Properties of the Velocity Field

Properties of the

Velocity Field

In a given flow situation, the determination, by experiment or theory, of the proper-

ties of the fluid as a function of position and time is considered to be the solution to

the problem. In almost all cases, the emphasis is on the space–time distribution of the

fluid properties. One rarely keeps track of the actual fate of the specific fluid parti-

cles.6 This treatment of properties as continuum-field functions distinguishes fluid

mechanics from solid mechanics, where we are more likely to be interested in the tra-

jectories of individual particles or systems.

Eulerian and Lagrangian 

Descriptions

There are two different points of view in analyzing problems in mechanics. The first

view, appropriate to fluid mechanics, is concerned with the field of flow and is called

the eulerian method of description. In the eulerian method we compute the pressure

field p(x, y, z, t) of the flow pattern, not the pressure changes p(t) that a particle expe-

riences as it moves through the field.

The second method, which follows an individual particle moving through the flow,

is called the lagrangian description. The lagrangian approach, which is more appro-

priate to solid mechanics, will not be treated in this book. However, certain numeri-

cal analyses of sharply bounded fluid flows, such as the motion of isolated fluid

droplets, are very conveniently computed in lagrangian coordinates [1].

Fluid dynamic measurements are also suited to the eulerian system. For example,

when a pressure probe is introduced into a laboratory flow, it is fixed at a specific

position (x, y, z). Its output thus contributes to the description of the eulerian pres-

sure field p(x, y, z, t). To simulate a lagrangian measurement, the probe would have

to move downstream at the fluid particle speeds; this is sometimes done in oceano-

graphic measurements, where flowmeters drift along with the prevailing currents.

The two different descriptions can be contrasted in the analysis of traffic flow along

a freeway. A certain length of freeway may be selected for study and called the field of

flow. Obviously, as time passes, various cars will enter and leave the field, and the iden-

tity of the specific cars within the field will constantly be changing. The traffic engi-

neer ignores specific cars and concentrates on their average velocity as a function of

time and position within the field, plus the flow rate or number of cars per hour pass-

ing a given section of the freeway. This engineer is using an eulerian description of the

traffic flow. Other investigators, such as the police or social scientists, may be interested

in the path or speed or destination of specific cars in the field. By following a specific

car as a function of time, they are using a lagrangian description of the flow.

The Velocity Field

Foremost among the properties of a flow is the velocity field V(x, y, z, t). In fact,

determining the velocity is often tantamount to solving a flow problem, since other properties follow directly from the velocity field. Chapter 2 is devoted to the calcu-lation of the pressure field once the velocity field is known. Books on heat transfer (for example, Ref. 20) are largely devoted to finding the temperature field from known velocity fields. 

One example where fluid particle paths are important is in water quality analysis of the fate of con- taminant discharges.

            In general, velocity is a vector function of position and time and thus has three components u, , and w, each a scalar field in itself:

The use of u, , and w instead of the more logical component notation Vx, Vy, and Vz

is the result of an almost unbreakable custom in fluid mechanics. Much of this text-

book, especially Chaps. 4, 7, 8, and 9, is concerned with finding the distribution of

the velocity vector V for a variety of practical flows.

The Acceleration Field

The acceleration vector, a -

 dV/dt, occurs in Newton’s law for a fluid and thus is very

important. In order to follow a particle in the Eulerian frame of reference, the final result

for acceleration is nonlinear and quite complicated. Here we only give the formula:

where (u, v, w) are the velocity components from Eq. (1.4). We shall study this

formula in detail in Chap. 4. The last three terms in Eq. (1.5) are nonlinear products

and greatly complicate the analysis of general fluid motions, especially viscous

flows.