Module 4 : Lecture 1 COMPRESSIBLE FLOWS (Fundamental Aspects: Part - I)

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1 Module 4 : Lecture COMPRESSIBLE FLOWS (Fundamental Asects: Part - I) Overview In general, the liquids and gases are the states of a matter that comes under the same category as fluids. The incomressible flows are mainly deals with the cases of constant density. Also, when the variation of density in the flow domain is negligible, then the flow can be treated as incomressible. Invariably, it is true for liquids because the density of liquid decreases slightly with temerature and moderately with ressure over a broad range of oerating conditions. Hence, the liquids are considered as incomressible. On the contrary, the comressible flows are routinely defined as variable density flows. Thus, it is alicable only for gases where they may be considered as incomressible/comressible, deending on the conditions of oeration. During the flow of gases under certain conditions, the density changes are so small that the assumtion of constant density can be made with reasonable accuracy and in few other cases the density changes of the gases are very much significant (e.g. high seed flows). Due to the dual nature of gases, they need secial attention and the broad area of in the study of motion of comressible flows is dealt searately in the subject of gas dynamics. Many engineering tasks require the comressible flow alications tyically in the design of a building/tower to withstand winds, high seed flow of air over cars/trains/airlanes etc. Thus, gas dynamics is the study of fluid flows where the comressibility and the temerature changes become imortant. Here, the entire flow field is dominated by Mach waves and shock waves when the flow seed becomes suersonic. Most of the flow roerties change across these waves from one state to other. In addition to the basic fluid dynamics, the knowledge of thermodynamics and chemical kinetics is also essential to the study of gas dynamics. Joint initiative of IITs and IISc Funded by MHRD Page of 57

2 Thermodynamic Asects of Gases In high seed flows, the kinetic energy er unit mass ( V ) is very large which is substantial enough to strongly interact with the other roerties of the flow. Since the science of energy and entroy is the thermodynamics, it is essential to study the thermodynamic asects of gases under the conditions comressible high seed flows. Perfect gas: A gas is considered as a collection of articles (molecules, atoms, ions, electrons etc.) that are in random motion under certain intermolecular forces. These forces vary with distances and thus influence the microscoic behavior of the gases. However, the thermodynamic asect mainly deals with the global nature of the gases. Over wide ranges of ressures and temeratures in the comressible flow fields, it is seen that the average distance between the molecules is more than the molecular diameters (about 0-times). So, all the flow roerties may be treated as macroscoic in nature. A erfect gas follows the relation of ressure, density and temerature in the form of the fundamental equation. R = ρrt; R = (4..) M Here, M is the molecular weight of the gas, R is the gas constant that varies from gas to gas and R ( = 834 J kg.k) is the universal gas constant. In a calorically erfect gas, the other imortant thermodynamic roerties relations are written as follows; h e c = ; cv = ; c cv = R T T c γ R R c = ; cv = ; γ = γ γ c v v (4..) In Eq. (4..), the arameters are secific heat at constant ressure ( c ), secific heat at constant volume ( c v ), secific heat ratio ( γ ), secific enthaly ( h ) and secific internal energy ( e ). Joint initiative of IITs and IISc Funded by MHRD Page of 57

3 First law of thermodynamics: A system is a fixed mass of gas searated from the surroundings by a flexible boundary. The heat added ( q ) and work done ( w ) on the system can cause change in energy. Since, the system is stationary, the change in internal energy. By definition of first law, we write, δq + δw = de (4..3) For a given de, there are infinite number of different ways by which heat can be added and work done on the system. Primarily, the three common tyes of rocesses are, adiabatic (no addition of heat), reversible (no dissiative henomena) and isentroic (i.e. reversible and adiabatic). Second law of thermodynamics: In order to ascertain the direction of a thermodynamic rocess, a new state variable is defined as entroy ( s ). The change in entroy during any incremental rocess ( ds) is equal to the actual heat added divided by the temerature ( dq T ), lus a contribution from the irreversible dissiative henomena ( dsirrev ) occurring within the system. δ q ds = + dsirrev (4..4) T Since, the dissiative henomena always increases the entroy, it follows that δ q ds ; ds 0 ( Adiabatic rocess) (4..5) T Eqs. (4..4 & 4..5) are the different forms of second law of thermodynamics. In order to calculate the change in entroy of a thermodynamic rocess, two fundamental relations are used for a calorically erfect gas by combining both the laws of thermodynamics; T s s = c ln Rln T T ρ s s = cv ln + Rln T ρ (4..6) Joint initiative of IITs and IISc Funded by MHRD Page 3 of 57

4 An isentroic rocess is the one for which the entroy is constant and the rocess is reversible and adiabatic. The isentroic relation is given by the following relation; γ γ ( γ ) ρ T = = ρ T (4..7) Imortant Proerties of Comressible Flows The simle definition of comressible flow is the variable density flows. In general, the density of gases can vary either by changes in ressure and temerature. In fact, all the high seed flows are associated with significant ressure changes. So, let us recall the following fluid roerties imortant for comressible flows; Bulk modulus ( E v ) : It is the roerty of that fluid that reresents the variation of density ( ρ ) with ressure ( ) at constant temerature ( T ). Mathematically, it is reresented as, E v ρ = v = ρ v T T T (4..8) In terms of finite changes, it is aroximated as, E v ( v v) ( ρ ρ) = = T T (4..9) Coefficient of volume exansion ( β ) : It is the roerty of that fluid that reresents the variation of density with temerature at constant ressure. Mathematically, it is reresented as, v ρ β = = v T ρ T (4..0) In terms of finite changes, it is aroximated as, ( v v) ( ρ ρ) β = = T T (4..) Joint initiative of IITs and IISc Funded by MHRD Page 4 of 57

