Introduction
We possess a range of concepts that prove to be valuable when contemplating the distribution of vorticity within the flowfield. Initially, we will focus on those concepts that are applicable to the typical realistic scenario where vorticity is continuously distributed.
In any region where vorticity is not equal to zero, it is possible to establish a vortex line as a spatial curve that runs parallel to the vorticity vector. This is akin to how a streamline aligns with the velocity vector. Consequently, within the vorticity field, a vortex line bears resemblance to a streamline within the velocity field. Just as we expanded the notion of a streamline to encompass a streamtube, we can similarly broaden the concept of a vortex line to encompass a vortex tube.
The vorticity flux across the boundary of a vortex tube is inherently zero as per its definition. Moreover, the divergence of the curl of a vector, specifically the velocity (whose curl represents the vorticity), is zero according to vector identity. Consequently, the flux remains constant across any cross-section of the tube, regardless of its position along the length.
The constancy of vorticity flux within a vortex tube governs the alterations in vorticity magnitude that must accompany the stretching of the vortex. When the cross-sectional area of a vortex tube diminishes, whether over time or along its length, the strength of the vorticity (the magnitude of the vorticity vector) must intensify. In the case of a segment of the vortex tube containing a fixed amount of fluid, a reduction in cross-sectional area typically necessitates an increase in length, or stretching. This stretching is particularly necessary if the fluid density remains constant, as we will explore later in relation to the conservation of mass. Consequently, the stretching of a vortex tube generally augments the local vorticity magnitude.
The constancy of vorticity flux in a vortex tube imposes the necessity for changes in the magnitude of vorticity when vortex stretching occurs. When the cross-sectional area of a vortex tube diminishes, whether over time or along its length, the intensity of vorticity (the magnitude of the vorticity vector) must augment. In order to accommodate a reduced cross-sectional area within a specific amount of fluid, an increase in length or stretching is typically required.
A vortex filament is a slender vortex tube with an extremely small maximum dimension in its cross section. The cross-sectional area of a vortex filament is also infinitesimally small, yet it is assumed to vary along the length of the filament, allowing it to meet the criteria of a vortex tube. In the case of a vortex filament, the vorticity flux across a cross-section is equal to the product of the vorticity magnitude and the cross-sectional area, known as the filament’s intensity. It is important to note that this definition of intensity, as the flux of vorticity through an infinitesimal area, differs from other familiar concepts of intensity, such as the intensity of a light beam, which is defined as the energy flux per unit area. Helmholtz’s second theorem states that the intensity of a vortex filament remains constant along its length. This conservation of intensity implies that a vortex filament cannot terminate within the fluid domain but must either form a closed loop (vortex loop) or terminate at the boundary of the domain.
Depending on the characteristics of the boundary, restrictions will be imposed on the possible ways in which vortex filaments or vortex lines can terminate there. Let us first examine the unique scenario of an individual vortex filament that is enclosed by irrotational flow. In the event that the flow remains constant and the boundary represents an interface through which the fluid is unable to pass, the vortex filament can only intersect the boundary in a perpendicular manner.
This requirement arises from the necessity of having a predominantly circular flow configuration in the vicinity of the filament, within planes that are perpendicular to the filament itself. Any deviation from this normal orientation would contradict the condition of no flow through the boundary.
Moreover, if the boundary is a fixed solid surface subject to a no-slip condition, the velocity components within planes perpendicular to the filament must diminish at the surface, while the vorticity magnitude must approach zero. Consequently, an isolated vortex filament is incapable of terminating at a solid surface characterized by a no-slip condition.
In the case of distributed vorticity, vortex lines can intersect a no-throughflow boundary with slip, and the intersection may not be in the normal direction. Conversely, on a stationary surface with no slip, the situation is more restricted. Since the tangential velocity is zero on the surface, the vorticity component normal to the surface must also be zero throughout. Therefore, if the vorticity magnitude is non-zero, the vortex lines must be tangent to the surface. This principle generally holds in the viscous flow around a stationary object, except for isolated singular points of separation or attachment where the vorticity magnitude on the surface is zero. In such cases, a vortex line may intersect the surface normally, but the normal vorticity component must still approach zero at the intersection point. Consequently, vortex lines can only intersect a no-slip surface at isolated singular points. It is a common misconception that vortex lines cannot intersect a no-slip surface at all, disregarding the exceptions mentioned above.
