a simple thermal problem response to a temperature step of a flow in a pipe:
from Lévêque to Graëtz: most of the ingredients.
2) Perturbation of a linear shear flow: Prandtl equations
- linear Airy solution [A=0 solution],
linear perturbation of shear stress and pressure in Fourier space
- non linear solution numerical solutions, flow over bumps. Separation.
- 3D extension, qualitative results for the flow over a bump in a linear shear flow
3) Quick Recall of classical Boundary Layer Theory: weak coupling
- The classical Blasius flow (incompressible and compressible), self similar solutions Falkner- Skan (incomp. and comp.) a key feature: the displacement thickness
- Integral methods, Falkner Skan closure
- second order boundary layer: the boundary layer retroaction on the ideal fluid
- three dead-end of B.L.T.:
* the singularity of separation [just to justify the need of the interaction]
* the paradox of upstream influence in supersonic flows
* the finite time singularity of the unsteady B.L.T.
4) The interacting Boundary Layer concept: strong coupling
- viewed from an extension of 2nd order BLT and as the step before Triple Deck
(the importance of the the Transpiration velocity
- solving the Prandtl equations in an inverse way
- recent view Mauss/ Cousteix
- examples Cebecci + Cousteix, Le Balleur...
- numerical examples in pipe, subcritical supercritical flows, in compressible and incompressible flows
5) Quick introduction to the The Triple Deck: strong coupling
- Main Deck: perturbation of the basic flow: the A function
- Lower Deck: perturbation of a linear shear flow (see above) matching A
- Upper Deck: the interaction between p and A
linear perturbations of an ideal fluid flow: (M<1) Hilbert integral,
(M>1) characteristic solution p=-A'
- The incompressible triple deck: linear solutions,
- the compressible triple deck: linear solutions
- Pipe Flow, A=0 and non symetrical case p and A''
- Other examples of coupling p and A in hydrolics (with Froude Number) and
compressible flows (hypersonic)
- comparisons of all the responses in cases of wedges and bumps
- upstream influence, self induced solution (p=-A', p=-A), no upstream influence (p=A etc)
6) Flows in pipes/ or in a 2D channel
- Flow in Pipes: the RNSP equations: Reduced Navier Stokes/ Prandtl equations
a simple model which recovers the TD and DD
- IBL in stenoses axi and 2D applications
- IBL in non symmetrical channel, interaction of the two boundary layers;
Course n°4 (pdf), and the slides
7) Other Examples and applications using the concepts
- Strong Hypersonic Interaction
- The bore: hydraulic jump
- The mixed convection thermal boundary layer: an example of self interacting flow with upstream influence
- Biological Flow like in Arteries (importance of pressure), Pedley's results on compliant walls,
- 1D fluid flows in arteries viewed as IBL, Womersley solution
- Example in shallow water flows: Flow over an erodible bed: formation of ripples in rivers Flow (importance of skin friction),