Category Basics of Aero – thermodynamics

The basics of fluid mechanics tell us, that supersonic flow will be accelerated (expanded), if the characteristic cross-section of the flow-domain is enlarged, and decelerated (compressed), if it is reduced. We call this expansion or compression by supersonic flow turning, see, e. g., [4].

Supersonic flow turning implies that small turning rates are involved, with basically isentropic changes of the flow. The Prandtl-Meyer expansion is an isentropic flow phenomenon in which the flow speed is increased. In a sense it can be considered as the counterpart of an oblique shock wave, where the flow speed is reduced. However, across the shock wave we have a discontinuity of speed, whereas with the Prandtl-Meyer expansion a gradual increase of the flow speed occurs.

The Prandtl-Meyer expansion is a “centered” expansion phenomenon, Fig. 6.25. Downstream of the expansion fan the flow again is parallel to the body surface. If a streamline (a) in the expansion fan is selected as a body contour, we have above it a “simple” expansion. However, in such a case the hidden Prandtl-Meyer corner can be used to describe the flow. The Laval – nozzle expansion between nozzle throat and contour inflection point, Fig. 5.7, can be considered as an example of such a simple expansion.

The real counterpart of the Prandtl-Meyer expansion is the isentropic compression of supersonic flow, which can be envisaged, for example, as the inverse flow in a Laval nozzle. Investigations have been made whether isen – tropic compression can be used in supersonic engine flow inlets [9]. Such a flow can be imagined as a compression by an infinite number of very weak shocks. In practice, however, such flow would be very sensitive to contour disturbances—the boundary layer already is such a disturbance—so that isen – tropic compression is considered today as not feasible.

Supersonic flow turning of a perfect gas is described by an expression obtained from eq. (6.116) for small ramp angles 5 (for the derivation see,

e. g., [4]):

Here S is the deflection angle and V the resultant flow velocity. Integrating this expression gives the Prandtl-Meyer angle v(M)

-6 + c= j /M2-ly = i/(M). (6.131)

After further manipulation we obtain the following relation for v(M)

.1/’ I dM

This relation integrated yields the Prandtl-Meyer function [3, 4, 11]:

v = vi + |S — Si I (6.134)

or compression turning:

v = vi — |S — Si I. (6.135)

Av can be computed as a function of Mi and y directly from eq. (6.133). No explicit relation is available to find the resulting Mach number for a given Prandtl-Meyer angle. An iterative solution procedure can be programmed.

Tabulated data for M and v for y = 1.4 can be found in [11]. In Fig. 6.27 we give the turning angle v [°] as function of the Mach number Mi for Y = 1.4.

Since both expansion and compression turning are isentropic, the relations given in Section 6.2 can be used to determine pressure, temperature and the like.

We consider now supersonic isentropic flow turning at very large Mach numbers. In such cases /M2 — 1 « M, and eq. (6.133) reduces to

The largest possible expansion turning angle is that, for which V2 = Vm. One has to keep in mind, that Vm is finite, eq. (6.15). If it is reached, T2 = 0 K, and hence M2 ^ ж. Because the Prandtl-Meyer relation is formulated in terms of the Mach number, we cannot determine from it the maximum expansion turning angle. The largest possible compression turning angle is that which leads to M2 = 1.

With these results we have attained insight into the two turning phe­nomena, of which the Prandtl-Meyer expansion is the more important one. However, it does not find a direct application in computation schemes, except for the shock-expansion scheme [4], which is used in approximate computa­tion methods for aerothermodynamic loads determination, see, e. g., [34].

We finally give examples for expansion and compression turning in Table 6.6. It is interesting to observe that for Mi = 20 despite the large Mach number changes (M2) with the given turning angle for expansion only a small velocity (V2 ) change occurs. The static pressure P2 however reacts strongly, less strongly the temperature T2.

We have seen in Sub-Chapters 6.3.1 and 6.3.2 that in inviscid iso-energetic flow, i. e., adiabatic flow without friction and heat conduction, the entropy increases across normal and oblique shock waves. Behind these shock waves the entropy is constant again along the streamlines. If the entropy is constant also from streamline to streamline, we call this flow a homentropic flow. This situation is found in the uniform free stream ahead of a shock wave, which we have assumed for our investigations, and behind it, if the shock wave is not curved.[67] If we have flow, where the entropy is constant only along streamlines, we call this isentropic flow. This is the situation behind a curved shock wave.

We have seen also that for a given free-stream Mach number the normal shock wave leads to the largest entropy increase, eq. (6.60). Across an oblique shock the entropy increase is smaller, eq. (6.95), and actually becomes smaller and smaller with decreasing shock angle в, until the shock angle approaches the Mach angle в ^ p.

The total pressure decreases if the entropy increases:

‘Ell = e-(s2-si)/R^ (6.127)

Ptі

Hence behind a curved shock surface we get a total pressure, which changes from streamline to streamline. The lowest total pressure is found on the streamline, which has passed the locally normal shock surface. If the total pressure changes from streamline to streamline, we have, with a given pressure field, a change of the flow velocity from streamline to streamline, see Bernoulli’s equation for compressible flow, eq. (6.21): the flow behind a curved shock surface is rotational.

The law which relates the vorticity со of this rotational inviscid flow to the entropy gradient across the streamlines is Crocco’s theorem, see, e. g., [4]

We illustrate the velocity profiles in the inviscid flow behind a curved bow-shock surface in Fig. 6.22 a) for the symmetric case, and in Fig. 6.22 b) for the asymmetric case.

In the symmetric case the profile resembles that of a slip-flow boundary layer, Fig. 9.26 on page 370. In the asymmetric case the entropy layer at the windward side has a wake-like appearance. This is due to the fact that the streamline crossing the normal portion of the bow-shock surface at P0 is not the stagnation-point streamline. That crosses the bow-shock surface at Pi. Hence the streamline crossing at P0 suffers a larger total-pressure loss or entropy rise than that crossing at Pi, which leads to the wake-like appearance.

The symmetric case is that considered usually in the literature. However, the asymmetric case appears to be that of larger importance in practice. Re­sults of numerical investigations point to the fact that it appears at the body surface which is strongest inclined against the free stream, see the discussion in [2]. This would be the windward side of a RV-W or the lower side of the forebody of an airbreathing CAV with forebody pre-compression.

The entropy-layer effect of general interest is the so-called entropy-layer swallowing of a boundary layer. To study it, we consider the development of the boundary layer at the blunt body in the symmetrical case. We assume that the characteristic Reynolds number is so large, that we can clearly distinguish between the inviscid flow field and the boundary layer, Sub-Section 4.3.1. We assume further laminar flow throughout, and neglect the Mangler effect, [7, 29], as well as the influence of the surface temperature, which with radiation cooling decreases along the body surface very fast from its maximum value in the stagnation-point area.[68]

The boundary layer has a thickness S, which grows with increasing running length along the body surface in the downstream direction.[69] The edge of the boundary layer is not coincident with a streamline.[70] On the contrary, it is crossed by the streamlines of the inviscid flow, i. e., the boundary layer entrains, or swallows, the inviscid flow.[71]

We have indicated at the lower side of Fig. 6.22 a) the point xi on the bow-shock surface. It is the point downstream of which the bow-shock surface can be considered as straight. This means that all streamlines crossing the shock downstream of xi have the same constant entropy behind the shock surface. The streamline, which crosses the shock surface at xi, enters the boundary layer at x2. Thus the boundary layer swallows the entropy-layer of the blunt body between the stagnation point Si and the point x2. The length on the body surface between the stagnation point and x2 is called the entropy-layer swallowing distance, see, e. g., [30], in which the body bluntness via the entropy-layer swallowing can have an effect on the boundary-layer parameters.

