3.1. Change of variables

Let us introduce the streamfunction $\psi _y = \overline \rho \tilde u$ and $\psi _x = - \overline \rho \tilde v$ and new variables by the formulas

(3.1) \begin{align} \psi = \rho _\infty u_\infty \delta \varPsi (\xi , \, \eta ), & \quad \tilde h = h_\infty + u_\infty ^2 \varPhi (\xi ,\, \eta ), \quad \overline \rho = \rho _\infty \omega (\xi ,\, \eta ), \nonumber \\ \xi = \ln {\textit{Re}}_\delta , &\quad {\textit{Re}}_\delta = \frac {\sqrt {\rho _\infty \rho _w}\, u_\infty \delta } {{\mu} _w}, \end{align}

where $\rho _w$ and ${\mu} _w$ are the averaged gas density and viscosity at the wall. Based on (3.1), we have

(3.2a–c) \begin{align} {\partial }_x = \frac {\varLambda } {\delta } ( {\partial }_\xi - \eta {\partial }_\eta ), \quad {\partial }_y = \frac {{\partial }_\eta } \delta , \quad \varLambda (\xi ) = \frac {\textrm {d} \delta } {\textrm {d} x}, \end{align}

(3.2d,e) \begin{align} \tilde u = u_\infty \omega ^{-1} \varPsi _\eta , \quad \tilde v = - u_\infty \varLambda \omega ^{-1} (\varPsi + \varPsi _\xi - \eta \varPsi _\eta ). \end{align}

The logarithm of the Reynolds number based on the boundary-layer thickness and the normalised distance from the wall are chosen as independent variables $\xi$ and $\eta$ . In addition to the dimensionless streamfunction $\varPsi (\xi ,\, \eta )$ and enthalpy $\varPhi (\xi ,\, \eta )$ , the third unknown function $\varLambda (\xi )$ – the growth rate of the boundary-layer thickness is introduced. From (2.1a ) and (2.1c ), for the functions $\varPsi$ , $\varPhi$ and $\varLambda$ , taking into account the closure relations (2.2), we obtain

(3.3a) \begin{align} &\varLambda \big [ \varPsi _\eta \big(\omega ^{-1} \varPsi _\eta \big)_\xi - (\varPsi + \varPsi _\xi ) \big(\omega ^{-1} \varPsi _\eta \big)_\eta \big ] = \big \{ \omega \big [\eta \big(\omega ^{-1} \varPsi _\eta \big)_\eta \big ]^2 \!S_e (\eta ,\, \varPi _1,\, \varPi _2,\, \varGamma )\big \}_\eta, \end{align}

(3.3b) \begin{align}&\qquad\quad \varLambda [\varPsi _\eta \varPhi _\xi - (\varPsi + \varPsi _\xi ) \varPhi _\eta ] = \big [ \omega \eta ^2 \varPhi _\eta \big(\omega ^{-1} \varPsi _\eta\big)_\eta H_e (\eta ,\, \varPi _1,\, \varPi _2,\, \varGamma ) \big ]_\eta \nonumber \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad + \omega \eta ^2 \big[\big(\omega ^{-1} \varPsi _\eta\big)_\eta\big]^3 S_e (\eta ,\, \varPi _1,\, \varPi _2,\, \varGamma ), \end{align}

(3.3c–e) \begin{align}&\qquad\qquad \omega ^{-1} = 1 + (\gamma - 1) M_\infty ^2 \varPhi , \quad \varPi _1 = \frac {\eta \big[\big(\omega ^{-1} \varPsi _\eta\big)_\eta\big]^2} {\varPhi _\eta }, \quad \varPi _2 = - \frac {{\partial } \ln \omega }{{\partial } \ln \eta }. \end{align}

Since (3.3) will be used only in the outer region of the boundary layer, the molecular viscosity and thermal conductivity terms are omitted, and the local Reynolds and Peclet numbers are set equal to infinity, which allows the notation (2.6) to be used.

Equations (3.3) are convenient for asymptotic analysis, since the boundary-layer thickness growth rate $\varLambda$ is a small quantity and the variable $\xi$ equal to the logarithm of the Reynolds number is a large quantity, which makes it possible to search for an asymptotic solution of the problem as $\xi \to \infty$ . The change of variables (3.1) and the equations for the desired functions (3.3) were first proposed by Vigdorovich (Reference Vigdorovich1993) for the boundary layer of an incompressible fluid.

3.2. Solution for the outer region

Let us consider the case of moderate free-stream Mach numbers $M_\infty = O(1)$ when, according to (3.3c ), the relative change in density across the boundary layer is of order one.

According to Vigdorovich (Reference Vigdorovich1993), the asymptotic expansions for the desired functions as $\xi \to \infty$ contain $\ln \xi$ in addition to inverse powers of $\xi$ . To simplify the form of the desired expansions and to avoid the appearance of logarithmic terms, we introduce a new independent variable $\zeta$ by the formula (Vigdorovich Reference Vigdorovich2017)

(3.4) \begin{align} \xi = \zeta + k \ln \zeta , \end{align}

where the coefficient $k$ will be chosen later. In the outer region of the boundary layer that is characterised by the condition $1/\eta = O(1)$ , we look for the solution of (3.3) in the form (Vigdorovich Reference Vigdorovich2017)

(3.5a) \begin{align} \varLambda &= \lambda _1 \zeta ^{-1} + O(\zeta ^{-2}), \end{align}

(3.5b) \begin{align} \frac {\tilde u}{u_\infty } &= \frac {\varPsi _\eta } \omega = 1 + \lambda _1 f_0 (\eta ) \zeta ^{-1} + O(\zeta ^{-2}), \end{align}

(3.5c) \begin{align} \varPhi &= \lambda _2 \varphi _0 (\eta ) \zeta ^{-1} + O(\zeta ^{-2}), \quad \zeta \to \infty , \end{align}

where $\lambda _1$ and $\lambda _2$ are constant coefficients. For the dimensionless density, equalities (3.3c ) and (3.5c ) give

(3.6) \begin{align} \omega ^{-1} = 1 + (\gamma - 1) M_\infty ^2 \lambda _2 \varphi _0 (\eta ) \zeta ^{-1} + O(\zeta ^{-2}). \end{align}

Substituting (3.5b )–(3.6) into (3.3d ) and (3.3e ), we obtain the estimates $\varPi _1 = O (\zeta ^{-1})$ and $\varPi _2 = O (\zeta ^{-1})$ , which on the basis of (2.6) and asymptotic representations (2.4a ) and (2.4b ) yield

(3.7) \begin{align} S_e (\eta ,\, \varPi _1,\, \varPi _2,\, \varGamma ) &= S_0 (\infty ,\, \eta ) + O (\zeta ^{-1}), \nonumber\\ H_e (\eta ,\, \varPi _1,\, \varPi _2,\, \varGamma ) &= H_0 (\infty ,\, \infty ,\, \eta ) + O (\zeta ^{-1}). \end{align}

By substituting the expansions (3.4)–(3.7) together with the equality $\varPsi = \eta + O (\zeta ^{-1})$ , which follows from (3.5b ) and (3.6), into (3.3) and taking the limit as $\zeta \to \infty$ with $1/\eta = O(1)$ , we obtain, for the functions $f_0$ and $\varphi _0$ , the ordinary differential equations

(3.8a) \begin{align} \left [\big(\eta f^{\prime}_0\big)^2 S_0 (\infty ,\, \eta ) \right ]^{\prime} + \eta f^{\prime}_0 = 0; \quad f_0(\infty ) = 0, \end{align}

(3.8b) \begin{align} \left [ \eta ^2 \varphi^{\prime}_{0} f^{\prime}_{0} H_0 (\infty ,\, \infty ,\, \eta ) \right ]^{\prime} + \eta \varphi^{\prime}_{0} = 0; \quad \varphi _0 (\infty ) = 0. \end{align}

The last term in (3.3b ) related to kinetic heating has a higher order of smallness and is not included in (3.8b ).

