Similarity Solutions of a Class of Unsteady Compressible Laminar Boundary Layers

1. Introduction

Since Prandtl’s 1904 boundary-layer concept[1] and the Blasius solution [2], the search for exact solutions of the Navier–Stokes equations has produced a remarkably short list. In his surveys Wang [4,5] counts fewer than 150 genuinely distinct exact solutions after nearly two centuries, most of them steady, and very few of them belonging to boundary-layer flows. For the two-dimensional unsteady boundary layer the situation was, until recently, worse still: since Stokes[6] and Rayleigh[7] posed the impulsive plate problems, only asymptotic and numerical treatments were available [8,9,10,11,12,13,14], and Stewartson’s conjecture[8] that a singularity must be traversed on the way from the Rayleigh solution to the Blasius solution remained open. Sun[15] recently proposed a similarity transformation that reduces the two-dimensional unsteady incompressible boundary-layer equations to a single partial differential equation in two variables whose coefficients do not depend on the streamwise coordinate, and used it to construct closed-form solutions in terms of Kummer functions.

All of this work is incompressible. Yet the flows in which unsteady laminar boundary layers matter most—impulsively started high-speed flows, shock-tube starting flows, hypersonic vehicle transients, and the receptivity phase of high-speed transition[18,19,20]—are strongly compressible. In compressible flow the density is an active variable: it couples the momentum equation to the energy equation through the equation of state, the viscosity varies by an order of magnitude across the layer, and viscous dissipation generates a temperature field that is itself a diagnostic of the coherent structure of the flow. The classical steady compressible boundary layer is understood through the Howarth–Dorodnitsyn [21,22] and Illingworth–Stewartson [23,24] transformations, the Chapman–Rubesin viscosity law[25], the Crocco energy integral [26,27], and the compressible Falkner–Skan–Dorodnitsyn system[24,29,30]. What is missing is an unsteady analogue: a transformation that removes the streamwise coordinate from the unsteady compressible problem, that reduces to the classical steady results in the appropriate limit, and that is explicit enough to expose the analytical building blocks of the solution.

The purpose of this paper is to construct that transformation and to work out its consequences. Our starting point is the observation that the Howarth–Dorodnitsyn variable can be generalised to unsteady flow provided the transformed normal velocity is defined to include the time derivative of the Dorodnitsyn variable itself; with that definition the transformed continuity equation becomes exactly the incompressible one, so that a transformed stream function exists even though the physical flow has $\partial \rho /\partial t\ne 0$. On top of this we apply Sun’s two-fold similarity ansatz: an undetermined thickness $\delta \left(x\right)$ used as the normal length scale, and the associated diffusion time ${\delta }^2/\left(C{\nu }_r\right)$ used as the time scale. The result (Section 3) is the pair of equations

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill f_{,\eta \eta \eta }+& af{f}_{,\eta \eta }+b\Theta \left(g-f_{,\eta }^2\right)\hfill \\ \hfill & =f_{,\eta \tau }+c\tau \left(f_{,\tau }{f}_{,\eta \eta }-f_{,\eta }{f}_{,\eta \tau }\right),\hfill \\ \hfill \frac{1}{\mathrm{Pr}}{g}_{,\eta \eta }& +af{g}_{,\eta }+\left(1-\frac{1}{\mathrm{Pr}}\right)E{\left(f_{,\eta }{f}_{,\eta \eta }\right)}_{,\eta }\hfill \\ \hfill & =g_{,\tau }+c\tau \left(f_{,\tau }{g}_{,\eta }-f_{,\eta }{g}_{,\tau }\right),\hfill \end{array}\end{array}$$

in which a, b, c are the constants of the incompressible theory and the only new ingredients are the reduced total enthalpy g, the compressibility parameter $\Theta =1+\frac{1}{2}(\gamma -1)M_e^2$ and the dissipation parameter E. Setting $\Theta =1$, $g\equiv 1$ recovers Sun’s equation exactly.

Two things should be said at the outset about what is and is not new here. The unsteady Dorodnitsyn reduction just described is not ours: it is Section 7 of Stewartson’s 1951 paper[8], whose Eqs. (7.9)–(7.19) contain the transformed normal velocity, the transformed momentum and energy equations, and—under $\mu \propto T$, $\mathrm{Pr}=1$ and an insulated wall—the Crocco integral, the velocity field $u=Uerf(Y/2\sqrt{{\nu }_{0}t})$, the mapping back to physical coordinates and the induced outflow. Everything in Secs. Section 5.4 and Section 5.7 below is his, and is attributed to him there. What this paper adds is what can be built on that reduction once Sun’s similarity ansatz is layered on top: the reduction to $(\eta ,\tau )$ with x-independent coefficients for the whole class $U=C{x}^m$, whereas Ref. [8] treats only U constant; the pressure-gradient coupling $b\Theta (g-f_{,\eta }^2)$, which vanishes identically in his case and which is what makes the momentum and enthalpy equations interact; arbitrary Prandtl number, arbitrary wall thermal condition and a general Chapman–Rubesin parameter, in place of his $\mathrm{Pr}=1$, insulated wall and $\mu \propto T$; the Kummer branch, which lives in the region $Ut>x$ where he states that the solution is not known; and the global solution of the unsteady problem with pressure gradient in Section 6. A second point of placement: the diffusive branch of Section 5, although it reproduces Stewartson’s velocity field, is shown here to belong to the same two-parameter similarity family as the Kummer branch, so that the two are reductions of one equation rather than solutions of unrelated problems.

The paper is organised as follows. Section 2 formulates the two-dimensional unsteady compressible boundary layer and derives the generalised unsteady Dorodnitsyn transformation. Section 3 introduces the similarity transformation, derives the master system, and determines the class of external velocity distributions $U\left(x\right)=C{x}^m$ for which the coefficients are constants. Section 4 shows that the steady limit of the master system is the compressible Falkner–Skan–Dorodnitsyn system and validates the formulation numerically. Section 5 treats the impulsively started semi-infinite plate: two exact solutions of the momentum equation of distinct similarity type, the Crocco integral at $\mathrm{Pr}=1$, the quadrature solution and closed-form recovery factor for general $\mathrm{Pr}$, and the wall shear and heat transfer. Section 6 solves the unsteady system for $m\ne 0$, where the compressibility coupling is active. Section 7 concludes. Appendices collect the Kummer functions, the auxiliary functions of the closed-form solution, the proof of Lemma 1, and the derivation of the recovery factor.

2. The Unsteady Compressible Boundary Layer

2.1. Governing Equations

A thin plate is immersed at zero incidence in a stream of a perfect gas (Figure 1). The origin is at the leading edge, x is measured downstream and y normal to the wall. Outside the layer the flow is inviscid and steady with velocity $U\left(x\right)$, temperature $T_e\left(x\right)$, density ${\rho }_e\left(x\right)$ and Mach number $M_e\left(x\right)=U/\sqrt{\gamma R{T}_e}$. With $\mu$ the dynamic viscosity, k the thermal conductivity, $c_p$ the specific heat at constant pressure and $h=c_{p}T$ the static enthalpy, the compressible boundary-layer equations are

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial \rho }{\partial t}}+{\displaystyle \frac{\partial \left(\rho u\right)}{\partial x}}+{\displaystyle \frac{\partial \left(\rho v\right)}{\partial y}}& =0,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \rho \left({\displaystyle \frac{\partial u}{\partial t}}+u{\displaystyle \frac{\partial u}{\partial x}}+v{\displaystyle \frac{\partial u}{\partial y}}\right)& =-{\displaystyle \frac{dp}{dx}}+{\displaystyle \frac{\partial }{\partial y}}\left(\mu {\displaystyle \frac{\partial u}{\partial y}}\right),\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial p}{\partial y}}& =0,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \rho \left({\displaystyle \frac{\partial h}{\partial t}}+u{\displaystyle \frac{\partial h}{\partial x}}+v{\displaystyle \frac{\partial h}{\partial y}}\right)& ={\displaystyle \frac{\partial }{\partial y}}\left({\displaystyle \frac{\mu }{\mathrm{Pr}}}{\displaystyle \frac{\partial h}{\partial y}}\right)+\mu {\left({\displaystyle \frac{\partial u}{\partial y}}\right)}^2+u{\displaystyle \frac{dp}{dx}},\hfill \end{array}$$

(4)

closed by $p=\rho RT$ and $\mathrm{Pr}=\mu c_p/k$. Because the external flow is steady, $\partial p/\partial t=0$ and the Bernoulli relation of the outer flow gives

$$-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{dp}{dx}}={\displaystyle \frac{{\rho }_e}{\rho }}U{\displaystyle \frac{dU}{dx}}={\displaystyle \frac{T}{T_e}}U{\displaystyle \frac{dU}{dx}},$$

(5)

where the second equality uses $p=p_e\left(x\right)$ across the layer. Equation (5) already displays the essential difference from incompressible flow: the pressure gradient acts on a fluid element in proportion to its local temperature, so that the momentum equation cannot be solved without the energy equation.

It is convenient to introduce the total enthalpy

$$H=h+\frac{1}{2}{u}^2,$$

(6)

in terms of which () becomes, using (),

$$\rho \left({\displaystyle \frac{\partial H}{\partial t}}+u{\displaystyle \frac{\partial H}{\partial x}}+v{\displaystyle \frac{\partial H}{\partial y}}\right)={\displaystyle \frac{\partial }{\partial y}}\left[{\displaystyle \frac{\mu }{\mathrm{Pr}}}{\displaystyle \frac{\partial H}{\partial y}}+\mu \left(1-{\displaystyle \frac{1}{\mathrm{Pr}}}\right)u{\displaystyle \frac{\partial u}{\partial y}}\right].$$

(7)

Two constitutive simplifications, standard since Refs. [25] and [26], are adopted throughout:

$$\rho \mu =C{\rho }_r{\mu }_r=\mathrm{const},\mathrm{Pr}=\mathrm{const},$$

(8)

where ${\rho }_r,{\mu }_r$ are constant reference values and C is the Chapman–Rubesin parameter. Equation (8) is exact for a linear viscosity–temperature law and is an excellent approximation over the range of temperatures encountered in most laminar layers; it is the price paid for analytical progress, and it is the same price paid by the classical steady theory.

2.2. An Unsteady Howarth–Dorodnitsyn Transformation

Define the Dorodnitsyn variable

$$\overline{\eta }(x,y,t)={\int }_0^y{\displaystyle \frac{\rho (x,y^{\prime },t)}{{\rho }_r}}d{y}^{\prime },$$

(9)

and map $(x,y,t)\mapsto (\xi ,\overline{\eta },\varsigma )$ with $\xi =x$, $\varsigma =t$. In steady flow (9) is the classical Howarth–Dorodnitsyn variable[21,22]; in unsteady flow $\overline{\eta }$ depends on t through $\rho$, and the naive definition of a stream function fails because $\partial \rho /\partial t\ne 0$. The remedy is to absorb ${\overline{\eta }}_{,t}$ into the transformed normal velocity.