Entropy-layer swallowing is an interesting and, for hypersonic vehicle design, potentially important phenomenon. It leads to a decrease of the boundary-layer thicknesses, and subsequently to an increase of the heat flux in the gas at the wall qgw, as well of the wall shear stress tw.[72] The changes in the characteristic boundary-layer thicknesses, i. e., of the form of the boundary – layer velocity profile, also influence the laminar-turbulent transition behavior of the boundary layer, see Section 8.2.

The effects of entropy layer swallowing have found attention quite early, see, e. g., [31, 32], however, obviously regarding only the symmetric case. In the following we consider entropy-layer swallowing from a purely qualitative and illustrative point of view.

We assume the symmetrical case of a blunt body in a given hypersonic free stream with Mach number Mand unit Reynolds number ReЦ,. In Fig. 6.22 a) we have indicated the stand-off distance A of the bow-shock surface. It is approximately a function of the normal-shock density ratio є, and the nose diameter Rs, eq. (6.122). We consider two limiting cases, assuming a comparable boundary-layer development.

1. The nose radius Rs is very large, and hence the boundary-layer thickness is very small compared to the shock stand-off distance: 6 ^ A. The boundary layer development is governed by the inviscid flow, which went through the normal-shock portion of the bow-shock surface. The entropy layer is not swallowed or only to a very small degree.

2. The nose radius Rs is small, and hence the boundary-layer thickness is not small compared to the shock stand-off distance: 6 « O(A). This situation is shown in the lower part of Fig. 6.22 a). The entropy layer is swallowed, and this means that increasingly inviscid flow with higher speed and lower temperature is entrained into the boundary layer.[73] This increases the effective local unit Reynolds number. As will be shown in Sub-Section 6.6, the unit Reynolds number directly behind the bow shock is smallest for the normal-shock part, and increases with decreasing shock angle в. This effect is directly significant for the boundary layer only away from the stagnation-point region, where the inviscid flow is weakly turned. The increased effective unit Reynolds number then increases the heat flux in the gas at the wall and the wall shear stress, as was mentioned above.

It is clear that we can make a similar consideration assuming constant M^ and constant nose radius Rs, and a changing unit Reynolds number Re Then we observe entropy-layer swallowing with decreasing Re^, because this increases the boundary-layer thickness.

We illustrate the results found qualitatively above with results of flow – field computations made with a first-order and a second-order coupled Euler/boundary-layer method for the second case, instead with solutions for two different nose radii.

A first-order boundary-layer scheme takes into account only the proper­ties of the stagnation-point streamline, and hence does not capture entropy – layer swallowing, which in reality happens. This is done only if we employ either a one-layer computation scheme, like a viscous shock-layer or a Navier – Stokes/RANS method, or a suitable two-layer computation scheme, i. e., a second-order boundary-layer method, which is coupled to an Euler method. We show results in Fig. 6.23 and Fig. 6.24 from the application of a second – order boundary-layer method with perturbation coupling, see Sub-Section 7.2.1, to the flow past a hyperbola at zero angle of attack [33], see also the symmetric case in Fig. 6.22 a).

T/W

Fig. 6.23. Illustration of the effect of entropy-layer swallowing at a hyperbola [33], case 1 of Table 6.3: a) typical velocity profiles u/Vref (z) of the inviscid flow and the boundary layer without (first-order solution), and with solution) entropy- layer swallowing, b) typical temperature profiles T/Tref (z) of the inviscid flow and the boundary layer without (first-order solution), and with (second-order solution) entropy-layer swallowing, adiabatic wall situation.

In Fig. 6.23 a) we find for a given location the rotational inviscid flow profile at the wall (see also Fig. 6.22 a), upper part), and the boundary – layer velocity profiles. The boundary-layer profile found with the first-order solution, i. e., without entropy-layer swallowing, is governed by the inviscid flow at the body surface, and hence by the stagnation-point entropy. The profile found with the second-order solution blends into the inviscid flow profile at a finite distance from the surface and hence swallows the entropy – layer as the boundary layer develops from the stagnation-point region in downstream direction.

In the second-order case, i. e., the case with entropy-layer swallowing, the boundary-layer edge velocity ue/Vref is larger than in the first-order case, i. e., without entropy-layer swallowing. Consequently we have with entropy – layer swallowing a larger velocity gradient d(u/Vref)/dz|w and because of approximately the same wall temperatures, Fig. 6.23 b), and hence the same viscosity at the wall, a larger wall shear stress tw.

Fig. 6.23 b) shows the temperature profiles for the adiabatic wall situation. In this case the adiabatic temperature Tr is a little smaller with entropy-layer swallowing than without. This is in contrast to the usually observed behavior of the wall-heat flux qgw, see also Fig. 6.24, where entropy-layer swallowing increases the heat flux at the wall. The reason for this probably is the overall

higher temperature level in the case without entropy-layer swallowing, be­cause of the already much higher boundary-layer edge temperature Te/Tref.

A typical cold-wall case is sketched in Fig. 6.24. There indicated is the much larger temperature gradient near the body surface, where dissipation work is larger in the case with entropy-layer swallowing due to the larger velocity gradient, Fig. 6.23 a). Consequently the temperature gradient at the wall is larger than in the case without entropy-layer swallowing. With the same wall temperature Tw in both cases we then find a larger qgw with entropy-layer swallowing than without.

6.4.1 Bow-Shock Stand-Off Distance at a Blunt Body

We have noted in the preceding sub-section that the shape of the bow-shock surface is governed by the free-stream properties and the effective body shape. These also govern the stand-off distance A0 of the bow-shock surface from the body surface. In particular the shape of the body nose has a strong influence on Ao, Fig. 6.18, [4], where 5/d = A0/2Rb.

The smallest stand-off distance is observed for the sphere, the largest for the two-dimensional cases. In general Ao is smaller at axisymmetric bodies, because there the stagnation-point flow is relieved three-dimensionally, in contrast to two-dimensional flow. With increasing free-stream Mach number all curves approach asymptotically limiting values of the bow-shock stand-off distance A0 (Mach number independence).

For the bow-shock stand-off distance at spheres and circular cylinders, ap­proximate analytical relations are available for sufficiently large Mach num­bers, say, Ыж ^ 5, see, e. g., [22].

We consider a sphere with radius Rb, Fig. 6.19. It is assumed that the bow-shock surface in the vicinity of the flow axis lies close and parallel to the body surface, and thus is a sphere with the radius Rs.

Further assumed is constant density in the thin layer between the body surface and the bow-shock surface. With these assumptions the shock stand­off distance A0 at a sphere with radius Rb is found [22]:

Here є is the inverse of the density rise across the normal shock portion, Fig. 6.19

Pco _ PI _ (7 – l)Mf + 2 Ps P‘2 (7+1 )Mf

Eq. (6.120) can be expanded in a power series to yield

^0 = Rs — Rb = є R

A similar relation for the circular cylinder reads [22]

For sufficiently large free-stream Mach numbers є can be approximated for both normal and oblique shock waves, eqs. (6.70) and (6.84), by the value for Mi ^ ж

where we must demand 7 > 1. For our purposes this is a fair approximation for Mi ^ 6 at least for 7 = 1.4, as can be seen from Fig. 6.20, where the inverse of є is shown as function of Mi, and for three values of 7.

