Category Theoretical and Applied Aerodynamics

In the first part of this chapter, inviscid hypersonic flows are discussed, starting with hypersonic similarity parameters. High Mach number flows over plates, wedges and cones are treated with hypersonic small disturbance theory as well as the Newtonian flow theory. Numerical methods to solve the blunt body problems are also covered.

To account for viscous effects, boundary layers and viscous/inviscid interaction procedures will be covered next. The main feature of hypersonic boundary layer is the high temperature effects and aerodynamic heating, while the main feature of the interaction problem is the vorticity and entropy in the inviscid layer.

Other important factors, including chemical nonequilibrium with dissociation and ionization as well as low density and rarefied gas dynamics will not be covered here and the reader is referred to more specialized references (see Park [12]).

The discussion is based on the continuum model, where the mean free path of the molecules A is much smaller than the characteristic length of the body, i. e. the Knudsen number is small (say less than 0.1), where

A

Kn =- (12.178)

From kinetic theory of gases, Kn = 1.26jYM0/Rel. For Kn > 10, free molec­ular regime exists and Boltzmann equations must be used.

For the continuum model, the flow is governed by the Navier-Stokes equations. At a solid surface, no penetration and no slip boundary conditions are imposed. Either the temperature or the heat flux are also specified (the effect of slip velocity and jump in temperature at the boundary will not be covered). Moreover, we will assume perfect gas. In general, real gas is governed by p = pRTZ, where Z is the compressibility factor and 7, the ratio of specific heat under constant pressure and constant volume (7 = Cp /Cv), is different before and after the shock.

Another important factor is the dependence of the viscosity coefficient on tem­perature, and we will assume in this discussion a power law, namely

and for simplicity, let w = 1. Also, radiation effects will not be considered.

Slender Body at Angle of Attack

Grimminger et al. [67], have derived general expressions for the lift and drag of inclined bodies of revolution. They assume that the streamlines remain parallel and leave the surface in a streamwise direction, see Fig. 12.7.

They used the Newtonian formula to calculate the pressure on the wetted part of the body.

Later, the Newtonian theory for slender bodies of revolution at small angle of attack was reported by Cole [66] and Cole and Brainerd [68], as a small perturba­tion on the solution at zero incidence. Linear equations are obtained with variable coefflcients depending on the zero incidence solution. The results are

where the body shape is given by r — 5F(x), F(1) — 1, as stated before.

Hayes and Probstein [2] extended Cole’s results for non slender bodies.

Cox and Crabtree [3] discuss the case of a cone of circular cross section and the appearance of what is called the vortical layer.

For a cone at zero incidence, the radial surfaces of constant entropy are shown in Fig. 12.8. In the same figure, the corresponding surfaces when a — 20° are shown indicating the existence of rotational flow.

Ferri [69, 71] and Ferri et al. [70] pointed out that the surface of the cone must have constant entropy and there exist a vortical singularity, where the entropy is multi-valued.

The thickness of the vortical layer is shown by Guiraud [72] to be

ev ~ YY—1 exp (——– Y + 1 ) (12.171)

v Y+ 1 a(Y – 1)

where y is a mean value of y downstream of the shock.

Fig. 12.8 Surfaces of constant entropy for flow past cone

Based on Guiraud’s analysis, the surface pressure distributions and the total forces and moment (relative to the nose) acting upon a body of revolution can be calculated in closed form. For slender bodies, Guiraud’s results agree with Cole’s.

Following Hayes and Probstein [2], singularities in solutions for an elliptic cone at small and large incidences are given in Fig. 12.9.

A topological law, based on a Poincare theory, relates the number of elementary singular points according to the relation

N + F – S + O = 2 (12.172)

where N is the number of nodes, F is the number of foci, S the number of saddle points and O the number of obstacles. For more details, see Ref. [74].

Over two decades, the problem of cones at small and large angles of attack has been studied by many authors, including Tsien [75], Lighthill [76], Young and Sisk [77], Stone [78, 79], Willet [80, 81], Holt [82], Cheng [83, 84], Sychev [85], Munson [86], Woods [87, 88], Melnick [89, 90]. Numerical simulations are discussed by Moretti [91], Dwyer [92] and Fletcher [93]. A comprehensive review can be found in the book by Bulakh [94].

Blunt Body Problems

There are two categories, the example for the first is cylinders or cones, while the second is a slender body with a blunt nose, see Fig. 12.10 from Ref. [2].

