And from energy equation dh = -VdV … (2) A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 29 CHAPTER TWO ISENTROPIC FLOW 2-1 Isentropic Flow in varying Area ducts For isentropic flow, from continuity dp dV dA p V A And from energy equation dh = -VdV … (2) We now introduce the property relation d Since flow is isentropic (ds = 0). Thus equation (3) becomes dp And from (2), so Or pV We introduce (6) into the differential form of the continuity equation (1) becomes dp dA dp Solve this for dp/ρ and show that + + = 0 … (1) d = d - … (3) p dh = … (4) p -VdV = … (5) p dV = - … (6) p A - pV2 + = 0 … (7)

Recall the definition of sonic velocity: ap ap s A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 30 d dA dp p A p Recall the definition of sonic velocity: ap ap s Since our flow is isentropic, we may drop the subscript and change the partial derivative to an ordinary derivative: dp Or dp = a2 dp … (11) Substituting this expression for dp into equation (8) yields dp V2 dA dp Introduce the definition of Mach number, V a dp dA dp p A p and combine the terms in dρ/ρ to obtain the following relation between density and area changes: dp M2 dA If we now substitute equation (14) into the differential form of the continuity equation (1), we can obtain a relation between velocity and area changes. Show that: dV 1 dA 1 - M2 A Now equation (6) can be divided by V to yield = V2 ( + ) … (8) a2 = ( ) … (9) a2 = … (10) p a2 ( A = + p ) … (12) M = = M2 ( + ) … (13) p = *1 - M2) A … (14) = - ( ) … (15) V

For convenience, we collect the three important relations that will be A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 31 dV dp V pV2 If we equate (15) and (16), we can obtain a relation between pressure and area changes. Show that 1 dA 1 - M2 A Dividing by p , dp pV2 1 dA 1 So dp kV2 1 dA Finally dp kM2 dA p 1 - M2 A For convenience, we collect the three important relations that will be referred to in the analysis that follows: dp kM2 dA p 1 - M2 A dp M2 dA dV 1 dA V 1 - M2 A = - … (16) dp = pV2 ( ) … (17) = ( - M2 ) … (18) p p A = ( - M2 ) … (19) p a2 A = ( ) … (20) = ( ) = - ( )

For subsonic flow, M < 1, then (1 - M2 ) is + ve. A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 32 Let us consider what is happening to fluid properties as it flows through a variable-area duct. For subsonic flow, M < 1, then (1 - M2 ) is + ve. When dA is negative (area is decreasing), then dp is negative (pressure decreases) and dp is negative (density decreases) and dV is positive (velocity increases) and vice versa. For supersonic flow, M > 1, then (1 - M2 ) is - ve. When dA is negative (area is decreasing), then dp is positive (pressure increases) and dp is positive (density increases) and is dV negative (velocity decreases) and vice versa. We summarize the above by saying that as the pressure decreases, the following variations occur: A similar chart could easily be made for the situation where pressure increases, but it is probably more convenient to express the above in an alternative graphical form, as shown in Figure2.1

From this equation we see that: A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 33 Combines equations (13) and (15) to eliminate the term dA/A with the following result: dp dV p V From this equation we see that: = -M2 … (21)

Figure 2.3 The cross section of a nozzle at the smallest flow area is A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi 34 At low Mach numbers, density variations will be quite small. This means that the density is nearly constant (dp � 0) in the low subsonic regime (M � 0. 3 ) and the velocity changes compensate for area changes. At a Mach number equal to unity, we reach a situation where density changes and velocity changes compensate for each another and thus no change in area is required (dA=0 ). At supersonic flow, the density decreases so rapidly that the accompanying velocity change cannot accommodate the flow and thus the area must increase. Consider the operation of devices such as nozzles and diffusers. A nozzle is a device that converts enthalpy (or pressure energy for the case of an incompressible fluid) into kinetic energy. From Figure 2.1 we see that an increase in velocity is accompanied by either an increase or decrease in area, depending on the Mach number. Figures 2.3 and 2.4 show what these devices look like in the subsonic and supersonic flow regimes. Figure 2.3 The cross section of a nozzle at the smallest flow area is called the throat.