2.1. Wavenumber domain (constant outer conditions)

The solution to the Laplace equation in this geometry is itself periodic and can be represented as a Fourier series

(2.2) \begin{align} \phi ^*(y^*,z^*) = \sum _{n=-\infty }^{\infty } \hat {\phi }^*_n(z^*) \exp ({\mathrm{i}} n \beta ^*_1 y^*), \end{align}

where $\beta ^*_1 = 2\pi /s^*$ , $\mathrm{i}$ is the imaginary unit and $n$ numbers the harmonics. Far from the periodic wall the mean ( $n=0$ ) component is given in equation (4b) of Luchini et al. (Reference Luchini, Manzo and Pozzi1991),

(2.3) \begin{align} \hat {\phi }^*_0 = a^*_0 + b^*_0 z^* = \hat {\phi }^*_{0,z^*}(L^*)(z^*+h^*), \end{align}

whereas all the $n\ne 0$ components decay exponentially as $\exp (-\beta ^*_1 z^*)$ or faster. It follows that in most of the domain (precisely for $\beta ^*_1 z^*\gg 1$ or $z^*\gg s^*$ ) the solution has a linear profile as if the wall were flat, and appears to cross zero at a virtual origin $z^*=-h^*$ . This virtual origin, or its distance from an arbitrary reference location that often is taken to be the riblet tip, by definition is the protrusion height, so named by Bechert & Bartenwerfer (Reference Bechert and Bartenwerfer1989) who first introduced and calculated it for the Laplace equation applied to longitudinal flow past riblets. Since the governing equation is linear, and the whole solution $\phi ^*$ is proportional to the strength of the imposed flux $\hat {\phi }^*_{0,z^*}(L^*)$ , the protrusion height is independent of $\hat {\phi }^*_{0,z^*}(L^*)$ and can be obtained once and for all for a given riblet geometry from a single numerical computation, several examples of which were given in Luchini et al. (Reference Luchini, Manzo and Pozzi1991). In addition, given the scale invariance of the Laplace equation, $h^*$ is proportional to $s^*$ and can be written as $h^*=h_1 s^*$ , where $h_1$ is a dimensionless protrusion height calculated for a period equal to unity.

It is also apparent, since the $n\ne 0$ Fourier components decay exponentially with distance and, thus, also with the ratio $L^*/s^*$ , that the approximation provided by substituting the linear profile (2.3) for the actual solution (2.2) is exponentially accurate (its error decays faster than any finite power of $s^*/L^*$ ), and a power expansion in the small parameter $\varepsilon =s^*/L^*$ at fixed $z^*/L^*$ , such as will be considered in the next section, contains no higher powers of $\varepsilon$ than the first-order term with coefficient $h_1$ already included in (2.3).

2.2. Physical domain (slowly varying outer conditions)

Let us now consider a situation where the outer imposed flux is not exactly constant with $y^*$ but slowly varying on a length scale $L^*\gg s^*$ . Under these conditions the solution of the Laplace equation will no longer be exactly periodic, and its expansion in powers of $s^*/L^*$ will contain non-trivial higher-order corrections. (Equivalently the period $s^*$ itself could be assumed to be slowly varying, but we shall not consider this extension at present.) When the main field is characterised by a length scale $L^*$ and the wall texture by a period $s^*$ , the problem lends itself to matched asymptotic expansions in the parameter $\varepsilon = s^*/L^*$ . Note that, even if the Laplace equation is linear with respect to forcing, it is not linear with respect to geometry, and therefore, the expansion is allowed to contain higher powers of $\varepsilon$ .

In order to introduce matched asymptotic expansions, in the style in which they are usually introduced in boundary layers (van Dyke Reference van Dyke1964), we need inner coordinates $Y, Z$ made dimensionless with the riblet period $s^*$ and outer coordinates $y=y_0+\varepsilon Y, z=\varepsilon Z$ made dimensionless with the outer scale $L^*$ in a neighbourhood of a generic boundary point $(y_0,0)$ . As a mnemonic, big $Z$ is a numerically big dimensionless ordinate, $z^*$ divided by the inner reference length, and small $z$ is a small dimensionless ordinate, $z^*$ divided by the outer reference length, so that $z=\varepsilon Z$ . The, now dimensionless, dependent variable $\phi$ can be expanded in an inner series

(2.4) \begin{align} \phi =\varPhi _0(Y,Z)+\varepsilon \varPhi _1(Y,Z)+ {\cdots} = \sum _{i=0}^\infty \varepsilon ^i\varPhi _i(Y,Z) , \end{align}

and an outer series

(2.5) \begin{align} \phi =\phi _0(y,z)+\varepsilon \phi _1(y,z)+ {\cdots} = \sum _{i=0}^\infty \varepsilon ^i\phi _i(y,z) . \end{align}

In consequence of the Laplace equation’s being linear and scale-invariant, each term of both the inner and outer expansions will independently obey its own Laplace equation:

(2.6) \begin{align} \Delta _2 \varPhi _i(Y,Z) = 0, \quad \Delta _2 \phi _i(y,z) = 0 \quad \forall i . \end{align}

In addition, $\varPhi _i$ will obey a Dirichlet boundary condition on a curve $Z=F(Y)$ , $\varPhi _i[Y,F(Y)]=0$ , with the riblet shape $F(Y)$ periodic of period $1$ (because $Y$ has been made dimensionless with the period $s^*$ ), and $\phi _i$ will obey a variable Neumann boundary condition at the outer wall $z=1$ (because $z$ has been made dimensionless with $L^*$ ):

(2.7) \begin{align} \phi _{0,z}(y,1)=q(y);\quad \ \phi _{i,z}(y,1)=0\quad\text{for }i\ne 0 . \end{align}

In order to match the inner to the outer solution in a neighbourhood of a generic boundary position $(y_0,0)$ , we can set $y=y_0 + \varepsilon Y$ , $z =\varepsilon Z$ and expand each term of the outer series (2.5) in a twofold Taylor series in $\varepsilon Y$ and $\varepsilon Z$ :

(2.8) \begin{align} \begin{split} \phi ={}&\phi _0(y_0,0)+\varepsilon Y\frac {\partial \phi _0}{\partial y}(y_0,0)+\varepsilon Z\frac {\partial \phi _0}{\partial z}(y_0,0)+\varepsilon \phi _1(y_0,0)+{}\\ &\varepsilon ^2\frac {Y^2}{2}\frac {\partial ^2\phi _0}{\partial y^2}(y_0,0)+\varepsilon ^2YZ\frac {\partial ^2\phi _0}{\partial y \partial z}(y_0,0)+\varepsilon ^2\frac {Z^2}{2}\frac {\partial ^2\phi _0}{\partial z^2}(y_0,0)+{}\\ &\varepsilon ^2 Y\frac {\partial \phi _1}{\partial y}(y_0,0)+\varepsilon ^2 Z\frac {\partial \phi _1}{\partial z}(y_0,0)+\varepsilon ^2\phi _2(y_0,0)+{} {\cdots} . \end{split} \end{align}

According to the principle of matched asymptotic expansions, as applied to the small-roughness limit, the inner and outer expansions will not be uniformly valid but only match in an intermediate distinguished limit, in which $Y$ and $Z$ tend to infinity at the same time as $z\rightarrow 0$ and $y\rightarrow y_0$ with $\varepsilon \rightarrow 0$ . In this limit we shall term by term equate coefficients of equal powers of $\varepsilon$ between (2.4) and (2.8). So doing yields the following hierarchy.

2.2.2. First order

At inner order $1$ from (2.8) we obtain

(2.9) \begin{align} \varPhi _1(y_0;Y,Z) \sim Z\phi _{0,z}(y_0,0)+\phi _1(y_0,0), \end{align}

where $y_0$ is to be treated as a parameter as far as the Laplace equation for $\varPhi _1$ is concerned, and the symbol ${}\sim {}$ (to be read ‘behaves as’) is here precisely defined to mean that the difference between its left-hand side and right-hand side must tend to zero when $Z\rightarrow \infty$ . This definition may imply more than one actual equation, since the left- and right-hand sides are asymptotically polynomials in $Z$ , and each coefficient of these polynomials must individually match for their difference to tend to zero. Note also that the second term in (2.8), $Y\phi _{0,y}(y_0,0)$ , has been omitted from (2.9) because $\phi _0(y,0) = 0$ for all $y$ , and its derivative is thus also zero. With ${}\sim {}$ given the above meaning, asymptotic condition (2.9) is equivalent to the two equations

(2.10) \begin{align} \varPhi _{1,Z}(y_0;Y,\infty ) = \phi _{0,z}(y_0,0) \end{align}

and

(2.11) \begin{align} \phi _1(y_0,0) = \lim _{Z\rightarrow \infty } \left [\varPhi _1(y_0;Y,Z) - Z\phi _{0,z}(y_0,0)\right ]\! .\end{align}

Equation (2.10) (the right-hand side of which will be a non-trivial function of $y_0$ unless, as in the previous section, $\phi _{0,z}(y,1)$ is constant) closes the Laplace problem for the order-1 inner solution $\varPhi _1$ . Equation (2.11) in turn provides the wall boundary condition for the first-order outer solution $\phi _1$ . Since $\varPhi _1\propto \phi _{0,z}(y_0,0)$ (because of (2.10) and the linearity of the Laplace problem), (2.11) can also be written as

(2.12) \begin{align} \phi _1(y_0,0) = h_1 \phi _{0,z}(y_0,0) ,\end{align}

where constant $h_1$ , independent of $y_0$ , is the same protrusion height as already introduced in § 2.1, and is the only quantity in (2.12) that depends on the actual wall texture $Z=F(Y)$ . In fact, introducing the normalised $\overline {\varPhi _{11}}=\varPhi _1(y_0;Y,Z)/\phi _{0,z}(y_0,0)$ , one realises that $\overline {\varPhi _{11}}$ is independent of $y_0$ , and obeys the Laplace equation $\Delta _2 \overline {\varPhi _{11}} = 0$ with the normalised boundary conditions

(2.13) \begin{align} \overline {\varPhi _{11}}[Y,F(Y)]=0,\quad \overline {\varPhi _{11}}_{,Z}(Y,\infty ) = 1 . \end{align}

Substituting $\overline {\varPhi _{11}}(Y,Z)$ in (2.11) and then comparing with (2.12), one can obtain

(2.14) \begin{align} h_1=\lim _{Z\rightarrow \infty } \left [\overline {\varPhi _{11}}(Y,Z) - Z\right ] . \end{align}

For theoretical purposes the inner–outer expansion is the preferred approach, but in numerical computation to have to deal with the expansion is inconvenient. We can, within the same order of approximation, sum the zeroth- and first-order boundary conditions to generate

(2.15) \begin{align} \phi (y,0)=\phi _0(y,0)+\varepsilon \phi _1(y,0) = \varepsilon h_1 \phi _{0,z}(y,0) +\text{O}(\varepsilon ^2), \end{align}

and, within a similar error O $(\varepsilon ^2)$ , replace $\varepsilon \phi _{0,z}(y,0)$ by $\varepsilon \phi _{z}(y,0)$ (analogously, in a sense, to what in aerodynamics is called the ‘interactive boundary-layer method’; see, e.g. Cebeci Reference Cebeci1999). There results a first-order equivalent boundary condition for the total $\phi$ without any remaining reference to its series expansion:

(2.16) \begin{align} \phi (y,0) = \varepsilon h_1 \phi _{z}(y,0) +\text{O}(\varepsilon ^2), \end{align}

or in dimensional form,

(2.17) \begin{align} \phi ^*(y^*,0) \simeq s^* h_1 \phi ^*_{z^*}(y^*,0), \end{align}

This is the slip-length boundary condition often encountered in the literature on the computational solution of the outer equation, with the slip length $h^*=s^*h_1$ being just another name for the protrusion height. But in addition, as will now be seen, the present matched asymptotic expansion allows the analysis to be continued to higher and higher orders.

2.2.3. Second order

Matching the second-order terms of (2.4) and (2.8) gives

(2.18) \begin{align} \varPhi _2(y_0;Y,Z) \sim YZ\phi _{0,yz}(y_0,0)+Y\phi _{1,y}(y_0,0)+Z\phi _{1,z}(y_0,0)+\phi _2(y_0,0) ,\end{align}

where $\phi _{0,yy}$ has been omitted because $\phi _0(y,0)$ is identically zero, and $\phi _{0,zz}$ has been omitted because from the Laplace equation it equals $-\phi _{0,yy}$ . We note also that, from the $y$ derivative of (2.12), $\phi _{1,y}(y_0,0)$ can be replaced by $h_1\phi _{0,yz}(y_0,0)$ .

A solution of the Laplace equation for $\varPhi _2$ such that it grows linearly in $Y$ at large $Y$ , like (2.18) requires, can be written as

(2.19) \begin{align} \varPhi _2 = \varPhi _{21}(y_0;Y,Z)+Y\,\varPhi _{211}(y_0;Y,Z), \end{align}

where $\varPhi _{21}$ and $\varPhi _{211}$ are periodic functions of $Y$ (of period $1$ ) obeying

(2.20) \begin{align} \Delta _2 \varPhi _{211}(y_0;Y,Z) = 0; \quad \Delta _2 \varPhi _{21}(y_0;Y,Z) = -2\varPhi _{211,Y}(y_0;Y,Z) \end{align}

and

(2.21) \begin{align} \varPhi _{211,Z}(y_0;Y,\infty ) & = \phi _{0,yz}(y_0,0) , \\[-9pt] \nonumber \end{align}

(2.22) \begin{align} \varPhi _{21,Z}(y_0;Y,\infty ) & = \phi _{1,z}(y_0,0) . \\[9pt] \nonumber \end{align}

This is the second-order inner solution. Owing to linearity of the Laplace equation with respect to its inhomogeneous boundary condition, $\varPhi _{211}$ can be written in normalised form as $\varPhi _{211}(y_0;Y,Z)=\overline {\varPhi _{211}}(Y,Z)\,\phi _{0,yz}(y_0,0)$ . But $\overline {\varPhi _{211}}$ turns out to obey the same equation and boundary conditions as $\overline {\varPhi _{11}}$ , and thus, coincides with it. In addition, using (2.12) to replace $\phi _{1,y}$ , and $\overline {\varPhi _{211}}=\overline {\varPhi _{11}}\sim Z+h_1$ from (2.14), it can be shown that $\varPhi _{211}$ cancels out on both sides of (2.18) as follows:

(2.23) \begin{align} \begin{split} &\lim _{Z\rightarrow \infty }\left [Y\varPhi _{211}(y_0;Y,Z) - YZ\phi _{0,yz}(y_0,0)-Y\phi _{1,y}(y_0,0)\right ] =\\ &\lim _{Z\rightarrow \infty }\left [ Y(Z+h_1) \phi _{0,yz}(y_0,0) - YZ\phi _{0,yz}(y_0,0)- Yh_1\phi _{0,yz}(y_0,0)\right ] = 0 . \end{split} \end{align}

The completion of (2.18) then gives

(2.24) \begin{align} \phi _2(y_0,0) = \lim _{Z\rightarrow \infty } \left [\varPhi _{21}(y_0;Y,Z) - Z\phi _{1,z}(y_0,0)\right ] \end{align}

as the wall boundary condition for the second-order outer solution.