For the bow shock stand-off distances for spheres and cylinders, approximate analytical relations are given by Hayes and Probstein [2]. For a sphere with a radius Rb and a shock of radius Rs, assuming constant density in the thin layer between the bow shock wave and the body surface

For the case of the circular cylinder

Y+1, and for a sphere, the stand-off distance becomes

Y — 1

A ~ Rs (12.177)

Y + 1

Recent work on theoretical models for stand-off distance is discussed by Olivier in [95].

At the axis, the normal shock wave leads to the largest entropy jump and the entropy decreases as the shock angle approaches the Mach angle. Hence, according to Crocco’s relation, vorticity is generated behind a curved shock.

For a blunt body at angle of attack, the entropy layer and the vortical layer will mix in the neighborhood of the body where the boundary layer is also growing.

The Flow field around a sphere at Mach 10 and Mach 50 are shown in Fig. 12.11.

The analogy of a bow shock on a blunt nosed body and unsteady one – dimensional explosion is sketched in Fig. 12.12.

These complicated flows have been studied over the years. In the series expansion methods, it is assumed that the shock shape is known. The conditions just downstream

of the shock can be obtained from the shock relations and the derivatives of the flow variables may be obtained using the governing equations. Taylor series in terms of the distance from the shock is used to describe the flow field between the shock and the body. Examples of this approach are given by Lin and Rubinov [96], Shen and Lin [97] and Cabannes [98, 99]. The convergence of the Taylor series, near the body is in general questionable as discussed by Van Dyke [100].

Numerical methods are more suitable for such a study as discussed by Hayes and Probstein [2].

Van Dyke [101] used an inverse method integrating the flow equations from the shock to the body by a marching technique, assuming the shock shape to be a conic section, see also Mangler [102]. Good results for stand-off distance for a sphere, in agreement with experimental measurements, are reported for different high Mach numbers. However, the problem is, in general, ill-posed as shown by Vaglio-Laurin [103], where almost indistinguishable shock shapes produce completely different body shapes. Special treatment is required to avoid numerical instabilities.

Another approach to solve the inverse problem was proposed by Garabedian and Lieberstein [104], based on the method of complex characteristics.

First the initial data for a known analytic shock curve are continued into a fictitious three-dimensional space which consists of the real value of one of the independent variables and the complex value of the other. The wave equation can be solved, in a stable manner, by a marching procedure, or by the method of characteristics.

However, any small change in the initial data in the real domain can result in a large change in the complex domain. Lin [105] noticed that the analytic continuation of the initial conditions from real values into the complex domain is itself the solution of an initial value problem for a Laplace equation. For simple closed form analytic shock shapes such as hyperbolas or parabolas for the hypersonic blunt body problem, the initial conditions are continued exactly into the complex domain avoiding any instabilities.

Moreover, Van Dyke [106], pointed out that the computational time of Garabe – dian and Lieberstein method was almost 200 times his straightforward marching procedure.

Inverse methods have been applied to asymmetric flows by Vaglio-Laurin and Ferri [107]. They treated bodies of revolution at small angles of attack by perturbation around a body at zero incidence.

Of course, the usual problem of interest is the direct problem, where the inverse methods can be used via trial and error. Hayes and Probstein [2], considered building up a catalog of solutions, a dangerous philosophy delaying the advances of direct methods.

In the late 50s, Dorodnitsyn proposed a direct method based on integral relations, see also Belotserkovskii and Chushkin [108]. See also Holt [109].

The basic idea is to solve a system of partial differential equations in a domain by dividing the domain into strips and obtain independent integral relations to obtain the solutions of the variables in each strip. Assuming a polynomial variation in one variable of the flow variables between the shock and the body and substituting into the equations of motion, a system of ordinary differential equations is generated and may be integrated numerically across the strips. An initial guess of the shock shape is assumed and the solution is obtained by iteration. Dorodnitsyn [110] has generalized his method, by multiplying the basic equations by a function to obtain the moments of the equations in an attempt to increase the accuracy of the calculations. Notice the analogy with the early boundary layer methods). Many variants exist and for more details see Holt [109].

Perhaps, more interesting are the applications of finite differences (and finite volumes) using relaxation techniques or unsteady approach to steady-state solutions of the governing equations.

