The Momentum Equation in Control-Volume Form

The Conservation of Momentum principle for a fluid system corresponds to the word statement of Newton’s Second Law of Motion that the net force acting on a system is equal to the time rate of change of momentum of the system. Because both force F and momentum P are vector quantities (i. e., magnitude and direction), the resulting equation is a vector equation. For brevity, we discuss only the details of the linear momentum balance in this subsection, which can be applied to the angular momentum H if the torque, M, is accounted for and applied to the control volume. The results to be derived for the linear momentum can be extended easily to handle problems in which angular momentum may be important. This is accomplished in the example applications presented herein.

To apply the Reynolds’ Transport Theorem to Eq. 3.3, we must first define the force vector, F, acting on the fluid in the control volume. That is, the rate of change of the system momentum is determined by this force field. Although the momentum equation often is referred to as “conservation of momentum,” momentum is con­served, strictly speaking, only if F is zero.

It is useful to distinguish between two types of forces that may act on the fluid. The first type acts on all of the fluid particles in the control volume—for example, the gravity force or the magnetic force acting on a conducting fluid. The differential body force on each element of volume enclosed by the control surface is given by pbdV (units are [slug/ft3][lbf/slug][ft3] = [lbf]), where b is the vector body force per unit mass. In general, b can change from point to point and also may be a function of time. Summing over the entire control volume:

body force = Fb = jjjpbdV (3.29)

V

The second type of force acts on the surface surrounding the control volume. In general, there is a force on each element of the surface of the control volume due to viscous and pressure stresses arising from the presence of surrounding parts of the flow field. Therefore, this force conveniently can be decomposed (see Fig. 2.2 and the related discussion) into components normal and tangential to the surface. Thus,

surface force = FS = jjzdS = jjтnndS + jjxf tdS, (3.30)

s S S

where т is the surface force per unit area (i. e., surface stress); Tn and Tt are the normal stress and tangential stress components, respectively. Thus, the total force acting on the system is:

F = Fb + F,. (3.31)

F also may involve mechanical forces applied directly to the control volume. To use this result, it is necessary to model the force system to represent the particular problem of interest. For example, the normal stress in Eq. 3.31 may be represented by the pressure; the tangential stress may be due to viscous shear of the type that is well represented by Newton’s Law of Viscous Force (see Eq. 2.4 and the related discussion). The body force may be due to a gravitational field, b = g, where the magnitude of vector g is the gravitational acceleration and its direction is along the gravitational field lines.

In applying the Reynolds’ Transport Theorem (Eq. 3.23), we see that the inten­sive property corresponding the system momentum, P, is now the velocity vector, V, as indicated in Table 3.1. Note that pV is the momentum per unit volume. For a fixed control volume, Newton’s Second Law can be stated in words as “The net force acting on the fluid within the control volume is equal to the time rate of change of momentum within the control volume (the unsteady contribution) plus the net momentum flux (outflow-inflow) through the surface of the control volume.” Thus, we must have:

F = jjj pb dV + jj TdS = d jjj pVdF+ jj pV (V ■ n) dS (3.32)

V Si 1 V S

The first term on the right side represents the rate of change of the linear momentum contained within the control volume. Because it is no longer necessary to distinguish between the system and the control volume, the more common notation for the volume and surfaces integrated over are used in Eq. 3.32. The net momentum flux across the surface of the control volume is equal to the vector momentum per unit mass, V multiplied by p(V• n)dS, the mass flow rate through the area dS.

Equation 3.32 is the most general form of the momentum equation. It may not be necessary to retain all of this generality in a particular application in aero­dynamics. Again, the student should acquire the necessary physical understanding to confidently drop unneeded terms. Consider the following simplifying assumptions.

It often is the case that flow may be assumed realistically to be steady. Then, the first term in Eq. 3.32 can be dropped. Most of the aerodynamic models in this book

Table 3.3. Momentum equation in control-volume form General form Eq. 3.32 Jt Ш P’VdV + Я P V(V ■ n)dS = ДО! pbdV + Я TdS V S v s Steady flow Я pV(V ■ n)dS = jjj pbdV + Я TdS S V S (3.33) Incompressible flow dt jjj VdV+ Я V(V • n)dS = jjj bdV + P Я T dS v S v P S (3.34) Flow without body forces dt jjj pVdV + Я p V(V • n)dS = Я T dS V[8] Si Si (3.35) Inviscid flow without body forces dt я) pVdv+я p v( V n)d. S=-jj pn dS Vі Si Si (3.36) Steady, incompressible, inviscid flow jj pV(V. n)dS = – jj pndS js Si (3.37)

use this approximation. Flow over an airplane wing flying at constant speed clearly is steady; however, aerodynamic modeling of a dragonfly wing, for example, requires full application of the unsteady formulation.

In this book, we concentrate on aerodynamics of incompressible flows. Then, the density can be treated as a constant and divided out of all but the last term in Eq. 3.32.

A commonly used simplification is to neglect the body force by setting b to zero. Body forces are negligible compared to other forces in most aerodynamic appli­cations. For example, the effect of gravitational forces on the motion of air around a body is important only if significant changes in elevation are involved. Review the model of the earth’s atmosphere presented in Chapter 2 to see that elevation changes like those involved in the flow over an airplane wing or fuselage are likely to involve only minor changes in fluid properties due to the gravitational acceleration. A possible exception is in airflow over a very large structure (e. g., a dirigible or a tall building). Even in these cases, the body-force effects often are negligible compared to pressure and viscous forces.

Other simplifications that often are appropriate include representing the sur­face tractions only by the pressure forces. Then, we set т = Tn = p, where p is the (vector) normal stress due to pressure. This amounts to the assumption that the flow is inviscid—that is, that viscous effects are negligible. Of course, no liquid or gas is truly inviscid but, in many cases, the viscous forces are not decisively large compared with other forces that dominate the problem. The assumption greatly simplifies the problem while still leading to useful and practical results, and we use it in several crucial analyses.

Special cases of the momentum equation are displayed in Table 3.3.

In the last two cases shown in the table, the surface force is represented only by the pressure. Then, the negative sign on the force term indicates that a positive pressure exerts a force normally inward at the surface. That is, the positive pressure force is in the opposite direction to the surface normal, n. The final special case, Eq. 3.37, has many applications and often is referred to as the Momentum Theorem. Expressed in words, it states that if the flow is steady, inviscid, and incompressible, then the momentum flux through the surface is controlled entirely by the pressure forces acting at the control surface. This is a useful form of the integral-momentum equation because, for steady flow, the volume-integral term drops out and the remaining terms require information only on the surface of the control volume, not within it. Eq. 3.37 is a vector equation and, in the most general case, it stands for three component equations in the coordinate directions.

Of course, there are other special cases of the momentum equation. Inspection of the forms shown in Table 3.3 reveals that it is a simple matter to adjust for any situation. It is only necessary to thoroughly understand and apply the meanings of the words steady, incompressible, and inviscid.