Category Basics of Aero – thermodynamics

Laminar-turbulent transition in high-speed flow is a phenomenon with a mul­titude of possible instability and receptivity mechanisms, which depend on a multitude of flow, surface and environment parameters. In the frame of this book only an overview over the most important issues can be given. Detailed introductions to the topic are found in, e. g., [11]—[14].

Possibly two basic transition scenarios can be distinguished, which, how­ever, may overlap to a certain degree:[119]

1. regular transition,

2. forced or by-pass transition.

These two scenarios can be characterized as follows:

— Regular transition occurs if—once a boundary layer is unstable—low – intensity level disturbances, which fit the receptivity properties of the un­stable boundary layer, undergo first linear, then non-linear amplification(s), until turbulent spots appear and actual transition to self-sustained turbu­lence happens. In [15] this is called “transition emanating from exponential instabilities”.

This scenario has been discussed in detail in the classical paper by M. V. Morkovin, [12], see also [13], who considers the (two-dimensional) laminar boundary layer as “linear and non-linear operator” which acts on small disturbances like free-stream vorticity, sound, entropy spots, but also high – frequency vibrations. This begins with linear amplification of Tollmien – Schlichting type disturbance waves, which can be modified by boundary – layer and surface properties like those which occur on real flight-vehicle configurations: pressure gradients, thermal state of the surface, three – dimensionality, small roughness, waviness and so on. It follow non-linear and three-dimensional effects, secondary instability and scale changes and finally turbulent spots and transition.

Probably this is the major transition scenario which can be expected to exist on CAV’s. An open question is how propulsion-system noise and airframe vibrations fit into this scenario.

— Forced transition is present, if large amplitude disturbances, caused, e. g., by surface irregularities, lead to turbulence without the boundary layer acting as convective exponential amplifier, like in the first scenario. Morkovin calls this “high-intensity bypass” transition.

Probably this is the major scenario on RV’s with a thermal protection system (TPS) consisting of tiles or shingles which pose a surface of large roughness. Forced transition can also happen at the junction, of, for in­stance, a ceramic nose cone and the regular TPS, if under thermal and mechanical loads a surface step of sufficient height appears. Similar sur­face disturbances, of course, also can be present on CAV’s, but must be avoided if possible. Attachment-line contamination, Sub-Section 8.2.4, also falls under this scenario.

Boundary-layer tripping on wind-tunnel models, if the Reynolds number is too small for regular transition to occur, is forced transition on purpose. However, forced transition can also be an—unwanted—issue in ground- simulation facilities, if a large disturbance level is present in the test section. Indeed Tollmien-Schlichting instability originally could only be studied and verified in (a low-speed) experiment after such a wind-tunnel disturbance level was discovered and systematically reduced in the classical work of G. B. Schubauer and H. K. Skramstadt [16]. In supersonic/hypersonic wind tunnels the sound field radiated from the turbulent boundary layers of the tunnel walls was shown by J. M. Kendall, [17], to govern transition at M = 4.5, see the discussion of L. M. Mack in [18].

In the following sub-sections we sketch basic issues of stability and tran­sition, and kind and influence of the major involved phenomena. We put emphasis on regular transition.

The state of the boundary layer, laminar or turbulent, strongly influences the thermal state of the surface, in particular if the surface is radiation cooled. The thermal state governs thermal surface effects and thermal loads, as well as the skin friction. Regarding the thermal state, a strong back-coupling exists to the state of the boundary layer. In particular the behavior of hydrodynamic stability, and hence laminar-turbulent transition, are affected by the thermal state of the surface, too.

We have seen that laminar-turbulent transition strongly rises the radia­tion-adiabatic temperature, Figs. 3.3 and 7.10. This is important on the one hand for the structure and materials layout of a hypersonic flight vehicle. On the other hand, transition rises also the wall shear stress, Fig. 7.11, to a large extent. Both the temperature and the shear stress rise are due to the fact that the characteristic boundary-layer thickness of the ensuing turbulent boundary layer, the thickness of the viscous sub-layer, Svs, is much smaller than the characteristic thickness of the—without transition—laminar boundary layer, Fig. 7.6. The latter is the (flow) boundary-layer thickness S.

Boundary-layer turbulence and its modeling is a wide-spread topic also in aerothermodynamics, but perhaps not so much—at least presently—the origin of turbulence, i. e., the phenomenon of laminar-turbulent transition and its modeling.

RV’s in general are not very sensitive to laminar-turbulent transition. Transition there concerns mainly thermal loads. Above approximately 60 to 40 km altitude the attached viscous flow is laminar, below that altitude it becomes turbulent, beginning usually at the rear part of the vehicle. Because the largest thermal loads occur at approximately 70 km altitude [1], the tra­jectory part with laminar flow is the governing one regarding thermal loads. Of course other trajectory patterns than the present baseline pattern, espe­cially also contingency abort trajectories, can change the picture. Transition phenomena, however, may also appear at high altitudes locally on RV’s, for instance, on deflected trim or control surfaces, due to shock/boundary-layer interaction and local separation.

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

DOI: 10.1007/978-3-319-14373-6 _8

Important is the observation that due to the large angles of attack, at least down to approximately 40 km altitude, Fig. 1.3 in Section 1.2, the boundary – layer edge Mach numbers are rather small at the windward side of RV’s.[116] At the windward side of the Space Shuttle Orbiter, the transition location lies at approximately 90 per cent vehicle length at approximately 50 km altitude while a « 35°, and has moved forward to approximately 10 per cent vehicle length at approximately 40 km altitude while a « 30° [2]. This means that on RV’s in general laminar-turbulent transition happens at the windward side actually not in a hypersonic boundary layer, but in an at most low supersonic boundary layer, however one with special properties, Section 1.2.

On (airbreathing) CAV’s, laminar-turbulent transition is not only a mat­ter of thermal loads, but also, since such vehicles are drag-sensitive in general, a matter of viscous drag and of airframe/propulsion integration, see, e. g., [3].

For the US National Aerospace Plane (NASP/X-30) it was reported that the uncertainty of the location of laminar-turbulent transition affects the take-off mass of the vehicle by a factor of two or more [4]. NASP/X-30 was an extremely ambitious project [5]. Strongly influenced by laminar-turbulent transition were mainly thermal loads, viscous drag, and the engine inlet onset flow (height of the boundary-layer diverter).

In the background of such a case looms a vicious snow-ball effect, see, e. g., [6]. Uncertainties in vehicle mass and total drag prediction lead to design margins, see, e. g., [7], which make, for instance, more engine thrust necessary. As a consequence bigger engines and a larger fuel tank volume are needed, hence a larger engine and tank mass, a larger airframe volume and a larger wetted vehicle surface, consequently a larger total drag, and finally a larger take-off mass ensues.

A flight vehicle is weight-critical, or mass-sensitive, if the take-off mass grows strongly with the ratio ‘empty-vehicle mass’ to ‘take-off mass’, see, e. g., [8]. Large mass-growth factors together with small payload fractions are typical for CAV’s. Such vehicles usually are viscous-effects dominated and especially transition sensitive. Laminar-turbulent transition definitely is the key problem in the design of CAV’s and ARV’s.

At CAV’s transition occurs indeed in hypersonic boundary layers. These vehicles typically fly at angles of attack which are rather small, see Fig. 1.3 for the SANGER space transportation system up to separation of the upper stage at about 35 km altitude. Because of the small angles of attack, the boundary-layer edge Mach numbers will be of the order of magnitude of the flight Mach number. At the windward side, with pre-compression in order to reduce the necessary inlet capturing area, the boundary-layer edge Mach number will be somewhat smaller, but in any case the boundary layer also here is a hypersonic boundary layer.

The problems with laminar-turbulent transition and with turbulence are the insufficient understanding of the involved phenomena on the one hand,

and the deficits of the ground-simulation means on the other hand. This holds for both ground-facility and computational simulation.

However, once hypersonic attached viscous flow can be considered as tur­bulent, i. e., if shape and location of the transition zone have been somehow established, it usually is possible to compute the properties of such flow to a fair degree of accuracy, see, e. g., [9, 10], and also Section 7.2. The situa­tion changes negatively if turbulent strong interaction phenomena and flow separation are present.

In hypersonic ground-simulation facilities basically the low attainable Reynolds numbers, the (in general wrong) disturbance environment, which the tunnel poses for the boundary layer on the model, and the thermal state of the model surface are the problems.[117] Either the Reynolds number (though lower than in flight) is large enough for laminar-turbulent transition to occur in a ground-simulation facility, although in general with wrong shape and location of the transition zone due to the wrong disturbance environment and the wrong thermal state of the surface, or artificial turbulence triggering must be em­ployed (where, again, shape and location of the transition zone somehow must have been guessed). In the latter case, too, the Reynolds number still must be large enough to sustain the artificially created turbulence. If that is the case, turbulent attached flow and strong interaction phenomena and separation can be simulated, however, in general without taking into account the proper ther­mal state of the surface. In general the model surface is cold, which is in con­trast to the actual flight situation, see, e. g., Figs. 3.3 and 7.10.

