Lecture 22, March 1, 2004 • Matlab Tutorial Today (3:30-4:30) (4:30-5:30) • Assignment 3 (will be posted today, due end of next

week?) • Quiz 2 (covers material after quiz1 to the end of

this week) Today • Average convection coefficient example • Start of Blasius Solution for flow over a flat plate

Boundary Layer Example Initially, the boundary layer on a flat plate will be laminar, and will transition to turbulence at some critical Reynolds number Rex,c. When the boundary layer becomes turbulent, it will of course become unsteady, and heat and mass transfer will be increased. Therefore we will consider the time average turbulent quantities. In reality this transition is not abrupt, but occurs over a transition region. We must fix a value of either Rex,c, or of xc in order to obtain a solution. Let’s start be examining some velocity profiles in the laminar portion of our plate, for air at 3 m/s. For these conditions xc corresponds to 3m, and so we will have a laminar boundary layer for the first 3m.

Clearly, the gradient of the velocity profile at the wall is decreasing with distance along the plate, and we therefore expect that the rate of heat transfer will decrease along the plate as well. Clearly, the highest heat transfer coefficient will exist at the leading edge of the plate. The rest of this example is posted separately. Blasius Solution for Laminar flow over a plat plate. Beginning with our equation for conservation of mass (2D Steady, constant properties),

And conservation of momentum in the x direction (with the boundary layer assumptions) and with zero pressure gradient as in the case of a flat plate

And, the boundary layer energy equation

In order to simplify these equations, we introduce the stream function

Substituting these definitions into mass conservation,

We see that it is identically satisfied and we need not consider mass conservation any further as long as we use the stream function. Next we want to simplify the momentum equation. We do this by postulating that there is some scaling parameter that will collapse all of the velocity profiles onto a single curve. This is a similarity parameter, and hence the solution is called a similarity solution. When we find this parameter, our two dimensional problem will be reduced to a one dimensional problem and will be much easier to solve. This parameter has been found, and is in fact the way the profiles develop (you can measure this).

Next, we want to define a non-dimensional stream function using our similarity parameter.

All we need to do is evaluate all of the terms in the momentum equation in terms of f and η, and then we will have our simplified ordinary differential equation (one dimensional). Beginning with u,

or,

And, more involved is v,

Finally, our three derivative terms,

We can now substitute all of these into the momentum equation to get,

or, in simplified notation,