Compressible Flow Introduction. Afshin J. Ghajar

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2 36 Comressible Flow Afshin J. Ghajar Oklahoma State University 36. Introduction he Mach Number and Flow Regimes Ideal Gas Relations Isentroic Flow Relations Stagnation State and Proerties Stagnation Proerty Relations Isentroic Flow with Area Changes Normal and Oblique Shock Waves Rayleigh Flow Fanno Flow Introduction Comressible flow is defined as variable density flow; this is in contrast to incomressible flow, where the density is assumed to be constant throughout. he variation in density is mainly caused by variations in ressure and temerature. We sometimes call the study of such fluids in motion gas dynamics. Fluid comressibility is a very imortant consideration in modern engineering alications. Knowledge of comressible fluid flow theory is required in the design and oeration of many devices commonly encountered in engineering ractice. A few imortant examles are the external flow over modern highseed aircrafts; internal flows through rocket, gas turbine, and recirocating engines; flow through natural gas transmission ielines; and flow in high-seed wind tunnels. he variation of fluid density and other fluid roerties for comressible flow gives rise to the occurrence of strange henomena in comressible flow not found in incomressible flow. For examle, with comressible flows we can have fluid deceleration in a convergent duct, fluid temerature decrease with heating, fluid acceleration due to friction, and discontinuous roerty changes in the flow. here are many useful comressible flow references that the reader can consult, such as Anderson [3], Oosthuizen and Carscallen [997], Hodge and Koenig [995], John [984], and Zucrow and Hoffman [976]. he objectives of this chater are to rimarily study comressibility effects by considering the steady, one-dimensional flow of an ideal gas. Although many real flows of engineering interest are more comlex, these restrictions will allow us to concentrate on the effects of basic flow rocesses. Another asect of this chater is the consistent formulation of the equations in a form suitable for comuter solution. he author and his associates have develoed interactive software for the calculation of the roerties of various comressible flows. he first version of the software called COMPROP was develoed to accomany the textbook by Oosthuizen and Carscallen [997]. A more recent version of the software called COMPROP was develoed to accomany the recent textbook by Anderson [3]. For more detail about the develoment of COMPROP and its caabilities see am et al. [] /5/$.+$.5 5 by CRC Press LLC 36-

3 36- he Engineering Handbook, Second Edition 36. he Mach Number and Flow Regimes he single most imortant arameter in the analysis of the comressible fluids is the Mach Number (M), named after the nineteenth century Austrian hysicist Ernst Mach. he Mach number (a dimensionless measure of comressibility) is defined as: M V a (36.) where V is the local flow velocity and a is the local seed of sound (the other common symbol used for the local seed of sound is c ). For an ideal gas the seed of sound is given by [Anderson, 3]: a gr (36.) where g is the secific heat ratio (.4 for air), R is the gas constant ( 87 J/kg K for air), and is the absolute fluid temerature. he seed of sound in a gas deends, therefore, only on the absolute temerature of the gas. For air at standard sea level conditions the seed sound is about 34 m/sec. he Mach number can be used to characterize flow regimes as follows (the numerical values listed are only rough guides): Incomressible flow he Mach number is very small comared to unity (M <.3). For ractical uroses the flow is treated as incomressible. For air at standard sea level conditions this assumtion is good for local flow velocities of about m/sec or less. Subsonic flow he Mach number is less than unity but large enough so that comressible flow effects are resent (.3 < M < ). Sonic flow he Mach number is unity (M ). he significance of the oint at which Mach number is equal to will be demonstrated in ucoming sections. ransonic flow he Mach number is very close to unity (.8 < M <.). Modern aircrafts are mainly owered by gas turbine engines that involve transonic flows. Suersonic flow he Mach number is larger than unity (M > ). For this flow, a shock wave is encountered. here are dramatic differences (hysical and mathematical) between subsonic and suersonic flows, as will be discussed in the future sections. Hyersonic flow he Mach number is larger than five (M > 5). When a sace shuttle reenters the earth s atmoshere, the flow is hyersonic. At very high Mach numbers the flowfield becomes very hot and dissociation and ionization of gases take lace. In these cases the assumtion of an ideal gas is no longer valid, and the flow must be analyzed by the use of kinetic theory of gases rather than continuum mechanics. In the develoment of the equations of the motion of a comressible fluid, much of the analysis will aear in terms of the Mach number Ideal Gas Relations Before we can roceed with the develoment of the equations of the motion of a comressible flow, we need to become familiar with the ideal gas fluid we will be working with. he ideal gas roerty changes can be evaluated from the following equation of state for an ideal gas: rr (36.3) where is the fluid absolute ressure, r is the fluid density, is the fluid absolute temerature, and R is the gas constant ( 87 J/kg K for air). he gas constant, R, reresents a constant for each distinct ideal gas, where