Mitchell and McCall [111], calculated the supersonic flow over blunt bodies using the stream function/vorticity formulation where the shock wave was assumed known from experimental measurements. A serious problem was the breakdown of the relaxation procedure in the neighborhood of the sonic line. Hayes and Probstein [2], discussed how to carry the calculations through the sonic region and somewhat beyond the limiting characteristics and established consistency relations used as boundary conditions on the sonic line. These conditions may indicate how the shock shape should be updated to obtain an improved solution. (The supersonic flow field can be always obtained via the method of characteristics).

In such calculations, the shock is fitted. A more attractive approach is the shock capturing, where conservative Euler equations, in primitive variables, are solved explicitly to reach steady-state. There are two main schools, the first is based on explicit artificial viscosity as in the Lax-Wendroff scheme and the second is based on upwind schemes as in the Godunov method. Early applications to hypersonic flows of Lax-Wendroff type scheme was reported by Burstein and Rusanov using his version of the Lax scheme. Applications of Godunov’ schemes are discussed in Holt’s book [109]. Implicit methods to solve the unsteady Euler equations have been developed since then, see for example Richtmeyer and Morton [112]. For recent review of numerical methods for hypersonic inviscid flows see Pletcher et al. [113]. Notice, some of the early methods applied for hypersonic flows have been extended to deal with lower transonic flow regime, for example Garabedian’s complex charac­teristics and the integral relation method as well as shock fitting and shock capturing techniques for the relaxation method of the stream function equation (see Emmons [114], Steger [115], Hafez and Lovell [116]), and for the unsteady Euler equations (see Hirsch [117], Van Leer [118], Roe [119], Osher [120] for upwind schemes).

To conclude this section, we will discuss briefly the works of Chernyi and Cheng for the slender bodies with blunt noses.

In Chernyi’s method [121], it is assumed that a plane piston starts to move from the nose with a velocity V = Uв, normal to the flow direction, to generate a wedge or a cone of semi-angle в (see also Cox and Crabtree [3]) for the governing inte­gral equations expressing conservation of energy and momentum in the equivalent unsteady flow problem.

The method of Cheng [122], includes also the effect of the entropy layer.

Guiraud [123] treated directly the effect of slight bluntness. Yakura [124] used the inverse problem as discussed in Van Dyke [100]. In Fig. 12.13, in the direct problem,

Fig. 12.14 Blunt axisymmetric slender body producing a paraboloidal shock with infinite Mach number and 7 = 1.4

the shock wave is forced away from the body due to bluntness, while in the inverse problem the body must move instead.

Yakura has applied his method also to the axisymmetric problems of the blunted cone that supports a hyperbolidal shock wave and the body that produces a paraboloidal shock wave at M = ro. Yakura’s results are close to those of Sychev [125], see Fig. 12.14.

The Newton-Busemann formulas can be obtained systematically from an asymp­totic hypersonic small disturbance theory following Cole [66], who introduced the following parameters

Y — 1

A = , (density ratio) (12.140)

Y + 1

1

H = , (hypersonic similarity parameter) (12.141)

M02-2

H y + 1 1

N = — = , (Newtonian flow parameter) (12.142)

t y — 1 M^S2

The small parameter A vanishes in the Newtonian limit y ^ 1 and

1 + A 2

Y = = 1 + 2A + 2 A2 + ••• (12.143)

1 — A

The small parameter S is the slope or maximum thickness ratio S = e/c.

For hypersonic small disturbances, the transverse perturbation velocity v is of order S, while the streamwise perturbation velocity u is of order S2. The density

perturbation is of order unity, while the perturbation in pressure is of order S2. These orders can be determined from shock wave relations or from Prandtl-Meyer expansion, provided M0S is not much less than unity.

Let the Newtonian flow parameter N be fixed as A ^ 0 and H ^ 0. The first order approximate equations obtained from the hypersonic small disturbance equations for the pressure, density and the transverse velocity are (a = 0 for two dimensional flow and a = 1 for axial symmetry)

The continuity equation is satisfied automatically. The momentum equation reads

1 дф д p

F" + = 0

Fa дr дr

By integration from the surface r = 0, one obtains

Fa F " ф + p = p0(x )

where ф = 0 at r = 0.

Notice that the surface pressures are independent of the Newtonian parameter N, although the flow field and shock shape depend on N (i. e. the limiting values of the surface pressures as N ^<x> should be a good approximation over a wide range of Mach numbers).

