CONSERVATION OF ENERGY

GOVERNING EQUATIONS

1.8 CONSERVATION OF ENERGY

The left-hand side of this equation may be further simpli¢ed if the two terms involved are expanded in which the quantityruju_kis considered to be the product ofrukanduj.

r@u_j

@t þu_j@r

@t þu_j @

@xk

ðrukÞ þru_k@u_j

@xk

¼@s_ij

@xi

þrf_j

The second and third terms on the left-hand side of this equation are now seen to sum to zero, since they amount to the continuity Eq. (1.3a) multiplied by the velocityuj. With this simpli¢cation, the equation that expresses con- servation of momentum becomes

r@u_j

@t þruk@u_j

@xk

¼@s_ij

@xi

þrfj ð1:4Þ It is useful to recall that this equation came from an application of Newton’s second law to an element of the £uid. The left-hand side of Eq. (1.4) repre- sents the rate of change of momentum of a unit volume of the £uid (or the inertia force per unit volume). The ¢rst term is the familiar temporal accel- eration term, while the second term is a convective acceleration and accounts for local accelerations (such as around obstacles) even when the

£ow is steady. Note also that this second term is nonlinear, since the velocity appears quadratically. On the right-hand side of Eq. (1.4) are the forces causing the acceleration. The ¢rst of these is due to the gradient of surface shear stresses while the second is due to body forces, such as gravity, which act on the mass of the £uid. A clear understanding of the physical signi¢cance of each of the terms in Eq. (1.4) is essential when approximations to the full governing equations must be made.The surface-stress tensorsijhas not been fully explained up to this point, but it will be investigated in detail in a later section.

apparent di⁄culty may be overcome by considering the instantaneous energy of the £uid to consist of two parts: intrinsic or internal energy and kinetic energy. That is, when applying the ¢rst law of thermodynamics, the energy referred to is considered to be the sum of the internal energy per unit masseand the kinetic energy per unit mass¹₂u

u. In this way the modi¢ed form of the ¢rst law of thermodynamics that will be applied to an element of the £uid states that the rate of change of the total energy (intrinsic plus kinetic) of the £uid as it £ows is equal to the sum of the rate at which work is being done on the £uid by external forces and the rate at which heat is being added by conduction.

With this basic law in mind, we again consider any arbitrary mass of

£uid of volumeVand follow it in a lagrangian frame of reference as it £ows.

The total energy of this mass per unit volume isreþ¹₂ru

u, so that the total energy contained inVwill beR

Vðreþ¹₂ru

^uÞdV. As was established in the previous section, there are two types of external forces that may act on the

£uid mass under consideration. The work done on the £uid by these forces is given by the product of the velocity and the component of each force that is colinear with the velocity. That is, the work done is the scalar product of the velocity vector and the force vector. One type of force that may act on the

£uid is a surface stress whose magnitude per unit area is represented by the vectorP. Then the total work done owing to such forces will beR

su

^PdS,

whereSis the surface area enclosingV. The other type of force that may act on the £uid is a body force whose magnitude per unit mass is denoted by the vectorf. Then the total work done on the £uid due to such forces will be R

Vu

^rfdV. Finally, an expression for the heat added to the £uid is required. Let the vectorqdenote the conductive heat £uxleavingthe control volume.Then the quantity of heat leaving the £uid mass per unit time per unit surface area will beq

^{n, wheren}is the unit outward normal, so that the net amount of heat leaving the £uid per unit time will beR

sq

^ndS.

Having evaluated each of the terms appearing in the physical law that is to be imposed, the statement may now be written down in analytic form. In doing so, it must be borne in mind that the physical law is being applied to a speci¢c, though arbitrarily chosen, mass of £uid so that lagrangian deriva- tives must be employed. In this way, the expression of the statement that the rate of change of total energy is equal to the rate at which work is being done plus the rate at which heat is beingaddedbecomes

D Dt

Z

V

ðreþ¹₂ru

^{uÞdV ¼R}su

^PdSþRVu

^{rfdV R}sq

^ndS

This equation may be converted to one involving eulerian derivatives only by use of Reynolds’ transport theorem, Eq. (1.2), in which the £uid propertyais

here the total energy per unit volumeðreþ¹₂ru

^uÞ. The resulting integro- di¡erential equation is

Z

V

@

@t

reþ1

2ru

^uþ_@^@_x

k

reþ1

2ru

^uuk

dV

¼ Z

s

u

^PdSþZ

V

u

^{rfdV Z}

s

q

^ndS

The next step is to convert the two surface integrals into volume integrals so that the arbitrariness ofVmay be exploited to obtain a di¡erential equation only. Using the fact that the force vectorPis related to the stress tensorsijby the equationPj¼sijni, as was shown in the previous section, the ¢rst surface integral may be converted to a volume integral as follows:

Z

s

u

^PdS¼Z

s

ujsijnidS¼ Z

V

@

@x_iðujsijÞdV

Here use has been made of Gauss’ theorem as documented in Appendix B.