4 Comressible Flow 36-3 R R M (36.4) with this notation, R is the universal gas constant ( 834 J/kg mol K) and M is the molecular weight of the ideal gas ( 8.97 for air). For an ideal gas, internal energy, u, and enthaly, h, are considered to be functions of temerature only, and where the secific heats at constant volume and ressure, c v and c, are also functions of temerature only. he changes in the internal energy and enthaly of an ideal gas are comuted for constant secific heats as: u - u c ( - ) v h - h c ( - ) (36.5) (36.6) For variable secific heats one must integrate du Ú cd v and dh Úcd or use the gas tables [Moran and Shairo, ]. Most modern thermodynamics texts now contain software for evaluating roerties of nonideal gases [Çengel and Boles, ]. From Equation (36.5) and Equation (36.6), we see that changes in internal energy and enthaly are related to the changes in temerature by the values of c v and c. We will now develo useful relations for determining c v and c. From Equation (36.5), Equation (36.6), and the definition of enthaly (h u + v u + R) it can be shown that [Moran and Shairo, ]: c - c R v (36.7) Equation (36.7) indicates that the difference between c v and c is constant for each ideal gas regardless of temerature. Also c > c v. If the secific heat ratio, g, is defined as (the other common symbol used for secific heat ratio is k ) g c cv (36.8) then combining Equation (36.7) and Equation (36.8) leads to R c g g - (36.9) and R c v g - (36.) For air at standard conditions, c 5 J/kg K and c v 78 J/kg K. Equation (36.9) and Equation (36.) will be useful in our subsequent treatment of comressible flow. For comressible flows, changes in the thermodynamic roerty entroy, s, are also imortant. From the first and the second laws of thermodynamics, it can be shown that the change in entroy of an ideal gas with constant secific heat values (c v and c ) can be obtained from [Anderson, 3]: s s c R - ln - ln (36.)

5 36-4 he Engineering Handbook, Second Edition and s s c R v - v ln + ln v (36.) For variable secific heats one must integrate Úcd and Úcd v or use the gas tables [Moran and Shairo, ]. Equation (36.) and Equation (36.) allow the calculation of the change in entroy of an ideal gas between two states with constant secific heat values in terms of either the temerature and ressure, or the temerature and secific volume. Note that entroy is a function of both and, or and v but not temerature alone (unlike internal energy and enthaly) Isentroic Flow Relations An adiabatic flow (no heat transfer) which is frictionless (ideal or reversible) is referred to as isentroic (constant entroy) flow. Such flow does not occur in nature. However, the actual changes exerienced by the large regions of the comressible flow field are often well aroximated by this rocess. his is the case in internal flows such as for nozzles and external flows such as around an airfoil. In the regions adjacent to the nozzle walls or the airfoil surface, a thin boundary layer is formed and isentroic flow aroximation fails. In this region flow is not adiabatic and reversible which causes the entroy to increase in the boundary layer. Imortant relations for an isentroic flow of an ideal gas with constant c v and c can be obtained directly from Equation (36.) and Equation (36.) by setting the left-hand side of these equations to zero (s s ) Ê Ë Á r ˆ Ê Ë Á ˆ r g g/( g -) (36.3) Equation (36.3) relates absolute ressure, density, and absolute temerature for an isentroic rocess, and is very frequently used in the analysis of comressible flows Stagnation State and Proerties Stagnation state is defined as a state that would be reached by a fluid if it were brought to rest isentroically (reversibly and adaibatically) and without work. Figure 36. shows a stagnation oint in comressible flow. he roerties at the stagnation state are refereed to as stagnation roerties (or total roerties). he stagnation state and the stagnation roerties are designated by the subscrit (or t). Stagnation roerties are very useful and are used as a reference state for comressible flows. Consider the steady flow of a fluid through a duct such as a nozzle, diffuser, or some other flow assage where the flow takes lace adiabatically and with no shaft or electrical work. Assuming the fluid exeri- V O ρ,, V,ρ,, FIGURE 36. Stagnation oint.

