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Computational Modeling of Complex Flows
Improvements in computing power and modeling sophistication over the past 15–20 years have made it possible for scientists to simulate increasingly complex physical processes. This, in turn, makes it possible for engineers to make significant progress in design. For example, computational improvements have let to advances in modeling of fluid flows, traditionally a very important area of research at JHU. Computational Fluid Dynamics (CFD) software is used to model the complex, turbulent flows encountered in IC engines, HVAC systems, fire safety applications, aircraft aerodynamics, and turbomachinery.
Computational fluid dynamics is typically approached in two very different ways. In the classical “top-down” approach, the field equations governing macroscopic flow phenomena are approximated by numerical techniques (like finite-differencing or spectral methods) that discretize these continuum equations in order to solve them on a computer. The “bottom-up” approach involves solving Newtonian laws of motion describing individual molecules and then working “upwards” to the large scale flow, a t
echnique known as Molecular Dynamics. Although this microscopic description is technically the most accurate, it strains even the fastest supercomputers. Models like this can only handle very small systems (10 million particles) and very short times (a few picoseconds). Professor Shiyi Chen, an expert in various CFD methodologies who came to Johns Hopkins from Los Alamos National Laboratory in 1999, solves various fluid flow problems using the Lattice Boltzmann Method (LBM), a technique that occupies an intermediate ground between typical “top-down” and “bottom-up” methods.
The LBM is constructed as a simplified kinetic molecular system in which single-particle distribution functions (i.e., very coarse histograms of how often a particle has a certain velocity) reside on discrete nodes of a lattice. During each time step, the particles move to the nearest lattice site along their direction of motion, where they “collide” with other particles that arrive at the same site. Only a few directions are allowed (e.g., up, down, left, right). The outcome of the collision is determined by solving the kinetic (Boltzmann) equation, and a new particle distribution function is determined for that site. This simplified molecular dynamics includes the essentials of the underlying microscopic processes, and so the averaged properties of LBM simulations obey macroscopic continuum equations, in this case, the classical Navier-Stokes equations. The fact that equations at each lattice node can be solved in parallel simplifies and speeds up the computation significantly. And because boundary conditions are imposed locally, lattice methods are ideal for simulating flows in complex geometry.
The data set that has to be crunched by LBM is impressively huge. But instead of taking the traditional approach and farming the data off to a supercomputer, Prof. Chen has put together a cluster of 64 networked PCs. This exploits the inherently parallel nature of the LBM technique by solving different parts of the problem simultaneously on different CPUs of the cluster and then reassembling them at various stages of the simulation, greatly reducing overall computing time. This system is used as well by Professor Joe Katz to analyze the huge amounts of data he collects in his Particle Image Velocimetry experiments (see Aerospace and Marine Systems, page 15).
Professor Chen has used LBM techniques to solve problems ranging from the flow of oil and water through sandstone (oil extraction), to flow over and around tires and automobiles for industry partners, and the complex flow patterns of granular materials, such as sand or snow.
Turbulence is another example of an area in which increased computer power translates directly into more complex and robust models. Hopkins has a long and illustrious history of turbulence research, including the work of Professor Stanley Corrsin, who was one of the first scientists to capture the dynamics of turbulence experimentally in the wind tunnel he built for that purpose. Professors Charles Meneveau, Joe Katz, Shiyi Chen, and Omar Knio are carrying on this tradition by conducting experiments and testing theories that may eventually give us a variety of reliable ways to model turbulent flows.
In 1883, British physicist Osborne Reynolds demonstrated that the transition from laminar to turbulent flow in a pipe depends on the ratio of inertial forces to viscous forces in the flow, a non-dimensional number now known as the Reynolds Number. The higher the Reynolds Number, the more complex the flow, and the more difficult it is to model. Realistic turbulent flows such as those encountered in many engineering and atmospheric applications have very high Reynolds numbers, and several different approaches are taken to try and quantify what is happening in the flow.
In turbulent flow, large-scale structures such as big vortices break down into smaller and smaller eddies, eventually being diffused by friction at the viscous scale. That range spans many orders of magnitude (e.g., for flow over aircraft fuselage, from tens of meters in the wake to tens of micrometers and less in the thin boundary layers). To further complicate matters, the equations governing turbulent fluid flow have a “closure problem”—meaning that the equations at a large scale contain unknown contributions from the smaller scales, which themselves are affected by even smaller scales, and so on. In addition, unlike smooth laminar flow, turbulent flow cannot be simplified by reducing the equations to two dimensions, since the eddies are inherently three-dimensional.
Direct Numerical Simulation (DNS) solves the Navier-Stokes equations without averaging any of the turbulent eddies. Professor Chen uses spectral methods to discretize the equations and solves them on parallel computers. DNS is limited to low and moderate Reynolds number flows in which the ratio of viscous to large-scale eddies is manageable, but it provides very detailed, three-dimensional and time-dependent information about the fundamental structure of turbulence.