MACH NUMBER AND HIGH VELOCITY FLOW EFFECTS

Basic concepts in high velocity flow

When fluids flow at speeds V significantly lower than the speed of sound a, we expect, as observers, a certain behavior of this fluid when it flows around bodies. A typical manifestation of subsonic flow is a smooth change in the flow trajectories already in front of the body being flowed around, as in Figure 520. This behavior is due to sound as a pressure disturbance propagating in a continuum from the source-S of pressure disturbance, which is the surface of the bodies. This means that in supersonic flow it is impossible for changes in pressure and other state quantities to propagate against the direction of flow. For this reason, we observe fundamental differences between the propagation of a pressure disturbance in a subsonic, sonic or supersonic flow in channels or when flowing around bodies and the formation of special phenomena (shock and expansion waves) that do not arise at subsonic speeds. The ratio of V to a is called the Mach number.

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Subsonic flow around

S-source of pressure disturbance.

Mach number

Mach number is defined as the ratio of the velocity of a fluid V to the speed of sound in a fluid a, see Equation 337. The speed of sound a is a function of the fluid composition and temperature.

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a [m·s^-1] speed of sound in continuum under investigation; M [Mach] Mach number; p [Pa] pressure; r [J·kg^-1·K^-1] specific gas constant; T [K] absolute gas temperature (static temperature); V [m·s^-1] velocity of body or flow; ρ [kg·m^-3] density; κ [1] heat capacity ratio. The derivation of the equation for the speed of sound is shown in Appendix 337.

Critical Mach number

The value of the Mach number in the investigated volume can vary, as the velocities of the fluid and sound change locally. For example, the fluid flowing around the airfoil in Figure 520 has a different velocity in front of the airfoil and a different velocity around the airfoil. This means that the Mach numbers at the inlet of the fluid into the investigated volume may be different from those at some points in the investigated volume. If we define a reference point at the inlet of the investigated volume at which we determine the Mach number, we can also determine the critical value of the Mach number at this point at which the fluid reaches the speed of sound somewhere inside the investigated volume, or M=1. The critical Mach number is less than 1 (M<1) if the velocity increases in the investigated volume, and greater than 1 (M>1) if the velocity decreases in the investigated area (for example, in diffusers).

Supersonic flow around bodies

In the case of supersonic speed (M≥1) the streamlines to not gradually rise in front of the flowed around body and the body is forced by its volume to displace the surrounding gas by abrupt compression - the energy to compress the gas is taken from the motion of the body. The gas thus compressed gradually expands away from the body. The boundary between the compressed gas and the surrounding gas is cone-shaped and is called a shock wave (Figure 339). The slope of the shock wave β_SW is always greater than the Mach angle μ - their equality would occur for the case of an infinite body.

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(a) source is moving at supersonic speed; (b) streamlines around oblique shock wave (compare with Figure 520, p. 3.3); (c) source is moving at speed of sound. SW-shock wave. β_SW [°] shock wave angle (μ<β_SW); μ [°] Mach angle. The figure does not deal with the situation of the shock waves at the time before τ=0, nor with the situation behind the shock wave and behind the body; this problem is discussed in the next section of the paper.

Sonic flow around bodies

In the case of sonic speeds (M=1), the pressure disturbance front is always at the source. The Mach angle μ is 90°, therefore the shock wave angle β_SW in front of the real body must be greater than 90°, see Figure 339c.

Principle of relativity of flow around flowing bodies

So far the propagation of sound waves or the generation of shock waves when the body is moving has been shown, but the same effect is achieved in the reverse case when the body is at rest and the gas is flowing.

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Shock wave

Compared to a sound wave, the shock wave is a permanent abrupt change in state variables (higher pressure, temperature, and density behind it). This situation resembles an expanding ball of compressed gas, with more gas added due to body movement. However, the cone's volume increases with the third power of time, while the amount of compressed gas grows linearly (at constant velocity). Thus, shock wave intensity decreases with distance from the tip of the shock cone.

Shock waves in channel flow

Machines in which supsonic speeds can occur can realistically only be designed for specific conditions (it can be shown that the ratio of the outlet flow area to the minimum flow area must also be different for different Mach numbers), changing the conditions would require changing the geometry of the machine to meet the requirements of the flow transition from supersonic to subsonic. This is often not possible to meet and the transition is made in the expanding part of the flow tube by the abrupt change in the state variables, i.e. the shock wave, only in this way can the conditions of Hugoniot's theorem be met ("smooth transition" is not possible in such channel). There are several basic types of shock waves according to the conditions under which they are generated, see the following chapters.

