Conservation of Energy
The first law of thermodynamics, known as the principle of conservation of energy, states that the change in energy stored within a Lagrangian fluid parcel is equal to the rate at which energy is added to it from external sources, such as heat or mechanical work. For a student of elementary thermodynamics, there are two aspects of this principle that may be new. Firstly, the motion of the fluid parcel plays a significant role in the overall energy balance, thus the bulk kinetic energy of the parcel must be considered as one of the forms of stored energy. Secondly, it is important to recognize that not only pressure, but also viscous forces, can contribute to the addition of mechanical work to the energy of the system.
Heat transfer within a parcel can occur through two main mechanisms: electromagnetic radiation and molecular conduction. Electromagnetic radiation involves the absorption or emission of energy within the parcel, while molecular conduction refers to the transfer of heat across the parcel’s boundary. It is important to note that radiation effects are volume-proportional, known as a “body” effect, while conduction effects are surface-related. In aerodynamics, surface effects are typically more significant than body effects. The mechanical work performed on the parcel is carried out by the same forces involved in momentum conservation. In aerodynamics, external body forces are often negligible, and the focus is on forces exerted by neighboring parcels.
However, the impact of internal fluid stresses on energy conservation is more complex than on momentum conservation. While momentum conservation only considers the net force on the parcel, energy conservation also takes into account the net force acting over the distance traveled by the parcel’s center of mass, contributing to changes in its bulk kinetic energy.
However, there are additional factors to consider. In the event that the parcel undergoes deformation, whether it be volumetric or in shear, certain portions of the parcel’s boundary will shift in relation to its center of mass. This movement can result in substantial work being performed on the parcel. Furthermore, the pressure exerted on the parcel, through compression or expansion, can lead to heating or cooling effects. Additionally, the parcel experiences heat due to viscous stresses, a phenomenon known as viscous dissipation.
Turbulence presents intriguing considerations in relation to the conservation of energy. Turbulent flows are typically analyzed through theoretical models that focus on time averages, effectively smoothing out the unsteady turbulence motions. Within the time-averaged flowfield, the kinetic energy associated with turbulence becomes a significant form of energy that requires consideration. Nevertheless, in numerous flow scenarios, the generation and dissipation of turbulence kinetic energy (TKE) are approximately balanced locally, allowing for the neglect of TKE:
- Conservation of Energy: The first law of thermodynamics, which states that energy cannot be created or destroyed but only transformed from one form to another, is discussed in the context of fluid parcels. This includes considering changes in kinetic energy, potential energy, and internal energy within the parcel.
- Forms of Energy: The various forms of energy within a fluid parcel are detailed, including bulk kinetic energy, internal energy (affected by pressure and viscous forces), and heat transfer mechanisms such as electromagnetic radiation and molecular conduction.
- Mechanical Work: The role of mechanical work on a fluid parcel is explained, noting that it can be performed by both external forces (such as pressure) and internal fluid stresses, which may arise from neighboring parcels or deformation of the parcel itself.
- Deformation and Compression Effects: The impact of parcel deformation and changes in pressure on the energy balance are discussed, highlighting how these factors can lead to substantial work being performed on the parcel and influence its thermal state.
- Viscous Dissipation: The phenomenon of viscous dissipation, where mechanical energy is converted into heat due to fluid viscosity, is mentioned as a factor contributing to energy conservation considerations.
- Turbulence: Turbulent flows introduce additional complexities to energy conservation, particularly concerning the generation and dissipation of turbulence kinetic energy (TKE). While TKE is significant in turbulent flows, it may be neglected in certain scenarios where generation and dissipation rates are approximately balanced.
Overall, we try to supply an explanation of how the principles of thermodynamics apply to fluid parcels, particularly in aerodynamic contexts, and highlight the various factors that must be considered when analyzing energy conservation in such systems.
Relationship Between Various Physical Quantities and Boundary Conditions
We have just examined the fundamental conservation laws in the Lagrangian reference frame. Whether these laws are applied in the Lagrangian or Eulerian frame, they yield five equations and we are faced with eight unknowns. These unknowns consist of three space coordinates (in Lagrangian) or velocity components (in Eulerian), as well as five local material and thermodynamic properties: pressure, density, temperature, and the coefficients of molecular viscosity and thermal conductivity. To fully define the system, three additional constitutive relations are required. In aerodynamics, these relations typically include the ideal gas equation of state, which correlates pressure, density, and temperature; the Sutherland law, which links viscosity to temperature alone; and Prandtl’s relation for thermal conductivity.
The comprehensive Navier-Stokes (NS) system encompasses all the internal fluid physics that are necessary for our analysis. When it comes to the boundaries of our flow domain, the specific boundary conditions we must apply are contingent upon the type of boundary in question. In the case of flow boundaries, no supplementary physics needs to be invoked as the NS equations themselves dictate which boundary conditions are permissible or obligatory, depending on the flow scenario. However, when dealing with boundaries that interface with another material, commonly referred to as “walls,” additional physical considerations are required to precisely define the boundary conditions.
