scholarly journals Low‐order preconditioning of the Stokes equations

2021 ◽  
Author(s):  
Alexey Voronin ◽  
Yunhui He ◽  
Scott MacLachlan ◽  
Luke N. Olson ◽  
Raymond Tuminaro
Keyword(s):  
Stokes Equations ◽  
Low Order

scholarly journals Examination of large-scale structures in a turbulent plane mixing layer. Part 2. Dynamical systems model

2001 ◽  
Vol 441 ◽  
pp. 67-108 ◽  
Author(s):  
L. UKEILEY ◽  
L. CORDIER ◽  
R. MANCEAU ◽  
J. DELVILLE ◽  
M. GLAUSER ◽  
...  
Keyword(s):  
Dynamical System ◽  
Velocity Field ◽  
Large Scale ◽  
Stokes Equations ◽  
Temporal Dynamics ◽  
Mixing Layer ◽  
Basis Set ◽  
Large Scale Structures ◽  
Scale Structures ◽  
Low Order

The temporal dynamics of large-scale structures in a plane turbulent mixing layer are studied through the development of a low-order dynamical system of ordinary differential equations (ODEs). This model is derived by projecting Navier–Stokes equations onto an empirical basis set from the proper orthogonal decomposition (POD) using a Galerkin method. To obtain this low-dimensional set of equations, a truncation is performed that only includes the first POD mode for selected streamwise/spanwise (k1/k3) modes. The initial truncations are for k3 = 0; however, once these truncations are evaluated, non-zero spanwise wavenumbers are added. These truncated systems of equations are then examined in the pseudo-Fourier space in which they are solved and by reconstructing the velocity field. Two different methods for closing the mean streamwise velocity are evaluated that show the importance of introducing, into the low-order dynamical system, a term allowing feedback between the turbulent and mean flows. The results of the numerical simulations show a strongly periodic flow indicative of the spanwise vorticity. The simulated flow had the correct energy distributions in the cross-stream direction. These models also indicated that the events associated with the centre of the mixing layer lead the temporal dynamics. For truncations involving both spanwise and streamwise wavenumbers, the reconstructed velocity field exhibits the main spanwise and streamwise vortical structures known to exist in this flow. The streamwise aligned vorticity is shown to connect spanwise vortex tubes.

Stabilization of Low-order Mixed Finite Elements for the Stokes Equations

2006 ◽  
Vol 44 (1) ◽  
pp. 82-101 ◽  
Author(s):  
Pavel B. Bochev ◽  
Clark R. Dohrmann ◽  
Max D. Gunzburger
Keyword(s):  
Finite Elements ◽  
Stokes Equations ◽  
Mixed Finite Elements ◽  
Low Order

Reduced-Order Modeling of Aeroacoustic Systems for Stability Analyses of Thermoacoustically Noncompact Gas Turbine Combustors

2015 ◽  
Vol 138 (5) ◽  
Author(s):  
Tobias Hummel ◽  
Constanze Temmler ◽  
Bruno Schuermans ◽  
Thomas Sattelmayer
Keyword(s):  
Gas Turbine ◽  
Time Domain ◽  
Stokes Equations ◽  
Order Reduction ◽  
High Order ◽  
Essential Information ◽  
Reduced Order ◽  
Stability Analyses ◽  
Modal Reduction ◽  
Low Order

A methodology is presented to model noncompact thermoacoustic phenomena using reduced-order models (ROMs) based on the linearized Navier–Stokes equations (LNSEs). The method is applicable to geometries with a complex flow field as in a gas turbine combustion chamber. The LNSEs, and thus the resulting ROM, include coupling effects between acoustics and mean fluid flow and are hence capable of describing propagation and (e.g., vortical) damping of the acoustic fluctuations within the considered volume. Such an ROM then constitutes the main building block for a novel thermoacoustic stability analysis method via a low-order hybrid approach. This method presents an expansion to state-of-the-art low-order stability tools and is conceptually based on three core features: First, the multidimensional and volumetric nature of the ROM establishes access to account spatial variability and noncompact effects on heat-release fluctuations. As a result, it is particularly useful for high-frequency phenomena such as screech. Second, the LNSE basis grants the ROM the capability to reconstruct complex acoustic performances physically accurate. Third, the formulation of the ROM in state-space allows convenient access to the frequency and time domain. In the time domain, nonlinear saturation mechanisms can be included, which reproduce the nonlinear stochastic limit cycle behavior of thermoacoustic oscillations. In order to demonstrate and verify the ROM's underlying methodology, a test case using an orifice-tube geometry as the acoustic volume is performed. The generation of the ROM of the orifice tube is conducted in a two-step procedure. As the first step, the geometrical domain is aeroacoustically characterized through the LNSE in frequency domain and discretized via the finite element method (FEM). The second step concerns the actual derivation of the ROM. The high-order dynamical system from the LNSE discretization is subjected to a modal reduction as order reduction technique. Mathematically, this modal reduction is the projection of the high-order (N∼ 200,000) system into its truncated left eigenspace. An order reduction of several magnitudes (ROM order: Nr∼ 100) is achieved. The resulting ROM contains all essential information about propagation and damping of the acoustic variables, and efficiently reproduces the aeroacoustic performance of the orifice tube. Validation is achieved by comparing ROM results against numerical and experimental benchmarks from LNSE–FEM simulations and test rig measurements, respectively. Excellent agreement is found, which grants the ROM modeling approach full eligibility for further usage in the context of thermoacoustic stability modeling. This work is concluded by a methodological demonstration of performing stability analyses of noncompact thermoacoustic systems using the herein presented ROMs.

The Correct Treatment of Harmonic Pressure-Flow Behavior in Microchannels

2000 ◽  
Author(s):  
Christopher J. Morris ◽  
Fred K. Forster
Keyword(s):  
Stokes Equations ◽  
Flow Behavior ◽  
Corner Frequency ◽  
Navier Stokes ◽  
Special Treatment ◽  
Correct Treatment ◽  
Pressure Flow ◽  
Navier Stokes Equations ◽  
Pump Output ◽  
Low Order

Abstract Advances in microfluidic technology give great potential for integrated systems with many components. Because continuum modeling is difficult even for simple systems, low-order modeling is essential to their design. Unsteady flow in microfluidic components such as channels and valves may require special treatment to achieve accurate low-order models of entire systems. One example is the fixed-valve micropump, in which the interaction between mechanical, electrical, and fluidic components produces maximum pump output at a resonant frequency near the corner frequency of the valves. In this frequency range the commonly-used value for fluid impedance was found to be in error by 30%, causing errors in predictions of pump membrane motion by as much as 300%. Proper treatment of the pressure-flow relations based on the exact solution of the Navier-Stokes equations for oscillatory flow corrected these deficiencies.

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