Here $\varPhi _{21}$ , on the other hand, is comprised of two normalised contributions, one from the right-hand side of (2.20) and one from the boundary condition (2.22), $\varPhi _{21}=\overline {\varPhi _{21}}\phi _{0,yz}(y_0,0)+\overline {\varPhi _{212}}\phi _{1,z}(y_0,0)$ . It turns out that $\overline {\varPhi _{212}}$ again coincides with $\overline {\varPhi _{11}}$ , whereas $\overline {\varPhi _{21}}$ is the solution of

(2.25) \begin{align} \Delta _2\overline {\varPhi _{21}} = - 2\overline {\varPhi _{11}}_{,Y} \end{align}

with homogeneous boundary conditions $\overline {\varPhi _{21}}[Y,F(Y)]=0$ , $\overline {\varPhi _{21}}_{,Z}(Y,\infty ) = 0$ . Upon defining a new constant

(2.26) \begin{align} h_2 = \lim _{Z\rightarrow \infty } \overline {\varPhi _{21}}(Y,Z) , \end{align}

the second-order outer boundary condition (2.24) can be rewritten as

(2.27) \begin{align} \phi _2(y,0)=h_1 \phi _{1,z}(y,0) + h_2 \phi _{0,yz}(y,0) \end{align}

and the compound equivalent boundary condition as

(2.28) \begin{align} \phi (y,0)= \varepsilon h_1 \phi _{z}(y,0) + \varepsilon ^2 h_2 \phi _{yz}(y,0), \end{align}

or in dimensional form as

(2.29) \begin{align} \phi ^*(y^*,0)= s^* h_1 \phi ^*_{z^*}(y^*,0) + s^{*2} h_2 \phi ^*_{y^*z^*}(y^*,0). \end{align}

The new dimensionless coefficient $h_2$ can be called the ‘second-order protrusion coefficient’ (or ‘second-order protrusion height’, but we refrain from this denomination because $s^{*2} h_2$ has physical dimensions of the square of a height, and subsequent $n$ th-order coefficients will be the $n$ th power of a height). Also, it can be noted that if $\overline {\varPhi _{21}}$ happens to be an odd function of $Y$ , a limit such as (2.26), which must then equal its opposite under the transformation $Y\leftarrow -Y$ , is necessarily zero. More about this symmetry is given at the end of § 2.2.5.

Now a pattern begins to appear.

2.2.4. Third order

The third-order matching condition is

(2.30) \begin{align} \varPhi _3(y_0;Y,Z) \sim \left (\!\frac {Y^2Z}{2}-\frac {Z^3}{6}\right ) \!\phi _{0,yyz} + \frac {Y^2-Z^2}{2}\phi _{1,yy} + YZ\phi _{1,yz}+Y\phi _{2,y}+Z\phi _{2,z}+\phi _3 , \end{align}

where second and higher $z$ derivatives of $\phi _0$ and $\phi _1$ have been substituted from the Laplace equation, and subsequently, $y$ derivatives of $\phi _0$ have been cancelled. Here $\phi _{1,yy}$ and $\phi _{2,y}$ can be replaced from (2.11) and (2.24) as

(2.31) \begin{align} \phi _{1,yy}(y_0,0) = h_1 \phi _{0,yyz}(y_0,0), \\[-28pt] \nonumber \end{align}

(2.32) \begin{align} \phi _{2,y}(y_0,0)=h_1 \phi _{1,yz}(y_0,0) + h_2 \phi _{0,yyz}(y_0,0) \\[-2pt] \nonumber \end{align}

to yield

(2.33) \begin{align} & \varPhi _3(y_0;Y,Z) \sim \left [\frac {Y^2}{2}(Z+h_1)+Yh_2-\frac {Z^3}{6} - \frac {Z^2}{2}h_1\right ]\phi _{0,yyz} + Y\left (Z+h_1\right )\phi _{1,yz} \nonumber \\ & \qquad +Z\phi _{2,z}+\phi _3 . \end{align}

By analogy with (2.19) one can make the following ansatz:

(2.34) \begin{align} \begin{split} \varPhi _3 = \left [\overline {\varPhi _{31}}(Y,Z)+Y\,\overline {\varPhi _{21}}(Y,Z)+(Y^2/2)\,\overline {\varPhi _{11}}(Y,Z)\right ]\phi _{0,yyz}+{}\\ \left [\,\overline {\varPhi _{21}}(Y,Z)+Y\overline {\varPhi _{11}}(Y,Z)\right ]\phi _{1,yz}+{}\\ \overline {\varPhi _{11}}(Y,Z)\,\phi _{2,z}. \end{split} \end{align}

The second and third line here are nothing else than the previous-order normalised solutions of the Laplace equation multiplied by $\phi _{1,yz}$ instead of $\phi _{0,yz}$ , and $\phi _{2,z}$ instead of $\phi _{0,z}$ . The first line, once inserted into the Laplace equation for $\varPhi _{3}$ , gives

(2.35) \begin{align} \Delta _2 \overline {\varPhi _{31}} + 2\overline {\varPhi _{21}}_{,Y} + \overline {\varPhi _{11}} = 0 \end{align}

as the determining equation for the new periodic function $\overline {\varPhi _{31}}$ , with boundary conditions

(2.36) \begin{align} \overline {\varPhi _{31}}[Y,F(Y)]=0,\quad \lim _{Z\rightarrow \infty }\big [\overline {\varPhi _{31}}_{,Z} + Z^2/2 + h_1 Z\big ] = 0 . \end{align}

From (2.35) and the already known asymptotic behaviour of $\overline {\varPhi _{21}}_{,Y}$ and $\overline {\varPhi _{11}}$ , one can get the asymptotic behaviour for $Z\rightarrow \infty$ of $\overline {\varPhi _{31}}$ as

(2.37) \begin{align} \overline {\varPhi _{31}} \sim -Z^3/6 - h_1 Z^2/2 + h_3. \end{align}

which, in addition to making (2.33) satisfied, of course implies the definition

(2.38) \begin{align} h_3 = \lim _{Z\rightarrow \infty } \big [\overline {\varPhi _{31}} +Z^3/6 + h_1 Z^2/2\big ] . \end{align}

Extracting $\phi _3(y_0,0)$ from (2.33) then gives the sought-after third-order outer boundary condition:

(2.39) \begin{align} \phi _3(y_0,0)=h_1 \phi _{2,z}(y_0,0) + h_2 \phi _{1,yz}(y_0,0) + h_3 \phi _{0,yyz}(y_0,0) . \end{align}

2.2.5. Compound equivalent boundary condition

The above process can be continued to higher and higher orders and, for each $n$ , will provide an outer boundary condition of the form

(2.40) \begin{align} \phi _n(y,0)= \sum _{i=1}^n h_i \frac {\partial ^{i}\phi _{n-i}}{\partial y^{i-1}\partial z}(y,0) . \end{align}

The boundary conditions of different orders can be multiplied by $\varepsilon ^n$ and summed together, if desired, to get the total outer $\phi$ at the virtual wall:

(2.41) \begin{align} \phi (y,0)=\sum _{n=0}^N \varepsilon ^n\phi _n(y,0)+\text{O} \big(\varepsilon ^{N+1} \big) =\sum _{n=1}^N \sum _{i=1}^n \varepsilon ^n h_i \frac {\partial ^{i}\phi _{n-i}}{\partial y^{i-1}\partial z}(y,0) +\text{O}\big(\varepsilon ^{N+1} \big) . \end{align}

This equation looks like the product of two power series. In fact, by reordering summations it can be rewritten as

(2.42) \begin{align} \phi (y,0)=\sum _{i=1}^N h_i \frac {\partial ^{i}}{\partial y^{i-1}\partial z} \sum _{n=i}^N \varepsilon ^{n} \phi _{n-i}(y,0) +\text{O} \big(\varepsilon ^{N+1} \big) . \end{align}

But $\sum _{n=i}^N \varepsilon ^{n} \phi _{n-i} =\varepsilon ^i\sum _{n=0}^{N-i} \varepsilon ^{n} \phi _{n} = \varepsilon ^i\phi +\text{O}(\varepsilon ^{N+1})$ , and therefore, within an error $\text{O}(\varepsilon ^{N+1})$ ,

(2.43) \begin{align} \phi (y,0)=\sum _{i=1}^N \varepsilon ^i h_i \frac {\partial ^{i}\phi }{\partial y^{i-1}\partial z} +\text{O} \big(\varepsilon ^{N+1} \big) , \end{align}

or, in dimensional form,

(2.44) \begin{align} \phi ^*(y^*,0)=\sum _{i=1}^N s^{*i} h_i \frac {\partial ^{i}\phi ^*}{\partial y^{*i-1}\partial z^*} +\text{O} \big(\varepsilon ^{N+1} \big) . \end{align}

Just as (2.16) was at first order, this single implicit $N$ th-order boundary condition is free of any reference to the outer series expansion, and may be preferred to (2.40) for computational purposes. One can also observe that if the wall shape $Z=F(Y)$ happens to be left–right symmetric, i.e. invariant under the mirror transformation $Y\leftarrow -Y$ , only even $y$ derivatives have the same symmetry and are allowed in (2.40)–(2.44). Odd $y$ derivatives are forbidden by symmetry, and the corresponding higher-order protrusion coefficients must automatically come out as zero. Note that this does not imply that the corresponding inner solutions are zero, only that they shall have the appropriate symmetry for these coefficients to vanish. For instance, when riblets are mirror symmetric, $\overline {\varPhi _{11}}$ is an even function of $Y$ and then $\overline {\varPhi _{21}}$ is an odd function of $Y$ , because so is $\overline {\varPhi _{11}}_{,Y}$ on the right-hand side of (2.25); its mean value, which when $Z\rightarrow \infty$ provides $h_2$ , is thus zero.

3.1. Wavenumber domain (constant outer conditions)

For constant $p_x$ and a constant velocity gradient at $Z=L=1/\varepsilon$ , the solution of (3.3) can be expanded in a Fourier series, just like the solution of the Laplace equation (2.1), i.e.

(3.4) \begin{align} U(Y,Z) = \sum _{n=-\infty }^{\infty } \hat {U}_n(Z) \exp ({\mathrm{i}} n 2\pi Y). \end{align}

The only difference is that now $\hat {U}_{0,ZZ}$ equals $\varepsilon ^2 p_x$ , and instead of (2.3) the mean ( $n=0$ ) Fourier component becomes

(3.5) \begin{align} \hat {U}_0 = a_0 + b_0 Z +\varepsilon ^2p_x Z^2/2. \end{align}

The coefficients $a_0$ and $b_0$ in this expression could be determined, as in § 2.1, from the value of the velocity gradient at a large distance. However, it must be noted that while in (2.3) $\hat \phi ^*_{0,z^*}$ is constant with $z^*$ , and $\hat \phi ^*_{0,z^*}(L^*)$ directly equals $b_0$ , in the presence of a pressure gradient the mean velocity profile becomes a quadratic, and to determine $b_0=\hat {U}_{0,Z}(0)$ is more complicated. If one wants to write an equivalent far-from-the-wall boundary condition transported in $Z=0$ ,

(3.6) \begin{align} a_0 = b_0 h_1 + \varepsilon ^2 p_x h^{(p_x)}_2 , \end{align}

by linearity, the two constants $h_1$ and $h^{(p_x)}_2$ can be determined separately: the first by applying a suitable non-zero boundary condition on $\hat {U}_{0,Z}$ at $Z=L$ with $p_x=0$ (and this shows that $h_1$ is precisely the same protrusion height as defined in § 2), and the second by applying the boundary condition $\hat {U}_{0,Z}=-\varepsilon ^2p_x L$ at $Z=L$ , to make $b_0=0$ with $p_x\ne 0$ . Since $ b_0 = 0 ,$ and one can obtain

(3.7) \begin{align} h^{(p_x)}_2 = \frac {\hat {U}_0(L)}{\varepsilon ^2p_x}- \frac {L^2}{2} ,\end{align}

and this value is actually independent of both $p_x$ and $L$ .

In outer coordinates, (3.6) provides an equivalent boundary condition

(3.8) \begin{align} u(y,0)=\varepsilon h_1 u_z(y,0)+ \varepsilon ^2 p_x h_2^{(p_x)} ,\end{align}

which includes the second-order effect of the pressure gradient. Again no higher powers of $\varepsilon$ are involved, and (3.8) is exponentially accurate, when outer conditions are constant and the inner solution is perfectly periodic.

3.2. Physical domain (slowly varying outer conditions)

In the presence of a slowly varying outer shear $u_{0,z}(y,1) = q(y)$ and dimensionless pressure gradient $p_x(y,z)$ , the outer problem becomes a boundary-value problem with an equivalent wall boundary condition at $z=0$ , which in full generality may be written by assigning $u(y,0)$ as a functional of $u_z(y,0)$ and $p_x(y,0)$ . In order to obtain the equivalent boundary condition without a priori solving the outer problem, one can observe that if $u(y,0)$ , $u_z(y,0)$ and $p_x$ were all simultaneously known, the outer problem could be in principle transformed into an initial-value problem and formally solved by a Cauchy series. As long as the solutions of the boundary-value problem and the initial-value problem are the same, it is therefore sufficient that the equivalent boundary condition be consistent with this Cauchy series.

