Streamlines, Streaklines, Timelines and Much More – Kinematics

introduction

In this post we explore the comprehensive and precise physical theory offered by the Navier-Stokes equations. These equations allow us to predict a wide range of phenomena in aerodynamics, encompassing even the flows of liquids like water. We begin by examining the fundamental representation of physics in these equations, the necessary assumptions made during their development, and the extent to which they remain valid. Subsequently, we delve into the intricate details of the equations and elucidate their significance.

The Continuum Assumption and Its Range of its Validity

In the NS formulation, the fluid is considered as a continuous substance, referred to as a continuum, with local physical characteristics that can be expressed through continuous functions in terms of space and time. These continuum properties are influenced by the properties of the individual molecules comprising the gas or liquid, as well as the underlying physics governing their movements and interactions. However, the continuum properties solely capture the overall effects of the underlying physics, rather than the specific details. As demonstrated in Post 1 – introductory Guide to Understanding the Foundations Concepts and Theoretical Frameworks and Post 2 – The Emergence of Aerodynamic Flows From The Molecular Level, this approach offers a representation that is not only sufficient but also remarkably precise across a diverse range of conditions.

The initial historical progression of the NS formulation took on a spontaneous method, presuming continuum behavior from the start and constructing a framework for viscosity effects through experimentation in basic flow scenarios. A significant portion of the effort put into this progression was focused on establishing the mathematical structure necessary to extend from straightforward flows to more complex ones.

The process of averaging provides us with precise definitions of fundamental continuum flow quantities, yet it does not lead us directly to the Navier-Stokes (NS) formulation. When we utilize the averaging process on the fundamental conservation laws for mass, momentum, and energy, we encounter two distinct types of terms that represent different sets of phenomena and necessitate different assumptions:

First, Terms that solely include the simple averages defining the continuum density, temperature, and velocity. No additional assumptions are necessary as these variables already form the basis of the NS formulation. These terms depict the local time rate of change of a conserved quantity or the convection of a conserved quantity by the local continuum velocity of the flow.

Second, Terms that incorporate averages of products of molecular velocities or products of a velocity component and the kinetic energy. Such terms signify the transport of a conserved quantity in relation to the local continuum motion of the flow. The transport of thermal energy corresponds to the heat flux resulting from molecular conduction. The transport of momentum mimics the effect of a continuum material experiencing internal stress, thereby giving rise to both the local continuum hydrostatic pressure and additional continuum “stresses” caused by viscous effects. The averaging process alone maintains these terms in a state that relies on statistical intricacies of molecular motions, necessitating further simplifying assumptions to transform them into expressions based on our fundamental continuum flow variables.

The NS equations involve terms that represent various transport phenomena, and these terms have straightforward functional dependencies on local continuum properties. The hydrostatic pressure is determined by an equilibrium thermodynamic relation, while the heat flux and viscous “stresses” are described by gradient-diffusion expressions, where the flux of a conserved quantity is proportional to its gradient. Fluids that exhibit this type of behavior for viscous stresses, as described in the NS equations, are commonly referred to as Newtonian fluids. However, reaching these simplified forms from the more general expressions obtained through the averaging process requires making certain simplifying assumptions about the physics involved. In the case of gases, it is necessary to assume that the fluid is in a state of local thermodynamic equilibrium throughout. This means that the probability distribution functions for molecular velocity, which appear in the complete transport expressions, must closely resemble their equilibrium forms. Achieving this requires that significant changes can only occur over length and time scales that are much larger than the mean-free path and time. When these conditions are met, meaning that local deviations from equilibrium are small, the terms related to transport can be accurately represented by the simple relationships used in the NS equations.

