Lecture 21, February 27, 2004 • Matlab Tutorial Monday – Signup sheets on my

door Last Day • Axioms of Mechanics / Governing Equations

Today • Boundary Layer Equations • Non-Dimensional Equations

Boundary Layer Equations We can assume for a boundary layer, that changes normal to the surface will be much greater than changes along the surface. We can also assume that the streamwise (u-component) of velocity will be much greater then the normal component (v-component)

These assumptions greatly simplify the governing equations, and even allow us in some instances to for analytic solutions of significant importance (Blasius solution next week). Conservation of mass is unchanged by these approximations. While x-momentum becomes

This may not seem greatly simplified, but in practice it is much easier to solve because the nature of the equation has changed considerably (from elliptic to parabolic). The y-momentum equation disappears altogether, stating that there is no pressure gradient normal to the surface.

And the energy equation becomes

which is also quite simplified. Non-Dimensional boundary layer equations This will enable us to uncover the form of the solutions even before we have any solutions, and enable us to determine what parameters will be important to consider. Simply non-dimensionalize each term, and substitute them into the boundary layer equations.

Substituting these into the boundary layer equations, we find

Now we can write the functional form of the solutions, before we formulate any solutions

We are often interested in the wall shear stress, which is often described using a friction coefficient, cf.

The pressure gradient comes from the geometry of the surface. Recall the cylinder.

The pressure is highest at the stagnation point where the flow first impinges on the cylinder. The flow then needs to accelerate around the cylinder. This acceleration results in a decrease in the pressure It is clear that the shape of the body has caused this acceleration and the corresponding pressure gradient. Another point can also be made here with regards to the boundary layer equations. Examining the velocity vectors around the

cylinder,

We can see that the vectors follow the surface of the cylinder over the first half of the cylinder, and then separate just past the top of the cylinder where the pressure should start increasing, again, due to the change in shape of the surface. Up to the point of separation, the flow ‘looks like a boundary layer’ in that there is a thin layer adjacent to the surface where the changes in velocity occur. After the point of separation, however, this region is no longer thin, and hence the boundary layer equations are no longer valid. Fortunately, Fluent solves the full Navier-Stokes equations which enables us to determine the solution even in separated regions. If you tried to do this with the boundary layer equations, the answers would be very wrong because the principal assumption of a thin boundary layer is violated. This is even more clear when we look closely at the surface around the point of separation.

It is also clear from the temperature distribution that the concept of a boundary layer is not valid in the wake of the cylinder (in the separated region),

where the region affected by the hot cylinder is very very large. The pressure field, and hence the pressure gradient is clearly determined by the geometry of the surface, and is therefore set for a fixed geometry. So, considering a single geometry, i.e. a cylinder, or an airfoil, or a flat plate. The functional form of our friction coefficient is thus simplified,

Similarly, for the energy equation

As with cf, we can describe the heat transfer with the Nusselt number, which is the non-dimensional temperature gradient at the wall.

And, for a fixed class of geometry (i.e. for a cylinder, or for a flat plate etc.),

Finally, we are usually interested in the total heat transfer which is simply an integration of the local value over the entire surface.

Which will give us the total heat transfer from a surface, or the total friction force on a surface. The functional forms of the solution now become,

This is really a very impressive result and achievement. We started with conservation of mass, momentum and energy, and we made various assumptions along the way. We thus found a rather formidable set of equations that need to be solved to characterize fluid flow and heat transfer. We continued however to non-dimensionalize the equations and found that if what we are interested in the friction forces and the heat transfer, then we can correctly quantify the results using functions of only Reynolds number and Prandtl number. This means that someone that has no interest in solving difficult sets on non-linear higher order partial differential equations can run experiments and measure drag and heat transfer AND they can express the results in a way that is very useful for the rest of us. This is precisely what has been done, and what will be presented in great detail in the remainder of the convection section of this course. For each class of geometry a correlation of experimental data has been made and expressed as above. Knowing these function, we can happily proceed to determine friction and heat transfer for our given surface as long as we can find an equation suited to that range of Re and Pr for that specific geometry. You can well imagine that this is a tremendous savings over the fellow who has not non-dimensionalized the equations and must run an experiment for every imaginable fluid over every single cylinder diameter at every desired flow speed.