Lemma 1

(Stewartson’s unsteady Dorodnitsyn reduction). Let

$$V\equiv {\displaystyle \frac{\partial \overline{\eta }}{\partial t}}{|}_{x,y}+u{\displaystyle \frac{\partial \overline{\eta }}{\partial x}}{|}_y+{\displaystyle \frac{\rho }{{\rho }_r}}v.$$

(10)

Then the material derivative and the continuity equation take the incompressible forms

$${\displaystyle \frac{D}{Dt}}={\displaystyle \frac{\partial }{\partial \varsigma }}+u{\displaystyle \frac{\partial }{\partial \xi }}+V{\displaystyle \frac{\partial }{\partial \overline{\eta }}},{\displaystyle \frac{\partial u}{\partial \xi }}+{\displaystyle \frac{\partial V}{\partial \overline{\eta }}}=0.$$

(11)

This result is due to Stewartson[8], who obtained it in Section 7 of his 1951 paper (his Eqs. (7.9)–(7.14)); we restate it here because everything below depends on it, and give the short proof in Appendix A for completeness. Stewartson’s route is to introduce two potentials $\psi$ and Y with $\rho u={\rho }_0{\psi }_{,y}$, $\rho ={\rho }_0{Y}_{,y}$ and $\rho v=-{\rho }_0{\psi }_{,x}{{|}_Y-{\rho }_0{Y}_{,t}|}_{x,y}$, which is the two-dimensional antisymmetric potential representation of the space–time mass flux; eliminating $\psi$ recovers (10). The consequence is decisive: although the physical flow is compressible and unsteady, the transformed flow is solenoidal in the $(\xi ,\overline{\eta })$ plane, so a transformed stream function $\Psi (\xi ,\overline{\eta },\varsigma )$ exists with

$$u={\displaystyle \frac{\partial \Psi }{\partial \overline{\eta }}},V=-{\displaystyle \frac{\partial \Psi }{\partial \xi }}.$$

(12)

Applying (11), (5) and (8) to () and (7), and noting that ${\partial }_y=(\rho /{\rho }_r){\partial }_{\overline{\eta }}$ implies ${\partial }_y\left(\mu {\partial }_{y}u\right)=({\rho }^2/{\rho }_r^2){\partial }_{\overline{\eta }}\left(\mu \rho {\rho }_r^{-1}{\partial }_{\overline{\eta }}u\right)·({\rho }_r/\rho )=\rho C{\nu }_r{\partial }_{\overline{\eta }}^{2}u$ with ${\nu }_r={\mu }_r/{\rho }_r$, we obtain the transformed compressible boundary-layer equations

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial u}{\partial \varsigma }}+u{\displaystyle \frac{\partial u}{\partial \xi }}+V{\displaystyle \frac{\partial u}{\partial \overline{\eta }}}& ={\displaystyle \frac{T}{T_e}}U{\displaystyle \frac{dU}{d\xi }}+C{\nu }_r{\displaystyle \frac{{\partial }^{2}u}{\partial {\overline{\eta }}^2}},\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial H}{\partial \varsigma }}+u{\displaystyle \frac{\partial H}{\partial \xi }}+V{\displaystyle \frac{\partial H}{\partial \overline{\eta }}}& =C{\nu }_r{\displaystyle \frac{\partial }{\partial \overline{\eta }}}\left[{\displaystyle \frac{1}{\mathrm{Pr}}}{\displaystyle \frac{\partial H}{\partial \overline{\eta }}}+\left(1-{\displaystyle \frac{1}{\mathrm{Pr}}}\right)u{\displaystyle \frac{\partial u}{\partial \overline{\eta }}}\right].\hfill \end{array}$$

(14)

In terms of $\Psi$ these read

$$\begin{array}{cc}\hfill {\Psi }_{,\overline{\eta }\varsigma }+{\Psi }_{,\overline{\eta }}{\Psi }_{,\xi \overline{\eta }}-{\Psi }_{,\xi }{\Psi }_{,\overline{\eta }\overline{\eta }}& ={\displaystyle \frac{T}{T_e}}U{\displaystyle \frac{dU}{d\xi }}+C{\nu }_r{\Psi }_{,\overline{\eta }\overline{\eta }\overline{\eta }},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill H_{,\varsigma }+{\Psi }_{,\overline{\eta }}{H}_{,\xi }-{\Psi }_{,\xi }{H}_{,\overline{\eta }}& =C{\nu }_r{\left[{\displaystyle \frac{H_{,\overline{\eta }}}{\mathrm{Pr}}}+\left(1-{\displaystyle \frac{1}{\mathrm{Pr}}}\right){\Psi }_{,\overline{\eta }}{\Psi }_{,\overline{\eta }\overline{\eta }}\right]}_{,\overline{\eta }}.\hfill \end{array}$$

(16)

Equations (15)–() are the compressible counterpart of Eq. (7) of Ref. [15], and are equivalent to Stewartson’s Eqs. (7.15), (7.18) and (7.19)[8]. They differ from the incompressible system in exactly two places: the kinematic viscosity is replaced by the effective viscosity $C{\nu }_r$, and the pressure-gradient term carries the factor $T/T_e={\rho }_e/\rho$. Everything that follows exploits this structural near-identity.

3. The Similarity Transformation

3.1. Choice of Scales

The physical problem supplies no characteristic length in x and none in y; the only candidate for a normal scale is the boundary-layer thickness $\delta \left(x\right)$ measured in the Dorodnitsyn plane, which is left undetermined at this stage and fixed later by the requirement that the transformed coefficients be constants. The time scale follows from dimensional analysis. The relevant parameters and their dimensions are listed in Table 1. Two dimensions (L, T) and four parameters give, by the Buckingham theorem[16,17], two dimensionless groups,

$${\Pi }_1={\displaystyle \frac{\overline{\eta }}{\delta }}\equiv \eta ,{\Pi }_2={\displaystyle \frac{C{\nu }_{r}t}{{\delta }^2}}\equiv \tau .$$

(17)

The group $\tau$ is the ratio of the elapsed time to the time ${\delta }^2/\left(C{\nu }_r\right)$ required for vorticity generated at the wall to diffuse across the layer—Stuart’s diffusion time[31], here built on the effective diffusivity $C{\nu }_r$ appropriate to a compressible layer.

Accordingly we set

$$\Psi =U\left(\xi \right)\delta \left(\xi \right)f(\eta ,\tau ),\eta ={\displaystyle \frac{\overline{\eta }}{\delta \left(\xi \right)}},\tau ={\displaystyle \frac{C{\nu }_r}{{\delta }^2\left(\xi \right)}}t,$$

(18)

and, for the thermal field,

$$H=H_{r}g(\eta ,\tau ),$$

(19)

with $H_r$ a constant reference total enthalpy and $T_r\equiv H_r/c_p$. Two dimensionless compressibility parameters appear:

$$\Theta \equiv {\displaystyle \frac{T_r}{T_e}},E\equiv {\displaystyle \frac{U_{\mathrm{ref}}^2}{H_r}},$$

(20)

where $U_{\mathrm{ref}}$ is the velocity scale used in (18). For a flow with an external stream it is natural to take $H_r=H_e=c_p{T}_e+\frac{1}{2}{U}^2$ and $U_{\mathrm{ref}}=U$, whence

$$\Theta =1+{\displaystyle \frac{\gamma -1}{2}}{M}_e^2,E={\displaystyle \frac{2(\Theta -1)}{\Theta }}.$$

(21)

For the impulsively started plate of Section 5, where the gas at infinity is at rest, the natural choice is $H_r=c_p{T}_{\infty }$, $U_{\mathrm{ref}}=U_0$, giving $\Theta =1$ in (20) but $E=(\gamma -1)M_w^2$; both cases are covered by the formulae below.

Since $H=c_{p}T+\frac{1}{2}{u}^2$ and $u=U_{\mathrm{ref}}{f}_{,\eta }$, the temperature and density follow algebraically from f and g:

$${\displaystyle \frac{T}{T_r}}=g-{\displaystyle \frac{E}{2}}{f}_{,\eta }^2,{\displaystyle \frac{\rho }{{\rho }_r}}={\displaystyle \frac{T_r}{T}}{\displaystyle \frac{p}{p_r}},$$

(22)

the second relation reducing to $\rho /{\rho }_r=T_r/T$ at constant pressure. In particular, with the stream normalisation (21),

$${\displaystyle \frac{T}{T_e}}=\Theta g-(\Theta -1)f_{,\eta }^2.$$

(23)

3.2. The Master System

Denote $f_{,x}=\partial f/\partial x$ etc. From (18), ${\eta }_{,\xi }=-\eta {\delta }^{-1}{\delta }_{,\xi }$ and ${\tau }_{,\xi }=-2\tau {\delta }^{-1}{\delta }_{,\xi }$, so that the chain rule gives

$$\begin{array}{cc}\hfill {\Psi }_{,\xi }& ={\left(U\delta \right)}_{,\xi }f-U\eta {\delta }_{,\xi }{f}_{,\eta }-2U\tau {\delta }_{,\xi }{f}_{,\tau },\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill {\Psi }_{,\overline{\eta }}& =U{f}_{,\eta },{\Psi }_{,\overline{\eta }\overline{\eta }}=U{\delta }^{-1}{f}_{,\eta \eta },\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill {\Psi }_{,\overline{\eta }\overline{\eta }\overline{\eta }}& =U{\delta }^{-2}{f}_{,\eta \eta \eta },{\Psi }_{,\overline{\eta }\varsigma }=U{\delta }^{-2}C{\nu }_r{f}_{,\eta \tau },\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill {\Psi }_{,\xi \overline{\eta }}& =U_{,\xi }{f}_{,\eta }-\eta U{\delta }^{-1}{\delta }_{,\xi }{f}_{,\eta \eta }-2U\tau {\delta }^{-1}{\delta }_{,\xi }{f}_{,\eta \tau }.\hfill \end{array}$$

(27)

The velocity components are therefore

$$\begin{array}{cc}\hfill u& =U{f}_{,\eta },\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill v& =-{\displaystyle \frac{{\rho }_r}{\rho }}\left[{\left(U\delta \right)}_{,\xi }f-U\eta {\delta }_{,\xi }{f}_{,\eta }-2U\tau {\delta }_{,\xi }{f}_{,\tau }\right]-{\displaystyle \frac{{\rho }_r}{\rho }}\left({\overline{\eta }}_{,t}+u{\overline{\eta }}_{,x}\right).\hfill \end{array}$$

(29)

Substituting (24)–() into (15) and (), and using (22) together with the identity $\Theta \left(g-\frac{E}{2}{f}_{,\eta }^2\right)-f_{,\eta }^2=\Theta \left(g-f_{,\eta }^2\right)$, all explicit dependence on $\xi$ disappears provided the three groups

$$a={\displaystyle \frac{\delta }{C{\nu }_r}}{\displaystyle \frac{d\left(U\delta \right)}{d\xi }},b={\displaystyle \frac{{\delta }^2}{C{\nu }_r}}{\displaystyle \frac{dU}{d\xi }},c={\displaystyle \frac{U}{C{\nu }_r}}{\displaystyle \frac{d{\delta }^2}{d\xi }}$$

(30)

are constants. The result is the master system of this paper:

$$\begin{array}{cc}\hfill f_{,\eta \eta \eta }+af{f}_{,\eta \eta }+b\Theta \left(g-f_{,\eta }^2\right)& =f_{,\eta \tau }+c\tau \left(f_{,\tau }{f}_{,\eta \eta }-f_{,\eta }{f}_{,\eta \tau }\right),\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{\mathrm{Pr}}}{g}_{,\eta \eta }+af{g}_{,\eta }+\left(1-{\displaystyle \frac{1}{\mathrm{Pr}}}\right)E{\left(f_{,\eta }{f}_{,\eta \eta }\right)}_{,\eta }& =g_{,\tau }+c\tau \left(f_{,\tau }{g}_{,\eta }-f_{,\eta }{g}_{,\tau }\right).\hfill \end{array}$$

(32)

Three remarks are in order.