Conversation Laws

The fundamental relationships found in the NS equations are the essential principles of conservation for mass, momentum, and energy. In order to establish a comprehensive set of equations, it is necessary to include an equation of state that connects temperature, pressure, and density, as well as expressions defining the remaining gas properties.
In the realm of aerodynamics, it is often a reasonable approximation to assume the ideal gas law, in conjunction with a fixed ratio of specific heats (γ) and viscosity and thermal conductivity coefficients (μ and k) that are contingent solely on temperature. It may seem counterintuitive that the transport coefficients μ and k are considered to be independent of density at a constant temperature. However, there is a straightforward explanation for this phenomenon.
With an increase in density, one might expect the transport coefficients to rise due to the greater mass per unit volume that needs to be transported in terms of momentum and thermal energy. Nevertheless, as density increases, the molecular mean free path decreases, thereby impeding molecular transport. At the ideal gas approximation level, the impacts of heightened mass per unit volume and reduced mean free path cancel each other out.
Consequently, in practical terms, the efficiency of molecular transport is contingent solely on the average velocity of the molecules, or in other words, the temperature. In certain formulations of the equations, the local speed of sound (“a”) is a factor, which, in the case of an ideal gas, is also solely dependent on temperature.

The Importance of Boundary Conditions

The Navier-Stokes (NS) equations, like any other field equations, require boundary conditions (BCs) to be properly solved. When it comes to flow boundaries, where the flow simply enters or exits the domain, the NS equations themselves dictate the possible combinations of BCs that can be imposed and the combinations that are necessary to “determine” the solution in different ways. However, when dealing with boundaries that interface with other materials, such as gas-solid or gas-liquid interfaces, the NS equations alone do not fully define the situation. In such cases, additional physics must be introduced. Based on theoretical models and experimental evidence, it has been observed that the interaction between most liquid and solid surfaces encountered in engineering practice and air at ordinary conditions is such that the velocity and temperature of the air adjust almost perfectly to the velocity and temperature of the surface. Therefore, assuming no slip (no relative motion between the fluid and the solid) and no temperature jump at the “wall,” and imposing BCs accordingly, provides an extremely accurate approximation.

However, it is important to interpret the no-slip BC correctly. In some descriptions, the fluid is described as “sticking” or “adhering” to the surface. While this description is not entirely inappropriate, it can be misleading, especially when considering gases. The term “adheres” implies the presence of a bond that can withstand both tension and shear. However, gases cannot be subjected to tension and cannot form tension-resisting bonds with other substances. Nevertheless, the no-slip condition assumes that there is no sliding between the fluid and the solid, so in terms of shear, the fluid behaves as if it were adhering to the surface.

The no-slip condition is applicable to both liquids and gases. The explanation for this phenomenon is more straightforward when considering gases. Although some gas molecules may temporarily stick to a solid surface or react chemically with it and remain attached, the majority of molecules that collide with the surface bounce off. The no-slip condition is a result of these bouncing interactions. If we envision gas molecules as smooth spheres bouncing off a smooth surface in a specular manner without losing any tangential momentum, there would be no exchange of shear force between the surface and the gas. Consequently, the gas would easily slide along the surface, and the concept of a no-slip condition would not exist. However, at the molecular level, no actual surface behaves like a perfectly smooth surface. All real surfaces are composed of atoms that are similar in size to gas molecules, making even the smoothest surface rough on the scale of a gas molecule. Additionally, most real surfaces exhibit significant roughness on larger scales. As a result, gas molecules colliding with real surfaces bounce off in random directions, leading to a very small average tangential velocity of molecules near the surface. By applying kinetic theory, one can estimate the effective slip velocity, which demonstrates that in practical scenarios, it is nearly zero. This holds true even for surfaces that may feel smooth to the touch, as our intuition incorrectly assumes that air can freely slide over them.

Hence, our comprehensive physical model comprises the NS equations in conjunction with the no-slip and no-temperature-jump boundary conditions. The scope of this formulation is remarkably extensive, with only a limited number of practical “aerodynamics” applications where it does not hold true. Instances that deviate from this formulation include gas flows at extremely low densities, such as those encountered at very high altitudes, as well as the intricate internal structure of shock waves. Even in cases where ionization, dissociation, or chemical reactions occur within the flow, they are not typically considered exceptions, as these effects can be integrated into our continuum formulation by incorporating appropriate variables for species concentration, reaction rates, and equations of state. Fortunately, in the field of aerodynamics, we are spared the complexities associated with non-Newtonian liquids, which play a significant role in biological systems and various industrial processes.

