# The shock polar

Although in practice plane-shock-wave data are used in the form of tables and curves based upon the shock relationships of the previous section, the study of shock waves is considerably helped by the use of a hodograph or velocity polar diagram set up for a given free-stream Mach number. This curve is the exit velocity vector displacement curve for all possible exit flows downstream of an attached plane shock in a given undisturbed supersonic stream, and to plot it out requires rearrangement of the equations of motion in terms of the exit velocity components and the inlet flow conditions. Reference to Fig. 6.25 shows the exit component velocities to be used. These are qt and qn, the radial and tangential polar components with respect to the free stream V direction taken as a datum. It is immediately apparent that the exit flow direction is given by arctan(<7t/<7n). For the wave angle /3 (recall the additional notation of

Fig. 6.25

Fig. 6.23), linear conservation of momentum along the wave front, Eqn (6.72), gives vi = V2, or, in terms of geometry: V cos /3 = V2 cos(/3 — 6) Expanding the right-hand side and dividing through:

V = V2 [cos S + tan /3 sin 6 or, in terms of the polar components:

y> =  which rearranged gives the wave angle

To express the conservation of momentum normal to the wave in terms of the polar velocity components, consider first the flow of unit area normal to the wave, i. e.  Pi +PlM? – P2 + Pl^i

Then successively, using continuity and the geometric relations:

P2 =P + Pi V sin/3[Fi sin/3 – qn sin/3 + qn cos 0 tan <5] P2=Pi+ Pi V sin/3[(V — qn) sin/3 + cos/3]

and, using Eqn (6.89):

P2 =Pi +PV{V – qa)

Again from continuity (expressed in polar components):

pi Vi sin /3 = p2 V2 sin(/3 – S) = p2?n(sin /3 — cos /3 tan <5)  or  Again recalling Eqn (6.91) to eliminate the wave angle and rearranging: P2 Vi – qn pi

P2 Viqn – ql-qi P1V1

Finally from the energy equation expressed in polar velocity components: up to the wave (6.96)

and downstream from the wave (6.97)

Substituting for these ratios in Eqn (6.93) and isolating the exit tangential velocity component gives the following equation: (6.98)

that is a basic form of the shock-wave-polar equation.

To make Eqn (6.98) more amenable to graphical analysis it may be made non­dimensional. Any initial flow parameters, such as the critical speed of sound a*, the ultimate velocity c, etc., may be used but here we follow the originator A. Busemann* and divide through by the undisturbed acoustic speed a: (6.99)   where q% = (qt/ai)2, etc. This may be further reduced to

where (6.101)

Inspection of Eqn (6.99) shows that the curve of the relationship between qi and qn is uniquely determined by the free-stream conditions (Mi) and conversely one shock-polar curve is obtained for each free stream Mach number. Further, since

A. Busemann, Stodola Festschrift, Ziirich, 1929.

the non-dimensional tangential component qt appears in the expression as a squared term, the curve is symmetrical about the qD axis.

Singular points will be given by setting qt = 0 and oo. For qt = 0,

(M, – qn)2(£ -Mi) = 0

giving intercepts of the qn axis at A:

qn = M (twice) (6.102)  at В

For qt oo, at C, qn = -4тМ і +Mi = Mi+ —2 — 7+1 (7+l)Afi

For a shock wave to exist Mi > 1. Therefore the three points B, A and C of the qD axis referred to above indicate values of qn < Mi, = Mi, and > Mi respectively. Further, as the exit flow velocity cannot be greater than the inlet flow velocity for a shock wave the region of the curve between A and C has no physical significance and attention need be confined only to the curve between A and B. Plotting Eqn (6.98) point by point confirms the values A, В and C above. Fig. 6.26 shows the shock polar for the undisturbed flow condition of Mi — 3. The upper

branch of the curve in Fig. 6.26 is plotted point by point for the case of flow at a free stream of Mi = 3. The lower half, which is the image of the upper reflected in the qa axis shows the physically significant portion, i. e. the closed loop, obtained by a simple geometrical construction. This is as follows:

(i) Find and plot points A, В and C from the equations above. They are all explicitly functions of Mi.

(ii) Draw semi-circles (for a half diagram) with AB and CB as diameters.

(iii) At a given value qn (Oa) erect ordinates to meet the larger semi-circle in c.

(iv) Join c to В intersecting the smaller semi-circle at b.

(v) The required point d is the intercept of bA and ac.

Geometrical proof Triangles Aad and acB are similar. Therefore

i. e. (6.105)

Again, from geometrical properties of circles,

(ac)2 = aB x aC

which substituted in Eqn (6.105) gives (ad)2 = (aA)2

Introduction of the scaled values, ad = qu

aB = Oa – OB = qa – Mi, aA — OA – Oa = Mi – qn, aC = OC – Oa = [2/(7 + l)]Mi +Mj – qn reveals Eqn (6.100), i. e.

Consider the physically possible flows represented by various points on the closed portion of the shock polar diagram shown in Fig. 6.27. Point A is the upper limiting value for the exit flow velocity and is the case where the free stream is subjected only to an infinitesimal disturbance that produces a Mach wave inclined at p to the free stream but no deflection of the stream and no change in exit velocity. The Mach wave angle is given by the inclination of the tangent of the curve at A to the vertical and this is the limiting case of the construction required to find the wave angle in general.

Point D is the second point at which a general vector emanating from the origin cuts the curve (the first being point E). The representation means that in going through a certain oblique shock the inlet stream of direction and magnitude given by OA is deflected through an angle 6 and has magnitude and direction given by vector OD (or Od in the lower half diagram). The ordinates of OD give the normal and tangential exit velocity components. The appropriate wave angle Д* is determined by the geometrical construction shown in the lower half of the curve, i. e. by the angle Ada. To establish this recall Eqn (6.99): The wave angle may be seen in better juxtaposition to the deflection 6, by a small extension to the geometrical construction. Produce Ad to meet the perpendicular from О in d’. Since AaAdlHA^d’O, 