5 Comressibility( κ ) : It is defined as the fractional change in the density of the fluid element er unit change in ressure. One can write the exression for κ as follows; d ρ κ = d ρ = ρκd ρ d (4..) In order to be more recise, the comression rocess for a gas involves increase in temerature deending on the amount of heat added or taken away from the gas. If the temerature of the gas remains constant, the definition is refined as isothermal comressibility( κ T ). On the other hand, when no heat is added/taken away from the gases and in the absence of any dissiative mechanisms, the comression takes lace isentroically. It is then, called as isentroic comressibility( κ s ). κ T ρ ρ ; κ = s = ρ ρ T s (4..3) Being the roerty of a fluid, the gases have high values of comressibility 5 ( κ T 0 m N for air at atm) = while liquids have low values of comressibility 0 much less than that of gases ( κ T 5 0 m N for water at atm) =. From the basic definition (Eq. 4..), it is seen that whenever a fluid exeriences a change in ressure d, there will be a corresonding change in dρ. Normally, high seed flows involve large ressure gradient. For a given change in d, the resulting change in density will be small for liquids (low values of κ ) and more for gases (high values of κ ). Therefore, for the flow of liquids, the relative large ressure gradients can create much high velocities without much change in densities. Thus, the liquids are treated to be incomressible. On the other hand, for the flow of gases, the moderate to strong ressure gradient leads to substantial changes in the density (Eq.4..) and at the same time, it can create large velocity changes. Such flows are defined as comressible flows where the density is a variable roerty and the fractional change in density ( dρ ρ ) is too large to be ignored. Joint initiative of IITs and IISc Funded by MHRD Page 5 of 57

6 Fundamental Equations for Comressible Flow Consider a comressible flow assing through a rectangular control volume as shown in Fig The flow is one-dimensional and the roerties change as a function of x, from the region to and they are velocity( u ), ressure ( ), temerature ( T ), density ( ρ ) and internal energy( e ). The following assumtions are made to derive the fundamental equations; Flow is uniform over left and right side of control volume. Both sides have equal area ( A ), erendicular to the flow. Flow is inviscid, steady and nobody forces are resent. No heat and work interaction takes lace to/from the control volume. Let us aly mass, momentum and energy equations for the one dimensional flow as shown in Fig Conservation of Mass: ρua+ ρ ua= 0 ρu = ρ u (4..4) Conservation of Momentum: ρ ( uau ) + ρ ( uau ) = ( A+ A) + ρu = + ρ u (4..5) Steady Flow Energy Conservation: u u u u + + = = + (4..6) ρ e e h h ρ Joint initiative of IITs and IISc Funded by MHRD Page 6 of 57

7 Here, the enthaly gas. h = e+ ρ is defined as another thermodynamic roerty of the Fig. 4..: Schematic reresentation of one-dimensional flow. Joint initiative of IITs and IISc Funded by MHRD Page 7 of 57

8 Module 4 : Lecture COMPRESSIBLE FLOWS (Fundamental Asects: Part - II) Wave Proagation in a Comressible Media Consider a gas confined in a long tube with iston as shown in Fig. 4..(a). The gas may be assumed to have infinite number of layers and initially, the system is in equilibrium. In other words, the last layer does not feel the resence of iston. Now, the iston is given a very small ush to the right. So, the layer of gas adjacent to the iston iles u and is comressed while the reminder of the gas remains unaffected. With due course of time, the comression wave moves downstream and the information is roagated. Eventually, all the gas layers feel the iston movement. If the ressure ulse alied to the gas is small, the wave is called as sound wave and the resultant comression wave moves at the seed of sound. However, if the fluid is treated as incomressible, the change in density is not allowed. So, there will be no iling of fluid due to instantaneous change and the disturbance is felt at all other locations at the same time. So, the seed of sound deends on the fluid roerty i.e. comressibility. The lower is its value; more will be the seed of sound. In an ideal incomressible medium of fluid, the seed of sound is infinite. For instance, sound travels about 4.3-times faster in water (484 m/s) and 5-times as fast in iron (50 m/s) than air at 0ºC. Let us analyze the iston dynamics as shown in Fig. 4..(a). If the iston moves at steady velocity dv, the comression wave moves at seed of sound a into the stationary gas. This infinitesimal disturbance creates increase in ressure and density next to the iston and in front of the wave. The same effect can be observed by keeing the wave stationary through dynamic transformation as shown in Fig. 4.. (b). Now all basic one dimensional comressible flow equations can be alied for a very small control enclosing the stationary wave. Joint initiative of IITs and IISc Funded by MHRD Page 8 of 57

9 Continuity equation: Mass flow rate ( m ) is conserved across the stationary wave. a m = ρa A = ( ρ+ d ρ)( a dv ) A dv = d ρ (4..) ρ Momentum equation: As long as the comression wave is thin, the shear forces on the control volume are negligibly small comared to the ressure force. The momentum balance across the control volume leads to the following equation; + d A A = m a m a dv dv = d (4..) ρa ( ) ( ) Fig. 4..: Proagation of ressure wave in a comressible medium: (a) Moving wave; (b) Stationary wave. Joint initiative of IITs and IISc Funded by MHRD Page 9 of 57

10 Energy equation: Since the comression wave is thin, and the motion is very raid, the heat transfer between the control volume and the surroundings may be neglected and the thermodynamic rocess can be treated as adiabatic. Steady flow energy equation can be used for energy balance across the wave. ( a dv ) a h + = ( h + dh) + dv = dh a (4..3) Entroy equation: In order to decide the direction of thermodynamic rocess, one can aly T ds relation along with Eqs (4.. & 4..3) across the comression wave. d T ds = dh = 0 ds = 0 (4..4) ρ Thus, the flow is isentroic across the comression wave and this comression wave can now be called as sound wave. The seed of the sound wave can be comuted by equating Eqs.(4.. & 4..). a d a = = = ρ ρa dρ ρ s (4..5) Further simlification of Eq. (4..5) is ossible by evaluating the differential with the use of isenroic equation. constant ln γ ln ρ constant γ ρ = = (4..6) Differentiate Eq. (4..6) and aly erfect gas equation ( ρ RT) exression for seed of sound. is obtained as below; = to obtain the γ γ = a = = ρ ρ ρ s γ RT (4..7) Joint initiative of IITs and IISc Funded by MHRD Page 0 of 57

