This drawing derived from potential flow theory shows how the streamlines around a combination of a flat plate and a cylinder would look if fluids did not have viscosity. When a real fluid, water, flows around the same shape the flow is quite different. Boundary layers, thin layers of fluid in which viscosity affects the flow dynamics, are formed along the surfaces of the object. The behavior of these boundary layers under the influence of pressure gradients can, in some cases, drastically alter the flow.
This streamlined pattern is very nearly what one would predict from inviscid flow theory. Because the Reynolds number is large, influences of viscosity are confined to a narrow region close to the surface of the wing. The primary effect of viscosity is simply to create a drag force on the wing through the integrated effect of surface shear stress. As the angle of attack is increased, viscous effects become more pronounced.
Pressure gradients imposed on the boundary layers become more severe and separation occurs on the upper surface. Now the airfoil is stalled. A region of recirculating flow is formed over the upper surface of the wing.
In order to understand how viscous forces can influence the entire flow field, we shall look at what causes boundary layers, how they grow, how they respond to pressure gradients, and the difference in behavior of laminar and turbulent layers. Before attempting to understand the boundary layer flow over this object, we shall look at the flow of a viscous fluid along the flat plate alone. These streamlines of water flowing in a channel are made visible with hydrogen bubbles generated by electrolysis. The flow is uniform and laminar. The fact that the timelines, the vertical patches of bubbles, remain perpendicular to the streamlines shows that the unobstructed flow is free of vorticity.
Here is the flow over a plate. The flow is two-dimensional and laminar. Upstream of the plate, the flow is still uniform, but in the fluid layers near the plate, the velocities are much lower and vorticity is present. In the narrow boundary layer region adjacent to the plate, the shearing forces and inertia forces are of equal importance in determining the motion of fluid elements.
Outside of this region, the shear stress can be neglected. The thickness of the boundary layer increases along the length of the plate. Fluid deceleration is transferred from one fluid layer to another by viscosity.
Moving the camera with the freestream velocity will give a better view of the boundary layer growth. The velocity throughout the boundary layer is less than that in the free stream. There is no slip between the plate and the layer of fluid immediately adjacent to it. At the plate, the relative velocity is zero. This is the no-slip boundary condition of viscous flow.
The displacement of the bubbles corresponds closely to the velocity profile in the boundary layer. The thickness of a boundary layer is sometimes defined as the distance delta from the surface to where the velocity u reaches some fixed percentage of the free stream value. For our purpose we will use 95% of the free stream velocity to define the thickness of our boundary layer. The gradient of the velocity normal to the wall, delta u by delta y, times the viscosity mu, is the wall's skin friction or shear stress tau.
The greater the velocity gradient normal to the wall, the greater the shear stress. Upstream, the velocity gradient is fairly large. Downstream, it is much less. This composite photograph compares the two velocity profiles. The smaller gradient downstream indicates the wall shear stress decreases along the plate.
The growth of the boundary layer thickness along the plate can be explained by considering the time history of the vorticity within the boundary layer. Stokes'theorem states that the area integral of the vorticity omega, bounded by a closed contour, is equal to the line integral of the velocity around the bounding contour. The circulation gamma is the sum of the vorticity enclosed by the contour. The contour at the upstream station is a unit length along the plate and is more than a boundary layer thickness in height.
At the top, the local velocity is parallel to the contour, but in the opposite sense. The components of velocity along the right and left parts of the contour are virtually zero. Because there is no slip, the velocity contribution to the circulation at the surface is in fact zero. Therefore, the circulation is minus the free stream velocity times a unit length.
Downstream, the circulation is also equal to minus U0 times a unit length. The total amount of vorticity within each contour is the same. Because there is no vorticity upstream of the plate, and because the circulation per unit length along it is constant, we conclude that all of the vorticity in slink.
Upstream, 95% of the total vorticity is contained within this area. Downstream, 95% of the total vorticity is contained within this area. Viscosity acts through the mechanism of molecular diffusion to spread out the vorticity as it is convected downstream. The local boundary layer thickness is a measure of the distance vorticity has diffused away from the plate. We can relate the factors controlling this growth process in the following simplified manner.
The diffusion length delta increases as the square root of the product of the kinematic viscosity nu and the time of diffusion t. At a distance L from the leading edge, the diffusion time is proportional to L divided by the free stream velocity U naught. We combine. So delta over L is proportional to 1 over the square root of the Reynolds number. This relationship is valid only at high Reynolds numbers when delta is small compared to L.
Increasing the flow velocity will decrease the boundary layer thickness at any given station along the plate. To see this effect, here is the normal flow. We will speed up the flow. Now, the flow velocity outside the boundary layer has increased. At any position along the plate, the boundary layer thickness is now less than before, because with a higher mainstream velocity, the boundary layer has had less time to grow.
We have seen the behavior of two-dimensional boundary layers on a flat plate in a uniform velocity field. So far, the pressure gradients in the flow direction have been negligibly small. However, in most flow situations, there are regions... of decreasing pressure and regions of increasing pressure in the flow direction.
We will examine these two flow situations separately. First, the case of a decreasing pressure or favorable pressure gradient in the flow direction. We will use this two-to-one flow contraction and watch the boundary layer along the flat side.
