Atmospheric Property Variations in Isothermal Layers

In an isothermal layer, the temperature is constant; therefore, the equation of state reduces to a relationship between pressure and density:

Thus, substituting for the density, the hydrostatics equation becomes:

dp p

where the combination in parentheses can be considered constant. This simple dif­ferential equation can be solved by direct integration, with the result:

where C2 is a constant of integration that can be evaluated by inserting the initial conditions for a particular isothermal layer from the information in Fig. 2.8 and
from results of the previous gradient layer evaluated at the initial altitude of the isothermal layer. Knowledge of the temperature allows evaluation of the speed of sound and the viscosity coefficients.

The Web site that accompanies this book includes a program labeled STDATM that evaluates the equations described in the preceding paragraphs to give atmos­pheric properties at any altitude. Running the program, the user is asked to choose the system of units (i. e., either English or SI) and to insert the altitude either in meters or feet. The program then returns the corresponding temperature, pressure, density, speed of sound, dynamic-viscosity, and kinematic-viscosity values. Also, the program can be called from within other program modules (described later) to determine automatically the required atmospheric data.

2.3 Summary

This chapter is a review of mathematical concepts for modeling fluid flows such as those experienced in most aerodynamics applications. Important ideas from con­tinuum mechanics and thermodynamics are reviewed, and modeling of the earth’s atmosphere is described using a simple application of fluid statics.

The most important part of this chapter is the introduction of the powerful method of dimensional analysis and the associated ideas of similarity. Application of these ideas reduces the basic problem of aerodynamics—namely, the estimation of the principal aerodynamic forces and moments—to its simplest form. We found that the key to solving this problem is the estimation of dimensionless coefficients such as lift and drag coefficients. An important discovery is that these coefficients are dependent on basic similarity parameters that can be identified as ratios of the fundamental set of forces that characterize the gas motion. In aerodynamics, by far the most important similarity parameters are the Mach and Reynolds numbers, which represent effects of compressibility and viscosity, respectively. Other forces frequently may enter an aerodynamics problem. By applying the examples given in this chapter, students can address such situations when they arise. The following problem set provides the opportunity for students to test their level of understanding.

Another important feature of the similarity approach is its great utility in planning efficient experiments. Understanding how the many variables interact through the dimensionless similarity parameters enables an experimenter to dras­tically reduce the number of tests needed to determine the significant interactions. It also provides a useful tool for classifying and interpreting various experimental results.

Finally, the stage is now set so that we can explore in more detail, both ana­lytically and numerically, the fascinating problem of flow over aerodynamic bodies and the attendant production of forces that can be used to design efficient flight vehicles.

REFERENCES AND SUGGESTED READING

Anderson, John D., Fundamental Aerodynamics, McGraw-Hill Book Company, New York, 1984.

Kuethe, A. M., and Schetzer, J. D., Foundations of Aerodynamics, 2nd ed., John Wiley & Sons, New York, 1961.

Liepmann, H. W., and Roshko, A., Elements of Gasdynamics, Dover Publications, Inc., New York, 1985.

Prandtl, L., and Tietjens, O. G., Fundamentals of Hydro – and Aeromechanics, Dover Publi­cations, Inc., New York, 1957.

Shevell, Richard S., Fundamentals of Flight, Prentice-Hall, Inc., NJ, 1989.

von Karman, Theodore, Aerodynamics, Selected Topics in the Light of Their Historical Devel­opment, Dover Publications, Inc., New York, 2004.

3.1 Introduction

To solve the fundamental problems of aerodynamics defined in Chapter 1, it is necessary to formulate a mathematical representation of the underlying fluid dynamics. The appropriate mathematical expressions or sets of equations may be algebraic, integral, or differential in character but will always represent basic physical laws or principles. In this chapter, the fundamental equations necessary for the solu­tion of aerodynamics problems are derived directly from the basic laws of nature. The resulting mathematical formulations represent a large class of fluid mechanics problems within which aerodynamics is an important subclass.

Some problems in aerodynamics require solutions for all of the variables needed to describe a moving stream of gas—namely, velocity, pressure, tempera­ture, and density. Because velocity is a vector quantity (i. e., with magnitude and direction), in a general case there are three scalar velocity components. Thus, in many cases of interest, there is a total of six unknowns: three velocity components and the scalar thermodynamic quantities of pressure, temperature, and density.[7] This requires six independent equations to be written to solve for the six unknowns. The physical laws of conservation of mass, momentum, and energy supply five such equations (i. e., the momentum equation is a vector equation; therefore, conser­vation of momentum leads in general to three component equations). For all of the subject matter in this book, the assumption of an ideal gas is physically realistic. Thus, the perfect gas law (i. e., equation of state) p = pRT, which relates pressure, density, and temperature, supplies the final equation needed to solve for the six unknowns.