LAB 8 - Velocity Profile above a Flat Plate 

Introduction 

In any viscous flow, the fluid in direct contact with a solid boundary has the same velocity as the boundary itself and the "no-slip condition" must be satisfied at the boundary. Since the fluid velocity at the stationary solid surface is zero, but the bulk fluid is moving, velocity gradients (and hence shear stress) must be present in the flow. For one-dimensional viscous flow, the shear stress t is given by 

      (1) 

where  = dynamic viscosity, u = stream wise velocity, and y = normal coordinate measured from the solid boundary. 

Thus, for external flow over a flat plate as shown in Figure 1, the wall shear stress can be determined from knowledge of the velocity profile.  

For an external flow in which the fluid is unbounded by walls, the viscous effects will grow and continually expand as flow moves further downstream along the solid surface. The viscous layers, either laminar or turbulent, are very thin, much thinner than the drawing above shows.  The boundary layer thickness is defined as the point where the velocity u parallel to the plate reaches 99% of the external free stream velocity U.  The accepted formulas offor flat-plate flow are [1] 

/x  5.0/Re_x^1/2  for laminar  flow      (2) 

In turbulent flow, the boundary layers expands more rapidly and the accepted correlation for is 

/x  0.16/Rex^1/7 for turbulent  flow     (3) 

A dimensionless quantity of particular interest in external flows is c_f, called the skin-friction coefficient, and is analogous to the friction factor f in internal flows.  c_f is defined by

c_f = 2_/U²       (4) 

where U = the free stream velocity outside the boundary layer, = the shear stress at the wall, and = the density of the fluid. 

Boundary layer theory, first formulated by Ludwig Prandtl in 1904, is one method for obtaining solutions for laminar flow inside a boundary layer.  By making certain order of magnitude assumptions the Navier-Stokes equations can be simplified into the boundary layer equations that may be solved relatively easily in some simple geometries.  Equation 3, Blassius's solution is a result of a boundary layer solution.  Blassius's solution results in an ordinary differential equation that must be numerically integrated. The tabular result is available in the fluid mechanics textbook [1]. It also provides a solution for c_f in laminar flow given by

c_f = 0.664/Re_x^1/2       (5)

There are three regions in the boundary layer in a turbulent flow: 1) the viscous or wall layer (viscous shear dominates), 2) outer layer (turbulent shear dominates), 3) overlap layer (both types of shear important).  There is no exact theory for turbulent flat-plate flow, although there are many computer solutions of the boundary-layer equations using various empirical models.  Another technique that produces a widely accepted result is an integral analysis where a logarithmic law is assumed for the velocity profile.  It is convenient to introduce a a parameter u*, called the shear velocity or friction velocity, to nondimensionalize the velocity profile. This is given by

u* = ()^1/2     (6)

The logarithmic law that we assume holds all the way across the boundary layer is

     (7) 

where the constants = 0.41 and B = 5.0.  This equation was obtained by examining boundary layers experimentally.  By applying equation (7) at the edged of the boundary layer where y = and u = U, and using the definition of skin-friction coefficient in equation (4) the following skin-friction law for turbulent flat-plate flow can be obtained.

        (8)

Experimental Apparatus 

A flat plate with a length of 14 inches has been secured inside the test section of the AERO lab wind tunnel. A pitot tube is mounted above the test section (O.D. = 0.10 in) at a distance of approximately 12 inches from the leading edge of the plate. The pitot tube can be moved in the y-direction above the plate using a traversing mechanism that is scaled in 0.001" increments (1 revolution = 0.025 in). Both the static and stagnation pressure taps on the pitot tube are connected to a manometer in order that the difference between these two pressures may be measured directly in inches of water. Other instrumentation includes a thermistor inserted into the flow stream at one of the ports on the side walls of the test section. The air velocity can be determined from the pressure difference data and the air density. The wall shear stress and skin-friction coefficent can be calculated from the velocity profile. 

Pitot Tube 

If Red > 1000, where d is the probe diameter, the flow around the probe is nearly frictionless and Bernoulli's equation applies with good accuracy. For incompressible flow, and neglecting any elevation head, the following expression relates the local velocity to the difference in stagnation and static pressure: 

      (9) 

where po = stagnation pressure, ps = static pressure, and  = fluid density. 

The difference between the stagnation and the static pressure can be measured in terms of h inches of water so that equation 9 can be rewritten as 

      (10) 

where k is a constant. 

Objectives 

1. To measure and plot the velocity profile. 
2. To compare the free stream velocity to that measured by the wind tunnel instrumentation. 
3. To calculate the wall shear stress and skin friction coefficient. 

Procedure 

1. Determine the room temperature and the barometric pressure. 
2. Set the pitot tube at a reference position at the surface of the plate and measure the distance from its tip to the leading edge. 
3. Zero the wind speed indicator on the AERO lab wind tunnel.  Be sure to start from a positive indication and decrease until it is zeroed. 
4. Increase the fan speed of the wind tunnel until a velocity as specified by the lab assistant is attained.
5. Move the pitot tube at suitable intervals and record the pressure difference (po - ps) in the manometer and the reading on the traversing mechanism at each interval. Try to obtain at least 10 data points within the boundary layer. 
   

Data Reduction & Points of Interest

1. Calculate the free stream velocity in the wind tunnel along with its uncertainty from the pitot tube measurement. Compare the pitot tube measurement of the velocity with that obtained from the AERO lab wind tunnel indicator.  Discuss possible reasons for any discrepancies.
2. Calculate the velocity at different y positions and plot the velocity profile for the experimental data. Compare and discuss this velocity profile to the expected velocity profile predicted by theory (Blassius solution for laminar flow or eqtn 7 for turbulent flow).    
3. Compute the skin-friction coefficient and wall shear stress and compare to values predicted from theory.  Discuss any differences.

References 

1. White, F.M., Fluid Mechanics, 3rd Ed., McGraw-Hill Book Co., New York, 1994. 
2. Holman, J.P., Experimental Methods for Engineers, 4th Ed., McGraw-Hill Book Co., New York, 1984.