After World War II the National Bureau of Standards started experimental investigations on the separation of boundary layers. Prior to this study there were very few experimental works on turbulent boundary layers, especially during separation. The studies that were made were limited to measuring mean velocity and surface pressure. To help rectify this lack of knowledge, an experiment was carried out by Schubauer and Klebanoff (1950) on a separating turbulent boundary layer generated over a surface in a wind tunnel (Fig 1). The main objective of this experiment was to quantify the velocity field structure of a turbulent boundary layer in an adverse pressure gradient sufficiently large to cause boundary layer separation.

The boundary layer was created over the surface of an airfoil body with the shape shown in Fig 1. The surface of the airfoil body was polished and smooth. The airfoil body was mounted between the two sidewalls of the wind tunnel 10 ft apart and the length of the airfoil body was 27.9 ft. The wall of the wind tunnel facing the working side of the airfoil body was modified by constructing the "blister" shown in Fig 1 to create an adverse pressure gradient along the tail end of the body (region III) sufficient to cause the boundary layer to separate. When the flow entered the space between the "blister" and the airfoil body, it first accelerated creating a negative pressure gradient (favorable pressure gradient) along the body surface in region I. It then entered the mid-section (region II) where the flow cross-section remained constant so that the pressure along the surface was approximately constant. The flow exited the test section where the flow cross sectional area was increasing (region III), decelerating the flow and creating a positive pressure gradient (adverse pressure gradient).

Boundary layer separation did not occur in the favorable pressure gradient region since the pressure gradient accelerated the flow there. Flow separation also did not occur in the mid section of the airfoil body where the pressure gradient (and therefore the pressure force) was zero. The boundary layer separated from the surface of the body near the trailing edge in the adverse pressure gradient region at the point where the pressure force brought the flow to rest near the surface. Beyond the point of separation, a recirculation zone is created.

In this test the Reynolds numbers of the flow were high but the Mach number sufficiently low so that the flow could be approximated as incompressible. Time averaged velocities were measured. Fig. 2 shows schematic diagram of the growth of a boundary layer formed over a surface similar to the entrance flow in Fig. 1. The thickness of the boundary layer d is the distance from the surface where most of the frictional stress and vorticity are confined. Let us define the boundary layer thicknessd as the distance from the surface to where the streamwise flow velocity reaches a peak, which we call the "edge velocity" Ue, as illustrated in Fig. 2.

There are other measures of boundary layer thickness that were evaluated in this experiment. The first is displacement thickness δ* is defined as:

where y is the distance perpendicular to the surface. A second measure of boundary layer thickness is the momentum thickness θ, defined from

Review the physical interpretations of these thicknesses in your undergraduate fluid mechanics text. Because δ* and θ are defined by integrals, for which the integrand vanishes in at the edge of the boundary layer, they can be evaluated more accurately with experimental data than can the boundary layer thickness δ. The corresponding Reynolds numbers based on displacement and momentum thickness are:

Note that Ue(x), δ*(x), and θ(x) all vary with distance along the surface (x) in our boundary layer

The test was conducted in an open-air wind tunnel at 20°C with a square cross section of 10 ft on each side (Figure 1). As mentioned in the introduction, an airfoil body spanned the two sidewalls. All measurements were made midway between the two sidewalls where the flow was closely two-dimensional. The top surface of the airfoil body was polished and smooth. The velocity field in the boundary layer was measured using the hot wire technique (see your undergraduate fluid mechanics text).

Air was pulled from the outside through the tunnel at a steady rate. x and y velocity components were measured as a function of time at different locations within the boundary layer using hot wires, where x is the distance along the surface and y is perpendicular to the surface. As you know, turbulent velocities fluctuate with time. The time-varying x and y velocity components u and v were averaged over long time to obtain the mean velocity components U and V. The differences u'=u-U and v'=v-V are called the "turbulent velocity fluctuations" (review your fluids text). The data were collected along the length of the body at almost even intervals as shown in Fig 1.

The boundary layer data for this experiment are given in the Excel document below. Three sets of velocity profile given in each region shown in Figure 1. The edge velocity Ue(x) and the boundary layer thicknesses δ, δ*, and θ are also given as a function of distance x along the surface of the airfoil body, as well as the velocity measured at the first two locations from the surface.

1. F.M. White, 1999, Fluid Mechanics, 4th Ed., McGraw-Hill, New York.
2. Schubauer, G. B. and Klebanoff, P.S., 1950, Investigation of the Separation of Turbulent Boundary Layer, NACA Technical Report 2133, Washington, D.C.
3. Kreith, F., Berger S.A., 1999, Mechanical Engineering Handbook, CRC Press LLC.

Your objectives are to analyze the differences and similarities between boundary layers formed on realistic object shapes with curved surfaces and the classic zero pressure gradient boundary layer that is described in detail in text books, and to evaluate the changes in boundary layer velocity profile over curved surfaces and their underlying causes.

