Linear and non-linear analysis of reaction-induced shear instabilities in navier-stokes flow

Abstract

Hydrodynamic instabilities created by a viscosity stratification-led shear are inevitable in the context of effective drag-reduction techniques used in food or chemical industries, aircraft wings, the famous polymer drag reduction, and lubricated pipe-lining for oil recovery processes. A chemical reaction can change the viscosity by modifying the chemical composition of underlying fluids; hence, generated instabilities are categorized into the bigger class called Chemo-Hydrodynamic instabilities. Mathematical modeling and solving the corresponding equations are essential to analyze these instabilities. From the industrial viewpoint, flow instability control and its transition to different states are important, too. It can be achieved by tuning the underlying dimensional groups representing various forces in the problem. We deal with incompressible, Newtonian, and miscible fluids flowing through a 2D channel. So the conservation of mass is expressed by the continuity equation, and the Navier-Stokes equations well describe the momentum’s conservation. Moreover, miscible reactive fluids satisfy a Convention-Diffusion-Reaction (CDR) equation for each solute concentration that explains how two physical processes, such as convection and diffusion, interact with the chemical reaction. Flow equations are coupled with these CDR equations to describe the reactive fluid flow in a channel. We solve the system numerically in the non-linear regime with the help of a finite volume method designed on a staggered grid. The scheme uses the WENO (Weighted Essentially Non-Oscillatory) scheme to handle the solute transport in highly convection-dominated regimes. The velocity is predicated for an intermediate time step using Adams-Bashforth and Crank-Nicolson methods. Then a pressure correction is executed following the continuity equation by the intermediate velocity to obtain the velocity at the new time step. A linear stability analysis (LSA) is performed in a layered flow configuration where one iso-viscous reactant fluid is assumed to flow parallelly in the axial direction, overlying on the other. At the contact zone, they produce a product of different viscosity in a fixed width of the reaction zone throughout the channel length. Thus the base state concentrations of two reactants and product fluid are assumed to follow a Reaction–Diffusion (RD) system till a freezing time for which the product is formed in a zone of that fixed width and made time-independent under Quasi-Streaty-State-Approximations. The RD system is numerically solved using a finite difference method that uses Crank-Nicolson method for diffusion term approximation and reaction terms are handled explicitly. The base state velocity is also assumed to be steady, satisfying the fully developed Navier-Stokes equation, where the viscosity is considered an exponential function of the product concentration. These basic states are perturbed with normal modes type perturbations . This converts the stability problem into a generalized Orr-Sommerfied equation under voriticity stream functions formulations for the velocity perturbations, which is further coupled to linear perturbed equations from the RD system. The Chebyshev spectral method is used to solve the final linear stability problem. The growth/decay of perturbations is obtained by finding eigenvalues with positive imaginary parts for a wide range of disturbance wave numbers. The flow goes most unstable for shorter waves because of the overlap mode, for which the phase speed of perturbations matches with the base velocity within the mixed layer. An energy budget analysis reveals that most energy is transferred to perturbations from the base viscosity profiles. If the reaction produces a less viscous product than the reactants, the growth rate is higher than that of a more viscous product except for very choppy/short waves. This is also shown through the critical log-mobility ratios (Rcrit c ) and Damköhler numbers (Da). Furthermore, it is found that if convection dominates more than solute diffusion, then the flow is more unstable. With an increasing Reynolds number (Re), the flow is found to be more unstable. However, flow can become more stable with increasing Reynolds number for a lesser rate of reaction. The inflectional instability mode is also found at this low reaction rate for a less viscous product case. Roll-up-like of Kelvin-Helmholtz instability (KHI) pattern develops at the upper reaction front in the linear regime if the product fluid is more viscous than reactants and flow is initiated from a perturbed basic state with a sinusoidal disturbance having a wavenumber predicated to be most unstable from LSA. However, the lower reaction front stably diffuses in a flat manner. One reason is that the base state velocity profile avoids infection at the lower reaction front while it possesses a favorable inflection point at the upper front. Secondly, roll-ups grow with time as the multiple streamlines oscillate synchronously, behaving like they are in a phase-locked system. A momentary analysis shows moments of reaction rate and product fluid shift oppositely in a transverse direction as the instability grows with time. The standard deviation in the transversely averaged product concentration profile from its mean increases in time at different rates, showing an intermediate convection-dominated regime sandwiched between two diffusion-dominated regimes. Severe interfacial deformations occur due to instability for greater Damköhler , Péclet (P e), and Reynolds numbers, which validates the LSA results. Furthermore, the instability onsets early for greater Damköhler and Reynolds numbers while delays for higher Péclet values. The KHI pattern grows for less viscous product fluids, contrasting to that of the previously observed more viscous product case. That is, roll-ups grow at the lower front, showing stabilizing effect. This is due to inflection in the base velocity at the lower reaction front, while the double derivative of the velocity gets away from zero at the upper reaction front. Since unstable roll-ups grow at the lower front, the front eventually collides with the lower reflective channel boundary. This makes streamlines out of phase, and the flow becomes more chaotic. The less viscous product spreads vertically more than a more viscous product. The instability onsets early for a less viscous reaction product compared to a more viscous product with equal log-mobility ratio magnitude. The (Da − Rcrit c ) phase plane obtained from computing the onset times shows similar asymmetric behavior around Rcrit c = 0 line as was observed from LSA. In the displacement flow, when a miscible solution of one reactant (A) displaces another iso-viscous reactant (B) and produces a more viscous product (C), KHI is found alongside an elongated finger. A local increase in viscosity gradient due to the formation of the more viscous product causes KHI only at one reaction front. While at the other front, the viscosity gradient decreases in the flow direction; hence no KHI is noticed there. A laminar Horse-Shoe vortex is also found to develop near the wall at the channel inlet, where the less viscous reactant pushes the more viscous product. These instabilities may not occur even at high reaction rates (sufficiently large Da), and viscosity contrast greater than a critical value is required to trigger them. The onset time and log-mobility ratio plots show unstable and stable time zones for each Da, P e, and Re value. Further, the onset time can be linearly scaled with the Péclet number; as such, the boundary curves between stable and unstable zones merge. This establishes proportionate dynamics with respect to P e in the early stages of the instability. Moreover, a reverse dependency of onset on lower Rc values for higher Reynolds numbers is observed. Like the layered flow system, the flow-directed KH roll-ups occur either at the A−C interface or the C−B interface in the displacement flow, depending on the product’s viscosity. However, as no initial perturbations are given, there are KH roll-ups at different axial stations with different amplitudes. Moreover, the number of KH roll-ups at the reactive interface is more when the product is less viscous, and full vortex completion of KH roll-ups is noticed. The total amount of product formation and reaction rate is more for a less viscous product, too. The product fluid mixing and vorticity strength are also more if the product is less viscous than the reactants. The instability onsets early for a less viscous product, with some exceptions near the critical log-mobility ratio. Keywords: Numerical method for partial differential equations, Linear stability analysis, Finite volume method, Finite difference, Chebyshev spectral collocation method, shear instability, channel flow, Chemo-Hydrodynamic instability, Kelvin-Helmholtz instability, Horse-Shoe Instability, control measure, layered flow, displacement flow.