Equations (3.8) coincide completely with those obtained by Vigdorovich (Reference Vigdorovich2017) for the incompressible boundary layer. A non-zero solution to (3.8a ) has the form (Vigdorovich Reference Vigdorovich2017)

(3.9) \begin{align} f_0 (\eta ) = - \int \limits _\eta ^\infty \frac {{\textrm{d}}\eta _1} {2 \eta _1 \sqrt {S_0 (\infty ,\, \eta _1)}} \int \limits _{\eta _1}^\infty \frac {{\textrm{d}} \eta _2} {\sqrt {S_0 (\infty ,\, \eta _2)}} \end{align}

and is uniquely determined by the supplied boundary condition (3.8a ). From (3.9), given the smoothness conditions for the universal function (2.5), we obtain the asymptotic representation (Vigdorovich Reference Vigdorovich2017)

(3.10a) \begin{align} f_0 (\eta ) &= \frac {I_0} \varkappa (\ln \eta + A_0 - \ln I_0) + O(\eta ^\alpha ), \quad \eta \to 0, \quad \alpha \gt 0, \end{align}

(3.10b) \begin{align} I_0 &= \left (- \int _0^\infty f_0 \,\textrm {d} \eta \right )^{1/2} = \int \limits _0^\infty \frac {{\textrm{d}} \eta }{2 \sqrt {S_0 (\infty ,\, \eta )}}. \end{align}

Here, $\varkappa = \sqrt {S_0 (\infty ,\, 0)}$ and $A_0$ is some constant.

The boundary-layer thickness $\delta (x)$ introduced by us can be chosen with a significant degree of arbitrariness. Let us introduce a new transverse scale by the formula $\hat \delta = a \delta$ , where $a$ is a constant factor. Then $\hat \eta = a^{-1} \eta$ and $\hat \varLambda = a \varLambda$ . Hence, based on (3.5a ) and (3.5b ), $\hat f_0 = a^{-1} f_0$ . Consequently, based on (3.10b ), $\hat I_0 = a^{-1} I_0$ . It is now clear that the constant $A_0$ in (3.10a ) is introduced in such a way that it does not depend on $a$ , i.e. on the choice of the transverse scale $\delta$ .

The solution to (3.8b ) is

(3.11) \begin{align} \varphi _0 (\eta ) = C_2 \int \limits _\eta ^\infty \frac 1 \eta \textrm {d} \exp \left [- \int \limits _0^\eta \frac {\textrm {d} \eta } {\eta f^{\prime}_0 H_0 (\infty ,\, \infty ,\, \eta )} \right ] \end{align}

and is determined with accuracy to the constant

(3.12) \begin{align} C_2 = - \int \limits _0^\infty \varphi _0 \,\textrm {d} \eta \end{align}

(this equality is obtained by integrating (3.11) and taking the integral by parts), which can be chosen arbitrarily. Putting the additional condition

(3.13) \begin{align} \sigma \int \limits _0^\infty \varphi _0 \,{\textrm{d}} \eta = I_0, \quad \sigma = \frac {S_0 (\infty ,\, 0)}{H_0 (\infty ,\, \infty ,\, 0)}, \end{align}

where $\sigma$ is the turbulent Prandtl number in the logarithmic region (Vigdorovich Reference Vigdorovich2023b ), we obtain from (3.11) and the smoothness conditions (2.5) that the temperature profile has the logarithmic asymptotics at the wall

(3.14) \begin{align} \varphi _0 (\eta ) = - \frac 1 \varkappa (\ln \eta + E_0 - \ln I_0) + O(\eta ^\alpha ), \quad \eta \to 0, \quad \alpha \gt 0, \end{align}

where $E_0$ is a constant, which is also independent of the choice of the transverse scale $\delta$ .

The numerical values of the von Kármán constant $\varkappa$ , turbulent Prandtl number in the logarithmic region $\sigma$ and constants $A_0$ and $E_0$ adopted in the present work are given in table 1.

Table 1.

Values of the empirical constants used.

Thus, the functions $f_0 (\eta )$ and $\varphi _0 (\eta )$ , specifying the dimensionless velocity and temperature defect profiles, can be taken from the solution for an incompressible fluid. The density variation in the outer region of the boundary layer is small and kinetic heating can be neglected.

3.3. Estimation of transverse pressure gradient

The Favre-averaged momentum equation in the $y$ direction has the form

(3.15) \begin{align} \frac {\partial } {{\partial } y} \big( \overline p + \overline \rho \tilde v^2 + \overline \rho \widetilde {v^{\prime \prime 2}} \big) = - \frac {\partial } {{\partial } x} \overline \rho \big( \tilde u \tilde v + \widetilde {u'' v''}\big). \end{align}

The molecular-viscosity terms in (3.15) are omitted. Integrating (3.15) over $y$ from zero to $\infty$ in view of the wall and free-stream conditions yields

(3.16) \begin{align} \overline p_w - p_\infty = \frac {\partial } {{\partial } x} \int _0^\infty \overline \rho \big( \tilde u \tilde v + \widetilde {u'' v''}\big) \textrm {d} y. \end{align}

Equality (3.2e ) and the expansions (3.5a ) and (3.5b ) imply that the dimensionless transverse velocity component is of order $\zeta ^{-2}$ , and the dimensionless shear stress is of the same order by virtue of (2.2a ) and (3.5b ). According to (3.2a ), differentiating by $x$ raises the order of the right-hand side of (3.16) by two more units. Thus, for the dimensionless pressure difference, we obtain the estimate

(3.17) \begin{align} \frac {\overline p_w - p_\infty }{\rho _\infty u_\infty ^2} = O (\zeta ^{-4}), \end{align}

which is enough to put $\overline p_w = p_\infty$ wherever necessary in further calculations.

5.1. Matching on the basis of the law of the wall for velocity

The law of the wall for the velocity profile (4.1b ) is an asymptotic expansion of the solution to the problem in the near-wall region as $y_+ \to \infty$ . Following the matching procedure (Van Dyke Reference Van Dyke1975), let us rewrite it in outer variables. The relation between the variables reads

(5.1) \begin{align} y_+ = {\textit{Re}}_\tau \eta , \quad {\textit{Re}}_\tau = \frac {\delta \sqrt {\rho _w \tau _w}}{{\mu} _w} = {\textit{Re}}_\delta \sqrt {\frac {1}{2} C_{\!f}}. \end{align}

Here, the definition of ${\textit{Re}}_\delta$  (3.1) is taken into account. According to (3.5b ) in the outer region

(5.2) \begin{align} \tilde u = u_\infty + u_\infty \lambda _1 f_0 (\eta ) \zeta ^{-1} + O(\zeta ^{-2}). \end{align}

Let us substitute (5.2) into the left-hand side of equality (4.1b ) and under the condition $1/\eta = O(1)$ , when $f_0 (\eta ) = O(1)$ , expand it in the small parameter $\zeta ^{-1}$ . The result is

(5.3) \begin{align} &\frac {\phi _\infty - \phi _w}{\tan \vartheta \sqrt {\frac {1}{2} C_{\!f}}} + \frac {u_\infty \phi ' (u_\infty ) \lambda _1 f_0 (\eta )} {\zeta \tan \vartheta \sqrt {\frac {1}{2} C_{\!f}}} + Q_1 (\phi _\infty ,\, \phi _w) + O(\zeta ^{-1}) + O \big(\sqrt {C_{\!f}}\big) \nonumber \\ &\quad = \frac 1 \varkappa \ln (\eta {\textit{Re}}_\tau ) + D_1 (\textit{Pr},\, n,\, Q_+) + O\big({\textit{Re}}_\tau ^{-\alpha }\big). \end{align}

Here we used (4.5a ), and $\phi _\infty = \phi (u_\infty )$ is the value of the variable $\phi$  (4.2a ) at the outer boundary of the boundary layer. From (4.2) we have

(5.4) \begin{align} \phi '(\tilde u) = \frac {\cos \psi }{\cos \phi } \sqrt {\frac \sigma {2 \tilde h}}, \end{align}

which based on (4.5b ) and (4.6) allows us to write

(5.5) \begin{align} u_\infty \phi ' (u_\infty ) = \tan \vartheta + O \big(\sqrt {C_{\!f}}\big). \end{align}

Let us now take in (5.3) the limit as $\eta \to 0$ . For this purpose, we replace the function $f_0 (\eta )$ by its asymptotic expansion (3.10a ). As a result, in view of (5.5), we obtain the matching condition (Van Dyke Reference Van Dyke1975)

