Towards scalable parallel-in-time turbulent flow simulations

Abstract

We present a reformulation of unsteady turbulent flow simulations. The initial condition is relaxed and information is allowed to propagate both forward and backward in time. Simulations of chaotic dynamical systems with this reformulation can be proven to be well-conditioned time domain boundary value problems. The reformulation can enable scalable parallel-in-time simulation of turbulent flows. I. NEED FOR SPACE-TIME PARALLELISMThe use of computational fluid dynamics (CFD) in science and engineering can be categorized into Analysis and Design. A CFD Analysis performs a simulation on a set of manually picked parameter values. The flow field is then inspected to gain understanding of the flow physics. Scientific and engineering decisions are then made based on understanding of the flow field. Analysis based on high fidelity turbulent flow simulations, particular Large Eddy Simulations, is a rapidly growing practice in complex engineering applications 12 .CFD based Design goes beyond just performing individual simulations, towards sensitivity analysis, optimization, control, uncertainty quantification and data based inference. Design is enabled by Analysis capabilities, but often requires more rapid turnaround. For example, an engineer designer or an optimization software needs to perform a series of simulations, modifying the geometry based on previous simulation results. Each simulation must complete within at most a few hours in an industrial design environment. Most current practices of design use steady state CFD solvers, employing RANS (Reynolds Averaged NavierStokes) models for turbulent flows. Design using high fidelity, unsteady turbulent flow simulations has been investigated in academia 3 . Despite their great potential, high fidelity design is infeasible in an industrial setting because each simulation typically takes days to weeks. FIG. 1:Exponential increase of high performance computing power, primarily sustained by increased parallelism in the past decade. Data originate from top500.org. GFLOPS, TFLOPS, PFLOPS and EFLOPS represent 10 9 , 10 12 , 10 15 and 10 18 FLoating point Operations Per Second, respectively.The inability of performing high fidelity turbulent flow simulations in short turnaround time is a barrier to the game-changing technology of high fidelity CFD-based design. Nevertheless, development in High Performance Computing (HPC), as shown in Figure 1, promises to delivery in about ten years computing hardware a thousand times faster than those available today. This will be achieved through extreme scale a) Corresponding

Towards high-fidelity aerospace design in the age of extreme scale supercomputing (Invited)

“…35 Mathematically, the Least Squares Shadowing method replaces the initial value problem with the wellconditioned Least Squares Shadowing problem. 8,10,35 The solution to the Least Squares Shadowing problem satisfies the model equation, and give correct gradient that can drive efficient optimization algorithms. The Least Squares Shadowing method has been applied to a range of dynamical systems, the largest being an isotropic homogeneous turbulent flow simulation at Taylor microscale Reynolds number Re λ = 33 by a 32 3 Fourier pseudo-spectral discretization.…”

“…[13][14][15] Parallel-in-time methods can scale a moderate-sized high-fidelity simulation to very large parallel computers, leading to faster simulations. 8 Better optimization algorithms, 16 including those that can exploit additional level of parallism, 17 can lead to faster design optimization. Acceleration achieved in these fields, together with improvements in computing hardware, have multiplicative effect.…”

“…These unsteady simulations are parallelized only in the spatial domain, because efforts towards parallelizing them in time is challenged by complex nonlinear dynamics these simulations can exhibit. 8,9 In addition, simulation-based design usually takes many iterations of such simulations, and therefore takes at least months to complete. Methods to accelerate CFD-based design, such as the adjoint method, is scarcely available, and is also challenged by complex nonlinear dynamics of these simulations.…”

Multiple Shooting Shadowing for Sensitivity Analysis of Chaotic Systems and Turbulent fluid flows

“…17 The "butterfly effect" ensures that u r (t) and u(t) will become decorrelated after some time, resulting in the non-commutation of the derivative discussed in the introduction. The poor conditioning of the initial value problem arises from the problem formulation: u r (t) and u(t) will only be close in phase space at t = T 0 where they share the initial condition u 0 ; nothing in the problem formulation requires u r (t) and u(t) to be correlated.…”

Output Error Estimation for Chaotic Flows

“…The advection term is added in order to ensure that the solution exhibits ergodic behavior. 40 The initial condition for the system is set to a triangular distribution on a discrete grid, where the solution is 0 at all locations except on the two middle elements, with u = 1 at x = 64, dropping linearly to zero at the adjacent nodes. The following boundary conditions were imposed: 40…”