With laminar-turbulent transition the situation is different to that of tur­bulence. Boundary-layer transition is a problem that has plagued several gen­erations of aerodynamicists. There are very few things about transition that are known with certainty, other than the fact that it happens if the Reynolds number is large enough (K. F. Stetson, 1992 [11]). Certainly we know much more about laminar-turbulent transition today than in 1992, but an empir­ical or semi-empirical transition prediction with the needed accuracy and reliability—if a hypersonic vehicle design is viscous-effects sensitive—or even a non-empirical transition prediction, is not yet possible. Unfortunately, an experimental determination of the transition location is also not possible in ground-simulation facilities (see above).

In the following Sections 8.1 to 8.3 we try to draw a picture of the dif­ferent instability and transition phenomena and their dependencies on flow – field parameters and vehicle surface properties, including the thermal state of the surface.[118] In Section 8.4 stability/transition methods and criteria for hypersonic flight-vehicle design purposes are given with due reservations re­garding their applicability and accuracy.

Turbulence in hypersonic flows and its modeling is treated rather briefly and with emphasis on computational simulation in Section 8.5.

We discuss the results of Navier-Stokes/RANS solutions ([41], based on [42, 43]) for the forebody of the lower stage of the SANGER TSTO space transportation system, Section 1.1, which is a CAV. The numerical simula­tions were made for a flight situation and a wind-tunnel situation (H2K is a hypersonic wind tunnel of DLR at Koln-Porz, Germany), Table 7.9. We check and interpret the computed data with the help of the approximate, reference-temperature extended boundary-layer relations for flat surface por­tions which we have presented in the preceding sub-sections.

In Figs. 7.8, [44], and 7.9, [45], the configuration is shown with computed skin-friction line patterns for the case under consideration. The first ramp of the inlet—not indicated in Fig. 7.8—lies at approximately 67 per cent length at the flat lower side. There the skin-friction line pattern indicates a flow between the primary attachment lines (not visible) which is to a good approximation parallel. This assures an effective pre-compression as well as the desired two-dimensionality of the inlet onset flow [36].

The presented and discussed results are the wall temperatures and the skin-friction coefficients at the lower, and partly at the upper symmetry line of the forebody. For the flight situation several assumptions were made re­garding surface-radiation cooling, gas model, and state of the boundary layer (laminar or turbulent).

We consider first the results for the lower (windward) symmetry line of the forebody. We see in Fig. 7.10 that the state of the boundary layer, laminar or turbulent, does not affect strongly the adiabatic (recovery) wall temperature Tw = Tr (cases e = 0). This can be understood by looking at the definition of the wall-heat flux in eq. (7.156). If the temperature gradient at the wall is zero by definition, the thermal conductivity к at the wall (always the laminar value), and in its vicinity (laminar or turbulent) can have any value with­out strong influence on the balance of thermal convection and conduction, compression and dissipation work, Sub-Section 4.3.2. The zero temperature gradient is also the reason why Tr does not depend on the inverse of some

power of the boundary-layer running length x, like the radiation-adiabatic temperature Tra or qgw.

Another explanation for the small influence of the state of the boundary layer can be obtained from a look at the relation for the estimation of the recovery temperature Tr at flat plates, eq. (3.7). If the recovery factor r is equal to VPr for laminar flow, and equal to /Pr for turbulent flow, Tr will not be much different for laminar and turbulent flow, because for air Pr = 0(1). The recovery temperature estimated with eq. (3.7) is constant. Note, however, that the computed Tw drops from the nose region, perfect gas Tw « 2,300 K (total temperature: Tt = 2,372.4 K, recovery temperature laminar with eq. (3.7): Tr = 2,069.4 K), by about 150 K, and only then is approximately constant.28

The flow past the lower side of the forebody is not exactly two-dimensional, Fig. 7.8. Hence all results are only more or less monotonic in x/L.

2500

Tw [K]

2000

e= 0 e« 0 o c – 0 c – 0.85 E – 0.85

1500 –

1000

500 –

Fig. 7.10. Wall temperatures along the lower and the upper symmetry line of the forebody of the lower stage of SANGER in the flight and the wind-tunnel (H2K) situation [41, 42] (configuration see Fig. 7.11). Influences of the state of the boundary layer (laminar or turbulent), the gas model, radiation cooling, and the location (windward side/leeward side) on the wall temperature. H2K-conditions: wall temperature in the Navier-Stokes/RANS solution [43].

High-temperature real-gas effects cannot be neglected at M= 6.8 in the flight situation, at least regarding Tr. The switch from the perfect-gas model to the equilibrium real-gas model results in a drop of the recovery temperature by approximately 250 to 300 K for both laminar and turbulent flow.

If we switch on radiation cooling (є = 0.85), the picture changes drasti­cally. The wall temperature is now the radiation-adiabatic temperature. In the case of laminar flow it drops from the stagnation-point temperature, now Tw = Tra « 1,600 K, very fast to temperatures between 600 K and 500 K. This drop follows quite well the proportionality Tw ж x-0’125 on flat surface portions, see eq. (3.25) and also Table 7.7. It is due to the inverse of the growth of the laminar boundary-layer thickness (ж x0’5), Table 7.3, and to the fact, that the surface-radiation flux is proportional to T^.

The case of turbulent flow reveals at once that the radiation-adiabatic temperature reacts much stronger on the state of the boundary layer, laminar or turbulent, than the adiabatic temperature. In our case we have an increase

of Tw (= Tra) behind the location of laminar-turbulent transition by about 400 K.

The transition location was chosen arbitrarily to lie between approxi­mately x/L = 0.08 and 0.12.[115] In reality a transition zone of larger extent in the x-direction is present with different positions at the windward and the lee­ward side, and with possible “tongues” extending upstream or downstream. (See in this regard for instance [1].)

The drop of the radiation-adiabatic wall temperature in the case of tur­bulent flow is weaker than for laminar flow. It follows quite well the propor­tionality Tw ж x-0 05, eq. (3.25), see also Table 7.7. This is due to the inverse of the growth of the thickness of the viscous sub-layer, respectively of the turbulent scaling length, of the turbulent boundary-layer, ж x0 2, Table 7.3, and of course also to the fact, that the surface-radiation flux is proportional

to TW.

In Fig. 7.10 also the radiation-adiabatic temperatures at the upper sym­metry line (leeward side of the forebody) are indicated. Qualitatively they behave like those on the windward side, however, the data for laminar flow are approximately 80 K lower, and for turbulent flow even 200 K lower than on the windward side. We observe that here the increase behind the laminar – turbulent transition zone is not as severe as on the windward side.

The explanation for the different temperature levels at the windward and the leeward side lies with the different unit Reynolds numbers at the two sides, and hence with the different boundary-layer thicknesses. We show this with data found with the help of an approximation of the vehicle by an infinitely thin flat plate. The ratio of the unit Reynolds numbers at the leeward (Ref ) and at the windward (ReW) symmetry lines at x/L = 0.5 for the flight Mach number Mо =6.8 and the angle of attack a = 6°, as well as the computed and the approximatively scaled, eq. (3.34), temperatures are given in Table 7.10.

The skin-friction coefficient along the symmetry line of the windward side of the forebody is given in Fig. 7.11. The influence of the wall temperature on the skin friction is very small, if the boundary layer is laminar, Table 7.5. The skin-friction coefficient drop follows well the primary proportionality rw a x-0 5, eq. (7.146), see also Table 7.5. This is due to the inverse of the growth of the laminar boundary-layer thickness, a x0 5, Table 7.3.

The influence of the wall temperature on the skin-friction coefficient in the cases of turbulent flow is very large. We note first for all three cases a non­monotonic behavior of cf behind the transition location in the downstream direction. The cause of this behavior is not clear. It can be the reaction on the enforced transition of the simple algebraic turbulence model employed in [42], as well as three-dimensional effects and also possible resolution problems of the discretization for the numerical solution. Only for x/L ^ 0.5 the skin – friction coefficient drops approximately proportional to x-0 2, which is the primary proportionality for turbulent flow, Table 7.5.

Important is the observation, that the turbulent skin friction is lowest for the perfect gas, adiabatic (є = 0) wall case, and largest (approximately 45 per cent larger) for the radiation-cooling (є = 0.85) case. For the equilibrium gas, adiabatic wall case cf is somewhat larger than for the perfect gas, adiabatic wall case, i. e., high-temperature real-gas effects can play a non-negligible role regarding turbulent skin friction.

The general result is: the smaller the wall temperature, the larger is the skin-friction of a turbulent boundary layer. This behavior follows with good approximation the dependence tw ж (T*/Tref )"(1+ш)-1, eq. (7.146), see also Table 7.5.