Normal shock wave

In normal shock wave, the velocity decreases almost stepwise (wave thickness is approximately 10^-7 m [Hloušek, 1992]) to subsonic without changing direction, as shown in Figure 519. Changes of state quantities in normal shock wave can be calculated using the Rankine-Hugoniot equations. Normal shock waves are generated in channels and around isolated bodies at the speed of sound.

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Passage of gas through normal shock wave

1-gas state before shock wave; 2-gas state behind shock wave. p [Pa] pressure; V* [m·s^-1] critical flow velocity. The derivation of the equations for normal shock wave is carried out, for example, in [Macur, 2010, p. 372].

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Change of gas state when passing through normal shock wave

The energy balance of normal shock wave was first established with a satisfactory result with satisfactory results in calculating changes in state quantities by the German physicist Ludwig Prandtl (1875-1953) by introducing the assumption that a stepwise change of state quantities in a shock wave also results in an increase in entropy, which can be clearly seen from the h-s diagram of the shock wave in Figure 338, p. 3.8. This also means that the shock wave generates a thermodynamic loss.

Oblique shock waves

The oblique shock wave is slope relative to the flow direction in front of it by the shock wave angle β_SW. In front of the oblique shock wave, velocity must be supersonic, but behind, it can be subsonic or supersonic. As the flow traverses the oblique shock wave, its direction is deflected by angle δ (Figure 107). Normal velocity components (V_1n, V_2n) share properties with those passing through a normal shock wave (see Problem 1007). The equality of tangential velocity components (V_1t=V_2t) can be proved (e.g., [Kadrnožka, 2004, p. 126-127]). Oblique shock waves are formed, for example, around surfaces when gases flow at supersonic speeds.

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Passage of gas through oblique shock wave

δ [°] velocity deflection post-shock from original direction. The index _n denotes the normal velocity components, the index _t denotes tangential velocity components.

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Shock wave angleand its effect on losses

Additionally, analyzing oblique shock wave properties reveals that when β_SW equals the Mach angle μ, V_1n=a₁ must be true, indicating only the sound wave—following the Mach angle definition. It's also demonstrated that the maximum energy loss (entropy increase) happens when β_SW=90°, making losses in the oblique shock wave less than in a normal shock wave for the same pressure ratio ahead and behind the wave.

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Crossed shock waves

If conditions are created behind an oblique shock wave for the formation of other oblique shock waves, then these other oblique shock waves will have a gradually increasing shock wave angle βSW. This means that these oblique shock waves will cross into one at a certain distance from the flown surface, see Figure 481. The resulting oblique shock wave has the same momentum as the crossed shock waves and also has a smaller angle β_SW than the first wave.

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(a) crossing of oblique shock waves on a stepped surface; (b) crossing of oblique shock waves on slowly rising surface (here the crossing is similar to compression wave). CW-set of compression waves.

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Shock wave splitting

In aviation, experiments have been carried out to reduce the sound effects caused by shock waves in supersonic flights based on the splitting of the shock wave into several partial waves, see Figure 905. This maximizes the angle of the resulting shock wave (behind all shock waves from the fuselage have met). This is because the larger the angle of the shock wave (preferably 90°), the smaller the sound effect from the wave [Hošek, 1962, p. 60] - which would allow transport aircraft to fly at high speeds even over populated areas.

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Project Quiet Spike

The project successfully explored the possibility of reducing the intensity of sound effects by using a stepped extension of the nose of aircraft. Here, testing of the telescoping nose of F-15B aircraft. NASA Photo by:Carla Thomas

Expansion waves

Increasing flow area also creates obtuse angles on bodies, like the trailing edge of projectiles, the beginning of fuselage taper, in nozzles etc., as seen in Figure 340, illustrating a typical supersonic obtuse angle flow. When wrapping around obtuse angles at supersonic velocity, gas must expand from pressure p₁ to p₂, flow velocity increases from V₁ to V₂, and the flow direction is deflected by angle δ from the original. In the expansion wave, there's a gradual change in state variables with very low losses (isentropic expansion).