Boundary conditions specify the behavior of a system at its boundaries or interfaces. They are essential for solving differential equations that govern physical phenomena and are often used to model interactions between different materials or regions within a system. Some common types of boundary conditions include:
- Dirichlet Boundary Conditions: These specify the values of the dependent variables (e.g., temperature, velocity) at the boundaries of a domain.
- Neumann Boundary Conditions: These specify the gradients or fluxes of dependent variables at the boundaries, rather than their absolute values.
- Robin Boundary Conditions: Also known as mixed boundary conditions, these specify a combination of prescribed values and gradients at the boundaries.
- Periodic Boundary Conditions: These are used to model systems where the boundaries wrap around to form a periodic domain, such as in simulations of periodic structures or fluid flows in channels.
- Interface Conditions: When modeling the interaction between different materials or phases, interface conditions specify how quantities such as stress, displacement, or heat flux are related across the interface.
Constitutive relations describe the relationship between various physical quantities within a material or fluid. These relations typically depend on the material properties and may vary based on the conditions the material or fluid is subjected to. Some common constitutive relations include:
- Stress-Strain Relations: In solid mechanics, these relations describe how stress (force per unit area) is related to strain (deformation) within a material. Different materials exhibit different stress-strain behaviors, such as elastic, plastic, or viscoelastic responses.
- Fluid Stress-Strain Relations: For fluids, constitutive relations often describe how stress (shear stress, normal stress) is related to strain rate (rate of deformation). These relations may involve parameters such as viscosity for Newtonian fluids, or more complex models for non-Newtonian fluids.
- Thermodynamic Relations: In thermodynamics, constitutive relations describe how properties such as pressure, temperature, and density are related under different conditions. Equations of state, such as the ideal gas law or more complex formulations for real gases, are examples of thermodynamic constitutive relations.
- Electromagnetic Relations: In materials science and electromagnetism, constitutive relations describe how electrical conductivity, permittivity, and permeability relate to the electric and magnetic fields.
The Mathematical Characteristics of Equations
The system of equations presented comprises five field PDEs and three algebraic constitutive relations, totaling eight unknowns. These equations exhibit a mixed hyperbolic/elliptic nature in space, necessitating boundary conditions for a solution across the entire domain. While numerical solutions can progress forward in time, spatial progression is not feasible. Due to the nonlinearity of the equations, solutions cannot be generally achieved through superposition. Even a steady-flow solution requires a method beyond a single matrix inversion, such as time-marching or iterative processes. These complexities will be further explored in the context of CFD methods. Solutions to the NS equations may not always be unique, particularly when multiple steady-flow solutions correspond to the same body geometry. While turbulence-free solutions exist theoretically, they are often dynamically unstable at high Reynolds numbers and are rarely observed in nature.
Due to the aforementioned challenges, finding analytic solutions to the NS equations is only possible for a limited number of simple cases with reduced dimensions and constant fluid properties. Even in these cases, the solutions are only applicable under certain limiting conditions where the effects of inertia can be disregarded. For instance, there exist one-dimensional solutions for steady, fully developed flow in planar two-dimensional or circular-cross-section ducts or pipes, as well as two-dimensional solutions for flow around a circular cylinder or sphere at low Reynolds numbers. In situations involving high Reynolds numbers, approximate solutions to the two-dimensional NS equations can be obtained using boundary-layer theory, which only requires solving a one-dimensional ordinary differential equation (ODE). However, for more general flows, numerical solutions are the only viable option unless simplifying assumptions can be made.
The Eulerian Frame
In the Eulerian framework, the observation is focused on the behavior of fluid as it moves past specific points within a designated spatial framework. Instead of monitoring the changes experienced by fixed portions of fluid, as done in the Lagrangian framework, the focus shifts to observing the behavior within infinitesimal volume elements integrated into the spatial coordinate system. These volume elements in the Eulerian framework have a continuous flow of fluid passing through them and over their boundaries. This flow mirrors the motion observed in the Lagrangian framework. The change in perspective necessitates a different treatment of the convection process when applying conservation laws. In the Lagrangian approach, convection is implicitly considered through the definition of fixed fluid parcels, with no terms in the conservation equations accounting for convection across parcel boundaries due to the absence of such movement by definition. In contrast, in the Eulerian approach, where fluid flux across volume element boundaries is common, the convection process must be explicitly included in the equations as additional terms.
Mathematically, the additional terms arise when we replace the time derivatives in the
Lagrangian equations with their Eulerian equivalents, using Equation:
The Eulerian equations incorporate terms that account for convection effects, which stem from the V · ∇ term on the right-hand side. To illustrate this concept, let’s examine the x component of the momentum of a Lagrangian parcel with a volume of dV, denoted as ρ udV. By applying the aforementioned equation to this specific quantity, we can further understand its implications:
The term appearing second on the right-hand side of the equation symbolizes the momentum convection in the Eulerian x-momentum equation in its most basic form. An alternative representation frequently encountered in scholarly works involves moving the density outside the derivative, resulting in a clearer connection to the Lagrangian acceleration Du/Dt.