We give cf-ratios computed at x/L = 0.5 in Table 7.11. We compare them with data found with the simple relation for flat surface portions with reference-temperature extension, eq. (7.146). This is done with a scaling by means of T*/Tref. The ratio T*/Tref, eq. (7.70), is computed with Tw taken from Fig. 7.10, with Ye = 1.4, Me = M= 6.8, and rturb = 0.896. We find at x/L = 0.5 that the skin-friction coefficient at the radiation-adiabatic surface is approximately 1.45 times larger than at the adiabatic surface with perfect gas assumption, and approximately 1.3 times at the adiabatic surface with equilibrium real-gas assumption. The scaled data are in fair agreement with the computed data. The scaling of the skin-friction coefficient for laminar flow gives an influence of the wall temperature of less than 10 per cent, which is barely discernible in the data plotted in Fig. 7.11.

The results in general show that for the estimation of the skin-friction drag of CAV’s the surface temperature, which is governed predominantly by radiation cooling, must be properly taken into account, at least for turbulent flow. In view of the fact that the skin-friction drag may account for a large part of the total drag, the vehicle surface should be flown as hot as possible. However, one has to take into account possible side effects: rise of the pressure (form) drag, degradation of the effectiveness of aerodynamic trim and control surfaces, Section 10.7.

For a typical cold hypersonic wind-tunnel situation, that of the H2K at DLR Koln-Porz, Germany, a computation has been performed for the

SANGER forebody, too [43]. The Mach number was the same as the flight Mach number, but the flow parameters were different, and in particular the surface temperature Tw was the typical ambient model temperature, Table 7.9. The resulting skin friction is quantitatively and qualitatively vastly dif­ferent from that computed for the flight situation. In the laminar regime just ahead of the location of enforced transition it is about eight times larger than that computed for the flight situation. The data do not scale properly.

In the turbulent regime the skin friction initially is about 2.2 times larger than for the flight situation. The scaling yields 2.65, which is a reasonable result. The slope of cf in the turbulent regime is much steeper in the wind- tunnel situation than in the flight situation. Both the unit Reynolds number, and the small body length may play a role in so far, as transition was enforced at a local Reynolds number which is too small to sustain turbulence. (In the wind tunnel experiment transition was not enforced and the state of the boundary layer was not clearly established.)

The large influence of the state of the boundary layer, laminar or tur­bulent, on the wall temperature and on the skin friction, if the surface is radiation cooled, poses big problems in hypersonic flight vehicle design. The transition location is very important in view of thermal-surface effects, as well as of thermal loads. The prediction and verification of the viscous drag (skin-friction drag plus viscosity-induced pressure drag (form drag)) is very problematic, if the flight vehicle is drag critical, which in general holds for CAV’s. Fig. 7.11 indicates that in such cases with present-day wind-tunnel techniques, especially with cold model surfaces, the viscous drag cannot be found with the needed degree of accuracy and reliability.

7.3 Problems

Problem 7.1. The flow past a flat plate has a unit Reynolds number ReU = 106 m-1. Assume incompressible flow and determine on the plate at x = 1 m for a) laminar and b) turbulent flow the boundary-layer thicknesses 5, 5i, 52, the shape factors H12, and for turbulent flow in addition 5vs, and 5sc.

Problem 7.2. The flow past a flat plate is that of Problem 7.1. Assume incompressible flow and determine on the plate, but now at x = 10 m for a) laminar and b) turbulent flow the boundary-layer thicknesses 5, 51, 52, the shape factors H12, and for turbulent flow in addition 5vs, and 5sc.

Problem 7.3. Compare and discuss the results from Problem 7.1 and 7.2 in view of Table 7.3.

Problem 7.4. Derive the formula (perfect gas) for the skin-friction drag Df (with reference-temperature extension) of one side of a flat plate with the length L and the width b (the reference area is Aref = bL). Write the formula for both fully laminar and fully turbulent flow.

Problem 7.5. Compute the skin-friction drag Df of a flat plate at zero angle of attack (both sides wetted!). The parameters are similar to those in Problem 6.8: Mсо = 6, H = 30 km, Aref = 1,860 m2, Lref = 80 m. Disregard possible hypersonic viscous interaction, Section 9.3. Use the power-law relation, eq. (4.15), for the viscosity, take in the relation for the reference temperature, eq. (7.70), у = 1.4, Pr* = 0.74. Assume laminar flow and two wall temperatures:

a) Tw = 1,000 K, b) Tw = 2,000 K.

Problem 7.6. Compute the skin-friction drag Df of a flat plate at zero angle of attack like in Problem 7.5, but now for turbulent flow. Mind the different recovery factors of laminar and turbulent flow.

Problem 7.7. Discuss the results from Problem 7.5 and 7.6, in particular also the influence of the wall temperature on the skin-friction drag. How large is the ratio ‘turbulent drag’ to ‘laminar drag’ for a) the lower temperature,

b) the higher temperature? How much in per cent is the drag reduced for the higher temperature for a) laminar flow, b) turbulent flow?

Problem 7.8. Compute the skin-friction forces Lf and Df for the CAV of Problem 6.8. On the windward (w) side we have the boundary-layer edge data: Mw, e = 5.182, ReW e = 3.123-106 m-1, vw, e = 1,773.63 m/s, pw, e = 0.0327 kg/m3, Tw, e = 291.52 K, and on the leeward (l) side: Mle = 6.997, Refe = 1.294-106’m-1, v, e = 1,840.01 m/s, p, e = 0.00297 kg/m3, Tle = 172.12 K. Consider each side separately as flat plate surface. Assume fully turbulent flow, wall temperatures Tw, w = 1,000 K, Tl, w = 800 K, and the other parameters like in Problem 7.6.

Problem 7.9. Add the inviscid parts, Problem 6.8, and the skin-friction parts, Problem 7.8, of the lift and the drag of the CAV, compute the lift to drag ratio and discuss the results.

The thermal state of the surface governs thermal-surface effects on wall and near-wall viscous and thermo-chemical phenomena, as well as the thermal loads on the structure, Section 1.4. Important is the fact that external surfaces of hypersonic flight vehicles basically are radiation cooled. Above we have seen, how the thermal state of the surface influences via Tw the boundary – layer thicknesses, and hence also the wall-shear stress of laminar or turbu­lent flow. In view of viscous thermal-surface effects, the thermal state of the surface is of large importance for CAV’s. Thermo-chemical thermal-surface effects in particular are important for RV’s. Thermal loads finally are of large importance for all vehicle classes.

We have defined the thermal state of a surface by two entities, the tem­perature of the gas at the wall which in the continuum-flow regime is the wall temperature Tgw = Tw, and the temperature gradient dT/dygw in the gas at the wall. For perfect gas or a mixture of thermally perfect gases in equilibrium the latter can be replaced by the heat flux in the gas at the wall

qgw.

In Section 3.1 several cases regarding the thermal state of the surface were distinguished. In the following we consider the first two cases.

If the vehicle surface is radiation cooled and the heat flux into the wall, qw, is small, the radiation-adiabatic temperature Tw = Tra is to be determined. This is case 1:

qgw(x, Z) « aeT^ai’X, z) t Tw = Tra(x, Z) = ?

Case 2 is the case with Tw prescribed directly, because, for instance, of design considerations, or because the situation at a cold-wall wind-tunnel model is studied. Hence the heat flux in the gas at the wall, qgw, is to be determined. This is case 2:

Tw Tw (x: z) t qgw (x: z) ?

To describe the thermal state of the surface we remain with the reference temperature extension, following [16, 39], and perfect-gas flow. We begin with the discussion of the situation at flat surface portions, fp’. The situation at the stagnation-point region and at attachment lines is considered in Sub­Section 7.2.6.

The basis of the following relations is the Reynolds analogy, respectively the Chilton-Colburn analogy [16][108]

Summary. We summarize the dependencies in Table 7.7. We substitute also here ‘o’ conditions by general ‘ref ’ conditions and chose ш = =

0.65 in the viscosity law, Section 4.2.[109] We break up the Reynolds numbers Re into the unit Reynolds number Reu and the running length x in order to show the explicit dependencies on these parameters. They reflect inversely the behavior of the boundary-layer thickness of laminar flow and the viscous sub­layer thickness of turbulent flow, Sub-Section 7.2.1. As was to be expected, the qualitative results from Sub-Section 3.2.1 are supported. We see also that indeed in case 2 with given Tw the heat flux in the gas at the wall qgw has the same dependencies as T4a.

— Dependence on the boundary-layer running length x.

Both T4a, case 1, and qgw, case 2, decrease with increasing x. This is stronger in the laminar, (ж x—0 5), than in the turbulent cases (ж x—0 2).

— Dependence on the unit Reynolds number ReUef.

The larger ReU, the larger T^a, case 1, and qgw, case 2, because the re­spective boundary-layer thicknesses become smaller with increasing ReЦ,. The thermal state of the surface with a laminar boundary layer reacts less strongly on changes of the unit Reynolds number, ж (Re%ef )0’5, than that of the turbulent boundary layer, ж (ReUef )0’8.