In a flow with large total enthalpy the temperature behind the bow-shock surface in the vicinity of the stagnation point region rises strongly. Accord­ingly the ratio of specific heats 7 decreases. If we take an effective mean ratio of the specific heats 7 as a measure for the bow-shock stand-off distance at large free-stream Mach numbers, we obtain for instance for the sphere, eq. (6.122), to first order

This result tells us that the bow-shock stand-off distance A0 at a body in hypersonic flight is largest for perfect gas, and with increasing high – temperature real-gas effects becomes smaller as the effective mean y is re­duced. A rigorous treatment of this phenomenon can be found in, e. g., [23, 24].

We illustrate this phenomenon with data computed by means of a cou­pled Euler/second-order boundary-layer method for a hyperbola of different dimensions at hypersonic Mach numbers, Table 6.3. (See also the example discussed in Sub-Section 5.5.1.)

The bow-shock stand-off distance in the case 1 (perfect gas computa­tion), is more than two times larger than that in the case 2 (equilibrium gas computation), Fig. 6.21 a), Table 6.4.

In reality, however, the situation can be more complex, depending on the size of the body, and hence on the first Damkohler number of the problem, Section 5.4. This is studied by means of true non-equilibrium computations for three different body sizes, cases 3 to 5 in Table 6.3. For the small body with L = 0.075 m (case 3) only a small degree of dissociation is found, Fig. 6.21 b). The reason is that due to the small absolute bow-shock stand-off distance and the large flow speed not enough time is available for dissociation to occur, DAM 1 ^ 0.

For the large body with L = 75 m, case 5, the situation is different. The absolute bow-shock stand-off distance is large, and the degree of dissociation, too, Fig. 6.21 b). At the stagnation point, X1A = 0, it compares well with the degree of dissociation m*N2 (= ^N2) found behind an isolated normal shock wave with the same flow conditions, hence we have an equilibrium situation, DAM 1 ^ to.

That mN2 at the stagnation point is a little smaller than m*N2, is due to the fact that between the bow-shock surface and the body surface a fur­ther compression happens, and hence a rise of the temperature from 5,600 K

behind the (isolated) normal shock to approximately 5,900 K at the stagna­tion point, which increases the degree of dissociation.

Case 4, L = 0.75 m, finally lies between the two other cases, and is a typical non-equilibrium case, DAM 1 = O(l), Table 6.4. Regarding the almost constant mass fractions for the cases 3 to 5 in Fig. 6.21 b), we note that an adiabatic wall was assumed throughout. Therefore the levels of surface temperature remain approximately the same along the hyperbola contours, and with them the respective mass fractions.

The results prove that with a coupled Euler/second-order boundary-layer computation formulated for non-equilibrium flow also the limiting cases of “frozen flow” and “equilibrium flow” can be handled. For purely inviscid flow computations of this kind, problems have been observed in the vicinity of the stagnation point of blunt bodies, where in any case the flow locally approaches equilibrium, which makes a special post-processing necessary [27].

The differences in the bow-shock stand-off distances between case 1 and case 3, as well as between case 2 and case 5, are due to the fact that we have viscous-flow cases. The approximate ratio of boundary-layer to shock-layer thickness S/A at 0.5 body length L is given for each case in Table 6.4. These ratios are approximately the same for cases 1 and 2, and also for the cases 3 and 4, because the Reynolds numbers are the same, respectively comparable for each pair. Of course the surface temperature levels are much higher in the cases 3 and 4, because of the much larger free-stream Mach number, and hence the boundary layers are thicker, Sub-Section 7.1. Case 5 finally, with a very large Reynolds number, has a very small boundary-layer thickness compared to the shock-layer thickness.

The results given in Table 6.4 point to the potential problem of a wrong determination of the shock stand-off distance compared to free flight. This can happen in a cold hypersonic tunnel, in a high-enthalpy facility, or with a computation method, because of the strong sensitivity of the shock stand­off distance on the high-temperature thermo-chemical behavior of the flow. Such a wrong determination can lead, for instance, to an erroneous localiza­tion of intersections of the bow-shock surface with the aft part of the flight vehicle—we discuss examples in Sub-Section 9.2.1—but also to errors in drag prediction etc., related to the interaction of the bow-shock surface with em­bedded shock waves and expansion phenomena.[66]

The absolute value of a potential error in the shock stand-off distance, of course, depends on the effective radius of the stagnation-point area, e. g., the blunt shape of a re-entry vehicle at large angle of attack. If we take as example a body at high Mach number with an effective nose radius of Rb =

1.5 m, we would get in flight, with an assumed mean 7 = 1.2 in the shock layer, a shock stand-off distance of approximately 0.1 m. For perfect gas with Y =1.4 this distance would be approximately 0.2 m, which is wrong by a factor of two.

On the basis of the above relations we obtain an overview of parameters in the shock layer in the nose region of a body, i. e., near the stagnation point, which are affected, if high-temperature real-gas effects are not properly taken into account there. The pressure coefficients cp2 and cp2t in that region for large free-stream Mach numbers M1 become independent of the Mach number, eqs. (6.72) and (6.73), see also Section 6.8, and also do not vary strongly with Y. Hence we can assume that for given large Mach numbers the pressure also does not vary strongly, and that the mean density in the layer between the bow shock and the body surface in the nose region will be proportional to the inverse of the mean temperature there:

pnieansh0ck layer ^ rpl ■ (6.126)

‘meanshock layer

With that result we get qualitatively the differences of flow parameters in the shock layer at the nose region between the flight situation and the perfect-gas situation, i. e., with an inadequate simulation of high-temperature real-gas effects, Table 6.5. The actual differences, of course, depend on the nose shape, and speed and altitude of flight compared to the parameters and the flow situation in ground-simulation. Whether they are of relevance in vehicle design, depends on the respective design margins [2]. However, our goal is to have uncertainties small in ground-simulation data in order to keep design margins small [28].

Shock waves have a definite structure with a thickness of only a few mean free paths. This would mean that for their computation the Boltzmann
equation must be employed [15]. In general the Boltzmann equation is valid in all four flow regimes, Section 2.3, i. e., from continuum to free molecular flow. In the free-molecular flow regime it is used as the so called collisionless Boltz­mann equation. However, due to the very large computational effort to solve it (recent developments in solution algorithms and computer power have im­proved the situation), discrete numerical continuum methods for the solution of the Euler and the Navier-Stokes/RANS equations and their derivatives will be employed in aerothermodynamics whenever possible. For computational cases, where a gap exists between the applicability range of the latter, and that of the Boltzmann equation, the bridging-function concept can be used [16].

The question is now how to solve the computational problem of shock waves in the continuum regime. In [17] it is shown that the structure of a normal shock wave can be computed with a continuum method, i. e., the Navier-Stokes/RANS equations, for pre-shock Mach numbers Mi ^ 2. This is done by comparing solutions of the Boltzmann equation, which employ the Bhatnagar-Gross-Krook model, with solutions of the Navier-Stokes equa­tions.

Since it is the Mach number MN normal to the front of the shock wave, which matters, also the structures of oblique shock waves can be treated with the Navier-Stokes equations, if MN ^ 2. We show this in Fig. 6.16 with the computation case of a reflecting shock wave at a surface without a boundary layer [18]. The case was set up such that the normal Mach number of the incoming shock wave, MN = M1 sin в = 1.5, was equal to that of a case treated in [17]. In Fig. 6.17 the solution of the Boltzmann equation (curve 1) [17] is compared at some station ahead of the reflection location, with solutions of a space-marching, shock-structure resolving scheme of a derivative of the Navier-Stokes equations [18].

