1. AMS 599 Special Topics in Applied Mathematics Lecture 3
James Glimm
Department of Applied Mathematics and Statistics,
Stony Brook University
Brookhaven National Laboratory

2. Fluid Transport The Euler equations neglect dissipative mechanisms
Corrections to the Euler equations are given by the Navier Stokes equations
These change order and type. The extra terms involve a second order spatial derivative (Laplacian). Thus the equations become parabolic. Discontinuities are removed, to be replaced by steep gradients. Equations are now parabolic, not hyperbolic. New types of solution algorithms may be needed.

3. Fluid Transport Single species Multiple species
Viscosity = rate of diffusion of momentum
Driven to momentum or velocity gradients
Thermal conductivity = rate of diffusion of temperature
Driven by temperature gradients: Fourier’s law
Multiple species
Mass diffusion = rate of diffusion of a single species in a mixture
Driven by concentration gradients
Exact theory is very complicated. We consider a simple approximation: Fickean diffusion

4. Comments
Why study the Euler equations if the Navier-Stokes equations are more exact (better)?
Often too expensive to solve the Navier-Stokes equations numerically
Often the Euler equations are “nearly” right, in that often the transport coefficients are small, so that the Euler equations provide a useful intellectual framework
Often the numerical methods have a hybrid character, part reflecting the needs of the hyperbolic terms and part reflecting the needs of the parabolic part.

5. Navier-Stokes Equations for Compressible Fluids

6. Incompressible Navier-Stokes Equation (3D)

14. Turbulent mixing for a jet in crossflow and plans for turbulent combustion simulations
James Glimm

15. The Team/Collaborators
Stony Brook University
James Glimm
Xiaolin Li
Xiangmin Jiao
Yan Yu
Ryan Kaufman
Ying Xu
Vinay Mahadeo
Hao Zhang
Hyunkyung Lim
College of St. Elizabeth
Srabasti Dutta
Los Alamos National Laboratory
David H. Sharp
John Grove
Bradley Plohr
Wurigen Bo
Baolian Cheng

16. Outline of Presentation
Problem specification and dimensional analysis
Experimental configuration
HyShot II configuration
Plans for combustion simulations
Fine scale simulations for V&V purposes
HyShot II simulation plans
Preliminary simulation results for mixing

17. Scramjet Project
Collaborated Work including Stanford PSAAP Center, Stony Brook University and University of Michigan

18. Proposed Plan on UQ/QMU
Decompose the large complex system into several subsystems
UQ/QMU on subsystems
Assemble UQ/QMU of subsystems to get the UQ/QMU for the full system
Sub-system analysis goal: UQ/QMU for the essential subsystem --- combustor

19. Proposed Plan on UQ/QMU (continued)
Our hypothesis is that an engineered system has a natural decomposition into subsystems, and the safe operation of the full system depends on a limited number of variables in the operation of the subsystems.
For the scramjet, with its supersonic flow velocity, a natural time like decomposition is achieved, with each subsystem getting information from the previous one and giving it to the next.
In this context, we hope that the number of variables to be specified at the boundaries between subsystems will be not too large. To show this in the scramjet context will be a research program, and central to the success of our objectives.
We call the boundaries between the subsystems to be gates. Or rather the boundary and the specification of the criteria to be satisfied there is the gate.