Weak Nonlinear Stability Research Articles

Laminar–turbulent transition in pipe flow for Newtonian and non-Newtonian fluids

A cylindrical pipe facility with a length of 32 m and a diameter of 40 mm has been designed. The natural transition Reynolds number, i.e. the Reynolds number at which transition occurs as a result of non-forced, natural disturbances, is approximately 60 000. In this facility we have studied the stability of cylindrical pipe flow to imposed disturbances. The disturbance consists of periodic suction and injection of fluid from a slit over the whole circumference in the pipe wall. The injection and suction are equal in magnitude and each distributed over half the circumference so that the disturbance is divergence free. The amplitude and frequency can be varied over a wide range.First, we consider a Newtonian fluid, water in our case. From the observations we compute the critical disturbance velocity, which is the smallest disturbance at a given Reynolds number for which transition occurs. For large wavenumbers, i.e. large frequencies, the dimensionless critical disturbance velocity scales according to Re−1, while for small wavenumbers, i.e. small frequencies, it scales as Re−2/3. The latter is in agreement with weak nonlinear stability theory. For Reynolds numbers above 30 000 multiple transition points are found which means that increasing the disturbance velocity at constant dimensionless wavenumber leads to the following course of events. First, the flow changes from laminar to turbulent at the critical disturbance velocity; subsequently at a higher value of the disturbance it returns back to laminar and at still larger disturbance velocities the flow again becomes turbulent.Secondly, we have carried out stability measurements for (non-Newtonian) dilute polymer solutions. The results show that the polymers reduce in general the natural transition Reynolds number. The cause of this reduction remains unclear, but a possible explanation may be related to a destabilizing effect of the elasticity on the developing boundary layers in the entry region of the flow. At the same time the polymers have a stabilizing effect with respect to the forced disturbances, namely the critical disturbance velocity for the polymer solutions is larger than for water. The stabilization is stronger for fresh polymer solutions and it is also larger when the polymers adopt a more extended conformation. A delay in transition has been only found for extended fresh polymers where delay means an increase of the critical Reynolds number, i.e. the number below which the flow remains laminar at any imposed disturbance.

A weak nonlinear stability analysis of double diffusive convection with cross-diffusion in a fluid-saturated porous medium

The effect of “Cross Diffusion” on the linear and nonlinear stability of double diffusive convection in a fluid-saturated porous medium has been studied analytically. In the case of linear theory, the normal mode technique has been used and the condition for the maintenance of “finger” and “diffusive” instabilities have been obtained. It has been found that fingers can form by taking cross diffusion terms of appropriate sign and magnitude even though both components make stabilizing contributions to the net vertical density gradient. It has also been shown that “finger” and “diffusive” instabilities can never occur simultaneously. The nonlinear theory is based on the truncated representation of Fourier series and it has been found that the finite amplitude convection may occur when both initial property gradients are stabilizing. Further, the region of finite amplitude instability always encloses the region of infinitesimal oscillatory instability. The effects of permeability and cross-diffusion terms on the heat and mass transports have also been clearly brought out.

STRONG NON-LINEAR DYNAMICS OF CUTTING PROCESSES

A strong non-linear dynamic model is developed to investigate the dynamic characteristics of cutting processes. First, the multiple scales method is applied to study the weak non-linear stability, and then the numerical method to solve the problems of strong non-linearity in cutting processes. The former shows that the subcritical bifurcation predicted by the weak non-linear theory is compatible with that predicted by the strong non-linear theory. The numerical study reveals that different cutting thicknesses result in qualitatively different behaviour of the finite amplitude instability. Going from small cutting thicknesses to the large ones, the behaviour of the finite amplitude instability can be divided into an unconditional stable region, a conditional stable region, a periodic region and a breakdown region.

Interfacial wave stability of concurrent two-phase flow in a horizontal channel

A wave front perturbation method is applied to a one-dimensional two-phase flow in a horizontal duct to analyze the interfacial wave stability. The governing equations are thus reduced to a first-order ordinary differential equation (ODE). The linear solution of this ODE agrees with the conclusion from the small amplitude wave perturbation approach. The non-linear theory suggests that the linear stability result is sufficient for predicting instability in wavy flow, and may over-predict the critical gas velocity for large disturbances. Accordingly, a weak non-linear stability criterion is obtained. The type of unstable wave, under-cut wave or roll wave, also can be predicted analytically.

A nonlinear dynamic model of cutting

A simple modified dynamic model of nonlinear cutting forces considering the geometry of orthogonal cutting, the relationship of cutting angles, the effects of tool cutting on a wavy surface of workpiece and the variations of cutting angles in the cutting process, is derived for predicting the nonlinear chatter phenomenon. Firstly, the linear stability of the cutting process was determined by normal mode analysis. The results show that the cutting feed presents a stabilizing effect, and that larger stiffness of the cutting tool structure gives a more stable region in the W- V (cutting thickness-cutting velocity) plane, secondly, the multiple scale method is used to study the weak nonlinear stability of the cutting process. The results show that the behavior of the cutting process in the linearly stable region is nonlinearly stable, but in the linearly unstable region the amplitude of chatter does not grow infinitely, i.e. supercritical stability exists in the nonlinear region. In the supercritical region, the nonlinear chatter frequency is smaller than that of linear prediction, and when cutting width increases the chatter frequency decreases slowly.