It is evident that as vortices come close to a solid surface with no-slip conditions, except at a single isolated point, the vortex lines are compelled to change direction to prevent intersection with the surface. This redirection often results in the vortices contributing to the vorticity within a viscous boundary layer formed on the surface.
1. Let us now explore the theoretical constructs that have been devised for idealized representations of flows characterized by highly concentrated vorticity. The presence of concentrated vorticity in specific regions plays a crucial role in the analysis of certain flows that will be discussed later on. For instance, in Chapter 8, we will delve into the vorticity patterns observed in the wake behind a lifting wing, where the vorticity initially exists in a concentrated form within a thin shear layer, eventually transitioning into two distinct, more or less axisymmetric vortices, all enveloped by nearly irrotational flow.
2. In theoretical models of such flow phenomena, these vortical structures are often simplified as mathematically thin concentrations, with the shear layers conceptualized as vortex sheets and the vortices as line vortices. Despite the vorticity being concentrated within regions of zero cross-sectional area, these idealized entities exhibit finite vorticity fluxes. Consequently, the vorticity distribution at the location of the sheet or line must be singular or infinite.
3. When dealing with a vortex sheet, the process typically involves integrating across a finite width of the sheet to determine a finite vorticity flux, even though the integrated area remains zero due to the infinitesimally thin nature of the sheet. On the other hand, for a line vortex, a single integration across the line (essentially a point) is adequate to calculate a finite flux. While there exists a formal mathematical framework that provides a rigorous treatment of these concepts, a detailed exploration of this theory is not necessary for a comprehensive understanding of the underlying principles.
The line vortex and the vortex filament, although they may appear similar at first glance, have significant distinctions. Firstly, the line vortex has a cross-sectional area of zero, whereas the filament has an infinitesimally small cross-sectional area. Additionally, the vorticity flux of a line vortex is finite, while that of a filament is infinitesimal. It is important to avoid confusing a line vortex, which represents a singular distribution of vorticity, with a vortex line, which is merely parallel to the vorticity vector and typically found in fields where vorticity is continuously distributed.
A point vortex, also known as a line vortex in a 2D planar flow, is characterized by a straight line that extends infinitely in both directions perpendicular to the 2D plane. This configuration gives the appearance of a single point within the 2D plane. The line vortex serves as one of the fundamental singularities that can be utilized as a fundamental component in constructing potential-flow theory solutions, as elaborated in Section 3.10. However, in more complex flows, the line vortex may exhibit curvature, which presents a unique challenge. At any given point along a curved line vortex, where the curvature is non-zero, the fluid velocity perpendicular to the vortex becomes infinite. Consequently, determining a realistic velocity at which the vortex line will be transported by the flow becomes impossible. In actual flows, vorticity is continuously distributed and possesses finite magnitude, thereby eliminating the occurrence of infinite velocities.
Associating the Velocity Field with Concentrations of Vorticity
The concept of highly concentrated vorticity is frequently simplified as either a vortex sheet or a line vortex. By utilizing Stokes’s theorem, we can now analyze the velocity distributions in the immediate vicinity that are required to correspond with these idealized vorticity distributions.
The figure above, labeled as (a), illustrates a vortex sheet in 2D flow. By applying Stokes’s theorem to a closed contour that encloses a short section of the sheet, it becomes evident that there is a jump in velocity magnitude across the sheet, which is equal to the local vorticity strength or the vorticity per unit distance along the sheet in the direction perpendicular to the vorticity vector. In this particular 2D case, the vorticity vector is perpendicular to the plane of the paper, and the distance along the sheet is measured in the flow direction. The physical flow associated with this idealized vortex sheet is a shear layer where the velocity jump is spread across a finite thickness, as depicted in the figure labeled as (b).
