Crocco's theorem

Last updated January 17, 2024

Crocco's theorem is an aerodynamic theorem relating the flow velocity, vorticity, and stagnation pressure (or entropy) of a potential flow. Crocco's theorem gives the relation between the thermodynamics and fluid kinematics. The theorem was first enunciated by Alexander Friedmann for the particular case of a perfect gas and published in 1922:^[1]

{\frac {D\mathbf {u} }{Dt}}=T\nabla \,s-\nabla \,h

However, usually this theorem is connected with the name of Italian scientist Luigi Crocco,^[2] a son of Gaetano Crocco.

Consider an element of fluid in the flow field subjected to translational and rotational motion: because stagnation pressure loss and entropy generation can be viewed as essentially the same thing, there are three popular forms for writing Crocco's theorem:

Stagnation pressure: $\mathbf {u} \times {\boldsymbol {\omega }}=v\nabla p_{0}$ ^[3]
Entropy (the following form holds for plane steady flows): $T{\frac {ds}{dn}}={\frac {dh_{0}}{dn}}+u\omega$ ^[4]
Momentum: ${\frac {\partial \mathbf {u} }{\partial t}}+\nabla \left({\frac {u^{2}}{2}}+h\right)=\mathbf {u} \times {\boldsymbol {\omega }}+T\nabla s+\mathbf {g} ,$

In the above equations, $\mathbf {u}$ is the flow velocity vector, $\omega$ is the vorticity, $v$ is the specific volume, $p_{0}$ is the stagnation pressure, $T$ is temperature, $s$ is specific entropy, $h$ is specific enthalpy, $\mathbf {g}$ is specific body force, and $n$ is the direction normal to the streamlines. All quantities considered (entropy, enthalpy, and body force) are specific, in the sense of "per unit mass".

Related Research Articles

Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a continuous medium rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century.

In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including aerodynamics and hydrodynamics. Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation.

<span class="mw-page-title-main">Navier–Stokes equations</span> Equations describing the motion of viscous fluid substances

The Navier–Stokes equations are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).

The vorticity equation of fluid dynamics describes the evolution of the vorticity $ω$ of a particle of a fluid as it moves with its flow; that is, the local rotation of the fluid. The governing equation is:

In fluid dynamics, Stokes' law is an empirical law for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. It was derived by George Gabriel Stokes in 1851 by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.

The stream function is defined for incompressible (divergence-free) flows in two dimensions – as well as in three dimensions with axisymmetry. The flow velocity components can be expressed as the derivatives of the scalar stream function. The stream function can be used to plot streamlines, which represent the trajectories of particles in a steady flow. The two-dimensional Lagrange stream function was introduced by Joseph Louis Lagrange in 1781. The Stokes stream function is for axisymmetrical three-dimensional flow, and is named after George Gabriel Stokes.

In fluid dynamics, the Euler equations are a set of quasilinear partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In particular, they correspond to the Navier–Stokes equations with zero viscosity and zero thermal conductivity.

In fluid mechanics, the Taylor–Proudman theorem states that when a solid body is moved slowly within a fluid that is steadily rotated with a high angular velocity $, the fluid velocity will be uniform along any line parallel to the axis of rotation. must be large compared to the movement of the solid body in order to make the Coriolis force large compared to the acceleration terms.$

Stokes flow, also named creeping flow or creeping motion, is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low, i.e. $. This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large, or the length-scales of the flow are very small. Creeping flow was first studied to understand lubrication. In nature, this type of flow occurs in the swimming of microorganisms and sperm. In technology, it occurs in paint, MEMS devices, and in the flow of viscous polymers generally.$

In fluid dynamics, the enstrophy $can be interpreted as another type of potential density; or, more concretely, the quantity directly related to the kinetic energy in the flow model that corresponds to dissipation effects in the fluid. It is particularly useful in the study of turbulent flows, and is often identified in the study of thrusters as well as in combustion theory and meteorology.$

In fluid mechanics, potential vorticity (PV) is a quantity which is proportional to the dot product of vorticity and stratification. This quantity, following a parcel of air or water, can only be changed by diabatic or frictional processes. It is a useful concept for understanding the generation of vorticity in cyclogenesis, especially along the polar front, and in analyzing flow in the ocean.

In fluid mechanics, Kelvin's circulation theorem states:

In a barotropic, ideal fluid with conservative body forces, the circulation around a closed curve moving with the fluid remains constant with time.

The derivation of the Navier–Stokes equations as well as its application and formulation for different families of fluids, is an important exercise in fluid dynamics with applications in mechanical engineering, physics, chemistry, heat transfer, and electrical engineering. A proof explaining the properties and bounds of the equations, such as Navier–Stokes existence and smoothness, is one of the important unsolved problems in mathematics.

In fluid dynamics, Faxén's laws relate a sphere's velocity $and angular velocity to the forces, torque, stresslet and flow it experiences under low Reynolds number conditions.$

The Clausius–Duhem inequality is a way of expressing the second law of thermodynamics that is used in continuum mechanics. This inequality is particularly useful in determining whether the constitutive relation of a material is thermodynamically allowable.

In fluid dynamics, Beltrami flows are flows in which the vorticity vector $and the velocity vector are parallel to each other. In other words, Beltrami flow is a flow where Lamb vector is zero. It is named after the Italian mathematician Eugenio Beltrami due to his derivation of the Beltrami vector field, while initial developments in fluid dynamics were done by the Russian scientist Ippolit S. Gromeka in 1881.$

In fluid mechanics, Helmholtz minimum dissipation theorem states that the steady Stokes flow motion of an incompressible fluid has the smallest rate of dissipation than any other incompressible motion with the same velocity on the boundary. The theorem also has been studied by Diederik Korteweg in 1883 and by Lord Rayleigh in 1913.

In fluid dynamics, Lamb vector is the cross product of vorticity vector and velocity vector of the flow field, named after the physicist Horace Lamb. The Lamb vector is defined as

In fluid dynamics, Prandtl–Batchelor theorem states that if in a two-dimensional laminar flow at high Reynolds number closed streamlines occur, then the vorticity in the closed streamline region must be a constant. A similar statement holds true for axisymmetric flows. The theorem is named after Ludwig Prandtl and George Batchelor. Prandtl in his celebrated 1904 paper stated this theorem in arguments, George Batchelor unaware of this work proved the theorem in 1956. The problem was also studied in the same year by Richard Feynman and Paco Lagerstrom and by W.W. Wood in 1957.

The streamline upwind Petrov–Galerkin pressure-stabilizing Petrov–Galerkin formulation for incompressible Navier–Stokes equations can be used for finite element computations of high Reynolds number incompressible flow using equal order of finite element space by introducing additional stabilization terms in the Navier–Stokes Galerkin formulation.

References

↑ Friedmann A. An essay on hydrodynamics of compressible fluid (Опыт гидромеханики сжимаемой жидкости), Petrograd, 1922, 516 p., reprinted Archived 2016-03-03 at the Wayback Machine in 1934 under the editorship of Nikolai Kochin (see the first formula on page 198 of the reprint).
↑ Crocco L. Eine neue Stromfunktion für die Erforschung der Bewegung der Gase mit Rotation. ZAMM, Vol. 17, Issue 1, pp. 1–7, 1937. DOI: 10.1002/zamm.19370170103. Crocco writes the theorem in the form $\scriptstyle \mathrm {rot} \,\mathbf {u} \times \mathbf {u} =T\mathrm {grad} \,S$ for perfect gas (the last formula on page 2).
↑ Shapiro, Ascher H. "National Committee for Fluid Mechanics Films Film Notes for 'Vorticity,'" 1969. Encyclopædia Britannica Educational Corporation, Chicago, Illinois. (retrieved from http://web.mit.edu/hml/ncfmf/09VOR.pdf (5/29/11)
↑ Liepmann, H. W. and Roshko, A. "Elements of Gasdynamics" 2001. Dover Publications, Mineola, NY (eq. (7.33)).

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Friedmann A. An essay on hydrodynamics of compressible fluid (Опыт гидромеханики сжимаемой жидкости), Petrograd, 1922, 516 p., reprinted Archived 2016-03-03 at the Wayback Machine in 1934 under the editorship of Nikolai Kochin (see the first formula on page 198 of the reprint).

[2] Crocco L. Eine neue Stromfunktion für die Erforschung der Bewegung der Gase mit Rotation. ZAMM, Vol. 17, Issue 1, pp. 1–7, 1937. DOI: 10.1002/zamm.19370170103. Crocco writes the theorem in the form $\scriptstyle \mathrm {rot} \,\mathbf {u} \times \mathbf {u} =T\mathrm {grad} \,S$ for perfect gas (the last formula on page 2).

[Shapiro-3] Shapiro, Ascher H. "National Committee for Fluid Mechanics Films Film Notes for 'Vorticity,'" 1969. Encyclopædia Britannica Educational Corporation, Chicago, Illinois. (retrieved from http://web.mit.edu/hml/ncfmf/09VOR.pdf (5/29/11)

[Liepmann-4] Liepmann, H. W. and Roshko, A. "Elements of Gasdynamics" 2001. Dover Publications, Mineola, NY (eq. (7.33)).

[1]

[2]

[3]

[4]