Chapter momentum thickness for an in-compressible boundary layer

Chapter 1
Boundary layer is a layer adjacent to a surface where viscous e?ects are important. Figure (1) depicts ?ow of a ?uid over a ?at plate.
The ?uid particles at the ?at plate surface have zero velocity and they act as a retardant to reduce velocity of adjacent particles in the vertical direction. Similar actions continue by other particles until at the edge of the boundary layer where the particles’ velocity is 99% of the free stream velocity. Boundary layers can also be measured by more signi?cant parameters. The main boundary layer parameters are as follows: The displacements thickness, ?? is de?ned as the distance by which the external streamlines are shifted due to the presence of the boundary layer:
The momentum thickness represents the height of the free-stream ?ow which would be needed to make up the de?ciency in momentum ?ux within the boundary layer due to the shear force at the surface. The momentum thickness for an in-compressible boundary layer is given by:
The skin-friction coe?cient is de?ned as:

The Reynolds number is a measure of the ratio of inertia forces to viscous forces. It can be used to characterize ?ow characteristics aver a ?at plate. Values under 500,000 are classi?ed as Laminar ?ow where values from 500,000 to 1,000,000 are deemed Turbulent ?ow. Is it important to distinguish between turbulent and non turbulent ?ow since the boundary layer thickness varies, as Fig. (2) shows.

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The basic assumption used in all following calculations is that the working ?uid, air, is an incompressible ?uid. This is a reasonable assumption for low speeds such as those involved in this testing. Standard day atmospheric conditions of air are also used within these calculations. All calculated data is presented within the Tables and Graphs section.

The e?ective center equation is used to measure the ?rst ?y distance on which data is taken at each location. This is a function of the outer and inner diameter of the Pitot tube. Measured values are D = 0.05″ and di = 0.025″.
yec = (0.131 + 0.82di D)?D = 0.69mm (5)

The recorded data for the experiment included Pressure readings with the units of in-H2O. This data had to be converted into Pascal’s for velocity calculations. Equations (6) and (7) were used for conversion and free stream velocity calculation.

Applying Equation (7), the free stream velocities for the conducted experiments ranged between 18.7±.3 m s and 19.6±.3 m s .
Having found the free stream velocity earlier it is possible to calculate the Reynolds number for all four ?ow conditions using the following relationship:

The length L was measured from the leading edge of the ?at plate at which the boundary layer distributions are being evaluated were measured in inches and were converted to meters.

Once the free stream velocity and velocities at each ?y interval are known, the displacement thickness ?? can be calculated according to equation (1). The following formula is used to get a linear approximation of the displacement thickness at all four pitot tube locations.

The thickness of the boundary layer itself is a function of Reynolds number. The boundary curve for turbulent ?ow is much steeper. These are the equations used to calculate ? for laminar and turbulent ?ow, respectively.

The momentum thickness for an in-compressible boundary layer is given by equation (2). The following formula is used to get a linear approximation of the momentum thickness at all four pitot tube locations.

With displacement and momentum thickness found, H can be calculated:
H =???
The skin-friction coe?cient can be evaluated using a variety of techniques:
? CLAUSER CHART = By evaluating the Clauser chart, the skin-friction coe?cients can be found. For all four locations the lowest values were taken for use in the Clauser chart, corresponding to equation (5). The corresponding skin-friction values Cf were read from the Clauser Chart.

? REYNOLDS NUMBER = The skin-friction coe?cients can be calculated using Reynolds number with these equations a laminar or turbulent boundary layer, respectively.

? MOMENTUM THICKNESS = Another way to calculate the skin friction coe?cient is to calculate the slope of ? vs the length L. With a zero pressure gradient,

the Von Karman integral equation

becomes, using the relationship of equation (3)

Fig. (4) shows the approximated, linear ? value using this method.

In order to get a con?dence interval of 95%, we can calculate the error around the mean from our raw data and multiply it by a factor of 2, according to equation (12). For all intervals for each Reynolds number, the maximum of these intervals is chosen to be the con?dence interval.

The CI for ?P varies between 6.42 Pascal and 7.18 Pascal. To simplify calculations, the value of 7.18 Pascal’s is used for all uncertainty calculations. The ?P measured uncertainty is 1.25 Pascal.


Chapter 2
Approximation solutions

The boundary layer equations are a set of partial differential equations which we have to solve to obtain results like the Blasius equation. In this chapter we will not obtain the exact solution, but we will approximate the solution of the boundary layer equations.

