Boundary Layers

(c) Srinivas and Auld 2009 --- www.aerodynamics4students.com

Introducton to Boundary Layers Viscous Effects in External Flows The analysis we have carried out so far are such that viscosity did not make a direct appearance. Then the potential flows we considered in the previous Chapters were inviscid, i.e., we deliberately ignored viscosity. In reality these flows are theoretical. we saw in the case of flow about a cylinder how viscosity alters the flow completely aft of the cylinder and its consequences. Any real flow in nature is viscous. In fact, it is viscosity that makes the flow interesting and of course challenging to understand and calculate. It is viscosity that gives rise to many of the interesting physical features of a flow. One other area that makes a flow exciting even though inviscid is that of compressible flow. The birth and development of a Boundary Layer, its transition, the way a flow handles a pressure gradient and a possible separation are some of the topics we consider in this chapter. We postpone any discussion of the calculation procedures to later chapters and deal with only qualitative features at present.

Boundary Layer Flow Recall our discussion in the very first chapter we had two parallel plates. One of the plates was stationary (the lower one) and the other one moving. We said that there was a No Slip condition, which meant that the fluid does not slip past the solid in contact. Needless to say that this is a typical effect of viscosity.

Figure 1: Formation of a Boundary Layer Let us now follow the effects as a flow approaches a solid body, to make it simple, a flat plate, Fig 1. Consider a uniform (inviscid) flow in front of a flat plate at a speed U ∞ . As soon as the flow 'hits' the plate No Slip Conditions gets into action. As a result, the velocity on the body becomes zero. Since the effect of viscosity is to resist fluid motion, the velocity close to the solid surface continuously decreases towards downstream. But away from the flat plate the speed is equal to the freestream value of U ∞ . Consequently a velocity gradient is set up in the fluid in a direction normal to flow. Thus a layer establishes itself close to the wall with a velocity gradient. This is what we call the Boundary Layer. We will find out later that this is a high Reynolds Number concept and is due to Prandtl, a leading German Aerodynamicist. The boundary layer is not a static phenomenon. It is dynamic. The thickness of boundary layer (the height from the solid surface where we first encounter 99% of free stream speed) continuously increases. A shear stress develops on the solid wall. It is this shear stress that causes drag on the plate. Boundary layer has a pronounced effect upon any object which is immersed and moving in a fluid. Drag on an aeroplane or a ship and friction in a pipe are some of the common manifestations of boundary layer. Understandably, boundary layer has become a very important branch of fluid dynamic research.

Laminar and Turbulent Boundary Layers A boundary layer may be laminar or turbulent. A laminar boundary layer is one where the flow takes place in layers, i.e., each layer slides past the adjacent layers. This is in contrast to Turbulent Boundary Layers shown in Fig. 2. where there is an intense agitation. In a laminar boundary layer any exchange of mass or momentum takes place only between adjacent layers on a microscopic scale which is not visible to the eye. Consequently molecular viscosity  is able predict the shear stress associated. Laminar boundary layers are found only when the Reynolds numbers are small.

Figure 2 : Typical velocity profiles for laminar and turbulent boundary layers A turbulent boundary layer on the other hand is marked by mixing across several layers of it. The mixing is now on a macroscopic scale. Packets of fluid may be seen moving across. Thus there is an exchange of mass, momentum and energy on a much bigger scale compared to a laminar boundary layer. A turbulent boundary layer forms only at larger Reynolds numbers. The scale of mixing cannot be handled by molecular viscosity alone. Those calculating turbulent flow rely on what is called Turbulence Viscosity or Eddy Viscosity, which has no exact expression. It has to be numerically modelled. Several models have been developed for the purpose.

