Chapter 6: Introduction to Convection UM-SJTU  Chapter 6: Introduction to Convection In the previous Chapters, h is assumed to be known   boundary condition  h = ?  

Conservation of Energy Approach: Continuum hypothesis (The molecular nature of the physical matter is ignored, and it is assumed that matter continuously fills a spatial domain) Apply fundamental laws: Conservation of Mass Conservation of Momentum Conservation of Energy The Equation of State Convection heat transfer is a field at the interface between heat transfer and fluid mechanics. In order to obtain solution for h, we need to find enough relations for

Navier Stokes Equations: Steady-state, under a Cartesian Coordinate system: Continuity equation (2D) --- Conservation of Mass

X- and Y- momentum equations – Conservation of momentum (The rate of change of momentum of the fluid within the C.V. + net rate of momentum flowing out of the C.V. = sum of all the forces on the fluid in the C.V.)

Energy equation – Conservation of energy (superposition of the heat fluxes) Equation of state for example, ideal gas law as above.

For , S.S., incompressible flow, no body forces, X=Y=0 (Difficulty in solving the above equations analytically)

Boundary Layer: Physical Observations: The fluid layer situated at y = 0 is stuck to the solid wall due to viscosity: Shear stress term such as cannot be neglected due to large at the wall Prandtl discovered that for most applications, the influence of viscosity is confined to an extremely thin region very close to the body surface and that the remainder of the flow field could, to a good approximation, be treated as inviscid ( )

The transition region from the wall to the main stream is very small ( ) In fact, the viscosity is very small for most fluids: Air: Water: Since , if is small, Outside the boundary layer, the flow is inviscid, and the analytical solution is generally available.

Let’s focus on the boundary layer region Let’s focus on the boundary layer region. Let be the order of magnitude of the distance in which u changes from 0 to roughly or 99%, and for locations not very close to the leading edge (swung dash  be of the same order of magnitude) Also: for most of fluids Some scale analysis rules: From the continuity equation:

Boundary layer equations:

Velocity boundary layer development on a flat plate: The critical Reynolds number is the value of Rex for which the transition begins. For flow over a flat plate, it is known to vary from approximately 105 to 3(106) , depending on surface roughness and the turbulence level of the free stream. A representative value is often used: