Introduction
The objective of this post is not to mathematically derive the equations, but rather to provide concise, intuitive explanations of the meanings of the different terms in the equations. Additionally, we aim to analyze some of the overarching concepts that can be deduced from the equations regarding the characteristics of flows.
The fundamental equations we utilize are representations of principles of conservation for mass, momentum, and energy. These principles are most effectively articulated and comprehended within the Lagrangian reference frame, where we depict the movement in relation to the paths of unchanging fluid parcels as they progress over time.
Nevertheless, the Eulerian reference frame, which involves describing the flow as it passes by points in a spatial reference frame independent of the fluid, is ultimately the favored option for both conceptual and quantitative reasons. The method I will adopt in this context is to provide a brief explanation of the significance of conservation laws in the Lagrangian frame before transitioning to a discussion on how they are articulated in the Eulerian frame.
In both the Lagrangian and Eulerian perspectives, we will examine the behavior of small fluid volumes, albeit with distinct definitions in each case. The derivation of our conservation laws as partial differential equations (PDEs) entails a formal process of approaching infinitesimally small dimensions for our fluid parcels. While we won’t delve into the specifics of this procedure in this discourse, it is important for the reader to bear in mind that fluid parcels in either reference frame should be conceptualized as arbitrarily minute.
Lamb (1932) provides a definition for a fixed Lagrangian parcel of fluid, stating that it consists solely of the same fluid particles throughout time. In order to maintain this consistency, the bounding surface of the parcel must move with the fluid in a manner that prevents any fluid particles from passing through it. However, it is important to acknowledge that this concept is an idealization that only holds true in our conceptual continuum world. In reality, molecules will inevitably diffuse across such a boundary in both directions, and the best we can do is ensure that the boundary follows the average motion of the fluid, resulting in no net flux of material across it. Regardless of the perspective taken, the parcel will always contain the same amount of material and exhibit no net flux of material across its bounding surface. This approach of disregarding mass diffusion serves well in the case of single-species fluids or multispecies fluids where relative species concentrations remain constant. However, if the relative species concentrations vary significantly, defining a Lagrangian fluid parcel becomes problematic. For the time being, we will overlook this minor limitation on the Lagrangian description and proceed with our discussion.
As previously mentioned, conservation laws are established for mass, momentum, and energy. The reason for this is that these quantities are fundamental in physics and thermodynamics, necessitating their conservation. Unlike pressure or viscous stresses, which do not have conservation laws due to their nature. Mass, momentum, and energy are intricately linked to the fluid material and are convected along with it. These convected quantities are associated with Lagrangian fluid parcels, meaning that any change in the quantity within a parcel can only occur due to physical processes within the parcel or at its boundaries. Conservation laws serve to quantify these changes, providing a framework for understanding the conservation of mass, momentum, and energy within the Lagrangian perspective.
Continuity of the Flow – our conservation of mass
According to our precise definition of a fluid parcel in the Lagrangian description, the conservation of mass within the parcel is inherently guaranteed. Nevertheless, the equation responsible for explicitly ensuring mass conservation must fulfill additional roles. The continuity equation establishes a connection between the fluid density at various points and the volume it occupies, thereby meeting two essential criteria:
- The conservation of mass is a fundamental principle within every Lagrangian parcel, in accordance with the parcel’s defined characteristics.
- There are no empty spaces between Lagrangian parcels, and neighboring parcels do not intersect. It is essential to view the entire fluid volume as being completely filled with Lagrangian parcels that maintain mass conservation.
The continuity equation in the Lagrangian description can be easily understood in a physical sense: When the volume of a fluid parcel changes, the density of the fluid must also change in order to maintain the constant mass of the parcel.
Although the basis for the continuity equation is physical (requirements 1 and 2 above), the requirements it imposes on the flow are not as direct in a cause-and-effect sense as those imposed by other equations. For example, in the conservation of momentum. forces directly cause accelerations; and in the conservation of energy.
Forces on Fluid Parcels and Conservation of Momentum
In the Lagrangian reference frame, the preservation of momentum is explicitly enforced through Newton’s second law, F = ma. Our Lagrangian fluid parcel possesses a constant mass, and its acceleration is determined by the cumulative effect of the forces exerted upon it.
External body forces such as gravitational and electromagnetic forces may act on a parcel, but in aerodynamics, these forces are typically considered insignificant. The focus is primarily on the forces exerted on the surface of the parcel by neighboring parcels. According to Newton’s third law, these surface forces must be equal and opposite across the shared boundary. These forces are referred to as apparent internal fluid “stresses.” It is recognized that these stresses can be viewed as distributed stresses in the idealized continuum world, while in reality, they are merely apparent stresses resulting from momentum transfer due to molecular motion. Nevertheless, moving forward, we will treat them as actual stresses.
In previous posts of the “All About CFD” -“Fundamentals of Aerodynamics” During our discussion, we explored the concept of representing these stresses as a tensor. This approach proves to be advantageous when it comes to mathematical manipulation. However, for the purpose of gaining a physical understanding, it is more intuitive to think in terms of force vectors. By contracting the stress tensor with the unit vector that is normal to the hypothetical boundary between parcels, we obtain a vector that represents the force per unit area acting across the boundary. Furthermore, we can decompose this vector into two components: one that is perpendicular to the boundary and another that is parallel to it. In the context of the NS equations, the perpendicular component is assumed to be the local hydrostatic pressure, often referred to as the static pressure. On the other hand, the parallel component is known as the shear stress, which arises solely from the effects of viscosity.
Understanding pressure intuitively presents challenges due to its inherent nature in continuum fluid mechanics. Pressure can be visualized as the normal stress exerted on hypothetical boundaries encompassing a specific point in space.
Despite being a scalar quantity, it exerts force uniformly in all directions at a given point. Initially, comprehending this concept can be difficult. Certain commentators, such as Anderson and Eberhardt (2001), have mistakenly defined static pressure as “the pressure measured parallel to the flow.”
However, this description contradicts the true essence of pressure, which remains unaffected by flow direction and uniformly acts in all directions. A more intuitive approach to comprehending pressure involves considering its impact on a small yet finite fluid parcel. Within a field of constant pressure, this parcel encounters equal inward forces from the surrounding fluid in all directions.
To induce any acceleration in the parcel, the total stresses acting on all faces of the parcel must result in a non-zero vector sum, indicating an unbalanced force. Stresses on opposite sides of the parcel act in opposite directions and cancel each other out if their magnitudes are equal. In a field of constant pressure, normal stresses cancel each other out, resulting in no unbalanced force. To have an unbalanced force, the magnitudes of stresses on opposite sides of the parcel must differ, requiring nonuniform pressure or viscous stress.
onsequently, the unbalanced force is not contingent on the stress per se but on the stress gradient, which is symbolized by ∇p in the context of pressure. This typically involves non-uniform fluid flow. Given that forces are influenced by the motion of the fluid parcel and its neighboring parcels, the cause-and-effect relationship between stresses and velocities becomes circular, thereby adding intricacy to our analysis. This subject matter will be further explored in an upcoming edition of our “All About CFD” series on the “Fundamentals of Aerodynamics.”
The acceleration of a parcel is governed by the momentum equation, therefore, to determine the parcel’s velocity, one must integrate the equation. Subsequent sections of the series will demonstrate how integrating the momentum equation for the steady flow of an inviscid fluid results in Bernoulli’s equation, a highly valuable flow relation.