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Supersonic flow near obtuse angle

ML-Mach line; Δ [m] distance traveled by fluid particle at velocity V before the pressure disturbance from source S reaches it.

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Flow deflection

The formation of the expansion wave in Figure 340 is initiated by a pressure disturbance at the edge S. This disturbance propagates at the speed of sound (so that it cannot propagate against the flow), which propagates with a Mach angle μ₁. The ML₁ boundary at which flow direction starts to change and the gas starts to expand is so-called Mach line or also the first expansion wave. It is obvious that the slope of this line is equal to the Mach angle μ₁. During expansion the Mach number changes and the expansion also changes its character because the Mach angle changes. The expansion stops at the Mach line ML₂ where the flowing gas reaches pressure p₂. The first and last Mach lines form the Mach wedge in which the gas expansion takes place. The value of the angle δ can be determined from the Prandtl-Meyer function ν [Anon., 2010], see Equation 521.

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Prandtl-Meyer function for flow deflection: ν(M) [°] Prandtl-Meyer function

Maximum flow deflection

The maximum angle of deflection of the flow when passing through the expansion wave δ_max and the maximum velocity V_2max is reached by the flow when expanding into the vacuum p₂=0. In vacuum expansion, M₂=∞. If the angle of inclination of the edge is greater than δ_max a vacuum will be created behind the edge S between the flow and the contact surface.

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Photo source [O’Farrell and Rieckhoff, 2011].

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Airfoils for high velocities

In a well-designed airfoil, the occurrence of high velocity effects and their effect on the aerodynamics of flight should be predictable. The rhombic airfoil is a good predictor in this respect. Expansion waves are only generated at the tips of the suction and pressure sides, and λ-shock waves are generated at the trailing edge of the airfoil. Of course, this profile is not suitable for low subsonic velocities, so various compromises of profile shapes are found depending on the velocities for which they are primarily intended, see Figure 894.

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Profile types suitable for high speeds in compressible flow

(a) transonic airfoil; (b) supersonic (lenticular shape); (c) supersonic (rhombic shape); (d) supersonic (trapezoidal shape); (e) hypersonic.

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Changes in values of aerodynamic parameters of airfoils at high flow speeds

To convert aerodynamic quantities obtained from measurements in incompressible flow to a compressible flow situation, the Glauert-Prandtl rule can be used, see Equation 906. These equations are valid only up to critical Mach or Reynolds numbers [Abbott and Doenhoff, 1959, p. 256 and pp. 283-287], [Hošek, 1949, p. 52]. Profiles with flows below the critical Reynolds number are termed laminar profiles. These equations align with experimental measurements.

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Glauert-Prandtl rule

(a) Glauert-Prandtl rule for pressure coefficient; (b) Glauert-Prandtl rule for lift coefficient; (c) effect of increasing velocity on drag coefficient. C_D [1] drag coefficient of airfoil; C_L [1] lift coefficient of airfoil; M [Mach] Mach number (before airfoil); C_P [1] pressure coefficient of airfoil. The index _i indicates incompressible flow, the index _c compressible flow. The derivation is shown in [Hošek, 1949, p. 49].

Aerodynamics of profile cascades in compressible flow

Compressible flow effects at high velocity also manifest in profile cascade, which has an impact on the resulting velocity triangles of turbomachines. The formation of these effects can be predicted by both numerical and analytical methods. However, there are supersonic wind tunnels with profile casacdes for measuring aerodynamic quantities and visualizing supersonic flow.

Visualization of supersonic flow in profiled cascade

Figure 636 shows an interferogram (photograph showing the changes in gas density) of supersonic flow through a turbine profile cascade, with inlet velocity at Mach 1,19 and outlet velocity at Mach 2,003 in isentropic flow. Supersonic flow at the inlet generates oblique shock waves at the leading edges of the profiles. Visible are expansion waves around the cascade outlet and shock waves forming at the trailing edge where two supersonic flows meet.

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left-scheme of situation recorded on interferogram; right-interferogram of supersonic flow through turbine profile cascade. Taken with Mach-Zehnder interferometer. Taken and images provided by Aerodynamic Laboratory in Nový Knín at Institute of Thermomechanics of AVČR, v.v.i.

Calculations of supersonic flow in profile cascades