Prandtl Meyer Expansion

3 Key excerpts on "Prandtl Meyer Expansion"

• 

  eBook - ePub

  Fundamentals of Gas Dynamics

  • Robert D. Zucker, Oscar Biblarz(Authors)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  Chapter 8

  Prandtl–Meyer Flow

  8.1 INTRODUCTION

  This chapter begins with an examination of weak shocks in supersonic flows. We show that for very weak oblique shocks the pressure change is related to the first power of the deflection angle, whereas the entropy change is related to the third power of the deflection angle. This will enable us to explain how a smooth turn can be accomplished isentropically—a situation known as Prandtl–Meyer Flow. Being reversible, such flows may represent expansions or compressions, depending on the circumstances.

  A detailed analysis of Prandtl–Meyer flow is made for the case of a perfect gas and, as usual, a tabular entry is developed to aid in problem solution. Typical flow fields involving Prandtl–Meyer flow are discussed. In particular, the entire performance of a converging–diverging nozzle can now be fully explained. We include a discussion of the flow around supersonic airfoils and introduce the aerospike nozzle.

  8.2 OBJECTIVES

  After completing this chapter successfully, you should be able to:

  1. State how entropy and pressure changes vary with deflection angles for weak oblique shocks.
  2. Explain how finite turns (with finite pressure ratios) can be accomplished isentropically in supersonic flow.
  3. Describe and sketch what occurs as fluid flows supersonically past a smooth concave corner and a smooth convex corner.
  4. Show Prandtl–Meyer flow (both expansions and compressions) on a T–s diagram.
  5. (Optional) Develop the differential relation between Mach number (M) and flow turning angle (v) for Prandtl–Meyer flow.
  6. Given the equation for the Prandtl–Meyer function (8.48) , show how tabular entries can be developed for Prandtl–Meyer flow. Explain the significance of the angle v.
  7. Explain the governing boundary conditions and show the results when shock waves and Prandtl–Meyer waves reflect off both physical and free boundaries.
  8. Draw the wave forms created by supersonic flows over rounded and/or wedge‐shaped wings as the angle of attack changes. Be able to solve for the flow properties in each region.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Introduction to Compressible Fluid Flow

  • Patrick H. Oosthuizen, William E. Carscallen(Authors)
  • 2013(Publication Date)
  • CRC Press

    (Publisher)

  Of course, in reality, the continuum and ideal gas assumptions cease to be valid before this situation is reached. Next, consider the application of Equation 7.14 to the calculation of the flow changes pro-duced by a Prandtl–Meyer expansion. Referring to Figure 7.10, the procedure is as follows. 1. From tables or graphs or from the equation find the value of θ corresponding to M 1 , i.e., θ 1 . This is equivalent to assuming that the initial flow was generated by an expansion around a hypothetical corner from Mach 1, the reference Mach number in the tables, to Mach M 1 . Expansion wave Vacuum 130.5° FIGURE 7.9 Expansion to zero pressure. M 1 , p 1 , ρ 1 M 2 , p 2 , ρ 2 Expansion wave δ FIGURE 7.10 Flow changes through a Prandtl–Meyer wave. 178 Introduction to Compressible Fluid Flow 2. Calculate the θ for the flow downstream of the corner. This will be given by θ 2 = θ 1 + δ 3. Find, using tables or graphs or the equation, the downstream Mach number, M 2 , corresponding to this value of θ 2 . 4. Any other required property of the downstream flow is obtained by noting that the expansion is isentropic and that the following relations therefore apply T T M M p p T T 2 1 1 2 2 2 2 1 2 1 1 1 2 1 1 2 = + -      + -      =   γ γ ,     =       --γ γ γ 1 2 1 2 1 1 1 , ρ ρ T T Alternatively, isentropic flow tables can be used, it being noted that the stagnation pressure remains constant across the wave. 5. If necessary, calculate the boundaries of the expansion wave. This is done by not-ing that these boundaries are the Mach lines corresponding to the upstream and downstream Mach numbers as shown in Figure 7.11. The various angles defined in Figure 7.11 are given by sin , sin , α α β α δ 1 2 2 2 1 1 = = = -M M M 1 M 2 β δ α 1 α 2 FIGURE 7.11 Angles associated with a centered expansion wave. 179 Expansion Waves: Prandtl–Meyer Flow Example 7.1 Air flows at Mach 1.8 with a pressure of 90 kPa and a temperature of 15°C down a wide channel.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Supercritical Fluid Technology in Materials Science and Engineering

  Syntheses: Properties, and Applications

  • Ya-Ping Sun(Author)
  • 2002(Publication Date)
  • CRC Press

    (Publisher)

  Nevertheless, work is un-derway to solve the two-dimensional problem for the rapid expansion of pure nonideal CO 2 through a RESS nozzle (35). More qualitative descriptions of the flow pattern of a freely expanding supersonic jet have been available since the early 1960s. Although this work is, strictly speaking, applicable only to ideal gases, it is useful for helping us understand the behavior of a rapidly expanding supercritical fluid in the free-jet region. In this section, we combine these classic, zero-dimensional models for the isentropic expansion of an ideal gas in both the supersonic and transsonic regions with the experimental and theoretical results of Ashkenas and Sherman (36), which describe the flow properties of a supersonic fluid between the nozzle exit and the Mach disk. Evidence will be presented that the pressure, temperature, and velocity profiles along the centerline are not the best choice to represent the entire flow regime. We perform all of our analyses for an ideal gas in which the ratio of the heat capacities, y = cp/ c v, is constant. Expansions are assumed to be isentropic, 410 Weber and Thies which is a reasonable assumption for a plain orifice (where L/D is small) and even more so for the subsequent expansion of the fluid on its way to the Mach disk. For expansions through capillaries with high L/D, the analysis given below can also be used for conditions from the capillary outlet outward. However, a method that accounts for the effect of friction must be used to calculate the conditions of temperature and pressure that would exist at the capillary inlet. Assuming uniform conditions across the entire Mach disk, the mass flow rate of the fluid passing through the disk can be calculated from the density and velocity that it has attained immediately before (p 3 X and W 3 X) passing through the disk, and from the area of the Mach disk, oim (see Figure 5a).

  Sign up to read

  Learn more about book