11 Mach number It may be seen that the seed of sound is the thermodynamic roerty that varies from oint to oint. When there is a large relative seed between a body and the comressible fluid surrounds it, then the comressibility of the fluid greatly influences the flow roerties. Ratio of the local seed ( V ) of the gas to the seed of sound ( a ) is called as local Mach number ( M ). There are few hysical meanings for Mach number; V V M = = (4..8) a γ RT (a) It shows the comressibility effect for a fluid i.e. M < 0.3 imlies that fluid is incomressible. (b) It can be shown that Mach number is roortional to the ratio of kinetic to internal energy. ( V ) v ( γ ) ( γ ) V ( γ ) ( ) V V γ γ = = = = M e c T RT a (4..9) (c) It is a measure of directed motion of a gas comared to the random thermal motion of the molecules. M V directed kinetic energy = = (4..0) a random kinetic energy Joint initiative of IITs and IISc Funded by MHRD Page of 57

12 Comressible Flow Regimes In order to illustrate the flow regimes in a comressible medium, let us consider the flow over an aerodynamic body (Fig. 4..). The flow is uniform far away from the body with free stream velocity ( V ) while the seed of sound in the uniform stream is a. Then, the free stream Mach number becomes M ( V a ) =. The streamlines can be drawn as the flow asses over the body and the local Mach number can also vary along the streamlines. Let us consider the following distinct flow regimes commonly dealt with in comressible medium. Subsonic flow: It is a case in which an airfoil is laced in a free stream flow and the local Mach number is less than unity everywhere in the flow field (Fig. 4..-a). The flow is characterized by smooth streamlines with continuous varying roerties. Initially, the streamlines are straight in the free stream, but begin to deflect as they aroach the body. The flow exands as it assed over the airfoil and the local Mach number on the to surface of the body is more than the free stream value. Moreover, the local Mach number ( M ) in the surface of the airfoil remains always less than, when the free stream Mach number ( M ) is sufficiently less than. This regime is defined as subsonic flow which falls in the range of free stream Mach number less than 0.8 i.e. M 0.8. Transonic flow: If the free stream Mach number increases but remains in the subsonic range close to, then the flow exansion over the air foil leads to suersonic region locally on its surface. Thus, the entire regions on the surface are considered as mixed flow in which the local Mach number is either less or more than and thus called as sonic ockets (Fig. 4..-b). The henomena of sonic ocket is initiated as soon as the local Mach number reaches and subsequently terminates in the downstream with a shock wave across which there is discontinuous and sudden change in flow roerties. If the free stream Mach number is slightly above unity (Fig. 4..-c), the shock attern will move towards the trailing edge and a second shock wave aears in the leading edge which is called as bow shock. In front of this bow shock, the streamlines are straight and arallel with a uniform suersonic free stream Mach number. After assing through the bow shock, the flow becomes subsonic close to the free stream value. Eventually, it further exands over the airfoil Joint initiative of IITs and IISc Funded by MHRD Page of 57

13 surface to suersonic values and finally terminates with trailing edge shock in the downstream. The mixed flow atterns sketched in Figs. 4.. (b & c), is defined as the transonic regime. Fig. 4..: Illustration of comressible flow regime: (a) subsonic flow; (b & c) transonic flow; (d) suersonic flow; (d) hyersonic flow. Suersonic flow: In a flow field, if the Mach number is more than everywhere in the domain, then it defined as suersonic flow. In order to minimize the drag, all aerodynamic bodies in a suersonic flow, are generally considered to be shar edged ti. Here, the flow field is characterized by straight, oblique shock as shown in Fig. 4..(d). The stream lines ahead of the shock the streamlines are straight, arallel and horizontal. Behind the oblique shock, the streamlines remain straight and arallel but take the direction of wedge surface. The flow is suersonic both ustream and downstream of the oblique shock. However, in some excetional strong oblique shocks, the flow in the downstream may be subsonic. Hyersonic flow: When the free stream Mach number is increased to higher suersonic seeds, the oblique shock moves closer to the body surface (Fig. 4..-e). At the same time, the ressure, temerature and density across the shock increase exlosively. So, the flow field between the shock and body becomes hot enough to ionize the gas. These effects of thin shock layer, hot and chemically reacting gases and many other comlicated flow features are the characteristics of hyersonic flow. In reality, these secial characteristics associated with hyersonic flows aear gradually as the free stream Mach numbers is increased beyond 5. Joint initiative of IITs and IISc Funded by MHRD Page 3 of 57

14 As a rule of thumb, the comressible flow regimes are classified as below; ( ) ( ) ( ) ( ) ( ) M < 0.3 incomressible flow M < subsonic flow 0.8 < M <. transonic flow M > suersonic flow M > 5and above hyersonic flow Rarefied and Free Molecular Flow: In general, a gas is comosed of large number of discrete atoms and molecules and all move in a random fashion with frequent collisions. However, all the fundamental equations are based on overall macroscoic behavior where the continuum assumtion is valid. If the mean distance between atoms/molecules between the collisions is large enough to be comarable in same order of magnitude as that of characteristics dimension of the flow, then it is said to be low density/rarefied flow. Under extreme situations, the mean free ath is much larger than the characteristic dimension of the flow. Such flows are defined as free molecular flows. These are the secial cases occurring in flight at very high altitudes (beyond 00 km) and some laboratory devices such as electron beams. Joint initiative of IITs and IISc Funded by MHRD Page 4 of 57

15 Module 4 : Lecture 3 COMPRESSIBLE FLOWS (Isentroic and Characteristics States) An isentroic rocess rovides the useful standard for comaring various tyes of flow with that of an idealized one. Essentially, it is the rocess where all tyes of frictional effects are neglected and no heat addition takes lace. Thus, the rocess is considered as reversible and adiabatic. With this useful assumtion, many fundamental relations are obtained and some of them are discussed here. Stagnation/Total Conditions When a moving fluid is decelerated isentroically to reach zero seed, then the thermodynamic state is referred to as stagnation/total condition/state. For examle, a gas contained in a high ressure cylinder has no velocity and the thermodynamic state is known as stagnation/total condition (Fig a). In a real flow field, if the actual conditions of ressure ( ), temerature ( T ), density ( ρ ), enthaly( h ), internal energy ( e ), entroy ( s) etc. are referred to as static conditions while the associated stagnation arameters are denoted as 0, T 0, ρ 0, h 0, e 0 and s 0, resectively. The stagnation state is fixed by using second law of thermodynamics where s = s0 as reresented in enthaly-entroy diagram called as the Mollier diagram (Fig b). Fig 4.3.: (a) Schematic reresentation of stagnation condition; (b) Mollier diagram. Joint initiative of IITs and IISc Funded by MHRD Page 5 of 57