You shall analyze these data in two parts. In part I you shall compare the experimental results with analytical solutions for laminar and turbulent flat plate boundary layers. In part II of your analysis you shall contrast the velocity profiles in these three regions of the boundary layer shown in Figure 1. Focus, in particular, in the different structures of the accelerating boundary layer (region I) vs. decelerating boundary layer (region III) vs. the zero pressure gradient boundary layer (region II).

YOUR WRITEUP

All equations in your writeup should be numbered on the right. All figures should be numbered underneath with a brief figure caption (eg, Figure 1. Static and dynamic pressure along airfoil surface outside the boundary layer.) Your report should be written in sections as per this case study description. Your discussion of each section should refer to the equations and figures. ALL EQUATIONS AND ALL FIGURES SHOULD BE REFERRED TO IN THE TEXT AT SOME APPROPRIATE PLACE IN YOUR DISCUSSION. Use my more extensive writeups for assignments as a guide. Your report should have a title page with your name on it. Reports should be typed in Word (for example). Figures may be included throughout the report, or all together at the end, numbered and with a caption. There should be no figure that has not been discussed in the text somewhere (otherwise there is no point to including the figure).

Part I. Comparisons with the Zero Pressure Gradient Flat Plate Boundary Layer

The boundary layer is a thin region near the surface in which viscous effects are important. Text books generally discuss the simplest boundary layer formed on a flat plate immersed in a uniform stream as illustrated in Fig. 3 where the flow begins as laminar, grows along the surface as the Reynolds number increases then transitions to a turbulent boundary layer. When the Reynolds number is sufficiently high, the boundary layer eventually becomes fully turbulent. In this simple flat plate flow, the edge velocity Ue is constant outside the boundary layer (y>δ) and equals the free stream velocity U∞. The boundary layer thickness δ is often defined as the distance y where

where Rex≡

For the turbulent flat plate boundary layer there is no exact solution as in the case of laminar boundary layer. However, the momentum integral method has been applied to obtain corresponding relationships for the turbulent boundary layer by approximating the experimental velocity profile by

Where n is a weak function of the flow Reynolds number. When n=7

Outside the Boundary Layer

(a) Explain why the stagnation pressure, Po, is consant outside the boundary layer (y>δ) where

and Pe(x) is static pressure at the boundary layer edge. Because Pois constant, it is given by the pressure at the inlet to the wind tunnel where the flow velocity is very low. We approximate Po therefore as atmospheric pressure (1.013×105 Pa).

Plot Pe(x)-Po and

(b) From this plot discuss how pressure and streamwise pressure gradient vary outside the boundary layer along the surface. Identify the regions of favorable, zero and adverse pressure gradient and relate these to the acceleration of the flow just outside the boundary layer.

Boundary Layer Growth and Separation

(c) Plot δ(x) and the correlations for the laminar and turbulent flat plate boundary layers together on the same plot.

(d) Plot δ(x), δ(x)*, and θ(x) on separate plots (with different scales) and all together on the same plot (with the same scale). Plot the ratios δ(x)*/δ(x), θ(x)/δ(x), and θ(x)/δ*(x)on separate plots from the data and from Eq. (6) for the flat plate boundary layer.

(e) Plot Rex vs. x on a linear scale.

(f) Use the plots in (a) - (e) to discuss the following:

• Examine the plots where data are compared with the correlations for the flat plate zero pressure gradient laminar and turbulent boundary layers. Over what part of the airfoil body is it reasonable to use these correlations, over what part is it not reasonable, and why? Is the boundary layer laminar turbulent, or a mix of first laminar then turbulent? Use more than one of your plots to justify your answers.n
• Why are the fluctuations in δ(x) so much greater than δ(x)* or θ(x)?
• " Identify in each plot the x-location where the curves suddenly change. Compare with the geometry of the airfoil body and discuss what property is causing these rapid changes

Part II. Velocity Profile Analysis

Streamwise Velocity Profile

(g) (g) On three separate graphs plot the normalized streamwise velocity profile u/Ue vs. y/δ at the three x locations given in each of the three regions along the surface (Fig.1). Describe, discuss and explain the differences in velocity profile structure within each region and among the three regions. Discuss and explain the differences. Does self-similarity exist in any of the three regions? Include in your discussion an explanation for the meaning of self-similar vs. non-self-similar velocity profiles (Review your text for a discussion of self-similar velocity profiles).

The Adverse Pressure Gradient Boundary Layer

You learned from part I that the boundary layer thickness increases rapidly in the adverse pressure gradient region. Another very important characteristic is the change in surface shear stress. We can estimate shear stress by fitting the velocities measured at the first two points off the surface plus a zero velocity at the surface to a second order ploynomial:

The data file contains the velocities u1 and u2 at the first two measurement points from the surface y1 and y2 at all x locations.

(h) For u(0)=0, u(y1)=u1, and u(y2)=u2, show that a=0, b=

(i) Use this result to determine the shear stress at the wall, τw=μ

Compare the data with the correlations and explain the differences and what causes them.

(j) Compare the data plots of τwvs. x, Pe vs. x, and δ(x) vs. x. Summarize and discuss all the effects of adverse pressure gradient on a turbulent boundary layer up to separation. Does the boundary layer actually separate from the surface in this experiment? Explain.