— Dependence on T*/Tref.

The larger T*/Tref, the smaller are T^a, case 1, and qgw, case 2. For a given Mref and a given Tref an increase of the wall temperature Tw would lead to a decrease of them. The effect is stronger for turbulent flow, ж (T*/Tref)-067, than for laminar flow, ж (T*/Tref)-0175. However, the major influence is that of the Tr — Tw, next item.

— Dependence on Tr — Tw.

at the attachment line of an infinite swept circular cylinder. We are aware that this is a more or less good approximation of the situation at stagnation points and (primary) attachment lines at a hypersonic flight vehicle in reality.

We find at the sphere, ‘sp’, respectively the circular cylinder (2-D case), for case 1 for perfect gas with the generalized reference-temperature formulation [16] like before

(7.160)

where

with C = 0.763 for the sphere and C = 0.57 for the circular cylinder. The velocity gradient due/dxx=0 is found with eq. (6.166). This is the laminar case.

The meanwhile classical formulation for qgw at the stagnation point of a sphere with given wall enthalpy hw, generalized case 2, in the presence of high-temperature real-gas effects is that of J. A. Fay and F. R. Riddell [40].[113] This result of an exact similar solution ansatz reads

kPr-0’b(pw Pw )0A(PePe)0A

The stagnation point values are denoted here as boundary-layer edge ‘e’ values. For the sphere k = 0.763 and for the circular cylinder k = 0.57 [39]. The term in square brackets contains the Lewis number Le, eq. (4.93). Its exponent is m = 0.52 for the equilibrium and m = 0.63 for the frozen case over a catalytic wall.

The term hD is the average atomic dissociation energy times the atomic mass fraction in the boundary-layer edge flow [40]. For perfect-gas flow the value in the square brackets reduces to one. Note that in eq. (7.162) the external flow properties pepe have a stronger influence on qgw than the prop­erties at the wall pw pw. This reflects the dependence of the boundary-layer thickness in the stagnation point on the boundary-layer edge parameters, eq. (7.133).

Because we wish to show the basic dependencies also here, we use the equivalent generalized reference-temperature formulation for case 2. The heat flux in the gas at the wall, qgw, for the sphere or the circular cylinder (2-D case) with given Tw, reads [16]

1 T

qgw, sp = kooPr1/3gsp—Tr(l – (7.163)

К T r

The radiation-adiabatic temperature, case 1, at the attachment line of an infinite swept circular cylinder, ‘scy’, is found from

where g*Scy is

1 T

4gw, scy = P)’3 kxgscy —Tr(l — – pjg – ). (7.166)

Like in Sub-Section 7.2.4, К is the radius of the cylinder, due/dxx=0 the gradient of the inviscid external velocity normal to the attachment line, eq. (6.167), and we = иж sin the inviscid external velocity along it, Fig. 6.37 b). For laminar flow C = 0.57, n = 0.5 and p* = p°’8pW2, = M°’8mW’2.

For turbulent flow C = 0.0345, n = 0.21. The reference-temperature values are found with eq. (7.151), and p* again with T* and the external pressure Pe.

Summary. In Table 7.8 the general dependencies of the radiation-adiabatic temperature, (case 1), are summarized for laminar and turbulent flow. Again we choose ш = = 0.65 in the viscosity law, Section 4.2. We break up,

like before, the Reynolds numbers Ke into the unit Reynolds number Keu and the radius К. The reference-temperature dependencies are also taken in simplified form. The dependencies of qgw (case 2) are the same, see eqs. (7.163) and (7.166) and are therefore not shown.

The results are:

— Dependence on the radius R.

The fourth power of the radiation adiabatic temperature T4a is the smaller, the larger К is. It depends (infinite swept circular cylinder) on К stronger for laminar, ж K—0 5, than for turbulent flow, ж K—0 21.

— Dependence on the sweep angle y>.

This dependence holds only for the infinite swept circular cylinder. For <p = 0° we have the case of the non-swept circular cylinder (2-D case). For <p ^ 90° T4a ^ 0. This means that we get the situation on an infinitely long cylinder aligned with the free-stream direction, where finally 5 and Svs ^ to, and hence T4a becomes zero.

— Dependence on the unit Reynolds number ReU.

T4a depends on some power of the unit Reynolds number in the same way as on flat surface portions with ж (Re^,)0 5 for the laminar and ж (ReU)0’79 for the turbulent case. The larger ReU, the larger is T4a, because 6 and 6vs become smaller with increasing ReЦ,.

— Dependence on T*/Tr ef.

The larger T*/Tref, the smaller is T4a. For a given Mref and a given Tref an increase in wall temperature Tw would lead to a decrease of them. The effect is stronger for turbulent flow, ж (T*/Tref )-0’67, than for laminar flow, ж (T*/Tref )-0175. However, the major influence is that of the Tr — Tw, next item.

— Dependence on Tr — Tw (not in Table 7.8).

The wall shear stress tw is the cause of the skin-friction drag, which is exerted by the flow on the flight vehicle. For a CAV this drag together with the form drag can be up to approximately 50 per cent of the total drag. For RV’s at large angle of attack, the skin-friction drag is almost negligible [36]. The wall-shear stress can also considerably influence lift forces, especially during flight at higher angles of attack, and moments around the pitch and the yaw axis. Finally it is a deciding factor regarding erosion phenomena of surface coatings of thermal protection systems of RV’s.

The wall shear stress in the continuum regime and in Cartesian coordi­nates is defined as, see also eq. (4.36)

(7.136)
Tw, lam
In terms of the unit Reynolds number Re^ = pTOuTO/^TO this reads
(7.137)
Tw, la
(7.140)
(7.141)
0.0592 (ЙЄ со,.)0-2
(7.142)
cf, turb
For the reference-temperature extension, density and viscosity in eqs. (7.136) and (7.140) are all to be interpreted as data at reference temper­ature conditions. The equation for the laminar flat-plate boundary layer then reads
	* Ч 0.5 0.5 P И
(7.143)
Tw, lam, c

We introduce p*T* = pcTc together with the power-law formulation of viscosity. With the nomenclature used for the thicknesses of compressible boundary layers, where the subscript ‘c’ stands for the compressible and ‘ic’ for the incompressible case, we obtain

with C = 0.332 and n = 0.5 for laminar flow, and C = 0.0296 and n = 0.2 for turbulent flow.

An alternative formulation is:

Before we summarize these results, we have a look at the result, which exact theory yields with the use of the Lees-Dorodnitsyn transformation for the wall shear stress of a self-similar compressible laminar boundary layer

[35].

We quote the result in a form given by J. D. Anderson Jr. [37]

The dependence of tw on the Reynolds number is like in eq. (7.144). The function f ‘(0) is the derivative of the velocity function f1 = u/ue at the wall. It implicitly is a function of the boundary-layer edge Mach number Me, the Prandtl number Pr, and the ratio of specific heats 7. Hence we have a dependence like in eq. (7.136) on these parameters, however implicitly and in different form.

Summary. Like for the boundary-layer thicknesses discrepancies can be found in the literature regarding the simple relations for the wall shear stress of two-dimensional incompressible and compressible flat-plate turbu­lent boundary layers. We do not pursue this problem further.

We summarize the results, eqs. (7.144), and (7.145), respectively eq. (7.146), in Table 7.5. We substitute also ‘TO’ conditions by general ‘ref’ con­ditions and choose ш = ш^ = 0.65 in the viscosity law, Section 4.2.[106] We in­troduce the dynamic pressure qref = 0.5 prefv2ef and break up all Reynolds numbers Re into the unit Reynolds number Reu and the running length x in order to show explicitly the dependencies on these parameters.

The third, fourth and the fifth column in Table 7.5 give the basic de­pendencies of the wall shear stress of both incompressible and compressible boundary layers on the running length x, the unit Reynolds number ReUef and the dynamic pressure qref, the sixth column the dependence of the wall shear stress of compressible boundary layers on the reference-temperature ratio T*/Tref, i. e., on the wall temperature Tw and the Mach number Mref, eq. (7.129).

These dependencies give us insight into the basic behavior of the wall shear stress of two-dimensional flat-plate boundary layers.

— Dependence on the boundary-layer running length x.

The wall shear stress decreases with increasing x. The wall shear stress of laminar boundary-layers reduces stronger (ж x-0 5) with x than that of turbulent boundary layers (ж x-0 2). Remember that the thickness of the viscous sub-layer grows only very weakly with x compared to the thickness of the laminar boundary layer.

— Dependence on the dynamic pressure qref.

The wall shear stress of laminar as well as of turbulent boundary layers increases linearly with increasing dynamic pressure.

— Dependence on the unit Reynolds number ReUef.

The wall shear stress depends on the inverse of some power of the unit Reynolds number. The larger ReUef, the smaller is the wall shear stress. The wall shear stress of laminar boundary layers reacts stronger on changes of the unit Reynolds number (ж (Re^^ )-0’5) than that of turbulent boundary layers (ж (Re^ef )-0 2). This result is not surprising, because the dynamic pressure qо was isolated, see, e. g., eq. (7.146). In the alternate formulation of eq. (7.147) we have ж (ReUef )0 5 for the laminar, and ж (ReUef )0 8 for the turbulent case.

— Dependence on T*/Tref.

Summary. In Table 7.6 the general dependencies of the wall shear stress tw in eq. (7.149) are summarized for laminar and turbulent flow. Again we choose ш = = 0.65 in the viscosity law, Section 4.2. We introduce, like

before, the dynamic pressure qref and break up the Reynolds number Re into the unit Reynolds number Reu and the radius R. Regarding the reference- temperature dependencies the differentiation recommended in [16] for the laminar and the turbulent case (see above) is not made. Instead we write on the basis of eq. (7.150)

The results are:

— Dependence on the dynamic pressure q, x.

— Dependence on the radius R.

The wall shear stress decreases with increasing R stronger for laminar, ж R-0’5, than for turbulent flow, ж R-0’21.

— Dependence on the sweep angle p.

For p = 0° we have the case of the non-swept circular cylinder (2-D case), where Tw, scy is zero in the stagnation point. For p ^ 90° also Tw, scy ^ 0. This means that the attachment line ceases to exist for p ^ 90° and we get the situation on an infinitely long cylinder aligned with the free-stream direction, where finally rw, scy becomes zero. In between we observe that Tw, scy first increases with increasing p and that stronger for laminar than for turbulent flow. This reflects the behavior of the component we of the external inviscid flow, Fig. 6.37 b), which grows stronger with increasing p than the velocity gradient due/dx across the attachment line declines with it. At large p finally the effect reverses, first for laminar then for turbulent flow, and Tw, scy drops to zero.

— Dependence on the unit Reynolds number ReЦ,.

The wall shear stress depends on the inverse of some power of the unit Reynolds number in the same way as on flat surface portions with ж (ReU)-0 5 for the laminar and ж (Re^)-0 21 for the turbulent bound­ary case. The larger ReU, the smaller is the wall shear stress. Again one has to keep in mind the dependence on the dynamic pressure, which has been isolated in this consideration.

— Dependence on T*/T, x.

At a general three-dimensional attachment line the velocity gradient due/dxx=0 and are not connected explicitly to a geometrical property of the configuration. In any case it can be stated that the larger due/dx, the smaller the relevant boundary-layer thickness, and the larger the wall shear stress. Of course also here the magnitude of we plays a role. tw depends also here inversely on the wall temperature and that, like in general, stronger for turbulent than for laminar flow.

The attached viscous flow, i. e., the boundary layer, has its origin at the forward stagnation point of a flight vehicle configuration [1].[105]

The primary attachment point, depending on the angle of attack, can lie away from the nose-point at the lower side of the configuration, as we have seen for the Blunt Delta Wing, Fig. 3.16. Also the primary attachment lines on a hypersonic flight vehicle with the typical large leading-edge sweep of the wing, can lie away from the blunt-wing leading edge at the lower side of the wing. Secondary and tertiary attachment lines can be present. We have shown this for the BDW in Figs. 3.16 and 3.17 (see also Figs. 3.19 and 3.20). In such cases we can have an infinite swept wing flow situation, i. e., zero or only weak changes of flow parameters along the attachment line, like on the (two) primary and the tertiary attachment line of the BDW. The same situation can be present at separation lines. This all is typical for RV’s and CAV’s.

The boundary layer at the forward stagnation point has a finite thickness despite the fact that there both the external inviscid velocity ue and the tangential boundary-layer velocity u(y) are zero [1]. The situation is similar at a three-dimensional attachment line, however there exists a finite velocity along it.

In the following discussions and also for the consideration of the wall shear stress and the thermal state of the surface we idealize the flow situation. We study the situation at the stagnation point of a sphere, and of a circular cylinder (2-D case), and at the attachment line of an infinite swept circular cylinder. The reason is that the velocity gradient due/dxx=0, which governs the flow there (at the swept cylinder it is the gradient across the attachment line), can be introduced explicitly as function of the radius R, Sub-Section 6.7.2. This may be a rather crude approximation of the situation found in reality on a RV, but it fits the situation more or less exactly for a CAV. Nev­ertheless, it permits us to gain insight into the basic parameter dependencies of the boundary-layer thickness, and later also of the wall-shear stress, Sub­Section 7.2.4, and of the thermal state of the surface, Sub-Section 7.2.6, at an attachment point and at a primary attachment line.

The classical approach to describe the boundary layer at a stagnation point is to replace in the boundary-layer equation, explicitly or implicitly, for instance the velocity ux=o by (du/dx)x=0, which, like (due/dx)x=o, is finite. This operation can be made, for instance, by differentiating the momentum equation in question with respect to the corresponding tangential coordinate [1].

We consider now the flow along an attachment line. At the attachment line we have a finite inviscid velocity component along it, and hence a boundary layer of finite thickness. Both may not or only weakly change in the direction of the attachment line.

In order to obtain the basic dependencies of the boundary-layer thickness, we assume that the attachment-line flow locally can be represented by the flow at the attachment line of an infinite swept circular cylinder, Fig. 6.37 b). There we have constant flow properties at the attachment line in the z-direction.

We only replace the velocity gradient for the stagnation point in eq. (7.133) by that for the swept infinite cylinder, eq. (6.167), and find for the laminar boundary layer:

For a subsonic leading edge, i. e., Mcos p < 1, exact theory shows that the dependence of S indeed is oc l/^cosy, but for a supersonic leading edge 5 is somewhat larger [34].

Summary. We summarize the dependencies in Table 7.4. We chose, like in Sub-Section 7.2.1, ш = ш^ = 0.65 in the viscosity law, Section 4.2. In all cases S is proportional to /Д, i. e., the larger the nose radius or the leading-edge radius, the thicker is the boundary layer. Also in all cases 5 increases with increasing reference-temperature ratio, this means in particular also with increasing wall temperature.

In the case of the infinite swept circular cylinder 5 increases ж (cos p)-0 5, at least for small sweep angles p. This result will hold also for turbulent flow. For p — 90° we get 5 —— ж. This is consistent with the situation on an infinitely long cylinder aligned with the free-stream direction.

We consider exclusively the relevant classical thicknesses of laminar and turbulent flow boundary layers. Their dependencies on the boundary-layer running length x, on the Reynolds number Reref, x, and on the reference temperature T* are studied. Thermal and mass-concentration boundary lay­ers can be treated likewise.

Boundary-Layer Thicknesses. The flow boundary-layer was introduced in Sub-Section 4.3.1. We call it—as usual—in the following simply “boundary layer”. It causes a virtual thickening of a body (via its displacement prop­erties), and it in particular prevents a full re-compression of the external inviscid flow at the aft part of the body, where we have a flow-off separation of the boundary layer, and the formation of a wake [1]. This is the cause of the viscosity-induced pressure or form drag of a body.[95] The boundary layer also reduces the aerodynamic effectiveness of lifting, stabilizing, and control surfaces. Its thickness finally governs the height of the boundary-layer di­verter in the case of airbreathing propulsion, Sub-Section 6.1. In all cases it holds: the thicker the boundary layer, the larger is the adverse effect.

On the other hand we have the effect that the boundary-layer causes and governs the wall-shear stress and the thermal state of the surface. The thicker the boundary layer (with turbulent flow the viscous sub-layer), the smaller the wall-shear stress, the smaller the heat-flux in the gas at the wall, and, in particular, the more effective is surface-radiation cooling.

Attached viscous flows, i. e., boundary layers, have finite thickness every­where on the body surface, also in stagnation points, and at swept leading edges, i. e., attachment lines in general, see, e. g., [1, 21, 22].

Of special interest is the thickness of a boundary layer as such, its dis­placement thickness, and the thickness of the viscous sub-layer, if the flow is turbulent.

The thickness 6 of a laminar boundary layer is not sharply defined. Boundary-layer theory yields the result that the outer edge of the bound­ary layer lies at y [21].

For the incompressible laminar flat-plate boundary layer the theory of Blasius gives for the thickness

with[96]

Poo Woo ж Q„s

‘ .., =—————- • (7.89)

In terms of the unit Reynolds number ReU — p^u^/y.^ it reads:

x0.5

Slam = C n 5 , (7.90)

(Re^o)0.5

A practical definition of 6 for laminar boundary layers is the distance at which locally the tangential velocity component u(y) has approached the inviscid external velocity ue by є ue (for three-dimensional boundary layers the resultant tangential velocity and the resultant external inviscid velocity are taken)[97]

Ue – u(y) A єие. (7.91)

Usually, although often not explicitly quoted, the boundary-layer thick­ness is defined with

є — 0.01. (7.92)

For the incompressible flat-plate boundary layer we find with this value from the Blasius solution the constant c in eq. (7.88):

є — 0.01: c — 5. (7.93)

If є — 0.001 is taken, the constant is c — 6. The exact Blasius data are c — 5: u — 0.99155 ue, c — 6: и — 0.99898 ue.

