Session L15: CFD: Algorithms II

Enhancing Data Assimilation through Omnidirectional Integration for Pressure Reconstruction in Noisy 2D Isotropic Turbulent Flow Observations

Numerical data assimilation plays a vital role in achieving optimal forecasts of dynamical processes by integrating theoretical model-based predictions with experimental observations. In our study, we focus on fluid mechanics and employ the forward Navier-Stokes solver as the theoretical model. The goal is to minimize the discrepancy between the computed velocity field and the measured velocity values using an adjoint-based data assimilation approach. To quantify this discrepancy, we utilize a cost function, which is minimized through the adjoint method, commonly employed in four-dimensional variational (4DVar) data assimilation. To validate our approach, we test the algorithm using 2D isotropic turbulent flow data with added noise to mimic real experimental conditions. The framework (Abassi et al. 2022) serves as a basis for applying data assimilation to noisy data, allowing us to reconstruct the velocity and pressure field under different pressure boundary conditions. Choosing an appropriate pressure boundary condition is a critical aspect of the process, as it ensures a divergence-free flow throughout. However, traditional Poisson solvers do not yield satisfactory pressure reconstructions when dealing with noisy data. To address this challenge, we implement the parallel ray omnidirectional integration technique developed by Liu et al. in 2016 to compute the pressure at the boundary at each time step. This approach allows us to impose Dirichlet boundary conditions for the pressure, thereby improving the accuracy of pressure computations. The validation of our algorithm using 2D turbulent noisy data will pave the way for extending the framework to handle 3D data. Ultimately, our objective is to establish a robust data assimilation method that can augment Time Resolved Tomo PIV measurements, enabling more accurate forecasting and analysis of complex fluid dynamics phenomena.   

An arbitrarily high-order, non-dissipative, and kinetic-energy conserving HDG numerical method to simulate incompressible flows in complex geometries and its application to PTV data assimilation

Non-dissipativity and kinetic energy conservation are two essential features of the base numerical method for a reliable and stable large-eddy simulation. Such numerical methods for incompressible flow simulation in complex geometries have been, at maximum, second-order accurate, until now. In this talk, we will, for the first time, present such a numerical method that is arbitrarily high-order accurate. This method is built using the Hybridized Discontinuous Galerkin (HDG) framework. It uses a simplicial mesh and is hence applicable to a real-world complex geometry. The details of the method and numerical experiments demonstrating its unique features will be presented and its broader implications will be discussed. We will also present a data assimilation application of the high-order method to reconstruct volumetric quantities from Particle Tracking Velocimetry (PTV) measurements. Using this assimilation strategy, a high-order approximation to the velocity gradient tensor and pressure can be obtained from the PTV measurements. The accuracy of the reconstructed quantities will be investigated for a few different flows.   

A Combined CFD and Machine Learning Technique for Efficient Prediction of Flow Behavior in Venturi Nozzle

Cavitation is the physical phenomenon when liquid evaporates into gas when the pressure drops below the saturation pressure.  This is observed in a wide variety of engineering processes and it can be used for flow control purposes.  In this research, cavitation of a flow stream in the venturi nozzle is simulated using a CFD software and the results are analyzed.  Cavitation may occur in the downstream of the throat region.  Simulations were conducted to identify the important parameters that affect the flow behavior. 

 

Machine learning is an efficient method that reduces the computational cost in many engineering problems. It has been shown that machine learning models can be used to speed up or even replace a part of CFD simulations. The primary goal is to maximize the steam quantity at the nozzle outlet for a given flow conditions.  The main difficulty is that a large number of simulations is needed due to the highly non-linear nature of the problem.  To reduce the computational cost, several artificial neural network models will be used to predict the cavitation efficiency based on the boundary conditions.  The results are then verified against the simulated flow field.  The ANN model is then combined with a genetic algorithm to find the optimized geometry.    

Stable nodal projection method on octree grids - Part I: Analytical Results

We introduce a novel projection method to solve the incompressible Navier-Stokes equations with arbitrarily shaped boundaries. Our method employs an adaptive mesh refinement strategy using non-graded quad/octree grids. The viscosity and projection steps are discretized using supra-convergent finite difference approximations with sharp boundary treatment. The novelty of our method comes from the fact that we collocate the velocity and pressure terms in the projection step. By collocating the projection step, we reduce the complexity in the underlying data structures used and streamlines the code development. In Part I of this two part presentation, we focus on the stability analysis of our method through a combination of proofs and numerical studies. We show that our projection operator yields second order accuracy and is stable in the presence of a variety of boundary and interface conditions.   