6 Comressible Flow 36-5 ences little or no change in its elevation and its otential energy, the energy equation between any two oints in the flow for this single-stream steady-flow system reduces to h V V + h + (36.4) In Equation (36.4), if we let one of the oints to be stagnation oint (V ), then h h V + (36.5) where h is the stagnation enthaly and h is the static enthaly of the fluid. Combining Equation (36.4) and Equation (36.5) h h (36.6) hat is, in the absence of any heat and work interactions and any changes in otential energy, the stagnation enthaly of a fluid remains constant during a steady-flow rocess. Flows through nozzles and diffusers usually satisfy these conditions, and any increase (or decrease) in fluid velocity in these devices will create an equivalent decrease (or increase) in the static enthaly of the fluid. Frequently, there is difficulty in understanding the difference between stagnation and static roerties. Stagnation roerties are those roerties exerienced by a fixed observer, the fluid being brought to rest at the observer (V ). Static roerties are those roerties exerienced by an observer moving with the same velocity as the stream. he difference between the static and stagnation roerties is due to the velocity (or kinetic energy) of the flow, see Equation (36.5). We may regard the stagnation conditions as local fluid roerties. Aside from analytical convenience, the definition of the stagnation state is useful exerimentally, since stagnation temerature,, and stagnation ressure,, are relatively easily measured. It is usually much more convenient to measure stagnation temerature than the static temerature Stagnation Proerty Relations Recall the definition of stagnation enthaly given by Equation (36.5), for an ideal gas with constant secific heats, its static and stagnation enthalies can be relaced by c or c, resectively, or c c V + V + c (36.7) In Equation (36.7), the stagnation temerature,, reresents the temerature an ideal gas will attain when it is brought to rest adiabatically. he term V /c corresonds to the temerature rise during such a rocess and is called the dynamic temerature (or imact temerature rise). Note that for low-seed flows, the stagnation and static temeratures are tyically the same. But for high-seed flows, the stagnation temerature (measured by a stationary robe, for examle) may be significantly higher than the static temerature of the fluid.

7 36-6 he Engineering Handbook, Second Edition Introducing Equation (36.) for M and Equation (36.9) for c into Equation (36.7) we obtain Ê g + - ˆ Á M Ë (36.8) Equation (36.8) gives the ratio of the stagnation to static temerature at a oint in a flow as a function of the Mach number at that oint. Equation (36.7) and Equation (36.8) are valid for any adiabatic flow whether thermodynamically reversible or not. hey are, therefore, valid across a shock wave which is irreversible. he ratio of the stagnation ressure to static ressure is obtained by substituting Equation (36.8) into the isentroic relation for ressure given by Equation (36.3) and letting state be the stagnation state: Ê M Ë Á ˆ Ê g + - ˆ Á Ë g/( g - ) g/( g - ) (36.9) he ratio of the stagnation density to static density is obtained by substituting Equation (36.8) into the isentroic relation for density given by Equation (36.3) and letting state be the stagnation state: r r /( g -) /( g -) g Ê Ë Á ˆ Ê + - ˆ Á M Ë (36.) Equation (36.9) and Equation (36.) give the ratios of stagnation to static ressure and density, resectively, at a oint in the flow field as a function of the Mach number at that oint. Equation (36.8), Equation (36.9), and Equation (36.) rovide imortant relations for stagnation roerties and are usually tabulated as a function of Mach number M for g.4 (corresonds to air at standard conditions) in most standard comressible flow (gas dynamics) textbooks [Anderson, 3; John, 984; Zucrow and Hoffman, 976]. Figure 36. was develoed using COMPROP [am et al., ] and shows the variation of these stagnation roerties as a function of Mach number for g.4. It is imortant to note that the local value of stagnation roerty deends only uon the local value of the static roerty and the local Mach number and is indeendent of the flow rocess. Equation (36.8), Equation (36.9), and Equation (36.) may be used to determine these local stagnation values, even for nonisentroic flow, assuming that the local static roerty and local Mach number are known. hese equations also allow us to relate stagnation roerties between any two oints (say oints and ) in the comressible flowfield. For examle, if the actual flow between oints and in the flowfield is reversible and adiabatic (isentroic), then,, and r have constant values at every oint in the FIGURE 36. Stagnation roerty ratios for an ideal gas with g.4. ρ ρ