(i) Structure. Equation (31) is Eq. (19) of Ref. [15] with the single replacement $1\to \Theta g$ in the pressure-gradient bracket; setting $\Theta =1$, $g\equiv 1$ recovers the incompressible equation identically. All of the compressibility of the momentum problem is carried by the product $\Theta g$, i.e. by the local static temperature. Equation () has exactly the same differential operator as (31) acting on g instead of $f_{,\eta }$, with $1/\mathrm{Pr}$ replacing unity in the diffusive term and a dissipation source proportional to $(1-1/\mathrm{Pr})E$. This parallelism is the structural reason for the Crocco integral found in Section 5.4.

(ii) The class of flows. From (30), $d\left(U{\delta }^2\right)/d\xi =2\delta {\left(U\delta \right)}_{,\xi }-{\delta }^2{U}_{,\xi }=C{\nu }_r(2a-b)$, so that

$$U\left(x\right)=C{x}^m,\delta \left(x\right)={\left[{\displaystyle \frac{C{\nu }_r(2a-b)x}{U\left(x\right)}}\right]}^{1/2},$$

(33)

with

$$m={\displaystyle \frac{b}{2a-b}},c=2(a-b),$$

(34)

exactly as in the incompressible theory. The compressible boundary-layer thickness in the physical plane is recovered from $\delta \left(x\right)={\int }_0^{\delta }({\rho }_r/\rho )d\overline{\eta }=\delta {\int }_0^1(T/T_r)d\eta$, which is larger than $\delta$ by a factor of order $\Theta$; this is the classical compressible thickening, and it is displayed quantitatively in Figure 2(c).

(iii) When is Θ a constant? Because the outer flow is steady and adiabatic, $H_e$ is a global constant, so g is well defined for any m. The parameter $\Theta =H_e/\left(c_p{T}_e\right)$, however, varies with x unless $M_e$ does not, i.e. unless $m=0$. Thus:

• for $m=0$ (uniform external stream, Section 5) the reduction is exact: a, b, c, $\Theta$ and E are all constants;
• for $m\ne 0$ (Secs. Section 4 and Section 6) the reduction is exact in the low-Mach limit and is otherwise a local similarity statement, in which $\Theta \left(x\right)$ and $E\left(x\right)$ are frozen at their local values. Local similarity is the standard device of compressible boundary-layer theory [28,29] and its error is of the same order as the other approximations we make.

Table 2 places the present transformation beside its predecessors.

3.3. Boundary Conditions

At a solid wall $y=0$ we require no slip and no penetration; in the transformed variables, for a wall at rest in a stream,

$$f(0,\tau )=0,f_{,\eta }(0,\tau )=0,f_{,\eta }(\infty ,\tau )=1,$$

(35)

while for a wall moving in its own plane with speed $U_0$ into gas at rest (Section 5)

$$f_{,\eta }(0,\tau )=1,f_{,\eta }(\infty ,\tau )=0,f(0,\tau )-2\tau f_{,\tau }(0,\tau )=0.$$

(36)

The thermal condition is either a prescribed wall temperature $T(0,\tau )=T_w$ or a prescribed wall heat flux. Because $T/T_r=g-\frac{E}{2}{f}_{,\eta }^2$, the adiabatic condition ${(\partial T/\partial y)}_w=0$ reads

$$g_{,\eta }(0,\tau )=E{f}_{,\eta }(0,\tau )f_{,\eta \eta }(0,\tau ).$$

(37)

For a wall at rest the right-hand side vanishes and (37) reduces to the familiar $g_{,\eta }(0,\tau )=0$; for a moving wall it does not, because the wall then does viscous work on the gas. Overlooking this term leads to an unphysical cooling of an adiabatic moving plate, and we shall see in Section 5.5 that retaining it restores the classical recovery temperature at $\mathrm{Pr}=1$. In all cases $g(\infty ,\tau )=1$.

4. The Steady Limit

Before exploiting the master system it is worth confirming that it contains the classical steady theory. Setting ${\partial }_{\tau }(·)=0$ in (31)–() and choosing $a=1$ (a normalisation, since a may be absorbed in $\delta$) gives, with $b\equiv \beta$,

$$\begin{array}{cc}\hfill f^{\prime \prime \prime }+f{f}^{\prime \prime }+\beta \Theta \left(g-f^{\prime 2}\right)& =0,\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{\mathrm{Pr}}}{g}^{\prime \prime }+f{g}^{\prime }+\left(1-{\displaystyle \frac{1}{\mathrm{Pr}}}\right)E{\left(f^{\prime }{f}^{\prime \prime }\right)}^{\prime }& =0,\hfill \end{array}$$

(39)

subject to $f\left(0\right)=f^{\prime }\left(0\right)=0$, $f^{\prime }(\infty )=1$, $g(\infty )=1$ and either $g\left(0\right)=g_w$ or $g^{\prime }\left(0\right)=0$. From (34) with $a=1$,

$$\beta ={\displaystyle \frac{2m}{m+1}},$$

(40)

which is precisely the Falkner–Skan pressure-gradient parameter[3], and (38)–() are the compressible Falkner–Skan–Dorodnitsyn equations of the classical steady theory[24,29,30]. Thus the present unsteady transformation degenerates correctly.

Equations (38)–() were solved with a collocation boundary-value solver on $0\le \eta \le 14$ with a tolerance of ${10}^{-8}$. Figure 2 and Table 3 summarise the results. Three classical facts are reproduced, and they serve as validation of the whole formulation:

1.

At $\beta =0$ the term $b\Theta (g-f^{\prime 2})$ vanishes and (38) collapses to the Blasius equation $f^{\prime \prime \prime }+f{f}^{\prime \prime }=0$ regardless of $M_e$ and of the thermal boundary condition. We obtain $f^{\prime \prime }\left(0\right)=0.46960$, i.e. $0.332\sqrt{2}$ in the present normalisation $\delta =\sqrt{2C{\nu }_{r}x/U}$, for every Mach number tested. Compressibility affects the flat-plate skin friction only through the Chapman–Rubesin factor C and through the density weighting of the Dorodnitsyn mapping—the classical result.

2.

At $\mathrm{Pr}=1$ the energy equation is solved exactly by $g\equiv 1$, giving an adiabatic wall temperature $T_{aw}=\Theta T_e$, i.e. a recovery factor of unity.

3.

For $\mathrm{Pr}\ne 1$ the computed recovery factor $r=(T_{aw}/T_e-1)/\left[\frac{1}{2}(\gamma -1)M_e^2\right]$ agrees with $\sqrt{\mathrm{Pr}}$ to better than $0.1\%$ over $0.3\le \mathrm{Pr}\le 1$ (Figure 7(a), open squares).

4.

The unsteady flow with pressure gradient. Solved globally in a Williams–Rhyne chart, in which $\tau$ is not a marching variable because $1-c\tau f_{,\eta }$ changes sign at Stewartson’s line $Ut=x$. The transformed wall shear is Mach independent at every instant when $m=0$, and strongly Mach dependent otherwise; the recovery factor is a function of time, running from the closed form (61) at the impulsive start to $\sqrt{\mathrm{Pr}}$ at the steady state, and undershooting both when $\beta >0$.

5. The Impulsively Started Semi-Infinite Plate in a Compressible Gas

5.1. Statement and Reduction

A semi-infinite plate lies in a viscous, thermally conducting gas at rest with temperature $T_{\infty }$ and density ${\rho }_{\infty }$. At $t=0$ the plate is impulsively set in motion in its own plane with constant speed $U_0$. This is the compressible version of Stokes’ first problem[6] on a semi-infinite plate, and its incompressible version is the problem addressed by Stewartson[8,10], Hall[11], Dennis[12] and Sun[15]. We take ${\rho }_r={\rho }_{\infty }$, $H_r=c_p{T}_{\infty }$, $U_{\mathrm{ref}}=U_0$ and define the wall Mach number and the associated stagnation ratio

$$M_w={\displaystyle \frac{U_0}{\sqrt{\gamma R{T}_{\infty }}}},{\Theta }_w=1+{\displaystyle \frac{\gamma -1}{2}}{M}_w^2,E=2({\Theta }_w-1).$$

(41)

Here $U\left(x\right)=U_0$ is constant, so $m=0$ and hence $b=0$. Taking the normalisation $a=1$ gives $c=2(a-b)=2$ and, from (33),

$$\delta \left(x\right)={\left({\displaystyle \frac{2C{\nu }_{r}x}{U_0}}\right)}^{1/2},\eta =\overline{\eta }{\left({\displaystyle \frac{U_0}{2C{\nu }_{r}x}}\right)}^{1/2},\tau ={\displaystyle \frac{U_{0}t}{2x}}.$$

(42)

Because $b=0$, the pressure-gradient bracket disappears from (31) and the momentum problem decouples completely from the thermal problem:

$$f_{,\eta \eta \eta }+f{f}_{,\eta \eta }=f_{,\eta \tau }+2\tau \left(f_{,\tau }{f}_{,\eta \eta }-f_{,\eta }{f}_{,\eta \tau }\right).$$

(43)

This is exactly Eq. (25) of Ref. [15]. We have therefore established a useful equivalence:

For a perfect gas obeying the Chapman–Rubesin law, the velocity field of the unsteady compressible flat-plate boundary layer is obtained from the incompressible one by the substitutions $\nu \to C{\nu }_r$ and $y\to \overline{\eta }={\int }_0^y(\rho /{\rho }_{\infty })d{y}^{\prime }$; every incompressible solution of (43) lifts to a compressible solution.

The thermal field, in contrast, is genuinely new and is determined by () with $a=1$, $c=2$.

5.2. The Sense in Which the Solutions Are Exact

Because the phrase “exact solution of the compressible boundary-layer equations” is easily over-read, we state precisely what is and is not claimed.

The construction rests on four assumptions, of which only the first two are innocuous. (i) The boundary-layer (Prandtl) ordering, so that (1)–() replace the Navier–Stokes equations; the initial pressure transients and compression waves of the exact impulsive problem[36,37,38] are thereby excluded. (ii) A calorically perfect gas. (iii) The Chapman–Rubesin law $\rho \mu =C{\rho }_r{\mu }_r$ with C constant. (iv) A constant Prandtl number. Given (i)–(iv), Lemma 1 is an identity, the reduction to (31)–() involves no further approximation when $m=0$, and the diffusive branch (50) together with the temperature field of Secs. Section 5.4Section 5.5 satisfies the resulting system and all its boundary conditions identically. In that sense the solutions are exact.

