Prandtl–Meyer expansion fan

A supersonic expansion fan, technically known as Prandtl–Meyer expansion fan, a two-dimensional simple wave, is a centered expansion process that occurs when a supersonic flow turns around a convex corner. The fan consists of an infinite number of Mach waves, diverging from a sharp corner. When a flow turns around a smooth and circular corner, these waves can be extended backwards to meet at a point. Consider the scenario shown in the adjacent figure. As a supersonic flow turns, the normal component of the velocity increases ( w 2 > w 1 {displaystyle w_{2}>w_{1}} ), while the tangential component remains constant ( v 2 = v 1 {displaystyle v_{2}=v_{1}} ). The corresponding change is the entropy ( Δ s = s 2 − s 1 {displaystyle Delta s=s_{2}-s_{1}} ) can be expressed as follows,Mach lines are a concept usually encountered in 2-D supersonic flows (i.e. M ≥ 1 {displaystyle Mgeq 1} ). They are a pair of bounding lines which separate the region of disturbed flow from the undisturbed part of the flow. These lines occur in pairs and are oriented at an angle A supersonic expansion fan, technically known as Prandtl–Meyer expansion fan, a two-dimensional simple wave, is a centered expansion process that occurs when a supersonic flow turns around a convex corner. The fan consists of an infinite number of Mach waves, diverging from a sharp corner. When a flow turns around a smooth and circular corner, these waves can be extended backwards to meet at a point. Each wave in the expansion fan turns the flow gradually (in small steps). It is physically impossible for the flow to turn through a single 'shock' wave because this would violate the second law of thermodynamics. Across the expansion fan, the flow accelerates (velocity increases) and the Mach number increases, while the static pressure, temperature and density decrease. Since the process is isentropic, the stagnation properties (e.g. the total pressure and total temperature) remain constant across the fan. The theory was described by Theodor Meyer on his thesis dissertation in 1908, along with his advisor Ludwig Prandtl, who had already discussed the problem a year before. The expansion fan consists of an infinite number of expansion waves or Mach lines. The first Mach line is at an angle μ 1 = arcsin ⁡ ( 1 M 1 ) {displaystyle mu _{1}=arcsin left({frac {1}{M_{1}}} ight)} with respect to the flow direction, and the last Mach line is at an angle μ 2 = arcsin ⁡ ( 1 M 2 ) {displaystyle mu _{2}=arcsin left({frac {1}{M_{2}}} ight)} with respect to final flow direction. Since the flow turns in small angles and the changes across each expansion wave are small, the whole process is isentropic. This simplifies the calculations of the flow properties significantly. Since the flow is isentropic, the stagnation properties like stagnation pressure ( p 0 {displaystyle p_{0}} ), stagnation temperature ( T 0 {displaystyle T_{0}} ) and stagnation density ( ρ 0 {displaystyle ho _{0}} ) remain constant. The final static properties are a function of the final flow Mach number ( M 2 {displaystyle M_{2}} ) and can be related to the initial flow conditions as follows, The Mach number after the turn ( M 2 {displaystyle M_{2}} ) is related to the initial Mach number ( M 1 {displaystyle M_{1}} ) and the turn angle ( θ {displaystyle heta } ) by, where, ν ( M ) {displaystyle u (M),} is the Prandtl–Meyer function. This function determines the angle through which a sonic flow (M = 1) must turn to reach a particular Mach number (M). Mathematically, By convention, ν ( 1 ) = 0. {displaystyle u (1)=0.,} Thus, given the initial Mach number ( M 1 {displaystyle M_{1}} ), one can calculate ν ( M 1 ) {displaystyle u (M_{1}),} and using the turn angle find ν ( M 2 ) {displaystyle u (M_{2}),} . From the value of ν ( M 2 ) {displaystyle u (M_{2}),} one can obtain the final Mach number ( M 2 {displaystyle M_{2}} ) and the other flow properties. As Mach number varies from 1 to ∞ {displaystyle infty } , ν {displaystyle u ,} takes values from 0 to ν max {displaystyle u _{ ext{max}},} , where