CASE STUDY TYPES OF FLOW

TABLES OF CONTENTPAGE

1) THEORY OF STEADY AND UNSTEADY FLOW2) THEORY OF LAMINAR AND TURBELANT FLOW3) IMAGES FOR STEADY AND UNSTEADY FLOW4) IMAGES FOR LAMINAR AND TURBELANT FLOW5) EXAMPLE OF APPLICATION LAMINAR AND TURBELANT FLOW (CANDLE SMOKE)6) EXAMPLE OF APPLICATION LAMINAR AND TURBELANT FLOW ( BLOOD VESSEL)2-3

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THEORY OF STEADY AND UNSTEADY FLOW

Hydrodynamics simulation of theRayleighTaylor instability steady: A steady flow is one in which the conditions (velocity, pressure and cross-section) may differ from point to point but DO NOT change with time. unsteady: If at any point in the fluid, the conditions change with time, the flow is described as unsteady. (In practise there is always slight variations in velocity and pressure, but if the average values are constant, the flow is considered steady.When all the time derivatives of a flow field vanish, the flow is considered to be asteady flow. Steady-state flow refers to the condition where the fluid properties at a point in the system do not change over time. Otherwise, flow is called unsteady (also called transient). Whether a particular flow is steady or unsteady, can depend on the chosen frame of reference. For instance, laminar flow over asphereis steady in the frame of reference that is stationary with respect to the sphere. In a frame of reference that is stationary with respect to a background flow, the flow is unsteady.Turbulentflows are unsteady by definition. A turbulent flow can, however, bestatistically stationary. According to Pope.The random fieldU(x,t) is statistically stationary if all statistics are invariant under a shift in time.This roughly means that all statistical properties are constant in time. Often, the mean field is the object of interest, and this is constant too in a statistically stationary flow.Steady flows are often more tractable than otherwise similar unsteady flows. The governing equations of a steady problem have one dimension fewer (time) than the governing equations of the same problem without taking advantage of the steadiness of the flow field.

THEORY OF LAMINAR AND TURBELANT FLOWTurbulenceis flow characterized by recirculation,eddies, and apparentrandomness. Flow in which turbulence is not exhibited is calledlaminar. It should be noted, however, that the presence of eddies or recirculation alone does not necessarily indicate turbulent flowthese phenomena may be present in laminar flow as well. Mathematically, turbulent flow is often represented via aReynolds decomposition, in which the flow is broken down into the sum of anaveragecomponent and a perturbation component.It is believed that turbulent flows can be described well through the use of theNavierStokes equations.Direct numerical simulation(DNS), based on the NavierStokes equations, makes it possible to simulate turbulent flows at moderate Reynolds numbers. Restrictions depend on the power of the computer used and the efficiency of the solution algorithm. The results of DNS have been found to agree well with experimental data for some flows. Most flows of interest have Reynolds numbers much too high for DNS to be a viable option,given the state of computational power for the next few decades. Any flight vehicle large enough to carry a human (L > 3 m), moving faster than 72km/h (20m/s) is well beyond the limit of DNS simulation (Re = 4 million). Transport aircraft wings (such as on anAirbus A300orBoeing 747) have Reynolds numbers of 40 million (based on the wing chord). In order to solve these real-life flow problems, turbulence models will be a necessity for the foreseeable future.Reynolds-averaged NavierStokes equations(RANS) combined withturbulence modellingprovides a model of the effects of the turbulent flow. Such a modelling mainly provides the additional momentum transfer by theReynolds stresses, although the turbulence also enhances theheatandmass transfer. Another promising methodology islarge eddy simulation(LES), especially in the guise ofdetached eddy simulation(DES)which is a combination of RANS turbulence modelling and large eddy simulation.From the standpoint of analysis of fluid flow, the distinction between laminar and turbulent is one of the most important. With the power of present-day computers essentially any laminar flow can be predicted with better accuracy than can be achieved with laboratory measurements. But for turbulent flow this is not the case. At present, except for the very simplest of flow situations, it is not possible to predict details of turbulent fluid motion. In fact, it is sometimes said that we do not even actually know what turbulence is. But certainly, at least from a qualitative perspective,we can readily recognize it, and on this basis it is clear that most flows of engineering importance are turbulent. It is our purpose in the present section to provide some examples that will help in developing intuition regarding the differences between laminar and turbulent flows. Probably our most common experience with the distinction between laminar and turbulent flow comes from observing the flow of water from a faucet as we increase the flow rate. We depict this in Fig. 2.19. Part (a) of the figure displays a laminar (and steady) relatively low-speed flow in which the trajectories followed by fluid parcels are very regular and smooth; furthermore, there is no indication that these trajectories might exhibit drastic changes in direction. In part (b) of the figure we present a flow that is still laminar, but one that results as we open the faucet more than in the previous case, permitting a higher flow speed. In such a case the surface of the stream of water begins to exhibit waves, and these will change in time (basically in a periodic way). Thus the flow has become time dependent, but there is still no apparent intermingling of trajectories. Finally, in part (c) of the figure we show a turbulent flow corresponding to much higher flow speed.We see that the paths followed by fluid parcels are now quite complicated and entangled indicating a high degree of mixing (in this case only of momentum). Such flows are three dimensional and time dependent, and very difficult to predict in detail.The most important single point to observe from the above figures and discussion is that as flow speed increases, details of the flow become more complicated and ultimately there is a transition

