6.1 Introduction 1. Definition of Oblique Shock - a straight compression shock wave inclined at an angle to the upstream flow direction - In general, the oblique shocks produce a change in flow direction as indicated in Fig. 6.1.
2. Occurrence 1] external flow - due to the presence of wedge in a supersonic flow - due to the presence of concave corner in a supersonic flow 2] internal flow - in supersonic flow through an over-expanded nozzle 3. Distinction 1] 2, 3 dimensional shock 1) 2-dimensional shock; due to the presence of wedge etc. 2) 3-dimensional shock; due to the presence of cone etc. 2] attached, detached shock 1) attached oblique shock ; = straight line for a given
2) detached shock ; = curved shock for a given where = deflection angle 4. Momentum Consideration 1] statement The oblique shock relations can be deduced from the normal shock relations by noting that the oblique shock can produce no momentum change parallel to the plane in which it lies. 2] proof 1) control volume (see p120 Fig. 6.2) 2) Because there is no momentum change parallel to the shock, must equal .
3) flow normal to an oblique shock wave (see p120 Fig. 6.3) ; All the properties of oblique shocks can be obtained by modification and manipulation of the normal shock relations provided that angle of the shock relative to the upstream flow is known.
6.2 Equations of Motion for a Straight Oblique Shock Wave 1./ Basic assumptions * frictionless surface * steady 2-dimensional planar adiabatic flow * no external work, negligible effect of body forces 2./ Governing Equations 1. control volume; (Fig. 6.2) - unit area parallel to the oblique shock wave - (=change in flow direction induced by the shock wave) 2. continuity equation
3. momentum equation 1] normal momentum equation 2] tangential momentum equation 4. energy equation
; If eqs. (6.1), (6.2), (6.3) and (6.4) are compared with the equations derived for normal shock waves it will be seen that they are identical in all respects except that and replace and respectively. ▣ Jump Conditions across an Oblique Shockwave
5. Rankine-Hugoniot Relations for Oblique Shock Waves ; exactly same as for normal shock
6. Relations between the Changes across the Shock Wave and the Upstream Mach Number 1] Geometric Relation
2] ; These are, of course, again identical to those used to study normal shocks, except that occurs in place of and occurs in place of . Hence, if in normal shock relations is replaced by and by , the following relations for oblique shocks are obtained using equations given in Ch.4.
3] Relations in terms of Upstream Mach Number and Wave Angle , Turning Angles
7. Limit Values of • 1) for normal shock ; → for oblique shock ; • 2) for normal shock ; → for oblique shock ;; Hence, for an oblique shock wave, can be greater than or less than 1. • 3) • The minimum value that can have is, therefore, i.e., the minimum shock wave angle is the Mach angle. When the shock has this angle, Eq. (6.10) shows that is equal to 1, i.e., the shock wave is a Mach wave. • The maximum value that can have is, of course, • , the wave then being a normal shock wave. Hence, the limits on • are : (6.15)
8. Relation between and (see p125 Fig. 6.6) Normal Shock Strong S.W.Sol. For M1=2, δ=15 δ for M2=1 Weak S.W.Sol. For M1=2, δ=15 Mach Wave Max δ for M1=2
1] formula 2] meaning of eq. (6.18) 1) The turning angle , is zero when and also when is equal to 1, i.e. , when : normal shock and : Mach wave ; Thus an oblique shock lies between a normal shock and a Mach wave. In both of these two limiting cases, there is no turning of the flow. Between these two limits reaches a maximum.
2) The normal shock limit and Mach wave limit on the oblique shock at a given value of are given by the intercepts of the curves with the vertical axis at (see p125 Fig. 6.8).
3] ; value of (= maximum turning angle) for a given 1) derivation ; p124 2) variation of maximum turning angle with upstream Mach number for (see p126 Fig. 6.8)
4] Remarks - For flow over bodies involving greater angles than this, a detached shock occurs. It should also be noted that as increases, increases so that if a body involving a given turning angle, accelerates from a low to a high Mach number, the shock can be detached at the low Mach numbers and become attached at the higher Mach numbers.
9. Strong and Weak (=non-strong) Shocks 1] two possible solutions for a value If is less than , there are two possible solutions,
i.e., two possible values for , for a given and • .(see p127 Fig. 6.11) • 2] classification • 1) strong shock ; larger : dotted line in Fig. 6.6 • 2) weak shock ; smaller • 3] experimental results • Experimentally, it is found that for a given and in external flows the shock angle, , is usually that corresponding to the weak or non strong shock solutions. • - Under some circumstance, the conditions downstream of the shock may cause the strong shock solution to exist in part of the flow. In the event of no other information being available, the non-strong shock solution should be used. • 4] physical meaning of • 1) meaning (physical interpretation)
2) remarks * if ; shock wave = Mach wave * greater greater discontinuity * intensity of shock 5] general relation of 1) for both cases : 2) * strong shock : * weak shock : 6] Occurrence of weak shock and strong shock 1) whether weak or strong shock = f (boundary condition)
2) weak shock * typically occurs in external aerodynamic flows * Of the two choice for , it is an experimental fact that the one corresponding to the weak shock usually occurs. 3) strong shock * The strong shock wave occurs if the downstream pressure is sufficiently high. The high downstream pressure may occur in flows in wind tunnels, in engine inlets, or in other ducts. 10. Characteristics of the Oblique Shock Wave 1] Reason for the deflection of stream direction * velocity component * So is deflected from the direction of , i.e., fluid stream is deflected toward the oblique shock wave.
2] Distinction between Mach wave and shock wave by normal velocity component • 1) Mach wave (= shock wave of zero intensity) ; • 2) shock wave ; • 3] deflection angle • 1) formula • 2) application • * applicable to conical shock as well as plane shock • * valid only for ; • 3) case of • a) (Mach angle) ; Mach wave • b) ; normal shock wave
4] • 1) • 2) if • ; the basic relation previously presented are not applicable. • 5] in 2, 3 dimensional shock wave • 2-dimensional shock (= plane shock) = angle of wedge • = angle of concave corner • 2) 3- dimensional shock • angle of cone • ; in this case streamlines after the conical shock must be curved in order that the 3-dimensional continuity eq. be satisfied.
6] corresponding to (= maximum flow deflection angle for a given ) 7] corresponding to
① ② ③ Example 6.2(from Compressible Fluid Flow) <Sol.> β1=26.6
① ② ③ θ1=26.6 θ2 -δ=24.5 M2=2.334, p2=77.8kPa, T2=272.6K, θ2=28.5
M1=2.5, p1=60kPa, T1=253K, θ1=26.6 M3=2.175, p399.4kPa, T3=292.6K M2=2.334, p2=77.8kPa, T2=272.6K, θ2=28.5
Reflected O.S.W. Mach Reflection