The Viscous Flow Energy Equation

The energy equation was derived in Section 2.7, where the first law of thermodynamics was applied to a finite control volume fixed in space. The resulting integral form of the energy equation was given by Equation (2.95), and differential forms were obtained in Equations (2.96) and (2.114). In these equations, the influence of viscous effects was expressed generically by such terms as Gviscous anC^ Viscous’ ft ft recommended that you review Section 2.7 before progressing further.

In the present section, we derive the energy equation for a viscous flow using as our model an infinitesimal moving fluid element. This will be in keeping with

our derivation of the Navier-Stokes equation in Section 15.4, where the infinitesimal element was shown in Figure 15.11. In the process, we obtain explicit expressions for Qviscous and Viscous in terms of the flow-field variables. That is, we once again derive Equation (2.114), except the viscous terms are now displayed in detail.

Consider again the moving fluid element shown in Figure 15.11. To this element, apply the first law of thermodynamics, which states

Rate of change net flux of rate of work

of energy inside = heat into + done on element fluid element element due to pressure and

stress forces on surface

or A = В + C [15.20] where A, B, and C denote the respective terms above.

Let us first evaluate C; that is, let us obtain an expression for the rate of work done on the moving fluid element due to the pressure and stress forces on the surface of the element. (Note that we are neglecting body forces in this derivation.) These surface forces are illustrated in Figure 15.11, which for simplicity shows only the forces in the x direction. Recall from Section 2.7 that the rate of doing work by a force exerted on a moving body is equal to the product of the force and the component of velocity in the direction of the force. Hence, the rate of work done on the moving fluid element by the forces in the x direction shown in Figure 15.11 is simply the a component of velocity и multiplied by the forces; for example, on face abed the rate of work done by zyx dx dz is мг„ dx dz., with similar expressions for the other faces. To emphasize these energy considerations, the moving fluid element is redrawn in Figure 15.12, where the rate of work done on each face by forces in the a direction is shown explicitly. Study this figure carefully, referring frequently to its companion in Figure 15.11, until you feel comfortable with the work terms given in each face. To obtain the net rate of work done on the fluid element by the forces in the a direction, note that forces in the positive x direction do positive work and that forces in the negative x direction do negative work. Hence, comparing the pressure forces on faces adhe and begf in Figure 15.12, the net rate of work done by pressure in the a direction is

‘ / 9 (up) 1 3(up)

up — up H——— ax) ddz =—————- ax dy dz

9 a ) ‘ 9 a

Similarly, the net rate of work done by the shear stresses in the a direction on faces abed and efgh is

Considering all the forces shown in Figure 15.12, the net rate of work done on the moving fluid element is simply

3(up) 3{utxx) 9(m tvv) B(u tzx)

9a + 9a + By + Bz

The above expression considers only forces in the x direction. When the forces in the у and г directions are also included, similar expressions are obtained (draw some pictures and obtain these expressions yourself). In total, the net rate of work done on the moving fluid element is the sum of all contributions in the x, y, and г directions; this is denoted by C in Equation (15.20) and is given by

Note in Equation (15.21) that the term in large parentheses is simply V • p.

Let us turn our attention to В in Equation (15.20), that is, the net flux of heat into the element. This heat flux is due to (1) volumetric heating such as absorption or emission of radiation and (2) heat transfer across the surface due to temperature gradients (i. e., thermal conduction). Let us treat the volumetric heating the same as was done in Section 2.7; that is, define q as the rate of volumetric heat addition per unit mass. Noting that the mass of the moving fluid element in Figure 15.12 is p dx dy dz, we obtain

Volumetric heating of element = pq dx dy dz [1 5.22]

Thermal conduction was discussed in Section 15.3. In Figure 15.12, the heat trans­ferred by thermal conduction into the moving fluid element across face adhe is

qxdydz, and the heat transferred out of the element across face bcgf is [qx + (dqx/dx) dx I dy dz,. Thus, the net heat transferred in the x direction into the fluid element by thermal conduction is

Taking into account heat transfer in the у and z directions across the other faces in Figure 15.12, we obtain

The term В in Equation (15.20) is the sum of Equations (15.22) and (15.23). Also, re­calling that thermal conduction is proportional to temperature gradient, as exemplified by Equation (15.2), we have

Finally, the term A in Equation (15.20) denotes the time rate of change of energy of the fluid element. In Section 2.7, we stated that the energy of a moving fluid per unit mass is the sum of the internal and kinetic energies, for example, e + Vz/2. Since we are following a moving fluid element, the time rate of change of energy per unit mass is given by the substantial derivative (see Section 2.9). Since the mass of the fluid element is p dx dy dz, we have

D ( v.

A = p — I e + — ) dx dy dz

The final form of the energy equation for a viscous flow is obtained by substituting Equations (15.21), (15.24), and (15.25) into Equation (15.20), obtaining

Equation (15.26) is the general energy equation for unsteady, compressible, three­dimensional, viscous flow. Compare Equation (15.26) with Equation (2.105); the viscous terms are now explicitly spelled out in Equation (15.26). [Note that the body force term in Equation (15.26) has been neglected.] Moreover, the normal and shear

stresses that appear in Equation (15.26) can be expressed in terms of the velocity field via Equations (15.5) to (15.10). This substitution will not be made here because the resulting equation would simply occupy too much space.

Reflect on the viscous flow equations obtained in this chapter—the Navier-Stokes equations given by Equations (15.19a to c) and the energy equation given by Equa­tion (15.26). These equations are obviously more complex than the inviscid flow equations dealt with in previous chapters. This underscores the fact that viscous flows are inherently more difficult to analyze than inviscid flows. This is why, in the study of aerodynamics, the student is first introduced to the concepts associated with inviscid flow. Moreover, this is why we attempt to model a number of practical aerodynamic problems in real life as inviscid flows—simply to allow a reasonable analysis of such flows. However, there are many aerodynamic problems, especially those involving the prediction of drag and flow separation, which must take into ac­count viscous effects. For the analysis of such problems, the basic equations derived in this chapter form a starting point.

Question: What is the form of the continuity equation for a viscous flow? To answer this question, review the derivation of the continuity equation in Section 2.4. You will note that the consideration of the viscous or inviscid nature of the flow never enters the derivation—the continuity equation is simply a statement that mass is conserved, which is independent of whether the flow is viscous or inviscid. Hence, Equation (2.52) holds in general.