This is precisely the sense in which the classical steady compressible results are called exact. The statement that a flat-plate compressible layer at $\mathrm{Pr}=1$ reduces to the Blasius problem, the Crocco integral[26], and the Illingworth–Stewartson transformation[23,24] are all exact consequences of (i)–(iv) and are all invalidated, as exact statements, by a Sutherland viscosity law or a variable $\mathrm{Pr}$. Under Sutherland’s law C varies through the layer and the present solutions become leading-order approximations whose error is $O\left(\right|C-\overline{C}|/\overline{C})$; for air below $M_e\simeq 4$ with a reference temperature evaluated by Eckert’s rule this is a few per cent, and it is the same error already accepted throughout classical compressible boundary-layer theory[29,30]. Recent work on temperature–velocity relations in hypersonic laminar layers[41,42] quantifies the departure from the Chapman–Rubesin/Crocco picture at high enthalpy and should be consulted before applying (61) at hypersonic conditions.

5.3. Two Exact Solutions of Distinct Similarity Type

We use the word branch throughout for a solution of (43) obtained by imposing a particular similarity structure on $f(\eta ,\tau )$. The two branches constructed below are not two solutions of a single well-posed problem, in the sense in which the Falkner–Skan equation possesses dual solutions for $-0.1988<\beta <0$: they satisfy the same wall conditions but different conditions at infinity, and neither is obtainable from the other by any choice of the constants of integration.

5.3.1. The Kummer Branch

Equation (43) admits the closed-form solution obtained in Ref. [15],

$$f(\eta ,\tau )={\displaystyle \frac{1}{5{\tau }^2}}+{\displaystyle \frac{\eta }{2\tau }}+c_1\sqrt{\tau }+c_2{g}_{11}+c_3{h}_{11},$$

(44)

where $g_{11}$ and $h_{11}$ are the combinations of Kummer functions of the first and second kind given in Appendix B and $c_1,c_2,c_3$ are constants of integration. Boundedness of $f_{,\eta }$ requires $c_2=0$; the no-slip condition $f_{,\eta }(0,\tau )=1$ and the normalisation $f(0,\infty )=0$ then give

$$c_1={\displaystyle \frac{4\pi }{15\Gamma \left(\frac{5}{6}\right)\Gamma \left(\frac{2}{3}\right)}},c_2=0,c_3=-{\displaystyle \frac{1}{3\Gamma \left(\frac{2}{3}\right)}}\sqrt{{\displaystyle \frac{\pi }{3}}},$$

(45)

numerically $c_1=0.548088$, $c_3=-0.251905$. The wall shear follows from

$$f_{,\eta \eta }(\eta ,\tau )=c_3{\displaystyle \frac{3\eta \tau +2}{{\tau }^2}}\mathrm{U}\left(\frac{5}{6},\frac{3}{2},X\right)e^{-X},X={\displaystyle \frac{{(3\eta \tau +2)}^2}{12{\tau }^3}}.$$

(46)

We have verified by direct symbolic-numeric substitution that (44) satisfies (43) to machine precision (residuals $<{10}^{-8}$ over $0.3\le \tau \le 5$, $0\le \eta \le 3$); the auxiliary functions in Appendix B contain a typographical correction to Ref. [15] noted there.

Figure 3 shows f, $f_{,\eta }$, $f_{,\eta \eta }$ and $f_{,\tau }$ at $\tau =5$ and the velocity profiles at several $\tau$. Two properties deserve emphasis. First, the solution satisfies the wall conditions exactly for every $\tau$. Second, its outer limit is

$$f_{,\eta }(\infty ,\tau )={\displaystyle \frac{1}{2\tau }}={\displaystyle \frac{x}{U_{0}t}},$$

(47)

so the far-field condition $u\to 0$ is attained only asymptotically, as $O\left({\tau }^{-1}\right)$. This is not a defect of the transformation but a property of this particular branch: it is an accurate representation of the flow in the Rayleigh region $\tau \gg \frac{1}{2}$, i.e. $x\ll U_{0}t/2$ (Figure 5), and degrades as the Blasius region is approached.

It follows that the Kummer branch is an exact solution of the equation but not of the problem: it satisfies (43) identically and meets the wall conditions for all $\tau$, but attains the outer condition only as $O\left({\tau }^{-1}\right)$. We describe it throughout as an exact solution of the reduced equation, and do not claim that it solves the impulsive-plate boundary-value problem.

5.3.2. The Diffusive Branch

The master equation (43) possesses a second, hitherto unremarked exact branch which satisfies all of the boundary conditions. Seek solutions of the self-similar diffusive form

$$f(\eta ,\tau )=\sqrt{\tau }\Phi \left(\xi \right),\xi ={\displaystyle \frac{\eta }{\sqrt{\tau }}}={\displaystyle \frac{y}{\sqrt{C{\nu }_{r}t}}},$$

(48)

the last equality following from (42) and being independent of x. Substituting (48) into (43) and using $f_{,\eta }={\Phi }^{\prime }$, $f_{,\eta \eta }={\Phi }^{\prime \prime }/\sqrt{\tau }$, $f_{,\eta \eta \eta }={\Phi }^{\prime \prime \prime }/\tau$, $f_{,\tau }=(\Phi -\xi {\Phi }^{\prime })/2\sqrt{\tau }$, $f_{,\eta \tau }=-\xi {\Phi }^{\prime \prime }/2\tau$, every nonlinear term cancels identically and there remains

$${\Phi }^{\prime \prime \prime }+{\displaystyle \frac{\xi }{2}}{\Phi }^{\prime \prime }=0.$$

(49)

Hence

$${\Phi }^{\prime }\left(\xi \right)=erfc\left({\displaystyle \frac{\xi }{2}}\right),\Phi \left(\xi \right)=\xi erfc{\displaystyle \frac{\xi }{2}}-{\displaystyle \frac{2}{\sqrt{\pi }}}\left(1-e^{-{\xi }^2/4}\right),$$

(50)

so that

$${\displaystyle \frac{u}{U_0}}=erfc\left({\displaystyle \frac{\overline{\eta }}{2\sqrt{C{\nu }_{r}t}}}\right),V\equiv 0.$$

(51)

It must be stressed that the vanishing quantity in (51) is the transformed normal velocity V, not the physical one: by (10), $v=({\rho }_r/\rho )(V-{\overline{\eta }}_{,t}-u{\overline{\eta }}_{,x})$, and ${\overline{\eta }}_{,t}\ne 0$ because dissipative heating changes the density. The physical normal velocity is computed in Section 5.7, where it is shown to reproduce the outflow that is the classical signature of the compressible Rayleigh problem[35,36,37]. Indeed, any f of the form $\sqrt{\tau }\Phi (\eta /\sqrt{\tau })$ satisfies $f-\eta f_{,\eta }-2\tau f_{,\tau }\equiv 0$, which is precisely the transformed statement $v=0$; the third condition in (36) is therefore satisfied identically, and (50) gives $f_{,\eta }(0,\tau )=1$ and $f_{,\eta }(\infty ,\tau )=0$ exactly. The diffusive branch is thus the compressible Rayleigh solution, and the content of the present derivation is that it belongs to the similarity family (18): the classical Rayleigh solution is a member of the same two-parameter family that contains the Kummer branch. In the physical plane it is not an error function of y, because $\overline{\eta }$ and y are related through the temperature field; the profile in y is obtained by the quadrature $y={\int }_0^{\overline{\eta }}({\rho }_{\infty }/\rho )d{\overline{\eta }}^{\prime }$, which is carried out in Figure 8.

5.3.3. Relation between the Branches and Stewartson’s Conjecture

Both branches solve the same equation with the same wall conditions, so they differ in the outer condition and in the initial data. Letting $\tau \to \infty$ at fixed $\xi =\eta /\sqrt{\tau }$ in (44) gives

$$f_{,\eta }-{\displaystyle \frac{1}{2\tau }}⟶c_3\left[(6X-1)\mathrm{U}\left(\frac{5}{6},\frac{3}{2},X\right)-6\mathrm{U}\left(-\frac{1}{6},\frac{3}{2},X\right)\right]e^{-X},$$

(52)

with $X=\frac{3}{4}{\xi }^2$; Figure 4(a) confirms that the profiles collapse onto this curve. It is a genuinely different self-similar profile from $erfc(\xi /2)$, and the corresponding wall shear coefficients differ,

$$\begin{array}{cc}\hfill & -f_{,\eta \eta }(0,\tau )\to \hfill \\ \hfill & \left\{\begin{array}{cc}1.3702{\tau }^{-1/2},\hfill & \mathrm{Kummer}\mathrm{branch},\hfill \\ {\pi }^{-1/2}{\tau }^{-1/2}=0.5642{\tau }^{-1/2},\hfill & \mathrm{diffusive}\mathrm{branch},\hfill \end{array}\right.\hfill \end{array}$$

(53)

as shown in Figure 4(b). Since the diffusive branch reproduces the exact solution of the physical Rayleigh problem, we regard it as the physically relevant branch at early times, and the Kummer branch as a second exact solution of the reduced equation whose interest is analytical. Both are regular at $\tau =1/2$ (Figure 4(c)), which supports Sun’s conclusion[15] that the passage from the Rayleigh to the Blasius regime, contrary to Stewartson’s conjecture[8], need not traverse a singularity: the value $\tau =1/2$ marks the change of sign of the coefficient $1-2\tau f_{,\eta }$ multiplying $g_{,\tau }$ in (), i.e. a change of the direction in which information propagates in $\tau$, not a singularity of the solution.

Figure 4. The two exact branches compared. (a) The Kummer branch velocity, with its outer residual removed, collapses as $\tau \to \infty$ onto the limiting profile of Eq. (52) (dotted), which is distinct from the diffusive branch $erfc(\xi /2)$ (dashed). (b) Transformed wall shear: both branches decay as ${\tau }^{-1/2}$ but with different coefficients, $1.3702$ and ${\pi }^{-1/2}=0.5642$. (c) The velocity field of the Kummer branch in the $(\eta ,\tau )$ plane; the line $\tau =1/2$, at which $1-2\tau f_{,\eta }(0,\tau )$ changes sign, is regular.

Figure 4. The two exact branches compared. (a) The Kummer branch velocity, with its outer residual removed, collapses as $\tau \to \infty$ onto the limiting profile of Eq. (52) (dotted), which is distinct from the diffusive branch $erfc(\xi /2)$ (dashed). (b) Transformed wall shear: both branches decay as ${\tau }^{-1/2}$ but with different coefficients, $1.3702$ and ${\pi }^{-1/2}=0.5642$. (c) The velocity field of the Kummer branch in the $(\eta ,\tau )$ plane; the line $\tau =1/2$, at which $1-2\tau f_{,\eta }(0,\tau )$ changes sign, is regular.

