Governing Equations for Quasi-OneDimensional Flow
Recall the one-dimensional flow treated in Chapter 8. There, we considered the flow – field variables to be a function of a only, that is, p = p(x), и = u(x), etc. Strictly speaking, a streamtube for such a flow must be of constant area; that is, the onedimensional flow discussed in Chapter 8 is constant-area flow, as sketched in Figure 10.4a.
In contrast, assume that the area of the streamtube changes as a function of x, that is, A = A(x), as sketched in Figure 10.4b. Strictly speaking, this flow is threedimensional; the flow-field variables are functions of x, y, and z, as can be seen simply by examining Figure 10.4b. In particular, the velocity at the boundary of the streamtube must be tangent to the boundary, and hence it has components in the у and z directions as well as the axial x direction. Flowever, if the area variation is moderate, the components in the у and z directions are small in comparison with the component in the x direction. In such a case, the flow-field variables can be assumed to vary with x only (i. e., the flow can be assumed to be uniform across any cross section at a given x station). Such a flow, where A = A (x), but p = p(x), p = p(x), и = u(x), etc., is defined as quasi-one-dimensionalflow, as sketched in Figure 10.4b. Such flow is the subject of this chapter. We have encountered quasi-one-dimensional flow earlier, in our discussion of incompressible flow through a duct in Section 3.3. Return to Section 3.3, and review the concepts presented there before progressing further.
Although the assumption of quasi-one-dimensional flow is an approximation to the actual flow in a variable-area duct, the integral forms of the conservation equations, namely, continuity [Equation (2.48)], momentum [Equation (2.64)], and energy [Equation (2.95)], can be used to obtain governing equations for quasi-onedimensional flow which are physically consistent, as follows. Consider the control volume given in Figure 10.5. At station 1, the flow across area A i is assumed to be uniform with properties p, p, и , etc. Similarly, at station 2, the flow across area An
is assumed to be uniform with properties p2, p2, u2, etc. The application of the integral form of the continuity equation was made to such a variable-area control volume in Section 3.3. The resulting continuity equation for steady, quasi-one-dimensional flow was obtained as Equation (3.21), which in terms of the nomenclature in Figure 10.5 yields
Consider the integral form of the momentum equation, Equation (2.64). For a steady, inviscid flow with no body forces, this equation becomes
Since Equation (10.2) is a vector equation, let us examine its jc component, given below:
(pV • dS)n = -0> (pdS)
where (pdS)x denotes the x component of the pressure force. Since Equation (10.3) is a scalar equation, we must be careful about the sign of the x components when evaluating the surface integrals. All components pointing to the right in Figure 10.5 are positive, and those pointing to the left are negative. The upper and lower surfaces of the control volume in Figure 10.5 are streamlines; hence, V • dS = 0 along these surfaces. Also, recall that across A , V and dS are in opposite directions; hence, V • dS is negative. Therefore, the integral on the left of Equation (10.3) becomes ~Pi 4] А і + p2uA2. The pressure integral on the right of Equation (10.2), evaluated over the faces A) and A2 of the control volume, becomes — (—pj At + p2A2). (The negative sign in front of p, A is because dS over A! points to the left, which is the
negative direction for the x components.) Evaluated over the upper and lower surface of the control volume, the pressure integral can be expressed as
where dA is simply the x component of the vector dS, that is, the area dS projected on a plane perpendicular to the x axis. The negative sign inside the integral on the left of Equation (10.4) is due to the direction of dS along the upper and lower surfaces; note that dS points in the backward direction along these surfaces, as shown in Figure
10.5. Hence, the x component of p dS is to the left, and therefore appears in our equations as a negative component. [Recall from Section 2.5 that the negative sign outside the pressure integral, that is, outside the integral on the left of Equation (10.4), is always present to account for the physical fact that the pressure force p dS exerted on a control surface always acts in the opposite direction of dS. If you are unsure about this, review the derivation of the momentum equation in Section 2.5. Also, do not let the signs in the above results confuse you; they are all quite logical if you keep track of the direction of the x components.] With the above results, Equation (10.3) becomes
Equation (10.5) is the momentum equation for steady, quasi-one-dimensional flow.
Consider the energy equation given by Equation (2.95). For inviscid, adiabatic, steady flow with no body forces, this equation becomes
Applied to the control volume in Figure 10.5, Equation (10.6) yields
Dividing Equation (10.7) by Equation (10.1), we have
Recall that h = e + pv = e + р/р. Hence, Equation (10.8) becomes
volume at station 1, where the area is A, has properties p, u, and p. In traversing the length dx, where the area changes by dA, the flow properties change by the corresponding amounts dp, dp, and du. Hence, the flow leaving at station 2 has the properties p + dp, u+du, and p + dp, as shown in Figure 10.6. For this case, Equation (10.5) becomes [recognizing that the integral in Equation (10.5) can be replaced by its integrand for the differential volume in Figure 10.6]
pA + pu2A + p dA = (p + dp)(A + dA) + (p + dp)(u + du)2(A + dA) [10.l 5]
In Equation (10.15), all products of differentials, such as dp dA, dp(du)2, are very small and can be ignored. Hence, Equation (10.15) becomes
A dp + Au2 dp + pu2 dA + 2puA du = 0 [10.16]
Expanding the continuity equation, Equation (10.14), and multiplying by u, we have
pu2 dA + puAdu + Au2 dp = 0 [10.17]
Subtracting Equation (10.17) from (10.16), we obtain
which is the differential form of the momentum equation for steady, inviscid, quasi- one-dimensional flow. Equation (10.18) is called Euler’s equation. We have seen it before—as Equation (3.12). In Section 3.2, it was derived from the differential form of the general momentum equation in three dimensions. (Make certain to review that derivation before progressing further.) In Section 3.2, we demonstrated that Equation
(3.12) holds along a streamline in a general three-dimensional flow. Now we see Euler’s equation again, in Equation (10.18), which was derived from the governing equations for quasi-one-dimensional flow.