The inability of our NS formulation to be applicable in exceptional circumstances may not solely be attributed to the extremely low densities at high altitudes or the small length scales in shock-wave problems that cause our averaging process to fail to converge. Although this situation can occur, it is not always the primary cause of the “failure.” To achieve convergence of a spatial average at a specific moment in time, it would be necessary to integrate over a sufficiently large volume that encompasses a significant number of molecules. Instantaneous spatial averages may not accurately capture the internal structure of a shock wave, for example. However, in many cases, flows are nearly steady, allowing us to define averages in small spatial volumes by averaging over an extended period. Most situations involving flight at extreme altitudes or detailed shock-wave physics can be addressed using this approach. In such instances, the failure of our continuum formulation is not due to the inability of our averaging process to converge, but rather arises from the breakdown of the assumption of local thermodynamic equilibrium that underlies our modeling of “transport” effects when flow gradients become significant on the scale of a mean-free path. Furthermore, errors associated with the no-slip and no-temperature-jump boundary conditions, which are typically negligible under normal conditions, become more significant fractions of the differences in flow quantities in the field under extreme conditions, leading to the breakdown of these approximations as well.

Let’s do some formal mathematics

Let us now examine some of the challenges that arise when we attempt to express our understanding of physics in mathematical terms. Our ultimate formulation will consist of a collection of partial-differential field equations (PDEs), accompanied by certain algebraic auxiliary relations. The choice of variables, as well as the determination of which variables are independent and which are dependent, hinges on how we opt to depict the flow. We have the option to describe it in terms of the behavior observed at fixed points in space and time, known as the Eulerian description, or we can instead define the paths followed by fixed parcels of fluid as they evolve over time, referred to as the Lagrangian formulation. In the Eulerian description, time and the coordinates within a spatial reference frame, which may or may not be inertial, serve as the independent variables, while the velocity, pressure, and other state variables of the fluid are considered dependent. On the other hand, in the Lagrangian description, the independent variables pertain to the fluid parcels themselves, typically identified by their spatial coordinates at an initial moment, and the dependent variables include the spatial coordinates of these parcels at subsequent moments. Although these two modes of description are theoretically equivalent in the sense that they can be employed to accurately model the same physics, they differ so significantly in their approach that they are not practically interchangeable.

The Eulerian framework is commonly preferred for various applications due to its convenience, serving as the foundation for the majority of quantitative studies in theoretical aerodynamics and computational fluid dynamics (CFD). This preference can be attributed to the fact that the Eulerian description offers a more intuitive approach for analyzing steady flows, which are the primary concern in aerodynamics. While the Eulerian description is utilized in higher-level conceptual modeling, there are instances where the Lagrangian description is also beneficial in discussing fundamental physical principles.

The Lagrangian derivative, denoted by the upper case D/Dt, represents the time rate of change of any physical quantity associated with a Lagrangian fluid parcel. This rate of change is influenced by two effects in the Eulerian frame. Firstly, the quantity may change with time at the points in space through which the parcel moves, indicated by the unsteady-flow term ∂/∂t or the Eulerian rate of change. Secondly, if the parcel moves with velocity V through a nonuniform field, it experiences a rate of change V • ∇ in addition to the unsteady-flow term. Therefore, the Lagrangian derivative is connected to derivatives in the Eulerian frame. Generally speaking, the Lagrangian derivative is connected to derivatives in the Eulerian frame through the equation (for the velocity):

Applying this transformation to the fluid velocity yields intriguing outcomes, particularly when determining the Lagrangian acceleration. In the case of a 1D steady flow, the application of the aforementioned equation to the velocity results in a reduction:

It can be observed that a specific material acceleration Du/Dt necessitates a significant spatial gradient ∂u/∂x when the velocity u is small, whereas only a minor ∂u/∂x is needed when u is large. This phenomenon arises from the motion of a Lagrangian fluid parcel within the velocity field.