The outer edge of a turbulent boundary layer in reality has a rugged unsteady pattern. In the frame of Reynolds-averaged turbulent boundary – layer theory it is defined as smooth time-averaged edge. The concept of the
thermal and the mass-concentration boundary layer, as we use it for laminar flow, is questionable with regard to turbulent flow.

For a low-Reynolds number, incompressible turbulent flat-plate boundary layer the boundary-layer thickness is found, by using the ^-th-power velocity distribution law [24], to[98]

x

Sturb = (7-94)

and in terms of the unit Reynolds number

x0.8

Sturb = 0:i7jm^- (7-95)

A very important thickness of turbulent boundary layers is that of the vis­cous sub-layer Svs, which is small compared to the boundary-layer thickness 5turb (Svs/ Slurb = O(0.01)), Fig. 7.5 b).

Fig. 7.5. Two-dimensional tangential velocity profile and boundary-layer thickness of a) laminar, b) turbulent boundary layer (time averaged).

In laminar boundary layers the thickness S is the characteristic thickness which governs shear stress and heat transfer at the wall. In turbulent flow the characteristic thickness is Svs, Fig. 7.6.

This thickness is found from the general definition of the viscous sub-layer:

y+ = < 5. (7.96)

A* ~

With the friction velocity uT = Jтш/p and the wall-shear stress of the ^-th-power boundary layer, eq. (7.140) in Sub-Section 7.2.3, we obtain (see also [25], where the different co-factor 72.91 results)

x

Svs = 29-ll(i n, , ■ (7-97)

G. Simeonides proposes for scaling purposes, in order to be consistent with the laminar approach, an alternative thickness, Ssc, which is slightly different from the viscous sub-layer thickness Svs [26]. With the definition of Ssc lying where the non-dimensional velocity u+ and the wall distance y+ are equal (u+ = u/uT = y+ = ypuT/у) he gets the scaling thickness 5sc for turbulent boundary layers

x

^=33-7v:,)o. s – (7-99)

We apply now the reference temperature concept in order to determine the dependence of the boundary-layer thicknesses on the wall temperature Tw and the Mach number, assuming perfect-gas flow.

We begin with a demonstration of the basic approach applied to the boundary-layer thickness of laminar flow Siam, c. We write eq. (7.88) in terms of the reference density and viscosity:

With

and the relations given in Sub-Section 7.1.6:

P^_Xо Moo _ fToo Ш

Poo _ T* ’ /V " V /

we find, with c = 5, the thickness 3lam, c of the compressible laminar flat-plate boundary layer[99]

If we assume T* ж Tw, as well as w ж 1, we obtain that the compressible boundary-layer thickness depends approximately directly on the wall tem­perature:

(7.101)

At an adiabatic wall the recovery temperature is:

Tr = Tw=T^(l + r^-±Mi2-

This relation says that for large Mach numbers Tr ж M2. Introducing this into eq. (7.101) we get the well known result for the compressible laminar adiabatic flat-plate boundary layer:

For the thickness of the compressible turbulent flat-plate ^-th-power boundary layer we obtain with the reference-temperature extension

and for the thickness of the viscous sub-layer eq. (7.97)

whereas the turbulent scaling thickness, eq. (7.99), is

Integral Parameters 5i and S2- Well defined integral parameters are the boundary-layer displacement thickness S1, the momentum thickness S2 and others.[100] These parameters appear on the one hand in boundary-layer solution methods (“integral” methods of Karman-Pohlhausen type [21]), and on the other hand in empirical criteria for laminar-turbulent transition, for separation, etc. Often also quotients of them are used, for instance the shape factor H12 = S1/S2.

In the following we define the more important integral parameters. We do it in all cases for compressible flow. The definitions are valid for both laminar and turbulent flow and contain the definitions for incompressible flow. Because of the importance of the displacement of the external inviscid flow by the boundary layer (weak interaction) we quote S1 and the equivalent inviscid source distribution for three-dimensional flow.

In the frame of first-order boundary-layer theory and in Cartesian coor­dinates the displacement thickness S1 of a three-dimensional boundary layer is defined by a linear partial differential equation of first order, see, e. g., [1]

In this definition surface suction or blowing in the frame of the boundary – layer assumptions is taken into account by p0v0, which in general can be a function of x and z. The symbols S1x and S1z denote “components” of S1 in x and z-direction, which are determined locally

with pu and pw being functions of y.

The displacement thickness S1 of three-dimensional boundary layers can become negative, although the two local components S1x and S1z are every­where positive [1].[101] The effect occurs especially at attachment lines, with a
steep negative bulging of the Si surface. Close to the beginning of separation lines a steep positive bulging is observed.[102] These effects are clues for the explanation of the hot-spot and the cold-spot situations along attachment and separation lines on radiation-cooled surfaces, Sub-Section 3.2.4.

For two-dimensional boundary layers eq. (7.107) reduces to

d

— peue{5 – d’ij] = p0v0 (x), (7.110)

which is the general definition of the boundary-layer displacement thickness in two dimensions. Only for p0v0 = 0, and peuex=0 = 0, we obtain the classical local formulation

In practice this two-dimensional formulation is almost always employed to determine the displacement properties of a boundary layer, usually without consideration of possible effects of three-dimensionality or, in two dimensions, of initial values, and of variations of the external flow. These can be relevant, for instance, at a wing with a round swept leading edge. Here a boundary layer with finite thickness exists already along the leading edge in span-wise direction. If the displacement thickness in chord-wise direction is to be de­termined, eq. (7.110) must be solved while properly taking into account the displacement thickness of the leading-edge boundary layer as initial value.

If boundary-layer criteria are correlated with the displacement thickness, the use of the proper definition is recommended, otherwise the correlation may have deficits. However, criteria correlated with a given displacement thickness should be applied with the same definition, even if the displacement thickness employed in the correlation is not fully representative.

The displacement properties of a boundary layer lead to a virtual thicken­ing of the body. In hypersonic computations by means of coupled Euler/boun – dary-layer methods this thickening must be taken into account. To change the body contour by a local superposition of the positive or negative displacement thickness would be a cumbersome procedure. A very effective alternative is to employ the equivalent inviscid source distribution pvisd [27]

This source distribution, which is a function of x and z, is found in a cou­pled solution method after the first Euler and boundary-layer solution has been performed. It is then employed at the body surface as boundary condi­tion for the next Euler solution. In this way the boundary-layer displacement

properties are iteratively taken into account. In practice only one coupling step (perturbation coupling) is sufficient [1, 28].

An integral parameter often used in correlations is the momentum thick­ness 62.[103] In three-dimensional boundary layer flow two such thicknesses ap­pear (62x and 62z), because the momentum flux is a vector [1]. In practice only the classical two-dimensional formulation

is employed either for the main-flow profile or assuming two-dimensional flow from the beginning.

So far we have discussed the definitions of integral parameters. We give now displacement and momentum thicknesses for two-dimensional laminar and turbulent boundary layers over flat surfaces, which we also extend by means of the reference temperature to compressible flow. We do this again in order to identify their dependencies on wall temperature and Mach number, and to permit estimations of these thicknesses for practical purposes.

For the laminar Blasius boundary layer we quote [21]

and for the turbulent ^-th-power boundary layer [24]

x

h, turb, ic = 0-0360 ‘ (7.117)

(Re^,x)

The respective shape factors are

H12,lam, ic = dl’lam’ic = 2.591, (7.118)

62,lam, ic

for the Blasius boundary layer, and

H12,turb, ic = dl’turb’ic = 1.286, (7.119)

62,turb, ic

for the ^-th-power boundary layer.

Because the tangential velocity profile of a turbulent boundary layer is much fuller than that of a laminar one, Fig. 7.5, its displacement thickness, but also its momentum thickness, are smaller in proportion to the boundary – layer thickness than those of a laminar boundary layer, Table 7.2.

This points to a very important property of two-dimensional and three­dimensional (main-flow profile) turbulent boundary layers, which in general holds also for hypersonic flow: close to the wall the flow momentum is propor­tionally larger than that of laminar boundary layers. A turbulent boundary layer hence can negotiate a stronger adverse pressure gradient than a laminar one, and therefore separates later than a laminar one. Although the skin fric­tion exerted by a turbulent boundary layer is larger, a body of finite volume may have a smaller total drag if the flow is turbulent, because with the same pressure field the laminar flow will separate earlier. The classical examples are the drag behavior as function of Reynolds number of the flat plate at zero angle of attack and of the sphere [21]. However, regarding the thermal state of a radiation, or otherwise cooled vehicle surface, the turbulent boundary layer has no such positive secondary effect.