To remind the reader what a Cauchy series is, it is in fact a Taylor series in a neighbourhood of a generic wall point $(y_0,0)$ , where $z$ derivatives higher than the first have been replaced using the differential equation, differentiated zero or more times in $z$ :

(3.9) \begin{align} u(y,z) & =u(y_0,z)+\frac {\partial u}{\partial y}(y-y_0)+\frac {\partial u}{\partial z}z + \frac {\partial ^2 u}{\partial y^2}\frac {(y-y_0)^2}{2}+\frac {\partial ^2 u}{\partial y\partial z}(y-y_0)z \nonumber \\ & \quad +\left (p_x - \frac {\partial ^2 u}{\partial y^2}\right )\frac {z^2}{2} + {\cdots} \end{align}

We, in fact, already did a similar replacement when, in (2.18), we obtained the second derivative $\phi _{0,zz}$ from the Laplace equation as $\phi _{0,zz}=-\phi _{0,yy}$ and, thus, determined it was zero.

This device was originally invented by Cauchy in order to express the general solution of an initial-value problem in time, but here it will be used to reformulate our boundary-value problem as an initial-value problem in $z$ . Whereas for an elliptic equation the initial-value problem is notoriously ill-posed, the trick here will be to formally apply the Cauchy series to initial values obtained from the two-sided boundary-value problem itself. This will allow us to reformulate the inner problem as an equivalent boundary condition, without (or before) actually solving the outer problem.

For the purposes of matched asymptotic expansions, the Cauchy series will replace the outer solution that the inner solution must match. If in (3.9) $y-y_0$ is replaced by $\varepsilon Y$ and  $z$ by $\varepsilon Z$ , (3.9) becomes

(3.10) \begin{align} u(y,z) & = u(y_0,0)+ \varepsilon \left (\frac {\partial u}{\partial y} Y+\frac {\partial u}{\partial z} Z\right ) \nonumber \\ & \quad +\varepsilon ^2\left [\frac {\partial ^2 u}{\partial y^2}\frac {Y^2}{2} + \frac {\partial ^2 u}{\partial yz}YZ +\left (p_x-\frac {\partial ^2 u}{\partial y^2}\right )\frac {Z^2}{2}\right ]+ {\cdots} \end{align}

and can be reordered as a power series in $\varepsilon$ . This is the asymptotic behaviour that the inner solution must match, term by term in $\varepsilon$ and uniformly in $Y$ (or at least for large $Y$ , but then automatically for any $Y$ ), when $Z\rightarrow \infty$ .

Since the problem depends linearly on $p_x$ , and the equivalent boundary condition can be written as a superposition of a $u(y_0,0)$ function of $u_z(y_0,0)$ and a $u(y_0,0)$ function of $p_x(y_0,0)$ , it is evident that the $u_z(y,0)$ -dependent part will be identical to the one calculated in § 2 and given by (2.28). The $p_x(y_0,0)$ -dependent part, unsurprisingly, first appears in the expansion at order $\varepsilon ^2$ in (3.10) and multiplies $Z^2/2$ . Therefore, the corresponding inner problem is

(3.11) \begin{align} \Delta _2 \overline {U_{22}} = 1 \end{align}

with boundary conditions $\overline {U_{22}}[Y,F(Y)]=0$ and $\lim _{Z\rightarrow \infty }[\overline {U_{22}}_{,Z}(Y,Z)- Z] =0$ . From the solution of this inner problem one extracts

(3.12) \begin{align} h_2^{(p_x)} = \lim _{Z\rightarrow \infty } \big[\overline {U_{22}}(Y,Z)-Z^2/2 \big] \end{align}

and (2.27) acquires an additional term $h_2^{(p_x)} p_{0,x}(y,0)$ , consistently with (3.8). Equation (2.28) then becomes

(3.13) \begin{align} u(y,0) = \varepsilon h_1 u_{z} + \varepsilon ^2 h_2 u_{yz} + \varepsilon ^2 h_2^{(p_x)} p_{x} . \end{align}

Consideration of the next order will be delayed until § 6 since it involves coupling the longitudinal flow with the cross-flow.

4.1. Wavenumber domain (constant outer conditions)

In Fourier space, the solution to the biharmonic equation is

(4.6) \begin{align} \psi ^*(y^*,z^*) = \sum _{n=-\infty }^{\infty } \hat \psi ^*_n(z^*) \exp (\mathrm{i} n \beta ^*_1 y^*), \end{align}

where the far-field (mean, $n=0$ ) component is a third-degree polynomial given in equation (5b) of Luchini et al. (Reference Luchini, Manzo and Pozzi1991),

(4.7) \begin{align} \begin{split} &\hat {\psi }^*_0 = A^*_0 + B^*_0 z^* + C^*_0 z^{*2} + D^*_0 z^{*3} = {}\\ & \hat {\psi }^*_0(L^*) +\hat {\psi }^*_{0,z^*}(L^*) (z^*-L^*) +\hat {\psi }^*_{0,z^*z^*}(L^*) \frac {1}{2}(z^*-L^*)^2 +\hat {\psi }^*_{0,z^*z^*z^*}(L^*) \frac {1}{6}(z^*-L^*)^3, \end{split} \end{align}

and all the other components decay at least as fast as $z^*\exp (-\beta ^*_1 z^*)=z^*\exp (-2\pi z^*/s^*)$ with $z^*\rightarrow \infty$ .

When the problem is linear and constant-coefficient, the $a^*,b^*,c^*,d^*$ coefficients can be exactly recovered from a set of two numerical calculations over a domain $0\le z^* \le L^*$ (with $L^*\gg s^*$ ):

1. (i) set $\psi ^*_{z^*z^*}(y^*,L^*) = 1$ (arbitrary constant), $\psi ^*_{z^*z^*z^*}(y^*,L^*) = 0$ , $\psi ^*_{n^*}|_{\text{w}} = 0$ (no slip), $\psi ^*|_{\text{w}} = 0$ (impermeable) and measure $\psi ^*(y^*,L^*)$ , $\psi _z(y^*,L^*)$ (constants with $y^*$ ), so we know $\boldsymbol{p}_1^T|_L \equiv (\hat {\psi }^*_0, \hat {\psi }^*_{0,z^*},\hat {\psi }^*_{0,z^*z^*},\hat {\psi }^*_{0,z^*z^*z^*})_1(L^*)$ ;

2. (ii) set $\psi ^*_{z^*z^*}(y^*,L^*) = 0$ , $\psi ^*_{z^*z^*z^*}(y^*,L^*) = 1$ (arbitrary constant), $\psi ^*_{n^*}|_{\text{w}} = 0$ (no slip), $\psi ^*|_{\text{w}} = 0$ (impermeable) and measure $\psi ^*(y^*,L^*)$ , $\psi _z(y^*,L^*)$ (constants with $y^*$ ), so we know $\boldsymbol{p}_2^T|_L \equiv (\hat {\psi }^*_0, \hat {\psi }^*_{0,z^*},\hat {\psi }^*_{0,z^*z^*},\hat {\psi }^*_{0,z^*z^*z^*})_2(L^*)$ .

We can get the far-field solution (4.7) and its derivatives propagated from $z^*=L^*$ to $z^*=0$ by

(4.8) \begin{align} \underbrace {\left [\begin{matrix} \hat {\psi }^*_0 \\[4pt] \hat {\psi }^*_{0,z^*} \\[4pt] \hat {\psi }^*_{0,z^*z^*} \\[4pt] \hat {\psi }^*_{0,z^*z^*z^*} \end{matrix}\right ](0)}_{\boldsymbol{p}|_0} = \left [\begin{matrix} 1 & -L^* & (-L^*)^2/2 & (-L^*)^3/6 \\[4pt] 0 & 1 & -L^* & (-L^*)^2/2 \\[4pt] 0 & 0 & 1 & -L^* \\[4pt] 0 & 0 & 0 & 1 \\[4pt] \end{matrix}\right ] \underbrace {\left [\begin{matrix} \hat {\psi }^*_0 \\[4pt] \hat {\psi }^*_{0,z^*} \\[4pt] \hat {\psi }^*_{0,z^*z^*} \\[4pt] \hat {\psi }^*_{0,z^*z^*z^*} \end{matrix}\right ](L^*)}_{\boldsymbol{p}|_L}, \end{align}

so we now know how to calculate $\boldsymbol{p}_1|_0$ and $\boldsymbol{p}_2|_0$ , which are linearly independent. From evaluating the two boundary condition equations in (4.3) for each one of $\boldsymbol{p}_1|_0$ and $\boldsymbol{p}_2|_0$ , we can write the four equations

(4.9) \begin{align} \left [\begin{matrix} (\hat {\psi }^*_{0,z^*z^*})_1|_0 & (\hat {\psi }^*_{0,z^*z^*z^*})_1|_0 \\[4pt] (\hat {\psi }^*_{0,z^*z^*})_2|_0 & (\hat {\psi }^*_{0,z^*z^*z^*})_2|_0 \end{matrix}\right ] \left [\begin{matrix} a^* \\[4pt] b^* \end{matrix}\right ] &=\left [\begin{matrix} (\hat {\psi }^*_{0,z^*})_1|_0 \\[4pt] (\hat {\psi }^*_{0,z^*})_2|_0 \end{matrix}\right ], \nonumber \\[4pt] \left [\begin{matrix} (\hat {\psi }^*_{0,z^*z^*})_1|_0 & (\hat {\psi }^*_{0,z^*z^*z^*})_1|_0 \\[4pt] (\hat {\psi }^*_{0,z^*z^*})_2|_0 & (\hat {\psi }^*_{0,z^*z^*z^*})_2|_0 \end{matrix}\right ] \left [\begin{matrix} c^* \\[4pt] d^* \end{matrix}\right ] &=\left [\begin{matrix} (\hat {\psi }^*_0)_1|_0\\[4pt] (\hat {\psi }^*_0)_2|_0 \end{matrix}\right ], \end{align}

and then solve for the four unknowns $a^*, b^*, c^*, d^*$ . The resulting equivalent boundary condition (4.3), containing terms up to third order, will be exponentially accurate, and never contain powers of $\varepsilon$ higher than the third, for constant outer boundary conditions.

4.2. Physical domain (slowly varying outer conditions)

Just like $\phi$ could in § 2.2, the dimensionless dependent variable $\psi$ can be expanded in inner coordinates $Y=(y^*-y_0^*)/s^*$ and $Z=z^*/s^*$ , $s^*$ being the wall-texture period, i.e.

(4.10) \begin{align} \psi =\varPsi _0(Y,Z)+\varepsilon \varPsi _1(Y,Z)+ {\cdots} = \sum _{i=0}^\infty \varepsilon ^i\varPsi _i(Y,Z) , \end{align}

and outer coordinates $y=y^*/L^*$ and $z=z^*/L^*$ , $L^*$ being an outer reference length (typically, the distance to the opposing wall):

(4.11) \begin{align} \psi =\psi _0(y,z)+\varepsilon \psi _1(y,z)+ {\cdots} = \sum _{i=0}^\infty \varepsilon ^i\psi _i(y,z) . \end{align}

The expansion parameter $\varepsilon$ is the length ratio $s^*/L^*$ .

Because the biharmonic equation is linear and scale-invariant, each term of both the inner and outer expansions will independently obey its own biharmonic equation:

(4.12) \begin{align} \Delta _2 \Delta _2 \varPsi _i(Y,Z) = 0, \quad \Delta _2 \Delta _2 \psi _i(y,z) = 0 \quad \forall i . \end{align}

In addition, $\varPsi _i$ will obey solid-wall boundary conditions on a curve $Z=F(Y)$ , i.e.

(4.13) \begin{align} \varPsi _i[Y,F(Y)]=\frac {\partial \varPsi _i}{\partial n}[Y,F(Y)]=0 , \end{align}

with $F(Y)$ periodic of period $1$ in inner coordinates, and $\psi _i$ will obey two generic boundary conditions at an outer wall $z=1$ .

The problem will be completed by matching conditions imposed in an intermediate distinguished limit where $Z\rightarrow \infty$ and $z\rightarrow 0$ as $\varepsilon \rightarrow 0$ .

4.2.1. Order zero

At outer order 0 we need to solve the biharmonic equation for $\psi _0(y,z)$ with whatever boundary conditions are imposed at the outer wall and $\psi _0(y,0)=\psi _{0,z}(y,0)=0$ at the flat inner wall $z=0$ , to which the periodic wall reduces for $\varepsilon =0$ . From the (probably numerical, when the outer boundary conditions are varying in a generic way) solution of this equation we obtain, among other things, the values of $\psi _{0,zz}(y,0)$ and $\psi _{0,zzz}(y,0)$ .

With $\psi (y,0)$ , $\psi _{z}(y,0)$ , $\psi _{zz}(y,0)$ and $\psi _{zzz}(y,0)$ all known, we can formally reconstruct the outer solution by Cauchy series (actually by Taylor series until, at fourth order, we begin to see the effect of the biharmonic differential equation in eliminating fourth and higher $z$ derivatives):

(4.14) \begin{align} \begin{split} &\psi (y,z) = \psi (y_0,0)+ (y-y_0)\psi _y(y_0,0)+z\psi _z(y_0,0) +{}\\ &\frac {(y-y_0)^2}{2}\psi _{yy}(y_0,0) +(y-y_0)z\psi _{yz}(y_0,0)+\frac {z^2}{2}\psi _{zz}(y_0,0)+{}\\ &\frac {(y-y_0)^3}{6}\psi _{yyy}(y_0,0) +\frac {(y-y_0)^2z}{2}\psi _{yyz}(y_0,0) +\frac {(y-y_0)z^2}{2}\psi _{yzz}(y_0,0)+{}\\ &\frac {z^3}{6}\psi _{zzz}(y_0,0)+ \frac {z^4}{24}\left [-\psi _{yyyy}(y_0,0)-2\psi _{yyzz}(y_0,0)\right ]+ {\cdots}. \end{split} \end{align}

For the purpose of matching, we insert the outer expansion (4.11) into (4.14), transform it in inner variables $Y=(y-y_0)/\varepsilon$ , $Z=z/\varepsilon$ , and reorder in powers of $\varepsilon$ :

(4.15)

The cancellations are due to $\psi _0$ and $\psi _{0,z}$ being zero, together with their $y$ derivatives, on the flat wall that corresponds to $\varepsilon =0$ . This is the sequence that must be matched, term by term according to powers of $\varepsilon$ , by the inner expansion (4.10).