The entropy equation states that p/p — K(ф). From the shock relation K — F’2 + N on the shock, and since rfshock is given before, K (ф) can be found parametrically.

At x — 0, ф — 0 and K(0) — F/2(0) + N, so that

The surface temperature is found from the equation of state, where

p0 — 1 + "02S’2 p — ІЇ

so that the surface temperature is constant

T F /2(0) K (0)

— +1 — + T0 N N

For cones, the first approximation is F (x) = x, 6 = tan вс and the shock is g(x) = Ax. The shock conditions are

p(x, Ax) = 1

p(x, Ax) = 1/(N + 1) (12.161)

v(x, Ax) = A – (N + 1)

The transverse momentum equation yields = 0, hence p = 1, 0 < r < Ax. From the entropy equation and the shock relations, it follows that the density is constant, p = 1/(N + 1).

The transverse component of the velocity can be found from the continuity equa­tion

(12.162)

and, using the boundary condition v = 0 at r = 0

This represents a flow filed whose streamlines are more inclined towards the cone surface as r increases.

The shock angle can be found from the boundary condition for v on the shock

The impact theory Newton proposed, is simply to calculate the normal force on a plate at angle в in terms of the change of normal momentum, i. e. (Fig. 12.3)

N = Ap0U2 sin2 в (12.121)

hence

Cp = P – P0 = 2sin2 в (12.122)

2 P0U 2

For a curved surface, the modified Newtonian theory is

Cp = Cp, max sin2 в (12.123)

where Cp, max is the stagnation pressure coefficient. Indeed, as M0 ^ ж, the flow across a normal shock gives Cp, max = 2. However, for finite M0, Cp, max < 2. See Fig. 12.4.

On the other hand, the oblique shock relations are given by

Fig. 12.3 Newton’s impact theory

Fig. 12.4 Modified Newton’s theory

since both в and в are small.

The pressure coefficient is given by

where U and R are the mean values of the velocity and of radius of curvature of the thin layer of thickness N, see Fig. 12.5.

The mass flow rate past a section of a body of revolution is given by

Ґ 2

m = 2nr pudn — nr zp0U (12.128)

0

Notice, rs — r. Hence

Fig. 12.5 Shock layer

и r r, r и

Ap = P0U2, and Cp = (12.129)

R 2 R U

As M0 and y ^ 1, a particle entering the shock with a tangential velocity

U cos в retains that velocity, therefore the mean velocity U is

1 Ґ

u= 2nrU cos edr (12.130)

nr2 0

The radius of curvature of the thin layer is governed by

r d в

= r sin в (12.131)

R dr

Hence,

sin в dв r

ACP = 2 cos вrdr (12.132)

r dr 0

Let A = nr2, the cross-sectional area; The modified surface pressure coefficient given by Busemann is

2 dв fA

Cp = 2 sin2 в + 2 sin в cos вdA (12.133)

dA J0

For a sphere, the above relation becomes

Cp = 2^1 – 4 sin2 ^ (12.134)

Lighthill [65] suggested that the shock wave separates from the surface of the sphere at the point of vanishing Cp, в = 60° measured from the forward stagnation point, Fig. 12.6.

Fig. 12.6 Sphere in Hypersonic Flow

The shape of the shock wave may be determined from

d – a

0 = 2sin2 в + 2sine – cos edA (12.135)

dA Jq

It can be shown that the shock shape tends asymtotically to be y ~ x1/3.

The modified Newton-Busemann formulas for slender bodies are

Cp = Cp0 (в2 + Г-‘J, for axisymmetric flow (12.136)

Cp = Cp0 (в2 + y-‘J, for plane flow (12.137)

For cones and wedges, the Busemann corrections vanish and the Newtonian for­mulas are consistent with the shock relations as M0 ^<x> and у ^ 1.

There is a certain class of self-similar motions for which the partial differential equa­tions reduce to ordinary differential equations, thus leading to a great simplification and in some cases to analytical solutions.

The concept of self-similarity refers here to flow in which the flow variables between the shock and the body are similar to each other at different stations along the body.

The use of unsteady analogy or equivalence principle, implies that a number of self-similar solutions obtained for unsteady flow problems become available for calculating steady hypersonic flows.