(5.6) \begin{align} &\frac {\phi _\infty - \phi _w} {\tan \vartheta \sqrt {\frac {1}{2} C_{\!f}}} + \frac {I_0 \lambda _1}{\varkappa \zeta \sqrt {\frac {1}{2} C_{\!f}}} (\ln \eta + A_0 - \ln I_0) + Q_1 (\phi _\infty ,\, \phi _w) + O(\zeta ^{-1}) + O \big(\sqrt {C_{\!f}}\big) \nonumber \\ &\quad = \frac 1 \varkappa \ln (\eta {\textit{Re}}_\tau ) + D_1 (\textit{Pr},\, n,\, Q_+). \end{align}

Equating in this equality the coefficients at $\ln \eta$ , we have

(5.7) \begin{align} \sqrt {\frac {1}{2} C_{\!f}} = \frac {I_0 \lambda _1} \zeta + O(\zeta ^{-2}). \end{align}

The second relation that follows from (5.6) is the friction law

(5.8) \begin{align} \ln (I_0 {\textit{Re}}_\tau ) = \frac {\varkappa (\phi _\infty - \phi _w)} {\tan \vartheta \sqrt {\frac {1}{2} C_{\!f}}} + \varkappa [Q_1 (\phi _\infty ,\, \phi _w) - D_1 (\textit{Pr},\, n,\, Q_+)] + A_0 + O \big(\sqrt {C_{\!f}}\big), \end{align}

i.e. a relationship between the skin-friction coefficient and Reynolds number. It is taken into account here that according to (5.7) and (4.5a ), $\sqrt {C_{\!f}} = O(\zeta ^{-1})$ .

Based on (5.1), (3.1), (3.4) and (5.7),

(5.9) \begin{align} \ln {\textit{Re}}_\tau = \zeta + k \ln \zeta + \ln \sqrt {\frac {1}{2} C_{\!f}} = \zeta + (k - 1) \ln \zeta + O(1). \end{align}

On the other hand, substitution of (5.7) into (5.8) yields

(5.10) \begin{align} \ln {\textit{Re}}_\tau = \frac {\varkappa (\phi _\infty - \phi _w) \zeta }{I_0 \lambda _1 \tan \vartheta } + O(1). \end{align}

From a comparison of (5.9) and (5.10), we obtain $k = 1$ and

(5.11) \begin{align} \lambda _1 = \frac {\varkappa (\phi _\infty - \phi _w)}{I_0 \tan \vartheta }. \end{align}

5.2. Matching on the basis of the Crocco integral

Let us now perform the matching based on the Crocco integral (4.1a ). On the basis of (3.1) and (3.5c ) in the outer region

(5.12) \begin{align} \tilde h = h_\infty + u_\infty ^2 \lambda _2 \varphi _0 (\eta ) \zeta ^{-1} + O(\zeta ^{-2}). \end{align}

Substitution of (5.2) and (5.12) into (4.1a ) and expanding in the small parameter $\zeta ^{-1}$ under the condition $1/\eta = O(1)$ , when $f_0 (\eta ) = O(1)$ and $\varphi _0 (\eta ) = O(1)$ , yield

(5.13) \begin{align} h_w - h_\infty - \frac {\sigma u_\infty ^2} 2 - \frac {\sigma u_\infty q_w}{\tau _w} &= \frac {u_\infty ^2 \lambda _2} \zeta \varphi _0 (\eta ) + \sigma u_\infty \left ( u_\infty + \frac {q_w}{\tau _w} \right ) \frac {\lambda _1} \zeta f_0(\eta ) \nonumber \\ &\quad + \frac {h_w \tan \vartheta P (\phi _\infty ,\, \phi _w)}{\cos ^2 \phi _w} \sqrt {\frac {1}{2} C_{\!f}} + O(C_{\!f}). \end{align}

Here, we also used (4.5a ) and took into account that $\sqrt {C_{\!f}} = O(\zeta ^{-1})$ . Taking in this equality the limit as $\eta \to 0$ , i.e. replacing the functions $f_0$ and $\varphi _0$ by their asymptotic expansions (3.10a ) and (3.14), gives the matching condition (Van Dyke Reference Van Dyke1975)

(5.14) \begin{align} h_w - h_\infty - \frac {\sigma u_\infty ^2} 2 - \frac {\sigma u_\infty q_w}{\tau _w} &= \frac {u_\infty ^2 \lambda _2}{\varkappa \zeta } (\ln \eta + E_0 - \ln I_0) \nonumber \\ &\quad + \sigma u_\infty \left ( u_\infty + \frac {q_w}{\tau _w} \right ) \frac {I_0 \lambda _1}{\varkappa \zeta } (\ln \eta + A_0 - \ln I_0) \nonumber \\ &\quad + \frac {h_w \tan \vartheta P (\phi _\infty ,\, \phi _w)}{\cos ^2 \phi _w} \sqrt {\frac {1}{2} C_{\!f}} + O(C_{\!f}). \end{align}

By equating the coefficients at $\ln \eta$ , in view of (5.7), we have

(5.15) \begin{align} \frac {\lambda _2} \zeta = \sigma \left ( 1 + \frac {q_w}{u_\infty \tau _w} \right ) \sqrt {\frac {1}{2} C_{\!f}} + O(C_{\!f}). \end{align}

The second relation, which follows from (5.14),

(5.16) \begin{align}& h_w - h_\infty - \frac {\sigma u_\infty ^2} 2 - \frac {\sigma u_\infty q_w}{\tau _w} \nonumber\\ &\quad= \sqrt {\frac {1}{2} C_{\!f}} \left [ \frac {\sigma u_\infty } \varkappa \left ( u_\infty + \frac {q_w}{\tau _w} \right ) (A_0 - E_0) + \frac {h_w \tan \vartheta P (\phi _\infty ,\, \phi _w)}{\cos ^2 \phi _w} \right ] + O(C_{\!f}), \end{align}

will allow us to calculate the recovery and Reynolds-analogy factors.

5.3. Matching on the basis of the law of the wall for temperature

Substituting (5.12) into (4.2b ), we get

(5.17) \begin{align} \psi = \psi _\infty + \psi ' (h_\infty ) u_\infty ^2 \lambda _2 \varphi _0 (\eta ) \zeta ^{-1} + O (\zeta ^{-2}). \end{align}

Here, $\psi _\infty = \psi (h_\infty )$ is the value of the variable $\psi$  (4.2b ) at the outer boundary of the boundary layer. Since based on (4.2b ),

(5.18) \begin{align} \psi '(\tilde h) = - \frac {\cot \psi }{2 \tilde h}, \end{align}

we obtain

(5.19) \begin{align} \psi = \psi _\infty - \frac 12 (\gamma - 1) M_\infty ^2 \cot \psi _\infty \lambda _2 \varphi _0 (\eta ) \zeta ^{-1} + O (\zeta ^{-2}), \end{align}

which after substitution into (4.1c ) and expanding in terms of the small parameter $\zeta ^{-1}$ yields

(5.20) \begin{align} \frac {\psi _\infty - \phi _w} {\tan \vartheta \sqrt {\frac {1}{2} C_{\!f}}} &- \frac {(\gamma - 1) M_\infty ^2 \cot \psi _\infty \lambda _2 \varphi _0 (\eta )} {2 \zeta \tan \vartheta \sqrt {\frac {1}{2} C_{\!f}}} + R (\psi _\infty ,\, \phi _w) + O \big(\sqrt {C_{\!f}}\big) \nonumber \\ &= \frac 1 \varkappa \ln ({\textit{Re}}_\tau \eta ) + D_1 (\textit{Pr},\, n,\, Q_+) + O\big({\textit{Re}}_\tau ^{-\alpha }\big). \end{align}

We obtain the matching condition (Van Dyke Reference Van Dyke1975) by replacing the function $\varphi _0 (\eta )$ in this equality by its asymptotic expansion (3.14):