The general matching principle is that the two higher $Z$ derivatives, $\partial ^2 \varPsi _{i}/\partial Z^2$ and $\partial ^3 \varPsi _{i}/\partial Z^3$ , associated with the dominant terms in the asymptotic expansion at large $Z$ , are set by the outer solution and drive the inner solution as boundary conditions in the limit of $Z\rightarrow \infty$ , just as $\partial \varPhi _i/\partial Z$ did in § 2.2, whereas the two lower $z$ derivatives, $\psi _i$ and $\partial \psi _{i}/\partial z$ , are set by the inner solution and drive the outer solution at $z=0$ , just as $\phi _i$ did in § 2.2.

At inner order 0, matching yields all zero boundary conditions on $\varPsi _0(Y,Z)$ ; as a result, $\varPsi _0$ is identically null (and the inner expansion could just as well have started from $i=1$ ).

4.2.2. First order

At inner order $1$ we obtain, from (4.15),

(4.16) \begin{align} \varPsi _1(y_0;Y,Z) \sim \psi _1(y_0,0), \end{align}

where, just as in § 2.2, $y_0$ is to be treated as a parameter insofar as the biharmonic equation for $\varPsi _1$ is concerned, and symbol ${}\sim {}$ (‘behaves as’) is precisely defined to mean that the difference between its left-hand side and right-hand side must tend to zero when $Z\rightarrow \infty$ . Second and third $Z$ derivatives of the right-hand side of (4.16) are zero, and therefore, $\varPsi _1$ , whose second and third derivatives must match those of the right-hand side, obeys all zero boundary conditions. For the same reasons as $\varPsi _0$ , $\varPsi _1$ is identically null.

At outer order 1, (4.16) consequently gives $\psi _1(y_0,0)=0$ . The biharmonic equation, however, requires two boundary conditions at each end, and in order to find the second boundary condition for $\psi _1$ we need to simultaneously examine the $\varepsilon ^2$ term of (4.15):

(4.17) \begin{align} \varPsi _2(y_0;Y,Z) \sim \frac {Z^2}{2}\psi _{0,zz}+Z\psi _{1,z}+\psi _2(y_0,0). \end{align}

Here $Y\psi _{1,y}(y_0,0)$ has been omitted because $\psi _1(y,0) = 0$ for all $y$ and its derivative is thus also zero.

The second $Z$ derivative of the right-hand side of (4.17), which determines the second $Z$ derivative of $\varPsi _2(y_0;Y,Z)$ at infinity, is non-zero, and therefore, $\varPsi _2$ is the first non-trivial term of the inner expansion. It must be found as a solution of the biharmonic equation with the boundary conditions

(4.18) \begin{align} \varPsi _2[Y,F(Y)]=\varPsi _{2,N}[Y,F(Y)]=0, \!\quad \varPsi _{2,ZZ}[Y,\infty ]= \psi _{0,zz}(y_0,0),\quad \! \varPsi _{2,ZZZ}[Y,\infty ] =0, \end{align}

or equivalently, because of linearity, by means of a normalised $\overline {\varPsi _{21}}$ obeying the biharmonic equation with boundary conditions

(4.19) \begin{align} \overline {\varPsi _{21}}[Y,F(Y)]=\overline {\varPsi _{21}}_{,N}[Y,F(Y)]=0,\quad \overline {\varPsi _{21}}_{,ZZ}[Y,\infty ]= 1,\quad \overline {\varPsi _{21}}_{,ZZZ}[Y,\infty ]=0 , \end{align}

as

(4.20) \begin{align} \varPsi _2(y_0;Y,Z)=\psi _{0,zz}(y_0,0)\,\overline {\varPsi _{21}}(Y,Z) . \end{align}

Once $\overline {\varPsi _{21}}$ has been determined, the first derivative of (4.17) provides the second wall boundary condition for the first-order outer solution $\psi _1$ ,

(4.21) \begin{align} \psi _{1,z}(y_0,0)=\lim _{Z\rightarrow \infty }\left [\varPsi _{2,Z} - Z\psi _{0,zz}(y_0,0) \right ] = a_1 \psi _{0,zz}(y_0,0), \end{align}

where

(4.22) \begin{align} a_1=\lim _{Z\rightarrow \infty }\left (\overline {\varPsi _{21}}_{,Z} - Z\right ). \end{align}

Constant $a_1$ , which coincides with $a_1$ in § 4.1 and is calculated in an equivalent manner, is the transverse protrusion height of Luchini et al. (Reference Luchini, Manzo and Pozzi1991). The present matching method embeds it into an expansion that can be continued to higher and higher orders.

4.2.3. Second order

Matching condition (4.17) did not yet exhaust its function. Enforcing (4.17) directly, in addition to its derivative, yields one of the boundary conditions for the second-order outer solution $\psi _2$ , i.e.

(4.23) \begin{align} \psi _2(y_0,0)=\lim _{Z\rightarrow \infty }\left [\varPsi _2(y_0;Y,Z) -\frac {Z^2}{2}\psi _{0,zz}-a_1 Z\psi _{0,zz}\right ] = c_2\psi _{0,zz}(y_0,0), \end{align}

where

(4.24) \begin{align} c_2=\lim _{Z\rightarrow \infty }\left [\overline {\varPsi _{21}}(Y,Z) -\frac {Z^2}{2}-a_1 Z\right ]\!. \end{align}

The other boundary condition for $\psi _2$ , as should by now be expected, is obtained from the third-order matching condition

(4.25) \begin{align} \varPsi _3 \sim \frac {YZ^2}{2}\psi _{0,yzz}+\frac {Z^3}{6}\psi _{0,zzz}+ YZ\psi _{1,yz}+\frac {Z^2}{2}\psi _{1,zz}+Y\psi _{2,y}+Z\psi _{2,z}+\psi _3(y_0,0). \end{align}

Preliminarily the third-order inner solution $\varPsi _3$ must be calculated, and it can be seen to be the superposition of several contributions. The first term of (4.25) proportional to $\psi _{0,yzz}$ can be combined with the third using (4.21) and the fifth using (4.23), to give $Y(Z^2/2+a_1Z+c_2)\psi _{0,yzz}$ . This asymptotic behaviour of $\varPsi _3$ can be satisfied, in analogy to (2.19), by a compound solution of the form

(4.26) \begin{align} \varPsi _{31} = \left [\overline {\varPsi _{31}}(Y,Z)+Y\overline {\varPsi _{21}}(Y,Z)\right ]\psi _{0,yzz}, \end{align}

where $\overline {\varPsi _{21}}(Y,Z)$ is the same as defined in (4.19), and $\overline {\varPsi _{31}}(Y,Z)$ is a new periodic function of $Y$ obeying the equation

(4.27) \begin{align} \Delta _2 \Delta _2\overline {\varPsi _{31}} + 4 \Delta _2\overline {\varPsi _{21}}_{,Y} = 0 \end{align}

with boundary conditions

(4.28) \begin{align} \overline {\varPsi _{31}}[Y,F(Y)]=\overline {\varPsi _{31}}_{,N}[Y,F(Y)]=0,\quad \overline {\varPsi _{31}}_{,ZZ}[Y,\infty ]= \overline {\varPsi _{31}}_{,ZZZ}[Y,\infty ] =0 . \end{align}

The second term of (4.25) proportional to $Z^3/6$ can be matched by a normalised contribution (here and in what follows we shall adopt a convention where contributions are numbered by a first digit denoting order and a second sequential digit) $\overline {\varPsi _{32}}$ that obeys the biharmonic equation with boundary conditions

(4.29) \begin{align} \overline {\varPsi _{32}}[Y,F(Y)]=\overline {\varPsi _{32}}_{,N}[Y,F(Y)]=0,\quad \overline {\varPsi _{32}}_{,ZZ}[Y,\infty ]\sim Z,\quad \overline {\varPsi _{32}}_{,ZZZ}[Y,\infty ] =1 . \end{align}

The fourth term of (4.25) is analogous to the term multiplying $\psi _{0,zz}$ in (4.17), and when used as a boundary condition produces the same normalised solution $\overline {\varPsi _{21}}$ . The sixth and seventh terms do not contribute to the second or third $Z$ derivatives, and thus, have no role in determining $\varPsi _3$ .

Combining all three contributions together provides

(4.30) \begin{align} \varPsi _3 =\left [\overline {\varPsi _{31}}(Y,Z)+Y\overline {\varPsi _{21}}(Y,Z)\right ]\psi _{0,yzz}+ \overline {\varPsi _{32}}(Y,Z)\psi _{0,zzz}+\overline {\varPsi _{21}}(Y,Z)\psi _{1,zz}. \end{align}

Injecting $\varPsi _3$ back into the $Z$ derivative of (4.25) now gives

(4.31) \begin{align} \psi _{2,z}(y_0,0)=a_2\psi _{0,yzz}+b_2\psi _{0,zzz}+a_1\psi _{1,zz}, \end{align}

where $a_1$ comes from (4.22) and

(4.32) \begin{align} a_2=\lim _{Z\rightarrow \infty }\overline {\varPsi _{31}}_{,Z}(Y,Z),\quad \ b_2=\lim _{Z\rightarrow \infty }\left ( \overline {\varPsi _{32}}_{,Z} -\frac {Z^2}{2}\right )\!. \end{align}

Equation (4.31) closes the problem for the second-order outer solution $\psi _2$ . Constants $b_2$ in (4.32) and $c_2$ in (4.24) have been so defined such that they coincide with $b_2$ and $c_2$ of § 4.1. Here $a_2$ on the other hand, the analogue of $h_2$ in § 2.2, did not exist in § 4.1, and can only be discovered by matched asymptotic expansions when $\psi _{0,zz}$ is a non-constant function of $y$ . Owing to variational properties of the biharmonic equation, discussed in Appendix A, it also turns out that $b_2=-c_2$ .

4.2.4. Third order

Just as happened for the second order, enforcing (4.25) as is yields one of the boundary conditions for the third-order outer solution $\psi _3$ :

(4.33) \begin{align} \begin{split} \psi _3(y_0,0)=&\lim _{Z\rightarrow \infty }\left [\varPsi _3 - \frac {YZ^2}{2}\psi _{0,yzz}-\frac {Z^3}{6}\psi _{0,zzz}- \frac {Z^2}{2}\psi _{1,zz}-Y\psi _{2,y}-Z\psi _{2,z}\right ] =\\ &c_3\psi _{0,yzz} +c_2\psi _{1,zz} + d_3\psi _{0,zzz}. \end{split} \end{align}

Here $c_2$ was defined above and

(4.34) \begin{align} c_3=\lim _{Z\rightarrow \infty }\left [\overline {\varPsi _{31}}(Y,Z) -a_2 Z\right ], \\[-28pt] \nonumber \end{align}

(4.35) \begin{align} d_3=\lim _{Z\rightarrow \infty }\left [\overline {\varPsi _{32}}(Y,Z) -\frac {Z^3}{6}-b_2 Z\right ]\!. \\[-2pt] \nonumber \end{align}

Constant $d_3$ , just as well as its previous sister constants, coincides with $d_3$ of § 4.1.

The second boundary condition, specifying $\psi _{3,z}(y_0,0)$ , must be obtained from the fourth-order matching condition

(4.36) \begin{align} \begin{split} &\varPsi _4(y_0;Y,Z) \sim \left (\frac {Y^2Z^2}{4}-\frac {Z^4}{12}\right )\psi _{0,yyzz}+\frac {YZ^3}{6}\psi _{0,yzzz}+\frac {Y^2Z}{2}\psi _{1,yyz}+\frac {YZ^2}{2}\psi _{1,yzz}+{}\\ &\frac {Z^3}{6}\psi _{1,zzz}+ \frac {Y^2}{2}\psi _{2,yy}+YZ\psi _{2,yz}+\frac {Z^2}{2}\psi _{2,zz}+Y\psi _{3,y}+Z\psi _{3,z}+\psi _4(y_0,0). \end{split} \end{align}

Two terms of this expression can be simplified using the previous boundary conditions, $\psi _{1,z}=a_1\psi _{0,zz}$ , $\psi _2=c_2\psi _{0,zz}$ , to give

(4.37) \begin{align} \begin{split} &\varPsi _4(y_0;Y,Z) \sim \left [\frac {Y^2}{2}\left (\frac {Z^2}{2}+a_1Z+c_2\right )-\frac {Z^4}{12}\right ]\psi _{0,yyzz}+\frac {YZ^3}{6}\psi _{0,yzzz}+{}\\ &\frac {YZ^2}{2}\psi _{1,yzz}+\frac {Z^3}{6}\psi _{1,zzz}+ YZ\psi _{2,yz}+\frac {Z^2}{2}\psi _{2,zz}+Y\psi _{3,y}+Z\psi _{3,z}+\psi _4(y_0,0). \end{split} \end{align}

In addition, the second line of (4.37) can be recognised to have the same structure as $\varPsi _3$ of (4.25), with only a shift from $\psi _0$ to $\psi _1$ , from $\psi _1$ to $\psi _2$ and from $\psi _3$ to $\psi _4$ , and the corresponding terms of $\varPsi _4$ can be written out accordingly.

The inner solution $\varPsi _{41}$ corresponding to the first term of (4.37) has the structure

(4.38) \begin{align} \varPsi _{41} = \left [\overline {\varPsi _{41}}(Y,Z)+Y\overline {\varPsi _{31}}(Y,Z)+\frac {Y^2}{2}\overline {\varPsi _{21}}(Y,Z)\right ]\psi _{0,yyzz}, \end{align}

where $\overline {\varPsi _{21}}$ and $\overline {\varPsi _{31}}$ were defined earlier, and $\overline {\varPsi _{41}}(Y,Z)$ is a new periodic function of $Y$ obeying the equation

(4.39) \begin{align} \Delta _2 \Delta _2\overline {\varPsi _{41}}+ 4\Delta _2\overline {\varPsi _{31}}_{,Y} + 6\overline {\varPsi _{21}}_{,YY}+2\overline {\varPsi _{21}}_{,ZZ} = 0 \end{align}

with boundary conditions

(4.40) \begin{align} \overline {\varPsi _{41}}[Y,F(Y)]=\overline {\varPsi _{41}}_{,N}[Y,F(Y)]=0,\quad\! \overline {\varPsi _{41}}_{,ZZ}[Y,\infty ]\sim -Z^2,\quad\! \overline {\varPsi _{41}}_{,ZZZ}[Y,\infty ] \sim -2Z , \end{align}

the latter two coming from the $-Z^4/12$ behaviour at $Z\rightarrow \infty$ expected of $\overline {\varPsi _{41}}$ in (4.37).

