Compressible flow is a crucial area of fluid mechanics that deals with fluids experiencing significant density changes due to pressure and temperature variations. This unit covers key concepts like Mach number, shock waves, and isentropic flow, which are essential for understanding high-speed fluid behavior.
The study of compressible flow has wide-ranging applications in aerospace engineering, propulsion systems, and high-speed vehicle design. By mastering these principles, engineers can analyze and optimize the performance of aircraft, rockets, and other supersonic systems, pushing the boundaries of speed and efficiency.
Compressible flow involves fluids with significant changes in density due to variations in pressure and temperature
Mach number (M) represents the ratio of the fluid velocity to the local speed of sound (a)
Subsonic flow: M<1
Sonic flow: M=1
Supersonic flow: M>1
Stagnation properties refer to the fluid properties that would exist if the fluid were brought to rest isentropically (stagnation temperature, pressure, and density)
Critical properties occur when the Mach number is equal to 1 (critical temperature, pressure, and density)
Isentropic flow assumes no heat transfer and no change in entropy, leading to reversible adiabatic processes
Shock waves are thin regions where abrupt changes in fluid properties occur, resulting in irreversible processes and entropy increase
Expansion waves are regions where fluid properties change gradually, allowing for isentropic flow
Thermodynamic Principles
First Law of Thermodynamics states that energy is conserved in a system, accounting for heat transfer and work done
Second Law of Thermodynamics introduces the concept of entropy and states that entropy always increases in irreversible processes
Ideal gas law (PV=nRT) relates pressure, volume, temperature, and the specific gas constant for ideal gases
Specific heat capacities (cpโ and cvโ) describe the amount of heat required to change the temperature of a substance per unit mass
cpโ: specific heat at constant pressure
cvโ: specific heat at constant volume
Isentropic relations connect fluid properties (pressure, density, and temperature) during isentropic processes using specific heat ratio (ฮณ)
Stagnation temperature (T0โ) is the temperature attained when a fluid is brought to rest adiabatically and isentropically
Stagnation pressure (p0โ) is the pressure attained when a fluid is brought to rest adiabatically and isentropically
Speed of Sound and Mach Number
Speed of sound (a) is the speed at which small pressure disturbances propagate through a fluid
For ideal gases: a=ฮณRTโ, where ฮณ is the specific heat ratio, R is the specific gas constant, and T is the absolute temperature
Mach number (M) is a dimensionless quantity that relates the fluid velocity (V) to the local speed of sound: M=aVโ
Mach number regimes:
Subsonic (M<1): flow velocity is less than the speed of sound
Transonic (Mโ1): flow velocity is near the speed of sound
Supersonic (M>1): flow velocity is greater than the speed of sound
Hypersonic (Mโซ1, typically M>5): flow velocity is much greater than the speed of sound
Mach angle (ฮผ) is the angle between the Mach wave and the flow direction in supersonic flow: sinฮผ=M1โ
Mach cone is a conical region formed by Mach waves emanating from a disturbance in supersonic flow
Isentropic Flow
Isentropic flow assumes no heat transfer and no change in entropy, resulting in reversible adiabatic processes
Isentropic flow relations connect fluid properties (pressure, density, and temperature) using the specific heat ratio (ฮณ):
p0โpโ=(1+2ฮณโ1โM2)โฮณโ1ฮณโ
ฯ0โฯโ=(1+2ฮณโ1โM2)โฮณโ11โ
T0โTโ=(1+2ฮณโ1โM2)โ1
Area-velocity relation in isentropic flow: AโAโ=M1โ[ฮณ+12โ(1+2ฮณโ1โM2)]2(ฮณโ1)ฮณ+1โ
A: local cross-sectional area
Aโ: critical area (area at Mach number 1)
Mass flow rate in isentropic flow: mห=ฯAV=T0โโp0โAโโRฮณโโ(ฮณ+12โ)2(ฮณโ1)ฮณ+1โ
Choked flow occurs when the Mach number reaches 1 at the minimum cross-sectional area (throat) in a converging-diverging nozzle
Normal Shock Waves
Normal shock waves are thin, planar regions where abrupt changes in fluid properties occur, resulting in irreversible processes and entropy increase
Fluid properties change discontinuously across a normal shock wave:
Pressure and density increase
Temperature increases
Velocity decreases
Mach number decreases (supersonic to subsonic)
Normal shock relations connect fluid properties before (subscript 1) and after (subscript 2) the shock using the upstream Mach number (M1โ) and specific heat ratio (ฮณ):
Density ratio: ฯ1โฯ2โโ=(ฮณโ1)M12โ+2(ฮณ+1)M12โโ
Temperature ratio: T1โT2โโ=(ฮณ+1)2M12โ[2ฮณM12โโ(ฮณโ1)][(ฮณโ1)M12โ+2]โ
Entropy increases across a normal shock wave, indicating an irreversible process
Stagnation pressure decreases across a normal shock wave due to the irreversible process
Oblique Shock Waves
Oblique shock waves are thin, inclined regions where abrupt changes in fluid properties occur, resulting in irreversible processes and entropy increase
Oblique shock waves form when a supersonic flow encounters a sharp corner or a compression corner
Fluid properties change discontinuously across an oblique shock wave:
Pressure and density increase
Temperature increases
Velocity decreases
Mach number decreases (supersonic to supersonic or subsonic)
Oblique shock angle (ฮฒ) depends on the upstream Mach number (M1โ) and the deflection angle (ฮธ)
Weak and strong shock solutions exist for a given deflection angle, with the weak shock being more common in practice
Oblique shock relations connect fluid properties before (subscript 1) and after (subscript 2) the shock using the upstream Mach number (M1โ), shock angle (ฮฒ), and specific heat ratio (ฮณ)
Entropy increases across an oblique shock wave, indicating an irreversible process
Stagnation pressure decreases across an oblique shock wave due to the irreversible process
Expansion Waves
Expansion waves are regions where fluid properties change gradually, allowing for isentropic flow
Expansion waves occur when a supersonic flow encounters a sharp convex corner or an expansion corner
Fluid properties change continuously through an expansion wave:
Pressure and density decrease
Temperature decreases
Velocity increases
Mach number increases (supersonic to supersonic)
Prandtl-Meyer function (ฮฝ) relates the Mach number to the flow deflection angle (ฮธ) in an isentropic expansion: