\unicode[STIX]{x2202}\{u_{a}\}}{\unicode[STIX]{x2202}y_{a}}\,\text{d}y_{a}}_{C_{f2}}\nonumber\\ \displaystyle & & \displaystyle +\,\underbrace{\frac{2}{\unicode[STIX]{x1D70C}_{\infty }u_{\infty }^{3}}\int _{0}^{\unicode[STIX]{x1D6FF}}\langle \unicode[STIX]{x1D70C}_{a}\rangle \frac{D\{K_{a}\}}{Dt_{a}}\,\text{d}y_{a}}_{C_{f3}}\nonumber\\ \displaystyle & & \displaystyle -\,\underbrace{\frac{2}{\unicode[STIX]{x1D70C}_{\infty }u_{\infty }^{3}}\int _{0}^{\unicode[STIX]{x1D6FF}}\{u_{a}\}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}(x_{a}+u_{\infty }t_{a})}(\langle \unicode[STIX]{x1D70F}_{xx}\rangle -\langle \unicode[STIX]{x1D70C}_{a}\rangle \{u_{a}^{\prime \prime }u_{a}^{\prime \prime }\}-\langle p_{a}\rangle )\,\text{d}y_{a}}_{C_{f4}}.\end{eqnarray}$$
Four contributing constituents are obtained:
$C_{f1}$
and
$C_{f2}$
represent the direct viscous dissipation and turbulence ‘dissipation’ into turbulent kinetic-energy production, respectively;
$C_{f3}$
represents the variation of mean streamwise kinetic energy with time, e.g. the time rate of fluid kinetic energy transferred from the moving wall in the absolute frame;
$C_{f4}$
is created by the streamwise heterogeneity, which had better be substituted with local information, in the case that streamwise derivatives are not contained in the database or are unfeasible to obtain (Mehdi & White Reference Mehdi and White2011).
Figure 12. Streamwise and spanwise two-point correlations for streamwise velocity fluctuation
$R_{u^{\prime }u^{\prime }}$
for five channel-flow cases.
Finally, equation (A 2) is allowed to be written equivalently in the wall-attached reference frame for the convenience of calculation
$$\begin{eqnarray}\displaystyle C_{f} & = & \displaystyle \underbrace{\frac{2}{\unicode[STIX]{x1D70C}_{\infty }u_{\infty }^{3}}\int _{0}^{\unicode[STIX]{x1D6FF}}\langle \unicode[STIX]{x1D70F}_{yx}\rangle \frac{\unicode[STIX]{x2202}\{u\}}{\unicode[STIX]{x2202}y}\,\text{d}y}_{C_{f1}}+\underbrace{\frac{2}{\unicode[STIX]{x1D70C}_{\infty }u_{\infty }^{3}}\int _{0}^{\unicode[STIX]{x1D6FF}}\langle \unicode[STIX]{x1D70C}\rangle \{-u^{\prime \prime }v^{\prime \prime }\}\frac{\unicode[STIX]{x2202}\{u\}}{\unicode[STIX]{x2202}y}\,\text{d}y}_{C_{f2}}\nonumber\\ \displaystyle & & \displaystyle +\,\underbrace{\frac{2}{\unicode[STIX]{x1D70C}_{\infty }u_{\infty }^{3}}\int _{0}^{\unicode[STIX]{x1D6FF}}\langle \unicode[STIX]{x1D70C}\rangle (\{u\}-u_{\infty })\frac{\text{D}\{u\}}{\text{D}t}\,\text{d}y}_{C_{f3}}\nonumber\\ \displaystyle & & \displaystyle -\,\underbrace{\frac{2}{\unicode[STIX]{x1D70C}_{\infty }u_{\infty }^{3}}\int _{0}^{\unicode[STIX]{x1D6FF}}(\{u\}-u_{\infty })\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}(\langle \unicode[STIX]{x1D70F}_{xx}\rangle -\langle \unicode[STIX]{x1D70C}\rangle \{u^{\prime \prime }u^{\prime \prime }\}-\langle p\rangle )\,\text{d}y}_{C_{f4}}.\end{eqnarray}$$
Appendix B. Two-point correlations of the streamwise velocity fluctuation
To prove the adequacy of the domain size, we check the two-point correlations of the streamwise velocity fluctuation,
$R_{u^{\prime }u^{\prime }}$
, both in the streamwise and spanwise directions, with the designated computational domain in § 3, as displayed in figure 12.
Figure 12(a,c,e,g,i) shows
$R_{u^{\prime }u^{\prime }}$
in the streamwise direction, and figure 12(b,d,f,h,j) shows
$R_{u^{\prime }u^{\prime }}$
in the spanwise direction. It can be seen that the correlations are limited to small values at large separations, indicating that the domain sizes are sufficiently long (wide).
