Lecture 18 February 13, 2004 Last Day • Intro to momentum and thermal boundary layers

Today • Steady/ Unsteady Flow • Laminar/Turbulent Flow • Flow Separation • Internal/External Flows • Governing Equations – Axioms of Mechanics

Introduction to convection continued If we think about a boundary layer, we can determine both the shear stress and the heat flux by manipulating the velocity and temperature gradients. Right at the wall the velocity is zero (no-slip) condition, and the heat flux is by conduction only. We can evaluate the heat flux using Fourier’s Law

The shear stress at the wall can analogously be determined using Newton’s law of viscosity

Note that these two equations have exactly the same form! We can also describe the convective heat transfer process using our now familiar Newton’s law of cooling,

All of the heat energy conducted through the surface is then carried away by conduction and convection by the fluid passing over the surface, and we can equate the two above expressions. First, lets non-dimensionalize our variables.

The Nusselt number, Nu, is a non dimensional number that describes the convective heat transfer process. It looks

very similar to the Biot number, but the conductivity is that of the fluid, not a solid. The interpretation is very different also – looking at the above equation we can see that Nu is the non-dimensional temperature gradient at the wall. We can now use Nu to solve convective heat transfer problems for any fluid, and that is the great advantage of non dimensional representations. Just from thinking about conservation of momentum we can reason • Pressure drops (shear stresses) will be greater in

turbulent flows compared to laminar flows • Heat transfer rates will be higher in turbulent flows

than in laminar flows. Recall the Reynolds number,

, which is another non-dimensional number that characterizes a flow. For low Re, we find laminar flows, while for high Re we find turbulent flows. The Reynolds number can be interpreted as a ratio of inertial forces to viscous forces. Let’s now think about flow separation.

Laminar/turbulent flow Turbulent flow is unsteady. Turbulent flow is three dimensional.