Besides the wedge and the cone, another useful solution is for cylindrical pistons having a motion given by r ~ tm, and from which the flow past power law bod­ies may be obtained, see Velesko et al. [52]. Other self-similar solutions include a piston expanding according to an exponential law, r = r0et0, see Sedov [53] and Gusev [54].

The analogy with steady hypersonic flows was developed by Grodzovskii [55], Chernyi [56] and Stanyukovich [57].

Lees and Kubota [58] and Mirles [59] obtained directly similarity solutions of the hypersonic small disturbance equation for axisymmetric bodies.

The earliest work on similar solutions of this type of problems was carried by Bechert [60] and Guderley [61]. Guderley derived the appropriate equations and boundary conditions and then reduced the problem to a single first order differential equation and applied his theory to the problem of implosions. In his work, he gave a detailed study of the various singular points of the governing differential equation.

Following Kubota [62], we shall consider steady solutions of the hypersonic small disturbance equations for axisymmetric bodies.

Let the nondimensional variables f, g, h for radial velocity, pressure and density be functions only of n = r /rs, where rs is the reduced shock radius rs/т, hence

and for flow similarity щ is constant, along the body rb/rs = constant, hence the body is given by rb = C2xm.

The ratio of the body to the shock radius and the variation of the flow variables between the shock and the body can be obtained from the solution of the differential equations.

It is possible, however, to find the values of m for which similar flows may exist by considering the drag of the body. Since the surface pressure pb is proportional to r’s, the drag is given by

(12.120)

D is finite if —2a < 1 + a, or m > 2/(3 + a), where a = 0 for plane and a = 1 for axisymmetric bodies.

The breakdown of similarity for m < 2/(3 + a) does not imply that no flow exists, but only that there is no similarity solution.

Kubota integrated the system of differential equations numerically, for 1 > m > 2/(3 + a), with the corresponding boundary conditions. The drag coefficient for bodies of revolution has a minimum value at m = 0.92, assuming y = 1.4. The distance between the shock and the body decreases as the value of 7 approaches unity. (For y = 1, the shock and the body coincide).

Mirles [59] developed an approximate method based on the asymptotic form of the flow in the neighborhood of the body surface. Both first and second order approximations are in agreement with the numerical solution obtained by Kubota.

The above similarity solutions for power law bodies are limited to the case of strong shocks (M2 в2 > 1).

For a finite Mach number, the solution may be expected to break down towards the rear of the slender body as the shock becomes weaker and the shock angle approaches the Mach angle. Also, the region near the nose of a power law body is excluded since the solutions are based on the use of the small disturbance equations. The similarity solutions for m < 1 predict a density which falls to zero at the body surface. Since the pressure is finite, the entropy will become infinite. Indeed, there is an entropy layer next to the surface in which the small disturbance equations are no longer valid and must be excluded from solutions obtained using the similarity method. For more details see Cox and Crabtree [3].

For steady two dimensional and axisymmetric compressible flows as well as for unsteady one dimensional flows, a stream function can be introduced to satisfy the continuity equation identically.

For example, in 2-D case, using Cartesian coordinates, one can use

дф дф

pu = , and pv = —-

dy dx

The equation for ф is given by

where w is the vorticity.

The equation for w, for steady isoenergetic flow is given by Crocco equation

(12.81)

or, simply w/p is constant along a streamline. The vorticity is related to the entropy gradient

(12.82)

where n is normal to the streamline. Notice S is constant along a streamline but it jumps across a shock.

Hence, the application of such a formulation requires the identification of the shock shape and position to find the jump of entropy.

There is another problem, even for irrotational flow, the density is not a single valued function of the mass flux. Indeed, for a given flux, there are two values, one for subsonic and the other for supersonic conditions. This is obvious considering the compressible flow in a stream tube (or a nozzle). Moreover, there is a square root singularity at the sonic point. The density is not defined (imaginary) for flux values higher than the sonic flux. The stream function equation is, in general, of mixed type as can be easily shown from the equivalent non-conservative form

u2 д2ф uv д2ф v2 d2 ф 22 ( 1 2 d ( S

— a2 dx2 — 2 a2 dxdy + 1 — a2 dy2 =~— a Y—1 + M° Лф Cp

where a is the speed of sound given by Bernoulli’s law

There is another form of the stream function equation introduced by Crocco, where the derivatives of ф is determined by the velocity component u and v only.