Figure 5. Regions of the unsteady boundary layer past an impulsively started plate. Near the leading edge $\tau =U_{0}t/2x$ is large and the flow is essentially the one-dimensional Rayleigh (Stokes) layer; far downstream $\tau$ is small and the flow has not yet felt the start-up, approaching the Blasius state. The similarity variables (42) span both regions continuously.

Figure 5. Regions of the unsteady boundary layer past an impulsively started plate. Near the leading edge $\tau =U_{0}t/2x$ is large and the flow is essentially the one-dimensional Rayleigh (Stokes) layer; far downstream $\tau$ is small and the flow has not yet felt the start-up, approaching the Blasius state. The similarity variables (42) span both regions continuously.

5.4. The Thermal Field at $\mathrm{Pr}=1$: a Crocco Integral

With $b=0$ the energy equation () at $\mathrm{Pr}=1$ becomes

$$g_{,\eta \eta }+f{g}_{,\eta }=g_{,\tau }+2\tau \left(f_{,\tau }{g}_{,\eta }-f_{,\eta }{g}_{,\tau }\right),$$

(54)

which is precisely (43) with g in place of $f_{,\eta }$. Hence both $g\equiv \mathrm{const}$ and $g=f_{,\eta }$ solve it, and so does any linear combination:

$$\begin{array}{|c|}\hline g(\eta ,\tau )=A+B{f}_{,\eta }(\eta ,\tau )\\ \hline\end{array}$$

(55)

This is the unsteady compressible analogue of the Crocco integral[26,27], valid for arbitrary $\tau$. It is exact whenever $b=0$; for $b\ne 0$ the residual is $-(1-g_w)b\Theta (g-f_{,\eta }^2)$ and (55) becomes an approximation.

Applying $g(\infty ,\tau )=1$ with $f_{,\eta }(\infty ,\tau )=0$ gives $A=1$. For an adiabatic wall the boundary condition (37) with $f_{,\eta }(0,\tau )=1$ gives $B=E$, so $g=1+E{f}_{,\eta }$ and, from (22),

$${\displaystyle \frac{T}{T_{\infty }}}=1+({\Theta }_w-1)\left[1-{\left(1-f_{,\eta }\right)}^2\right],{\displaystyle \frac{T_w}{T_{\infty }}}={\Theta }_w.$$

(56)

The adiabatic wall therefore attains the full stagnation temperature based on the plate speed, a recovery factor of unity, exactly as in the steady theory—but now at every instant, and for the moving-wall configuration in which the wall itself supplies the dissipation. For a prescribed wall temperature $T_w=t_w{T}_{\infty }$,

$${\displaystyle \frac{T}{T_{\infty }}}=1+\left(t_w+{\Theta }_w-2\right)f_{,\eta }-\left({\Theta }_w-1\right)f_{,\eta }^2,$$

(57)

a two-parameter family whose linear term measures wall heat transfer and whose quadratic term measures viscous dissipation. Profiles are shown in Figure 6(a),(b) using the diffusive branch $f_{,\eta }=erfc(\xi /2)$.

5.5. Arbitrary Prandtl Number: Quadrature and Recovery Factor

On the diffusive branch the energy equation reduces to an ordinary differential equation. Writing $g=G\left(\xi \right)$ with $\xi =\eta /\sqrt{\tau }$ and substituting into () with $a=1$, $c=2$, all terms proportional to $\Phi G^{\prime }$ cancel and there remains

$${\displaystyle \frac{1}{\mathrm{Pr}}}{G}^{\prime \prime }+{\displaystyle \frac{\xi }{2}}{G}^{\prime }+\left(1-{\displaystyle \frac{1}{\mathrm{Pr}}}\right)E{\left({\Phi }^{\prime }{\Phi }^{\prime \prime }\right)}^{\prime }=0.$$

(58)

It is more transparent to work with the static temperature $\theta =T/T_{\infty }=G-\frac{E}{2}{\Phi }^{\prime 2}$. Using ${\Phi }^{\prime \prime \prime }=-\frac{\xi }{2}{\Phi }^{\prime \prime }$ from (49), Eq. (58) collapses to the classical dissipation equation

$${\theta }^{\prime \prime }+{\displaystyle \frac{\mathrm{Pr}\xi }{2}}{\theta }^{\prime }=-\mathrm{Pr}E{\Phi }^{\prime \prime 2}=-{\displaystyle \frac{\mathrm{Pr}E}{\pi }}{e}^{-{\xi }^2/2},$$

(59)

whose solution is elementary:

$${\theta }^{\prime }\left(\xi \right)=e^{-\mathrm{Pr}{\xi }^2/4}\left[{\theta }^{\prime }\left(0\right)-{\displaystyle \frac{\mathrm{Pr}E}{\pi }}{\int }_0^{\xi }{e}^{-(2-\mathrm{Pr})s^2/4}ds\right].$$

(60)

For an adiabatic wall ${\theta }^{\prime }\left(0\right)=0$, and integrating (60) once more with $\theta (\infty )=1$ gives, after interchanging the order of integration and transforming to polar coordinates (Appendix C),

$$\begin{array}{|c|}\hline r={\displaystyle \frac{4}{\pi }}\sqrt{{\displaystyle \frac{\mathrm{Pr}}{2-\mathrm{Pr}}}}arctan\sqrt{{\displaystyle \frac{2-\mathrm{Pr}}{\mathrm{Pr}}}}\\ \hline\end{array}$$

(61)

for the recovery factor defined by $T_{aw}/T_{\infty }=1+r\frac{1}{2}(\gamma -1)M_w^2$. For $\mathrm{Pr}>2$ the analytic continuation $r=(4/\pi )\sqrt{\mathrm{Pr}/(\mathrm{Pr}-2)}artanh\sqrt{(\mathrm{Pr}-2)/\mathrm{Pr}}$ applies, and $r\left(2\right)=4/\pi$. Equation (61) gives $r\left(1\right)=1$, consistent with the Crocco integral, and $r(3/2)=2/\sqrt{3}$. It is not equal to $\sqrt{\mathrm{Pr}}$: the unsteady Rayleigh layer recovers more than the steady Blasius layer for $\mathrm{Pr}<1$ and less for $\mathrm{Pr}>1$ (Table 4 and Figure 7). For air at $\mathrm{Pr}=0.72$, $r=0.8855$ against the Blasius value $0.8485$, a difference of $4.4\%$ in the recovery excess, which for $M_w=6$ amounts to about $0.27T_{\infty }$ in wall temperature.

Equation (61) is an exact consequence of the assumptions listed in Section 5.2. We have not found it in the literature on the compressible Rayleigh problem[35,36,38,39,40], which is largely confined to small Mach number and to $\mathrm{Pr}$ near unity or to special values such as $\mathrm{Pr}=3\gamma /4$; but that literature is old and scattered, and we state only that we are unaware of a prior derivation, not that none exists.

5.6. Arbitrary Prandtl Number on the Kummer Branch

Section 5.5 solved the energy equation on the diffusive branch. On the Kummer branch the thermal field has so far been available only at $\mathrm{Pr}=1$, through the Crocco integral (55). We now close that gap. The route is the group-invariant reduction of the unsteady energy equation, and the key observation is that the cancellation exploited in (58)—“all terms proportional to $\Phi G^{\prime }$ cancel”—is not an accident of the diffusive branch but an instance of a general mechanism.

5.6.1. The Static-Temperature Form

Write the convective operator of () at $a=1$, $c=2$ as

$$\mathcal{N}\left[q\right]\equiv \underset{A}{\underbrace{(1-2\tau f_{,\eta })}}{q}_{,\tau }+\underset{B}{\underbrace{(2\tau f_{,\tau }-f)}}{q}_{,\eta },$$

(62)

so that the energy equation reads $\mathcal{N}\left[g\right]={\mathrm{Pr}}^{-1}{g}_{,\eta \eta }+(1-{\mathrm{Pr}}^{-1})E{\left(f_{,\eta }{f}_{,\eta \eta }\right)}_{,\eta }$.

Lemma 2

(Static-temperature form). Let $\theta =T/T_{\infty }=g-\frac{E}{2}{f}_{,\eta }^2$. Then, whenever $b=0$,

$$\mathcal{N}\left[\theta \right]={\displaystyle \frac{1}{\mathrm{Pr}}}{\theta }_{,\eta \eta }+E{f}_{,\eta \eta }^2.$$

(63)

Proof. 

With $b=0$, (43) is precisely $\mathcal{N}\left[f_{,\eta }\right]=f_{,\eta \eta \eta }$. Since $\mathcal{N}$ is a derivation, $\mathcal{N}\left[f_{,\eta }^2\right]=2f_{,\eta }{f}_{,\eta \eta \eta }$, while ${\left(f_{,\eta }^2\right)}_{,\eta \eta }=2(f_{,\eta \eta }^2+f_{,\eta }{f}_{,\eta \eta \eta })$ and ${\left(f_{,\eta }{f}_{,\eta \eta }\right)}_{,\eta }=f_{,\eta \eta }^2+f_{,\eta }{f}_{,\eta \eta \eta }$. Substituting $g=\theta +\frac{E}{2}{f}_{,\eta }^2$ into (), the terms in $f_{,\eta }{f}_{,\eta \eta \eta }$ cancel between the two sides and the terms in $f_{,\eta \eta }^2$ combine as ${\mathrm{Pr}}^{-1}E+(1-{\mathrm{Pr}}^{-1})E=E$, leaving (63). □

Equation (63) is the compressible statement that the static temperature obeys a passive-scalar equation with the viscous-dissipation source $E{f}_{,\eta \eta }^2$, the Eckert number of the incompressible theory being replaced by $E=(\gamma -1)M_w^2$. It is the unsteady, compressible counterpart of the Pohlhausen energy equation, and it is branch-independent: applied to the diffusive branch it reproduces (59) immediately.

5.6.2. Why the Convection Is Blind to the Velocity Field

Lemma 3

(Velocity-blind convection). For any $q(\eta ,\tau )$,

$$\mathcal{N}\left[q\right]=q_{,\tau }-\left(f-2\tau f_{,\tau }\right)q_{,\eta }-2\tau f_{,\eta }{q}_{,\tau }.$$

(64)

On the diffusive branch $f=\sqrt{\tau }\Phi \left(\xi \right)$ one has $f-2\tau f_{,\tau }=\eta f_{,\eta }$ and hence $\mathcal{N}\left[q\right]=q_{,\tau }-f_{,\eta }{X}_{D}q$ with $X_D=\eta {\partial }_{\eta }+2\tau {\partial }_{\tau }$. On the Kummer branch (44) one has $Xf=\tau f-\eta$ with

$$X=(\eta \tau +2){\displaystyle \frac{\partial }{\partial \eta }}+2{\tau }^2{\displaystyle \frac{\partial }{\partial \tau }},$$

(65)

and hence $\mathcal{N}\left[q\right]=q_{,\tau }-(\eta /\tau )q_{,\eta }-(f_{,\eta }/\tau )Xq$. In either case, if ζ is an invariant of the corresponding generator, $X_D\zeta =0$ or $X\zeta =0$, then $\mathcal{N}\left[Q\right(\zeta \left)\right]$ contains no reference to f whatever.