The second term of (4.37) asymptotically proportional to $YZ^3/6$ in turn can be written as

(4.41) \begin{align} \varPsi _{42} = \left [\overline {\varPsi _{42}}(Y,Z)+Y\overline {\varPsi _{32}}(Y,Z)\right ]\psi _{0,yzzz}, \end{align}

where $\overline {\varPsi _{32}}$ was defined earlier, and $\overline {\varPsi _{42}}(Y,Z)$ is yet another periodic function of $Y$ obeying the equation

(4.42) \begin{align} \Delta _2 \Delta _2\overline {\varPsi _{42}}+ 4\Delta _2\overline {\varPsi _{32}}_{,Y} = 0 \end{align}

with boundary conditions

(4.43) \begin{align} \overline {\varPsi _{42}}[Y,F(Y)]=\overline {\varPsi _{42}}_{,N}[Y,F(Y)]=0,\quad \overline {\varPsi _{42}}_{,ZZ}[Y,\infty ]= \overline {\varPsi _{42}}_{,ZZZ}[Y,\infty ] =0 , \end{align}

the latter two coming from the lack of any $Y$ -independent asymptotic part in the second term of (4.37).

Combining all contributions together gives

(4.44) \begin{align} \begin{split} \varPsi _4 =& \left [\overline {\varPsi _{41}}(Y,Z)+Y\overline {\varPsi _{31}}(Y,Z)+\frac {Y^2}{2}\overline {\varPsi _{21}}(Y,Z)\right ]\psi _{0,yyzz}+{}\\ &\left [\overline {\varPsi _{42}}(Y,Z)+Y\overline {\varPsi _{32}}(Y,Z)\right ]\psi _{0,yzzz}+{}\\ &\left [\overline {\varPsi _{31}}(Y,Z)+Y\overline {\varPsi _{21}}(Y,Z)\right ]\psi _{1,yzz}+ \overline {\varPsi _{32}}(Y,Z)\psi _{1,zzz}+\overline {\varPsi _{21}}(Y,Z)\psi _{2,zz} .\end{split} \end{align}

Injecting $\varPsi _4$ back into the $Z$ derivative of (4.37) now gives

(4.45) \begin{align} \psi _{3,z}(y_0,0)=a_3\psi _{0,yyzz}+b_3\psi _{0,yzzz}+a_2\psi _{1,yzz}+b_2\psi _{1,zzz}+a_1\psi _{2,zz}, \end{align}

where the two new coefficients are

(4.46) \begin{align} a_3=\lim _{Z\rightarrow \infty }\big ( \overline {\varPsi _{41}}_{,Z}(Y,Z)+2Z^3/3\big ),\quad b_3=\lim _{Z\rightarrow \infty }\overline {\varPsi _{42}}_{,Z}(Y,Z) . \end{align}

Equation (4.45), together with (4.33), closes the problem for the third-order outer solution $\psi _3$ . Owing to special symmetries of the biharmonic equation, it also turns out that $b_3=c_3$ .

4.2.5. Higher orders and equivalent boundary condition

No apparent obstacle, aside from becoming overwhelmed with calculations, exists to continuing the procedure to higher and higher orders. Each inner contribution $\varPsi _n$ must in turn be asymptotically matched to an expression derived from the Cauchy series of the outer solution, in the precise sense that the difference between the two must tend to zero for $Z\rightarrow \infty$ . This matching implies a growing number of numerical equalities, because both sides are asymptotically polynomials of growing degree and every coefficient of these polynomials must individually match for the two polynomials to coincide.

Each $n$ th matching condition will contain two new contributions, one proportional to $\partial ^n\psi _0/\partial y^{n-2}\partial z^2$ and one proportional to $\partial ^n\psi _0/\partial y^{n-3}\partial z^3$ , in addition to a sequence of other terms that mimic the $(n-1){\text{th}}$ matching condition with $\psi _i$ replaced by $\psi _{i+1}$ . The two new contributions will produce two independent normalised inner solutions $\overline {\varPsi _{n1}}(Y,Z)$ and $\overline {\varPsi _{n2}}(Y,Z)$ , and eventually produce constant numerical coefficients $a_{n-1}$ , $b_{n-1}$ , $c_{n}$ and $d_{n}$ for the $n{\text{th}}$ outer boundary condition

(4.47) \begin{align} \left [\begin{matrix} \displaystyle \frac {\partial \psi _{n}}{\partial z}(y,0) \\[9pt] \psi _n(y,0) \end{matrix}\right ] =\left [\begin{matrix} \displaystyle \sum _{i=1}^n a_i\frac {\partial ^{i+1}\psi _{n-i}}{\partial y^{i-1}\partial z^2}(y,0) + \sum _{i=2}^n b_i \frac {\partial ^{i+1}\psi _{n-i}}{\partial y^{i-2}\partial z^3}(y,0)\\[15pt] \displaystyle \sum _{i=2}^n c_i\frac {\partial ^{i}\psi _{n-i}}{\partial y^{i-2}\partial z^2}(y,0) + \sum _{i=3}^n d_i \frac {\partial ^{i}\psi _{n-i}}{\partial y^{i-3}\partial z^3}(y,0) \\[15pt] \end{matrix}\right ] . \end{align}

In a similar manner to (2.41), (4.47) can be multiplied by $\varepsilon ^n$ and summed over $n=[0,N]$ , so as to yield $\psi = \sum _{n=0}^{N} \varepsilon ^n\psi _n + \mathrm{O}(\varepsilon ^{N+1})$ , and then recast as the formal product of two power series that only contains the total $\psi$ on the right-hand side as well as the left-hand side:

(4.48) \begin{align} \left [\begin{matrix} \displaystyle \frac {\partial \psi }{\partial z}(y,0) \\[9pt] \psi (y,0) \end{matrix}\right ] =\left [\begin{matrix} \displaystyle \sum _{i=1}^N \varepsilon ^i a_i\frac {\partial ^{i+1}\psi }{\partial y^{i-1}\partial z^2}(y,0) + \sum _{i=2}^N \varepsilon ^i b_i \frac {\partial ^{i+1}\psi }{\partial y^{i-2}\partial z^3}(y,0)\\[15pt] \displaystyle \sum _{i=2}^N \varepsilon ^i c_i\frac {\partial ^{i}\psi }{\partial y^{i-2}\partial z^2}(y,0) + \sum _{i=3}^N \varepsilon ^i d_i \frac {\partial ^{i}\psi }{\partial y^{i-3}\partial z^3}(y,0) \\[15pt] \end{matrix}\right ] + \mathrm{O} \big(\varepsilon ^{N+1}\big) . \end{align}

The implicit boundary condition (4.48), analogous to (2.43) for the Laplace problem, may be preferred in numerical computations. In dimensional form it can immediately be written using the wall-texture period $s^*$ in place of $\varepsilon$ as

(4.49) \begin{align} \left [\begin{matrix} \displaystyle \frac {\partial \psi ^*}{\partial z^*}(y^*,0) \\[9pt] \psi ^*(y^*,0) \end{matrix}\right ] =\left [\begin{matrix} \displaystyle \sum _{i=1}^N s^{*i} a_i\frac {\partial ^{i+1}\psi ^*}{\partial y^{*i-1}\partial z^{*2}}(y^*,0) + \sum _{i=2}^N s^{*i} b_i \frac {\partial ^{i+1}\psi ^*}{\partial y^{*i-2}\partial z^{*3}}(y^*,0)\\[15pt] \displaystyle \sum _{i=2}^N s^{*i} c_i\frac {\partial ^{i}\psi ^*}{\partial y^{*i-2}\partial z^{*2}}(y^*,0) + \sum _{i=3}^N s^{*i} d_i \frac {\partial ^{i}\psi ^*}{\partial y^{*i-3}\partial z^{*3}}(y^*,0) \\[15pt] \end{matrix}\right ] \! . \end{align}

This is the generalisation to slowly varying outer conditions of (4.5), which was valid for constant outer conditions.

Also, as was true in § 2.2.5, when the geometry is left–right symmetric, (4.47)–(4.49) must enforce the same symmetry, which means that only even $y$ derivatives of $\psi$ can appear. The $a_i,b_i,c_i,d_i$ coefficients corresponding to odd derivatives will automatically turn out to be zero in this case.

5.1. Outer expansion

Just as in § 4.2, the outer solution will see equivalent boundary conditions assigned at a virtual flat wall that is chosen as the origin $z=0$ . The outer expansion is written as

(5.4) \begin{align} \psi (y,z,t) = \sum _{i=0}^\infty \varepsilon ^i \psi _i(y,z,t), \end{align}

where the small parameter is, as already discussed, $\varepsilon =s^*/\ell ^*=s^+$ . Upon inserting (5.4) into (5.3) and collecting equal powers of $\varepsilon$ , $\psi _0$ still obeys the same (5.3), but with no-slip and impermeability conditions on the virtual flat wall, $\psi _0(y,0,t) =\psi _{0,z}(y,0,t) =0$ (as explained near (4.2)). From the (numerical, say) solution of the $\psi _0$ equation we can obtain, among other things, the values of $\psi _{0,zz}(y,0,t)$ and $\psi _{0,zzz}(y,0,t)$ . With this information available, we can formally reconstruct the solution by Cauchy series. The beginning of this series will be identical to (4.14) up to third order, and only start to differ in the fourth-order term that is derived from the governing differential equation:

(5.5) \begin{align} &\psi (y,z,t) = \psi (y_0,0)+ (y-y_0)\psi _y(y_0,0)+z\psi _z(y_0,0) \nonumber \\& + \frac {(y-y_0)^2}{2}\psi _{yy}(y_0,0) +(y-y_0)z\psi _{yz}(y_0,0)+\frac {z^2}{2}\psi _{zz}(y_0,0) \nonumber \\& + \frac {(y-y_0)^3}{6}\psi _{yyy}(y_0,0) +\frac {(y-y_0)^2z}{2}\psi _{yyz}(y_0,0) +\frac {(y-y_0)z^2}{2}\psi _{yzz}(y_0,0) \nonumber \\& +\frac {z^3}{6}\psi _{zzz}(y_0,0) + \frac {z^4}{24}\left [-\psi _{yyyy}(y_0,0)-2\psi _{yyzz}(y_0,0)+ \psi _z \Delta _2\psi _y - \psi _y \Delta _2\psi _z + \Delta _2 \psi _t\right ] \nonumber \\ & + {\cdots}. \end{align}

5.2. Inner expansion

The inner expansion is $\psi (y_0+\varepsilon Y,\varepsilon Z,t)=\varPsi (y_0,t;Y,Z)$ , where

(5.6) \begin{align} \varPsi (y_0,t;Y,Z) = \sum _{i=0}^\infty \varepsilon ^i \varPsi _i(y_0,t; Y,Z) . \end{align}

There is no rescaling in the time coordinate insofar as the riblet shape does not depend on time; in the inner solution, time plays the role of a parameter just like $y_0$ does.

Since (5.3) is homogeneous with respect to space coordinates, when this equation is transformed in inner variables it remains visibly unchanged save the time derivative:

(5.7) \begin{align} \Delta _2 \Delta _2 \varPsi = \varepsilon ^2\Delta _2 \varPsi _t + \varPsi _Z \Delta _2\varPsi _Y - \varPsi _Y \Delta _2\varPsi _Z. \end{align}

Equation (5.7) may then give the impression that $\varPsi _0$ shall obey a nonlinear equation. However, it must be remembered that in § 4.2 $\varPsi _0$ and $\varPsi _1$ turned out to be identically zero, and the expansion effectively started with $\varPsi _2$ . There is a simple qualitative explanation for this behaviour: owing to the boundary conditions $\psi =\psi _z=0$ , the streamfunction $\psi$ is O $(z^2)$ near the wall $z=0$ , and thus, proportional to $\varepsilon ^2 Z^2$ in inner coordinates, and of the second order in $\varepsilon$ . This estimate is unchanged in the Navier–Stokes problem. With the expansion (5.6) starting to be non-zero from $i=2$ , (5.7) produces the following hierarchy:

(5.8a) \begin{align} \Delta _2 \Delta _2 \varPsi _2 &= 0, \\[-10pt] \nonumber \end{align}

(5.8b) \begin{align} \Delta _2 \Delta _2 \varPsi _3 &= 0, \\[-10pt] \nonumber \end{align}

(5.8c) \begin{align} \Delta _2 \Delta _2 \varPsi _4 &= \Delta _2 \varPsi _{2,t} + \varPsi _{2,Z} \Delta _2\varPsi _{2,Y} - \varPsi _{2,Y} \Delta _2\varPsi _{2,Z}, \\[9pt] \nonumber \end{align}

It can be observed that each equation in (5.8) has the same, linear, left-hand side, nonlinearity being confined to the right-hand side that at every stage is a known function of previous terms.

5.3. Orders 0, 1, 2

It begins to appear that, (5.5) and (5.8) being identical to the corresponding equations in § 4.2 up to third order, nothing changes in the formulation up to inner order 3 and outer order 2 (we recall that outer order 3 partially depends upon inner order 4). This has a number of consequences.

1. (i) Expected: all the previous literature that used the reduced, steady Stokes equations to describe the viscous sublayer of an otherwise general and unsteady flow was rational in doing so: this reduction can be justified by a formal asymptotic expansion, and indeed correctly represents the Navier–Stokes equations at leading order in feature size $\varepsilon = s^+$ .

2. (ii) Maybe less expected: the approximation is not only good at leading order (inner order 2, outer order 1), but also at inner order 3, outer order 2. Therefore, second-order outer boundary conditions (4.23) and (4.31) are linear and Stokes-like as well.

3. (iii) However: the above second-order boundary condition involves several different contributions on equal footing, and may be more complex than previously thought (and even more complex it will become in § 6).

