Dr. Kamel Mohamed Guedri Umm Al-Qura University, Room H1091 ME804306-2 Fluid Mechanics Chapter 4 Real Fluids Dr. Kamel Mohamed Guedri Mechanical Engineering Department, The College of Engineering and Islamic Architecture, Umm Al-Qura University, Room H1091 Website: https://uqu.edu.sa/kmguedri Email: kmguedri@uq.edu.sa

Outline Introduction Objectives 5.1 Real Fluid 5.2 Types of Flow 5.3 Boundary Layers 5.3.1 Development of Boundary Layer 5.3.2 Laminar Boundary Layer Over Flat Plate 5.3.3 Turbulent Boundary Layer 5.3.4 Laminar Sub-Layer Summary

Introduction In the earlier chapter, the basic equations of continuity, energy and momentum were introduced and applied to fluid flow cases where the assumption of frictionless flow (or ideal fluid flow) was made. It is now necessary to introduce concepts which enable the extension of the previous work to real fluids in which viscosity is accepted and frictional effects cannot be ignored. The concept of Reynolds number as an indication of flow type will be used extensively.

Objectives Describe the appearance of laminar and turbulent flow. Compute Reynolds number and identify the type of flow. State the characteristics laminar, turbulent and transitional flow. Define boundary layers. Calculate the properties of laminar and turbulent boundary layer.

4.1 Real Fluid In a real fluid viscosity produces resistance to motion by causing shear or friction forces between fluid particles and between these and boundary walls. Due to this viscous effects, fluid tends to ‘stick’ to solid surfaces and have stresses within their body. The inclusion of viscosity allows the existence of two physically distinct flow regimes, known as laminar and turbulent flow.

5.2 Types of Flow Theoretically the physical nature of fluid flow can be categorized into three types, i.e. laminar, transition and turbulent flow. To predict whether the flow will be laminar, transition or turbulent, it is necessary to explore the characteristics of flow in each of these region. This phenomenon has been studied in detailed by Osborne Reynolds (1883) using the apparatus shown in figure 4.1

Schematic diagram of Reynolds apparatus Figure 4.1: Schematic diagram of Reynolds apparatus In this experiment, a filament of dye was injected to the flow of water. The discharge was carefully controlled, and passed through a glass tube so that observations could be made. Reynolds discovered that the dye filament would flow smoothly along the tube as long as the discharge is low. By gradually increased the discharge, a point is reached where the filament became wavy. A small further increase in discharge will cause vigorous eddying motion, and the dye mixed completely with water.

3 (THREE) distinct patterns of flow were recognized: Viscous or Laminar – in which the fluid particles appear to move in definite smooth parallel path with no mixing, and the velocity only in the direction of flow. Transitional – in which some unsteadiness becomes apparent (the wavy filament). Turbulent – in which the flow incorporates an eddying or mixing action. The motion of a fluid particle within a turbulent flow is complex and irregular, involving fluctuations in velocity and directions.

Figure 5.2: Flow Pattern

(4.1) where  = density  = dynamic viscosity V = mean velocity Reynolds experiment also revealed that the initiation of turbulence was a function of fluid velocity, viscosity, and a typical dimension. This led to the formation of the dimensionless Reynolds Number (Re). (4.1) where  = density  = dynamic viscosity V = mean velocity D = pipe diameter

It can be seen that it has no units It can be seen that it has no units. A quantity that has no units is known as a non-dimensional (or dimensionless) quantity. Thus the Reynolds number, Re, is a non-dimensional number. Realizing that the kinematic viscosity can be represented with the dynamic viscosity over density , the Re can also be written as ; (4.2)

Example 4.1 If the pipe and the fluid have the following properties, at what velocity the flow in a pipe stops being laminar? water density,  = 1000 kg/m3 pipe diameter, D = 0.5m (dynamic) viscosity,  = 0.55x103 Ns/m2 Solution: We want to know the maximum velocity for laminar flow, i.e. when Re =2000.

Example 4.2 Oil of viscosity 0.05 kg/m.s and density 860 kg/m3 flows in a 0.1 m diameter pipe with a velocity of 0.6 m/s. Determine the type of flows. Solution: Re = 1032 < 2000  laminar flow

Laminar flow Transitional flow Turbulent flow Re < 2000; 'low' velocity; Dye does not mix with water; Fluid particles move in straight lines; Simple mathematical analysis possible; and Rare in practice in water systems. 2000 > Re < 4000; 'medium' velocity; and Dye stream wavers in water - mixes slightly. Re > 4000; 'high' velocity; Dye mixes rapidly and completely; Particle paths completely irregular; Average motion is in the direction of the flow; Changes/fluctuations are very difficult to detect; Mathematical analysis very difficult - so experimental measures are used; and Most common type of flow.

4.3 Boundary Layers Whether a flow is in general laminar or turbulent, the effects of the viscosity of the fluid are greatest in regions close to solid boundaries. It is a characteristic of all real fluid, where the viscous effect causes it to "stick" to solid surfaces and have stresses within their body. When a fluid flows over a stationary surface, e.g. the bed of a river, or the wall of a pipe, the fluid touching the surface is brought to rest by the shear stress o at the wall. The velocity increases from zero at the wall to a maximum in the main stream of the flow. Looking at this two-dimensionally we get the velocity profile from the wall to the centre of the flow as shown in the figure.

A typical velocity profile Figure 5.3: A typical velocity profile This profile doesn't just exit, it must build up gradually from the point where the fluid starts to flow past the surface - e.g. when it enters a pipe. If we consider a flat plate in the middle of a fluid, we will look at the build up of the velocity profile as the fluid moves over the plate. This lead to the development of a thin layer of fluid adjacent to the surface in which the fluid velocity varies from zero at the surface to the free stream value a short distance out from the surface. This layer has been called the boundary layer for the surface.

4.3.1 Development of Boundary Layer To develop the boundary layer concept, let’s consider a flow pass over a stationary which lies parallel to the flow. The flow just upstream the plate has a uniform free stream velocity, Us. As the flow comes into contact with the plate, the layer of fluid immediately adjacent to the plate decelerates due to viscous friction and comes to rest. The fluid in contact with the plate surface has zero velocity, known as ‘no slip’ condition. A velocity gradient exists between the fluid in the free stream and the plate surface. Newton’s law of viscosity tells us that the shear stress, , in a fluid is proportional to the velocity gradient-the rate of change of velocity across the fluid path. (1.1) Note: This is introduced earlier in Chap 1

Shearing action between the layer of fluid in contact with the plate and second layer of the fluid forced the second layer to decelerate. This creates a shearing action between the third layer of fluid and so on. As the fluid passes along the plate, the zone in which shearing action occurs tend to spread further outwards. This zone is known as ‘boundary layer’. Outside the boundary layer, the flow remains effectively free of shear, so the fluid is not subjected to viscosity related forces. Thus, the fluid outside the boundary layer may be assumed to act like an ideal fluid.

Figure 5.4: Development of boundary layer along a flat plate (after Douglas et. al, 2001)

The boundary layer, which starts at the leading edge of the surface, is a laminar layer which grows in thickness along the surface. Eventually, at some distance downstream, this laminar layer become unstable, transforming into a turbulent layer. As with any other sheared flow, flow in boundary layer can be laminar or turbulent, depending on the local Re number, (5.3)

For smooth polished plate, translation to turbulent flow occurs when the local value of Reynolds Number is about 500,000. However, for rough plates transition may occur at much lower values. The thickness of the boundary layer, , is defined as the distance from the wall to the point where the local velocity is 99% of the "free stream" velocity,  = distance from wall to point where u = 0.99 ufreestream The value of  will increase with distance from the point where the fluid first starts to pass over the boundary - the flat plate in our example. It increases to a maximum in fully developed flow. Correspondingly, the drag force FD on the fluid due to shear stress o at the wall increases from zero at the start of the plate to a maximum in the fully developed flow region where it remains constant.

Boundary Layer development at pipe entrance As flow enters a pipe the boundary layer will initially be of the laminar form. This will change depending on the ratio of initial and viscous forces; i.e. whether we have laminar (viscous forces high) or turbulent flow (inertial forces high). For pipe flow it is normal practice to base the Reynolds number, Re, on the mean flow velocity and pipe diameter. Generally, for values of Re < 2000, the flow may be assumed to be laminar, although it has been shown possible to maintain laminar flow at higher values of Reynolds number under specialized laboratory conditions. Above Re = 2000 it is, however, reasonable to suppose that the flow will be turbulent and that the boundary layer development will include a transition and a turbulent region, as described for the flat plate. The only major difference is that, in the pipe flow case, there is a limit to the growth of the boundary layer thickness, namely the pipe radius.

(a) laminar flow; (b) turbulent flow If, therefore, this limit is reached before transition occurs, i.e. if laminar boundary layers meet at the pipe centre, the flow in the remainder of the pipe will be laminar. On the other hand, if transition within the boundary layer occurs before they fill the pipe, the flow in the rest of the pipe will be turbulent. These two cases are illustrated in the figure below. Figure 5.5 : Development of fully developed laminar and turbulent flow in circular pipe (a) laminar flow; (b) turbulent flow (after Douglas et. al, 2001)

If we only have laminar flow the profile is parabolic as only the first part of the boundary layer growth diagram is used. So we get the top diagram in the above figure. If turbulent (or transitional), both the laminar and the turbulent (transitional) zones of the boundary layer growth diagram are used. The growth of the velocity profile is thus like the bottom diagram in the above figure. Once the boundary layer, whether laminar or turbulent in nature, has grown to fill the whole pipe cross-section, the flow may be said to be fully developed and no further changes in velocity profile are to be expected downstream, provided that the pipeline characteristics (i.e. diameter, surface roughness) remain constant.

Theoretically, the entry length for a particular pipe (i. e Theoretically, the entry length for a particular pipe (i.e. the distance from entry at which a laminar or turbulent boundary layer ceases to grow) is infinite. However, it is normally assumed that the flow has become fully developed when the maximum velocity, at the pipe centreline, becomes 0.99 of the theoretical maximum. Using this approximation, typical entry lengths for the establishment of fully developed laminar or turbulent flow may be taken as 120 and 60 pipe diameters, respectively. The entry length characteristic of turbulent flow is the shorter owing to the higher growth rate of the turbulent boundary layer. Thus, the assumption of steady, uniform flow restricts the application of the equations derived for pipe flow to that part of a conduit beyond the entry length. Normally, this is not a serious restriction as the entry length is usually small compared with the total length of the pipeline. At points very close to the boundary the velocity gradients become very large and the velocity gradients become very large with the viscous shear forces again becoming large enough to maintain the fluid in laminar motion. This region is known as the laminar sub-layer. This layer occurs within the turbulent zone and is next to the wall and very thin - a few hundredths of a mm.

5.3.2 Laminar boundary layer over flat plate The velocity distribution in a laminar boundary layer may be expressed in a number of forms. A typical equation is (4.4) For laminar flow over a flat plate. Blasius solved the basic boundary layer equations and obtain analytical solution which have been verified experimentally to be remarkably accurate. The classic Blasius solutions for laminar layer are; (4.5)

The boundary shear is given by (5.6) where The boundary shear is given by (5.6) The total drag force on one side of a plate of length L and width B is defined as, (5.7) and (5.7a) where and Cfd = total frictional drag coefficient.

5.3.3 Turbulent boundary layer The majority of boundary layers met in engineering practice are turbulent over most of their length, and so the study of this section of the development of the boundary layer is usually regarded as of greater fundamental importance than that of the laminar section. The turbulent boundary layer will have much steeper velocity gradients at the boundary than the laminar boundary layer. The velocity distribution is logarithmic and could be conveniently expressed in the form of a power law, over a range of Reynolds number.

For between 5x106 and 2x107, the velocity distribution can be expressed by the 1/7 power law, By assuming that the boundary layer to be turbulent from the leading edge (i.e. x = 0) the boundary layer thickness can be written as (4.8) The shear stress at the boundary can be expressed by (4.9)

The drag force on one side of the plate (4.10) where (4.10a)

5.3.4 Laminar Sublayer In a turbulent boundary, a laminar sub-layer forms close to the solid surface. In this regimes, the flow remains laminar. The laminar sub-layer is usually very thin and its thickness is found by experiments to be; (4.11) where = shear velicity Despite its thinness, the laminar sub-layer can play a vital role in the friction characteristics of the surface. This is particularly relevant when defining pipe friction. In turbulent flow if the height of the roughness of a pipe is greater than the thickness of the laminar sub-layer then this increases the amount of turbulence and energy losses in the flow. If the height of roughness is less than the thickness of the laminar sub-layer the pipe is said to be smooth and it has little effect on the boundary layer. In laminar flow the height of roughness has very little effect.

Example 4.3 A flat plate 15 cm wide and 45 cm long is placed longitudinally in a stream of oil flowing with a free stream velocity of 6.0 m/s. Find the thickness of the boundary layer and shear stress at the trailing edge. the drag force. Assume that the transition from laminar to turbulent occurs at the Re= 500,000. Solution: Therefore the flow is laminar.

At trailing edge, x = L = 0.45m Total drag force on both sides of the plate

Summary a) extensive used of Reynolds Number (Re) in order to identify the type of flows b) characteristics of types of flow c) the development of a boundary layer of a flow which concentrate on turbulent boundary layer and laminar sub-layer