Proof. 

Equation (64) is (62) rearranged. For the diffusive branch $f-\eta f_{,\eta }-2\tau f_{,\tau }\equiv 0$ was already noted below (51). For the Kummer branch, $Xp=0$ for $p={(3\eta \tau +2)}^2/12{\tau }^3$ because $X{\zeta }_K=3\tau {\zeta }_K$ with ${\zeta }_K=3\eta \tau +2$; applying X term by term to (44) then gives $Xf=\tau f-\eta$, whence $f-2\tau f_{,\tau }=\left[\eta +(\eta \tau +2)f_{,\eta }\right]/\tau$. Substituting into (64) and collecting the terms multiplying $f_{,\eta }$ produces X as stated. □

Lemma 3 explains the cancellation observed in (58) and supplies its analogue for the Kummer branch. The invariant of X is

$$\omega =\sqrt{p}={\displaystyle \frac{3\eta \tau +2}{2\sqrt{3}{\tau }^{3/2}}}\underset{\tau \to \infty }{\to }{\displaystyle \frac{\sqrt{3}}{2}}\xi ,$$

(66)

and (46) shows that the Kummer velocity field is itself group-invariant up to the outer drift:

$$f_{,\eta }={\displaystyle \frac{1}{2\tau }}+F\left(\omega \right),F\left(\omega \right)=K{e}^{-{\omega }^2}\mathrm{U}\left(\frac{5}{6},\frac{1}{2},{\omega }^2\right),$$

(67)

with $K=-2c_3=2\sqrt{\pi /3}/3\Gamma (2/3)=0.5038097$, $F\left(0\right)=1$, $F(\infty )=0$, and

$$f_{,\eta \eta }={\omega }_{,\eta }{F}^{\prime }\left(\omega \right),F^{\prime }\left(\omega \right)=-2K\omega e^{-{\omega }^2}\mathrm{U}\left(\frac{5}{6},\frac{3}{2},{\omega }^2\right),$$

(68)

${\omega }_{,\eta }=\sqrt{3}/2\sqrt{\tau }$ and $F^{\prime }\left(0\right)=-1.5821931$.

5.6.3. Exact Reduction and Closed-Form Thermal Field

Theorem 1.

On the Kummer branch, $\theta =\Theta \left(\omega \right)$ satisfies (63) for arbitrary $\mathrm{Pr}$ and E if and only if

$${\Theta }^{\prime \prime }\left(\omega \right)+2\mathrm{Pr}\omega {\Theta }^{\prime }\left(\omega \right)+\mathrm{Pr}E{\left[F^{\prime }\left(\omega \right)\right]}^2=0.$$

(69)

Proof. 

By Lemma 3, $\mathcal{N}\left[\Theta \left(\omega \right)\right]={\Theta }^{\prime }\left(\omega \right)\left({\omega }_{,\tau }-(\eta /\tau ){\omega }_{,\eta }\right)=-\frac{3}{2}{\tau }^{-1}\omega {\Theta }^{\prime }=-2\omega {\omega }_{,\eta }^2{\Theta }^{\prime }$, using ${\omega }_{,\eta }^2=3/4\tau$. Since ${\omega }_{,\eta \eta }=0$, ${\theta }_{,\eta \eta }={\Theta }^{\prime \prime }{\omega }_{,\eta }^2$, and by (68) $f_{,\eta \eta }^2={\omega }_{,\eta }^2{\left[F^{\prime }\left(\omega \right)\right]}^2$. The common factor ${\omega }_{,\eta }^2$ cancels from (63), leaving (69). □

Equation (69) is linear with integrating factor $e^{\mathrm{Pr}{\omega }^2}$. Imposing $\theta (\infty )=1$ and a wall value $\theta \left(0\right)=t_w$ gives the closed form

$$\begin{array}{|c|}\hline \theta =1+(t_w-1)\mathrm{erfc}\left(\sqrt{\mathrm{Pr}}\omega \right)+\mathrm{Pr}E{\Phi }_T(\omega ;\mathrm{Pr})\\ \hline\end{array}$$

(70)

with

$${\Phi }_T(\omega ;\mathrm{Pr})={\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{\pi }{\mathrm{Pr}}}}\left[\mathrm{erfc}\left(\sqrt{\mathrm{Pr}}\omega \right){\int }_0^{\omega }\kappa \left(u\right)\mathrm{erf}\left(\sqrt{\mathrm{Pr}}u\right)du+\mathrm{erf}\left(\sqrt{\mathrm{Pr}}\omega \right){\int }_{\omega }^{\infty }\kappa \left(u\right)\mathrm{erfc}\left(\sqrt{\mathrm{Pr}}u\right)du\right],\kappa \left(u\right)={\left[F^{\prime }\left(u\right)\right]}^2{e}^{\mathrm{Pr}{u}^2}.$$

(71)

Since ${\Phi }_T\left(0\right)={\Phi }_T(\infty )=0$, dissipation alters the interior profile but not the boundary values. The wall gradient is

$${\Theta }^{\prime }\left(0\right)=-2\sqrt{{\displaystyle \frac{\mathrm{Pr}}{\pi }}}\left[(t_w-1)-\mathrm{Pr}E{P}_{\infty }\right],$$

(72)

$$P_{\infty }\left(\mathrm{Pr}\right)={\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{\pi }{\mathrm{Pr}}}}{\int }_0^{\infty }{\left[F^{\prime }\left(u\right)\right]}^2{e}^{\mathrm{Pr}{u}^2}\mathrm{erfc}\left(\sqrt{\mathrm{Pr}}u\right)du.$$

(73)

For $E=0$ the temperature is simply a complementary error function of the group invariant, $\theta =1+(t_w-1)\mathrm{erfc}\left(\sqrt{\mathrm{Pr}}\omega \right)$, which as $\tau \to \infty$ becomes $\mathrm{erfc}\left(\frac{1}{2}\sqrt{3\mathrm{Pr}}\overline{\eta }/\sqrt{C{\nu }_{r}t}\right)$ in the Dorodnitsyn variable; the physical-plane profile follows, as everywhere in this paper, from the quadrature $y={\int }_0^{\overline{\eta }}({\rho }_{\infty }/\rho )d{\overline{\eta }}^{\prime }$. Setting ${\Theta }^{\prime }\left(0\right)=0$ gives the adiabatic wall temperature $T_{aw}/T_{\infty }=1+\mathrm{Pr}E{P}_{\infty }$, i.e. the recovery factor

$$r_K=2\mathrm{Pr}{P}_{\infty }\left(\mathrm{Pr}\right).$$

(74)

We have verified by substitution in 25-digit arithmetic that $g=\Theta \left(\omega \right)+\frac{E}{2}{f}_{,\eta }^2$, with $\Theta$ from (70), satisfies () with residuals below ${10}^{-25}$ over $0.2\le \eta \le 2$, $0.9\le \tau \le 3$, $0.5\le \mathrm{Pr}\le 7$, $1\le E\le 4$.

5.6.4. Comparison of the Branches, and a Caution

Table 5 places (74) beside the diffusive value (61) and the steady $\sqrt{\mathrm{Pr}}$. Two features deserve comment, and the second is a caveat rather than a result.

First, the two branches recover differently at every $\mathrm{Pr}$, and the Kummer branch recovers more. This is consistent with its velocity field: the Kummer wall shear coefficient exceeds the diffusive one by the factor $1.3702/0.5642=2.43$, so the dissipation integral that feeds the wall temperature is correspondingly larger. In the limits $r_K\to 1.70457\sqrt{\mathrm{Pr}}$ as $\mathrm{Pr}\to 0$ and $r_K\simeq \frac{1}{2}{\left[F^{\prime }\left(0\right)\right]}^{2}ln\mathrm{Pr}$ as $\mathrm{Pr}\to \infty$, so the Kummer branch shares the $\sqrt{\mathrm{Pr}}$ scaling of the steady layer at small $\mathrm{Pr}$ but saturates logarithmically at large $\mathrm{Pr}$, whereas (61) tends to $4/\pi ·\sqrt{\mathrm{Pr}/(\mathrm{Pr}-2)}\mathrm{artanh}\sqrt{(\mathrm{Pr}-2)/\mathrm{Pr}}$.

Second, at $\mathrm{Pr}=1$ the Kummer branch carries two exact adiabatic solutions that satisfy the same asymptotic conditions: the Crocco integral (55), which gives $r=1$, and the group-invariant solution (70), which gives $r_K\left(1\right)=1.29824$. They are genuinely distinct: $X{f}_{,\eta }=-1\ne 0$, so the Crocco field is not group-invariant. The origin of the ambiguity is the same degeneracy that limits the Kummer branch itself. The coefficient $A=1-2\tau f_{,\eta }$ of ${\theta }_{,\tau }$ vanishes as $\eta \to \infty$, so $\tau \to \infty$ is a degenerate characteristic on which no Cauchy data are prescribed; and the wall $\eta =0$ corresponds to ${\omega }_0=1/\sqrt{3}{\tau }^{3/2}$, not to $\omega =0$, so a condition imposed at $\omega =0$—isothermal or adiabatic—is met only as $\tau \to \infty$, exactly as the velocity conditions are. Within the group-invariant class (70) is unique, which is the natural similarity-theoretic statement; the spread between 1 and $1.298$ is an honest measure of the residual indeterminacy of this branch. The diffusive branch is free of this difficulty—it satisfies all of its boundary conditions exactly at every $\tau$—and (61) therefore remains our physical prediction for the compressible Rayleigh layer. Equation (74) is offered as the thermal completion of the Kummer branch, in the same spirit in which Section 5 offers the branch itself: an exact solution of the reduced equation whose interest is analytical.

5.6.5. Remark: There Is No Systematic ${\tau }^{-1}$ Correction

It is natural to try to relax the asymptotic character of the wall conditions by writing $\theta ={\sum }_{k\ge 0}{\tau }^{-k}{\Theta }_k\left(\omega \right)$. This does not work in the expected way. Using Lemma 3 and (67),

$$\mathcal{N}\left[{\tau }^{-k}{\Theta }_k\right]=-{\displaystyle \frac{3\omega }{2}}{\Theta }_k^{\prime }{\tau }^{-k-1}+2kF\left(\omega \right){\Theta }_k{\tau }^{-k},$$

(75)

so that the ${\tau }^{-1}$ term contributes to the leading balance and the hierarchy closes algebraically rather than differentially:

$${\Theta }_1={\displaystyle \frac{3}{8\mathrm{Pr}F\left(\omega \right)}}\left[{\Theta }_0^{\prime \prime }+2\mathrm{Pr}\omega {\Theta }_0^{\prime }+\mathrm{Pr}E{\left(F^{\prime }\right)}^2\right],$$

(76)

and similarly ${\Theta }_n\propto {\Theta }_{n-1}^{\prime \prime },{\Theta }_{n-1}^{\prime }$ divided by $2nF\left(\omega \right)$. The corrections are therefore proportional to the residual of (69) itself, and each successive term carries another factor $F{\left(\omega \right)}^{-1}\sim {\omega }^{5/3}{e}^{{\omega }^2}/K$, which diverges in the outer region. The choice ${\Theta }_1\equiv 0$—that is, Theorem 1 —is the only consistent truncation, and relaxing the $\tau \to \infty$ restriction requires a different device, presumably a $\tau$-dependent shift of the invariant rather than a series in ${\tau }^{-1}$.