One of the challenges in mathematics arises from the presence of vectors and tensors among the quantities we need to handle. The velocity, for instance, is a vector, and the equation for the conservation of momentum is a vector equation. In three-dimensional space, this leads to three variables and three equations, making it relatively easy to comprehend intuitively.

The issue of representing the transfer of forces through “contact” between neighboring fluid parcels is not immediately obvious. From a physical standpoint, these forces arise from the transfer of momentum through molecular movements. However, in the continuum formulation, the cumulative effects of numerous molecular motions are depicted as apparent internal stresses within the fluid or as forces exerted per unit area along the boundary of a parcel.

The mathematical challenge that we encounter pertains to the overarching issue of representing the stress state within a continuous material. Initially, we must familiarize ourselves with the concept of hypothetical boundaries that separate neighboring portions of the material. Subsequently, we need to mentally visualize how two adjacent portions of the material exert equal and opposite stresses on each other across their shared boundary surface. Our explanation must possess the capability to accurately determine the stress state at any given point within the fluid, accounting for the appropriate magnitude of opposing forces regardless of the orientation of the hypothetical boundary. Stress, in this context, refers to a vector quantity denoting the force per unit area, which is contingent upon the orientation of an imaginary dividing surface. This dividing surface can be defined by the direction of its normal vector.

The stress is a tensor, which has led to the development of tensor analysis, a field of mathematics dedicated to providing rigorous methods for manipulating such quantities. This mathematical framework is not only applicable in continuum mechanics but also in various branches of physics. Alongside tensor analysis, shorthand notations have been devised to express these manipulations efficiently. Tensor notation offers the most reliable approach to handle stress terms and convection terms in equations, particularly when transforming them into different coordinate systems. While it is possible to perform these manipulations without tensor notation, the likelihood of errors significantly increases. Regardless of the use of tensor notation, these manipulations quickly become exercises in symbol manipulation, making it challenging to retain a clear understanding of their physical significance.

So far, we have talked about the NS equations only in their local or differential form,
which is the form that will relate most directly to most of our succeeding discussions.
However, in some applications, a more global view of the flow suffices and can be easier to
deal with. For these situations, we have the control-volume form of the equations, in which
the equations have been integrated over a volume and the surfaces bounding the volume.
The control-volume equations are “exact” in the sense that there is no loss of accuracy
relative to the differential equations, but they are “simplified” in the sense that they can tell
us only what happens to integrated quantities and nothing about how the local quantities are
distributed over the volume and bounding surfaces.

In conventional approaches to the NS equations, all flow variables exhibit continuity and differentiability, even in the presence of shocks. This advantageous characteristic allows us to leverage significant mathematical principles without the need for incorporating any “physical” considerations. Consequently, this leads us to delve into the subject matter of the subsequent section.

Kinematics 1: Streamlines and Streaklines

Fundamental to comprehending flowfields is the utilization of k;inematic descriptions. It is imperative to grasp the kinematic structure of a flow in order to delve into the fundamental dynamics at play. The characteristics of a flowfield’s kinematic structure are inherently bound by the nature of the velocity field as a continuous vector field.
Two commonly used kinematic concepts are streamlines and streaklines. Streamlines are 3D space curves that are parallel to the velocity vector at every point. On the other hand, streaklines are also 3D space curves, but they are defined by the positions of a series of fluid parcels that have all passed through a specific “point of origin” somewhere upstream in the flowfield. While the point of origin for a streakline is typically considered to be a fixed point in space, it can also be allowed to move with time. It is important to note that a streamline is a mathematical construct that can only be defined by solving a mathematical problem, specifically by constructing a curve that is parallel to a given vector field. In contrast, a streakline can be observed, at least to some extent, in real flows that are marked by a passive contaminant such as dye in liquids or smoke in air.
In the case of a constant flow, the streamlines and streaklines originating from specific points will align and match the paths of individual particles, known as Lagrangian parcels. Despite the steadiness of the flow, there may still be intriguing complexities in deciphering the flow patterns.
Timelines are the lines formed by a set of fluid particles that were marked at a previous instant in time, creating a line or a curve that is displaced in time as the particles move.