Let

(12.85)

effects is given by

Pai found analytical solution for attached shocks to a plane ogive. He also found analytical solution to the axially symmetric case. He concluded that the first order linearized theory failed at certain points in the weak shock region and he suggested to use the method of characteristics to solve the full equations in the neighborhood of these points.

On the other hand, Kogan [50] obtained solutions for Crocco’s stream function equations for both two dimensional and axisymmetric flows with shocks attached to the leading edges. His results for biconvex parabolic airfoil are in good agreement with the method of characteristics near the airfoil leading edge. using successive approximation, taking the flow behind a plane shock wave as zero-order, the shock wave profile obtained in each step is used to determine the domain and the boundary conditions of the next step.

For axially symmetric flow, the flow past a cone is used a zero-order approxima­tion for the ogive problem. Successive approximations with appropriate boundary conditions are solved analytically. The first order solution for V contains a |-power branch point at в = в0 (the cone surface) and no logarithmic term as reported by Shen and Lin [51]. Shen and Lin results seem to give infinite pressure gradient at the ogive surface, although solutions by the method of characteristics give no such indication.

The stream function formulation for the reduced problem was used by Van Dyke in his hypersonic small disturbance theory.

and boundary conditions

The tangency condition at the surface of the cone is f (b) = 0.

Van Dyke solved the nonlinear ordinary differential equation for f numerically, step by step, starting from the shock surface inward until f vanishes to produce the solution at the solid surface. The pressure coefficient is given by

(12.106)

The surface pressure agrees well with the results of the full equations (with error O (52) as expected).

Van Dyke calculated also the flow over plane ogive and ogive of revolution. Again, his results agree with Kogan’s.

It is clear, in the case of the nonlinear transonic small disturbance theory, that subsonic and supersonic theories are special cases.

For hypersonic flow, a connection with the adjoining supersonic range will be advantageous. The difficulty arises in the continuity equation. In the hypersonic theory, it becomes

(12.75)

while in the linearized supersonic theory it reduces instead to

dp du d v dw

UdX + P0 dX + dy + Tz = 0

The term p0 dU must be retained in linearized supersonic theory, and it must be neglected in the hypersonic theory in order to achieve similitude.

Van Dyke proved, however, that the small disturbance hypersonic theory covers the linearized supersonic theory if it is interpreted in accordance with the similarity rule of the latter. The solutions of the hypersonic small disturbance theory remain valid at small values of the parameter M0t provided that the latter is replaced by pr,

where в = m2 – 1 and the pressure and density are scaled as follows

p = p()(yM02r2p – , and p = p0 ^в0p – _L^ (12.77)

The pressure coefficient becomes

The error in the unified theory is O (r2) or O (r/в) whichever is the greater. See Van Dyke [45].

Extension to unsteady motions involving small time dependent oscillations of a thin body exposed to a steady uniform stream is straightforward. The full problem has a total derivative including local time derivative term, i. e.

with a reduced time introduced where t = Ut/L, the continuity equation for example becomes

(12.72)

The equation for u remains uncoupled from the other equations. However, u can no longer be found in terms of the other variables since there is no counterpart of the Bernoulli’s equation. If required, U can be obtained form solving the x-momentum equation.

Notice that x and Ї derivatives appear only in the combination

д d д Ї + dx

hence, introducing the coordinates

x = x – Ut, or x = x – Ut

reduces the unsteady small disturbance problem to exactly the form of the steady problem. This means that Hayes’ analogy remains valid for unsteady motion if the variation of the contour with time is taken into account. For more details on this point, see Hamaker and Wong [48], and Lighthill [49].

Van Dyke considered the shock as an internal boundary (shock fitting) and wrote the equations in non-conservative form for the smooth flow downstream of the shock wave.

He also replaced the energy equation by the condition that the entropy is constant along streamlines (but not across the shock)

(12.69)

It is clear now, that the x-momentum equation for u is uncoupled from the others. The continuity, the y – and z-momentum equations and the entropy equation can be solved for p, v, w and p, independently of її. If required, її can be determined from Bernoulli’s law

(12.70)

Hence, Van Dyke confirmed Hayes’ observation that the reduced problem is com­pletely equivalent to a full problem for unsteady flow in one less space dimension.