Gauss’ theorem may be applied directly to the heat-£ux term to give Z

s

q

^ndS¼Z

s

qjnjdS¼ Z

V

@q_j

@xj

dV

Since the stress tensorsijhas been brought into the energy equation, it is necessary to use the tensor notation from this point on. Then the expression for conservation of energy becomes

Z

V

@

@tðreþ¹₂rujujÞ þ @

@x_k½ðreþ¹₂rujujÞuk

dV

¼ Z

V

@

@x_iðujs_ijÞdVþ Z

V

u_jrf_jdV Z

V

@qj

@x_j dV

Having converted each term to volume integrals, the conservation equation may be considered to be of the formR

V f gdV ¼0, where the choice ofVis arbitrary.Then the quantity inside the brackets in the integrand must be zero, which results in the following di¡erential equation:

@

@tðreþ¹₂ru_ju_jÞ þ @

@xk

½ðreþ¹₂ru_ju_jÞuk ¼ @

@xi

ðujs_ijÞ þu_jrf_j@q_j

@xj

This equation may be made considerably simpler by using the equations which have already been derived, as will now be demonstrated.The ¢rst term on the left-hand side may be expanded by consideredreand¹₂rujujto be the products (r)(e) andðrÞð¹₂ujujÞ, respectively. Then

@

@tðreþ¹₂rujujÞ ¼r@e

@t þe@r

@t þr@

@tð¹₂ujujÞ þ¹₂ujuj@r

@t

Similarly, the second term on the left-hand side of the basic equation may be expanded by consideringreukto be the product (e)(ruk) and¹₂rujujukto be the productð¹₂ujujÞðrukÞ. Thus,

@

@xk

½ðreþ¹₂ru_ju_jÞuk ¼e @

@xk

ðrukÞ þru_k @e

@xk

þ¹₂ujuj @

@x_kðrukÞ þruk @

@x_kð¹₂ujujÞ

In this last equation, the quantityð@=@xkÞðrukÞ,which appears in the ¢rst and third terms on the right-hand side, may be replaced by@r=@tin view of the continuity Eq. (1.3a). Hence it follows that

@

@x_k½ðreþ¹₂rujujÞuk ¼ e@r

@t þruk @e

@x_k¹₂ujuj@r

@t þruk @

@x_kð¹₂ujujÞ Now when the two components constituting the left-hand side of the basic conservation equation are added, the two terms with minus signs above are canceled by corresponding terms with plus signs to give

@

@tðreþ¹₂ru_ju_jÞ þ @

@xk

½ðreþ¹₂ru_ju_jÞuk

¼r@e

@tþru_k @e

@xk

þr@

@tð¹₂u_ju_jÞ þru_k @

@xk

ð¹₂u_ju_jÞ

¼r@e

@tþru_k @e

@x_kþru_j@uj

@t þru_ju_k@uj

@x_k Then, noting that

@

@x_iðujs_ijÞ ¼u_j@sij

@x_i þs_ij@uj

@x_i

the equation that expresses the conservation of energy becomes r@e

@tþruk @e

@x_kþruj@uj

@t þrujuk@uj

@x_k¼uj@sij

@x_i þsij@uj

@x_iþujrfj@qj

@x_j Now it can be seen that the third and fourth terms on the left-hand side are canceled by the ¢rst and third terms on the right-hand side, since these terms

collectively amount to the product ofujwith the momentum Eq. (1.4). Thus the equation expressing conservation of thermal energy becomes

r@e

@tþru_k @e

@x_k¼s_ij@uj

@x_i@qj

@x_j ð1:5Þ

The terms that were dropped in the last simpli¢cation were the mechanical- energy terms. The equation of conservation of momentum, Eq. (1.4), may be regarded as an equation of balancing forces with j as the free subscript.

Therefore, the scalar product of each force with the velocity vector, or the multiplication byuj, gives the rate of doing work by the mechanical forces, which is the mechanical energy. On the other hand, Eq. (1.5) is a balance of thermal energy, which is what is left when the mechanical energy is sub- tracted from the balance of total energy, and is usually referred to as simply theenergy equation.

As with the equation of momentum conservation, it is instructive to interpret each of the terms appearing in Eq. (1.5) physically. The entire left- hand side represents the rate of change of internal energy, the ¢rst term being the temporal change while the second is due to local convective changes caused by the £uid £owing from one area to another. The entire right-hand side represents the cause of the change in internal energy. The ¢rst of these terms represents the conversion of mechanical energy into thermal energy due to the action of the surface stresses. As will be seen later, part of this con- version is reversible and part is irreversible. The ¢nal term in the equation represents the rate at which heat is being added by conduction from outside.