In the case of an unsteady flow, the complexity of the situation increases significantly, resulting in distinct variations in streamlines, streaklines, and particle paths. Merely observing the pattern formed by any one of these elements provides an inadequate and often deceptive representation of the flow. The accompanying figures below illustrate the contrasting appearances of unsteady flow in the wake of a circular cylinder when depicted in terms of streaklines (a) and streamlines (b). Additionally, timelines (c), which will be defined shortly, offer an entirely different perspective.

Kinematics 2: Streamtubes, Stream Surfaces, and the Stream Function

The concept of a streamtube is typically applied exclusively to steady flows. A streamtube is defined by a closed curve in the flowfield, with steady streamlines or streaklines passing through all points on the curve. This closed curve forms the boundary of a curvilinear tube, with the bounding surface being parallel to the velocity vector. As a result, no continuous fluid parcel passes through this surface. In a steady flow, according to the principle of continuity, the mass flux in a streamtube remains constant at any cross section along its length. In a two-dimensional flowfield, we can still define a streamtube in the same manner as in three dimensions, using a closed curve to establish the boundary. However, a more practical definition is to allow the closed curve defining the streamtube to degenerate into two points. This transforms the streamtube into a two-dimensional layer of flow, defined by one streamline passing through each point.

The boundary surface of a streamtube represents a specific instance of the broader concept of a stream surface, typically associated with steady flows. The curve in space that gives rise to a stream surface does not necessarily have to be a closed curve, and the resulting stream surface does not have to form a closed tube. A general stream surface is a surface that no continuous fluid parcel passes through. In three-dimensional flows, stream surfaces that initially appear flat can become highly distorted as the flow progresses downstream. The concept of a stream function is applicable only to two-dimensional flows. When considering two points A and B within a two-dimensional flow, the mass flux across any curve connecting these points depends solely on the positions of the points and time, assuming the flow is either incompressible or steady. For instance, in the scenario depicted in the figure below, the mass flux across any contour connecting the points corresponds to the mass flux within the shaded streamtube. Consequently, if point A is fixed, the mass flux calculated in this manner for all other points B defines a unique function known as a stream function. Consequently, the stream function remains constant along streamlines, and the discrepancy in its value between two streamlines equates to the mass flux within the streamtube bounded by them. The stream function was more commonly utilized in the past than it is presently. It was frequently employed in earlier theoretical discussions of incompressible flows and occasionally utilized in numerical techniques for solving the Navier-Stokes equations in two dimensions.

Kinematics 3: Timelines

Timelines are a valuable kinematic concept that finds its most common application in 2D flows, although it can be defined in any type of flow, whether steady or unsteady. The process of defining a timeline begins by marking a series of Lagrangian fluid parcels arranged across the flow at a specific initial moment. Subsequently, a timeline is formed by tracing the path of these parcels at a later moment in time. Timelines prove to be particularly useful when established as a collection of multiple lines, with each line’s initial moment separated by equal time intervals. In practical scenarios, timelines can be approximated by passive-contaminant markers, typically originating from a thin wire stretched across the flow. In the case of air flows, the wire is coated with oil, and a pulsed electric current applied to the wire generates short bursts of smoke, which serve as markers for the cross-stream lines that move downstream. In water flows, electric pulses can generate lines composed of small hydrogen or oxygen bubbles, effectively marking the flow.

The figure bellow serves as an illustrative example of timelines within a turbulent boundary layer, highlighting a crucial characteristic of timelines in turbulent flows:

Within a fully turbulent boundary layer, the magnitude of turbulent velocity fluctuations is not a significant fraction of the average velocity. Consequently, the younger timelines situated near the left edge of the image maintain a sense of order and gradually accumulate distortions, resembling a smoother flow compared to the rest of the photograph. As the flow progresses from left to right, these distortions accumulate until the right half of the image portrays a chaotic and disordered collection of timelines that are entirely within the boundary layer. In this fully turbulent flow, the timeline depiction falsely suggests an increasing intensity of turbulent motions from left to right.

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