(5.21) \begin{align} \frac {\psi _\infty - \phi _w}{\tan \vartheta \sqrt {\frac {1}{2} C_{\!f}}} &+ \frac {\cot \psi _\infty \tan \vartheta \lambda _2} {\sigma \varkappa \zeta \sqrt {\frac {1}{2} C_{\!f}}} (\ln \eta + E_0 - \ln I_0) + R (\psi _\infty ,\, \phi _w) + O \big(\sqrt {C_{\!f}}\big) \nonumber \\ &= \frac 1 \varkappa \ln ({\textit{Re}}_\tau \eta ) + D_1 (\textit{Pr},\, n,\, Q_+). \end{align}

Here we have taken into account equality (4.5b ). Now, equating in (5.21) the coefficients at $\ln \eta$ , we have

(5.22) \begin{align} \frac {\lambda _2} \zeta = \frac {\sigma \tan \psi _\infty }{\tan \vartheta } \sqrt {\frac {1}{2} C_{\!f}} + O(C_{\!f}). \end{align}

Using (4.6) and then (4.2) with the accuracy of $O(\sqrt {C_{\!f}})$ yields

(5.23) \begin{align} \frac {\tan \psi _\infty }{\tan \vartheta } = \frac {\sin \phi _\infty }{\cos \psi _\infty \tan \vartheta } = \frac {\sqrt \sigma (u_\infty + q_w/\tau _w)}{\tan \vartheta \sqrt {2 h_\infty }}. \end{align}

Hence, given (4.5b ), we can conclude that the expression for $\lambda _2$  (5.22) coincides with (5.15).

The second relation, which follows from the matching condition (5.21), is another formulation of the friction law

(5.24) \begin{align} \ln (I_0 {\textit{Re}}_\tau ) = \frac {\varkappa (\psi _\infty - \phi _w)} {\tan \vartheta \sqrt {\frac {1}{2} C_{\!f}}} + \varkappa [R (\psi _\infty ,\, \phi _w) - D_1 (\textit{Pr},\, n,\, Q_+)] + E_0 + O \big(\sqrt {C_{\!f}}\big). \end{align}

6.1. Recovery factor

In the case of an adiabatic plate, when $q_w = 0$ , $\phi _w = 0$ and $h_w = h_r$ , (5.16) takes the form

(6.1) \begin{align} u_\infty ^2 (r - \sigma ) = \sqrt {2 C_{\!f}}\, \left [ \frac {\sigma u_\infty ^2} \varkappa (A_0 - E_0) + h_r \tan \vartheta P (\phi _\infty ,\, 0) \right ] + O(C_{\!f}). \end{align}

At zero heat flux, (4.2a ) implies that

(6.2) \begin{align} \sin ^2 \phi _\infty = \frac {\sigma u_\infty ^2}{2 h_r}, \end{align}

which allows us to write (6.1) as

(6.3) \begin{align} \frac r \sigma = 1 + \sqrt {\frac {1}{2} C_{\!f}} \left [ \frac 2 \varkappa (A_0 - E_0) + \frac {\tan \vartheta P (\phi _\infty ,\, 0)} {\sin ^2 \phi _\infty } \right ] + O(C_{\!f}). \end{align}

This equality confirms the estimate given by Van Driest (Reference Van Driest1956, Reference Van Driest1959) for the recovery factor $r = \sigma + O (\sqrt {C_{\!f}})$ , using which, based on (4.5b ), we write (6.2) in the form

(6.4) \begin{align} \sin ^2 \phi _\infty = \frac {\sigma u_\infty ^2}{2 h_\infty + r u_\infty ^2} = \sin ^2 \vartheta \left [ 1 + O \big(\sqrt {C_{\!f}}\big) \right ] \end{align}

or $\phi _\infty = \vartheta + O(\sqrt {C_{\!f}})$ , which in view of the expression (4.1d ) for the function $P(\phi _\infty ,\, 0)$ , allows us to finally write the recovery factor as

(6.5) \begin{align} \frac r \sigma = 1 + \sqrt {2 C_{\!f}} \left ( \frac {A_0 - E_0} \varkappa + A_2 + \frac {2 A_1 \vartheta } {\sin 2 \vartheta } \right ) + O(C_{\!f}). \end{align}

Thus, just like Van Driest (Reference Van Driest1956, Reference Van Driest1959), we have obtained the recovery factor for the boundary layer on a flat plate in the form of an asymptotic expansion in the small parameter – the root of the skin-friction coefficient. The leading term of the expansion is the turbulent Prandtl number in the logarithmic region $\sigma$ , and the coefficient of the small parameter, in view of the expression for $\vartheta$ (4.5b ), is a function of the Mach number and heat capacity ratio. In addition, relation (6.5) contains five more universal constants, three of which -- $\varkappa$ , $A_0$ and $E_0$ -- are known for the incompressible boundary layer, while $A_1$ and $A_2$ enter the Crocco integral for a compressible gas (Vigdorovich Reference Vigdorovich2023b ).

Since according to (4.5b ), $\vartheta \to \pi /2$ as $M_\infty \to \infty$ , the expression (6.5) has a singularity at $M_\infty = \infty$ and, hence, is valid only at moderate free-stream Mach numbers. On the other hand, the limit at $M_\infty = 0$ exists and is equal to

(6.6) \begin{align} \frac r \sigma = 1 + \sqrt {2 C_{\!f}} \left ( \frac {A_0 - E_0} \varkappa + A_1 + A_2 \right ) + O(C_{\!f}). \end{align}

This equality can be compared with the expression obtained by Rotta (Reference Rotta1964) for an incompressible fluid, $r = \sigma + B_0 \sqrt {{C_f /2}} + O(C_{\!f})$ , which differs only in the value of the coefficient $B_0 \approx 1$ . It is about half of the corresponding coefficient in (6.6).

Note that according to (6.5), the recovery factor with the accuracy of $O(C_{\!f})$ is independent of the molecular Prandtl number. Furthermore, this conclusion is supported by the expressions obtained by Van Driest (Reference Van Driest1956, Reference Van Driest1959) and Rotta (Reference Rotta1964).

6.2. Reynolds-analogy factor

By introducing the Stanton number $St = q_w /\rho _\infty u_\infty (h_w - h_r)$ and the Reynolds-analogy factor $\alpha = 2 St/C_{\!f}$ , we can write

(6.7) \begin{align} \frac {u_\infty q_w}{\tau _w} = \alpha (h_w - h_r), \end{align}

which after substitution into (5.16) gives

(6.8) \begin{align} & (1 - \sigma \alpha )(h_w - h_r) + (r - \sigma ) \frac {u_\infty ^2} 2 \nonumber \\ &\quad = \sqrt {\frac {1}{2} C_{\!f}} \left \{ \frac \sigma \varkappa \left [ u_\infty ^2 + \alpha (h_w - h_r) \right ] (A_0 - E_0) + \frac {h_w \tan \vartheta P (\phi _\infty ,\, \phi _w)}{\cos ^2 \phi _w} \right \} + O(C_{\!f}). \end{align}

The second term on the left-hand side of (6.8), in view of the expression for the recovery factor (6.5), has the form

(6.9) \begin{align} (r - \sigma ) \frac {u_\infty ^2} 2 = \sigma u_\infty ^2 \sqrt {\frac {1}{2} C_{\!f}} \left ( \frac {A_0 - E_0} \varkappa + A_2 + \frac {2 A_1 \vartheta } {\sin 2 \vartheta } \right ) + O(C_{\!f}), \end{align}

which after substitution into (6.8) yields

(6.10) \begin{align} (1 - \sigma \alpha )(h_w - h_r) &= \sqrt {\frac {1}{2} C_{\!f}} \left [ \frac {\sigma \alpha } \varkappa (h_w - h_r)(A_0 - E_0) - \sigma u_\infty ^2 \left ( A_2 + \frac {2 A_1 \vartheta } {\sin 2 \vartheta } \right ) \right . \nonumber \\ &\quad +\left . \, \frac {h_w \tan \vartheta P (\phi _\infty ,\, \phi _w)}{\cos ^2 \phi _w} \right ] + O(C_{\!f}). \end{align}

In view of the expression (4.1d ) for the function $P(\phi _\infty ,\, \phi _w)$ , the last term on the right-hand side of (6.10) is

