Similarity solutions of a class of unsteady compressible laminar boundary layers

The unsteady compressible laminar boundary layer is the natural setting in which the earliest, pre-transitional history of a high-speed viscous flow is written, yet almost no exact solutions of the two-dimensional unsteady compressible boundary-layer equations are known. We extend a recently proposed incompressible similarity transformation [B. H. Sun, Phys. Fluids 36, 083616 (2024)] to compressible flow. Building on the unsteady Howarth--Dorodnitsyn reduction introduced by Stewartson [Q. J. Mech. Appl. Math. 4, 182 (1951), Section 7], and combining it with an undetermined boundary-layer thickness δ(x) and a diffusion time τ=Cνᵣ t/δ², we show that the compressible mass, momentum and energy equations collapse onto a pair of partial differential equations in two variables (η,τ) whose coefficients are independent of the streamwise coordinate. Remarkably, the compressible momentum equation differs from its incompressible counterpart only through the replacement 1→Θg, where g is the reduced total enthalpy and Θ=1+1/2(γ-1)Mₑ². Three applications are developed. (i) The steady limit of the system is shown to be exactly the compressible Falkner--Skan--Dorodnitsyn system, which we solve numerically as a validation. (ii) For the impulsively started semi-infinite plate we obtain two exact solutions of the momentum equation of distinct similarity type---one in Kummer functions, one diffusive---and, at Pr=1, a closed-form Crocco integral for the temperature; for arbitrary Prandtl number the energy equation is integrated in quadrature and yields the closed-form adiabatic recovery factor r=(4/π)√Pr/(2-Pr)√(2-Pr)/Pr, which differs from the classical steady value √Pr. (iii) For m ≠ 0 we solve the unsteady system globally in a Williams--Rhyne chart, in which both ends are characteristic, and show that the transformed wall shear is Mach independent for all time when $m=0$ but not otherwise, and that the recovery factor migrates continuously from the unsteady closed form to √Pr. Throughout, the sense in which the solutions are exact---exact under the Chapman--Rubesin and constant-Prandtl-number assumptions, in the same sense as the classical steady results---is stated explicitly, and the transformation is validated against the Blasius, Falkner--Skan, Crocco and compressible-Rayleigh results of the classical literature, including the induced outflow v_∞=(γ-1)Mw²√Cνᵣ/2π t.