We perform the linear modal stability analysis of a pipe flow subjected to a step-like increment in the flow rate from a steady initial flow with flow rate, $Q_i$, to a final flow with flow rate, $Q_f$, at the time, $t_c$. A step-like increment in the flow rate induces a non-periodic unsteady flow for a definite time interval. The ratio, $\Gamma_a={Q}_i/{Q}_f$, parameterizes the increase in the flow rate, and it ranges between $0$ to $1$. The stability analysis for a pipe flow subjected to a step-like increment in the flow rate from the steady laminar flow ($\Gamma_a>0$) is not reported in the literature. The present work investigates the effect of varying $\Gamma_a$ on the stability characteristics of an unsteady pipe flow. The step-like increment in the flow rate for $0\leq\Gamma_a\leq0.72$ induces a viscous type instability for a definite duration and the flow is modally unstable. The non-axisymmetric disturbance with azimuthal wavenumber, $m=1$ is the most unstable mode. The flow is highly-unstable for $\Gamma_a=0$ and the flow becomes less unstable with an increase in $\Gamma_a$. The flow becomes stable before it attains the steady-state condition for all $\Gamma_a$.

A one-dimensional mathematical model is developed for an unsteady incompressible laminar flow in a circular pipe subjected to a rapid change in the flow rate from an initial flow with flow rate, Qi, to a final flow with flow rate, Qf, in a step-like fashion at an arbitrary time, tc. The change in the flow rate may either be an increment, Qf > Qi, or a decrement, Qf < Qi. The change time, tc, may either belong to the initial flow remaining in a temporally developing state or temporally developed state. The developed model is solved using the Laplace transform method to deduce generalized analytical expressions for the flow characteristics, viz., velocity, pressure gradient, wall shear stress, and skin friction factor, CfRe, where Re is Reynolds number based on the cross-sectional area-averaged velocity and pipe radius. Exact solutions for λa=Qi/Qf=0 and λd=Qf/Qi=0 with tc≥tsi are available in the literature and the present generalized analytical solutions fill the remaining range of parameters, 0<λa<1 and 0<λd<1 with 0<tc<tsi and tc≥tsi, where tsi is the time at which the initial flow reaches the temporally developed state. Exact solutions for canonical pipe flow problems reported in the literature are deduced as subsets of the derived generalized solutions. The parametric study reveals the effects of varying λa or λd and tc on the quantities of practical importance, viz., τs and CfRe, where τs is the time required for the final flow to reach the temporally developed state.

<div class="section abstract"><div class="htmlview paragraph">In order to meet the stringent emission norms like EU6 and EU7 together with CAFÉ/CAFC norms, down-sizing of the engine is one of the thrust areas of focus among the OEMs. To this end, keeping the engine size small but to achieve the required power output, advanced Turbo charged Gasoline Direct Injection engine technology (TGDI) has emerged. However, TGDI technology is susceptible to an abnormal combustion phenomenon termed as Low Speed Pre-Ignition (LSPI) event. This event happens prior to the intended combustion, which causes the catastrophic engine damage. Several studies in terms of simulation and experiments to understand this phenomenon are reported in the literature. The main factors influencing this occurrence are found to be engine design and calibration, fuel types and engine oil formulation (in terms of calcium content). In this paper, advanced methods to handle the LSPI occurrence severity and component level advances in design robustness to avoid the engine damage are reported. The developed techniques include robust piston design, Piston ring design and coating technologies. In overall, the techniques related to engine design to predict the LSPI occurrence in terms of peak cylinder pressure (PCP) and number of events are summarized.</div></div>

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