(6.11) \begin{align} \frac {h_w \tan \vartheta P (\phi _\infty ,\, \phi _w)}{(h_w - h_r) \cos ^2 \phi _w} = \frac {h_w \tan \vartheta }{(h_w - h_r)} \left \{ \frac {2 A_1 (\phi _\infty - \phi _w)}{\cos ^2 \phi _w} \right . \nonumber \\ \left . + \frac {A_2 \sin 2 \phi _\infty }{\cos ^2 \phi _w} + 2 [D (\textit{Pr}) - C - A_2] \tan \phi _w \right \}\!. \end{align}

Equality (6.10) implies the estimate given by Van Driest (Reference Van Driest1956, Reference Van Driest1959) for the Reynolds-analogy factor $\alpha = 1/\sigma + O (\sqrt {C_{\!f}})$ , by using which along with (4.2), (4.3), (4.5b ), (4.6) and (6.7), we have with the accuracy of $O(\sqrt {C_{\!f}})$ :

(6.12) \begin{align} \frac {h_w \tan \vartheta \sin 2 \phi _\infty }{(h_w - h_r) \cos ^2 \phi _w} = 1 + \frac {\sigma u_\infty ^2}{h_w - h_r}, \quad \frac {2 h_w \tan \vartheta \tan \phi _w}{(h_w - h_r)} = \sqrt {\frac {h_w}{h_\infty }}. \end{align}

Substitution of (6.11) and (6.12) into (6.10) yields

(6.13) \begin{align} \sigma \alpha = 1 & + \sqrt {\frac {1}{2} C_{\!f}} \left [ \frac {E_0 - A_0} \varkappa - A_2 + [C + A_2 - D (\textit{Pr})] \sqrt {\frac {h_w}{h_\infty }} \right . \nonumber \\ & \left . +\, \frac {2 A_1}{h_w - h_r} \left ( \frac {\sigma u_\infty ^2 \vartheta }{\sin 2 \vartheta } - \frac {(\phi _\infty - \phi _w) h_w \tan \vartheta }{\cos ^2 \phi _w} \right ) \right ] + O(C_{\!f}). \end{align}

Replacing in (6.7) $r$ and $\alpha$ by $\sigma$ and $1/\sigma$ , respectively, we get

(6.14) \begin{align} \frac {u_\infty q_w}{\tau _w} = \left ( \frac {h_w - h_\infty } \sigma - \frac {u_\infty ^2} 2 \right ) \left [1 + O \big(\sqrt {C_{\!f}}\big) \right ]\!. \end{align}

The quantities $\phi _\infty$ and $\phi _w$ on the right-hand side of (6.13) as well as $\psi _\infty$ can be calculated to the accuracy of $O(\sqrt {C_{\!f}})$ using (4.2), (4.3) and (6.14) in terms of the free-stream parameters and wall enthalpy. Denoting the corresponding quantities by $\phi ^{\prime}_\infty$ , $\psi ^{\prime}_\infty$ and $\phi ^{\prime}_w$ , we have

(6.15a) \begin{align} \psi ^{\prime}_\infty &= \phi ^{\prime}_\infty = \arctan \left [ \frac {(2 + \sigma Ec) \tan \vartheta }{2 \sigma Ec} \right ]\!, \end{align}

(6.15b) \begin{align} \phi ^{\prime}_w &= \arctan \left [ \frac {(2 - \sigma Ec) \tan \vartheta }{2 \sigma Ec} \sqrt {\frac {h_\infty }{h_w}} \right ]\!, \quad Ec = \frac {u_\infty ^2}{h_w - h_\infty } = \frac {(\gamma - 1) M_\infty ^2}{h_w/h_\infty - 1}, \\[9pt] \nonumber \end{align}

where $Ec$ is the Eckert number characterising gas kinetic heating. Then, we finally obtain

(6.16) \begin{align} \sigma \alpha = 1 + \sqrt {\frac {1}{2} C_{\!f}} \left [ \frac {E_0 - A_0} \varkappa - A_2 + [C + A_2 - D (\textit{Pr})] \sqrt {\frac {h_w}{h_\infty }} \right . \nonumber\\ \left . + A_1 \cot \phi ^{\prime}_w \left (\frac \vartheta {\cos ^2 \vartheta } - \frac {\psi ^{\prime}_\infty - \phi ^{\prime}_w} {\cos ^2 \psi ^{\prime}_\infty } \right ) \sqrt {\frac {h_\infty }{h_w}} \right ] + O(C_{\!f}). \end{align}

Formula (6.16) specifies the Reynolds-analogy factor in the form of an asymptotic expansion in the small parameter $\sqrt {C_{\!f}}$ , the leading term of which is the reciprocal of the turbulent Prandtl number, which coincides in structure with the expression obtained by Van Driest (Reference Van Driest1956, Reference Van Driest1959). Compared with the recovery factor, the number of parameters on which the Reynolds-analogy factor depends is two more due to the enthalpy ratio $h_w/h_\infty$ and the molecular Prandtl number. The latter quantity enters (6.16) via the function $D (\textit{Pr})$ , which is known from the logarithmic law for temperature in an incompressible fluid (Vigdorovich Reference Vigdorovich2017, Reference Vigdorovich2023b ). Equality (6.16) also includes the constant $C$ from the logarithmic law for the velocity profile in an incompressible fluid (Vigdorovich Reference Vigdorovich2017, Reference Vigdorovich2023b ).

Since when $M_\infty \to \infty$ , $\vartheta \to \pi /2$ according to (4.5b ) and $\phi ^{\prime}_\infty \to \pi /2$ according to (6.15a ), the expression (6.16) has a singularity at $M_\infty = \infty$ and, therefore, like the expression (6.5) for the recovery factor, is valid only at moderate free-stream Mach numbers. In the case of incompressible flow taking in (6.16) the limit as $\vartheta \to 0$ and $h_w/h_\infty \to 1$ , we have the expression

(6.17) \begin{align} \sigma \alpha = 1 + \sqrt {\frac {1}{2} C_{\!f}} \left [ \frac {E_0 - A_0} \varkappa + C - D (\textit{Pr}) \right ] + O(C_{\!f}), \end{align}

which was obtained by Vigdorovich (Reference Vigdorovich2017) for the incompressible boundary layer.

The case of an adiabatic plate ( $h_w = h_r$ ) corresponds to the limit as $Ec \to 2/\sigma$ , for which from (6.15) we have

(6.18) \begin{align} \phi ^{\prime}_\infty = \vartheta + \frac {2 - \sigma Ec}{4 \sigma Ec} \sin 2 \vartheta + \ldots , \quad \phi ^{\prime}_w = \frac {2 - \sigma Ec} {2 \sigma Ec} \sin \vartheta + \ldots . \end{align}

Here, we take into account that $h_r/h_\infty = \sec ^2 \vartheta + O(\sqrt {C_{\!f}})$ . Substitution of (6.18) into (6.16) yields

(6.19) \begin{align} \sigma \alpha = 1 &+ \sqrt {\frac {1}{2} C_{\!f}} \left \{ \frac {E_0 \!-\! A_0} \varkappa - A_2 \!+\! [C + A_1 + A_2 - D (\textit{Pr})] \sec \vartheta - A_1 (1 + 2 \vartheta \tan \vartheta ) \right \} \nonumber \\ & +\, O(C_{\!f}). \end{align}

At $\vartheta = 0$ , this expression gives (6.17) for an incompressible fluid.

Formulas (6.5) and (6.16) allow us to calculate the wall heat flux from the known free-stream parameters, wall temperature and skin-friction coefficient. The results of calculations using the values of the constants from table 1 are presented in table 2 and yield values of $Q_+$ that are very close to those obtained by DNS (Zhang, Duan & Choudhari Reference Zhang, Duan and Choudhari2018). Moreover, as can be seen from table 2, good agreement is achieved despite the rather large free-stream Mach numbers.

Table 2.

Heat-flux values calculated with the help of formulas for the recovery and Reynolds-analogy factors (6.5) and (6.16) and the DNS data by Zhang et al. (Reference Zhang, Duan and Choudhari2018).

7.1. About the averaging methods

The proposed theory is based on Favre’s averaging (Favre Reference Favre1969), since it preserves the form of the continuity equation (2.1b ). In experiments, Reynolds’ averaging is used to determine the velocity and temperature profiles. Therefore, before proceeding to the comparison with measurement data, we evaluate how the averaging method affects the magnitude of the parameters in the considered outer region of the boundary layer.