For the coarse discretization of the latter solution with about ten nodes in the shock wave in stream-wise direction (curve 2) the comparison is not satisfactory. The fine discretization with about twenty nodes (curve 3) yields a good agreement on the high-pressure side (behind the shock wave). On the low-pressure side (ahead of the shock wave) the agreement is only fair. This could be due to the fact, that in [17] the Prandtl number Pr = 1 was used, and in [18] the actual value.

This brings us back to the original question of how to treat shock waves in flow fields past flight vehicles, which globally are fully in the continuum regime. Obviously a discretization with twenty nodes across the shock struc­ture is beyond the performance range of computers for a long time to come. Even if this would be possible, the problem remains, that shock-wave struc­tures can be described by means of the Navier-Stokes equations only if the relevant shock-wave Mach number is small enough.

In Euler and Navier-Stokes/RANS solutions shock waves therefore are treated as discontinuity surfaces in the frame of the “weak-solution con­cept”, see, e. g., [19]. This “shock-capturing” approach demands to employ the

governing equations in conservative formulation, Appendix A, see also Section 4.3. The “shock-fitting” approach, see, e. g., [20], which is applicable more or less only to describe the bow-shock surface of a vehicle, is now seldom used.

The shock-capturing approach is well proven. In the slip-flow regime, how­ever, where over a blunt body a “thick” shock wave can even merge with the also “thick” boundary layer (merged-layer regime [21]) the shock-capturing approach is somewhat questionable, also in view of the fact that the shock wave has a “secondary” structure due to thermochemical relaxation effects, Sub-Section 5.5.1.

Consider streamline (1) in Fig. 6.1 a). It penetrates the bow-shock surface at a location, where the latter is inclined by approximately 45° against the free-stream direction. The streamline is deflected slightly behind the shock towards the shock surface.

We employ the same reasoning as for the normal shock wave, and define the phenomenological model “oblique shock wave”, which is a plane shock surface, which lies inclined (shock angle 9) to a parallel and uniform super­sonic stream. It is induced for example by a ramp with ramp or deflection angle S, Fig. 6.9. The flow is inviscid, and the ramp surface is a streamline. Behind the oblique shock wave the flow is again parallel and uniform,[62] but now it is deflected by S towards the shock surface, Fig. 6.9.

For a shock wave, like for any wave phenomenon, changes of flow param­eters happen only in the direction normal to the shock wave. This means that the governing entity at any point of a shock surface is the normal Mach number M1n, i. e., the Mach number of the flow velocity component normal to the shock surface u1 at that point, Fig. 6.9:

u Vi sin9 ,, . „ .

MN = — = ————– = Msin6. (6.81)

The shock relations (again for perfect gas) are in principle the same as those for the normal shock, only that now Ui1 replaces u. Prandtl’s relation now reads

uyui = a2 – А“Тг~’2’ (6.82)

у + 1

where V is the component of Vi tangential to the oblique shock wave, Fig. 6.9, which does not change across it[63]

V = Vi = v>2 = VicosO. (6.83)

The Rankine-Hugoniot relations are identical to those for the normal shock wave.

We give now, as we did for the normal shock, some relations for the change of the flow parameters across the oblique shock, Fig. 6.9, in terms of the upstream Mach number M1, the shock-inclination angle O, and the ratio of specific heats у, again see, e. g., [11]:

We can at once take over qualitatively the results for the normal shock wave, when we deal with an oblique compression shock. This means that also across the oblique shock pressure p, density p, temperature T are increas­ing instantly, whereas speed V decreases instantly. The important difference is, that we get subsonic flow behind the shock too, but only for the flow component normal to the shock surface:

M2n oblique shock < 1 (OOO)

Then the question arises, whether the resultant flow behind the oblique shock is subsonic or supersonic, i. e., whether the Mach number M2 = V2/a2 < 1 (strong shock wave) or > 1 (weak shock wave). In fact, as we have indicated in Figures 6.1 to 6.4, behind a detached shock surface at a blunt or a sharp nosed body we find a subsonic pocket, which can be rather large, Fig. 6.3 b). This implies that in the vicinity of the point, where the shock surface is just a normal shock, the flow behind the oblique shock surface is subsonic, i. e., M2 < 1, while farther away it becomes supersonic.

With the help of eq. (6.87) we determine, at what angle в this happens. We set M2 = 1 and obtain the critical angle в* (sonic limit) in terms of the upstream Mach number Mi, and the ratio of specific heats 7

1

47M2 ‘

(7 + 1)M2 + (7 – 3) + 07 + 1)[(7 + mi + 2(7 – 3)M2 + (7 + 9)] .

(6.1O1)

For —в* в +в* we have subsonic, and above these values supersonic

flow behind the oblique shock surface. Of course we cannot determine on this basis the shape of the sonic surface, i. e., the shape of the surface on which between a detached shock and the body surface М =1.

For large upstream Mach numbers, i. e., Мі ^ ж, we find the limiting values of в

For a perfect gas with y = 1.4 they are 9*Ml^^ = ± 67.79°. In a large Mach number range 9* varies only weakly, Fig. 6.10. For 7 = 1.4 the value of 9* drops from 90° at Mi = 1 to a minimum of approximately 61° at Mi « 1.8, and rises then monotonously to the limiting value 9*Ml= 67.79°, see also Figs. 6.12 and 6.13.

Because for a given Mach number MTO, 90° ^ 9* ^ pTO, Fig. 6.19, the sonic line can meet the bow shock, especially at low Mach numbers, far away from the body surface, Fig. 6.11.

Now to the flow parameters at large upstream Mach numbers, i. e., M1 ^ ж. Generally we obtain the same limiting values across the oblique shock for the ratios of densities, temperatures etc. in the same way as those across the normal shock. We note here only the following exceptions

and finally

^ ж indeed is larger than that behind the normal shock portion. This holds for all finite Mach numbers Mi > 1, too.

The relations for the velocity components at M1 ^ ж behind the oblique shock are

V2 4y sin2 в

——- >■ 1—– — ——

П2 (7 +1)2′

We note finally that the shape of the bow-shock surface, either detached or attached, depends for a perfect-gas flow only on the free-stream Mach number Mi, the ratio of specific heats 7, and the effective body shape with its embedded shock wave and expansion phenomena.[64] All shock relations given above for normal and oblique shocks are valid always locally on bow shock and embedded shock surfaces. If real-gas effects, in particular high-temperature effects, are present, they influence the bow-shock shape, Sub-Section 6.4.1, but also the shock layer, Section 5.4.

Explicit relations, which connect the shock properties to the body shape, are available only for the two-dimensional inviscid ramp flow. These are the oblique-shock relations discussed above.

In Fig. 6.9 we have indicated the angle 5, by which the streamlines are deflected behind the oblique shock. If we select such a streamline, we obtain the two-dimensional ramp flow, Fig. 6.1 d). We find for it the relation between the shock-wave angle в and the deflection angle 5, in terms of the upstream Mach number M1, and the ratio of specific heats 7

If the deflection angle 5 is given, the shock-wave angle в can be determined with an iterative scheme. For a quick look-up we give in Figs. 6.12 and 6.13 the shock-wave angle as function of the deflection angle for selected upstream Mach numbers M1 for air with 7 = 1.4 [11].