To obtain this alternative form, it is necessary to invoke the principle of conservation of mass, which, in its Lagrangian formulation, asserts that the mass of a Lagrangian parcel remains constant over time. If we again make use of the substantial derivative definition, we arrive at:
Convection Process in Detail
Let us now delve deeper into the convection process. It is a well-established fact that convection is reciprocal across a shared boundary between two parcels. This concept can be likened to Newton’s third law in mechanics, which dictates that the forces exerted by two bodies in contact must be equal and opposite. This equilibrium is necessary because there is no mechanism at the interface to sustain an imbalanced force. When considering two fluid parcels sharing a boundary, there is no mechanism to alter the flux of a conserved quantity, thus the flux leaving one parcel must be equivalent to the flux entering the other. In our general Navier-Stokes formulation, we do not need to explicitly enforce this reciprocity as it is inherently maintained by the continuity of all flow variables. However, in specialized theories involving discontinuities, such as shock modeling in inviscid solutions, additional equations are required to ensure the conservation relationships are upheld across the discontinuity.
The convection terms in the conservation equations have a clear physical interpretation. When there is an imbalance between the rate of convection into a volume element and the rate of convection out, convection acts as a source of the conserved quantity and needs to be considered in the conservation law. These terms indicate the overall rate at which a conserved quantity is being transported into or out of a volume element. In the case of mass conservation, this net convection is the sole factor contributing to the change in total mass within the element over time, especially in steady flow conditions where the mass flux into and out of the element must be equal. This principle extends beyond local parcels to larger volumes, such as a steady-flow streamtube, where the mass flux through any surface intersecting the tube must remain constant. Net convection plays a crucial role in maintaining balance in the conservation of momentum and energy, but it is essential to also consider the effects of external forces acting on the fluid (momentum and energy) and heat conduction.
The momentum and energy balances in our formulation can be influenced by external sources, such as gravitational or electromagnetic forces acting on the fluid, as well as heat transfer through absorption and emission of radiation. These contributions can be easily accounted for. Additionally, exchanges of force or energy between fluid parcels that are not in direct contact with each other, known as internal nonlocal effects, can also affect the balances. However, in aerodynamics, these external effects and internal nonlocal effects are typically considered to be negligible. Therefore, the only effects that need to be represented in our equations are those transmitted through direct parcel-to-parcel contact. This includes the interparcel forces represented by the apparent internal stresses and the heat fluxes due to conduction, which are exchanged between adjacent fluid parcels. It is important to note that these quantities are not physically tied to the fluid material and are not convected with it. They remain unaffected by changes in the velocity of our reference frame and appear the same in both the Eulerian and Lagrangian frames.
In the typical scenario observed in aerodynamics, the primary transmission of forces within the fluid occurs between neighboring fluid parcels. Similarly, convection effects in an Eulerian framework also operate solely between adjacent Eulerian parcels. Consequently, in our conventional aerodynamic flows, there is no mechanism for any form of “force at a distance” exchange, thereby ruling out remote “induction” or similar effects. Although the Biot-Savart law may suggest a remote induction effect, it is erroneous to perceive the velocity at one point as being “induced” or “caused” by the vorticity at another point. This serves as just one illustration of the challenges associated with attributing cause and effect in the realm of fluid mechanics.
Motivation The role of the aeronautical engineer has undergone significant changes in recent years and will continue to evolve. The advent of advanced computational tools has revolutionized the design processes for various types of flight vehicles, enabling the achievement of unprecedented levels of design technology. As performance targets become more demanding, the engineer’s role in…
In the realm of computational fluid dynamics (CFD) analysis, Reynolds-averaged Navier Stokes (RANS) methods have traditionally been the go-to choice for studying turbulent flows in practical engineering scenarios. RANS-based techniques fall on one end of the spectrum of turbulent calculation methods, wherein turbulent fluid dynamic effects are substituted with a turbulence model. On the opposite…
Motivation The notion of partitioning the motion surrounding an object into an external inviscid motion and an internal viscous motion, each governed by simplified equations, was initially introduced by Prandtl in 1904. Since then, this concept has undergone extensive development and has been widely employed as the primary approach for making quantitative predictions of viscous…
Motivation Machine learning presents many chances to further the subject of computational fluid dynamics and is quickly emerging as a fundamental tool for scientific computing. We highlight some of the areas with the greatest potential influence in this perspective, such as enhancing turbulence closure modeling, speeding up direct numerical simulations, and to create improved lower-order models.…
“Every mathematician believes that he is ahead of the others. The reason none state this belief in public is because they are intelligent people” – Andrey Kolmogorov PDF Download: The k-ω Family of Turbulence Models (by Tomer Avraham) Abstract Three versions of the k-omega two-equation turbulence model will be presented. The first is the original…