8 Comressible Flow 36-7 A V ρ V A * FIGURE 36.3 Converging diverging nozzle. flowfield. On the other hand, if the flow is irreversible and adiabatic, then only will remain constant at every oint in the flow. However, for the case where the actual flow is irreversible and nonadiabatic, none of the stagnation roerties stay constant between oints and in the flowfield ( π, π, and r π r ) Isentroic Flow with Area Changes Nozzles are flow assages which accelerate the fluid to higher seeds. Diffusers accomlish the oosite, that is, they are used to decelerate the flow to lower seeds. hese devices are quite common in gas turbines, rockets, and flow metering devices. In incomressible flow (r constant), the volumetric flow rate (roduct of flow velocity, V, and the cross-sectional area, A) is constant. hus, any assage which converges (causes A to decrease in the flow direction) is a nozzle, and any diverging assage (causes A to increase in the flow direction) is a diffuser. In fact, in any subsonic flow (M < ), a converging channel accelerates and a diverging channel decelerates the flow. As will be shown in this section, just the oosite is true in suersonic flow (M > ). For the converging diverging nozzle shown in Figure 36.3, the conservation of mass (continuity) under steady state conditions for this one-dimensional flow can be written as ṁ rva constant (36.) he above equation can be used to relate the mass flow rate ( ṁ ) at different sections of the channel. aking the logarithm of Equation (36.) and then differentiating the resulting equation, we get dr da dv r + A + V (36.) he differential form of the frictionless momentum equation for our steady one-dimensional flow is d + rvdv (36.3) Equation (36.3) could have also been obtained from the steady one-dimensional energy equation for an isentroic flow with no work interactions and no otential energy. Combining Equation (36.3) with Equation (36.) and introducing the definition of Mach number, Equation (36.), one can obtain or da A dv Ê V ˆ dv V - Á - V Ë d d - Ê Á / r V Ë - a ˆ da dv A V ( M - ) (36.4)

9 36-8 he Engineering Handbook, Second Edition Diffuser M < da > dv < Subsonic flow M < M < da < dv > Nozzle Nozzle M < da > dv < Suersonic flow M > M < da < dv > Diffuser FIGURE 36.4 Area and velocity changes for subsonic and suersonic gas flow. ABLE 36. Variation of Flow Proerties in Converging or Diverging Channels ye of Flow Passage M da dm dv d d dr Subsonic converging nozzle < + + Subsonic diverging diffuser < Suersonic converging diffuser > Suersonic diverging nozzle > where in the above equation for an isentroic flow, the seed of sound can be exressed as a d/ dr [John, 984]. Insection of Equation (36.4), without actually solving it, shows a fascinating asect of comressible flow. As mentioned at the beginning of this section, roerty changes are of oosite sign for subsonic and suersonic flow. his is because of the term (M ) in Equation (36.4). here are four combinations of area change and Mach number summarized in Figure he variation of velocity, ressure, temerature, and density in converging diverging channels in both subsonic and suersonic flow is tabulated in able 36.. Equation (36.4) and Figure 36.4 have many ramifications. For M (sonic flow), Equation (36.4) yields da/dv. Mathematically this result suggests that the area associated with sonic flow (M ) is either a minimum or a maximum amount. he minimum in area is the only hysically realistic solution. A convergent divergent channel involves a minimum area (see Figure 36.3). hese results indicate that the sonic condition (M ) can occur in a converging diverging duct at the minimum area location, often referred to as the throat of a converging diverging channel. herefore, for the steady flow of an ideal gas to exand isentroically from subsonic to suersonic seeds, a convergent divergent channel must be used. his is why rocket engines, in order to exand the exhaust gases to high-velocity, suersonic seeds, use a large bell-shaed exhaust nozzle. Conversely, for an ideal gas to comress isentroically from suersonic to subsonic seeds, it must also flow through a convergent divergent channel, with a throat where M occurs. he Mach number area variation in a nozzle can be determined by combining the continuity relation [Equation (36.)] with the ideal gas and isentroic flow relations. For this urose, equate the mass flow rate at any section of the nozzle in Figure 36.3 to the mass flow rate under sonic conditions (at the throat, the flow is sonic and the conditions are denoted by an asterisk and are referred to as critical conditions): or r V A r*v*a* A r* V * r* M* a* Ê r* ˆ */ A * V M a M Á Ê r ˆ Á r r Ë r Ë r / (36.5)