The equation of state for a perfect gas implies the equality

(7.22) \begin{align} \frac {p'}{\overline p} = \frac {\rho '}{\overline \rho } + \frac {h''}{\tilde h} + \frac {\rho ' h''}{\overline \rho \, \tilde h}, \end{align}

and from the definitions of Favre’s and Reynolds’ averaging we have

(7.23) \begin{align} \tilde u = \overline u \left ( 1 + \frac {\overline {\rho ' u'}}{\overline \rho \, \overline u} \right )\!, \quad \tilde h = \overline h \left ( 1 + \frac {\overline {\rho ' h'}}{\overline \rho \, \overline h} \right )\!. \end{align}

We will proceed from the estimates

(7.24) \begin{align} u' \thicksim u'' \thicksim \sqrt {\widetilde {u^{\prime \prime 2}}}, \quad h' \thicksim h'' \thicksim \sqrt {\widetilde {h^{\prime \prime 2}}}. \end{align}

Relations (2.3a ) and (2.3b ) allow us to estimate the magnitude of the root-mean-square fluctuations through the gradients of the mean quantities

(7.25) \begin{align} \sqrt {\widetilde {u^{\prime \prime 2}}} \thicksim y \frac {{\partial } \tilde u}{{\partial } y}, \quad \sqrt {\widetilde {h^{\prime \prime 2}}} \thicksim y \frac {{\partial } \tilde h} {{\partial } y}, \end{align}

which after calculation of these gradients from the velocity and enthalpy defect laws (7.1) and (7.3) yields

(7.26a,b) \begin{align} \frac {u'}{u_\infty } \thicksim \sqrt {C_{\!f}}, \quad \frac {h'}{h_\infty } \thicksim M_\infty ^2 \sqrt {C_{\!f}}. \end{align}

Based on the pressure-fluctuation estimate $p' \thicksim \overline \rho u^{\prime 2}$ and relation (7.26a ), we obtain

(7.27) \begin{align} \frac {p'}{p_\infty } \thicksim M_\infty ^2 C_{\!f}. \end{align}

Hence, (7.22) and (7.26b ) give

(7.28) \begin{align} \frac {\rho '}{\overline \rho } \thicksim M_\infty ^2 \sqrt {C_{\!f}}. \end{align}

After substituting (7.26) and (7.28) into (7.23), we have in the outer region

(7.29) \begin{align} \tilde u = \overline u \left [1 + M_\infty ^2 O (C_{\!f}) \right ]\!, \quad \tilde h = \overline h \left [1 + M_\infty ^4 O (C_{\!f}) \right ]\!. \end{align}

Based on similar estimates of flow parameters in the logarithmic region, Vigdorovich (Reference Vigdorovich2023b ) obtained the relations that can be written as

(7.30) \begin{align} \tilde u = \overline u \left [1 + M_\infty ^2 O (C_{\!f}) \right ]\!, \quad \tilde h = \overline h \left [1 + M_\infty ^2 O (C_{\!f}) \right ]\!. \end{align}

Since the velocity and enthalpy defect laws (7.7) and (7.11) are formulated with an accuracy of $O(\sqrt {C_{\!f}})$ , relations (7.29) and (7.30) imply that at moderate Mach numbers, when $M_\infty = O(1)$ , these laws remain valid if Reynolds’ averaging is used instead of Favre’s averaging.

Conclusions about the small difference between the results of Favre’s and Reynolds’ averaging based on estimates partly employing empirical information are given by Cebeci & Bradshaw (Reference Cebeci and Bradshaw1984), and as a result of the analysis of the channel-flow DNS data are given by Huang et al. (Reference Huang, Coleman and Bradshaw1995).

Figure 1.

Calculated and experimental velocity and temperature profiles in subsonic and supersonic turbulent boundary layers on an adiabatic plate plotted in similarity variables (7.7) and (7.11). The description of the calculated curves is given in table 3. (c,d) Data from (Fernholz & Finley Reference Fernholz and Finley1977, CAT 7302), (e) data from (Fernholz & Finley Reference Fernholz and Finley1977, CAT 5501), (f) data from (Fernholz & Finley Reference Fernholz and Finley1977, CAT 6502), (g) data from (Fernholz, Finley & Mikulla Reference Fernholz, Finley and Mikulla1981, CAT 7801). The Reynolds number ${\textit{Re}}_\theta$ is calculated based on the free-stream parameters and momentum thickness.

7.2. Comparison with calculated and experimental data

In figure 1 the velocity and temperature profiles in the turbulent boundary layer are plotted in the similarity variables (7.7) and (7.11), respectively. The quantities (7.5) are used as transverse scales. According to (7.7) and (7.11), the velocity and temperature profiles in the outer region and logarithmic sublayer are described by universal curves and, thus, do not depend on the free-stream Mach number (if, according to the assumptions made, it remains moderate and does not reach hypersonic values), Reynolds number and wall-heat-transfer conditions. The data in figure 1, obtained at zero wall heat flux, fully confirm the conclusions of the theory. In figure 1(a) the calculated velocity profiles in the incompressible (curve 2) (Schlatter & Örlü Reference Schlatter and Örlü2010) and compressible supersonic boundary layer at $M_\infty = 2$ (curves 3–5) (Pirozzoli & Bernardini Reference Pirozzoli and Bernardini2011, Reference Pirozzoli and Bernardini2013), outside the viscous sublayer, practically, describe a single curve, which can also be specified by the Coles formula for the incompressible boundary layer (7.19) (curve 1), while the Reynolds number affects only the logarithmic section extent. The temperature profiles obtained at $M_\infty = 2$ (figure 1 b) behave in the same way; in the outer region and logarithmic sublayer, they are very close to the empirical approximation of the temperature profile in the incompressible boundary layer (7.21). Complete universality in Mach and Reynolds numbers is demonstrated by the experimental data (Fernholz & Finley Reference Fernholz and Finley1977, CAT 7302) in figures 1(c,d), where all experimental points obtained at Mach numbers from $0.8$ to $2.8$ and different Reynolds numbers, outside the viscous sublayer lie on the calculated curves corresponding to $M_\infty = 2$ . The velocity profiles shown in figures 1(e–g), of which the profile in figure 1(f), in particular, corresponds to a very high Reynolds number ${\textit{Re}}_\theta = 7.34 \boldsymbol{\cdot }10^5$ and $M_\infty = 2.67$ , demonstrate the same behaviour.

Table 3.

Description of curves in figures 1 and 2.

Figure 2.

Calculated and experimental velocity and temperature profiles in supersonic turbulent boundary layers on a plate with heat transfer plotted in similarity variables (7.7) and (7.11). The calculated curves description is given in figure 1 and table 3. (c,d) Data from (Fernholz & Finley Reference Fernholz and Finley1977, CAT 7702) at $h_w/h_r = 0.58$ , (e, f) data from (Spina & Smits Reference Spina and Smits1987) at $h_w/h_r = 1.11$ and $M_\infty = 2.9$ .

The validity of the universal velocity and temperature defect laws (7.7) and (7.11) is confirmed by computational and experimental data obtained at non-zero wall heat transfer. Curve 7 in figure 2(a), which specifies the velocity profile on a cooled plate at $Q_+ = -0.01285$ , $h_w/h_r = 0.75$ and $M_\infty = 2$ (Cogo et al. Reference Cogo, Salvadore, Picano and Bernardini2022), outside the viscous sublayer is in perfect agreement with the velocity profile in the incompressible boundary layer (Schlatter & Örlü Reference Schlatter and Örlü2010). Furthermore, the corresponding temperature profile in figure 2(b) is quite close to the supersonic profile at zero heat flux (Pirozzoli & Bernardini Reference Pirozzoli and Bernardini2013) as well as the curve given by the empirical formula for an incompressible fluid (7.21). The experimental velocity and temperature profiles in figures 2(c,d), corresponding to $h_w/h_r = 0.58$ (Fernholz & Finley Reference Fernholz and Finley1977, CAT 7702), also follow the similarity rules (7.7) and (7.11). Across a cooled boundary layer, the temperature varies non-monotonically, so the temperature profiles plotted in similarity variables (7.11) in figures 2(b,d) have a local maximum in the logarithmic region. In figures 2(e, f) the velocity and temperature profiles measured in different sections of the boundary layer on a heated plate follow universal curves close to the profiles of the corresponding quantities in a zero-heat-flux boundary layer. Since the wall heat flux was not measured in this experiment, it is determined from the free-stream parameters, wall enthalpy and skin friction using (6.5) and (6.16) for the recovery and Reynolds-analogy factors. Figures 1 and 2 show that relations (7.7) and (7.11) provide universal velocity and temperature distributions up to $M_\infty = 3$ .