We do not show the derivation of the reference-temperature extensions of the above displacement and the momentum thicknesses to compressible flow, because the effort would be too large. Instead we quote them from [16, 29], taking Pr = 0.72, and rfam = /Pr = 0.848, r*turb = fPr = 0.896.

Summary. We discuss the relations for the boundary-layer thicknesses and the integral parameters together. First, however, we consider the different factors, which we quoted for the displacement thickness of incompressible 4-th-power turbulent boundary layers, eqs. (7.116) and (7.123), and the dif­ferent values of the shape factor in eqs. (7.119) and (7.127).

Discrepancies can be found in the literature regarding all simple rela­tions for two-dimensional incompressible and compressible flat-plate turbu­lent boundary layers. Possible reasons in such cases are that not enough information is given:

— The relations in general are valid only in a certain Reynolds-number range. The thicknesses for the th-power turbulent boundary layer, which we gave above, Sturb, ic, eq. (7.94), Svs, eq. (7.97), as well as the integral pa­rameters Si, turb, ic, eq. (7.116), and S2,turb, ic, eq. (7.117), are valid only for moderate Reynolds-numbers of « O(105) to « O(107) [24]. For larger Reynolds numbers other factors in these relations or completely other re­lations are required, which can be found in the literature.

— The above relations assume that the boundary layer is turbulent from the leading edge of the flat plate onwards. Relations or factors derived from measurement data may not take this into account and hence differ from these relations. The classical shape-parameter data of Schubauer and Klebanoff [30] vary in the transition region of a flat plate in incompressible flow from H12 = 2.6 in the laminar part to H12 = 1.4 in the turbulent part. They are of course locally determined data. The value for the laminar part compares well with the value given in eq. (7.118), whereas the value for the turbulent part is larger than that given in eq. (7.119). Obviously it was employed in eq. (7.123).

We mention that the location of laminar-turbulent transition, if desired, can easily be taken into account in the determination of the boundary-layer thicknesses with a simple approach. In [31] a procedure is given which works with the virtual origin of the turbulent boundary layer.

The virtual origin is found from the matching of the momentum deficits of the two boundary layers on both sides of the transition location xtr: xi, j = X2,j, Fig. 7.7:

(peu2e52)2 = (PeU2e52)\, (7.128)

with S2 being the momentum-loss thickness.

This matching can also be used at flat plate/ramp junctions etc. [16]. A detailed description of the procedure can be found in [1].

— The wall temperature (thermal state of the surface) and the Mach number influence as well as the surface properties may have not been taken properly into account.

It is not attempted here to rate the available relations or to recommend one or the other. Basically it is intended to show the dependencies of thicknesses and integral parameters on typical flow and other parameters, which the engineer or researcher should know. Besides that the relations should permit to obtain an estimation of the magnitude of these entities with reasonable accuracy.

We combine now the above results in Table 7.3. We substitute ‘TO’ condi­tions by general ‘ref’ conditions and chose ш = ш^ = 0.65 in the viscosity law, Section 4.2.[104] We break up all Reynolds numbers Re into the unit Reynolds number Reu and the running length x in order to show the explicit depen­dencies on these parameters.

An inspection of the reference-temperature relation eq. (7.70) and the reference-temperature extensions eqs. (7.100), (7.104), (7.105) and (7.106), (7.120) to (7.124) shows, that we can write all temperature-extension terms symbolically as)

This form shows the desired limit behavior, since for incompressible flow with Mref = 0 the last term in the large brackets vanishes, and with Tw = Tref the two first terms in sum have the value ‘1′.

The reference-temperature extension relations of the integral parameters 51,iam, c, eq. (7.120), and S1iturb, c, eq. (7.122), have different factors in the large brackets, but these factors are quite close to those in the original reference temperature relation, eq. (7.70). Therefore we use eq. (7.129) in order to contract these relations for an easier discussion. In Table 7.3 they are marked accordingly with ‘m’.

The third and the fourth column in Table 7.3 give the basic dependencies of the thicknesses of both incompressible and compressible boundary layers on the running length x and the unit Reynolds number ReUef, the fifth column the dependence of the thicknesses of compressible boundary layers on the reference-temperature ratio T*/Tref, i. e., implicitly on the wall temperature Tw and the Mach number Mref, eq. (7.129).

The dependencies summarized in Table 7.3 give us insight into the basic behavior of thicknesses and integral parameters of two-dimensional flat-plate boundary layers. The results also hold for not too strongly three-dimensional flow over flat surfaces.

— Dependence on the boundary-layer running length x.

All boundary-layer thicknesses increase with increasing x. The thicknesses of turbulent boundary-layers grow stronger (ж x0 8) with x than those of laminar boundary layers (ж x0 5). The thickness of the viscous sub-layer grows only very weakly with x, ж x01, and also the turbulent scaling thickness, ж x0 2.

— Dependence on the unit Reynolds number ReUef.

All thicknesses of boundary-layers depend on the inverse of some power of the unit Reynolds number. The larger ReUef, the smaller are the boundary – layer thicknesses. Thicknesses of laminar boundary layers react stronger on changes of the unit Reynolds number (ж (ReUef )-0’5) than those of turbulent boundary layers (ж (ReUef )-0 2). Strongest reacts the thickness of the viscous sub-layer, ж (ReUef )-0 9, and that of the turbulent scaling thickness, ж (Re%ef )-0’8.

— Dependence on T*/Tref.

relations are available to prescribe them. We note only the dependencies of

the boundary-layer thickness S, which holds for laminar and turbulent flow.

— Pressure gradient dp/dx in main-stream direction.

In general a negative pressure gradient (accelerated flow) reduces S, a positive (decelerated flow) increases it. The cross-flow pressure gradient dp/dz governs boundary-layer profile skewing and the form of the cross­flow boundary-layer profile (see Sub-Sections 7.1.2 and 7.1.3).

— Change of the body cross-section dA/dx in main-stream direction.

If the cross-section grows in the main-stream direction (dA/dx > 0), see, e. g., Fig. 6.22 a), the boundary layer gets “stretched” in circumferential direction, because the wetted surface grows in the main-stream direction. With the same pressure field and the same boundary conditions S becomes smaller at such a body than over a flat plate [1]. This effect is the Man – gler effect [32]. Both wall-shear stress and heat flux in the gas at the wall are correspondingly larger. The inverse Mangler effect enlarges S. It oc­curs where the body cross-section in the main-stream direction is reduced (dA/dx < 0). A typical example is the boat-tailed after-body of a flight vehicle (Fig. 6.22 a), if the flow would come from the right-hand side).

— 3-D effects.

We mention here only the most obvious 3-D effects, which we find at at­tachment and separation lines [1]. We have discussed them in the context of surface-radiation cooling already in Sub-Section 3.2.4. At an attachment line, due to the diverging flow pattern, the boundary layer is effectively thinned, compared to that in the vicinity, Fig. 3.5. At a separation line, the flow has a converging pattern and hence the tendency is the other way around, Fig. 3.6. Consequently the boundary-layer thickness S is reduced at attachment lines, and enlarged at separation lines.

— Surface parameter k.

Basic concepts and results are discussed. Quantitative relations are given for the important properties of attached viscous flow, i. e., boundary-layer thick­nesses, wall shear stress, and the thermal state of the surface (either the heat flux in the gas at the wall qgw or the radiation-adiabatic temperature Tra) in two-dimensional flow. The relations for incompressible flow are extended by means of the reference-temperature concept to compressible perfect-gas flow. This suffices to show their basic dependencies on overall flow parameters and on the wall temperature. With these relations also fair estimations of the different properties can be made, if the boundary layer under consideration is two-dimensional or only weakly three-dimensional, and the stream-wise pressure gradient is not too large.

We study now the equations of motion for hypersonic attached viscous flow. We find these equations in the same way in which we found the boundary- layer equations in Sub-Section 7.1.3, but now making the pressure dimen­sionless with pref instead of pref v2ef. We assume perfect gas and keep the bulk viscosity к because the terms containing it will not drop out, unless we go to the boundary-layer limit. For convenience we consider only the two­dimensional case and keep again x as coordinate tangential to the surface and y normal to it. The equations are written dimensionless, but we leave

the prime away. Higher order terms regarding the reference Reynolds number are neglected. The extension to three dimensions is straight forward.

In analogy to eqs. (7.5), (7.16), (7.17), and (7.20) we find first the (un­changed) continuity equation

dpu dpv

dx R dy

then the Navier-Stokes equations

and finally the energy equation, where on purpose the Eckert number is not employed

It is obvious that these equations reduce to the boundary-layer equations, see (7.36) to (7.40), albeit with some different co-factors, if the Reynolds number Reref is very large and the ratio ‘M^ef ’ to ‘Reref’ very small:

M? ef

-^«F

Reref

This means that if eq. (7.78) is true, the flow is of boundary-layer type, because the pressure is constant in the direction normal to the surface.