Stable nodal projection method on octree grids - Part II: Computational Results

This is the second part of a two part presentation reviewing our novel collocated projection method for solving the incompressible Navier-Stokes equations with arbitrarily shaped boundaries. Our method utilizes entirely collocated variables (pressure and velocity), which greatly simplifies the complexity of the underlying data structures and streamlines code development. In this second part, we will focus on verification and validation of our numerical scheme by presenting the results of our solver for a variety of canonical two- and three-dimensional fluid flows. We further demonstrate that our collocated method is a competitive tool for studying complex fluid flows by presenting results from real-world science and engineering problems.   

Sharp, stable, and collocated numerical simulation of incompressible, multi-phase fluid flows

Motivated by atmospheric and oceanic applications, we present a novel collocated projection method for simulating incompressible multi-phase fluid flows in two and three spatial dimensions. This work is an extension of our single-phase, collocated projection method that employs an adaptive mesh refinement strategy using non-graded quad/octrees and supra-convergent finite difference approximations for the viscosity and projection operators. The moving interface is captured using a coupled level set-reference map method, which provides a sharp representation of the interface position. In this presentation, we will focus on the verification and validation of our solver using a variety of two and three dimensional multi-phase flows.   

A Localized Artificial Diffusivity Method for High Density Ratio Multimaterial Flows

High-order compact finite difference schemes’ spectral-like accuracy is well suited for capturing turbulent mixing, but additional stabilization is needed to capture jumps in material properties in multimaterial flows. Localized artificial diffusivity (LAD) is a type of diffuse interface method where artificial fluid properties are added at material interfaces and shocks for stabilization. LAD has shown success with high-order compact finite difference schemes to maintain accuracy in multimaterial flows, but traditional methods struggle for very large density ratios because of the offset between gradients in mass fraction and density. We present a novel LAD method that uses both the volume and mass fractions to compute species diffusivities. This ensures stability and accuracy for simulations with very large density ratios. The strength of the proposed scheme is validated on a suite of test problems.   

Adaptive Interface Capturing Approach for Multicomponent Flows

Compressible flows may exhibit two different types of discontinuities: shocks and contact discontinuities (material interfaces). These discontinuities are often treated together in numerical simulations by methods such as weighted essentially non-oscillatory (WENO) or Monotonicity preserving (MP) schemes. Capturing these various discontinuities in the context of numerical simulations is challenging since the properties of these discontinuities differ, and their numerical treatment is key to their accurate representation. In this presentation, we propose a novel approach whereby the material interfaces (contact discontinuities) are reconstructed by using the THINC scheme via a novel contact discontinuity detector. The acoustic waves are captured by the WENO or MP schemes. The proposed approach significantly reduces the numerical diffusion across the material interfaces. Several benchmark test cases for the compressible multicomponent flows are presented, indicating the advantages of the proposed approach.   

Reproducing utility of positive definiteness for Stokes flow with AMR

Many important physical problems require numerical methods that use a range of solution resolution scales, usually realized through the use of adaptive mesh refinement (AMR). The introduction of AMR into a solver can result in lower accuracy or a loss of symmetric positive definiteness (SPD). This is especially relevant in fluctuating hydrodynamics, as the structure of the Laplace operator is exploited in satisfying the fluctuation-dissipation-balance by setting the noise porportional to the square root of the viscous operator. This work describes two second order accurate methods for constructing a Laplace operator using the marker and cell (MAC) method for Stokes flow with adaptive mesh refinement that reproduce the utility of the SPD operator for uniform meshes. In one approach, Galerkin differencing is used to force the resulting system to by SPD, yielding a result with a nontrivial square root. And in the other, the resulting operator is not SPD, but is constructed in a way that defines a matrix that can be used to satisfy the fluctuation-dissipation-balance.   

Adaptive Conservative Time Integration (ACTI) for Compressible Flow

For time dependent simulations the time steps have to be chosen appropriately small to guarantee stability and to minimize time integration errors. Opposed to classical time stepping schemes, which employ the same time step everywhere, time adaptive solution algorithms allow for large time steps wherever possible, while resorting to high temporal resolution only where needed. Regarding computational  efficiency such schemes are very attractive, but it is not straight forward to also achieve stability, high accuracy, and conservation.

Recently, an adaptive conservative time integration (ACTI) scheme for finite volume methods was devised. Besides being adaptive in time, ACTI is conservative, all operations are local, and it is of high order in space and time. The performance of ACTI has been demonstrated for various hyperbolic conservation laws.

Here, an ACTI scheme for the compressible Navier-Stokes-Fourier system is introduced, which is particularly attractive for unsteady flows involving boundary layers. Unlike in inviscid flow simulations, the time step size not only is restricted by a CFL criterion, but also by the viscous terms. Further, in order to preserve 2nd order in space and time, a novel discretization scheme for the viscous fluxes is devised.

Results of various unsteady flows, including shock boundary layer interaction, demonstrate efficiency and accuracy of the new ACTI scheme. Further, given that all operations are local, it is an ideal candidate for massive parallel computations.