Implementing A General Fluid Model In Oomph-Lib A Comprehensive Guide

Introduction

This article delves into the implementation of a general fluid model within the Oomph-Lib framework, drawing parallels with the existing PVD (Prager-Drucker Viscoplastic) solid mechanics framework. The primary goal is to develop a versatile fluid model that can handle both compressible and incompressible fluid behaviors, offering users flexibility in their simulations. We will explore the core concepts, constitutive laws, element types, and considerations for multi-physics applications, aiming to provide a comprehensive understanding of the proposed implementation strategy.

Key Concepts and Objectives

At the heart of this endeavor lies the desire to create a unified framework for fluid dynamics within Oomph-Lib. This framework should seamlessly integrate with existing solid mechanics capabilities, leveraging shared infrastructure and methodologies. The main objective is to develop a robust and flexible fluid model that allows users to simulate a wide range of fluid behaviors, from highly incompressible liquids to compressible gases. To achieve this, we will focus on several key aspects:

• Constitutive Laws: Implementing constitutive laws that relate stress to strain rate and spin rate, potentially based on the rate of strain energy in terms of strain invariants. This allows for capturing complex fluid behaviors and material properties.
• Compressibility: Providing options for both compressible and incompressible elements, enabling users to accurately model different fluid types and flow regimes. The choice between compressible and incompressible formulations is crucial for the accuracy and efficiency of the simulations.
• Element Types: Exploring different element types, such as fully continuous, Crouzeix-Raviart, and Taylor-Hood elements, to determine the most suitable approach for representing pressure or density fields. The element type plays a significant role in the stability and accuracy of the numerical solution.
• Multi-physics Integration: Ensuring seamless integration with other physical phenomena, such as solid mechanics and heat transfer, to enable complex multi-physics simulations. This requires careful consideration of how different physical quantities interact and how they are represented within the framework.
• Body Forces: Implementing a consistent and scalable approach for handling body forces, such as gravity, while minimizing user effort and ensuring compatibility with multi-physics scenarios. The treatment of body forces is essential for simulating realistic fluid flows in various applications.

By addressing these key aspects, we aim to create a general fluid model that is both powerful and user-friendly, expanding the capabilities of Oomph-Lib and enabling a wider range of simulations.

Constitutive Laws: Defining Fluid Behavior

In the context of fluid mechanics, constitutive laws play a pivotal role in defining the behavior of the fluid under various conditions. These laws establish the relationship between stress and strain, effectively dictating how the fluid will respond to applied forces. Our approach involves passing the strain rate tensor and spin rate tensor to a constitutive law, allowing for a flexible and accurate representation of fluid behavior. The constitutive law can then be formulated based on the rate of strain energy, potentially expressed in terms of strain invariants. This approach provides a robust framework for capturing a wide range of fluid behaviors, from simple Newtonian fluids to complex non-Newtonian fluids.

Rate of Strain Energy

The concept of the rate of strain energy is central to our formulation. By basing the constitutive law on this principle, we can ensure that the fluid model adheres to fundamental thermodynamic principles. The rate of strain energy represents the rate at which energy is stored within the fluid due to deformation. This energy is related to the internal stresses and the rate at which the fluid is deforming. By expressing the rate of strain energy in terms of strain invariants, we can obtain a concise and physically meaningful representation of the fluid's behavior. Strain invariants are scalar quantities that characterize the state of strain in a material, independent of the coordinate system. Using strain invariants in the constitutive law allows us to capture the essential features of the fluid's deformation without being tied to a specific coordinate frame.

Compressible vs. Incompressible Constitutive Laws

A crucial distinction in fluid mechanics lies between compressible and incompressible fluids. Compressible fluids, such as gases, exhibit significant density changes under pressure variations, while incompressible fluids, such as liquids, maintain a nearly constant density. Our implementation will accommodate both types of fluids by allowing users to specify whether the constitutive law should enforce incompressibility or not. This flexibility is essential for accurately modeling a wide range of fluid phenomena. For instance, simulations involving high-speed gas flows or shock waves require a compressible formulation, while simulations of liquid flows in pipes or channels can often be simplified using an incompressible formulation.

The choice between compressible and incompressible constitutive laws has a direct impact on the governing equations and the numerical methods used to solve them. Incompressible formulations typically involve a constraint on the velocity field, ensuring that the fluid density remains constant. This constraint is often enforced using a pressure variable as a Lagrange multiplier. Compressible formulations, on the other hand, directly solve for the density field, which can vary significantly throughout the fluid domain. The selection of the appropriate constitutive law is therefore a critical step in setting up a fluid simulation.

Implications for Element Design

The choice of constitutive law also has implications for the design of the finite elements used in the simulation. In particular, the incompressibility constraint can lead to numerical instabilities if not handled carefully. Techniques such as mixed finite element methods, which use different interpolation orders for velocity and pressure, are often employed to address this issue. Furthermore, the element formulation must be consistent with the chosen constitutive law. For example, elements designed for incompressible fluids typically include a pressure degree of freedom, while elements designed for compressible fluids may not. We will explore these considerations in more detail when discussing element types.

Element Types and Compressibility Flags

Oomph-Lib's existing solid mechanics elements provide a useful framework for understanding how to handle compressibility in fluid elements. Solid elements are classified as either with pressure or without pressure. Elements with pressure are designed to solve either incompressible or nearly incompressible PVD equations, while elements without pressure solve for the compressible PVD equations. This distinction is crucial for accurately modeling different material behaviors and ensuring numerical stability. We aim to extend this concept to fluid elements, providing users with similar flexibility in choosing the appropriate element type for their simulations. The interplay between the element incompressibility flag and the constitutive law incompressibility flag will determine the specific equations being solved.

Element Incompressibility Flag

The element incompressibility flag serves as a switch that dictates the behavior of the element with respect to compressibility. If the flag is set to