16 The simlified form of energy equation for steady, one-dimensional flow with no heat addition, across two regions and of a control volume is given by, u u h+ = h + (4.3.) For a calorically erfect gas, relacing, h= ct, so the Eq. (4.3.) becomes, u u ct + = ct + (4.3.) If the region refers to any arbitrary real state in the flow field and the region refers to stagnation condition, then Eq. (4.3.) becomes, It can be solved for ( T0 T ) as, u ct + = ct 0 (4.3.3) T0 u u u = + = + = + T c T γrt γ a γ T0 γ u or, ( ) ( ) γ = + = + M T a (4.3.4) For an isentroic rocess, the thermodynamic relation is given by, ρ T = = ρ T γ γ γ (4.3.5) From, Eqs (4.3.4) and (4.3.5), the following relations may be obtained for stagnation ressure and density. 0 γ = + M ρ0 γ = + M ρ γ γ γ (4.3.6) In general, if the flow field is isentroic throughout, the stagnation roerties are constant at every oint in the flow. However, if the flow in the regions and is non-adiabatic and irreversibile, thent 0 T 0 ; 0 0 ; ρ 0 ρ 0 Joint initiative of IITs and IISc Funded by MHRD Page 6 of 57

17 Characteristics Conditions Consider an arbitrary flow field, in which a fluid element is travelling at some Mach number ( M ) and velocity ( V ) at a given oint A. The static ressure, temerature and density are T, and ρ, resectively. Now, imagine that the fluid element is adiabatically slowed down ( if M > ) or seeded u ( if ) M < until the Mach number at A reaches the sonic state as shown in Fig Thus, the temerature will change in this rocess. This imaginary situation of the flow field when a real state in the flow is brought to sonic state is known as the characteristics conditions. The associated arameters are denoted as, T, ρ, a etc. Fig. 4.3.: Illustration of characteristics states of a gas. Now, revisit Eq. (4.3.) and use the relations for a calorically erfect gas, by relacing, below; γ R c = and a = γ RT. Another form of energy equation is obtained as γ a u a u γ + = γ + (4.3.7) At the imagined condition (oint ) of Mach, the flow velocity is sonic and u Then the Eq. (4.3.7) becomes, = a. a u a a + = + γ γ a u γ + or, + = a γ ( γ ) (4.3.8) Joint initiative of IITs and IISc Funded by MHRD Page 7 of 57

18 Like stagnation roerties, these imagined conditions are associated roerties of any fluid element which is actually moving with velocity u. If an actual flow field is non-adiabatic from A B, then aa ab.on the other hand if the general flow field is adiabatic throughout, then a is a constant value at every oint in the flow. Dividing u both sides for Eq. (4.3.8) leads to, ( au) γ + a + = γ ( γ ) u or, M = ( γ + / ) M ( γ ) (4.3.9) This equation rovides the relation between actual Mach number ( M ) and characteristics Mach number ( M ). It may be shown that when M aroaches infinity, M reaches a finite value. From Eq. (4.3.9), it may be seen that M M M M = M = < M < > M > γ + M γ (4.3.0) Relations between stagnation and characteristics state The stagnation seed and characteristics seed of sound may be written as, a RT a γ = = RT (4.3.) γ ; 0 0 Rewrite Eq. (4.3.7) for stagnation conditions as given below; a o a u + = γ γ (4.3.) Equate Eqs. (4.3.8) and (4.3.), γ + a 0 a T a = = = ( γ ) γ a0 T0 γ + (4.3.3) Joint initiative of IITs and IISc Funded by MHRD Page 8 of 57

19 More useful results may be obtained for Eqs. (4.3.4) & (4.3.6), when we define = ; T = T ; ρ = ρ ; a= a for Mach 0 0 γ γ γ ρ = ; = γ + ρ γ + (4.3.4) With γ =.4 (for air), the Eqs (4.3.3) & (4.3.4) reduces to constant value. a T ρ = = 0.833; = 0.58; = a0 T0 0 ρ0 (4.3.5) Critical seed and Maximum seed The critical seed of the gas ( u ) is same as that seed of sound ( a ) u a at M = =. A gas can attain the maximum seed ( ) max at sonic state i.e. u when it is hyothetically exanded to zero ressure. The static temerature corresonding to this state is also zero. The maximum seed of the gas reresents the seed corresonding to the comlete transformation of kinetic energy associated with the random motion of gas molecules into the directed kinetic energy. Rearranging Eq. (4.3.3), one can obtain the following equation; γ γ RT T = T + u ; At T = 0; u = u = γr γ 0 0 max or, u max = a 0 γ (4.3.6) Now, the Eqs (4.3.3) & (4.3.6) can be simlified to obtain the following relation; u a max γ + = γ (4.3.7) Joint initiative of IITs and IISc Funded by MHRD Page 9 of 57

20 Steady Flow Adiabatic Ellise It is an ellise in which all the oints have same total energies. Each oint differs from the other owing to relative roortions of thermal and kinetic energies corresonding to different Mach numbers. Now, rewrite Eq. (4.3.3) by relacing γ R c = and a = γ RT ; γ γ γ u γ R T c + = u + a = c (4.3.8) When, T = 0, u = u so that the constant aearing in Eq. (4.3.8) can be considered max as, c= u max. Then, Eq. (4.3.8) is written as follows; u a u + a = umax + = γ umax γ umax (4.3.9) Relacing the value of u max following exression; from Eq. (4.3.6) in Eq. (4.3.9), one can write the u u a + = (4.3.0) a max 0 This is the equation of an ellise with major axis as u max and minor axis as a 0 as shown in Fig Now, rearrange Eq. (4.3.0) in the following form; a a u a = 0 0 umax (4.3.) Now, differentiate Eq. (4.3.) with resect to u and simlify; da γ u γ = M M da = = du a γ du (4.3.) Joint initiative of IITs and IISc Funded by MHRD Page 0 of 57