The latter is the unsteady motion in the (y, z) plane due to a moving solid boundary described by B = 0, where x is interpreted as the time. For example, the steady hypersonic flow over a slender pointed body of revolution is equivalent to the prob­lem of unsteady planar motion due to a circular cylinder whose radius varies with time, growing from zero at time x = 0 (see also Goldsworthy [46]).

In fact, for two dimensional steady flows, the equations of motion reduce to those of unsteady one dimensional flow if one of the velocity components remains constant everywhere (see Bird [47]).

Dorrance [43] obtained analytical formulas for two-dimensional profiles via expand­ing the expression of the pressure coefficient as power series in K where the change of flow angle Лш is taken positive for compression and negative for expansion, hence for oblique shocks

For expansion fans, the expansion becomes

The first two terms of the two expressions are identical, the third terms differ slightly (-10%) for K = O(1).

Dorrance worked out in a manner like that of the linear Ackeret theory, simple forms for different profile shapes. For symmetric double wedge, the results are

12.2.3 Bodies of Revolution

The applicability of hypersonic similarity for cylindrical bodies with conical or ogival noses is demonstrated in Ref. [44]. For fitness ratio

F = length of the nose section/maximum diameter of the nose section (12.32)

the similarity parameter, K = M0/F

and for F > 2 and M0 > 2, the validity of the hypersonic similarity was estab­lished.

Hypersonic Flow over Cones

For large M0, the shock angle is close to the semi-vertex angle of the cone, в, hence

Cp = – L – – 1 = (Y + 1) в2 (12.33)

Y M2 -0

therefore

p – p0 – Y (Y + ^ K2 (12.34)

p0 2

with boundary condition

V. VB = 0, at B = 0

where B(x, y, z) = 0 represents the body shape.

The far field is uniform supersonic flow with M0 > 1. The shock wave is described by S(x, y, z) = 0 and it is not known.

The shock jump conditions can be obtained from the weak solution of the conser­vation laws. Conservation of tangential momentum requires that the velocity compo­nent tangent to the shock surface be continuous. The tangential velocity component is

Vt = (n л V) л n (12.42)

where n is the unit vector normal to the surface S, and

< u > < v > < w >

dS = dS = dS ’ at S =0

dx dy dz

The component normal to the shock is

and the other jump conditions at S = 0 are

dS dS dS

< pu > + < pv > + < pw > = 0

dx dy dz

The second law of thermodynamics requires that entropy does not decrease across shock waves, hence

< — >> 0, at S = 0 (12.48)

pY

The body is assumed to be thin where the streamwise slope of its surface is everywhere small compared to unity. A small parameter т is introduced which may be the thickness ratio or the angle of attack for inclined bodies.

From the tangency condition, all cross wind velocities are, in general, of order т. From the approximate solution for thin plane wedge, the pressure coefficient and the streamwise velocity perturbation are of order т2.

Such considerations suggest introducing new independent and dependent vari­ables as follows

yz

X = x, y = -, Z = —, B = B(X, y, Z), S= S(X, y, z) (12.49)

тт

u = U ^1 + т2й^ , v = UtV, w = UтW, p = p0p, p = yM^p0т2p

(12.50)

The above new quantities are dimensionless and of order unity for a body of unit length. The reduced first order equations in т are

(12.51) (12.52)

d p d y	(12.53) (12.54) (12.55) (12.56) (12.57) (12.58) (12.59) (12.53) (12.61) (12.62) (12.63)
dpv + dpv2 + dpvw
dx dy	d z
dpw dpv w d pw2 dX + dy + dZ	d p d Z
dpH + dpv HI + dpw HI ^
d x	d y	d z
with boundary conditions
d B _ д В _ д В + v – + w = 0, at В = 0 dx dy dz
or
d p В + d pv В + d pw В ^
d x	d y	d z
and
as x ^ —<x
Notice,
, __ . 7 and H — + 2u + —2 + w2 , 2 y — 1 P 2
The jump conditions at the shock wave become
dS dS dS < V > + < Pv > + < Vw > = 0 dx dy dv

(12.64)

Notice also, that the parameters M0 and т of the full problem appear only in the upstream condition on p, and not anywhere else, in the combination M0t.

Based on the above formulation, ignoring all т2 terms, the hypersonic similarity is justified.

It is interesting to notice that the error of first order small disturbance theories are of O (т2/3), O (t) and O (t2) for transonic, linearized supersonic and hypersonic flows respectively. A second order theory for hypersonic speeds is needed the least!