In the case of a 3D flow, the velocity jump across a vortex sheet, in a vector sense, must still be perpendicular to the vorticity vector. In aerodynamics, it is common to encounter a sheet with no jump in velocity magnitude, only in direction. In such cases, the jump in the velocity vector is perpendicular to the vorticity vector, which is parallel to the direction of the mean of the velocity vectors on the two sides of the sheet, as illustrated in the figure labeled as (c). It can be demonstrated that if the vorticity vector were not parallel to the mean of the two velocity vectors, there would have to be a jump in velocity magnitude.
Vortex sheets resembling the one depicted in the figure labeled as (c) are frequently modeled in 3D potential-flow theory. It is evident from the definition of the velocity potential that the jump in the velocity vector necessitates a jump in the velocity potential as well.
If a physical shear layer is effectively thin, meaning that the flow changes across the layer occur much faster than changes in other directions, the velocity jump will be approximately equal in magnitude and perpendicular to the integral of the vorticity across the layer.
Is velocity induction by vorticity a fallacy?
Every engineering student inevitably encounters the Biot-Savart law during their undergraduate studies, whether it pertains to fluid mechanics or classical electromagnetics. This law suggests that understanding the curl of a vector field at a specific point provides insights into the vector field’s behavior at a different point.
Despite its initial appeal, the concept can be deceptive as it commonly results in ambiguity regarding the relationship between cause and effect. Additionally, the ability to convert Navier-Stokes equations from velocity to vorticity formulation and the utilization of potential flow models to introduce obstacles to the flow further support the widely held belief that vorticity causes velocity, as suggested by the Biot-Savart principle.
The fallacy lies here. In the absence of gravitational or electromagnetic forces, there is no action at a distance in ordinary fluid flows. Expressing the equations in different forms and referring to the Bio-Savart law as a calculus relationship between a vector field and its curl does not imply that a vortex at point A can induce a velocity at a distant point B. While it is true that a mathematical relationship like the Biot-Savart law enables us to deduce both quantitative and qualitative details about the velocity field at a remote point, in fluid mechanics, it does not accurately depict the physics. Therefore, the direct cause and effect relationship is somewhat misleading in this context compared to its counterpart in classical electromagnetics.
The Biot-Savart law proves to be beneficial for quantitative computations. However, the qualitative concept that understanding the vorticity at a specific point enables us to deduce information about the velocity at another point holds its own value. This concept serves as one of the most influential tools for comprehending flowfields. Nevertheless, despite its potency, it can also be a double-edged sword as it often causes confusion when it comes to determining cause and effect.
The issue arises due to the fact that vorticity is considered the “input” while velocity is seen as the “output”, leading to the common practice of referring to the velocity deduced from vorticity as induced velocity. This can easily lead one to believe that vorticity somehow “causes” the velocity it “determines.” However, this line of thinking is incorrect. In the absence of significant gravitational or electromagnetic body forces, there is no action at a distance in regular fluid flows. Significant forces are only transmitted through direct contact between neighboring fluid parcels.
Therefore, a vortex at point A cannot directly “cause” a velocity at a distant point B, and terms like “caused by,” “induced,” and even “due to” misrepresent the physics involved. It is crucial to remember that Biot-Savart is simply a mathematical relation between a vector field and its curl, and in fluid mechanics, it does not indicate a direct physical cause-and-effect relationship. This point is of utmost importance and yet has not been emphasized enough in the literature. It is intriguing to explore the perspectives of other authors on this matter. Aerodynamicists have contributed to the confusion by liberally using terms like “induced velocity” and “induction.” These terms originate from another field, classical electromagnetics, where the Biot-Savart law is applicable, and it is stated that the magnetic field is “induced” by the electric current. In electromagnetics, this terminology is suitable as there is believed to be genuine action at a distance occurring, making the term “induction” physically fitting. However, in fluid mechanics, there is no direct causal connection. We understand that vorticity is generated, transported, and diffused, which explains why vorticity in our flowfields exists: it serves more as an indication of the overall flow pattern rather than a cause of it.
In order to elucidate the presence of a flow pattern, it is necessary to reference the actual physics involved, specifically the equilibrium of forces within fluid elements at a given location.