The upstream boundary layer is growing slowly as it enters the contraction. After the contraction, the boundary layer is thinner. Note that the distance from the wall to the streamline downstream is about half what it is upstream, because the two-dimensional flow contracts two to one. In the contraction, the effect of the pressure gradient in the boundary layer is to enhance the already high shear stress or vorticity concentration very near the wall.
It is as though a new boundary layer were being generated within the old. The combined profile at the exit is relatively thinner because there has been little time for growth of the new boundary layer by viscous diffusion. Using vorticity arguments, we can see that the vorticity added to the boundary layer by the contraction has had little time to diffuse. So downstream, a larger percentage of the total vorticity is nearer the wall.
This results in a relatively thinner boundary layer and higher wall shear stress. On the other hand, in a divergent channel, the boundary layer becomes thicker, with a corresponding decrease in wall shear stress. In this case, we have an unfavorable pressure gradient. The increasing pressure associated with a decreasing mainstream velocity decreases the velocity in the boundary layer, so velocity gradients and shear stress are less, and the boundary layer is thicker.
At a larger angle, the decrease in velocity is so great that somewhere along the wall, the velocity gradient normal to it becomes zero, accompanied by a local reversal of flow immediately downstream of the point. The fluid, which was in the upstream boundary layer, is no longer in contact with the wall and is separated from it by a region of reversed or recirculating flow. The boundary layer is said to have separated. The point on the wall where the fluid in the upstream boundary layer meets the fluid from the region of flow reversal is called the separation point.
The wall shear stress is zero there. Here, the point of separation of the laminar boundary layer is near the first wire. The boundary layers we have seen so far have all been laminar. However, in most practical high Reynolds number situations, the boundary layers are not laminar. Rather, they are turbulent.
Flow along a cylinder filmed with a high-speed camera can show the stages in the transition of the boundary layer from a laminar one to a turbulent one. A slight adverse pressure gradient causes transition to occur within the field of view. The steps in the transition are very complicated and interdependent.
...of nearly two-dimensional waves, the Tomein-Schlichting waves, followed by the appearance and growth of three-dimensional disturbances associated with streamwise vorticity. Then turbulent spots can be seen. And finally, fully turbulent flow appears. Here, the exact position where growth of two-dimensional disturbances begins depends on random small-scale fluctuations in the flow, and therefore varies with time in a random way. This boundary layer around the channel bend is made turbulent by placing an obstruction in the layer upstream.
The obstruction stimulates the naturally occurring processes and hastens the onset of transition. The flow here is all downstream. Removing the trip rod results in separation and backflow. Bubbles accumulate at the separation point. In this diffuser, which was used before, we have made the bottom boundary layer turbulent by inserting a trip rod upstream.
The turbulent boundary layer is able to withstand the adverse pressure gradient in the diffuser. While the laminar layer along the top wall is separated, allowing reverse flow. These timelines show that in the turbulent boundary layer, there is no separation.
The overall flow is downstream. To see why a turbulent boundary layer can withstand a larger unfavorable pressure gradient without separating, we shall again examine the flow along a flat plate. The boundary layer on the bottom side is laminar and two-dimensional.
On the top side, the boundary layer has been tripped by a wire placed well upstream. Unsteady motions in the turbulent boundary layer are three-dimensional. Some of the motions are perpendicular to the plane of view.
These timelines correspond closely to the instantaneous velocity profiles for the two types of boundary layers. Superimposing a number of displacement lines enables us to obtain a mean velocity profile for the turbulent layer and the laminar layer, and at the same time gives an experimental notion as to where the fluctuations occur and how large they are in the plane of mean motion. In this photograph we can compare mean laminar and turbulent profiles.
Here is the laminar one, the turbulent one, and here they are superimposed. The velocity gradient normal to the plate is larger for the turbulent layer and it therefore has a larger wall shear stress or drag. The circulation is the same for both layers. Both boundary layers therefore contain the same amount of vorticity per unit length of the plate.
However, the distribution of vorticity in the two layers is very different. There is more vorticity near the plate in the turbulent layer. The distribution of momentum in the two layers is also different.
In the turbulent layer, high momentum fluid is moved toward the plate, and low momentum fluid is moved away from the plate by unsteady random motions. There are larger shear stresses and more momentum in the turbulent boundary layer than in the laminar layer. And the turbulent boundary layer is slightly thicker. In the diffuser, the turbulent boundary layer along the bottom wall does not separate, while the laminar one does, because the three-dimensional interchange between regions of high and low momentum fluid in the turbulent layer is more effective than the molecular diffusion in the laminar layer.
Similarly, a turbulent boundary layer on the upper surface of this airfoil prevents large-scale separation or stall until a very high angle of attack. The blades along this wing are called vortex generators. They introduce axial vorticity which enhances the naturally occurring rotary momentum interchange in the already turbulent boundary layer.
Such mixing delays or prevents separation. In this film, we looked at flow along a flat plate and then saw the effect of pressure gradient on boundary layer flow. Now we can understand how this cylinder influences the flow on this plate. Without the plate attached, there is a static pressure increase along the stagnation streamline with a corresponding decrease in velocity.
When the plate is attached, a positive pressure gradient is imposed along the plate in the flow direction. This unfavorable pressure gradient causes boundary layer separation, and the flow field is radically altered. Thank you.