If on the other hand

we must expect a pressure not constant in direction normal to the surface. If this happens on a flat vehicle configuration element, in the limiting case on a flat plate, we speak of hypersonic viscous interaction, which we treat in Section 9.3, where we will meet again the term МД, f /Reref, respectively its square root Mref j y/Reref.

We study now in more detail the characteristic properties of eqs. (7.74) to (7.77) in order to get clues regarding the kind of possible numerical com­putation schemes for such flows.

We introduce again characteristic manifolds p [13] for derivatives like in eq. (7.41). To make the problem treatable, we simplify the governing equa­tions by keeping only the leading terms in the equations, omitting the co­factors containing Mref, Reref, 7, and Prref, and by assuming constant transport properties p, к and heat capacity cp:

du dp dv dp

Ртг + !1V + Pw~ + v— = 0,

dx dx dy dy

du du dp dp d2u

Pu~:—— к pvw~ = ~w~t.—– к

dx dy dp dx dy2

dv dv dp dp d2v

PlW к Pvw~ = — к A‘Tyvy,

dx dy dp dy dy2

( dT dT d2T

cp pu— + pv— = k— + CWT + DWT,

dx dy dy2

with CWT and DWT being abbreviations of the compression and dissipation work terms, respectively.

The characteristic matrix then reads, with px and py the characteristic directions of the problem [13]

upx + vpy. From this we obtain:

and finally with A = upx + vpy = 0, eq. (7.46), and by identifying dp/dp = a2 with the speed of sound

C = p p к ip у [(г/2 – gr_)ip2c + 2 uvipxspy + (v2 – a2)y>2] = 0. (7.86)

The term in this equation coming from the pressure gradient term in eq. (7.75) is underlined. Because we have assumed constant viscosity, no coupling of the continuity equation and the momentum equations with the energy equation exists. Hence from the latter only the convective term and the thermal conduction term in у-direction are reflected in eqs. (7.85) and (7.86), respectively.

We find thus a four-fold characteristic in y-direction and in the angular brackets elliptic characteristics for subsonic flow and hyperbolic ones for su­personic flow. If the underlined pressure-gradient/speed-of-sound term would be omitted, we would get in the angular brackets for all flow velocities hyper­bolic characteristics [18]. The system of equations (7.74) to (7.77) without the terms, which are of smaller order of magnitude, and without the pressure – gradient term in eq. (7.75), thus would constitute a linearized system of equations for the description of a weakly disturbed hypersonic flow with u2 ^ a2, v = O(a). If the pressure-gradient term in eq. (7.76) would be zero, the whole equation would disappear, and we would get the boundary-layer equations for hypersonic flows.

If we accept the simplifications made in the equations of motion and those made additionally in order to investigate the characteristic properties of the system of equations, we get the result that the equations of motion without the second-order terms in ж-direction are essentially of elliptic nature for subsonic flows and of hyperbolic nature for supersonic flows.

If the problem at hand permits it, the pressure-gradient term in eq. (7.75) can be omitted. This is possible for flows where

(7’87)

In this case the system of equations is of hyperpolic/parabolic type in the whole flow domain and can be solved as initial/boundary value problem with a space-marching numerical scheme, Section 9.3. If the term cannot be omitted, a parabolization scheme, for instance that of Vigneron et al. [19], must be employed, otherwise the solution process will become unstable.11 [93] [94]

In [14] it is shown that for laminar high-speed flows boundary-layer skin fric­tion and wall heat transfer can be obtained with good accuracy by employing the relations for incompressible flow, if the fluid property density p and the transport property viscosity p are determined at a suitable reference temper­ature T *. The approach is based on the observation that the results of the investigated boundary-layer methods depend strongly on the exponent ш of the employed viscosity relation, see Section 4.2, on the wall temperature Tw, and on the boundary-layer edge Mach number Me, which is representative for the ratio ‘boundary-layer edge temperature’ to ‘total temperature of the flow’, Te/Tt. The dependence on the Prandtl number Pr is weak.

The reference temperature approach is interesting, although the complex interactions of convective and molecular heat transfer, compression and dis­sipation work, see eq. (4.58) or (7.40), with the boundary conditions at the body surface and at the boundary-layer’s outer edge do not suggest it at a first glance. We will see later, that theory based on the Lees-Dorodnitsyn transformation to a certain degree supports this approach.

The reference temperature concept was extended to the reference enthalpy concept [15] in order to take high-temperature real-gas effects into account. In [15] it is shown that it can be applied with good results to turbulent boundary layers, too.

Reference temperature (T*) for perfect gas and reference enthalpy (h*) for high-temperature real gas, respectively, combine the actual values of the temperature T or enthalpy h of the gas at the boundary layer’s outer edge (e), at the wall (w), and the recovery value (r) in the following way:

T * =0.28Te + 0.5Tw +0.22Tr, (7.62)

h* =0.28he + 0.5hw + 0.22hr. (7.63)

Eq. (7.63) contains eq. (7.62) for perfect gas.

Because T*, respectively h*, depend on boundary-layer edge data and in particular on wall data, we note

on general vehicle surfaces : T* = T*(x, z), resp. h* = h*(x, z),

i. e., reference temperature or enthalpy are not constant on a vehicle surface, especially if this surface is radiation cooled.

The recovery values are found in terms of the boundary-layer edge data:

for laminar flow, and with acceptable accuracy

r* = r*h = VPr*, (7.68)

for turbulent flow, with Pr* being the Prandtl number at reference-tempe­rature conditions:

P c*

Pr* = (7.69)

With these definitions the reference temperature, eq. (7.62), becomes in terms of the boundary-layer edge Mach number Me

T* = 0.5Te + 0.5TW + 0.22r*le~l M;Te. (7.70)

Eqs. (7.62) and (7.63) were found with the help of comparisons of results from solutions of the boundary-layer equations with data from experiments [14, 15]. They are valid for air, and for both laminar and turbulent boundary layers. Other combinations have been proposed, for instance, for boundary layers at swept leading edges, Sub-Section 7.2.4.

We employ now and in the following sub-sections only the reference tem­perature approach to demonstrate thermal surface effects in connection with attached viscous flow. This is permitted, because today materials of thermal protection systems or hot primary structures allow temperatures at most up to 1,800-2,000 K, see the examples given in Section 5.6 (RV’s) and in Sec­tion 7.3 (CAV). The power-law relations given in Section 4.2 for the viscosity and the thermal conductivity can be strained up to these temperatures. Vi­brational excitation, Chapter 5, possibly can be neglected. This all holds for qualitative considerations. For quantitative considerations one anyway has to establish first whether an approximate relation can be used for a given problem and how large the immanent errors are.

We use now ‘ *’ to mark reference-temperature data and relate them to overall reference flow parameters, which we mark with ‘ref ’. For the Reynolds number Re*x we thus find

4= P VrefX ___ PrefVrefX p Pref ______ p P Pref

^ex ~ I ~ 7~ ~ xi-eref, x —

P Pref Pref P Pref P

In attached viscous flow the pressure to first order is constant through the boundary layer in direction normal to the surface, and hence we have locally in the boundary layer:

P*T* = Pref Tref.

We introduce this into eq. (7.71) together with the approximate relation p = cTu for the viscosity, Sub-Section 4.2, and finally obtain[92]

T 1+ш

Ret = Reref}X ( /./ j. (7.73)

For flat plates at zero angle of attack, and approximately for slender bodies at small angle of attack, except for the blunt nose region, we can choose ‘ref’ = ‘TO’, whereas in general the conditions at the outer edge of the boundary layer are the reference conditions: ‘ref’ = ‘e’.

The reference-temperature/enthalpy extension of incompressible boun­dary-layer relations is not only a simple and effective method to demonstrate thermal-surface effects on attached viscous flow. In its generalized formula­tion given by G. Simeonides, [16], it is also an effective tool for the actual determination—with sufficient accuracy—of properties of attached compress­ible laminar or turbulent viscous flows, even for flows with appreciable high – temperature thermo-chemical effects, see, e. g., [17]. In Sub-Section 7.2.1 we give the generalized relations for the determination of boundary-layer thick­nesses and integral parameters and in Sub-Sections 7.2.3 to 7.2.6 for the determination of skin friction and thermal state of the surface.

The generalized relations can be applied on generic surfaces with either inviscid flow data found from impact methods or in combination with invis­cid flow field data found by means of Euler solutions. Of course only weak three-dimensionality of the flow is permitted. The stream-wise pressure gra­dient in principle also must be weak, but examples in [17] show that flows with considerable pressure gradients can be treated with good results. Flow separation and re-attachment, see Section 9.1, strong interaction phenomena and hypersonic viscous interaction, see Sections 9.2 and 9.3, must be absent, also slip-flow, Section 9.4. To describe such phenomena Navier-Stokes/RANS methods must be employed. Slip flow, however, can also be treated in the frame of boundary-layer theory.