The temperature profiles in figures 1 and 2 follow empirical formula (7.21) and, thus, confirm the conclusion made by Vigdorovich (Reference Vigdorovich2017) that the temperature profile constructed in similarity variables can be obtained from the velocity profile by simply stretching the transverse coordinate by a factor of $1/\sigma$ . This empirical observation is consistent with the exact equality (7.6) that relates the integral parameters of the velocity and temperature profiles, and is confirmed by the DNS results – the calculated velocity and temperature profiles given in similarity variables in figure 3 are quite close.

Figure 3.

Velocity and temperature profiles in similarity variables (7.7) and (7.11), respectively. The DNS data by Pirozzoli & Bernardini (Reference Pirozzoli and Bernardini2011) at $M_\infty = 2$ .

9.1. Adiabatic plate

In the case of a thermally insulated plate, relations (9.2), taking into account the expressions for the functions $Q_1 (\phi _\infty ,\, 0)$ and $R (\psi _\infty ,\, 0)$  (4.1e ) and (4.1f ) and the equality $D_1 (\textit{Pr},\, n,\, 0) = D (\textit{Pr})$ , become

(9.6) \begin{align} \ln {\textit{Re}}_u &= \frac {\varkappa \phi _\infty \cot \vartheta }{\sqrt {\frac {1}{2} C_{\!f}}} + \varkappa A_1 \phi _\infty \tan \phi _\infty + (\varkappa A_3 - 1) \ln \cos \phi _\infty - \varkappa C + A_0 \nonumber\\ &\quad +\, O \big(\sqrt {C_{\!f}}\big), \nonumber\\ \ln (\sigma {\textit{Re}}_h)& = \frac {\varkappa \psi _\infty \cot \vartheta }{\sqrt {\frac {1}{2} C_{\!f}}} - \varkappa A_1 \psi _\infty \cot \psi _\infty + (\varkappa A_3 - 1) \ln \cos \psi _\infty - \varkappa (A_2 + C) \nonumber\\ &\quad +\, E_0 + O \big(\sqrt {C_{\!f}}\big). \end{align}

For $h_w = h_r$ , relations (4.2) on the basis of (4.5b ) yield

(9.7) \begin{align} \phi _\infty = \arcsin \frac {\sin \vartheta }{\sqrt {1 + (r/\sigma - 1) \sin ^2 \vartheta }}, \quad \psi _\infty = \arccos \frac {\cos \vartheta }{\sqrt {1 + (r/\sigma - 1) \sin ^2 \vartheta }}. \end{align}

Substituting here the expression for the recovery factor (6.5), after expanding in the small parameter $\sqrt {C_{\!f}}$ , we find that both relations (9.6) give the same expression, i.e.

(9.8) \begin{align} \ln {\textit{Re}}_u = \ln \sigma {\textit{Re}}_h &= \frac {\varkappa \vartheta \cot \vartheta }{\sqrt {\frac {1}{2} C_{\!f}}} + (\varkappa A_3 - 1) \ln \cos \vartheta \nonumber\\ &\quad +\, A_0 \cos ^2 \vartheta + (E_0 - \varkappa A_2) \sin ^2 \vartheta - \varkappa C + O \big(\sqrt {C_{\!f}}\big), \end{align}

while (9.5) is written as

(9.9) \begin{align} \ln {\textit{Re}}_x &= \frac {\varkappa \vartheta \cot \vartheta }{\sqrt {\frac {1}{2} C_{\!f}}} - \ln \frac {C_{\!f}} 2 + (\varkappa A_3 - 1) \ln \cos \vartheta \nonumber\\ &\quad + A_0 \cos ^2 \vartheta + (E_0 - \varkappa A_2) \sin ^2 \vartheta - \varkappa C + O \big(\sqrt {C_{\!f}}\big). \end{align}

For incompressible flow in the limit as $M_\infty \to 0$ ( $\vartheta \to 0$ ), relations (9.8) and (9.9) become

(9.10) \begin{align} \ln {\textit{Re}}_{\delta ^*} &= \frac \varkappa {\sqrt {\frac {1}{2} C_{\!f}}} - \varkappa C + A_0 + O \big(\sqrt {C_{\!f}}\big),\nonumber \\ \ln {\textit{Re}}_x &= \frac \varkappa {\sqrt {\frac {1}{2} C_{\!f}}} - \ln \frac {C_{\!f}} 2 - \varkappa C + A_0 + O \big(\sqrt {C_{\!f}}\big), \end{align}

where ${\textit{Re}}_{\delta ^*}$ is the Reynolds number based on the displacement thickness. These formulas for incompressible flow are given by Vigdorovich (Reference Vigdorovich2015).

Thus, when $M_\infty$ ranges from zero (incompressible flow) to moderate supersonic values, the skin-friction coefficient on an adiabatic wall is a known function of the Reynolds number ${\textit{Re}}_u$ , ${\textit{Re}}_h$ or ${\textit{Re}}_x$ , free-stream Mach number and heat capacity ratio (see (4.5b ) for the quantity $\vartheta$ ), as well as seven universal constants, five of which, i.e. $\varkappa$ , $C$ , $\sigma$ , $A_0$ and $E_0$ , are known for incompressible flow, and the two $A_2$ and $A_3$ are included in the laws of the wall for velocity and temperature for a compressible gas (Vigdorovich Reference Vigdorovich2024). Here, it should be emphasised again that according to (9.8) and (9.9), the skin-friction coefficient on an adiabatic plate does not depend on the molecular Prandtl number.

Expressing the skin-friction coefficient from equality (9.8), we obtain

(9.11) \begin{align} \sqrt {\frac {1}{2} C_{\!f}} = \frac {\varkappa \vartheta \cot \vartheta }{\ln {\textit{Re}}_u + (1 - \varkappa A_3) \ln \cos \vartheta - A_0 \cos ^2 \vartheta + (\varkappa A_2 - E_0) \sin ^2 \vartheta + \varkappa C}. \end{align}

Figure 5.

Skin-friction law for compressible turbulent boundary layers on an adiabatic plate. The description of the experimental and calculated points is given in table 4. The bisector of the right angle corresponds to (9.11), while $X_1$ is the right-hand side of equality (9.11).

In figure 5, this formula is compared with a large body of experimental data collected by Fernholz & Finley (Reference Fernholz and Finley1977), Fernholz et al. (Reference Fernholz, Finley and Mikulla1981) and the DNS data of Pirozzoli & Bernardini (Reference Pirozzoli and Bernardini2011, Reference Pirozzoli and Bernardini2013). As can be seen in table 4, which shows the range of Mach and Reynolds numbers in which the experiments were performed, it describes well the measurement results up to $M_\infty = 5$ .

Table 4.

Description of experimental and calculated points in figure 5.

9.2. Heat transfer at the wall

In the case of non-zero wall heat flux, the friction law in the form (9.2) contains one empirical function $D_1 (\textit{Pr},\, n,\, Q_+)$ , which depends on the dimensionless heat flux $Q_+$ , molecular Prandtl number and viscosity law (Vigdorovich Reference Vigdorovich2024). According to (4.2) and (4.4), the quantities $\phi _w$ , $\phi _\infty$ , $\psi _\infty$ and $Q_+$ included in (9.2) are calculated based on the wall-heat-flux value $q_w$ . If this quantity is unknown or is not determined accurately enough, which is quite typical of available experimental data (Fernholz & Finley Reference Fernholz and Finley1977; Fernholz et al. Reference Fernholz, Finley and Mikulla1981; Spina & Smits Reference Spina and Smits1987), it can be determined from the known free-stream parameters, skin friction and wall enthalpy using the formulas for the recovery and Reynolds-analogy factors (6.5) and (6.16).