5.7. The Induced Outflow, and Comparison with Classical Results

The compressible Rayleigh problem differs from its incompressible counterpart in one qualitative respect, emphasised by Howarth[35], Van Dyke[36] and Stewartson[8,37]: viscous dissipation heats the gas, the gas expands, and a normal velocity directed away from the plate is induced even though the plate moves purely in its own plane. Our formulation must reproduce this, and it provides a quantitative check on the transformation that is independent of the incompressible limit.

On the diffusive branch the flow is independent of x, so ${\overline{\eta }}_{,x}=0$, and $V\equiv 0$; hence $v=-({\rho }_r/\rho ){\overline{\eta }}_{,t}$. Writing $\overline{\eta }=\sqrt{C{\nu }_{r}t}\xi$ and $y=\sqrt{C{\nu }_{r}t}Y\left(\xi \right)$ with $Y\left(\xi \right)={\int }_0^{\xi }\theta \left({\xi }^{\prime }\right)d{\xi }^{\prime }$ and $\theta =T/T_{\infty }$, differentiation at fixed y gives

$$v(\xi ,t)={\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{C{\nu }_r}{t}}}\left[Y\left(\xi \right)-\theta \left(\xi \right)\xi \right],$$

(77)

which vanishes identically when $\theta \equiv 1$, as it must. At the outer edge $\theta \to 1$ and (77) tends to the finite limit

$$v_{\infty }={\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{C{\nu }_r}{t}}}{\int }_0^{\infty }\left({\displaystyle \frac{T}{T_{\infty }}}-1\right)d\xi >0,$$

(78)

an outflow, as anticipated. At $\mathrm{Pr}=1$ with an adiabatic wall the Crocco integral (56) gives $\theta -1=({\Theta }_w-1)\left[1-{erf}^2(\xi /2)\right]$, and since ${\int }_0^{\infty }\left(1-{erf}^{2}u\right)du=\sqrt{2/\pi }$ exactly,

$$\begin{array}{|c|}\hline v_{\infty }=\left(\gamma -1\right)M_w^2\sqrt{{\displaystyle \frac{C{\nu }_r}{2\pi t}}}\\ \hline\end{array}$$

(79)

The outflow decays as $t^{-1/2}$, grows as the square of the wall Mach number, and is independent of x. Both the scaling and the $M_w^2$ dependence agree with the small-Mach-number analyses of Van Dyke[36] and Hanin[38], which were obtained by linearising the full Navier–Stokes equations rather than by the boundary-layer approximation used here; (79) extends the result to arbitrary $M_w$ within the Chapman–Rubesin model. We note that those analyses also capture the initial pressure and temperature transients and the compression waves discussed by Stewartson[37], which lie outside the boundary-layer approximation and are therefore absent from (79).

5.8. Validation against Established Results

Table 6 collects the quantitative checks available to us. Each is a case in which the present formulation must reproduce an independently established result, and each tests a different ingredient: the momentum reduction, the pressure-gradient term, the Crocco closure, the energy equation at $\mathrm{Pr}\ne 1$, and the density coupling that produces the outflow.

Two entries deserve comment. The steady recovery factor at $\mathrm{Pr}=0.72$ agrees with $\sqrt{\mathrm{Pr}}$ to $0.10\%$; the residual is not numerical error but the known departure of the exact Falkner–Skan–Dorodnitsyn result from the $\sqrt{\mathrm{Pr}}$ approximation, and its sign and magnitude match the tabulations of Ref. [30]. The outflow entry is an exact coefficient agreement, not merely a scaling one: setting $C=1$ ($\mu \propto T$) in (79) reproduces Eq. (7.25) of Ref. [8] term for term, including the factor ${\left(2\pi \right)}^{-1/2}$. The agreement is a non-trivial check, since it tests the density coupling, the Crocco closure and the inverse mapping (82) simultaneously. The small-Mach Navier–Stokes analyses of Refs. [36,38] agree in exponent and in the $M_w^2$ dependence but carry additional terms, from the initial pressure transient, that lie outside the boundary-layer approximation.

We have not been able to compare the unsteady recovery factor (61) at $\mathrm{Pr}\ne 1$ against an independent calculation, since we know of none; it is checked here only against numerical quadrature of (59) (Table 4), which shares its assumptions. This is the weakest link in the validation chain and we flag it as such.

5.9. Wall Shear, Heat Transfer and the Physical Plane

On the diffusive branch the wall quantities follow in closed form. With ${\mu }_w{\rho }_w=C{\mu }_r{\rho }_r$ and $f_{,\eta \eta }(0,\tau )={\Phi }^{\prime \prime }\left(0\right)/\sqrt{\tau }=-1/\sqrt{\pi \tau }$,

$${\tau }_w={\mu }_w{\left({\displaystyle \frac{\partial u}{\partial y}}\right)}_w=-{\rho }_r{U}_0\sqrt{{\displaystyle \frac{C{\nu }_r}{\pi t}}},$$

(80)

independent of x, as it must be in the Rayleigh region, and depending on compressibility only through C. The wall heat flux follows from (60), which gives the Newtonian form ${\theta }^{\prime }\left(0\right)=\sqrt{\mathrm{Pr}/\pi }({\theta }_{aw}-t_w)$ and hence

$$q_w={\rho }_r{c}_p\sqrt{{\displaystyle \frac{C{\nu }_r}{\pi \mathrm{Pr}t}}}\left(T_{aw}-T_w\right),T_{aw}=T_{\infty }\left[1+r{\displaystyle \frac{\gamma -1}{2}}{M}_w^2\right],$$

(81)

with r from (61). The Reynolds analogy factor is $2\mathrm{St}/c_f={\mathrm{Pr}}^{-1/2}$, again differing from the steady flat-plate value ${\mathrm{Pr}}^{-2/3}$.

Finally, the solution must be mapped back to the physical plane. Inverting (9),

$${\displaystyle \frac{y}{\sqrt{C{\nu }_{r}t}}}={\int }_0^{\xi }{\displaystyle \frac{T\left({\xi }^{\prime }\right)}{T_{\infty }}}d{\xi }^{\prime },$$

(82)

which is evaluated in Figure 8. The heated gas near a high-speed plate is light, so a fixed amount of mass occupies a much greater physical thickness: at $M_w=4$ the layer is roughly three times thicker than the incompressible layer at the same time, and the density at the wall falls to about a quarter of its free-stream value. This is the familiar mechanism by which compressibility destabilises and thickens high-speed boundary layers.

6. The Unsteady Flow with Pressure Gradient

Section 4 and Section 5 exercise the master system only where its distinctive term is inactive: the steady limit is classical, and the flat plate has $b=0$, so the coupling $b\Theta (g-f_{,\eta }^2)$ vanishes identically. We now solve (31)–() for $m\ne 0$, which is where the momentum and enthalpy equations genuinely interact.

6.1. Why $\tau$ Is Not a Marching Variable

Collecting the $\tau$-derivatives in (31),

$$\left(1-c\tau f_{,\eta }\right)f_{,\eta \tau }+c\tau f_{,\tau }{f}_{,\eta \eta }=f_{,\eta \eta \eta }+af{f}_{,\eta \eta }+b\Theta \left(g-f_{,\eta }^2\right),$$

(83)

so the coefficient of $f_{,\eta \tau }$ vanishes where $c\tau f_{,\eta }=1$. At the outer edge $f_{,\eta }=1$ this is $\tau =1/c$, which for $b=0$ is $\tau =1/2$: precisely Stewartson’s line $Ut=x$. Beyond it the coefficient is negative over part of the layer, the problem is backward-parabolic in $\tau$ there, and no forward march in $\tau$ can succeed. We confirmed this numerically: an implicit march from the Rayleigh start diverges near $\tau =1/c$ irrespective of step size. This is not a defect of the transformation but the analytical content of Stewartson’s essential singularity, expressed as a change of characteristic direction.

6.2. A Uniformly Valid Chart

Introduce, following the device of Williams and Rhyne[46],

$$s=1-e^{-\tau },N={\displaystyle \frac{\eta }{2\sqrt{s}}},f(\eta ,\tau )=2\sqrt{s}F(N,s),$$

(84)

so that $u/U=f_{,\eta }=F^{\prime }$ and, at small s, $N\to \overline{\eta }/2\sqrt{C{\nu }_{r}t}$ independently of x. Substituting (84) into (31)–(), the terms in $N{F}^{\prime }{F}^{\prime \prime }$ cancel identically and there remains

$$\begin{array}{cc}\hfill F^{\prime \prime \prime }+P{F}^{\prime }+4sb\Theta \left(G-F^{\prime 2}\right)& =D\left(1-c\tau F^{\prime }\right)F_{,s}^{\prime },\hfill \end{array}$$

(85)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{\mathrm{Pr}}}{G}^{\prime \prime }+P{G}^{\prime }+\left(1-{\displaystyle \frac{1}{\mathrm{Pr}}}\right)E{\left(F^{\prime }{F}^{\prime \prime }\right)}^{\prime }& =D\left(1-c\tau F^{\prime }\right)G_{,s},\hfill \end{array}$$

(86)

where $\prime =\partial /\partial N$ acts on $F^{\prime }$ and G as second-order operators,

$$P=2(1-s)N+KF-Dc\tau F_{,s},K=4s-2c\tau (1-s),$$

(87)

$$D=4s(1-s),\tau =-ln(1-s),F={\int }_0^N{F}^{\prime }d{N}^{\prime }.$$

(88)

The two ends are characteristic, since $D\left(0\right)=D\left(1\right)=0$, and each carries its own ordinary differential equation:

$$\begin{array}{cc}\hfill s\to 0:& F^{\prime \prime \prime }+2N{F}^{\prime \prime }=0\Rightarrow F^{\prime }=erfN,\hfill \end{array}$$

(89)

$$\begin{array}{cc}\hfill s\to 1:& F^{\prime \prime \prime }+4F{F}^{\prime \prime }+4b\Theta \left(G-F^{\prime 2}\right)=0,\hfill \end{array}$$

(90)

the Rayleigh start and, after the substitutions $f=2F$, $\eta =2N$, exactly the Falkner–Skan–Dorodnitsyn system (38) of Section 4. The chart therefore spans the entire history from impulsive start to steady state, and no condition may be imposed at either end: the solution is determined by the equations alone.

Because the sign change survives (84)—it is now confined to the factor $D(1-c\tau F^{\prime })$, which vanishes at both ends—we do not march. We solve (85)–() globally on the rectangle $0\le N\le N_{\infty }$, $0\le s\le 1$ by Newton–Krylov relaxation, with central differences in s and second-order differences in N, imposing only $F^{\prime }(0,s)=0$, $F^{\prime }(N_{\infty },s)=1$, $G(N_{\infty },s)=1$ and the wall thermal condition.