10 Comressible Flow A A * M FIGURE 36.5 Variation of A/A * with Mach number in isentroic flow for g.4. where from Equation (36.), a g R, and at the throat, the area of the throat is A* and M* ( V* a* gr* ), and for an isentroic flow r and is constant throughout the flow. Substituting Equation (36.8) and Equation (36.) into Equation (36.5) and recognizing that the critical temerature ratio (*/ ) and the critical density ratio (r*/r ) is obtained from Equation (36.8) and Equation (36.), resectively with M, we obtain A È Ê + - ˆ Í Á M A* M Î + Ë g g g + ( g -) (36.6) Equation (36.6) and COMPROP [am et al., ] were used to generate the lot of A/A* shown in Figure 36.5 for g.4. Numerical values of A/A* versus M are also usually tabulated alongside stagnation roerties given by Equation (36.8) through Equation (36.) in most standard comressible flow textbooks, see, for examle, Anderson [3]. As can be seen from Figure 36.5, for each value of A/A*, there are two ossible isentroic solutions, one subsonic and the other suersonic. For examle, from Equation (36.6) with g.4 or Figure 36.5, with A/A*, M.3 and also M.. he minimum area (throat) occurs at M. his agrees with the results of Equation (36.4), illustrated in Figure hat is, to accelerate a slow moving fluid to suersonic velocities, a converging diverging nozzle is needed. Under steady state conditions, the mass flow rate through a nozzle can be calculated from Equation (36.) exressed in terms of M and the stagnation roerties and r from Equation (36.8) and Equation (36.9) as: P m AV ( ) R AM R PAM PAM Ê M Ë Á ˆ g g + - Ê g ˆ r g Á R gr Ë ( g + ) - ( g -) (36.7) hus the mass flow rate of a articular fluid through a nozzle is a function of the stagnation roerties of a fluid, the flow area, and the Mach number. he above relationshi is valid at any location along the length of the nozzle. For a secified flow area A and stagnation roerties and r, the maximum mass flow rate through a nozzle can be determined by differentiating Equation (36.7) with resect to M and setting the result equal to zero. It yields M. As discussed above, the only location in a nozzle where M is at the throat (minimum flow area). herefore, the maximum ossible mass flow asses through a nozzle when its throat is at the critical or sonic condition. he nozzle is then said to be choked and can carry no additional mass flow unless the throat is widened. If the throat is constricted further, the mass flow rate through the nozzle must decrease. We can obtain an exression for the maximum mass flow rate by substituting M in Equation (36.7):

11 36- he Engineering Handbook, Second Edition C.V. Normal Shock V, ρ,, V, ρ,, FIGURE 36.6 Stationary normal shock wave. g + * * Ê ˆ ( g -) PA m mmax g Á Ë g + gr (36.8) hus, for isentroic flow of a articular ideal gas through a nozzle, the maximum mass flow rate ossible with a given throat area is fixed by the stagnation ressure and temerature of the inlet flow Normal and Oblique Shock Waves When the flow velocity exceeds the seed of sound (M > ), adjustments in the flow often take lace through abrut discontinuous surfaces called shock waves. his is one of the most interesting and unique henomena that occurs in suersonic flow. A shock wave can be considered as a discontinuity in the roerties of the flowfield. he rocess is irreversible. A shock wave is extremely thin, usually only a few molecular mean free aths thick (for air ª 5 cm). A shock wave is, in general, curved. However, many shock waves that occur in ractical situations are straight, being either at right angles to the flow ath (termed a normal shock) or at an angle to the flow ath (termed an oblique shock). In case of a normal shock, the velocities both ahead (i.e., ustream) of the shock and after (i.e., downstream) the shock are at right angles to the shock wave. However, in the case of an oblique shock there is a change in the flow direction across the shock. Attention will be given first to the changes that occur through a stationary normal shock. Fluid crossing a normal shock exeriences a sudden increase in ressure, temerature, and density, accomanied by a sudden decrease in velocity from a suersonic flow to a subsonic flow. Consider an ideal gas flowing in a duct as shown in Figure For steady state flow through a stationary normal shock, with no direction change, area change (shock is very thin), heat transfer (shock is adiabatic), or work done, the mass (continuity), momentum, and energy equations are: Mass: r V r V (36.9) Momentum: - r V( V -V ) (36.3) V V Energy: + + (36.3) c c hese equations, together with the definition of Mach number, Equation (36.), the equation for seed of sound, Equation (36.), the equation of state, Equation (36.3), and the equation for stagnation temerature, Equation (36.8), will yield two solutions. One solution, which is trivial, states that there is no change and hence no shock wave. he other solution, which corresonds to the change across a stationary normal shock wave, can be exressed in terms of the ustream Mach number:

