Linear instability of Poiseuille flows with highly non-ideal fluids

Abstract

The objective of this work is to investigate linear modal and algebraic instability in Poiseuille flows with fluids close to their vapour-liquid critical point. Close to this critical point, the ideal gas assumption does not hold and large non-ideal fluid behaviours occur. As a representative non-ideal fluid, we consider supercritical carbon dioxide (CO 2 ) at pressure of 80 bar, which is above its critical pressure of 73.9 bar. The Poiseuille flow is characterized by the Reynolds number (Re = ρ * w u * r h * /µ * w ), the product of Prandtl (Pr = µ * w C * pw /κ * w ) and Eckert number (Ec = u * 2 r /C * pw T * w ), and the wall temperature that in addition to pressure determines the thermodynamic reference condition. For low Eckert numbers, the flow is essentially isothermal and no difference with the well-known stability behaviour of incompressible flows is observed. However, if the Eckert number increases, the viscous heating causes gradients of thermodynamic and transport properties, and non-ideal gas effects become significant. Three regimes of the laminar base flow can be considered, subcritical (temperature in the channel is entirely below its pseudo-critical value), transcritical, and supercritical temperature regime. If compared to the linear stability of an ideal gas Poiseuille flow, we show that the base flow is more unstable in the subcritical regime, inviscid unstable in the transcritical regime, while significantly more stable in the supercritical regime. Following the corresponding states principle, we expect that qualitatively similar results will be obtained for other fluids at equivalent thermodynamic states.

Boundary-layer stability of supercritical fluids in the vicinity of the Widom line

“…The derivatives of the properties, such as ∂µ/∂ρ, ∂ 2 p/∂T 2 are also important in building the stability operator. The gradients are determined using a second-order finite differences method (numerical details can be found in Ren et al 2019).…”

“…Here µ is the dynamic viscosity, λ = µ b − 2/3µ the second viscosity, µ b the bulk viscosity and κ is the thermal conductivity. It is known that µ b has a very limited effect on the linear stability of channel flows (Ren et al 2019). Results presented in the following sections are subjected to µ b = 0.…”

“…The linear stability equations for the non-ideal gas have been derived and documented in Ren et al (2019). It is known that for simple compressible systems (e.g.…”

“…Here q = (ρ , u , v , w , T ) T is the perturbation vector and the detailed expressions for the matrices L t , L x , L y , L z , L q , V xx , V yy , V zz , V xy , V yz and V xz are functions of the dimensionless parameter (2.4), the base flow and thermodynamic and transport properties, and their detailed expressions can be found in Ren et al (2019). The perturbation is assumed to have the normal-mode form,…”

“…In terms of heat transfer, as concluded in a recent review (Pizzarelli 2018), future applications are still limited by the poor understanding and prediction of heat transfer deterioration in supercritical fluids. The linear stability of flows with such highly non-ideal fluids has only been considered recently by Ren, Fu & Pecnik (2019) for plane Poiseuille flows. It was found that compared to ideal gases at the same conditions, the non-ideal gas can become more stable/unstable, or even inviscid unstable in different thermodynamic regimes.…”