Eq. (6.114) can be rearranged to yield

For small deflection angles 5 it becomes

—Л:/2 tail 0 ] 5,
M12 sin2 в – 1	(6.116)

01<5 —л0 = A4 = sin

In Figs. 6.1 b) and c) we have noted that at a given free-stream Mach number a critical cone angle exists, at which the bow-shock surface becomes detached (this is called the strong shock wave). Such an angle, 5max, exists also for ramp flow. The (largest possible) shock-wave angle Qsmaxrelated to it—the weak shock-wave limit—is found with

This relation is similar to that for the sonic limit, eq. (6.101). The curves in the respective chart, Figs. 6.12 and 6.13, are similar, too.

Cone flow, of course, is different from ramp flow, because it has, in a sense, three-dimensional aspects [7]. For a given upstream Mach number Mi and equal values of ramp angle 6 and (circular) cone half-angle a we find a larger shock-wave angle в for the ramp flow, Fig. 6.14.

The flow behind the attached oblique ramp shock is uniform and parallel to the ramp surface. The flow behind the attached shock surface of the cone at zero angle of attack of the cone is also circular, see, e. g., [4, 11]. It is first, i. e., directly behind the shock surface, deflected according to eq. (6.114). The streamlines then approach asymptotically the direction of the cone surface, i. e., the semi-vertex angle a. All flow quantities are constant on each con­centric conical surface, which lies between the shock surface and the body surface. The Mach number on the body surface, and hence also away from it, can be subsonic or supersonic, depending on the pairing Mand a. Graphs of the flow parameters for circular cone flow can be found in [11].

In closing this sub-section we have a look at shock intersections and re­flections. Such phenomena can occur, see Chapter 9, in flows past ramps, at
the interaction of embedded shocks and the bow shock but also in flows in propulsion inlets etc., Section 6.1.

Intersections of shock waves belonging to opposite families in uniform and parallel flow are symmetric, if the incoming shocks are symmetric, i. e., of equal strength, Fig. 6.15 a), and asymmetric, if not, Fig. 6.15 b).[65] In the symmetric case the streamlines are first deflected towards the shock waves. Behind the intersected shocks they are again parallel and uniform. Indeed, this is the governing condition for the shock intersection.

This also holds for the asymmetric case. However, because the involved shocks are of different strength, we get behind the intersection two parallel and uniform streams with different total-pressure loss. Since the static pres­sures are the same, these two streams have different speeds. The result is, in terms of inviscid flow, a slip surface, which in reality is a vortex sheet, Fig. 6.15b). This vortex sheet can become unstable via the Kelvin-Helmholtz mechanism [13, 14].

a)

Fig. 6.15. Schematic of intersections of shocks belonging to opposite families : a) symmetric intersection, b) asymmetric intersection.

Shock reflection at a wall can be interpreted, in inviscid flow, as, for instance, one half of a symmetric shock intersection (the upper half of Fig. 6.15 a)). In viscous flow the interaction with the boundary layer must be taken into account. If the effective shock strength is small, a thickening of the boundary layer will happen, otherwise boundary-layer separation will occur, Section 9.1.

We note finally that intersecting shocks of the same family in a triple point T form a single stronger shock [4]. In T also a slip line and an expansion fan originate, see Fig. 9.6 c) in Sub-Section 9.2.1.

In the following two sections we study normal and oblique shock waves and discuss the Rankine-Hugoniot conditions, which connect the flow properties upstream and downstream of them. We generally do not give tables and charts for the determination of the flow properties. For oblique shock waves, however, a chart is given with the shock-wave angle в as function of the flow-deflection angle S for a number of upstream Mach numbers Mi. Further given is a figure with the Prandtl-Meyer angle v as function of the upstream Mach number M1. Both the chart and the data for that figure—taken from [11]—are for perfect gas (air, у = 1.4).

6.3.1 The Normal Shock Wave

Consider the shock surface in Fig. 6.1 a). On the axis, the symmetric shock surface lies orthogonal to the supersonic free stream. Its thickness is very small compared to its two principle radii. We approximate the shock surface locally by a plane surface. If the body lies very far downstream, we can
consider the flow behind the normal portion of the shock wave as parallel and uniform, like the flow ahead of the shock wave.

With these assumptions we have defined the phenomenological model of the “normal shock wave”, that is a plane shock surface, which lies orthogonal to a parallel and uniform supersonic stream. Behind it the flow is again parallel and uniform, but subsonic: the normal shock wave is a strong shock wave.

We study now the normal shock wave, like any aerodynamic object, in an object-fixed frame, Fig.6.8 (see also Fig. 5.4).[59] We know that the shock wave is a viscous layer of small thickness with strong molecular transport across it. In continuum flow it usually can be considered approximately as a discontinuity in the flow field. The flow parameters, which are constant ahead of the shock surface, change instantly across that discontinuity, Fig. 5.4 b), and are constant again downstream of it. How can we relate them to each other across the discontinuity?

Obviously there are entities, which are constant across the discontinuity, as is shown in the following. These are the mass flux, the momentum flux and the total enthalpy flux. We find the describing equations by reduction of the conservative formulations of the governing equations, eq. (4.83), eq. (4.33), and eq. (4.64) to one-dimensional inviscid flow (we omit dx)

d(pu) = 0, (6.39)

d(pu2 + p)=0, (6.40)

d(puh) — udp = 0. (6.41)

Eq. (6.41) can be combined with eqs. (6.3) and (6.39) to yield a form, which immediately can be integrated

dpu(h + – u2)] = 0. (6-42)

The integration then yields the constant entities postulated above:

Piui = P2U2, (6.43)

Piu2 + pi = P2u2 + P2, (6.44)

ріпі (hi + 5/2) = p2’U2 (ft? + (6.45)

The relation for the total enthalpy flux, eq. (6.45), reduces with eq. (6.43) to

hi + qiq = I12 + ^«2- (6.46)

This relation says, that the total enthalpy ht is constant across shock waves:

ht! = ht2 = ht. (6.47)

This important result is in contrast to that for the total pressure. We will see below, that pt is not constant across shock waves.

Relation (6.47) indicates also, that the maximum speed Vm is constant across the shock wave. For a perfect gas the total temperature Tt, the total speed of sound at, and the critical speed of sound a* are constant across the shock wave, too.

For the following considerations we assume perfect gas flow. We combine eqs. (6.43) and (6.46) and find

Pi, P‘2 , ,r.

——– b ‘Mi =——— h гг-2 • (6.48)

Piui P2U2

After introducing the critical speed of sound a*, we arrive at Prandtl’s relation[60]

and at the Rankine-Hugoniot relations

P2_ _ (7 + 1)P2 ~ (7 ~ l)pi Pi (7 + !)Pi – (7 – l)/>2 ’

P2 _ (7 + 1)P2 ~ (7 ~ l)Pi Pi (7 + l)Pi – (7 – 1)P2 ’

P2 – Pi P2 + P

———- = 7——- :—– •

P2 – Pi P2 + Pi

Eq. (6.49) can be written in terms of a Mach number related to the critical speed of sound M* = u/a* as

MФ1 M*2 = 1. (6.53)

The Mach number M* is related to the Mach number M by eq. (6.20). Both have similar properties, which are shown in Table 6.1.

Prandtl’s relation in either form implies the two solutions, that the flow speed ahead of the normal shock is either supersonic or subsonic, and behind the normal shock accordingly subsonic or supersonic, i. e., that we have either a compression shock or an expansion shock.