21 Fig : Steady flow adiabatic ellise. Thus, the change of sloe from oint to oint on the ellise indicates the change in Mach number and hence the seed of sound and velocity. So, it gives the direct comarison of the relative magnitudes of thermal and kinetic energies. Different comressible flow regimes can be obtained with the knowledge of sloe in Fig The following imortant inferences may be drawn; - In high Mach numbers flows, the changes in Mach number are mainly due to the changes in seed of sound. - At low Mach numbers flows, the changes in Mach number are mainly due to the changes in the velocity. - When the flow Mach number is below 0.3, the changes in seed of sound is negligible small and the flow is treated as incomressible. Joint initiative of IITs and IISc Funded by MHRD Page of 57

22 Module 4 : Lecture 4 COMPRESSIBLE FLOWS (One-Dimensional Analysis) Mach Waves Consider an aerodynamic body moving with certain velocity ( V ) in a still air. When the ressure at the surface of the body is greater than that of the surrounding air, it results an infinitesimal comression wave that moves at seed of sound ( a ). These disturbances in the medium sread out from the body and become rogressively weaker away from the body. If the air has to ass smoothly over the surface of the body, the disturbances must warn the still air, about the aroach of the body. Now, let us analyze two situations: (a) the body is moving at subsonic seed ( V am ; ) (b) the body is moving at suersonic seed ( V am ; ) > >. < < ; Case I: During the motion of the body, the sound waves are generated at different time intervals ( t ) as shown in Fig The distance covered by the sound waves can be reresented by the circle of radius ( at, at,3 at...soon). During same time intervals ( t ), the body will cover distances reresented by, Vt, Vt, 3 Vt...so on. At subsonic seeds ( V am ; ) < <, the body will always remains inside the family of circular sound waves. In other words, the information is roagated through the sound wave in all directions. Thus, the surrounding still air becomes aware of the resence of the body due to the disturbances induced in the medium. Hence, the flow adjusts itself very much before it aroaches the body. Case II: Consider the case, when the body is moving at suersonic seed ( V am ; ) > >. With a similar manner, the sound waves are reresented by circle of radius ( at, at,3 at...soon) after different time ( t) intervals. By this time, the body would have moved to a different location much faster from its initial osition. At any oint of time, the location of the body is always outside the family of circles of sound waves. The ressure disturbances created by the body always lags behind the body that created the disturbances. In other words, the information reaches the surrounding Joint initiative of IITs and IISc Funded by MHRD Page of 57

23 air much later because the disturbances cannot overtake the body. Hence, the flow cannot adjust itself when it aroaches the body. The nature induces a wave across which the flow roerties have to change and this line of disturbance is known as Mach wave. These mach waves are initiated when the seed of the body aroaches the seed of sound ( V am ; ) in the Mach number. = =. They become rogressively stronger with increase Fig. 4.4.: Sread of disturbances at subsonic and suersonic seeds. Some silent features of a Mach wave are listed below; - The series of wave fronts form a disturbance enveloe given by a straight line which is tangent to the family of circles. It will be seen that all the disturbance waves lie within a cone (Fig. 4.4.), having a vertex/aex at the body at time considered. The locus of all the leading surfaces of the waves of this cone is known as Mach cone. - All disturbances confine inside the Mach cone extending downstream of the moving body is called as zone of action. The region outside the Mach cone and extending ustream is known as zone of silence. The ressure disturbances are largely concentrated in the neighborhood of the Mach cone that forms the outer limit of the zone of action (Fig. 4.4.). Joint initiative of IITs and IISc Funded by MHRD Page 3 of 57

24 - The half angle of the Mach cone is called as the Mach angle ( µ m ) that can be easily calculated from the geometry of the Fig ( ) ( ) ( ) ( ) at a t a 3t a sin µ m = = =... = = µ m = sin Vt V t V 3t V M M (4.4.) Fig. 4.4.: Illustration of a Mach wave. Shock Waves Let us consider a subsonic and suersonic flow ast a body as shown in Fig In both the cases, the body acts as an obstruction to the flow and thus there is a change in energy and momentum of the flow. The changes in flow roerties are communicated through ressure waves moving at seed of sound everywhere in the flow field (i.e. both ustream and downstream). As shown in Fig (a), if the incoming stream is subsonic i.e. M < ; V < a, the sound waves roagate faster than the flow seed and warn the medium about the resence of the body. So, the streamlines aroaching the body begin to adjust themselves far ustream and the flow roerties change the attern gradually in the vicinity of the body. In contrast, when the flow is suersonic, (Fig b) i.e. M > ; V > a, the sound waves overtake the seed of the body and these weak ressure waves merge themselves ahead of the body leading to comression in the vicinity of the body. In other words, the flow medium gets comressed at a very short distance ahead of the body in a very thin region that may be comarable to the mean free ath of the molecules in the medium. Since, these comression waves roagate ustream, so they tend to merge as shock wave. Ahead of the shock wave, the flow has no idea of resence of the body and immediately behind the shock; the flow is subsonic as shown in Fig (b). Joint initiative of IITs and IISc Funded by MHRD Page 4 of 57

25 The thermodynamic definition of a shock wave may be written as the instantaneous comression of the gas. The energy for comressing the medium, through a shock wave is obtained from the kinetic energy of the flow ustream the shock wave. The reduction in kinetic energy is accounted as heating of the gas to a static temerature above that corresonding to the isentroic comression value. Consequently, in flowing through the shock wave, the gas exeriences a decrease in its available energy and accordingly, an increase in entroy. So, the comression through a shock wave is considered as an irreversible rocess. Normal Shock Waves Fig : Illustration of shock wave henomena. A normal shock wave is one of the situations where the flow roerties change drastically in one direction. The shock wave stands erendicular to the flow as shown in Fig The quantitative analysis of the changes across a normal shock wave involves the determination of flow roerties. All conditions of are known ahead of the shock and the unknown flow roerties are to be determined after the shock. There is no heat added or taken away as the flow traverses across the normal shock. Hence, the flow across the shock wave is adiabatic ( q = 0). Fig : Schematic diagram of a standing normal shock wave. Joint initiative of IITs and IISc Funded by MHRD Page 5 of 57