Momentum-Integral Equation

Approximation solutions of the boundary layer equations can be obtained from the Momentum-Integral Equation in combination with making an educated guess for the velocity profile. First we will obtain the Momentum-Integral Equation4. We start with the boundary layer equations from equation and the definition for the pressure distribution from equation

Let us rewrite the continuity equation in order to obtain the velocity in y-direction inside the boundary layer:

Where we used . Here y is an arbitrary value. Also, we rewrite the x-momentum equation using like:

Now substitute (3.60) in (3.61) in order to obtain:

Integrate both sides, with h an arbitrary location in the free stream where and

We can carry out the integral for the term on the right side of the equation:

It follows from equation (3.34) that the partial derivative can be rewritten to obtain:

Taking this into account the right side of equation (3.64) yields:

Returning to equation (3.63), we can rewrite the second term inside the integral with the method of integration by parts:

Substituting equations (3.66) and (3.67) in equation (3.63), we get:

Now we need some more mathematical rewriting to derive the Momentum-Integral Equation:

We recall the definitions of the displacement thickness and the momentum thickness:

Now we have gathered all the information for the Momentum-Integral Equation. We will multiply equation (3.69) with -1, and substitute equations (3.70), and we find the Momentum-Integral Equation for plane, incompressible boundary layers:

Approximation velocity profile

The Momentum-Integral Equation initially contains too many unknowns to solve the equation. We need to approximate the velocity profile and assume that this velocity profile has the same shape everywhere in the boundary layer, i.e. is self-similar. We will begin with a velocity profile depending on , wich is a dimensionless parameter ( ). Note that this ? is different from the one used in section 3.1. We will work out all the results for a cubic velocity profile and then for a quartic velocity profile 5.

Cubic velocity profile

We take the following cubic velocity profile:

We have 4 yet unknown constants . To find these constants, we need 4 boundary conditions. On the plate (? = 0)due to the no-slip condition the velocity has to be zero. But also the derivative with respect to x of the velocity in x-direction are zero, so this transforms the x-momentum boundary layer equation (3.39) applied at y=0, with the definition for the pressure distribution from equation (3.41) in:

So we obtain 2 boundary conditions for ? = 0. The other boundary conditions can be found at the edge of the boundary layer (? = 1). There the velocity is equal to the free stream velocity and the first derivative, second derivative, and so on with respect to y are zero on the edge of the boundary layer. So the 4 boundary conditions for the cubic velocity profile are:

To obtain the unknown constants in the velocity profile we substitute the boundary conditions in the equation for the velocity. We present the results from this procedure in matrix form:

This results in the following:

Substituting these results in the equation for the velocity we get:

Here ? is a dimensionless velocity gradient
Numerical solutions

Up to this point we calculated the solutions of the quantities characterizing the boundary layer analytically by solving the boundary layer equations using the shooting method or by choosing an approximation for the velocity profile inside the boundary layer. The scope of problems that can be handled by these methods is limited, since this approach is based on a number of assumptions. To expand the possibilities for solving the boundary layer equations, we consider numerical solutions.

Explicit Discretisation

Numerical solutions of the boundary layer equations are based on replacing derivatives by finite-difference approximations. This approximation is called discretisation. We will use this to find numerical solutions for the boundary layer equations. We will start with the boundary layer equations for a steady free stream flow, where we consider again a constant free stream flow:

The domain containing the boundary layer is divided in small elements: the so-called grid. This grid consists of grid points which are formed by the coordinate lines. This grid is illustrated in figure 3.4. At the grid points we want to determine the unknown velocity components and .
We can use this method since we know the x- and y-components of the velocity at a starting line namely the leading edge of the horizontal plate. The first step will be to find the new components of the velocity at the grid points on the line . We will obtain this result by using equation (3.101). Next step is to calculate the vertical velocity at this next grid point.
Let us examine the difference equations in terms of this grid. We will use the centered space difference expressions to approximate the partial derivative of u with respect to y.

We cannot use this method for the partial derivative with respect to x since the only data available is known for the initial station, and not for a station downstream of this station. Therefore we will use the forward space difference method to approximate the partial derivative with respect to x:

Now we substitute these three equations in equation (3.101). We obtain:

We can transform this equation to obtain an expresion for the x-component of the velocity at the next station:

Now we have the values known. With this we can determine the values . We will use equation (3.100) for this. First, we use a backward difference for since we know the value of one step back in , but not one step forward in .

To obtain good accuracy, we can see this as a central difference at a location between grid points j and j-1 . We use this fact to approximate not far from this point at the location between i and i+1 and j and j-1 :

Substituting equations (3.107) and (3.108) in equation (3.101) we get:

Re-arranging this equation leads to the expression for :
Step size
We now have the two explicit relations to calculate the x- and y-component of the velocity at the next grid point. The stability of the numerical scheme is an important point to consider. The step sizes en cannot be chosen arbitrarily. The stability criterion for this difference scheme is2:

We did the calculation for the case for which the maximum – the free stream velocity 10 m/s – is and the kinematic viscosity is 1.4 x 10-5 m2/s . Using this criterion from equation (3.111) we choose and . This means that it takes 2000 steps to cover the whole length of the flat plate when its length is 1m . The extent of the computational domain in y-direction is 0.025.