12 Comressible Flow 36- M ( g - ) M + gm -( g -) (36.3) gm -( g -) ( g + ) (36.33) g/( g -) /( g -) È ( g + ) M È g + Í Î + ( g -) M Í ÎgM -( g -) (36.34) M M g -( g -) [ + ( g -) ] ( g + ) M (36.35) r r V ( g + ) M V + ( g -) M (36.36) he variations of /, r /r, /, /, and M with M as obtained from Equation (36.3) through Equation (36.36) and COMPROP [am et al., ] are lotted in Figure 36.7 for g.4, and they are normally tabulated in standard comressible flow texts in Normal Shock ables, see, for examle, Oosthuizen and Carscallen [997]. From Figure 36.7 we can see that rather large losses of stagnation ressure occur across the normal shock. For an adiabatic rocess (flow across the normal shock is adiabatic but irreversible), the stagnation ressure reresents a measure of available energy of the flow in a given state. A decrease in stagnation ressure, or increase in entroy (s > s ), reresents an energy dissiation or loss of available energy. he increase in entroy across the normal shock can be related to the stagnation ressure ratio across the shock by substituting Equation (36.8) and Equation (36.9) in Equation (36.) and recognizing that stagnation temerature remains constant across the normal shock, [see energy Equation (36.3)]. he result is: s s R - - ln (36.37) ρ ρ 3 M M 4 FIGURE 36.7 Variation of flow roerties across a stationary normal shock wave for g.4.

13 36- he Engineering Handbook, Second Edition From the second law of thermodynamics we must have s > s. In order to minimize the loss of available energy across a normal shock, we need to have a small change in the stagnation ressure across the normal shock. Examination of Figure 36.7 shows that in order for this to haen the Mach number at the ustream of the normal shock (M ) must be near unity. If a lane shock is not erendicular to the flow but inclined at an angle (termed an oblique shock), the shock will cause the fluid assing through it to change direction, in addition to increasing its ressure, temerature, and density, and decreasing its velocity. An oblique shock is illustrated in Figure 36.8; in this case the fluid flow is deflected through an angle d, called the deflection angle. he angle q shown in the figure is referred to as the shock or wave angle and the subscrits n and t indicate directions normal and tangent to the shock, resectively. he conservation of mass, momentum, and energy equations for the indicated control volume, see Figure 36.8, are: Mass: r V r V (36.38) n n Momentum: - r V - r V (normal to the shock) (36.39) n n rv n Vt - V t or Vt V t ( ) (tangent to the shock) (36.4) V n Vn Energy: + + (36.4) c c Since the conservation equations [Equation (36.38), Equation (36.39), and Equation (36.4)] for the oblique shock contain only the normal comonent of the velocity, they are identical to the normal shock conservation equations [Equation (36.9) through Equation (36.3)]. In other words, an oblique shock acts as a normal shock for the comonent normal to the shock, while the tangential velocity remains unchanged, [Equation (36.4)]. his fact ermits the use of normal shock equations to calculate oblique shock arameters. o use normal shock equations for oblique shock calculations, in normal shock equations relace M by M n and M by M n where M n M sinq and M n M sin(q d). Equations for M n and M n are deendent on d and their values cannot be determined until the deflection angle is obtained. However, d is a unique function of M and q. From the geometry of Figure 36.8 and some trigonometric maniulation, the following d q M relationshi can be obtained: cot q( M sin q -) tan d + M ( g + cos q) (36.4) Equation (36.4) secifies d as a unique function of M and q. his relation is vital to the analysis of oblique shocks. he results obtained from Equation (36.4) are usually resented in the form of a grah as shown in Figure Detailed oblique shock grahs (at times referred to as oblique shock charts) may be found in Oosthuizen and Carscallen [997]. hese charts along with the modified form of the normal shock equations are used for determination of the oblique shock roerties. Figure 36.9 is a lot of shock angle versus deflection angle, with the ustream Mach number as a arameter. It is interesting stream line shock V M > θ M < M control V volume δ V n V V t θ V V n (θ-δ) δ V t V t V n <V n V V <V t FIGURE 36.8 Oblique shock wave.