In view of (4.5b ) and (6.7), (4.3) can be written as

(9.12) \begin{align} \tan \phi _w = \alpha \tan \vartheta \frac {2 - r Ec}{2 Ec} \sqrt {\frac {h_\infty }{h_w}}. \end{align}

Substituting here the expressions for the recovery and Reynolds-analogy factors (6.5), (6.16) and expanding in the small parameter $\sqrt {C_{\!f}}$ , we have

(9.13) \begin{align} \tan \phi _w &= \tan \phi ^{\prime}_w - N \sqrt {\frac {1}{2} C_{\!f}} + O(C_{\!f}), \\N &= \frac 12 \sec \psi ^{\prime}_\infty \sec \phi ^{\prime}_w \left [ P(\psi ^{\prime}_\infty ,\,\phi ^{\prime}_w) + \frac 1 \varkappa (A_0 - E_0) \sin 2 \psi ^{\prime}_\infty \right ]\!. \nonumber \end{align}

Here, we have taken into account the expression (6.15a ) and the representations (4.2). From (9.13), we obtain

(9.14) \begin{align} \phi _w = \phi ^{\prime}_w - \cos ^2 \phi ^{\prime}_w N \sqrt {\frac {1}{2} C_{\!f}} + O(C_{\!f}). \end{align}

Relations (4.2b ) and (9.14) imply that

(9.15) \begin{align} \psi _\infty = \psi ^{\prime}_\infty - \frac 12 \sin 2 \phi ^{\prime}_w \cot \psi ^{\prime}_\infty N \sqrt {\frac {1}{2} C_{\!f}} + O(C_{\!f}). \end{align}

Equalities (9.14) and (9.15) yield

(9.16) \begin{align} \frac {\psi _\infty - \phi _w} {\sqrt {\frac {1}{2} C_{\!f}}} = \frac {\psi ^{\prime}_\infty - \phi ^{\prime}_w}{\sqrt {\frac {1}{2} C_{\!f}}} + \frac {\cos \phi ^{\prime}_w \sin \big(\psi ^{\prime}_\infty - \phi ^{\prime}_w\big) N}{\sin \psi ^{\prime}_\infty } + O\big(\sqrt {C_{\!f}}\big). \end{align}

Substituting this relation into (9.2b ), we obtain the skin-friction law, which is valid at moderate free-stream Mach numbers over the entire range of wall heat flux,

(9.17) \begin{align} \varkappa D_1 \big(\textit{Pr},\, n,\, Q_{+}^{\prime }\big) = &\, \frac {\varkappa \big(\psi ^{\prime}_\infty - \phi ^{\prime}_w\big)}{\tan \vartheta \sqrt {\frac {1}{2} C_{\!f}}} - \ln {\textit{Re}}_u + \frac {\varkappa A_1 \big(\psi ^{\prime}_\infty - \phi ^{\prime}_w\big)}{\tan \vartheta } \left ( \sqrt {\frac {h_w}{h_\infty }} - 1 \right ) \nonumber\\ & +\, \left \{ A_0 - E_0 + \varkappa A_2 + \varkappa [D (\textit{Pr}) - C - A_2] \sqrt {\frac {h_w}{h_\infty }} \right \} \frac {\sin \big(\psi ^{\prime}_\infty - \phi ^{\prime}_w\big)}{\tan \vartheta }\nonumber \\ & +\, \frac 12 (1 - \varkappa A_3) \ln \frac {h_w}{h_\infty } + E_0 + O(\sqrt {C_{\!f}}), \end{align}

where in view of (9.1), ${\textit{Re}}_u$ can be replaced by $\sigma {\textit{Re}}_h$ . According to this equality, its right-hand side, which can be calculated from the Reynolds number, skin-friction coefficient, wall enthalpy and free-stream parameters, for fixed $\textit{Pr}$ and $n$ , is a function of only the quantity

(9.18) \begin{align} Q_{+}^{\prime } = \frac 2 \sigma \tan \phi ^{\prime}_w \tan \vartheta \sqrt {\frac {1}{2} C_{\!f}}, \end{align}

which based on (4.3) and (4.4) represents the quantity $Q_+$ with the accuracy $O (\sqrt {C_{\!f}})$ and may also be calculated in terms of the skin-friction coefficient, wall enthalpy and free-stream parameters. In other words, the skin-friction distribution can be described by a single universal curve that is specified by the function $D_1 (\textit{Pr},\, n,\, Q_{+}^{\prime })$ .

Since in all available experiments and calculations $Q_{+}^{\prime }$ is a sufficiently small quantity (see table 5), after expanding in the series

(9.19) \begin{align} D_1 (\textit{Pr},\, n,\, Q_{+}^{\prime }) = D (\textit{Pr}) - d (\textit{Pr},\, n) Q_{+}^{\prime } + O(Q_{+}^{\prime 2}), \quad d (\textit{Pr},\, n) = - \frac {{\partial } D_1 (\textit{Pr},\, n,\, 0)}{{\partial } Q_+} \end{align}

the left-hand side of (9.17) taking into account (9.18) can be written as

(9.20) \begin{align} \varkappa D (\textit{Pr}) - \frac {2 \varkappa } \sigma d (\textit{Pr},\, n) \tan \phi ^{\prime}_w \tan \vartheta \sqrt {\frac {1}{2} C_{\!f}}. \end{align}

This transforms (9.17) into an equation for the skin-friction coefficient, which contains one unknown parameter $d (\textit{Pr},\, n)$ , the value of the derivative of the function $D_1 (\textit{Pr},\, n,\, Q_+)$ at $Q_+ = 0$ . As a result of processing the velocity profiles in the logarithmic region of boundary layers based on the law of the wall (4.1b ), Vigdorovich (Reference Vigdorovich2024) obtained the estimate for this quantity $d (\textit{Pr},\, n) \approx 22$ , which will be used further.

Table 5.

Description of experimental and calculated points in figure 6.

Figure 6.

Skin-friction law for supersonic turbulent boundary layers on a plate with heat transfer. The description of the experimental and calculated points is given in table 5. The bisector of the right angle corresponds to (9.21), while $X_2$ is the right-hand side of equality (9.21).

The solution of (9.17) with the right-hand side (9.20) is

(9.21) \begin{align} &\qquad\sqrt {\frac {1}{2} C_{\!f}} = \frac {2 a}{\sqrt {c^2 + 4 a b} + c}, \end{align}

(9.22) \begin{align} &\qquad\qquad a = \varkappa \big(\psi ^{\prime}_\infty - \phi ^{\prime}_w\big), \quad b = - \frac {2 \varkappa } \sigma d (\textit{Pr},\, n) \tan \phi ^{\prime}_w \tan ^2 \vartheta , \end{align}

(9.23) \begin{align} c &= \tan \vartheta \left [ \ln {\textit{Re}}_u - \frac 12 (1 - \varkappa A_3) \ln \frac {h_w}{h_\infty } + \varkappa D (\textit{Pr}) - E_0 \right ] \nonumber \\ &\quad - \varkappa A_1 \big(\psi ^{\prime}_\infty - \phi ^{\prime}_w\big) \left ( \sqrt {\frac {h_w}{h_\infty }} - 1 \right )\nonumber \\ &\quad - \left \{ A_0 - E_0 + \varkappa A_2 + \varkappa [D (\textit{Pr}) - C - A_2] \sqrt {\frac {h_w}{h_\infty }} \right \} \sin \big(\psi ^{\prime}_\infty - \phi ^{\prime}_w\big). \end{align}

In figure 6 this formula is compared with available experimental and DNS data for the boundary layer on a cooled and heated plate. For the derivative $d (\textit{Pr},\, n)$ , the above value is adopted, also given in table 1. It can be seen that (9.21) generally describes the experimental and calculated data well. The maximum inaccuracy is observed at the largest heat-flux values (see table 5), which is natural given the assumption under which this formula is obtained from exact relation (9.17) where $D_1 (\textit{Pr},\, n,\, Q_+)$ is approximated by a linear function. Although even at the heat-flux value $Q_{+}^{\prime } = -0.1$ , for which $h_w/h_r = 0.24$ , the discrepancy does not exceed $10\, \%$ . The agreement between the theory and available experimental and calculated data is observed up to $M_\infty = 6$ .