26 The basic one dimensional comressible flow equations can be written as below; u u ρu= ρu; + ρu = + ρu; h+ = h+ (4.4.) For a calorically erfect gas, thermodynamic relations can be used, = ρrt; h = c T; a = γ / ρ (4.4.3) The continuity and momentum equations of Eq. (4.4.) can be simlified to obtain, Since, a = γ RT and M a γu a u u γu = (4.4.4) V =, the energy equation is written as, a a u γ + γ + γ + = a a = a u γ ( γ ) (4.4.5) Both a and a can now be exressed as, a γ + ( a ) γ u; a γ ( a ) γ + = = u (4.4.6) Substitute Eqs. (4.4.6) in Eq. (4.4.4) and solve for Recall the relation for a uu M M a = = (4.4.7) M and M and substitute in Eq. (4.4.7), M ( γ ) ( γ ) M = (4.4.8) + + M + Substitute Eq. (4.4.8) in Eq. (4.4.7) and solve for M M γ + M = γ γ M Using continuity equation and Prandtl relation, we can write, (4.4.9) ρ u u u = = = = ( M ) (4.4.0) ρ u uu a Joint initiative of IITs and IISc Funded by MHRD Page 6 of 57

27 Substitute Eq. (4.4.8) in Eq. (4.4.0) and solve for density and velocity ratio across the normal shock. ( γ + ) ( γ ) ρ M = = (4.4.) ρ + u u M The ressure ratio can be obtained by the combination of momentum and continuity equations i.e. u u ( ) = ρu u u = ρu ; = γ M u u (4.4.) u Substituting the ratio from Eq. (4.4.0) in Eq. (4.4.) and simlifying for the u ressure ratio across the normal shock, we get, γ = + γ + ( M ) (4.4.3) For a calorically erfect gas, equation of state relation (Eq ) can be used to obtain the temerature ratio across the normal shock i.e. + ( γ ) ( ) h T ρ γ M = = = ( M ) h T ρ + γ + γ + M (4.4.4) Thus, the ustream Mach number is the owerful tool to dictating the shock wave roerties. The stagnation roerties across the normal shock can be comuted as follows; ( 0 ) ( ) 0 = 0 0 (4.4.5) Joint initiative of IITs and IISc Funded by MHRD Page 7 of 57

28 0 0 Here, the ratios and can be obtained from the isentroic relation for the regions and resectively. Knowing the ustream Mach number M, Eq. (4.4.9) gives the downstream Mach number M. Further, Eq. (4.4.3) can be used to obtain the static ressure ratio to,. After substitution of these ratios, Eq. (4.4.5) reduces γ γ γ + M 0 γ 0 γ γ + M γ = + ( M ) γ + (4.4.6) 0 Many a times, another significant ressure ratio is imortant for a normal shock which is normally called as Rayleigh Pitot Tube relation. γ γ γ γ = = + M + ( M ) γ + Recall the energy equation for a calorically erfect gas: (4.4.7) u u ct + = ct + ct 0 = ct 0 (4.4.8) Thus, the stagnation temeratures do not change across a normal shock. Entroy across a normal shock The comression through a shock wave is considered as irreversible rocess leading to an increase in entroy. The change in entroy can be written as a function of static ressure and static temerature ratios across the normal shock. T s s = c ln Rln T (4.4.9) Mathematically, it can be seen that the entroy change across a normal shock is also a function of the ustream Mach number. The second law of thermodynamics uts the limit that entroy must increase ( s s ) for a rocess to occur in a certain 0 Joint initiative of IITs and IISc Funded by MHRD Page 8 of 57

29 direction. Hence, the ustream Mach number ( M ) must be greater than (i.e. suersonic). It leads to the fact that M ( ) ( ρ ρ ) ( T T ) ; ; ;. The entroy change across a normal shock can also be calculated from another simle way by exressing the thermodynamic relation in terms of total ressure. Referring to Fig , it is seen that the discontinuity occurs only in the thin region across the normal shock. If the fluid elements is brought to rest isentroically from its real state (for both ustream and downstream conditions), then they will reach an imaginary state a and a. The exression for entroy change between the imaginary states can be written as, T s s = c ln Rln a a a a Ta a Since, a a a a 0 a 0 a 0 to, (4.4.0) s = s ; s = s ; T = T = T ; = and =, the Eq.(4.4.0) reduces = ln = ( ) 0 0 s s R s s R e 0 0 (4.4.) Because of the fact s > s, Eq. (4.4.) imlies that 0 < 0. Hence, the stagnation ressure always decreases across a normal shock. Joint initiative of IITs and IISc Funded by MHRD Page 9 of 57

30 Module 4 : Lecture 5 COMPRESSIBLE FLOWS (Two-Dimensional Analysis) Oblique Shock Wave The normal shock waves are straight in which the flow before and after the wave is normal to the shock. It is considered as a secial case in the general family of oblique shock waves that occur in suersonic flow. In general, oblique shock waves are straight but inclined at an angle to the ustream flow and roduce a change in flow direction as shown in Fig. 4.5.(a). An infinitely weak oblique shock may be defined as a Mach wave (Fig b). By definition, an oblique shock generally occurs, when a suersonic flow is turned into itself as shown in Fig. 4.5.(c). Here, a suersonic flow is allowed to ass over a surface, which is inclined at an angle ( θ ) to the horizontal. The flow streamlines are deflected uwards and aligned along the surface. Since, the ustream flow is suersonic; the streamlines are adjusted in the downstream an oblique shock wave angle ( β ) with the horizontal such that they are arallel to the surface in the downstream. All the streamlines exerience same deflection angle across the oblique shock. Fig. 4.5.: Schematic reresentation of an oblique shock. Joint initiative of IITs and IISc Funded by MHRD Page 30 of 57