The derivation of the inner solutions $\varPsi _2=\overline {\varPsi _{21}}\,\psi _{0,zz}$ and $\varPsi _3=(\overline {\varPsi _{31}}+Y\overline {\varPsi _{21}})\,\psi _{0,yzz}+\overline {\varPsi _{32}}\,\psi _{0,zzz}$ is identical to (4.19), (4.27), (4.29) and will not be repeated. The corresponding equivalent boundary conditions for the outer problem are

(5.9a) \begin{align} \text{First order: }&\left \{ \begin{array}{l} \psi _z(y,0,t)=\varepsilon a_1 \psi _{zz}, \\[4pt]\psi (y,0,t)=0, \end{array}\right . \\[-10pt] \nonumber \end{align}

(5.9b) \begin{align} \text{Second order: }&\left \{ \begin{array}{l} \psi _z(y,0,t)=\varepsilon a_1 \psi _{zz}+\varepsilon ^2 a_2 \psi _{yzz}+\varepsilon ^2 b_2 \psi _{zzz}, \\[4pt]\psi (y,0,t)=\varepsilon ^2 c_2 \psi _{zz} ,\end{array}\right . \\[9pt] \nonumber \end{align}

where numerical constants $a_1$ , $a_2$ , $b_2$ and $c_2=-b_2$ are defined in (4.22), (4.32), (4.24), respectively, and $a_2$ is expected to be zero for left–right-symmetric riblets (and is zero in fact for any riblets, as will later be shown to be the case).

It can be practically useful to reformulate (5.9) in primitive variables. Using $v=\psi _z$ and $w=-\psi _y$ , a straightforward transformation of (5.9) gives

(5.10) \begin{align} \text{First order: }&\left \{ \begin{array}{l} v(y,0,t)=\varepsilon a_1 v_{z}, \\[4pt]w(y,0,t)=0, \end{array}\right . \nonumber \\ \text{Second order: }&\left \{ \begin{array}{l} v(y,0,t)=\varepsilon a_1 v_{z}-\varepsilon ^2 a_2 w_{zz}+\varepsilon ^2 b_2 v_{zz} ,\\[4pt]w(y,0,t)=\varepsilon ^2 c_2 w_{zz}. \end{array}\right . \end{align}

This is not the most useful form, however. Whereas in (5.9) up to the third $z$ derivative of $\psi$ is allowed to appear, in primitive variables $v_{zz}$ and $w_z$ can be substituted from the momentum and continuity equations (5.2), and actually must be substituted in order to maintain compatibility with the Cauchy expansion in the developments to be presented later in the 3D extensions of this argument. (When the continuity equation, for instance, becomes 3D as $u_x+v_y+w_z=0$ , $v_y$ and $-w_z$ no longer become interchangeable; $u_x,v_y$ , without $w_z$ , is the pair that appears in the Cauchy expansion and formulas derived from it.)

In this minimal- $z$ -derivative form, the equivalent boundary conditions become

(5.11a) \begin{align} \text{First order: }&\left \{ \begin{array}{l} v(y,0,t)=\varepsilon a_1 v_{z}, \\[4pt] w(y,0,t)=0 ,\end{array}\right . \\[-10pt] \nonumber \end{align}

(5.11b) \begin{align} \text{Second order: }&\left \{ \begin{array}{l} v(y,0,t)=\varepsilon a_1 v_{z}+\varepsilon ^2 a_2 v_{yz}+\varepsilon ^2 b_2 p_{y}, \\[4pt] w(y,0,t)=-\varepsilon ^2 c_2 v_{yz}, \end{array}\right . \\[9pt] \nonumber \end{align}

where $v_{0,zz}$ has been eliminated through the momentum equation $v_{0,zz}+v_{0,yy}=p_{0,y}$ combined with $v_{0,yy}=0$ , and $w_{0,zz}$ has been eliminated from the $z$ derivative of the continuity equation, $w_{0,zz}=-v_{0,yz}$ .

The first-order condition contains, as already stressed, the classical transverse protrusion height $a_1$ for the wall-parallel velocity and zero for the wall-normal velocity. The wall-parallel second-order condition contains the wall-parallel pressure gradient in a similar manner to the longitudinal condition (3.13); however, just as (3.13) contains an additional second-order term proportional to $u_{yz}$ , both conditions in (5.11b ) contain an additional term proportional to $v_{yz}$ . One of the coefficients of these terms, $a_2$ , vanishes for left–right-symmetric riblets (and in reality for any kind of riblets, as will be discovered later), but not $-c_2$ , which always equals $b_2$ (see Appendix A).

The same boundary conditions can be cast in dimensional form by simply replacing $\varepsilon$ by the riblet period $s^*$ .

5.4. Third order

We recall from § 4.2.4 that one of the third-order conditions, the one for $\psi _{3}(y,0)$ , was obtained from the third-order inner solution. This condition remains unchanged in the Navier–Stokes problem, and is still

(5.12) \begin{align} \psi _3(y,0,t)=c_2\psi _{1,yzz} + c_3\psi _{0,yzz} +d_3\psi _{0,zzz}, \end{align}

with constants $c_2$ , $c_3$ and $d_3$ defined by (4.24), (4.34) and (4.35), respectively.

The other boundary condition, specifying $\psi _{3,z}(y,0,t)$ , must be obtained from the fourth-order inner solution, which now obeys (5.8c ), i.e. on account of (4.20):

(5.13) \begin{align} \begin{split} &\Delta _2 \Delta _2 \varPsi _4(y_0,t;Y,Z) =\Delta _2 \overline {\varPsi _{21}}(Y,Z)\,\psi _{0,zzt}(y_0,0,t)+{} \\ &\left [\overline {\varPsi _{21}}_{,Z}(Y,Z) \Delta _2\overline {\varPsi _{21}}_{,Y}(Y,Z)- \overline {\varPsi _{21}}_{,Y}(Y,Z) \Delta _2\overline {\varPsi _{21}}_{,Z}(Y,Z)\right ]\psi ^2_{0,zz}(y_0,0,t) . \end{split} \end{align}

Since the left-hand side of this equation is still linear, its solution can be obtained by simply adding to the expression (4.44) of $\varPsi _4$ the two new contributions $\overline {\varPsi _{43}}(Y,Z)\,\psi _{0,zzt}(y_0,0,t)+\overline {\varPsi _{44}}(Y,Z)\,\psi ^2_{0,zz}(y_0,0,t)$ (with an incremental second subscript according to the convention set forth before), where $\overline {\varPsi _{43}}$ obeys the equation

(5.14) \begin{align} \Delta _2 \Delta _2 \overline {\varPsi _{43}} = \Delta _2 \overline {\varPsi _{21}} \end{align}

with boundary conditions

(5.15) \begin{align} \overline {\varPsi _{43}}[Y,F(Y)]=\overline {\varPsi _{43}}_{,N}[Y,F(Y)]=0,\quad \overline {\varPsi _{43}}_{,ZZ}[Y,\infty ]\sim Z^2/2,\quad \overline {\varPsi _{43}}_{,ZZZ}[Y,\infty ] \sim Z , \end{align}

and $\overline {\varPsi _{44}}$ obeys the equation

(5.16) \begin{align} \Delta _2 \Delta _2 \overline {\varPsi _{44}} = \overline {\varPsi _{21}}_{,Z}(Y,Z) \Delta _2\overline {\varPsi _{21}}_{,Y}(Y,Z)- \overline {\varPsi _{21}}_{,Y}(Y,Z) \Delta _2\overline {\varPsi _{21}}_{,Z}(Y,Z) \end{align}

with boundary conditions

(5.17) \begin{align} \overline {\varPsi _{44}}[Y,F(Y)]=\overline {\varPsi _{44}}_{,N}[Y,F(Y)]=0,\quad \overline {\varPsi _{44}}_{,ZZ}[Y,\infty ]= \overline {\varPsi _{44}}_{,ZZZ}[Y,\infty ] =0 . \end{align}

Injecting $\varPsi _4$ back into the $Z$ derivative of the matching condition (4.37) now gives, instead of (4.45),

(5.18) \begin{align} \psi _{3,z}(y_0,0) & = a_3\psi _{0,yyzz}+b_3\psi _{0,yzzz}+a_2\psi _{1,yzz}+b_2\psi _{1,zzz}+a_1\psi _{2,zz}+e_3^{(v_{zt})}\psi _{0,zzt} \nonumber \\& \quad + e_3^{(nl)}\psi _{0,zz}^2 , \end{align}

where the two new numerical constants are

(5.19) \begin{align} e_3^{(v_{zt})}=\lim _{Z\rightarrow \infty } \big [\overline {\varPsi _{43}}_{,Z}(Y,Z)-Z^3/6\big ],\quad e_3^{(nl)}=\lim _{Z\rightarrow \infty }\overline {\varPsi _{44}}_{,Z}(Y,Z) . \end{align}

In primitive-variable implicit form, the nonlinear third-order boundary condition is now

(5.20) \begin{align} \left \{ \begin{array}{l} v(y,0,t)=\varepsilon a_1 v_{z}+\varepsilon ^2 ( a_2 v_{yz}+b_2 p_y)+\varepsilon ^3 (a_3 v_{yyz}+b_3p_{yy}+e_3^{(v_{zt})}v_{zt}+e_3^{(nl)}v_{z}^2) ,\\[5pt]w(y,0,t)=-\varepsilon ^2 c_2 v_{yz}-\varepsilon ^3 (c_3 v_{yyz}+ d_3 p_{yy}) .\end{array}\right . \end{align}

Note that $e_3^{(nl)}$ , together with $a_2$ , $b_3$ and $c_3$ , is one of those coefficients that are forbidden by symmetry, and thus, zero for left–right-symmetric riblets, because when $\overline {\varPsi _{21}}$ is an even function of $Y$ , the right-hand side of (5.16) is odd symmetric in $Y$ and so becomes $\overline {\varPsi _{44}}$ . It follows that for left–right-symmetric riblets, nonlinearity only intervenes at an even higher than third order in $\varepsilon$ .

6.1. First order

Nothing new to add to the already acquired first-order solution. The first-order equations are homogeneous just as they were in the previous sections, and driven by the matching conditions only. They lead to the longitudinal and transverse protrusion heights of Luchini et al. (Reference Luchini, Manzo and Pozzi1991) according to (2.14) and (4.22).

6.2. Second order

Here the pressure gradient makes its appearance in the $U_2$ equation (6.4a ), just as it was taken into account in § 3. However, rather than being constant with $Y,Z$ , the pressure gradient is shaped by the first-order transverse Stokes solution (6.3b – d ). More precisely

(6.6) \begin{align} P_0=\overline {P_{01}}(Y,Z)v_{0,z}(x,y_0,0,t)+p_0(x,y_0,0,t), \end{align}

where $\overline {P_{01}}(Y,Z)$ satisfies the Cauchy–Riemann relations

(6.7) \begin{align} \overline {P_{01}}_{,Y}=(\Delta _2\overline {\varPsi _{21}})_Z,\quad \overline {P_{01}}_{,Z}=-(\Delta _2\overline {\varPsi _{21}})_Y, \end{align}

with $\overline {P_{01}}(Y,\infty )=0$ , so that $P_0(Y,\infty )=p_0(x,y_0,0,t)$ , and $\overline {\varPsi _{21}}$ being defined by (4.19); $v_{0,z}(x,y_0,0,t), p_0(x,y_0,0,t)$ are known components of the zeroth-order outer solution at the wall (and the gradient of $p_0(x,y_0,0,t)$ is the same pressure gradient as formerly considered in § 3). Therefore,

(6.8) \begin{align} U_2 & = \overline {\varPhi _{21}}(Y,Z)u_{0,yz}(x,y_0,0,t)+\overline {U_{22}}(Y,Z)p_{0,x}(x,y_0,0,t) \nonumber \\ & \quad +\overline {U_{23}}(Y,Z)v_{0,xz}(x,y_0,0,t), \end{align}

where $\overline {\varPhi _{21}}$ was defined in (2.19), $\overline {U_{22}}$ was defined in (3.11) and the new $\overline {U_{23}}$ obeys

(6.9) \begin{align} \Delta _2 \overline {U_{23}} = \overline {P_{01}}, \end{align}

with boundary conditions $ \overline {U_{23}}[Y,F(Y)]=0,\;\overline {U_{23}}_{,Z}[Y,\infty ]=0 .$

The corresponding second-order outer boundary condition (3.8) now becomes

(6.10) \begin{align} u_2(x,y,0,t) = h_1 u_{1,z} + h_2 u_{0,yz} + h^{(p_x)}_2 p_{0,x} + h_2^{(v_{xz})} v_{0,xz}, \end{align}

where $h_1,h_2,h^{(p_x)}_2$ were defined in (2.14), (2.26), (3.12), and

(6.11) \begin{align} h_2^{(v_{xz})}=\lim _{Z\rightarrow \infty }\overline {U_{23}}(Y,Z) . \end{align}

As to the right-hand side of (6.4d ), it makes a streamfunction impossible to use to derive the velocity field, and the simplest choice is to solve (6.4b – d ) numerically in primitive form. Nonetheless, their solution can be written as the linear superposition of a solution with non-zero forcing from $v_{0,z}(x,y_0,0,t)$ and zero right-hand side, which can be derived from a streamfunction and is exactly the one already dealt with in § 4.2.3, and a solution with $v_{0,z}(x,y_0,0,t)=0$ and non-zero right-hand side of (6.4d ), which can be represented through a new normalised triplet $\overline {V_{23}},\overline {W_{23}},\overline {P_{13}}$ that solves

(6.12a) \begin{align} \overline {V_{23}}_{,YY}+\overline {V_{23}}_{,ZZ} - \overline {P_{13}}_{,Y} &=0, \\[-10pt] \nonumber \end{align}

(6.12b) \begin{align} \overline {W_{23}}_{,YY}+\overline {W_{23}}_{,ZZ} - \overline {P_{13}}_{,Z} &=0, \\[-10pt] \nonumber \end{align}

(6.12c) \begin{align} \overline {V_{23}}_{,Y} + \overline {W_{23}}_{,Z} &= -\overline {\varPhi _{11}} ; \\[9pt] \nonumber \end{align}

with $\overline {\varPhi _{11}}$ taken from (2.13) and boundary conditions $ \overline {V_{23}}[Y,F(Y)]=\overline {W_{23}}[Y,F(Y)]=0,\;\overline {V_{23}}_{,Z}(Y,\infty )=0,\;\overline {P_{13}}(Y,\infty )\sim -Z .$

Then, since $U_{1,x}(x,y_0,t; Y,Z)=\overline {\varPhi _{11}}(Y,Z)\,u_{0,xz}(x,y_0,0,t)$ , the second-order boundary conditions (5.11) become

(6.13) \begin{align} \left \{\begin{array}{l} v(x,y,0,t)=\varepsilon a_1 v_{z}+\varepsilon ^2 a_2 v_{yz}+\varepsilon ^2 b_2 p_y +\varepsilon ^2 f_2 u_{xz}, \\[5pt] w(x,y,0,t)=-\varepsilon ^2 c_2 v_{yz} +\varepsilon ^2 g_2 u_{xz}, \end{array}\right . \end{align}

where the two newly introduced numerical constants are

(6.14) \begin{align} f_2 =\lim _{Z\rightarrow \infty } \overline {V_{23}}(Y,Z),\quad g_2 =\lim _{Z\rightarrow \infty } [\overline {W_{23}}(Y,Z)+Z^2/2+h_1 Z]. \end{align}

6.3. Third order

The procedure should by now be self-evident. Each stage of the $\varepsilon$ expansion entails a growing number of contributions, which superpose linearly and can be handled independently of each other. All of the terms already included in the 2D analysis are present. As compared with the 2D Navier–Stokes equations discussed in § 5.4, the third order (6.5) of the 3D Navier–Stokes problem produces the following additions, which we shall label according to the convention already adopted in the previous sections, a first digit denoting order and a second sequential digit.

6.3.1. Longitudinal strain

Each component of the momentum equation contains a second $x$ derivative, representing viscous longitudinal strain, in a position analogous to the time derivative, as highlighted below:

(6.15) \begin{align} U_{3,YY}+U_{3,ZZ} &= P_{1,x}\,\fbox{$-U_{1,xx}+U_{1,t}$} + V_1U_{1,Y} + W_1U_{1,Z},\nonumber \\V_{3,YY}+V_{3,ZZ} - P_{2,Y} &= \fbox{$-V_{1,xx}+V_{1,t}$} + V_1V_{1,Y} + W_1V_{1,Z},\nonumber \\W_{3,YY}+W_{3,ZZ} - P_{2,Z} &= \fbox{$-W_{1,xx}+W_{1,t}$} + V_1W_{1,Y} + W_1W_{1,Z},\nonumber \\ V_{3,Y} + W_{3,Z} &= - U_{2,x} . \end{align}

Indeed, the analogy is so precise that in the transverse equivalent boundary condition (5.20) it suffices to replace $v_{0,zt}$ by $v_{0,zt}-v_{0,xxz}$ (and $w_{0,zt}$ by $w_{0,zt}-w_{0,xxz}$ if they were present; but, by continuity and the no-slip conditions on $u_0$ and $v_0$ , $w_{0,zt} = w_{0,xxz} = 0$ ). In the longitudinal momentum equation, where the time derivative was not included in the analysis of § 3, this contribution to $U_3$ can be written as $\overline {U_{33}}(Y,Z)[u_{0,zt}(x,y_0,0,t)-u_{0,xxz}(x,y_0,0,t)]$ , where the new function $\overline {U_{33}}$ is the solution of

(6.16) \begin{align} \Delta _2 \overline {U_{33}} = \overline {\varPhi _{11}}, \end{align}

$\overline {\varPhi _{11}}$ being defined in (2.13), with boundary conditions $ \overline {U_{33}}[Y,F(Y)]=0,\;\overline {U_{33}}_{,Z}(Y,\infty )\sim Z^2/2+h_1 Z .$

There follows an additional term to the equivalent boundary condition (3.13) of the form $\varepsilon ^3 h_3^{(u_{zt})} (u_{zt}-u_{xxz})$ , where

(6.17) \begin{align} h_3^{(u_{zt})}=\lim _{Z\rightarrow \infty }[\overline {U_{33}}(Y,Z)-Z^3/6-h_1 Z^2/2] . \end{align}

6.3.2. Nonlinear terms

The nonlinear contributions to the third-order inner solution,

(6.18) \begin{align} U_{3,YY}+U_{3,ZZ} &= P_{1,x}-U_{1,xx}+U_{1,t} + \fbox{$V_1U_{1,Y} + W_1U_{1,Z}$}\,,\nonumber \\V_{3,YY}+V_{3,ZZ} - P_{2,Y} &= -V_{1,xx}+V_{1,t} + \fbox{$V_1V_{1,Y} + W_1V_{1,Z}$}\,,\nonumber \\W_{3,YY}+W_{3,ZZ} - P_{2,Z} &= -W_{1,xx}+W_{1,t} + \fbox{$V_1W_{1,Y} + W_1W_{1,Z}$}\,,\nonumber \\V_{3,Y} + W_{3,Z} &= - U_{2,x} , \end{align}

can be subdivided into a transverse part, which is identical to the one already studied in § 5.4, and a longitudinal part that represents transport of longitudinal momentum caused by the cross-flow velocities. The right-hand side for the longitudinal part can be written as $[\overline {\varPsi _{21}}_{,Z}\overline {\varPhi _{11}}_{,Y}-\overline {\varPsi _{21}}_{,Y}\overline {\varPhi _{11}}_{,Z}]v_{0,z}u_{0,z}$ . Therefore, the corresponding contribution to $U_3$ is a new function $\overline {U_{34}}(Y,Z)v_{0,z}u_{0,z}$ , where

(6.19) \begin{align} \Delta _2\overline {U_{34}} = \overline {\varPsi _{21}}_{,Z}\overline {\varPhi _{11}}_{,Y}-\overline {\varPsi _{21}}_{,Y}\overline {\varPhi _{11}}_{,Z} \end{align}

with boundary conditions $ \overline {U_{34}}[Y,F(Y)]=0,\;\overline {U_{34}}_{,Z}(Y,\infty )=0 .$

There ensues an additional term to the equivalent boundary condition (3.13) of the form $\varepsilon ^3 h^{(nl)}_3v_{0,z}u_{0,z}$ , where

(6.20) \begin{align} h^{(nl)}_3=\lim _{Z\rightarrow \infty }\overline {U_{34}}(Y,Z) . \end{align}

It can be observed that, just as its transverse counterpart, this nonlinear contribution turns out to be forbidden (the right-hand side of (6.20) is an odd function of $Y$ and so is $\overline {U_{34}}$ ) for left–right-symmetric wall features.

6.3.3. Cross-coupling

Finally, (6.5) contains the same cross-coupling terms between longitudinal and transverse components that made their first appearance at the second order:

(6.21) \begin{align} U_{3,YY}+U_{3,ZZ} &= \fbox{$P_{1,x}$}-U_{1,xx}+U_{1,t} + V_1U_{1,Y} + W_1U_{1,Z},\nonumber \\ V_{3,YY}+V_{3,ZZ} - P_{2,Y} &= -V_{1,xx}+V_{1,t} + V_1V_{1,Y} + W_1V_{1,Z},\nonumber \\ W_{3,YY}+W_{3,ZZ} - P_{2,Z} &= -W_{1,xx}+W_{1,t} + V_1W_{1,Y} + W_1W_{1,Z},\nonumber \\ V_{3,Y} + W_{3,Z} &= \fbox{$- U_{2,x} $}\,. \end{align}

These cross-couplings ingenerate several third-order contributions, one for every second-order term that takes the stage on the right-hand side in turn.

Here $P_{1,x}$ can be obtained from $\varPsi _3$ of (4.30), with additional contributions from $\overline {P_{13}}$ of (6.12) and from the order-1 outer pressure gradient at the wall. Summing the $Z$ derivative of (4.30) with the term involving $\overline {V_{23}}$ of (6.12) gives

(6.22) \begin{align} V_2&=\left [\overline {\varPsi _{31}}_{,Z}(Y,Z)+Y\overline {\varPsi _{21}}_{,Z}(Y,Z)\right ]v_{0,yz}(x,y_0,0,t)+ \overline {\varPsi _{32}}_{,Z}(Y,Z)p_{0,y}(x,y_0,0,t)\nonumber \\[4pt]& \quad + \overline {V_{23}}(Y,Z)u_{0,xz}(x,y_0,0,t)+\overline {\varPsi _{21}}_{,Z}(Y,Z)v_{1,z}(x,y_0,0,t), \end{align}

and correspondingly,

(6.23) \begin{align} P_{1,x} &=\left [\overline {P_{11}}(Y,Z)+Y\overline {P_{01}}(Y,Z)\right ]v_{0,xyz}(x,y_0,0,t)+ \overline {P_{12}}(Y,Z)p_{0,xy}(x,y_0,0,t) \nonumber \\[5pt]& \quad + \overline {P_{13}}(Y,Z)u_{0,xxz}(x,y_0,0,t)+\overline {P_{01}}(Y,Z)v_{1,xz}(x,y_0,0,t)+p_{1,x}(x,y_0,0,t) . \end{align}

To convert (6.22) into (6.23) it mostly suffices to apply Cauchy–Riemann relationships analogous to (6.7) term by term. This is true, in particular, for $\overline {P_{01}}$ , already available from (6.7) itself, and for $\overline {P_{12}}$ , obtained from a similar set of equations applied to $\overline {\varPsi _{32}}$ , whereas $\overline {P_{13}}$ directly comes from (6.12). Only $\overline {P_{11}}$ requires special care, owing to the presence of the term proportional to $Y$ in the square brackets, as follows.

Let us write Cauchy–Riemann relationships for the entire square brackets multiplying $v_{0,yz}$ in (6.22) and (6.23):

(6.24) \begin{align} \left [\overline {P_{11}}+Y\overline {P_{01}}\right ]_Y = \Delta _2\left [\overline {\varPsi _{31}}+Y\overline {\varPsi _{21}}\right ]_Z,\quad \left [\overline {P_{11}}+Y\overline {P_{01}}\right ]_Z = - \Delta _2\left [\overline {\varPsi _{31}}+Y\overline {\varPsi _{21}}\right ]_Y . \end{align}

Expanding the derivatives, and taking into account that $\overline {P_{01}},\overline {\varPsi _{21}}$ already obey their own Cauchy–Riemann relationships, gives

(6.25) \begin{align} \overline {P_{11}}_{,Y}+\overline {P_{01}} = \Delta _2\overline {\varPsi _{31}}_{,Z}+2\overline {\varPsi _{21}}_{,YZ},\quad \overline {P_{11}}_{,Z} = - \Delta _2\overline {\varPsi _{31}}_{,Y}-3\overline {\varPsi _{21}}_{,YY}-\overline {\varPsi _{21}}_{,ZZ} . \end{align}

Therefore $\overline {P_{11}}$ must be obtained from (6.25), with a boundary condition $\overline {P_{11}}(Y,\infty )\sim -Z$ .

The contribution to $U_3$ resulting from all the terms of (6.23) together, to be added to (2.34), can therefore be written as

(6.26) \begin{align} &\left [\overline {U_{35}}(Y,Z)+Y\overline {U_{23}}(Y,Z)\right ]v_{0,xyz}(x,y_0,0,t) \nonumber \\ &+ \left [\overline {U_{36}}(Y,Z)+Y\overline {U_{22}}(Y,Z)\right ]p_{0,xy}(x,y_0,0,t) + \overline {U_{37}}(Y,Z)u_{0,xxz}(x,y_0,0,t) \nonumber \\ &+\overline {U_{23}}(Y,Z)v_{1,xz}(x,y_0,0,t) +\overline {U_{22}}p_{1,x}(x,y_0,0,t), \end{align}

where $\overline {U_{22}}$ and $\overline {U_{23}}$ were already defined in (3.11) and (6.9), respectively, and the new functions $\overline {U_{35}}$ , $\overline {U_{36}}$ , $\overline {U_{37}}$ obey

(6.27) \begin{align} \Delta _2\overline {U_{35}}+2\overline {U_{23}}_{,Y}=\overline {P_{11}},\quad \Delta _2\overline {U_{36}}+2\overline {U_{22}}_{,Y}=\overline {P_{12}},\quad \Delta _2\overline {U_{37}}=\overline {P_{13}} ,\end{align}

with boundary conditions $\overline {U_{35}}_{,Z}(Y,\infty )\sim -Z^2/2$ , $\overline {U_{36}}_{,Z}(Y,\infty )=0$ , $\overline {U_{37}}_{,Z}(Y,\infty )=0$ .

The equivalent boundary condition for $u(x,y,0,t)$ , (3.13), will then acquire new contributions

(6.28) \begin{align} \varepsilon ^3 h_3^{(v_{xyz})} v_{xyz}+\varepsilon ^3 h_3^{(p_{xy})} p_{xy}+\varepsilon ^3 h_3^{(u_{xxz})}u_{xxz}, \end{align}

where

(6.29) \begin{align} \begin{split} &h_3^{(v_{xyz})}=\lim _{Z\rightarrow \infty }\big [\overline {U_{35}}(Y,Z)+Z^3/6\big ],\quad h_3^{(p_{xy})}=\lim _{Z\rightarrow \infty }\overline {U_{36}}(Y,Z),\\ & h_3^{(u_{xxz})}=\lim _{Z\rightarrow \infty }\overline {U_{37}}(Y,Z). \end{split} \end{align}

As far as the right-hand side of the continuity equation (6.5d ) is concerned, the $x$ derivative of the second-order longitudinal velocity, $U_2$ of (6.8), is

(6.30) \begin{align} U_{2,x}&= \overline {\varPhi _{11}}(Y,Z)u_{1,xz}(x,y_0,0,t) +\big [\overline {\varPhi _{21}}(Y,Z)+Y\,\overline {\varPhi _{1}(Y,Z)}\big ]u_{0,xyz}(x,y_0,0,t) \nonumber \\[4pt]& \quad + \overline {U_{22}}(Y,Z)p_{0,xx}(x,y_0,0,t)+\overline {U_{23}}(Y,Z)v_{0,xxz}(x,y_0,0,t) . \end{align}

Each term in turn must be inserted into four problems similar to (6.12), one with the right-hand side $-\overline {\varPhi _{11}}(Y,Z)$ yielding the same $\overline {V_{23}}, \overline {W_{23}}, \overline {P_{13}}$ as (6.12), one with the right-hand side $-\overline {\varPhi _{21}}(Y,Z)-Y\overline {\varPhi _{11}}(Y,Z)$ yielding a new triplet $\overline {V_{35}}, \overline {W_{35}}, \overline {P_{25}}$ , one with the right-hand side $-\overline {U_{22}}(Y,Z)$ yielding a new triplet $\overline {V_{36}}, \overline {W_{36}}, \overline {P_{26}}$ and one with the right-hand side $-\overline {U_{23}}(Y,Z)$ yielding a new triplet $\overline {V_{37}}, \overline {W_{37}}, \overline {P_{27}}$ .

