Fluid Mechanics and Energy Transport BIEN 301 Lecture 14 Introduction to Compressible Flows Juan M. Lopez, E.I.T. Research Consultant LeTourneau University Adjunct Lecturer Louisiana Tech University

2/6/2007BIEN 301 – Winter Compressible Flow  When density changes become significant, we have compressible flow. Occurs in flows moving at sufficient speed Occurs in flows moving at sufficient speed Extremely high pressure gradients in compressible fluids (gases, blood) Extremely high pressure gradients in compressible fluids (gases, blood)  Most important effects of compressible flows: Choking Choking Shock Waves Shock Waves

2/6/2007BIEN 301 – Winter Compressible Flow  Mach Number (White 9.1) Awesome transitional picture, p. 598 White. Awesome transitional picture, p. 598 White. Crossing Mach 1.0, but what is this number? Crossing Mach 1.0, but what is this number? Fractional portion of speed of sound in a mediumFractional portion of speed of sound in a medium Defined as:Defined as:

2/6/2007BIEN 301 – Winter Compressible Flow Mach Regimes Mach Regimes Ma < 0.3Incompressible FlowMa < 0.3Incompressible Flow 0.3 < Ma < 0.8Subsonic Flow0.3 < Ma < 0.8Subsonic Flow 0.8 < Ma < 1.2Transonic Flow0.8 < Ma < 1.2Transonic Flow 1.2 < Ma < 3.0Supersonic Flow1.2 < Ma < 3.0Supersonic Flow 3.0 > MaHypersonic Flow3.0 > MaHypersonic Flow These are only guides, the individual flow scenarios affect how shock waves might develop.These are only guides, the individual flow scenarios affect how shock waves might develop.

2/6/2007BIEN 301 – Winter Compressible Flow Specific Heat Ratio Specific Heat Ratio Usually Tabulated for you alreadyUsually Tabulated for you already Variations have small effects on compressible flow calculations, so we will stick with air, k = 1.40 for most calculations in this chapter.Variations have small effects on compressible flow calculations, so we will stick with air, k = 1.40 for most calculations in this chapter.

2/6/2007BIEN 301 – Winter Compressible Flow Ideal Gas Ideal Gas Simple state equation we often use to set up boundary behavior calculations.Simple state equation we often use to set up boundary behavior calculations.

2/6/2007BIEN 301 – Winter Isentropic Process Deals with change in entropy Deals with change in entropy The solved integral for constant c p:The solved integral for constant c p:

2/6/2007BIEN 301 – Winter Isentropic Process Setting entropy equal (isentropic) Setting entropy equal (isentropic) The solved integral for constant cp reduces to the power law relationship:The solved integral for constant cp reduces to the power law relationship:

2/6/2007BIEN 301 – Winter Speed of Sound  Adiabatic Process Expanding our control volume analysis to a wave moving through a medium, we can see how friction and heat transfer effects are limited to the interior of the wave (White, Fig. 9.1) Expanding our control volume analysis to a wave moving through a medium, we can see how friction and heat transfer effects are limited to the interior of the wave (White, Fig. 9.1) Speed of sound relates the propagation of a pressure pulse of infinitessimal length through a still fluid. Speed of sound relates the propagation of a pressure pulse of infinitessimal length through a still fluid. The speed of sound of a fluid is given by: The speed of sound of a fluid is given by:

2/6/2007BIEN 301 – Winter Steady Flow  Adiabatic and Isentropic Steady Flow (White 9.3) Through these simplifications, we are able to reduce our steady flow to a few simple relationships Through these simplifications, we are able to reduce our steady flow to a few simple relationships

2/6/2007BIEN 301 – Winter Rigid Body Motion  Definition (White 2.9) All particles are in combined translation and rotation, and there is no relative motion between the particles. All particles are in combined translation and rotation, and there is no relative motion between the particles. Recalling our last lecture, what terms drop out? Recalling our last lecture, what terms drop out?