We developed the phenomenological model “normal shock wave” starting from a supersonic upstream situation. In our mathematical model we did not make an assumption about the upstream Mach number. The question is now, whether both or only one of the two solutions is viable.

Before we decide this, we give some relations for the change of the flow parameters across the normal shock wave in terms of the upstream Mach number Mi, and the ratio of specific heats 7, see, e. g., [11]:

6.65)

We make a plausibility check by putting M = 1 into the above relations. We find for the ratios of density, velocity, temperature, speed of sound, pres­sure, and total pressure, as well as for the Mach number M2 the values “1”, as was to be expected. The pressure quotient cp2 is zero, whereas the ratios of pressure to total pressure remain, also as to be expected, smaller than one.

The pressure coefficient at the stagnation point, cpt2, is the same as that for shock-free flow.

Now we come back to the question, which solution, that with supersonic or that with subsonic flow upstream of the shock wave, is viable. To decide this, we consider the change of entropy across the shock wave. We note from eq. (6.60) that a total pressure decrease is equivalent to an entropy rise.

We just plug in numbers into eq. (6.60), choosing 7 = 1.4. The results are given in Table 6.2. We find that only for Mach numbers Mi > 1 the total pressure drops across the shock, i. e., only for supersonic flow ahead of the normal shock we observe a total pressure decrease, i. e., an entropy rise. Hence the above relations for the change across the normal shock wave are only valid, if the upstream flow is supersonic.

Expansion shocks are not viable, because they would, Table 6.2 (the first four Mach numbers M1 < 1), lead to a rise of the total pressure, i. e., an entropy reduction, which is ruled out by the second law of thermodynam­ics. Of course, we should have known this from the beginning, because we introduced the shock surface as a thin viscous layer.

The results given in Table 6.2 demand a closer inspection. We find from the second bracket of eq. (6.60) a singularity at

Ml = (6.66)

2Y

which with y =1.4 becomes M2 = 1/7.

Below that value the total pressure ratio across the normal shock is un­defined. From eq. (6.55) we see that for Mi = — l)/27 the temperature

behind the shock wave becomes zero, i. e., that for this upstream Mach num­ber the maximum velocity Vm, eq. (6.15), is reached.

The result thus is: if one assumes an expansion shock, the smallest pos­sible upstream Mach number is that, for which behind the shock the largest possible speed is reached. Behind this result lies the fact that in our mathe­matical model the conservation of the total enthalpy is assured also for the physically not viable expansion shock.

Just plugging in numbers as we did above is somewhat unsatisfactory. The classical way to show that only the supersonic compression shock is viable,
which however is restricted to Mach numbers not much different from Mi = 1, is to make a series expansion, which yields

— = eMS2~Sl)/fi = і———— ——(Mі —l)3 + terms of higher order. (6.67)

Pti 3(Y + 1)2

Because of the odd exponent of the bracket of the second term on the right-hand side of the equation we see that indeed a total-pressure decrease, and hence an entropy increase, across the normal shock happens only for Mi > 1.

With this result we obtain from eqs. (6.54) to (6.57) that across the normal shock wave pressure p, density p, temperature T are increasing instantly, whereas speed u and Mach number M decrease instantly. The most important result is, that we always get subsonic flow behind the normal shock wave:

M2 I normal shock < 1 (6.68)

For large upstream Mach numbers, i. e., M1 ^ ж, we find some other in­teresting results.[61] They permit us, for instance, to judge in hypersonic flow— at large temperatures—the influence of a changed 7 on the flow parameters downstream of a normal shock.

We obtain finite values for the ratios of velocities, and densities, as well as for the Mach number and the pressure coefficient behind the normal shock for M1 ^ ж

and for the pressure coefficients

(6.72)

and

Static temperature, speed of sound, and pressure ratios, however, for M1 tend to infinite values

obtained

For this and the following sections we assume steady, inviscid, iso-energetic, one-dimensional flow of a perfect gas.[57] Used is the nomenclature (x, u) given in Fig. 4.1, Section 4.1. The reader should note that we use the the same sym­bol “cpv for both the specific heat at constant pressure and for the pressure coefficient.

We treat in this section continuous shock-free flow as a prerequisite, and then in the Sub-Sections 6.3.1 and 6.3.2 normal and oblique shock waves.

The governing equations in Section 4.3 for mass, momentum and energy transport, eqs. (4.83), (4.27), (4.63) reduce for one-dimensional flow to (we omit dx)

The energy equation in differential form, eq. (6.4), can be rewritten by introducing the perfect-gas law, eq. (5.1), and some of the relations from eq. (5.10):

——— d — ] T и du = 0. (6-5)

Y- 1 pj

This, combined with eq. (6.3), yields the pressure-density relation for isentropic flow

— = constant = -^7. P7 Pt

The subscript ‘t indicates the reservoir or ‘total’ condition. pt is the total pressure. Eq. (6.6) can be generalized to

Here a is the speed of sound, the speed at which disturbances propagate through the fluid. Sound waves have so small amplitudes, that they can be considered as isentropic [4]: s = constant. Hence for perfect gas

In adiabatic flow the second law of thermodynamics demands:

s — sref ^ 0. (6.12)

Isentropic flow processes are defined by

s — sref = 0 : s = constant. (6.13)

The integrated energy equation eq. (6.5) in the familiar form for perfect gas reads, with cp the specific heat at constant pressure[58]

CpT + У«2 = cpTt. (6.14)

From this relation we see, that with a given total temperature Tt by expansion only a maximum speed Vm can be reached. This is given when the “static” temperature T reaches zero (expansion limit):

Vm = v72cpTt. (6.15)

The maximum speed can be expressed as function of at and y:

Locally the “critical” or “sonic” condition is reached, when the speed u is equal to the speed of sound a

Eq. (6.14) combined with eq. (6.7) gives the equation of Bernoulli for compressible isentropic flow of a perfect gas:

12

-pur + p=pt.

The first term on the left-hand side is the dynamic pressure q (p is the “static” pressure)

Ч=ри2. (6.23)

The concept of dynamic pressure is used also for compressible flow. There, however, it is no more simply the difference of total and static pressure. Eq. (6.23) can be re-written for perfect gas as, for instance

4 = pu2 = lpM2. (6.24)

The free-stream dynamic pressure q= 0.5 pv^ is used also in aerother – modynamics to non-dimensionalize pressure as well as aerodynamic forces and moments.

For perfect gas we can relate temperature, density and pressure to their total values, and to the flow Mach number:

Tt=T, (6.25)

1

Pt=P (l + ^M2)7" , (6.26)

Pt=P (і + ^-М2У’ 1 . (6.27)

Similarly we can relate temperature, density, pressure and the speed u to their free-stream properties ‘to’:

with ^$TO UTO>/aTO •

The pressure coefficient Cp is

where qTO is the dynamic pressure of the free-stream. For perfect gas the pressure coefficient reads with eq. (6.24)

= (то1) (,U2)

The expansion limit (see above) is reached with p ^ 0. With this we get

the vacuum pressure coefficient

The pressure coefficient can be expressed in terms of the ratio local speed to free-stream value u, and the free-stream Mach number M„o

as well as in terms of the local Mach number M and its free-stream value

With the help of eq. (6.27) cp can be related to the total pressure pt

In subsonic compressible flow we get for the stagnation point (isentropic compression) with p = pt, u = (M =) 0

In the case of supersonic flow of course the total-pressure loss across the shock must be taken into account (see next section).