Just as with (6.25), the term $-\overline {\varPhi _{21}}(Y,Z)-Y\overline {\varPhi _{11}}(Y,Z)$ requires special care for the presence of a part proportional to $Y$ . Inserting this binomial into a Stokes problem like (6.12) gives

(6.31) \begin{align} (\overline {V_{35}}+Y\overline {V_{23}})_{YY}+(\overline {V_{35}}+Y\overline {V_{23}})_{ZZ} - (\overline {P_{25}}+Y\overline {P_{13}})_{Y} &=0,\nonumber \\ (\overline {W_{35}}+Y\overline {W_{23}})_{YY}+(\overline {W_{35}}+Y\overline {W_{23}})_{ZZ} - (\overline {P_{25}}+Y\overline {P_{13}})_{Z} &=0,\nonumber \\ (\overline {V_{35}}+Y\overline {V_{23}})_{Y} + (\overline {W_{35}}+Y\overline {W_{23}})_{Z} &=-\overline {\varPhi _{21}}-Y\overline {\varPhi _{11}} , \end{align}

i.e. upon remembering that $\overline {V_{23}},\overline {W_{23}},\overline {P_{13}}$ themselves satisfy (6.12),

(6.32a) \begin{align} \overline {V_{35}}_{,YY}+\overline {V_{35}}_{,ZZ} - \overline {P_{25}}_{,Y} &=\overline {P_{13}}-2\overline {V_{23}}_{,Y}, \\[-10pt] \nonumber \end{align}

(6.32b) \begin{align} \overline {W_{35}}_{,YY}+\overline {W_{35}}_{,ZZ} - \overline {P_{25}}_{,Z} &=-2\overline {W_{23}}_{,Y}, \\[-10pt] \nonumber \end{align}

(6.32c) \begin{align} \overline {V_{35}}_{,Y} + \overline {W_{35}}_{,Z} &=-\overline {\varPhi _{21}}-\overline {V_{23}} . \\[9pt] \nonumber \end{align}

This is the system of equations to obtain $\overline {V_{35}},\overline {W_{35}},\overline {P_{25}}$ from.

There will result three new contributions to the $v$ and $w$ equivalent boundary conditions (5.20),

(6.33) \begin{align} \left \{ \begin{array}{l} \varepsilon ^3 f_3^{(u_{xyz})} u_{xyz}+\varepsilon ^3 f_3^{(p_{xx})}p_{xx}+\varepsilon ^3 f_3^{(v_{xxz})}v_{xxz} ,\\[5pt]\varepsilon ^3 g_3^{(u_{xyz})} u_{xyz}+\varepsilon ^3 g_3^{(p_{xx})}p_{xx}+\varepsilon ^3 g_3^{(v_{xxz})}v_{xxz}, \\ \end{array}\right . \end{align}

where

(6.34) \begin{align} \begin{split} &f_3^{(u_{xyz})}=\lim _{Z\rightarrow \infty }[\overline {V_{35}}(Y,Z)+Z^3/6],\quad f_3^{(p_{xx})}=\lim _{Z\rightarrow \infty }\overline {V_{36}}(Y,Z),\\ &f_3^{(v_{xxz})}=\lim _{Z\rightarrow \infty }\overline {V_{37}}(Y,Z) ,\end{split} \end{align}

and

(6.35) \begin{align} \begin{split} &g_3^{(u_{xyz})}=\lim _{Z\rightarrow \infty } \big[\overline {W_{35}}(Y,Z)+(h_2+f_2 ) Z \big], \\&g_3^{(p_{xx})}=\lim _{Z\rightarrow \infty }\big[\overline {W_{36}}(Y,Z)+Z^3/6+h_2^{(p_x)}Z \big], \\&g_3^{(v_{xxz})}=\lim _{Z\rightarrow \infty }\big[\overline {W_{37}}(Y,Z)+h_2^{(v_{xz})}Z \big]. \end{split} \end{align}

6.4. Fourth and higher orders

Although we shall not pursue here the detailed calculation of fourth- and higher-order contributions, the road should now be clear. The next order in the expansion of (6.2), for instance, is as follows.

Fourth order:

(6.36a) \begin{align} U_{4,YY}+U_{4,ZZ} & = P_{2,x}-U_{2,xx}+U_{2,t} +U_1U_{1,x}+ V_2U_{1,Y}+ V_1U_{2,Y} \nonumber \\ & \quad + W_2U_{1,Z} + W_1U_{2,Z}, \\[-10pt] \nonumber \end{align}

(6.36b) \begin{align} V_{4,YY}+V_{4,ZZ} - P_{3,Y} & = -V_{2,xx}+V_{2,t} +U_1V_{1,x} + V_2V_{1,Y} + V_1V_{2,Y} \nonumber \\ & \quad + W_2V_{1,Z} + W_1V_{2,Z}, \\[-10pt] \nonumber \end{align}

(6.36c) \begin{align} W_{4,YY}+W_{4,ZZ} - P_{3,Z} &= -W_{2,xx}+W_{2,t} +U_1W_{1,x} + V_2W_{1,Y} + V_1W_{2,Y} \nonumber \\ & \quad + W_2W_{1,Z} + W_1W_{2,Z}, \\[-10pt] \nonumber \end{align}

(6.36d) \begin{align} &V_{4,Y} + W_{4,Z} = - U_{3,x} . \\[9pt] \nonumber \end{align}

As can be seen, linear terms follow a similar pattern at every order, whereas nonlinear, advective terms produce a convolution of previous orders stemming from their product structure. When these systems are solved in a cascade they will produce a general $n$ th-order equivalent boundary condition for the velocity vector $(u,v,w)$ at the equivalent wall $z=0$ akin to (4.47) and (4.48). Just as (4.47) and (4.48) contained higher and higher lateral derivatives of $\psi _{zz}$ and $\psi _{zzz}$ , this boundary condition will contain partial derivatives of order up to $n-1$ with respect to $x,y$ and $t$ of the triplet $(u_z,v_z,p)$ . As can be imagined, the number of terms goes up rapidly with order, though not as rapidly as it might, because many of them will have zero coefficients. Rather than trying to provide a tensorial notation for the general formula, which would be cumbersome, we prefer to name each relevant term individually in the next section.

7.1. Comparison with previous literature

Up to the second order, the asymptotic expansion (7.1) is consistent with the two-dimensional $(y,z)$ , two-component $(v,w)$ rough-wall equations (7) and (8) and table 5 of Sudhakar et al. (Reference Sudhakar, Lācis, Pasche and Bagheri2021), written here in the present notation and wall scaling (but without changing the numerical subscripts of the coefficients):

(7.2) \begin{align} v = \varepsilon [\mathcal{L}_{11}(v_z+w_y)] + \varepsilon ^2[\mathcal{K}_{11} (-p_y + 2 w_{zy})], \quad w = \varepsilon ^2 [\mathcal{M}_{211}(v_{zy} + w_{yy})]. \end{align}

Following Appendix A of Ahmed & Bottaro (Reference Ahmed and Bottaro2025), we note that $w = \mathrm{O}(\varepsilon ^2)$ to neglect the $y$ derivatives $w_y$ , $w_{zy}$ and $w_{yy}$ on the right-hand sides, and then identify, by matching with (7.1) presently, that $\mathcal{L}_{11} = a_1$ , $-\mathcal{K}_{11} = b_2$ , $\mathcal{M}_{211} = -c_2$ . Noting that $a_2= 0$ (Appendix B) and ignoring the third component $u$ or third dimension $x$ in (7.1), it can be seen that (7.2) and (7.1) are identical up to the second order.

Likewise, the present (7.1) is consistent with the two-dimensional $(y,z)$ , two-component $(v,w)$ third-order effective boundary conditions in equations (54) and (55) of Bottaro & Naqvi (Reference Bottaro and Naqvi2020), written in the present notation and wall scaling (without changing the numerical subscripts of the coefficients and dropping the superscripts of the coefficients):

(7.3) \begin{align} &v = \varepsilon [\lambda (v_z + w_y)] + \varepsilon ^2[m_{12} (-p_y + 2w_{zy})] \nonumber \\ &\quad + \varepsilon ^3[\theta (v_{zyy} + w_{yyy}) + q_{12} (v_{zt} + w_{yt})], \nonumber \\ &w = \varepsilon ^2[m_{21}(v_{zy}+w_{yy})] + \varepsilon ^3[q_{21}(-p_{yy} + 2w_{zyy})]. \end{align}

First, up to second order, it can be verified that this is consistent with (7.2), i.e. the equation of Sudhakar et al. (Reference Sudhakar, Lācis, Pasche and Bagheri2021), with $\lambda =\mathcal{L}_{11}$ , $m_{12} = \mathcal{K}_{11}$ and $m_{21} = \mathcal{M}_{211}$ and, therefore, also consistent with (7.1). Furthermore, it has been recognised, see, e.g. table 3 of Bottaro & Naqvi (Reference Bottaro and Naqvi2020), that $m_{21} = -m_{12}$ , i.e. $\mathcal{K}_{11} = -\mathcal{M}_{211}$ , consistent with the numerical values in table 5 of Sudhakar et al. (Reference Sudhakar, Lācis, Pasche and Bagheri2021), and also, in the present notation, $-b_2 = c_2$ , also verified in the present table 1 and now proved in Appendix A. Focusing now on the third-order terms, noting again that $w = \mathrm{O}(\varepsilon ^2)$ , such that the $y$ derivatives $w_{yyy}$ , $w_{yt}$ and $w_{zyy}$ can be neglected, we then identify by comparing with (7.1) that $\theta = a_3$ , $q_{12} = e_3^{(v_{zt})}$ and $q_{21} = d_3$ in addition to $\lambda = a_1$ , $m_{12} = -b_2$ and $m_{21} = -c_2$ already established at second order. The remaining unmatched, extra terms in (7.1) are either zero for symmetric shapes (those indicated in grey) or involve the third dimension $x$ or the third component $u$ . The third-order coefficient $e_3^{(nl)}$ is zero (Appendix B), but the third-order coefficient $b_3 = c_3$ (cf. Appendix A), multiplying $p_{yy}$ and $v_{yyz}$ , is non-zero for asymmetric shapes (table 1), but not previously discussed, e.g. by Bottaro & Naqvi (Reference Bottaro and Naqvi2020). The claim that $q_{12} = q_{21}$ (Bottaro & Naqvi Reference Bottaro and Naqvi2020), i.e. that $e_3^{(v_{zt})}$ and $d_3$ are equal, is not supported by the present data (table 1).

So far, no 3D coefficients have been compared numerically with previous literature. We address this now through the 3D second-order boundary condition for a rough wall, equation (2.21) of Ahmed & Bottaro (Reference Ahmed and Bottaro2025) written with the present notation and wall scaling:

(7.4) \begin{align} &u = \varepsilon [\lambda _x (u_z+w_x)] + \varepsilon ^2 \big[\mathcal{K}_{xz}^{\textit{it f}}(-p_x+2w_{zx})\big],\nonumber \\[5pt]&v = \varepsilon [\lambda _y (v_z+w_y)] + \varepsilon ^2 \big[\mathcal{K}_{yz}^{\textit{it f}}(-p_y+2w_{zy}) \big],\nonumber \\[5pt]&w = \varepsilon ^2 \big[ -\mathcal{K}_{xz}^{\textit{it f}}(u_{zx}+w_{xx}) -\mathcal{K}_{yz}^{\textit{it f}}(v_{zy}+w_{yy})\big]; \end{align}

cf. equations (68) and (69) of Naqvi & Bottaro (Reference Naqvi and Bottaro2021). For a rough wall, $\mathcal{K}_{zz} = 0$ (Ahmed & Bottaro Reference Ahmed and Bottaro2025, Appendix C). These four coefficients $(\lambda _x, \lambda _y, \mathcal{K}_{xz}^{\textit{it f}},\mathcal{K}_{yz}^{\textit{it f}})$ are to be compared with $(h_1, a_1, h_2^{(p_x)}=g_2, b_2=-c_2)$ of (7.1), numerically tabulated presently in table 1 ( $60^\circ$ triangular, rectangular, trapezoidal, sawtooth riblets) corresponding respectively to table 4 of Ahmed & Bottaro (Reference Ahmed and Bottaro2025) (equilateral triangle, thick blade, trapezoidal, right asymmetric triangle), showing that the numbers agree. At second order in three dimensions and three components, the new coefficient is $f_2 = -h_2^{(v_{xz})}$ , which is only non-zero for asymmetric shapes, multiplying $u_{xz}$ and $-v_{xz}$ , previously undocumented by either Ahmed & Bottaro (Reference Ahmed and Bottaro2025) or Sudhakar et al. (Reference Sudhakar, Lācis, Pasche and Bagheri2021).

In summary, compared with previous literature, (7.1) highlights missing coefficients for asymmetric shapes. In particular, $f_2 = -h_2^{(v_{xz})}$ was missing at second order for three-dimensional and three-component flow, and $b_3 = c_3$ was missing at third order for two-dimensional and two-component flow. Moreover, a third-order boundary condition for three-dimensional and three-component flow had not been attempted, now addressed by (7.1), revealing an additional seven new coefficients $h_3$ , $h_3^{(u_{zt})}$ , $h_3^{(v_{xyz})}$ , $h_3^{(u_{xxz})}$ , $f_3^{(u_{xyz})}$ , $f_3^{(v_{xxz})}$ and $g_3^{(p_{xx})}$ for symmetric shapes plus a further two, $h_3^{(p_{xy})} = -g_3^{(u_{xyz})}$ and $f_3^{(p_{xx})}= -g_3^{(v_{xxz})}$ (Appendix A), for asymmetric shapes. In (7.1), regardless of shape symmetry, $h_2=a_2=0$ and surprisingly $h_3^{(nl)} = e_3^{(nl)} = 0$ (Appendix B), i.e. nonlinear terms are absent at third order.