2/6/2007BIEN 301 – Winter Steady Flow  Mach Number Relationships

2/6/2007BIEN 301 – Winter Bernoulli’s  The isentropic, adiabatic process is actually a process that can be described via Bernoulli’s.

2/6/2007BIEN 301 – Winter Examples, Homework  Examples for this chapter P9.10, P9.33 P9.10, P9.33  No homework from this section for next time.  Homework from Ch. 5 will be due next Thursday (will discuss in a minute)

2/6/2007BIEN 301 – Winter Homework, Reports  Due Date: Remember, last two homework assignments (Ch. 7, Ch. 8), as well as the Individual and Group Projects are due next week: Tuesday, Feb. 13. Remember, last two homework assignments (Ch. 7, Ch. 8), as well as the Individual and Group Projects are due next week: Tuesday, Feb. 13.  Any questions thus far? I’ll be posting the numerical analysis method file soon I’ll be posting the numerical analysis method file soon

2/6/2007BIEN 301 – Winter Chapter 5 Assignment  Because there is not class on Thursday, we are going to be reading Chapter 5, and answering a few questions (DUE THURSDAY, FEB. 15). Define “Dimensional Analysis”, and describe how this is important for our work in fluid mechanics. Define “Dimensional Analysis”, and describe how this is important for our work in fluid mechanics. Describe the Principle of Dimensional Homogeneity (PDH) Describe the Principle of Dimensional Homogeneity (PDH) Explain the Pi theorem Explain the Pi theorem Explain why nondimensionalization is important for some types of analyses. Explain why nondimensionalization is important for some types of analyses. Explain the following terms: Explain the following terms: Geometric SimilarityGeometric Similarity Kinematic SimilarityKinematic Similarity Dynamic SimilarityDynamic Similarity In as much detail as possible, describe the steps you would consider in designing a test stand to study a scaled-up version of a blood flow obstruction to determine wall shear stresses and stagnation points at the obstruction. In as much detail as possible, describe the steps you would consider in designing a test stand to study a scaled-up version of a blood flow obstruction to determine wall shear stresses and stagnation points at the obstruction. Which of the items above would be most important?Which of the items above would be most important? How would you go about ensuring the validity of your scaled-up experiment?How would you go about ensuring the validity of your scaled-up experiment? Why would doing this scaled-up test be useful in the first place?Why would doing this scaled-up test be useful in the first place?

2/6/2007BIEN 301 – Winter Exam 2 and Homework  No curve on exam Definitely none needed. Definitely none needed. Exam worked as planned: Exam worked as planned: If you have been faithfully doing ALL of your homework and your OWN work without taking ANY shortcuts or giving up on problems in homework, you did very well.If you have been faithfully doing ALL of your homework and your OWN work without taking ANY shortcuts or giving up on problems in homework, you did very well. If you did not do well, consider how much of your homework habits may have hindered you.If you did not do well, consider how much of your homework habits may have hindered you. Rewarded those who have been working extremely faithfully at getting their homework done RIGHT. Offset the cheating that was going on.Rewarded those who have been working extremely faithfully at getting their homework done RIGHT. Offset the cheating that was going on.

2/6/2007BIEN 301 – Winter Remember  No study session tomorrow night.  I’ll be leaving for Houston this afternoon. Will not be available for office hours. You may reach me via .

2/6/2007BIEN 301 – Winter Flow Visualizations  Came across some amazing flow visualizations (computational and actual video). I’ll post the link on Blackboard. I’ll play them now while I hand back the exams and homework, and the presenter for today sets up. I’ll play them now while I hand back the exams and homework, and the presenter for today sets up. After the presentation, I’ll review the exam for anyone that would like to stay behind. After the presentation, I’ll review the exam for anyone that would like to stay behind. Your grades are recorded in my notebook, I’ll post them to Blackboard later on today. Your grades are recorded in my notebook, I’ll post them to Blackboard later on today.

2/6/2007BIEN 301 – Winter Questions?