## Fluid MechanicsProperties of Fluids Fluid Statics Control Volume Analysis, Integral Methods Applications of Integral Methods Potential Flow Theory Examples of Potential Flow Dimensional Analysis Introduction to Boundary Layers Viscous Flow in Pipes |
## Introducton to Boundary Layers## Viscous Effects in External FlowsIn previous sections viscosity was neglected in most of the analyses. The potential flows that were considered in the previous sections are inviscid, i.e., ignored viscosity. In reality these flows are purely theoretical. In the case of flow about a cylinder viscosity will alter the flow completely in the aft of the cylinder. Any real flow in nature incorporates viscosity. It is viscosity that gives rise to many of the interesting physical features of a flow. The layer of flow near to a surface is dominated by viscosity. This layer is called a boundary layer and will be the focus of this section. Boundary layer growth, transition between smooth and turbulent flow, changes due to pressure gradient and boundary layer separation will be covered in this section. ## Boundary Layer FlowFor flow over a plate, right a the surface a "No Slip" condition occurs. This means that that the fluid is stuck to the surfce and does not move relative to the surface. This is a typical effect of viscosity. Figure 1: Formation of a Boundary Layer
In The boundary layer is not a static phenomenon. It is dynamic. The thickness of boundary layer (the height from the solid surface where the flow is again 99% of free stream speed) continuously increases downstream along the plate. A shear stress develops on the solid surface. It is this shear stress that causes drag on the plate. The Boundary Layer has a pronounced effect upon any object which is immersed and moving in a fluid. Drag on an aircraft or a ship and friction in a pipe are some of the common manifestations of boundary layer. Understandably, boundary layer analysis has become a very important branch of fluid dynamic research. ## Laminar and Turbulent Boundary LayersA boundary layer may be laminar or turbulent. A laminar boundary layer
is one where the flow is smooth and layered, i.e., each layer slides
past the adjacent layers. This is in contrast to a turbulent boundary
layer as shown in 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 Figure 2 :
Typical velocity profiles for laminar and
turbulent boundary layers
A turbulent boundary layer on the other hand is marked by random mixing of fluid across
several layers. The mixing is on a macroscopic scale.
Packets of fluid may be seen moving up and down with the layer. 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 at
larger Reynolds numbers. The scale of mixing cannot be handled by
molecular viscosity alone. Calculation of turbulent flow relies on
a convection parameter called Figure 3 :
Typical velocity profiles for laminar and
turbulent boundary layers
The turbulent layer can be described using an average velocity profile. 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 high. Typical
laminar and turbulent boundary layer profiles are shown in where For a turbulent flow it is given by Wall shear stress is another important parameter for boundary layers. It
is usually expressed as where $τ_w$ is the wall shear stress given by and $U_∞$ is the free stream speed. Skin friction for laminar and turbulent flows are $$C_f={0.0594}/{{Re}_x}^{0.2}\text" -Turbulent Flow"$$ ## Separation of FlowPressure gradient is another of the factors that influences the growth of the boundary layer flow.
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 applied to the
flow. A negative pressure gradient is termed a A positive pressure gradient has
the opposite effect and is termed an Figure 4 :
Separation of flow over a curved surface
One of the severe effects of an adverse pressure gradient is to separate
the flow away from the body surface. For the flow past a curved surface as shown in The adverse pressure gradient begins to retard the flow. 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 Depending on the flow conditions the recirculating flow may terminate and then the flow may reattach to the body. A separation bubble is formed. There are a variety of factors that could influence this reattachment. The pressure gradient may become favourable due to further changes in body geometry. The other factor is that the flow which was initially laminar may undergo transition within the bubble and may become turbulent. A turbulent flow produces more energy and momentum in the near wall region. This can remove the recirculating flow region and the flow may reattach. Figure 5 : Separation bubble over an aerofoil
On aerofoils at low angles the separation may occur near the leading edge and give rise to a short bubble by transition to turbulence. At high angles the separated flow may not reattach and simply merge with the wake region. This will result in stall of the aerofoil (loss of lift, increase in drag). ## DragDrag is the 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 the agent that causes drag. It
gives rise to boundary layers on solid surfaces. The is shear
stress in boundary layer applies a retarding force to the body
. This is shown for an aerofoil surface in
Figure 6 : Shear stress on a body
There is another mechanism that can cause drag. This is the pressure difference
upon the flow. It can come about due to geometric effects and due to
separation. This is called The sum of pressure drag and skin friction drag constitutes Drag about
the body or 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 Figure 8 : Drag about a flat plate
## Drag CoefficientDrag force is non-dimensionalised as where 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 shown in 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 10
A laminar boundary layer exists up to the vertical
centreline of the cylinder (^{5}Fig.10). The flow separates at around this point S,
which makes an angle of about 80^{0}
with the centre of the cylinder. A wide wake is seen downstream. The
pressure in the separated region is almost constant. The observed C_{P}
distribution is shown in Fig 31 in the section on Potential Flow.
The net pressure difference P_{A}
- P_{B}
contributes to pressure drag.A dramatic change takes place when the
Reynolds number is around 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 those who bowl "swings" would like to have one side of the ball more smooth than the other. The idea is to keep flow on one side of the ball laminar and the other one turbulent. The ball will swing from the laminar to the turbulent side. 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. In nature, bird wings and fish are examples of reduced drag bodies.
Figure 11 : C_{D}values for familiar two-dimensional objects.Figure 12 : C_{D }Values for familiar three-dimensional objects |