Chapter 15

Compressible Flow

Abstract

A flow is considered compressible when changes in fluid momentum produce important variations in fluid pressure and density, and the fluid’s thermodynamic characteristics play a direct role in the flow’s development. When the pressure variations are small enough, linear acoustic theory may apply. However, larger finite-amplitude pressure disturbances produce nonlinear effects. Compressible flows in ducts and nozzles may reach limiting mass-flow-rate values that cannot be exceeded even when the downstream pressure is decreased. Here friction and heat addition or extraction may have unexpected consequences. Supersonic flows (Mach number > 1) may also contain shock waves that induce nearly discontinuous changes in the flow’s state. In supersonic flow, downstream pressure disturbances cannot propagate upstream and oblique expansion or compression waves emanate from locations where the flow changes direction. Thus, supersonic flows are often easier to analyze than subsonic flows because the various influences of geometric features of the flow’s boundaries need not be assessed simultaneously as would be the case in subsonic flow.

Keywords

Compressible flow; Supersonic flow; Gas dynamics; Mach number; Perfect gas; Isentropic flow; Acoustics; One-dimensional flow; Shock wave; Jump conditions; Shock structure; Nozzle flow; Friction; Heating; Choked flow; Unsteady flow; Mach angle; Prandtl-Meyer expansion; Supersonic aerodynamics
Chapter Objectives

15.1. Introduction

Several startling and fascinating phenomena, which defy intuition and expectations developed from incompressible flows, arise in compressible flows and are described in this chapter. Near discontinuities (shock waves) may appear within the flow. An increase (or decrease) in flow area may accelerate (or decelerate) a uniform stream. Friction may increase a flow's speed. And, heat addition may lower a flow's temperature. These phenomena are therefore worthy of our attention because they either have no counterpart or act oppositely in low-speed flows. Except for the treatment of friction in constant-area ducts in Section 15.6, the material presented here is limited to that of frictionless flows outside boundary layers. In spite of this simplification, the results presented here have a great deal of practical value because boundary layers are especially thin in high-speed flows. Gravitational effects, which are of minor importance in compressible flows, are also neglected.
As discussed in Section 4.11, the importance of compressibility for moving fluids can be assessed by considering the Mach number M, defined as:

MU/c,

image (4.111)

where U is a representative flow speed, and c is the speed of sound, a thermodynamic quantity defined by:

c2(p/ρ)s.

image (1.25)

Here, the subscript s signifies that the partial derivative is taken at constant entropy. In particular, the dimensionless scaling (4.109) of the compressible-flow continuity equation for isentropic conditions leads to:

·u=M2(ρ0ρ)DDt(pp0ρ0U2),

image (4.110)

where ρ0 and p0 are appropriately chosen reference values for density and pressure. In (4.110), the pressure is scaled by fluid inertia parameters as is appropriate for primarily frictionless high-speed flow. In engineering practice, the incompressible flow assumption is presumed valid if M < 0.3, but not at higher Mach numbers. Equation (4.110) suggests that M = 0.3 corresponds to ∼10% departure from perfectly incompressible flow behavior when the remainder of the right side of (4.110) is of order unity.
Although the significance of the ratio U/c was known for a long time, the Swiss aerodynamist Jacob Ackeret introduced the term Mach number, just as the term Reynolds number was introduced by Sommerfeld many years after Reynolds’ experiments. The name of the Austrian physicist Ernst Mach (1836–1916) was chosen because of his pioneering studies on supersonic motion and his invention of the so-called Schlieren method for optical visualization of flows involving density changes; see von Karman (1954, p. 106). (Mach distinguished himself equally well in philosophy. Einstein acknowledged that his own thoughts on relativity were influenced by “Mach’s principle,” which states that properties of space had no independent existence but are determined by the mass distribution within it. Strangely, Mach never accepted either the theory of relativity or the atomic structure of matter.)
Using the Mach number, compressible flows can be nominally classified as follows:
(i) Incompressible flow: M = 0. Fluid density does not vary with pressure in the flow field. The flowing fluid may be a compressible gas but its density may be regarded as constant.
(ii) Subsonic flow: 0 < M < 1. The Mach number does not exceed unity anywhere in the flow field. Shock waves do not appear in the flow. In engineering practice, subsonic flows for which M < 0.3 are often treated as being incompressible.
(iii) Transonic flow: The Mach number in the flow lies in the range 0.8–1.2. Shock waves may appear. Analysis of transonic flows is difficult because the governing equations are inherently nonlinear, and also because a separation of the inviscid and viscous aspects of the flow is often impossible. (The word “transonic” was invented by von Karman and Hugh Dryden, although the latter argued in favor of spelling it “transsonic.” von Karman [1954] stated, “I first introduced the term in a report to the U.S. Air Force. I am not sure whether the general who read the word knew what it meant, but his answer contained the word, so it seemed to be officially accepted” [p. 116].)
(iv) Supersonic flow: M > 1. Shock waves are generally present. In many ways analysis of a flow that is supersonic everywhere is easier than analysis of a subsonic or incompressible flow as we shall see. This is because information propagates along certain directions, called characteristics, and a determination of these directions greatly facilitates the computation of the flow field.
(v) Hypersonic flow: M > 3. Very high flow speeds combined with friction or shock waves may lead to sufficiently large increases in a fluid's temperature so that molecular dissociation and other chemical effects occur.

Perfect Gas Thermodynamic Relations

As density changes are accompanied by temperature changes, thermodynamic principles are constantly used throughout this chapter. Most of the necessary concepts and relations have been summarized in Sections 1.81.9, which may be reviewed before proceeding further. The most frequently used relations, valid for a perfect gas with constant specific heats, are listed here for quick reference:

Internalenergy:e=cvT,Enthalpy:h=cpT,Thermalequationofstate:p=ρRT,

image (15.1a– c)

Specificheats:cv=Rγ1,cp=γRγ1,cpcv=R,γ=cp/cv,

image (15.1d– g)

SpeedofSound:c=γRT=γp/ρ,andEntropychange:s2s1=cpln(T2T1)Rln(p2p1)=cvln(T2T1)Rln(ρ2ρ1).

image (15.1h, 15.1i)

Equation (15.1h) implies that c is larger in monotonic and low-molecular weight gases (where γ and R are larger), and that it increases with increasing temperature. An isentropic process involving a perfect gas between states 1 and 2 obeys the following relations:

p2p1=(ρ2ρ1)γ,andT2T1=(ρ2ρ1)γ1=(p2p1)(γ-1)/γ.

image (15.2a, 15.2b)

Some important properties of air at ordinary temperatures and pressures are:

R=287m2/(s2K),cv=717m2/(s2K),cp=1004m2/(s2K),andγ=1.40;

image (15.3a– d)

these values are useful for solution of the exercises at the end of this chapter.
Example 15.1
Which of the following are compressible flows? a) A weather balloon rises at 5 m/s from sea level to an altitude of more than 15 km. b) Water flows through the nozzle of a water jet cutter and the gage pressure drops from 100 MPa to zero. c) The piston of an internal combustion engine moves at 15 m/s and compresses air and gaseous fuel from 40 kPa to 1300 kPa. d) Liquid nitrogen at 100 kPa with density 807 kg/m3 evaporates from a stationary dewar in a 1 m/s airflow to become nitrogen gas with density 1.16 kg/m3. e) A 1.0 MPa compression wave travels at 1.0 km/s into air at 15°C and 1.0 atm.
Solution

Δρ/ρΔp/ρc2100MPa/(1000kgm3·(1480ms1)2)=0.046.

image

 
This flow might qualify as compressible depending on the level of accuracy that is sought. For c), the pressure and density of the combustion gases rise by factors of more than 30 and 10, respectively. However, like the weather balloon, such changes primarily depend on piston location and are largely independent of the low-Mach number piston speed, so this is not a compressible flow. For d), the nitrogen accelerates to 1 m/s and its density drops by a factor ∼700, but there are no significant pressure variations, so this flow is not compressible. For e), the wave speed is supersonic and the increase in air pressure (a factor of ∼10) and density (a factor of ∼4) across the wave are significant. This is a compressible flow.