14 Comressible Flow Strong shock solution θ( ) Weak shock solution M. FIGURE 36.9 Oblique shock angle versus deflection angle and ustream Mach number for g.4. to note that in the oblique shock figures, for a given initial Mach number (M ) and a given deflection angle (d), there are either two solutions (solid line and the dashed line in the figure) or none at all. Figure 36.9 shows that for a given M (in this case M.), a maximum deflection angle (d max ) can be found (in this case d max 3 ). his maximum varies from at M to about 45 as M Æ. If d max is exceeded, the shock detaches, that is, moves ahead of the turning surface and becomes curved. For such a case, no solution exists on the oblique shock figures. In case of the attached shock, where two solutions are ossible, the weak shock solution (solid line in Figure 36.9) is most common. A weak shock is the solution farthest removed from the normal shock case Rayleigh Flow δ max 5 δ( ) Ideal gas flow in a constant area duct with heating or cooling (stagnation temerature change) and without friction is referred to as Rayleigh flow. Heat can be added or removed to the gas by heat exchange through the duct walls, by radiative heat transfer, by combustion, or by evaoration and condensation. Although the frictionless (inviscid) assumtion may aear unrealistic, Rayleigh flow is nevertheless useful for the analysis of jet engine combustors and flowing gaseous lasers. In these devices, the heat addition rocess dominates the viscous effects. he conservation equations for the Rayleigh flow combined with the equation of state, the Mach number equation, and the definitions of the stagnation temerature and ressure, leads to the following equations; a comlete analysis is given in Anderson [3]: M M * ( g + ) ( M ) [ + + ( g - g ) ] Ê + g Á * Ë + gm ( + g) M * ( + gm ) ( + g) * ( + gm ) ˆ È + ( g -) M Í Î g + g/( g -) (36.43) (36.44) (36.45) (36.46) For convenience of calculation, in Equation (36.43) through Equation (36.46) sonic flow has been used as a reference condition, where the suerscrit asterisk (*) signifies roerties at M. In this way, the fluid roerties for Rayleigh flow have been resented as a function of a single variable, the local Mach number. Equation (36.43) through Equation (36.46) and COMPROP [am et al., ] were used to

15 36-4 he Engineering Handbook, Second Edition * * * * *. * * * Mach number, M FIGURE 36. Rayleigh flow roerty variations with Mach number for g.4. generate Figure 36. for g.4. hese roerty variations are also normally tabulated in standard comressible flow texts in Rayleigh Flow ables, see, for examle, John [984]. Note that for a given flow no matter what the local flow roerties are, the reference sonic conditions (the starred quantities) are constant values. Given articular inlet conditions to the duct (,, M ) and q (the rate of heat transfer er unit mass of the flowing fluid), we can obtain the duct exit conditions after a given change in stagnation temerature (heating or cooling) as follows: he value of inlet Mach number (M ) fixes the value of / * from Equation (36.43) and thus the value of *, since we know. he exit state / * can then be determined from the following conservation of energy equation for this flow (for a given value of c or g): or c( ) ( V V ) q c( - ) cd q + * * c* (36.47) By the value of / *, then M, /*, /*, and / * are all fixed from Equation (36.43) through Equation (36.46). Referring to Figure 36. and comaring the flow roerties with the / * curve, several interesting facts about Rayleigh flow are evident. Considering the case of heating (increasing / *), we notice that the increase in the stagnation temerature drives the Mach number toward unity for both subsonic and suersonic flow. After the Mach number has reached the sonic condition (M ), any further increase in heating (increase in stagnation temerature) is ossible only if the initial conditions at the inlet of the duct are changed. herefore, a maximum amount of heat can be added to flow in a duct, this maximum is determined by the attainment of Mach. hus, flow in a duct can be chocked by heat addition. In this case flow is referred to as thermally chocked. Another interesting observation from Figure 36. is