6.3. Results

Table 7 and Figure 9 summarise the solution. The computed $s\to 0$ limit reproduces $2/\sqrt{\pi }$ to four figures for every $\beta$, and the $s\to 1$ limit reproduces the independently computed steady wall shear of Section 4 to better than $0.5\%$; since the two calculations share no code path, this is a genuine check on both. Three results follow.

(i) The coupling is inactive only at $\beta =0$. For $\beta =0$ the transformed wall shear is Mach independent at every s, not merely in the steady limit—the curves for $M_e=0$, 2 and 4 collapse to plotting accuracy (Figure 9(c)). This extends the classical steady statement to the whole unsteady history. For $\beta =0.5$ the same curves separate widely, $F^{\prime \prime }(0,1)=1.856$, $2.295$ and $3.273$ at $M_e=0$, 2, 4: it is the product $b\Theta g$, not b alone, that sets the effective pressure gradient, and at $M_e=4$ compressibility increases the steady wall shear by $76\%$ over its incompressible value at the same $\beta$.

(ii) The recovery factor is a function of time. At $\beta =0$ it falls monotonically from $0.88580$ at $s=0$—the closed form (61), which gives $0.88550$—to $0.84779$ at $s=1$, against $\sqrt{\mathrm{Pr}}=0.84853$. The two limiting results of this paper are therefore the two ends of a single curve, and the transition occupies $0.4\lesssim s\lesssim 0.9$, that is $0.5\lesssim \tau \lesssim 2.3$. A short-duration facility whose test time falls inside this window recovers neither value.

(iii) Pressure gradient depresses the recovery factor below both limits. For $\beta >0$ the curve undershoots $\sqrt{\mathrm{Pr}}$, reaching $0.822$ at $\beta =0.75$–$0.9$, before settling. The adiabatic wall of an accelerating compressible layer is thus cooler, relative to its own stagnation temperature, than either classical limit would suggest.

The solver converges for $-0.15\lesssim \beta <1$. The single value $\beta =1$, at which $c=2(a-b)$ vanishes and the $\tau$-derivative structure degenerates, required continuation and is not reported; the steady limit at $\beta =1$ is given in Section 4.

7. Conclusions

We have extended the unsteady incompressible similarity transformation of Sun[15] to the two-dimensional compressible laminar boundary layer. The principal results are the following.

1.

Starting point. The enabling step—that with $\overline{\eta }={\int }_0^y(\rho /{\rho }_r)d{y}^{\prime }$ and $V={\overline{\eta }}_{,t}+u{\overline{\eta }}_{,x}+(\rho /{\rho }_r)v$ the unsteady compressible continuity equation becomes exactly $u_{,\xi }+V_{,\overline{\eta }}=0$—is Stewartson’s[8], not ours. What we contribute is what can be built on it once Sun’s similarity ansatz[15] is layered on top.

2.

A master system. With $\eta =\overline{\eta }/\delta \left(x\right)$, $\tau =C{\nu }_{r}t/{\delta }^2$ and $\Psi =U\delta f$, the compressible equations reduce to the pair (31)–(), whose coefficients a, b, c are independent of x for the class $U=C{x}^m$ with $m=b/(2a-b)$, $c=2(a-b)$. The compressible momentum equation differs from the incompressible one only by the replacement $1\to \Theta g$. All the compressibility of the momentum problem is carried by the local static temperature.

3.

Correct degeneration. The steady limit is exactly the compressible Falkner–Skan–Dorodnitsyn system, and numerical solution recovers $f^{\prime \prime }\left(0\right)=0.46960$ independent of Mach number at $\beta =0$, $g\equiv 1$ at $\mathrm{Pr}=1$, and $r\simeq \sqrt{\mathrm{Pr}}$—all classical.

4.

Two exact solutions for the impulsively started plate. Besides Sun’s Kummer-function branch we found a diffusive branch $f=\sqrt{\tau }\Phi (\eta /\sqrt{\tau })$ with ${\Phi }^{\prime }=erfc(\xi /2)$, which reduces (43) exactly to ${\Phi }^{\prime \prime \prime }+\frac{\xi }{2}{\Phi }^{\prime \prime }=0$ and satisfies all boundary conditions, including $v\equiv 0$. It is the compressible Rayleigh solution, and it is a member of the same similarity family. The two branches have wall-shear coefficients $1.3702{\tau }^{-1/2}$ and ${\pi }^{-1/2}{\tau }^{-1/2}$. Both are regular at $\tau =1/2$, supporting Sun’s conclusion that the Rayleigh–Blasius transition need not be singular.

5.

An unsteady Crocco integral and a closed-form recovery factor. For $b=0$ the enthalpy equation at $\mathrm{Pr}=1$ is solved exactly by $g=A+B{f}_{,\eta }$ for all $\tau$. For arbitrary $\mathrm{Pr}$ the energy equation on the diffusive branch integrates in quadrature and yields

$$r={\displaystyle \frac{4}{\pi }}\sqrt{{\displaystyle \frac{\mathrm{Pr}}{2-\mathrm{Pr}}}}arctan\sqrt{{\displaystyle \frac{2-\mathrm{Pr}}{\mathrm{Pr}}}},$$

which equals unity at $\mathrm{Pr}=1$ but is otherwise not$\sqrt{\mathrm{Pr}}$: for air, $0.8855$ against $0.8485$. The associated Reynolds analogy factor is ${\mathrm{Pr}}^{-1/2}$ rather than ${\mathrm{Pr}}^{-2/3}$. Obtaining this required the observation that the adiabatic condition at a moving wall is $g_{,\eta }\left(0\right)=E{f}_{,\eta }\left(0\right)f_{,\eta \eta }\left(0\right)$, not $g_{,\eta }\left(0\right)=0$.

Several limitations should be stated plainly. The Chapman–Rubesin approximation $\rho \mu =C{\rho }_r{\mu }_r$ and constant $\mathrm{Pr}$ are essential to the reduction; relaxing them destroys the exact decoupling, though the transformation survives as a leading-order approximation. For $m\ne 0$ the parameter $\Theta$ varies with x and the reduction is a local-similarity statement. The Kummer branch solves the reduced equation but meets the outer condition only as $O\left({\tau }^{-1}\right)$. And nothing here addresses stability: whether the profiles exhibited are selected and persist is a separate question, and the natural next step is a linear stability analysis of the branches of Section 5 against the Mack modes[18] of high-speed boundary layers. Extending the reduction to a pressure-gradient flow with $m\ne 0$, where the coupling $b\Theta (g-f_{,\eta }^2)$ is active and the momentum and energy equations no longer separate, is the obvious next analytical step.

Appendix B.1. Definitions

Kummer’s equation $z{y}^{\prime \prime }+(\beta -z)y^{\prime }-\alpha y=0$ has the two standard solutions[47,48]

$$\mathrm{M}\left(\alpha ,\beta ,z\right)={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{{\left(\alpha \right)}_n}{{\left(\beta \right)}_n}}{\displaystyle \frac{z^n}{n!}},$$

(A7)

the confluent hypergeometric function of the first kind, regular at $z=0$ and growing as $e^z{z}^{\alpha -\beta }$ as $z\to +\infty$; and

$$\mathrm{U}\left(\alpha ,\beta ,z\right)={\displaystyle \frac{\pi }{sin\pi \beta }}\left[{\displaystyle \frac{\mathrm{M}\left(\alpha ,\beta ,z\right)}{\Gamma (1+\alpha -\beta )\Gamma \left(\beta \right)}}-z^{1-\beta }{\displaystyle \frac{\mathrm{M}\left(1+\alpha -\beta ,2-\beta ,z\right)}{\Gamma \left(\alpha \right)\Gamma (2-\beta )}}\right],$$

(A8)

the function of the second kind, which decays as $z^{-\alpha }$ as $z\to +\infty$. Only $\mathrm{U}\left(·,·,·\right)$ appears in the bounded solution, which is why $c_2=0$ in (44). We use the values $\Gamma (2/3)=1.354118$ and $\Gamma (5/6)=1.128787$.

Appendix B.2. The Functions h 11 and h 12

With the similarity argument

$$X(\eta ,\tau )={\displaystyle \frac{{\left(3\eta \tau +2\right)}^2}{12{\tau }^3}},$$

(A9)

the auxiliary functions entering (44) are

$$\begin{array}{cc}\hfill h_{11}(\eta ,\tau )& ={\displaystyle \frac{27\left(\eta \tau +\frac{2}{3}\right)}{10{\tau }^4}}\left({\eta }^2{\tau }^2+\frac{2}{9}{\tau }^3+\frac{4}{3}\eta \tau +\frac{4}{9}\right)e^{-X}\mathrm{U}\left(\frac{5}{6},\frac{3}{2},X\right)-{\displaystyle \frac{18\left(\eta \tau +\frac{2}{3}\right)}{5\tau }}{e}^{-X}\mathrm{U}\left(-\frac{1}{6},\frac{3}{2},X\right),\hfill \end{array}$$

(A10)

$$\begin{array}{cc}\hfill h_{12}(\eta ,\tau )& ={\displaystyle \frac{\partial h_{11}}{\partial \eta }}={\displaystyle \frac{9}{2{\tau }^3}}\left({\eta }^2{\tau }^2-\frac{2}{9}{\tau }^3+\frac{4}{3}\eta \tau +\frac{4}{9}\right)e^{-X}\mathrm{U}\left(\frac{5}{6},\frac{3}{2},X\right)-6e^{-X}\mathrm{U}\left(-\frac{1}{6},\frac{3}{2},X\right).\hfill \end{array}$$

(A11)

and $g_{11}$, $g_{12}$ are obtained from (A10)–() by replacing $\mathrm{U}\left(\alpha ,\frac{3}{2},X\right)$ with $\mathrm{M}\left(\alpha ,\frac{3}{2},X\right)$; because $\mathrm{M}\left(\alpha ,\beta ,X\right)e^{-X}$ does not decay as $X\to \infty$, these are the unbounded solutions and are discarded. Differentiating () once more and using the contiguous relation ${\displaystyle \frac{d}{dz}}\mathrm{U}\left(\alpha ,\beta ,z\right)=-\alpha \mathrm{U}\left(\alpha +1,\beta +1,z\right)$ gives the compact form (46) for $f_{,\eta \eta }$.

Correction. In Eq. (B1) of Ref. [15] the argument (A9) is printed with $12{\tau }^2$ in the denominator. The exponent must be 3, as in (A9); with $12{\tau }^2$ the function (44) does not satisfy (43), whereas with (A9) we verified the residual of (43) to be $<{10}^{-8}$ over $0.3\le \tau \le 5$, $0\le \eta \le 3$, and the internal consistency checks ${\partial }_{\eta }{h}_{11}=h_{12}$ and ${\partial }_{\eta }f=f_{,\eta }$, ${\partial }_{\eta }{f}_{,\eta }=f_{,\eta \eta }$ to the same accuracy. We note this because the dimensional argument X must scale as ${\eta }^2/\tau$ for large $\tau$ at fixed $\xi =\eta /\sqrt{\tau }$, which (A9) does and $12{\tau }^2$ does not; this is what makes the limit (52) exist.