For incompressible flow we find the pressure coefficient with the help of eq. (6.22) and constant pt

u2

cP = 1——- —, (6.38)

uL

and note finally that at a stagnation point Cp for compressible flow is always larger than that for incompressible flow with cp = 1, see, e. g., [2].

Shock waves can occur if the speed in a flow field is larger than the speed of sound. Across shock waves the flow speed is drastically reduced, and density, pressure and temperature rise strongly.[53] The phenomenon is connected to the finite propagation speed of pressure disturbances, the speed of sound. At sea level this is аж « 330 m/s or « 1,200 km/h. If the vehicle is flying slower, that is with subsonic speed Ыж < 1, the pressure disturbances travel ahead of the vehicle: the air then gives gradually way to the approaching vehicle. If the vehicle flies faster than the speed of sound, with supersonic Ыж > 1 or hypersonic Ыж ^ 1, speed, the air gives way only very shortly ahead of the vehicle, and almost instantaneously, through a shock wave. In this case we speak about the (detached) bow shock, which envelopes the body at a certain distance with a generally convex shape. This bow shock moves in steady flight with the speed of the flight vehicle, i. e., it is fixed to the vehicle-reference frame, and (for steady flight) it is a steady phenomenon.

However, shock waves can also be embedded in the vehicle’s flow field, if locally supersonic flow is present. Embedded shocks may occur already in the transonic flight regime, where the free-stream speed still is subsonic, but super-critical, and in the supersonic and hypersonic flight regimes. If the shock wave lies orthogonal to the local speed direction (normal shock), we have subsonic speed behind it, if it is sufficiently oblique, the speed behind it is supersonic. We treat these phenomena in Sub-Sections 6.3.1 and 6.3.2, but refer for a deeper treatment to the literature, for instance [3]-[6].

Depending on the flight-vehicle configuration a flow field pattern with either subsonic or supersonic, or mixed, speed is resulting behind the bow

shock, Fig. 6.1. At a blunt-nosed body, Fig. 6.1 a), we always have a detached bow shock with a small subsonic (M < 1) pocket behind it. The flow then is expanding to supersonic speed again.

The bow shock is dissipating far downstream of the vehicle, while its inclination against the free-stream direction approaches asymptotically the free-stream Mach angle pTO. The Mach angle p reads

1

sm и = —. h M

It is defined in the interval 1 A M A to, Fig.

At a sharp-nosed cone, if the opening angle is small enough (sub-critical opening angle), the bow shock is attached, Fig. 6.1 b), the speed behind

it is subsonic/supersonic or supersonic, Sub-Section 6.3.2.[54] With sufficiently large opening angle (super-critical opening angle), Fig. 6.1 c), the bow shock is detached, the speed behind it is initially subsonic, and then, in the figure at the following cylinder, supersonic again.

Lastly, at the ramp we have an embedded shock wave. If the ramp angle is sub-critical, the shock will be “attached”, with supersonic speed behind the oblique shock, Fig. 6.1 d). If the ramp angle is sufficiently large (super­critical), the shock will be a “detached” normal shock, with a subsonic pocket behind it. Then the shock will bend around, like shown for the cone in Fig. 6.1 c).

The shape of the bow shock and the subsonic pocket are in principle different for RV’s and (airbreathing) CAV’s, Fig. 6.3. The slender CAV flies at small angle of attack, with a small subsonic pocket ahead of the small­bluntness nose, Fig. 6.3 a). The RV with its anyway blunt shape during re-entry flies at large angle of attack, and has a large subsonic pocket over the windward side, Fig. 6.3 b).[55]

Why are bow-shock shapes of interest? They are of interest because the entropy rises across them, which is equivalent to a loss of total pressure. The consequence is the wave drag, which is a another form of aerodynamic drag besides induced drag, skin-friction drag, and viscosity-induced pressure drag (form drag) [7].

The entropy rise is largest where the shock is normal to the free-stream flow (normal shock), and becomes smaller with decreasing inclination of the shock against the free-stream (oblique shock). In this respect we note two fundamentally different cases, Sub-Section 6.4.2:

1. For a symmetric body at zero angle of attack, Fig. 6.1 a), the—stagnation – point—streamline on the axis carries the largest entropy rise As, because the bow shock is orthogonal to it.

2. For an asymmetric body at angle of attack, Fig. 6.22, the streamline through the locally normal bow-shock surface carries the largest entropy rise, Aso = Asmax. The stagnation-point streamline, however, which penetrates the bow-shock surface at at a certain distance from the locally normal bow-shock surface, carries a lower entropy rise, Asi < Asmax. Hence in this case the stagnation-pressure loss is smaller than in the symmetric case.

In general it can be stated: the larger the portion of the bow-shock surface with large inclination against the free stream, the larger is at a given free – stream Mach number the wave drag.

We understand now why a RV, which actually flies a braking mission, is blunt and flies at a large angle of attack, see also [2]. In addition, as we have seen in Section 3.2, large bluntness leads to thick boundary layers, and hence to a large radiation-cooling efficiency. The blunt vehicle shape thus serves both large drag and low surface temperatures, and hence thermal loads, which passive thermal protection systems can cope with, Fig. 3.2.

The situation is different with (airbreathing) CAV’s. Such vehicles must have a low total drag, therefore they are slender, have a small nose bluntness, and fly at small angles of attack. Here the small nose bluntness has an adverse effect regarding radiation cooling. Small nose radii lead to thin boundary layers, and hence to small radiation-cooling efficiency. In CAV design trade­offs are necessary to overcome the contradicting demands of small wave drag, and sufficient radiation-cooling efficiency.

Bow-shock shapes and shock locations in general are of interest also for another reason. They can lead to shock/boundary-layer and shock/shock/ boundary-layer interactions, which we treat in Section 9.2. These “strong” interactions—locally—can cause large mechanical and thermal loads (hot spots) with possible severe consequences for the airframe integrity. They go along also with local separation and unsteadiness phenomena.

In Fig. 6.4 we show such a strong-interaction situation, which can arise at both CAV and RV wings with a second delta part, which is needed to enhance the aspect ratio of the wing in order to assure proper flight characteristics at low-speed and high-angle of attack flight.[56] Other locations where these interactions can occur, are vertical stabilizers, trim and control surfaces, inlet elements, and pylons.

Finally we look at the role, which shock waves play in hypersonic air­breathing propulsion and in aerothermodynamic airframe/propulsion inte­gration of CAV’s [9]. It is recalled that the flight-speed range of turbojet engines is up to Ыж = 3 to 4, that of ramjet engines 3 ^ M^ 6, and of scramjet engines 6 ^ M^ 12.

Important in our context is the fact that the pre-compressor Mach number of turbojet engines, like the combustion-chamber Mach number of ramjet engines is M = 0.4 to 0.6, and the combustion-chamber Mach number of scramjet engines M = 2 to 3. These numbers show that the characteristic pre­engine Mach numbers are much lower than the actual flight Mach numbers.

The only way to attain these engine Mach numbers is to decelerate the air stream via shock waves [9]. Normal shock waves would give too large total-pressure losses, hence one or more oblique shock waves are employed. In Fig. 6.5 we show the schematic of a three-ramp inlet, which is typical for a ramjet engine.

The inlet-onset flow a) is pre-compressed by the lower side of the forebody [2]. The forebody boundary layer must be diverted by the boundary-layer diverter, in order to avoid unwanted distortion of the inlet flow, and finally of the flow entering the engine. In the outer compression regime b) the three oblique ramp shocks, together with the vehicle’s bow shock, are centered on the lip of the inlet cowl (shock-on-lip situation at the design flight Mach number and angle of attack).

