1. Specificity of Closure Laws and UQ
Vincent A Mousseau

2. Outline Define Closure Laws Generic Form First Order Analysis
Multi-Physics, Non-linearity, Transients, and Flow Regimes
Steady State (Time Scales) and Fully Developed (Length Scales)
Ramps, Clipping, Smoothing, and Under-relaxation
Stability, Non-oscillatory, and Monotonicity
Conclusions

3. Closure Laws
Closure laws describe the sub-grid physics that is not resolved by the grid/mesh/nodalization and the conservation laws
By definition, sub-grid physics depends on the mesh size
All physics has length scales and time scales
Time scales can be removed by equilibrium assumptions or resolved when employing kinetic physics
This discussion will focus on the closure laws that exchange mass, momentum and energy.

4. Generic Form All closure laws can be linearized to the following form
This can be described in terms of chemistry as a reaction rate times a driving potential
The equilibrium solution occurs when the driving potential is driven to zero
The reaction rated describes how fast one goes to the equilibrium solution
Steady state can be achieved with a non-zero potential when there is a balance between competing physics

5. First Order Analysis
A single closure law applied to a large data set will have more error than multiple closure laws applied to subsets of the data
More closure laws tend to be more complex and therefore a higher probability of bugs
More experimental data is required to calibrate more closure laws
Specific closures provide more accuracy for their intended use but have the potential for larger error outside of the range of applicability
General closure laws apply to a wider range of applications but have a larger error everywhere

6. Multi-Physic, Non-linearity, Transients, and Flow Regimes
Closure laws are much easier for single physics, linear, steady state problems.
In multi-physics each physics has it own length scales and time scales and the closure laws can create new length scales and time scales
Non-linearity requires new solution algorithms that require initial guesses to be within the ball of convergence
Transients require that all of the physical length scales and time scales (including closure laws) are resolved by the mesh and time step
Flow regimes describe changes in topology that have significant impact on the solution and are not resolved by the mesh or the conservation laws

7. Steady State and Fully Developed
Most closure laws assume steady state and fully developed flow.
Steady state means infinitely fast time scales
Fully developed means infinitely short length scales
These assumptions lead to discontinuities in space and time
Space and time integration methods employ Taylor series which require smooth (no discontinuities) solutions

8. Ramps, Clipping, Smoothing, and Under-relaxation
To remove the discontinuities that we caused by assuming steady state and fully developed we “modify” the physics
Ramps throw away good experimental data and replace it with an arithmetic average of two correlations
Clipping is a way to create a continuous solution by taking the min and max of closure laws applied outside of their range of applicability
Smoothing is the process of arithmetic averaging closure laws in space to remove discontinuities
Under-relaxation smooths the physics over multiple time steps. Linear – approx 20: Exponential - approximately 200

9. Stability, Non-oscillatory, and Monotone
Kinetic Closure laws are designed to drive the physics toward the equilibrium solution
Numerically, kinetic reactions can drive the answer past the equilibrium value, this is unphysical
It is also possible for the reaction to drive the volume fraction greater than one or less than zero
There is a constraint on the time step to keep the reaction from exceeding physical limits

10. Conclusions
A larger number of well designed closure laws can reduce uncertainty.
A larger number of poorly designed closure laws can increase uncertainty.
The best approach is to build a small number of well designed closure laws and then add new ones as uncertainty requirements demand.