31 Oblique Shock Relations Unlike the normal shocks, the analysis of oblique shocks is revalent mainly in the two-dimensional suersonic flows. The flow field roerties are the functions of x and yas shown in Fig In the ustream of the shock, the streamlines are horizontal where, the Mach number and velocity of the flow are M and V, resectively. The flow is deflected towards the shock in the downstream by angle θ such that the Mach number and velocity becomes M and V, resectively. The comonents of V, arallel and erendicular to the shock are u and v, resectively. Similarly, the analogous comonents for V are, u and v resectively. The normal and tangential Mach numbers ahead of the shock are Mn and Mtwhile the corresonding Mach numbers behind the shock are, Mn and Mtresectively. Fig. 4.5.: Geometrical reresentation of oblique shock wave. The continuity equation for oblique shock is, ρu = ρ u (4.5.) Considering steady flow with no body forces, the momentum equation can be resolved in tangential and normal directions. ( ρu) v+ ( ρu) v= ( ρu ) u + ( ρ u ) u = ( + ) Tangential comonent: 0 Normal comonent: (4.5.) Substitute Eq. (4.5.) in Eq. (4.5.), v = v ; + ρu = + ρ u (4.5.3) Joint initiative of IITs and IISc Funded by MHRD Page 3 of 57

32 Thus, it is seen that the tangential comonent of flow velocity does not change across an oblique shock. Finally, the energy equation gives, ( ) V V V V u + u = ρ e+ u+ ρ e+ u h+ = h+ (4.5.4) From the geometry of the Fig. 4.5., V = u + v and v = v, hence So, the energy equation becomes, ( ) ( ) V V = u + v u + v = u u (4.5.5) u u h+ = h + (4.5.6) Examining the Eqs (4.5., and 4.5.6), it is noted that they are identical to governing equations for a normal shock. So, the flow roerties changes in the oblique shock are governed by the normal comonent of the ustream Mach number. So, the similar exressions can be written across an oblique shock in terms of normal comonent of free stream velocity i.e. ( γ + ) ( ) + / ( γ ) ( γ ) M M n Mn = Msin β; Mn = γ / n ρ M T ρ γ γ ρ n γ ρ = ; = + ( M n ; ) = + Mn + T M n T M = ; s s = c ln Rln sin ( β θ) T (4.5.7) Thus, the changes across an oblique shock are function of ustream Mach number ( M) and oblique shock angle ( β ) while the normal shock is a secial case when π β =. Joint initiative of IITs and IISc Funded by MHRD Page 3 of 57

33 Referring to geometry of the oblique shock (Fig b), u tan β ( β θ) v Since, v = v, Eq. (4.5.8) reduces to, u v = ; tan = (4.5.8) tan ( β θ) u ρ tan β = = (4.5.9) u ρ Use the relations given in Eq. (4.5.7) and substituting them in Eq. (4.5.9), the trigonometric equation becomes, M sin β tanθ = cot β M ( γ + cos β) + (4.5.0) It is a famous relation showing θ as the unique function of β and M. Eq. (4.5.0) is used to obtain the θ β M curve (Fig ) for γ =.4. Fig : θ β M curves for an oblique shock. Joint initiative of IITs and IISc Funded by MHRD Page 33 of 57

34 The following inferences may be drawn from θ β M curves. It is seen that there is a maximum deflection angle θ max. - For any given M, if, θ < θmax, the oblique shock will be attached to the body (Fig a). Whenθ > θ, there will be no solution and the oblique shock max will be curved and detached as shown in Fig (b). The locus of θmax obtained by joining the oints (a, b, c, d, e and f ) in the Fig can be - Again, if θ < θmax, there will be two values of β redicted from θ β M relation. Large value of β corresonds to strong shock solution while small value refers to weak shock solution (Fig c). In the strong shock solution, M is subsonic while in the weak shock region, M is suersonic. The locus of such oints (a, b, c, d, e and f ) as shown in Fig , is a curve that also signifies the weak shock solution. The conditions behind the shock could be subsonic if θ becomes closer to θ max. - If θ = 0, it corresonds to a normal shock when becomes a Mach wave when β = µ m. π β = and the oblique shock Fig : (a) Attached shock; (b) Detached shock; (c) Strong and weak shock. Joint initiative of IITs and IISc Funded by MHRD Page 34 of 57

35 Oblique Exansion Waves Another class of two dimensional waves occurring in suersonic flow shows the oosite effects of oblique shock. Such tyes of waves are known as exansion waves. When the suersonic flow is turned away from itself, an exansion wave is formed as shown in Fig (a). Here, the flow is allowed to ass over a surface which is inclined at an angle ( θ ) to the horizontal and all the flow streamlines are deflected downwards. The change in flow direction takes lace across an exansion fan centered at oint A. The flow streamlines are smoothly curved till the downstream flow becomes arallel to the wall surface behind the oint A. Here, the flow roerties change smoothly through the exansion fan excet at oint A. An infinitely strong oblique exansion wave may be called as a Mach wave. An exansion wave emanating from a shar convex corner is known as a centered exansion which is commonly known as Prandtl-Meyer exansion wave. Few features of PM exansion waves are as follows; - Streamlines through the exansion wave are smooth curved lines. - The exansion of the flow takes lace though an infinite number of Mach waves emitting from the center A. It is bounded by forward and rearward Mach lines as shown in Fig (b). These Mach lines are defined by Mach angles i.e. Forward Mach angle: µ = sin ( M ) ( M ) m Rearward Mach angle: µ = sin m (4.5.) - The exansion takes lace through a continuous succession of Mach waves such that there is no change in entroy for each Mach wave. Thus, the exansion rocess is treated as isentroic. - The Mach number increases while the static roerties such as ressure, temerature and density decrease during the exansion rocess. Joint initiative of IITs and IISc Funded by MHRD Page 35 of 57

36 Fig : Schematic reresentation of an exansion fan. The quantitative analysis of exansion fan involves the determination of M,, T and ρfor the given ustream conditions of M,, T, ρand θ. Consider the infinitesimal changes across a very weak wave (Mach wave) as shown in Fig From the law of sine, Fig : Infinitesimal change across a Mach wave. π sin + µ m V + dv dv = + = V V π sin µ m dθ (4.5.) Joint initiative of IITs and IISc Funded by MHRD Page 36 of 57