The cowl lip, which is blunt in order to withstand thermal and mechanical loads, causes an own bow shock, which interacts with the incoming shocks, Fig. 6.6. This in principle is a shock/shock/boundary-layer interaction of Edney IV type, Sub-section 9.2.2. For practical reasons usually the exact shock-on-lip situation is avoided, because of the associated adverse interaction effects [2].

In Fig. 6.5 the part of the inlet is indicated, where the internal compression by a train of oblique shocks, region c), finally leads to sub-sonic flow, which is further decelerated in the diffuser d) to the above mentioned small pre-engine Mach number of the flow entering the combustion chamber.

A fully oblique shock train is shown in Fig. 6.7 for a two-strut scramjet geometry at a low (off-design) flight Mach number [10]. In both the outer and in inner flow path we see shock reflections, shock intersections, Sub-Section 6.3.2, expansion waves, Sub-Section 6.5, and slip surfaces at the sharp strut trailing edges.

In Sub-Section 4.3.1 we have seen that if the Reynolds number characterizing a flow field is large enough, we can separate the flow field into inviscid and viscous portions. From Fig. 2.3, Section 2.1, we gather that the unit Reynolds numbers in the flight domain of interest are indeed sufficiently large. This does not mean that aerodynamic properties of hypersonic vehicles can be described fully by means of inviscid theory. This is at best possible for the longitudinal motion of re-entry vehicles.

At transonic, supersonic and hypersonic flight we observe the important phenomenon of shock waves. The shock wave is basically a viscous phe­nomenon, but in general can be understood as a flow discontinuity embedded in an flow field. In this chapter we look at shock waves as compressibility phenomena occurring in the inviscid flow fields past hypersonic flight vehi­cles. We treat their basic properties, and also the properties of the isentropic Prandtl-Meyer expansion. Of importance in hypersonic flight-vehicle design is the stand-off distance of the bow-shock surface at blunt vehicle noses. We investigate this phenomenon as well as the effects of entropy-layer swallowing by the vehicle’s boundary layers.

Also of interest for the development of boundary layers is the change of the unit Reynolds number across shock waves. We will see that an increase of the unit Reynolds number only will occur, if the shock wave is sufficiently oblique. Then a boundary layer behind a shock wave close to the body surface will be thinned. This subsequently changes the thermal state of the surface which influences the skin friction, the heat flux in the gas at the wall and the radiation-adiabatic temperature of the radiation-cooled surface.

Basics of Newtonian flow are considered then. Newtonian flow is an inter­esting limiting case, and the related computation method, with appropriate corrections, is an effective and cheap tool to estimate (inviscid) surface pres­sure and velocity fields. Related to Newtonian flow is the hypersonic shadow effect, which can appear in hypersonic flow past flight vehicles, in particular if they fly at large angle of attack. It is characterized by a concentration of the aerodynamic forces with increasing Mach number at the windward side of the vehicle, the “pressure side” of classical aerodynamics. The leeward side, the “suction side”, is in the “hypersonic shadow” of the body, and loses its

(C Springer International Publishing Switzerland 2015 E. H. Hirschel, Basics of Aerothermodynamics,

DOI: 10.1007/978-3-319-14373-6 _6 role as a force-generating surface. We do not discuss this phenomenon, but refer to [1], where several examples are given, also regarding control-surface efficiency.

The chapter closes with the discussion of a principle being very important for the aerodynamic shape definition and the ground-facility simulation of hypersonic flight vehicles, the Mach-number independence principle.

In this chapter we assume throughout flow with perfect gas. That may be a diatomic gas (air) with 7 = 1.4 or a monatomic gas with 7 = 1.666, Section 5.2. If high-temperature real-gas effects are present in the flow, closed rela­tions like those given in this chapter for perfect gas are not available. However, the Lighthill gas with 7 = 1.333 and the Yeff-approach, see, e. g., [2], to a cer­tain degree permit to treat such flows. Discrete numerical methods of fluid mechanics, however, fully permit to simulate flows with high-temperature real-gas effects, see the examples in Chapter 5.

For the computation of equilibrium or non-equilibrium flows different air models are available. If ionization, Section 2.1, can be neglected, one can work with the five species (N2, O2, N, O, NO) model with 17 reactions. If ionization is to be taken into account, the eleven species model with 33 reactions, [7], can be employed. In this section we give some references to liter­ature on detailed computation models for the thermo-chemical and transport properties of air in the temperature and density/pressure range of interest for both thermo-chemical equilibrium and non-equilibrium flow, see also [6].

Equilibrium Flow. The transport properties of air can be determined with the models presented in Chapter 4. Characteristic data for thermo-chemical properties can be found, e. g., in [7], and for transport properties in [29]-[32]. Regarding the general state of the art, including modeling problems, we refer to the review paper [9].

It is recommended to work in computation methods with state surfaces of thermodynamic and transport properties, because their use is computa­tionally more efficient than that of basic formulations. As was mentioned in Sub-Section 4.2.5 such state surfaces are available in the literature, e. g., [33] (only up to 6,000 K), [34, 35].

In [36] approximations of such state surfaces are given. We show as exam­ple in Fig. 5.14 the temperature T as function of the two variables density p (here in the form lg(p/po)), and internal energy e, and in Fig. 5.15 the ratio of specific heats 7 as function of lg(p/po) and lg(pp0/pp0). Note that van der Waals effects, Section 5.1, are not included.

Examples of state surfaces of the viscosity and the thermal conductivity are given in Sub-Section 4.2.5.

Non-equilibrium Flow. Flows with thermo-chemical non-equilibrium must be modeled with the help of the basic formulations. Characteristic data again can be found, e. g., in [7] and [11]. We refer also to the review paper [9] regarding the general state of the art, including modeling problems.

Transport properties are treated like in equilibrium flow by taking into account the momentarily present thermo-chemical state of the gas with ap­propriate formulations, Sub-Section 4.2.5.

Special regard must be given to the boundary conditions of the species continuity equations, if finite catalytic surfaces are to be modeled. In [19], for instance, we find the formulations for the five and the eleven species gas model, and also general slip-flow boundary conditions.

5.3 Problems

Problem 5.1. A gas has the ratio of specific heats 7 = 1. Show that this is equivalent to f = to.

Problem 5.2. The reservoir enthalpy of a ground-simulation facility with air as test gas is ht = 20 MJ/kg. What are a) the maximum possible speed and the Mach number at the exit of the nozzle, b) the speed and the Mach number, if the static temperature at the nozzle exit is Texit = 1,000 K (assume a Lighthill gas at the exit), c) the speed and the Mach number, if in addition 20 per cent of the reservoir enthalpy is frozen? d) What is the general result?

Problem 5.3. The stagnation temperature is the temperature at the stag­nation point of a blunt body. For inviscid flow it is the total temperature Tt. A CAV is approximated by a flat plate and flies with v= 2 km/s at 30 km altitude. What is a) Tt, if we assume 7 = 1.4 throughout? What is Tt, if we mimic high-temperature real-gas effects with b) Yeff = 1.3, and c) with Yeff = 1.1? d) What y is in a realistic range?

Problem 5.4. How does the uppermost curve of q in Fig. 5.11 scale with the ж-dependence indicated in eq. (3.27)? The result in Fig. 5.11 was obtained for a constant wall temperature, eq. (3.27) holds for a constant wall temper­ature, too. Measure q at ж = 1 m, and compare with the measured value at ж = 6 m.