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

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.

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

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.

Navier-Stokes Equations for Compressible Fluids

Incompressible Navier-Stokes Equation (3D)

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

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

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

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

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

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.