Figure 3 : Typical velocity profiles for laminar and turbulent boundary layers As a consequence of intense mixing a turbulent boundary layer has a steep gradient of velocity at the wall and therefore a large shear stress. In addition heat transfer rates are also high. Typical laminar

and turbulent boundary layer profiles are shown in Fig 3.. Typical velocity profiles for laminar and turbulent boundary layers Growth Rate (the rate at which the boundary layer thickness  of a laminar boundary layer is small. For a flat plate it is given by

 5.0 = x  Rex

(1)

where Rex is the Reynolds Number based on the length of the plate. For a turbulent flow it is given by

 0.385 = 0.2 x Rex

(2)

Wall shear stress is another parameter of interest in boundary layers. It is usually expressed as Skin friction defined as

w

C f=

where

1 U 2∞ 2

(3)

w is the wall shear stress given by w =

  ∂u ∂y

(4) y=0

and U ∞ is the free stream speed. Skin friction for laminar and turbulent flows are given by

C f=

C f=

0.664 , Laminar Flow  Rex

0.0594 ,Turbulent Flow R0.2 ex

(5)

Separation of Flow Pressure gradient is an is one of the factors that influences a flow immensely. It is easy to see that the shear stress caused by viscosity has a retarding effect upon the flow. This effect can however be overcome if there is a negative pressure gradient offered to the flow. A negative pressure gradient is termed a Favourable pressure gradient. Such a gradient enables the flow. A positive pressure gradient has the opposite effect and is termed the Adverse Pressure Gradient. Fluid might find it difficult to negotiate an adverse pressure gradient. Sometimes, we say the the fluid has to climb the pressure hill.

Figure 4 : Separation of flow over a curved surface One of the severe effects of an adverse pressure gradient is to separate the flow. Consider flow past a curved surface as shown in Fig. 4. The geometry of the surface is such that we have a favourable gradient in pressure to start with and up to a point P. The negative pressure gradient will counteract the retarding effect of the shear stress (which is due to viscosity) in the boundary layer. For the geometry considered we have a an adverse pressure gradient downstream of P. Now the adverse pressure gradient begins to retard. This effect is felt more strongly in the regions close to the wall where the momentum is lower than in the regions near the free stream. As indicated in the figure, the velocity near the wall reduces and the boundary layer thickens. A continuous retardation of flow brings the wall shear stress at the point S on the wall to zero. From this point onwards the shear stress becomes negative and the flow reverses and a region of recirculating flow develops. We see that the flow no longer follows the contour of the body. We say that the flow has separated. The point S where the shear stress is zero is called the Point of Separation. Depending on the flow conditions the recirculating flow terminate and the flow may become reattached to the body. A separation bubble is formed. There are a variety of factors that could influence this reattachment. The pressure gradient may be now favourable due to body geometry and other reasons. The other factor is that the flow initially laminar may undergo transition within the bubble and may become turbulent. A turbulent flow has more energy and momentum than a laminar flow. This can kill separation and the flow may reattach. A short bubble may not be of much consequence.

Figure 5 : Separation bubble over an aerofoil On aerofoils sometimes the separation occurs near the leading edge and gives rise to a short bubble. What can be dangerous is the separation occurring more towards the trailing edge and the flow not reattaching. In this situation the separated region merges with the wake and may result in stall of the aerofoil (loss of lift).

Drag Drag is a force that opposes motion. An aircraft flying has to overcome the drag force upon it, a ball in flight, a sailing ship and an automobile at high speed are some of the other examples. It is clear that viscosity is an agent that causes drag. We have seen that it gives raise to boundary layers on solid

surfaces. There is shear stress in boundary layers that do tend to retard the motion of fluid past the solid surface. This is sketched for an aerofoil surface in Fig 6. This is termed Skin friction Drag .

Figure 6 : Shear stress on a body There is another agent that can cause drag. This is the pressure difference upon the flow. This could come about due to geometrical effects that induce separation as happens with a cylinder to be discussed later. This is called Pressure Drag or Form Drag, since it is due to the body geometry. The sum of pressure drag and skin friction drag constitutes Drag about the body or Profile Drag.

