Version, Date, POC Name 1 Purpose: To investigate multiscale flow discretizations that represent both the geometry and solution variable using variable-order polynomial and spline-based (NURBS) approximations allow variable inter-element regularity Product/Results: Implement high-order polynomial and NURBS- approximations in existing frameworks for Navier-Stokes (NS) and shallow water (SW) equations Initial adaptive strategies that exploit approximation order, mesh size, and regularity for NS and SW Application of adaptive strategies to challenging SW problems and initial free- surface isogeometric (NURBS) capability Payoff: High-order, accurate models for free surface flows in complex geometries New, efficient adaptive strategies for modeling multiscale flows. Schedule & Cost Total $750K MILESTONES FY10 FY11 FY12 Army ($K) Implementation of High-Order FEMs High-Order, multiscale flow simulation Adaptive, High-Order FEMs for NS and SW Hybrid FEM strategies for SW and free-surface NS 2 1 Ending TRL/SRLs Beginning TRL/SRLs Variational multiscale simulation for 3-D backward facing step WP # Status: New

Version, Date, POC Name 2 1. What is the problem? It remains very difficult to represent many complex free-surface flows accurately with existing low-order approximations given available computational resources. The objective of our research is to study multiscale flow discretizations that represent problem geometry and solution using variable-degree polynomial and spline-based (NURBS) approximations. Our approaches will also allow varying types and degrees of inter-element regularity. 2. What are the barriers to solving the problem? The flow problems we are interested in are complex, turbulent free- surface flows (hydrostatic or nonhydrostatic) in which capturing small scale variation is critical to overall accuracy. Unfortunately, gaining sufficient accuracy can be infeasible using standard low-order approximations. Higher order approximations can be more accurate, but traditional or naïve approaches have not been adequate for complex, engineering- scale problems. 3. How will you overcome these barriers? Recent advancements have found effective ways to incorporate high- order approximations into robust finite element frameworks for CFD. We will explore two of these approaches: Isogeometric analysis and discontinuous Galerkin approximations. We will investigate high-order versions of each and investigate novel hybrid combinations to test our hypothesis: High-order approximations in the correct finite element framework will be much more efficient than existing low-order methods for many free- surface flows 4. What are the anticipated results and value of this research? Isogeometric finite element approximations for laminar and turbulent channel flows. High-order, adaptive DG methods for shallow water flows. Hybrid Isogeometric-DG strategies that exploit variable inter-element regularity for improved efficiency. 5. What is innovative about this work? We will be exploring state-of-the-art computational methods for CFD that have shown much promise, but have not been applied to free surface problems or real-world channel flows. We will be advancing methods that are very new and developing hybrid, adaptive approaches to make these techniques practical for engineering-scale problems. If we are successful, our approaches could revolutionize the way free-surface flows are modeled in the Corps and the community as a whole. 6. What is your publication plan? Oct Hybrid isogeometric discontinuous Galerkin methods for open channel flows. CMAME May Isogeometric analysis for microscale models of flow through permeable media. Physics of Fluids. Dec Mixed element hp refinement strategies for DG approximations of the shallow water equations. CMAME Oct Isogeometric level-set formulations for free-surface flows. JCP. 7. Transition plan: We will be implementing and extending high-order isogeometric and DG methods in our research codes. A natural transition would be introduction of these techniques into production codes like ADH 8. Collaboration across ERDC, commercial firms and/or academia: CHL researchers will be collaborating academic researchers (Yuri Bazilevs at UCSD) High-Order, multiscale flow simulation WP # Status: New

Version, Date, POC Name 3 High-Order, multiscale flow simulation WP # Status: New Q1. What is the research problem? It remains very difficult to represent many complex free-surface flows accurately with existing low-order approximations given available computational resources. Q2. Describe the research objective. To advance classes of high-order accurate finite element approximations that have shown great promise for multiscale flow simulation. Q3. What is the toughest technical challenge in the work? High-order finite element approximations for nonlinear flow problems are complex. Isogeometric (NURBS-based) methods are in their infancy, and we are looking at developing strategies that are efficient for engineering problems. Q4. If successful, what will be the significance and impact of this research? Our approaches would add wholly new computational capabilities for hydrostatic and non-hydrostatic free surface flows. Q5. What future applications could result from this research? New software tools for flow modeling which can move directly from CAD geometries to analysis Q6. What other organizations are pursuing this research? These methods are actively being pursued in academia Q7. What makes this research innovative, original and high risk? We will be exploring state-of-the-art computational methods for CFD that have shown much promise, but have not been applied to free surface problems or real-world channel flows. We will be advancing methods that are very new and developing hybrid, adaptive approaches to make these techniques practical for engineering-scale problems. Q8. How could this research develop future capabilities for ERDC? If we are successful, our approaches could revolutionize the way free-surface flows are modeled in the Corps and the community as a whole Q9. Describe the research team and plan. The research team consists of PI’s from CHL and a leading expert in finite element method from academia. The research plan consists of three phases where begin by implementing methods for laminar flows then transition to adaptive techniques and turbulent problems. Q10. Is there adequate funding, equipment, and facilities to complete this research as planned? If funded at the expected level.