A closed-form solution is presented for internal compressible flows with an arbitrary combination of area change, heat addition, friction and non-uniform flow. The approach assumes steady flow of a calorically perfect ideal gas using the integral form of the governing equations. It is shown that the equations reduce to a simple, bi-quadratic equation in exit Mach number. This solution is compared to previously known solutions for Rayleigh flow, Fanno flow, isentropic flow and normal shock relations. Theoretical solutions for sudden expansion, sudden contraction, and supersonic–supersonic two-stream constant area mixing are also presented. These solutions were found to be in agreement with data available in the literature.

The study of entropy in the context of quasi-one-dimensional flow is expanded in this work. Specifically, a new entropy transport equation is derived and integrated into a closed-form algebraic expression for entropy change. The derivation identifies the respective components of the entropy change and is valid for flows with an arbitrary combination of area change, heat transfer, and friction. The irreversibility is identified and found to be composed of frictional dissipation, irreversible heat transfer, and irreversible flow work. The irreversible flow work is a new term that results from the restriction to quasi-one-dimensional flows. The algebraic expression for each component of the entropy change is first validated using several canonical flows (i.e., isentropic, Fanno, Rayleigh, and a normal shock). A unique entropy production mechanism is identified for each of the entropy producing canonical flows (e.g., Fanno, Rayleigh, and a normal shock). Two additional cases, sudden expansion and contraction, are then considered and show that irreversible flow work is the sole entropy production mechanism. Finally, simultaneous friction and heat transfer are examined, and the overall entropy change is decomposed into the respective contributions from frictional dissipation and the heat transfer terms. In all cases, the net entropy change from the newly derived expressions agrees with known solutions to within numerical precision.

An entropy-constrained closure is developed for steady quasi-one-dimensional flow of a calorically perfect ideal gas. Net irreversibility is represented by a prescribed entropy change between the inlet and outlet or, equivalently, by a stagnation pressure ratio together with the stagnation temperature ratio implied by the energy balance equation. The thermodynamic constraint, enforced through Gibbs relation together with conservation of mass and energy, yields an algebraic mapping from the inlet flow state, area ratio and prescribed stagnation property ratios to the outlet flow state. The momentum balance is retained only as an auxiliary post-processing relation and may be used to infer integral quantities such as a net wall force or an equivalent mean wall pressure. For an ideal gas with integer number of molecular degrees of freedom $f$ , the resulting equation reduces to a polynomial of degree $f+1$ in the squared outlet Mach number. The $f=3$ case reduces to a quartic, and the $f\to \infty$ limit admits an explicit Lambert- $W$ solution; existence, multiplicity, choking and admissibility conditions consistent with the second law of thermodynamics are obtained for general $f$ . Classical isentropic nozzle, Rayleigh and Fanno flows, normal shocks, and simultaneous friction and heat transfer in a constant-area duct are recovered in the respective limiting cases. Comparisons with experimental measurements of nozzle wall pressures, sudden expansion outlet Mach numbers and sudden contraction stagnation pressure ratios show good agreement with predictions across a wide range of inlet Mach numbers and area ratios.

The spatial and temporal density variations that are characteristic of a compressible free shear layer give rise to significant optical aberrations, leading to a reduction in performance of an optical system viewing through the flow. Previous experimental, computational, and analytical studies into the aero-optical effects of a compressible shear layer have shown disagreement in the magnitude and spatial distribution of density variations in the flow and, thus, the thermodynamic origin of the optical distortions. In this study, we leverage advanced computational fluid dynamics simulations to effectively investigate the thermofluidic interactions present in a compressible free shear layer and provide insight into the driving mechanism for the spatial variations that are observed in numerical and experimental realizations of density, static temperature, and total temperature.

This paper presents an explicit Lambert-W inversion of the classical Fanno relation for steady, adiabatic, quasi-one-dimensional (Q1D) flow of a calorically perfect ideal gas in a constant-area duct with wall friction. The formulation isolates the downstream Mach number as a function of the upstream Mach number, the specific-heat ratio, and the prescribed friction-length parameter 4fL/D. The appropriate real branch of the Lambert-W function is selected from the inlet sonic regime: the W−1 branch for subsonic inlet flow and the W0 branch for supersonic inlet flow. The resulting expression eliminates the conventional iterative inversion of the implicit Fanno equation for admissible, unchoked cases. Numerical comparisons with the classical implicit Fanno relation show agreement to numerical precision for representative subsonic and supersonic cases. The expression may be useful for instruction, code verification, preliminary sizing and rapid evaluation of idealized compressible duct systems, and reduced-order analysis of microchannels where wall friction is a dominant effect.