Figure 7 : Effect of thickness of body on drag The shape of the body determines the relative magnitude of the drag components. A thin body (small t/l ratio) as shown in Fig. 7. obviously causes less pressure drag. Almost all drag comes from skin friction. A thick body (large t/l ratio) is readily prone for separation and produces considerable pressure drag. Streamlining a body to avoid separation will enable to decrease pressure drag considerably. It is obvious that a bluff body like a cylinder or a sphere or a flat plate placed normal to flow will cause separation and lead to pressure drag which may far more than the skin friction drag. In case of a flat plate placed parallel to flow (Fig.8 ), it is the skin friction drag that dominates. On the other hand, in case of a flat plate placed normal to flow, it is the pressure drag that dominates. In the latter case, the plate behaves like a blunt body and gives raise to separation behind it which contributes to pressure drag.

Figure 8 : Drag about a flat plate Drag Coefficient Drag force is non-dimensionalised as

C D=

Drag 1  U2 A 2 ∞ ∞

(6)

where CD is defined as Drag Coefficient. U ∞ is the free stream speed, ∞ is the free stream density, A is the area. What area to use depends upon the application. In case of a cylinder it is the projected area normal to flow. For a flow past a thin flat plate, it will be the area of plate exposed to flow. The relative importance of the two kinds of drag is very apparent in case of flow over a circular cylinder or a sphere. The flow depends strongly upon Reynolds number as is clear from Fig. 9. When the Reynolds numbers are small (1 and below)the flow behaves like a potential flow. There is no separation. The drag is all due to skin friction. As the Reynolds number is increased this drag decreases. At Reynolds numbers around 2 - 30, there is a separation of boundary layer, but the wake is of a limited length. The eddies formed seem fixed behind the cylinder. For Reynolds numbers close to 40 -70, there is a periodic oscillation of the wake. For higher Reynolds numbers the eddies break off from the cylinder. As the Reynolds number is increased, the eddies are continuously shed from the cylinder and washed downstream. Two rows of vortices are formed called the Vortex Street. Now the pressure drag contributes to almost 90% of the total drag. The value of CD reaches a minimum of around 0.9 at a Reynolds number of around 2000. Increasing the Reynolds numbers further results in large angular velocities and a degeneration of vortices into turbulence.

Figure 9 : Flow past a Circular Cylinder at various Reynolds Numbers

Figure 10 : Flow past a Circular Cylinder at various Reynolds Numbers, continued. In the Reynolds number range 104 to 105 one sees a laminar boundary layer to the left of the vertical centreline of the cylinder (Fig.10.). The flow separates at point S, which makes an angle of about 800 with the centre of the cylinder. A wide wake is seen downstream. The pressure in the separated regions is almost constant. The observed CP distribution is shown in Fig 31 in the section on Potential Flow.. The net pressure difference PA - PB contributes to pressure drag. A dramatic change takes place when the Reynolds number is around 2x105 when the boundary layer becomes turbulent before separation. Now the separation is postponed since a turbulent boundary layer is able to sustain for a longer time than a laminar flow. The point of separation S now is found at 1300 as shown in Fig.10. Notice that the wake has now narrowed. The CP distribution indicates that the pressure in the wake is now higher than that for the laminar case (Fig 31 Potential Flow Section). The consequence is that CD is now reduced to about 0.3. This reduction in drag around the cylinder is exploited in golf. The purpose of providing dimples on golf ball is to trip turbulence in order to decrease drag. Bowlers in cricket, especially the ones that bowl swings would like to have one side of the ball shining than the other. The idea is to keep flow on one side of the ball laminar and the other one turbulent. The ball is to swing from the laminar to the turbulent side. It is clear that decrease in pressure drag can be achieved by delaying or stopping separation of flow. One of the strategies developed is to streamline the body. An aerofoil surface is an excellent example while the birds and fish are natural examples of this. Figure 11 gives the numerical values of CD for some of the familiar two-dimensional shapes. It is clear that CD depends upon the orientation of the object to the flow. CD values for some of the threedimensional objects are given in Fig. 12.

Figure 11 : CDvalues for familiar two-dimensional objects.

Figure 12 : CD Values for familiar three-dimensional objects

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