16 Comressible Flow 36-5 that the stagnation ressure always decreases for heat addition to the stream. Hence, combustion will cause a loss in stagnation ressure. Conversely, cooling tends to increase the stagnation ressure. 36. Fanno Flow Flow of an ideal gas through a constant-area adiabatic duct with wall friction is referred to as Fanno flow. In effect, this is similar to a Moody-tye ie flow but with large changes in kinetic energy, enthaly, and ressure in the flow. he conservation equations for the Fanno flow combined with the definition of friction factor, equation of state, the Mach number equation, and the definitions of the stagnation temerature and ressure, leads to the following equations; a comlete analysis is given in Anderson [3]: 4f D L Ê - M ˆ ( ) M * Á ln Ë M + g + È g + Í g g Î + ( g -) M (36.48) ( g + ) * + ( g -) M (36.49) È ( g + ) * M Í Î + ( g -) M (36.5) È + ( g -) M Í * M Î ( g + ) ( g + )/[ ( g -)] (36.5) Analogous to our discussion of Rayleigh flow, in Equation (36.48) through Equation (36.5) sonic flow (M ) has been used as a reference condition, where the flow roerties are denoted by *, *, and *. L* is defined as the length of the duct necessary to change the Mach number of the flow from M to unity and f is an average friction factor. Equations (36.48) through Equation (36.5) and COMPROP [am et al., ] were used to generate Figure 36. for g.4. hese equations are also normally tabulated in standard comressible flow texts in Fanno Flow ables, see, for examle, John [984]. Consider a duct of given cross-sectional area and variable length. If the inlet, mass flow rate, and average friction factor are fixed, there is a maximum length of the duct that can transmit the flow. Since the Mach number is unity at the duct exit in that case, the length is designated L* and the flow is said to be friction-chocked. In other words, friction always derives the Mach number toward unity, decelerating a suersonic flow and accelerating a subsonic flow. From Equation (36.48) we can see that at any oint in the duct (say oint ), the variable f L*/D deends only on the Mach number at that oint (M ) and g. Since the diameter (D) is constant and f is assumed constant, then at some other oint (say ) a distance L (L < L*) downstream from oint, we have Ê 4f 4 4 D L * ˆ f D L * f Á Ê Ë Ë Á ˆ - Ê Ë Á D L ˆ (36.5) From Equation (36.5), we can determine M. If in a given situation M was fixed, then Equation (36.5) can be rearranged to determine the length of duct required (L) for Mach number M to change to Mach number M.

17 36-6 he Engineering Handbook, Second Edition * f L * D 4 f L * D * * FIGURE 36. Fanno flow roerty variations with Mach number for g.4. Defining erms Comressible flow Flow in which the fluid density varies. Isentroic flow An adiabatic flow (no heat transfer) which is frictionless (ideal or reversible). For this flow the entroy is constant. Stagnation state A state that would be reached by a fluid if it were brought to rest isentroically (reversibly and adaibatically) and without work. he roerties at the stagnation state are referred to as stagnation roerties (or total roerties). Shock wave A fully develoed comression wave of large amlitude, across which density, ressure, temerature, and article velocity change drastically. A shock wave is, in general, curved. However, many shock waves that occur in ractical situations are straight, being either at right angles to the flow ath (termed a normal shock) or at an angle to the flow ath (termed an oblique shock). Rayleigh flow An idealized tye of gas flow in which heat transfer may occur, satisfying the assumtions that the flow takes lace in constant-area cross-section and is frictionless and steady, that the gas is ideal and has constant secific heat, that the comosition of the gas does not change, and that there are no devices in the system that deliver or receive mechanical work. Fanno flow An ideal flow used to study the flow of fluids in long ies; the flow obeys the same simlifying assumtions as Rayleigh flow excet that the assumtion there is no friction is relaced by the requirement the flow be adiabatic. References Mach number, M Anderson, J. D. 3. Modern Comressible Flow with Historical Persective, 3rd ed. McGraw-Hill, New York. Çengel, Y. A. and Boles, M. A.. hermodynamics: An Engineering Aroach, 4th ed. McGraw-Hill, New York. Hodge, B. K. and Koenig, K Comressible Fluid Dynamics with Personal Comuter Alications, Prentice Hall, Englewood Cliffs, NJ. John, J. E. A Gas Dynamics, nd ed. Prentice Hall, Englewood Cliffs, NJ.

18 Comressible Flow 36-7 Moran, M. J. and Shairo, H. N.. Fundamentals of Engineering hermodynamics, 4th ed. John Wiley & Sons, New York. am, L. M., Ghajar, A. J., and Pau, C. W.. Comressible Flow Software for Proerties Calculations and Airfoil Analysis, Proceedings of the Eight International Conference of Enhancement and Promotion of Comuting Methods in Engineering and Science (EPMESC VIII), eds. L. Shaoei et al. July 5 8, Shanghai, China. Oosthuizen, P. H. and Carscallen, W. E Comressible Fluid Flow, McGraw-Hill, New York. Zucrow, M. J. and Hoffman, J. D Gas Dynamics, Volumes I and II, John Wiley & Sons, New York.