Fluid Mechanics

Compressible flow

A flow is said to be compressible if the flow is characterized by significant changes in fluid density. This happens when the density of the fluid varies due to its velocity. Gases, but not liquids, display such behavior. To distinguish between compressible and incompressible flow in gases, the Mach number (the ratio of the speed of the flow to the speed of sound) must be greater than about 0.3 before significant compressibility occurs. For a compressible flow, it is necessary that the fluid is compressible. For example, air is compressible since its density will change if temperature changes, or if some external force is applied (e.g. a child squeezing a balloon). However, the fluid is compressible doesn't mean the flow is compressible.

Pathlines

Pathlines are the trajectories that individual fluid particles follow. These can be thought of as "recording" the path of a fluid element in the flow over a certain period. The direction the path takes will be determined by the streamlines of the fluid at each moment in time: dx/dt=u,dy/dt=v,dz/dt=w.

Kinematic viscosity

Ratio of dynamic viscosity to the fluid density.

Fluid

State of matter/substance that continuously deforms under the influence of shear stress no matter how small it is. Liquids and gases are grouped under fluids.

Streaklines

Streaklines are the locus of points of all the fluid particles that have passed continuously through a particular spatial point in the past. Dye steadily injected into the fluid at a fixed point extends along a streakline.

Streamlines

Streamlines are a family of curves that are instantaneously tangent to the velocity vector of the flow. These show the direction a fluid element will travel in at any point in time. If ds=dxi ̂+dyj ̂+dzk ̂ is a small line segment along a streamline, then dx/u=dy/v=dz/w.

Streamtube

Streamtube is a group of streamlines that start from a closed contour, i.e. a bundle of streamlines that form a tube-like shape.

Mechanics

Study of the relationship between motion and forces/moments/torques.

Knudsen number

The Knudsen number (Kn) is a dimensionless number defined as the ratio of the molecular mean free path length to a representative physical length scale (e.g. vehicle dimension). For small Knudsen numbers (≪1), in other words if the mean free path of the molecules is small relative to the dimensions of the flow, then we may employ the continuum hypothesis and treat the fluid as though it were a single homogeneous material instead of being composed of molecules. On the other hand, if Kn<1, or Kn~1, then the mean free path of the molecules is on the order of the flow dimensions and it may be inappropriate to employ the continuum assumption. A kinetic treatment of the fluid may be required. Knudsen number is thus very low at low altitudes.

Continuum

The continuum assumption, considers fluids to be continuous i.e. properties such as density, pressure, temperature, and velocity are taken to be well-defined at "infinitely" small points, defining a REV (Reference Element of Volume), at the geometric order of the distance between two adjacent molecules of fluid. Properties are assumed to vary continuously from one point to another, and are statistically averaged values in the REV. The fact that the fluid is made up of discrete molecules that collide with one another and solid objects is ignored. Concept of continuum works well at low altitudes. But at high altitudes, under rarefied conditions, this concept fails, and the molecules must be treated as individual particles.

Viscosity

The viscosity of a fluid is a measure of its resistance to gradual deformation by shear stress or tensile stress.

Fluid particle/element

a (large) collection of molecules/particles that are closely packed, which can be treated as a continuum.

Temperature

a measure of the kinetic energy associated with the random motion of the molecules that form the continuous matter.

Compressibility

a measure of the relative volume change of a fluid or solid as a response to a pressure (or mean stress) change. β=-1/V ∂V/∂p

Fluid parcel

an infinitesimal volume that moves with the velocity of the fluid.

Stress

force per unit area.

Ideal fluid

inviscid and incompressible fluid. Inviscid means not oppose a change in its shape with any internal resistance. Air and water have very low viscosity thus high Re. In general, tangential forces are very small. But no-slip condition does hold good even for very low viscosity fluids. Separation and drag originate from no-slip boundary condition. So inviscid fluid model is useless to calculate drag (e.g. D'Alembert's paradox).

Very viscous fluid

oil, glycerine (C3H8O3 甘油，丙三醇).

Gauge pressure

p-p_atmosphere.

Buoyancy

phenomenon by which an upward force is exerted by a fluid that opposes the weight of an immersed object. The magnitude of that force is proportional to the difference in the pressure between the top and the bottom of the column, and (as explained by Archimedes' principle) is also equivalent to the weight of the fluid that would otherwise occupy the column, i.e. the displaced fluid. For this reason, an object whose density is greater than that of the fluid in which it is submerged tends to sink.

Mean free path

the average distance that an (air) molecule will travel before it collides with another particle. In gases, under normal conditions (e.g. low altitudes), the mean free path is very low (of the order of microns) compared to the characteristic dimensions of the vehicle.

Pressure

the normal force exerted by the continuous matter on a plane placed in the fluid, per unit area of the plane.

Mildly viscous fluid

water, alcohol, air.

Viscosity

For liquids, the (dynamic) viscosity, μ, is a property of the fluid (and not flow) that quantifies the fluid's resistance to deformation under the action of shear stresses (or tensile stresses), and tangential stress is proportional to viscosity. It is a quantitative measure of the friction between adjacent fluid layers due to the cohesive forces between fluid molecules. It is associated with dissipative fluid forces and can be characterized in a number of ways. From a macroscopic point of view, it can be viewed as the "stickiness" of a fluid. It is what imposes the no-slip boundary condition for fluids moving near solid surfaces. In general, viscosity is a function of T and p, but T effects dominate (Sutherland's formula). In liquids, viscosity is caused by intermolecular forces. When a liquid is heated, the molecules move apart, and the intermolecular forces decrease. Thus viscosity of a liquid decreases with temperature. For gases, from a kinetic theory point of view, the macroscopic concept of viscosity is related to the statistical average of the linear momentum exchange occurring between molecules by random collision. Consider a fluid with nonuniform macroscopic mean flow velocity. The random component of the velocity of individual molecules will tend to move some molecules from regions of high mean momentum to regions of low mean momentum, and vice versa. The net effect of this diffusion of momentum is viscosity. In gases, as temperature increases, the energy of the molecules associated with this random motion increases. Thus viscosity increases in gases with temperature.

Vorticity

In continuum mechanics, the vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate ), as would be seen by an observer located at that point and traveling along with the flow.

Vortex

In fluid dynamics, a vortex (plural vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved.

Incompressible flow

In fluid mechanics or more generally continuum mechanics, incompressible flow (isochoric flow) refers to a flow in which the material density is constant within a fluid parcel (and not necessarily within the entire flow field). An equivalent statement that implies incompressibility is that the divergence of the fluid velocity is 0. A flow can assumed to be incompressible if the change in density due to the velocity/speed of the flow is negligible. Incompressible flow does not imply that the fluid itself is incompressible. Incompressible flow implies that the density remains constant within a parcel of fluid that moves with the fluid velocity.

Incompressible flow

In fluid mechanics or more generally continuum mechanics, incompressible flow (isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An equivalent statement that implies incompressibility is that the divergence of the flow velocity is zero.

Streamline vs. Streakline vs. Pathline

In unsteady flows, different particles that start at the same spatial location may travel along different paths. In steady flows, particles that start from the same starting locations will follow the same path, thus streamlines and pathlines are the same.

Dynamic viscosity

It is the proportionality constant in Newton's law of friction. It is the shear stress experienced on the plane between two adjacent fluid layers when the velocity gradient between the layers is maintained unity.