A differential form of the energy equation follows directly from Equation (10.9), which states that
Differentiating this equation, we have
In summary, Equations (10.14), (10.18), and (10.19) are differential forms of the continuity, momentum, and energy equations, respectively, for a steady, inviscid, adiabatic, quasi-one-dimensional flow. We have obtained them from the algebraic forms of the equations derived earlier, applied essentially to the picture shown in Figure 10.6. Now you might ask the question, Since we spent some effort obtaining partial differential equations for continuity, momentum, and energy in Chapter 2, applicable to a general three-dimensional flow, why would we not simply set d/dy = 0 and 9/3z = 0 in those equations and obtain differential equations applicable to the one-dimensional flow treated in the present chapter? The answer is that we certainly could perform such a reduction, and we would obtain Equations (10.18) and (10.19) directly. [Return to the differential equations, Equations (2.113a) and (2.114), and prove this to yourself.] However, if we take the general continuity equation, Equation
(2.52) , and reduce it to one-dimensional flow, we obtain d(pu) = 0. Comparing this result with Equation (10.14) for quasi-one-dimensional flow, we see an inconsistency. This is another example of the physical inconsistency between the assumption of quasi-one-dimensional flow in a variable-area duct and the three-dimensional flow which actually occurs in such a duct. The result obtained from Equation (2.52), namely, d(pu) = 0, is a truly one-dimensional result, which applies to constant – area flows such as considered in Chapter 8. [Recall in Chapter 8 that the continuity equation was used in the form pu = constant, which is compatible with Equation
(2.52) .] However, once we make the quasi-one-dimensional assumption, that is, that uniform properties hold across a given cross section in a variable-area duct, then Equation (10.14) is the only differential form of the continuity equation which insures mass conservation for such an assumed flow.
Let us now use the differential forms of the governing equations, obtained above, to study some physical characteristics of quasi-one-dimensional flow. Such physical information can be obtained from a particular combination of these equations, as follows. From Equation (10.14),
We wish to obtain an equation which relates the change in velocity du to the change in area dA. Hence, to eliminate dp/p in Equation (10.20), consider Equation (10.18) written as
dp dp dp
p dp p
Keep in mind that we are dealing with inviscid, adiabatic flow. Moreover, for the time being, we are assuming no shock waves in the flow. Hence, the flow is isentropic. In particular, any change in density dp with respect to a change in pressure dp takes place isentropically; that is,
Substituting Equation (10.24) into (10.20), we have
Equation (10.25) is the desired equation which relates dA to du; it is called the area-velocity relation.
Equation (10.25) is very important; study it closely. In the process, recall the standard convention for differentials; for example, a positive value of du connotes an increase in velocity, a negative value of du connotes a decrease in velocity, etc. With this in mind, Equation (10.25) tells us the following information:
1. For 0 < M < 1 (subsonic flow), the quantity in parentheses in Equation (10.25) is negative. Hence, an increase in velocity (positive du) is associated with a decrease in area (negative dA). Likewise, a decrease in velocity (negative du) is associated with an increase in area (positive dA). Clearly, for a subsonic compressible flow, to increase the velocity, we must have a convergent duct, and to decrease the velocity, we must have a divergent duct. These results are illustrated at the top of Figure 10.7. Also, these results are similar to the familiar trends for incompressible flow studied in Section 3.3. Once again we see that subsonic compressible flow is qualitatively (but not quantitatively) similar to incompressible flow.
2. For M > 1 (supersonic flow), the quantity in parentheses in Equation (10.25) is positive. Hence, an increase in velocity (positive du) is associated with an increase in area (positive dA). Likewise, a decrease in velocity (negative du) is associated with a decrease in area (negative dA). For a supersonic flow, to
Figure 10*7 Compressible flow in converging and diverging ducts.
increase the velocity, we must have a divergent duct, and to decrease the velocity, we must have a convergent duct. These results are illustrated at the bottom of Figure 10.7; they are the direct opposite of the trends for subsonic flow.
3. ForM = 1 (sonic flow), Equation (10.25) shows thatrM = 0 even though a finite du exists. Mathematically, this corresponds to a local maximum or minimum in the area distribution. Physically, it corresponds to a minimum area, as discussed below.
Imagine that we want to take a gas at rest and isentropically expand it to supersonic speeds. The above results show that we must first accelerate the gas subsonically in a convergent duct. However, as soon as sonic conditions are achieved, we must further expand the gas to supersonic speeds by diverging the duct. Hence, a nozzle designed to achieve supersonic flow at its exit is a convergent-divergent duct, as sketched at the top of Figure 10.8. The minimum area of the duct is called the throat. Whenever an isentropic flow expands from subsonic to supersonic speeds, the flow must pass through a throat; moreover, in such a case, M = 1 at the throat. The converse is also true; if we wish to take a supersonic flow and slow it down isentropically to subsonic speeds, we must first decelerate the gas in a convergent duct, and then as soon as sonic flow is obtained, we must further decelerate it to subsonic speeds in a divergent duct. Here, the convergent-divergent duct at the bottom of Figure 10.8 is operating as a diffuser. Note that whenever an isentropic flow is slowed from supersonic to subsonic speeds, the flow must pass through a throat; moreover, in such a case, M = 1 at the throat.
As a final note on Equation (10.25), consider the case when M = 0. Then we have dA/A — —du/u. which integrates to Au = constant. This is the familiar continuity equation for incompressible flow in ducts as derived in Section 3.3 and as given by Equation (3.22).