Effects of relaxation time on start-up time for starting flow of Maxwell fluid in a pipe

Although the analytical solution of the starting flow of Maxwell fluid in a pipe has been derived for a long time, the effect of relaxation time λ on start-up time ts of this flow is still not well understood. Especially, there exist a series of jumps on the ts-λ. curve. In this paper we introduce a normalized mechanical energy by mode decomposition and mathematical analogy to describe the start-up process. An improved definition of start-up time is presented based on the normalized mechanical energy. It is proved that the ts-λ. curve contains a series of jumps if λ is larger than a critical value. The exact positions of the jumps are determined and the physical reason of the jumps is discussed.

Starting flow in a rotating parallel plate channel

The starting flow due to a suddenly applied pressure gradient in a parallel plate channel which is rotating as a system is studied. Exact analytic series solutions to the unsteady Navier-Stokes equations are found by both the Laplace transform method and the separation of parameters method, the latter is shown to be superior. Rotation not only induces a secondary transverse flow but also alters the character of the transient flow rate and velocity profiles. Back flow and inertial oscillations occur, especially at higher rota- tion rates.

Hypersonic starting flow at high angle of attack

Compressible starting flow at small angle of attack （AoA） involves small amplitude waves and time-dependent lift coefficient and has been extensively studied before. In this paper we consider hypersonic starting flow of a two-dimensional flat wing or airfoil at large angle of attack involving strong shock waves. The flow field in some typical regions near the wing is solved analytically. Simple expressions of time-dependent lift evolutions at the initial and final stages are given. Numerical simulations by compuational fluid dynamics are used to verify and complement the theoretical results. It is shown that below the wing there is a straight oblique shock （OSW） wave, a curved shock wave （CSW） and an unsteady horizontal shock wave （USW）, and the latter moves perpendicularlly to the wing. The length of these three parts of waves changes with time. The pres- sure above OSW is larger than that above USW, while across CSW there is a significant drop of the pressure, making the force nearly constant during the initial period of time. When, however, the Mach number is very large, the force coefficient tends to a time-independent constant, proportional to the square of the sine of the angle of attack.

Subsonic impulsively starting flow at a high angle of attack with shock wave and vortex interaction

Impulsively starting flow, by a sudden attainment of a large angle of attack, has been well studied for incompressible and supersonic flows, but less studied for subsonic flow. Recently,a preliminary numerical study for subsonic starting flow at a high angle of attack displays an advance of stall around a Mach number of 0.5, when compared to other Mach numbers. To see what happens in this special case, we conduct here in this paper a further study for this case, to display and analyze the full flow structures. We find that for a Mach number around 0.5, a local supersonic flow region repeatedly splits and merges, and a pair of left-going and right-going unsteady shock waves are embedded inside the leading edge vortex once it is sufficiently grown up and detached from the leading edge. The flow evolution during the formation of shock waves is displayed in detail. The reason for the formation of these shock waves is explained here using the Laval nozzle flow theory. The existence of this shock pair inside the vortex, for a Mach number only close to 0.5, may help the growing of the trailing edge vortex responsible for the advance of stall observed previously.

AN EXPERIMENTAL AND NUMERICAL INVESTIGATION OF THE STARTING FLOW OF A UNIFORMLY ACCELERATING CIRCULAR CYLIINDER

The starting flow around a uniformly accelerated circular cylinder is investigated by flow visualization techncque and solving N-S equations numerically in detail.The results of the starting flow are similar with those of the impulsively started one.At lower Re_α,numbers the “phenomenon a”is revealed;the“phenomenon β”appears at higher Re_α numbers,but with only one secondary vortex.At the early stage of the starting vortex formation,the flow is laminar;at the later stage,the flow transforms into turbulent flow.The velocity distribution in the flow field is measured with the multiwires installment.The calculational results are compared with the results of experiment,and both are in agreement with each other.