from laminar to turbulent flow.Identification of turbulence as a class of fluid flow was first made by Leonardo da Vinci more than 500 years ago as indicated by his now famous sketches, one of which we present here in Fig. 2.20. In fact, da Vinci was evidently the first to use the word turbulence to describe this type of flow behavior. Despite this early recognition of turbulence, little formal investigation wascarried out until the late 19th Century when experimental facilities were first becoming sufficiently sophisticated to permit such studies. The work of Osbourne Reynolds in the 1880s and 1890s is still widely used today, and in some sense little progress has been made over the past 100 years. In Fig.2.21 we display a rendition of Reynolds original experiments that indicated in a semi-quantitative way the transition to turbulence of flow in a pipe as the flow speed is increased. What is evident from this figure is analogous to what we have already seen with flow from a faucet, but now in the context of an actual experiment; namely, as long as the flow speed is low the flow will be laminar, but as soon as it is fast enough turbulent flow will occur. Details as to how and why this happens are not completely understood and still constitute a major area of research in fluid dynamics, despite the fact that the problem has been recognized for five centuries and has been the subject of intense investigation for the past 120 years.

Figure 2.20: da Vinci sketch depicting turbulent flow.

IMAGES FOR STEADY AND UNSTEADY

IMAGES FOR LAMINAR AND TURBELANT

EXAMPLE OF APPLICATION LAMINAR AND TURBELANT FLOW (CANDLE SMOKE)If you have been around smokers, you probably noticed that the cigarette smoke rises in a smooth plume for the first few centimeters and then starts fluctuating randomly in all directions as it continues its rise. Other plumes behave similarly in below figure The flow regime in the first case is said to be laminar, characterized by smooth streamlines and highly ordered motion, and turbulent in the second case, where it is characterized by velocity fluctuations and highly disordered motion.

EXAMPLE OF APPLICATION LAMINAR AND TURBELANT FLOW (BLOOD VESSEL)LAMINAR FLOWLaminar flow is the normal condition for blood flow throughout most of the circulatory system. It is characterized by concentric layers of blood moving in parallel down the length of a blood vessel. The highest velocity (Vmax) is found in the center of the vessel. The lowest velocity (V=0) is found along the vessel wall. The flow profile is parabolic oncelaminar flowis fully developed. This occurs in long, straight blood vessels, under steady flow conditions. One practical implication of parabolic,laminar flowis that when flow velocity is measured using a Doppler flowmeter, the velocity value represents the average velocity of a cross-section of the vessel, not the maximal velocity found in the center of the flow stream.

TURBELANT FLOWGenerally in the body, blood flow is laminar. However, under conditions of high flow, particularly in the ascending aorta, laminar flow can be disrupted and becometurbulent. When this occurs, blood does not flow linearly and smoothly in adjacent layers, but instead the flow can be described as being chaotic.The Turbulent flow also occurs in large arteries at branch points, in diseased and narrowed (stenotic) arteries (see figure below). Turbulence increases the energy required to drive blood flow because turbulence increases the loss of energy in the form of friction, which generates heat. Turbulence does not begin to occur until the velocity of flow becomes high enough that the flow lamina break apart. Therefore, as blood flow velocity increases in a blood vessel or across a heart valve, there is not a gradual increase in turbulence. Instead,turbulenceoccurs when a critical Reynolds number (Re) is exceeded.

FLUID MECHANICSPage 2