7.3 Normal and Oblique Shock Waves

Normal and oblique shock waves are crucial phenomena in supersonic flows. These discontinuities cause abrupt changes in flow properties, playing a key role in compressible fluid dynamics and aerospace engineering.

Understanding shock waves is essential for designing supersonic aircraft, rocket nozzles, and wind tunnels. We'll explore their formation, characteristics, and the equations governing their behavior, connecting theory to real-world applications in high-speed fluid flow.

Normal Shock Wave Formation

Characteristics and Physics of Normal Shock Waves

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• Normal shock waves form discontinuities in flow properties occurring in supersonic flows when flow decelerates to subsonic speeds
• Accumulation of pressure disturbances unable to propagate upstream in supersonic flow causes normal shock wave formation
• Normal shock waves create abrupt changes in flow properties (pressure, temperature, density, velocity)
• Thickness of normal shock wave measures extremely small (few mean free paths of gas molecules)
• Flow properties across normal shock wave experience following changes:
  • Increase in pressure, temperature, and density
  • Decrease in flow velocity
• Mach number transitions from supersonic (M > 1) upstream to subsonic (M < 1) downstream
• Entropy increases across normal shock wave indicating irreversible process

Examples and Applications

• Normal shock waves observed in supersonic wind tunnels (test section)
• Occur in supersonic nozzles operating at off-design conditions
• Found in front of blunt objects in supersonic flow (aircraft nosecones)
• Appear in overexpanded rocket nozzles

Rankine-Hugoniot Equations for Normal Shocks

Fundamental Equations and Relationships

• Rankine-Hugoniot equations consist of conservation equations relating flow properties across normal shock wave
• Equations incorporate conservation of mass, momentum, and energy
• Express Rankine-Hugoniot equations using upstream Mach number and ratio of specific heats of gas
• Key relationships derived from equations include ratios across shock wave:
  • Pressure ratio: $\frac{p_2}{p_1} = 1 + \frac{2\gamma}{\gamma+1}(M_1^2 - 1)$
  • Temperature ratio: $\frac{T_2}{T_1} = \frac{[2\gamma M_1^2 - (\gamma-1)][\frac{2}{\gamma+1}]}{M_1^2}$
  • Density ratio: $\frac{\rho_2}{\rho_1} = \frac{(\gamma+1)M_1^2}{2+(\gamma-1)M_1^2}$
  • Velocity ratio: $\frac{u_2}{u_1} = \frac{2+(\gamma-1)M_1^2}{(\gamma+1)M_1^2}$
• Normal shock function relates upstream and downstream Mach numbers: $M_2^2 = \frac{1 + \frac{\gamma-1}{2}M_1^2}{\gamma M_1^2 - \frac{\gamma-1}{2}}$

Practical Applications and Problem-Solving

• Normal shock tables based on Rankine-Hugoniot equations provide quick determination of flow properties
• Apply equations to find unknown flow properties given known conditions before or after shock wave
• Solve problems involving normal shocks in various engineering scenarios:
  • Supersonic inlet design for jet engines
  • Shock tube experiments in laboratory settings
  • Analysis of flow in supersonic wind tunnels

Oblique Shock Wave Concept

Formation and Characteristics

• Oblique shock waves form at angle to flow direction when supersonic flow encounters deflection or compression
• Occur in supersonic flow over wedges, cones, or surfaces causing flow direction change
• Characterized by shock angle (β) and deflection angle (θ) related to upstream Mach number
• Oblique shock strength depends on upstream Mach number and flow deflection angle
• Can be attached (to surface) or detached (bow shock) based on flow conditions and geometry
• Two possible oblique shock solutions for given upstream Mach number and deflection angle:
  • Weak shock (smaller β, supersonic flow behind shock)
  • Strong shock (larger β, subsonic flow behind shock)

Types and Examples

• Attached oblique shocks form on sharp leading edges of supersonic aircraft wings
• Detached bow shocks appear in front of blunt bodies in supersonic flow (spacecraft reentry)
• Multiple oblique shocks occur in supersonic inlets of ramjet engines
• Shock diamonds in overexpanded jet exhaust consist of series of oblique shocks and expansion fans

Oblique Shock Wave Properties

Oblique Shock Relations and Analysis

• Derive oblique shock relations from Rankine-Hugoniot equations modified for oblique nature
• θ-β-M relation connects deflection angle, shock angle, and upstream Mach number: $\tan\theta = 2\cot\beta\frac{M_1^2\sin^2\beta - 1}{M_1^2(\gamma + \cos2\beta) + 2}$
• Property ratios across oblique shock depend on normal component of upstream Mach number (Mn1)
• Calculate normal component of Mach number: $M_{n1} = M_1\sin\beta$
• Use Mn1 to determine oblique shock strength and flow properties
• Prandtl-Meyer expansion fans may occur with oblique shocks when flow turns away from itself

Practical Applications and Tools

• Oblique shock charts and tables based on relations aid in determining flow properties and shock angles
• Analyze multiple oblique shocks in series by applying relations sequentially
• Consider new flow conditions after each shock in multi-shock systems
• Applications of oblique shock analysis:
  • Design of supersonic aircraft inlets
  • Optimization of hypersonic vehicle shapes
  • Analysis of shock wave interactions in scramjet engines