37 Use trigonometric identities and Taylor series exansion, Eq. (4.5.) can be simlified as below; Since, ( dv V ) dθ = (4.5.3) tan µ m sin µ m = tan µ m =, so the Eq. (4.5.3) can be simlified and M M integrated further from region to, M dv dv dθ = M dθ = M V V θ (4.5.4) θ M From the definition of Mach number, dv dm da V = Ma = + (4.5.5) V M a For a calorically erfect gas, the energy equation can be written as, γ da γ γ ao = + M = M + M dm a a Use Eqs (4.5.5 & 4.5.6) in Eq. (4.5.4) and integrate from θ = 0 to θ, (4.5.6) θ M M dm dθ = θ 0 = (4.5.7) γ M M M θ + The integral in the Eq. (4.5.8) is known as Prandtl-Meyer function, ν ( M ). M dm γ + γ ν ( M) = = ( M ) M γ + M M γ γ + Finally, Eq. (4.5.7) reduces to, tan tan (4.5.8) ( M ) ( M ) θ = ν ν (4.5.9) Thus, for a given ustream Mach number calculate using given ν ( M ) M, one can obtain ν ( ) M, subsequently and θ. Since, the exansion rocess is isentroic, the flow roerties can be calculated from isentroic relations. Joint initiative of IITs and IISc Funded by MHRD Page 37 of 57

38 Module 4 : Lecture 6 COMPRESSIBLE FLOWS (Hyersonic Flow: Part - I) Introduction to Hyersonic Flow The hyersonic flows are different from the conventional regimes of suersonic flows. As a rule of thumb, when the Mach number is greater than 5, the flow is classified as hyersonic. However, the flow does not change its feature all of a sudden during this transition rocess. So, the more aroriate definition of hyersonic flow would be regime of the flow where certain hysical flow henomena become more imortant with increase in the Mach number. One of the hysical meanings may be given to the Mach number as the measure of the ordered motion of the gas to the random thermal motion of the molecules. In other words, it is the ratio of ordered energy to the random energy as given in Eq. (4.6.). M ( ) ( ) V Ordered kinetic energy = = (4.6.) a Random kinetic energy In the case of hyersonic flows, it is the directed/ordered kinetic energy that dominates over the energy associated with random motion of the molecules. Now, recall the energy equation exressed in the form of flow velocity ( V ), seed of sound ( a = γ RT ) and stagnation seed of sound ( a0 γ RT0) =. a0 a V a γ V = + + = γ γ a a 0 0 (4.6.) Eq. (4.6.) forms an adiabatic ellise which is obtained for steady flow energy equation. When the flow aroaches the hyersonic limit, the ratio becomes Then, Eq. (4.6.) simlifies to the following exression. a a. 0 V a γ RT γ γ 0 0 (4.6.3) In other words, the entire kinetic energy of the flow gets converted to internal energy of the flow which is a function total temerature ( T0 ) of the flow. Joint initiative of IITs and IISc Funded by MHRD Page 38 of 57

39 The study/research on hyersonic flows revels many exciting and unknown flow features of aerosace vehicles in the twenty-first century. The resence of secial features in a hyersonic flow is highly deendent on tye of trajectory, configuration of the vehicle design, mission requirement that are decided by the nature of hyersonic atmoshere encountered by the flight vehicle. Therefore, the hyersonic flight vehicles are classified in four different tyes, based on the design constraints imosed from mission secifications. - Reentry vehicles (uses the rocket roulsion system) - Cruise and acceleration vehicle (air-breathing roulsion such as ramjet/scramjet) - Reentry vehicles (uses both air-breathing and rocket roulsion) - Aero-assisted orbit transfer vehicle (resence of ions and lasma in the vicinity of sacecraft) Characteristics Features of Hyersonic Flow There are certain hysical henomena that essentially differentiate the hyersonic flows as comared to the suersonic flows. Even though, the flow is treated as suersonic, there are certain secial features that aear when the seed of the flow is more than the seed of sound tyically beyond the Mach number of 5. Some of these characteristics features are listed here; Thin shock layer: It is known from oblique shock relation ( θ β M ) that the shock wave angle ( β ) decreases with increase in the Mach number ( M ) for weak shock solution. With rogressive increase in the Mach number, the shock wave angle reaches closer to the flow deflection angle ( θ ). Again, due to increase in temerature rise across the shock wave, if chemical reaction effects are included, the shock wave angle will still be smaller. Since, the distance between the body and the shock wave is small, the increase in the density across the shock wave results in very high mass fluxes squeezing through small areas. The flow region between the shock wave and the body is known as thin shock layer as shown in Fig. 4.6.(a). It is the basic characteristics of hyersonic flows that shock waves lie closer to the body and shock layer is thin. Further, the shock wave merges with the thick viscous boundary layer growing from the body surface. The comlexity of flow field increases due to thin Joint initiative of IITs and IISc Funded by MHRD Page 39 of 57

40 shock layer where the boundary layer thickness and shock layer thickness become comarable. Fig. 4.6.: Few imortant henomena in a hyersonic flow: (a) Thin shock layer; (b) Entroy layer; (c) Temerature rofile in a boundary layer; (d) High temerature shock layer; (e) Low density effects. Entroy layer: The aerodynamic body configuration used in hyersonic flow environment is tyically blunt to avoid thin shock layers to be closer to the body. So, there will be a detached bow shock standing at certain distance from the nose of the body and this shock wave is highly curved (Fig b). Since, the flow rocess across the shock is a non-isentroic henomena, an entroy gradient is develoed that varies along the distance of the body. At the nose ortion of the blunt body, the bow shock resembles normal to the streamline and the centerline of the flow will exerience a larger entroy gradient while all other neighboring streamlines undergo the entroy changes in the weaker ortion of the shock. It results in an entroy layer that ersists all along the body. Using the classical Crocco s theorem, the entroy layer may be related to vorticity. Hence, the entroy layer in high Mach number flows, exhibits strong gradient of entroy which leads to higher vorticity at higher magnitudes. Due to the resence of entroy layer, it becomes difficult to redict the boundary layer roerties. This henomenon in the hyersonic flow is called as vortcity generation. In addition to thin shock layer, the entroy layer also interacts Joint initiative of IITs and IISc Funded by MHRD Page 40 of 57